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\begin{document}
\title{The Six Circles Theorem revisited}
\author{D. Ivanov\footnote{Moscow, Russia. e-mail: [email protected]}
\ and
S. Tabachnikov\footnote{
Department of Mathematics,
Penn State,
University Park, PA 16802,
and ICERM, Brown University,
Box 1995, Providence, RI 02912.
e-mail: [email protected]}
}
\date{}
\maketitle
{\bf Introduction}. Given a triangle $P_1 P_2 P_3$, construct a chain of circles: $C_1$, inscribed in the angle $P_1$; $C_2$, inscribed in the angle $P_2$ and tangent to $C_1$; $C_3$, inscribed in the angle $P_3$ and tangent to $C_2$; $C_4$, inscribed in the angle $P_1$ and tangent to $C_3$, and so on. The claim of The Six Circles Theorem is that this process is 6-periodic: $C_7=C_1$, see Figure \ref{inside}.
\begin{figure}
\caption{The Six Circles Theorem: the centers of the consecutive circles are labeled $1,2,\dots,7$.}
\label{inside}
\end{figure}
This beautiful theorem is one of many in the book \cite{EMT} which is a result of collaboration of three geometry enthusiasts, C. Evelyn, G. Money-Coutts, and J. Tyrrell. The following is a quotation from John Tyrrell's obituary \cite{LS}:
\begin{quote}
John also worked with two amateur mathematicians, C. J. A. Evelyn and G. B. Money-Coutts, who found theorems by using outsize drawing instruments to draw large figures. They then looked for concurrencies, collinearities, or other special features. The three men used to meet for tea at the Cafe Royal and talk about mathematics, and then go to the opera at Covent Garden, where Money-Coutts had a box.
\end{quote}
We refer to \cite{TP,Ri,FT,Ta,Tr} for various proofs and generalizations and to \cite{Ty} for a brief biography of C. J. A. Evelyn. See also \cite{Bo,wi,Wo} for Internet resources.
{\bf A refinement}. The formulation of the Six Circles Theorem needs clarification. Firstly, there are two choices for each next circle; we assume that each time the smaller of the two circles tangent to the previous one is chosen (that is, the one which is closer to the respective vertex of the triangle).
Secondly it well may happen that the next circle is tangent not to a side of the triangle but rather to its extension.
The Six Circles Theorem, as stated at the beginning, holds for a chain of circles for which all tangency points lie on the sides of the triangle, not their extensions. And what about the latter case? Figure \ref{preper} shows what may happen.
\begin{figure}
\caption{The chain of circles is eventually 6-periodic with pre-period of length two: $C_{9}
\label{preper}
\end{figure}
\begin{theorem} \label{main}
Assume that, for the initial circle, at least one of the tangency points lies on a side of the triangle.
Then the chain of circles is eventually 6-periodic. One can choose the shape of a triangle and an initial circle so that the pre-period is arbitrarily long.
\end{theorem}
The existence of pre-periods is due to the fact that the map assigning the next circle to the previous one is not 1-1, that is, the inverse map is multi-valued.
Concerning the assumption that at least one of the tangency points of a circle with the sides of the angle of a tirangle
lies on a side of the triangle, and not its extension, we observe the following.
\begin{lemma} \label{tang}
If the first circle in the chain satisfies this assumption then so do all the consecutive circles.
\end{lemma}
\paragraph{Proof.}
If circle $C_1$ touches side $P_1 P_2$ then circle $C_2$ also touches this side, at a point closer to $P_2$ than the previous tangency point. Shifting the index by one, if circle $C_2$ does not touch side $P_2 P_3$ but
touches side $P_1 P_2$ then it intersects side $P_1 P_3$, and the next circle $C_3$ touches side $P_1 P_3$, at a point closer to $P_3$ than the intersection points. See Figure \ref{case2} below for an illustration.
$\Box$
What about the case when the initial circle touches the extensions of both sides, $P_1 P_2$ and $P_1 P_3$? If the circle does not intersect side $P_2 P_3$ then the next circle in the chain cannot be constructed, so this case is not relevant to us. If the first circle intersects side $P_2 P_3$ then the next circle touches side $P_2 P_3$, and thus satisfies the assumption of Theorem \ref{main}, see Figure \ref{out}. Hence this assumption holds, starting with the second circle in the chain, and we may make it without loss of generality.
\begin{figure}
\caption{When the initial circle touches the extensions of both sides of the triangle.}
\label{out}
\end{figure}
{\bf Beginning of the proof}.
The proof consists of reducing the system to iteration of a piecewise linear function; this is achieved by a trigonometric change of variables (see \cite{EMT,Ta,Tr,FT} for versions of this approach). The choices of coordinates and various manipulations may look somewhat unmotivated; they are merely justified by the fact that they work. The reader interested in a coordinate-free, but less elementary, approach is referred to \cite{Ta}.
Let us introduce notations. The angles of the triangle are $2\alpha_1, 2\alpha_2$ and $2\alpha_3$; its side lengths are $a_1, a_2, a_3$ (with the usual convention that $i$th side is opposite $i$th vertex). Let $p=(a_1+a_2+a_3)/2$. We note that $p>a_i$ for $i=1,2,3$: this is the triangle inequality.
We denote the radii of the circles $C_i$ by $r_i,\ i=1,2,\ldots$ and assume that $C_i$ is a circle that is inscribed into ($i$ mod $3$)-rd angle.
\begin{figure}
\caption{The first case of equation (\ref{nextrad}
\label{case1}
\end{figure}
\begin{figure}
\caption{The second case of equation (\ref{nextrad}
\label{case2}
\end{figure}
\begin{figure}
\caption{Also the second case of equation (\ref{nextrad}
\label{minus}
\end{figure}
If two circles of radii $r_1$ and $r_2$ are tangent externally then the length of their common tangent segment (segment $AB$ in Figures \ref{case1}, \ref{case2}, \ref{minus}) is
$$
\sqrt{(r_1+r_2)^2-(r_1-r_2)^2} = 2\sqrt{r_1 r_2}.
$$
Thus, depending on the mutual positions of the consecutive circles, as shown in Figures \ref{case1}, \ref{case2} and \ref{minus}, we obtain the equations
\begin{equation} \label{nextrad}
r_1\cot \alpha_1 + 2\sqrt{r_1 r_2} + r_2\cot \alpha_2=a_3\ \ {\rm or}\ \ r_1\cot \alpha_1 - 2\sqrt{r_1 r_2} + r_2\cot \alpha_2=a_3,
\end{equation}
or the cyclic permutation of the indices $1,2,3$ thereof. Specifically, if $C_1$ is tangent to the side $P_1P_2$ then we have the first equation (\ref{nextrad}), and if $C_1$ is tangent to the extension side $P_1P_2$ then we have the second equation.
{\bf Solving the equations}.
Equations (\ref{nextrad}) determine the new radius $r_2$ as a function of the previous one, $r_1$. We shall solve these equations in two steps. First, introduce the notations
$$
u_1=\sqrt{r_1 \cot \alpha_1},\ \ e_3=\sqrt{\tan \alpha_1 \tan \alpha_2},
$$
and their cyclic permutations. Then (\ref{nextrad}) is rewritten as
\begin{equation} \label{inu}
u_1^2 \pm 2e_3u_1u_2+u_2^2=a_3,
\end{equation}
or
\begin{equation} \label{expu}
u_1(u_1\pm e_3u_2) + u_2(u_2\pm e_3u_1)=a_3.
\end{equation}
Solve (\ref{inu}) for $u_2$:
\begin{equation} \label{solve}
u_2=-e_3u_1+\sqrt{a_3-(1-e_3^2)u_1^2},\ \ {\rm or}\ \ u_2=e_3u_1-\sqrt{a_3-(1-e_3^2)u_1^2},
\end{equation}
according as the sign in (\ref{inu}) is positive or negative. The minus sign in front of the radical in the second formula (\ref{solve}) is because our construction chooses the smaller of the two circles tangent to the previous one. Likewise, solve for $u_1$:
$$
u_1=-e_3u_2+\sqrt{a_3-(1-e_3^2)u_2^2},\ \ {\rm or}\ \ u_1=e_3u_2+\sqrt{a_3-(1-e_3^2)u_2^2},
$$
again depending on the sign in (\ref{inu}). The plus sign in front of the radical in the second formula is due to the fact that, going in the reverse direction, from $C_2$ to $C_1$, one chooses the greater of the two circles.
Substitute to (\ref{expu}) to obtain
\begin{equation} \label{ufin}
u_1 \sqrt{a_3-(1-e_3^2)u_2^2} \pm u_2 \sqrt{a_3-(1-e_3^2)u_1^2} =a_3.
\end{equation}
The sign depends on whether $u_1^2$ is smaller or greater than $a_3$ (and if $u_1^2=a_3$ then $u_2=0$ in (\ref{solve})).
{\bf Trigonometric substitution}.
We shall rewrite the previous formula as the formula for sine of the sum or difference of two angles. To do so, we need a lemma.
Given a triangle $ABC$, let $a,b,c$ be its sides, $p$ its semi-perimeter, and $2\alpha, 2\beta, 2{\gamma}amma$ its angles.
\begin{lemma} \label{tri}
One has
$$
1-\tan\alpha \tan\beta=\frac{c}{p}.
$$
\end{lemma}
\begin{figure}
\caption{To proof of Lemma \ref{tri}
\label{inscr}
\end{figure}
\paragraph{Proof.}
Let $R$ be the inradius and $S$ the area of the triangle. Let
$$
T_A=AF=AG,\ T_B=BG=BE,\ T_C=CE=CF,
$$
see Figure \ref{inscr}.
Then $p=T_A+T_B+T_C$, and $S=Rp$. By Heron's formula, $S=\sqrt{p T_AT_BT_C}$. Therefore
$R^2 p = T_AT_BT_C$.
On the other hand, $\tan \alpha=R/T_A, \tan\beta=R/T_B$, hence
$$
1-\tan \alpha\tan\beta=1-\frac{R^2}{T_AT_B}=1-\frac{T_C}{p}=\frac{T_A+T_B}{p}=\frac{c}{p},
$$
as claimed.
$\Box$
Using the lemma, we rewrite (\ref{ufin}) as
\begin{equation} \label{uff}
\frac{u_1}{\sqrt{p}}\sqrt{1-\frac{u_2^2}{p}}\pm \frac{u_2}{\sqrt{p}}\sqrt{1-\frac{u_1^2}{p}} = \sqrt{\frac{a_3}{p}}.
\end{equation}
We are ready for the final change of variables. Let
$$
\varphi_i = \arcsin\left(\frac{u_i}{\sqrt{p}}\right),\ \beta_i=\arcsin\left(\sqrt{\frac{a_i}{p}}\right).
$$
To justify the second formula, we note that $a_i < p$. Likewise, each circle is tangent to a side of the triangle, so $u_i^2$ is not greater than some side, and hence less than $p$. This justifies the first formula.
In the new variables, (\ref{uff}) rewrites as $\sin(\varphi_1 \pm \varphi_2) = \sin \beta_3$, where one has plus sign for $\varphi_1 < \beta_3$ and minus sign otherwise. Hence
\begin{equation} \label{module}
\varphi_2 = |\varphi_1-\beta_3|.
\end{equation}
This equation describes the dynamics of the chain of circles.
Before studying the dynamics of this function we note that the angles $\beta_i$ satisfy the triangle inequality, as the next lemma asserts. Assume that $\beta_1 \le \beta_2 \le \beta_3$.
\begin{lemma} \label{ineq}
One has $\beta_3 < \beta_1+\beta_2.$
\end{lemma}
\paragraph{Proof.}
We start by noting that $\sin \beta_i <1$ for $i=1,2,3$, and that
$$
\sin^2 \beta_1+\sin^2 \beta_2+\sin^2 \beta_3=2\ \ {\rm or}\ \ \sin^2 \beta_3=\cos^2 \beta_1 + \cos^2 \beta_2.
$$
Assume that the triangle inequality is violated for some triangle. Since the inequality holds for an equilateral triangle, one can deform it to obtain a triangle for which $\beta_3 =\beta_1+\beta_2.$ Then
$$
\cos^2 \beta_1 + \cos^2 \beta_2=\sin^2 \beta_3 = (\sin \beta_1\cos\beta_2+\sin\beta_2\cos\beta_1)^2.
$$
It follows, after some manipulations, that
$$
\sin \beta_1 \sin \beta_2 \cos\beta_1 \cos\beta_2=\cos^2\beta_1 \cos^2\beta_2\ \ {\rm or}\ \ \sin \beta_1 \sin \beta_2=\cos\beta_1 \cos\beta_2.
$$
Therefore
$$
\cos(\beta_1+\beta_2)=0\ \ {\rm or}\ \ \beta_1+\beta_2=\frac{\pi}{2}.
$$
Hence $\sin^2 \beta_1+\sin^2 \beta_2=1$, and thus $\sin \beta_3 =1$. This is a contradiction.
$\Box$
{\bf Piecewise linear dynamics}.
We are ready to investigate the function (\ref{module}). Although the dynamics of a piecewise linear function can be very complex \cite{Na}, ours is quite simple.
Iterating the map three times, with the values of the index $i=1,2,3$, yields the function
$y=|||x-\beta_1|-\beta_2|-\beta_3|$. We scale the $xy$ plane so that $\beta_1=1$ and rewrite the function as
\begin{equation} \label{funct}
f(x)=|||x-1|-a|-b|
\end{equation}
where $a\le b$ and $b<a+1$. We will show that every orbit of the map $f$ is eventually 2-periodic, see Figure \ref{web}.
\begin{figure}
\caption{Iteration of function $f(x)$ for $a=3.6, b=4.2$.}
\label{web}
\end{figure}
The graph of $f(x)$ is shown in Figure \ref{graph} with the characteristic points marked.
\begin{figure}
\caption{The graph $y=f(x)$. The segment $[b-a,1]$ consists of 2-periodic points.}
\label{graph}
\end{figure}
It is clear that iterations of the function $f$ take every orbit to the segment $[0,b]$, and this segment is mapped to itself. Indeed, if $x{\gamma}eq a+b+1$ then $f(x)=x-a-b-1$, and if $x \leq a+b+1$ then $f(x) \leq b$.
Thus iterations of the function $f$ will keep decreasing $x$ until it lands on $[0,b]$.
Let
$$
I_1=[0,b-a],\ I_2=[b-a,1],\ I_3=[1,b].
$$
Then $I_2$ consists of 2-periodic points, and we need to show that every orbit lands on this interval. Indeed, $f(I_1)=[1,b-a+1]\subset I_3$. On the other hand, each iteration of $f$ ``chops off" from the left a segment of length $1+a-b$ from $I_3$ and sends it
to $I_2$. It follows that every orbit eventually reaches $I_2$.
If $|I_2|=a+1-b$ is small, it may take an orbit a long time to reach $I_2$. For example, take $a=1$ and $b=2-\varepsilon$. Then, choosing $\varepsilon$ sufficiently small, one can make the pre-period of point $x=\varepsilon$ arbitrarily long. This choice corresponds to an isosceles triangle with the obtuse angle close to $\pi$ and a small initial circle $C_1$, compare with Figure \ref{preper}.
{\bf Final comments}. \\
1) Although our considerations are close to those in \cite{EMT}, the authors of this book did not consider the pre-periodic behavior of the chain of circles. They addressed the issue of the two choices in each step of the construction and noted:
\begin{quote}
... we may make the first three sign choice quite arbitrarily provided that, thereafter, we make `correct' choices ...
\end{quote}
so that the chain becomes 6-periodic.\\
2) For a parallelogram, a similar phenomenon holds: the chain of circles is eventually 4-periodic but with a pre-period, see \cite{Tr}. Our analysis is similar to that of Troubetzkoy.\\
3) For $n>3$, the chain of circles inscribed in an $n$-gon is generically chaotic, see \cite{Tr} for a proof when $n=4$ and Figure \ref{penta} for an illustration when $n=5$. However, for every $n$, there is a class of $n$-gons enjoying $2n$-periodicity, see \cite{Ta}. Presumably, this periodicity is also eventual, with an arbitrary long pre-period.\\
\begin{figure}
\caption{A chain of circles in a pentagon.}
\label{penta}
\end{figure}
4) A version of the Six Circles Theorem holds for curvilinear triangles made of arcs of circles \cite{EMT,TP,Ri}, and a generalization to $n$-gons is available as well \cite{Ta}. Again, one expects eventual periodicity with arbitrarily long pre-periods.\\
5) Constructing the chains of circles, we consistently chose the smaller of the two circles tangent to the previous one. It is interesting to investigate what happens when other choices are made; for example, one may toss a coin at each step. See Figure \ref{hist} for an experiment with a randomly chosen triangle.\\
6) The Six Circles Theorem is closely related with the Malfatti Problem: to inscribe three pairwise tangent circles into the three angles of a triangle; see, e.g., \cite{Gu} and the references therein. This 3-periodic chain of circles exists and is unique for every triangle; it corresponds to the fixed point of the function $f(x)$. See \cite{BZ} for a discussion of the Malfatti Problem close to our considerations.
\begin{figure}
\caption{The histogram represents 3000 chains of circles in a generic triangle. The selection, out of two, of each next circle in a chain is random. The horizontal axis represents the length of the pre-period, and the vertical the number of chains having this pre-period.}
\label{hist}
\end{figure}
{\bf Acknowledgments}. Most of the experiments that inspired this note and of the drawings were made in GeoGebra. The second author was supported by the NSF grant DMS-1105442. We are grateful to the referees for their criticism and advice.
\end{document} | math | 15,713 |
\begin{equation}gin{document}
\begin{equation}gin{center}{\Large \bf
A return to observability near exceptional points in a schematic
${\cal PT}-$symmetric model
}\end{center}
\begin{equation}gin{center}
{\bf Miloslav Znojil}
\'{U}stav jadern\'e fyziky AV \v{C}R,
250 68 \v{R}e\v{z},
Czech Republic
{e-mail: [email protected]}
\end{center}
\section*{Abstract}
Many indefinite-metric (often called pseudo-Hermitian or ${\cal
PT}-$symmetric) quantum models $H$ prove ``physical" (i.e.,
Hermitian with respect to an innovated, {\em ad hoc} scalar
product) inside a characteristic domain of parameters ${\cal D}$.
This means that the energies get complex (= unobservable) beyond
the boundary $\partial {\cal D}$ (= Kato's ``exceptional points", EPs).
In a solvable example we detect an enlargement of ${\cal D}$
caused by the emergence of a new degree of freedom. We conjecture
that such a beneficial mechanism of a return to the real spectrum
near EPs may be generic and largely model-independent.
\section{Introduction and summary}
Over virtually any model in quantum phenomenology one initially
feels urged to consider {\em all} the relevant degrees of freedom
in the corresponding Langangian ${\cal L}$ or Hamiltonian ${\cal
H}$. This tendency is limited by the imperatives of the
tractability of calculations and of a feasibility of making
measurable predictions. Thus, for example, for an electron moving
in a very strong external Coulomb field, an exhaustive theoretical
analysis requires the full-fledged formalism of relativistic
quantum field theory but some of the measurable properties of the
bound states are still very satisfactorily predicted by the mere
quantum-mechanical, exactly solvable Dirac-equation model
\cite{Greiner}.
One of the most characteristic features of many ``reduced" models
of the latter type is that their reliability (i.e., in an abstract
formulation, the negligibility of relevance of their ``frozen"
degrees of freedom) may vary with some of their dynamical
parameters. Thus, in the same illustration one reveals that when
the external field becomes strong enough, some of the
Dirac-field-excitation components of the system enter the scene
and become directly coupled to the motion of the electron itself
\cite{Greiner}. In such a dynamical regime the Dirac-equation
predictions fail and, formally and typically, the energies of the
electron itself become complex.
On a simplified model-building level the similar ``paradoxes" in
the behavior of the energies may be explained using the
parity-pseudo-Hermitian Hamiltonians $H$. They are often called
${\cal PT}-$symmetric, with the defining property $H \neq
H^\dagger= {\cal T}\,H\,{\cal T}= {\cal P}\,H\,{\cal P}$ and with
a formal operator-conjugation ${\cal T}$ mimicking the time
reversal and with ${\cal P}$ representing the parity (cf. also
Appendix A for more details).
The latter Hamiltonians still can be re-interpreted as
self-adjoint (i.e., observable) but one must restrict their set of
dynamical parameters (i.e., of couplings etc) to a certain
subdomain ${\cal D}^{(physical)}$ on which their spectrum remains
real. In the context of Quantum Mechanics, more details may be
found in the review paper \cite{Geyer}, while an immediate and
inspiring extension of such a recipe to Field Theory has only been
proposed much more recently, in ref.~\cite{BM}.
In our recent letter \cite{Hendrik} we introduced, for
illustrative purposes, a two-state model with the generic
one-parametric ${\cal PT}-$symmetric Hamiltonian
\begin{equation}
H^{(2)} = \left (
\begin{equation}gin{array}{cc}
-1&a\\
-a&1
\end{array}
\right )\,,
\ \ \ \ \ \
\ \ \ \
{\cal P}^{(2)} = \left (
\begin{equation}gin{array}{cc}
1&0\\
0&-1
\end{array}
\right )\,.
\label{pseudoh}
\end{equation}
We also explained there the existence of a nontrivial formal
relationship between our simple model (\ref{pseudoh}) and its more
standard (and phenomenologically ambitious) differential-operator
predecessors or analogues $H^{({\cal PT})}$ (say, of
refs.~\cite{others}). In essence, this relationship is based on
the replacement of $H^{({\cal PT})}$ by their equivalent
infinite-dimensional matrix representants $H^{(\infty)}$ (in a
suitable basis) and, subsequently, by their variational,
$N-$dimensional truncated-matrix approximants $H^{(N)}$ with $N <
\infty$. In such a context it still makes sense to coin the name
``parities" for the corresponding ``indefinite-metric" matrices
${\cal P}^{(N)}$ which enter the finite-dimensional
pseudo-Hermiticity property $H^\dagger\,{\cal P}^{(N)}={\cal
P}^{(N)}H$ of the matrix toy Hamiltonians $H=H^{(N)}$.
The key purpose of our present short paper is to show that the
simple matrix models of the form (\ref{pseudoh}) can say a lot
about the interpretation of the general ${\cal PT}-$symmetric
Hamiltonians in the critical regime where their energies are about
to complexify. In this sense we intend to complement now our
remark \cite{Hendrik} on the schematic model $H^{(2)}$ by a few
new and interesting observations based on a tentative immersion of
the two-dimensional system in a generic three-dimensional one,
\begin{equation}
H^{(3)}=
\left (\begin{equation}gin {array}{cc|c}
-1&a&0
\\
-a&1&b\\
\hline
0&-b&3+c
\end {array}\right )\,\ \ \ \ \
\ \ \ \ \ \ \ \ \ \
{\cal P}^{(3)}=
\left (\begin{equation}gin {array}{cc|c} 1&0&0\\
0&-1&0
\\
\hline
0&0&1
\end {array}\right )\,.
\label{sekoust}
\end{equation}
Here, the standard Jacobi rotation has been employed in setting
zeros in the corners of $H^{(3)}$ so that the three-parametric
example (\ref{sekoust}) preserves the full generality of its
one-parametric predecessor (\ref{pseudoh}).
In a continuation of the study \cite{Hendrik} we shall be able to
show that and how a fairly satisfactory insight in several
qualitative though, up to now, not too well understood properties
of the general ${\cal PT}-$symmetric models can be deduced from
the mere comparison of the virtually elementary models
(\ref{pseudoh}) and (\ref{sekoust}). First of all, such a
comparison will enable us to study some of the aspects of the
above-mentioned conflict between the use of the models with ``too
few" and ``too many" degrees of freedom by choosing the models
$H^{(2)}$ and $H^{(3)}$ as their respective representatives. In
section \ref{kolik} we emphasize that the vanishing of one of the
two coupling constants in the ``universal-like" model $H^{(3)}$
leads directly to the sample ``reduced" model of the form
$H^{(2)}$. A few relevant results on $H^{(2)}$ of
ref.~\cite{Hendrik} are summarized there for the sake of
completeness as well.
Our mathematical encouragement lies in the exact non-numerical
tractability of the ``universal" model $H^{(3)}$. In section
\ref{core} the feasibility of quantitative calculations will
enable us to extend the results of ref.~\cite{Hendrik} to the
richer model $H^{(3)}$ where we set $c=0$ for the sake of
simplicity. In particular, we shall show that the formula for the
boundary $\partial {\cal D}(H^{(3)})$ of the domain where all the
energies remain real can be written in closed form.
A core of our message will be formulated in
section~\ref{generalka} where we show how the growth of ${\cal
D}(H^{(N)})$ from $N=2$ to $N=3$ shifts the Kato's exceptional
points \cite{Kato}. In subsection \ref{subsef} we emphasize that
in the closest vicinity of such an exceptional point at $N=2$,
even the weakest coupling to an ``observer channel" induces a
steady growth of the modified quasi-Hermiticity domain ${\cal
D}(H^{(2)})$. In subsection \ref{tripll}, this observation is
extended to the manifestly non-perturbative regime with the
strongest couplings near the doubly exceptional points where all
the three energy levels coincide.
One could summarize our present message as opening the possibility
of a systematic amendment of various ${\cal PT}-$symmetric models
near exceptional points via a re-activation of certain ``frozen"
degrees of freedom. In this sense, more work is still needed to
confirm that our qualitative observations might stay valid far
beyond the range of the present study.
Let us add that in Appendix A we complemented our discussion by a
concise review of literature showing the physical background and,
perhaps, broader relevance and possible impact of our schematic
models. In a more technical remark of Appendix B we finally show
that the role of $c\neq 0$ in our model $H^{(3)}$ can be
rightfully ignored as not too essential.
\section{Simulated changes of degrees of freedom\label{kolik}}
\subsection{Decoupling an observer state: $b \to
0$ and $H^{(3)} \to H^{(2)}$
\label{dvakratdva} }
Once we start from the illustrative example (\ref{pseudoh}),
letter \cite{Hendrik} tells us that
\begin{equation}gin{itemize}
\item
the eigenenergies remain real and non-degenerate whenever $a^2<
1$, $$ E_\partialm = \partialm \sqrt{1-a^2}$$ so that we may set $ a = \cos
\alpha$ with $ \alpha \in (0,\partiali)$ in ${\cal D}(H^{(2)})$;
\item
the necessary {\it ad hoc} scalar products of Appendix A are
obtainable via a metric operator in (\ref{newpr}). The choice of
this operator is ambiguous, with its elements numbered by an
overall multiplicative constant and by another real parameter
$\gamma \in [0, \partiali/2)$,
\begin{equation}
\Theta \sim
\left (
\begin{equation}gin{array}{cc}
1+\xi&-\cos \alpha\\
-\cos \alpha&1-\xi
\end{array}
\right )\,,\ \ \ \ \ \
\xi = \sin \alpha \sin \gamma\,;
\label{finalre}
\end{equation}
\item
at {\em both} the exceptional points $\alpha^{(EP)} = 0, \partiali$ of
the boundary $\partial {\cal D}$, {\em all} the matrices
$\Theta=\Theta(\gamma)$ cease to be invertible so that
$\Theta^{-1}$ [needed in definition (\ref{newconj}) below] ceases
to exist.
\end{itemize}
\noindent
One may note that at the EP singularities the geometric and
algebraic multiplicities of eigenvalues become different. Some
energies complexify immediately beyond these points. Near the
points of the boundary $\partial {\cal D}$, all the predictions of
quantum mechanics may be more sensitive to perturbations and must
be examined particularly carefully.
\subsection{A re-activated degree of freedom:
$b \neq 0$ and $H^{(2)} \to H^{(3)}$ \label{trikrattri}}
In the language of physics, one should contemplate introducing
some new degree(s) of freedom near every exceptional point. The
majority of the current Hamiltonians $H$ [say, of the
differential-operator form (\ref{ptsp}) discussed in Appendix A]
does not offer a feasible option of this type. In contrast, the
finite-dimensional matrix models can incorporate a new degree of
freedom very easily, via an elementary increase of their
dimension.
In an illustration let us first recollect that in the
two-dimensional model of preceding paragraph we have $a^{(EP)}=\partialm
1$. In the vicinity of these exceptional points we may set
$a=\partialm(1-\varepsilon)$ with a real and sufficiently small
$\varepsilon$ which remains positive inside ${\cal D}(H^{(2)})$,
vanishes in the EP regime and gets negative outside the domain.
An increase of the dimension $N$ in $H^{(N)}$ from 2 to 3 should
be accompanied by a coupling of the submatrix $H^{(2)}$ to a new,
``observer" element of the basis. The resulting ${\cal
PT}-$symmetric three-state matrix model (\ref{sekoust}) contains a
new real coupling $b$ and another real parameter $c \neq -2, -4 $.
Note that the presence of the two vanishing elements in $H^{(3)}$
does not weaken its generality since the corresponding two-by-two
submatrix remains Hermitian and is assumed pre-diagonalized.
\section{Exceptional points in the model $H^{(3)}$
\label{core} }
A pairwise attraction of the energy levels mediated by the
variations of the couplings $a$ and $b$ in $H^{(3)}$ should
control the changes of the spectrum in full analogy with the
generic two-state model. The role of the third parameter $c$ is
less essential and the discussion of its influence is postponed to
Appendix B. Now we set $c=0$ and insert our toy Hamiltonian
(\ref{sekoust}) in the three-state Schr\"{o}dinger equation. Its
determinantal secular equation for energies
\begin{equation}
-{{\it E}}^{3}+3\,{{\it E}}^{2}+\left (-{a}^{2}+1-{b}^{2}\right
){ \it E}-3+3\,{a}^{2}-{b}^{2}=0
\label{sekouspro}
\end{equation}
is solvable in closed form, via the well known Cardano formulae.
Cardano formulae offer the roots of eq.~(\ref{sekouspro}) in the
compact and non-numerical form which is, unfortunately, not too
suitable for the specification of the domain ${\cal D}$. In a
preparatory step, let us analyze a few simpler special cases of
eq.~(\ref{sekouspro}), therefore.
\subsection{Boundary
$\partial {\cal D}$ at $a=c=0$ or $b=c=0$ \label{hladka}}
A quick inspection of eq. (\ref{sekouspro}) reveals that the
cheapest information about $\partial {\cal D}$ becomes available when
$ab=0$. We choose $b=0$ and decouple $H^{(3)}=H^{(2)}\bigoplus
H^{(1)}$. The analysis degenerates to the two-dimensional problem
and restricts the admissible values of $a$ to the following open
interval,
\begin{equation}n
a \in \left . {\cal D}\right |_{\,b=0}=(-1,1).
\end{equation}n
This means that at $b=0$ the ``observer" level $E_2=3$ stays
decoupled and it does not vary with $a$ at all, while the two
other levels (i.e., $E_0=-1$ and $E_1=1$ at $a=0$) become
attracted in proportion to the strength $a\neq 0$ of the
non-Hermiticity.
A completely analogous situation is encountered at $a=0$. In this
case it is comfortable to shift $E \to E- 2$ and get another
section of quasi-Hermiticity domain in closed form,
\begin{equation}n
b \in \left . {\cal D}\right |_{\,a=0}=(-1,1).
\end{equation}n
The genuine three-state phenomena may only occur when {\em both}
$a$ and $b$ remain non-zero, making all the three energy levels
{\em mutually} attracted.
\subsection{The regime of simultaneous attraction, $a \neq 0 \neq b$
\label{tasmhle} }
The set of the exceptional points forms the boundary
$\partialartial{\cal D}$ which connects the above-mentioned four
exceptional points in the $a - b$ plane. Its shape (see Figure 1)
may be deduced from secular eq.~(\ref{sekouspro}) by a
``brute-force" numerical technique. There also exists its
non-numerical description replacing the non-degenerate triplet of
energies $E$ by the doubly degenerate energy $z=1+\begin{equation}ta$ {\em
plus} a separate, ``observer" third energy value $y=-1+2\alpha$.
In the case of $b>a>0$ the value of $z$ should result from a
merger of $E_1=1$ with $E_2=3$ so that we may expect that $\begin{equation}ta
\in (0,1)$. Similarly, one may discuss the other orderings of $a$
and $b$. In parallel, the necessary universality of the attraction
of the levels (as observed above in the two-dimensional model)
implies that we must always have $\alpha > 0$. Thus, once we
replace eq. (\ref{sekouspro}) by its adapted polynomial EP version
of the same (third) degree in $E$,
\begin{equation}
-(E-z)^2(E-y)=
-{{\it E}}^{3}+(2z+y)\,{{\it E}}^{2}-\left (z^2+2yz\right
){ \it E}+yz^2
=0\,
\label{sekousjo}
\end{equation}
the comparison of the quadratic terms in these two alternatives
gives us the constraint $\alpha+\begin{equation}ta=1$. Similarly, the
reparametrization of the linear and constant contributions leads
to the set of the two equations
\begin{equation}n
a^2+b^2=4-3\begin{equation}ta^2, \ \ \ \ \ \
3a^2-b^2=4-3\begin{equation}ta^2-2\begin{equation}ta^3
\end{equation}n
which may be re-read as the desired one-parametric definition of
the star-like shape of the curve forming the boundary
$\partialartial{\cal D}(H^{(3)})$, with $\begin{equation}ta \in (-1,1)$,
\begin{equation}
a=a_\partialm = \partialm \sqrt{\frac{1}{2}\left (
4-3\begin{equation}ta^2-\begin{equation}ta^3
\right )}, \ \ \ \
b=b_\partialm = \partialm \sqrt{\frac{1}{2}\left (
4-3\begin{equation}ta^2+\begin{equation}ta^3
\right )}\,.
\label{paramo}
\end{equation}
We may notice that all the four above-mentioned special EP cases
are reproduced by this formula at $\begin{equation}ta=\partialm 1$. Our analytic
description of the boundary $\partial {\cal D}(H^{(3)})$ at $c=0$ is
complete.
\section{Beneficial effects of the growth of $b \neq 0$
\label{generalka} }
Having the parametric definition (\ref{paramo}) of boundary $\partial
{\cal D}(H^{(3)})$ at our disposal we know precisely where the
energy spectrum remains real. This observation has several
mathematically easy but physically appealing and relevant
consequences.
\subsection{A return of energies from complex to real
\label{subsef} }
We originally started from the two-level model $H^{(2)}$
containing a single parameter $a$. This means that in the language
of the ``complete" three-state model $H^{(3)}$ we worked in the
regime $b=0$ where the ``spectator" degree of freedom stayed
decoupled. Critical EP values were $a=a^{(EP)}=\partialm 1$ so that the
energies lost their observability (i.e., the system collapsed) in
arbitrarily small vicinities of these EPs.
The situation changes when the real and, say, not too large
coupling $b \neq 0$ is switched on.
\subsection*{Lemma}
Whenever our two-level model $H^{(2)}$ becomes coupled to a
``spectator" state with $c=0$ and $b\neq 0$ in $H^{(3)}$, energies
remain real for $a\in (-1-\eta,1+\eta)$ at certain $\eta=\eta(b)>
0$.
\subsection*{Proof}
From the definition (\ref{paramo}) we may infer that, say, near
the EP where $(a,b)=(1,0)$ we may set $\begin{equation}ta = -1+ \varepsilon^2
+{\cal O}(\varepsilon^3) $ and deduce that
\begin{equation}n
b^{(EP)}=
\sqrt{\frac{1}{2}\left [
4-3(1-2\varepsilon^2)-(1-3\varepsilon^2)
+{\cal O}(\varepsilon^3)
\right ]}=
\frac{3\varepsilon}{\sqrt{2}} +{\cal O}(\varepsilon^2).
\end{equation}n
In parallel we have
\begin{equation}n
a^{(EP)}=
\sqrt{\frac{1}{2}\left [
4-3(1-2\varepsilon^2)+(1-3\varepsilon^2)
+{\cal O}(\varepsilon^3)
\right ]}=
1+ \frac{3\varepsilon^2}{4}+{\cal O}(\varepsilon^3)
\end{equation}n
so that we come to the conclusion that the EP value of $a$ grows
with $|\,b|$,
\begin{equation}n
a^{(EP)}=
1+ \frac{\left [b^{(EP)}\right ]^2}{6}+{\cal O}
\left \{ \left [b^{(EP)}\right ]^3 \right \}
.
\label{paramoto}
\end{equation}n
This means that we are allowed to choose a positive $\eta(b)={\cal
O}\left ( b^2\right )$.
{\bf QED.}
\noindent
We see that whenever we introduce a new ``degree of freedom" by
setting $b \neq 0$, our system becomes stable in a non-empty
vicinity of any of the two original exceptional points $a=\partialm 1$.
In the light of a ``generic" character of our example $H^{(3)}$,
one may expect similar behaviour of parametric dependence of the
reality of the spectrum in {\em all} the other (or at least
``many") ${\cal PT}-$symmetric models, irrespectively of their
particular matrix or differential-operator realization.
\subsection{Doubly exceptional character of
the strongest acceptable couplings
\label{tripll}}
Due to the mutual attraction of the energy levels in our
``generic" three-by-three example one may expect that there exist
certain ``doubly exceptional" points (DEPs) of the boundary $\partial
{\cal D}(H^{(3)})$ where {\em all} the three energies coincide at
a triple root $E=z$ of the secular equation,
\begin{equation}
(E-z)^3=0\,.
\label{sekousne}
\end{equation}
The comparison of the coefficients in eqs. (\ref{sekouspro}) and
(\ref{sekousne}) at $E^2$ gives $z=1$. The subsequent two
comparisons provide the two other coupled polynomial equations,
\begin{equation}n
-3=1-{a}^{2}-{b}^{2}, \ \ \ \ \ \ 1 =-3+3\,{a}^{2}
-{b}^{2}\,.
\end{equation}n
We get quickly $b^2=4-a^2$ from the first equation while the
assignment $a^2=2$ follows from the second one, in agreement with
eq.~(\ref{paramo}) at $\begin{equation}ta=0$.
We may conclude that in the light of Figure 1 and formulae
(\ref{paramo}), the boundary of the domain ${\cal D}(H^{(3)})$ of
the allowed real matrix elements $a$ and $b$ remains smooth not
only in the perturbative vicinity of the four points of subsection
\ref{hladka},
$$(a,b) \in \{\, (1,0),
\, (0,1), \,(-1,0), \,(0,-1) \,\},$$
but also at all the pairwise mergers of the real energies. The
spikes are encountered at the four ``maximal-coupling" vertices of
a circumscribed square,
$$(a,b) \in \{\,
(\sqrt{2},\sqrt{2}), \, (-\sqrt{2},\sqrt{2}),
\,(-\sqrt{2},-\sqrt{2}), \,(\sqrt{2},-\sqrt{2}) \,\}$$
where one locates the DEP triple-energy mergers. The fourfold
symmetry of the whole boundary $\partial {\cal D}^{(3)}$ of the
quasi-Hermiticity domain is just an accidental consequence of our
simplifying choice of the vanishing spectral shift~$c=0$.
\section*{Figure captions}
\subsection*{Figure 1. Domain of quasi-Hermiticity at $c=0$}
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\section*{Appendix A: A concise review of the
origin of the present schematic model
}
The concept of quasi-Hermiticity has been introduced in nuclear
physics \cite{Geyer} where variational calculations of complicated
nuclei proved facilitated by the replacement of the common inner
product $ \langle \partialsi|\,\partialhi \rangle$ in Hilbert space by its
generalization
\begin{equation}
\langle \partialsi|\,
\Theta\,|\,\partialhi \rangle
\,,\ \ \ \ \ \ \Theta=\Theta^\dagger>0\,.
\label{newpr}
\end{equation}
Indeed, quantum mechanics can be formulated using {\em any}
invertible and positive definite metric operator $\Theta$ in
(\ref{newpr}). One can feel free to choose any nonstandard
$\Theta\neq I$ and to select observables (i.e., Hamiltonians $H$
etc) represented by operators which are Hermitian with respect to
the new product (\ref{newpr}). Whenever $\Theta\neq I$ we may call
such observables {\em quasi-Hermitian}. In the language of algebra
this means
\begin{equation}
H=H^\ddagger\ \equiv\ \Theta^{-1}\,H^\dagger\,\Theta \,.
\label{newconj}
\end{equation}
This condition may be compatible with the manifest non-Hermiticity
of $H$, provided only that the spectrum remains real.
Incidentally, such a reality condition has been found satisfied by
the quartic anharmonic oscillator ``with wrong sign" \cite{BG} (in
this case the coordinate ceases to be observable \cite{Jones}) as
well as by the ``wrong-coupling" cubic oscillator \cite{Caliceti}
(in this model the non-observability concerns its purely imaginary
potential \cite{postc}). Still, a real boom of interest in the
manifestly non-Hermitian quantum Hamiltonians with real spectra
has only been inspired by the well written letter by Bender and
Boettcher in 1998 \cite{BB}. Very persuasive numerical and WKB
arguments have been given there supporting the reality of spectrum
for a broad class of Hamiltonians, with a remarkable impact on
field theory \cite{BBJ}.
The latter class involves the manifestly non-Hermitian
one-dimensional models
\begin{equation}
H = -\frac{d^2}{dx^2} + U(x) + {\rm i}\,W(x) \neq H^\dagger\,
\label{ptsp}
\end{equation}
defined on $I\!\!L_2(I\!\!R)$ and containing the two real
components of the potential exhibiting the property called ${\cal
PT}-$symmetry \cite{BG},
\begin{equation}
{\cal P}\,U(x)\,{\cal P}
\ (\ \equiv\ U(-x)\ )
=
+U(x)\,,\ \ \ \ \ \
{\cal P}\,W(x)\,{\cal P}=
-W(x)\,.
\end{equation}
The rigorous proofs \cite{DDT} of the reality of the spectra (or,
in the present language, of the quasi-Hermiticity) of many
non-Hermitian toy Hamiltonians $H \neq H^\dagger$ proved
complicated but the inconvenience has been circumvented by the
turn of attention to exactly solvable ${\cal PT}-$symmetric
potentials~\cite{solvable}. Their use simplified the proofs and
mathematics but still, our understanding of the ``correct"
assignment of the physical interpretation to a given model
remained incomplete \cite{reviews}. Another simplifying reduction
of the problem was needed and finite-dimensional matrix
Hamiltonians entered the scene \cite{Wang}. In particular, our
recent discussion of the ambiguity of $\Theta$ \cite{Hendrik}
proved best illustrated by the replacement of both the ${\cal
PT}-$symmetric Hamiltonian (\ref{ptsp}) and the operator of parity
${\cal P}$ by the mere two-dimensional, highly schematic matrices
with real elements.
\section*{Appendix B: An irrelevance of the
shift $c$ }
Even though we confirmed the fourfold symmetry of $\partial {\cal
D}(H^{(3)})$ in paragraph \ref{tasmhle}, this symmetry must be
interpreted as a mere artifact attributed to our choice of the
most comfortable specific $c=0$. Numerical experiments indicate
that the curve $\partial {\cal D}(H^{(3)})$ gets deformed and distorted
in proportion to the degree of violation of the equidistance of
the diagonal matrix elements in $H^{(3)}$. Moreover, as long as
all the $c\neq 0$ models (\ref{sekoust}) are characterized by a
not too much more complicated secular equation
\begin{equation}
-{{{E}}}^{3}+\left (3+c\right ){{{E}}}^{2}+\left (
1-{a}^{2}-{b}^{2}\right ){{E}}-3+3\,{a}^{2}
-c+c{a}^{2}-{b}^{2}=0\,
\label{sekous}
\end{equation}
it would still be feasible to quantify the effect non-numerically.
The method employed in paragraph \ref{tripll} remains applicable
and it leads to an elementary shift of the DEP energy, $3z=3+c$.
The parallel analytic analysis of the $c \neq 0$ problem is left
to the reader. It remains straightforward though a bit boring. For
example, in the most interesting triple-confluence regime the mere
slightly more complicated pair of equations
\begin{equation}n
-3=1-{a}^{2}-{b}^{2}, \ \ \ \ \ \ 1 =-3+3\,{a}^{2}
-c+c{a}^{2}-{b}^{2}\,
\end{equation}n
gives the mere rescaled formulae which relate the DEP matrix
elements in~$H^{(3)}$,
$$
a^2=2-\frac{c}{4+c} = 4-b^2. $$
Obviously, the fourfold symmetry of Figure~1 will be broken in a
way which is continuous in the limit of $c \to 0$.
\end{document} | math | 29,611 |
\begin{document}
\title[On an upper bound for the global dimension of ADR algebras]{On an upper bound for the global dimension of \\ Auslander--Dlab--Ringel algebras}
\author[Mayu Tsukamoto]{Mayu Tsukamoto}
\mathsf{add}\hspace{.01in}ress{Department of Mathematics, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan}
\curraddr{Graduate school of Sciences and Technology for Innovation, Yamaguchi University, 1677-1 Yoshida, Yamaguchi 753-8511, Japan}
\email{[email protected]}
\subjclass[2010]{Primary 16G10; Secondary 16E10}
\keywords{Left-strongly quasi-hereditary algebras, Auslander--Dlab--Ringel algebras, Global dimension, Rejective subcategories}
\begin{abstract}
Lin and Xi introduced Auslander--Dlab--Ringel (ADR) algebras of semilocal modules as a generalization of original ADR algebras and showed that they are quasi-hereditary.
In this paper, we prove that such algebras are always left-strongly quasi-hereditary.
As an application, we give a better upper bound for global dimension of ADR algebras of semilocal modules.
Moreover, we describe characterizations of original ADR algebras to be strongly quasi-hereditary.
\end{abstract}
\maketitle
\section*{Introduction}
Quasi-hereditary algebras were introduced by Cline, Parshall and Scott to study highest weight categories which arise in the representation theory of semisimple complex Lie algebras and algebraic groups \cite{{CPS}, {S}}.
Dlab and Ringel intensely studied quasi-hereditary algebras from the viewpoint of the representation theory of artin algebras \cite{{DR2}, {DR}, {DR6}}.
Motivated by Iyama's finiteness theorem, Ringel introduced the notion of left-strongly quasi-hereditary algebras in terms of highest weight categories \cite{R}.
One of the advantages of left-strongly quasi-hereditary algebras is that they have better upper bound for global dimension than that of general quasi-hereditary algebras.
Moreover, Ringel studied a special class of left-strongly quasi-hereditary algebras called strongly quasi-hereditary algebras.
Let $A$ be an artin algebra with Loewy length $m$.
In \cite{A}, Auslander studied the endomorphism algebra $B:=\mathcal{E}nd_{A}(\bigoplus_{j=1}^{m}A/J(A)^{j})$ and proved that $B$ has finite global dimension.
Furthermore, Dlab and Ringel showed that $B$ is a quasi-hereditary algebra \cite{DR3}.
Hence $B$ is called an Auslander--Dlab--Ringel (ADR) algebra.
Recently, Conde gave a left-strongly quasi-hereditary structure on ADR algebras \cite{C}.
Moreover, ADR algebras were studied in \cite{{C2}, {CEr}} and appeared in \cite{{Co}, {KK}}.
In this paper, we study ADR algebras of semilocal modules introduced by Lin and Xi \cite{LX}.
Recall that a module $M$ is called semilocal if $M$ is a direct summand of modules which have a simple top.
Since any artin algebra is a semilocal module, the ADR algebras of semilocal modules are a generalization of the original ADR algebras.
In \cite{LX}, they proved that ADR algebras of semilocal modules are quasi-hereditary.
We refine this result in Section \ref{ADR}.
\begin{mainthm} [Theorem \ref{thm1}] \label{thma}
The Auslander--Dlab--Ringel algebra of any semilocal module is left-strongly quasi-hereditary.
\end{mainthm}
As an application, we give a tightly upper bound for global dimension of an ADR algebra (see Corollary \ref{cor}).
In Section \ref{SQHADR}, we study a connection between ADR algebras and strongly quasi-hereditary algebras.
An ADR algebra is a left-strongly quasi-hereditary algebra but not necessarily strongly quasi-hereditary.
We give characterizations of original ADR algebras to be strongly quasi-hereditary.
\begin{mainthm} [Theorem \ref{thm2}] \label{thmb}
Let $A$ be an artin algebra with Loewy length $m \geq 2$ and $J$ the Jacobson radical of $A$.
Let $B:=\mathcal{E}nd_A(\bigoplus_{j=1}^{m} A/J^j)$ be the ADR algebra of $A$.
Then the following statements are equivalent.
\begin{itemize}
\item[{\rm (i)}] $B$ is a strongly quasi-hereditary algebra.
\item[{\rm (ii)}] $\operatorname{op}\nolimitseratorname{gldim}\nolimits B =2$.
\item[{\rm (iii)}] $J \in \mathsf{add}\hspace{.01in} (\bigoplus_{j=1}^{m} A/J^j)$.
\end{itemize}
\end{mainthm}
It is known that if $B$ is strongly quasi-hereditary, then the global dimension of $B$ is at most two \cite[Proposition A.2]{R}.
We note that algebras with global dimension at most two are not necessarily strongly quasi-hereditary.
However, for original ADR algebras, the converse is also true.
\section{Preliminaries}
\subsection*{Notation}
Let $A$ be an artin algebra, $J(A)$ the Jacobson radical of $A$ and $\mathrm{D}$ the Matlis dual.
We denote by $\operatorname{op}\nolimitseratorname{gldim}\nolimits A$ the global dimension of $A$.
We fix a complete set of representatives of isomorphism classes of simple $A$-modules $\{ S(i) \; | \; i \in I \}$.
We denote by $P(i)$ the projective cover of $S(i)$ and $E(i)$ the injective hull of $S(i)$ for any $i \in I$.
We write $\mathsf{mod}\hspace{.01in} A$ for the category of finitely generated right $A$-modules and $\mathsf{proj}\hspace{.01in} A$ for the full subcategory of $\mathsf{mod}\hspace{.01in} A$ consisting of finitely generated projective $A$-modules.
For $M \in \mathsf{mod}\hspace{.01in} A$, we denote by $\mathsf{add}\hspace{.01in} M$ the full subcategory of $\mathsf{mod}\hspace{.01in} A$ whose objects are direct summands of finite direct sums of $M$.
The composition of two maps $f:X \to Y$ and $g: Y \to Z$ is denoted by $g \circ f$.
For a quiver $Q$, we denote by $\alpha \beta$ the composition of two arrows $\alpha: x \to y$ and $\beta: y \to z$ in $Q$.
We denote by $K$ an algebraically closed field.
In this section, we quickly review a relationship between strongly quasi-hereditary algebras and rejective chains.
For more detail, we refer to \cite{{I2}, {T}}.
We start this section with recalling the definition of left-strongly quasi-hereditary algebras.
Let $\leq$ be a partial order on the index set $I$ of simple $A$-modules.
For each $i \in I$, we denote by $\nabla(i)$ the maximal submodule of $E(i)$ whose composition factors have the form $S(j)$ for some $j \leq i$.
The module $\nabla(i)$ is called the {\it costandard module} corresponding to $i$.
Let $\nabla:= \{ \nabla(i) \; | \; i \in I \}$ be the set of costandard modules.
We denote by $\mathcal{F}(\nabla)$ the full subcategory of $\mathsf{mod}\hspace{.01in} A$ whose objects are the modules which have a $\nabla$-filtration, that is, $M \in \mathcal{F}(\nabla)$ if and only if there exists a chain of submodules
\[
M=M_0 \supseteq M_1 \supseteq \cdots \supseteq M_l=0
\]
such that $M_i/M_{i+1}$ is isomorphic to a module in $\nabla$.
For $M \in \mathcal{F}(\nabla)$, we denote by $(M: \nabla(i))$ the filtration multiplicity of $\nabla(i)$, which dose not depend on the choice of $\nabla$-filtrations.
\begin{definition}[{\cite[\S 4]{R}}]
Let $A$ be an artin algebra and $\leq$ a partial order on $I$.
\begin{itemize}
\item[{\rm (1)}] A pair $(A, \leq)$ (or simply $A$) is called {\it left-strongly quasi-hereditary} if there exists a short exact sequence
\begin{equation*}
0 \to \nabla(i) \to E(i) \to E(i) /\nabla(i) \to 0
\end{equation*}
for any $i \in I$ with the following properties:
\begin{enumerate}
\item [{\rm (a)}] $E(i)/\nabla(i) \in \mathcal{F}(\nabla)$ for any $i \in I$;
\item [{\rm (b)}] if $(E(i)/\nabla(i) :\nabla(j)) \not= 0$, then we have $i < j$;
\item [{\rm (c)}] $E(i)/\nabla(i)$ is an injective $A$-module, or equivalently, $\nabla(i)$ has injective dimension at most one.
\end{enumerate}
\item[{\rm (2)}] We say that a pair $(A, \leq)$ (or simply $A$) is {\it right-strongly quasi-hereditary} if $(A^{\operatorname{op}\nolimits}, \leq)$ is left-strongly quasi-hereditary.
\item[{\rm (3)}] We say that a pair $(A, \leq)$ (or simply $A$) is {\it strongly quasi-hereditary} if $(A, \leq)$ is left-strongly quasi-hereditary and right-strongly quasi-hereditary.
\end{itemize}
\end{definition}
By definition, strongly quasi-hereditary algebras are left-strongly quasi-hereditary algebras.
Since a pair $(A, \leq)$ satisfying the conditions (a) and (b) is a quasi-hereditary algebra, left-strongly quasi-hereditary algebras are quasi-hereditary.
Left-strongly (resp.\ right-strongly) quasi-hereditary algebras are characterized by total left (resp.\ right) rejective chains, which are chains of certain left (resp.\ right) rejective subcategories.
We recall the notion of left (resp.\ right) rejective subcategories.
Let $\mathcal{C}$ be an additive category, and put $\mathcal{C}(X, Y):=\operatorname{op}\nolimitseratorname{Hom}\nolimits_{\mathcal{C}}(X,Y)$.
In this section, {\it we assume that any subcategory is full and closed under isomorphisms, direct sums and direct summands.}
\begin{definition} [{\cite[2.1(1)]{I}}] \label{rejsub}
Let $\mathcal{C}$ be an additive category.
A subcategory $\mathcal{C}'$ of $\mathcal{C}$ is called
\begin{itemize}
\item[{\rm (1)}] a \emph{left $($resp.\ right$)$ rejective subcategory} of $\mathcal{C}$ if, for any $X\in\mathcal{C}$, there exists an epic left $($resp.\ monic right$)$ $\mathcal{C}'$-approximation $f^X \in \mathcal{C}\left(X,Y\right)$ $($resp.\ $f_X \in \mathcal{C}\left(Y,X\right))$ of $X$,
\item[{\rm (2)}] a \emph{rejective subcategory} of $\mathcal{C}$ if $\mathcal{C}'$ is a left and right rejective subcategory of $\mathcal{C}$.
\end{itemize}
\end{definition}
To define a total left (resp.\ right) rejective chain, we need the notion of cosemisimple subcategories.
Let $\mathcal{J}_{\mathcal{C}}$ be the Jacobson radical of $\mathcal{C}$.
For a subcategory $\mathcal{C}'$ of $\mathcal{C}$, we denote by $[\mathcal{C}']$ the ideal of $\mathcal{C}$ consisting of morphisms which factor through some object of $\mathcal{C}'$, and by $\mathcal{C}/[\mathcal{C}']$ the factor category (\emph{i.e.}, $\mathit{ob}(\mathcal{C}/[\mathcal{C}']):=\mathit{ob}(\mathcal{C})$ and $(\mathcal{C}/[\mathcal{C}'])(X,Y):= \mathcal{C}(X,Y)/[\mathcal{C}'](X,Y)$ for any $X, Y \in \mathcal{C}$).
Recall that an additive category $\mathcal{C}$ is called a {\it Krull--Schmidt} category if any object of $\mathcal{C}$ is isomorphic to a finite direct sum of objects whose endomorphism rings are local.
We denote by $\mathsf{ind}\hspace{.01in} \mathcal{C}$ the set of isoclasses of indecomposable objects in $\mathcal{C}$.
\begin{definition}
Let $\mathcal{C}$ be a Krull--Schmidt category.
A subcategory $\mathcal{C}'$ of $\mathcal{C}$ is called {\it cosemisimple} in $\mathcal{C}$ if $\mathcal{J}_{\mathcal{C}/[\mathcal{C'}]}=0$ holds.
\end{definition}
We give a characterization of cosemisimple left rejective subcategories.
\begin{proposition} [{\cite[1.5.1]{I2}}] \label{crrs}
Let $\mathcal{C}$ be a Krull--Schmidt category and let $\mathcal{C}'$ be a subcategory of $\mathcal{C}$.
Then $\mathcal{C}'$ is a cosemisimple left $($resp.\ right$)$ rejective subcategory of $\mathcal{C}$ if and only if, for any $X \in \mathsf{ind}\hspace{.01in} \mathcal{C} \setminus \mathsf{ind}\hspace{.01in} \mathcal{C}'$,
there exists a morphism $\varphi: X \to Y$ $($resp.\ $\varphi : Y \to X)$ such that $Y \in \mathcal{C}'$ and $\mathcal{C}(Y, -) \xrightarrow{-\circ \varphi} \mathcal{J}_{\mathcal{C}}(X, -)$ $($resp.\ $\mathcal{C}(-, Y) \xrightarrow{\varphi \circ -} \mathcal{J}_{\mathcal{C}}(-, X))$ is an isomorphism on $\mathcal{C}$.
\end{proposition}
Now, we introduce the following key notion in this paper.
\begin{definition}[{\cite[2.1(2)]{I}}] \label{rejch}
Let $\mathcal{C}$ be a Krull--Schmidt category.
A chain
\begin{equation*}
\mathcal{C}= \mathcal{C}_0 \supset \mathcal{C}_1 \supset \cdots \supset \mathcal{C}_n =0
\end{equation*}
of subcategories of $\mathcal{C}$ is called
\begin{itemize}
\item[{\rm (1)}] a \emph{rejective chain} if $\mathcal{C}_i$ is a cosemisimple rejective subcategory of $\mathcal{C}_{i-1}$ for $1 \leq i \leq n$,
\item[{\rm (2)}] a \emph{total left $(${\rm resp.}\ right$)$ rejective chain} if the following conditions hold for $1 \leq i \leq n$:
\begin{enumerate}
\item[(a)] $\mathcal{C}_i$ is a left (resp.\ right) rejective subcategory of $\mathcal{C}$;
\item[(b)] $\mathcal{C}_{i}$ is a cosemisimple subcategory of $\mathcal{C}_{i-1}$.
\end{enumerate}
\end{itemize}
\end{definition}
The following proposition gives a connection between left-strongly quasi-hereditary algebras and total left rejective chains.
\begin{proposition} [{\cite[Theorem 3.22]{T}}] \label{thm0}
Let $A$ be an artin algebra.
Let $M$ be a right $A$-module and $B:= \mathcal{E}nd_A(M)$.
Then the following conditions are equivalent.
\begin{itemize}
\item[{\rm (i)}] $B$ is a left-strongly $($resp.\ right-strongly$)$ quasi-hereditary algebra.
\item[{\rm (ii)}] $\mathsf{proj}\hspace{.01in} B$ has a total left $($resp.\ right$)$ rejective chain.
\item[{\rm (iii)}] $\mathsf{add}\hspace{.01in} M$ has a total left $($resp.\ right$)$ rejective chain.
\end{itemize}
In particular, $B$ is strongly quasi-hereditary if and only if $\mathsf{add}\hspace{.01in} M$ has a rejective chain.
\end{proposition}
We end this section with recalling a special total left rejective chain, which plays an important role in this paper.
\begin{definition}[{\cite[Definition 2.2]{I2}}]
Let $A$ be an artin algebra and $\mathcal{C}$ a subcategory of $\mathsf{mod}\hspace{.01in} A$.
A chain
\begin{equation*}
\mathcal{C}= \mathcal{C}_0 \supset \mathcal{C}_1 \supset \cdots \supset \mathcal{C}_n =0
\end{equation*}
of subcategories of $\mathcal{C}$ is called an \emph{$A$-total left $(${\rm resp.}\ right$)$ rejective chain of length $n$} if the following conditions hold for $1 \leq i \leq n$:
\begin{enumerate}
\item[(a)] for any $X \in \mathcal{C}_{i-1}$, there exists an epic $(${\rm resp.}\ monic$)$ in $\mathsf{mod}\hspace{.01in} A$ left $(${\rm resp.}\ right$)$ $\mathcal{C}_i$-approximation of $X$;
\item[(b)] $\mathcal{C}_{i}$ is a cosemisimple subcategory of $\mathcal{C}_{i-1}$.
\end{enumerate}
\end{definition}
All $A$-total left rejective chains of $\mathcal{C}$ are total left rejective chains.
Moreover, If $\mathrm{D}A \in \mathcal{C}$, then the converse also holds.
We can give an upper bound for global dimension by using $A$-total left rejective chains.
\begin{proposition}[{\cite[Theorem 2.2.2]{I2}}] \label{iygl}
Let $A$ be an artin algebra and $M$ a right $A$-module.
If $\mathsf{add}\hspace{.01in} M$ has an $A$-total left $($resp.\ right$)$ rejective chain of length $n>0$, then $\operatorname{op}\nolimitseratorname{gldim}\nolimits \mathcal{E}nd_A(M) \leq n$ holds.
\end{proposition}
\section{ADR algebras of semilocal modules} \label{ADR}
The aim of this section is to show Theorem \ref{thma}.
First, we recall the definition of semilocal modules.
\begin{definition}
Let $M$ be an $A$-module.
\begin{itemize}
\item[(1)] $M$ is called a {\it local} module if $\operatorname{top}\nolimits M$ is isomorphic to a simple $A$-module.
\item[(2)] $M$ is called a {\it semilocal} module if $M$ is a direct sum of local modules.
\end{itemize}
\end{definition}
Clearly, any local module is indecomposable and any projective module is semilocal.
Throughout this section, suppose that $\displaystyle M$ is a semilocal module with Loewy length $\ell \ell (M)=m$.
We denote by $\widetilde{M}$ the basic module of $\operatorname{op}\nolimitslus_{i=1}^{m} M/MJ(A)^i$ and call $\mathcal{E}nd_A(\widetilde{M})$ the {\it Auslander--Dlab--Ringel algebra} (ADR algebra) of $M$.
Note that $\mathcal{E}nd_A(\widetilde{A})$ is an ADR algebra in the sense of \cite{C}.
Lin and Xi showed that the ADR algebras of semilocal modules are quasi-hereditary (see \cite[Theorem]{LX}).
In this section, we refine this result.
\begin{theorem}\label{thm1}
The ADR algebra of any semilocal module is left-strongly quasi-hereditary.
\end{theorem}
Observe that Theorem \ref{thm1} gives a better upper bound for global dimension of ADR algebras (see Remark \ref{rem}).
In the following, we give a proof of Theorem \ref{thm1}.
Let $\mathsf{F}$ be the set of pairwise non-isomorphic indecomposable direct summands of $\widetilde{M}$ and $\mathsf{F}_i$ the subset of $\mathsf{F}$ consisting of all modules with Loewy length $m-i$.
We denote by $\mathsf{F}_{i, 1}$ the subset of $\mathsf{F}_i$ consisting of all modules $X$ which do not have a surjective map in $\mathcal{J}_{\mathsf{mod}\hspace{.01in} A}(X,N)$ for all modules $N$ in $\mathsf{F}_i$. For any integer $j>1$, we inductively define the subsets $\mathsf{F}_{i,j}$ of $\mathsf{F}_{i}$ as follows: $\mathsf{F}_{i, j}$ consists of all modules $X\in\mathsf{F}_{i}\setminus \bigcup_{1 \leq k \leq j-1}\mathsf{F}_{i,k}$ which do not have a surjective map in $\mathcal{J}_{\mathsf{mod}\hspace{.01in} A}(X,N)$ for all modules $N\in\mathsf{F}_{i}\setminus \bigcup_{1 \leq k \leq j-1}\mathsf{F}_{i,k}$.
We set $n_{i}:=\min\{j\mid \mathsf{F}_{i}=\bigcup_{1 \leq k \leq j}\mathsf{F}_{i,k}\}$ and $n_{M}:=\sum_{i=0}^{m-1} n_i$.
For $0 \leq i \leq m-1$ and $1 \leq j \leq n_i$, we set
\begin{align}
\mathsf{F}_{>(i,j)}&:=\mathsf{F}\setminus ((\cup_{-1\le k \le i-1}\mathsf{F}_{k})\cup (\cup_{1\le l \le j}\mathsf{F}_{i,l})),\notag \\
\mathcal{C}_{i,j} &:= \mathsf{add}\hspace{.01in} \bigoplus_{N \in \mathsf{F}_{> (i,j)}} N, \notag
\end{align}
where $\mathsf{F}_{-1}:=\emptyset$.
Now, we display an example to explain how the subsets $\mathsf{F}_{i, j}$ are given.
\begin{example} \label{eg}
Let $A$ be the $K$-algebra defined by the quiver
\[
\xymatrix@=15pt{ 1 \ar[r] & 2 \ar[r] \ar [d] & 3 \\
& 4
}
\]
and $M:= P(1) \operatorname{op}\nolimitslus P(1)/S(3) \operatorname{op}\nolimitslus P(1)/S(4) \operatorname{op}\nolimitslus P(2)/S(3)$.
We can easily check that $M$ is a semilocal module.
The ADR algebra $B$ of $M$ is given by the quiver
\[
\xymatrix@=15pt{ P(1)/S(4) \ar[r]^{a} & P(1) & P(1)/S(3) \ar[l]_{b} \ar[rd]^{c} \\
& P(1)/ P(1) J(A)^2 \ar[lu]^{d} \ar[ru]_{e} \ar[rd]_{f} & &P(2)/ S(3) \\
& S(1) \ar[u]^{g} & S(2) \ar [ru]_{h}
}
\]
with relations $da-eb, ec-fh$ and $gf$.
Then $\mathsf{F}_{0,1}=\{ P(1)/S(4), P(1)/S(3) \}$, $\mathsf{F}_{0,2}=\{ P(1) \}$, $\mathsf{F}_{1,1}=\{ P(1)/P(1)J(A)^2, P(2)/S(3) \}$, $\mathsf{F}_{2,1} = \{ S(1), S(2) \}$.
\end{example}
To prove Theorem \ref{thm1}, we first show the following proposition.
\begin{proposition} \label{prop}
Let $A$ be an artin algebra and $M$ a semilocal $A$-module.
Then $\mathsf{add}\hspace{.01in} \widetilde{M}$ has the following $A$-total left rejective chain with length $n_{M}$.
\begin{equation*}
\mathsf{add}\hspace{.01in} \widetilde{M} =:\mathcal{C}_{0,0}\supset \mathcal{C}_{0,1} \supset \cdots \supset \mathcal{C}_{0,n_0} \supset \mathcal{C}_{1,1} \supset \cdots \supset \mathcal{C}_{m-1,n_{m-1}}=0.
\end{equation*}
\end{proposition}
To show Proposition \ref{prop}, we need the following lemma.
\begin{lemma} \label{lem2}
For any $M' \in \mathsf{F}_{0,1}$, the canonical surjection $\rho : M' \twoheadrightarrow M'/M' J(A)^{m-1}$ induces an isomorphism
\begin{equation*}
\varphi : \operatorname{op}\nolimitseratorname{Hom}\nolimits_A (M'/M' J(A)^{m-1}, \widetilde{M}) \xrightarrow{- \circ \rho} \mathcal{J}_{\mathsf{mod}\hspace{.01in} A}(M', \widetilde{M}).
\end{equation*}
\end{lemma}
\begin{proof}
Since $\varphi$ is a well-defined injective map, we show that $\varphi$ is surjective.
Let $N$ be an indecomposable summand of $\widetilde{M}$ with Loewy length $k$ and let $f:M' \to N$ be any morphism in $\mathcal{J}_{\mathsf{mod}\hspace{.01in} A}(M', N)$.
Then we show $f(M' J(A)^{m-1})=0$.
(i) Assume that $\operatorname{top}\nolimits M' \not \cong \operatorname{top}\nolimits N$ or $k=m$.
Then we have $\operatorname{op}\nolimitseratorname{Im}\nolimits f\subset NJ(A)$, and hence
\begin{align}
f(M' J(A)^{m-1})=f(M')J(A)^{m-1}\subset (NJ(A))J(A)^{m-1}=0. \notag
\end{align}
(ii) Assume that $\operatorname{top}\nolimits M' \cong \operatorname{top}\nolimits N$ and $k<m$.
Since $m-k>0$ holds, we obtain
\begin{align}
f(M' J(A)^{m-1})=f(M')J(A)^{m-1}\subset NJ(A)^{m-1}= (NJ(A)^{k})J(A)^{m-k-1}=0. \notag
\end{align}
Since $f(M' J(A)^{m-1})=0$ holds, there exists $g: M' /M' J(A)^{m-1}\to N$ such that $f=g \circ \rho$.
\begin{align}
\xymatrix{
0\ar[r]&M' J(A)^{m-1}\ar[r]\ar[rd]_{0}&M' \ar[r]^{\rho\hspace{10mm}}\ar[d]^{f}&M' /M' J(A)^{m-1}\ar[r]\ar@{-->}[dl]^{\exists{g}}&0\\
&&N&&
}\notag
\end{align}
Hence the assertion follows.
\end{proof}
Now, we are ready to prove Proposition \ref{prop}.
\begin{proof}[Proof of Proposition \ref{prop}]
We show by induction on $n_{M}$.
If $n_{M}=1$, then this is clear.
Assume that $n_{M} >1$.
By Proposition \ref{crrs} and Lemma \ref{lem2}, $\mathcal{C}_{0,1}$ is a cosemisimple left rejective subcategory of $\mathsf{add}\hspace{.01in} \widetilde{M}$.
Since $N:=M/(\operatorname{op}\nolimitslus_{X \in \mathsf{F}_{0,1}} X) \operatorname{op}\nolimitslus (\operatorname{op}\nolimitslus_{X \in \mathsf{F}_{0,1}} X/ X J(A)^{m-1})$ is a semilocal module satisfying $\widetilde{N}=\widetilde{M}/\operatorname{op}\nolimitslus_{X\in\mathsf{F}_{0,1}}X$ and $n_{N}<n_{M}$, we obtain that
\begin{align}
\mathsf{add}\hspace{.01in} \widetilde{N}= \mathcal{C}_{0,1} \supset \cdots \supset \mathcal{C}_{0,n_{0}}\supset \mathcal{C}_{1,1} \supset \cdots \supset \mathcal{C}_{m-1, n_{m-1}} =0\notag
\end{align}
is an $A$-total left rejective chain by induction hypothesis.
By composing $\mathcal{C}_{0,0} \supset \mathcal{C}_{0,1}$ and it, we have the desired $A$-total left rejective chain.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm1}]
By Proposition \ref{thm0}, it is enough to show that $\mathsf{add}\hspace{.01in} \widetilde{M}$ has a total left rejective chain.
Hence the assertion follows from Proposition \ref{prop}.
\end{proof}
We give some remark on partial orders for left-strongly quasi-hereditary algebras
\begin{remark}
We define two partial orders on the isomorphism classes of simple $B$-modules.
One is $\{ \mathsf{F}_{0,1} < \cdots < \mathsf{F}_{0,n_0} < \mathsf{F}_{1,1} < \cdots < \mathsf{F}_{m-1, n_{m-1}} \}$, called the {\it ADR order}.
Another one is $\{ \mathsf{F}_{0} < \mathsf{F}_{1} < \cdots < \mathsf{F}_{m-1} \}$, called the {\it length order}.
By Proposition \ref{prop}, ADR algebras of semilocal modules are left-strongly quasi-hereditary with respect to the ADR order.
On the other hand, Conde shows that original ADR algebras are left-strongly quasi-hereditary with respect to the length order \cite{C}.
Since, for an original ADR algebra, the length order coincides with the ADR order, we can recover Conde's result.
However, the ADR algebra of a semilocal module is not necessarily left-strongly quasi-hereditary with respect to the length order, as shown by the following example.
\end{remark}
\begin{example}
Let $A$ and $M$ be in Example \ref{eg}.
Then we can check that the ADR algebra $B$ of $M$ is left-strongly quasi-hereditary with respect to the ADR order
\begin{equation*}
\{ \mathsf{F}_{0,1} < \mathsf{F}_{0,2} < \mathsf{F}_{1,1} < \mathsf{F}_{2,1} \}.
\end{equation*}
However, we can also check that $B$ is not left-strongly quasi-hereditary with respect to the length order
\begin{align}
\{ \{P(1)/S(3), P(1)/S(4), P(1)\} < \{P(1)/P(1)J(A)^2, P(2)/S(3)\} < \{S(1), S(2)\} \}.\notag
\end{align}
\end{example}
As an application, we give an upper bound for global dimension of ADR algebras.
\begin{corollary} \label{cor}
Let $A$ be an artin algebra and $M$ a semilocal $A$-module.
Then
\begin{align}
\operatorname{op}\nolimitseratorname{gldim}\nolimits \mathcal{E}nd_{A}(\widetilde{M})\le n_{M}. \notag
\end{align}
\end{corollary}
\begin{proof}
By Proposition \ref{prop}, $\mathsf{add}\hspace{.01in}\widetilde{M}$ has an $A$-total left rejective chain with length $n_{M}$.
Hence the assertion follows from Proposition \ref{iygl}.
\end{proof}
\begin{remark} \label{rem}
In \cite{LX}, they showed that the ADR algebra of a semilocal module $M$ is quasi-hereditary.
This implies $\operatorname{op}\nolimitseratorname{gldim}\nolimits \mathcal{E}nd_{A}(\widetilde{M})\le 2(n_{M}-1)$ by \cite[Statement 9]{DR}.
By Corollary \ref{cor}, we can obtain a better upper bound for global dimension of ADR algebras.
This can be seen by the following example.
\end{remark}
The following example tells us that the upper bound for the global dimension in Corollary \ref{cor} is tightly.
Let $n \geq 2$. Let $A$ be the $K$-algebra defined by the quiver
\[
\def\scriptstyle{\scriptstyle}
\def\scriptstyle{\scriptstyle}
\vcenter{
\hbox{
$
\xymatrix@C=6pt{
& & 1 \ar[lld] \ar[ld] \ar@{}[d]|(.6){\dots} \ar[rd] \ar[rrd]
& & \\
2 & 3 & \dots\dots & n-1 & n
}
$
}
}
\]
and $M$ a direct sum of all factor modules of $P(1)$. Clearly, $M$ is semilocal and $n_M=n$.
Let $B$ be its ADR algebra. Then we have
\begin{align*}
\operatorname{op}\nolimitseratorname{gldim}\nolimits B =
\begin{cases}
n-1 & (n \geq 3)\\
2 & (n=2).
\end{cases}
\end{align*}
Indeed, the assertion for $n=2$ clearly holds.
Assume $n \geq 3$.
It is easy to check that, for $X \in \mathsf{F}_{0,l}$ ($1 \leq l \leq n_0$),
\begin{align*}
\operatorname{pd}\nolimits_{B^{\operatorname{op}\nolimits}} \operatorname{top}\nolimits (\operatorname{op}\nolimitseratorname{Hom}\nolimits_A(X, \widetilde{M})) =l. \notag
\end{align*}
Thus we have
\begin{align*}
\max \{ \operatorname{pd}\nolimits_{B^{\operatorname{op}\nolimits}} \operatorname{top}\nolimits (\operatorname{op}\nolimitseratorname{Hom}\nolimits_A(X, \widetilde{M})) \mid X \in \mathsf{F} \} = n_0 = n_M -1. \notag
\end{align*}
Hence the assertion for $n \geq 3$ holds.
\section{Strongly quasi-hereditary ADR algebras} \label{SQHADR}
In this section, we prove Theorem \ref{thmb}.
We keep the notation of the previous section.
Throughout this section, $A$ is an artin algebra with Loewy length $m$ and $B:=\mathcal{E}nd_{A}(\widetilde{A})$ the ADR algebra of $A$.
Then $n_j =1$ holds for any $0 \leq j \leq m-1$.
Hence we obtain the following $A$-total left rejective chain by Proposition \ref{prop}.
\begin{equation} \label{rej}
\mathsf{add}\hspace{.01in} \widetilde{A} \supset \mathcal{C}_{0,1} \supset \mathcal{C}_{1,1} \supset \cdots \supset \mathcal{C}_{m-1,1} = 0.
\end{equation}
Note that if $m=1$, then $B$ is semisimple.
Hence we always assume $m\ge 2$ in the rest of section.
\begin{theorem}\label{thm2}
Let $A$ be an artin algebra with Loewy length $m \geq 2$ and $B$ the ADR algebra of $A$.
Then the following statements are equivalent.
\begin{itemize}
\item[{\rm (i)}] $B$ is a strongly quasi-hereditary algebra.
\item[{\rm (ii)}] The chain \eqref{rej} is a rejective chain of $\mathsf{add}\hspace{.01in} \widetilde{A}$.
\item[{\rm (iii)}] $\operatorname{op}\nolimitseratorname{gldim}\nolimits B =2$.
\item[{\rm (iv)}] $J(A) \in \mathsf{add}\hspace{.01in} \widetilde{A}$.
\end{itemize}
\end{theorem}
To prove Theorem \ref{thm2}, we need the following lemma.
\begin{lemma}\label{lem3}
Let $A$ be an artin algebra.
If $P(i)J(A) \in \mathsf{add}\hspace{.01in} \widetilde{A}$ for any $i \in I$, then $P(i)J(A)/P(i)J(A)^j \in \mathsf{add}\hspace{.01in} \widetilde{A}$ for $1 \leq j \leq m$.
\end{lemma}
\begin{proof}
Since $P(i)J(A) \in \mathsf{add}\hspace{.01in} \widetilde{A}$, we have $P(i)J(A) \cong \displaystyle{\bigoplus_{k, l} P(k)/P(k)J(A)^l}$.
For simplicity, we write $P(i)J(A) \cong P(k)/P(k)J(A)^l$.
Then we have $P(i)J(A)/P(i)J(A)^j \cong (P(k)/P(k)J(A)^l)/(P(k)J(A)^j/P(k)J(A)^l) \cong P(k)/P(k)J(A)^j \in \mathsf{add}\hspace{.01in} \widetilde{A}$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm2}]
(ii) $\mathcal{R}ightarrow$ (i): The assertion follows from Proposition \ref{thm0}.
(i) $\mathcal{R}ightarrow$ (iii): It follows from \cite[Proposition A.2]{R} that the global dimension of $B$ is at most two.
It is enough to show that there exists a $B$-module such that its projective dimension is two.
Let $S$ be a simple $A$-module.
Then we have the following short exact sequence.
\begin{equation*}
0 \to \mathcal{J}_{\mathsf{mod}\hspace{.01in} A}(\widetilde{A},S) \to \operatorname{op}\nolimitseratorname{Hom}\nolimits_A (\widetilde{A},S) \to \operatorname{top}\nolimits \operatorname{op}\nolimitseratorname{Hom}\nolimits_A(\widetilde{A},S) \to 0.
\end{equation*}
Assume that $\mathcal{J}_{\mathsf{mod}\hspace{.01in} A}(\widetilde{A},S)$ is a projective right $B$-module.
Then there exists an $A$-module $Y \in \mathsf{add}\hspace{.01in} \widetilde{A}$ such that $\mathcal{J}_{\mathsf{mod}\hspace{.01in} A}(\widetilde{A},S) \cong \operatorname{op}\nolimitseratorname{Hom}\nolimits_A(\widetilde{A},Y)$.
By $S \in \mathsf{add}\hspace{.01in} \widetilde{A}$, there exists a non-zero morphism $f : Y \to S$ such that $\operatorname{op}\nolimitseratorname{Hom}\nolimits_A(\widetilde{A},f) : \operatorname{op}\nolimitseratorname{Hom}\nolimits_A(\widetilde{A}, Y) \to \operatorname{op}\nolimitseratorname{Hom}\nolimits_A(\widetilde{A}, S)$ is an injective map.
Since the functor $\operatorname{op}\nolimitseratorname{Hom}\nolimits_{A}(\widetilde{A},-)$ is faithful, $f$ is an injective map.
Hence $f$ is an isomorphism.
This is a contradiction since $\mathcal{J}_{\mathsf{mod}\hspace{.01in} A}(\widetilde{A},S) \cong \operatorname{op}\nolimitseratorname{Hom}\nolimits_A(\widetilde{A},S)$.
Therefore, we obtain the assertion.
(iii) $\Leftrightarrow$ (iv): This follows from \cite[Proposition 2]{Sm}.
(iv) $\mathcal{R}ightarrow$ (ii):
First, we show that $\mathcal{C}_{0,1}$ is a cosemisimple rejective subcategory of $\mathsf{add}\hspace{.01in} \widetilde{A}$.
By Proposition \ref{prop}, it is enough to show that $\mathcal{C}_{0,1}$ is a right rejective subcategory of $\mathsf{add}\hspace{.01in} \widetilde{A}$.
For any $X \in \mathsf{ind}\hspace{.01in}(\mathsf{add}\hspace{.01in} \widetilde{A}) \setminus \mathsf{ind}\hspace{.01in}(\mathcal{C}_{0,1})$, there exists an inclusion map $\varphi : XJ(A) \hookrightarrow X$ with $XJ(A) \in \mathcal{C}_{0,1}$ by the condition (iv).
Since $X$ is a projective $A$-module such that its Loewy length coincides with the Loewy length of $A$, the map $\varphi$ induces an isomorphism
\begin{equation*}
\operatorname{op}\nolimitseratorname{Hom}\nolimits_A(\widetilde{A}, XJ(A)) \xrightarrow{\varphi \circ -} \mathcal{J}_{\mathsf{mod}\hspace{.01in} A}(\widetilde{A}, X).
\end{equation*}
It follows from Proposition \ref{crrs} that $\mathcal{C}_{0,1}$ is a cosemisimple right rejective subcategory of $\mathsf{add}\hspace{.01in} \widetilde{A}$.
Hence we obtain that $\mathcal{C}_{0,1}$ is a cosemisimple rejective subcategory of $\mathsf{add}\hspace{.01in} \widetilde{A}$.
Next, we prove that $\mathsf{add}\hspace{.01in} \widetilde{A}$ has a rejective chain
\begin{equation*}
\mathsf{add}\hspace{.01in} \widetilde{A} \supset \mathcal{C}_{0,1} \supset \mathcal{C}_{1,1} \supset \cdots \supset \mathcal{C}_{m-1,1} = 0
\end{equation*}
by induction on $m$.
If $m=2$, then the assertion holds.
Assume that $m \geq 3$.
Let $X \in \mathsf{ind}\hspace{.01in}(\mathcal{C}_{0,1}) \setminus \mathsf{ind}\hspace{.01in}(\mathcal{C}_{1,1})$.
Then $X=P(i)/P(i)J(A)^{m-1}$ for some $i \in I$ and we have
\begin{equation*}
(P(i)/P(i)J(A)^{m-1})J(A/J^{m-1}(A)) \cong P(i)J(A)/P(i)J(A)^{m-1}.
\end{equation*}
Since $P(i)J(A) \in \mathsf{add}\hspace{.01in} \widetilde{A}$, we obtain $P(i)J(A)/P(i)J(A)^{m-1} \in \mathcal{C}_{0,1}$ by Lemma \ref{lem3}.
By induction hypothesis, $\mathcal{C}_{0,1}$ has the following rejective chain.
\begin{equation*}
\mathcal{C}_{0,1} \supset \mathcal{C}_{1,1} \supset \cdots \supset \mathcal{C}_{m-1,1} = 0.
\end{equation*}
Composing it with $\mathsf{add}\hspace{.01in} \widetilde{A} \supset \mathcal{C}_{0,1}$, we obtain a rejective chain of $\mathsf{add}\hspace{.01in} \widetilde{A}$.
\end{proof}
By Theorem \ref{thm2}(i) $\mathcal{R}ightarrow$ (ii), a strongly quasi-hereditary structure of the ADR algebra $B$ can be always realized by the ADR order. However, for a semilocal module, such an assertion does not necessarily hold. In fact, we give an example that the ADR algebra of a semilocal module is strongly quasi-hereditary but not strongly quasi-hereditary with respect to the ADR order.
\begin{example}
Let $A$ be the $K$-algebra defined by the quiver
\[
\xymatrix@=15pt{ 1 \ar@(lu,ld)_{\alpha} \ar[r]^{\beta} & 2
}
\]
with relations $\alpha \beta$ and $\alpha^3$.
Clearly, $M:= P(1) \operatorname{op}\nolimitslus P(1)/\operatorname{soc}\nolimits P(1) \operatorname{op}\nolimitslus P(2)$ is a semilocal module.
The ADR algebra $B$ of $M$ is given by the quiver
\[
\xymatrix@=15pt{ & P(1) \ar@<-1.5ex>[ldd]_{a} \ar[rd]^{b} & \\
& P(1)/P(1)J(A)^2 \ar[u]^{c} \ar[d]_{d} &P(1)/\operatorname{soc}\nolimits P(1) \ar[l]^{e} \\
P(2) &S(1) \ar[ru]_{f} &
}
\]
with relations $eca, fed$ and $cb-df$.
Then $B$ is not strongly quasi-hereditary with respect to the ADR order $\{ \mathsf{F}_{0,1} < \mathsf{F}_{1,1} < \mathsf{F}_{1,2} < \mathsf{F}_{2,1} \}$, but $B$ is strongly quasi-hereditary with respect to $\{ P(1) < P(1)/P(1)J(A)^2 < P(1)/\operatorname{soc}\nolimits P(1) < \{P(2), S(1) \} \}$.
\end{example}
\subsection*{Acknowledgment}
The author wishes to express her sincere gratitude to Takahide Adachi and Professor Osamu Iyama.
The author thanks Teresa Conde and Aaron Chan for informing her about the reference \cite[Proposition 2]{Sm}, which greatly shorten her original proof.
\end{document} | math | 33,242 |
\begin{document}
\textwidth 150mm \textheight 225mm
\title{On the $A_\alpha$ spectral radius and $A_\alpha$ energy of non-strongly connected digraphs \thanks{\small Supported by the National Natural Science Foundation of China (No. 11871398), and the Natural Science Foundation of Shaanxi Province (No. 2020JQ-107).}}
\author{{Xiuwen Yang$^{a,b}$, Ligong Wang$^{a,b,}$\footnote{Corresponding author.}}\\
{\small $^{a}$ School of Mathematics and Statistics, Northwestern Polytechnical University,}\\{\small Xi'an, Shaanxi 710129, P.R. China.}\\ {\small $^{b}$ Xi'an-Budapest Joint Research Center for Combinatorics, Northwestern Polytechnical University,}\\{\small Xi'an, Shaanxi 710129, P.R. China.}\\{\small E-mail: [email protected], [email protected]}}
\date{}
\maketitle
\begin{center}
\begin{minipage}{135mm}
\vskip 0.3cm
\begin{center}
{\small {\bf Abstract}}
\end{center}
{\small Let $A_\alpha(G)$ be the $A_\alpha$-matrix of a digraph $G$ and $\lambda_{\alpha 1}, \lambda_{\alpha 2}, \ldots, \lambda_{\alpha n}$ be the eigenvalues of $A_\alpha(G)$. Let $\rho_\alpha(G)$ be the $A_\alpha$ spectral radius of $G$ and $E_\alpha(G)=\sum_{i=1}^n \lambda_{\alpha i}^2$ be the $A_\alpha$ energy of $G$ by using second spectral moment. Let $\mathcal{G}_n^m$ be the set of non-strongly connected digraphs with order $n$, which contain a unique strong component with order $m$ and some directed trees which are hung on each vertex of the strong component. In this paper, we characterize the digraph which has the maximal $A_\alpha$ spectral radius and the maximal (minimal) $A_\alpha$ energy in $\mathcal{G}_n^m$.
\vskip 0.1in \noindent {\bf Key Words}: \ $A_\alpha$ spectral radius; $A_\alpha$ energy; non-strongly connected digraphs \vskip
0.1in \noindent {\bf AMS Subject Classification (2020)}: \ 05C20, 05C50 }
\end{minipage}
\end{center}
\section{Introduction }
Let $G=(\mathcal{V}(G),\mathcal{A}(G))$ be a digraph which $\mathcal{V}(G)=\{v_1,v_2,\ldots,v_n\}$ is the vertex set of $G$ and $\mathcal{A}(G)$ is the arc set of $G$. For an arc from vertex $v_i$ to $v_j$, we denote by $(v_i, v_j)$, and $v_i$ is the tail of $(v_i, v_j)$ and $v_j$ is the head of $(v_i, v_j)$. The outdegree $d_i^+=d_G^+(v_i)$ of $G$ is the number of arcs whose tail is vertex $v_i$ and the indegree $d_i^-=d_G^-(v_i)$ of $G$ is the number of arcs whose head is vertex $v_i$. We denote the maximum outdegree and the maximum indegree of $G$ by $\displaystyleelta^+(G)$ and $\displaystyleelta^-(G)$, respectively. A walk $\pi$ of length $l$ from vertex $u$ to vertex $v$ is a sequence of vertices $\pi$: $u=v_0,v_1,\ldots,v_l=v$, where $(v_{k-1},v_k)$ is an arc of $G$ for any $1\leq k\leq l$. If $u=v$ then $\pi$ is called a closed walk. Let $c_2$ denote the number of all closed walks of length $2$. A directed path $P_{n}$ with $n$ vertices is a digraph which the vertex set is $\{v_i|\ i=1,2,\ldots,n\}$ and the arc set is $\{(v_i,v_{i+1})|\ i=1,2,\ldots,n-1\}$. A directed cycle $C_n$ with $n\geq2$ vertices is a digraph which the vertex set is $\{v_i|\ i=1,2,\ldots,n\}$ and the arc set is $\{(v_i,v_{i+1})|\ i=1,\ldots,n-1\}\cup\{(v_n,v_1)\}$. A digraph $G$ is connected if its underlying graph is connected. A digraph $G$ is strongly connected if for each pair of vertices $v_i,v_j\in\mathcal{V}(G)$, there is a directed path from $v_i$ to $v_j$ and one from $v_j$ to $v_i$. A strong component of $G$ is a maximal strongly connected subdigraph of $G$. Throughout this paper, we only consider a connected digraph $G$ containing neither loops nor multiple arcs.
For a digraph $G$ with order $n$, the adjacency matrix $A(G)=(a_{ij})_{n\times n}$ of $G$ is a $(0,1)$-square matrix whose $(i,j)$-entry equals to $1$, if $(v_i, v_j)$ is an arc of $G$ and equals to $0$, otherwise. The Laplacian matrix $L(G)$ and the signless Laplacian matrix $Q(G)$ of $G$ are $L(G)=D^+(G)-A(G)$ and $Q(G)=D^+(G)+A(G)$, respectively, where $D^+(G)=diag(d_1^+,d_2^+,\ldots,d_n^+)$ is a diagonal outdegree matrix of $G$. In 2019, Liu et al. \cite{LiWC} defined the $A_\alpha$-matrix of $G$ as
$$A_\alpha(G)=\alpha D^+(G)+(1-\alpha)A(G),$$
where $\alpha\in[0,1]$. It is clear that if $\alpha=0$, then $A_0(G)=A(G)$; if $\alpha=\frac{1}{2}$, then $A_\frac{1}{2}(G)=\frac{1}{2}Q(G)$; if $\alpha=1$, then $A_1(G)=D^+(G)$. Since $D^+(G)$ is not interesting, we only consider $\alpha\in[0,1)$. The eigenvalue of $A_\alpha(G)$ with largest modulus is called the $A_\alpha$ spectral radius of $G$, denoted by $\rho_\alpha(G)$.
Actually, in 2017, Nikiforov \cite{Ni} first proposed the $A_\alpha$-matrix of a graph $H$ with order $n$ as
$$A_\alpha(H)=\alpha D(H)+(1-\alpha)A(H),$$
where $D(H)=diag(d_1,d_2,\ldots,d_n)$ is a diagonal degree matrix of $H$ and $\alpha\in[0,1]$. After that, many scholars began to study the $A_\alpha$-matrices of graphs. Nikiforov et al. \cite{NiPR} gave several results about the $A_\alpha$-matrices of trees and gave the upper and lower bounds for the spectral radius of the $A_\alpha$-matrices of arbitrary graphs. Let $\lambda_1(A_\alpha(H))\geq\lambda_2(A_\alpha(H))\geq\cdots\geq\lambda_n(A_\alpha(H))$ be the eigenvalues of $A_\alpha(H)$. Lin et al. \cite{LiXS} characterized the graph $H$ with $\lambda_k(A_\alpha(H))=\alpha n-1$ for $2 \leq k\leq n$ and showed that $\lambda_n(A_\alpha(H))\geq2\alpha-1$ if $H$ contains no isolated vertices. Liu et al. \cite{LiDS} presented several upper and lower bounds on the $k$-th largest eigenvalue of $A_\alpha$-matrix and characterized the extremal graphs corresponding to some of these obtained bounds. More results about $A_\alpha$-matrix of a graph can be found in \cite{LiSu,LiHX,LiLX,LiLi,NiRo,WaWT}. Recently, Liu et al. \cite{LiWC} characterized the digraph which had the maximal $A_\alpha$ spectral radius in $\mathcal{G}_n^r$, where $\mathcal{G}_n^r$ is the set of digraphs with order $n$ and dichromatic number $r$. Xi et al. \cite{XiSW} determined the digraphs which attained the maximum (or minimum) $A_\alpha$ spectral radius among all strongly connected digraphs with given parameters such as girth, clique number, vertex connectivity or arc connectivity. Xi and Wang \cite{XiWa} established some lower bounds on $\displaystyleelta^+-\rho_\alpha(G)$ for strongly connected irregular digraphs with given maximum outdegree and some other parameters. Ganie and Baghipur \cite{GaBa} obtained some lower bounds for the spectral radius of $A_\alpha(G)$ in terms of the number of vertices, the number of arcs and the number of closed walks of the digraph $G$.
It is well-known that the energy of the adjacency matrix of a graph $H$ first defined by Gutman \cite{Gut} as $E_A(H)=\sum_{i=1}^{n}\nu_i$, where $\nu_i$ is an eigenvalue of the adjacency matrix of $H$. Pe\~{n}a and Rada \cite{PeRa} defined the energy of the adjacency matrix of a digraph $G$ as $E_A(G)=\sum_{i=1}^{n} |\textnormal{Re}(z_i)|$, where $z_i$ is an eigenvalue of the adjacency matrix of $G$ and $\textnormal{Re}(z_i)$ is the real part of eigenvalue $z_i$. Some results about the energy of the adjacency matrices of graphs and digraphs have been obtained in \cite{Bru,CvDS,GuLi}. Lazi\'c \cite{La} defined the Laplacian energy of a graph $H$ as $LE(H)=\sum_{i=1}^n \mu_i^2$ by using second spectral moment, where $\mu_i$ is an eigenvalue of $L(H)$. Perera and Mizoguchi \cite{PeMi} defined the Laplacian energy $LE(G)$ of a digraph $G$ as $LE(G)=\sum_{i=1}^n \lambda_i^2$ by using second spectral moment, where $\lambda_i$ is an eigenvalue of $L(G)$. Yang and Wang \cite{YaWa} defined the signless Laplacian energy as $E_{SL}(G)=\sum_{i=1}^n q_i^2$ of a digraph $G$ by using second spectral moment, where $q_i$ is an eigenvalue of $Q(G)$. In this paper, we study the $A_\alpha$ energy as $E_\alpha(G)=\sum_{i=1}^n \lambda_{\alpha i}^2$ of a digraph $G$ by using second spectral moment, where $\lambda_{\alpha i}$ is an eigenvalue of $A_\alpha(G)$.
\begin{figure}
\caption{An out-star $\stackrel{\rightarrow}
\label{fi:ch-1}
\end{figure}
\begin{figure}
\caption{Two different in-trees}
\label{fi:ch-2}
\end{figure}
Next, we will introduce some concepts of digraphs.
An arc $(v_i,v_j)$ is said to be simple if $(v_i,v_j)$ is an arc in $G$ but $(v_j,v_i)$ is not an arc in $G$. A digraph $G$ is simple if every arc in $G$ is simple. An arc $(v_i,v_j)$ is said to be symmetric if both $(v_i,v_j)$ and $(v_j,v_i)$ are arcs in $G$. A digraph $G$ is symmetric if every arc in $G$ is symmetric.
Let $T$ be a directed tree with $n$ vertices and $e$ arcs without cycles and $n=e+1$. If $n=1$, then the directed tree is a vertex.
Let $\stackrel{\rightarrow}{K}_{1,n-1}$ be an out-star with $n$ vertices which has one vertex with outdegree $n-1$ and other vertices with outdegree $0$ (see $\stackrel{\rightarrow}{K}_{1,n-1}$ in Figure \ref{fi:ch-1}). And the vertex with outdegree $n-1$ is called the centre of $\stackrel{\rightarrow}{K}_{1,n-1}$.
Let $\stackrel{\leftarrow}{K}_{1,n-1}$ be an in-star with $n$ vertices which has one vertex with indegree $n-1$ and other vertices with indegree $0$ (see $\stackrel{\leftarrow}{K}_{1,n-1}$ in Figure \ref{fi:ch-1}).
Let $\stackrel{\leftrightarrow}{K}_{1,n-1}$ be a symmetric-star with $n$ vertices which all the arcs are symmetric and have a common vertex (see $\stackrel{\leftrightarrow}{K}_{1,n-1}$ in Figure \ref{fi:ch-1}).
Let in-tree be a directed tree with $n$ vertices which the outdegree of each vertex of the directed tree is at most one. Then the in-tree has exactly one vertex with outdegree $0$ and such vertex is called the root of the in-tree (see Figure \ref{fi:ch-2}). Let $\infty[m_1,m_2,\ldots,m_t]$ be a generalized $\infty$-digraph with $n=\sum_{i=1}^t m_i-t+1$ $(m_i\geq2)$ vertices which has $t$ directed cycles $C_{m_i}$ with exactly one common vertex (see $\infty[m_1,m_2,\ldots,m_t]$ in Figure \ref{fi:ch-3}).
A $p$-spindle with $n$ vertices is the union of $p$ internally disjoint $(x,y)$-directed paths for some vertices $x$ and $y$. The vertex $x$ is said to be the initial vertex of spindle and $y$ its terminal vertex. A $(p+q)$-bispindle with $n$ vertices is the internally disjoint union of a $p$-spindle with initial vertex $x$ and terminal vertex $y$ and a $q$-spindle with initial vertex $y$ and terminal vertex $x$. Actually, it is the union of $p$ $(x,y)$-directed paths and $q$ $(y,x)$-directed paths. We denote the $(p+q)$-bispindle by $B[p,q]$ (see $B[p,q]$ in Figure \ref{fi:ch-3}).
\begin{figure}
\caption{A generalized $\infty$-digraph and a $(p+q)$-bispindle}
\label{fi:ch-3}
\end{figure}
\begin{figure}
\caption{A digraph $G\in\mathcal{G}
\label{fi:ch-4}
\end{figure}
Let $\mathcal{G}_n^m$ be the set of non-strongly connected digraphs with order $n$, which contain a unique strong component with order $m$ and some directed trees which are hung on each vertex of the strong component. For a non-strongly connected digraph $G\in\mathcal{G}_n^m$, we assume that $G^*$ is the unique strong component of $G$ with $m$ vertices and $T^{(i)}$ is the directed tree with $n_i$ vertices which hangs on each vertex of $G^*$, where $n=\sum_{i=1}^m n_i$ and $i=1,2,\ldots,m$. Then the vertex set of $G$ is $\mathcal{V}(G)=\bigcup_{i=1}^m \mathcal{V}(T^{(i)})$, where $\mathcal{V}(T^{(i)})=\{u_1^{(i)},u_2^{(i)},\ldots,u_{n_i}^{(i)}\}$, $\mathcal{V}(G^*)=\{v_1,v_2,\ldots,v_m\}$ and $v_i=u_1^{(i)}$, $i=1,2,\ldots,m$. Let $d_{G}^+(u_j^{(i)})$ be the outdegree of vertex $u_j^{(i)}$ of $G$ and $d_{G^*}^+(v_1)\geq d_{G^*}^+(v_2)\geq \cdots\geq d_{G^*}^+(v_m)$ be the outdegrees of vertices of $G^*$, where $i=1,2,\ldots,m$ and $j=1,2,\ldots,n_i$. We take an example in Figure \ref{fi:ch-4}.
\noindent\begin{definition}\label{de:ch-1.1} Let $G\in\mathcal{G}_n^m$ be a non-strongly connected digraph with $n$ vertices.
(i) Let
$$G'=G-\sum_{i=1}^m\sum_{s,t=1}^{n_i} (u_s^{(i)},u_t^{(i)})+\sum_{i=1}^m\sum_{j=2}^{n_i} (u_1^{(i)},u_j^{(i)}),$$
where $(u_s^{(i)},u_t^{(i)})\in\mathcal{A}(G)$, $i=1,2,\ldots,m$ and $s,t,j=1,2,\ldots,n_i$. Then $G'\in\mathcal{G}_n^m$ is a non-strongly connected digraph, which each directed tree $T^{(i)}$ is an out-star $\stackrel{\rightarrow}{K}_{1,n_i-1}$ and the centre of $\stackrel{\rightarrow}{K}_{1,n_i-1}$ is $v_i$ of $G^*$, where $i=1,2,\ldots,m$ (see $G'$ in Figure \ref{fi:ch-5}).
(ii) Let
\begin{align*}
G''&=G-\sum_{i=1}^m\sum_{s,t=1}^{n_i} (u_s^{(i)},u_t^{(i)})+\sum_{i=1}^m\sum_{j=2}^{n_i} (u_1^{(1)},u_j^{(i)})\\
&=G'-\sum_{i=2}^m\sum_{j=2}^{n_i} (u_1^{(i)},u_j^{(i)})+\sum_{i=2}^m\sum_{j=2}^{n_i} (u_1^{(1)},u_j^{(i)}),
\end{align*}
where $(u_s^{(i)},u_t^{(i)})\in\mathcal{A}(G)$, $i=1,2,\ldots,m$ and $s,t,j=1,2,\ldots,n_i$. Then $G''\in\mathcal{G}_n^m$ is a non-strongly connected digraph, which only has an out-star $\stackrel{\rightarrow}{K}_{1,n-m}$ and the centre of $\stackrel{\rightarrow}{K}_{1,n-m}$ is $v_1$ of $G^*$, $v_1$ is the maximal
outdegree vertex of $G^*$ and other directed tree $T^{(i)}$ is just a vertex $v_i$ of $G^*$ for $i=2,3,\ldots,m$ (see $G''$ in Figure \ref{fi:ch-5}).
(iii) Let $G'''\in\mathcal{G}_n^m$ be a non-strongly connected digraph by changing each directed tree in $G$ to an in-tree which the root of the in-tree is $v_i$ of $G^*$, where $i=1,2,\ldots,m$. We take an example in Figure \ref{fi:ch-6}.
\end{definition}
\begin{figure}
\caption{Digraphs $G',G''\in\mathcal{G}
\label{fi:ch-5}
\end{figure}
\begin{figure}
\caption{A digraph $G'''\in\mathcal{G}
\label{fi:ch-6}
\end{figure}
The arrangement of this paper is as follows. In Section 2, we characterize the digraph which has the maximal $A_\alpha$ spectral radius in $\mathcal{G}_n^m$. In Section 3, we characterize the digraph which has the maximal (minimal) $A_\alpha$ energy in $\mathcal{G}_n^m$.
\section{The maximal $A_\alpha$ spectral radius of non-strongly connected digraphs}
In this section, we will consider the maximal $A_\alpha$ spectral radius of non-strongly connected digraphs in $\mathcal{G}_n^m$. Firstly, we list some known results used for later.
\noindent\begin{lemma}\label{le:ch-2.1}(\cite{HoJo}) Let $M$ be an $n\times n$ nonnegative irreducible matrix with spectral radius
$\rho(M)$ and row sums $R_1,R_2,\ldots,R_n$. Then
$$\underset{1\leq i\leq n}{\mathrm{min}} R_i\leq \rho(M)\leq \underset{1\leq i\leq n}{\mathrm{max}} R_i.$$
Moreover, one of the equalities holds if and only if the row sums of $M$ are all equal.
\end{lemma}
\noindent\begin{definition}\label{de:ch-2.2}(\cite{BePl}) Let $A =(a_{ij})$, $B=(b_{ij})$ be two $n\times n$ matrices. If $a_{ij}\leq b_{ij}$ for all $i$ and $j$, then $A\leq B$. If $A\leq B$ and $A\neq B$, then $A<B$. If $a_{ij}<b_{ij}$ for all $i$ and $j$, then $A\ll B$.
\end{definition}
\noindent\begin{lemma}\label{le:ch-2.3}(\cite{BePl}) Let $A =(a_{ij})$, $B=(b_{ij})$ be two $n\times n$ matrices with the spectral radius $\rho(A)$ and $\rho(B)$, respectively. If $0\leq A\leq B$, then $\rho(A)\leq\rho(B)$. Furthermore, If $0\leq A<B$ and $B$ is irreducible, then $\rho(A)<\rho(B)$.
\end{lemma}
\noindent\begin{lemma}\label{le:ch-2.4}(\cite{LiWC}) Let $G$ be a digraph with the $A_\alpha$ spectral radius $\rho_\alpha(G)$ and maximal outdegree $\displaystyleelta^+(G)$. If $H$ is a subdigraph of $G$, then $\rho_\alpha(H)\leq\rho_\alpha(G)$, especially, $\rho_\alpha(G)\geq\alpha\displaystyleelta^+(G)$. If $G$ is strongly connected and $H$ is a proper subdigraph of $G$, then $\rho_\alpha(H)<\rho_\alpha(G)$.
\end{lemma}
\noindent\begin{theorem}\label{th:ch-2.5} Let $G\in\mathcal{G}_n^m$ be a non-strongly connected digraph with $\mathcal{V}(G)=\{v_1,v_2,\ldots,v_n\}$. Let $G^*$ be a unique strong component of $G$ with $\mathcal{V}(G^*)=\{v_1,v_2,\ldots,v_m\}$. Let $\lambda_{\alpha1}, \lambda_{\alpha2}, \ldots, \lambda_{\alpha n}$ be the eigenvalues of $A_\alpha(G)$ and $d_1^+, d_2^+, \ldots, d_n^+$ be the outdegrees of vertices of $G$. Then
$$\lambda_{\alpha i}=\alpha d_i^+,$$
for $i=m+1,m+2,\ldots,n$.
\end{theorem}
\begin{proof}
Let $A_\alpha(G)=\alpha D^+(G)+(1-\alpha)A(G)$ be the $A_\alpha$-matrix of $G$. Let $\mathcal{V}(G)=V_1\bigcup V_2$ be the vertex set of $G$, where $V_1=\mathcal{V}(G^*)=\{v_1,v_2,\ldots,v_m\}$ and $V_2=\mathcal{V}(G-G^*)=\{v_{m+1},v_{m+2},\ldots,v_n\}$. According to the partition of vertex set of $G$, we partition $A_\alpha(G)$ into
$$
A_\alpha(G)=\left[
\begin{array}{c|c}
A_{11} & A_{12}\\ \hline
A_{21} & A_{22}\\
\end{array}
\right].
$$
The characteristic polynomial $\phi_{A_\alpha(G)}(x)$ of $G$ is $\phi_{A_\alpha(G)}(x)=|xI_n-A_\alpha(G)|$. Since the vertices of $V_2$ are not on the strong component, there must exist a vertex with indegree $0$ or outdegree $0$. Then the elements of column or row of $A_\alpha(G)$ corresponding to that vertex are all $0$, except the diagonal element. So by the property of the determinant, we have $\phi_{A_\alpha(G)}(x)=|xI_n-A_\alpha(G)|=|xI_n-A_{11}|\prod_{i=m+1}^n (x-\alpha d_i^+)$. Hence, $\lambda_{\alpha i}=\alpha d_i^+$, for $i=m+1,m+2,\ldots,n$.
\end{proof}
With the above theorem, we can get a more general result.
\noindent\begin{corollary}\label{co:ch-2.6}
Let $G$ be any digraph with $n$ vertices. Let $\lambda_{\alpha1}, \lambda_{\alpha2}, \ldots, \lambda_{\alpha n}$ be the eigenvalues of $A_\alpha(G)$ and $d_1^+, d_2^+, \ldots, d_n^+$ be the outdegrees of vertices of $G$. For any vertex $v_i$ which is not on the strong components of $G$, we have
$$\lambda_{\alpha i}=\alpha d_i^+.$$
\end{corollary}
Next, we give our main results.
\noindent\begin{theorem}\label{th:ch-2.7} Let $G,G'\in\mathcal{G}_n^m$ be two non-strongly connected digraphs as defined in Definition \ref{de:ch-1.1}. Then $\rho_\alpha(G')\geq\rho_\alpha(G)$.
\end{theorem}
\begin{proof}
By the definition of $G'$, we know $G'\in\mathcal{G}_n^m$ is a non-strongly connected digraph, which each directed tree $T^{(i)}$ is an out-star $\stackrel{\rightarrow}{K}_{1,n_i-1}$ and the centre of $\stackrel{\rightarrow}{K}_{1,n_i-1}$ is $v_i$ of $G^*$, where $i=1,2,\ldots,m$. Then $d_{G'}^+(v_i)=d_{G'}^+(u_1^{(i)})=d_{G^*}^+(v_i)+n_i-1$, $d_{G'}^+(u_j^{(i)})=0$, where $i=1,2,\ldots,m$ and $j=2,3,\ldots,n_i$.
First, we consider the $A_\alpha$-eigenvalues of $G'$. From Theorem \ref{th:ch-2.5}, for the vertex $u_j^{(i)}$ which is not on the strong component $G^*$, we have
$$\lambda_{\alpha j}^{(i)}(G')=\alpha d_{G'}^+(u_j^{(i)})=0,$$
where $i=1,2,\ldots,m$ and $j=2,3,\ldots,n_i$. For the vertex $v_i=u_1^{(i)}$ which is on the strong component $G^*$, the $A_\alpha$-eigenvalues $\lambda_{\alpha 1}^{(i)}(G')$ are equal to the eigenvalues of $A_{11}'$, where
$$A_{11}'=\alpha diag\left(d_{G^*}^+(v_1)+n_1-1,d_{G^*}^+(v_2)+n_2-1,\ldots,d_{G^*}^+(v_m)+n_m-1\right)+(1-\alpha)A(G^*).$$
Obviously, $\rho_\alpha(G')=\rho(A_{11}')$.
Next, we consider the $A_\alpha$-eigenvalues of $G$. From Theorem \ref{th:ch-2.5}, for the vertex $u_j^{(i)}$ which is not on the strong component $G^*$, we have
$$\lambda_{\alpha j}^{(i)}(G)=\alpha d_{G}^+(u_j^{(i)}),$$
where $i=1,2,\ldots,m$ and $j=2,3,\ldots,n_i$. For the vertex $v_i=u_1^{(i)}$ which is on the strong component $G^*$, the $A_\alpha$-eigenvalues $\lambda_{\alpha 1}^{(i)}(G)$ are equal to the eigenvalues of $A_{11}$, where
$$A_{11}=\alpha diag\left(d_{G}^+(v_1),d_{G}^+(v_2),\ldots,d_{G}^+(v_m)\right)+(1-\alpha)A(G^*).$$
Hence, $\rho_\alpha(G)=\underset{1\leq i\leq m,2\leq j\leq n_i}{\mathrm{max}}\left\{\rho(A_{11}), \alpha d_{G}^+(u_j^{(i)})\right\}$.
Finally, we prove
$$\rho_\alpha(G')=\rho(A_{11}')\geq\rho_\alpha(G)=\underset{1\leq i\leq m,2\leq j\leq n_i}{\mathrm{max}}\left\{\rho(A_{11}), \alpha d_{G}^+(u_j^{(i)})\right\}.$$
From Lemma \ref{le:ch-2.3}, since
$$d_{G^*}^+(v_i)+n_i-1\geq d_{G}^+(v_i),$$
we have $A_{11}'\geq A_{11}$. Then $\rho(A_{11}')\geq\rho(A_{11})$. From Lemma \ref{le:ch-2.4}, we have
$$\rho_\alpha(G')\geq\alpha \displaystyleelta^+(G') \geq \alpha \displaystyleelta^+(G) \geq \alpha d_{G}^+(u_j^{(i)}).$$
Therefore, we have $\rho_\alpha(G')\geq\rho_\alpha(G)$.
\end{proof}
\noindent\begin{theorem}\label{th:ch-2.8} Let $G',G''\in\mathcal{G}_n^m$ be two non-strongly connected digraphs as defined in Definition \ref{de:ch-1.1}. If $\alpha\in\left[\frac{d_{G^*}^+(v_1)}{d_{G^*}^+(v_1)+n-m-n_1+1},1\right)$, then $\rho_\alpha(G'')\geq\rho_\alpha(G')$; if $\alpha=0$, then $\rho_\alpha(G'')=\rho_\alpha(G')$.
\end{theorem}
\begin{proof}
By the definition of $G''$, we know $G''\in\mathcal{G}_n^m$ is a non-strongly connected digraph, which only has an out-star $\stackrel{\rightarrow}{K}_{1,n-m}$ and the centre of $\stackrel{\rightarrow}{K}_{1,n-m}$ is $v_1$ of $G^*$, $v_1$ is the maximal
outdegree vertex of $G^*$ and each other directed tree $T^{(i)}$ is just a vertex $v_i$ of $G^*$ for $i=2,3,\ldots,m$. The vertex set $\mathcal{V}(G'')=\mathcal{V}(\stackrel{\rightarrow}{K}_{1,n-m})\bigcup(\mathcal{V}(G^*)-v_1)$, where $\mathcal{V}(\stackrel{\rightarrow}{K}_{1,n-m})=\{u_1^{(1)},u_2^{(1)},\ldots,u_{n-m+1}^{(1)}\}$, $\mathcal{V}(G^*)=\{v_1,v_2,\ldots,v_m\}$ and $u_1^{(1)}=v_1$. Then $d_{G''}^+(v_1)=d_{G^*}^+(v_1)+n-m$, $d_{G''}^+(u_j^{(1)})=0$ and $d_{G''}^+(v_i)=d_{G^*}^+(v_i)$, where $i=2,3,\ldots,m$ and $j=2,3,\ldots,n-m+1$. Since $d_{G^*}^+(v_1)\geq d_{G^*}^+(v_2)\geq \cdots\geq d_{G^*}^+(v_m)$, by Lemma \ref{le:ch-2.4}, we have
$$\rho_\alpha(G'') \geq \alpha \displaystyleelta^+(G'')=\alpha\left(d_{G^*}^+(v_1)+n-m\right).$$
From the proof of Theorem \ref{th:ch-2.7}, we have $\rho_\alpha(G')=\rho(A_{11}')$. By Lemma \ref{le:ch-2.1}, we have
$$\underset{1\leq i\leq m}{\mathrm{min}} R_i(A_{11}')\leq \rho(A_{11}')\leq \underset{1\leq i\leq m}{\mathrm{max}} R_i(A_{11}').$$
Then
$$\rho_\alpha(G')\leq \underset{1\leq i\leq m}{\mathrm{max}} \left\{(1-\alpha)d_{G^*}^+(v_i)+\alpha\left(d_{G^*}^+(v_i)+n_i-1\right)\right\}=\underset{1\leq i\leq m}{\mathrm{max}} \left\{d_{G^*}^+(v_i)+\alpha(n_i-1)\right\}.$$
Without loss of generality, let $\underset{1\leq i\leq m}{\mathrm{max}} \left\{d_{G^*}^+(v_i)+\alpha(n_i-1)\right\}=d_{G^*}^+(v_t)+\alpha(n_t-1)$. That is $d_{G^*}^+(v_t)+\alpha(n_t-1)\geq d_{G^*}^+(v_1)+\alpha(n_1-1)$.
If $\alpha\neq0$, we have $n_t\geq\frac{d_{G^*}^+(v_1)-d_{G^*}^+(v_t)+\alpha n_1}{\alpha}$. Next, we prove
$$\rho_\alpha(G'')\geq \alpha\left(d_{G^*}^+(v_1)+n-m\right)\geq d_{G^*}^+(v_t)+\alpha(n_t-1)\geq \rho_\alpha(G').$$
We only need to prove
$$\alpha\left(d_{G^*}^+(v_1)+n-m\right)\geq d_{G^*}^+(v_t)+\alpha\left(\frac{d_{G^*}^+(v_1)-d_{G^*}^+(v_t)+\alpha n_1}{\alpha}-1\right).$$
That is, $1>\alpha\geq\frac{d_{G^*}^+(v_1)}{d_{G^*}^+(v_1)+n-m-n_1+1}$.
If $\alpha=0$, we have $\rho_0(G')=\rho(A(G^*))$ and $\rho_0(G'')=\rho(A(G^*))$, then $\rho_\alpha(G'')=\rho_\alpha(G')$.
Therefore, if $\alpha\in\left[\frac{d_{G^*}^+(v_1)}{d_{G^*}^+(v_1)+n-m-n_1+1},1\right)$, then $\rho_\alpha(G'')\geq\rho_\alpha(G')$; if $\alpha=0$, then $\rho_\alpha(G'')=\rho_\alpha(G')$.
\end{proof}
From Theorems \ref{th:ch-2.7} and \ref{th:ch-2.8}, we have the following theorem.
\noindent\begin{theorem}\label{th:ch-2.9} Among all digraphs in $\mathcal{G}_n^m$, if $\alpha\in\left[\frac{d_{G^*}^+(v_1)}{d_{G^*}^+(v_1)+n-m-n_1+1},1\right)$ or $\alpha=0$, then $G''$ is a digraph which has the maximal $A_\alpha$ spectral radius.
\end{theorem}
We only obtain $G''$ is a digraph having the maximal $A_\alpha$ spectral radius in $\mathcal{G}_n^m$ when $\alpha\in\left[\frac{d_{G^*}^+(v_1)}{d_{G^*}^+(v_1)+n-m-n_1+1},1\right)$ or $\alpha=0$. However, we know $\frac{d_{G^*}^+(v_1)}{d_{G^*}^+(v_1)+n-m-n_1+1}=\frac{d_{G^*}^+(v_1)}{d_{G^*}^+(v_1)+\sum_{i=2}^{m}(n_i-1)}$. The bigger $\sum_{i=2}^{m}(n_i-1)$ is, the smaller $\frac{d_{G^*}^+(v_1)}{d_{G^*}^+(v_1)+n-m-n_1+1}$ is. And if $\sum_{i=2}^{m}(n_i-1)\rightarrow0$, then $n_1-1\rightarrow n-m$ and $G'\rightarrow G''$. So we think the result is true for all $\alpha\in[0,1)$. Therefore, we give the following conjecture.
\noindent\begin{conjecture}\label{co:ch-2.10} Among all digraphs in $\mathcal{G}_n^m$, for any $\alpha\in[0,1)$, $G''$ is a digraph which has the maximal $A_\alpha$ spectral radius.
\end{conjecture}
\section{The maximal $A_\alpha$ energy of non-strongly connected digraphs}
In this section, we will consider the maximal $A_\alpha$ energy of non-strongly connected digraphs in $\mathcal{G}_n^m$. Firstly, we will introduce some basic concepts of $A_\alpha$ energy of digraphs.
Let $E_\alpha(G)$ be $A_\alpha$ energy of a digraph $G$. By using second spectral moment, Xi \cite{Xi} defined the $A_\alpha$ energy as $E_\alpha(G)=\sum_{i=1}^n \lambda_{\alpha i}^2$, where $\lambda_{\alpha i}$ is an eigenvalue of $A_\alpha(G)$. She also obtained the following result.
\noindent\begin{lemma}\label{le:ch-3.1}(\cite{Xi}) Let $G$ be a connected digraph with $n$ vertices and $c_2$ be the number of all closed walks of length $2$. Let $d_1^+,d_2^+,\ldots,d_n^+$ be the outdegrees of vertices of $G$. Then
$$E_\alpha(G)=\sum_{i=1}^n \lambda_{\alpha i}^2=\alpha^2\sum_{i=1}^n(d_i^+)^2+(1-\alpha)^2c_2.$$
\end{lemma}
Obviously, we can get the following results.
\noindent\begin{theorem}\label{th:ch-3.2} Let $G$ be a connected digraph with $n$ vertices and $e$ arcs. Let $d_1^+,d_2^+,\ldots,d_n^+$ be the outdegrees of vertices of $G$.
(i) If $G$ is a simple digraph, then
$$E_\alpha(G)=\alpha^2\sum_{i=1}^n (d_i^+)^2.$$
(ii) If $G$ is a symmetric digraph, then
$$E_\alpha(G)=\alpha^2\sum_{i=1}^n (d_i^+)^2+(1-\alpha)^2 e.$$
\end{theorem}
\begin{proof}
From Lemma \ref{le:ch-3.1}, the conclusion is obvious.
\end{proof}
Let $c_2$ be the number of all closed walks of length $2$ of a digraph. From Theorem \ref{th:ch-3.2}, we have the following results.
\noindent\begin{example}\label{ex:ch-3.3} We give some $A_\alpha$ energies of special digraphs as follows: \\
(1) $E_\alpha(P_n)=\alpha^2(n-1);$\\
(2) $E_\alpha(C_n)=
\begin{cases}
\alpha^2n, & \textnormal{if}\ n>2,\\
2(\alpha^2-2\alpha+1), & \textnormal{if}\ n=2;
\end{cases}$\\
(3) $E_\alpha(\stackrel{\rightarrow}{K}_{1,n-1})=\alpha^2(n-1)^2;$\\
(4) $E_\alpha(\stackrel{\leftarrow}{K}_{1,n-1})=\alpha^2(n-1);$\\
(5) $E_\alpha(\stackrel{\leftrightarrow}{K}_{1,n-1})=\alpha^2 n(n-1)+2(1-\alpha)^2(n-1);$\\
(6) $E_\alpha(\infty[m_1,m_2,\ldots,m_t])=\alpha^2(t^2+n-1)+(1-\alpha)^2c_2;$\\
(7) $E_\alpha(B[p,q])=\alpha^2(p^2+q^2+n-2)+(1-\alpha)^2c_2.$
\end{example}
\noindent\begin{lemma}\label{le:ch-3.4}(\cite{Xi}) Let $T$ be a directed tree with $n$ vertices. Then
$$\alpha^2(n-1)\leq E_\alpha(T)\leq\alpha^2(n-1)^2.$$
Moreover, $E_\alpha(T)=\alpha^2(n-1)$, if and only if $T$ is an in-tree with $n$ vertices; $E_\alpha(T)=\alpha^2(n-1)^2$ if and only if $T$ is an out-star $\stackrel{\rightarrow}{K}_{1,n-1}$.
\end{lemma}
Next, we give our main results.
\noindent\begin{theorem}\label{th:ch-3.5} Let $G,G'\in\mathcal{G}_n^m$ be two non-strongly connected digraphs as defined in Definition \ref{de:ch-1.1}. Then $E_\alpha(G')\geq E_\alpha(G)$ with equality holds if and only if $G\cong G'$.
\end{theorem}
\begin{proof}
By the definition of $G$, we know $G\in\mathcal{G}_n^m$ is a non-strongly connected digraph with order $n$, which contains a unique strong component with order $m$ and some directed trees which are hung on each vertex of the strong component. From Lemma \ref{le:ch-3.4}, we know the maximal $A_\alpha$ energy of $T^{(i)}$ is
$$\left(E_\alpha(T^{(i)})\right)_{\mathrm{max}}=\alpha^2(n_i-1)^2,$$
where $i=1,2,\ldots,m$. Then we have
\begin{align*}
E_\alpha(G)&=\alpha^2\sum_{i=1}^m\sum_{j=1}^{n_i}\left(d_G^+(u_j^{(i)})\right)^2+(1-\alpha)^2c_2\\
&=\alpha^2\sum_{i=1}^m\left(d_{G^*}^+(u_1^{(i)})+d_{T^{(i)}}^+(u_1^{(i)})\right)^2+ \alpha^2\sum_{i=1}^m\sum_{j=2}^{n_i}\left(d_G^+(u_j^{(i)})\right)^2+(1-\alpha)^2c_2\\
&=\alpha^2\sum_{i=1}^m\left(\left(d_{G^*}^+(v_i)\right)^2+\left(d_{T^{(i)}}^+(u_1^{(i)})\right)^2+
2d_{G^*}^+(v_i)d_{T^{(i)}}^+(u_1^{(i)})\right)\\
&\ \ \ \ +\alpha^2\sum_{i=1}^m\sum_{j=2}^{n_i}\left(d_{T^{(i)}}^+(u_j^{(i)})\right)^2+(1-\alpha)^2c_2\\
&=\alpha^2\sum_{i=1}^m\left(d_{G^*}^+(v_i)\right)^2+\alpha^2\sum_{i=1}^m\sum_{j=1}^{n_i}\left(d_{T^{(i)}}^+(u_j^{(i)})\right)^2
+2\alpha^2\sum_{i=1}^md_{G^*}^+(v_i)d_{T^{(i)}}^+(v_i)+(1-\alpha)^2c_2\\
&\leq \alpha^2\sum_{i=1}^m\left(d_{G^*}^+(v_i)\right)^2+\alpha^2\sum_{i=1}^m(n_i-1)^2
+2\alpha^2\sum_{i=1}^md_{G^*}^+(v_i)(n_i-1)+(1-\alpha)^2c_2\\
&=\alpha^2\sum_{i=1}^m\left(d_{G^*}^+(v_i)+(n_i-1)\right)^2+(1-\alpha)^2c_2\\
&=E_\alpha(G').
\end{align*}
The equality holds if and only if
$$\sum_{j=1}^{n_i}\left(d_{T^{(i)}}^+(u_j^{(i)})\right)^2+2d_{G^*}^+(v_i)d_{T^{(i)}}^+(v_i)= (n_i-1)^2+2d_{G^*}^+(v_i)(n_i-1),$$
for all $i=1,2,\ldots,m$. Anyway, the strong component $G^*$ does not change, so $d_{G^*}^+(v_i)$ does not change. That is, $d_G^+(u_1^{(i)})=d_{T^{(i)}}^+(v_i)=n_i-1$, and $d_G^+(u_j^{(i)})=0$, where $i=1,2,\ldots,m$ and $j=2,3,\ldots,n_i$. Then each directed tree $T^{(i)}$ is an out-star $\stackrel{\rightarrow}{K}_{1,n_i-1}$.
Hence, we have $E_\alpha(G')\geq E_\alpha(G)$ with equality holds if and only if $G\cong G'$.
\end{proof}
\noindent\begin{theorem}\label{th:ch-3.6} Let $G',G''\in\mathcal{G}_n^m$ be two non-strongly connected digraphs as defined in Definition \ref{de:ch-1.1}. Then $E_\alpha(G'')\geq E_\alpha(G')$ with equality holds if and only if $G'\cong G''$.
\end{theorem}
\begin{proof}
By the definition of $G''$, we know $G''\in\mathcal{G}_n^m$ is a non-strongly connected digraph which only has an out-star $\stackrel{\rightarrow}{K}_{1,n-m}$ and the centre of $\stackrel{\rightarrow}{K}_{1,n-m}$ is $v_1$ of $G^*$, $v_1$ is the maximal outdegree vertex of $G^*$ and other directed tree $T^{(i)}$ is just a vertex $v_i$ of $G^*$ for $i=2,3,\ldots,m$. Then we have
$$E_\alpha(G'')=\alpha^2\left(d_{G^*}^+(v_1)+n-m\right)^2+\alpha^2\sum_{i=2}^m(d_{G^*}^+(v_i))^2+(1-\alpha)^2c_2.$$
Since
\begin{align*}
E_\alpha(G')&=\alpha^2\sum_{i=1}^m\left(d_{G^*}^+(v_i)+(n_i-1)\right)^2+(1-\alpha)^2c_2\\
&=\alpha^2\left(\sum_{i=1}^m(d_{G^*}^+(v_i))^2+\sum_{i=1}^m(n_i-1)^2+2\sum_{i=1}^md_{G^*}^+(v_i)(n_i-1)\right)+(1-\alpha)^2c_2\\
&\leq\alpha^2\left(\sum_{i=1}^m(d_{G^*}^+(v_i))^2+\left(\sum_{i=1}^m(n_i-1)\right)^2+2\sum_{i=1}^md_{G^*}^+(v_1)(n_i-1)\right)+(1-\alpha)^2c_2\\
&=\alpha^2\left(\sum_{i=1}^m(d_{G^*}^+(v_i))^2+(n-m)^2+2d_{G^*}^+(v_1)(n-m)\right)+(1-\alpha)^2c_2\\
&=\alpha^2\left(d_{G^*}^+(v_1)+n-m\right)^2+\alpha^2\sum_{i=2}^m(d_{G^*}^+(v_i))^2+(1-\alpha)^2c_2\\
&=E_\alpha(G'').
\end{align*}
The equality holds if and only if
$$\sum_{i=1}^m(n_i-1)^2+2\sum_{i=1}^md_{G^*}^+(v_i)(n_i-1)=\left(\sum_{i=1}^m(n_i-1)\right)^2+2\sum_{i=1}^md_{G^*}^+(v_1)(n_i-1).$$
Anyway, the strong component $G^*$ does not change, so $d_{G^*}^+(v_i)$ does not change. That is, $n_i-1=0$ for all $i=2,3,\ldots,m$ and $n_1=n-m+1$. Then the directed tree $T^{(1)}$ is an out-star $\stackrel{\rightarrow}{K}_{1,n-m}$, and each other directed tree is a vertex $v_i$, where $i=2,3,\ldots,m$. Hence, we have $E_\alpha(G'')\geq E_\alpha(G')$ with equality holds if and only if $G'\cong G''$.
\end{proof}
\noindent\begin{theorem}\label{th:ch-3.7} Let $G,G'''\in\mathcal{G}_n^m$ be two non-strongly connected digraphs as defined in Definition \ref{de:ch-1.1}. Then $E_\alpha(G)\geq E_\alpha(G''')$ with equality holds if and only if $G\cong G'''$.
\end{theorem}
\begin{proof}
From Lemma \ref{le:ch-3.4}, we know the minimal $A_\alpha$ energy of $T^{(i)}$ is
$$\left(E_\alpha(T^{(i)})\right)_{\mathrm{min}}=\alpha^2(n_i-1),$$
where $i=1,2,\ldots,m$. Similar to the proof of Theorem \ref{th:ch-3.5}, we can get the result easily. And
$$E_\alpha(G''')=\alpha^2\sum_{i=1}^n(d_{G^*}^+(v_i))^2+\alpha^2(n-m)+(1-\alpha)^2c_2.$$
\end{proof}
From Theorems \ref{th:ch-3.5} and \ref{th:ch-3.7}, we have the following results.
\noindent\begin{theorem}\label{th:ch-3.8} Among all digraphs in $\mathcal{G}_n^m$, $G''$ is the unique digraph which has the maximal $A_\alpha$ energy and $G'''$ is the unique digraph which has the minimal $A_\alpha$ energy.
\end{theorem}
\noindent\begin{corollary}\label{co:ch-3.9} Let $G\in\mathcal{G}_n^m$ be a non-strongly connected digraph with $n$ vertices. Then
\begin{align*}
&\alpha^2\sum_{i=1}^m(d_{G^*}^+(v_i))^2+\alpha^2(n-m)+(1-\alpha)^2c_2\leq E_\alpha(G)\\
&\leq\alpha^2\left(d_{G^*}^+(v_1)+n-m\right)^2+\alpha^2\sum_{i=2}^m(d_{G^*}^+(v_i))^2+(1-\alpha)^2c_2.
\end{align*}
Moreover, the first equality holds if and only if each directed tree is in-tree which the root of the in-tree is $v_i$ of $G^*$, where $i=1,2,\ldots,m$; the second equality holds if and only if $G\in\mathcal{G}_n^m$ only has an out-star $\stackrel{\rightarrow}{K}_{1,n-m}$ and the centre of $\stackrel{\rightarrow}{K}_{1,n-m}$ is $v_1$ of $G^*$, $v_1$ is the maximal outdegree vertex of $G^*$ and each other directed tree $T^{(i)}$ is just a vertex $v_i$ of $G^*$ for $i=2,3,\ldots,m$.
\end{corollary}
From Corollary \ref{co:ch-3.9}, we can get some bounds of $A_\alpha$ energies of special non-strongly connected digraphs.
\noindent\begin{corollary}\label{ex:ch-3.10} The bounds of $A_\alpha$ energies of special non-strongly connected digraphs $\widehat{U}_n^{m}$, $\widehat{\infty}[m_1,m_2,\ldots,m_t]$ and $\widehat{B}[p,q]$.
(i) Let $\widehat{U}_n^{m}\in\mathcal{G}_n^m$ be a unicyclic digraph with order $n$ which contains a unique directed cycle $C_m$ and some directed trees which are hung on each vertex of $C_m$, where $m\geq2$. Then
$$2\alpha^2+\alpha^2(n-2)+2(1-\alpha)^2\leq E_\alpha(\widehat{U}_n^{2})\leq\alpha^2(n-1)^2+\alpha^2+2(1-\alpha)^2,$$
and
$$\alpha^2m+\alpha^2(n-m)\leq E_\alpha(\widehat{U}_n^{m})\leq\alpha^2(n-m+1)^2+\alpha^2(m-1)\ (m\geq3).$$
Moreover, the first equality holds if and only if each directed tree is in-tree which the root of the in-tree is connected with $C_m$; the second equality holds if and only if $\widehat{U}_n^{m}\in\mathcal{G}_n^m$ only has an out-star $\stackrel{\rightarrow}{K}_{1,n-m}$ and the centre of $\stackrel{\rightarrow}{K}_{1,n-m}$ is an any vertex of $C_m$.
(ii) Let $\widehat{\infty}[m_1,m_2,\ldots,m_t]\in\mathcal{G}_n^m$ be a generalized $\widehat{\infty}$-digraph with order $n$ which contains $\infty[m_1,m_2,\ldots,m_t]$ and some directed trees which are hung on each vertex of $\infty[m_1,m_2,\ldots,m_t]$, where $2=m_1\cdots=m_s<m_{s+1}\leq\cdots\leq m_t$, $m=\sum_{i=1}^tm_i-t+1$ and the common vertex of $t$ directed cycles $C_{m_i}$ is $v$. Then
\begin{align*}
&\alpha^2(m-1+t^2)+\alpha^2(n-m)+2s(1-\alpha)^2\leq E_\alpha(\widehat{\infty}[m_1,m_2,\ldots,m_t])\\
&\leq\alpha^2\left(n-m+t\right)^2+\alpha^2(m-1)+2s(1-\alpha)^2.
\end{align*}
Moreover, the first equality holds if and only if each directed tree is in-tree which the root of the in-tree is connected with $\infty[m_1,m_2,\ldots,m_t]$; the second equality holds if and only if $\widehat{\infty}[m_1,m_2,\ldots,m_t]\in\mathcal{G}_n^m$ only has an out-star $\stackrel{\rightarrow}{K}_{1,n-m}$ and the centre of $\stackrel{\rightarrow}{K}_{1,n-m}$ is $v$.
(iii) Let $\widehat{B}[p,q]\in\mathcal{G}_n^m$ be a digraph with order $n$ vertices which contains $B[p,q]$ and some directed trees which are hung on each vertex of $B[p,q]$, where $\mathcal{V}(B[p,q])=m$ and $p\geq q$. If both $(x,y)$ and $(y,x)$ are arcs in $\widehat{B}[p,q]$, then
\begin{align*}
&\alpha^2(m-2+p^2+q^2)+\alpha^2(n-m)+2(1-\alpha)^2\leq E_\alpha(\widehat{B}[p,q])\\
&\leq\alpha^2\left(n-m+p\right)^2+\alpha^2(m-2+q^2)+2(1-\alpha)^2.
\end{align*}
Otherwise,
$$\alpha^2(m-2+p^2+q^2)+\alpha^2(n-m)\leq E_\alpha(\widehat{B}[p,q])\leq\alpha^2\left(n-m+p\right)^2+\alpha^2(m-2+q^2).$$
Moreover, the first equality holds if and only if each directed tree is in-tree which the root of the in-tree is connected with $B[p,q]$; the second equality holds if and only if $\widehat{B}[p,q]\in\mathcal{G}_n^m$ only has an out-star $\stackrel{\rightarrow}{K}_{1,n-m}$ and the centre of $\stackrel{\rightarrow}{K}_{1,n-m}$ is $x$.
\end{corollary}
\end{document} | math | 35,622 |
\begin{document}
\title[]{The Hurwitz-type theorem for the regular Coulomb wave function via Hankel determinants}
\author[\'A. Baricz]{\'Arp\'ad Baricz}
\address{Department of Economics, Babe\c{s}-Bolyai University, Cluj-Napoca, Romania}
\address{Institute of Applied Mathematics, \'Obuda University, Budapest, Hungary}
\email{[email protected]}
\author[F. \v{S}tampach]{Franti\v{s}ek \v{S}tampach}
\address{Department of Mathematics, Stockholm University, Stockholm, Sweden}
\address{Department of Applied Mathematics, Faculty of Information Technology, Czech Technical University in~Prague,
Th{\' a}kurova~9, 160~00 Praha, Czech Republic}
\email{[email protected]}
\keywords{Hankel determinant, Coulomb wave function, Bessel function, Rayleigh function}
\subjclass[2010]{15A15, 33C15, 33C10}
\begin{abstract}
We derive a closed formula for the determinant of the Hankel matrix whose entries are given by sums of negative powers of the zeros of the regular Coulomb wave function.
This new identity applied together with results of Grommer and Chebotarev allows us to prove a Hurwitz-type theorem about the zeros of the regular
Coulomb wave function. As a particular case, we obtain a new proof of the classical Hurwitz's theorem from the theory of Bessel functions that is based on algebraic
arguments. In addition, several Hankel determinants with entries given by the Rayleigh function and Bernoulli numbers are also evaluated.
\end{abstract}
\maketitle
\section{Introduction}
The Bessel functions as well as the Coulomb wave functions belong to the very classical special functions
that appear frequently at various places in mathematics and physics.
Since the regular Coulomb wave function represents a generalization of the Bessel function of the first kind,
its properties are often reminiscent to the respective properties of the Bessel function. On the other hand,
not all methods applicable to Bessel functions admit a straightforward generalization to the case of Coulomb wave functions.
A significant part of the research on Bessel functions and their generalizations is devoted to the study of their zeros.
Naturally, methods of mathematical, in particular, complex analysis turn out to be the very well suited techniques in
the analysis of the zeros that prove their usefulness many times during the last century. However, one of the main aim of this article
is to stress the importance of linear algebra in the analysis of zeros of entire functions. By using primarily linear algebraic techniques,
we prove a theorem on the reality and the exact number of possible complex zeros of the regular Coulomb wave function. The proof consists
of two main ingredients. First, with the aid of certain properties of a particularly chosen family of orthogonal polynomials which is intimately
related to the Coulomb wave function and was studied in~\cite{stampachstovicek_jmaa14}, we evaluate the determinant of the Hankel matrix whose
entries are given by sums of negative powers of the zeros. Second, we combine the formula for the Hankel determinant with general results of
Grommer~\cite{grommer_14} and Chebotarev~\cite{chebotarev_ma28}, which straightforwardly yields the desired goal.
The obtained result generalizes the well-known theorem of Hurwitz about the zeros of the Bessel function of the first kind; see Theorem~\ref{thm:hurwitz} below
for its formulation. Consequently, the present approach provides an alternative proof of this classical result, which is rather algebraic and simple.
Let us remark that the original proof from Hurwitz~\cite{hurwitz_ma88}, based on the connection between Lommel polynomials and Bessel functions,
need not be easy to read nowadays since the modern terms were not introduced in his time. In addition, the original proof contained some gaps that were corrected
later by Watson~\cite{watson_plms20}. Nonetheless, the Hurwitz's result was very influential and many other proofs were found; see, for example, the articles of
Hilb~\cite{hilb_mz22}, Obreschkoff~\cite{obreschkoff_jdmv29}, P{\' o}lya~\cite{polya_jdmv29}, Hille and Szeg\H{o}~\cite{hilleszego_bams43}, Peyerimhoff~\cite{peyerimhoff_mmj13},
Runckel~\cite{runckel_tams69}, and Ki and Kim~\cite{kikim_dmj00}.
Let us also point out that the methods developed in the papers cited above seem not to be readily applicable to the case of the Coulomb wave function treated here.
A reason for this can be that, unlike Lommel polynomials, no explicit expression for the coefficients of the orthogonal polynomials associated with the Coulomb
wave function is known. Rather than that, only a recurrence relation for these coefficients is available,
see~\cite[Prop.~10]{stampachstovicek_jmaa14}.
The main results of this paper are Theorems~\ref{thm:det_hankel_coulomb} and~\ref{thm:coulomb_zeros}, and the paper is organized as follows.
In Section~\ref{sec:coulomb_func}, we briefly recall necessary definitions and basic properties of the regular Coulomb wave
function and an associated spectral zeta function which, in a special case, simplifies to the well-known Rayleigh function.
In Section~\ref{sec:hankel_det}, we prove a closed formula for the determinant of the Hankel matrix whose entries are
given by the zeta function associated with the regular Coulomb wave function. As a corollary, we obtain formulas for
determinants of two Hankel matrices with entries given by the Rayleigh function. These two Hankel determinants
can be viewed as two particular instances of a more general Hankel matrix. Although no simple formula for the determinant
of this more general matrix is expected, we provide some partial results on the account of its evaluation.
Yet another corollary on some Hankel determinants with Bernoulli and Genocchi numbers is presented.
Finally, as an application of the main result of Section~\ref{sec:hankel_det} and the results of Grommer and Chebotarev,
we prove the Hurwitz-type theorem for the zeros of the regular Coulomb wave function in Section~\ref{sec:zeros_coulomb}.
\section{The regular Coulomb wave function and the associated zeta function}\label{sec:coulomb_func}
Recall that regular and irregular Coulomb wave functions,
$F_{L}(\eta,\rho)$ and $G_{L}(\eta,\rho)$, are two linearly
independent solutions of the second-order differential equation
\[
\frac{d^{2}u}{d\rho^{2}}+\left(1-\frac{2\eta}{\rho}
- \frac{L(L+1)}{\rho^{2}}\right)\!u = 0,
\]
see, for instance, \cite[Chp.~14]{abramowitz64}. The function $F_{L}(\eta,\rho)$ admits the decomposition
\cite[Eqs.~14.1.3 and 14.1.7]{abramowitz64}
\begin{equation}
F_{L}(\eta,\rho)
= C_{L}(\eta)\rho^{L+1}\phi_{L}(\eta,\rho),
\label{eq:F_L_decomp}
\end{equation}
where
\[
C_{L}(\eta):=\frac{2^{L}{\rm e}^{-\pi\eta/2}\,|\Gamma(L+1+i\eta)|}{\Gamma(2L+2)},
\]
and
\begin{equation}
\phi_{L}(\eta,\rho) := {\rm e}^{-i\rho}\,{}_{1}F_{1}(L+1-i\eta;2L+2;2i\rho).
\label{eq:phi_rel_1F1}
\end{equation}
The confluent hypergeometric function $\,{}_{1}F_{1}$ is defined by the power series~\cite[Chp.~13]{abramowitz64}
\[
{}_{1}F_{1}(a;b;z):=\sum_{n=0}^{\infty}\frac{(a)_{n}}{(b)_{n}}\frac{z^{n}}{n!},
\]
for $a,b,z\in{\mathbb{C}}$, such that $b\notin-\mathbb{N}_{0}$, where the Pochhammer symbol $(a)_{0}=1$ and $(a)_{n}=a(a+1)\dots(a+n-1)$, for $n\in\mathbb{N}$.
Here and below, $\mathbb{N}$ is the set of all positive integers and $\mathbb{N}_{0}:=\{0\}\cup\mathbb{N}$.
For the particular values of parameters, $L=\nu-1/2$ and $\eta=0$, one
gets \cite[Eqs. 14.6.6 and 13.6.1]{abramowitz64}
\[
F_{\nu-1/2}(0,\rho) = \sqrt{\frac{\pi\rho}{2}}\, J_{\nu}(\rho)
\]
and
\begin{equation}
\phi_{\nu-1/2}(0,\rho) = {\rm e}^{-i\rho}\,
{}_{1}F_{1}(\nu+1/2;2\nu+1;2i\rho)
\,=\,\Gamma(\nu+1)\left(\frac{2}{\rho}\right)^{\!\nu}
J_{\nu}(\rho),
\label{eq:phi_part}
\end{equation}
where $J_{\nu}$ is the Bessel function of the first kind. From this point of view, the regular Coulomb wave function represents a one-parameter generalization of the Bessel function of the first kind.
One can see from~\eqref{eq:F_L_decomp} that, with the possible exception of the origin, the zeros of $F_{L}(\eta,\cdot)$ are the same as of $\phi_{L}(\eta,\cdot)$.
If $L\notin-(\mathbb{N}+1)/2$, the function $\phi_{L}(\eta,\cdot)$ is well-defined for all $\eta\in{\mathbb{C}}$. Even for $L\in-(\mathbb{N}+1)/2$, the function $\phi_{L}(\eta,\cdot)$ is well-defined,
if we additionally require ${\rm i}\eta\in{\mathbb{Z}}$ and $L+2+{\rm i}\eta\leq0$, in which case the confluent hypergeometric series in~\eqref{eq:phi_rel_1F1} is terminating. In general,
the function $F_{L}(\eta,\rho)$ can be continued analytically to the complex values of all variables $L$, $\eta$, and $\rho$; the interested reader is referred to~\cite{dzieciol-etal_jmp99,humblet_ap84,barnett-thompson_jcp86}.
As one can readily verify by using the Taylor coefficients of the confluent hypergeometric function in~\eqref{eq:phi_rel_1F1}, $\phi_{L}(\eta,\cdot)$ is an entire function of order $1$ for $L\notin-(\mathbb{N}+1)/2$ and $\eta\in{\mathbb{C}}$.
Consequently, the series
\[
\zeta_{L}(k):=\sum_{n=1}^{\infty}\frac{1}{\rho_{L,n}^{k}},
\]
where $\rho_{L,1},\rho_{L,2},\dots$ are the zeros of $\phi_{L}(\eta,\cdot)$, is absolutely convergent for $k\geq2$, $L\notin-(\mathbb{N}+1)/2$, and $\eta\in{\mathbb{C}}$. Here we use the notation
from~\cite{stampachstovicek_jmaa14} where $\zeta_{L}$ is referred to as the spectral zeta function since the zeros of $\phi_{L}(\eta,\cdot)$ are eigenvalues of a certain Jacobi
operator. Although $\zeta_{L}$ as well as $\rho_{L,n}$ depend also on $\eta$, the dependence is not explicitly designated for brevity.
Let us remark that $\zeta_{L}(k)$ is a polynomial in $\eta$ and a rational function in $L$ with singularities in the set $-(\mathbb{N}+1)/2$, as it can be seen from the recurrence relation
\begin{equation}
\zeta_{L}(2) = \frac{1}{2L+3}\left(1+\frac{\eta^{2}}{(L+1)^{2}}\right)
\label{eq:zeta_F_2}
\end{equation}
and
\begin{equation}
\zeta_{L}(k+1) = \frac{1}{2L+k+2}\left(\frac{2\eta}{L+1}\,\zeta_{L}(k)+\sum_{l=1}^{k-2}\zeta_{L}(l+1)\zeta_{L}(k-l)\right)\!,\ \ k\geq2,
\label{eq:zeta_F_genrecur}
\end{equation}
see \cite[Eqs.~(78) and~(79)]{stampachstovicek_jmaa14} or \cite[Eq.~(2.18)]{baricz_jmaa15}.
In the special case of $L=\nu-1/2$ and $\eta=0$, one has
\begin{equation}
\zeta_{\nu-1/2}(k)=\sum_{n\in{\mathbb{Z}}\setminus\{0\}}\frac{1}{j_{\nu,n}^{k}},
\label{eq:zeta_bessel_zeros}
\end{equation}
where $j_{\nu,1}, j_{\nu,2},\dots$ are the zeros of $J_{\nu}$ for which either ${\mathbb{R}}e j_{\nu,n}>0$ or $\mathop\mathrm{Im}\nolimits j_{\nu,n}>0$, if ${\mathbb{R}}e j_{\nu,n}=0$.
The remaining zeros are $j_{\nu,-n}:=-j_{\nu,n}$, for $n\in\mathbb{N}$, which follows from the fact that
that the function $\rho\mapsto\rho^{-\nu}J_{\nu}(\rho)$ is even and the origin is not a zero.
Consequently, for $k\in\mathbb{N}$, one gets
\begin{equation}
\zeta_{\nu-1/2}(2k+1)=0
\label{eq:zeta_vanish}
\end{equation}
and
\begin{equation}
\zeta_{\nu-1/2}(2k)=2\sigma_{2k}(\nu)=2\sum_{n=1}^{\infty}\frac{1}{j_{\nu,n}^{2k}},
\label{eq:zeta_rayleigh_spec}
\end{equation}
where $\sigma_{2k}(\nu)$ is the Rayleigh function of order $2k$ introduced by Kishore in~\cite{kishore_pams63}.
\section{Hankel determinants}\label{sec:hankel_det}
For $L\notin-(\mathbb{N}+1)/2$, $\eta\in{\mathbb{C}}$, and $n\in\mathbb{N}$, we define the Hankel matrix
\begin{equation}
H_{n}(L,\eta):=\begin{pmatrix}
\zeta_{L}(2) & \zeta_{L}(3) & \dots & \zeta_{L}(n+1)\\
\zeta_{L}(3) & \zeta_{L}(4) & \dots & \zeta_{L}(n+2)\\
\vdots & \vdots & \ddots & \vdots\\
\zeta_{L}(n+1) & \zeta_{L}(n+2) & \dots & \zeta_{L}(2n)
\end{pmatrix}\!.
\label{eq:def_hankel_mat_coulomb}
\end{equation}
By making use of the properties of a particular family of orthogonal polynomials associated with the regular Coulomb wave function studied in~\cite{stampachstovicek_jmaa14}, we
may deduce a simple formula for the determinant of $H_{n}(L,\eta)$.
\begin{thm}\label{thm:det_hankel_coulomb}
For $L\notin-(\mathbb{N}+1)/2$, $\eta\in{\mathbb{C}}$, and $n\in\mathbb{N}$, one has
\begin{equation}
\det H_{n}(L,\eta)=\prod_{k=0}^{n-1}\frac{1}{(2L+2n-2k+1)^{2k+1}}\left(1+\frac{\eta^{2}}{(L+n-k)^{2}}\right)^{\!k+1}.
\label{eq:det_hankel_coulomb}
\end{equation}
\end{thm}
\begin{proof}
The proof is based on the well-known relation between the recurrence coefficients from the three-term recurrence for a family of orthogonal polynomials and the
determinant of the corresponding moment Hankel matrix. Namely, we use that for the orthogonal polynomials generated by the recurrence
\begin{equation}
p_{n+1}(z)=(z-b_{n})p_{n}(z)-a_{n}p_{n-1}(z), \quad n\in\mathbb{N},
\label{eq:recur_monic_ogp}
\end{equation}
with the initial conditions $p_{-1}(z)=0$ and $p_{0}(z)=1$, it holds
\begin{equation}
\Delta_{n}=\prod_{m=1}^{n-1}\prod_{j=1}^{m}a_{j}, \quad n\in\mathbb{N},
\label{eq:delta_n_rel_a_n}
\end{equation}
where
\[
\Delta_{0}:=1, \quad \Delta_{n}:=\det\left[\mathcal{L}\left(z^{i+j}\right)\right]_{i,j=0}^{n-1}, \quad n\in\mathbb{N},
\]
and $\mathcal{L}$ is the corresponding normalized moment functional; see~\cite[Chp.~I, Thm.~4.2(a)]{chihara78}.
To obtain the formula from the statement, we make use of the particular family of orthogonal polynomials studied in~\cite{stampachstovicek_jmaa14},
for which the moment Hankel matrix coincides with $H_{n}(L,\eta)$ up to an unimportant multiplicative factor. These polynomials satisfies~\eqref{eq:recur_monic_ogp}
with
\begin{equation}
a_{n} = \frac{(n+L+1)^{2}+\eta^{2}}{(n+L+1)^{2}(2n+2L+1)(2n+2L+3)},
\quad b_{n}=-\frac{\eta}{(n+L+1)(n+L+2)},
\label{eq:lambda_w_coulomb}
\end{equation}
and the corresponding moment sequence is given by
\[
\mathcal{L}\left(z^{n}\right)=\frac{\zeta_{L}(n+2)}{\zeta_{L}(2)}, \quad n\in\mathbb{N}_{0},
\]
see \cite[Rem.~19]{stampachstovicek_jmaa14}. Consequently, by taking~\eqref{eq:delta_n_rel_a_n} into account, one gets
\begin{equation}
\det H_{n}(L,\eta)=\left(\zeta_{L}(2)\right)^{n}\prod_{m=1}^{n-1}\prod_{j=1}^{m}a_{j}, \quad n\in\mathbb{N}.
\label{eq:det_H_n_rel_moment_det}
\end{equation}
Now, it suffices to use~\eqref{eq:zeta_F_2} and~\eqref{eq:lambda_w_coulomb} to deduce the formula from the statement.
\end{proof}
\begin{rem}
The functions $\zeta_{L}(n)$, for $n\geq 2$, appear as coefficients in the power series expansion of the logarithmic derivative of $\phi_{L}(\eta,\rho)$ with respect to $\rho$,
see, for example, the first unlabeled equation above \cite[Eq.~(77)]{stampachstovicek_jmaa14}.
As a consequence, Theorem~\ref{thm:det_hankel_coulomb} could be equivalently proved by using the connection between the Hankel determinant~\eqref{eq:def_hankel_mat_coulomb} and
the known coefficients from the continued fraction expansion of the logarithmic derivative of $\phi_{L}(\eta,\rho)$, see \cite[Thm.~7.14]{jones_thron-80} and~\cite{barnett-etal_cpc74}.
On the other hand, one may observe by comparing the formula in \cite[Eq.~7.2.18a]{jones_thron-80} with~\cite[Chp.~I, Thm.~4.2(a)]{chihara78}
that the coefficient $k_{1}$ from the continued fraction expansion in \cite[Eq.~7.2.16]{jones_thron-80} actually coincides with $\zeta_{L}(2)$ and
$k_{n}$ coincides with $a_{n-1}$ from \eqref{eq:lambda_w_coulomb} for $n\geq2$. By using \cite[Eq.~7.2.29]{jones_thron-80}, one arrives
at~\eqref{eq:det_H_n_rel_moment_det} again.
\end{rem}
As a corollary of Theorem~\ref{thm:det_hankel_coulomb}, we compute determinants of two Hankel matrices with entries given by the Rayleigh function of even order. For this purpose, we define
\[
H_{n}^{(\ell)}(\nu):=\begin{pmatrix}
\sigma_{2\ell+2}(\nu) & \sigma_{2\ell+4}(\nu) & \dots & \sigma_{2n+2\ell}(\nu)\\
\sigma_{2\ell+4}(\nu) & \sigma_{2\ell+6}(\nu) & \dots & \sigma_{2n+2\ell+2}(\nu)\\
\vdots & \vdots & \ddots & \vdots\\
\sigma_{2n+2\ell}(\nu) & \sigma_{2n+2\ell+2}(\nu) & \dots & \sigma_{4n+2\ell-2}(\nu)
\end{pmatrix}\!,
\]
for $n\in\mathbb{N}$, $\ell\in\mathbb{N}_{0}$, and $\nu\notin-\mathbb{N}$.
\begin{cor}\label{cor:det_hankel_bessel}
For $n\in\mathbb{N}$, $\nu\notin-\mathbb{N}$, and $\ell\in\{0,1\}$, one has
\begin{equation}
\det H_{n}^{(\ell)}(\nu)=2^{-2n(n+\ell)}\prod_{k=1}^{2n+\ell-1}\left(\nu+k\right)^{k-2n-\ell}.
\label{eq:det_hankel_bessel}
\end{equation}
\end{cor}
\begin{proof}
By taking the particular parameters $L=\nu-1/2$ and $\eta=0$ in~\eqref{eq:def_hankel_mat_coulomb} and using~\eqref{eq:zeta_bessel_zeros}, \eqref{eq:zeta_vanish}, and~\eqref{eq:zeta_rayleigh_spec},
one sees that the $(i,j)$-th element of the matrix $H_{n}(\nu-1/2,0)$ coincides with $2\sigma_{i+j}(\nu)$, if the parity of $i$ and $j$ is the same,
while it vanishes, if the parity of $i$ and $j$ is different. The latter observation and simple manipulations with the determinants, see for
example~\cite[Lem.~1.34]{holtz-tyaglov_siam12}, yield the identities
\begin{equation}
\det H_{2n+1}(\nu-1/2,0)=2^{2n+1}\det H_{n+1}^{(0)}(\nu)\det H_{n}^{(1)}(\nu),
\label{eq:det_col_det_bes_odd}
\end{equation}
and
\begin{equation}
\det H_{2n}(\nu-1/2,0)=2^{2n}\det H_{n}^{(0)}(\nu)\det H_{n}^{(1)}(\nu),
\label{eq:det_col_det_bes_even}
\end{equation}
for $n\in\mathbb{N}$.
From~\eqref{eq:det_col_det_bes_odd} and~\eqref{eq:det_col_det_bes_even}, one deduces that
\[
\det H_{n}^{(0)}(\nu)=\frac{\det H_{2n-1}(\nu-1/2,0)}{2\det H_{2n-2}(\nu-1/2,0)}\det H_{n-1}^{(0)}(\nu)
\]
and
\[
\det H_{n}^{(1)}(\nu)=\frac{\det H_{2n}(\nu-1/2,0)}{2\det H_{2n-1}(\nu-1/2,0)}\det H_{n-1}^{(1)}(\nu),
\]
for $n\geq2$, which further implies that
\begin{equation}
\det H_{n}^{(0)}(\nu)=\sigma_{2}(\nu)\prod_{k=2}^{n}\frac{\det H_{2k-1}(\nu-1/2,0)}{2\det H_{2k-2}(\nu-1/2,0)}
\label{det_bes_0_inproof}
\end{equation}
and
\[
\det H_{n}^{(1)}(\nu)=\sigma_{4}(\nu)\prod_{k=2}^{n}\frac{\det H_{2k}(\nu-1/2,0)}{2\det H_{2k-1}(\nu-1/2,0)},
\]
for $n\in\mathbb{N}$. At this point, one can use Theorem~\ref{thm:det_hankel_coulomb} and the
well-known formulas
\begin{equation}
\sigma_{2}(\nu)=\frac{1}{4(\nu+1)} \quad \mbox{ and } \quad \sigma_{4}(\nu)=\frac{1}{16(\nu+1)^{2}(\nu+2)},
\label{eq:sig2_sig4}
\end{equation}
which can be computed, for example, from~\eqref{eq:zeta_F_2} and~\eqref{eq:zeta_F_genrecur}, to verify the formulas from the statement.
Alternatively, one can use~\eqref{eq:det_H_n_rel_moment_det} which particularly reads
\[
\det H_{n}(\nu-1/2,0)=\left(2\sigma_{2}(\nu)\right)^{n}\Delta_{n}, \quad n\in\mathbb{N},
\]
together with the identity
\[
\frac{\Delta_{n+1}}{\Delta_{n}}=\prod_{j=1}^{n}a_{j},
\]
as it follows from~\eqref{eq:delta_n_rel_a_n}, to rewrite the right-hand side of~\eqref{det_bes_0_inproof} getting
\[
\det H_{n}^{(0)}(\nu)=\left(\sigma_{2}(\nu)\right)^{n}\prod_{k=2}^{n}\prod_{j=1}^{2k-2}a_{j}, \quad n\in\mathbb{N}.
\]
Here $a_{j}$ is given by~\eqref{eq:lambda_w_coulomb} where $L=\nu-1/2$ and $\eta=0$, i.e.,
\[
\det H_{n}^{(0)}(\nu)=\left(\sigma_{2}(\nu)\right)^{n}\prod_{k=2}^{n}\prod_{j=1}^{2k-2}\frac{1}{4(\nu+j)(\nu+j+1)}, \quad n\in\mathbb{N}.
\]
Analogically, one shows that
\[
\det H_{n}^{(1)}(\nu)=\sigma_{4}(\nu)\left(\sigma_{2}(\nu)\right)^{n-1}\prod_{k=2}^{n}\prod_{j=1}^{2k-1}\frac{1}{4(\nu+j)(\nu+j+1)}, \quad n\in\mathbb{N}.
\]
Finally, recalling~\eqref{eq:sig2_sig4}, simple algebraic manipulations yield the identities from the statement.
\end{proof}
\begin{rem}
An incorrect formula for $\det H_{n}^{(0)}(\nu)$ appeared earlier in \cite[Thm.~1]{zhangchen_aaa14}. To obtain the correct formula,
one should write $\sigma_{\nu}^{(k+j-1)}$ instead of $\sigma_{\nu}^{(k+j+1)}$ and $2^{-(n+1)(2n+1)}$ instead of $2^{(n+1)(2n+1)}$
in the identity therein.
\end{rem}
\begin{rem}
The identity~\eqref{eq:det_hankel_bessel} is no longer true if $\ell\geq2$. We do not have a formula for $\det H_{n}^{(\ell)}(\nu)$ for general $\ell\in\mathbb{N}_{0}$.
Nevertheless, in principle, the determinant can be computed recursively for a fixed $\ell$. Indeed, with the aid of the Desnanot--Jacobi identity, one gets
\begin{equation}
\det H_{n+1}^{(\ell)}(\nu)\det H_{n-1}^{(\ell+2)}(\nu)=\det H_{n}^{(\ell+2)}(\nu)\det H_{n}^{(\ell)}(\nu)-\left(\det H_{n}^{(\ell+1)}(\nu)\right)^{\!2},
\label{eq:desnanot-jacobi_hankel_det_bessel}
\end{equation}
for $\ell\in\mathbb{N}_{0}$ and $n\geq 2$. Observe that~\eqref{eq:desnanot-jacobi_hankel_det_bessel} is a first-order difference equation in~$n$ for $\det H_{n}^{(\ell+2)}(\nu)$,
which can be readily solved. If we temporarily denote $d_{n}^{(\ell)}:=\det H_{n}^{(\ell)}(\nu)$ and ignore for a while a possible division by zero, then the solution of
the difference equation~\eqref{eq:desnanot-jacobi_hankel_det_bessel} is given by a somewhat cumbersome formula
\[
d_{n}^{(\ell+2)}=d_{n+1}^{(\ell)}\left(\frac{\sigma_{2\ell+6}(\nu)}{\sigma_{2\ell+2}(\nu)\sigma_{2\ell+6}(\nu)-\sigma_{2\ell+4}^{2}(\nu)}
+\sum_{k=2}^{n}\frac{\left(d_{k}^{(\ell+1)}\right)^{\!2}}{d_{k}^{(\ell)}d_{k+1}^{(\ell)}}\right)\!, \quad n\in\mathbb{N}.
\]
On the other hand, it is not difficult to use~\eqref{eq:desnanot-jacobi_hankel_det_bessel} and Corollary~\ref{cor:det_hankel_bessel} in order to verify that the Hankel determinants for $\ell=2,3$ read
\begin{equation}
\det H_{n}^{(2)}(\nu)=2^{-2n(n+2)}(n+1)(n+\nu+1)\prod_{k=1}^{2n+1}\left(\nu+k\right)^{k-2n-2}
\label{eq:det_H_n_2_nu}
\end{equation}
and
\begin{align*}
\det H_{n}^{(3)}(\nu)&=2^{-2n(n+3)}\prod_{k=1}^{2n+2}\left(\nu+k\right)^{k-2n-3}\\
&\times\frac{1}{6}(n+1)(n+2)(n+\nu+1)(n+\nu+2)\left[2n^{2}+6n+3+\nu(2n+3)\right]\!.
\end{align*}
\end{rem}
\begin{rem}
Besides the obvious positivity of $\det H_{n}^{(\ell)}(\nu)$ for $n\in\mathbb{N}$, $\ell\in\{0,1\}$, and $\nu>-1$, one can also use the formula~\eqref{eq:det_hankel_bessel}
to prove that $\det H_{n}^{(\ell)}(\nu)$, as function of~$\nu$, is completely monotone for $\nu>-1$, with $n\in\mathbb{N}$ and $\ell\in\{0,1\}$ fixed.
This can be easily checked by using the fact that a product of completely monotone functions is a completely monotone function. Similarly, one can use~\eqref{eq:det_H_n_2_nu}
to verify the complete monotonicity of $\det H_{n}^{(2)}(\nu)$ for $\nu>-1$, with $n\in\mathbb{N}$ fixed. It seems that the same holds true even for $\ell\geq3$, however,
the verification would require a more sophisticated analysis.
\end{rem}
At last, we formulate yet another corollary which is obtained by even more special choice of parameters taking $\eta=0$ and either $L=0$ or $L=-1$. Recall that $\sigma_{2n}(\pm 1/2)$ can be expressed
in terms of even values of Riemann zeta function, which, in its turn, can be evaluated with the aid of Bernoulli numbers. Namely, one has \cite[Eqs.~(4) and~(5)]{kishore_pams63}
\begin{equation}
\sigma_{2n}\left(\frac{1}{2}\right)=\frac{\zeta(2n)}{\pi^{2n}}=(-1)^{n+1}\frac{2^{2n-1}}{(2n)!}B_{2n}, \quad n\in\mathbb{N},
\label{eq:sigma_2n_1/2}
\end{equation}
and
\begin{equation}
\sigma_{2n}\left(-\frac{1}{2}\right)=\left(2^{2n}-1\right)\frac{\zeta(2n)}{\pi^{2n}}=(-1)^{n}\frac{2^{2n-2}}{(2n)!}G_{2n}, \quad n\in\mathbb{N},
\label{eq:sigma_2n_-1/2}
\end{equation}
where $G_{n}=2(1-2^{n})B_{n}$ are the Genocchi numbers.
The special case of Corollary~\ref{cor:det_hankel_bessel} yields formulas for determinants of Hankel matrices
with entries given by either $B_{2n+2\ell}/(2n+2\ell)!$ or $G_{2n+2\ell}/(2n+2\ell)!$ for $\ell\in\{0,1\}$.
Formulas for determinants of Hankel matrices with entries given just by $B_{2n+2\ell}$, for $\ell\in\{0,1\}$,
can be found in~\cite[Eqs.~(3.59) and~(3.60)]{krattenthaler_slc99}. The determinant of the Hankel matrix with
entries given by $G_{2n}$ can be deduced from~\cite[Eq.~(3.12)]{dumontzeng-am94}.
\begin{cor}
For all $n\in\mathbb{N}$ and $\ell\in\{0,1\}$, one has
\begin{equation}
\det\left(\frac{B_{2j+2i+2\ell-2}}{(2j+2i+2\ell-2)!}\right)_{i,j=1}^{n}=(-1)^{n\ell}2^{-n(4n+4\ell-1)}\prod_{k=1}^{2n+\ell-1}\left(k+\frac{1}{2}\right)^{\!k-2n-\ell}
\label{eq:hankel_det_bernoulli}
\end{equation}
and
\begin{equation}
\det\left(\frac{G_{2j+2i+2\ell-2}}{(2j+2i+2\ell-2)!}\right)_{i,j=1}^{n}=(-1)^{n(\ell+1)}2^{-n(4n+4\ell-2)}\prod_{k=1}^{2n+\ell-1}\left(k-\frac{1}{2}\right)^{\!k-2n-\ell}.
\label{eq:hankel_det_genocchi}
\end{equation}
\end{cor}
\begin{proof}
First, observe that if $H_{n}$ and $\tilde{H}_{n}$ are two Hankel matrices with $(H_{n})_{i,j}=h_{i+j}$ and $(\tilde{H}_{n})_{i,j}=\tilde{h}_{i+j}$ such that
$h_{m}=\alpha^{m} \tilde{h}_{m}$, for some $\alpha\in{\mathbb{C}}$, then $H_{n}=D_{n}(\alpha)\tilde{H}_{n}D_{n}(\alpha)$, where $D_{n}(\alpha)=\mathop\mathrm{diag}\nolimits(\alpha,\alpha^{2}\dots,\alpha^{n})$.
In particular, one has $\det H_{n}=\alpha^{n(n+1)}\det \tilde{H}_{n}$.
By using the above observation together with~\eqref{eq:sigma_2n_1/2} and~\eqref{eq:sigma_2n_-1/2}, one obtains
\[
\det\left(\frac{B_{2j+2i+2\ell-2}}{(2j+2i+2\ell-2)!}\right)_{i,j=1}^{n}=(-1)^{n\ell}2^{-n(2n+2\ell-1)}\det H_{n}^{(\ell)}\left(\frac{1}{2}\right)
\]
and
\[
\det\left(\frac{G_{2j+2i+2\ell-2}}{(2j+2i+2\ell-2)!}\right)_{i,j=1}^{n}=(-1)^{n(\ell+1)}2^{-n(2n+2\ell-2)}\det H_{n}^{(\ell)}\left(-\frac{1}{2}\right)\!,
\]
for $n\in\mathbb{N}$. Now, it suffices to apply Corollary~\ref{cor:det_hankel_bessel}.
\end{proof}
\begin{rem}
For $\ell=0$, the formula~\eqref{eq:hankel_det_bernoulli} is a correct version of the expression that appeared in~\cite[Cor.~2]{zhangchen_aaa14}. In addition,
it shows that the determinant $\Delta_{m}'$ considered in~\cite[Ex.~3]{kytmanovkhodos_caop17} is indeed positive for all $m\in\mathbb{N}$.
\end{rem}
\begin{rem}
It might be also of interest that both determinants~\eqref{eq:hankel_det_bernoulli} and~\eqref{eq:hankel_det_genocchi} are equal to a reciprocal value of an integer
which is easy to check.
\end{rem}
\section{An application to the zeros of the regular Coulomb wave function}\label{sec:zeros_coulomb}
In \cite[Prop.~13]{stampachstovicek_jmaa14}, it was proved that the zeros of $\phi_{L}(\eta,\cdot)$ are all real if $-1\neq L>-3/2$ and $\eta\in{\mathbb{R}}\setminus\{0\}$ or $L>-3/2$
and $\eta=0$. This was also observed earlier by Ikebe~\cite{ikebe_mc75} for integer values of $L$ (which is an unnecessary restriction). As an application of
Theorem~\ref{thm:det_hankel_coulomb} combined with general results due to Grommer~\cite{grommer_14} and Chebotarev~\cite{chebotarev_ma28}, we can provide an
alternative proof of the above statement and, moreover, complement it by adding an information on the exact number of complex zeros of the regular Coulomb wave
function.
For this purpose, we recall the results of Grommer and Chebotarev in a special form adjusted to the situation
concerning the entire functions of order~$1$, which is the case of interest here. Chebotarev generalized the theorem of Grommer which, in its turn, is a
generalization of an analogous statement known for polynomials and attributed to Hermite~\cite{hermite_56}. For further research on this account, the reader may also
consult~\cite{chebotarev-meiman_49,krein_62,krein-langer_mn77}. Some of these classical results were recently rediscovered by Kytmanov and Khodos~\cite{kytmanovkhodos_caop17}
not mentioning the references~\cite{grommer_14,chebotarev_ma28}.
\begin{thm}\label{thm:grommer-chebotarev}
Let $f$ be an entire function of order $1$ with real Taylor coefficients. Denote
\[
D_{-1}:=1 \quad \mbox{ and } \quad D_{n}:=\det\left(s_{i+j}\right)_{i,j=0}^{n-1}, \; \mbox{ for } n\in\mathbb{N}_{0},
\]
where
\[
s_{k}:=\sum_{j=1}^{\infty}\frac{1}{z_{j}^{k+2}}, \quad k\in\mathbb{N}_{0},
\]
and $z_{1},z_{2},\dots$ are the zeros of $f$. Then the following statements hold true.
\begin{enumerate}[{\upshape i)}]
\item $\mathrm{(Grommer)}$ All the zeros of $f$ are real if and only if $D_{n}>0$ for all $n\in\mathbb{N}_{0}$.
\item $\mathrm{(Chebotarev)}$ If the sequence $\{D_{n-1}D_{n}\}_{n\in\mathbb{N}_{0}}$ contains exactly $m$ negative numbers,
then the function $f$ has $m$ distinct pairs of complex conjugate zeros and an infinite number of real zeros.
\end{enumerate}
\end{thm}
Now we are ready to prove the following statement.
\begin{thm}\label{thm:coulomb_zeros}
Suppose $\eta,L\in{\mathbb{R}}$. The the following claims hold true.
\begin{enumerate}[{\upshape i)}]
\item If $-1\neq L>-3/2$ for $\eta\neq0$ and $L>-3/2$ for $\eta=0$, then all zeros of $F_{L}(\eta,\cdot)$ are real.
\item If $L<-3/2$ and $L \notin-\mathbb{N}/2$ for $\eta\neq0$ and $L \notin-\mathbb{N}-1/2$ for $\eta=0$, then $F_{L}(\eta,\cdot)$
has $\lfloor-L-1/2\rfloor$ distinct pairs of complex conjugate zeros and an infinite number of real zeros.
\end{enumerate}
\end{thm}
\begin{proof}
Recall that the zeros of $F_{L}(\eta,\cdot)$ are the same as the zeros of $\phi_{L}(\eta,\rho)$ with the possible exception of the origin
as it follows from~\eqref{eq:F_L_decomp}. We apply Theorem~\ref{thm:grommer-chebotarev} to the function $f(\rho):=\phi_{L}(\eta,\rho)$ which is entire of order $1$.
It is by no means obvious from \eqref{eq:phi_rel_1F1} that the Taylor coefficients of $f$ are real for $\eta, L\in{\mathbb{R}}$.
To see that this is indeed the case, one can apply the Kummer transform \cite[Eq.~13.1.27]{abramowitz64}
\[
{}_{1}F_{1}(a;b;z)=e^{z}\,_{1}F_{1}(b-a;b;-z), \quad a,b,z\in{\mathbb{C}},\ b\notin-\mathbb{N}_{0},
\]
in~\eqref{eq:phi_rel_1F1}. This shows that $\overline{f(\rho)}=f(\overline{\rho})$, and hence the assumptions of Theorem~\ref{thm:grommer-chebotarev}
are fulfilled.
Denote $D_{n}:=\det H_{n+1}(L,\eta)$ for $n\in\mathbb{N}_{0}$. It is obvious from the identity~\eqref{eq:det_hankel_coulomb} that, for $-1\neq L>-3/2$ and $\eta\neq0$,
$D_{n}>0$ for all $n\in\mathbb{N}_{0}$. If $\eta=0$, then by~\eqref{eq:det_col_det_bes_odd} and~\eqref{eq:det_col_det_bes_even}, $D_{n}$ is equal to the product
of the expressions given in~\eqref{eq:det_hankel_bessel} where $\nu=L+1/2$. It can be readily checked that both these expressions are positive for $L>-3/2$
and so the value $L=-1$ need not be excluded in this special case. In total, the claim (i) follows from the part (i) of Theorem~\ref{thm:grommer-chebotarev}.
Further, it follows from~\eqref{eq:det_hankel_coulomb} that
\begin{align}
D_{n-1}D_{n}=\frac{1}{2L+2n+3}&\left(1+\frac{\eta^{2}}{(L+n+1)^{2}}\right)\nonumber\\
&\times\prod_{k=0}^{n-1}\frac{1}{(2L+2n-2k+1)^{4k+4}}\left(1+\frac{\eta^{2}}{(L+n-k)^{2}}\right)^{\!2k+3},
\label{eq:DD_eta_neq_0}
\end{align}
for $\eta\neq0$, $L\notin-(\mathbb{N}+1)/2$, and $n\in\mathbb{N}_{0}$. Similarly as above, in the particular case when $\eta=0$, $\phi_{L}(0,\rho)$ is to be understood as the Bessel function in the sense of~\eqref{eq:phi_part}, and
the negative integer values of $L$ need not be excluded. In this case, the formula~\eqref{eq:DD_eta_neq_0} takes the form
\begin{equation}
D_{n-1}D_{n}=\frac{1}{2L+2n+3}\prod_{k=0}^{n-1}\frac{1}{(2L+2n-2k+1)^{4k+4}},
\label{eq:DD_eta_eq_0}
\end{equation}
and holds true for $L\notin-\mathbb{N}-1/2$ and $n\in\mathbb{N}_{0}$. Recall that, in~\eqref{eq:DD_eta_neq_0} and~\eqref{eq:DD_eta_eq_0}, $D_{-1}=1$ and the empty product is set $1$ by definition.
In any case, it is clear from~\eqref{eq:DD_eta_neq_0} and~\eqref{eq:DD_eta_eq_0} that the sign of $D_{n-1}D_{n}$ equals the sign of the factor $2L+2n+3$.
Consequently, the number of negative elements in $\{D_{n-1}D_{n}\}_{n\in\mathbb{N}_{0}}$ coincides with the the number of elements of the set $\{n\in\mathbb{N}_{0} \mid 2L+2n+3<0\}$.
This observation together with the part (ii) of Theorem~\ref{thm:grommer-chebotarev} implies the claim~(ii) and the statement is proved.
\end{proof}
Apart from the claim on the purely imaginary zeros, the particular case of Theorem~\ref{thm:coulomb_zeros} with $\eta=0$
and $L=\nu-1/2$ implies Hurwitz's theorem about the zeros of the Bessel function of the first kind, which can be formulated as follows.
\begin{thm}[Hurwitz] \label{thm:hurwitz} Then following statements hold true.
\begin{enumerate}[{\upshape i)}]
\item If $\nu>-1$, then all zeros of $J_{\nu}$ are real.
\item If $-2s-2<\nu<-2s-1$ for $s\in\mathbb{N}_0$, then $J_{\nu}$ has $4s+2$ complex zeros, of which two are purely imaginary.
\item If $-2s-1<\nu<-2s$ for $s\in\mathbb{N}$, then $J_{\nu}$ has $4s$ complex zeros, of which none are purely imaginary.
\end{enumerate}
\end{thm}
\begin{rem}
The occurrence of a pair of purely imaginary zeros seems to be a special feature of the Bessel function.
No similar phenomenon was observed in the general case of the Coulomb wave function. The latter statement, however, is based on numerical experiments only.
For instance, with the aid of Wolfram Mathematica 11, we found the following numerical values for the non-real zeros of $\phi_{L}(3/2,\cdot)$:
\begin{align*}
0.1500\pm {\rm i}0.2520, &\quad\mbox{ for } L=-7/4,\\
-0.2147\pm{\rm i}0.8230,\ 0.5887\pm{\rm i}0.4090, &\quad\mbox{ for } L=-11/4,\\
-0.8719\pm{\rm i}1.2916,\ 0.3538 \pm{\rm i}1.2646,\ 1.1374\pm{\rm i}0.5345, &\quad\mbox{ for } L=-15/4.
\end{align*}
\end{rem}
\section*{Acknowledgments}
The research of \' Arp\'ad Baricz was supported by the STAR-UBB Advanced Fellowship-Intern of the Babe\c{s}-Bolyai
University of Cluj-Napoca. This author is also grateful to Mourad E.~H.~Ismail and \'Agoston R\'oth for useful discussions.
The authors thank both referees for useful comments, in particular, for providing the original references for Theorem~\ref{thm:grommer-chebotarev}.
\end{document} | math | 33,466 |
\begin{document}
\title[The commuting probability]{On the commuting probability for subgroups of a finite group}
\author{Eloisa Detomi}
\address{Dipartimento di Ingegneria dell'Informazione - DEI, Universit\`a
di Padova, Via G. Gradenigo 6/B, 35121 Padova, Italy}
\email{[email protected]}
\author{Pavel Shumyatsky}
\address{Department of Mathematics, University of Brasilia\\
Brasilia-DF \\ 70910-900 Brazil}
\email{[email protected]}
\thanks{
The first author is a member of GNSAGA (Indam).
The second author was supported by FAPDF and CNPq.}
\keywords{Commutativity degree, Conjugacy classes, Nilpotent subgroups}
\subjclass[2020]{20E45, 20D60, 20P05}
\begin{abstract} Let $K$ be a subgroup of a finite group $G$. The probability that an element of $G$ commutes with an element of $K$ is denoted by
$Pr(K,G)$. Assume that $Pr(K,G)\geq\epsilon$ for some fixed $\epsilon>0$.
We show that there is a normal subgroup $T\leq G$ and a subgroup $B\leq K$ such that the indexes $[G:T]$ and $[K:B]$ and the order of the commutator subgroup $[T,B]$ are $\epsilon$-bounded. This extends the well known theorem, due to P. M. Neumann, that covers the case where $K=G$.
\noindent We deduce a number of corollaries of this result. A typical application is that if $K$ is the generalized Fitting subgroup $F^*(G)$ then
$G$ has a class-2-nilpotent normal subgroup $R$ such that both the index $[G:R]$ and the order of the commutator subgroup $[R,R]$ are $\epsilon$-bounded. In the same spirit we consider the cases where $K$ is a term of the lower central series of $G$, or a Sylow subgroup, etc.
\end{abstract}
\maketitle
\section{Introduction}
The probability that two randomly chosen elements of a finite group $G$ commute is given by $$Pr(G)=\frac{|\{(x,y)\in G\times G\ :\ xy=yx \}|}{|G|^2}.$$ The above number is called the {\it commuting probability} (or
the {\it commutativity degree}) of $G$. This is a well studied concept. In the literature one can find publications dealing with problems on the set of possible values of $Pr(G)$ and the influence of $Pr(G)$ over the structure of $G$ (see \cite{eberhard,guraro,gustaf,lescot1,lescot2} and references therein). The reader can consult \cite{mann,shalev} and references therein for related developments concerning probabilistic identities in
groups.
P. M. Neumann \cite{neumann} proved the following theorem (see also \cite{eberhard}).
\begin{theorem}\lambdabel{neumann} Let $\epsilon>0$, and let $G$ be a finite group such that $Pr(G)\geq\epsilon$. Then $G$ has a nilpotent normal subgroup $R$ of nilpotency class at most $2$ such that both the index $[G:R]$
and the order of the commutator subgroup $[R,R]$ are $\epsilon$-bounded.
\end{theorem}
Throughout the article we use the expression ``$(a,b,\dots)$-bounded" to mean that a quantity is bounded from above by a number depending only on the parameters $a,b,\dots$.
If $K$ is a subgroup of $G$, write $$Pr(K,G)=\frac{|\{(x,y)\in K\times G\ :\ xy=yx \}|}{|K||G|}.$$ This is the probability that an element of $G$ commutes with an element of $K$ (the relative commutativity degree of
$K$ in $G$).
This notion has been studied in several recent papers (see in particular \cite{lescot3,nath}). Here we will prove the following proposition.
\begin{proposition}\lambdabel{main} Let $\epsilon>0$, and let $G$ be a finite
group having a subgroup $K$ such that $Pr(K,G)\geq\epsilon$. Then there is a normal subgroup $T\leq G$ and a subgroup $B\leq K$ such that the indexes $[G:T]$ and $[K:B]$, and the order of the commutator subgroup $[T,B]$
are $\epsilon$-bounded.
\end{proposition}
Theorem \ref{neumann} can be easily obtained from the above result taking
$K=G$.
Proposition \ref{main} has some interesting consequences. In particular, we will establish the following results.
Recall that the generalized Fitting subgroup $F^*(G)$ of a finite group $G$ is the product of the Fitting subgroup $F(G)$ and all subnormal quasisimple subgroups; here a group is quasisimple if it is perfect and its quotient by the centre is a non-abelian simple group. Throughout, by a class-$c$-nilpotent group we mean a nilpotent group whose nilpotency class is
at most $c$.
\begin{theorem}\lambdabel{fitt}
Let $G$ be a finite group such that $Pr(F^*(G),G)\geq\epsilon$. Then $G$ has a class-$2$-nilpotent normal subgroup $R$ such that both the index $[G:R]$ and the order of the commutator subgroup $[R,R]$ are $\epsilon$-bounded.
\end{theorem}
A somewhat surprising aspect of the above theorem is that information on the commuting probability of a subgroup (in this case $F^*(G)$) enables one to draw a conclusion about $G$ as strong as in P. M. Neumann's theorem. Yet, several other results with the same conclusion will be established in this paper.
Our next theorem deals with the case where $K$ is a subgroup containing $\gamma_i(G)$ for some $i\geq1$. Here and throughout the paper $\gamma_i(G)$ denotes the $i$th term of the lower central series of $G$.
\begin{theorem}\lambdabel{main1} Let $\epsilon>0$, and let $K$ be a subgroup of a finite group $G$ containing $\gamma_i(G)$ for some $i\geq1$. Suppose
that $Pr(K,G)\geq\epsilon$. Then $G$ has a nilpotent normal subgroup $R$
of nilpotency class at most $i+1$ such that both the index $[G:R]$ and the order of $\gamma_{i+1}(R)$ are $\epsilon$-bounded.
\end{theorem}
P. M. Neumann's theorem is a particular case of the above result (take $i=1$).
In the same spirit, we conclude that $G$ has a nilpotent subgroup of $\epsilon$-bounded index if $K$ is a verbal subgroup corresponding to a word implying virtual nilpotency such that $Pr(K,G)\geq\epsilon$. Given a group-word $w$, we write $w(G)$ for the corresponding verbal subgroup of a group $G$, that is the subgroup generated by the values of $w$ in $G$.
Recall that a group-word $w$ is said to imply virtual nilpotency if every
finitely generated metabe\-li\-an group $G$ where $w$ is a law, that is $w(G)=1$, has a nilpotent subgroup of finite index. Such words admit several important characterizations (see \cite{black,bume,groves}).
In particular, by a result of Gruenberg~\cite{Gr53}, all Engel words imply virtual nilpotency. Burns and Medvedev proved that for any word $w$ implying virtual nilpotency there exist integers $e$ and $c$ depending only on $w$ such that every finite group $G$, in which $w$ is a law, has a class-$c$-nilpotent normal subgroup $N$ such that $G^e\leq N$ \cite{bume}. Here $G^e$ denotes the subgroup generated by all $e$th powers of elements of $G$. Our next theorem provides a probabilistic variation of this result.
\begin{theorem}\lambdabel{virtunil} Let $w$ be a group-word implying virtual nilpotency. Suppose that $K$ is a subgroup of a finite group $G$ such that $w(G)\leq K$ and $Pr(K,G)\geq\epsilon$. There is an $(\epsilon,w)$-bounded integer $e$ and a $w$-bounded integer $c$ such that $G^e$ is nilpotent of class at most $c$.
\end{theorem}
We also consider finite groups with a given value of $Pr(P,G)$, where $P$
is a Sylow $p$-subgroup of $G$.
\begin{theorem} \lambdabel{sylow}
Let $P$ be a Sylow $p$-subgroup of a finite group $G$ such that $Pr(P,G) \ge \epsilon$. Then $G$ has a class-$2$-nilpotent normal $p$-subgroup $L$
such that both the index $[P:L]$ and the order of $[L,L]$ are $\epsilon$-bounded.
\end{theorem}
Once we have information on the commuting probability of all Sylow subgroups of $G$, the result is as strong as in P. M. Neumann's theorem.
\begin{theorem} \lambdabel{allsylow}
Let $\epsilon >0$, and let $G$ be a finite group such that $Pr(P,G) \ge \epsilon$ whenever $P$ is a Sylow subgroup. Then $G$ has a nilpotent normal subgroup $R$ of nilpotency class at most $2$ such that both the index $[G:R]$ and the order of the commutator subgroup $[R,R]$ are $\epsilon$-bounded.
\end{theorem}
If $\phi$ is an automorphism of a group $G$, the centralizer $C_G(\phi)$ is the subgroup formed by the elements $x\in G$ such that $x^\phi=x$. In the case where $C_G(\phi)=1$ the automorphism $\phi$ is called fixed-point-free. A famous result of Thompson \cite{thompson} says that a finite group admitting a fixed-point-free automorphism of prime order is nilpotent. Higman proved that for each prime $p$ there exists a number $h=h(p)$ depending only on $p$ such that whenever a nilpotent group $G$ admits
a fixed-point-free automorphism of order $p$, it follows that $G$ has nilpotency class at most $h$ \cite{higman}. Therefore a finite group admitting a fixed-point-free automorphism of order $p$ is nilpotent of class at most $h$. Khukhro obtained the following ``almost fixed-point-free" generalization of this fact \cite{khukhro}: if a finite group $G$ admits an automorphism $\phi$ of prime order $p$ such that $C_G(\phi)$ has order $m$, then
$G$ has a nilpotent subgroup of $p$-bounded nilpotency class and $(m,p)$-bounded index. We will establish a probabilistic variation of the above results. Recall that an automorphism $\phi$ of a finite group $G$ is called coprime if $(|G|,|\phi|)=1$.
\begin{theorem}\lambdabel{auto} Let $G$ be a finite group admitting a coprime
automorphism $\phi$ of prime order $p$ such that $Pr(C_G(\phi),G)\geq\epsilon$. Then $G$ has a nilpotent subgroup of $p$-bounded nilpotency class
and $(\epsilon,p)$-bounded index.
\end{theorem}
An even stronger conclusion will be derived about groups admitting an elementary abelian group of automorphisms of rank at least 2.
\begin{theorem}\lambdabel{auto2} Let $\epsilon>0$, and let $G$ be a finite group admitting an elementary abelian coprime group of automorphisms $A$ of
order $p^2$ such that $Pr(C_G(\phi),G)\geq\epsilon$ for each nontrivial $\phi\in A$. Then $G$ has a class-$2$-nilpotent normal subgroup $R$ such that both the index $[G:R]$ and the order of $[R,R]$ are $(\epsilon,p)$-bounded.
\end{theorem}
Proposition \ref{main}, which is a key result of this paper, will be proved in the next section. The other results will be established in Sections 3--5.
\section{The key result}
A group is said to be a BFC-group if its conjugacy classes are finite and
of bounded size. A famous theorem of B. H. Neumann says that in a BFC-group the commutator subgroup $G'$ is finite \cite{bhn}. It follows that if $|x^G|\leq m$ for each $x\in G$, then the order of $G'$ is bounded by a number depending only on $m$. A first explicit bound for the order of $G'$ was found by J. Wiegold \cite{wie}, and the best known was obtained in \cite{gumaroti} (see also \cite{neuvoe} and \cite{sesha}). The main technical tools employed in this paper are provided by the recent results \cite{cri,dms,glasgow,dieshu} strengthening B. H. Neumann's theorem.
A well known lemma due to Baer says that if $A,B$ are normal subgroups of
a group $G$ such that $[A:C_A(B)]\leq m$ and $[B:C_B(A)]\leq m$ for some integer $m\geq1$, then $[A,B]$ has finite $m$-bounded order (see \cite[14.5.2]{Rob}).
We will require a stronger result. Here and in the rest of the paper, given an element $x\in G$ and a subgroup $H\leq G$, we write $x^H$ for the set of conjugates of $x$ by elements from $H$.
\begin{lemma}\lambdabel{normal}
Let $m\geq1$, and let $G$ be a group containing normal subgroups $A,B$ such that $[A:C_A(y)]\leq m$ and $[B:C_B(x)]\leq m$ for all $x\in A$, $y\in B$. Then $[A,B]$ has finite $m$-bounded order.
\end{lemma}
\begin{proof}
We first prove that, given $x\in A$ and $y\in B$, the order of $[x,y]$ is $m$-bounded. Let $H=\lambdangle x,y\rangle$. By assumptions, $[A:C_A(y)]\leq m$ and $[B:C_B(x)]\leq m$. Hence there exists an $m$-bounded number $l$ such that $x^l$ and $y^l$ are contained in $Z(H)$ (for example we can take $l=m!$). Let $D=A\cap B \cap H$ and $N=\lambdangle D,x^l,y^l\rangle$. Then $H/N$ is abelian of order at most $l^2$. Both $x$ and $y$ have centralizers of index at most $m$ in $N$. Moreover every element of $N$ has centralizer of index at most $m$ in $N$. Indeed $|d^N| \le |d^A|\le m $ for every $d \in D \le A\cap B$. So, every element of $H$ is a product
of at most $l^2+1$ elements each of which has centralizer of index at most $m$ in $N$. Therefore each element of $H$ has centralizer of $m$-bounded index in $H$. We conclude that $H$ is a BFC-group in which the sizes of
conjugacy classes are $m$-bounded. Hence $|H'|$ is $m$-bounded and so the
order of $[x,y]$ is $m$-bounded, too.
Now we claim that for every $x \in A$, the subgroup $[x,B]$ has finite $m$-bounded order. Indeed, $x$ has at most $m$ conjugates $\{x^{b_1}, \dots , x^{b_m} \}$ in $B$, so $[x,B]$ is generated by as most $m$ elements.
Let $C$ be a maximal normal subgroup of $B$ contained in $C_B(x)$. Clearly $C$ has $m$-bounded index in $B$ and centralizes $[x,B]$. Thus, the centre of $[x,B]$ has $m$-bounded index in $[x,B]$. It follows from Schur's
theorem \cite[10.1.4]{Rob} that the derived subgroup of $[x,B]$ has finite $m$-bounded order. Since $[x,B]$ is generated by as most $m$ elements of $m$-bounded order, we deduce that the order of $[x,B]$ is finite and $m$-bounded.
Choose $a\in A$ such that $[B:C_B(a)]=\max_{x \in A} [B:C_B(x)]$ and set $n=[B:C_B(a)]$, where $n \le m$. Let $b_1,\dots, b_n$ be elements of
$B$ such that $a^B=\{a^{b_1},\dots, a^{b_n}\}$ is the set of (distinct)
conjugates of $a$ by elements of $B$. Set $U=C_A(b_1,\dots,b_n)$ and note that $U$ has $m$-bounded index in $A$.
Given $u\in U$, the elements $(ua)^{b_1},\dots, (ua)^{b_n}$ form the conjugacy class $(ua)^B$ because they are all different and their number is the allowed maximum. So, for an arbitrary element $y\in B$ there exists
$i$ such that $(ua)^y=(ua)^{b_i}=u a^{b_i}$. It follows that $u^{-1}u^y=a^{b_i}a^{-y}$, hence
\[[u,y]=a^{b_i}a^{-y} =[a, b_i^{a^{-1}}][ y^{a^{-1}},a] \in [a, B].\] Therefore $[U,B]\leq[a,B]$. Let $a_1,\dots,a_s$ be coset representatives of $U$ in $A$ and note that $s$ is $m$-bounded. As each $[x,B]$ is normal in $B$ and $[U,B]\leq[a,B]$, we deduce that $[A,B]=[a,B]\prod[a_i,B]$. So $[A,B]$ is a product of $m$-boundedly many subgroups of $m$-bounded order. These subgroups are normal in $B$ and therefore their product has finite $m$-bounded order.
\end{proof}
In the next lemma $B$ is not necessarily normal. Instead, we require that $B$ is contained in an abelian normal subgroup. Throughout, $\lambdangle H^G\rangle$ denotes the normal closure of a subgroup $H$ in $G$.
\begin{lemma}\lambdabel{lem2} Let $m\geq1$, and let $G$ be a group containing
a normal subgroup $A$ and a subgroup $B$ such that $[A:C_A(y)]\leq m$ and $[B:C_B(x)]\leq m$ for all $x\in A$, $y\in B$. Assume further that $\lambdangle B^G\rangle$ is abelian. Then $[A, B]$ has finite $m$-bounded order.
\end{lemma}
\begin{proof} Without loss of generality we can
assume that $G=AB$. Set $L=\lambdangle B^G\rangle=\lambdangle B^A\rangle$.
Let $x\in A$. There is an $m$-bounded number $l$ such that $x$ centralizes $y^l$ for every $y\in B$. Since $L$ is abelian, $[x,y]^i=[x,y^i]$ for
each $i$ and therefore the order of $[x,y]$ is at most $l$. Thus $[x,B]$ is an abelian subgroup generated by at most $m$ elements of $m$-bounded order, whence $[x,B]$ has finite $m$-bounded order.
Now we choose $a\in A$ such that $[B:C_B(a)]$ is as big as possible. Let $b_1,\dots,b_{m}$ be elements of $B$ such that $a^B=\{a^{b_1},\dots,a^{b_{m}}\}$. Set $U=C_A(b_1,\dots,b_m)$ and note that $U$ has $m$-bounded
index in $A$. Arguing as in the previous lemma, we see that for arbitrary $u\in U$ and $y\in B$, the conjugate $(ua)^y$ belongs to the set
$\{(ua)^{b_1},\dots, (ua)^{b_m}\}$. Let $(ua)^y=(ua)^{b_i}$. Then $u^{-1}u^y=a^{b_i}a^{-y}$ and hence $[u,y]=a^{b_i}a^{-y}\in[a, B]$. Therefore $[U,B]\leq[a,B]$.
Let $V=\cap_{x\in A}U^x$ be the maximal normal subgroup of $A$ contained in $U$. We know that $[V,B]$ has $m$-bounded order, since $[V,B]\le[a,B]$. Denote the index $[A:V]$ by $s$. Evidently, $s$ is $m$-bounded. Let $a_1,\dots,a_s$ be a transversal of $V$ in $A$. As $[V,B] \le L=\lambdangle B^A\rangle$ is abelian, we have
$$\lambdangle[V,B]^G\rangle=\lambdangle[V,B]^A\rangle=\prod_{i=1}^s[V,B]^{a_i}.$$
Thus $[V,L]=[V, B^A] = \lambdangle[V,B]^A\rangle$ is a product of $m$-boundedly many subgroups of $m$-bounded order, and hence it has $m$-bounded order. Write
$$L=\lambdangle B^A\rangle \leq \lambdangle B^{V a_i} \mid i=1, \dots s \rangle \leq [V,L]\prod_{i=1}^sB^{a_i}.$$ Thus, it becomes clear that $L$ is a product of $m$-boundedly many conjugates of $B$. Say $L $ is a product of $t$ conjugates of $B$. Then, every $y \in L$ can be written as a product of at most $t$ conjugates of elements of $B$ and consequently $[A: C_A(y)] \le m^t.$ Moreover, as $A$ is normal in $G$ and $|a^B| \le m$ for every $a\in A$, the conjugacy class $x^L$ of an element $x \in A$ has size at
most $m^t$. Now Lemma \ref{normal} shows that $[A,B] \le [A,L]$ has finite $m$-bounded order.
\end{proof}
We will now show that if $K$ is a subgroup of a finite group $G$ and $N$ is a normal subgroup of $G$, then $Pr(KN/N,G/N)\geq Pr(K,G)$. More precisely, we will establish the following lemma.
\begin{lemma}\lambdabel{quoti} Let $N$ be a normal subgroup of a finite group
$G$, and let $K\leq G$. Then $Pr(K,G)\leq Pr(KN/N,G/N)Pr(N\cap K,N)$.
\end{lemma}
This is an improvement over \cite[Theorem 3.9]{lescot3} where the result was obtained under the additional hypothesis that $N\leq K$.
\begin{proof} In what follows $\bar{G}=G/N$ and $\bar{K}=KN/N$. Write
$\bar{K_0}$ for the set of cosets $(N\cap K)h$ with $h\in K$. If $S_0=(N\cap K)h\in\bar{K_0}$, write $S$ for the coset $Nh\in\bar{K}$. Of course, we have a natural one-to-one correspondence between $\bar{K_0}$ and $\bar{K}$.
Write
$$|K||G|Pr(K,G)=\sum_{x\in K}|C_G(x)|=\sum_{S_0\in\bar{K_0}}\sum_{x\in S_0}\frac{|C_G(x)N|}{|N|}|C_N(x)| $$
$$\leq\sum_{S_0\in\bar{K_0}}\sum_{x\in S_0}|C_{\bar{G}}(xN)||C_N(x)|=\sum_{S\in\bar{K}}|C_{\bar{G}}(S)|\sum_{x\in S_0}|C_N(x)|= $$
$$=\sum_{S\in\bar{K}}|C_{\bar{G}}(S)|\sum_{y\in N}|C_{S_0}(y)|.$$
If $C_{S_0}(y)\neq\emptyset$, then there is $y_0\in C_{S_0}(y)$ and so $S_0=(N\cap K)y_0$. Therefore $$C_{S_0}(y)=(N\cap K)y_0\cap C_G(y)=C_{N\cap K}(y)y_0,\text{ whence }|C_{S_0}(y)|=|C_{N\cap K}(y)|.$$
Conclude that $$|K||G|Pr(K,G)\leq \sum_{S\in\bar{K}}|C_{\bar{G}}(S)|\sum_{y\in N}|C_{N\cap K}(y)|.$$ Observe that $$\sum_{S\in\bar{K}}|C_{\bar{G}}(S)|=\frac{|K|}{|N\cap K|}\frac{|G|}{|N|}Pr(\bar{K},\bar{G})$$ and $$\sum_{y\in N}|C_{N\cap K}(y)|=|N\cap K||N|Pr(N\cap K,N).$$
It follows that $Pr(K,G)\leq Pr(\bar{K},\bar{G})Pr(N\cap K,N)$, as required.
\end{proof}
The following theorem is taken from \cite{cri}. It plays a crucial role in the proof of Proposition \ref{main}.
\begin{theorem}\lambdabel{cristi} Let $m$ be a positive integer, $G$ a group having a subgroup $K$
such that $|x^G| \le m$ for each $x\in K$, and let $H=\lambdangle K^G\rangle$. Then the order of the commutator subgroup $[H,H]$ is finite and $m$-bounded.
\end{theorem}
A proof of the next lemma can be found in Eberhard \cite[Lemma 2.1]{eberhard}.
\begin{lemma}\lambdabel{lem}
Let $G$ be a finite group and $X$ a symmetric subset of $G$ containing the identity. Then $\lambdangle X \rangle= X^{3r}$ provided $(r+1)|X| > |G|$.
\end{lemma}
We are now ready to prove Proposition \ref{main} which we restate here for the reader's convenience:
\noindent{\it Let $\epsilon>0$, and let $G$ be a finite group having a subgroup $K$ such that $Pr(K,G)\geq\epsilon$. Then there is a normal subgroup $T\leq G$ and a subgroup $B\leq K$ such that the indexes $[G:T]$ and $[K:B]$ and the order of $[T,B]$ are $\epsilon$-bounded.
}
\begin{proof}[Proof of Proposition \ref{main}] Set
$$X=\{x\in K \mid |x^G|\leq 2/\epsilon\} \text{ and } B=\lambdangle X\rangle.$$
Note that $K \setminus X=\{ x\in K \mid |C_G(x)|\leq (\epsilon/2) |G| \}$, whence
\begin{eqnarray*}
\epsilon |K||G| &\le & | \{ (x,y) \in K\times G \mid xy=yx \} |=\sum_{x\in K}|C_G(x)|\\
&\le & \sum_{x \in X} |G| + \sum_{x \in K \setminus X} \frac{\epsilon}{2} |G| \\
&\le & |X| |G| +(|K| - |X|)\frac{\epsilon}{2}|G|.
\end{eqnarray*}
Therefore
$ \epsilon |K| \le |X|+ ({\epsilon}/{2}) (|K| - |X|)$,
whence $({\epsilon}/{2}) |K| < |X|$.
Clearly, $|B| \ge |X| > ({\epsilon}/{2}) |K|$ and so the index of
$B$ in $K$ is at most $2/\epsilon$.
As $X$ is symmetric and $(2/\epsilon ) |X| > |K|$, it follows from Lemma
\ref{lem} that every element of $B$ is a product of at most $6/\epsilon$ elements of $X$.
Therefore $|b^G| \le (2/\epsilon)^{6/\epsilon}$ for every $b \in B$.
Let $L=\lambdangle B^G\rangle$. Theorem \ref{cristi} tells us that the commutator subgroup $[L,L]$ has $\epsilon$-bounded order. Let us use the bar
notation for the images of the subgroups of $G$ in $G/[L,L]$. By Lemma \ref{quoti},
$$ Pr(\bar K, \bar G) \ge Pr(K,G)\geq\epsilon. $$
Moreover,
$ [\bar K: \bar B] \le [K:B] \le {\epsilon}/{2} $ and $
|\bar b^{ \bar G}| \le |b^G| \le (2/\epsilon)^{6/\epsilon} .$
Thus we can pass to the quotient over $[L,L]$ and assume that $L$ is abelian.
Now we set
\[ Y= \{ y \in G \mid |y^K| \le 2/\epsilon\} =
\{ y \in G \mid |C_K(y)| \ge ({\epsilon}/{2}) |K|\}. \]
Note that
\begin{eqnarray*}
\epsilon |K||G| & \le & | \{ (x,y) \in K \times G \mid xy=yx \} |\\
&\le & \sum_{y \in Y} |K| + \sum_{y \in G \setminus Y} \frac{\epsilon}{2} |K| \\
&\le & |Y| |K| +(|G| - |Y|)\frac{\epsilon}{2}|K| \le |Y| |K| +\frac{\epsilon}{2}|G| |K|.
\end{eqnarray*}
Therefore $({\epsilon}/{2}) |G| < |Y|.$
Set $E= \lambdangle Y \rangle$. Thus $|E| \ge |Y| > ({\epsilon}/{2}) |G|$,
and so the index of $E$ in $G$ is at most $2/\epsilon$. As $Y$ is symmetric and $(2/\epsilon ) |Y| > |G|$, it follows from Lemma \ref{lem} that every element of $E$ is a product of at most $6/\epsilon$ elements of $Y$.
Since $|y^K|\le 2/\epsilon$ for every $y\in Y$, we conclude that $|e^K|\le (2/\epsilon)^{6/\epsilon}$ for every $e\in E$. Let $T$ be the maximal normal subgroup of $G$ contained in $E$. Clearly, the index $[G:T]$ is $\epsilon$-bounded.
So, now $|b^G| \le (2/\epsilon)^{6/\epsilon} $ for every $b \in B$ and $|e^B| \le (2/\epsilon)^{6/\epsilon}$ for every $e\in T$.
As $L$ is abelian, we can apply Lemma \ref{lem2} to conclude that $[T,B]$ has $\epsilon$-bounded order and the result follows.
\end{proof}
\begin{remark}\lambdabel{remark} Under the hypotheses of Proposition \ref{main} the subgroup $N=\lambdangle[T,B]^G\rangle$ has $\epsilon$-bounded order.
\end{remark}
\begin{proof} Since $[T,B]$ is normal in $T$, it follows that there are only boundedly many conjugates of $[T,B]$ in $G$ and they normalize each other. Since $N$ is the product of those conjugates, $N$ has $\epsilon$-bounded order.
\end{proof}
As usual, $Z_i(G)$ stands for the $i$th term of the upper central series of a group $G$.
\begin{remark}\lambdabel{remark2} Assume the hypotheses of Proposition \ref{main}. If $K$ is normal, then the subgroup $T$ can be chosen in such a way
that $K\cap T\leq Z_3(T)$.
\end{remark}
\begin{proof} According to Remark \ref{remark}, $N=\lambdangle[T,B]^G\rangle$ has $\epsilon$-bounded order. Let $B_0=\lambdangle B^G\rangle$ and note that
$B_0 \le K$ and $[T,B_0]\leq N$. Since the index $[K:B_0]$ and the order of $N$ are $\epsilon$-bounded, the stabilizer in $T$ of the series $$1\leq N\leq B_0\leq K,$$ that is, the subgroup $$H=\{g\in T\ \vert\ [N,g]=1\ \& \ [K,g]\leq B_0\}$$ has $\epsilon$-bounded index in $G$. Note that $K\cap H\leq Z_3(H)$, whence the result.
\end{proof}
\section{Probabilistic almost nilpotency of finite groups}
Our first goal in this section is to furnish a proof of Theorem \ref{fitt}. We restate it here.
\noindent{\it Let $G$ be a finite group such that $Pr(F^*(G),G)\geq\epsilon$. Then $G$ has a class-$2$-nilpotent normal subgroup $R$ such that both the index $[G:R]$ and the order of the commutator subgroup $[R,R]$ are $\epsilon$-bounded.}
As mentioned in the introduction, the above result yields a conclusion about $G$ which is as strong as in P. M. Neumann's theorem.
\begin{proof}[Proof of Theorem \ref{fitt}.]
Set $K=F^*(G)$. In view of Proposition \ref{main} there is a normal subgroup $T\leq G$ and a subgroup $B\leq K$ such that the indexes $[G:T]$ and $[K:B]$, and the order of the commutator subgroup $[T,B]$ are $\epsilon$-bounded.
As $K$ is normal in $G$,
according to Remark \ref{remark2} the
subgroup $T$ can be chosen in such a way that $K\cap T\leq Z_3(T)$. By \cite[Corollary X.13.11(c)]{hb3} we have $K\cap T=F^*(T)$. Therefore $F^*(T)\leq Z_3(T)$ and in view of \cite[Theorem X.13.6]{hb3} we conclude that $T=F^*(T)$ and so $T\leq K$. It follows that the index of $K$ in $G$ is $\epsilon$-bounded. By Remark \ref{remark} the subgroup $N=\lambdangle[T,B]^G\rangle$ has $\epsilon$-bounded order. Conclude that $R=\lambdangle B^G\rangle\cap C_G(N)$ has $\epsilon$-bounded index in $G$. Moreover $R$ is nilpotent of class at most 2 and $[R,R]$ has $\epsilon$-bounded order. This completes the proof.
\end{proof}
Now focus on Theorem \ref{main1}, which deals with the case where $\gamma_i(G)\leq K$. Start with a couple of remarks on the result. Let $G$ and $R$ be as in Theorem \ref{main1}. The fact that both the index $[G:R]$ and the order of $\gamma_{i+1}(R)$ are $\epsilon$-bounded implies that for any $x_1,\dots,x_i\in R$ the centralizer of the long commutator $[x_1,\dots,x_i]$ has $\epsilon$-bounded index in $G$. Therefore there is an $\epsilon$-bounded number $e$ such that $G^e$ centralizes all commutators $[x_1,\dots,x_i]$ where $x_1,\dots,x_i\in R$. Then $G_0=G^e\cap R$ is a nilpotent normal subgroup of nilpotency class at most $i$ with $G/G_0$ of $\epsilon$-bounded exponent (recall that a positive integer $e$ is the exponent of a finite group $G$ if $e$ is the minimal number for which $G^e=1$).
If $G$ is additionally assumed to be $m$-generator for some $m\geq1$, then $G$ has a nilpotent normal subgroup of nilpotency class at most $i$ and $(\epsilon,m)$-bounded index. Indeed, we know that for any $x_1,\dots,x_i\in R$ the centralizer of the long commutator $[x_1,\dots,x_i]$ has $\epsilon$-bounded index in $G$. An $m$-generator group has only $(j,m)$-boundedly many subgroups of any given index $j$ \cite[Theorem 7.2.9]{mhall}. Therefore $G$ has a subgroup $J$ of $(\epsilon,m)$-bounded index that centralizes all commutators $[x_1,\dots,x_i]$ with $x_1,\dots,x_i\in R$. Then $J\cap R$ is a nilpotent normal subgroup of nilpotency class at most $i$ and $(\epsilon,m)$-bounded index in $G$.
These observations are in parallel with Shalev's results on probabilistically nilpotent groups \cite{shalev}.
Our proof of Theorem \ref{main1} requires the following result from \cite{glasgow}.
\begin{theorem}\lambdabel{glas} Let $G$ a group such that $|x^{\gamma_k(G)}|\leq n$ for any $x\in G$. Then $\gamma_{k+1}(G)$ has finite $(k,n)$-bounded order.
\end{theorem}
We can now prove Theorem \ref{main1}.
\begin{proof}[Proof of Theorem \ref{main1}] Recall that $K$ is a subgroup
of the finite group $G$ such that $\gamma_k(G)\leq K$ and $Pr(K,G)\geq\epsilon$. In view of \cite[Theorem 3.7]{lescot3} observe that $Pr(\gamma_k(G),G)\geq\epsilon$. Therefore without loss of generality we can assume that $K=\gamma_k(G)$.
Proposition \ref{main} tells us that there is a normal subgroup $T\leq G$ and a subgroup $B\leq K$ such that the indexes $[G:T]$ and $[K:B]$ and the order of $[T,B]$ are $\epsilon$-bounded. In particular, $|x^{B}|$ is $\epsilon$-bounded for every $x\in T$. Since $B$ has $\epsilon$-bounded index in $K$, we deduce that $|x^{\gamma_k(G)}|$ is $\epsilon$-bounded for every $x\in T$. Now Theorem \ref{glas} implies that $\gamma_{k+1}(T)$ has $\epsilon$-bounded order. Set $R=C_T(\gamma_{k+1}(T))$. It follows that $R$ is as required.
\end{proof}
Our next goal is a proof of Theorem \ref{virtunil}. As mentioned in the introduction, a group-word $w$ implies virtual nilpotency if every finitely generated metabe\-li\-an group $G$ where $w$ is a law, that is $w(G)=1$, has a nilpotent subgroup of finite index. A theorem, due to Burns and Medvedev, states that for any word $w$ implying virtual nilpotency there exist integers $e$ and $c$ depending only on $w$ such that every finite group $G$, in which $w$ is a law, has a nilpotent of class at most $c$ normal subgroup $N$ with $G^e\leq N$ \cite{bume}.
\begin{proof}[Proof of Theorem \ref{virtunil}.] Recall that $w$ is a group-word implying virtual nilpotency while $K$ is a subgroup of a finite group $G$ such that $w(G)\leq K$ and $Pr(K,G)\geq\epsilon$. We need to show that there is an $(\epsilon,w)$-bounded integer $e$ and a $w$-bounded integer $c$ such that $G^e$ is nilpotent of class at most $c$.
As in the proof of Theorem \ref{main1} without loss of generality we can assume that $K=w(G)$. Proposition \ref{main} tells us that there is a normal subgroup $T\leq G$ and a subgroup $B\leq K$ such that the indexes $[G:T]$ and $[K:B]$ and the order of the commutator subgroup $[T,B]$ are $\epsilon$-bounded. According to Remark \ref{remark2} the subgroup $T$ can
be chosen in such a way that $K\cap T\leq Z_3(T)$. In particular $w(T)\leq Z_3(T)$. Taking into account that the word $w$ implies virtual nilpotency, we deduce from the Burns-Medvedev theorem that there are $w$-bounded numbers $i$ and $c$ such that the subgroup generated by the $i$th powers of elements of $T$ is nilpotent of class at most $c$. Recall that the index of $T$ in $G$ is $\epsilon$-bounded. Hence there is an $\epsilon$-bounded integer $e$ such that every $e$th power in $G$ is an $i$th power of an element of $T$. The result follows.
\end{proof}
If $[x^i,y_1,\dots,y_j]$ is a law in a finite group $G$, then $\gamma_{j+1}(G)$ has $\{i,j\}$-bounded exponent (the case $j=1$ is a well-known result, due to Mann \cite{M}; see \cite[Lemma 2.2]{CS} for the case $j\geq2$). If the $j$-Engel word $[x,y,\dots,y]$, where $y$ is repeated $j$ times, is a law in a finite group $G$, then $G$ has a normal subgroup $N$ such that the exponent of $N$ is $j$-bounded while $G/N$ is nilpotent with $j$-bounded class \cite{bume2}. Note that both words $[x^i,y_1,\dots,y_j]$ and $[x,y,\dots,y]$ imply virtual nilpotency.
Therefore, in addition to Theorem \ref{virtunil}, we deduce
\begin{theorem}\lambdabel{exp} Assume the hypotheses of Theorem \ref{virtunil}.
\begin{enumerate}
\item If $w=[x^n,y_1,\dots,y_k]$, then $G$ has a normal subgroup $T$ such that the index $[G:T]$ is $\epsilon$-bounded and the exponent of $\gamma_{k+4}(T)$ is $w$-bounded.
\item There are $k$-bounded numbers $e_1$ and $c_1$ with the property that if $w$ is the $k$-Engel word, then $G$ has a normal subgroup $T$ such that the index $[G:T]$ is $\epsilon$-bounded and the exponent of $\gamma_{c_1}(T)$ divides $e_1$.
\end{enumerate}
\end{theorem}
\begin{proof} By \cite[Theorem 3.7]{lescot3}, without loss of generality we can assume that $K=w(G)$. Proposition \ref{main} tells us that there is a normal subgroup $T\leq G$ and a subgroup $B\leq w(G)$ such that the indexes $[G:T]$ and $[w(G):B]$ and the order of $[T,B]$ are $\epsilon$-bounded. Since $K$ is normal in $G$, according to Remark \ref{remark2} the
subgroup $T$ can be chosen in such a way that $w(G)\cap T\leq Z_3(T)$. If $w=[x^n,y_1,\dots,y_k]$, then $[x^n,y_1,\dots,y_{k+3}]$ is a law in $T$, whence the exponent of $\gamma_{k+4}(T)$ is $w$-bounded. If $w$ is the $k$-Engel word, then the $(k+3)$-Engel word is a law in $T$ and the theorem follows from the Burns-Medvedev theorem \cite{bume2}.
\end{proof}
\section{Sylow subgroups}
As usual, $O_p(G)$ denotes the maximal normal $p$-subgroup of a finite group $G$. For the reader's convenience we restate Theorem \ref{sylow}:
\noindent{\it Let $P$ be a Sylow $p$-subgroup of a finite group $G$ such that $Pr(P,G) \ge \epsilon$. Then $G$ has a class-$2$-nilpotent normal $p$-subgroup $L$ such that both the index $[P:L]$ and the order of the commutator subgroup $[L,L]$ are $\epsilon$-bounded. }
\begin{proof}[Proof of Theorem \ref{sylow}] Proposition \ref{main} tells us that there is a normal subgroup $T\leq G$ and a subgroup $B\leq P$ such that the indexes $[G:T]$ and $[P:B]$ and the order of the commutator subgroup $[T,B]$ are $\epsilon$-bounded. In view of Remark \ref{remark} the
subgroup $N=\lambdangle[T,B]^G\rangle$ has $\epsilon$-bounded order. Therefore $C=C_T(N)$ has $\epsilon$-bounded index in $G$. Set $B_0=B\cap C$
and note that $[C,B_0]\leq Z(C)$. It follows that $B_0\leq Z_2(C)$ and we
conclude that $B_0\leq O_p(G)$. Let $L=\lambdangle {B_0}^G\rangle$.
As $B_0 \le L \le O_p(G)$, it is clear that
$L$ is contained in $P$ as a subgroup of $\epsilon$-bounded index. Moreover $[L,L]\leq N$ and so the order of $[L,L]$ is $\epsilon$-bounded. Hence the result.
\end{proof}
We will now prove Theorem \ref{allsylow}.
\begin{proof}[Proof of Theorem \ref{allsylow}] Recall that $G$ is a finite group such that $Pr(P,G) \ge \epsilon$ whenever $P$ is a Sylow subgroup. We wish to show that $G$ has a nilpotent normal subgroup $R$ of nilpotency class at most $2$ such that both the index $[G:R]$ and the order of
the commutator subgroup $[R,R]$ are $\epsilon$-bounded.
For each prime $p\in\pi(G)$ choose a Sylow $p$-subgroup $S_p$ in $G$. Theorem \ref{sylow} shows that $G$ has a normal $p$-subgroup $L_p$ of class at most $2$ such that both $[S_p:L_p]$ and $|[L_p,L_p]|$ are $\epsilon$-bounded. Since the bounds on $[S_p:L_p]$ and $|[L_p,L_p]|$ do not depend on $p$, it follows that there is an $\epsilon$-bounded constant $C$ such that $S_p=L_p$ and $[L_p,L_p]=1$ whenever $p\geq C$. Set $R=\prod_{p\in\pi(G)}L_p$.
Then all Sylow subgroups of $G/R$ have $\epsilon$-bounded order and therefore the index of $R$ in $G$ is $\epsilon$-bounded. Moreover, $R$ is of class at most $2$ and $|[R,R]|$ is $\epsilon$-bounded, as required.
\end{proof}
\section{Coprime automorphisms and their fixed points}
If $A$ is a group of automorphisms of a group $G$, we write $C_G(A)$ for the centralizer of $A$ in $G$. The symbol $A^{\#}$ stands for the set of nontrivial elements of the group $A$.
The next lemma is well-known (see for example \cite[Theorem 6.2.2 (iv)]{go}). In the sequel we use it without explicit references.
\begin{lemma}\lambdabel{cc}
Let $A$ be a group of automorphisms of a finite group $G$ such that $(|G|,|A|)=1$. Then
$C_{G/N}(A)=NC_G(A)/N$ for any $A$-invariant normal subgroup $N$ of $G$.
\end{lemma}
\begin{proof}[Proof of Theorem \ref{auto}.] Recall that $G$ is a finite group admitting a coprime automorphism $\phi$ of prime order $p$ such that
$Pr(K,G)\geq\epsilon$, where $K=C_G(\phi)$. We need to show that $G$ has a nilpotent subgroup of $p$-bounded nilpotency class and $(\epsilon,p)$-bounded index.
By Proposition \ref{main} there is a normal subgroup $T\leq G$ and a subgroup $B\leq K$ such that the indexes $[G:T]$ and $[K:B]$ and the order of
the commutator subgroup $[T,B]$ are $\epsilon$-bounded. Let $T_0$ be the maximal $\phi$-invariant subgroup of $T$. Evidently, $T_0$ is normal and the index $[G:T_0]$ is $(\epsilon,p)$-bounded. Since $\lambdangle[T_0,B]^G\rangle\leq\lambdangle[T,B]^G\rangle$, Remark \ref{remark} implies that $M=\lambdangle[T_0,B]^G\rangle$ has $\epsilon$-bounded order. Moreover, $M$ is $\phi$-invariant. Set $D=C_G(M)\cap T_0$ and $\bar{D}=D/Z_2(D)$, and note that $D$ is $\phi$-invariant.
In a natural way $\phi$ induces an automorphism of $\bar{D}$ which we will denote by the same symbol $\phi$.
We note that $C_{\bar{D}}(\phi)=C_D(\phi) Z_2(D)/Z_2(D) $, so its order is $\epsilon$-bounded because $B\cap D\leq Z_2(D)$. The Khukhro theorem
\cite{khukhro} now implies that $\bar{D}$ has a nilpotent subgroup of $p$-bounded class and $(\epsilon,p)$-bounded index. Since $\bar{D}=D/Z_2(D)$ and since the index of $D$ in $G$ is $(\epsilon,p)$-bounded, we deduce
that $G$ has a nilpotent subgroup of $p$-bounded class and $(\epsilon,p)$-bounded index. The proof is complete.
\end{proof}
A proof of the next lemma can be found in \cite{gushu}.
\begin{lemma}\lambdabel{eee} If $A$ is a noncyclic elementary abelian $p$-group acting on a finite $p'$-group $G$ in such a way that $|C_G(a)|\leq m$ for each $a\in A^{\#}$, then the order of $G$ is at most $m^{p+1}$.
\end{lemma}
We will now prove Theorem \ref{auto2}.
\begin{proof}[Proof of Theorem \ref{auto2}] By hypotheses, $G$ is a finite group admitting an elementary abelian coprime group of automorphisms $A$ of order $p^2$ such that $Pr(C_G(\phi),G)\geq\epsilon$ for each $\phi\in A^{\#}$. We need to show that $G$ has a nilpotent normal subgroup $R$ of nilpotency class at most $2$ such that both the index $[G:R]$ and the order of the commutator subgroup $[R,R]$ are $(\epsilon,p)$-bounded.
Let $A_1,\dots,A_{p+1}$ be the subgroups of order $p$ of $A$ and set $G_i=C_G(A_i)$ for $i=1,\dots,p+1$. According to Proposition \ref{main} for each $i=1,\dots,p+1$ there is a normal subgroup $T_i\leq G$ and a subgroup $B_i\leq G_i$ such that the indexes $[G:T_i]$ and $[G_i:B_i]$ and the order of the commutator subgroup $[T_i,B_i]$ are $\epsilon$-bounded. We let $U_i$ denote the maximal $A$-invariant subgroup of $T_i$ so that each $U_i$ is a normal subgroup of $(\epsilon,p)$-bounded index. The intersection of all $U_i$ will be denoted by $U$. Further, we let $D_i$ denote the maximal $A$-invariant subgroup of $B_i$ so that each $D_i$ has $(\epsilon,p)$-bounded index in $G_i$. Note that a modification of Remark \ref{remark} implies that $N_i=\lambdangle[U_i,D_i]^G\rangle$ is $A$-invariant and has $\epsilon$-bounded order. It follows that the order of $N=\prod_iN_i$ is $(\epsilon,p)$-bounded. Let $V$ denote the minimal ($A$-invariant) normal subgroup of $G$ containing all $D_i$ for $i=1,\dots,p+1$. It is easy to see that $[U,V]\leq N$.
Obviously, $U$ has $(\epsilon,p)$-bounded index in $G$. Let us check that
this also holds with respect to $V$. Let $\bar{G}=G/V$. Since $V$ contains $D_i$ for each $i=1,\dots,p+1$ and since $D_i$ has $(\epsilon,p)$-bounded index in $G_i$, we conclude that the image of $G_i$ in $\bar{G}$ has $(\epsilon,p)$-bounded order. Now Lemma \ref{eee} tells us that the order of $\bar{G}$ is $(\epsilon,p)$-bounded and we conclude that indeed $V$ has $(\epsilon,p)$-bounded index in $G$. Also note that since $N$ has $(\epsilon,p)$-bounded order, $C_G(N)$ has $(\epsilon,p)$-bounded index in
$G$. Let $$R=U\cap V\cap C_G(N).$$ Then $R$ is as required since the subgroups $U,V,C_G(N)$ have $(\epsilon,p)$-bounded index in $G$ while $[R,R]\leq N\leq C_G(R)$. The proof is complete.
\end{proof}
\end{document} | math | 38,475 |
\begin{document}
\title{Homological aspects of branching laws}
\author{Dipendra Prasad}
{\rm ad\, }dress{
Indian Institute of Technology Bombay \\
Powai \\
Mumbai - 400 076 \\ \
India}
\email{[email protected]}
\thanks{}
\subjclass[2020]{Primary 11F70; Secondary 22E55}
\keywords{Branching laws, Ext groups, Euler-Poincar\'e characteristic, GGP conjectures, classical groups}
\maketitle
\newcommand{\mathcal{S}}{\mathcal{S}}
\newcommand{\mathcal{V}}{\mathcal{V}}
\newcommand{\mathfrak{R}}{\mathfrak{R}}
\newcommand{\mathcal{W}}{\mathcal{W}}
\newcommand{\mathcal{H}}{\mathcal{H}}
\newcommand{\mathcal{P}}{\mathcal{P}}
\newcommand{\mathbb{Q}}{\mathbb{Q}}
\newcommand{\mathbb{R}}{\mathbb{R}}
\newcommand{\mathbb{Z}}{\mathbb{Z}}
\newcommand{\mathbb{G}_m}{\mathbb{G}_m}
\newcommand{\mathbb{C}}{\mathbb{C}}
\newcommand{\mathfrak{F}}{\mathfrak{F}}
\newcommand{\mathbb{N}}{\mathbb{N}}
\newcommand{\mathbb{R}R}{\mathcal{R}}
\newcommand{{\rm Tor}}{{\rm Tor}}
\newcommand{{\rm Hom}}{{\rm Hom}}
\newcommand{{\rm EP}}{{\rm EP}}
\newcommand{{\rm Bes}}{{\rm Bes}}
\newcommand{{\rm PD}}{{\rm PD}}
\newcommand{{\rm Ps}}{{\rm Ps}}
\newcommand{{\rm Ext}}{{\rm Ext}}
\newcommand{{\rm Ind}}{{\rm Ind}}
\newcommand{{\rm ind}}{{\rm ind}}
\def{\,^\circ G}{{\,^\circ G}}
\def{\rm G}{{\rm G}}
\def{\rm Aut}{{\rm Aut}}
\def{\rm SL}{{\rm SL}}
\def{\rm Spin}{{\rm Spin}}
\def{\rm PSp}{{\rm PSp}}
\def{\rm PSO}{{\rm PSO}}
\def{\rm PGSO}{{\rm PGSO}}
\def{\rm PSL}{{\rm PSL}}
\def{\rm G}Sp{{\rm GSp}}
\def{\rm PGSp}{{\rm PGSp}}
\def{\rm Sp}{{\rm Sp}}
\def{\rm sc}{{\rm sc}}
\def{\rm St}{{\rm St}}
\def{\rm G}U{{\rm GU}}
\def{\rm SU}{{\rm SU}}
\def{\rm U}{{\rm U}}
\def\mathcal{W}h{{\rm Wh}}
\def{\rm G}O{{\rm GO}}
\def{\rm G}L{{\rm GL}}
\def{\rm PGL}{{\rm PGL}}
\def{\rm G}SO{{\rm GSO}}
\def{\rm G}al{{\rm Gal}}
\def{\rm SO}{{\rm SO}}
\def{\rm O}{{\rm O}}
\def{\rm Out}{{\rm Out}}
\def{\mathcal O}{{\mathcal O}}
\def{\rm Sym}{{\rm Sym}}
\def{\rm tr\,}{{\rm tr\,}}
\def{\rm ad\, }{{\rm ad\, }}
\def{\rm Ad\, }{{\rm Ad\, }}
\def{\rm rank\,}{{\rm rank\,}}
\begin{abstract}
In this mostly expository article, we consider certain homological aspects
of branching laws for representations of a group restricted to its subgroups in the context of $p$-adic groups.
We follow our earlier paper \cite{Pr3} updating it with some more recent works. In particular,
following Chan and Chan-Savin, see many of their papers listed in the bibliography,
we have emphasized in this work that the restriction of a (generic) representation
$\pi$ of a group $G$ to a closed subgroup $H$ (most of the paper is written in the context of GGP)
turns out to be a projective representation
on most Bernstein blocks of the category of
smooth representations of $H$. Further, once $\pi|_H$ is a projective module in a particular
Bernstein block, it has a simple structure.
\end{abstract}
\maketitle
\setcounter{tocdepth}{1}
\tableofcontents
\section{Introduction}
If $H$ is a subgroup of a group $G$, $\pi_1$ an irreducible representation
of $G$, one is often interested in decomposing the representation $\pi_1$ when restricted to $H$,
called the branching laws. In this paper, we will be dealing mostly with infinite dimensional representations of a group $G$
which when restricted to $H$ are usually not completely reducible and there is often no obvious meaning to
``decomposing the representation restricted to $H$'', or a meaning has to be assigned in some precise way,
such as the Plancherel decomposition for unitary representations of $G$ restricted to $H$.
Unless otherwise mentioned, we will say that a representation $\pi_2$ of $H$ appears in a representation $\pi_1$ of $G$ if
\[{\rm Hom}_H[\pi_1,\pi_2] \not = 0.\]
The local GGP conjectures (which are all theorems now!) are about such branching laws for certain pairs of classical groups
$(G,H)$, which in this paper we will often take to be $({\rm G}L_{n+1}(F), {\rm G}L_n(F))$, or $({\rm SO}_{n+1}(F), {\rm SO}_n(F))$, where $F$
is a local field which will be non-archimedean unless otherwise mentioned.
For an irreducible admissible representation $\pi_1$ of ${\rm SO}_{n+1}(F)$, and $\pi_2$ of ${\rm SO}_n(F)$, the question of interest for GGP is the understanding of the Hom spaces,
\begin{eqnarray*}
{\rm Hom}_{{\rm SO}_n(F)}[\pi_1,\pi_2] & \cong & {\rm Hom}_{{\rm SO}_n(F)}[\pi_1 \otimes \pi_2^\vee, \mathbb{C} ] \\
&\cong& {\rm Hom}_{{\rm SO}_{n+1}(F) \times {\rm SO}_n(F)}[\mathcal{S}(X), \pi_1^\vee \otimes \pi_2],\end{eqnarray*}
where
$X= {\rm SO}_n(F)\backslash [{\rm SO}_n(F) \times {\rm SO}_{n+1}(F)],$ and $\mathcal{S}(X)$ denotes the space of compactly supported
smooth functions on $X$.
The first important result about branching laws considered by GGP
is the multiplicity one property:
\[m(\pi_1,\pi_2):=\dim {\rm Hom}_{{\rm SO}_n(F)}[\pi_1,\pi_2] \leq 1.\]
This is due to A. Aizenbud, D. Gourevitch, S. Rallis and G. Schiffmann in \cite{AGRS} in the non-archimedean case,
and B. Sun and C. Zhu in \cite{Sun-Zhu} in the archimedean case.
It may be mentioned that before
the full multiplicity one theorem was proved, even finite dimensionality of the multiplicity spaces was not
known, which were later answered in greater generality in the work of Y. Sakellaridis and A. Venkatesh
in \cite{Sak-Ven}. For infinite dimensional representations which is what we are mostly dealing with,
there is also the possibility that $m(\pi_1,\pi_2)$ could be identically 0 for a particular $\pi_1$!
With the multiplicity one theorems proved,
one then goes on to prove a more precise
description of the set of irreducible admissible representations $\pi_1$ of ${\rm SO}_{n+1}(F)$ and $\pi_2$ of ${\rm SO}_n(F)$
with
\[{\rm Hom}_{{\rm SO}_n(F)}[\pi_1,\pi_2] \not = 0.\]
Precise theorems about ${\rm Hom}_{{\rm SO}_n(F)}[\pi_1,\pi_2]$ have become available in a series of papers due to
Waldspurger and Moeglin-Waldspurger, cf. \cite{Wa}, \cite{Wa1}, \cite{Wa2}, \cite{Mo-Wa} for orthogonal groups. These
were followed by a series of papers by
Beuzart-Plessis for unitary groups, cf. \cite{Ra1}, \cite{Ra2}, \cite{Ra3}.
Given the interest in the space
\[{\rm Hom}_{{\rm SO}_n(F)}[\pi_1,\pi_2] \cong {\rm Hom}_{{\rm SO}_{n+1}(F) \times {\rm SO}_n(F)}[\mathcal{S}(X), \pi_1^\vee\otimes \pi_2],\]
it is natural
to consider the related spaces
\[{\rm Ext}^i_{{\rm SO}_n(F)}[\pi_1,\pi_2] \cong {\rm Ext}^i_{{\rm SO}_{n+1}(F) \times {\rm SO}_n(F)}[\mathcal{S}(X), \pi_1^\vee \otimes \pi_2],\] and in fact
homological algebra methods suggest that the simplest answers are not for
these individual spaces, but for the
alternating sum of their dimensions:
$${\rm EP}[\pi_1,\pi_2] = \sum_{i=0}^{\infty}(-1)^i\dim {\rm Ext}^i_{{\rm SO}_n(F)}[\pi_1,\pi_2];$$
these hopefully more manageable
objects - certainly more flexible - when coupled with vanishing of higher ${\rm Ext}$'s (when available)
may give theorems about
$${\rm Hom}_{{\rm SO}_n(F)}[\pi_1,\pi_2].$$
We hasten to add that before we can define ${\rm EP}[\pi_1,\pi_2]$,
${\rm Ext}^i_{{\rm SO}_n(F)}[\pi_1,\pi_2]$
needs to be proved to be finite dimensional
for $\pi_1$ and $\pi_2$ finite length admissible representations
of ${\rm SO}_{n+1}(F)$ and ${\rm SO}_n(F)$ respectively, and also
proved to be 0 for $i$ large.
Vanishing of
\[{\rm Ext}^i_{{\rm SO}_n(F)}[\pi_1,\pi_2]\]
for large $i$ is a well-known generality:
for reductive $p$-adic groups $G$ considered here, it is
known that
\[{\rm Ext}^i_G[\pi,\pi'] = 0 \]
for any two smooth representations $\pi$ and $\pi'$ of $G$ when $i$ is greater than the $F$-split rank of $G$.
This is a standard application of the projective resolution of the trivial representation
$\mathbb{C}$ of $G$ provided by the (Bruhat-Tits) building associated to $G$.
For the proof of the finite dimensionality
of ${\rm Ext}^i_G[\pi_1,\pi_2]$
we note
that unlike the Hom spaces, ${\rm Hom}_{G}[\pi_1,\pi_2]$, where we will have no idea how to
prove finite dimensionality of ${\rm Hom}_{G}[\pi_1,\pi_2]$ if both $\pi_1$ and $\pi_2$ are cuspidal,
for ${\rm Ext}^i_G[\pi_1,\pi_2]$ exactly
this case can be handled a priori, for $i> 0$, as almost by the very definition of cuspidal
representations, they are both projective and injective objects in the category of smooth representations
(and projective objects remain projective on restriction to a closed subgroup).
The finite dimensionality of ${\rm Ext}^i_{{\rm SO}(n)}[\pi_1,\pi_2]$ when one of the representations
$\pi_1,\pi_2$ is a full principal series representation,
is achieved by an inductive argument
both on $n$ and on the split rank of the Levi from which the principal
series arises. The resulting analysis needs the notion of {\it Bessel models}, which is also
a restriction problem involving a subgroup which has both reductive and unipotent parts.
Recently, there is a very general
finiteness theorem for ${\rm Ext}_G^i[\pi_1,\pi_2]$ (for spherical varieties) due to
A. Aizenbud and E. Sayag in \cite{AS}. However, the approach via Bessel models which intervene when
analyzing principal series representations of ${\rm SO}_{n+1}(F)$ when
restricted to ${\rm SO}_n(F)$ has, as a bonus, explicit answers about
Euler-Poincar\'e characteristics (at least in some cases).
The definition and the theorem below are due to Aizenbud and Sayag.
\begin{definition}
(Locally finitely generated representations) Suppose $G$ is a $p$-adic group and $\pi$ is a smooth represetation of $G$. Then
$\pi$ is said to be a {\rm locally finitely generated representation} of $G$ (or, also, just locally finite representation) if it satisfies one of the following equivalent conditions.
\begin{enumerate}
\item For each compact open subgroup $K$ of $G$, $\pi^K$ is a finitely generated module over the Hecke-algebra ${\mathcal H}(K\backslash G /K)$.
\item For each cuspidal datum $(M,\rho)$, i.e., $M$ a Levi subgroup of $G$, and $\rho$ a cuspidal representation of $M$, $\pi[M,\rho]$, the corresponding
component of $\pi$ in the Bernstein decomposition of the category of smooth representations of $G$, is a finitely generated $G$-module.
\end{enumerate}
\end{definition}
\begin{thm} \label{AS} (Aizenbud-Sayag)
For $\pi$ an irreducible admissible representation of ${\rm G}L_{n+1}(F)$, the restriction
of $\pi$ to ${\rm G}L_n(F)$ is locally finite (and true more generally for {\it spherical pairs} where
finite multiplicity is known).
\end{thm}
As a consequence of this theorem due to Aizenbud and Sayag, note that the restriction of an irreducible
representation $\pi$
of ${\rm G}L_{n+1}(F)$ to ${\rm G}L_n(F)$
is finitely generated in any Bernstein component of ${\rm G}L_n(F)$, hence $\pi|_{{\rm G}L_n(F)}$
has nonzero irreducible quotients by generalities (a statement which we will not know how to prove
for a general restriction problem as we said earlier).
The following corollary is an easy consequence of standard homological algebra where we also use the fact that
if a module is finitely generated over a noetherian ring $R$ (which need not be commutative but contains 1),
then it has a resolution by finitely generated projective
$R$-modules.
\begin{cor}
For $\pi_1$ an irreducible representation of ${\rm G}L_{n+1}(F)$, and
$\pi_2$ of $H={\rm G}L_n(F)$ (and true more generally for {\it spherical pairs} where
finite multiplicity is known),
\[ {\rm Ext}^i_H [\pi_1,\pi_2]\]
are finite dimensional, and zero beyond the split rank of $H$.
\end{cor}
We end the introduction by suggesting that although in this work we discuss exclusively
the restriction problems arising
in the GGP context, the notion of a locally finitely generated representation,
and its becoming a projective module on restriction to suitably chosen subgroups -- which is one of the properties
emphasized in this work -- should work
well in many other situations involving finite multiplicities, such as the Weil representation and its restriction to dual reductive pairs which we
briefly mention now. A criterion on locally finitely generated, and projectivity, would be very welcome
in the geometric context, say when a ($p$-adic) group $G$ acts on a ($p$-adic) space $X$ with an equivariant sheaf
$\psi$, where one would like to understand these questions for the action of $G$ on the Schwartz space $\mathcal{S}(X,\psi)$.
In the context of the Howe correspondence for a dual reductive pair $(G_1,G_2)$ with $G_1$ ``smaller than or equal to'' $G_2$, with $K_1$, $K_2$ compact open subgroups in $G_1$ and $G_2$,
it appears that the Weil representation $\omega$ of the ambient group, $\omega^{K_1 \times K_2}$ is a finitely generated module over both ${\mathcal H}(K_1\backslash G_1/K_1)$ and ${\mathcal H}(K_2\backslash G_2/K_2)$
and is a projective module over ${\mathcal H}(K_1\backslash G_1/K_1)$, and that
one can use $\omega^{K_1 \times K_2}$ as a bimodule to construct
an embedding of the category of smooth representations of the smaller pair among the dual reductive pair to
the bigger pair. Investigations on this ``functorial approch'' to the Howe correspondence seems not to have been undertaken so far.
\section{Branching laws from ${\rm G}L_{n+1}(F)$ to ${\rm G}L_n(F)$}
Recall
the following basic result which is proved as a consequence of the Rankin-Selberg theory, cf. \cite{Pr2}.
\begin{thm} \label{duke93}Given
an irreducible generic representation $\pi_1$ of ${\rm G}L_{n+1}(F)$, and an
irreducible generic representation $\pi_2$ of ${\rm G}L_{n}(F)$,
\[{\rm Hom}_{{\rm G}L_n(F)}[\pi_1,\pi_2] \cong \mathbb{C}.\]
\end{thm}
The following theorem can be considered as the Euler-Poincar\'e version of the above theorem and is much more flexibile than the previous theorem, and proved more easily!
\begin{thm}\label{whittaker} Let
$\pi_1$ be an admissible representation of ${\rm G}L_{n+1}(F)$ of finite length, and
$\pi_2$ an admissible representation of ${\rm G}L_{n}(F)$ of finite length.
Then, ${\rm Ext}^i_{{\rm G}L_n(F)}[\pi_1,\pi_2]$ are finite dimensional vector spaces over $\mathbb{C}$, and
$${\rm EP}_{{\rm G}L_n(F)}[\pi_1,\pi_2]
= \dim {\rm Wh}(\pi_1) \cdot \dim {\rm Wh}(\pi_2),$$
where ${\rm Wh}(\pi_1)$, resp. ${\rm Wh}(\pi_2)$, denotes the space of Whittaker models for $\pi_1$, resp. $\pi_2$,
with respect to fixed non-degenerate characters on a maximal unipotent subgroup
in ${\rm G}L_{n+1}(F)$ and ${\rm G}L_n(F)$ respectively.
\end{thm}
Here is a curious corollary!
\begin{cor}
{ If
$\pi_1$ is an irreducible admissible representation of ${\rm G}L_{n+1}(F)$, and
$\pi_2$ an irreducible admissible representation of ${\rm G}L_{n}(F)$,
then the only values taken by ${\rm EP}_{{\rm G}L_n(F)}[\pi_1,\pi_2]$ is 0 and 1,
in particular it is $\geq 0$.}
\end{cor}
{\bf Proof of Theorem \ref{whittaker}}: The proof of the Theorem \ref{whittaker} is accomplished using some results of Bernstein and Zelevinsky, cf.
\S3.5 of \cite{BZ1}, regarding the
structure of representations of ${\rm G}L_{n+1}(F)$ restricted to the mirabolic subgroup.
Recall that $E_{n}$, the mirabolic subgroup of ${\rm G}L_{n+1}(F)$,
consists of matrices
in ${\rm G}L_{n+1}(F)$
whose last row is equal to $(0, 0,\cdots, 0, 1)$.
For a representation $\pi$ of ${\rm G}L_{n+1}(F)$, Bernstein-Zelevinsky define
\[ \pi^i = \text{ the {\it i}-th derivative of $\pi$}, \]
which is a representation of ${\rm G}L_{n+1- i}( F)$.
Of crucial importance is the fact that if $\pi$ is of finite length for ${\rm G}L_{n+1}(F)$,
then $\pi^i$ are representations of finite length of ${\rm G}L_{n+1-i}(F)$.
Bernstein-Zelevinsky prove that the restriction of an admissible representation $\pi$ of ${\rm G}L_{n+1}(F)$ to the mirabolic $E_n$ has a finite filtration whose successive quotients are described by the derivatives $\pi^i$ of $\pi$.
Using the Bernstein-Zelevinsky filtration, and a form of Frobenius reciprocity
for Ext groups, Theorem \ref{whittaker} eventually follows from the following easy lemma. We refer to \cite{Pr3} for more details.
\begin{lemma} \label{vanishing} If $V$ and $W$ are any two finite length representations of ${\rm G}L_d(F)$, then
if $d>0$, $${\rm EP}[V,W] = 0.$$
If $d=0$, then of course
\[{\rm EP}[V,W] = \dim V \cdot \dim W.\]
\end{lemma}
The following result conjectured by the author some years ago, cf. \cite{Pr3},
and recently proved by Chan and Savin in \cite{CS2},
is at the root of why the simple and general result in Theorem \ref{whittaker} above
translates into a simple result about Hom spaces for generic representations
in Theorem \ref{duke93}.
\begin{thm} \label{vanishing} Let $\pi_1$ be an irreducible generic representation of ${\rm G}L_{n+1}(F)$, and $\pi_2$ an
irreducible generic representation of ${\rm G}L_{n}(F)$. Then,
\[{\rm Ext}^i_{{\rm G}L_n(F)}[\pi_1,\pi_2] = 0, \]
for all $i > 0$.
\end{thm}
On the other hand, Theorem \ref{whittaker} also has implications for non-vanishing of (higher) Ext groups in certain cases that we discuss now in the following remark.
\begin{remark} \label{rem1}
One knows, cf. \cite{Pr2}, that there are irreducible generic representations of ${\rm G}L_{3}(F)$ which have the trivial
representation of ${\rm G}L_2(F)$ as a quotient; similarly, there are
irreducible non-generic representations of ${\rm G}L_{3}(F)$ with irreducible generic
representations of ${\rm G}L_2(F)$ as a quotient. For such pairs $(\pi_1,\pi_2)$ of representations, it
follows from Theorem \ref{whittaker} on Euler-Poincar\'e characteristic that
\[{\rm EP}_{{\rm G}L_2(F)}[\pi_1, \pi_2]=0,\]
whereas
\[{\rm Hom}_{{\rm G}L_2(F)}[\pi_1, \pi_2] \not = 0.\]
Therefore, for such pairs $(\pi_1,\pi_2)$
of irreducible representations, we must have
\[{\rm Ext}^i_{{\rm G}L_2(F)}[\pi_1, \pi_2] \not =0,\]
for some $i>0$. The paper \cite{GGP2} studies more generally branching problem ${\rm Hom}_{{\rm G}L_n(F)}[\pi_1,\pi_2]$
when one of the irreducible representations, $\pi_1$ of ${\rm G}L_{n+1}(F)$ or $\pi_2$ of ${\rm G}L_n(F)$, is not generic, and both are Speh modules on discrete series representations, i.e., belongs to A-packets, thus leading to
non-vanishing of higher Ext groups.
\end{remark}
\section{Bessel subgroup} \label{bessel}
We will use Bessel subgroups, and Bessel models without defining them referring the reader to \cite{GGP},
except to recall that
these are defined for the classical groups ${\rm G}L(V), {\rm SO}(V), {\rm U}(V)$, through a subspace $W\subset V$,
with $V/W$ odd dimensional which in the case of ${\rm SO}(V)$ will be a
split quadratic space. In this paper we will use these subgroups only for ${\rm SO}(V)$.
The Bessel subgroup ${\rm Bes}(V,W)$ (shortened to ${\rm Bes}(W)$ if $V$ is understood)
is a subgroup of ${\rm SO}(V)$ of the form
${\rm SO}(W) \cdot U$ where $U$ is a unipotent subgroup of ${\rm SO}(V)$ which comes
with a character $\psi: U \rightarrow \mathbb{C}^\times$ normalized by ${\rm SO}(W)$.
The Bessel subgroup ${\rm Bes}(V,W) = {\rm SO}(W)$ if $ \dim (V/W)=1$. For a representation $\rho$ of ${\rm SO}(W)$,
we denote by $\rho \otimes \psi$ the corresponding representation of ${\rm Bes}(W) = {\rm SO}(W) \cdot U $.
The representation ${\rm ind}_{{\rm Bes}(W)}^{{\rm SO}(V)} (\rho \otimes \psi)$ of ${\rm SO}(V)$
will be called a Gelfand-Graev-Bessel representation, and plays a prominent role in analysing the
restriction problem from ${\rm SO}(V^+)$ to ${\rm SO}(V)$ for $V^+$ a quadratic space containing $V$ as a subspace
of codimension 1 such that $V^+/W$ is a split quadratic space of even dimension.
\begin{prop} \label{proj} If $\rho$ is a finite length representation of ${\rm SO}(W)$,
then the Gelfand-Graev-Bessel representation, \[ {\rm ind}_{{\rm Bes}(W)}^{{\rm SO}(V)} (\rho \otimes \psi),\]
is a
locally finitely generated representation of ${\rm SO}(V)$ which is
projective if, further, $\rho$ is cuspidal.
\end{prop}
\begin{proof}
Projectivity of the Gelfand-Graev representation for any quasi-split group is due to Chan and Savin in the appendix to the
paper \cite{CS3}. Let us remind ourselves a slightly delicate point.
By exactness of $U$-coinvariants, what is obvious is that ${\rm Ind}_{U}^{G} ( \psi)$ is an injective module for
$U$ any unipotent subgroup of a reductive group $G$.
That the dual of a projective module is an injective module is a generality, but this does not prove that
${\rm ind}_{U}^{G} ( \psi)$ is projective!
Instead of directly proving that ${\rm ind}_{U}^{G} ( \psi)$ is projective, Chan and Savin
prove that ${\rm Ext}_G^i[{\rm ind}_{U}^{G} ( \psi), \sigma] = 0$ for all $\sigma$ and all $i > 0$. By generalities, for algebras ${\mathcal H}$ containing a finitely generated $\mathbb{C}$-algebra $Z$ in its center over which
${\mathcal H}$ is finitely generated as a $Z$-module,
${\rm Ext}_{\mathcal H}^i[M,N] = 0$ for $i>0$ and for all $N$, if and only if this is true for
finitely generated $N$ and eventually ${\rm Ext}_{\mathcal H}^i[M,N] = 0$ for $i>0$ and for all $N$, if and only if ${\rm Ext}_{\mathcal H}^i[M,N] = 0$ for $i>0$ and for all $N$ of finite length. (Clearly, only irreducible $N$ are adequate!)
Going from finitely generated to finite length is a generality that Chan and Savin discuss, and is also in Proposition 5.2 of
\cite{NP} according to which
\[{\rm Ext}_{\mathcal H}^i[M,N]
\otimes_Z \widehat{Z} \cong {\rm Ext}_{\mathcal H}^i[M,\widehat{N}]
\cong \lim_{\leftarrow}{\rm Ext}_{\mathcal H}^i[M,N/{\mathfrak m} ^nN],\]
where $\widehat{N} = \displaystyle{ \lim_{\leftarrow}}(N/{\mathfrak m}^nN)$. For all this, finite generation of
$M$ is essential for which Chan and Savin quote the paper \cite{Bu-He} which proves that the Gelfand-Graev
representations are locally finitely generated.
In our case, we can appeal to Theorem \ref{AS} of Aizenbud-Sayag to prove that
the Gelfand-Graev-Bessel representation $ {\rm ind}_{{\rm Bes}(W)}^{{\rm SO}(V)} (\rho \otimes \psi)$ are locally
finitely generated which we now elaborate upon; the rest of the argument of Chan-Savin in \cite{CS3} goes verbatim.
Let $V^+= V + L$ where $L$ is a one dimensional quadratic space such that
$V^++ L = X + W + Y$ for $X,Y$ isotropic, perpendicular to $W$. Consider the representation $\tau \times \rho$ of
${\rm SO}(V^+)$, a
parabolically induced representation of ${\rm SO}(V^+)$ from the parabolic with Levi subgroup ${\rm G}L(X) \times {\rm SO}(W)$ of the representation
$\tau \boxtimes \rho$ where $\tau$ is any cuspidal representation of ${\rm G}L(X)$. Then it follows from the analogue of
Bernstein-Zelevinsky filtration for the restriction of the representation $\tau \times \rho$ of
${\rm SO}(V^+)$ to ${\rm SO}(V)$
due to Moeglin-Waldspurger, cf. \cite{Mo-Wa},
that $ {\rm ind}_{{\rm Bes}(W)}^{{\rm SO}(V)} (\rho \otimes \psi)$
is a submodule of the representation $\tau \times \rho$ of
${\rm SO}(V^+)$ restricted to ${\rm SO}(V)$. Since the rings which govern a Bernstein block are Noetheriam rings, submodules of
locally finitely generated representations are locally finitely generated, proving the proposition. \end{proof}
Note a particular case of this proposition.
\begin{cor}
If $W\subset V$ is a codimension one subspace of $V$, a quadratic space, and $\rho$
a finite length representation of ${\rm SO}(W)$, then ${\rm ind}_{{\rm SO}(W)}^{{\rm SO}(V)} (\rho)$ is a locally finitely
generated representation of ${\rm SO}(V)$
which is projective if $\rho$ is cuspidal
(and if $\dim(W)=2$, $W$ is not split). Also, similar assertions
for ${\rm G}L_n(F), {\rm U}_n$.
\end{cor}
\section{What does the restriction really looks like!}
So far, we have been discussing the question: which representations of ${\rm G}L_n(F)$ appear as a quotient of an irreducible representation
of ${\rm G}L_{n+1}(F)$. It is possible to have a more complete understanding of what a representation of ${\rm G}L_{n+1}(F)$ restricted to ${\rm G}L_n(F)$ looks like.
Vanishing of Ext groups in many but not in all cases, suggests that
the restriction to ${\rm G}L_n(F)$ of
an irreducible admissible
(generic) representation $\pi$ of ${\rm G}L_{n+1}(F)$ is close to being a projective module without
being one in all the cases.
Since the category of smooth representations of ${\rm G}L_n(F)$ is decomposed
into blocks parametrized by the inertial equivalence classes of cuspidal datum $(M,\rho)$ in ${\rm G}L_n(F)$, one can
ask if the projection of $\pi$ to the particular block, call it $\pi[M,\rho]$, is a projective module in that block.
This appears
to be an important question to understand: given an irreducible representation $\pi$ of ${\rm G}L_{n+1}(F)$, for which
blocks $(M,\rho)$ in ${\rm G}L_n(F)$, is $\pi[M,\rho]$ a projective module.
The following proposition is a direct consequence of the Bernstein-Zelevinski filtration which describes
the restriction of a representation $\pi$ of ${\rm G}L_{n+1}(F)$ to the mirabolic subgroup of ${\rm G}L_{n+1}(F)$
in terms of the derivatives $\pi^i$ of $\pi$ which are finite length smooth representations of ${\rm G}L_{n+1-i}(F)$.
Recall that the derivatives satisfy the Leibnitz rule (in the Grothendieck group of representations
of ${\rm G}L_{n+1}(F)$):
\[ (\pi_1 \times \pi_2)^d = \sum_{i=0}^{d} \pi_1^{d-i} \times \pi_2 ^i,\]
and that for an irreducible cuspidal representation $\pi$ of ${\rm G}L_d(F)$, the only nonzero derivatives are $\pi^0=\pi$, and
$\pi^d = \mathbb{C}$.
\begin{prop}\label{gln}
Let $\pi$ be a generic representation of ${\rm G}L_{n+1}(F)$. Let $(M,\rho)$ be a cuspidal datum in ${\rm G}L_n(F)$, thus
$M= {\rm G}L_{n_1}(F) \times \cdots \times {\rm G}L_{n_k}(F)$ with $n = n_1+\cdots + n_k$,
is a Levi subgroup inside ${\rm G}L_n(F)$, and $\rho = \rho_1 \boxtimes \cdots \boxtimes \rho_k$ is a tensor product of irreducible
cuspidal representations of ${\rm G}L_{n_i}(F)$. Assume that none of the cuspidal representations $\rho_i$ of ${\rm G}L_{n_i}(F)$
appear in the cuspidal support of $\pi$ even after an unramified twist.
Then $\pi|_{{\rm G}L_n(F)} [M,\rho]$ is a projective representation and is the
$[M,\rho]$ component of the Gelfand-Graev representation ${\rm ind}_N^{{\rm G}L_n(F)} \psi$.
\end{prop}
Here is the corresponding result for classical groups, asserted for simplicity of notation
only for ${\rm SO}(W) \subset {\rm SO}(V)$ where
$W \subset V$ is a codimension 1 nondegenerate subspace of a quadratic space $V$ with $\dim(V)=n+1$. This
result like Proposition \ref{gln} is also a
consequence of a Bernstein-Zelevinski like filtration (due to Moeglin and Waldspurger in \cite{Mo-Wa})
on the restriction of a representation of ${\rm SO}(V)$ to ${\rm SO}(W)$ when the representation of ${\rm SO}(V)$ is
induced from a maximal parabolic
with Levi of the form ${\rm G}L_m(F) \times {\rm SO}(W')$ of a representation of the form $\mu_1 \boxtimes \mu_2$, and using the
Bernstein-Zelevinski filtration for $\mu_1$ restricted to a mirabolic in ${\rm G}L_m(F)$. The proposition below uses
the representation ${\rm ind}_{{\rm Bes}(W_0)}^{{\rm SO}(W)} (\rho_0 \otimes \psi)$, for
$\rho_0$ a cuspidal representation of ${\rm SO}(W_0)$, which we called a Gelfand-Graev-Bessel representation
in section \ref{bessel}, and which is a projective representation by Proposition \ref{proj}.
Here, ${\rm Bes}(W_0)$ is the Bessel subgroup inside ${\rm SO}(W)$, introduced in section \ref{bessel},
where $W_0\subset W$ is a
nondegenerate subspace of a quadratic space $W$ with $W_0^\perp$ an odd dimensional hyperbolic space,
\begin{prop} \label{son}
Let $\pi$ be an admissible representation of ${\rm SO}(V)$ which is the full induction of a cuspidal representation
of a Levi subgroup of ${\rm SO}(V)$. Let $(M,\rho)$ be a cuspidal datum in ${\rm SO}(W)$, thus,
$M= {\rm G}L_{n_1}(F) \times \cdots \times {\rm G}L_{n_k}(F) \times {\rm SO}(W_0)$ with $n = 2n_1+\cdots + 2n_k+ \dim(W_0)$,
is a Levi subgroup inside ${\rm SO}(W)$, and $\rho = \rho_1 \boxtimes \cdots \boxtimes \rho_k \boxtimes \rho_0$
is a tensor product of irreducible cuspidal
representations of ${\rm G}L_{n_i}(F)$, and $\rho_0$ is an irreducible cuspidal representation of ${\rm SO}(W_0)$.
Assume that none of the cuspidal representations $\rho_i$ of ${\rm G}L_{n_i}(F)$
appear in the cuspidal support of $\pi$ even after an unramified twist (no condition on $\rho_0$).
Then $\pi|_{{\rm SO}(W)} [M,\rho]$ is a projective representation and is the
$[M,\rho]$ component of the Gelfand-Graev-Bessel representation ${\rm ind}_{{\rm Bes}(W_0)}^{{\rm SO}(W)} (\rho_0 \otimes \psi)$.
\end{prop}
\begin{remark}
We assumed $\pi$ in Proposition \ref{gln} to be generic as otherwise the assertion in the Proposition \ref{gln} will become
empty, i.e., $\pi|_{{\rm G}L_n(F)} [M,\rho]$ will be zero if $\pi$ is nongeneric. However, in Proposition \ref{son}
we do not assume that $\pi$ is generic. Neither of the two propositions require $\pi$ to be irreducible, and
in Proposition \ref{son} we do not require the inducing data for $\pi$ to be irreducible.
\end{remark}
The following theorem is due to Chan and Savin, cf. \cite{CS1}, \cite{CS2}, especially section 5 of
\cite{CS2}.
\begin{thm} \label{CS}
\begin{enumerate}
\item Restriction of an irreducible admissible representation
$\pi$ of ${\rm G}L_{n+1}(F)$ to ${\rm G}L_n(F)$ is projective in a
particular Bernstein block of smooth representations of ${\rm G}L_n(F)$
if and only if $\pi$ itself is generic and
all irreducible ${\rm G}L_n(F)$-quotients of $\pi$,
in that particular Bernstein block of smooth representations of ${\rm G}L_n(F)$,
are generic.
\item If $\pi_1,\pi_2$ are any two irreducible representations
of ${\rm G}L_{n+1}(F)$ whose restrictions to ${\rm G}L_n(F)$ are projective in a
particular Bernstein block of smooth representations of ${\rm G}L_n(F)$, then
$\pi_1$ and $\pi_2$ are isomorphic in that particular Bernstein block of smooth representations of ${\rm G}L_n(F)$.
\item For $\pi$ an irreducible generic representation of ${\rm G}L_{n+1}(F)$ which is projective when
restricted to the Iwahori block of ${\rm G}L_n(F)$,
\[ \pi|_{{\rm G}L_n(F)}[I,1] \cong {\rm ind}_{G(O_F)}^{G(F)} ({\rm St}).\]
\item More generally, by theorems of Bushnell and Kutzko, cf. \cite{B-K}, \cite{B-K2},
a general block for ${\rm G}L_n(F)$ arising out of a cuspidal datum
$(M,\rho)$ is equivalent to the Iwahori block of a product of general linear groups. Therefore,
there is an analogue of the representation ${\rm ind}_{G(O_F)}^{G(F)} ({\rm St})$ for each block in
${\rm G}L_n(F)$ and which the restriction problem from ${\rm G}L_{n+1}(F)$ to ${\rm G}L_n(F)$ picks up when the
restriction is projective in that block.
\end{enumerate}
\end{thm}
\begin{remark}
After this theorem of Chan and Savin, the unfinished tasks are:
\begin{enumerate}
\item Given an irreducible generic representation $\pi$ of ${\rm G}L_{n+1}(F)$, can we classify exactly the
Bernstein blocks of ${\rm G}L_n(F)$ in which $\pi|_{{\rm G}L_n(F)}$ is not projective?
\item More generally, if $\pi$ is an irreducible representation of ${\rm G}L_{n+1}(F)$ which may or may not be generic,
can one understand projective dimension (i.e., the minimal length of a projective resolution) of
$\pi|_{{\rm G}L_n(F)}$ in a particular Bernstein block?
\end{enumerate}
As is often the case in representation theory of $p$-adic groups,
dealing with discrete series which are non-cuspidal is often the most difficult part. In Proposition
\ref{restriction} in the next section,
we prove that for $\pi$ a generic representation of ${\rm G}L_{n+1}(F)$, $\pi|_{{\rm G}L_n(F)}$ is a projective
representation in those Bernstein blocks of ${\rm G}L_n(F)$ which contain no non-cuspidal discrete series representations.
Both Proposition
\ref{restriction} and Proposition
\ref{gln} can be considered as the simplest blocks where there is a nice answer.
\end{remark}
The following theorem of Chan, cf. \cite{Chan}, gives a complete classification of the irreducible
representations of ${\rm G}L_{n+1}(F)$ which when restricted to ${\rm G}L_n(F)$ are projective modules, thus remain projective
in {\it all} blocks.
\begin{thm} \label{thmchan}
Let $\pi$ be an irreducible representation of ${\rm G}L_{n+1}(F)$. Then $\pi$ restricted to ${\rm G}L_n(F)$ is a projective representation
if and only if
\begin{enumerate}
\item Either $\pi$ is essentially square integrable, or,
\item $(n+1) =2d$, $\pi = \pi_1 \times \pi_2$ where $\pi_i$ are cuspidal on ${\rm G}L_{d}(F)$.
\end{enumerate}
\end{thm}
\begin{remark} \label{rem3}
The non-tempered GGP, conjectured in \cite{GGP2} and proved for ${\rm G}L_n(F)$ in \cite{Chan2}, \cite{Gur}, describes irreducible representations $\pi$ of ${\rm G}L_{n+1}(F)$ and $\pi'$ of ${\rm G}L_n(F)$ (which are Speh representations on discrete series representations, i.e., have $A$-parameters) with
\[{\rm Hom}_{{\rm G}L_n(F)}[\pi,\pi'] \not = 0.\]
Thus from the list in Theorem \ref{thmchan}, we see that for the tempered representation $\pi= {\rm St}_d \times \chi {\rm St}_d$
of ${\rm G}L_{2d}(F)$ where $\chi$ is a unitary character of $F^\times$, and ${\rm St}_d$ is the Steinberg representation
of ${\rm G}L_d(F)$, although $\pi$ has no non-generic quotient with an $A$-parameter, it does have other
non-generic quotients.
\end{remark}
\begin{remark}
From theorems of Chan, just like cuspidal representations, discrete series representations of ${\rm G}L_{n+1}(F)$
are always projective representations when restricted to ${\rm G}L_n(F)$. This seems a general feature of all
the GGP pairs for which there is no proof yet.
\end{remark}
\section{A theorem of Roche and some consequences}
In the last section we discussed some situations where restriction of irreducible admissible
representations of ${\rm G}L_{n+1}(F)$ (resp., other classical groups) to ${\rm G}L_n(F)$ (resp., subgroups of other
classical groups) give rise to projective modules and which are for ${\rm G}L_{n+1}(F)$, by theorems of Chan and Savin,
very explicit compactly induced representations. In this section, we use a theorem of Alan Roche to one more such situation for both
${\rm G}L_{n+1}(F)$ as well as for classical groups where the restriction gives rise to projective modules. In this case,
however, the projective modules are {\it universal principal series} representations.
\begin{thm} \label{Roche} (Alan Roche) Let $G$ be a reductive $p$-adic group, $(M,\rho)$, a cuspidal datum. Let $M^0$ be the subgroup of $M$ generated by
compact elements in $M$. Assume
that no nontrivial element of $N_G(M)/M$ preserves $\rho$ up to an unramified twist. Then
the induced representation,
\[ {\rm Ind}_P^G(\rho),\]
is irreducible. Furthermore, the parabolic induction from $P$ (with Levi $M$) to $G$ gives an equivalence of categories
\[ { \mathbb{R}R}
[M]{[\rho]} \rightarrow {\mathbb{R}R}[G]{[M, \rho]}.\]
In particular, since the category of representations ${\mathbb{R}R}[M]{[\rho]}$ in the Bernstein component of $M$ corresponding to the cuspidal representation $\rho$ of $M$ is the same as the category of modules over an Azumaya algebra with center
the ring of functions on the complex torus consisting of the unramified twists of $\rho$, the same is true
of the Bernstein component ${\mathbb{R}R}[G]{[M, \rho]}$ of $G$.
\end{thm}
\begin{remark}
By the Geometric Lemma (which calculates Jacquet modules of full principal series representations), the assertion
that no nontrivial element of $N_G(M)/M$ preserves $\rho$ up to an unramified twist is equivalent to say that
the Jacquet module with respect to the parabolic $P$ of the principal series representation
${\rm Ind}_P^G(\rho)$ contains $\rho$ with multiplicity 1, and
no unramified twist of it distinct from itself.
\end{remark}
\begin{prop} \label{restriction} Let ${\rm G}_{n+1}$ be any of the classical groups ${\rm G}L_{n+1}(F),$ ${\rm SO}_{n+1}(F),$
${\rm U}_{n+1}(F)$. Let $\pi_1$ be an irreducible representation of ${\rm G}_{n+1}(F)$ belonging to a generic $L$-packet of ${\rm G}_{n+1}(F)$,
and let $(M,\rho)$ be a cuspidal
datum for ${\rm G}_n(F)$. Assume that no nontrivial element of $N_{{\rm G}_n(F)}(M)/M$ preserves $\rho$ up to an unramified twist.
Let $\rho^0$ be an irreducible representation of $M^0$, the subgroup of $M$ generated by compact
elements of $M$, with $\rho^0 \subset \rho|_{M^0}$.
Then,
the $(M,\rho)$ Bernstein component of $\pi_1$ restricted to ${\rm G}_n(F)$ is the
``universal principal series''
representation, i.e.,
\[ \pi_1|_{{\rm G}_n(F)} [M,\rho] \cong {\rm ind}_{P^0}^{{\rm G}_n}(\rho^0) \cong {\rm Ind}_{P}^{{\rm G}_n}{\rm ind}_{M^0}^M(\rho^0) ,\]
where $P^0=M^0N$. In particular, the $(M,\rho)$ Bernstein component of $\pi_1|_{{\rm G}_n(F)}$ is a projective representation which
is independent of $\pi_1$.
\end{prop}
\begin{proof}
Since $M$ is a Levi subgroup of ${\rm G}_n(F)$, $M$ is a product of the groups ${\rm G}L_{n_i}(F)$ with
${\rm G}_m(F)$ for some $m \geq 0$ (which are semisimple if $m>2$),
it is easy to see that any irreducible representation of $M$ when restricted to $M^0$,
is a finite direct sum of irreducible representations of $M^0$ with multiplicity 1. By the second adjointness combined with a form of Frobenius reciprocity for open subgroups,
for $\pi$ any irreducible representation of ${\rm G}_n(F)$,
\[ {\rm Hom}_{{\rm G}_n(F)}[ {\rm Ind}_{P}^{{\rm G}_n(F)} {\rm ind}_{M^0}^M(\rho^0), \pi] \cong {\rm Hom}_M [{\rm ind}_{M^0}^M(\rho^0), \pi_{\bar{N}} ] \cong
{\rm Hom}_{M^0} [\rho^0, \pi_{\bar{N}}].\]
Thus any irreducible representation of ${\rm G}_n(F)$ which appears as a quotient of ${\rm Ind}_{P^0}^{{\rm G}_n(F)}(\rho^0)$ appears with multiplicity atmost one, and appears with multiplicity one if and only if it belongs to the Bernstein block $[M,\rho]$, and is
a full principal series. Further, because of this multiplicity 1, the Azumaya algebra appearing in Theorem \ref{Roche}, is the ring $R$ of Laurent polynomials
$R=\mathbb{C}[X_1,X_1^{-1}, \cdots , X_d, X_d^{-1}]$
which is the ring of regular functions on a $d$-dimensional complex torus.
Now we analyse $\pi_1|_{{\rm G}_n(F)}[M,\rho]$ considered as a module, call it ${\mathcal M}$ over the ring,
$R=\mathbb{C}[X_1,X_1^{-1}, \cdots , X_d, X_d^{-1}]$. In the category of
$R$-modules, irreducible = simple modules are of the form $R/{\mathfrak m}$ where ${\mathfrak m}$ are maximal
ideals in $R$, and therefore irreducible quotients of $\pi_1|_{{\rm G}_n(F)}[M,\rho]$ are homomorphism of $R$-modules
${\mathcal M} \rightarrow R/{\mathfrak m} =\mathbb{C}$, equivalently, homomorphism of $R$-modules
${\mathcal M}/{\mathfrak m}{\mathcal M} \rightarrow \mathbb{C}$.
As every irreducible representation in this Bernstein block is a full principal series,
in particular they are all generic, therefore by Theorem \ref{gln} in the case of ${\rm G}L_{n+1}(F)$ and by GGP conjectures (theorems!) in other cases, each have
these irreducible principal series representations arise as a quotient of $\pi_1$ with multiplicity 1 (multiplicity identically zero is a possibility too for ${\rm SO}_{n+1}(F),{\rm U}_{n+1}(F)$; the important thing is that the multiplicity
is constant among all irreducible representations in this block).
This analysis can then be summarized to say that for the module ${\mathcal M}$ over $R=\mathbb{C}[X_1,X_1^{-1}, \cdots , X_d, X_d^{-1}]$ corresponding to $\pi_1|_{{\rm G}_n(F)}[M,\rho]$, ${\mathcal M}/{\mathfrak m}{\mathcal M} \cong \mathbb{C}$ for all maximal ideals ${\mathfrak m}$ in $R$.
By
Theorem \ref{AS} of Aizenbud-Sayag we know the finite generation of ${\mathcal M}$ over $R=\mathbb{C}[X_1,X_1^{-1}, \cdots , X_d, X_d^{-1}]$. Thus all the assumptions in the Lemma \ref{commutative} below are satisfied, and hence
${\mathcal M}$ is a projective module of rank 1 over
the ring $R$ of Laurent polynomials
$R=\mathbb{C}[X_1,X_1^{-1}, \cdots , X_n, X_n^{-1}]$. This is also the case for the universal principal series
representation ${\rm ind}_{P^0}^{{\rm G}_n}(\rho^0)$. Since
any rank 1 projective module over a Laurent polynomial ring is free,
this concludes the proof of the proposition.
\end{proof}
\begin{lemma} \label{commutative} Let $R$ be a finitely generated $k$-algebra where $k$ is a field. Suppose that $R$ has no nilpotent
elements. Let ${\mathcal M}$ be a finitely generated module over $R$ such that for each maximal ideal
${\mathfrak m}$ of $R$, ${\mathcal M}/{\mathfrak m}{\mathcal M}$ is free of rank 1 over $R/{\mathfrak m}$, then ${\mathcal M}$ is a
projective module of rank 1
over $R$. \end{lemma}
\begin{remark}
Proposition \ref{restriction} applies to all Bernstein blocks of ${\rm G}L_n(F)$ which do not contain a
non-cuspidal discrete series representation of ${\rm G}L_n(F)$, in particular it applies to
Bernstein blocks of ${\rm G}L_n(F)$ which contain a
cuspidal representation of ${\rm G}L_n(F)$!
\end{remark}
\section{ Euler-Poincar\'e characteristic for classical groups}
In the next few sections we will discuss Euler-Poincar\'e characteristic for branching laws for
classical groups, restricting ourselves to the case of $G={\rm SO}_{n+1}(F)$ and $H={\rm SO}_n(F)$.
In the following theorem, so as to simplify notation, if $\lambda_1$ is a representation of ${\rm SO}(V_1)$
and $\lambda_2$ is a representation of ${\rm SO}(V_2)$, then by ${\rm EP}_{{\rm Bes}}[\lambda_1, \lambda_2]$ we will give it the
usual meaning if $V_2 \subset V_1$ with $V_1/V_2$ odd dimensional split quadratic space, whereas
${\rm EP}_{{\rm Bes}}[\lambda_1, \lambda_2]$
will stand for
${\rm EP}_{{\rm Bes}}[\lambda_2, \lambda_1]$ if $V_1 \subset V_2$ with $V_2/V_1$ odd dimensional split quadratic space.
The notation ${\rm EP}_{{\rm Bes}}[\lambda_1, \lambda_2]$ will presume that we are in one of the two cases. This notation
has the utility of being able to add hyperbolic spaces of arbitrary dimension to $V_1$ or $V_2$, and by Theorem
15.1 of \cite{GGP},
\[ {\rm EP}_{{\rm Bes}}[\lambda_1, \lambda_2] = {\rm EP}_{{\rm Bes}}[\tau_1 \times \lambda_1, \tau_2 \times \lambda_2],\]
where $\tau_1,\tau_2$ are any cuspidal representations on general linear groups of arbitrary dimensions.
The following theorem is the analogue of Theorem \ref{whittaker}
which was for representations ${\rm G}L_{n+1}(F),{\rm G}L_n(F)$, now for classical groups, but as in the rest of the paper, we assert it only for special orthogonal groups.
\begin{thm} \label{EP} Let $V$ be a quadratic space over $F$, $V'$ a nondegenerate subspace of
codimension 1 inside $V$.
Let $\sigma = \pi_0 \times \sigma_0,$
be a representation for ${\rm SO}(V)$
where $\pi_0$ is a finite length representation of ${\rm G}L_{n_0}(F)$, and
$\sigma_0$ is a finite length representation of ${\rm SO}(V_0)$ where $V_0\subset V$
is a quadratic subspace of $V$ such that the quadratic space $V/V_0$ is a
hyperbolic space of dimensions $2n_0$. Similarly, let
$\sigma' = \pi'_0 \times \sigma'_0,$
be a representation for ${\rm SO}(V')$, then:
\[{\rm EP}_{{\rm SO}(V')}[\sigma,\sigma'] = \dim \mathcal{W}h(\pi_0) \cdot \dim \mathcal{W}h(\pi_0') \cdot \dim {\rm EP}_{{\rm Bes}}[\sigma_0, \sigma'_0].\]
\end{thm}
\begin{proof}
The proof of this theorem is very analogous to the proof of Theorem \ref{whittaker} for representations $\pi_1,\pi_2$ of ${\rm G}L_{n+1}(F),{\rm G}L_n(F)$,
replacing the Bernstein-Zelevinsky exact sequence describing the restriction of the representation $\pi_1$
to the mirabolic subgroup in ${\rm G}L_{n+1}(F)$, by a similar exact sequence describing the restriction of the representation $\sigma_0$
to the subgroup ${\rm SO}(V')$ due to Moeglin-Waldspurger in \cite{Mo-Wa}. The essential part of the proof of Theorem \ref{whittaker} was Lemma \ref{vanishing}
about vanishing of ${\rm EP}[V,W]$ when $V,W$ are finite length representations of ${\rm G}L_m(F)$, $m \geq 1$. This continues
to be the case here. We give some details of the proof here.
According to \cite{Mo-Wa}, the restriction of $\sigma = \pi_0 \times \sigma_0,$ to ${\rm SO}(V')$ has a filtration
with one sub-quotient equal to
\[ {\rm Ind}_{P'}^{{\rm SO}(V')} ( \pi_0 \times \sigma_0|_{{\rm SO}(V'_{0})}), \tag{1} \]
where $V'_{0} = V_0 \cap V'$ is a codimension one subspace of $V_0$, and $P'$ the parabolic in ${\rm SO}(V')$
with Levi ${\rm G}L_{n_0}(F) \times {\rm SO}(V'_{0}) $. (If there is no such parabolic in ${\rm SO}(V')$, then this term will not be there.)
The other subquotients of $\sigma|_{{\rm SO}(V')}$ are the principal series representations in ${\rm SO}(V')$ induced from
maximal parabolics with Levi whose ${\rm G}L$ part is of dimension $n_0-i$ (and we do not describe
the ${\rm SO}$ part of the Levi just calling it $V_i'$)
\[ \pi_0^i \times {\rm ind}_{{\rm Bes}(V_0)}^{{\rm SO}(V_i')} (\sigma_0 \otimes \psi) , \,\,\,\,\,\, n_0\geq i \geq 1 \tag{2} \]
Given the filtration on $\sigma|_{{\rm SO}(V')}$ with successive quotients as in (1) and (2),
and as ${\rm EP}[\sigma, \sigma']$ is additive in exact sequences,
one applies the 2nd adjointness theorem of Bernstein together with Lemma \ref{vanishing}
about vanishing of ${\rm EP}[V,W]$ when $V,W$ are finite length representations of ${\rm G}L_m(F)$, $m \geq 1$, and the Kunneth theorem, cf. \cite{Pr3}.
This implies that the only non-vanishing contribution to ${\rm EP}[\sigma, \sigma']$ will come from the
term in (2) corresponding to the highest derivative of $\pi_0$ giving rise to the representation
of ${\rm G}L(0) = \{1\}$ of dimension $\dim \mathcal{W}h(\pi_0)$. This needs to be multiplied by
\[{\rm EP} [{\rm ind}_{{\rm Bes}(V_0)}^{{\rm SO}(V_{n_0}')} (\sigma_0 \otimes \psi), \sigma'] = {\rm EP}_{{\rm Bes}(V_0)}[\sigma', \sigma_0].\]
Thus, \[{\rm EP}_{{\rm SO}(V')}[\sigma,\sigma'] = \dim \mathcal{W}h(\pi_0) \cdot \dim {\rm EP}_{{\rm Bes}(V_0)}[\sigma',\sigma_0]. \tag{3}\]
Doing this once more, using now the representation $\sigma' = \pi'_0 \times \sigma'_0,$ we get,
\[ {\rm EP}_{{\rm Bes}(V_0)}[\sigma',\sigma'_0] = \dim \mathcal{W}h(\pi'_0) \cdot \dim {\rm EP}_{{\rm Bes}(V'_0)}[\sigma_0,\sigma'_0]. \tag{4}\]
By (3) and (4), we find:
\[{\rm EP}_{{\rm SO}(V')}[\sigma,\sigma'] = \dim \mathcal{W}h(\pi_0) \cdot \dim \mathcal{W}h(\pi'_0) \cdot \dim
{\rm EP}_{{\rm Bes}}[\sigma_0,\sigma'_0], \tag{5}\]
completing the proof of Theorem \ref{EP}.
\end{proof}
\section{Euler-Poincar\'e characteristic for the group case: Kazhdan orthogonality}
The branching laws considered in this paper are for $H \hookrightarrow G \times H$,
where $H \subset G$, eventually interpreted as the $(G \times H)$ spherical variety
$\Delta(H) \backslash (G \times H)$,
in which we try to understand ${\rm Ext}^i_H(\pi_1,\pi_2)$
for an irreducible representation
$\pi_1 \boxtimes \pi_2$ of $G \times H$. A special case of this branching problem is for the ``group case''
where $H=G$, so in the case of
$G \hookrightarrow G \times G$ where we will be considering ${\rm Ext}^i_G(\pi_1,\pi_2)$ where $\pi_1, \pi_2$
are representations of the same group $G$. This could be considered as a precursor of the more general branching
for $H \hookrightarrow G \times H$, and has played an important role in the subject.
Explicit calculation of ${\rm Ext}_G^i(\pi_1,\pi_2)$ has been carried out in several cases for $G$, a reductive $p$-adic group, and $\pi_1,\pi_2$
irreducible representations of $G$. In particular, if both $\pi_1$ and $\pi_2$ are tempered representations of $G$,
there are general results in \cite{Op-Sol} using the formulation of $R$-groups, and there are some independent
specific calculations in \cite{Ad-Dp}.
Ext groups for certain non-tempered representations are considered in \cite{Or} as well as in \cite{Dat}.
In the archimedean case, $H^i({\mathfrak g}, K, \pi) = {\rm Ext}^i(\mathbb{C}, \pi)$ has been much studied, but not ${\rm Ext}^i(\pi_1,\pi_2)$ as far as I know.
The following theorem was conjectured by Kazhdan and was proved by Schneider-Stuhler, cf. \cite{Sch-Stu}, and by Bezrukavnikov. It is known only in characteristic zero.
\begin{theorem}
Let $\pi$ and $\pi'$ be finite-length, smooth
representations of a
reductive
$p$-adic
group $G$.
Then \[{\rm EP}_G[\pi,\pi'] =
\int_{C_{ellip}} \Theta(c)\bar{\Theta}'(c)\, dc,\]
where $\Theta$ and $\Theta'$ are the characters of $\pi$ and $\pi'$,
and $dc$ is a natural measure on the set
$C_{ellip}$ of regular elliptic conjugacy classes in $G$, and is given by
\[ dc = {W(G(F), T(F))}^{-1} \cdot \| \det (1- Ad(\gamma))_{{\frak g}/{\frak g}_\gamma} \| dt,\]
where $dt$ is the normalized Haar measure on the elliptic torus $T= G_\gamma$ giving it measure 1.
\end{theorem}
\section{An integral formula of Waldspurger}
In this section we review an integral formula of Waldspurger, cf. \cite{Wa}, \cite{Wa1},
which we then propose to be the
integral formula for the Euler-Poincar\'e pairing for
\[{\rm EP}_{{\rm Bes}(V,W)}[\sigma,\sigma' ]\]
for $\sigma$
any finite length
representation of $ {\rm SO}(V)$, and $\sigma'$ any finite length representation of ${\rm SO}(W)$,
where $V$ and $W$ are quadratic spaces over $F$ with
\[V = X + D + W + Y\]
with $W$ a quadratic subspace of $V$ of codimension $2k+1$
with $X$ and $Y$ totally isotropic subspaces of $V$ of dimension $k$, in duality with each other under the underlying bilinear form,
and $D$ an anisotropic line in $V$. Let $Z=X+Y$.
Let $\underline{\mathcal T}$ denote the set of elliptic tori $T$ in ${\rm SO}(W)$ such that there exist quadratic subspaces $W_T,W'_T$ of $W$ such that:
\begin{enumerate}
\item $W= W_T \oplus W'_T$, and $V=W_T \oplus W'_T \oplus D \oplus Z$.
\item $\dim (W_T)$ is even, and ${\rm SO}(W'_T)$ and
${\rm SO}(W'_T \oplus D \oplus Z)$ are quasi-split.
\item $T$ is a maximal (elliptic) torus in ${\rm SO}(W_T)$.
\end{enumerate}
Let
${\mathcal T}$ denote a set of orbits for the action of ${\rm SO}(W)$ on $\underline{\mathcal T}$.
For our purposes we note the
most important elliptic torus $T= \langle e \rangle$
corresponding to $W_T= 0$.
For $\sigma$ an admissible representation of ${\rm SO}(V)$ of finite length, define a function
$c_\sigma(t)$ for regular elements of a torus $T$ belonging to $\underline{\mathcal T}$ by the germ
expansion of the character $\theta_\sigma(t)$ of $\sigma$ on the centralizer of $t$ in the Lie algebra
of ${\rm SO}(V)$, and picking out `the' leading term.
Similarly, for $\sigma'$ an admissible representation of ${\rm SO}(W)$ of finite length,
one defines a function
$c_{\sigma'}(t)$ for regular elements of a torus $T$ belonging to $\underline{\mathcal T}$ by the germ
expansion of the character $\theta_{\sigma'}(t)$ of $\sigma'$.
Define a function $\Delta_T$ on an elliptic torus $T$ belonging to
$\underline{\mathcal T}$ with $W= W_T \oplus W'_T$, by
\[\Delta(t) = |\det(1-t)|_{W_T}|,\] and let $D^H$ denote the function
on $H(F) = {\rm SO}(W)$ defined by:
\[ D^H(t) = |\det(Ad(t) -1)_{h(F)/h_t(F)}|_F,\]
where $h(F)$ is the Lie algebra of $H$ and $h_t(F)$ is the Lie algebra of the centralizer of $t$ in $H$.
For a torus $T$ in $H$, define the Weyl group $W(H,T)$ by the usual normalizer divided by the centralizer:
\[W(H,T) = N_{H(F)}(T)/Z_{H(F)}(T).\]
The following theorem of Waldspurger could be considered as the analogue
of Kazhdan orthogonality for the group case encountered earlier.
\begin{thm}
Let $V = X + D + W + Y$ be a quadratic space over the non-archimedean local field $F$ with $W$ a quadratic subspace of codimension $2k+1$ as above.
Then for any irreducible admissible representation $\sigma$ of ${\rm SO}(V)$
and irreducible admissible representation $\sigma'$ of ${\rm SO}(W)$,
\[c(\sigma,\sigma'): =\sum _{T \in {\mathcal T}} |W(H,T)|^{-1} \int_{T(F)} c_\sigma(t) c_{\sigma'}(t) D^H(t) \Delta^k(t) dt ,\]
is a finite sum of absolutely convergent integrals. (The Haar measure on $T(F)$ is normalized to have volume 1.) If either $\sigma$ is a supercuspidal representation of
${\rm SO}(V)$,
and $\sigma'$ is arbitrary irreducible admissible representation of ${\rm SO}(W)$,
or both $\sigma$ and $\sigma'$ are tempered representations, then
$$c(\sigma,\sigma') = \dim {\rm Hom}_{{\rm Bes}(V,W)}[\sigma,\sigma' ].$$
\end{thm}
\section{Conjectured EP formula}
Given the theorem of Waldspurger, it is most natural to propose the following conjecture on
Euler-Poincar\'e pairing following the earlier notation of
$V = X + D + W + Y$, a quadratic space over the non-archimedean local field $F$ with $W$ a quadratic subspace of
$V$ of codimension $2k+1$.
\begin{conj} \label{integral}
\begin{enumerate}
\item If $\sigma$ and $\sigma'$ are irreducible tempered representations of ${\rm SO}(V), {\rm SO}(W)$ respectively with
$W\subset V$, a nondegenerate subspace with $V/W$ a split quadratic space of odd dimension, then
\[{\rm Ext}^i_{{\rm Bes}(V,W)}[\sigma,\sigma'] = 0\] for $i > 0$.
\item For finite length representations $\sigma$ of ${\rm SO}(V)$
and $\sigma'$ of ${\rm SO}(W)$, we have:
\begin{eqnarray*}
{\rm EP}_{{\rm Bes}(V,W)}[\sigma, \sigma' ] & : =&
\sum_i (-1)^i \dim {\rm Ext}^i_{{\rm Bes}(V,W)}[\sigma,\sigma'], \\
& = & \sum _{T \in {\mathcal T}} |W(H,T)|^{-1} \int_{T(F)} c_\sigma(t) c_{\sigma'}(t) D^H(t) \Delta^k(t) dt.
\end{eqnarray*}
\end{enumerate}
\end{conj}
\begin{remark}
\begin{itemize}
\item Waldspurger's theorem
is equivalent to the
conjectural statement on Euler-Poincar\'e characteristic if $\sigma$ or $\sigma'$ is supercuspidal (except that it is not proved if $\sigma'$ is supercuspidal, but $\sigma$ is arbitrary).
\item Waldspurger integral formula is available also in the work of R. Beuzart-Plessis for unitary groups.
\item A general integral formula for spherical varieties has been formulated by Chen Wan in \cite{Wan}.
\end{itemize}
\end{remark}
The following theorem asserts that once ${\rm Ext}_{{\rm Bes}}^i[\pi_1,\pi_2]$ are known to be zero for tempered
representations for $i \geq 1$, the Conjecture \ref{integral} on Waldspurger integral formula giving
Euler-Poincar\'e characteristic for all finite length representations holds. There is the further assertion
on vanishing of ${\rm Ext}_{{\rm Bes}}^i[\pi_1,\pi_2]$ for $i \geq 1$ for $\pi_1,\pi_2$ standard modules assuming that ${\rm Ext}_{{\rm Bes}}^i[\pi_1,\pi_2]$ are known to be zero for tempered
representations for $i \geq 1$.
\begin{thm}
For an irreducible tempered representation
$\sigma$ of ${\rm SO}(V)$
and $\sigma'$ of ${\rm SO}(W)$, assume that,
\[{\rm Ext}^i_{{\rm Bes}(V,W)}[\sigma,\sigma'] = 0 {\rm ~~for~~} i > 0,\]
then
\[{\rm Ext}^i_{{\rm Bes}(V,W)}[\sigma,\sigma'] = 0 {\rm ~~for~~} i > 0,\]
for all standard modules $\sigma$ of ${\rm SO}(V)$
and $\sigma'$ of ${\rm SO}(W)$. In particular, as irreducible representations of an orthogonal group
which belong to a generic $L$-packet are standard modules,
\[{\rm Ext}^i_{{\rm Bes}(V,W)}[\sigma,\sigma'] = 0 {\rm ~~for~~} i > 0,\]
if $\sigma$ is an irreducible representation belonging to a generic $L$-packet of ${\rm SO}(V)$ and $\sigma'$ of ${\rm SO}(W)$.
Further (assuming vanishing of higher Ext groups for tempered representations), we have the
Euler-Poincar\'e formula
for any finite length representation $\sigma$ of ${\rm SO}(V)$
and any finite length representation $\sigma'$ of ${\rm SO}(W)$:
\begin{eqnarray*} {\rm EP}_{{\rm Bes}(V,W)}[\sigma, \sigma' ] & : = &
\sum_i (-1)^i \dim {\rm Ext}^i_{{\rm Bes}(V,W)}[\sigma,\sigma'] \\ &=&
\sum _{T \in {\mathcal T}} |W(H,T)|^{-1} \int_{T(F)} c_\sigma(t) c_{\sigma'}(t) D^H(t) \Delta^k(t) dt.
\end{eqnarray*}
\end{thm}
\begin{proof}
Since both sides of the proposed equality:
\begin{eqnarray*} {\rm EP}_{{\rm Bes}(V,W)}[\sigma, \sigma' ] & : = &
\sum_i (-1)^i \dim {\rm Ext}^i_{{\rm Bes}(V,W)}[\sigma,\sigma'] \\ &=&
\sum _{T \in {\mathcal T}} |W(H,T)|^{-1} \int_{T(F)} c_\sigma(t) c_{\sigma'}(t) D^H(t) \Delta^k(t) dt,
\end{eqnarray*}
are bilinear forms, it suffices to prove it for
$\sigma$ belonging to a set of generators for the Grothendieck group of
finite length representations of ${\rm SO}(V)$, and $\sigma'$ belonging to a set of generators
for the Grothendieck group of
finite length representations of ${\rm SO}(W)$. It is well known that
standard modules form a generator, in fact a basis, of the Grothendieck group of finite length representations
of any reductive $p$-adic group. Therefore if we can prove:
\begin{eqnarray*} {\rm EP}_{{\rm Bes}(V,W)}[\sigma, \sigma' ] & : = &
\sum_i (-1)^i \dim {\rm Ext}^i_{{\rm Bes}(V,W)}[\sigma,\sigma'] \\ &=&
\sum _{T \in {\mathcal T}} |W(H,T)|^{-1} \int_{T(F)} c_\sigma(t) c_{\sigma'}(t) D^H(t) \Delta^k(t) dt,
\end{eqnarray*}
for $\sigma, \sigma'$ standard modules, we would know it for all finite length modules.
Let,
\[\sigma = \pi_1|\cdot|_F^{b_1}\times \cdots \times \pi_t|\cdot|_F^{b_t} \times \sigma_0,\]
be a standard module for ${\rm SO}(V)$, thus, we have:
\begin{enumerate}
\item For $i=1,\cdots, t$, $\pi_i$ is an irreducible, admissible, tempered representation of ${\rm G}L_{n_i}(F)$.
\item $\sigma_0$ is an irreducible, admissible, tempered representation of ${\rm SO}(V_0)$ where $V_0\subset V$
is a quadratic subspace of $V$ such that the quadratic space $V/V_0$ is an orthogonal direct sum
of hyperbolic spaces of dimensions $2n_i$.
\item The $b_i$ are real with $b_1\geq b_2\geq \cdots \geq b_t \geq 0$.
\end{enumerate}
Similarly, let
\[\sigma' = \pi'_1|\cdot|_F^{b'_1}\times \cdots \pi'_{t'}|\cdot|_F^{b'_{t'}} \times \sigma'_0,\]
be a standard module for ${\rm SO}(W)$.
We recall that by Proposition 1.1 of \cite{Mo-Wa},
\[ \dim {\rm Hom}[\sigma, \sigma'] = \dim {\rm Hom}[\sigma_0, \sigma_0'] .\]
Since the representations $\sigma_0$ of ${\rm SO}(V_0)$ and $\sigma'_0$ of ${\rm SO}(V'_0)$ are
irreducible tempered representations, by our assumption,
\[ {\rm Ext}^i[\sigma_0, \sigma_0'] = 0 {\rm ~~for~~} i>0.\]
Therefore, the Euler-Poincar\'e formula
\begin{eqnarray*} {\rm EP}_{{\rm Bes}(V,W)}[\sigma, \sigma' ] & : = &
\sum_i (-1)^i \dim {\rm Ext}^i_{{\rm Bes}(V,W)}[\sigma,\sigma'] \\ &=&
\sum _{T \in {\mathcal T}} |W(H,T)|^{-1} \int_{T(F)} c_\sigma(t) c_{\sigma'}(t) D^H(t) \Delta^k(t) dt,
\end{eqnarray*}
is valid for $\sigma_0$ and $\sigma'_0$ by the work of Waldspurger from \cite{Wa2}.
Now Proposition 1.1 of \cite{Mo-Wa} relating
$\dim {\rm Hom}[\sigma, \sigma']$ and $ \dim {\rm Hom}[\sigma_0, \sigma_0']$
is proved in two steps, proving $\dim {\rm Hom}[\sigma, \sigma'] \leq \dim {\rm Hom}[\sigma_0, \sigma_0']$ and then proving
$\dim {\rm Hom}[\sigma, \sigma'] \geq \dim {\rm Hom}[\sigma_0, \sigma_0']$. The first step uses relationships
of certain central exponents which works equally well to allow one to conclude
that $\dim {\rm Ext}^i[\sigma, \sigma']
\leq \dim {\rm Ext}^i[\sigma_0, \sigma_0']$ for all $i \geq 0$, and therefore as
we assume that ${\rm Ext}^i[\sigma_0, \sigma_0'] =0$ for all $i \geq 1$ (for tempered representations),
the same holds for ${\rm Ext}^i[\sigma, \sigma']$ for all $i \geq 1$.
We do not give more details here.
Once $ {\rm Ext}^i[\sigma, \sigma']$ are proved to be zero for $i\geq 1$, it then suffices to prove that the sum:
\[ \sum _{T \in {\mathcal T}} |W(H,T)|^{-1} \int_{T(F)} c_\sigma(t) c_{\sigma'}(t) D^H(t) \Delta^k(t) dt,\]
is the same for $\sigma_0$ and $\sigma'_0$ as it is for $\sigma$ and $\sigma'$. This is a consequence of the
van Dijk formula for principal series representations, see Lemma 2.3 of \cite{Wa1}. \end{proof}
\section{The Schneider-Stuhler duality theorem}
The following theorem is a mild generalization of a
duality theorem of Schneider and Stuhler in \cite{Sch-Stu}, see \cite{NP}; it turns
questions on ${\rm Ext}^i[\pi_1,\pi_2] $ to ${\rm Ext}^j[\pi_2,\pi_1]$, and is of considerable
importance to our theme.
\begin{thm} \label{SS}
Let $G$ be a reductive $p$-adic group, and $\pi$ an irreducible admissible representation of $G$.
Let $d(\pi) $ be the split rank of the center of the Levi subgroup $M$ of $G$ which carries the cuspidal support of $\pi$,
$D(\pi)$ be the Aubert-Zelevinsky involution of $\pi$. Then,
\begin{enumerate}
\item ${\rm Ext}_G^{d(\pi)}[\pi, D(\pi)] \cong \mathbb{C}$, and
\item For any smooth representation $\pi'$ of $G$, the bilinear pairing
\[(*) \,\,\,\,\, {\rm Ext}^{i}_G[\pi, \pi'] \times {\rm Ext}^{j}_G[\pi', D(\pi)] \rightarrow
{\rm Ext}^{i+j = d(\pi)}_G[\pi, D(\pi)] \cong \mathbb{C}, \]
is non-degenerate.
\end{enumerate}
\end{thm}
\section{An example: triple products for ${\rm G}L_2(F)$}
As suggested earlier, we expect that for all the GGP pairs $(G,H)$, when the irreducible representation
$\pi_1$ of $G$ and $\pi_2$ of $H$ are tempered,
${\rm Ext}^i_H[\pi_1,\pi_2]$ is non-zero only for $i=0$.
On the other hand, by the duality theorem discussed in the last section,
we expect that ${\rm Ext}^i_H[\pi_2,\pi_1]$ is typically zero for $i=0$, i.e., ${\rm Hom}_H[\pi_2,\pi_1]=0$ (so no wonder branching is usually not considered as a subrepresentation!), and shows up only for $i$ equals the split rank of the center of the Levi
from which $\pi_2$ arises through parabolic induction of a
supercuspidal representation. This is not completely correct as we will see.
The purpose of this section is to do an explicit restriction problem as
an example of what happens for classical groups
in one specific instance: the restriction problem from split ${\rm SO}(4)$ to split ${\rm SO}(3)$.
Thus we calculate ${\rm Ext}^i_{{\rm SO}_3(F)}[V,V']$, $i\geq 0$, and ${\rm EP}[V,V']$,
for $V$ a representation of ${\rm SO}_4(F)$ of finite length,
and $V'$ a representation of ${\rm SO}_3(F)$ of finite length, and then investigate when the restriction of $V$ to
${\rm SO}(3)$ is a projective module. As a consequence of what we do here, we will have constructed a projective module
in the Iwahori block of ${\rm SO}(3)={\rm PGL}_2(F)$ which is different from what we encountered earlier in the restriction
problem from ${\rm G}L_3(F)$ to ${\rm G}L_2(F)$ which all had only generic representations as a quotient, but here there will
be another possibility. (In fact Lemma 2.4 of \cite{CS3} has two options for projective modules which have multiplicity 1, and this other possibility which we will see here is the second option for projective modules in Lemma 2.4 of \cite{CS3}.)
Since
${\rm SO}_4(F)$ and ${\rm SO}_3(F)$ are closely related to ${\rm G}L_2(F) \times {\rm G}L_2(F)$
and ${\rm G}L_2(F)$ respectively, we equivalently consider $V \cong \pi_1 \otimes \pi_2$
for admissible representations $\pi_1, \pi_2$ of ${\rm G}L_2(F)$, and
$V' = \pi_3$ of ${\rm G}L_2(F)$. Our aim then is to calculate
$${\rm Ext}^i_{{\rm G}L_2(F)}[\pi_1 \otimes \pi_2, \pi_3],$$
or since we will prefer not to bother with central characters, we assume
that $\pi_1 \otimes \pi_2$ and $\pi_3$ have trivial central characters, and we will
then calculate,
$${\rm Ext}^i_{{\rm PGL}_2(F)}[\pi_1 \otimes \pi_2, \pi_3].$$
The following proposition follows from more general earlier results on Euler-Poincar\'e characteristic for principal
series representations, or can be deduced directly from Mackey theory. If at least one of the $\pi_i$ is cuspidal, then it is easy to see
that $ {\rm Ext}^1_{{\rm PGL}_2(F)}[\pi_1 \otimes \pi_2, \pi_3] =0$, and the proposition is equivalent to by-now well-known
results, cf. \cite{Pr1} about $ {\rm Hom}_{{\rm PGL}_2(F)}[\pi_1 \otimes \pi_2, \pi_3]$. The proposition in case one of the $\pi_i$'s is a twist of the Steinberg representation of ${\rm G}L_2(F)$ follows by embedding the Steinberg representation
in the corresponding principal series, and using additivity of EP in exact sequences.
\begin{prop} \label{trilinear} Let
$\pi_1, \pi_2$ and $\pi_3$
be either irreducible, infinite dimensional representations of ${\rm G}L_2(F)$, or (reducible)
principal series representations
of ${\rm G}L_2(F)$ induced from one dimensional representations. Assume that the product of the central characters of $\pi_1$ and $\pi_2$ is trivial, and $\pi_3$ is
of trivial central character. Then,
\[ {\rm EP}_{{\rm PGL}_2(F)}[\pi_1 \otimes \pi_2, \pi_3] =1,\]
except when all the representations $\pi_i$ are irreducible discrete series representations, and there is
a $D^\times$ invariant linear form on $\pi'_1 \otimes \pi'_2 \otimes \pi'_3$ where $\pi_i'$ denotes the
representation of $D^\times$ associated to $\pi_i$ by Jacquet-Langlands.
\end{prop}
Since for ${\rm PGL}_2(F)$, the only nonzero ${\rm Ext}^i$ can be for $i=0,1$, ${\rm EP}$ together with knowledge of ${\rm Hom}$ spaces implies
the following corollary.
\begin{cor} \label{ext-vanish} If $\pi_1, \pi_2$ and $\pi_3$ are any three irreducible, infinite dimensional representations
of ${\rm G}L_2(F)$,
with the product of the central characters of $\pi_1$ and $\pi_2$ trivial, and $\pi_3$
of trivial central character, then ${\rm Ext}^i_{{\rm PGL}_2(F)}[\pi_1 \otimes \pi_2,\pi_3] = 0$ for $i>0$.
\end{cor}
\begin{remark}
The authors Cai and Fan in \cite{CF} prove more generally that
\[ {\rm Ext}^i_{{\rm G}L_2(F)}[\Pi, \mathbb{C}] = 0 ~~~{\rm~~ for ~~~} i \geq 1,\]
where $\Pi$ is an irreducible generic representation of ${\rm G}L_2(E)$
whose central character restricted to $F^\times$ is trivial for $E$ a cubic \'etale extension of $F$.
\end{remark}
We now use Proposition \ref{trilinear} and Corollary \ref{ext-vanish}
to study the restriction problem from $[{\rm G}L_2(F) \times {\rm G}L_2(F)]/\Delta (F^\times)$ to ${\rm PGL}_2(F)$, and to understand when $\pi= \pi_1 \otimes \pi_2$ where $\pi_1, \pi_2$ are any two irreducible, infinite dimensional representations
of ${\rm G}L_2(F)$,
with the product of the central characters of $\pi_1$ and $\pi_2$ trivial, is a projective representation of ${\rm PGL}_2(F)$
As discussed
earlier, if a smooth representation $\pi$ of ${\rm PGL}_2(F)$ is locally finitely generated, then it is a projective module in the category
of smooth representations of ${\rm PGL}_2(F)$ if and only if
\[ {\rm Ext}^1_{{\rm PGL}_2(F)}[\pi, \pi'] = 0, \]
for all smooth finitely generated representations $\pi'$ of ${\rm PGL}_2(F)$ which is the case
if and only if
\[ {\rm Ext}^1_{{\rm PGL}_2(F)}[\pi, \pi'] = 0, \]
for all finite length representations $\pi'$ of ${\rm PGL}_2(F)$, which is the case
if and only if
\[ {\rm Ext}^1_{{\rm PGL}_2(F)}[\pi, \pi'] = 0, \]
for all irreducible representations $\pi'$ of ${\rm PGL}_2(F)$.
In our case, $\pi= \pi_1 \otimes \pi_2$ where $\pi_1, \pi_2$ are any two irreducible, infinite dimensional representations
of ${\rm G}L_2(F)$,
with the product of the central characters of $\pi_1$ and $\pi_2$ trivial. Therefore if $\pi'$ is any
infinite dimensional irreducible representation
of ${\rm PGL}_2(F)$, the desired vanishing of ${\rm Ext}^1[\pi,\pi']$ is a consequence of Corollary \ref{ext-vanish}. Therefore to
prove projectivity of $\pi= \pi_1 \otimes \pi_2$ as a representation of ${\rm PGL}_2(F)$, it suffices to check that,
\[ {\rm Ext}^1_{{\rm PGL}_2(F)}[\pi_1\otimes \pi_2, \chi] = 0, \]
for $\chi: F^\times /F^{\times 2} \rightarrow \mathbb{C}^{\times}$, treated as a character of ${\rm PGL}_2(F)$. Now,
\[ {\rm Ext}^1_{{\rm PGL}_2(F)}[\pi_1\otimes \pi_2, \chi] = {\rm Ext}^1_{{\rm PGL}_2(F)}[\pi_1, \chi \pi_2^\vee].\]
It is easy to see that if $\pi_1,\pi_2$ are irreducible and infinite dimensional representations of ${\rm G}L_2(F)$, then
${\rm Ext}^1_{{\rm PGL}_2(F)}[\pi_1, \chi \pi_2^\vee]$ is not zero if and only if $\pi_1,\pi_2^\vee$ are irreducible principal series representations
of ${\rm G}L_2(F)$ such that $\pi_1 \cong \chi \pi_2^\vee$ with $\chi$ a quadratic character. (Vanishing of ${\rm Ext}^1_{{\rm PGL}_2(F)}[{\rm St}_2, \chi {\rm St}_2]$ is a well-known generality about discrete series representations; for a proof in this case, see Lemma 7 of \cite{Pr2}.)
We summarize this analysis in the following proposition.
\begin{prop} \label{gl2} Let
$\pi_1, \pi_2$
be irreducible, infinite dimensional representations of ${\rm G}L_2(F)$ such that the product of the central
characters of $\pi_1$ and $\pi_2$ is trivial. Then the representation $\pi_1\otimes \pi_2$ of ${\rm PGL}_2(F)$ is a
projective module unless $\pi_1,\pi_2$ are irreducible principal series representations
of ${\rm G}L_2(F)$ such that $\pi_1 \cong \chi \pi_2^\vee$ with $\chi$ a quadratic character, in which case it is
not a projective module exactly in the block of ${\rm PGL}_2(F)$ containing the character $\chi$. In particular,
if at least one of $\pi_1$ or $\pi_2$ is a twist of the Steinberg representation, $\pi_1\otimes \pi_2$ is a projective
module in the category of smooth representations of ${\rm PGL}_2(F)$.
\end{prop}
\begin{cor}
Let ${\rm St}_2$ be the Steinberg representation of ${\rm PGL}_2(F)$, and $T$ the diagonal split torus of ${\rm PGL}_2(F)$.
Then ${\rm St}_2 \otimes {\rm St}_2$ is a projective representation
of ${\rm PGL}_2(F)$, and ${{\rm ind}}_T^{{\rm PGL}_2(F)}(\mathbb{C})$ which is not a projective module but which contains the Steinberg
representation by Lemma 5.4 of \cite{Pr1} and has the property that (using Lemma 5.4 of
\cite{Pr1} for the isomorphism)
\[ {{\rm ind}}_T^{{\rm PGL}_2(F)}(\mathbb{C})/{\rm St}_2 \cong {\rm St}_2 \otimes {\rm St}_2\]
is a projective representation of
${\rm PGL}_2(F)$.
\end{cor}
In earlier sections we saw the construction of projective modules in the Iwahori block given by
${\rm ind}_{G(O_F)}^{G(F)} {\rm St}$.
Proposition \ref{gl2} allows one to construct another projective module
in the Iwahori block of ${\rm PGL}_2(F)$ which are the same outside the reducible principal series which also arises
from the restriction problem (from ${\rm G}L_2(F) \times {\rm G}L_2(F)$ to the diagonal ${\rm G}L_2(F)$); it is of course
the projective representation ${\rm ind}_{G(O_F)}^{G(F)} \mathbb{C}$ given by Lemma 2.4 of \cite{CS3}.
For the Steinberg representation ${\rm St}_2$ of ${\rm PGL}_2(F)$, ${\rm St}_2 \otimes {\rm St}_2$ is a projective module,
does not have ${\rm St}_2$ as a quotient, but has the trivial representation as a quotient. Further, ${\rm St}_2 \otimes {\rm St}_2$ has all other
irreducible principal series as a unique quotient. On the other hand, as ${\rm St}_2 \otimes {\rm St}_2$
is a projective module,
and the principal series ${\rm Ps}(\nu^{1/2}, \nu^{-1/2})$ has the trivial repesentation of ${\rm PGL}_2(F)$ as a quotient, there is a surjective map from ${\rm St}_2 \otimes {\rm St}_2$ to ${\rm Ps}(\nu^{1/2}, \nu^{-1/2})$.
Thus ${\rm St}_2 \otimes {\rm St}_2$ as a module $M$ over the Iwahori Hecke algebra,
hence over its center $Z $, is a 2-dimensional free module, which at the maximal ideals $\mathfrak{m}$ of $Z$ corresponding points away from
the character $(\nu^{1/2}, \nu^{-1/2})$ has $M/\mathfrak{m}M$ as two dimensional complex vector space corresponding to the Iwahori
fixed vectors in an irreducible unramified principal series representation of ${\rm PGL}_2(F)$ whereas at the maximal ideal corresponding
the character $(\nu^{1/2}, \nu^{-1/2})$, $M/\mathfrak{m}M$ as two dimensional complex vector space corresponding to the Iwahori
fixed vectors in the reducible principal series representation ${\rm Ps}(\nu^{1/2}, \nu^{-1/2})$ of ${\rm PGL}_2(F)$.
On the other hand, for distinct irreducible cuspidal representations $\pi_1,\pi_2$ of ${\rm PGL}_2(F)$, $\pi_1 \otimes \pi_2$ is a projective module, which in the Iwahori block
of ${\rm PGL}_2(F)$, has ${\rm St}_2$ as a quotient, but not the trivial representation as a quotient, and has all
irreducible principal series as a unique quotient. As $\pi_1 \otimes \pi_2$
is a projective module,
and the principal series ${\rm Ps}(\nu^{-1/2}, \nu^{1/2})$ has the Steinberg repesentation of ${\rm PGL}_2(F)$ as a quotient, there is a surjective map from $\pi_1 \otimes \pi_2$ to ${\rm Ps}(\nu^{-1/2}, \nu^{1/2})$.
Summarizing, the restriction problem from ${\rm G}L_2(F) \times {\rm G}L_2(F)$ to the diagonal ${\rm G}L_2(F)$), when it is projective in the Iwahori block,
gives rise to the two projective modules ${\rm ind}_{{\rm PGL}_2(O_F)}^{{\rm PGL}_2(F)} {\rm St}$ and ${\rm ind}_{{\rm PGL}_2(O_F)}^{{\rm PGL}_2(F)} \mathbb{C}$,
and also gives rise to a module which is not projective in a very rare case as described
in Proposition \ref{gl2}, and when non-projective, it contains a submodule (a twist of the Steinberg) as we will presently see. All these three options are explicitly described in Proposition \ref{gl2}, or are explicitly describable!
It may be hoped that this kind of complete explicit description can be made for the branching problems around GGP.
Here is an application of the calculation on Ext groups which when combined with the
duality theorem leads to existence of submodules.
The
following proposition gives a complete classification
of irreducible submodules $\pi$ of the tensor product $\pi_1 \otimes \pi_2$ of two (irreducible, infinite dimensional)
representations $\pi_1,\pi_2$ of ${\rm G}L_2(F)$ with the product of their central characters trivial. A more general result is available in \cite{CF}.
\begin{prop}
Let $\pi_1, \pi_2$ be two irreducible admissible infinite dimensional
representations of ${\rm G}L_2(F)$ with product of their
central characters trivial. Then the following is a complete list of irreducible sub-representations
$\pi$ of $\pi_1 \otimes \pi_2$ as ${\rm PGL}_2(F)$-modules.
\begin{enumerate}
\item $\pi$ is a supercuspidal representation of ${\rm PGL}_2(F)$, and appears as a quotient of $\pi_1 \otimes \pi_2$.
\item $\pi$ is a twist of the Steinberg representation, which we assume by absorbing the twist in
$\pi_1$ or $\pi_2$ to be the Steinberg representation ${\rm St}$ of ${\rm PGL}_2(F)$. Then ${\rm St}$ is a
submodule of $\pi_1 \otimes \pi_2$ if and only if
$\pi_1, \pi_2$ are both irreducible
principal series representations, and $\pi_1 \cong \pi_2^\vee$.
\end{enumerate}
\end{prop}
\begin{remark}
Unlike the case of triple products above, Chan in \cite{Chan} has proved that in the case of the pair $({\rm G}L_{n+1}(F),{\rm G}L_n(F))$,
if ${\rm Hom}_{{\rm G}L_n(F)}[\pi_2,\pi_1] \not = 0$, for $\pi_1$ an irreducible representation of ${\rm G}L_{n+1}(F)$ and $\pi_2$ of ${\rm G}L_n(F)$, then both $\pi_1,\pi_2$ must be one dimensional. Thus in this case, even supercuspidals of ${\rm G}L_n(F)$ do not arise as submodules which is related to the non-compact center of the subgroup ${\rm G}L_n(F)$
(and which is not contained in the center of the ambient ${\rm G}L_{n+1}(F)$).
\end{remark}
\section{Template from algebraic geometry}
We enumerate some of the basic theorems in algebraic geometry which seem to have
closely related analogues in our context, although for no obvious reason! For the analogy, we consider
$H^0(X,{\mathfrak F})$,
for $X$ a
smooth projective varieties (or sometimes more general varieties) equipped
with a coherent sheaf ${\mathfrak F}$ versus ${\rm Hom}[\pi_1,\pi_2]$, and corresponding $H^i$
and ${\rm Ext}^i$.
\begin{enumerate}
\item Finite dimensionality of $H^i(X,{\mathfrak F})$
and vanishing for $i> \dim X$.
\item Semi-continuity theorems available both in algebraic geometry for
$H^i(X,{\mathfrak F}_\lambda)$, and ${\rm Ext}^i[\pi_{1, \lambda}, \pi_{2,\mu}]$ for families of sheaves or
of representations.
\item Riemann-Roch theorem expressing $EP(X,{\mathfrak F})$ in terms of simple
invariants associated to $X$ and the sheaf ${\mathfrak F}$. In our case, these are the integral formulae
which go into the Kazhdan conjecture and in the work of Waldspurger, involving invariants of the
space $X$, certain elliptic tori, and invariants associated to sheaves= representations through character
theory.
\item Kodaira vanishing for $H^i(X,{\mathfrak F})$, $i> 0$ for an ample sheaf ${\mathfrak F}$.
\item Serre duality
\[{\rm Ext}^i({\mathcal O}_X,{\mathfrak F}) \times {\rm Ext}^{d-i}({\mathfrak F}, \omega_X)
\rightarrow {\rm Ext}^d({\mathcal O}_X, \omega_x) = F.\]
\item Special role played by $X = {\mathbb P}^d(F)$ in Algebraic geometry, and here,
we have our own, {\it
her all-embracing majesty}, ${\rm G}L_n(F)$.
\end{enumerate}
{\bf Acknowledgement:}
This paper is an expanded and written version of my lecture
in the IHES Summer School on the Langlands Program in July 2022. The author would like to thank the organizers for putting together a wonderful program. The author especially thanks R. Beuzart-Plessis and Kei Yuen Chan for all their
helpful remarks.
\end{document}
(By Theorem \ref{AS}, this means that the Gelfand-Graev representation
${\rm ind}_N^{{\rm G}L_n(F)} \psi$ is locally finitely generated, an
assertion which is known for general split reductive groups.)
\end{document} | math | 78,309 |
\begin{document}
\begin{abstract}
We study the Minkowski length $L(P)$ of a
lattice polytope $P$, which is defined to be the largest number of
non-trivial primitive segments whose Minkowski sum lies in $P$. The
Minkowski length represents the largest possible number of factors in a
factorization of polynomials with exponent vectors in $P$, and shows up in
lower bounds for the minimum distance of toric codes. In this paper we give a
polytime algorithm for computing $L(P)$ where $P$ is a 3D lattice
polytope.
We next study 3D lattice polytopes of Minkowski length 1. In particular, we show that
if $Q$, a subpolytope of $P$, is the Minkowski sum of $L=L(P)$ lattice polytopes $Q_i$,
each of Minkowski length 1, then the total number of interior lattice points of the polytopes
$Q_1,\cdots, Q_L$ is at most 4. Both results extend previously known results for lattice polygons.
Our methods differ substantially from those used in the two-dimensional case.
\end{abstract}
\title{Minkowski Length of 3D Lattice Polytopes}
\section*{Introduction}
Let $P$ be a convex lattice polytope in $\mathbb{R}^n$. Then $P$ defines $\mathcal{L}_{\mathbb{F}}(P)$, a vector space over a field $\mathbb{F}$ spanned by the monomials in $P$. That is,
$$\mathcal{L}_{\mathbb{F}}(P)
=\operatorname{span}_{\mathbb{F}}\{t^m\ |\ m\in P\cap\mathbb{Z}^n\},$$
where $t^m=t_1^{m_1}\cdots t_n^{m_n}.$ In this paper we address the following question: What is the largest number of factors that a polynomial in $\mathcal{L}_{\mathbb{F}}(P)$ could have?
We also study those factors and obtain results regarding their Newton polytopes. Although these questions are interesting on its own,
our motivation comes from studying toric codes.
The {\it toric code} $\mathcal{C}_P$, first introduced by Hansen in
\cite{Han}, is defined by evaluating the polynomials in $\mathcal{L}_{\mathbb{F}_q}(P)$ at all the points
$t$ in the algebraic torus $(\mathbb{F}_q^*)^n$. That is, $\mathcal{C}_P$ is a linear code whose codewords
are the strings $(f(t)\ |\ t\in (\mathbb{F}_q^*)^n)$ for
$f\in\mathcal{L}_{\mathbb{F}_q}(P)$.
It is convenient to assume that $P$ is contained in the square $[0,q-2]^n$,
so that all the monomials in $\mathcal{L}_{\mathbb{F}_q}(P)$ are linearly independent
over $\mathbb{F}_q$~\cite{Ru}. Thus $\mathcal{C}_P$ has block length $(q-1)^n$ and
dimension equal to the number of the lattice points in $P$.
Note that the weight of each non-zero codeword in $\mathcal{C}_P$ is
the number of points $t\in(\mathbb{F}_q^*)^n$ where the corresponding polynomial does not
vanish. Therefore, the minimum distance of $\mathcal{C}_P$ (which is the minimum weight for linear codes)
equals
$$d(\mathcal{C}_P)=(q-1)^n-\max_{0\neq f\in\mathcal{L}_{\mathbb{F}_q}(P)}Z(f),$$
where $Z(f)$ is the number of zeroes (i.e. points of vanishing) in $(\mathbb{F}_q^*)^n$
of $f$.
For toric surface codes, that is, in the case $n=2$, Little and Schenck in \cite{LSch} used Hasse-Weyl bound and the intersection theory on toric surfaces to come up with the following general idea: If $q$ is sufficiently large, then polynomials $f\in\mathcal{L}_{\mathbb{F}_q}(P)$ with more absolutely irreducible factors will
necessarily have more zeroes in $(\mathbb{F}_q^*)^2$ (\cite{LSch}, Proposition 5.2). In \cite{SS1} this idea was expanded to produce explicit bounds for the
minimum distance of $\mathcal{C}_P$ in terms of a certain geometric invariant $L(P)$, (full) Minkowski length of $P$, which was introduced in that paper.
This invariant $L(P)$ reflects the largest possible number of
absolutely irreducible factors a polynomial $f\in\mathcal{L}_{\mathbb{F}_q}(P)$ can have. A polytime algorithm for computing $L(P)$ for polygons was provided in \cite{SS1}.
In this paper, we extend this result to dimension 3 (Theorem~\ref{T:alg}, based on Theorem~\ref{T:3d2}).
Moreover, \cite{SS1} provides a description of the factorization $f=f_1\cdots f_{L(P)}$
for $f\in\mathcal{L}_{\mathbb{F}_q}(P)$ with the largest number of factors: it turns out that in such a factorization
the Newton polygon $P_{f_i}$ (which is the convex hull of the exponents of the
monomials in $f_i$) is either a primitive segment, a unit simplex, or
a triangle with exactly 1 interior point. It is also shown in \cite{SS1} that a triangle with an interior point
can occur in such a factorization at most once. This implies that the total number $I$ of interior
lattice points of $P_{f_i}$ is at most 1. This result is essential for establishing bounds on the minimum distance of toric surface codes in~\cite{SS1}.
This argument is not directly extendable to dimension 3, as it does not seem feasible to obtain a description of the Newton polytopes $P_{f_i}$ in dimension 3 (See~\cite{MSRI} for some examples).
Nevertheless, in this paper we show that in the 3D case $I \leq 4$ (Theorem~\ref{T:main}). Our methods
differ substantially from those used in the two-dimensional case. An initial version of our argument relied heavily on the classification of 3D Fano tetrahedra \cite{Kas}. Although we were able to completely get rid of this dependency in our final argument, the classification helped us significantly in our explorations.
In \cite{SS1} combinatorial results about Minkowski length of polygons lead to lower bounds on the minimum distance of surface toric codes. We hope that our (entirely combinatorial) paper will in the future lead to similar bounds for 3D toric codes.
\section{Minkowski Length of Lattice Polytopes}
Here we recall the definition of the (full) Minkowski length introduced in \cite{SS1} as well as reproduce and refine some results
from that paper using new methods which will later be applied to the 3D case.
\subsection{Minkowski sum}
Let $P$ and $Q$ be convex polytopes in $\mathbb{R}^n$. Their {\it Minkowski sum} is
$$P+Q=\{p+q\in\mathbb{R}^n\ |\ p\in P,\ q\in Q\},$$
which is again a convex polytope. Figure \ref{F:sum} shows the Minkowski sum of a triangle and a square.
\begin{figure}
\caption{The Minkowski sum of two polygons}
\label{F:sum}
\end{figure}
Let $f$ be a Laurent polynomial in $\mathbb{F}_q[t_1^{\pm 1},\dots,t_n^{\pm 1}]$. Then its
{\it Newton polytope} $P_f$ is the convex hull of the exponent vectors
of the monomials appearing in $f$. Thus $P_f$ is a {\it lattice polytope} as its vertices belong to the integer lattice $\mathbb{Z}^n\subset\mathbb{R}^n$. Note that if $f,g\in \mathbb{F}_q[t_1^{\pm 1},\dots,t_n^{\pm 1}]$ then the Newton polytope of their product $P_{fg}$ is the Minkowski sum $P_f+P_g$. A {\it primitive segment} $E$
is a lattice segment whose only lattice points are its endpoints. The difference of the endpoints
is a vector $v_E$ whose coordinates are relatively prime ($v_E$ is defined up to sign).
A polytope which is the Minkowski sum of primitive segments is called a {\it (lattice) zonotope}.
We say that two lattice polytopes are {\it equal} if they are the same up to translation.
The automorphism group of the lattice is the group of {affine unimodular transformations}, denoted
by $\A{n}$, which consists of translations by an integer vector and linear transformations in $\operatorname{GL}(n,\mathbb{Z})$. It is a standard fact from lattice-point geometry
that any two primitive segments in $\mathbb{R}^n$ are $\A{n}$-equivalent (\cite{Newman}, Theorem II.1).
\subsection{Minkowski length}
Let $P$ be a lattice polytope in $\mathbb{R}^n$.
\begin{definition}\label{D:maximal}
The (full) Minkowski length $L=L(P)$ of a lattice polytope $P$ is the largest number
of primitive segments $E_1, E_2,\dots, E_L$ whose Minkowski sum is in $P$.
Equivalently, $L=L(P)$ is the largest number of non-trivial lattice polytopes $Q_1,\dots, Q_L$ whose
Minkowski sum is in $P$. Any collection of $L=L(P)$ non-trivial lattice polytopes $Q_1,\dots, Q_L$ whose
Minkowski sum is in $P$ will be referred to as a {\it maximal
(Minkowski) decomposition in P}. The {\it dimension} of a maximal decomposition is the dimension of the Minkowski sum
$Q_1+\cdots+Q_L$.
\end{definition}
\begin{example}
In the figure below, the first polygon, called $T_0$, has Minkowski length 1. For the second one, the Minkowski length is 2. Notice that this triangle has many maximal decompositions: the sum of two horizontal segments, the sum of two vertical segments, the sum of two diagonal segments, the sum of two standard 2-simplices, etc.
For the last polygon, the Minkowski length is 3, as there is a parallelogram inside that is a sum of three lattice segments.
\begin{figure}\label{F:MinkLength}
\end{figure}
\end{example}
Clearly, $L(P)$ is an $\A{n}$-invariant and the summands of every maximal decomposition in $P$ are polytopes of
Minkowski length 1.
It does not seems feasible to describe
polytopes of Minkowski length 1 in general. However, in
dimension 2 such a description is given in \cite{SS1} and we
reproduce it here.
\begin{figure}
\caption{Polygons of Minkowski length 1}
\label{F:indecomp}
\end{figure}
\begin{theorem}\label{T:indecomp}\cite{SS1}
Let $P$ be a convex lattice polygon in the plane with $L(P)=1$. Then $P$ is $\A2$-equivalent
to a primitive segment, the standard 2-simplex $\mathcal{D}$ or the triangle $T_0$ with
vertices $(1,0)$, $(0,1)$ and $(2,2)$.
\end{theorem}
It is also proved in \cite{SS1} that a maximal decomposition $Q\subseteq P$ can have at most one summand $\A2$-equivalent to $T_0$, and if this is the case, the
remaining summands are $[0,e_1]$, $[0,e_2]$, and $[0,e_3]$, that is, $Q$ is $\A{2}$-equivalent to $Q=T_0+n_1[0,e_1]+n_2[0,e_2]+n_3[0,e_1+e_2]$. Here $e_1,e_2$ are the standard basis vectors.
We will recover this result (using a new method that will be later applied to the 3D case) and will also show that if a triangle $\Delta$ is a summand of a maximal decomposition of a polygon $P$, then the other summands are either primitive segments, or exactly that triangle $\Delta$. That is, if $Q_1$ and $Q_2$ are triangles and $L(Q_1+Q_2)=2$, then $Q_1=Q_2$. This refinement will be important for our 3D discussion.
Before we state the result, we set notation and prove a lemma which will also be important for our future discussion in dimension 3.
Let $P$ be a lattice polytope in $\mathbb{R}^n$. For each segment whose endpoints are lattice points in $P$, consider its direction vector reduced modulo 3. Since $v$ and $-v$ define the same segment, we identify such vectors. Using this equivalence relation, we obtain the set $\mathbb{Z}_3 \mathbb{P}^{n-1}$ of equivalence classes.
\begin{lemma}\label{L:mult} Let $L(P)=L(Q)=1$ and $L(P+Q)=2$, where $P$ and $Q$ are lattice polytopes in $\mathbb{R}^n$. Then if $P$ and $Q$ each have a segment of some class $a$, then those two segments are equal (are the same up to translation). If $P$ has at least two segments of class $a$, then $Q$ has no segments from that class.
\end{lemma}
\begin{proof}
If $P$ and $Q$ both have lattice segments from the same equivalence class, then, unless these segments are equal, their Minkowski sum contains a segment of Minkowski length 3 (since either sum or difference of the direction vectors is a multiple of 3). If $P$ has multiple segments from one class, then these segments cannot be translates of each other, as they would form a parallelogram in $P$ of Minkowski length at least 2. If $Q$ has a segment from that class, it would be not a translate of at least one of the two segments in $P$ and we again conclude $L(P+Q)\geq 3$.
\end{proof}
\begin{theorem} Let $P$ be a convex lattice polygon. If one of the summands of a maximal decomposition $Q$ in $P$ is $\A{2}$-equivalent to $T_0$, then $Q$ is $\A{2}$-equivalent to $Q=T_0+n_1[0,e_1]+n_2[0,e_2]+n_3[0,e_1+e_2]$. If one of the summands $\Delta$ of $Q$ is $\A{2}$-equivalent to the standard 2-simplex , then the remaining summands that are not primitive segments, are equal to $\Delta$.
\end{theorem}
\begin{proof}
We have four equivalence classes in $\mathbb{Z}_3 \mathbb{P}^1$:
$$(1,1), (1,-1), (1,0), (0,1).
$$
Notice that if $a$ and $b\in\mathbb{Z}_3\mathbb{P}^1$ are linearly independent (that is, $a\neq \pm b$), then they generate all the classes:
$$\langle a,b \rangle= \{ a, b, a+b, a-b\}.$$
Now let one of the summands in a maximal decomposition in $P$ be $\A{2}$-equivalent to $T_0$. Then we can assume this summand is exactly $T_0$. The direction vectors of the lattice segments in $T_0$ are $(1,0)$, $(0,1)$, $(1,1)$, $(1,-1)$, $(1, 2)$, and $(2,1)$. The last three are all from the same class, so by Lemma~\ref{L:mult} segments from this class cannot show up in other summands. The first three are all from distinct classes, hence by the lemma only segments with direction vectors $(1,0)$, $(0,1)$, and $(1,1)$ can show up in other summands. Since all four classes are covered, we have shown that the direction vectors of lattice segments in other summands can only be $(1,0)$, $(0,1)$, and $(1,1)$. One can use such segments to form a triangle in four different ways. The result will be the standard 2-simplex and its reflections. In each of these four cases, it's easy to check that the Minkowski sum of such a triangle with $T_0$ is 3, which proves our first statement.
Next, let one of the summands be equivalent to the standard 2-simplex $\Delta$, so we can assume it's exactly $\Delta$. The direction vectors $(1,0), (0,1), $ and $(-1,1)$ are all from distinct classes, hence if other summands have lattice segments from these classes, they would have to have these direction vectors. If there is another triangle in the maximal decomposition, it would have to be equivalent to the standard 2-simplex, as $T_0$ is not possible by the above argument. The direction vectors would have to belong to three distinct classes, so at least two of the sides would have to have direction vectors $(1,0), (0,1),$ or $(-1,1)$. Here are eight triangles that could be formed in this way:
The last four have a segment with a direction vector either $(-2, 1)$ or $(-1, 2)$, which are from the same class with $(1,1)$, so the Minkowski sum of any of these
triangles with $\Delta$ is at least 3. For all the remaining triangles, except $\Delta$ itself we easily check that their sum with $\Delta$ has Minkowski length 3.
\end{proof}
\begin{corollary}\label{C:sharetriang}\ Let $P, Q$ be lattice polytopes in $\mathbb{R}^3$ with $L(P+Q)=2$. Consider the intersection of a plane $\pi$ with each $P$ and $Q$.
If each $\pi\cap P$ and $\pi\cap Q$ contains a lattice triangle, then
these lattice triangles are the same up to translation.
\end{corollary}
\begin{proof}
Let $u$ be a primitive normal vector to $\pi$. Let $A\in {\rm GL}(3,\mathbb{Z})$ be a matrix whose last row is $u$. (It is shown, for example, in \cite{Newman}, Theorem II.1, why such a matrix exists.)
Then $A$ maps $\pi$ to the $(x,y)$-plane and the result follows from the previous theorem.
\end{proof}
\section{Algorithm for Computing $L(P)$ for 3D polytopes.}
It was shown in \cite{SS1} that in the plane case there always exists a maximal decomposition in $P$ of a very simple form. Namely, there exists a maximal decomposition that is equivalent to $n_1[0,e_1]+n_2[0,e_2]+n_3[0,e_1+e_2]$ for some $n_1, n_2, n_3\in \mathbb{N}$. This fact was used in \cite{SS1} to build an algorithm for finding $L(P)$. To extend this result to the 3D case, we first make a definition.
\begin{definition}
Let $P\subset\mathbb{R}^n$ be a convex lattice polytope. Then the set of its maximal decompositions is partially ordered by inclusion. That is, we say that
$$A+P_1+\cdots+P_k< B+Q_1+\cdots+Q_l$$
if $A+P_1+\cdots+P_k\subsetneq B+Q_1+\cdots+Q_l$. Here $A$ and $B$ are points in $\mathbb{Z}^n$. A maximal decomposition is called a {\it smallest maximal decomposition} if it is minimal with respect to
this partial order. Note that a smallest maximal decomposition is a Minkowski sum of segments.
\end{definition}
\begin{proposition}\label{T:2d1}
Let $P\subseteq \mathbb{R}^2$ be a lattice polygon. Consider a smallest maximal decomposition $Z$ in $P$:
$$P\supseteq Z = A+ n_1 E_1+\cdots+n_l E_l.
$$
Then ${\rm Area}(E_i+E_j)\leq 1$ for any choice of $1\leq i,j\leq l$.
\end{proposition}
\begin{proof}
Let $v_1$ and $v_2$ be the primitive direction vectors of the segments $E_i$ and $E_j$ and assume that the area of the parallelogram spanned by $v_1$ and $v_2$ is at least 2. Applying an $\A{2}$ transformation, we can assume that $A$ is the origin, $v_1=e_1=(1,0)$ and $v_2=(a,b)$, where $0\leq a <b$ and $b>1$, which implies that $(1,1)\in\mathbb{P}i=[0,e_1]+[0,(a,b)]$. We show now that there is always a segment $I$ of Minkowski length 2 that lies strictly inside $\mathbb{P}i$ and hence, we can pass from $\mathbb{P}i$ to $2I$ and get a smaller maximal decomposition.
If both $a$ and $b$ are even then
$2[0,(a/2,b/2)]$ is strictly inside of $\mathbb{P}i$; if $a$ is odd and $b$ is even then
$2[0,((a+1)/2,b/2)]\subsetneq \mathbb{P}i$; if $a$ is even and $b$ is odd, $(1,1)+2[0,(a/2,(b-1)/2)]\subsetneq \mathbb{P}i$; if $a$ and $b$ are both odd, $(1,1)+2[0,((a-1)/2),(b-1)/2)]\subsetneq \mathbb{P}i$.
\end{proof}
\begin{theorem}\label{T:2d2}
Let $P\subseteq \mathbb{R}^2$ be a lattice polygon. If $Z$ is a smallest maximal decomposition in $P$, then it is $\A{2}$-equivalent to
$$P\supseteq Z = n_1 [0,e_1]+n_2[0,e_2]+n_3 [0,e_1+e_2].
$$
\end{theorem}
\begin{proof}
Let $P\supseteq Z = n_1 E_1+\cdots+n_l E_l$ with $v_1,\dots, v_l$ distinct primitive direction vectors of the segments $E_1,\dots, E_l$. Applying an $\A{2}$ transformation we can assume that $v_1=e_1$. Next, since $\det(v_1,v_2)=\pm 1$, we can assume that $v_2=e_2$. Then by the previous proposition, any other $v_k$ is either $(1,1)$ or $(1,-1)$ as we can always switch a vector to its negative. Notice that these two vectors cannot simultaneously appear in a smallest decomposition, as the sum of the corresponding segments would contain a segment of Minkowski length 2. Finally, these two remaining cases are $\A{2}$-equivalent.
\end{proof}
We next treat the 3D case.
\begin{proposition}\label{T:3d1}
Let $P\subset\mathbb{R}^3$ be a lattice polytope. Consider a smallest maximal decomposition $Z$ in $P$
$$P\supseteq Z = A+n_1 E_1+\cdots+n_l E_l.
$$
Then ${\rm Vol}(E_i+E_j+E_k)\leq 2$ for any choice of $1\leq i,j,k\leq l$.
\end{proposition}
\begin{proof}
Let $v_1$, $v_2$, and $v_3$ be the primitive vectors that go along the segments $E_i$, $E_j$, $E_k$, and assume that the volume of the parallelepiped spanned by $v_1$, $v_2$, and $v_3$ is at least 3. Applying an $\A{2}$ transformation (and using Proposition~\ref{T:2d1}), we can assume that $A$ is the origin, $v_1=e_1=(1,0,0)$, $v_2=e_2=(0,1,0)$, and $v_3=(s,t,u)$, where $0\leq s \leq t <u$. The volume of the parallelepiped spanned by $e_1,e_2,v_3$ is $|u|$. If $s=0$, then the area spanned by $e_2$ and $v_3$ is $|u|\geq 3$, which would contradict the minimality of $Z$. We next observe that the parallelepiped spanned by $e_1,e_2$, and $v_3$ is defined by
$$\mathbb{P}i=\left\{(x,y,z) \in \mathbb{R}^3\mid 0 \leq z \leq u,\ \frac{s}{u} z \leq x \leq \frac{s}{u} z + 1, \ \frac{t}{u} z \leq y \leq \frac{t}{u} z + 1\right\}
$$
and consider the following three cases.
\begin{enumerate}
\item[Case 1.] $s\leq u/2$, $t\leq u/2$ \\
Using the description of $\mathbb{P}i$ above, we can easily check that $(1,1,2)$ and $(s,t,u-2)$ are both in $\mathbb{P}i$. Hence a parallelogram with the vertices
$(1,1,0), (1,1,2), (s,t, u-2)$, and $(s, t, u)$ is inside $\mathbb{P}i$. This parallelogram is a Minkowski sum of three segments
$$(1,1,0)+2[0,(0,0,1)]+[0,(s-1,t-1,u-2)]\subsetneq \mathbb{P}i ,$$
which contradicts the minimality of $Z$. Notice that $u\geq 3$ ensures that the segments involved in the decomposition are non-trivial.
\item[Case 2.] $s> u/2$, $t>u/2$ \\
Then $(2,2,2)$ and $(s-1,t-1,u-2)$ are in $\mathbb{P}i$, so a parallelogram with the vertices $(0,0,0), (2,2,2), (s-1,t-1,u-2)$, and $(s+1, t+1, u)$ is inside $\mathbb{P}i$.
This parallelogram is a Minkowski sum of three segments
$$2[0, (1,1,1)]+[0, (s-1,t-1,u-2)]\subsetneq\mathbb{P}i,$$ which contradicts the minimality of $Z$.
\item[Case 3.] $s\leq u/2$, $t>u/2$ \\
Then $(1,2,2)$ and $(s,t-1,u-2)$ are in $\mathbb{P}i$, so a parallelogram with the vertices $(1,0,0), (1,2,2), (s,t-1,u-2)$, and $(s, t+1, u)$ is inside $\mathbb{P}i$.
This parallelogram is a Minkowski sum of three segments
$$(1,0,0)+2[0, (0,1,1)]+[0, (s-1,t-1,u-2)]\subsetneq\mathbb{P}i,$$
and we get the same contradiction again.
\end{enumerate}
\end{proof}
\begin{theorem}\label{T:3d2}
Let $P\subset \mathbb{R}^3$ be a lattice polytope. Let $Z$ be a smallest maximal decomposition in $P$, then it is $\A{2}$-equivalent to either
$$n_1[0, e_1] + n_2[0, e_2] + n_3[0, e_1+e_2+2e_3] + n_4[0, e_1+ e_2+e_3] + n_5[0, e_1 + e_3]
+n_6[0, e_2 + e_3] + n_7[0, e_3]
$$
or
$$n_1 [0,e_1] + n_2 [0,e_2] + n_3 [0,e_3] + n_4 [0,e_1 + e_2 + e_3] + n_5 [0,e_1 \pm e_2] + n_6 [0,e_1+e_3] + n_7 [0,e_2 + e_3].
$$
\end{theorem}
\begin{proof} Assume first that there are three segments in the maximal decomposition $Z$ whose direction vectors generate a parallelepiped of volume 2.
We can then assume that the first direction vector is $e_1$. By Proposition \ref{T:2d1} we can assume that the second vector is $e_2$. Next, we can assume that the third direction vector $v$ is of the form $(s,t,u)$ where $0 \leq s \leq t <u$. Since $2=|\det(e_1,e_2,v)|$, we know that $u=2$ and the only options for the third vector
are $(0,1,2)$ and $(1,1,2)$. The first of these two options is impossible, as the sum of $(0,1,2)$ and $(0,1,0)$ is not primitive, which contradicts the minimality of $Z$. We have shown that the third vector is $(1, 1, 2)=e_1+e_2+2e_3$.
Let $v=(a,b,c)$ be a direction vector of some other segment in the maximal decomposition $Z$. We know that $|\det(e_1, e_2, v)|\leq 2$, $|\det(e_1, e_1+e_2+2e_3, v)|\leq 2$, and $|\det(e_2,e_1+e_2+2e_3, v)|\leq 2$, which gives us the following restraints on the components of $v$: $|c - 2 b| \leq 2$, $|c - 2 a| \leq 2$, and $|c|\leq 2$. By flipping the direction vector $v$ if necessary, we can assume that $c\geq 0$.
If $c=0$, then $v=(1,1,0)$ or $(1,-1,0)$. Both options are impossible as then the sum of $v$ with $(1,1,2)$ is $(2,2,2)$ or $(2,2,0)$, so we can pass to a smaller maximal decomposition. If $c=1$, then $v=(0,0,1)$, $(0,1,1)$, $(1,0,1)$, $(1,1,1)$.
If $c=2$, then $v=(0,1,2)$, $(1,0,2)$, $(1,2,2)$, or $(2,1,2)$. Adding either $e_1$ or $e_2$ to each of these four vectors we can get a non-primitive vector,
so none of these vectors occur in our maximal decomposition. We have shown that in the case when there are three segments in the maximal decomposition $Z$ that generate a parallelepiped of volume 2, then $Z$ is $\A{2}$-equivalent to
$$n_1[0, e_1] + n_2[0, e_2] + n_3[0, e_1+e_2+2e_3] + n_4[0, e_1+ e_2+e_3] + n_5[0, e_1 + e_3]
+n_6[0, e_2 + e_3] + n_7[0, e_3].
$$
Next, assume that any three segments in the maximal decomposition $Z$ generate a parallelepiped of volume at most 1.
If for any three vectors the volume is zero, then we are in the plane case and we are done. Otherwise, we can assume that
first three vectors are $e_1$, $e_2$, and $e_3$. Let $v=(a,b,c)$ be any other direction vector in the maximal decomposition $Z$.
Then we have $|a|\leq 1$, $|b|\leq 1$ and $|c|\leq 1$. By flipping the direction vector we can assume that $c\geq 0$.
Here are the options for $v$ that we get, written in four lines:
$$(1,1, 0), (1,-1, 0),$$
$$(0,1,1), (0,-1,1),$$
$$(1, 0, 1), (-1,0,1),$$
$$(1, 1,1), (1,-1,1), (-1,1,1), (-1,-1,1).$$
Notice that no two vectors from the same line here can occur in $Z$ together as their sum is not primitive, which would contradict the minimality of $Z$.
By flipping the direction of basis vectors, we can assume that if any of the four vectors in the last line occur in $Z$, then it is $(1,1,1)$. Let's assume that this is the case and $(1,1,1)$ occurs in $Z$.
We notice next $(-1,0,1)$ and $(0,-1, 1)$ can not occur in $Z$ together as if we add these two vectors together with $(1,1,1)$, we get $(0,0,3)$.
We can make the same observation about $(-1,0,1)$ and $(1,-1,0)$ and then about $(0,-1,1)$ and $(1,-1,0)$. This implies that only one of $(1,-1,0)$, $(0,-1,1)$,
and $(-1,0,1)$ occurs in $Z$. By permuting $e_1,e_2$, and $e_3$ we can assume that the one that occurs is $(1,-1,0)$.
In the case when none of of the four vectors $(1, 1,1), (1,-1,1), (-1,1,1), (-1,-1,1)$ occur in $Z$, by applying a diagonal change of basis with $\pm 1$'s on the main diagonal (which will not change $[0,e_1], [0,e_2]$, $[0,e_3]$), we can turn any pair of vectors from the set $(1,-1,0), (0,-1,1), (-1,0,1)$ into corresponding vectors
with positive entries. For example, a matrix with the diagonal entries $-1,-1,1$ will turn $(-1,0,1)$ and $(0,-1,1)$ into $(1,0,1)$ and $(0,1,1)$. Hence we will be able to get rid of all the vectors with negative entires except, possibly, one. By permuting $e_1$, $e_2$, and $e_3$, we can assume that the vector with a negative entry is $(1,-1,0)$. We have shown that if any three segments in $Z$ generate a parallelepiped of volume at most 1, then $Z$ is $\A{3}$-equivalent to either
$$n_1 [0,e_1] + n_2 [0,e_2] + n_3 [0,e_3] + n_4 [0,e_1 + e_2 + e_3] + n_5 [0,e_1+ e_2] + n_6 [0,e_1+e_3] + n_7 [0,e_2 + e_3]
$$
or
$$n_1 [0,e_1] + n_2 [0,e_2] + n_3 [0,e_3] + n_4 [0,e_1 + e_2 + e_3] + n_5 [0,e_1-e_2] + n_6 [0,e_1+e_3] + n_7 [0,e_2 + e_3].
$$
\end{proof}
Notice that in 2D a smallest maximal decomposition has at most 3 distinct summands; in 3D, as we have just shown, such a decomposition has at most 7 distinct summands. It turns out that in dimension $n$ a smallest maximal decomposition has at most $2^{n}-1$ distinct summands.
\begin{proposition}\label{P:bound} Let $P\subset\mathbb{R}^n$ be a convex lattice polytope. Let $Z$ be a smallest maximal decomposition in $Z$.
Then $Z$ has at most $2^n-1$ distinct summands.
\end{proposition}
\begin{proof} Reduce all the summands in $Z$ modulo 2. Since the summands are primitive segments, there will be $2^n-1$ possibilities for a reduced segment.
If the number of distinct segments in $Z$ is at least $2^n$, we will have two summands that are equal modulo 2. Then their sum is non-primitive, which contradicts the minimality of $Z$.
\end{proof}
Although we expect that the sum of the $2^n-1$ segments with $0,1$ components mentioned in the proof of the above proposition has Minkowski length $2^n-1$, we do not have a proof of this statement, which would have implied that the bound of the proposition is sharp.
Let a lattice polytope $P$ be described by its facets equations. Then Barvinok's algorithm \cite{Bar, Koppe} counts the number of lattice points in $P$ in polynomial time.
We will assume that the list $P\cap\mathbb{Z}^3$ of the lattice points in $P\subset\mathbb{R}^3$ is given, and will explain how to find the Minkowski length of $P$ in polynomial time in $P\cap\mathbb{Z}^3$.
\begin{theorem}\label{T:alg}
Let $P\subset \mathbb{R}^3$ be a lattice polytope with the given set of its lattice points $P\cap\mathbb{Z}^3$. Then the Minkowski length $L(P)$ can be
found in polynomial time in $P\cap\mathbb{Z}^3$.
\end{theorem}
\begin{proof}
This algorithm relies on Theorem~\ref{T:3d2}. We search for all possible decompositions of the form described in the theorem.
For every quadruple of points $\{A,B,C,D\}\subseteq P\cap\mathbb{Z}^3$, where it is
important which point goes first and the order of the other three does
not matter, we check if $[0,B-A]$, $[0,C-A]$, and $[0,D-A]$ generate a
parallelepiped of volume one or two. If the volume is one, these segments are equivalent
to $[0,e_1]$, $[0,e_2]$, $[0,e_3]$ and we look for maximal decompositions equivalent to
$$n_1 [0,e_1] + n_2 [0,e_2] + n_3 [0,e_3] + n_4 [0,e_1 + e_2 + e_3] + n_5 [0,e_1 \pm e_2] + n_6 [0,e_1+e_3] + n_7 [0,e_2 + e_3],
$$
that is, maximal decompositions of the form
$$n_1E_1+ n_2 E_2+ n_3 E_3 + n_4 E_4+ n_5 E_5 + n_6 E_6 + n_7 E_7,
$$
where $E_1=[0,B-A]$, $E_2=[0,C-A]$, $E_3=[0,D-A]$, $E_4=[0,B+C+D-3A]$, $E_5=[0,B+C-2A]$ or $[0, B-C]$, $E_6=[0,B+D-2A]$,
$E_7=[0, C+D-2A]$.
If the volume is two, we check if the segments $[0,B-A]$, $[0,C-A]$, and $[0,D-A]$ are primitive and if any two of them generate
a parallelogram whose only lattice points are the vertices. If this is the case, these three segments are equivalent to $[0,e_1]$, $[0,e_2]$,
and $[0, e_1+e_2+2e_3]$. We then let $E_1=[0,B-A]$, $E_2=[0,C-A]$, $E_3=[0,D-A]$, $E_4=[0,(B+C+D-3A)/2]$, $E_5=[0,(B+D-C-3A)/2]$,
$E_6=[0,(C+D-B-3A)/2]$, $E_7=[0,(D-A-B-C)/2]$.
Next, for every $1\leq i\leq 7$, we find $M_i$, the largest integer such that there is some lattice point $F$ in $P$ with $F+M_iE_i\subseteq P$.
For each $7$-tuple of integers $m=(n_1, \dots, n_7)$ where $0\leq
n_i\leq M_i$,
we check if some lattice translate of the zonotope
$Z_m=n_1E_1+\cdots+n_7E_7$ is contained in $P$ (we run through lattice
points $F$ in $P$ to check if $F+Z_m$ is contained in $P$). For all
such zonotopes that fit into $P$ we look at $n_1+\cdots+n_7$ and find the
maximal possible value $N$ of this sum.
Finally, the largest such sum $N$ over all choices of
$\{A,B,C, D\}\subseteq P\cap\mathbb{Z}^3$ is $L(P)$. Clearly, this algorithm is polynomial in $P\cap\mathbb{Z}^3$.
\end{proof}
A group of REU students (Ian Barnett, Benjamin Fulan, and Candice Quinn) at Kent State University in Summer 2011 tried to generalize this algorithm to dimension 4.
Their first step was to obtain a 4D version of Proposition~\ref{T:3d1}. They showed that if $P\subseteq\mathbb{R}^4$ is a lattice polytope
and $Z= A+n_1 E_1+\cdots+n_l E_l$ is a smallest maximal decomposition in $P$, then
${\rm Vol}(E_i+E_j+E_k+E_m)\leq 14$ for any choice of $1\leq i,j,k,m\leq l$, and this bound is sharp.
Unfortunately, this bound is too high to obtain a description of smallest maximal decompositions in 4D, similar to the one of
Proposition~\ref{T:3d2}.
\section{Lattice Polytopes of Minkowski Length 1}
It was shown in Theorem 1.6 of \cite{SS1} that if $P\supseteq Q=Q_1+\cdots+Q_l$ is a maximal decomposition of a polygon $P$ then at most one of $Q_i$ has an integer lattice point in its interior, that is, $\sum I(Q_i)\leq 1$. This fact was crucial in~\cite{SS1} for establishing bounds on the minimum distance of the toric surface code defined by $P$. We expect that in order to extend these bounds to 3D codes, one needs to explore similar questions in dimension 3. As it was mentioned above, a description of polytopes of Minkowski length 1 in dimension 3 does not seem feasible. We will instead reduce the lattice segments contained in a 3D lattice polytope modulo 3, which will help us show that if $P$ is a 3D polytope with $L(P)=1$, then $\sum I(Q_i)$ is at most 4.
Let $P$ be a lattice polytope in $\mathbb{R}^3$ of Minkowski length 1. Then $P$ has at most 8 lattice points. Indeed, otherwise there would have been two lattice points in $P$ that are congruent modulo 2, and hence the segment connecting them would have Minkowski length of at least 2.
For each segment whose endpoints are lattice points in $P$, we consider its direction vector reduced modulo 3. Since $v$ and $-v$ define the same segment, we identify such vectors. For thus defined modulo 3 segments there are 13 equivalence classes:
$$(1,1,1), (1,1,-1), (1,1,0), (1,-1,1),(1,-1,-1), (1,-1,0),$$
$$ (1,0,1),(1,0,-1),(1,0,0),(0,1,1), (0,1,-1), (0,1,0), (0,0,1),
$$
that is, we are dealing with the projective space $\mathbb{Z}_3 \mathbb{P}^2$. Notice that if $a, b,$ and $c\in\mathbb{Z}_3\mathbb{P}^2$ are linearly independent (that is, $a\neq b$, and $c\notin \langle a,b\rangle=\{ a, b, a\pm b\}$), they generate all the classes:
$$a, b, c, a+b, a-b, a+c, a-c, b+c, b-c, a+b+c, a+b-c, a-b+c, -a+b+c.
$$
Let $S$ be a five-point lattice set contained in a polytope of Minkowski length 1. There are ten lattice segments that connect lattice points in $S$. We will classify such sets $S$ according to the numbers of segments from distinct classes in $\mathbb{Z}_3 \mathbb{P}^2$.
\begin{proposition}\label{T:class} If $L(P)=1$, then any 5-point lattice set $S$ in $P$ is of one of the following types.
\begin{itemize}
\item ${\rm 4+2+2+2}$ The segments are from classes $4a, 2b, 2(a+b), 2(a-b)$. Here a 4 or a 2 in front of segment's class denotes its multiplicity, which is the number of times it occurs among the lattice segments in $S$.
\item ${\rm 3+3+2+2}$ The segments are from classes $3a,3b, 2(a+b), 2(a-b)$.
\item ${\rm 3+(7)}$ The segments are from classes $3a, b, a+b, a-b, c, a+c, a-c, a+b-c$.
\item ${\rm 2+2+(6)}$ The segments are from classes $2a, 2b, a+b,a-b, a+c, b+c, a+b+c, c$.
\item ${\rm(10)}$ All ten lattice segments connecting points in $S$ are from distinct classes in $\mathbb{Z}_3 \mathbb{P}^2$.
\end{itemize}
The elements $a,b,c\in\mathbb{Z}_3 \mathbb{P}^2$ in each of the type descriptions are linearly independent. All types except for the last one have segments from classes $a, b, a+b, a-b$.
\end{proposition}
\begin{proof}
We assign direction to the segments by picking a standard representative from each of the classes. If two segments from the same class share a vertex, the arrows cannot both point to or away from the vertex as in this case the third side in the triangle is of Minkowski length at least 3. We also notice that if two sides in a triangle are from the same class, then the third one is also from that class and we get the triangle diagram below.
{\noindent}No other segment starting in one of these three vertices can be of class $a$, so the only remaining segment in $S$ that could be of class $a$, is the one connecting two remaining points of $S$. Hence the largest number of segments of the same class in $S$ is 4. If we have 4 segments of the same class we get the diagram below.
\begin{figure}
\caption{4+2+2+2}
\end{figure}
{\noindent} We call this type 4+2+2+2 as there are 4 segments of one type and 2 segments of each of the three other types.
Next, assume we only have 3 segments from class $a$. Then they would have to form a triangle. We could also have another 3 segments of class $b$, forming a triangle sharing a vertex with the first triangle. Then there are 3 segments of class $a$, 3 of class $b$, and 2 of each of $a+b$ and $a-b$. We call this type 3+3+2+2.
Assume next there is no other triangle. Connect one of the vertices of the triangle whose sides are of class $a$ to a fourth lattice point in $S$. Let this segment be of class $b$. The segment connecting the fourth lattice point to the fifth cannot be from classes $a,b, a+b,$ or $a-b$, as this would give either another triangle or four segments of the same class. Hence that segment is of class $c$, such that the set $\{a,b,c\}$ is linearly independent and we get the diagram below.
\begin{figure}
\caption{3+(7)}
\end{figure}
We call this type 3+(7).
If there are no 3s but there is a 2, we get a configuration of type $2+2+(6)$.
\begin{figure}
\caption{2+2+(6)}
\end{figure}
Finally, it is possible that there are no repeats among classes of segments. An example of this situation is a tetrahedron with the vertices $(1,0,0), (0,1,0), (0,0,1), (-1,-1,-1)$ with one lattice point, the origin, strictly inside. We call this type (10).
\end{proof}
\begin{lemma}\label{L:int} Let $a,b,c,d\in \mathbb{Z}_3 \mathbb{P}^2$ where $a\neq b$ and $c\neq d$. Then $\langle a,b\rangle\cap \langle c,d\rangle\neq\emptyset$.
\end{lemma}
\begin{proof}
If $c\in\langle a,b\rangle$ the conclusion is obvious, so we can assume that $\langle a,b,c\rangle= \mathbb{Z}_3 \mathbb{P}^2$. One of $d,c+d, c-d$ does not have $c$ in its expression in terms of $a,b,c$, hence it belongs to $\langle a,b\rangle$.
\end{proof}
\begin{proposition} Let $P$ and $Q$ be 3D lattice polytopes of Minkowski length 1 with at least five lattice points each. If there exists a 5-point lattice subset $S$ of $P$ of type {\rm 4+2+2+2} or {\rm 3+3+2+2}, then $L(P+Q)\geq 3$.
\end{proposition}
\begin{proof}
Pick any 5-point lattice subset $T$ of $Q$. Since in $S$ we have used up four classes with multiplicities greater than 1, by Lemma~\ref{L:mult}, $T$ cannot be of type $(10)$, since the overall number of classes is 13. In $S$, we have multiple segments of each of the classes $a$, $b$, $a+b$, $a-b$ for some $a,b\in\mathbb{Z}_3\mathbb{P}^2$. Since $T$ is not of type (10), we also have segments of classes $c$, $d$, $c+d$, $c-d$ in $T$ for some $c,d\in\mathbb{Z}_3\mathbb{P}^2$, not necessarily with multiplicities. By Lemmas \ref{L:int} and \ref{L:mult} we conclude $L(P+Q)\geq 3$.
\end{proof}
\begin{proposition}
Let $P$ and $Q$ be 3D lattice polytopes of Minkowski length 1. If there exist 3-point lattice subsets $S$ and $T$ of $P$ and $Q$ correspondingly, each of which forms a triangle with sides of the same class, then $L(P+Q)\geq 3$. In particular, if both $S$ and $T$ are of type $3+(7)$, then $L(P+Q)\geq 3$.
\end{proposition}
\begin{proof}
We first notice that if a lattice polytope contains a lattice triangle with sides of the same class, then this triangle is equivalent to $T_0$.
Indeed, we can easily map this triangle to one in the $(x,y)$-plane by creating a matrix of determinant 1 whose last row is a primitive vector
orthogonal to the plane of the triangle. We know that in the $(x,y)$-plane, up to the equivalence, there are only two triangles of length one, the unit triangle and $T_0$. The unit triangle has sides that belong to three distinct classes and the sides of $T_0$ are all from the same class.
Hence the initial triangle is equivalent to $T_0$ and, therefore, has a lattice point inside and all four points are in the same plane.
We have such a configuration in both $P$ and $Q$. Let the sides of the triangle in $Q$ be of class $c$
and one of the segments inside this triangle be of class $d$. By Lemma~\ref{L:int}, $P$ and $Q$ share a segment. By Lemma~\ref{L:mult},
they cannot share $a$ or $c$, so they have a common lattice segment inside the triangles. We can assume that $b$ and $d$ represent parallel segments.
Notice that when extended to the intersection with the opposite side of $T_0$, these segments have Minkowski length 1.5, that is, if we add them up
we get a segment of Minkowski length 3. Hence $L(P+Q)\geq 3$. For example, if $P=Q=T_0$ we get the diagram below.
\end{proof}
\begin{proposition} Let $P$ and $Q$ be 3D lattice polytopes of Minkowski length 1. If there exist 5-point lattice subsets $S$ and $T$ of $P$ and $Q$
of types $2+2+(6)$ and $3+(7)$ correspondingly, then $L(P+Q)\geq 3$.
\end{proposition}
\begin{proof}
We assume that $L(P+Q)=2$.
Let the multiple classes in $S$ be $a$ and $b$ and the class of multiplicity 3 in $T$ be $d$.
Let the triangle in $T$ with all sides of class $d$ be $ABC$ and the remaining two lattice points in $T$ be $D$ and $E$ with $AD$ of class $e$ and $BE$ of class $f$, as depicted in the diagram below.
By Lemma~\ref{L:int}, $\langle a, b\rangle \cap \langle d, e\rangle\neq\emptyset$ and $\langle a, b\rangle \cap \langle d, f\rangle\neq\emptyset$.
Since classes $a$ and $b$ cannot occur in $T$ and class $d$ cannot occur in $S$, segments of classes $a+b$ and $a-b$ have to appear in $T$, and we can assume that $e=a+b$ and $f=a-b$. This is because these two segments in $T$ cannot share a vertex, as the third side in the triangle formed by $a+b$ and $a-b$ would have to be of class either $a$ or $b$.
By Proposition~\ref{T:class}, the segments connecting the lattice points in $S$ are of classes $a$, $b$, $a+b$, $a-b$, $a+c$, $b+c$, $a+b+c$, and $c$ for some linearly independent $a,b,c\in\mathbb{Z}_3 \mathbb{P}^2$.
Hence there are five options left for $d$: $a-c$, $b-c$, $a+b-c$, $a-b-c$, $a-b+c$. Notice that in $T$ we have segments of classes $d+a+b$, $d-a-b$, $d-a+b$, and $d+a-b$.
Going through the five options for $d$, we observe that every time there are four lattice segments that are shared between $S$ and $T$. Two of them are $a+b$ and $a-b$. The remaining two for each of the five cases are listed in the table below.
\begin{center}
\begin{tabular}{c|c}
$d$& shared segments in $S$ and $T$\\
\hline
$a-c$&$a-c-(a+b)=-(b+c), a-c+(a-b)=- (a+b+c)$\\
$b-c$&$b-c-(a+b)=-(a+c), b-c-(a-b)=-(a+b+c)$\\
$a+b-c$&$a+b-c-(a-b)=-(b+c), a+b-c+(a+b)=-(a+b+c)$\\
$a-b-c$&$a-b-c-(a-b)=-c$, $a-b-c+(a+b)=-(a+c)$\\
$a-b+c$&$a-b+c-(a-b)=c$, $a-b+c-(a+b)=b+c$
\end{tabular}
\end{center}
We have checked that there are always at least four segments shared between $S$ and $T$, with the extra condition that in $T$ none of these four segments is $ED$.
In $T$, three of these segments cannot all have $E$ as an endpoint, as this would imply that two of these segments in $S$ also share an endpoint, so by Corollary~\ref{C:sharetriang} there is a shared triangle, one of whose sides is $d$, which is impossible. Similarly, three of the shared segments cannot all have $D$ as an endpoint. Hence there are two possible scenarios, depicted in the diagram below. The shared segments are marked by $a+b,a-b,x$, and $y$.
In the first scenario, the triangle formed by $x$ and $y$ in $S$ is shared, so its third side $b$ is also shared, which leads to a contradiction.
In the second scenario, two triangles, one formed by $a+b$ and $y$, and another by $a-b$ and $x$ are shared, so their third side $u$ appears twice in $S$, which is impossible.
\end{proof}
\begin{proposition}
Let $P$ and $Q$ be 3D lattice polytopes of Minkowski length 1. If there exist 5-point lattice subsets $S$ and $T$ of $P$ and $Q$ correspondingly, of type {\rm 2+2+(6)} each, then $L(P+Q)\geq 3$.
\end{proposition}
\begin{proof}
We assume that $L(P+Q)=2$. Let the $a$ and $b$ be the classes of lattice segments in $S$ of multiplicity 2. Then $S$ also has segments of classes $a+b$ and $a-b$. By Lemma~\ref{L:mult} segments in $T$ of multiplicity 2 cannot be of classes $a,b,a+b,a-b$, so we can assume that one of them is $c$ and the other is $a+b+c$, by switching direction vectors of $a$, $b$, and $c$, if needed. Since we have 13 classes total, $S$ and $T$ overlap in at least 3 segments. One of them is of class $a+b$. We will show that $S$ and $T$ share three segments that form a triangle one of whose sides is of class either $a+b$ or $a-b$.
Assume that the segment of class $a-b$ is also shared. A segment of class $a-b$ in $T$ cannot share a vertex with a segment of class $a+b$ (then either the sum or difference of those classes would also be represented in $S$, but classes $a$ and $b$ cannot appear in $S$).
Changing direction of $c$ and/or both $a$ and $b$, we can assume that $a-b$ is as in the diagram below.
Let the class of the third shared segment be $x$. Assume that in $S$ the segment of class $x$ shares a vertex with $a-b$. Then in $T$ the segment of class $x$ would have to share
a vertex with $a-b$ as the only ones that don't are $c$, $a+b$, and $a+b+c$ and $x$ cannot be one of them. Hence $a-b$ and $x$ share a vertex in both $S$ and $T$ and therefore by Corollary~\ref{C:sharetriang} $S$ and $T$ share a triangle with sides $a-b$, $a+b-c$, and $a+c$. Hence $x=a+b-c$ and the diagrams for $S$ and $T$ are below.
We see that both $S$ and $T$ have triangles with $a-c$ and $a+c$ as sides, but those triangles are not identical, which contradicts $L(P+Q)=2$.
Next, let next $x$ share a vertex with $a+b$ in $S$. If $x$ does not share a vertex with $a+b$ in $T$ as well, then the options for $x$ are $a+c$ and $a+b-c$. If $x=a+c$ then either $x-a=c$ or $x+b=a+b+c$ is in $S$, which is impossible. If $x=a+b-c$, then either $x+a+b=a+b+c$ or $x-(a+b)=c$ is in $S$, which is also impossible.
Hence a triangle with base $a+b$ is shared between $S$ and $T$, which leads to the same diagram and the same contradiction as before.
It remains to consider the case when $a-b$ is not shared between $S$ and $T$. We can also assume that $a+b-c$ is not shared. (If it is shared replace $S$ and $T$ in the above argument.) Then $S$ and $T$ share two segments both of which have the fifth point as an endpoint in both $S$ and $T$. Then the triangle formed by these two segments is shared and the base of that triangle is $a+b$. Notice that there is no room for other shared segments as they would have to have a fifth point
as a vertex and $a+b$ is the only option for shared base. Let's denote one of the shared segments by $d$. We get the following diagram below.
We next search for the expression of $d$ in terms of $a,b$ and $c$ so that the only common segments between $S$ and $T$ are $a+b$, $d$, and $a+b+d$. Below is the list of segments used in $S$ and $T$.
\begin{center}
\begin{tabular}{c|c}
$S$& $T$\\
\hline
$2a$&$2c$\\
$2b$&$2(a+b+c)$\\
$a+b$&$a+b$\\
$a-b$&$a+b-c$\\
$d$&$d$\\
$d+a$&$d-c$\\
$d+b$&$d+a+b+c$\\
$d+a+b$&$d+a+b$
\end{tabular}
\end{center}
We clearly have $d\neq \pm a, \pm b, \pm (a+b), \pm (a-b), \pm c, \pm (a+b+c), \pm (a+b-c)$. All the remaining options are also very easy to get rid of. If $d=\pm b+c, \pm a+c, \pm (a-b)+c$, then $d-c\in\langle a+b \rangle$. If $d=-a-c$, then $d+a$ appears in both $S$ and $T$. If $d=-b-c$, then $d+b$ appears in both $S$ and $T$. If $d=a+b-c$, then $d+a+b$ appears in both $S$ and $T$. If $d=a-b-c$, then $d+a+b+c$ appears in both $S$ and $T$. If $d=a-c$, then $d+a$ is of the same class as $d-c$. Finally, if $d=b-c$, then $d+b$ is of the same class as $d-c$. Every time we arrive at a contradiction, which proves the proposition.
\end{proof}
\begin{proposition} Let $P$ and $Q$ be two 3D lattice polytopes of Minkowski length 1. If both $P$ and $Q$ are of type (10) and $L(P+Q)=2$, then $P$ and $Q$ are equal (the same up to translation).
\end{proposition}
\begin{proof}
Since there are 13 classes of segments total, $P$ and $Q$ share at least 7 segments. Among these shared segments we can find three that have a common endpoint in $P$. At least two of these three shared segments have a common endpoint in $Q$. Hence $P$ and $Q$ share a triangle $ABC$. Let the two remaining lattice points in $P$ be $D$ and $E$.
At least one of the two tetrahedra $ABCD$ and $ABCE$ with base $ABC$, say, $ABCD$, has at least two lateral edges that are shared. Together with a segment in the base these two edges form a triangle with shared sides. At least two of these three shared sides share a vertex in $Q$, so by Corollary~\ref{C:sharetriang} we get a shared triangle. We have shown that $P$ and $Q$ have two pairs of shared triangles that have a common edge, so $P$ and $Q$ share a tetrahedron $ABCD$. Let the fifth vertices in $P$ and $Q$ be $E$ and $E'$.
Since $P$ and $Q$ share at least seven lattice segments, there is a segment in $P$ connecting one of $A, B, C, D$ to $E$ which is shared with $Q$.
Let this segment be $AE$. The parallel segment in $Q$ is of the form $E'X$ where $X=A, B, C,$ or $D$. In any of these cases, $AE$ and $E'X$ are adjacent
to a shared segment $AX$, so triangles $AEX$ and $AE'X$ are translates of each other, which implies that $P$ and $Q$ are the same up to a translation .
\end{proof}
\begin{proposition} Let $P$ and $Q$ be 3D lattice polytopes of Minkowski length 1. If there exist 5-point lattice subsets $S$ and $T$ of $P$ and $Q$
of types $3+(7)$ and $(10)$ correspondingly, then $L(P+Q)\geq 3$.
\end{proposition}
\begin{proof}
We assume that $L(P+Q)=2$.
There are 13 classes of segments total, so $S$ and $T$ have to share at least 5 segments.
Let $S$ have lattice points $A$, $B$, $C$, $D$, and $E$, where $ABC$ is a triangle with sides of the same class and $D$ a point inside
that triangle. If $AE$, $BE$, and $CE$ are all shared with $T$, then two of them share a vertex in $T$ and hence one of the triangles $ABE$, $ACE$, or $BCE$ is shared, which is impossible since sides of $ABC$ cannot be shared. Hence one of $AE$, $BE$, $CE$ is not shared. Similarly, one of $AD$, $BD$, $CD$ is not shared.
This implies that $ED$ is shared, and we can assume that $ADE$ is a shared triangle, $DB$ and $EC$ are shared, while $DC$ and $EB$ are not shared.
Then $b$ and $b+c-a$ in $T$ should not be adjacent to $ED$ (or either $DC$ or $EB$ would be shared).
Hence one of $b$, $b+c-a$ has $A$ as an endpoint in $T$. If it is $b$, we have either $a$ or $a-b$ in $T$;
if it is $b+c-a$, then either $b+c$ or $a$ is in $T$. Both of these options are impossible.
\end{proof}
\begin{proposition} Let $P$ and $Q$ be 3D lattice polytopes of Minkowski length 1. If there exist 5-point lattice subsets $S$ and $T$ of $P$ and $Q$
of types $2+2+(6)$ and $(10)$ correspondingly, then $L(P+Q)\geq 3$.
\end{proposition}
\begin{proof}
We assume that $L(P+Q)=2$.
There are 13 classes of segments total, so $S$ and $T$ have to share at least 5 segments. Let $S$ have lattice points $A$, $B$, $C$, $D$, and $E$, where $AB$ and $CD$ are from the same class, and $BC$ and $AD$ are from the same class.
At least 3 segments starting at $E$ are shared. In $T$ at least two of them share a vertex, hence $S$ and $T$ share a triangle with vertex $E$. We can assume that that triangle is $AEC$. One of $EB$, $ED$ is also shared, let's assume it's $EB$.
This segment in $T$ cannot share a vertex with $AE$ or $EC$ (this would imply that either $AB$ or $BC$ is shared), so it connects two remaining vertices $X$ and $Y$.
If $ED$ is also shared, same would be true, but there are only five vertices, so $ED$ is not shared. Hence $BD$ is shared. But it would have to have either $X$ or $Y$ as one of the vertices, hence $EBD$ is shared, a contradiction.
\end{proof}
\begin{theorem}\label{T:threeP} Let $P$, $Q$, $R$ be three 3D lattice polytopes of Minkowski length 1 with at least five lattice points each. Then $L(P+Q+R)\geq 4$.
\end{theorem}
\begin{proof}
Let $S, T,$ and $U$ be any 5-point lattice subsets of $P$, $Q$, and $R$ correspondingly. By the above propositions, $S$, $T$, and $U$ are all of type $(10)$.
If any of the three polytopes has more than five points, then there are at least 15 lattice segments connecting them. Hence there are multiple segments of the same class. Reducing the number of points, we get sets $S$, $T$, and $U$ where at least one of these lattice sets is not of type $(10)$.
Hence we can assume that each of $P$, $Q$, and $R$ has exactly five lattice points and is of type $(10)$. Furthermore, all three polytopes are translates of each other.
It remains to show that $L(3P)\geq4$.
Let us assume first that $P$ has four lattice vertices $A, B, C, D$ and a lattice point $E$ inside. Let $G$ be the centroid of $P$. Draw through $G$ four planes, each parallel to one of the facets of $P$. Each of the planes cuts off a tetrahedron off of $P$, which is similar to $P$ with a coefficient of $3/4$. These four tetrahedra
cover $P$, so the interior lattice point $E$ belongs to at least one of them, say, to the tetrahedron one of whose vertices is $A$. Then if we continue $AE$ to the point of intersection $F$ with the plane $BCD$, then $AE/AF\leq 3/4$ and $3AF\geq 4AE$, so in $3P$ we have a segment $3AF$ whose lattice length is at least 4. We have shown that in this case $L(3P)\geq 4$.
Next, let $P$ of type (10) have 5 lattice vertices and no other lattice points. It was shown in \cite{Scarf}, Theorem~3.5 that $P$ is $\A{3}$ equivalent to a polytope with the vertices $(0,0,0)$, $(1, 0, 0)$, $(0,1,0)$, $(0,0,1)$, and $(1, a, b)$ where $\gcd(a, b)=1$ and $0\leq a\leq b$. The sum of the two segments, $[0, a, b]$ and $[0,1,0]$ is a parallelogram of area $b$.
If $b\geq 3$ then by Lemma~1.7 of \cite{SS1} the Minkowski length of this parallelogram is at least $3$ and hence $L(2P)\geq 3$. We are left with two cases $a=b=1$ and $a=1$, $b=2$ (if $a=0$ then $L(P)\neq 1$). In the second case $P$ is not of type $(10)$ as $(0,a,b)=(0,1,-1)$. It remains to deal with the case $a=b=1$. Let $I$ be the vertical segment of length 1 and let $J$ be the segment connecting the origin to $(1,1,0)$. Then $I+1/2J\subseteq P$ and hence $2I+J\subseteq 2P$, so we again have $L(2P)\geq 3$.
\end{proof}
Notice that we have also proved the following Corollary.
\begin{corollary} If $P$ and $Q$ are 3D lattice polytopes of Minkowski length 1 with at least 5 lattice points each, then $L(P+Q)\geq 3$ unless $P$ and $Q$ are of type $(10)$ and are the same up to translation.
\end{corollary}
Next example demonstrates that one could have $L(2P)=2$ for a 3D lattice polytope $P$ of type $(10)$.
\begin{example} Let $P$ have the vertices $(-1,-1,-1)$, $(1,0, 0)$, $(0,1,0)$, and $(0,0,1)$. Then $P$ is of type (10), $L(P)=1$, and $L(2P)=2$.
\end{example}
\begin{proposition}\label{P:5and6} Let $P$ and $Q$ be 3D lattice polytopes of Minkowski length 1. If $P$ has at least 6 lattice points and $Q$ has at least 5, then $L(P+Q)\geq 3$.
\end{proposition}
\begin{proof}
Since $P$ has $6$ lattice points, there are 15 lattice segments in $P$. Since there are 13 classes total, some of the segments will repeat and we will either have two segments
of the same class not sharing endpoints, or three segments of the same class forming a triangle. Hence we can pick $5$ points in $P$ that form a lattice subset of type other than (10), which implies $L(P+Q)\geq 3$.
\end{proof}
\begin{theorem}\label{T:main} Let $Q_1+\cdots+Q_N\subseteq P$ be a maximal decomposition. Let $I(Q_i)$ denote the number of interior lattice points with respect to the 3D topology.
Then the overall number of interior lattice points
$$I=\sum_{i=1}^{N}I(Q_i)\leq 4.
$$
Furthermore, if more than one $Q_i$ has interior lattice points with respect to the 3D topology, we have $I\leq 2$.
\end{theorem}
\begin{proof}
In order to have an interior lattice point with respect to the 3D topology, $Q_i$ has to have at least 5 lattice points. Hence, by Theorem \ref{T:threeP}, at most two of the $Q_i$'s could have interior lattice points and be three-dimensional. Let's assume that this holds for two of the $Q_i$'s. If one of these two $Q_i$'s has at least 2 interior lattice points, we get a contradiction with Proposition~\ref{P:5and6}. Hence in this case the total number of interior lattice points is at most 2. If only one of the $Q_i$'s has interior lattice points with respect to the 3D topology, then there are at most 4 of them, as $L(Q_i)=1$ and $Q_i$ has at most 8 lattice points total.
\end{proof}
The following example shows that there exists a Minkowski length one 3D polytope that has 4 interior lattice points, so the bound of Theorem~\ref{T:main} is sharp.
\begin{example} Consider a simplex $P$ with the vertices $(0,0,0)$, $(1,3,0)$, $(0,2,3)$, and $(4,1,3)$. Then the interior lattice points of $P$ are
$(1,2,1)$, $(1,2,2)$, $(1,1,1)$, and $(2,1,2)$. This can be checked by hand or using Polymake~\cite{Polymake}. It is easy to verify that there are no parallel
lattice segments connecting lattice points in $P$, which implies that $L(P)=1$.
\end{example}
An example of a 3D lattice polytope of Minkowski length one with 8 lattice points (all of them on the boundary) was given in an MSRI-UP project directed by John Little \cite{MSRI}. The number of lattice points in a lattice polytope in the $n$-space is at most $2^n$. A group of REU students at Kent in Summer 2011 constructed a lattice $n$-dimensional Minkowski length one polytope with $2^n$ points. It would be interesting to see if there exists an $n$-dimensional simplex that has $2^n$ lattice points and Minkowski length one.
\subsection*{Acknowledgments}
We are thankful to the anonymous referee for pointing out a gap in one of the arguments as well as for numerous corrections and suggestions.
\end{document} | math | 55,741 |
\begin{document}
\title[An explicit formula for coefficients]{An explicit formula for\\ the coefficients in Laplace's method}
\begin{abstract}
Laplace's method is one of the fundamental techniques in the asymptotic approximation of integrals. The coefficients appearing in the resulting asymptotic expansion, arise as the coefficients of a convergent or asymptotic series of a function defined in an implicit form. Due to the tedious computation of these coefficients, most standard textbooks on asymptotic approximations of integrals do not give explicit formulas for them. Nevertheless, we can find some more or less explicit representations for the coefficients in the literature: Perron's formula gives them in terms of derivatives of an explicit function; Campbell, Fr\"oman and Walles simplified Perron's method by computing these derivatives using an explicit recurrence relation. The most recent contribution is due to Wojdylo, who rediscovered the Campbell, Fr\"oman and Walles formula and rewrote it in terms of partial ordinary Bell polynomials. In this paper, we provide an alternative representation for the coefficients, which contains ordinary potential polynomials. The proof is based on Perron's formula and a theorem of Comtet. The asymptotic expansions of the gamma function and the incomplete gamma function are given as illustrations.
\end{abstract}
\maketitle
\section{Introduction}
Laplace's method is one of the best-known techniques developing asymptotic approximation for integrals. The origins of the method date back to Pierre-Simon de Laplace (1749 -- 1827), who studied the estimation of integrals, arising in probability theory, of the form
\begin{equation}\label{eq1}
I\left( \lambda \right) = \int_a^b {\mathrm{e}^{ - \lambda f\left( x \right)} g \left( x \right)\mathrm{d}x} \quad \left(\lambda \to +\infty\right).
\end{equation}
Here $\left(a,b\right)$ is a real (finite or infinite) interval, $\lambda$ is a large positive parameter and the functions $f$ and $g$ are continuous. Laplace made the observation that the major contribution to the integral $I\left( \lambda \right)$ should come from the neighborhood of the point where $f$ attains its smallest value. Observe that by subdividing the range of integration at the minima and maxima of $f$, and by reversing the sign of $x$ whenever necessary, we may assume, without loss of generality, that $f$ has only one minimum in $\left[a, b\right]$ occuring at $x = a$. With certain assumptions on $f$, Laplace's result is
\[
\int_a^b {\mathrm{e}^{ - \lambda f\left( x \right)} g \left( x \right)\mathrm{d}x} \sim g\left( {a} \right)\mathrm{e}^{ - \lambda f\left( {a} \right)} \sqrt {\frac{{\pi }}{{2 \lambda f''\left( {a} \right)}}} .
\]
The sign $\sim$ is used to mean that the quotient of the left-hand side by the right-hand side approaches $1$ as $\lambda \to +\infty$. This formula is now known as Laplace's approximation. A heuristic proof of this formula may proceed as follows. First, we replace $f$ and $g$ by the leading terms in their Taylor series expansions around $x = a$, and then we extend the integration limit to $+\infty$. Hence,
\begin{align*}
\int_a^b {\mathrm{e}^{ - \lambda f\left( x \right)} g\left( x \right)\mathrm{d}x} & \approx \int_a^b {\mathrm{e}^{ - \lambda \left( {f\left( {a} \right) + \frac{{f''\left( {a} \right)}}{2}\left( {x - a} \right)^2 } \right)} g\left( {a} \right)\mathrm{d}x} \\
& \approx g\left( {a} \right)\mathrm{e}^{ - \lambda f\left( {a} \right)} \int_{a}^{ + \infty } {\mathrm{e}^{ - \lambda \frac{{f''\left( {a} \right)}}{2}\left( {x - a} \right)^2 } \mathrm{d}x} \\
& = g\left( {a} \right)\mathrm{e}^{ - \lambda f\left( {a} \right)} \sqrt {\frac{{\pi }}{{2 \lambda f''\left( {a} \right)}}} .
\end{align*}
The modern version of the method was formulated in 1956 by Arthur Erd\'elyi (1908 -- 1977), who applied Watson's lemma to obtain a complete asymptotic expansion for the integral \eqref{eq1}. His method requires some assumptions on $f$ and $g$. Suppose, again, that $f$ has only one minimum in $\left[a, b\right]$ which occurs at $x = a$. Assume also that, as $x \to a^+$,
\begin{equation}\label{exp1}
f\left( x \right) \sim f\left( a \right) + \sum\limits_{k = 0}^\infty {a_k \left( {x - a} \right)^{k + \alpha } } ,
\end{equation}
and
\begin{equation}\label{exp2}
g \left( x \right) \sim \sum\limits_{k = 0}^\infty {b_k \left( {x - a} \right)^{k + \beta - 1} }
\end{equation}
with $\alpha>0$, $\Re \left( \beta \right) > 0$, and that the expansion of $f$ can be term-wise differentiated, that is,
\begin{equation}\label{exp3}
f'\left( x \right) \sim \sum\limits_{k = 0}^\infty {a_k \left( {k + \alpha } \right)\left( {x - a} \right)^{k + \alpha - 1} }
\end{equation}
as $x \to a^+$. We may also assume, without loss of generality, that $a_0 \ne 0$ and $b_0 \ne 0$. The following theorem is Erd\'elyi's formulation of Laplace's classical method.
\begin{theorem}\label{theorem1} For the integral
\[
I\left( \lambda \right) = \int_a^b {\mathrm{e}^{ - \lambda f\left( x \right)} g \left( x \right)\mathrm{d}x} ,
\]
we assume that
\begin{enumerate}[(i)]
\item $f(x)>f(a)$ for all $x \in (a,b)$, and for every $\delta>0$ the infimum of $f(x)-f(a)$ in $\left[a+\delta,b\right)$ is positive;
\item $f'(x)$ and $g(x)$ are continuous in a neighborhood of $x=a$, expect possibly at $a$;
\item the expansions \eqref{exp1}, \eqref{exp2} and \eqref{exp3} hold; and
\item the integral $I\left( \lambda \right)$ converges absolutely for all sufficiently large $\lambda$.
\end{enumerate}
Then
\begin{equation}\label{eq2}
I\left( \lambda \right) \sim \mathrm{e}^{ - \lambda f\left( a \right)} \sum\limits_{n = 0}^\infty {\varGamma \left( {\frac{{n + \beta }}{\alpha }} \right)\frac{{c_n }}{{\lambda ^{\left( {n + \beta } \right)/\alpha } }}} ,
\end{equation}
as $\lambda \to +\infty$, where the coefficients $c_n$ are expressible in terms of $a_n$ and $b_n$.
\end{theorem}
The first three coefficients $c_n$ are given explicitly by
\[
c_0 = \frac{{b_0 }}{{a_0^{\beta /\alpha } \alpha }},\quad \quad c_1 = \frac{1}{{a_0^{\left( {\beta + 1} \right)/\alpha } }}\left( {\frac{{b_1 }}{\alpha } - \frac{{\left( {\beta + 1} \right)a_1 b_0 }}{{\alpha ^2 a_0 }}} \right),
\]
and
\[
c_2 = \frac{1}{{a_0^{\left( {\beta + 2} \right)/\alpha}}}\left( {\frac{{b_2 }}{\alpha } - \frac{{\left( {\beta + 2} \right)a_1 b_1 }}{{\alpha ^2 a_0 }} + \left( {\left( {\beta + \alpha + 2} \right)a_1^2 - 2\alpha a_0 a_2 } \right)\frac{{\left( {\beta + 2} \right)b_0 }}{{2\alpha ^3 a_0^2 }}} \right).
\]
To understand the origin of these coefficients, we sketch the proof of Erd\'elyi's theorem. Detailed proofs are given in many standard textbooks on asymptotic analysis, e.g., in Erd\'elyi's original monograph \cite[p. 38]{Erdelyi} or the classical books of Olver \cite[p. 81]{Olver2} and Wong \cite[p. 58]{Wong}. By conditions (ii) and (iii) there exists a number $c \in (a,b)$ such that $f'(x)$ and $g(x)$ are continuous in $(a,c]$, and $f'(x)$ is also positive in this interval. Let $T=f(c)-f(a)$, and define the new variable $t=t(x)$ as
\[
t=f(x)-f(a).
\]
Since $f(x)$ is increasing in $(a,c)$, we can write
\begin{equation}\label{eq3}
\int_a^c {\mathrm{e}^{ - \lambda f\left( x \right)} g\left( x \right)\mathrm{d}x} = \mathrm{e}^{-\lambda f\left( a \right)} \int_0^T {\mathrm{e}^{ - \lambda t} h\left( t \right)\mathrm{d}t}
\end{equation}
with $h(t)$ being the continuous function in $\left(0,T\right]$ given by
\begin{equation}\label{eq4}
h(t) = g(x)\frac{\mathrm{d}x}{\mathrm{d}t}= \frac{g(x)}{f'(x)}.
\end{equation}
By assumption,
\[
t \sim \sum\limits_{k = 0}^\infty {a_k \left( {x - a} \right)^{k + \alpha } } \quad\text{as } x\to a^+.
\]
And so, by series reversion, we obtain
\[
x - a \sim \sum\limits_{k = 1}^\infty {d_k t^{k/\alpha } }\quad\text{as } t\to 0^+ .
\]
Substituting this into \eqref{eq4} and using the asymptotic expansions \eqref{exp2}-\eqref{exp3} yields
\[
h\left( t \right) \sim \sum\limits_{k = 0}^\infty {c_k t^{\left( {k + \beta } \right)/\alpha - 1} }
\]
as $t\to 0^+$. (In the common case where $g \equiv 1$, we have $\beta = 1$ and $c_k = \left(k + 1\right)d_{k+1}/\alpha$.) We now apply Watson's lemma to the integral on the right-hand side of \eqref{eq3} to obtain
\[
\int_a^c {\mathrm{e}^{ - \lambda f\left( x \right)} g\left( x \right)\mathrm{d}x} \sim \mathrm{e}^{ - \lambda f\left( a \right)} \sum\limits_{n = 0}^\infty {\varGamma \left( {\frac{{n + \beta }}{\alpha }} \right)\frac{{c_n }}{{\lambda ^{\left( {n + \beta } \right)/\alpha } }}} ,
\]
as $\lambda \to +\infty$. To complete the proof, one shows that the integral on the remaining range $(c,b)$ is negligible.
The coefficients $c_n$ are traditionally calculated manually in the way sketched above: by computing the reversion coefficients $d_n$, substituting the resulting series into the asymptotic expansion of \eqref{eq4}, expanding the series in powers of $t$, and equating the coefficients in each specific application of Laplace's method. Due to this somewhat tedious computation, most standard textbooks on asymptotic approximations of integrals do not give explicit formulas for the $c_n$'s. Nevertheless, there are certain formulas in different degrees of explicitness for the coefficients $c_n$ in the literature. When $\left( {x - a} \right)^{ - \alpha } \left( {f\left( x \right) - f\left( a \right)} \right)$ and $\left( {x - a} \right)^{1 - \beta } g\left( x \right)$ are analytic at $x=a$, Perron's formula gives the coefficients in terms of derivatives of an explicit function involving $f$ and $g$. Campbell, Fr\"oman and Walles rediscovered Perron's method and went further by computing these derivatives using an explicit recurrence formula \cite{Campbell}. The most recent contribution is given by Wojdylo, who rediscovered the Campbell, Fr\"oman and Walles formula and rewrote it in terms of partial ordinary Bell polynomials \cite{Wojdylo1}\cite{Wojdylo2}. Using new ideas of combinatorial analysis, he was able to simplify and systematize the computation of the $c_n$'s. The definition of the partial ordinary Bell polynomials provides a direct way of expanding the higher derivatives in Perron's formula. However, there is a formula by Comtet allowing us to obtain a representation for those derivatives in terms of ordinary potential polynomials. This gives a new and alternative method for the computation of the coefficients $c_n$ (see Corollary \ref{maintheorem}). We discuss these formulas in details in Section \ref{section2}.
In Section \ref{section3}, we give two illustrative examples to demonstrate the application of our new method and to compare the results with those given by Wojdylo's formula. In the first example, we obtain several explicit expressions for the Stirling coefficients appearing in the asymptotic expansion of the gamma function. In the second example, we investigate certain polynomials related to the coefficients in the uniform asymptotic expansion of the incomplete gamma function.
The definition and basic properties of the partial ordinary Bell polynomials and the ordinary potential polynomials are collected in Appendix \ref{appendix}. For further discussion on these polynomials, we refer the reader to the book of Comtet \cite[pp. 133--153]{Comtet} or Riordan \cite[pp. 189--191]{Riordan}.
\section{Explicit formulas for the coefficients $c_n$}\label{section2}
The asymptotic theory of integrals of type \eqref{eq1} is also well established when $\lambda$ is complex and $f$, $g$ are holomorphic functions in a domain of the complex plane containing the path of integration $\mathscr{C}$ joining $a$ to $b$. A well-known method for obtaining asymptotic expansions for such integrals is the method of steepest descents (for a detailed discussion of this method, see, e.g., \cite[pp. 5--99]{Paris2} or \cite[pp. 84--103]{Wong}). This method requires the deformation of $\mathscr{C}$ into a specific path that passes through one or more saddle points of $f$ such that the function $\Im\left(f\right)$ is constant on it. (Recall that $z_0$ is a saddle point of $f$ iff $f'\left(z_0\right)=0$.) This new path is called the path of steepest descent. However, in many specific cases, the construction of such a path can be extremely complicated. This problem may be bypassed using Perron's method which -- by requiring some extra assumptions -- avoids the computation of the path of steepest descent, and provides an explicit expression for the coefficients in the resulting asymptotic series \cite{Perron}\cite[p. 103]{Wong}. A direct adaptation of Erd\'{e}lyi's theorem to complex integrals was formulated by Olver, where the coefficients in the asymptotic expansion can be computed formally in the same way as in the real case \cite[p. 121]{Olver2}. In addition, both Perron's and Olver's method allow the functions $f$ and $g$ to have algebraic singularities at the endpoint $a$ with convergent expansions of the form \eqref{exp1} and \eqref{exp2}.
Based on Wojdylo's work, we show below that Perron's explicit formula also holds formally in the case of Erd\'{e}lyi's theorem, when the expansions \eqref{exp1} and \eqref{exp2} may be merely asymptotic. Starting from the equation $h\left(t\right)\mathrm{d}t = g\left(x\right)\mathrm{d} x$, Wojdylo showed that
\[
c_n^\ast \stackrel{\mathrm{def}}{=} \frac{{a_0^{\left( {n + \beta } \right)/\alpha } \alpha }}{{b_0 }}c_n = \sum\limits_{k = 0}^n {\frac{{b_{n - k} }}{{b_0 }}\frac{1}{{k!}}\left[ {\frac{\mathrm{d}^k}{\mathrm{d}x^k}\left( {1 + \sum\limits_{k = 1}^\infty {\frac{{a_k }}{{a_0 }}x^k } } \right)^{ - \left( {n + \beta } \right)/\alpha } } \right]_{x = 0} },
\]
where $\mathrm{d}^k / \mathrm{d}x^k$ is the formal $k$th derivative with respect to $x$. Through his analysis, he used these scaled coefficients $c_n^\ast$. We shall not, however, use them in our paper. His expression can be simplified to
\begin{align}
c_n & = \frac{1}{{\alpha a_0^{\left( {n + \beta } \right)/\alpha } }}\sum\limits_{k = 0}^n {\frac{{b_{n - k} }}{{k!}}\left[ {\frac{{\mathrm{d}^k }}{{\mathrm{d}x^k }}\left( {1 + \sum\limits_{k = 1}^\infty {\frac{{a_k }}{{a_0 }}x^k } } \right)^{ - \left( {n + \beta } \right)/\alpha } } \right]_{x = 0} } \label{eq5} \\
& = \frac{1}{{\alpha a_0^{\left( {n + \beta } \right)/\alpha } }}\sum\limits_{k = 0}^n {\frac{{b_{n - k} }}{{k!}}\left[ {\frac{{\mathrm{d}^k }}{{\mathrm{d}x^k }}\left( {\frac{{a_0 x^\alpha }}{{\sum\nolimits_{k = 0}^\infty {a_k x^{k + \alpha } } }}} \right)^{\left( {n + \beta } \right)/\alpha } } \right]_{x = 0} }\nonumber \\
& = \frac{1}{{\alpha a_0^{\left( {n + \beta } \right)/\alpha } }}\sum\limits_{k = 0}^n {\frac{{b_{n - k} }}{{k!}}\left[ {\frac{{\mathrm{d}^k }}{{\mathrm{d}x^k }}\left( {\frac{{a_0 \left( {x - a} \right)^\alpha }}{{\sum\nolimits_{k = 0}^\infty {a_k \left( {x - a} \right)^{k + \alpha } } }}} \right)^{\left( {n + \beta } \right)/\alpha } } \right]_{x = a} } .\nonumber
\end{align}
If we identify $f$ with its asymptotic expansion \eqref{exp1}, we arrive at the final forms
\begin{gather}\label{eq6}
\begin{split}
c_n & = \frac{1}{{\alpha a_0^{\left( {n + \beta } \right)/\alpha } }}\sum\limits_{k = 0}^n {\frac{{b_{n - k} }}{{k!}}\left[ {\frac{{\mathrm{d}^k }}{{\mathrm{d}x^k }}\left( {\frac{{a_0 \left( {x - a} \right)^\alpha }}{{f\left( x \right) - f\left( a \right)}}} \right)^{\left( {n + \beta } \right)/\alpha } } \right]_{x = a} } \\
& = \frac{1}{{\alpha n!a_0^{\left( {n + \beta } \right)/\alpha } }}\left[ {\frac{{\mathrm{d}^n }}{{\mathrm{d}x^n }}\left\{ {G\left( x \right)\left( {\frac{{a_0 \left( {x - a} \right)^\alpha }}{{f\left( x \right) - f\left( a \right)}}} \right)^{\left( {n + \beta } \right)/\alpha } } \right\}} \right]_{x = a} ,
\end{split}
\end{gather}
where $G\left(x\right)$ is the formal power series $\sum\nolimits_{k = 0}^\infty {b_k \left( {x - a} \right)^k }$. An alternative form of this expression -- using the formal residue operator -- is stated in \cite[p. 45]{NIST}. If $\left( {x - a} \right)^{ - \alpha } \left( {f\left( x \right) - f\left( a \right)} \right)$ and $\left( {x - a} \right)^{1 - \beta } g\left( x \right)$ are analytic at $x=a$, this is the Perron formula. Applying the definition of the ordinary potential polynomials to \eqref{eq5}, we obtain
\begin{align}
c_n & = \frac{1}{{\alpha a_0^{\left( {n + \beta } \right)/\alpha } }}\sum\limits_{k = 0}^n {b_{n - k} \mathsf{A}_{ - \left( {n + \beta } \right)/\alpha ,k} \left( {\frac{{a_1 }}{{a_0 }},\frac{{a_2 }}{{a_0 }}, \ldots ,\frac{{a_k }}{{a_0 }}} \right)} \label{eq9} \\
& = \frac{1}{{\alpha a_0^{\left( {n + \beta } \right)/\alpha } }}\sum\limits_{k = 0}^n {b_{n - k} \sum\limits_{j = 0}^k { \binom {- \frac{{n + \beta }}{\alpha }}{j}\frac{1}{{a_0^j }}\mathsf{B}_{k,j} \left( {a_1 ,a_2 , \ldots ,a_{k - j + 1} } \right)} }. \nonumber
\end{align}
This is essentially Wojdylo's formula. To make it more convenient to use, we apply the identity
\[
\binom{ - \rho }{j}= \left( { - 1} \right)^j \binom{j + \rho - 1}{j} = \left( { - 1} \right)^j \frac{{\varGamma \left( {j + \rho } \right)}}{{j!\varGamma \left( \rho \right)}},
\]
which yields
\begin{equation}\label{eq13}
c_n = \frac{1}{{\alpha \varGamma \left( {\frac{{n + \beta }}{\alpha }} \right)}}\sum\limits_{k = 0}^n {b_{n - k} \sum\limits_{j = 0}^k {\frac{{\left( { - 1} \right)^j }}{{a_0^{\left( {n + \beta } \right)/\alpha + j} }}\frac{{\mathsf{B}_{k,j} \left( {a_1 ,a_2 , \ldots ,a_{k - j + 1} } \right)}}{{j!}}\varGamma \left( {\frac{{n + \beta }}{\alpha } + j} \right)} }.
\end{equation}
This is an even more compact form of Wojdylo's formula. In the common case, when $g \equiv 1$ (and therefore $\beta = 1$), formula \eqref{eq13} is reduced to
\begin{equation}\label{eq17}
c_n = \frac{1}{{\alpha \varGamma \left( {\frac{{n + 1}}{\alpha }} \right)}}\sum\limits_{k = 0}^n {\frac{{\left( { - 1} \right)^k }}{{a_0^{\left( {n + 1} \right)/\alpha + k} }}\frac{{\mathsf{B}_{n,k} \left( {a_1 ,a_2 , \ldots ,a_{n - k + 1} } \right)}}{{k!}}\varGamma \left( {\frac{{n + 1}}{\alpha } + k} \right)} .
\end{equation}
Wojdylo's formula \eqref{eq13}, together with the recurrence \eqref{eq32} for the partial ordinary Bell polynomials, provides a systematic way to calculate the coefficients $c_n$.
Comtet \cite[p. 142]{Comtet} showed that an ordinary potential polynomial can be written explicitly in terms of the values of the polynomial at non-negative integers. His formula takes the form
\[
\mathsf{A}_{ - z,k} \left( {\frac{{a_1 }}{{a_0 }},\frac{{a_2 }}{{a_0 }}, \ldots ,\frac{{a_k }}{{a_0 }}} \right) = \frac{{\varGamma \left( {z + k + 1} \right)}}{{k!\varGamma \left( z \right)}}\sum\limits_{j = 0}^k {\frac{\left( { - 1} \right)^j}{z + j} \binom{k}{j}\mathsf{A}_{j,k} \left( {\frac{{a_1 }}{{a_0 }},\frac{{a_2 }}{{a_0 }}, \ldots ,\frac{{a_k }}{{a_0 }}} \right)} .
\]
Applying this in \eqref{eq9}, we obtain the following result.
\begin{corollary}\label{maintheorem}
The coefficients $c_n$ appearing in \eqref{eq2} are given explicitly by
\begin{gather}\label{eq8}
\begin{split}
c_n & = \frac{1}{{\alpha \varGamma \left( {\frac{{n + \beta }}{\alpha }} \right)}}\sum\limits_{k = 0}^n { \frac{\varGamma \left( {\frac{{n + \beta }}{\alpha } + k + 1} \right)b_{n - k}}{k! a_0^{\left( {n + \beta } \right)/\alpha }}\sum\limits_{j = 0}^k {\frac{\left( { - 1} \right)^j}{\frac{{n + \beta }}{\alpha } + j} \binom{k}{j}\mathsf{A}_{j,k}}}\\
& = \frac{1}{{\alpha a_0^{\left( {n + \beta } \right)/\alpha } }}\sum\limits_{k = 0}^n {\binom{ - \frac{{n + \beta }}{\alpha }}{k} b_{n - k} \sum\limits_{j = 0}^k {\left( { - 1} \right)^{k + j} \frac{{n + \beta + \alpha k}}{{n + \beta + \alpha j}} \binom{k}{j}\mathsf{A}_{j,k}} },
\end{split}
\end{gather}
where $\mathsf{A}_{j,k} = \mathsf{A}_{j,k} \left( {a_1 a_0^{-1},a_2 a_0^{-1},\ldots,a_k a_0^{-1}} \right)$.
\end{corollary}
In the common case when $g \equiv 1$, we have $\beta = 1$ and formula \eqref{eq8} is reduced to
\begin{gather}\label{eq14}
\begin{split}
c_n & = \frac{{\varGamma \left( {\frac{{n + 1 }}{\alpha } + n + 1} \right)}}{{\alpha a_0^{\left( {n + 1 } \right)/\alpha } n!\varGamma \left( {\frac{{n + 1 }}{\alpha }} \right)}}\sum\limits_{k = 0}^n {\frac{{\left( { - 1} \right)^k }}{{\frac{{n + 1 }}{\alpha } + k}}\binom{n}{k}\mathsf{A}_{k,n} }\\
& = \frac{1}{{\alpha a_0^{\left( {n + 1 } \right)/\alpha } }} \binom{- \frac{{n + 1 }}{\alpha }}{n}\sum\limits_{k = 0}^n {\left( { - 1} \right)^{n + k} \frac{{n + 1 + \alpha n}}{{n + 1 + \alpha k}}\binom{n}{k}\mathsf{A}_{k,n} } .
\end{split}
\end{gather}
From the recurrence relation \eqref{eq33} for the ordinary potential polynomials it follows that
\[
\mathsf{A}_{j,k} = \sum\limits_{i = 0}^k {\frac{{a_i }}{{a_0 }}\mathsf{A}_{j - 1,k - i} } .
\]
This, together with \eqref{eq8}, provides a simple and efficient way to compute the coefficients $c_n$.
In the next section we shall use the following connection formula between the partial ordinary Bell polynomials and the ordinary potential polynomials (cf. Comtet \cite[p. 156, Exercise 3]{Comtet}).
\begin{equation}\label{eq16}
\mathsf{B}_{k,j} = \left( { - 1} \right)^j a_0^j\sum\limits_{i = 0}^j {\left( { - 1} \right)^i \binom{j}{i} \mathsf{A}_{i,k} } .
\end{equation}
\section{Examples}\label{section3}
In this section, we give two illustrative examples to demonstrate the application of Corollary \ref{maintheorem} as well as to compare the results with that given by Wojdylo's formula. In the first example, we derive explicit expressions for the so-called Stirling coefficients appearing in the asymptotic expansion of the gamma function. In the second example, we obtain an explicit formula for certain polynomials related to the coefficients in the uniform asymptotic expansion of the incomplete gamma function.
\subsection*{Example 1} The gamma function can be defined by the following integral
\begin{equation}\label{eq19}
\varGamma \left( {\lambda + 1} \right) = \int_0^{ + \infty } {\mathrm{e}^{ - t} t^\lambda \mathrm{d}t}, \quad \lambda >0.
\end{equation}
If we put $t = \lambda\left(1 + x\right)$, we obtain
\[
\varGamma \left( {\lambda + 1} \right) = \lambda ^{\lambda + 1} \mathrm{e}^{ - \lambda } \int_{ - 1}^{ + \infty } {\mathrm{e}^{ - \lambda \left( {x - \log \left( {1 + x} \right)} \right)} \mathrm{d}x} ,
\]
and hence, using the identity $\varGamma \left( {\lambda + 1} \right) = \lambda \varGamma \left( \lambda \right)$,
\begin{equation}\label{eq10}
\frac{{\varGamma \left( \lambda \right)}}{{\lambda ^\lambda \mathrm{e}^{ - \lambda } }} = \int_0^{ + \infty } {\mathrm{e}^{ - \lambda \left( {x - \log \left( {1 + x} \right)} \right)} \mathrm{d}x} + \int_0^1 {\mathrm{e}^{ - \lambda \left( { - x - \log \left( {1 - x} \right)} \right)} \mathrm{d}x}
\end{equation}
follows. Consider the first integral. Let $f\left(x\right) = x - \log (1 + x)$, $x \geq 0$. Then $f$ has a global minimum at $x=0$ and the expansion
\[
f\left( x \right) = \sum\limits_{k = 0}^\infty {\left( { - 1} \right)^k \frac{{x^{k + 2} }}{{k + 2}}}
\]
holds as $x \to 0^+$. We can apply Theorem \ref{theorem1} with $g \equiv 1$, $\alpha = 2$, $\beta = 1$, $b_0 = 1$, $b_k = 0$ for $k>0$, and $a_k = \left( { - 1} \right)^k /\left(k+2\right)$. The result is
\begin{equation}\label{eq28}
\int_0^{ + \infty } {\mathrm{e}^{ - \lambda \left( {x - \log \left( {1 + x} \right)} \right)} \mathrm{d}x} \sim \sum\limits_{n = 0}^\infty {\varGamma \left( {\frac{{n + 1}}{2}} \right)\frac{{c_n }}{{\lambda ^{\left( {n + 1} \right)/2} }}} ,
\end{equation}
where, by Perron's formula,
\begin{equation}\label{eq29}
c_n = \frac{1}{{2n!}}\left[ {\frac{{\mathrm{d}^n }}{{\mathrm{d}x^n }}\left( {\frac{{x^2 }}{{x - \log \left( {1 + x} \right)}}} \right)^{\left( {n + 1} \right)/2} } \right]_{x = 0} .
\end{equation}
Similarly, one finds that
\[
\int_0^1 {\mathrm{e}^{ - \lambda \left( {- x - \log \left( {1 - x} \right)} \right)} \mathrm{d}x} \sim \sum\limits_{n = 0}^\infty {\varGamma \left( {\frac{{n + 1}}{2}} \right)\frac{{\tilde{c}_n }}{{\lambda ^{\left( {n + 1} \right)/2} }}} ,
\]
where
\begin{multline*}
\tilde{c}_n = \frac{1}{{2n!}}\left[ {\frac{{\mathrm{d}^n }}{{\mathrm{d}x^n }}\left( {\frac{{x^2 }}{{- x - \log \left( {1 - x} \right)}}} \right)^{\left( {n + 1} \right)/2} } \right]_{x = 0}\\ = \frac{\left(-1\right)^n}{{2n!}}\left[ {\frac{{\mathrm{d}^n }}{{\mathrm{d}x^n }}\left( {\frac{{x^2 }}{{x - \log \left( {1 + x} \right)}}} \right)^{\left( {n + 1} \right)/2} } \right]_{x = 0} = \left(-1\right)^n c_n.
\end{multline*}
By substituting these asymptotic series into \eqref{eq10} and performing a simple rearrangement, we deduce
\begin{equation}\label{eq20}
\varGamma \left( \lambda \right) \sim \sqrt {2\pi } \lambda ^{\lambda - 1/2} \mathrm{e}^{ - \lambda } \sum\limits_{n = 0}^\infty {\left( { - 1} \right)^n \frac{{\gamma _n }}{{\lambda ^n }}}
\end{equation}
as $\lambda \to +\infty$. Here
\begin{equation}\label{eq18}
\gamma _n = \left( { - 1} \right)^n \sqrt {\frac{2}{\pi }} \varGamma \left( {n + \frac{1}{2}} \right)c_{2n} = \frac{{\left( { - 1} \right)^n }}{{2^n n!}}\left[ {\frac{{\mathrm{d}^{2n} }}{{\mathrm{d}x^{2n} }}\left( {\frac{1}{2}\frac{{x^2 }}{{x - \log \left( {1 + x} \right)}}} \right)^{n + 1/2} } \right]_{x = 0}
\end{equation}
are the so-called Stirling coefficients \cite[p. 26]{Paris2}. This representation is well known (see, e.g., \cite{Brassesco}\cite[p. 111]{Wong}). The first few are $\gamma_0 = 1$ and
\[
\gamma _1 = - \frac{1}{12},\quad \gamma _2 = \frac{1}{288},\quad \gamma _3 = \frac{139}{51840},\quad \gamma _4 = - \frac{571}{2488320}.
\]
In this case, we are able to obtain explicit formulas for both $\mathsf{B}_{n,k}$ and $\mathsf{A}_{k,n}$. Due to the simpler generating function, we use our new formula \eqref{eq14} and compute the ordinary potential polynomials. Here the generating function takes the form
\[
\left( {2\frac{{x - \log \left( {1 + x} \right)}}{{x^2}}} \right)^k = \sum\limits_{n = 0}^\infty {\mathsf{A}_{k,n} x^n } ,
\]
where $\mathsf{A}_{k,n} = \mathsf{A}_{k,n} \left( {a_1 a_0^{-1},a_2 a_0^{-1},\ldots,a_n a_0^{-1}} \right)$ and $a_n = \left( { - 1} \right)^n /\left(n+2\right)$. Using the generating function of the (signless) Stirling numbers of the first kind yields
\begin{align*}
\left( {x - \log \left( {1 + x} \right)} \right)^k & = \sum\limits_{j = 0}^k {\binom{k}{j}x^{k - j} \left( { - \log \left( {1 + x} \right)} \right)^j } \\
& = \sum\limits_{j = 0}^k {\binom{k}{j}x^{k - j} \sum\limits_{n = 0}^\infty {\left( { - 1} \right)^n j!s\left( {n,j} \right)\frac{{x^n }}{{n!}}} } \\
& = \sum\limits_{n = 0}^\infty {\left( {\sum\limits_{j = 0}^k {\left( { - 1} \right)^{n - k + j}\binom{k}{j} j!\frac{{s\left( {n - k + j,j} \right)}}{{\left( {n - k + j} \right)!}}} } \right)x^n } ,
\end{align*}
which gives
\begin{equation}\label{eq15}
\mathsf{A}_{k,n} = 2^k \sum\limits_{j = 0}^k {\left( { - 1} \right)^{n + k + j}\binom{k}{j} j!\frac{{s\left( {n + k + j,j} \right)}}{{\left( {n + k + j} \right)!}}} .
\end{equation}
Substituting this into \eqref{eq14} produces
\begin{align*}
c_{2n} & = \frac{{2^{n-1/2}\varGamma \left( {3n + \frac{3}{2}} \right)}}{{ \left( {2n} \right)!\varGamma \left( {n + \frac{1}{2}} \right)}}\sum\limits_{k = 0}^{2n} {\frac{{2^k }}{{n + \frac{1}{2} + k}}\binom{2n}{k} \sum\limits_{j = 0}^k {\left( { - 1} \right)^{j}\binom{k}{j}j!\frac{{s\left( {2n + k + j,j} \right)}}{{\left( {2n + k + j} \right)!}}}} \\
& = \frac{1}{{\varGamma \left( {n + \frac{1}{2}} \right)}}\sum\limits_{k = 0}^{2n} {\frac{{2^{n + k + 1/2} \varGamma \left( {3n + \frac{3}{2}} \right)}}{{\left( {2n + 2k + 1} \right)\left( {2n - k} \right)!}}\sum\limits_{j = 0}^k {\frac{{\left( { - 1} \right)^j s\left( {2n + k + j,j} \right)}}{{\left( {k - j} \right)!\left( {2n + k + j} \right)!}}} } .
\end{align*}
Finally, by \eqref{eq18},
\begin{equation}\label{eq23}
\gamma_n = \sum\limits_{k = 0}^{2n} {\frac{{\left( { - 1} \right)^n 2^{n + k + 1} \varGamma \left( {3n + \frac{3}{2}} \right)}}{{\sqrt \pi \left( {2n + 2k + 1} \right)\left( {2n - k} \right)!}}\sum\limits_{j = 0}^k {\frac{{\left( { - 1} \right)^j s\left( {2n + k + j,j} \right)}}{{\left( {k - j} \right)!\left( {2n + k + j} \right)!}}} } .
\end{equation}
As far as we know, this representation of the Stirling coefficients is entirely new. Using expression \eqref{eq16} together with \eqref{eq15}, Wojdylo's formula \eqref{eq17} yields the more complicated formula
\begin{equation}\label{eq21}
\gamma _n = \sum\limits_{k = 0}^{2n} {\frac{{\left( { - 1} \right)^n 2^n \varGamma \left( {n + k + \frac{1}{2}} \right)}}{{\sqrt \pi }}} \sum\limits_{j = 0}^k {\frac{{2^j }}{{\left( {k - j} \right)!}}\sum\limits_{i = 0}^j {\frac{{\left( { - 1} \right)^i s\left( {2n + j + i,i} \right)}}{{\left( {j - i} \right)!\left( {2n + j + i} \right)!}}} },
\end{equation}
which is the result of L\'{o}pez, Pagola and P\'{e}rez Sinus\'\i a \cite{Lopez1}.
We remark that the substitution $x = \log(t/\lambda)$ in \eqref{eq19} leads to the form
\[
\frac{{\varGamma \left( \lambda \right)}}{{\lambda ^\lambda \mathrm{e}^{ - \lambda } }} = \int_0^{ + \infty } {\mathrm{e}^{ - \lambda \left( {\mathrm{e}^x - x - 1} \right)} \mathrm{d}x} + \int_0^{ + \infty } {\mathrm{e}^{ - \lambda \left( {\mathrm{e}^{ - x} + x - 1} \right)}\mathrm{d}x} .
\]
Similar procedures to the one described above give \eqref{eq20} with
\begin{align}
\gamma _n & = \frac{\left( { - 1} \right)^n}{{2^n n!}}\left[ {\frac{{\mathrm{d}^{2n} }}{{\mathrm{d}x^{2n} }}\left( {\frac{1}{2}\frac{{x^2 }}{{\mathrm{e}^x - x - 1}}} \right)^{n + 1/2} } \right]_{x = 0} \nonumber \\
& = \sum\limits_{k = 0}^{2n} {\frac{{\left( { - 1} \right)^n 2^{n + k + 1} \varGamma \left( {3n + \frac{3}{2}} \right)}}{{\sqrt \pi \left( {2n + 2k + 1} \right)\left( {2n - k} \right)!}}\sum\limits_{j = 0}^k {\frac{{\left( { - 1} \right)^j S\left( {2n + k + j,j} \right)}}{{\left( {k - j} \right)!\left( {2n + k + j} \right)!}}} } \label{eq24} \\
& = \sum\limits_{k = 0}^{2n} {\frac{{\left( { - 1} \right)^n 2^n \varGamma \left( {n + k + \frac{1}{2}} \right)}}{{\sqrt \pi }}} \sum\limits_{j = 0}^k {\frac{{2^j }}{{\left( {k - j} \right)!}}\sum\limits_{i = 0}^j {\frac{{\left( { - 1} \right)^i S\left( {2n + j + i,i} \right)}}{{\left( {j - i} \right)!\left( {2n + j + i} \right)!}}} } \label{eq22},
\end{align}
using formula \eqref{eq6}, \eqref{eq14} and \eqref{eq17}, respectively. Here $S(n,k)$ denotes the Stirling numbers of the second kind. The first representation is known (see, e.g., \cite{Brassesco} \cite{De Angelis}). The second formula is a simplified form of the one derived in \cite{Nemes}. To our knowledge, the third one is entirely new. Note the remarkable similarity between expressions \eqref{eq23} and \eqref{eq24}, and expressions \eqref{eq21} and \eqref{eq22}.
\subsection*{Example 2} The incomplete gamma function is traditionally defined by the following integral:
\[
\varGamma \left( {a,x} \right) = \int_x^\infty {\mathrm{e}^{ - t} t^{a - 1} \mathrm{d}t} ,\quad a > 0,\quad x \ge 0.
\]
For large values of $x$, the function admits the following asymptotic expansion
\[
\varGamma \left( {a,x} \right)x^{1 - a} \mathrm{e}^x \sim 1 + \frac{{a - 1}}{x} + \frac{{\left( {a - 1} \right)\left( {a - 2} \right)}}{{x^2 }} + \cdots ,
\]
which is useful only when $a =o(x)$ \cite[p. 179]{NIST}. However, in exponentially improved asymptotics, we need the asymptotic properties of $\varGamma \left( {a,x} \right)$ as $a \to +\infty$ and $x = \lambda a$, where $\lambda \neq 0$ is a constant. (Do not confuse it with the variable of the integral \eqref{eq1}.) Starting from the integral representation
\[
\frac{{\varGamma \left( {a ,x} \right)}}{{\varGamma \left( a \right)}} = \frac{{\mathrm{e}^{- a \varphi \left( \lambda \right)} }}{{2\pi \mathrm{i}}}\int_{c - \mathrm{i}\infty }^{c + \mathrm{i}\infty } {\mathrm{e}^{ a \varphi \left( t \right)} \frac{1}{{\lambda - t}}\mathrm{d}t} ,\quad 0 < c < \lambda ,
\]
\[
\varphi \left( t \right) = t - \log t - 1,
\]
Temme \cite{Temme1} gave the following uniform asymptotic expansion of the (normalized) incomplete gamma function as $a \to +\infty$
\[
\frac{{\varGamma \left( {a,x} \right)}}{{\varGamma \left( a \right)}} \sim \frac{1}{2}\mathrm{erfc}\left( {\eta \sqrt {\frac{1}{2}a} } \right) + \frac{{\mathrm{e}^{ - \frac{1}{2}a\eta ^2 } }}{{\sqrt {2\pi a} }}\sum\limits_{n = 0}^\infty {\frac{{C_n \left( \eta \right)}}{{a^n }}} .
\]
Here $\eta = \sqrt {2 \varphi \left( \lambda \right)}$ and $\mathrm{erfc}$ denotes the complementary error function. He gave a recurrence for the coefficients $C_n \left( \eta \right)$ and showed that they have the general structure
\begin{equation}\label{eq27}
C_n \left( \eta \right) = \left( { - 1} \right)^n \left( {\frac{{Q_n \left( \mu \right)}}{{\mu ^{2n + 1} }} - \frac{{2^n \varGamma \left( {n + \frac{1}{2}} \right)}}{{\sqrt \pi \eta ^{2n + 1} }}} \right),
\end{equation}
where $\mu = \lambda - 1$ and $Q_n$ is a polynomial in $\mu$ of degree $2n$ \cite{Temme2}. The first few of the $Q_n$'s are given by
\begin{align*}
Q_0 \left( \mu \right) & = 1,\\
Q_1 \left( \mu \right) & = 1 + \mu + \frac{1}{{12}}\mu ^2 ,\\
Q_2 \left( \mu \right) & = 3 + 5\mu + \frac{{25}}{{12}}\mu ^2 + \frac{1}{{12}}\mu ^3 + \frac{1}{{288}}\mu ^4 .
\end{align*}
As far as we know, no simple explicit formula for these polynomials exists in the literature. Dunster et al. \cite{Dunster} derived the integral representation
\[
C_n \left( \eta \right) = \frac{{\left( { - 1} \right)^{n+1} \varGamma \left( {n + \frac{1}{2}} \right)}}{{\left( {2\pi } \right)^{3/2} \mathrm{i}}}\oint_{\mathscr{C}} {\frac{{\mathrm{d}z}}{{\left( {z - \lambda } \right)\left( {z - \log z - 1} \right)^{n + 1/2} }}} .
\]
Here, $\mathscr{C}$ is a loop in the $z$ plane enclosing the poles $z = 1$ and $z = \lambda$. The term $\left( {z - \log z - 1} \right)^{n + 1/2}$ is real and positive for $\arg z = 0$ and $z > 1$ and is defined by continuity elsewhere. We suppose that $\lambda \neq 1$ ($\eta \neq 0$) and use the Residue Theorem to obtain
\begin{align*}
& C_n \left( \eta \right) = \frac{{\left( { - 1} \right)^{n + 1} \varGamma \left( {n + \frac{1}{2}} \right)}}{{\sqrt {2\pi } }}\left( {\mathop {\mathrm{Res}}\limits_{z = 1} \frac{1}{{\left( {z - \lambda } \right)\left( {z - \log z - 1} \right)^{n + 1/2} }} + \frac{1}{{\left( {\lambda - \log \lambda - 1} \right)^{n + 1/2} }}} \right)\\
& = \left( { - 1} \right)^n \left( { - \frac{{\varGamma \left( {n + \frac{1}{2}} \right)}}{{\sqrt {2\pi } }}\left[ {\frac{{\mathrm{d}^{2n} }}{{\mathrm{d}z^{2n} }}\left\{ {\frac{1}{{\left( {z - \lambda } \right)}}\left( {\frac{{\left( {z - 1} \right)^2 }}{{z - \log z - 1}}} \right)^{n + 1/2} } \right\}} \right]_{z = 1} - \frac{{2^n \varGamma \left( {n + \frac{1}{2}} \right)}}{{\sqrt \pi \eta ^{2n + 1} }}} \right)\\
& = \left( { - 1} \right)^n \left( {\frac{1}{{2^{2n + 1/2} n!}}\left[ {\frac{{\mathrm{d}^{2n} }}{{\mathrm{d}x^{2n} }}\left\{ {\frac{1}{{\left( {\mu - x} \right)}}\left( {\frac{{x^2 }}{{x - \log \left( {1 + x} \right)}}} \right)^{n + 1/2} } \right\}} \right]_{x = 0} - \frac{{2^n \varGamma \left( {n + \frac{1}{2}} \right)}}{{\sqrt \pi \eta ^{2n + 1} }}} \right) .
\end{align*}
Upon comparing this with \eqref{eq27}, we see that
\[
Q_n \left( \mu \right) = \frac{{\mu ^{2n + 1} }}{{2^{2n + 1/2} n!}}\left[ {\frac{{\mathrm{d}^{2n} }}{{\mathrm{d}x^{2n} }}\left\{ {\frac{1}{{\left( {\mu - x} \right)}}\left( {\frac{{x^2 }}{{x - \log \left( {1 + x} \right)}}} \right)^{n + 1/2} } \right\}} \right]_{x = 0} ,
\]
and from Perron's formula \eqref{eq6},
\begin{equation}\label{eq26}
Q_n \left( \mu \right) = \frac{{\left( {2n} \right)!\mu ^{2n + 1} }}{{2^{2n - 1/2} n!}}c_{2n} = \sqrt {\frac{2}{\pi }} \varGamma \left( {n + \frac{1}{2}} \right)\mu ^{2n + 1} c_{2n} ,
\end{equation}
where the $c_n$'s are the coefficients appearing in the asymptotic expansion of the integral \eqref{eq1} with $a=0$, $b>0$, $f\left( x \right) = x - \log \left( {1 + x} \right)$ and $g\left( x \right) = \left( {\mu - x} \right)^{ - 1}$.
Now, using Corollary \ref{maintheorem}, we derive an explicit formula for the polynomials $Q_n$ in terms of the Stirling numbers of the first kind. Near $x=0$, $g\left( x \right) = \sum\nolimits_{k = 0}^\infty {\mu ^{ - k - 1} x^k }$, therefore $b_k = \mu ^{ - k - 1}$. The corresponding ordinary potential polynomials are given by \eqref{eq15}. Hence, by Corollary \ref{maintheorem},
\[
c_{2n} = \frac{1}{{\varGamma \left( {n + \frac{1}{2}} \right)}}\sum\limits_{k = 0}^{2n} {\frac{{2^{n + 1/2} \varGamma \left( {n + k + \frac{3}{2}} \right)\mu ^{ - 2n + k - 1} }}{{k!}}\sum\limits_{j = 0}^k {\frac{{\left( { - 1} \right)^j }}{{2n + 2j + 1}}\binom{k}{j}\mathsf{A}_{j,k} } } .
\]
This, together with \eqref{eq26}, yields
\[
Q_n \left( \mu \right) = \sum\limits_{k = 0}^{2n} {\frac{{2^{n + 1} \varGamma \left( {n + k + \frac{3}{2}} \right)\mu ^k }}{{\sqrt \pi k!}}\sum\limits_{j = 0}^k {\frac{{\left( { - 1} \right)^j }}{{2n + 2j + 1}}\binom{k}{j}\mathsf{A}_{j,k} } } ,
\]
or $Q_n \left( \mu \right) = \sum\nolimits_{k = 0}^{2n} {q_k^{\left( n \right)} \mu ^k }$, where, by \eqref{eq15},
\begin{equation}\label{eq31}
q_k^{\left( n \right)} = \sum\limits_{j = 0}^k {\frac{{ \left( { - 1} \right)^k 2^{n + j + 1} \varGamma \left( {n + k + \frac{3}{2}} \right)}}{{\sqrt \pi \left( {2n + 2j + 1} \right)\left( {k - j} \right)!}}\sum\limits_{i = 0}^j {\frac{{\left( { - 1} \right)^i s\left( {k + j + i,i} \right)}}{{\left( {j - i} \right)!\left( {k + j + i} \right)!}}} } .
\end{equation}
We remark that $q_{2n}^{\left( n \right)} = \left( { - 1} \right)^n \gamma _n$, where $\gamma_n$ is the $n$th Stirling coefficient. Naturally, we could use Wojdylo's formula \eqref{eq13} to obtain an alternative representation for the coefficients $c_n$, and thus for the polynomials $Q_n \left( \mu \right)$ themselves. The corresponding partial ordinary Bell polynomials can be given explicitly using expression \eqref{eq16} together with \eqref{eq15}, the resulting formula is, however, even more elaborate than \eqref{eq31}.
When $\lambda = 1$ ($\eta = 0$), we have $C_n \left( 0 \right) = \lim _{\eta \to 0} C_n \left( \eta \right)$ and Temme's expansion takes the form
\[
\frac{{\varGamma \left( {a,a} \right)}}{{\varGamma \left( a \right)}} \sim \frac{1}{2} + \frac{1}{{\sqrt {2\pi a} }}\sum\limits_{n = 0}^\infty {\frac{{C_n \left( 0 \right)}}{{a^n }}}
\]
as $a \to +\infty$. However, the value of the limit $\lim _{\eta \to 0} C_n \left( \eta \right)$ is not obvious from \eqref{eq27}. To derive an explicit formula for the $C_n \left( 0 \right)$'s, we proceed as follows: for every positive integer $m$
\begin{align*}
\frac{{\varGamma \left( {m,m} \right)}}{{\varGamma \left( m \right)}} = \mathrm{e}^{ - m} \sum\limits_{k = 0}^{m - 1} {\frac{{m^k }}{{k!}}} & = - \frac{{m^m \mathrm{e}^{ - m} }}{{m!}} + \frac{{m^m \mathrm{e}^{ - m} }}{{m!}}\sum\limits_{k = 0}^m {\binom{m}{k}\frac{{k!}}{{m^k }}}\\
& = - \frac{{m^m \mathrm{e}^{ - m} }}{{m!}} + \frac{{m^m \mathrm{e}^{ - m} }}{{m!}}\sum\limits_{k = 0}^m {\binom{m}{k}m\int_0^{ + \infty } {\mathrm{e}^{ - mx} x^k \mathrm{d}x} } \\
& = \frac{{m^m \mathrm{e}^{ - m} }}{{m!}}\left( { - 1 + m\int_0^{ + \infty } {\mathrm{e}^{ - m\left( {x - \log \left( {1 + x} \right)} \right)} \mathrm{d}x} } \right).
\end{align*}
It is well known that
\begin{equation}\label{eq30}
\frac{{m^m \mathrm{e}^{ - m} }}{{m!}} \sim \frac{1}{{\sqrt {2\pi m} }}\sum\limits_{n = 0}^\infty {\frac{{\gamma _n }}{{m^n }}},
\end{equation}
where $\gamma_n$ denotes the $n$th Stirling coefficient \cite[p. 26]{Paris2}. By \eqref{eq28}, we have
\[
- 1 + m\int_0^{ + \infty } {\mathrm{e}^{ - m\left( {x - \log \left( {1 + x} \right)} \right)} \mathrm{d}x} \sim - 1 + \sqrt m \sum\limits_{n = 0}^\infty {\varGamma \left( {\frac{{n + 1}}{2}} \right)\frac{{c_n }}{{m^{n/2} }}} ,
\]
where $c_n$ is given by \eqref{eq29}. From these we deduce
\begin{align*}
& \frac{{\varGamma \left( {m,m} \right)}}{{\varGamma \left( m \right)}} \sim - \frac{1}{{\sqrt {2\pi m} }}\sum\limits_{n = 0}^\infty {\frac{{\gamma _n }}{{m^n }}} + \frac{1}{{\sqrt {2\pi } }}\sum\limits_{n = 0}^\infty {\frac{{\gamma _n }}{{m^n }}} \sum\limits_{n = 0}^\infty {\varGamma \left( {\frac{{n + 1}}{2}} \right)\frac{{c_n }}{{m^{n/2} }}} \\
& = - \frac{1}{{\sqrt {2\pi m} }}\sum\limits_{n = 0}^\infty {\frac{{\gamma _n }}{{m^n }}} + \frac{1}{{\sqrt {2\pi } }}\sum\limits_{n = 0}^\infty {\left( {\sum\limits_{k = 0}^{\left\lfloor {n/2} \right\rfloor } {\gamma _k \varGamma \left( {\frac{{n - 2k + 1}}{2}} \right)c_{n - 2k} } } \right)\frac{1}{{m^{n/2} }}} \\
& = \frac{1}{2} - \frac{1}{{\sqrt {2\pi m} }}\sum\limits_{n = 0}^\infty {\frac{{\gamma _n }}{{m^n }}} + \frac{1}{{\sqrt {2\pi m} }}\sum\limits_{n = 0}^\infty {\underbrace {\left( {\sum\limits_{k = 0}^{\left\lfloor {\left( {n + 1} \right)/2} \right\rfloor } {\gamma _k \varGamma \left( {\frac{n}{2} - k + 1} \right)c_{n - 2k + 1} } } \right)}_{\nu _n }\frac{1}{{m^{n/2} }}} .
\end{align*}
The well-known identity $\sum\nolimits_{k = 0}^n {\left( { - 1} \right)^{n - k} \gamma _k \gamma _{n - k} } = 0$ ($n \geq 1$) \cite[p. 33]{Paris1} gives
\[
\nu _{2n - 1} = \sum\limits_{k = 0}^n {\gamma _k \varGamma \left( {n - k + \frac{1}{2}} \right)c_{2n - 2k} } = \sqrt {\frac{\pi }{2}} \sum\limits_{k = 0}^n {\left( { - 1} \right)^{n - k} \gamma _k \gamma _{n - k} } = 0,
\]
from which we obtain
\[
\frac{{\varGamma \left( {m,m} \right)}}{{\varGamma \left( m \right)}} \sim \frac{1}{2} + \frac{1}{{\sqrt {2\pi m} }}\sum\limits_{n = 0}^\infty {\left( { - \gamma_n + \sum\limits_{k = 0}^n {\gamma _k \varGamma \left( {n - k + 1} \right)c_{2n - 2k + 1} } } \right)\frac{1}{{m^n }}} .
\]
From the uniqueness theorem on asymptotic series we finally have
\begin{align*}
C_n \left( 0 \right) & = - \gamma_n + \sum\limits_{k = 0}^n {\gamma _k \varGamma \left( {n - k + 1} \right)c_{2n - 2k + 1} } \\ & = - \frac{1}{3}\gamma _n + \sum\limits_{k = 0}^{n - 1} {\left(n - k\right)!\gamma _k c_{2n - 2k + 1} } .
\end{align*}
The first few values are given by
\[
C_0 \left( 0 \right) = - \frac{1}{3},\quad C_1 \left( 0 \right) = -\frac{1}{540},\quad C_2 \left( 0 \right) = \frac{25}{6048},\quad C_3 \left( 0 \right) = \frac{101}{155520}.
\]
These are in agreement with those given in \cite[p. 181]{NIST}.
We remark that one can start with the representation
\[
\frac{{\varGamma \left( {m,m} \right)}}{{\varGamma \left( m \right)}} = \frac{{m^m \mathrm{e}^{ - m} }}{{\varGamma \left( m \right)}}\int_0^{ + \infty } {\mathrm{e}^{ - m\left( {x - \log \left( {1 + x} \right)} \right)} \frac{1}{{1 + x}}\mathrm{d}x} ,
\]
which follows from the definition of the incomplete gamma function. The factor before the integral can be expanded into an asymptotic series using \eqref{eq30}. The asymptotic expansion of the integral can be deduced from Erd\'{e}lyi's theorem. In this way, we get a similar representation for the coefficients $C_n \left( 0 \right)$ to the one above. However, in this case $g\left(x\right) = \left(1+x\right)^{-1}$, whereas in the previous one $g \equiv 1$, which leads to a somewhat complicated representation.
\section{Conclusion and future work}
Laplace's method is one of the classical techniques in the theory of asymptotic expansion of real integrals. The coefficients $c_n$ appearing in the resulting asymptotic expansion, arise as the coefficients of a convergent or asymptotic series of a function defined in an implicit form. Traditionally, series inversion and composition have been used to compute these coefficients, which can be extremely complicated. Nevertheless, there are certain formulas of varying degrees of explicitness for $c_n$'s in the literature. One of them is Perron's formula which gives the coefficients in terms of derivatives of an explicit function. The most explicit formula is given by Wojdylo, who rewrote Perron's formula in terms of some combinatorial objects, called the partial ordinary Bell polynomials.
In this paper we have given an alternative way for simplifying Perron's formula. The new representation involves the ordinary potential polynomials.
We have applied the method to the two important examples of the gamma function and the incomplete gamma function. We have obtained new and explicit formulas for the so-called Stirling coefficients appearing in the asymptotic expansion of the gamma function. We have also derived explicit formulas for certain polynomials related to the coefficients in the uniform asymptotic expansion of the incomplete gamma function.
It turns out that formally the same coefficients appear in the asymptotic expansion of contour integrals with a complex parameter. Our new method, hence, can be applied to give explicit formulas for the coefficients arising in the saddle point method \cite[p. 125]{Olver2}\cite[p. 47]{NIST}, the method of steepest descents \cite{Lopez2}\cite[p. 12]{Paris2} and even in the principle of stationary phase \cite{Olver1}\cite[p. 45]{NIST}.
A fruitful direction for research would be to characterize those integrals that can be transformed into the form \eqref{eq1} and have asymptotic expansions where the ordinary potential polynomials, corresponding to the coefficients, are polynomial-time-computable functions. In the case of the gamma function and the incomplete gamma function, Corollary \ref{maintheorem} produced simpler expressions for the coefficients in their asymptotic expansions than that of Wojdylo's. The question arises whether the ordinary potential polynomial representation generically leads to a simpler form, compared to the partial ordinary Bell polynomial representation, or not. If not, an interesting problem would be to characterize the exceptional cases.
\section*{Acknowledgment}
The author would like to thank the two anonymous referees for their thorough, constructive and helpful comments and suggestions on an earlier version of this paper. Especially, he wishes to thank the second referee, for bringing the relevance of Comtet's work to his attention and giving suggestions for future research.
\appendix
\section{Powers of Power Series}\label{appendix}
Let $F\left( x \right) = 1 + \sum\nolimits_{n = 1}^\infty {f_n x^n }$ be a formal power series. For any nonnegative integer $k$, we define the partial ordinary Bell polynomials $\mathsf{B}_{n,k} \left( {f_1 ,f_2 , \ldots ,f_{n - k + 1} } \right)$ (associated with $F$) by the generating function
\[
\left( {F\left( x \right) - 1} \right)^k = \left( {\sum\limits_{n = 1}^\infty {f_n x^n } } \right)^k = \sum\limits_{n = k}^\infty {\mathsf{B}_{n,k} \left( {f_1 ,f_2 , \ldots ,f_{n - k + 1} } \right)x^n },
\]
so that $\mathsf{B}_{0,0} = 1$, $\mathsf{B}_{n,0} = 0$ ($n\geq1$), $\mathsf{B}_{n,1} = f_n$ and $\mathsf{B}_{n,2} = \sum\nolimits_{j = 1}^{n-1} {f_j f_{n - j} }$. From the simple identity, $\left( {F\left( x \right) - 1} \right)^{k + 1} = \left( {F\left( x \right) - 1} \right)\left( {F\left( x \right) - 1} \right)^k$, we obtain the recurrence relation
\begin{equation}\label{eq32}
\mathsf{B}_{n,k + 1} \left( {f_1 ,f_2 , \ldots ,f_{n - k} } \right) = \sum\limits_{j = 1}^{n - k} {f_j \mathsf{B}_{n - j,k} \left( {f_1 ,f_2 , \ldots ,f_{n - j - k + 1} } \right)} .
\end{equation}
An explicit representation, that follows from the definition, is given by
\[
\mathsf{B}_{n,k} \left( {f_1 ,f_2 , \ldots ,f_{n - k + 1} } \right) = \sum {\frac{{k!}}{{k_1 !k_2 ! \cdots k_{n - k + 1} !}}f_1^{k_1 } f_2^{k_2 } \cdots f_{n - k + 1}^{k_{n - k + 1} } },
\]
where the sum runs over all sequences $k_1,k_2, \ldots , k_n$ of non-negative integers such that $k_1 + 2k_2 + \cdots + \left(n-k+1\right)k_{n-k+1} = n$ and $k_1 + k_2 + \cdots + k_{n-k+1} = k$.
For any complex number $\rho$, we define the ordinary potential polynomials $\mathsf{A}_{\rho ,n} \left( {f_1 ,f_2 , \ldots ,f_n } \right)$ (associated to $F$) by the generating function
\[
\left( {F\left( x \right)} \right)^\rho = \left( {1 + \sum\limits_{n = 1}^\infty {f_n x^n } } \right)^\rho = \sum\limits_{n = 0}^\infty {\mathsf{A}_{\rho ,n} \left( {f_1 ,f_2 , \ldots ,f_n } \right)x^n },
\]
hence,
\[
\mathsf{A}_{\rho ,n} \left( {f_1 ,f_2 , \ldots ,f_n } \right) = \sum\limits_{k = 0}^n {\binom{\rho}{k} \mathsf{B}_{n,k} \left( {f_1 ,f_2 , \ldots ,f_{n - k + 1} } \right)}.
\]
The first few are $\mathsf{A}_{\rho ,0} = 1$, $\mathsf{A}_{\rho ,1} = \rho f_1$, $\mathsf{A}_{\rho ,2} = \rho f_2 + \binom{\rho}{2}f_1^2$, and in general
\[
\mathsf{A}_{\rho ,n} \left( {f_1 ,f_2 , \ldots ,f_n } \right) = \sum {\binom{\rho}{k}\frac{{k!}}{{k_1 !k_2 ! \cdots k_n !}}f_1^{k_1 } f_2^{k_2 } \cdots f_n^{k_n } } ,
\]
where the sum extends over all sequences $k_1,k_2, \ldots , k_n$ of non-negative integers such that $k_1 + 2k_2 + \cdots + nk_n = n$ and $k_1 + k_2 + \cdots + k_n = k$. Since $\left( {F\left( x \right)} \right)^\rho = \left( {F\left( x \right)} \right)^{\rho - 1} F\left( x \right)$, we have the recurrence
\begin{equation}\label{eq33}
\mathsf{A}_{\rho ,n} \left( {f_1 ,f_2 , \ldots ,f_n } \right) = \mathsf{A}_{\rho - 1,n} \left( {f_1 ,f_2 , \ldots ,f_n } \right) + \sum\limits_{k = 1}^n {f_k \mathsf{A}_{\rho - 1,n - k} \left( {f_1 ,f_2 , \ldots ,f_{n - k} } \right)} .
\end{equation}
In general, if $G\left( x \right) = \sum\nolimits_{n = 0}^\infty {g_n x^n }$ is a formal power series, then by definition, we have
\[
G\left( {y\left( {F\left( x \right) - 1} \right)} \right) = \sum\limits_{n = 0}^\infty {\left( {\sum\limits_{k = 0}^n {g_k \mathsf{B}_{n,k} \left( {f_1 ,f_2 , \ldots ,f_{n - k + 1} } \right)y^k } } \right)x^n } .
\]
Specially,
\[
\exp \left( {y\sum\limits_{n = 1}^\infty {f_n x^n } } \right) = \sum\limits_{n = 0}^\infty {\left( {\sum\limits_{k = 0}^n {\frac{{\mathsf{B}_{n,k} \left( {f_1 ,f_2 , \ldots ,f_{n - k + 1} } \right)}}{{k!}}y^k } } \right)x^n } .
\]
For more details see, e.g., Comtet's book \cite[pp. 133--153]{Comtet}.
\end{document} | math | 49,724 |
\begin{document}
\title{Two-solvable and two-bipolar knots with large four-genera}
\author{Jae Choon Cha}
\address{
Department of Mathematics\unskip, \ignorespaces
POSTECH\unskip, \ignorespaces
Pohang Gyeongbuk 37673\unskip, \ignorespaces
Republic of Korea\linebreak
School of Mathematics\unskip, \ignorespaces
Korea Institute for Advanced Study \unskip, \ignorespaces
Seoul 02455\unskip, \ignorespaces
Republic of Korea
}
\email{[email protected]}
\author{Allison N.~Miller}
\address{
Department of Mathematics\unskip, \ignorespaces
Rice University\unskip, \ignorespaces
Houston, TX, USA}
\email{[email protected]}
\author{Mark Powell}
\address{
Department of Mathematical Sciences\unskip, \ignorespaces
Durham University\unskip, \ignorespaces
United Kingdom}
\email{[email protected]}
\def\textup{2010} Mathematics Subject Classification{\textup{2010} Mathematics Subject Classification}
\expandafter\let\csname subjclassname@1991\endcsname=\textup{2010} Mathematics Subject Classification
\expandafter\let\csname subjclassname@2000\endcsname=\textup{2010} Mathematics Subject Classification
\subjclass{57M25, 57M27, 57N70.}
\keywords{four-genus, knot concordance, grope, solvable filtration,
bipolar filtration, $L^{(2)}$-signature, Casson-Gordon invariant}
\begin{abstract}
For every integer $g$, we construct a $2$-solvable and $2$-bipolar knot whose
topological $4$-genus is greater than~$g$. Note that $2$-solvable knots are in
particular algebraically slice and have vanishing Casson-Gordon obstructions.
Similarly all known smooth 4-genus bounds from gauge theory and Floer homology
vanish for $2$-bipolar knots. Moreover, our knots bound smoothly embedded height
four gropes in $D^4$, an a priori stronger condition than being $2$-solvable. We use
new lower bounds for the $4$-genus arising from $L^{(2)}$-signature defects
associated to meta-metabelian representations of the fundamental
group.
\end{abstract}
\maketitle
\section{Introduction}
A knot $K$ in $S^3$ is \emph{slice} if there exists a locally flat proper embedding
$D^2 \hookrightarrow D^4$ such that the boundary of $D^2$ is the knot $K$. This idea of `4-dimensional triviality' can be
generalized in a number of ways, perhaps most easily by approximating a disc by
a small genus surface. The \emph{$4$-genus} $g_4(K)$ of a knot $K$ in $S^3$ is
the minimal possible genus $g(\Sigma)$ of an orientable surface~$\Sigma$ with a locally flat proper embedding $\Sigma
\hookrightarrow D^4$ in the 4-ball, where
$\Sigma$ has a single boundary component whose image coincides with~$K$. From this point of view, a knot is approximately
slice if it has small 4-genus. However, this perspective does not give successively closer approximations to sliceness; there also exist many knots of $4$-genus one, such as the trefoil, which intuitively seem far from slice.
An alternative approach is to approximate the slice disc exterior $X_D := D^4 \ssm
\nu(D^2)$, a compact $4$-manifold with the three key properties that
(i)~$\partial X_D=M_K$, the 0-surgery of $S^3$ along~$K$; (ii)~the inclusion
induces an isomorphism $H_1(M_K) \cong H_1(X_D)$; and (iii)~$H_2(X_D)=0$. We
therefore think of a compact 4-manifold $W$ with $\partial W= M_K$ such that $i_* \colon H_1(M_K)\to
H_1(W)$ is an isomorphism and some condition on $H_2(W)$ is satisfied
as an approximation to a slice disc exterior. One might ask that $H_2(X_D)$ is of small rank, but a little thought shows that this essentially recovers the
4-genus condition, besides again not yielding arbitrarily refined
approximations.
In~\cite{Cochran-Orr-Teichner:1999-1}, Cochran, Orr, and
Teichner introduced a new perspective, motivated by surgery theory, in which one
allows $H_2(W)$ to be arbitrarily large but requires that it is generated by
almost disjointly embedded surfaces with a condition on the image of their fundamental groups in~$\pi_1(W)$. See
Section~\ref{section:solvable} for the precise definition. In fact, they give
an infinite family of increasingly strict conditions, indexed by $h \in \frac12
\mathbb{N}$: a knot is said to be \emph{$h$-solvable} if its 0-surgery bounds a
slice disc exterior approximation satisfying the $h$th such condition. It is an open question whether any knot which
is $h$-solvable for all $h$ must be slice, and in general knots which are
$h$-solvable for large $h$ are hard to distinguish from slice knots.
The idea of solvability is closely related to the more geometric notion of bounding a \emph{grope} of large height. A grope of height 1 is defined to be an orientable surface of arbitrary genus and a single boundary component, and a grope of height $n$ is obtained by attaching boundaries of gropes of height $n-1$ to an orientable surface along standard basis curves. We refer to
\cite{Freedman-Quinn:1990-1,Cochran-Orr-Teichner:1999-1}, or our
Section~\ref{section:height-four-gropes} for the precise definition.
A grope of larger height is a better approximation to a disc.
Gropes are ingredients of fundamental importance for the topological disc embedding technology of Freedman and Quinn~\cite{Freedman:1984-1,Freedman-Quinn:1990-1} on 4-manifolds, and also in the work of Cochran, Orr
and Teichner~\cite{Cochran-Orr-Teichner:1999-1} discussed above, where it was shown that if a knot $K$ bounds an embedded framed grope of height $h$ in $D^4$ then $K$ is $(h-2)$-solvable. The converse remains an open question.
It is natural to ask whether the 4-genus and grope/$n$-solvability approximations to sliceness have any relationship.
\begin{quest}[{\cite[Remark~5.6]{Cha:2006-1}}]
\label{quest:main}
For a fixed $h$, do there exist $h$-solvable knots, i.e.\ knots which are
close to slice in the sense of \cite{Cochran-Orr-Teichner:1999-1}, which have
arbitrarily large 4-genera, and hence are far from slice in the first sense?
\end{quest}
This question seems to be difficult, one reason for which is that existing methods for extracting lower bounds for the
topological 4-genus are not effective for $h$-solvable knots with $h\ge 2$. The simplest lower bounds are the Tristram-Levine signature function and Taylor's bound~\cite{Taylor:1979}, the best possible bound for the 4-genus coming from the Seifert form. For algebraically slice knots these lower bounds vanish.
In~\cite{Gilmer:1982-1}, Gilmer showed that there are algebraically slice knots with arbitrarily large 4-genus using Casson-Gordon
signatures~\cite{Casson-Gordon:1978-1, Casson-Gordon:1986-1}.
In~\cite{Cha:2006-1}, Cha showed that there exist knots with arbitrarily large
4-genus which are algebraically slice and have vanishing Casson-Gordon
signatures, using Cheeger-Gromov Von Neumann $L^{(2)}$ $\rho$-invariants
corresponding to metabelian fundamental group representations. The above
abelian and metabelian lower bounds can be used to give affirmative answers to
Question~\ref{quest:main} for the initial cases $h=0$,~$1$, but these lower bounds vanish for $h$-solvable knots with $h\ge 2$. Extending Cha's $\rho$-invariant approach beyond the metabelian level to give further lower bounds for the 4-genus was left open, essentially because of difficulties arising from non-commutative algebra.
In this paper, we present a new method that avoids the non-commutative algebra problem. It enables us to go one step further than Gilmer and Cha, by combining a Casson-Gordon type approach and $L^{(2)}$-signatures associated with representations to 3-solvable groups i.e.\ solvable groups with length 3 derived series. Here is our main result.
\begin{thm}
\label{thm:mainthm-intro}
For each $g \in \mathbb{N}$, there exists a 2-solvable knot $K$
with $g_4(K) > g$. Moreover, $K$ bounds an embedded framed grope of height 4 in
$D^4$.
\end{thm}
Moreover, the knots of Theorem~\ref{thm:mainthm-intro} are \emph{2-bipolar} in the sense of Cochran, Harvey
and Horn~\cite{Cochran-Harvey-Horn:2012-1}. We give the definition in
Section~\ref{section:solvable}, noting for now that the notion of bipolarity is
an approximation to being smoothly slice, which combines the idea of Donaldson's
diagonalization theorem with fundamental group information related to gropes
and derived series. Also, for a 2-bipolar knot, the invariants $\tau$,
$\Upsilon$, $\varepsilon$, $\nu^+$ from Heegaard-Floer homology, as well as the
$d$-invariants of $p/q$ surgery, all cannot prove that the knot is not smoothly
slice, and consequently cannot bound the smooth
$4$-genus~\cite{Cochran-Harvey-Horn:2012-1}. This also holds for gauge theoretic
obstructions such as those arising from Donaldson's theorem and the $10/8$
theorem.
Theorem~\ref{thm:mainthm-intro} answers the $h=2$ case of
Question~\ref{quest:main}, and prompts us to conjecture that the answer is `yes' in general. In fact, we make a bolder conjecture.
\begin{conj}\label{conj:stablegenus}
Let $K$ be an $h$-solvable knot which is not torsion in $\mathcal{C}$. Then
$\{\#^n K\}$ is a collection of $h$-solvable knots containing knots with
arbitrarily large 4-genera.
\end{conj}
A knot which did not satisfy the second sentence would be an example of a
non-torsion knot with stable 4-genus zero i.e.\ $\lim_{n \to \infty} g_4(nK)/n =
0$, and it is unknown whether any such knots exist~\cite{Livingston:2010}. Thus
a counterexample to this conjecture would also be very interesting.
One might also wonder whether there exist highly bipolar knots with large smooth
4-genus, especially with the additional requirement that they be topologically
slice. The following question seems to be unknown even in the case $h=0$.
\begin{quest}
Do there exist topologically slice $h$-bipolar knots with large smooth
4-genus?
\end{quest}
As above, there are many reasonable candidate knots with which one might hope to answer `yes.'
The main result of~\cite{Cha-Kim:2017} gave many examples of
topologically slice, $h$-bipolar knots $K$ which are of infinite order, even
modulo the subgroup of $(h+1)$-bipolar knots, and a smooth/bipolar analogue of
Conjecture~\ref{conj:stablegenus} suggests we should expect $\#^n K$ to have
arbitrarily large smooth 4-genus as $n \to \infty$.
\subsubsection*{Summary of the construction and proof}
In order to construct 2-bipolar knots bounding height four gropes, we
take connected sums of sufficiently many copies of the
seed ribbon knot $R:=11_{n74}$, and perform satellite operations on a collection
of judiciously chosen infection curves $\{\alpha_i^+,\alpha_i^-\}$, with
$\alpha_i^{\pm}$ lying in the second derived subgroup $\pi_1(S^3 \ssm R)^{(2)}$
of the knot group of the $i$th copy of $R$. Our choice of $(R, \alpha^+, \alpha^-)$ is depicted on the right side of Figure~\ref{fig:11n742}.
We use knots $\{J_i^+,J_i^-\}$ with Arf invariant zero for the companions of the satellite operations, chosen so that
the $\{J_i^+\}$ have increasingly large negative Tristram-Levine signature
functions and the $\{J_i^-\}$ have increasingly large positive signature
functions.
Let $K$ be the result of these satellite operations. In
Proposition~\ref{prop:infection2solvable}, we show that $K$ is 2-solvable; in
Proposition~\ref{prop:buildingbipolar}, we show that $K$ is 2-bipolar; and in
Proposition~\ref{prop:gropebounding}, we show that $K$ bounds a grope of
height 4 in~$D^4$. Writing $K_i$ for the knot resulting from the satellite
construction on $(R,\alpha_i^{\pm}, J_i^{\pm})$, we have $K = \#_{i=1}^N K_i$.
Let $M_{K_i}$ be the zero-surgery manifold of $K_i$ and write $Y:=
\bigsqcup_{i=1}^N M_{K_i}$.
The main idea of our proof is as follows. If there were a surface $\Sigma$ of genus $g$ embedded in $D^4$ with boundary $K$, then there would be an associated 4-manifold $Z$ with boundary $Y$ and a quotient $\Gamma$ of $\pi_1(Z)$ such that the $L^{(2)}$ $\rho$-invariant \[\rho^{(2)}(Y,\Gamma) := \rho^{(2)}(Y,\phi \colon \pi_1(Y) \to \pi_1(Z) \to \Gamma)\] would be bounded above by a constant depending only on $g$ and the base knot $R$. However, by choosing the infection knots $\{J_i^{\pm}\}$ to have suitably large Tristram-Levine signature functions, $L^{(2)}$-induction will imply that $\rho^{(2)}(Y,\Gamma)$ must be very large so long some curve $\alpha_i^{\pm}$ represents an element of $\pi_1(Y)$ mapping nontrivially to $\Gamma$. The key difficulty is to show that this must always be the case, recalling that $\Gamma$ depends on the hypothesized surface $\Sigma$.
In Example~\ref{subsection:example2solvable} we
present a slightly simpler construction of a family of $2$-solvable knots with
arbitrary 4-genera, starting with connected sums of the ribbon knot
$8_8$ and performing a single satellite construction on each copy of~$8_8$ as indicated in Figure~\ref{fig:88}.
\subsubsection*{Coefficient systems: comparison with earlier methods}
To show the nontriviality of some $\alpha_i^{\pm}$ in $\Gamma$, we use twisted
homology over a metabelian representation to define the coefficient system.
Although the representation is non-abelian, we use the ideas of Casson and
Gordon~\cite{Casson-Gordon:1986-1} to define finitely generated twisted homology
modules over a \emph{commutative} principal ideal domain. The commutativity enables us to consider the ``size'' of the twisted homology modules in terms of the minimal number of generators, generalizing the abelian representation case in
e.g.~\cite{Cha:2006-1}. Supposing that the 4-genus is small compared to the
size of the twisted first homology, we show that there is a \emph{meta-metabelian}
quotient $\Gamma$ of $\pi_1(Z)$, i.e. a quotient whose third derived subgroup vanishes, in which one of the $\alpha_i^{\pm}$ is nontrivial in order to eventually obtain a contradiction. In previous approaches to slice obstructions using $L^{(2)}$-signature defects corresponding to representations to groups with nontrivial $n$th derived subgroups for $n \geq 2$, the homology modules associated to non-abelian representations were over non-commutative rings, for which it is still unknown how to implement an analogous generating rank argument.
In our method, it is also crucial to use $L^{(2)}$-signatures
over amenable groups that are \emph{not torsion-free}, which were developed
in~\cite{Cha-Orr:2009-01, Cha:2014-1} and deployed in a similar context in~\cite{MilPow17}.
\subsubsection*{The smooth slice genus}
We remark that concordance obstructions predicated on a smooth embedding cannot
be used to draw conclusions about locally flat surfaces, and hence cannot be
used to prove our result. On the other hand our knots have arbitrarily large
smooth 4-genus, since a smooth embedding of a surface in $D^4$ is in particular
a locally flat embedding.
If we were interested in the smooth 4-genus version
of Theorem~\ref{thm:mainthm-intro}, currently known techniques using Heegaard
Floer homology or gauge theory would not apply. It is unknown whether the
Rasmussen $s$-invariant, which does provide a lower bound on the smooth 4-genus
of a knot, must vanish for 2-bipolar knots. Our knots are even the first
examples in the literature of 1-bipolar knots with large 4-genus, though for that result one could use a simpler Casson-Gordon signature argument analogous to~\cite{Gilmer:1982-1}.
\subsubsection*{Horn's results}
The fact that $g_4(K)$ is large implies that the base surface of any embedded
grope in $D^4$ with boundary $K$ must have large genus. The main theorem of
Horn~\cite{Horn:2011} gives examples, for each $g$ and each $n$, of knots
bounding height $n$ gropes such that the base surface of any height $n$ grope
must have genus at least $g$. However, Horn's example knots are not known to
have large 4-genera: he was only able to provide lower bounds on the genera of
surfaces that extend to an embedding of a height $n$ grope.
\subsubsection*{Organization of the paper}
The next four sections are concerned with background theory.
Section~\ref{section:solvable} recalls the definitions of the derived series of
a group, a useful variation called the local derived series, and what it means
for a knot to be $h$-solvable or $h$-bipolar. We also explain here how to
construct $h$-solvable and $h$-bipolar knots using the satellite construction.
Section~\ref{section:disconnected} introduces some conventions for dealing with
disconnected manifolds, in particular as relates to representations of their
fundamental groupoids and associated twisted homology groups.
Section~\ref{section:L2-invariants} recalls the Cheeger-Gromov von Neumann
$L^{(2)}$ $\rho$-invariant $\rho^{(2)}(Y,\phi)$ of a closed 3-manifold $Y$
together with a homomorphism of its fundamental group $\pi_1(Y) \to \Gamma$ to a
group $\Gamma$, and gives the facts about this invariant that we will need.
Section~\ref{section:metabelian-homology} describes homology twisted with
metabelian representations. In particular we consider coefficient systems
inspired by Casson-Gordon invariants~\cite{Casson-Gordon:1986-1}.
Section~\ref{section:statement} begins the proof of
Theorem~\ref{thm:mainthm-intro}, by precisely stating the criteria that will
imply certain knots have large topological 4-genus, giving a brief outline of
the proof, and providing examples meeting those criteria.
Section~\ref{section:controlling-homology-groups} proves some technical lemmas
that are vital in arranging that the representation used for our
$\rho$-invariant computation is suitably nontrivial. For this, we control the
size of the homology groups of certain covering spaces. In
Section~\ref{section:a-standard-cobordism} we review a standard cobordism used
in the proof of Theorem~\ref{thm:mainthm-intro}, and carefully investigate the
way metabelian representations extend over this cobordism.
Section~\ref{section:proof-of-main-theorem} proves the main theorem by bounding
the $\rho$-invariant in two different ways as described above.
Section~\ref{section:height-four-gropes} proves that our knots bound height four
embedded gropes.
\section{The solvable and bipolar filtrations}\label{section:solvable}
In this section we recall the definitions of the solvable and bipolar
filtrations, and how to construct highly solvable or bipolar knots. We will
also need, later in the article, not just the standard derived series of a group
but also the local derived series~\cite{Cochran-Harvey:2004-1,
Cochran-Harvey:2007-01, Cha:2014-1}.
\begin{defn}\label{defn:nderived}
Let $G$ be a group. The \emph{$h$th derived subgroup} $G^{(h)}$ of $G$ is
defined recursively via $G^{(0)}:=G$ and $G^{(h)}= [G^{(h-1)}, G^{(h-1)}]$ for
$h\geq1$. Moreover, for any sequence $\mathcal{S}= (S_i)_{i \in \mathbb{N}}$ of
abelian groups, define the \emph{$h$th $\mathcal{S}$-local derived subgroup} of
$G$ recursively by $G_\mathcal{S}^{(0)}:=0$ and, for $h \geq 1$,
\[
G_{\mathcal{S}}^{(h)}:= \ker\Bigl\{ G^{(h-1)}_\mathcal{S}
\to G^{(h-1)}_\mathcal{S} / [ G^{(h-1)}_\mathcal{S} , G^{(h-1)}_\mathcal{S} ]
\to \bigl( G^{(h-1)}_\mathcal{S} /
[ G^{(h-1)}_\mathcal{S} , G^{(h-1)}_\mathcal{S} ] \bigr)
\otimes_{\Z} S_{h} \Bigr\}.
\]
\end{defn}
We remark that a group $G$ is called \emph{metabelian} if $G^{(1)} \neq 0$ but $G^{(2)}=0$ and analogously \emph{meta-metabelian} if $G^{(2)} \neq 0$ but $G^{(3)}=0$. This explains some language from the introduction.
For any sequence $\mathcal{S}$ and any $h \in \mathbb{N}$ we have
that $G^{(h)} \subseteq G^{(h)}_\mathcal{S}$. Note that since for fixed $h \in \mathbb{N}$ the subgroup $G^{(h)}_\mathcal{S}$ only depends on the first $h$ terms of $\mathcal{S}$,
we will often take $\mathcal{S}=(S_1, \dots, S_h)$ to be a partial sequence.
We will be particularly interested
in $\mathcal{S}= (\Q, \Z_{p}, \Q)$ for
a prime $p$.
For $h \in \mathbb{N}_{\geq 0}$, we now define $h$-solvability of a knot. As indicated in the
introduction, there is an extension of this definition to $h \in \frac{1}{2} \mathbb{N}_{\geq 0}$.
We do not require this more general definition, and refer the reader to~\cite[Definition 1.2]{Cochran-Orr-Teichner:1999-1}
for details.
\begin{defn}\label{defn:solvable}
A knot $K$ is \emph{$h$-solvable} if there exists a compact spin 4-manifold
$W$ such that $\partial W= M_K$, the inclusion induced map $H_1(M_K) \to
H_1(W)$ is an isomorphism, and there exist embedded surfaces with trivial
normal bundle $D_1, \dots, D_k$ and $L_1, \dots, L_k$ in $W$ such that
\begin{enumerate}
\item The surfaces are pairwise disjoint except for $D_j$ and $L_j$, which
for each $j =1,\dots,k$ intersect transversely in a single point.
\item The second homology classes represented by $D_1, \dots, D_k, L_1,
\dots, L_k$ generate $H_2(W)$.
\item The inclusion induced maps $\pi_1(D_i) \to \pi_1(W)$ and $\pi_1(L_i)
\to \pi_1(W)$ have image contained in $\pi_1(W)^{(h)}$.
\end{enumerate}
\end{defn}
This gives a filtration of the knot concordance group by subgroups
$\mathcal{F}_h$ consisting of the concordance classes of $h$-solvable knots,
explored in \cite{Cochran-Orr-Teichner:1999-1, Cochran-Orr-Teichner:2002-1,
Cochran-Teichner:2003-1, Cochran-Harvey-Leidy:2009-1}, among others. Every
$1$-solvable knot is algebraically slice and every $2$-solvable knot has
vanishing Casson-Gordon invariant sliceness obstruction. In particular, as
mentioned in the introduction, the traditional 4-genus lower bounds of
Tristram-Levine and Casson-Gordon signatures cannot be usefully employed with
$2$-solvable knots.
The satellite operation interacts particularly nicely with the solvable
filtration. We remind the reader that given a knot $R$, \emph{infection curves}
$\alpha_1, \dots, \alpha_k$ in $S^3 \ssm \nu(R)$ that form an unlink in $S^3$,
and infection knots $J_1, \dots, J_k$, the \emph{satellite of $R$ by $\{J_i\}$
along $\{\alpha_i\}$} is defined to be the image of $R$ in
\[
\Big( S^3 \ssm \bigsqcup_{i=1}^k \nu(\alpha_i)\Big) \cup
\bigsqcup_{i=1}^k E_{J_i} \cong S^3,
\]
where $E_{J_i}$ is the exterior of $J_i$ and the identification is made so that
a $0$-framed longitude of $\alpha_i$, denoted by $\lambda(\alpha_i)$, is
identified with a meridian of $J_i$ and vice versa. We denote this knot
by~$R_{\alpha}(J)$. The next proposition comes from
\cite[Proposition~3.1]{Cochran-Orr-Teichner:2002-1}. We will apply it with
$h=2$ to see that the knots we construct are $2$-solvable.
\begin{prop}\label{prop:infection2solvable}
Let $R$ be a slice knot and $\{\alpha_i\}_{i=1}^k$ be a collection of
unknotted, unlinked curves in $S^3 \ssm R$ such that $[\alpha_i] \in
\pi_1(M_R)^{(h)}$ for all $i=1, \dots, k$. If for each $i=1, \dots, k$ the
knot $J_i$ has $\Arf(J_i)=0$, then $R_{\alpha}(J)$ is $h$-solvable.
\end{prop}
While our discussions have been thus far focused on the topological category, there are analogous notions of smooth sliceness, concordance, and 4-genera of knots. There is considerable interest in understanding the structure of $\mathcal{T}$, the collection of topologically slice knots modulo smooth concordance.
Here the $h$-solvable filtration is of no use, since every topologically slice knot lies in $\bigcap_{h=0}^{\infty} \mathcal{F}_h$. This prompted Cochran-Harvey-Horn to define the \emph{bipolar filtration} as follows.
\begin{defn}\label{defn:positive}
A knot $K$ is $h$-positive (respectively $h$-negative) if there exists a
smoothly embedded disc $D$ in a smooth simply connected 4-manifold $V$ such
that $\partial(V, D)= (S^3, K)$ and such that there exist disjointly embedded
surfaces $S_1, \dots, S_k$ in $V \ssm \nu(D)$ which form a basis for $H_2(V)$
such that for each $i=1, \dots k$,
\begin{enumerate}
\item The surface $S_i$ has $S_i \cdot S_i=+1$ (respectively $S_i \cdot
S_i=-1$).
\item The inclusion induced map $\pi_1(S_i) \to \pi_1(V \ssm D)$ has image
contained in~$\pi_1(V\ssm D)^{(h)}$.
\end{enumerate}
\end{defn}
Note that smoothly slice knots are $h$-positive for all $h \in \mathbb{N}$, that
the connected sum of two $h$-positive knots is $h$-positive, and that any knot
that can be unknotted by changing crossings from positive to negative (negative
to positive) is $0$-positive ($0$-negative)~\cite{Cochran-Harvey-Horn:2012-1}.
\begin{defn}
We say that a knot $K$ is \emph{$h$-bipolar} if it is both $h$-positive and $h$-negative.
\end{defn}
The following proposition, inspired by \cite[Lemma 2.3]{Cha-Kim:2017}, gives us
a way to construct $h$-bipolar knots; we will apply it when $h=2$.
\begin{prop}\label{prop:buildingbipolar}
Let $R$ be a smoothly slice knot and let $\eta^+$ and $\eta^-$ be curves in
the complement of $R$ that form an unlink in~$S^3$. Suppose that each
$\eta^{\pm}$ represents a class in $\pi_1(S^3 \ssm R)^{(h)}$, and that for
any knot $J$ we have that $R_{\eta^+}(J)$ and $R_{\eta^-}(J)$ are both
smoothly slice. Then for any $0$-positive knot $J^+$ and 0-negative knot
$J^-$, the satellite knot $R_{\eta^+, \eta^-}(J^+, J^-)$ is $h$-bipolar.
\end{prop}
\begin{proof}
Since $R_{\eta^+}(J^+)$ is slice and $J^-$ is 0-negative, the knot
\[
R_{\eta^+, \eta^-}(J^+, J^-)= \bigl(R_{\eta^+}(J^+)\bigr)_{\eta_-}(J^-)
\]
is $h$-negative by \cite[Proposition 3.3]{Cochran-Harvey-Horn:2012-1}. We see
that
\[
R_{\eta^+, \eta^-}(J^+, J^-) = \bigl(R_{\eta^-}(J^-)\bigr)_{\eta_+}(J^+)
\]
is $h$-positive by a symmetric argument.
\end{proof}
\section{Disconnected manifolds, fundamental groups and twisted homology}\label{section:disconnected}
We will need to understand the twisted homology of a connected 4-manifold $X$
with disconnected boundary $Y$. In this section, we establish some technical
details in this setting, for example by defining inclusion maps from the twisted
homology of $Y$ to that of $X$ and showing that there is a long exact sequence of
the homology of the pair $(X, Y)$. On a first reading we encourage the reader to
skim this section, focusing on the paragraph leading into
Definition~\ref{defn:disconnextension} and the statement of
Proposition~\ref{prop:les-pairs}. A similar discussion can be found in
\cite[Section~2.1]{FK06}.
We note once and for all that manifolds are oriented and either compact or
arising as an infinite cover of a compact manifold. For a manifold $V$, we
write $p \colon \widetilde{V} \to V$ for the universal cover.
Let $Y = \bigsqcup_{i=1}^N Y_i$ be a compact $\ell$-dimensional manifold with
$N$ connected components. Let $y_i \in Y_i$ be a basepoint for each connected
component. Let $S$ be a ring with unity and let $A$ be a left $S$-module. A
representation $\Phi$ of the fundamental groupoid of $Y$ into $\Aut(A)$ is
equivalent to a homomorphism
\[
\Phi = \coprod_{i=1}^N \Phi_i \colon
\coprod_{i=1}^N \pi_1(Y_i,y_i) \to \Aut(A)
\]
from the free product of the fundamental groups of the connected components to
$\Aut(A)$. We will use the following examples.
\begin{enumerate}[(a)]
\item Let $\Gamma$ be a group. Then we will take $A = S= \Z\Gamma$, with
$\Phi_i \colon \pi_1(Y_i,y_i) \to \Gamma \subseteq \Aut(A)$, where $g \in
\Gamma$ acts on $A$ by left multiplication. We will also take
$A=\mathcal{N}\Gamma$, the group Von Neumann algebra of $\Gamma$, discussed in
Section~\ref{section:L2-invariants}.
\item The ring $S$ is a commutative PID and $A= S^r$, together with a
homomorphism
\[
\Phi_i \colon \pi_1(Y_i,y_i) \to GL_r(S) = \Aut(S^r).
\]
\end{enumerate}
For each $i$ we use the representation $\Phi_i\colon \pi_1(Y_i,y_i) \to \Aut(A)$
to give $A$ a right $\Z[\pi_1(Y_i,y_i)]$-module structure. Then we let
$C_*(\widetilde{Y}_i)$ be a cellular chain complex obtained by lifting some CW decomposition of $Y_i$
(or a CW complex homotopy equivalent to $Y_i$ in the case that $Y_i$ is a
topological 4-manifold), and define the homology of $Y$ twisted by $\Phi$ to be
\[
H_*(Y;A) := \bigoplus_{i=1}^N
H_*\bigl(A \otimes_{\Z[\pi_1(Y_i,y_i)]} C_*(\widetilde{Y}_i)\bigr).
\]
Now suppose that $Y = \partial X$, where $X$ is a compact connected
$(\ell+1)$-dimensional manifold with $\partial X=Y = \bigsqcup_{i=1}^N Y_i$. A
schematic of a similar situation is shown in Figure~\ref{fig:diagramzz}. Let $x
\in X$ be a basepoint and let $\tau_i \colon [0,1] \to X$ be a path from $x$ to
$y_i$. The paths $\tau_i$ induce homomorphisms $\iota_i \colon \pi_1(Y_i,y_i)
\to \pi_1(X,x)$, by $\gamma \mapsto \tau_i \gamma \overline{\tau_i}$.
\begin{defn}\label{defn:disconnextension}
We say that $\Phi \colon \coprod_{i=1}^N \pi_1(Y_i,y_i) \to \Aut(A)$
\emph{extends over $X$} if there is a homomorphism $\Psi \colon \pi_1(X,x) \to
\Aut(A)$ such that $\Psi \circ \iota_i = \Phi_i$ for each $i=1,\dots,N$.
\end{defn}
Use the inclusion $j_i\colon Y_i \to X$ to define the pullback cover of $Y_i$ in
terms of the universal cover of $X$ via the diagram:
\[
\xymatrix{
\widetilde{Y}_i^X \ar[r] \ar[d] & \widetilde{X} \ar[d]^p \unskip, \ignorespaces
Y_i \ar[r]_{j_i} & X
}
\]
The pullback $\widetilde{Y}_i^X$ is given by pairs $\{(y, \widetilde{x}) \in Y_i \times
\widetilde{X} \mid j_i(y) = p(\widetilde{x})\}.$ Apply the action of the group
$\pi_1(X,x)$ on $\widetilde{X}$ to the second factor to obtain an action of
$\pi_1(X,x)$ on $\widetilde{Y}_i^X$. This is defined since the action on $\widetilde{X}$ is
equivariant with respect to~$p$. The action of $\pi_1(X,x)$ on $\widetilde{Y}_i^X$
induces an action of $\Z[\pi_1(X,x)]$ on the chain complex $C_*(\widetilde{Y}_i^X)$.
\begin{lem}\label{lemma:pullback-covers}
We have a homeomorphism
\[
\pi_1(X,x) \times_{\pi_1(Y,y_i)} \widetilde{Y}_i \cong \widetilde{Y}_i^X,
\]
where by definition the left hand side means:
\[
\pi_1(X, x) \times \widetilde{Y_i}/ \big((\gamma, \widetilde{y}) \sim (\gamma', \widetilde{y}')
\text{ if there is } g \in \pi_1(Y, y_i) \text{ such that }
\gamma \iota_i(g) = \gamma' \text{ and } g \cdot \widetilde{y}= \widetilde{y}'\big).
\]
\end{lem}
\begin{proof}
Start with the covering space $\pi_1(Y_i,y_i) \to \widetilde{Y}_i \to Y_i$ with fibre
$\pi_1(Y_i,y_i)$, and then apply the `product over $\pi_1(Y_i,y_i)$'
construction to obtain a covering space $\pi_1(X,x) \times_{\pi_1(Y_i,y_i)}
\pi_1(Y_i,y_i) \to \pi_1(X,x) \times_{\pi_1(Y_i,y_i)} \widetilde{Y_i} \to Y_i.$ Since
$\pi_1(X,x) \times_{\pi_1(Y_i,y_i)} \pi_1(Y_i,y_i) \cong \pi_1(X,x)$ as
discrete spaces and affine sets over $\pi_1(X,x)$, this fibre bundle is
homeomorphic to
\[
\pi_1(X,x) \to \pi_1(X,x) \times_{\pi_1(Y_i,y_i)} \widetilde{Y_i} \to Y_i.
\]
Since both $\pi_1(X,x) \times_{\pi_1(Y_i,y_i)} \widetilde{Y_i}$ and $\widetilde{Y}^X$ are
covering spaces of $Y_i$ corresponding to the homeomorphism $\iota_i \colon
\pi_1(Y_i,y_i) \to \pi_1(X,x)$, they are homeomorphic by the classification of
covering spaces.
\end{proof}
It follows from Lemma~\ref{lemma:pullback-covers} that we have an chain
isomorphism $\Z[\pi_1(X,x)] \otimes_{\Z[\pi_1(Y_i,y_i)]} C_*(\widetilde{Y}_i) \cong
C_*(\widetilde{Y}_i^X)$. Consider the sequence of chain maps:
\begin{align*}
A \otimes_{\Z[\pi_1(Y_i,y_i)]} C_*(\widetilde{Y}_i) & \xrightarrow{\cong}
A \otimes_{\Z[\pi_1(X,x)]} \Z[\pi_1(X,x)] \otimes_{\Z[\pi_1(Y_i,y_i)]}
C_*(\widetilde{Y}_i)
\unskip, \ignorespaces
& \xrightarrow{\cong}
A \otimes_{\Z[\pi_1(X,x)]} C_*(\widetilde{Y}^X_i)
\unskip, \ignorespaces
& \xrightarrow{\hphantom{\cong}}
A \otimes_{\Z[\pi_1(X,x)]} C_*(\widetilde{X}).
\end{align*}
The first map sends $a \otimes c \mapsto a \otimes 1 \otimes c$. The second map
uses the isomorphism discussed above, and the third map is induced by~$j_i$.
This chain level map induces a map on homology $(j_i)_*\colon H_*(Y_i;A) \to
H_*(X;A)$, which in turn induces
\[
\bigoplus_{i=1}^N (j_i)_* \colon \bigoplus_{i=1}^N H_*(Y_i;A)
\cong H_*(Y;A) \to H_*(X;A).
\]
Let $\widetilde{Y}^X := \bigsqcup_{i=1}^N \widetilde{Y}_i^X$. Then by identifying $\widetilde{Y}^X$
with its image in $\widetilde{X}$ we also have relative twisted homology groups
\[
H_*(X,Y;A) := H_*(A \otimes_{\Z[\pi_1(X,x)]} C_*(\widetilde{X},\widetilde{Y}^X)).
\]
The chain maps above fit into a short exact sequence of chain complexes
\[
0 \to \bigoplus_{i=1}^N A \otimes_{\Z[\pi_1(Y_i,y_i)]} C_*(\widetilde{Y}_i)
\to A \otimes_{\Z[\pi_1(X,x)]} C_*(\widetilde{X})
\to A \otimes_{\Z[\pi_1(X,x)]} C_*(\widetilde{X},\widetilde{Y}^X) \to 0.
\]
That this is exact follows from the chain isomorphism
\[
A \otimes_{\Z[\pi_1(Y_i,y_i)]} C_*(\widetilde{Y}_i)
\cong A \otimes_{\Z[\pi_1(X,x)]} C_*(\widetilde{Y}^X_i).
\]
The short exact sequence of chain complexes gives rise to a long exact sequence
in homology, which we record in the next proposition.
\begin{prop}\label{prop:les-pairs}
With a fixed choice of paths $\{\tau_i\}$ and a representation $\Phi\colon
\coprod_{i=1}^n \pi_1(Y_i, y_i) \to \Aut(A)$ that extends over $X$, there is a
long exact sequence in twisted homology
\[
\cdots \to H_k(Y;A) \to H_k(X;A) \to H_k(X,Y;A) \to H_{k-1}(Y;A) \to \cdots
\]
with $H_k(Y;A) \to H_k(X;A)$ the inclusion induced map discussed above.
\end{prop}
In later sections we work with many different representations of a given
fundamental group(oid), and so we emphasize the representation $\Phi$ rather
than the module $A$ by writing $H_k^\Phi(Y)$ for $H_k(Y; A)$.
\section{$L^{(2)}$-signature invariants}\label{section:L2-invariants}
In this section we introduce the Von Neumann $L^{(2)}$ $\rho$-invariant of a
closed (not necessarily connected) 3-manifold equipped with a representation of
its fundamental group or groupoid, and we recall the key properties of this
invariant required for the proof of Theorem~\ref{thm:mainthm-intro}. In
particular, we review the Cheeger-Gromov bound, a satellite formula, and an
upper bound in terms of the second Betti number of a bounding $4$-manifold.
\begin{defn}\label{defn:rho-inv}
Let $Y$ be a closed oriented 3-manifold, let $\Gamma$ be a discrete group, and
let $\phi \colon \pi_1(Y) \to \Gamma$ be a representation. Note that $Y$ might
be disconnected, in which case we use the conventions of
Section~\ref{section:disconnected}. Suppose that $\phi$ extends to $\Phi\colon
\pi_1(W) \to \Gamma$ where $W$ is a compact oriented 4-manifold with $\partial
W = Y$. The \emph{von Neumann $L^{(2)}$ $\rho$-invariant} of $(Y, \phi)$ is
the signature defect
\[
\rho^{(2)}(Y,\phi) = \sigma^{(2)}_\Gamma(W, \Phi)- \sigma(W),
\]
where $\sigma^{(2)}_\Gamma(W, \Phi)$ is the $L^{(2)}$-signature of the
intersection form $\lambda_{\Gamma} \colon H_2(W, \mathcal{N}\Gamma) \times
H_2(W, \mathcal{N}\Gamma) \to \mathcal{N}\Gamma$ and $\sigma(W)$ is the
ordinary signature of the intersection form on $H_2(W;\Q)$. Here the
$L^{(2)}$-signature is defined via the completion $\Z\Gamma \to
\mathbb{C}\Gamma \to \mathcal{N}\Gamma$ to the Von Neumann algebra, and the
spectral theory of operators on $\mathcal{N}\Gamma$-modules. We refer to
\cite[Section~5]{Cochran-Orr-Teichner:1999-1} and
\cite[Section~3.1]{Cha:2014-1} for more details. In particular,
$\rho^{(2)}(Y,\phi)$ only depends on the pair $(Y,\phi)$ since both the
$L^{(2)}$ signature and the ordinary signature satisfy Novikov additivity and
also $\sigma^{(2)}_\Gamma(V, \Phi)= \sigma(V)$ for a \emph{closed}
4-manifold~$V$. (See~\cite[p.~323]{Chang-Weinberger:2003-1}
and~\cite[Lemma~5.9]{Cochran-Orr-Teichner:1999-1}.)
\end{defn}
This invariant was originally defined by Cheeger and Gromov via Riemannian
geometry and $\eta$-invariants, independently of any bounding 4-manifold, so the
above definition could be taken as a proposition that the two definitions
coincide. For our purposes, as is common in the knot concordance literature, it
is simpler to take the above as the definition; for a discussion, see
\cite[Section~5]{Cochran-Orr-Teichner:1999-1} and
\cite{Cochran-Teichner:2003-1}.
\begin{exl}\label{example:abelian-rep-integral}
Let $M_J$ be the zero-framed surgery manifold of a knot $J \subset S^3$ and
let $\phi \colon \pi_1(M_J) \to \Z$ be the abelianization map. Then
\[\rho_0(J):= \rho^{(2)}(M_J,\Z) = \int_{\omega \in S^1} \sigma_{\omega}(J)
\,d \omega,\] where $\sigma_{\omega}(J)$ is the Tristram-Levine signature of
$J$ at $\omega \in S^1$, that is the signature of $(1-\omega)V +
\overline{(1-\omega)}V^T$ for $V$ a Seifert matrix of~$J$. See
\cite[Lemma~5.4]{Cochran-Orr-Teichner:1999-1} for the proof.
\end{exl}
We will need the following theorem of Cheeger and Gromov, establishing a
universal bound for the $\rho$-invariants of a fixed closed 3-manifold $Y$.
\begin{thm}[\cite{Cheeger-Gromov:1985-1}]\label{thm:cheeger-gromov-thm}
Let $Y$ be a closed oriented 3-manifold. Then there exists a constant $C$ such
that $ | \rho^{(2)}(Y, \phi) | \leq C$ for any discrete group $\Gamma$ and any
representation $\phi \colon \pi_1(Y) \to \Gamma$.
\end{thm}
We will refer to the infimum of all such constants $C$ as the Cheeger-Gromov
constant of $Y$, denoted $C(Y)$. We note that \cite{Cha:2016-CG-bounds} has
given a proof of Theorem~\ref{thm:cheeger-gromov-thm} using the signature defect
definition of $\rho^{(2)}(Y,\phi)$ given above, and has given explicit bounds
for $C(Y)$ in terms of the triangulation complexity of~$Y$.
The following proposition comes from~\cite{Cochran-Harvey-Leidy:2009-1}.
\begin{prop}\label{prop:additivity}
Let $K= R_{\alpha}(J)$ be the result of a satellite operation on a knot $R$ by
infection knots $\{J_k\}$ along infection curves $\{\alpha_k\}$. Let $\phi
\colon \pi_1(M_K) \to \Gamma$, and suppose that for some $h \in \mathbb{N}$
we have $\alpha_k \in \pi_1(M_R)^{(h)}$ for all $k$ and $\Gamma^{(h+1)}=1$.
Suppose that for all $k$, either $\phi(\alpha_k) = 1$ or $\phi(\alpha_k)$ is
infinite order in $\Gamma$. Then the restriction induced maps $\pi_1(M_R \ssm
\bigsqcup \nu(\alpha_k)) \to \Gamma$ and $\pi_1(E_{J_k}) \to \Gamma$ extend
uniquely to $\phi_0 \colon \pi_1(M_R) \to \Gamma$ and $\phi_k\colon
\pi_1(M_{J_k}) \to \Gamma$ and we have
\begin{align*}
\rho^{(2)}(M_K, \phi)&= \rho^{(2)}(M_R, \phi_0)+
\sum_{k} \rho^{(2)}(M_{J_k}, \phi_k).
\end{align*}
\end{prop}
\begin{proof}
The proof of \cite[Lemma~2.3]{Cochran-Harvey-Leidy:2009-1} applies, with the
following modification. The original statement of this proposition assumes the
additional hypothesis that $\Gamma$ is a poly-torsion-free-abelian (PTFA)
group. However, an inspection of the proof shows that we need only assume that
for each $k$ either $\phi(\alpha_k) = 1 \in \Gamma$ or $\phi(\alpha_k)$ is
infinite order. In the case that $\phi(\alpha_k) \neq 1$ in $\Gamma$, they
need in the proof of \cite[Lemma~2.3]{Cochran-Harvey-Leidy:2009-1} that
$H_1(\alpha_k;\Z\Gamma) =0$. But since $\phi(\alpha_k)$ is infinite order,
$H_1(\alpha_k;\Z\Gamma)$ is the first homology of $\mathbb{R} \times
\Gamma/\langle \alpha_k \rangle$, which vanishes.
\end{proof}
We will apply Proposition~\ref{prop:additivity} when $\Gamma= G/ G^{(3)}_{(\Q,
\Z_p, \Q)}$ for some group $G$ and $h=2$. For such $\Gamma$, any curves
$\alpha_k \in \pi_1(M_R)^{(2)}$ satisfy the hypothesis of the proposition, since
then $\phi(\alpha_k) \in \Gamma^{(2)}$ and $\Gamma^{(2)}/ \Gamma^{(3)}$ is
torsion-free.
Under the assumptions of Proposition~\ref{prop:additivity}, we have that the map
$\phi_k \colon \pi_1(M_{J_k}) \to \Gamma$ factors through the abelianization
map. To see this, note that each meridian of $J_k$ is identified with a
longitude of $\alpha_k$, which lies in $\pi_1(M_R)^{(h)}$ and hence is sent to
$\Gamma^{(h)}$. So the image of $\phi_k\colon \pi_1(M_{J^k})\to \Gamma$ is
contained in $\Gamma^{(h)}$, which is an abelian group since $\Gamma^{(h+1)}=1$.
When $\Gamma^{(h)}/ \Gamma^{(h+1)}$ is torsion-free, as occurs when $\Gamma= G/
G^{(3)}_{(\Q, \Z_p, \Q)}$ and $h=2$, we therefore have that $\phi_k$ is either
the zero map or maps onto a copy of $\Z$ in $\Gamma$. By the principle of
$L^{(2)}$-induction \cite[Proposition~5.13]{Cochran-Orr-Teichner:1999-1} and
Example~\ref{example:abelian-rep-integral}, we have that
\begin{align*}
\rho^{(2)}(M_{J_k}, \phi_k)= \begin{cases}
\rho_0(J_k) & \text{ if } \phi_k \neq 0 \unskip, \ignorespaces
0 & \text{ if } \phi_k =0.
\end{cases}
\end{align*}
Finally, note that since a meridian of $J_k$ is identified with a longitude
$\lambda(\alpha_k)$ of $\alpha_k$ in $M_K$, we have that $\phi_k$ is the zero
map if and only if $\phi(\lambda(\alpha_k))=0$. We summarize the results of the
above discussion for later use.
\begin{prop}\label{prop:ouradditivity}
Let $K=R_{\alpha}(J)$ be the result of a satellite operation on $R$ by
infection knots $\{J_k\}$ along infection curves $\{\alpha_k\}$ lying
in~$\pi_1(M_R)^{(2)}$. Let $\Gamma= G/ G_{(\Q, \Z_p, \Q)}^{(3)}$ for some
group $G$ and prime~$p$, and let $\phi\colon \pi_1(M_K) \to \Gamma$. Then the
restriction induced maps $\pi_1(M_R \ssm \bigsqcup \nu(\alpha_k)) \to \Gamma$
and $\pi_1(E_{J_k}) \to \Gamma$ extend uniquely to maps $\phi_0\colon
\pi_1(M_R) \to \Gamma$ and $\phi_k\colon \pi_1(M_{J_k}) \to \Gamma$. Moreover,
\[
\rho^{(2)}(M_K, \phi) = \rho^{(2)}(M_R, \psi_0) +
\sum_k \rho^{(2)}(M_{J_k}, \phi_k) =\rho^{(2)}(M_R, \phi_0) +
\sum_k \delta_k(\psi) \rho_0(J_k),
\]
where
\[
\delta_k(\psi)= \begin{cases} 0 &\text{ if } \psi(\lambda(\alpha_k))=0 \unskip, \ignorespaces
1 &\text{ if } \psi(\lambda(\alpha_k)) \neq 0. \end{cases}
\]
\end{prop}
The following straightforward consequence of \cite[Theorem~3.11]{Cha:2014-1}
will provide our key upper bound on $L^{(2)}$-signatures. Strebel's class of
groups $D(\Z_p)$ was defined in \cite{Strebel:1974-1}; we will not recall the
definition. We will use the fact that for any group $G$ and any $h \in
\mathbb{N}$, we have that $\Lambda = G/G^{(h)}_\mathcal{S}$ is amenable and lies
in $D(\Z_p)$ provided $S_i$ is either $\Q$ or $\Z_{p^{a_i}}$ for every $i \in
\mathbb{N}$~\cite[Lemma~6.8]{Cha-Orr:2009-01}.
\begin{thm}\label{thm:upperbound}
Let $Z$ be a 4-manifold with boundary $\partial Z=Y$ and let $\phi \colon
\pi_1(Y) \to \pi_1(Z) \to \Lambda$ be a homomorphism, where $\Lambda$ is
amenable and in Strebel's class $D(\mathbb{Z}_p)$. Then $|\rho^{(2)}(Y, \phi)|
\leq 2 \dim_{\Z_p} H_2(Z, \mathbb{Z}_p).$
\end{thm}
\begin{proof}[Proof of Theorem~\ref{thm:upperbound}]
Let $\widetilde{Z}$ be the cover of $Z$ induced by the homomorphism $\pi_1(Z)
\to \Lambda$. Since $Z$ is a compact 4-manifold with boundary, it has the
homotopy type of a finite 3-dimensional CW complex. This follows from
\cite[\textsection 1(III)]{KS69} to get a finite CW complex, combined with
\cite[Corollary 5.1]{Wa66} to restrict the dimension of the CW complex to
three. Let $C_*$ be the corresponding chain complex, and let
$\widetilde{C_*}:= C_*(\widetilde{Z})$ denote the chain complex of
$\widetilde{Z}$. Since $\Lambda$ is amenable and in
Strebel's~\cite{Strebel:1974-1} class $D(\Z_p)$,
\cite[Theorem~3.11]{Cha:2014-1} tells us that
\begin{align*}
\dim^{(2)} H_2(Z; \mathcal{N} \Lambda)
& = \dim_{\Z_p} H_2( \Z_p \otimes_{\Z \Lambda} \widetilde{C_*})
\unskip, \ignorespaces
&= \dim_{\Z_p} H_2( \Z_p \otimes_{\Z} C_*)= \dim_{\Z_p} H_2(Z; \Z_p).
\end{align*}
It follows that
\begin{align*}
|\rho^{(2)}(Y, \phi)|&= | \sigma_\Lambda^{(2)}(Z)- \sigma(Z)|
\leq \dim^{(2)} H_2(Z; \mathcal{N} \Lambda) + \dim_{\Q} H_2(Z;\Q)
\leq 2\dim_{\Z_p} H_2(Z; \Z_p).
\end{align*}
We use the universal coefficient theorem to deduce that $\dim_{\Q} H_2(Z;\Q)
\leq \dim_{\Z_p} H_2(Z; \Z_p)$ for the final inequality.
\end{proof}
\section{Metabelian twisted homology}\label{section:metabelian-homology}
In this section we review Casson-Gordon type metabelian representations of knot
groups, and the resulting twisted homology. The behavior of infection curves in
this twisted homology will be key to our proof of
Theorem~\ref{thm:mainthm-intro}.
We now let $S$ denote a commutative PID and let $Q$ denote its quotient field.
We will often take $S= \F[t^{\pm1}]$ and $Q= \F(t)$ for some field $\F$, as
well as $S= \Z$ and $Q=\Q$.
Let $X$ be a space homotopy equivalent to a finite CW-complex and let $A$ be a
left $S$-module given the structure of a right $\Z[\pi_1(X)]$-module by a
homomorphism $\phi\colon \pi_1(X) \to \Aut(A)$. Note that $\pi_1(X)$ naturally
acts on $\widetilde{C_*}$, the chain complex of the universal cover
$\widetilde{X}$ of $X$, on the left. Then as in
Section~\ref{section:disconnected}, the twisted homology $H_*(X; A)$ is defined
to be
\begin{align*}
H_*^\phi (X):= H_*(A \otimes_{\Z[ \pi_1(X)]} \widetilde{C_*}).
\end{align*}
We will be particularly interested in the following metabelian representations.
Suppose that we have a preferred surjection $\varepsilon \colon H_1(X) \to \Z$.
For every $r \in \mathbb{N}$, we let $p_r\colon \Z \to \Z_r$ be the usual
projection map and let $X^r$ denote the $r$-fold cyclic cover of $X$
corresponding to $\ker(p_r \circ \varepsilon)$. Note that covering
transformations give $H_1(X^r)$ the structure of a
$\mathbb{Z}[\mathbb{Z}_r]$-module.
Choosing a preferred element $\gamma_0 \in \pi_1(X)$ with
$\varepsilon(\gamma_0)=+1$ then gives us a map
\begin{align*}
\rho_{\gamma_0}\colon \pi_1(X) \to \Z \ltimes H_1(X^r) \quad\text{by }
\gamma \mapsto \big(t^{\varepsilon(\gamma)},
[\gamma_0^{-\varepsilon(\gamma)} \gamma]\big),
\end{align*}
where $\gamma_0^{-\varepsilon(\gamma)} \gamma \in \pi_1(X_r) \leq \pi_1(X)$ and
$[\gamma_0^{-\varepsilon(\gamma)} \gamma]$ denotes the image of
$\gamma_0^{-\varepsilon(\gamma)} \gamma$ under the Hurewicz map.
Given any choice of a homomorphism $\chi\colon H_1(X^r) \to \Z_m$, we let
$\xi_m= e^{2 \pi i/ m}$ and obtain a map $\theta_\chi \colon \Z \ltimes
H_1(X^r) \to \GL_r(\Q(\xi_m)[t^{\pm1}])$ by
\[
(t^j,a) \mapsto
\begin{bmatrix}
0& \dots &0&t \vphantom{\xi_m^{\chi(a)}}\unskip, \ignorespaces
1&0& \dots &0 \vphantom{\xi_m^{\chi(a)}}\unskip, \ignorespaces
\vdots &\ddots & \ddots & \vdots \unskip, \ignorespaces
0&\dots & 1 & 0 \vphantom{\xi_m^{\chi(a)}}
\end{bmatrix}^j
\begin{bmatrix}
\xi_m^{\chi(a)} & 0 & \dots & 0 \unskip, \ignorespaces
0&\xi_m^{\chi(t\cdot a)} & \dots& 0\unskip, \ignorespaces
\vdots&\vdots&\ddots & \vdots\unskip, \ignorespaces 0&0&\dots &\xi_m^{\chi(t^{k-1} \cdot a)}
\end{bmatrix}.
\]
We then let $S=\Q(\xi_m)[t^{\pm1}]$ and $A= \Q(\xi_m)[t^{\pm1}]^r$, noting that
$\theta_\chi \circ \rho_{\gamma_0}$ gives $A$ a right $\Z[\pi_1(X)]$-module
structure. These representations appear in \cite{Kirk-Livingston:1999-2, Let00,
Friedl:2003-4, HKL08, Friedl-Powell:2010-1}, modelled on the covering spaces
used in the definition of Casson-Gordon invariants~\cite{Casson-Gordon:1986-1}.
We refer to such representations as \emph{Casson-Gordon type representations}.
In particular, given an oriented knot $K$ and a preferred meridian $\mu \in
\pi_1(X_K)$, the canonical abelianization map $\varepsilon\colon \pi_1(X_K) \to
\Z$ has $\varepsilon(\mu)= +1$. Note that since the zero-framed longitude
$\lambda_K$ of~$K$ is an element of $\pi_1(X_K)^{(2)}$, for every $r \in
\mathbb{N}$ the map $\rho_\mu\colon \pi_1(X_K) \to \Z \ltimes H_1(X^r_K)$
extends uniquely over $\pi_1(M_K)$. The homology $H_1(X^r)$ splits canonically
as $H_1(\Sigma_r(K)) \oplus \Z$, where $\Sigma_r(K)$ is the $r$th cyclic
branched cover of $S^3$ along $K$. Our map $\chi\colon H_1(X^r) \to \Z_m$ will
always be chosen to factor through the projection map to $H_1(\Sigma_r(K))$.
In the case $r=2$ we have that $t$ must act by $-1$ on $H_1(\Sigma_2(K))~$, as discussed in the first paragraphs of~\cite{Davis95}, and so
we can conveniently decompose $\theta_\chi \circ \rho_\mu$ differently as
$\theta \circ f_\chi$, where
\begin{align*}
f_\chi \colon \pi_1(M_K) &\to \Z \ltimes \Z_m \unskip, \ignorespaces
\gamma &
\mapsto (t^{\varepsilon(\gamma)}, \chi([\mu^{-\varepsilon(\gamma)} \gamma]))
\end{align*}
and
\begin{align*}
\theta\colon \Z \ltimes \Z_m &\to \GL_2(\Q(\xi_m)[t^{\pm1}]) \unskip, \ignorespaces
(t^j, a) &\mapsto
\begin{bmatrix}
0 & t \unskip, \ignorespaces
1 & 0 \unskip, \ignorespaces
\end{bmatrix}^j
\begin{bmatrix}
\xi_m^{a} & 0 \unskip, \ignorespaces
0 & \xi_m^{-a}
\end{bmatrix}.
\end{align*}
The following proposition is a slight modification of a result of
\cite[Prop.~7.1]{MilPow17}, and gives the key connection between a certain
derived series and metabelian homology, when $m=q^s$ is a prime power.
\begin{prop}\label{prop:7-1-MP}
Let $W$ be a 4-manifold with boundary $\partial W= Y$. Let $\Phi \colon
\pi_1(W) \to \Aut(A)$ be a representation that factors through $\Z \ltimes
\Z_{q^s}$ for some prime $q$, and let $\phi\colon \pi_1(Y) \to \Aut(A)$ be the
composition of $\Phi$ with the inclusion map $\pi_1(Y) \to \pi_1(W)$. Let
$\eta \in \pi_1(Y)^{(2)}$ and suppose $\eta$ is sent to the identity in
$\pi_1(W)/\pi_1(W)^{(3)}_{(\Q, \Z_{q^s}, \Q)}$. Then for any $v \in A$ and any
$\widetilde{\eta}$, a lift of $\eta$ to the cover of $W$ induced by $\Phi$, we
have that the class $[v \otimes \widetilde{\eta}]$ in $H_1^{\phi}(Y)$ maps to
$0$ in $H_1^{\Phi}(W)$.
\end{prop}
\begin{proof}
The proof of \cite[Prop.~7.1]{MilPow17} (with its first and last sentences
deleted) applies verbatim.
\end{proof}
Finally, we recall the twisted Blanchfield form. In analogy to the linking form
on the torsion part of the ordinary first homology of a closed oriented
3-manifold, if $\phi\colon \pi_1(M_K) \to GL_r(\Q(\xi_m)[t^{\pm1}])$ arises as
above then there is a metabelian twisted Blanchfield form~\cite{MilPow17}
\[
\Bl^{\phi}\colon H_1^{\phi}(M_K) \times H_1^{\phi}(M_K)
\to \Q(\xi_m)(t)/ \Q(\xi_m)[t^{\pm1}].
\]
Note that in the above circumstance $H_1^\phi(M_K)$ is a torsion
$\Q(\xi_m)[t^{\pm1}]$-module, by the corollary
to~\cite[Lemma~4]{Casson-Gordon:1986-1}; see also~\cite{Friedl-Powell:2010-1}.
In Section~\ref{subsection:example2solvable}, we will need to know that this
form is \emph{sesquilinear}~\cite{Powell:2016-1}. That is, letting
$\widebar{\cdot}$ denote the involution of $\Q(\xi_m)[t^{\pm1}]$ induced by
sending $t \to t^{-1}$ and $a+bi \mapsto a-bi$, we have
\[
\Bl^{\phi}(px, qy)= p \widebar{q} \Bl^{\phi}(x,y)
\quad\text{for every } p, q \in \Q(\xi_m)[t^{\pm1}]
\text{ and } x, y \in H_1^{\phi}(M_K).
\]
\section{Main theorem and examples}\label{section:statement}
Here is the result that we use to show that certain satellite knots have large 4-genus.
\begin{thm}\label{thm:general}
Let $R$ be a ribbon knot and let $\eta^1, \dots, \eta^r$ be curves in $S^3
\ssm \nu(R)$ that form an unlink in $S^3$ such that each $\eta^j$ represents
an element of $\pi_1(M_R)^{(2)}$. Suppose that there is a prime $p$ such that
for every nontrivial character $\chi\colon H_1(\Sigma_2(R)) \to \Z_p$ we have
\begin{enumerate}[\upshape\selectfont(1)]
\item The module $H_1^{\theta \circ f_{\chi}}(M_R):= H_1 \left(\Q(\xi_p)[t^{\pm1}]^2 \otimes_{\Z[\pi_1(M_R)]} C_*(\widetilde{M_R}) \right)$ is nontrivial and
generated by the collection $\{[1,0] \otimes [\eta^j]\}_{j=1}^r$.
\item The order of $H_1^{\theta \circ f_{\chi}}(M_R)$ is relatively prime to
$\Delta_R(t)$ over $\Q(\xi_{p^a})(t)$ for all $a>0$.
\end{enumerate}
Let $m_R>0$ denote the generating rank of the $p$-primary part of $H_1(\Sigma_2(R))$ and let $d_R$ denote the number of distinct orders of $H_1^{\theta \circ f_{\chi}}(M_R)$ as $\chi$ ranges over all nontrivial characters from $H_1(\Sigma_2(R))$ to $\Z_p$.
Now fix $g>0$ and suppose that $N\geq \frac{4g(d_R+1)+2}{m_R}$ and
that the collection of knots $\{J_i^j\mid 1 \leq i \leq N, 1 \leq j \leq r\}$
satisfy
\begin{align*}
|\rho_0(J^j_i)|> 2(2g+N-1) + N C_R
+ \sum_{k=1}^{i-1} \sum_{\ell=1}^r |\rho_0(J^\ell_k)|
+ \sum_{\ell=1}^{j-1} |\rho_0(J^\ell_i)|
\end{align*}
for each $1 \leq i \leq N$, $1 \leq j \leq r$. Then the knot $K= \#_{i=1}^N
R_{\eta^1, \dots, \eta^r}(J^1_i, \dots, J^r_i)$ has 4-genus at least $g+1$.
\end{thm}
We remark that both $m_R$ and $d_R$ depend not only on the ribbon knot $R$ but also on the choice of prime $p$, though for convenience we suppress this from the notation. We remark also that
Theorem~\ref{thm:general} can be generalized to consider higher prime power branched covers by appropriately changing the constants; we leave the details of that to the interested reader.
For convenience, we write $\eta^j_i$ for the curve $\eta^j$ in the $i$th copy of
$R$ in $\#_{i=1}^N R$. Note that we can also write $K= \#_{i=1}^N K_i$, where
$K_i:= R_{\eta^1, \dots, \eta^j}(J^1_i, \dots, J^r_i)$. We will prove
Theorem~\ref{thm:general} in Section~\ref{section:proof-of-main-theorem} by
assuming that $g_4(K) \leq g$ and obtaining a contradiction as follows.
Under the assumption that $g_4(K) \leq g$, we construct a manifold $Z$ with $\partial Z=Y:= \bigsqcup_{i=1}^{N}M_{K_i}$, $\chi(Z)=2g$, and a few other nice properties. We then let
\[\psi \colon\pi_1(Y) \to \pi_1(Z) \to \Lambda:= \pi_1(Z)/ \pi_1(Z)^{(3)}_{(\Q, \Z_{p^a}, \Q)}\] be the map induced by inclusion. Since $\Lambda$ is amenable and in $D(\Z_p)$~\cite[Lemma~4.3]{Cha:2014-1}, Proposition~\ref{thm:upperbound} gives an upper bound on $|\rho^{(2)}(Y, \psi)|$ in terms of $g$. Our result follows from obtaining a contradictory lower bound on $|\rho^{(2)}(Y, \psi)|$. By Proposition~\ref{prop:ouradditivity} and our choices of the $J^j_i$ knots, we will obtain a contradiction if for some $i$ and $j$ we have $\lambda(\eta_i^j) \not \in \pi_1(Z)^{(3)}_{(\Q, \Z_{p^a}, \Q)}$. By Proposition~\ref{prop:7-1-MP}, this will be implied if we can construct some representation $\phi \colon \pi_1(Y) \to \Z \ltimes \Z_{p^a}$ which extends over $\pi_1(Z)$ to $\Phi$ such that for some $i$ and $j$, the
element $[1,0] \otimes [\lambda(\eta_i^j)] \in H_1^{\phi}(Y)$ is not in the kernel of the inclusion induced map $H_1^{\phi}(Y) \to H_1^{\Phi}(Z)$.
The technical work of the proof consists of showing that such a map $\phi$ must exist under the assumption that $g_4(K) \leq g$ together with our construction of $K$.\unskip, \ignorespaces
We will first give some examples of knots satisfying the hypotheses of the
theorem and then prove some technical lemmas in the next two sections. In
particular we will need to gain some control over the size of certain homology
groups, in order to show that some curve $\eta_i^j$ always survives into a
suitable 3-solvable quotient of the fundamental group of the complement of a
hypothesized locally flat embedded surface of genus~$g$. Of course
Theorem~\ref{thm:mainthm-intro} follows immediately from
Theorem~\ref{thm:general} together with the examples exhibited in
Section~\ref{subsection:example2bipolar} below and (for the grope bounding result) Proposition~\ref{prop:gropebounding}.
It is relatively easy to find examples of seed ribbon knots satisfying the
hypotheses of Theorem~\ref{thm:general}, at least with the help of a computer
program to compute twisted metabelian homology, as developed in \cite{MilPow17}.
In Section~\ref{subsection:example2solvable} we give one such example of a pair
$(R,\alpha)$, and describe the appropriate infection by knots with Arf invariant
zero and large signature in order to obtain 2-solvable large 4-genus knots.
It is a little harder to find suitable seed knots $R$ that also satisfy
Proposition~\ref{prop:buildingbipolar}, and therefore produce 2-solvable and
2-bipolar large 4-genus examples for the proof of
Theorem~\ref{thm:mainthm-intro}.. Nonetheless, we exhibit such a seed knot $R$
with suitable infection curves in Section~\ref{subsection:example2bipolar}, and
describe the appropriate infection by 0-bipolar knots with large signature in
order to obtain 2-bipolar, 2-solvable large 4-genus knots.
\subsection{Example 1: a 2-solvable knot with large 4-genus}
\label{subsection:example2solvable}
Let $R$ denote the ribbon knot $8_8$, with the unknotted curve $\alpha$ in $S^3
\ssm \nu(R)$ illustrated in Figure~\ref{fig:88}.
\begin{figure}
\caption{The knot $R=8_8$ with a grey Seifert surface and a red
infection curve~$\alpha$.}
\label{fig:88}
\end{figure}
This is the same knot-curve pair $(R,\alpha)$ as in
\cite[Example~8.1]{MilPow17}, with a slight isotopy to make it more obvious
that the curve $\alpha$ bounds a surface in the complement of a Seifert surface
for $R$, and hence lies in $\pi_1(M_R)^{(2)}$. We will need a few computations
from that paper. First, $H_1(\sr) \cong \Z_{25}$. Note that rescaling a
character $\chi: H_1(\sr) \to \Z_5$ by a nonzero element of $\Z_5$ does not
change the underlying $(\Z \ltimes \Z_5)$-covering space of $M_{R}$, and hence
preserves the isomorphism type of the twisted homology. It is therefore not
hard to check that given any nontrivial character $\chi\colon H_1(\Sig_2(R))
\to \Z_5$, the corresponding twisted homology $H_1^{\theta \circ
f_\chi}(M_{R})$ is isomorphic to $\Q(\xi_5)[t^{\pm1}]/ \langle t^2-3t+1
\rangle$. Since $\alpha$ is in $\pi_1(M_R)^{(2)}$, it lifts to a curve
$\widetilde{\alpha}$ in the $f_\chi$ covering space of $M_R$, and hence
$x:=[1,0] \otimes \widetilde{\alpha}$ is an element of $H_1^{\theta \circ
f_\chi}(M_{R})$. Finally, the metabelian twisted Blanchfield pairing
$Bl^{\theta \circ f_\chi}(x, x)$ is non-zero in $\Q(\xi_5)(t)/
\Q(\xi_5)[t^{\pm1}]$, as was computed in \cite[Example~8.1]{MilPow17}.
\begin{lem}
The element $x$ generates $H_1^{\theta\circ f_{\chi}}(M_R)$.
\end{lem}
\begin{proof}
Supposing for a contradiction that $x$ does not generate. Let $y$ be some
generator for $H_1^{\theta\circ f_{\chi}}(M_R)$. Note that $\Bl^{\theta \circ
f_\chi}(y,y)$ is of the form $\frac{q(t)}{t^2-3t+1}$ for some $q(t) \in
\Q(\xi_5)[t^{\pm1}]$. The polynomial $t^2-3t+1$ factors as $(t-w_+)(t-w_-)$
for $w_{\pm}=\frac{3\pm \sqrt{5}}{2} \in \Q(\xi_5)$. Therefore, if $x$ does
not generate then it must be homologous to $c(t-w_{*}) y$ for some $c \in
\Q(\xi_5)[t^{\pm1}]$ and $*\in \{ \pm\}$; without loss of generality, say
$*=+$. But then we can obtain a contradiction, since
\begin{align*}
\Bl^{\theta \circ f_\chi}(x,x)&= \Bl^{\theta \circ f_\chi}( c(t-w_{+}) y, c(t-w_{+}) y)\unskip, \ignorespaces
&= (t-w_+)\overline{(t-w_+)} c \overline{c} \Bl^{\theta \circ f_\chi}( y, y) \unskip, \ignorespaces
&= (t-w_+)(t^{-1}-w_+)c \overline{c} \frac{q(t)}{t^2-3t+1} \unskip, \ignorespaces
&=-t^{-1}w_+(t-w_+)(t-w_-) c \overline{c} \frac{q(t)}{t^2-3t+1} \unskip, \ignorespaces
&= -t^{-1} w_+ c \overline{c} q(t)=0 \in \Q(\xi)(t)/ \Q(\xi_5)[t^{\pm1}].\qedhere
\end{align*}
\end{proof}
Now, fix some $g \geq 0$ and let $N=8g+2$, noting that $m_R$, the generating rank of the 5-primary part of $H_1(\Sigma_2(R))$, is 1 and that there is only one isomorphism class of $H_1^{\theta \circ f_\chi}(M_R)$ and so $d_R=1$ as well. Note that $t^2-3t+1$ is relatively prime to $\Delta_R(t)=
2-6t+9t^2-6t^3+2t^4$, even considered over $\mathbb{C}$.
By \cite[Theorem~1.9]{Cha:2016-CG-bounds}, $C_R =10^9$ is an upper bound for the Cheeger-Gromov constant $C(M_R)$ of the 0-surgery on $R$. For $k=1, \dots, N$,
let $J_k$ be a knot with $\Arf(J_k)=0$ and
\begin{align} \label{eqn:choosingjknots}
\int_{\omega \in S^1} \sigma_{\omega}(J_k) \,d \omega > 2(2g+N-1)+ C_RN +
\sum_{i=1}^{k-1}\Big( \int_{\omega \in S^1} \sigma_{\omega}(J_i) \, d \omega\Big).
\end{align}
We can achieve this by taking $J_k$ to be a sufficiently large even connected
sum of negative trefoils, since for the negative trefoil
\[
\int_{\omega \in S^1} \sigma_{\omega}(J_i) \, d \omega = 4/3>0.
\]
In fact, the numerically minded reader can easily verify that
Equation~(\ref{eqn:choosingjknots}) is satisfied if we define $J_k$ to be the
connected sum of $10^{k+10} g$ negative trefoils.
We note that $K=\#_{i=1}^N R_\alpha(J_i)$ is a 2-solvable knot (by
Proposition~\ref{prop:infection2solvable}) which satisfies the hypotheses of
Theorem~\ref{thm:general}, and hence has topological 4-genus at least~$g+1$.
\subsection{Example 2: a 2-solvable, 2-bipolar knot with large 4-genus}
\label{subsection:example2bipolar}
Let $R$ be the knot depicted on the left of Figure~\ref{fig:11n74}. On the
right of Figure~\ref{fig:11n74} we see a genus 2 Seifert surface $F$ for $R$
along with two sets of \emph{derivative curves} for $R$: each of the two component links
$D=D_1 \cup D_2$ (blue) and $L= L_1 \cup L_2$ (red)
generates a half-rank summand of $H_1(F)$, forms a
slice link (in fact, an unlink) in $S^3$, and is 0-framed by~$F$.
\begin{figure}
\caption{The knot $R=11_{n74}
\label{fig:11n74}
\end{figure}
Now let $\beta_1, \beta_2 ,\gamma_1, \gamma_2$ be the curves indicated on the
left of Figure~\ref{fig:11n742}. These curves are disjoint from $F$ and hence
lie in $\pi_1(M_R)^{(1)}$. Note that the indicated basepoints should be thought
of as living in a plane `far below' the plane of the diagram; in that plane
they can be connected, uniquely up to homotopy, to a single preferred basepoint
for~$\pi_1(M_R)$.
\begin{figure}
\caption{Unknotted curves $\beta_1, \beta_2 ,\gamma_1, \gamma_2$ in
$\pi_1(M_R)^{(1)}
\label{fig:11n742}
\end{figure}
Let $\alpha^-=[\beta_1, \beta_2]$ and $\alpha^+=[\gamma_2, \gamma_2]$, where
$[v,w]=vwv^{-1}w^{-1}$; unknotted representatives for $\alpha^{\pm}$ are shown
on the right of Figure~\ref{fig:11n742}. Note that $\alpha^{\pm} \in
\pi_1(M_R)^{(2)}$. Since $\alpha^+$ has no geometric linking with either component of
the link $D$, for any knot $J$ the satellite knot
$R_{\alpha^+}(J)$ still has a smoothly slice derivative, and hence is itself
smoothly slice. Similarly, since $\alpha^-$ has no geometric linking with the
either component of the link $L$, the satellite knot $R_{\alpha^-}(J)$ is slice for every knot $J$.
Therefore, by Proposition~\ref{prop:buildingbipolar}, for any 0-positive knot
$J^+$ and 0-negative knot $J^-$, the knot $R_{\alpha^+, \alpha^-}(J^+, J^-)$ is
2-bipolar.
We now proceed to verify the conditions of Theorem~\ref{thm:general}, so that
for appropriate choices of $N \in \mathbb{N}$ and of $J^{\pm}_i$, $i=1, \dots,
N$, the knot $\#_{i=1}^N R_{\alpha^+, \alpha^-}(J_i^+, J_i^-)$ will have large
topological 4-genus.
Note that $H_1(\Sigma_2(R)) \cong \Z_3 \oplus \Z_3$, and so (up to rescaling by
a nonzero constant), there are four nontrivial characters $\chi\colon
H_1(\Sigma_2(R)) \to \Z_3$. We compute that for three of these characters, which
we call $\chi_1, \chi_2,$ and $\chi_3$, the resulting twisted homology is
\[
H_1^{\theta \circ f_{\chi_i}}(M_R)
\cong \Q(\xi_3)[t^{\pm1}]/ \langle(t-1)^2 \rangle=:A_1.
\]
For the fourth character, denoted by $\chi_4$, we compute that the twisted
homology is
\[
H_1^{\theta \circ f_{\chi_4}}(M_R) \cong \Q(\xi_3)[t^{\pm1}]
/ \langle t^2-14t+1 \rangle=:A_2.
\]
Crucially, for any nontrivial $\chi_i$ the lifts of $\alpha^+$ and $\alpha^-$ to
the cover of $M_R$ induced by $f_{\chi_i}$ generate $H_1^{\theta \circ
f_{\chi_i}}(M_R)$. (More precisely, $[1,0] \otimes [\alpha^+]$ and $[1,0]
\otimes [\alpha^-]$ generate.)
As in \cite{MilPow17}, we computed the twisted homology using a Maple program,
available for download on the authors' websites. The program obtains a
presentation for the twisted homology using the Wirtinger presentation, taking
the Fox derivatives, and then applying the representation. It then simplifies
the presentation by row and column operations to obtain a diagonal matrix.
Keeping track of how the original generators, which can be identified in the
knot diagram, are modified under the sequence of row and column operations, we
not only compute the twisted homology $H_1^{\theta \circ f_{\chi_i}}(M_R)$ but
also can identify which elements the curves $[1,0] \otimes [\alpha^{\pm}]$
represent in $H_1^{\theta \circ f_{\chi_i}}(M_R)$. Note also that the orders of
$A_1$ and $A_2$ are both relatively prime to $\Delta_R(t)= (t^2-t+1)^2$ even over $\mathbb{C}$.
Now let $g \in \mathbb{N}$ be given. By our discussion above, we have $m_R=d_R=2$, and so we let
\[N:=\frac{4g(2+1)+2}{2}= 6g+1.\]
Let $C_R$ denote an
upper bound for the Cheeger-Gromov constant $C(M_R)$. For each $i=1, \dots, N$,
successively pick $m_i$ to be even and large enough that $J_i^+:= \#^{m_i}
T_{2,3}$ satisfies
\begin{align*}
\rho_0(J_i^+)&<-2(2g+N-1)- N C_R
+ \sum_{j=1}^{i-1} \rho_0(J_i^+)- \sum_{j=1}^{i-1} \rho_0(J_i^-),
\end{align*}
and then pick $m_i'$ to be even and large enough that $J_i^-:=-\#^{m_i'}T_{2,3}$
satisfies
\begin{align*}
\rho_0(J_i^-)&> 2(2g+N-1)+ NC_R-\sum_{j=1}^{i} \rho_0(J_i^+)
+ \sum_{j=1}^{i-1} \rho_0(J_i^-) .
\end{align*}
Note that in particular $\rho_0(J_i^+)<0< \rho_0(J_i^-)$ for all~$i$.
We now let $K_i= R_{\alpha^+, \alpha^-}(J_i^+, J_i^-)$ and $K= \#_{i=1}^N
K_i$. Observe that $K$ is 2-solvable by
Proposition~\ref{prop:infection2solvable} and 2-bipolar by
Proposition~\ref{prop:buildingbipolar}; $K$ also satisfies the hypotheses of
Theorem~\ref{thm:general}, so $g_4(K) \geq g+1$. In
Section~\ref{section:height-four-gropes}, we will show that $K$ bounds an
embedded grope of height four in the 4-ball. The knot $K$ therefore gives the
example claimed in Theorem~\ref{thm:mainthm-intro}.
It now remains to prove Theorem~\ref{thm:general}.
\section{Controlling the size of some homology groups}
\label{section:controlling-homology-groups}
This section contains some technical results needed for the proof of
Theorem~\ref{thm:general}, with the theme that we need to control the size of
certain homology groups of some covering spaces.
We start this section with an elementary algebraic lemma. This lemma and the one
after it are very similar to, and are inspired by, results of Levine in
\cite{Levine:1994-1}, in particular Lemma 4.3 of Part I and Proposition~3.2 of
Part II. To avoid citing lemmas that were written for a different situation,
and for the edification of the reader, we provide short self-contained proofs.
\begin{lem}\label{lemma:algebraic-fact-square-matrices}
Let $F \colon M \to M$ be an endomorphism of a finitely generated free
$\Z[\Z_2]$-module $M$ such that
\[
\Id \otimes F \colon \Z \otimes_{\Z[\Z_2]} M \to \Z \otimes_{\Z[\Z_2]} M
\]
is an isomorphism, where $\Z$ is a $\Z[\Z_2]$-module via the trivial action of $\Z_2$ on $\Z$.
Then
\[
\Id \otimes F \colon \Q[\Z_2] \otimes_{\Z[\Z_2]} M
\to \Q[\Z_2] \otimes_{\Z[\Z_2]} M
\]
is also an isomorphism.
\end{lem}
\begin{proof}
Let $A+BT \in \Z[\Z_2]$ be the determinant of $F$, where $A,B \in \Z$ and $T
\in \Z_2$ denotes the generator. Then $A+B = \pm 1$ by hypothesis. Thus
$A^2-B^2 = (A+B)(A-B) = \pm (A-B) \neq 0$ since $A+B \equiv A-B$ modulo~$2$.
Now \[(A+BT)\cdot \frac{1}{A^2-B^2}(A-BT) = 1,\] so over $\Q[\Z_2]$ the
determinant of $F$ is invertible, and hence $\Id_{\Q[\Z_2]} \otimes F$ is an
isomorphism as desired.
\end{proof}
Next we apply this lemma to obtain some control on the size of the homology of
double covering spaces.
\begin{lem}\label{lem:chain-homotopy-lifting}
Let $f \colon X \to Y$ be a map of finite CW complexes such that
\[
f_* \colon H_i(X;\Z) \to H_i(Y;\Z)
\]
is an isomorphism for $i=0$ and a surjection for $i=1$. Let $\varphi \colon
\pi_1(Y) \to \Z_2$ be a surjective homomorphism and let $X^2,Y^2$ be the
induced 2-fold covers. Then
\[
f_* \colon H_i(X^2;\Q) \cong H_i(X;\Q[\Z_2])
\to H_i(Y^2;\Q) \cong H_i(Y;\Q[\Z_2])
\]
is also an isomorphism for $i=0$ and a surjection for $i=1$.
\end{lem}
\begin{proof}
The zeroth and first relative homology groups vanish, that is $H_i(Y,X;\Z)=0$
for $i=0,1$. Thus the cellular chain complex $(C_*(Y,X;\Z),\partial_*)$
admits a partial chain contraction: writing $C_*$ to abbreviate $C_*(Y,X;\Z)$,
the partial chain homotopy comprises maps $s_0 \colon C_0 \to C_1$ and $s_1
\colon C_1 \to C_2$ such that $\partial \circ s_0 = \Id \colon C_0 \to C_0$
and $\partial \circ s_1 + s_0 \circ \partial = \Id \colon C_1 \to C_1$.
To see this, follow the proof of the fundamental lemma of homological algebra:
for each basis element $x_i \in C_0$, choose a lift $y_i \in C_1$ with
$\partial y_i = x_i$, and define $s_0(x_i) =y_i$, and then extend linearly.
Such a $y_i$ exists since $\partial \colon C_1 \to C_0$ is surjective. Then
for each generator $z_i \in C_1$, consider $\Id(z_i) - s_0 \circ
\partial(z_i)$. Since
\[
\partial (\Id(z_i) -s_0\circ \partial(z_i)) = \partial (z_i)
- \partial \circ s_0 \circ \partial (z_i) = \partial (z_i) - \Id \circ
\partial (z_i) =0,
\]
we have that $\Id(z_i) -s_0\circ \partial(z_i)$ is a cycle. Hence there is a
$v_i \in C_2$ such that $\partial v_i = \Id(z_i) -s_0\circ \partial(z_i)$.
Define $s_1(z_i):= v_i$, and extend linearly to define $s_1$ on all of $C_1$.
Then $\partial \circ s_1 (z_i)= \partial v_i = \Id(z_i) -s_0\circ
\partial(z_i)$ for every generator $z_i$ of $C_1$. This completes the
construction of a partial chain homotopy.
Now consider the chain complex $D_*:= C_*(Y,X;\Z[\Z_2]) \cong C_*(Y^2,X^2)$,
the relative chain complex of the 2-fold covering spaces. Since the cellular
chain groups are finitely generated free modules, the partial chain
contraction $s_0,s_1$ lifts to maps $\widetilde{s}_0 \colon D_0 \to D_1$ and
$\widetilde{s}_1 \colon D_1 \to D_2$.
We claim that these maps induce a partial chain contraction after tensoring
over $\Q[\Z_2]$. To see the claim, the maps
\[
F:= \widetilde{s}_0 \circ \partial \colon D_0 \to D_0 \quad\text{and}\quad
G:= \partial \circ \widetilde{s}_1 + \widetilde{s}_0 \circ \partial \colon D_1 \to D_1
\]
are endomorphisms of the free modules $D_0$ and $D_1$ respectively, that
become automorphisms when tensored over~$\Z$. That is,
\begin{align*}
\Id \otimes F \colon \Z \otimes_{\Z[\Z_2]} D_0 &\to \Z \otimes_{\Z[\Z_2]} D_0,
\unskip, \ignorespaces
\Id \otimes G \colon \Z \otimes_{\Z[\Z_2]} D_1 &\to \Z \otimes_{\Z[\Z_2]} D_1
\end{align*}
are isomorphisms. By Lemma~\ref{lemma:algebraic-fact-square-matrices},
\begin{align*}
\Id \otimes F \colon \Q[\Z_2] \otimes_{\Z[\Z_2]} D_0
& \to \Q[\Z_2] \otimes_{\Z[\Z_2]} D_0,
\unskip, \ignorespaces
\Id \otimes G \colon \Q[\Z_2] \otimes_{\Z[\Z_2]} D_1
& \to \Q[\Z_2] \otimes_{\Z[\Z_2]} D_1
\end{align*}
are also isomorphisms. Thus \[\Id \otimes \widetilde{s}_i \colon \Q[\Z_2]
\otimes_{\Z[\Z_2]} D_i \to \Q[\Z_2] \otimes_{\Z[\Z_2]} D_{i+1}\] is a partial
chain contraction and
\[
H_i(\Q[\Z_2] \otimes_{\Z[\Z_2]} D_*)
= H_i(Y^2,X^2;\Q) \cong H_i(Y,X;\Q[\Z_2])=0
\]
for $i=0,1$. The lemma follows from the long exact sequence of the pair
~$(Y,X)$ (Proposition~\ref{prop:les-pairs}).
\end{proof}
Our next lemma requires some facts about finitely generated modules over
commutative PIDs, which we remind the reader of in order to establish notation.
\begin{defn}
Let $S$ be a commutative PID with quotient field $Q$, and let $A$ be a
finitely generated module over~$S$.
\begin{enumerate}
\item $TA:=\{ a \in A \text{ such that } sa= 0 \text{ for some } s \in S\}$,
the $S$-torsion submodule of~$A$.
\item $A^\wedge:= \Hom_S(A, Q/S)$. If $A$ is torsion (i.e.\ $A=TA$), then $A$
and $A^\wedge$ are non-canonically isomorphic.
\item Given a map of $S$ modules $f\colon A \to B$, we abbreviate
$f|_{TA}\colon TA \to TB$ by $f|_T$. We emphasize that $\coker(f|_T)$ is
therefore isomorphic to $TB/ \Imm(f|_{TA})$, not $B/ \Imm(f|_{TA})$.
\item We say that $A$ has \emph{generating rank $k$ over $S$} if $A$ is
generated as an $S$-module by $k$ elements but not by $k-1$ elements, and
write $\grr_S A=k$. It follows immediately from the definition that if $A$
surjects onto $B$ then $\grr_S B \leq \grr_S A$. It is also true and easy to
check that if $B \leq A$ then $\grr_S B \leq \grr_S A$, though this is less
obvious and uses that $S$ is a commutative~PID.
\item By the fundamental theorem of finitely generated modules over PIDs,
there exist $j, k \in \mathbb{N}$ and elements $s_1, \dots, s_k \in S$ such
that there is a (non-canonical) isomorphism \[A \cong S^j \oplus TA \cong
S^j \oplus \bigoplus_{i=1}^k S/ \langle s_i \rangle.\] When $j>0$ we say
that the \emph{order} of $A$ is $|A|=0$ and when $j=0$ we say that the order
of $A$ is $|A|= \prod_{i=1}^k s_i$. This is well-defined up to
multiplication by units in $S$. The key property of order we use is that if
$f\colon A \to B$ is a map of $S$-modules with $\ker(f)$ torsion, then
$|\Imm(f)|= |A|/|\ker(f)|$.
\end{enumerate}
\end{defn}
We will need the following basic lemma in the proof of Theorem~\ref{thm:general}, noting for future use that $\mathbb{F}[t^{\pm1}]$ is a Euclidean domain whenever $\mathbb{F}$ is a field.
\begin{lem}\label{lem:intersecttorsion}
Let $A$ be a finitely generated module over a Euclidean domain $S$, hence non-canonically isomorphic to $S^{m} \oplus TA$ for some $m \geq 0$.
Suppose that $B$ is a submodule of $A$ such there exists $C \subseteq A/B$ of generating rank $\ell>m$.
Then there exists a module $C' \subseteq TA/ (B \cap TA)$ of generating rank $\ell-m$ such that the order of $C'$ divides the order of $C$.
\end{lem}
\begin{proof}
Let $a_1, \dots, a_{\ell}$ be elements of $A$ such that $[a_1], \dots, [a_\ell]$ generate $C \subseteq A/B$. Pick a decomposition of $A\cong S^m \oplus TA$ and use it to write each $a_i= (s_i^j)_{j=1}^m \oplus \alpha_i$ for $(s_i^j)_{j=1}^m \in S^m$ and $\alpha_i \in TA$.
Since $S$ is a Euclidean domain, row-reduction of the $\ell \times m$ matrix $M$ with $M_{i,j}:=s_{i}^j$ yields a matrix $M'$ in Hermite normal form. Since $\ell>m$, we have that $M'$ contains at least $\ell-m$ rows of zeros.
By taking the corresponding linear combinations of the $a_i$, we obtain a new collection of elements $a'_i= (t_i^j)_{j=1}^m \oplus \alpha'_i$ such that the collection of $[a'_i]$ generate $C$. Moreover, for $i>m$ we have that $t_i^j=0$ for all $j=1, \dots, m$ and so $a'_i= \alpha'_i \in TA$. Note that for $i>m$ we have that $\alpha_i' \neq 0$, since if $\alpha_i=a'_i=0$ then the generating rank of $C$ would be strictly less than $\ell$. For similar reasons, we see that the generating rank of the submodule of $TA/ (B \cap TA)$ generated by $\alpha'_{m+1}, \dots, \alpha'_{\ell}$ is exactly $\ell-m$. So let $C'$ be this submodule. Finally, the order of $C'$ divides the order of $C$ because $C'$ is a submodule of $C$.
\end{proof}
We will use the next lemma twice, once in the proof of
Proposition~\ref{prop:homology}, and then again in Step 3 of the proof of
Theorem~\ref{thm:general} in Section~\ref{section:proof-of-main-theorem}.
\begin{lem}\label{lemma:moregeneralhomology}
Let $X$ be a 4-manifold with boundary $\partial X=Y$. Let $S$ be a commutative
PID with quotient field $Q$ and let $A := S^n$ for some $n \in \mathbb{N}$.
Suppose there is a representation $\Phi$ of the fundamental groupoid of $Y$
into $\Aut(A)$ that extends over~$X$, as in
Section~\ref{section:disconnected}. Consider the long exact sequence
(Proposition~\ref{prop:les-pairs}) of $S$-modules of the pair, with
coefficients taken in~$A$:
\begin{align*}
\cdots \to H_2(X) \xrightarrow{j_2} H_2(X,Y) \xrightarrow{\partial} H_1(Y) \xrightarrow{i_1} H_1(X) \xrightarrow{j_1} H_1(X,Y) \to \cdots.
\end{align*}
Suppose that $H_1(X,Y)$ is torsion. Then $\ker(j_1|_T) \cong \coker(j_2|_T)$.
\end{lem}
\begin{proof}
Unless otherwise specified, all homology groups are taken with coefficients in
$A$. First, we argue that the Bockstein homomorphism $\beta$ is an isomorphism.
This Bockstein arises in the long exact sequence of $\Ext$
groups~\cite[IV,~Prop.~7.5]{HS97} associated to the short exact sequence $0
\to S \to Q \to Q/S \to 0$, as follows:
\[
\Ext^0_S(H_1(X,Y), Q) \to \Ext^0_S(H_1(X,Y), Q/S)
\xrightarrow{\beta} \Ext^1_S(H_1(X,Y),S)\to \Ext^1_S(H_1(X,Y), Q).
\]
Since $Q$ is an injective $S$-module, $\Ext^1_S(H_1(X,Y),Q)$ vanishes, and
$\Ext^0_S(H_1(X,Y),Q)=0$ because $H_1(X,Y)$ is torsion. Thus $\beta$ is an
isomorphism.
Therefore Poincar{\'e} duality, universal coefficients, and the
(inverse of the) Bockstein homomorphism together induce natural isomorphisms
fitting into a commutative diagram:
\[
\xymatrix@[email protected]{
TH_2(X) \ar[r]^-{P.D.}_-{\cong} \ar[d]^{j_2|_T} & TH^2(X,Y) \ar[r]^-{U.C.}_-{\cong} \ar[d] &\Ext^1_S(H_1(X,Y),S) \ar[r]^-{\beta^{-1}}_-{\cong} \ar[d] &\Ext^0_S(H_1(X,Y), Q/S) \ar[r]_-{\cong} \ar[d] & TH_1(X,Y)^\wedge \ar[d]_-{(j_1|_T)^\wedge} \unskip, \ignorespaces
TH_2(X,Y) \ar[r]^-{P.D.}_-{\cong} & TH^2(X) \ar[r]^-{U.C.}_-{\cong} & \Ext^1_S(H_1(X), S)
\ar[r]^-{\beta^{-1}}_-{\cong} & \Ext^0_S(H_1(X), Q/S) \ar[r]_-{\cong} & TH_1(X)^{\wedge}
}
\]
By the naturality of the above sequence of maps, the following square commutes
and so $ \coker(j_2|_T) \cong \coker((j_1|_T)^\wedge)$.
\begin{equation}\label{eqn:squaregeneral}
\vcenter{\xymatrix @R+0.3cm @C+0.3cm {
TH_2(X) \ar[r]^{j_2|_T} \ar[d]_{\cong} & TH_2(X,Y) \ar[d]^{\cong}\unskip, \ignorespaces
TH_1(X,Y)^{\wedge} \ar[r]^-{(j_1|_T)^\wedge} & TH_1(X)^{\wedge}
}}
\end{equation}
Now let $H := \ker(j_1|_T) \leq TH_1(X)$ and define $\Phi\colon
\coker((j_1|_T)^\wedge) \to H^\wedge$ by $\Phi([f])= f|_{H}.$ Observe that
$\Phi$ is well-defined, since for any $g \in TH_1(X,Y)^\wedge$ we have
\[
(j_1|_T)^\wedge(g)(x)= g(j_1(x))= g(0)=0
\quad\text{for all } x \in H= \ker(j_1|_T).
\]
Also, $\Phi$ is onto: given any $f \in H^\wedge$ (i.e.\ a map $f\colon H \to
Q/S$), since $H \leq TH_1(X)$, and using that $Q/S$ is an injective
$S$-module, we can extend $f$ to a map $g\colon TH_1(X) \to Q/ S$, and will
have that $\Phi([g])=f$. Therefore, in order to show that $\Phi$ is an
isomorphism it is enough to show that $ |\coker((j_1|_T)^\wedge)|=
|H^\wedge|$. Note that
\begin{equation}\label{eqn:coker1general}
| \coker((j_1|_T)^\wedge)|= \frac{|TH_1(X)^\wedge|}{|\Imm((j_1|_T)^\wedge)|}
= \frac{ |TH_1(X)^\wedge|\,|\ker((j_1|_T)^\wedge)|}{|TH_1(X,Y)^\wedge)|}.
\end{equation}
Also, $(j_1|_T)^{\wedge}(f)=0$ if and only if $f(j_1|_T(x))=0$ for all $x \in
TH_1(X)$ if and only if $f$ vanishes on $\Imm(j_1|_T)$, so
\[
|\ker((j_1|_T)^\wedge)|= \frac{|TH_1(X,Y)^\wedge)|}{|\Imm(j_1|_T)|}.
\]
Therefore we can rewrite Equation~(\ref{eqn:coker1general}) as
\[
|\coker((j_1|_T)^\wedge)|= \frac{|TH_1(X)^\wedge|}{|\Imm(j_1|_T)|}
=\frac{|TH_1(X)|}{|\Imm(j_1|_T)|} = |H|= |H^{\wedge}|.
\]
So $\Phi\colon \coker((j_1|_T)^\wedge) \to \ker(j_1|_T)^\wedge$ is an
isomorphism. Since $\ker(j_1|_T)$ is a torsion $S$-module, there is an (albeit
non-canonical) identification $\ker(j_1|_T) \cong (\ker(j_1|_T))^\wedge$, and
so we have as desired that
\[
\coker(j_2|_T) \cong \coker((j_1|_T)^\wedge)
\cong_\Phi \ker(j_1|_T)^\wedge \cong \ker(j_1|_T).
\qedhere
\]
\end{proof}
Note that in particular this implies that $\ker(j_1|_T)_p \cong
\coker(j_2|_T)_p$ for any prime $p$, where for a $\Z$-module $G$ we write $G_p$
for the $p$-primary part.
Next we apply the control on homology of double covers gained in Lemma~\ref{lem:chain-homotopy-lifting} along with the homological algebra of Lemma~\ref{lemma:moregeneralhomology} to manifolds $M^3$ and $V^4$.
\begin{prop}\label{prop:homology}
Let $M$ be a homology $S^1 \times S^2$, and let $V$ be a connected 4-manifold with
boundary $M$ such that the inclusion induced map $H_1(M) \xrightarrow{i_*}
H_1(V)$ an isomorphism. Suppose that $H_1(M^2) \cong \Z \oplus G$, where $G$
is torsion. Then for any prime $p$ the $p$-primary part of $TH_1(M^2)/
\ker(TH_1(M^2) \to TH_1(V^2))$ has generating rank at least
$n:=\frac{m-2\chi(V)}{2}$, where $m$ denotes the generating rank of the
$p$-primary part of $TH_1(M^2)$.
\end{prop}
\begin{proof}
Observe that $H_i(V, M; \Z)=0$ for $i=0, 1$. It follows from
Lemma~\ref{lem:chain-homotopy-lifting} that
\[
H_i(V^2, M^2; \Q) \cong H_i(V, M; \Q[\Z_2]) =0 \quad\text{for } i=0,1.
\]
Therefore $\dim H_1(V^2, \Q) \leq \dim H_1(M^2, \Q) =1$ and
\[
\dim H_3(V^2; \Q)= \dim H^3(V^2; \Q)= \dim H_1(V^2, M^2; \Q)=0.
\]
Also note that
\[
2\chi(V)=\chi(V^2)= 1 - \dim H_1(V^2, \Q)+ \dim H_2(V^2, \Q).
\]
We therefore have that $\dim H_2(V^2, \Q) = 2\chi(V) + \dim H_1(V^2, \Q) - 1$
is at most~$2\chi(V)$. Now consider the following long exact sequence:
\[
\dots \to H_2(V^2) \xrightarrow{j_2} H_2(V^2, M^2)
\xrightarrow{\partial} H_1(M^2) \xrightarrow{i_1} H_1(V^2)
\xrightarrow{j_1} H_1(V^2, M^2) \dots.
\]
From above we have $H_1(V^2, M^2; \Q)=0$, and so by
Lemma~\ref{lemma:moregeneralhomology} we have that $\grr \ker(j_1|_T)_p= \grr
\coker(j_2|_T)_p$. Moreover, for any finitely generated abelian group $A$, we have that
$A \cong \Z^{a} \oplus \, TA$ for some $a \in \mathbb{N}_{\geq 0}$ and hence
that $\grr(A \otimes \Z_p)= a + \grr(A_p)$. In particular $\grr(\ker(j_1|_T) \otimes
\Z_p)= \grr (\coker(j_2|_T) \otimes \Z_p)$.
Combining $\grr(A \otimes \Z_p)= a + \grr(A_p)$ with $H_1(M^2) \cong \Z \oplus
TH_1(M^2)$, we have that in order to show as desired that the generating rank of the
$p$-primary part of $\left(TH_1(M^2)/ \ker(i_1|_T)\right)$ is at least $n$, it
suffices to show
\[
\grr \left( (H_1(M^2)/ \ker(i_1)) \otimes \Z_p \right) \geq n+1.
\]
Note that $H_1(M^2)/ \ker(i_1)\cong \Imm(i_1) \cong \ker(j_1)$, so if
$\grr\left(\ker(j_1) \otimes \Z_p\right) \geq n+1$ then we are done.
Similarly, since $\grr\left(H_1(M^2) \otimes \Z_p\right) = m+1$ and
\[
\ker(i_1)= \Imm(\partial) \cong H_2(V^2, M^2)/ \ker(\partial)
= H_2(V^2, M^2)/ \Imm(j_2)= \coker(j_2),
\]
if $\grr \left(\coker(j_2) \otimes \Z_p\right) \leq m-n$, then we are also
done.
So suppose for a contradiction $\grr\left( \coker(j_2) \otimes \Z_p\right)>
m-n$ and $\grr\left( \ker(j_1)\otimes \Z_p \right) \leq n$. Note that since
$H_3(V^2,M^2) \cong H^1(V^2) \cong H^1(M^2) \cong H_2(M^2)$, the ranks of
$H_2(V^2,M^2)$ and $H_2(V^2)$ coincide and so $H_2(V^2,M^2)$ splits (albeit
non-canonically) as $\Z^{b_2(V^2)} \oplus TH_2(V^2,M^2)$. Thus we obtain our
desired contradiction as follows:
\begin{align*}
2\chi(V)+n=m-n&< \grr \left( \coker(j_2) \otimes \Z_p \right) \unskip, \ignorespaces
&\leq \grr \left((H_2(V^2,M^2)/ \Imm(j_2|_{T})) \otimes \Z_p \right)\unskip, \ignorespaces
&= \grr \left(\Z^{b_2(V^2)} \otimes \Z_p \right)
+ \grr \left( (TH_2(V^2, M^2)/ \Imm(j_2|_T)) \otimes \Z_p\right) \unskip, \ignorespaces
&= b_2(V^2) + \grr \left(\coker(j_2|_T )\otimes \Z_p\right) \unskip, \ignorespaces
&= b_2(V^2)+ \grr \left(\ker(j_1|_T) \otimes \Z_p\right)\unskip, \ignorespaces
&\leq b_2(V^2)+ \grr \left(\ker(j_1) \otimes \Z_p\right)\leq 2\chi(V)+n.
\qedhere
\end{align*}
\end{proof}
\section{A standard cobordism}\label{section:a-standard-cobordism}
In this section we study a standard cobordism $U$ between the zero-framed
surgery manifold of a connected sum of knots $M_K=M_{\#_{i=1}^N K_i}$ and the
disjoint union $Y:=\bigsqcup_{i=1}^N M_{K_i}$ of the zero-framed surgery
manifolds of the summands $K_i$. In particular we need to understand the
behavior of certain representations of the fundamental groups. We will also
explicitly choose the basepoints $\{x_i\}$ and paths $\{\tau_i\}$ necessary to
define twisted homology for disconnected manifolds, as discussed in
Section~\ref{section:disconnected}.
\begin{figure}
\caption{A Kirby diagram for $U$.}
\label{fig:upart1}
\end{figure}
Let $U'$ be $M_K \times [0,1]$ with $(N-1)$ 0-framed 2-handles attached along
`longitudes of $K_i$.' A schematic of a relative Kirby diagram for $U'$ is given
by the black and blue curves of Figure~\ref{fig:upart1}. Note that we depict
each $K_i$ as the boundary of a Seifert surface $G_i$, and hence $K=\#_{i=1}^N
K_i$ as the boundary of $\natural_{i=1}^N G_i$. Since repeatedly sliding the
black 0-framed curve over the blue curves gives the standard surgery diagram for
$Y'$, we have $\partial_+(U')= Y':= \#_{i=1}^N M_{K_i}$. Now let $U''$ be $Y'
\times [1,2]$ together with $(N-1)$ 3-handles attached along 2-spheres (whose
outline is indicated in green in Figure~\ref{fig:upart1}) so that $\partial_+
U''=Y= \bigsqcup_{i=1}^N M_{K_i}$. Let $U= U' \cup_{Y'} U''$.
We now consider the points, arcs, and closed curves shown in
Figure~\ref{fig:ualex}.
\begin{figure}
\caption{Basepoints $x_i$, arcs $\kappa_i$, and meridians $\mu_i$ for
$i=1, \dots, N$ and closed curves $\delta_j$ for $j=1, \dots, m$ in $M_K$.}
\label{fig:ualex}
\end{figure}
Note that the curves $\delta_j$ for $j=1, \dots, m$ form a normal generating set
for the first commutator subgroup of $\pi_1(M_K, x_N)$, when suitably based
using the arcs $\kappa_i$.
The attaching regions for the 2-handles of $U'$ avoid
\[
\bigcup_{i=1}^N \mu_{K_i} \cup \bigcup_{i=1}^{N-1} \kappa_i
\cup \bigcup_{j=1}^m \delta_j \subset M_K,
\]
and so the points $x_i':= x_i \times \{1\}$, arcs $\kappa_i':= \kappa_i \times
\{1\}$, and loops $\mu_{K_i}':= \mu_{K_i} \times \{1\}$ and $\delta_j':=
\delta_j \times \{1\}$ lie in $\partial_+U'=Y'$ for all $i=1, \dots, N$ and
$j=1, \dots, m$. Similarly, the attaching regions for the 3-handles of $U''$
avoid
\[
\Big( \bigcup_{i=1}^N \mu_{K_i}' \cup \bigcup_{j=1}^m \delta_j' \Big)
\subset Y',
\]
and so the loops $\mu_{K_i}'':= \mu_{K_i}' \times \{2\}$ and
$\delta_j'':=\delta_j' \times \{2\}$ lie in $\partial_+U''=Y$ for all $i=1,
\dots, N$ and $j=1, \dots, m$. For each $i=1, \dots N$ we have an
inclusion-induced map
\[
\iota_i \colon \pi_1(M_{K_i}, x_i' \times \{2\}) \to \pi_1(U'', x_N')
\quad\text{by }
\beta\mapsto \kappa_i' \cdot (x_i' \times [1,2]) \cdot \beta \cdot
\overline{(x_i' \times [1,2])} \cdot \overline{\kappa_i'}.
\]
Let $U=U' \cup_{Y'} U''$, and note that we also have an inclusion-induced map
\[
\iota\colon \pi_1(U'', x_N') \to \pi_1(U, x_N)
\quad\text{by }
\gamma \mapsto (x_N \times [0,1]) \cdot \gamma
\cdot \overline{(x_N \times [0,1])}.
\]
In the language of Section~\ref{section:disconnected}, $\iota \circ \iota_i$ is
induced by the path from $x_N$ to $x_i''$ given by
\[
\tau_i= (x_N \times [0,1]) \cdot \kappa_i' \cdot (x_i' \times [1,2]).
\]
We return to using the notation from Section~\ref{section:metabelian-homology} in order to state and prove the following.
\begin{prop}\label{prop:standardu}
Let $K= \#_{i=1}^N K_i$ and $U$ be the standard cobordism from $M_K$ to
$\bigsqcup_{i=1}^N M_{K_i}$ as above. Let $p \in \mathbb{N}$ and choose maps
$\chi_i\colon H_1(\Sigma_2(K_i)) \to \mathbb{Z}_p$ for $i=1$, \dots,~$N$, so
$(\chi_i)_{i=1}^N\colon H_1(\Sigma_2(K)) \to \Z_p.$ Let $\mu_{K_N}$ be the
preferred meridian for $K$ and for $i=1$, \dots,~$N$ let $\mu_{K_i}''$ be
the preferred meridian for $K_i$. Then the map
\[
f^{K}_{(\chi_i)_{i=1}^N}\colon \pi_1(M_K, x_N) \to \Z \ltimes \Z_p
\]
extends uniquely to a map $F\colon \pi_1(U, x_N) \to \Z \ltimes \Z_p$. Also,
the composition
\[
f_i\colon \pi_1(M_{K_i}, x_i' \times \{2\}) \xrightarrow{\iota_i}
\pi_1(U'', x_N') \xrightarrow{\iota} \pi_1(U, x_N) \xrightarrow{F}
\Z \ltimes \Z_p
\]
satisfies $f_i= f^{K_i}_{\chi_i}$.
\end{prop}
\begin{proof}
Notice that $\pi_1(U)= \pi_1(M_K)/ \langle \lambda_{K_1}, \dots,
\lambda_{K_{N-1}} \rangle$. Therefore, since each $\lambda_{K_i}$ bounds a
subsurface of $\natural_{i=1}^N G_i$ and hence lies in $ \pi_1(M_K)^{(2)}$,
the map $f^K_{(\chi_i)_{i=1}^N}$ extends uniquely as desired.
Observe that for all $i=1, \dots, N$ we have
\begin{align*}
\iota(\iota_i( \mu_{K_i}''))&= \iota( \kappa_i' \cdot (x_i' \times [1,2])
\cdot \mu_{K_i}'' \cdot \overline{(x_i' \times [1,2])}
\cdot \overline{\kappa_i'})
\unskip, \ignorespaces
& = \iota(\kappa_i' \cdot \mu_{K_i}'\cdot \overline{\kappa_i'})
\unskip, \ignorespaces
&=(x_N \times [0,1]) \cdot \kappa_i'\cdot \mu_{K_i}'
\cdot \overline{\kappa_i'} \cdot \overline{(x_N \times [0,1])}
= \mu_{K_N} \in \pi_1(U, x_N).
\end{align*}
Therefore
\[
f_i(\mu_{K_i}'')= F(\mu_{K_N})= f^{K}_{(\chi_i)_{i=1}^N}(\mu_{K_N})
= (t,0)= f_{\chi_i}^{K_i}(\mu_{K_i}'') \in \Z \ltimes \Z_p.
\]
For each $i=1, \dots N$, every element $\gamma \in \pi_1(M_{K_i}, x_i'')$ can
be written as $\gamma= (\mu_{K_i}'')^{\varepsilon(\gamma_i)} a$ for some
element $a \in \pi_1(M_{K_i}, x_i'')^{(1)}$. Moreover, the collection of
$\delta_j''$ corresponding to $K_i$ in Figure~\ref{fig:ualex} normally
generate $ \pi_1(M_{K_i}, x_i'')^{(1)}$. It therefore suffices to check that
$f_i$ agrees with $f^{K_i}_{\chi_i}$ on $\mu_{K_i}''$ (as done above) and on
the collection of $\delta_j''$ corresponding to~$K_i$.
Supposing that $\delta_j''$ corresponds to $K_{i}$, we have that
\begin{align*}
\iota(\iota_i( \delta_j''))&= \iota( \kappa_i' \cdot (x_i' \times [1,2])
\cdot \delta_j'' \cdot\overline{(x_i' \times [1,2])} \cdot\overline{\kappa_i'})
\unskip, \ignorespaces
& = \iota(\kappa_i' \cdot \delta_j'\cdot \overline{\kappa_i'})
\unskip, \ignorespaces
&=(x_N \times [0,1]) \cdot \kappa_i'\cdot \delta_j'
\cdot \overline{\kappa_i'} \cdot \overline{(x_N \times [0,1])}
\unskip, \ignorespaces
&= \kappa_i \delta_j \overline{\kappa_i} \in \pi_1(U, x_N).
\end{align*}
Now fix a lift $\widetilde{x_N}$ of $x_N$ to $M_K^2$, the double cover of
$M_K$. Since $x_N$ does not lie in a tubular neighborhood of $K$, we can
think of $\widetilde{x_N}$ as lying in $E_K^2\subseteq \Sigma_2(K)$ as well.
The inclusion induced maps $M_K \to U$ and $M_{K_i} \to U$ induce isomorphisms
on first homology, and so the double cover $U^2$ is a cobordism from $M_K^2$
to $\bigsqcup_{i=1}^N M_{K_i}^2$. For each $i=1$, \dots, $N$, lifting the arc
\[
\kappa_i \cdot (x_i \times [0,1]) \cdot (x_i' \times [1,2])
\]
to $U^2$ starting at $\widetilde{x_N}$ gives a preferred basepoint
$\widetilde{x_i''}$ in $M_{K_i}^2$. As before, we also think of this basepoint
as lying in $E_{K_i}^2 \subseteq \Sigma_2(K_i)$. We can therefore speak of
\emph{the} lift $\widetilde{\gamma}$ of a curve $\gamma$ based at $x_N$
(respectively, $x_i''$) to $\Sigma_2(K)$ (respectively, $\Sigma_2(K_i)$) by
choosing the lift with basepoint $\widetilde{x_N}$ (respectively,
$\widetilde{x_i''}$).
\begin{remark}
A choice of basepoint is technically always necessary to define $f^K_\chi$,
though this was suppressed in Section~\ref{section:metabelian-homology} in
our discussion of the connected case, where it was less important.
\end{remark}
Therefore
\begin{align*}
f_i(\delta_j'') = F( \kappa_i \delta_j \overline{\kappa_i})
= f^{K}_{(\chi_k)_{k=1}^N}(\kappa_i \delta_j \overline{\kappa_i})
=(0, (\chi_k)_{k=1}^N(\widetilde{\kappa_i \delta_j \overline{\kappa_i}})).
\end{align*}
Similarly,
\begin{align*}
f_{\chi_i}^{K_i}(\delta_j'')= (0, \chi_i(\widetilde{\delta_j''})).
\end{align*}
It therefore only remains to show that
\[
(\chi_k)_{k=1}^N(\widetilde{\kappa_i \delta_j \overline{\kappa_i}})=
\chi_i(\widetilde{\delta_j''}).
\]
First, note that $\chi_k(\widetilde{\kappa_i \delta_j \overline{\kappa_i}})=0$
unless $k=i$. Also, the homology class of $\widetilde{\kappa_i \delta_j
\overline{\kappa_i}}$ in
\[
H_1(\Sigma_2(K_i)) \subset \bigoplus_{i=1}^N H_1(\Sigma_2(K_i))
\cong H_1(\Sigma_2(K)
\]
is exactly the same as that of $\widetilde{\delta_j''}$ in
$H_1(\Sigma_2(K_i))$, and so we have that
\[
\chi_i(\widetilde{\kappa_i \delta_j \overline{\kappa_i}})
= \chi_i(\widetilde{\delta_j''}).\qedhere
\]
\end{proof}
We note for later use that the inclusion induced maps $M_K \to U$ and $M_{K_i}
\to U$ give isomorphisms on first homology, and that $H_2(U) \cong \Z^{N}$ and
$H_3(U) \cong \Z^N$.
\section{Proof of Theorem~\ref{thm:general}}\label{section:proof-of-main-theorem}
Since the proof of Theorem~\ref{thm:general} is rather long, for the reader's convenience we outline the main steps of the argument, with references to key results from elsewhere in the paper.
\begin{enumerate}
\item (Proposition~\ref{prop:step1}.) Construct a 4-manifold $V$ with boundary $\partial V=M_K$ such that the inclusion
induced map $H_1(M_K) \to H_1(V)$ is an isomorphism and $H_2(V) \cong \Z^{2g}$.
Let $U$ denote the standard cobordism between $M_K$ and $Y:=
\bigsqcup_{i=1}^{N}M_{K_i}$ discussed in Section~\ref{section:a-standard-cobordism}
and let $Z:= V \cup_{M_K} U$. Note for later use that $H_2(Z)=\Z^{2g+N-1}$ and $\chi(Z)=2g$.
\item (Propositions~\ref{prop:standardu} and~\ref{prop:step2}.) Show that we can choose maps $\chi_i\colon H_1(M_{K_i}) \to \Z_p$ such
that the corresponding map $\phi\colon \coprod_{i=1}^N \pi_1(M_{K_i}) \to \Z
\ltimes \Z_p$ extends to a map $\Phi\colon \pi_1(Z) \to \Z \ltimes \Z_{p^a}$
for some $a \geq 1$ and such that at least $n:=\frac{m_RN-4g}{2}$ of the $\chi_i$ are
nonzero.
\item (Claim~\ref{prop:step-3}.) Show that for some $1 \leq i \leq N$ and $1 \leq j \leq r$, the element $ [1,0] \otimes [\lambda(\eta_i^j)] $ in $H_1^\phi(Y) = H_1 \bigl(\Q(\xi_p)[t^{\pm1}]^2 \otimes_{\Z[\pi_1(Y)]} C_*(\widetilde{Y}) \bigr)$ does not map to $0$ in $H_1^{\Phi}(Z)$. (Recall that $\lambda(\eta_i^j)$ is a longitude of the infection curve $\eta_i^j$ and lies in $M_{K_i} \subset Y$.) This step, which contains much of the technical work of the theorem, crucially relies on our
assumption that for every nontrivial $\chi\colon H_1(\Sigma_2(R)) \to \Z_p$ we
know that the collection $\{[1,0] \otimes [\eta^j]\}$ generates $H_1^{\theta
\circ f_\chi}(R)$ and that the order of $H_1^{\theta \circ f_\chi}(R)$ is
relatively prime to $\Delta_R(t)$.
\item (Last two paragraphs of Section~\ref{section:proof-of-main-theorem}.) Construct a local coefficient derived series representation $\pi_1(Y)
\to \Lambda$ extending over $\pi_1(Z)$ and bound the $L^{(2)}$
$\rho$-invariant $\rho^{(2)}(Y,\Lambda)$ in two different ways to get a
contradiction. Essentially, since $\chi(Z)=2g$ and our representation
extends over $\pi_1(Z)$, Theorem~\ref{thm:upperbound} implies that $|\rho^{(2)}(Y,\Lambda)|$ is small, while our assumptions on $|\rho_0(J^j_i)|$ together with Step 3, Proposition~\ref{prop:additivity}, and Proposition~\ref{prop:7-1-MP} will imply
that $|\rho^{(2)}(Y,\Lambda)|$ is very large.
\end{enumerate}
We now prove the two propositions crucial to Steps 1 and 2, respectively.
\begin{prop}\label{prop:step1}
Let $N \in \mathbb{N}$ be arbitrary and $K= \#_{i=1}^N K_i$ be a knot with $g_4(K) \leq g$.
Then there exists a compact connected 4-manifold $V$ such that, letting $U$ denote the standard cobordism between $M_K$ and $Y:=\sqcup_{i=1}^N M_{K_i}$ from Section~\ref{section:a-standard-cobordism}, $Z:= U \cup_{M_K} V$ satisfies:
\begin{enumerate}[(i)]
\item $\partial Z= Y$,
\item $H_2(Z)=\Z^{2g+N-1}$, and
\item $\chi(Z)= 2g$.
\end{enumerate}
\end{prop}
\begin{proof}
Let $F'$ be a locally flat surface embedded in $D^4$ with $\partial F'=K$ and $g(F')=g$.
Following \cite[Proposition~5.1]{Cha:2006-1}, we construct a topological
4-manifold $V$ with boundary $\partial V=M_K$, $H_1(M_K) \to H_1(V)$ an
isomorphism, and $H_2(V)\cong \Z^{2g}$, as follows. Let $X=X_0(K)$ denote the
0-trace of $K$, the 4-manifold obtained from $D^4$ by attaching a 0-framed
2-handle along a neighborhood of $K$. Let $F$ be the closed surface in $X$
obtained by taking the union of $F'$ with a core of the 2-handle. Note that
since $F$ is locally flat it has a normal bundle by
\cite[Section~9.3]{Freedman-Quinn:1990-1}. Observe that $F \cdot F=0$, and so
$\nu(F) \cong F \times D^2$. Now, let $V= \left(X \ssm \nu(F)\right)
\cup_{\partial \nu(F)} H \times S^1$, where $H$ is any handlebody with $\partial
H= F$. A Mayer-Vietoris argument shows that $H_1(V) \cong \Z$, with generator a
meridian to $F$, and that $H_2(V) \cong \Z^{2g}$. Note that by Poincar{\'e}
duality and universal coefficients, we have
\[
H_3(V) \cong H^1(V, M_K) \cong \Hom(H_1(V,M_K),\Z) \cong \Hom(0,\Z) \cong 0.
\]
So in particular the Euler characteristic of $V$ is $\chi(V)= 1-1+2g-0+0=2g$.
Let $U$ be the standard cobordism between $M_K$ and $\bigsqcup_{i=1}^N M_{K_i}$
discussed rather extensively in Section~\ref{section:a-standard-cobordism}. Now let $Z= V \cup_{M_K} U$, as
illustrated schematically in Figure~\ref{fig:diagramzz}.
\begin{figure}
\caption{A schematic diagram of $Z= U \cup_{M_K}
\label{fig:diagramzz}
\end{figure}
Note that $H_2(Z) \cong \Z^{2g} \oplus \Z^{N-1}$, and the inclusion induced map
$H_1(M_K) \to H_1(Z)$ is an isomorphism, as are each of the maps $H_1(M_{K_i})
\to H_1(Y) \to H_1(Z)$ for $i=1, \dots, N$. Also, $H_3(Z) \cong \Z^{N-1}$ and
$H_4(Z)=0$. So the Euler characteristic of $Z$ is
\[
\chi(Z)= 1-1 + (2g + N-1)- (N-1))= 2g. \qedhere
\]
\end{proof}
\begin{prop}\label{prop:step2}
Let $R$ be a ribbon knot and $p$ be a prime, and let $m_R$ denote the generating rank of the $p$-primary part of $H_1(\Sigma_2(R))$.
Fix $N \in \mathbb{N}$, and for each $i=1, \dots, N$ let $K_i$ be a knot obtained by infection along an unlink $\{\eta^j\}_{j=1}^r$ in the complement of $R$ such that each $\eta^j$ represents an element of $\pi_1(M_R)^{(1)}$.
Let $K= \#_{i=1}^N K_i$, and suppose that $M_K$ bounds a compact connected 4-manifold $V$ such that $H_1(M_K) \to H_1(V)$ is an isomorphism.
Then there exist $\chi_i \colon H_1(\Sigma_2(R)) \to \Z_p$, for $i=1, \dots, N$ such that:
\begin{enumerate}[(a)]
\item at least $\frac{m_RN-2 \chi(V)}{2}$ of the $\chi_i$ are nonzero, and
\item for some $a>0$, there exists a map $\pi_1(V) \to \Z \ltimes \Z_{p^a}$ such that the composition $\pi_1(M_K) \to \pi_1(V)\to \Z \ltimes \Z_{p^a}$ is given by the post-composition of $f_{\oplus_{i=1}^N \chi^{K_i}_i}$
with the inclusion $\Z \ltimes \Z_p \hookrightarrow \Z \ltimes \Z_{p^a}$.
\end{enumerate}
\end{prop}
\begin{proof}
For convenience, let $n=\frac{m_R N-2 \chi(V)}{2}$.
There is a canonical identification $H_1(M_K^2) \cong \Z \oplus H_1(\Sig_2(K))$,
and so given any $(\chi_i^{K_i})_{i=1}^N$ we obtain not just a map $\chi\colon
H_1(\Sig_2(K)) \to \Z_p$ but also a map $\overline{\chi}\colon H_1(M_K^2) \to \Z_p$
by sending the $\Z$ coordinate to zero. Since the inclusion $H_1(M_K) \to
H_1(V)$ is an isomorphism, it therefore suffices to show that there are
homomorphisms $(\chi^R_i)_{i=1}^N\colon H_1(\Sig_2(R)) \to \Z_p$, at least $n$ of which are
nonzero, such that the map
\[
\overline{\chi}:=\overline{(\chi^{K_i}_i)_{i=1}^N}\colon H_1(M_K^2) \to \Z_p
\]
extends over $H_1(V^2)$, perhaps after expanding its codomain to $\Z_{p^a}$ for
some $a>0$. Note that $\overline{\chi}$ extends over $H_1(V^2)$ up to enlarging its
codomain if and only if $\overline{\chi}$ vanishes on
\[
H:=\ker(H_1(M_K^2) \xrightarrow{i_1} H_1(V^2)).
\]
The group of characters $TH_1(M_K^2) \to \Z_p$ is isomorphic to $H_1(\Sig_2(K),
\Z_p)$, which is in turn congruent to $(\Z_p^{m_R})^N$, where we recall that $m_R$
denotes the generating rank of the $p$-primary part of $H_1(\Sig_2(R))$. The
subgroup of characters vanishing on $H$ is in bijective correspondence with
$(TH_1(M_K^2)/H) \otimes \Z_p$.
Note that $M_K$ is a homology $S^1 \times S^2$ with
$\grr (TH_1(M^2) \otimes \Z_p)=m_RN$ Therefore, by Proposition~\ref{prop:homology}, the $p$-primary part of $TH_1(M_K^2)/H$ has generating rank at least~$\frac{m_RN- 2 \chi(V)}{2}=n$. Therefore $TH_1(M_K^2)/H$ has a subgroup
isomorphic to~$\Z_p^n$. Our desired result now follows from a linear algebra argument (see the proof of \cite[Theorem~6.1]{KimLivingston:2005}): every subgroup of $\Z_p^{M}$ isomorphic to $\Z_p^{\ell}$ ($0 \leq \ell \leq M$) has an element at least $\ell$ of whose coordinates are nonzero.
\end{proof}
Now we prove Theorem~\ref{thm:general}.
\begin{proof}[Proof of Theorem~\ref{thm:general}]
Suppose for the sake of contradiction that there is some locally flat surface
$F'$ embedded in $D^4$ with $\partial F'=K$ and $g(F')=g$.
Let $U, V,$ and $Z$ be as in Proposition~\ref{prop:step1}. Note that as discussed in Section~\ref{section:a-standard-cobordism} we have a standard choice of basepoints and paths
inducing inclusion maps; for the rest of the proof, these choices will remain
fixed though not explicitly discussed.
We pause to establish notation. For a knot $J$ in $S^3$, we denote its exterior
by~$E_J$. For a manifold $X$ with $H_1(X) \cong \Z$, we denote its canonical
double cover by $X^2$. The choice of a meridian $\mu_J$ determines a splitting
$\pi_1(M_J) \cong \Z \ltimes \mathcal{A}(J)$, where $\mathcal{A}(J)$ denotes the
Alexander module of $J$. Note that $H_1(\Sig_2(J))$ is naturally identified with
$\mathcal{A}(J)/ \langle t+1 \rangle$, and so a map $\chi\colon H_1(\Sig_2(J))
\to \Z_p$ induces a map
\[
f_\chi\colon \pi_1(M_J) \xrightarrow{\cong} \Z \ltimes \mathcal{A}(J)
\to \Z \ltimes H_1(\Sig_2(J)) \xrightarrow{Id \ltimes \chi} \Z \ltimes \Z_p.
\]
Note that in the setting of Proposition~\ref{prop:step1}, since $H_1(M_K) \to H_1(Z) \cong \Z$ is an isomorphism,
$Z$ also has a canonical double cover $Z^2$. It is easy to check that $Z^2= V^2
\cup_{M_K^2} U^2$ and that $\partial Z^2= \bigsqcup_{i=1}^{N} M^2_{K_i}$.
For each $i=1, \dots, N$, we have a canonical, linking form--preserving
identification of $H_1(\Sig_2(K_i))$ with $H_1(\Sigma_2(R))$ coming from the
degree one maps $E_{J_i^j} \to E_{\text{unknot}}$. Given a map $\chi^R\colon
H_1(\Sig_2(R)) \to \Z_p$ we will use $\chi^{K_i}$ to denote the corresponding
map from $H_1(\Sig_2(K_i)) \to \Z_p$, and vice versa. We will also always
identify $H_1(\Sigma_2(K))$ with $\bigoplus_{i=1}^N H_1(\Sigma_2(K_i))$ in the
canonical, linking form--preserving way.
Define $n := \frac{m_RN-4g}{2}$. (Note that with $\chi(V)=2g$ this agrees with the definition of $n$ used above.) We wish to show that there exist $\chi^R_i\colon
H_1(\Sigma_2(R)) \to \Z_p$, for $i=1, \dots, N$, such that at least $n$ of the
$\chi^R_i$ are nonzero and for some $a>0$, there exists a map $\pi_1(Z) \to \Z
\ltimes \Z_{p^a}$ such that the composition $\pi_1(M_{K_i}) \xrightarrow{\iota
\circ \iota_i} \pi_1(Z)\to \Z \ltimes \Z_{p^a}$ is given by the postcomposition of $f_{\chi^{K_i}_i}$
with the inclusion $\Z \ltimes \Z_p \hookrightarrow \Z \ltimes \Z_{p^a}$.
Henceforth, we will implicitly take the usual inclusion of
$\Z_p$ in $\Z_{p^a}$ without further comment.
We will accomplish this in a somewhat indirect fashion, by focusing on
constructing an appropriate map on $\pi_1(M_K)$ which extends over $\pi_1(U)$
and $\pi_1(V)$ separately. By Proposition~\ref{prop:standardu}, given any choice
of $\chi^R_1, \dots, \chi^R_N\colon H_1(\Sigma_2(R)) \to \Z_p$, the map
$f_{(\chi^{K_i}_i)_{i=1}^N}\colon \pi_1(M_K) \to \Z \ltimes \Z_p$ extends
uniquely to a map $F\colon \pi_1(U) \to \Z \ltimes \Z_p$ such that when we
consider the composition
\[
F \circ\iota \circ \iota_i\colon \pi_1(M_{K_i}) \to \Z \ltimes \Z_p,
\]
we have $F \circ \iota \circ\iota_i= f_{\chi^{K_i}_i}$.
By applying Proposition~\ref{prop:step2} to our $K$ and $V$ and extending over $U$ as discussed above, we obtain $\chi= (\chi^R_1, \dots, \chi^R_{N})$ with at least $\frac{m_R N-4g}{2}=:n$ of the $\chi_i$ nonzero together with a map
$F\colon \pi_1(Z) \to \Z \ltimes \Z_{p^a}$ such that the composition
\[
\pi_1(M_{K_i}) \xrightarrow{\iota \circ \iota_i} \pi_1(Z) \to \Z \ltimes \Z_{p^a}
\]
is given by $f_{\chi^{K_i}_i}$.\unskip, \ignorespaces
As described in Section~\ref{section:metabelian-homology}, we have a fixed map $\theta \colon \Z
\ltimes \pi_1(\Sigma_2(K)) \to \GL_2(\Q(\xi_{p^a})[t^{\pm1}])$. By
post-composing $F$ and each $f_{\chi^{K_i}_i}$ with this map, we obtain
\begin{align*}
\Phi= \theta \circ F\colon& \pi_1(Z) \to \GL_2(\Q(\xi_{p^a})[t^{\pm1}]),
\unskip, \ignorespaces
\phi_i= \theta \circ f_{\chi_i^{K_i}}\colon& \pi_1(M_{K_i})
\to \GL_2(\Q(\xi_{p^a})[t^{\pm1}]).
\end{align*}
We let
\[
\phi= \coprod_{i=1}^N \phi_i\colon \coprod_{i=1}^N \pi_1(M_{K_i}) \to
\GL_2(\Q(\xi_{p^a})[t^{\pm1}]).
\]
For convenience, let $\F= \Q(\xi_{p^a})$, $S= \F[t^{\pm1}]$,
$Q= \F(t)$, and $S/ p$ be shorthand for $S/ \langle p(t) \rangle$ for any
polynomial $p(t) \in S$. Since our infection curves $\alpha_i$ live in the
second derived subgroup of $M_{R_0}$, the degree one maps $f_i\colon E_{J_i^j}
\to E_{\text{unknot}}$ give us an identification
\[
f_*\colon H_1^{\phi}(Y, S) =
\bigoplus_{i=1}^{N} H_1^{\phi_i}(M_{K_i}, S)
\xrightarrow{\cong} \bigoplus_{i=1}^{N} H_1^{\theta \circ f_{\chi_i^R},S}(M_{R})
\]
where the maps $\chi^R_i\colon H_1(\Sigma_2(R)) \to \Z_p \hookrightarrow
\Z_{p^a}$ are as above.
We now work towards proving the following claim.
\begin{claim}\label{claim:not-contained}
\[ H:= \bigoplus_{\{i \mid \chi_i^{R} \neq 0\}} H_1^{\phi_i}(M_{K_i})
\text{ is not contained in }\ker( H_1^{\phi}(Y; \F[t^{\pm1}]) \to H_1^{\Phi}(Z; \F[t^{\pm1}])).\]
\end{claim}
\begin{proof}[Proof of Claim~\ref{claim:not-contained}]
First, note that $H$ has generating rank at least $\lceil k/d_R \rceil $, since for some nontrivial $\chi_0 \colon H_1(\Sigma_2(R)) \to \Z_p$ there is a submodule of $H$ isomorphic to $\left( H_1^{\theta \circ f_{\chi_0}}(M_R) \right)^{\lceil k/{d_R}\rceil}$.
Note that if $\chi_i^R=0$ then
\[
H_1^{\theta \circ f_{\chi_i^R}}(M_{R} )
\cong\mathcal{A}_{\Q} (R) \otimes_{\Q[t^{\pm1}]} \F[t^{\pm1}].
\]
We therefore have that
\[
H_1^{\phi}(Y, S) \cong (\mathcal{A}_{\Q} (R) \otimes_{\Q[t^{\pm1}]} \F[t^{\pm1}])^{N-k}
\oplus H
\]
where $k \geq \frac{m_R N-4g}{2}$ is the number of nonzero $\chi_i^R$.
We now compute the rank of $H_2^{\Phi}(Z; Q)$. We can immediately see that
$H_0^{\Phi}(Z; Q)) \cong H_0^{\phi}(Y; Q) \cong 0$, since $H_0^{\Phi}(Z)$ and
$H_0^{\phi}(Y)$ are annihilated by $t-1$. Note that for each $i=1, \dots, N$
the inclusion map $Y_i \to Z$ induces an isomorphism on $H_0(-; \Z)$ and
$H_1(-;\Z)$. By the proof of \cite[Proposition~4.1]{Friedl-Powell:2010-1}, modified to use only a partial chain contraction for $C_*^{\Phi}(Z,Y_i;Q)$ in degrees $0,1$, as in \cite[Proposition~2.10]{Cochran-Orr-Teichner:1999-1}, this implies that the map
$H_1^{\phi_i}(Y_i; Q) \to H_1^{\Phi}(Z; Q)$ is onto. We have already observed
that $H_1^{\phi_i}(Y_i)$ is a torsion $S$-module and so $H_1^{\phi_i}(Y_i;
Q)=0$; it follows that $H_1^{\Phi}(Z; Q)=0$ as well. Consideration of the long exact sequence of the
pair $(Z,Y)$ then allows us to conclude that $H_1^{\Phi}(Z, Y; Q)=0$.
By Poincar{\'e}-Lefschetz
duality, universal coefficients, and the long exact sequence of $(Z, Y)$ with
$Q$-coefficients we have that
\[
H_3^{\Phi}(Z;Q) \cong H^1_{\Phi}(Z,Y; Q) \cong
\Hom(H_1^{\Phi}(Z, Y; Q), Q) \cong 0.
\]
Finally, since $Z$ is a topological 4-manifold and hence homotopy equivalent to a finite
CW complex with cells of dimension at most 3 (see the proof of
Theorem~\ref{thm:upperbound} for references for this fact), we have that
$H_j^{\Phi}(Z; \F(t))=0$ for all $j \geq 4$. Re-computing $\chi(Z)$ with
$Q$-coefficients, we obtain
\[
2g= \chi(Z)= 0 - 0 + \dim_{Q} H_2^{\Phi}(Z; Q) -0+0
=\dim_{Q} H_2^{\Phi}(Z; Q).
\]
We now return to working with $S = \mathbb{F}[t^{\pm 1}]$-coefficients and
consider the long exact sequence of Proposition~\ref{prop:les-pairs}
\[
\dots \to H_2^{\phi}(Y) \xrightarrow{i_2} H_2^\Phi(Z)
\xrightarrow{j_2} H_2^\Phi(Z,Y) \xrightarrow{\partial} H_1^{\phi}(Y)
\xrightarrow{i_1} H_1^\Phi(Z) \xrightarrow{j_1} H_1^\Phi(Z,Y) \to \dots.
\]
Suppose now for a contradiction that
$H \leq \ker(i_1)$. Since
\begin{align*}
\ker(i_1) =
\Imm(\partial) &\cong H_2^\Phi(Z,Y)/ \ker(\partial) =
H_2^\Phi(Z,Y)/ \Imm(j_2) \cong \left( S^{2g} \oplus TH_2^\Phi(Z,Y) \right)/ \Imm(j_2),
\end{align*}
it follows that $\left( S^{2g} \oplus TH_2^\Phi(Z,Y) \right)/ \Imm(j_2)$ has a submodule $H'$ isomorphic to $H$.
By applying Lemma~\ref{lem:intersecttorsion} with $A= H_2^{\Phi}(Z,Y)$, $B= \Imm(j_2)$, and $C= H'$ we obtain that $TH_2^{\Phi}(Z,Y) / (\Imm(j_2) \cap TH_2^{\Phi}(Z,Y) )$ contains a submodule $H''$ of generating rank at least $\lceil k/{d_R}\rceil-2g$ and of order which divides the order of $H'$ and so is relatively prime to $\Delta_R$.
Since $\Imm(j_2|_T) \subseteq \Imm(j_2) \cap TH_2^{\Phi}(Z,Y)$, it follows immediately that $\coker(j_2|_T)=TH_2^\Phi(Z,Y)/ \Imm(j_2|_T)$ contains a submodule of generating rank at least $\lceil k/{d_R}\rceil-2g$ and of order relatively prime to $\Delta_R$.
As argued above, we have that $H_1^{\Phi}(Z,Y;Q) =0$, i.e. that $H_1^{\Phi}(Z,Y)$ is torsion, and so we can apply Lemma~\ref{lemma:moregeneralhomology} to conclude that
\begin{align*}
\coker(j_2|_T)) \cong \ker(j_1|_T)= \ker(j_1)= \Imm(i_1).
\end{align*}
Therefore $\Imm(i_1)$ has a submodule of generating rank at least $\lceil k/{d_R}\rceil-2g$ and of order that is relatively prime to $\Delta_R$.
Since $k \geq n= \frac{m_R N-4g}{2}$ and $N \geq \frac{4g(d_R+1)+2}{m_R}$, we obtain that $\lceil k/d_R\rceil -2g >0$
and so there is a submodule of
$\Imm(i_1)$ isomorphic to $S/s$ for some nontrivial polynomial $s$ relatively
prime to $\Delta_R$. This is our desired contradiction, since
$H \leq \ker(i_1)$
also implies that $\Imm(i_1)$ is a quotient of $\left(\mathcal{A}_{\Q} (R) \otimes_{\Q[t^{\pm1}]} \F[t^{\pm1}]\right)^{N-k}$, which has order $\Delta_R^{N-k}$ and therefore cannot contain a submodule isomorphic
to~$S/s$.
This completes the proof of the claim. \end{proof}
\begin{claim}\label{prop:step-3}
For some $1 \leq i \leq N$ and $1 \leq j \leq r$, the element $ [1,0] \otimes [\lambda(\eta_i^j)] $ does not map to $0$ in $H_1^{\Phi}(Z)$.
\end{claim}
\begin{proof}[Proof of Claim~\ref{prop:step-3}]
Observe that since the longitude $\lambda(\eta_i^j)$ of $\eta_i^j$ is in the
second derived subgroup of $\pi_1(M_{R})$ it must lift to a curve $l_i^j$ in
the cover $\widetilde{M_{R}}$ of $M_{R}$ determined by $\phi_i$. (In fact, it
lifts to $\Z \ltimes \Z_{p^a}$ copies -- pick one.)
Since whenever $\chi_i^R \neq 0$ we have that the collection $\{[1,0] \otimes [l_i^j]\}_{j=1}^r$ generates $H_1^{\phi_i}(M_R)$, our argument that $H \not \leq \ker(i_1)$ in fact implies that
for at least one $i$ and $j$ with $1 \leq i \leq N$ and $1\leq j \leq r$, we have $i_1([1,0] \otimes
[l_i^j]) \neq 0$ in $H_1^{\Phi}(Z).$ This completes the proof of Claim~\ref{prop:step-3} and of Step~3.
\end{proof}
We are now ready to complete the proof of Theorem~\ref{thm:general}, as described in Step 4, by constructing a new representation of $\pi_1(Y)$ and bounding $\rho^{(2)}(Y, \psi)$ in two different ways to derive a contradiction.
Let
\[\psi \colon\pi_1(Y) \to \pi_1(Z) \to \Lambda:= \pi_1(Z)/ \pi_1(Z)^{(3)}_{(\Q, \Z_{p^a}, \Q)}\]
be the map induced by inclusion. Since
$\Lambda$ is amenable and in $D(\Z_p)$~\cite[Lemma~4.3]{Cha:2014-1} and
$\psi$ evidently extends over $\pi_1(Z)$, Theorem~\ref{thm:upperbound} and the
fact from Step 1 that $H_2(Z) \cong \Z^{2g+N-1}$ tells us that
\begin{align}\label{equation:upper-bound}
|\rho^{(2)}(Y, \psi)| \leq 2 \dim_{\Z_p}H_2(Z, \Z_p) = 2(2g+ N-1).
\end{align}
Let $(i_0, j_0)$ be the maximal tuple (with respect to the lexicographic ordering) such that $[1,0] \otimes [l_i^j]$ does not map to $0$ in $H_1^\Phi(Z)$.
Proposition~\ref{prop:7-1-MP} implies that $\lambda(\eta_{i_0}^{j_0}) \not \in
\pi_1(Z)^{(3)}_{(\Q, \Z_{p^a}, \Q)}$.
Moreover, Proposition~\ref{prop:ouradditivity} tells us that, letting $\delta_i^j(\psi)=\begin{cases} 1, & \psi(\lambda(\eta_i^j)) \neq 0 \unskip, \ignorespaces 0, &\psi(\lambda(\eta_i^j))=0 \end{cases}$, we have
\begin{align}
\rho^{(2)}(Y, \psi)&=\sum_{i=1}^{N}\rho^{(2)}(M_{R_{\alpha}(J_i)}, \psi|_{M_{R_{\alpha}(J_i)}}) = \sum_{i=1}^{N}\Big(\rho^{(2)}(M_{R}, \psi^0_{i})+\sum_{j=1}^r \delta_i^j(\psi)
\rho_0(J_i^j) \Big).
\label{eqn:siglargept2}
\end{align}
Since $|\rho^{(2)}(M_{R}, \psi^0_{i})| \leq C_R$ for all $i$, the tuple $(i_0, j_0)$ is maximal such that $ \delta_i^j(\psi) \neq 0$, and $J_{i_0}^{j_0}$ satisfies
\[ |\rho_0(J^{j_0}_{i_0})|> 2(2g+N-1) + N C_R
+ \sum_{k=1}^{i_0-1} \sum_{\ell=1}^r |\rho_0(J^\ell_k)|
+ \sum_{\ell=1}^{j_0-1} |\rho_0(J^\ell_i)|,
\]
Equation~\ref{eqn:siglargept2} gives the desired contradiction with Equation~\ref{equation:upper-bound}, which completes the proof of Theorem~\ref{thm:general}.
\end{proof}
\section{Height four gropes}\label{section:height-four-gropes}
In Proposition~\ref{prop:gropebounding} below, we will show the following: the knot $K$ in
Section~\ref{subsection:example2bipolar} bounds a framed grope of height 4
embedded in~$D^4$. For the reader's convenience, we begin by recalling the
definition of a (capped) grope, a certain type of 2-complex.
\begin{defn}
[Grope of height $h$~{\cite{Freedman-Quinn:1990-1,
Cochran-Orr-Teichner:1999-1}}] A \emph{capped surface}, or a \emph{capped
grope of height 1,} is an oriented surface of genus $g>0$ with nonempty
connected boundary, together with discs attached along the $2g$ curves of a
standard symplectic basis for the surface. The discs are called \emph{caps}.
If $G$ is a capped grope of height $h-1$, then a 2-complex obtained by
replacing each cap of $G$ with a capped surface is called a \emph{capped grope
of height $h$}. A \emph{grope of height $h$} is obtained by removing caps
from a capped grope of height $h$. It is also called the \emph{body} of the
capped grope. The initial surface that the inductive construction starts with
is called the \emph{base surface}, and the \emph{boundary of a grope},
$\partial G$, is the boundary of its base surface.
\end{defn}
A (capped) grope defined above is often called \emph{disc-like}. An
\emph{annulus-like} (capped) grope is defined in the same way, starting from a
base surface with two boundary components.
\begin{remark}
It is not a priori obvious that a 2-complex $G$ known to be a grope has a
well-defined height, but it is true. For the reader's convenience, we give a
quick argument. Let $\tau \subset G$ be the singular set of the grope union
its boundary, i.e.\ the 1-complex consisting of the points where~$G$ is not
locally homeomorphic to an open disc. Then $G \ssm \tau$ consists of a
collection of open surfaces, many of which are planar. Removing the subset of
$G$ corresponding to the non-planar surfaces (the interior of the `top stage'
of $G$) gives a new grope with a strictly smaller singular set; we can then
repeat the above procedure. In this perspective, the height of a grope is
exactly the number of such steps needed to reduce the grope to a circle. We
leave to the reader the analogous argument that the height of a capped grope
is well-defined, as well as the intrinsic definition of the $i$th stage of a
grope, $1 \leq i \leq h$.
\end{remark}
A (capped) grope admits a standard embedding in the upper half 3-space
$\mathbb{R}^3_+=\{z\ge 0\}$ which takes the boundary to $\mathbb{R}^2$. Compose
it with $\mathbb{R}^3_+\hookrightarrow \mathbb{R}^4_+$, take a regular
neighborhood in $\mathbb{R}^4_+$, and possibly perform finitely many plumbings.
An embedding of the result in a 4-manifold is called an \emph{immersed framed
(capped) grope}. If no plumbing is performed, then we say that it is
\emph{embedded}. Often we will regard an immersed/embedded (capped) grope as a
2-complex, but it is always assumed to be framed in this sense. In addition, we
assume that each intersection in an immersed capped grope is always between a
cap and a surface in the body, following the convention
of~\cite{Cha-Kim:2016-1}. Note that in a simply connected 4-manifold, an
embedded grope without caps can be promoted to an immersed capped grope.
Returning to our case, recall that the knot $K$ in
Section~\ref{subsection:example2bipolar} is the connected sum of satellite knots.
We will use the following terminology and results
from~\cite{Cha:2012-1,Cha-Kim:2016-1}, which also consider link versions.
\begin{defn}[Satellite capped
grope~{\cite[Definition~4.2]{Cha:2012-1}},
{\cite[Definition~4.2]{Cha-Kim:2016-1}}]
\label{definition:satellite-capped-grope}
Suppose $K$ is a knot in $S^3$ and $\alpha$ is an unknotted circle in $S^3$
disjoint from~$K$. Let $E_\alpha$ be the exterior of $\alpha$, and let
$\lambda_\alpha$ be a zero linking longitude on~$\partial E_\alpha$. A
\emph{satellite capped grope} for $(K,\alpha)$ is a disc-like capped grope $G$
immersed in $E_\alpha\times I$ such that the boundary of $G$ is
$\lambda_\alpha\times 0$, the body of $G$ is disjoint from $K\times I$, and
the caps are transverse to $K\times I$.
\end{defn}
\begin{defn}[Capped grope concordance~{\cite[Definition~4.3]{Cha-Kim:2016-1}}]
\label{definintion:grope-concordance}
A \emph{capped grope concordance} between two knots $J$ and $J'$ is an
annulus-like capped grope immersed in $S^3\times I$ such that the base surface
is bounded by $J\times 0 \cup -J'\times 1$.
\end{defn}
\begin{prop}[{\cite[Section~4.1]{Cha-Kim:2016-1}}]
\label{proposition:product-satellite-grope}
Suppose that there is a satellite capped grope of height $h$ for $(K,\alpha)$
and a capped grope concordance of height $\ell$ between two knots $J$
and~$J'$. Then there is a capped grope concordance of height $h+\ell$ between
the satellite knots $K_{\alpha}(J)$ and~$K_{\alpha}(J')$.
\end{prop}
The height $h+\ell$ capped grope concordance in
Proposition~\ref{proposition:product-satellite-grope} is obtained by a
``product'' construction described in \cite[Definition~4.4]{Cha-Kim:2016-1}.
The last ingredient we need is the following result
from~\cite{Cochran-Orr-Teichner:1999-1}.
\begin{prop}[{\cite[Remark~8.14]{Cochran-Orr-Teichner:1999-1}}]
\label{proposition:height-two-grope}
A knot in $S^3$ with trivial Arf invariant bounds a capped grope of height two
immersed in~$D^4$.
\end{prop}
We can now prove the following.
\begin{prop} \label{prop:gropebounding}
Let $K= \#_{i=1}^n R_{\alpha^+,\alpha^-}(J_i^+, J_i^-)$ be a connected sum of satellite knots, where $(R,\alpha^+,\alpha^-)$ is as in the right of Figure~\ref{fig:11n742} and $\{J_i^+, J_i^{-}\}_{i=1}^n$ a collection of knots with vanishing Arf invariant.
Then $K$ bounds an embedded grope of height 4 in~$D^4$.
\end{prop}
\begin{proof}
First, note that it suffices to show that each $ R_{\alpha^+,\alpha^-}(J_i^+, J_i^-)$ bounds
an embedded grope of height 4, since we can then take the boundary connected sum of
such gropes to obtain one with boundary $K$. We therefore show that under the hypothesis that $\Arf(J^+)= \Arf(J^-)=0$ the knot $R_{\alpha^+,\alpha^-}(J^+, J^-)$ bounds a grope of height 4.
Observe that the curve $\alpha^-$ in
Figure~\ref{fig:11n742} bounds a disjoint capped grope of height two embedded
in~$S^3$, where the body surfaces are disjoint from the knot $R$ but the caps
are allowed to intersect~$R$. This is a geometric analogue of the commutator
relation $\alpha^-=[\beta_1, \beta_2]$ where the curves $\beta_1$ and $\beta_2$
shown in the left of Figure~\ref{fig:11n742} are again commutators in the
fundamental group.
\begin{figure}
\caption{An embedded height 2 grope with boundary $\alpha^-$ in $S^3 \ssm (R \sqcup \alpha^+)$.}
\label{fig:height2grope}
\end{figure}
Indeed, in the planar diagram in the right of Figure~\ref{fig:11n742}, the
bounded region enclosed by $\alpha^-$ is the projection of an obviously seen
embedded disc which intersects $R$ in four points, and by tubing on this disc,
one obtains a genus one surface, shown in red in Figure~\ref{fig:height2grope},
which is disjoint from~$R$. This surface is the base surface of the promised
height two grope bounded by~$\alpha^-$. The curves $\beta_1$ and $\beta_2$ are
parallel to standard basis curves of the base surface, and they bound disjoint
genus one surfaces obtained by tubing the obviously seen discs along the
knot~$R$, as illustrated in Figure~\ref{fig:height2grope}. Attach them to the
base stage surface to obtain a height two grope.
Note that all the surfaces used above are disjoint from the other curve
$\alpha^+$, so by performing the satellite construction, we obtain a height two
grope in $S^3\ssm R_{\alpha_+}(J^+)$ bounded by~$\alpha^-$. Identify $S^3$ with
$S^3\times 0 \subset S^3\times I$, push the interior of the grope into the
interior of $S^3\times I$, and add caps using the simple connectedness of
$S^3\times I$ as noted above. Apply general position to make the caps
transverse to $R_{\alpha_+}(J^+)\times I$, to obtain a satellite capped grope
for~$(R_{\alpha_+}(J^+),\alpha^-)$.
Since the knot $J^-$ has trivial Arf invariant, $J^-$ bounds a capped
grope of height two in~$D^4$, by Proposition~\ref{proposition:height-two-grope}.
Remove, from $D^4$, a small open 4-ball which intersects the capped grope in an
unknotted 2-disc lying in the interior of the base surface, to obtain a capped
grope concordance of height two between $J^-$ and the trivial knot. By
Proposition~\ref{proposition:product-satellite-grope} and the above paragraph,
the satellite knot $R_{\alpha^+,\alpha^-}(J^+,J^-) =
(R_{\alpha^+}(J^+))_{\alpha^-}(J^-)$ is height 4 capped grope concordant to
the knot~$R_{\alpha^+}(J^+)$. Forget the caps of this capped grope concordance,
and attach a slicing disc for the knot $R_{\alpha^+}(J^+)$, to obtain a grope of
height 4 bounded by~$R_{\alpha^+,\alpha^-}(J^+,J^-)$.
\end{proof}
\begin{remark}
A similar argument shows the existence of a bounding grope of height 4 for the
simpler example in Section~\ref{subsection:example2solvable}. In this case,
the height two surfaces constructed by ``tubing along the knot $R$'' in the
3-space are not disjoint, but the intersection can be removed by pushing the
surfaces into 4-space. We omit the details.
\end{remark}
\def\MR#1{}
\end{document} | math | 121,947 |
\begin{document}
\title[Rigidity of the \'{A}lvarez class]{Rigidity of the \'{A}lvarez class of Riemannian foliations with nilpotent structure Lie algebras}
\author{Hiraku Nozawa}
\address{Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan}
\email{[email protected]}
\begin{abstract}
We show that if the structure algebra of a Riemannian foliation $\mathcal{F}$ on a closed manifold $M$ is nilpotent, the integral of the \'{A}lvarez class of $(M,\mathcal{F})$ along every closed path is the exponential of an algebraic number. As a corollary, we prove that the \'{A}lvarez class and geometrically tautness of Riemannian foliations on a closed manifold $M$ are invariant under deformation, if the fundamental group of $M$ has polynomial growth.
\end{abstract}
\maketitle
\section{Introduction}
A Riemannian foliation $\mathcal{F}$ on a closed manifold $M$ is geometrically taut if there exists a bundle-like metric $g$ on $M$ such that every leaf of $\mathcal{F}$ is a minimal submanifold of $(M,g)$. Geometrically tautness of Riemannian foliations is a purely differential geometric property, but it is known that it has remarkable relations with the cohomological properties of $(M,\mathcal{F})$. For examples, Masa~\cite{Mas} characterized the geometrically tautness of $(M,\mathcal{F})$ by the nontriviality of the top degree part of the basic cohomology of $(M,\mathcal{F})$ and \'{A}lvarez L\'{o}pez~\cite{Alv} defined a cohomology class $[\kappa_b]$ of degree $1$ for $(M,\mathcal{F})$ which vanishes if and only if $(M,\mathcal{F})$ is geometrically taut. We call $[\kappa_b]$ the \'{A}lvarez class of $(M,\mathcal{F})$. In this paper, we show that \'{A}lvarez classes of Riemannian foliations have a rigidity property, if the structure Lie algebra is nilpotent. As a corollary of the rigidity of \'{A}lvarez classes, we obtain the invariance of geometrically tautness of Riemannian foliations under deformation, if the fundamental group of the ambient manifold has polynomial growth.
The main result in this paper is the following theorem:
\begin{thm}\label{expalg}
Let $M$ a closed manifold and $\mathcal{F}$ be a Riemannian foliation on $M$ with nilpotent structure Lie algebra. Then $e^{\int_{\gamma} [\kappa_b]}$ is an algebraic number for every $\gamma$ in $\pi_1(M)$ where $[\kappa_b]$ is the \'{A}lvarez class of $(M,\mathcal{F})$.
\end{thm}
\noindent
Theorem~\ref{expalg} is shown by computation of \'{A}lvarez classes in terms of the holonomy of the basic fibration in Section 2 and application of Mal'cev theory in Section 3.
We state two corollaries of Theorem~\ref{expalg}. Let $M$ be a closed manifold whose fundamental group has polynomial growth. Then the structure Lie algebra of every Riemannian foliation on $M$ is nilpotent according to Carri\`{e}re~\cite{Car}. By Theorem~\ref{expalg}, the \'{A}lvarez classes of Riemannian foliations on $M$ are contained in a countable subset of $H^1(M;\mathbb{R})$ which is independent of foliations. On the other hand, if we have a smooth family $\{\mathcal{F}^{t}\}_{t \in [0,1]}$ of Riemannian foliation on $M$, their \'{A}lvarez classes varies continuously in $H^1(M;\mathbb{R})$ as shown in~\cite{Noz}. Hence we have the following corollary:
\begin{cor}\label{alvinv}
Let $M$ be a closed manifold whose fundamental group has polynomial growth and $\{\mathcal{F}^{t}\}_{t \in [0,1]}$ be a smooth family of Riemannian foliation on $M$ over $[0,1]$. Then we have $[\kappa_{b}^{t}]=[\kappa_{b}^{t'}]$ in $H^1(M;\mathbb{R})$ for any $t$ and $t'$ in $[0,1]$ where $[\kappa_{b}^{t}]$ is the \'{A}lvarez class of $(M,\mathcal{F}^{t})$.
\end{cor}
Since $(M,\mathcal{F}^{t})$ is geometrically taut if and only if the \'{A}lvarez class of $(M,\mathcal{F}^{t})$ vanishes according to \'{A}lvarez L\'{o}pez~\cite{Alv}, we have the following corollary of Corollary~\ref{alvinv}:
\begin{cor}\label{definv}
Let $M$ be a closed manifold whose fundamental group has polynomial growth and $\{\mathcal{F}^{t}\}_{t \in [0,1]}$ be a smooth family of Riemannian foliation on $M$. Then one of the following is true:
\begin{enumerate}
\item $(M,\mathcal{F}^{t})$ is geometrically taut for every $t$ in $[0,1]$.
\item $(M,\mathcal{F}^{t})$ is not geometrically taut for every $t$ in $[0,1]$.
\end{enumerate}
\end{cor}
If the dimension of $\mathcal{F}$ is $1$, the structure Lie algebra of $\mathcal{F}$ is always abelian according to Caron-Carri\`{e}re~\cite{CaCa} or Carri\`{e}re~\cite{Car}. Hence the conclusion of Corollary~\ref{alvinv} and Corollary~\ref{definv} follow from Theorem~\ref{expalg} without the assumption on the growth of the fundamental group of $M$ for the cases where $\mathcal{F}$ is $1$-dimensional.
Note that we can show Theorem~\ref{expalg} by more direct computation in~\cite{Noz} if the dimension of $\mathcal{F}$ is $1$, using a theorem of Caron-Carri\`{e}re~\cite{CaCa} which claims $1$-dimensional Lie foliations with dense leaves are linear flows on tori with irrational slopes and computation of the mapping class group of the group of diffeomorphisms preserving a linear foliation with dense leaves on tori by Molino and Sergiescu~\cite{MoSe}.
The author expresses his gratitude to Jes\'{u}s Antonio \'{A}lvarez L\'{o}pez, Jos\'{e} Royo Prieto Ignacio and Steven Hurder. The conversation with \'{A}lvarez L\'{o}pez and Royo Prieto on \'{A}lvarez classes gave the author a definite clue to carry out the computation of \'{A}lvarez classes in Section 2. The idea on conditions for the growth of the fundamental group of the ambient manifold came from the conversation with Hurder.
\section{Computation of \'{A}lvarez classes in terms of the basic fibration}
We prepare the notation to state the main result in this section. Let $(M,\mathcal{F})$ be a closed manifold with an orientable transversely parallelizable foliation. Let $\pi \colon M \longrightarrow W$ be the basic fibration of $(M,\mathcal{F})$. Fix a point $x$ on $M$. We denote a fiber of $\pi$ which contains $x$ by $N$ and the restriction of $\mathcal{F}$ to $N$ by $\mathcal{G}$. Then $(N,\mathcal{G})$ is a Lie foliation by Molino's structure theorem. We denote the dimension and codimension of $(N,\mathcal{G})$ by $k$ and $l$ respectively. Let $\mathfrak{g}$ be the structure algebra of $(N,\mathcal{G})$, which has dimension $l$. Throughout Section 2, we assume the unimodularity of $\mathfrak{g}$, that is,
\begin{equation}\label{unimod}
H^{l}(\mathfrak{g}) \cong \mathbb{R}.
\end{equation}
Note that assumption~\eqref{unimod} is satisfied if $\mathfrak{g}$ is nilpotent and may not be satisfied if $\mathfrak{g}$ is solvable.
For the basic definition of the spectral sequences of foliated manifolds, we refer to a paper by Kamber and Tondeur~\cite{KaTo2}. Let $E_{2}^{0,k}$ be the $(0,k)$-th $E_2$ term of the spectral sequence of $(N,\mathcal{G})$. Then the dimension of $E_{2}^{0,k}$ is $1$ by the assumption~\eqref{unimod}, since we have
\begin{equation}
H^{l}(\mathfrak{g}) \cong H_{B}^{l}(N/\mathcal{G}) \cong E_{2}^{0,k}
\end{equation}
where the first isomorphism follows from the denseness of the leaves of $(N,\mathcal{G})$ and the second isomorphism follows from the duality theorem of Masa in~\cite{Mas}. Hence $\Aut (E_{2}^{0,k})$ is canonically identified with $\mathbb{R}-\{0\}$. We denote the orientation preserving automorphism group of $E_{2}^{0,k}$ by $\Aut_{+} (E_{2}^{0,k})$, which is identified with the set of positive real numbers $\mathbb{R}_{>0}$.
Let $\Diff(N,\mathcal{G})$ be the group of diffeomorphisms on $N$ which map each leaf of $\mathcal{G}$ to a leaf of $\mathcal{G}$ and denote its mapping class group in the $C^{\infty}$ topology by $\pi_{0}(\Diff(N,\mathcal{G}))$. We have a canonical action $\Phi \colon \pi_{0}(\Diff(N,\mathcal{G})) \longrightarrow \Aut(E_{2}^{0,k})$ defined in the following way: Let $H^{k}(\mathcal{G})$ be the $k$-th leafwise cohomology group of $(N,\mathcal{G})$, which is identified with $E_{1}^{0,k}$ in the spectral sequence of $(N,\mathcal{G})$. $E_{2}^{0,k}$ is identified with the kernel of $d_{1,0} \colon H^{k}(\mathcal{G}) \longrightarrow E_{1}^{1,k}$ where $d_{1,0}$ is the map induced on $E_1$ terms from the composition of the de Rham differential and the projection $C^{\infty}(\wedge^{k+1} T^{*}N) \longrightarrow C^{\infty}((T\mathcal{G}^{\perp})^{*} \otimes \wedge^{k} T^{*}\mathcal{G})$ determined by a Riemann metric $g$ of $(N,\mathcal{G})$ (See~\cite{KaTo2}.). $\Diff(N,\mathcal{G})$ acts to $E_{2}^{0,k}$, since $\Diff(N,\mathcal{G})$ acts to the leafwise cohomology group by pulling back leafwise volume forms and this action preserves $E_{2}^{0,k}$. This action descends to an action of $\pi_{0}(\Diff(N,\mathcal{G}))$, since the action of the identity component of $\Diff(N,\mathcal{G})$ to the leafwise cohomology group is trivial according to the integrable homotopy invariance of leafwise cohomology shown by El Kacimi Alaoui~\cite{ElK}.
We show the following proposition which computes the period of the \'{A}lvarez class $[\kappa_b]$ of $(M,\mathcal{F})$ in terms of the holonomy of the basic fibration:
\begin{prop}\label{holonomyformula}
The diagram
\begin{equation}\label{diag1}
\xymatrix{ \pi_1(M,x) \ar[rr]^{\int [\kappa_b]} \ar[d]_{\pi_{*}} & & \mathbb{R} \\
\pi_1(W,\pi(x)) \ar[r]^{hol_{\pi}} & \pi_{0}(\Diff(N,\mathcal{G})) \ar[r]^{\Phi} & \Aut_{+}(E_{2}^{0,k}) \ar[u]_{\log}}
\end{equation}
is commutative where $\int [\kappa_b]$ is the period map of $[\kappa_b]$, $hol_{\pi}$ is the holonomy map of the basic fibration $\pi \colon M \longrightarrow W$ , $\log$ is defined through the identification of $\Aut(E_{2}^{0,k})$ with $\mathbb{R}_{>0}$ and $\Phi$ is the canonical action described above.
\end{prop}
\begin{proof}
To show the commutativity of the diagram~\eqref{diag1} for an element $[\gamma]$ of $\pi_1(M,x)$ which is represented by a smooth path $\gamma$, it suffices to show the case of $W=S^1$ pulling back the fibration $\pi$ by $\gamma$. By the assumption~\eqref{unimod}, the \'{A}lvarez class of $(N,\mathcal{G})$ cannot be nontrivial. Hence the restriction of \'{A}lvarez class of $(M,\mathcal{F})$ to a fiber of $\pi$ is zero. We will compute the integration of \'{A}lvarez class of $(M,\mathcal{F})$ along a path which gives a generator of $\pi_1(S^1,\pi(x))$. We denote the holonomy of the $(N,\mathcal{G})$-bundle $\pi$ over $S^1$ along a path which gives a generator of $\pi_1(S^1,\pi(x))$ by $f$ and its action on $E_{2}^{0,k}$ by $f^*$. We write $\overline{\mathcal{F}}$ for the foliation defined by the fibers of $\pi$.
We fix a bundle-like metric $g'$ on $(M,\mathcal{F})$. Let $E$ be the vector bundle of rank $1$ over $S^1$ whose fiber $E_t$ over $t$ in $S^1$ is the $(0,k)$-th $E_2$ term of the spectral sequence of $(\pi^{-1}(t),\mathcal{F}|_{\pi^{-1}(t)})$. We define an affine connection $\nabla$ on $E$ by
\begin{equation}
\nabla s = \int_{\pi} (d_{1,0} s)
\end{equation}
for $s$ in $C^{\infty}(E)$ where $d_{1,0}$ is the map induced on $E_1$ terms from the composition of the de Rham differential and the projection $C^{\infty}(\wedge^{k+1} T^{*}M) \longrightarrow C^{\infty}((T\mathcal{F}^{\perp})^{*} \otimes \wedge^{k} T^{*}\mathcal{F})$ determined by a Riemann metric $g$ of $(M,\mathcal{F})$, and $\int_{\pi}$ is the integration on fibers on $\pi$ with respect to the first component of $C^{\infty}(T^{*}M) \otimes C^{\infty}(E)$ using the fiberwise volume form $vol_{\pi}$ of $\pi$ determined by $g'$ which is defined by $\int_{\pi} (\alpha \otimes h[\chi]) = \big(\int_{\pi} h\alpha \wedge vol_{\pi} \big) \otimes [\chi]$ for $\alpha$ in $C^{\infty}(T^{*}M)$ and $h$ in $C^{\infty}(M)$. The Rummler's formula
\begin{equation}\label{Rum}
d_{1,0} \chi = -\kappa \wedge \chi
\end{equation}
in~\cite{Rum} implies $\nabla$ is an connection on $E$. Note that $d_{1,0}$ coincides with the differential on $E_1$ terms of the spectral sequence of $(M,\mathcal{F})$ which is determined only by $\mathcal{F}$ and hence $\nabla$ is independent of the metric $g'$.
The connection $\nabla$ is flat, since every connection on a vector bundle over $S^1$ is flat. The holonomy of $(E,\nabla)$ which corresponds to a generator of $\pi_1(S^1,\pi(x))$ is shown to be equal to $f^{*}$ in the following way: We pull back the $(N,\mathcal{G})$ bundle $\pi$ by the canonical map $\iota \colon [0,1] \longrightarrow [0,1]/ \{0\} \sim \{1\}=S^1$ and denote the total space of $\iota^{*}\pi$ by $(M',\mathcal{F}')$. Fix a trivialization $(M',\mathcal{F}') \cong (N,\mathcal{G}) \times [0,1]$ as a $(N,\mathcal{G})$ bundle, then we have an induced trivialization of $\iota^{*}E$. Since $\iota^{*}\nabla$ is independent of the metric, we can assume that $\iota^{*}\nabla$ is defined by a product metric. Then the parallel section of $(\iota^{*}E,\iota^{*}\nabla)$ is the constant sections with respect to the trivialization. Since $(E,\nabla)$ is obtained by identifying the boundaries of $(\iota^{*}E,\iota^{*}\nabla)$ by $f^*$, the holonomy of $(E,\nabla)$ is equal to $f^{*}$.
We will show that the holonomy of $(E,\nabla)$ which corresponds to a generator of $\pi_1(S^1,\pi(x))$ is equal to $e^{\int_{S^1} \kappa_b}$ where the holonomy of $(E,\nabla)$ is regarded as a real number. For this purpose, we construct a bundle-like metric $g$ on $(M,\mathcal{F})$ such that each leaf of $\overline{\mathcal{F}}$ is a minimal manifold of $(M,g)$ and the leafwise volume form $\chi^{t}$ of $(\pi^{-1}(t),\mathcal{F}|_{\pi^{-1}(t)})$ determined by $g$ satisfies $d_{1,0}^{t} \chi^{t}=0$ where $d_{1,0}^{t}$ is the transverse component of de Rham differential of $({\pi^{-1}(t)},\mathcal{F}|_{{\pi^{-1}(t)}})$ defined in the same way as $d_{1,0}$ for $(N,\mathcal{G})$. By the duality theorem of Masa in~\cite{Mas} and the assumption~\eqref{unimod}, we have a bundle-like metric $g_0$ on $(N,\mathcal{G})$ such that each leaf of $\mathcal{G}$ is a minimal submanifold of $(N,g_0)$. Note that the leafwise cohomology class $[\chi_0]$ of the characteristic form $\chi_0$ of $(N,\mathcal{G},g_0)$ is an eigenvector of $f^*$, since $[\chi_0]$ is a generator of $E_{2}^{0,k}$ and $f^*$ preserves $E_{2}^{0,k}$. Hence we can write $f^*[\chi_0]=c[\chi_0]$ for some real number $c$. By the Moser's argument in~\cite{Ghy}, we can isotope $f$ to $f_1$ in $\Diff(N,\mathcal{G})$ so that $f^{*}_{1}\chi_0=c\chi_0$. Let $\omega_0$ be a basic transverse volume form of $(N,\mathcal{G})$. Then we have $f^{*}_{1}\omega_0=b\omega_0$ for some real number $b$, the leaves of $(N,\mathcal{G})$ are dense. Since $f_{1}$ induces the identity map on $H^{k+l}(N ; \mathbb{R})$ and a pairing
\begin{equation}\label{pair}
E_{2}^{0,k} \times E_{2}^{l,0} \longrightarrow E_{2}^{l,k}=H^{k+l}(N ; \mathbb{R})
\end{equation}
is natural with respect to $\Diff(N,\mathcal{G})$, we have $d=\frac{1}{c}$. Hence we have $f^{*}_{1} (\chi_0 \wedge \omega_0)=\chi_0 \wedge \omega_0$. We define a bundle-like metric $g$ of $(M,\mathcal{F})$ by $g= \rho(t)g_{0} +(1-\rho(t))f^{*}_{1}g_{0} + dt \otimes dt$ where $\rho$ is a smooth function on $[0,1]$ which satisfies $\rho(t)=0$ near $0$ and $\rho(t)=1$ near $1$. We denote the characteristic forms of $(M,\mathcal{F},g)$ by $\chi$. Then we have $\chi = \rho(t)\chi_{0} +(1-\rho(t))f^{*}_{1}\chi_{0}$. By the Rummler formula, we have $d_{1,0}\chi_0=0$ and hence $d_{1,0}^{t}(\chi|_{\pi^{-1}(t)})=0$ for every $t$ in $S^1$. Note that each leaf of $\overline{\mathcal{F}}$ is a minimal submanifold of $(M,g)$, since $f_1$ preserves the volume form of $(N,g_0)$.
We calculate the holonomy of $(E,\nabla)$ using $g$. We denote the map which corresponds the leafwise cohomology class $[\chi|_{\pi^{-1}(t)}]$ to $t$ in $S^1$ by $[\chi]$. Then $[\chi]$ is a global section of $E$, since $d_{1,0}^{t} \chi|_{\pi^{-1}(t)}=0$ is satisfied by the construction of $\chi$. By the Rummler's formula~\eqref{Rum}, we have
\begin{equation}\label{conn}
\nabla [\chi] = -\Big(\int_{\pi} \kappa \Big) \otimes [\chi].
\end{equation}
We show that $\int_{\pi} \kappa$ is a closed $1$-form on $S^1$ which satisfies $\kappa_b(\frac{\partial}{\partial t})=\int_{\pi} \kappa(\frac{\partial}{\partial t})$ on $S^1$. By the minimality of the fibers of $\pi$ with respect to $g$ and Rummler's formula for $(M,\overline{\mathcal{F}})$, $dvol_{\pi}$ has no component which has $m$-form tangent to the fibers of $\pi$ where $m$ is the dimension of the fibers of $\pi$. Hence we have
\begin{equation}
\begin{array}{rl}
d\int_{\pi} \kappa & = d\int_{\pi} \kappa_b \\
& =\int_{\pi} \Big(d\kappa_b \wedge vol_{\pi} - \kappa_b \wedge dvol_{\pi} \Big) \\
& = \int_{\pi} \Big(d\kappa_b \wedge vol_{\pi} - \kappa_b \wedge dvol_{\pi} \Big) \\
& = \int_{\pi} \Big(d\kappa_b \wedge vol_{\pi}\Big) \\
& = \int_{\pi} d\kappa_b.
\end{array}
\end{equation}
Since $\kappa_b$ is closed by Corollary~3.5 of \'{A}lvarez L\'{o}pez~\cite{Alv}, we have $d\int_{\pi} \kappa=0$. $\kappa_b(\frac{\partial}{\partial t})=\int_{\pi} \kappa(\frac{\partial}{\partial t})$ is clear by the definition of $\kappa_b$.
Hence $\nabla$ is a flat connection defined by a closed form $\int_{\pi} \kappa$ and we can show the holonomy of $(E,\nabla)$ is equal to $e^{\int_{S^1} \int_{\pi} \kappa}=e^{\int_{S^1} \kappa_b}$ in a standard way.
\end{proof}
\section{Application of Mal'cev Theory}
Let $(M,\mathcal{F})$ be a closed manifold with a Riemannian foliation with nilpotent structure Lie algebra. Since the \'{A}lvarez class of $(M,\mathcal{F})$ is defined by the integration along fibers of the \'{A}lvarez class of $(M^{1},\mathcal{F}^{1})$ where $M^{1}$ is the transverse orthonormal frame bundle of $(M,\mathcal{F})$ and $\mathcal{F}^{1}$ is the lift of $\mathcal{F}$, which is transversely parallelizable, Theorem~\ref{expalg} is reduced to the case where $(M,\mathcal{F})$ is transversely parallelizable. Moreover we can assume the orientability of $\mathcal{F}$, since a square root of an algebraic number is algebraic. To show Theorem~\ref{expalg} in the cases where $(M,\mathcal{F})$ is orientable and transversely parallelizable, it suffices to show the following Proposition~by Proposition~\ref{holonomyformula}:
\begin{prop}\label{algebraicaction}
Let $(N,\mathcal{G})$ be a closed manifold with a Lie foliation with nilpotent structure Lie algebra of which every leaf is dense in $N$. Then the image of $\Phi \colon \pi_{0}(\Diff(N,\mathcal{G})) \longrightarrow \Aut(E_{2}^{0,k})$ is contained in the set of algebraic numbers where $\Aut(E_{2}^{0,k})$ is canonically identified with $\mathbb{R}-\{0\}$.
\end{prop}
We show a lemma which reduces Proposition~\ref{algebraicaction} to a problem on nilpotent Lie groups and will apply Mal'cev theory to complete the proof of Proposition~\ref{algebraicaction}.
Let $N$ be a closed manifold and $\mathcal{G}$ be a Lie foliation on $N$ of dimension $k$ and codimension $l$ of which every leaf is dense in $N$. We denote the structure Lie algebra of $(N,\mathcal{G})$ by $\mathfrak{g}$. We fix a point $x$ on $N$. Then the holonomy homomorphism $hol \colon \pi_1(N,x) \longrightarrow G$ of $(N,\mathcal{G})$ is determined where $G$ is the simply connected structure Lie group determined by $\mathfrak{g}$.
We denote the canonical action $\Diff(N,\mathcal{G}) \longrightarrow \Aut(E_{2}^{0,l})$ by $\Psi$ where $E_{2}^{l,0}$ is the $(0,l)$-th $E_2$ term of the spectral sequence of $(N,\mathcal{G})$. Note that $E_{2}^{l,0}$ is isomorphic to the $l$-th basic cohomology group of $(N,\mathcal{G})$~\cite{Hae} and hence isomorphic to $H^{l}(\mathfrak{g})$, since the leaves of $\mathcal{G}$ are dense in $N$.
\begin{lem}\label{liefol}
If we assume
\begin{equation}\label{unimod2}
H^{l}(\mathfrak{g}) \cong \mathbb{R},
\end{equation}
then we have the following:
\begin{enumerate}
\item The diagram
\begin{equation}
\xymatrix{ \Diff(N,\mathcal{G}) \ar[r]^{\Phi} \ar[rd]_{\Psi} & \Aut(E_{2}^{0,k}) \\
& \Aut(E_{2}^{l,0}) \ar[u]_{i}}
\end{equation}
commutes where $i$ is the map which takes the inverse through identification of $\Aut(E_{2}^{0,k})$ and $\Aut(E_{2}^{l,0})$ with $\mathbb{R}-\{0\}$.
\item \label{2} We denote the image of $hol$ by $\Gamma$. A foliation preserving diffeomorphism $f \colon (N,\mathcal{G}) \longrightarrow (N,\mathcal{G})$ which fixes $x$ induces an automorphism of $G$ which preserves $\Gamma$.
\item We denote the group of diffeomorphisms which fix $x$ and preserve $\mathcal{G}$ by $\Diff(N,x,\mathcal{G})$, the group of automorphisms of $G$ which preserve $\Gamma$ by $\Aut(G, \Gamma)$ and the homomorphism $\Diff(N,x,\mathcal{G}) \longrightarrow \Aut(G)$ which is obtained by~\eqref{2} by $\iota$. Then the diagram
\begin{equation}
\xymatrix{ \Diff(N,x,\mathcal{G}) \ar[r]^{\Phi} \ar[rd]_{\iota} & \Aut(E_{2}^{l,0}) \\
& \Aut(G,\Gamma) \ar[u]_{A} }
\end{equation}
commutes where $A$ is the canonical action of the automorphism group of Lie group to the Lie algebra cohomology under the identification of $E_{2}^{l,0}$ with $H^{l}(\mathfrak{g})$.
\end{enumerate}
\end{lem}
\begin{proof}
We prove (1). Let $f$ be an element of $\Diff(N,\mathcal{G})$. Since the pairing~\eqref{pair} is natural with respect to $\Diff(N,\mathcal{G})$ and the action of $f$ to $H^{k+l}(N;\mathbb{R})$ is trivial, we have the conclusion.
We prove (2). Let $X$ be a basic transverse vector field on $(N,\mathcal{G})$ which we regard as an element of $\mathfrak{g}$. Then $f_{*}X$ is again an element of $\mathfrak{g}$ where $f_{*} \colon C^{\infty}(TN/T\mathcal{G}) \longrightarrow C^{\infty}(TN/T\mathcal{G})$ is the map induced by $f$, since every leafwise constant function of $(N,\mathcal{G})$ is constant. Hence we have an automorphism of $\mathfrak{g}$ induced by $f_{*}$, which we denote by $f_{*}$ again. Let $(\tilde{N},\tilde{\mathcal{G}})$ be the universal cover of $(N,\mathcal{G})$. We fix a point $\tilde{x}$ on $\tilde{N}$ such that $u(\tilde{x})=x$ where $u$ is the projection of the universal covering $\tilde{N} \longrightarrow N$. We can lift $f$ to $\tilde{f} \colon (\tilde{N},\tilde{\mathcal{G}}) \longrightarrow (\tilde{N},\tilde{\mathcal{G}})$ so that $\tilde{f}$ fixes $\tilde{x}$. $\tilde{f}$ induces a diffeomorphism $\overline{f}$ on $G=\tilde{N}/\tilde{\mathcal{G}}$. Let $g$ be an automorphism of $G$ which is induced by an element $(f_{*})^{-1}$ of $\Aut(\mathfrak{g})$. Then $\overline{f} \circ g = L_{h}$ for some $h$ in $G$, since $d(\overline{f} \circ g)$ preserves every left-invariant vector field on $G$. But since $\overline{f} \circ g$ fixes $e$, we have $\overline{f} \circ g = id_{G}$.
We prove the latter part. By the definition of $hol$, we have $hol(\gamma)=p \circ \tilde{\gamma}(1)$ for each element $\gamma$ in $\pi_1(N,x)$ where $p$ is the canonical projection $p \colon \tilde{N} \longrightarrow \tilde{N}/\tilde{\mathcal{G}}=G$ and $\tilde{\gamma}$ is the lift of $\gamma$ to $\tilde{N}$ such that $\tilde{\gamma}(0)=\tilde{x}$. Then we have $\overline{f}(hol(\gamma))=\overline{f} \circ p \circ \tilde{\gamma}(1)= p \circ \tilde{f} \circ \tilde{\gamma}(1) = p \circ \tilde{f \circ \gamma}(1) = hol(f \circ \gamma)$, since $\tilde{f}$ fixes $\tilde{x}$.
(3) is clear from the construction.
\end{proof}
Note that every element of $\pi_{0}(\Diff(N,\mathcal{G}))$ is represented by an element of $\Diff(N,x,\mathcal{G})$. In fact, $(N,\mathcal{F})$ is a Lie foliation and the identity component of $\Diff(N,\mathcal{G})$ acts to $N$ transitively. Hence Proposition~\ref{algebraicaction} is reduced to the algebraicity of $A \colon \Aut(G,\Gamma) \longrightarrow \Aut(E^{0,l}_2)$.
We prove the following proposition applying Mal'cev theory.
\begin{prop}\label{malcev}
Let $G$ be an $l$-dimensional simply connected nilpotent Lie group and $\Gamma$ be a finitely generated dense subgroup of $G$. We put $\mathfrak{g}=\Lie(G)$. If we denote the group of automorphisms of $G$ which preserves $\Gamma$ by $\Aut(G,\Gamma)$, then the image of the canonical action
\begin{equation}
A \colon \Aut(G,\Gamma) \longrightarrow \Aut(H^{l}(\mathfrak{g}))
\end{equation}
is contained in the set of algebraic numbers where $\Aut(H^{l}(\mathfrak{g}))$ is canonically identified with $\mathbb{R}-\{0\}$.
\end{prop}
\begin{proof}
At first, we apply Mal'cev theory following Ghys~\cite{Ghy}. We refer to~\cite{Mal} and Chapter II of~\cite{Rag} on Mal'cev theory. We have a simply connected nilpotent Lie group $H$ and an embedding $i \colon \Gamma \longrightarrow H$ such that $i(\Gamma)$ is a uniform lattice of $H$ by Mal'cev theory. The homomorphism $i^{-1} \colon i(\Gamma) \longrightarrow G$ can be extend to a homomorphism $\pi \colon H \longrightarrow G$ again by Mal'cev theory. $\pi$ is clearly surjective, since $\Gamma$ is dense in $G$. We also have a lift $\tilde{f}$ of $f$ which is an automorphism of $H$ and preserves $i(\Gamma)$ again by Mal'cev theory, since $i(\Gamma)$ is a uniform lattice of $H$. Then we have a diagram:
\begin{equation}\label{diag2}
\xymatrix{
H \ar[dd]_{\pi} \ar[rrr]^{\tilde{f}} & & & H \ar[dd]^{\pi} \\
& \Gamma \ar[ul]^{i} \ar[ld] \ar[r]^{f} & \Gamma \ar[ur]_{i} \ar[rd] \\
G \ar[rrr]^{f} & & & G }
\end{equation}
which satisfies the following conditions:
\begin{description}
\item[(a)] $H$ is nilpotent,
\item[(b)] $\pi$ is surjective and
\item[(c)] $i$ is an embedding of a uniform lattice.
\end{description}
Proposition~\ref{malcev} follows from the following lemma:
\begin{lem}\label{ind}
If we have a diagram~\eqref{diag2} which satisfies conditions (a), (b) and (c), then $A(f) \colon H^{l}(\mathfrak{g}) \longrightarrow H^{l}(\mathfrak{g})$ is algebraic under the canonical identification of $\Aut(H^{l}(\mathfrak{g}))=\mathbb{R}-\{0\}$.
\end{lem}
We prove Lemma~\ref{ind} inductively on the rank of $H$ as a nilpotent Lie group.
If $H$ is abelian, $(H,\Gamma)$ is isomorphic to $(\mathbb{R}^{m},\mathbb{Z}^{m})$. Then $f$ is an element of $\SL(m;\mathbb{Z})$. $f$ induces a homomorphism $\hat{f} \colon \wedge^{l} (\mathbb{R}^{m})^{*} \longrightarrow \wedge^{l} (\mathbb{R}^{m})^{*}$ and $\hat{f}$ has integral entries with respect to the standard basis, since the entries of $\hat{f}$ are minor determinants of $f$. $H^{l}(\mathfrak{g})$ is generated by an element represented by left invariant volume forms of $G$ and left invariant volume forms of $G$ are eigenvectors of $\hat{f}$ by the diagram~\eqref{diag2}. Then $A(f)$ is algebraic, since $A(f)$ is the eigenvalue of $\hat{f}$ with respect to left invariant volume forms of $G$.
Assume that we have the diagram~\eqref{diag2} which satisfies the conditions (a),(b) and (c) where the rank of $H$ is $n$ and the claim of Lemma~\ref{ind} is correct if the rank of $H$ is less than $n$. Then $A([f,f])$ and $A(H_1(f))$ are algebraic under the identification of $H^{l'}([\mathfrak{g},\mathfrak{g}])$ and $H^{l''}(\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])$ with $\mathbb{R}-\{0\}$ where $l'=\dim [\mathfrak{g},\mathfrak{g}]$ and $l''=\dim \mathfrak{g}/[\mathfrak{g},\mathfrak{g}]$ by the following diagrams:
\begin{equation}
\xymatrix{
[H,H] \ar[dd]_{[\pi,\pi]} \ar[rrr]^{[\tilde{f},\tilde{f}]} & & & [H,H] \ar[dd]^{[\pi,\pi]} \\
& \Gamma \cap [H,H] \ar[ul]^{[i,i]} \ar[ld] \ar[r]^{[f,f]} & \Gamma \cap [H,H] \ar[ur]_{[i,i]} \ar[rd] \\
[G,G] \ar[rrr]^{[f,f]} & & & [G,G] }
\end{equation}
and
\begin{equation}
\xymatrix{
H/[H,H] \ar[dd]_{H_1(\pi)} \ar[rrr]^{H_1(\tilde{f})} & & & H/[H,H] \ar[dd]^{H_1(\pi)} \\
& p(\Gamma) \ar[ul]^{H_1(i)} \ar[ld] \ar[r]^{H_1(f)} & p(\Gamma) \ar[ur]_{H_1(i)} \ar[rd] \\
G/[G,G] \ar[rrr]^{H_1(f)} & & & G/[G,G] }
\end{equation}
where $p$ is the canonical projection $H \longrightarrow [H,H]$. Then we have the algebraicity of $A(f)$, since we have $A(f)=A([f,f])\cdot A(H_1(f))$ under the identification of $\Aut(H^{l}(\mathfrak{g})), H^{l'}([\mathfrak{g},\mathfrak{g}])$ and $H^{l''}(\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])$ with $\mathbb{R}-\{0\}$ by the following diagram:
\begin{equation}
\xymatrix{ 0 \ar[r] & [G,G] \ar[r] \ar[d]^{[f,f]} & G \ar[r] \ar[d]^{f} & G/[G,G] \ar[r] \ar[d]^{H_1(f)} & 0 \\
0 \ar[r] & [G,G] \ar[r] & G \ar[r] & G/[G,G] \ar[r] & 0.}
\end{equation}
Hence Lemma~\ref{ind} and Proposition~\ref{malcev} are proved.
\end{proof}
\section{Examples}
\subsection{Torus fibration over $S^1$}
Let $A$ be an element of $\SL(n;\mathbb{Z})$ with an eigenvector $v$ with respect to an eigenvalue $\lambda$. Assume that the components of $v$ are linearly independent over $\mathbb{Q}$ and $\lambda$ is a positive real number. For an example, take
\begin{equation}
A=\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}, v= \begin{pmatrix} \frac{\sqrt{5}-1}{2} \\ -1 \end{pmatrix}, \lambda=\frac{3-\sqrt{5}}{2}.
\end{equation}
$A$ induces a diffeomorphism $\overline{A}$ on $T^{n}=\mathbb{R}^{n}/\mathbb{Z}^{n}$. We denote the mapping torus $T^{n} \times [0,1]/(\overline{A}w,0) \sim (w,1)$ of $\overline{A}$ by $M$ and define a map $\pi \colon M \longrightarrow S^1$ by $\pi([(w,t)])=t$, which gives a $T^n$ fibration over $S^1$. Since $v$ is an eigenvector of $A$, we have a foliation $\mathcal{F}$ on $M$ formed by the lines parallel to $v$ in each $T^{n}$ fiber of $\pi$. By the assumption on $v$, the leaves of $\mathcal{F}$ are dense in the fibers of $\pi$. $(M,\mathcal{F})$ is Riemannian since we can construct a bundle-like metric $g$ of $(M,\mathcal{F})$ by $g = \rho(t)g_0 + (1-\rho(t))g_0 + dt \otimes dt$ where $g_0$ is the flat metric on $T^n$ and $\rho(t)$ is a smooth function which satisfies $\rho(t)=0$ near $0$ and $\rho(t)=1$ near $1$. Note that the structure Lie algebra of $(M,\mathcal{F})$ is abelian.
In this case, the mean curvature form of $(M,\mathcal{F},g)$ is a closed form on $S^1$ and we can calculate the \'{A}lvarez class $[\kappa_b]$ of $(M,\mathcal{F})$ directly to obtain
\begin{equation}
\int_{S^1}[\kappa_b]= \log \lambda.
\end{equation}
\subsection{A Riemannian foliation on a solvmanifold}
We present an example of a $2$-dimensional Riemannian foliation on a $6$-dimensional nilmanifold bundle over $S^1$ with nonabelian nilpotent structure Lie algebra and nontrivial \'{A}lvarez class.
Let $p$ be a prime and $\alpha$ be an element of $\mathbb{Z}(\sqrt{p})$ which has an inverse $\beta$ in $\mathbb{Z}(\sqrt{p})$. We put $k=\GCD(\alpha_2,\beta_2)$ where $\alpha_2$ and $\beta_2$ are integers which satisfies $\alpha=\alpha_1+\alpha_2\sqrt{p}$ and $\beta=\beta_1+\beta_2\sqrt{p}$ for some integers $\alpha_1$ and $\beta_1$. Let $G$ be a nilpotent Lie group which is defined by
\begin{equation}
G=\Big\{ \begin{pmatrix} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{pmatrix} \Big| x,y,z \in \mathbb{R} \Big\}
\end{equation}
and $\Gamma$ be a subgroup of $G$ which is generated by
\begin{equation}
A_1=\begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix},
A_2=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{pmatrix},
A_3=\begin{pmatrix} 1 & k\sqrt{p} & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix},
A_4=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & k\sqrt{p} \\ 0 & 0 & 1 \end{pmatrix}.
\end{equation}
$\Gamma$ is dense in $G$, since we have $[A_1,A_2]=\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$, $[A_1,A_4]=\begin{pmatrix} 1 & 0 & k\sqrt{p} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ and the Lie algebra of the closure of $\Gamma$ is equal to the Lie algebra of $G$.
We define a Lie group $H$ by $H=G \oplus \mathbb{R}^{2}$. Let $\iota'$ be a homomorphism $\iota' \colon \Gamma \longrightarrow \mathbb{R}^{2}$ which is defined by
\begin{equation}
\iota'\Big(\begin{pmatrix} 1 & x_1 + x_2 \sqrt{p} & z \\ 0 & 1 & y_1 + y_2 \sqrt{p} \\ 0 & 0 & 1 \end{pmatrix} \Big)=(x_2,y_2)
\end{equation}
where $x_1,x_2,y_1$ and $y_2$ are integers and we define an embedding $\iota \colon \Gamma \longrightarrow H$ by $\iota(g)=(g,\iota'(g))$ for every $g$. Then $\iota(\Gamma)$ is a uniform lattice of $H$. The fibers of the first projection $H \longrightarrow G$ are preserved by the right multiplication of $\iota(\Gamma)$ to $H$ and define a $G$-Lie foliation $\mathcal{G}$ of dimension $2$ and codimension $3$ on $H/\iota(\Gamma)$.
Let $\begin{pmatrix} a' & b' \\ c' & d' \end{pmatrix}$ be an element of $\SL(2;\mathbb{Z})$ and put $\begin{pmatrix} a & b \\ c & d \end{pmatrix}=\begin{pmatrix} a'\alpha & b'\alpha \\ c'\alpha & d'\alpha \end{pmatrix}$. Let $f$ be a map $G \longrightarrow G$ defined by
\begin{equation}
f\begin{pmatrix} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{pmatrix}=\begin{pmatrix} 1 & ax+by & \alpha^{2} z+acx^2+bdy^2+bcxy \\ 0 & 1 & cx+dy \\ 0 & 0 & 1 \end{pmatrix}.
\end{equation}
Then $f$ is a homomorphism from $G$ to $G$ by the definition of $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ and $\alpha$. Since $f$ is bijective, $f$ is an automorphism. Clearly $f$ preserves $\Gamma$. Moreover $f(\Gamma)=\Gamma$ follows from the definition of $k$ and $\Gamma$. Note that there exists an element $g$ of $\Gamma$ which satisfies $g=\begin{pmatrix} 1 & 0 & \delta \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ for every $\delta$ in $\mathbb{Z}(\sqrt{p})$ which has a form $\delta=\delta_1+\delta_2 k\sqrt{p}$ where $\delta_1$ and $\delta_2$ are integers.
Since $\iota(\Gamma)$ is a uniform lattice of $H$, the automorphism $f|_{\Gamma}$ of $\Gamma$ is uniquely extended to an automorphism $\tilde{f}$ of $H$ and induces a diffeomorphism $\overline{f}$ on $H/\iota(\Gamma)$. Since $\overline{f}$ preserves $\mathcal{G}$, we have a foliation $\mathcal{F}$ on the mapping torus $M$ of $\overline{f}$ which is Riemannian. By $f^{*}(dx \wedge dy \wedge dz)=\alpha^{3} dx \wedge dy \wedge dz$ and Proposition~\ref{holonomyformula}, we have
\begin{equation}
\int_{S^1}[\kappa_b]= \log \alpha^3
\end{equation}
where $S^1$ is the base space of the canonical fibration $M \longrightarrow S^1$ of the mapping torus and $[\kappa_b]$ is the \'{A}lvarez class of $(M,\mathcal{F})$.
\end{document} | math | 33,779 |
\begin{document}
\maketitle
\begin{abstract}
This paper contains a stronger version of a final identification theorem for the
`generic' groups of finite Morley rank.
\end{abstract}
\section{Introduction}
This paper contains a version of a generic identification theorem for simple groups of finite Morley rank adapted from the main result of \cite{BB04} by weakening its assumptions. It is published because it is being used in the authors' study of actions of groups of finite Morley rank \cite{BB11}.
A general discussion of the subject can be found in the books \cite{bn} and \cite{abc}.
A group of finite Morley rank is said to be of {\em $p'$-type}, if
it contains no infinite abelian subgroup of exponent $p$. Notice
that a simple algebraic group over an algebraically closed field $K$ is of $p'$-type if
and only if ${\rm char}\, K \ne p$. Other definitions can be found in
the next section.
The aim of this work is to prove the following.
\begin{theorem}
Let\/ $G$\/ be a simple group of finite
Morley rank and\/ $D$ a maximal\/ $p$-torus in $G$
of Pr\"{u}fer rank at least\/ $3$. Assume that
\begin{itemize}
\item[{\rm (A)}] Every proper connected definable subgroup of $G$ which contains $D$ is a $K$-group.
\item[{\rm (B)}] For every element\/ $x$ of order\/
$p$ in $D$, the group $C^\circ_G(x)$ is of\/ $p'$-type and
$C^\circ_G(x)=F^\circ(C^\circ_G(x))E(C^\circ_G(x)) $.
\item[{\rm (C)}] $\langle C^\circ_G(x) \mid x \in D, \;|x|=p\rangle = G$.
\end{itemize}
Then $G$ is a Chevalley group over an algebraically closed field of characteristic distinct
from
$p$.
\label{sis-kebap}
\end{theorem}
Notice that, under assumption (B) of Theorem~\ref{sis-kebap},
$C^\circ_G(x)$ is a central product of $F^\circ(C^\circ_G(x))$
and $E(C^\circ_G(x))$.
The predecessor of this theorem, Theorem 1.2 of \cite{BB04}, was based on a stronger assumption than our assumption (A), namely, that \emph{every} proper definable subgroup of $G$ is a $K$-group, a condition that was difficult to check in its actual applications.
The proof of Theorem~\ref{sis-kebap} is given in
Section~\ref{sec:sis-kebap}.
\subsection{Definitions}
All definitions related to groups of finite Morley rank in general
can be found in \cite{bn} and \cite{abc}.
{}From now on $G$ is a group of finite Morley rank. The group $G$
is called
a {\it $K$-group}, if every
infinite simple definable and connected section of the group is an
algebraic group over an algebraically closed field.
A {\em $p$-torus} $S$ is a direct product of finitely many copies
of the quasicyclic group ${{\Bbb Z}}_{p^\infty}$. The number of
copies is called the {\em Pr\"{u}fer $p$-rank} of $S$ and is
denoted by ${\rm pr}(S)$. For a definable group $H$, ${\rm
pr}(H)$ is the maximum of the Pr\"{u}fer ranks of $p$-subgroups
in $H$. It is easy to see that the Pr\"{u}fer $p$-rank of any
subgroup of a group of finite Morley rank is finite.
A group $H$ is called {\it quasi-simple} if $H'=H$ and $H/Z(H)$
is simple and non-abelian. A quasi-simple subnormal subgroup of
$G$ is called a {\it component} of $G$. The product of all
components of $G$ is called the {\it layer} of $G$ and denoted by
$L(G)$, and $E(G)$ stands for $L^\circ(G)$. It is known (see
Lemmas 7.6 and 7.10 in \cite{bn}) that $G$ has finitely many
components and that they are definable and are normal in $E(G)$.
$F(H)$ is the
{\em Fitting subgroup} of $H$, that is, the maximal normal definable
nilpotent subgroup.
If $H$ is a group of finite Morley rank then $B(H)$ is the
subgroup generated by all definable connected $2$-subgroups of
bounded exponent in $H$. Note that $B(H)$ is connected by
Assertion~\ref{zilb}.
\section{Background Material}
\subsection{Algebraic Groups}
For a discussion of the model theory of algebraic groups, the reader
might like to see Section 3.1 in \cite{berkman}.
The basic structural facts and definitions related to algebraic groups
can be easily found in the standard references
such as \cite{cartk,hump}.
First note that a connected algebraic group $G$ is called {\em
simple} if it has no proper normal connected and closed
subgroups. Such a group turns out to have a finite center, the
quotient group being simple as an abstract group. The classical
classification theorem for simple algebraic groups states that
simple algebraic groups over algebraically closed fields are
Chevalley groups, that is, groups constructed from Chevalley
bases in simple complex Lie algebras as described, for example, in
\cite{cartk}.
Now fix a maximal torus $T$ in a connected algebraic group
$G$ and denote the corresponding
root system by $\Phi$, and for each $\alpha\in \Phi$, denote the
corresponding root subgroup by $X_\alpha$. The subgroup $\langle
X_\alpha,X_{-\alpha}\rangle$ is known to be isomorphic to $SL_2$
or $PSL_2$ and is called a {\em root $SL_2$-subgroup}.
If $G$ is simple,
the roots can have at most two different lengths, and the terms `short
root $SL_2$-subgroup' and `long root $SL_2$-subgroup'
have the obvious meanings.
A simple algebraic group is generated by its root $SL_2$-subgroups. In a
simple algebraic group, all long root
$SL_2$-subgroups are conjugate to each other, and similarly all short
root $SL_2$-subgroups are conjugate to each other.
\begin{fact}
\label{prop:class-in-algebraic} Suppose that\/ $G$ is a simple algebraic
group over an algebraically closed field. Let\/ $T$ be a
maximal torus in $G$ and $K$, $L$ closed subgroups of $G$
that are isomorphic to $SL_2$ or\/ $PSL_2$ and are
normalised by $T$. Then the following hold.
\begin{enumerate}
\item Either\/ $[K,L]=1$ or\/ $\langle
K,L\rangle$ is a simple algebraic group of rank $2$;
that is of type $A_2$, $B_2$ or\/ $G_2$.
\item The subgroups $K$ and $L$
are embedded in $\langle K,L\rangle$ as root $SL_2$-subgroups.
\item If $\langle K,L\rangle$ is of type $G_2$, then $G=\langle K,L\rangle$.
\end{enumerate}
\end{fact}
\paragraph{Proof.} The proof follows from the description of closed
subgroups in simple algebraic groups normalised by a maximal torus
\cite[2.5]{seitz}; see also \cite[Section~3.1]{seitz2}.
$\Box$
\begin{fact}
Let\/ $G$ be a simple algebraic group over an algebraically
closed field of characteristic $\neq p$,
and let $D$ be a maximal $p$-torus in $G$. Then
$C_G(D)$ is a maximal torus in $G$. \label{fact:p-torus}
\end{fact}
\begin{proof} The proof follows from the description of
centralisers of subgroups of commuting semisimple elements in
simple algebraic groups \cite[Theorem~5.5.8]{ste1}.
\end{proof}
\subsection{Groups of Finite Morley Rank}
\begin{fact}
\label{zilb} {\rm (Zil'ber's Indecomposability Theorem)}
A subgroup of a
group of finite Morley rank which is generated by a family
of definable connected subgroups is also definable
and connected.
\end{fact}
\begin{proof} See \cite{zilber} or Corollary~5.28 in \cite{bn}.
\end{proof}
\begin{fact} {\rm \cite[Theorem~8.4]{bn}}
Let\/ $G\rtimes H$ be a group of finite Morley rank, where $G$
and\/ $H$ are definable, $G$ is an infinite simple algebraic group
over an algebraically closed field and\/ $C_H(G)=1$. Then $H$ can be viewed as a
subgroup of the group of automorphisms of\/ $G$, and moreover\/
$H$ lies in the product of the group of inner automorphisms and
the group of graph automorphisms of $G$ (which preserve root
lengths). In particular, when $H$ is connected, then $H$ consists
of inner automorphisms only. \label{defaut}
\end{fact}
\begin{fact}
{\rm \cite{ac}} \label{centext}
Suppose $G$ is a group of finite
Morley rank, $G=G'$, and\/ $G/Z(G)$ is a simple algebraic group
over an algebraically closed field, and is of finite Morley rank, then $Z(G)$ is
finite and\/ $G$ is also algebraic.
\end{fact}
\begin{lemma}
Let\/ $G$ be a connected $K$-group of\/ $p'$-type and\/ $D$ a
maximal\/ $p$-torus in $G$. If\/ $L \lhd G$ is a component in $G$,
then $D\cap L$ is a maximal\/ $p$-torus in $L$ and\/ $D=C_D(L)(D\cap L)$.
\label{lm:torus-in-component}
\end{lemma}
\begin{proof} As $G$ is connected, $L \triangleleft G$. Now the lemma
immediately follows from Assertions~\ref{centext}, \ref{defaut} and
\ref{fact:p-torus}. \end{proof}
\begin{lemma}
\label{algcomps} Under the assumptions of Theorem~{\rm
\ref{sis-kebap}}, we have, for every $p$-element\/ $t\in D$,
\[C^\circ_G(t)=F\cdot L_1\cdots L_r,\]
where $F=F^\circ(C_G^\circ(t))$ and each\/ $L_i$ is a simple
algebraic group over an algebraically closed field of
characteristic $ \ne p$.
\end{lemma}
\begin{proof} For every $p$-element $t$ in $G$, $C^\circ_G(t)=F\cdot
E(C^\circ_G(t))$. And assumption (C) ensures that
$C^\circ_G(t)$ is a $K$-group, hence its components are algebraic
groups by Assertion~\ref{centext}. \end{proof}
\subsection{Lyons's Theorem}
A detailed discussion of this particular version
of Lyons's theorem can be found in \cite{berkman}.
\begin{fact}[(Lyons \cite{gls,gls3})]
\label{Lyons} Suppose that\/ ${\Bbb F}$ is an algebraically closed
field, $I$ is one of the connected Dynkin diagrams of the simple
algebraic groups of Tits rank at least\/ $3$ and $\tilde{G}$ is
the simply connected simple algebraic group of type $I$ over
${\Bbb F}$. Let\/ $G$ be an arbitrary group and for each $i\in I$,
$K_i$ stand for a subgroup of $G$ which is centrally isomorphic to
$PSL_2({\Bbb F})$, and\/ $T_i < K_i$ denote a maximal torus in
$K_i$. Also assume that the following statements hold.
\begin{enumerate}
\item The group $G$ is generated by $K_i$ where $i\in I$.
\item For all\/ $i,j \in I$, $[T_i,T_j]=1$.
\item If\/ $i\ne j$ and $(i,j)$ is not an edge
in $I$, then $[K_i,K_j]=1$.
\item If\/ $(i,j)$ is a single edge in $I$, then
$G_{ij} =\langle K_i,K_j\rangle$ is isomorphic to
$(P)SL_3({\Bbb F})$.
\item If\/ $(i,j)$ is a double edge in $I$, then
$G_{ij}=\langle K_i,K_j\rangle$ is isomorphic to
$(P)Sp_4({\Bbb F})$.
Moreover, in that case, if\/ $r_i\in N_{K_i}(T_iT_j)$ and
$r_j\in N_{K_j}(T_iT_j)$ are involutions, then
the order of\/ $r_ir_j$ in $N_{G_{ij}}(T_iT_j)/T_iT_j$
is $4$.
\item For all $i,j\in I$, $K_i$ and $K_j$
are root\/ $SL_2$-subgroups of\/ $G_{ij}$
corresponding to the maximal torus $T_iT_j$ of\/ $G_{ij}$.
\end{enumerate}
Then there is a homomorphism from $\tilde{G}$ onto
$G$, under which the root $SL_2$-subgroups of $\tilde{G}$
{\rm (}for some simple root system{\rm )} correspond to the subgroups $K_i$.
\end{fact}
\subsection{Reflection Groups}
A linear semisimple transformation of finite order is called a
{\it reflection} if it has exactly one eigenvalue which is not 1.
\begin{theorem}
Let\/ $W$ be a finite group and assume that the following
statements hold.
\begin{enumerate}
\item There is a normal subset $S\subseteq W$ consisting of involutions
and generating $W$.
\item Over $\Bbb C$, $W$ has a faithful irreducible
representation of dimension $n\geqslant 3$ in which involutions from $S$ act as reflections.
\item For almost all prime numbers $q$, $W$ has faithful irreducible
representations {\rm(}possibly of different dimensions{\rm )} over
fields\/ ${\Bbb F}_q$. Moreover, for every such representation,
involutions in $S$ act as reflections.
\end{enumerate}
Then $W$ is one of the
groups $A_n$, $B_n$, $C_n$, $D_n$, $E_6$, $E_7$, $E_8$, $F_4$ for
$n\geqslant 3$.
\label{reflection}
\end{theorem}
\begin{proof} A proof, based on the classification of irreducible
complex reflection groups \cite{shephard-todd}, can be found in
\cite{berkman}. \end{proof}
\section{Proof of Theorem~\ref{sis-kebap}}
\label{sec:sis-kebap}
The strategy is to construct the Weyl group and the root system
of $G$, and then to apply Lyons's Theorem. So from now on $G$ is
a simple group of finite Morley rank and $D$ is a maximal $p$-torus in $G$
of Pr\"{u}fer rank $\geqslant 3$ and such that every proper definable connected subgroup contatining $D$ is a $K$-group. We also assume that
$C^\circ_G(x)$ is of $p'$-type for every element $x \in D$ of
order $p$, $C^\circ_G(x)=F^\circ(C^\circ_G(x))E(C^\circ_G(x)) $
and
\[G = \langle C^\circ_G(x) \mid x \in D, \; |x|=p\rangle.\]
Notice that if $M$ is a proper definable subgroup in $G$ normalised by $D$ then $MD$ is a proper definable subgroup of $G$, for otherwise $M$ would be normal in $G$, which contradicts simplicity of $G$. Therefore, $MD$ and hence $M$ are $K$-groups by assumption (A). This observation will be used throughout the proof.
We also shall systematically use the following observation.
\begin{lemma}
$F^\circ(C^\circ_G(x))$ centralises
$D$ for every element $x \in D$ of order $p$. \label{lm:F}
\end{lemma}
\begin{proof} The result immediately follows from the fact that a
$p$-torus in a definable nilpotent group belongs to the center of
this group \cite[Theorem~6.9]{bn}. \end{proof}
\subsection{Root Subgroups}
{}From now on, $SL_2$ will be used instead of $SL_2({\Bbb F})$, etc.
Denote by $\Sigma$ the set of all definable subgroups isomorphic
to $(P)SL_2$ and normalised by $D$. These are our future root
$SL_2$-subgroups.
If $N$ is a subgroup of $G$ which is normalised by $D$, then set
$H_N:=C_N(D\cap N)$. Note that if $K\in\Sigma$, then $H_K$ is
an algebraic torus in $K$.
\begin{lemma}
The set\/ $\Sigma$ is non-empty.
\end{lemma}
\begin{proof} Assume the contrary. If $L=E(C^\circ_G(x)) \ne 1$ for
some element $x$ of order $p$ from $D$, then, $L$ being a central
product of simple algebraic groups, contains a definable
$SL_2$-subgroup normalised by $D$. Therefore $C^\circ_G(x) =
F^\circ(C^\circ_G(x))$ centralises $D$ by Lemma~\ref{lm:F}. But
then \[G = \langle C^\circ_G(x) \mid x \in D, \; |x|=p\rangle\]
centralises $D$ which contradicts the assumption that $G$ is
simple. \end{proof}
\begin{lemma}
\label{pairs} Let $K,L\in \Sigma$ be distinct and set
$M = \langle K, L\rangle$. Then the following statements hold.
\begin{itemize}
\item[{\rm (1)}] The subgroup
$C_D(K) \cap C_D(L) \ne 1$ and $M$ is a $K$-group.
\item[{\rm (2)}] Either\/ $K$ and\/ $L$ commute or\/
$M$ is an algebraic group of type $A_2$, $B_2$ or $G_2$.
\item[{\rm (3)}] $D\cap M = (D\cap K)(D\cap L)$ is a maximal\/ $p$-torus in
$M$.
\item[{\rm (4)}] If\/ $K$ and\/ $L$
do not commute then $H_M$ is a
maximal algebraic torus of the algebraic group $M$, and
$K$ and\/ $L$ are root\/ $SL_2$-subgroups of the algebraic
group $M$ with respect to the maximal torus $H_M$.
\item[{\rm (5)}] For all $K, L\in\Sigma$, we have $[H_K,H_L]=1$.
\item[{\rm (6)}] For any $K,L\in\Sigma$, if the $p$-tori
$D\cap K$ and\/ $D\cap L$ have
intersection of order $>2$, then $K=L$.
\end{itemize}
\label{lm:tori}
\end{lemma}
\begin{proof} For the proof of $C_D(K) \cap C_D(L) \ne 1$ we refer the reader to the proof of \cite[Lemma~9.3]{berkman}. After that $M \leqslant C_G(C_D(K) \cap C_D(L))$ is a proper definable subgroup of $G$ and is a $K$-group since $D$ normalizes $M$.
(2)-(3) For $L\in \Sigma$ set $R_L = C^\circ_D(L)$. If $n =
{\rm pr}(D)$, then the Pr\"{u}fer $p$-rank of $R_L$ is $n-1$.
Note that since $D$ is maximal and $D$ centralises a $p$-torus in
$L$, $D \cap L$ is a maximal $p$-torus in $L$. Now let $x$ be an
element of order $p$ in $R_K\cap R_L$. Then $K,L\leqslant
E(C_G(x))$ by the assumptions of the theorem. Set $E =
E(C_G(x))$. It follows from Lemma~\ref{lm:torus-in-component}
that the subgroup $D \cap E$ is a maximal $p$-torus of $E$, and
the subgroups $K$ and $L$, being $D$-invariant, lies in
components of $E$. If $K$ and $L$ belong to different components
of $E$, then they commute. Otherwise the component $A$ that
contains both $K$ and $L$ is a simple algebraic group, and
moreover $D\cap A$ is a maximal $p$-torus in $A$. Hence
the results follow from Assertion~\ref{prop:class-in-algebraic}.\\
(4)-(6) These follow by inspecting case by case and Assertion~\ref{prop:class-in-algebraic}. \end{proof}
\begin{lemma}
The subgroups in $\Sigma$ generate $G$.
\label{lm:sigmagenerates}
\end{lemma}
\begin{proof} Let $x\in D$ be of order $p$, then by assumption (B) of
the theorem
\[C^\circ_G(x) = F \cdot L_1\cdots L_n,\]
where $F = F^\circ(C^\circ_G(x))$ and $L_i \lhd C^\circ_G(x)$ is
a simple algebraic group, for each $i=1,\ldots,n$.
The first step is to prove that $L_1\cdots
L_n\leqslant\langle\Sigma\rangle$. Note that $D\leqslant
C^\circ_G(x)$ and $D\cap L_i$ is a maximal $p$-torus in $L_i$ by
Lemma~\ref{lm:torus-in-component}. Let $H_i$ stand for the
maximal algebraic torus in $L_i$ containing $D\cap L_i$ and
$\Gamma_i$ be the collection of root $SL_2$-subgroups in $L_i$
normalised by $H_i$. Since $D\cap L_i\leqslant H_i$, $D\cap L_i$
normalises the subgroups in $\Gamma_i$. By Lemma~\ref
{lm:torus-in-component}, we have $D=C_D(L_i)(D\cap L_i)$, hence
$D$ normalises $\langle\Gamma_i\rangle=L_i$; that is
$\Gamma_i\subseteq \Sigma$ for each $i=1,\ldots,n$. This proves
the first step.
Hence for each $x\in D$ of order $p$, $C^\circ_G(x)=F\cdot
E(C^\circ_G(x))\leqslant F\langle\Sigma\rangle \leqslant
C_G(D)\langle\Sigma\rangle$. Therefore
\[G=\langle C^\circ_G(x)\mid x\in D, |x|=p\rangle\leqslant C_G(D)
\langle\Sigma\rangle.\] Since $C_G(D)$ normalises
$\langle\Sigma\rangle$, we have $\langle\Sigma\rangle\unlhd G$.
Now the result follows, since $G$ is simple. \end{proof}
We make $\Sigma$ into a graph by taking $SL_2$-subgroups $L \in
\Sigma$ for vertices and connecting two vertices $K$ and $L$ by
an edge if $K$ and $L$ do not commute.
\begin{lemma}
The graph $\Sigma$ is connected. \label{lm:connected}
\end{lemma}
\begin{proof} Otherwise consider a decomposition $\Sigma = \Sigma'
\cup \Sigma''$ of $\Sigma$ into the union of two non-empty sets
such that no vertex in $\Sigma'$ is connected to a vertex in
$\Sigma''$. Then we have
\[G = \langle \Sigma \rangle = \langle \Sigma' \rangle \times \langle \Sigma'' \rangle,
\]
which contradicts the assumption that $G$ is simple. \end{proof}
\begin{lemma}
If\/ $L \in \Sigma$ then $L = E(C_G(C_D(L)))$. \label{lm:L-unique}
\end{lemma}
\noindent {\bf Proof.} $\,$ Let ${\rm pr}(D) = n$, then ${\rm pr}(C_G(C_D(L)))=n$
and ${\rm pr}(E(C_G(C_D(L)))) =1$. Since $L \leqslant
E(C_G(C_D(L)))$ and $E(C_G(C_D(L)))$ is a central product of
simple algebraic groups over algebraically closed fields of
characteristic $\ne p$, we immediately conclude that $L =
E(C_G(C_D(L)))$.
$\Box$
\subsection{Weyl Group}
Recall that when $L\in\Sigma$, $H_L$ stands for the maximal
algebraic torus $H_L:=C_L(D\cap L)$ in $L\cong SL_2$. Now set $H =
\langle H_L \mid L \in \Sigma \rangle$ and call it the {\em
natural torus associated with $D$}. It easily follows from
Lemma~\ref{lm:tori}(5) that $H$ is a divisible abelian group.
For any $L \in \Sigma$, $W(L):=N_L(H)H/H=N_L(H_L)/H_L$ is the
Weyl group of $SL_2$ and has order $2$; hence $W(L)$ contains a
single involution, which will be denoted by $r_L$. Note that the
subgroup $L$ is uniquely determined by $r_L$, since $C_D(L) =
C^\circ_D(r_L)$ and $L = E(C_G(C_D(L)))$ by
Lemma~\ref{lm:L-unique}.
\begin{lemma}
\label{refgp} Consider a graph\/ $\Delta$ with the set of vertices
$\Sigma$, in which two vertices $K$ and\/ $L$ are connected by an edge
if\/ $[r_K, r_L] \ne 1$. If\/ $K$ and\/ $L$ belong to different
connected components of\/ $\Delta$, then $[K,L] =1$.
\end{lemma}
\noindent {\bf Proof.} $\,$ It suffices to check this statement in the subgroup
$M=\langle K,L\rangle$, where it is obvious by Lemma~\ref{pairs}(2).
$\Box$
Notice that $D$ is a $p$-torus, the subgroups $N_G(D)$ and
$C_G(D)$ are definable and the factor group $N_G(D)/C_G(D)$ is
finite. Set $W:=N_G(D)/C_G(D)$. The images of involutions $r_L$,
for $L\in \Sigma$, in $W$ generate a subgroup which we denote by
$W_0$. Since, by their construction, involutions $r_L$ normalise
$D$, there is a natural action of $W_0$ on $D$.
\begin{lemma} The $p$-torus $D$ lies in the natural torus $H$.
In particular, $D$ is the Sylow $p$-subgroup of\/ $H$. \label{lm:D<H}
\end{lemma}
\noindent {\bf Proof.} $\,$ Set $D' = \langle D\cap L \mid L \in \Sigma \rangle$. It suffices
to prove that $D'=D$. First note that $D'\leqslant D \cap H$.
If $D'<D$ then, since $[D, r_L] = D\cap L$,
all involutions $r_L$ act trivially on the factor group $D/D'$
which is divisible.
Let us take an element $d \in D$ which has sufficiently big order
so that the image of $d^{|W_0|}$ in $D/D'$ has order at least
$p^2$. Then the element $$z=\prod_{w\in W_0} d^w$$ has the same
image in $D/D'$ as $d^{|W_0|}$ and thus $z$ has order at least
$p^2$. Since $D = C_D(L)(D\cap L)$ and $|C_D(L) \cap (D\cap L)|
\leqslant |Z(L)| \leqslant 2$, we see that $|C_D(r_L):
C_D(L)|\leqslant 2$. Of course, the equality is possible only if
$p=2$. In any case, since $z \in C_D(r_L)$, $z^p \in C_D(L)$ for
all $L \in \Sigma$ and $z^p \ne 1$. Hence $z^p \in C_G(\langle
\Sigma \rangle) = Z(G)$. This contradiction shows that $D = D'$
and $D \leqslant H$.
$\Box$
\begin{lemma}
$N_G(D)= N_G(H)$. \label{lm:N=N}
\end{lemma}
\noindent {\bf Proof.} $\,$ The embedding $N_G(H) \leqslant N_G(D)$ follows from
Lemma~\ref{lm:D<H}. Vice versa, if $x \in N_G(D)$ then the action
of $x$ by conjugation leaves the set $\Sigma$ invariant, hence it
leaves invariant the set of algebraic tori $\{H_L \mid L \in
\Sigma\}$ which generates $H$. Therefore $x \in N_G(H)$.
$\Box$
\begin{lemma}
$C_G(D)= C_G(H)$. \label{lm:C=C}
\end{lemma}
\noindent {\bf Proof.} $\,$ Let $x \in C_G(D)$, then, for every $L \in \Sigma$, $x$
centralises $C_D(L)$ and thus, by Lemma~\ref{lm:L-unique},
normalises $L = E(C_G(C_D(L)))$. Since $x$ centralises a maximal
$p$-torus $D\cap L$ of $L$, it centralises the maximal torus $H_L
= C_L(D\cap L)$. Hence $x\in C_G(H)$. This proves $C_G(D)
\leqslant C_G(H)$. The reverse inclusion follows from
Lemma~\ref{lm:D<H}.
$\Box$
In view of Lemmata~\ref{lm:N=N} and \ref{lm:C=C}, we can refer to
$W_0$ either as the subgroup generated by the images of
involutions $r_L$ in the factor group $N_G(D)/C_G(D)$ or as the
subgroup generated by the images of involutions $r_L$ in
$N_G(H)/C_G(H)$. Also, we now know that the group $W_0$ acts on
$D$ faithfully.
\subsection{Tate Module}
Now the aim is to construct a $\Bbb Z$-lattice on which $W_0$
acts as a crystallographic reflection group. For that purpose we
shall associate with $D$ the {\em Tate module} $T_p$. It is
constructed in the following way.
Let $E_{p^k}$ be the subgroup of $D$ generated by elements of order
$p^k$. Notice that every $r_L$ acts on $E_{p^k}$ as a reflection,
that is, $E_{p^k} = C_{E_{p^k}}(r_L)\times [E_{p^k}, r_L]$, $[E_{p^k},
r_L]\leqslant D \cap L$ is a cyclic group and $r_L$
inverts every element in $[E_{p^k}, r_L]$.
Consider the sequence of subgroups
\[E_p \longleftarrow E_{p^2} \longleftarrow E_{p^3} \longleftarrow
\cdots\] linked by the homomorphisms $x \mapsto x^p$. The
projective limit of this sequence is the free module $T_p$ over
the ring ${\Bbb Z}_p$ of $p$-adic integers. The action of $W_0$
on $D$ can be lifted to $T_p$, where it is still an irreducible
reflection group. By construction, $T_p /pT_p$ is isomorphic to
$E_p$ as a $W_0$-module. Notice also that $W_0$ acts on the
tensor product $T_p \otimes_{{\Bbb Z}_p} {\Bbb C}$ as a (complex)
reflection group, and that the dimension of $T_p
\otimes_{{\Bbb Z}_p} {\Bbb C}$ over $\Bbb C$
coincides with the Pr\"{u}fer
$p$-rank of $D$, hence is at least $3$.
\subsection{More Reflection Representations for ${W_0}$}
Now let us focus on odd primes $q\ne p$.
Consider the elementary abelian $q$-subgroups $E_q$ generated in
$H$ by all elements of the fixed prime order $q$. For the sake of
complete reducibility of the action of $W$ on $E_q$, one can
consider only $q > |W|$.
Lemmas~\ref{kerdcc}, \ref{long} and \ref{equiv} below are
similar to some results in \cite{berkman}.
We include the proofs here for the sake of completeness of exposition.
\begin{lemma} {\rm \cite[Lemma 9.7]{berkman}}
\label{kerdcc} Let\/ $N=N_G(H)$, then $C_N(E_q) = C_G(H)$.
\end{lemma}
\noindent {\bf Proof.} $\,$ It is clear that $C_G(H)\leqslant C_N(E_{q})$. To see the
converse, let $x\in C_N(E_{q})$. Since $x\in N$,
it acts on the elements of $\Sigma$ by conjugation.
First let us prove that $x$ normalises each subgroup in $\Sigma$.
To get a contradiction, assume that there is some subgroup
$L\in\Sigma$ such that $L^x\neq L$. But then by
Lemma~\ref{pairs}, $L$ and $L^x$ either commute or generate a
semisimple group as root $SL_2$-subgroups. Hence $|L\cap
L^x|\leqslant 2$. But this gives a contradiction since $q$ is an
odd prime and $L\cap E_{q}=L^x\cap E_{q} \leqslant L\cap L^x$.
Hence for each $L\in \Sigma$, $L^x=L$ and $x$ acts on $H\cap L$ as
an element from $N_L(H\cap L)$, since $SL_2$ does not have any
definable outer automorphisms. Note that the Weyl group of $SL_2$
is generated by an involution which inverts the torus $H\cap L$.
Since $x$ centralises $E_{q}\cap H$, $x$ centralises $H\cap L$ for
each $L\in\Sigma$, and hence $x$ centralises $H=\langle H\cap
L\mid L\in \Sigma\rangle$ and $x\in C_G(H)$. This proves the
equality.
$\Box$
Now notice that $[E_q,r_L]$ is generated by a $q$-element in $H_L$
and thus has order $q$. Hence $E_q$ is a finite dimensional
vector space over ${\Bbb F}_q$ on which $W_0$ acts as a group
generated by reflections.
\begin{lemma}
\label{wirr} The group $W_0$ acts irreducibly on $E_q$.
\end{lemma}
\noindent {\bf Proof.} $\,$ Note that $W_0$ acts on $E_q$ faithfully by
Lemma~\ref{kerdcc}. Since $q > |W|$, the action of $W_0$ on $E_q$
is completely reducible. So if the action is reducible, then we can write
$E_q = E' \oplus E''$ for two proper $W_0$-invariant subspaces.
Assume that $W_0$ acts trivially on one of the subspaces $E'$ or
$E''$, say on $E'$. If $L\in \Sigma$, then $E_q = C_{E_q}(L)
\times (E_q \cap L)$, and, obviously, $C_{E_q}(L) = C_{E_q}(r_L)$.
Hence all $L \in \Sigma$ centralise $E'$ and $E' \leqslant
C_G(\langle \Sigma\rangle) = Z(G) =1$. Therefore $W_0$ acts
nontrivially on both $E'$ and $E''$.
For $L \in \Sigma$, the
$-1$-eigenspace $[E_q, r_L]$ of $r_L$ belongs to one of the
subspaces $E'$ or $E''$ and hence $r_L$ acts as a reflection on
one of the subspaces $E'$ or $E''$ and centralises the other.
Set $\Sigma'=\{\,L\in\Sigma \mid [E_q,r_L] \leqslant E'\,\}$
and $\Sigma'' = \{\,L\in\Sigma \mid [E_q,r_L] \leqslant E''\,\}$. It
is easy to see that $[r_K,r_L] =1$ for $K \in \Sigma'$ and $L \in
\Sigma''$. By Lemma~\ref{refgp}, $K$ and $L$ commute for all $K
\in \Sigma'$ and $L \in \Sigma''$, which contradicts
Lemma~\ref{lm:connected}. Hence $W_0$ is irreducible on $E_q$.
$\Box$
\begin{lemma}
The group $W_0$ acts irreducibly on $T_p \otimes_{{\mathbb{Z}}_p}
\mathbb{C}$.
\end{lemma}
\noindent {\bf Proof.} $\,$ The proof is analogous to that of the previous lemma.
$\Box$
\subsection{Root System}
The aim of this subsection is to construct a root system on which $W_0$
acts as a crystallographic reflection group. The existence of such a
root system is guaranteed by the following lemma.
\begin{lemma}
\label{rootsys} There exists an irreducible root system on which\/
$W_0$ acts as a crystallographic reflection group.
\end{lemma}
\noindent {\bf Proof.} $\,$ Recall that $n \geqslant 3$. By Theorem~\ref{reflection},
the quotient group $W_0$ is one of the crystallographic reflection
groups $A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4$ and acts on the
corresponding root system.
$\Box$
Let now $R= \{\, {r}_i \mid i \in I \,\}$ be a simple
system of reflections in $W_0$. We shall identify $I$ with the
set of nodes of the Dynkin diagram for $W_0$. It is well known
that every reflection in an irreducible reflection group $W_0$ is
conjugate to a reflection in $R$.
\begin{lemma} {\rm \cite[Lemma~9.9]{berkman}}
\label{long} Every reflection\/ $r \in W_0$ has the form $r_K$ for
some $SL_2$-subgroup $K \in \Sigma$.
\end{lemma}
\noindent {\bf Proof.} $\,$ Let $ r\in W_0$ be a reflection. Working our way back
through the construction of the module $T_p$, one can easily see
that the Pr\"{u}fer $p$-rank of $[D,r]$ is $1$.
Let $r_L\in W_0$ be a reflection which corresponds to a
$SL_2$-subgroup $L\in \Sigma$. By comparing the Pr\"{u}fer
$p$-ranks of the groups $C_D(r_L)$ and $C_D(r)$, we see that
$Z=(C_H(r_L) \cap C_H(r))^\circ$ has Pr\"{u}fer $p$-rank at least $1$.
Hence the subgroup $\langle L,H, r \rangle$ contains a non-trivial
central $p$-torus; also note that $\langle L,H, r \rangle$ is a
is a $K$-group, since $D$ lies in $H$. It is well known that a finite irreducible reflection
group contains at most two conjugate classes of reflections.
Therefore, after replacing $r$ and $r_L$ by their appropriate
conjugates in $W_0$, we can assume without loss of generality that
the images of $r_L$ and $r$ in $W_0$ correspond to adjacent nodes
of the Dynkin diagram. Now we can easily see that $\langle L,H, r
\rangle = Y * Z$ for some simple algebraic group $Y$ of Lie rank
$2$, and that $r=r_K$ for some root $SL_2$-subgroup $K$ of $Y$
such that $K\in \Sigma$.
$\Box$
\subsection{Final Step}
The next task is to prove that the conditions of Lyons's Theorem
(Theorem~\ref{Lyons}) are satisfied.
\begin{lemma} {\rm \cite[Lemma~9.10]{berkman}}
\label{equiv} Attach an $SL_2$-subgroup $L_i \in \Sigma$ to each
vertex of the Dynkin diagram $I$ in such a way that the simple
reflection ${r}_i$ corresponding to this vertex is $r_{L_i}$ in
$W_0$. Then the following statements hold.
{\rm (1)} $[L_i,L_j]=1$
if and only if\/ $| {r}_i {r}_j|=2$.
{\rm (2)} $\langle
L_i,L_j\rangle$ is isomorphic to $(P)SL_3$ if and only if $| {r}_i
{r}_j|=3$.
{\rm (3)} $\langle L_i,L_j\rangle$ is isomorphic to
$(P)Sp_4$ if and only if\/ $| {r}_i {r}_j|=4$.
{\rm (4)} $L_i$
and $L_j$ are embedded in $\langle L_i,L_j\rangle$ as root\/
$SL_2$-subgroups.
\end{lemma}
\begin{proof} It is well known that for each $i, j\in I$, the order
$| {r}_i {r}_j|$ takes the values $2$, $3$ or $4$, in a Dynkin
diagram of type $A_n, B_n, C_n, D_n, E_6, E_7, E_8$ or $F_4$. By
Lemma~\ref{pairs}, $L_i$ and $L_j$ either commute or generate
$(P)SL_3$, $(P)Sp_4$ or $G_2$. However $\langle
L_i,L_j\rangle\cong G_2$ is not possible in our case since $|
{r}_i {r}_j|=6$ does not occur in $I$.
The `only if' parts of (1) and (2) are easy to see.
In the case of part (3), that is when $L_i$ and $L_j$
generate $(P)Sp_4$, we have to show that $| {r_i} {r_j}|\neq 2$.
To get a contradiction, assume that $L_i$ and $L_j$
generate $(P)Sp_4$ and $| {r_i} {r_j}|=2$.
But then
$L_i$ and $L_j$ are both
short root $SL_2$-subgroups. Note that ${r_i}$ and ${r_j}$ are
simple reflections, and it can be checked by inspection that one
of them must be a long reflection. This proves the `only if' part
of (3). Now parts (1), (2) and (3) follow from Lemma~\ref{pairs}
and the previous discussion. Part (4) is a direct consequence of
Lemma~\ref{pairs}. \end{proof}
\begin{lemma}
Each subgroup in $\Sigma$ is isomorphic to $(P)SL_2({\Bbb F})$ for
the same field $\Bbb F$.
\end{lemma}
\begin{proof} By Lemma~\ref{lm:connected} any two subgroups of
$\Sigma$ are connected by a sequence of edges. Note that each pair
which is connected by a single edge generates a simple group of
Lie rank 2 by Lemma~\ref{pairs}(2), hence their underlying fields
coincide. Thus the underlying fields of any two subgroups in
$\Sigma$ coincide. \end{proof}
Finally, we are in a position to apply Lyons's Theorem. Set $G_0$
to be the subgroup of $G$ generated by the subgroups $L_i$ for $ i
\in I$. By Lyons's Theorem, $G_0$ is a simple algebraic group over
$\Bbb F$ with the Dynkin diagram $I$. Its Weyl group, with respect
to the torus $T$, is $W_0$, hence $G_0$ contains all subgroups
from $\Sigma$. Therefore by Lemma~\ref{lm:sigmagenerates}, $G_0=G$
is a Chevalley group over $\Bbb F$. Since $G$ is of $p'$-type,
$\Bbb F$ is of characteristic different from $p$.
$\Box$
$\Box$
\small
\end{document} | math | 31,691 |
\betagin{document}
\title[Gelfand-Kirillov Dimensions of the $\mathbb{Z}$-graded
Oscillator Representations]{ Gelfand-Kirillov Dimensions of the $\mathbb{Z}$-graded
Oscillator Representations of $\mathfrak{o}(n,\mathbb{C})$ and $\mathfrak{sp}(2n,\mathbb{C})$}
\author{Zhanqiang Bai}
\mathrm{ad}dress{ School of Mathematics and Statistics , Wuhan University , Wuhan, 430072, P.R. China}
\email{Zhanqiang\[email protected]}
\thanks{2010 Mathematics Subject Classification. Primary 17B10; Secondary 22E47 }
\maketitle
\betagin{abstract}
In this paper, we give a method to compute the Gelfand-Kirillov dimensions of some polynomial type weight modules. These modules are
infinite-dimensional irreducible $\mathfrak{o}(n, \mathbb C)$-modules and $\mathfrak{sp}(2n, \mathbb C)$-modules that appeared in the
$\mathbb{Z}$-graded oscillator generalizations of the classical theorem on harmonic polynomials established by Luo
and Xu. We also found that some of these modules have the secondly minimal GK-dimension, and some of them have the larger GK-dimension than the maximal GK-dimension apearing in unitary highest-weight modules.
{\bf Key Words:} Gelfand-Kirillov dimension; Weight module; Oscillator representation.
\end{equation}d{abstract}
\section{Introduction}
Fifty years ago, Gelfand and Kirillov \cite{Ge-Ki} introduced a quantity to
measure the rate of growth of an algebra in terms of any generating
set, which is now known as Gelfand-Kirillov dimension. Since then the Gelfand-Kirillov dimension has become a very useful and powerful tool for people to measure the size of infinite-dimensional irreducible modules of Lie algebras and Lie groups. However, usually it is not easy to compute the Gelfand-Kirillov dimensions of explicit modules.
A module of a finite-dimensional simple Lie algebra is called a {\it
weight module} if it is a direct sum of its weight subspaces. The classification of weight modules had been completed by Mathieu \cite{Ma} after the contributions of many mathematicians. But we don't have many results about the distribution of Gelfand-Kirillov dimensions of weight modules. Let
$M$ be an irreducible highest-weight module for a finite-dimensional
simple Lie algebra $\mathfrak{g}$. Then $M$ is naturally a weight
module with finite-dimensional weight subspaces.
Denote by $d_{M}$
its Gelfand-Kirillov dimension. We fix a Cartan subalgebra
$\mathfrak{h}$, a root system $\Delta\subset \mathfrak{h}^*$ and a
set of positive roots $\Delta_+ \subset \Delta$. Let $\rho$ be half
the sum of all positive roots. Suppose that $\betata$ is the highest
root. It is well known that $d_M=0$ if and only if $M$ is
finite-dimensional, in which case irreducible modules are classified
by the highest-weight theory. From Vogan \cite{Vogan-81} and Wang
\cite{wang-99}, we know that the next smallest integer occurring is
$d_M=(\rho,\betata^{\vee})$. We call them the \emph{minimal
Gelfand-Kirillov dimension module}. These small modules are of great
interest in representation theory. A general introduction can be
found in Vogan \cite{Vogan-81}. Recently we \cite{Bai-Hu} studied the GK-dimensions of unitary highest-weight modules. We found that the secondly minimal GK-dimension of a unitary highest-weight module is $2((\rho,\betata^{\vee})-C)$ and the maximal GK-dimension of a unitary highest-weight module is $r((\rho,\betata^{\vee})-(r-1)C)$, where $C$ and $r$ are constants only depending on the type of Lie algebras and given by Enright, Howe and Wallach in \cite{EHW}. Does any irreducible weight module have larger GK-dimension than $r((\rho,\betata^{\vee})-(r-1)C)$? We will confirm the answer in this paper.
In classical harmonic analysis, a fundamental theorem says that
the spaces of homogeneous harmonic polynomials are irreducible
modules of the corresponding orthogonal Lie group (algebra) and the
whole polynomial algebra is a free module over the invariant
polynomials generated by harmonic polynomials. Bases of these
irreducible modules can be obtained easily (e.g., cf. \cite{Xu-08}).
The algebraic beauty of the above theorem is that the Laplace equation
characterizes the irreducible submodules of the polynomial algebra
and the corresponding quadratic invariant gives a decomposition of
the polynomial algebra into a direct sum of irreducible submodules,
namely, complete reducibility.
Recently Luo and Xu \cite{Luo-Xu} established the $\mathbb{Z}^2$-graded
oscillator generalizations of the above theorem for
$\mathfrak{sl}(n,\mathbb C)$, where the irreducible submodules are $\mathbb
Z^2$-graded homogeneous polynomial solutions of deformed Laplace
equations. In \cite{Bai-sl}, we find an exact formula of
Gelfand-Kirillov dimensions for these $\mathfrak{sl}(n,\mathbb C)$-modules. It turns out that their
Gelfand-Kirillov dimensions are independent of the double grading
and three infinite subfamilies of these modules have the minimal
Gelfand-Kirillov dimension. In \cite{Luo-Xu-lie,Luo-Xu-algebra}, by using the results in \cite{Luo-Xu}, Luo and Xu established the structure of the corresponding two-parameter $\mathbb{Z}$-graded oscillator representations of $\mathfrak{o}(n,\mathbb{C})$ and $\mathfrak{sp}(2n,\mathbb{C})$. It turned out that these modules are irreducible weight modules. In this paper, we will compute the Gelfand-Kirillov dimensions of these modules. Below we give a more detailed introduction for theses modules.
For convenience, we will use the notion
$\overline{{i,i+j}}=\{i,i+1,i+2,...,i+j\}$ for integers $i$ and $j$
with $i\leq j$. Denote by $\mathbb{N}$ the additive semigroup of
nonnegative integers. Let $E_{r,s}$ be the square matrix with $1$ as
its $(r,s)$-entry and $0$ as the others.
The orthogonal Lie algebra
$$\mathfrak{o}(2n,\mathbb{C})=\sum_{i,j=1}^n\mathbb{C}(E_{i,j}-E_{n+j,n+i})+\sum_{1\leq
i<j\leq
n}[\mathbb{C}(E_{i,n+j}-E_{j,n+i})+\mathbb{C}(E_{n+j,i}-E_{n+i,j})]$$
Denote
${\mathcal{B}}=\mathbb{C}[x_1,...,x_n,y_1,...,y_n]$. Fix
$n_1,n_2\in\overline{1,n}$ with $n_1\leq n_2$. We have the following non-canonical oscillator representation of $\mathfrak{o}(2n,\mathbb{C})$ on
${\mathcal{B}}$ determined by
\begin{equation}\label{xy} (E_{i,j}-E_{n+j,n+i})|_{\mathcal{B}}=E_{i,j}^x-E_{j,i}^y\qquad \text{for}~ i,j\in\overline{1,n}\end{equation} with
\begin{equation} E_{i,j}^x|_{\mathcal{B}}=\left\{\betagin{array}{ll}-x_j\partial_{x_i}-\delta_{i,j}&\mathbfox{if}\;
i,j\in\overline{1,n_1},\\
\partial_{x_i}\partial_{x_j}&\mathbfox{if}\;i\in\overline{1,n_1},\;j\in\overline{n_1+1,n},\\
-x_ix_j &\mathbfox{if}\;i\in\overline{n_1+1,n},\;j\in\overline{1,n_1},\\
x_i\partial_{x_j}&\mathbfox{if}\;i,j\in\overline{n_1+1,n}
\end{equation}d{array}\right.\end{equation}
and
\begin{equation} E_{i,j}^y|_{\mathcal{B}}=\left\{\betagin{array}{ll}y_i\partial_{y_j}&\mathbfox{if}\;
i,j\in\overline{1,n_2},\\
-y_iy_j&\mathbfox{if}\;i\in\overline{1,n_2},\;j\in\overline{n_2+1,n},\\
\partial_{y_i}\partial_{y_j} &\mathbfox{if}\;i\in\overline{n_2+1,n},\;j\in\overline{1,n_2},\\
-y_j\partial_{y_i}-\delta_{i,j}&\mathbfox{if}\;i,j\in\overline{n_2+1,n}
\end{equation}d{array}\right.\end{equation}
and
\begin{equation} E_{i,n+j}|_{\mathcal{B}}=\left\{\betagin{array}{ll}
\partial_{x_i}\partial_{y_j}&\mathbfox{if}\;
i\in\overline{1,n_1}, \,j\in\overline{1,n_2},\\
-y_j\partial_{x_i}&\mathbfox{if}\;i\in\overline{1,n_1},\, j\in\overline{n_2+1,n},\\
{x_i}\partial_{y_j} &\mathbfox{if}\;i\in\overline{n_1+1,n}, \,j\in\overline{1,n_2},\\
-x_iy_j &\mathbfox{if}\; i\in \overline{n_1+1,n}, \, j\in\overline{n_2+1,n}
\end{equation}d{array}\right.\end{equation}
and
\begin{equation}\label{xy-} E_{n+i,j}|_{\mathcal{B}}=\left\{\betagin{array}{ll}
-{x_j}{y_i}&\mathbfox{if}\;
j\in\overline{1,n_1}, \,i\in\overline{1,n_2},\\
-x_j\partial_{y_i}&\mathbfox{if}\;j\in\overline{1,n_1}, \,i\in\overline{n_2+1,n},\\
{y_i}\partial_{x_j} &\mathbfox{if}\;j\in\overline{n_1+1,n}, \,i\in\overline{1,n_2},\\
\partial_{x_j}\partial_{y_i} &\mathbfox{if}\; j\in \overline{n_1+1,n}, \, i\in\overline{n_2+1,n}.
\end{equation}d{array}\right.\end{equation}
The related variated Laplace operator becomes
\begin{equation} {\mathcal{D}}=\sum_{i=1}^{n_1}x_i\partial_{y_i}-\sum_{r=n_1+1}^{n_2}\partial_{x_r}\partial_{y_r}+\sum_{s=n_2+1}^n
y_s\partial_{x_s}.\end{equation}
Set
\begin{equation}{\mathcal{B}}_{\langle
k'\ranglengle}=\mathbfox{Span}\{x^\alphapha
y^\betata\mid\alphapha,\betata\in\mathbb{N}\:^n,\sum_{r=n_1+1}^n\alphapha_r-\sum_{i=1}^{n_1}\alphapha_i+
\sum_{i=1}^{n_2}\betata_i-\sum_{r=n_2+1}^n\betata_r=k'\}\end{equation} for
$k'\in\mathbb{Z}$. Define
\begin{equation}{\mathcal{H}}_{\langle k'\ranglengle}=\{f\in
{\mathcal{B}}_{\langle k'\ranglengle}\mid
{\mathcal{D}}(f)=0\}.\end{equation}
The following is the first main theorem of this paper.
\betagin{Thm}\label{main} For any $k'\in\mathbb{Z}$, if
the $\mathfrak{o}(2n,\mathbb{C})$-module ${\mathcal{H}}_{\langle
k' \ranglengle}$ is irreducible, then it has the
Gelfand-Kirillov dimension
\begin{equation} \label{formula}
d=\left\{
\betagin{array}{ll}
2n-1, & \hbox{\text{\emph{if~}} $1=n_1< n_2<n-1$, \emph{or~} $3\leq n_1<n_2=n$,} \\
& \hbox{\emph{or~}$1<n_1< n_2\leq n-1$, \emph{or~}$n_1=n_2$ \emph{when~} $n{\cal G}({\cal A})mmaeq 5$;}\\
2n-2, & \hbox{\emph{if~} $1=n_1< n_2=n-1,n$, \emph{or~} $2=n_1<n_2=n$} \\
& \hbox{\emph{or~} $n_1=n_2$ \emph{when~} $n=4$;}\\
2n-3, & \hbox{\emph{if~} $n_1=n_2$ \emph{when~} $n=2,3$.}
\end{equation}d{array}
\right. \end{equation}
\end{equation}d{Thm}
\betagin{Rem}For this case, the minimal GK-dimension is $2n-3$. From our paper \cite{Bai-Hu}, the secondly minimal GK-dimension is max($4n-10, 2n-2$),
and the maximal GK-dimension is $\varphirac{n(n-1)}{2}$. So $2n-1$ is larger than the secondly minimal GK-dimension of any unitary highest-weight modules when $n\leq 4$ and smaller then the secondly minimal GK-dimension of any unitary highest-weight modules when $n>4$.
\end{equation}d{Rem}
We observe that the orthogonal Lie algebra
$$\mathfrak{o}(2n+1,\mathbfb{C})=\mathfrak{o}(2n,\mathbfb{C})\opluslus\bigoplus_{i=1}^n
[\mathbfb{C}(E_{0,i}-E_{n+i,0})+\mathbfb{C}(E_{0,n+i}-E_{i,0})].$$
Let ${\mathcal B}'=\mathbfb{C}[x_0,x_1,...,x_n,y_1,...,y_n]$. We define a
non-canonical oscillator representation of $\mathfrak{o}(2n+1,\mathbfb{C})$ on
${\mathcal B}'$ by the differential operators in (\ref{xy})-(\ref{xy-}) and
$$ E_{0,i}|_{{\mathcal
B}'}=\left\{\betagin{array}{ll}-x_0x_i&\mathbfox{if}\;i\in\overline{1,n_1},\\
x_0\partial_{x_i}&\mathbfox{if}\;i\in\overline{n_1+1,n},\\
x_0\partial_{y_{i-n}}&\mathbfox{if}\;i\in\overline{n+1,n+n_2},\\
-x_0y_{i-n}&\mathbfox{if}\;i\in\overline{n+n_2+1,2n}\end{equation}d{array}\right.$$
and
$$ E_{i,0}|_{{\mathcal
B}'}=\left\{\betagin{array}{ll}\partial_{x_0}\partial_{x_i}&\mathbfox{if}\;i\in\overline{1,n_1},\\
x_i\partial_{x_0}&\mathbfox{if}\;i\in\overline{n_1+1,n},\\
y_{i-n}\partial_{x_0}&\mathbfox{if}\;i\in\overline{n+1,n+n_2},\\
\partial_{x_0}\partial_{y_{i-n}}&\mathbfox{if}\;i\in\overline{n+n_2+1,2n}.\end{equation}d{array}\right.$$
Now the variated Laplace operator becomes
$${\mathcal
D}'=\partial_{x_0}^2-2\sum_{i=1}^{n_1}x_i\partial_{y_i}+2\sum_{r=n_1+1}^{n_2}\partial_{x_r}\partial_{y_r}-2\sum_{s=n_2+1}^n
y_s\partial_{x_s}.$$
Set
$${\mathcal B}'_{\langle k\ranglengle}=\sum_{i=0}^\infty {\mathcal B}_{\langle
k-i\ranglengle}x_0^i,\qquad {\mathcal H}'_{\langle k\ranglengle}=\{f\in {\mathcal B}'_{\langle
k\ranglengle}\mid {\mathcal D}'(f)=0\}.$$
The following is the second main theorem of this paper.
\betagin{Thm}\label{main} For any $k'\in\mathbb{Z}$, the irreducible $\mathfrak{o}(2n+1,\mathbb{C})$-module ${\mathcal{H}'}_{\langle
k' \ranglengle}$ has the
Gelfand-Kirillov dimension
\begin{equation} \label{formula}
d=\left\{
\betagin{array}{ll}
2n, & \hbox{\text{\emph{if~}} $1\leq n_1< n_2<n-1$, \emph{or~} $3\leq n_1<n_2=n$,} \\
& \hbox{\emph{or~}$1<n_1< n_2= n-1$, \emph{or~}$n_1=n_2$ \emph{when~} $n{\cal G}({\cal A})mmaeq 5$,}\\
& \hbox{\emph{or~}$n_1=n_2=n=3$, \emph{or~}$n_1=n_2>1$ \emph{when~} $n=4$;}\\
2n-1, & \hbox{\emph{if~} $1=n_1< n_2=n-1,n$, \emph{or~} $2=n_1<n_2=n$}, \\
& \hbox{\emph{or~} $n_1=n_2=2$ \emph{when~} $n=2,3$, \emph{or~} $n_1=n_2=1$ \emph{when~} $n=1,4$;}\\
2n-2, & \hbox{\emph{if~} $1=n_1=n_2<n=2,3$.}
\end{equation}d{array}
\right. \end{equation}
\end{equation}d{Thm}
\betagin{Rem}For this case, the minimal GK-dimension is $2n-2$. From our paper \cite{Bai-Hu}, the secondly minimal GK-dimension and the maximal GK-dimension are the same, i.e., $2n-1$. So $2n$ is larger than the maximal GK-dimension of any unitary highest-weight modules.
\end{equation}d{Rem}
The symplectic Lie
algebra\betagin{eqnarray*}\hspace{1cm}\mathfrak{sp}(2n,\mathbfb{C})&=&
\sum_{i,j=1}^n\mathbfb{C}(E_{i,j}-E_{n+j,n+i})+\sum_{i=1}^n(\mathbfb{C}E_{i,n+i}+\mathbfb{C}E_{n+i,i})\\
& &+\sum_{1\leq i<j\leq n
}[\mathbfb{C}(E_{i,n+j}+E_{j,n+i})+\mathbfb{C}(E_{n+i,j}+E_{n+j,i})].\end{equation}d{eqnarray*}
We define the two-parameter $\mathbb{Z}$-graded oscillator representation of $\mathfrak{sp}(2n,\mathbfb{C})$ on
${\mathcal B}$ via (\ref{xy})-(\ref{xy-}).
The related variated Laplace operator becomes
\begin{equation} {{D}}=\sum_{r=n_1+1}^{n}x_r\partial_{x_r}-\sum_{i=1}^{n_1}x_i\partial_{x_i}+\sum_{i=1}^{n_2}y_i\partial_{y_i}
-\sum_{r=n_2+1}^{n}y_r\partial_{y_r}.\end{equation}
Set
\begin{equation}{\mathcal{B}}_{\langle
k'\ranglengle}=\mathbfox{Span}\{x^\alphapha
y^\betata\mid\alphapha,\betata\in\mathbb{N}\:^n,\sum_{r=n_1+1}^n\alphapha_r-\sum_{i=1}^{n_1}\alphapha_i+
\sum_{i=1}^{n_2}\betata_i-\sum_{r=n_2+1}^n\betata_r=k'\}\end{equation} for
$k' \in\mathbb{Z}$. Then ${\mathcal{B}}_{\langle k'\ranglengle}=\{f\in{\mathcal{B}}\mid{D}(f)=k'f\}$.
The following is the third main theorem of this paper.
\betagin{Thm}\label{main} For any $k'\in\mathbb{Z}$, if
the $\mathfrak{sp}(2n,\mathbb{C})$-module ${\mathcal{B}}_{\langle
k' \ranglengle}$ is irreducible, then it has the
Gelfand-Kirillov dimension
\begin{equation}
d=2n-1. \end{equation}
When $n_1=n_2$, the $\mathfrak{sp}(2n,\mathbfb{C})$-module ${\mathcal B}_{\langle 0\ranglengle}$ also has the
Gelfand-Kirillov dimension $d=2n-1.$ When $n_1=n_2=n$, the two irreducible components of ${\mathcal B}_{\langle 0\ranglengle}$ also have the
Gelfand-Kirillov dimension $d=2n-1.$
\end{equation}d{Thm}
\betagin{Rem}For this case, the minimal GK-dimension is $n$. From our paper \cite{Bai-Hu}, the secondly minimal GK-dimension and is $2n-1$. So all the modules in the above theorem have the secondly minimal GK-dimension.
\end{equation}d{Rem}
{\bf Acknowledgements.}The author is partially supported by
This work is partially supported by NSFC Grant No. 11601394 and the Fundamental Research Funds for the Central Universities Grant No. 2042016kf0041 from Wuhan University. We would like to thank the referee for the comments on an earlier version
of this paper.
\section{Preliminaries on Gelfand-Kirillov Dimension}
We recall some definitions and properties of the Gelfand-Kirillov dimension. Details may be found in Refs.\cite{Bo-Kr, Ja, Kr-Le, NOTYK, Vogan-78, Vogan-91}.
\betagin{definition}
Let $A$ be an algebra (not necessarily associative) generated by a
finite-dimensional subspace $V$. Let $V^n$ denote the linear span of
all products of length at most $n$ in elements of $V$. The
\emph{Gelfand-Kirillov dimension} of $A$ is defined by:
$$GKdim(A) = \limsup_{n\rightarrow \infty} \varphirac{\log\mathrm{dim}( V^{n} )}{\log n}.$$
\end{equation}d{definition}
\betagin{Rem}It is well-known that the above definition is independent of the choice of the
finite dimensional generating subspace $V$ (see Ref.\cite{Bo-Kr, Kr-Le}).
Clearly $GKdim(A)=0$ if and only if $\mathrm{dim}(A) <\infty$.
\end{equation}d{Rem}
The notion of Gelfand-Kirillov dimension can be extended for left $A$-modules. In fact, we have the following definition.
\betagin{definition} Let $A$ be an algebra (not necessarily associative) generated by a finite-dimensional subspace $V$. Let $M$ be a left $A$-module generated by a finite-dimensional subspace $M_{0}$. Let $V^n$ denote the linear span of all products of length at most $n$ in elements of $V$.
The \emph{Gelfand-Kirillov dimension} $GKdim(M)$ of $M$ is defined by
$$GKdim(M) = \limsup_{n\rightarrow \infty}\varphirac{\log\mathrm{dim}( V^{n}M_{0} )}{\log n}.$$
\end{equation}d{definition}
In particular, let $\mathfrak{g}$ be a complex Lie algebra. Let $A=\mathcal{U}(\mathfrak g)$ be the enveloping algebra of $\mathfrak g$, with the standard filtration given by $A_{n}=\mathcal{U}_{n}(\mathfrak g)$, the subspace of $\mathcal{U}(\mathfrak g)$ spanned by products of at most $n$-elements of $\mathfrak g$. By the Poincar\'{e}-Birkhoff-Witt theorem (see Knapp \cite[Prop. 3.16]{Knapp}), the graded algebra $\text{gr} (\mathcal{U}(\mathfrak g))$ is canonically isomorphic to the symmetric algebra $S(\mathfrak g)$. Suppose $M$ is a $\mathcal{U}(\mathfrak g)$-module generated by a finite-dimensional subspace $M_{0}$. We set $M_{n}=\mathcal{U}_{n}(\mathfrak g)M_{0}$. Denote $\text{gr}M=\bigoplus\limits_{n=0}^{\infty}\text{gr}_{n}M$, where $\text{gr}_{n}M=M_{n}/M_{n-1}$. Then $\text{gr}M$ becomes a graded $S(\mathfrak g)$-module. We denote $\dim(M_{n})$ by $\varphi_{M}(n)$. Then we have the following lemma.
\betagin{Lem}\label{hi-se}
\emph{(Hilbert-Serre \cite[Chapter VII. Th.41]{Za-Sa} )}
With the notations as above, there exists a unique polynomial $\tilde{\varphi}_{M}(n)$ such that $\varphi_{M}(n)=\tilde{\varphi}_{M}(n)$ for large $n$. The leading term of $\tilde{\varphi}_{M}(n)$ is $$\varphirac{c(M)}{(d_{M})!}n^{d_{M}},$$ where $c(M)$ is an integer.
\end{equation}d{Lem}
\betagin{Rem}
From the definition of Gelfand-Kirillov dimension, we know $$GKdim(M)=\limsup_{n\rightarrow \infty} \varphirac{\log\dim (U_{n}(\mathfrak g)M_{0} )}{\log n}=\limsup_{n\rightarrow \infty} \varphirac{\log\tilde{\varphi}_{M}(n)}{\log n}=d_{M}=\dim \mathscr{V}(M).$$
\end{equation}d{Rem}
\betagin{ex}
Let $M=\mathbb{C}[x_{1},...,x_{k}]$. Then $M$ is an algebra generated by the finite-dimensional subspace $V=Span_{\mathbb{C}}\{x_{1},...,x_{k}\}$. So $M_{n}=V^{n}=\bigoplus\limits_{0\leq q \leq n}P_{q}[x_{1},...,x_{k}]$ is the subset of homogeneous polynomials of degree $\leq n$.
Then \betagin{align*}\varphi_{M}(n)=&\sum\limits_{0\leq q \leq n}\dim_{\mathbb{C}}(P_{q}[x_{1},...,x_{k}])\\
=&\sum\limits_{0\leq q\leq n}\binom{k+q-1}{q}\\
=&\binom{k+n}{n}\\
=&\varphirac{n^k}{k!}+O(n^{k-1}).\end{equation}d{align*}
Then we have $GKdim(M)=k.$
\end{equation}d{ex}
\section{Proof of the main theorem for $\mathfrak{o}(2n,\mathbfb{C})$}
We keep the same notations with the introduction. Through the paper we always take $\mathcal {K}=\sum\limits_{i,j=1}^{n}\mathbb{C}(E_{i,j}-E_{n+j,n+i})$, and $\mathcal{K}_{+}=\sum_{1\leq i<j\leq n}\mathbfb{C}(E_{i,j}-E_{n+j,n+i})$.
A weight vector $v$ in $\mathcal{B}$ is called a \emph{$\mathcal{K}$-singular vector} if $\mathcal{K}_{+}(v)=0$.
We simply write $E_{i,j}|_{\mathcal{B}}$ as $E_{i,j}$.
Take
\begin{equation} \mathfrak{h}=\sum_{i=1}^{n}\mathbfb{C}(E_{i,i}-E_{n+i,n+i})\end{equation}
as a Cartan subalgebra of $\mathfrak{o}(2n,\mathbfb{C})$ and the subspace spanned
by positive root vectors:
\begin{equation} \mathfrak{o}(2n,\mathbfb{C})_+=\sum_{1\leq i<j\leq n}\mathbfb{C}(E_{i,j}-E_{n+j,n+i})+\sum_{1\leq i<j\leq n}\mathbfb{C}(E_{i,n+j}-E_{j,n+i}).\end{equation}
Correspondingly, we have \begin{equation} \mathfrak{o}(2n,\mathbfb{C})_-=\sum_{1\leq i<j\leq n}\mathbfb{C}(E_{j,i}-E_{n+i,n+j})+\sum_{1\leq i<j\leq n}\mathbfb{C}(E_{n+j,i}-E_{n+i,j}).\end{equation}
If we take $\mathcal{P}_{+}=\sum_{1\leq i<j\leq n}\mathbfb{C}(E_{i,n+j}-E_{j,n+i})$, then $\mathfrak{o}(2n,\mathbfb{C})_+=\mathcal{K}_{+}+\mathcal{P}_{+}$. From the PBW theorem we know that the irreducible $\mathfrak{o}(2n,\mathbb{C})$-module ${\mathcal{H}}_{\langle k \ranglengle}=U(\mathfrak{g})v_{\mathcal{K}}=U(\mathfrak{g_{-}+\mathcal{P}_{+}})v_{\mathcal{K}}$ for any $\mathcal K$-singular vector $v_{\mathcal{K}}$. In the following we will compute the Gelfand-Kirillov dimension of $\mathcal{H}_{\langle k \ranglengle}$ in a case-by-case way.
Firstly we need the following two well-known lemmas.
\betagin{Lem}\emph{(Multinomial theorem)}\\
Let $n,m$ be two positive integers,then
\begin{equation} \left|\{(k_1,k_2,...,k_m)\in \mathbb{N}^{m}|\sum\limits_{i=1}^{m}k_{i}=n\}\right|={n+m-1 \choose m-1}.
\end{equation}
\end{equation}d{Lem}
\betagin{Lem}Let $p, n$ be two positive integers, then
$$\sum\limits_{i=0}^{n}i^p=\varphirac{(n+1)^{p+1}}{p+1}+\sum\limits_{k=1}^{p}\varphirac{B_k}{p-k+1}{p \choose k}(n+1)^{p-k+1},$$
where $B_k$ denotes a Bernoulli number.
\end{equation}d{Lem}
From these two lemmas, we can get several propositions.
\betagin{Prop}\label{ak}Let $k\in \mathbb{N}$ and we denote $$M_{k}=\left\{\prod\limits_{\substack{1\leq i\leq n_1 \\ n_{1}+1\leq t\leq n}} ({x_i x_{t}})^{p_{it}}| \sum\limits_{\substack{1\leq i\leq n_1 \\ n_{1}+1\leq t\leq n}} p_{it}=k, p_{it}\in\mathbb{N} \right\}.$$ Then $$d_k=\dim Span_{\mathbb{R}}M_{k}={n_{1}+k-1 \choose k}{n-n_{1}+k-1 \choose k}\approx ak^{n-2},$$ for some constant a.
\end{equation}d{Prop}
\betagin{proof}From the definition of $M_k$, we know that all the elements in $M_k$ are monomials and they must form a basis for $Span_{\mathbb{R}}M_{k}$. Thus
\betagin{align*}d_k=&\dim Span_{\mathbb{R}}M_{k}=\#\left\{\prod\limits_{\substack{1\leq i\leq n_1 \\ n_{1}+1\leq t\leq n}} ({x_i x_{t}})^{p_{it}}| \sum\limits_{\substack{1\leq i\leq n_1 \\ n_{1}+1\leq t\leq n}} p_{it}=k, p_{it}\in\mathbb{N} \right\}\\
=&\#\left\{\prod\limits_{\substack{1\leq i\leq n_1 }} (x_i)^{\sum_{n_{1}+1\leq t\leq n}p_{it}}\prod\limits_{\substack{n_{1}+1\leq t\leq n}} (x_t)^{\sum_{1\leq i\leq n_1 }p_{it}}| \sum\limits_{\substack{1\leq i\leq n_1 \\ n_{1}+1\leq t\leq n}} p_{it}=k, p_{it}\in\mathbb{N} \right\}\\
=&{n_{1}+k-1 \choose k}{n-n_{1}+k-1 \choose k}\approx ak^{n-2}, \emph{~~for some constant~} \emph{a}.
\end{equation}d{align*}
\end{equation}d{proof}
The idea of the proof for the following propositions are very simple: Denote ${\mathcal{B}}=\mathbb{C}[x_1,...,x_n,y_1,...,y_n]$. We define a partial order (i.e., dictionary order) on the monomials of ${\mathcal{B}}$:
$$x_{1}^{p_{1}}...x_n^{p_n}y_{1}^{p_{n+1}}...y_n^{p_{2n}}\preceq x_{1}^{p'_{1}}...x_n^{p'_n}y_1^{p'_{n+1}}...y_n^{p'_{2n}}$$
if there exists $1\leq m \leq 2n$, such that $p_i=p'_i$ for any $i<m$ and $p_m<p'_m$. We can also interchange the place of $x_1$ and some $x_j$ (or $y_j$), then define a similar partial order.
Suppose $I=\cup\{i\}$ is a given index set and $P$ is a set of homogeneous polynomials which are products of some binomials $(f_i-g_i)^{a_i}$ ($f_i$ and $g_i$ are monomials of degree $2$ in ${\mathcal{B}}$, and $\sum \limits_{i\in I}a_i=k$ is a constant), i.e., $P=\{\prod\limits_{i\in I}(f_i-g_i)^{a_i}| \sum a_i=k\}$. We fix $i\in \overline{1,n}.$ We choose two subsets $I_1$ and $I_2$ in $I$, such that $I_1\cap I_2=\emptyset$, and $I_1\cup I_2=\cup \{i\}=I$. Suppose $f_{i_p}$ is a multiple of $x_i$ and $g_{i_p}$ is not a multiple of $x_i$ when $i_p\in I_1$, and $g_{i_l}$ is a multiple of $x_i$ and $f_{i_l}$ is not a multiple of $x_i$ when $i_l\in I_2$.
Then we have $$\dim Span_{\mathbb{R}}P{\cal G}({\cal A})mmaeq \#\{\prod\limits_{i_p\in I_1}(f_{i_p})^{a_{i_p}}\cdot\prod\limits_{i_l\in I_2}(g_{i_l})^{a_{i_l}}|I=I_1\sqcup I_2, \sum a_{i_p}+\sum a_{i_l}=k\},$$
here we interchange the place of $x_1$ and $x_i$. So the monomial $\prod\limits_{i_p\in I_1}(f_{i_p})^{a_{i_p}}\cdot\prod\limits_{i_l\in I_2}(g_{i_l})^{a_{i_l}}$ which contain the largest power of $x_i$ is the leading term in the expression of $\prod\limits_{i\in I}(f_i-g_i)^{a_i}$.
\betagin{Prop}\label{n=n}
\betagin{enumerate}
\item $(n_1=n_2=1)$ Let $k\in \mathbb{N}$ and we denote $$T_k=\left\{\prod\limits_{\substack{2\leq p<t\leq n}} ({x_p y_{t}}-x_{t}y_{p})^{g_{pt}}\cdot \prod\limits_{2\leq t \leq n}(x_1x_t-y_1y_t)^{g_{1t}}| \sum\limits_{1\leq p<t\leq n} g_{pt}=k, g_{pt}\in\mathbb{N} \right\}.$$
Then we have \betagin{align*} d_k=\dim Span_{\mathbb{R}}T_{k}\approx\left\{
\betagin{array}{ll}
c_0k^{2n-4}, & \text{\emph{if}~} {n=2 \mathrm{~or~} n=3;} \\
c_1k^{2n-3}, & \text{\emph{if}~} {n=4;} \\
c_2k^{2n-2}, & \text{\emph{if}~} {n{\cal G}({\cal A})mmaeq 5.}
\end{equation}d{array}
\right. \end{equation}d{align*}
Here $c_0, c_1$ and $c_2$ are some positive constants which are independent of $k$.
\item $(n_1=n_2=n-1)$ Let $k\in \mathbb{N}$ and we denote $$S_k=\left\{\prod\limits_{\substack{1\leq i<r\leq n-1}} ({x_i y_{r}}-x_{r}y_{i})^{f_{ir}}\cdot \prod\limits_{1\leq i \leq n-1}(x_ix_n-y_iy_n)^{f_{in}}| \sum\limits_{1\leq i<r\leq n} f_{ir}=k, f_{ir}\in\mathbb{N} \right\}.$$
Then we have \betagin{align*} d_k=\dim Span_{\mathbb{R}}S_{k}\approx\left\{
\betagin{array}{ll}
a_0k^{2n-4}, & \text{\emph{if}~} {n=2 \mathrm{~or~} n=3;} \\
a_1k^{2n-3}, & \text{\emph{if}~} {n=4;} \\
a_2k^{2n-2}, & \text{\emph{if}~} {n{\cal G}({\cal A})mmaeq 5.}
\end{equation}d{array}
\right. \end{equation}d{align*}
Here $a_0, a_1$ and $a_2$ are some positive constants which are independent of $k$.
\item $(n_1=n_2=n)$Let $k\in \mathbb{N}$ and we denote $$R_k=\left\{\prod\limits_{\substack{1\leq i<r\leq n}} ({x_i y_{r}}-x_{r}y_{i})^{f_{ir}}| \sum\limits_{1\leq i<r\leq n} f_{ir}=k, f_{ir}\in\mathbb{N} \right\}.$$
Then we have \betagin{align*} d_k=\dim Span_{\mathbb{R}}R_{k}\approx\left\{
\betagin{array}{ll}
b_0k^{2n-4}, & \text{\emph{if}~} {n=2 \mathrm{~or~} n=3;} \\
b_1k^{2n-3}, & \text{\emph{if}~} {n=4;} \\
b_2k^{2n-2}, & \text{\emph{if}~} {n{\cal G}({\cal A})mmaeq 5.}
\end{equation}d{array}
\right. \end{equation}d{align*}
Here $b_0, b_1$ and $b_2$ are some positive constants which are independent of $k$.
\item $(1<n_1=n_2<n-1)$Suppose $1<n_1<n-1$.
Let $k\in \mathbb{N}$ and we denote \betagin{align*}U_k&=\left\{\prod\limits_{\substack{n_1+1\leq p<t\leq n}} ({x_p y_{t}}-x_{t}y_{p})^{g_{pt}}\cdot \prod\limits_{\substack{1\leq i<r\leq n_1}} ({x_i y_{r}}-x_{r}y_{i})^{g_{ir}}\right.\\
&\left.\cdot \prod\limits_{\substack{1\leq i\leq n_1\\ n_1+1\leq t \leq n}}(x_ix_t-y_iy_t)^{g_{1t}}| \sum\limits_{1\leq p<t\leq n} g_{pt}=k, g_{pt}\in\mathbb{N} \right\}.\end{equation}d{align*}
Then we have \betagin{align*} d_k=\dim Span_{\mathbb{R}}U_{k}\approx\left\{
\betagin{array}{ll}
e_1k^{2n-3}, & \text{\emph{if}~} {n=4;} \\
e_2k^{2n-2}, & \text{\emph{if}~} {n{\cal G}({\cal A})mmaeq 5.}
\end{equation}d{array}
\right. \end{equation}d{align*}
Here $e_0, e_1$ and $e_2$ are some positive constants which are independent of $k$.
\end{equation}d{enumerate}
\end{equation}d{Prop}
\betagin{proof}The statements $(1)$ and $(2)$ are dual to each other. The proof of $(4)$ is similar to $(1)$. So we only need to give the proof for $(1)$ and $(3)$.
\text{Proof of $(1)$:}
When $n=3$, we have
\betagin{align*}d_k&=\dim Span_{\mathbb{R}}T_{k}\\
&=\dim Span_{\mathbb{R}}\left\{ ({x_1 x_{2}}-y_{1}y_{2})^{g_{12}}({x_1 x_{3}}-y_{1}y_{3})^{g_{13}}(x_2y_3-x_3y_2)^{g_{23}}|\sum g_{pt}=k\right\}\\
&{\cal G}({\cal A})mmaeq \dim Span_{\mathbb{R}}\left\{({x_1 x_{2}})^{g_{12}}(x_1x_3)^{g_{13}}(x_2y_3)^{g_{23}}|\sum g_{pt}=k \right\}\\
&=\dim Span_{\mathbb{R}}\left\{({ x_{1}})^{g_{12}+g_{13}}(y_3)^{g_{23}}\right.\\
&\quad\quad\quad\quad\quad\quad\cdot(x_2)^{g_{12}+g_{23}}(x_3)^{g_{13}}
\left.|\sum g_{pt}=k \right\}\\
&\approx c_{00}k^2, \emph{~for some constant }c_{00}.
\end{equation}d{align*}
On the other hand, we have $d_k=\dim Span_{\mathbb{R}}T_{k}\leq c_{01}k^{3-1}=c_{01}k^{2},$ for some positive constant $c_{01}$. So we must have $d_k=\dim Span_{\mathbb{R}}T_{k}\approx c_{0}k^{2}=c_{0}k^{2n-4},$ for some positive constant $c_{0}$.
When $n=4$, we have
\betagin{align*}d_k&=\dim Span_{\mathbb{R}}T_{k}\\
&=\dim Span_{\mathbb{R}}\left\{ ({x_1 x_{2}}-y_{1}y_{2})^{g_{12}}({x_1 x_{3}}-y_{1}y_{3})^{g_{13}}({x_1 x_{4}}-y_{1}y_{4})^{g_{14}}\right.\\
&\quad\quad\quad\quad\quad\quad \left.(x_2y_3-x_3y_2)^{g_{23}}(x_2y_4-x_4y_2)^{g_{24}}(x_3y_4-x_4y_3)^{g_{34}}|\sum g_{pt}=k\right\}\\
&{\cal G}({\cal A})mmaeq \dim Span_{\mathbb{R}}\left\{({y_1 y_{2}})^{g_{12}}(x_1x_3)^{g_{13}}(x_1x_4)^{g_{14}}(x_3y_2)^{g_{23}}(x_2y_4)^{g_{24}}(x_4y_3)^{g_{34}}|\sum g_{pt}=k \right\}\\
&=\dim Span_{\mathbb{R}}\left\{({ y_{2}})^{g_{12}+g_{23}}(x_1)^{g_{13}+g_{14}}(y_3)^{g_{34}}(y_4)^{g_{24}}\right.\\
&\quad\quad\quad\quad\quad\quad\cdot({y_1 })^{g_{12}}(x_3)^{g_{13}+g_{23}}(x_4)^{g_{14}+g_{34}}(x_2)^{g_{24}}
\left.|\sum g_{pt}=k \right\}\\
&\approx c_{10}k^5, \emph{~for some constant }c_{10}.
\end{equation}d{align*}
On the other hand, we have $d_k=\dim Span_{\mathbb{R}}T_{k}\leq c_{11}k^{6-1}=c_{11}k^{5},$ for some positive constant $c_{11}$. So we must have $d_k=\dim Span_{\mathbb{R}}T_{k}\approx c_{1}k^{5}=c_{1}k^{2n-3},$ for some positive constant $c_{1}$.
When $n{\cal G}({\cal A})mmaeq 5$, we have\betagin{align*}d_k&=\dim Span_{\mathbb{R}}T_{k}\\
&=\dim Span_{\mathbb{R}}
\left\{\prod\limits_{\substack{2\leq p<t\leq 4}} ({x_p y_{t}}-x_{t}y_{p})^{g_{pt}}\cdot \prod\limits_{2\leq t \leq 4}(x_1x_t-y_1y_t)^{g_{1t}}\right.\\
& \quad\quad\quad\quad\quad\quad\left.\cdot\prod\limits_{\substack{2\leq p<t\leq n \\ t{\cal G}({\cal A})mmaeq 5}} ({x_p y_{t}}-x_{t}y_{p})^{g_{pt}}\cdot \prod\limits_{ t {\cal G}({\cal A})mmaeq 5}(x_1x_t-y_1y_t)^{g_{1t}} | \sum\limits_{1\leq p<t\leq n} g_{pt}=k, \right\}\\
&{\cal G}({\cal A})mmaeq \dim Span_{\mathbb{R}}\left\{({ y_{2}})^{g_{12}+g_{23}}(x_1)^{g_{13}+g_{14}}(y_3)^{g_{34}}(y_4)^{g_{24}}\prod\limits_{5\leq t\leq n}y_{t}^{g_{it}+g_{2t}}\prod\limits_{3\leq p<n}y_{p}^{\sum\limits_{p<t\leq n} g_{pt}}
\right.\\
&\quad\quad\quad\quad\quad\quad\cdot({y_1 })^{g_{12}+\sum\limits_{5\leq t\leq n} g_{1t}}(x_3)^{g_{13}+g_{23}}(x_4)^{g_{14}+g_{34}}(x_2)^{g_{24}+\sum\limits_{5\leq t\leq n} g_{2t}}\prod\limits_{5\leq t\leq n} x_{t}^{\sum\limits_{3\leq p<t}g_{pt}}\\
&\quad\quad\quad\quad\quad\quad\left.|\sum g_{pt}=k \right\}\\
&\approx c_{20}k^{2n-2}, \emph{~for some constant }c_{20}.
\end{equation}d{align*}
On the other hand, we have
\betagin{align}\nonumber
d_k=&\dim Span_{\mathbb{R}}T_{k}\\\nonumber
\leq &\dim Span_{\mathbb{R}}\left\{\prod\limits ({x_1 })^{p_{1}}\prod({ y_{t}})^{q_{t}}\cdot\prod({ x_{t}})^{l_{t}}\prod(y_{1})^{f_{1}} |\right.\\\nonumber
&\quad\quad\quad\quad\quad\left. p_{1}+\sum\limits_{2\leq t\leq n} q_{t}=f_{1}+\sum\limits_{2\leq t\leq n} l_{t}=k \right\}\\
\approx & c_{21} k^{2n-2}, \emph{~for some constant }c_{21}.\nonumber
\end{equation}d{align}\nonumber
So we must have $d_k=\dim Span_{\mathbb{R}}T_{k}\approx c_{2}k^{2n-2},$ for some positive constant $c_2$.
\hspace{1mm}
\text{Proof of $(3)$:}
When $n=2$, we have $d_k=\dim Span_{\mathbb{R}}R_{k}=\dim Span_{\mathbb{R}}\left\{ ({x_1 y_{2}}-x_{2}y_{1})^{k}\right\}=1$.
When $n=3$, we have
\betagin{align*}d_k&=\dim Span_{\mathbb{R}}R_{k}\\
&=\dim Span_{\mathbb{R}}\left\{ ({x_1 y_{2}}-x_{2}y_{1})^{f_{12}}(x_1y_3-x_3y_1)^{f_{13}}(x_2y_3-x_3y_2)^{f_{23}}|f_{12}+f_{13}+f_{23}=k\right\}\\
&{\cal G}({\cal A})mmaeq \dim Span_{\mathbb{R}}\left\{({x_1 y_{2}})^{f_{12}}(x_3y_1)^{f_{13}}(x_2y_3)^{f_{23}}|f_{12}+f_{13}+f_{23}=k \right\}\\
&={3+k-1 \choose k}\\
&\approx\varphirac{1}{2}k^2.
\end{equation}d{align*}
On the other hand, we have $d_k=\dim Span_{\mathbb{R}}R_{k}\leq b_{00}k^{3-1}=b_{00}k^{2},$ for some positive constant $b_{00}$. So we must have $d_k=\dim Span_{\mathbb{R}}R_{k}\approx b_{0}k^{2}=b_{0}k^{2n-4},$ for some positive constant $b_{0}$.
When $n=4$, we have
\betagin{align*}d_k&=\dim Span_{\mathbb{R}}R_{k}\\
&=\dim Span_{\mathbb{R}}\left\{\prod\limits_{\substack{1\leq i<r\leq 4}} ({x_i y_{r}}-x_{r}y_{i})^{f_{ir}}| \sum\limits_{1\leq i<r\leq 4} f_{ir}=k, f_{ir}\in\mathbb{N} \right\}\\
&{\cal G}({\cal A})mmaeq \dim Span_{\mathbb{R}}\left\{({x_1 y_{2}})^{f_{12}} ({x_3 y_{1}})^{f_{13}} ({x_1 y_{4}})^{f_{14}} ({x_2 y_{3}})^{f_{23}} ({x_4 y_{2}})^{f_{24}} ({x_3 y_{4}})^{f_{34}}\right.\\
& \quad\quad\quad\quad\quad \quad \left.| \sum\limits_{1\leq i<r\leq 4} f_{ir}=k, f_{ir}\in\mathbb{N} \right\}\\
&=\dim Span_{\mathbb{R}}\left\{({x_1 })^{f_{12}+f_{14}} ({x_3 })^{f_{13}+f_{34}} ({x_4 })^{f_{24}} ({x_2})^{f_{23}}
\cdot ({ y_{2}})^{f_{12}+f_{24}} ({y_{1}})^{f_{13}} ({ y_{4}})^{f_{14}+f_{34}} ({y_{3}})^{f_{23}} \right.\\
& \quad\quad\quad\quad\quad \quad \left.| \sum\limits_{1\leq i<r\leq 4} f_{ir}=k, f_{ir}\in\mathbb{N} \right\}\\
&\approx b_{11}k^5, \emph{~for some constant }b_{11}.
\end{equation}d{align*}
On the other hand, we have $d_k=\dim Span_{\mathbb{R}}R_{k}\leq b_{12}k^{6-1}=b_{12}k^{5},$ for some positive constant $b_{12}$. So we must have $d_k=\dim Span_{\mathbb{R}}R_{k}\approx b_{1}k^{5}=b_{1}k^{2n-3},$ for some positive constant $b_{1}$.
When $n{\cal G}({\cal A})mmaeq 5$, we have
\betagin{align*}d_k&=\dim Span_{\mathbb{R}}R_{k}\\
&=\dim Span_{\mathbb{R}}\left\{\prod\limits_{\substack{1\leq i<r\leq 4}} ({x_i y_{r}}-x_{r}y_{i})^{f_{ir}}\cdot \prod\limits_{\substack{1\leq i<r\leq n \\ r{\cal G}({\cal A})mmaeq 5}}({x_i y_{r}}-x_{r}y_{i})^{f_{ir}} | \sum\limits_{1\leq i<r\leq n} f_{ir}=k, f_{ir}\in\mathbb{N} \right\}\\
&{\cal G}({\cal A})mmaeq \dim Span_{\mathbb{R}}\left\{\prod\limits_{\substack{1\leq i<r\leq 4}} ({x_i y_{r}}-x_{r}y_{i})^{f_{ir}}\cdot \prod\limits_{5\leq r\leq n}({x_1 y_{r}})^{f_{1r}} ({x_2 y_{r}})^{f_{2r}} ({x_r y_{3}})^{f_{3r}} ({x_r y_{4}})^{f_{4r}} \right.\\
& \quad\quad\quad\quad\quad\quad \quad \left.| \sum\limits_{1\leq i<r\leq n} f_{ir}=k, f_{ir}\in\mathbb{N} \right\}\\
&=\dim Span_{\mathbb{R}}\left\{({x_1 })^{f_{12}+f_{14}+\sum f_{1r}} ({x_3 })^{f_{13}+f_{34}} ({x_4 })^{f_{24}} ({x_2})^{f_{23}+\sum f_{2r}}\prod\limits_{5\leq r\leq n}(x_r)^{f_{3r}+f_{4r}}\right.\\
&\quad\quad\quad\quad\quad \quad \quad\left. \cdot ({ y_{2}})^{f_{12}+f_{24}} ({y_{1}})^{f_{13}} ({ y_{4}})^{f_{14}+f_{34}+\sum f_{4r}} ({y_{3}})^{f_{23}+\sum f_{3r}} \prod\limits_{5\leq r\leq n}(y_r)^{f_{1r}+f_{2r}}\right.\\
& \quad\quad\quad\quad\quad \quad \quad\left.| \sum\limits_{1\leq i<r\leq n} f_{ir}=k, f_{ir}\in\mathbb{N} \right\}\\
&\approx b_{21}k^{2n-2}, \emph{~for some constant }b_{21}.
\end{equation}d{align*}
On the other hand, we have
\betagin{align*}d_k&=\dim Span_{\mathbb{R}}R_{k}\\
&\leq \dim Span_{\mathbb{R}} \left\{\prod\limits_{\substack{1\leq i\leq n \\1 \leq r\leq n }} {x_i}^{a_i} y_{r}^{b_{r}}
| \sum a_{i}=\sum b_{r}=k \right\}\\
&\approx b_{22}k^{2n-2},\end{equation}d{align*} for some positive constant $b_{22}$. So we must have $d_k=\dim Span_{\mathbb{R}}R_{k}\approx b_{2}k^{2n-2},$ for some positive constant $b_{2}$.
\end{equation}d{proof}
\betagin{Prop}\label{n<m}
\betagin{enumerate}
\item $(1<n_1<n_2=n-1)$Suppose $1<n_1< n-1$. Let $k\in \mathbb{N}$ and we denote
\betagin{align*} \nonumber V_k&=\left\{\prod\limits(x_ix_s)^{p_{is}} \prod\limits(x_iy_s)^{l_{is}}\prod\limits(x_sy_n)^{u_{s}}\prod\limits(y_{s}y_{n})^{q_{s}}\prod\limits({x_{i} x_{n}}-y_{i}y_{n})^{h_{i}}\prod\limits({x_{i} y_{r}}-x_{r}y_{i})^{f_{ir}}|\right.\\
&\quad \left.\sum\limits_{\substack{1\leq i\leq n_1 \\n_1+1\leq s\leq n-1}}p_{is}+\sum\limits_{\substack{1\leq i\leq n_1 \\n_1+1\leq s\leq n-1}}l_{is}+\sum\limits_{\substack{n_1+1\leq s\leq n-1 }}u_{s}+ \sum\limits_{\substack{n_1+1\leq s\leq n-1 }}q_{s}\right.\\
&\quad\quad \left.+\sum\limits_{\substack{1\leq i\leq n_1 }}h_{i}+\sum\limits_{1\leq i<r\leq n_1}=k \right\},\\
\end{equation}d{align*}
then
$$d_k=\dim Span_{\mathbb{R}}V_{k}\approx bk^{2n-2},$$ for some constant $b$.
\item $(1=n_1<n_2<n-1)$Suppose $1<n_2< n-1$. Let $k\in \mathbb{N}$ and we denote
\betagin{align*}
W_k&=\left\{\prod\limits(x_1x_s)^{p_{s}} \prod\limits(x_1y_s)^{l_{s}}\prod\limits(x_sy_t)^{u_{st}}\prod\limits(y_{s}y_{t})^{q_{st}}\prod\limits({x_{1} x_{t}}-y_{1}y_{t})^{h_{t}}\prod\limits({x_{p} y_{t}}-x_{t}y_{p})^{g_{pt}}|\right.\\
&\quad \left.\sum\limits_{2\leq s\leq n_2}p_{s}+\sum\limits_{2\leq s\leq n_2}l_{s}
+\sum\limits_{\substack{2\leq s\leq n_2 \\ n_2+1\leq t\leq n}}u_{st}+ \sum\limits_{\substack{2\leq s\leq n_2 \\ n_2+1\leq t\leq n }}q_{st}+\sum\limits_{\substack{n_2+1\leq t\leq n }}h_{t}+\sum\limits_{n_2+1\leq p<t\leq n}g_{pt}=k \right\},\\
\end{equation}d{align*}
then
$$d_k=\dim Span_{\mathbb{R}}W_{k}\approx \betata k^{2n-2},$$ for some constant $\betata$.
\item$(1<n_1<n_2=n)$
Suppose $1<n_1<n$. Let $k\in \mathbb{N}$ and we denote
\betagin{align*} \nonumber Z_k&=\left\{\prod\limits(x_ix_s)^{p_{is}}\prod\limits(x_{i}y_{s})^{l_{is}}\prod\limits({x_{i} y_{r}}-x_{r}y_{i})^{f_{ir}}|\right.\\
&\quad\quad \left.\sum\limits_{\substack{1\leq i\leq n_1 \\n_1+1\leq s\leq n}}p_{is}+\sum\limits_{\substack{1\leq i\leq n_1 \\n_1+1\leq s\leq n}}l_{is}+ \sum\limits_{\substack{1\leq i<r\leq n_1 }}f_{ir}=k \right\},
\end{equation}d{align*}
then
\betagin{align*} d_k=\dim Span_{\mathbb{R}}Z_{k}\approx\left\{
\betagin{array}{ll}
\alphapha_0k^{2n-3}, & \text{\emph{if}~} {n_1=2 <n;} \\
\alphapha_1k^{2n-2}, & \text{\emph{if}~} {3\leq n_1<n.} \end{equation}d{array}
\right. \end{equation}d{align*}
Here $\alphapha_0$ and $\alphapha_1$ are some positive constants which are independent of $k$.
\item$(1<n_1<n_2<n-1)$
Let $k\in \mathbb{N}$. Suppose $1<n_1< n_2<n-1$ and we denote
\betagin{align*} \nonumber N^{\prime}_k&=\left\{\prod\limits(x_ix_s)^{p_{is}}\prod\limits(y_{s}y_{t})^{q_{st}}\prod\limits(x_iy_s)^{l_{is}}
\prod\limits(x_sy_t)^{u_{st}}\right.\\
& \quad \left.\cdot \prod\limits({x_{i} x_{t}}-y_{i}y_{t})^{h_{it}}\prod\limits({x_{i} y_{r}}-x_{r}y_{i})^{f_{ir}}\prod\limits({x_{p} y_{t}}-x_{t}y_{p})^{g_{pt}}\right.\\
&\quad \left. \mid \sum\limits_{\substack{1\leq i\leq n_1 \\n_1+1\leq s\leq n_2}}(p_{is}+l_{is})+\sum\limits_{\substack{n_1+1\leq s\leq n-1\\ n_2+1\leq t\leq n }}(u_{st}+q_{st})\right.\\
&\quad \left.+ \sum\limits_{\substack{1\leq i\leq n_1 \\ n_2+1\leq t\leq n}}h_{it}+\sum\limits_{1\leq i<r\leq n_1}f_{ir}+\sum\limits_{n_2+1\leq p<t\leq n_1}g_{pt}=k \right\},
\end{equation}d{align*}
then
$$d_k=\dim Span_{\mathbb{R}}N'_{k}\approx ck^{2n-2},$$ for some constant $c$.
\end{equation}d{enumerate}
\end{equation}d{Prop}
\betagin{proof}The statements $(1)$ and $(2)$ are dual to each other. Their proofs are similar to the proof of $(4)$. So we only need to prove $(3)$ and $(4)$.
\text{Proof of $(3)$:}
When $n_1=2$, we have
\betagin{align*}
d_{k}&=\dim Span_{\mathbb{R}}\left\{\prod\limits(x_ix_s)^{p_{is}}\prod\limits(x_{i}y_{s})^{l_{is}}({x_{1} y_{2}}-x_{2}y_{1})^{f_{12}}|\right.\\
&\quad\quad \quad \quad \quad \quad \left.\sum\limits_{\substack{1\leq i\leq 2 \\n_1+1\leq s\leq n}}p_{is}+\sum\limits_{\substack{1\leq i\leq 2 \\3\leq s\leq n}}l_{is}+ f_{12}=k \right\}\\
=&\dim Span_{\mathbb{R}}\left\{\prod\limits_{1\leq i\leq 2}(x_i)^{\sum\limits_{3\leq s\leq n}p_{is}+l_{is}}\cdot\prod\limits_{3\leq s\leq n}(x_s)^{\sum\limits_{1\leq i\leq 2}p_{is}}\prod\limits_{3\leq s\leq n}(y_{s})^{\sum\limits_{1\leq i\leq 2}l_{is}}\right.\\
&\quad\quad \quad \quad \quad \quad \left.\cdot
\prod\limits({x_{1} y_{2}}-x_{2}y_{1})^{f_{12}}
|\sum\limits_{\substack{1\leq i\leq 2 \\3\leq s\leq n}}p_{is}+\sum\limits_{\substack{1\leq i\leq 2 \\3\leq s\leq n}}l_{is}+ f_{12}=k \right\}\\
&\approx \alphapha_{0}k^{2n-3}, \emph{~for some constant }\alphapha_{0}.
\end{equation}d{align*}
When $n_1>2$, we have
\betagin{align*}
d_{k}&=\dim Span_{\mathbb{R}}\left\{\prod\limits(x_ix_s)^{p_{is}}\prod\limits(x_{i}y_{s})^{l_{is}}\prod\limits({x_{i} y_{r}}-x_{r}y_{i})^{f_{ir}}|\right.\\
&\quad\quad \left.\sum\limits_{\substack{1\leq i\leq n_1 \\n_1+1\leq s\leq n}}p_{is}+\sum\limits_{\substack{1\leq i\leq n_1 \\n_1+1\leq s\leq n}}l_{is}+ \sum\limits_{\substack{1\leq i<r\leq n_1 }}f_{ir}=k \right\}\\
{\cal G}({\cal A})mmaeq &\dim Span_{\mathbb{R}}\left\{\prod\limits_{1\leq i\leq n_1}(x_i)^{\sum\limits_{n_1+1\leq s\leq n}p_{is}+l_{is}}\cdot\prod\limits_{n_1+1\leq s\leq n}(x_s)^{\sum\limits_{1\leq i\leq n_1}p_{is}}\prod\limits_{n_1+1\leq s\leq n}(y_{s})^{\sum\limits_{1\leq i\leq n_1}l_{is}}\right.\\
& \quad \quad \quad \left.\cdot
\prod\limits_{2\leq i<r\leq n_1}(x_{r}y_{i})^{f_{ir}}\cdot (x_1y_{n_1})^{f_{1n_1}}\cdot\prod\limits_{2\leq r<n_1}(x_ry_1)^{f_{1r}}\right.\\
&\left.\quad\quad |\sum\limits_{\substack{1\leq i\leq n_1 \\n_1+1\leq s\leq n}}p_{is}+\sum\limits_{\substack{1\leq i\leq n_1 \\n_1+1\leq s\leq n}}l_{is}+ \sum\limits_{\substack{1\leq i<r\leq n_1 }}f_{ir}=k \right\}\\
&=\dim Span_{\mathbb{R}}\left\{\prod\limits_{1\leq i\leq n_1}(x_i)^{\sum\limits_{n_1+1\leq s\leq n}p_{is}+l_{is}}\cdot \prod\limits_{2<r\leq n_1}(x_{r})^{\sum\limits_{i<r}f_{ir}}\cdot (x_1)^{f_{1n_1}}\cdot\prod\limits_{2\leq r<n_1}(x_r)^{f_{1r}}
\right.\\
& \quad \quad \quad\quad\quad \left.\cdot\prod\limits_{n_1+1\leq s\leq n}(x_s)^{\sum\limits_{1\leq i\leq n_1}p_{is}}\prod\limits_{n_1+1\leq s\leq n}(y_{s})^{\sum\limits_{1\leq i\leq n_1}l_{is}}\cdot
\prod\limits_{2\leq i< n_1}(y_{i})^{\sum\limits_{i<r\leq n_1}f_{ir}}\right.\\
&\quad\quad\quad\quad\quad \left.\cdot\prod\limits_{2\leq r<n_1}(y_1)^{f_{1r}}\cdot (y_{n_1})^{f_{1n_1}}
|\sum\limits_{\substack{1\leq i\leq 2 \\3\leq s\leq n}}p_{is}+\sum\limits_{\substack{1\leq i\leq 2 \\3\leq s\leq n}}l_{is}+ f_{12}=k \right\}\\
&\approx \alphapha_{10}k^{2n-2}, \emph{~for some constant }\alphapha_{10}.
\end{equation}d{align*}
On the other hand, we have
\betagin{align}\nonumber
d_k=&\dim Span_{\mathbb{R}}Z_{k}\\\nonumber
\leq &\dim Span_{\mathbb{R}}\left\{\prod\limits ({x_i })^{p_{i}}\cdot\prod(x_s)^{a_s}\prod(y_s)^{b_s}\prod(y_{i})^{f_{i}} \right.\\\nonumber
&\quad\quad\quad\quad\quad\left. |\sum p_{i}=\sum f_{i}+\sum a_s+\sum b_s=k \right\}\\
\approx & \alphapha_{11} k^{2n-2}, \emph{~for some constant }\alphapha_{11}.\nonumber
\end{equation}d{align}\nonumber
So we must have $d_k=\dim Span_{\mathbb{R}}Z_{k}\approx \alphapha_{1}k^{2n-2},$ for some positive constant $\alphapha_{1}$.
\text{Proof of $(4)$:}
When $n_1=2<n_2<n-1$, we have
\betagin{align*}&d_{k}=\dim Span_{\mathbb{R}}\left\{\prod\limits(x_ix_s)^{p_{is}}\prod\limits(y_{s}y_{t})^{q_{st}}\prod\limits(x_iy_s)^{l_{is}}
\prod\limits(x_sy_t)^{u_{st}}\right.\\
& \quad \quad\quad \quad \quad \quad\quad \left.\prod\limits({x_{i} x_{t}}-y_{i}y_{t})^{h_{it}}\prod\limits({x_{i} y_{r}}-x_{r}y_{i})^{f_{ir}}\prod\limits({x_{p} y_{t}}-x_{t}y_{p})^{g_{pt}}| \right.\\
&\left.\quad\quad \quad \quad \quad \quad \quad\sum\limits p_{is}+\sum\limits q_{st}+ \sum\limits l_{is}+\sum\limits u_{st}+\sum\limits h_{it}+\sum\limits f_{ir}+\sum\limits g_{pt}=k \right\}\\
&{\cal G}({\cal A})mmaeq \dim Span_{\mathbb{R}}\left\{\prod\limits(x_ix_s)^{p_{is}}\prod\limits(y_{s}y_{t})^{q_{st}}\prod\limits(x_iy_s)^{l_{is}}
\prod\limits(x_sy_t)^{u_{st}}\right.\\
& \quad \quad\quad \quad \quad \quad\quad \left.\prod\limits_{n_2+2\leq t\leq n}({x_{1} x_{t}})^{h_{1t}}\cdot(y_1y_{n_2+1})^{h_{1,n_2+1}}\cdot(x_2x_{n_2+1})^{h_{2,n_2+1}}\cdot\prod\limits_{n_2+2\leq t\leq n} (y_2y_t)^{h_{2t}}\right.\\
&\left.\quad\quad \quad \quad \quad \quad \quad
\prod\limits({x_{1} y_{2}}-x_{2}y_{1})^{f_{12}}\prod\limits(x_{t}y_{n_2+1})^{g_{n_2+1,t}}| \right.\\
&\left.\quad\quad \quad \quad \quad \quad \quad\sum\limits p_{is}+\sum\limits q_{st}+ \sum\limits l_{is}+\sum\limits u_{st}+\sum\limits h_{it}+\sum\limits f_{ir}+\sum\limits g_{pt}=k \right\}\\
&{\cal G}({\cal A})mmaeq \dim Span_{\mathbb{R}} \left\{\left(({x_2})^{h_{2,n_2+1}+\sum (p_{2s}+l_{2s})}(x_1)^{\sum (p_{1s}+l_{1s})+f_{12}}(\prod\limits_{n_2+2\leq t\leq n}(x_1)^{ h_{1t}})\right.\right.\\
&\quad \quad\quad\quad\quad\quad\left.\left.\prod\limits_{n_2+2\leq t\leq n} (y_t)^{h_{2t}} (y_{t})^{\sum (q_{st}+u_{st})}({y_{n_2+1}})^{h_{1,n_2+1}+\sum g_{n_2+1,t}}\right)\right. \\
& \quad \quad\quad\quad\quad\quad \left. \cdot \left( \prod\limits(x_s)^{p_{1s}+p_{2s}+\sum u_{st}}(x_{n_2+1})^{h_{2,n_2+1}}(\prod\limits_{n_2+2\leq t\leq n}(x_t)^{h_{1t}+g_{n_2+1,t}})\right.\right.\\
&\quad \quad\quad\quad\quad\quad\left.\left.(\prod\limits(y_{s})^{\sum q_{st}+\sum l_{is}})({y_{1}})^{h_{1,n_2+1}}\prod\limits_{n_2+2\leq t\leq n}(y_2)^{ h_{2t}}y_{2}^{f_{12}}\right)\right.\\
&\quad \quad\quad\quad\quad\quad \left. | \sum\limits p_{is}+\sum\limits q_{st}+ \sum\limits l_{is}+\sum\limits u_{st}+\sum\limits h_{it}+\sum\limits f_{ir}+\sum\limits g_{pt}=k \right\}\\
&\approx c_0k^{2n-2}, \emph{~for some constant }c_0.
\end{equation}d{align*}
On the other hand, we have
\betagin{align}\label{d'k<}
d_k=&\dim Span_{\mathbb{R}}N'_{k}\\
\leq &\dim Span_{\mathbb{R}}\left\{\prod\limits ({x_i })^{p_{i}}\prod({ y_{t}})^{q_{t}}\cdot\prod(x_s)^{a_s}\prod(y_s)^{b_s}\prod({ x_{t}})^{l_{t}}\prod(y_{i})^{f_{i}} |\right.\\
&\quad\quad\quad\quad\quad\left. \sum p_{i}+\sum q_{t}=\sum l_{t}+\sum f_{i}+\sum a_s+\sum b_s=k \right\}\\
\approx & c_{00} k^{2n-2}, \emph{~for some constant }c_{00}.
\end{equation}d{align}
So we must have $d_k=\dim Span_{\mathbb{R}}N'_{k}\approx ck^{2n-2},$ for some positive constant $c$.
When $n_1>2$, we have a similar argument. And for these cases we still have $d_k=\dim Span_{\mathbb{R}}N'_{k}\approx ck^{2n-2},$ for some positive constant $c$.
\end{equation}d{proof}
Next, we will compute the Gelfand-Kirillov dimensions of our modules in a case-by-case way.
Luo and Xu \cite{Luo-Xu-lie} proved that
for any
$n_1-n_2+1-\delta_{n_1,n_2}{\cal G}({\cal A})mmaeq k'\in\mathbb{Z}$,
${\mathcal{H}}_{\langle k' \ranglengle}$ is an irreducible
$\mathfrak{o}(2n,\mathbb{C})$-module. Moreover, the
homogeneous subspace $\mathcal{
B}_{\langle k'\ranglengle}=\bigoplus_{i=0}^\infty\eta^i(\mathcal{
H}_{\langle k'-2i\ranglengle})$ is a direct sum of
irreducible submodules. The module ${\mathcal{H}}_{\langle k' \ranglengle}$ under the assumption is of highest-weight type only if $n_2=n$, in which case $x_{n_1}^{-k'}$ is a highest-weight
vector with weight
$-k'\lambda_{n_1-1}+(k'-1)\lambda_{n_1}+[(k'-1)\delta_{n_1,n-1}-2k'\delta_{n_1,n}]\lambda_n$.
\subsection{Case 1. $n_1+1\leq n_2$ \text{and} $n_{1}-n_{2}+1{\cal G}({\cal A})mmaeq k' \in \mathbb{Z}$.}
\hspace{1cm}
In this case we have:
\betagin{align} \label{ri}(E_{r,i}-E_{n+i,n+r})|_{\mathcal{B}}&=-x_i\partial_{x_r}-y_i\partial_{y_r} &\text{for~}& 1\leq i<r\leq
n_1,\\
\label{si}(E_{s,i}-E_{n+i,n+s})|_{\mathcal{B}}&=-{x_i x_{s}}-{y_i}\partial_{y_{s}} &\text{for} ~&i\in\overline{1,n_1},\;s\in\overline{n_1+1,n_2},
\\
\label{ti}(E_{t,i}-E_{n+i,n+t})|_{\mathcal{B}}&=-{x_i x_{t}}+y_{i}y_{t} &\text{for} ~&i\in\overline{1,n_1},\;t\in\overline{n_2+1,n},\\
\label{sj}(E_{s,j}-E_{n+j,n+s})|_{\mathcal{B}}&=x_s\partial_{x_j}-y_j\partial_{y_s} &\text{for}~& n_1<
j<s\leq n_2,\\
\label{ts}(E_{t,s}-E_{n+s,n+t})|_{\mathcal{B}}&=x_{t}\partial_{x_{s}}+y_{s} y_{t} &\text{for} ~&s\in\overline{n_1+1,n_2},\;t\in\overline{n_{2}+1,n},\\
\label{tp}(E_{t,p}-E_{n+p,n+t})|_{\mathcal{B}}&=x_t\partial_{x_p}+y_t\partial_{y_p}&\text{for}~& n_2+1\leq p<t\leq
n,\\
\label{inr}(E_{i,n+r}-E_{r,n+i})|_{\mathcal{B}}&=\partial_{x_i}\partial_{y_r}-\partial_{x_r}\partial_{y_i} &\text{for~}& 1\leq i<r\leq
n_1,\\
\label{nri}(E_{n+r,i}-E_{n+i,r})|_{\mathcal{B}}&=-{x_i}{y_r}+{x_r}{y_i} &\text{for~}& 1\leq i<r\leq
n_1,\\
\label{ins}(E_{i,n+s}-E_{s,n+i})|_{\mathcal{B}}&=\partial_{x_i}\partial_{y_s}-{x_s}\partial_{y_i} &\text{for} ~&i\in\overline{1,n_1},\;s\in\overline{n_1+1,n_2},
\\
\label{nsi}(E_{n+s,i}-E_{n+i,s})|_{\mathcal{B}}&=-{x_i}{y_s}-{y_i}\partial_{x_s} &\text{for} ~&i\in\overline{1,n_1},\;s\in\overline{n_1+1,n_2},
\\
\label{int}(E_{i,n+t}-E_{t,n+i})|_{\mathcal{B}}&=-y_t\partial_{x_i}-x_t\partial_{y_i} &\text{for} ~&i\in\overline{1,n_1},\;t\in\overline{n_2+1,n},\\
\label{nti}(E_{n+t,i}-E_{n+i,t})|_{\mathcal{B}}&=-x_i\partial_{y_t}-y_i\partial_{x_t} &\text{for} ~&i\in\overline{1,n_1},\;t\in\overline{n_2+1,n},\\
\label{jns}(E_{j,n+s}-E_{s,n+j})|_{\mathcal{B}}&=x_j\partial_{y_s}-x_s\partial_{y_j} &\text{for}~& n_1<j<s\leq n_2,\\
\label{njs}(E_{n+j,s}-E_{n+s,j})|_{\mathcal{B}}&=-y_s\partial_{x_j}+y_j\partial_{x_s} &\text{for}~& n_1<j<s\leq n_2,\\
\label{snt}(E_{s,n+t}-E_{t,n+s})|_{\mathcal{B}}&=-x_s{y_t}-x_t\partial_{y_s} &\text{for} ~&s\in\overline{n_1+1,n_2},\;t\in\overline{n_{2}+1,n},\\
\label{nst}(E_{n+s,t}-E_{n+t,s})|_{\mathcal{B}}&=-\partial_{x_s}\partial_{y_t}-y_s\partial_{x_t} &\text{for} ~&s\in\overline{n_1+1,n_2},\;t\in\overline{n_{2}+1,n},\\
\label{pnt}(E_{p,n+t}-E_{t,n+p})|_{\mathcal{B}}&=-x_p{y_t}+x_t{y_p} &\text{for}~& n_2+1\leq p<t\leq n,\\
\label{npt}(E_{n+p,t}-E_{n+t,p})|_{\mathcal{B}}&=-\partial_{x_p}\partial_{y_t}+\partial_{x_t}\partial_{y_p} &\text{for}~& n_2+1\leq p<t\leq n.
\end{equation}d{align}
Then the above root elements form a basis for the subalgebra $\mathfrak{g}(\mathcal{P}_{+})_{-}:=\mathfrak{o}(2n,\mathbfb{C})_-+\mathcal{P}_{+}$.
From Luo-Xu \cite{Luo-Xu-lie} we know that the $\mathcal{K}$-singular vectors in ${\mathcal H}_{\langle k'\ranglengle}$
are:
\begin{equation} x_{n_1}^{m_1}y_{n_2+1}^{m_2}\qquad\mathbfox{with}\;-(m_1+m_2)=k',\end{equation}
\begin{equation} x_{n_1+1}^{m_1}y_{n_2+1}^{m_2}\qquad\mathbfox{with}\;m_1-m_2=k',\end{equation}
\begin{equation}\label{n} x_{n_1}^{m_1}y_{n_2}^{m_2}\qquad\mathbfox{with}\;-m_1+m_2=k',\end{equation}
for all possible $m_1,m_2\in\mathbb{N}$. When $n_1+1=n_2=n$, the $\mathcal{K}$-singular vectors in ${\mathcal H}_{\langle k'\ranglengle}$ are those in $(\ref{n})$.
Let $\mathfrak{g}_1$ be the subalgebra of $\mathfrak{o}(2n,\mathbb{C})$ spanned by the root vectors in the following set:$$I_1:=\{(\ref{ri}),(\ref{sj}),(\ref{tp}),(\ref{inr}),(\ref{ins}),(\ref{int}),(\ref{nti}),(\ref{jns}),(\ref{njs}),(\ref{nst}),(\ref{npt})\}.$$
Let $\mathfrak{g}_2$ be the subalgebra of $\mathfrak{o}(2n,\mathbb{C})$ spanned by the root vectors in the following set:$$I_2:=\{(\ref{si}),(\ref{ti}),(\ref{ts}),(\ref{nri}),(\ref{nsi}),(\ref{snt}),(\ref{pnt})\}.$$
So we get $U(\mathfrak{g}(\mathcal{P}_{+})_{-})=U(\mathfrak{g}_{2})U(\mathfrak{g}_{1}).$ From the construction of the root vectors we have the following lemma.
\betagin{Lem}Every root vector in $\mathfrak{g}_1$ acts locally nilpotently on $\mathcal{H}_{\langle k' \ranglengle}$ and every root vector in $\mathfrak{g}_2$ acts torsion-freely (injectively) on $\mathcal{H}_{\langle k' \ranglengle}$.
\end{equation}d{Lem}
We take a $\mathcal K$-singular vector $v_{\mathcal{K}}=x_{n_1}^{-k'}$, and set $M_{0}=U(\mathfrak{g}_1)x_{n_1}^{-k'}$.
Then $M_{0}$ is finite-dimensional from the above lemma.
Thus
$${\mathcal{H}}_{\langle k' \ranglengle}=U(\mathfrak{g})v_{\mathcal{K}}=U(\mathfrak{g_{-}+\mathcal{P}_{+}})v_{\mathcal{K}}=U(\mathfrak{g}_{2})M_0.$$
Let $k$ be any positive integer. We want to compute $\mathrm{dim}(U_{k}(\mathfrak{g}_{2})M_{0})$, and then get the Gelfand-Kirillov dimension of $U(\mathfrak{g}_{2})M_{0}$.
The cardinality of a set $A$ is usually denoted by $|A|$.
We use $E_{p}$ to stand for the root vector in the equation $(p)$. And denote
\betagin{align*}N_{0}(k)=&\left\{\left(\prod E_{\ref{si}}^{p_{is}}\prod E_{\ref{ti}}^{h_{it}}\prod E_{\ref{ts}}^{q_{st}}\prod E_{\ref{nri}}^{f_{ir}}\prod E_{\ref{nsi}}^{l_{is}}\prod E_{\ref{snt}}^{u_{st}}\prod E_{\ref{pnt}}^{g_{pt}}\right)v_{\mathcal{K}}\right.\\
& \left.| \sum p_{is}+\sum h_{it}+\sum q_{st}+\sum f_{ir}+\sum l_{is}+\sum u_{st}+\sum g_{pt}=k,\right.\\
&\left.p_{is}, h_{it}, q_{st}, f_{ir},l_{is}, u_{st}, g_{pt}\in \mathbb{N} \right\}.\end{equation}d{align*}
From the definition we know
\betagin{align*}&\left(\prod E_{\ref{si}}^{p_{is}}\prod E_{\ref{ti}}^{h_{it}}\prod E_{\ref{ts}}^{q_{st}}\prod E_{\ref{nri}}^{f_{ir}}\prod E_{\ref{nsi}}^{l_{is}}\prod E_{\ref{snt}}^{u_{st}}\prod E_{\ref{pnt}}^{g_{pt}}\right)v_{\mathcal{K}}\\
=&\left(\prod (-{x_i x_{s}}-{y_i}\partial_{y_{s}})^{p_{is}}\prod(-{x_{i} x_{t}}+y_{i}y_{t})^{h_{it}} \prod(y_{s}y_{t})^{q_{st}}\prod(-x_iy_r+x_ry_i)^{f_{ir}}\right.\\
&\left.\cdot\prod(-x_iy_s-y_i\partial_{x_s})^{l_{is}}\prod(-x_sy_t-x_t\partial_{y_s})^{u_{st}} \right)\cdot x_{n_1}^{-k'}\\
=&\left(\prod (-{x_i x_{s}})^{p_{is}}\prod(-{x_{i} x_{t}}+y_{i}y_{t})^{h_{it}} \prod(y_{s}y_{t})^{q_{st}}\prod(-x_iy_r+x_ry_i)^{f_{ir}}\right.\\
&\left.\cdot\prod(-x_iy_s)^{l_{is}}\prod(-x_sy_t)^{u_{st}} \right)\cdot x_{n_1}^{-k'}\\
&+\text{lower degree polynomials of }~y_{s} \text{~and~} x_s.
\end{equation}d{align*}
Now we suppose $1<n_1<n_2<n-1$. Then we must have $$\dim Span_{\mathbb{R}}N_{0}(m){\cal G}({\cal A})mmaeq d_m=\dim Span_{\mathbb{R}}N'_{m}.$$
Using the same idea with the proof of Proposition \ref{n<m} (4), we can also get $$\dim Span_{\mathbb{R}}N_{0}(m)\leq d_{m}=\dim Span_{\mathbb{R}}N'_{m}.$$
Thus $\dim Span_{\mathbb{R}}N_{0}(m)= d_{m}=\dim Span_{\mathbb{R}}N'_{m}.$
Then using Proposition \ref{n<m} (4), we can get
\betagin{align*}
&\dim Span_{\mathbb{R}}(\bigcup\limits_{0\leq m \leq k}N_{0}(m))\\
=& \sum\limits_{0\leq m \leq k} \dim Span_{\mathbb{R}}N'_{m}\\
=&c\sum\limits_{0\leq m\leq k}m^{2n-2}\\
=&c'k^{2n-1}
\end{equation}d{align*}
Also we have
$$
\dim Span_{\mathbb{R}}(\bigcup\limits_{0\leq m\leq k }N_{0}(m))\leq \mathrm{dim}(U_{k}(\mathfrak{g}_{2})M_{0})\leq \dim{M_0}\dim Span_{\mathbb{R}}(\bigcup\limits_{0\leq m\leq k }N_{0}(m)).
$$
Then from the definition, we know that the Gelfand-Kirillov dimension of $U(\mathfrak{g}_{2})M_{0}$ is
$$d=2n-1, \text{if $1<n_1<n_2<n-1$.}$$
When $n_1=1<n_2<n-1$ or $1<n_1<n_2=n-1$, through a similar argument and using Proposition \ref{n<m} (1) and \ref{n<m}(2),
we find that the Gelfand-Kirillov dimension of $U(\mathfrak{g}_{2})M_{0}$ is $$d=2n-1.$$
When $1=n_1<n_2=n-1$ or $1=n_1<n_2=n$, through a similar argument and using Proposition \ref{ak},
we find that the Gelfand-Kirillov dimension of $U(\mathfrak{g}_{2})M_{0}$ is $$d=2n-2.$$
When $1<n_1<n_2=n$, through a similar argument and using Proposition \ref{n<m} (3),
we find that the Gelfand-Kirillov dimension of $U(\mathfrak{g}_{2})M_{0}$ is
$$d=\left\{
\betagin{array}{ll}
2n-2, & \hbox{if $n_1=2<n_2=n$;}\\
2n-1, & \hbox{if $3\leq n_1<n_2=n$.}
\end{equation}d{array}
\right.
$$
Therefore, for this case $n_1+1\leq n_2$, the Gelfand-Kirillov dimension of $U(\mathfrak{g}_{2})M_{0}$ is
\betagin{align*}d=\left\{
\betagin{array}{ll}
2n-2, & \hbox{if $n_1=1, n_2=n-1$ ~or~ $n_1=1,2,n_2=n$;}\\
2n-1, & \hbox{if $n_1=1<n_2<n-1$ ~or~ $1<n_1<n_2\leq n-1$ ~or~ $3\leq n_1<n_2=n$.}
\end{equation}d{array}
\right.
\end{equation}d{align*}
\hspace{1cm}
\subsection{Case 2. $n_1=n_2$ \emph{and} $0{\cal G}({\cal A})mmaeq k'\in \mathbb{Z}$.}
\hspace{1cm}
In this case we have:
\betagin{align} \label{ri-}(E_{r,i}-E_{n+i,n+r})|_{\mathcal{B}}&=-x_i\partial_{x_r}-y_i\partial_{y_r} &\text{for~}& 1\leq i<r\leq
n_1,\\
\label{ti-}(E_{t,i}-E_{n+i,n+t})|_{\mathcal{B}}&=-{x_i x_{t}}+y_{i}y_{t} &\text{for} ~&i\in\overline{1,n_1},\;t\in\overline{n_1+1,n},\\
\label{tp-}(E_{t,p}-E_{n+p,n+t})|_{\mathcal{B}}&=x_t\partial_{x_p}+y_t\partial_{y_p}&\text{for}~& n_1+1\leq p<t\leq
n,\\
\label{inr-}(E_{i,n+r}-E_{r,n+i})|_{\mathcal{B}}&=\partial_{x_i}\partial_{y_r}-\partial_{x_r}\partial_{y_i} &\text{for~}& 1\leq i<r\leq
n_1,\\
\label{nri-}(E_{n+r,i}-E_{n+i,r})|_{\mathcal{B}}&=-{x_i}{y_r}+{x_r}{y_i} &\text{for~}& 1\leq i<r\leq
n_1,\\
\label{int-}(E_{i,n+t}-E_{t,n+i})|_{\mathcal{B}}&=-y_t\partial_{x_i}-x_t\partial_{y_i} &\text{for} ~&i\in\overline{1,n_1},\;t\in\overline{n_1+1,n},\\
\label{nti-}(E_{n+t,i}-E_{n+i,t})|_{\mathcal{B}}&=-x_i\partial_{y_t}-y_i\partial_{x_t} &\text{for} ~&i\in\overline{1,n_1},\;t\in\overline{n_1+1,n},\\
\label{pnt-}(E_{p,n+t}-E_{t,n+p})|_{\mathcal{B}}&=-x_p{y_t}+x_t{y_p} &\text{for}~& n_1+1\leq p<t\leq n,\\
\label{npt-}(E_{n+p,t}-E_{n+t,p})|_{\mathcal{B}}&=-\partial_{x_p}\partial_{y_t}+\partial_{x_t}\partial_{y_p} &\text{for}~& n_1+1\leq p<t\leq n.
\end{equation}d{align}
Then the above root elements form a basis for the subalgebra $\mathfrak{g}(\mathcal{P}_{+})_{-}:=\mathfrak{o}(2n,\mathbfb{C})_-+\mathcal{P}_{+}$.
Suppose $n_1=n_2<n-1$. From Luo-Xu \cite{Luo-Xu-lie} we know that the $\mathcal{K}$-singular vectors in ${\mathcal H}_{\langle k'\ranglengle}$
are:
\begin{equation} x_{n_1}^{m_1}y_{n_2+1}^{m_2}\qquad\mathbfox{with}\;-(m_1+m_2)=k',\end{equation}
\begin{equation} x_{n_1}^{-k'}\zeta_{1}^{m+1}\qquad\mathbfox{with}\;m\in \mathbb{N},\end{equation}
\begin{equation}\label{n-} y_{n_1+1}^{-k'}\zeta_{2}^{m+1}\qquad\mathbfox{with}\;m\in \mathbb{N},\end{equation}
where $\zeta_1=x_{n_1-1}y_{n_1}-x_{n_1}y_{n_1-1}$ and $\zeta_2=x_{n_1+1}y_{n_1+2}-x_{n_1+2}y_{n_1+1}$.
Let $\mathfrak{g}_1$ be the subalgebra of $\mathfrak{o}(2n,\mathbb{C})$ spanned by the root vectors in the following set:$$I_1:=\{(\ref{ri-}),(\ref{tp-}),(\ref{inr-}),(\ref{int-}),(\ref{nti-}),(\ref{npt-})\}.$$
Let $\mathfrak{g}_2$ be the subalgebra of $\mathfrak{o}(2n,\mathbb{C})$ spanned by the root vectors in the following set:$$I_2:=\{(\ref{ti-}),(\ref{nri-}),(\ref{pnt-})\}.$$
So we get $U(\mathfrak{g}(\mathcal{P}_{+})_{-})=U(\mathfrak{g}_{2})U(\mathfrak{g}_{1}).$
We take a $\mathcal K$-singular vector $v_{\mathcal{K}}=x_{n_1}^{-k'}$, and set $M_{0}=U(\mathfrak{g}_1)x_{n_1}^{-k'}$.
Then $M_{0}$ is finite-dimensional.
Thus
$${\mathcal{H}}_{\langle k' \ranglengle}=U(\mathfrak{g})v_{\mathcal{K}}=U(\mathfrak{g_{-}+\mathcal{P}_{+}})v_{\mathcal{K}}=U(\mathfrak{g}_{2})M_0.$$
Let $k$ be any positive integer. We want to compute $\mathrm{dim}(U_{k}(\mathfrak{g}_{2})M_{0})$, and then get the Gelfand-Kirillov dimension of $U(\mathfrak{g}_{2})M_{0}$.
The argument for this case is similar to case 1, and using Proposition \ref{n=n},
we find that the Gelfand-Kirillov dimension of $U(\mathfrak{g}_{2})M_{0}$ is
\betagin{align}\label{n1=n2}d=\left\{
\betagin{array}{ll}
2n-3, & \hbox{if $n=2, 3$;}\\
2n-2, & \hbox{if $n=4$;}\\
2n-1, & \hbox{if $n{\cal G}({\cal A})mmaeq 5$.}
\end{equation}d{array}
\right.
\end{equation}d{align}
When $n_1=n_2=n-1,n$, the arguments are similar with the above, and the conclusion is the same with \ref{n1=n2}.
\section{Proof of the main theorem for $\mathfrak{o}(2n+1,\mathbfb{C})$}
We keep the same notations with the introduction. We know
$$\mathfrak{o}(2n+1,\mathbfb{F})=\mathfrak{o}(2n,\mathbfb{C})\opluslus\bigoplus_{i=1}^n
[\mathbfb{C}(E_{0,i}-E_{n+i,0})+\mathbfb{C}(E_{0,n+i}-E_{i,0})]$$
and $\mathcal{B}'=\mathbfb{C}[x_0,x_1,...,x_n,y_1,...,y_n]$.
Luo and Xu \cite{Luo-Xu-lie} proved that
for any
$ k'\in\mathbb{Z}$,
${\mathcal{H}'}_{\langle k' \ranglengle}$ is an irreducible
$\mathfrak{o}(2n+1,\mathbb{C})$-module. Moreover, the
homogeneous subspace ${\mathcal B}'=\bigoplus_{k\in \mathbb{Z}}\bigoplus_{i=0}^\infty(\eta')^i({\mathcal H}'_{\langle k'\ranglengle})$ is a
decomposition of irreducible submodules. The module ${\mathcal{H}'}_{\langle k' \ranglengle}$ under the assumption is of highest-weight type only if $n_2=n$, in which case $x_{n_1}^{-k'}$ is a highest-weight vector
with weight
$-k'\lambda_{n_1-1}+(k'-1)\lambda_{n_1}+[(k'-1)\delta_{n_1,n-1}-2k'\delta_{n_1,n}]\lambda_n.$
\subsection{Case 1. $n_1<n_2$ \emph{and} $k'\in \mathbb{N}$.}
\hspace{1mm}
The representation of $\mathfrak{o}(2n+1,\mathbfb{C})$ on $\mathcal{B}'$ by the differential operators in (\ref{ri})-(\ref{npt}) and $\mathcal{K}_{+}$ with $|_{\mathcal{B}}$ is replaced by $|_{\mathcal{B}'}$ and also contains the following:
\betagin{align} \label{0i}(E_{0,i}-E_{n+i,0})|_{\mathcal{B}'}&=-x_0x_i-y_i\partial_{x_0} &\text{for} ~&i\in\overline{1,n_1},\\
\label{0s}(E_{0,s}-E_{n+s,0})|_{\mathcal{B}'}&=x_0\partial_{x_s}-y_s\partial_{x_0} &\text{for} ~&s\in\overline{n_1+1,n_2},\\
\label{0t}(E_{0,t}-E_{n+t,0})|_{\mathcal{B}'}&=x_0\partial_{x_t}-\partial_{x_0}\partial_{y_t} &\text{for} ~&t\in\overline{n_2+1,n},\\
\label{0ni}(E_{0,n+i}-E_{i,0})|_{\mathcal{B}'}&=x_0\partial_{y_i}-\partial_{x_0}\partial_{x_i} &\text{for} ~&i\in\overline{1,n_1},\\
\label{0ns}(E_{0,n+s}-E_{s,0})|_{\mathcal{B}'}&=x_0\partial_{y_s}-{x_s}\partial_{x_0} &\text{for} ~&s\in\overline{n_1+1,n_2},\\
\label{0nt}(E_{0,n+t}-E_{t,0})|_{\mathcal{B}'}&=-x_0{y_t}-{x_t}\partial_{x_0} &\text{for} ~&t\in\overline{n_2+1,n}.
\end{equation}d{align}
Now we want to compute the Gelfand-Kirillov dimensions of the $\mathfrak{o}(2n+1,\mathbb{C})$-module ${\mathcal{H}'}_{\langle k'\ranglengle}$ and ${\mathcal{H}'}_{\langle -k'\ranglengle}$ for this case.
From Luo-Xu \cite{Luo-Xu-lie} we know that ${\mathcal{H}'}_{\langle k'\ranglengle}$ is an irreducible $\mathfrak{o}(2n+1,\mathbfb{C})$-submodule generated by $x_{n_1+1}^{k'}$, and ${\mathcal{H}'}_{\langle -k'\ranglengle}$ is an irreducible $\mathfrak{o}(2n+1,\mathbfb{C})$-submodule generated by $x_{n_1}^{k'}$. Then similar to the computation of
$\mathfrak{o}(2n+1,\mathbb{C})$, the Gelfand-Kirillov dimension of ${\mathcal{H}'}_{\langle k'\ranglengle}$ is
\betagin{align*}d=\left\{
\betagin{array}{ll}
2n-1, & \hbox{if $2=n_1<n_2=n$ or $1=n_1<n_2=n-1,n$;}\\
2n, & \hbox{if $3\leq n_1<n_2=n$ or $1<n_1<n_2=n-1$ or $1\leq n_1<n_2<n-1$.}
\end{equation}d{array}
\right.
\end{equation}d{align*}
${\mathcal{H}'}_{\langle -k'\ranglengle}$ has the same Gelfand-Kirillov dimension with ${\mathcal{H}'}_{\langle k'\ranglengle}$.
\subsection{Case 2. $n_1=n_2$ \emph{and} $k'\in \mathbb{N}$.}
\hspace{1mm}
The representation of $\mathfrak{o}(2n+1,\mathbfb{C})$ on $\mathcal{B}'$ by the differential operators in (\ref{ri})-(\ref{npt}) and $\mathcal{K}_{+}$ with $|_{\mathcal{B}}$ is replaced by $|_{\mathcal{B}'}$ and also contains the following:
\betagin{align} \label{0i-}(E_{0,i}-E_{n+i,0})|_{\mathcal{B}'}&=-x_0x_i-y_i\partial_{x_0} &\text{for} ~&i\in\overline{1,n_1},\\
\label{0t-}(E_{0,t}-E_{n+t,0})|_{\mathcal{B}'}&=x_0\partial_{x_t}-\partial_{x_0}\partial_{y_t} &\text{for} ~&t\in\overline{n_2+1,n},\\
\label{0ni-}(E_{0,n+i}-E_{i,0})|_{\mathcal{B}'}&=x_0\partial_{y_i}-\partial_{x_0}\partial_{x_i} &\text{for} ~&i\in\overline{1,n_1},\\
\label{0nt-}(E_{0,n+t}-E_{t,0})|_{\mathcal{B}'}&=-x_0{y_t}-{x_t}\partial_{x_0} &\text{for} ~&t\in\overline{n_2+1,n}.
\end{equation}d{align}
Now we want to compute the Gelfand-Kirillov dimensions of the $\mathfrak{o}(2n+1,\mathbb{C})$-module ${\mathcal{H}'}_{\langle k'\ranglengle}$ and ${\mathcal{H}'}_{\langle -k'\ranglengle}$ for this case.
From Luo-Xu \cite{Luo-Xu-lie} we know that ${\mathcal{H}'}_{\langle -k'\ranglengle}$ is an irreducible $\mathfrak{o}(2n+1,\mathbfb{C})$-submodule generated by $x_{n_1}^{k'}$, and ${\mathcal{H}'}_{\langle k'\ranglengle}$ is an irreducible $\mathfrak{o}(2n+1,\mathbfb{C})$-submodule generated by $T_{1}(y_{n_1}^{k'-1})$ (here $T_1=\sum\limits_{i=0}^{\infty}\varphirac{(-2)^ix_0^{2i+1}{
\mathcal{D}}^i}{(2i+1)!}$ and $\mathcal{D}=-\sum\limits_{i=1}^{n_1}x_i\partial_{y_i}+\sum\limits_{s=n_1+1}^{n_2}\partial_{x_s}\partial_{y_s}-\sum\limits_{t=n_2+1}^n
y_t\partial_{x_t}$). Then similar to the computation of
$\mathfrak{o}(2n+1,\mathbb{C})$, the Gelfand-Kirillov dimension of ${\mathcal{H}'}_{\langle -k'\ranglengle}$ is
\betagin{align*}d=\left\{
\betagin{array}{ll}
2n-2, & \hbox{if $1=n_1=n_2<n=2,3$;}\\
2n-1, & \hbox{if $n_1=n_2=2$ when $ n=2,3$ or $n_1=n_2=1$ when $n=1,4$;}\\
2n, & \hbox{if $n_1=n_2=n=3$ or $2\leq n_1=n_2\leq n=4$ or $1\leq n_1=n_2\leq n$ when $n{\cal G}({\cal A})mmaeq 5$.}
\end{equation}d{array}
\right.
\end{equation}d{align*}
${\mathcal{H}'}_{\langle k'\ranglengle}$ has the same Gelfand-Kirillov dimension with ${\mathcal{H}'}_{\langle -k'\ranglengle}$.
\section{Proof of the main theorem for $\mathfrak{sp}(2n,\mathbfb{C})$}
We keep the same notations with the introduction.
Recall the symplectic Lie
algebra\betagin{eqnarray*}\hspace{1cm}\mathfrak{sp}(2n,\mathbfb{C})&=&
\sum_{i,j=1}^n\mathbfb{C}(E_{i,j}-E_{n+j,n+i})+\sum_{i=1}^n(\mathbfb{C}E_{i,n+i}+\mathbfb{C}E_{n+i,i})\\
& &+\sum_{1\leq i<j\leq n
}[\mathbfb{C}(E_{i,n+j}+E_{j,n+i})+\mathbfb{C}(E_{n+i,j}+E_{n+j,i})].\end{equation}d{eqnarray*}
Again we take the Cartan subalgebra
$\mathfrak{h}=\sum_{i=1}^n\mathbfb{C}(E_{i,i}-E_{n+i,n+i})$ and the subspace
spanned by positive root vectors
$$\mathfrak{sp}(2n,\mathbfb{C})_+=\sum_{1\leq i<j\leq n}[\mathbfb{C}(E_{i,j}-E_{n+j,n+i})
+\mathbfb{C}(E_{i,n+j}+E_{j,n+i})]+\sum_{i=1}^n\mathbfb{C}E_{i,n+i}.$$
Correspondingly, we have $$\mathfrak{sp}(2n,\mathbfb{C})_-=\sum_{1\leq i<j\leq n}[\mathbfb{C}(E_{j,i}-E_{n+i,n+j})
+\mathbfb{C}(E_{n+i,j}+E_{n+j,i})]+\sum_{i=1}^n\mathbfb{C}E_{n+i,i}.$$
Fix $1\leq n_1\leq n_2\leq n$. We have the following two-parameter $\mathbb{Z}$-graded oscillator representation of $\mathfrak{sp}(2n,\mathbfb{C})$ on $\mathcal{B}=\mathbb{C}[x_1,...,x_n,y_1,...,y_n]$ determined by $$(E_{i,j}-E_{n+j,n+i})|_{\mathcal{B}}=E_{i,j}^{x}-E_{j,i}^{y}.$$
In particular we have
\betagin{align}
\label{si3}(E_{s,i}-E_{n+i,n+s})|_{\mathcal{B}}&=-{x_i x_{s}}-{y_i}\partial_{y_{s}} &\text{for} ~&i\in\overline{1,n_1},\;s\in\overline{n_1+1,n_2},
\\
\label{ti3}(E_{t,i}-E_{n+i,n+t})|_{\mathcal{B}}&=-{x_i x_{t}}+y_{i}y_{t} &\text{for} ~&i\in\overline{1,n_1},\;t\in\overline{n_2+1,n},\\
\label{ts3}(E_{t,s}-E_{n+s,n+t})|_{\mathcal{B}}&=x_{t}\partial_{x_{s}}+y_{s} y_{t} &\text{for} ~&s\in\overline{n_1+1,n_2},\;t\in\overline{n_{2}+1,n},\\
\label{nri3}(E_{n+r,i}+E_{n+i,r})|_{\mathcal{B}}&=-{x_i}{y_r}-{x_r}{y_i} &\text{for~}& 1\leq i<r\leq
n_1,\\
\label{nsi3}(E_{n+s,i}+E_{n+i,s})|_{\mathcal{B}}&=-{x_i}{y_s}+{y_i}\partial_{x_s} &\text{for} ~&i\in\overline{1,n_1},\;s\in\overline{n_1+1,n_2},\\
\label{snt3}(E_{s,n+t}+E_{t,n+s})|_{\mathcal{B}}&=-x_s{y_t}+x_t\partial_{y_s} &\text{for} ~&s\in\overline{n_1+1,n_2},\;t\in\overline{n_{2}+1,n},\\
\label{pnt3}(E_{p,n+t}+E_{t,n+p})|_{\mathcal{B}}&=-x_p{y_t}-x_t{y_p} &\text{for}~& n_2+1\leq p<t\leq n,\\
\label{ini}(E_{n+i,i})|_{\mathcal{B}}&=-x_i{y_i} &\text{for}~& i\in\overline{1,n_1},\\
\label{tnt}(E_{t,n+t})|_{\mathcal{B}}&=-x_t{y_t} &\text{for}~& t\in\overline{n_2+1,n}.
\end{equation}d{align}
Then the above root elements form a subalgebra for $\mathfrak{sp}(2n,\mathbfb{C})$, denoted by $\mathfrak{g}_2$. The remaining root elements form another subalgebra for $\mathfrak{sp}(2n,\mathbfb{C})$, denoted by $\mathfrak{g}_1$.
Luo and Xu \cite{Luo-Xu-lie} proved that
for any
$ k'\in\mathbb{Z}$, when $n_1<n_2$ or $k'\neq 0$,
${\mathcal{B}}_{\langle k' \ranglengle}$ is an irreducible weight
$\mathfrak{sp}(2n,\mathbb{C})$-module. Moreover, the module ${\mathcal{B}}_{\langle k' \ranglengle}$ under the assumption is of highest-weight type only if $n_2=n$, in which case for
$m\in\mathbfb{N}$, $x_{n_1}^{-m}$ is a highest-weight vector of ${\mathcal
B}_{\langle -m\ranglengle}$ with weight $-m\lambda_{n_1-1}+(m-1)\lambda_{n_1}$,
$x_{n_1+1}^{m+1}$ is a highest-weight vector of ${\mathcal B}_{\langle
m+1\ranglengle}$ with weight
$-(m+2)\lambda_{n_1}+(m+1)\lambda_{n_1+1}+(m+1)\delta_{n_1,n-1}\lambda_n$ if
$n_1<n_2=n$, and $y_n^{m+1}$ is a highest-weight vector of ${\mathcal B}_{\langle
m+1\ranglengle}$ with weight $(m+1)\lambda_{n-1}-2(m+1)\lambda_n$ if $n_1=n_2=n$. When $n_1=n_2$, the subspace ${\mathcal B}_{\langle 0\ranglengle}$ is a direct sum
of two irreducible weight $\mathfrak{sp}(2n,\mathbfb{C})$-submodules. If $n_1=n_2=n$, they
are highest-weight modules with a highest-weight vector $1$ of
weight $-2\lambda_n$ and with a highest-weight vector
$x_{n-1}y_n-x_ny_{n-1}$ of weight $(1-\delta_{n,2})\lambda_{n-2}-4\lambda_n$,
respectively.
We take $\mathcal {K}=\sum\limits_{i,j=1}^{n}\mathbb{C}(E_{i,j}-E_{n+j,n+i})$, and $\mathcal{K}_{+}=\sum_{1\leq i<j\leq n}\mathbfb{C}(E_{i,j}-E_{n+j,n+i})$.
A weight vector $v$ in $\mathcal{B}$ is called a \emph{$\mathcal{K}$-singular vector} if $\mathcal{K}_{+}(v)=0$.
From the PBW theorem we have
$${\mathcal{B}}_{\langle k' \ranglengle}=U(\mathfrak{g})v_{\mathcal{K}}=U(\mathfrak{g}_{2})U(\mathfrak{g}_{1})v_{\mathcal{K}}$$ for any fixed $\mathcal K$-singular vector $v_{\mathcal{K}}$. If we denote $M_0:=U(\mathfrak{g}_{1})v_{\mathcal{K}}$, then $M_0$ is finite-dimensional and ${\mathcal{B}}_{\langle k' \ranglengle}=U(\mathfrak{g}_{2})M_0$.
Similar to the $\mathfrak{o}(2n,\mathbfb{C})$ case, we can compute the Gelfand-Kirillov dimension of $\mathcal{B}_{\langle k' \ranglengle}$ in a case-by-case way. Actually the Gelfand-Kirillov dimension is equal to $$2n-1$$ for any irreducible $\mathfrak{sp}(2n,\mathbb{C})$-module $\mathcal{B}_{\langle k' \ranglengle}$.
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\end{equation}d{document} | math | 73,927 |
\mbox{$\B(E)$}\/gin{document}
\noindent
{\Large\bf Explicit recurrence criteria for symmetric gradient type Dirichlet forms satisfying a Hamza type condition}
\noindent
{\bf by Minjung Gim} \footnote{The research of Minjung Gim was supported by NRF (National Research Foundation of Korea) Grant funded by the Korean Government(NRF-2012-Global Ph.D. Fellowship Program).}
{\bf and Gerald Trutnau} \footnote{The research of Gerald Trutnau was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(MEST)(2012006987)
and Seoul National University Research Grant 0450-20110022.}
\noindent
{\it Department of Mathematical Sciences and Research Institute of Mathematics of Seoul National University,
599 Gwanak-Ro, Gwanak-Gu, Seoul 151-747, South Korea}
\noindent
{E-mails: [email protected], [email protected]}
\noindent
{\small{\bf Abstract.} In this note, we present explicit conditions for symmetric gradient type Dirichlet forms to be recurrent.
This type of Dirichlet form is typically strongly local and hence associated to a diffusion. We consider the one dimensional case
and the multidimensional case, as well as the case with reflecting boundary conditions. Our main achievement is that the explicit results are obtained under quite weak assumptions on the closability, hence regularity of the underlying coefficients. Especially in dimension one, where a Hamza type condition is assumed, the construction of the sequence of functions $(u_n)_{n\in {\mathbb N}}$ in the Dirichlet space that determine
recurrence works for quite general Dirichlet forms but is still explicit.}
\noindent
{\bf Mathematics Subject Classification (2000)}: 31C25, 60J60, 60G17.
\noindent
{\bf Key words:} Symmetric Dirichlet forms, diffusion processes, recurrence criteria.
\section{Introduction}\label{1}
For all notions and results that may not be defined or cited in this introduction we refer to \cite{FOT}. \\
Let $E$ be a locally compact separable metric space and $\mu$ be a $\sigma$-finite measure on the Borel $\sigma$-algebra of $E$.
Consider a regular symmetric Dirichlet form $({\cal E},D({\cal E}))$ on $L^2(E,\mu)$ with associated resolvent $(G_\alpha)_{\alpha > 0}$.
Then it is well-known that there exists a Hunt process
$\mathbb{M}=(\Omega,({\cal F}_t)_{t\ge 0},(X_t)_{t\ge 0}, (P_x)_{x\in E_{\cal D}elta})$ with lifetime $\zeta$ such that the resolvent
$R_\alpha f(x)=E_{x}[\int_0^{\infty}e^{-\alpha t}f(X_t)dt]$ of $\mathbb{M}$ is a q.c. $\mu$-version of $G_\alpha f$ for all $\alpha >0$, $f\in L^2(E,\mu)$, $f$ bounded (see\cite[Theorems 4.2.3 and 7.2.1]{FOT}). Here $E_x$ denotes the expectation w.r.t. $P_x$. Then $({\cal E},D({\cal E}))$ is (called) {\it recurrent}, iff for
any $f\in L^1(E,\mu)$ which is strictly positive, we have
\mbox{$\B(E)$}\/gin{eqnarray}\label{recurrent}
E_x[\int_0^{\infty}f(X_t)dt]=\infty \mbox{ for } \mu\mbox{-a.e. } x\in E.
\mbox{${\cal E }$}nd{eqnarray}
The last is equivalent to the existence of a sequence $(u_n)_{n\in {\mathbb N}}\subset D({\cal E})$ with $0\leq u_n \leq 1$, $n\in {\mathbb N}$, $u_n \nearrow 1 $ as $n \rightarrow \infty$ $\mu$-a.e. and
$\lim_{n\rightarrow \infty}{\cal E}(u_n , u_n)=0$ (cf. \cite[Lemma 1.6.4 (ii) and Theorem 1.6.3]{FOT} and \cite[Corollary 2.4 (iii)]{o92r}).
In particular, recurrence implies conservativeness, which means that $P_x(\zeta=\infty)=1$ for quasi every $x\in E$ (see \cite[Corollary 2.4]{o92r}). If the Dirichlet form $({\cal E},D({\cal E}))$ is additionally {\it irreducible}, then some more refined recurrence statements than (\ref{recurrent}) can be made
(cf. \cite[Chapter 4.7]{FOT2} and \cite{ge80}), and if the Dirichlet form $({\cal E},D({\cal E}))$ satisfies the absolute continuity condition (cf. \cite[(4.2.9)]{FOT}), then the statements hold pointwise and not only for quasi every $x\in E$ (see \cite[Problem 4.6.3]{FOT}).\\
A sufficient condition for recurrence determined by the $\mu$-volume growth of balls is derived in \cite[Theorem 3]{Stu1} for energy forms, i.e. Dirichlet forms that are given by a carr\'e du champs. The condition is very general, but in some cases not explicit, i.e. difficult to verify (cf. e.g. \cite[Proposition 3.3. and Theorem 3.11]{oukrutr} where the determination of the $\mu$-volume of balls is tedious). Moreover, in \cite{Stu1} it is throughout assumed that the underlying Dirichlet form is irreducible, which we do not assume. In \cite{o92r} and \cite[Theorem 1.6.7]{FOT} the recurrence determining sequence $(u_n)_{n\in {\mathbb N}}\subset D({\cal E})$ is explicitly constructed, but the underlying Dirichlet form fulfills stronger (explicit) assumptions than in \cite{Stu1}, and moreover \cite{o92r} and \cite[Theorem 1.6.7]{FOT} concern only variants of the $(a_{ij})$-case. \\
This note concerns an intermediate approach, which resembles the construction in \cite[Theorem 1.6.7]{FOT}, but under quite more general assumptions.
In particular, we consider reference measures that are different to the Lebesgue measure.
The conditions for the existence of a recurrence determining sequence $(u_n)_{n\in {\mathbb N}}$ given here are (as in \cite{FOT} and \cite{Stu1}) sufficient.
However, in dimension one they turn out to be equivalent, whenever there exists a natural scale (see e.g. \cite[Lemma 3.1]{o92r} and \cite[Theorem 3.11]{oukrutr}).
\section{The one dimensional non-reflected case}\label{2}
Let $\varphi \in L^1_{loc}({\mathbb R},dx)$ with $\varphi>0$ $dx$-a.e. and let $\mu$ be the $\sigma$-finite measure defined by $d\mu=\varphi dx$. In particular, it then follows that $\mu$ is $\sigma$-finite and $\mu $ has full support, i.e.
$$\int_V \varphi(x) dx>0\qquad for\; all\; V \subset {\mathbb R},\; V\;non-empty\;and\;open.$$
Let $\sigma$ be a (measurable) function such that $\sigma>0$ $dx$-a.e. and such that $\sigma \varphi \in L^1_{loc}({\mathbb R}, dx)$. Consider the symmetric bilinear form
$${\cal E}(f,g):= \frac{1}{2} \int_{\mathbb R} \sigma(x) f'(x)g'(x) \mu(dx), \qquad f,g\in C^\infty_0({\mathbb R})$$
on $L^2 ({\mathbb R},\mu)$. Here $f'$ denotes the derivative of a function $f$ and $C_0^\infty ({\mathbb R})$ denotes the set of all infinitely differentiable function on ${\mathbb R}$ with compact support. Let $U$ be the largest open set in ${\mathbb R}$ such that
$$
\frac{1}{\sigma\varphi}\in L^1_{loc}(U,dx)
$$
and assume
\mbox{$\B(E)$}\/gin{equation}\label{Hamza}
dx({\mathbb R} \setminus U)=0.
\mbox{${\cal E }$}nd{equation}
\centerline{}
Furthermore we suppose $({\cal E},C_0^\infty ({\mathbb R} ))$ is closable on $L^2({\mathbb R},d\mu)$. (By the results of (\cite[II, 2 a)]{mr}), if there is some open set $\tilde{U}\subset {\mathbb R}$ such that $1/\varphi \in L^1_{loc}(\tilde{U},dx)$ and $dx({\mathbb R} \setminus(U\cap \tilde{U}))=0$, then $({\cal E},C_0^\infty({\mathbb R}))$ is closable on $L^2({\mathbb R},\mu)$.) Denote the closure by $({\cal E},D({\cal E}))$. We want to present sufficient conditions for the recurrence of the symmetric Dirichlet form $({\cal E},D({\cal E}))$. Here we recall $({\cal E},D({\cal E}))$ is called recurrent (in the sense of \cite{FOT}) if there exists a sequence $(u_n)_{n\in {\mathbb N}}\subset D({\cal E})$ with $0\leq u_n \leq 1$, $n\in {\mathbb N}$, $u_n \nearrow 1 $ as $n \rightarrow \infty $ $dx$-a.e. and
$$ \lim_{n\rightarrow \infty}{\cal E}(u_n , u_n)=0.$$
\centerline{}
\mbox{$\B(E)$}\/gin{rem}
Under quite weak regularity assumptions on $\sigma$ and $\varphi$ one can show using integration by parts that the generator $L$ corresponding to $({\cal E},D({\cal E}))$ (for the definition of generator see \cite{FOT} or \cite{mr}) is given by
$$
Lf= \frac{\sigma}{2}f'' + \frac{1}{2}{\cal B}ig ( \sigma ' + \sigma \frac{\varphi '}{\varphi} {\cal B}ig) f',
$$
i.e. for $f\in D(L)\subset D({\cal E})$ we have
$$-\int_{\mathbb R} Lf \cdot g d\mu ={\cal E}(f,g).$$
In particular (assuming again that everything is sufficiently regular) choosing
$$\varphi(x)=\frac{1}{\sigma(x)}e^{\int_0^x \frac{2b}{\sigma}(s)ds}$$
for some (freely chosen) function $b$, we get
$$Lf= \frac{\sigma}{2}f''+bf'.$$
Therefore our framework is suitable for the description of nearly any diffusion type operator in dimension one with concrete coefficients.
\mbox{${\cal E }$}nd{rem}
\centerline{}
Since $U\subset {\mathbb R}$ is open, $U$ is the disjoint union (we use the symbol $\cupdot$ to denote this) of countably (finite or infinite) many open intervals.\\
There are five possible cases that we summarize in the following theorem.
\mbox{$\B(E)$}\/gin{theo}\label{nonreflected}
$({\cal E},D({\cal E}))$ is recurrent if one of the following conditions holds:\\
(i) $U=(-\infty, \infty)$ and
$$\int_{-\infty}^0 \frac{1}{\sigma \varphi}(s) ds = \int_0^\infty \frac{1}{\sigma \varphi}(s) ds =\infty.$$
(ii) $U=(-\infty,a) \cupdot V \cupdot (b,\infty)$ where $V$ is some open set (so either $V$ is empty if $a=b$, or $V$ is non-empty if $a<b$) and
$$\int ^{a-1}_{-\infty} \frac{1}{\sigma \varphi}(s) ds = \int^\infty_{b+1}\frac{1}{\sigma \varphi}(s) ds = \infty.$$
(iii) $U= \bigcup_{n\in {\mathbb N}} I_{-n} \cupdot V \cupdot \bigcup_{n\in {\mathbb N}}I_n$, $I_n=(x_n, x_{n+1})$, $x_n < x_{n+1}$, $I_{-n}=(x_{-n}, x_{-n+1})$, $x_{-n} < x_{-n+1}$, $n\in {\mathbb N}$, $V$ some open set, and
\mbox{$\B(E)$}\/gin{align*}
a_n&:= \int_{c_n}^{d_n} \frac{1}{\sigma \varphi}(s) ds \; \longrightarrow \infty \\
a_{-n}=b_n&:=\int^{c_{-n}}_{d_{-n}} \frac{1}{\sigma \varphi}(s) ds \; \longrightarrow \infty
\mbox{${\cal E }$}nd{align*}
as $n \rightarrow \infty$, where
$$c_n= \frac{x_{n+1}+x_n}{2}, \quad n\in {\mathbb Z} \setminus \{0\},$$
and for $n\in {\mathbb N}$
$$
d_n:= \mbox{$\B(E)$}\/gin{cases}\;x_{n+1}-\frac{1}{n}\qquad \;\qquad &\;\;x_{n+1}-c_n>1,\\
\; x_{n+1}-\frac{x_{n+1}-c_n}{n} &\;\;x_{n+1}-c_n\leq 1,
\mbox{${\cal E }$}nd{cases}
$$
$$
d_{-n}:= \mbox{$\B(E)$}\/gin{cases}\;x_{-n}+\frac{1}{n}\qquad \;\qquad &\;\;c_{-n}-x_{-n}>1,\\
\; x_{-n}+\frac{c_{-n}-x_{-n}}{n} &\;\;c_{-n}-x_{-n}\leq 1.
\mbox{${\cal E }$}nd{cases}
$$
(iv) $U=\bigcup_{n\in {\mathbb N}}I_{-n}\cupdot V \cupdot (b,\infty),$ $I_{-n}=(x_{-n}, x_{-n+1}),$ $x_{-n}<x_{-n+1}$, $n\in {\mathbb N},$ $V$ some open set,
$$
\int^\infty_{b+1}\frac{1}{\sigma \varphi}(s) ds = \infty,
$$
and
$$
a_{-n}=b_n:=\int^{c_{-n}}_{d_{-n}} \frac{1}{\sigma \varphi}(s) ds \; \longrightarrow \infty
$$
as $n \rightarrow \infty$ where for $n\in {\mathbb N}$
$$
c_{-n}= \frac{x_{-n}+x_{-n+1}}{2}
$$
and
$$
d_{-n}(x):= \mbox{$\B(E)$}\/gin{cases}\;x_{-n}+\frac{1}{n}\qquad \;\qquad &\;\;c_{-n}-x_{-n}>1,\\
\; x_{-n}+\frac{c_{-n}-x_{-n}}{n} &\;\;c_{-n}-x_{-n}\leq 1.
\mbox{${\cal E }$}nd{cases}
$$
(v) $U=(-\infty,a)\cupdot V \cupdot \bigcup_{n\in {\mathbb N}} I_n$, $I_{n}=(x_{n}, x_{n+1}),$ $x_{n}<x_{n+1}$, $n\in {\mathbb N},$ $V$ some open set,
$$
\int ^{a-1}_{-\infty} \frac{1}{\sigma \varphi}(s) ds=\infty,
$$
and
$$
a_n:= \int_{c_n}^{d_n} \frac{1}{\sigma \varphi}(s) ds \; \longrightarrow \infty
$$
as $n\rightarrow \infty$ where for $n\in {\mathbb N}$
$$
c_n= \frac{x_{n+1}+x_n}{2}
$$
and
$$
d_n:= \mbox{$\B(E)$}\/gin{cases}\;x_{n+1}-\frac{1}{n}\qquad \;\qquad &\;\;x_{n+1}-c_n>1,\\
\; x_{n+1}-\frac{x_{n+1}-c_n}{n} &\;\;x_{n+1}-c_n\leq 1.
\mbox{${\cal E }$}nd{cases}
$$
\mbox{${\cal E }$}nd{theo}
\centerline{}
\centerline{}
\mbox{$\B(E)$}\/gin{rem}\label{openset}
With the obvious modifications Theorem \ref{nonreflected} can be reformulated for Dirichlet forms that are given as the closure of
$$ \frac{1}{2}\int_V \sigma(x) f'(x) g'(x) \mu(dx), \qquad f,g\in C^\infty_0(V)$$
where $V$ is an arbitrary open and connected set in ${\mathbb R}$. We omit this here to avoid trivial complications.
\mbox{${\cal E }$}nd{rem}
\centerline{}
\centerline{}
\mbox{$\B(E)$}\/gin{proof}
(i) By \cite[Theorem 1.6.3]{FOT}, it suffices to find a sequence $(u_n)_{n\in {\mathbb N}}\subset D({\cal E})$ with $0\leq u_n \leq 1$, $n\in {\mathbb N}$, $u_n \nearrow 1 $ as $n \rightarrow \infty $ and $ \lim_{n\rightarrow \infty}{\cal E}(u_n , u_n)=0.$ Define sequences $a_n$ and $b_n$ by
$$
a_n:= \int_0^n \frac{1}{\sigma \varphi}(s)ds, \qquad b_n:= \int_{-n}^0 \frac{1}{\sigma \varphi}(s) ds.
$$
Since $1/\sigma\varphi \in L^1_{loc}({\mathbb R},dx)$, these are well defined and converge to $\infty$. Let\\
$$
u_n(x):= \mbox{$\B(E)$}\/gin{cases}\; 1-\frac{1}{a_n}\int_0^x \frac{1}{\sigma\varphi}(t)dt\qquad \; \;&x\in[0,n],\\
\; 1-\frac{1}{b_n}\int_x^0 \frac{1}{\sigma \varphi}(t)dt&x\in[-n,0], \\
\;0&elsewhere.
\mbox{${\cal E }$}nd{cases}
$$
For each $n\in {\mathbb N}$, $u_n$ has compact support and is bounded. Moreover, since $1/\sigma\varphi \in L^1_{loc}({\mathbb R},dx)$, $u_n(x)$ is differentiable at every Lebesgue point of $1/\sigma \varphi$, hence $dx$-a.e. Furthermore, we can easily check $u_n \nearrow 1$ $dx$-a.e. as $n \rightarrow \infty$ and $u_n'(x)$ exists $dx$-a.e. with
$$
u_n'(x)= \mbox{$\B(E)$}\/gin{cases}\; -\frac{1}{a_n} \frac{1}{\sigma\varphi}(x)\qquad \; \;&x\in[0,n],\\
\;\frac{1}{b_n}\frac{1}{\sigma \varphi}(x)&x\in[-n,0], \\
\; 0&elsewhere.
\mbox{${\cal E }$}nd{cases}
$$
Thus, it remains to show that $(u_n)_{n\in {\mathbb N}} \subset D({\cal E})$. Let $\mbox{${\cal E }$}ta$ be a standard mollifier on ${\mathbb R}$. Set $\mbox{${\cal E }$}ta_\mbox{${\cal E }$}psilon(x)= \frac{1}{\mbox{${\cal E }$}psilon} \mbox{${\cal E }$}ta(\frac{x}{\mbox{${\cal E }$}psilon})$ so that $\int_{\mathbb R} \mbox{${\cal E }$}ta_\mbox{${\cal E }$}psilon dx=1$ and so that the support of $\mbox{${\cal E }$}ta_\mbox{${\cal E }$}psilon$ is in $(-\mbox{${\cal E }$}psilon,\mbox{${\cal E }$}psilon)$. Then $\mbox{${\cal E }$}ta_\mbox{${\cal E }$}psilon*u_n\in C^\infty_0({\mathbb R})$ and $ (\mbox{${\cal E }$}ta_\mbox{${\cal E }$}psilon*u_n)'=\mbox{${\cal E }$}ta_\mbox{${\cal E }$}psilon*u_n',$ $n\in {\mathbb N}.$ We have
$$ (\mbox{${\cal E }$}ta_\mbox{${\cal E }$}psilon*u_n(x)-u_n(x))= \int_{\mathbb R} [u_n(x-y)-u_n(x)] \mbox{${\cal E }$}ta_\mbox{${\cal E }$}psilon(y) dy,$$
$$| \mbox{${\cal E }$}ta_\mbox{${\cal E }$}psilon*u_n(x)-u_n(x)|^2 \leq \int_{\mathbb R} |u_n(x-y)-u_n(x)|^2 \mbox{${\cal E }$}ta_\mbox{${\cal E }$}psilon(y) dy,$$
\mbox{$\B(E)$}\/gin{align*}
\int_{\mathbb R} (\mbox{${\cal E }$}ta_\mbox{${\cal E }$}psilon*u_n (x)-u_n(x))^2 \varphi(x) dx &\leq \int_{\mathbb R} \int_{\mathbb R} |u_n(x-y)-u_n(x)|^2 \mbox{${\cal E }$}ta_\mbox{${\cal E }$}psilon(y) dy \varphi(x)dx \\
&= \int_{\mathbb R} \int_{\mathbb R} |u_n(x-y) - u_n(x)|^2 \varphi(x)dx \mbox{${\cal E }$}ta_\mbox{${\cal E }$}psilon(y) dy\\
&=\int_{\mathbb R} \int_{\mathbb R} |u_n(x-\mbox{${\cal E }$}psilon y)-u_n(x)|^2 \varphi(x) dx \mbox{${\cal E }$}ta(y)dy.
\mbox{${\cal E }$}nd{align*}
\centerline{}
If we let $g(y):= \int_{\mathbb R} |u_n(x-y)-u_n(x)|^2 \varphi(x) dx$, then $g(0)=0$ and $g$ is continuous and bounded. Thus by the Dominated Convergence Theorem $\lim_{\mbox{${\cal E }$}psilon \rightarrow 0} \mbox{${\cal E }$}ta_\mbox{${\cal E }$}psilon*u_n = u_n$ in $L^2({\mathbb R},\mu)$. Furthermore for $0<\mbox{${\cal E }$}psilon<1$
\mbox{$\B(E)$}\/gin{align*}
{\cal E}(\mbox{${\cal E }$}ta_\mbox{${\cal E }$}psilon*u_n,\mbox{${\cal E }$}ta_\mbox{${\cal E }$}psilon*u_n)&= \int_{\mathbb R} {\cal B}ig (\mbox{${\cal E }$}ta_\mbox{${\cal E }$}psilon*u_n'(x) {\cal B}ig )^2 \sigma(x)\varphi(x) dx \\
&\leq \| u_n' \|^2_{L^1({\mathbb R},dx)} \int_{-n-1}^{n+1} \sigma(x)\varphi (x) dx
\mbox{${\cal E }$}nd{align*}
Since $| \mbox{${\cal E }$}ta_\mbox{${\cal E }$}psilon*u_n'(x)| \leq \|u_n'\|_{L^1({\mathbb R}, dx)} < \infty$, we have
$$ \sup_{0<\mbox{${\cal E }$}psilon<1} {\cal E}(\mbox{${\cal E }$}ta_\mbox{${\cal E }$}psilon*u_n, \mbox{${\cal E }$}ta_\mbox{${\cal E }$}psilon*u_n) < \infty.$$
Thus from \cite[Lemma 2.12]{mr}, $u_n \in D({\cal E})$ for all $n\in {\mathbb N}.$
\mbox{$\B(E)$}\/gin{align*}
{\cal E}(u_n,u_n) &= \frac{1}{2} \int_{\mathbb R} u_n'(x) u_n'(x) \sigma(x)\varphi (x) dx\\
&=\frac{1}{2} {\cal B}ig[ \frac{1}{a_n^2} \int_0^n \frac{1}{\sigma\varphi}(x) dx +\frac{1}{b_n^2}\int_{-n}^0 \frac{1}{\sigma \varphi}(x) dx {\cal B}ig]\\
&= \frac{1}{2} \big[\frac{1}{a_n}+\frac{1}{b_n}\big].
\mbox{${\cal E }$}nd{align*}
Therefore $\lim_{n\rightarrow \infty}{\cal E}(u_n,u_n)= 0$. i.e. $({\cal E},D({\cal E}))$ is recurrent. \\
\centerline{}
(ii) Let
$$
u_n(x):= \mbox{$\B(E)$}\/gin{cases}\;1 \qquad \qquad \qquad \qquad &x\in [a-1,b+1],\\
\; 1-\frac{1}{a_n}\int_{b+1}^x \frac{1}{\sigma \varphi}(t)dt&x\in[b+1,b+1+n],\\
\; 1-\frac{1}{b_n}\int_x^{a-1} \frac{1}{\sigma \varphi}(t)dt&x\in[a-1-n,a-1], \\
\;0&\quad elsewhere,
\mbox{${\cal E }$}nd{cases}
$$
where $a_n=\int^{b+1+n}_{b+1} \frac{1}{\sigma\varphi}(s) ds$, $ b_n=\int^{a-1}_{a-1-n} \frac{1}{\sigma \varphi}(s) ds$. Then $(u_n)_{n\in {\mathbb N}}$ satisfies the desired properties and determines recurrence.\\
\centerline{}
(iii) Let
$$
u_n(x):= \mbox{$\B(E)$}\/gin{cases}\;1 \qquad \qquad \qquad \qquad &x\in [c_{-n},c_n],\\
\; 1-\frac{1}{a_n}\int_{c_n}^x \frac{1}{\sigma\varphi}(t) dt&x\in[c_n,d_n],\\
\; 1-\frac{1}{b_n}\int_x^{c_{-n}} \frac{1}{\sigma\varphi}(t) dt&x\in[d_{-n},c_{-n}], \\
\;0&\quad elsewhere,
\mbox{${\cal E }$}nd{cases}
$$
where $a_n= \int_{c_n}^{d_n} \frac{1}{\sigma \varphi}(s) ds$, $b_n= \int_{c_{-n}}^{d_{-n}} \frac{1}{\sigma \varphi}(s) ds$. Then $(u_n)_{n\in {\mathbb N}}$ satisfies the desired properties and determines recurrence.\\
\centerline{}
(iv) and (v) are combinations of (ii) and (iii) and are proved by combining the proofs of (ii) and (iii).
\mbox{${\cal E }$}nd{proof}
\centerline{}
\mbox{$\B(E)$}\/gin{rem}\label{notirreducible}
Note that we do not assume that $({\cal E},D({\cal E}))$ is irreducible. As a non-trivial example consider the following:
Let $S=\{ x_i \in {\mathbb R} |\, i\in {\mathbb Z}\}$ with $x_i < x_{i+1}$ for all $i\in {\mathbb Z}$ and assume $S$ does not have an accumulation point in ${\mathbb R}$. For $\alpha \geq 1$, define a function $\varphi$ by
$$
\varphi(x)= |x-x_i|^\alpha, x\in \big[ \frac{x_i+x_{i-1}}{2}, \frac{x_i+x_{i+1}}{2} \big], i\in {\mathbb Z}.$$
Then $\varphi>0$ on ${\mathbb R}\setminus S$, hence $dx$-a.e. Assume $\sigma \mbox{${\cal E }$}quiv 1$, then since $1/\varphi \in L^1_{loc}({\mathbb R} \setminus S,dx),$ (\ref{Hamza}) is also satisfied. Thus, the symmetric bilinear form $({\cal E},C_0^\infty({\mathbb R}))$ defined by
$$
{\cal E}(f,g):=\frac{1}{2} \int_{\mathbb R} f'(x)g'(x) \mu(dx), \qquad f,g\in C^\infty_0({\mathbb R})
$$
is closable on $L^2({\mathbb R},d\mu)$ where $d\mu=\varphi dx$. Define the sequences $a_n$, $b_n$, $c_n$ and $d_n$ as in the Theorem \ref{nonreflected} (iii), then
$$
a_n= \int_{c_n}^{d_n} \frac{1}{\varphi}(s) ds=\int_{c_n}^{d_n} \frac{1}{(x_{n+1}-s)^\alpha} ds.
$$
(i) If $\alpha=1$, then
\mbox{$\B(E)$}\/gin{align*}a_n&=-\log(x_{n+1}-s)|^{d_n}_{c_n} \\
&=-\log(x_{n+1}-d_n) + \log (x_{n+1}-c_n).
\mbox{${\cal E }$}nd{align*}
In this case, if $x_{n+1}-c_n>1$, then $x_{n+1}-d_n=\frac{1}{n}$ and
\mbox{$\B(E)$}\/gin{align*}a_n&=-\log\big(\frac{1}{n}\big) +\log(x_{n+1}-c_n) \\
&>\log n.
\mbox{${\cal E }$}nd{align*}
And if $x_{n+1}-c_n\leq 1$, then $x_{n+1}-d_n=\frac{x_{n+1}-c_n}{n}$ and
\mbox{$\B(E)$}\/gin{align*}a_n&=-\log\big(\frac{x_{n+1}-c_n}{n}\big) +\log(x_{n+1}-c_n) \\
&=\log n.
\mbox{${\cal E }$}nd{align*}
Thus $\lim_{n\rightarrow \infty}a_n=\infty$ if $\alpha=1$. \\ \\
(ii) If $\alpha>1$, then
\mbox{$\B(E)$}\/gin{align*}a_n&=\int_{c_n}^{d_n} \frac{1}{(x_{n+1}-s)^\alpha} ds \\
&=\frac{-1}{1-\alpha}(x_{n+1}-s)^{1-\alpha}|^{d_n}_{c_n} \\
&=\frac{1}{\alpha-1}{\cal B}ig[ (x_{n+1}-d_n)^{1-\alpha}-(x_{n+1}-c_n)^{1-\alpha} {\cal B}ig] .
\mbox{${\cal E }$}nd{align*}
In this case, if $x_{n+1}-c_n>1$, then $x_{n+1}-d_n=\frac{1}{n}$ and
\mbox{$\B(E)$}\/gin{align*}a_n&= \frac{1}{\alpha-1}{\cal B}ig[ \big (\frac{1}{n}\big)^{1-\alpha}-(x_{n+1}-c_n)^{1-\alpha} {\cal B}ig] \\
&>\frac{1}{\alpha-1}(n^{\alpha-1}-1).
\mbox{${\cal E }$}nd{align*}
And if $x_{n+1}-c_n\leq 1$, then $x_{n+1}-d_n=\frac{x_{n+1}-c_n}{n}$ and
\mbox{$\B(E)$}\/gin{align*}a_n&= \frac{1}{\alpha-1}{\cal B}ig[ \big(\frac{x_{n+1}-c_n}{n}\big)^{1-\alpha}-(x_{n+1}-c_n)^{1-\alpha} {\cal B}ig] \\
&= \frac{1}{\alpha-1}{\cal B}ig[ (x_{n+1}-c_n)^{1-\alpha} \big\{ \big(\frac{1}{n}\big)^{1-\alpha} -1 \big\} {\cal B}ig] \\
&\geq \frac{1}{\alpha-1} (n^{\alpha-1}-1).
\mbox{${\cal E }$}nd{align*}
Thus $\lim_{n\rightarrow \infty}a_n=\infty$ if $\alpha \geq 1$. In the same way, $\lim_{n\rightarrow \infty} b_n=\infty$.
Therefore, $({\cal E},D({\cal E}))$ is recurrent, thus in particular conservative (cf. \cite[Theorems 1.6.5 and 1.6.6]{FOT}). Since $({\cal E},D({\cal E}))$ is also strongly local, the process associated to $({\cal E},D({\cal E}))$ is a conservative diffusion (cf. \cite{FOT}). Moreover, since by \cite[Example 3.3.2]{FOT} Cap$(\{x_i\})=0,$ the sets $(x_i, x_{i+1})$ are all invariant, i.e.
$$ p_t 1_{(x_i,x_{i+1})}(x) =0,\quad x\notin(x_i,x_{i+1}),\;\forall i \in {\mathbb Z}$$
where $p_t$ is the transition semigroup of (the process associated to) the Dirichlet form $({\cal E},D({\cal E}))$. But
$$ d\mu((x_i,x_{i+1}))\neq 0, \qquad d\mu({\mathbb R}\setminus (x_i,x_{i+1}))\neq0 \quad \forall i\in {\mathbb Z}.$$
Therefore $({\cal E},D({\cal E}))$ is not irreducible (in the sense of \cite{FOT}).
\mbox{${\cal E }$}nd{rem}
\centerline{}
\section{The one dimensional reflected case}\label{3}
Let $I=[0,\infty)$ and $C_0^\infty(I):=\{f:I \rightarrow {\mathbb R} \; |\; \mbox{${\cal E }$}xists g \in C_0^\infty ({\mathbb R})$ with $g=f$ on $I \}$. Let $\varphi \in L^1_{loc}(I,dx)$ with $\varphi>0$ $dx$-a.e. Furthermore assume $\sigma$ be a (measurable) function such that $\sigma>0$ $dx$-a.e. and such that $\sigma \varphi \in L^1_{loc}(I, dx)$. Consider the symmetric bilinear form
$${\cal E}(f,g):= \frac{1}{2} \int_I \sigma(x) f'(x)g'(x) \mu(dx), \qquad f,g\in C^\infty_0(I)$$
on $L^2 (I,\mu)$ where as before $\mu:=\varphi dx$. As in the Section \ref{2}, we suppose $U$ is the largest open set in $I$ such that
$$\frac{1}{\sigma\varphi}\in L^1_{loc}(U,dx)$$
and assume
\mbox{$\B(E)$}\/gin{equation}\label{Hamza2}
dx(I \setminus U)=0.
\mbox{${\cal E }$}nd{equation}
\centerline{}
We suppose $({\cal E},C_0^\infty (I ))$ is closable on $L^2(I,\mu)$. (By the results of \cite[Lemma 1.1]{tr}, if there is some open set $\tilde{U}\subset I$ such that $1/\varphi \in L^1_{loc}(\tilde{U},dx)$ and $dx(I \setminus(U\cap \tilde{U}))=0$, then $({\cal E},C_0^\infty(I))$ is closable on $L^2(I,\mu)$). Denote the closure by $({\cal E},D({\cal E}))$. \\
There are two possible cases that we summarize in the following theorem.
\mbox{$\B(E)$}\/gin{theo}\label{reflected}
$({\cal E},D({\cal E}))$ is recurrent if one of the following conditions holds:\\
(i) $U= V \cupdot (a,\infty)$ where $V$ is some open set (so either $V$ is empty if $a=0$, or $V$ is non-empty if $a>0$) and
$$
\int^\infty_{a+1}\frac{1}{\sigma \varphi}(s) ds = \infty.
$$
(ii) $U= V \cupdot \bigcup_{n\in {\mathbb N}}I_n$, $I_n=(x_n, x_{n+1})$, $x_n < x_{n+1}$, $n\in {\mathbb N}$, $V$ some open set, and
\mbox{$\B(E)$}\/gin{align*}
a_n&:= \int_{c_n}^{d_n} \frac{1}{\sigma \varphi}(s) ds \; \longrightarrow \infty
\mbox{${\cal E }$}nd{align*}
as $n \rightarrow \infty$, where for $n\in {\mathbb N}$
$$
c_n= \frac{x_{n+1}+x_n}{2}
$$
and
$$
d_n:= \mbox{$\B(E)$}\/gin{cases}\;x_{n+1}-\frac{1}{n}\qquad \;\qquad &\;\;x_{n+1}-c_n>1,\\
\; x_{n+1}-\frac{x_{n+1}-c_n}{n} &\;\;x_{n+1}-c_n\leq 1.
\mbox{${\cal E }$}nd{cases}
$$
\mbox{${\cal E }$}nd{theo}
\centerline{}
\mbox{$\B(E)$}\/gin{proof}
(i) Let
$$
u_n(x):= \mbox{$\B(E)$}\/gin{cases}\;1 \qquad \qquad \qquad \qquad &x\in [0,a+1],\\
\; 1-\frac{1}{a_n}\int_{a+1}^x \frac{1}{\sigma \varphi}(t)dt&x\in[a+1,a+1+n],\\
\;0&\quad elsewhere,
\mbox{${\cal E }$}nd{cases}
$$
where $a_n=\int^{a+1+n}_{a+1} \frac{1}{\sigma\varphi}(s) ds$. Then
$(u_n)_{n\in {\mathbb N}}\subset D({\cal E})$ with $0\leq u_n \leq 1$, $n\in {\mathbb N}$,
$u_n \nearrow 1 $ as $n \rightarrow \infty $ and $ \lim_{n\rightarrow \infty}{\cal E}(u_n , u_n)=0.$\\
(ii) Let
$$
u_n(x):= \mbox{$\B(E)$}\/gin{cases}\;1 \qquad \qquad \qquad \qquad &x\in [0,c_n],\\
\; 1-\frac{1}{a_n}\int_{c_n}^x \frac{1}{\sigma\varphi}(t) dt&x\in[c_n,d_n],\\
\;0&\quad elsewhere,
\mbox{${\cal E }$}nd{cases}
$$
where $a_n= \int_{c_n}^{d_n} \frac{1}{\sigma \varphi}(s) ds$. Then $(u_n)_{n\in {\mathbb N}}$ satisfies the desired properties and determines recurrence.
\mbox{${\cal E }$}nd{proof}
\centerline{}
\centerline{}
\mbox{$\B(E)$}\/gin{exam}\label{Bessel}
If $\varphi(x)=x^{\delta-1}$ with $\delta>0$ and $\sigma(x)\mbox{${\cal E }$}quiv 1$ on $I$, then clearly (\ref{Hamza2}) is satisfied. In this case the process associated to the regular Dirichlet form $({\cal E},D({\cal E}))$ (cf. \cite{FOT}) is the well-known Bessel process of dimension $\delta$. We are going to find a sufficient condition on the dimension $\delta$ for recurrence. Since
$$
\int_1^\infty \frac{1}{\varphi}(s) ds =\int_1^{\infty} s^{1-\delta}ds =\mbox{$\B(E)$}\/gin{cases}\;\lim_{n\rightarrow \infty}\log\,n \qquad \qquad &\delta=2,\\
\; \lim_{n\rightarrow \infty}\frac{1}{2-\delta}[n^{2-\delta}-1]&\delta \neq 2,\\
\mbox{${\cal E }$}nd{cases}
$$
we see by Theorem \ref{reflected} (i) with $a=0$ that the Bessel processes of dimension $\delta$ is recurrent if $\delta\in(0,2]$. Note that using \cite[Theorem 3]{Stu1} we obtain the same calculations up to a constant. However, in \cite{Stu1} the Dirichlet form is supposed to be irreducible throughout which we do not demand.
\mbox{${\cal E }$}nd{exam}
\mbox{$\B(E)$}\/gin{rem}\label{closedset}
Of course Theorem \ref{reflected} can be easily reformulated for Dirichlet forms defined on more general closed sets (cf. Remark \ref{openset}).
\mbox{${\cal E }$}nd{rem}
\centerline{}
\section{The multidimensional case}\label{4}
We have found sufficient conditions for recurrence of Dirichlet forms (${\cal E},D({\cal E}))$ with one-dimensional state space. Now we will extend our results to multi-dimensional Dirichlet forms of gradient type.\\
Let $d\geq 2$ and $\varphi \in L_{loc}^1({\mathbb R}^d,dx)$ such that $\varphi>0$ $dx$-a.e. Let further $a_{ij}$ be measurable functions such that $a_{ij}=a_{ji}$, $1\leq i,j \leq d$ and such that for each compact set $K\subset {\mathbb R}^d$, there exists $c_K >0$ such that
\mbox{$\B(E)$}\/gin{equation}\label{locallyelliptic}
\sum_{i,j=1}^d a_{ij}(x) \xi_i \xi_j \leq c_K \sum_{i=1}^d \xi_i ^2, \qquad \forall x\in K,\quad \forall\xi\in{\mathbb R}^d.
\mbox{${\cal E }$}nd{equation}
Assume that the bilinear form
$${\cal E}(f,g)=\sum_{i,j=1}^d\int_{{\mathbb R}^d} a_{ij}(x)\partial_i f(x) \partial_j g(x) \varphi(x) dx, \quad f,g \in C_0^\infty({\mathbb R}^d)$$
is well-defined, positive and closable on $L^2({\mathbb R}^d,\mu)$ where $d\mu=\varphi dx$. By (\ref{locallyelliptic}), we can define an increasing function $b(r)$ on $[0,\infty)$ by
$$
b(r):=c_{\overline{B_r(0)}}
$$
where $\overline{B_r(0)}:=\{x\in {\mathbb R}^d\,|\,|x|\leq r\}$, and $|\cdot |$ denotes the Euclidean norm. Let $S$ be the $(d-1)$-dimensional surface measure. Define
$$\psi(r):=\int_{\partial \overline{B_r(0)}} \varphi (x) S(dx),$$ and assume $\psi(r) \in (0,\infty)$ $dr$-a.e. Let $U\subset [0,\infty)$ be the largest open set such that
$$
\frac{1}{\psi}\in L^1_{loc}(U,dx).
$$
We suppose $dx([0,\infty) \setminus U)=0$.
\centerline{}
\mbox{$\B(E)$}\/gin{rem}
Note that $\psi(r)>0$ for almost every $r>0$ follows easily from \cite[Theorem, page 38]{maz} since $\varphi>0$ $dx$-a.e.
Hence our assumption $\psi(r) \in (0,\infty)$ for a.e. $r$ is satisfied whenever $\varphi\in L^1(\partial \overline{B_r(0)},dS)$ for a.e. $r$. The latter is for instance satisfied if $\varphi \in H^{1,1}_{loc}({\mathbb R}^d,dx)$.
\mbox{${\cal E }$}nd{rem}
\centerline{}
Again we will present sufficient conditions for the recurrence of $({\cal E},D({\cal E}))$, i.e. for the existence of $(u_n)_{n\in {\mathbb N}}\subset D({\cal E})$ with $0\leq u_n \leq 1$, $n\in {\mathbb N}$, $u_n \nearrow 1 $ as $n \rightarrow \infty $ and $ \lim_{n\rightarrow \infty}{\cal E}(u_n , u_n)=0.$
\centerline{}
\mbox{$\B(E)$}\/gin{theo}\label{multidim1}
$({\cal E},D({\cal E}))$ is recurrent if one of the following conditions holds: \\
(i) $U= V \cupdot (a,\infty)$ where $V$ is some open set (so either $V$ is empty if $a=0$, or $V$ is non-empty if $a>0$) and
$$a_n:=\int_{a+1}^{a+1+n}\frac{1}{\psi}(s) ds \longrightarrow \infty,\quad \frac{b(a+1+n)}{a_n} \longrightarrow 0$$
as $n \rightarrow \infty.$\\
(ii) $U= V \cupdot \bigcup_{n\in {\mathbb N}}I_n$, $I_n=(x_n, x_{n+1})$, $x_n < x_{n+1}$, $n\in {\mathbb N}$, $V$ some open set, and
$$a_n:= \int_{c_n}^{d_n} \frac{1}{\psi}(s) ds \; \longrightarrow \infty,\quad \frac{b(d_n)}{a_n}\longrightarrow 0$$
as $n \rightarrow \infty$, where for $n\in {\mathbb N}$
$$c_n= \frac{x_{n+1}+x_n}{2}$$
and
$$d_n:= \mbox{$\B(E)$}\/gin{cases}\;x_{n+1}-\frac{1}{n}\qquad \;\qquad &\;\;x_{n+1}-c_n>1,\\
\; x_{n+1}-\frac{x_{n+1}-c_n}{n} &\;\;x_{n+1}-c_n\leq 1.
\mbox{${\cal E }$}nd{cases}$$
\mbox{${\cal E }$}nd{theo}
\centerline{}
\mbox{$\B(E)$}\/gin{proof}
(i) Let
$$u_n(x):= \mbox{$\B(E)$}\/gin{cases}\;1 \qquad \qquad \qquad \qquad &|x|\in [0,a+1],\\
\; 1-\frac{1}{a_n}\int_{a+1}^{|x|} \frac{1}{\psi}(t)dt&|x|\in[a+1,a+1+n],\\
\;0&\quad elsewhere,
\mbox{${\cal E }$}nd{cases}$$
where $a_n=\int^{a+1+n}_{a+1} \frac{1}{\psi}(s) ds$. Then $(u_n)_{n\in {\mathbb N}}\subset D({\cal E})$ with $0\leq u_n \leq 1$, $n\in {\mathbb N}$, $u_n \nearrow 1 $ as $n \rightarrow \infty.$ Note that
\mbox{$\B(E)$}\/gin{align*}
{\cal E}(u_n,u_n)&=\sum_{i,j=1}^d\int_{{\mathbb R}^d} a_{ij}(x)\partial_i u_n(x) \partial_j u_n(x) \varphi(x) dx\\
&= \int_{\overline{B_{a+1+n}(0)} \setminus B_{a+1}(0)} \sum_{i,j=1}^d a_{ij}(x)x_i x_j \frac{1}{|x|^2}\frac{1}{a_n^2} \frac{1}{\psi(|x|)^2} \varphi(x)dx\\
&\leq \frac{b(a+1+n)}{a_n^2} \int_{a+1}^{a+1+n}\,\int_{\partial\overline{B_r(0)}} \varphi(x) S(dx)\,\frac{1}{\psi(r)^2}dr\\
&=\frac{b(a+1+n)}{a_n}.
\mbox{${\cal E }$}nd{align*}
where $B_r(0):=\{x\in {\mathbb R}^d\,|\,|x|<r\}$. Therefore $ \lim_{n\rightarrow \infty}{\cal E}(u_n , u_n)=0.$\\
(ii) Let
$$u_n(x):= \mbox{$\B(E)$}\/gin{cases}\;1 \qquad \qquad \qquad \qquad &|x|\in [0,c_n],\\
\; 1-\frac{1}{a_n}\int_{c_n}^{|x|} \frac{1}{\psi}(t) dt&|x|\in[c_n,d_n],\\
\;0&\quad elsewhere,
\mbox{${\cal E }$}nd{cases}$$
where $a_n= \int_{c_n}^{d_n} \frac{1}{\psi}(s) ds$. Then $(u_n)_{n\in {\mathbb N}}$ satisfies desired the properties and determines recurrence.
\mbox{${\cal E }$}nd{proof}
\centerline{}
Finally, let us consider a case which is possibly easier to calculate than the previous one. Let $\varphi \in L^1_{loc}({\mathbb R}^d,dx)$, $\varphi>0$ $dx$-a.e. and $a_{ij}$ be measurable functions such that $a_{ij}=a_{ji}$, $1\leq i,j \leq d$. We assume that the bilinear form
$${\cal E}(f,g)=\sum_{i,j=1}^d\int_{{\mathbb R}^d} a_{ij}(x)\partial_i f(x) \partial_j g(x) \mu(dx), \quad f,g \in C_0^\infty({\mathbb R}^d)$$
is well-defined, positive and closable on $L^2({\mathbb R}^d,\mu)$ where $\mu:=\varphi dx$. \\
\centerline{}
\mbox{$\B(E)$}\/gin{theo}\label{multidim2}
Suppose that there is some compact set $K\subset {\mathbb R}^d$ and a function $\phi$ with
$$
\|A(x)\|\varphi(x) \leq \phi(| x |) \qquad \forall x\in {\mathbb R}^d\setminus K
$$
where $\|A(x)\|= \big[ \sum_{i,j=1}^d a_{ij}(x)^2 \big]^{\frac{1}{2}}$ and $\phi \in L^1_{loc}({\mathbb R}^d \setminus K,dx)$. Let
$$
a_n:= \int_{\overline{B_n(0)}\setminus B_\rho(0)} \frac{|y|^{2-2d}}{\phi (|y|)} dy
$$
where $\rho>0$ is such that $K\subset B_\rho (0)$ and $n>\rho$. If $a_n$ is finite for any $n>\rho$ and converges to $\infty$ as $n\rightarrow \infty$, then $({\cal E},D({\cal E}))$ is recurrent.
\mbox{${\cal E }$}nd{theo}
\mbox{$\B(E)$}\/gin{proof}
Since $\frac{|y|^{2-2d}}{\phi (|y|)}$ is rotationally invariant, we can rewrite $a_n$ as
$$
a_n=d\cdot vol_d(B_1(0)) \int_\rho^n \frac{s^{2-2d}}{\phi(s)}s^{d-1}ds =d\cdot vol_d(B_1(0)) \int_\rho^n \frac{s^{1-d}}{\phi(s)}ds
$$
where $vol_d(B_1(0))$ is the volume of $B_1(0)$ in ${\mathbb R}^d$.
Define
$$
\psi_n(r):= \mbox{$\B(E)$}\/gin{cases}\;1 \qquad \qquad \qquad \qquad \qquad \qquad &r\in [0,\rho],\\
\; 1-\frac{1}{a_n}\int_{\overline{B_r(0)}\setminus B_\rho(0)} \frac{|y|^{2-2d}}{\phi (|y|)} dy &r \in [\rho,n], \\
\; 0 &elsewhere.
\mbox{${\cal E }$}nd{cases}
$$
Then, we can rewrite $\psi_n$ as
$$
\psi_n(r)= \mbox{$\B(E)$}\/gin{cases}\;1 \qquad \qquad \qquad \qquad \qquad \qquad &r\in [0,\rho],\\
\; 1-\frac{d\cdot vol_d(B_1(0))}{a_n}\int_\rho^r \frac{s^{1-d}}{\phi(s)}ds &r \in [\rho,n],\\
\; 0 &elsewhere.
\mbox{${\cal E }$}nd{cases}
$$
Set $u_n(x)=\psi_n(|x|)$, $n\in {\mathbb N}$ sufficiently large. Then $u_n$ is rotationally invariant, differentiable $dx$-a.e. and in $D({\cal E})$. Note that
\mbox{$\B(E)$}\/gin{align*}
{\cal E}(u_n,u_n)&=\sum_{i,j=1}^d\int_{{\mathbb R}^d} a_{ij}(x)\partial_i u_n(x) \partial_j u_n(x) \mu(dx)\\
&= \sum_{i,j=1}^d\int_{\overline{B_n(0)}\setminus B_\rho(0)} \frac{d^2\cdot vol_d(B_1(0))^2}{a_n^2}a_{ij}(x)x_i x_j|x|^{-2d} \frac{1}{\phi^2 (|x|)} \varphi (x) dx \\
&\leq \frac{d^2\cdot vol_d(B_1(0))^2 }{a_n^2} \int_{\overline{B_n(0)}\setminus B_\rho(0)} |x|^{2-2d} \frac{\|A(x)\|}{\phi^2 (|x|)} \varphi (x) dx \\
&\leq \frac{d^2\cdot vol_d(B_1(0))^2}{a_n^2}\int_{\overline{B_n(0)}\setminus B_\rho(0)} |x|^{2-2d} \frac{1}{\phi (|x|)} dx \\
&= \frac{d^2\cdot vol_d(B_1(0))^2}{a_n}.
\mbox{${\cal E }$}nd{align*}
Since $d\cdot vol_d(B_1(0))$ is independent on $n$, ${\cal E}(u_n,u_n) \longrightarrow 0$ as $n \rightarrow \infty$. Therefore $({\cal E},D({\cal E}))$ is recurrent.
\mbox{${\cal E }$}nd{proof}
\mbox{$\B(E)$}\/gin{rem}
Of course, Theorems \ref{multidim1} and \ref{multidim2} can be easily reformulated for more general open sets and in the reflected case (cf. Remarks \ref{openset} and \ref{closedset}, and Section \ref{3}).
\mbox{${\cal E }$}nd{rem}
\mbox{$\B(E)$}\/gin{thebibliography}{XXX}
\bibitem{FOT} Fukushima, M. , Oshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov processes, Walter de Gruyter 1994. x+392 pp.
\bibitem{FOT2} Fukushima, M. , Oshima, Y., Takeda, M.: Dirichlet forms and
Symmetric Markov processes. Berlin-New York: Walter de Gruyter 2011.
\bibitem{ge80} Getoor, R. K.: Transience and recurrence of Markov processes, Seminar on Probability, XIV, pp. 397--409, Lecture Notes in Math., 784, Springer, Berlin, 1980.
\bibitem{mr} Ma, Z.M., R\"ockner, M.: Introduction to the Theory of (Non-Symmetric)
Dirichlet Forms. Berlin: Springer 1992.
\bibitem{maz} Maz'ya, V.: Sobolev spaces with applications to elliptic partial differential equations. Second, revised and augmented edition. Grundlehren der Mathematischen Wissenschaften, Vol. 342. Springer, Heidelberg, 2011.
\bibitem{oukrutr} Ouknine, Y., Russo, F., Trutnau, G.: On countably skewed Brownian motion with accumulation point, arXiv, 2013.
\bibitem{o92r} Oshima, Y.: On conservativeness and recurrence criteria for Markov processes, Potential Anal. 1 (1992), no. 2, 115--131.
\bibitem{Stu1} Sturm. K. T.: Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and $L^p$-Liouville properties, J. Reine Angew. Math. 456 (1994), 173-196.
\bibitem{tr} Trutnau, G.: Skorokhod decomposition of reflected diffusions on bounded Lischitz domains with singular non reflection part, Probab. Theory Relat. Fields 127 (2003), No.4, pp. 455-495.
\mbox{${\cal E }$}nd{thebibliography}
\mbox{${\cal E }$}nd{document} | math | 35,312 |
\begin{document}
\begin{frontmatter}
\title{Bi-Discriminator GAN For Tabular Data Synthesis\footnote{{Supplementary materials and source codes are available at} \href{https://github.com/EsmaeilpourMohammad/BCT-GAN.git}{this github-Repo.}}}
\author{Mohammad Esmaeilpour$^{\mathsection \dagger}$\corref{}}
\ead{[email protected]}
\author{Nourhene Chaalia$^\dagger$}
\ead{[email protected]}
\author{Adel Abusitta$^\ddagger$}
\ead{[email protected]}
\author{François-Xavier Devailly$^\dagger$}
\ead{[email protected]}
\author{Wissem~Maazoun$^\dagger$}
\ead{[email protected]}
\author{Patrick Cardinal$^\mathsection$}
\ead{[email protected]}
\address{$^{\mathsection}$\'{E}cole de Technologie Sup\'{e}rieure (\'{E}TS), Universit\'{e} du Qu\'{e}bec and \\ IVADO Institution, Montr\'{e}al, Qu\'{e}bec, Canada\\
$^{\dagger}$The Fédération des Caisses Desjardins du Québec\\
$^{\ddagger}$\'{E}cole Polytechnique de Montr\'{e}al and McGill University\\
1100 Notre-Dame W, Montr\'{e}al, H3C 1K3, Qu\'{e}bec, Canada}
\begin{abstract}
This paper introduces a bi-discriminator GAN for synthesizing tabular datasets containing continuous, binary, and discrete columns. Our proposed approach employs an adapted preprocessing scheme and a novel conditional term using the $\chi^{2}_{\beta}$ distribution for the generator network to more effectively capture the input sample distributions. Additionally, we implement straightforward yet effective architectures for discriminator networks aiming at providing more discriminative gradient information to the generator. Our experimental results on four benchmarking public datasets corroborates the superior performance of our GAN both in terms of likelihood fitness metric and machine learning efficacy.
\end{abstract}
\begin{keyword}
Generative adversarial network (GAN) , tabular data synthesis, bi-discriminator GAN, conditional generator, variational Gaussian mixture model (VGM).
\end{keyword}
\end{frontmatter}
\section{Introduction}
\label{sec:intro}
Tabular data is among the most common modalities which has been widely used for maintaining massive databases of financial institutions, insurance corporations, networking companies, healthcare industries, etc. \cite{even2007economics,shwartz2021tabular,clements2020sequential,buczak2015survey,ulmer2020trust}. These databases include immense combination of personal, confidential, and general records for every customer, client, and patient in different formats (e.g., continuous and discrete data types). Semantic patterns derivable in such records efficiently contribute to extract meaningful information for the benefit of companies in various aspects such as large-scale decision-making \cite{xu2020synthesizing}, risk management \cite{aven2010risk}, long-term investment \cite{kornfeld1998automatically}, fraud or unusual activity detection \cite{cartella2021adversarial}, etc. However, exploiting these patterns is a challenging task since tabular datasets are heterogeneous \cite{sheth1990federated,wang1990polygen} and they contain sparse representations of discrete and continuous records with low correlation compared to homogeneous datasets (e.g., speech, environmental audio, image, etc.) \cite{borisov2021deep}. Unfortunately, extracting semantic relational patterns from heterogeneous datasets requires implementing costly data-driven algorithms \cite{loorak2016exploring,khan2020toward}.
During the last decade and especially after the proliferation of deep learning (DL) algorithms, various cutting-edge approaches have been introduced for processing tabular datasets in different frameworks \cite{socher2012deep,traquair2019deep,gorishniy2021revisiting}, particularly for synthesis purposes \cite{bourou2021review}. Presumably, this is due to two major applications. Firstly, complex DL algorithms configured in the generative adversarial network (GAN) \cite{goodfellow2014generative} synthesis platforms can be used for augmenting sparse datasets with low cardinality and poor sample quality \cite{shanmugam2020and}. Often, this data augmentation procedure effectively triggers the semantic pattern extraction operations. Secondly, GAN-based synthesis approaches yield models capable of generating new records similar (and non-identical) to the ground-truth samples available in the original databases. The synthesized records can be used for development purposes such as extracting relational patterns \cite{tsechansky1999mining} without publicly releasing the original dataset. This efficiently contributes to protect the privacy of people and clients whose their information is stored in the tabular datasets of companies. Our focus in this paper is on the latter application since it has been among the most demanding appeals of some large-scale financial institutions towards avoiding data leakage \cite{shabtai2012survey,alneyadi2016survey}. Briefly, we make the following contributions in this paper:
\begin{enumerate}[(i)]
\item developing a novel data preprocessing scheme and defining a new conditional term (vector) for the generator network configured in a GAN synthesis setup,
\item implementing a bi-discriminator GAN for providing more gradient information to the generator network in order to improve its performance in runtime,
\item designing straightforward architectures for generator and discriminator networks.
\end{enumerate}
The organization of this paper is as the following. Section~\ref{sec:background} provides a summary of synthesis approaches based on the state-of-the-art GANs for tabular datasets. In Section~\ref{sec:proposed}, we explain the details of our proposed synthesis approach and finally in Section~\ref{sec:experiment}, we report and analyze our conducted experiments on four benchmarking databases.
\section{Background: Tabular Data Synthesis}
\label{sec:background}
Over the past years, variational autoencoder (VAE\cite{kingma2013auto}) and GAN frameworks have been recognized as the state-of-the-art approaches for data fusion, particularly in the context of tabular data synthesis \cite{xu2020synthesizing}. Fundamentally, these two frameworks are similar to the baseline classical Bayesian network (CLBN) \cite{chow1968approximating} and its variants such as private Bayesian network (PrivBN) \cite{zhang2017privbayes}. Thus far, many modern forms of these generative models (e.g., \cite{ma2020vaem,park2018data}) have been introduced and practically implemented for real-life applications. However, they both suffer from some major technical limitations and instability side-effects \cite{genevay2017gan}. In terms of comparison, there are some debates on their relative performance over another \cite{mi2018probe}, nevertheless we do not address them herein since they are out of the scope of the current work.
In this section, we briefly review the background of tabular data synthesis with a focus on GANs since we are mostly interested in developing generative models capable of synthesizing new records (a collection of fields) directly from random latent distributions \cite{goodfellow2014generative}. This is one of the pivotal characteristics of GANs which normally provides a wider synthesis domain and yields a more comprehensive generative model \cite{feizi2017understanding}. In general, a standard GAN configuration (i.e., vanilla GAN) employs two DL architectures organized in a mini-max setup as the following \cite{goodfellow2016deep}.
\begin{equation}
\min_{G} \max_{D} \mathbb{E}_{\mathbf{x}\sim p_{r}}\left [ \log D(\mathbf{x}) \right ]+
\mathbb{E}_{\mathbf{z}\sim p_{z}}\left [ \log \left ( 1-D(G(\mathbf{z})) \right ) \right ]
\label{eq:gan}
\end{equation}
\noindent where the aforementioned architectures are denoted by $G(\cdot)$ and $D(\cdot)$ which represent the generator and discriminator networks, respectively. For $\mathbf{z}_{i} \in \mathbb{R}^{d_{z}}$ with dimension $d_{z}$ drawn from the multivariate random distribution $p_{z}$, the generator network synthesizes a record similar to the ground-truth sample indicated by $\mathbf{x}$. The control of this synthesis process is on the discriminator which samples from the real distribution $p_{r}$. Specifically, this process binds $D(\cdot)$ to measure and tune the convergence of the generator's distribution ($p_{g}$) to $p_{r}$ in an iterative pipeline.
There are several outstanding variants for Eq.~\ref{eq:gan} employing different DL architectures, optimization properties, and loss functions. For instance, MedGAN \cite{choi2017generating} is among the premier generative models developed for synthesizing tabular datasets particularly for multi-label discrete records. This model exploits an autoencoder on top of the generator network to learn salient features of the continuous medical records. Recently, an extension for MedGAN has been released which imposes a regularization scheme on the generator using an adversarially training-based autoencoding policy \cite{camino2018generating}. Regarding the conducted experiments and reported results, although this scheme improves the performance of the generator for multi-categorical continuous records, it increases the chance of training instability and mode collapse.
A simpler yet effective introduced approach for synthesizing highly heterogeneous tabular datasets is Table-GAN \cite{park2018data} which is inspired from the conventional deep convolutional GAN configuration \cite{radford2015unsupervised}. Unlike MedGAN, it can be adapted for any tabular datasets with wide ranges of multi-categorical records. This generalizability mainly originates from the distinctive loss function of Table-GAN as follows.
\begin{equation}
L_{G}(\cdot) = \mathbb{E} \Big[ \left | \ell_{app}(G(\mathbf{z}_{i}))-C(\ell_{rm} (G(\mathbf{z}_{i}))) \right | \Big]_{z_{i}\sim p_{z}(z)}
\label{eq:tableGAN}
\end{equation}
\noindent where $\ell_{app}(\cdot)$ and $\ell_{rm}(\cdot)$ denote the analytical appending and removing functions for the label attributes (fields) of each column, respectively. Additionally, $C(\cdot)$ retrieves the output logit (predicted column label) of the generator network. In a nutshell, $L_{G}(\cdot)$ synchronizes the output logits of the generator to smoothly match the order of the original records and consequently improves the accuracy of the entire model. However, it might be computationally prohibitive for tabular datasets with numerous columns (features).
Another straightforward approach for generating complex combinations of continuous and discrete records without employing such a costly loss function in Eq.~\ref{eq:tableGAN} is introduced by Mottini {\it et al.}~\cite{mottini2018airline}. They propose a noble similarity measure using the Cram\'{e}r integral probablity metric (IPM \cite{mroueh2017sobolev}) \cite{bellemare2017cramer} in order to minimize the distribution discrepancies among original and synthesized samples. Following this technique, they incorporate a decomposition policy for sampling $\mathbf{z}_{i}$ as:
\begin{equation}
\delta (\mathbf{z}_{i}) \doteq \mathrm{KL}_{div} \Big[ p_{r} \left (\rho_{1} |\mathbf{z}_{i} \right)\parallel \rho_{2} \Big], \quad \rho_{1},\rho_{2} \sim \mathcal{B}\Big(\frac{1}{2}\Big)
\end{equation}
\noindent where $\mathrm{KL}_{div}$ refers to the statistical Kullback-Leibler divergence, $\mathcal{B}$ indicates the Bernoulli's distribution, and $ \delta (\mathbf{z}_{i})$ is the normalized sampled vector for further reducing potential local discrepancy among the synthesized records. Mottini's GAN (MT-GAN) has been successfully evaluated on public tabular databases and it has been relatively outperformed other GANs with standard $\varphi$-divergence IPMs such as vanilla GAN~\cite{goodfellow2014generative}.
ITS-GAN which stands for the incomplete table synthesis GAN \cite{chen2019faketables} is another reliable cutting-edge variant for Eq.~\ref{eq:gan}. One of the major novelties in this approach is the possibility of making a trade-off between accuracy and generalizability for the generator network. Technically, ITS-GAN exploits a massive combination of functional dependencies (FD: similar to the critic functions \cite{muller1997integral} in the sequence-to-sequence GAN platforms \cite{mroueh2017fisher}) for the continuous records using an independent pretrained autoencoder. However, this setting is primarily designed for small-scale datasets with relatively high correlation such as US Census \cite{kohavi1996scaling}.
The class-conditional tabular GAN (CT-GAN) \cite{xu2019modeling} is a more sophisticated approach compared to ITS-GAN. For improving computational complexity it does not employ any costly FD modeling procedure. However, it implements a conditional generator (i.e., $G(\cdot)$ receives partial information of the columns in addition to $\mathbf{z}_{i}$) with mask vectors sampled form the original records. According to the published experimental results, CT-GAN confidently outperforms other advanced generative models for the public tabular datasets. In the following section, we extend this generative model to a bi-discriminator configuration with a novel definition for the conditional term and employing fully convolutional architectures. Finally in Section~\ref{sec:experiment}, we experimentally prove the superior performance of our proposed synthesis GAN on four popular benchmarking datasets.
\section{Proposed Approach: Bi-Discriminator class-conditional tabular GAN (BCT-GAN)}
\label{sec:proposed}
Our proposed tabular synthesis approach is based on the CT-GAN \cite{xu2019modeling}, however with three major improvements in normalizing categorical records (preprocessing), defining the conditional term for the generator, and designing the architecture of the generator and two discriminator networks. The motivation behind employing double discriminators in our synthesis setup is the possibility of gaining more gradient information for the benefit of the generator during training. However, we do not provide the gradient analysis in this paper and we only demonstrate the performance of our GAN setup by measuring the likelihood fitness and machine learning efficacy metrics \cite{xu2019modeling}. The general overview of our proposed BCT-GAN is shown in Fig.~\ref{fig:overview-ECT}.
\subsection{Data Preprocessing: Continuous Column Normalization}
\label{sec:preprocess}
\begin{figure*}
\caption{Overview of the proposed BCT-GAN. Herein, the random input vector is denoted by the multivariate $\mathbf{z}
\label{fig:overview-ECT}
\end{figure*}
Tabular databases contain continuous, binary (boolean), and discrete columns with variable lengths. Therefore, they should be correctly represented and meaningfully distributed before using them for training purposes. Representing boolean and discrete fields are straightforward since they can be transformed into a block of one-hot floating-point vectors adjusted in the range of $\left [ -1,1 \right ]$ using the conventional $\tanh$ function \cite{choi2017generating}. Conversely, continuous columns are often non-Gaussian and resemble a multimodal distribution. This obliges to implement a fundamental preprocessing operation for transforming each column of the dataset according to a ordinal policy. The policy which we use herein is based on the variational Gaussian mixture model (VGM) \cite{bishop2007pattern} since its superior performance has been demonstrated for a variety of comprehensive tabular datasets \cite{xu2019modeling}. Inspired from CT-GAN, we also fit a VGM with $\varrho$ modes on every continuous column $C_{i}$ of the given dataset but we impose the following straightforward probability fairness condition \cite{hogg2005introduction}:
\begin{equation}
p_{C_{i}}(c_{i,j})=\sum_{\tau=1}^{\varrho}\hat{\mu}_{\tau} \bigg( \mathcal{N}\Big(c_{i,j}\mid \mu_{\tau},\sigma_{\tau}^{2}\Big) \bigg), \quad \hat{\mu}_{\tau} \sim \mathcal{N}(0,sI)
\label{eq:probuniformity}
\end{equation}
\noindent where $c_{i,j}$ denotes the $j$-th field of $C_{i}$. For supporting the smoothness in computing the probability density function of every mode (i.e., every single Gaussian model of the VGM) in the mixture setup (see a relevant discussion in \cite{nualart2006malliavin}), we empirically set $s \rightarrow 0.5$. Statistically, Eq.~\ref{eq:probuniformity} centralizes all the modes, around the most dominant probability distribution with measurable displacements and avoids discarding others with short skewness \cite{friedman2017elements}. Herein, $\mu_{\tau}$ and $\sigma_{\tau}$ indicate the mean and standard deviation parameters of every mode in the actual VGM model. Our proposed condition in Eq.~\ref{eq:probuniformity} acts fairly since the term $\hat{\mu}_{\tau}$ is employed to resist against marginalizing Gaussian models with potentially trivial $\mu_{\tau}$ and $\sigma_{\tau}$ parameters (in terms of resembling low entropy \cite{hogg2005introduction,frenken2007entropy}). The accuracy of this action is highly dependent to the domain of $s$ and $\varrho$. For instance, setting $s \rightarrow 0$ might negatively affect the entire VGM with the possibility of eliminating non-trivial modes \cite{dineen2005non}.
To the best of our knowledge, there is no analytical approach for finding the optimal value for $\varrho$ and it should be approximated according to the properties of the training dataset. Finally, we transform every record ($\mathbf{r}_{i}$) of the dataset into a one-hot vector of probabilities coming from the achieved VGM, linearly appended with the vector of normalized discrete values as follows \cite{xu2019modeling}:
\begin{equation}
\mathbf{r}_{i} \equiv \left \langle \vec{\Omega}_{\mathrm{cont}_{i}}\oplus \vec{\Omega}_{\mathrm{disc}_{i}} \right \rangle, \quad \forall i \in \mathbb{N}^{\left | \mathrm{cont} \right |}\cdot \mathbb{N}^{\left | \mathrm{disc} \right |}
\label{eq:records}
\end{equation}
\noindent where $\vec{\Omega}_{\mathrm{cont}_{i}}$ and $\vec{\Omega}_{\mathrm{disc}_{i}}$ denote the vectors of preprocessed continuous and discrete columns, respectively. Moreover, $\oplus$ is a mathematical symbol for appending operation (element-wise) \cite{cohn1981universal}. This operation provides a solid representation for successfully training a GAN \cite{xu2019modeling}. The major challenge of this preprocessing transformation is marginalizing the less frequent discrete columns while $\vec{\Omega}_{\mathrm{disc}_{i}}$ are in minority compared to $\vec{\Omega}_{\mathrm{cont}_{i}}$ \cite{mclachlan1988mixture}. For resolving such an issue, we can either weight the discrete components of $\mathbf{r}_{i}$ using the Cauchy operation (probability distribution function) or redistributing $p_{C_{i}}(\cdot)$ with smaller $\varrho$ in the VGM model (Eq.~\ref{eq:probuniformity}) \cite{ming2003background}.
\subsection{Defining the Conditional term for the Generator}
The motivation behind employing the class-conditional platform over regular GANs is twofold. Firstly, it has been shown that the conditional platforms demonstrates a relatively higher performance over the conventional configurations (e.g., vanilla GAN) \cite{brock2018large}. Secondly, tabular datasets are still heterogeneous even after running the aforementioned preprocessing and one-hot vectorization procedures (Section~\ref{sec:preprocess}). However, the performance of a conditional GAN is highly dependent to correctly defining the conditional term. We propose to bind the generator network according to a non-Gaussian probability distribution with asymmetric density function and wider variance. This operation is designed to expand the learning domain of the generator without biasing it toward any columns $C_{i}$ \cite{westfall2013understanding}. Hence, we empirically opt for the $\chi^{2}_{\beta}$ distribution with $\beta$ degree of freedom (this is a hyperparameter) and define the following binary mask function \cite{gupta2020fundamentals}:
\begin{equation}
\varpi_{i}(\alpha_{i}) = \left\{\begin{matrix}
1 & \mathrm{if}\quad \alpha_{i} <\frac{1}{2} \\
0 & \mathrm{otherwise.}
\end{matrix}\right. \quad \mathrm{and} \quad \alpha_{i} \in p_{\alpha} \sim \chi^{2}_{\beta} \quad
\label{eq:maskmask}
\end{equation}
\noindent where $\alpha_{i}$ is a conditioning parameter randomly drawn from a predefined probability distribution $p_{\alpha}$ and $\varpi_{i}(\alpha)$ refers to \textit{the moment-generating function} of $\chi^{2}_{\beta}$ \cite{casella1990statistical}. Given the fact that, this function is not symmetric, imposing the condition of $\alpha_{i} <0.5$ in Eq.~\ref{eq:maskmask} smoothly increases the chance of yielding sparse vector for $\varpi_{i}(\alpha_{i})$. Not only this contributes to improving the computational complexity of the entire GAN model during training, but also forces the generator to be slightly more independent to the conditional vector (the prior information). Finally, we define our novel conditional term for the generator network as follows.
\begin{equation}
\oplus_{i=1}^{n_{t}} := \varpi_{1}(\alpha_{1}) \oplus \varpi_{2}(\alpha_{2}) \oplus \cdots \oplus \varpi_{n_{t}}(\alpha_{n_{t}})
\label{eq:compoundMask}
\end{equation}
\noindent where $n_{t}$ refers to the total number of discrete fields in every record (see Eq.~\ref{eq:records}). In fact, this term is a compound mask function \cite{xu2019modeling} for $\vec{\Omega}_{\mathrm{disc}_{i}}$ which binds $\mathbf{r}_{i}$ to discrete vectors. In other words, Eq.~\ref{eq:compoundMask} selects which discrete fields should be shown to the generator network during training aiming at avoiding memorizing the order of original records. The motivation behind this binding is the simplicity of conducting numerical operations on these vectors compared to the Gaussian models of the VGM. Moreover, similar to the CT-GAN, it enables training-by-sampling procedure \cite{xu2019modeling}. In the following subsection, we provide more details about architecture of the generator.
\subsection{Designed Architecture for the BCT-GAN}
As shown in Fig.~\ref{fig:overview-ECT}, our proposed BCT-GAN employs one generator and two independent discriminator networks. In one hand, such configuration relatively improves stability of the entire model since ideally, multiple discriminators should provide more gradients to the generator and, to some extent, avoid extreme instabilities, especially at early iterations \cite{hardy2019md}. On the other hand, exploiting multiple discriminators dramatically increases the total number of training parameters. Thus, we opt for two discriminators in order to circumvent such a potential side-effect and obtain a reasonable balance.
The designed architecture for the generator is a fully convolutional deep neural network (DNN) with five hidden layers. The input layer deploys $\mathbf{z}_{i} \in \mathbb{R}^{\left | \mathbf{r}_{i} \right |}$ where $\left | \mathbf{r}_{i} \right |$ denotes the total length of the discrete and continuous fields in every record. The first hidden layer implements $256$ filters padded with $5\times 5$ receptive field and $8 \times 32$ channels followed by batch normalization and ReLU activation function. Subsequent layers exploit $512$ filters plus $16\times 64$ channels, skip-$z$ \cite{brock2018large}, weight normalization \cite{salimans2016weight} and $\tanh$ activation function. Finally, the output layer exploits a fully connected layer with transposed convolution \cite{mao2018effectiveness} and gumbel softmax (with ratio $0.2$) \cite{jang2016categorical}. Furthermore, we implement the chordal distance minimization operation \cite{esmaeilpour2020class} to partially avoid potential extreme instabilities.
For simplicity and avoiding unnecessary complication during training, we implement an identical architecture for both discriminator networks. This unique architecture is also fully convolutional and requires a vector with dimension $\left | \mathbf{r}_{i} \right |$ in the input layer. There are three hidden layers with $256$, $512$, and $1024$ filters followed by skip-$z$, weight normalization, and leaky ReLU activation function. All these layers are distributed over $16\times 64$ channels without dropout. The output layer is fully connected which yields a logit vector for updating the generator weights according to the training policy of the least-square GAN \cite{hong2019generative}.
\begin{table*}[h]
\centering
\scriptsize
\caption{Comparison of the GANs on four tabular datasets. Herein, $\mathcal{L}_{val}$ and $\mathcal{L}_{test}$ measure the correlation between each synthesized dataset and its associated ground-truth oracle \cite{xu2019modeling} during the 5-fold cross validation and test phases, respectively.}
\rotatebox{90}{
\begin{tabular}{ccccccccccccc}
\hline
\multicolumn{1}{c||}{\multirow{2}{*}{Method}} & \multicolumn{3}{c||}{Adult} & \multicolumn{3}{c||}{Census} & \multicolumn{3}{c||}{Credit} & \multicolumn{3}{c}{News} \\ \cline{2-13}
\multicolumn{1}{c||}{} & \multicolumn{1}{c|}{$\mathcal{L}_{val}$} & \multicolumn{1}{c|}{$\mathcal{L}_{test}$} & \multicolumn{1}{c||}{$F_{1}$} & \multicolumn{1}{c|}{$\mathcal{L}_{val}$} & \multicolumn{1}{c|}{$\mathcal{L}_{test}$} & \multicolumn{1}{c||}{$F_{1}$} & \multicolumn{1}{c|}{$\mathcal{L}_{val}$} & \multicolumn{1}{c|}{$\mathcal{L}_{test}$} & \multicolumn{1}{c||}{$F_{1}$} & \multicolumn{1}{c|}{$\mathcal{L}_{val}$} & \multicolumn{1}{c|}{$\mathcal{L}_{test}$} & \multicolumn{1}{c}{$R^{2}$} \\ \hline \hline
\multicolumn{1}{l||}{Ground-truth} & \multicolumn{1}{c|}{$-$} & \multicolumn{1}{c|}{$-$} & \multicolumn{1}{c||}{$0.667$} & \multicolumn{1}{c|}{$-$} & \multicolumn{1}{c|}{$-$} & \multicolumn{1}{c||}{$0.486$} & \multicolumn{1}{c|}{$-$} & \multicolumn{1}{c|}{$-$} & \multicolumn{1}{c||}{$0.741$} & \multicolumn{1}{c|}{$-$} & \multicolumn{1}{c|}{$-$} & \multicolumn{1}{r}{$0.156$} \\ \hline \hline
\multicolumn{1}{l||}{CLBN \cite{chow1968approximating}} & \multicolumn{1}{c|}{$-2.763$} & \multicolumn{1}{c|}{$-3.184$} & \multicolumn{1}{c||}{$0.341$} & \multicolumn{1}{c|}{$-5.172$} & \multicolumn{1}{r|}{$-6.970$} & \multicolumn{1}{c||}{$0.308$} & \multicolumn{1}{c|}{$-7.502$} & \multicolumn{1}{c|}{$-8.106$} & \multicolumn{1}{c||}{$0.411$} & \multicolumn{1}{c|}{$-5.129$} & \multicolumn{1}{c|}{$-8.081$} & \multicolumn{1}{c}{$-6.607$} \\ \hline
\multicolumn{1}{l||}{PrivBN \cite{zhang2017privbayes}} & \multicolumn{1}{c|}{$-2.631$} & \multicolumn{1}{c|}{$-4.152$} & \multicolumn{1}{c||}{$0.416$} & \multicolumn{1}{c|}{$-4.525$} & \multicolumn{1}{c|}{$-5.149$} & \multicolumn{1}{c||}{$0.109$} & \multicolumn{1}{c|}{$-6.317$} & \multicolumn{1}{c|}{$-6.942$} & \multicolumn{1}{c||}{$0.189$} & \multicolumn{1}{c|}{$-4.306$} & \multicolumn{1}{c|}{$-6.449$} & \multicolumn{1}{r}{$-4.710$} \\ \hline
\multicolumn{1}{l||}{MedGAN \cite{choi2017generating}} & \multicolumn{1}{c|}{$-2.985$} & \multicolumn{1}{c|}{$-3.126$} & \multicolumn{1}{c||}{$0.351$} & \multicolumn{1}{c|}{$-4.711$} & \multicolumn{1}{c|}{$-5.542$} & \multicolumn{1}{c||}{$0.011$} & \multicolumn{1}{c|}{$-5.820$} & \multicolumn{1}{c|}{$-6.171$} & \multicolumn{1}{c||}{$0.027$} & \multicolumn{1}{c|}{$-4.451$} & \multicolumn{1}{c|}{$-5.993$} & \multicolumn{1}{r}{$-8.632$} \\ \hline
\multicolumn{1}{l||}{Table-GAN \cite{park2018data}} & \multicolumn{1}{c|}{$-3.899$} & \multicolumn{1}{c|}{$-5.270$} & \multicolumn{1}{c||}{$0.511$} & \multicolumn{1}{c|}{$-4.753$} & \multicolumn{1}{c|}{$-5.381$} & \multicolumn{1}{c||}{$0.154$} & \multicolumn{1}{c|}{$-5.776$} & \multicolumn{1}{c|}{$-5.992$} & \multicolumn{1}{c||}{$0.036$} & \multicolumn{1}{c|}{$-3.525$} & \multicolumn{1}{c|}{$-4.766$} & \multicolumn{1}{c}{$-3.663$} \\ \hline
\multicolumn{1}{l||}{MT-GAN \cite{mottini2018airline}} & \multicolumn{1}{c|}{$-3.704$} & \multicolumn{1}{c|}{$-4.086$} & \multicolumn{1}{c||}{$0.432$} & \multicolumn{1}{c|}{$-4.661$} & \multicolumn{1}{c|}{$-4.748$} & \multicolumn{1}{c||}{$0.163$} & \multicolumn{1}{c|}{$-5.106$} & \multicolumn{1}{c|}{$-6.355$} & \multicolumn{1}{c||}{$0.022$} & \multicolumn{1}{c|}{$-3.235$} & \multicolumn{1}{c|}{$ -5.633$} & \multicolumn{1}{r}{$-4.239$} \\ \hline
\multicolumn{1}{l||}{ITS-GAN \cite{chen2019faketables}} & \multicolumn{1}{c|}{$-3.393$} & \multicolumn{1}{c|}{$-4.001$} & \multicolumn{1}{c||}{$0.446$} & \multicolumn{1}{c|}{$-4.670$} & \multicolumn{1}{c|}{$-5.118$} & \multicolumn{1}{c||}{$0.172$} & \multicolumn{1}{c|}{$-5.251$} & \multicolumn{1}{c|}{$-5.995$} & \multicolumn{1}{c||}{$0.012$} & \multicolumn{1}{c|}{$-3.289$} & \multicolumn{1}{c|}{$-4.488$} & \multicolumn{1}{r}{$-5.415$} \\ \hline
\multicolumn{1}{l||}{CT-GAN \cite{xu2019modeling}} & \multicolumn{1}{c|}{$-2.442$} & \multicolumn{1}{c|}{$-2.968$} & \multicolumn{1}{c||}{$0.592$} & \multicolumn{1}{c|}{$-4.119$} & \multicolumn{1}{c|}{$-4.615$} & \multicolumn{1}{c||}{$0.387$} & \multicolumn{1}{c|}{$\mathbf{-4.248}$} & \multicolumn{1}{c|}{$\mathbf{-4.703}$} & \multicolumn{1}{c||}{$\mathbf{0.642}$} & \multicolumn{1}{c|}{$-3.614$} & \multicolumn{1}{c|}{$-4.210$} & \multicolumn{1}{r}{$0.018$} \\ \hline
\multicolumn{1}{l||}{BCT-GAN (ours)} & \multicolumn{1}{c|}{$\mathbf{-2.197}$} & \multicolumn{1}{c|}{$\mathbf{-2.622}$} & \multicolumn{1}{c||}{$\mathbf{0.618}$} & \multicolumn{1}{c|}{$\mathbf{-3.722}$} & \multicolumn{1}{c|}{$\mathbf{-3.805}$} & \multicolumn{1}{c||}{$\mathbf{0.419}$} & \multicolumn{1}{c|}{$-4.379$} & \multicolumn{1}{c|}{$-4.897$} & \multicolumn{1}{c||}{$0.619$} & \multicolumn{1}{c|}{$\mathbf{-3.142}$} & \multicolumn{1}{c|}{$\mathbf{-3.589}$} & \multicolumn{1}{r}{$\mathbf{0.131}$} \\ \hline
\multicolumn{13}{c}{Statistically, $R^{2}$ might result in negative values for underfitted regression models or those not capable of achieving accurate decision boundaries \cite{hogg2005introduction}.}
\label{table:results}
\end{tabular}
}
\end{table*}
\section{Experiments}
\label{sec:experiment}
This section provides the details of our conducted experiments on four public datasets which have been benchmarked for tabular data processing, particularly for synthesis purposes \cite{xu2019modeling}. Adult, Census, and News are among the selected datasets from the UCI online repository \cite{dua2017uci} and they contain extensive combinations of continuous, binary, and complex discrete records. Following the baseline approaches \cite{xu2019modeling}, we have also carried out some experiments on the popular Credit dataset taken from the \href{https://www.kaggle.com/mlg-ulb/creditcardfraud}{Kaggle machine learning archive.}
These four datasets not only are independent and different in terms of statistical distribution and latent properties, but they also have various number of continuous, boolean, and discrete columns in several formats. For instance, the average number of continuous columns in the Credit dataset is 29 while this number for Adult, Census, and News is 6, 7, and 45, respectively. Moreover, the dataset with the highest number of discrete columns is Census (with 31 features) while the News dataset does not contain any discrete column. Another difference among these datasets is their benchmarking capacity. Specifically, three datasets as of Adult, Census, and Credit are benchmarked for classification tasks while News fits in the regression category. This diversity helps to effectively evaluate the performance of the generative models in different scales and capacities.
For training the GANs, we firstly transform and normalize all the records of the aforementioned datasets into the designated format as mentioned in Eq.~\ref{eq:records}. Toward this end, we empirically set $\varrho \rightarrow 10$ during fitting the VGM on the preprocessed datasets and fill up the potential empty fields with the null value. For fairness in comparison we constantly use Adam optimizer \cite{kingma2014adam} with $\left [ 0, 0.9 \right ]$ parameters and $1.8\cdot 10^{-3}$ learning rate. Similar to \cite{brock2018large}, we also implement an exploratory operation for tuning the optimal number of steps for training the generators over discriminators. Since our proposed synthesis approach employs two discriminators, we opt for two and three steps over $D_{1}(\cdot)$ and $D_{2}(\cdot)$, respectively. For other generative models we exploit two steps with the constant decay rate of $0.99$ on seven NVIDIA GTX-1080-Ti GPU with $9\times 11$ memory. Moreover, we use the static batch size of 500 with orthogonal regularization \cite{SaxeMG13} and spectral normalization \cite{miyato2018spectral} for both the generator and discriminator networks. We stop the training procedure as early as observing signs of instability or mode collapse \cite{brock2018large} (in average around 400 epochs). Eventually, the generator should craft a dataset with dimensions identical to the associated ground-truth oracle. For evaluation purposes, we organize the generated samples into two subsets of validation and test with ratio of 0.7 and 0.3, respectively.
For evaluating the performance of the generative models we measure two metrics. Firstly, we compute the likelihood fitness metric ($\mathcal{L}$) \cite{xu2019modeling} which measures the relative correlation ($\mathrm{RC}$) of the synthesized and ground-truth datasets as the following \cite{xu2019modeling,eghbal2017likelihood}.
\begin{equation}
\mathcal{L} \propto \left \lfloor \mathrm{RC} \Big(p_{g}(\mathbf{z}_{i}; \theta_{g}),p_{r}( \mathbf{r}_{i})\Big) \right \rfloor, \quad \forall i \in \left \{ 1,2,\cdots, \left | \mathbf{r}_{i} \right | \right \}
\end{equation}
\noindent where $p_{g}$ denotes the generator's distribution. We compute this metric separately for the validation and test subsets denoted respectively by $\mathcal{L}_{val}$ and $\mathcal{L}_{test}$. Results of this experiment on four benchmarking datasets are summarized in Table~\ref{table:results}.
The second evaluation metric is about the benchmarking capacity which is also known as the machine learning efficacy \cite{xu2019modeling}. In other words, we compare the classification or regression performance of some front-end algorithms on the ground-truth and the synthesized datasets. Technically, the performance of such algorithms on a synthesized dataset should be very close to the associated ground-truth oracle. Thus for such a comparison, we employ $F_{1}$ and $R^{2}$ scores for the classification and regression tasks, respectively \cite{xu2019modeling}. Regarding the choice of the front-end algorithms, we follow the pipeline suggested by Xu {\it et al.}~\cite{xu2019modeling}. Specifically, we implement Adaboost (with 50 estimators), decision tree (with depth 30), DNN (with 40 hidden layers) for Adult, Census, and Credit datasets. Additionally, we fit linear regression and DNN (with 100 hidden layers) for the News dataset. Average $F_{1}$ and $R^{2}$ scores over these front-end algorithms are shown in Table~\ref{table:results}. As shown in this table, for three datasets, our proposed BCT-GAN outperforms other generative models as it achieves higher $\mathcal{L}_{val}$, $\mathcal{L}_{test}$, $F_{1}$, and $R^{2}$ values. However, for the Credit dataset, our approach competitively loses against CT-GAN. We conjecture that this is due to the dependency of our BCT-GAN to discrete columns for yielding $\mathbf{r}_{i}$s (see Eq.~\ref{eq:records}). Since the Credit dataset is entirely continuous and binary, hence this negatively affects the performance of our proposed GAN. We are determined to resolve this issue via separating discrete and continuous embeddings of Eq.~\ref{eq:records} in our future works.
\section{Conclusion}
In this paper, we introduced a bi-discriminator GAN for tabular data synthesis. The major novelties of our approach is firstly the development of a preprocessing scheme and secondly defining a solid conditional term for the generator network to improve the entire performance of the generative model. This term is a vector based on a masked function using $\chi^{2}_{\beta}$ probability density function for more effectively constraining over the generator and consequently better capturing the input sample distributions. We experimentally demonstrated that, for the majority of the cases, our proposed BCT-GAN outperforms the state-of-the-art approaches both in terms of likelihood fitness metric and machine learning efficacy.
\section*{Acknowledgment}
This work was funded by Fédération des Caisses Desjardins du Québec and Mitacs accelerate program with agreement number IT25105.
\end{document} | math | 37,984 |
\begin{document}
\title{Modeling double slit interference via anomalous diffusion: independently
variable slit widths}
\author{Johannes \surname{Mesa Pascasio}\textsuperscript{1,2}}
{\rm e}mail[E-mail: ]{[email protected]}
\homepage[Visit: ]{http://www.nonlinearstudies.at/}
\author{Siegfried \surname{Fussy}\textsuperscript{1}}
{\rm e}mail[E-mail: ]{[email protected]}
\homepage[Visit: ]{http://www.nonlinearstudies.at/}
\author{Herbert \surname{Schwabl}\textsuperscript{1}}
{\rm e}mail[E-mail: ]{[email protected]}
\homepage[Visit: ]{http://www.nonlinearstudies.at/}
\author{Gerhard \surname{Grössing}\textsuperscript{1}}
{\rm e}mail[E-mail: ]{[email protected]}
\homepage[Visit: ]{http://www.nonlinearstudies.at/}
\affiliation{\textsuperscript{1}Austrian Institute for Nonlinear Studies, Akademiehof\\
Friedrichstr.~10, 1010 Vienna, Austria}
\affiliation{\textsuperscript{2}Institute for Atomic and Subatomic Physics, Vienna
University of Technology\\
Operng.~9, 1040 Vienna, Austria \vspace*{4cm}
}
\begin{abstract}
Based on a re-formulation of the classical explanation of quantum
mechanical Gaussian dispersion (Grössing~\textit{et~al}. 2010 \cite{Groessing.2010emergence})
as well as interference of two Gaussians (Grössing~\textit{et~al}.
2012 \cite{Groessing.2012doubleslit}), we present a new and more
practical way of their simulation. The quantum mechanical ``decay
of the wave packet'' can be described by anomalous sub-quantum diffusion
with a specific diffusivity varying in time due to a particle's changing
thermal environment. In a simulation of the double-slit experiment
with different slit widths, the phase with this new approach can be
implemented as a local quantity. We describe the conditions of the
diffusivity and, by connecting to wave mechanics, we compute the exact
quantum mechanical intensity distributions, as well as the corresponding
trajectory distributions according to the velocity field of two Gaussian
wave packets emerging from a double-slit. We also calculate probability
density current distributions, including situations where phase shifters
affect a single slit's current, and provide computer simulations thereof.
\begin{lyxgreyedout}
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\global\long\,\mathrm{d}ef\partial{\partialartial}
{\rm e}nd{lyxgreyedout}
{\rm e}nd{abstract}
\keywords{quantum mechanics, ballistic diffusion, Gaussian dispersion, zero-point
field, finite differences}
\maketitle
\section{Introduction}
In reference \cite{Groessing.2010emergence} we presented a classical
model for the explanation of quantum mechanical dispersion of a free
Gaussian wave packet. In accordance with the classical model, we shall
now relate it more directly to a ``double solution'' analogy gleaned
from \cite{Couder.2012probabilities}. For, as is shown, e.g., in
\cite{Holland.1993,Elze.2011general}, one can construct various forms
of classical analogies to quantum mechanical Gaussian dispersion.
Originally, the expression of a ``double solution'' refers to an
early idea of \cite{DeBroglie.1960book} to model quantum behavior
by a two-fold process, i.e., by a the movement of a hypothetical point-like
``singularity solution'' of the Schrödinger equation, and by the
evolution of the usual wave function that would provide the empirically
confirmed statistical predictions. Recently, \cite{Couder.2012probabilities}
used this ansatz to describe the behaviors of their ``bouncer''-
(or ``walker''-) droplets: On an individual level, one observes
particles surrounded by circular waves they emit through the phase-coupling
with an oscillating bath, which provides, on a statistical level,
the emergent outcome in close analogy to quantum mechanical behavior
(like, e.g., diffraction or double-slit interference). The simulation
of interference in the double-slit experiment was in \cite{Groessing.2012doubleslit}
easily achieved by assuming the simple case where the two slits have
equal aperture. Instead, in this paper, we discard that simplification
and show in a more detailed analysis that one can i) find a formulation
applicable to independently variable slit widths, and ii) provide
computer simulations thereof.
\section{From classical phase-space distributions to quantum mechanical dispersion}
In the context of the double solution idea, which is related to correlations
on a statistical level between individual uncorrelated particle positions
$x$ and momenta $p$, respectively, we consider the free Liouville
equation
\begin{equation}
\frac{\partial f}{\partial t}+\sum_{i=1}^{3}\frac{p_{i}}{m}\frac{\partial f}{\partial x_{i}}-\sum_{i=1}^{3}\frac{\partial V}{\partial x_{i}}\frac{\partial f}{\partial p_{i}}=0\label{eq:td.1}
{\rm e}nd{equation}
with potential $V$ and mass $m$. For simplicity, we restrict ourselves
to the 1-dimensional space coordinate $x$ further on. Liouville's
equation~(\ref{eq:td.1}) implies the spatial conservation law and
has the property that precise knowledge of initial conditions is not
lost in the course of time. That is, it provides a phase-space distribution
$f\left(x,p,t\right)$ that shows the emergence of correlations between
$x$ and $p$ from an initially uncorrelated product function of non-spreading
(``classical'') Gaussian position distributions as well as momentum
distributions,
\begin{equation}
f_{0}\left(x,p\right)=\frac{1}{2\partiali\sigma_{0}\partiali_{0}}{\rm e}xp\left\{ -\frac{x^{2}}{2\sigma_{0}^{2}}\right\} {\rm e}xp\left\{ -\frac{p^{2}}{2\partiali_{0}^{2}}\right\} ,\label{eq:td.2}
{\rm e}nd{equation}
where $\sigma_{0}$ is the initial space deviation, i.e., $\sigma_{0}=\sigma(t=0)$,
and $\partiali_{0}:=mu_{0}$ is the momentum deviation. Then the phase-space
distribution reads as
\begin{equation}
f\left(x,p,t\right)=\frac{1}{2\partiali\sigma_{0}mu_{0}}{\rm e}xp\left\{ -\frac{\left(x-pt/m\right)^{2}}{2\sigma_{0}^{2}}\right\} {\rm e}xp\left\{ -\frac{p^{2}}{2m^{2}u_{0}^{2}}\right\} .\label{eq:td.3}
{\rm e}nd{equation}
\noindent The above-mentioned correlations between $x$ and $p$ emerge
when one considers the probability density in $x$--space, which is
given by the integral
\begin{equation}
P\left(x,t\right)=\int f\,\mathrm{d} p=\frac{1}{\sqrt{2\partiali}\sigma}{\rm e}xp\left\{ -\frac{x^{2}}{2\sigma^{2}}\right\} ,\label{eq:td.4}
{\rm e}nd{equation}
with the variance at time $t$ given by
\begin{equation}
\sigma^{2}=\sigma_{0}^{2}+u_{0}^{2}\, t^{2}.\label{eq:td.5}
{\rm e}nd{equation}
In other words, the distribution~(\ref{eq:td.4}) describing a spreading
Gaussian is obtained from a continuous set of classical, non-spreading,
Gaussian position distributions whose momenta also have a non-spreading
Gaussian distributions. One thus obtains the exact quantum mechanical
dispersion formula for a Gaussian, as we have obtained also previously
from our classical ansatz by relating different kinetic energy terms
in our diffusion model~\cite{Groessing.2010emergence}. For confirmation
with respect to that model we use the Einstein relation
\begin{equation}
D=\frac{\hbar}{2m}\,,\label{eq:td.6}
{\rm e}nd{equation}
with the reduced Planck constant $\hbar=h/(2\partiali)$ and $m$ being
the particle's mass, and we note that with (\ref{eq:td.4}), $\nabla P=\frac{\partial}{\partial x}P=-\frac{x}{\sigma^{2}}P$
and the usual definition of the ``osmotic'' velocity $u$ one obtains
\begin{equation}
u=u(x,t)=-D\frac{\nabla P}{P}=\frac{xD}{\sigma^{2}}\,.\label{eq:td.7}
{\rm e}nd{equation}
For the average initial value we find
\begin{equation}
u_{0}:=\left.\vphantom{\int}\meanx u\right|_{t=0}=u(\sigma_{0},0)=\frac{D}{\sigma_{0}}\,,\label{eq:td.8}
{\rm e}nd{equation}
which turns out to be the same as the initial velocity at position
$x=\sigma_{0}$ at $t=0$. This is a characteristic value for Gaussians,
which we simply called ``initial velocity'' in our recent papers.
It corresponds exactly to the velocity $u$ at starting time $t=0$
at the trajectory that has distance $\xi(0)=\partialm\sigma_{0}$ from the
maximum of the Gaussian (Fig.~\ref{fig:td.1}). With Eq.~(\ref{eq:td.8})
one can rewrite Eq.~(\ref{eq:td.5}) in the more familiar form
\begin{equation}
\sigma^{2}=\sigma_{0}^{2}\left(1+\frac{D^{2}t^{2}}{\sigma_{0}^{4}}\right).\label{eq:td.9}
{\rm e}nd{equation}
\noindent Note also that by using the Einstein relation~(\ref{eq:td.6})
the norm in~(\ref{eq:td.3}) becomes the invariant expression
\noindent
\begin{equation}
\frac{1}{2\partiali\sigma_{0}mu_{0}}=\frac{1}{2\partiali mD}=\frac{2}{h}\label{eq:td.10}
{\rm e}nd{equation}
reflecting the ``exact uncertainty relation''~\cite{Hall.2002schrodinger}.
\section{Spreading of the wave packet due to a path excitation field}
\noindent We note that $\sigma/\sigma_{0}$ is a spreading ratio for
the wave packet independent of $x$. This functional relationship
is thus not only valid for the particular point $\xi(t)=\sigma(t)$,
but for all $x$ of the Gaussian. Therefore, one can generalize (\ref{eq:td.9})
for all $x$, i.e.,
\begin{equation}
\xi(t)=\xi(0)\frac{\sigma}{\sigma_{0}},\qquad\text{where }\quad\frac{\sigma}{\sigma_{0}}=\sqrt{1+\frac{D^{2}t^{2}}{\sigma_{0}^{4}}}\;.\label{eq:td.11}
{\rm e}nd{equation}
In other words, one derives also the time-invariant ratio for the
spreading
\begin{equation}
\frac{\xi(t)}{\sigma}=\frac{\xi(0)}{\sigma_{0}}=\text{const.}\label{eq:td.12}
{\rm e}nd{equation}
\begin{figure}[th]
\centering{}\includegraphics{td-fig1}\caption{Bohm-type trajectories for a quantum particle with initial Gaussian
distribution exhibiting the characteristics of ballistic diffusion\label{fig:td.1}}
{\rm e}nd{figure}
In Fig.~\ref{fig:td.1} the spreading according to Eq.~(\ref{eq:td.11})
is sketched. We can now try to implement our previous assumption that
the ``bouncer'' particle is phase locked with its nonlocal diffusion
wave field such that the Gaussian describing the diffusion process
has long undulatory tails representing the locking in with the undulations
of the zero-point field. In other words, we can now re-consider our
classical simulations of Gaussian dispersion and double slit interference
\cite{Groessing.2012doubleslit}, respectively, by constructing from
(\ref{eq:td.4}) a description of our ``path excitation field''
via the introduction of the amplitude $R$ as product of a Gaussian
(at rest in the $x$--direction) and a plane wave (in the $y$--direction),
\begin{equation}
R\left(x,t\right)=\left(2\partiali\sigma^{2}\right)^{-1/4}{\rm e}xp\left\{ -\frac{x^{2}}{4\sigma^{2}}\right\} \cos\left(k_{y}y\right).\label{eq:td.13}
{\rm e}nd{equation}
The product (\ref{eq:td.13}) is factorisable for all $t$ into $x$--
and $y$--dependent functions, due to the motion in the $y$--direction
by $y\left(t\right)=\hbar kt/m$. According to our principle of path
excitation \cite{Groessing.2012doubleslit}, we deal with a single,
classical particle of velocity $v=p/m$ following the propagations
of waves of equal amplitude $R$ comprising a wave-like thermal bath
that permanently provides some momentum fluctuations $\,\mathrm{d}elta p$.
The latter are reflected in Eq.~(\ref{eq:td.9}) via the r.m.s.~deviation
$\sigma(t)$ from the classical path. In other words, one has to do
with a wave packet with an overall uniform velocity $v$, where the
position $x_{0}=vt$ moves like a free classical particle, as indicated
in Fig.~\ref{fig:td.1}. As the packet spreads according to Eq.~(\ref{eq:td.9}),
$\xi(t)$ describes the result of the motion along a trajectory of
a point of this packet that was initially at $\xi(0)$. The smaller
the initial value of $\left|\xi(0)\right|$, i.e., the distance from
$x_{0}$ of the center point of the packet, the slower said spreading
takes place. In our model picture, this is easy to understand: For
a particle exactly at the center of the packet, $x_{{\rm tot}}=x_{0}\Leftrightarrow\xi(0)=0$
, the momentum contributions from the ``heated up'' environment
on average cancel each other for symmetry reasons. However, the further
off a particle is from that center, the stronger this symmetry will
be broken, i.e., leading to a position-dependent net acceleration
or deceleration, respectively, or, in effect, to the ``decay of the
wave packet''. The actual decay of the wave packet starts, roughly
spoken, at a time $t_{{\rm k}}$, indicated by a ``kink'' in Fig.~\ref{fig:td.1}
which is due to the squared time-behavior in Eq.~(\ref{eq:td.9}).
From Fig.~\ref{fig:td.1} we find $x_{{\rm tot}}=x_{0}+v(t)t+\xi(t)$
and $\xi(t)=\xi(0)+u(t)t.$ Without loss of generality we set $v={\rm const.}$
and $x_{0}=0$ further on. With the use of Eq.~(\ref{eq:td.11})
we obtain
\begin{equation}
x_{{\rm tot}}(t)=vt+\xi(t)=vt+\xi(0)\frac{\sigma}{\sigma_{0}}=vt+\xi(0)\sqrt{1+\frac{u_{0}^{2}t^{2}}{\sigma_{0}^{2}}}\;.\label{eq:td.14}
{\rm e}nd{equation}
In our model picture, $x_{{\rm tot}}$ is the position of the ``smoothed
out'' \textit{trajectories}, i.e., those averaged over a very large
number of Brownian motions.
Moreover, one can now also calculate the \textit{average total velocity
field of a Gaussian wave packet} as
\begin{equation}
v_{{\rm tot}}(t)=\frac{\,\mathrm{d} x_{{\rm tot}}(t)}{\,\mathrm{d} t}=v+\xi(0)\,\frac{u_{0}^{2}t/\sigma_{0}^{2}}{\sqrt{1+u_{0}^{2}t^{2}/\sigma_{0}^{2}}}\;,\label{eq:td.15}
{\rm e}nd{equation}
which describes the velocity field $v_{{\rm tot}}$ of a point along
a trajectory (i.e, the residue of the ``path excitation field''
to be explicated further below).
Finally, we derive the \textit{average total acceleration field of
a Gaussian wave packet} as
\begin{equation}
a_{{\rm tot}}(t)=\frac{\,\mathrm{d} v_{{\rm tot}}(t)}{\,\mathrm{d} t}=\xi(0)\,\frac{u_{0}^{2}/\sigma_{0}^{2}}{\sqrt{\left(1+u_{0}^{2}t^{2}/\sigma_{0}^{2}\right)^{3}}}\;,\label{eq:td.16}
{\rm e}nd{equation}
describing the acceleration of a point along the trajectory at time
$t$. Eqs.~(\ref{eq:td.14}) to (\ref{eq:td.16}) allow us to calculate
the quantities along a trajectory only from a given starting point,
indicated by $\xi(0)$.
Actually, however, we are interested in the dynamics at any given
position $(x,t)$ directly. Using
\begin{equation}
\xi(t)=x-vt\label{eq:td.17}
{\rm e}nd{equation}
and Eq.~(\ref{eq:td.11}) we rewrite
\begin{equation}
\xi(0)=\frac{x-vt}{\sqrt{1+u_{0}^{2}t^{2}/\sigma_{0}^{2}}}
{\rm e}nd{equation}
which leads to the generalized fields,
\begin{align}
x_{{\rm tot}}(x,t) & =x,\vphantom{\intop_{0}^{0}}\label{eq:td.19}\\
v_{{\rm tot}}(x,t) & =v+\xi(t)\,\frac{u_{0}^{2}t/\sigma_{0}^{2}}{1+u_{0}^{2}t^{2}/\sigma_{0}^{2}}=v+(x-vt)\,\frac{u_{0}^{2}t}{\sigma^{2}}\,,\vphantom{\intop_{0}^{0}}\label{eq:td.20}\\
a_{{\rm tot}}(x,t) & =\xi(t)\,\frac{u_{0}^{2}/\sigma_{0}^{2}}{\left(1+u_{0}^{2}t^{2}/\sigma_{0}^{2}\right)^{2}}=(x-vt)\,\frac{u_{0}^{2}\sigma_{0}^{2}}{\sigma^{4}}\;,\vphantom{\intop_{0}^{0}}\label{eq:td.21}
{\rm e}nd{align}
which will be used in the simulations later on.
\section{The derivation of $D_{{\rm t}}$}
We derive a solution for a diffusion equation with a time-dependent
diffusion coefficient $kt^{\alpha}$ for a generalized diffusion equation
(cf. \cite{Mesa.2012classical})
\begin{equation}
\frac{\partialartial P}{\partialartial t}=kt^{\alpha}\frac{\partialartial^{2}P}{\partialartial x^{2}}\;,\quad\alpha>0.\label{eq:tdde.1}
{\rm e}nd{equation}
Here, $t$ and $k$ denote the time and a constant factor, respectively.
Inserting $P(x,t)$ of Eq.~(\ref{eq:td.4}) as a solution into Eq.~(\ref{eq:tdde.1})
yields
\begin{align}
\frac{P\,\mathrm{d}ot{\sigma}}{\sigma}\left(\frac{x^{2}}{\sigma^{2}}-1\right) & =kt^{\alpha}\frac{P}{\sigma^{2}}\left(\frac{x^{2}}{\sigma^{2}}-1\right)\;,
{\rm e}nd{align}
and, after integration,
\begin{align}
\frac{\sigma^{2}}{2} & =k\frac{t^{\alpha+1}}{\alpha+1}+\frac{c_{0}}{2}\;.\label{eq:tdde.4}
{\rm e}nd{align}
Substitution of (\ref{eq:td.9}) into (\ref{eq:tdde.4}) yields $c_{0}=\sigma_{0}^{2}$
and
\begin{align}
k\frac{2t^{\alpha+1}}{\alpha+1} & =\frac{D^{2}t^{2}}{\sigma_{0}^{2}}\;,
{\rm e}nd{align}
which can only be fulfilled by $\alpha=1$, so that
\begin{align}
k & =\frac{D^{2}}{\sigma_{0}^{2}}\;.
{\rm e}nd{align}
The time-dependent diffusion coefficient $D_{{\rm t}}$ is with (\ref{eq:td.8})
identified as
\begin{equation}
D_{{\rm t}}:=\frac{D^{2}}{\sigma_{0}^{2}}\, t=u_{0}^{2}\, t=\frac{\hbar^{2}}{4m^{2}\sigma_{0}^{2}}\, t.\label{eq:td.22}
{\rm e}nd{equation}
Finally, Eq.~(\ref{eq:tdde.1}) reads as
\begin{align}
\frac{\partialartial P}{\partialartial t} & =\frac{D^{2}t}{\sigma_{0}^{2}}\,\frac{\partialartial^{2}P}{\partialartial x^{2}}\label{eq:ballisticDE}
{\rm e}nd{align}
and turns out to be a \textit{ballistic diffusion equation,} defined
by $\alpha=1$, as the special case of an anomalous diffusion where
the diffusion coefficient $D_{{\rm t}}$ grows linearly with time
$t$.
Essentially, the ``decay of the wave packet'' thus simply results
from sub-quantum diffusion with a diffusivity varying in time due
to the particle's changing thermal environment: as the heat initially
concentrated in a narrow spatial domain gets gradually dispersed,
so must the diffusivity of the medium change accordingly.
Now we look at the time $t_{{\rm k}}$ of the kink (Fig.~\ref{fig:td.1}).
The wave packet begins to spread differently at the kink, which is,
according to Eq.~(\ref{eq:td.11}), obviously at that time $t=t_{{\rm k}}$
where the influence of the right term is equal to the left term under
the square root and hence $D^{2}t^{2}\overset{!}{=}\sigma_{0}^{4}$
(i.e., $\sigma=\sqrt{2}\sigma_{0}$). Then we find with (\ref{eq:td.22})
that
\begin{equation}
D_{{\rm t}}=\frac{t}{t_{{\rm k}}}D.\label{eq:td.23}
{\rm e}nd{equation}
As one can see, $t=t_{{\rm k}}$ is the time when $D_{{\rm t}}=D$.
Note that the diffusivity $D$ is constant for all times $t$ and
has to be distinguished from the diffusion coefficient $D_{{\rm t}}$.
In a different approach, one could also start out with the ``exact''
uncertainty relation, $mu_{0}^{2}t_{{\rm k}}=\hbar/2$, with $u_{0}=D/\sigma_{0}$.
This again leads to $D_{{\rm t}}=D^{2}t/\sigma_{0}^{2}=u_{0}^{2}t=t/t_{k}D$.
We recall Boltzmann's relation $\Delta Q=2\omega_{0}\,\mathrm{d}elta S$ \cite{Groessing.2008vacuum,Groessing.2009origin}
between the heat applied to an oscillating system and a change in
the action function $\,\mathrm{d}elta S=\frac{1}{2\partiali}\,\mathrm{d}elta\int_{0}^{\tau}E_{{\rm kin}}\,\mathrm{d} t$,
respectively, providing
\begin{equation}
\nabla Q=2\omega_{0}\nabla(\,\mathrm{d}elta S)\;.\label{eq:td.24}
{\rm e}nd{equation}
Here, $\,\mathrm{d}elta S$ relates to the momentum fluctuation via
\begin{equation}
\nabla(\,\mathrm{d}elta S)=\,\mathrm{d}elta\mathbf{p}=:m\mathbf{u}=-\frac{\hbar}{2}\frac{\nabla P}{P}\;,\label{eq:td.25}
{\rm e}nd{equation}
and therefore, with $P=P_{0}{\rm e}^{-\,\mathrm{d}elta Q/kT_{0}}$ and $\Delta Q=kT=\hbar\omega$,
\begin{equation}
m\mathbf{u}=\frac{\nabla Q}{2\omega}\;.\label{eq:td.26}
{\rm e}nd{equation}
Using the initial velocity (\ref{eq:td.8}) together with Eq.~(\ref{eq:td.23})
we find
\begin{equation}
D_{{\rm t}}=u_{0}^{2}t=\frac{2}{\hbar}mu_{0}^{2}tD=\frac{2}{\hbar}\frac{(\,\mathrm{d}elta p)^{2}}{2m}tD=\frac{2}{\hbar}\left[\,\mathrm{d}elta S(t)-\,\mathrm{d}elta S(0)\right]D=\frac{\Delta Q(t)}{\hbar\omega}D.\label{eq:td.27}
{\rm e}nd{equation}
Actually, $\,\mathrm{d}elta S(0)=0$, since there are no initial fluctuations.
Substitution of (\ref{eq:td.23}) into (\ref{eq:td.27}) leads then
to
\begin{equation}
D_{{\rm t}}=\frac{t}{t_{{\rm k}}}D=\frac{\Delta Q}{kT}D=-\ln\frac{P(t)}{P(0)}D=-D\left[\ln P(t)-\ln P(0)\right].\label{eq:td.28}
{\rm e}nd{equation}
One can also derive a condition that does not require to know the
diffusion coefficient at $t=0$,
\begin{equation}
\Delta D=D_{{\rm t}}(t_{2})-D_{{\rm t}}(t_{1})=-D\left[\ln P(t_{2})-\ln P(t_{1})\right],\label{eq:td.29}
{\rm e}nd{equation}
by choosing two arbitrary time steps $t_{1}$ and $t_{2}$ as suggested
in Fig.~\ref{fig:td.1}.
From condition (\ref{eq:td.22}) one can immediately see that
\begin{equation}
\frac{\partial D_{{\rm t}}}{\partial t}=\frac{D^{2}}{\sigma_{0}^{2}}=\textrm{const.}\label{eq:td.31}
{\rm e}nd{equation}
Thus, one can also rewrite Eq.~(\ref{eq:td.29}) as
\begin{equation}
\Delta D=D_{{\rm t}}(t_{1})+\frac{D^{2}}{\sigma_{0}^{2}}(t_{2}-t_{1}),\label{eq:td.32}
{\rm e}nd{equation}
which is only valid for equal slit widths $x_{01}=x_{02}$ and thus
$\sigma_{1}=\sigma_{2}$. In order to compute the distribution of
$P(x,t)$, one starts with Eq.~(\ref{eq:td.32}) and takes the local
properties of the diffusivity into account. For given times $t_{1}$
and $t_{2}=t_{1}+\Delta t$ one obtains with (\ref{eq:td.8}),
\begin{align}
D_{{\rm t}}(x,t_{1}) & =-D\ln P(x,t_{1}),\label{eq:td.36}\\
D_{{\rm t}}(x,t_{2}) & =-D\ln P(x,t_{2})+D_{{\rm t}}(x,t_{1}),\label{eq:td.37}
{\rm e}nd{align}
thereby constituting a rule to numerically compute the distribution
$P(x,t)$.
\section{Finite difference scheme}
Starting with the ballistic diffusion equation (\ref{eq:ballisticDE})
with time-dependent diffusivity $D_{{\rm t}}$ we use an explicit
finite difference forward scheme (cf. \cite{Schwarz.2009numerische}),
\begin{align}
\frac{\partialartial P}{\partialartial t} & \rightarrow\frac{1}{\Delta t}\left(P[x,t+1]-P[x,t]\right),\\
\frac{\partialartial^{2}P}{\partialartial x^{2}} & \rightarrow\frac{1}{\Delta x^{2}}\left(P[x+1,t]-2P[x,t]+P[x-1,t]\right),
{\rm e}nd{align}
with 1-dimensional cells. In case $D_{{\rm t}}$ is independent of
$x$, the complete equation after reordering leads to
\begin{equation}
P[x,t+1]=P[x,t]+\frac{D[t+1]\Delta t}{\Delta x^{2}}\left\{ P[x+1,t]-2P[x,t]+P[x-1,t]\right\} \label{eq:tdde.1.2.3}
{\rm e}nd{equation}
with space $x$ and time $t$, and initial Gaussian distribution $P(x,0)$
with standard deviation $\sigma_{0}$.
As can be seen, calculation of a cell's value at time $t$ only depends
on cell values at the previous time. The time-dependence of the diffusion
coefficient can also be calculated without any knowledge of neighboring
cells. The diffusion coefficient represents the underlying physics
of the current cell and is calculated for the evaluated time step
$t+1$.
The stability condition for the scheme of Eq.~(\ref{eq:tdde.1.2.3})
is that
\begin{equation}
\left|\frac{D_{{\rm t}}\Delta t}{\Delta x^{2}}\right|\leq\frac{1}{2}\label{eq:tdde.1.4.1}
{\rm e}nd{equation}
be satisfied for all values of the cells $[x,t]$ in the domain of
computation. The general procedure is that one considers each of the
\textit{frozen coefficient problems} arising from the scheme. The
frozen coefficient problems are the constant coefficient problems
obtained by fixing the coefficients at their values attained at each
point in the domain of the computation (cf.~\cite{Strikwerda.2004finite}).
Substituting Eq.~(\ref{eq:td.22}) into (\ref{eq:tdde.1.4.1}) leads
to
\begin{equation}
\Delta t\leq\frac{\Delta x^{2}\sigma_{0}^{2}}{2D_{{\rm t}}^{2}t}\:.\label{eq:tdde.1.4.2}
{\rm e}nd{equation}
This shows that the finite difference scheme (\ref{eq:tdde.1.2.3})
is suitable to solve the ballistic diffusion equation~(\ref{eq:ballisticDE})
as long as the spreading is not too big. Beyond that, the computations
are no longer economically practical due to the necessarily enormous
number of cells. Then, one has to replace the explicit scheme~(\ref{eq:tdde.1.2.3})
by an implicit scheme, for example, which has less stringent restrictions
on the stability conditions, but needs linear equation solvers instead.
For our simulations we employed the explicit scheme introduced above
as well as implicit schemes with an open source software for numerical
computation, Scilab \cite{Scilab}, on a standard personal computer.
\section{The connection to wave mechanics: The double-slit experiments with
different slit widths}
For a more generalized picture, we now take a closer look at the double-slit
experiment.
\begin{figure}[th]
\centering{}\includegraphics{td-fig2}\caption{Sketch of a double-slit with two different widths and Bohm-type trajectories
(and same-widths scenario indicated by gray lines)\label{fig:td.2}}
{\rm e}nd{figure}
Consider a scenario as shown in Fig.~\ref{fig:td.2} with two slits
of different widths. We assume the initial Gaussians passing through
a slit have a standard deviation value matching the slit width, e.g.,
$\sigma_{01}=\sigma_{0}$ and $\sigma_{02}=\sigma_{0}/2$, respectively,
with $\sigma_{0i}$ then being also the width of slit~$i$. The resulting
Bohm-type trajectories of the two decaying Gaussians are sketched
in Fig.~\ref{fig:td.2} with red lines. Thus
\begin{equation}
t_{{\rm k}2}=t_{{\rm k}1}/4
{\rm e}nd{equation}
while the spreading is doubled (compare with the grayed out spreading
of slit~2 for the case of $\sigma_{0}$ for both slits). According
to Eq.~(\ref{eq:td.22}), the diffusion coefficients of the two slits
yield
\begin{equation}
D_{{\rm t},1}(t)=\frac{D^{2}t}{\sigma_{02}^{2}}\neq D_{{\rm t},2}(t)=\frac{D^{2}t}{\sigma_{01}^{2}},\quad t>0.
{\rm e}nd{equation}
The advantage of Eq.~(\ref{eq:td.28}) lies in it's local fit due
to its dependence on $P(x,t)$ instead on $\sigma(t)$, since the
latter is just a global statistical value of too less local relevance.
For the general case, we have to deal with a diffusion coefficient
$D_{{\rm t}}(x,t)$ further on. The time-dependent diffusion equation
reads then
\begin{equation}
\frac{\partialartial P}{\partialartial t}=\frac{\partial}{\partial x}\left(D_{{\rm t}}(x,t)\frac{\partialartial P}{\partialartial x}\right)\;.\label{eq:td.35}
{\rm e}nd{equation}
We have now all the tools necessary to consider the inclusion of wave
mechanics in our model. We define the phase as
\begin{equation}
\varphi=S/\hbar\label{eq:td.38}
{\rm e}nd{equation}
with the general action $S$. Identifying $v_{{\rm tot}}$ of Eq.~(\ref{eq:td.20})
with
\begin{equation}
v_{{\rm tot}}=\frac{\nabla S}{m}\label{eq:td39}
{\rm e}nd{equation}
we find for the action
\begin{equation}
S=\int mv_{{\rm tot}}(t)\,\mathrm{d} x-\int E\,\mathrm{d} t=m\int\left(v+\frac{u_{0}^{2}t}{\sigma_{0}^{2}+u_{0}^{2}t^{2}}\,\xi(t)\right)\,\mathrm{d} x-\int E\,\mathrm{d} t,\label{eq:td.40}
{\rm e}nd{equation}
with $E$ being the system's total energy. As $v$ does not depend
on $x$ we can solve the first integral, and for the conservative
case also the second integral, providing with (\ref{eq:td.12})
\begin{equation}
S=mvx+\frac{mu_{0}^{2}}{2}\left(\frac{\xi(t)}{\sigma(t)}\right)^{2}t-Et=mvx+\frac{mu_{0}^{2}}{2}\left(\frac{\xi(0)}{\sigma_{0}}\right)^{2}t-Et.\label{eq:td.41}
{\rm e}nd{equation}
Here, the action $S$ \textit{along a trajectory} is given by the
sum of the usual momentum-related term and a term depending on the
kinetic energy, or kinetic temperature, respectively, of the ``heated
up'' environment, weighted by a factor that solely depends on a particular
trajectory indicated by the initial location $\xi(0)$ in the Gaussian.
Finally, we rewrite the phase with the help of Eqs.~(\ref{eq:td.38})
and (\ref{eq:td.17}) as
\begin{equation}
\varphi=\frac{1}{\hbar}\left[mvx+\frac{mu_{0}^{2}}{2}\left(\frac{\xi(0)}{\sigma_{0}}\right)^{2}t-Et\right]=\frac{1}{\hbar}\left[mvx+\frac{mu_{0}^{2}}{2}\left(\frac{x-vt}{\sigma(t)}\right)^{2}t-Et\right].\label{eq:td.42}
{\rm e}nd{equation}
The expression containing $\xi(0)$ indicates a phase $\varphi$ along
a trajectory, while the r.h.s.~sticks to our coordinate system and
is thus the better choice to do interference calculations.
Instead of following just one Gaussian, we extend our simulation scheme
to include two possible paths of a particle which eventually cross
each other. For this, we use two Gaussians approaching each other.
Following our earlier approach in \cite{Groessing.2012doubleslit}
we simulate a double-slit experiment by independent numerical computation
of two Gaussian wave packets with total distribution given by
\begin{equation}
P_{{\rm tot}}:=P_{1}+P_{2}+2\sqrt{P_{1}P_{2}}\cos\varphi_{12}.\label{eq:td.43}
{\rm e}nd{equation}
Since each Gaussian has its own phase (\ref{eq:td.42}) we are free
to add a phase shifter $\Delta\varphi$ for one of the slits of the
two-slit experiment, say slit 1, which modifies the phase to
\begin{equation}
\varphi{}_{1}=\frac{S_{1}}{\hbar}+\Delta\varphi\label{eq:td.46}
{\rm e}nd{equation}
and yields for the phase difference
\begin{align}
\varphi_{12}=\varphi_{2}-\varphi{}_{1}= & \frac{m}{\hbar}\left[\vphantom{\intop}v_{2}(x-x_{02})-v_{1}(x-x_{01})\right]\label{eq:td.44}\\
& +\frac{mt}{2\hbar}\left[\frac{u_{02}^{2}(x-x_{02}-v_{2}t)^{2}}{\sigma_{2}^{2}(t)}-\frac{u_{01}^{2}(x-x_{01}-v_{1}t)^{2}}{\sigma_{1}^{2}(t)}\right]-\Delta\varphi.\nonumber
{\rm e}nd{align}
The two slits at positions $x_{01}$ and $x_{02}$ have different
slit widths and hence different parameters, $\sigma_{01}$, $\sigma_{1}$,
$u_{01}$ and $\sigma_{02}$, $\sigma_{2}$, $u_{02}$, respectively,
as illustrated by the red trajectories in Fig.~\ref{fig:td.2} for
the example of $\sigma_{01}=\sigma_{0}$ and $\sigma_{02}=\sigma_{0}/2$,
respectively. One can observe several characteristics of the averaged
particle trajectories, which, just because of the averaging, are identical
with the Bohmian trajectories. As one can see, the phase difference
(\ref{eq:td.44}) is at any time defined for the whole domain, and
hence $\varphi_{12}$ is intrinsically nonlocal.
Finally, we recall our derivation of the total average density current
\cite{Groessing.2012doubleslit,Grossing.2012quantum}, i.e., the most
general expression (including weights $P_{i}$) for our ``path excitation
field'',
\begin{equation}
J_{{\rm tot}}=P_{1}v_{1}+P_{2}v_{2}+\sqrt{P_{1}P_{2}}\left(v_{1}+v_{2}\right)\cos\varphi_{12}+\sqrt{P_{1}P_{2}}\left(u_{1}-u_{2}\right)\sin\varphi_{12},\label{eq:td.45}
{\rm e}nd{equation}
where
\begin{equation}
v_{{\rm tot}}=\frac{J_{{\rm tot}}}{P_{{\rm tot}}}\:,
{\rm e}nd{equation}
with osmotic velocities $u_{i}$ of Eq.~(\ref{eq:td.7}) and total
velocities $v_{i}$ of Eq.~(\ref{eq:td39}) applied to both slits,
1 and 2, and with the phases~(\ref{eq:td.44}). Note that the last
term on the r.h.s~in Eq.~(\ref{eq:td.45}) is termed ``entangling
current'' $J_{\rm e}$ by us \cite{Groessing.2012doubleslit}, which is of a genuinely
``quantum'' nature in that the velocities $u_{i}$ are generally
entangled with the velocities $v_{i}$.
\section{Simulation results}
In Figs.~\ref{fig:td.3} to \ref{fig:td.5}, the graphical results
of a classical computer simulation of the interference pattern in
double-slit experiments are shown, including the trajectories. In
Fig.~\ref{fig:td.3a} the maximum of the intensity is distributed
along the symmetry line exactly in the middle between the two slits.
\cite{Groessing.2012doubleslit}
\begin{figure}[th]
\subfloat[equal slit widths\label{fig:td.3a}]{\centering{}\includegraphics{td-fig3a}}\subfloat[$\sigma_{01}=2\sigma_{02}$\label{fig:td.3b}]{\centering{}\includegraphics{td-fig3b}}\caption{Classical computer simulation of the interference pattern with different
slit widths: intensity distribution with increasing intensity from
white through yellow and orange, with trajectories (red) for two Gaussian
slits ($v_{x,1}=v_{x,2}=0$)\label{fig:td.3}}
{\rm e}nd{figure}
In the examplary figures, trajectories according to Eq.~(\ref{eq:td.43})
for the two Gaussian slits are shown. The interference hyperbolas
for the maxima characterize the regions where the phase difference
$\varphi=2n\partiali$, and those with the minima lie at $\varphi=(2n+1)\partiali$,
$n=0,\partialm1,\partialm2,\ldots$ Note in particular the ``kinks'' of trajectories
moving from the center-oriented side of one relative maximum to cross
over to join more central (relative) maxima. In our classical explanation
of double slit interference, a detailed ``micro-causal'' account
of the corresponding kinematics can be given. The trajectories are
in full accordance with those obtained from the Bohmian approach,
as can be seen by comparison with \cite{Holland.1993,Bohm.1993undivided,Sanz.2009context,Sanz.2012trajectory},
for example.
We use the same double-slit arrangements in Figs.~\ref{fig:td.4}
and \ref{fig:td.5}, but include a phase shifter affecting the current
from slit~1, as sketched by the dashed red lines or rectangles on
the left hand side, respectively. Even though the total applied phase
shift is either $3\partiali$ or $5\partiali$ in Figs.~\ref{fig:td.4a} and
\ref{fig:td.5a}, respectively, one recognizes the effective phase
difference of $\Delta\varphi_{{\rm mod}}=\Delta\varphi\mod2\partiali=\partiali$
in each case, which eventually results in equal shifts of the interference
fringes. By comparing with Fig.~\ref{fig:td.3a} we now observe a
minimum of the resulting distribution along the central symmetry line,
in full accordance with the Aharonov-Bohm effect, independently of
the times $t_{1}$ and $t_{2}$ during which the shift has been applied.
To bring out the shifting of the interference fringes more clearly,
we apply in Fig.~\ref{fig:td.5} the phase shift in between the indicated
times $t_{1}$ and $t_{2}$, respectively. Note that the phase shift
applied only to a single slit's current at a time $t$ when the decaying
Gaussians are already overlapping (Fig.~\ref{fig:td.5}) is shown
here for didactic reasons only. In this highly idealized scenario,
then, one can see an illustration of the immediate effectiveness of
$\Delta\varphi$ over the whole domain according to Eq.~(\ref{eq:td.44}),
i.e., of the nonlocality of the relative phase.
\begin{figure}[!tbh]
\subfloat[Probability density $P$\label{fig:td.4a}]{\centering{}\includegraphics{td-fig4a}}\subfloat[Phase shift\label{fig:td.4b}]{\centering{}\includegraphics{td-fig4b}}\subfloat[Entangling current~$J_{{\rm e}}$\label{fig:td.4c}]{\centering{}\includegraphics{td-fig4c}}\caption{Classical computer simulation as in Fig.~\ref{fig:td.3a}, but with
additional phase shift $\Delta\varphi=3\partiali$ accumulated during the
time interval between $t_{1}$ and $t_{2}$ at slit~1\label{fig:td.4}}
{\rm e}nd{figure}
\begin{figure}[!tbh]
\subfloat[Probability density $P$\label{fig:td.5a}]{\centering{}\includegraphics{td-fig5a}}\subfloat[Phase shift\label{fig:td.5b}]{\centering{}\includegraphics{td-fig5b}}\subfloat[Entangling current~$J_{{\rm e}}$\label{fig:td.5c}]{\centering{}\includegraphics{td-fig5c}}\caption{Classical computer simulation as in Fig.~\ref{fig:td.4}, but with
different times $t_{i}$ and with accumulated additional phase $\Delta\varphi=5\partiali$.
This results in the same distributions of $P$ and $J_{{\rm e}}$
for times $t>t_{2}$ and shows the effect of the shifting of the interference
fringes more clearly than Fig.~\ref{fig:td.4}\label{fig:td.5}}
{\rm e}nd{figure}
To conclude, we have in this paper provided a detailed description
of the velocity fields involved in the analytical calculations as
well as the computer simulations illustrating Gaussian dispersion
and interference at a double slit. We have arrived at an expression
for the local value of the phase, Eq.~(\ref{eq:td.42}), which made
it possible also to extend our previous model to slit systems with
independently variable slit widths. With the computer simulations
of the latter, the nonlocal nature of the relative phase can be clearly
demonstrated.
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{\rm e}nd{document} | math | 40,255 |
\begin{document}
\topical[Essential entanglement]{Essential entanglement for atomic and molecular physics}
\author{Malte C. Tichy$^{1}$, Florian Mintert$^{1,2}$, Andreas Buchleitner$^{1}$}
\address{(1) Physikalisches Institut - Albert-Ludwigs-Universit\"at Freiburg - Hermann-Herder-Strasse 3, D-79104 Freiburg im Breisgau, Germany}
\address{(2) Freiburg Institute for Advanced Studies (FRIAS), Albert-Ludwigs-Universit\"at Freiburg, Albertstrasse 19, D-79104 Freiburg, Germany}
\ead{[email protected]}
\date{\today}
\begin{abstract}
Entanglement is nowadays considered as a key quantity for the understanding of correlations, transport properties, and phase transitions in composite quantum systems, and thus receives interest beyond the engineered applications in the focus of quantum information science. We review recent experimental and theoretical progress in the study of quantum correlations under that wider perspective, with an emphasis on rigorous definitions of the entanglement of identical particles, and on entanglement studies in atoms and molecules.
\end{eqnarray}d{abstract}
\date{\today}
\submitto{\JPB}
\maketitle
\tableofcontents
\section{Introduction}
While first considered as an indicator of the incompleteness of quantum physics \cite{PhysRev.47.777}, entanglement \cite{Schrodinger:1935ly} is today understood as one of the quantum world's most important and glaring properties. It contradicts the intuitive assumption that any physical object has distinctive individual properties that completely define it as an independent entity and that the result of measurement outcomes on one system are independent of any operations performed on another space-like separate system, an attitude also known as \emph{local realism}.
Thereby it poses important epistemological questions \cite{Schrodinger:1935fk}.
Since an experimental test of the scenario suggested in \cite{PhysRev.47.777} to prove that incompleteness was long considered unfeasible, the interest in entanglement was long rather restricted to the philosophical domain. Not less than 30 years after the formulation of the Einstein Podolsky Rosen (EPR) paradox \cite{PhysRev.47.777,Bohm:1957vn}, a proposal for the direct, experimental violation of local realism, paraphrased in terms of a simple inequality, re-anchored the discussion on physical grounds. The \emph{Bell inequality} \cite{Bell:1964pt,Clauser:1969qa}, which sets strict thresholds on classical correlations of measurement results, was then proven to be violated in experiments which employed entangled states of photons. Stronger correlations than permitted by local realism were thus testified \cite{Freedman:1972fk,Aspect:1981zr,Salart:2008uq,Groblacher:2007kx,Tittel:1998ve}.
Possible technological applications have triggered enormous interest in quantum correlations. With the discovery of Shor's factoring algorithm \cite{shor:303}, which relies on entanglement, quantum correlations became a topic of the information sciences, since they hold the potential to very considerably speed up quantum computers with respect to classical supercomputing facilities \cite{Jozsa:2003fk} and may thus jeopardize classical data encryption. Also other quantum technologies such as quantum imaging \cite{al:2002uq}, certain key-distribution schemes in quantum cryptography \cite{Ekert:1991kx} or quantum teleportation \cite{Bennett:1993hc} rely on entanglement. Extensive research activity in these diverse areas by now led to considerable progress in our understanding of quantum correlations when associated with engineered systems with well-defined substructures.
More recently, the theory of entanglement has penetrated into other fields of physics, to gain a fresh perspective on naturally occurring, often, rather complex systems, and to understand the role of quantum correlations for their spectral and dynamical properties, or for their functionality, even in biological structures \cite{Scholak:2011fk,scholak3560,Scholak:2010fk,Cai:2010qf,Briegel:2008ff}. It was shown, {\it e.g.}, that entanglement yields a versatile characterization of quantum phase transitions in many-body systems \cite{Amico:2008ph,Wu:2004ly,Osterloh:2002ve,MintertRey:2009uq,Gu:2004bs}, and that simple concepts like the \emph{area law} are indicative of an efficient numerical treatment of certain types of many-particle systems \cite{Eisert:2010fk}.
In atomic and molecular physics experimental progress nowadays permits the detailed analysis and control of various coherent phenomena in few-body dynamics where entanglement is once again a potentially very useful tool. For example, it was recently proposed to solve the long-standing problem of a core vacancy (de)localization during a molecular ionization process by analyzing the entanglement between the photo- and Auger electrons born in such process \cite{al.:2007df}.
The physical objects encountered in these latter fields are, however, much more difficult to control and to describe than the designed and engineered systems familiar from quantum information science. Atoms and molecules do not naturally exhibit definite and clear entanglement properties, nor well-separated entities such as photons in different optical modes \cite{Wieczorek:2009ff,PhysRevLett.101.010503} or strongly repelling ions in radio frequency traps \cite{Blatt:2008cl} do: The identification of subsystems which can carry entanglement therefore becomes a non-trivial question, possibly complicated by the indistinguishability of particles, and typical Hilbert space dimensions tend to be rather large. Additionally, the interaction between system constituents is typically of long-range type, thus rendering entanglement a dynamical quantity, difficult to grasp, and without any unambiguous straightforward definition. Moreover, it remains to be clearly defined to which extent phenomena like macroscopic quantum superpositions imply the existence of entanglement \cite{PhysRevA.81.010101,PhysRevLett.65.1838}, and vice-versa. This defines a challenging task for conceptual research in quantum information theory, which ought to respond to novel experimental challenges.
The scope of the present Topical Review is threefold: First, we will present an overview over the different facets entanglement can have, together with a conceptual framework which permits to compare the entanglement properties of distinct physical systems. Second, we will illustrate these theoretical concepts with specific examples from experiments. Finally, we will discuss recent experimental and theoretical developments, in areas where entanglement receives attention only very recently. In order to keep the presentation most intuitive, we will focus on specific physical realizations of entanglement rather than on abstract mathematical properties, thus always stressing the \emph{physical} impact of entanglement on measurement results.
To avoid redundancies with earlier reviews, we will not cover studies of entanglement in established fields, but refer the interested reader to \cite{RevModPhys.81.865,Bengtsson:2006fu}, where the mathematical and quantum-information aspects of entanglement are reviewed extensively. More specialized reviews focus on the specific properties of multipartite entanglement \cite{Eisert:2006il}, on the entanglement in continuous-variable systems \cite{Adesso:2007ul,Braunstein:2005mb}, and on the interconnection between entanglement and violations of local realism \cite{Genovese:2005cr,Werner:2001dz}. An introduction to the quantification of entanglement, {\it i.e.}~on entanglement measures, can be found in \cite{Plenio:2006uq}, an overview on approximations to such measures of entanglement, especially on efficient lower bounds, together with applications to general scenarios of open system entanglement dynamics, is given in \cite{Mintert:2005rc}. Applications of entanglement to the simulation of many-body-systems are reviewed in \cite{Amico:2008ph,Latorre:2009fv}, focus on area laws can be found in \cite{Eisert:2010fk}. The relevance of entanglement for decoherence theory is touched upon in \cite{Buchleitner:2009fk,abucoherent}, entanglement in trapped ion systems is discussed in \cite{Haffner:2008wo}, and entanglement between electronic spins in solid-state devices in \cite{Burkard:2007ly}.
The text is organized as follows: In the next Section, we introduce fundamental notions of entanglement and related concepts, such as to define a common language for the subsequent Sections. In Section \ref{secentities}, we discuss the different possibilities to subdivide physical systems into subunits, and the influence of the specific choice of the partition on the resulting entanglement. Contact is made to state-of-the-art quantum optics experiments, for illustration. Finally, in Section \ref{atommol}, we move on to theoretical and experimental studies of quantum correlations in atomic and molecular physics which bear virtually all of the aforementioned complications. We conclude with an outlook on possible future directions for studies of atomic, molecular and biological systems from a quantum-information perspective.
\section{Subsystem structures and entanglement quantifiers}
Let us now shortly recollect the required basic notions of entanglement. Since several reviews and introductory articles are available \cite{RevModPhys.81.865,Plenio:2006uq,Mintert:2005rc}, we introduce only those concepts necessary for the understanding of the subsequent Sections, to make this review self-containted. In particular, we define the different entanglement measures employed in atomic and molecular physics. We also explicitly discuss the requirements on the subsystem-structure, which are usually implicitly assumed. This is necessary to establish a general formalism that will allow us to classify the many diverse approaches to entanglement in different systems.
\subsection{Quantum and classical correlations}
The central property that attracts broad attention are the ``nonclassical'' correlations of measurement results on different subsystems of an entangled state \cite{PhysRev.47.777}. The situation is easiest illustrated by a two-body system with two spatially well-separated subsystems. The Hilbert space $\mathscr{H}$ of the two-body system then naturally decomposes into a tensor product $\mathscr{H}_1 \otimes \mathscr{H}_2$
of the two Hilbert-spaces $\mathscr{H}_1$ and $\mathscr{H}_2$ of the two subsystems, which we both assume to be of dimension $d$.
Now, consider the following exemplary bipartite state:
\begin{eqnarray}
\ket{\Psi}=\frac{1}{\sqrt{d}}\sum_{j=1}^{d}\ket{j}\otimes\ket{j} \, .
\label{eq:psimm}
\end{eqnarray}
A projective measurement in the (orthonormal) basis $\{\ket{j}\}$, performed on the first subsystem, has completely unpredictable results:
each of the possible outcomes will occur with the same probability $1/d$.
The same is true for the analogous measurement on the second subsystem. Yet, the measurement results on \emph{both} subsystems are perfectly correlated:
once a measurement result -- say $j=j_0$ -- is obtained on one subsystem, the two-body state is projected on the state $\ket{j_0}\otimes\ket{j_0}$,
so that a subsequent measurement on the other subsystem will yield the result $j_0$ with certainty.
Such correlations of measurement results might seem surprising. They could, however, be explained rather simply: Think, for example, of an experiment in which both subsystems are always prepared in the same (random) state $\ket{j}$.
In subsequent runs of the experiment (which are required to obtain reliable measurement statistics) the choice of $j$ is completely random.
The experimentalist thus creates a mixed state
\begin{eqnarray} \rho=\frac 1 d \sum_{j=1}^d \ket{\psi_j} \bra{\psi_j}\ , \ \mathrm{ with }\ket{\psi_j}=\ket{j}\otimes\ket{j}\ . \label{sepmixed} \end{eqnarray}
This mixed state gives rise to exactly the same correlations of measurement results as found for our initial example (\ref{eq:psimm}) above. In fact, one does not even need a quantum mechanical system to observe such correlations --
also colored socks \cite{Bell:1987uq} or marbles will do.
The situation changes completely, however, if a measurement in a second, \emph{distinct} orthonormal basis set $\{\ket{\alpha_j}\}$ is performed: To be specific, let us assume that this measurement is performed on the first subsystem.
Given the measurement outcome $\alpha_p$, the two-body state (\ref{eq:psimm}) is projected on the state
\begin{eqnarray}
\ket{\alpha_p}\otimes \sum_j \braket{\alpha_p}{j}\ket{j}=\ket{\alpha_p}\otimes\ket{\tilde\alpha_p}\ ,
\end{eqnarray}
that is to say, the second subsystem is left in the state $\ket{\tilde\alpha_p}=\sum_i\langle\alpha_p|j\rangle|j\rangle$.
Quite strikingly, the states $\ket{\tilde\alpha_p}$ ($p=1,...,d$) are mutually orthogonal:
\begin{eqnarray}
\langle\tilde\alpha_q|\tilde\alpha_p\rangle=
\sum_{k,j} \langle j|\alpha_q\rangle \langle j|k\rangle \langle\alpha_p|k\rangle=\langle\alpha_p|\alpha_q\rangle=\delta_{pq}\ .
\end{eqnarray}
The outcomes of a subsequent measurement on the second subsystem in the basis $\ket{\tilde\alpha_p}$ can therefore be predicted with certainty, although they were completely undetermined before the prior measurement on the first subsystem.
If the same is attempted with the state (\ref{sepmixed}), the first measurement projects the state on
\begin{eqnarray}
\ket{\alpha_p}\bra{\alpha_p}\otimes \left(\sum_{j=1}^d\left|\langle\alpha_p\right|j\rangle|^2\ket{j}\bra{j}\right)\ .
\end{eqnarray}
In other words, the second subsystem is left in a mixed state, with incoherently added components $\ket{j}\bra{j}$. Unless the basis $\{\ket{\alpha_p}\}$ coincides with the basis $\{\ket{j}\}$, results of a projective measurement on this second system component remain uncertain.
In conclusion, a mixed state such as (\ref{sepmixed}) can only explain correlations that are observed in \emph{one specific} single-particle basis, whereas the entangled state (\ref{eq:psimm}) exhibits correlations for \emph{all} possible choices of orthogonal local basis settings.
We therefore see that quantum physics hosts a type of correlations which cannot be classically described. These correlations are often referred to as \emph{quantum correlations} and understood as the ultimate physical manifestation of ``entanglement''.
\subsection{Separable and entangled states} \label{sepaent}
In order to assign quantum states the label ``entanglement'' in a systematic fashion, we first need to define this new quality on a formal level, to subsequently introduce measures for the amount of entanglement that is carried by a quantum state.
\subsubsection{Pure states}
A bipartite quantum state \begin{eqnarray} \ket{\Phi} \in \mathscr{H}_1 \otimes \mathscr{H}_2 \end{eqnarray} is \emph{separable} if it can be written as a product state, {\it i.e.}~if one can find single particle states $\ket{\phi_i} \in \mathscr{H}_i$ such that \begin{eqnarray} \ket{\Phi}=\ket{\phi_1} \otimes \ket{\phi_2}. \label{separable} \end{eqnarray}
Separable states are completely determined by the single-particle states
$\ket{\phi_1}$ and $\ket{\phi_2}$, which contain all information on possible measurement outcomes.
Unlike the situation described by (\ref{eq:psimm}), a measurement performed on one subsystem has no effect on the other subsystem, {\it i.e.}~the subsystems are uncorrelated. Consequently, the \emph{reduced density matrices},
\begin{eqnarray}
\varrho_1=\mathrm{Tr}_2\left( \ket{\Phi}\bra{\Phi} \right)\hspace{1cm}\mathrm{and}\hspace{1cm}\varrho_2=\mathrm{Tr}_1 \left( \ket{\Phi}\bra{\Phi} \right) \ ,\label{eq:reddmat}
\end{eqnarray}
where
\begin{eqnarray} \mathrm{Tr}_{1(2)}(\rho) = \sum_{j=1}^d \bra{\chi_j}_{1(2)} \rho \ket{\chi_j}_{1(2)} , \label{partialtrace} \end{eqnarray} denotes the partial trace over the first (second) subsystem, describe pure states, and
\begin{eqnarray} \varrho_1=\ket{\phi_1} \bra{\phi_1} , \hspace{1cm} \varrho_2=\ket{\phi_2} \bra{\phi_2} , \end{eqnarray}
such that the compound state can be written as a tensor product, $ \ket{\Phi}\bra{\Phi}=\varrho_1 \otimes \varrho_2 . $ This is equivalent to the statement that the first particle is prepared in $\ket{\phi_1}$, while the second particle is prepared in $\ket{\phi_2}$, which corresponds to the assignment of a \emph{physical reality}, as will be discussed in more detail in Section \ref{properties} below.
A state that is not separable is called \emph{entangled}.
The information carried by an \emph{entangled} state $\ket{\Psi}$, which cannot be written as tensor product as in (\ref{separable}), is not completely specified in terms of the states of the subsystems: For the reduced density matrices, we have $\mathrm{Tr}{\varrho_{1/2}^2}\neq 1$, {\it i.e.}~the subsystems' states are mixed. Furthermore, one finds $\ket{\Psi}\bra{\Psi}\neq\varrho_1\otimes\varrho_2$, {\it i.e.}~the two-particle state $\ket{\Psi}$ contains more information on measurement outcomes than is contained in the two single-particle states $\varrho_1$ and $\varrho_2$ together, in contrast to the above separable state (\ref{separable}).
Moreover, distinct entangled states can give rise to the same reduced density matrices: The states $\ket{\Phi^{+}}=\left( \ket{1,1}+\ket{0,0} \right)/\sqrt{2}$ and $\ket{\Psi^{+}}=\left( \ket{0,1}+\ket{1,0} \right)/\sqrt{2}$ lead to the same, maximally mixed, reduced density matrices, $\varrho_{1}=\varrho_{2}=\mathbbm{1}_2/2$, where $\mathbbm{1}_l$ denotes the identity in $l$ dimensions. The state $\ket{\Phi^{+}}$, however, describes a \emph{correlated} pair, whereas $\ket{\Psi^+}$ is \emph{anti-correlated.}
Formally, the attribute of separability (and, correspondingly, entanglement) boils down to the question whether the coefficient matrix $c_{j,k}$ in the state representation
\begin{eqnarray} \ket{\Psi}=\sum_{j,k=1}^d c_{j,k} \ket{j,k} , \label{generalrepresent} \end{eqnarray}
admits a product representation, {\it i.e.}~$c_{j,k}=c^{(1)}_j \cdot c^{(2)}_k$
-- in this case, the state is separable.
\subsubsection{Mixed states}
Separable and entangled states can also be defined for mixed states of a bipartite quantum system, which have to be described in terms of density matrices. In this case, the mixedness of the reduced density matrices is not equivalent to entanglement.
A state that can be expressed as a tensor product of single-body density matrices,
\begin{eqnarray}
\rho_p=\varrho_1\otimes\varrho_2\ , \label{productstate}
\end{eqnarray}
bears no correlations between local measurement results at all, and is called a \emph{product} state.
\emph{Separable} states are defined by sets of single particle states $\varrho_1^{(i)}$ and $\varrho_2^{(i)}$ of the first and second subsystem, respectively, and by associated probabilities $p_i$ ({\it i.e.}~$p_i\ge 0$, and $\sum_ip_i=1$), such that \cite{Werner:1989ve}
\begin{eqnarray}
\rho_s=\sum_i\ p_i\ \varrho_1^{(i)}\otimes\varrho_2^{(i)}\ .\label{separho}
\end{eqnarray}
Such separable states imply correlations between measurement results on the different subsystems, but these correlations can be explained in terms of the probabilities $p_i$, and, therefore, do not qualify as quantum correlations.
The tag \emph{entanglement} is, thus, reserved for those states that \emph{cannot} be described in terms of product states of single-particle states and of classical probabilities as in (\ref{separho}). They thus need to be described as
\begin{eqnarray}
\rho_{\mathrm{ent}} = \sum_j p_j \ket{\Psi_j}\bra{\Psi_j} \label{nonsepdd}
,\end{eqnarray}
where, in \emph{any} pure-state decomposition of $\varrho_{\mathrm{ent}}$, at least one state $\ket{\Psi_j}$ is entangled.
\subsection{Bell inequality violation and nonlocality} \label{Bellin}
So far, we identified the difference between classical and quantum correlations with the help of the exemplary states given by Eqs.~(\ref{eq:psimm}) and (\ref{sepmixed}), and we have given formal definitions for entangled and separable states. On the other hand, we still need to establish how to unambiguously identify and quantify the exceptional, ``non-classical'', correlations inscribed in (\ref{eq:psimm}) in an experimental setting. This is required to provide a connection to probability theory and to provide a benchmark for the verification of entanglement in an experiment.
For this purpose, we first need to specify what is understood as ``classical'' in our present context. We call a theory ``classical'' if it is local and realistic. The principle of \emph{locality} states that any object is influenced directly only by its immediate surroundings, and that there can be no signals between space-like separated events. \emph{Realism} denotes the assumption that any physical system possesses intrinsic properties, {\it i.e.}~an experimentalist who measures the value of an observable merely reads off a predefined value \cite{PhysRev.47.777} (as, {\it e.g.}, for the mixed state (\ref{sepmixed}) that is created by a random, however, realistic mechanism). That is to say, although one may ignore the value of a certain observable, each observable still possesses a definite value at any moment. In practice, ``local realism'' implies that measurement outcomes at one subsystem are independent of the measurements performed on the other subsystem, provided both subunits are spatially separated. Theories that obey local realism can be described by classical random variables.
A rigorous way to establish a correspondence between ``non-classical'' correlations and entanglement is provided by \emph{Bell inequalities} \cite{Bell:1964pt}. These are defined in terms of correlations between measurement results of different single-partice observables, and they give threshold values for the correlations to be describable in terms of classical probability theory. An experimental violation, {\it i.e.}~any excess beyond the threshold value, indicates that the corresponding observables cannot be described as classical random variables. The community jargon also speaks of the unavailability of a ``local realistic description''.
A widely used Bell inequality is the one presented by Clauser, Horne, Shimony and Holt (CHSH) \cite{Clauser:1969qa}. It is formulated for \emph{dichotomic observables} such as polarization, {\it i.e.}~observables which only take the values $+1$ and $-1$, and reads
\begin{eqnarray} | \langle A_1 \cdot B_1 \rangle + \langle A_1 \cdot B_2 \rangle + \langle A_2 \cdot B_1 \rangle - \langle A_2 \cdot B_2 \rangle | \leq 2 \label{CHSH}, \end{eqnarray}
where $A_j$ and $B_k$ are different observables acting on the two subsystems. This inequality is derived under the assumption of local realism, which means that measurement outcomes at one subsystem are independent of the measurements performed on the other subsystem, provided both subunits are spatially separated. In order to violate the inequality, the local observables need to be non-commuting, $[A_1,A_2] \neq 0 \neq [B_1,B_2]$. It is hence necessary to implement a rotation of the measurement basis to assess such non-commuting observables. This is also illustrated in the polarization rotation effectuated on the photons in Figure \ref{Bellillu}:
\begin{figure}[h] \center \includegraphics[width=12.5cm,angle=0]{fig1.pdf} \caption{Bell-type experiment. A source creates photons in an entangled state, here the maximally entangled $\ket{\Psi^+}$ Bell state (\ref{Psimp}). The polarization of the photons is rotated at polA and polB, where at least two different settings, $\hat A_{1}/\hat A_2$ and $\hat B_{1}/\hat B_2$, are available. The photons then fall onto polarizing beam splitters (PBS) where horizontally polarized photons are reflected and vertically polarized ones are transmitted. The subsequent detection of the photons at either one of the two detectors yields the distinct values $\pm 1$.} \label{Bellillu} \end{eqnarray}d{figure}
In a polarization-entanglement experiment, it is not sufficient to measure the correlations in a given basis, say the $z$-basis, but the orientation of the quantization axis needs to be chosen locally. Similarly, correlations in the position or in the momentum alone \emph{do not} rigorously prove that a two-particle state is entangled in the external degrees of freedom (see also the discussion in Sections \ref{atomphotonent},\ref{twoelectronsmol}). The difficulty of the implementation of mutually distinct measurement bases is an impediment for the direct assessment of the entanglement properties of many naturally occurring, multicomponent quantum systems such as atoms \cite{Grobe:1994fk,Fedorov:2004kx}, or biological structures \cite{Arndt:2009uq}. It is then necessary -- and still largely open an issue -- to conceive alternative indicators that distinguish entanglement from classical correlations \cite{PhysRevLett.106.210501}.
For a \emph{maximally entangled bipartite qubit} (or \emph{Bell-}) \emph{state} as defined below in (\ref{Psimp}), (\ref{Phimp}) below, the expectation value of the left hand side of (\ref{CHSH}) can reach values up to $2 \sqrt 2\approx 2.82$, for a suitable choice of the measurement settings $A_j$, $B_k$. When such experimental conditions are met, the notion of non-locality as defined by Bell inequalities is qualitatively in agreement with the definition of separability and entanglement in Section \ref{sepaent} above: A violation of a Bell inequality proves that the state under consideration is entangled. The reverse is, however, not true: There are states which are entangled according to Section \ref{sepaent}, but do not violate any Bell inequality \cite{PhysRevLett.72.797}, since a description in terms of classical probabilities is available \emph{despite their non-separbility according to} (\ref{nonsepdd}) \cite{Werner:1989ve}. An example is given by the Werner-state $\rho_{\mathrm{W}}(p)$ \cite{Werner:1989ve}. For bipartite qubit systems, it reads
\begin{eqnarray}
\rho_{\mathrm{W}}(p)=(1-p) \ket{\Psi^-}\bra{\Psi^-} + \frac{p}{4} \mathbbm{1}_4 , \label{WernerState}
\end{eqnarray}
where $p\in [0,1]$, {\it i.e.}~the state is a mixture of the maximally entangled antisymmetric state $\ket{\Psi^-}$ given below in (\ref{Psimp}) and the maximally mixed, fully uncorrelated state $\mathbbm{1}_4/4= \mathbbm{1}_2/2 \otimes \mathbbm{1}_2/2$ (see (\ref{productstate})). The entanglement and the non-local properties of the state depend on the parameter $p$: $\rho_{\mathrm{W}}(0)$ is a pure, maximally entangled state, which also violates (\ref{CHSH}) maximally. This violation persists for $0<p<1-1/\sqrt{2}$. For $p\ge 1-1/\sqrt{2}$, a local realistic description is available, but the state is still entangled as long as $p<2/3$. In other words, for $1-1/\sqrt{2}<p<2/3$, the state is entangled according to (\ref{nonsepdd}), but it does not violate local realism. This enforced qualitative distinction between non-locality and entanglement needs to be uphold for mixed states, whereas for pure bipartite qubit states, any non-product state violates a Bell inequality \cite{Gisin1991201}.
When we conclude from the violation of a Bell inequality that no local realistic description of a given experiment exists, we rely, among others, on the strong assumptions that the measurements which are performed on the subsystems are space-like separated \cite{PhysRevLett.81.5039}, and that possible detector inefficiencies are unbiased with respect to the measurement outcomes \cite{PhysRevD.2.1418,PhysRevD.35.3831}. These requirements represent serious challenges for experiments, such that tests of Bell inequalities are often plagued by \emph{loopholes}: The failure to fulfill the aforementioned assumptions may allow a description of the experiment outcomes by classical theories, and additional experimental effort is required to ``close the loopholes'' \cite{PhysRevLett.93.130409,Rowe:2001vn,Merali18032011}.
\subsection{Entanglement witnesses} \label{witnessesslabl}
Since the above discussion implies that non-locality is a stronger criterion than entanglement, Bell inequalities are not a universal means to experimentally detect quantum correlations. A general method to verify entanglement is given by
\emph{entanglement witnesses} \cite{Horodecki:1996vn,Terhal:2002fk}, {\it i.e.}~operators which detect, or ``witness'', entanglement \cite{Barbieri:2003kx,Bourennane:2004bh}. A hermitian operator $\hat W$ is an entanglement witness if it fulfills
\begin{eqnarray}
\mathrm{Tr} (\rho_s \hat W ) \ge 0 \label{eq:witness_pure},\end{eqnarray}
for all separable states $\rho_s$, and
\begin{eqnarray}
\mathrm{Tr} (\rho_e \hat W ) < 0 \label{eq:witness_purev},\end{eqnarray}
for at least one entangled state $\rho_e$.
Consequently, any quantum state $\rho$ for which Tr$(\rho \hat W)<0$, {\it i.e.}~which yields a negative expectation value of the witness in the experiment, is thereby verified to be entangled. Witnesses are universal in the sense that one can find a suitable witness for any entangled state. Bell inequalities can be seen as a specific class of entanglement witnesses \cite{Hyllus:2005dq}, which only detect states that violate local realism. We summarize the concept of separable and entangled, pure and mixed states in Figure \ref{witnessfig}.
\begin{figure}[h] \center \includegraphics[width=8.5cm,angle=0]{fig2.pdf} \caption{Illustration of the structure of the set of density matrices and of the concept of entanglement witnesses. Pure states lie on the surface of the convex set of density matrices; all states that are not on the surface are mixed. Separable states are a subset which is itself convex. The set of entangled states is shaded in gray, and not convex. In this geometrical picture, an entanglement witness defines a hyperplane (dotted line): States above that plane are detected to be entangled, while states below that plane cannot be unambiguously classified by this specific choice of witness/hyperplane. Adapted from \cite{RevModPhys.81.865}.} \label{witnessfig} \end{eqnarray}d{figure}
\subsection{Local operations and classical communication} \label{measures}
The distinction between separable and entangled states has attracted substantial interest both from the theoretical \cite{Peres:1996ys,Horodecki:1997zr,Horodecki:1996vn,Lewenstein:1998ve,Alcaraz:2002ij} and from the experimental side \cite{Kampermann:2010ly,Barbieri:2004qf} (see, {\it e.g.}, \cite{Ghne:2009ys}, for a recent review), but this distinction alone is largely insufficient to draw a complete picture of the physics of entanglement.
The next step towards a deeper understanding is the ability to compare the entanglement content of different states. Given the rather abstract nature of entanglement, it is, however, not obvious how such comparison should work.
By now, some consensus has been reached in the literature that the concept of {\em local operations and classical communication} (LOCC) provides an appropriate framework \cite{Nielsen:1999uq}.
\emph{Local operations} include all manipulations that are allowed by the laws of quantum mechanics -- including measurements, coherent driving, interactions with auxiliary degrees of freedom -- under the condition that they are restricted to either one of the individual subsystems. Operations that require an interaction between the subsystems do not fall into this class, and quantum correlations thus cannot be generated by local operations.
Only classical communication is here admissible to create correlations: The result of a measurement on one subsystem can be communicated to a receiver which is ready to execute a local operation on the other subsystem, and this subsequent operation can be conditioned on the prior measurement result.
One can thereby indeed induce correlations: For example, the exemplary state (\ref{sepmixed}) could have been prepared by randomly preparing one of the basis states $\ket{i}$ of the first subsystem, and subsequent communication of the choice of $i$ to a receiver which controls the second subsystem.
If this person prepares its subsystem in the same state, then many repetitions of this procedure yield (\ref{sepmixed}). It is, however, impossible to create an entangled state through the application of LOCC, starting out from an initially separable state.
Therefore, it is justified to consider a state $\rho_2$ equally or less entangled than a state $\rho_1$,
if $\rho_2$ can be obtained from $\rho_1$ through the application of LOCC.
\subsubsection{Maximally entangled states} \label{maxienttt}
This immediately implies the notion of a \emph{maximally entangled state}, from which any other state can be generated through LOCC: For bipartite qubit states, the four Bell states
\begin{eqnarray}
\ket{\Psi^\pm} = \frac{1}{\sqrt 2} \left( \ket{0,1}\pm \ket{1,0} \right) \label{Psimp} ,\\
\ket{\Phi^\pm} = \frac{1}{\sqrt 2} \left( \ket{0,0}\pm \ket{1,1} \right) \label{Phimp},
\end{eqnarray}
are maximally entangled, since any bi-qubit state can be realized through LOCC applied to either one of them \cite{Nielsen:2000fk,Kaye:2007uq}.
Also on higher dimensional subsystems such states can be constructed, and, indeed, are precisely of the form (\ref{eq:psimm}).
The application of suitable LOCC allows to convert (\ref{eq:psimm}) into arbitrary bipartite $d$-level states.
If, however, the number of system components is increased,
the concept of a maximally entangled state cannot be generalized unambiguously.
The most illustrative example is that of the tripartite Greenberger-Horne-Zeilinger (GHZ) state
\begin{eqnarray} \ket{GHZ}=\frac{1}{\sqrt 2}\left(\ket{0,0,0} + \ket{1,1,1} \right) \label{GHZstate} , \end{eqnarray}
on the one hand, and of the W-state
\begin{eqnarray} \ket{W}=\frac{1}{\sqrt 3}\left( \ket{0,0,1}+ \ket{0,1,0} + \ket{1,0,0} \right)\ , \label{Wstate} \end{eqnarray} on the other.
Both states are strongly entangled, and are certainly candidates to qualify as maximally entangled.
Though, neither can a $\ket{W}$ state be obtained through the application of LOCC on a $\ket{GHZ}$ state, nor is the inverse possible \cite{Duerr:2000pj}. Consequently, since there is no state in a tri-qubit system from which all other states can be obtained, one has to get acquainted with the idea that there is no unique maximally entangled state in multi-partite systems, but that there are inequivalent classes, or families, of entangled states. The classification of these is subject to active research (see \cite{RevModPhys.81.865} for an overview, and \cite{Hiesmayr:2008dz,Huber:2010kx,PhysRevLett.104.020504,Salwey:2010fk} for recent results), and still far from being accomplished.
\subsection{Entanglement quantification} \label{quantif}
In order to define a \emph{measure} of entanglement, we need to specify under which conditions two quantum states can be regarded as equivalent, {\it i.e.}~when they carry the same amount of entanglement. This can be certified if the states are related to each other via invertible LOCC, {\it i.e.}~via unitary operations that are applied locally and independently on the subsystems:
\begin{eqnarray} \ket{\Phi_1} = {\cal U}_{\mathrm{local}} \ket{\Phi_2} = {\cal U}_1 \otimes {\cal U}_2 \otimes \dots \otimes {\cal U}_N \ket{\Phi_2} , \nonumber \\
\ket{\Phi_2} = {\cal U}_{\mathrm{local}}^\dagger \ket{\Phi_1} = {\cal U}_1^\dagger \otimes {\cal U}_2^\dagger \otimes \dots \otimes {\cal U}_N^\dagger \ket{\Phi_1} , \end{eqnarray}
where local unitaries $ {\cal U}_{\mathrm{local}}= {\cal U}_1 \otimes {\cal U}_2 \otimes \dots {\cal U}_N$ are induced by single-particle Hamiltonians $ H_1, \dots H_N$, such that (with the convention $\hbar=1$)
\begin{eqnarray}
{\cal U}_l={\cal U}_1\otimes{\cal U}_2 \otimes \dots \otimes {\cal U}_N =e^{-i\tau(H_1\otimes{\mathbbm 1}\otimes\dots \otimes {\mathbbm 1}+{\mathbbm 1}\otimes H_2\otimes\dots \otimes {\mathbbm 1} + \dots )}\ . \label{hamilnointeraction}
\end{eqnarray}
This equivalence relation is motivated by the fact that the parties which are in possession of the subsystems can perform such local invertible operations without any mutual communication or other infrastructure. Any entanglement quantifier, therefore, ought to be independent of such local unitaries.
\subsubsection{Entanglement monotones for bipartite pure states}
In the particular case of a pure state of a bipartite system, the invariants under local unitaries are precisely given by the state's \emph{Schmidt coefficients} $\lambda_j$, which are the squared weights of the state's \emph{Schmidt decomposition}
\begin{eqnarray} \ket{\Psi}=\sum_{i=1}^s \sqrt{\lambda_i} \ket{\tilde \phi_{1,i}} \otimes \ket{\tilde \phi_{2,i}}\ \label{SchmidtDecomp} , \end{eqnarray}
with the characteristic trait that one summation index suffices, in contrast to a representation of $\ket{\Psi}$ in arbitrary basis sets on $\mathscr{H}_1$ and $\mathscr{H}_2$, as in (\ref{generalrepresent}). The Schmidt coefficients coincide with the eigenvalues of the reduced density matrices (\ref{eq:reddmat}), which can be deduced from the specific form of (\ref{SchmidtDecomp}): The partial trace (\ref{partialtrace}) on either subsystem directly yields \begin{eqnarray}\varrho_1=\sum_j\lambda_j \ket{\tilde \phi_{1,j}} \bra{\tilde \phi_{1,j}}, \hspace{.71cm}\mathrm{and}\hspace{.71cm}\varrho_2=\sum_j\lambda_j \ket{\tilde \phi_{2,j}} \bra{\tilde \phi_{2,j}} \ . \label{reducedden} \end{eqnarray} Since the states $\{\ket{\tilde \phi_{1,j}}\}$ and $\{\ket{\tilde \phi_{2,j}}\}$ form orthogonal bases, respectively, they are indeed the eigenstates of $\varrho_1$ and $\varrho_2$, and the $\lambda_j$ are the corresponding eigenvalues.
The $\lambda_j$ fully determine the entanglement of $\ket{\Psi}$, and functions $M(\ket{\Psi})$ of the $\lambda_j$ that are non-increasing under LOCC are called \emph{entanglement monotones} \cite{Vidal:2000kx}, which quantify the state's entanglement. Under some additional requirements \cite{Vedral:1997uq,Plenio:2006uq} beyond the scope of our present discussion, entanglement monotones are also called \emph{entanglement measures}. The following entanglement monotones for bipartite systems will appear in the course of this review:
\begin{itemize}
\item The \emph{Schmidt rank} $s$ \cite{Terhal:2000uq}, which is the number of non-vanishing \emph{Schmidt coefficients} $\lambda_j$ in the expansion (\ref{SchmidtDecomp}). It ranges from unity (separable) to $d$. The matrix rank of the reduced density matrix of either subsystem equals $s$ (see (\ref{reducedden})).
\item The \emph{Schmidt number} \cite{Grobe:1994fk}, \begin{eqnarray} K = \frac{1}{\sum_{i=1}^s \lambda_i^2} =\frac{1}{\mathrm{Tr}(\varrho_1^2) }\ , \label{Schmidtnumber}\end{eqnarray}
which estimates the number of states involved in the Schmidt decomposition. It can also be seen as the inverse participation ratio, and ranges from unity for separable states to $d$, with $d$ the dimension of the subsystems.
\item The \emph{concurrence} \cite{Wootters:1998fk,PhysRevA.64.042315}, defined by
\begin{eqnarray} C(\ket{\Psi})= \sqrt{\frac{d}{d-1}\left(1-\sum_{j=1}^d \lambda_j^2 \right) }\ , \label{concurrence} \end{eqnarray}
is directly related to the \emph{linear entropy} of a probability distribution defined by the weights $\lambda_j$,
\begin{eqnarray}
S_{\mathrm{lin}}(\{ \lambda_j \})=1-\sum_{j=1}^d \lambda_j^2 . \end{eqnarray}
\item[-] The \emph{entanglement of formation} or \emph{entanglement entropy} \cite{Bennett:1996fk}, is the von Neumann-entropy \cite{von-Neumann:1955ys} of one of the reduced density matrices $\varrho_j$, \begin{eqnarray} E(\ket{\Psi})=S(\varrho_j)=-\mathrm{Tr}( \varrho_j ~ \mathrm{Log}_2(\varrho_j) ) , \label{eoff} \end{eqnarray}
which, given the spectral decomposition of $\varrho_{j}$, boils down to the Shannon entropy \cite{Shannon:1948fk} of a probability distribution defined by the Schmidt coefficients $\lambda_j$,
\begin{eqnarray} E(\ket{\Psi})= H(\{ \lambda_1 \dots \lambda_j \}) = -
\sum_j \lambda_j \mathrm{Log}_2(\lambda_j) .\end{eqnarray}
\end{eqnarray}d{itemize}
Since all these quantities are given in terms of the Schmidt coefficients $\lambda_j$, they are readily evaluated for arbitrary bipartite pure states.
In addition, the Schmidt coefficients allow us to decide whether a state $\ket{\Phi}$ can be prepared deterministically from an initially given state $\ket{\Psi}$ via LOCC: This is possible if and only if the Schmidt coefficients of $\ket{\Phi}$ are \emph{majorized} by the Schmidt coefficients of $\ket{\Psi}$ \cite{Nielsen:1999uq}. Majorization is defined by
\begin{eqnarray} \forall k, 1\le k \le d: \sum_{j=1}^k \lambda^\Phi_j \le \sum_{j=1}^k \lambda^{\Psi}_j ,\end{eqnarray}
where the $\lambda^\alpha_j$ are the entries of the Schmidt vectors $\vec \lambda^\alpha$, $\alpha=\Phi,\Psi$, sorted in increasing order.
Consistently with our definition of maximally entangled states (see Section \ref{maxienttt}), maximally entangled states as the one given by (\ref{eq:psimm}) majorize any other less entangled state.
\subsubsection{Entanglement monotones for mixed states}
It is more difficult to evaluate the entanglement content of a mixed state given by its pure state decomposition $\rho=\sum_jp_j\ket{\Psi_j}\bra{\Psi_j}$. Since this decomposition is not unique \cite{Hughston:1993ly}, a simple average over its pure states' entanglement $M(\ket{\Psi_j})$, with weights $p_j$, does not provide an unambiguous result (also see \cite{Nha:2004ve,Viviescas:2010qf,Carvalho:2007bh}). The problem is cured by taking the infimum over all pure-state decompositions \cite{Bennett:1996kx},
\begin{eqnarray} M(\rho)= \mathrm{inf}_{\left\{ \Psi_j, p_j \right\} } \sum_j p_j M(\ket{\Psi_j}) , \label{purestatedecomp} \end{eqnarray}
which, indeed, implies a variation over states $\ket{\Psi_j}$ \emph{and} weights $p_j$, since the cardinality of the sum is itself variable. The thus defined \emph{mixed state entanglement monotone} guaranties in particular that $M(\rho)$ vanishes on the separable states.
A closed formula is available for the concurrence (\ref{concurrence}) of a mixed two-qubit system $\rho$ \cite{Hill:1997eu,Wootters:1998fk},
\begin{eqnarray}
C(\rho)=\mathrm{max}\left\{ 0, \lambda_1 - \lambda_2 - \lambda_3 - \lambda_4 \right\} , \label{concurrencemixedstate}
\end{eqnarray}
where the $\lambda_j$ are the eigenvalues of the matrix
\begin{eqnarray} R= \sqrt{\sqrt{\rho} (\sigma_y \otimes \sigma_y) \rho^\star (\sigma_y \otimes \sigma_y) \sqrt{\rho}}~ , \end{eqnarray}
in decreasing order. In practice, the optimization problem implicit in (\ref{purestatedecomp}) renders its quantitative evaluation a challenging task for larger systems, beyond two qubits, and there is only limited insight and literature on approximations and rigorous bounds \cite{Mintert:2007ys,Borras:2009zr,Mintert:2005rc,Ma:2010vn,Ghne:2009ys,Guhne:2008zr,Gao:2006ly}.
For systems with more than two subunits, no straightforward generalization of the Schmidt decomposition is available. Other concepts of entanglement measures have thus been designed, which can be generalized to such \emph{multipartite} states, {\it i.e.}~states with more than two subsystems. One example is the distance to the set of separable states \cite{Bengtsson:2006fu}, \begin{eqnarray} E_D(\rho) = \mathrm{min}_{\sigma \in S}D(\rho,\sigma) , \label{metricdistanceent} \end{eqnarray} where $D(\rho,\sigma)$ is a distance measure between two states \cite{Reed:1980ly}, and the minimum is taken over all states $\sigma$ within the set of separable states $S$. This quantity possesses a straightforward geometrical interpretation, illustrated in Figure \ref{metricdistance}.
\begin{figure}[h] \center \includegraphics[width=5.5cm,angle=0]{fig3.pdf} \caption{Illustration of the concept of the metric distance entanglement measure $E_D$, defined in (\ref{metricdistanceent}): The separable state $\sigma$ is chosen such that the metric distance to the entangled state of interest is minimal. } \label{metricdistance} \end{eqnarray}d{figure}
The evaluation of this and any other multipartite measure for mixed, multipartite states is, in general, computationally demanding, due to the reasons described above.
\subsection{Modeling physical systems}
\subsubsection{Subsystem structures} \label{substruct}
In the modeling of entanglement inscribed into real physical systems, it is natural to ask how to partition the Hilbert space, {\it i.e.}~how to choose the subsystem structure on which entanglement is defined. This choice can be largely variable \cite{Zanardi:2004fk}: Consider, {\it e.g.}, a 16-dimensional space, $\mathscr{H}=\mathbbm{C}^{16}$. This can be seen as the tensor product of four Hilbert spaces that represent a two-level system each ($16=2^4, \mathbbm{C}^{16}=\mathbbm{C}^2\otimes \mathbbm{C}^2\otimes \mathbbm{C}^2\otimes \mathbbm{C}^2$, Figure \ref{partitions}a), or as
the tensor product of two Hilbert spaces that each represent a particle with four discrete eigenstates ($16=4^2, \mathbbm{C}^{16}=\mathbbm{C}^4 \otimes \mathbbm{C}^4$, see Figure \ref{partitions}b), or as one, indivisible, Hilbert space of dimension 16 (Figure \ref{partitions}c). The specific physical situation is then clearly distinct, as evident from the illustrations.
\begin{figure}[h] \center \includegraphics[width=12.5cm,angle=0]{fig4.pdf} \caption{Illustration of possible subsystem-structures of $\mathscr{H}=\mathbbm{C}^{16}$. } \label{partitions} \end{eqnarray}d{figure}
In general, a natural partition is induced by the definition of the system degrees of freedom: A 16-dimensional Hilbert space is spanned by two four-level atoms, a four-ion quantum register where each ion bears two computational levels, or, {\it e.g.}, by a quantized single mode resonator field populated by maximally 15 photons.
The very definition of the system degrees of freedom thus also affects the expected entanglement between them. Consider, {\it e.g.}, the hydrogen atom, as an example for a continuous variable system with infinite-dimensional subsystem dimension: When partitioned into center-of-mass degree of freedom and relative coordinates, both degrees of freedom separate completely, and no entanglement is exhibited. In contrast, if we choose the partition into electron- and proton-coordinate, the subunits remain coupled, and exhibit non-vanishing entanglement \cite{Tommasini:1998qf}.
Very distinct entanglement properties can thus be ascribed to the same physical system \cite{Barnum:2004wc,Barnum:2005xz,Zanardi:2004fk,Zanardi:2001kx} -- according to the choice of the system partition. As long as the partition is itself invariant under the system dynamics, it can be defined a priori once and forever, while the situation may turn more complicated if this condition is not fulfilled -- {\it e.g.} for identical particles that are scattered by a beam splitter (see Section \ref{entextr}), or in electrons bound by atoms (see Section \ref{elecelec})).
\subsubsection{Superselection rules} \label{secSSR}
Even when given a fixed subsystem structure, constraints are possible on realizable operations and measurements on the system, which also restrict the verifiable or exploitable entanglement. The impossibility to measure coherent superpositions of eigenstates of certain operators, either due to fundamental reasons like, {\it e.g.}, charge conservation, or as a consequence of the lack of a shared reference frame that prevents the experimentalists to gauge their measurement instruments, is described by the formulation of \emph{superselection rules} (SSR) \cite{PhysRev.88.101,Greenberger:2009uq,Bartlett:2007bf}. Such rules strongly restrict the nature and outcome of possible measurements. In contrast to selection rules which give statements on the system evolution generated by some Hamilton with certain symmetries, SSRs postulate a much more strict behavior. Two states $\ket{\Psi_1}$ and $\ket{\Psi_2}$ are said to be \emph{separated by a SSR} if for \emph{any} physically realizable observable $ A$ (and not just for a specific Hamiltonian),
\begin{eqnarray} \bra{\Psi_1} A \ket{\Psi_2} =0 .\end{eqnarray}
It is important to note that SSR are not equivalent to conservation laws and also do not restrict the accessible states of a system. Instead, a SSR is a postulated statement on the \emph{physical realizability of operators} \cite{Bartlett:1991ys}. For example, the direct implementation of an operator which projects on a coherent superposition of an electron and a proton is impossible.
When a SSR applies, the executable operations and measurements are restricted, and the entanglement of a system possibly cannot be fully accessed. Entanglement constrained by SSR is indeed bounded from above by the entanglement evaluated according to the rules of the above sections \cite{Bartlett:1991ys,Schuch:2004kx,Schuch:2004xl}. A SSR that prevents a local basis rotation, {\it e.g.}, may enforce that the entanglement which is formally present in a state effectively reduces to a classical correlation. Suppose two parties share the state $\left(\ket{e,p}+\ket{p,e} \right)/\sqrt{2}$ where $\ket{e}$ and $\ket{p}$ indicate the wave-function of an electron and a proton, respectively. While this state is formally entangled, it is effectively impossible to measure a coherent superposition of an electron and a proton. Hence, measurements in other bases, {\it e.g.} in the basis $\left(\ket{e}\pm\ket{p}\right)/\sqrt{2}$, are unfeasible, and no \emph{quantum} correlations can be verified. Consequently, the very classification of pure entangled states becomes more complex due to the impossibility of performing certain measurements \cite{Bartlett:2006gb}.
Effective SSR do not only appear due to fundamental conservation rules, but also when two parties in possession of a compound quantum state do not share a perfect reference frame \cite{Bartlett:2007bf,Enk:2006hc}, {\it i.e.}~a convention on time, phase, and spatial directions. This connection between SSRs and reference frames can be seen in the following example: Suppose one party, A, prepares a quantum state $\rho$ and sends it to another party, B. The reference frames of A and B are related to each other by a transformation $T(g)$, where $g$ is an element of the transformation group, and $T(g)$ is its unitary realization. If, however, B does not know the relative orientation of the individual reference frames, the state B will effectively have access to is the one send by A, but averaged over all possible transformations. Such effectively accessible quantum state for B reads
\begin{eqnarray} \bar \rho = \int \mathrm{d} g ~T(g) \rho T^{-1}(g) \begin{eqnarray}uiv G[\rho] ,\end{eqnarray}
where the integral is taken over all elements of the group, and $G$ is baptized the \emph{twirling operation} \cite{Bartlett:2007bf}. In general, $G$ is not injective,\footnote{In other words, $G$ does not preserve distinctness: There can be two distinct states $\rho_1\neq \rho_2$ for which $G[\rho_1]=G[\rho_2]$.} hence the accessible space becomes smaller under the twirling operation. Only quantum states that are fully invariant under arbitrary transformations, {\it i.e.}~states $ \rho$ with $[ \rho, T(g)]=0$ for any $g$, can be distinguishable building blocks, and only operations that commute with all transformations $g$ can be reliably performed on the subsystems \cite{Bartlett:1991ys}. The transformation group can, {\it e.g.}, represent spatial rotations and generate invariance under $O(3)$, as illustrated in Figure \ref{referenceframe}: If party A sends a particle in the spin-up state, $\ket{\uparrow}$, to party B, the latter has access to this pure state only when A and B share a convention regarding the spatial quantization axis, {\it i.e.}~the axis in space with respect to which the state has been prepared.
\begin{figure}[h] \center \includegraphics[width=8.5cm,angle=0]{fig5.pdf} \caption{Transmission of the quantum state $\ket{\uparrow}$ between two parties which share a spatial reference frame (upper panel), or do not (lower panel). The party B has access only to an unbiased mixture when no such convention is available. } \label{referenceframe} \end{eqnarray}d{figure}
Without such convention, B will observe a fully mixed state and cannot extract any information. In this case, B cannot distinguish any single-particle state by its spin, because all are equivalent to the completely unbiased mixture $G[\ket{\uparrow}]=\mathbbm{1}_2/2$.
On the other hand, given a fundamental SSR generated by a compact group,\footnote{In other words, a group whose elements are elements of a compact set, in contrast to, {\it e.g.} the Lorentz or the Poincar\'e group.} like the very $U(1)$ group which describes charge conservation \cite{Aharonov:1967ys}, the possibility to access a shared reference frame in which the underlying symmetry is broken can actually enable the preparation and measurement of superposition states \cite{Bartlett:2007bf}, and thereby to circumvent the SSR. The indirect observation of such coherences may be possible with the help of ancilla\footnote{\label{ancillad} Ancillae denote particles which help to perform certain tasks or operations, inspired by the latin translation of ``maiden''. Often, they fall in oblivion when the desired final state is obtained.} particles which interact with the system and thereby provide such symmetry breaking \cite{White:2009vn}.
The problem of entanglement constrained by SSR will turn saliant in Section \ref{modenen}, where we will discuss in more detail the problem of assigning entanglement to mode-entangled states, {\it i.e.}~states in which the local particle number is not fixed.
\section{Subunits and degrees of freedom} \label{secentities}
In the last chapter, we discussed entanglement as a property of a compound system that characterizes correlations between the system's subunits, which, from now on, will also be called \emph{entities}. In general, certain degrees of freedom of these subunits can be quantum-mechanically correlated, {\it i.e.}~entangled, to those of other entities, and the tensorial Hilbert space structure directly reflects this, possibly hierarchical, subsystem structure.
For identical particles, however, the invariance of all physical observables with respect to the exchange of any two particles and the (anti)symmetry of the (fermionic) bosonic multiparticle wave-function lead to a new conceptual challenge, and to the failure of the above strategy. The strong constraints on allowed states and on possible measurements imply that direct access to the ``first'' or ``second'' particle is not possible when dealing with identical particles, since particle \emph{labels} are no physical degrees of freedom. Therefore, the identification of Hilbert spaces with subsystems, which we explicitly assumed in Section \ref{sepaent}, fails when dealing with identical particles, and a different formalism is required to characterize the subunits between which entanglement is considered. As we will see below, upon closer inspection of several, partially irreconcilable entanglement concepts \cite{Ghirardi:2004fk,Wiseman:2003mz,Dowling:2006kx,schliemann-cirac,Eckert:2002vn}, the debate on what is the proper formalism is still not completely settled in the literature.
In order to find a suitable concept for the entanglement of identical particles, we start with the conceptually simpler case of non-identical or effectively non-identical particles. Once we will have established the physical ingredients of entanglement, and after relating these to the formal definitions introduced in Section 2, we will proceed towards more elaborate subsystem structures with identical particles involved. Finally, we give an overview of the diverse degrees of freedom that can exhibit entanglement in a real experimental setting.
\subsection{Non-identical particles}
We first shortly revisit the entanglement of distinguishable particles in order to establish the \emph{physical} meaning of entanglement and of separability in the spirit of the original EPR approach \cite{PhysRev.47.777}.
We will thus establish the terminology which we also need for a rigorous definition of the entanglement of identical particles. We aim at an approach which allows for a continuous transition from the entanglement properties of indistinguishable to those of distinguishable particles (as quantified in Section \ref{measures}), {\it e.g.} under a dynamical evolution which transforms the system constituents from an initially indistinguishable to a finally distinguishable state (as possibly induced, {\it e.g.}, by an atomic or molecular fragmentation process).
In the case of non-identical particles, the entities that carry entanglement are well defined: They correspond to the very particles, which exhibit some properties such as rest mass or charge, which allows to distinguish them without ambiguity. Other degrees of freedom of these particles that can be coherently superposed then eventually give rise to the particles' entanglement.
In this situation, we can directly associate Hilbert space $\mathscr{H}_1$ with the Hilbert space of the first particle, and Hilbert space $\mathscr{H}_2$ with that of the second. Vectors in these Hilbert spaces directly describe the state of the dynamical degree of freedom, such as spin or momentum, for the respective particle.
\subsubsection{Entanglement and subsystem properties according to the EPR approach} \label{properties}
A quantum state is not entangled \cite{PhysRev.47.777} if we can assign \emph{a complete set of properties} to each individual subsystem \cite{Ghirardi-statphys}, {\it i.e.}~if we can design a projective measurement on each subsystem with an outcome that can be predicted with certainty. In other words, there exists an observable such that its measurement on the given non-entangled state reveals the ``pre-existing'' system values, which we intuitively ascribe to it, as in classical mechanics. In the jargon, this is equivalent to finding a \emph{realistic description} of the system's constituents \cite{PhysRev.47.777}.
Specifically, for a pure state $\ket{\Psi}$ which describes a composite two-particle system, the subsystem $S_1$ is non-entangled with the subsystem $S_2$ if there exists a one-dimensional projection operator $P$ (with eigenvalue 1, by definition) such that
\begin{eqnarray} \bra{\Psi} P \otimes \mathbbm{1} \ket{\Psi}= 1 , \end{eqnarray} where $P$ acts on the Hilbert space of the first particle, and the identity $\mathbbm{1}$ on the second particle \cite{Ghirardi-statphys}. In this case, we can assert that the first subsystem possesses the physical properties defined by the operator $P$. The outcome of the measurement of $P$ is not subject to any uncertainty, the first subsystem was necessarily prepared in the unique eigenstate of $P$.
This notion of ``possession of a complete set of properties'' is fully equivalent to the notion of separability that we fixed in (\ref{separable}): The properties of any separable state are naturally given by projections on the quantum states of its constituent particles; as soon as a state requires more than one product state for its representation (and thus is an entangled state), such well-defined constituent properties cannot be identified anymore.
Besides separability, we can also define \emph{partial} and \emph{total entanglement} in the above framework \cite{Ghirardi-statphys}. For example, the state
\begin{eqnarray} \frac{1}{\sqrt 2} \left(\ket{\uparrow}_1 \ket{\downarrow}_2+ \ket{\downarrow}_1 \ket{\uparrow}_2 \right) \ket{L}_1 \ket{R}_2 ~,\label{partialent} \end{eqnarray}
where $\ket{\uparrow}, \ket{\downarrow}$ denote internal degrees of freedom, while $\ket{L}$ and $\ket{R}$ denote orthogonal spatial wave-functions, is partially entangled: Whereas system 1 possesses properties associated with it being prepared in the $\ket{L}$ state, we cannot specify all of its properties, since the particles' internal degrees of freedom remain fully correlated and locally unknown. This corresponds to the situation in most experiments: Particles do possess properties by which they can be distinguished -- {\it e.g.}, their position -- while their entanglement manifests in some other degree of freedom. In such a situation, the ``labeling'' degree of freedom (position $\ket{L}$ or $\ket{R}$ in (\ref{partialent})) can be simply dropped, and the entangled states, {\it e.g.}, of a photon's polarization and of the electronic degree of freedom of an atom, lives in the two-particle Hilbert-space
\begin{eqnarray} \ket{\Psi} \in \mathbbm{C}_1^2 \otimes \mathbbm{C}_2^2 \label{qubithsp} .\end{eqnarray} Each factor $\mathbbm{C}^2$ in (\ref{qubithsp}) describes the two-dimensional state-space of a polarized photon or of a two-level atom.
This is the typical structure of a two-qubit state so frequently encountered in the quantum information context.
\subsection{Creation and dynamics of entanglement} \label{entint}
\subsubsection{Interaction and entanglement}
For distinguishable particles, the strict rule that \begin{quotation}``Only interaction between particles\footnote{The interaction can be direct or through an intermediate, ancilla degree of freedom (see footnote on p. \pageref{ancillad} above).} can lead to an entangled state.''\end{eqnarray}d{quotation} can be formulated, in contrast to indistinguishable particles, as we will see in Section \ref{measurementinduced}. Distinguishable particles, even if prepared in a pure separable state, will be entangled for almost all times \cite{Durt:2004tg}, if the unitary evolution which describes their common evolution contains parts which do not factorize as in (\ref{hamilnointeraction}), {\it i.e.}~if the Hamiltonian contains an interaction part. Given a many-particle Hamiltonian, one can deduce from its interaction part which degrees of freedom will be entangled. A Hamiltonian which only couples external degrees of freedom will, {\it e.g.}, not induce any entanglement in the particles' spin.
In every many-particle bound state such as that of a simple hydrogen atom, the constituents are necessarily entangled in their external degrees of freedom \cite{Tommasini:1998qf}: The binding potential contains the operator $\vec r_1 - \vec r_2$, {\it i.e.}~it couples the position operators of the constituents (proton and electron, for the hydrogen atom). Inclusion of the spin-orbit interaction additionally induces entanglement between the external degrees of freedom and the spin, as we will discuss in more detail in Section \ref{ionizedelectrons}. The analogous reasoning applies for unbound systems: In the course of a scattering process, particles naturally entangle under the specified interaction \cite{Law:2004qf}.
We retain the intuitive picture that, for distinguishable particles, entanglement is a direct consequence of interaction.
\subsubsection{Open system entanglement}
If a quantum system is closed, \emph{i.e.} if it is decoupled from uncontrolled degrees of freedom lumped together under the term ``environment'', all entanglement properties are encoded in the Hamiltonian and in its eigenstates, or in the dynamical evolution it generates.
The time evolution of entanglement under such strictly Hamiltonian dynamics can then be evaluated by application of pure state entanglement measures on the time dependent state vector $\ket{\Psi(t)}$.
Purely Hamiltonian evolution is, however, untypical for experiments, where
the environment cannot be screened away completely. Therefore, decoherence, {\it i.e.}~the gradual loss of the off-diagonal entries of the density matrix \cite{Breuer:2006ud}, makes entanglement to fade away and limits its possible harvesting.
To account for such -- detrimental -- environmental influence, one can evaluate mixed state entanglement measures as defined in (\ref{purestatedecomp}) on the system density matrix $\rho(t)$, for all $t$. However, this quickly turns into a tedious, if not unfeasible task with increasing system size, due to the optimization problem implied by (\ref{purestatedecomp}). Efficiently evaluable lower bounds of entanglement alleviate the computational challenge \cite{Huber:2010kx,Mintert:2005rc,Mintert:2007ys,Borras:2009zr}, though cannot fully compensate the unfavorable scaling of optimization space with system size.
Alternative approaches try to circumvent this problem by incorporating the specific type of environment coupling into the analysis, to extract the entanglement evolution of representative ``benchmark'' states \cite{Konrad:2008vn,Tiersch:2008kx,Tiersch:2009uq,Li:2009ys,unravelling, vogelsberger}. The very knowledge of the source of decoherence effectively reduces the complexity of the problem (however, it remains to be quantified to which extent). For the simplest scenario of open system entanglement evolution, a qubit pair with only one qubit coupled to the environment, even an exact \emph{entanglement evolution equation} is available: Given the pure two-qubit's initial state $\ket{\chi}$, and the completely positive map\footnote{A positive map $\Lambda$ maps positive operators -- defined by a spectrum with strictly non-negative eigenvalues -- on positive operators, and, in particular, density matrices onto density matrices. For $\Lambda_C$ to be completely positive, also all possible extensions of the map to larger systems of the form $\mathbbm{1}_N \otimes \Lambda_C$ need to be positive.} \$ \cite{Breuer:2006ud} that one qubit is exposed to, the system's final state reads
\begin{eqnarray} \rho^\prime = (\mathbbm{1}\otimes \$ ) \ket{\chi}\bra{\chi} .\end{eqnarray}
The entanglement of the evolved quantum state $\rho^\prime$, quantified by the concurrence (as defined in (\ref{concurrence})), is then given by the entanglement evolution equation \cite{Konrad:2008vn} \begin{eqnarray} C\left[ (\mathbbm{1} \otimes \$) \ket{\chi} \bra{\chi} \right] = C( \ket{\chi}) C\left[(\mathbbm{1}\otimes \$) \ket{\Phi} \bra{\Phi} \right] \ , \label{factorevoo} \end{eqnarray}
where $\ket{\Phi}$ is the (maximally entangled, see (\ref{Psimp},\ref{Phimp})) benchmark state, and $\ket{\chi}$ an \emph{arbitrary} initial state. The entanglement evolution of $\ket{\chi}$ thus factorizes into a contribution given by the benchmark state's entanglement upon the action of the map \$, and a second term given by the initial state's concurrence $C(\ket{\chi})$. This result can be generalized for bipartite systems with $d$ dimensional subspaces \cite{Tiersch:2008kx}, with $C$ in (\ref{factorevoo}) replaced by G-concurrence $G_d(\rho)$ \cite{Gour:2005zr}. The measure $G_d(\rho)$ quantifies entanglement of rank $d$ states, {\it i.e.}~of states that are obtained as a coherent superposition that exhausts all basis states. Whenever the Schmidt rank (see Section \ref{quantif}) of the evolved state drops below $d$, $G_d$ drops to 0, while lower-ranked entanglement may still be present in the system. For $k<d$, upper bounds for $G_k$ and thereby clear disentanglement criteria can be derived within the same formalism. Since the evolution of the quantum state is continuous, it is, however, guaranteed that any state with initially non-vanishing $G_d$ will remain $G_d$-entangled at least for short times.
\subsubsection{Entanglement statistics}
Given the difficulty to characterize the open system entanglement evolution for individual initial states, because of the exponential scaling of state space with system size, it is suggestive to employ statistical tools. It can then be shown, under rather general assumptions on the open system dynamics described by some time-dependent map $\Lambda_t$ and on the employed entanglement measure $E$,\footnote{The measure $E$ needs to be Lipschitz-continuous \cite{Ruskai:1994fk}, and the system and environment need to be initially uncorrelated or at most classically correlated such that $\Lambda_t$ does not depend on the initial state.} that deviations $\epsilon$, of the final entanglement of an arbitrary pure initial state $\ket{\Psi}$ under the action of $\Lambda_t$ from the mean entanglement $\langle E \rangle(t)$ of all pure states acted upon by $\Lambda_t$, are exponentially suppressed in $\epsilon$ and in the system dimension $d$ \cite{tiersch}:
\begin{eqnarray} P\left( \left|E(\Lambda_t \ket{\Psi} \bra{\Psi} )-\langle E \rangle (t) \right| > \epsilon \right) \le 4 \mathrm{exp}\left( - \frac{2d-1}{96 \pi^2 \eta_E^2 \eta_{\Lambda_t}^2} \epsilon^2 \right) , \label{boundedeps} \end{eqnarray}
where $P$ quantifies the probability of the event specified by its argument, and $\eta_E$ and $\eta_{\Lambda_t}$ are parameters that characterize $E$ and $\Lambda_t$, with $\eta_{\Lambda_t}<1$.
Whereas the efficient evaluation of the entanglement evolution of individual initial states of large composite systems will at some point turn prohibitive, (\ref{boundedeps}) provides a statistical estimate which becomes ever tighter with increasing system size. In other words, the vast majority of initially pure multipartite states in a high-dimensional Hilbert space share the same entanglement properties \cite{tiersch}. Consequently, the entanglement evolution is similar for almost all initial states, and an asymptotic behavior emerges with typical traits that are independent of the exact quantum state, reminiscent of thermodynamic quantities.
\subsection{Identical particles} \label{idparticlesee}
As anticipated above, a consistent treatment of the entanglement of identical particles is much more subtle than for distinguishable particles. The Hilbert-space structure of two or more identical particles does not reflect any more a physical partition into subsystems, due to the (anti)symmetrization of the many-particle wave-function, as a result of the symmetrization postulate \cite{Ballentine:1998vn}. The principle of indistinguishability \cite{Messiah:1964ys} that applies to all physical operators leads to an uncircumventable super-selection rule (see Section \ref{secSSR}). This problem has been at the origin of a long debate \cite{Ghirardi-statphys,Ghirardi:2004fk,Ghirardi:2003uq,Cavalcanti:2007vn,Dowling:2006kx,Shi:2003ys,Paskauskas:2001ly,Ghirardi:2004kx,al:2009cr,Zhou:2009fk,Li:2001uq} on how to define a useful measure for a given state of identical particles. In the following, we describe the current state of affairs, and elaborate on how to refine the above criterium of a complete set of properties (Section \ref{properties}) for the case of identical particles \cite{Ghirardi-statphys}.
\subsubsection{Entanglement of particles} \label{entidp}
In many cases, when two identical particles are well separated -- as in typical experiments with photons in different optical modes, or with strongly repelling trapped ions -- no ambiguity is possible, and the physical subsystem-structure is apparent from the preparation of the state and the accessible observables \cite{Herbut:1987hb}. This is already realized in quantum mechanics textbooks, {\it e.g.} \cite{Peres:1993jt} states that \begin{quotation}``No quantum prediction, referring to an atom located in our laboratory, is affected by the mere presence of similar atoms in remote parts of the universe.'' \end{eqnarray}d{quotation}
Still, the formal notion of entanglement which we introduced for the case of distinguishable subsystems in Section \ref{measures} ought to be adjusted, since its naive application yields an unphysical form of entanglement, as can be seen by closer scrutiny of the following, exemplary state:
\begin{eqnarray} \ket{\Psi_{AB}}= \frac{1}{\sqrt{2}} \left( \ket{A, 1} \otimes \ket{B, 0} \pm \ket{B,0} \otimes \ket{A,1} \right) \label{ABstate} , \end{eqnarray}
where $\ket{A}$ and $\ket{B}$ describe orthogonal wave-functions. On a first glance, the wave-function appears to be entangled, since it cannot be written as product state, and it has Schmidt rank two (see (\ref{SchmidtDecomp})). However, a more careful analysis of the situation shows that neither particle in the system is affected by any \emph{physical} uncertainty: The wave-function $\ket{\Psi_{AB}}$ describes two particles, with their positions in space described by $\ket{A}$ and $\ket{B}$, which are prepared in the internal states $\ket 1$ and $\ket 0$, respectively. Physically speaking, we can easily assign a physical reality and thereby \emph{properties} (in the sense of Section \ref{properties}) to the particles: A measurement of the particle located at $\ket{A}$ ($\ket B$) will always yield the internal state $\ket{1}$ ($\ket{0}$). On the other hand, no physical operator can be conceived which refers unambiguously to the ``first'' or the ``second'' particle, since the particles are - by assumption - indistinguishable, which implies the permutation symmetry of all operators. In other words, the merely \emph{formal} entanglement in the unphysical particle labels \emph{cannot} be exploited directly, and \emph{does not} correspond to a lack of information about the physical preparation of the system's constituents -- indeed we have just established that there are two particles in the system which both possess a physical reality \cite{Ghirardi:2004fk}. Hence, instead of the Hilbert spaces of the particles which do not allow any more to address the particles individually, some other, \emph{physical}, degrees of freedom need to be identified with the distinctive properties of the entities that carry entanglement.
\subsubsection{Slater decomposition and rank} \label{slaterdecomp}
In order to differentiate between physical correlations and mere correlations in the particle labels, we use the \emph{Slater decomposition} instead of the Schmidt decomposition (\ref{SchmidtDecomp}): Two fermions which occupy an $n$-dimensional Hilbert space can always be described by the following quantum state \cite{schliemann-cirac,Schliemann:2001uq}
\begin{eqnarray} \ket{\Psi_{\mathrm{fermion}}}=\sum_{a,b} w_{a,b} f^\dagger_a f^\dagger_b \ket{0}, \end{eqnarray}
with antisymmetric coefficients $w_{a,b}=-w_{b,a}$, and fermionic creation operators $f^{\dagger}_a$ ($f^{\dagger}_b$), which act on the vacuum state $\ket{0}$ and create a particle in the single-particle state $\ket{a}$ ($\ket{b}$). In analogy to any bipartite state of distinguishable particles that can be written in the Schmidt-decomposition (\ref{SchmidtDecomp}), the above state can be represented in the \emph{Slater decomposition} \cite{schliemann-cirac},
\begin{eqnarray} \ket{\Psi_{\mathrm{fermion}}}= \sum_{i} z_i f^\dagger_{a^{(i)}} f^\dagger_{b^{(i)}} \ket{0} ,\end{eqnarray}
where the single particle states $\ket{a^{(i)}}=f^\dagger_{a^{(i)}} \ket 0 $ and $\ket{b^{(i)}}=f^\dagger_{b^{(i)}} \ket 0 $ fulfill $\braket{a^{(i)}}{b^{(j)}}=\delta_{a,b} \delta_{i,j}$. The number $r$ of non-vanishing expansion coefficients $z_i$ defines the {\it Slater rank}, which is unity for non-entangled states, and larger than unity for entangled states. In other words, elementary Slater determinants that describe fermions are the analogues of product states in systems that consist of distinguishable particles. The state (\ref{ABstate}) represents -- in the antisymmetric case for fermions -- a single Slater determinant and is, therefore, correctly recognized as \emph{non-entangled}. The entanglement measures introduced in Section \ref{quantif}, based on the distribution of Schmidt coefficients, can thus, in general, be recovered for identical particles by consideration of the Slater coefficients instead.
The convex roof construction (\ref{purestatedecomp}) for distinguishable particles can be imported directly, and allows the computation of entanglement measures for mixed state of identical particles. For example, the Schmidt rank of a mixed state of fermions can be obtained as follows: Given a mixed state of fermions decomposed into pure states, \begin{eqnarray} \rho=\sum_{j} p_j \ket{\Psi_j^{(r_j)}}\bra{\Psi_j^{(r_j)}} ,\end{eqnarray} where $r_j$ is the Slater rank of the respective pure state $\ket{\Psi_j^{(r_j)}}$, the Slater rank of $\rho$ is defined as $k=\mathrm{min}(r_{\mathrm{max}})$, where $r_{\mathrm{max}}$ is the maximal Slater rank within one decomposition, and the minimum is taken over all decompositions, in strict analogy to the case of distinguishable particles \cite{Terhal:2000uq}. Witnesses (see section \ref{witnessesslabl}) for the minimal number of Schmidt coefficients \cite{Sanpera:2001fk} for distinguishable particles can be imported to Slater witnesses \cite{schliemann-cirac} which witness states that require a certain minimal Slater rank.
While the analogies between Slater rank and Schmidt rank, worked out in \cite{schliemann-cirac}, also suggest similarities for the properties and the interpretation of the reduced density matrix of one particle (see Section \ref{quantif}), it is important to note that the reduced density matrix of one particle, $\varrho_{1}$, still exhibits some intrinsic uncertainty due to the formal entanglement in the particle label. The relationship between Schmidt coefficients and the eigenvalues of the reduced density matrix (see (\ref{SchmidtDecomp}),(\ref{reducedden}) in Section \ref{quantif}) therefore breaks down in the case of identical particles: The Slater coefficients are not directly related to the eigenvalues of the reduced density matrix. Entanglement measures based on the reduced density matrix therefore need to be interpreted carefully here, since they do not yield a result in full analogy to the case of distinguishable particles. Such interpretation follows below in Section \ref{propertiesidpa}.
\
Similarly as for fermions, a quantum state of two bosons \cite{Paskauskas:2001ly,Eckert:2002vn,Li:2001uq} can be written as
\begin{eqnarray} \ket{\Psi_{\mathrm{boson}}}=\sum_{i,j=1}^n v_{i,j} b_i^\dagger b_j^\dagger \ket {0} .\end{eqnarray} The symmetric coefficient matrix $v_{i,j}=v_{j,i}$ can be diagonalized such that the state can be written as a combination of doubly-occupied quantum states,
\begin{eqnarray} \ket{\Psi_{\mathrm{boson}}}=\sum_{j=1}^n \tilde v_{j} \left( \tilde b_j^\dagger \right)^2 \ket {0} \end{eqnarray}
Again, we call the minimal number $r$ of non-vanishing $\tilde v_j$ the \emph{bosonic Slater rank}. As we will see below, the interpretation of the Slater rank is not directly analogous to the case of distinguishable particles, as it was for fermions. The reason lies in the fact that two bosons can populate the very same quantum state, which is impossible for fermions and cannot be modeled within the usual quantum-information framework laid out in Section 2.
\subsubsection{Subsystem properties and identical particles} \label{subpropid}
In physical terms, the concept of properties of particles, discussed in Section \ref{properties}, can be adapted to the case of identical particles by taking into account that the particle label itself is \emph{not} a physical property \cite{Ghirardi:2004kx,Ghirardi:2004fk,Ghirardi:2003uq}. This fact needs to be included in the design of the projection operators that define a complete set of properties \cite{Ghirardi:2004fk}, in the terminology of Section \ref{properties}. Given two particles prepared in a quantum state $\ket{\Psi}$ of identical bosons or fermions, we therefore say that one of the constituents \emph{possesses a complete set of properties} if, and only if, there is a one-dimensional projection operator $P$ on the single-particle Hilbert space $\mathcal{H}$ such that
\begin{eqnarray} \bra{\Psi} \mathscr{E}_P \ket{\Psi}=1 \label{sepghirardi}, \end{eqnarray}
for
\begin{eqnarray} \mathscr{E}_P=P \otimes ( \mathbbm{1}-P) + (\mathbbm{1}-P) \otimes P~, \label{sepghirardi2} \end{eqnarray} where the order of the operators reflects the respective Hilbert spaces they act on.
The above projection operator can be interpreted as the projection on the subspace in which one particle possesses the properties given by $P$, while the other particle is projected onto a subspace orthogonal to $P$. Similarly to the case of non-identical particles, a particle cannot be entangled to any other particle if it possesses a complete set of properties.\footnote{We avoid the use of the term ``separable'', instead of ``non-entangled'', in the context of identical particles, since the term ``separable'' is often used to refer to the mere mathematical structure of the state, rather than to its possession of well-defined single-particle properties.} This definition rigorously adapts our discussion of Section \ref{properties} to the example (\ref{ABstate}) in Section \ref{entidp}. Indeed, for the state vector (\ref{ABstate}), the projection operator $P=\ket{A}\bra{A} \otimes \ket{1}\bra{1}$ has the required properties (\ref{sepghirardi},\ref{sepghirardi2}).
Thus we come to the following conclusion \cite{Ghirardi:2003uq}: \begin{quotation}Identical fermions of a composite quantum state are non-entangled if their state is given by the antisymmetrization of a factorized state. \end{eqnarray}d{quotation}
The case of bosons is more subtle to treat \cite{Ghirardi:2003uq}: \begin{quotation}Identical bosons are non-entangled if either the state is obtained by symmetrization of a product of two orthogonal states, or if the bosons are prepared in the same, identical state. \end{eqnarray}d{quotation}
A special case arises when a product of two non-orthogonal states is symmetrized, as illustrated by the following example: Consider the state
\begin{eqnarray} \ket{\Psi} &=& N(\alpha) \left( \ket{\phi}\otimes \left( \cos \alpha \ket{\phi} + \sin \alpha \ket{\psi} \right) \right. \nonumber \\ &&\left. + \left( \cos \alpha \ket{\phi} + \sin \alpha \ket{\psi} \right) \otimes \ket{\phi} \right) \label{bosonentan} \end{eqnarray}
where we assume $\braket{\phi}{\psi}=0$, and $N(\alpha)$ is an appropriate normalization constant. The parameter $\alpha$ interpolates between a symmetrized state of a product of two identical single-particle quantum states, and the symmetrized product of two orthogonal states. For $\alpha=0$, it corresponds to the state $\ket{\phi}\otimes \ket{\phi}$, {\it i.e.}~a non-entangled state, since we can attribute the property $\ket{\phi}\bra{\phi}$ to both particles. For $0>\alpha>\pi/2$, two non-orthogonal states are symmetrized, and no projection operator $P$ which satisfies (\ref{sepghirardi}) can be found. In other words, no statement about ``at least one particle possesses a certain set of properties'' is possible. Thus, a physical reality can be attributed to neither one of the particles, and the state has to be considered entangled. For $\alpha=\pi/2$, the state corresponds to the symmetrized product of two orthogonal states, one particle possesses the property $P=\ket{\psi}\bra{\psi}$, the other particle possesses $Q=\ket{\phi}\bra{\phi}$, and the state is hence not entangled.
\subsubsection{Subsystem properties and entanglement measures} \label{propertiesidpa}
The physical criteria based on the possession of realistic properties, as imported above from the case of distinguishable particles, can be directly related to entanglement measures such as the Slater rank introduced in Section \ref{slaterdecomp} \cite{Ghirardi:2004kx,Ghirardi:2004fk,Ghirardi:2003uq}. Also the von Neumann entropy of the reduced density matrix of either one of the particles can be used for the characterization of entanglement, though with some caution, as already mentioned in Section \ref{slaterdecomp}. Here, the interpretation for identical particles is distinct from the one established for distinguishable ones. One can finally formulate \cite{Ghirardi:2004fk} for fermions:
\begin{itemize}
\item[(i)] Slater rank of $\ket{\Psi}=1 \Leftrightarrow S(\varrho^{(1)})=1 \Leftrightarrow \ket{\Psi}$ is \emph{non-entangled} (the state is obtained by antisymmetrization of a product of two states).
\item[(ii)] Slater rank of $\ket{\Psi}>1 \Leftrightarrow S(\varrho^{(1)})>1 \Leftrightarrow \ket{\Psi}$ is \emph{entangled} (the state is obtained by antisymmetrization of a sum of products of states).
\end{eqnarray}d{itemize}
In contrast to the case of distinguishable particles, the entropy $S(\varrho^{(1)})$ is bounded from below by unity instead of zero: \emph{This residual value reflects the mere uncertainty in the particle label.} The analogy between the first case (i) and product states of distinguishable particles, and between the second case (ii) and entangled states of distinguishable particles is apparent.
Note that the antisymmetrization of a product of two non-orthogonal states gives rise to an unnormalized and non-entangled state.
For bosons, due to the possibility that the state is obtained by symmetrization of a product of two non-orthogonal states, the criterion is more elaborate \cite{Ghirardi-statphys}
\begin{itemize}
\item[(iii)] Slater rank of $\ket{\Psi}=1 \Leftrightarrow S(\varrho^{(1)})=0 \Rightarrow \ket{\Psi}$ is \emph{non-entangled} (both particles are in the same quantum state).
\item[(iv)] Slater rank of $\ket{\Psi}=2, 0<S(\varrho^{(1)}) <1 \Rightarrow \ket{\Psi}$ is \emph{entangled} (the state is obtained by symmetrization of a product state of non-orthogonal single-particle states).
\item[(v)] Slater rank of $\ket{\Psi}=2 , S(\varrho^{(1)})=1 \Rightarrow \ket{\Psi}$ is \emph{non-entangled} (the state is obtained by symmetrization of a product state of two orthogonal single-particle states).
\item[(vi)] Slater rank of $\ket{\Psi}>2 \Rightarrow \ket{\Psi}$ is \emph{entangled} (the state is composed by symmetrizing a sum of more than one product states).
\end{eqnarray}d{itemize}
Again, the entropy of the reduced density matrix $S(\varrho^{(1)})$ reflects the uncertainty in the particle label. Due to the higher occupation numbers allowed for bosons, it is, however, not bounded from below as for fermions. Note that the entanglement of a state is directly reflected neither by the Slater rank, nor by the entropy alone: A Slater rank of 2 can correspond to, both, an entangled and a non-entangled state. The two latter cases (v) and (vi) are, again, analogous to the case of non-entangled and entangled distinguishable particles, as for fermions. The first two situations (iii) and (iv), however, can only occur for bosons and do not possess any analogy with distinguishable particles. They are realized by the example (\ref{bosonentan}) discussed above.
\subsection{Measurement-induced entanglement} \label{measurementinduced}
In the preceding Section, we elaborated on a notion of entanglement for identical particles that follows the spirit of the original EPR approach \cite{PhysRev.47.777}: If it is possible to assign a physical reality (subsystem properties) to the constituent particles of a composite quantum system, they are considered as \emph{not entangled}. When dealing with identical particles, however, another source of quantum correlations emerges. In measurement setups that delete which-way information \cite{Englerta:1999uq,Walborn:2002oj}, entanglement can be created between identical particles without any interaction, in contrast to distinguishable particles (see Section \ref{entint}). In order to understand this, consider, once again, the quantum state
\begin{eqnarray} \ket{\Psi_{AB}}= \frac{1}{\sqrt{2}} \left( \ket{A, 1} \otimes \ket{B, 0} \pm \ket{B,0} \otimes \ket{A,1} \right) . \nonumber \hspace{2.4cm} (\ref{ABstate}) \end{eqnarray}
As we have shown above, this state is not entangled according to the criterion of \emph{particles which possess a complete set of properties.} Indeed, we can unambiguously assign the property $P=\ket{A, 1}\bra{A, 1}$ to one of the particles.
Let us now assume, however, that we use two detectors to measure the particles, and that these detectors do not spatially project onto $\ket{A}$ or $\ket{B}$, but on a linear combination of these states: We decide to measure a particle in the external state $\ket{L}:=(\ket{A}+\ket{B})/\sqrt{2}$, and another one in the external state $\ket{R}:=(\ket{A}-\ket{B})/\sqrt{2}$, orthogonal to $\ket{L}$. Such measurement is not deterministic, {\it i.e.}~not in each realization of the experiment does one find one particle in each detector. The internal states of the particles which are possibly registered by the detectors are not determined any more, {\it i.e.}~no assignment of a complete set of properties is possible for particles located in $\ket{L}$ or $\ket{R}$ -- while \emph{before} the measurement the state (\ref{ABstate}) did not exhibit entanglement, according to our criterion defined earlier. A posteriori, the internal states of the particles detected in $\ket{L}$ and $\ket{R}$ are \emph{perfectly anti-correlated}, and the Bell inequality (\ref{CHSH}) can be violated \cite{Tichy:2009kl}.
We call such choice of detectors $ O_L=\ket{L}\bra{L}$ and $ O_R=\ket{R}\bra{R}$ \emph{ambiguous}, since both particles initially prepared in the state given by (\ref{ABstate}) have a finite probability to trigger each detector, {\it i.e.}~the detectors have no one-to-one relationship to the external states of the particles. This situation is also illustrated in Figure \ref{ambiguoussetting}: The quantum state is initially non-entangled according to the above (EPR) criteria, but the individual spins of the particles measured by the two detectors are maximally uncertain, and strictly anti-correlated. Thereby, erasure of which-way information takes place: By the measurement of a particle in the state $\ket{L}$, the initial preparation of the particle -- whether in $\ket A$ or $\ket B$ -- is completely obliterated.
\begin{figure}[h] \center \includegraphics[width=6.5cm,angle=0]{fig6.pdf} \caption{Two particles are initially prepared in two orthogonal quantum states $\ket{A}, \ket{B}$ with definite spin values $\ket{\uparrow}, \ket{\downarrow}$, which permits the attribution of unambiguous properties to each of them. They hence both possess a physical reality. However, for ambiguous detector settings with $\ket{B}\neq \ket{L}\neq \ket{A}$ and $\ket{B} \neq \ket{R} \neq \ket{A}$, the detection of one particle in each detector, $\ket{L}\bra{L}$ or $\ket{R}\bra{R}$, will come along with wide uncertainty regarding the state of the measured particles' spin, due to the deletion of which-way information in the course of the measurement.} \label{ambiguoussetting} \end{eqnarray}d{figure}
An ambiguous choice of the detector setting can thus induce quantum correlations between the measurement results at these detectors, even if the initial state (like the one in (\ref{ABstate})) is non-entangled, and no interaction between the particles has taken place. For an initially entangled state, the detected particles can result to be more, but also to be less entangled \cite{Tichy:2009kl}, depending on the details of the setup. Furthermore, the statistical behavior of identical particles upon detection strongly depends on whether and how they are entangled. For example, photons prepared in a maximally entangled $\ket{\Psi^-}$-state, see (\ref{Psimp}), behave like fermions when scattered on an unbiased beam splitter \cite{PhysRevLett.61.2921}: When one photon enters at each input port, they will always exit at different ports and never occupy the same output mode.
We denote this phenomenon -- non-vanishing entanglement upon detection of particles initially prepared in a non-entangled state (according to \cite{Ghirardi:2004kx} and (\ref{sepghirardi})) -- \emph{measurement-induced} entanglement. This is the characteristic additional feature encountered when dealing with identical particles and their entanglement properties.
We retain that, in contrast to distinguishable particles, which can be entangled by deterministic procedures mediated by mutual interaction, measurement-induced entanglement is intrinsically probabilistic, {\it i.e.}~the success rate to find one particle in each detector is strictly smaller than unity.
\subsubsection{Entanglement extraction} \label{entextr}
The abstract principle of deleting which-way information constitutes the basis for many applications which exploit the indistinguishability of particles to create entangled states. Such schemes are widely used, for example in today's quantum optics experiments with photons (see \cite{Prevedel:2009ec,Wieczorek:2009ff,Wieczorek:2009fe,Deb:2008xr,PhysRevLett.101.010503}, for an inexhaustive list of very recent state-of-the-art applications). For massive particles, however, no experiments have been performed yet which implement analogous ideas, but recent advances in current technology (see, {\it e.g.}~\cite{Karski:2009kx}) feed the hope that a realization analogous to the photon experiments will be performed in the near future.
An ambiguous choice of detectors which implements the scheme of Section \ref{measurementinduced} is realized by a simple beam splitter in a Hong-Ou-Mandel configuration \cite{Hong:1987mz}: In the original experiment, two identical photons fall onto the opposite input ports of a balanced beam splitter and, due to two-particle interference, they always leave the setup together, at one output port. Instead of two identical photons, one can inject a horizontally and a vertically polarized photon \cite{PhysRevLett.61.2921}, as illustrated in Figure \ref{HOMfig}, {\it i.e.}~the initial state
\begin{eqnarray} \ket{\Psi_{\mathrm{ini}}}= \hat a^\dagger_{1,H}\hat a^\dagger_{2,V} \ket{0} \label{HOMINI} .\end{eqnarray}
\begin{figure}[h] \center \includegraphics[width=6.5cm,angle=0]{fig7.pdf} \caption{Entanglement creation with the Hong-Ou-Mandel (HOM) setup \cite{Hong:1987mz}. Two photons that are initially prepared in the modes $\hat a_{1(2)}^\dagger$ and which are horizontally (vertically) polarized fall onto the beam splitter (BS). Upon detection of one photon in either one of the two detectors, the polarization state of this photon is unknown, but fully anti-correlated to the polarization of the other one. Purely quantum correlations are detected when the photons have perfect overlap in space, time, and frequency. Slight deviations from this ideal case, as suggested in the figure where the overlap of the transmitted and reflected wave packets is not optimal, jeopardize the quantum nature of the correlations \cite{Tichy:2009kl}.} \label{HOMfig} \end{eqnarray}d{figure}
The final state after the scattering on the beam splitter reads
\begin{eqnarray} \ket{\Psi_{\mathrm{fin}}}=\frac 1 2 \left(b_{1,H}^\dagger b_{1,V}^\dagger -b_{2,H}^\dagger b_{2,V}^\dagger + b_{1,H}^\dagger b_{2,V}^\dagger -b_{2,H}^\dagger b_{1,V}^\dagger \right) \ket 0 \label{HOMFIN} .\end{eqnarray}
Disregarding measurement results with two photons in the same port -- described by the first two terms in $\ket{\Psi_{\mathrm{fin}}}$ --, which is known as the \emph{post-selection} of those detection events with one particle in each output port (the two last terms in $\ket{\Psi_{\mathrm{fin}}}$), one effectively performs a projection onto the maximally entangled $\ket{\Psi^-}$ Bell state.
Consequently, when analyzing the postselected subset of the total measurement record, perfect anti-correlations between the measured photons are encountered \cite{Giovannetti:2006jh}. Such a combination of two-particle interferometry and which-way detection can be used to entangle any degree of freedom of indistinguishable particles, whether bosons or fermions \cite{Bose:2002le,Bose:2002vf}, or to distinguish fermionic from bosonic quantum states \cite{Bose:2003kx}. Recent experiments have shown that this scheme works even if the photons are created independently \cite{Beugnon:2006ec,Halder:2007th} or measured at very large distances from each other \cite{Marcikic:2004qr}. These effects lend themselves to manipulate entangled states by purely quantum-statistical means, {\it i.e.}~to change their entanglement properties without the need of any interaction between the constituents \cite{Omar:2002zr,al:2010lh,Paunkovic:2002jt}.
The two-particle Hong-Ou-Mandel effect on which the above scheme relies can be generalized to an arbitrary number of input and output ports \cite{Lim:2005qt,Tichy:2010kx}. This also allows to generalize the above scheme to create entanglement within multipartite systems \cite{Zhang:2006yq,PhysRevA.71.013809}. Again, one prepares a non-entangled state of photons in which not all photons share the same internal state. The particles are scattered simultaneously on a multiport beam splitter, as illustrated in Figure \ref{multiport}. Detection is conditioned on one particle at each output port \cite{Lim:2005bf,Wang:2009ud}. The which-path information and thereby the information on the internal state of the photons that are found at each output port is deleted in the course of the scattering process, and the state obtained upon detection of the predefined event (one particle in each output port) is a many-particle entangled state. Such situation is illustrated for four particles in Figure \ref{multiport}.
\begin{figure}[h] \center \includegraphics[width=10.5cm,angle=0]{fig8.pdf} \caption{Multiport beam splitter setup to create a four-particle W-state \cite{Lim:2005bf}. The beam splitters (short and thin horizontal lines) have reflectivities chosen such that the probability for any incoming photon to exit at any output port is always 1/4. When one particle is found at each output port, the four-photon polarization state is described by a four-particle W-state \cite{Lim:2005bf}. Note the strict analogy to the scheme in Figure \ref{HOMfig}.} \label{multiport} \end{eqnarray}d{figure}
It is thus possible to create a multitude of different multipartite entangled states \cite{Lim:2005bf,Sagi:2003qf}. Similar schemes, where the photons propagate in free space rather than through a beam splitter array, were shown to permit the preparation of multipartite entangled states through suitable settings and detection strategies \cite{PhysRevLett.99.193602,PhysRevLett.102.053601,Maser:2010fu,Maser:2009pi}. Indeed, with the local variation of the polarization states onto which the photons are projected, it is possible to tune through several families of entangled states \cite{PhysRevLett.101.010503}, including the large subclass of states that are symmetric upon exchange of any two subsystems \cite{PhysRevLett.99.193602,PhysRevLett.102.053601}. The induced correlations are not restricted to the polarization, but can also be created in motional degrees of freedom \cite{Guo:2008hc}.
All the above scenarios rely on the fact that all particles are identical, and only distinguished through the mode and the internal state they are initially prepared in, {\it i.e.}~only through the property used for discrimination, and the degree of freedom in which they will be entangled in the final state. Whenever some discriminating information is available on output, {\it i.e.}~when the particles are partially distinguishable, {\it e.g.} due to different arrival times or different energies, the entanglement in the final state is jeopardized \cite{Velsen:2005ve,branning1999eia,Velsen:2005nx,Jha:2008bs}. Instead of a fully quantum-mechanically correlated state, the state exhibits more and more classical correlations, and the more so the more distinguishable the particles are. Indeed, the transition from fully indistinguishable to fully distinguishable particles used in these schemes corresponds to a transition from purely quantum-mechanically entangled to purely classically correlated final states \cite{Tichy:2009kl}.
Besides the intrinsic indistinguishability which is the very basis of the entanglement strategies described above, it is also possible to exploit the Pauli principle, and thereby use the quantum-statistical behavior of fermions \cite{Cavalcanti:2007vn}, to create entangled states. Quantum correlations between the spins of two independent fermions in the conduction band of a semiconductor are enforced by the Pauli principle, and selective electron-hole recombination then transfers this entanglement to the polarization of the emitted photons.
\subsubsection{Entanglement in the Fermi gas} \label{fermigasentanglement}
Different types of entanglement can be considered within a Fermi gas. On the one hand, electron-hole entanglement can be defined \cite{Beenakker:2006zt}. Here, we focus, however, on the entanglement between identical particles, specifically the electrons, in their spin degree of freedom. This entanglement can be understood as measurement-induced, similar to what we saw above. Different studies illustrate how measurement-induced entanglement is ubiquitous in a scenario in which the particles are measured in states different from those they were prepared in, in full analogy to the generic situation described in Section \ref{measurementinduced}: The Fermi gas is constituted of electrons prepared in momentum eigenstates, which are then detected in position eigenstates.
The ground state of a Fermi gas of non-interacting particles at zero temperature reads
\begin{eqnarray} \ket{\phi_0}=\prod_{s=\uparrow,\downarrow} \prod_{|p|\le k_F} b_s^\dagger(p) \ket{0} , \end{eqnarray}
where $\ket{0}$ represents the vacuum and $b_s^\dagger(p)$ creates an electron with spin $s$ and momentum $p$. This state corresponds to a single Slater determinant, and it is therefore not entangled according to our reasoning in Section \ref{subpropid}, but may exhibit measurement-induced entanglement. Indeed, when two particles are detected at positions $r$ and $r^\prime$, one observes entanglement between the spins of the detected electrons \cite{Vedral:2003uq}. The spin correlations found between the fermions depend on their relative distance $x=|r-r^\prime|$: The two-body reduced density matrix of the particle pair reads
\begin{eqnarray} \rho_{12} = \left( \begin{tabular}{cccc}
$1-f(x)^2 $& 0 & 0 & 0 \\
0 & 1 & $-f(x)^2$ & 0 \\
0 & $-f(x)^2$ &1& 0 \\
0 & 0 & 0 &$1-f(x)^2 $
\end{eqnarray}d{tabular} \right), \label{fermistate}
\end{eqnarray}
in the basis \begin{eqnarray} \{ \ket{\uparrow}_r \otimes \ket{\uparrow}_{r^\prime}, \ \ket{\uparrow}_{r}\otimes \ket{\downarrow}_{r^\prime}, \ \ket{\downarrow}_{r}\otimes \ket{\uparrow}_{r^\prime},\ \ket{\downarrow}_{r}\otimes \ket{\downarrow}_{r^\prime} \}, \end{eqnarray} which describes the spin orientations of the two electrons found at the two locations, $r$ and $r^\prime$, with
\begin{eqnarray} f(x) = \frac{3 j_1(k_F x)}{k_F x}, \end{eqnarray} where $j_1(x)$ is the Bessel function of the first kind.
We can interpret (\ref{fermistate}) as a state of two effectively distinguishable particles \cite{Tichy:2009kl}, since it describes the spin state of the two electrons detected by two distinct detectors. Thereby, we can apply the notions of Section 2.
The two-particle state (\ref{fermistate}) is a Werner state (see (\ref{WernerState})) with the particular property that it cannot violate any Bell inequality for a large range of $f$, although it exhibits entanglement \cite{Werner:1989ve}. For inter-particle distances $x$ of the order or smaller than $\pi/(8 k_F)$, we find $f(x)^2\ge 1/2$, and the detected particles result to be entangled, by virtue of (\ref{purestatedecomp}). Physically, this corresponds to the situation in which the Pauli principle inhibits the detection of both particles with the same spin, and forces the electrons into an anti-correlated state.
The above consideration allows the extraction of a many-particle density matrix that describes the spin state of the detected particles, and permits to tackle entanglement between electrons in the Fermi gas under numerous distinct perspectives. Instead of detecting only two particles, it is immediate to study multipartite entanglement between many electrons at different locations \cite{Jie:2008oq,Lunkes:2005ao,Vertesi:2007qu,Habibian:2010ys}. Finite temperatures can be considered \cite{Oh:2004yq,Lunkes:2005et}, and also electron-electron interactions were included \cite{Hamieh:2009zr,Hamieh:2010ly}, although only as screened Coulomb interactions in an effective treatment that provides an approximation to the Fermi liquid. It is also possible to relax the assumption that the particles be projected onto position eigenstates \cite{Cavalcanti:2005vl}, and consider more realistic, coarse-grained measurement devices. All studies in this area share the conclusion that a rich variety of entangled states can be extracted, purely due to measurement-induced entanglement, and that such quantum correlations are not substantially affected when finite temperatures or screened Coulomb interactions are incorporated in the model.
Hence, the measures for particle entanglement introduced in Section \ref{idparticlesee} are satisfactory in concept, but many dynamical situations occur in which the detection process itself indeed induces entanglement between particles, and a treatment that takes into account measurement-induced entanglement, as discussed in Section \ref{measurementinduced}, is more appropriate. This conclusion is not restricted to the case of a Fermi gas, but the scheme can be applied to any system of many identical particles.
\subsection{Mode entanglement} \label{modenen}
While the above discussion of the entanglement of identical particles was centered around the correlations between degrees of freedom carried by \emph{particles}, one can also define the entanglement between modes. In this case, it is the very number of particles which becomes the degree of freedom in which the entities -- the modes -- can be entangled. For example, the state \begin{eqnarray} \frac{1}{2} \left( a^\dagger a^\dagger \otimes \mathbbm{1}_{(B)}+ \mathbbm{1}_{(A)} \otimes \hat b^\dagger \hat b^\dagger \right) \ket {0}_A \otimes \ket {0}_B \nonumber \\ = \frac{1}{\sqrt 2} \left( \ket{0}_A\otimes \ket{2}_B + \ket{2}_A\otimes\ket{0}_B \right) ,\end{eqnarray}
where $\hat a^\dagger$ and $\hat b^\dagger$ represent particle creation operators of mode A and B,
is a state which exhibits entanglement between modes A and B (we make the tensor product structure explicit here): It describes a coherent superposition of two particles located in A and two particles located in B. The local particle number in $A$ and $B$ is unknown and exhibits correlations with the number of particles in the respective other mode.
\subsubsection{The definition of modes and the problem of locality}
The Pauli principle results in a maximal mode occupation number one for fermions, so that each mode can only reside in the state $\ket 0$ (no fermion) or $\ket 1$ (one fermion). That is to say, the Fock-space of $m$ fermionic modes is isomorphic to an $m$-qubit Hilbert-space, which is formalized by a map $\Lambda$ with the property \cite{Zanardi:2002uq}
\begin{eqnarray}
\Lambda: \prod_{l=1}^L \left(\hat c^\dagger_l \right)^{n_l} \ket{0}^{2nd quant.} \mapsto \otimes_{l=1}^L (\sigma_l^+)^{n_l} \ket{0,\dots,0}^{1st quant.}
,\end{eqnarray}
where $n_l \in \{0,1\}$, and we made the first and second-quantized formulation explicit: $c^\dagger_l$ is the fermionic creation operator for mode $l$, and $\sigma^+_l$ is the raising operator of the $l$th qubit, {\it i.e.}~$\sigma^+_l \ket{0}_l=\ket{1}_l$. On the level of operators, the Jordan-Wigner transformation \cite{Jordan:1928uq} maps fermionic creation operators onto raising operators, such that
\begin{eqnarray}
\hat c_l^\dagger \mapsto \sigma_l^+ \prod_{k=1}^{l-1}(-\sigma_k^z) , \ \ \sigma_l^\dagger \mapsto \hat c_l^\dagger \prod_{k=1}^{l-1} e^{i \pi \hat c_k^\dagger \hat c_k } .
\end{eqnarray}
In other words, in order to take into account the fermionic anti-commutation relations for \emph{distinct} sites, $\{\hat c_k, \hat c^\dagger_l \}=\delta_{k,l}$, the mapping from the $l$th creation operator to the raising operator is \emph{non-local in the modes}, since it depends on the occupation numbers of all modes $k$ with $k<l$. Consequently, also the action on one mode depends on the occupation of other modes, and cannot be considered ``local'' anymore. The difficulties that arise due to this non-trivial mapping are aggravated by the impossibility of arbitrary rotations of qubits due to particle-number conservation (for massive and/or charged fermions, $\sum_{k} n_k= \mathrm{const.}$), or total parity conservation (for excitations, $\mathrm{mod}(\sum_{k} n_k,2)=\mathrm{const.}$). Fermionic mode entanglement can still be exploited, {\it e.g.}~for quantum computation purposes when suitable protocols are designed \cite{Bravyi2002210}, allowing applications in close analogy to the unrestricted form of particle entanglement.
Fermionic mode entanglement also possesses a physical interpretation, since it can be related to different types of superconductivity \cite{Zanardi:2002fk}. For example, in the BCS-model \cite{Bardeen:1957kx}, a finite value of the superconductivity order parameter is a necessary condition for entanglement, superconductivity and entanglement are thus related.
Mode entanglement can be defined analogously for bosons \cite{Sanders:1992kx,Huang:1994uq}, with the difference that, due to the unconstrained occupation number, the resulting Hilbert space is much larger than in the case of fermions.
In general, mode entanglement is strongly dependent on the way the modes are defined themselves, since any transformation of mode creation operators \begin{eqnarray} \hat c^\dagger_i \rightarrow \tilde c^\dagger_i = \sum_j U_{ij} \hat c_j^\dagger \label{modetransfo} ,\end{eqnarray} with unitary $U$ leaves the commutation relations of the creation operators invariant, while the reduced density matrix for a single mode will in general change considerably \cite{Zanardi:2002uq}. Indeed, a transformation as (\ref{modetransfo}) ought to be considered \emph{non-local}, since different modes are brought into coherent superposition of each other.
While central to the definition of the entanglement of distinguishable particles, where the term ``local'' denotes operations which act on one subsystem alone, the very concept of locality is indeed difficult to define, let alone to apply consistently, when dealing with the entanglement of identical particles or with the entanglement of modes: The approaches to particle entanglement proposed in \cite{schliemann-cirac} -- particle entanglement according to complete sets of properties, as in Section \ref{entidp} -- and in \cite{Zanardi:2002uq} -- mode entanglement -- are compared in \cite{Gittings:2002gf}. The authors come to the conclusion that the definition based on the Slater rank presented in \cite{schliemann-cirac} does not meet the requirements for an entanglement measure, and is thus inappropriate for the quantification of entanglement of identical particles, since it is invariant under non-local operations, while it changes under operations restricted to one mode. This argument is based on the observation that a single-particle two-site unitary transformation does not affect the Slater rank of any non-entangled state and, thus, does not change its entanglement according to \cite{schliemann-cirac}, while such operation can be considered as nonlocal in the sites/modes. This line of thought shows that particle entanglement is a concept which does not go well along with the very concept of locality, since particles can be strongly delocalized and yet unentangled, in the sense that they individually possess a complete set of properties (in the EPR sense of Section \ref{properties}), as we show in the following: Consider, {\it e.g.}, the passage of two orthogonally polarized photons through a beam splitter, as already discussed in Section \ref{entextr} with Eqs.~(\ref{HOMINI}),(\ref{HOMFIN}), and illustrated in Figure \ref{HOMfig}. The quantum state and its evolution read \begin{eqnarray} \ket{\Psi_{\mathrm{ini}}}=\hat a^\dagger_{1,H}\hat a^\dagger_{2,V} \ket 0 \mapsto \frac 1 2 (b_{1,H}^\dagger - b_{2,H}^\dagger)(b_{1,V}^\dagger + b_{2,V}^\dagger) \ket 0 \nonumber \\ =: \tilde a^\dagger_{1,H} \tilde a^\dagger_{2,V} \ket 0 \label{moderedefi}, \end{eqnarray}
where $\hat a_{i,X}^\dagger$ ($\hat b_{i,X}^\dagger$) denote creation operators for particles in mode A$_i$ (B$_i$) with polarization $X$. While the initial state is clearly non-entangled, since the modes are spatially separated, the final state is entangled in the modes defined by the creation operators $b^\dagger_{k,Y}$. Since the photons never interact, the functional form of the initial state is preserved in the final state, which therefore can still be rewritten as a non-entangled state in terms of transformed modes $\tilde a_{k,Y}^\dagger$, as done in the second line of (\ref{moderedefi}). While the particles in the modes populated by $\tilde a^\dagger_j$ can still be attributed a full set of properties, local measurements in the modes $b_k$ exhibit quantum correlations. The choice of modes is clearly induced by the experimental setting: measurements take place on the modes defined by $\tilde b_j^\dagger$, and since entanglement is fundamentally defined by the correlations of measurement results (see Section \ref{Bellin}) the application of an entanglement measure ought to be performed on the state as described in terms of the $\tilde b_j^\dagger$, in the first line of (\ref{moderedefi}), while recasting the state into the second line's form is nothing but a formal manipulation, without a physical counterpart.
When entanglement is \emph{not} considered within a measurement setup, the question arises of which is the natural mode substructure to be considered in order to quantify the entanglement induced \emph{solely} by interaction, in a system of identical particles. A natural choice is given by the eigenstates of the single-particle Hamiltonian, in the absense of the interaction part \cite{Shi:2004la}. The resulting mode-entanglement, when the interaction is turned on again, then originates uniquely from the interaction terms of the system Hamiltonian, and is not a by-product of the above redefinition of modes. However, given the indispensable, defining anchoring of entanglement to correlations of measurement results, such considerations have a rather formal flavor.
\subsubsection{Particle entanglement}
The physical relevance of mode entanglement has been an object of intense debate \cite{Enk:2005gd,Enk:2006re,Drezet:2006gf,Drezet:2007by,Heaney:2009jl}, since a particle number superselection rule (SSR, see Section \ref{secSSR}) forbids, in principle, the detection of coherent superpositions of states of massive particles with different local particle numbers. While we show in Section \ref{modeentdet} that schemes exist which allow to circumvent this problem in some cases for both massless and massive particles, given a suitable experimental infrastructure, let us assume for the moment that such equipment is not present, and that particle-number correlations of massive particles cannot be regarded as entanglement, as proposed in \cite{Wiseman:2003mz}.
The system under consideration allows its modes to have a variable number of particles, and the particles to carry an internal degree of freedom. The mode entanglement of a given state stems from both, the entanglement in the particle-number degree of freedom, {\it i.e.}~intrinsic mode entanglement, and entanglement in the internal degree of freedom of the particles. In order to assure that only the latter be capitalized as bona fide entanglement, only the entanglement from the state's projection onto states with locally well-defined particle numbers is computed \cite{Wiseman:2003mz}, thus neglecting the coherent superposition of different particle numbers. The result then reduces to the entanglement one would assign to distinguishable spins \cite{Shi:2003ys}. Simple models of two bosons or two spinless fermions distributed over four modes, the Hubbard dimer, and non-interacting electrons on a lattice are treated following this convention in \cite{Dowling:2006kx}, with a comparison to the application of a mode-entanglement measure. For the electrons on a lattice, the resulting entanglement in the spin degree of freedom agrees well with studies of entanglement in the Fermi gas as mentioned in Section \ref{fermigasentanglement} above. In general, the degree of particle entanglement is found to differ from the degree of mode entanglement. For example, the entanglement of particles for two non-interacting bosons in four sites vanishes, while the entanglement of modes of the same system adopts a finite value. Such discrepancy can be expected, since the very carriers of entanglement are different.
\subsubsection{Detection of mode-entanglement} \label{modeentdet}
The discussion of the previous Section followed the assumption \cite{Wiseman:2003mz} that coherent superpositions of different particle numbers of massive particles cannot be detected. This assumption itself is, however, under debate \cite{Enk:2005gd}, as we will discuss in the following. The key issue is best illustrated by the case of a single, delocalized particle. This simple setting has attracted much attention during the last decades, from very different perspectives \cite{Hardy:1994fu,Revzen:1996zt,Aharonov:2000mi,Dunningham:2007pi,Cooper:2008ff,Peres:1995mi,Pawlstrokowski:2006pi,Lo-Franco:2005ff,Lee:2003lh,Bjork:2001fu,Moussa:1998il}. The debate on whether such state bears entanglement or not has been intense, and only in the case of photons the issue can be regarded as thoroughly settled by experimental verification. Note, however, the fundamental difference between massive particle for which conservation laws imply a SSR, {\it e.g.} of charge, and massless particles such as photons, which can be locally created or destroyed \cite{Ashhab:2007udd}, under the condition that energy, momentum and angular momentum are conserved.
The controversy arises from the fact that a state of a delocalized photon, which one obtains, {\it e.g.}, by simple scattering on a beam splitter, can be written, in the occupation number basis, as \cite{Enk:2005gd}
\begin{eqnarray} \ket{\Psi}_{A,B}=\frac{1}{\sqrt{2}}\left( \ket{0}_A \otimes \ket{1}_B + \ket{1}_A \otimes \ket{0}_B \right) \label{singlephotonnonlocal} \end{eqnarray}
where $A,B$ denote the modes, and $\ket{k}_{A/B}$ the occupation number, or Fock state. The state seems to exhibit entanglement between mode $A$ and mode $B$, but it is fully equivalent to the application of the following linear combination of creation operators on the vacuum
\begin{eqnarray} \ket{\Psi}_{A,B}=\frac{1}{\sqrt 2}\left( \hat a^\dagger_A \otimes \mathbbm{1} + \mathbbm{1} \otimes \hat a^\dagger_B \right) \ket{\mathrm{0}}_A \otimes \ket{\mathrm{0}}_B ,\end{eqnarray}
for which the entanglement is not as obvious. When expressed in first quantization, there is, due to the irreducible structure of the Hilbert space, no place for entanglement at all:
\begin{eqnarray} \ket{\Psi}_{A,B}=\frac{1}{\sqrt 2} \left( \ket{A}+\ket{B} \right) .\end{eqnarray} The choice of modes as the carriers of entanglement suggests, however, that indeed (\ref{singlephotonnonlocal}) captures the essential physics of the problem.
The apparently intuitive argument brought up against the feasibility to test the non-locality of such a state was that the detection of the particle at one mode -- a necessary requirement in the experiment -- would prohibit the measurement of any property associated with this very same particle at the other mode. However, by its construction does the verification of non-locality rely rather on the \emph{wave}-like rather than the \emph{particle}-like features, which indeed can be probed, as we will see now.
Proposals to verify a single particle's mode entanglement mainly follow two lines, and we review one exemplary setup for each. \begin{figure}[h] \center \includegraphics[width=8.5cm,angle=0]{fig9.pdf} \caption{Mode entanglement detection with coherent states \cite{Hardy:1994fu,PhysRevLett.66.252}. The non-local single photon state is superposed with the coherent states $\ket{\alpha_1}$ and $\ket{\alpha_2}$, respectively, at the two beam splitters BS1 and BS2. The four detectors are depicted as triangles. } \label{coherentStateModeentdet} \end{eqnarray}d{figure}
\paragraph{Phase reference via external ancilla states}
The first method relies on coherent states as phase reference \cite{PhysRevLett.66.252}: one of the modes in (\ref{singlephotonnonlocal}) is superposed, at a beam splitter, with a coherent state $\ket{\alpha}$ in the other mode. A photon detector that is located at one output port of the beam splitter cannot distinguish photons which originate from either input mode. Therefore, the amplitudes for a single photon detection event, which stem from the single photon mode and from that fed with the coherent state, are added. Variation of the phase of the coherent state allows to project, upon detection of a photon at one output port of the beam splitter, onto different coherent superpositions of one and no photon in the input mode. Thus, the relative phase of the one-particle component $\ket{1}$ with respect to the no-particle component $\ket{0}$ can be measured, {\it i.e.}~one is not restricted to measurements in the basis $\{ \ket{0}, \ket{1} \}$.
This procedure can be performed on both modes and thereby permits a full Bell experiment, as illustrated in Figure \ref{coherentStateModeentdet}.
The setup works analogously when single photon states are used instead of the coherent states \cite{Lee:2000dk,Sciarrino:2002kl}, provided a phase reference is available.
While the very implementation of that setup was not considered an issue and also acknowledged by critics, it was argued that the exhibited non-locality is not a property of the single photon, but intrinsically relies on the coupling to the coherent states which may bring in additional non-locality \cite{Vaidman:1995ye,Greenberger:1995qo,Hardy:1995tw}. This problem can be circumvented by modifying the setup such that the coherent states that are used are totally uncorrelated \cite{Cooper:2008ff}: If the reference states are created by the observers themselves in an independent way, no additional quantum correlations can be induced. The phase relation between the coherent states is crucial for the functioning of the scheme, but can also be achieved by purely classical communication.
Experimentally, mode entanglement was finally verified, closely following the original proposal \cite{PhysRevLett.66.252}, in \cite{Hessmo:2004bs,Babichev:2004fv} with photons delocalized in space, and in \cite{DAngelo:2006dz} for photons delocalized in time.
Note, however, that this verification does not rely on the fact that one uses massless particles: Instead of a coherent state of light, a reservoir such as a BEC can also be used as a reference state, when massive particles are under consideration. Thereby, the fundamental SSR for the particle number which states that no coherent superpositions of a different number of particles can be directly measured, can be circumvented \cite{Ashhab:2007if,Heaney:2009jl,Heaney:2009kh,Goold:2009ys}.
\paragraph{Transfer of mode entanglement onto ancilla particles}
The second method for the verification of mode-entanglement does not require coherent states as a reference, but introduces ancilla particles, which acquire the information whether the mode was populated or not, in a coherent way. A subsequent readout of the state of the ancilla particles can then violate locality. In contrast to the above setup using coherent reference states, the issue of creation or annihilation of massive particles (ruled out by a fundamental SSR) is more intricate here, as we will see below. The basic idea is strongly influenced by a proposal for entangling gates in quantum computation \cite{Cirac:1994qf}, and can be described as follows:
\begin{figure}[h] \center \includegraphics[width=6.5cm,angle=0]{fig10.pdf} \caption{Detection of mode-entanglement with ancilla particles. A photon is scattered by the beam splitter (solid horizontal line), and excites coherently either the upper ancilla particle, or the lower one, which are both initially in their ground state $\ket{g}$. A subsequent read-out of the states of the ancilla particle in different bases reveals the quantum correlations between them.} \label{Modeentanglementillu} \end{eqnarray}d{figure}
Consider the state \cite{Enk:2005gd,Cunha:2007fk,Ashhab:2007udd}
\begin{eqnarray} \frac{1}{\sqrt 2}\left( \ket{0,1}_{A,B}+\ket{1,0 }_{A,B} \right)\otimes \ket{g,g} ,\end{eqnarray} where $\ket{i,j}_{A,B}$ represent the state of the modes in which $i$ particles are prepared in mode $A$, and $j$ particles are prepared in $B$. The state $\ket{g,g}$ describes two ancilla particles in their ground state, locally coupled to the modes $A$ and $B$, respectively. If an interaction between the particles prepared in the modes and the ancilla particles takes place, such that the latter become excited upon the presence of a particle in the respective mode, the full system's quantum state will evolve into
\begin{eqnarray} \frac{1}{\sqrt 2}\ket{0,0}_{A,B} \otimes \left( \ket{g,e} + \ket{e,g} \right) , \end{eqnarray}
due to the coherent absorption of the delocalized particle by the ancilla particle coupled to A or to B. The setup is schematically depicted in Figure \ref{Modeentanglementillu}.
The entanglement which was present as mode-entanglement is transferred to entanglement between the two ancilla particles which can be verified by the usual means, under the assumption that no SSR inhibits the measurement of coherent superpositions of $\ket{e}$ and $\ket{g}$ \cite{Benatti:2010fv}. Hence, by simple interaction with ancillae, the inaccessible mode entanglement can be transferred to an unconstrained degree of freedom. A realistic scenario based on this idea was proposed for cavity-QED experiments in \cite{Gerry:1996qa}, and successfully implemented, {\it e.g.}, in \cite{Maitre:1997bh} (see \cite{Raimond:2001bs} for a review of related cavity-QED experiments). Other methods for the realization of the proposal include the storage and retrieval of quantum information of one single delocalized photon in an atomic ensemble \cite{Choi:2008fk}.
As insinuated above, it is important to note that the absorption of the particle is indeed crucial for the entanglement between the ancilla particles. If the particle is not absorbed, the final state reads
\begin{eqnarray} \frac{1}{\sqrt 2}\left( \ket{0,1}_{A,B} \otimes \ket{g,e} + \ket{1,0}_{A,B} \otimes \ket{e,g} \right) , \end{eqnarray}
and the density matrix that describes the state of the ancilla particles is a fully mixed state which contains only classical correlations, due to the remaining entanglement with the delocalized particle. One may argue that this problem can be circumvented if that particle is again measured in a coherent superposition of the modes $\frac{1}{\sqrt 2}\left( \ket{A} \pm \ket{B} \right)$, \emph{after} it has interacted with the ancilla particles. In this case, the ancilla state again remains in an entangled state $\frac{1}{\sqrt{2}} \left(\ket{e,g} \pm \ket{g,e} \right)$. The last projective measurement, however, can only be performed after the modes have been superposed at a beam splitter, and constitutes a \emph{global} measurement on the system. Hence, the requirements for a rigorous proof of non-locality by the violation of a Bell inequality are not given \cite{Ashhab:2007if}.
It is to be retained that a rigorous experimental proof for the non-locality of mode-entanglement of photons has been achieved, as discussed above \cite{Hessmo:2004bs,Babichev:2004fv}. Moreover, single-photon entanglement has also been purified \cite{PhysRevLett.104.180504}, and theoretical schemes for the detection of entanglement in one-particle many-mode entangled states are available \cite{al:2009oq}.
For massive particles, the issue is more intricate, since the SSR which inhibits the local rotation of bases is of fundamental nature (see \cite{Cunha:2007fk} for an illustration of the analogies between massless and massive particles). As a way out, a pure reservoir state with broad particle distribution can be shown to be a good reference frame to circumvent the particle number SSR \cite{Ashhab:2009rq,White:2009vn,Heaney:2009jl}, in general. Since the reservoir state in the scheme in \cite{Heaney:2009jl} is shared by the two parties to ensure phase coherence, no strict non-locality conditions are established. Furthermore, the quantum correlations that are measured may have their origin in the entanglement between spatial regions of the reference state, and not necessarily in the coherent delocalization of a single massive particle. Also in this latter case, however, a Bell inequality violation signals the mode-entanglement of massive particles, either stemming from the single delocalized particle, or from the reservoir. Other proposals circumvent the particle-number SSR by using two identical copies of the state under consideration, such that one copy provides a phase reference for the other one \cite{Heaney:2010cr}, and no coupling to a particle reservoir is necessary. The advancement of experimental techniques in the field of ultracold atoms \cite{Bakr:2009oq,Sherson:2010kl,Karski:2009tg,Treutlein:2006ij}, where mode entanglement is naturally present \cite{Heaney:2007nx}, feed the hope that the issue of mode entanglement of massive particles will be resolved in the near future.
\subsubsection{Creation of mode entanglement}
Since a simple transformation of modes can create a mode-entangled state, it is not surprising that a panoply of scenarios have been designed to produce highly mode entangled multipartite states \cite{PhysRevA.55.2564,Pryde:2003fu,Fiurasek:2002ye,Lee:2002rt,Fiurasek:2003lh,PhysRevA.55.2564}. All these schemes build on linear optics and on single-photon detectors, hence consist of transformations of the modes and of projective measurements. In principle, the dimensionality of the subsystem states and the number of parties of a multipartite quantum state which can be obtained with a limited number of particles is unbounded \cite{Fiurasek:2003lh}. Indeed, entanglement was verified, {\it e.g.}, for a four-partite state created with only one single photon \cite{Papp:2009kl}.
\subsection{Entangled degrees of freedom} \label{photontypesent}
Given the physical carriers of entanglement, it remains to choose the degrees of freedom which define the subsystem Hilbert spaces. The spectra associated with these degrees of freedom can be finite and discrete, infinite and discrete, continuous, or composed of, both, a discrete and a continuous part. In general, any degree of freedom which can be measured in different bases, {\it i.e.}~which is not subjected to an uncircumventable SSR (see Section \ref{secSSR}), is a candidate to exhibit measurable entanglement. The actual choice is, however, often determined by the actually achievable experimental control over the respective degrees of freedom.
Photons are particularly versatile carriers of entanglement, since they were shown to exhibit entanglement in various, distinct degrees of freedom. The most prominent example is certainly their polarization degree of freedom. Already before the prevalence of parametric down-conversion \cite{Kwiat:1995mw} which represents today's most versatile source for polarization-entangled photons, the implementation of quantum-informational protocols \cite{Bouwmeester:1997dz} was mainly tested with polarization-entangled photons stemming from the two-photon decay of excited atomic states \cite{Freedman:1972fk,Aspect:1981zr}. Other degrees of freedom which support higher dimensional Hilbert spaces can equally well exhibit entanglement: The orbital angular momentum \cite{Mair:2001fv,Calvo:2007oq}, the position and momentum \cite{Howell:2004fu}, the time \cite{Marcikic:2004qr}, or the frequency axis \cite{Olislager:2010fc}.
Entanglement needs not involve the same degree of freedom in each subsystem, even though these may be represented by identical physical entities (such as photons, ions, molecules). If the degree of freedom that is entangled differs between the subsystems, one talks of \emph{hybrid entanglement}. For example, the polarization of a photon can be entangled to the momentum of another one \cite{PhysRevA.80.042322,PhysRevA.79.042101}, the arrival time to the polarization \cite{fujiwara}, or the orbital angular momentum to the polarization \cite{Nagali:2010ss}.
\emph{Hyperentanglement} (or \emph{multiparameter entanglement}) denotes quantum correlations that involve several, or even all different degrees of freedom of a \emph{single} physical particle: Any particle in a hyperentangled state is correlated not only in one degree of freedom such as polarization, but also in several ones. Hyperentanglement was successfully demonstrated for photons, including two photons that are hyperentangled in three degrees of freedom, namely in their polarization, in their longitudinal momentum, and in their mode \cite{Vallone:2009qc}. Analogous experiments with photons entangled in frequency, wave vector and polarization were reported in \cite{PhysRevA.66.023822,Barreiro:2005ve}. The use of several degrees of freedom of each particle naturally allows to realize high-dimensional states with few particles, hence more information can be encoded in few carriers. One impressive example is given by the 1024-dimensional, ten-qubit hyperentangled state of photons in \cite{Gao:2010tg}, where both the polarization and the mode degree of freedom define the subsystem structure.
\section{Atoms and molecules} \label{atommol}
In terms of the different facets entanglement can exhibit, as well as of the related conceptual issues which need to be dealt with, atoms and molecules represent fascinating physical objects which bear all of the difficulties that were discussed in Sections 2 and 3. In most hitherto existing experiments, however, these systems were isolated and shielded from their environment, and most degrees of freedom were neglected and decoupled from the ones of interest, such as to realize the quantum-information abstraction of two qubits, with great success. However, atoms and molecules naturally offer several other ways to encode entanglement, ranging from atoms that are entangled with each other in their center of mass degree of freedom (see Section \ref{twoatomsent}), over the photons emitted in the course of a de-excitation or of a recombination process (see Sections \ref{atomphotonent}, \ref{photonphoton}), to the very constituents of the atoms, {\it i.e.}~to electrons entangled with nuclei (Section \ref{electronsnuclei}), or electrons entangled with each other (Section \ref{elecelec}). Thereby, they provide unique testing grounds for the direct application of the conceptual tools we discussed above.
The studies that we will review hereafter employ the same physical object, but they illuminate very different aspects of entanglement, and they are motivated by different incitements. For unbound systems such as the products of a decay process, means were established to quantify correlations and experimentally accessible observables were proposed. For bound systems, the direct verification of entanglement is out of reach: Prior to any measurement, the fragmentation of the system has to be induced by some mechanism which necessarily impacts heavily on the quantum correlations of the constituents, and thereby strongly changes their entanglement properties. Alternatively, consequences of entanglement between the constituents of bound systems for other physical phenomena such as the bosonic or fermionic compound behavior were identified, and thereby indirect means for the verification of entanglement were established. Finally, entanglement provides a benchmark for approximation techniques in bound many-body systems, {\it e.g.} electronic entanglement assesses the strength of the exchange-correlation energy in density functional theory \cite{Grobe:1994fk}.
A particular case is the motional entanglement between atomic or molecular fragments. The conservation of momentum and energy, and the isotropy of the system enforces highly correlated fragmentation dynamics in three-dimensional configuration space. The kinematic nature of these correlations in atomic and molecular fragmentation processes provides a rather intuitive understanding of their very existence and their strength. On the other hand, the advances of experimental technologies have brought coherent phenomena in atomic processes on the experimentalists' agenda \cite{Becker:2009oq,Corkum:2007fk,Krausz:2009ve}. Today, a complete mapping of the momenta of all fragmentation products is possible, and very strong and particular correlations have indeed been recorded (see Section \ref{elecexp}). The question whether the interaction between fragments or between fragments and the environment cause the deterioration of entanglement to merely classical correlations in the final (detected) state, or whether \emph{rigorously quantum} correlations persist is, however, open. It constitutes both a challenge for the theory which needs to account for decoherence, and for the experiment, in which one currently cannot perform a rotation of basis in momentum/position space. While several means are available for quantifying correlations, the \emph{nature} of those, classical or quantum, has to be assessed with Bell inequalities or by novel proposals such as the persistence of Cohen-Fano interferences for fragmenting molecules, discussed in Section \ref{elecelec} \cite{Chelkowski:2010hc}.
\subsection{Entanglement between atoms} \label{twoatomsent}
Dichotomic, or qubit-like, entanglement between the \emph{internal} (electronic) states of trapped ions is well established and documented (see {\it e.g.} \cite{Hagley:1997bd}, or \cite{Haffner:2008wo} for a review on the entanglement of trapped ions), while entanglement in the motional degrees of freedom has only started to receive substantial attention \cite{Opatrny:2001ye,Opatrny:2003fk,Savage:2007qo,Guo:2008hc,Sancho:2009zt}.
One proposal to create and verify entanglement in atomic momenta involves the Feshbach dissociation of a diatomic, ultracold molecule \cite{Gneiting:2008bh,Gneiting:2010qf}. Two subsequent magnetic field pulses coherently prepare a molecule in a dissociating state which is given as the superposition of an ``early'' and a ``late'' particle pair (one single pulse does not suffice to dissociate the molecule deterministically), as illustrated in Figure \ref{Gneitingfigure}. The resulting quantum state of the atomic fragments after the two pulses can thus be written as
\begin{eqnarray} \ket{\Psi}=\frac{1}{\sqrt 2} \left(\ket{e,e}+\ket{l,l} \right), \end{eqnarray}
where $\ket{e}$ and $\ket{l}$ denote the quantum states of an atom released by the early, or by the late pulse, respectively. This state has the form of the Bell state given by (\ref{Phimp}), and can be considered as \emph{time-bin} entangled, a type of entanglement that was already verified for photons \cite{Marcikic:2004qr}. This \emph{dissociation time entangled} state can be probed by correlation measurements in different bases, in a Mach-Zehnder interferometer \cite{springerlink:10.1134/S0030400X10020062,springerlink:10.1007/s00340-009-3457-4}, such as to test a Bell inequality.
\begin{figure}[h] \center \includegraphics[width=10cm,angle=0]{fig11.pdf} \caption{Courtesy of C. Gneiting and K. Hornberger \cite{Gneiting:2008bh}. Dissociation-time entanglement scheme \cite{Gneiting:2010qf,Gneiting:2008bh}. (a) A series of pulses creates a coherent superposition of particle pairs with early and late dissociation times. (b,c) With the help of a switch which deflects the early wave component into the long arm of an unbalanced Mach-Zehnder interferometer and leaves the late wave component in the short arm, the wave packets are brought to interference. By changing the phase $\varphi$ and the splitting ratio $\vartheta$, projective measurements onto arbitrary basis states can be performed. The detection of a particle in either one of the detectors is recorded as $\sigma_1=\pm 1$ and corresponds to the projection onto one of the states of the basis specified by the choice of $
\vartheta$ and $\varphi$. Copyright 2008 by The American Physical Society. } \label{Gneitingfigure} \end{eqnarray}d{figure} The interferometer can be substituted by proper adjustment of the magnetic field pulse shapes, such that the late pair has a larger momentum than the early one, and interference between the early and late wave packets occurs naturally. By increasing the delay between the dissociating pulses and thereby between the wave-packets, the technique bears the potential to investigate the length scales over which massive particles can be coherently delocalized. With more intricate pulse protocols, {\it e.g.} the dissociation of the molecule with three or more subsequent pulses into three or more pairs, also entanglement of massive particles involving larger dimensions should in principle be accessible. A bottleneck for the experimental realization of such protocols is the reproducibility of the magnetic pulse sequence for the dissociation of the molecules: This is a crucial ingredient, to ensure a stable relative phase between the early and the late component. On the other hand, the two interferometers do not need to be have a fixed relative phase, which allows macroscopic separations. Additionally, no spatial nor temporal resolution is required for the particle detection. For the proposed molecular BEC produced from a balanced spin mixture of fermionic $^6$Li, a temporal separation of $\tau=1$s between the early and the late component and a dissociation velocity of $v_{\mathrm{rel}}=1$cm/s are feasible \cite{Gneiting:2010qf}.
\subsection{Atom-photon entanglement} \label{atomphotonent}
While the entanglement between particles of different kinds represents the conceptually simplest case (see Section \ref{properties}) and naturally occurs in all interacting systems, it was experimentally verified only recently. A paradigmatic example is provided by two electronic energy levels of an atom (a ``two-level atom'') and a photon which populates a single mode of the quantized electromagnetic field. Such bipartite qubit-entanglement has been realized in cavity-QED \cite{Raimond:2001bs} and in cold atom experiments \cite{PhysRevLett.93.090410,Nature428,PhysRevLett.96.030404,PhysRevLett.101.260403,Matsukevich10222004,PhysRevLett.95.040405}. Since experiment and theory of this conceptually (not experimentally, though!) rather elementary scenario is well established and documented \cite{Raimond:2001bs}, we focus here on advances in high-dimensional entanglement in the external degrees of freedom, {\it e.g.} in the momenta of atoms and photons.
In contrast to molecules that first need to be dissociated by some external field, in order to provide two separate subunits, already the simple spontaneous decay of an electronically excited atom under emission of a single photon constitutes an elementary scattering process in which entanglement between the light field and the atom can be studied.
\subsubsection{Decoherence of atoms due to photon emission}
Such correlations in the momentum of a spontaneously emitted photon with the momentum of the remaining atom, which originate from momentum conservation in the spontaneous-emission process, were measured in \cite{Kurtsiefer:1997kh}. In general, when photons are spontaneously emitted by atoms, the latter effectively loose their coherence, which leads to an observable loss of fringe visibility \cite{Pfau:1994dq,Chapman:1995cr} in interference experiments. This decoherence of the atomic states can be explained by the entanglement of the atoms with the emitted photons and the subsequent loss of these photons and of the information they carry. Formally, this loss corresponds to a partial trace over the photonic degree of freedom, leaving the atom in a mixed state without the potential to exhibit interference. This situation should be contrasted to the scenario discussed in Section \ref{measurementinduced}, in which the emitted photons are detected in a way that their which-way information is lost, and the atoms remain mutually entangled.
\subsubsection{Occurrence and strength of atom-photon entanglement} Under the assumption that the atom has infinite mass and that it is described by a $\delta$-localized wave-function \cite{springerlink:10.1007/BF01336768}, no motional entanglement between the photon and the atom can arise, simply because the atom is fixed in space. A theoretical treatment of the spontaneous decay of an excited atom with emission of one photon shows that entanglement is naturally present in the atom-photon system \cite{Chan:2002gb,Chan:2003ly} when the aforementioned idealization is given up. Like in the case of a bound hydrogen atom discussed in Section \ref{substruct}, the two-particle wave-function can be written in a factorized, {\it i.e.}~unentangled, form when a suitable transformation to collective coordinates is performed \cite{Fedorov:2005bh}:
\begin{eqnarray}
\ket{\Psi_{\mathrm{sys}}} &= & \ket{\Phi_{\mathrm{CM}}} \otimes \ket{\phi_{\mathrm{rel}}} \in \mathcal{H}_{\mathrm{CM}} \otimes \mathcal{H}_{\mathrm{rel}} \nonumber \\
&=& \sum_{j} \sqrt{\lambda_j } \ket{\chi_{\mathrm{at}}^{(j)}}\otimes \ket{\chi_{\mathrm{ph}}^{(j)}} \in \mathcal{H}_{\mathrm{at}} \otimes \mathcal{H}_{\mathrm{ph}} \label{choiceofsub} ,
\end{eqnarray}
where $\ket{\Phi_{\mathrm{CM}}} $ and $\ket{\phi_{\mathrm{rel}}}$ denote wave-functions that describe the center-of-mass-like and relative coordinates of the atom-photon system, respectively (the collective coordinates). On the other hand, $\ket{\chi_{\mathrm{at}}^{(j)}}$ and $\ket{\chi_{\mathrm{ph}}^{(j)}}$ refer to the atom and the photon wave-function, respectively. For a physical choice of subsystems, namely of the atom and the photon, however, entanglement is exhibited, as also immediate from the second line of (\ref{choiceofsub}). Once again, the choice of subsystems is the key issue, as already discussed in Section \ref{substruct}. Such motional entanglement, as quantified by the Schmidt number $K$, (see Eq.~(\ref{Schmidtnumber})), is, in principle, unbounded since position and momentum are continuous degrees of freedom \cite{Keyl:2003uq}. In practice, however, only a finite number of discrete Schmidt modes are occupied, {\it i.e.}~the entanglement is typically rather low (in terms of Section \ref{quantif}, only few Schmidt coefficients $\lambda_j$ do not vanish). This can be understood from a kinematic argument which takes into account the mass discrepancy between the atom and the emitted photon, and the restriction of available phase-space due to momentum conservation. The Schmidt number of the resulting atom-photon entangled state turns out to be inversely proportional to the line width of the transition \cite{Chan:2002gb}. This can be understood intuitively: The narrower the transition, the better the energy of the compound photon-atom system is defined, and the stronger the photon is correlated to the recoiling atom. Specific three-level scattering schemes \cite{Zhu:1995nx} could, in principle, enhance the resulting entanglement between atom and photon \cite{Guo:2006qf,Chan:2003ly,Guo:2007fv}.
As an elementary scattering process, the atom-photon system constitutes hence a rather well understood scenario in which correlations are naturally present, and their strength can be understood from kinematic arguments and conservation laws. Due to this kinematic nature of their genesis, it is, again, the probing of the coherences of the many-particle state that constitutes the biggest experimental challenge.
\subsubsection{Coincidence measurements and wave-packet narrowing}
Under the assumption that all observed correlations are of quantum nature, {\it i.e.}~that the quantum state under consideration is a pure state, correlations in position can be used to quantify the entanglement between the atom and the photon \cite{Fedorov:2005bh,al:2006fu}. This very assumption is, however, debatable, since an analysis based on correlations recorded in one specific basis does not allow to exclude a merely classically correlated state (see Section \ref{Bellin}).
Anomalous narrowing and broadening of the atomic and photon wave packets can be quantified by coincidence and single-particle measurement schemes \cite{Fedorov:2005bh}. In other words, when the position of the nucleus is known, the state of the photon is largely determined, whereas without the information on the nucleus' state, the width of the observed photonic wavepacket is considerably larger.
This argument can be made quantitative by studying the coincident probability distribution $P(r_{\mathrm{ph}}, r_{\mathrm{at}},t)$ of the photon (ph) and the atom (at), which are obtained by fixing the position of one detector while varying the position of the other, and by counting only coincident signals from both detectors. From the joined atom-photon probability distribution $P(r_{\mathrm{ph}}, r_{\mathrm{at}},t)$, the marginal distributions can be obtained,
\begin{eqnarray} P(r_{\mathrm{ph}},t)=\int \mathrm{d} r_{\mathrm{at}} P(r_{\mathrm{ph}}, r_{\mathrm{at}},t) , \\
P(r_{\mathrm{at}},t) =\int \mathrm{d} r_{\mathrm{ph}} P(r_{\mathrm{ph}}, r_{\mathrm{at}},t) , \end{eqnarray}
which describe the wavepackets of the photon and the atom, respectively, when no assumptions on the remaining particle is made, such that an integration over the coordinate not under consideration is effectuated.
We denote the width, {\it i.e.}~the variance of the probability distributions, by
\begin{eqnarray}
\Delta r_{\mathrm{ph}}^{(c)/(s)}(t) , \ \ \ \Delta r_{\mathrm{at}}^{(c)/(s)}(t) ,
\end{eqnarray}
for the photon and the atom, respectively, in the coincidence scheme (c) or for the marginal distribution (s). The ratio
\begin{eqnarray} R(t)=\frac{\Delta r_{\mathrm{ph}}^{(s)}(t) }{\Delta r_{\mathrm{ph}}^{(c)}(t)} =\frac{\Delta r_{\mathrm{at}}^{(s)}(t) }{\Delta r_{\mathrm{at}}^{(c)}(t)}, \end{eqnarray}
then quantifies the correlations of the wave-functions and, thereby, the \emph{spatial wave-packet narrowing}: A smaller width in coincident detection as compared to the marginal distribution is a consequence of correlations; by construction, $R(t) \ge 1$. Wave-packet narrowing can also be understood as ``measurement-induced localization'' \cite{Freyberger:1999fk}. For $t=0$, $R$ corresponds to the Schmidt number $K$ (see Eq.~(\ref{Schmidtnumber})): $K=R(t=0)$. The entanglement inscribed into the system, and thereby the value of $K$, necessarily remain constant, since the particles are non-interacting. However, the spreading of wave packets leads to temporally increasing $R(t)$ \cite{al:2006fu}. The presented scheme probes, however, no coherences, but only correlations, as immediate from the probability distributions given only in terms of position coordinates. Only if both the momentum and the position can be measured, a full realization of the EPR paradox becomes feasible \cite{Reid:2009bs}.
\subsubsection{Decoherence of the photon-atom system}
The interaction of the atom with other background photons subsequent to photon emission leads to decoherence of the entangled atom-photon system \cite{Guo:2007fv}. The timescale $\Delta t_{\mathrm{dis}}$ of disentanglement can be estimated by modeling the environment as background photon bath. It turns out to be inversely proportional to the average number of resonant photons in the heat bath and can thus be increased by decreasing the temperature of the environment. On the other hand, the disentanglement time $\Delta t_{\mathrm{dis}}$ is also inversely proportional to the Schmidt number $K$. Hence, the timescale depends itself on the initial entanglement of the state, and the more entangled the state, the faster its entanglement is lost. The inclusion of other single body coherent effects like the dispersion of the atomic wave packet are shown to be of no relevance for the entanglement in the system \cite{Chan:2003ly}.
\subsection{Photon-photon entanglement} \label{photonphoton}
Two-photon emission in de-excitation transitions in atoms and ions has been the first source of polarization-entangled photons \cite{Freedman:1972fk,Aspect:1981zr,Aspect:1982ly,Haji-Hassan:1989cr,Perrie:1985fk}. Due to conservation of energy, momentum, and angular momentum, photon pairs that are emitted in such processes are, however, not only entangled in polarization, but also highly correlated in energy and angular direction ({\it i.e.}~in frequency and linear momentum).
Correlations are not only present within identical degrees of freedom, but the system also exhibits hybrid entanglement (see Section \ref{photontypesent}), {\it i.e.}~entanglement between different degrees of freedom such as the polarization of one photon and the angular direction of the other. Furthermore, the strength of polarization-entanglement may depend on the emission angle, on the fraction of the available energy carried by one photon, and on the nuclear charge, as shown in the theoretical studies reviewed hereafter \cite{Radtke:2008eu,2010arXiv1006.4799F,Surzhykov:2001ly,Surzhykov:2005ly,Borowska:2006nx}.
\subsubsection{Experimental progress}
Experimental studies of the correlations of photons emitted in a two-photon decay have recently concentrated on highly charged ions, for which relativistic effects become important. For such systems, only the total decay rates and excited state life times were available until two decades ago \cite{Dunford:1989ve,Drake:1986qf}. The spectral distribution of the emitted photons was resolved later \cite{Schaffer:1999bh,Derevianko:1997dq,Jentschura:2008dq}, and today the spectrum of the two photons emitted, {\it e.g.}, in the recombination of $K$-shell vacancies of silver \cite{Mokler:2004bh} and gold \cite{Dunford:2003tg} can be measured, and are found in agreement with theoretical predictions \cite{Tong:1990ve}. Indeed, the experimental accuracy is sufficient to verify relativistic effects, {\it e.g.} in the two-photon spectrum of He-like tin \cite{Trotsenko:2010nx}.
This important progress in the experimental capabilities \cite{al:2005ys,Mokler:2004cr} permits today to detect photons in a wide range of energies, and to extract differential cross-sections as a function of the energy redistribution among the photons, and of the opening angle, even for photons in the X-ray regime. Also the polarization of hard X-rays, for example in radiative electron capture into the K-shell of highly-charged uranium ions \cite{al:2009vn,Tashenov:2006fk}, becomes accessible. In the next years, the observation of polarization-correlations of photons in the X-ray domain is therefore likely to become possible. Thus, experiments analogous to those which measure correlations of low-energetic photons in transition processes of light atomic species, as already performed three decades ago \cite{Freedman:1972fk,Aspect:1981zr,Aspect:1982ly}, will soon be feasible for heavy nuclei, and for photons in the X-ray regime. These developments have triggered an extensive research agenda with the goal to fully characterize the emitted photon pair \cite{al:2007kx} in general atomic two-photon decay processes.
\subsubsection{Theoretical studies}
The first issue that was addressed by theoretical studies is the dependence of the photon polarization properties on the emission properties, {\it i.e.}, instead of back-to-back emission, an arbitrary geometry was considered. Already the polarization of a single emitted photon depends on its emission angle with respect to the polarization axis of the atom \cite{Surzhykov:2001ly}. If two photons are emitted, the entanglement between their polarization degrees of freedom strongly depends on the relative angle of emission \cite{Radtke:2008eu,2010arXiv1006.4799F}, as also shown in Figure \ref{surzy}. It is also apparent that the photons also exhibit angular correlations, {\it i.e.}~they are not emitted isotropically.
\begin{figure}[h] \center \includegraphics[width=14.5cm,angle=0]{fig12.pdf} \caption{Courtesy of T. Radtke \cite{Radtke:2008eu}. Two-photon angular correlation (upper panel) and polarization entanglement quantified by the concurrence (lower panel), in the $2s_{1/2}\rightarrow 1s_{1/2}$-decay of initially polarized atomic hydrogen, {\it i.e.}~for which only one magnetic sublevel is populated. All values are given as a function of the angle $\Theta_2$ between the polarization axis of the atom (the quantization axis) and the first photon. Data are shown for different values of $\Theta_1$ -- the angle between the second photon and the polarization axis of the atom, and $\varphi_2$ -- the emission angle of the second photon with respect to the axis perpendicular to the plane spanned by the atom polarization and the emission direction of the first photon. Angular correlations and the entanglement depend strongly on the particular decay geometry defined by the three angles $\Theta_1, \Theta_2, \varphi_2$. The probability to find photons at a given emission angle is proportional to the angular correlation function, which is shown here in arbitrary units. Copyright 2008 by The American Physical Society.} \label{surzy} \end{eqnarray}d{figure}
In the non-relativistic dipole approximation, the Schr\"odinger-equation is solved under the assumption of the interaction of the electrons with a spatially homogeneous electric field (the wavelength of the photon is much larger than the dimension of the atom) \cite{Goppert-Mayer:1931fk}. In this model, the
polarization-state of the two photons can simply be written as \cite{2010arXiv1006.4799F,1011.5816}
\begin{eqnarray} \ket{\Psi_{2\gamma}} = \frac{1}{1+\cos^2 \theta} \left( \ket{H,H}+\cos \theta \ket{V,V} \right) \label{nonrel} \end{eqnarray}
where $\ket{H}$ and $\ket{V}$ denote horizontal and vertical polarization with respect to the emission plane of the photons, $\theta$ is their opening angle, and the factor $1/(1+\cos^2 \theta)$ ensures normalization. For back-to-back emission, {\it i.e.}~$\theta=\pi$, the state is maximally entangled, while perpendicular emission, {\it i.e.}~$\theta=\pi/2$, leads to a non-entangled two-photon state. The entanglement properties of a photon pair emitted in such extreme configuration ($\theta=\pi$ or $\theta=\pi/2$) can also be derived by taking into account uniquely the conservation of angular momentum, they hence do not provide information about microscopic details of the decay process.
Whereas, in practice, the total two-photon decay rates are dominated by the dipole-transition and higher multipoles do not contribute significantly, the latter can have a considerable influence on the \emph{correlations} that are carried by the emitted photons. The inclusion of relativistic and non-dipole effects -- in an approach based on the relativistic Dirac-equation in which the electric field is not assumed to be spatially constant \cite{Au:1976zr} -- leads to a prediction that significantly differs from (\ref{nonrel}) \cite{Surzhykov:2005ly,Borowska:2006nx,Radtke:2005oq}. In particular, the symmetry of the emission with respect to $\theta=\pi/2$, evident from (\ref{nonrel}), is broken by non-dipole contributions. This effect is further enhanced for large nuclear charges $Z$. Hence, multipole and relativistic effects do not only manifest in strictly dipole-forbidden decay channels \cite{1011.5816}.
In an experiment, the initial state preparation of the atoms or ions is typically not under perfect control, and one needs to account for an incoherent mixture of the magnetic sublevels. This mixedness can jeopardize the entanglement in the final photonic state. The 2$s_{1/2}\rightarrow 1s_{1/2}$ transition, however, is unaffected by the incoherent population of magnetic sublevels: Due to angular momentum conversation, the final two-photon state remains pure. In contrast, the incoherent preparation of the initial state leads to a mixed final state for the $3d_{5/2}\rightarrow 1s_{1/2}$ transition. The resulting, mixed two-photon sate is of Werner type (see Section \ref{Bellin}), under certain geometries \cite{Radtke:2008gf}.
Beyond correlations in the polarization degree of freedom, entanglement also prevails in other degrees of freedom. For example, the photon-photon emission angle is correlated with the polarization of one photon - independently of the polarization of the other one \cite{Surzhykov:2009ve}, {\it i.e.}~the system exhibits hybrid entanglement, similarly as in the studies discussed in Section \ref{photontypesent}. As concerns the verification of the quantum nature of such correlations, a change of basis is achievable in the polarization degree of freedom, though in the angle a change of basis was realized so far only for photons in the optical range \cite{DAngelo:2004fk,Howell:2004uq}.
The above studies have shown that photon-photon entanglement may display a much more intricate behavior than one would expect by considering the conservation of angular momentum alone. Relativistic effects significantly alter the symmetry properties of the system, and thereby lead to observable consequences in the angular correlations and the polarization entanglement. An inclusion of many-body effects \cite{Surzhykov:2010uq} into the theory promises the future design of probes of effects like parity violation \cite{Dunford:1996dq}.
\subsection{Entanglement between electrons and nuclei} \label{electronsnuclei}
The choice of electrons and nuclei as carriers of entanglement allows to consider bound systems as well as fragmented ones. The fragmented systems exhibit close analogies to the arguments presented in Section \ref{atomphotonent}, where photon-atom entanglement is discussed, adding final-state interaction and of the finite mass of the electron to the problem. Again, kinematic intuition can explain the strength of momentum-correlations: The discrepancy of the masses of the electron and the nucleus leads to a rather low value of entanglement, whereas an analogous scenario of dissociating molecules with balanced fragment masses results in stronger entanglement \cite{Fedorov:2004kx}. In this investigation, besides the momentum of the incident photon, also the Coulomb interaction in the final state is neglected, hence the effect of the possible deterioration of entanglement due to interaction is not taken into account. The evolution of the state and of the entanglement between the two particles is thereby constrained by the free-particle two-body momentum and energy conservation.
\subsubsection{Bound systems: Composite behavior}
Electrons bound by a nucleus are naturally entangled with it \cite{Tommasini:1998qf} in their external degrees of freedom. As we will discuss hereafter, such entanglement between the constituents of any bound system can be related to the compound's bosonic or fermionic behavior and thereby provides an important indicator for the strength of effects due to the composite nature of particles. With the experimental realization of Bose Einstein Condensation (BEC) with bosonic atoms, the question naturally arises to which extent composite particles can be treated as elementary bosons \cite{Avancini}, and under which conditions the underlying constituents, {\it i.e.}~the electrons, will manifest themselves and jeopardize the bosonic behavior of the compound.
Intuitively, one would relate the \emph{density} of such composite particles to their bosonic or fermionic behavior: Roughly speaking, if the wave-functions of the electrons of different atoms start to overlap considerably, one would expect their fermionic character to inhibit the composite system to behave as a boson. An upper limit for the occupation number of composite boson states was derived formalizing this idea \cite{Rombouts:2002uq}, and applied to the problem of trapped hydrogen atoms. Typical maximal occupation numbers of the ground state are shown to lie in the range of $10^{13}$, while current experiments reach $10^6-10^9$, well below this bound. Composite bosonic behavior is, on the other hand, not restricted to systems in which the constituents are close in space and bound by interaction: Also non-interacting and spatially separated biphotons can exhibit composite-particle properties, as demonstrated by experiments measuring the de Broglie wave-length of an ensemble of entangled photons, defined as $\lambda/n$ where $\lambda$ is the average wave-length of the photons, and $n$ the average number of photons in the ensemble \cite{Fonseca:1999uq,Jacobson:1995kx}. Hence, compositeness is not limited to spatially bounded particles; the density of particles and the overlap of the wave-functions of the constituents cannot constitute a universal criterion for composite behavior.
The entanglement between the constituent particles provides such a key quantity that defines to what extent the bosonic/fermionic commutation relations for creation operators of composite particles are valid \cite{Law:2005dq,Chudzicki:2010bh}. Given a composite system $C$ of two particles of type $A$ and $B$, which are either both fermions or bosons, the compound quantum state can be written in Schmidt decomposition as
\begin{eqnarray} \ket{\Psi_C}= \sum_{n=0}^\infty \sqrt{\lambda_n} \ket{\phi_{A,n}} \ket{\phi_{B,n}} .\end{eqnarray}
The creation operator for a composite particle then reads
\begin{eqnarray} \hat c^\dagger = \sum_{n=0}^\infty \sqrt{\lambda_n} \hat a^\dagger_n \hat b^\dagger_n ,\end{eqnarray}
where $\hat a^\dagger_n$ creates a particle in the state $\ket{\phi_{A,n}}$, and analogously for $\hat b^\dagger_n$.
The commutation relation of creation and annihilation operators $\hat c$ and $\hat c^\dagger$ becomes \cite{Law:2005dq}
\begin{eqnarray} [\hat c, \hat c^\dagger ] = 1 + s \Delta ,\end{eqnarray}
where $s=1 (-1)$ when $A$ and $B$ are bosons (fermions). Due to the term \begin{eqnarray} \Delta= \sum_{n=0}^\infty \lambda_n(\hat a_n^\dagger \hat a_n + \hat b_n^\dagger \hat b_n)\ , \label{DeltaD}\end{eqnarray} the creation/annihilation operators $\hat c^\dagger$ and $\hat c$ are not strictly speaking bosonic operators. By analyzing quantum states of many composite $C$-particles, it turns out that the Schmidt number $K$, (\ref{Schmidtnumber}), is a direct indicator for the composite behavior: The quotient $N/K$, where $N$ is the number of composite C-particles in one quantum state, needs to be very small in comparison to unity in order for the composite particles to behave as bosons, {\it i.e.}~the larger the entanglement, the better the composite particle behaves as a boson. This can also be retraced intuitively from the structure of $\Delta$ in (\ref{DeltaD}): This expression is indeed a combination of the Schmidt coefficients $\lambda_n$ with the number operators $\hat a^\dagger_n \hat a_n$ and $\hat b^\dagger_n \hat b_n$. The larger the Schmidt number $K$, the smaller are the individual Schmidt coefficients and, consequently, also the expectation value of $\Delta$.
For the hydrogen atom, it can be shown that the particle density above which bosonic behavior breaks down, as obtained by such entanglement-based analysis \cite{Chudzicki:2010bh} roughly corresponds to the value previously derived in \cite{Rombouts:2002uq}. The formalism can also be applied to other composite bosons such as Cooper pairs \cite{PhysRevA.75.043613}, and to the behavior of non-elementary fermions \cite{Sancho:2006nx}.
The above treatment of the composite behavior has shown that certain problems can have a universal solution under a quantum-information perspective that is not restricted to a particular situation. Furthermore, the composite behavior can, in principle, be probed experimentally, and thereby the entanglement between constituents becomes accessible even without breakup of the whole system.
\subsection{Electron-electron entanglement} \label{elecelec}
Electrons as potential carriers of entanglement provide a rich Hilbert space structure with bound and unbound spectra of compounds. They can carry entanglement in their spin and spatial degrees of freedom, and entanglement can result from interactions, or it can be induced by measurements \cite{Tichy:2009kl,Vedral:2003uq}. Due to the prevalence of the long-range, inter-electronic Coulomb-interaction over spin-spin effects, however, most theoretical studies concentrate on the entanglement in spatial (external) degrees of freedom. Since the electron-electron-interaction varies in relative strength with respect to the electron-nucleus-interaction, depending on the charge of the nucleus, both theory and experiment can, in principle, access very different regimes. Indistinguishability plays an important role, since the electronic wave-functions typically overlap in space and the (anti)symmetrization of the spatial part of the wave-function thereby becomes relevant.
Both the rigorous theoretical description and the experimental verification of entanglement remains, however, a difficult challenge in this setting: In the experiment, the spins of ejected electrons cannot be measured since the methods known from solid-state physics \cite{Burkard:2007ly} cannot be directly implemented for free electrons. For entanglement verification in the external degrees of freedom, a measurement in a second basis in addition to momentum cannot be implemented with state-of-the-art technology, since the realization of a basis rotation analogous to the one realized for photons in \cite{DAngelo:2004fk,Howell:2004uq} needs the implementation of electron lenses with strengths much superior than currently available. One is therefore restricted to measurements of mere momentum-correlations.
In theory, the many-body wave-function for many-electron atoms is necessary for entanglement studies. While first steps were performed which allow to compute the degree of entanglement via the electron density in special models such as the fermionic Hubbard model \cite{Franc-ca:2008zr,franca}, such treatment is, up to now, not available for atoms. The computation of the realistic many-particle wave-function requires, however, large if not prohibitive numerical effort. Due to the non-integrability of any non-hydrogenlike atom, theoretical studies of multi-electron systems have, so far, mainly focused on exactly solvable model atoms. While such models differ strongly from real multielectron atoms - as concerns the interelectronic interaction and the definition of the confining potential, they allow insight in some qualitative features. They also constitute a testing ground for approximate techniques \cite{Amovilli:2003zr,Coe:2008kx,Maiolo:2006kx,Dehesa:2010dq} which may help to study entanglement in more realistic atomic models in the future.
\subsubsection{Harmonic confinement and interaction: The Moshinsky atom}
One exactly solvable model-system for two bound electrons is the Moshinsky atom \cite{Moshinsky:1968nx,Moshinsky:1968ly} which was initially introduced to characterize the reliability of the Hartree-Fock approximation in interacting systems. In the model, both the confinement and the interaction are harmonic. The Hamiltonian of the system reads (with masses and actions measured in units of $m$ and $\hbar$, respectively)
\begin{eqnarray}
\hat H=\frac{1}{2} \left(\Omega^2 \hat x_1^2 + \hat p_1^2 \right) + \frac{1}{2} \left(\Omega^2 \hat x_2^2 + \hat p_2^2 \right) + \frac{\omega^2}{2} (\hat x_1-\hat x_2)^2 ,
\label{MoshinskyHamiltonian}\end{eqnarray}
where we denote the spring constant of the confining (inter-electronic) interaction with $\Omega$ ($\omega$). In center-of-mass and relative coordinates,
\begin{eqnarray}
\hat X=\frac{1}{\sqrt 2} \left( \hat x_1+\hat x_2 \right), \ \ \hat x=\frac{1}{\sqrt 2}\left(\hat x_1- \hat x_2\right) ,
\end{eqnarray}
the Hamiltonian factorizes in two independent harmonic oscillators in $\hat X$ and $\hat x$, with frequencies $\omega$ and $\sqrt{2\Omega^2+\omega^2}$, respectively, which permits a closed analytic solution of the problem: The wave-function factorizes in these coordinates, and can be written as
\begin{eqnarray} \ket{n_{\mathrm{rel}}, n_{\mathrm{CM}}} = \ket{n_{\mathrm{rel}}} \otimes \ket{n_{\mathrm{CM}}} ,\end{eqnarray}
where the quantum numbers $n_{\mathrm{rel}}$ and $n_{\mathrm{CM}}$ denote the excitation in the relative and in the center-of-mass coordinate, respectively. The relative strength $\tau=\omega/\Omega$ of the interelectronic interaction determines the reliability of the Hartree-Fock calculation. For $\tau=1$, the ground-state energy in the Hartree-Fock approximation amounts to approximately 95\% of the exact value \cite{Moshinsky:1968nx}. This is consistent with an overlap of 94\% between the exact and the Hartree-Fock wave-function \cite{Moshinsky:1968ly}.
Some basic intuition for the particle entanglement in this system is gained under the assumption that the electrons are distinguishable and do not carry spin \cite{Amovilli:2004fk}. Then only the relative strength of the interelectronic interaction $\tau$ affects the entanglement of the electrons. The entropy of the one-electron reduced density matrix, (\ref{eoff}), can be extracted for the ground state by purely analytical means and grows monotonically with $\tau$.
A slightly more complex model is realized by taking into account the spin degree of freedom and the indistinguishability of electrons \cite{Yauez:2010fk}. Due to vanishing spin-spin and spin-orbit interaction, the total wave-function factorizes in a spatial, $\phi(x_1,x_2)$, and a spin-component, $\chi(\sigma_1,\sigma_2)$. The symmetry of the spin-part of the wave-function directly governs the symmetry of the spatial part, since the two terms must be of opposite parity in order to obtain an antisymmetric compound wave-function. Due to this dependence, one would expect that the parity of the spin part thereby governs the entanglement of the system, but it turns out that it is the relative orientation of the spins (parallel or antiparallel) which is a relevant parameter for entanglement, as evaluated with the help of the properties of the single-particle reduced density matrix. The electron-electron entanglement vanishes in the ground state for $\tau \rightarrow 0$, and it is maximal for $\tau \rightarrow \infty$. For the excited states, however, the entanglement shows a discontinuous behavior at $\tau=0$: In the limit of very small interactions, {\it i.e.}~$\tau \rightarrow 0$, the entanglement remains finite in the case of antiparallel spins, only for $\tau=0$ the system is non-entangled again. In this case, the system is non-entangled due to the degeneracy of energy-eigenstates. In contrast, for parallel spins, the system shows always a continuous behavior in the limit of small relative interaction strengths and vanishes for $\tau\rightarrow 0$. As demonstrated by this study \cite{Yauez:2010fk}, the indistinguishability of particles adds qualitatively new features as compared to the model of distinguishable particles. A physical explanation for the exhibited, rather intricate behavior has, however, not yet been achieved.
\subsubsection{Coulombic interactions}
The harmonic interactions considered in the last Section permitted an analytical treatment of the model-atoms and a straightforward evaluation of entanglement measures. The next step towards a more realistic scenario consists in including the Coulomb interaction into the model. Already for two electrons bound by one nucleus, however, an analytical treatment is impossible, and the numerical treatment is difficult due to the triple collision singularity, {\it i.e.}~the attractive fixed point of the classical phase-space flow for which both electrons fall into the nucleus, symmetrically, along the collinear axis. Only strongly simplified models have been in the focus of recent studies. Since the nonregularizable triple collision singularity constitutes one of the dynamical key features of the three-body Coulomb problem \cite{0953-4075-40-8-F01,Tanner:2000vn,Choi:2004ys,Byun:2007zr}, its neglect eliminates one of the sources of classically chaotic dynamics in the three body dynamics. Chaos, however, implies the abundance of avoided crossings on the spectral level, and the latter have been shown to be associated with entanglement in interacting many-particle systems \cite{Venzl:2009kl}.
One-dimensional helium is studied in \cite{Carlier:2007uq}, where the two electrons are restricted to a one-dimensional space, such that they are always on opposite sides of the nucleus (the $eZe$-configuration). Effectively, one deals with a one-dimensional atom with charge $Z=2$ and two spin-free electrons. The Hamiltonian thus reads
\begin{eqnarray}
H= \frac{\hat p_x^2}{2}-\frac 2 {\hat x} + \frac{\hat p_y^2}{2} - \frac 2 {\hat y} + \frac{1}{\hat x+\hat y} ,
\end{eqnarray}
where $x,y>0$ denote the distance of the two electrons to the nucleus, and the Coulomb interaction between the electrons is described by $1/(\hat x+\hat y)$. The two-particle wave-functions, on which the analysis of the entanglement in the system is based, are obtained by using the analytically exact solutions of the single-electron problem, \begin{eqnarray}
\left( \frac{\hat p_x^2}{2} - \frac{2}{x} \right) \phi_n(x) = E_n \phi_n(x) .
\end{eqnarray}
The Hamiltonian $H$ is then evaluated in the truncated product basis,
\begin{eqnarray}
\Psi_{i,j}(x,y)=\phi_i(x) \phi_j(y) , \end{eqnarray}
and it is diagonalized numerically. The electrons always remain spatially separated, no overlap of their wave-functions in space can occur, and thereby the indistinguishability does not play any role. The singularity between the electrons and their vanishing overlap does not prevent entanglement to be generated, and, again, the entanglement grows with the state's energy.
Other simplified model atoms include the Crandall atom (harmonic confinement and ${1}/{r^2}$-interaction), and the Hooke atom (harmonic confinement and Coulomb interaction, a good description for two electrons in a quantum dot) \cite{al:2010ys}. The indistinguishability and the spin degree of freedom of the particles were included, but no spin-orbit interaction. The features already encountered for the Moshinsky atom in \cite{Yauez:2010fk} were reproduced qualitatively by both models: The entanglement grows with relative interaction strength and with the excitation. For antiparallel spins, the limit of vanishing interaction strengths does not necessarily yield a non-entangled state. The entanglement measures in \cite{al:2010ys} are identical to those used in \cite{Yauez:2010fk}, and it remains open whether this discontinuity effect has to be considered an artifact of the entanglement measures that are used, or whether a physical explanation will be provided in future.
A numerical study of entanglement \cite{al:2010ys} can also be performed on helium-like atoms using eigenfunctions of the Kinoshita type \cite{PhysRev.105.1490}, which are a modification of the Hylleraas expansion \cite{Hylleraas:1928uq}. In the study \cite{al:2010ys}, the wave-function is approximated with functions of the following form \cite{Koga:1996ij},
\begin{eqnarray}
\ket{\Psi_N}=e^{- \chi s} \sum_{j=1}^N c_j s^{l_j/2} \left( \frac{t}{u} \right)^{m_j} \left(\frac u s\right)^{n_j/2} ,
\end{eqnarray}
where $s=|\vec r_1|+|\vec r_2|$, $t=|\vec r_1|-|\vec r_2|$ and $u=|\vec r_1-\vec r_2|$ naturally satisfy the triangle inequality, $s\ge u \ge |t|$. The exponent $\chi$, the mixing coefficients $\{c_j \}$ and the non-negative integers $\{l_j, m_j, n_j \}$ are parameters that are determined numerically \cite{Koga:1996ij}. The entanglement of the ground state is again lower than the entanglement of the triplet or singlet excited states. The ground-state entanglement decreases with increasing nuclear charge $Z$, which can be understood, again, as caused by the relative decrease of the electron-electron-interaction strength.
\subsubsection{Quantum critical points}
A connection between the behavior of entanglement at quantum critical points and at the ionization threshold in a few-electron atom is drawn in \cite{Osenda:2007kx} for the entanglement in the ground state of a spherical helium model in which the Coulombic repulsion between electrons is replaced by its spherical average. In this, again, effectively one-dimensional model, the von Neumann entropy (see (\ref{eoff})) of the reduced density matrix of one electron in the Helium ground state exhibits singular behavior at the critical point, {\it i.e.}~for the value of $Z$ for which the system becomes unbound. For the excited triplet state \cite{Osenda:2008fk}, the scaling properties of the von Neumann entropy is shown to be qualitatively different: Due to the opposite symmetry of the spatial part of the wave-function (antisymmetric ground state and symmetric triplet state), the entanglement is a continuous function of the nuclear charge. This can be understood from the fact that the ionized system possesses the same symmetry as the triplet state, while the symmetry of the ground state and the unbound system are distinct.
\subsubsection{Conclusions}
In general, while the quantum information tools presented in Section 2 are today available to study the entanglement of electrons in atoms, the complexity of many-electron atoms has not yet permitted a quantitative and realistic treatment. Still, a consistent picture that provides kinematic intuition emerges from the available studies: The entanglement grows, as expected, with the interaction between the electrons and with the energy, or principal quantum number, of the state. This observation is consistent with the fact that the Hartree-Fock approximation deteriorates when one considers highly excited states \cite{Moshinsky:1968nx}. These properties are contained in all models that were considered so far. Also, a growing spatial separation of two particles is not a direct indicator for a smaller degree of entanglement they exhibit: Indeed, in the above models, the typical distances grows with larger energies, and so does the entanglement. Given the qualitative agreement of all existing studies this tendency is expected to persist in future studies with more accurate electronic wave-functions. However, with the onset of chaotic dynamics, qualitatively new features can be expected \cite{Venzl:2009kl,Garcia-Mata:2007kl,Benenti:2009oq,Gopar:2008tg,Wang:2004ij}.
The state energy and entanglement were also shown to be not always monotonously dependent on each other, even in these simplified models \cite{Yauez:2010fk}. Entanglement thus offers an additional, independent analytical quantity in the study of atoms. The interplay of interaction and indistinguishability of particles has lead to interesting first results which we, however, do not yet have intuitive understanding for. It is unclear how the indistinguishability will manifest in systems for which more degrees of freedom, spin-orbit-coupling and other relativistic effects are incorporated.
\subsubsection{Entanglement of ionized electrons} \label{ionizedelectrons}
In the process of single-photon double-ionization of atoms \cite{Chandra:2004ij}, a Werner state in the spins \cite{Werner:1989ve} emerges (see Eq.~(\ref{WernerState})). In the case of vanishing coupling between spins and angular momenta, the entanglement of the final state is fully determined by conservation laws and thereby totally independent of the details of the process. It can thus be inferred by the sole knowledge of the electrons' energies. In contrast, spin-orbit interaction induces dependences of the entanglement on the incident photon's polarization \cite{Chandra:2006kl}. The same argument applies also for the spin correlations of electrons ejected in the photoionization of linear molecules \cite{Chandra:2004dq}: They turn out to be widely independent of their spatial correlations, and, in the absence of spin-dependent interactions, the degree of entanglement can be predicted purely from conservation laws. It does, however, sensitively depend on the kinematic details of the process when spin-orbit interactions are included and thereby spatial symmetries are broken.
\subsection{Experiments with electrons: From two-center interference to two-particle correlations} \label{elecexp}
The above theoretical studies on entanglement between the spins of ejected electrons show again that a simple picture based on conservation laws has to be refined when a more realistic and intricate interaction is considered. The verification of spin-correlations between ejected electrons is, however, out of reach for current technology. In contrast, experimental progress has permitted detailed measurements of momentum-correlations between electrons, as we discuss hereafter.
\subsubsection{Cohen-Fano interference}
The two-center interference of \emph{single} electrons has been a subject of intense theoretical and experimental studies. With the advance of reaction microscopes, the implementation of the original proposal of the Cohen-Fano interference in the photo-ionization of molecules has finally become possible. Oscillations were predicted in the cross-section which depend on the ejection angle of the photoelectron with respect to the molecular axis \cite{PhysRev.150.30}. These can be interpreted as the result of the interference of two inequivalent paths induced by the presence of the two nuclear centers and the previous delocalization of the electron at these two centers.
Interference patterns, predicted by theoretical calculations of the photoionization of homonuclear molecules (see, {\it e.g.}, \cite{al:2009kl,Fernndez:2009qa} for recent discussions) could be verified in experiments that use photoionization \cite{Okunishi:2009fu} or other scattering mechanisms such as impact ionization \cite{al:2010mi,Hargreaves:2009dz,Galassi:2002ly,Sarkadi:2003qf}.
The mechanisms which lead to the breakdown of the interference pattern are also rather well understood: For asymmetric molecular configurations, partial localization of the emitted electrons takes place \cite{al:2010fv}, as also probed experimentally for CO$_2$ in \cite{Sturm:2009ij}. Due to the breaking of the symmetry, a preferred direction emerges which also leads to an asymmetric electron emission. Such asymmetry can also be induced for homonuclear molecules simply by the polarization of the absorbed photon \cite{Martin:2007pi}.
In such processes, also decoherence of the single-electron states plays an important role \cite{Zimmermann:2008dq}. In order to observe the interference pattern, the wavelengths of the directly ejected and the scattered wave need to coincide, and an interaction with one center which affects one of the pathways can jeopardize this coherence. Indeed, for large kinetic energies, the interaction at one center is strong enough to inhibit interference of the two pathways, and the interference pattern is lost.
Despite being a single-particle interference effect, a connection of Cohen-Fano-interference to entanglement was recently proposed in \cite{Chelkowski:2010hc}. The scheme aims at probing the delocalization of the electronic wave-function over separated nuclei and thereby at probing the persistence of mode entanglement. The pump-probe-protocol first dissociates a $H_2^+$ molecule. After a certain delay time, in which the fragments fly apart, the probe pulse eventually also photoionizes the fragments and the ejected electron is recorded. The persistence of Cohen-Fano interferences for large delay times would show that even at large internuclear distance the electron can be well delocalized over two separated nuclei, whereas its absence would indicate that the state needs to be described by a classical mixture. In this sense, Cohen-Fano interferences witness mode-entanglement of a single delocalized electron: Such state of two separated nuclei and a single electron that is in a coherent superposition of being bound by either one of them realizes single-particle mode entanglement (see Section \ref{modenen}), this time, however, with a massive, charged particle. The experimental proposal hence opens a route to probe whether there are bounds to the distances over which massive particles can be delocalized, and its realization bears the potential to contribute to the debate on mode-entanglement mentioned in Section \ref{modenen}.
\subsubsection{Two-electron measurements} \label{twoelectronsmol}
The success in the verification of coherent \emph{single}-particle effects \cite{Zimmermann:2008dq} has recently also lead to a consideration of the impact of coherent \emph{many}-particle phenomena. The treatment of the photo-induced breakup of the H$_2$-molecule was proposed in \cite{W.-Vanroose:2005rt} and experimentally performed in \cite{al.:2007df} using the Cold Target Recoil Ion Momentum Spectroscopy (COLTRIMS) \cite{Drner:2000kl} technique. In this experiment, molecular hydrogen is doubly photoionized, and all reaction products are collected, such that a full reconstruction of all momenta takes place. This permits also to measure the orientation of the molecule in space, and thereby the emission angles of the electrons with respect to the molecular axis. The data show that when energy sharing between the electrons is very unbalanced, {\it i.e.}~one electron acquires a much larger kinetic energy than the other one, both electrons exhibit a Cohen-Fano-like interference pattern in their direction of emission. If, however, the electrons share the kinetic energy in a more balanced way, the single-electron interference pattern vanishes. This can be understood from the interaction between the electrons that is important in such range of energy sharing. If the data is filtered (postselected) to a given emission angle of one electron, an interference pattern re-emerges for the other one, {\it i.e.}~the electrons are strongly correlated. The data, reproduced here in Figure \ref{doerner}, thereby suggests that entanglement between the outgoing electrons is responsible for the correlations that are visible between their emission angles.
\begin{figure}[h] \center \includegraphics[width=10.5cm,angle=0]{fig13.pdf}\caption{Courtesy of R. D\"orner \cite{al.:2007df}. Electron-electron correlations in the double-photo-ionization of molecular hydrogen \cite{al.:2007df}, for photon-energy $E_\gamma=160$ eV, and the energy of the second electron conditioned in the range 5 eV$<E_2<$25 eV, corresponding to $E_1\approx$ 85 eV to 105 eV.
Panel A: Event number as a function of the angle between molecular axis and fast electron $\Phi_{\mathrm{e-mol}}$, and between both electrons $\Phi_{\mathrm{e-e}}$. Panel B: Event distribution in $\Phi_{\mathrm{e-mol}}$ conditioned on 50$^{\circ}< \Phi_{\mathrm{e-e}} < 80^{\circ} $. Panel D: Same for -80$^{\circ}< \Phi_{\mathrm{e-e}} < -50^{\circ} $. Panels C,E: Polar data corresponding to B and D, respectively. Note that when one selects the whole range of $\Phi_{\mathrm{e-e}}$, the resulting distribution in $\Phi_{\mathrm{e-mol}}$ results to be homogeneous and looses the structure exhibited in B-D. From \cite{al.:2007df}. Reprinted with permission from AAAS.} \label{doerner} \end{eqnarray}d{figure}
Such interference was observed for the same setting also in the sum of electron momenta \cite{Kreidi:2008bs}. It is visible in the momentum distribution of the individual electrons only for extremely asymmetric energy sharing, while it always persists for the sum of moments.
The evidence of entanglement in these experiments is rather strong: Instead of probing a purely kinematic effect, the persistence of single-particle interference is assessed. Indeed, particles loose their ability to exhibit perfect interference fringes when they are entangled to other particles. By the selection of the electrons in certain ranges of energy sharing, one can effectively select different regimes of mutual interaction strength. Thereby, it is probed how the interaction between the electrons affects the interference pattern. On the other hand, only observables which commute with the momenta of the electrons are measured, and, strictly speaking, the correlations that are found can also be reproduced by local realistic theories, similarly as discussed below in (\ref{localrealisticion}).
\subsubsection{Entanglement and symmetry breaking}
An entanglement-based study proposes a resolution to the question whether a core vacancy created in a diatomic homonuclear molecule by ionization is localized at one center, or delocalized. By photo double ionization of N$_2$ and collection of both the photoelectron and the subsequently ejected Auger electron, it was shown \cite{Schoffler:2008cr} that the two electrons are, again, highly correlated: If conditioned on the photoelectron to be emitted in the direction of the molecular axis, the remaining Auger-electron is left distributed asymmetrically. In contrast, if the photoelectron is detected perpendicularly to the molecular axis, {\it i.e.}~without any preferred direction, also the Auger electron seems to be ejected from a delocalized state, as suggested by its angular distribution. This is clearly observed in the data shown in Figure \ref{corehole}.
\begin{figure}[h] \center \includegraphics[width=5cm,angle=0]{fig14.pdf}\caption{Courtesy of M. Sch\"offler \cite{Schoffler:2008cr}. Auger electron and photoelectron angular distributions in the molecular frame, for circularly polarized light with $E_\gamma=419$ eV. Dots denote experimental data, lines the theoretical prediction, Eq. 1 in \cite{Schoffler:2008cr}. The molecular axis is depicted by the barbell, the photon propagates into the figure plane. A and F show the non-conditioned data for Auger electron and photoelectron. By conditioning on selected angles of the Auger electron as in A, the distributions G-J result for the photoelectron. On the other hand, by selecting the photoelectron as shown in F, the Auger electron adopts the distributions B-E. From \cite{Schoffler:2008cr}. Reprinted with permission from AAAS.
} \label{corehole} \end{eqnarray}d{figure}
Hence, the electrons are anti-correlated, independently of the choice of the emission direction of the first one. The filter that fixes the emission direction to a certain angle can be interpreted as a choice of basis from either left and right localized states $\left\{ \ket{L}, \ket{R} \right\}$ or even and odd, delocalized states $\{ \ket{E}, \ket{O} \}$. The latter are defined as follows for Auger (A) and photo (P) electrons:
\begin{eqnarray} \ket{E}_{A/P}&=&\frac{1}{\sqrt 2} \left( \ket{L}_{A/P}+ \ket{R}_{A/P} \right) ,\\
\ket{O}_{A/P}&=&\frac{1}{\sqrt 2} \left( \ket{L}_{A/P}- \ket{R}_{A/P} \right) .\end{eqnarray}
The electrons seem to be neither localized nor delocalized, but in an entangled state such that the condition of (de)localization imposed on one electron implies the (de)localization of the other. As a simple model, we can consistently describe the observed data with the following quantum state
\begin{eqnarray} \ket{\Psi}&=& \frac{1}{\sqrt{2}} \left( \ket{R}_A\ket{L}_P -\ket{L}_A\ket{R}_P \right) \nonumber \\
&=& \frac{1}{\sqrt{2}} \left( \ket{E}_A\ket{O}_P - \ket{O}_A\ket{E}_P \right) .\end{eqnarray}
One has to retain, however, that this ``change of basis'' performed by the condition on certain emission angles relies on strong model-assumptions: Effectively, only momenta are, again, measured. No information on any expectation value of observables that do not commute with the momentum is obtained, and the a-posteriori conditioning on a certain angle does not correspond to an active choice of different, non-commuting observables. The requirements for the violation of a Bell inequality (see Section \ref{Bellin}) are therefore \emph{not} met. Strictly speaking, the data currently cannot rule out a separable mixed state of the following form:
\begin{eqnarray} \rho &\propto & p_1 \ket{L}_A\ket{R}_P \bra{L}_{A}\bra{R}_P + p_2 \ket{R}_A\ket{L}_P \bra{R}_{A}\bra{L}_P \nonumber \\
&+& p_3 \ket{O}_A\ket{E}_P \bra{O}_{A}\bra{E}_P + p_4 \ket{E}_A\ket{O}_P \bra{E}_{A}\bra{O}_P , \label{localrealisticion} \end{eqnarray}
where the $p_i$ are the probability weights of the respective pure states. This state describes a purely probabilistic mixture of anti-correlated electron pairs, and corresponds to a local-realistic model which simulates the acquired data, and does not require entanglement. These equations reflect again very directly the problem of distinguishing a classical mixture, as given in (\ref{eq:psimm}), from a coherent superposition as in (\ref{sepmixed}), when only the correlations in one basis are measured.
\section{Conclusions and outlook}
In fact deeply rooted in atomic physics that initially stimulated the very development of quantum mechanics, entanglement has now largely overcome its restriction to the reductionist form of bipartite qubit correlations of some internal degree of freedom. It is a phenomenon sought under natural conditions which has become a tool for the deeper understanding for naturally occurring, typically complex systems, and thereby returns to its very conceptual origins. Simultaneously, it provides an analytic tool for many-body phenomena that are hard to understand in terms of single-particle observables.
Given a fixed subsystem structure, a complete mathematical apparatus, insinuated in Section 2, is available today for the characterization of entanglement, despite the computational difficulties for the characterization of entanglement in mixed states with many parties and dimensions. \emph{Conceptual} difficulties like the problem of identical particles, discussed in Section 3, turned out not to possess a universal solution which can be applied like an all-purpose tool to all possible situations. Knowledge on the physical setup and the observables under consideration and, especially, on the restrictions which potentially apply for measurements are necessary to find the suitable treatment in a given scenario.
First studies on entanglement in bound systems of electrons have emerged, in which, however, the combined effects of the particles' indistinguishability, the long-range character of the Coulomb-interaction and the spin-orbit coupling have not yet been fully incorporated.
Where the electrons' indistinguishability was taken into account, the very intrinsic feature of identical particles, however, {\it i.e.}~measurement-induced entanglement (see Section \ref{measurementinduced}), has not yet found applications in interacting systems apart from a Fermi gas model with screened Coulomb interaction which does not affect the spins, {\it i.e.}~the degree of freedom in which entanglement is considered \cite{Hamieh:2009zr,Hamieh:2010ly}. While, conceptually, the possession of a complete set of properties defines a physical reality and characterizes a separable subunit, measurement-induced entanglement beyond the engineered examples mentioned in Section \ref{entextr} can be expected in many situation, {\it e.g.}, between electrons ejected from the same orbital quantum state. In systems in which both the particles' indistinguishability and their mutual interaction play a prominent role, effects that can not be explained in terms of one of these aspects alone and hence require the understanding of their subtle interplay can be expected \cite{Bose:2005cl}.
As opposed to bound systems in which entanglement is present due to a permanent binding interaction, the unbound decaying systems we have reviewed typically share the same physical reason for the existence of quantum correlations: Conservation laws for energy and momentum leave the fragments entangled in these very degrees of freedom, at any range of energy, from Feshbach-resonance induced decay of ultracold molecules \cite{Gneiting:2010qf} in the very low energy range, to electron-positron pairs created at very high photon energies \cite{Krekora:2005rq}. A quite intuitive feature is that the degree of entanglement between products which are very unbalanced in mass tends to be smaller than between constituents of similar properties, simply due to kinematics and conservation of momentum. This rule of thumb can, however, be circumvented with suitable schemes \cite{Guo:2006qf,Chan:2003ly}, which shows that entanglement adds a qualitatively new feature to the description of dynamical systems which \emph{cannot} be completely reduced to kinematical quantities. The understanding of entanglement is hence by far not completed by considering the kinematics of processes: The very distinction between classical and quantum correlations and the mechanisms which lead to the breakdown of coherences remain the most important issues that need to be addressed. The quantification of wave-packet narrowing emerged as tool for the verification of correlations between the fragments. It holds the disadvantage that it only provides a clear signature for entanglement under the assumption of pure quantum states, {\it i.e.}~it cannot distinguish classical and quantum correlations. Correlations were indeed recently verified \cite{Schoffler:2008cr,al.:2007df}, however, in the momentum probability distribution rather than position.
Beyond the application in atomic and molecular physics, the conceptual issues we have discussed also arise in the presently very active field of quantum effects in biological systems. The question whether coherent effects play a relevant role for biological phenomena has arisen and is under debate \cite{EisertBio,Briegel:2008ff,Arndt:2009uq}. Experimental evidence for the role of coherence in light-harvesting complexes, responsible for the functioning of photosynthesis, was recently obtained \cite{Ishizaki:2010kx,Collini:2010vn,Sarovar:2010ys} and has triggered intensive research activities in theory \cite{Fassioli:2010gb,Caruso:2009ys,scholak3560,Scholak:2011fk,Scholak:2010fk,Mulken:2010uq,Muhlbacher:2009kx,Hoyer:2010vn,Thorwart:2009ys}. Simultaneously, theoretical results support the idea of surviving dynamical entanglement at room temperature in an environment prohibitive for entanglement in static systems \cite{Cai:2010qf,Cai:2010fu,Galve:2010kx}, which feeds the hope to encounter coherent effects in other noisy, wet and warm systems. The importance of multipartite entanglement \cite{scholak3560} as well as of molecular vibrations \cite{asadian} for efficient energy transport in networks was also shown recently, and contributes to the picture that quantum effects in biology could play an important functional role.
All this is evidence that the fields of quantum information and of atomic and molecular physics have only started to interact. The reason is twofold: On the one hand, the experimental capabilities that permit to resolve and verify coherent phenomena have only emerged recently \cite{Becker:2009oq}, and, on the other hand, conceptual issues inhibited the direct application of the highly abstract and idealized notions of quantum information science to this field. First successful applications of the concepts borrowed from quantum information yielded interesting results and new insight in the dynamics of atoms and molecules and provided answers to fundamental questions, {\it e.g.} regarding the delocalization and entanglement of massive particles in nature. The study of many-particle quantum coherence in atomic and molecular physics promises further interesting results in the next years, and will eventually lead to vital feedback to the field of quantum information itself.
\ack The authors enjoyed fruitful and stimulating discussions with Celsus Bouri, Vivian Fran\c ca, Dominik H\"orndlein, Pierre Lugan, Fernando de Melo, Benno Salwey, Torsten Scholak, Markus Tiersch and Hannah Venzl, and are indebted to Vivian Fran\c ca and Pierre Lugan for careful proofreading. Financial support by the German National Academic Foundation (MCT), by DFG (FM), and by DAAD withiin the PROCOPE program (MCT and AB) is gratefully acknowledged.
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\begin{document}
\title{Geometric Firefighting in the Half-plane\thanks{This work has been supported by DFG grant Kl 655/19 as part of a DACH project.} \thanks{This is a pre-print of an article published in \textit{Algorithms and Data Structures - 16th International Symposium, {WADS} 2019}.
}}
\author{Sang-Sub Kim
\and Rolf Klein
\and David Kübel
\and Elmar Langetepe
\and Barbara Schwarzwald
}
\authorrunning{S. Kim, R. Klein, D. Kübel, E. Langetepe, and B. Schwarzwald}
\institute{Department of Computer Science, University of Bonn, 53115 Bonn, Germany \\
\email{\{sang-sub,rolf.klein,dkuebel,schwarzwald\}@uni-bonn.de}\\
\email{[email protected]}}
\maketitle
\begin{abstract}
In 2006, Alberto Bressan~\cite{b-dicff-07} suggested the following problem. Suppose a circular fire spreads in the Euclidean plane at unit speed. The task is to build, in real time, barrier curves to contain the fire. At each time $t$ the total length of all barriers built so far must not exceed $t \cdot v$, where $v$ is a speed constant. How large
a speed $v$ is needed? He proved that speed $v>2$ is sufficient, and that $v>1$ is necessary.
This gap of $(1,2]$ is still open. The crucial question seems to be the following. {\em When trying to contain a fire, should one build, at maximum speed, the enclosing barrier, or does it make sense to spend some time on placing extra delaying barriers in the fire's way?}
We study the situation where the fire must be contained in the upper $L_1$ half-plane by an infinite horizontal barrier to which vertical line segments may be attached as delaying barriers.
Surprisingly, such delaying barriers are helpful when properly placed. We prove that speed $v=1.8772$ is sufficient, while $v >1.66$ is necessary.
\keywords{barrier, firefighting, geodesic circle}
\end{abstract}
\section{Introduction and problem statement}
Fighting wildfires is a difficult problem, involving many parameters one can neither foresee nor control. But there seem to be two main techniques firefighters employ, namely to extinguish the fire by dropping water or chemicals from aircraft, and to prevent the fire from spreading further by firebreaks.
In 2006, Alberto Bressan~\cite{b-dicff-07} developed a rather general model for containing a fire by means of barrier curves that must be built in real time, subject to velocity constraints. Barriers are impenetrable by fire, they do not burn and cannot be moved once built.
In addition to general optimality results \cite{bbfj-bsfcp-08,bw-msbph-09,bw-osibp-12}, in~\cite{b-dicff-07} Bressan proposed the following problem. Suppose a circular fire spreads in the plane at unit speed. In real time, barrier curves must be built to contain it. At each time $t$, the total length of barriers built so far must not exceed $t$ times $v$, for some velocity constant $v$.
The question is how large a velocity is needed to contain the fire.
Bressan showed that $v>1$ is necessary and that $v>2$ is sufficient; see also~\cite{kl-cgc-16} for short proofs.
He conjectured that speed $v=2$ is necessary.
But the gap $(1,2]$ is still open, even though a 500 USD reward has been offered~\cite{b-awa-11} in 2011.
It seems that the difficulty lies with the following question. {\em To contain a fire, should one build an enclosing barrier at maximum speed, or is it better to invest some time in building extra delaying barriers that will not be part of the final enclosure but can slow the fire down during construction?}
If delaying barriers could be shown to be useless, Bressan's proof of the lower bound~1 could be easily extended to prove his conjecture, the lower bound of~2.
In fact they consider a special variant in \cite{bw-msbph-09}, where the fire spreads in a half plane. In that case they can construct an optimal strategy without delaying barriers, that encloses the fire between the boundary of the half plane and the barrier curve.
To study the effectiveness of delaying barriers we study a different setting where an infinite horizontal barrier\xspace has to be built to contain the fire in the upper half-plane, instead of the interior of a closed barrier curve.
To this horizontal barrier\xspace, vertical line segments may be attached as delaying barriers. Without vertical barriers speed $v=2$ is necessary and sufficient to build the horizontal barrier\xspace.
While it takes extra time to build vertical barriers, they offer some respite because the expanding fire has to overcome them before it reaches the horizontal barrier\xspace again.
To simplify matters further we are working in the $L_1$ norm, so that distances are free of square roots. Also, all intersections of the fire's boundary with the barriers advance at unit speed.
Our main result is the following. In our setting, speed $v>1.66$ is necessary, and,
with a careful placement of delaying barriers, speed $v=1.8772$ is sufficient.
While this result does not disprove Bressan's conjecture it casts a new light on the problem by showing that building delaying barriers can be helpful.
Also, the gap we leave open is smaller than the one for the original containment problem.
Previous, but weaker results have been presented at EuroCG'18~\cite{kkls-ffp-18}.
\subsection{Related work}
Among theoretical work on {\em extinguishing} a fire, the ``lion and man'' problem stands out~\cite{dsz-ovlmp-08,bkns-eosit-09,bggk-hmlnc-09,k-rpffp-18}. Here, $r$ fighters are tasked with quenching a fire in an $n \times n$ grid. In every step, fighters and fire move simultaneously to adjacent cells, subject to certain rules.
While $r=n$ fighters can easily extinguish the fire, $\lfloor \nf{n}{2} \rfloor$ fighters are not enough.
The gap in between is still open, despite serious efforts.
How to {\em contain} a fire has received a lot of attention in graph theory, see, e.\,g.,\xspace \cite{fm-fpsrd-07,fkmr-fpgmd-07,fhl-fpgc-16}.
In quite a few examples, in each round, a stationary guard can be placed in a vertex not on fire, then the fire spreads to all unguarded adjacent vertices.
This continues until the fire cannot spread any further.
The problem to determine the maximum number of vertices that can be protected is NP-hard, even in trees of degree~3.
Similar in spirit is a geometric firefighting problem in simple polygons~\cite{kll-aagfb-18}, where barriers must be chosen from a set of pairwise disjoint diagonals, to save an area of maximum size. Even for convex polygons, the problem is NP-hard, but a 0.086 approximation algorithm exists.
It is interesting to see what happens when building a barrier along the boundary of an expanding circular fire~\cite{bbfj-bsfcp-08,bw-msbph-09,kll-ffp-15,kllls-ffp-18}. A spiraling curve results that closes on itself, and thus contains the fire, if the speed of building is larger than $2.6144$. Then the number of rounds to completion
can be determined by residue calculus. Below this threshold, the curve keeps winding forever.
The rest of this paper is organized as follows.
Section~\ref{section:modelDefinition} formally introduces the problem as well as terms and definitions required for the analysis.
In Section~\ref{section:lowerBounds} we develop a lower bound of $v > 1.66$.
In Section~\ref{section:upperBounds} we show that $v=\nf{17}{9}=1.\overline{8}$ is sufficient and discuss how this value can even be reduced to $v=1.8772$.
\section{Model}
\label{section:modelDefinition}
In our model, the fire\xspace spreads from the origin and continuously expands over time with speed $1$ according to the $L_1$ metric.
To prevent the fire from immediately spreading into the lower half-plane, we allow an arbitrarily small head-start of barrier of length $\ensuremath{s}\xspace$ into both directions along the $x$-axis.
\begin{figure}
\caption{Fire spreading along delaying barriers. The dashed line shows the fire front at different times $t$, solid points represent consumption\xspace points, while empty points represent places, where the fire burns along the back of already consumed parts of the barrier\xspace $b_i$. In a) there is one consumption point, so there is a $1$-interval in the right direction. In b) there are three consumption\xspace points and in c) there is a $0$-interval in the right direction as there are no consumption points.}
\label{fig:delayingBarriers}
\end{figure}
Assume that a system of barrier\xspaces has been built.
The barrier system\xspace consists of a horizontal barrier\xspace containing the fire in the upper half-plane and several vertical delaying barriers attached to it.
To describe a barrier system\xspace, we denote the $i$-th delaying barrier\xspace to the right by $b_i$.
The part of the horizontal barrier\xspace between $b_{i-1}$ and $b_i$ is denoted by $a_{i}$.
For simplicity, we also refer to their length by $a_i$ and $b_i$.
For the other direction, we use $c_i$ and $d_i$ respectively.
For convenience, $A_i := \sum_{j=1}^i a_j$ will denote the total length of horizontal barrier\xspaces in the right direction up till and including $a_i$ and $B_i := \sum_{j=1}^i b_j$ will denote the total length of vertical barrier\xspaces in the right direction up till and including $b_i$.
Equivalently for the left direction we define $C_i$ and $D_i$.
As the fire spreads over the barrier system\xspace, it represents a geodesic $L_1$ circle, which consumes the barrier\xspaces when burning along them.
The fire-front is the set of all points in the plane, which shortest non-barrier\xspace-crossing path to the fire origin has length $t$.
We consider a point $x$ on a barrier\xspace as {\em consumed} at time $t$ if the fire\xspace has reached this point at time $t$. That means there exists a non-barrier\xspace-crossing path of length at most $t$ from the fire origin to the point $x$.
Hence, any piece of the barrier is not consumed all at once, but as the fire burns along it.
We call a point on a barrier\xspace, which shortest non-barrier\xspace-crossing path to the fire has exactly length $t$ a consumption\xspace point at time $t$, so the consumption\xspace points are a subset of the fire front.
We call the number of consumption\xspace points at time $t$ the {\em current consumption\xspace} and a time interval with constant $k$ consumption\xspace points at all times a {\em $k$-interval}.
The fire front, consumption\xspace points and the effect of vertical delaying barrier\xspaces are illustrated in \autoref{fig:delayingBarriers}.
As one can see, after the fire reaches a delaying barrier for the first time, it may burn along multiple barrier\xspaces at multiple points.
However, after reaching both ends and passing the top of a barrier\xspace there might be no consumption\xspace for a while as the delaying barrier\xspace has already been burned along from the other side.
We define the \emph{total consumption\xspace} $bF{}$ and \emph{consumption-ratio\xspace} $\QF{}$ for a time interval $\left[t_1, t_2\right]$ in a barrier system\xspace:
\begin{equation*}
\begin{array}{r l}
bF{}(t_1, t_2)
& := \text{length of barrier\xspace pieces consumed by the fire between } t_1 \text{ and } t_2\\
\QF{}(t_1, t_2)
& := \frac{ bF{}(t_1, t_2)}{t_2 - t_1}\,.
\end{array}
\end{equation*}
For the consumption\xspace in a time interval $[0, t]$, we will also write $bF{}(t)$ and $\QF{}(t)$ for short.
In our setting, if $[t_0, t_1]$ is a $k$-interval, then $bF{}(t_1) = bF{}(t_0) + (t_1-t_0)\cdot k$.
Note that all these definitions can easily be applied to either side of the barrier system\xspace, denoted by \consumptionRatio[l], \consumptionRatio[r] and $bF{}^l(t)$, $bF{}^r(t)$ equivalently.
Obviously, $\consumptionRatio = \consumptionRatio[l] + \consumptionRatio[r]$ and $bF{}(t) = bF{}^l(t)+ bF{}^r(t)$.
It is clear that when building a barrier system\xspace simultaneously to the fire\xspace spreading, then every piece of barrier should be build before the fire\xspace reaches it.
For a limited build speed $v$, it is necessary and sufficient to have $bF{}(t) \leq v \cdot t$ for all times $t$, which means $v \geq \sup_t \QF{}(t)$.
The question then obviously is: What is the minimum speed $v$ for which such a barrier system\xspace exists?
\section{Prerequisites}\label{sec:prereq}
Observe that a vertical barrier\xspace which is shorter than the predecessor in the same direction does not delay the fire. Hence, we can assume that vertical barrier\xspaces in one direction increase strictly in length, so $b_i > b_{i-1}$ and $d_i > d_{i-1}$ for all $i>1$. But we can show an even stronger bound on the growth of successive vertical barrier\xspaces.
\begin{lemma}\label{lemma:prereq}
If there exists a barrier system\xspace with $bF{}(t) \leq v \cdot t$ at all times $t$, then there also exists such a barrier system\xspace in which any vertical barrier\xspace $b_i$ (or $d_i$) is more than twice as long as the previous barrier\xspace $b_{i-1}$ (or $d_{i-1}$) in the same direction.
\end{lemma}
\begin{proof}
Assume we are given any barrier system\xspace $\mathcal{S}$ with $bF{S}(t) \leq v\cdot t$, not fulfilling both properties $b_i > 2 b_{i-1}$ and $d_i > 2 d_{i-1}$. Then we can transform it to a new barrier system\xspace $\mathcal{S}'$ that fulfils both properties $b_i > 2 b_{i-1}$ and $d_i > 2 d_{i-1}$ while $bF{S'}(t) < bF{S}(t) \leq v\cdot t$ for all $t$.
The construction is identical for both directions, so we just consider the right direction.
Let $b_k$ ($k>1$) be the first vertical barrier\xspace in the right direction with $b_k < 2 b_{k-1}$. Then we can remove $b_k$ and move all following vertical barrier\xspaces away from the fire by $2delta = 2(b_{k} - b_{k-1})$.
So, more precisely the right side of our barrier system\xspace $\mathcal{S}'$ consists of $b'_i$ and $a'_i$ as follows:
\begin{eqnarray}
\text{for } i<k \quad & \quad b'_i = b_i \quad & \quad a'_i = a_i \\
\text{for } i = k \quad & \quad b'_i = b_{i+1} \quad & \quad a'_i = a_i + a_{i+1} + 2delta \\
\text{for } i \geq k \quad & \quad b'_i = b_{i+1} \quad & \quad a'_i = a_{i+1}
\end{eqnarray}
If $b_k$ is the last vertical delaying barrier\xspace in the right direction, it can just be removed instead.
\begin{figure}
\caption{ The situation in $\mathcal{S}
\label{prereq-fig}
\end{figure}
To sketch the proof, let us assume that $a_k \geq b_{k-1}$, $a_{k+1} \geq b_k$ and $b_{k+1} \geq 2b_k$ hold.
The consumption-ratio\xspace in the right direction is identical for $\mathcal{S}$ and $\mathcal{S}'$ until time $A_k + b_{k-1}$ when the fire reaches $b_k$ in $\mathcal{S}$.
\autoref{prereq-fig} shows the next sequences of consumption intervals in $\mathcal{S}$ and $\mathcal{S}’$ until the fire reaches the top of $b'_k$ in $\mathcal{S}'$. Due to the linearity of consumption\xspace within each $k$-interval, only the
points where intervals change can attain maximal values.
Direct comparison shows that, $\mathcal{S}'$ has a smaller consumption at all such points in time.
Once the fire has overcome the gap between $b'_{k-1}$ and $b'_k$ in $\mathcal{S}'$,
each configuration $K'$ at time $t'$ in $\mathcal{S}'$ corresponds to a configuration $K$ at time $t = t-2 delta$ in $\mathcal{S}$. But, due to the presence of vertical barrier $b_k$ and the missing horizontal extension by $2 delta$, in $K$ the consumption differs by $b_k - 2 delta = 2 b_{k-1} - b_k$, which is positive by assumption.
Thus, $K’$ has a lower consumption ratio than $K$.
All other cases work similarly: The additional consumption contributed by the added $2delta$ of horizontal barrier between $b'_{k-1}$ and $b'_k$ is always covered by the removal of the vertical barrier of length $b_k > 2 delta$.
Note, that these arguments require that no part of $a_k$ is covered by the head-start~\ensuremath{s}\xspace. We can assume so by a similar argument. Let $b_s$ ($s>1$) be the last vertical barrier\xspace in the right direction with $A_s \leq s$, which means that all horizontal barriers $a_1, a_2, \ldots, a_s$ are covered by the head-start. Then combining all barriers $b_1$ to $b_s$ into one barrier $b_s$ at the end of $\ensuremath{s}\xspace$ does not increase $bF{}^r(t)$ for any $t$.
This concludes the proof. \qed
\end{proof}
This means that when given an arbitrary barrier system\xspace, we can assume $b_i > 2 b_{i-1}$ and $d_i > 2 d_{i-1}$ for all $i>1$. From this we can derive a helpful observation about the order of consumption\xspace of vertical and horizontal barrier\xspaces in a barrier system\xspace: when the fire reaches the top of a vertical barrier\xspace $b_i$ at some time $t$ (compare \autoref{RHsituation-fig}), every barrier\xspace $a_k$ and $b_k$ with $k \leq i$ has been completely consumed, as for every point on $a_k$ or $b_k$ the shortest non-barrier-crossing path has length smaller than $A_i + b_i = t$. Hence, a $0$-interval in the right direction will begin at such times $t$ and $bF{}^r(t) = A_i + B_i - s$, where \ensuremath{s}\xspace denotes the length of the head-start not contributing to the consumption.
This observation holds equivalently for both directions.
\section{A lower bound of \ensuremath{v > 1.66}}\label{section:lowerBounds}
Assume there exists a barrier system\xspace $\mathcal{S}$ consisting of horizontal barrier\xspaces along the $x$-axis and vertical barrier\xspaces attached to it. Further assume for $\mathcal{S}$ that $bF{}(t) \leq v \cdot t$ at all times $t$ for some $v = (1+V)$ with $V \leq \frac{2}{3}$.
For this we will construct a contradiction by identifying a specific time $t_\mathcal{S}$, for which $bF{}(t_\mathcal{S}) > (1+V) \cdot t_\mathcal{S}$.
By \autoref{lemma:prereq}, we can assume $b_i > 2 b_{i-1}$ and $d_i > 2 d_{i-1}$ for all $i>1$ in $\mathcal{S}$.
\begin{figure}
\caption{At some time $t = A_i + b_i$ the fire will reach the top of a vertical barrier $b_i$.}
\label{RHsituation-fig}
\end{figure}
As without vertical delaying barriers, the consumption-ratio\xspace just goes towards $2$, $\mathcal{S}$ has an unbounded number of vertical barriers in at least one direction. W.\,l.\,o.\,g.\xspace assume this is the right direction.
Consider a moment when the fire reaches the end of some barrier $b_i$ as illustrated in \autoref{RHsituation-fig}.
As explained in \autoref{sec:prereq}, this happens at time $t = b_i + A_i$ and Lemma~\ref{lemma:prereq} implies we have $bF{}^r(t) = A_i + B_i - \ensuremath{s}\xspace$.
\begin{eqnarray}
bF{}^r(t) &= & A_i + B_i - \ensuremath{s}\xspace = A_i + b_i + B_{i-1} - \ensuremath{s}\xspace \quad\quad \mid B_{i-1} > 2 \ensuremath{s}\xspace \text{ for $i$ large enough}\nonumber\\
&>& A_i + b_i + \ensuremath{s}\xspace > t + \ensuremath{s}\xspace > t \label{leftRequirement}
\end{eqnarray}
Hence for $t$ large enough, $\QF{}^r(t) > 1$ at times $t$, when the fire reaches the top of a vertical barrier.
Therefore, $\mathcal{S}$ has repeated $0$-intervals in the left direction as well, or else $\QF{}^l(t)$ would go towards $1$ and $\QF{}(t) > 2$ at such times $t$.
\begin{figure}
\caption{All three possible situations for the left side to be in at time $t$. Note that in case 1) and 2) the fire might have reached $d_{j+1}
\label{AllCases-fig}
\end{figure}
We now consider the situation in the left direction at time $t = b_i + A_i$. Let $d_j$ denote the last vertical barrier, whose upper end was reached by the fire, so $t = d_j + C_j + \delta$ with $0 \leq \delta < c_{j+1} + d_{j+1} - d_{j}$.
W.\,l.\,o.\,g.\xspace we assume that $b_{i+1} + A_{i+1} \geq d_{j+1} + C_{j+1}$. Otherwise, there must be multiple vertical barriers in the right direction whose upper ends are reached by the fire after it reaches the upper end of $d_j$ and before it reaches the upper end of $d_{j+1}$. In that case, we can assume that $b_i$ is the last among those, such that $b_{i+1} + A_{i+1} \geq d_{j+1} + C_{j+1}$ holds.
We split our consideration in three cases, which are all illustrated in \autoref{AllCases-fig}:
\begin{enumerate}
\item $0 \leq \delta < d_j$ \label{LH1}
\item $d_j \leq \delta < d_j + c_{j+1}$ \label{LH2}
\item $d_j + c_{j+1} \leq \delta < c_{j+1} + d_{j+1} - d_{j}$ \label{LH3}
\end{enumerate}
In the first case, the fire has not reached the horizontal barrier $c_{j+1}$ yet after passing over $d_j$; in the second case, it has reached $c_{j+1}$, but not its end; in the third case the fire has completely consumed $c_{j+1}$.
In Case~\ref{LH3}, $\delta = d_j + c_{j+1} + \epsilon$ and then $bF{}^l(t) \geq C_{j+1} + D_{j} + 2d_j + \epsilon - \ensuremath{s}\xspace > (d_j + C_j) + (d_j + c_{j+1}) + \epsilon = t$, which together with Inequality~(\ref{leftRequirement}) already gives $bF{} (t) > 2 t > (1+V) \cdot t$ which is a contradiction.
For both remaining cases, we will derive a lower bound for $d_j$. We will then consider the moment $t_1 = 2 d_j + C_{j+1}$, when the fire reaches the end of the horizontal barrier $c_{j+1}$.
Using the lower bound on $d_j$, we will prove $bF{}(t_1) > (1+V) \cdot t_1$.
\subsection{Case 1: \ensuremath{0 \leq \delta < d_j}}
In Case \ref{LH1},
$bF{}^l(t) > C_{j} + D_{j} - \ensuremath{s}\xspace = C_{j} + d_{j} + D_{j-1} - \ensuremath{s}\xspace > C_{j} + d_{j}$, since $D_{j-1} > s$ for $j$ large enough.
Now at time $t$, it must hold:
\begin{eqnarray}
bF{}(t) = bF{}^r(t) + bF{}^l(t) & < & (1+V) \cdot t \quad\quad\quad\quad\quad\quad\,\,\,\, \mid \text{Inequality~(\ref{leftRequirement})}\nonumber \\
rightarrow \,\quad\quad\quad\quad\quad\quad C_{j} + d_{j} & < & V (d_j + C_{j} + \delta) \nonumber \\
rightarrow \quad\quad\quad\quad\quad (V-1) C_{j} & >& (1-V) d_j - V \delta \quad\quad\quad\quad \mid \left(V < 1 \right) \nonumber \\
\Leftrightarrow \,\,\quad\quad\quad\quad \quad\quad\quad\quad C_j & < & \nicefrac{V}{(1-V)} \cdot \delta - d_j \label{lastIn1}
\end{eqnarray}
$V\leq\frac{2}{3}$ implies $\nicefrac{V}{(1-V)} \leq 2$ by direct calculation, which gives bounds for $C_j, d_j$:
\begin{eqnarray}\label{central}
C_j & < & 2 \delta - d_j < d_j \quad \mid \delta < d_j \text{ in Case~\ref{LH1}}\nonumber\\
rightarrow \,\,\,\quad 2 d_j &>& C_j + \delta \nonumber \\
\Leftrightarrow \quad\quad d_j &>& \nicefrac{1}{2} (C_j + \delta) \label{LH1-D-Bound}
\end{eqnarray}
\subsection{Case 2: \ensuremath{d_j \leq \delta < d_j + c_{j+1}}}
In Case~\ref{LH2} a part of $c_{j+1}$ of length $(\delta-d_j)$ has already been consumed, so $bF{}^l(t) \geq D_j + C_j + (\delta-d_j) - \ensuremath{s}\xspace > d_j + C_j + (\delta-d_j) = C_j + \delta$, as $D_{j-1} > s$ for $j$ large enough.
Now at time $t$ it must hold
\begin{eqnarray}
bF{}(t) = bF{}^r(t) + bF{}^l(t) & < & (1+V) \cdot t \quad\quad\quad\quad\quad\quad\,\,\,\, \mid \text{Inequality~(\ref{leftRequirement})}\nonumber \\
rightarrow \quad\quad\quad\quad\quad\:\:\:\:\: C_j + \delta & < & V( d_j + C_j + \delta) \nonumber\\
rightarrow \quad\quad (1-V) (C_j + \delta) & < & V d_j \nonumber\\
rightarrow \quad\quad\quad\quad\quad\quad\quad\quad d_j & > & \nicefrac{(1-V)}{V} ( C_j + \delta )
\end{eqnarray}
$V\leq\frac{2}{3}$ implies $\nicefrac{(1-V)}{V} \geq \frac12$ by direct calculation, which gives the bound:
\begin{equation}
d_j > \nicefrac{1}{2}(C_j + \delta) \label{LH2-D-Bound}
\end{equation}
This is the same bound as found for Case~\ref{LH1} in Inequality~(\ref{LH1-D-Bound}).
\subsection{Deriving the contradiction \ensuremath{bF{}(t_1) > (1+V) \cdot t_1}}
\begin{figure}
\caption{After $d_{j}
\label{LH1situation-fig}
\end{figure}
Now we consider time $t_1 = C_{j+1} + 2 d_j > t$, when the fire reaches the end of the horizontal barrier\xspace $c_{j+1}$.
As for any time, at time $t_1$, it must hold
\begin{eqnarray}
bF{}(t_1) &=& bF{}^r(t_1) + bF{}^l(t_1) \leq (1+V) \cdot t_1 \nonumber \\
\Leftrightarrow \quad\quad\:\:\: bF{}^l(t_1) &\leq & (1+V) \cdot t_1 - bF{}^r(t_1) \nonumber\\
&=& (1+V) \cdot t + (1+V) (t_1-t) - (bF{}^r(t) + bF{}^r(t, t_1)) \nonumber\\
&\leq& V t + (1+V) (t_1-t) + t - bF{}^r(t) \quad \mid \text{Ineq.~(\ref{leftRequirement})}\label{TightRequirement} \\
rightarrow \quad bF{}^l(t_1) + s &<& V t + (1+V) (t_1-t)\label{Case1leftRequirement}
\end{eqnarray}
By construction, $t_1 = C_{j+1} + 2 d_j$.
As $t = d_j + C_{j} + \delta$, this means $t_1 = t + (d_j + c_{j+1} - \delta)$.
Due to Lemma~\ref{lemma:prereq}, we know that the fire has not reached the end of $d_{j+1}$ yet, hence $bF{}^l(t_1) \geq 3 d_j + C_{j+1} -\ensuremath{s}\xspace$.
Hence, we arrive at the following inequalities:
\begin{eqnarray}
3 d_j + C_{j+1} & < & V(d_j + C_{j} + \delta) + (1+V) (d_j + c_{j+1}-\delta) \nonumber \\
\Leftrightarrow \quad -V (c_{j+1}-\delta) & < & (V-1)\delta + (V-1) C_{j} + (2V-2)d_j \quad\quad\quad \mid \left(1>V\right) \nonumber\\
\Leftrightarrow \quad\quad\quad\,\, c_{j+1}-\delta & > & \frac{1-V}{V} \delta + \frac{1-V}{V} C_{j} + 2\frac{1-V}{V} d_j \,. \label{last2}
\end{eqnarray}
$V\leq\frac{2}{3}$ implies $\nicefrac{(1-V)}{V} \geq \frac12$ by direct calculation, which gives the bound:
\begin{eqnarray}
c_{j+1}-\delta & > & \frac{1}{2} \delta + \frac{1}{2} C_j + d_j\, \nonumber\\
\Leftrightarrow \qquad d_j + c_{j+1} - \delta &>& \frac{1}{2} \delta + \frac{1}{2} C_j + 2 d_j\label{finalconcl}
\end{eqnarray}
Now in both cases we got $d_j > \nicefrac{1}{2} (C_j + \delta)$ (Inequalities~(\ref{LH1-D-Bound}) and (\ref{LH2-D-Bound})), so we can apply that and conclude:
\begin{eqnarray}
t_1-t = d_j + c_{j+1} - \delta & > & C_j + d_j + \delta = t = A_i + b_i \label{ineq:final}
\end{eqnarray}
So we know, that in both cases $t_1 - t > b_i + A_i$.
Now consider the situation in the right direction again (compare \autoref{RHsituation-fig}). At $t + b_i$ the fire reaches the horizontal barrier\xspace $a_{i
+1}$ behind $b_i$. Additionally, by assumption $b_{i+1} + A_{i+1} \geq d_{j+1} + C_{j+1}$, the fire has not reached the top of the next barrier $b_{i+1}$ at $t_1$. This means, that between $t + b_i$ and $t_1$, there is always at least consumption $1$ in the right direction, which means the fire has consumed barrier\xspaces of length at least $A_i$, hence $bF{}^r(t,t_1) \geq A_i$.
As our whole consideration is based on inequalities, we will consider an edge case with a contradiction that can be extended to our given barrier system\xspace $\mathcal{S}$.
More precisely, assume, that Inequality~(\ref{TightRequirement}) is tight for some $t_1^*$, so:
\begin{eqnarray*}
bF{}^l(t_1^*) &=& V t + (1+V) (t_1^*-t) + t - bF{}^r(t) \\
\Leftrightarrow \quad bF{}^r(t) + bF{}^l(t_1^*) &=& (1+V)t_1^*
\end{eqnarray*}
By our arguments above, $bF{}^r(t,t_1^*) \geq A_i$ and hence $bF{}(t_1^*) = bF{}^r(t,t_1^*) + bF{}^r(t) + bF{}^l(t_1^*) \geq (1+V)t_1^* + A_i > (1+V)t_1^*$, which is a contradiction for this edge case.
Now in our given barrier system\xspace $\mathcal{S}$ it holds $t_1 = t_1^* + x$ for some $x>0$. As everything except $c_{j+1}$ is fixed at $t$, this additional time results in additional consumption of at least horizontal barrier\xspaces of length $x$ in both directions in comparison to the edge case. Hence we can extend the contradiction:
\begin{eqnarray*}
bF{}(t_1) &=& bF{}^l(t_1) + bF{}^r(t) + bF{}^r(t,t_1) \\
&=& bF{}^l(t_1^*) + bF{}^r(t) + bF{}^r(t,t_1^*) + 2x \\
&=& (1+V)t_1^* + 2x + A_i > (1+V) (t_1^* + x) = (1+V) t_1.
\end{eqnarray*}
\begin{theorem}
The fire can not be contained in the upper half-plane with speed $v \leq 1.66$ by a barrier system\xspace consisting of a horizontal barrier\xspace along the $x$-axis and vertical barrier\xspaces attached to it.
\end{theorem}
\section{Upper bounds}
\label{section:upperBounds}
We prove the upper bound by defining a barrier system\xspace with bounded consumption-ratio\xspace.
Before we present the construction, we give some intuition.
We choose the following conditions:
\begin{equation}
\label{equation:conditionsOfRecursion}
\begin{array}{r c c c l}
& a_{i+1} \geq b_{i} & \text{and} & b_{i+1} \geq 2 b_{i} & \forall i \geq 1, \\
\text{similarly }
& c_{i+1} \geq d_{i} & \text{and} & d_{i+1} \geq 2 d_{i} & \forall i \geq 1.
\end{array}
\end{equation}
This forces the $0$-intervals generated by $b_i$ to be of length of $b_i$.
For a single direction this results in a repeating sequence of $k$-intervals of specific lengths and $k$ as shown in \autoref{fig:interval-cycle}.
\begin{figure}
\caption{
A sequence of $k$-intervals to the right of $(0,0)$.
The length is given above each interval and the current consumption\xspace below.
}
\label{fig:interval-cycle}
\end{figure}
The idea is to construct the barrier system\xspace in such a way that the $0$-intervals always appear in an alternating fashion, so the local maxima in the consumption-ratio\xspace of one direction can be countered by the $0$-intervals of the other direction.
To show that this idea can be realized, we consider the periodic interlacing of time intervals as illustrated in \autoref{fig:recursiveInterlacing}.
There, the ends of the $0$-intervals in one direction coincide with the ends of the \mbox{$3$-intervals} in the other direction, that is, at $t_3$ and $t_6$.
\begin{figure}
\caption{The periodic interlacing of time intervals.}
\label{fig:recursiveInterlacing}
\end{figure}
The current consumption\xspace is always greater than $1$, since the $0$-intervals do not overlap.
Also, the combined consumption-ratio\xspace $\QF{}(t)$ must be smaller than $2$ at all times.
This also implies that $t_3$ is no local maximum and the consumption-ratio\xspace grows towards 2 between $t_3$ and $t_4$.
Hence, by setting $d_i > 2b_i$ we make $t_1$, $t_4$, $t_7$ the local maxima and $t_2$, $t_5$ the local minima of $\QF{}(t)$.
\newcommand{\ensuremath{\alpha}\xspace}{\ensuremath{\alpha}\xspace}
\newcommand{\ensuremath{\beta}\xspace}{\ensuremath{\beta}\xspace}
Let us now consider the consumption-ratio\xspace $\QF{}(t_1, t_4)$ of the cycle from $t_1$ to $t_4$.
There are two $1$-intervals involved in this cycle in the right direction.
The first one, where the fire burns along $a_{i+1}$, is of length $a_{i+1}-b_i$ and lies partially in this cycle.
The second one, where the fire crawls up along $b_{i+1}$, is of length $b_{i+1}-2b_i$ and lies completely in this cycle.
As the beginning of this cycle is given by the start of the $0$-interval on one side and the end is given by the end of the second $1$-interval on the other side, we know that the length of this cycle is $d_i + (b_{i+1} - 2 b_i)$.
The total consumption\xspace in this cycle is $1 \cdot d_i + 2 \cdot b_i + 2(b_{i+1} - 2 b_i)$.
Now we define $d_i = \ensuremath{\beta}\xspace\cdot b_{i}$, $b_{i+1} = \ensuremath{\beta}\xspace \cdot d_i$, and $d_i = \ensuremath{\alpha}\xspace + 2b_i$ for some $\ensuremath{\alpha}\xspace, \ensuremath{\beta}\xspace \in \mathbb{R}_{>0}$.
Note that this choice satisfies all our conditions, including $d_i > 2 b_i$, and that $\ensuremath{\alpha}\xspace = (\ensuremath{\beta}\xspace -2)b_i$ and $b_{i+1} = \ensuremath{\beta}\xspace ^2b_i$.
Then the consumption-ratio\xspace $\QF{}(t_1, t_4)$ of the cycle is given by
\begin{equation*}
\frac{bF{}(t_1, t_4)}{t_4 - t_1}
= \frac{(\ensuremath{\alpha}\xspace + 2b_i) + 2b_i + 2(b_{i+1}-2b_i)}{(\ensuremath{\alpha}\xspace + 2 b_i) +b_{i+1}-2b_i}
= \frac{\ensuremath{\alpha}\xspace + 2b_{i+1}}{\ensuremath{\alpha}\xspace + b_{i+1}}
= \frac{(\ensuremath{\beta}\xspace-2) + 2\ensuremath{\beta}\xspace^2}{(\ensuremath{\beta}\xspace -2)+ \ensuremath{\beta}\xspace^2}
\end{equation*}
and attains a minimal value of \nf{17}{9} for $\ensuremath{\beta}\xspace = 4$.
Note that by design, $\QF{}(t_1, t_2)$ and $\QF{}(t_1, t_3)$ stay below \nf{17}{9}, as well.
Moreover, if the consumption-ratio\xspace has a maximum of \nf{17}{9} at the beginning of the cycle at $t_1$, this will also be the case at the end at $t_4$ as
\begin{equation*}
\QF{}(t_4)
= \frac{bF{}(t_1)+bF{}(t_1, t_4)}{t_4}
= \frac{t_1}{t_4} \cdot \frac{bF{}(t_1)}{t_1} + \frac{t_4 - t_1}{t_4} \cdot \frac{bF{}(t_1, t_4)}{t_4 - t_1}
\leq \frac{17}{9}.
\end{equation*}
Since the cycles change their roles at $t_4$ such that the $0$-interval occurs on the right side of $(0,0)$, the same argument can be used to bound the local consumption-ratio\xspace in the following interval and for all subsequent cycles, recursively.
Note that by looking at the time interval from $t_3$ to $t_6$, we can derive a closed form for $c_{i+1}$.
Similarly we proceed for $a_{i+1}$.
To prove the final theorem, it remains to find initial values to get the interlacing started, while maintaining $\QF{}(t) \leq \nf{17}{9}$.
Suitable values are
\begin{equation*}
\begin{array}{r l r l r l r l r l}
a_1 &:= \ensuremath{s}\xspace
& \hspace{.5cm}b_1 &:= 17 \ensuremath{s}\xspace
& \hspace{.5cm} a_2 &:= 34 \ensuremath{s}\xspace
& \hspace{.5cm}a_{i+1} &:= 7.5 b_{i}
& \hspace{.5cm}b_{i+1} &:= 4 d_{i}\\
c_1 & := \ensuremath{s}\xspace
& d_1 &:= 34 \ensuremath{s}\xspace
& c_2 &:= 238 \ensuremath{s}\xspace
& c_{i+1} &:= 7.5 d_{i}
& d_{i+1} &:= 4 b_{i+1},\\
\end{array}
\end{equation*}
which results in the starting intervals given in \autoref{fig:intervalStart}.
The local maxima at $t_1$ and $t_4$ then have consumption-ratio\xspace exactly $\nf{17}{9}$. The interval between $t_2$ and $t_3$ is set up equivalent to the one between $t_3$ and $t_6$ in \autoref{fig:recursiveInterlacing}, which means the interlacing construction can be applied to all intervals beyond.
Note that all barriers scale with \mbox{$s$.}
An example of this construction for $\ensuremath{s}\xspace = 1$ is given in \autoref{fig:exampleBarrierSystem}.
\begin{figure}
\caption{Illustration of time intervals at the start. Due to their growth, the sizes of the intervals are not true to scale.}
\label{fig:intervalStart}
\end{figure}
\begin{figure}
\caption{Example for the final barrier system\xspace for $\ensuremath{s}
\label{fig:exampleBarrierSystem}
\end{figure}
\begin{theorem}
The fire can be contained in the upper half-plane with speed $v = \frac{17}{9} = 1.\overline{8}$
\end{theorem}
\newcommand{\ensuremath{\delta}\xspace}{\ensuremath{\delta}\xspace}
\subsection{Improving the upper bound}
It is possible to reduce the upper bound of $v=1.\overline{8}$ slightly.
As shown in \autoref{fig:recursiveInterlacing}, the end of the $3$-interval in one direction coincides with the end of the $0$-interval in the other direction, which makes $t_4$ the only local maximum of the interval $[t_1, t_4]$.
We introduce a regular shift by a factor of \ensuremath{\delta}\xspace , see \autoref{fig:recursiveInterlacingImproved}.
This allows the $3$-interval in one direction to lie completely inside the $0$-interval of the other direction, as shown in \autoref{fig:recursiveInterlacingImproved}.
Then, there are two local maxima in the equivalent interval $[t_1, t_5]$, namely at $t_3$ and $t_5$.
We force both maxima to attain the same value to minimize both at the same time.
\begin{figure}
\caption{A general periodic interlacing of time intervals.}
\label{fig:recursiveInterlacingImproved}
\end{figure}
Again, we set $d_i = \ensuremath{\beta}\xspace \cdot b_i$ and $b_{i+1} = \ensuremath{\beta}\xspace \cdot d_i$, for some $\ensuremath{\beta}\xspace \geq 1$ determined below.
Then the value of the first local maximum can be expressed as
\begin{equation*}
\QF{}(t_1, t_3)
= \frac{bF{}(t_1, t_3)}{t_3 - t_1}
= \frac{1 \cdot (\ensuremath{\delta}\xspace \cdot b_i) + 3 \cdot b_i}{\ensuremath{\delta}\xspace \cdot b_i + b_i}
= \frac{\ensuremath{\delta}\xspace + 3}{\ensuremath{\delta}\xspace + 1}
= 1 + \frac{2}{\ensuremath{\delta}\xspace + 1}.
\end{equation*}
Considering the cycle from $t_1$ to $t_5$ in \autoref{fig:recursiveInterlacingImproved}, we can conclude that $c_{i+1} = b_{i+1} - b_i + \ensuremath{\delta}\xspace b_i + \ensuremath{\delta}\xspace d_i$.
Similarly, we can proceed on the interval from $t_5$ to $t_9$ to express $a_{i+1}$ in terms of \ensuremath{\beta}\xspace, \ensuremath{\delta}\xspace and $b_i$.
Using these identities, we obtain for the second local maximum
\begin{equation*}
\def1.4{1.4}
\begin{array}{rl}
\QF{}(t_1, t_5)
& = \frac{bF{}(t_1, t_5)}{t_5 - t_1}
= \frac{1 \cdot (\ensuremath{\delta}\xspace \cdot b_i) + 3b_i + 1\cdot (b_{i+1} - 2 b_i ) + 1 \cdot (c_{i+1} - d_i - \ensuremath{\delta}\xspace \cdot d_i)}{d_i + (c_{i+1} - d_i) - \ensuremath{\delta}\xspace \cdot d_i}\\
& = \frac{c_{i+1} - \ensuremath{\delta}\xspace \cdot d_i}{c_{i+1} - \ensuremath{\delta}\xspace \cdot d_i} + \frac{b_{i+1} + \ensuremath{\delta}\xspace \cdot b_i + b_i - d_i}{c_{i+1} - \ensuremath{\delta}\xspace \cdot d_i}
= 1 + \frac{b_{i+1} - b_i + \ensuremath{\delta}\xspace \cdot b_i + 2b_i - d_i}{b_{i+1} - b_i + \ensuremath{\delta}\xspace b_i}\\
& = 2 + \frac{2 b_i - d_i}{b_{i+1} - b_i + \ensuremath{\delta}\xspace b_i} = 2 + \frac{2 - \ensuremath{\beta}\xspace}{\ensuremath{\beta}\xspace^2 - 1 + \ensuremath{\delta}\xspace}.
\end{array}
\end{equation*}
As mentioned above, we set both local maxima to be equal, solve for \ensuremath{\delta}\xspace and obtain
\begin{equation*}
\delta = \frac{1}{2} \left( \ensuremath{\beta}\xspace - \ensuremath{\beta}\xspace^2 + \sqrt{-12 + 4 \ensuremath{\beta}\xspace + 5 \ensuremath{\beta}\xspace^2 - 2 \ensuremath{\beta}\xspace^3 + \ensuremath{\beta}\xspace^4}\right).
\end{equation*}
Plugging this into either one of the two local maxima and minimizing the resulting function for $\ensuremath{\beta}\xspace\geq 1$, we obtain
\begin{equation*}
\ensuremath{\beta}\xspace = \frac{3}{2} + \frac{1}{6} \left( 513 - 114 \sqrt{6} \right)^{\nf{1}{3}} + \frac{\left( 19 (9 + 2 \sqrt{6}) \right)^{\nf{1}{3}}}{2 \cdot 3^{\nf{2}{3}}} \approx 4.06887
\end{equation*}
for the optimal value of \ensuremath{\beta}\xspace, $\ensuremath{\delta}\xspace \approx 1.2802$ and
\begin{equation*}
v = \frac{1}{6} \left( 10 - \frac{19^{\nf{2}{3}}}{\sqrt[3]{2 (4 + 3 \sqrt{6})}} + \frac{\sqrt[3]{19 (4 + 3 \sqrt{6}))}}{2^{\nf{2}{3}}} \right) \approx 1.8771
\end{equation*}
as the minimum speed.
Note that the optimal value for \ensuremath{\beta}\xspace satisfies our conditions given in \autoref{equation:conditionsOfRecursion}, so that the barrier system\xspace can in fact be realized.
Finally, we give suitable values to get the interlacing started:
\begin{equation*}
\begin{array}{r l r l}
b_1 & := 1
& d_1 &:= 2 b_1\\
\ensuremath{s}\xspace & := \frac{(4\ensuremath{\beta}\xspace + 2 \delta + 1) - v(2\ensuremath{\beta}\xspace + \delta + 1)}{v} \cdot b_1
& a_1 & := c_1 := \ensuremath{s}\xspace\\[1em]
a_2 & := (\delta + 1) \cdot b_1
& c_2 &:= (2 \ensuremath{\beta}\xspace + 3 \delta - 1) \cdot b_1\\
a_{i+1} &:= (\delta - 1) d_i + (\ensuremath{\beta}\xspace + \delta) b_{i+1}
& \hspace{1cm}b_{i+1} &:= \ensuremath{\beta}\xspace \cdot d_i\\
c_{i+1} &:= (\delta - 1) b_i + (\ensuremath{\beta}\xspace + \delta) d_i
& d_{i+1} &:= \ensuremath{\beta}\xspace \cdot b_{i+1}.\\
\end{array}
\end{equation*}
To keep the expression simple, we fixed the value of $b_1$ and scaled the value of $\ensuremath{s}\xspace$ as listed above.
These values can be rescaled to work for any given \ensuremath{s}\xspace.
\begin{theorem}
The fire can be contained in the upper half-plane with speed $v = 1.8772$.
\end{theorem}
\section{Conclusion}\label{section:conclusion}
We have shown non-trivial bounds for the problem of protecting the lower half-plane from fire with an infinite horizontal barrier\xspace.
Our results show that delaying barriers -- in this case vertical segments attached to the horizontal barrier\xspace -- can help to break the obvious upper bound of~2 for the building speed.
More complex delaying barriers, e.\,g.,\xspace free-floating ones, were not analysed specifically, however it is hard to imagine a way for those to have improving effects.
It will be interesting to see if such an effect can also be achieved for the problem of containing the fire by a closed barrier curve, i.\,e.\xspace, for Bressan's original problem.
As a intermediate result in that direction, one ought to extend these results to the Euclidean metric first, where the effect of delaying barriers is less pronounced and harder to analyse.
\paragraph*{Acknowledgements}
We thank the anonymous referees for their valuable input.
\footnotesize
\end{document} | math | 40,055 |
\begin{document}
\begin{center}
\textbf{Homomorphisms from Functional Equations: The Goldie Equation, II. }
\textbf{by}
\textbf{N. H. Bingham and A. J. Ostaszewski.}
\end{center}
\noindent \textbf{Abstract. }In this sequel to [Ost3] by the second author,
we extend to multi-dimensional (or infinite-dimensional) settings the Goldie
equation arising in the general regular variation of [BinO5]. The theory
focusses on extension of the treatment there of Popa groups, permitting a
characterization of Popa homomorphisms (in two complementary theorems, 4A
and 4B\ below). This in turn enables a characterization of the (real-valued)
solutions of the multivariate Goldie equation, to be presented in the
further sequel [BinO7]. The Popa groups here contribute to a structure
theorem describing Banach-algebra valued versions of the Goldie equation in
[BinO8].
\noindent \textbf{Keywords. }Regular variation, general regular variation,
Popa groups, Go\l \k{a}b-Schinzel equation\textit{,} Goldie functional
equation.
\noindent \textbf{Classification}: 26A03, 26A12, 33B99, 39B22, 62G32.
\textbf{1. Introduction. }The \textit{Goldie functional equation} $(GFE)$ in
its simplest form, involving as unknowns a primary function $K$ called a
\textit{kernel} and an \textit{auxiliary} $g$, both \textit{continuous},
reads
\begin{equation}
K(x+y)=K(x)+g(x)K(y). \tag{$GFE$}
\end{equation}
The auxiliary should be interpreted as a local `addition accelerator' -- see
the more general form of $(GFE)$ given shortly below in Section 2. This
equation (a special case of a Levi-Civita equation [Lev] -- see [Stet, Ch.
5]; cf. [AczD, Ch.14], [BinO3], [Ost3]) plays a key role both in univariate
classical and Beurling regular variation [BinGT, Ch. 3], and for its
connections with extreme-value theory see \S 7.2. In the original context of
$\mathbb{R}$ the equation is very closely linked (cf. Cor. 2 in Section 6)
to the better known \textit{Go\l \k{a}b-Schinzel functional equation}:
\begin{equation}
\eta (x+y\eta (x))=\eta (x)\eta (y), \tag{$GS$}
\end{equation}
and it emerged most clearly in [BinO5] that $(GFE)$ is best studied by
reference to a group structure on $\mathbb{R}$ first introduced by Popa
[Pop] in 1965, as that enables $(GS)$ to be restated as homomorphy with the
multiplicative group of positive reals. It was also noticed in [Ost3], again
in the context of $\mathbb{R},$ that $(GFE)$ itself can be equivalently
formulated as a homomorphy between a pair of Popa groups on $\mathbb{R}.$
Here and in the companion paper [BinO7] (which is motivated by the continued
role of $(GFE)$ in general regular variation with vector-valued primary
function and scalar auxiliary) we pursue, in the wider context of a
topological vector space (tvs, see Section 2), the question whether
solutions of the $(GFE)$ may be reformulated as some kind of homomorphy. It
is natural to turn again to $(GS)$ and to use a natural extension of Popa's
definition in a vector space, suggested earlier by Javor in 1968 as a
formalization of contemporaneous work by Wo\l od\'{z}ko on solutions to $
(GS) $. This approach via multivariate Popa groups is reinforced by the more
recent characterization in that context of the continuous solutions of $(GS)$
by Brzd\k{e}k in 1992 in [Brz1] based on [BriD, Prop. 3]. A homomorphism
between multivariate Popa groups on two tvs necessarily satisfies $(GFE).$
In this paper we develop radial properties of multivariate Popa groups in
order to characterize Popa homomorphisms. In the companion paper [BinO7]
analogous radial properties of `Goldie kernels' are established under the
additional hypothesis that the auxiliary is `smooth' at the origin; this
regularity assumption is relatively innocuous, given that $g$ is necessarily
multiplicative and continuous, as continuity in the solution of a functional
equation typically promotes itself to differentiability, see J\'{a}rai
[Jar]. These radiality properties alongside the characterizations of this
paper show that such Goldie kernels are also Popa homomorphisms, thereby
giving them a characterization. In the companion paper we then go one step
further to show a similar result for a more general variant of $(GFE)$ with
two auxiliaries, i.e. two `addition accelerators' one for each of the
addition signs in the equation above.
The remainder of the paper is organized as follows. In Section 2 we consider
the Goldie functional equation of our title in our present multivariate
setting. In Section 3 we extend to a multi-dimensional (or
infinite-dimensional) setting the \textit{Popa group} structure previously
used successfully in univariate regular variation, in [BinO4] and especially
in the study of $(GFE)$ (cf. [BinO5], [Ost3]), distinguishing this from the
closely related \textit{Javor group}. Here we characterize radial subgroups
of Popa groups as also being Popa groups (Theorem 1 and its corollary
Theorem1Q for the field of rational $\mathbb{Q}$). In Section 4 we
specialize a theorem of Chudziak to homomorphisms from Javor groups to Popa
groups, and then study homomorphisms between Popa groups, which ultimately
characterize the kernel functions introduced above. Here we prove the
Abelian Dichotomy Theorem (Theorem 2), which helps to establish two theorems
characterizing in Section 5 the radial behaviours of kernel functions
(Theorems 3A and 3B corresponding to direction vectors outside and inside
the null space $\mathcal{N(\rho )}$), all three of which are needed to
establish our main result in Section 6: a dichotomy is satisfied by
homomorphisms between Popa groups. They can only be of one of two types, as
characterized in the First and Second Popa Homomorphism Theorems (Theorems
4A and 4B). This dichotomy is driven by the Abelian Dichotomy above. In
Corollary 2 we establish the close connection between $(GFE)$ and Popa
groups. That the solution functions $K$ to $(GFE)$ (for some auxiliary $g$)
and homomorphisms between Popa groups are identical is established in
[BinO7]; that too is driven by a further dichotomy: the intersection of two
hyperplanes through the origin has co-dimension 1 or 2. We close in Section
7 with complements, including a brief discussion of multivariate
extreme-value theory.
When convenient we relegate routine calculations to \S 8 (Appendix) of this
fuller arXiv version of the paper (under its initial title: `Multivariate
general regular variation: Popa groups on vector spaces' 1910.05816v2).
\textbf{2. The multivariate Goldie functional equation. }For $X$ a real
topological vector space (tvs), write $\langle u\rangle _{X},$ or $\langle
u\rangle ,$ for the \textit{linear span} of $u\in X$ and following [Ost1]
call a function $\varphi :X\rightarrow \mathbb{R}$ \textit{self-equivarying}
over $X,$ $\varphi \in SE_{X},$ if for each $u\in X$ both $\varphi (tu)=O(t)$
and
\begin{equation*}
\varphi (tu+v\varphi (tu))/\varphi (tu)\rightarrow \eta _{u}^{\varphi
}(v)\qquad (v\in \langle u\rangle _{X},t\rightarrow \infty )
\end{equation*}
locally uniformly in $v.$ This appeals to the underlying uniformity
structure on $X$ generated by the neighbourhoods of the origin. As in [Ost1]
(by restriction to the linear span $\langle u\rangle _{X})$ the limit
function $\eta =\eta _{u}^{\varphi }$ satisfies $(GS)$ for $x,y\in \langle
u\rangle _{X}.$ When the limit function $\eta _{u}$ is continuous, one of
the forms it may take is
\begin{equation*}
\eta _{u}(x)=1+\rho _{u}x\qquad (x\in \langle u\rangle _{X})
\end{equation*}
for some $\rho _{u}\in \mathbb{R}$, the alternative form being $\eta
(x)=\max \{1+\rho _{u}x,0\}.$ A closer inspection of the proof in [Ost1]
shows that in fact the restriction $x,y\in \langle u\rangle _{X}$ above is
unneccessary. Consequently, one may apply the Brillou\"{e}t-Dhombres-Brzd
\k{e}k theorem [BriD, Prop. 3], [Brz1, Th. 4], on the continuous solutions
of $(GS)$ with $\eta :X\rightarrow \mathbb{R}$, to infer that $\eta $ here
takes the form
\begin{equation*}
\eta (x)=1+\rho (x)\qquad (x\in X),
\end{equation*}
for some continuous linear functional $\rho :X\rightarrow \mathbb{R}$, the
alternative form being $\eta (x)=\max \{1+\rho (x),0\}$. On this matter, see
also [Bar], [BriD], [Brz1]; cf. [Chu2,3], the former cited in detail below.
(For the same conclusion under assumptions such as radial continuity, or
Christensen measurability, see [Jab1,2], and [Brz2] under boundedness on a
non-meagre set.)
Below we study the implications of replacing $\rho _{u}$ by a continuous
linear function $\rho (x).$ For this we now need to extend the definition of
\textit{general regular variation} [BinO5] from the real line to a
multivariate setting. For real topological vector spaces $X,Y,$ a function $
f:X\rightarrow Y$ is $\varphi $-\textit{regularly varying} for $\varphi \in
SE_{X}$ relative to the (auxiliary)\textit{\ norming }function $
h:X\rightarrow \mathbb{R}$ if the\textit{\ kernel} function $K$ below is
well defined for all $x\in X$ by
\begin{equation}
K(x):=\lim_{t\rightarrow \infty }[f(tx+x\varphi (tx))-f(tx)]/h(tx)\qquad
(x\in X). \tag{$GRV$}
\end{equation}
For later use, we note the underlying \textit{radial dependence}: for $u\in
X $ put
\begin{equation*}
K_{u}(x):=\lim_{s\rightarrow \infty }f(su+x\varphi (su))-f(su)]/h(su)\qquad
(x\in \langle u\rangle _{X}).
\end{equation*}
Writing $x=\xi u$ with $\xi >0$ and $s:=t\xi >0,$
\begin{eqnarray*}
K(x) &=&K(\xi u)=\lim_{t\rightarrow \infty }f(t\xi u+x\varphi (t\xi
u))-f(t\xi u)]/h(t\xi u) \\
&=&\lim_{s\rightarrow \infty }f(su+x\varphi (su))-f(su)]/h(su)=K_{u}(x).
\end{eqnarray*}
So here $K_{u}=K|\langle u\rangle _{X},$ as $K(\xi u)=K_{u}(\xi u).$
We work radially: above with half-lines $(0,\infty )$ and below with those
of the form $(-1/\rho ,\infty )$ for $\rho >0$ (on $\langle u\rangle $ with
context determining $u$), see [BinO5].
\noindent \textbf{Proposition 1.1.} \textit{Let }$h$\textit{\ and }$\varphi
\in SE_{X}$\textit{\ be such that the limit}
\begin{equation*}
g(x):=\lim_{t\rightarrow \infty }h(tx+x\varphi (tx))/h(tx)\qquad (x\in X)
\end{equation*}
\textit{exists. Then the kernel }$K:X\rightarrow Y$\textit{\ in }$(GRV)$
\textit{satisfies the Goldie functional equation:}
\begin{equation}
K(x+\eta ^{\varphi }(x)y)=K(x)+g(x)K(y) \tag{$GFE$}
\end{equation}
\textit{for }$y\in \langle x\rangle _{X}.$ \textit{Furthermore, }$g$\textit{
\ satisfies }$(GFE)$\textit{\ in the alternative form}
\begin{equation}
g(x+\eta ^{\varphi }(x)y)=g(x)g(y)\qquad (y\in \langle x\rangle _{X}).
\tag{$GFE_{\times }$}
\end{equation}
\noindent \textbf{Proof. }Fix $x$ and $y.$ Writing $s=s_{x}:=t+\varphi (tx),$
so that $sx=tx+x\varphi (tx),$
\begin{equation*}
\frac{f(tx+(x+y)\varphi (tx))-f(tx)}{h(tx)}\hspace{3.5in}
\end{equation*}
\begin{eqnarray*}
&=&\frac{f(sx+y[\varphi (tx)/\varphi (sx)]\varphi (sx))-f(sx)}{h(sx)}\cdot
\frac{h(tx+x\varphi (tx))}{h(tx)} \\
&&+\frac{f(tx+x\varphi (tx))-f(tx)}{h(tx)}.
\end{eqnarray*}
Here $\varphi (sx)/\varphi (tx)=\varphi (tx+x\varphi (tx))/\varphi
(tx)\rightarrow \eta (x).$ Passage to the limit yields $(GFE),$ since $
\varphi (tx)=O(t).$ The final assertion is similar but simpler.
$
\square $
See Theorem BO below for the form that continuous radial kernels $K(\xi u)$
take as functions of $\xi $. This includes the alternative form as a
limiting case.
We will achieve a characterization of the kernel function $K$ by identifying
the dependence between the different \textit{radial restrictions} $K|\langle
u\rangle _{X}.$
\textbf{3. Popa-Javor circle groups and their radial subgroups. }For a real
topological vector space $X$ and a continuous linear function $\rho
:X\rightarrow \mathbb{R},$ the associated function
\begin{equation*}
\varphi (x)=\eta _{\rho }(x):=1+\rho (x)
\end{equation*}
satisfies $(GS)$, as may be routinely checked. The associated circle
operation $\circ _{\rho }:$
\begin{equation*}
x\circ _{\rho }y=x+y\varphi (x)=x+y+\rho (x)y
\end{equation*}
(which gives for $\rho (x)=x$ and $X=\mathbb{R}$ the \textit{circle
operation }of ring theory: cf. [Jac, II.3], [Coh, 3.1], and [Ost3, \S 2.1]
for the historical background) is due to Popa in 1965 on the line and by
Javor in 1968 in a vector space ([Pop], [Jav], cf. [BinO4]). It is
associative as noted in [Jav]. As in [BinO5] we need the open sets
\begin{equation*}
\mathbb{G}_{\rho }=\mathbb{G}_{\rho }(X):=\{x\in X:\eta _{\rho }(x)=1+\rho
(x)>0\}.
\end{equation*}
Note that if $x,y\in \mathbb{G}_{\rho }$, then $x\circ _{\rho }y\in \mathbb{G
}_{\rho }$, as
\begin{equation*}
\eta _{\rho }(x\circ _{\rho }y)=\eta _{\rho }(x)\eta _{\rho }(y)>0.
\end{equation*}
Javor [Jav] shows that
\begin{equation*}
\mathbb{G}_{\rho }^{\ast }:=\{x\in X:\eta _{\rho }(x)\neq 0\}
\end{equation*}
is a group under $\circ _{\rho };$ we will call this the \textit{Javor group}
. The result remains true under the additional restriction $\eta _{\rho
}(y)>0,$ giving rise to what we term \textit{Popa groups}:
\noindent \textbf{Theorem J }(after Javor [Jav])\textbf{.}\textit{\ For }$X$
\textit{\ a topological vector space and }$\rho ~:~X~\rightarrow ~\mathbb{R}$
\textit{\ a continuous linear function,} $(\mathbb{G}_{\rho }(X),\circ
_{\rho })$ \textit{is a group.}
\noindent \textbf{Proof. }This is routine, and one argues just as in [Jav],
but must additionally check positivity of $\eta _{\rho }.$ Here $0\in
\mathbb{G}_{\rho }$ and is the neutral element; the inverse of $x\in \mathbb{
G}_{\rho }$ is $x_{\rho }^{-1}:=-x/(1+\rho (x)),$ which is in $\mathbb{G}
_{\rho }$ since $1=\eta _{\rho }(0)=\eta _{\rho }(x)\eta _{\rho }(x_{\rho
}^{-1}),$ so that $\eta _{\rho }(x_{\rho }^{-1})>0.$ For other details see
\S 8 (Appendix).
$\square $
\noindent \textbf{Definitions.} 1. For $u\in \mathbb{G}_{\rho }(X)$, put
\begin{equation*}
\langle u\rangle _{\rho }:=\langle u\rangle \cap \mathbb{G}_{\rho
}(X)=\{tu:\eta _{\rho }(tu)=1+t\rho (u)>0,t\in \mathbb{R}\}.
\end{equation*}
(If $\rho (u)\neq 0,$ $\langle u\rangle _{\rho }=\{tu:t>-1/\rho (u)\}$ which
is a half-line in $\langle u\rangle _{X}$; otherwise $\langle u\rangle
_{\rho }=\langle u\rangle _{X}.$ Note that $\mathbb{G}_{\rho }(X)$ is an
affine half-space in $X.$)
\noindent 2. For $K$ with domain $\mathbb{G}_{\rho }(X)$ we will write $
K_{u}=K|\langle u\rangle _{\rho }$. (This will not clash with the radial
notation of \S 1.)
\noindent \textbf{Lemma.} \textit{The one-dimensional subgroup} $\langle
u\rangle _{\rho }$ \textit{is an abelian subgroup of }$\mathbb{G}_{\rho }(X)$
\textit{\ isomorphic with }$\mathbb{G}_{\rho (u)}(\mathbb{R}).$
\noindent \textbf{Proof. }We check closure under multiplication and
inversion. As before $\varphi (su\circ _{\rho }tu)=\varphi (su)\varphi
(tu)>0;$ also $\varphi (r(tu))>0$ for $\varphi (tu)>0,$ as $1=\varphi
(0)=\varphi (tu\circ _{\rho }r(tu))=\varphi (tu)\varphi (r(tu)).$ Further,
since
\begin{equation*}
su\circ _{\rho }tu=su+tu+st\rho (u)u=(s\circ _{\rho (u)}t)u,
\end{equation*}
the operation $\circ _{\rho }$ is abelian on $\langle u\rangle _{\rho }.$
$\square $
\noindent \textbf{Remark. }Despite the lemma above, unless $\rho \equiv 0$
or $X=\mathbb{R},$ the group $\mathbb{G}_{\rho }(X)$ itself is non-abelian.
(In the commutative case, except when $X=\mathbb{R},$ one may select $x\neq
0 $ with $\rho (x)=0;$ then $x\rho (y)=y\rho (x)=0$ and so $\rho (y)=0$ for
all $y.)$ We return to this matter in detail in Theorem 2 below.
The assumption on $\Sigma $ below is effectively that all its radial
subgroups are closed and dense in themselves. Key to the proof is the
observation that if $1+\rho (u)<0$, then a fortiori $1+\rho (-u)>1-\rho
(u)>0,$ i.e. if $u\notin \langle u\rangle _{\rho },$ then its negative $
-u\in \langle u\rangle _{\rho }$ and likewise its $\mathbb{G}_{\rho }(X)$
-inverse $(-u)_{\rho }^{-1}\in \langle u\rangle _{\rho }$.
\noindent \textbf{Theorem 1 (Radial Subgroups Theorem).}\newline
\textit{Radial subgroups of Popa groups are Popa. That is, for }$\Sigma $
\textit{\ a subgroup of} $\mathbb{G}_{\rho }(X)$ \textit{with }$\langle
u\rangle _{\rho }\subseteq \Sigma $\textit{\ for each }$u\in \Sigma :$
\begin{equation*}
\Sigma =\mathbb{G}_{\rho }(\langle \Sigma \rangle ).
\end{equation*}
\noindent \textbf{Proof. }With $\langle \Sigma \rangle $ the linear span, $
\Sigma \subseteq \mathbb{G}_{\rho }(\langle \Sigma \rangle )$ follows from $
\Sigma \subseteq \langle \Sigma \rangle $, as $\Sigma $ and $\mathbb{G}
_{\rho }(\langle \Sigma \rangle )$ are subgroups of $\mathbb{G}_{\rho }(X)$.
For the converse, we first show that $\alpha x+\beta y\in \Sigma $ for $
x,y\in \Sigma $ and scalars $\alpha ,\beta $ whenever $\alpha x+\beta y\in
\mathbb{G}_{\rho }(\langle \Sigma \rangle )$. As a preliminary, notice that
one at least of $\alpha x,\beta y$ is in $\Sigma .$ Otherwise, $1+\rho
(\alpha x)<0,$ as $x\in \Sigma $ and $\alpha x\in \langle x\rangle
\backslash \Sigma ,$ and likewise $1+\rho (\beta y)<0.$ Summing,
\begin{equation*}
2+\rho (\alpha x)+\rho (\beta y)<0.
\end{equation*}
But $\alpha x+\beta y\in \mathbb{G}_{\rho }(X),$ so
\begin{equation*}
0<1+\rho (\alpha x+\beta y)=1+\rho (\alpha x)+\rho (\beta y)<-1,
\end{equation*}
a contradiction. We proceed by cases.
\noindent \textit{Case 1.} \textit{Both }$u:=\alpha x$ \textit{and} $
v:=\beta y$\textit{\ are in} $\Sigma .$ Here
\begin{equation*}
\alpha x+\beta y=u+v=u\circ _{\rho }[v/(1+\rho (u))]\in \Sigma ;
\end{equation*}
indeed, by assumption $1+\rho (u))>0$ and $1+\rho (u+v)>0,$ so by linearity
\begin{equation*}
1+\rho (v/(1+\rho (u)))=[1+\rho (u+v)]/(1+\rho (u))>0,
\end{equation*}
and so $v/(1+\rho (u))\in \langle v\rangle _{\rho }\subseteq \Sigma .$
\noindent \textit{Case 2. One of} $u:=\alpha x,v=:\beta y$ \textit{is not in}
$\Sigma $\textit{\ (`off the half-line }$\langle x\rangle _{\rho }$\textit{\
or }$\langle y\rangle _{\rho }$\textit{').}
By commutativity of addition, w.l.o.g. $v\notin \Sigma .$ Then $-v\in \Sigma
.$ As $\Sigma $ is a subgroup, $(-v)^{-1}=v/(1-\rho (v))\in \Sigma $ and,
setting
\begin{equation*}
\delta :=(1-\rho (v))/[1+\rho (u)],
\end{equation*}
\begin{equation*}
\alpha x+\beta y=u+v=u\circ _{\rho }\delta (-v)^{-1}=u+\delta v[1+\rho
(u)]/(1-\rho (v))\in \Sigma .
\end{equation*}
Indeed, $\delta (-v)^{-1}=\delta v/(1-\rho (v))\in \langle v\rangle _{\rho
}\subseteq \Sigma ,$ since by assumption $1+\rho (u))>0$ and $1+\rho
(u+v)>0, $ so
\begin{equation*}
1+\rho (\delta (-v)^{-1})=1+\rho \left( \frac{v}{1+\rho (u)}\right) =\frac{
1+\rho (u+v)}{1+\rho (u)}>0.
\end{equation*}
Thus in all the possible cases $\alpha x+\beta y\in \Sigma $ for $x,y\in
\Sigma $ with $\alpha x+\beta y\in \mathbb{G}_{\rho }(\langle \Sigma \rangle
).$
Next we proceed by induction, with what has just been established as the
base step, to show that for all $n\geq 2,$ if $\alpha _{1}u_{1}+\alpha
_{2}u_{2}+...+\alpha _{n}u_{n}\in \mathbb{G}_{\rho }(\langle \Sigma \rangle
),$ for $u_{1},u_{2},...,u_{n}\in \Sigma $ and scalars $\alpha _{1},\alpha
_{2},...,\alpha _{n},$ then $\alpha _{1}u_{1}+\alpha _{2}u_{2}+...+\alpha
_{n}u_{n}\in \Sigma .$
Assuming the above for $n,$ we pass to the case of $u_{1},u_{2},...,u_{n+1}
\in \Sigma $ and scalars $\alpha _{1},\alpha _{2},...,\alpha _{n+1}$ with $
\alpha _{1}u_{1}+\alpha _{2}u_{2}+...+\alpha _{n+1}u_{n+1}\in \mathbb{G}
_{\rho }(\langle \Sigma \rangle ).$
Again as a preliminary, notice that, by permuting the subscripts as
necessary, w.l.o.g. $x:=\alpha _{1}u_{1}+...+\alpha _{n}u_{n}\in \mathbb{G}
_{\rho }(\langle \Sigma \rangle );$ otherwise, for $j=1,...,n+1$
\begin{equation*}
1+\rho (\sum\nolimits_{i\neq j}\alpha _{i}u_{i})<0,
\end{equation*}
and again as above, on summing, this leads to the contradiction
\begin{equation*}
0<n[1+\rho (\alpha _{1}u_{1}+\alpha _{2}u_{2}+...+\alpha _{n+1}u_{n+1})]<-1.
\end{equation*}
So we suppose w.l.o.g. that $\alpha _{1}u_{1}+\alpha _{2}u_{2}+...+\alpha
_{n}u_{n}\in \mathbb{G}_{\rho }(\langle \Sigma \rangle );$ by the inductive
hypothesis, $x:=\alpha _{1}u_{1}+\alpha _{2}u_{2}+...+\alpha _{n}u_{n}\in
\Sigma .$ Take $y:=u_{n+1}\in \Sigma $ and apply the base case $n=2$ to $x$
and $y.$ Then, since $w:=\alpha _{1}u_{1}+\alpha _{2}u_{2}+...+\alpha
_{n+1}u_{n+1}=x+\alpha _{n+1}y\in \mathbb{G}_{\rho }(\langle \Sigma \rangle
) $, $w\in \Sigma .$ This completes the induction, showing $\mathbb{G}_{\rho
}(\langle \Sigma \rangle )\subseteq \Sigma .$
$\square $
In view of the role in quantifier weakening of countable subgroups dense in
themselves [BinO3,5], we note in passing that the proof above may be
relativized to the subfield of \textit{rational} scalars to give (with $
\langle \cdot \rangle ^{_{\mathbb{Q}}}$ below the rational linear span):
\noindent \textbf{Theorem 1Q.} \textit{For }$\Sigma $\textit{\ a countable
subgroup of} $\mathbb{G}_{\rho }(X)$ \textit{with }$\langle u\rangle _{\rho
}^{\mathbb{Q}}\subseteq \Sigma $\textit{\ for each }$u\in \Sigma ,$ \textit{
if} $\rho (\Sigma )\subseteq \mathbb{Q}:$
\begin{equation*}
\Sigma =\mathbb{G}_{\rho }(\langle \Sigma \rangle ^{_{\mathbb{Q}}}).
\end{equation*}
\noindent \textbf{Proof. }For rational coefficients $\alpha ,\beta $ etc.
the preceding proof generates only rational quotients.
$\square $
\textbf{4. Abelian dichotomy and homomorphisms. }Our first result here,
Theorem 2, allows us to characterize in Theorems 4A and 4B homomorphisms
between Popa groups in vector spaces. We recall that
\begin{equation*}
\eta _{1}(t):=1+t
\end{equation*}
takes $\mathbb{G}_{1}(\mathbb{R)}\overset{\eta _{1}}{\rightarrow }(\mathbb{R}
_{+},\times \mathbb{)}$ ($=\mathbb{G}_{\infty }(\mathbb{R)}$, see below and
[BinO5]), isomorphically. Note that
\begin{equation*}
\eta _{\rho }(x)=\eta _{1}(\rho (x))=1+\rho (x).
\end{equation*}
In the case of $X=\mathbb{R}$, where $\rho (x)=\rho x,$ this reduces to
\begin{equation*}
1+\rho x.
\end{equation*}
\noindent \textbf{Theorem 2 (Abelian Dichotomy Theorem).\newline
}\textit{A commutative subgroup }$\Sigma $\textit{\ of} $\mathbb{G}_{\rho
}(X)$ \textit{is either}\newline
\noindent (i)\textit{\ a subspace of the null space }$\mathcal{N}(\rho ),$
\textit{so a subgroup of }$(X,+),$\textit{\ or}\newline
\noindent (ii)\textit{\ for some }$u\in \Sigma $ \textit{a subgroup of }$
\langle u\rangle _{\rho }$\textit{\ isomorphic under }$\rho $\textit{\ to a
subgroup of} $\mathbb{G}_{1}(\mathbb{R)}:$
\begin{equation*}
\rho (x\circ _{\rho }y)=\rho (x)\circ _{1}\rho (y).
\end{equation*}
\textit{Thus the image of }$\Sigma $\textit{\ under }$\eta _{\rho }$\textit{
\ is a subgroup of }$(\mathbb{R}_{+},\times \mathbb{)}$.
\noindent \textbf{Proof.} Either $\rho (z)=0$ for each $z\in \Sigma ,$ in
which case $\Sigma $ is a subgroup of $(X,+),$ or else there is $z\in \Sigma
\backslash \{0\}$ with $\rho (z)\neq 0.$ In this case take $u=u_{\rho
}(z):=z/\rho (z)\neq 0.$ Then $\rho (u)=1$ so $u\in \Sigma ,$ and for all $
x\in \Sigma $ by commutativity $x=\rho (u)x=\rho (x)u,$ i.e. $\Sigma $ is
contained in the linear span $\langle u\rangle _{X}$ and so in $\langle
u\rangle _{\rho }$. So the operation $\circ _{\rho }$ on $\Sigma $ takes the
form
\begin{equation*}
x\circ _{\rho }y=\rho (x)u+\rho (y)u+\rho (\rho (x)u)\rho (y)u.
\end{equation*}
But $x\circ _{\rho }y=\rho (x\circ _{\rho }y)u,$ so as $u\neq 0$ the
asserted isomorphism follows from
\begin{equation*}
\rho (x\circ _{\rho }y)u=[\rho (x)+\rho (y)+\rho (x)\rho (y)]u.
\end{equation*}
In turn this implies
\begin{equation*}
\eta _{\rho }(x\circ _{\rho }y)=1+\rho (x\circ _{\rho }y)=(1+\rho
(x))(1+\rho (y))=\eta _{\rho }(x)\eta _{\rho }(y),
\end{equation*}
so that $\eta _{\rho }$ is a homomorphism into $(\mathbb{R}_{+},\times
\mathbb{)}$, where $\mathbb{R}_{+}:=(0,\infty ).$
$\square $
Before we pass to a study of radial behaviours in \S 4, we recall the
following result [Ost3, Prop. A], [Chu1] (cf. [BinO5, Th. 3]) for the
context $\mathbb{G}_{\rho }(\mathbb{R})$ with $\rho (x)=\rho x.$ To
accommodate alternative forms of $(GFE),$ the matrix includes the
multiplicative group $(\mathbb{R}_{+},\times )$ as $\rho =\infty $ ; for a
derivation via a passage to the limit see [BinO5], but note that
\begin{equation*}
\rho x+\rho y+\rho x\rho y=[\rho x\cdot \rho y](1+o(\rho ))\qquad (x,y\in
\mathbb{R}_{+},\rho \rightarrow \infty ).
\end{equation*}
\noindent \textbf{Theorem BO.} \textit{Take }$\psi :\mathbb{G}_{\rho
}\rightarrow \mathbb{G}_{\sigma }$\textit{\ a homomorphism with }$\rho
,\sigma \in \lbrack 0,\infty ].$\textit{\ Then the lifting }$\Psi :\mathbb{R}
\rightarrow \mathbb{R}$\textit{\ \ of }$\psi $\textit{\ to }$\mathbb{R}$\
\textit{defined by the canonical isomorphisms }$\log ,\exp ,$\textit{\ }$
\{\eta _{\rho }:\rho >0\}$ \textit{is bounded above on }$\mathbb{G}_{\rho }$
\textit{\ iff }$\Psi $\textit{\ is bounded above on }$\mathbb{R}$\textit{,
in which case }$\Psi $ \textit{and }$\psi $\textit{\ are continuous. Then
for some }$\kappa \in \mathbb{R}$ \textit{one has} $\psi (t)$ \textit{as
below:}
\renewcommand{1}{1.25}
\begin{equation*}
\begin{tabular}{|l|l|l|l|}
\hline
Popa parameter & $\sigma =0$ & $\sigma \in (0,\infty )$ & $\sigma =\infty $
\\ \hline
$\rho =0$ & $\kappa t$ & $\eta _{\sigma }^{-1}(e^{\sigma \kappa t})$ & $
e^{\kappa t}$ \\ \hline
$\rho \in (0,\infty )$ & $\log \eta _{\rho }(t)^{\kappa /\rho }$ & $\eta
_{\sigma }^{-1}(\eta _{\rho }(t)^{\sigma \kappa /\rho })$ & $\eta _{\rho
}(t)^{\kappa /\rho }$ \\ \hline
$\rho =\infty $ & $\log t^{\kappa }$ & $\eta _{\sigma }^{-1}(t^{\sigma
\kappa })$ & $t^{\kappa }$ \\ \hline
\end{tabular}
\end{equation*}
\newline
\renewcommand{1}{1}
After linear transformation, all the cases reduce to some variant (mixing
additive or multiplicative structures) of the Cauchy functional equation.
(The parameters are devised to achieve continuity across cells, see [BinO5].)
We next show how this theorem is related to the current context of $(GFE)$.
As a preliminary we note a result of Chudziak in which $\circ _{\rho }$ is
applied to all of $X$, so in practice to Javor groups -- i.e. without
restriction to $\mathbb{G}_{\rho }(X).$ We repeat his proof, amending it to
the range context of $\mathbb{G}_{\sigma }(Y).$\textit{\ }Here one fixes $u$
\ with $\rho (u)=1,$\ obtaining constants $\kappa =\kappa (u),$ and $\tau
=\tau (u)$.
\noindent \textbf{Theorem Ch} (\textbf{Javor Homomorphism Theorem}, [Chu2,
Th. 1]). \textit{A continuous function} $K:X\rightarrow $ $\mathbb{G}
_{\sigma }(Y)$ \textit{with }$X,Y$\textit{\ real topological vector spaces,
satisfying}
\begin{equation*}
K(x\circ _{\rho }y)=K(x)\circ _{\sigma }K(y)\qquad (x,y\in X)
\end{equation*}
\textit{with }$\rho $ \textit{not identically zero takes the form}
\begin{equation*}
K(x)=A_{u}(x)+[1+\sigma (A_{u}(x))][(1+\rho (x))^{\tau \kappa }-1]K(u)/\tau ,
\end{equation*}
\textit{for any }$u$\textit{\ with }$\rho (u)=1$\textit{\ and corresponding
constants }$\kappa =\kappa (u),\tau =\sigma (K(u)),$\textit{\ and continuous
}$A_{u}$\textit{\ satisfying}
\begin{equation}
A_{u}(x+y)=A_{u}(x)\circ _{\sigma }A_{u}(y)\qquad (x,y\in X) \tag{$A$}
\end{equation}
\textit{(so with abelian range).} \textit{The relevant factor for }$\tau =0$
\textit{is to be read as} $K(u)\log (1+\rho (x))/\log 2.$
\noindent \textbf{Proof. }Take any $u\in X$ with $\rho (u)=1$ and set
\begin{equation*}
A_{u}(x):=K\left( x-\rho (x)u\right) ,\qquad \mu _{u}(t):=K((t-1)u).
\end{equation*}
The former is continuous and additive over $X$ (for $v_{i}=x_{i}-\rho
(x_{i})u,$ apply $K$ to $v_{1}+v_{2}=v_{1}\circ _{\rho }v_{2})$, hence with
image an abelian subgroup of $\mathbb{G}_{\sigma }(Y).$ The latter is an
isomorphism between $(\mathbb{R}_{+},\times )$ and a subgroup of $\mathbb{G}
_{\sigma }(Y)$ with
\begin{equation*}
\mu _{u}(st)=\mu _{u}(s)\circ _{\sigma }\mu _{u}(t).
\end{equation*}
This last follows from the identity
\begin{equation*}
(st-1)u=(s-1)u+[1+\rho ((s-1)u)](t-1)u.
\end{equation*}
Now the image subgroup under $\mu _{u}$, being abelian, is a subgroup of $
\langle K(u)\rangle _{\sigma }$ so isomorphic to a subgroup of $\mathbb{G}
_{\tau }(\mathbb{R})$ for $\tau :=\sigma (K(u))\in \mathbb{R}$, by Theorem
2. So $\mu _{u}$ is an isomorphism from $(\mathbb{R}_{+},\times )=\mathbb{G}
_{\infty }(\mathbb{R})$ to $\mathbb{G}_{\tau }(\mathbb{R}),$ for $\tau
=\sigma (K(u)),$ and so by Theorem BO for some $\kappa =\kappa (u)$
\begin{equation*}
\mu _{u}(t)=\eta _{\sigma (K(u))}^{-1}(t^{\sigma (K(u))\kappa (u)})K(u).
\end{equation*}
So, as $\rho ([x-\rho (x)u])=0,$
\begin{eqnarray*}
K(x) &=&K([x-\rho (x)u])\circ _{\rho }\rho (x)u)=A_{u}(x)\circ _{\sigma
}K(\rho (x)u) \\
&=&A_{u}(x)\circ _{\sigma }\mu _{u}(1+\rho (x)).
\end{eqnarray*}
For $\sigma (K(u))=0$ the above result should be amended to its limiting
value as $\tau \rightarrow 0,$ namely $K([x-\rho (x)u])+K(u)\log (1+\rho
(x))/\log 2$ (since $\kappa (u)=1/\log 2).$
$\square $
\textbf{5. Radial behaviours. }Our next two results help establish in \S 5
Theorems 4A and 4B two not entirely dissimilar representations for the
circle groups, including the case $\rho \equiv 0$ from which the form of $
A_{u}$ above may be deduced in view of equation $(A)$ in Th. Ch.
\noindent \textbf{Theorem 3A (Radial behaviour outside} $\mathcal{N}(\rho )$
\textbf{).}\newline
\textit{If }$K:\mathbb{G}_{\rho }(X)\rightarrow \mathbb{G}_{\sigma }(Y)$
\textit{is continuous and satisfies}
\begin{equation}
K(x\circ _{\rho }y)=K(x)\circ _{\sigma }K(y)\qquad (x,y\in \mathbb{G}_{\rho
}(X)), \tag{$K$}
\end{equation}
\textit{then, for }$x$ \textit{with }$\rho (x)\neq 0$ \textit{and }$\sigma
(K(x))\neq 0,$\textit{\ there is }$\kappa =\kappa (x)\in \mathbb{R}$ $
\backslash \{0\}$ \textit{with }
\begin{equation*}
K(z)=\eta _{\sigma }^{-1}(\eta _{\rho }(z)^{\sigma (K(x))\kappa })\qquad
(z\in \langle x\rangle _{\rho }).
\end{equation*}
\textit{Moreover, the index }$\gamma (x):=\sigma (K(x))\kappa (x)$\textit{\
is then continuous and extends to satisfy the equation}
\begin{equation*}
\gamma (a\circ _{\rho }b)=\gamma (a)+\gamma (b)\qquad (a,b\in \mathbb{G}
_{\rho }(X)).
\end{equation*}
\noindent \textbf{Proof.} For $x$ as above, take $u=u_{\rho }(x)\neq 0$ and $
v=u_{\sigma }(K(x))\neq 0,$ both well-defined as $\rho (x)$ and $\sigma
(K(x))$ are non-zero (also $u$ $\in \langle x\rangle _{\rho }$ and $v$ $\in
\langle K(x)\rangle _{\sigma }$, as $\rho (u)=\sigma (v)=1$). The
restriction $K_{u}=K|\langle u\rangle _{\rho }$ yields a continuous
homomorphism into $\mathbb{G}_{\sigma }(Y).$ As $\langle u\rangle _{\rho }$
is an abelian group under $\circ _{\rho }$, its image under $K_{u}$ is an
abelian subgroup of $\mathbb{G}_{\sigma }(Y).$ So, as in Theorem 2, it is a
\textit{non-trivial} subgroup of $\langle v\rangle _{\sigma }$. As noted, $
\rho (u)=\sigma (v)=1,$ so we have the following \textit{isomorphisms}:
\begin{eqnarray*}
&&\langle u\rangle _{\rho }\overset{\rho }{\rightarrow }\mathbb{G}_{1}(
\mathbb{R)}\overset{\eta _{1}}{\rightarrow }(\mathbb{R}_{+},\times \mathbb{)}
, \\
&&\langle v\rangle _{\sigma }\overset{\sigma }{\rightarrow }\mathbb{G}_{1}(
\mathbb{R)}\overset{\eta _{1}}{\rightarrow }(\mathbb{R}_{+},\times \mathbb{)}
\end{eqnarray*}
(writing $\rho ,\sigma =$ for $\rho |_{\langle u\rangle }$ and $\sigma
|_{\langle v\rangle }),$ which combine to give
\begin{equation*}
k(t):=\eta _{1}\sigma K_{u}\rho ^{-1}\eta _{1}^{-1}(t)=\eta _{\sigma
}K_{u}\eta _{\rho }^{-1}(t)
\end{equation*}
as a \textit{non-trivial} homomorphism of $(\mathbb{R}_{+},\times \mathbb{)}$
into itself:
\begin{equation*}
k(st)=k(s)k(t).
\end{equation*}
Solving this Cauchy equation for a non-constant continuous $k$ yields
\begin{equation*}
k(t)\equiv t^{\gamma }\qquad (t\in \mathbb{R}_{+}),
\end{equation*}
for some $\gamma =\gamma (u)\in \mathbb{R}\backslash \{0\}$; so $k$ is
bijective. Write $\gamma =\gamma (u)=\sigma (K(u))\kappa (u)$, then, as
asserted (abbreviating the symbols),
\begin{eqnarray*}
K_{u}(z) &=&\eta _{\sigma }^{-1}k\eta _{\rho }(z)=\eta _{\sigma }^{-1}(\eta
_{\rho }(z)^{\sigma \kappa }) \\
&=&\sigma ^{-1}(\eta _{1}^{-1}(1+\rho (z))^{\sigma \kappa })),\qquad (z\in
\langle u\rangle _{\rho }).
\end{eqnarray*}
In particular, $K_{u}$ is injective. As $u\neq 0,$ $0\neq K(u)\in \langle
v\rangle _{\sigma },$ so $K(u)=sv$ for some $s\neq 0.$ Hence $\sigma
(K(u))=s\sigma (v)=s\neq 0.$ Since $\sigma (tK(u))=t\sigma (K(u)),$
\begin{equation*}
K(z)=K_{u}(z)=[((1+\rho (z))^{\sigma (K(u))\kappa (u)}-1)/\sigma
(K(u))]K(u)\qquad (z\in \langle u\rangle _{\rho }).
\end{equation*}
Here $\rho (z)=t$ for $z=tu,$ as $\rho (u)=1$ by choice. Taking $z=u$ gives
\begin{equation*}
(2^{\sigma (K(u))\kappa (u)}-1)/\sigma (K(u))=1:\qquad \kappa (u)=\log
(1+\sigma (K(u))/[\sigma (K(u))\log 2],
\end{equation*}
and so $\gamma (u):=\sigma (K(u))\kappa (u)$ is continuous and satisfies the
equation
\begin{equation*}
\gamma (a\circ _{\rho }b)=\gamma (a)+\gamma (b)\qquad (a,b\in \mathbb{G}
_{\rho }(X)).
\end{equation*}
Indeed, write $\alpha =K(a),\beta =K(b);$ then as $K(a\circ _{\rho
}b)=\alpha \circ _{\sigma }\beta ,$ by linearity of $\sigma $
\begin{eqnarray*}
\log (1+\sigma (K(a\circ _{\rho }b)) &=&\log (1+\sigma (\alpha +\beta
+\sigma (\alpha )\beta )) \\
&=&\log (1+\sigma (\alpha )+\sigma (\beta )+\sigma (a)\sigma (\beta )) \\
&=&\log (1+\sigma (\alpha ))+\log (1+\sigma (\beta )).
\end{eqnarray*}
For $\sigma (K(x))=0,$ the map $\langle v\rangle _{\sigma (v)}$ $\overset{
\eta _{\sigma }}{\rightarrow }$ $(\mathbb{R}_{+},\times \mathbb{)}$ above
must be intepreted as exponential. A routine adjustment of the argument
yields
\begin{equation*}
K(z)=K_{u}(z)=K(u)\log (1+\rho (z))/\log 2\qquad (z\in \langle u\rangle
_{\rho }),
\end{equation*}
justifying hereafter a \textit{L'Hospital convention} (of taking limits $
\sigma (K(u))\rightarrow 0$ in the `generic' formula).
$\square $
We consider now the case $\rho (x)=0,$ which turns out as expected, despite
Theorem 2 being of no help here. We state the result we need and omit the
proof (which is in \S 8 (Appendix)), citing
instead the more detailed variant from [BinO7, Th. 2].
\noindent \textbf{Theorem 3B\ }(Radial behaviour inside $\mathcal{N}(\rho ))$
\textbf{.\newline
}\textit{For continuous }$K$\textit{\ satisfying }$(K)$ \textit{and }$u\neq
0 $\textit{, if }$\rho (u)=0,$\textit{\ then }
\begin{equation*}
K(\langle u\rangle _{\rho })\subseteq \langle K(u)\rangle _{\sigma }\sim
\mathbb{G}_{\sigma (K(u))}\mathbb{(R)};
\end{equation*}
\textit{in fact}
\begin{equation*}
K(\xi u)=\lambda _{u}(\xi )K(u).
\end{equation*}
\textit{Moreover, if }$K(u)\neq 0,$\textit{\ then }$\lambda _{u}:(\mathbb{R}
,+)\rightarrow \mathbb{G}_{\sigma (K(u))}\mathbb{(R)}$ \textit{is for some }$
\kappa =\kappa (u)$ \textit{the isomorphism}
\begin{equation*}
\lambda _{u}(t)=(e^{\sigma (K(u))\kappa (u)t}-1)/\sigma (K(u))\text{ }
\end{equation*}
\textit{or just} $t$ \textit{if }$\sigma (K(u))=0.$
\noindent \textbf{Proof}. This follows from [BinO7, Th. 2], since $x\mapsto
\rho (x)$ and $y\mapsto \sigma (y)$ are Fr\'{e}chet differentiable (at $0)$
.
$\square $
\noindent \textbf{Corollary 1.}\textit{\ In Theorem 3B, if }$\rho (u)=0$
\textit{\ and }$K(u)\neq 0,$ \textit{then either}\newline
\noindent (i)\textit{\ }$\sigma (K(u))=0$ \textit{and }$\kappa (u)=1,$
\textit{\ or\newline
}\noindent (ii) $\sigma (K(u))>0,$ $\kappa (u)=\log [1+\sigma (K(u))]/\sigma
(K(u))$ \textit{and the index }$\gamma (u):=\sigma (K(u))\kappa (u)$\textit{
\ is additive on }$\mathcal{N}(\rho )$\textit{:}
\begin{equation*}
\gamma (u+v)=\gamma (u)+\gamma (v)\qquad (u,v\in \mathcal{N}(\rho )).
\end{equation*}
\noindent \textbf{Proof. }As $\rho (u)=0,$ the notation in the proof above
is valid, so $\lambda _{u}(1)=1,$ as $0\neq K(u)=\lambda _{u}(1)K(u).$
\textbf{\ }If \textit{\ }$\sigma (K(u))=0,$ then $\kappa (u)=1,$ by Theorem
3B. Otherwise,
\begin{equation*}
(e^{\sigma (K(u))\kappa (u)}-1)/\sigma (K(u))=1:\qquad \kappa (u)=\log
(1+\sigma (K(u)))/\sigma (K(u)),
\end{equation*}
and, as $\gamma (u)=\log (1+\sigma (K(u))),$ the concluding argument is as
in Theorem 3A (with $\circ _{\rho }=+$ on $\mathcal{N}(\rho )$).
$
\square $
\textbf{6. Homomorphism dichotomy. }The paired Theorems 4A and 4B below, our
main contribution, amalgamate the earlier radial results according to the
two forms identified by Theorem 2 that an \textit{abelian} Popa subgroup may
take (see below). Theorem 4A covers $\sigma \equiv 0$ as $\mathcal{N}(\sigma
)=\mathbb{G}_{\sigma }(Y)=Y,$ whereas $\rho \equiv 0$ may occur in the
context of either theorem. Relative to Theorem Ch., new here is Theorem 4B
exhibiting an additional source of regular variation (a `Fr\'{e}chet' term
-- \S 7.2).
We begin by noting that, since $\circ _{\rho }$ on $\mathcal{N}(\rho )$ is
addition, $\mathcal{N}(\rho )$ is an abelian subgroup of $\mathbb{G}_{\rho
}(X)$ and so
\begin{equation*}
\Sigma :=K(\mathcal{N}(\rho )),
\end{equation*}
as a homeomorph, is also an abelian subgroup of $\mathbb{G}_{\sigma }(Y).$
By Theorem 2 there are now two cases to consider, differing only in their
treatment of radial behaviour (in or out of $\mathcal{N}(\rho ))$. These two
cases are dealt with in Theorems 4A and 4B below, which we apply in [BinO7].
\noindent \textbf{Theorem 4A }(\textbf{First Popa Homomorphism Theorem})
\textbf{.} \textit{If }$K$ \textit{is a continuous function }$K$\textit{\
satisfying }$(K)$ \textit{with}
\begin{equation*}
\mathit{\ }K(\mathcal{N}(\rho ))\subseteq \mathcal{N}(\sigma ),
\end{equation*}
\textit{then:}\newline
\noindent $K|\mathcal{N}(\rho )$\textit{\ is linear, and either}\newline
\noindent (i) $K$ \textit{is linear, or}
\noindent (ii)\textit{\ for some constant }$\tau $ \textit{and} \textit{each
}$u$ \textit{with }$\rho (u)=1$\textit{,} $\sigma (K(u))=\tau $ \textit{and}
\begin{equation*}
K(x)=K(\pi _{u}(x))+[(1+\rho (x))^{\log (1+\tau )/\log 2}-1]K(u)/\tau ,
\end{equation*}
\textit{with }$\pi _{u}(x):=x-\rho (x)u$ \textit{projection onto }$\mathcal{N
}(\rho )$ \textit{parallel to }$u,$ \textit{with a L'Hospital convention
applying for }$\tau =0.$ \textit{That is,\ for }$\tau =0$ \textit{the
relevant formula reads}
\begin{equation*}
K(u)\log (1+\rho (x))/\log 2.
\end{equation*}
\textit{In particular, }$x\mapsto K(\pi _{u}(x))$\textit{\ is linear.}
\noindent \textbf{Proof.} If $\rho \equiv 0,$ then $K(X)=K(\mathcal{N}(\rho
))\subseteq \mathcal{N}(\sigma ).$ Here $\sigma (K(x))=0$ for all $x$ so
since $\circ _{\sigma }=+$ on $\mathcal{N}(\sigma )$, $K$ is linear.
Otherwise, fix $u\in X$ with $\rho (u)=1;$ then $x\mapsto \pi _{u}(x)=x-\rho
(x)u$ is a (linear) projection onto $\mathcal{N}(\rho )$ and so
\begin{equation*}
x=(x-\rho (x)u)\circ _{\rho }\rho (x)u.
\end{equation*}
(So $\mathbb{G}_{\rho }(X)$ is generated by $\mathcal{N}(\rho )$ and any $
u\notin \mathcal{N}(\rho ).$)
By assumption $\sigma (\pi _{u}(x))=0$ and as $K|\mathcal{N}(\rho )$ is
linear
\begin{equation*}
K(x)=K(\pi _{u}(x))\circ _{\sigma }K(\rho (x)u)=K(\pi _{u}(x))+K(\rho (x)u).
\end{equation*}
If $\tau :=\sigma (K(u))\neq 0,$ then by Theorem 3A
\begin{equation*}
K(\rho (x)u)=[(1+\rho (x))^{\log (1+\tau )/\log 2}-1]K(u)/\tau .
\end{equation*}
Now consider $u,v\in \mathbb{G}_{\rho }(X)$ with $\rho (u)=1=\rho (v).$ As $
v-u\in \mathcal{N}(\rho ),$ also $\sigma (K(v-u))=0.$ But
\begin{equation*}
v=(v-u)+u=(v-u)\circ _{\rho }u,
\end{equation*}
and as $\sigma (K(v-u))=0,$
\begin{equation*}
K(v)=K(v-u)\circ _{\sigma }K(u)=K(v-u)+K(u):\qquad K(v-u)=K(v)-K(u).
\end{equation*}
So, by linearity of $\sigma $,
\begin{equation*}
0=\sigma (K(v-u))=\sigma (K(v))-\sigma (K(u)):\qquad \sigma (K(v))=\sigma
(K(u))=\tau .
\end{equation*}
Thus also
\begin{equation*}
K(\rho (x)v)=[(1+\rho (x))^{\log (1+\tau )/\log 2}-1]K(v)/\tau .
\end{equation*}
If $\tau :=\sigma (K(u))=0,$ then as in Theorem 3A,
\begin{equation*}
K(\rho (x)u)=K(u)\log (1+\rho (x))/\log 2,
\end{equation*}
again justifying the L'Hospital convention in force (the formula follows
from the main case taking limits as $\tau \rightarrow 0$).
$\square $
\noindent \textbf{Theorem 4B }(\textbf{Second Popa Homomorphism Theorem})
\textbf{.} \textit{If }$K$ \textit{is a continuous function }$K$\textit{\
satisfying }$(K)$ \textit{with }
\begin{equation*}
K(\mathcal{N}(\rho ))=\langle K(w)\rangle _{\sigma }
\end{equation*}
\textit{for some }$w$\textit{\ with }$\rho (w)=0$ \textit{and }$\sigma
(K(w))=1,$ \textit{then}\newline
\noindent (i) $V_{0}:=\mathcal{N}(\rho )\cap K^{-1}(\mathcal{N}(\sigma ))$
\textit{\ is a vector subspace and }$K_{0}=K|V_{0}$ \textit{is linear;
\newline
\noindent }(ii) \textit{for any subspace }$V_{1}$\textit{\ with }$w\in V_{1}$
\textit{\ complementary to }$V_{0}$ \textit{in }$\mathcal{N}(\rho ),$\textit{
\ and any }$u\in X$ \textit{with }$\rho (u)=1$\textit{, if any, there are a
constant }$\kappa =\kappa (u)$\textit{\ and a linear map }$\kappa
_{w}:V_{1}\rightarrow \mathbb{R}$ \textit{with\ }
\begin{eqnarray*}
K(x) &=&K_{0}(\pi _{0}(x))+[e^{\kappa _{w}(\pi _{1}(x))}-1]K(w) \\
&&+[1+\sigma (K(\pi _{1}(x)))]\cdot \lbrack (1+\rho (x))^{\log (1+\sigma
(K(u)))/\log 2}-1]K(u)/\sigma (K(u)).
\end{eqnarray*}
\textit{Here }$\pi _{i}$\textit{\ denotes projection from }$X$\textit{\ onto}
$V_{i}$ \textit{so that }$K_{0}\pi _{0}$\textit{\ is linear, and }$\sigma
(K(\pi _{1}(x)))\neq 0$ \textit{unless }$\pi _{1}(x)=0$\textit{. (The case }$
\sigma (K(u))=0$ \textit{is interpreted again by the L'Hospital convention,
but the term is excluded when there are no }$u\in X$ \textit{with }$\rho
(u)=1$\textit{.)}
\noindent \textbf{Proof.} The assumption on $K\ $here is taken in the
initially more convenient form: $K(\mathcal{N}(\rho ))\subseteq \langle
w\rangle _{\sigma },$ \textit{for some} $w\in \Sigma =K(\mathcal{N}(\rho )),$
and of course w.l.o.g $\sigma (w)\neq 0$, as otherwise this case is covered
by Theorem 4A.
To begin with $V_{0}:=\mathcal{N}(\rho )\cap K^{-1}(\mathcal{N}(\sigma ))$
is a subgroup of $\mathbb{G}_{\rho }(X),$ as $K$ is a homomorphism.
Similarly as in Theorem 4A, we work with a linear map, namely $
K_{0}:=K|V_{0} $, as we claim $V_{0}$ to be a subspace of $\mathcal{N}(\rho
).$ (Then $V_{0}=\mathbb{G}_{0}(V_{0}).$)
The claim follows by linearity of $\sigma $ and Theorem 3B. Indeed, if $\rho
(x)=\rho (y)=0$ and $\sigma (K(x))=\sigma (K(y))=0,$ then $K(\alpha
x)=\lambda _{x}(\alpha )K(x)$ and $K(\beta y)=\lambda _{y}(\beta )K(y),$ and
since $\mathcal{N}(\rho )$ is a vector subspace on which $+$ agrees with $
\circ _{\rho }:$
\begin{eqnarray*}
K(\alpha x+\beta y) &=&K(\alpha x\circ _{\rho }\beta y) \\
&=&\lambda _{x}(\alpha )K(x)+\lambda _{y}(\beta )K(y)+\lambda _{x}(\alpha
)\lambda _{y}(\beta )\sigma (K(x))K(y):
\end{eqnarray*}
\begin{equation*}
\sigma (K(\alpha x+\beta y))=\lambda _{x}(\alpha )\sigma (K(x))+\lambda
_{y}(\beta )\sigma (K(y))=0.
\end{equation*}
Hence $V_{0}$ is a subspace of $\mathcal{N}(\rho )$ and $K_{0}:V_{0}
\rightarrow \mathcal{N}(\sigma )$ is linear with $K(V_{0})\subseteq \mathcal{
N}(\sigma ),$ as in Theorem 4A.
Since $K(\mathcal{N}(\rho ))\subseteq \mathcal{N}(\sigma )$ does not hold,
choose in $\mathcal{N}(\rho )$ a subspace $V_{1}$ complementary to $V_{0},$
and let $\pi _{i}:X\rightarrow V_{i}$ denote projection onto $V_{i}$. For $
v\in \mathcal{N}(\rho )$ and $v_{i}=\pi _{i}(v)\in V_{i},$ as $K(v_{0})\in
\mathcal{N}(\sigma ),$
\begin{equation*}
K(v)=K(\pi _{0}(v)\circ \pi _{1}(v))=K(\pi _{0}(v))+_{\sigma }K(\pi
_{1}(v))=K_{0}(\pi _{0}(v))+K(\pi _{1}(v)).
\end{equation*}
Here $K_{0}\pi _{0}$ is linear and $\sigma (K(v_{1}))\neq 0$ unless $v_{1}=0$
. Recalling that $V_{1}$ is a subgroup of $\mathbb{G}_{\rho }(X),$ re-write
the result of Theorem 3B as $K(v_{1})=\lambda _{w}(v_{1})w$ with $\lambda
_{w}:V_{1}\rightarrow \mathbb{G}_{\sigma (w)}(\mathbb{R})$ and
\begin{equation*}
\lambda _{w}(v_{1}+v_{1}^{\prime })=\lambda _{w}(v_{1})\circ _{\sigma
(w)}\lambda _{w}(v_{1}^{\prime }).
\end{equation*}
With $w$ fixed, $\lambda _{w}$ is continuous (as $K$ is), with $1+\sigma
(w)\lambda _{w}(v_{1})>0$.
So as in Theorem 4A, for $v\in V_{1}$ and some $\kappa =\kappa _{w}(v)$
\begin{equation*}
K(tv)=\lambda _{w}(tv)w=\sigma (w)^{-1}[e^{\sigma (w)\kappa
_{w}(v)t}-1]w\qquad (t\in \mathbb{R}).
\end{equation*}
Taking $t=1$ gives
\begin{equation*}
\sigma (w)\kappa _{w}(v)=\log [1+\sigma (w)\lambda _{w}(v)].
\end{equation*}
As $\lambda _{w}$ is continuous, so is $\kappa _{w}:V_{1}\rightarrow \mathbb{
R}$. But, as in Theorem 2 but with $\sigma (w)$ fixed, $\kappa _{w}$ is
additive and so by continuity linear on $V_{1}$. So, as $t\kappa
_{w}(v)=\kappa _{w}(tv),$
\begin{equation*}
K(v)=\sigma (w)^{-1}[e^{\sigma (w)\kappa _{w}(v)}-1]w\qquad (v\in V_{1}).
\end{equation*}
For $x\in X$ take $v_{i}:=\pi _{i}(x)\in V_{i}$ and $v:=v_{0}+v_{1}.$ If $
\rho $ is not identically zero, again fix $u\in X$ with $\rho (u)=1,$ and
then $x\mapsto \pi _{u}(x)=x-\rho (x)u$ is again (linear) projection onto $
\mathcal{N}(\rho )$. If $\rho \equiv 0,$ set $u$ below to $0.$ Then, whether
or not $\rho \equiv 0,$ as $\rho (x-\rho (x)u)=0,$
\begin{equation*}
x=v_{0}+v_{1}+\rho (x)u=v\circ _{\rho }\rho (x)u.
\end{equation*}
So, as $\sigma (K(v_{0}))=0$ and $\rho (\rho (x)u)=\rho (x)\rho (u)=\rho
(x), $with $\tau =\sigma (K(u))\neq 0$
\begin{equation*}
K(x)=K(v)\circ _{\sigma }K(\rho (x)u)=K(v)\circ _{\sigma }\eta _{\sigma
(K(u))}^{-1}(\eta _{\rho }(\rho (x)u)^{\kappa }),
\end{equation*}
which we expand as
\begin{eqnarray*}
&&K(v_{0})+K(v_{1})+[1+\sigma (K(v_{0}+v_{1}))][(1+\rho (x))^{\log (1+\tau
)/\log 2}-1]K(u)/\tau \\
&=&K_{0}(\pi _{0}(v))+K(\pi _{1}(v))+[1+\sigma (K(v_{1}))][(1+\rho
(x))^{\log (1+\tau )/\log 2}-1]K(u)/\tau :
\end{eqnarray*}
\begin{eqnarray*}
K(x) &=&K_{0}(\pi _{0}(x))+[e^{\sigma (w)\kappa _{w}(\pi
_{1}(x))}-1]w/\sigma (w) \\
&&+[1+\sigma (K(\pi _{1}(x)))][(1+\rho (x))^{\log (1+\sigma (K(u)))/\log
2}-1]K(u)/\sigma (K(u)).
\end{eqnarray*}
For $v_{1}\neq 0,$ $\sigma (K(v_{1}))\neq 0,$ as otherwise $v_{1}\in
\mathcal{N}(\rho )\cap K^{-1}(\mathcal{N}(\sigma ))=V_{0},$ contradicting
complementarity of $V_{1}$.
Here $\sigma (w/\sigma (w))=1.$ Finally, as $w\in \Sigma =K(\mathcal{N}(\rho
)),$ we replace $w$ by $K(w)$ with $\rho (w)=0$ and $\sigma (K(w))=1.$ If $
\tau =\sigma (K(u))=0,$ then, as in Theorem 4A, the final term is to be
interpreted by the L'Hospital convention (limiting value as $\sigma
(K(u))\rightarrow 0).$
If $\rho \equiv 0,$ then $u=0$ so that $K(u)=0,$ and the final term
vanishes.
$\square $
For the final d\'{e}noument, which is the connection between $(GFE)$ and
Popa groups, we work here under a simplifying assumption. We return to these
matters in [BinO7], where we use Theorems 4A and 4B to characterize the
continuous solutions of $(GFE)$ as homomorphisms between Popa groups $
\mathbb{G}_{\rho }(X)$ and $\mathbb{G}_{\sigma }(Y)$.
\noindent \textbf{Corollary 2 }(After [Ost3, Th. 1])\textbf{.} \textit{For }$
K:\mathbb{G}_{\rho }(X)\rightarrow Y$\textit{\ as in Prop. 1.1,\ if }$K$
\textit{\ is injective, then for some linear} $\sigma :Y\rightarrow \mathbb{R
}$
\begin{equation*}
K(u\circ _{\rho }v)=K(u)+g(u)K(v)=K(u)\circ _{\sigma }K(v).
\end{equation*}
\noindent \textbf{Proof. }Set $\varphi (y):=g(K^{-1}(y))$ for $y\in K(
\mathbb{G}_{\rho }(X)).$ Then $\varphi $ is continuous and by $(GFE_{\times
})$ satisfies $(GS):$ $u=K^{-1}(a)$ and $v=K^{-1}(b)$ above yield that
\begin{equation*}
\varphi (a\circ _{\varphi }b)=g(K^{-1}(a)\circ _{\rho
}K^{-1}(b))=g(K^{-1}(a))g(K^{-1}(b))=\varphi (a)\varphi (b).
\end{equation*}
By [Brz1, Cor. 4], $\sigma (y):=\varphi (y)-1$ is linear for $\varphi (y)>0.$
$\square $
\textbf{7. Complements}
\noindent 7.1 \textit{Univariate and Beurling regular variation}
Regular variation in one dimension (widely used in analysis, probability and
elsewhere -- cf. [Bin2]) explores the ramifications of limiting relations
such as
\begin{equation}
f(\lambda x)/f(x)\rightarrow K(\lambda )\equiv \lambda ^{\gamma }
\tag{$Kar_{\times }$}
\end{equation}
or its additive variant, more thematic here:
\begin{equation}
f(x+u)-f(x)\rightarrow K(u)\equiv \kappa u \tag{$Kar_{+}$}
\end{equation}
[BinGT, Ch. 1], and
\begin{equation}
\lbrack f(x+u)-f(x)]/h(x)\rightarrow K(u)\equiv (u^{\gamma }-1)/\gamma
\tag{$BKdH$}
\end{equation}
(Bojani\'{c} \& Karamata, de Haan, [BinGT, Ch. 3]). Beurling regular
variation similarly explores the ramifications of relations such as
\begin{equation}
\varphi (x+t\varphi (x))/\varphi (x)\rightarrow 1\text{ or }\eta (t)
\tag{$Beu$}
\end{equation}
[BinGT, \S\ 2.11] and [Ost1]. For background and applications, see the
standard work [BinGT] and e.g. [BinO1-5], [Bin1,2,3]. Both theory and
applications prompt the need to work in higher dimensions, finite or
infinite. This is the ultimate motivation for the present paper.
\noindent 7.2 \textit{Functional equations in probability}
The identification of some specific distribution characterized by the
asymptotic behaviour of empirical distributions is routinely performed by
solving some specific functional equation. For an introduction and survey of
particularly important examples see e.g. [Ost4]. For a telling example (of
stable laws) see [Ost2]. We note here that Beurling aspects play a key role
in probability in both the theory of \textit{addition }of independent random
variables (as in [Ost2]) and that of taking the \textit{maximum }(as in \S
7.4 below). For background on similarities (and contrasts) between these two
settings, see e.g. [BinGT, Ch. 8].
In the case of extreme-value theory, to find the limiting distribution $G$
of the laws of sample maxima drawn from a law $F$, one considers the inverse
function $U$ of $1/(1-F)$ and studies the kernel function
\begin{equation*}
E(x):=\lim_{s\rightarrow \infty }U(sx)-U(s)]/a(s)\qquad (x\in \mathbb{R}
_{+}),
\end{equation*}
for some auxiliary $a$ with well-defined limit function $A(y):=\lim_{s
\rightarrow \infty }a(sy)/a(s).$ As in [HaaF, p. 7 ], by Prop. 1.1 this
leads to the Goldie equation
\begin{equation*}
E(xy)=E(x)A(y)+E(y)\qquad (x,y\in \mathbb{R}_{+}).
\end{equation*}
All the functions here being measurable, one concludes directly via Theorem
BO (measurable solutions of the Cauchy functional equation being
continuous), or via [BinO3]), that for some constants $\kappa ,\gamma $ (the
latter the `extreme-value index')
\begin{equation*}
E(t)=\kappa (t^{\gamma }-1)/\gamma ,\qquad A(t)=t^{\gamma }.
\end{equation*}
After some algebraic manipulations this yields the generalised extreme-value
(GEV) distribution
\begin{equation*}
G(x)=G_{\gamma }(x):=\exp (-(1+\gamma x)^{-1/\gamma })\qquad \text{for }
1+\gamma x>0.
\end{equation*}
(`G' for Goldie; this includes the three types: Gumbel: $\gamma >0$,
Weibull: $\gamma <0,$ and Fr\'{e}chet $\gamma =0$, when $G(x)=\exp
(-e^{-x}), $ as identified by the celebrated theorem due to Fisher and
Tippett and also Gnedenko characterizing extreme-value distributions -- see
[BinGT, \S 8.13], [HaaF, Th.1.1.3], [BeiGST, Ch. 8].)
\noindent 7.3 \textit{Popa Haar-measure}\textbf{\ }
With the induced Euclidean topology, $\mathbb{G}_{\rho }(\mathbb{R}^{d})$ is
an open subspace of $\mathbb{R}^{d},$ so by the argument in Hewitt and Ross
[HewR, 15.18], for $\lambda _{d}$ Lebesgue measure, the Popa Haar-measure on
$\mathbb{G}_{\rho }$ is (as in [BinO5]) proportional to
\begin{equation*}
\frac{\lambda _{d}(\mathrm{d}x)}{1+\rho (x)}.
\end{equation*}
\noindent 7.4 \textit{Multivariate extreme-value theory}
Extreme-value theory in probability and statistics (stemming from work of
Fisher and Tippett in 1928) actually pre-dates regular variation (stemming
from Karamata in 1930). The profound importance of regular variation to
probability was realised by Sudakov in the 50s and Feller in the 60s. The
profound importance of a proper theoretical understanding of floods, tidal
surges and the like was underlined by the tragic North Sea floods of 31
January - 1 February 1953 and the death tolls in the Netherlands and the UK.
The Netherlands probabilist Laurens de Haan has dedicated much of his career
from 1970 on to extreme-value theory (EVT). For background, see e.g. de Haan
and Ferreira [HaaF], [BinO6].
While EVT originated in one dimension, consideration of, for instance, the
numerous measuring stations along the Netherlands coast suggested the need
for a multi-dimensional theory (and the entire coastline the need of an
infinite-dimensional one). This is discussed briefly in [HaaF], and at
greater length in Balkema and Embrechts [BalE]. We refer to [DavPR] and the
recent [RooSW] and [KirRSW] for recent treatments and references to the
relevant background (see e.g. the treatment of multivariate GEV and
generalised Pareto (GP) distributions, and property (T2), in [RooSW]).
\noindent 7.5 \textit{Hidden regular variation.}
The radial dependence in the theory above is needed for both theory and
applications. The applications include Resnick's \textit{hidden regular
variation }([Res1,2]; [DasR], [LinRR]), so called because (cf. \S 7.2)
norming (or auxiliary) functions $a$ of different orders of magnitude may
occur in different directions, and norming by the largest will swamp (or
`hide') the behaviour relevant to smaller ones.
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some functional equation. \textsl{Aequationes Math.} 81 (2011), 155--165.
\newline
\noindent \lbrack Jac] N. Jacobson, \textsl{Lectures in Abstract Algebra}.
vol. I. Van Nostrand, New York, 1951.\newline
\noindent \lbrack Jar] A. J\'{a}rai, \textsl{Regularity properties of
functional equations in several variables.} Springer, 2005.\newline
\noindent \lbrack Jav] P. Javor, On the general solution of the functional
equation $f(x+yf(x))=f(x)f(y)$. \textsl{Aequationes Math.} \textbf{1}
(1968), 235--238.\newline
\noindent \lbrack KirRSW] A. Kiriliouk, H. Rootz\'{e}n, J. Segers and J. L.
Wadsworth, Peaks over thresholds modeling with multivariate generalized
Pareto distributions. \textsl{Technometrics} \textbf{61} (2019), 123--135.
\newline
\noindent \lbrack LinRR] F. Lindskog, S. I. Resnick and J. Roy, Regularly
varying measures on metric spaces: hidden regular variation and hidden
jumps. \textsl{Prob. Surveys} \textbf{11} (2014), 270-314.\newline
\noindent \lbrack Mil] H. I. Miller, Generalization of a result of Borwein
and Ditor. \textsl{Proc. Amer. Math. Soc.} \textbf{105} (1989), 889--893.
\newline
\noindent \lbrack MilMO] H. I. Miller, L. Miller-van Wieren and A. J.
Ostaszewski, Beyond Erd\H{o}s-Kunen-Mauldin: Shift-compactness properties
and singular sets, \textsl{Topology Appl.}, to appear (arXiv:1901.09654).
\newline
\noindent \lbrack MilO] H. I. Miller and A. J. Ostaszewski, Group action and
shift-compactness. \textsl{J. Math. Anal. App.} \textbf{392} (2012), 23--39.
\newline
\noindent \lbrack Ost1] A. J. Ostaszewski, Beurling regular variation, Bloom
dichotomy, and the Go\l \k{a}b-Schinzel functional equation. \textsl{
Aequationes Math.} \textbf{89} (2015), 725-744. \newline
\noindent \lbrack Ost2] A. J. Ostaszewski, Stable laws and Beurling kernels.
\textsl{Adv. Appl. Probab.} \textbf{48A} (2016) (N. H. Bingham Festschrift),
239--248. \newline
\noindent \lbrack Ost3] A. J. Ostaszewski, Homomorphisms from Functional
Equations: The Goldie Equation. \textsl{Aequationes Math. }\textbf{90}
(2016), 427-448 (arXiv: 1407.4089).\newline
\noindent \lbrack Ost4] A. J. Ostaszewski, Homomorphisms from Functional
Equations in Probability, in: \textsl{Developments in Functional Equations
and Related Topics}, ed. J. Brzd\k{e}k et al., Springer (2017), 171-213.
\newline
\noindent \lbrack Pop] C. G. Popa, Sur l'\'{e}quation fonctionelle $
f[x+yf(x)]=f(x)f(y).$ \textsl{Ann. Polon. Math.} \textbf{17} (1965), 193-198.
\newline
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order regular variation and asymptotic independence. \textsl{Extremes}
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Probabilistic and statistical modelling.} Springer, 2007.\newline
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Multivariate peaks over thresholds models. \textsl{Extremes} \textbf{21}
(2018), 115--145.\newline
\noindent Mathematics Department, Imperial College, London SW7 2AZ;
[email protected] \newline
Mathematics Department, London School of Economics, Houghton Street, London
WC2A 2AE; [email protected]
\textbf{8. Appendix}
Below, in checking routine details omitted in the main text, we offer a
fuller treatment of Theorems J\ and 3B.
\noindent 8.1 \textit{Proof of Theorem J}
We argue as in [Jav], but must additionally check positivity of $\eta _{\rho
}.$ Here $0\in \mathbb{G}_{\rho }$ and is the neutral element, as $\rho (0)=0
$ so that both $\eta _{\rho }(0)>0$ and
\begin{equation*}
x\circ _{\rho }0=x\text{ and }0\circ _{\rho }y=y.
\end{equation*}
For fixed $x,$ the solution for $y$ to
\begin{equation*}
x\circ _{\rho }y=x+y+\rho (x)y=0
\end{equation*}
uniquely gives the right inverse of $x,$ written as either $r(x)$ or $
x_{\rho }^{-1},$ to be
\begin{equation*}
r(x):=x_{\rho }^{-1}:=-x/(1+\rho (x))\in \mathbb{G}_{\rho },
\end{equation*}
with membership in $\mathbb{G}_{\rho }$, as $1=\eta _{\rho }(0)=\eta _{\rho
}(x\circ _{\rho }r(x))=\eta _{\rho }(x)\eta _{\rho }(r(x))$ and $\eta _{\rho
}(x)>0.$ By linearity of $\rho ,$
\begin{equation*}
\rho (r(x))=\frac{-\rho (x)}{1+\rho (x)},
\end{equation*}
whence it emerges that
\begin{equation*}
r(x)\circ _{\rho }x=\frac{-x}{1+\rho (x)}+\frac{x(1+\rho (x))}{1+\rho (x)}+
\frac{-\rho (x)x}{1+\rho (x)}=0,
\end{equation*}
i.e. $r(x)$ is also a left inverse for $x.$ But for any left inverse $\ell $
of $x,$ by associativity,
\begin{equation*}
\ell =\ell \circ _{\rho }x\circ _{\rho }r(x)=r(x),
\end{equation*}
i.e. $r(x)$ is also a unique left inverse. So $(\mathbb{G}_{\rho }(X),\circ
_{\rho })$ is a group.
$\square $
\noindent 8.2 \textit{Proof of Theorem 3B.}
As $\xi u+\xi u=\xi u\circ _{\rho }\xi u,$ notice that
\begin{equation*}
K(2u)=K(u)+K(u)+\sigma (K(u))K(u)=(2+\sigma (K(u))K(u).
\end{equation*}
By induction,
\begin{equation*}
K(nu)=a_{n}(u)K(u)\in \langle K(u)\rangle _{Y},
\end{equation*}
where $a_{n}=a_{n}(u)$ solves
\begin{equation*}
a_{n+1}=1+(1+\sigma (K(u))a_{n},
\end{equation*}
since
\begin{equation*}
K(u+nu)=K(u)+a_{n}K(u)(1+\sigma (K(u)).
\end{equation*}
Suppose w.l.o.g. $\sigma (K(u))\neq 0,$ the case $\sigma (K(u))=0$ being
similar, but simpler (with $a_{n}=n).$ So
\begin{equation*}
a_{n}=((1+\sigma (K(u))^{n}-1)/\sigma (K(u))\neq 0\qquad (n=1,2,...).
\end{equation*}
Replacing $u$ by $u/n,$
\begin{equation*}
K(u)=K(nu/n)=a_{n}(u/n)K(u/n),
\end{equation*}
giving
\begin{equation*}
K(u/n)=a_{n}(u/n)^{-1}K(u)\in \langle K(u)\rangle _{Y}.
\end{equation*}
So
\begin{eqnarray*}
K(mu/n) &=&a_{m}(u/n)K(u/n) \\
&=&a_{m}(u/n)a_{n}(u/n)^{-1}K(u) \\
&=&\frac{((1+\sigma (K(u/n))^{m}-1)/\sigma (K(u/n)}{((1+\sigma
(K(u/n))^{n}-1)/\sigma (K(u/n)}K(u) \\
&=&\frac{((1+\sigma (K(u/n))^{m}-1)}{((1+\sigma (K(u/n))^{n}-1)}K(u)\in
\langle K(u)\rangle _{Y}.
\end{eqnarray*}
By continuity of $K$ (and of scalar multiplication), this implies that $
K(\xi u)\in \langle K(u)\rangle _{Y}$ for any $\xi \in \mathbb{R}.$ So we
may uniquely define $\lambda (s)=\lambda _{u}(s)$ via
\begin{equation*}
K(su)=\lambda _{u}(s)K(u).
\end{equation*}
(In the case $\sigma (K(u))=0$ with $a_{n}=n,$ $K(mu/n)=(m/n)K(u),$ so that $
K(su)=sK(u).)$ Then, as $\rho (u)=0,$
\begin{eqnarray*}
\lambda (\xi +\eta )K(u) &=&K((\xi +\eta )u)=K(\xi u\circ _{\rho }\eta
u)=K(\xi u)+K(\eta u)+\sigma (K(\xi u))K(\eta u) \\
&=&K(\xi u)+K(\eta u)+\sigma (\lambda (\xi )K(u))\lambda (\eta )K(u) \\
&=&\lambda (\xi )K(u)+\lambda (\eta )K(u)+\lambda (\xi )\lambda (\eta
)\sigma (K(u))K(u) \\
&=&[\lambda (\xi )+\lambda (\eta )+\lambda (\xi )\lambda (\eta )\sigma
(K(u))]K(u).
\end{eqnarray*}
So if $K(u)\neq 0$
\begin{equation*}
\lambda _{u}(\xi +\eta )=\lambda _{u}(\xi )+\lambda _{u}(\eta )+\lambda
_{u}(\xi )\lambda _{u}(\eta )\sigma (K(u))=\lambda _{u}(\xi )\circ _{\sigma
(K(u))}\lambda _{u}(\eta ).
\end{equation*}
Thus $\lambda _{u}:(\mathbb{R},+)\rightarrow \mathbb{G}_{\sigma (K(u))}(
\mathbb{R}).$ By Theorem BO, with $\tau =\sigma (K(u))$ for some $\kappa
=\kappa (u)$
\begin{equation*}
\lambda _{u}(t)=(e^{\tau \kappa (u)t}-1)/\tau \text{ or }\kappa (u)t=t,\text{
if }\sigma (K(u))=0.\qquad \hfil\square \break
\end{equation*}
\end{document} | math | 64,899 |
\begin{document}
\title[\tiny{Grothendieck ring of semialgebraic formulas }]{\rm Grothendieck ring of semialgebraic formulas and motivic real Milnor fibres}{}
\author{Georges COMTE}
\address{Laboratoire de Math\'ematiques de l'Universit\'e de Savoie, UMR CNRS 5127,
B\^atiment Chablais, Campus scientifique,
73376 Le Bourget-du-Lac cedex, France}
\email{[email protected]}
\urladdr{http://gc83.perso.sfr.fr/}
\author{Goulwen FICHOU}
\address{IRMAR, UMR 6625 du CNRS, Campus de Beaulieu, 35042 Rennes cedex, France}
\email{[email protected]}
\urladdr{http://perso.univ-rennes1.fr/goulwen.fichou/}
\begin{abstract}
We define a Grothendieck ring for basic
real semialgebraic formulas,
that is for systems of real algebraic equations and inequalities.
In this ring the class of a formula takes into consideration
the algebraic nature of the set of points satisfying this formula and this ring
contains as a subring the usual Grothendieck ring of real algebraic formulas.
We give a realization of our ring that allows us to express a class as a
${ \mathbb Z}[\mathbb F_2rac{1}{2}]$-linear combination of
classes of real algebraic formulas, so this realization gives rise to a notion of
virtual Poincar\'e polynomial for basic semialgebraic
formulas. We then define zeta functions with coefficients in our ring, built on
semialgebraic
formulas in arc spaces. We show that they are rational
and relate them to the topology of real Milnor fibres.
\end{abstract}
\maketitle
\renewcommand{\partname}{}
\section*{Introduction}
Let us consider the category $SA({ \mathbb R})$ of real semialgebraic sets,
the morphisms being the semialgebraic maps. We denote by
$(K_0(SA({ \mathbb R})),+,\cdot)$, or simply $K_0(SA({ \mathbb R}))$,
the Grothendieck ring of $SA({ \mathbb R})$, that is to say the free ring
generated by all semialgebraic sets A, denoted by $[A]$ as viewed as
element of
$K_0(SA({ \mathbb R}))$, in such a way that for all objects $A,B$ of $SA({ \mathbb R})$
one has:
$[A\times B]=[A]\cdot[B]$ and for all closed semialgebraic set $F$ in $A$
one has: $[A\setminus F]+[F]=[A]$ (this implies that for every
semialgebraic sets $A,B$, one has: $[A\cup B]=[A]+[B]-[A\cap B]$).
When furthermore an equivalence relation for semialgebraic
sets is previously considered for the definition of $K_0(SA({ \mathbb R}))$,
one has to be aware that the induced quotient ring,
still denoted for simplicity by $K_0(SA({ \mathbb R}))$, may dramatically collapse.
For instance let us consider the equivalence relation $A\sim B$ if and only if
there exists a semialgebraic bijection from $A$ to $B$.
In this case we simply say that $A$ and $B$ are isomorphic.
Then for the definition of $K_0(SA({ \mathbb R}))$, starting from
classes of isomorphic sets instead of simply sets, one obtains a quite
trivial Grothendieck ring, namely $K_0(SA({ \mathbb R}))={ \mathbb Z}$.
Indeed, denoting
$[{ \mathbb R}]$ by ${ \mathbb L}$ and $[\{*\}]$ by ${ \mathbb P}$, from the fact that
$\{*\}\times \{*\}\sim \{*\}$, one gets
$$ { \mathbb P}^k={ \mathbb P}, \ \mathbb F_2orall k\in { \mathbb N}^*,$$
and from the fact that ${ \mathbb R}=
]-\infty, 0[ \cup \{0\} \cup ]0,+\infty[$ and that intervals
of the same type are isomorphic, one gets
$$ { \mathbb L}=-{ \mathbb P}. $$
On the other hand, by the semialgebraic cell decomposition theorem,
we obtain that a real semialgebraic set is a finite union of disjoint
open cells, each of which is isomorphic to ${ \mathbb R}^k$, with $k\in { \mathbb N}$
(with the convention that ${ \mathbb R}^0=\{*\}$). It follows that
$K_0(SA({ \mathbb R}))=<{ \mathbb P}>$, the ring generated by ${ \mathbb P}$.
At this point, the ring $<{ \mathbb P}>$ could be trivial.
But one knows that the Euler-Poincar\'e characteristic with compact
supports $\chi_c:SA({ \mathbb R})\to { \mathbb Z}$ is surjective. Let us recall that the Euler-Poincar\'e characteristic with compact
supports is a topological invariant defined on locally compact semialgebraic sets and uniquely extended to an additive
invariant on all semialgebraic sets (see for instance \cite{Coste}, Theorem 1.22). Since $\chi_c$ is additive,
multiplicative and invariant under isomorphims, it factors through
$K_0(SA({ \mathbb R}))$, giving a surjective morphism of rings, and finally
an isomorphism of rings, still denoted for simplicity by
$\chi_c$ (cf also \cite{Q}):
\vskip0mm
$$ \shorthandoff{;:!?}
\xymatrix{
SA({ \mathbb R})\ar[d] \ar[r]^{\chi_c}& { \mathbb Z} \\
<{ \mathbb P}> =K_0(SA({ \mathbb R})) \ar[ru]_{\chi_c} & \\} $$
The characteristic $\chi_c(A)$ of a semialgebraic set $A$ is in fact
defined in the same way, so we obtain the equality
$K_0(SA({ \mathbb R}))=<{ \mathbb P}>$,
that is from a specific cell decomposition of $A$, where $<{ \mathbb P}>$ is
replaced by $\chi_c(\{*\})=1$. The difficulty in the definition of $\chi_c$ is
then to show that $\chi_c$ is
independent of the choice of the cell decomposition of $A$ (it technically
consists in showing that the definition of $\chi_c(A)$ does not depend on the
isomorphism class of $A$, see \cite{Dri} for instance).
When one starts from the category of real algebraic varieties ${\rm Var}_{ \mathbb R}$ or from
the category of
real algebraic sets ${ \mathbb R} {\rm Var}$,
as we do not have algebraic cell decompositions, we could expect
that the induced Grothendieck ring $K_0({\rm Var}_{ \mathbb R})$
is no longer trivial. This is indeed the case, since for instance
the virtual Poincar\'e polynomial morphism
factors through $K_0({\rm Var}_{ \mathbb R})$ and has image ${ \mathbb Z}[u]$ (see \cite{MCP}).
The first part of this article is devoted to the construction of
non-trivial
Gro\-then\-dieck ring $K_0(BSA_{{ \mathbb R}})$ associated to $SA({ \mathbb R})$, with a
canonical inclusion
$$K_0({\rm Var}_{ \mathbb R}) \hookrightarrow K_0(BSA_{{ \mathbb R}}),$$
that gives rise to a notion of virtual
Poincar\'e polynomial
for basic real semialgebraic formulas extending the virtual Poincar\'e polynomial
of real algebraic sets
and that allows factorization of the Euler-Poincar\'e
characteristic of real semialgebraic sets of points satisfying the formulas.
To be more precise, we first construct
$K_0(BSA_{{ \mathbb R}}))$, the Grothendieck ring of basic real semialgebraic
formulas (which are quantifier free real semialgebraic formulas
or simply systems of real algebraic equations and inequalities) where the class
of basic formulas without inequality is considered up to algebraic isomorphism
of the underlying real algebraic varieties.
In general a class in $K_0(BSA_{{ \mathbb R}})$
of a basic real semialgebraic
formula depends strongly on the formula itself rather than
only on the geometry of the real
semialgebraic set of points satisfying this formula. This construction is achieved in
Section $2$.
In order to make some computations more convenient we present a realization, denoted $\chi$,
of the ring
$K_0(BSA_{{ \mathbb R}})$ in the somewhat more simple ring $K_0({\rm Var}_{ \mathbb R})\otimes { \mathbb Z}[\mathbb F_2rac{1}{2}]$,
that is a
morphism of rings
$\chi : K_0(BSA_{{ \mathbb R}}) \to K_0({\rm Var}_{ \mathbb R})\otimes { \mathbb Z}[\mathbb F_2rac{1}{2}],$
that restricts to the identity map on $K_0({\rm Var}_{ \mathbb R})\hookrightarrow K_0(BSA_{{ \mathbb R}})$.
The morphism $\chi$ provides an explicit computation (see Proposition \ref{prop-alg})
presenting a class of
$K_0(BSA_{{ \mathbb R}})$ as a ${ \mathbb Z}[\mathbb F_2rac{1}{2}]$-linear combination of classes of $K_0({\rm Var}_{ \mathbb R})$.
When one wants to further simplify the computation of a class of a basic real semialgebraic
formula,
one can shrink the original ring $K_0(BSA_{{ \mathbb R}})$
a little bit more from
$K_0({\rm Var}_{ \mathbb R})\otimes { \mathbb Z}[\mathbb F_2rac{1}{2}]$ to $K_0({ \mathbb R}{\rm Var})\otimes { \mathbb Z}[\mathbb F_2rac{1}{2}]$, where
for instance algebraic formulas with empty set of real points have trivial class.
However as noted in point \ref{nontrivial} of Remark \ref{rmks}
the class of a basic real semialgebraic formula
with empty set of real points may be not trivial in
$K_0({ \mathbb R}{\rm Var})\otimes { \mathbb Z}[\mathbb F_2rac{1}{2}]$.
The ring $K_0(BSA_{{ \mathbb R}})$ is not defined with a prior notion of isomorphism relation
contrary to the ring $K_0({\rm Var}_{ \mathbb R})$ where algebraic isomorphism classes of varieties
are generators. Nevertheless we indicate a notion of isomorphism for basic semialgebraic
formulas that factors through $K_0(BSA_{{ \mathbb R}})$ (see Proposition \ref{iso}).
This is done in Section $2$.
The realization $ \chi : K_0(BSA_{{ \mathbb R}}) \to K_0({\rm Var}_{ \mathbb R}))\otimes { \mathbb Z}[\mathbb F_2rac{1}{2}]$ naturally
allows us to define in Section $4$ a notion of virtual Poincar\'e polynomial for basic real semialgebraic
formulas: for a class $[F]$ in $K_0(BSA_{{ \mathbb R}})$ that is written as a
${ \mathbb Z}[\mathbb F_2rac{1}{2}]$-linear combination $ \sum_{i=1}^q a_i[A_i]$
of classes $[A_i] \in K_0({\rm Var}_{ \mathbb R})$ of real algebraic varieties $A_i$,
we simply define
the virtual Poincar\'e polynomial of $F$ as the corresponding
${ \mathbb Z}[\mathbb F_2rac{1}{2}]$-linear combination $ \sum_{i=1}^q a_i\beta(A_i)$ of virtual Poincar\'e
polynomials $\beta(A_i)$ of the varieties $A_i$. The virtual Poincar\'e polynomial of
$F$ is thus a polynomial $\beta(F)$ in ${ \mathbb Z}[\mathbb F_2rac{1}{2}][u]$.
It is then shown that the evaluation at $-1$ of $\beta(F)$ is the Euler-Poincar\'e
characteristic of the real semialgebraic set of points satisfying the basic formula $F$
(Proposition \ref{eval}).
These constructions are summed up in the following commutative diagram
\vskip-3mm
$$ \shorthandoff{;:!?}
\xymatrix{
Var_{ \mathbb R} \ar[d] \ar[rd] \ar@{^{(}->}[rrr]& & &BSA_{{ \mathbb R}} \ar[ddd]^{\chi_c} \ar[lld] \\
K_0({\rm Var}_{ \mathbb R}) \ar[dd]_\beta \ar@{^{(}->}[r] \hskip2mm\ar@{^{(}->}[rd]
& K_0(BSA_{{ \mathbb R}}) \ar[d]^\chi \ar[rd]^\chi& &\\
& K_0({\rm Var}_{ \mathbb R}) \otimes { \mathbb Z}[\mathbb F_2rac{1}{2}] \ar[d]^\beta \ar[r] & K_0({ \mathbb R}{\rm Var}) \otimes { \mathbb Z}[\mathbb F_2rac{1}{2}]
\ar[ld]^\beta& \\
{ \mathbb Z}[u] \ar@{^{(}->}[r] & { \mathbb Z}[\mathbb F_2rac{1}{2}] [u] \ar[rr]^{u=-1}& &{ \mathbb Z} \\}
$$
\vskip0mm
The second and last part of this article concerns the real Milnor fibres of a given
polynomial function $f\in { \mathbb R}[x_1,\cdots, x_d]$.
As geometrical objects, we consider real semialgebraic Milnor fibres of the following
types: $f^{-1}(\pm c)\cap \bar B(0,\alpha) $,
$f^{-1}(]0,\pm c[)\cap \bar B(0,\alpha) $,
$f^{-1}(]0,\pm\infty[)\cap S(0,\alpha) $, for
$0<\vert c \vert \ll\alpha\ll 1$, $ \bar B(0,\alpha)$ the closed ball of
${ \mathbb R}^d$ of centre $0$ and radius $\alpha$ and $ S(0,\alpha)$ the sphere of centre $0$
and radius $\alpha$. The topological types of these fibres are easily comparable, and
in order to present a motivic version of these real semialgebraic Milnor fibres we define appropriate
zeta functions
with coefficients in $(K_0({\rm Var}_{ \mathbb R})\otimes { \mathbb Z}[\mathbb F_2rac{1}{2}])[{ \mathbb L}^{-1}]$
(the localization of the ring $K_0({\rm Var}_{ \mathbb R})\otimes { \mathbb Z}[{1\over 2}]$ with respect to
the multiplicative set generated by ${ \mathbb L}$). As
in the complex context (see \cite{DL1}, \cite{DL2}), we prove that these zeta functions
are rational functions expressed in terms of an embedded resolution of $f$ (see Theorem
\ref{Zeta function}).
For a complex hypersurface $f$, the rationality of the corresponding zeta function
allows
the
definition of the motivic Milnor fibre $S_f$, defined as the negative of the limit at
infinity
of the rational expression of the zeta function. In the real semialgebraic case, the same
definition makes sense
but we obtain a class $S_f$ in $K_0({\rm Var}_{ \mathbb R}))\otimes { \mathbb Z}[\mathbb F_2rac{1}{2}]$ having a realization under
the Euler-Poincar\'e characteristic of greater
combinatorial complexity in terms of the data of the resolution of $f$ than in the
complex case. Indeed, all the strata
of the natural stratification of the exceptional divisor of the resolution of $f$ appear in
the expression of $\chi_c(S_f)$ in the real case.
Nevertheless we show that the motivic real semialgebraic Milnor fibres have
for value under the Euler-Poincar\'e characteristic morphism the Euler-Poincar\'e
characteristic
of the corresponding set-theoretic real semialgebraic Milnor fibres
(Theorem \ref{Milnor}).
In what follows we sometimes simply say {\sl measure} for
the class of an object in a given Grothendieck ring. The term {\sl inequation} refers to the symbol $\not=$,
and the term {\sl inequality} refers to the symbol $>$.
\tableofcontents
\section{The Grothendieck ring of basic semialgebraic formulas.}
\subsection{Affine real algebraic varieties.}
By an affine algebraic variety over ${ \mathbb R}$ we mean an affine reduced and
separated scheme of finite type over ${ \mathbb R}$. The category of
affine algebraic varieties over ${ \mathbb R}$ is denoted by ${\rm Var}_{ \mathbb R}$.
An affine real algebraic variety $X$ is then defined by a subset of $\mathbb A^n$ together
with a finite number of polynomial equations. Namely, there exist $P_i
\in { \mathbb R}[X_1,\ldots,X_n]$, for $i=1,\ldots,r$, such that the real points $X({ \mathbb R})$
of $X$ are given by
$$X({ \mathbb R})=\{x\in \mathbb A^n | P_i(x)=0,~i=1,\ldots,r\}.$$
A Zariski-constructible subvariety $Z$ of $\mathbb A^n$ is similarly defined by real
polynomial equations and inequations. Namely there exist $P_i, Q_j
\in { \mathbb R}[X_1,\ldots,X_n]$, for $i=1,\ldots,p$ and $j=1,\ldots,q$, such
that the real points $Z({ \mathbb R})$ of $Z$ are given by
$$Z({ \mathbb R})=\{x\in \mathbb A^n | P_i(x)=0, Q_j(x) \neq 0,~i=1,\ldots,p,~j=1,\ldots,q
\}.$$
As an abelian group, the Grothendieck ring $K_0({\rm Var}_{{ \mathbb R}})$ of affine real algebraic
varieties is formally generated by isomorphism classes $[X]$ of
Zariski-constructible real algebraic varieties,
subject to the additivity relation
$$[X]=[Y]+[X\setminus Y],$$
in case $Y\subset X$ is a closed subvariety of $X$. Here $X\setminus Y$
is the Zariski-constructible variety defined by combining
the equations and inequations that define $X$ together with the
equations and inequations obtained by
reversing the equations and inequations that define $Y$.
The product of
constructible sets induces a ring structure on $K_0({\rm Var}_{{ \mathbb R}})$. We
denote by ${ \mathbb L}$ the class in $K_0({\rm Var}_{{ \mathbb R}})$ of $\mathbb A^1$.
\subsection{Real algebraic sets.}
The real points $X({ \mathbb R})$ of an affine algebraic variety $X$ over ${ \mathbb R}$ form a real
algebraic set (in the sense of \cite{BCR}). The Grothendieck
ring $K_0({ \mathbb R}{\rm Var})$ of affine real algebraic sets
\cite{MCP} is defined in a similar way than that of real algebraic
varieties over ${ \mathbb R}$. Taking the real points of an affine real
algebraic variety over ${ \mathbb R}$ gives a ring morphism from $K_0({\rm Var}_{{ \mathbb R}})$
to $K_0({ \mathbb R}{\rm Var})$. A great advantage of $K_0({ \mathbb R}{\rm Var})$ from a geometrical point of view is that the additivity property implies that the measure of
an algebraic set without real point is zero in $K_0({ \mathbb R}{\rm Var})$.
We already know some realizations of $K_0({ \mathbb R}{\rm Var})$ in simpler rings,
such as the Euler characteristics with compact supports in ${ \mathbb Z}$ or the virtual
Poincar\'e polynomial in ${ \mathbb Z}[u]$ (cf. \cite{MCP}). We obtain therefore similar
realizations for $K_0({\rm Var}_{{ \mathbb R}})$ by composition with the realizations of
$K_0({\rm Var}_{{ \mathbb R}})$ in $K_0({ \mathbb R}{\rm Var})$.
\subsection{Basic semialgebraic formulas.}
Let us now specify the definition of the Grothendieck ring
$K_0(BSA_{{ \mathbb R}})$
of basic semialgebraic formulas. This definition is inspired by \cite{DL3}.
The ring $K_0(BSA_{{ \mathbb R}})$ will
contain $K_0({\rm Var}_{{ \mathbb R}})$ as a subring (Proposition \ref{incl}) and will be projected
on the ring $K_0({\rm Var}_{{ \mathbb R}})\otimes { \mathbb Z}[\mathbb F_2rac{1}{2}]$
(Theorem \ref{thm-prin}) by
an explicit computational process.
A basic semialgebraic formula $A$ in $n$ variables is defined as a
finite
number of equations, inequations and inequalities, namely there exist
$P_i, Q_j, R_k
\in { \mathbb R}[X_1,\ldots,X_n]$, for $i=1,\ldots,p$, $j=1,\ldots,q$ and
$k=1,\ldots,r$, such that $A({ \mathbb R})$ is equal to the set of points
$x\in \mathbb A^n$ such that
$$P_i(x)=0, Q_j(x) \neq
0,R_k(x)>0,~i=1,\ldots,p,~j=1,\ldots,q, ~k=1,\ldots,r.$$
The relations $Q_j(x) \neq 0$ are called inequations and the relations
$R_k(x)>0$ are called inequalities.
We will simply denote a basic semialgebraic formula by
$$A=\{P_i=0, Q_j\neq 0, R_k>0,~i=1,\ldots,p,~j=1,\ldots,q, ~k=1,\ldots,r \}.$$
In particular $A$ is not characterized by its real points $A({ \mathbb R})$, that is
by the real
solutions of these equations, inequations and inequalities, but by these
equations, inequations and inequalities themselves.
We will consider basic semialgebraic formulas up to algebraic
isomorphisms, when the basic semialgebraic formulas are defined without
inequality.
\begin{remark} In the sequel, we will allow ourselves to use the notation
$\{P<0\}$ for the basic semialgebraic formula $\{-P>0\}$ and similarly $\{P>1\}$ instead of
$\{P-1>0\}$, where $P$ denotes a polynomial with real coefficients. Furthermore given two basic semialgebraic formulas $A$ and $B$, the notation $\{A,B\}$ will denote the basic formula
with equations, inequations and inequalities coming from $A$
and $B$ together.
\end{remark}
We define the Grothendieck ring $K_0(BSA_{{ \mathbb R}})$ of basic semialgebraic
formulas as the free abelian ring generated by basic semialgebraic formulas $[A]$,
up to algebraic isomorphim when the formula $A$ has no
inequality, and subject to the three following relations
\begin{enumerate}
\item (\textit{algebraic additivity}) $$[A]=[A,S=0]+[A,
\{S\neq 0\}]$$ where $A$ is a basic semialgebraic formula in $n$ variables and
$S\in { \mathbb R}[X_1,\ldots,X_n]$.
\item (\textit{semialgebraic additivity}) $$[A,R\neq 0]=[A, R>0]+
[A,-R>0]$$ where $A$ is a basic semialgebraic
formula in $n$ variables and $R\in { \mathbb R}[X_1,\ldots,X_n]$.
\item (\textit{product}) The product of basic semialgebraic formulas, defined by taking the
conjonction of the formulas with disjoint sets of free variables, induces the ring
product on
$K_0(BSA_{{ \mathbb R}})$. In other words we consider the relation
$$ [A,B]=[A]\cdot[B], $$
for $A$ and $B$ basic real semialgebraic formulas with disjoint set of variables.
\end{enumerate}
\begin{remark}\label{rmk-iso}
\begin{enumerate}
\item Contrary to the Grothendieck ring of algebraic varieties or algebraic sets,
we do not consider isomorphism classes of basic real semialgebraic formulas in
the definition of $K_0(BSA_{{ \mathbb R}})$. As a consequence the realization
we are interested in does depend in a crucial way on the
description of the basic semialgebraic set as a basic semialgebraic formula.
For instance $\{X-1>0\}$ and $\{X>0,X-1>0\}$ will have different measures.
\item One may decide to enlarge the basic semialgebraic formulas with non-strict inequalities
by imposing, by convention, that the measure of $\{ A,R \geq 0\}$, for $A$ a basic
semialgebraic
formula in $n$ variables and $R\in { \mathbb R}[X_1,\ldots,X_n]$, is the sum of the measures of
$\{A, R > 0\}$ and of $\{A, R = 0\}$.
\end{enumerate}
\end{remark}
\begin{prop}\label{incl} The natural map $i$ from $K_0({\rm Var}_{{ \mathbb R}})$ that associates to
an affine real algebraic variety its value in the Grothendieck ring $K_0(BSA_{{ \mathbb R}})$
of basic real semialgebraic formulas is an injective morphism
$$i:K_0({\rm Var}_{{ \mathbb R}}) \longrightarrow K_0(BSA_{{ \mathbb R}}).$$
\end{prop}
We therefore identify $K_0({\rm Var}_{{ \mathbb R}})$ with a subring of $K_0(BSA_{{ \mathbb R}}).$
\begin{proof} We construct a left inverse $j$ of $i$ as follows. Let
$a \in K_0(BSA_{{ \mathbb R}})$ be a sum of products of measures of
basic semialgebraic formulas. If there exist Zariski
constructible real algebraic sets $Z_1,\ldots,Z_m$ such that
$[Z_1]+\cdots+[Z_m]$ is equal to $a$ in
$K_0(BSA_{{ \mathbb R}})$, then we define the image of $a$ by
$j$ to be
$$j(a)=[Z_1]+\cdots+[Z_m] \in K_0({\rm Var}_{{ \mathbb R}}).$$
Otherwise, the image of $a$ by
$j$ is defined to be zero in $K_0({\rm Var}_{{ \mathbb R}})$.
The map $j$ is well-defined. Indeed, if $Y_1,\ldots, Y_l$ are other
Zariski constructible sets such that $[Y_1]+\cdots+[Y_l]$ is equal to
$a$ in $K_0(BSA_{{ \mathbb R}})$, then
$$[Y_1]+\cdots+[Y_l]=[Z_1]+\cdots+[Z_m]$$
in $K_0(BSA_{{ \mathbb R}})$. This equality still holds in $K_0({\rm Var}_{{ \mathbb R}})$ by
definition of the structure ring of $K_0({\rm Var}_{{ \mathbb R}})$ and the fact that $j$ defines a left inverse of $i$ is immediate.
\end{proof}
\begin{remark} Note however that the map $j$ constructed in the proof
of Proposition \ref{incl} is not a group
morphism. For instance $j([X>0])=j([X<0])=0$ whereas
$j([X\neq 0])={ \mathbb L}-1$.
\end{remark}
\section{A realization of $K_0(BSA_{{ \mathbb R}})$}
An example of a ring morphism from $K_0(BSA_{{ \mathbb R}})$ to $\mathbb Z$ is given by
the Euler characteristic with compact supports $\chi_c$. We construct in this section a
realization for elements in $K_0(BSA_{{ \mathbb R}})$ with values in the ring of
polynomials with coefficient in ${ \mathbb Z}[\mathbb F_2rac{1}{2}]$. This realization specializes to the
Euler characteristic with compact supports. To
this aim, we construct a ring morphism from
$K_0(BSA_{{ \mathbb R}})$ to the tensor product of $K_0({\rm Var}_{{ \mathbb R}})$ with
${ \mathbb Z}[\mathbb F_2rac{1}{2}]$.
\subsection{The realization.}
We define a morphism $\chi$ from the ring $K_0(BSA_{{ \mathbb R}})$ to the ring $K_0({\rm Var}_{{ \mathbb R}})\otimes { \mathbb Z}[\mathbb F_2rac{1}{2}]$ as follows. Let
$A$ be a basic semialgebraic formula without inequality. We assign to $A$ its value $\chi(A)=[A]$ in $K_0({\rm Var}_{{ \mathbb R}})$ as a
constructible set. We proceed now by induction on the number of
inequalities in the description of the basic semialgebraic formulas.
Assuming that we have defined $\chi$ for basic
semialgebraic formulas with at most $k$ inequalities, $k\in { \mathbb N}$, let $A$
be a basic real semialgebraic formula with $n$ variables and at most $k$ inequalities and let us consider $R\in
{ \mathbb R}[X_1,\ldots,X_n]$. Define $\chi([A, R>0])$ by
$$\chi([A, R>0]):=\mathbb F_2rac{1}{4}\big(\chi([A, Y^2=R])-\chi([A,
Y^2=-R])\big)+\mathbb F_2rac{1}{2} \chi([A, R\neq 0]),$$
where $\{A, Y^2=\pm R\}$ is a basic real semialgebraic formula with $n+1$ variables,
with at most $k$ inequalities and $\{A, R\neq 0\}$ is a
basic semialgebraic formula with $n$ variables with at most $k$ inequalities.
\begin{remark}\label{rmk-ori}
The way to define $\chi$ may be seen as an average of two different
natural ways of understanding a basic semialgebraic formula as a quotient
of algebraic varieties. Namely, for a basic semialgebraic formula in
$n$ variables of the form $\{R>0\}$, we may see its set of real
points as the projection, with fibre two points, of
$\{Y^2=R\}$ minus the zero set of $R$, or as the complement
of the projection of $Y^2=-R$. The algebraic
average of these two possible points of view is
$$\mathbb F_2rac{1}{2}\Big(\big(\mathbb F_2rac{1}{2}[Y^2=R]-[R=0]\big)+ \big( { \mathbb L}^n-\mathbb F_2rac{1}{2}[Y^2=-R]\big) \Big),$$
which, considering that $ { \mathbb L}^n-[R=0]= [R\not=0]$, gives for $\chi(R>0)$ the expression
just defined above.
\end{remark}
We give below the general formula that computes the measure of a basic
semialgebraic formula in terms of the measure of real algebraic varieties.
\begin{prop}\label{prop-alg} Let $Z$ be a constructible set in ${ \mathbb R}^n$ and take $R_k\in
{ \mathbb R}[X_1,\ldots,X_n]$, with $k=1,\ldots,r$. For $I\subset
\{1,\ldots,r\}$ a subset of cardinal $\sharp I=i$ and $ vari\'et\'eepsilon \in \{\pm
1\}^i$, we denote by $R_{I, vari\'et\'eepsilon}$ the real constructible set defined by
$$R_{I, vari\'et\'eepsilon}=\{Y_j^2= vari\'et\'eepsilon_jR_j(X),j\in I;~~R_k(X)\neq 0, k\notin I\}.$$
Then
$\chi([Z,R_k>0,~k=1,\ldots,r])$ is equal to
$$\sum_{i=0}^r \mathbb F_2rac{1}{2^{r+i}}\sum_{I\subset
\{1,\ldots,r\},\sharp I=i}\sum_{ vari\'et\'eepsilon \in \{\pm 1\}^i}(\prod_{j\in I} vari\'et\'eepsilon_j)
[Z,R_{I, vari\'et\'eepsilon}]$$
\end{prop}
\begin{proof} If $r=1$ it follows from the
definition of $\chi$. We prove the general result by induction on
$r\in { \mathbb N}$. Assume $Z={ \mathbb R}^n$ to simplify notation. Take $R_k\in { \mathbb R}[X_1,\ldots,X_n]$, with
$k=1,\ldots,r+1$.
Denote by $A$ the formula $R_1>0,\ldots, R_r>0$. By definition of $\chi$ we obtain
$$\chi([A,R_{r+1}>0])=\mathbb F_2rac{1}{4}(\chi([A, Y^2=R_{r+1}])-\chi([A, Y^2=-R_{r+1}]))+\mathbb F_2rac{1}{2}\chi([A, R_{r+1}\neq 0]).$$
Now we can use the induction assumption to express the terms in the right hand side of the formula upstair as
$$\sum_{i=0}^r \mathbb F_2rac{1}{2^{r+i}}\sum_{I\subset
\{1,\ldots,r\},\sharp I=i}\sum_{ vari\'et\'eepsilon \in \{\pm 1\}^i}(\prod_{j\in I} vari\'et\'eepsilon_j) \big(\mathbb F_2rac{1}{4}([R_{I, vari\'et\'eepsilon}, Y^2=R_{r+1}]-[R_{I, vari\'et\'eepsilon}, Y^2=-R_{r+1}])$$
$$+\mathbb F_2rac{1}{2}[R_{I, vari\'et\'eepsilon}, R_{r+1}\neq 0] \big) $$
Choose $I\subset
\{1,\ldots,r\}$ a subset of cardinal $\sharp I=i$ and $ vari\'et\'eepsilon \in \{\pm
1\}^i$. Then, we obtain from the definition of $\chi$ that
$$\mathbb F_2rac{1}{4}([R_{I, vari\'et\'eepsilon}, Y^2=R_{r+1}]-[R_{I, vari\'et\'eepsilon}, Y^2=-R_{r+1}])+\mathbb F_2rac{1}{2}[R_{I, vari\'et\'eepsilon}, R_{r+1}\neq 0]$$
is equal to
$$\mathbb F_2rac{1}{4}([R_{I\cup \{r+1\}, vari\'et\'eepsilon^+}]-[R_{I\cup \{r+1\}, vari\'et\'eepsilon^-}])+\mathbb F_2rac{1}{2}[R_{\tilde I, vari\'et\'eepsilon}]$$
where $ vari\'et\'eepsilon^+=( vari\'et\'eepsilon_1,\ldots, vari\'et\'eepsilon_r,1)$,
$ vari\'et\'eepsilon^-=( vari\'et\'eepsilon_1,\ldots, vari\'et\'eepsilon_r,-1)$ and $\tilde I$ denotes $I$ as a
subset of $\{1,\ldots,r+1\}$. Therefore
$$\mathbb F_2rac{1}{2^{r+i}}(\prod_{j\in I} vari\'et\'eepsilon_j)[R_{r+1}>0,
R_{I, vari\'et\'eepsilon}]$$
is equal to
$$\mathbb F_2rac{1}{2^{(r+1)+(i+1)}}(\prod_{j\in I} vari\'et\'eepsilon_j)([R_{I\cup
\{r+1\}, vari\'et\'eepsilon^+}]-[R_{I\cup
\{r+1\}, vari\'et\'eepsilon^-}])+\mathbb F_2rac{1}{2^{(r+1)+i}}(\prod_{j\in
I} vari\'et\'eepsilon_j)[R_{\tilde I, vari\'et\'eepsilon}]$$
which gives the result.
\end{proof}
The morphism $\chi$ is then defined on $K_0(BSA_{{ \mathbb R}})$.
\begin{thm}\label{thm-prin} The map $$\chi: K_0(BSA_{{ \mathbb R}}) \longrightarrow
K_0({\rm Var}_{{ \mathbb R}})\otimes { \mathbb Z}[\mathbb F_2rac{1}{2}]$$ is a ring morphism that is
the identity on $K_0({\rm Var}_{{ \mathbb R}})\subset K_0(BSA_{{ \mathbb R}})$.
\end{thm}
\begin{proof} We must prove that the given definition of $\chi$ is
compatible with the algebraic and semialgebraic additivities. However the semialgebraic additivity follows directly from the definition of
$\chi$. Indeed, if $A$ is a basic semialgebraic formula and $R$ a real
polynomial, then the sum of $\chi([A ,R>0])$ and $\chi([A ,-R>0])$ is equal to
$$\mathbb F_2rac{1}{4}\big(\chi([A,Y^2=R])-\chi([A,Y^2=-R])\big)+\mathbb F_2rac{1}{2}\chi([A,R\neq 0])$$
$$+\mathbb F_2rac{1}{4}\big(\chi([A,Y^2=-R])-\chi([A,Y^2=R])\big)+\mathbb F_2rac{1}{2}
\chi([A,-R\neq 0])$$
$$=\chi([A,-R\neq 0]).$$
The algebraic additivity as well as the multiplicativity follow from Proposition \ref{prop-alg} that
enables to express the measure of a basic semialgebraic formula in terms
of algebraic varieties for which additivity and multiplicativity hold.
We conclude by noting that we may construct a left inverse to $\chi$
restricted to $K_0({\rm Var}_{{ \mathbb R}})$ in the same way as in the proof of
Proposition \ref{incl}.
\end{proof}
\begin{ex}\label{ex1}
\begin{enumerate}
\item A half-line defined by $X>0$ has measure in $K_0({\rm Var}_{{ \mathbb R}})\otimes
{ \mathbb Z}[\mathbb F_2rac{1}{2}]$ half of the value of the line minus one point, as expected, since
by definition
$$\chi([X>0])=\mathbb F_2rac{1}{4}({ \mathbb L}-{ \mathbb L})+\mathbb F_2rac{1}{2}\big({ \mathbb L}-1)=\mathbb F_2rac{1}{2}\big({ \mathbb L}-1).$$
However, if we add one more inequality, like $\{X>0,X>-1\}$, then
the measure has more complexity. We will see in section \ref{sect-virt} that, evaluated in the polynomial ring ${ \mathbb Z}[\mathbb F_2rac{1}{2}][u]$ we
obtain in that case
$$\beta([X>0,X>-1])=\mathbb F_2rac{5u-11}{16}.$$
\item Using the multiplicativity, we find the measure of the
half-plane and the measure of the quarter
plane as expected
$$\chi([X_1>0])=\mathbb F_2rac{1}{2}({ \mathbb L}^2-{ \mathbb L})$$
and
$$\chi([X_1>0,X_2>0])=\mathbb F_2rac{1}{4}({ \mathbb L}-1)^2.$$
\end{enumerate}
\end{ex}
\begin{remark}\label{rmks}
\begin{enumerate}
\item Let $R\in { \mathbb R}[X_1,\ldots,X_n]$ be odd. Then
$$\chi([R>0])=\chi([R<0])=\mathbb F_2rac{[R\neq 0]}{2}.$$
Indeed, the varieties $Y^2=R(X)$ and $Y^2=-R(X)$
are isomorphic via $X\mapsto -X$, and the result follows from the
definition of $\chi$.
\item\label{nontrivial} The ring morphism from $K_0({\rm Var}_{{ \mathbb R}})$ to
$K_0({ \mathbb R}{\rm Var})$ gives a realization from the ring $K_0(BSA_{{ \mathbb R}})$ to the ring
$K_0({ \mathbb R}{\rm Var})\otimes { \mathbb Z}[\mathbb F_2rac{1}{2}]$
for which the measure of a
real algebraic variety without real point is zero, this is why it is often
convenient to push the computations to the ring
$K_0({ \mathbb R}{\rm Var})\otimes { \mathbb Z}[\mathbb F_2rac{1}{2}]$ rather than staying at the higher level
of $K_0({\rm Var}_{{ \mathbb R}})\otimes { \mathbb Z}[\mathbb F_2rac{1}{2}]$. However we have to notice that the
measure of a basic real semialgebraic formula without real point is not
necessarily zero in $K_0({ \mathbb R}{\rm Var})\otimes { \mathbb Z}[\mathbb F_2rac{1}{2}]$.
For instance, let us compute the measure of $X^2+1>0$ in
$K_0({ \mathbb R}{\rm Var})\otimes { \mathbb Z}[\mathbb F_2rac{1}{2}]$.
By definition of $\chi$ we obtain that $\chi([X^2+1>0])$ is equal to
$$\mathbb F_2rac{1}{4}\big(\chi([Y^2=X^2+1])-\chi([Y^2=-X^2-1])\big)+\mathbb F_2rac{1}{2}\chi([X^2+1
\neq 0])$$
$$=\mathbb F_2rac{1}{4}({ \mathbb L}-1)+\mathbb F_2rac{1}{2}{ \mathbb L}=\mathbb F_2rac{1}{4}(3{ \mathbb L}-1).$$
By additivity we have
$$ \chi([X^2+1<0])=\chi([X^2+1\not=0])-\chi([X^2+1>0])$$
$$ ={ \mathbb L}-\chi([X^2+1=0])-\chi([X^2+1>0]). $$
But since $\chi([X^2+1=0])=0$ in $K_0({ \mathbb R}{\rm Var})\otimes { \mathbb Z}[\mathbb F_2rac{1}{2}]$,
we obtain that the measure of $\{X^2+1<0\}$ in $K_0({ \mathbb R}{\rm Var})\otimes { \mathbb Z}[\mathbb F_2rac{1}{2}]$, whose real points set is empty, is
$$\chi([X^2+1<0])=\mathbb F_2rac{1}{4}({ \mathbb L}+1).$$
\item In a similar way, the basic semialgebraic formula $\{P>0,-P>0\}$ with
$P(X)=1+X^2$, whose set of real points is empty, has measure
$$\chi([P>0,-P>0])=\mathbb F_2rac{1}{8}({ \mathbb L}+1).$$
\end{enumerate}
\end{remark}
\subsection{Isomorphism between basic semialgebraic formulas}
In this section we give a condition for two basic semialgebraic formulas to have the same realization by $\chi$. It deals with the
complexification of the algebraic liftings of the basic semialgebraic
formulas.
Let $X$ be a
real algebraic subvariety of $\mathbb R^n$ defined by $P_i
\in { \mathbb R}[X_1,\ldots,X_n]$, for $i=1,\ldots,r$. The complexification
$X_{{ \mathbb C}}$ of $X$ is defined to be the complex algebraic subvariety of
${ \mathbb C}^n$ defined by the same polynomials $P_1,\ldots,P_r$. We define
similarly the complexification of a real algebraic map.
Let $Y\subset { \mathbb R}^n$ be a Zariski
constructible subset of ${ \mathbb R}^n$ and take $R_1,\ldots,R_r
\in { \mathbb R}[X_1,\ldots,X_n]$. Let $A$ denote the basic
semialgebraic formula of ${ \mathbb R}^n$ defined by $Y$ together with the inequalities
$R_1>0,\ldots,R_r>0$, and $V$ denote the Zariski constructible subset
of ${ \mathbb R}^{n+r}$ defined by
$$V=\{Y,Y_1^2=R_1,\ldots,Y_r^2=R_r\}.$$
Note that $V$ is endowed with an action
of $\{\pm 1\}^r$ defined by multiplication by $-1$ on the
indeterminates $Y_1,\ldots,Y_r$.
Let $Z\subset { \mathbb R}^n$ be a Zariski constructible subset of ${ \mathbb R}^n$ and
take similarly $S_1,\ldots,S_r
\in { \mathbb R}[X_1,\ldots,X_n]$. Let $B$ denote the basic
semialgebraic formula of ${ \mathbb R}^n$ defined by $Z$ together with the inequalities
$S_1>0,\ldots,S_r>0$, and $W$ denote the Zariski constructible subset
of ${ \mathbb R}^{n+r}$ defined by
$$W=\{Z,Y_1^2=S_1,\ldots,Y_r^2=S_r\}.$$
\begin{definition}
We say that the basic semialgebraic formulas $A$ and $B$ are isomorphic
if there exists a real algebraic isomorphism $\phi:V \longrightarrow
W$ between $V$ and $W$ which is equivariant with respect to the action
of $\{\pm 1\}^r$ on $V$ and $W$, and whose complexification
$\phi_{{ \mathbb C}}$ induces a complex algebraic isomorphism between the
complexifications $V_{{ \mathbb C}}$ and $W_{{ \mathbb C}}$ of $V$ and $W$.
\end{definition}
\begin{remark}\label{rmk-1} Let us consider first the particular case $Y={ \mathbb R}^n$, $Z={ \mathbb R}^n$ and $r=1$.
Change moreover the notation as
follows. Put $V^+=V$ and $W^+=W$, and define $V^-=\{y^2=-R(x)\}$ and
$W^-=\{y^2=-S(x)\}$.
Then the complex points $V_{\mathbb C}^+$ and $V_{\mathbb C}^-$ of
$V^+$ and $V^-$ are isomorphic via the complex (and not real)
isomorphism $(x,y)\mapsto (x,iy)$. Now, suppose that the basic
semialgebraic formula $\{R>0\}$ is isomorphic to $\{S>0\}$. Let $\phi=(f,g):(x,y)\mapsto (f(x,y),g(x,y))$ be the real isomorphism involved in the definition (that is $f$ and $g$ are defined by real equations,
and moreover $f(x,-y)=f(x,y)$ and $g(x,-y)=-g(x,y)$). Then the following diagram
$$ \begin{matrix}
V^+_{\mathbb C} &\buildrel{(f,g)}\over \longrightarrow & W^+_{\mathbb C}\\
\buildrel{(x,y)\mapsto (x,iy)}\over \downarrow & & \buildrel{(x,y)\mapsto (x,iy)}\over \downarrow \\
V^-_{\mathbb C} & & W^-_{\mathbb C}\\
\end{matrix} $$
induces a complex isomorphism $(F,G)$ between $ V^-_{\mathbb C}$ and $W^-_{\mathbb C}$ given by
$$(x,y) \mapsto (f(x,-iy),ig(x,-iy)).$$
In fact, this isomorphism is defined over $\mathbb R$ since
$$\overline {F(x,y)}=\overline {f(x,-iy)}=f(\overline x, \overline {-iy})=f(\overline x,i \overline y)=f(\overline x,-i \overline y)=F(\overline x,\overline y)$$ and
$$\overline {G(x,y)}=\overline {ig(x,-iy)}=-ig(\overline x, \overline
{-iy})=-ig(\overline x,i \overline y)=ig(\overline x,-i \overline
y)=G(\overline x,\overline y),$$
where the bar denotes complex conjugation. Therefore it induces a real
algebraic isomorphism between $V^-$ and $W^-$.
Moreover $g(x,0)=-g(x,0)$ so $g(x,0)=0$ and then the real algebraic sets $\{R=0\}$ and $\{S=0\}$ are also isomorphic.
\end{remark}
\begin{prop}\label{iso} If the basic semialgebraic formulas $A$ and $B$ are
isomorphic, then $\chi([A])=\chi([B])$.
\end{prop}
\begin{proof} Thanks to Proposition \ref{prop-alg}, we only need to prove that
the real algebraic varieties $R_{I, vari\'et\'eepsilon}$ corresponding to $A$ and
$B$ are isomorphic two by two, which is a direct generalization of
Remark \ref{rmk-1}.
\end{proof}
\section{Virtual Poincar\'e polynomial}
\subsection{Polynomial realization}\label{sect-virt}
The best realization known (with respect to the highest
algebraic complexity of the realization ring) of the Grothendieck ring of real algebraic varieties is
given by the virtual Poincar\'e polynomial \cite{MCP}. This
polynomial, whose coefficients coincide with the Betti numbers with
coefficients in $\mathbb F_2rac{{ \mathbb Z}}{2{ \mathbb Z}}$ when sets are compact and nonsingular,
has coefficient in ${ \mathbb Z}$. As a corollary of Theorem \ref{thm-prin} we
obtain the following realization of $K_0(BSA_{{ \mathbb R}})$ in ${ \mathbb Z}[\mathbb F_2rac{1}{2}][u]$.
\begin{prop} There exists a ring morphism
$$\beta:K_0(BSA_{{ \mathbb R}}) \longrightarrow { \mathbb Z}[\mathbb F_2rac{1}{2}][u]$$
whose restriction to $K_0({\rm Var}_{{ \mathbb R}})\subset K_0(BSA_{{ \mathbb R}})$ coincides with
the virtual Poincar\'e polynomial.
\end{prop}
The interest of such a realization is that it enables to make concrete computations.
\begin{ex}
\begin{enumerate}
\item The virtual Poincar\'e polynomial of the open disc $X_1^2+X_2^2<1$ is equal to
$$\mathbb F_2rac{1}{4}\big(\beta([Y^2=1-(X_1^2+X_2^2)])-\beta([Y^2=X_1^2+X_2^2-1])\big)+
\mathbb F_2rac{1}{2}\beta([X_1^2+X_2^2\neq 1])$$
$$=\mathbb F_2rac{1}{4}(u^2+1-u(u+1))+\mathbb F_2rac{1}{2}(u^2-u-1)=\mathbb F_2rac{1}{4}(2u^2-3u-1).$$
\item Let us compute the measure of the formula $X>a,X>b$ with $a\neq b \in \mathbb R$. By Proposition \ref{prop-alg}, we are lead to compute the virtual Poincar\'e polynomial of the real algebraic subsets of $\mathbb R^3$ defined by $\{y^2=\pm (x - a),~~z^2=\pm (x - b)\}$. These sets are isomorphic to $\{y^2 \pm z^2 = \pm (a - b)\}$, and we recognise either a circle, a hyperbola or the emptyset.
In particular, using the formula in Proposition \ref{prop-alg}, we obtain
$$\beta([X>a,X>b])=\mathbb F_2rac{1}{16}(2(u-1)-(u+1))+\mathbb F_2rac{1}{8}(2u-2u)+\mathbb F_2rac{1}{8}(2-2)+\mathbb F_2rac{1}{4}(u-2)=\mathbb F_2rac{5u-11}{16}$$
\end{enumerate}
\end{ex}
\begin{remark} In case the set of real points of a basic semialgebraic
formula is a real algebraic set (or even an arc symmetric set \cite{KK,F}), its
virtual Poincar\'e polynomial does not coincide in general with the
virtual Poincar\'e polynomial of the real algebraic set. For
instance, the basic semialgebraic formula $X^2+1>0$, considered in
Remark \ref{rmks}, has virtual Poincar\'e
polynomial equal to
$\mathbb F_2rac{1}{4}(3u-1)$ whereas its set of points is a real line whose
with virtual Poincar\'e polynomial equals $u$ as a real algebraic set.
\end{remark}
Evaluating $u$ at an integer gives another realization, with
coefficient in ${ \mathbb Z}[\mathbb F_2rac{1}{2}]$.
The virtual Poincar\'e polynomial of a real algebraic variety, evaluated
at $u=-1$, coincides with its Euler characteristic with compact
supports \cite{MCP}. Indeed, evaluating the virtual Poincar\'e
polynomial of a basic semialgebraic formula gives also the Euler characteristic with compact
supports of its set of real points, and therefore has its values in ${ \mathbb Z}$.
\begin{prop}\label{eval} The virtual Poincar\'e
polynomial $\beta(A)$ of a basic semialgebraic formula $A$ is equal to
the Euler characteristic with compact supports of its set of real
points $A({ \mathbb R})$
when evaluated at $u=-1$. In other words
$$\beta(A)(-1)=\chi_c(A({ \mathbb R})).$$
\end{prop}
\begin{proof} We recall that in Proposition \ref{prop-alg} we explain how to express the class of $A$ as a
linear combination
of classes of real algebraic varieties for which the virtual Poincar\'e polynomial evaluated
at $u=-1$ coincides with the Euler characteristic with compact
supports. At each step of our inductive process to obtain such a linear combination, we introduce a new
variable and a double covering of the set of points satisfying one less inequality. The inductive formula
$$\chi([B, R>0]):=\mathbb F_2rac{1}{4}\big(\chi([B, Y^2=R])-\chi([B,
Y^2=-R])\big)+\mathbb F_2rac{1}{2} \chi([B, R\neq 0]),$$
used at this step to eliminate one inequality by replacing the system $\{B,R>0\}$ by other systems $\{B, Y^2=R\}, \{B, Y^2=-R\},
\{B, R\neq 0\}$ is compatible with the Euler characteristic of the underlying sets of points, that is to say that
our induction formula is true for $\chi=\chi_c$. The geometric reason for this fact is explained in Remark \ref{rmk-ori}, and is
the intuitive motivation to define the realization $\chi$ by induction precisely as it is defined.
\end{proof}
\subsection{Homogeneous case}
We propose some computations of the virtual Poin\-ca\-r\'e polynomial of
basic real semialgebraic formulas of the form $\{R>0\}$ where $R$ is
homogeneous. Looking at Euler characteristic with compact supports, it
is equal to the product of the Euler characteristics with compact
supports of $\{X>0\}$ with
$\{R=1\}$. We investigate the case of virtual Poincar\'e polynomial. A
key point in the proofs will be the invariance of the virtual Poincar\'e polynomial
of constructible sets under regular homeomorphisms (see \cite{MCP2}, Proposition 4.3).
\begin{prop}\label{homo-odd} Let $R\in { \mathbb R}[X_1,\ldots,X_n]$ be a homogeneous polynomial
of degre $d$. Assume $d$ is odd. Then
$$\beta([R>0])=\beta([X>0])\beta([R=1]).$$
\end{prop}
\begin{proof} The algebraic varieties defined by $Y^2=R(X)$ and
$Y^2=-R(X)$ are isomorphic since $R(-X)=-R(X)$, therefore
$$\beta([R>0])=\mathbb F_2rac{\beta([R\neq 0])}{2}.$$
The map $(\lambda,x)\mapsto \lambda x$ from $\mathbb R^* \times
\{R=1\}$ to $R\neq 0$ is a regular homeomorphism with inverse $y
\mapsto (R(y)^{1/d}, \mathbb F_2rac{y}{R(y)^{1/d}})$ therefore
$$\beta([R\neq 0])=\beta(\mathbb R^*)\beta([R=1]),$$
so that
$$\beta([R>0])=\mathbb F_2rac{\beta(\mathbb R^*)}{2}\beta([R=1])=\beta([X>0])\beta([R=1]).$$
\end{proof}
The result is no longer true when the degre is even. However, in the
particular case of the square of a homogeneous polynomial of odd
degre, the relation of Proposition \ref{homo-odd} remains valid.
\begin{prop} Let $P\in { \mathbb R}[X_1,\ldots,X_n]$ be a homogeneous polynomial
of degre $k$. Assume $k$ is odd, and define $R\in
{ \mathbb R}[X_1,\ldots,X_n]$ by $R=P^2$. Then
$$\beta([R>0])=\beta([X>0])\beta([R=1]).$$
\end{prop}
\begin{proof} Note first that $\{Y^2-R\}$ can be factorized as
$(Y-P)(Y+P)$ therefore the virtual Poincar\'e polynomial of
$Y^2-R$ is equal to
$$\beta(Y-P=0 )+\beta(Y+P=0)-\beta(P=0).$$
However the algebraic varieties $Y-P=0$ and $Y+P=0$ are
isomorphic to a $n$-dimensional affine space, whereas $Y^2+R=0$
is isomorphic to $P=0$ since $R=P^2$ is positive, so
that the virtual Poincar\'e polynomial of $R>0$ is equal to
$$\mathbb F_2rac{1}{4}(2\beta(\mathbb
R^n)-2\beta([P=0]))+\mathbb F_2rac{1}{2}\beta([P\neq 0])=\beta([P\neq 0]).$$
To compute $\beta([P\neq 0]$, note that the map
$(\lambda,x)\mapsto \lambda x$ from
$\mathbb R^* \times \{P=1\}$ to $\{P\neq 0\}$ is a regular homeomorphism with inverse $y
\mapsto (R(y)^{1/k}, \mathbb F_2rac{y}{R(y)^{1/k}})$ therefore
$$\beta([P\neq 0])=\beta(\mathbb R^*)\beta([P=1]).$$
We achieve the proof by noticing that $R-1=(P-1)(P+1)$ so that
$\beta([P=1])=\mathbb F_2rac{\beta([R=1])}{2}$ because the degree of the
homogeneous polynomial $P$ is odd.
Finally
$$\beta([R>0])=\mathbb F_2rac{\beta(\mathbb R^*)}{2}\beta([R=1])$$
and the proof is achieved.
\end{proof}
More generally, for a homogeneous polynomial $R$ of degre twice a odd number, we can
express the
virtual Poincar\'e polynomial of $[R>0]$ in terms of that of
$[R=1]$, $[R=-1]$ and $[R\neq 0]$ as follows.
\begin{prop}\label{homo-even} Let $k\in \mathbb N$ be odd and put $d=2k$.
Let $R\in { \mathbb R}[X_1,\ldots,X_n]$ be a homogeneous polynomial
of degre $d$. Then
$$\beta([R>0])=\mathbb F_2rac{1}{4}\beta(\mathbb R^*)(\beta([R=1])-\beta([R=-1]))+\mathbb F_2rac{1}{2}\beta([R\neq 0]).$$
\end{prop}
\begin{ex} We cannot do better in general as illustrated by the
following examples. For $R_1=X_1^2+X_2^2$ one obtain
$$\beta([R_1>0])=\mathbb F_2rac{3}{2}\beta([X>0])\beta([R_1=1])$$
whereas for $R_2=X_1^2-X_2^2$ one has
$$\beta([R_2>0])=\beta([X>0])\beta([R_2=1]).$$
\end{ex}
The proof of Proposition \ref{homo-even} is a direct consequence of the next lemma.
\begin{lemma} Let $k\in \mathbb N$ be odd and put $d=2k$. Let $R\in { \mathbb R}[X_1,\ldots,X_n]$ be a homogeneous polynomial
of degre $d$. Then
$$\beta([Y^2=R])=\beta([R= 0])+\beta(\mathbb R^*)\beta([R=1]).$$
\end{lemma}
\begin{proof} Note first that the algebraic varieties $Y^2=R$ and
$Y^d=R$ have the same virtual Poincar\'e polynomial. Indeed the
map $(x,y)\mapsto (x,y^k)$ realizes a regular homeomorphism between
$Y^2=R$ and
$Y^d=R$, whose inverse is given by $(x,y)\mapsto (x,y^{1/k})$.
However the polynomial $Y^d-R$ being homogeneous, we obtain a regular homeomorphism
$$\mathbb R^* \times (\{R=1\}\cap \{Y^d=R\}) \longrightarrow \{R\neq 0\}\cap \{Y^d=R\}$$
defined by $(\lambda,x,y) \mapsto (\lambda x,\lambda y)$. As a consequence
$$\beta([Y^d-R=0])=\beta([R= 0])+\beta(\mathbb R^*)\beta([R=1]).$$
\end{proof}
\section{Zeta functions and Motivic real Milnor fibres}
We apply in this section the preceding construction of
$\chi : K_0(BSA_{{ \mathbb R}})\to K_0({\rm Var}_{ \mathbb R})\otimes
{ \mathbb Z}[\mathbb F_2rac{1}{2}]$ in defining, for a given polynomial $f\in { \mathbb R}[X_1,\cdots, X_d]$,
zeta functions whose coefficients are classes
in
$(K_0({\rm Var}_{ \mathbb R})\otimes { \mathbb Z}[\mathbb F_2rac{1}{2}])[{ \mathbb L}^{-1}]$
of real semialgebraic formulas in truncated arc spaces.
We then show that these
zeta functions are deeply related to the topology of some
corresponding set-theoretic real semialgebraic Milnor fibres of~$f$.
\subsection{Semialgebraic zeta functions and real Denef-Loeser formulas.}
Let $f:{ \mathbb R}^d \to { \mathbb R}$ be a polynomial function with coefficients in ${ \mathbb R}$ sending $0$
to $0$.
We denote by ${\cal L}$ or $\cal L({ \mathbb R}^d,0)$ the space of formal arcs
$\gamma(t)=(\gamma_1(t), \cdots, \gamma_d(t))$ on ${ \mathbb R}^d$, with $\gamma_j(0)=0$ for all
$j\in \{1, \cdots, d\}$, by $\cal L_n$ or $\cal L_n({ \mathbb R}^d,0)$ the space of
truncated arcs ${\cal L}/(t^{n+1})$ and by $\pi_n : \cal L \to \cal L_n$
the truncation map. More generally, for $M$ a variety and $W$ a closed subset of $M$,
$\cal L(M,W)$ (resp. $\cal L_n(M,W)$) will denote the
space of arcs on $M$ (resp. the $n$-th jet-space on $M$)
with endpoints in $W$.
Let $ vari\'et\'eepsilonilon$ be one of the symbols in the set
$\{ \hbox{\sl naive}, -1, 1, >, < \}$.
For such a symbol $ vari\'et\'eepsilonilon$, via the
realization of $K_0(BSA_{{ \mathbb R}})$ in
$K_0({\rm Var}_{ \mathbb R})\otimes { \mathbb Z}[\mathbb F_2rac{1}{2}]$, we
define a zeta function $Z_f^ vari\'et\'eepsilonilon(T)\in (K_0({\rm Var}_{ \mathbb R})\otimes
{ \mathbb Z}[\mathbb F_2rac{1}{2}])[{ \mathbb L}^{-1}][[T]]$ by
$$ Z_f^ vari\'et\'eepsilonilon(T):=\sum_{n\ge 1}\ [X_{n,f}^ vari\'et\'eepsilonilon]{ \mathbb L}^{-nd}T^n,$$
where $X_{n,f}^ vari\'et\'eepsilonilon$ is defined in the following way:
\vskip0,3cm
- $X_{n,f}^{naive}=\{\gamma \in {\cal L}_n;
\ f(\gamma(t))=at^n+\cdots, a\not=0\}$,
\vskip0,1cm
- $X_{n,f}^{-1}=\{\gamma \in {\cal L}_n;
\ f(\gamma(t))=at^n+\cdots, a=-1\}$,
\vskip0,1cm
- $X_{n,f}^{1}=\{\gamma \in {\cal L}_n;
\ f(\gamma(t))=at^n+\cdots, a=1\}$,
\vskip0,1cm
- $X_{n,f}^>=\{\gamma \in {\cal L}_n;
\ f(\gamma(t))=at^n+\cdots, a >0\}$,
\vskip0,1cm
- $X_{n,f}^<=\{\gamma \in {\cal L}_n;
\ f(\gamma(t))=at^n+\cdots, a<0\}$.
\vskip3mm
Note that $X_{n,f}^ vari\'et\'eepsilonilon$ is a real algebraic variety for $ vari\'et\'eepsilonilon = -1$ or $1$,
a real algebraic constructible set for $ vari\'et\'eepsilonilon =naive$
and a semialgebraic set, given by an explicit description involving one inequality, for $ vari\'et\'eepsilonilon$ being the symbol $ >$ or the symbol $<$.
Consequently,
$Z_f^ vari\'et\'eepsilonilon (T)\in K_0({\rm Var}_{ \mathbb R})[{ \mathbb L}^{-1}][[T]]$ for $ vari\'et\'eepsilonilon \in \{naive, -1, 1\}$ and
$Z_f^ vari\'et\'eepsilonilon (T)\in (K_0({\rm Var}_{ \mathbb R})\otimes
{ \mathbb Z}[\mathbb F_2rac{1}{2}])[{ \mathbb L}^{-1}][[T]]$ for $ vari\'et\'eepsilonilon \in \{ > , <\}$.
We show in this section that $Z_f^ vari\'et\'eepsilonilon(T)$ is a rational function expressed in terms
of the combinatorial data of a resolution of $f$. To define those data let us consider
$\sigma : (M,\sigma^{-1}(0))\to
({ \mathbb R}^d,0)$ a proper birational map which is an isomorphism over the complement
of $\{f=0\}$ in $({ \mathbb R}^d,0)$, such that $f\circ \sigma$ and the jacobian determinant
$\jac\ \sigma$ are normal crossings and $\sigma^{-1}(0)$ is a union of components of the exceptional divisor. We denote by
$ E_j$, for $ j\in \cal J$, the irreducible components of
$(f\circ \sigma)^{-1}(0)$ and
assume that $E_k$ are the irreducible components of
$\sigma^{-1}(0)$ for $k\in \cal K \subset \cal J$. For $j\in \cal J$
we denote by $N_j$ the multiplicity
$mult_{E_j}f\circ \sigma $ of
$f\circ \sigma$ along $E_j$ and for $k\in \cal K$ by $\nu_k$ the number $\nu_k=
1+mult_{E_k}\jac \ \sigma$. For any $I\subset \cal J$, we put
$E^0_I=(\bigcap_{i\in I} E_i)\setminus (\bigcup_{j\in \cal J\setminus I}E_j)$.
These sets $E^0_I$ are constructible sets and the collection $(E_I^0)_{I\subset \cal J}$ gives a canonical stratification of the divisor
$f\circ \sigma=0$, compatible with $\sigma=0$ such that in some affine open subvariety $U$
in $M$ we have $f\circ \sigma (x)=u(x) \prod_{i\in I} x_i^{N_i}$, where
$u$ is a unit, that is to say a rational function which does not vanish on $U$, and
$x=(x',(x_i)_{i\in I})$ are local coordinates.
Finally for $ vari\'et\'eepsilonilon \in \{-1, 1, >,<\}$ and $I\subset \cal J$,
we define $\widetilde E^{0, vari\'et\'eepsilonilon}_I$ as the gluing along $E^0_I$ of the sets
$$ R_U^ vari\'et\'eepsilonilon=\{ (x,t)\in (E^0_I\cap U)\times { \mathbb R}; \ t^m \cdot u(x)\ ?_ vari\'et\'eepsilonilon \ \},$$
where $?_ vari\'et\'eepsilonilon$ is $=-1$, $=1$, $>0$ or $<0$ in case $ vari\'et\'eepsilonilon$ is $-1,1, >$ or
$< $ and $m=gcd_{i\in I}(N_i)$.
\begin{remark}\label{gluing} The definition of the $R_U^ vari\'et\'eepsilonilon$'s is
independent of the choice of the coordinates, as well as
the gluing of the $R^ vari\'et\'eepsilonilon_U$ is allowed, up to isomorphism, since when in some Zariski neighborhood of
$E^0_I$ one has in another coordinate system $z=z(x)=(z', (z_i)_{i\in I})$ the expression $f\circ \sigma (z)= v(z)\prod_{i\in I}
z^{N_i}$, there exist non-vanishing functions $\alpha_i$
so that $z_i=\alpha_i(z)\cdot x_i$. We thus obtain
$v(z)\prod_{i\in I}\alpha_i^{N_i}(z)=u(x)$, and the transformation
$$ \begin{matrix}
& \{(x,t)\in (E_I^0\cap U)\times { \mathbb R}; t^m \cdot u(x)\ ?_{ vari\'et\'eepsilonilon} \} & \to &
\{(z,s)\in (E_I^0\cap U)\times { \mathbb R}; s^m\cdot v(z) \ ?_{ vari\'et\'eepsilonilon} \} \\
&
(x,t) & \mapsto & (z,s=t\prod_{i\in I}\alpha_i(z)^{N_i/m})
\\
\end{matrix} $$
is an isomorphism in case $?_ vari\'et\'eepsilonilon$ is $=1$ or $=-1$, and induces an isomorphism between the associate double covers
$\cal R^ vari\'et\'eepsilonilon_U =\{(x,t,y)\in (E^0_I\cap U)\times { \mathbb R}\times { \mathbb R}; t^m\cdot u(x)\cdot y^2= \eta( vari\'et\'eepsilonilon) \} $ and
$\cal R^{' vari\'et\'eepsilonilon}_U=\{(z,s,w)\in (E_I^0\cap U)\times { \mathbb R}\times { \mathbb R}; s^m\cdot v(z)\cdot w^2=\eta( vari\'et\'eepsilonilon)\} $, with $\eta( vari\'et\'eepsilonilon)=1$ when $ vari\'et\'eepsilonilon$ is the symbol $>$
and $\eta( vari\'et\'eepsilonilon)=-1$ when $ vari\'et\'eepsilonilon$ is the symbol $<$, the induced isomorphism simply being
$$ \begin{matrix}
&
\cal R^ vari\'et\'eepsilonilon_U & \to &
\cal R^{' vari\'et\'eepsilonilon}_U
\\
&
(x,t,y) & \mapsto & (z,s, w=y ).
\\
\end{matrix}$$
Also notice that $\widetilde E_I^{0, vari\'et\'eepsilonilon}$ is a constructible set when
$ vari\'et\'eepsilonilon$ is $-1$ or $1$ and a semialgebraic set with explicit description
over the constructible set $E_I^0$ when $ vari\'et\'eepsilonilon$ is $<$ or $>$.
\end{remark}
We can thus
define the class $[\widetilde E_I^{0, vari\'et\'eepsilonilon}]\in \chi(K_0(BSA_{{ \mathbb R}}))$ as follows. Choosing a finite covering $(U_l)_{l\in L}$ of $M$ by affine open subvarieties $U_l$, for $l\in L$, we set
$$[\widetilde E_I^{0, vari\'et\'eepsilonilon}]=\sum _{S\subset L} (-1)^{|S|+1}[R^ vari\'et\'eepsilonilon_{\cap_{s\in S}U_{s}}].$$
The class $[\widetilde E_I^{0, vari\'et\'eepsilonilon}]$ does not depend on the choice of the covering thanks to Remark \ref{gluing} and the algebraic additivity in $K_0(BSA_{{ \mathbb R}})$.
With this notation one can give the expression of $Z_f^ vari\'et\'eepsilonilon(T)$ in terms of
$[\widetilde E^{0, vari\'et\'eepsilonilon}_I]$, as,
for instance, in \cite{DL1}, \cite{DL2}, \cite{DL4}, \cite{Loo}, essentially
using the Kontsevitch change of variables formula in motivic integration (\cite{Kon},
\cite{DL2} for instance).
\begin{theorem}\label{Zeta function}
With the notation above, one has
$$Z_f^ vari\'et\'eepsilonilon(T)= \sum_{I\cap \cal K \not= \emptyset}({ \mathbb L}-1)^{\vert I\vert -1 }[\widetilde E^{0, vari\'et\'eepsilonilon}_I ]\prod_{i\in I}
\mathbb F_2rac{{ \mathbb L}^{-\nu_i}T^{N_i}}{1-{ \mathbb L}^{-\nu_i}T^{N_i}}$$
for $ vari\'et\'eepsilonilon$ being $-1,1,>$ or $<$.
\end{theorem}
\begin{remark} Classically, the right hand side of equality
of Theorem
\ref{Zeta function} does not depend, as a formal series
in $(K_0({\rm Var}_{ \mathbb R})\otimes { \mathbb Z}[\mathbb F_2rac{1}{2}])[{ \mathbb L}^{-1}][[T]]$, on the
choice of the resolution $\sigma$,
as the definition of $Z_f^ vari\'et\'eepsilonilon(T)$ does not depend itself on
any choice of resolution.
\end{remark}
To prove this theorem, we first start with a lemma that needs the following notation. We denote by
$$\sigma_*: \cal L(M,\sigma^{-1}(0))\to \cal L({ \mathbb R}^d,0),$$ and for $n\in { \mathbb N}$, by
$$\sigma_{n,*} : \cal L_n(M,\sigma^{-1}(0))\to \cal L_n({ \mathbb R}^d,0)$$
the natural mappings induced by $\sigma : (M,\sigma^{-1}(0)) \to ({ \mathbb R}^d,0)$. Let
$$Y_{n,f}^ vari\'et\'eepsilonilon=\pi_n^{-1}(X_{n,f}^ vari\'et\'eepsilonilon).$$
Then $Y_{n,f\circ \sigma}^ vari\'et\'eepsilonilon= \{\gamma\in \cal L(M,\sigma^{-1}(0)); \
f(\sigma(\pi_n(\gamma)))(t)=at^n+\cdots, \ a \ ?_ vari\'et\'eepsilonilon \}$, where $?_ vari\'et\'eepsilonilon$ is $=-1$, $=1$, $>0$ or $<0$ in case $ vari\'et\'eepsilonilon$ is $-1,1, >$ or
$< $, and note also that $Y_{n,f\circ \sigma}^ vari\'et\'eepsilonilon = \sigma_*^{-1}(Y_{n,f}^ vari\'et\'eepsilonilon)$.
Finally for $e\ge 1$, let
$$\Delta_e=\{ \gamma\in \cal L (M,\sigma^{-1}(0)); \ mult_t \ (\jac \ \sigma)(\gamma(t))=e \}
\hbox{ and } Y_{e,n,f\circ \sigma}^ vari\'et\'eepsilonilon=Y_{n,f\circ \sigma}^ vari\'et\'eepsilonilon\cap \Delta_e.$$
\begin{lem}\label{lemme calculatoire}
With the notation above, there exists $c\in { \mathbb N}$ such that
$$ Z_f^ vari\'et\'eepsilonilon(T)
= \displaystyle { \mathbb L}^d\sum_{n\ge 1} T^n
\sum_{e\le cn} { \mathbb L}^{-e}\sum_{I\not=\emptyset}{ \mathbb L}^{-(n+1)d} [{\cal L}_n(M,E_I^0\cap \sigma^{-1}(0)) \cap \pi_n(\Delta_e)
\cap X_{n,f\circ \sigma}^ vari\'et\'eepsilonilon].$$
\end{lem}
\begin{proof}
As usual in motivic integration, the class of the cylinder $Y_{n,f}^ vari\'et\'eepsilonilon=\pi_n^{-1}(X_{n,f}^ vari\'et\'eepsilonilon)$, $n\ge 1$, is
an element of $(K_0({\rm Var}_{ \mathbb R})\otimes { \mathbb Z}[{1\over 2}])[{ \mathbb L}^{-1}]$,
the localization of the ring $K_0({\rm Var}_{ \mathbb R})\otimes { \mathbb Z}[{1\over 2}]$ with respect to
the multiplicative set generated by ${ \mathbb L}$, and defined
by $[Y_{n,f}^ vari\'et\'eepsilonilon]:={ \mathbb L}^{-(n+1)d}[X_{n,f}^ vari\'et\'eepsilonilon]$, since the truncation morphisms $\pi_{k+1,k}:\cal L_{k+1}({ \mathbb R}^d,0)\to \cal L_k({ \mathbb R}^d,0)$, $k\ge 1$, are locally trivial
fibrations with fibre ${ \mathbb R}^d$. Hence $Z_f^ vari\'et\'eepsilonilon(T)= \displaystyle { \mathbb L}^d\sum_{n\ge 1} [Y_{n,f}^ vari\'et\'eepsilonilon]T^n$.
Take now $\gamma \in \sigma_*^{-1}(Y_{n,f}^ vari\'et\'eepsilonilon)$,
and let $I\subset \cal J$ such that $\gamma(0)\in E_I^0$. In some neighbourhood of
$E_I^0$, one has coordinates such that $f\circ \sigma (x)=u(x)\prod_{i\in I}x_i^{N_i}$ and $\jac(\sigma)(x)=v(x)\prod_{i\in I}x_i^{\nu_i-1}$, with $u$ and
$v$ units.
If one denotes $\gamma=(\gamma_1, \cdots, \gamma_d)$ in these coordinates, with $k_i$ the multiplicity of $\gamma_i$ at $0$ for $i\in I$,
then we have
$mult_t(f\circ \sigma\circ \gamma(t))=\sum_{i\in I} k_i N_i=n$. Now
$$mult_t(\jac \sigma)(\gamma(t))=\sum_{i\in I}k_i(\nu_i-1)\le \max_{i\in I} (\mathbb F_2rac{\nu_i-1}{N_i})\sum_{i\in I}N_ik_i
=\max_{i\in I} (\mathbb F_2rac{\nu_i-1}{N_i})n.$$
Therefore if one sets $c= \max_{i\in I} (\mathbb F_2rac{\nu_i-1}{N_i})$, one has
$$Y_{n,f\circ \sigma}^ vari\'et\'eepsilonilon=\bigcup_{e\ge 1}Y^ vari\'et\'eepsilonilon_{e,n,f\circ \sigma} =\bigcup_{1\le e\le cn} Y^ vari\'et\'eepsilonilon_{e,n,f\circ \sigma}, $$
as disjoint unions. Now we can apply the change of variables theorem (see \cite{DL2}, \cite{Kon}) to compute $[Y^ vari\'et\'eepsilonilon_{n,f}]$ in terms of
$[Y^ vari\'et\'eepsilonilon_{e,n,f\circ \sigma}]$:
$$ [Y^ vari\'et\'eepsilonilon_{n,f}]= \sum_{e\le cn} { \mathbb L}^{-e}[Y^ vari\'et\'eepsilonilon_{e,n,f\circ \sigma}], $$
and summing over the subsets $I$ of $\cal J$, as $Y^ vari\'et\'eepsilonilon_{e,n,f\circ \sigma}$ is the disjoint union
$$\bigcup_{I\not =\emptyset} Y^ vari\'et\'eepsilonilon_{e,n,f\circ \sigma}
\cap \pi_0^{-1}(E_I^0\cap \sigma^{-1}(0)),$$
we obtain
$$Z_f^ vari\'et\'eepsilonilon(T)= \displaystyle { \mathbb L}^d\sum_{n\ge 1} [Y_{n,f}^ vari\'et\'eepsilonilon]T^n=\displaystyle { \mathbb L}^d\sum_{n\ge 1} T^n
\sum_{e\le cn} { \mathbb L}^{-e}\sum_{I\not=\emptyset}
[Y^ vari\'et\'eepsilonilon_{e,n,f\circ \sigma} \cap \pi_0^{-1}
(E_I^0\cap \sigma^{-1}(0))]$$
$$=\displaystyle { \mathbb L}^d\sum_{n\ge 1} T^n
\sum_{e\le cn} { \mathbb L}^{-e}\sum_{I\not=\emptyset} { \mathbb L}^{-(n+1)d}
[\pi_n(Y^ vari\'et\'eepsilonilon_{e,n,f\circ \sigma} \cap \pi_0^{-1}
(E_I^0\cap \sigma^{-1}(0)))]= $$
$$=\displaystyle { \mathbb L}^d\sum_{n\ge 1} T^n
\sum_{e\le cn} { \mathbb L}^{-e}\sum_{I\not=\emptyset}{ \mathbb L}^{-(n+1)d} [{\cal L}_n(M,E_I^0\cap \sigma^{-1}(0)) \cap \pi_n(\Delta_e)
\cap X_{n,f\circ \sigma}^ vari\'et\'eepsilonilon].$$
\end{proof}
\vskip5mm
\begin{proof}[Proof of Theorem \ref{Zeta function} ]
Considering the expression of $Z_f^ vari\'et\'eepsilonilon(T)$ given by Lemma \ref{lemme calculatoire},
we have to compute the class of $[{\cal L}_n(M,E_I^0\cap \sigma^{-1}(0)) \cap \pi_n(\Delta_e)
\cap X_{n,f\circ \sigma}^ vari\'et\'eepsilonilon]$. For this we notice that on
some neighbourhood $U$ of the end point
$\gamma(0)\in E_I^0\cap \sigma^{-1}(0)$,
one has coordinates such that
$$ f\circ \sigma (x)=u(x)\prod_{i\in I}x_i^{N_i} \hbox{ and }
\jac(\sigma)(x)=v(x)\prod_{i\in I}x_i^{\nu_i-1},$$
with $u$ and $v$ units.
As a consequence ${\cal L}_n(M,E_I^0\cap U\cap \sigma^{-1}(0)) \cap \pi_n(\Delta_e) \cap X_{n,f\circ \sigma}^ vari\'et\'eepsilonilon$ is isomorphic to
$$\{\gamma\in {\cal L}_n(M,\sigma^{-1}(0)); \gamma(0)\in E_I^0\cap U\cap \sigma^{-1}(0), \sum_{i\in I}N_i k_i=n, \sum_{i\in I}
k_i(\nu_i-1)=e,$$
$$ f\circ \sigma (\gamma(t))=at^n+\cdots, a \ ?_ vari\'et\'eepsilonilon \},$$
where $?_ vari\'et\'eepsilonilon$ is $=-1$, $=1$, $>0$ or $<0$ in case $ vari\'et\'eepsilonilon$ is $-1,1, >$ or
$< $ and $k_i$ is the multiplicity of $\gamma_i$ for $i\in I$.
Now denoting by $A(I,n,e)$ the set
$$A(I,n,e):=\{k=(k_1, \cdots,k_d)\in { \mathbb N}^d ; \sum_{i\in I}N_i k_i=n, \sum_{i\in I} k_i(\nu_i-1)=e \},$$
and identifying for simplicity
$x$ and $((x_i)_{i\not\in I},(x_i)_{i\in I})$,
the set
$${\cal L}_n(M,E_I^0\cap U\cap \sigma^{-1}(0)) \cap \pi_n(\Delta_e) \cap X_{n,f\circ \sigma}^ vari\'et\'eepsilonilon$$
is isomorphic to the product
$$({ \mathbb R}^n)^{d-\vert I\vert}\times \bigcup_{k\in A(I,n,e)} \{ x\in (E_I^0\cap U \cap \sigma^{-1}(0))\times ({ \mathbb R}^*)^{\vert I\vert };
u((x_i)_{i\not\in I},0) \prod_{i\in I} x_i^{N_i}\ ?_ vari\'et\'eepsilonilon\}
\times \prod_{i\in I}({ \mathbb R}^{n-k_i})$$
Indeed, denoting $\gamma=(\gamma_1, \dots, \gamma_d)$ by
$\gamma_i(t)=a_{i,0}+\cdots + a_{i,n}t^n$ for $i\not \in I$
and $\gamma_i(t)=a_{i,k_i}t^{k_i}+\cdots + a_{i,n} t^n$ for $i\in I$, an arc of $\cal L_n(M,E_I^0\cap U \cap \sigma^{-1}(0))$, the first
factor of the product comes from the free choice of the coefficients $a_{i,j}$, $i\not\in I$, $j=1, \cdots, n$, the last factor of the product
comes from the free choice of the coefficients $a_{i,j}$, $i\in I$, $j= k_i+1, \dots, n $
and the middle factor of the product comes from the choice of the coefficients $a_{i,0}\in E_I^0\cap U \cap \sigma^{-1}(0)$, $i\not\in I$ and from
the choice of the coefficients $a_{i,k_i}$, $i\in I$, subject to $f\circ \sigma (\gamma(t))=u(\gamma(t))\prod_{i\in I}\gamma^{N_i}_i(t)=
u((a_{i,0})_{i\not\in I},0) (\prod_{i\in I}a_{i,k_i}^{N_i})t^n+\cdots=at^n+\cdots, a\ ?_ vari\'et\'eepsilonilon$.
\vskip2mm
We now choose $n_i\in { \mathbb Z}$ such that $\sum_{i\in I}n_iN_i=m=gcd_{i\in I}(N_i)$ and consider the two semialgebraic sets
$$W_U^ vari\'et\'eepsilonilon = \{ x\in (E_I^0\cap U \cap \sigma^{-1}(0))\times ({ \mathbb R}^*)^{\vert I\vert };
u((x_i)_{i\not\in I},0) \prod_{i\in I} x_i^{N_i}\ ?_ vari\'et\'eepsilonilon\}$$
and
$$W_U^{' vari\'et\'eepsilonilon}= \{ (x',t)\in (E_I^0\cap U \cap \sigma^{-1}(0))\times ({ \mathbb R}^*)^{\vert I\vert }\times { \mathbb R}^*; u((x'_i)_{i\not\in I},0)t^m\ ?_ vari\'et\'eepsilonilon, \
\prod_{i\in I}x_i'^{N_i/m}=1 \}, $$
where $?_ vari\'et\'eepsilonilon$ is $=-1$, $=1$, $>0$ or $<0$ in case $ vari\'et\'eepsilonilon$ is $-1,1, >$ or
$< $.
In case $?_ vari\'et\'eepsilonilon=1$ or $?_ vari\'et\'eepsilonilon=-1$, the mapping
$$ \begin{matrix} &W^{' vari\'et\'eepsilonilon}_U &\to &W^ vari\'et\'eepsilonilon_U
\\
&(x',t) &\mapsto &x=((x'_i)_{i\not\in I}, (t^{n_i}x'_i)_{i\in I} )\\
\end{matrix}$$
is an isomorphism of inverse
$$ \begin{matrix} &W_U^ vari\'et\'eepsilonilon&\to &W_U^{' vari\'et\'eepsilonilon}
\\
&x &\mapsto &(x'=((x_i)_{i\not\in I},
((\prod_{\ell\in I} x_\ell^{N_\ell/m})^{-n_i}x_i)_{i\in I}),
t=\prod_{\ell\in I} x_\ell^{N_\ell/m} ). \\
\end{matrix}$$
In the semialgebraic case, this isomorphism induces a natural isomorphism
on the double-covers $\cal W_U^ vari\'et\'eepsilonilon$ and $\cal W_U^{' vari\'et\'eepsilonilon}$
associated to $ W_U^ vari\'et\'eepsilonilon$ and $W_U^{' vari\'et\'eepsilonilon}$ and defined by
$$\cal W_U^ vari\'et\'eepsilonilon=\{(x,y)\in (E_I^0\cap U \cap \sigma^{-1}(0))\times
({ \mathbb R}^*)^{\vert I\vert }\times { \mathbb R};
y^2u((x'_i)_{i\not\in I},0)\prod_{i\in I} x_i^{N_i}
=\eta( vari\'et\'eepsilonilon) \}$$ and
$$\cal W_U^{' vari\'et\'eepsilonilon}=\{ (x,t,w)\in (E_I^0\cap U \cap \sigma^{-1}(0))\times ({ \mathbb R}^*)^{\vert I\vert }\times { \mathbb R}^*\times { \mathbb R};$$
$$w^2u((x'_i)_{i\not\in I},0)t^m =\eta( vari\'et\'eepsilonilon), \
\prod_{i\in I}x_i'^{N_i/m}=1\},$$
where $\eta( vari\'et\'eepsilonilon)=1$ when $ vari\'et\'eepsilonilon$ is the symbol $>$ and
$\eta( vari\'et\'eepsilonilon)=-1$ when $ vari\'et\'eepsilonilon$ is the symbol $<$. In consequence, $[W_U^ vari\'et\'eepsilonilon]=[W_U^{' vari\'et\'eepsilonilon}]$ in the algebraic
case ($ vari\'et\'eepsilonilon=-1$ or $1$) as well as in the semialgebraic case ($ vari\'et\'eepsilonilon=<$ or $>$) considering our realization formula for basic semialgebraic
formulas in
$ K_0({\rm Var}_{ \mathbb R})\otimes { \mathbb Z}[\mathbb F_2rac{1}{2}]$.
Now we observe in the case where $ vari\'et\'eepsilonilon$ is $-1$ or $1$ that $W_U^{' vari\'et\'eepsilonilon}$ is isomorphic to
$R^ vari\'et\'eepsilonilon_U \times ({ \mathbb R}^*)^{\vert I\vert -1}$ (see \cite{DL4}, Lemma 2.5) whereas in the case where
$ vari\'et\'eepsilonilon$ is $<$ or $>$, we obtain that the class of $W_U^{' vari\'et\'eepsilonilon}$ is equal to the class of
$R^ vari\'et\'eepsilonilon_U \times ({ \mathbb R}^*)^{\vert I\vert -1}$, considering again the double coverings associated to the
basic semialgebraic formulas defining these two sets.
We finally obtain
$$[{\cal L}_n(M,E_I^0\cap \sigma^{-1}(0)) \cap \pi_n(\Delta_e)
\cap X_{n,f\circ \sigma}^ vari\'et\'eepsilonilon]=\displaystyle\sum_{k\in A(I,n,e)}
{ \mathbb L}^{nd-\sum_{i\in I}k_i}[W_U^{' vari\'et\'eepsilonilon}]=$$
$$\displaystyle\sum_{k\in A(I,n,e)}
{ \mathbb L}^{nd-\sum_{i\in I}k_i}\times[R_U^{ vari\'et\'eepsilonilon}]\times
({ \mathbb L}-1)^{\vert I\vert -1}.$$
Summing over the charts $U$, the expression of $Z_f^ vari\'et\'eepsilonilon(T)$ given by Lemma \ref{lemme calculatoire} is now
$$ Z_f^ vari\'et\'eepsilonilon(T)=\displaystyle \sum_{I\cap \cal K \not= \emptyset} { \mathbb L}^d\sum_{n\ge 1} T^n
\sum_{e\le cn} { \mathbb L}^{-e}({ \mathbb L}-1)^{\vert I \vert-1 }{ \mathbb L}^{-(n+1)d}
[{\widetilde E}_I^{0, vari\'et\'eepsilonilon}]\displaystyle\sum_{k\in A(I,n,e)} { \mathbb L}^{nd-\sum_{i\in I}k_i}$$
$$=\displaystyle \sum_{I\cap \cal K \not= \emptyset} ({ \mathbb L}-1)^{\vert I \vert-1 }
[{\widetilde E}_I^{0, vari\'et\'eepsilonilon} ]
\sum_{n\ge 1} T^n
\sum_{e\le cn}\sum_{k\in A(I,n,e)} { \mathbb L}^{-e-\sum_{i\in I}k_i}$$
Noticing that the $(k_i)_{i\in I}$'s such that $k=((k_i)_{i\not\in I}),
(k_i)_{i\in I})\in \displaystyle \bigcup_{e\le cn, n\ge 1} A(I,n,e)$ are
in bijection with ${ \mathbb N}^{*\vert I \vert }$, we have
$$Z_f^ vari\'et\'eepsilonilon(T)=
\displaystyle \sum_{I\cap \cal K\not= \emptyset} ({ \mathbb L}-1)^{\vert I \vert-1 }
[{\widetilde E}_I^{0, vari\'et\'eepsilonilon}]
\sum_{(k_i)_{i\in I}\in { \mathbb N}^{\vert I \vert }}
\prod_{i\in I}({ \mathbb L}^{-\nu_i}T^{N_i})^{k_i}$$
$$= \displaystyle \sum_{I\cap \cal K\not= \emptyset} ({ \mathbb L}-1)^{\vert I \vert-1 }
[{\widetilde E}_I^{0, vari\'et\'eepsilonilon}]
\prod_{i\in I}\mathbb F_2rac{{ \mathbb L}^{-\nu_i}T^{N_i}}{1-{ \mathbb L}^{-\nu_i}T^{N_i}}. $$
\end{proof}
\subsection{Motivic real Milnor fibres and their realizations.}
We can now define a motivic real Milnor fibre by taking the constant term
of the rational function $Z_f^ vari\'et\'eepsilonilon(T)$ viewed as a power series in $T^{-1}$.
This process formally consists in letting $T$ going to $\infty$ in the rational
expression of $Z_f^ vari\'et\'eepsilonilon(T)$ given by Theorem \ref{Zeta function} and using
the usual computation rules as in the convergent case (see for instance \cite{DL1}, \cite{DL4}).
\begin{definition}\label{real D-L}
Let $f:{ \mathbb R}^d\to { \mathbb R}$ be a polynomial function and $ vari\'et\'eepsilonilon$ be one of the symbols $naive, 1, -1, >$ or $<$.
Consider a resolution of $f$ as above and let us adopt the same notation $(E_I^0)_I$ for the stratification of the exceptional
divisor of this resolution, leading to the notation $\widetilde E_I^{0, vari\'et\'eepsilonilon}$.
The real motivic
Milnor $ vari\'et\'eepsilonilon$-fibre $S^ vari\'et\'eepsilonilon_f$ of $f$ is defined as (see \cite{DL4} for the complex case)
$$ S^ vari\'et\'eepsilonilon_f:=-\lim_{T\to \infty} Z^ vari\'et\'eepsilonilon_f(T):=\displaystyle
-\sum_{I\cap \cal K\not= \emptyset} (-1)^{\vert I \vert}
[{\widetilde E}_I^{0, vari\'et\'eepsilonilon}]({ \mathbb L}-1)^{\vert I \vert-1 }
\in K_0({\rm Var}_{ \mathbb R})\otimes { \mathbb Z}[\mathbb F_2rac{1}{2}].$$
It does not depend on the choice of the resolution $\sigma$.
\end{definition}
For $ vari\'et\'eepsilonilon$ being the symbol $1$ for instance, we have $S_f^1\in K_0({\rm Var}_{ \mathbb R})$.
We can consider, first in the complex case, the realization
of $S_f^1$ via the Euler-Poincar\'e characteristic ring morphism $\chi_c : K_0({\rm Var}_{ \mathbb C})\to { \mathbb Z}$. Note that in the complex case, the Euler characteristics with and without compact supports are equal. For $f:{ \mathbb C}^d\to { \mathbb C}$, since $\chi_c({ \mathbb L}-1)=0$, we obtain
$$\chi_c(S_f^1)=\displaystyle\sum_{\vert I\vert=1, I\subset \cal K}
\chi_c({\widetilde E}_I^{0,1}) =\displaystyle\sum_{\vert I\vert=1, I\subset \cal K}
N_I\cdot \chi_c(E_I^0\cap \sigma^{-1}(0)).$$
Now denoting by
$F$ the set-theoretic Milnor fibre of the fibration $f_{\vert B(0,\alpha)\cap f^{-1}(D^\times_\eta)}:
B(0,\alpha)\cap f^{-1}(D^\times_\eta)
\to D_\eta^\times $, with $B(0,\alpha)$ the open ball in ${ \mathbb C}^d$ of radius $\alpha$
centred at
$0$, $D_\eta$ the disc in ${ \mathbb C}$
of radius $\eta$ centred at $0$ and $D_\eta^\times=D_\eta\setminus\{0\} $, with
$0<\eta\ll \alpha\ll 1$,
comparing the above expression $\chi_c(S_f^1)=\displaystyle
\sum_{\vert I\vert=1, I\subset \cal K}
N_I\cdot \chi_c(E_I^0)$ with the following A'Campo formula of \cite{ACA}
for the first Lefschetz number of the iterates of the monodromy
$M:H^*(F,{ \mathbb C})\to H^*(F,{ \mathbb C})$ of $f$,
that is for the Euler-Poincar\'e characteristic of the fibre $F$:
$$ \chi_c(F)=\displaystyle\sum_{\vert I\vert=1, I\subset \cal K} N_I\cdot
\chi_c(E_I^0\cap \sigma^{-1}(0)) $$
we simply observe that
$$\chi_c(S_f^1)= \chi_c(F).$$
The closure $f^{-1}(c)\cap \bar B(0,\alpha)$, $0<\vert c\vert \ll \alpha \ll 1$, of the
Milnor fibre $F$ being denoted by $\bar F$ and the boundary of $\bar F$
being the odd dimensional compact
manifold $f^{-1}(c)\cap S(0,\alpha)$,
$\chi_c(f^{-1}(c)\cap S(0,\alpha) )=0$ and we finally have
$$\chi_c(S_f^1)=\chi_c(F)=\chi_c(\bar F).$$
\begin{remark} There is {\sl a priori} no hint in the definition of $Z_f^ vari\'et\'eepsilonilon(T)$
that the opposite of the constant term $S_f^1$ of the power series in $T^{-1}$ induced by the
rationality of $Z_f^ vari\'et\'eepsilonilon(T)$
could be the motivic version of the Milnor fibre of $f$ (as well
as, for instance, there is no evident hint that the expression of $Z_f^ vari\'et\'eepsilonilon$ in Theorem \ref{Zeta function}
does not depend on the resolution $\sigma$).
As mentionned above, in the complex case, we just observe that the
expression of $\chi_c(S_f^1)$ is the expression of $\chi_c(F)$ provided
by the A'Campo formula. Exactly in the same way there is no
{\sl a priori} reason for $\chi_c(S_f^ vari\'et\'eepsilonilon)$, regarding the definition of $Z_f^ vari\'et\'eepsilonilon$, to be so acurately related to
the topology of $f^{-1}( vari\'et\'eepsilonilon \vert c \vert)\cap B(0,\alpha)$. Nevertheless we prove
that it is actually the case (Theorem \ref{Milnor}).
\end{remark}
In order to establish this result we start hereafter by a
geometrical proof of the formula in the complex
case (compare with \cite{ACA} where only ${ \mathbb L}ambda(M^0)$ is considered, $M^k$ being
the $k$th iterate of the monodromy $M:H^*(F,{ \mathbb C})\to H^*(F,{ \mathbb C})$ of $f$).
We will then extend to the reals this computational proof in
the proof of Theorem \ref{Milnor}, allowing us interpret the complex proof
as the first complexity level of its real extension.
\begin{remark} Note that in the complex case a proof of the fact that ${ \mathbb L}ambda(M^k)=\chi_c(X^1_{k,f})$,
for $k\ge 1$, is given in \cite{HL}
without the help of resolution of singularities, that is to say without help of
A'Campo's formulas (see Theorem 1.1.1 of \cite{HL}). As a direct corollary it is thus proved that $\chi_c(S^1_f)=\chi_c(F)$
in the complex case, without using
A'Campo formulas.
\end{remark}
\vskip5mm
\noindent
{\bf Realization of the complex motivic Milnor fibre under $\chi_c$. }
The fibre $F=\{ f=c\}\cap B(0,\alpha)$ is homeomorphic to the fibre
$\cal F=\{f\circ \sigma = c\}\cap \sigma^{-1}(B(0,\alpha))$, with $\sigma^{-1}(S(0,\alpha))$ viewed as the boundary of a
tubular neighbourhood
of $\sigma^{-1}(0)=\bigcup_{E_J^0\subset \sigma^{-1}(0)}E_J^0$,
keeping the same notation $(E_J^0)_J$ as before for the natural stratification of the strict transform $\sigma^{-1}(\{f=0\})$ of $f=0$.
Now the formula may be established for
$\cal F$ in some chart of $M\cap \sigma^{-1}(B(0,\alpha))$, by additivity.
In such a chart, where $f\circ \sigma$ is normal crossing, we consider
\begin{enumerate}
\item[-] the set $E_J=\bigcap_{i\in J}E_i \subset \sigma^{-1}(0)$,
given by $x_i=0$, $ i\in J$,
\item[-] a closed small enough tubular neighbourhood $V_J$ in $M$ of
$\bigcup_{J\subset K, K\not= J} E^0_K$, that is
a tubular neighbourhood of all the $E^0_K$'s bounding $E^0_J$, such that
$E_J^0\setminus V_J$ is homeomorphic to $E_J^0$,
\item[-] and $\pi_J$ the projection onto $E_J$ along the
$x_j$'s coordinates, for $j\in J$.
\item[-] an open neighbourhood $\cal E_J$
of $E_J^0\setminus V_J$ in $\sigma^{-1}(B(0,\alpha)) $ given by
$\pi_J^{-1}(E_J^0\setminus V_J), \vert x_j\vert \le \eta_J$,
$j\in J $, with $\eta_J>0$ small enough,
\end{enumerate}
\begin{remark}\label{Thom}
For $I=\{i \}$, we remark that $\cal F \cap \cal E_I$ is homeomorphic
to $N_i$ copies of $E_I^0\cap \cal E_I$, and thus to $N_i$ copies of
$E_I^0$. Indeed, assuming $f\circ \sigma = u(x)x_i^{N_i}$
in $\cal E_I$, we observe that the family $(f_t)_{t\in [0,1]}$, with
$f_t=u((x_j)_{j\not\in I}, t\cdot x_i) x_i^{N_i}-c$,
has homeomorphic fibres $\{f_t=0\}\cap \cal E_J$,
$t\in [0,1]$, by Thom's isotopy lemma, since
$$\mathbb F_2rac{\partial f_t}{\partial x_i}(x)=t\mathbb F_2rac{\partial u}{\partial x_i}(x)
x_i^{N_i}+
u(x) x_i^{N_i-1}=0, $$
would imply $\displaystyle t\mathbb F_2rac{\partial u}{\partial x_i}(x)x_i+
u(x)=0$. But the first term in this sum goes to $0$ as $x_i$ goes to $0$, since the
derivatives of $u$ are
bounded on the compact $adh(\cal E_I)$, although the norm of the
second term is bounded from below on $\cal E_I$ by a non zero constant, since $u$ is a unit.
Finally, as
$\{ f_1=0\}\cap \cal E_I$ is homeomorphic to $\{ f_0=0\}\cap \cal E_I$ and
$\{ f_0=0\}\cap \cal E_I$
is a $N_i$-graph
over $E_I^0\cap \cal E_I$, $\cal F\cap \cal E_I$ is homeomorphic to $N_i$ copies of $E_I^0$.
\end{remark}
By this remark, $\cal F$ covers maximal dimensional stratum $E_I^0$,
$\vert I \vert=1$, $I \subset \cal K$, with $N_i$ copies of a leaf $\cal F_I$ of $\cal F$.
To be more accurate, with the notation introduced above,
$\cal F_I$ covers the neighbourhood $E_I^0 \cap \cal E_I$ of $E_I^0\setminus V_I$.
Moreover the $\cal F_I$'s overlap in $\cal F$
over the open set $E_J^0 \cap \cal E_J$ of the strata $E^0_J$ that bound
the $E^0_I$'s, for $\vert I \vert =1$, $\vert J\vert =2$
and $I \subset J$, in bundles over the $E_J^0 \cap \cal E_J$'s of fibre ${ \mathbb C}^*$.
Those sub-leaves
$\cal F_J$ of $\cal F$
overlap in turn over the open $E_Q^0 \cap \cal E_Q$ of the strata $E_Q^0$,
$\vert Q\vert=3, J\subset Q $,
that bound the $E_J^0$'s, in bundles over the $E_Q^0 \cap \cal E_Q$'s of fibres
$({ \mathbb C}^*)^2$ and so forth...
For instance when $f\circ \sigma = u(x)\prod_{i\in I}x_i^{N_i}$ in
$\cal E_I$, $I=\{i\}$,
and
$f\circ \sigma = v(x) x_i^{N_i}x_j^{N_j}$ in $\cal E_J$, $J=\{i,j\}$, the $N_i$ leaves
$\cal F_I$, homeomorphic to the $N_i$ copies $x_i^{N_i}=c /u(x)$ of $E_I^0$, overlap
over $E_J^0 \cap \cal E_J$
in sub-leaves $\cal F_J$ of $\cal F_I$, given by $v(x)x_i^{N_i}x_j^{N_j}=c$, fibering over $E_J^0$ with
fibre $GCD(\{N_i,N_j\})$ copies of $({ \mathbb C}^*)^{\vert J\vert -1}$ and so forth... (see figure 1).
\vskip1,0cm
\hskip1,5cm \includegraphics[height=8cm]{fig1.eps}
\vskip-7,7cm
\hskip11,4cm$f\circ \sigma=c$
\vskip-0,4cm
\hskip1,0cm $\cal F_{I'}$
\vskip0,8cm
\hskip0,9cm $E^0_K$
\vskip0,5cm
\hskip1,0cm $\cal F_K$
\vskip0,8cm
\hskip1,1cm $\cal F_J$
\vskip-0,5cm
\hskip11,7cm $E^0_I$
\vskip0,7cm
\hskip10,9cm $\cal F_I$
\vskip-0,2cm
\hskip0,9cm $E^0_J $
\vskip3,0cm
\centerline{figure 1}
\vskip1cm
\begin{remark}\label{retract}
Note that the topology of $\cal F=\{f\circ \sigma =c\} \cap \sigma^{-1}(B(0,\alpha))$
is the same as the topology of
$\bigcup_{J\cap \cal K \not=0}\cal F_J$ (that is the topology of $\cal F$ above the strata
$E^0_J$ of $\sigma^{-1}(0)$) since
the retraction of $\cal F$ onto $\bigcup_{J\cap \cal K \not=\emptyset} \cal F_J$,
as $\alpha$ goes to $0$, induces
a homeomorphism from $\cal F$ to $\bigcup_{J\cap \cal K \not=\emptyset} \cal F_J$.
\end{remark}
From Remark \ref{retract}, by additivity, it follows that the Euler-Poincar\'e characteristic of $\cal F$ (in our chart) is the sum
$$\displaystyle\sum_{\vert I\vert=1, I\subset \cal K}
N_I\cdot \chi_c(E_I^0\cap \sigma^{-1}(0)) + L,\eqno(*)$$ where $L$ is some ${ \mathbb Z}$-linear
combination of
Euler-Poincar\'e characteristics of bundles
over the open sets $E_J\cap \cal E_J^0$, $\vert J \vert>1 $, of fibre a power of tori ${ \mathbb C}^*$. Now the A'Campo formula
$$ \chi_c(F)=\displaystyle\sum_{\vert I\vert=1, I\subset \cal K}
N_I\cdot \chi_c(E_I^0\cap \sigma^{-1}(0))$$
for
the Milnor number follows from the fact that $\chi_c({ \mathbb C}^*)=0$ implies $L=0$.
\vskip5mm
\noindent
{\bf Realization of the real motivic Milnor fibres under $\chi_c$. }
The partial covering of $\cal F$ by the pieces $\cal F_J$, for
$J\cap \cal K \not= \emptyset$,
over the strata of the stratification $(E^0_J)_{J\cap \cal K \not= \emptyset}$ of
$\sigma^{-1}(0)$ allows us to compute the Euler-Poincar\'e characteristic
of the Milnor fibre $\cal F$ in terms of the Euler-Poincar\'e characteristic of
the strata $E_J^0$, in the complex as well as in the real case. In the complex case, as noted above, for $J$ with $\vert J\vert >1$,
one has $\chi_c(\cal F_J)=0$. This cancellation provides a quite simple formula for
$\chi_c(F)$: only the strata of the maximal
dimension of the divisor $\sigma^{-1}(0)$ appear in this formula, as expected
from the A'Campo formula.
In the real case one does not have such cancellations: on one hand
the expression of $\chi_c(F)$
in terms of $\chi_c(\widetilde E_J^{0, vari\'et\'eepsilonilon})$ is no more trivial
(the remaining term $L$ of equation $(*)$ is not zero and consequently terms
$\chi_c(\widetilde E_J^{0, vari\'et\'eepsilonilon})$, for $\vert J\vert >1 $ and
$E_j \cap \sigma^{-1}(0)\not=\emptyset$, appear), and on the other hand the
expression
of $\chi_c(S_f^ vari\'et\'eepsilonilon)$ given by the real Denef-Loeser formula in Definition
\ref{real D-L} have terms $2^{\vert J \vert -1}\chi_c(\widetilde E_J^{0, vari\'et\'eepsilonilon})$
, for $\vert J\vert >1 $ and $J\cap \cal K \not=\emptyset $
(since $\chi_c({ \mathbb L}-1) =-2$ in the real case).
Nevertheless, in the real case we show that
$\chi_c(S_f^ vari\'et\'eepsilonilon)$ is again $\chi_c(\bar F)$, justifying the terminology of
motivic real semialgebraic Milnor fibre of $f$ at $0$ for $S_f^ vari\'et\'eepsilonilon$. The formula
stated in Theorem \ref{Milnor} below
is the real analogue of the A'Campo-Denef-Loeser formula for complex hypersurface
singularities and thus appears as the extension to the
reals of this complex formula, or, in other words, the complex formula is the notably
first level
of complexity of the more general real formula.
\begin{notation}\label{Khim}
Let $f: { \mathbb R}^d\to { \mathbb R}$ be a polynomial function such that $f(0)=0$ and with isolated
singularity at $0$, that is $\grad f(x)=0$ only for $x=0$ in some open neighbourhood of
$0$.
Let $0<\eta \ll \alpha$ be such that the topological type of
$f^{-1}(c)\cap B(0,\alpha)$ does
not depend on $c$ and $\alpha$, for $0<c<\eta$ or for $-\eta<c<0$.
- Let us denote, for $ vari\'et\'eepsilonilon\in \{-1,1\}$ and $ vari\'et\'eepsilonilon\cdot c>0$, this topological type by $F_ vari\'et\'eepsilonilon$, by $\bar F_ vari\'et\'eepsilonilon$ the topological type of the closure of the Milnor fibre $F_ vari\'et\'eepsilonilon$ and by
$Lk(f)$ the link $f^{-1}(0)\cap S(0,\alpha)$ of $f$ at the origin.
We recall that the topology of $Lk(f)$ is the same as the topology of the boundary
$f^{-1}(c)\cap S(0,\alpha)$ of the Milnor fibre $\bar F_ vari\'et\'eepsilonilon$, when $f$ has an
isolated singularity at $0$.
- Let us denote, for $ vari\'et\'eepsilonilon\in \{<,>\}$, the topological type
of $f^{-1}(]0,c_ vari\'et\'eepsilonilon[)\cap B(0,\alpha)$ by $F_ vari\'et\'eepsilonilon$, and
the topological type of $f^{-1}(]0,c_ vari\'et\'eepsilonilon[)\cap \bar B(0,\alpha)$
by $\bar F_ vari\'et\'eepsilonilon$,
where $c_<\in ]-\eta, 0[$ and $c_>\in ]0,\eta[$.
- Let us denote, for $ vari\'et\'eepsilonilon\in \{<,>\}$, the topological type
of $\{f \ \bar vari\'et\'eepsilonilon \ 0\}\cap S(0,\alpha)$ by $G_ vari\'et\'eepsilonilon$, where
$\bar vari\'et\'eepsilonilon$ is $\le$ when $ vari\'et\'eepsilonilon$ is $<$ and $\bar vari\'et\'eepsilonilon$ is $\ge$
when $ vari\'et\'eepsilonilon$ is $>$.
\end{notation}
\begin{remark}\label{bord}
When $d$ is odd, $Lk(f)$ is a smooth odd-dimensional submanifold of ${ \mathbb R}^d$ and
consequently $\chi_c(Lk(f))=0$. For
$ vari\'et\'eepsilonilon \in \{-1, 1, <, >\}$, we thus have in this situation,
$\chi_c(F_ vari\'et\'eepsilonilon)=\chi_c(\bar F_ vari\'et\'eepsilonilon)$. This is the situation in the complex
setting. When $d$ is even and for $ vari\'et\'eepsilonilon \in \{-1, 1\}$ since
$\bar F_ vari\'et\'eepsilonilon$ is a compact manifold with boundary
$Lk(f)$, one knows that
$$\chi_c(\bar F_ vari\'et\'eepsilonilon)=-\chi_c (F_ vari\'et\'eepsilonilon)=\mathbb F_2rac{1}{2}\chi_c(Lk(f)).$$
For general $d\in { \mathbb N}$ and for $ vari\'et\'eepsilonilon \in \{-1, 1, <, >\}$, we thus have
$$ \chi_c(\bar F_ vari\'et\'eepsilonilon)=(-1)^{d+1}\chi_c (F_ vari\'et\'eepsilonilon).$$
On the other hand we recall that for $ vari\'et\'eepsilonilon \in \{ <, >\}$
$$ \chi_c(G_ vari\'et\'eepsilonilon )=\chi_c(\bar F_{\delta_ vari\'et\'eepsilonilon}),$$
where $\delta_>$ is $1$ and $\delta_<$ is $-1$
(see \cite{Ar}, \cite{Wa}).
\end{remark}
\begin{theorem}\label{Milnor}
With notation \ref{Khim}, we have, for $ vari\'et\'eepsilonilon\in \{-1, 1, <, >\}$
$$ \chi_c(S_f^ vari\'et\'eepsilonilon)=\chi_c(\bar F_ vari\'et\'eepsilonilon)=(-1)^{d+1}\chi_c (F_ vari\'et\'eepsilonilon), $$
and for $ vari\'et\'eepsilonilon\in \{<, >\}$
$$ \chi_c(S_f^ vari\'et\'eepsilonilon)=-\chi_c(G_ vari\'et\'eepsilonilon).$$
\end{theorem}
\begin{proof} Assume first that $ vari\'et\'eepsilonilon\in \{-1, 1\}$. We denote
by $\cal F$ the fibre
$\sigma^{-1}(F_ vari\'et\'eepsilonilon)$ and recall that $\cal F$ and $F_ vari\'et\'eepsilonilon$ have the same
topological type.
Let us denote $\bar{\cal K}$ the set of multi-indices
$J\subset \cal I$ such that
$\bar E_J \cap \sigma^{-1}(0)\not=\emptyset$, with
$\bar E_J$ the closure of $E_J=\bigcap_{i\in J} E_i$. In what follows
only $J\in\bar{\cal K}$ are concerned,
since we study the
local Milnor fibre at $0$.
The proof consists in the computation
of the Euler-Poincar\'e characteristic of $\cal F$ using the decomposition of $\cal F$ by the overlapping components $\cal F_I$
introduced just before figure 1 and illustrated on figure 1. We simply count the number of these overlapping components in the decomposition of $\cal F$ they provide.
Note that a connected component of $E^0_J$ (still denoted $E_J^0$ for simplicity in the sequel), for $J\subset \cal J$, is covered by
$n_J:=M_J\cdot 2^{\vert J\vert -1}$ connected components $\cal G$
of $\cal F$, where $M_J$ is
$0$, $1$ or $2$ depending on the fact that the multiplicity $m_J=gcd_{j\in J}(N_j)$ defining $\widetilde E^{0, vari\'et\'eepsilonilon}_J$ is
odd or even, and on sign condition on $c$ (remember from figure 1 how $E^0_J$ is covered by $\cal F_J$. Here the term covered simply means that one can naturally project the component $\cal F_J$ onto $E^0_J$).
Note furthermore that $M_J$ is the degree of the covering $\widetilde E_J^{0, vari\'et\'eepsilonilon}$ of $E_J^0$.
Now expressing a connected component $\cal G$ of $\cal F$ as the union
$\displaystyle\bigcup_{\vert I \vert=1, \cal F_I\subset \cal G} \cal F_I$, where the
(connected) leaves $\cal F_I$
cover (the open subset $E^0_I\cap \cal E_I^0$ of $E^0_I$ homeomorphic to)
$E^0_I$,
and using the additivity of $\chi_c$, one has
that $\chi_c(\cal G)$ is expressed as a sum of characteristics of the overlapping
connected sub-leaves
$\cal F_J$ of the $\cal F_I$'s, each of them with sign coefficient
$s_J:=(-1)^{\vert J \vert -1 }$ .
Note that (a connected component of)
$E_J^0$ is covered by $n_J$ copies of such
a $\cal F_J$, coming from the $n_J$ connected components of $\cal F$ above $E_J^0 \cap \cal E_J^0$,
and that a connected sub-leaf $\cal F_J$ has the topology of
$(E_J^0\cap \cal E_J^0)\times { \mathbb R}^{\vert J\vert -1}$. We denote by $t_J$ the characteristic
$t_J:=\chi_c({ \mathbb R}^{\vert J\vert -1})=(-1)^{\vert J\vert -1}$.
With this notation, summing over all the connected components $\cal G$ of $\cal F$, one
gets
$$\chi_c(\cal F)=\sum_{J\in \bar{\cal K}}
s_J \times n_J \times \chi_c(E^0_J)\times t_J$$
$$=\sum_{J\in \bar{\cal K}}
(-1)^{\vert J \vert -1} \times 2^{\vert J\vert -1}M_J \times \chi_c(E^0_J)
\times (-1)^{\vert J \vert -1}$$
$$=
\sum_{J\in \bar{\cal K}} 2^{\vert J\vert-1}
\chi_c(\widetilde E_J^{0, vari\'et\'eepsilonilon}) $$
$$=\sum_{J\cap\cal K\not=\emptyset} 2^{\vert J\vert-1} \chi_c(\widetilde E_J^{0, vari\'et\'eepsilonilon})
+ \sum_{J\cap\cal K=\emptyset, J\in \bar{\cal K}} 2^{\vert J\vert-1} \chi_c(\widetilde E_J^{0, vari\'et\'eepsilonilon}) $$
$$ = \chi_c(S_f^ vari\'et\'eepsilonilon)
+\sum_{J\cap\cal K=\emptyset,J\in \bar{\cal K}} 2^{\vert J\vert-1}
\chi_c(\widetilde E_J^{0, vari\'et\'eepsilonilon})$$
$$ = \chi_c(S_f^ vari\'et\'eepsilonilon)
+\chi_c(\bigcup_{J\cap\cal K=\emptyset,J\in \bar{\cal K}} \cal F_J). $$
Note that the sub-leaves $\cal F_J$ for $J\cap \cal K=\emptyset$ and $J\in \bar{\cal K}$ cover
the set $\{f\circ \sigma = c\}\cap \hat S(0,\alpha)$, for $ vari\'et\'eepsilonilon\cdot c >0$,
where $\hat S(0,\alpha)$
is a neighbourhood $\sigma^{-1}(S(0,\alpha)\times ]0,\beta[)$ of
$\sigma^{-1}(S(0,\alpha))$, with $0<\beta \ll \alpha$.
It follows that
$$\chi_c(\bigcup_{J\cap\cal K=\emptyset,J\in \bar{\cal K}} \cal F_J)=
\chi_c(F_ vari\'et\'eepsilonilon\cap (S(0,\alpha)\times ]0,\beta[))=\chi_c(Lk(f)\times ]0,\beta[)=
-\chi_c(Lk(f)). $$
We finally obtain
$$ \chi_c(F_ vari\'et\'eepsilonilon)=\chi_c(S_f^ vari\'et\'eepsilonilon)-\chi_c(Lk(f)),$$
and
$$ \chi_c(\bar F_ vari\'et\'eepsilonilon)= \chi_c(F_ vari\'et\'eepsilonilon)+\chi_c(Lk(f))=\chi_c(S_f^ vari\'et\'eepsilonilon).$$
This proves the first equality of our statement, the equality
$\chi_c(\bar F_ vari\'et\'eepsilonilon)=(-1)^{d+1}\chi_c (F_ vari\'et\'eepsilonilon)$ being proved
in Remark \ref{bord}.
Assume now that $ vari\'et\'eepsilonilon \in \{<,>\}$, and denote $\delta_<:=-1$ and $\delta_>:=1$, like
in Remark \ref{bord}.
With this notation
$\bar F_ vari\'et\'eepsilonilon=\bar F_{\delta_ vari\'et\'eepsilonilon}\times ]0,1[$,
and by the formula proved above in the case $ vari\'et\'eepsilonilon\in \{-1,1\}$, we obtain
$$ \chi_c(\bar F_ vari\'et\'eepsilonilon)=\chi_c(\bar F_{\delta_ vari\'et\'eepsilonilon})\chi_c(]0,1[)
=-\chi_c(\bar F_{\delta_ vari\'et\'eepsilonilon})=-\chi_c(S_f^{\delta_ vari\'et\'eepsilonilon})=-
\sum_{J\cap \cal K \not=\emptyset } 2^{\vert J\vert-1}
\chi(\widetilde E_J^{0,\delta_ vari\'et\'eepsilonilon}).$$
But since $\widetilde E_J^{0, vari\'et\'eepsilonilon}=
\widetilde E_J^{0,\delta_ vari\'et\'eepsilonilon}\times { \mathbb R}_+$, it follows that
$$\chi_c( \bar F_ vari\'et\'eepsilonilon)= \sum_{J\cap \cal K \not=\emptyset } 2^{\vert J\vert-1}
\chi(\widetilde E_J^{0,\delta_ vari\'et\'eepsilonilon})\chi_c({ \mathbb R}_+)=
\sum_{J\cap \cal K \not=\emptyset} 2^{\vert J\vert-1} \chi(\widetilde E_J^{0, vari\'et\'eepsilonilon})
=\chi_c(S_f^ vari\'et\'eepsilonilon).$$
This proves the first equality of our statement. The equality
$\chi_c(\bar F_ vari\'et\'eepsilonilon)=(-1)^{d+1}\chi_c (F_ vari\'et\'eepsilonilon)$
is the consequence of $\chi_c(\bar F_ vari\'et\'eepsilonilon)=\chi_c(\bar F_{\delta_ vari\'et\'eepsilonilon})\chi_c(]0,1[)$,
$\chi_c( F_ vari\'et\'eepsilonilon)=\chi_c( F_{\delta_ vari\'et\'eepsilonilon})\chi_c(]0,1[)$
and
$\chi_c(\bar F_{\delta_ vari\'et\'eepsilonilon})=(-1)^{d+1}\chi_c (F_{\delta_ vari\'et\'eepsilonilon})$.
To finish, equality $\chi_c(S_f^ vari\'et\'eepsilonilon)=-\chi_c(G_ vari\'et\'eepsilonilon)$ comes from the
equality $ \chi_c(G_ vari\'et\'eepsilonilon )=\chi_c(\bar F_{\delta_ vari\'et\'eepsilonilon})$
recalled in Remark \ref{bord} and from
$\chi_c(\bar F_ vari\'et\'eepsilonilon)=-\chi_c(\bar F_{\delta_ vari\'et\'eepsilonilon})$,
$\chi_c(S_f^ vari\'et\'eepsilonilon)=\chi_c(\bar F_ vari\'et\'eepsilonilon)$.
\end{proof}
\begin{remark}
As stated in Theorem \ref{Milnor}, the realization via $\chi_c$
of the motivic Milnor fibre $S_f^ vari\'et\'eepsilonilon$, for $ vari\'et\'eepsilonilon\in \{-1,1,<,>\}$, gives the
Euler-Poincar\'e characteristic of the corresponding set theoretic semialgebraic
closed Milnor fibre $\bar F_ vari\'et\'eepsilonilon$.
Nevertheless it is worth noting that this equality is in general not true at the higher
level of
$\chi(K_0[BSA_{ \mathbb R}])$. Even computed in $K_0({\rm Var}_{ \mathbb R})\otimes { \mathbb Z}[\mathbb F_2rac{1}{2}]$, we may have
$S_f^ vari\'et\'eepsilonilon\not=[A_{f, vari\'et\'eepsilonilon}]$, for a given semialgebraic formula
$A_{f, vari\'et\'eepsilonilon}$ with real points $\bar F_ vari\'et\'eepsilonilon$.
Let us illustrate this remark by the following quite trivial example.
\end{remark}
\begin{example}
Let us consider the simple case where $f:{ \mathbb R}^2\to { \mathbb R}$ is given by $f(x,y)=xy$.
After one blowing-up $\sigma: M\to { \mathbb R}^2$ of the origin of ${ \mathbb R}^2$, the situation is as required by Theorem \ref{Zeta function}.
We denote by $E_1$ the exceptional divisor $\sigma^{-1}(0)$ (which is isomorphic to ${ \mathbb P}_1$) and by
$E_2,E_3$ the irreducible components of the strict transform $\sigma^{-1}(\{f=0\})$.
The induced
stratification of $E_1$ is given by $E_{1,2}^0=E_1\cap E_2$, $E_{1,3}^0=E_1\cap E_3$, and
the two connected components ${E}_1^{'0}, {E}_1^{''0}$ of $E_1\setminus (E_2\cup E_3)$.
We consider a chart $(X,Y)$ of $M$ such that $\sigma(X,Y)=(x=Y,y=XY)$. In this chart
$(f\circ \sigma)(X,Y)=XY^2$. The multiplicity of $f\circ \sigma$ along $E_1$ is $N_1=2$,
and the multiplicity of $\jac\sigma$ along $E_1$ is $1$, thus $\nu_1=2$. Assuming that
$ E_1^{'0}$ corresponds to $X>0$ and $ E_1^{''0}$ corresponds to $X<0$,
it follows that
$$\widetilde E^{'0, vari\'et\'eepsilonilon}_1=\{(X,t); X\in E_1^{'0}, t\in { \mathbb R}, Xt^2?_ vari\'et\'eepsilonilon\}
\hbox{ and } \widetilde E^{''0, vari\'et\'eepsilonilon}_1=\{(X,t); X\in E_1^{''0}, t\in { \mathbb R}, Xt^2?_ vari\'et\'eepsilonilon\},$$
where $?_ vari\'et\'eepsilonilon$ is
$=1$, $=-1$, $>$ or $<0$ in case $ vari\'et\'eepsilonilon$ is $1$, $-1$, $>$ or $<$.
In case $ vari\'et\'eepsilonilon=1$ we obtain
$$[\widetilde E^{'0,1}_1]={ \mathbb L}-1 \hbox{ and } [\widetilde E^{''0,1}_1]=0$$
since $\widetilde E^{'0,1}_1$ has a one-to-one projection onto
$\{(X,Y); X=0, Y\not=0\}$) and $\widetilde E^{''0,1}_1$ is
empty.
Now in a neighbourhood of $E^0_{1,2}$, $f\circ \sigma (X,Y)=XY^2$,
giving $N_1=1$, $N_2=2$ and $m=gcd(N_1,N_2)=1$. We also have $\nu_1=2$ and $\nu_2=1$.
It follows that
$$\widetilde E^{0,1}_{1,2}=\{(0,t); t\in { \mathbb R}, t=1\} \hbox{ thus } [\widetilde E^{0,1}_{1,2}]=1.$$
In the same way, using another chart, one finds
$$ \ [\widetilde E^{0,1}_{1,3}]=1.$$
By Theorem \ref{Zeta function} we then have
$$ Z_f^1(T)=({ \mathbb L}-1)^{1-1}({ \mathbb L}-1)\Big(\mathbb F_2rac{{ \mathbb L}^{-2}T^2}{1-{ \mathbb L}^{-2}T^2}\Big)
+
2({ \mathbb L}-1)^{2-1}\Big(\mathbb F_2rac{{ \mathbb L}^{-2}T^2}{1-{ \mathbb L}^{-2}T^2}\Big)
\Big(\mathbb F_2rac{{ \mathbb L}^{-1}T}{1-{ \mathbb L}^{-1}T}\Big),$$
$$Z^1_f(T)=\mathbb F_2rac{{ \mathbb L}-1}{({ \mathbb L} T^{-1}-1)^2} \hbox{ and } S_f^1=-({ \mathbb L}-1).$$
Of course we find that $\chi_c(S_f)=\chi_c(\{ f=c \}\cap \bar B(0,1))=2$,
$0<c\ll 1$.
Now let us for instance choose $\{ xy=c,1-x^2-y^2>0\}$, for $0<c\ll 1$, as a basic semialgebraic formula
to represent the open Milnor fibre of $f=0$ and let us compute
$\beta([xy=c,1-x^2-y^2>0])$ (rather than $[xy=c,1-x^2-y^2>0]$ itself, since we use regular homeomorphims in our computations).
By definition of the realization $\beta:K_0(BSA_{ \mathbb R})\to { \mathbb Z}[\mathbb F_2rac{1}{2}][u]$,
we have
$$\beta([xy=c,1-x^2-y^2>0])=\mathbb F_2rac{1}{4}\beta([xy=c,z^2=1-x^2-y^2])$$
$$
-\mathbb F_2rac{1}{4}\beta([xy=c,z^2=x^2+y^2-1])+
\mathbb F_2rac{1}{2}\beta([xy=c,1-x^2-y^2\not=0]).$$
Projecting the algebraic set $ \{ xy=c,z^2=1-x^2-y^2\}$ orthogonally to the plane $x=-y$ with
coordinates $(X=1/\sqrt 2(x-y),z)$ one finds
twice the quadric $z^2+2X^2=1-2c$ that is, up to regular homeomorphism, two circles. A circle having
class $u+1$, we have
$$\beta([xy=c,z^2=1-x^2-y^2])=2(u+1). $$
Projecting the algebraic set $ \{ xy=c,z^2=x^2+y^2-1\}$ to the plane $x=-y$ with
coordinates $(X=1/\sqrt 2(x-y),z)$ one finds
twice the hyperbola $2X^2-z^2=1-2c$. Projecting orthogonally again the hyperbola onto one of its asymptotic
axes we see that this hyperbola has class $u-1$. It gives
$$\beta([xy=c,z^2=x^2+y^2-1])=2(u-1).$$
Finally the constructible set $\{xy=c,1-x^2-y^2\not=0\}$ being the hyperbola without
$4$ points, we have
$$\beta([xy=c,1-x^2-y^2>0])=\mathbb F_2rac{1}{2}(u+1)-\mathbb F_2rac{1}{2}(u-1)+\mathbb F_2rac{1}{2}(u-1)-2=
\mathbb F_2rac{u-3}{2}. $$
Of course $\chi_c(\chi([xy=c,1-x^2-y^2>0]))=\chi_c(\{f=c\}\cap B(0,1)) =-2$.
The simple semialgebraic formula representing the set theoretic closed Milnor fibre is
$\{xy=c,1-x^2-y^2\ge0\}$, it has class $\displaystyle \beta([xy=c,1-x^2-y^2>0])+4\beta([\{*\}])=\mathbb F_2rac{u+5}{2}$
in $ { \mathbb Z}[\mathbb F_2rac{1}{2}][u]$. But although
$$\chi_c(\chi([xy=c,1-x^2-y^2\ge0]))=\chi_c(S_f^1)=\chi_c(\{f=c\}\cap \bar B(0,1))=2$$
as expected from Theorem \ref{Milnor},
we observe that
$$\mathbb F_2rac{u+5}{2}=\beta([xy=c,1-x^2-y^2\ge0]) \not=\beta(S_f^1)=-(u-1).$$
As a final consequence, we certainly cannot have
this equality between $\chi([xy=c,1-x^2-y^2\ge0])$ and $S_f^1 $ at the level of $K_0({\rm Var}_{ \mathbb R})\otimes { \mathbb Z}[\mathbb F_2rac{1}{2}]$.
\end{example}
\vskip2cm
\end{document} | math | 94,319 |
\begin{document}
\title{Besov Spaces and Frames on Compact Manifolds }
\author{Daryl Geller\\
\footnotesize\texttt{{[email protected]}}\\
Azita Mayeli \\
\footnotesize\texttt{{[email protected]}}
}
\maketitle
\begin{abstract}
We show that one can characterize the Besov
spaces on a smooth compact oriented Riemannian manifold, for the full range of indices, through a knowledge of the size of frame
coefficients, using the smooth, nearly tight frames we have constructed in \cite{gm2}.
\footnotesize{Keywords and phrases: \textit{Wavelets, Frames, Spectral Theory, Besov Spaces,
Manifolds, Pseudodifferential Operators.}}
\end{abstract}
\section{Introduction}
In \cite{gm2}, we have constructed smooth, nearly tight frames on $({\bf M},g)$, a general smooth, compact oriented
Riemannian manifold without boundary.
Our goal in this article is to show that one can characterize the (inhomogeneous) Besov
spaces on ${\bf M}$, for the full range of indices, through a knowledge of the size of frame
coefficients, using the frames we have constructed. (We hope to consider Triebel-Lizorkin spaces in a
forthcoming article.) Our methods, in addition to using the results of \cite{gm2},
are largely adapted from those of Frazier and Jawerth \cite{FJ1},
who gave a similar characterization of Besov spaces on $\RR^n$. However, as we shall explain below,
some new ideas are needed on manifolds.
Let us briefly review our construction of smooth, nearly tight frames on ${\bf M}$.
Say $f^0 \in {\cal S}(\RR^+)$ (the space of restrictions to $\RR^+$ of functions in ${\mathcal S}(\RR)$).
Say $f^0 \not\equiv 0$, and let
\[f(s) = sf^0(s).\]
One then has
the {\em Calder\'on formula}: if $c \in (0,\infty)$ is defined by
\[c = \int_0^{\infty} |f(t)|^2 \frac{dt}{t} = \int_0^{\infty} t|f^0(t)|^2 dt,\]
then for all $s > 0$,
\begin{equation}
\label{cald}
\int_0^{\infty} |f(ts)|^2 \frac{dt}{t} = c < \infty.
\end{equation}
Discretizing (\ref{cald}), if $a \neq 1$ is sufficiently close to $1$,
one obtains a special form of
{\em Daubechies' condition}:
for all $s > 0$,
\begin{equation}
\label{daub}
0 < A_a \leq \sum_{j=-\infty}^{\infty} |f(a^{2j} s)|^2 \leq B_a < \infty,
\end{equation}
where
\begin{equation}
\label{daubest}
A_a = \frac{c}{2|\log a|} \left(1 - O(|(a-1)^2 (\log|a-1|)|\right),\:\:\: B_a=
\frac{c}{2|\log a|}\left(1 + O(|(a-1)^2 (\log|a-1|)|)\right).
\end{equation}
((\ref{daubest}) was proved in \cite{gm1}, Lemma
7.6) In particular, $B_a/A_a$ converges nearly quadratically to $1$ as $a \rightarrow 1$. For example,
Daubechies calculated that if $f(s) = se^{-s}$ and $a=2^{1/3}$, then $B_a/A_a = 1.0000$ to four
significant digits.
Our general program is to construct (smooth, nearly tight) frames, and analogues of continuous wavelets, on
much more general spaces, by replacing the positive number $s$ in (\ref{cald}) and
(\ref{daub}) by a positive self-adjoint operator $T$ on a Hilbert space ${\cal H}$.
If $P$ is the projection onto the null space of $T$, by the spectral theorem
we obtain the relations
\begin{equation}
\label{gelmay1}
\int_0^{\infty} |f|^2(tT) \frac{dt}{t} = c(I-P)
\end{equation}
and
\begin{equation}
\label{gelmay2}
A_a(I-P) \leq \sum_{j=-\infty}^{\infty} |f|^2(a^{2j} T) \leq
B_a(I-P).
\end{equation}
(The integral in (\ref{gelmay1}) and the sum in (\ref{gelmay2}) converge strongly.
In (\ref{gelmay2}), $\sum_{j=-\infty}^{\infty} :=
\lim_{M, N \rightarrow \infty} \sum_{j=-M}^N$, taken in the strong operator
topology.)
(\ref{gelmay1}) and (\ref{gelmay2}) were justified in section 2 of our earlier article
\cite{gmcw}.
In \cite{gmcw} and \cite{gm2}, we looked at the
situation in which $T$ is the Laplace-Beltrami operator $\Delta$ on $L^2({\bf M})$,
We constructed smooth, nearly tight frames
in this context. Here $P$ is the projection onto the
one-dimensional space of constant functions.
We constructed continuous wavelets on ${\bf M}$ in \cite{gmcw}.
To see how frames can be obtained from (\ref{gelmay2}), suppose that, for any $t > 0$,
$K_t$ is the Schwartz kernel of $f(t^2T)$. Thus, if $F \in L^2({\bf M})$,
\begin{equation}
\label{schkerf}
[f(t^2T)F](x) = \int_{\bf M} F(y) K_t(x,y) d\mu(y),
\end{equation}
here $\mu$ is the measure on ${\bf M}$ arising from integration with respect to the
volume form on ${\bf M}$.
Say now that $\int_{\bf M} F = 0$, so that $F = (I-P)F$.
By (\ref{gelmay2}),
\begin{equation}
\label{aasumba0}
A_a \langle F,F \rangle \leq \langle \sum_j |f|^2(a^{2j} T)F, F \rangle \leq B_a \langle F,F \rangle .
\end{equation}
Thus
\begin{equation}
\label{aasumba}
A_a \langle F,F \rangle \leq \sum_j \langle f(a^{2j} T)F, f(a^{2j} T)F \rangle \leq B_a \langle F,F \rangle ,
\end{equation}
so that
\begin{equation}
\label{kerfaj}
A_a \langle F,F \rangle \leq \sum_j \int|\int K_{a^j}(x,y) F(y)d\mu(y)|^2 d\mu(x) \leq B_a \langle F,F \rangle .
\end{equation}
Now, pick $b > 0$, and for each $j$, write ${\bf M}$
as a disjoint union of measurable sets $E_{j,k}$ with diameter at most $ba^j$.
Take $x_{j,k} \in E_{j,k}$.
It is then reasonable to expect that, for any $\epsilon > 0$, if
$b$ is sufficiently small, and if $x_{j,k} \in E_{j,k}$, then
\begin{equation}
\label{kerfajap}
(A_a-\epsilon) \langle F,F \rangle \leq
\sum_j \sum_{k}|\int K_{a^j}(x_{j,k},y) F(y)d\mu(y)|^2 \mu(E_{j,k}) \leq (B_a + \epsilon) \langle F,F \rangle ,
\end{equation}
which means
\begin{equation}
\label{mtvest}
(A_a-\epsilon) \langle F,F \rangle \leq
\sum_j \sum_{k} |(F,\phi_{j,k})|^2 \leq (B_a + \epsilon) \langle F,F \rangle ,
\end{equation}
where
\begin{equation}
\label{phjkdf}
\phi_{j,k}(x)=[\mu(E_{j,k})]^{1/2} \overline{K_{a^j}}(x_{j,k},x).
\end{equation}
In our earlier article \cite{gm2}, we showed that (\ref{mtvest}) indeed holds,
provided the $E_{j,k}$ are also ``not too small'' (precisely, if they satisfy (\ref{measgeq}) directly below).
In fact, in Theorem \ref{framainfr} of that article, we showed (a more general form of) the following result:
\begin{theorem}
\label{framain}
Fix $a >1$, and say $c_0 , \delta_0 > 0$. Suppose $f \in {\mathcal S}(\RR^+)$,
and $f(0) = 0$. Suppose that the Daubechies condition $(\ref{daub})$ holds.
Then there exists a constant $C_0 > 0$ $($depending only on ${\bf M}, f, a, c_0$ and $\delta_0$$)$
as follows:\\
For $t > 0$, let $K_t$ be the kernel of $f(t^2\Delta)$.
Say $0 < b < 1$.
Suppose that, for each $j \in \ZZ$, we can write ${\bf M}$ as a finite disjoint union of measurable sets
$\{E_{j,k}: 1 \leq k \leq N_j\}$,
where:
\begin{equation}
\label{diamleq}
\mbox{the diameter of each } E_{j,k} \mbox{ is less than or equal to } ba^j,
\end{equation}
and where:
\begin{equation}
\label{measgeq}
\mbox{for each } j \mbox{ with } ba^j < \delta_0,\: \mu(E_{j,k}) \geq c_0(ba^j)^n.
\end{equation}
$($In \cite{gm2} we show that such $E_{j,k}$ exist
provided $c_0$ and $\delta_0$ are sufficiently small, independent of the
values of $a$ and $b$.$)$
For $1 \leq k \leq N_j$, define $\phi_{j,k}$ by (\ref{phjkdf}). Then
if $P$ denotes the projection in $L^2({\bf M})$ onto the space of constants, we have
\[(A_a-C_0b) \langle F,F \rangle \leq
\sum_j \sum_{k} |(F,\phi_{j,k})|^2 \leq (B_a + C_0b) \langle F,F \rangle ,\]
for all $F \in (I-P)L^2({\bf M})$. In particular, if $A_a - C_0b > 0$, then
$\left\{\phi_{j,k}\right\}$ is a frame for $(I-P)L^2({\bf M})$,
with frame bounds $A_a - C_0b$ and $B_a + C_0b$.
\end{theorem}
Thus, in these circumstances, if $b$ is sufficiently small,
$\{\phi_{j,k}\}$ is a frame, in fact a smooth, nearly tight frame, since
\[ \frac{B_a + C_0b}{A_a - C_0b} \sim \frac{B_a}{A_a} =
1 + O(|(a-1)^2 (\log|a-1|)|). \]
To justfiy the formal argument leading from (\ref{aasumba0}) to (\ref{phjkdf}),
and to go beyond the $L^2$ theory, one needs the following information about the kernel
$K_t$, which we established in Lemma \ref{manmolcw} of our earlier paper \cite{gmcw} (see also the remark following
the proof of that lemma):
\begin{lemma}
\label{manmol}
Say $f(0) = 0$. Then
for every pair of
$C^{\infty}$ differential operators $X$ $($in $x)$ and $Y$ $($in $y)$ on ${\bf M}$,
and for every integer $N \geq 0$, there exists $C_{N,X,Y}$ as follows. Suppose
$\deg X = j$ and $\deg Y = k$. Then
\begin{equation}
\label{diagest}
t^{n+j+k} \left|\left(\frac{d(x,y)}{t}\right)^N XYK_t(x,y)\right| \leq C_{N,X,Y}
\end{equation}
for all $t > 0$ and all $x,y \in {\bf M}$. (The result holds even without the hypothesis
that $f(0) = 0$, provided we look only at $t \in (0,1]$.)
\end{lemma}
The main results are Theorems \ref{besmain2} and \ref{besmain3} below, whose precise statements can be read now.
To summarize them: fix any $M_0 > 0$.
We study frame expansions for the space $B_{p,0}^{\alpha q}$, consisting
of distributions $F$ in the Besov space $B_{p}^{\alpha q}$ on ${\bf M}$ for which $F1 = 0$.
We assume that the $E_{jk}$ satisfy the conditions of Theorem \ref{framain} above
for $b$ sufficiently small, and also that, if $0 < p < 1$, and if
$ba^j \geq \delta_0$, then $\mu(E_{j,k}) \geq {\cal C}$ (for some
${\cal C} > 0$). (Such sets $E_{j,k}$ are easily constructed.) We assume that $f(s) = s^l f_0(s)$ for some $f_0
\in {\cal S}(\RR^+)$, and for $l$ sufficiently large, depending on the indices $p,q, \alpha$
(so that $f(t^2\Delta) = t^{2l}\Delta^l f_0(t^2\Delta)$).
We let the
$\phi_{j,k}$ be as in Theorem \ref{framain}, and let
$\varphi_{j,k}(x)= \phi_{j,k}(x)/[\mu(E_{j,k})]^{1/2} = \overline{K_{a^j}}(x_{j,k},x)$. We then show that
a distribution $F$, of order at most $M_0$ and satisfying $F1 = 0$, is in $B_{p,0}^{\alpha q}$ if and only if
\[ (\sum_{j = -\infty}^{\infty} a^{-j\alpha q} [\sum_k \mu(E_{j,k})
|\langle F, \varphi_{j,k} \rangle |^p]^{q/p})^{1/q} < \infty;\]
further this
expression furnishes a norm on $B_{p,0}^{\alpha q}$ which is equivalent to the usual norm. Moreover,
if $F \in B_{p,0}^{\alpha q}$, there exist constants $r_{j,k}$ with
\begin{equation}
\label{rjknrmdf}
(\sum_{j = -\infty}^{\infty} a^{-j\alpha q} [\sum_k \mu(E_{j,k})|r_{j,k}|^p]^{q/p})^{1/q} < \infty
\end{equation}
such that
\begin{equation}
\label{besexpF}
F = \sum_{j=-\infty}^{\infty} \sum_k \mu(E_{j,k}) r_{j,k} \varphi_{j,k}
\end{equation}
with convergence in $B_p^{\alpha q}$; and the infimum of the sums in (\ref{rjknrmdf}), taken over
all collections of numbers $\{r_{j,k}\}$ for which (\ref{besexpF}) holds, defines a norm on
$B_{p,0}^{\alpha q}$, which is equivalent to the usual norm.\\
In addition to Lemma \ref{manmol},
our main tools will be the characterization of Besov spaces on $\RR^n$ by Frazier and Jawerth
\cite{FJ1}, and the characterization of these spaces on ${\bf M}$ by Seeger and
Sogge \cite{SS}. Our methods are largely adapted from those of Frazier and Jawerth.
There are, however, at least three major differences:\\
\ \\
1. We need to find replacements, on ${\bf M}$, for the condition
that a function have numerous vanishing moments. Specifically, note that if $g \in C_c^{\infty}(\RR^n)$,
then $\Delta^l g$ has $2l-1$ vanishing moments for any $l \geq 1$, if $\Delta$ is the usual Laplacian.
In order to make effective use of our frames in Besov spaces,
we need an analogue of this on ${\bf M}$. Say $g \in C^{\infty}({\bf M})$; what replacement condition
does $\Delta^l g$ satisfy, if, as usual, $\Delta$ is the Laplace-Beltrami operator on ${\bf M}$?
We will find an effective replacement in Lemma \ref{fjan2} of the next section. These considerations
explain why we need to
use functions $f$ of the form $f(s) = s^l f_0(s)$ for $l$ sufficiently large, depending on the indices.\\
\ \\
2. If one knows appropriate information
about the size of the frame coefficients of a function $F$, then, by adapting the methods of
Frazier-Jawerth and by using the results of Seeger-Sogge, we learn only that
$SF$ (not $F$) is in the desired Besov space, where
\begin{equation}
\label{sover}
SF = \sum_j \sum_k \mu(E_{j,k}) \langle F,\phi_{j,k} \rangle \phi_{j,k};
\end{equation}
another step is then required. Although, for $b$ sufficiently small, $SF$ is an excellent approximation
to a multiple of $F$ (since the $\{\phi_{j,k}\}$ are a nearly tight frame),
it generally does not {\em equal} a multiple of $F$. To conclude that $F$ itself is in the desired Besov space,
we will need to use the theory of pseudodifferential operators (in Theorem \ref{besmain1} below).\\
\ \\
3. We need to show that $S$ is bounded on the Besov spaces, and a technical issue arises when the index $p$
lies between $0$ and $1$. Since the $p$-triangle inequality generally becomes more and more wasteful when one splits
quantities more finely (e.g. if we write $x > 0$ as $\frac{x}{N} + \ldots + \frac{x}{N}$ ($N$ terms), then
we find the wasteful estimate $x^p = (\frac{x}{N} + \ldots + \frac{x}{N})^p \leq N\frac{x^p}{N^p} = N^{1-p}x^p$),
and since we must use fine grids (that is, we must take $b$ to be small),
a rather subtle ``regrouping'' (or ``amalgamation'') argument is
needed at one point (Theorem \ref{sumopbes} below).\\
\ \\
In order for the notation to be fully analogous to that in \cite{FJ1}, we shall need
to adapt the notation that we used in our earlier article \cite{gm2};
through much of this article, we will write $E^j_k = E_{-j,k}$,
$x^j_k = x_{-j,k}$, $\varphi^j_k = \varphi_{-j,k}$. (The notations on the right sides of these
equations were used frequently in \cite{gm2}.)
\subsection{Historical Comments}
Although we are adapting the methods of Frazier and Jawerth \cite{FJ1}, we should note that
they were not working with nearly tight frames, but rather with the $\varphi$-transform. Characterizations
of Besov spaces on $\RR^n$, which are similar to ours,
were obtained by Gr\"ochenig \cite{groch} (see also \cite{fg0}, \cite{fg1}, \cite{fg2}) through use of frames, and by
Meyer \cite{meyer}, through use of bases of orthonormal wavelets.
In \cite{dahsch}, Dahmen and Schneider
used parametric liftings from standard bases on the unit cube
to obtain biorthogonal wavelet bases on manifolds which are the disjoint union of smooth parametric images of
the standard cube. Using these bases, they obtained characterizations of the Besov spaces
$B_p^{\alpha q}({\bf M})$, for $0 < p \leq \infty$, $q \geq 1$, and $\alpha > 0$. Their results
hold on manifolds with less than $C^{\infty}$ regularity (for a range of $\alpha$);
also, they applied their methods to the discretization of elliptic operator equations. We consider
neither of these topics here. However, our methods have the advantage of holding for all ${\bf M}$,
and all $p,q, \alpha$. Our frames have the advantage of being nearly tight, and admitting a space-frequency
analysis. Moreover our results are coordinate-free, in the sense that our frames are constructed
without patching the manifold with charts. We presume that this would lead to greater stability in applications,
if data is moving around the manifold in time, since one does not have to worry about data moving from
chart to chart, although this presumed advantage has not been established.
In \cite{narc2}, Narcowich, Petrushev and Ward obtain a characterization of both Besov and Triebel-Lizorkin
spaces, through the size of frame coefficients, in the special case ${\bf M} = S^n$. As frames they use
the ``needlets'' that they constructed in \cite{narc1}. We discussed the similarities, advantages and
disadvantages of these frames as compared to ours on $S^n$, in section 3 of our earlier article \cite{gm2}.
They proved and used a result similar to our Lemma \ref{manmol}, and our methods (based on adapting the ideas in
\cite{FJ1}) are rather similar to theirs. However, on the sphere, they constructed tight frames, so they did not need to
deal with the issues \#1,2 and 3 above.
Han and Sawyer \cite{hansaw} define Besov spaces on general spaces of homogeneous type, for a range of
indices. In \cite{hanyang}, Han and Yang give a characterization of these spaces using frames which they
construct. These frames cannot be expected to be nearly tight, nor (on ${\bf M}$) have they been shown
to admit a space-frequency analysis. Further, in the very general situation of \cite{hanyang}, there are no
derivatives, so results are obtained there only for smoothness index $\alpha \in (0,1)$.
\section{Integrating Products}
We shall need the following basic facts, from section 3 of \cite{gmcw}, about ${\bf M}$ and its
geodesic distance $d$.
For $x \in {\bf M}$, we let $B(x,r)$ denote the ball $\{y: d(x,y) < r\}$.
\begin{proposition}
\label{ujvj}
Cover ${\bf M}$ with a finite collection of open sets $U_i$ $(1 \leq i \leq I)$,
such that the following properties hold for each $i$:
\begin{itemize}
\item[$(i)$] there exists a chart $(V_i,\phi_i)$ with $\overline{U}_i
\subseteq V_i$; and
\item[$(ii)$] $\phi_i(U_i)$ is a ball in $\RR^n$.
\end{itemize}
Choose $\delta > 0$ so that $3\delta$ is a Lebesgue number for the covering
$\{U_i\}$. Then, there exist $c_1, c_2 > 0$ as follows:\\
For any $x \in {\bf M}$, choose any $U_i \supseteq B(x,3\delta)$. Then, in
the coordinate system on $U_i$ obtained from $\phi_i$,
\begin{equation}
\label{rhoeuccmp2}
d(y,z) \leq c_2|y-z|
\end{equation}
for all $y,z \in U_i$; and
\begin{equation}
\label{rhoeuccmp}
c_1|y-z| \leq d(y,z)
\end{equation}
for all $y,z \in B(x,\delta)$.
\end{proposition}
We fix collections $\{U_i\}$, $\{V_i\}$, $\{\phi_i\}$ and also $\delta$ as in
Proposition \ref{ujvj}, once and for all.
\begin{itemize}
\item Notation as in Proposition \ref{ujvj},
there exist $c_3, c_4 > 0$, such that, whenever $x \in {\bf M}$ and $0 < r \leq \delta$,
\begin{equation}
\label{ballsn}
c_3r^n \leq \mu(B(x,r)) \leq c_4r^n
\end{equation}
and such that, whenever $x \in {\bf M}$ and $r > \delta$,
\begin{equation}
\label{ballsn1}
c_3 \delta^n \leq \mu(B(x,r)) \leq \mu({\bf M}) \leq c_4r^n.
\end{equation}
\item
For any $N > n$ there exists $C_N$ such that, for all $x \in {\bf M}$ and $t > 0$,
\begin{equation}
\label{ptestm}
\int_{\bf M} [1 + d(x,y)/t]^{-N} d\mu(y) \leq C_N t^n.
\end{equation}
\item
For any $N > n$ there exists $C_N^{\prime}$ such that, for all $x \in {\bf M}$ and $t > 0$,
\begin{equation}
\label{ptestm1}
\int_{d(x,y) \geq t} d(x,y)^{-N} d\mu(y) \leq C_N^{\prime} t^{n-N}.
\end{equation}
\end{itemize}
In Lemma 3.3 of \cite{FJ1}, Frazier and Jawerth proved, in essence, the following key
lemma on $\RR^n$,
for which we must find analogues on ${\bf M}$.
\begin{lemma}
\label{fjrn}
Say $L, M$ are integers with $L \geq -1$ and $M \geq L+n+1$. Then there exists $C > 0$ as follows.
Supppose $\varphi_1 \in C(\RR^n)$ and $\varphi_2 \in C^{L+1}(\RR^n)$ satisfy, for some $\sigma, \nu \in \ZZ$ with
$\sigma \geq \nu$,
\[ |\varphi_1(x)| \leq (1+2^{\sigma}|x|)^{-M}, \]
\begin{equation}
\label{momfjrn}
\int x^{\alpha} \varphi_1(x) dx = 0 \mbox{ whenever } |\alpha| \leq L,
\end{equation}
and
\[ |\partial^{\gamma}\varphi_2(x)| \leq (1+2^{\nu}|x|)^{L+n+1-M} \mbox{ whenever } |\gamma| = L+1. \]
Then
\[ |(\varphi_1 * \varphi_2)(x)| \leq C 2^{-\sigma(L+n+1)}(1+2^{\nu}|x|)^{L+n+1-M}. \]
\end{lemma}
To clarify, (\ref{momfjrn}) is the empty condition if $L = -1$.
We will need two different analogues of this lemma. The first is a straightforward adaptation of
Lemma (\ref{fjrn}) and its proof.
\begin{lemma}
\label{fjan1}
Say $L, M$ are integers with $L \geq -1$ and $M \geq L+n+1$. Then there exists $C > 0$ as follows.
Say $\sigma \in \ZZ$, $\nu \in \RR$ with $2^{\sigma} \geq a^{\nu}$.
Say $x_0 \in {\bf M}$. Select one of the charts $U_i$ (as in Proposition \ref{ujvj}) with
$B(x_0,3\delta) \subseteq U_i$. Suppose, that in local coordinates on $U_i$, $Q$
is a dyadic cube of side $2^{-\sigma}$, and $3Q \subseteq B(x_0,\delta)$. (Here $3Q$ is the cube
with the same center as $Q$ and $3$ times the side length $l(Q)$ of $Q$.)
Suppose also that
$\varphi_1 \in C({\bf M})$ satisfies the following conditions:
\[ \supp \varphi_1 \subseteq 3Q; \]
\[ |\varphi_1(y)| \leq 1 \mbox{ for all } y \in {\bf M}; \]
\begin{equation}
\label{momfjm}
\int y^{\alpha} \varphi_1(y) d\mu(y) = 0 \mbox{ whenever } |\alpha| \leq L,
\end{equation}
Also suppose $x_1 \in {\bf M}$, that $\varphi_2 \in C^{L+1}({\bf M})$, and that for all $y \in 3Q$,
\[ |\partial^{\gamma}\varphi_2(y)| \leq (1+a^{\nu}d(y,x_1))^{L+n+1-M} \mbox{ whenever } |\gamma| = L+1. \]
Then,
\[ |\int_{\bf M}(\varphi_1 \varphi_2)(y)d\mu(y)| \leq C 2^{-\sigma(L+n+1)}(1+a^{\nu}d(x_0,x_1))^{L+n+1-M}. \]
\end{lemma}
To clarify, (\ref{momfjm}) is the empty condition if $L = -1$.\\
{\bf Proof} In this proof, $C$ will always denote a constant which depends only on $L$ and $M$ (and $n$, $a$ and
$({\bf M},g)$, of course); it may change from one line to the next.
Surely $|\int_{\bf M}(\varphi_1 \varphi_2)(y)d\mu(y)| = |\int_{3Q}(\varphi_1 \varphi_2)(y)d\mu(y)|$.
We write
\begin{eqnarray*}
|\int_{3Q}(\varphi_1 \varphi_2)(y)d\mu(y)| & = &
|\int_{3Q}\varphi_1(y)[\varphi_2(y) -
\sum_{|\gamma| \leq L} \frac{\partial^{\gamma} \varphi_2(x_0)(y-x_0)^{\gamma}}{\gamma!}] d\mu(y)|\\
& \leq & C(\int_{\{y \in 3Q: d(y,x_0) \leq c_1d(x_0,x_1)/2c_2\}} +
\int_{\{y \in 3Q: d(y,x_0) > c_1d(x_0,x_1)/2c_2\}})|x_0-y|^{L+1} \Phi(y) d\mu(y)\\
& = & I + II,
\end{eqnarray*}
where $c_1$ and $c_2$ are as in Proposition \ref{ujvj}, and
where $\Phi(y) = \sup_{0 < \epsilon < 1} \sum_{|\gamma|=L+1}|\partial^{\gamma}\varphi_2(x_0 + \epsilon(y-x_0))|$.
In I, we have that whenever $0 < \epsilon < 1$, then
\begin{equation}
\label{linsegest}
d(x_0 + \epsilon(y-x_0),x_0) \leq c_2|(x_0 + \epsilon(y-x_0)-x_0| \leq c_2|y-x_0| \leq (c_2/c_1)d(y,x_0) \leq
d(x_0,x_1)/2,
\end{equation}
so that in I, by the hypotheses,
\[ |\Phi(y)| \leq C (1+a^{\nu}d(x_0,x_1))^{L+n+1-M}. \]
In II we just note that
\[ |\Phi(y)| \leq 1. \]
Accordingly
\[ I \leq C (1+a^{\nu}d(x_0,x_1))^{L+n+1-M}\int_{3Q} |x_0-y|^{L+1} d\mu(y) \leq
C 2^{-\sigma(L+n+1)}(1+a^{\nu}d(x_0,x_1))^{L+n+1-M}, \]
while if $a^{\nu}d(x_0,x_1) \leq 1$, we have
\[ II \leq \int_{3Q} |x_0-y|^{L+1}d\mu(y) \leq C 2^{-\sigma(L+n+1)}
\leq C 2^{-\sigma(L+n+1)}(1+a^{\nu}d(x_0,x_1))^{L+n+1-M}. \]
Finally, say $a^{\nu}d(x_0,x_1) > 1$. Note that $1 \leq C[2^{\sigma}|x_0-y|]^{-1}$ for all $y \in 3Q$. Raising this
to the $M$th power, and then using (\ref{ptestm1}) and the assumption that $2^{\sigma} \geq a^{\nu}$, we see that
\begin{eqnarray}
II & \leq & C 2^{-\sigma M}\int_{\{y \in 3Q: d(y,x_0) > c_1 d(x_0,x_1)/2c_2\}}|x_0-y|^{L-M+1}d\mu(y)\nonumber\\
& \leq & C 2^{-\sigma M}\int_{d(y,x_0) > c_1 d(x_0,x_1)/2c_2}d(y,x_0)^{L-M+1}d\mu(y)\label{IIstrt}\\
& \leq & C 2^{-\sigma M} d(x_0,x_1)^{L+n+1-M} \nonumber\\
& = & C 2^{-\sigma(L+n+1)} [2^{\sigma} d(x_0,x_1)]^{L+n+1-M} \nonumber\\
& \leq & C 2^{-\sigma(L+n+1)}(1+a^{\nu}d(x_0,x_1))^{L+n+1-M},\label{IIend}
\end{eqnarray}
as claimed.\\
In our second analogue of Lemma \ref{fjrn}, instead of assuming that $\varphi_1$
is supported in a chart and satisfies familiar moment conditions there, as we did in Lemma \ref{fjan1},
we will instead allow $\varphi_1$ to be supported anywhere in ${\bf M}$. The moment conditions will be
replaced by an assumption that $\varphi = \Delta^l \Phi$ for another well-behaved function $\Phi$.
Formally, in this lemma,
the role of $L$ in Lemma \ref{fjan1} will be played by $2l-1$.
\begin{lemma}
\label{fjan2}
Say $l, M$ are integers with $l \geq 0$ and $M > n$.
Then there exists $C > 0$ as follows.
Say $\sigma, \nu \in \RR$ with $\sigma \geq \nu$.
Say $x_0 \in {\bf M}$, and suppose that
$\varphi_1 = \Delta^l\Phi$, where $\Phi \in C^{2l}({\bf M})$ satisfies:
\[ |\Phi(y)| \leq (1+a^{\sigma}d(y,x_0))^{-M}. \]
Also suppose $x_1 \in {\bf M}$, that $\varphi_2 \in C^{2l}({\bf M})$, and that for all $y \in {\bf M}$,
\[ |\Delta^l \varphi_2(y)| \leq (1+a^{\nu}d(y,x_1))^{n-M}.\]
Then,
\[ |\int_{\bf M}(\varphi_1 \varphi_2)(y)d\mu(y)| \leq C a^{-\sigma n}(1+a^{\nu}d(x_0,x_1))^{n-M}. \]
\end{lemma}
{\bf Proof} In this proof, $C$ will always denote a constant which depends only on $l, M$ and $n$ (and
$a, ({\bf M},g)$, of course); it may change from one line to the next.
First we observe that we may take $l = 0$. Indeed, if this case were known, then we could
prove the general result simply by noting that $\Delta$ is self-adjoint, so that
\[ \int_{\bf M}(\varphi_1 \varphi_2)(y)d\mu(y) = \int_{\bf M}(\Delta^l \Phi \varphi_2)(y)d\mu(y) =
\int_{\bf M}(\Phi \Delta^l \varphi_2)(y)d\mu(y). \]
So we may assume $l=0$.
We have
\[ |\int_{\bf M}(\varphi_1 \varphi_2)(y)d\mu(y)| \leq
I+II, \]
where we are setting
\[ I = \int_{d(y,x_0) \leq d(x_0,x_1)/2}|\varphi_1(y)\varphi_2(y)| d\mu(y), \]
\[ II = \int_{d(y,x_0) > d(x_0,x_1)/2}|\varphi_1(y)\varphi_2(y)| d\mu(y). \]
We shall show that I and II are less than or equal to $C a^{-\sigma n}(1+a^{\nu}d(x_0,x_1))^{n-M}$,
and then we will be done.
For I and II we need to estimate $\varphi_2(y)$. We shall use the evident estimates:
\begin{equation}
\label{estIphi}
\mbox{In I, } |\varphi_2(y)| \leq (1+a^{\nu}d(x_0,x_1))^{n-M},
\end{equation}
and
\begin{equation}
\label{estIIphi}
\mbox{In II, } |\varphi_2(y)| \leq 1.
\end{equation}
From (\ref{estIphi}) and the hypotheses on $\varphi_1 = \Phi$, we find that
\[ I \leq
(1+a^{\nu}d(x_0,x_1))^{n-M}\int_{\bf M} (1+a^{\sigma}d(y,x_0))^{-M}d\mu(y) \leq
Ca^{-\sigma n}(1+a^{\nu}d(x_0,x_1))^{n-M} \]
as needed. (We have used (\ref{ptestm}).)
For II, we have the estimate
\[ II \leq \int_{d(y,x_0) > d(x_0,x_1)/2}(1+a^{\sigma}d(y,x_0))^{-M}d\mu(y)\]
Suppose first that $a^{\nu}d(x_0,x_1) \leq 1$. Then we can just note that, by (\ref{ptestm}),
\[ II \leq C a^{-\sigma n } \leq C a^{-\sigma n}(1+a^{\nu}d(x_0,x_1))^{n-M}. \]
If, instead, $a^{\nu}d(x_0,x_1) > 1$, we find that
\begin{eqnarray*}
II & \leq & C a^{-\sigma M}\int_{d(y,x_0) > d(x_0,x_1)/2}d(y,x_0)^{-M}d\mu(y)\\
& \leq & C a^{-\sigma n} (1+a^{\nu}d(x_0,x_1))^{n-M}
\end{eqnarray*}
as an argument just like the one
beginning with (\ref{IIstrt}) and ending with (\ref{IIend}) shows. This completes the proof.\\
Next we need analogues of Lemma 3.4 of \cite{FJ1}. After one multiplies by certain constants,
that lemma states:
\begin{lemma}
\label{fjrn2}
If $Q$ is a dyadic cube of $\RR^n$, let $x_Q$ denote its center.
Let $1 \leq p \leq \infty$, and suppose $\sigma, \eta \in \ZZ$, $\eta \leq \sigma$. Suppose $F(x) =
\sum_{l(Q) = 2^{-\sigma}} s_Q f_Q(x)$, where
\[ |f_Q(x)| \leq 2^{-\sigma n}(1+2^{\eta}|x-x_Q|)^{-n-1}. \]
Then
\[ \|F\|_{L^p} \leq C 2^{-\eta n} (\sum_{l(Q) = 2^{-\sigma}} 2^{-\sigma n}|s_Q|^p)^{1/p}. \]
\end{lemma}
Here is our first analogue.
\begin{lemma}
\label{fjan3}
Let $1 \leq p \leq \infty$, and as usual fix $a > 1$. Also fix $b > 0$.
Then there exists $C > 0$ as follows.
Suppose $\eta \in \RR$, $j \in \ZZ$, $\eta \leq j$.
Write ${\bf M}$ as a finite disjoint union of
measurable subsets $\{E^j_k: 1 \leq k \leq {\cal N}_j\}$, each of diameter less than $ba^{-j}$.
For each $k$ with $1 \leq k \leq {\cal N}_j$, select any $x^j_k \in E^j_k$.
Suppose that, for $x \in {\bf M}$, $F(x) =
\sum_{k=1}^{{\cal N}_j} s_{j,k} f_{j,k}(x)$, where
\[ |f_{j,k}(x)| \leq \mu(E^j_k)(1+a^{\eta}d(x,x^j_k))^{-n-1}. \]
Then
\[ \|F\|_{L^p} \leq C a^{-\eta n} (\sum_{k} \mu(E^j_k)|s_{j,k}|^p)^{1/p}. \]
\end{lemma}
{\bf Proof}
We have
\begin{equation}
\label{sjkamu}
\|F\|_p \leq (\int_{\bf M}
[\sum_{k=1}^{{\cal N}_j} |s_{j,k}| \mu(E^j_k)(1+a^{\eta}d(x,x^j_k))^{-n-1}]^p d\mu(x))^{1/p}
\end{equation}
Let $Y_j$ be the finite measure space $\{1,\ldots,{\cal N}_j\}$ with measure $\lambda$
where $\lambda(\{k\}) = \mu(E^j_k)$ for each $k \in Y_j$. and define
${\cal K}: L^p(Y_j) \rightarrow L^p({\bf M})$ by ${\cal K} r(x) = \int_{Y_j} K(x,k)r(k) d\lambda(k)$ for $r \in L^p(Y_j)$,
where
\[ K(x,k) = (1+a^{\eta}d(x,x^j_k))^{-n-1}. \]
By standard arguments, $\|{\cal K} r\|_p \leq M\|r\|_p$, where
$M$ is any number satisfying
\begin{equation}
\label{mkxkmu}
M \geq \int_{\bf M} K(x,k) d\mu(x)
\end{equation}
for all $k \in Y_j$ and also
\begin{equation}
\label{mkxkla}
M \geq \int_{Y_j} K(x,k) d\lambda(k)
\end{equation}
for all $x \in {\bf M}$.
By (\ref{sjkamu}), $\|F\|_p \leq \|{\cal K}r\|_p$ if $r(k) = |s_{j,k}|$.
Thus we need only show that we may take $M = C a^{-\eta n}$. But, for this $M$, (\ref{mkxkmu})
holds by (\ref{ptestm}). As for (\ref{mkxkla}), choose $B > \max(2b,1)$. If $x \in {\bf M}$, we have
\begin{eqnarray}
\int_{Y_j} |K(x,k)| d\lambda(k) & \leq & C\sum_{k} \mu(E^j_k)(B+a^{\eta}d(x,x^j_k))^{-n-1} \label{m2stt}\\
& \leq & C \sum_{k} \int_{E^j_k}(B+a^{\eta}d(x,y))^{-n-1} d\mu(y) \nonumber \\
& \leq & C\int_{\bf M}(1+a^{\eta}d(x,y))^{-n-1} d\mu(y) \nonumber \\
& \leq & Ca^{-\eta n} \label{m2end}
\end{eqnarray}
as desired. Here we have used the assumption that $\eta \leq j$, the fact that the diameter
of $E^j_k$ is at most $ba^{-j}$, and (\ref{ptestm}). This completes the proof.\\
Our second analogue deals only with $L^p$ norms of functions defined on finite sets.
\begin{lemma}
\label{fjan4}
Let $1 \leq p \leq \infty$, and as usual fix $a > 1$. Also fix $b > 0$.
Then there exists $C > 0$ as follows.
Say $j \in \ZZ$. Select sets $E^j_k$ and points $x^j_k$ as in Lemma \ref{fjan3}. Let $Y_j$
again be the finite measure space $\{1,\ldots,{\cal N}_j\}$ with measure $\lambda$
where $\lambda(\{k\}) = \mu(E^j_k)$ for each $k \in Y_j$.
Suppose $U_i$ is one of the charts of Proposition \ref{ujvj}. Say $B(x_0,3\delta) \subseteq U_i$.
Say $\sigma \in \ZZ$, and let
${\cal Q}_{\sigma}$ denote the set of dyadic cubes of length
$2^{-\sigma}$ in $U_i$ which are contained in $B(x_0,\delta)$. (We are using local coordinates on
$U_i$.) \\
(a) Say $2^{\sigma} \geq a^{j}$. Suppose that, for $k \in Y_j$, $F(k) =
\sum_{Q \in {\cal Q}_{\sigma}} s_Q f_{Q}(k)$, where
\[ |f_Q(k)| \leq 2^{-\sigma n}(1+a^{j}d(x_Q,x^j_k))^{-n-1}. \]
Then
\[ \|F\|_{L^p(Y_j)} \leq C a^{-jn}(\sum_{l(Q) = 2^{-\sigma}} 2^{-\sigma n}|s_{Q}|^p)^{1/p}. \]
(b) Suppose instead $2^{\sigma} \leq a^{j}$. Suppose that, for $k \in Y_j$, $F(k) =
\sum_{Q \in {\cal Q}_{\sigma}} s_Q f_{Q}(k)$, where
\[ |f_Q(k)| \leq 2^{-\sigma n}(1+2^{\sigma}d(x_Q,x^j_k))^{-n-1}. \]
Then
\[ \|F\|_{L^p(Y_j)} \leq C 2^{-\sigma n}(\sum_{l(Q) = 2^{-\sigma}} 2^{-\sigma n}|s_{Q}|^p)^{1/p}. \]
\end{lemma}
{\bf Proof} Say ${\cal Q}_{\sigma} = \{Q_1,\ldots,Q_{I(\sigma)}\}$, and let
$X_{\sigma} = \{1,\ldots,I(\sigma)\}$, with measure $\tau$, where $\tau(m) = 2^{-\sigma n}$
for each $m \in X_{\sigma}$.
In either (a) or (b), we define an operator
${\cal K}: L^p(X_{\sigma}) \rightarrow L^p(Y_j)$ by
${\cal K} r(k) = \int_{X_{\sigma}} K(k,m)r(m) d\tau(m)$ for $r \in L^p(X_{\sigma})$,
where in (a),
\begin{equation}
\label{kkma}
K(k,m) = (1+a^{j}d(x_{Q_m},x^j_k))^{-n-1},
\end{equation}
while in (b)
\begin{equation}
\label{kkmb}
K(k,m) = (1+2^{\sigma}(x_{Q_m},x^j_k))^{-n-1}.
\end{equation}
With these definitions of $K(k,m)$, in either (a) or (b),
$\|F\|_p \leq \|{\cal K}r\|_p$ if $r(m) = |s_{Q_m}|$.
On the other hand, $\|{\cal K} r\|_p \leq M\|r\|_p$,
where $M$ is any number satisfying
\begin{equation}
\label{mkkmla}
M \geq \int_{Y_j} K(k,m) d\lambda(k)
\end{equation}
for all $m \in X_{\sigma}$ and also
\begin{equation}
\label{mkkmtau}
M \geq \int_{X_{\sigma}} K(k,m) d\tau(m)
\end{equation}
for all $k \in Y_j$.
In (a), where $K(k,m)$ is given by (\ref{kkma}), we need only show that we may take
$M = C a^{-jn}$. The fact that (\ref{mkkmla}) then holds follows just as in the argument starting with
(\ref{m2stt}) and ending with (\ref{m2end}).
Similarly, for (\ref{mkkmtau}), say $k \in Y_j$. Since $Q_m \subseteq B(x_0,\delta)$, the diameter of
$Q_m$ is at most $c2^{-\sigma}$ for some $c$. Choose $B > \max(2c,1)$. Then
\begin{eqnarray*}
\int_{X_{\sigma}} K(k,m) d\tau(m) & \leq & C\sum_{m} 2^{-\sigma n}(B+a^{j}d(x_{Q_m},x^j_k))^{-n-1}\\
& \leq & C \sum_{m} \int_{Q_m}(B+a^{j}d(x,x^j_k))^{-n-1} d\mu(y)\\
& \leq & C B \int_{\bf M}(1+a^{j}d(x,x^j_k))^{-n-1} d\mu(y) \\
& \leq & C a^{-jn}
\end{eqnarray*}
as desired. Here we have used the assumption that $2^{-\sigma} \leq a^{-j}$, the fact that the diameter
of $Q_m$ is at most $c2^{-\sigma}$, and (\ref{ptestm}).
In (b), where $K(k,m)$ is given by (\ref{kkmb}),
we need only show that we may take $M = C 2^{-\sigma n}$.
This, however, follows in a similar manner to (a), if one now uses the assumption that
$a^{-j} \leq 2^{-\sigma}$. This completes the proof.
\section{Besov spaces}
We will need the following simple fact about operators on $l^q(\NN)$, for $0 < q \leq 1$,
which is again adapted from arguments in \cite{FJ1}.
\begin{proposition}
\label{lqconv}
Suppose $0 < q \leq \infty$.
Say $K: \NT \times \NT \rightarrow \RR$ is nonnegative. If $z$ is a nonnegative
sequence, define the nonnegative sequence ${\cal K}z$ by
\[ ({\cal K}z)(r) = \sum_{s=0}^{\infty} K(r,s) z(s). \]
Let $M_q$ be a number satisfying
\[ M_q \geq [\sum_{s=0}^{\infty} K(r,s)^q]^{1/q} \]
for all $r$, and also
\[ M_q \geq [\sum_{r=0}^{\infty} K(r,s)^q]^{1/q} \]
for all $s$.
Then:\\
(a) If $1 \leq q \leq \infty$, then
for every nonnegative sequence $z$, $\|{\cal K}z\|_q \leq M_1\|z\|_q$.\\
(b) If $0 < q < 1$, then
for every nonnegative sequence $z$, $\|{\cal K}z\|_q \leq M_q\|z\|_q$.\\
(Here, $\|z\|_q$ denotes the $l^q(\NT)$ ``norm'' of $z$, which could be $\infty$;
and all nonnegative numbers here are allowed to be $\infty$. Also, here and elsewhere,
we follow the usual rules for interpreting the expressions when $q=\infty$.)
\end{proposition}
{\bf Proof} (a) is of course well known. For (ii),
note that, by the $q$-triangle inequality,
\[ ({\cal K}z)(r)^q \leq \sum_{s=0}^{\infty} K(r,s)^q z(s)^q. \]
By the known case $q=1$ of the proposition, we now see that
\[ \|({\cal K}z)^q\|_1 \leq M_q^q \|z^q\|_1. \]
Raising both sides to the $1/q$ power, we obtain the desired result.\\
For the rest of this section, we fix $a > 1$. We also fix $\alpha, p, q$ with
$-\infty < \alpha < \infty$ and $0 < p,q \leq \infty$. \\
We use the notation for inhomogeneous Besov spaces $B_p^{\alpha q}$ on $\RR^n$ from \cite{FJ1}.
Thus, on $\RR^n$, one takes
any $\Phi \in {\cal S}$ supported in the closed unit ball, which does not vanish anywhere in the ball of
radius $5/6$ centered at $0$. One also takes functions $\varphi_{\nu} \in {\cal S}$ for $\nu \geq 1$,
supported in the annulus $\{\xi: 2^{\nu-1} \leq |\xi| \leq 2^{\nu+1}\}$, satisfying
$|\varphi_{\nu}(\xi)| \geq c > 0$ for $3/5 \leq 2^{-\nu}|\xi| \leq 5/3$ and also
$|\partial^{\gamma} \varphi_{\nu}| \leq c_{\gamma} 2^{-\nu \gamma}$ for every multiindex $\gamma$.
The Besov space $B_p^{\alpha q}(\RR^n)$ is then the space of $F \in {\cal S}'(\RR^n)$ such that
\[ \|F\|_{B_p^{\alpha q}} = \|\check{\Phi}*F\|_{L^p} +
\left(\sum_{\nu = 0}^{\infty} (2^{\nu \alpha}\|\check{\varphi}_{\nu}*F\|_{L^p})^q \right)^{1/q} < \infty. \]
(Here we use the usual conventions if $p$ or $q$ is $\infty$.
The definition of $B_p^{\alpha q}(\RR^n)$ is independent of the choices of
$\Phi, \varphi_{\nu}$ (\cite{Peet}, page 49). Moreover, $B_p^{\alpha q}(\RR^n)$ is a quasi-Banach
space, and the inclusion $B_p^{\alpha q} \subseteq {\cal S}'$ is continuous
(\cite{Trieb}, page 48). In particular the space $B^{\alpha}_{\infty,\infty}(\RR^n) = {\cal C}^{\alpha}(\RR^n)$,
which is the usual H\"older space if $0 < \alpha < 1$, or in general a H\"older-Zygmund space for $\alpha > 0$
(\cite{Trieb}, page 51). It is not hard to see, by using the definition and the Fourier transform, that
if $K \subseteq \RR^n$ is compact, and if $N$ is sufficiently large, then
\begin{equation}
\label{ccNbes}
\{F \in C^N: \mbox{ supp}F \subseteq K\} \subseteq B_p^{\alpha q}
\end{equation}
where the inclusion map is continuous if we regard the left side as a subspace of $C^N$.
Pseudodifferential operators of order $0$ are bounded on the Besov spaces (\cite{P}); in particular,
if $\psi \in C_c^{\infty}(\RR^n)$, the mapping $F \rightarrow \psi F$ is a bounded map on
$B_p^{\alpha q}(\RR^n)$. Moreover, if $\eta: \RR^n \rightarrow \RR^n$ is a diffeomorphism which equals
the identity outside a compact set, then one can define $F \circ \eta$ for $F \in B_p^{\alpha q}(\RR^n)$,
and the map $F \rightarrow F \circ \eta$ is bounded on the Besov spaces (\cite{Trieb}, chapter 2.10).
These facts then enable one
to define $B_p^{\alpha q}({\bf M})$: let $(W_i, \chi_i)$ be a finite atlas on ${\bf M}$ with charts $\chi_i$
mapping $W_i$ into the unit ball on $\RR^n$, and suppose $\{\zeta_i\}$ is a partition of unity subordinate
to the $W_i$. Then one defines $B_p^{\alpha q}({\bf M})$ to be the space of distributions $F$ on
${\bf M}$ for which
\[ \|F\|_{B_p^{\alpha q}({\bf M})} = \sum_i \|(\zeta_i F) \circ \chi_i^{-1}\|_{B_p^{\alpha q}({\bf R}^n)} < \infty. \]
This definition does not depend on the choice of charts or partition of unity (\cite{Triebman}).\\
It will be convenient to fix a spanning set of the differential operators on ${\bf M}$ of degree less
than or equal to $J$ (for any fixed $J$).
Recall that we have already fixed a finite set ${\mathcal P}$ of real $C^{\infty}$ vector fields on ${\bf M}$,
whose elements span the tangent space at each point. For any integer $L \geq 1$, we let
\begin{equation}
\label{pmdfopdf}
{\mathcal P}^J = \{X_1\ldots X_M: X_1,\ldots,X_M \in {\mathcal P}, 1 \leq M \leq J\} \cup
\{\mbox{the identity map}\}.
\end{equation}
(In particular, ${\mathcal P}^1$ is what we have previously called ${\mathcal P}_0$.) \\
\begin{lemma}
\label{besov1}
Fix $b > 0$. Also fix
an integer $l \geq 1$ with
\begin{equation}
\label{lgqgint}
2l > \max(n(1/p-1)_+ - \alpha, \alpha).
\end{equation}
where here $x_+ = \max(x,0)$. Fix $M$ with $(M-2l-n)p > n+1$ if $0 < p < 1$, $M-2l-n > n+1$
otherwise.
Then there exists $C > 0$ as follows.
Say $j \in \ZZ$. Select sets $E^j_k$ and points $x^j_k$ as in Lemma \ref{fjan3}.
Suppose that, for each $j \geq 0$, and each $k$,
\begin{equation}
\label{var2lph}
\varphi^j_k = (a^{-2j}\Delta)^l\Phi^j_k,
\end{equation}
where $\Phi^j_k \in C^{\infty}({\bf M})$ satisfies the following conditions:
\begin{equation}
\label{xphip}
|X\Phi^j_k(y)| \leq a^{j (\deg X + n)}(1+a^{j}d(y,x^j_k))^{-M} \mbox{ whenever } X \in {\mathcal P}^{4l}.
\end{equation}
Then, for every $F$ in the inhomogeneous Besov space $B_p^{\alpha q}({\bf M})$, if we let
\[ s_{j,k} = \langle F, \varphi^j_k \rangle, \]
then
\begin{equation}
\label{besov1way}
(\sum_{j = 0}^{\infty} a^{j\alpha q} [\sum_k \mu(E^j_k)|s_{j,k}|^p]^{q/p})^{1/q} \leq C\|F\|_{B_p^{\alpha q}}.
\end{equation}
\end{lemma}
{\bf Proof}
Cover ${\bf M}$ by a finite collection of open sets $\{W_r\}$, where each $W_r$ has the form
$B(x_r,\delta)$ for some $x_r \in {\bf M}$. Let $\{\zeta_r\}$ be a partition of unity subordinate to
the $\{W_r\}$. Let $Z_r = \supp \zeta_r \subseteq W_r$.
Then \cite{Trieb} the map $F \rightarrow \zeta_r F$ is continuous from $B_p^{\alpha q}$ to
itself. Without loss, we may therefore assume that $\supp F \subseteq Z$ where $Z$ is a compact subset
of $W=B(x_0,\delta)$ for some $x_0$. Choose a chart $U_i$, as in Proposition \ref{ujvj}, with
$B(x_0,3\delta) \subseteq U_i$. Select $\zeta \in C_c^{\infty}(W)$ with $\zeta \equiv 1$
in a neighborhood of $Z$.
In local coordinates on $U_i$, $U_i$ is a ball in $\RR^n$. By \cite{FJ1}
(changing their notation slightly by multiplying by certain constants) we may write
\[ F = \sum_{m \in \ZZ^n} s_m b_m + \sum_{\sigma=0}^{\infty} \sum_{l(Q)=2^{-\sigma}} s_Q a_Q \]
with convergence in $B_p^{\alpha q}$. Here, if $Q_{0m}$ is the dyadic cube of side $1$ with ''lower left corner'' $m$,
\[ \mbox{supp} b_m \subseteq 3Q_{0m}, \]
\[ |\partial ^{\gamma} b_m| \leq 1 \mbox{ if } |\gamma| \leq 2l, \]
\[ \mbox{supp} a_Q \subseteq 3Q, \]
\[ |\partial ^{\gamma} a_Q| \leq |Q|^{-|\gamma|/n} \mbox{ if } |\gamma| \leq 2l, \]
\[ \int x^{\gamma} a_Q(x)dx = 0 \mbox{ if } |\gamma| \leq 2l-1,\]
and finally
\[ (\sum_{m \in \ZZ^n} |s_m|^p)^{1/p} +
[\sum_{\sigma=0}^{\infty} 2^{\sigma \alpha q}[\sum_{l(Q)=2^{-\sigma}} 2^{-\sigma n}|s_Q|^p]^{q/p}]^{1/q} \leq C\|F\|_{B_p^{\alpha q}}. \]
Since $F = \zeta F$, we have
\begin{equation}
\label{fsmsq}
F = \sum_{m \in \ZZ^n} s_m \zeta b_m + \sum_{\sigma=0}^{\infty} \sum_{l(Q)=2^{-\sigma}} s_Q \zeta a_Q
\end{equation}
with convergence in $B_p^{\alpha q}$, hence in ${\cal E}'({\bf M})$.
Now, the (Euclidean) distance from supp$\zeta$ to $W^c$ is positive.
Thus there exists $\sigma_0$ with the property
that if $\sigma \geq \sigma_0$, $l(Q) = 2^{-\sigma}$ and $3Q \cap \mbox{supp}\zeta \neq \oslash$, then
supp$\zeta + 3Q \subseteq W$. (Note that $\sigma_0$ does not depend on $F$.)
Accordingly, if $l(Q) \leq 2^{-\sigma_0}$, then either $\zeta a_Q \equiv 0$ or $3Q \subseteq W$.
Moreover, only finitely many cubes $3Q$ with $2^{-\sigma_0} < l(Q) \leq 1$ intersect the compact set supp$\zeta$;
let ${\cal Q}_0$ denote the collection of such cubes. Thus we may write
\begin{equation}
\label{ff0sq}
\zeta F = F_0 + \sum_{\sigma=\sigma_0}^{\infty} \sum_{l(Q)=2^{-\sigma},3Q \subseteq W} s_Q \zeta a_Q
\end{equation}
where
\begin{equation}
\label{foqodf}
F_0 = \sum_{m \in \ZZ^n, Q_{0m} \in {\cal Q}_0} s_m \zeta b_m +
\sum_{\sigma=0}^{\sigma_0-1} \sum_{l(Q)=2^{-\sigma}, Q \in {\cal Q}_0} s_Q \zeta a_Q
\end{equation}
Let $c = \log_2 a$. Then, since the series in (\ref{fsmsq}) converges to $F$ in ${\cal E}'({\bf M})$,
\[ s_{j,k} = \langle F_0,\varphi^j_k \rangle +
\sum_{\sigma_0 \leq \sigma \leq jc}\: \sum_{l(Q)=2^{-\sigma},3Q \subseteq W} s_Q \langle \zeta a_Q, \varphi^j_k \rangle
+ \sum_{\sigma > jc}\: \sum_{l(Q)=2^{-\sigma},3Q \subseteq W} s_Q \langle \zeta a_Q, \varphi^j_k \rangle. \]
For each $j \geq 0$,
let $Y_j$ be the finite measure space $\{1,\ldots,{\cal N}_j\}$ with measure $\lambda$
where $\lambda(\{k\}) = \mu(E^j_k)$ for each $k \in Y_j$.
For each $j \geq 0$, define $s_j: Y_j \rightarrow \CC$ by $s_j(k) = s_{j,k}$. Also, for each $Q$ with
$3Q \subseteq W$, define $u^Q_j: Y_j \rightarrow \CC$ by
$u^Q_j(k) = \langle \zeta a_Q, \varphi^j_k \rangle$. Finally, define $u^0_j: Y_j \rightarrow \CC$ by
$u^0_j(k) = \langle F_0,\varphi^j_k \rangle$. Then
\begin{equation}
\label{sjsumal}
s_j = u^0_j +
\sum_{\sigma_0 \leq \sigma \leq jc}\: \sum_{l(Q)=2^{-\sigma},3Q \subseteq W} s_Q u^Q_j
+ \sum_{\sigma > jc}\: \sum_{l(Q)=2^{-\sigma},3Q \subseteq W} s_Q u^Q_j.
\end{equation}
Define $h$ on $U_i$ by $d\mu = h dx$ there. Note that if $3Q \subseteq W$ then $\int x^{\gamma} (a_Q/h) d\mu = 0$
if $|\gamma| \leq 2l-1$.
Now say that $\sigma > jc$, i.e., that $2^{\sigma} \geq a^{j}$. Then by Lemma \ref{fjan1}
(replacing ``$\nu$'' in that lemma by $j$, and with
$a_Q/h = \varphi_1$ and $h\zeta\varphi^j_k = a^{j(2l+n)}\varphi_2$), then
\begin{equation}
\label{uqjest1}
|u^Q_j(k)| = |\langle \zeta a_Q, \varphi^j_k \rangle| = |\langle a_Q/h , h\zeta\varphi^j_k \rangle|
\leq Ca^{j(2l+n)} 2^{-\sigma(2l+n)}(1+a^{j}d(x_Q,x^j_k))^{2l+n-M}.
\end{equation}
Say instead $\sigma_0 \leq \sigma \leq jc$, so that $2^{\nu} < a^{j}$. Then by Lemma \ref{fjan2}
(replacing ``$\nu$'' in that lemma by
$\sigma/c$, replacing ``$\sigma$'' in that lemma by $j$,
and with $\Phi^j_k = a^{jn}\Phi$, $\varphi^j_k = a^{jn-2jl}\varphi_1$, and $\zeta a_Q = 2^{2\sigma l}\varphi_2$)
we have
\begin{equation}
\label{uqjest2}
|u^Q_j(k)| = |\langle \zeta a_Q, \varphi^j_k \rangle| \leq C2^{2\sigma l} a^{-2jl}(1+2^{\sigma}d(x_Q,x^j_k))^{n-M}.
\end{equation}
Moreover, if $Q$ is one of the finitely many cubes in ${\cal Q}_0$, then again by Lemma \ref{fjan2}
(taking ``$\nu$'' in that lemma to be $0$, replacing ``$\sigma$'' in that lemma by $j$,
and with $\Phi^j_k = a^{jnl}\Phi$, $\varphi^j_k = a^{jn-2jl}\varphi_1$,
and $C\zeta b_m = \varphi_2$ if $Q = Q_{0m}$ has side length $1$, or $C\zeta a_Q = \varphi_2$ otherwise) we have
\begin{equation}
\label{uqjest3}
|u^0_j(k)| \leq C{\cal T} a^{-2jl}(1+d(x_Q,x^j_k))^{n-M},
\end{equation}
where
\[ {\cal T} =
\sum_{m \in \ZZ^n, Q_{0m} \in {\cal Q}_0} |s_m| +
\sum_{\sigma=0}^{\sigma_0-1} \sum_{l(Q)=2^{-\sigma}, Q \in {\cal Q}_0} |s_Q|. \]
Say now $p \geq 1$, and let $\|\:\|_p$ denote $L^p(Y_j)$ norm. From (\ref{sjsumal}) we obtain
\[ \|s_j\|_p \leq \|u^0_j\|_p +
\sum_{\sigma_0 \leq \sigma \leq jc}\: \|\sum_{l(Q)=2^{-\sigma},3Q \subseteq W} s_Q u^Q_j\|_p
+ \sum_{\sigma > jc}\: \|\sum_{l(Q)=2^{-\sigma},3Q \subseteq W} s_Q u^Q_j\|_p. \]
Let
\[ A_j = \|s_j\|_p = [\sum_{k=1}^{{\cal N}_j} \mu(E^j_k)|s_{j,k}|^p]^{1/p},\:\:
B_{\sigma} = [\sum_{l(Q)=2^{-\sigma},3Q \subseteq W} 2^{-\sigma n}|s_Q|^p]^{1/p}. \]
Then by (\ref{uqjest1}), (\ref{uqjest2}), (\ref{uqjest3}) and Lemma \ref{fjan4}, we see that
\begin{equation}
\label{ajbnuest}
A_j \leq C({\cal T}a^{-2jl} + \sum_{\sigma_0 \leq \sigma \leq jc} a^{-2jl} 2^{2\sigma l}B_{\sigma}
+ \sum_{\sigma > jc} a^{2jl} 2^{-2\sigma l}B_{\sigma}).
\end{equation}
(In using Lemma \ref{fjan4}, we have noted that $3Q \subseteq W \Rightarrow Q \subseteq W$.)
Now also write
\[ A^{\alpha}_j = a^{j\alpha} A_j,\:\: B^{\alpha}_{\sigma} = 2^{\sigma \alpha} B_{\sigma}. \]
Then, by (\ref{ajbnuest}),
\begin{equation}
\label{ajbnalest}
A_j^{\alpha} \leq C({\cal T} a^{-j(2l-\alpha)}+ \sum_{\sigma=\sigma_0}^{\infty} K(j,\sigma) B^{\alpha}_{\sigma}),
\end{equation}
where
\[ K(j,\sigma) = a^{-j(2l-\alpha)}2^{\sigma(2l-\alpha)} \mbox{ if } \sigma_0 \leq \sigma \leq jc, \]
\[ K(j,\sigma) = a^{j(2l+\alpha)}2^{-\sigma(2l+\alpha)} \mbox{ if } \sigma_0 \geq jc. \]
By (\ref{lgqgint}), $2l$ is more than $\max(\alpha,-\alpha)$.
Recall also that $c = \log_2 a$. Thus, by Proposition \ref{lqconv},
$\|(A_j^{\alpha})\|_q \leq C({\cal T}+\|(B_{\sigma}^{\alpha})\|_q)$. Consequently
\begin{equation}
\label{bescmp1}
(\sum_{j = 0}^{\infty} a^{j\alpha q} [\sum_k \mu(E^j_k)|s_{j,k}|^p]^{q/p})^{1/q}
\leq C[(\sum_{m \in \ZZ^n} |s_m|^p)^{1/p} +
(\sum_{\sigma=0}^{\infty} 2^{\sigma \alpha q}[\sum_{l(Q)=2^{-\sigma}} 2^{-\sigma n}|s_Q|^p]^{q/p})^{1/q}]
\leq C\|F\|_{B_p^{\alpha q}}
\end{equation}
as desired, at least if $p \geq 1$.
If instead $0 < p < 1$, we evaluate each side of (\ref{sjsumal}) at $k$ (for $1 \leq k \leq {\cal N}_j$), and
use the $p$-triangle inequality, to obtain
\begin{equation}
\label{sjsumalp}
|s_j(k)|^p \leq |u^0_j(k)|^p +
\sum_{\sigma_0 \leq \sigma \leq jc}\: \sum_{l(Q)=2^{-\sigma},3Q \subseteq W} |s_Q|^p |u^Q_j(k)|^p
+ \sum_{\sigma > jc}\: \sum_{l(Q)=2^{-\sigma},3Q \subseteq W} |s_Q|^p |u^Q_j(k)|^p.
\end{equation}
Let $A_j$, $B_{\sigma}$ be as above.
Integrating both sides of (\ref{sjsumalp}) over $Y_j$, using (\ref{uqjest1}), (\ref{uqjest2}), (\ref{uqjest3}), and using
Lemma \ref{fjan4} (taking ``$p$'' in that lemma to be $1$), we find
\[ A_j^p \leq C({\cal T}^p a^{-2jlp} + \sum_{\sigma_0 \leq \sigma \leq jc} a^{-2jlp} 2^{2\sigma lp}B_{\sigma}^p
+ \sum_{\sigma > jc} a^{j(2l+n)p-jn} 2^{-\sigma(2l+n)p+\sigma n}B_{\sigma}^p). \]
Let $A^{\alpha}_j$, $B^{\alpha}_{\sigma}$ be as above. We find that
\begin{equation}
\label{ajbnalpest}
(A_j^{\alpha})^p \leq C({\cal T}^p a^{-j(2l-\alpha)p}+ \sum_{\sigma=\sigma_0}^{\infty} K(j,\sigma) (B^{\alpha}_{\sigma})^p),
\end{equation}
where now
\[ K(j,\sigma) = a^{-j(2l-\alpha)p}2^{\sigma(2l-\alpha)p} \mbox{ if } \sigma_0 \leq \sigma \leq jc, \]
\[ K(j,\sigma) = a^{j[(2l+n+\alpha)p -n]}2^{-\sigma[(2l+n+\alpha)p -n]} \mbox{ if } \sigma_0 \geq jc. \]
By (\ref{lgqgint}), $2l$ is more than $\max(\alpha,n/p-n-\alpha)$.
Recall also that $c = \log_2 a$. Thus, by Proposition \ref{lqconv},
$\|(A_j^{\alpha})^p\|_{q/p} \leq C({\cal T}^q+\|(B_{\sigma}^{\alpha})^p\|_{q/p})$.
Upon raising both sides to the $1/q$ power, one again obtains (\ref{bescmp1}), as desired.\\
\ \\
{\bf Remark} Presumably the assumptions in the last lemma can be weakened: assuming only
$2m > n(1/p-1)_+ - \alpha$, we conjecture that one should be able to replace the assumption (\ref{var2lph})
by the assumption that $\varphi^j_k = (a^{-2j}\Delta)^m\Phi^j_k$, where $\Phi^j_k$ satisfies
(\ref{xphip}) for all $X \in {\mathcal P}^{2(l+m)}$ (not ${\mathcal P}^{4l}$).
\begin{lemma}
\label{besov2}
Fix $b > 0$. Also
fix an integer $l \geq 1$ with
\begin{equation}
\label{lgqgintac}
2l > n(1/p-1)_+ - \alpha.
\end{equation}
where here $x_+ = \max(x,0)$. Fix $M$ with $(M-n)p > n+1$ if $0 < p < 1$, $M-n > n+1$
otherwise.
If $0 < p < 1$, we also fix a number $\rho > 0$.
Then there exists $C > 0$ as follows.
Say $j \in \ZZ$. Select sets $E^j_k$ and points $x^j_k$ as in Lemma \ref{fjan3}.
If $0 < p < 1$, we assume that, for all $j,k$,
\begin{equation}
\label{ejkro}
\mu(E^j_k) \geq \rho a^{-jn}
\end{equation}
Suppose that, for each $j \geq 0$, and each $k$,
$\varphi^j_k = (a^{-2j}\Delta)^l\Phi^j_k$, where $\Phi^j_k \in C^{\infty}({\bf M})$ satisfies the following conditions:
\[ |X\Phi^j_k(y)| \leq a^{j (\deg X + n)}(1+a^{j}d(y,x^j_k))^{-M} \mbox{ whenever } X \in {\mathcal P}^{4l}. \]
Suppose that $\{s_{j,k}: j \geq 0, 1 \leq k \leq {\cal N}_j\}$ satisfies
\[ (\sum_{j = 0}^{\infty} a^{j\alpha q} [\sum_k \mu(E^j_k)|s_{j,k}|^p]^{q/p})^{1/q} < \infty. \]
Then $\sum_{j=0}^{\infty} \sum_k \mu(E^j_k) s_{j,k} \varphi^j_k$ converges in $B_p^{\alpha q}({\bf M})$, and
\begin{equation}
\label{besov2way}
\|\sum_{j=0}^{\infty} \sum_k \mu(E^j_k) s_{j,k} \varphi^j_k\|_{B_p^{\alpha q}} \leq
C(\sum_{j = 0}^{\infty} a^{j\alpha q} [\sum_k \mu(E^j_k)|s_{j,k}|^p]^{q/p})^{1/q}.
\end{equation}
\end{lemma}
{\bf Proof} In \cite{SS}, Seeger and Sogge gave an equivalent characterization of $B_p^{\alpha q}$.
(We change their notation a little; what we shall call $\beta_{k-1}(s^2)$, they called $\beta_k(s)$.)
Choose $\beta_0 \in C_c^{\infty}((1/4,16))$, with the property that
for any $s > 0$, $\sum_{\nu=-\infty}^{\infty} \beta^2_0(2^{-2\nu}s) = 1$. For $\nu \geq 1$, define
$\beta_{\nu} \in C_c^{\infty}((2^{2\nu-2},2^{2\nu+4}))$, by $\beta_{\nu}(s) = \beta_0(2^{-2\nu}s)$.
Also, for $s > 0$, define
the smooth function $\beta_{-1}(s)$ by
$\beta_{-1}(s) = \sum_{\nu=-\infty}^{-1} \beta(2^{-2\nu}s)$. (Note that $\beta_{-1}(s) = 0$
for $s > 4$.)
Then (\cite{SS}), for $F \in C^{\infty}({\bf M})$, $\|F\|_{B_p^{\alpha q}}$ is equivalent to
the $l^q$ norm of the sequence $\{ 2^{\nu\alpha}\|\beta_{\nu}(\Delta)F\|_p: -1 \leq \nu \leq \infty\}$.
For $\nu \geq -1$, let $J_{\nu}$ be the kernel of $\beta_{\nu}(\Delta)$. Using the eigenfunction
expansion of $\beta_{\nu}(\Delta)$ (see (\ref{kerexp}) of \cite{gmcw}), one
sees at once that $J_{-1}(x,y)$ is smooth in $(x,y)$. Moreover, for $\nu \geq 0$, $\beta_{\nu} \in {\cal S}(\RR^+)$
and $\beta_{\nu}(0) = 0$; so $J_{\nu}$ is smooth as well.
For any integer $I \geq 0$, we set
$\beta_0^I(s) = \beta_0(s)/s^I$, and define $\beta_{\nu}^I$ by
$\beta_{\nu}^I(s) = \beta_0^I(2^{-2\nu}s)$. Then $\beta_{\nu}^I(s) = 2^{2\nu I}\beta_{\nu}(s)/s^I$, so that
$\beta_{\nu}(\Delta) = 2^{-2\nu I} \beta_{\nu}^I(\Delta)\Delta^I $. Thus, if $J_{\nu}^I$ is the kernel of $\beta_{\nu}^I(\Delta)$,
we have
\begin{equation}
\label{jjdyi}
J_{\nu}(x,y) = 2^{-2{\nu}I}\Delta_y^I J_{\nu}^I(x,y),
\end{equation}
where $\Delta_y$ means $\Delta$ as applied in the $y$ variable. Also, by Lemma \ref{manmol},
since $\beta_{\nu}^I(\Delta) = \beta_0^I(2^{-2\nu}\Delta)$, we know the following:
for every pair of
$C^{\infty}$ differential operators $X$ (in $x$) and $Y$ (in $y$) on ${\bf M}$,
and for every integer $N \geq 0$, and for any fixed $I$, there exists $C$ such that for all $\nu$,
\begin{equation}
\label{jliest}
|XY J_{\nu}^I(x,y)| \leq C 2^{\nu(n+\deg X + \deg Y)}(1+2^\nu d(x,y))^{-N}
\end{equation}
for all $x,y \in {\bf M}$.
In proving (\ref{besov2way}), we may assume that for all but finitely many $j$, all $s_{j,k} = 0$. For if we can
prove the inequality (\ref{besov2way}) in that case, it will follow at once that the partial sums of
$\sum_{j=0}^{\infty} \sum_k \mu(E^j_k) s_{j,k} \varphi^j_k$ form a Cauchy sequence in the quasi-Banach space
$B_p^{\alpha q}({\bf M})$. Thus the series will converge in that quasi-Banach space, and moreover the inequality
(\ref{besov2way}) will hold in full generality. In fact (\ref{besov2way}) shows that the convergence is unconditional.
With this assumption, we may let $F = \sum_{j=0}^{\infty} \sum_k \mu(E^j_k) s_{j,k} \varphi^j_k$.
Let $c = \log_a 2$. For $\nu \geq -1$, we have
\begin{equation}
\label{beftot}
\beta_{\nu}(\Delta)F =
\sum_{0 \leq j \leq \nu c}\: \sum_k s_{j,k} \mu(E^j_k)\beta_{\nu}(\Delta)\varphi^j_k
+ \sum_{j > \nu c}\: \sum_k s_{j,k} \mu(E^j_k)\beta_{\nu}(\Delta)\varphi^j_k.
\end{equation}
Of course, in each term, $[\beta_{\nu}(\Delta)\varphi^j_k](z) = \int J_{\nu}(x,y) \varphi^j_k(y) d\mu(y)$.
Suppose that $x \in {\bf M}$.
Now say that $j > \nu c$, i.e., that $a^j \geq 2^{\nu}$. Then by Lemma \ref{fjan2}
(replacing ``$\sigma$'' in that lemma by $j$, ``$x_0$'' by $x^j_k$, replacing
``$\nu$'' in that lemma by $\nu c$,
and with $\Phi^j_k = a^{jn} \Phi$, $\varphi^j_k = a^{jn-2jl}\varphi_1$, and
$J_{\nu}(x,y) = 2^{\nu (n+2l)}\varphi_2(y)$, then
\begin{equation}
\label{bdfest1}
|\mu(E^j_k)\beta_{\nu}(\Delta)\varphi^j_k(x)|
\leq Ca^{-2jl} 2^{\nu(n+2l)}\mu(E^j_k)(1+2^{\nu}d(x,x^j_k))^{n-M}.
\end{equation}
Say instead $0 \leq j \leq \nu c$, so that $a^j \leq 2^{\nu}$. Select $I$ with
\[ 2I > \max(\alpha,2l). \]
Then by Lemma \ref{fjan2}
(replacing ``$\sigma$'' in that lemma by
$\nu c$, replacing ``$\nu$'' in that lemma by $j$,
and with $J^I_{\nu}(x,y) = 2^{\nu n}\Phi(y)$, $J_{\nu}(x,y) = 2^{\nu n -2\nu I}\varphi_1(y)$,
and $\varphi^j_k = a^{j(n+2l)}\varphi_2$)
we have
\begin{equation}
\label{bdfest2}
|\mu(E^j_k)\beta_{\nu}(\Delta)\varphi^j_k(x)|
\leq Ca^{j(n+2I)} 2^{-2\nu I}\mu(E^j_k)(1+a^j d(x,x^j_k))^{n-M},
\end{equation}
since $a^{2jl} \leq a^{2jI}$.
Say now $p \geq 1$. From (\ref{beftot}) we obtain
\[ \|\beta_{\nu}(\Delta)F\|_p \leq
\sum_{0 \leq j \leq \nu c}\: \|\sum_k s_{j,k} \mu(E^j_k)\beta_{\nu}(\Delta)\varphi^j_k\|_p
+ \sum_{j > \nu c}\: \|\sum_k s_{j,k} \mu(E^j_k)\beta_{\nu}(\Delta)\varphi^j_k\|_p. \]
Let
\[ A_{\nu} = \|\beta_{\nu}(\Delta)F\|_p,\:\:
B_{j} = [\sum_k \mu(E^j_k)|s_{j,k}|^p]^{1/p}. \]
Then by (\ref{bdfest1}), (\ref{bdfest2}), and Lemma \ref{fjan3}, we see that
\begin{equation}
\label{ajbnuest3}
A_{\nu} \leq C(\sum_{0 \leq j \leq \nu c} a^{2jI} 2^{-2\nu I}B_{j}
+ \sum_{j > \nu c} a^{-2jl} 2^{2\nu l}B_{j}).
\end{equation}
Now also write
\[ A^{\alpha}_{\nu} = 2^{\nu \alpha} A_{\nu},\:\: B^{\alpha}_{j} = a^{j \alpha} B_{j}. \]
Then, by (\ref{ajbnuest3}),
\begin{equation}
\label{ajbnalest3}
A_{\nu}^{\alpha} \leq C(\sum_{j=0}^{\infty} K(\nu,j) B^{\alpha}_{j}),
\end{equation}
where
\[ K(\nu,j) = 2^{-\nu(2I-\alpha)}a^{j(2I-\alpha)} \mbox{ if } 0 \leq j \leq \nu c, \]
\[ K(\nu,j) = 2^{\nu(2l+\alpha)}a^{-j(2l+\alpha)} \mbox{ if } j > \nu c. \]
By (\ref{lgqgintac}), $2l$ is more than $-\alpha$, and we have also taken $I$ to satisfy $2I > \alpha$.
Recall also that $c = \log_a 2$. Thus, by Proposition \ref{lqconv},
$\|(A_{\nu}^{\alpha})\|_q \leq C\|(B_{j}^{\alpha})\|_q$. Consequently the $l^q$ norm of the
sequence $\{2^{\nu \alpha} \|\beta_{\nu}(\Delta)F\|_p\}$ is less than or equal to
$C(\sum_{j = 0}^{\infty} a^{j\alpha q} [\sum_k \mu(E^j_k)|s_{j,k}|^p]^{q/p})^{1/q}$, which, because
of the result of Seeger-Sogge, gives the lemma, at least if $p \geq 1$.
If instead $0 < p < 1$, we evaluate each side of (\ref{beftot}) at $x$ (for each $x \in {\bf M}$), and
use the $p$-triangle inequality, to obtain
\[ |\beta_{\nu}(\Delta)F(x)|^p \leq
\sum_{0 \leq j \leq \nu c}\: \sum_k |s_{j,k}|^p \mu(E^j_k)^p|\beta_{\nu}(\Delta)\varphi^j_k(x)|^p
+ \sum_{j > \nu c}\: \sum_k |s_{j,k}|^p \mu(E^j_k)^p|\beta_{\nu}(\Delta)\varphi^j_k(x)|^p \]
so, by the assumption (\ref{ejkro}),
\begin{align}
&|\beta_{\nu}(\Delta)F(x)|^p \nonumber \\
&\leq C (\sum_{0 \leq j \leq \nu c}\: \sum_k |s_{j,k}|^p a^{jn(1-p)}\mu(E^j_k)|\beta_{\nu}(\Delta)\varphi^j_k(x)|^p
+ \sum_{j > \nu c}\: \sum_k |s_{j,k}|^p a^{jn(1-p)}\mu(E^j_k)|\beta_{\nu}(\Delta)\varphi^j_k(x)|^p) \label{beftotp}
\end{align}
Let $A_{\nu}$, $B_{j}$ be as above.
Integrating both sides of (\ref{beftotp}) over ${\bf M}$, using (\ref{bdfest1}), (\ref{bdfest2}), and Lemma \ref{fjan3}
(taking ``$p$'' in that lemma to be $1$), we find
\begin{equation}
\label{anupbjp}
A_{\nu}^p \leq C(\sum_{0 \leq j \leq \nu c} a^{2jIp} 2^{-2\nu Ip}B_{j}^p
+ \sum_{j > \nu c} a^{-j(2l+n)p+jn} 2^{\nu(2l+n)p-\nu n}B_{j}^p).
\end{equation}
Let $A^{\alpha}_{\nu}$, $B^{\alpha}_{j}$ be as above. We find that
\begin{equation}
\label{ajbnalpest2}
(A_{\nu}^{\alpha})^p \leq C(\sum_{j=0}^{\infty} K(\nu,j) (B^{\alpha}_{j})^p),
\end{equation}
where now
\[ K(\nu,j) = 2^{-\nu(2I-\alpha)p}a^{j(2I-\alpha)p} \mbox{ if } 0 \leq \sigma \leq \nu c, \]
\[ K(\nu,j) = 2^{\nu[(2l+n+\alpha)p -n]}a^{-j[(2l+n+\alpha)p -n]} \mbox{ if } j \geq \nu c. \]
By (\ref{lgqgintac}), $2l$ is more than $n/p-n-\alpha$; also $2I > \alpha$.
Recall also that $c = \log_2 a$. Thus, by Proposition \ref{lqconv},
$\|(A_{\nu}^{\alpha})^p\|_{q/p} \leq C(\|(B_{j}^{\alpha})^p\|_{q/p})$.
Upon raising both sides to the $1/q$ power, one again obtains (\ref{besov2way}), as desired.\\
For any $x \in {\bf M}$, and any integer $I, J \geq 1$, we let
\begin{equation}
\label{mxdefL}
{\mathcal M}_{x,t}^{IJ} = \{\varphi \in C_0^{\infty}({\bf M}):
t^{n+\deg Y} |(\frac{d(x,y)}{t})^N Y\varphi(y)|\leq 1
\mbox{ whenever } y \in {\bf M}, \: 0 \leq N \leq J \mbox{ and }
Y \in {\mathcal P}^I\}.
\end{equation}
(This space is a variant of
a space of molecules, as defined earlier in \cite{gil} and \cite{han2}.
In the notation of our earlier article \cite{gm2}, ${\mathcal M}_{x,t}^{n+2,1} = {\mathcal M}_{x,t}$.)
Note that, if for each $x \in {\bf M}$, we
define the functions $\varphi^t_x, \psi^t_x$ on ${\bf M}$ by
$\varphi^t_x(y) = K_t(x,y)$ and $\psi^t_x(y) = K_t(y,x)$ (notation as in
Lemma \ref{manmol}), then by Lemma \ref{manmol}, for each
$I, J \geq 1$,
there exists $C_{IJ} > 0$ such that $\varphi^t_x$ and $\psi^t_x$ are in $C_{IJ}{\mathcal M}_{x,t}^{IJ}$
{\em for all} $x \in {\bf M}$ and all $t > 0$.\\
We recall Theorem \ref{sumopthmfr} of our earlier article \cite{gm2}:
\begin{theorem}
\label{sumopthm}
Fix $a>1$. Then there exists $C_1, C_2 > 0$ as follows.
For each $j \in \ZZ$, write ${\bf M}$ as a finite disjoint union of
measurable subsets $\{E^j_k: 1 \leq k \leq {\cal N}_j\}$, each of diameter less than $a^{-j}$.
For each $j,k$, select any $x^j_k \in E^j_k$, and select $\varphi^j_k$,
$\psi^j_k$ with \\$\varphi^j_k , \psi^j_k\in \mathcal{M}^{n+2,1}_{x^j_k, a^{-j}}$.
For $F \in C^1({\bf M})$, we claim that we may define
\begin{equation}
\label{sumopdf2}
SF = S_{\{\varphi^j_k\},\{\psi^j_k\}}F =
\sum_{j}\sum_k \mu(E^j_k) \langle F, \varphi^j_k \rangle \psi^j_k.
\end{equation}
$($Here, and in similar equations below, the sum in $k$ runs from $k = 1$ to $k = {\cal N}_j$.$)$
Indeed:
\begin{itemize}
\item[$(a)$] For any $F\in C^1({\bf M})$, the series defining $SF$ converges absolutely, uniformly on $\bf{M}$,
\item[$(b)$] $\parallel SF\parallel_2\leq C_2\parallel F\parallel_2$ for all
$F\in C^1(\bf{M})$.\\
Consequently, $S$ extends to be a bounded operator on $L^2({\bf M})$, with norm less than
or equal to $C_2$.
\item[$(c)$] If $F \in L^2({\bf M})$, then
\begin{equation}
\label{uncndl2}
SF = \sum_{j}\sum_k \mu(E^j_k) \langle F, \varphi^j_k \rangle \psi^j_k
\end{equation}
where the series converges unconditionally.
\item[$(d)$] If $F,G \in L^2({\bf M})$, then
\begin{equation}\label{eq:wk}
\langle SF,G \rangle =
\sum_{j}\sum_{k}\mu(E^j_k) \langle F, \varphi^j_k \rangle \langle \psi^j_k,G \rangle ,
\end{equation}
where the series converges absolutely.\\
\end{itemize}
\end{theorem}
This result, which was proved by use of the $T(1)$ theorem, explains the $L^2$ theory of the summation
operator $S$. On Besov spaces, we have the following result for summation operators, where now we consider the sum over
nonnegative $j$:
\begin{theorem}
\label{sumopbes}
Fix an integer $l \geq 1$ with $2l > \max(n(1/p-1)_+ - \alpha, \alpha)$. Also fix $J$
with $(J-2l-n)p > n+1$ if $0 < p < 1$, $J-2l-n > n+1$ if $p \geq 1$.
Then there exists $C > 0$ as follows.
For each integer $j \geq 0$, write ${\bf M}$ as a finite disjoint union of
measurable subsets $\{E^j_k: 1 \leq k \leq {\cal N}_j\}$, each of diameter less than $a^{-j}$,
select $x^j_k \in E^j_k$, select
$\Phi^j_k, \Psi^j_k \in \mathcal{M}_{x^j_k, a^{-j}}^{4l,J}$ and set
$\varphi^j_k = (a^{-2j}\Delta)^l\Phi^j_k$, $\psi^j_k = (a^{-2j}\Delta)^l\Psi^j_k$.
By Theorem \ref{sumopthm}, we may then define
\begin{equation}
\label{sumopdf4}
S^{\prime}F = S^{\prime}_{\{\varphi^j_k\},\{\psi^j_k\}}F =
\sum_{j=0}^{\infty}\sum_k \mu(E^j_k) \langle F, \varphi^j_k \rangle \psi^j_k,
\end{equation}
at first for $F \in C^1({\bf M})$, and Theorem \ref{sumopthm} applies.\\
Then, if $F \in B_p^{\alpha q}$, the series in (\ref{sumopdf4}) converges in $B_p^{\alpha q}$, to
a distribution $S'F \in B_p^{\alpha q}$, such that $(S'F)(1) = 0$. Moreover, $\|S'F\|_{B_p^{\alpha q}}
\leq C\|F\|_{B_p^{\alpha q}}$.
\end{theorem}
{\bf Proof} Setting $s_{j,k} = \langle F, \varphi^j_k \rangle$, we see by Lemmas \ref{besov1} and
\ref{besov2} that the series in (\ref{sumopdf4}) converges to a distribution $S'F$ in $B_p^{\alpha q}$,
and that
\begin{equation}
\label{sumbes1}
\|S'F\|_{B_p^{\alpha q}} = \|\sum_{j=0}^{\infty}\sum_k \mu(E^j_k) \langle F, \varphi^j_k \rangle \psi^j_k\|_{B_p^{\alpha q}}
\leq C(\sum_{j = 0}^{\infty} a^{j\alpha q}
[\sum_k \mu(E^j_k)|\langle F, \varphi^j_k \rangle|^p]^{q/p})^{1/q} \leq C\|F\|_{B_p^{\alpha q}},
\end{equation}
{\em provided} that $p \geq 1$, or alternatively that $0 < p < 1$ and (\ref{ejkro}) holds. If $0 < p < 1$ and
(\ref{ejkro}) does not hold, we at least know that the second inequality in (\ref{sumbes1}) holds. For the
first inequality, we need to regroup.
Say then that $0 < p < 1$.
By the discussion before Theorem \ref{framainfr} of \cite{gm2}, for each $j \geq 0$,
there exists a finite covering of ${\bf M}$ by
disjoint measurable sets ${\cal F}^j_1,\ldots,{\cal F}^j_{L(j)}$, such that whenever $1 \leq i \leq L(j)$, there
is a $y^j_i \in {\bf M}$ with $B(y^j_i,2a^{-j}) \subseteq {\cal F}^j_i \subseteq B(y^j_i,4a^{-j})$.
Fix $j$ for now. For each $k$ with $1 \leq k \leq {\cal N}_j$, select a number $i_{j,k}$ with $1 \leq i_{j,k} \leq L(j)$,
such that ${\cal F}^j_{i_{j,k}} \cap E^j_k \neq \oslash$. For $1 \leq i \leq L(j)$, let $S_{j,i} = \{k: i_{j,k} = i\}$,
and then let
\[ {\cal E}^j_i = \cup_{k \in S_{j,i}} E^j_k. \]
Then we have that
\[ B(y^j_i,a^{-j}) \subseteq {\cal E}^j_i \subseteq B(y^j_i,5a^{-j}). \]
The second inclusion here is evident from the facts that, firstly, ${\cal F}^j_i \subseteq B(y^j_i,4a^{-j})$,
secondly, that ${\cal F}^j_{i} \cap E^j_k \neq \oslash$ whenever $k \in S_{j,i}$, and thirdly, that each
$E^j_k$ has diameter less than $a^{-j}$. For the first inclusion, say $x \in B(y^j_i,a^{-j})$; we need
to show that $x \in {\cal E}^j_i$. Choose $k$ with $x \in E^j_k$. Since $E^j_k$ has diameter less than $a^{-j}$,
surely $E^j_k \subseteq B(y^j_i,2a^{-j}) \subseteq {\cal F}^j_i$. Thus ${\cal F}^j_i$ is the only
one of the sets ${\cal F}^j_1,\ldots,{\cal F}^j_{L(j)}$ which $E^j_k$ intersects, and so $k$ must be in
$S_{j,i}$. Accordingly $x \in E^j_k \subseteq {\cal E}^j_i$, as claimed.
For each $j, i$ let
\[ r^{j,i} = \max_{k \in S_{j,i}} |\langle F, \varphi^j_k \rangle|. \]
Then, for some $k \in S_{j,i}$, $r^{j,i} = |\langle F, \varphi^j_k \rangle|$.
Now $d(x^j_k, y^j_i) \leq 5a^{-j}$, so it is evident from (\ref{mxdefL}) that, for some
absolute constant $C_0$, $\mathcal{M}_{x^j_k, a^{-j}}^{4l,J} \subseteq C_0\mathcal{M}_{y^j_i, a^{-j}}^{4l,J}$;
in particular, $\Phi^j_k \in C_0\mathcal{M}_{y^j_i, a^{-j}}^{4l,J}$.
Also, $\mbox{diam}{\cal E}^j_i \leq 10a^{-j}$. Thus, by Lemma \ref{besov1},
\begin{equation}
\label{rjifbpq}
(\sum_{j = 0}^{\infty} a^{j\alpha q} [\sum_i \mu({\cal E}^j_i)|r^{j,i}|^p]^{q/p})^{1/q} \leq C\|F\|_{B_p^{\alpha q}}.
\end{equation}
Let ${\bf F}$ be any finite subset of $\{(j,k): j \geq 0, 1 \leq k \leq {\cal N}_j\}$. Define
$s_{j,k} = \langle F, \varphi^j_k \rangle$ if $(j,k) \in {\bf F}$; otherwise, let $s_{j,k} = 0$.
Also, for each $j,i$ with $1 \leq i \leq L(j)$, let
\[ s^{j,i} = \max_{k \in S_{j,i}} |s_{j,k}|. \]
Then only finitely many of the $s^{j,i}$ are nonzero, and we always have $0 \leq s^{j,i} \leq r^{j,i}$.
Therefore by (\ref{rjifbpq}),
in order to show the convergence of the series for $S'F$ in $B_p^{\alpha q}$, and that
$\|S'F\|_{B_p^{\alpha q}} \leq C\|F\|_{B_p^{\alpha q}}$, it is sufficient to show that
\begin{equation}
\label{sjksjiok}
\|\sum_{j=0}^{\infty}\sum_k s_{j,k} \mu(E^j_k) \psi^j_k\|_{B_p^{\alpha q}} \leq
C(\sum_{j = 0}^{\infty} a^{j\alpha q} [\sum_i \mu({\cal E}^j_i)|s^{j,i}|^p]^{q/p})^{1/q},
\end{equation}
where here $C$ is independent of ${\bf F}$ (and our choices of $E^j_k, x^j_k, \Phi^j_k,
\Psi^j_k, {\cal E}^j_i$).
Let $G = \sum_{j=0}^{\infty}\sum_k s_{j,k} \mu(E^j_k) \psi^j_k$.
Let $\beta_{\nu}$ and $c$ be as in the proof of Lemma \ref{besov2}.
We have from (\ref{beftot}) that, for each $x \in {\bf M}$,
\begin{equation}
\label{beftotabs}
|\beta_{\nu}(\Delta)G(x)| \leq
\sum_{0 \leq j \leq \nu c}\: \sum_{i=1}^{L(j)}\sum_{k \in S_{j,i}} \mu(E^j_k)|s_{j,k}| |\beta_{\nu}(\Delta)\psi^j_k(x)|
+ \sum_{j > \nu c}\: \sum_{i=1}^{L(j)}\sum_{k \in S_{j,i}}\mu(E^j_k)|s_{j,k}| \beta_{\nu}(\Delta)\psi^j_k(x)|.
\end{equation}
For each $j, i$, and each $x \in {\bf M}$, let
\[ H^{j,i}(x) = \max_{k \in S_{j,i}}|\beta_{\nu}(\Delta)\psi^j_k(x)|. \]
From (\ref{beftotabs}), we see that
\begin{equation}
\label{beftotamlg}
|\beta_{\nu}(\Delta)G(x)| \leq
\sum_{0 \leq j \leq \nu c}\: \sum_{i=1}^{L(j)} s^{j,i} \mu({\cal E}^j_i)H^{j,i}(x)
+ \sum_{j > \nu c}\: \sum_{i=1}^{L(j)}s^{j,i} \mu({\cal E}^j_i)H^{j,i}(x).
\end{equation}
Note once again that if $k \in S_{j,i}$, then $d(x^j_k, y^j_i) \leq 5a^{-j}$; and therefore,
$\Psi^j_k \in C_0\mathcal{M}_{y^j_i, a^{-j}}^{4l,J}$ for some absolute constant $C_0$. Note also that
$H^{j,i}(x) = |\beta_{\nu}(\Delta)\psi^j_k(x)|$ for some $k \in S_{j,i}$. Thus, the reasoning leading to
(\ref{bdfest1}) and (\ref{bdfest2}) shows that if $j > \nu c$, then
\begin{equation}
\label{bdfest3}
\mu({\cal E}^j_i) H^{j,i}(x)
\leq Ca^{-2jl} 2^{\nu(n+2l)}\mu({\cal E}^j_i)(1+2^{\nu}d(x,y^j_i))^{n-J},
\end{equation}
while if we select $I$ with $2I > \max(\alpha, 2l)$, and if $0 \leq j \leq \nu c$, then
\begin{equation}
\label{bdfest4}
\mu({\cal E}^j_i) H^{j,i}(x)
\leq Ca^{j(n+2I)} 2^{-\nu(2I)}\mu({\cal E}^j_i)(1+a^j d(x,y^j_i))^{n-J}.
\end{equation}
Since $B(y^j_i,a^{-j}) \subseteq {\cal E}^j_i$ for all $j, i$, there exists $\rho > 0$ such that
$\mu({\cal E}^j_i) \geq \rho a^{-jn}$ for all $j \geq 0$. Consequently, the reasoning leading to (\ref{beftotp})
shows that
\[ |\beta_{\nu}(\Delta)G(x)|^p
\leq C (\sum_{0 \leq j \leq \nu c}\: \sum_i |s^{j,i}|^p a^{jn(1-p)}\mu({\cal E}^j_i) |H^{j,i}(x)|^p
+ \sum_{j > \nu c}\: \sum_k |s^{j,i}|^p a^{jn(1-p)}\mu({\cal E}^j_i) |H^{j,i}(x)|^p). \]
Now set
\[ A_{\nu} = \|\beta_{\nu}(\Delta)G\|_p,\:\:
B_{j} = [\sum_i \mu({\cal E}^j_i)|s^{j,i}|^p]^{1/p},\:\:
A^{\alpha}_{\nu} = 2^{\nu \alpha} A_{\nu},\:\: B^{\alpha}_{j} = a^{j \alpha} B_{j}. \]
Just as in the proof of Lemma \ref{besov2},
(\ref{bdfest3}), (\ref{bdfest4}) and Lemma \ref{fjan3} (taking ''$p$'' in that lemma to be $1$) show
that (\ref{anupbjp}) and (\ref{ajbnalpest2}) hold, with $K(\nu,j)$ as just after (\ref{ajbnalpest2}).
This again gives $\|(A_{\nu}^{\alpha})^p\|_{q/p} \leq C(\|(B_{j}^{\alpha})^p\|_{q/p})$.
Upon raising both sides to the $1/q$ power, one obtains (\ref{sjksjiok}), as claimed.
Finally, the fact that $(S'F)(1) = 0$ follows from the fact that each term of the summation in
(\ref{sumopdf4}) vanishes when applied to $1$ (since the assumption that $l \geq 1$ implies that
each $\psi^j_k$ has integral zero),
and the fact that the series in (\ref{sumopdf4}) converges in $B_p^{\alpha q}({\bf M})$, and hence in
${\cal E}'({\bf M})$. This completes the proof.
\begin{definition}
\label{bes0}
We let $B_{p,0}^{\alpha q}({\bf M}) = \{F \in B_p^{\alpha q}({\bf M}): F1 = 0\}$. (Here $F1$ is the
result of applying the distribution $F$ to the constant function $1$.)
\end{definition}
\begin{theorem}
\label{besmain1}
Say $c_0 , \delta_0 > 0$. Fix an integer $l \geq 1$ with $2l > \max(n(1/p-1)_+ - \alpha, \alpha)$.
Say $f_0 \in {\cal S}(\RR^+)$, and let $f(s) = s^l f_0(s)$.
Suppose also that the Daubechies condition (\ref{daub}) holds.
Then there exist constants $C > 0$ and $0 < b_0 < 1$ as follows:\\
Say $0 < b < b_0$.
Suppose that, for each $j$, we can write ${\bf M}$ as a finite disjoint union of measurable sets
$\{E_{j,k}: 1 \leq k \leq N_j\}$, and that (\ref{diamleq}), (\ref{measgeq}) hold.
(In the notation of Theorem \ref{framainfr} of \cite{gm2}, this is surely possible if
$c_0 \leq c_0^{\prime}$ and $\delta_0 \leq 2\delta$.)
Select $x_{j,k} \in E_{j,k}$ for each $j,k$.
For $t > 0$, let $K_t$ be the kernel of $f(t^2\Delta)$. Set
\[ \varphi_{j,k}(y) = \overline{K}_{a^j}(x_{j,k},y). \]
By Lemma (\ref{manmol}), there is a constant $C_0$ (independent of the
choice of $b$ or the $E_{j,k}$), such that
$\varphi_{j,k} \in C{\mathcal M}_{x_{j,k},a^j}$
for all $j,k$. Thus, we may form the summation operator $S$ with
\begin{equation}
\label{sumjopdfbes}
SF = S_{\{\varphi_{j,k}\},\{\varphi_{j,k}\}}F =
\sum_{j}\sum_k \mu(E_{j,k}) \langle F, \varphi_{j,k} \rangle \varphi_{j,k},
\end{equation}
at first for $F \in C^1({\bf M})$, and Theorem \ref{sumopthm} applies.
Then:\\
If $F \in B_{p,0}^{\alpha q}$, then the series in (\ref{sumjopdfbes}) converges in $B_{p,0}^{\alpha q}$,
and $S: B_{p,0}^{\alpha q} \rightarrow B_{p,0}^{\alpha q}$ is bounded and {\em invertible}. We have
$\|SF\|_{B_p^{\alpha q}} \leq C\|F\|_{B_p^{\alpha q}}$
and $\|S^{-1}F\|_{B_p^{\alpha q}} \leq C\|F\|_{B_p^{\alpha q}}$
for all $F \in B_{p,0}^{\alpha q}$.
\end{theorem}
{\bf Proof} By the last sentence of Lemma \ref{manmol},
if we let $K^0_t(x,y)$ be the kernel of $f_0(t^2\Delta)$,
and if we let $\eta_{j,k}(y) = \overline{K}^0_{a^j}(x_{j,k},y)$ for $j \leq 0$, then for every $I,J$ there exists
$C_{IJ}$ with $\eta_{j,k} \in C_{IJ}{\mathcal M}^{IJ}_{x_{j,k},a^j}$. Also (for instance, by
looking at eigenfunction expansions, as in (\ref{kerexp}) of \cite{gmcw}), one has that
$\varphi_{j,k} = (a^{2j}\Delta)^l \eta_{j,k}$. Thus, by Theorem \ref{sumopbes}, if $F \in B_p^{\alpha q}$, the series
\begin{equation}
\label{sumjopdfbesneg}
\sum_{j \leq 0}\sum_k \mu(E_{j,k}) \langle F, \varphi_{j,k} \rangle \varphi_{j,k} := S'F
\end{equation}
converges in $B_{p,0}^{\alpha q}$, and $\|S'F\|_{B_{p,0}^{\alpha q}} \leq C\|F\|_{B_p^{\alpha q}}$.
We shall next show that, if $F \in B_p^{\alpha q}$, then the series
\begin{equation}
\label{sprprf}
\sum_{j > 0}\sum_k \mu(E_{j,k}) \langle F, \varphi_{j,k} \rangle \varphi_{j,k}
\end{equation}
also converges in $B_{p,0}^{\alpha q}$, and if we call the sum of this series $S''F$, then
$\|S''F\|_{B_{p,0}^{\alpha q}} \leq C\|F\|_{B_p^{\alpha q}}$.
Since $S = S' + S''$,
this will tell us that $\|SF\|_{B_{p,0}^{\alpha q}} \leq C\|F\|_{B_p^{\alpha q}}$.
More generally, let us show the following:\\
\ \\
(*) There exist $N, C_0$ (independent of our choices of $b$,
$E_{j,k}$, $x_{j,k}$) as follows. Suppose $\psi_{j,k}, \Psi_{j,k}$ are smooth functions
on ${\bf M}$ (for all $j > 0$ and $1 \leq k \leq N(j)$), which satisfy
\begin{equation}
\label{jposway}
\|\psi_{j,k}\|_{C^N} \leq C_1 a^{-2j},\:\: \|\Psi_{j,k}\|_{C^N} \leq C_1 a^{-2j},
\end{equation}
for all $j,k$. Then, if $F \in B_p^{\alpha q}$, the series
\begin{equation}
\label{jposway1}
\sum_{j > 0}\sum_k \mu(E_{j,k}) \langle F, \psi_{j,k} \rangle \Psi_{j,k}
\end{equation}
converges in $B_{p,0}^{\alpha q}$, and if we call the sum of this series $S''F =
S^{\prime \prime}_{\{\psi_{j,k}\},\{\Psi_{j,k}\}}F$, then
\begin{equation}
\label{jposway2}
\|S''F\|_{B_{p,0}^{\alpha q} }\leq C_0 C_1\|F\|_{B_p^{\alpha q}}.
\end{equation}
Note that our $\varphi_{j,k}$ do satisfy (\ref{jposway}), since, as we noted in section 4 of \cite{gmcw},
$\lim_{t \rightarrow \infty} t K_{\sqrt t}(x,y) = 0$ in $C^{\infty}({\bf M} \times {\bf M})$.\\
To prove (*), we need only note that, by (\ref{ccNbes}) and a partition of unity argument,
one has (for some $N$) a continuous inclusion $C^N({\bf M}) \subseteq B_p^{\alpha q}({\bf M})$.
Also, since the inclusion $B_p^{\alpha q} \subseteq {\cal S}'(\RR^n)$ is continuous, we have a
continuous inclusion $B_p^{\alpha q}({\bf M}) \subseteq {\cal E}'(\RR^n)$. In particular, for some $N$,
\begin{equation}
\label{bescn}
\|G\|_{B_p^{\alpha q}} \leq C\|G\|_{C^N}
\end{equation}
for all $G \in C^N$,
while
\begin{equation}
\label{besepcn}
|\langle F, G \rangle| \leq C\|F\|_{B_p^{\alpha q}}\|G\|_{C^N}
\end{equation}
for all $F \in B_p^{\alpha q}$ and all
$G \in C^{\infty}$. In particular, in (\ref{jposway1}),
\[\sum_k \mu(E_{j,k}) |\langle F, \psi_{j,k} \rangle|\|\Psi_{j,k}\|_{C^N}
\leq CC^2_1 a^{-4j}\|F\|_{B_p^{\alpha q}}\sum_k \mu(E_{j,k})
\leq CC^2_1 \mu({\bf M}) a^{-4j}\|F\|_{B_p^{\alpha q}}.\]
Therefore the series in (\ref{jposway1}) converges
absolutely in $C^N$, hence in $B_p^{\alpha q}$, and we have the estimate
(\ref{jposway2}) as well.\\
To complete the proof, we return to the notation we used in the proof of Theorem \ref{framainfr}
of \cite{gm2} (taking
``${\mathcal J}$'' in that proof to be $\ZZ$, and setting $Q = Q^{\ZZ}$). We wish to show that
$Q$, when restricted to $C^{\infty}({\bf M})$, has a bounded extension to an operator
$Q: B_p^{\alpha q} \rightarrow B_{p,0}^{\alpha q}$, and that, as operators on $B_p^{\alpha q}$,
\[ \|Q-S\| \leq Cb \]
(where as usual $C$ is independent of our choices of $b$, the $E_{j,k}$ and the $x_{j,k}$).
Now, $C^{\infty}$ is dense in $B_p^{\alpha q}$ (for instance, by Theorem 7.1 (a) of \cite{FJ1};
the constructions in that paper show that the building blocks can be taken to be smooth). Thus
it is enough to show that
$\|(Q-S)F\| \leq Cb\|F\|$ for all $F \in C^{\infty}$ (where, until further notice, $\|\:\|$ means
the $B_p^{\alpha q}$ norm).
Say then that $F \in C^{\infty}$. As in the proof of Theorem \ref{framainfr} of \cite{gm2}, we may assume that
$\delta_0 \leq \delta$. Again put $\Omega_b = \log_a(\delta_0/b)$.
In the proof of Theorem \ref{framainfr} of \cite{gm2}, we have obtained a formula
for $\langle (Q-S)F, F \rangle$. This expression may be polarized, and a formula for $(Q-S)F$ may be
obtained from it. We see that $(Q-S)F = \sum_{i=1}^N I_i + II$, where in the notation of the proof of Theorem
\ref{framainfr} of \cite{gm2},
\[ I_i = b \int_{\mathcal B}
\int_0^1 [ S_{\{\varphi_{j,k}^{w,s}\},\{\psi_{j,k}^{w,s}\}}F
+ S_{\{\psi_{j,k}^{w,s}\},\{\varphi_{j,k}^{w,s}\}}F ] dsdw \]
and where
\[ II = \sum_{j \geq \Omega_b} \int_{\bf M} \langle F, \Phi_{x,a^j} \rangle \Phi_{\cdot,a^j} d\mu(x)
+ \sum_{j \geq \Omega_b} \sum_k \mu(E_{j,k}) \langle F, \varphi_{j,k} \rangle \varphi_{j,k}. \]
(Note that $I_i$ and $II$ are functions. In the first term of $II$ we have represented the independent
variable as $\cdot$.)
In $I_i$ we note, from the explicit expressions for the $\varphi_{j,k}^{w,s}$ and $\psi_{j,k}^{w,s}$,
and by using $K^0_t$ as in the first paragraph of this proof,
that we can write $\varphi_{j,k}^{w,s} = (a^{-2j}\Delta)^l\eta_{j,k}^{w,s}$,
$\psi_{j,k}^{w,s} = (a^{-2j}\Delta)^l\Psi_{j,k}^{w,s}$, where for any $I,J$ there exists $C_{IJ}$ with
$\eta_{j,k}^{w,s}, \Psi_{j,k}^{w,s} \in C_{IJ}\mathcal{M}_{x_{j,k}, a^j}^{IJ}$. By what we have already
seen in this proof, this implies that $\|I_i\| \leq Cb\|F\|$. Also, $I_i(1)=0$.
In $II$ we note that, for some $C$, $\|\Phi_{z,a^j}\|_{C^N} \leq Ca^{-2j}$ (for all $j \geq \Omega_b$ and
all $z \in {\bf M}$), while $\|\varphi_{j,k}\|_{C^N} \leq Ca^{-2j}$ (for all $j \geq \Omega_b$ and all $k$).
By (\ref{bescn}) and (\ref{besepcn}), we see that
\[ \|II\| \leq C \|F\| \sum_{j \geq \Omega_b}a^{-4j} \leq Cb \|F\|, \]
for $0 < b < 1$, since $\Omega_b = \log_a(\delta_0/b)$. Moreover, $II(1) = 0$.
We have, then, that for $F \in C^{\infty}$, $\|QF\| \leq \|SF\| + Cb\|F\|$. So
$Q: B_p^{\alpha q} \rightarrow B_p^{\alpha q}$ is bounded.
Also, we see that $\|Q-S\| \leq Cb$. Further, since $Q=S+\sum_{i=1}^N I_i + II$, we know $Q(1)=0$.
To complete the proof, it suffices to show that
$Q: B_{p,0}^{\alpha q} \rightarrow B_{p,0}^{\alpha q}$ is invertible. Indeed, if $\|\:\|$ now
denotes the norm of an operator on $B_{p,0}^{\alpha q}$, we will then have that
$\|I-Q^{-1}S\| \leq Cb\|Q^{-1}\|$, and the theorem will follow for $b$ sufficiently small.
For $\lambda \in \RR$, let $H(\lambda) = \sum_{j=-\infty}^{\infty} |f|^2(a^{2j} \lambda^2)
= \sum_{j=-\infty}^{\infty} (a^{j}\lambda)^{4l}g_0(a^j \lambda)$, if we write $g_0(\xi) = |f_0|^2(\xi^2)$.
Since $g_0$ and all its derivatives are bounded and decay rapidly
at $\infty$, $H$ is a smooth even function on $\RR^+ \setminus \{0\}$.
By Daubechies' criterion, $1/H = G$, say, is a smooth even function on $\RR^+ \setminus \{0\}$.
Now, let $u$ equal either $G$ or $H$. Note that $u(\lambda) = u(a^{-1}\lambda)$,
so that $u$ is actually a bounded smooth function on $\RR^+ \setminus \{0\}$. Moreover,
if $\lambda > 0$, we may choose an integer $m$ with $a^{m} \leq \lambda \leq a^{m+1}$.
Since $u(\lambda) = u(a^{-m}\lambda)$, for any $k$ we have
\[ |u^{(k)}(\lambda)| = a^{-km} |u^{(k)}(a^{-m}\lambda)| \leq a^{k}\lambda^{-k}M, \]
where $M = \max_{1 \leq \lambda \leq a} |u^{(k)}(\lambda)|$.
This implies that $\|\lambda^k u^{(k)}(\lambda)\|_{\infty} < \infty$
for any $k$.
Now choose an even function $v \in C_c^{\infty}(\RR)$ with $\supp v \subseteq (-\lambda_1,\lambda_1)$
and $v \equiv 1$ in a neighborhood of $0$. (Here $\lambda_1$ is the smallest positive
eigenvalue of $\Delta$). Then the even function $u_1 := (1-v)u \in C^{\infty}(\RR)$ is in
$S_1^0(\RR)$, so $u_1(\sqrt{\Delta}) \in OPS^0_{1,0}({\bf M})$. Moreover, $u_1(\sqrt{\Delta}) =
u(\sqrt{\Delta})$ on $C_0^{\infty}$ (:= $(I-P)C^{\infty}({\bf M})$). Recall that, here, $u= G$ or $H$;
set $G_1 = (1-v)G$, $H_1 = (1-v)H$.
Note further that $Q = H(\sqrt{\Delta}) = H_1(\sqrt{\Delta})$,
first when acting on finite linear combinations of non-constant
eigenfunctions of $\Delta$, then
on $C_0^{\infty}$, since such finite linear combinations are dense in $C_0^{\infty}$,
and the operators are bounded on $L^2$. Moreover $G_1(\sqrt{\Delta})H_1(\sqrt{\Delta}) =
H_1(\sqrt{\Delta})G_1(\sqrt{\Delta}) = (H_1G_1)(\sqrt{\Delta}) = I$.
first when acting on finite linear combinations of non-constant
eigenfunctions of $\Delta$, then on $C_0^{\infty}$, and finally on $B_{p,0}^{\alpha q}$, since by
\cite{P}, operators in $OPS^0_{1,0}({\bf M})$ are bounded on $B_p^{\alpha q}$. This shows that
$Q$ is indeed invertible on $B_{p,0}^{\alpha q}$, and establishes the theorem.\\
In Theorem \ref{besmain1}, we have again required the condition (\ref{measgeq}), that
$\mu(E_{j,k}) \geq c_0(ba^j)^n$, whenever $ba^j < \delta_0$.
In the next theorem, if $0 < p < 1$, we will require a mild condition on $\mu(E_{j,k})$ if $ba^j \geq \delta_0$,
namely that
\begin{equation}
\label{measgeq2}
\mu(E_{j,k}) \geq {\cal C} \mbox{ whenever } ba^j \geq \delta_0,
\end{equation}
(for some ${\cal C} > 0$). Sets $E_{j,k}$ which satisfy (\ref{diamleq}), (\ref{measgeq}) and
(\ref{measgeq2}) are easily constructed. Indeed, set $t = ba^j/2 \geq \delta_0/2$, and,
as in the second bullet point prior to Theorem \ref{framainfr} of \cite{gm2}, select a
finite covering of ${\bf M}$ by
disjoint measurable sets $E_1,\ldots,E_N$, such that whenever $1 \leq k \leq N$, there
is a $y_k \in {\bf M}$ with $B(y_k,t) \subseteq E_k \subseteq B(y_k,2t)$. Then, by (\ref{ballsn})
and (\ref{ballsn1}), there is a constant $c_0^{\prime}$, depending only on ${\bf M}$, such that
$\mu(E_k) \geq c_0^{\prime}\min(\delta^n,(\delta_0/2)^n)$, as desired.
\begin{theorem}
\label{besmain2}
Say $c_0 , \delta_0, M_0, {\cal C} > 0$. Fix an integer $l \geq 1$ with $2l > \max(n(1/p-1)_+ - \alpha, \alpha)$.
Say $f_0 \in {\cal S}(\RR^+)$, and let $f(s) = s^l f_0(s)$.
Suppose also that the Daubechies condition (\ref{daub}) holds.
Then there exist constants $C_1 > 0$ and $0 < b_0 < 1$ as follows:\\
Say $0 < b < b_0$. Then there exists a constant $C_2 > 0$ as follows:\\
Suppose that, for each $j$, we can write ${\bf M}$ as a finite disjoint union of measurable sets
$\{E_{j,k}: 1 \leq k \leq N_j\}$, and that (\ref{diamleq}), (\ref{measgeq}) hold. If $0 < p < 1$,
we suppose that (\ref{measgeq2}) holds as well. Select $x_{j,k} \in E_{j,k}$ for each $j,k$.
For $t > 0$, let $K_t$ be the kernel of $f(t^2\Delta)$. Set
\[ \varphi_{j,k}(y) = \overline{K}_{a^j}(x_{j,k},y). \]
Suppose $F$ is a distribution on ${\bf M}$ of order at most $M_0$, and that $F1=0$.
Then the following are equivalent:\\
(i) $F \in B_{p,0}^{\alpha q}$;\\
(ii) $(\sum_{j = -\infty}^{\infty} a^{-j\alpha q} [\sum_k \mu(E_{j,k})
|\langle F, \varphi_{j,k} \rangle |^p]^{q/p})^{1/q} < \infty$. Further\\
\begin{equation}
\label{nrmeqiv1}
\|F\|_{B_p^{\alpha q}}/C_2 \leq (\sum_{j = -\infty}^{\infty} a^{-j\alpha q} [\sum_k \mu(E_{j,k})
|\langle F, \varphi_{j,k} \rangle |^p]^{q/p})^{1/q} \leq C_1\|F\|_{B_p^{\alpha q}}.
\end{equation}
Moreover, if $p \geq 1$, then $C_2$ may be chosen to be independent of the choice of $b$
with $0 < b < b_0$.
\end{theorem}
{\bf Proof} Note first that, if $\eta_{j,k}$ is as in the first paragraph of the proof of Theorem
\ref{besmain1}, then as we noted there for every $I,J$ there exists
$C_{IJ}$ with $\eta_{j,k} \in C_{IJ}{\mathcal M}^{IJ}_{x_{j,k},a^j}$, and
$\varphi_{j,k} = (a^{2j}\Delta)^l \eta_{j,k}$.
Say now that $F \in B_{p,0}^{\alpha q}$. As we noted in the first paragraph of the proof of
Theorem \ref{sumopbes}, it follows from Lemma \ref{besov1} that
\[ (\sum_{j = -\infty}^{0} a^{-j\alpha q} [\sum_k \mu(E_{j,k})
|\langle F, \varphi_{j,k} \rangle |^p]^{q/p})^{1/q} \leq C\|F\|_{B_p^{\alpha q}}. \]
As for the terms in (ii) with $j > 0$, we note that, as in section 4 of \cite{gmcw},
$\lim_{t \rightarrow \infty} t^M K_{\sqrt t}(x,y) = 0$ in $C^{\infty}({\bf M} \times {\bf M})$,
for any $M$. Consequently, for any $M,N$,
\begin{equation}
\label{vphcnm}
\|\varphi_{j,k}\|_{C^N} \leq C a^{-Mj}.
\end{equation}
We choose any $M > -\alpha$. Then,
by (\ref{bescn}) and (\ref{besepcn}), we see that
\[ (\sum_{j = 1}^{\infty} a^{-j\alpha q} [\sum_k \mu(E_{j,k})
|\langle F, \varphi_{j,k} \rangle |^p]^{q/p})^{1/q} \leq
C\|F\|_{B_p^{\alpha q}} \mu(M)^{1/p} (\sum_{j = 1}^{\infty} a^{-j\alpha q - jMq})^{1/q} \leq
C\|F\|_{B_p^{\alpha q}}. \]
This proves that $(i) \Rightarrow (ii)$, and also establishes the rightmost inequality in
(\ref{nrmeqiv1}).
Say, conversely, that $(ii)$ holds. Our first step will be to show that the series for $SF$
(as in (\ref{sumjopdfbes}))
converges in $B_{p,0}^{\alpha q}$. For this,
it will be enough to show that the series for $S'F$ and $S''F$ (as in (\ref{sumjopdfbesneg})
and (\ref{sprprf}))each converge in $B_{p,0}^{\alpha q}$.
Lemma \ref{besov2}, with $s_{j,k}$ in that Lemma being our $\langle F, \varphi_{-j,k} \rangle$,
implies that the series for $S'F$ does converge in $B_{p,0}^{\alpha q}$, and,
moreover, that
\begin{equation}
\label{sprfiii}
\|S'F\|_{B_p^{\alpha q}} \leq C(\sum_{j = -\infty}^{0} a^{-j\alpha q} [\sum_k \mu(E_{j,k})
|\langle F, \varphi_{j,k} \rangle |^p]^{q/p})^{1/q}.
\end{equation}
(If $0 < p < 1$, we need to note that, since we are assuming (\ref{measgeq}) and (\ref{measgeq2}),
we have that (\ref{ejkro})
holds for some $\rho > 0$. Since $\rho$ depends on $b$, the constant $C$ in (\ref{sprfiii}) depends on
$b$ as well, if $0 < p < 1$.)
As for $S''F$, by (\ref{bescn}), it is enough to show that the series for it converges absolutely in $C^N$ (for any
fixed $N$). By (\ref{vphcnm}), we need only show:\\
\ \\
(*) Suppose that $\{r_{j,k}: 1 \leq j < \infty, 1 \leq k \leq N(j)\}$ are constants.
If $M$ is sufficiently large, then
\begin{equation}
\label{cnmjpq}
\sum_{j=1}^{\infty}a^{-Mj} \sum_k \mu(E_{j,k})|r_{j,k}| \leq
C(\sum_{j = 1}^{\infty} a^{-j\alpha q} [\sum_k \mu(E_{j,k}) |r_{j,k}|^p]^{q/p})^{1/q}.
\end{equation}
Here $C$ depends only on $c_0, \delta_0$ if $p \geq 1$ and only on $c_0, \delta_0, b$ if
$0 < p < 1$.\\
\ \\
To see this, let $a_j = \sum_k \mu(E_{j,k})|r_{j,k}|$,
$d_j = [\sum_k \mu(E_{j,k}) |r_{j,k}|^p]^{1/p}$; we begin by showing that $a_j \leq C d_j$.
If $p \geq 1$, we let $Y_j$ be the finite measure space $\{1,\ldots,N_j\}$ with measure
$\lambda$, where $\lambda(\{k\}) = \mu(E_{j,k})$. Applying H\"older's inequality on $Y_j$,
we find that $a_j \leq \mu({\bf M})^{1/p'} d_j$, as claimed. If, instead, $0 < p < 1$, the
$p$-triangle inequality implies that
\[ a_j^p \leq \sum_k \mu(E_{j,k})^p|r_{j,k}|^p \leq C\sum_k \mu(E_{j,k}) |r_{j,k}|^p = Cd_j^p \]
where now $C$ depends on $b$. (We have noted that, if $ba^j < \delta$ then
$1 \leq \mu(E_{j,k})^{1-p}/[c_0 (ba^j)^n]^{1-p} \leq
\mu(E_{j,k})^{1-p}/[c_0 b^n]^{1-p}$
by (\ref{measgeq}), while if
$ba^j \geq \delta$, then $1 \leq \mu(E_{j,k})^{1-p}/[{\cal C}]^{1-p}$ by (\ref{measgeq2}).)
To prove (\ref{cnmjpq}), it is enough, then, to show that, for $M$ sufficiently large,
\begin{equation}
\label{cnmjpq1}
\sum_{j=1}^{\infty}a^{-Mj} a_j \leq C(\sum_{j = 1}^{\infty} a^{-j\alpha q} a_j^q)^{1/q},
\end{equation}
for the right side of (\ref{cnmjpq1}) is less than or equal to
$C(\sum_{j = 1}^{\infty} a^{-j\alpha q} d_j^q)^{1/q}$, as we have seen, which gives (\ref{cnmjpq})
at once. But (\ref{cnmjpq1}) is true for {\em any} nonnegative constants $a_j$ (for $M$ sufficiently
large), for the following reason. If $0 < q \leq 1$, the $q$-triangle inequality tells us
that $(\sum_{j=1}^{\infty}a^{-j} a_j)^q \leq \sum_{j = 1}^{\infty} a^{-j\alpha q} a_j^q$, as claimed.
On the other hand, if $q > 1$, and $(-M + \alpha)q' < -1$, then (\ref{cnmjpq1}) follows by writing
$a^{-Mj} a_j = a^{(-M+\alpha)j}(a^{-\alpha j}a_j)$, and using H\"older's inequality.
In all, then, the series for $SF$
converges in $B_{p,0}^{\alpha q}$. Moreover, if we call the sum of this series $S_0F$, we have from
(\ref{sprfiii}) and (\ref{cnmjpq}) that
\begin{equation}
\label{sofbpq}
\|S_0F\|_{B_p^{\alpha q}} \leq C(\sum_{j = -\infty}^{\infty} a^{-j\alpha q} [\sum_k \mu(E_{j,k})
|\langle F, \varphi_{j,k} \rangle |^p]^{q/p})^{1/q},
\end{equation}
with $C$ independent of $b$ if $p \geq 1$.
To complete the proof we need only prove that if (ii) holds, and $b$ is sufficiently small,
then $F \in B_{p,0}^{\alpha q}$. For then we will surely have that $S_0 F = SF$. By Theorem \ref{besmain1},
$S: B_{p,0}^{\alpha q} \rightarrow B_{p,0}^{\alpha q}$ is invertible, with $\|S^{-1}\|$
independent of $b$ (for $b$ sufficiently small.) Thus the
leftmost inequality in (\ref{nrmeqiv1}) will follow from (\ref{sofbpq}), with $C_2$ independent of $b$ if
$p \geq 1$ (and $b$ is sufficiently small).
We have assumed that $F$ is a distribution of order at most $M_0$, and we claim that this implies that
$F \in B_p^{\gamma q}$ for some $\gamma \in \RR$. To see this, let $\beta_{\nu}$ be as in the proof of
Lemma \ref{besov2}; we need to show that
$\{ 2^{\nu\gamma}\|\beta_{\nu}(\Delta)F\|_p: -1 \leq \nu \leq \infty\}$ is in $l^q$ for some $\gamma \in
\RR$. As in the proof of Lemma \ref{besov2}, for $\nu \geq -1$, let $J_{\nu}$ be the kernel of $\beta_{\nu}(\Delta)$.
The arguments in the second paragraph of the proof of Lemma \ref{besov2} (specifically (\ref{jliest}), with $I = 0$,
in which case $J_{\nu}^I = J_{\nu}$), show that $\|J_{\nu}\|_{C^{M_0}({\bf M} \times {\bf M})} \leq C_02^{\nu(n+M_0)}$
for some $C_0 > 0$.
For any fixed $x \in {\bf M}$, let $J_{\nu,x}(y) = J_{\nu}(x,y)$. Then, for any $x \in {\bf M}$,
\[ |[\beta_{\nu}(\Delta)F](x)| = |F(J_{\nu,x})| \leq C_12^{\nu(n+M_0)}, \]
for some $C_1 > 0$. Thus
\[ |[\beta_{\nu}(\Delta)F](x)|_p \leq C_1\mu({\bf M})^{1/p} 2^{\nu(n+M_0)}. \]
Therefore
$\{ 2^{\nu\gamma}\|\beta_{\nu}(\Delta)F\|_p: -1 \leq \nu \leq \infty\}$ is in $l^q$ if $\gamma + n + M_0 < 0$,
that is, if $\gamma < -n -M_0$. Fix $\gamma < \min(-n-M_0, \alpha)$; then $F \in B_p^{\gamma q}$,
and $\gamma \leq \alpha$. In fact, since we are assuming that $F(1) = 0$, $F \in B_{p,0}^{\gamma q}$. \\
By Theorem \ref{besmain1}, we may choose $b_0$ sufficiently small that $S: B_{p,0}^{\alpha q}
\rightarrow B_{p,0}^{\alpha q}$ and $S: B_{p,0}^{\gamma q} \rightarrow B_{p,0}^{\gamma q}$ are both invertible if
$0 < b < b_0$. For such $b$, since $S_0F \in B_{p,0}^{\alpha q} \subseteq B_{p,0}^{\gamma q}$, there is a
unique $F_1 \in B_{p,0}^{\alpha q}$ with $SF_1 = S_0F$, and a unique $F_2 \in B_{p,0}^{\gamma q}$ with
$SF_2 = S_0F$. Now $S_0F$ is the sum (in $B_{p,0}^{\alpha q}$) of the series in (\ref{sumjopdfbes}). Since
$F \in B_{p,0}^{\gamma q}$, that series converges in $B_{p,0}^{\gamma q}$ to $SF$. Thus $S_0F = SF$
(as elements of $B_{p,0}^{\gamma q}$),
so $F_2 = F$. Also
$F_1 \in B_{p,0}^{\alpha q} \subseteq B_{p,0}^{\gamma q}$, and $SF_1 = S_0F$, so $F_1 = F_2 = F$.
But then $F = F_1 \in B_{p,0}^{\alpha q}$, as desired. This completes the proof.
\begin{theorem}
\label{besmain3}
Say $c_0 , \delta_0, {\cal C} > 0$. Fix an integer $l \geq 1$ with $2l > \max(n(1/p-1)_+ - \alpha, \alpha)$.
Say $f_0 \in {\cal S}(\RR^+)$, and let $f(s) = s^l f_0(s)$.
Suppose also that the Daubechies condition (\ref{daub}) holds.
Then there exist constants $C_1 > 0$ and $0 < b_0 < 1$ as follows:\\
Say $0 < b < b_0$. Then there exists a constant $C_2 > 0$ as follows:\\
Suppose that, for each $j$, we can write ${\bf M}$ as a finite disjoint union of measurable sets
$\{E_{j,k}: 1 \leq k \leq N_j\}$, and that (\ref{diamleq}), (\ref{measgeq}) hold. If $0 < p < 1$, we
assume that (\ref{measgeq2}) holds as well.
Select $x_{j,k} \in E_{j,k}$ for each $j,k$.
For $t > 0$, let $K_t$ be the kernel of $f(t^2\Delta)$. Set
\[ \varphi_{j,k}(y) = \overline{K}_{a^j}(x_{j,k},y). \]
Then:\\
If $F \in B_{p,0}^{\alpha q}$, there exist constants $r_{j,k}$ with
$(\sum_{j = -\infty}^{\infty} a^{-j\alpha q} [\sum_k \mu(E_{j,k})|r_{j,k}|^p]^{q/p})^{1/q} < \infty$
such that
\begin{equation}
\label{dfexp}
F = \sum_{j=-\infty}^{\infty} \sum_k \mu(E_{j,k}) r_{j,k} \varphi_{j,k},
\end{equation}
with convergence in $B_p^{\alpha q}$. Further\\
\begin{equation}
\label{nrmeqiv2}
\|F\|_{B_p^{\alpha q}}/C_2 \leq
\inf\{(\sum_{j = -\infty}^{\infty} a^{-j\alpha q} [\sum_k \mu(E_{j,k})|r_{j,k}|^p]^{q/p})^{1/q}:(\ref{dfexp}) \mbox{ holds}\}
\leq C_1\|F\|_{B_p^{\alpha q}}.
\end{equation}
Moreover, if $p \geq 1$, then $C_2$ may be chosen to be independent of the choice of $b$
with $0 < b < b_0$.
\end{theorem}
{\bf Proof} We choose $b_0 > 0$ sufficiently small that $S: B_{p,0}^{\alpha q} \rightarrow B_{p,0}^{\alpha q}$
is invertible for $0 < b < b_0$. For such $b$, if $F \in B_{p,0}^{\alpha q}$, then
$S^{-1}F \in B_{p,0}^{\alpha q}$, so
\begin{equation}
\label{ssinvf}
F = S(S^{-1}F) = \sum_{j}\sum_k \mu(E_{j,k}) \langle S^{-1}F, \varphi_{j,k} \rangle \varphi_{j,k}.
\end{equation}
so that (\ref{dfexp}) holds with $r_{j,k} = \langle S^{-1}F, \varphi_{j,k} \rangle$. By Theorem
\ref{besmain2} and Theorem \ref{besmain1}, $(\sum_k \mu(E_{j,k})|r_{j,k}|^p]^{q/p})^{1/q} \leq
C_1^{\prime} \|S^{-1}F\|_{B_p^{\alpha q}} \leq C_1\|F\|_{B_p^{\alpha q}}.$
That leaves only the leftmost inequality in (\ref{nrmeqiv2}) to prove. We need only show that, for any $F$
as in (\ref{dfexp}) (convergence in $B_p^{\alpha q}$), we have the inequality
\begin{equation}
\label{frjkbpq}
\|F\|_{B_p^{\alpha q}} \leq C_2\sum_{j = -\infty}^{\infty} a^{-j\alpha q} [\sum_k \mu(E_{j,k})|r_{j,k}|^p]^{q/p})^{1/q}.
\end{equation}
with $C_2$ independent of $0 < b < b_0$ if $p \geq 1$.
It is enough to do this in each of two cases: (i) if $r_{j,k} = 0$ whenever $j > 0$; and (ii) if
$r_{j,k} = 0$ whenever $j \leq 0$. Case (i) follows at once from Lemma \ref{besov2} (and, if $0 < p < 1$, the hypotheses
(\ref{measgeq}), (\ref{measgeq2}), which imply (\ref{ejkro})). Case (ii) follows from (*) of the proof of
Theorem \ref{besmain2}, since the inequality (\ref{cnmjpq}) there shows that $\|F\|_{C^N}$ is less than
or equal to the right side of (\ref{frjkbpq}), for any fixed $N$. This completes the proof.
\ \\
STONY BROOK UNIVERSITY
\end{document} | math | 92,291 |
\begin{document}
\title{
{\begin{flushleft}
\vskip 0.45in
\end{flushleft}
\vskip 0.45in
\bfseries\scshape Optimal control and numerical software: an overview}}
\author{\bfseries Helena Sofia Rodrigues$^1$
\and \bfseries M. Teresa T. Monteiro$^2$
\and \bfseries Delfim F. M. Torres$^3$
\thanks{E-mail addresses: [email protected], [email protected], [email protected]}\\
$^1$CIDMA and School of Business,\\Viana do Castelo Polytechnic Institute, Portugal\\
$^2$R\&D Centre Algoritmi, Department of Production and Systems,\\University of Minho, Portugal\\
$^3$CIDMA, Department of Mathematics,\\University of Aveiro, Portugal}
\date{}
\maketitle
\thispagestyle{empty}
\setcounter{page}{1}
\thispagestyle{fancy}
\fancyhead{}
\fancyhead[L]{This is a preprint of a paper whose final and definite form will appear in the book\\
'Systems Theory: Perspectives, Applications and Developments',\\
Nova Science Publishers, Editor: Francisco Miranda.\\
Submitted 21/Sept/2013; Revised 01/Dec/2013; Accepted 22/Jan/2014.}
\fancyhead[R]{}
\fancyfoot{}
\renewcommand{0pt}{0pt}
\begin{abstract}
Optimal Control (OC) is the process of determining control
and state trajectories for a dynamic system, over a period
of time, in order to optimize a given performance index.
With the increasing of variables and complexity, OC problems can no longer be solved
analytically and, consequently, numerical methods are required. For this purpose,
direct and indirect methods are used. Direct methods consist in the discretization of the OC problem,
reducing it to a nonlinear constrained optimization problem. Indirect methods are based
on the Pontryagin Maximum Principle, which in turn reduces to a boundary value problem.
In order to have a more reliable solution, one can solve the same problem through
different approaches. Here, as an illustrative example, an epidemiological application
related to the rubella disease is solved using several software packages,
such as the routine ode45 of Matlab, OC-ODE, DOTcvp toolbox, IPOPT and Snopt,
showing the state of the art of numerical software for OC.
\end{abstract}
\noindent \textbf{Key Words}: optimal control, numerical software, direct methods, indirect methods, rubella.
\noindent \\
{\textbf AMS Subject Classification:} 49K15, 90C30, 93C15, 93C95.
\section{Introduction}
Historically, optimal control (OC) is an extension of the calculus of
variations. The first formal results of the calculus of variations
can be found in the seventeenth century.
Johann Bernoulli challenged other famous contemporary mathematicians---such
as Newton, Leibniz, Jacob Bernoulli, L'H\^{o}pital and von Tschirnhaus---with the
\emph{brachistochrone} problem: ``if a particle moves, under the
influence of gravity, which path between two fixed points enables
the trip of shortest time?'' (see, e.g., \cite{Bryson1996}).
This and other specific problems were solved, and a general mathematical
theory was developed by Euler and Lagrange. The most fruitful
applications of the calculus of variations have been to
theoretical physics, particularly in connection with Hamilton's
principle or the Principle of Least Action. Early applications to
economics appeared in the late 1920s and early 1930s by Ross,
Evans, Hottelling and Ramsey, with further applications published
occasionally thereafter \cite{Sussmann1997}.
\pagestyle{fancy}
\fancyhead{}
\fancyhead[EC]{H. S. Rodrigues, M. T. T. Monteiro, D. F. M. Torres}
\fancyhead[EL,OR]{\thepage}
\fancyhead[OC]{Optimal control and numerical software: an overview}
\fancyfoot{}
\renewcommand0pt{0.5pt}
The generalization of the calculus of variations to optimal
control theory was strongly motivated by military applications and has
developed rapidly since 1950. The decisive breakthrough was achieved
by the Russian mathematician Lev S. Pontryagin (1908-1988) and his
co-workers (V. G. Boltyanskii, R. V. Gamkrelidz and E. F.
Misshchenko) with the formulation and proof of the Pontryagin Maximum
Principle \cite{Pontryagin1962}. This principle has provided research with
suitable conditions for optimization problems with differential
equations as constraints. The Russian team generalized variational problems by
separating control and state variables and admitting control
constraints. The two approaches use a different point of view
and the OC approach often affords insight into a
problem that might be less readily apparent through the calculus
of variations. OC is also applied to problems where
the calculus of variations is not convenient, such as those
involving constraints on the derivatives of functions \cite{Leitmann1997}.
The theory of OC brought new approaches to Mathematics with Dynamic Programming.
Introduced by R. E. Bellman, Dynamic Programming makes use of the principle
of optimality and it is suitable for solving discrete problems, allowing for a significant
reduction in the computation of the optimal controls (see \cite{Kirk1998}). It is also possible
to obtain a continuous approach to the principle of optimality that leads to the solution
of a partial differential equation called the Hamilton-Jacobi-Bellman equation.
This result allowed to bring new connections between the OC problem and the Lyapunov stability theory.
Before the computer age, only fairly simple OC problems could be solved.
The arrival of the computer enabled the application of OC theory and its methods
to many complex problems. Selected examples are as follows:
\begin{itemize}
\item Physical systems, such as stable performance of motors and
machinery, robotics, and optimal guidance of rockets \cite{Goh2008,Molavi2008};
\item Aerospace, including driven problems, orbits transfers, development of
satellite launchers and recoverable problems of atmospheric
reentry \cite{Bonnard2008,Hermant2010};
\item Economics and management, such as optimal exploitation of natural
resources, energy policies, optimal investment of production
strategies \cite{Munteanu2008,Sun2008};
\item Biology and medicine, as regulation of physiological functions, plants growth,
infectious diseases, oncology, radiotherapy \cite{Joshi2002,Joshi2006,Lenhart2007,Nanda2007}.
\end{itemize}
Nowadays, OC is an extensive theory with several approaches.
One can adjust controls in a system
to achieve a certain goal, where the underlying system can include:
ordinary differential equations, partial differential equations,
discrete equations, stochastic differential equations,
integro-difference equations, combination of discrete and
continuous systems. In this work we restrict ourselves to deterministic
OC theory of ordinary differential equations in a fixed time interval.
\section{Optimal control problems}
\label{sec:1:1}
A typical OC problem requires a performance index or cost functional,
$J[x(\cdot),u(\cdot)]$; a set of state variables, $x(\cdot) \in X$;
and a set of control variables, $u(\cdot) \in U$.
The main goal consists in finding a piecewise continuous control $u(t)$,
$t_0 \leq t \leq t_f$, and the associated state variable $x(t)$,
to maximize the given objective functional.
\begin{definition}[Basic OC Problem in Lagrange form]
\label{OC_Lagrange}
\index{Lagrange form}
\noindent An OC problem is:
\begin{equation}
\label{OC_Lagrange_eq}
\begin{gathered}
\underset{u(\cdot)}{\max} \ J[x(\cdot),u(\cdot)]
=\int_{t_0}^{t_f}f(t,x(t),u(t))dt,\\
\text{s.t. }\ \dot{x}(t)=g(t,x(t),u(t)),\\
x(t_0)=x_0.
\end{gathered}
\end{equation}
\end{definition}
\begin{remark}
\label{rem:1}
The value of $x(t_f)$ in \eqref{OC_Lagrange_eq} is free,
which means that the value of $x(t_f)$ is unrestricted.
Sometimes one also considers problems with $x(t_f)$ fixed,
\emph{i.e}, $x(t_f)=x_f$ for a certain given $x_f$.
\end{remark}
For our purposes, $f$ and $g$ will always be continuously differentiable
functions in all three arguments. The controls will always
be piecewise continuous, and the associated states will always
be piecewise differentiable. Note that we can switch back and forth
between maximization and minimization,
by simply negating the cost functional:
$$
\min\{J\}=-\max\{-J\}.
$$
An OC problem can be presented in many different, but equivalent ways,
depending on the purpose or the software to be used.
\subsection{Lagrange, Mayer and Bolza formulations}
\label{sec:1:2}
There are three well known equivalent formulations to describe an
OC problem, which are the Lagrange (Definition~\ref{OC_Lagrange}),
Mayer and Bolza forms \cite{Chachuat2007,Zabczyk2008}.
\begin{definition}[Basic OC Problem in Bolza form]
\label{OC_Bolza}\index{Bolza form}
Bolza's formulation of the OC problem is:
\begin{equation}
\label{OC_Bolza_eq}
\begin{gathered}
\underset{u(\cdot)}{\max} \ J[x(\cdot),u(\cdot)]
=\phi\left(t_{0},x(t_{0}),t_f,x(t_f)\right)+\int_{t_0}^{t_f}f(t,x(t),u(t))dt,\\
\text{s.t. } \ \dot{x}(t)=g(t,x(t),u(t)),\\
x(t_0)=x_0,
\end{gathered}
\end{equation}
where $\phi$ is a continuously differentiable function.
\end{definition}
\begin{definition}[Basic OC Problem in Mayer form]
\label{OC_Mayer}\index{Mayer form}
Mayer's formulation of the OC problem is:
\begin{equation}
\label{OC_Mayer_eq}
\begin{gathered}
\underset{u(\cdot)}{\max} \ J[x(\cdot),u(\cdot)]
=\phi\left(t_{0},x(t_{0}),t_f,x(t_f)\right),\\
\text{s.t. } \ \dot{x}(t)=g(t,x(t),u(t)),\\
x(t_0)=x_0.
\end{gathered}
\end{equation}
\end{definition}
\begin{theorem}
\label{thm:1}
The three formulations, Lagrange (Definition~\ref{OC_Lagrange})\index{Lagrange form},
Bolza (Definition~\ref{OC_Bolza})\index{Bolza form} and Mayer
(Definition~\ref{OC_Mayer}\index{Mayer form}), are equivalent.
\end{theorem}
\begin{proof}
See, e.g., \cite{Chachuat2007,Zabczyk2008}.
\end{proof}
The proof of Theorem~\ref{thm:1} gives a method to rewrite an optimal control problem
in any of the three forms to any other of the three forms.
Note that, from a computational perspective,
some of the OC problems, often presented in the Lagrange form,
should be converted into the equivalent Mayer form. Hence, using a standard procedure,
one rewrites the cost functional, augmenting the state vector with an extra component
(\textrm{cf.}, e.g., \cite{Lewis1995}). More precisely,
the Lagrange formulation \eqref{OC_Lagrange_eq} is rewritten as
\begin{equation}
\label{cap1_lagrange-mayer}
\begin{gathered}
\underset{u(\cdot)}{\max} \ x_c(t_f),\\
\text{s.t. } \ \dot{x}(t)=g(t,x(t),u(t)),\\
\dot{x}_c(t)=f(t,x(t),u(t)),\\
x(t_0)=x_0,\\
x_c(t_0)=0.
\end{gathered}
\end{equation}
\subsection{Pontryagin's Maximum Principle}
\label{sec:1:3}
\index{Pontryagin's Maximum Principle}
Necessary first order optimality conditions were developed by
Pontryagin and his co-workers. The result is considered as one of the most important
results of Mathematics in the 20th century.
Pontryagin introduced the idea of adjoint functions to append the
differential equation to the objective functional. Adjoint
functions have a similar purpose as Lagrange multipliers in
multivariate calculus, which append constraints to the function of
several variables to be maximized or minimized.
\begin{definition}[Hamiltonian]
\label{Hamiltonian}\index{Hamiltonian}
Consider the OC problem \eqref{OC_Lagrange_eq}. The function
\begin{equation}
\label{eq:H}
H(t, x, u,\lambda)=f(t,x,u)+\lambda \, g(t,x,u)
\end{equation}
is called the Hamiltonian (function), and $\lambda$
is the adjoint variable.
\end{definition}
We are ready to formulate the Pontryagin Maximum Principle (PMP)
for problem \eqref{OC_Lagrange_eq}.
\begin{theorem}[Pontryagin's Maximum Principle for \eqref{OC_Lagrange_eq}]
\label{PMP}\index{Pontryagin's Maximum Principle}
If $u^{*}(\cdot)$ and $x^{*}(\cdot)$ are optimal for problem \eqref{OC_Lagrange_eq},
then there exists a piecewise differentiable adjoint variable
$\lambda(\cdot)$ such that
$$
H(t,x^{*}(t),u(t),\lambda(t))
\leq H(t,x^{*}(t),u^{*}(t),\lambda(t))
$$
for all controls $u(t)$ at each time $t$, where $H$ is the
Hamiltonian \eqref{eq:H}, and
\begin{equation*}
\begin{split}
\lambda'(t) &=-\frac{\partial H(t,x^{*}(t),u^{*}(t),\lambda(t))}{\partial x},\\
\lambda(t_f)&=0.
\end{split}
\end{equation*}
\end{theorem}
The proof of Theorem~\ref{PMP} follows classical variational methods
and can be found, e.g., in \cite{Lenhart2007}. The original
Pontryagin's book \cite{Pontryagin1962}
or Clarke's book \cite{Clarke1990} are good references
to find more general results and their detailed proofs.
\begin{remark}
The last condition of Theorem~\ref{PMP}, $\lambda(t_f)=0$, is
called the transversality condition\index{Transversality condition},
and is only used when the OC problem does not have the terminal value in the state variable,
\emph{i.e.}, $x(t_f)$ is free (cf. Remark~\ref{rem:1}).
\end{remark}
Theorem~\ref{PMP} converts the problem of finding a control which
maximizes the objective functional subject to the state ODE and
initial condition, into the problem of optimizing the Hamiltonian
pointwise. As a consequence, we have
\begin{equation}
\label{Hu}
\displaystyle{\frac{\partial H}{\partial u}}=0
\end{equation}
at $u^{*}$ for each $t$, that is, the Hamiltonian\index{Hamiltonian} has a
critical point at at $u^{*}$. Usually this condition is called the \emph{optimality
condition}\index{Optimality condition}.
\begin{remark}
If the Hamiltonian\index{Hamiltonian} is linear in the control variable $u$, it can
be difficult to calculate $u^{*}$ from the optimality equation,
since $\frac{\partial H}{\partial u}$ would not contain $u$.
Specific ways to solve such kind of problems can be found, for example,
in \cite{Lenhart2007}.
\end{remark}
Until here we have shown necessary conditions to solve basic optimal control problems.
Now, it is important to study some conditions that can guarantee the existence
of a finite objective functional value at the optimal control and state variables
\cite{Fleming1975,Kamien1991,Lenhart2007,Macki1982}.
The following is an example of a sufficient condition.
\begin{theorem}
\label{thm:suf}
Consider the following problem:
\begin{equation*}
\begin{gathered}
\underset{u(\cdot)}{\max}\ J\left[x(\cdot),u(\cdot)\right]
=\int_{t_0}^{t_f}f(t,x(t),u(t))dt,\\
\text{s.t. } \ \dot{x}(t)=g(t,x(t),u(t)),\\
x(t_0)=x_0.
\end{gathered}
\end{equation*}
Suppose that $f(t,x,u)$ and $g(t,x,u)$ are both continuously differentiable
functions in their three arguments and concave in $x$ and $u$. If $u^{*}(\cdot)$
is a control with associated state $x^{*}(\cdot)$ and $\lambda(\cdot)$
a piecewise differentiable function such that $u^{*}(\cdot)$, $x^{*}(\cdot)$
and $\lambda(\cdot)$ together satisfy
\begin{gather*}
f_u+\lambda g_u=0 \Leftrightarrow \frac{\partial H}{\partial u} = 0,\\
\lambda'=-(f_x+\lambda g_x) \Leftrightarrow \lambda'= - \frac{\partial H}{\partial x},\\
\lambda(t_f)=0,\\
\lambda(t)\geq 0
\end{gather*}
on $t_0\leq t \leq t_f$, then
$$
J[x^{*}(\cdot),u^{*}(\cdot)] \geq J[x(\cdot),u(\cdot)]
$$
for any admissible pair $\left(x(\cdot),u(\cdot)\right)$.
\end{theorem}
\begin{proof}
The proof of this theorem can be found in \cite{Lenhart2007}.
\end{proof}
Theorem~\ref{thm:suf} is not strong enough to guarantee
that $J[x^{*}(\cdot),u^{*}(\cdot)]$ is finite.
Such results usually require some conditions on $f$ and/or $g$.
Next theorem is an example of an existence result from \cite{Fleming1975}
(cf. \cite[Theorem~2.2]{Lenhart2007}).
\begin{theorem}
\label{thm:ex}
Let the set of controls for problem \eqref{OC_Lagrange_eq}
be Lebesgue integrable functions on $t_0\leq t\leq t_f$ in $\mathbb{R}$.
Suppose that $f(t,x,u)$ is concave in $u$, and there exist constants
$C_1, C_2, C_3 >0$, $C_4$, and $\beta >1$ such that
\begin{gather*}
g(t,x,u)=\alpha(t,x)+\beta(t,x)u,\\
|g(t,x,u)|\leq C_1(1+|x|+|u|),\\
|g(t,x_1,u)-g(t,x,u)|\leq C_2 |x_1-x|(1+|u|),\\
f(t,x,u)\leq C_3|u|^{\beta}-C_4
\end{gather*}
for all $t$ with $t_0\leq t\leq t_1$, $x$, $x_1$, $u$ in $\mathbb{R}$.
Then there exists an optimal pair $\left(x^{*}(\cdot),u^{*}(\cdot)\right)$
maximizing $J$, with $J[x^{*}(\cdot),u^{*}(\cdot)]$ finite.
\end{theorem}
\begin{proof}
The proof is given in \cite{Fleming1975}.
\end{proof}
\begin{remark}
For a minimization problem, $f$ would have a convex property
and the inequality on $f$ would be reversed (coercivity).
\end{remark}
It is important to note that the necessary conditions developed to this point deal
with piecewise continuous optimal controls, while the existence
Theorem~\ref{thm:ex} guarantees an optimal control which is only Lebesgue integrable.
This gap can be overcome by studying regularity conditions
\cite{my:reg1,my:reg2}.
\subsection{Optimal control with bounded controls}
\label{sec:1:4}
Many problems, to be realistic, require bounds on the controls.
\begin{definition}[OC problem with bounded controls]
An OC problem with bounded control, in Lagrange form, is:
\begin{equation}
\label{OC_bounded_control}
\begin{gathered}
\underset{u(\cdot)}{\max} J[x(\cdot),u(\cdot)]=\int_{t_0}^{t_f} f(t,x(t),u(t)) dt,\\
\text{s.t. }\ \dot{x}(t)=g(t,x(t),u(t)),\\
x(t_0)=x_0,\\
a\leq u(t)\leq b,
\end{gathered}
\end{equation}
where $a$ and $b$ are fixed real constants with $a<b$.
\end{definition}
The Pontryagin Maximum Principle (Theorem~\ref{PMP})
remains valid for problems with bounds on the control,
except the maximization is over all admissible controls,
that is, $a\leq u(t)\leq b$ for all $t \in [t_0, t_f]$.
\begin{theorem}[Pontryagin's Maximum Principle for \eqref{OC_bounded_control}]
\label{thm:PMP:bc}
If $u^{*}(\cdot)$ and $x^{*}(\cdot)$ are optimal for problem \eqref{OC_bounded_control},
then there exists a piecewise differentiable adjoint variable $\lambda(\cdot)$ such that
$$
H(t,x^{*}(t),u(t),\lambda(t))\leq H(t,x^{*}(t),u^{*}(t),\lambda(t))
$$
for all admissible controls $u$ at each time $t$, where $H$ is the Hamiltonian \eqref{eq:H}, and
\begin{gather}
\lambda'(t)=-\frac{\partial H(t,x^{*}(t),u^{*}(t),\lambda(t))}{\partial x} \tag{adjoint condition},\\
\lambda(t_f)=0 \tag{transversality condition}\index{Transversality condition}.
\end{gather}
\end{theorem}
The following proposition is a direct consequence of Theorem~\ref{thm:PMP:bc}.
The proof can be found, e.g., in \cite{Kamien1991} or \cite{Lenhart2007}.
\begin{proposition}
The optimal control $u^{*}(\cdot)$ to problem \eqref{OC_bounded_control}
must satisfy the following optimality condition\index{Optimality condition}:
\begin{equation}
\label{eq:opt:cond}
u^{*}(t)=
\begin{cases}
a & \textrm{if } \frac{\partial H}{\partial u}<0\\
\tilde{u} & \textrm{if } \frac{\partial H}{\partial u}=0\\
b & \textrm{if } \frac{\partial H}{\partial u}>0,
\end{cases}
\end{equation}
where $a\leq \tilde{u} \leq b$,
is obtained by the expression $\frac{\partial H}{\partial u}=0$.
In particular, the optimal control $u^{*}(\cdot)$ maximizes $H$ pointwise with respect to $a \leq u \leq b$.
\end{proposition}
\begin{remark}
If we have a minimization problem instead of maximization,
then $u^{*}$ is instead chosen to minimize $H$ pointwise.
This has the effect of reversing $<$ and $>$ in the first and third lines
of the optimality condition\index{Optimality condition} \eqref{eq:opt:cond}.
\end{remark}
So far, we have only examined problems with one control
and one state variable. Often,
it is necessary to consider more variables.
Below, one such optimal control problem, related to rubella, is presented.
The PMP continues valid for problems with several state and several
control variables.
\begin{example}
\label{example_rubeola}
Rubella, commonly known as German measles, is most common in child age,
caused by the rubella virus. Children recover more quickly than adults.
Rubella can be very serious during pregnancy. The virus is contracted through
the respiratory tract and has an incubation period of 2 to 3 weeks.
The primary symptom of rubella virus infection is the appearance
of a rash on the face which spreads to the trunk and limbs
and usually fades after three days. Other symptoms include low grade fever,
swollen glands, joint pains, headache and conjunctivitis.
We present an optimal control problem to study the dynamics
of rubella over three years, using a vaccination process ($u$)
as a measure to control the disease. More details can be found in \cite{Buonomo2011}.
Let $x_1$ represent the susceptible population, $x_2$ the proportion
of population that is in the incubation period, $x_3$ the proportion
of population that is infected with rubella, and $x_4$ the rule
that keeps the population constant. The optimal control problem can be defined as:
\begin{equation}
\label{cap1:ode_rubeola}
\begin{gathered}
\min \ \int_{0}^{3}(Ax_3+u^2)dt\\
\text{s.t. } \ \dot{x}_1=b-b(p x_2+q x_2)-b x_1-\beta x_1 x_3 - u x_1,\\
\dot{x}_2=b p x_2 +\beta x_1 x_3 -(e+b)x_2,\\
\dot{x}_3=e x_2-(g+b)x_3,\\
\dot{x}_4=b-b x_4,
\end{gathered}
\end{equation}
with initial conditions $x_1(0)=0.0555$, $x_2(0)=0.0003$,
$x_3(0)=0.0004$, $x_4(0)=1$ and the parameters $b=0.012$, $e=36.5$,
$g=30.417$, $p=0.65$, $q=0.65$, $\beta=527.59$ and $A=100$.
The control $u$ is defined as taking values in the interval $[0,0.9]$.
\end{example}
It is not easy to solve analytically the problem of Example~\ref{example_rubeola}.
For the majority of real OC applications, it is necessary to employ numerical methods.
\section{Numerical methods to solve optimal control problems}
In the last decades, the computational power has been developed in an amazing way.
Not only in hardware issues, such as efficiency, memory capacity, speed,
but also in terms of software robustness. Ground breaking achievements
in the field of numerical solution techniques for differential and integral
equations have enabled the simulation of highly complex real world scenarios.
OC also won with these improvements, and numerical methods
and algorithms have evolved significantly.
\subsection{Indirect methods}
\label{sec:2:2}
\index{Indirect methods}
In an indirect method, the PMP\index{Pontryagin's Maximum Principle}
is used. Therefore, the indirect approach leads to a multiple-point boundary-value
problem that is solved to determine candidate optimal trajectories, called extremals.
To apply it, it is necessary to explicitly get the adjoint equations,
the control equations, and all the transversality conditions\index{Transversality condition},
if they exist. A numerical approach using the indirect method,
known as the \emph{backward-forward sweep method}, is now presented.
This method is described in \cite{Lenhart2007}.
The process begins with an initial guess on the control variable. Then, simultaneously,
the state equations are solved forward in time and the adjoint
equations are solved backward in time. The control is updated
by inserting the new values of states and adjoints into its characterization, and
the process is repeated until convergence occurs.
Let us consider $\vec{x}=(x_1,\ldots,x_N+1)$ and $\vec{\lambda}=(\lambda_1,\ldots,\lambda_N+1)$
the vector of approximations for the state and the adjoint.
The main idea of the algorithm is described as follows.
\begin{description}
\item[Step 1.] Make an initial guess for $\vec{u}$
over the interval ($\vec{u}\equiv 0$ is almost always sufficient).
\item[Step 2.] Using the initial condition $x_1=x(t_0)=a$
and the values for $\vec{u}$, solve $\vec{x}$ forward
in time according to its differential equation in the optimality system.
\item[Step 3.] Using the transversality condition $\lambda_{N+1}=\lambda(t_f)=0$
and the values for $\vec{u}$ and $\vec{x}$, solve $\vec{\lambda}$ backward
in time according to its differential equation in the optimality system.
\item[Step 4.] Update $\vec{u}$ by entering the new $\vec{x}$ and $\vec{\lambda}$
values into the characterization of the optimal control.
\item[Step 5.] Verify convergence: if the variables are sufficiently close
to the corresponding ones in the previous iteration, then output the current
values as solutions, otherwise return to Step 2.
\end{description}
For Steps 2 and 3, Lenhart and Workman \cite{Lenhart2007} use, for the state
and adjoint systems, the Runge--Kutta fourth order
procedure to make the discretization process.
On the other hand, Wang \cite{Wang2009} applies the same philosophy
but solving the differential equations with the \texttt{ode45}\index{ode45 routine}
solver of \texttt{Matlab}\index{Matlab}. This solver is based on an explicit
Runge--Kutta (4,5) formula, the Dormand--Prince pair. That means
that the \texttt{ode45} numerical solver combines a fourth and a fifth order method,
both of which being similar to the classical fourth order Runge--Kutta method.
These vary the step size, choosing it at each step, in an attempt to achieve the desired
accuracy. Therefore, the \texttt{ode45} solver is suitable for a wide variety
of initial value problems in practical applications. In general, \texttt{ode45}
is the best method to apply, as a first attempt, to most problems \cite{Houcque}.
\begin{example}
\label{ex_rubeola_Lenhart}
Let us consider the problem of Example~\ref{example_rubeola} about rubella disease.
With $\vec{x}(t)=\left(x_1(t),x_2(t),x_3(t),x_4(t)\right)$
and $\vec{\lambda}(t)=\left(\lambda_1(t),\lambda_2(t),\lambda_3(t),\lambda_4(t)\right)$,
the Hamiltonian of this problem can be written as\index{Hamiltonian}
\begin{multline*}
\label{cap2:hamiltonian_rubeola}
H(t,\vec{x}(t),u(t),\vec{\lambda}(t))= Ax_3+u^2
+\lambda_1\left(b-b(p x_2+q x_2)-b x_1-\beta x_1 x_3 - u x_1\right)\\
+\lambda_2\left(b p x_2 +\beta x_1 x_3 -(e+b)x_2\right)
+\lambda_3\left(e x_2-(g+b)x_3\right)
+\lambda_4\left(b-b x_4\right).
\end{multline*}
Using the PMP, the optimal control problem can be studied with the control system
\begin{equation*}
\label{cap2:ode2_rubeola}
\begin{cases}
\dot{x}_1=b-b(p x_2+q x_2)-b x_1-\beta x_1 x_3 - u x_1\\
\dot{x}_2=b p x_2 +\beta x_1 x_3 -(e+b)x_2\\
\dot{x}_3=e x_2-(g+b)x_3\\
\dot{x}_4=b-b x_4
\end{cases}
\end{equation*}
subject to initial conditions $x_1(0)=0.0555$, $x_2(0)=0.0003$,
$x_3(0)=0.0004$, $x_4(0)=1$, and the adjoint system
\begin{equation*}
\label{cap2:ode3_rubeola}
\begin{cases}
\dot{\lambda}_1=\lambda_1(b+u+\beta x_3) - \lambda_2\beta x_3\\
\dot{\lambda}_2=\lambda_1 b p + \lambda_2(e+b+p b)-\lambda_3 e\\
\dot{\lambda}_3=-A+\lambda_1(b q +\beta x_1)-\lambda_2\beta x_1+\lambda_3(g+b)\\
\dot{\lambda}_4=\lambda_4 b
\end{cases}
\end{equation*}
with transversality conditions\index{Transversality condition}
$\lambda_i(3)=0$, $i=1,\ldots,4$. The optimal control is given by
\begin{equation*}
u^{*}=\left\{
\begin{array}{lll}
0 & \textrm{ if } & \frac{\partial H}{\partial u}<0,\\[0.25cm]
\frac{\lambda_1 x_1}{2} & \textrm{ if } & \frac{\partial H}{\partial u}=0,\\[0.25cm]
0.9 & \textrm{ if } & \frac{\partial H}{\partial u}>0.
\end{array}
\right.
\end{equation*}
We present here the main part of the code for the backward-forward
sweep method with fourth order Runge--Kutta. The complete
code can be found in the website \cite{SofiaSITE}.
The obtained optimal curves for the states variables
and optimal control are shown in Figure~\ref{cap2_rubeola}.
{\tiny{
\begin{verbatim}
for i = 1:M
m11 = b-b*(p*x2(i)+q*x3(i))-b*x1(i)-beta*x1(i)*x3(i)-u(i)*x1(i);
m12 = b*p*x2(i)+beta*x1(i)*x3(i)-(e+b)*x2(i);
m13 = e*x2(i)-(g+b)*x3(i);
m14 = b-b*x4(i);
m21 = b-b*(p*(x2(i)+h2*m12)+q*(x3(i)+h2*m13))-b*(x1(i)+h2*m11)-...
beta*(x1(i)+h2*m11)*(x3(i)+h2*m13)-(0.5*(u(i) + u(i+1)))*(x1(i)+h2*m11);
m22 = b*p*(x2(i)+h2*m12)+beta*(x1(i)+h2*m11)*(x3(i)+h2*m13)-(e+b)*(x2(i)+h2*m12);
m23 = e*(x2(i)+h2*m12)-(g+b)*(x3(i)+h2*m13);
m24 = b-b*(x4(i)+h2*m14);
m31 = b-b*(p*(x2(i)+h2*m22)+q*(x3(i)+h2*m23))-b*(x1(i)+h2*m21)-...
beta*(x1(i)+h2*m21)*(x3(i)+h2*m23)-(0.5*(u(i) + u(i+1)))*(x1(i)+h2*m21);
m32 = b*p*(x2(i)+h2*m22)+beta*(x1(i)+h2*m21)*(x3(i)+h2*m23)-(e+b)*(x2(i)+h2*m22);
m33 = e*(x2(i)+h2*m22)-(g+b)*(x3(i)+h2*m23);
m34 = b-b*(x4(i)+h2*m24);
m41 = b-b*(p*(x2(i)+h2*m32)+q*(x3(i)+h2*m33))-b*(x1(i)+h2*m31)-...
beta*(x1(i)+h2*m31)*(x3(i)+h2*m33)-u(i+1)*(x1(i)+h2*m31);
m42 = b*p*(x2(i)+h2*m32)+beta*(x1(i)+h2*m31)*(x3(i)+h2*m33)-(e+b)*(x2(i)+h2*m32);
m43 = e*(x2(i)+h2*m32)-(g+b)*(x3(i)+h2*m33);
m44 = b-b*(x4(i)+h2*m34);
x1(i+1) = x1(i) + (h/6)*(m11 + 2*m21 + 2*m31 + m41);
x2(i+1) = x2(i) + (h/6)*(m12 + 2*m22 + 2*m32 + m42);
x3(i+1) = x3(i) + (h/6)*(m13 + 2*m23 + 2*m33 + m43);
x4(i+1) = x4(i) + (h/6)*(m14 + 2*m24 + 2*m34 + m44);
end
for i = 1:M
j = M + 2 - i;
n11 = lambda1(j)*(b+u(j)+beta*x3(j))-lambda2(j)*beta*x3(j);
n12 = lambda1(j)*b*p+lambda2(j)*(e+b-p*b)-lambda3(j)*e;
n13 = -A+lambda1(j)*(b*q+beta*x1(j))-lambda2(j)*beta*x1(j)+lambda3(j)*(g+b);
n14 = b*lambda4(j);
n21 = (lambda1(j) - h2*n11)*(b+u(j)+beta*(0.5*(x3(j)+x3(j-1))))-...
(lambda2(j) - h2*n12)*beta*(0.5*(x3(j)+x3(j-1)));
n22 = (lambda1(j) - h2*n11)*b*p+(lambda2(j) - h2*n12)*(e+b-p*b)-(lambda3(j) - h2*n13)*e;
n23 = -A+(lambda1(j) - h2*n11)*(b*q+beta*(0.5*(x1(j)+x1(j-1))))-...
(lambda2(j) - h2*n12)*beta*(0.5*(x1(j)+x1(j-1)))+(lambda3(j) - h2*n13)*(g+b);
n24 = b*(lambda4(j) - h2*n14);
n31 = (lambda1(j) - h2*n21)*(b+u(j)+beta*(0.5*(x3(j)+x3(j-1))))-...
(lambda2(j) - h2*n22)*beta*(0.5*(x3(j)+x3(j-1)));
n32 = (lambda1(j) - h2*n21)*b*p+(lambda2(j) - h2*n22)*(e+b-p*b)-(lambda3(j) - h2*n23)*e;
n33 = -A+(lambda1(j) - h2*n21)*(b*q+beta*(0.5*(x1(j)+x1(j-1))))-...
(lambda2(j) - h2*n22)*beta*(0.5*(x1(j)+x1(j-1)))+(lambda3(j) - h2*n23)*(g+b);
n34 = b*(lambda4(j) - h2*n24);
n41 = (lambda1(j) - h2*n31)*(b+u(j)+beta*x3(j-1))-(lambda2(j) - h2*n32)*beta*x3(j-1);
n42 = (lambda1(j) - h2*n31)*b*p+(lambda2(j) - h2*n32)*(e+b-p*b)-(lambda3(j) - h2*n33)*e;
n43 = -A+(lambda1(j) - h2*n31)*(b*q+beta*x1(j-1))-...
(lambda2(j) - h2*n32)*beta*x1(j-1)+(lambda3(j) - h2*n33)*(g+b);
n44 = b*(lambda4(j) - h2*n34);
lambda1(j-1) = lambda1(j) - h/6*(n11 + 2*n21 + 2*n31 + n41);
lambda2(j-1) = lambda2(j) - h/6*(n12 + 2*n22 + 2*n32 + n42);
lambda3(j-1) = lambda3(j) - h/6*(n13 + 2*n23 + 2*n33 + n43);
lambda4(j-1) = lambda4(j) - h/6*(n14 + 2*n24 + 2*n34 + n44);
end
u1 = min(0.9,max(0,lambda1.*x1/2));
\end{verbatim}}}
\begin{figure}
\caption{\label{cap2_rubeola}
\label{cap2_rubeola}
\end{figure}
\end{example}
There are several difficulties to overcome when
an optimal control problem is solved by indirect methods\index{Indirect methods}.
Firstly, it is necessary to calculate the Hamiltonian\index{Hamiltonian},
adjoint equations, the optimality\index{Optimality condition}
and transversality conditions\index{Transversality condition}.
Besides, the approach is not flexible, since each time a new problem is formulated,
a new derivation is required. In contrast, a direct method\index{Direct method}
does not require explicit derivation of necessary conditions.
Due to its simplicity, the direct approach has been gaining popularity
in numerical optimal control over the past three decades \cite{Betts2001}.
\subsection{Direct methods}
\label{sec:2:3}
A new family of numerical methods for dynamic
optimization has emerged, referred as direct methods.\index{Direct method}
This development has been driven by the industrial need to solve large-scale
optimization problems and it has also been supported by the rapidly increasing
computational power. A direct method\index{Direct method} constructs a sequence of points
$x_1, x_2,\ldots,x^{*}$, such that the objective function $F$ to be minimized
satisfies $F(x_1)>F(x_2)> \cdots >F(x^{*})$. Here the state and/or control
are approximated using an appropriate function approximation
(e.g., polynomial approximation or piecewise constant parameterization).
Simultaneously, the cost functional is approximated as a cost function.
Then, the coefficients of the function approximations are treated
as optimization variables and the problem is reformulated
as a standard nonlinear optimization problem (NLP):
\begin{gather*}
\underset{x}{\min} \ F(x) \\
\text{s.t. } \ c_{i}(x)=0, \quad i\in E, \\
c_{j}(x)\geq 0, \quad j \in I,
\end{gather*}
where $c_{i}$, $i\in E$, and $c_{j}$, $j\in I$,
are the set of equality and inequality constraints, respectively.
In fact, the NLP is easier to solve than the boundary-value problem,
mainly due to the sparsity of the NLP and the many well-known software programs
that can handle it. As a result, the range of problems that
can be solved via direct methods\index{Direct method} is significantly larger
than the range of problems that can be solved via indirect methods. Direct methods
have become so popular these days that many people have written sophisticated
software programs that employ these methods.
Here we present two types of codes/packages: specific solvers
for OC problems and standard NLP solvers used after a discretization process.
\subsubsection{Specific optimal control software}
\subsubsection*{OC-ODE}
\index{OC-ODE}
The \texttt{OC-ODE} \cite{Matthias2009}, \emph{Optimal Control of Ordinary-Differential Equations},
by Matthias Gerdts, is a collection of \texttt{Fortran 77} routines for optimal control problems subject
to ordinary differential equations. It uses an automatic direct discretization method
for the transformation of the OC problem into a finite-dimensional NLP.
\texttt{OC-ODE} includes procedures for numerical adjoint estimation and sensitivity analysis.
\begin{example}
\label{ex_rubeola_OC-ODE}
Considering the same problem of Example~\ref{example_rubeola}, we show the main part
of the code in \texttt{OC-ODE}. The complete code can be found in the website \cite{SofiaSITE}.
The achieved solution is similar to the indirect approach plotted in Figure~\ref{cap2_rubeola},
and therefore is omitted.
\scriptsize{
\begin{verbatim}
c Call to OC-ODE
c OPEN( INFO(9),FILE='OUT',STATUS='UNKNOWN')
CALL OCODE( T, XL, XU, UL, UU, P, G, BC,
+ TOL, TAUU, TAUX, LIW, LRW, IRES,
+ IREALTIME, NREALTIME, HREALTIME,
+ IADJOINT, RWADJ, LRWADJ, IWADJ, LIWADJ, .FALSE.,
+ MERIT,IUPDATE,LENACTIVE,ACTIVE,IPARAM,PARAM,
+ DIM,INFO,IWORK,RWORK,SOL,NVAR,IUSER,USER)
PRINT*,'Ausgabe der Loesung: NVAR=',NVAR
WRITE(*,'(E30.16)') (SOL(I),I=1,NVAR)
c CLOSE(INFO(9))
c READ(*,*)
END
c-------------------------------------------------------------------------
c Objective Function
c-------------------------------------------------------------------------
SUBROUTINE OBJ( X0, XF, TF, P, V, IUSER, USER )
IMPLICIT NONE
INTEGER IUSER(*)
DOUBLEPRECISION X0(*),XF(*),TF,P(*),V,USER(*)
V = XF(5)
RETURN
END
c-------------------------------------------------------------------------
c Differential Equation
c-------------------------------------------------------------------------
SUBROUTINE DAE( T, X, XP, U, P, F, IFLAG, IUSER, USER )
IMPLICIT NONE
INTEGER IFLAG,IUSER(*)
DOUBLEPRECISION T,X(*),XP(*),U(*),P(*),F(*),USER(*)
c INTEGER NONE
DOUBLEPRECISION B, E, G, P, Q, BETA, A
B = 0.012D0
E = 36.5D0
G = 30.417D0
P = 0.65D0
Q = 0.65D0
BETA = 527.59D0
A = 100.0D0
F(1) = B-B*(P*X(2)+Q*X(3))-B*X(1)-BETA*X(1)*X(3)-U(1)*X(1)
F(2) = B*P*X(2)+BETA*X(1)*X(3)-(E+B)*X(2)
F(3) = E*X(2)-(G+B)*X(3)
F(4) = B-B*X(4)
F(5) = A*X(3))+U(1)**2
RETURN
END
\end{verbatim}
}
\end{example}
\subsubsection*{DOTcvp}
\index{DOTcvp}
The \texttt{DOTcvp} \cite{Dotcvp}, \emph{Dynamic Optimization Toolbox with Vector Control Parametrization},
is a dynamic optimization toolbox for \texttt{Matlab}\index{Matlab}. The toolbox provides
an environment for a \texttt{Fortran} compiler to create the '.dll' files of the ODE, Jacobian, and sensitivities.
However, a \texttt{Fortran} compiler has to be installed in the \texttt{Matlab} environment.
The toolbox uses the control vector parametrization approach for the calculation
of the optimal control profiles, giving a piecewise solution for the control.
The OC problem has to be defined in Mayer form\index{Mayer form}.
For solving the NLP, the user can choose several deterministic solvers
--- \texttt{Ipopt}\index{Ipopt}, \texttt{Fmincon}, \texttt{FSQP} ---
or stochastic solvers --- \texttt{DE}, \texttt{SRES}.
The modified \texttt{SUNDIALS} tool \cite{Hindmarsh2005} is used for solving
the IVP and for the gradients and Jacobian automatic generation. Forward integration
of the ODE system is ensured by CVODES, a part of \texttt{SUNDIALS}, which is able to perform
the simultaneous or staggered sensitivity analysis too. The IVP can be solved with the Newton
or Functional iteration module and with the Adams or BDF linear multistep method. Note that the
sensitivity equations are analytically provided and the error control strategy for the sensitivity variables
can be enabled. \texttt{DOTcvp} has a user friendly graphical interface (GUI).
\begin{example}
Considering the same problem of Example~\ref{example_rubeola},
we present here a part of the code used in \texttt{DOTcvp}. The complete
code can be found in the website \cite{SofiaSITE}. The solution,
despite being piecewise continuous, follows the curves plotted in Figure~\ref{cap2_rubeola}.
\tiny{
\begin{verbatim}
data.odes.Def_FORTRAN = {''};
e.g. {'double precision k10, k20, ..'}
data.odes.parameters = {'b=0.012',' e=36.5',' g=30.417',' p=0.65',' q=0.65',' beta=527.59',
' d=0',' phi1=0','phi2=0','A=100 '};
data.odes.Def_MATLAB = {''};
data.odes.res(1) = {'b-b*(p*y(2)+q*y(3))-b*y(1)-beta*y(1)*y(3)-u(1)*y(1)'};
data.odes.res(2) = {'b*(p*y(2)+q*phi1*y(3))+beta*y(1)*y(3)-(e+b)*y(2)'};
data.odes.res(3) = {'b*q*phi2*y(3)+e*y(2)-(g+b)*y(3)'};
data.odes.res(4) = {'b-b*y(4)'};
data.odes.res(5) = {'A*y(3)+u(1)*u(1)'};
data.odes.black_box = {'None','1.0','FunctionName'};
for all constraints],...
[a black box model function name]
data.odes.ic = [0.0555 0.0003 0.0004 1 0];
data.odes.NUMs = size(data.odes.res,2);
data.odes.t0 = 0.0;
data.odes.tf = 3;
data.odes.NonlinearSolver = 'Newton';
Functional for non-stiff problems
data.odes.LinearSolver = 'Dense';
['GMRES'|'BiCGStab'|'TFQMR'] /for the Newton NLS
data.odes.LMM = 'Adams';
BDF for stiff problems
data.odes.MaxNumStep = 500;
data.odes.RelTol = 1e-007;
data.odes.AbsTol = 1e-007;
data.sens.SensAbsTol = 1e-007;
data.sens.SensMethod = 'Staggered';
data.sens.SensErrorControl= 'on';
data.nlp.RHO = 10;
data.nlp.problem = 'min';
data.nlp.J0 = 'y(5)';
data.nlp.u0 = [0 ];
data.nlp.lb = [0 ];
data.nlp.ub = [0.9];
data.nlp.p0 = [];
data.nlp.lbp = [];
data.nlp.ubp = [];
data.nlp.solver = 'IPOPT';
data.nlp.SolverSettings = 'None';
for NLP solver, if does not exist use ['None']
data.nlp.NLPtol = 1e-005;
data.nlp.GradMethod = 'FiniteDifference';
data.nlp.MaxIter = 1000;
data.nlp.MaxCPUTime = 60*60*0.25;
(60*60*0.25) = 15 minutes
data.nlp.approximation = 'PWC';
free time problem
data.nlp.FreeTime = 'off';
data.nlp.t0Time = [data.odes.tf/data.nlp.RHO];
data.nlp.lbTime = 0.01;
data.nlp.ubTime = data.odes.tf;
data.nlp.NUMc = size(data.nlp.u0,2);
data.nlp.NUMi = 0;
control variables,
if not equal to 0 you need to use some MINLP solver ['ACOMI'|'MISQP'|'MITS']
data.nlp.NUMp = size(data.nlp.p0,2);
\end{verbatim}
}
\normalsize
\end{example}
\subsubsection*{Muscod-II}
\index{Muscod-II}
In NEOS\index{NEOS platform} platform \cite{NEOS}, there is a large set
of software packages. NEOS is considered as the state of the art in optimization.
One recent solver is \texttt{Muscod-II} \cite{Muscod} (Multiple Shooting CODe for Optimal Control)
for the solution of mixed integer nonlinear ODE or DAE constrained optimal
control problems in an extended \texttt{AMPL}\index{AMPL} format.
\texttt{AMPL} is a modelling language for mathematical programming
created by Fourer, Gay and Kernighan \cite{AMPL}. The modelling languages organize
and automate the tasks of modelling, which can handle a large volume of data and,
moreover, can be used in machines and independent solvers, allowing the user
to concentrate on the model instead of the methodology to reach the solution.
However, the \texttt{AMPL} modelling language itself does not allow the formulation
of differential equations. Hence, the \texttt{TACO Toolkit} has been designed
to implement a small set of extensions for easy and convenient modeling
of optimal control problems in \texttt{AMPL}, without the need for explicit encoding
of discretization schemes. Both the \texttt{TACO Toolkit} and the
NEOS interface to \texttt{Muscod-II} are still under development.
\begin{example}
\ \
\scriptsize{
\begin{verbatim}
include OptimalControl.mod;
var t ;
var x1, >=0 <=1;
var x2, >=0 <=1;
var x3, >=0 <=1;
var x4, >=0 <=1;
var u >=0, <=0.9 suffix type "u0";
minimize
cost: integral (A*x3+u^2,3);
subject to
c1: diff(x1,t) = b-b*(p*x2+q*x3)-b*x1-beta*x1*x3-u*x1;
c2: diff(x2,t) = b*p*x2+beta*x1*x3-(e+b)*x2;
c3: diff(x3,t) = e*x2-(g+b)*x3;
c4: diff(x4,t) = b-b*x4;
\end{verbatim}
}
\normalsize
\end{example}
\subsubsection{Nonlinear optimization software}
The three nonlinear optimization software packages presented here,
were used through the NEOS platform with codes formulated in \texttt{AMPL}\index{AMPL}.
\subsubsection*{Ipopt}
The \texttt{Ipopt} \cite{Ipopt}, \emph{Interior Point OPTimizer},\index{Ipopt}
is a software package for large-scale nonlinear optimization.
It is written in \texttt{Fortran} and \texttt{C}. \texttt{Ipopt} implements
a primal-dual interior point method and uses a line search strategy based on the filter method.
\texttt{Ipopt} can be used from various modeling environments.
It is designed to exploit 1st and 2nd derivative information,
if provided, usually via automatic differentiation routines in modeling
environments such as \texttt{AMPL}\index{AMPL}. If no Hessians are provided,
\texttt{Ipopt} will approximate them using a quasi-Newton method, specifically a BFGS update.
\begin{example}
\label{ex_rubeola_IPOPT}
Continuing with problem of Example~\ref{example_rubeola}, the \texttt{AMPL} code
is here shown for \texttt{Ipopt}. The Euler discretization was selected.
This code can also be implemented in other nonlinear software packages
available in NEOS platform, reason why the code for the next two software packages
will not be shown. The full version can be found on the website \cite{SofiaSITE}.
\scriptsize{
\begin{verbatim}
#### OBJECTIVE FUNCTION ###
minimize cost: fc[N];
#### CONSTRAINTS ###
subject to
i1: x1[0] = x1_0;
i2: x2[0] = x2_0;
i3: x3[0] = x3_0;
i4: x4[0] = x4_0;
i5: fc[0] = fc_0;
f1 {i in 0..N-1}: x1[i+1] = x1[i] + (tf/N)*(b-b*(p*x2[i]+q*x3[i])-b*x1[i]
-beta*x1[i]*x3[i]-u[i]*x1[i]);
f2 {i in 0..N-1}: x2[i+1] = x2[i]+(tf/N)*(b*p*x2[i]+beta*x1[i]*x3[i]-(e+b)*x2[i]);
f3 {i in 0..N-1}: x3[i+1] = x3[i] + (tf/N)*(e*x2[i]-(g+b)*x3[i]);
f4 {i in 0..N-1}: x4[i+1] = x4[i] + (tf/N)*(b-b*x4[i]);
f5 {i in 0..N-1}: fc[i+1] = fc[i] + (tf/N)*(A*x3[i]+u[i]^2);
\end{verbatim}
}
\normalsize
\end{example}
\subsubsection*{Knitro}
\index{Knitro}
\texttt{Knitro} \cite{Knitro}, short for ``Nonlinear Interior point Trust Region Optimization'',
was created primarily by Richard Waltz, Jorge Nocedal, Todd Plantenga and Richard Byrd.
It was introduced in 2001 as a derivative of academic research at Northwestern,
and has undergone continual improvement since then.
\texttt{Knitro} is also a software for solving large scale mathematical optimization problems
based mainly on the two Interior Point (IP) methods and one active set algorithm.
\texttt{Knitro} is specialized for nonlinear optimization, but also solves linear programming problems,
quadratic programming problems, and systems of nonlinear equations. The unknowns in these problems
must be continuous variables in continuous functions. However, functions can be convex or nonconvex.
The code also provides a multistart option for promoting the computation of the global minimum.
This software was tested through the NEOS platform\index{NEOS platform}.
\subsubsection*{Snopt}
\index{Snopt}
\texttt{Snopt} \cite{Snopt}, by Philip Gill, Walter Murray and Michael Saunders,
is a software package for solving large-scale optimization problems (linear and nonlinear programs).
It is specially effective for nonlinear problems whose functions and gradients are expensive to evaluate.
The functions should be smooth but do not need to be convex.
\texttt{Snopt} is implemented in \texttt{Fortran 77} and distributed as source code.
It uses the SQP (Sequential Quadratic Programming) philosophy,
with an augmented Lagrangian approach combining a trust region
adapted to handle the bound constraints. \texttt{Snopt} is also available
in NEOS platform\index{NEOS platform}.
\section{Conclusion}
Choosing a method for solving an optimal control problem depends largely on the type of problem to be
solved and the amount of time that can be invested in coding.
An indirect shooting method has the advantage of being simple to understand
and produces highly accurate solutions when it converges \cite{Rao2009}.
The accuracy and robustness of a direct method\index{Direct method}
is highly dependent upon the method used. Nevertheless, it is easier
to formulate highly complex problems in a direct way and standard
NLP solvers can be used, which is an extra advantage. This last feature has the benefit of converging
with poor initial guesses and being extremely computationally efficient since most
of the solvers exploit the sparsity of the derivatives in the constraints and objective function.
\label{lastpage-01}
\end{document} | math | 46,129 |
\begin{document}
\title{Phase gadget compilation for diagonal qutrit gates}
\begin{abstract}
Phase gadgets have proved to be an indispensable tool for reasoning about ZX-diagrams, being used in optimisation and simulation of quantum circuits and the theory of measurement-based quantum computation. In this paper we study phase gadgets for qutrits. We present the \emph{flexsymmetric} variant of the original qutrit ZX-calculus, which allows for rewriting that is closer in spirit to the original (qubit) ZX-calculus. In this calculus phase gadgets look as you would expect, but there are non-trivial differences in their properties. We devise new qutrit-specific tricks to extend the graphical Fourier theory of qubits, resulting in a translation between the `additive' phase gadgets and a `multiplicative' counterpart we dub \emph{phase multipliers}. This enables us to build different types of qutrit multiple-controlled phase gates.
As an application of these results we find a construction for \emph{emulating} arbitrary qubit diagonal unitaries, and specifically find an emulation for the qubit CCZ gate that only requires three single-qutrit non-Clifford gates to implement --- provably lower than the four $T$ gates needed using just qubit gates.
\end{abstract}
\section{Introduction}
Most quantum computing theory developed thus far has focussed on qubits --- two-level quantum systems.
However, there has been a recent surge of interest in studying the more general case of $d$-level quantum systems, called \emph{qudits}. This has led to applications of qudits for quantum algorithms~\cite{WangY2020quditsreview}, improving magic state distillation noise thresholds~\cite{CampbellE2014quditmsdthresholds}, and communication noise resilience~\cite{CozzolinoD2019quditcommunication}.
Qudits have been experimentally demonstrated on quantum processors based on ion traps~\cite{RingbauerM2021quditions} and superconducting devices~\cite{BlokM2021scrambling,YeB2018cphasephoton,YurtalanM2020Walsh-Hadamard,HillA2021doublycontrolled}.
The specific case of qu\emph{trits}, where $d=3$, has been used to improve qubit readout~\cite{MalletF2009qubitreadout2state}, but most notably, qutrits have been used to study \emph{emulation}: where qubit computation is emulated inside a subspace of the qudits to enable more resource-efficient gate implementations. While using qudits to emulate qubit computation can lead to efficiency advantages, the converse is not the case~\cite{BullockS2005qubitemuqudit}.
Most work on qutrits and emulation has focussed on \emph{classical} functions: those that come from a map of classical trits.
For instance, using qutrits we can build logarithmic-depth Toffolis~\cite{GokhaleP2019asymptotic,NikolaevaAS2022mctqutrit} and binary AND gates on superconducting qutrits~\cite{ChuJ2021superconductingand}.
This leaves open the question of whether there is any advantages to emulation by studying `truly' quantum gates such as diagonal unitaries.
For qubits a useful tool for understanding diagonal unitaries has been the concept of a \emph{phase gadget}~\cite{KissingerA2020reduc}. This is a type of symmetric multi-qubit interaction that occurs naturally in many hardware architectures~\cite{SheldonS2016IBMCRgate,PinoJM2021honeywellarchitecture,PetitL2020silicontwoqubitgates}, and serves as a good basis for optimising quantum circuits~\cite{phaseGadgetSynth,cowtan2020generic,deBeaudrapN2020reducepifourphase,deBeaudrapN2020treducspidernest,vandeWeteringJ2021globalgates,Backens2020extraction}. Any diagonal qubit unitary can be expressed as a product of phase gadgets by writing the unitary as a \emph{phase polynomial}~\cite{AmyVerification,deGriendA2020architecturephasepoly}.
In this paper we study the generalisation of phase gadgets to the qutrit setting. We do this by adapting the qutrit ZX-calculus of Refs.~\cite{GongX2017equivalence,WangQ2018qutrit} and transforming it into a \emph{flexsymmetric} calculus~\cite{Carette2021OTM} where the spiders have more desirable symmetry properties. We find this calculus has a simple set of rules for the Clifford fragment. We define phase gadgets analogously to the qubit case, meaning that as diagrams they look nearly identical. There are however significant differences between the qubit and qutrit gadgets. We will show that we can nevertheless use qutrit phase gadgets for compiling qutrit diagonal unitaries, such as controlled phase gates, and a type of gate we dub a \emph{phase multiplier}. This last one is possible by generalising the formula that leads to the \emph{graphical Fourier theory} for qubit diagonal unitaries~\cite{GraphicalFourier2019}.
As an application of our results we show how we can emulate an arbitrary qubit diagonal unitary using qutrit phase gadgets. This leads us to a construction of the emulated qubit CCZ gate that requires only three non-Clifford qutrit \emph{$R$ gates}~\cite{GlaudellA2022qutritmetaplecticsubset}. This is surprising because using just qubits, we would require at least four $T$ gates to implement the CCZ~\cite{HowardM2017resourcetheorymagic}.
We start the paper by reviewing the basics of qutrit quantum computation in Section~\ref{sec:qutrit-computing}. Then we introduce the flexsymmetric qutrit ZX-calculus in Section~\ref{sec:qutrit-ZX}. Diagonal qutrit unitaries, phase gadgets, controlled phase gates, and phase multipliers are studied in Section~\ref{sec:diagonal-gates}. We show how to use these to emulate diagonal qubit unitaries in Section~\ref{sec:emulation-application} and end with some discussion on future work in Section~\ref{sec:conclusion}.
\section{Qutrit quantum computation}\label{sec:qutrit-computing}
A qubit is a two-dimensional Hilbert space. Similarly, a qutrit is a three-dimensional Hilbert space. We will write $\ket{0}$, $\ket{1}$, and $\ket{2}$ for the standard computational basis states of a qutrit.
Any normalised qutrit state can then be written as $\ket{\psi} = \alpha \ket{0} + \beta \ket{1} + \gamma \ket{2}$ where $\alpha,\beta,\gamma\in \mathbb{C}$ and $|\alpha|^2 + |\beta|^2 + |\gamma|^2 = 1$.
Several concepts for qubits extend to qutrits, or more generally to qu\emph{dits}, which are $d$-dimensional quantum systems. In particular, the concept of Pauli's and Cliffords.
For a $d$-dimensional qudit, we define the respective Pauli $X$ and $Z$ gates as
\begin{equation}
X\ket{k} = \ket{k+1} \qquad\qquad Z\ket{k} = \omega^k \ket{k}
\end{equation}
where $\omega:= e^{2\pi i/d}$ is such that $\omega^d = 1$, and the addition $\ket{k+1}$ is taken modulo $d$~\cite{GottesmanD1999ftqudit,HowardM2012quditTgate}.
We define the \emph{Pauli group} as the set of unitaries generated by tensor products of the $X$ and $Z$ gate.
For qubits this $X$ gate is just the NOT gate, while $Z=\text{diag}(1,-1)$. In this paper we will work solely with qutrits, so we take $\omega$ to always be equal to $e^{2\pi i/3}$. Note that $\omega^{-1} = \omega^2 = \bar{\omega}$ where $\bar{z}$ denotes the complex conjugate of $z$.
For a qubit there is only one non-trivial permutation of the standard basis states, implemented by the $X$ gate.
For qutrits there are five non-trivial permutations of the basis states. By analogy we will call them all ternary $X$ gates. These gates are $X_{+1}$, $X_{-1}$, $X_{01}$, $X_{12}$, and $X_{02}$.
$X_{\pm 1}$ sends $\ket{t}$ to $\ket{(t \pm 1) \text{ mod } 3}$ for $t \in \{0, 1, 2\}$; $X_{01}$ is just the qubit $X$ gate which is the identity when the input is $\ket{2}$; $X_{12}$ sends $\ket{1}$ to $\ket{2}$ and $\ket{2}$ to $\ket{1}$, and likewise for $X_{02}$.
Note that the qutrit Pauli $X$ gate is the $X_{+1}$ gate, while $X^\dagger = X_{-1} = X^2$.
Another concept that translates to qutrits (or more generally qudits) is that of Clifford unitaries.
\begin{definition}
Let $U$ be a qudit unitary acting on $n$ qudits. We say it is \emph{Clifford} when every Pauli is mapped to another Pauli under conjugation by $U$. I.e.~if $UPU^\dagger$ is in the Pauli group for any Pauli $P$.
\end{definition}
The set of $n$-qudit Cliffords forms a group under composition. For qubits, this group is generated by the $S$, Hadamard and CX gates. The same is true for qutrits, for the right generalisation of these gates.
\begin{definition}
The qutrit $S$ gate is $S:= \text{diag}(1,1,\omega)$. I.e.~it multiplies the $\ket{2}$ state by the phase $\omega$.
\end{definition}
For qubits, the Hadamard gate interchanges the $Z$ eigenbasis $\{\ket{0},\ket{1}$ and the $X$ eigenbasis consisting of the states $\ket{\pm} := \frac{1}{\sqrt{2}}(\ket{0}\pm \ket{1})$. The same holds for the qutrit Hadamard.
In this case the $X$ basis consists of the following states:
\begin{equation*}
\ket{+} \ := \ \frac{1}{\sqrt{3}} (\ket{0}+\ket{1}+\ket{2}) \qquad
\ket{\omega} :=\ \frac{1}{\sqrt{3}} (\ket{0}+\omega\ket{1}+\bar{\omega}\ket{2}) \qquad
\ket{\bar{\omega}} \ :=\ \frac{1}{\sqrt{3}} (\ket{0}+\bar{\omega}\ket{1}+\omega\ket{2})
\end{equation*}
\begin{definition}
The \emph{qutrit Hadamard gate} $H$ is the unitary mapping $\ket{0} \mapsto \ket{+}$, $\ket{1}\mapsto \ket{\omega}$ and $\ket{2} \mapsto \ket{\bar{\omega}}$.
\begin{equation}\label{eq:hgatedef}
H \ := \ \frac{1}{\sqrt{3}}\begin{pmatrix}
1 & 1 & 1 \\
1 & \omega & \bar{\omega} \\
1 & \bar{\omega} & \omega
\end{pmatrix}
\end{equation}
\end{definition}
Note that, unlike the qubit Hadamard, the qutrit Hadamard is \emph{not} self-inverse.
Instead we have $H^2 = -X_{12}$ so that $H^4 = \mathbb{I}$. This means that $H^\dagger = H^3$.
The final Clifford gate we need is the qutrit CX.
\begin{definition}
The qutrit CX is defined such that $\text{CX}\ket{i,j} = \ket{i,(i+j)~\text{mod }3}$, where $i,j\in \{0,1,2\}$.
\end{definition}
Any qutrit Clifford unitary can be written as a composition of $S$, $H$ and CX gates (up to global phase).
Clifford gates are efficiently classically simulable, so we need to add another gate to get a (approximately) universal gate set for quantum computing~\cite{GottesmanD1999ftqudit}. This will be a phase gate.
\begin{definition}\label{def:Z-phase-gate}
We write $Z(a,b)$ for the \emph{phase gate} that acts as $Z(a,b)\ket{0} = \ket{0}$, $Z(a,b)\ket{1} = \omega^a\ket{1}$ and $Z(a,b)\ket{2} = \omega^b\ket{2}$ where we take $a,b\in \mathbb{R}$.
\end{definition}
We define $Z(a,b)$ in this way, taking $a$ and $b$ to correspond to phases that are multiples of $\omega$, because then $Z(a,b)$ is Clifford iff $a$ and $b$ are both integers, so that we can easily see from the parameters whether the gate is Clifford or not. The group of $Z(a,b)$ phase gates constitutes the group of diagonal single-qutrit unitaries modded out by a global phase. Composition of these gates is given by $Z(a,b)\cdot Z(c,d)=Z(a+b,c+d)$. Note that $S = Z(0,1)$.
This brings us to the definition of the qutrit $T$ gate.
\begin{definition}
The qutrit $T$ gate is defined as $T := Z(\frac{1}{3}, -\frac{1}{3}) = \text{diag}(1,\omega^{\frac{1}{3}},\omega^{-\frac{1}{3}})$~\cite{PrakashS2018normalform,CampbellE2012tgatedistillation,HowardM2012quditTgate}.
\end{definition}
Like the qubit $T$ gate, the qutrit $T$ gate belongs to the third level of the Clifford hierarchy, can be injected into a circuit using magic states, and its magic states can be distilled by magic state distillation. This means that we can fault-tolerantly implement this qutrit $T$ gate on many types of quantum error correcting codes.
Also as for qubits, the qutrit Clifford+$T$ gate set is approximately universal, meaning that we can approximate any qutrit unitary using just Clifford gates and the $T$ gate~\cite[Theorem 1]{CuiS2015universalmetaplectic}.
There is another useful single-qutrit non-Clifford gate.
\begin{definition}
The qutrit \emph{reflection} gate is defined as $R:= Z(0,3/2) = \text{diag}(1,1,-1)$.
\end{definition}
Like the $T$ gate, the $R$ gate can be added to the Clifford gate set to attain universality~\cite{GottesmanD1999ftqudit}, as explicitly proved in Ref.~\cite[Theorem~2]{CuiS2015universalmetaplectic}.
It can be exactly synthesized fault-tolerantly in three known ways: magic state distillation followed by repeat-until-success injection~\cite{AnwarH2012r2distillation}, braiding and topological measurement of weakly-integral non-Abelian anyons~\cite{CuiS2015universalweakly,CuiS2015universalmetaplectic} followed by repeat-until-success injection~\cite{AnwarH2012r2distillation}, or unitarily in qutrit Clifford+$T$~\cite{GlaudellA2022qutritmetaplecticsubset}.
\subsection{Controlled unitaries}
When we have an $n$-qubit unitary $U$, we can speak of the controlled gate that implements $U$. This is the $(n+1)$-qubit gate that acts as the identity when the first qubit is in the $\ket{0}$ state, and implements $U$ on the last $n$ qubits if the first qubit is in the $\ket{1}$ state.
For qutrits there are multiple notions of control.
\begin{definition}\label{def:ket2-controlled}
Let $U$ be a qutrit unitary. Then the \emph{$\ket{2}$-controlled $U$} is the unitary $\ket{2}$-$U$ that acts as
\begin{equation*}
\ket{0}\otimes \ket{\psi} \mapsto \ket{0}\otimes \ket{\psi} \qquad
\ket{1}\otimes \ket{\psi} \mapsto \ket{1}\otimes \ket{\psi} \qquad
\ket{2}\otimes \ket{\psi} \mapsto \ket{2}\otimes U\ket{\psi}
\end{equation*}
I.e.~it implements $U$ on the last qutrits if and only if the first qutrit is in the $\ket{2}$ state.
\end{definition}
Note that by conjugating the first qutrit with $X_{+1}$ or $X_{-1}$ gates we can make the gate also be controlled on the $\ket{1}$ or $\ket{0}$ state.
A different notion of qutrit control was introduced in Ref.~\cite{BocharovA2017ternaryshor} where if the control is in the $\ket{x}$ state, then it should apply $U^x$ on the target, i.e.~apply $U$ once iff $x=1$ and $U^2$ iff $x=2$.
An example of this is the Clifford CX gate defined earlier, which applies $X_{+1}$ when the control is $\ket{1}$ and $X_{+2}$ when it is $\ket{2}$.
Note that we can get this latter notion of control from the former: just apply a $\ket{1}$-controlled $U$, followed by a $\ket{2}$-controlled $U^2$.
A number of Clifford+$T$ constructions for controlled qutrit unitaries are already known. For instance, all the $\ket{2}$-controlled permutation $X$ gates can be built from the constructions given in Ref.~\cite{BocharovA2016ternaryarithmetics}.
In our previous work, we provided ancilla-free explicit constructions for any multiple-controlled Clifford+$T$ unitary in the Clifford+$T$ gate set, with gate count polynomial in the number of controls~\cite{YehL2022qutritcontrolledcliffordplust}.
In this work, by using the qutrit ZX-calculus, we will build upon our previous results and show how to construct multiple-controlled phase gates for an arbitrary phase.
\section{The qutrit ZX-calculus}\label{sec:qutrit-ZX}
We will assume the reader has some familiarity with the original qubit ZX-calculus~\cite{CoeckeB2011interacting}. For a review see Ref.~\cite{vandewetering2020zxcalculus}.
A qutrit ZX-calculus was presented and used in Refs.~\cite{RanchinA2014quditzx,WangQ2014qutritcalculus,GongX2017equivalence,WangQ2018qutrit}. While quite similar to the qubit one, it loses some of the properties that make the original easy to work with. In particular, for each X-spider, the distinction between its input wires and output wires becomes important. This means we can no longer treat qutrit ZX-diagrams as undirected graphs with the spiders as vertices. This makes intuitive reasoning about these diagrams harder, and also complicates the implementation of software for dealing with these diagrams.
Here we will present a variation on the qutrit ZX-calculus of Refs.~\cite{GongX2017equivalence,WangQ2018qutrit} where the spiders do enjoy this additional symmetry between inputs and outputs. The way we do this is by redefining the X-spider. In the original qutrit ZX-calculus we have
\begin{equation}
\tikzfig{Xsp} \ \ \propto \sum_{\substack{\vec x, \vec y\\x_1+\cdots+x_n = y_1+\cdots + y_m}} \!\!\!\!\!\!\!\!\ketbra{\vec y}{\vec x}.
\end{equation}
Here the sum $x_1+\cdots+x_n = y_1+\cdots + y_m$ is taken modulo 3. If we put a cup on one of the wires to turn an output into an input, then this has the effect of introducing a minus sign on that variable, changing for instance $x_1+x_2 = y_1+y_2$ into $x_1+x_2-y_2 = y_1$. For qubits this is not a problem since $-x = x$ modulo $2$, but for qutrits this changes the map. We fix this by defining a new X-spider as
\begin{equation}\label{eq:XDsp}
\tikzfig{XDsp} \ \ \propto \sum_{\substack{\vec x, \vec y\\x_1+\cdots+x_n+y_1+\cdots + y_m = 0}} \!\!\!\!\!\!\!\!\ketbra{\vec y}{\vec x}.
\end{equation}
We see that in this definition the inputs and outputs are treated on equal footing. In order to prevent confusion with earlier work, we will denote this new X-spider in pink, instead of in red.
Let's now give the full definition of the spiders. We define the Z-spider as
\[\tikzfig{Zsp-phase} \ = \ \ketbra{0\cdots 0}{0\cdots 0} \ +\ \omega^\alpha \ketbra{1\cdots 1}{1\cdots 1} \ +\ \omega^\beta \ketbra{2\cdots 2}{2\cdots 2}\]
Here we have two phase angles $\alpha$ and $\beta$, as opposed to just the one angle in qubit ZX. In general, for a $d$-dimensional spider, you will need to specify $d-1$ phases. In particular, when written in a spider $\frac{\alpha}{\beta}$ should be interpreted as two different phases and not as the fraction $\alpha/\beta$.
Note that we define the phase angles as $\omega^\alpha$ and $\omega^\beta$ so that these correspond to the complex phases $e^{2\pi/3 \alpha}$ and $e^{2\pi/3 \beta}$. This means that when $\alpha$ and $\beta$ are integers, that the spiders correspond to the Clifford fragment of the calculus.
We define the X-spider similarly, but with respect to the X-basis:
\[\tikzfig{XDsp-phase} \ = \ \ketbra{+\cdots +}{+\cdots +} \ +\ \omega^\alpha \ketbra{\bar{\omega}\cdots \bar{\omega}}{\omega\cdots \omega} \ +\ \omega^\beta \ketbra{\omega\cdots \omega}{\bar{\omega}\cdots \bar{\omega}}\]
This requires some explanation, because this does not look symmetric in the inputs and outputs. However, note that
$\bra{\omega} = (\ket{\omega})^\dagger = (\ket{0}+\omega \ket{1}+\omega \ket{2})^\dagger = \bra{0} + \bar{\omega} \bra{1} + \omega \bra{2}$.
Hence, if we take the transpose of $\ket{\omega}$ we actually get $\bra{\bar{\omega}}$.
It is straightforward to check that with $\alpha=\beta=0$ we get back Eq.~\eqref{eq:XDsp}. These definitions of the Z-spider and X-spider satisfy the symmetry properties we want, namely:
\begin{equation*}
\scalebox{0.9}{\tikzfig{spider-symmetries}}
\end{equation*}
These symmetries mean our spiders are \emph{flexsymmetric}, as defined by Carette~\cite{Carette2021OTM}, and as a result we may treat our ZX-diagrams as undirected graphs with the spiders as vertices. Note that here the cups and caps are defined with respect to the $Z$ basis: $\subset \ =\ \ket{00}+\ket{11}+\ket{22}$. As usual, our calculus also formally has generators for the identity wire and the swap.
It will be useful to introduce an additional graphical generator for the Hadamard:
\begin{equation}
\left\llbracket\tikzfig{RGgenerator//RGg_Hada}\right\rrbracket=\ket{+}\bra{0}+ \ket{\omega}\bra{1}+\ket{\bar{\omega}}\bra{2}=\ket{0}\bra{+}+ \ket{1}\bra{\bar{\omega}}+\ket{2}\bra{\omega}
\end{equation}
We write the Hadamard as a slanted box, because it is self-transpose, but not self-adjoint, and so should be denoted in a way that is symmetric under a rotation, but not a reflection.
Our redefinition of the X-spider comes at a `cost'. Namely, the 1-input, 1-output X-spider is no longer the identity:
$
\tikzfig{XD1-1} \ = \ \ket{0}\bra{0}+ \ket{2}\bra{1}+\ket{1}\bra{2} \ = \ \ket{+}\bra{+}+\ket{\bar{\omega}}\bra{\omega}+ \ket{\omega}\bra{\bar{\omega}}
$.
This map is implementing $\ket{x} \mapsto \ket{-x}$ where $-x$ is taken modulo 3, and is equal to $X_{12}$. Additionally, the X-spider is not really a spider any more in the sense that it doesn't satisfy the standard spider-fusion equation. Instead it satisfies the `harvestman equation'~\cite{Carette2021OTM} that also holds for for instance the W-spider~\cite{hadzihasanovic2015diagrammatic} and H-box~\cite{EPTCS287.2}:
\[\tikzfig{XD_rules/spidernewprime}\]
In Figure~\ref{fig:qutritphaseexactflexsymmetricrules}, we present a full set of rewrite rules for this qutrit ZX-calculus.
We have accounted for the global phase for each rule here as a complex number, as those will be relevant to us. Note however that the rewrite rules are not scalar-accurate as we are ignoring factors of $\sqrt{3}$.
\begin{figure}
\caption{Rules for the flexsymmetric qutrit ZX-calculus. These hold for all $\alpha,\beta,\eta,\theta\in\mathbb{R}
\label{fig:qutritphaseexactflexsymmetricrules}
\end{figure}
Using these rules, other useful qutrit ZX-calculus rewrite rules may be derived.
In particular, we can use these rules to prove the derived rules presented in Figure~\ref{fig:qutritphaseexactflexsymmetricderivedrules}.
\begin{figure}
\caption{These rules are derivable from the rules of Figure~\ref{fig:qutritphaseexactflexsymmetricrules}
\label{fig:qutritphaseexactflexsymmetricderivedrules}
\end{figure}
As these rules are (a slight variation) on the non-flexsymmetric qutrit rules of Ref.~\cite{WangQ2018qutrit}, our calculus is also complete for the qutrit Clifford fragment (when ignoring non-zero scalars).
The proofs of the derived rules of Figure~\ref{fig:qutritphaseexactflexsymmetricderivedrules} are given in Appendix~\ref{appendix:derivedrules}. We show in Appendix~\ref{app:necessity-of-rules} that most rules of Figure~\ref{fig:qutritphaseexactflexsymmetricrules} are necessary (i.e.~not derivable from the others).
We see in these rules that there is a special role for phases of the form $\frac{x}{2x}$ where $x\in \{0,1,2\}$. This is because $\tikzfig{ket-x-2x} \propto \ket{x}$ and $\tikzfig{Z-x-phase} = Z^x$. These relations can be derived by using the identity $1+\omega+\bar{\omega} = 0$ together with $\omega^2 = \omega^{-1} = \bar{\omega}$. In general we will see a lot of $\tikzfig{Z-alpha-2alpha}$ phases because they implement the $\ket{x}\mapsto e^{i\alpha x}\ket{x}$ phase gate.
Additionally, note that the flexsymmetric $(P2)$ rule on the qubit subspace is exactly the familiar qubit ZX rule \tikzfig{XD_rules/qubitk2}, since the red $\pi$ is the qubit Pauli $X$ while the pink $\frac{1}{2}$ phase is the qutrit Pauli $X$.
\section{Diagonal qutrit gates}\label{sec:diagonal-gates}
\subsection{Phase gadgets}
For qubits the concept of a \emph{phase gadget} has proven very useful. There's several different ways we can define a qubit phase gadget.
One way is to consider it as the diagonal gate $\ket{x,y}\mapsto e^{i\alpha (x\oplus y)} \ket{x,y}$ (for simplicity we are only considering a two-qubit phase gadget). This applies a phase of $e^{i\alpha}$ when $x\oplus y = 1$. Here $\oplus$ is the XOR operation, which is the addition on $\mathbb{Z}_2$. This suggests that we should define the qutrit phase gadget as $\ket{x,y}\mapsto e^{i\alpha (x+y)}\ket{x,y}$ where now we take $x+y$ to be modulo 3.
We could also define a phase gadget by its circuit realisation or diagrammatic representation. For qubits~\cite{KissingerA2020reduc}:
\[\tikzfig{phase-gadget-qubit}\]
We claim the qutrit variant of this construction is given by the following circuit which can be simplified to a similar diagrammatic representation:
\begin{equation}\label{eq:phase-gadget-qutrit-simp}
\tikzfig{phase-gadget-qutrit-simp}
\end{equation}
Indeed, inputting $\ket{x,y}$ into this diagram allows us to calculate its action:
\begin{equation}\label{eq:phase-gadgets-qutrit-calc}
\tikzfig{phase-gadget-qutrit-calc}
\end{equation}
This `floating scalar' expression evaluates to $\sqrt{3} e^{i\alpha (x+y \text{ mod }3)}$, so that this diagram indeed implements the operation we want, and we see that these three ways to define a qubit phase gadget, via the action, via the circuit, or via the diagrammatic representation are also equal for qutrits.
We can easily generalise this construction to an arbitrary number of qutrits:
\begin{equation}\label{eq:phase-gadget-qutrit-multi}
\tikzfig{phase-gadget-qutrit-multi} \ \ :: \ \ \ket{x,y,z,w} \ \mapsto \ e^{i\alpha (x+y+z+w \text{ mod }3)} \ket{x,y,z,w}
\end{equation}
We can also define more general phase gadgets where the phases don't have to be related to each other, i.e. we can replace $Z(\alpha, 2\alpha)$ with $Z(\alpha, \beta)$.
In this case we would still be calculating the value $x+y+z+w$ modulo 3, but then we apply a different phase depending on the value of this sum: if it is $0$ we don't apply any phase; if it is $1$ we apply $e^{i\alpha}$; and if it is $2$ we apply $e^{i\beta}$.
A particularly relevant choice of phases here is when $\alpha=\beta$. In this case, we apply the phase iff the sum value is not $0$. For a trit $x$ it turns out that $x^2 = 0$ if $x=0$ and $x^2 = 1$ otherwise --- this is actually a consequence of Fermat's little theorem and generalises to $x^{p-1} = 1$ modulo $p$ when $x\neq 0$ for $p$ prime. Hence:
\begin{equation}\label{eq:Z-alpha-alpha}
\tikzfig{Z-alpha-alpha} :: \ket{x} \mapsto e^{i\alpha (x^2 \text{ mod }3)} \ket{x}
\end{equation}
There is a complication with the phase gadget circuit representation that doesn't arise in the qubit setting, which is that the CNOT gate is self-inverse while the CX qutrit gate is not. In Eq.~\eqref{eq:phase-gadget-qutrit-simp} we needed to pair a CX with a CX$^\dagger$ to make the construction work. If we instead have a pair of CX gates, we get something a bit more complicated:
\begin{equation}\label{eq:phase-gadget-wrong-cnot}
\tikzfig{phase-gadget-wrong-cnot}
\end{equation}
\begin{remark}
Another way to view qubit phase gadgets is as an exponentiated Pauli $e^{i\alpha Z\otimes Z}$~\cite{phaseGadgetSynth,cowtan2020generic}. This however does \emph{not} generalise to qutrits, as the qutrit Pauli $Z$ is not self-adjoint, and hence cannot be exponentiated to give a unitary. In fact, a qutrit phase gadget cannot be represented as the exponential of a `pure tensor' like $e^{i\alpha A\otimes B}$. This does suggest that there could be another suitable generalisation of a phase gadget that is the exponential of a tensor of \emph{Gell-Mann matrices}, a qutrit basis of self-adjoint matrices.
\end{remark}
\subsection{Controlled phase gates}
The other type of useful diagonal gate for qubits is the \emph{controlled phase gate}. Such a gate applies a $Z(\alpha)$ gate on a qubit controlled on the value of a control.
There are multiple ways in which we can generalise these to the qutrit setting. The type of control we will consider first is the $\ket{2}$-control of Definition~\ref{def:ket2-controlled}.
To see how we can build a $\ket{2}$-controlled $Z$ phase gate, we will take inspiration from the qubit construction. Recall that there we have:
\begin{equation}\label{eq:controlled-Z-gate-decomp}
\tikzfig{controlled-Z-gate} \ = \ \tikzfig{controlled-Z-decomp}
\end{equation}
We can `port' the right-hand side to the qutrit setting, by taking each of the phases to be a $Z(\alpha,\beta)$. However, we then run into some problems. It is easy to check that when the top qutrit (the control) is $\ket{0}$ that the diagram indeed acts as the identity on the bottom qutrit (the target). However, it implements a different phase gate on the target depending on whether the control is in $\ket{1}$ or $\ket{2}$:
\begin{equation}\label{eq:controlled-Z-qutrit}
\tikzfig{controlled-Z-qutrit} \ \rightsquigarrow \
\begin{cases}
Z(0,0) \ &\text{if control is}\ \ket{0} \\
Z(2\alpha-\beta,\alpha+\beta) \ &\text{if control is}\ \ket{1} \\
Z(\alpha+\beta,2\beta-\alpha) \ &\text{if control is}\ \ket{2}
\end{cases}
\end{equation}
Seeing as we want to construct the $\ket{2}$-controlled gate that should act as the identity when the control is $\ket{1}$ this is a problem.
We solve this issue by `doubling up' the construction, with the second construction being conjugated by $X_{12}$ on the control in order to interchange the role of $\ket{1}$ and $\ket{2}$:
\begin{equation}\label{eq:controlled-Z-qutrit-paired}
\tikzfig{controlled-Z-qutrit-paired}
\end{equation}
By referring to Eq.~\eqref{eq:controlled-Z-qutrit} we see then that in order for Eq.~\eqref{eq:controlled-Z-qutrit-paired} to be equal to the $\ket{2}$-controlled $Z(\theta,\phi)$ gate it needs to satisfy a set of linear equations.
We can solve these to get a (unique up to some Clifford phases) solution:
\begin{equation}
\alpha \ =\ \frac{\theta - \phi}{3} \quad\qquad \beta \ =\ \frac{\theta}{3} \quad\qquad \gamma \ =\ \frac{\phi}{3} \quad\qquad \delta \ =\ \frac{\phi - \theta}{3}
\end{equation}
We can hence write any $\ket{2}$-controlled phase gate using at most four CX gates and four phase gates. For example, if we pick $\theta = 2\pi/3$ and $\phi = 4\pi/3$ (so that we are constructing the controlled $Z$ gate) we get:
\begin{equation}\label{eq:ket2-Z}
\tikzfig{ket2-Z-decomp}
\end{equation}
Here we write this blue dot with a $2$ in it to denote a $\ket{2}$-control.
We see then that our construction in the special case of $\ket{2}$-controlled Paulis indeed achieves the lowest known $T$-count of $3$~\cite{BocharovA2016ternaryarithmetics}. By conjugating the control wire by either $X_{+1}$ or $X_{-1}$ we can make the gate also be controlled on either $\ket{1}$ or $\ket{0}$. We can also construct tritstring-controlled phase gates by decomposing Eq.~\eqref{eq:controlled-Z-qutrit-paired} into CX, $X_{12}$ and phase gates and adding a control to each of these gates --- combining this $\ket{2}$-controlled phase gate construction with our previous work on $\ket{2...2}$-controlling any Clifford+$T$ unitary in Ref.~\cite{YehL2022qutritcontrolledcliffordplust}. Note that this construction is not efficient as it requires an exponential number of gates as the number of controls increases, and additionally the sizes of the phases involved get exponentially smaller.
\subsection{Phase multipliers}
The $\ket{2}$-controlled phase gate is just one possible way to extend the idea of a controlled-phase gate from qubits. Another way is to realise that for qubits we can describe the action of a controlled phase gate as $\ket{x,y} \mapsto e^{i\alpha \,x\cdot y} \ket{x,y}$. Indeed, if the control qubit is in the state $x=0$, then this is just the identity, while if $x=1$, we apply $e^{i\alpha y}$ which corresponds to a $Z(\alpha)$ gate on the $\ket{y}$ qubit.
We see then that while a phase gadget is based on the addition operation of $\mathbb{Z}_2$, controlled phase gates are based on the multiplication operation of $\mathbb{Z}_2$. This suggests that the controlled phase gate equivalent for qutrits should be $\ket{x,y}\mapsto e^{i\alpha x\cdot y}\ket{x,y}$ where now we take $x\cdot y$ modulo 3. We will show how we can construct this operation using phase gadgets. In order to distinguish this type of gate from the previously considered controlled phase gates, we will refer to a gate where the phase depends on $x\cdot y$ as a \emph{phase multiplier}. Before we show how to build phase multipliers for qutrits, we first need to understand how to build them for qubits.
For bits $x$ and $y$ we have the relation
\begin{equation}\label{eq:bit-plus-rel}
x\cdot y = \frac12 (x+y- (x\oplus y)).
\end{equation}
Importantly, we are considering the~$+$ operation here not modulo $2$, but just as an action on real numbers, and we are writing $\oplus$ for addition modulo $2$. Using this relation we can write $e^{i\alpha (x\cdot y)} = e^{i\frac12\alpha (x+y-(x\oplus y))} = e^{i\frac12 \alpha x} e^{i\frac12\alpha y} e^{-i\frac12\alpha (x\oplus y)}$. This is where the circuit decomposition of Eq.~\eqref{eq:controlled-Z-gate-decomp} comes from.
This relation between additive and multiplicative phase gates follows from a Fourier-type duality that exists for semi-Boolean functions, which is explored in detail in Ref.~\cite{GraphicalFourier2019}.
It turns out that a similar decomposition is possible for qutrits. Note that we can derive Eq.~\eqref{eq:bit-plus-rel} by starting with the equation $(x+y)^2 = x^2 + y^2 + 2x\cdot y$ and then realising that $x^2 = x$ for $x\in \{0,1\}$ so that this reduces to $x\oplus y = x + y + 2x\cdot y$ for bits. When working with trits we can't remove these squares, but we can still get a useful relation. Bring terms to the other side to get $-2x\cdot y = x^2 + y^2 - (x+y)^2$ and then use the fact that modulo 3 we have $-2 = 1$ to get $x\cdot y = x^2 + y^2 - (x+y)^2$. It is now straightforward to check that this continues to hold when we interpret the outer $+$ and $-$ here not modulo 3, but as operations on the real numbers, so that we get the relation:
\begin{equation}\label{eq:trit-plus-rel}
x\cdot y~\text{mod } 3 = (x^2~\text{mod } 3) + (y^2~\text{mod } 3) - ((x+y)^2~\text{mod } 3)
\end{equation}
Hence, using Eq.~\eqref{eq:Z-alpha-alpha} we get the following decomposition:
\begin{equation}\label{eq:controlled-qutrit-phase}
\tikzfig{controlled-qutrit-phase-decomp} \qquad :: \qquad \ket{x,y} \mapsto e^{i\alpha (x\cdot y \text{ mod }3)} \ket{x,y}
\end{equation}
We can easily generalise Eq.~\eqref{eq:trit-plus-rel} to as many variables as desired by iterating it. For three trits:
\begin{align}
(x\cdot y)\cdot z\ &=\ x^2\cdot z + y^2\cdot z - (x+y)^2\cdot z \nonumber \\
&=\ x^4 + z^2 - (x^2 + z)^2 + y^4 + z^2 - (y^2 + z)^2 - (x+y)^4 - z^2 + ((x+y)^2+z)^2 \nonumber \\
&=\ x^2 + y^2 + z^2 - (x^2 + z)^2 - (y^2 + z)^2 - (x+y)^2 + ((x+y)^2+z)^2 \label{eq:triple-product}
\end{align}
Here we used that $x^4 = x^2$ modulo 3.
Note that Eq.~\eqref{eq:Z-alpha-alpha} shows how to apply a phase proportional to the input trit squared modulo $3$. However, in order to use this trick to apply a phase proportional to a higher order term such as $(y+x^2)^2$, we need a way to compute $y+x^2$ and store it ``on the wire''. In other words, we need to construct a circuit for the unitary defined by $\ket{x,y}\mapsto \ket{x,y+x^2}$. Because this simply adds $1$ (modulo 3) to $y$ iff $x\neq 0$, we construct it by adding $1$ to the second qubit, and then applying a $\ket{0}$-controlled $X_{-1}$ gate. To build this gate, we use the $\ket{2}$-controlled $Z$ gate that we built from two phase gadgets in Eq.~\eqref{eq:ket2-Z}:
\begin{equation}\label{eq:square-control}
\tikzfig{square-control} \quad :: \quad \ket{x,y} \mapsto \ket{x,y+x^2}
\end{equation}
The above trick was also described in Ref.~\cite{BocharovA2016ternaryarithmetics}. We further use this to build the type of phase gate below.
\begin{equation}\label{eq:square-phase}
\tikzfig{square-phase} \quad :: \quad \ket{x,y} \mapsto e^{i\alpha (y+x^2)^2} \ket{x,y}.
\end{equation}
We now have all the ingredients necessary to build the phase multiplier corresponding to the formula~\eqref{eq:triple-product}.
What is interesting about this is that we do not have to use smaller factors of $\alpha$. This is in contrast to the qubit counterpart of the formula~\eqref{eq:triple-product} where we get a factor of $\frac{1}{4}$, due to the factor of $\frac{1}{2}$ in Eq.~\eqref{eq:bit-plus-rel}. In fact, for qubits, the generalisation to $n$ variables will have a prefactor of $(1/2)^{n-1}$ so that for instance the three-qubit-controlled $Z$ and controlled $T$ gates cannot be constructed without ancillae in Clifford+$T$~\cite{GilesB2013multiqubitcliffordplustsynthesis}, as we need $\pi/8$ phase gates. Instead, no matter the number of qutrits, we do not get such a prefactor and can iteratively construct it as in the formula~\eqref{eq:triple-product}, as we did for qutrit Clifford+$T$ in Ref.~\cite{YehL2022qutritcontrolledcliffordplust}.
The circuit~\eqref{eq:square-phase}, alongside the square phase of Eq.~\eqref{eq:Z-alpha-alpha} suffices to generalise Eq.~\eqref{eq:controlled-qutrit-phase} to any number of qutrits.
\begin{proposition}\label{prop:phasemultiplier}
We can construct, without ancillae and using $O(2^n)$ Clifford+$T$, $Z(\alpha,\alpha)$, and $Z(-\alpha,-\alpha)$ gates, the $n$-qutrit phase multiplier gate defined by $\ket{x_1,\ldots, x_n} \mapsto e^{i\alpha \left(\left(x_1\cdots x_n\right) \text{ mod } 3\right)} \ket{x_1,\ldots, x_n}$.
\end{proposition}
See Appendix~\ref{app:phase-multipliers} for the details.
\begin{remark}
In Ref.~\cite{CuiS2017Diagonalhierarchy} the diagonal gates at all levels of the Clifford hierarchy are analysed for any qudit of prime dimension. They show for instance that the gate implementing $\ket{x_1\cdots x_n}\mapsto \omega^{x_1\cdots x_n} \ket{x_1\cdots x_n}$ (which is the $n$-controlled $2\pi/3$ phase multiplier gate) is in the $n$th level of the Clifford hierarchy. This might be surprising as our construction shows how to build this gate, for any $n$, only using gates from the third level of the Clifford hierarchy (namely Clifford gates and the $T$ gate). However, note that while the \emph{diagonal} gates on a level of the hierarchy form a group, the full set of (not necessarily diagonal) gates is not closed under composition, and hence we can build higher-level unitaries using lower-level ones.
\end{remark}
\section{Applications}\label{sec:emulation-application}
We've now seen that we can use phase gadgets to build a number of useful diagonal unitaries. In this section we will see how we can build more general diagonal qutrit unitaries, and specifically those that \emph{emulate} qubit operations.
Qudit emulation of qubit operations can result in efficiency gains, by using higher level states rather than ancillae. While there has been significant work on emulating qubits using qutrits and qudits, much of this has been limited to realising gates within classical reversible computing such as multiple-controlled Toffolis.
In contrast, fewer works have addressed qutrit gate sets containing arbitrary phases. Examples include a $\ket{2}$-controlled $Z(0,\phi)$ decomposition in terms of qutrit-controlled qubit $\phi/3$ rotations~\cite{DiY2013synthesis} or quantum multiplexers and uniformly controlled Givens rotations from the cosine-sine decomposition~\cite{KhanF2006synthesisqudit}.
Throughout this section, we will write $\,\stackrel{\mathclap{\normalfont\mbox{e}}}{=}\,$ to denote that a qubit unitary is emulated by a qutrit unitary.
We will first see how to emulate arbitrary qubit diagonal unitaries.
Note that when we restrict to the $\{\ket{0},\ket{1}\}$ subspace, that a qutrit phase multiplier $\ket{x_1,\ldots, x_n} \mapsto e^{i\alpha \left(\left(x_1\cdots x_n\right) \text{ mod } 3\right)} \ket{x_1,\ldots, x_n}$ only applies a phase $\alpha$ if and only if all $n$ qubits are in the $\ket{1}$ state.
Hence, for instance, the two-qubit C$Z(\alpha)$ gate is directly emulated by its two-qutrit counterpart of Eq.~\eqref{eq:controlled-qutrit-phase} with action $\ket{x,y} \mapsto e^{i\alpha x\cdot y} \ket{x,y}$. Consequently, by Proposition~\ref{prop:phasemultiplier} we see that using qutrit Clifford+$T$ gates along with $Z(\alpha,\alpha)$ and $Z(-\alpha,-\alpha)$ we can emulate the multiple-controlled $Z(\alpha)$ qubit gate without ancillae.
Now, by conjugating a multiple-controlled C$Z(\alpha)$ gate by the appropriate $X_{01}$ gates, we can decide on which input the $e^{i\alpha}$ phase is applied. Using multiple of these gates we can then arbitrarily decide for each input which phase should be applied to it. This then allows us to emulate an arbitrary diagonal qubit gate.
\begin{proposition}
We can emulate the $\text{diag}(e^{i\alpha_1},\ldots, e^{i\alpha_{2^n}})$ qubit unitary using qutrit Clifford+$T$, $Z(\alpha_j,\alpha_j)$ and $Z(-\alpha_j,-\alpha_j)$ gates and without using ancillae.
\end{proposition}
When using a standard qubit unitary synthesis algorithm, the desired phases $e^{i\alpha_j}$ would be implemented using many-controlled phase gates that require exponentially small angles $e^{i\alpha_j/2^n}$, which is problematic when the use-case is in fault-tolerant computing where non-Clifford phase gates must be constructed using magic state distillation and injection.
Our construction could hence lead to some benefits in synthesising diagonal qubit unitaries using less non-Clifford resources.
It turns out that for the specific case of a qubit CCZ gate, that we can emulate it using qutrits in an even more efficient way.
While we could use the emulation construction above, it turns out to be better to consider an altered construction.
\begin{lemma}\label{lemma:qubitccu}
Given a qutrit $\ket{2}$-controlled $U$ gate for an emulated qubit unitary $U$, we can construct a qutrit emulation of the qubit CCU gate with the same non-Clifford cost as the $\ket{2}$-controlled $U$ gate.
\end{lemma}
\begin{proof}
One can readily verify, by initializing the top two qubits to $\{\ket{00},\ket{01},\ket{10},\text{and}\ket{11}\}$, that the below qutrit decomposition of Ref.~\cite{GokhaleP2019asymptotic} emulates the qubit CCU gate.
\begin{equation}\label{eq:qubitccu}
\tikzfig{qubitccuemu}
\end{equation}
We can then replace the two non-Clifford gates $\ket{1}$-controlled $X_{+1}$ and $\ket{1}$-controlled $X_{-1}$ by CX and CX$^\dagger$, preserving correctness of the emulation as the action on the $\{\ket{0},\ket{1}\}$ subspace is unchanged~\cite{BocharovA2017ternaryshor}.
\end{proof}
Using this lemma we see that to get an efficient emulation of the qubit CCZ, it remains to find an efficient qutrit emulation of the $\ket{2}$-controlled qubit $Z$ gate.
\begin{lemma}
Let $U=\text{diag}(1,e^{i\frac{2\pi}{3}\eta})$ be an arbitrary qubit $Z$ phase gate.
Then we can build the $\ket{2}$-controlled emulated $U$ using the controlled phase gate of Eq.~\eqref{eq:controlled-Z-qutrit}:
\begin{equation}
\tikzfig{phase-gadget-emu}
\end{equation}
Here $\alpha$ and $\beta$ satisfy, for some $k \in \mathbb{Z}$, $2\alpha - \beta = 3 k$ and $\alpha + \beta = \eta$
and the questionmarks $?$ denote that these phases are irrelevant for the emulation.
\end{lemma}
If we choose $\alpha = 3/2$ and $\beta = 0$ in this construction we are emulating the $\ket{2}$-controlled qubit $Z$ gate, because the phase of $\omega^{3/2} = -1$ applies iff the control qutrit is $\ket{2}$ and the target qubit is $\ket{1}$.
Note that a $Z(3/2,0)$ phase is equal to $X_{12} \ R \ X_{12}$. Hence:
\begin{corollary}\label{cor:tcqubitz}
The $\ket{2}$-controlled qubit $Z$ gate can be emulated with $R$-count $3$.
\end{corollary}
Combine this corollary with Lemma~\ref{lemma:qubitccu} we arrive at our result.
\begin{proposition}
The qubit CCZ gate can be emulated ancilla-free in qutrit Clifford+$R$ with $R$-count 3.
\end{proposition}
As shown in Ref.~\cite{HowardM2017resourcetheorymagic}, any implementation of a CCZ gate requires at least four qubit $T$ gates. Additionally any unitary implementation requires at least seven~\cite{GossetD2014opttoffolit}. Here we see that surprisingly, by embedding the CCZ into qutrit space, we can construct it using just three non-Clifford single-qutrit gates and that moreover this is unitary and ancilla-free. This is also a new minimum amongst qudit emulations: for instance, in Ref.~\cite{HeyfronL2019quditcompiler}, they needed four qudit (for prime $d > 3$) $T$ gates to emulate qubit CCZ.
\section{Conclusion}
\label{sec:conclusion}
We introduced phase gadgets in the qutrit ZX-calculus.
To do this, we adapted the original qutrit ZX-calculus to be flexsymmetric so that the phase gadgets' behaviour would not depend on the directionality of their edges.
Using phase gadgets we showed how to build two types of qutrit controlled phase gates: tritstring-controlled phase gates and phase multipliers. This allowed us to emulate the qubit CCZ gate using just three single-qudit non-Clifford gates.
While some of our constructions will naturally generalise to arbitrary qudit dimension, some things are qutrit specific.
It seems to be a coincidence that for qutrits, in contrast with other dimension qudits, you can derive a relation between modular multiplication and addition~\eqref{eq:trit-plus-rel} from the same binomial as for qubits~\eqref{eq:bit-plus-rel}, which comes from having a natural way to express $x^2$ mod 3 thanks to Fermat's little theorem. As a result, qubit and qutrit phase multipliers admit constructions which are structurally similar, despite the fact that for qubits it applies a phase of $\alpha$ on only one possible input --- where all $n$ qubits are $\ket{1}$ --- while for qutrits it applies a phase, which can be $\alpha$ or $2\alpha$, for $2^n$ of the $3^n$ possible input basis states. Moreover, it seems quite special that Eq.~\eqref{eq:trit-plus-rel} does not have any factors making the size of the phases internal to the decomposition decrease (in contrast to the qubit case).
We believe we could use these results as a stepping stone towards defining a qutrit \emph{ZH-calculus}~\cite{EPTCS287.2}.
In the qubit ZH-calculus, the H-boxes represent matrices with coefficients $a^{i_1 ... i_m j_1 ... j_n}$ for a complex number~$a$ and $i_1,...,i_m,j_1,...,j_n \in \{0,1\}$. Therefore, the obvious generalisation to qutrits (at least for $a$ a complex phase) corresponds to our qutrit phase multipliers. Phase gadgets and phase multipliers could then be related in the same way as they are for qubit ZX and ZH~\cite{GraphicalFourier2019}.
An open problem is to find a suitable qutrit equivalent of exponentiated Paulis. The canonical self-adjoint generalisation of qubit Paulis to qutrits, the Gell-Mann matrices, can be exponentiated to unitaries, but it is not clear how they are related to the qutrit Paulis exactly. A starting point to find the proper relation here is to express exponentiations using a Hermitian operator basis constructed from the qutrit Paulis~\cite{AsadianA2015hermitianhwops}.
Finally, let us mention that based on work on an earlier draft of this paper, a proposed scheme for physically implementing a qutrit phase gadget in superconducting qutrit hardware was made~\cite{CaoS2022qutritzxsuperconducting}.
\textbf{Acknowledgements}: JvdW is supported by an NWO Rubicon personal fellowship. LY is supported by an Oxford - Basil Reeve Graduate Scholarship at Oriel College with the Clarendon Fund. The authors wish to thank Aleks Kissinger, Shuxiang Cao, Alex Cowtan and Will Simmons for valuable discussions.
\appendix
\section{Qutrit ZX-calculus}
\subsection{Necessity of rules}\label{app:necessity-of-rules}
We can show that most of the rules in Figure~\ref{fig:qutritphaseexactflexsymmetricrules} are \emph{necessary}, meaning that they cannot be derived from the other rules. We do this by adapting the reasoning of Ref.~\cite{VilmartR2019nearminzx}.
Namely, the following rules are definitely necessary:
\begin{itemize}
\item $(SZ)$: this is the only rule which can decompose a generator with four or more legs into generators with fewer legs.
\item $(P)$: this is the only rule which resolves diagrams containing generators to the identity.
\item $(B1)$: this is the only rule that can transform a connected diagram into a disconnected one.
\item $(EU)$: this is necessary per the argument of Ref.~\cite[Proposition 3.2]{WangQ2014qutritcalculus}.
\item At least one of $(H)$ and $(H')$ is necessary as these are the only ones that can convert a diagram containing a $X$ generator with a non-integer phase into one containing a $Z$ generator with a non-integer phase.
\end{itemize}
We do not know whether the other rules are necessary, although we do suspect this is the case.
\subsection{Proofs of the derived rules}\label{appendix:derivedrules}
\begin{lemma}
The $(ID)$ rule can be derived from the $(SZ)$ and $(SP)$ rules.
\end{lemma}
\begin{proof}~
\[\tikzfig{XD_rules/derived_rules/id} \qedhere\]
\end{proof}
\begin{lemma}
The $(H2)$ rule can be derived from the $(H')$ and $(ID)$ rules.
\end{lemma}
\begin{proof}~
\[\tikzfig{XD_rules/derived_rules/h2} \qedhere\]
\end{proof}
\begin{lemma}
The $(H4)$ rule can be derived from the $(H)$, $(ID)$, and $(H2)$ rules.
\end{lemma}
\begin{proof}~
\[\tikzfig{XD_rules/derived_rules/h4} \qedhere\]
\end{proof}
\begin{lemma}
The $(SX)$ rule can be derived from the $(SZ)$, $(H')$, $(H2)$, and $(H4)$ rules.
\end{lemma}
\begin{proof}
\[\tikzfig{XD_rules/derived_rules/sx} \qedhere\]
\end{proof}
\begin{lemma}
The $(P1')$ rules can be derived from the $(P1)$, $(SX)$, $(H)$, $(H')$, $(H2)$, and $(H4)$ rules.
\end{lemma}
\begin{proof}
Let's first derive the rule for $x=0$:
\begin{equation}\label{eq:k1_0}
\tikzfig{XD_rules/derived_rules/k1_0}
\end{equation}
From that, we can derive the below rule:
\begin{equation}
\tikzfig{XD_rules/derived_rules/k1_2}
\end{equation}
The two above rules, along with the $(P1)$ rule, are captured by the following rule where $x \in \{0,1,2\}$:
\begin{equation}\label{eq:k1}
\tikzfig{XD_rules/derived_rules/k1}
\end{equation}
We now colour-change the above rule to finish deriving all the $(P1')$ rules:
\begin{equation}
\tikzfig{XD_rules/derived_rules/k1_p}
\end{equation}
\end{proof}
\begin{lemma}
The $(P2')$ rules can be derived from the $(P2)$, $(H)$, $(H')$, $(SZ)$, $(SX)$, $(H2)$, and $(H4)$ rules.
\end{lemma}
\begin{proof}
We prove them one by one:
\begin{equation}
\tikzfig{XD_rules/derived_rules/k2_2}
\end{equation}
\begin{equation}\label{eq:k2_p1}
\tikzfig{XD_rules/derived_rules/k2_p1}
\end{equation}
\begin{equation}
\tikzfig{XD_rules/derived_rules/k2_p2}
\end{equation}
\end{proof}
\begin{lemma}
The $(EU')$ rules can be derived from the $(EU)$, $(H)$, $(H')$, $(SZ)$, $(SX)$, $(H2)$, and $(H4)$ rules.
\end{lemma}
\begin{proof}
We show the first equation directly:
\begin{equation}\label{eq:eu_p}
\tikzfig{XD_rules/derived_rules/eu_p}
\end{equation}
For the second one we first note that:
\begin{equation}\label{eq:eu_p1}
\tikzfig{XD_rules/derived_rules/eu_p1}
\end{equation}
Then:
\begin{equation}\label{eq:eu1_p1}
\tikzfig{XD_rules/derived_rules/eu1_p1}
\end{equation}
And finally, we find the different decomposition of $H$:
\begin{equation}\label{eq:eu1}
\tikzfig{XD_rules/derived_rules/eu1}
\end{equation}
\end{proof}
\section{Constructing general phase multipliers}\label{app:phase-multipliers}
When we have two variables we use the formula
\begin{equation}\label{app:trit-plus-rel}
x\cdot y~\text{mod } 3 = (x^2~\text{mod } 3) + (y^2~\text{mod } 3) - ((x+y)^2~\text{mod } 3)
\end{equation}
to construct the two-qutrit phase multiplier. We generalised this to three variables in the following way:
\begin{equation}\label{app:three-variables}
(x\cdot y)\cdot z\ =\ x^2\cdot z + y^2\cdot z - (x+y)^2\cdot z
\ =\ x^2 + y^2 + z^2 - (x^2 + z)^2 - (y^2 + z)^2 - (x+y)^2 + ((x+y)^2+z)^2.
\end{equation}
To see how we go to $4$ variables and beyond, we start with the expression $(x\cdot y\cdot z)\cdot w$ and decompose $x\cdot y\cdot z$ with the above formula resulting in terms $t_1^2,\ldots, t_n^2$. Each of these terms is a square, because that is the case for all the terms in Eq.~\eqref{app:trit-plus-rel}. Since we are working with qutrits we have $(t_j^2)^2 = t_j^2$.
The terms in our formula are now of the form $t_j^2\cdot w$. We apply Eq.~\eqref{app:trit-plus-rel} to each of these. This gives us terms $t_j^4$, $w^2$ and $(t_j^2+w)^2$. The first of these is just $t_j^2$, and by induction we already know how to construct the appropriate phase term on the circuit for this term. The second of these is $w^2$, and hence corresponds to a simple phase gate. Note that this is the same for each $t_j^2\cdot w$ we are decomposing. Furthermore, the plus signs and minus signs on the terms are such that they almost all cancel, and we will have one copy of $w^2$.
The only `interesting' new term we then get is hence $(t_j^2+w)^2$. For instance, in Eq.~\eqref{app:three-variables} the terms of this form are $(x^2+z)^2$, $(y^2+z)^2$ and $((x+y)^2 +z)^2$. The corresponding phase terms are constructed by using the gadget of Eq.~\eqref{eq:square-control} to store $t_j^2+w$ ``on the wire'' and then applying a $Z(\alpha,\alpha)$ phase gate.
Hence, if we go from 3 to 4 variables we get each of the original terms $t_j^2$, plus a $w^2$ term and and a $(t_j^2 + w)^2$ term for each $j$. This straightforwardly generalises to $n$ variables, and it is then easy to check that we will have $2^n-1$ terms.
We can build a circuit for the $n>2$ qutrit phase multiplier by first building the circuit for $n-1$ qutrits, and then inserting the gadget of Eq.~\eqref{eq:square-phase} after every application of a $Z(\alpha,\alpha)$ phase with as the target the $n$th qutrit. The reason this works is because the $n$-qutrit phase multiplier still contains every term of the $n-1$ qutrit multiplier, but now also needs to combine those terms with the $n$th variable. In the four variable case, we would first store on a wire the value of the term $t_j$ we need, and then apply a $Z(\alpha,\alpha)$ gate in order to get the phase $e^{i\alpha t_j^2}$. Then we would apply Eq.~\eqref{eq:square-phase} on the qutrit of $w$ in order to get the phase $e^{i\alpha (t_j^2+w)^2}$. This construction involves temporarily storing $t_j^2+w$ on the wire of $w$, so we can use this term if we want to construct the five-qutrit phase multiplier as well.
We then see that the cost of the $n$-qutrit phase multiplier in terms of (non-Clifford) gates is the cost of the $n-1$ qutrit phase multiplier plus the cost of $2^{n-1}-1$ applications of the Eq.~\eqref{eq:square-phase} gadget. In particular, each phase term requires precisely one of either a $Z(\alpha,\alpha)$ or a $Z(-\alpha,-\alpha)$ gate, so that we need $2^n-1$ of them. The circuit of Eq.~\eqref{eq:square-phase} requires 6 $T$ gates to construct, and hence the $T$-count of the $n$-qutrit phase multiplier is $6 (2^{n-1}-1) = 3\cdot 2^n - 6$ (for $n>2$).
\end{document} | math | 54,264 |
\begin{document}
\title{Conditional linear-optical measurement schemes generate effective photon nonlinearities}
\author{G.G.\ Lapaire$^1$, Pieter Kok$^2$, Jonathan P.\ Dowling$^2$, and J.E.\ Sipe$^1$}
\affiliation{$^1$Department of Physics, University of Toronto,
60 St.~George St., Toronto, ON M5S 1A7, Canada \\
$^2$Quantum Computing Technologies Group, Section 367, Jet Propulsion Laboratory,
California Institute of Technology, Mail Stop 126-347, 4800 Oak Grove Drive, Pasadena, CA 91109}
\pacs{03.67.-a, 03.67.Lx, 42.50.-p, 42.65.-k}
\begin{abstract}
We provide a general approach for the analysis of optical state
evolution under conditional measurement schemes, and identify the
necessary and sufficient conditions for such schemes to simulate
unitary evolution on the freely propagating modes. If such unitary
evolution holds, an effective photon nonlinearity can be
identified. Our analysis extends to conditional measurement schemes
more general than those based solely on linear optics.
\end{abstract}
\maketitle
\section{\protect
Introduction}
One of the main problems that optical quantum computing has to overcome is
the efficient construction of two-photon gates \cite{kok00}. We can use Kerr
nonlinearities to induce a phase shift in one mode that depends on the
photon number in the other mode, and this nonlinearity is sufficient to
generate a universal set of gates \cite{kerr}. However, passive Kerr media
have typically small nonlinearities (of the order of $10^{-16}\,\text{cm}
^{2}\,\text{sV}^{-1}$ \cite{boyd99}). We can also construct large Kerr
nonlinearities using slow light, but these techniques are experimentally
difficult \cite{lukin00}.
On the other hand, we can employ linear optics with projective measurements.
The benefit is that linear optical schemes are experimentally much easier to
implement than Kerr-media approaches, but the downside is that the
measurement-induced nonlinearities are less versatile and the success rate
can be quite low (especially when inefficient detectors are involved).
However, Knill, Laflamme and Milburn \cite{klm} showed that with sufficient
ancilla systems, these linear-optical quantum computing (LOQC) devices can
be made near-deterministic with only polynomial resources. This makes linear
optics a viable candidate for quantum computing. Indeed, many linear optical
schemes and approaches have been proposed since \cite
{gottesman,franson,simple,mathis,kyi,snmk}, and significant experimental
progress has already been made \cite{feedfwd,franson2}.
The general working of a device that implements linear optical processing
with projective measurements is shown in Fig.~1. The computational input and
the ancilla systems add up to $N$ optical modes that are subjected to a
unitary transformation $U$, which is implemented with beam splitters, phase
shifters, \textit{etc}. This is called an optical $N$-port device. In order
to induce a transformation of interest on the computational input, the
output is conditioned on a particular measurement outcome of the ancilla
system. For example, one can build a single-photon quantum nondemolition
detector with an optical $N$-port device \cite{kok02}. In general, $N$-port
devices have been studied in a variety of applications \cite{nport}.
The class of such devices of interest here is that in which a unitary
evolution on the computational input is effected. To date these devices have
been proposed and studied on a more-or-less case by case basis. Our approach
is to address this class in a more general way, and identify the conditions
that such a device must satisfy to implement a unitary evolution on the
computational input. Once that unitary evolution is established, an
effective photon nonlinearity associated with the device can be identified.
In this paper, we present necessary and sufficient conditions for the
unitarity of the optical transformation of the computational input, and we
derive the effective nonlinearities that are associated with some of the
more common optical gates in LOQC. We begin section \ref{sec:formalism} by
introducing the formalism. In sections \ref{sec:consec}-\ref{sec:nscon}, we
examine the transformation equation under the assumption that it is unitary.
We show that there are two necessary and sufficient conditions for the
transformation to be unitary and we provide a simple test condition. In
section \ref{sec:post}, we expand the formalism and conditions to include
measurement dependent output processing (see Fig.~2), which is used in
several schemes. In section \ref{sec:examples}, we show how the formalism
can be applied to quantum computing gates. We choose as examples two quantum
gates already proposed, the conditional sign flip of Knill, Laflamme, and
Milburn \cite{klm}, and the polarization-encoded CNOT of Pittman \textit{et
al}.\ \cite{franson}. Our concluding remarks are presented in section \ref
{sec:conclusions}, where we note that our main results extend to devices
where the unitary transformation $U$ is more general than those
implementable with linear optics alone.
\section{The general formalism}
\label{sec:formalism}
We consider a class of optical devices that map the computational input
state onto an output state, conditioned on a particular measurement outcome
of an ancilla state (see Fig.~1). We introduce a factorization of the entire
Hilbert space into a space $\mathcal{H}_{C}$ involving the input computing
channels (\textit{i.e.}, both ``target'' and ``control'' in a typical
quantum gate), and a Hilbert space $\mathcal{H}_{A}$ involving the input
ancilla channels,
\[
\mathcal{H}=\mathcal{H}_{C}\otimes \mathcal{H}_{A}\; .
\]
We assume that the input computing and ancilla channels are uncorrelated and
unentangled, so we can write the full initial density operator as $\rho
\otimes \sigma $, where $\rho $ is the initial density operator for the
computing channels, and $\sigma $ the initial density operator for the
ancilla channels.
Let $U$ be the unitary operator describing the pre-measurement evolution of
the optical multi-port device. At the end of this process we have a full
density operator given by $U\left( \rho \otimes \sigma \right) U^{\dagger }$
. In anticipation of the projective measurement, it is useful to introduce a
new factorization of the full Hilbert space into an output computing space $
\mathcal{H}_{\bar{C}}$ and a new ancilla space $\mathcal{H}_{\bar{A}}$,
\[
\mathcal{H=H}_{\bar{C}}\otimes \mathcal{H}_{\bar{A}}\; .
\]
The Von Neumann projective measurements of interest are described by
projector-valued measures (or PVMs) of the type $\left\{ \bar{P},{I-}\bar{P}
\right\} $, where ${I}$ is the identity operator for the whole Hilbert
space, and the projector $\bar{P}$ is of the form
\begin{equation}
\bar{P}={I}_{\bar{C}}\otimes \sum_{\bar{k}}s_{\bar{k}}\left| \bar{k}
\right\rangle \left\langle \bar{k}\right| \; , \label{projector}
\end{equation}
where ${I}_{\bar{C}}$ is the identity operator in $\mathcal{H}_{\bar{C}}$,
and we use Roman letters with an overbar, \textit{e.g., }$\left| \bar{k}
\right\rangle $, to label a set of orthonormal states, $\left\langle \bar{k}|
\bar{l}\right\rangle =\delta _{\bar{k}\bar{l}}$, spanning the Hilbert space $
\mathcal{H}_{\bar{A}}$; each $s_{\bar{k}}$ is equal to zero or unity. The
number of nonzero $s_{\bar{k}}$ identifies the rank of the projector $\bar{P}
$ in $\mathcal{H}_{\bar{A}}$. ``Success'' is defined as a measurement
outcome associated with the projector $\bar{P}$, and the probability of
success is thus
\begin{equation}
d(\rho )\equiv \mathrm{Tr}_{\bar{C},\bar{A}}\left( U\left( \rho \otimes
\sigma \right) U^{\dagger }\bar{P}\right)\; . \label{Ddef}
\end{equation}
Clearly, in general $d(\rho )$ depends on the ancilla density operator $
\sigma $, the unitary evolution $U$, and the projector $\bar{P}$, as well as
on $\rho $. However, we consider the first three of these quantities fixed
by the protocol of interest and thus only display the dependence of the
success probability on the input density operator $\rho $. In the event of a
successful measurement, the output of the channels associated with $\mathcal{
H}_{\bar{C}}$ is identified as the computational result, and it is described
by the reduced density operator
\begin{equation}
\bar{\rho}=\frac{\mathrm{Tr}_{\bar{A}}\left( \bar{P}U\left( \rho \otimes
\sigma \right) U^{\dagger }\bar{P}\right) }{\mathrm{Tr}_{\bar{C},\bar{A}
}\left( U\left( \rho \otimes \sigma \right) U^{\dagger }\bar{P}\right) }.
\label{rhobar}
\end{equation}
For any $\rho $ with $d(\rho )\neq 0$, this defines a so-called completely
positive (CP), trace preserving map $\mathcal{T}$ that takes each $\rho $ to
its associated $\bar{\rho}$: $\bar{\rho} = \mathcal{T}(\rho )$, relating
density operators in $\mathcal{H}_{C}$ to density operators in $\mathcal{H}_{
\bar{C}}$. It will be convenient to write $\mathcal{T}(\rho )=\mathcal{V}
(\rho )/d(\rho )$, where
\begin{equation}
\mathcal{V}(\rho )\equiv \mathrm{Tr}_{\bar{A}}\left( \bar{P}U\left( \rho
\otimes \sigma \right) U^{\dagger }\bar{P}\right) \label{Vdef}
\end{equation}
is a linear (non-trace preserving) CP map of density operators in $\mathcal{H
}_{C}$ to positive operators in $\mathcal{H}_{\bar{C}}$ that is defined for
all density operators $\rho $ in $\mathcal{H}_{C}$. We restrict ourselves to
density operators $\rho $ over a subspace $\mathcal{S}_{C}$ of $\mathcal{H}
_{C}$. This is usually the subspace in which the quantum gate operates.
As an example, consider the gate that turns the computational basis into the
Bell basis. In terms of polarization states, the subspace $\mathcal{S}_{C}$
might be spanned by the computational basis $\{|H,H\rangle ,|H,V\rangle
,|V,H\rangle ,|V,V\rangle \}$ (whereas $\mathcal{H}_{C}$ is spanned by the
full Fock basis). The Bell basis on $\mathcal{S}_{C}$ is then given by $\{
|\Psi^+\rangle, |\Psi^-\rangle, |\Phi^+\rangle, |\Phi^-\rangle \}$, where
\[
|\Psi ^{\pm }\rangle =\frac{1}{\sqrt{2}}\left( |H,V\rangle \pm |V,H\rangle
\right) \text{~and~} |\Phi ^{\pm }\rangle =\frac{1}{\sqrt{2}}\left(
|H,H\rangle \pm |V,V\rangle \right) \;.
\]
This gate is very important in quantum information theory, because it
produces maximal entanglement, and its inverse can be used to perform Bell
measurements. Both functions are necessary in, \textit{e.g.,} quantum
teleportation \cite{bennett93}. However, it is well known that such gates
cannot be constructed deterministically, and we therefore need to include an
ancilla state $\sigma$ and a projective measurement. We consider gates such
as these in this paper.
Suppose the subspace $\mathcal{S}_{C}$ is spanned by a set of vectors
labeled by Greek letters, \textit{e.g., }$\left| \alpha \right\rangle $. We
can then write
\begin{equation}
\rho =\sum_{\alpha ,\beta }\left| \alpha \right\rangle \rho ^{\alpha \beta
}\left\langle \beta \right| , \label{rhodecompose}
\end{equation}
where $\rho ^{\alpha \beta }\equiv \left\langle \alpha |\rho |\beta
\right\rangle $. We identify a convex decomposition of the ancilla density
operator $\sigma $ as
\[
\sigma =\sum_{i}p_{i}\left| \chi _{i}\right\rangle \left\langle \chi
_{i}\right| ,
\]
where the normalized (but not necessarily orthogonal) vectors $\left| \chi
_{i}\right\rangle $ are elements of $\mathcal{H}_{A}$, and the $p_{i}$ are
all non-negative and sum to unity,
\[
\sum_{i}p_{i}=1\; .
\]
We can then use (\ref{rhobar}) to write down an expression for the matrix
elements of $\bar{\rho}$. Note that it is possible to work with the
eigenkets of $\sigma $ so that $\left\{ \left| \chi _{i}\right\rangle
\right\} $ is an orthonormal set; however, this does not simplify the
analysis so we do not introduce the restriction. Furthermore, dealing with
non-orthogonal states in the ancilla convex decomposition may be more
convenient, depending on the system of interest. Choosing an orthonormal
basis of $\mathcal{H}_{\bar{C}}$ that we label by Greek letters with
overbars, \textit{e.g., }$\left| \bar{\alpha}\right\rangle $, we find
\begin{equation}
\bar{\rho}^{\bar{\alpha}\bar{\delta}}=\sum_{\beta ,\gamma }\sum_{i,\bar{k}
}\left( W_{\bar{k},i}^{\bar{\alpha}\beta }(\rho )\right) \rho ^{\beta \gamma
}\left( W_{\bar{k},i}^{\bar{\delta}\gamma }(\rho )\right) ^{*},
\label{rhobarwork}
\end{equation}
where
\[
W_{\bar{k},i}^{\bar{\alpha}\beta }(\rho )=s_{\bar{k}}\sqrt{\frac{p_{i}}{
d(\rho )}}\left( \left\langle \bar{k}\right| \left\langle \bar{\alpha}
\right| \right) U\left( \left| \beta \right\rangle \left| \chi
_{i}\right\rangle \right) .
\]
Note that
\[
\sum_{\bar{\alpha}}\sum_{i,\bar{k}}\left( W_{\bar{k},i}^{\bar{\alpha}\gamma
}(\rho )\right) ^{*}\left( W_{\bar{k},i}^{\bar{\alpha}\beta }(\rho )\right)
=\delta _{\gamma \beta }\;,
\]
which is confirmed by
\begin{equation}
\mathrm{Tr}_{\bar{C}}(\bar{\rho})=\sum_{\bar{\alpha}}\bar{\rho}^{\bar{\alpha}
\bar{\alpha}}=\sum_{\beta }\rho ^{\beta \beta }=\mathrm{Tr}_{C}(\rho ),
\label{tracepreserve}
\end{equation}
This last equation follows immediately from (\ref{rhobar}), since $\mathcal{T
}$ is a trace preserving CP map and $\mathrm{Tr}_{C}(\rho )=1$.
In this paper, we consider a special class of maps that constitute a unitary
transformation on the computational subspace $\mathcal{S}_{C}$. In
particular, such transformations include the CNOT, the C-SIGN, and the
controlled bit flip. These are not the only useful maps in linear optical
quantum computing, but they arguably constitute the most important class.
Before we continue, we introduce the following definition:
\begin{description}
\item[Definition:] We call a CP map $\rho \rightarrow \bar{\rho}=\mathcal{T}
(\rho )$ an \textit{operationally unitary transformation }on density
operators $\rho $ over a subspace $\mathcal{S}_{C}$ if and only if:
\begin{itemize}
\item For each $\rho $ over the subspace $\mathcal{S}_{C}$ we have $d(\rho
)\neq 0$, and
\item For each $\rho $ defined by Eq.~(\ref{rhodecompose}) over the
subspace $\mathcal{S}_{C}$, the map $\mathcal{T}(\rho )$ yields a $\bar{\rho}
$ given by
\begin{equation}
\bar{\rho}=\sum_{\alpha ,\beta }\left| \bar{\nu}_{\alpha }\right\rangle \rho
^{\alpha \beta }\left\langle \bar{\nu}_{\beta }\right| , \label{rhobareu}
\end{equation}
where the $\left| \bar{\nu}_{\alpha }\right\rangle $ are fixed vectors in $
\mathcal{H}_{\bar{C}}$ satisfying $\left\langle \bar{\nu}_{\alpha }|\bar{\nu}
_{\beta }\right\rangle =\left\langle \alpha |\beta \right\rangle =\delta
_{\alpha \beta }$.
\end{itemize}
\end{description}
This forms the obvious generalization of usual unitary evolution, since it
maintains the inner products of vectors under the transformation. Much of
our concern in this paper is in identifying the necessary and sufficient
conditions for a general map $\mathcal{T}(\rho )$ of Eqs.~(\ref{rhobar}) and
(\ref{rhobarwork}) to constitute an operationally unitary map. We begin in
the next section by considering what can be said about such maps.
\section{Consequences of operational unitarity}
\label{sec:consec}
In this section we restrict ourselves to CP maps $\mathcal{T}(\rho )$ that
are operationally unitary [see Eqs.~(\ref{rhodecompose}) and (\ref{rhobareu}
)] for density operators $\rho $ over a subspace $\mathcal{S}_{C}$ of $
\mathcal{H}_{C}$. The linearity of such maps implies that the convex sum of
two density operators is again a density operator:
\[
\rho _{c}=x\rho _{a}+(1-x)\rho _{b},
\]
with $0\leq x\leq 1$. Applying Eqs.~(\ref{rhodecompose}) and (\ref{rhobareu}
) to the three density operators $\rho _{a}$, $\rho _{b}$, and $\rho _{c}$
it follows immediately that
\begin{equation}
\bar{\rho}_{c}=x\bar{\rho}_{a}+(1-x)\bar{\rho}_{b}. \label{tranfirst}
\end{equation}
Now a second expression for $\bar{\rho}_{c}$ can be worked out by using the
defining relation (\ref{rhobar}) directly,
\begin{eqnarray}
\bar{\rho}_{c} &=&\mathcal{T}(\rho _{c})=\frac{\mathcal{V}(\rho _{c})}{
d(\rho _{c})} \label{transecond} \\
&=&\frac{x\mathcal{V}(\rho _{a})+(1-x)\mathcal{V}(\rho _{b})}{xd(\rho
_{a})+(1-x)d(\rho _{b})} \nonumber \\
&=&\frac{xd(\rho _{a})\bar{\rho}_{a}+(1-x)d(\rho _{b})\bar{\rho}_{b}}{
xd(\rho _{a})+(1-x)d(\rho _{b})} \nonumber
\end{eqnarray}
where in the second line we have used the linearity of $\mathcal{V}(\rho )$ (
\ref{Vdef}) and $d(\rho )$ (\ref{Ddef}), and in the third line we have used
the corresponding relations for $\bar{\rho}_{a}$ in terms of $\rho _{a}$,
and $\bar{\rho}_{b}$ in terms of $\rho _{b}$. Setting the right-hand-sides
of Eqs.~(\ref{tranfirst}) and (\ref{transecond}) equal, we find
\begin{equation}
x(1-x)\left[ d(\rho _{b})-d(\rho _{a})\right] (\bar{\rho}_{a}-\bar{\rho}
_{b})=0. \label{eureka}
\end{equation}
Since it is easy to see from Eqs.~(\ref{rhodecompose}) and (\ref{rhobareu})
that if $\rho _{a}$ and $\rho _{b}$ are distinct then $\bar{\rho}_{a}$ and $
\bar{\rho}_{b}$ are as well; choosing $0<x<1$ it is clear that the only way
the operator equation (\ref{eureka}) can be satisfied is if $d(\rho
_{a})=d(\rho _{b})$. But since this must hold for \textit{any} two density
operators acting over $\mathcal{S}_{C}$, we have established that:
\begin{itemize}
\item If a map $\mathcal{T}(\rho )$ is operationally unitary for $\rho $
(acting on a subspace $\mathcal{S}_{C}$), then $d(\rho )$ is independent of $
\rho $: $d(\rho )=d$, for all $\rho $ acting on that subspace.
\end{itemize}
With this result in hand we can simplify Eq.~(\ref{rhobarwork}) for a map
that is operationally unitary, writing
\begin{equation}
\bar{\rho}^{\bar{\alpha}\bar{\delta}}=\sum_{\beta ,\gamma }\sum_{J}w_{J}^{
\bar{\alpha}\beta }\rho ^{\beta \gamma }\left( w_{J}^{\bar{\delta}\gamma
}\right) ^{*}, \label{tsimpform}
\end{equation}
where now
\[
w_{J}^{\bar{\alpha}\beta }=w_{\bar{k},i}^{\bar{\alpha}\beta }=s_{\bar{k}}
\sqrt{\frac{p_{i}}{d}}\left( \left\langle \bar{k}\right| \left\langle \bar{
\alpha}\right| \right) U\left( \left| \beta \right\rangle \left| \chi
_{i}\right\rangle \right)
\]
is independent of $\rho $; we have also introduced a single label $J$ to
refer to the pair of indices $\bar{k},i$. A further simplification arises
because the condition of operational unitarity guarantees that the subspace $
\mathcal{S}_{\bar{C}}$ of $\mathcal{H}_{\bar{C}}$, over which the range of
density operators $\bar{\rho}$ generated by $\mathcal{T}(\rho )$ act as $
\rho $ ranges over $\mathcal{S}_{C}$, has the same dimension as $\mathcal{S}
_{C}$. We can thus adopt a set of orthonormal vectors $\left| \bar{\alpha}
\right\rangle $ that span that subspace $\mathcal{S}_{\bar{C}}$, and the
matrices $w_{J}^{\bar{\alpha}\beta }$ are square.
At this point we can formally construct a unitary map on $\mathcal{S}_{C}$: $
\tilde{\rho}\equiv \mathcal{U}(\rho )$, which is isomorphic in its effect on
density operators $\rho $ with our operationally unitary map $\bar{\rho}=
\mathcal{T}(\rho )$. We do this by associating each $\left| \bar{\alpha}
\right\rangle $ with the corresponding $\left| \alpha \right\rangle $,
introducing a density operator $\tilde{\rho}$ acting over $\mathcal{S}_{C}$,
and putting
\begin{eqnarray}
\tilde{\rho}^{\alpha \delta } &\equiv &\bar{\rho}^{\bar{\alpha}\bar{\delta}},
\label{correspondence} \\
M_{J}^{\alpha \beta } &\equiv &w_{J}^{\bar{\alpha}\beta }. \nonumber
\end{eqnarray}
The unitary map $\tilde{\rho}\equiv \mathcal{U}(\rho )$ is defined by the CP
map
\[
\tilde{\rho}^{\alpha \delta }=\sum_{\beta ,\gamma }\sum_{J}M_{J}^{\alpha
\beta }\rho ^{\beta \gamma }(M_{J}^{\delta \gamma })^{*},
\]
or simply
\begin{equation}
\tilde{\rho}=\sum_{J}M_{J}\rho M_{J}^{\dagger }. \label{unitransform}
\end{equation}
This is often what is done implicitly when describing an operationally
unitary map, and we will see examples later in section \ref{sec:examples};
here we find this strategy useful to simplify our reasoning below.
Since the map $\tilde{\rho}\equiv \mathcal{U}(\rho )$ is unitary it can be
implemented by a unitary operator $M$,
\[
\tilde{\rho}=M\rho M^{\dagger },
\]
where $M^{\dagger }=M^{-1}$. Thus $(M_{1},M_{2},....)$ and $(M,0,0,....)$,
where we add enough copies of the zero operator so that the two lists have
the same number of elements, constitute two sets of Kraus operators that
implement the same map $\tilde{\rho}\equiv \mathcal{U}(\rho )$. From Nielsen
and Chuang \cite{NandC} we have the following theorem:
\begin{description}
\item[Theorem:] Suppose $\{E_{1},\ldots ,E_{n}\}$ and $\{F_{1},\ldots
,F_{m}\}$ are Kraus operators giving rise to CP linear maps $\mathcal{E}$
and $\mathcal{F}$ respectively. By appending zero operators to the shorter
list of elements we may ensure that $m=n$. Then $\mathcal{E}=\mathcal{F}$ if
and only if there exists complex numbers $u_{jk}$ such that $
E_{j}=\sum_{k}u_{jk}F_{k}$, and $u_{jk}$ is an $m\times m$ unitary matrix.
\end{description}
Hence, $(M_{1},M_{2},....)$ must be related to $(M,0,0,....)$ by a unitary
matrix, and each $M_{J}$ is proportional to the single operator $M$. This
proof carries over immediately to the operationally unitary map $\mathcal{
T(\rho )}$ under consideration, and we have
\begin{itemize}
\item If a map $\mathcal{T}(\rho )$ is operationally unitary for $\rho $
acting over a subspace $\mathcal{S}_{C}$, then for fixed $\bar{k}$ and $i$
the square matrix defined by
\[
w_{\bar{k},i}^{\bar{\alpha}\beta }=s_{\bar{k}}\sqrt{\frac{p_{i}}{d}}\left(
\left\langle \bar{k}\right| \left\langle \bar{\alpha}\right| \right) U\left(
\left| \beta \right\rangle \left| \chi _{i}\right\rangle \right) ,
\]
with $\bar{\alpha}$ labeling the row and $\beta $ the column, either
vanishes or is proportional to all other nonvanishing matrices identified by
different $\bar{k}$ and $i$. We can thus define a matrix $w^{\bar{\alpha}
\beta }$ proportional to all the nonvanishing $w_{\bar{k},i}^{\bar{\alpha}
\beta }$ such that we can write our map (\ref{tsimpform}) as
\begin{equation}
\bar{\rho}^{\bar{\alpha}\bar{\delta}}=\sum_{\beta ,\gamma }w^{\bar{\alpha}
\beta }\rho ^{\beta \gamma }\left( w^{\bar{\delta}\gamma }\right) ^{*}.
\label{tfinal}
\end{equation}
\end{itemize}
It is in fact easy to show that the two \textit{necessary} conditions we
have established here for a map $\mathcal{T}(\rho )$ to be an operationally
unitary transformation are also \textit{sufficient }conditions to guarantee
that it is. We show this in section \ref{sec:nscon}. First, however, we
establish a simple way of identifying whether or not $d(\rho )$ is
independent of $\rho $.
\section{The test condition}
In this section we consider a general map $\mathcal{T}(\rho )$ of the form
of Eq.~(\ref{rhobar}), and seek a simple condition equivalent to the
independence of $d(\rho )$ on $\rho $ for all $\rho $ acting over $\mathcal{S
}_{C}$. To do this we write $d(\rho )$ of Eq.~(\ref{Ddef}) by taking the
complete trace over $\mathcal{H}_{C}$ and $\mathcal{H}_{A}$ rather than over
$\mathcal{H}_{\bar{C}}$ and $\mathcal{H}_{\bar{A}}$,
\begin{eqnarray*}
d(\rho ) &=&\mathrm{Tr}_{C,A}\left( U\left( \rho \otimes \sigma \right)
U^{\dagger }\bar{P}\right) \\
&=&\mathrm{Tr}_{C,A}\left( \left( \rho \otimes \sigma \right) U^{\dagger }
\bar{P}U\right) \\
&=&\mathrm{Tr}_{C}(\rho\, T)
\end{eqnarray*}
where we have introduced a \textit{test operator }$T$ over the Hilbert space
$\mathcal{H}_{C}$ as
\[
T=\mathrm{Tr}_{A}\left( \sigma\, U^{\dagger }\bar{P}U\right) ,
\]
which does not depend on $\rho$. The operator $T$ is clearly Hermitian; it
is also a positive operator, since the probability for success $d(\rho )\geq
0$ for all $\rho $. We can now identify a condition for $d(\rho )$ to be
independent of $\rho$:
\begin{description}
\item[Theorem:] $d(\rho )$ is independent of $\rho $, for density operators
$\rho $ acting over a subspace $\mathcal{S}_{C}$ of $\mathcal{H}_{C}$, if
and only if the test operator $T\,$ is proportional to the identity operator
${I}_{\mathcal{S}_{C}}$ over the subspace $\mathcal{S}_{C}.$ We refer to
this condition on $T$ as the \textit{test condition.}
\item[Proof:] The sufficiency of the test condition for a $d(\rho )$
independent of $\rho $ is clear. Necessity is easily established by
contradiction: Suppose that $d(\rho )$ were independent of $\rho $ but $T$
not proportional to ${I}_{\mathcal{S}_{C}}$. Then at least two of the
eigenkets of $T$ must have different eigenvalues; call those eigenkets $
\left| \mu _{a}\right\rangle $ and $\left| \mu _{b}\right\rangle $. It
follows that $d(\rho _{a})\neq d(\rho _{b})$, where $\rho _{a}=$ $\left| \mu
_{a}\right\rangle \left\langle \mu _{a}\right| $ and $\rho _{b}=\left| \mu
_{b}\right\rangle \left\langle \mu _{b}\right| $, in contradiction with our
assumption.
$\square $
\end{description}
When the test condition is satisfied we denote the single eigenvalue of $T$
over $\mathcal{S}_{C}$ as $\tau $, i.e., $T=\tau {I}_{\mathcal{S}_{C}}$.
Then $d(\rho )=\tau $, and $\tau $ is identified as the probability that the
measurement indicated success. For any given protocol the calculation of the
operator $T$ gives an easy way to identify whether or not $d(\rho )$ is
independent of $\rho $.
\section{Necessary and sufficient conditions}
\label{sec:nscon}
We can now identify necessary and sufficient conditions for a map $\bar{\rho}
=\mathcal{T}(\rho )$, to be an operationally unitary map for $\rho $ acting
on a subspace $\mathcal{S}_{C}$ of $\mathcal{H}_{C}$. They are:
\begin{enumerate}
\item The test condition is satisfied: Namely, the operator
\[
T=\mathrm{Tr}_{A}\left( \sigma \,U^{\dagger }\bar{P}U\right)
\]
is proportional to the identity operator ${I}_{\mathcal{S}_{C}}$ over the
subspace $\mathcal{S}_{C}$.
\item Each matrix
\[
w_{\bar{k},i}^{\bar{\alpha}\beta }=s_{\bar{k}}\sqrt{\frac{p_{i}}{\tau }}
\left( \left\langle \bar{k}\right| \left\langle \bar{\alpha}\right| \right)
U\left( \left| \beta \right\rangle \left| \chi _{i}\right\rangle \right) ,
\]
identified by the indices $\bar{k}$ and $i$, with row and column labels $
\bar{\alpha}$ and $\beta $ respectively, either vanishes or is proportional
to all other such nonvanishing matrices; here $\tau $ is the eigenvalue of $T
$.
\end{enumerate}
The necessity of the first condition follows because it is equivalent to the
independence of $d(\rho )$ on $\rho $, which was established above as a
necessary condition for the transformation to be operationally unitary, as
was the second condition given here. So we need only demonstrate
sufficiency, which follows immediately: If the first condition is satisfied
then $d(\rho )=\tau $ is independent of $\rho $, and if the second is
satisfied then, from Eq.~(\ref{tsimpform}), we can introduce a single matrix
$w^{\bar{\alpha}\beta }$ such that (\ref{tfinal}) is satisfied. Then
\[
\sum_{\bar{\alpha}}\bar{\rho}^{\bar{\alpha}\bar{\alpha}}=\sum_{\beta ,\gamma
}\rho ^{\beta \gamma }\sum_{\bar{\alpha}}\left( w^{\bar{\alpha}\gamma
}\right) ^{*}w^{\bar{\alpha}\beta }.
\]
Now the Hermitian matrix
\[
Y^{\gamma \beta }\equiv \sum_{\bar{\alpha}}\left( w^{\bar{\alpha}\gamma
}\right) ^{*}w^{\bar{\alpha}\beta }
\]
must in fact be the unit matrix: $Y^{\gamma \beta }=\delta _{\gamma \beta }$
, otherwise we would not have
\[
\sum_{\bar{\alpha}}\bar{\rho}^{\bar{\alpha}\bar{\alpha}}=\sum_{\beta }\rho
^{\beta \beta }
\]
for an arbitrary $\rho $ over $\mathcal{S}_{C}$, and we know our general map
$\bar{\rho}=\mathcal{T}(\rho )$ satisfies that condition [see Eq.~(\ref
{tracepreserve})]. Thus $w^{\bar{\alpha}\beta }$ is a unitary matrix, and
from the form of Eq.~(\ref{tfinal}) of the map from $\rho $ to $\bar{\rho}$
it follows immediately that the map is operationally unitary [see Eqs.~(\ref
{rhodecompose}) and (\ref{rhobareu})].
The physics of the two necessary and sufficient conditions given above is
intuitively clear, and indeed the results we have derived here could have
been guessed beforehand. For if the probability for success $d(\rho )$ of
the measurement were dependent of the input density operator $\rho $, by
monitoring the success rate in an assembly of experiments all characterized
by the same input $\rho $, one could learn something about $\rho $, and we
would not expect operationally unitary evolution in the presence of this
kind of gain of information. And the independence of the nonvanishing
matrices $w_{\bar{k},i}^{\bar{\alpha}\beta }$ on $\bar{k}$ and $i$, except
for overall factors, can be understood as preventing the `mixedness' of both
the input ancilla state $\sigma $ and the generally high rank projector $
\bar{P}$, from degrading the operationally unitary transformation and
leading to a decrease in purity.
If a map is found to be operationally unitary, we can introduce the formally
equivalent unitary operator $M$ on $\mathcal{H}_{C}$, as in Eq.~(\ref
{correspondence}), which can then be written in terms of an effective action
operator $Q$,
\begin{equation}
M=e^{-iQ/\hbar }\; . \label{action}
\end{equation}
The operator $Q$ can be determined simply by diagonalizing $M$, and its form
reveals the nature of the Hamiltonian evolution simulated by the conditional
measurement process. We can define an effective Hamiltonian $H_{eff}$ that
characterizes an effective photon nonlinearity acting through a time $
t_{eff}\,$by putting $H_{eff}\equiv Q/t_{eff}$, where $t_{eff}$ can be taken
as the time of operation of the device.
In a special but common case, the input ancilla state is pure and the
projector $\bar{P}$ is of unit rank in $\mathcal{H}_{\bar{A}}$. For cases
such as this there is only one matrix $w^{\bar{\alpha}\beta }$ in the
problem, and thus there is only a single necessary and sufficient condition
for the map to be operationally unitary:
\begin{itemize}
\item In the special case of a projector $\bar{P}$ of rank 1 in $\mathcal{H}
_{\bar{A}}$, where $\bar{P}=$ ${I}_{\bar{C}}\otimes \left| \bar{K}
\right\rangle \left\langle \bar{K}\right| $, and a pure input ancilla state,
$\sigma =\left| \chi \right\rangle \left\langle \chi \right| $, then map $
\bar{\rho}=\mathcal{T}(\rho )$ is operationally unitary for $\rho $ acting
on a subspace $\mathcal{S}_{C}$ of $\mathcal{H}_{C}$ if and only if $T$
satisfies the test condition. Here
\[
T=\left\langle \chi |U^{\dagger }\bar{P}U|\chi \right\rangle ,
\]
which is an operator in $\mathcal{H}_{C}$. If it does satisfy this
condition, then the transformation is given by
\begin{equation}
\bar{\rho}^{\bar{\alpha}\bar{\delta}}=\sum_{\beta ,\gamma }w^{\bar{\alpha}
\beta }\rho ^{\beta \gamma }\left( w^{\bar{\delta}\gamma }\right) ^{*},
\label{spectrans}
\end{equation}
where
\[
w^{\bar{\alpha}\beta }=\sqrt{\frac{1}{\tau }}\left( \left\langle \bar{K}
\right| \left\langle \bar{\alpha}\right| \right) U\left( \left| \beta
\right\rangle \left| \chi \right\rangle \right) ,
\]
and $\tau $ is the single eigenvalue of $T$ over $\mathcal{S}_{C}$.
\end{itemize}
\section{Generalization to include feed-forward processing}
\label{sec:post}
Suppose that the measurement outcome of the ancilla does not yield the
desired result, but that it signals that the output can be transformed by
simply applying a (deterministic) unitary mode transformation on the output
(see Fig. 2). This is called feed-forward processing
and is widely used. For example, in teleportation, Alice sends Bob a
classical message which allows him to correct for `wrong' outcomes of
Alice's Bell measurement. Here, we can explicitly take into account
feed-forward processing.
Suppose the projective measurement is characterized by a set of projectors,
each identifying a different detection signature, $\left\{ \bar{P}_{(1)},
\bar{P}_{(2)},...\bar{P}_{(N)},\bar{P}_{\perp }\right\} $, where
\[
\bar{P}_{\perp }={I-}\sum_{L=1}^{N}\bar{P}_{(L)},
\]
and
\[
\bar{P}_{(L)}={I}_{\bar{C}}\otimes \sum_{\bar{k}}s_{L,\bar{k}}\left| \bar{k}
\right\rangle \left\langle \bar{k}\right| .
\]
All the $s_{L,\bar{k}}$ are equal to zero or unity, such that
\[
\bar{P}_{(L)}\bar{P}_{(L^{\prime })}=\bar{P}_{(L)}\delta _{LL^{\prime }}.
\]
Here success arises if the measurement outcome is associated with
\textit{any }of the operators $\bar{P}_{(L)}$. And if outcome $L$ is
achieved, then the computational output is processed by application of
the unitary operator $\bar{V}_{(L)}$ acting over $\mathcal{H}_{\bar{C}}$.
The probability of achieving outcome $L$ is
\[
d_{(L)}(\rho )\equiv \mathrm{Tr}_{\bar{C},\bar{A}}\left( U\left( \rho
\otimes \sigma \right) U^{\dagger }\bar{P}_{(L)}\right)
\]
and if outcome $L$ is achieved the feed-forward processed computational output is
then
\[
\bar{\rho}_{(L)}=\frac{\bar{V}_{(L)}\left[ \mathrm{Tr}_{\bar{A}}\left( \bar{P
}_{(L)}U\left( \rho \otimes \sigma \right) U^{\dagger }\bar{P}_{(L)}\right)
\right] \bar{V}_{(L)}^{\dagger }}{\mathrm{Tr}_{\bar{C},\bar{A}}\left(
U\left( \rho \otimes \sigma \right) U^{\dagger }\bar{P}_{(L)}\right) }.
\]
which defines a map $\bar{\rho}_{(L)}=\mathcal{T}_{(L)}(\rho )$ for those $
\rho $ for which $d_{(L)}(\rho )\neq 0$. In this more general case we define
the \textit{set} of maps $\left\{ \mathcal{T}_{(L)}\right\} $ to be
operationally unitary for density operators $\rho $ over the subspace $
\mathcal{S}_{C}$ when:
\begin{itemize}
\item For each $\rho $ over the subspace $\mathcal{S}_{C}$ at least one of
the $d_{(L)}(\rho )\neq 0$, and
\item For each $\rho $ over the subspace $\mathcal{S}_{C}$, for each $L$
for which $d_{(L)}(\rho )\neq 0$ the map $\mathcal{T}_{(L)}(\rho )$ yields a
$\bar{\rho}_{(L)}$ of the form of Eq.~(\ref{rhobareu}), independent of $L$.
\end{itemize}
The kind of arguments we have presented above can be extended to show that
the necessary and sufficient conditions for such a set of maps to be
operationally unitary for density operators $\rho $ over the subspace $
\mathcal{S}_{C}$ are:
\begin{enumerate}
\item Test conditions are satisfied: The operators
\[
T_{(L)}=\mathrm{Tr}_{A}\left( \sigma U^{\dagger }\bar{P}_{(L)}U\right)
\]
are each proportional to the identity operator ${I}_{\mathcal{S}C}$ over the
subspace $\mathcal{S}_{C}$. The proportionality constants $\tau _{(L)}$ need
not be the same for all $L$.
\item Omitting matrices associated with any $L$ for which $\tau _{(L)}=0$,
each matrix
\[
w_{L,\bar{k},i}^{\bar{\alpha}\beta }=s_{L,\bar{k}}\sqrt{\frac{p_{i}}{\tau
_{(L)}}}\sum_{\bar{\lambda}}\bar{V}_{(L)}^{\bar{\alpha}\bar{\lambda}}\left(
\left\langle \bar{k}\right| \left\langle \bar{\lambda}\right| \right)
U\left( \left| \beta \right\rangle \left| \chi _{i}\right\rangle \right) ,
\]
identified by the indices $L,\bar{k},$ and $i$, with row and column labels $
\bar{\alpha}$ and $\beta $ respectively, either vanishes or is proportional
to all other such nonvanishing matrices.
\end{enumerate}
The probability of success is $\sum_{L}\tau _{(L)}=\tau .$ This expanded
formalism applies to the feed-forward schemes discussed by Pittman
\textit{et al}.\ \cite{franson2} and the teleportation schemes of Gottesman
and Chuang \cite{gottesman}. In devices such as these, a measurement
provides classical information that is used in the subsequent evolution of
the output state.
In a common special case, the input ancilla state is pure, $\sigma =\left|
\chi \right\rangle \left\langle \chi \right| $, and each of the projectors $
\bar{P}_{(L)}$ is of unit rank in $\mathcal{H}_{\bar{A}}$, $\bar{P}_{(L)}={I}
_{\bar{C}}\otimes \left| \overline{k_{L}}\right\rangle \left\langle
\overline{k_{L}}\right| $. Here the two necessary and sufficient conditions
for the set of maps to be operationally unitary for density operators $\rho $
over the subspace $\mathcal{S}_{C}$ simplify to:
\begin{enumerate}
\item All the operators
\[
T_{(L)}=\left\langle \chi |U^{\dagger }\bar{P}_{(L)}U|\chi \right\rangle
\]
over $\mathcal{H}_{C}$ satisfy the test condition.
\item Omitting matrices associated with any $L$ for which $\tau _{(L)}=0$,
each matrix
\[
w_{L}^{\bar{\alpha}\beta }=\frac{1}{\sqrt{\tau _{(L)}}}\sum_{\bar{\lambda}}
\bar{V}_{(L)}^{\bar{\alpha}\bar{\lambda}}\left( \left\langle \overline{k_{L}}
\right| \left\langle \bar{\lambda}\right| \right) U\left( \left| \beta
\right\rangle \left| \chi \right\rangle \right) ,
\]
identified by the indices $L,$ with row and column labels $\bar{\alpha}$ and
$\beta $ respectively, either vanishes or is proportional to all other such
nonvanishing matrices.
\end{enumerate}
If these conditions are met, then the operationally unitary transformation
is given by
\[
\bar{\rho}^{\bar{\alpha}\bar{\delta}}=\sum_{\beta ,\gamma }w_{L}^{\bar{\alpha
}\beta }\rho ^{\beta \gamma }\left( w_{L}^{\bar{\delta}\gamma }\right) ^{*},
\]
which is independent of $L$.
Another extension of the standard Von Neumann, or projection, measurements
is to the class of measurements described by more general positive operator
valued measures, or POVMs. \textit{\ }These can be used to describe more
complicated measurements, often resulting from imperfections in a designed
PVM. Our analysis can be generalized to POVMs by expanding the ancilla
space, and then describing the POVMs by PVMs in this expanded space. In some
instances operationally unitarity might still be possible; in others, the
extension would allow us to study of the effect of realistic limitations
such as detector loss and the lack of single-photon resolution.
\section{Examples}
\label{sec:examples}
In this section we will apply the formalism developed above to two proposed
optical quantum gates for LOQC. The straightforward calculation of the
effects of these gates presented in the original publications make it clear
that they are operationally unitary; our purpose here is merely to
illustrate how the approach we have introduced here is applied.
To evaluate the test operators $T_{(L)}$ and matrix elements $w_{L,\bar{k}
,i}^{\bar{\alpha}\beta }$ it is useful to have expression for quantities
such as $Ua_{\Omega }U^{\dagger }$, where we use capital Greek letters as
subscripts on the letter $a$ to denote annihilation operators for input
(computing and ancilla) channels; similarly, we use $a_{\bar{\Delta}}$ to
denote annihilation operators for output (computing and ancilla) channels.
We now characterize the unitary transformation $U$ by a set of quantities
\textsf{U} $_{\Omega \bar{\Delta}}^{*}$ that give the complex amplitude for
an output photon in mode $\bar{\Delta}$ given an input photon in mode $
\Omega $. That is,
\begin{equation}
U\left( a_{\Omega }^{\dagger }\left| \text{vac}\right\rangle \right) =\sum_{
\bar{\Delta}}\mathsf{U}_{\Omega \bar{\Delta}}^{*}\left( a_{\bar{\Delta}
}^{\dagger }\left| \text{vac}\right\rangle \right) , \label{Uadagger}
\end{equation}
where $\left| \text{vac}\right\rangle $ is the vacuum of the full Hilbert
space $\mathcal{H}$. Since only linear optical elements are involved we have
$U^{\dagger }\left| \text{vac}\right\rangle =\left| \text{vac}\right\rangle $
, and it further follows from (\ref{Uadagger}) that
\begin{equation}
Ua_{\Omega }^{\dagger }U^{\dagger }=\sum_{\bar{\Delta}}\mathsf{U}_{\Omega
\bar{\Delta}}^{*}a_{\bar{\Delta}}^{\dagger }, \label{Udaggerevol}
\end{equation}
or
\begin{equation}
Ua_{\Omega }U^{\dagger }=\sum_{\bar{\Delta}}\mathsf{U}_{\Omega \bar{\Delta}
}a_{\bar{\Delta}}. \label{Ufund}
\end{equation}
Using the commutation relations satisfied by the creation and annihilation
operators, it immediately follows that the matrix $\mathsf{U}_{\Omega \bar{
\Delta}}$ , which identifies the unitary transformation $U$, is itself a
unitary matrix. Certain calculations can be simplified by its
diagonalization, but for the kind of analysis of few photon states that we
require this is not necessary. We will need to express, in terms of few
photon states with respect to the decomposition $\mathcal{H}_{\bar{C}}$ $
\otimes $ $\mathcal{H}_{\bar{A}}$, the result of acting with $U$ on few
photon states of the decomposition $\mathcal{H}_{C}$ $\otimes \mathcal{H}_{A}
$; this follows directly from (\ref{Udaggerevol}). For example, denoting by $
\left| 1_{\Omega _{1}}2_{\Omega _{2}}\right\rangle $ the state with one
photon in mode $\Omega _{1}$ and two in mode $\Omega _{2}$, we have
\begin{eqnarray}
U\left| 1_{\Omega _{1}}2_{\Omega _{2}}\right\rangle &=&Ua_{\Omega
_{1}}^{\dagger }\frac{\left( a_{\Omega _{2}}^{\dagger }\right) ^{2}}{\sqrt{2}
}\left| \text{vac}\right\rangle \label{Uuse} \\
&=&\frac{1}{\sqrt{2}}\left( Ua_{\Omega _{1}}^{\dagger }U^{\dagger }\right)
\left( Ua_{\Omega _{2}}^{\dagger }U^{\dagger }\right) \left( Ua_{\Omega
_{2}}^{\dagger }U^{\dagger }\right) \left| \text{vac}\right\rangle
\nonumber \\
&=&\frac{1}{\sqrt{2}}\sum_{\bar{\Delta}_{1},\bar{\Delta}_{2},\bar{\Delta}
_{3}}\mathsf{U}_{\Omega _{1}\bar{\Delta}_{1}}^{*}\mathsf{U}_{\Omega _{2}\bar{
\Delta}_{2}}^{*}\mathsf{U}_{\Omega _{2}\bar{\Delta}_{3}}^{*}\left( a_{\bar{
\Delta}_{1}}^{\dagger }a_{\bar{\Delta}_{2}}^{\dagger }a_{\bar{\Delta}
_{3}}^{\dagger }\right) \left| \text{vac}\right\rangle , \nonumber
\end{eqnarray}
and doing the sums in the last line allow us to indeed accomplish our goal.
\subsection{\protect
KLM conditional sign flip}
The first example we consider is the conditional sign flip discussed by
Knill, Laflamme, and Milburn \cite{klm}. Note that in this case the input
ancilla state is pure, there is no feed-forward processing, and the projector $\bar{P
}$ is of unit rank in $\mathcal{H}_{\bar{A}}$. The necessary and sufficient
conditions for the transformation to be operationally unitary are those of
the special case discussed in section \ref{sec:nscon}. The gate consists of one
computational input port (labeled 1) and two ancilla input ports (2 and 3).
The projective measurement is performed on two output ports (b,c) and the
one remaining port is the computational output (a). The subspace $\mathcal{S}
_{C}$ is spanned by the Fock states $|0\rangle$, $|1\rangle$, and $|2\rangle$
in each optical mode.
The pre-measurement evolution, which is done \textit{via }beam splitters and
a phase shifter, is given by the unitary transformation $U$ and
characterized by the matrix
\begin{equation}
\mathsf{U}=\mathsf{U}^{*}=\left[
\begin{array}{lll}
1-\sqrt{2} & 2^{-1/4} & (3/\sqrt{2}-2)^{1/2} \\
2^{-1/4} & 1/2 & 1/2-1/\sqrt{2} \\
(3/\sqrt{2}-2)^{1/2} & 1/2-1/\sqrt{2} & \sqrt{2}-1/2
\end{array}
\right] .
\end{equation}
The ancilla input state is
\begin{equation}
\left| \chi \right\rangle =a_{2}^{\dagger }\left| \text{vac}
_{A}\right\rangle , \label{ancilla input}
\end{equation}
denoting a single photon in the 2 mode, where $\left| \text{vac}
_{A}\right\rangle $ denotes the vacuum of $\mathcal{H}_{A}$. The projective
measurement operator is given by
\[
\bar{P}={I}_{\bar{C}}\otimes \left| \bar{K}\right\rangle \left\langle \bar{K}
\right| ={I}_{\bar{C}}\otimes a_{b}^{\dagger }\left| \text{vac}_{\bar{A}
}\right\rangle \left\langle \text{vac}_{\bar{A}}\right| a_{b}
\]
which corresponds to the detection of one and only one photon in mode b, and
zero photons in mode c. The basis states that define the subspace $\mathcal{S
}_{C}$ are
\[
\left| 0\right\rangle =\left| \text{vac}_{C}\right\rangle , \quad \left|
1\right\rangle =a_{1}^{\dagger }\left| \text{vac}_{C}\right\rangle , \quad
\left| 2\right\rangle =\frac{\left( a_{1}^{\dagger }\right) ^{2}}{\sqrt{2}}
\left| \text{vac}_{C}\right\rangle ,
\]
and the basis states of $\mathcal{H}_{\bar{C}}$ are
\[
\left| \overline{0}\right\rangle =\left| \text{vac}_{\bar{C}}\right\rangle ,
\quad \left| \overline{1}\right\rangle =a_{a}^{\dagger }\left| \text{vac}_{
\bar{C}}\right\rangle , \quad \left| \overline{2}\right\rangle =\frac{\left(
a_{a}^{\dagger }\right) ^{2}}{\sqrt{2}}\left| \text{vac}_{\bar{C}
}\right\rangle ,
\]
In order to evaluate the test function, we first write
\[
U^{\dagger }\bar{P}U=\sum_{\overline{\alpha }}U^{\dagger }\left(
a_{b}^{\dagger }\left| \text{vac}_{\bar{A}}\right\rangle \otimes \left|
\overline{\alpha }\right\rangle \right) \left( \left\langle \overline{\alpha
}\right| \otimes \left\langle \text{vac}_{\bar{A}}\right| a_{b}\right) U
\]
and look at the matrix elements
\begin{eqnarray}
&&\left( \left\langle \alpha \right| \otimes \left\langle \chi\right|
\right) U^{\dagger }\bar{P}U\left( \left| \chi\right\rangle \otimes \left|
\beta \right\rangle \right) \label{alphasum} \\
&=&\sum_{\overline{\alpha }}\left( \left\langle \alpha \right| \otimes
\left\langle \text{vac}_{A}\right| a_{2}\right) U^{\dagger }\left(
a_{b}^{\dagger }\left| \text{vac}_{\bar{A}}\right\rangle \otimes \left|
\overline{\alpha }\right\rangle \right) \left( \left\langle \overline{\alpha
}\right| \otimes \left\langle \text{vac}_{\bar{A}}\right| a_{b}\right)
U\left( a_{2}^{\dagger }\left| \text{vac}_{A}\right\rangle \otimes \left|
\beta \right\rangle \right) \nonumber
\end{eqnarray}
over the computational subspace, $\mathcal{S}_{C}$. The calculation is
straightforward. Applying the operator $U$ on each of the states $
a_{2}^{\dagger }\left| \text{vac}_{A}\right\rangle \otimes \left| \beta
\right\rangle $ gives the following states in the $\mathcal{H}_{\bar{C}}$ $
\otimes $ $\mathcal{H}_{\bar{A}}$ decomposition
\begin{eqnarray*}
U\left( a_{2}^{\dagger }\left| \text{vac}_{A}\right\rangle \otimes \left|
0\right\rangle \right) &=&\left( 2^{-1/4}a_{a}^{\dagger }+\frac{1}{2}
a_{b}^{\dagger }+\left[ \frac{1}{2}-\frac{1}{\sqrt{2}}\right] a_{c}^{\dagger
}\right) \left| \text{vac}\right\rangle \\
U\left( a_{2}^{\dagger }\left| \text{vac}_{A}\right\rangle \otimes \left|
1\right\rangle \right) &=&\left( 2^{-1/4}a_{a}^{\dagger }+\frac{1}{2}
a_{b}^{\dagger }+\left[ \frac{1}{2}-\frac{1}{\sqrt{2}}\right] a_{c}^{\dagger
}\right) \\
&&\times \left( \left[ 1-\sqrt{2}\right] a_{a}^{\dagger
}+2^{-1/4}a_{b}^{\dagger }+\left[ \frac{3}{\sqrt{2}}-2\right]
^{1/2}a_{c}^{\dagger }\right) \left| \text{vac}\right\rangle \\
U\left( a_{2}^{\dagger }\left| \text{vac}_{A}\right\rangle \otimes \left|
2\right\rangle \right) &=&\frac{1}{\sqrt{2}}\left( 2^{-1/4}a_{a}^{\dagger }+
\frac{1}{2}a_{b}^{\dagger }+\left[ \frac{1}{2}-\frac{1}{\sqrt{2}}\right]
a_{c}^{\dagger }\right) \\
&&\times \left( \left[ 1-\sqrt{2}\right] a_{a}^{\dagger
}+2^{-1/4}a_{b}^{\dagger }+\left[ \frac{3}{\sqrt{2}}-2\right]
^{1/2}a_{c}^{\dagger }\right) ^{2}\left| \text{vac}\right\rangle .
\end{eqnarray*}
and we can then separately evaluate the terms in the sum (\ref{alphasum}),
noting that the non-zero elements are
\begin{eqnarray*}
\left| \left( \left\langle \overline{0}\right| \otimes \left\langle \text{vac
}_{\bar{A}}\right| a_{b}\right) U\left( a_{2}^{\dagger }\left| \text{vac}
_{A}\right\rangle \otimes \left| 0\right\rangle \right) \right| ^{2} &=&
\frac{1}{4} \\
\left| \left( \left\langle \overline{1}\right| \otimes \left\langle \text{vac
}_{\bar{A}}\right| a_{b}\right) U\left( a_{2}^{\dagger }\left| \text{vac}
_{A}\right\rangle \otimes \left| 1\right\rangle \right) \right| ^{2} &=&
\frac{1}{4} \\
\left| \left( \left\langle \overline{2}\right| \otimes \left\langle \text{vac
}_{\bar{A}}\right| a_{b}\right) U\left( a_{2}^{\dagger }\left| \text{vac}
_{A}\right\rangle \otimes \left| 2\right\rangle \right) \right| ^{2} &=&
\frac{1}{4}
\end{eqnarray*}
The test operator $T$ is then
\begin{eqnarray*}
T &=&\frac{1}{4}\left[ \left| \text{vac}_{C}\right\rangle \left\langle \text{
vac}_{C}\right| +a_{1}^{\dagger }\left| \text{vac}_{C}\right\rangle
\left\langle \text{vac}_{C}\right| a_{1}+\frac{\left( a_{1}^{\dagger
}\right) ^{2}}{\sqrt{2}}\left| \text{vac}_{C}\right\rangle \left\langle
\text{vac}_{C}\right| \frac{\left( a_{1}\right) ^{2}}{\sqrt{2}}\right] \\
&=&\frac{1}{4}{I}_{\mathcal{S}C},
\end{eqnarray*}
and is indeed a multiple of the unit operator in the computational input
space. The probability of a success-indicating measurement is 1/4,
independent of the computational input state. Since this test condition is
satisfied, the transformation (\ref{spectrans}) is operationally unitary.
The terms of the transformation matrix $w^{\bar{\alpha}\beta }$ can be
calculated noting that the non-zero $\left\langle \bar{K}\right|
\left\langle \bar{\alpha}\right| U\left| \beta \right\rangle \left|
\chi\right\rangle $ terms are
\begin{eqnarray*}
\left\langle \bar{K}\right| \left\langle \overline{0}\right| U\left|
0\right\rangle \left| \chi\right\rangle &=&\frac{1}{2}, \\
\left\langle \bar{K}\right| \left\langle \overline{1}\right| U\left|
1\right\rangle \left| \chi\right\rangle &=&\frac{1}{2}, \\
\left\langle \bar{K}\right| \left\langle \overline{2}\right| U\left|
2\right\rangle \left| \chi\right\rangle &=&-\frac{1}{2},
\end{eqnarray*}
and since $\tau =1/4$ the non-zero elements of the transformation matrix are
\begin{eqnarray*}
w^{\overline{0}0} &=&1, \\
w^{\overline{1}1} &=&1, \\
w^{\overline{2}2} &=&-1,
\end{eqnarray*}
which corresponds to the conditional sign flip, since with probability 1/4
the gate takes the input state $\left| \psi \right\rangle =\alpha _{0}\left|
0\right\rangle +\alpha _{1}\left| 1\right\rangle +\alpha _{2}\left|
2\right\rangle $ and produces the state $\left| \bar{\psi}\right\rangle
=\alpha _{0}\left| \overline{0}\right\rangle +\alpha _{1}\left| \overline{1}
\right\rangle -\alpha _{2}\left| \overline{2}\right\rangle $.
This map can be seen to exhibit an effective nonlinear interaction between
the photons, since the formally equivalent unitary map (see section \ref{sec:consec}) is
characterized by the unitary operator $M$ (\ref{correspondence}),
\[
\left| \tilde{\psi}\right\rangle =\alpha _{0}\left| 0\right\rangle +\alpha
_{1}\left| 1\right\rangle -\alpha _{2}\left| 2\right\rangle =M\left( \alpha
_{0}\left| 0\right\rangle +\alpha _{1}\left| 1\right\rangle +\alpha
_{2}\left| 2\right\rangle \right) ,
\]
which can be written in terms of an effective action operator $Q$ (\ref
{action}), where we can take
\[
Q=\frac{\pi \hbar }{2}\left( 5\widehat{n}-\widehat{n}^{2}\right) ,
\]
with $\hat{n}$ the photon number operator. But such an effective action
operator exists only if we restrict ourselves to the three-dimensional
subspace $\mathcal{S}_{C}$, spanned by the kets $\left| 0\right\rangle $, $
\left| 1\right\rangle $, and $\left| 2\right\rangle $. For consider an
attempt to expand this subspace to that spanned by the kets $\left( \left|
0\right\rangle ,\left| 1\right\rangle ,\left| 2\right\rangle ,\left|
3\right\rangle \right) $. The device guarantees that a computational input
of three photons can only produce a computational three-photon output, since
a successful measurement requires the detection of one and only one photon
in the ancilla space. The test operator is therefore still diagonal in the
photon number basis. However, we find
\[
\left| \left( \left\langle \overline{3}\right| \otimes \left\langle \text{vac
}_{\bar{A}}\right| a_{b}\right) U\left( a_{2}^{\dagger }\left| \text{vac}
_{A}\right\rangle \otimes \left| 3\right\rangle \right) \right| ^{2}=\left( 2
\sqrt{2}-\frac{5}{2}\right) ^{2},
\]
and thus the test operator $T$ is no longer a multiple of the unit operator
in this enlarged subspace. In this larger space the probability of a
success-indicating measurement is dependent on the input, and the map is not
operationally unitary.
\subsection{Polarization encoded CNOT}
The second example is the polarization-encoded Gottesman-Chuang protocol
discussed by Pittman \textit{et al}.\ \cite{franson}. In this case the input
ancilla state is pure, there is feed-forward processing, and there are several
projectors $\bar{P}_{(L)}$ of unit rank in $\mathcal{H}_{\bar{A}}$. The
necessary and sufficient conditions for the transformation to be
operationally unitary are therefore those of the special case discussed in
section \ref{sec:post}. The device has two computational input ports (labeled $a$ and $b$
) and four ancilla input ports (1-4). A projective measurement is made on
four output ports ($p$,$q$,$n$,$m$) while the two remaining ports are the
computational output (5 and 6). A photon of horizontal polarization
represents a logical 0, and a vertically polarized photon represents a
logical 1. We use the same notation as Pittman \textit{et al. }\cite{franson}
. For example, $\left| H(V)_{a}\right\rangle $ represents a
horizontally(vertically) polarized photon in port `a' and the Hadamard
transformed modes are $\left| F(S)_{a}\right\rangle =\frac{1}{2}\left[
\left| H_{a}\right\rangle \pm \left| V_{a}\right\rangle \right] $. The four
basis states of the computational input are $\left| 00\right\rangle =$ $
\left| H_{a}\right\rangle \left| H_{b}\right\rangle ,$ $\left|
01\right\rangle =$ $\left| H_{a}\right\rangle \left| V_{b}\right\rangle ,$ $
\left| 10\right\rangle =$ $\left| V_{a}\right\rangle \left|
H_{b}\right\rangle ,$ $\left| 11\right\rangle =$ $\left| V_{a}\right\rangle
\left| V_{b}\right\rangle $ and the output states are labeled as $\left|
\overline{00}\right\rangle =\left| H_{5}\right\rangle \left|
H_{6}\right\rangle ,\left| \overline{01}\right\rangle =\left|
H_{5}\right\rangle \left| V_{6}\right\rangle ,\left| \overline{10}
\right\rangle =\left| V_{5}\right\rangle \left| H_{6}\right\rangle ,\left|
\overline{11}\right\rangle =\left| V_{5}\right\rangle \left|
V_{6}\right\rangle .$ The input ancilla state is
\begin{eqnarray*}
\left| \chi \right\rangle &=&\frac{1}{2}\left( \left| H_{1}\right\rangle
\left| H_{4}\right\rangle \left| H_{2}\right\rangle \left|
H_{3}\right\rangle +\left| H_{1}\right\rangle \left| V_{4}\right\rangle
\left| H_{2}\right\rangle \left| V_{3}\right\rangle \right) \\
&&+\frac{1}{2}\left( \left| V_{1}\right\rangle \left| H_{4}\right\rangle
\left| V_{2}\right\rangle \left| V_{3}\right\rangle +\left|
V_{1}\right\rangle \left| V_{4}\right\rangle \left| V_{2}\right\rangle
\left| H_{3}\right\rangle \right) ,
\end{eqnarray*}
and the measurement projectors, $\bar{P}_{(L)}={I}_{\bar{C}}\otimes \left|
\overline{k_{L}}\right\rangle \left\langle \overline{k_{L}}\right| ,$
represent the 16 possible success outcomes:
\begin{eqnarray*}
\left| \overline{k_{1}}\right\rangle &=&\left| F_{p}\right\rangle \left|
F_{q}\right\rangle \left| F_{n}\right\rangle \left| F_{m}\right\rangle \\
&=&\frac{1}{4}\left( \left| H_{p}\right\rangle +\left| V_{p}\right\rangle
\right) \left( \left| H_{q}\right\rangle +\left| V_{q}\right\rangle \right)
\left( \left| H_{n}\right\rangle +\left| V_{n}\right\rangle \right) \left(
\left| H_{m}\right\rangle +\left| V_{m}\right\rangle \right) \\
\left| \overline{k_{2}}\right\rangle &=&\left| F_{p}\right\rangle \left|
F_{q}\right\rangle \left| F_{n}\right\rangle \left| S_{m}\right\rangle \\
&=&\frac{1}{4}\left( \left| H_{p}\right\rangle +\left| V_{p}\right\rangle
\right) \left( \left| H_{q}\right\rangle +\left| V_{q}\right\rangle \right)
\left( \left| H_{n}\right\rangle +\left| V_{n}\right\rangle \right) \left(
\left| H_{m}\right\rangle -\left| V_{m}\right\rangle \right) \\
&& \\
&&\vdots \\
&& \\
\left| \overline{k_{15}}\right\rangle &=&\left| S_{p}\right\rangle \left|
S_{q}\right\rangle \left| S_{n}\right\rangle \left| F_{m}\right\rangle \\
&=&\frac{1}{4}\left( \left| H_{p}\right\rangle -\left| V_{p}\right\rangle
\right) \left( \left| H_{q}\right\rangle -\left| V_{q}\right\rangle \right)
\left( \left| H_{n}\right\rangle -\left| V_{n}\right\rangle \right) \left(
\left| H_{m}\right\rangle +\left| V_{m}\right\rangle \right) \\
\left| \overline{k_{16}}\right\rangle &=&\left| S_{p}\right\rangle \left|
S_{q}\right\rangle \left| S_{n}\right\rangle \left| S_{m}\right\rangle \\
&=&\frac{1}{4}\left( \left| H_{p}\right\rangle -\left| V_{p}\right\rangle
\right) \left( \left| H_{q}\right\rangle -\left| V_{q}\right\rangle \right)
\left( \left| H_{n}\right\rangle -\left| V_{n}\right\rangle \right) \left(
\left| H_{m}\right\rangle -\left| V_{m}\right\rangle \right)
\end{eqnarray*}
The polarizing beam splitters perform a unitary evolution on the input
ports, characterized by the set of quantities \textsf{U}$_{\Omega \bar{\Delta
}}^{*}$. One can summarize the evolution of modes in $\mathcal{H}_{C}$ $
\otimes$ $\mathcal{H}_{A}$ to modes in $\mathcal{H}_{\bar{C}}$ $\otimes $ $
\mathcal{H}_{\bar{A}}$ with the following linear map
\begin{eqnarray*}
\left| H_{1}\right\rangle &\rightarrow &\left| H_{p}\right\rangle ,\left|
V_{1}\right\rangle \rightarrow -i\left| V_{q}\right\rangle , \\
\left| H_{2}\right\rangle &\rightarrow &\left| H_{5}\right\rangle ,\left|
V_{2}\right\rangle \rightarrow \left| V_{5}\right\rangle , \\
\left| H_{3}\right\rangle &\rightarrow &\left| H_{6}\right\rangle ,\left|
V_{3}\right\rangle \rightarrow \left| V_{6}\right\rangle , \\
\left| H_{4}\right\rangle &\rightarrow &\left| H_{m}\right\rangle ,\left|
V_{4}\right\rangle \rightarrow -i\left| V_{n}\right\rangle , \\
\left| H_{a}\right\rangle &\rightarrow &\left| H_{q}\right\rangle ,\left|
V_{a}\right\rangle \rightarrow -i\left| V_{p}\right\rangle , \\
\left| H_{b}\right\rangle &\rightarrow &\left| H_{n}\right\rangle ,\left|
V_{b}\right\rangle \rightarrow -i\left| V_{m}\right\rangle ,
\end{eqnarray*}
since \textsf{U}$_{H_{1}H_{p}}^{*}=1,$ \textsf{U}$_{V_{1}V_{q}}^{*}=-i,$
\textit{etc}. As in the previous example, to evaluate the test operators, we
first look at the terms
\begin{eqnarray*}
U\left( \left| 00\right\rangle \left| \chi \right\rangle \right) &=&\frac{
\left| H_{q}\right\rangle \left| H_{n}\right\rangle }{2}\left[
\begin{array}{c}
\left| H_{p}\right\rangle \left| H_{m}\right\rangle \left|
H_{5}\right\rangle \left| H_{6}\right\rangle -i\left| H_{p}\right\rangle
\left| V_{n}\right\rangle \left| H_{5}\right\rangle \left|
V_{6}\right\rangle \\
-i\left| V_{q}\right\rangle \left| H_{m}\right\rangle \left|
V_{5}\right\rangle \left| V_{6}\right\rangle -\left| V_{q}\right\rangle
\left| V_{n}\right\rangle \left| V_{5}\right\rangle \left|
H_{6}\right\rangle
\end{array}
\right] \\
U\left( \left| 01\right\rangle \left| \chi \right\rangle \right) &=&\frac{
-i\left| H_{q}\right\rangle \left| V_{m}\right\rangle }{2}\left[
\begin{array}{c}
\left| H_{p}\right\rangle \left| H_{m}\right\rangle \left|
H_{5}\right\rangle \left| H_{6}\right\rangle -i\left| H_{p}\right\rangle
\left| V_{n}\right\rangle \left| H_{5}\right\rangle \left|
V_{6}\right\rangle \\
-i\left| V_{q}\right\rangle \left| H_{m}\right\rangle \left|
V_{5}\right\rangle \left| V_{6}\right\rangle -\left| V_{q}\right\rangle
\left| V_{n}\right\rangle \left| V_{5}\right\rangle \left|
H_{6}\right\rangle
\end{array}
\right] \\
U\left( \left| 10\right\rangle \left| \chi \right\rangle \right) &=&\frac{
-i\left| V_{p}\right\rangle \left| H_{n}\right\rangle }{2}\left[
\begin{array}{c}
\left| H_{p}\right\rangle \left| H_{m}\right\rangle \left|
H_{5}\right\rangle \left| H_{6}\right\rangle -i\left| H_{p}\right\rangle
\left| V_{n}\right\rangle \left| H_{5}\right\rangle \left|
V_{6}\right\rangle \\
-i\left| V_{q}\right\rangle \left| H_{m}\right\rangle \left|
V_{5}\right\rangle \left| V_{6}\right\rangle -\left| V_{q}\right\rangle
\left| V_{n}\right\rangle \left| V_{5}\right\rangle \left|
H_{6}\right\rangle
\end{array}
\right] \\
U\left( \left| 11\right\rangle \left| \chi \right\rangle \right) &=&\frac{
-\left| V_{p}\right\rangle \left| V_{m}\right\rangle }{2}\left[
\begin{array}{c}
\left| H_{p}\right\rangle \left| H_{m}\right\rangle \left|
H_{5}\right\rangle \left| H_{6}\right\rangle -i\left| H_{p}\right\rangle
\left| V_{n}\right\rangle \left| H_{5}\right\rangle \left|
V_{6}\right\rangle \\
-i\left| V_{q}\right\rangle \left| H_{m}\right\rangle \left|
V_{5}\right\rangle \left| V_{6}\right\rangle -\left| V_{q}\right\rangle
\left| V_{n}\right\rangle \left| V_{5}\right\rangle \left|
H_{6}\right\rangle
\end{array}
\right]
\end{eqnarray*}
The matrix elements of interest are now
\begin{eqnarray}
&&\left( \left\langle \alpha \right| \otimes \left\langle \chi \right|
\right) U^{\dagger }\bar{P}_{(L)}U\left( \left| \chi \right\rangle \otimes
\left| \beta \right\rangle \right) \label{alphasum2} \\
&=&\sum_{\overline{\alpha }}\left( \left\langle \alpha \right| \otimes
\left\langle \chi \right| \right) U^{\dagger }\left| \overline{k_{L}}
\right\rangle \left| \overline{\alpha }\right\rangle \left\langle \overline{
\alpha }\right| \left\langle \overline{k_{L}}\right| U\left( \left| \chi
\right\rangle \otimes \left| \beta \right\rangle \right) \nonumber
\end{eqnarray}
and the non-zero terms of the sum in (\ref{alphasum2}) are
\begin{eqnarray*}
\left| \left\langle \overline{00}\right| \left\langle \overline{k_{L}}
\right| U\left( \left| \chi \right\rangle \otimes \left| 00\right\rangle
\right) \right| ^{2} &=&\frac{1}{16} \\
\left| \left\langle \overline{01}\right| \left\langle \overline{k_{L}}
\right| U\left( \left| \chi \right\rangle \otimes \left| \overline{01}
\right\rangle \right) \right| ^{2} &=&\frac{1}{16} \\
\left| \left\langle \overline{11}\right| \left\langle \overline{k_{L}}
\right| U\left( \left| \chi \right\rangle \otimes \left| \overline{10}
\right\rangle \right) \right| ^{2} &=&\frac{1}{16} \\
\left| \left\langle \overline{10}\right| \left\langle \overline{k_{L}}
\right| U\left( \left| \chi \right\rangle \otimes \left| \overline{11}
\right\rangle \right) \right| ^{2} &=&\frac{1}{16}
\end{eqnarray*}
for all $L$. The test functions, $\left\{ T_{(L)}\right\} $ , are then
\begin{eqnarray*}
T_{(L)} &=&\frac{1}{64}\left[
\begin{array}{c}
\left| H_{a}\right\rangle \left| H_{b}\right\rangle \left\langle
H_{b}\right| \left\langle H_{a}\right| +\left| H_{a}\right\rangle \left|
V_{b}\right\rangle \left\langle V_{b}\right| \left\langle H_{a}\right| \\
+\left| V_{a}\right\rangle \left| H_{b}\right\rangle \left\langle
H_{b}\right| \left\langle V_{a}\right| +\left| V_{a}\right\rangle \left|
V_{b}\right\rangle \left\langle V_{b}\right| \left\langle V_{a}\right|
\end{array}
\right] \\
&=&\frac{1}{64}\;{I}_{\mathcal{S}C}
\end{eqnarray*}
and are indeed multiples of the unit operator in the computational input
space. In this scheme $\tau _{(L)}=1/64$, and the probability of success is
the sum of the individual probabilities of the 16 detection outcomes, $
\sum_{L}\tau _{(L)}=1/4.$ The terms of the transformation matrices $w_{L}^{
\bar{\alpha}\beta }$ can be calculated noting that the non-zero $
\left\langle \overline{k_{L}}\right| \left\langle \bar{\lambda}\right|
U\left| \beta \right\rangle \left| \chi \right\rangle $ terms are
\begin{eqnarray*}
\left\langle \overline{k_{L}}\right| \left\langle \overline{00}\right|
U\left| 00\right\rangle \left| \chi \right\rangle &=&e^{i\phi _{L,0}}/8, \\
\left\langle \overline{k_{L}}\right| \left\langle \overline{01}\right|
U\left| 01\right\rangle \left| \chi \right\rangle &=&e^{i\phi _{L,1}}/8, \\
\left\langle \overline{k_{L}}\right| \left\langle \overline{11}\right|
U\left| 10\right\rangle \left| \chi \right\rangle &=&e^{i\phi _{L,2}}/8, \\
\left\langle \overline{k_{L}}\right| \left\langle \overline{10}\right|
U\left| 11\right\rangle \left| \chi \right\rangle &=&e^{i\phi _{L,3}}/8,
\end{eqnarray*}
where $e^{i\phi _{L,0}}=1,e^{i\phi _{1,1}}=-1,e^{i\phi _{2,1}}=1,\ldots
,e^{i\phi _{16,3}}=1$ are phase factors of $\pm $1$.$ For this
transformation to be operationally unitary, the $w_{L}^{\bar{\alpha}\beta }$
matrices must all be proportional to each other. In certain outcomes,
single-qubit operations ($\pi $-phase shifts) are required to correct the
phase factors so that the transformation is operationally unitary and the
desired output is produced. The feed-forward processing matrices, $\bar{V}_{(L)}^{
\bar{\alpha}\bar{\lambda}}$, represent these single qubit operations.
Setting
\[
\bar{V}_{(L)}^{\overline{00},\overline{00}}=e^{i\phi _{L,0}},\quad \bar{V}
_{(L)}^{\overline{01},\overline{01}}=e^{i\phi _{L,1}},\quad \bar{V}_{(L)}^{
\overline{11},\overline{11}}=e^{i\phi _{L,2}},\quad \bar{V}_{(L)}^{\overline{
10},\overline{10}}=e^{i\phi _{L,3}}\;,
\]
with all other elements equal to zero gives the appropriate corrections. The
non-zero transformation matrix elements, are then
\begin{eqnarray*}
w_{L}^{\overline{00},00} &=&1 \\
w_{L}^{\overline{01},01} &=&1 \\
w_{L}^{\overline{11},10} &=&1 \\
w_{L}^{\overline{10},11} &=&1
\end{eqnarray*}
for all $L$. Since the 16 evolution matrices are identical, the
proportionality condition is satisfied. The transformation is then
\[
\bar{\rho}^{\bar{\alpha}\bar{\delta}}=\sum_{\beta ,\gamma }w_{L}^{\bar{\alpha
}\beta }\rho ^{\beta \gamma }\left( w_{L}^{\bar{\delta}\gamma }\right) ^{*}
\]
which is the CNOT operation. This gate takes the input state $\left| \psi
\right\rangle =\alpha _{0}\left| 00\right\rangle +\alpha _{1}\left|
01\right\rangle +\alpha _{2}\left| 10\right\rangle +\alpha _{3}\left|
11\right\rangle $ and produces the state $\alpha _{0}\left| \overline{00}
\right\rangle +\alpha _{1}\left| \overline{01}\right\rangle +\alpha
_{2}\left| \overline{11}\right\rangle +\alpha _{3}\left| \overline{10}
\right\rangle $ with probability 1/4. Again, this map exhibits an effective
nonlinear interaction between the photons since the formally equivalent
unitary map is characterized by a nonlinear effective action operator $Q$ (
\ref{action}). In this case one could choose
\[
Q=\frac{\pi \hbar }{2}(3+a_{b}^{\dagger }(1-\hat{n}_{b})+(1-\hat{n}
_{b})a_{b})\hat{n}_{a}.
\]
Again, however, the operational unitarity is restricted to the subspace.
Suppose we expand the computational subspace to include an extra photon in
one of the input modes. As an example, consider the special state $\left|
S\right\rangle =$ $\left| H_{a}\right\rangle \left| H_{b}\right\rangle
\left| H_{b}\right\rangle $. The form of the projectors indicates that the
detection events involve one and only one photon in the appropriate modes.
Evaluating the corresponding test operator elements we find
\[
\left| \left\langle \overline{\alpha }\right| \left\langle \overline{k_{L}}
\right| U\left( \left| \chi \right\rangle \otimes \left| S\right\rangle
\right) \right| ^{2}=0\;,
\]
since the extra photon inhibits a success-indicating measurement result. The
evolution cannot be operationally unitary in this expanded subspace because
the test operator is no longer proportional to the unit operator.
\section{Conclusion}
\label{sec:conclusions}
In this paper we introduced a general approach to the investigation of
conditional measurement devices. We considered an important class of optical
$N$-port devices, including those employing projectors of rank greater than
unity, mixed input ancilla states, multiple success outcomes, and
feed-forward processing. We also sketched how more general POVMs, rather than PVMs,
could be included. The necessary and sufficient conditions for these devices
to simulate unitary evolution have been derived. They are not surprising,
and indeed from a physical point of view are fairly obvious. But to our
knowledge they have not been discussed in this general way before. One of
the conditions is that the probability of each successful outcome must be
independent of the input density operator. Whether or not this holds can be
checked by evaluating a set of test operators over the input computational
Hilbert space, which is easily done for any proposed device. In the special
case of only one successful outcome there is only one test operator to be
computed; furthermore, if the ancilla state is pure and the success
projector of rank one, then the passing of a test condition by that single
test operator guarantees that the map is operationally unitary. In the case
of more than one successful outcome it is a necessary consequence of
operational unitarity that each of the test operators pass the test
condition. This is not sufficient to imply operational unitarity in the
multiple projector case unless the proportionality condition is also
satisfied. The proportionality condition can often be satisfied by
introducing feed-forward processing.
Besides application in the analysis of particular proposed devices, we
believe the general framework presented here will be useful in exploring the
different types of pre-measurement evolution and measurements that might be
useful in the design, optimization, and characterization of such devices. In
particular, the conditional sign flip and polarization-encoded CNOT devices
we considered functioned as operationally unitary maps only over the input
computational subspaces for which they were originally proposed. So while
effective photon nonlinearities could be introduced, the degree to which
they are physically meaningful is somewhat limited. An outstanding issue,
perhaps even of interest more from the general perspective of nonlinear
optics than from that of quantum computer design, is the study of potential
devices that provide effective photon nonlinearities over much larger input
computational subspaces. The question remains: to what extent are such
devices possible in theory and feasible in practice?
Finally, we note that only in section \ref{sec:examples} did we assume that
the pre-measurement unitary evolution $U$ is associated with linear elements
in an optical system. The more general framework of the earlier sections may
find application in describing other proposed devices for quantum
information processing that involve conditional measurement schemes in the
presence of more complicated interactions \cite{dfs}.
\pagebreak
\begin{picture}(400,150)(0,0)
\put(150,0){\framebox(60,120){U}}
\put(75,90){\vector(1,0){75}}
\put(75,30){\vector(1,0){75}}
\put(210,90){\vector(1,0){75}}
\put(210,30){\vector(1,0){25}}
\put(235,10){\framebox(70,40)}
\put(30,75){Computational Input}
\put(220,75){Computational Output}
\put(40,15){Ancilla Input}
\put(240,33){Measurement}
\put(260,18){on $\mathcal{H}_{\bar{A}}$}
\put(290,90){$\mathcal{H}_{\bar{C}}$}
\put(55,30){$\mathcal{H}_{A}$}
\put(55,90){$\mathcal{H}_{C}$}
\end{picture}
\begin{figure}
\caption{A schematic diagram of a basic conditional measurement
device.
The input computational channels, $\mathcal{H}
\end{figure}
\begin{picture}(400,200)
\put(120,0){\framebox(60,120){U}}
\put(75,90){\vector(1,0){45}}
\put(75,30){\vector(1,0){45}}
\put(275,90){\vector(1,0){20}}
\put(180,30){\vector(1,0){25}}
\put(180,90){\vector(1,0){25}}
\put(205,70){\framebox(70,40)}
\put(205,10){\framebox(70,40)}
\put(210,33){Measurement}
\put(229,18){on $\mathcal{H}_{\bar{A}}$}
\put(212,93){Feed-forward}
\put(212,78){processing}
\put(300,90){$\mathcal{H}_{\bar{C}}$}
\put(55,30){$\mathcal{H}_{A}$}
\put(55,90){$\mathcal{H}_{C}$}
\put(242,50){\line(0,1){20}}
\put(238,50){\line(0,1){20}}
\end{picture}
\begin{figure}
\caption{A schematic diagram of a conditional measurement
device that incorporates
feed-forward processing. The double line connecting the two small boxes
represents a classical channel that carries the measurement result. Based
on the outcome, the appropriate processing
is performed on the output channel.}
\end{figure}
\end{document} | math | 70,467 |
\begin{document}
\title{ extbf{Large butterfly Cayley graphs and digraphs}
{\let\thefootnote\relax\footnotetext
{${}^\dagger$Department of Computer and Information Sciences, University of Strathclyde, Glasgow, Scotland.}}
{\let\thefootnote\relax\footnotetext
{2010 Mathematics Subject Classification:
05C25,
05C20,
05C35.
}}
\begin{abstract}
\noindent
We present
families of
large
undirected and directed
Cayley graphs
whose construction is related to butterfly networks.
One approach yields,
for every large~$k$ and for values of~$d$ taken from a large interval,
the largest known Cayley
graphs and digraphs
of diameter~$k$ and
degree~$d$.
Another
method
yields, for sufficiently large~$k$ and infinitely many values of~$d$,
Cayley
graphs and digraphs
of diameter~$k$ and
degree~$d$ whose order is exponentially
larger in~$k$
than any previously constructed.
In the directed case, these are within a linear factor in $k$ of the Moore bound.
\end{abstract}
\section{Introduction}
The goal of the \emph{degree--diameter problem} is to determine the largest possible order of
a graph or digraph,
perhaps restricted to some special class,
with given maximum (out)degree and diameter.
For an overview of progress on a wide variety of approaches to this problem, see the
survey by Miller \& {\v{S}}ir{\'a}{\v{n}}~\cite{MS2013}.
Our concern here is with
large
\emph{Cayley} graphs and digraphs.
Recall that, for a group $G$ and a unit-free generating subset $S$ of $G$, the \emph{Cayley digraph} of $G$ generated by $S$ has vertex set $G$ and a directed edge from $g$ to $gs$ for all $g\in G$ and $s\in S$.
If $S$ is symmetric, i.e. $S=S^{-1}$, then
the corresponding undirected simple graph is the \emph{Cayley graph} of $G$ generated by $S$.
The Cayley (di)graph is thus regular of (out)degree $|S|$
and vertex-transitive.
We are interested in graphs and digraphs
of
degree~$d$ and diameter $k$,
for arbitrary large $k$ and varying $d$.
If a construction yields graphs of order $n_{d,k}$, we say that it
has \emph{asymptotic order} $f(d,k)$ if,
for fixed $k$,
$$
\liminfty[d]
\frac{n_{d,k}}{f(d,k)}
\;=\;
1
.
$$
No graph or digraph can be larger than the corresponding \emph{Moore bound}.
For undirected graphs, this bound is
$\mathrm{M}_{d,k}=1+ \frac{d}{d-2}\big((d-1)^k-1\big)$ if $d>2$.
In the directed case, it is
$\mathrm{DM}_{d,k}=\frac{1}{d-1}\big(d^{k+1}-1\big)$ if $d>1$.
In both cases, the Moore bound has asymptotic order $d^k$.
Previously, for arbitrary degree and diameter,
the largest known directed Cayley graphs were obtained by Vetr\'ik~\cite{Vetrik2012} and Abas \& Vetr\'ik~\cite{AV2017}, whose constructions have asymptotic order
$k\big(\frac{d}{2}\big)^{\!k}$ for even $k$, and $2k\big(\frac{d}{2}\big)^{\!k}$ for odd $k$.
Our construction yields Cayley digraphs whose order is asymptotically $k\hspace{0.07 em}d^{k-1}$.
For fixed diameter $k\geqslant8$, these digraphs are larger than those in~\cite{Vetrik2012} and~\cite{AV2017} for every value of $d$ in a large interval.
We also construct,
for fixed $k$ and infinitely many values
of $d$,
Cayley digraphs whose asymptotic order is
$\frac{d^k}{e^2k}$,
a factor of $\frac{2^{k-1}}{e^2k^2}$ larger than those of Abas \& Vetr\'ik, and
within a linear factor in $k$ of the Moore bound.
The undirected case is similar. Previously, the largest known Cayley graphs
were obtained by Macbeth, {\v{S}}iagiov{\'a}, {\v{S}}ir{\'a}{\v{n}} \& Vetr\'ik~\cite{MSSV2010}, whose construction has asymptotic order $k\big(\frac{d}{3}\big)^{\!k}$.
For $d-k\not\equiv 3\!\pmod{4}$,
we construct Cayley graphs whose order is asymptotically $k\big(\frac{d}{2}\big)^{\!k-1}$.
For sufficiently large diameter~$k$, these graphs are larger than those in~\cite{MSSV2010} for every suitable value of $d$ in a large interval.
We also construct,
for given $k$ and infinitely many values
of $d$,
Cayley graphs whose asymptotic order is
$\frac{1}{e^2k}\big(\frac{d}{2}\big)^{\!k}$,
a factor of $\frac{1}{e^2k^2}\big(\frac{3}{2}\big)^{\!k}$ larger than those
in~\cite{MSSV2010}.
Our constructions are based on a two-parameter family of groups.
For $t\geqslant2$, let $\bbZ_t=\mathbb{Z}/t\mathbb{Z}$ be the additive group of integers modulo $t$, and for $r\geqslant2$, let $\bbZ_tr$ denote the product $\bbZ_t\times\ldots\times\bbZ_t$, where $\bbZ_t$ occurs $r$ times,
considered as an additive group of vectors.
Let $\alpha$ be the automorphism of $\bbZ_tr$, defined by $\alpha(v_0,\ldots,v_{r-1})=(v_{r-1},v_0,\ldots,v_{r-2})$,
that cyclically shifts coordinates rightwards by one, and
consider the semidirect product $G=\bbZ_tr\rtimes\bbZ_r$, of order $r\hspace{0.07 em}t^r$,
with the group operation given by $(u,s)\!\cdot\!(v,s')=(u+\alpha^s(v),s+s')$,
for $u,v\in\bbZ_tr$ and $s,s'\in\bbZ_r$.
We write elements of $G$ in the form $(v_0,\ldots,v_{r-1};s)$, where each $v_i\in\bbZ_t$ and $s\in\bbZ_r$.
Using this notation, the group operation is
\begin{multline*}
(u_0,\ldots,u_{r-1};\,s) \!\cdot\! (v_0,\ldots,v_{r-1};\,s') \\
\;\,=\;\,
(u_0+v_{r-s},\,
\ldots,\, u_{s-1}+v_{r-1},\, u_s+v_0,\,
\ldots,\, u_{r-1}+v_{r-1-s};\, s+s' ) ,
\end{multline*}
arithmetic in the subscripts being performed modulo $r$.
The group $G$ is used to create all our Cayley graphs and digraphs.
The Cayley digraph generated by elements of $G$ of the form $(a,0,\ldots,0;1)$, $a\in\bbZ_t$ is isomorphic to the base-$t$ order-$r$ (wrapped) \emph{butterfly network}, $B_t(r)$,
so called because it is composed
of
$rt^{r-1}$ edge-disjoint
\mbox{$t$-\emph{butterflies}} (copies of the complete bipartite graph $K_{t,t}$); see~\cite[Figure~2]{ABR1990}.
Butterfly networks are closely related to the \emph{de~Bruijn graphs}~\cite{deBruijn1946},
the directed base-$t$ order-$r$ de~Bruijn graph being a coset graph of
$B_t(r)$~\cite[Theorem~4.4]{ABR1990}.
Cayley graphs and digraphs of $G$ were used previously by
Macbeth, {\v{S}}iagiov{\'a}, {\v{S}}ir{\'a}{\v{n}} \& Vetr\'ik~\cite{MSSV2010}
and Vetr\'ik~\cite{Vetrik2012}
in the constructions mentioned above, though in neither case is
the connection to the
butterfly networks
made explicit.
Each of our results is a consequence of choosing an appropriate set of generators for $G$.
We make use of two distinct constructions.
\Needspace*{7\baselineskip}
\section{The first construction}
We present the directed case first, since it is slightly simpler.
\thmbox{
\begin{thmO}\label{thm1stDir}
For
any $k \geqslant 4$ and $d \geqslant k-1$, there exist Cayley digraphs that have
diameter $k$, outdegree~$d$, and order $(k-1)(d-k+3)^{k-1}$.
\end{thmO}
}
\begin{proof}
Let $r=k-1$ and $t=d-k+3$, and let the underlying group of the Cayley digraph be $G=\bbZ_tr\rtimes\bbZ_r$.
The order of $G$ is $r\hspace{0.07 em}t^r=(k-1)(d-k+3)^{k-1}.$
As generators for the Cayley digraph
we use
the $t$ \emph{shift and add} elements
$(a,0,\ldots,0;1)$, for each $a\in\mathbb{Z}_t$,
together with the remaining $r-2$ nonzero \emph{cyclic shift} elements
$(0,\ldots,0;s)$, for $2\leqslant s\leqslant r-1$.
Thus the digraph has outdegree $t+r-2=d$.
It also has diameter $r+1=k$. Every element
is the product of
$r$ shift and add operations (establishing the vector) and possibly a single cyclic shift (to establish the final shift value if it is nonzero).
On the other hand, if $s\neq0$ then $(1,\ldots,1;s)$ cannot be obtained as a product of fewer than $k$ generators.
\end{proof}
Clearly, the butterfly network $B_t(r)$ is a subdigraph of the Cayley digraph of Theorem~\ref{thm1stDir}.
The additional edges in our construction, corresponding to the cyclic shift elements, consist of $t^r$ vertex-disjoint copies of the complete digraph on $r$ vertices with a directed $r$-cycle removed.
Vetr\'ik~\cite{Vetrik2012} presents, for any $k\geqslant3$ and $d\geqslant4$,
a family of Cayley digraphs
of diameter~$k$, degree~$d$, and order $k\!\floor{\frac{d}{2}}^{\!k}$.
For
odd diameters,
Abas \& Vetr\'ik~\cite{AV2017} improve this result by a factor of two, constructing Cayley digraphs
of diameter at most~$k$ and degree~$d$ of order $2k\!\floor{\frac{d}{2}}^{\!k}$.
Clearly, for large enough $d$, these digraphs are bigger than those of Theorem~\ref{thm1stDir}.
However,
for any given diameter $k\geqslant8$,
it can be confirmed (using a computer algebra system, or otherwise)
that the digraphs of Theorem~\ref{thm1stDir} are larger than those of Vetr\'ik and Abas \& Vetr\'ik if
$$
2\hspace{0.07 em}k + 2\ln k
\;\;<\;\; d \;\;<\;\;
2^{k-1}\big(1-\tfrac{1}{k}\big) -k^2.
$$
For specific values of the degree, we can do much better.
If we set $d=k^2-3k$, then the digraphs of Theorem~\ref{thm1stDir} have orders
at least $\mathrm{DM}_{d,k}/ek$,
within a linear factor of the Moore bound, and
exceeding those of Abas \& Vetr\'ik by a factor of at least $2^{k-1}/{e k^2}$, which exceeds $1$ for $k\geqslant9$.
For the undirected case, we simply add elements to
the generating set to
make it symmetric.
\thmbox{
\begin{thmO}\label{thm1stUndir}
For any $k \geqslant 5$ and $d \geqslant k$ such that $d-k\not\equiv 3\!\pmod{4}$,
there exist Cayley graphs that have diameter $k$, degree $d$, and order $(k-1)\big(\!\floor{\frac{d-k}{2}}+2\big)^{\!k-1}$.
\end{thmO}
}
\begin{proof}
Let $r=k-1$ and $t=\floor{\frac{d-k}{2}}+2$, and let
$G=\bbZ_tr\rtimes\bbZ_r$.
As generators for the Cayley graph of $G$ we use
the $t$ elements $(a,0,\ldots,0;1)$, along with their inverses $(0,\ldots,0,-a;-1)$,
and the remaining $r-3$ nonzero elements $(0,\ldots,0;s)$ for $2\leqslant s\leqslant r-2$.
In addition, if $d-k\equiv 1\!\pmod{4}$, in which case $t$ is even, then the involution $(0,\ldots,0,\frac{t}{2};0)$ is also included as a generator.
Thus the graph has degree $2t+r-3 +(d-k\!\mod2)= d$.
As in the directed case,
it has diameter $r+1=k$.
Every element
is the product of
$k-1$ shift and add operations and possibly a single cyclic shift.
On the other hand, if $s\notin\{-1,0,1\}$ then $(1,\ldots,1;s)$ cannot be obtained as a product of fewer than $k$ generators,
and $G$ has such an element since $r\geqslant4$.
\end{proof}
Macbeth, {\v{S}}iagiov{\'a}, {\v{S}}ir{\'a}{\v{n}} \& Vetr\'ik~\cite{MSSV2010} present, for any $k \geqslant 3$ and $d \geqslant 5$,
a family of Cayley graphs with diameter at most $k$, degree $d$, and
order no greater than
$k\big(\frac{d+1}{3}\big)^{\!k}$.\footnote{
The graphs in~\cite{MSSV2010} are slightly larger than those of Macbeth, {\v{S}}iagiov{\'a} \& {\v{S}}ir{\'a}{\v{n}}~\cite{MSS2012}, whose order is at most $k\big(\frac{d+1}{3}\big)^{\!k}-k$.}
Their constructions also use the group $G$, with a different generating set.
For large enough $d$, these graphs are bigger than those of Theorem~\ref{thm1stUndir}.
However,
for $k\geqslant27$,
the graphs of Theorem~\ref{thm1stUndir} are larger than those of Macbeth, {\v{S}}iagiov{\'a}, {\v{S}}ir{\'a}{\v{n}} \& Vetr\'ik
for any $d-k\not\equiv 3\!\pmod{4}$ satisfying
$$
3\hspace{0.07 em}k + 6\ln k
\;\;<\;\; d \;\;<\;\;
2\big(\tfrac{3}{2}\big)^{\!k}\big(1-\tfrac{1}{k}\big)-k^2 .
$$
For specific values of the degree, we can do much better.
If we set $d=k^2-2k$, then the graphs of Theorem~\ref{thm1stUndir} have orders
exceeding those in~\cite{MSSV2010} by a factor of at least $\frac{2}{e k^2}\big(\tfrac{3}{2}\big)^{\!k}$, which exceeds $1$ for $k\geqslant14$.
\section{The second construction}
In our second construction, we conceive of the
vectors of length $r$
as being partitioned into $k-1$ \emph{long} blocks, each of length $\ell$, and a single \emph{short} block, of length $m$.
Again, the directed case is presented first, since it is simpler.
\thmbox{
\begin{thmO}\label{thm2ndDirConstruct}
For any $k,\ell,t\geqslant2$ and positive $m<\ell$, there exist Cayley digraphs that have
diameter $k$, outdegree~$t^\ell+(r-1)t^m-1$, and order $r\hspace{0.07 em}t^r$, where $r=(k-1)\ell+m$.
\end{thmO}
}
\begin{proof}
As before, let
$G=\bbZ_tr\rtimes\bbZ_r$, of order
$r\hspace{0.07 em}t^r$.
As generators for the Cayley digraph,
we use the $t^\ell$ \emph{long} elements $(a_1,\ldots,a_\ell,0,\ldots,0;\ell)$, $a_i\in\bbZ_t$,
together with the additional $(r-1)t^m-1$ nonzero \emph{short} elements $(a_1,\ldots,a_m,0,\ldots,0;s)$, $a_i\in\bbZ_t$, $s\neq\ell$.
Thus the digraph has outdegree $t^\ell+(r-1)t^m-1$.
Long elements shift by $\ell$ and modify a long block; short elements shift arbitrarily and modify a short block.
The digraph has diameter $k$.
Every element
is the product of
a single short element (establishing $m$ components of the vector and guaranteeing the final shift value)
and
$k-1$ long elements (establishing the remaining $(k-1)\ell=r-m$ components of the vector).
On the other hand, $(1,\ldots,1;0)$ cannot be obtained as a product of fewer than $k$ generators.
\end{proof}
The Cayley digraph of Theorem~\ref{thm2ndDirConstruct}
contains both of the butterfly networks $B_{t^\ell}(r)$ and $B_{t^m}(r)$ as subdigraphs.
Its edges can be partitioned into
$rt^{r-\ell}$ copies of the
\mbox{$t^\ell$-butterfly}, from the long elements,
$r(r-2)t^{r-m}$ copies of the
\mbox{$t^m$-butterfly}, from the short elements that have nonzero shift,
and a collection of directed cycles from the short elements with zero shift.
Given $k$, $\ell$ and $t$, for judicious choice of $m$, these digraphs are larger than those of
Abas \& Vetr\'ik~\cite{AV2017}.
For example,
if we let $t=2$, then for all $k\geqslant31$ and sufficiently large $\ell$, the order of our digraphs
is greater than that of those in~\cite{AV2017}
if
$$
\ell - k - \log_2\! \ell + 2
\;\;<\;\;
m
\;\;<\;\;
\ell - \log_2\! k\ell -\tfrac{2}{k}(\log_2\! k + 2)
.
$$
If $m$ is chosen optimally, we can do much better than that.
\thmbox{
\begin{corO}\label{cor2ndDir}
For any $k \geqslant 3$, there are arbitrarily large values of $d$ for which there exist Cayley digraphs that have diameter $k$, outdegree $d$, and order at least $\tfrac{1}{k}\big(\frac{k}{k+2}(d+1)\big)^{\!k}$.
\end{corO}
}
\begin{proof}
We use the construction of Theorem~\ref{thm2ndDirConstruct}.
Let $t=2$, and let $\ell$ be any sufficiently large positive integer such that $\log_2k^2\ell\leqslant\tfrac{3}{4}\ell$.
Let $r=\ceil{k\ell - \log_2 k^2\ell}$, and $m=r-(k-1)\ell$, so $r=(k-1)\ell+m$.
Note that $0<m<\ell$.
The digraph has diameter $k$ and order $r \hspace{0.07 em} 2^r$, which (rounding $r$ down) is at least
$$
n_0
\;=\;
\big(k\ell - \log_2 k^2\ell\big)2^{k\ell - \log_2 k^2\ell}
\;=\;
\left(\frac{1}{k} - \frac{\log_2k^2\ell}{k^2\ell}\right)2^{k\ell} .
$$
Its degree is $d=2^\ell+(r-1)2^m-1$, which (substituting for $m$ and rounding $r$ up) is less than
$$
d^+
\;=\;
2^\ell \:+\: \big(k\ell - \log_2 k^2\ell\big) 2^{k\ell - \log_2 k^2\ell + 1 - (k-1)\ell} \:-\: 1
\;=\;
\left(1+\frac{2}{k}-\frac{2\log_2k^2\ell}{k^2\ell}\right)2^\ell \:-\: 1 .
$$
Let $\theta = \frac{\log_2k^2\ell}{k\ell}$.
Note that the condition on $\ell$ implies that $\theta \leqslant \frac{3}{4k}\leqslant\frac{1}{4}$, since $k\geqslant3$.
Now,
$$
k \hspace{0.07 em} n_0 \left(\frac{k}{k+2}(d^++1)\right)^{\!-k}
\;=\;
\left(1 - \theta\right)
\left(1 + \frac{2\theta}{k+2-2\theta} \right)^{\!\!k}
\;>\;
\left(1 - \theta\right)
\left(1 + \frac{2k\theta}{k+2-2\theta} \right)
,
$$
which is at least 1 if $k\geqslant2$ and $0\leqslant\theta\leqslant\frac{k-2}{2k-2}$.
Since $k\geqslant3$ and $\theta \leqslant \frac{1}{4}$, the result follows.
\end{proof}
These digraphs have asymptotic order exceeding
$\tfrac{d^k}{e^2k}$,
a factor of $\tfrac{2^{k-1}}{e^2k^2}$ larger than those of Abas \& Vetr\'ik, and within a linear factor in $k$ of the Moore bound.
It is worth briefly explaining the choice of values for $t$ and $r$ in the proof of Corollary~\ref{cor2ndDir}.
Suppose we fix $t$ and $r$ (and hence the order $rt^r$), and also fix the diameter $k$.
What is the optimal choice for $\ell$, that minimises the degree $t^\ell+(r-1)t^{r - (k - 1)\ell}-1$?
Differentiating with respect to $\ell$ and equating to zero yields $\ell=\frac{1}{k}\big(r+\log_t(k - 1) (r - 1)\big)$.
Solving for $r$ then gives
$$
r \;=\;
\frac{1}{\ln t}
\hspace{0.07 em}
W\!\left(\frac{t^{k \ell-1} \ln t}{k-1}\right) \:+\: 1,
$$
where $W$ is the \emph{Lambert W function}, defined implicitly by $W(z)e^{W(z)}=z$. Asymptotically, $W(z) = \ln z-\ln\ln z+o(1)$. Applying this approximation for $W$ then yields $r\approx k\ell - \log_t\! k^2\ell$.
Using this value for $r$ results in a digraph whose order is asymptotically at least
$\tfrac{1}{k}\big(\frac{k}{k+t}(d+1)\big)^{\!k}$. Setting $t=2$ makes this maximal.
The results in the undirected case are similar.
As before, we just add elements to
the generating set to
make it symmetric.
\thmbox{
\begin{thmO}\label{thm2ndUndirConstruct}
For any $k,\ell,t\geqslant2$ and positive $m<\ell$, there exist Cayley graphs that have
diameter $k$, degree~$2t^\ell + (2 r - 3) t^m - r$, and order $r\hspace{0.07 em}t^r$, where $r=(k-1)\ell+m$.
\end{thmO}
}
\begin{proof}
Let
$G=\bbZ_tr\rtimes\bbZ_r$.
As generators for the Cayley graph of $G$ with these parameters, we use:
\begin{bullets}
\item the $t^\ell$ long elements $(a_1,\ldots,a_\ell,0,\ldots,0;\ell)$, $a_i\in\bbZ_t$
\item their $t^\ell$ inverses $(0,\ldots,0,a_1,\ldots,a_\ell;-\ell)$
\item the $(r-2)(t^m-1)$ short elements $(a_1,\ldots,a_m,0,\ldots,0;s)$, $a_i\in\bbZ_t$ not all zero, $s\notin\{0,\ell\}$
\item their $(r-2)(t^m-1)$ inverses $(0,\ldots,0,\overbrace{a_1,\ldots,a_m,0,\ldots,0}^s;-s)$
\item the $t^m-1$ nonzero short elements $(a_1,\ldots,a_m,0,\ldots,0;0)$; this set is symmetric
\item the $r-3$ short elements $(0,\ldots,0;s)$, $s\notin\{0,\pm\ell\}$; this set is also symmetric
\end{bullets}
Thus the graph has degree $2t^\ell + (2 r - 3) t^m - r$.
As in the directed case, it has order $r\hspace{0.07 em}t^r$ and diameter $k$.
\end{proof}
Given $k$, $\ell$ and $t$, for appropriate choice of $m$, these graphs are larger than those of Macbeth, {\v{S}}iagiov{\'a}, {\v{S}}ir{\'a}{\v{n}} \& Vetr\'ik~\cite{MSSV2010}.
For example,
if we let $t=2$, then for all $k\geqslant69$ and sufficiently large $\ell$, the order of our graphs
is greater than that of those in~\cite{MSSV2010}
if
$$
\ell +k - \log_2\! 3^k\ell + 1
\;\;<\;\;
m
\;\;<\;\;
\ell - \log_2\! k\ell -\tfrac{3}{k}(\log_2\! k + 2) - 1
.
$$
If $m$ is chosen optimally, we have the following.
\thmbox{
\begin{corO}\label{cor2ndUndir}
For any $k \geqslant 3$, there are arbitrarily large values of $d$ for which there exist Cayley graphs that have diameter $k$, degree $d$, and order at least
$$
\tfrac{1}{k}\Big(\tfrac{k}{2k+4}\big(d \:+\: k\log_2 \tfrac{d}{2} \:-\: \log_2\log_2 d \:-\: \log_2 8k^2 \big)\!\Big)^{\!k}
.
$$
\end{corO}
}
\begin{proof}
We use the construction of Theorem~\ref{thm2ndUndirConstruct}.
As in the proof of Corollary~\ref{cor2ndDir},
let $t=2$, and let~$\ell$ be any sufficiently large positive integer such that $\log_2k^2\ell\leqslant\tfrac{3}{4}\ell$.
Let $r=\ceil{k\ell - \log_2 k^2\ell}$, and $m=r-(k-1)\ell$, so $r=(k-1)\ell+m$.
The graph has diameter $k$ and order $r \hspace{0.07 em} 2^r$, which is at least
$$
n_0
\;=\;
\big(k\ell - \log_2 k^2\ell\big)2^{k\ell - \log_2 k^2\ell}
\;=\;
\left(\frac{1}{k} - \frac{\log_2k^2\ell}{k^2\ell}\right)2^{k\ell} .
$$
Its degree is $d=2^{\ell+1} + (2 r - 3) 2^m - r$, which (substituting for $m$ and rounding $r$ up in the second term) is less than
$$
2^{\ell+1} \:+\: \big(2k\ell - 2\log_2 k^2\ell - 1\big) 2^{k\ell - \log_2 k^2\ell+1-(k-1)\ell} \:-\: r \\
\;=\;
\left(2+\frac{4}{k}-\frac{1+4\log_2k^2\ell}{k^2\ell}\right)2^\ell \:-\: r
.
$$
Thus, $\thalf(d+r)$ is less than
$
q =
\left(1+\frac{2}{k}-\frac{2\log_2k^2\ell}{k^2\ell}\right)2^\ell ,
$
and by the argument in the proof of Corollary~\ref{cor2ndDir} (with $q=d^++1$), we know that
$
k n_0
>
\big(\frac{kq}{k+2}\big)^{\!k}
>
\big(\frac{k}{2k+4}(d+r)\big)^{\!k}
.
$
It remains to establish the appropriate lower bound for $r$.
Now, $k n_0<2^{k\ell}$ and $q>\tfrac{d}{2}$, so $2^\ell>\frac{k d}{2k+4}$ and thus
$
\ell
>
\log_2\tfrac{k d}{2k+4}
=
\log_2 \tfrac{d}{2}-\log_2\big(1+\tfrac{2}{k}\big)
.
$
Since
$\big(1+\frac{2}{k}\big)^k < e^2 < 2^3$, we have
$\log_2\big(1+\tfrac{2}{k}\big)<\frac{3}{k}$ and thus
$\ell > \log_2 \tfrac{d}{2}-\frac{3}{k}$.
Now, $r \geqslant {k\ell - \log_2 k^2\ell}$, so
$$
r
\;>\;
k\log_2 \tfrac{d}{2} \:-\: 3 \:-\: \log_2k^2 \:-\: \log_2 \!\big(\log_2 \tfrac{d}{2} -\tfrac{3}{k}\big)
,
$$
which
is greater than
$
k\log_2 \tfrac{d}{2} - \log_2\log_2 d - \log_2 8k^2 ,
$
as required.
\end{proof}
These graphs have asymptotic order exceeding
$\tfrac{1}{e^2k}\big(\tfrac{d}{2}\big)^{\!k}$,
a factor of $\tfrac{1}{e^2k^2}\big(\tfrac{3}{2}\big)^{\!k}$ larger than those of Macbeth, {\v{S}}iagiov{\'a}, {\v{S}}ir{\'a}{\v{n}} \& Vetr\'ik.
{\footnotesize
}
\end{document} | math | 20,715 |
\begin{document}
\title
[Location of concentrated vortices]{Location of concentrated vortices in planar steady Euler flows}
\author{Guodong Wang, Bijun Zuo}
\address{Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, P.R. China}
\email{[email protected]}
\address{College of Mathematical Sciences, Harbin Engineering University, Harbin {\rm150001}, PR China}
\email{[email protected]}
\begin{abstract}
In this paper, we study two-dimensional steady incompressible Euler flows in which the vorticity is sharply concentrated in a finite number of regions of small diameter in a bounded domain. Mathematical analysis of such flows is an interesting and physically important research topic in fluid mechanics.
The main purpose of this paper is to prove that in such flows the locations of these concentrated blobs of vorticity must be in the vicinity of some critical point of the Kirchhoff-Routh function, which is determined by the geometry of the domain. The vorticity is assumed to be only in $L^{4/3},$ which is the optimal regularity for weak solutions to make sense. As a by-product, we prove a nonexistence result for concentrated multiple vortex flows in convex domains.
\end{abstract}
\maketitle
\section{Introduction}
Let $D\subset\mathbb R^2$ be a simply-connected bounded domain with smooth boundary $\partial D$. Consider in $D$ an ideal fluid in steady state, the motion of which is described by the famous Euler equations
\begin{equation}\lambdabel{euler}
\begin{cases}
(\mathbf v\cdot\nabla)\mathbf v=-\nabla P&\mathbf x=(x_1,x_2)\in D,\\
\nabla\cdot\mathbf v=0&\mathbf x\in D,\\
\mathbf v\cdot\mathbf n =g&\mathbf x\in\partial D,
\end{cases}
\end{equation}
where $\mathbf v=(v_1,v_2)$ is the velocity field, $P$ is a scalar function that represents the pressure, $\mathbf n$ is the unit outward normal on $\partial D,$ and $g$ is a given function satisfying the following compatibility condition
\begin{equation}\lambdabel{g}
\int_{\partial D}gdS=0.
\end{equation}
Here we assume that the fluid is of unit density. The first two equations in \eqref{euler} are the momentum conservation and mass conservation respectively, and the boundary condition in \eqref{euler} means that the rate of mass flow across the boundary per unit area is $g$. In particular, if $g\equiv0,$ then there is no matter flow through the boundary.
The scalar vorticity $\omega$, defined as the signed magnitude of curl$\mathbf v,$ that is,
\[\omega=\partial_{x_1} v_2-\partial_{x_2}v_1,\]
is one of the fundamental physical quantities and plays an important role in the study of two-dimensional flows.
Below we reformulate the Euler equations \eqref{euler} as a single equation of $\omega$, which is much easier to handle mathematically.
First we show that $\mathbf v$ can be recovered from $\omega$. In fact, since $\mathbf v$ is divergence-free and $D$ is simply-connected, we can apply the Green's theorem to show that there is a scalar function $\psi,$ called the \emph{stream function}, such that
\begin{equation}\lambdabel{psi}
\mathbf v=(\partial_{x_2}\psi,-\partial_{x_1}\psi).
\end{equation}
For convenience, throughout this paper we will use the symbol
$\mathbf b^\perp$ to denote the clockwise rotation through $\pi/2$ of any planar vector $\mathbf b=(b_1,b_2)$, that is, $\mathbf b^\perp=(b_2,-b_1)$, and $\nabla^\perp f$ to denote $(\nabla f)^\perp$ for any scalar function $f$, that is, $\nabla^\perp f=(\partial_{x_2}f,-\partial_{x_1}f)$. Thus \eqref{psi} can also be written as
\begin{equation}\lambdabel{psi2}
\mathbf v=\nabla^\perp \psi.
\end{equation}
It is easy to check that $\psi$ and $\omega$ satisfy
\begin{equation}\lambdabel{poisson}
\begin{cases}
-\Delta\psi=\omega&\text{in }D,\\
\nabla^\perp\psi\cdot\mathbf n=g&\text{on }\partial D.
\end{cases}
\end{equation}
To deal with the boundary condition in \eqref{poisson}, we consider the following elliptic problem
\begin{equation}\lambdabel{q}
\begin{cases}
-\Delta \psi_0=0&\text{in }D,\\
\nabla^\perp \psi_0\cdot\mathbf n=g&\text{on }\partial D.
\end{cases}
\end{equation}
To solve \eqref{q}, we first solve the following Laplace equation with standard Neumann boundary condition
\begin{equation*}
\begin{cases}
-\Delta \psi_1=0&\text{in }D,\\
\frac{\partial \psi_1}{\partial\mathbf n}=g&\text{on }\partial D,
\end{cases}
\end{equation*}
then the harmonic conjugate of $\psi_1$ solves \eqref{q}. Note that by the maximum principle the solution to \eqref{q} is unique up to a constant.
Now it is easy to see that $\psi-\psi_0$ satisfies
\begin{equation}\lambdabel{gw}
\begin{cases}
-\Delta(\psi-\psi_0)=\omega&\text{in }D,\\
\nabla^\perp (\psi-\psi_0)\cdot\mathbf n=0&\text{on }\partial D.
\end{cases}
\end{equation}
The boundary condition in \eqref{gw} implies that $\psi-\psi_0$ is a constant on $\partial D$ (recall that $D$ is simply-connected). Without loss of generality by adding a suitable constant we assume that $\psi-\psi_0=0$ on $\partial D,$ thus $\psi-\psi_0$ can be expressed in terms of the Green's operator as follows
\begin{equation}\lambdabel{exp}
\psi-\psi_0=\mathcal G\omega:=\int_DG(\cdot,\mathbf y)\omega(\mathbf y)d\mathbf y,
\end{equation}
where $G(\cdot,\cdot)$ is the Green's function for $-\Delta$ in $D$ with zero boundary condition. Combining \eqref{psi2} and \eqref{exp}, we have recovered $\psi$ from $\omega$ in the following
\begin{equation}\lambdabel{bs}
\mathbf v=\nabla^\perp(\mathcal G\omega+\psi_0),
\end{equation}
which is usually called the Biot-Savart law in fluid mechanics.
On the other hand, taking the curl on both sides of the momentum equation in \eqref{euler} we get
\begin{equation}\lambdabel{ve1}
\mathbf v\cdot\nabla \omega=0.
\end{equation}
From \eqref{bs} and \eqref{ve1}, the Euler equations \eqref{euler} are reduced to a single equation of $\omega$
\begin{equation}\lambdabel{ve}
\nabla^\perp(\mathcal G\omega+\psi_0)\cdot\nabla \omega=0\quad\text{ in }D,
\end{equation}
which is usually called the \emph{vorticity equation}.
\begin{remark}
When $D$ is multiply-connected, the above discussion is still valid. The only difference is that one needs to replace the usual Green's function $G$ by the hydrodynamic Green's function (see \cite{Flu}, Definition 15.1), which does not cause any essential difficulty for the problem discussed in this paper.
\end{remark}
In the rest of this paper, we will be focused on the study of \eqref{ve}. Note that once we have obtained a solution $\omega$ to \eqref{ve}, we immediately get a solution to \eqref{euler} with
\[\mathbf v=\nabla^\perp(\mathcal G\omega+\psi_0),\quad P(\mathbf x)=\int_{L_{\mathbf x_0,\mathbf x}}\omega(\mathbf y)\mathbf v^\perp(\mathbf y)\cdot d\mathbf y-\frac{1}{2}|\mathbf v(x)|^2,\]
where $\mathbf x_0$ is a fixed point in $D$ and $L_{\mathbf x_0,\mathbf x}$ is any $C^1$ curve joining $\mathbf x_0$ and $\mathbf x$ (one can easily check that the above line integral is well defined by using Green's theorem and the fact that $\omega$ is a solution).
Since in many physical problems the vorticity is of low regularity, not even continuous, it is necessary to define the notion of weak solutions to \eqref{ve}. In the rest of this paper, we regard $\psi_0$ as a given function.
\begin{definition}\lambdabel{wsve}
Let $\omega\in L^{4/3}(D)$. If for any $\phi\in C_c^\infty(D)$ it holds that
\begin{equation}\lambdabel{int}
\int_D\omega\nabla^\perp(\mathcal G\omega+\psi_0)\cdot\nabla\phi d\mathbf x=0,\end{equation}
then $\omega$ is called a weak solution to the vorticity equation \eqref{ve}.
\end{definition}
The above definition is reasonable from the fact that one can multiply any test function
$\phi$ on both sides of \eqref{ve} and integrate by parts formally to get \eqref{int}.
\begin{remark}
Since $\psi_0$ is harmonic (thus smooth) and $\phi$ has compact support in $D$, we see that the integral $\int_D\omega \nabla^\perp \psi_0\cdot\nabla\phi d\mathbf x$ in \eqref{int} makes sense. Note that throughout this paper we do not impose any condition on the boundary value of $\psi_0$.
\end{remark}
\begin{remark}
By the Calderon-Zygmund inequality and Sobolev inequality, $\omega\in L^{4/3}(D)$ is the optimal regularity for the integral $\int_D\omega\nabla^\perp\mathcal G\omega\cdot\nabla\phi d\mathbf x$ in \eqref{int} to be well-defined.
\end{remark}
In the literature, there has been extensive study on the existence of weak solutions to \eqref{ve}. See \cite{B1,B2,CLW,CPY1,CPY2,CW1,CWZ2,CWZu,EM,LP,LYY,SV,T,W,WZ} for example. The solutions obtained in these papers have one common feature, that is, the vorticity is the function of the stream function ``locally". In this regard, Cao and Wang \cite{CW2} proved a general criterion for an $L^{4/3}$ function to be a weak solution.
\begin{theorem}[Cao--Wang, \cite{CW2}]\lambdabel{cwthm}
Let $k$ be a positive integer and $\psi_0\in C^1(\bar D)$.
Suppose that $\omega\in L^{4/3}(D)$ satisfies
\begin{equation}\lambdabel{fo}
\omega=\sum_{i=1}^k\omega_i, \,\,\min_{1\leq i< j\leq k}\{\text{dist}(\text{supp}\omega_i,\text{supp}\omega_j)\}>0,\,\,\omega_i=f^i(\mathcal G\omega+\psi_0), \text{a.e. in (supp}\omega_i)_\delta,
\end{equation}
where $\delta$ is a positive number,
\[\text{(supp}\omega_i)_\delta:=\{\mathbf x\in D\mid \text{dist}(\mathbf x,\text{supp}\omega_i)<\delta\},\]
and each $f^i$ is either monotone from $\mathbb R$ to $\mathbb R\cup\{\pm\infty\}$ or Lipschitz from $\mathbb R$ to $\mathbb R$.
Then $\omega$ is a weak solution to the vorticity equation \eqref{ve}.
\end{theorem}
Some examples of such flows are as follows. When $f_i$ in Theorem \ref{cwthm} is a Heaviside function, the solutions are called vortex patches, and related existence results can be found in \cite{CPY1,CW1,CWZu,T,WZ}. When $f_i$ is a power function, related papers are \cite{CLW,CPY2,LP,LYY,SV}. In \cite{B1,B2,EM}, the authors obtained some steady vortex flows by maximizing or minimizing the kinetic energy of the fluid on the rearrangement class of some given function. The solutions obtained in \cite{B1,B2,EM} still have the form \eqref{fo}, where each $f_i$ is a monotone function, but the precise expression of $f_i$ is unknown. Recently Cao, Wang and Zhan \cite{CWZ2,W} modified Turkington's method \cite{T} and proved the existence of a large class of solutions of the form \eqref{fo}, where each $f_i$ is a given function with few restrictions.
Among the flows mentioned above, some are of particular interest and attract more attention, that is, flows in which the vorticity is sharply concentrated in a finite number of small regions and vanishes elsewhere, just like a finite sum of Dirac measures. Mathematically, the vorticity in such flows has the form
\begin{equation}\lambdabel{cv}
\omega_\varepsilon=\sum_{i=1}^k\omega_{\varepsilon,i},\quad
{\rm supp}(\omega_{\varepsilon,i})\subset B_{o(1)}(\bar x_i),\quad\int_D\omega_{\varepsilon,i} d\mathbf x=\kappa_i+o(1),\quad i=1,\cdot\cdot\cdot,k,
\end{equation}
where $\varepsilon$ is a small positive parameter, $k$ is a positive integer, $\bar x_i\in D$, $\kappa_i$ is a fixed non-zero real number, $i=1,\cdot\cdot\cdot,k$, and $o(1)\to0$ as $\varepsilon \to0^+$. Papers concerning the existence of such solutions include \cite{CLW,CPY1,CPY2,CW1,CWZ2,CWZu,SV,T,W}. Note that all the flows constructed in these papers have bounded vorticity.
Euler Flows with vorticity of the form \eqref{cv} is closely related to a very famous Hamiltonian system in $\mathbb R^2$, the point vortex model (see \cite{L}), which describes the evolution of a finite number of point vortices with their locations being the canonical variables. The point vortex model is only an approximate model, and its precise connection with the 2D Euler equations with concentrated vorticity in the evolutionary case is a tough and unsolved problem. For a detailed discussion, we refer the interested readers to \cite{MP1,MP2,MP3,MPa,T2}. According to the point vortex model, the locations of concentrated blobs of vorticity in steady Euler flows are not arbitrary, but \emph{should} be near a critical point of the following Kirchhoff-Routh function
\begin{equation}\lambdabel{krf}
W(\mathbf x_1,\cdot\cdot\cdot,\mathbf x_k)=-\sum_{1\leq i<j\leq k}\kappa_i\kappa_jG(\mathbf x_i,\mathbf x_j)+\frac{1}{2}\sum_{i=1}^k\kappa_i^2H(\mathbf x_i)+\sum_{i=1}^k\kappa_i\psi_0(\mathbf x_i),
\end{equation}
where
\[(\mathbf x_1,\cdot\cdot\cdot,\mathbf x_k)\in \underbrace{D\times\cdot\cdot\cdot\times D}_{k\text { times}}\setminus \{(\mathbf x_1,\cdot\cdot\cdot,\mathbf x_k)\mid \mathbf x_i\in D, \mathbf x_i=\mathbf x_j \text{ for some }i\neq j\}\]
and $H(\mathbf x)=h(\mathbf x,\mathbf x)$ with $h$ being the regular part of Green's function, that is,
\[h(\mathbf x,\mathbf y):=-\frac{1}{2\pi}\ln|\mathbf x-\mathbf y|-G(\mathbf x,\mathbf y),\quad \mathbf x,\mathbf y\in D.\]
However, to our knowledge there is no complete and rigorous proof on this issue in the literature, although the solutions of the form \eqref{cv} constructed in \cite{CLW,CPY1,CPY2,CW1,CWZ2,EM,SV,T} are all based on the hypothesis that $(\bar {\mathbf x}_1,\cdot\cdot\cdot,\bar {\mathbf x}_k)$ is a critical point of $ W$.
The aim of paper is prove that such a hypothesis is necessary.
This paper is organized as follows. In Section 2, we state our main results (Theorems \ref{mthm} and \ref{none}) and give some comments. In Sections 3 and 4 we provide the proofs of them.
\section{Main results}
In this section, we present our two main results.
The first result is about the necessary condition about the locations of concentrated vortices.
\begin{theorem}\lambdabel{mthm}
Let $k$ be a positive integer, $\bar {\mathbf x}_1,\cdot\cdot\cdot,\bar {\mathbf x}_k\in D$ be $k$ different points and $\kappa_1,\cdot\cdot\cdot,\kappa_k$ be $k$ non-zero real numbers.
Assume that there exists a sequence of weak solutions $\{\omega_n\}_{n=1}^{+\infty}$ to the vorticity equation \eqref{ve}, satisfying
$\omega_n=\sum_{i=1}^k\omega_{n,i}$ with $\omega_{n,i}\in L^{4/3}(D)$ and
\[{\rm supp}(\omega_{n,i})\subset B_{o(1)}(\bar {\mathbf x}_i),\quad
\int_D\omega_{n,i} d\mathbf x=\kappa_i+o(1),\quad i=1,\cdot\cdot\cdot,k,\]
where $o(1)\to0$ as $n\to+\infty$.
Then $(\bar {\mathbf x}_1,\cdot\cdot\cdot,\bar {\mathbf x}_k)$ must be a critical point of $W$ defined by \eqref{krf}.
\end{theorem}
Here we compare Theorem \ref{mthm} with two related results in \cite{CGPY} and \cite{CM}.
In \cite{CGPY}, Cao, Guo, Peng and Yan studied planar Euler flows with vorticity of the following patch form
\begin{equation}\lambdabel{ceq}
\omega^\lambdambda=\sum_{i=1}^k\omega^\lambdambda_i,\quad
\omega^\lambdambda_i=\lambdambda\chi_{\{\mathbf x\in D\mid \mathcal G\omega^\lambdambda(\mathbf x)>\mu_i^\lambdambda\}\cap B_{\delta}(\bar {\mathbf x}_i)},\quad \int_D\omega_i^\lambdambda d\mathbf x=\kappa_i, \quad i=1,\cdot\cdot\cdot,k,
\end{equation}
where $\lambdambda$ is a large positive parameter, $\chi$ denotes the characteristic function, each $\mu^\lambdambda_i$ is a real number depending on $\lambdambda$ and each $\kappa_i$ is a given non-zero number.
They proved that if supp$\omega^\lambdambda_i$ ``shrinks" to $\bar {\mathbf x}_i$ as $\lambdambda\to+\infty$, then $\bar {\mathbf x}_1\cdot\cdot\cdot,\bar {\mathbf x}_k$ must necessarily constitute a critical point of $W$ (see Theorem 1.1 in \cite{CGPY} for the precise statement) . Compared with their result, we consider more general flows and only impose very weak regularity on the vorticity in Theorem \ref{mthm}. Moreover, as we will see in the next section, the proof we provide is shorter and more elementary. The other relevant work is \cite{CM}. In \cite{CM}, Caprini and Marchioro studied the evolution of a finite number of blobs of vorticity in $\mathbb R^2$ and proved the finite-time localization property (see Theorem 1.2 in \cite{CM} for the precise statement). In their result, each $\omega_{n,i}$ is required additionally to have a definite sign and satisfy the growth condition
\begin{equation}\lambdabel{gwth}
\|\omega_{n,i}\|_{L^\infty}\leq M(\text{diam(supp}\omega_{n,i}))^{-\delta},
\end{equation}
where $M$ and $\delta$ are both fixed positive numbers. As a consequence of their result, Theorem \ref{mthm} holds true if the additional growth condition \eqref{gwth} is satisfied (although they only considered the whole plane case, similar result for a bounded domain can also be proved without any difficulty). In this sense, our result can be regarded as a strengthened version of Caprini and Marchioro's result in the steady case.
\begin{remark}
In Theorem 1.1 in \cite{CGPY}, for vorticity of the form \eqref{ceq}, $\bar{\mathbf x}_i\in D$ and $\bar {\mathbf x}_i\neq \bar {\mathbf x}_j$ for $ i\neq j$ are not assumptions but can be proved as conclusions. However, in the very general setting of this paper, these two conclusions may be false. For example, we can regard
a single blob of vorticity as two artificially, thus they may concentrate on the same point. Also, Cao, Wang and Zuo \cite{CWZu} constructed a pair steady vortex patches with opposite rotation directions in the unit disk (Theorem 5.1, \cite{CWZu}), and it can be checked that as the ratio of the circulations of the two patches goes to infinity, the patch with smaller circulation will approach the boundary of the disk.
\end{remark}
Our second result is about the nonexistence of concentrated multiple vortex flows in convex domains, which can be seen as a by-product of Theorem \ref{mthm}.
\begin{theorem}\lambdabel{none}
Let $\delta_0>0$ be fixed, $D$ be a smooth convex domain, $k\geq 2$ be a positive integer, $ \kappa_1,\cdot\cdot\cdot,\kappa_k$ be $k$ positive numbers and $f_1,\cdot\cdot\cdot,f_k$ be $k$ real functions satisfying
\[\lim_{t\to0^+}f_i(t)=0,\quad i=1\cdot\cdot\cdot,k.\]
If $\psi_0\equiv0,$ then there exists $\varepsilon_0>0$, such that for any
$\varepsilon\in(0,\varepsilon_0),$ there is no weak solution $\omega_\varepsilon$ to the vorticity equation \eqref{ve} satisfying
\begin{itemize}
\item[(1)] $\omega_\varepsilon=\sum_{i=1}^k\omega_{\varepsilon,i},$ $\omega_{\varepsilon,i}\in L^{4/3}(D), i=1\cdot\cdot\cdot,k;$
\item[(2)] $\text{dist(supp}\omega_{\varepsilon,i},\text{supp}\omega_{\varepsilon,j}) >\delta_0 \,\, \forall\,1\leq i<j\leq k$ and
$\text{dist(supp}\omega_{\varepsilon,i},\partial D)>\delta_0\,\, \forall\,1\leq i\leq k;$
\item[(3)] diam(supp$\omega_{\varepsilon,i}$)<$\varepsilon,\,\,i=1,\cdot\cdot\cdot,k.$
\item[(4)] $\int_D\omega_{\varepsilon,i}d\mathbf x=\kappa_i+f_i(\varepsilon),\,\,i=1,\cdot\cdot\cdot,k.$
\end{itemize}
\end{theorem}
\section{Proof of Theorem \ref{mthm}}
First we need the following lemma.
\begin{lemma}\lambdabel{lem}
Let $\omega\in L^{4/3}(\mathbb R^2)$ with compact support. Define
\begin{equation*}
f(\mathbf x)=\int_{\mathbb R^2}\ln|\mathbf x-\mathbf y|\omega(\mathbf y)d\mathbf y.
\end{equation*}
Then $f\in W^{2,4/3}_{\rm loc}(\mathbb R^2)$ and the distributional partial derivatives of $f$ can be expressed as
\begin{equation}\lambdabel{deri}
\partial_{x_i} f(\mathbf x)=\int_{\mathbb R^2}\frac{x_i-y_i}{|\mathbf x-\mathbf y|^2}\omega(\mathbf y)d\mathbf y\quad \text{a.e. }\,\mathbf x\in\mathbb R^2, \,\,i=1,2.
\end{equation}
\end{lemma}
\begin{proof}
By the Calderon-Zygmund estimate we have $f\in W^{2,4/3}_{\rm loc}(\mathbb R^2)$. The expression \eqref{deri} follows from Theorem 6.21 on page 157, \cite{LL}.
\end{proof}
Now we are ready to prove Theorem \ref{mthm}. The key point of the proof is to use the anti-symmetry of the singular part of the Biot-Savart kernel.
\begin{proof}[Proof of Theorem \ref{mthm}]
Fix $l\in\{1,\cdot\cdot\cdot,k\}$. It is sufficient to show that
\[\nabla_{{\mathbf x}_l}W(\bar {\mathbf x}_1,\cdot\cdot\cdot,\bar {\mathbf x}_k)=\mathbf 0.\]
Let $r_0$ be a small positive number such that
\[r_0<\text{dist}(\bar{\mathbf x}_i,\partial D)\quad \forall\,1\leq i\leq k,\quad r_0<\frac{1}{2}\text{dist}(\bar{\mathbf x}_i,\bar{\mathbf x}_j)\quad \forall\,1\leq i<j\leq k.\]
Choose $\phi(\mathbf x)=\rho(\mathbf x)\mathbf b\cdot \mathbf x$ in Definition \ref{wsve}, where $\mathbf b$ is a constant planar vector and $\rho$ satisfies
\[\rho\in C_c^\infty( D),\,\,\rho\equiv 1\text{ in } B_{r_0}(\bar {\mathbf x}_l),\,\,\rho\equiv 0\text{ in } B_{r_0}(\bar {\mathbf x}_i)\,\,\forall\,i\neq l.\]
Existence of such $\rho$ can be easily obtained by mollifying a suitable patch function.
Then we have
\[\int_D\omega_n\nabla^\perp\left(\mathcal G\omega_n+\psi_0\right)\cdot\nabla\phi d\mathbf x
=0.\]
Denote
\[A_n=\int_D\omega_n\nabla^\perp\mathcal G\omega_n\cdot\nabla\phi d\mathbf x,\quad B_n=\int_D\omega_n\nabla^\perp\psi_0\cdot\nabla\phi d\mathbf x.\]
Then \begin{equation}\lambdabel{ab}
A_n+B_n=0,\quad n=1,2,\cdot\cdot\cdot.
\end{equation}
Below we analyze $A_n$ and $B_n$ separately.
For $A_n$, we have
\begin{align*}
A_n=&\int_D\omega_n(\mathbf x)\nabla^\perp_{\mathbf x}\int_D\left(-\frac{1}{2\pi}\ln|\mathbf x-\mathbf y|-h(\mathbf x,\mathbf y)\right)\omega_n(\mathbf y)d\mathbf y\cdot\nabla\phi d\mathbf x\\
=&-\frac{1}{2\pi}\int_D\omega_n(\mathbf x)\int_D\frac{(\mathbf x-\mathbf y)^\perp}{|\mathbf x-\mathbf y|^2}\omega_n(\mathbf y)d\mathbf y\cdot\nabla\phi d\mathbf x-\int_D\omega_n\int_D\nabla^\perp_{\mathbf x}h(\mathbf x,\mathbf y)\omega_n(\mathbf x)(\mathbf y)d\mathbf y\cdot\nabla\phi d\mathbf x.
\end{align*}
Here we used Lemma \ref{lem} and the facts that $h\in C^\infty(D\times D)$ and $\omega_n$ has compact support in $D$.
Since $\omega_n\in L^{4/3}(D)$, by the Hardy-Littlewood-Sobolev inequality (see Theorem 0.3.2 in \cite{SO}) we have
\[\int_D\frac{|\omega_n(\mathbf y)|}{|\mathbf x-\mathbf y|}d\mathbf y\in L^4(D).\]
Now we can apply Fubini's theorem (see \cite{ru}, page 164) to obtain
\[\frac{(\mathbf x-\mathbf y)^\perp}{|\mathbf x-\mathbf y|^2}\cdot\nabla\phi\omega_n(\mathbf x)\omega_n(\mathbf y)\in L^1(D\times D)\]
and
\[\int_D\omega_n(\mathbf x)\int_D\frac{(\mathbf x-\mathbf y)^\perp}{|\mathbf x-\mathbf y|^2}\omega_n(\mathbf y)d\mathbf y\cdot\nabla\phi d\mathbf x=\int_D\int_D\frac{(\mathbf x-\mathbf y)^\perp\cdot\nabla\phi}{|\mathbf x-\mathbf y|^2}\omega_n(\mathbf x)\omega_n(\mathbf y) d\mathbf xd\mathbf y.\]
Thus we have obtained
\begin{align*}
A_n=-\frac{1}{2\pi}\int_D\int_D\frac{(\mathbf x-\mathbf y)^\perp\cdot\nabla\phi}{|\mathbf x-\mathbf y|^2}\omega_n(\mathbf x)\omega_n(\mathbf y) d\mathbf xd\mathbf y-\int_D\int_D\nabla^\perp_{\mathbf x}h(\mathbf x,\mathbf y)\cdot\nabla\phi\omega_n(\mathbf x) \omega_n(\mathbf y)d\mathbf xd\mathbf y.
\end{align*}
Substituting $\phi(\mathbf x)=\rho(\mathbf x)\mathbf b\cdot\mathbf x$ in $A_n$, for sufficiently large $n$ we have
\begin{align*}
A_n&=-\frac{1}{2\pi}\int_D\int_{D}\frac{(\mathbf x-\mathbf y)^\perp\cdot\mathbf b}{|\mathbf x-\mathbf y|^2}\omega_{n,l}(\mathbf x)\omega_n(\mathbf y) d\mathbf xd\mathbf y-\int_D\int_D\nabla^\perp_{\mathbf x}h(\mathbf x,\mathbf y)\cdot\mathbf b\omega_{n,l}(\mathbf x) \omega_n(\mathbf y)d\mathbf xd\mathbf y\\
=&-\frac{1}{2\pi}\sum_{j=1}^k\int_D\int_D\frac{(\mathbf x-\mathbf y)^\perp \cdot\mathbf b}{|\mathbf x-\mathbf y|^2}\omega_{n,l}(\mathbf x)\omega_{n,j}(\mathbf y)d\mathbf xd\mathbf y
-\sum_{j=1}^k\int_D\int_D\nabla_{\mathbf x}^\perp h(\mathbf x,\mathbf y)\cdot\mathbf b\omega_{n,l}(\mathbf x)\omega_{n,j}(y)d\mathbf xd\mathbf y\\
=&-\frac{1}{2\pi}\int_D\int_D\frac{(\mathbf x-\mathbf y)^\perp \cdot\mathbf b}{|\mathbf x-\mathbf y|^2}\omega_{n,l}(\mathbf x)\omega_{n,l}(\mathbf y)d\mathbf xd\mathbf y
-\frac{1}{2\pi}\sum_{j=1,j\neq l}^k\int_D\int_D\frac{(\mathbf x-\mathbf y)^\perp \cdot\mathbf b}{|\mathbf x-\mathbf y|^2}\omega_{n,l}(\mathbf x)\omega_{n,j}(\mathbf y)d\mathbf xd\mathbf y\\
&-\sum_{j=1}^k\int_D\int_D\nabla_{\mathbf x}^\perp h(\mathbf x,\mathbf y)\cdot\mathbf b\omega_{n,l}(\mathbf x)\omega_{n,j}(\mathbf y)d\mathbf xd\mathbf y\\
=&-\frac{1}{2\pi}\int_D\int_D\frac{(\mathbf x-\mathbf y)^\perp \cdot\mathbf b}{|\mathbf x-\mathbf y|^2}\omega_{n,l}(\mathbf x)\omega_{n,l}(\mathbf y)d\mathbf xd\mathbf y
+\sum_{j=1,j\neq l}^k\int_D\int_D\nabla_{\mathbf x}^\perp G(\mathbf x,\mathbf y)\cdot\mathbf b\omega_{n,l}(\mathbf x)\omega_{n,j}(\mathbf y)d\mathbf xd\mathbf y\\
&-\int_D\int_D\nabla_{\mathbf x}^\perp h(\mathbf x,\mathbf y)\cdot\mathbf b\omega_{n,l}(\mathbf x)\omega_{n,l}(\mathbf y)d\mathbf xd\mathbf y\\
:=&C_n+D_n,
\end{align*}
where
\[C_n=-\frac{1}{2\pi}\int_D\int_D\frac{(\mathbf x-\mathbf y)^\perp \cdot\mathbf b}{|\mathbf x-\mathbf y|^2}\omega_{n,l}(\mathbf x)\omega_{n,l}(\mathbf y)d\mathbf xd\mathbf y,\]
\[D_n=\left(\sum_{j=1,j\neq l}^k\int_D\int_D\nabla_{\mathbf x}^\perp G(\mathbf x,\mathbf y)\omega_{n,l}(\mathbf x)\omega_{n,j}(\mathbf y)d\mathbf xd\mathbf y
-\int_D\int_D\nabla_{\mathbf x}^\perp h(\mathbf x,\mathbf y)\omega_{n,l}(\mathbf \mathbf x)\omega_{n,l}(\mathbf y)d\mathbf xd\mathbf y\right)\cdot\mathbf b.\]
By the anti-symmetric property of the integrand in $C_n$, we see that
\[C_n=0 \quad \text{for sufficiently large $n$}.\]
For $D_n,$ it is clear that
\[\lim_{n\to+\infty}D_n=\left(\sum_{j=1,j\neq l}^k \kappa_l\kappa_j\nabla_{\mathbf x}^\perp G(\bar {\mathbf x}_l,\bar {\mathbf x}_j)
-\kappa_l^2\nabla_{\mathbf x}^\perp h(\bar {\mathbf x}_l,\bar {\mathbf x}_l)\right)\cdot\mathbf b.\]
To conclude, we have obtained
\begin{equation}\lambdabel{a}
\lim_{n\to+\infty}A_n=\left(\sum_{j=1,j\neq l}^k \kappa_l\kappa_j\nabla_x^\perp G(\bar {\mathbf x}_l,\bar {\mathbf x}_j)
-\kappa_l^2\nabla_{\mathbf x}^\perp h(\bar {\mathbf x}_l,\bar {\mathbf x}_l)\right)\cdot\mathbf b.
\end{equation}
For $B_n,$ it is also clear that
\begin{equation}\lambdabel{b}
\lim_{n\to+\infty}B_n=\kappa_l\nabla^\perp\psi_0(\bar{\mathbf x}_l)\cdot\mathbf b.
\end{equation}
Combining \eqref{ab}, \eqref{a} and \eqref{b} we immediately get
\[\left(\sum_{j=1,j\neq l}^k \kappa_l\kappa_j\nabla_{\mathbf x}^\perp G(\bar {\mathbf x}_l,\bar {\mathbf x}_j)
-\kappa_l^2\nabla_{\mathbf x}^\perp h(\bar {\mathbf x}_l,\bar {\mathbf x}_l)+\kappa_l\nabla^\perp\psi_0(\bar{\mathbf x}_l)\right)\cdot \mathbf b= 0\]
for sufficiently large $n$.
Since $\mathbf b$ can be any constant vector, we deduce that
\[\sum_{j=1,j\neq l}^k \kappa_l\kappa_j\nabla_{\mathbf x}^\perp G(\bar {\mathbf x}_l,\bar {\mathbf x}_j)
-\kappa_l^2\nabla_{\mathbf x}^\perp h(\bar {\mathbf x}_l,\bar {\mathbf x}_l)+\kappa_l\nabla^\perp\psi_0(\bar{\mathbf x}_l)= \mathbf 0,\]
which is exactly
\[\nabla_{{\mathbf x}_l}W(\bar {\mathbf x}_1,\cdot\cdot\cdot,\bar {\mathbf x}_k)=\mathbf 0.\]
\end{proof}
\section{Proof of Theorem \ref{none}}
In this section we give the proof of Theorem \ref{none}. To begin with, we need an important property of the Kirchhoff-Routh function in a convex domain proved by Grossi and Takahashi. We only state the following simple version of their result which is enough for our use.
\begin{theorem}[Grossi--Takahashi, Theorem 3.2, \cite{GT}]\lambdabel{gtt}
Let $D$ be a smooth convex domain, $k\geq 2$ be a positive integer and $\kappa_1,\cdot\cdot\cdot,\kappa_k$ be $k$ positive numbers. If $\psi_0\equiv0,$ then the Kirchhoff-Routh function $W$ defined by \eqref{krf} has no critical point in
\[\underbrace{D\times\cdot\cdot\cdot\times D}_{k\text { times}}\setminus \{(\mathbf x_1,\cdot\cdot\cdot,\mathbf x_k)\mid \mathbf x_i\in D, \mathbf x_i=\mathbf x_j \text{ for some }i\neq j\}.\]
\end{theorem}
\begin{proof}[Proof of Theorem \ref{none}]
Suppose, by contradiction, that there exist a sequence of positive numbers $\{\varepsilon_{n}\}_{n=1}^{+\infty}$, $\varepsilon_n\to0^+$ as $n\to+\infty$, and a sequence of weak solutions $\{\omega_n\}_{n=1}^{+\infty}$ to the vorticity equation \eqref{ve} such that
\begin{itemize}
\item[(i)] $\omega_n=\sum_{i=1}^k\omega_{n,i},$ $\omega_{n,i}\in L^{4/3}(D), i=1\cdot\cdot\cdot,k;$
\item[(ii)] $\text{dist(supp}\omega_{n,i},\text{supp}\omega_{n,j}) >\delta_0 \,\,\,\forall\,1\leq i<j\leq k$ and
$\text{dist(supp}\omega_{n,i},\partial D)>\delta_0\,\,\, \forall\,1\leq i\leq k;$
\item[(iii)] diam(supp$\omega_{n,i}$)<$\varepsilon_n,\,\,i=1,\cdot\cdot\cdot,k.$
\item[(iv)] $\int_D\omega_{n,i}d\mathbf x=\kappa_i+f_i(\varepsilon_n),\,\,i=1,\cdot\cdot\cdot,k.$
\end{itemize}
Define
\[{\mathbf x}_{n,i}=\left(\int_{D}\omega_{n,i}d\mathbf x\right)^{-1}\int_{D}\mathbf x\omega_{n,i}d\mathbf x,\quad i=1\cdot\cdot\cdot,k.\]
By (ii) and (iii) we see that
\[\text{dist}(\mathbf x_{n,i},\mathbf x_{n,j}) \geq\frac{\delta_0}{2} \,\,\, \forall\,1\leq i<j\leq k,\quad
\text{dist}(\mathbf x_{n,i},\partial D)\geq\frac{\delta_0}{2}\,\,\, \forall\,1\leq i\leq k\]
if $n$ is large enough.
Thus we can choose a subsequence $\{\mathbf x_{n_m,i}\}$ such that $\mathbf x_{n_m,i}\to\bar {\mathbf x}_i$ as $m\to+\infty, i=1,\cdot\cdot\cdot,k,$ where $\bar{\mathbf x}_1,\cdot\cdot\cdot,\bar{\mathbf x}_k$ satisfy
\[\text{dist}(\bar{\mathbf x}_{i},\bar {\mathbf x}_{j}) \geq\frac{\delta_0}{2} \,\,\, \forall\,1\leq i<j\leq k,\quad
\text{dist}(\bar{\mathbf x}_{i},\partial D)\geq\frac{\delta_0}{2}\,\,\, \forall\,1\leq i\leq k.\]
Now we can see that the sequence of solutions$\{\omega_{n_m}\}$ satisfies the assumptions in Theorem \ref{mthm}, and therefore $(\bar{\mathbf x}_1,\cdot\cdot\cdot,\bar{\mathbf x}_k)$ must be a critical point of $W$ (with $\psi_0\equiv 0$). This is a contradiction to Theorem \ref{gtt}.
\end{proof}
{\bf Acknowledgements:}
{G. Wang was supported by National Natural Science Foundation of China (12001135, 12071098) and China Postdoctoral Science Foundation (2019M661261).}
\phantom{s}
\thispagestyle{empty}
\end{document} | math | 29,441 |
\begin{document}
\title{Corrections for\ ``Efficient Active Learning of Halfspaces:\ an Aggressive Approach''}
Dear Editor and Reviewers,
\vspace*{2em}
We are glad to be informed that our paper has been accepted with minor revisions. We have addressed your comments and corrected the manuscript accordingly. Please see below our responses to your comments and suggestions.
\begin{flushright}
Sincerely Yours, $\qquad\qquad\qquad\qquad\qquad$\\
A.~Gonen, S.~Sabato, S.~Shalev-Shwartz
\end{flushright}
\section{Reviewer 1}
\begin{enumerate}
\item
We have added a relevant discussion after the (sketch of the) proof of
theorem 5. In addition we have included, in Section 6.2, our
experimental results showing the advantage of selecting a majority
hypothesis over a random one (see Figure 2).
\item
$P(\cdot )$ is the probability measure defined for $\mathcal{H}$, while $\mathbb{P}[ \cdot]$ is notation
for a probability of events. We have made a second pass to make sure
notations are consistent.
\item
In example 2 we do wish to demonstrate also that the label complexity
can be exponential in the dimension, even when the points are on the
unit sphere, thus we elect to pack as many points as possible.
\item
Fixed.
\item
Equation 3 is fixed. We thank the reviewer for the suggested changes
in presentation of the proof of theorem 5, however we feel that
providing separate results for the different parts of the proof, and
providing background on submodularity, would be more beneficial for
the reader who wishes to understand the problem deeply.
\item
Fixed.
\item
Regarding Example 22, we have clarified the discussion above the
example. The point of this example is that even when CAL and active
learning have access to the same pool size which is required by
passive learning, there can still be an exponential gap in their label
complexity. Example 22 does not use the full class of separating
hyperplanes but a subset of this class, and only shows a marked
difference when the dimension is very high. Thus we feel that Example
19 is still important as it uses the full halfspace class and works in
a low dimension.
\item
We have accidentally left out the normalization in the margin
definition. This is now fixed.
\item
Fixed.
\item
Redundant P(.) was removed.
\item
Fixed.
\item
Fixed.
\item
Fixed.
\item
Fixed.
\end{enumerate}
\section{Reviewer 2}
\begin{enumerate}
\item
We added further explanations as suggested.
\item
There is no i.i.d. assumption: the lower bounds for active learning
hold for the examples in Section 2 even if the algorithm is allowed to
query any point in the support of the distribution.
\item
\begin{enumerate}
\item
We have clarified the definition of the backwards arrow in the first
paragraph of section 3.
\item
Fixed.
\item
Fixed.
\item
Indeed we mean the union of all equivalence classes. We have added a
clarification on the first paragraph of page 6.
\end{enumerate}
\item
\begin{enumerate}
\item
We have added a comment to that effect.
\item
The volume estimation algorithm, proposed by Kannan et al. (1997), is quite involved and it might be misleading to describe it in a few short lines. We hope that the interested reader can find all the details in the original reference.
\item
In general P is not necessarily uniform. For a uniform distribution Vol(V) and P(V) are equivalent (up to a constant factor). We have added a clarification in the text after eq. (4).
\end{enumerate}
\item
\begin{enumerate}
\item
We have moved the algorithms for non-separable data and kernels from the appendix to section 5.2.
\item
Indeed this is correct. We have rephrased the result.
\end{enumerate}
\item
\begin{enumerate}
\item
The test errors were added.
\item
The lambda/norm parameter was chosen to yield reasonable passive
learning loss, and H was selected to be an upper bound to this
loss. The dimension was selected to generate a separable distribution
in the transformed space. As we mention in the text, having to guess
an upper bound on the loss is a disadvantage of our method.
\end{enumerate}
\end{enumerate}
\end{document} | math | 4,086 |
\begin{document}
\baselineskip 16pt
\title{On generalized $\sigma$-soluble groups\thanks{Research is supported by an NNSF grant of China (Grant No.
11401264) and a TAPP of Jiangsu Higher Education Institutions (PPZY 2015A013)}}
\author{Jianhong Huang\\\
{\small School of Mathematics and Statistics, Jiangsu Normal University,}\\ {\small Xuzhou, 221116, P.R. China}\\
{\small E-mail: [email protected]}\\ \\
Bin Hu \thanks{Corresponding author}\\
{\small School of Mathematics and Statistics, Jiangsu Normal University,}\\
{\small Xuzhou 221116, P. R. China}\\
{\small E-mail: [email protected]}\\ \\
{ Alexander N. Skiba }\\
{\small Department of Mathematics and Technologies of Programming, Francisk Skorina Gomel State University,}\\
{\small Gomel 246019, Belarus}\\
{\small E-mail: [email protected]}}
\date{}
\maketitle
\begin{abstract} Let $\sigma =\{\sigma_{i} | i\in I\}$ be a partition of
the set of all primes $\Bbb{P}$ and $G$ a finite group. Let $\sigma
(G)=\{\sigma _{i} : \sigma _{i}\cap \pi (G)\ne \emptyset$.
A set ${\cal H}$ of subgroups of $G$ is said to be a \emph{complete Hall
$\sigma $-set} of $G$ if every member $\ne 1$ of
${\cal H}$ is a Hall $\sigma _{i}$-subgroup of $G$ for some $i\in I$ and
$\cal H$ contains exactly one Hall $\sigma _{i}$-subgroup of $G$ for
every $i$ such that $\sigma _{i}\in \sigma (G)$.
We say that $G$ is \emph{$\sigma$-full} if $G$ possesses a complete Hall
$\sigma $-set.
A complete Hall $\sigma $-set $\cal H$ of $G$ is said to be a
\emph{$\sigma$-basis} of $G$ if every two subgroups $A, B \in\cal H$ are
permutable, that is, $AB=BA$.
In this paper, we study properties of finite groups having a $\sigma$-basis.
In particular, we prove that if $G$ has a a $\sigma$-basis, then $G$ is
\emph{generalized $\sigma$-soluble}, that is, $G$ has a complete Hall
$\sigma $-set and for every chief factor $H/K$ of
$G$ we have $|\sigma (H/K)|\leq 2$. Moreover, answering to Problem 8.28 in \cite{commun},
we prove the following
{\bf Theorem A.} {\sl Suppose that $G$ is $\sigma$-full. Then every complete
Hall $\sigma$-set of $G$ forms a $\sigma$-basis of $G$ if and only if $G$ is
generalized $\sigma$-soluble and for the automorphism group $G/C_{G}(H/K)$,
induced by $G$ on any its
chief factor $H/K$, we have either $\sigma (H/K)=\sigma (G/C_{G}(H/K))$
or $\sigma (H/K) =\{\sigma _{i}\}$ and $G/C_{G}(H/K)$ is a
$\sigma
_{i} \cup \sigma _{j}$-group for some $i\ne j$.
}
\end{abstract}
\footnotetext{Keywords: finite group, Hall subgroup, $\sigma$-semipermutable
subgroup, $\sigma$-basis, generalized ${\sigma}$-soluble group.}
\footnotetext{Mathematics Subject Classification (2010): 20D10, 20D15}
\let\thefootnote\thefootnoteorig
\section{Introduction}
Throughout this paper, all groups are finite and $G$ always denotes
a finite group. Moreover, $\mathbb{P}$ is the set of all primes,
$\pi= \{p_{1}, \ldots , p_{n}\} \subseteq \Bbb{P}$ and $\pi' = \Bbb{P} \setminus \pi$.
If
$n$ is an integer, the symbol $\pi (n)$ denotes
the set of all primes dividing $n$; as usual, $\pi (G)=\pi (|G|)$, the set of all
primes dividing the order of $G$.
In what follows, $\sigma$ is some partition of
$\Bbb{P}$, that is, $\sigma =\{\sigma_{i} |
i\in I \}$, where $\Bbb{P}=\bigcup_{i\in I} \sigma_{i}$
and $\sigma_{i}\cap
\sigma_{j}= \emptyset $ for all $i\ne j$.
By the analogy with the notation $\pi (n)$, we write $\sigma (n)$ to denote
the set $\{\sigma_{i} |\sigma_{i}\cap \pi (n)\ne
\emptyset \}$; $\sigma (G)=\sigma (|G|)$.
A \emph{complete set of Sylow subgroups $\cal S$ of $G$}
contains exactly one Sylow $p$-subgroup for each prime $p$, that is,
${\cal S}=\{1, P_{1}, \ldots P_{t}\}$, where $P_{i}$ is a Sylow $p_{i}$-subgroup of $G$ and
$\pi (G)=\{p_{1}, \ldots , p_{t}\}$.
The set $\cal S$ is said to be a \emph{ Sylow basis of $G$} provided every two
subgroups $P, Q \in {\cal S}$ are permutable, that is, $PQ=QP$.
In general, we say that a set ${\cal H}$ of subgroups of $G$ is a
\emph{complete Hall $\sigma $-set} of $G$ \cite{commun} if
every member $\ne 1$ of ${\cal H}$ is a Hall $\sigma _{i}$-subgroup of $G$
for some $\sigma _{i} \in \sigma$ and ${\cal H}$ contains exactly one Hall
$\sigma _{i}$-subgroup of $G$ for every $\sigma _{i}\in \sigma (G)$.
We say that $G$ is \emph{$\sigma$-full} \cite{commun} if $G$ possesses a complete Hall
$\sigma $-set. A complete Hall $\sigma $-set $\cal H$ of $G$ is said to be a
\emph{$\sigma$-basis} of $G$ \cite{2} if every two subgroups $A, B \in\cal H$ are
permutable, that is, $AB=BA$.
In this paper we deal with the following two generalizations of
solubility.
{\bf Definition 1.1.} We say that $G$ is:
(i) \emph{$\sigma$-soluble}
\cite{2} if $|\sigma (H/K)|= 1$ for every chief factor $H/K$ of $G$.
(i) \emph{generalized$\sigma$-soluble}
if $G$ is $\sigma$-full and $|\sigma (H/K)|\leq 2$ for every
chief factor $H/K$ of $G$.
Before continuing, consider three classical cases.
{\bf Example 1.2.} (i) In the classical case,
when $\sigma =\sigma ^{0}=\{\{2\}, \{3\}, \ldots
\}$, $G$ is $\sigma ^{0}$-soluble if
and only if it is soluble. Note also that in view of the Burnside's
$p^{a}q^{b}$-theorem, $G$ is $\sigma ^{0}$-soluble if
and only if it is generalized $\sigma ^{0}$-soluble.
(ii) In the other classical case, when $\sigma =\sigma ^{\pi}=\{\pi,
\pi'\}$ $G$ is: $\sigma ^{\pi}$-soluble if and only if $G$ is
$\pi$-separable; generalized $\sigma ^{\pi}$-soluble if and only if
$G$ has both a Hall $\pi$-subgroup and a Hall $\pi'$-subgroup.
(iii) In fact, in the theory of $\pi$-soluble groups ($\pi= \{p_{1}, \ldots , p_{n}\}$)
we deal with the partition
$\sigma =\sigma ^{0\pi }=\{\{p_{1}\}, \ldots , \{p_{n}\}, \pi'\}$ of $\Bbb{P}$.
Note that $G$ is $\sigma ^{0\pi }$-soluble if and only if it is
$\pi$-soluble. Note also that $G$ is generalized $\sigma ^{0\pi
}$-soluble if and only if the following hold: (a) $G$ has both
a Hall $\pi$-subgroup $E$ and a Hall $\pi'$-subgroup, and (b) $(H\cap E)K/K$ is a
Sylow subgroup of $H/K$ for every chief factor $H/K$ of $G$.
The well-known Hall's theorem states that $G$ is
soluble if and only if it has a Sylow basis. This classical result makes
natural to ask: {Does it true that $G$ is
$\sigma$-soluble if and only if it has a $\sigma$-basis?}
A partial confirmation of this hypothesis is given by the following
{\bf Theorem 1.3} (See Theorem A in \cite{2}). {\sl Every $\sigma$-soluble group
possesses a $\sigma$-basis}
Now consider the following
{\bf Example 1.4.} Let $\sigma =\{\{2, 3\}, \{5\}, \{2, 3, 5\}'\}$, and
let $G=A_{5}\times (C_{11}\rtimes \text{Aut} (C_{11}))$, where $A_{5}$ is the
alternating group of degree 5 and $C_{11}$ is a group of order 11. Let
$A_{4}\simeq A\leq A_{5}$ and $B$ a Sylow 5-subgroup of $A_{5}$.
Let $H_{1}=AC_{2}$, $H_{2}=BC_{5}$ and $H_{3}=C_{11}$,
where $C_{5}\times C_{2}=\text{Aut}
(C_{11}))$.
Then the set
$\{H_{1}, H_{2}, H_{3}\}$ is a $\sigma$-basis of $G$. Nevertheless,
$G$ is not $\sigma$-soluble. This example shows that in general the
answer to above question is negative.
Nevertheless, the following result
is true.
{\bf Theorem A.} {\sl Suppose that $G$ possesses a complete Hall $\sigma
$-set $\cal H$. }
(i) {\sl If $\cal H$ is a $\sigma$-basis
of $G$, then $G$ is generalized $\sigma$-soluble. }
(ii) {\sl If $H\leq A\in {\cal H}$ and
$HV^{x}=V^{x}H$ for all $x\in G$ and all $V\in \cal H$ such that
$(|H|, |V|)=1$, then $H^{G}$ is generalized $\sigma$-soluble.
}
In view of the Burnside's $p^{a}q^{b}$-theorem, in the case where
$\sigma =\{\{2\}, \{3 \}, \ldots \}$ we get from Theorem A(ii) the following
{\bf Corollary 1.5} (Isaaks \cite{isaaks}). {\sl If a $p$-subgroup $H$
of $G$ permutes with all Sylow subgroups $P$ of $G$ such that $(p, |P|)=1$, then $H^{G}$ is soluble. }
{\bf Corollary 1.4.} {\sl If $H\in {\cal H}$ and $HV^{x}=V^{x}H$ for all $x\in G$ and all $V\in \cal H$ such that
$(|H|, |V|)=1$, then $G$ is $\sigma _{i}'$-soluble where
$\sigma (H)= \{\sigma _{i}\}$.
}
{\bf Corollary 1.6} (Borovikov \cite{borov}). {\sl If a Sylow $p$-subgroup $P$
of $G$ is permutes with all Sylow subgroups $Q$ of $G$ such that $(p, |Q|)=1$, then $G$
is $p'$-soluble. }
The integers $n$ and $m$ are called \emph{$\sigma$-coprime} if $\sigma (n)
\cap \sigma (m)=\emptyset$.
In fact, Theorem A(i) is a corollary of the following fact.
{\bf Theorem B.} {\sl Suppose that $G=A_{1}A_{2}=A_{2}A_{3}=A_{1}A_{3},$
where $A_{1}$ is $\sigma$-soluble and $A_{2}$ and $A_{3}$
are generalized $\sigma$-soluble subgroups of $G$.
If for some $i, j, k\in I$ the three
indices
$|G:N_{G}(O^{\sigma _{i}}(A_{1}))|$, $|G:N_{G}(O^{\sigma _{j}}(A_{2}))|$,
$|G:N_{G}(O^{\sigma _{k}}(A_{3}))|$ are pairwise $\sigma$-coprime,
then
$G$ is generalized $\sigma$-soluble. }
{\bf Corollary 1.7.} {\sl Suppose that $A_{1}$ is a
$\sigma$-soluble subgroup and $A_{2}$ and $A_{3}$
are generalized $\sigma$-soluble subgroups of $G$.
If the three
indices
$|G:A_{1}|$, $|G:A{2}|$,
$|G:A_{3}|$ are pairwise $\sigma$-coprime,
then
$G$ is generalized $\sigma$-soluble. }
{\bf Corollary 1.8.} {\sl
Suppose that $G=A_{1}A_{2}=A_{2}A_{3}=A_{1}A_{3},$
where $A_{1}$, $A_{2}$ and $A_{3}$ are soluble subgroups of $G$.
If the three indices
$|G:N_{G}(A_{1})|$, $|G:N_{G}(A_{2})|$, $|G:N_{G}(A_{3})|$ are pairwise coprime, then
$G$ is soluble. }
{\bf Corollary 1.8.} (H. Wielandt). {\sl If $G$ has
three soluble subgroups $A_{1}$, $A_{2}$ and $A_{3}$ whose indices
$|G:A_{1}|$, $|G:A_{2}|$, $|G:A_{3}|$ are pairwise coprime, then
$G$ is itself soluble. }
Following \cite{commun}, we use $\frak{H}_{\sigma}$
to denote the class of all $\sigma$-full groups $G$ such that every
complete Hall $\sigma$-set of $G$ forms a $\sigma$-basis of $G$.
In \cite[VI, Section 3]{hupp}, Huppert described soluble groups $G$ in which
every complete Sylow set of $G$ forms a Sylow basis of $G$. The results
in \cite[VI, Section 3]{hupp} are motivations for
the following two questions.
{\bf Question 1.9} (See Problem 8.29 in \cite{commun}). {\sl Describe groups in
$\frak{H}_{\sigma}$.}
{\bf Question 1.10} (See Problem 8.30 in \cite{commun}). {\sl Describe
$\sigma$-soluble groups groups in $\frak{H}_{\sigma}\cap \frak{S}_{\sigma}$
}
As one more application of Theorem A, we prove the
following result, which gives the answer to Question 1.9.
{\bf Theorem C.} {\sl Suppose that $G$ is $\sigma$-full. Then every complete
Hall $\sigma$-set of $G$ forms a $\sigma$-basis of $G$ if and only if $G$ is
generalized $\sigma$-soluble and for the automorphism group $G/C_{G}(H/K)$,
induced by $G$ on any its
chief factor $H/K$, we have $|\sigma (G/C_{G}(H/K))|\leq 2$ and also
$\sigma(H/K)\subseteq \sigma (G/C_{G}(H/K))$ in the case $|\sigma (G/C_{G}(H/K))|= 2$.
}
ON the base of Theorem C we prove the following result which gives the
answer to Question 1.10.
{\bf Theorem D.} {\sl Suppose that $G$ is $\sigma$-soluble.
Then the following statements are equivalent:}
(i) {\sl Every complete Hall $\sigma$-set of $G$ forms a $\sigma$-basis of $G$. }
(ii) {\sl The automorphism group $G/C_{G}(H/K)$, induced by $G$ on any its chief factor
$H/K$ with $\sigma (H/K)=\{\sigma _{i}\}$, is either a $\sigma _{j}$-group or
a $\sigma _{i}\cup \sigma _{j}$-group for
some $j\ne i$. }
(iii) {\sl $G\simeq G^{*}/R$, where $G^{*}\leq A_{1}\times \cdots \times A_{t}$ for some
$\sigma$-biprimary $\sigma$-soluble groups $A_{1}, \ldots , A_{t}$. }
Note that Satz 3.1 in \cite[VI]{hupp} can be obtained as
a special case of Theorem B, when $\sigma =\{\{2\}, \{3 \}, \ldots \}$.
Finally, we prove the following
{\bf Theorem E.}
{\sl If $G$ possesses a complete Hall $\sigma$-set $\cal H$ with
$|G:N_{G}(H)|$ is $\sigma$-primary for all $H\in \cal H$, then
$G$ is generalized $\sigma$-soluble.}
{\bf Corollary 1.11} (See Zhang \cite {zhang} or Guo \cite[Theorem 3]{Gu}).
{\sl If for every Sylow subgroup $P$ of $G$
the number $|G:N_{G}(P)|$ is
a prime power, then $G$ is soluble.}
\section{Preliminaries}
We use ${\mathfrak{S}}_{g\sigma}$ to denote the class of all
$\sigma$-soluble groups.
The direct calculations show that the following lemma is true
{\bf Lemma 2.1.} (i) {\sl The class ${\mathfrak{S}}_{g\sigma}$ is
closed under taking
products of normal subgroups, homomorphic images and subgroups.
Moreover, any extension of the generalized
$\sigma$-soluble group by a generalized
$\sigma$-soluble group is generalized
a $\sigma$-soluble group as well. }
(ii) {\sl If $G/R, G/N \in {\mathfrak{S}}_{g\sigma}$, then $G/R\cap N
\in {\mathfrak{S}}_{g\sigma}$.}
(iii) {\sl ${\mathfrak{S}}_{g\sigma} \subseteq {\mathfrak{S}}_{g\sigma ^{*}}$ for any
partition
${\sigma ^{*}}=\{\sigma^{*}_{j}\ |\ j\in J\}$ of $\Bbb{P}$ such that
$J\subseteq I$ and $\sigma _{j}\subseteq \sigma ^{*}_{j}$ for all $j\in J$.}
Recall that $G$ is said to be: a $D_{\pi}$-group if $G$ possesses a Hall
$\pi$-subgroup $E$ and every $\pi$-subgroup of $G$ is contained in some
conjugate of $E$; a \emph{$\sigma$-full group
of Sylow type} \cite{1} if every subgroup of $G$ is a $D_{\sigma _{i}}$-group for every
$\sigma _{i}\in \sigma$.
In view of Theorem B in \cite{2}, the following fact is true.
{\bf Lemma 2.1.} {\sl If $G$ is $\sigma$-soluble, then $G$ is a $\sigma$-full group
of Sylow type.
}
Let $\Pi \subseteq \sigma$.
A natural number $n$ is said to be a \emph{$\Pi$-number}
if $\sigma (n)\subseteq \Pi$. A subgroup $A$ of $G$ is said to be: a
\emph{ $\Pi$-subgroup} of $G$ if $\sigma (G)\subseteq \Pi$;
a \emph{Hall $\Pi$-subgroup} of $G$ \cite{1} if $|A|$ is a $\Pi$-number
and $|G:A|$ is a $\Pi'$-number
We use $\frak{H}_{\sigma}$ to denote the class of all
groups $G$ such that $G$ has a complete Hall
$\sigma$-set ${\cal H}=\{H_{1}, \ldots , H_{t} \}$ satisfying the condition
$H_{i}^{x}H_{j}^{y}=H_{j}^{y}H_{i}^{x}$ for all $x, y\in G$ and all
$i\ne j$. Note that in view of Lemma 2.1, each $\sigma$-biprimary
$\sigma$-soluble group belongs to the class $\frak{H}_{\sigma}$.
{\bf Lemma 2.2.} {\sl The class
$\frak{H}_{\sigma}$ is closed under taking homomorphic images and
and direct products. Moreover, if $G$ is a $\sigma$-full group
of Sylow type and $E\leq G\in \frak{H}_{\sigma}$, then $E\in \frak{H}_{\sigma}$.}
{\bf Proof.}
Let $R$ be a normal subgroup
of $G\in \frak{H}_{\sigma}$.
By hypothesis, $G$ has a complete Hall
$\sigma$-set ${\cal H}=\{H_{1}, \ldots , H_{t} \}$ satisfying the condition
$H_{i}^{x}H_{j}^{y}=H_{j}^{y}H_{i}^{x}$ for all $x, y\in G$ and all
$i\ne j$.
Then ${\cal H}_{0}=\{H_{1}R/R, \ldots , H_{t}R/R \}$ is a
complete Hall $\sigma$-set of $G/R$ such that
$$ (H_{i}R/R)^{xR}(H_{j}R/R)^{yR}=
H_{i}^{x}H_{j}^{y}R/R=H_{j}^{y}H_{i}^{x}R/R$$$$=(H_{j}R/R)^{yR}(H_{i}R/R)^{xR}.$$
Thus $G/R\in
\frak{H}_{\sigma}$. Therefore the class $\frak{H}_{\sigma}$ is closed
under taking homomorphic images.
Now we show that the class
$\frak{H}_{\sigma}$ is closed under taking direct products. It is enough
to show that if
$A, B \in \frak{H}_{\sigma}$, then $G=A\times B \in
\frak{H}_{\sigma}$. Let ${\cal A}=\{A_{1}, \ldots , A_{n}\}$ be a complete Hall $\sigma$-set of $A$
and ${\cal B}=\{B_{1}, \ldots , B_{m}\}$ be a complete Hall $\sigma$-set of $B$ such that
$A_{i}^{a_{1}}A_{j}^{a_{2}}=A_{j}^{a_{2}}A_{i}^{a_{1}}$ for all $a_{1}, a_{2}\in A$ and all
$i\ne j$ and $B_{i}^{b_{1}}B_{j}^{b_{2}}=B_{j}^{b_{2}}B_{i}^{b_{1}}$ for all
$b_{1}, b_{2}\in B$ and all
$i\ne j$. We can assume without loss of generality that $1 \in {\cal
A}\cap {\cal B} $ and that $\sigma (G)= \{\sigma _{1}, \ldots \sigma
_{t}\}$. Therefore for every $i$ there are indices $a_{i}$ and $b_{i}$ such that
$A_{a_{i}} \times B_{b_{i}}$ is a Hall $\sigma _{i}$-subgroup of $G$.
Moreover, if $x=a_{1}b_{1}$ and $y=a_{2}b_{2} $, where $a_{1}, a_{2}\in A$
and $b_{1}, b_{2}\in B$, then $$(A_{a_{i}}
\times B_{b_{i}})^{x}(A_{a_{j}} \times B_{b_{j}})^{y}=(A_{a_{i}}^{a_{1}}
\times B_{b_{i}}^{b_{1}})(A_{a_{j}}^{a_{2}} \times B_{b_{j}}^{b_{2}})$$$$=
(A_{a_{j}}^{a_{2}} \times B_{b_{j}}^{b_{2}})(A_{a_{i}}^{a_{1}}
\times B_{b_{i}}^{b_{1}})=(A_{a_{j}} \times B_{b_{j}})^{y}(A_{a_{i}}
\times B_{b_{i}})^{x}.$$ Hence
$A_{a_{1}} \times B_{b_{1}}, \ldots , A_{a_{t}} \times B_{b_{t}}$ is a
$\sigma$-basis of $G$.
Thus $G \in \frak{H}_{\sigma}$.
Finally, assume that $G$ is a $\sigma$-full group
of Sylow type. Then for every complete Hall
$\sigma$-set ${\cal E}=\{E_{1}, \ldots , E_{r} \}$ of $E$ there is a complete Hall
$\sigma$-set ${\cal H}=\{H_{1}, \ldots , H_{t} \}$ of $G$ such that
$E_{i}=H_{i}\cap E$ for all $i=1, \ldots t$. We can assume without loss
of generality that $E_{i}$ is a $\sigma _{i}$-group.
Now let $\Pi =\{\sigma _{i}, \sigma _{j}\}$.
Then for $x, y\in E$ we have
$\langle E_{i}^{x},
E_{j}^{y} \rangle \leq E\cap H_{i}^{x}H_{j}^{y}$, where $E\cap
H_{i}^{x}H_{j}^{y}$ is a $Pi$-subgroup of $E$ and so $|E\cap
H_{i}^{x}H_{j}^{y}|\leq |E_{i}^{x}||
E_{j}^{y}|$. Hence $\langle E_{i}^{x},
E_{j}^{y} \rangle=E_{i}^{x}E_{j}^{y}$.
Thus $E \in \frak{H}_{\sigma}$.
The lemma is proved.
The next lemma is evident.
{\bf Lemma 2.3.} {\sl If the chief factors $H/K$ and $T/L$ of $G$ are $G$-isomorphic, then
$C_{G}(H/K)=C_{G}(T/L)$.}
We use $\frak{X}_{\sigma}$ to denote
the class of all generalized $\sigma$-soluble groups $G$ such that
for the automorphism group $G/C_{G}(H/K)$, induced by $G$ on any its
chief factor $H/K$, we have $|\sigma (G/C_{G}(H/K))|\leq 2$ and also
$\sigma(H/K)\subseteq \sigma (G/C_{G}(H/K))$ in the case $|\sigma (G/C_{G}(H/K))|= 2$.
{\bf Lemma 2.4.} {\sl The class
$\frak{X}_{\sigma}$ is closed under taking homomorphic images, direct products
and subgroups.}
{\bf Proof.} Let $R$ be a normal subgroup
of $G\in \frak{X}_{\sigma}$. Then $G/R$ is generalized $\sigma$-soluble and
for any chief factor $(H/R)/(K/R)$ of
$G/R$ we have $C_{G/R}((H/R)/(K/R))=C_{G}(H/K)/R$, where $H/K$ is a chief
factor of $G$. Hence $G/R\in \frak{X}_{\sigma}$, so
the class $\frak{X}_{\sigma}$ is closed
under taking homomorphic images.
Now let $G=A_{1}\times A_{2}$, where $A_{1}, A_{2}\in \frak{X}_{\sigma}$. The
Jordan-H\"{o}lder
theorem for groups with operators \cite[A, 3.2]{DH} implies that for every chief factor $H/K$
of $G$, there is $i$ such that some chief factor $T/L$ of $G$ below $A_{i}$
is $G$-isomorphic to $H/K$. Moreover, $T/L $ is a chief factor of
$A_{i}$ and $G/C_{G}(T/L)\simeq A_{i}/C_{A_{i}}(T/L)$. Hence, by Lemma 2.3,
$G\in \frak{X}_{\sigma}$, so the class $\frak{X}_{\sigma}$ is closed
under taking direct products.
Finally, we show that if $G\in \frak{X}_{\sigma}$ and
$E\leq G$, then $E\in \frak{X}_{\sigma}$.
Let $1=G_{0} < G_{1} < \cdots < G_{t-1} < G_{t}=G$ be a chief series of $G$.
Let $H/K$ be any chief factor
of $E$ such that $G_{l-1}\cap E \leq K < H \leq G_{l}\cap E$ for some $l$.
Since $$C_{G}(G_{l}/G_{l-1})\cap
E\leq C_{E}(G_{l}\cap E/G_{l-1}\cap E)\leq C_{E}(H/K),$$
$|\sigma (E/C_{E}(H/K))|\leq 2$. Moreover, if $|\sigma
(E/C_{E}(H/K))|= 2$, then $|\sigma
(G/C_{G}(G_{l}/G_{l-1}))|= 2$, $\sigma
(G/C_{G}(G_{l}/G_{l-1}))=\sigma (E/C_{E}(H/K)))$
and $\sigma(G_{l}/G_{l-1})\subseteq
\sigma (G/C_{G}(G_{l}/G_{l-1}))$. Hence from the isomorphism
$E\cap G_{l}/E\cap G_{l-1}\simeq (E\cap G_{l})G_{l-1}/G_{l-1}$ we get
that $\sigma(H/K)\subseteq
\sigma (E/C_{E}(H/K))$. Now
applying the Jordan-H\"older theorem for
groups with operators \cite[A, 3.2]{DH} we get that $E\in
\frak{X}_{\sigma}$. Therefore the class $\frak{X}_{\sigma}$ is closed
under taking subgroups.
The lemma is proved.
A \emph{class} of groups is a collection $\mathfrak{X}$ of groups with
the property that if $G\in \mathfrak{X}$ and if $H\simeq G$, then
$H\in \mathfrak{X}$. The symbol $(\mathfrak{Y})$ \cite[p. 264]{DH} denotes the smallest
class of groups containing $\mathfrak{Y}$.
For a class $\mathfrak{X}$ of groups we define, following \cite[p. 264]{DH}:
$\text{S}(\mathfrak{X})=(G: G\leq H$ for some $H\in \mathfrak{X});$
$\text{Q}(\mathfrak{X})=(G: G$ is an epimorphic image of some $H\in \mathfrak{X});$
$\text{D}_{0}(\mathfrak{X})=(G: G=H_{1}\times \cdots \times H_{r}$ for some
$H_{1}, \ldots , H_{r}\in \mathfrak{X});$
$\text{R}_{0}(\mathfrak{X})=(G: G$ has normal subgroups $N_{1}, \ldots ,
N_{r}$ with all $G/N_{i}\in \mathfrak{X})$
and $N_{1}\cap \cdots \cap N_{r}=1 )$.
We say, following \cite{shem}, that a class $\mathfrak{X}$ of groups is a
\emph{semiformation} if
$$\text{S}(\mathfrak{X})=\mathfrak{X}=\text{Q}(\mathfrak{X}).$$
The following fact is well-known (see, for example, \cite[p. 57]{malc}).
{\bf Lemma 2.5.} {\sl If $A\simeq B\leq G$, then for some $G^{*}\simeq G$ we
have $A\leq G^{*}$.}
{\bf Lemma 2.6.} {\sl Let $\mathfrak{X}$ be a class of groups.}
(1) {\sl If $\mathfrak{X}$ is a semiformation, then}
$\text{S}(\text{D}_{0}(\mathfrak{X}))\text{=}R_{0}(\mathfrak{X}).$
(2) $\text{Q}(\text{R}_{0}(\mathfrak{X}))\subseteq \text{R}_{0}(\text{Q}(\mathfrak{X}))$.
(3) $\text{R}_{0}(\text{R}_{0}(\mathfrak{X})) = \text{R}_{0}(\mathfrak{X})$.
{\bf Proof.} (1) Let
$\mathfrak{M} =\text{S}(\text{D}_{0}(\mathfrak{X}))$ and $\mathfrak{H} =
\text{R}_{0}(\mathfrak{X})$. First suppose that $G\in \mathfrak{M}$.
Then, in view of Lemma 2.5, $G\leq H_{1}\times \cdots \times H_{r}$ for some
$H_{1}, \ldots , H_{r}\in \mathfrak{X}$. We show that $G\in
\mathfrak{H}$. If $r=1$, it is clear. Now assume that $r > 1$ and let
$N_{i}=H_{1}\times
\cdots H_{i-1}H_{i+1}\times \cdots \times H_{r}$ for all $i=1, \ldots , r$.
Then $$G/G\cap N_{i}\simeq GN_{i}/N_{i} \leq (H_{1}\times \cdots \times
H_{r})/N_{i}\simeq H_{i} \in \mathfrak{X}.$$ It is clear also that $G\cap N_{1}, \ldots G\cap N_{r}$
are normal subgroups of $G$ with $(G\cap N_{1})\cap \cdots \cap (G\cap N_{r})=1$. Hence
$G\in \mathfrak{H}$. Therefore $\mathfrak{M} \subseteq \mathfrak{H}$.
Now suppose that $G\in \mathfrak{H}$, that is, $G$ has normal subgroups $N_{1}, \ldots ,
N_{r}$ with all $G/N_{i}\in \mathfrak{X}$
and $N_{1}\cap \cdots \cap N_{r}=1$. Then, by Lemma 4.17 in \cite[A]{DH},
$G$ is isomorphic with a subgroup of
$ (G/N_{1})\times \cdots \times (G/N_{r}).$ Hence
$G\in \mathfrak{M}$, so
$\mathfrak{H} \subseteq \mathfrak{M}$.
Therefore we have (1).
(2), (3) See respectively Lemmas 1.18(b) and Lemma 1.6 in \cite[II]{DH}.
The lemma is proved.
{\bf Lemma 2.7.} {\sl Let $\mathfrak{X}$ be a semiformation of groups. Suppose that
$\frak{F}$ is the class of groups $A$ which can be represented in the form
$A\simeq A^{*}/R$, where $A^{*}\leq A_{1}\times \cdots \times A_{t}$ for some
$A_{1}, \ldots , A_{t}\in \frak{X}$. If $G/R, G/N\in\frak{F}$, then
$G/(R \cap N)\in\frak{F}$.
}
{\bf Proof. } We can assume without loss of generality that $R\cap N=1$.
Then, by Lemma 2.5,
$$G\in \text{R}_{0}(\text{Q}(\text{S}(\text{D}_{0}(\mathfrak{X}))))=
\text{R}_{0}(\text{Q}(\text{R}_{0}(\mathfrak{X})))\subseteq
\text{Q}(\text{R}_{0}(\text{R}_{0}(\mathfrak{X})))=\text{Q}(\text{R}_{0}(\mathfrak{X})
=\text{Q}(\text{D}_{0}(\mathfrak{X})))\subseteq \frak{F}.$$
The lemma is proved.
{\bf Lemma 2.8.} {\sl Let $G=RH_{1} \cdots H_{n}$, where
$[H_{i}^{G}, H_{j}^{G}]=1$ for all $i\ne j$ and $R$ is normal in $G$.
Then $G\simeq G^{*}$, where $G^{*}\leq (RH_{1})\times \cdots \times (RH_{n})$. }
{\bf Proof.} See pages 671--672 in \cite{hupp}.
{\bf Lemma 2.9} (See \cite[2.2.8]{15}). {\sl If $\frak{F}$ is a non-empty
formation and
$N$, $R$ be subgroups of $G$, where $N$ is normal in $G$.}
(i) {\sl
$(G/N)^{\frak{F}}=G^{\frak{N}}N/N.$ }
(ii) {\sl If $G=RN$, then $G^{\frak{N}}N=R^{\frak{N}}N$}.
{\bf Lemma 2.10.} {\sl Suppose that $G$ has a $\sigma _{i}$-subgroup
$A\ne 1$ and a $\sigma _{j}$-subgroup $B\ne 1$ such that
$AB^{x}=B^{x}A$ for all $x\in G$. If $O_{\sigma _{i}\cup \sigma _{j}}(G)= 1$, then
$[A^{G}, B^{G}]= 1$.
}
{\bf Proof.} By hypothesis,
$A(B^{x})^{y}=(B^{x})^{y}A$ for all $x, y\in G$ and $$D=\langle
A^{B^{x}} \rangle \cap \langle (B^{x})^{A}\rangle\leq AB^{x},$$ where $D$ is
subnormal in $G$ by \cite[1.1.9(2)]{prod}. Then $D\leq O_{\sigma _{i}\cup \sigma
_{j}}(G)= 1$, so
Then $ [A, B^{x}] \leq [\langle A^{B^{x}}\rangle , \langle B^{x})^{A}\rangle ]\leq D = 1$.
Therefore $[A^{x}, B^{y}]=1$ for all $x, y\in G$ and so $[A^{G}, B^{G}]= 1$.
The lemma is proved.
\section{Proofs of the results}
{\bf Proof of Theorem B.}
Suppose that this theorem is false and let $G$ be a
counterexample with $|G|$ minimal. We can assume without loss of
generality that $i=1$, $j=2$ and $k=3$.
(1) {\sl If $R$ is a minimal normal subgroup of $G$, then $G/R$ is generalized
$\sigma$-soluble.
Hence $R$ is the unique minimal normal subgroup of $G$ and $R$ is not generalized
$\sigma$-soluble.
Thus $C_{G}(R)=1$. }
First note that if $A_{i}\leq R$, then for $j\ne i$ we have $G= A_{i}A_{j}
=RA_{j}$, so $G/R\simeq A_{j}/A_{j}\cap R$ is generalized
$\sigma$-soluble.
Now suppose that $A_{i}\nleq R$ for all $i=1, 2, 3$.
For any group $A$ and any $l\in I$ we have
$O^{\sigma _{l}}(A)=A^{\frak{G}_{\sigma _{l}}}$, where
$\frak{G}_{\sigma _{l}}$ is the class of all $\sigma _{l}$-groups.
Therefore $O^{\sigma _{l}}(A)R/R=O^{\sigma _{l}}(AR/R)$ by Lemma 2.10(ii).
Hence
$$N_{G}(O^{\sigma _{l}}(A))R/R\leq
N_{G}(O^{\sigma _{l}}(A)R)/R = N_{G/R}(O^{\sigma _{l}}(AR/R).$$
Hence the three indices
$|(G/R):N_{G/R}((O^{\sigma _{1}}(A_{1}))|$, $
|(G/R):N_{G/R}(O^{\sigma _{2}}(A_{2}))|,$ and
$ |(G/R):N_{G/R}(O^{\sigma _{3}}(A_{3}))|$ are pairwise $\sigma$-coprime.
On the other hand, clearly, $A_{1}R/R\simeq A_{1}/A_{1}\cap R$ is $\sigma$-soluble
and
$A_{i}R/R\simeq A_{i}/A_{i}\cap R$ is generalized $\sigma$-soluble
for all $i=2, 3$.
Therefore the hypothesis holds
for $G/R$, so $G/R$ is generalized $\sigma$-soluble
by the choice of $G$. Hence $R$ is not generalized $\sigma$-soluble
and
$R$ is the unique minimal normal subgroup
of $G$ by Lemma 2.1(ii). Therefore, since $C_{G}(R)$ is normal in $G$, $C_{G}(R)=1$.
(2) {\sl At least two of
the subgroups $A_{1}, A_{2}, A_{3}$ are not $\sigma$-primary.}
Indeed, if $A_{i}$ and $A_{j}$ are $\sigma$-primary for some $i\ne j$,
then $G=A_{i}A_{j}$ is generalized $\sigma$-soluble, which contradicts the choice of $G$.
(3) {\sl If $A_{l} $ is $\sigma$-soluble and $L$ is a minimal normal
subgroup of $A_{l}$, then for some $t\ne l$ we have $O^{\sigma
_{t}}(A_{t})=1$, so $A_{t}$ is $\sigma$-primary.}
We can assume without loss of generality that $l=2$.
Let $L$ be a minimal normal subgroup of $A_{l}=A_{2}$. Then $L$
is a $\sigma _{i}$-group for
some prime $i$ since $A_{2}$ is $\sigma$-soluble by hypothesis.
On the other
hand, again by hypothesis,
$|G:N_{G}(O^{\sigma _{1}}(A_{1}))|$ and
$|G:N_{G}(O^{\sigma _{3}}(A_{3}))|$ are $\sigma$-coprime, so
at least one of the numbers,
$|G:N_{G}(O^{\sigma _{3}}(A_{3}))|$ say, is a $\sigma _{i}'$-number.
But $G=A_{2}A_{3}=A_{2}N_{G}(O^{\sigma _{3}}(A_{3}))$, so
$|A_{2}:A_{2}\cap N_{G}(O^{\sigma _{3}}(A_{3}))|$ is a $\sigma _{i}'$-number.
Hence $$L\leq A_{2}\cap N_{G}(O^{\sigma _{3}}(A_{3}))\leq N_{G}(O^{\sigma _{3}}(A_{3})).$$
Therefore
$$R\leq L^{G}=L^{A_{2}N_{G}(O^{\sigma _{3}}(A_{3}))}=
L^{N_{G}(O^{\sigma _{3}}(A_{3}))}\leq N_{G}(O^{\sigma _{3}}(A_{3}))$$ by
Claim (1).
Suppose that $O^{\sigma _{3}}(A_{3})\ne 1$ and let $H=O^{\sigma _{3}}(A_{3}) \cap R$.
If $H\ne 1$, then $H$ is a non-identity normal generalized $\sigma$-soluble
subgroup of $R$ since $A_{3}$ is generalized $\sigma$-soluble by hypothesis. Hence $R$ is generalized $\sigma$-soluble, contrary to Claim (1).
Thus $H=1$, so $O^{\sigma _{3}}(A_{3})\leq
C_{G}(R)\leq R$, so $O^{\sigma _{3}}(A_{3}) =1$ by Claim (1). This
contradiction completes the proof of (3).
{\sl Final contradiction.} Since $A_{1}$ is $\sigma$-soluble by
hypothesis, Claim (3) implies that one of the subgroups $A_{2}$ or $A_{3}$, $A_{2}$ say, is
$\sigma$-primary. Then $A_{2}$ is $\sigma$-soluble and so, again by Claim (3), one
of the subgroups $A_{1}$ or $A_{3}$ is also $\sigma$-primary. But this
is impossible by Claim (2).
This contradiction completes the proof of
the result.
{\bf Proof of Theorem A.}
We
can assume without loss of generality that $H$ is a $\sigma _{1}$-group, $2\in \sigma _{1}$
and $ {\cal H}=\{H_{1}, \ldots , H_{t}\}$, where $H_{i}$ is a $\sigma _{i}$-group for all
$i=1, \ldots, t$. Then $ t > 2$ since otherwise $G$ and $H^{G}$ are
generalized $\sigma$-soluble.
(i) This assertion, in fact, is a corollary of Theorem D. Indeed,
let
$E_{i}=H_{1}\cdots H_{i-1}H_{i+1} \cdots H_{t}$
for all $i= 1, \ldots , t$. Then, by
induction, $E_{1}, \ldots , E_{t}$ are generalized $\sigma$-soluble and, by the
Feit-Thompson theorem, $E_{1}=H_{2}\cdots H_{t}$ is soluble. Since $t >
2$, and evidently, the indeces $|G:E_{1}|$, $|G:E_{2}|$,
$|G:E_{3}|$ are pairwise $\sigma$-coprime, $G$ is generalized
$\sigma$-soluble by Corollary 1.6.
(ii) Assume that this assertion is falls and let $G$ be a
counterexample of minimal order. By hypothesis, $HA^{x}=A^{x}H$
for all $x\in G$ and all $A \in \cal H$ such that $(|H|, |A|)=1$.
(1) {\sl For some $i > 1$ and $x\in G$ we have $H_{i}^{x}\nleq N_{G}(H)$.}
Indeed, suppose that for all $i > 1$ and all $x\in G$ we have $H_{i}^{x}\leq
N_{G}(H)$. Then $$E=(H_{2})^{G} \cdots (H_{t})^{G}\leq N_{G}(H)$$
and so $H^{G}=H^{H_{1}E}=H^{H_{1}}\leq H_{1}$ is generalized
$\sigma$-soluble, a contradiction. Hence we have (1).
(2) {\sl $H^{G}R/R$ is generalized $\sigma$-soluble. }
It is clear that $ {\cal H}_{0}=\{H_{1}R/R, \ldots , H_{t}R/R\}$ is a
complete Hall $\sigma$-set of $G/R$ and $HR/R\leq H_{1}R/R$. Moreover,
$$(HR/R)(H_{i}R/R)^{xR}=HH_{i}^{x}R/R=H_{i}^{x}HR/R=(H_{i}R/R)^{xR}(HR/R) $$ for all
$ i > 1$ and all $xR\in G/R$. Therefore $HR/R$
is ${\sigma}$-semipermutable
in $G/R$
with respect to ${\cal H}_{0}$, so $(HR/R)^{G}=H^{G}R/R$ is generalized $\sigma$-soluble
by the choice of $G$.
(3) {\sl Every minimal normal subgroup of $G$ is not generalized $\sigma$-soluble. }
Indeed, if $R$ is generalized $\sigma$-soluble, then from the isomorphism $H^{G}R/R\simeq
H^{G}/H^{G}\cap R$ and Claim (2) we get that $H^{G}$ is
generalized $\sigma$-soluble, contrary to the choice of $G$.
{\sl Final contradiction for (ii).} Let $i > 1$. By hypothesis,
$HH_{i}^{x}=H_{i}^{x}H$ for all $x\in G$. Om the other hand, $O_{\sigma
_{1}\cup \sigma _{i}}(G)=1$ by Claim (3). Therefore $[H^{G}, H_{i}^{G}]= 1$ by Lemma 2.11, contrary to Claim (1). Hence
Assertion (ii) holds.
The theorem is proved.
In fact, Theorem C is a corollary of the following
{\bf Proposition 3.1.} {\sl If $G$ is $\sigma$-full, then
$ \frak{H}_{\sigma}=\frak{X}_{\sigma} $.}
{\bf Proof.} First we show $ \frak{H}_{\sigma}\subseteq \frak{X}_{\sigma} $.
Assume that this is false and let $G$ be a group of
minimal orderin $ \frak{H}_{\sigma}\setminus \frak{X}_{\sigma} $.
Then $|\sigma (G)| > 2$. By hypothesis, $G$ has a complete Hall
$\sigma$-set ${\cal H}=\{H_{1}, \ldots , H_{t} \}$ satisfying the condition
$H_{i}^{x}H_{j}^{y}=H_{j}^{y}H_{i}^{x}$ for all $x, y\in G$ and all
$i\ne j$.
We can assume without loss of
generality that $H_{i}$ is a $\sigma _{i}$-group for all $i=1, \ldots , t$.
Let $R$ be a minimal normal subgroup of $G$
(1) {\sl $G/R\in \frak{X}_{\sigma}$. Hence $R$ is the only minimal normal subgroup of $G$. }
First note that since $G\in \frak{H}_{\sigma}$, we have $G/R\in \frak{H}_{\sigma}$ by Lemma
2.2 and so
$G/R\in \frak{X}_{\sigma}$ by the choice of $G$. Now assume that $G$ has a minimal normal
subgroup $N\ne R$. Then $G/N\in \frak{X}_{\sigma}$, so from
the $G$-isomorphism $RN/N\simeq R$, Lemma 2.3
and the Jordan-H\"older theorem for
groups with operators \cite[A, 3.2]{DH} we get that $G\in \frak{X}_{\sigma}$, a
contradiction. Hence we have (1).
(2) {\sl $G$ is generalized $\sigma$-soluble. Hence $R$ is
a $R$ is a $\sigma _{i}\cup \sigma _{j}$-group for some $i, j$}
(This follows from Theorem C(i)).
(3) {\sl $O_{\sigma _{l}\cup \sigma _{k}}(G)= 1$ for all $k, l$ such that
$\sigma (R) \not \subseteq O_{\sigma _{i}, \sigma _{j}}$. } (This
follows from Claims (1) and (2)).
(4) $ \frak{H}_{\sigma}\subseteq \frak{X}_{\sigma} $.
Let
$\sigma _{k}\in \sigma (G)$ such that $i\ne k\ne j$.
First assume that $\sigma (R)=\{\sigma _{i}, \sigma _{j}\}$. Then
$R=(R\cap H_{i})(R\cap H_{j})$. Claim (1) implies that $O_{\sigma
_{i}\cup \sigma _{k}}(G)= 1$ and $O_{\sigma
_{j}\cup \sigma _{k}}(G)= 1$. Then, by Lemma 2.11, $H_{k}^{G}\leq C_{G}(H_{i})$ and
$H_{k}^{G}\leq C_{G}(H_{j})$. Therefore $H_{k}^{G}\leq C_{G}(R)$.
Therefore $G/C_{G}(R)$ is a $\sigma _{i}\cup \sigma _{j}$-group and so
$G\in \frak{X}_{\sigma}$ since $G/R\in \frak{X}_{\sigma}$ by Claim (1), a contradiction.
Hence we can assume that $\sigma (R)=\{\sigma _{i}\}$.
Let $l\ne k$ and $l\ne i\ne k$. Then $O_{\sigma _{l}\cup \sigma
_{k}}(G)= 1$ by Claim (3). Hence $[H_{l}^{G}, H_{k}^{G}]=1$ by Lemma 2.11.
Therefore from Claim (1) we get that $H_{l}\leq C_{G}(R)$ for all $l\ne
i$ and so $G/C_{G}(R)$ is a $\sigma _{i}$-group. Therefore we again get
that $G\in \frak{X}_{\sigma}$.
This contradiction completes the proof of the inclusion
$ \frak{H}_{\sigma}\subseteq \frak{X}_{\sigma} $.
Now we show that if $G\in \frak{X}_{\sigma}$, then
$VW=WV$ for every Hall $\sigma _{i}$-subgroup $V$,
every Hall $\sigma _{j}$-subgroup $W$ of $G$ and all $i\ne j$.
Assume that this is false and let $G$ be a counterexample of minimal
order. Then $|\sigma (G)| > 2$.
(a) {\sl $G=RVW$. Hence $G/R$ is a $\sigma _{i}\cup \sigma _{j}$-group}
Lemma 2.4 implies that the hypothesis holds for $G/R$, so the choice
of $G$ implies that $$(VR/R)(WR/R)=(WR/R)(VR/R)=VWR/R.$$ Hence $RVW$ is a subgroup
of $G$. Suppose that $RVW < G$. For each $k\ne i, j$, $H_{k}\cap R$ is
a Hall $\sigma _{k}$ subgroup of $RVW$.
Therefore the hypothesis holds for $RVW$ by Lemma 2.4 and so
$VW=WV$, a contradiction. Thus we have (a).
(b) {\sl $R$ is the unique minimal normal subgroup of $G$. Hence $R$ is a $p$-group for some
$p\in \sigma _{i}$. }
Indeed, suppose that $G$ has a minimal normal subgroup $N\ne R$. Then
$G/N$ is a $\sigma _{i}\cup \sigma _{j}$-group by Claim (1),
so $G\simeq G/R\cap N$ is a $\sigma _{i}\cup \sigma _{j}$-group. Hence
$|\sigma (G)|=2$, a contradiction. Therefore $R$ is the unique minimal normal
subgroup of $G$. Suppose that $R$ is non-abelian. Then $C_{G}(R)=1$,
since $C_{G}(R)$ is normal in $G$. Hence $G\simeq G/C_{G}(R)$ is $\sigma
$-biprimary, a contradiction. Thus we have (b).
(c) {\sl $R$ is a Sylow $p$-subgroup of $G$.}
Assume that $R$ is not a Sylow $p$-subgroup of $G$. Since $G=RVW$, by
Claim (5), but $VW\ne WV$, it follows that $R\nleq VW$. Hence for a Sylow
$p$-subgroup $P$ of $G$ we have $P\cap V=1=P\cap W$. Hence $P\nleq RVW=G$,
a
contradiction. Thus we have (c).
(d) $ \frak{X}_{\sigma}\subseteq \frak{H}_{\sigma} $.
In view of Claims (b) and (c), there is a
maximal subgroup $M$ of $G$ such that $G=R M$ and $M_{G}=1$.
Hence, for some $k$, we have $R=C_{G}(R)=O_{p}=H_{k}$ by \cite[A, 15.6]{DH}.
Then $\sigma _{k} \not \in \sigma (G/C_{G}(R))$, so $G/C_{G}(R))=G/R$ is a
$\sigma _{l}$-group for some $l$ since $G\in {\frak{G}}_{\sigma}$ by
hypothesis. But then $|\sigma (G)| \leq 2$.
This contradiction completes the proof of the inclusion
$ \frak{X}_{\sigma}\subseteq \frak{H}_{\sigma} $.
From Claims (c) and (d) we get that $ \frak{X}_{\sigma} =
\frak{H}_{\sigma} $.
The theorem is proved.
{\bf Proof of Theorem D.} The implications (i) $\Rightarrow$ (ii) and (ii) $\Rightarrow$
(i) directly follow from Theorem A.
Let $\frak{X}$
be the class of all $\sigma$-biprimary $\sigma$-soluble groups.
Then from Lemma
2.2 it follows that if $A$ is a group satisfying $A\simeq A^{*}/R$ for
some $A^{*}\leq A_{1}\times \cdots \times A_{t}$ and $A_{1}, \ldots , A_{t}\in \frak{X}$,
then $A$ satisfies also Condition (i).
Thus (iii) $\Rightarrow$ (i).
Now we show that (i) $\Rightarrow$ (iii). Assume that this is false and
let $G$ be a group of minimal order among the groups which satisfy
Condition (i) but do not satisfy Condition (iii). Since $G$ is $\sigma$-soluble, it has
a complete Hall $\sigma$-set $ {\cal H}=\{H_{1}, \ldots , H_{t}\}$ and, by hypothesis,
$H_{i}^{x}H_{j}^{y}=H_{j}^{y}H_{i}^{x}$ for all $x, y\in G$ and all
$i\ne j$. We can assume without loss of generality that
$H_{i}$ is a $\sigma _{i}$-group for all
$i=1, \ldots, t$. Then $ t > 2$ since otherwise Condition (iii) holds
for $G$.
Let $L$ be a minimal normal
subgroup of $G$.
Since $G$ is $\sigma$-soluble, $L$ is
$\sigma$-primary, $L$ is a $\sigma _{1}$-group say. Let
$\frak{F}$ be the class of groups $A$ which can be represented in the form
$A\simeq A^{*}/R$, where $A^{*}\leq A_{1}\times \cdots \times A_{t}$ for some
$A_{1}, \ldots , A_{t}\in \frak{X}$.
Lemma 2.2 implies that Condition (i) holds for $G/L$, so $G/L \in
\frak{F}$ by the choice of $G$. If $G$ has a minimal normal subgroup $N\ne
L$, then also we have $G/N \in \frak{F}$, so $G\simeq G/1 =G/L\cap N \in \frak{F}$
by Lemma
2.7. Thus $L$ is the unique minimal normal subgroup of $G$.
Now let $j\ne i\ne 1\ne j$, and let $A=H_{i}$ and $B=H_{j}$.
Then
$AB^{x}=B^{x}A$ for all $x\in G$. It is clear also that
$O_{\sigma _{i}\cup \sigma _{j}}(G)=1$. Then $[A^{G}, B^{G}]= 1$ by Lemma 2.10.
Now using Lemma 2.8, we get that $G\in \frak{F}$. This
contradiction completes the proof of the result.
{\bf Proof of Theorem E.} Assume that this is false and let $G$ be a counterexample of
minimal order. Let
${\cal H}=\{H_{1}, \ldots , H_{t} \}$.
Then $ t > 2$. We can assume without loss of generality that $H_{i}$ is a
non-identity $\sigma _{i}$-group for all $i=1,
\ldots , t$. Let $N_{i}=N_{G}(H_{i})$ for all $i=1,
\ldots , t$.
First we show that $G$ is not simple.
Assume that $G$ is a simple non-abelian group. Then
$|G:N_{i}|\ne 1$ is $\sigma$-primary for all
$i=1, \ldots , t$.
Let $|G:N_1|$
be a $\sigma_{i}$-number and $I_{0}=\{1, \ldots , i\} \setminus \{1, i\}$.
Then $I_{0} \ne \emptyset$ since $t > 2$. We can assume without loss of generality that
$i=t$.
If for some
$k\in I_{0}$, $|G:N_{k}|$ is not a $\sigma_{t}$-number, then
$G=N_{1}N_{k}$ and $|G:N_{1}\cap N_{k}|=|G:N_{1}||G:N_{k}|$ is a $\sigma
_{k}'$-number, so $H_{k}\leq N_{1}\cap N_{k}$ since $N_{1}\cap N_{k}\leq N_{G}(H_{k})$.
Therefore $$H_{k}^{G}= H_{k}^{N_{k}N_{1}}=H_{k}^{N_{1}}\leq N_{1},$$
so $G$ is not simple since $N_{1}\ne G$, a contradiction.
Therefore
$|G:N_{k}|$ is a $\sigma_{t}$-number for all $k=1, \ldots , t-1$.
Let $|G:N_{t}|$ be a $\sigma_{k}$-number and $j\in \{1, \ldots , t\}\setminus
\{k, t \}$. Then $G=N_{j}N_{t}$ and $H_{j}\leq N_{j}\cap N_{t}$,
Hence $H_{j}^{G}= H_{j}^{N_{j}N_{t}}=H_{j}^{N_{t}}\leq N_{t}$,
so $G$ is not simple.
Now let $R$ be any non-identity normal subgroup of $G$. Then
$\{H_{1}R/R, \ldots ,
H_{t}R/R \}$ is a complete Hall $\sigma$-set of
$G/R$. On the other hand, since $N_{i}R/R\leq
N_{G/R}(H_{i}R/R)$ and $|G:N_{i}|$ is $\sigma$-primary,
$N_{G/R}(H_{i}R/R)$ is $\sigma$-primary.
Therefore the hypothesis holds on $G/R$ for any minimal normal
subgroup $R$ of $G$. Hence the choice of $G$ implies that $G/R$ is
generalized
$\sigma$-soluble, and consequently $R$ is not generalized
$\sigma$-soluble.
It is clear that $\{R_{1}, \ldots ,
R_{t}\}$ is a complete Hall $\sigma$-set of
$R$, where $R_{i}=H_{i}\cap R$ for all $i=1, \ldots , t$.
Moreover, we have also $N_{i}\leq N_{G}(R_{i})$ and so from $|N_{i}R:N_{i}|=|R:R\cap N_{i}|$
we get that $|R:N_{R}(R_{i})|$ is $\sigma$-primary.
Therefore the hypothesis holds for $R$ and $R < G$ since $G$ is not
simple. Therefore $R$ is generalized $\sigma$-soluble by the choice of
$G$. This contradiction completes the proof of the result.
\end{document} | math | 41,233 |
\begin{equation}gin{document}
\title[Short Title]{Dissipation-Assisted Quantum Information Processing with Trapped Ions }
\author{A. Bermudez}
\affiliation{Institut f\"ur Theoretische Physik, Albert-Einstein Alle 11, Universit\"at Ulm, 89069 Ulm, Germany}
\author{T. Schaetz}
\affiliation{Albert-Ludwigs-Universit\"at Freiburg, Physikalisches Institut, Hermann-Herder-Strasse 3, 79104 Freiburg, Germany}
\author{M. B. Plenio}
\affiliation{Institut f\"ur Theoretische Physik, Albert-Einstein Alle 11, Universit\"at Ulm, 89069 Ulm, Germany}
\pacs{ 03.67.Lx, 37.10.Ty, 32.80.Qk}
\begin{equation}gin{abstract}
We introduce a scheme to perform dissipation-assisted quantum information processing in ion traps considering realistic decoherence rates, for example, due to motional heating. By means of continuous sympathetic cooling, we overcome the trap heating by showing that the damped vibrational excitations can still be exploited to mediate coherent interactions, as well as collective dissipative effects. We describe how to control their relative strength experimentally, allowing for protocols of coherent or dissipative generation of entanglement. This scheme can be scaled to larger ion registers for coherent or dissipative many-body quantum simulations.
\end{abstract}
\maketitle
Among the platforms for quantum computation (QC) and simulations (QS)~\cite{qc_qs}, trapped ions~\cite{qc_qs_ions} stand out as excellent small-scale prototypes, which have a well-defined roadmap towards large-scale devices based on micro-fabrication~\cite{microtraps, many_body_review}. The success of this technology depends on the impact of various sources of decoherence, such as the anomalous heating induced by the electric noise emanating from the trap electrodes~\cite{anomalous_heating}. The strong ion-ion couplings, required for scalable QC/QS, demand that the ions lie closer to the electrodes of these miniaturized traps, where the heating is critical and must be carefully considered. A strategy to minimize it is to cryogenically cool the setup~\cite{electrode_cooling}, or to clean the electrodes by laser ablation~\cite{laser_cleaning} or ion bombardment~\cite{ion_bombardment_cleaning}. Although these approaches are promising, a substantial residual noise still exists. We propose to minimize it by applying sympathetic laser cooling {\it continuously} during the whole QC/QS protocol.
Sympathetic cooling requires active laser cooling of a subset of ions, and passive cooling of the remaining ions by Coulomb interaction. This technique may overcome the motional heating~\cite{sympathetic_kielpinski,gs_sympathetic_cooling_focusing}, and has already been implemented between sequential gates for QC~\cite{sc_quantum_logic}. However, the larger heating rates of surface traps would require to cool also during the gates. There are different schemes along these lines: {\it (i)} In the absence of fluctuating electric gradients, interactions can be mediated by vibrational modes robust to the heating, while continuously cooling the remaining modes~\cite{sympathetic_kielpinski}. {\it (ii)} By using far-detuned state-dependent forces~\cite{ms}, the mediated interactions do not rely on the motional coherence, and can thus withstand a heating/cooling that is considerably weaker than the interactions. {\it (iii)} For ground-state cooled crystals, resolved-sideband cooling may provide a dissipative force that improves the success/fidelity of protocols that are shorter than the inverse of the heating rate~\cite{Beige}. Unfortunately, these requirements are not met in current surface traps: {\it (i)} Since ions lie close to the electrodes, electric gradients prevent the isolation of robust modes. {\it (ii-iii)} Heating rates in room-temperature setups ($1$ phonon/ms~\cite{phonon_hopping}) coincide with the above protocols speed~\cite{ms,Beige}.
In this work, we propose a {\it dissipation-assisted protocol} based on an always-on sympathetic cooling that overcomes the anomalous heating for surface traps. We identify regimes where the sympathetically-cooled vibrational modes can be used as mediators of both coherent interactions and collective dissipation. Since we only require Doppler cooling, this proposal can be applied to larger registers for QC/QS.
\begin{equation}gin{figure}
\centering
\includegraphics[width=1\columnwidth]{fig1.pdf}
\caption{ {\bf (a)} Coulomb crystal in a surface trap. The laser-cooled ions (red) assist the coherent/dissipative dynamics of the spins of the physical ions (blue). Both isotopes may be stored in the same individual minima without affecting the resultant lattice geometry. {\bf (b)} Proof-of-principle experiment with three ions in a conventional rf-trap. The red arrows correspond to a standing wave providing the Doppler cooling of the central ion, whereas the blue arrows lead to a running wave tuned to the axial red sideband of the outer ions. }
\label{fig_scheme}
\end{figure}
{\it The model.--} We consider an array of two types of ions $\{\sigma,\tau\}$ confined in a radio-frequency (rf) trap (Fig.~\ref{fig_scheme}{\bf (a)}). Two hyperfine ground-states $\{\ket{{\uparrow}},\ket{{\downarrow}}\}$ of the $\sigma$-ions provide the playground for QC/QS, whereas the $\tau$-ions act as an auxiliary gadget to sympathetically cool the crystal. In particular, the $\tau$-ions are Doppler cooled by using a standing wave red-detuned from a dipole-allowed transition with decay rate $\Gamma_{\tau}$~\cite{comment_sw_cooling}, while the $\sigma$-ions are subjected to a spin-phonon coupling obtained from a two Raman beam in a traveling-wave configuration~\cite{wineland_review}. When the laser cooling is strong, the atomic degrees of freedom of the $\tau$-ions can be traced out~\cite{sup_mat}, and one obtains a master equation for the reduced dynamics of the $\sigma$-spins and the collective vibrations
\begin{equation}gin{equation}
\label{laser_cooling}
\dot{\mu}=-{\rm i}[H_{\sigma}+H_{\rm ph}+V_{\sigma}^{{\rm ph}},\mu]+\mathcal{\tilde{D}}(\mu),\hspace{2ex}\mu={\rm Tr}_{\tau,{\rm atomic}}\{\rho\}.
\end{equation}
Here, we have introduced the bare spin and phonon Hamiltonians $H_{\sigma}=\frac{1}{2}\sum_{i}\omega_{0}^{\!\sigma}\sigma_i^z$, $H_{\rm ph}=\sum_{n}\omega_{n}b_n^{\dagger}b_n,$ where $\omega_{0}^{\!\sigma}$ and $\omega_n$ are the electronic and longitudinal normal-mode frequencies~\cite{comment}. Additionally, $\sigma_i^z=\ket{{\uparrow_i}}\bra{{\uparrow_i}}-\ket{{\downarrow_i}}\bra{{\downarrow_i}}$, and $b_n^{\dagger},b_n^{\phantom{\dagger}}$ are the operators that create-annihilate phonons. The two crucial ingredients in~\eqref{laser_cooling} for our dissipation-assisted protocol are:
{\it (i)} A {\it spin-phonon coupling}, provided by the Raman beams tuned to the so-called red sideband~\cite{sup_mat}, leads to
\begin{equation}gin{equation}
\nonumber
V_{\sigma}^{{\rm ph}}\!=\!\sum_{i,n}\mathcal{F}^{\sigma}_{in}\sigma_i^+b_n\end{equation}^{-{\rm i}\omega_{\sigma}t}+\text{H.c.},\hspace{1ex} \mathcal{F}^{\sigma}_{in}=\textstyle{\frac{{\rm i}\Omega_{{\sigma}}}{2}\eta^{\sigma}_n\mathcal{M}_{in}\end{equation}^{{\rm i}\phi_{i}}},
\end{equation}
where $\sigma_i^+=\ket{{\uparrow_i}}\bra{{\downarrow_i}}$, and the sum is extended to all $\sigma$-ions and normal modes. Here, $\Omega_{\sigma}$ is the Rabi frequency of the Raman beams, $\omega_{\sigma} ({\bf k}_{\sigma})$ is its frequency (wavevector), and $\phi_i={\bf k}_{\sigma}\cdot{{\bf r}_i^{\sigma}}$ is defined in terms of the ion position ${{\bf r}_i^{\sigma}}$. The Lamb-Dicke parameter $\eta^{\sigma}_n={\bf k}_{\sigma}\cdot{\bf e}_{\rm d}/\sqrt{2m_{\sigma}\omega_n}$ describes the laser coupling to the $n$-th normal mode, where the $i$-th ion displacement along the direction ${\bf e}_{\rm d}$ is given by $\mathcal{M}_{in}$, and $m_{\sigma}$ is the ion mass.
{\it (ii)} An {\it effective phonon damping}, provided by the sympathetic Doppler cooling~\cite{sup_mat}, which can be described by
\begin{equation}gin{equation}
\nonumber
\mathcal{\tilde{D}}(\mu)\!=\!\sum_{n}\!\begin{itemize}g\{\Gamma_{n}^+\!(b_n^{\dagger}\mu b_n^{\phantom{\dagger}}\!-b_n^{\phantom{\dagger}}b_n^{\dagger}\mu)\!+\!\Gamma_{n}^-\!(b_n^{\phantom{\dagger}}\mu b_n^{\dagger}\!-b_n^{\dagger}b_n^{\phantom{\dagger}}\mu)\!\begin{itemize}g\}\!+\!\text{H.c.}
\end{equation}
Here, we have introduced Lorentzian-shaped cooling/heating couplings, which allow for an experimental control fo the damping of the vibrational modes, and have the expression $\Gamma_{n}^{\mp}=\sum_l(\textstyle\frac{1}{2}\Omega_{{\tau}}\eta^{\tau}_{n}\mathcal{M}_{ln})^2/(\textstyle\frac{1}{2}\Gamma_{\tau}+{\rm i}(-\Delta_{\tau}\pm\omega_n))$, where we sum over all the $\tau$-ions. In these expressions, we have introduced the laser Rabi frequency $\Omega_{\tau}$, its detuning $\Delta_{\tau}$, and its wavevector ${\bf k}_{\tau}$ that determines $\eta^{\tau}_n={\bf k}_{\tau}\cdot{\bf e}_{\rm p}/\sqrt{2m_{\tau}\omega_n}$.
The master equation~\eqref{laser_cooling} describes an array of spins coupled to a set of damped vibrational modes. The idea now is to use the quanta of these modes, i.e. the phonons, as mediators of a coherent spin-spin interaction. However, in addition to the coherent dynamics, the phonons also provide an indirect coupling to the electromagnetic reservoir leading to some collective dissipation on the spins. Our goal is to find suitable regimes where these collective effects still allow for QC/QS. To guide this search, note that the two-qubit gates implemented in different laboratories~\cite{gates_review} use nearly-resonant spin-dependent forces, and rely on the motional coherence to suppress the residual spin-phonon entanglement. Since the motional coherence is absent in our case, we must work in the far off-resonant regime~\cite{ms,porras_qs}, where $|\mathcal{F}_{in}^{\sigma}|\ll|\delta_n|\ll\omega_n$, such that $\delta_n=\omega_{\sigma}-(\omega_0^{\sigma}-\omega_n)$. In this regime, motional excitations by the spin-phonon coupling are negligible. We identify below the additional conditions to tailor the coherent/dissipative phonon-mediated processes in presence of laser cooling.
{\it Collective Liouvillian.--} For the values considered below, the laser-cooling rates reach $W_n\approx 10^{-2}\omega_n$. In this case, the cooling is very strong, and the vibrations reach the steady state very fast. Hence, we can apply the theory of Schrieffer-Wolff (SW) transformations for open systems~\cite{open_sw} to trace out the phonons from Eq.~\eqref{laser_cooling}, and obtain an effective Liouvillian
\begin{equation}gin{equation}
\label{eff_me}
\dot{\mu}_{\sigma}=\mathcal{L}_{\rm eff}(\mu_{\sigma})=-{\rm i}[H_{\rm eff},\mu_{\sigma}]+\mathcal{D}_{\rm eff}(\mu_{\sigma}),
\end{equation}
where $\mu_{\sigma}={\rm Tr}_{\rm ph}\{\mu\}$. Here, the coherent Hamiltonian is
\begin{equation}gin{equation}
\nonumber
H_{\rm eff}=\sum_{i>j}\begin{itemize}g(J_{ij}^{\rm eff}\sigma_i^+\sigma_j^-+\text{H.c.}\begin{itemize}g)+\sum_{in}\textstyle\frac{1}{2} B_{in}^{\rm eff}\sigma_i^z,
\end{equation}
which contains the phonon-mediated interactions of strength $J_{ij}^{\rm eff}$, which describe processes where a phonon is virtually created, and then absorbed elsewhere in the chain. These interactions can be used to implement gates for QC, or to explore spin models for QS. Additionally, we also find an effective ac-Stark shift, which can be interpreted as an effective magnetic field $B_{in}^{\rm eff}$, arising from the processes where the phonon is created and absorbed by the same ion. Note that the same virtual phonon exchange also introduces an indirect dissipation in~\eqref{eff_me}
\begin{equation}gin{equation}
\nonumber
\begin{equation}gin{split}
\mathcal{D}_{\rm eff}(\mu_{\sigma})=\sum_{i,j}{\Gamma^{\prime}}_{ij}^{\rm eff}&(\sigma_i^+\mu_{\sigma}\sigma_j^--\sigma_j^-\sigma_i^+\mu_{\sigma}+\text{H.c.})\\
+\sum_{i,j}(\Gamma_{ij}^{\rm eff}+{\Gamma'}_{ij}^{\rm eff})&(\sigma_i^-\mu_{\sigma}\sigma_j^+-\sigma_j^+\sigma_i^-\mu_{\sigma}+\text{H.c.}),
\end{split}
\end{equation}
where $\Gamma^{\rm eff}_{ij},\Gamma'^{\rm eff}_{ij}$ are the strengths of the collective processes of spontaneous and stimulated dissipation, respectively.
To find the correct regime for QC/QS purposes, we must compare the time-scales derived from the expressions
\begin{equation}gin{equation}
\nonumber
\label{parameters}
\begin{equation}gin{split}
J_{ij}^{\rm eff}&\!=\!-\!\sum_n\!\frac{\mathcal{F}_{in}^{\sigma}(\mathcal{F}_{jn}^{\sigma})^*}{\tilde{\delta}_n^2+W_n^2}\tilde{\delta}_n,\hspace{2.5ex} B_{in}^{\rm eff}\!=\!-\!\frac{\mathcal{F}_{in}^{\sigma}(\mathcal{F}_{in}^{\sigma})^*}{\tilde{\delta}_n^2+W_n^2}\tilde{\delta}_n(2\begin{align}r{n}_n+1),\\
\Gamma_{ij}^{\rm eff}&\!=\!+\!\sum_n\!\frac{\mathcal{F}_{in}^{\sigma}(\mathcal{F}_{jn}^{\sigma})^*}{\tilde{\delta}_n^2+W_n^2}W_n,\hspace{2.ex}{\Gamma'}_{ij}^{\rm eff}\!=\!\sum_n\!\frac{\mathcal{F}_{in}^{\sigma}(\mathcal{F}_{jn}^{\sigma})^*}{\tilde{\delta}_n^2+W_n^2}W_n\begin{align}r{n}_n.
\end{split}
\end{equation}
Here, the laser cooling leads to the effective cooling rates $
W_n={\rm Re}\{\Gamma^{-}_n-\Gamma^{+}_n\}$ that damp the ion vibrations, and to a Lamb-type shift of the vibrational frequencies leading to $\tilde{\delta}_n=\delta_n+{\rm Im}\{\Gamma_n^+-(\Gamma_n^{-})^*\}$. Additionally $\begin{align}r{n}_n={\rm Re}(\Gamma^{+}_n)/W_n$ are the mean phonon numbers in the steady state of the laser cooling. From these expressions, it is clear that by tuning the ratio $
\mathcal{R}_n=W_n (\begin{align}r{n}_n+1)/|\tilde{\delta}_n|$, we control if the spin interactions prevail over the dissipation $\mathcal{R}_n\ll1$, or vice versa $\mathcal{R}_n\gg1$.
{\it Coherent and dissipative generation of entanglement.--} We consider the simplest scenario to test our scheme: a three-ion chain in a linear Paul trap (Fig.~\ref{fig_scheme}{\bf (b)}). To use realistic parameters, we consider $^{25}$Mg$^{+}$-$^{24}$Mg$^{+}$-$^{25}$Mg$^{+}$, and set the axial trap frequency to $\omega_z/2\pi=4.1$\hspace{0.2ex}MHz, which is possible by optimizing the trap voltages. The dipole-allowed transition $\ket{g}=\ket{3S_{1/2}}\leftrightarrow\ket{e}=\ket{3P_{1/2}}$ of $^{24}$Mg$^{+}$, which is characterized by $\lambda_{\tau}\approx 280.3$nm and $\Gamma_{\tau}/2\pi\approx 41.4$\hspace{0.2ex}MHz, shall be used for continuous sympathetic cooling. By applying an external magnetic field, we can encode the spins in a couple of Zeeman sub-levels $\ket{F,M}$ of the ground-state manifold of $^{25}$Mg$^{+}$, e.g. $\ket{{\uparrow}}=\ket{2,2}$ and $\ket{{\downarrow}}=\ket{3,3}$. This leads to a resonance frequency of $\omega^{\sigma}_0/2\pi\approx 1.79$\hspace{0.2ex}GHz, and a negligible decay rate of $\Gamma_{\sigma}/2\pi\approx 10^{-14}$Hz. Finally, a pair of off-resonant lasers drive the axial red-sideband through an excited state in the $3P_{3/2}$ manifold of $^{25}$Mg$^{+}$, such that $\eta_1^{\sigma}\approx 0.16$.
The isotopic mass ratio $m_{\tau}/m_{\sigma}\approx 0.96$ implies that the axial vibrational modes are almost unchanged with respect to the homogeneous chain, $\omega_n/2\pi\approx\{4.1,7.1,10.1\}$\hspace{0.2ex}MHz. To attain a wide range of values for the ratio $\mathcal{R}_n$, we tune the Raman lasers closer to the highest-frequency mode, the so-called egyptian mode, such that the detunings are $\delta_n/2\pi\in\{6.2,3.2,0.3\}$\hspace{0.2ex}MHz. Note that due to the large detuning from the remaining modes, the collective effects will be mediated by the egyptian mode. To sympathetically cool it, we place the cooling isotope at the middle of the chain, such that it coincides with the node of a standing-wave laser~\cite{comment_sw_cooling}. This laser has frequency that is red-detuned from the transition, and we set the detuning to be $\Delta_{\tau}=-\Gamma_{\tau}/2$. This leads to a steady-state mean phonon number $\begin{align}r{n}_3=0.65$ independent of the standing-wave Rabi frequency. Therefore, we can modify the Rabi frequency $0.2\leq\Omega_{\tau}/\Gamma_{\tau}\leq 2$ in order to control the cooling rate $W_3$, and thus the ratio $\mathcal{R}_3$, thus exploring regimes of either dominant dissipation or interactions. We remark that the laser used for cooling $^{24}$Mg$^{+}$ will be highly detuned from the cooling transition of $^{25}$Mg$^{+}$ (i.e. $\Delta/2\pi\approx 2.7$\hspace{0.2ex}THz), such that the induced decay rate for the considered regime fulfills $\Gamma_{\tau}(\Omega_{\tau}/\Delta)^2/(2\pi)\leq10^{-2}$Hz. Therefore, this laser only contributes with off-resonant ac-Stark shifts that shall be considered later on. We now explore two possible applications.
{ (a)} Coherent generation of entanglement: Our goal is to use the coherent phonon-mediated interaction in Eq.~\eqref{eff_me} to generate entanglement between the $^{25}$Mg$^{+}$ ions. By setting $\Omega_{\tau}=0.15\Gamma_{\tau}$, we obtain a cooling rate of $W_3/\omega_3\approx 4.3\cdot10^{-3}$, such that $\mathcal{R}_3\approx 7\cdot 10^{-3}$, and the Hamiltonian part of the Liouvillian~\eqref{eff_me} thus dominates. Initializing the spin state in $\ket{\psi_{\sigma}(0)}=\ket{{\uparrow_1\downarrow_3}}$, and setting $\Omega_{\sigma}\eta_3^{\sigma}\approx 10W_{3}$, such that the distance between the ions is an integer multiple of the effective Raman wavelength, we obtain the Bell state $\ket{\psi_{\rm B}}=\frac{1}{\sqrt{2}}(\ket{{\uparrow_1\downarrow_3}}-{\rm i}\ket{{\downarrow_1\uparrow_3}})$
for $t_{\rm f}\approx 4$\hspace{0.2ex}ms (Fig.~\ref{fig_ent}{\bf (a)}). In the numerical simulations, we have considered a realistic heating rate for macroscopic rf-traps of $\Gamma_{\rm ah}\approx 0.1$phonon/ms, by substituting $\Gamma_{n}^+\to\Gamma_{n}^++\Gamma_{\rm an}$ in the dissipator of~\eqref{laser_cooling}. From this figure, we observe that, even if the process is slower than the usual gates~\cite{gates_review}, it prevails over the phonon-mediated decoherence leading to errors as low as $\epsilon_{\rm B}\sim 10^{-2}$. Note that such errors are not sufficient for fault-tolerance QC, which require $\epsilon_{\rm ft}\sim 10^{-2}$-$10^{-4}$. On the one hand, we can achieve lower error rates by working with larger detunings. On the other hand, this leads to slower gates, which require an additional scheme to decouple from other sources of decoherence that shall be introduced below. Finally, for the anomalous heating rates in micro-fabricated surface traps $\Gamma_{\rm ah}\approx 1$phonon/ms, the same parameters lead to errors $\epsilon_{\rm B}\approx 2\cdot 10^{-2}$ for $t_{\rm f}\approx 5$\hspace{0.2ex}ms, which illustrates the robustness of our scheme with respect to motional heating. Let us also advance that our protocol might be scaled directly to many ions for QS~\cite{sup_mat}, which do not require such small error rates.
\begin{equation}gin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{fig2.pdf}
\caption{ {\bf (a)} In the main panel, we show the coherent flip-flop dynamics for the three-ion setup, when the red sideband is tuned close to the highest-frequency vibrational mode. The solid lines represent the spin populations ($P_{\uparrow,1}$: blue, $P_{\uparrow,3}$: red) given by the original Liouvillian~\eqref{laser_cooling}, while the symbols ($P_{\uparrow,1}$: circles, $P_{\uparrow,3}$: squares) correspond to the effective description~\eqref{eff_me}. In the left lower panel, the fidelity $\mathcal{F}_{\rm B}=|\langle\psi_{\rm B}|\mu_{\sigma}|\psi_{\rm B}\rangle|$ with the Bell state $\ket{\psi_{\rm B}}=\frac{1}{\sqrt{2}}(\ket{{\uparrow_1\downarrow_3}}-{\rm i}\ket{{\downarrow_1\uparrow_3}})$, for a single flip-flop exchange is displayed. An optimization of the fidelity for different gate times $t_{\rm B}$ is shown in the right lower panel. In both cases, green solid lines represent the complete Liouvillian~\eqref{laser_cooling}, whereas the stars follow from the effective description~\eqref{eff_me}. {\bf (b)} In the upper panel, the dissipative dynamics under equation~\eqref{laser_cooling} ($P_{\uparrow,1}$: blue solid line, $P_{\uparrow,3}$: red solid line) and equation~\eqref{eff_me} ($P_{\uparrow,1}$: circles, $P_{\uparrow,3}$: squares) is shown. In the lower panel, we display the fidelity $\mathcal{F}_-=|\langle\phi_{-}|\mu_{\sigma}|\phi_{-}\rangle|$ with the Bell state $\ket{\phi_{-}}=\frac{1}{\sqrt{2}}(\ket{{\uparrow_1\downarrow_3}}-\ket{{\downarrow_1\uparrow_3}})$. Again, we use green solid lines for the complete Liouvillian~\eqref{laser_cooling}, and stars for the effective description~\eqref{eff_me}.}
\label{fig_ent}
\end{figure}
{ (b)} Dissipative generation of entanglement: A different possibility would be to exploit the collective dissipation in Eq.~\eqref{eff_me} to generate entanglement in the steady state. The idea is to set $\Omega_{\tau}=2\Gamma_{\tau}$, such that the dissipative part of the Liouvillian~\eqref{eff_me} becomes dominating $\mathcal{R}_3\approx 2.3$, and we can exploit a superradiant/subradiant phenomenon~\cite{superradiance}. By controlling the ion-ion distance with respect to the Raman wavelength such that ${\bf k}_{\sigma}\cdot ({\bf r}_1^{\sigma}-{\bf r}_3^{\sigma})=p\pi$, where $p\in\mathbb{Z}$, the decay rates fulfill $\Gamma_{11}^{\rm eff}=\Gamma_{33}^{\rm eff}=\Gamma_{\rm eff}$, and $\Gamma_{13}^{\rm eff}=\Gamma_{31}^{\rm eff}=(-1)^p\Gamma_{\rm eff}$ (equally for ${\Gamma^{\prime}}_{ij}^{\rm eff}$). In this limit, the dissipator in~\eqref{eff_me} can be written as
\begin{equation}gin{equation}
\nonumber
\mathcal{D}_{\rm eff}(\mu_{\sigma})=L_-\mu_{\sigma}L_-^{\dagger}+L_+\mu_{\sigma}L_+^{\dagger}-L_-^{\dagger}L_-\mu_{\sigma}-L_+^{\dagger}L_+\mu_{\sigma}+\text{H.c.},
\end{equation}
where we have introduced the collective jump operators $L_-=\sqrt{\Gamma_{\rm eff}(\begin{align}r{n}_3+1)}(\sigma_1^-+(-1)^p\sigma_3^-)$, and $L_+=\sqrt{\Gamma_{\rm eff}\begin{align}r{n}_3}(\sigma_1^++(-1)^p\sigma_3^+)$. One can check that the symmetric/antisymmetric Bell states $\ket{\phi_{\pm}}=\frac{1}{\sqrt{2}}(\ket{{\uparrow_1\downarrow_3}}\pm\ket{{\downarrow_1\uparrow_3}})$ are dark states of these jump operators for $p$ odd, or $p$ even, respectively. These are the so-called sub-radiant decay channels~\cite{superradiance_review}, which allows us to get a mixed stationary state that is partially entangled. In particular, starting
from $\ket{\psi_{\sigma}(0)}=\ket{{\uparrow_1\downarrow_3}}$ for $p$ even, and $\Omega_{\sigma}\eta_3^{\sigma}\approx W_{3}$, we obtain a decoherence-free entangled steady-state $\ket{\phi_-}$ for $t\gg t_{\rm ss}\approx50$\hspace{0.2ex}$\mu$s with fidelities around 30$\%$ (Fig.~\ref{fig_ent}{\bf (b)})\cite{comment_entanglement} . Note that this {\it phononic subradiance} is not affected by limitations in the ratio of the ion-ion distance with respect to the wavelength of the emitted light. The collective nature of the vibrations that mediate the subradiance allows to surpass the limitations of the pioneering trapped-ion experiments~\cite{sup_ions}. We also note that the ultimate limit of $50\%$ cannot be achieved due to the thermal contribution to Eq.~\eqref{eff_me}. However, schemes originally formulated for cavities~\cite{steady_entanglement} can be adapted for our trapped-ion setting to reach unit fidelities.
Let us emphasize that, although we have considered a particular example, the scheme is also applicable to other ion species. For the regime of dominant dissipation, any ion will work equally well. Conversely, for dominant coherent interactions, the required strong sympathetic-cooling strengths and detunings are likely to be optimized for crystals with two light isotopes. At this point, it is also worth commenting that the strong rates provided by standing-wave laser cooling are required to obtain the target states in time-scales which are not prohibitively large. In light of the results shown in Fig.~\ref{fig_ent}, the regime of coherent interactions necessarily requires standing-wave cooling. Conversely, the regime of leading dissipation is faster, and may also work with the more standard traveling-wave cooling. Let us note, however, that the experiments~\cite{slowly_moving_standing_wave} show that standing-wave cooling with a precise positioning of the ions with respect to the standing wave is possible.
{\it Sources of noise.--} In addition to the motional heating, other sources of noise become relevant for the time-scales of the above protocols $0.1$-$10$\hspace{0.2ex}ms. In fact, fluctuating magnetic fields and laser intensities, together with thermal noise, lead to the dephasing term $H_{\rm n}=\sum_i\frac{1}{2}\begin{itemize}g(\sum_nB^{\rm eff}_{in}+F_i(t)\begin{itemize}g)\sigma_i^z$.
Here, $B^{\rm eff}_{in}$ in Eq.~\eqref{eff_me} introduces noise via fluctuations over the phonon steady state, and $F_i(t)$ is a random process that models the noise of external magnetic fields, or uncompensated ac-Stark shifts. This random noise, which typically has a short correlation time $\tau_{\rm c}$, leads to a dephasing rate $\Gamma_{\rm d}/2\pi\sim$0.1-1$\,$kHz~\cite{sup_mat}.
For any practical implementation of the proposed protocol, this dephasing should be carefully considered. The standard approach for prolonging the coherence of a system consists of a sequence of refocusing pulses, a technique known as pulsed dynamical decoupling~\cite{pdd}. Another approach, known as continuous dynamical decoupling, produces a similar effect by continuous drivings~\cite{cdd_trapped_ions, cdd_ions}. As recently demonstrated experimentally~\cite{cdd_ions_exp}, this techniques allows to implement robust 2-qubit gates exploiting a single sideband~\cite{cdd_ions}. As a byproduct, we show that this tool
allows for slower gates, and thus smaller errors in principle. Moreover, it also provides a new gadget to tailor the collective Liouvillian~\eqref {eff_me}.
We apply a continuous driving resonant with the spins, such that the bare spin Hamiltonian reads
$
H_{\sigma}=\textstyle\frac{1}{2}\sum_{i}\omega^{\sigma}_0\sigma_i^z+\left(\Omega_{\rm d}\sigma_i^+\cos(\omega_{\rm d}t)+\text{H.c.}\right)$ with $\omega_{\rm d}=\omega^{\sigma}_0$. In this regime, a modified SW transformation leads to $\dot{\mu}_{\sigma}=\tilde{\mathcal{L}}_{\rm eff}(\mu_{\sigma})$, where
\begin{equation}gin{equation}
\label{eff_me_ising}
\tilde{\mathcal{L}}_{\rm eff}(\mu_{\sigma})=-{\rm i}[\tilde{H}_{\rm eff}+\tilde{H}_{\rm n},\mu_{\sigma}]+\tilde{\mathcal{D}}_{\rm eff}(\mu_{\sigma})+\tilde{\mathcal{D}}_{\rm n}(\mu_{\sigma}).
\end{equation}
In the limit of a strong driving~\cite{sup_mat}, the Hamiltonian above corresponds to an interacting Ising model
\begin{equation}gin{equation}
\nonumber
\tilde{H}_{\rm eff}=\sum_{i>j}\tilde{J}_{ij}^{\rm eff}\sigma_i^x\sigma_j^x+\sum_{in}\textstyle\frac{1}{2} \Omega_{\rm d}\sigma_i^x,
\end{equation}
and we obtain a collective phonon-mediated dephasing
\begin{equation}gin{equation}
\nonumber
\tilde{\mathcal{D}}_{\rm eff}(\mu_{\sigma})=\sum_{i,j}\tilde{\Gamma}_{ij}^{\rm eff}(2\sigma_i^x\mu_{\sigma}\sigma_j^x-\sigma_j^x\sigma_i^x\mu_{\sigma}-\mu_{\sigma}\sigma_j^x\sigma_i^x).
\end{equation}
Assuming that the surface-trap array is designed such that ${\bf k}_{\rm L}\cdot{\bf r}_i^{\sigma}=2\pi p$ with $p\in\mathbb{Z}$, we emphasize that the interaction strengths and dissipation rates,
\begin{equation}gin{equation}
\nonumber
\tilde{J}_{ij}^{\rm eff}\!=\!-\!\sum_n\!\frac{|\mathcal{F}_{in}^{\sigma}\mathcal{F}_{jn}^{\sigma}|}{\tilde{\delta}_n^2+W_n^2}\frac{\tilde{\delta}_n}{2}, \hspace{2ex}
\tilde{\Gamma}_{ij}^{\rm eff}\!=\!\sum_n\!\frac{|\mathcal{F}_{in}^{\sigma}\mathcal{F}_{jn}^{\sigma}|}{\tilde{\delta}_n^2+W_n^2}\frac{W_n}{2}(\begin{align}r{n}_n+\textstyle\frac{1}{2}),
\end{equation}
lead to a very similar control parameter $
\tilde{\mathcal{R}}_n=W_n (\begin{align}r{n}_n+\textstyle\frac{1}{2})/|\tilde{\delta}_n|$. Accordingly, under the same assumptions as we considered above, we can interpolate between regimes where the coherent Ising interactions dominate over the collective dephasing, or vice versa.
In addition to the new range of possibilities offered by this collective Liouvillian, note that the dephasing noise terms contribute to the new Liouvillian~\eqref{eff_me_ising} with
\begin{equation}gin{equation}
\nonumber
\tilde{H}_{\rm n}=\sum_i\frac{1}{2}\tilde{\Omega}_{\rm d}\sigma_i^x, \hspace{2ex}\tilde{\mathcal{D}}_{\rm n}(\mu_{\sigma})=\sum_{i}\sum_{\alpha=y,z}\textstyle\frac{1}{2}\tilde{\Gamma}_{\rm d}\left(\sigma_i^{\alpha}\mu_{\sigma}\sigma_i^{\alpha}-\mu_{\sigma}\right),
\end{equation}
where we have assumed that the noise is local, and introduced $\tilde{\Omega}_{\rm d}=({B}_{jn}^{\rm eff}\tilde{\delta}_n\tau_{\rm c}+\Gamma_{\rm d})/2\Omega_{\rm d}\tau_{\rm c}$, and $\tilde{\Gamma}_{\rm d}=\Gamma_{\rm d}/(\Omega_{\rm d}\tau_{\rm c})^2$ in the limit of a strong driving $\Omega_{\rm d}\tau_{\rm c}\gg 1$~\cite{sup_mat}.
According to the above constraints, these noisy terms are suppressed by a sufficiently-strong driving. To be more specific, for noise correlation times on the order of $\tau_{\rm c}=10^{-2}/\Gamma_{\rm d}$ and detunings $\delta_n/2\pi\sim$0.1-1\hspace{0.2ex}MHz, it suffices to apply drivings with $\Omega_{\rm d}/2\pi\approx 10$\hspace{0.2ex}MHz to reduce the noise by more than two orders of magnitude. Therefore, we can decouple from the noise efficiently, while preserving the collective part of the Liouvillian for QC/QS. In contrast to Fig.~\ref{fig_ent}{\bf (a)}, the new Liouvillian~\eqref{eff_me_ising} allows for the coherent generation of all four Bell states.
{\it Conclusions.--} We have proposed a scheme based on sympathetic cooling to overcome the anomalous heating in surface traps, while allowing for QC/QS. The sympathetic cooling becomes a tool to tailor the collective effects of the Liouvillian. By controlling a single parameter, namely the laser-cooling power, we have shown how the Liouvillian interpolates between regimes of dominating coherent interactions/collective dissipation, both of which allow for generation of entanglement. Moreover, we have this control can also be exploited for coherent/dissipative many-body QS.
{\it Acknowledgements.--} This work was supported by PICC, DFG (SCHA973/1-6), and by the Alexander von Humboldt Foundation. A.B. thanks FIS2009-10061, and QUITEMAD. We thank A. Albrecht and J. Almeida for useful discussions.
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N. Timoney, et {\it al.}, \href{http://www.nature.com/nature/journal/v476/n7359/full/nature10319.html}{Nature {\bf 476,} 185 (2011)}; A. Noguchi, S. Haze, K. Toyoda, and S. Urabe, \href{http://prl.aps.org/abstract/PRL/v108/i6/e060503}{Phys. Rev. Lett. {\bf 108,} 060503 (2012).}
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\vspace*{150ex}
\hypertarget{sm}{\section{Supplemental Material: \\ {\it Dissipation-Assisted Quantum Information Processing with Trapped Ions }}}
{\it Dissipative ion crystals.--} We consider a collection of atomic ions confined in radio-frequency traps~\cite{wineland_review_sm}. In particular, we will explore dissipation-assisted protocols for an array of ${N}_{\rm t}=N_{\sigma}+N_{\tau}$ trapped ions in equilibrium positions $\{{\bf r}_i^{\sigma},{\bf r}_l^{\tau}\}$. The geometry of the array will depend on the particular trap under consideration. For micro-fabricated surface traps, we will consider arbitrary geometries (Fig.~\ref{fig_scheme}{\bf (a)}), whereas for the more standard rf-traps, we restrict to one-dimensional chains (Fig.~\ref{fig_scheme}{\bf (b)}). A fraction of the ions, $N_{\tau}$, is laser cooled via a dipole-allowed transition $\ket{g}\leftrightarrow\ket{ e}$ with frequency ${\omega}_0^{\tau}$ and decay rate $\Gamma_{\tau}$, providing the sympathetic cooling of the remaining ions, $N_{\sigma}$. Two hyperfine ground-states of the $\sigma$-ions $\{\ket{{\uparrow}},\ket{{\downarrow}}\}$ form the spins/qubits for QS/QC purposes, such that their decay rate is negligible, and their energy spacing is $\omega_0^{\sigma}$ ($\hbar=1$). To minimize the action of the cooling laser on the spins, one may use beams tightly focused on the ${\tau}$-species, or exploit two different ion species/isotopes.
As customary in the theory of open quantum-optical systems~\cite{breuer_book}, one obtains the master equation
\begin{equation}gin{equation}
\dot{\rho}=-{\rm i}[H,\rho]+\mathcal{D}(\rho),
\end{equation} after tracing out the electromagnetic bath. Here, we have introduced the Hamiltonian $H$ and dissipative $\mathcal{D}(\rho)$ parts. Let us start by describing the Hamiltonian
\begin{equation}gin{equation}
H=H_{\tau}+H_{\sigma}+H_{\rm ph}+V_{\sigma}^{{\rm ph}}+V_{\tau}^{{\rm ph}}.
\end{equation}
Here, $H_{\tau}=\frac{1}{2}\sum_l\omega^{\!{\tau}}_{0}\tau_{l}^z$, $H_{\sigma}=\frac{1}{2}\sum_{i}\omega_{0}^{\!\sigma}\sigma_i^z$, and $H_{\rm ph}=\sum_{n}\omega_{n}b_n^{\dagger}b_n,$
represent the atomic degrees of freedom of the laser-cooled $\tau$-ions, the (pseudo)spins of the $\sigma$-ions, and the vibrational excitations of the ion crystal, respectively. We have introduced $\tau_l^z=\ket{{e}_l}\bra{e_l}-\ket{g_l}\bra{g_l}$, $\sigma_i^z=\ket{{\uparrow_i}}\bra{{\uparrow_i}}-\ket{{\downarrow_i}}\bra{{\downarrow_i}}$, and the creation/annihilation operators $b_n^{\dagger},b_n^{\phantom{\dagger}}$ for a particular phonon branch with frequencies $\omega_n$. Additionally, we include a spin-phonon coupling that is provided by a stimulated Raman transition~\cite{wineland_review_sm} tuned to the so-called vibrational red-sideband
\begin{equation}gin{equation}
\label{resolved_sideband_sm}
V_{\sigma}^{{\rm ph}}\!=\!\sum_{in}\mathcal{F}^{\sigma}_{in}\sigma_i^+b_n\end{equation}^{-{\rm i}\omega_{\sigma}t}+\text{H.c.},\hspace{1ex} \mathcal{F}^{\sigma}_{in}=\textstyle{\frac{{\rm i}\Omega_{{\sigma}}}{2}\eta^{\sigma}_n\mathcal{M}_{in}\end{equation}^{{\rm i}\phi_{i}}},
\end{equation}
where we re-write the definitions already used in the main text for convenience. Here, $\sigma_i^+=\ket{{\uparrow_i}}\bra{{\downarrow_i}}$. Here, $\Omega_{\sigma}$ is the two-photon Rabi frequency, $\omega_{\sigma} ({\bf k}_{\sigma})$ is the Raman frequency (wavevector), and $\phi_i={\bf k}_{\sigma}\cdot{{\bf r}_i^{\sigma}}$. The Lamb-Dicke parameter $\eta^{\sigma}_n={\bf k}_{\sigma}\cdot{\bf e}_{\rm d}/\sqrt{2m_{\sigma}\omega_n}$ describes the laser coupling to the $n$-th mode with displacements $\mathcal{M}_{in}$ along ${\bf e}_{\rm d}$, where $m_{\sigma}$ is the ion mass. This spin-phonon coupling arises from the dipole laser-ion interaction after expanding $\eta_{n}^{\sigma}\ll 1$, such that $\omega_{\sigma}\approx \omega_0^{\sigma}-\omega_n$. We set $|\Omega_{\sigma}|\ll\omega_n$ to neglect the contribution of other terms (i.e. carrier and blue sideband) from the laser-ion interaction.
We laser cool the $\tau$-species in the node of a standing wave~\cite{laser_cooling_cirac}, which shall be red-detuned with respect to the atomic transition, and can be described by
\begin{equation}gin{equation}
\label{non_resolved_sideband_sm}
V_{\tau}^{{\rm ph}}=\sum_{ln}\mathcal{F}^{\tau}_{ln}\tau_l^+Q_n\end{equation}^{-{\rm i}\omega_{\tau}t}+\text{H.c.}, \hspace{1ex} \mathcal{F}^{\tau}_{ln}=-\textstyle{\frac{\Omega_{{\tau}}}{2}\eta^{\tau}_n\mathcal{M}_{ln}},
\end{equation}
where $\tau_l^+=\ket{{e_l}}\bra{{g_l}}=(\tau_l^-)^{\dagger}$, $Q_n=b_n^{\phantom{\dagger}}+b_n^{\dagger}$, and the remaining parameters are defined as below Eq.~\eqref{resolved_sideband_sm}. Note that the differences between Eqs.~\eqref{resolved_sideband_sm} and~\eqref{non_resolved_sideband_sm} are due to the different regimes $\Gamma_{\sigma}\ll\omega_n\ll\Gamma_{\tau}$, which forbid resolving the sidebands of the dipole-allowed transition of the $\tau$-ions (i.e. Doppler cooling regime). Additionally, the component of the laser-ion interaction that drives the carrier vanishes at the node of the standing wave. This allows us to consider strong Rabi frequencies $\Omega_{\sigma}$ to optimize the cooling rates in the regime of interest.
Finally, the dissipator including recoil effects~\cite{recoil}, can be described as the sum of two terms
\begin{equation}gin{equation}
\mathcal{D}(\rho)=\mathcal{D}_0(\rho)+\mathcal{D}_1(\rho).
\end{equation}
Here, $\mathcal{D}_0(\rho)$ is the usual dissipation super-operator in Lindblad form~\cite{lindblad} for a two-level atom at rest
\begin{equation}gin{equation}
\mathcal{D}_0(\rho)=\sum_l\textstyle\frac{1}{2}\Gamma_{\tau}\left(\tau_l^{-}\rho\tau_l^{+}-\tau_l^{+}\tau_l^{-}\rho\right)+\text{H.c.},
\end{equation}
where the typical ion distances forbid collective dissipative effects. Additionally $
\mathcal{D}_1(\rho)$ which describes recoil effects
\begin{equation}gin{equation}
\mathcal{D}_1(\rho)=\sum_{lnm}\textstyle\frac{1}{2}\Gamma_{\tau,nm}\tau_l^{-}\begin{itemize}g(Q_n\rho Q_m-Q_nQ_m\rho\begin{itemize}g)\tau_l^{+}+\text{H.c.},
\end{equation}
where we have introduced $\Gamma_{\tau,nm}=2\Gamma_{\tau}\alpha_q\eta^{\tau}_n\eta^{\tau}_m\mathcal{M}_{ln}\mathcal{M}_{lm}$, and $\alpha_q=(1+3q^2)/10(1+q^2)$, such that $q=0,\pm1$ depends on the linear/circular polarization of the emitted photon.
In addition to the Doppler-regime condition $\omega_n\ll\Gamma_{\tau}$, we further impose that $\mathcal{F}^{\tau}_{ln}\ll\omega_n,\Omega_{\tau}$. In this case, the laser-cooled ions reach the steady state very fast, and can be integrated out~\cite{laser_cooling_cirac}, which leads to the master equation
\begin{equation}gin{equation}
\label{laser_cooling_sm}
\dot{\mu}=-{\rm i}[H_{\sigma}+H_{\rm ph}+V_{\sigma}^{{\rm ph}},\mu]+\mathcal{\tilde{D}}(\mu),\hspace{2ex} \mu={\rm Tr}_{\tau,{\rm at}}\{\rho\}
\end{equation}
which is the starting point of our work in Eq.~\eqref{laser_cooling}. The effective dissipator describing the laser cooling of the vibrational modes is
\begin{equation}gin{equation}
\mathcal{\tilde{D}}(\mu)\!=\!\sum_{n}\!\begin{itemize}g\{\Gamma_{n}^+\!(b_n^{\dagger}\mu b_n^{\phantom{\dagger}}\!-b_n^{\phantom{\dagger}}b_n^{\dagger}\mu)\!+\!\Gamma_{n}^-\!(b_n^{\phantom{\dagger}}\mu b_n^{\dagger}\!-b_n^{\dagger}b_n^{\phantom{\dagger}}\mu)\!\begin{itemize}g\}\!+\!\text{H.c.},
\end{equation}
where the effective rates are expressed as $\Gamma^{\pm}_n=S(\mp\omega_n)$, and $S(\omega_n)=\int_0^{\infty} ds\end{equation}^{{\rm i}\omega_ns}\langle F_n(s)F_n(0)\rangle_{\rm ss}$ is the steady-state fluctuation spectrum of $F_n=\sum_l\mathcal{F}_{ln}^{\tau}(\tau^{+}_l+\tau^-_l)$. At this level, we introduce the heating by $\Gamma^{+}_n\to\Gamma^{+}_n=S(-\omega_n)+\Gamma^{\rm ah}_n$, where $\Gamma^{\rm ah}_n$ is the anomalous heating rate. The cooling rates and mean phonon numbers are
$
W_n={\rm Re}\{\Gamma^{-}_n-\Gamma^{+}_n\},\hspace{1ex} \begin{align}r{n}_n={\rm Re}(\Gamma^{+}_n)/W_n$. Therefore, it is straightforward to see that we can overcome the heating by shaping the laser-cooling fluctuation spectrum such that $W_n>0$, and obtain an overall cooling.
{\it Modeling the noisy dynamics.--} Let us now describe in more details how to take into account possible sources of noise, which appear for the time-scales of interest in addition to the anomalous heating. We will consider three possible sources: {\it (i)} The fluctuations around the state-state mean phonon number $\begin{align}r{n}_n$ will cause a pure dephasing of a thermal origin. {\it (ii)} Fluctuations of non-shielded Zeeman shifts will induce a pure dephasing of a magnetic origin. {\it (iii)} Fluctuations of non-compensated ac-Stark shifts also induce a pure dephasing, whose major contribution may be cuased by instabilities in the laser intensity of the cooling lasers. These three terms can be modeled by
\begin{equation}gin{equation}
H_n=\sum_i\frac{1}{2}\begin{itemize}g(\sum_nB^{\rm eff}_{in}+F_i(t)\begin{itemize}g)\sigma_i^z,
\end{equation}
where $B^{\rm eff}_{in}$ in Eq.~\eqref{eff_me} yields the thermal noise, and $F_i(t)$ is a random process for the magnetic/laser-intensity dephasing. We assume a local Gaussian noise with a short correlation time $\tau_{\rm c}$, which determines the stochastic average of two-time correlators $\langle F_j(s)F_i\rangle_{\rm st}=\frac{2\Gamma_{\rm d}}{\tau_{\rm c}}\end{equation}^{-s/\tau_{\rm c}}\delta_{ji}$, and in turn the dephasing rate of the spin dynamics $\Gamma_{\rm d}=\textstyle\frac{1}{2}\int_0^{\infty}{\rm d}s\langle F_j(s)F_j\rangle_{\rm st}=1/2T_2$. Let us note that typical decoherence times in trapped-ion experiments are $T_2\approx 1$-10\hspace{0.2ex}ms ($\Gamma_{\rm d}\approx 0.05$-$0.5 $\hspace{0.2ex}kHz), and that the regimes considered above yield $B_{in}^{\rm eff}/\begin{align}r{n}_n\approx 0.3$-$3$\hspace{0.2ex}kHz. Comparing these values to the time-scales of the dissipation-assisted protocols ($\sim$$0.1$-$10$\hspace{0.2ex}ms), it emphasizes that we need a scheme to actively decouple from this noise.
A partial solution would be the use of states that are insensitive to linear Zeeman shifts, such as $\ket{{\uparrow}}=\ket{2,1}$ and $\ket{{\downarrow}}=\ket{3,1}$ at a field of $B_0=213\hspace{0.2ex}$G for $^{25}$Mg$^{+}$. However, since we still need to mitigate the other sources of dephasing, we will exploit a different mechanism. We introduce a continuous driving of the spins, such that the bare spin Hamiltonian
\begin{equation}gin{equation}
H_{\sigma}=\textstyle\frac{1}{2}\sum_{i}\omega^{\sigma}_0\sigma_i^z+\left(\Omega_{\rm d}\sigma_i^+\cos(\omega_{\rm d}t)+\text{H.c.}\right),
\end{equation}
may be provided by a microwave source.
Here, the driving parameters are $\omega_{\rm d}=\omega^{\sigma}_0$, and $|\Omega_{\rm d}|\ll\omega_{\sigma}$.
Since the resonance frequencies are $\omega_{\sigma}/2\pi\approx 1$\hspace{0.2ex}GHz, the driving can still be strong enough to fulfill {\it strong-driving conditions}
\begin{equation}gin{equation}
\{W_n,\tilde{\delta_n}\}\ll |\Omega_{\rm d}|, \hspace{2ex} \Gamma_{\rm d},\tau_{\rm c}^{-1}\ll |\Omega_{\rm d}|.
\end{equation}
The first of these conditions allows us to use a modified Schriefer-Wolff transformation, which leads us to the new phonon-mediated terms $\tilde{H}_{\rm eff}$ and $\tilde{\mathcal{D}}_{\rm eff}$ of the Liouvillian in Eq.~\eqref{eff_me_ising} of the main text, after neglecting of-resonant contributions for such a strong driving. The second condition allows us to obtain the effect of the residual noise $\tilde{H}_{\rm n}$ and $\tilde{\mathcal{D}}_{\rm n}$ in Eq.~\eqref{eff_me_ising}, by using a Born-Markov approximation
\begin{equation}gin{equation}
\dot{\hat{\rho}}=-\int_0^{\infty}{\rm d}s\langle[\hat{H}_{\rm n}(t),[\hat{H}_{\rm n}(t-s),\hat{\rho}(t)]]\rangle_{\rm st},
\end{equation}
where we perform an stochastic average, and work in the interaction picture with respect to the resonant diving $\hat{H}_{\rm n}(t)=UH_{\rm n}U^{\dagger}$, where $U={\rm exp}\{{\rm i} t\sum_i\textstyle\frac{1}{2}\Omega_{\rm d}\sigma_i^x\}$.
{\it Many-body physics.--} The collective Liouvillians in Eqs.~\eqref{eff_me} and~\eqref{eff_me_ising} define our toolbox for dissipation-assisted QS. In the main text, we have considered quantum information processing by means of an isolated vibrational mode that mediates the collective dynamics. In the many-ion scenario, this would lead to fully-connected spin networks that become very interesting in the presence of magnetic frustration~\cite{network}. However, to achieve strongly-correlated models, it is better to work with a full vibrational branch of a small frequency width. The small ratios $\mathcal{R}_n\sim 10^{-3}$ obtained for trapping frequencies $\omega_{\rm t}/2\pi\approx 10$MHz, indicate that the coherent dynamics can be dominating also in this case, while minimizing the heating. This would allow for the QS of exotic models with 3-body interactions~\cite{exotic} or topological order~\cite{kitaev} in surface ion traps.
These many-body QS focus on the Hamiltonian while minimizing the influence of the environment. However, the dissipative dynamics may also lead to interesting many-body phenomena~\cite{dissipative_many_body}. The possibility to control the relative strength of the coherent/dissipative parts in~\eqref{eff_me} and~\eqref{eff_me_ising} is very appealing in this respect. In particular, we note that for intermediate driving strengths $\Omega_{\rm d}$, competing dissipative terms in~\eqref{eff_me_ising} may lead to purely-dissipative quantum phase transitions.
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\begin{itemize}bitem{wineland_review_sm}
{\it See} D. J. Wineland, et {\it al.}, \href{http://nvlpubs.nist.gov/nistpubs/jres/103/3/cnt103-3.htm}{J. Res. Natl. I. St. Tech. {\bf 103,} 259 (1998)}, {\it and references therein.}
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H. P. Breuer and F. Petruccione, The theory of open quantum
systems, (Oxford University Press, Oxford, 2003); A. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction (Springer, Heidelberg, 2012).
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J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, \href{http://pra.aps.org/abstract/PRA/v46/i5/p2668_1}{Phys. Rev. A {\bf 46,} 2668 (1992).}
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J. Dalibard and C. Cohen-Tannoudji, \href{http://iopscience.iop.org/0022-3700/18/8/019}{J. Phys. B: At. Mol. Phys. {\bf 18,} 166 (1985)}; Y. Castin, H. Wallis, and J. Dalibard, \href{http://www.opticsinfobase.org/josab/abstract.cfm?uri=josab-6-11-2046}{J. Opt. Soc. Am. B. {\bf 6,} 2046 (1989).}
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G. Lindblad, \href{http://link.springer.com/article/10.1007/BF01608499?LI=true}{Commun. Math. Phys. {\bf 48,} 119 (1976).}
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G.-D. Lin, C. Monroe, L.-M. Duan, \href{http://prl.aps.org/abstract/PRL/v106/i23/e230402}{Phys. Rev. Lett. {\bf 106,} 230402 (2011).}
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A. Bermudez, D. Porras, and M. A. Martin-Delgado, \href{http://pra.aps.org/abstract/PRA/v79/i6/e060303}{Phys. Rev. A {\bf 79,} 060303(R) (2009)}.
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R. Schmied, J. H. Wesenberg, and D. Leibfried, \href{http://iopscience.iop.org/1367-2630/13/11/115011/article/}{New J. Phys. {\bf 13,} 115011 (2011).}
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M.B. Plenio and S.F. Huelga, \href{http://prl.aps.org/abstract/PRL/v88/i19/e197901}{Phys. Rev. Lett. {\bf 88,} 197901 (2002)}; S.F. Huelga and M.B. Plenio, \href{http://prl.aps.org/abstract/PRL/v98/i17/e170601}{Phys. Rev. Lett. {\bf 98,} 170601 (2007)}; M. M\"uller, S. Diehl, G. Pupillo, P. Zoller, \href{http://www.sciencedirect.com/science/article/pii/B9780123964823000016}{Adv. At. Mol. Opt. Phys. {\bf 61,} 1 (2012)}.
\end{references}
\end{document} | math | 53,829 |
\begin{document}
\begin{abstract}
A long-standing conjecture of Berge suggests that every
bridgeless cubic graph can be expressed as a union of at most
five perfect matchings. This conjecture trivially holds for
$3$-edge-colourable cubic graphs, but remains widely open for
graphs that are not $3$-edge-colourable. The aim of this paper
is to verify the validity of Berge's conjecture for cubic
graphs that are in a certain sense close to $3$-edge-colourable
graphs. We measure the closeness by looking at the colouring
defect, which is defined as the minimum number of edges left
uncovered by any collection of three perfect matchings. While
$3$-edge-colourable graphs have defect $0$, every bridgeless
cubic graph with no $3$-edge-colouring has defect at least $3$.
In 2015, Steffen proved that the Berge conjecture holds for
cyclically $4$-edge-connected cubic graphs with colouring
defect $3$ or $4$. Our aim is to improve Steffen's result in
two ways. First, we show that all bridgeless cubic graphs with
defect $3$ satisfy Berge's conjecture irrespectively of their
cyclic connectivity. Second, we prove that if the graph in
question is cyclically $4$-edge-connected, then four perfect
matchings suffice, unless the graph is the Petersen graph. The
result is best possible as there exists an infinite family of
cubic graphs with cyclic connectivity $3$ which have defect $3$
but cannot be covered with four perfect matchings.
\end{abstract}
\subjclass[2020]{05C15, 05C70; 05C75}
\maketitle
\begin{center}
\textit{Dedicated to professor J\'an Plesn\i\'ik, our teacher
and colleague.}
\end{center}
\section{Introduction}
\noindent{}In 1970's, Claude Berge made a conjecture that every
bridgeless cubic graph $G$ can have its edges covered by at
most five perfect matchings (see \cite{Mazz}). The
corresponding set of matchings is called a \textit{Berge cover}
of $G$.
Berge's conjecture relies on the fact that every preassigned
edge of $G$ belongs to a perfect matching, and hence there is a
set of perfect matchings that cover all the edges of $G$. The
smallest number of perfect matchings needed for this purpose is
the \textit{perfect matching index} of $G$, denoted by
$\pi(G)$. Clearly, $\pi(G)=3$ if and only if $G$ is
$3$-edge-colourable, so if $G$ has chromatic index~$4$, then
the value of $\pi(G)$ is believed to be either $4$ or $5$. In
this context it may be worth mentioning that $\pi(Pg)=5$, where
$Pg$ denotes the Petersen graph, and that cubic graphs with
$\pi=5$ are very rare \cite{GGHM, MS-pmi}.
Berge's conjecture is closely related to a stronger and
arguably more famous conjecture of Fulkerson \cite{F},
attributed also to Berge and therefore often referred to as the
Berge-Fulkerson conjecture \cite{Seymour-multi}. The latter
conjecture suggests that every bridgeless cubic graph contains
a collection of six perfect matchings such that each edge
belongs to precisely two of them. Fulkerson's conjecture
clearly implies Berge's. Somewhat surprisingly, the converse
holds as well, which follows from an ingenious construction due
to Mazzuoccolo \cite{Mazz}.
Very little is known about the validity of either of these
conjectures. Besides the $3$-edge-colourable cubic graphs, the
conjectures are known to hold only for a few classes of graphs,
mostly possessing a very specific structure (see
\cite{Chen-Fan, FV-pmi, HaoCQZ-2009, HaoCQZ-2018,
LiuHaoCQZ-2021, ManSh}). On the other hand, it has been proved
that if Fulkerson's conjecture is false, then the smallest
counterexample would be cyclically $5$-edge-connected (M\'a\v
cajov\'a and Mazzuoccolo~\cite{MM}).
In this paper we investigate Berge's conjecture under more
general assumptions, not relying on any specific structure of
graphs: our aim is to verify the conjecture for
all cubic graphs that are, in a certain sense, close to
$3$-edge-colourable graphs. In our case, the proximity to
$3$-edge-colourable graphs will be measured by the value of
their colouring defect. The \textit{colouring defect} of a
cubic graph, or just \textit{defect} for short, is the smallest
number of edges that are left uncovered by any set of three
perfect matchings. This concept was introduced and thoroughly
studied by Steffen \cite{S2} in 2015 who used the notation
$\mu_3(G)$ but did not coin any term for it. Since a cubic
graph has defect $0$ if and only if it is $3$-edge-colourable,
colouring defect can serve as one of measures of
uncolourability of cubic graphs, along with resistance,
oddness, and other similar invariants recently studied by a
number of authors, see for example \cite{Allie, AlMaS,
FMS-survey}.
In \cite[Corollary~2.5]{S2} Steffen proved that the defect of
every bridgeless cubic graph that cannot be $3$-edge-coloured
is at least $3$, and can be arbitrarily large. He also proved
that every cyclically $4$-edge-connected cubic graph with
defect $3$ or $4$ satisfies Berge's conjecture
\cite[Theorem~2.14]{S2}.
We strengthen the latter result in two ways.
First, we show that every bridgeless
cubic graph with defect $3$ satisfies the Berge
conjecture irrespectively of its cyclic connectivity.
\begin{theorem}\label{thm:1}
Every bridgeless cubic graph with colouring defect $3$
admits a Berge cover.
\end{theorem}
Second, we prove that if the graph in question is cyclically
$4$-edge-connected, then four perfect matchings are enough to
cover all its edges, except for the Petersen graph. The
existence of a cover with four perfect matchings is known to
have a number of important consequences. For example, such a
graph satisfies the Fan-Raspaud conjecture \cite{FR}, admits a
$5$-cycle double cover and has a cycle cover of length
$4/3\cdot m$, where $m$ is the number of edges
\cite[Theorem~3.1]{S2}.
\begin{theorem}\label{thm:2}
Let $G$ be a cyclically $4$-edge-connected cubic graph with
defect~$3$. Then $\pi(G)=4$, unless $G$ is the Petersen graph.
\end{theorem}
Theorem~\ref{thm:2} is best possible in the sense that there
exist infinitely many $3$-connected cubic graphs with colouring
defect $3$ that cannot be covered with four perfect matchings.
Our proofs use a wide range of methods. Both main results rely
on Theorem~\ref{thm:6cuts} which describes the structure of a
subgraph resulting from the removal of a $6$-edge-cut from a
bridgeless cubic graph. Moreover, the proof of
Theorem~\ref{thm:2} establishes an interesting relationship
between the colouring defect and the bipartite index of a cubic
graph. We recall that the bipartite index of a graph is the
smallest number of edges whose removal yields a bipartite
graph. This concept was introduced by Thomassen in \cite{Th1}
and was applied in \cite{Th1, Th2} in a different context.
In closing, we would like to mention that our
Theorem~\ref{thm:1} can be alternatively derived from a recent
result published in \cite{SW}, which states that every
bridgeless cubic graph containing two perfect matchings that
share at most one edge admits a Berge cover. The two results
are connected via an inequality between two uncolourability
measures proved in \cite[Theorem~2.2]{JS} (see also
\cite[Proposition~4.2]{KMNSgirth}). Although the main result of
\cite{SW} is important and interesting, its proof is very
technical and difficult to comprehend, which is why we have not
been able to verify all its details. We believe that more
effort is needed to clarify the arguments presented
in~\cite{SW}.
Our paper is organised as follows. The next section summarises
definitions and results needed for understanding the rest of
the paper. Section~\ref{sec:arrays} provides a brief account of
the theory surrounding the notion of colouring defect. Specific
tools needed for the proofs of Theorems~\ref{thm:1}
and~\ref{thm:2} are established in Section~\ref{sec:tools}. The
two theorems are proved in Sections~\ref{sec:thm1}
and~\ref{sec:thm2}, respectively. The final section illustrates
that the condition on cyclic connectivity in
Theorem~\ref{thm:2} cannot be removed.
\section{Preliminaries}
\noindent{}\textbf{2.1. Graphs.} Graphs studied in this paper
are finite and mostly cubic (that is, 3-valent). Multiple edges
and loops are permitted. The \emph{order} of a graph $G$,
denoted by $|G|$, is the number of its vertices. A
\emph{circuit} in $G$ is a connected $2$-regular subgraph of
$G$. A $k$-\emph{cycle} is a circuit of length~$k$. The
\emph{girth} of $G$ is the length of a shortest circuit in $G$.
An \emph{edge cut} is a set $R$ of edges of a graph whose
deletion yields a disconnected graph. A common type of an edge
cut arises by taking a subset of vertices or an induced
subgraph $H$ of $G$ and letting $R$ to be the set $\delta_G(H)$
of all edges with exactly one end in $H$. We omit the subscript
$G$ whenever $G$ is clear from the context.
A connected graph $G$ is said to be \emph{cyclically
$k$-edge-connected} for some integer $k\ge 1$ if the removal of
fewer than $k$ edges cannot leave a subgraph with at least two
components containing circuits. The \emph{cyclic connectivity}
of $G$ is the largest integer $k$ not exceeding the cycle rank
(Betti number) of $G$ such that $G$ is cyclically
$k$-edge-connected. An edge cut $R$ in $G$ that separates two
circuits from each other is \emph{cycle-separating}. It is not
difficult to see that the set $\delta_G(C)$ leaving a shortest
circuit $C$ of a cubic graph $G$ is cycle-separating unless
$G$ is the complete bipartite graph $K_{3,3}$, the complete
graph $K_4$, or the \emph{$3$-dipole}, the graph which consists
of two vertices and three parallel edges joining them. Observe
that an edge cut formed by a set of independent edges is always
cycle-separating. Conversely, a cycle-separating edge cut of
minimum size is independent.
\noindent{}\textbf{2.2. Edge colourings and flows.} An
\emph{edge colouring} of a graph $G$ is a mapping from the edge
set of $G$ to a set of colours. A colouring is \emph{proper} if
any two edge-ends incident with the same vertex receive
distinct colours. A \emph{$k$-edge-colouring} is a proper edge
colouring where the set of colours has $k$ elements. Unless
specified otherwise, our colouring will be assumed to be proper
and graphs to be \emph{subcubic}, that is, with vertices of
valency $1$, $2$, or~$3$.
There is a standard method of transforming a
$3$-edge-colouring to another $3$-edge-colouring: it uses
so-called Kempe switches: Let $G$ be a subcubic graph endowed
with a proper $3$-edge-colouring $\sigma$. Take two distinct
colours $i$ and $j$ from $\{1,2,3\}$. An \emph{$(i,j)$-Kempe
chain} in $G$ (with respect to $\sigma$) is a non-extendable
walk $L$ that alternates edges coloured $i$ with those coloured
$j$. It is easy to see that $L$ is either a bicoloured circuit or
path starting and ending at the vertex of valency smaller than
$3$. The \emph{Kempe switch} along a Kempe chain produces a new
$3$-edge-colouring of $G$ by interchanging the colours~on~$L$.
It is often useful to regard $3$-edge-colourings of cubic
graphs as nowhere-zero flows. To be more precise, one can
identify each colour from the set $\{1,2,3\}$ with its binary
representation; thus $1=(0,1)$, $2=(1,0)$, and $3=(1,1)$.
Having done this, the condition that the three colours meeting
at every vertex $v$ are all distinct becomes equivalent to
requiring the sum of the colours at $v$ to be $0=(0,0)$. The
latter is nothing but the Kirchhoff law for nowhere-zero
$\mathbb{Z}_2\times\mathbb{Z}_2$-flows. Recall that an
\textit{$A$-flow} on a graph $G$ is a pair $(D,\phi)$ where
$\phi$ is an assignment of elements of an abelian group $A$ to
the edges of $G$, and $D$ is an assignment of one of two
directions to each edge in such a way that, for every vertex
$v$ in $G$, the sum of values flowing into $v$ equals the sum
of values flowing out of~$v$ (\emph{Kirchhoff's law}). A
\emph{nowhere-zero} $A$-flow is one which does not assign $0\in
A$ to any edge of $G$. If each element $x\in A$ satisfies
$x=-x$, then $D$ can be omitted from the definition. It is well
known that the latter is satisfied if and only if $A\cong
\mathbb{Z}_2^n$ for some $n\ge 1$.
The following well-known statement is a direct consequence of
Kirchhoff's law.
\begin{lemma}{\rm (Parity Lemma)}\label{lem:par}
Let $G$ be a cubic graph endowed with a
$3$-edge-colouring~$\xi$. The following holds for every edge
cut $\delta_G(H)$, where $H$ is a subgraph of $G$:
$$\sum_{e\in\delta_G(H)}\xi(e)=0.$$
Equivalently, the number of edges in $\delta_G(H)$ carrying any
fixed colour has the same parity as the size of the cut.
\end{lemma}
A cubic graph $G$ is said to be \emph{colourable} if it admits
a $3$-edge-colouring. A $2$-connected cubic graph that admits
no $3$-edge-colouring is called a \emph{snark}. Our definition
agrees with that of Cameron et al. \cite{CCW}, Nedela and \v
Skoviera \cite{NS-decred}, Steffen \cite{S1}, and others, and
leaves the concept of a snark as wide as possible. A more
restrictive definition requires a snark to be to be cyclically
$4$-edge-connected, with girth at least~$5$, see for
example~\cite{FMS-survey}. We call such snarks
\emph{nontrivial}.
\noindent{}\textbf{2.3. Perfect matchings.} The classical
theorems of Tutte \cite{Tutte} and Plesn\'ik \cite{Plesnik},
stated below,
will be repeatedly used throughout the paper, albeit in a
slightly modified form permitting parallel edges and loops.
These extensions can be proved easily by using the standard
versions of the corresponding theorems.
Let $\operatorname{odd}\xspace(G)$
denote the number of \emph{odd components} of $G$, that is, the
components with an odd number of vertices.
\begin{theorem} \label{thm:Tutte}
{\rm (Tutte, 1947)} A graph $G$, possibly containing parallel
edges and loops, has a perfect matching if and only if
$$\operatorname{odd}\xspace(G-S)\le |S| \quad \text{for all }S\subseteq V(G).$$
\end{theorem}
\begin{theorem} \label{thm:Plesnik}
{\rm (Plesn\'\i k, 1972)} Let $G$ be an $(r-1)$-edge-connected
$r$-regular graph with $r\ge 1$, and let $A$ be an arbitrary
set of $r-1$ edges in $G$. If $G$ has even order, then $G - A$
has a perfect matching.
\end{theorem}
\section{Colouring defect of a cubic graph}
\label{sec:arrays}
\noindent{}In this section we discuss a number of structures
related to the concept of colouring defect, which will be
used in the proofs of Theorems~\ref{thm:1} and~\ref{thm:2}.
More details on this matter can be found in \cite{KMNSred} and
\cite{S2}.
For a bridgeless cubic graph $G$ we define a \emph{$k$-array of
perfect matchings}, or briefly a \emph{$k$-array} of $G$, as an
arbitrary collection $\mathcal{M}=\{M_1, M_2, \ldots, M_k\}$ of
$k$ perfect matchings of $G$, not necessarily pairwise
distinct. The concept of a $k$-array unifies a number of
notions, such as Berge covers, Fulkerson covers, Fan-Raspaud
triples, and others. Our main concern here are $3$-arrays,
which we regard as approximations of $3$-edge-colourings.
Let $\mathcal{M}=\{M_1, M_2, M_3\}$ be a $3$-array of perfect
matchings of a cubic graph $G$. An edge of $G$ that belongs to
at least one of the perfect matchings of $\mathcal{M}$ will be
considered to be \emph{covered}. An edge will be called
\emph{uncovered}, \emph{simply covered}, \emph{doubly covered},
or \emph{triply covered} if it belongs, respectively, to zero,
one, two, or three distinct members of~$\mathcal{M}$.
Given a cubic graph $G$, it is natural task to maximise the
number of covered edges, or equivalently, to minimise the
number of uncovered ones. A $3$-array that leaves the minimum
number of uncovered edges will be called \emph{optimal}. The
number of edges left uncovered by an optimal $3$-array is the
\emph{colouring defect} of $G$, or the \emph{defect} for short,
denoted by $\df{G}$.
Let $\mathcal{M}=\{M_1, M_2, M_3\}$ be a $3$-array of perfect
matchings of a cubic graph $G$. One way to describe
$\mathcal{M}$ is based on regarding the indices $1$, $2$, and
$3$ as colours. Since the same edge may belong to more than one
member of $\mathcal{M}$, an edge of $G$ may receive more than
one colour. To each edge $e$ of $G$ we can therefore assign the
list $\phi(e)$ of all colours in lexicographic order it
receives from $\mathcal{M}$. We let $w(e)$ denote the number of
colours in the list $\phi(e)$ and call it the \emph{weight} of
$e$ (with respect to $\mathcal{M}$). In this way $\mathcal{M}$
gives rise to a colouring
$$\phi\colon E(G)\to\{\emptyset, 1, 2, 3, 12, 13, 23, 123\}$$
where $\emptyset$ denotes the empty list. Obviously, a mapping
$\sigma$ assigning subsets of $\{1,2,3\}$ to the edges of $G$
determines a $3$-array if and only if, for each vertex $v$ of
$G$, each element of $\{1,2,3\}$ occurs precisely once in the
subsets of $\{1,2,3\}$ assigned by $\sigma$ to the edges
incident with $v$. In general, $\phi$ need not be a proper
edge-colouring. However, since $M_1$, $M_2$, and $M_3$ are
perfect matchings, the only possibility when two edges equally
coloured under $\phi$ meet at a vertex is that both of them
receive colour $\emptyset$. In this case the third edge
incident with the vertex is coloured $123$.
A different but equivalent way of representing a $3$-array uses
a mapping
\[
\chi\colon E(G)\to \mathbb{Z}_2^3,
\quad e\mapsto \chi(e)=(x_1, x_2, x_3)
\]
defined by setting $x_i = 0$ if and only if $e\in M_i$, where
$i\in\{1,2,3\}$. Since the complement of each $M_i$ in $G$ is a
$2$-factor, it is easy to see that $\chi$ is a
$\mathbb{Z}_2^3$-flow. We call $\chi$ \emph{the characteristic
flow} for $\mathcal{M}$. Again, $\chi$ is a nowhere-zero
$\mathbb{Z}_2^3$-flow if and only if $G$ contains no triply
covered edge. In the context of $3$-arrays the characteristic
flow was introduced in~\cite[p.~166]{JSM}. Observe that the
characteristic flow $\chi$ of a $3$-array and the colouring
$\phi$ determine each other. In particular, the
condition on $\phi$ requiring all three indices from
$\{1,2,3\}$ to occur precisely once in a colour around any
vertex is equivalent to Kirchhoff's law.
The following result characterises $3$-arrays with no triply
covered edge.
\begin{proposition}\label{prop:notriply}
Let $\mathcal{M}$ be a $3$-array of perfect matchings of a
cubic graph $G$. The following three statements are equivalent.
\begin{enumerate}[{\rm (i)}]
\item $G$ has no triply covered edge with respect to
$\mathcal{M}$.
\item The associated colouring $\phi\colon
E(G)\to\{\emptyset, 1,2, 3, 12, 13, 23, 123\}$ is
proper.
\item The characteristic flow $\chi$ for $\mathcal{M}$,
with values in $\mathbb{Z}_2^3$, is nowhere-zero.
\end{enumerate}
\end{proposition}
The next important structure associated with a $3$-array is its
core. The \emph{core} of a $3$-array $\mathcal{M}=\{M_1, M_2,
M_3\}$ of $G$ is the subgraph of $G$ induced by all the edges
of $G$ that are not simply covered; we denote it by
${\mathrm{core}}(\mathcal{M})$. The core is called \emph{optimal}
whenever $\mathcal{M}$ is optimal.
It is worth mentioning that if $G$ is $3$-edge-colourable and
$\mathcal{M}$ consists of three pairwise disjoint perfect
matchings, then ${\mathrm{core}}(\mathcal{M})$ is empty. If $G$ is not
$3$-edge-colourable, then every core must be nonempty.
Figure~\ref{fig:petersen_core} shows the Petersen graph endowed
with a $3$-array whose core is its ``outer'' $6$-cycle. The
hexagon is in fact an optimal core.
\begin{figure}
\caption{An optimal $3$-array of the Petersen graph}
\label{fig:petersen_core}
\end{figure}
The following proposition, due to Steffen \cite[Lemma~2.2]{S2},
describes the structure of optimal cores in the general case.
\begin{proposition}\label{prop:core}
Let $\mathcal{M}=\{M_1, M_2, M_3\}$ be an optimal $3$-array of
perfect matchings of a snark $G$. Then every component of
${\mathrm{core}}(\mathcal{M})$ is either an even circuit of length at
least $6$ or a subdivision of a cubic graph. Moreover, the
union of doubly and triply covered edges forms a perfect
matching of ${\mathrm{core}}(\mathcal{M})$.
\end{proposition}
The next theorem characterises snarks with minimal possible
colouring defect. The lower bound for the defect of a snark --
the value $3$ -- is due to Steffen~\cite[Corollary~2.5]{S2}.
\begin{theorem}\label{thm:main}
Every snark $G$ has $\df{G}\ge 3$. Furthermore, the following
three statements are equivalent.
\begin{enumerate}[{\rm(i)}]
\item $\df{G}=3$.
\item The core of any optimal $3$-array of $G$ is a
$6$-cycle.
\item $G$ contains an induced $6$-cycle $C$ such that the
subgraph $G-E(C)$ admits a proper $3$-edge-colouring
under which the six edges of $\delta(C)$ receive
colours $1,1,2,2,3,3$
with respect to the cyclic order induced by
an orientation of $C$.
\end{enumerate}
\end{theorem}
\begin{figure}
\caption{The hexagonal core and its vicinity.}
\label{fig:core3}
\end{figure}
If $G$ is an arbitrary snark with $\df{G} = 3$, then, by
Theorem~\ref{thm:main}~(iii), $G$ contains an induced $6$-cycle
$C=(e_0e_1e_2e_3e_4e_5)$ such that $G-E(C)$ is
$3$-edge-colourable. We say that that $C$ is a \emph{hexagonal
core} of $G$.
Let $f_i$ denote the edge of $\delta(C)$ which is incident with
$e_{i-1}$ and $e_i$, where $i\in\{0,1,\ldots,5\}$ and the
indices are reduced modulo $6$; see Figure~\ref{fig:core3}.
Since $G-E(C)$ is $3$-edge-colourable but $G$ is not, it is not
difficult to see that for every $3$-edge-colouring
of $G-E(C)$ the cyclic order of colours around $C$ is
$(1,1,2,2,3,3)$ up to permutation of colours. Moreover, we can
assume that the values of the associated colouring $\phi$ of
$G$ induced by $\mathcal{M}$ in the vicinity of $C$ are those
as shown in Figure~\ref{fig:core3}, or can be obtained from
them by the rotation one step clockwise. Note that the two
possibilities only depend on the position of the uncovered
edges. In any case, there are no triply covered edges, and so
$\phi$ is a proper edge colouring due to
Proposition~\ref{prop:core}.
We finish this section by stating the following property of a
hexagonal core, which will be needed in Sections~\ref{sec:c4c}.
We leave the proof to the reader or refer to \cite{KMNSred}.
\begin{lemma}\label{lem:3+4}
Let $G$ be a snark with $\df{G}=3$. If a hexagonal core of $G$
intersects a triangle or a quadrilateral, then the intersection
consists of a single uncovered edge.
\end{lemma}
\section{Tools}\label{sec:tools}
\noindent{}In this section we establish tools for proving
Theorems~\ref{thm:1} and~\ref{thm:2}. Its main results are
stated as Theorems~\ref{thm:6cuts} and~\ref{thm:abg_class1},
both having somewhat bipartite flavour. The first theorem
suggests that if a subgraph $H$ of a bridgeless cubic graph is
separated from the rest by a $6$-edge-cut and has no perfect
matching, then the structure of $H$ is -- essentially -- that
of a bipartite cubic graph. Moreover, only one of the two parts
is incident with the cut.
\begin{theorem}\label{thm:6cuts}
Let $G$ be a bridgeless cubic graph and let $H\subseteq G$ be a
subgraph with $|\delta_G(H)|=6$. Then $H$ has a perfect
matching, or else $H$ contains an independent set $S$ of
trivalent vertices such that
\begin{itemize}
\item[(i)] every component of $H-S$ is odd,
\item[(ii)] $\operatorname{odd}\xspace(H-S) =|S|+ 2$, and
\item[(iii)] $|\delta_G(L)| = 3$ for each component $L$ of
$H-S$.
\end{itemize}
\end{theorem}
\begin{proof}
Set $K=G-V(H)$. Assume that $H$ has no perfect matching. Tutte's
Theorem tells us that there exists a set $S\subseteq V(H)$ such
that $\operatorname{odd}\xspace(H-S)>|S|$. As $H$ has an even number of vertices,
the numbers $|S|$ and $\operatorname{odd}\xspace(H-S)$ have the same parity, so
\begin{equation}\label{eq:loc1}
\operatorname{odd}\xspace(H-S)-|S|\ge 2.
\end{equation}
Set $a=|\delta_G(S)\cap\delta_G(K)|$. Clearly,
$a\in\{0,1,\ldots,6\}$. Since $|\delta_H(S)|+a=|\delta_G(S)|\le
3|S|$, we get
\begin{equation}\label{eq:dS-le}
|\delta_H(S)|\le 3|S|-a.
\end{equation}
To bound $|\delta_H(S)|$ from below, we first realise that each
odd component of $H-S$ is incident with at least three edges of
$\delta_G(H-S)$ because $G$ is bridgeless. Moreover, there are
$6-a$ edges joining $H-S$ to $K$ (see Figure~\ref{fig:tutte}).
Therefore
\begin{equation}\label{eq:dS-ge}
|\delta_H(S)|=|\delta_H(H-S)|=|\delta_G(H-S)|-(6-a)\ge 3\cdot\operatorname{odd}\xspace(H-S)-(6-a).
\end{equation}
If we combine \eqref{eq:dS-le} with \eqref{eq:dS-ge}, we get
\begin{equation}\label{eq:loc2}
3\cdot\operatorname{odd}\xspace(H-S)-6+a \le |\delta_H(S)| \le 3|S|-a.
\end{equation}
Since $3\cdot\operatorname{odd}\xspace(H-S)-3|S|\ge 6$ according to \eqref{eq:loc1}, we
can rewrite \eqref{eq:loc2} as
$$-2a\ge 3\cdot\operatorname{odd}\xspace(H-S)-3|S|-6\ge 0,$$
which implies that $a=0$.
Now we insert $a=0$ into
\eqref{eq:loc2} and get $3\cdot\operatorname{odd}\xspace(H-S)-3|S|\le 6$. Together
with \eqref{eq:loc1}, multiplied by $3$, this yields that
$$3\cdot\operatorname{odd}\xspace(H-S)-6\le |\delta_H(S)| \le 3|S|\le 3\cdot\operatorname{odd}\xspace(H-S)-6.$$
\begin{figure}
\caption{The structure of $G$ as described in the proof of
Theorem~\ref{thm:6cuts}
\label{fig:tutte}
\end{figure}
\noindent
Hence,
\begin{equation}\label{eq:S+2}
|\delta_H(S)|=3|S|=3\cdot\operatorname{odd}\xspace(H-S)-6,
\end{equation}
which implies that $S$ is an independent set of $H$, with all its
vertices $3$-valent, and that each component $K$ of $H-S$ is
odd with $|\delta_G(K)|=3$. Thus we have proved Statements
(i) and (iii). Moreover, Equation~\eqref{eq:S+2} implies
statement~(ii). The proof is complete.
\end{proof}
\begin{example}
We illustrate Theorem~\ref{thm:6cuts} in two simple instances.
First, if $G$ is the $3$-dimensional cube $Q_3$ and $H=G-V(C)$,
where $C$ is an induced $6$-cycle of $G$, then
Theorem~\ref{thm:6cuts} holds with $S=\emptyset$. If $G$ is the
Petersen graph $Pg$ and $H= G-V(C)$, where $C$ is again a
$6$-cycle, then $H$ is isomorphic to the complete bipartite
graph $K_{1,3}$ and $S$ is constituted by its central vertex.
\end{example}
We proceed to our second tool, which describes cubic graphs
just one step away from being bipartite. Recall that if a cubic
graph is bipartite, then it is obviously bridgeless. Moreover,
if it is connected, then its bipartition is uniquely
determined. On the other hand, if a cubic graph is not
bipartite, then, clearly, at least two edges have to be removed
in order to produce a bipartite graph. Motivated by these two
facts we define a cubic graph $G$ to be \emph{almost bipartite}
if it is bridgeless, not bipartite, and contains two edges $e$
and $f$ such that $G-\{e,f\}$ is a bipartite graph. The edges
$e$ and $f$ are said to be \emph{surplus edges} of $G$. If a
cubic graph is almost bipartite, then there exists a component
$K$ of $G$ such that the surplus edges connect vertices within
different partite sets of $K$.
Our aim is to show that every almost bipartite graph is
$3$-edge-colourable. We start with the following.
\begin{proposition}\label{prop:almost_bip}
Every almost bipartite cubic graph has a perfect matching that
contains both surplus edges.
\end{proposition}
\begin{proof}
Let $G$ be an almost bipartite cubic graph with surplus edges
$e$ and $f$. Since $e$ and $f$ belong to the same component of
$G$, we may assume that $G$ is connected. Let $\{A,B\}$ be the
bipartition of $G'=G-\{e,f\}$. As said before, the endvertices
of one of the surplus edges, say $e$, belong to $A$ and those
of $f$ then belong to $B$. In particular, this means that
$|A|=|B|=n$ for some positive integer $n$. By
Theorem~\ref{thm:Plesnik}, there exists a perfect matching $M$
containing the edge~$e$. There are $n-2$ edges of $M$ that
match $n-2$ vertices of $A$ to $n-2$ vertices of $B$. It
follows that $f\in M$.
\end{proof}
We would like to mention that our
Proposition~\ref{prop:almost_bip} has been inspired by
Lemma~3~(iii) of \cite{RLER}. In principle, the mentioned lemma
could be used to prove Proposition~\ref{prop:almost_bip},
however, the proof would not be anything close to being
straightforward.
We are now ready for the following theorem.
\begin{theorem}\label{thm:abg_class1}
Every almost bipartite cubic graph is $3$-edge-colourable.
\end{theorem}
\begin{proof}
Let $G$ be an almost bipartite graph with surplus edges $e$ and
$f$. By Proposition~\ref{prop:almost_bip}, $G$ has a perfect
matching $M$ which includes both $e$ and $f$. Now, $G-M$ is
bipartite $2$-regular spanning subgraph of $G$, so each
component of $G-M$ is an even circuit. It means that $G-M$ can
be decomposed into two disjoint perfect matchings $N_1$ and
$N_2$. Put together, $M$, $N_1$, and $N_2$ are colour classes
of a proper $3$-edge-colouring of $G$.
\end{proof}
The previous theorem links the problem of
$3$-edge-colourability of cubic graphs to an important concept
of a bipartite index, which has been introduced to measure how
far a graph is from being bipartite. Following \cite[Definition
2.4]{Th2}, we define the \emph{bipartite index} {$\operatorname{bi}\xspace(G)$ of a
graph $G$ to be the smallest number of edges that must be
deleted in order to make the graph bipartite. In other words,
the bipartite index of a graph is the number of edges outside a
maximum edge cut.
Our Theorem~\ref{thm:abg_class1} states that all bridgeless
cubic graphs with bipartite index at most two are $3$-edge
colourable. This is, in fact, best possible as there exist
infinitely many nontrivial snarks whose bipartite index equals
$3$: this is true, for example, for the Petersen graph or for
all Isaacs flower snarks $J_{2n+1}$; see \cite{Isaacs} for the
definition and Figure~\ref{fig:j7} for $J_7$.
\begin{figure}
\caption{The flower snark $J_7$}
\label{fig:j7}
\end{figure}
Threorem~\ref{thm:abg_class1} thus suggests that, along with
oddness, resistance, flow resistance, and other similar
invariants extensively studied in \cite{FMS-survey}, bipartite
index can serve as another measure of uncolourability of cubic
graphs. It is easy to see, for example, that for a cubic graph
$\operatorname{bi}\xspace(G)\geq \omega(G)$, where $\omega(G)$ denotes the
\emph{oddness} of a cubic graph, that is, the smallest number
of odd circuits in a $2$-factor of $G$. Since there exist
snarks of arbitrarily large oddness \cite{LMMS}, there are
snarks of arbitrarily large bipartite index.
\section{Proof of Theorem~\ref{thm:1}}\label{sec:thm1}
\noindent{}The purpose of this section is to establish the existence of a
Berge cover for every bridgeless cubic graph of defect $3$
(Theorem~\ref{thm:1}) and for every cyclically
$4$-edge-connected cubic graph of defect $4$. We begin with the
following auxiliary result.
\begin{lemma}\label{lem:M4}
Let $G$ be a cubic graph of defect $3$ and let
$\mathcal{M}=\{M_1,M_2,M_3\}$ be an optimal $3$-array of
perfect matchings for $G$. Then $G$ has a fourth perfect
matching $M_4$ which covers at least two of the three edges
left uncovered by $\mathcal{M}$.
\end{lemma}
\begin{proof}
Consider the $6$-cycle $C=(e_0e_1\ldots e_5)$ constituting the
core of $\mathcal{M}$. We adopt the notation introduced in
Figure~\ref{fig:core3}; in particular, $e_1$, $e_3$, and $e_5$
are the three uncovered edges of~$C$. We also assume that the
edge colouring $\phi$ associated with $\mathcal{M}$ takes
values as indicated in Figure~\ref{fig:core3}. By
Proposition~\ref{prop:notriply}, the colouring is proper.
An uncovered edge $e_i$ of $C$ will be considered \emph{bad} if
$G$ has a cycle-separating $3$-edge-cut $R_i$ containing the
edges $e_{i-1}$ and $e_{i+1}$. We claim that not all of the
edges $e_1$, $e_3$, and $e_5$ can be bad. Suppose to the
contrary that all of them are bad. For $i\in\{1,3,5\}$ let
$h_i$ denote the third edge of the cut $R_i$. Since the edges
$f_i$ and $f_{i+1}$ of $\delta_G(C)$ are both adjacent to
$e_i$, the set $R_i'=\{f_{i}, f_{i+1},h_i\}$ is also a
$3$-edge-cut. Moreover, all three edges of $R_i'$ are simply
covered. Let $L_i$ be the component of $G-R_i'$ that does not
contains $e_i$.
\begin{figure}
\caption{Constructing a $3$-edge-colouring of $G$ provided
that $G$ has three bad edges.
The stage preceding Kempe switches.}
\label{fig:constr}
\end{figure}
Recall that the restriction of $\phi$ to $G-E(C)$ is a proper
$3$-edge-colouring. We show that this colouring can be modified
and extended to a $3$-edge-colouring $\psi$ of the entire~$G$.
By Kirchhoff's law, $\phi(f_1)+\phi(f_2)+\phi(h_1)=0$. Since
$\phi(f_1)=3$, and $\phi(f_2)=2$, we have $\phi(h_1)=1$.
Similarly, $\phi(h_3)=3$ and $\phi(h_5)=2$. To define $\psi$,
we first set $\psi(e_1)=1$, $\psi(e_3)=3$, and $\psi(e_5)=2$,
and extend this definition to the uncovered edges of $G$ in
such a way that the hexagon $C$ is properly coloured. The
values of $\psi$ on $C$ now cause a conflict with the values of
$\phi$ on $\delta_G(C)$ at each vertex of $C$. However, this
problem can easily be resolved by the use of suitable Kempe
switches. Indeed, take the $(3,2)$-Kempe chain in $G-E(C)$ with
respect to $\phi$ that starts with the edge $f_1$ coloured~$3$;
denote the Kempe chain by $B_1$. Clearly, $B_1$ leaves $L_1$
through either $h_1$ or $f_2$. However, $\phi(h_1)=1$, so $B_1$
ends with $f_2$ (see Figure~\ref{fig:constr}).
After switching the colours of $\phi$ on $B_1$
we produce a new $3$-edge-colouring of $G-E(C)$, in which the
colouring conflicts at the vertices $v_1$ and~$v_2$ have been
removed. Proceeding similarly with Kempe chains stating at
$f_3$ and $f_5$ we resolve the conflicts at the remaining
vertices of $C$ and eventually obtain the required proper
$3$-edge-colouring $\psi$ of $G$. Since $G$ is a snark, this
cannot occur. Therefore, at least one uncovered edge of $C$
must not be bad.
Without loss of generality we may assume that the edge $e_1$ is
not bad. Take the path $P=e_3e_4e_5\subseteq C$ and set
$H=G-V(P)$. Clearly, $|\delta_G(H)|=6$, so
Theorem~\ref{thm:6cuts} applies. We claim that $H$ admits a
perfect matching. Suppose not. Then, by
Theorem~\ref{thm:6cuts}, $H$ contains an independent set $S$ of
vertices such that each component $L$ of $H-S$ has
$|\delta_G(L)|=3$ and each edge of $\delta_G(H)$ joins a vertex
of $H-S$ to a vertex of $P$. Since the edge $e_1$ belongs to
$H-S$, the component containing $e_1$ is nontrivial. It follows
that $e_1$ is bad, and we have arrived at a contradiction. Thus
$H$ admits a perfect matching, and this matching is readily
extended to a perfect matching $M_4$ of $G$ that covers the
uncovered edges $e_3$ and $e_5$. The proof is complete.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:1}]
Take an optimal $3$-array $\{M_1,M_2,M_3\}$ for $G$. It leaves
three uncovered edges. According to Lemma~\ref{lem:M4}, there
is a perfect matching $M_4$ that covers at least two of them.
The remaining uncovered edge (if any) can be covered by a
perfect matching $M_5$ guaranteed by Theorem~\ref{thm:Plesnik}.
Together these perfect matchings a Berge cover of $G$.
\end{proof}
In the rest of this section we prove that every cyclically
$4$-edge-connected cubic graph with defect $4$ admits a Berge
cover. Our proof significantly differs from the one provided by
Steffen \cite[Theorem~2.14]{S2} in that it avoids the use of
Seymour's results on $p$-tuple multicolourings
\cite{Seymour-multi} by employing Theorem~\ref{thm:6cuts}
instead.
First we establish the following lemma.
\begin{lemma}\label{lem:cycliccore}
Let $G$ be a cyclically $4$-edge-connected cubic graph. If $G$
has an optimal $3$-array whose core is a circuit of length $d$,
then $\pi(G)\le 3+\lceil d/4\rceil$.
\end{lemma}
\begin{proof}
Let $\mathcal{M}$ be an optimal $3$-array of $G$ whose core is
a circuit $C$ of length $d$. By Proposition~\ref{prop:core},
$C$ alternates uncovered and doubly covered edges; in
particular, $d$ is even. Note that $d\ge 6$ by
Proposition~~\ref{prop:core}.
Let $P=efg\subseteq C$ be an arbitrary path of length $3$ whose
middle edge is doubly covered. We show that $G$ has a perfect
matching containing both uncovered edges of $P$. Set
$H=G-V(P)$. If $H$ has a perfect matching, say $M$, then
$M\cup\{e,g\}$ is a perfect matching of $G$ containing both $e$
and $g$, and we are done. Thus we may assume that $H$ has no
perfect matching. By Theorem~\ref{thm:6cuts}, there exists an
independent set $S\subseteq V(H)$ such that $H-S$ satisfies
Items (i)--(iii) of the theorem. In particular, each component
$L$ of $H-S$ is odd and has $|\delta_G(L)|=3$. Since $G$ has no
non-trivial $3$-cuts, each component is just a vertex, implying
that $H$ is a bipartite graph. It follows that $C\cap H$ is a
path of even length, and hence $|E(C)|=|E(C\cap H)|+5$ is an
odd number, which is impossible. We have thus proved that there
exists a perfect matching containing any two uncovered edges of
$C$ joined by a doubly covered edge.
Now we finish the proof. It is easy to see that $C$ contains
$\lfloor d/4\rfloor$ pairwise edge-disjoint paths of length 3
that begin and end with an uncovered edge. At most one
uncovered edge remains outside these paths, which means that
the uncovered edges of $C$ can be covered by at most $\lceil
d/4\rceil$ paths of length 3 that begin and end with an
uncovered edge. In other words, there exists a set
$\mathcal{S}$ of $\lceil d/4 \rceil$ perfect matchings that
collectively contain all uncovered edges. Hence, $\pi(G)\le
|\mathcal{M}\cup\mathcal{S}|\le 3+\lceil d/4 \rceil$, as
claimed.
\end{proof}
\begin{theorem} Every cyclically $4$-edge-connected
cubic graph of defect $4$ admits a Berge cover.
\end{theorem}
\begin{proof}
Let $G$ be a cyclically $4$-edge-connected cubic graph of
defect $4$. Proposition~\ref{prop:core} implies that any core
with four uncovered edges must be either an 8-cycle or a
subgraph consisting of two disjoint triangles joined by a
triply covered edge. Since $G$ is cyclically
$4$-edge-connected, the latter possibility does not occur.
Hence, the core of any optimal $3$-array for $G$ is an 8-cycle,
and the result follows from Lemma~\ref{lem:cycliccore}.
\end{proof}
\section{Proof of Theorem~\ref{thm:2}}\label{sec:thm2}
\label{sec:c4c}
\noindent{}We prove that every cyclically $4$-edge-connected cubic graph
$G$ of defect $3$ has $\pi(G)=4$ with the only exception of the
Petersen graph.
\begin{proof}[Proof of Theorem~\ref{thm:2}]
Let $G$ be a cyclically $4$-edge-connected cubic graph with
$\df{G}=3$ and $\pi(G)>4$. Our aim is to show that $G$ is
isomorphic to the Petersen graph.
Let $C=(e_0e_1\ldots e_5)$ be the hexagonal core of an optimal
$3$-array $\mathcal{M}=\{M_1,M_2,M_3\}$ of $G$, and let
$H=G-V(C)$. We may assume that the colouring $\phi$ associated
with $\mathcal{M}$ and the names of the vertices and edges in
the vicinity of $C$ are as in Figure~\ref{fig:core3}. In
particular, the endvertices of the edges of the cut
$\delta_G(C)$ not lying on $C$ are $u_0$, $u_1$, \ldots, $u_5$
and are listed in a cyclic order around $C$. Set
$U=\{u_0,\ldots,u_5\}$.
\noindent Claim 1. \emph{The subgraph $H$ is bipartite.}
\noindent
\emph{Proof of Claim 1.} If $H$ contains a perfect matching,
then the matching can be extended to a perfect matching $M_4$
of $G$ which covers the three uncovered edges of $C$. It
follows that $\pi(G)=4$, contrary to our assumption. Therefore $H$ has
no perfect matching. In this situation Theorem~\ref{thm:6cuts} tells
us that there exists a set $S\subseteq V(H)$ such that each
component $L$ of $H-S$ has $|\delta_G(L)|=3$ and each edge of
$\delta_G(H)$ joins a vertex of $H-S$ to a vertex of $C$. As
$G$ is cyclically 4-edge-connected, each component of $H-S$
is a single vertex. Consequently, $H$
is a bipartite graph with bipartition $\{S, V(H)-S\}$ and edge set
$\delta_G(S)$. This establishes Claim~1.
Next we explore the colouring properties of the subgraph
$H^+=G-E(C)\subseteq G$. Note that
$H^+=H+\{f_0,f_1,\ldots,f_5\}$ and $H^+\cup C=G$. Set
$C^+=C+\{f_0,f_1,\ldots,f_5\}$. In $C^+$, the endvertices
$u_0$, $u_1$, \dots, $u_5$ are assumed as pairwise distinct.
Before proceeding further we need several definitions. A
\emph{colour vector} is any sequence $\alpha=c_0c_1\ldots c_5$
of six colours $c_i\in\{1,2,3\}$ satisfying Parity Lemma; that
is, $c_0+\ldots+c_5=0$. Every $3$-edge-colouring $\sigma$ of
$H^+$ thus \emph{induces} the colour vector
$\alpha_{\sigma}=\sigma(f_0)\sigma(f_1)\ldots\sigma(f_5)$. By
permuting the colours $1$, $2$, and $3$ in $\alpha$ we obtain a
new colour vector, nevertheless, the difference between the two
is insubstantial. Therefore each of the six permutations of
colours produces a sequence that encodes essentially `the same
colour vector'. In order to have a canonical representative, we
choose from them the lexicographically minimal sequence and
call it the \emph{type} of $\alpha$. A \emph{colouring type} is
the type of some colour vector~$\alpha$. The \emph{type} of a
$3$-edge-colouring $\sigma$ of $H^+$ is the type of the induced
colour vector
$\sigma(f_0)\sigma(f_1)\ldots\sigma(f_5)$. A
colouring type is said to be \emph{admissible} for $H^+$ if it
is the type of a certain $3$-edge-colouring of $H^+$. Similar
definitions apply to $C^+$.
\noindent
Claim 2. \emph{The colour vector induced by any
$3$-edge-colouring of $H^+$ involves all three~colours.}
\noindent
\emph{Proof of Claim 2.} Consider the colour vector induced by
a $3$-edge-colouring of $H^+$. If there was a colour that does
not appear in it, then the corresponding colour class could be
extended to a perfect matching of the entire $G$ that covers
the three uncovered edges of $C$. Consequently, $G$ could be
covered with four perfect matchings, contrary to our
assumption. Claim~2 is proved.
There exist exactly fifteen colouring types involving all three
colours; they are listed as
$\alpha_1,\alpha_2,\ldots,\alpha_{15}$ in the first column of
Table~\ref{tbl:types} following the lexicographic order. Let
$\mathcal{A}$ denote the set comprising all of them. We now
specify three subsets of $\mathcal{A}$:
\begin{align*}
\mathcal{B} & = \{123132, 121323, 123213\},\\
\mathcal{C} & = \{112332, 122133, 123123, 123321\},\quad\text{ and}\\
\mathcal{P} & = \{112233, 112323, 121233, 121332, 122313, 122331,
123231, 123312\}.
\end{align*}
Clearly, $\mathcal{B}$, $\mathcal{C}$, and $\mathcal{P}$ form a
partition of $\mathcal{A}$. The sets $\mathcal{C}$ and
$\mathcal{P}$ have a straightforward interpretation if we
choose our graph $G$ to be the Petersen graph $Pg$ and $C$ the
hexagonal core of an arbitrary optimal $3$-array. In this case,
$\mathcal{C}$ and $\mathcal{P}$ are simply the sets of all
colouring types including all three colours that are admissible
for $C^+$ and \mbox{$H^+=(Pg-V(C))^+$}, respectively.
\begin{table}[htb!]
\centering
\begin{tabular}{ccc}
\toprule
colouring type & & contained in\\
\midrule
$\alpha_1=112233$ & & $\mathcal{P}$\\
$\alpha_2=112323$ & & $\mathcal{P}$\\
$\alpha_3=112332$ & & $\mathcal{C}$\\
$\alpha_4=121233$ & & $\mathcal{P}$\\
$\alpha_5=121323$ & & $\mathcal{B}$\\
$\alpha_6=121332$ & & $\mathcal{P}$\\
$\alpha_7=122133$ & & $\mathcal{C}$\\
$\alpha_8=122313$ & & $\mathcal{P}$\\
$\alpha_9=122331$ & & $\mathcal{P}$\\
$\alpha_{10}=123123$ & & $\mathcal{C}$\\
$\alpha_{11}=123132$ & & $\mathcal{B}$\\
$\alpha_{12}=123213$ & & $\mathcal{B}$\\
$\alpha_{13}=123231$ & & $\mathcal{P}$\\
$\alpha_{14}=123312$ & & $\mathcal{P}$\\
$\alpha_{15}=123321$ & & $\mathcal{C}$\\
\bottomrule
\end{tabular}
\operatorname{bi}\xspacegskip
\caption{The fifteen colouring types involving all three
colours.} \label{tbl:types}
\end{table}
\noindent
Claim 3. \emph{Every colouring type admissible for $H^+$
belongs to $\mathcal{P}$.}
\noindent
\emph{Proof of Claim 3.} To prove the claim it is sufficient to
exclude the existence of colourings of $H^+$ whose type belongs
to $\mathcal{B}\cup\mathcal{C}$. We first observe that $H^+$
cannot have a colouring of any type from
$\mathcal{C}=\{\alpha_3,\alpha_7,\alpha_{10},\alpha_{15}\}$.
Indeed, if it had, then such a colouring could easily be
extended to a $3$-edge-colouring of the entire $G$. Since $G$
is a snark, this cannot happen. Thus it remains to deal with
colourings whose type belongs to
$\mathcal{B}=\{\alpha_5,\alpha_{11},\alpha_{12}\}$.
Suppose that $H^+$ admits a colouring of type
$\alpha_5=121323$. Take the $(1,3)$-Kempe chain starting
at~$f_0$. It must terminate at $f_2$, $f_3$, or $f_5$, and the
corresponding Kempe switches yield the colour vectors $323323$,
$321123$, or $321321$, respectively. The first of them has a
missing colour, and therefore cannot occur, by Claim~2.
The latter two have the type that belongs to $\mathcal{C}$,
which we have already excluded. Therefore $H^+$ has no
colouring of type type $\alpha_5$.
Next, consider the colouring type $\alpha_{11}=123132$.
Consider the $(2,3)$-Kempe chain starting at $f_1$. The chain
necessarily terminates at $f_2$, $f_4$, or $f_5$, and the
corresponding switches yield the colour vectors $132132$,
$133122$, or $133133$, respectively. The first two colour vectors
have the type that belongs to $\mathcal{C}$, while the last
one has a missing colour. Hence, the colouring type
$\alpha_{11}$ is also excluded.
Finally we deal with the colouring type $\alpha_{12}=123213$.
This time we consider the $(1,2)$-Kempe chain starting at
$f_0$. Its terminal edge must be one of $f_1$, $f_3$, or $f_4$,
and the corresponding switches produce the vectors $213213$,
$223113$, or $223223$, respectively. Again, the first two
colour vectors have the type that belongs
to $\mathcal{C}$, while last one has a missing colour.
The colouring type $\alpha_{12}$ is thus excluded as well.
Summing up, we have shown that $H^+$ admits no colouring whose
type belongs to $\mathcal{B}\cup\mathcal{C}$. It follows that
$H^+$ only admits colourings whose type belongs to
$\mathcal{P}$. Claim~3 is proved.
We continue with the proof by analysing the set
$U=\{u_0,\ldots,u_5\}$. In general, the vertices $u_0$, $u_1$,
\ldots, $u_5$ need not be pairwise distinct, and some of them
may coincide. Clearly, it cannot happen that $u_j=u_{j+1}$ for
some index $j$, for otherwise $G$ would contain a triangle
$(u_jv_jv_{j+1})$, which is impossible because $G$ is
cyclically $4$-edge-connected. If $u_j=u_{j+2}$ for some
index~$j$, then $G$ contains a quadrangle
$(u_jv_jv_{j+1}v_{j+2})$ whose intersection with $C$ is a path
of length two. However, this contradicts Lemma~\ref{lem:3+4}.
Thus, the only possibility is that some of the pairs $u_i$ and
$u_{i+3}$ coincide. Let $x$ be the number of such pairs;
clearly $x\in\{0,1,2,3\}$. We prove that $x=3$.
\begin{figure}
\caption{Constructing $H^{\sharp}
\label{obr:HPQ}
\end{figure}
Suppose to the contrary that $x\le 2$. Without loss of
generality, we may assume that $u_2\neq u_5$. We now take the
graph $H^+$ and create from it a new cubic graph $H^{\sharp}$
as follows. First, we remove the edges $f_2$ and $f_5$ and add
an edge $e$ between $u_2$ and $u_5$. Then we identify the
endvertices $v_0$ and $v_1$ of $f_0$ and $f_1$ into a new
vertex $s$. Similarly, we identify the endvertices $v_3$ and
$v_4$ of $f_3$ and $f_4$ into a vertex $t$. Finally, we add an
edge $f$ between $s$ and $t$; see Figure~\ref{obr:HPQ}.
We prove that $H^{\sharp}$ is not $3$-edge-colourable. If
$H^{\sharp}$ had a $3$-edge-colouring, then such a colouring
would result from a $3$-edge-colouring $\sigma$ of $H^+$ whose
colour type $\alpha=c_0c_1\ldots c_5$ satisfies all of the
following conditions:
\begin{itemize}
\item[(i)] $c_0\ne c_1$ \quad ($\sigma$ is proper at $s$),
\item[(ii)] $c_3\ne c_4$ \quad ($\sigma$ is proper at
$t$), and
\item[(iii)] $c_2=c_5$ \quad ($e$ is properly coloured).
\end{itemize}
Moreover, Claim~3 tells us that $\alpha\in\mathcal{P}$.
However, by checking the elements of $\mathcal{P}$ we see that
the colouring types $\alpha_1$ and $\alpha_2$ violate (i),
$\alpha_6$ and $\alpha_9$ violate~(ii), and $\alpha_4$,
$\alpha_8$, $\alpha_{13}$, and $\alpha_{14}$ violate (iii). In
other words, no element of $\mathcal{P}$ satisfies all the
conditions (i)-(iii), and therefore $H^{\sharp}$ admits no
$3$-edge-colouring.
Observe that $H^{\sharp}-\{e,f\}$ is bipartite with
bipartition $\{S\cup \{s,t\},(V(H)-S)\}$. Since $H^{\sharp}$ is
not $3$-edge-colourable, from Theorem~\ref{thm:abg_class1} we
deduce that $H^{\sharp}$ must have a bridge, say $b$. A simple
counting argument shows that in $H^{\sharp}$ the bridge $b$
separates $e$ from~$f$. Let $Q$ be the component of
$H^{\sharp}-b$ containing $e$.
Now, $Q-e$ is a subgraph of $G$ separated from the rest of $G$
by the cut $\delta_G(Q-e)=\{b,f_2,f_5\}$. Since $Q$ has at
least two vertices and so does the other component of
$G-\delta_G(Q-e)$, we conclude that $\delta_G(Q-e)$ is a
cycle-separating $3$-edge-cut in $G$. This contradicts the
assumption that $G$ is cyclically $4$-edge-connected, and
proves that $x=3$.
We have thus proved that $u_0=u_3$, $u_1=u_4$, and $u_2=u_5$.
Consider the subgraph $J\subseteq G$ induced by the set
$C\cup\{u_0,u_1,u_2\}$. Each of the vertices $u_0$, $u_1$, and
$u_2$ is $2$-valent in $J$, which means that $\delta_G(J)$ is a
$3$-edge-cut. As $G$ is cyclically $4$-edge-connected,
$G-(C\cup\{u_0,u_1,u_2\})$ must be a single vertex. Now it is
immediate that $G$ is isomorphic to the Petersen graph.
Summing up, we have proved that every cyclically
$4$-edge-connected cubic graph $G$ with $\df{G}=3$ and
$\pi(G)\ge 5$ is isomorphic to the Petersen graph, as required.
This completes the proof.
\end{proof}
\section{Family of snarks with defect 3 and perfect matching index 5}
\noindent{}The requirement of cyclic connectivity at least $4$
in Theorem~\ref{thm:2} is essential, because there exist
infinitely many $3$-edge-connected snarks with defect $3$ and
perfect matching index~$5$. They can be constructed as follows:
Take the Petersen graph $Pg$ and remove a vertex $v$ from it
together with the three incident edges, leaving a graph $P$
containing three $2$-valent vertices. Further, take an
arbitrary connected bipartite cubic graph on at least four
vertices, and similarly remove a vertex $u$, leaving a graph
$B$ with tree $2$-valent vertices. Create a cubic graph $H$ by
joining every $2$-valent vertex of $P$ to a $2$-valent vertex
of $B$. Clearly, the set $R$ consisting of the newly added
edges is a cycle-separating $3$-edge-cut in $H$. We claim that
the graph $H$ has defect 3 and perfect matching index $5$.
We first observe that $\df{H}=3$. Parity Lemma implies that $P$
is uncolourable, so $H$ is a snark, and therefore $\df{H}\ge3$.
Since $Pg$ is vertex-transitive, one can choose an optimal
array for $Pg$ in such a way that its hexagonal core
avoids the vertex~$v$.
As every bipartite graph is $3$-edge-colourable, one can easily
extend the $3$-array of $Pg$ to a $3$-array of $H$ which
retains the original core of $Pg$. Hence, $\df{H}=3$.
Now we want to prove that $\pi(H)=5$. Suppose to the contrary
that $\pi(H)\le4$. Then $H$ has a covering
$\mathcal{C}=\{M_1,M_2,M_3,M_4\}$ with four perfect matchings.
Since both $P$ and $B$ contain an odd number of vertices, we
conclude that each $M_i$ has an odd number of common edges with
$R$. Moreover, since each edge of $H$ is in at most two of the
four perfect matchings, the weights of edges in the edge cut
$R$ are either $1,1,2$, or $2,2,2$. The former possibility is
excluded, because contracting $B$ to a single vertex would
produce a covering of $Pg$ with four perfect matchings,
contradicting the fact that $\pi(Pg)=5$. Therefore the weight
of each edge in $R$ equals $2$. Now, each vertex of $B$ is
incident with precisely one edge of weight $2$, including those
in $R$. It follows that the simply covered edges form a
$2$-factor of $B$, say $F$. However, $B$ has an odd number of
vertices, so at least one circuit of $F$ is odd, which is
impossible because $B$ is bipartite. Hence, $\pi(H)\ge 5$, and
from Theorem~\ref{thm:1} we infer that $\pi(H)=5$.
\end{document} | math | 51,711 |
\begin{document}
\title{On the degree of non-Markovianity of quantum evolution}
\author{Dariusz Chru\'sci\'nski$^1$ and Sabrina Maniscalco$^{2}$\\
$^1$Institute of Physics, Nicolaus Copernicus University,
Grudzi\c{a}dzka 5/7, 87--100 Toru\'n, Poland\\
$^2$SUPA, EPS/Physics, Heriot-Watt University, Edinburgh, EH14 4AS, United Kingdom }
\begin{abstract}
We propose a new characterization of non-Markovian quantum evolution based on the concept of non-Markovianity degree. It provides an analog of a Schmidt number in the entanglement theory and reveals the formal analogy between quantum evolution and the entanglement theory: Markovian evolution corresponds to a separable state and non-Markovian one is further characterized by its degree. It enables one to introduce a non-Markovinity witness -- an analog of an entanglement witness -- and a family of measures -- an analog of Schmidt coefficients -- and finally to characterize maximally non-Markovian evolution being an analog of maximally entangled state.
Our approach allows to classify the non-Markovianity measures introduced so far in a unified rigorous mathematical framework.
\end{abstract}
\pacs{03.65.Yz, 03.65.Ta, 42.50.Lc}
\maketitle
{\em Introduction} --- Open quantum systems and their dynamical features are
attracting increasing attention nowadays. They are of paramount
importance in the study of the interaction between a quantum
system and its environment, causing dissipation, decay, and
decoherence \cite{Breuer,Weiss,Alicki}. On the other hand, the robustness of
quantum coherence and entanglement against the detrimental
effects of the environment is one of the major focuses in
quantum-enhanced applications, as both entanglement and
quantum coherence are basic resources in modern quantum
technologies, such as quantum communication, cryptography,
and computation \cite{QIT}. Recently, much effort was devoted to {the} description, analysis and classification of non-Markovian quantum evolution (see e.g. \cite{R1}--\cite{Pater} and the collection of papers in \cite{RECENT}). In particular various concepts of non-Markovianity were introduced and several so called non-Markovianity measures were proposed. The main approaches to the problem of (non)Markovian evolution are based on divisibility \cite{Wolf,RHP,Hou}, distinguishability of states \cite{BLP},
quantum entanglement \cite{RHP}, quantum Fisher information flow \cite{Fisher}, fidelity \cite{fidelity}, mutual information \cite{Luo1,Luo2}, channel capacity \cite{Bogna}, and geometry of the set of accessible states \cite{Pater}.
In this Letter we accept the definition based on divisibility \cite{Wolf}: the quantum evolution is Markovian if the corresponding dynamical map $\Lambda_t$ is CP-divisible, that is,
\begin{equation}\label{CP}
\Lambda_t = V_{t,s} \Lambda_s \ ,
\end{equation}
and $V_{t,s}$ provides a family of legitimate (completely positive and trace-preserving) propagators for all $t\geq s\geq 0$.
The essential property of $V_{t,s}$ is the following composition law
$ V_{t,s} \, V_{s,u} = V_{t,u}$,
for all $t\geq s\geq u$. It provide{s} a natural generalization of a semigroup law $e^{tL} e^{sL} = e^{(t+s)L}$. Interestingly, the very property of CP-divisibility is fully characterized in terms of the time-local generator $L_t$: if $\Lambda_t$ satisfies time-local master equation $\dot{\Lambda}_t = L_t \Lambda_t$, then $\Lambda_t$ is CP-divisible iff $L_t$ has the standard Lindblad form for all $t \geq 0$, i.e.
$$ L_t\rho = -i[H(t),\rho] + \sum_\alpha \left( V_\alpha(t) \rho V_\alpha^\dagger(t) - \frac 12 \{ V_\alpha^\dagger(t) V_\alpha(t),\rho\} \right), $$
with time-dependent Lindblad (noise) operators $V_\alpha(t)$ and time-dependent effective system Hamiltonian $H(t)$ \cite{GKS,Lindblad,Alicki}. A very appealing concept of Markovianity was proposed by Breuer, Lane and Piilo (BLP) \cite{BLP}: $\Lambda_t$ is Markovian if
\begin{equation}\label{BLP}
\sigma(\rho_1,\rho_2;t) = \frac{d}{dt}||\Lambda_t(\rho_1-\rho_2)||_1 \leq 0 \ ,
\end{equation}
for all pairs of initial states $\rho_1$ and $\rho_2$. BLP call $\sigma(\rho_1,\rho_2;t)$ an information flow and interpret $\sigma(\rho_1,\rho_2;t)> 0$ as a backflow of information from the environment to the system which clearly indicates the non-Markovian character of the evolution. As usual $||X||_1$ denotes the trace norm of $X$, i.e. $||X||_1 = {\rm Tr}\sqrt{XX^\dagger}$. It turns out that CP-divisibility implies (\ref{BLP}) but the converse needs not be true \cite{versus1,versus2,versus3}.
In this Letter we propose {a} more refine{d} approach to non-Markovian evolution. We reveal the formal analogy with the entanglement theory: Markovian evolution corresponds to separable state and non-Markovian evolution is characterized by {a} positive integer --- {the} non-Markovianity degree --- corresponding to the Schmidt number of an entangled state. The notion of non-Markovianity degree enables one to introduce a family of measures and finally to characterize maximally non-Markovian evolution being an analog of maximally entangled state.
{\em Schmidt number and $k$-positive maps} --- Let us recall that a state of {a} composite quantum system may be uniquely characterized by its Schmidt number \cite{HHHH,Pawel}: for any normalized vector $\psi \in \mathcal{H} {\,\otimes\,} \mathcal{H}$ let ${\rm SR}(\psi)$ denote the Schmidt rank of $\psi$, i.e. a number of non-vanishing Schmidt coefficients in the decomposition $\psi = \sum_k s_k e_k {\,\otimes\,} f_k$, with $s_k > 0$ and $\sum_k s_k^2=1$. Now, for any density operator $\rho$ one defines its Schmidt number by
\begin{equation}\label{}
{\rm SN}(\rho) = \min_{p_k,\psi_k}\, \{ \max_k \, {\rm SR}(\psi_k) \}\ ,
\end{equation}
where the minimum is performed over all decompositions $\rho = \sum_k p_k |\psi_k\rangle\langle\psi_k|$ with $p_k >0$ and $\sum_k p_k=1$. Let $S_k= \{ \, \rho\, |\, {\rm SN}(\rho) \leq k\, \}$. One has
\begin{equation}\label{SSS}
S_1 \subset S_2 \subset \ldots \subset S_n \ ,
\end{equation}
where $S_1$ denotes a set of separable states and $S_n$ denotes a set of all states in $\mathcal{H} {\,\otimes\,} \mathcal{H}$. Note that a maximally entangled state $\psi$ satisfies $\lambda_1 = \ldots = \lambda_n$ and the corresponding projector $|\psi\rangle\langle\psi|$ defines an element of $S_n$. The Schmidt number does not increase under local operation, i.e. ${\rm SN}([\mathcal{E}_1 {\,\otimes\,} \mathcal{E}_2]\rho) \leq {\rm SN}(\rho)$, where $\mathcal{E}_1$ and $\mathcal{E}_2$ are arbitrary quantum channels. Moreover, if $\Phi$ is a $k$-positive map, i.e. $\oper_k {\,\otimes\,} \Phi$ is positive, then for any $\rho \in S_k$ one has $[\oper_k {\,\otimes\,} \Phi](\rho) \geq 0$ ($\oper_k$ denotes an identity map acting in $M_k$ --- the space of $k\times k$ complex matrices). This simple property establishes a duality between $k$-positive maps and quantum bipartite states with the Schmidt number bounded by $k$.
{\em Non-Markovianity degree} --- The notion of $k$-positive maps enables one to provide a natural generalization of CP-divisibility: we call a dynamical map $\Lambda_t$ $k$-divisible iff $V_{t,s}$ is $k$-positive for all $t\geq s\geq 0$. Hence, $n$-divisible maps are CP-divisible and 1-divisible are simply P-divisible, i.e. $V_{t,s}$ is positive. Now, we introduce a degree of non-Markovianity which is an analog of a Schmidt number: a dynamical map $\Lambda_t$ has a non-Markovianity degree ${\rm NMD}[\Lambda_t]=k$ iff $\Lambda_t$ is $(n-k)$ but not $(n+1-k)$--divisible.
It is clear that $\Lambda_t$ is Markovian iff ${\rm NMD}[\Lambda_t]=0$ and essentially non-Markovian iff ${\rm NMD}[\Lambda_t]=n$. Denoting by
$ \mathcal{N}_k = \{ \, \Lambda_t\ | \ {\rm NMD}[\Lambda_t] \leq k \, \}$,
one has a natural chain of inclusions
\begin{equation}\label{NNN}
\mathcal{N}_0 \subset \mathcal{N}_1 \subset \ldots \subset \mathcal{N}_{n-1} \subset \mathcal{N}_n\ ,
\end{equation}
where $\mathcal{N}_0$ denotes Markovian maps and $\mathcal{N}_n$ all dynamical maps. The characterization of $k$-divisible maps is provided by the following
\begin{Theorem}
If $\Lambda_t$ is $k$-divisible, then
\begin{equation}\label{contr-k}
\frac {d}{dt}\, || [\oper_k {\,\otimes\,} \Lambda_t](X) ||_1 \leq 0\ ,
\end{equation}
for all operators $X \in M_k {\,\otimes\,} \mathcal{B}(\mathcal{H})$.
\end{Theorem}
For the proof see Supplementary material. In particular all $k$-divisible maps ($k=1,\ldots,n$) satisfy
\begin{equation}\label{contr-1}
\frac {d}{dt}\, || \Lambda_t(X) ||_1 \leq 0\ ,
\end{equation}
for all $X \in \mathcal{B}(\mathcal{H})$. Note, that BLP condition (\ref{BLP}) is a special case of (\ref{contr-1}) with $X$ being traceless Hermitian operator. It is, therefore, clear that BLP condition is weaker than all conditions in the hierarchy (\ref{contr-k}) and it is satisfied for all $k$-divisible maps not necessarily CP-divisible. According to our definition of Markovianity (Markovianity = CP-divisibility) $k$-divisible maps which are not CP-divisible are clearly non-Markovian. However, such non-Markovian evolution always satisfy (\ref{contr-1}). We propose to call such dynamical maps {\em weakly non-Markovian}. Dynamical map which is even not P-divisible will be called {\em essentially non-Markovian}. Hence, $\Lambda_t$ is weakly non-Markovian iff $\Lambda_t \in \mathcal{N}_{n-1} - \mathcal{N}_0$ and it is essentially non-Markovian iff $\Lambda_t \in \mathcal{N}_{n} - \mathcal{N}_{n-1}$. Using the notion of degree of non-Markovianity $\Lambda_t$ is weakly non-Markovian iff $0< {\rm NMD}[\Lambda_t] \leq n-1$ and it is essentially non-Markovian iff ${\rm NMD}[\Lambda_t] =n$. Note that maps which violate BLP condition are always essentially non-Markovian. Similarly, if $\Lambda_t$ is at least 2-divisible, then the relative entropy satisfies the following monotonicity property \cite{Petz}
\begin{equation}\label{RE}
\frac {d}{dt}\, S( \Lambda_t(\rho_1) || \Lambda_t(\rho_2)) \leq 0\ ,
\end{equation}
for any pair $\rho_1$ and $\rho_2$. The violation of (\ref{RE}) means that $\Lambda_t$ is at most P-divisible or essentially non-Markovian.
{\em Non-Markovianity witness} --- Actually, if $\Lambda_t$ is invertible, then it is $k$-divisible if and only if (\ref{contr-k}) holds. Clearly, a generic map is invertible (all its eigenvalues are different from zero) and hence this result is true for a generic dynamical map (a notable exception is Jaynes-Cummings model on resonance \cite{Breuer,JC}).
Hence, if (\ref{contr-k}) is violated for some $t > 0$, then $\Lambda_t$ is not $k$-divisible or equivalently ${\rm NMD}[\Lambda_t] > n-k$. It is, therefore, natural to call such $X$ a non-Markovianity witness in analogy to the well known concept of an entanglement witness. Recall, that a Hermitian operator $W$ living in $\mathcal{H} {\,\otimes\,} \mathcal{H}$ is an entanglement witness \cite{HHHH} iff $i)$ $\langle\Psi|W|\Psi\rangle \geq 0$ for all product vectors $\Psi=\psi{\,\otimes\,} \phi$ and $ii)$ $W$ is not a positive operator, i.e. it possesses at least one negative eigenvalue. Similarly, $W$ is a $k$-Schmidt witness \cite{Sanpera} if $\langle\Psi|W|\Psi\rangle \geq 0$ for all vectors $\Psi=\psi_1{\,\otimes\,} \phi_1 + \ldots + \psi_k {\,\otimes\,} \phi_k$, that is, if ${\rm Tr}(\rho W) <0$, then $\rho$ is entangled and moreover ${\rm SN}(\rho) > k$. Note, that if $X \geq 0$, then (\ref{contr-k}) is always satisfied due to the fact that
$|| [\oper_k {\,\otimes\,} \Lambda_t](X) ||_1 = ||X||_1$. Hence, similarly as $W$, a non-Markovianity witness $X$ has to possess a negative eigenvalue.
{\em Non-Markovianity measures} --- The above construction allows to define a series of natural measures measuring departure from $k$-divisibility:
\begin{equation}\label{}
{\cal M}_k[\Lambda_t] = \sup_{X} \frac{N_k^+[X]}{|N_k^-[X]|} \ ,
\end{equation}
where
\begin{eqnarray*}\label{}
N^+_k[X] &=& \int_{\lambda_k(X;t) >0} \lambda_k(X;t) dt\ ,
\end{eqnarray*}
and similarly for $ N^+_k[X]$ (where now one integrates over time intervals such that $\lambda_k(X;t) < 0$), and
\begin{equation}\label{}
\lambda_k(X;t) = \frac {d}{dt}\, || [\oper_k {\,\otimes\,} \Lambda_t](X) ||_1\ .
\end{equation}
The supremum is taken over all Hermitian $X \in M_k {\,\otimes\,} \mathcal{B}(\mathcal{H})$.
Note that
\begin{eqnarray*}
&& \int_0^\infty \frac {d}{dt}\, || [\oper_k {\,\otimes\,} \Lambda_t](X) ||_1 \, dt \\ && = ||[\oper_k {\,\otimes\,} \Lambda_\infty](X) ||_1 - ||X||_1 \leq 0\ ,
\end{eqnarray*}
and hence $|N_-[\Lambda_t]| \geq N_+[\Lambda_t|$ which proves that $\mathcal{M}_k[\Lambda_t] \in [0,1]$. Clearly, if $l >k$, then $\mathcal{M}_l[\Lambda_t] \geq \mathcal{M}_k[\Lambda_t]$ and hence
\begin{equation*}
0 \leq \mathcal{M}_1[\Lambda_t] \leq \ldots \leq \mathcal{M}_n[\Lambda_t] \leq 1\ ,
\end{equation*}
which provides an analog of a similar relation among the Schmidt coefficients $s_1\geq \ldots \geq s_n$. Now, following the analogy with an entanglement theory, we may call $\Lambda_t$ maximally non-Markovian iff $\mathcal{M}_1[\Lambda_t]=1$ which immediately implies
\begin{equation}\label{}
\mathcal{M}_1[\Lambda_t] = \ldots = \mathcal{M}_n[\Lambda_t] =1\ ,
\end{equation}
in a perfect analogy with maximally entangled state corresponding to $s_1 = \ldots = s_n$.
{\em Examples} --- Let us illustrate the above introduced notions by a few simple examples.
\begin{Example}
Consider pure decoherence of a qubit system described by the following local generator
\begin{equation}\label{pure}
L_t(\rho) = \frac 12 \gamma(t) (\sigma_z \rho \sigma_z - \rho) \ ,
\end{equation}
The corresponding evolution of the density matrix reads
\begin{equation}\label{}
\rho_t = \left( \begin{array}{cc} \rho_{11} & \rho_{12} e^{-\Gamma(t)} \\ \rho_{12} e^{-\Gamma(t)} & \rho_{22} \end{array} \right) \ ,
\end{equation}
where $\Gamma(t) = \int_0^t \gamma(\tau) d\tau$. The evolution is completely positive iff $\Gamma(t) \geq 0$ and it is $k$-divisible ($k=1,2$) iff $\gamma(t) \geq 0$. Taking $X = \sigma_x $ one finds $||\Lambda_t(X)||_1 = 2e^{-\Gamma(t)}$. Observe that
\begin{equation}\label{}
|N_-[\Lambda_t]| = N_+[\Lambda_t] + e^{-\Gamma(\infty)} - 1 \ ,
\end{equation}
and hence if $\Gamma(\infty)=0$ the evolution is maximally non-Markovian. Note, that $\Gamma(\infty)=0$ implies that $\rho_t \rightarrow \rho$, that is, asymptotically one always recovers an initial state -- perfect recoherence. Actually, this example may be immediately generalized as follows: let $L$ be a Lindblad generator and consider a time-dependent generator defined by $L_t = \gamma(t) L$. Now, $L_t$ gives rise to a legitimate quantum dynamical map iff $\Gamma(t) \geq 0$ and it is $k$-divisible ($k=1,2,\ldots,n$) iff $\gamma(t)\geq 0$. The corresponding dynamics is maximally non-Markovian if $\Gamma(\infty)=0$.
\end{Example}
\begin{Example} Consider the qubit dynamics governed by the time-dependent generator
\begin{equation}\label{III}
L_t(\rho) = \frac 12 \sum_{k=1}^3 \gamma_k(t) (\sigma_k \rho \sigma_k - \rho) \ .
\end{equation}
It is clear that (\ref{III}) provides simple generalization of (\ref{pure}) by introducing two additional decoherence channels.
The corresponding dynamical map reads
\begin{equation}\label{rud}
\Lambda_t(\rho) = \sum_{\alpha=0}^3 p_\alpha(t) \sigma_\alpha \rho \sigma_\alpha \ ,
\end{equation}
where $\sigma_0 = \mathbb{I}$, and the probability distribution $p_\alpha(t)$ is defined as follows
\begin{eqnarray*}
p_0(t) &=&\frac{1}{4}\, [1+ \lambda_3(t) + \lambda_2(t) + \lambda_1(t)] \ , \\
p_1(t) &=&\frac{1}{4}\, [1- \lambda_3(t) - \lambda_2(t) + \lambda_1(t)] \ ,\\
p_2(t) &=&\frac{1}{4}\, [1- \lambda_3(t) + \lambda_2(t) - \lambda_1(t)] \ ,\\
p_3(t) &=&\frac{1}{4}\, [1+ \lambda_3(t) - \lambda_2(t) - \lambda_1(t)] \ ,
\end{eqnarray*}
where $\lambda_1(t) = e^{-[\Gamma_2(t) + \Gamma_3(t)]}$ and similarly for $\lambda_2(t)$ and $\lambda_3(t)$. Finally, $\Gamma_k(t) = \int_0^t \gamma_k(\tau) d\tau$. Again, the map $\Lambda_t$ is completely positive iff $\Gamma_k(t) \geq 0$ for $k=1,2,3$. Interestingly, in this example there is an essential difference between CP-divisibility (= Markovianity) and only P-divisibility: CP-divisibility is equivalent to
\begin{equation}\label{123}
\gamma_1(t) \geq 0 \ , \ \ \gamma_2(t) \geq 0 \ , \ \ \gamma_3(t) \geq 0 \ ,
\end{equation}
whereas P-divisibility is equivalent to much weaker conditions \cite{PLA}
\begin{eqnarray}\label{2gamma}
\gamma_1(t) + \gamma_2(t) &\geq& 0 \ , \nonumber \\
\gamma_1(t) + \gamma_3(t) &\geq& 0 \ , \\
\gamma_2(t) + \gamma_3(t) &\geq& 0 \ , \nonumber
\end{eqnarray}
for all $t\geq 0$. Actually, the BLP condition reproduces (\ref{2gamma}). Now, violation of at least one inequality from (\ref{2gamma}) implies essential non-Markovianity. Suppose for example that $\gamma_2(t) + \gamma_3(t) \ngeq 0$. Assuming that $\Gamma_2(\infty)=\Gamma_3(\infty)=0$ one finds that $ \mathcal{M}_1[\Lambda_t] =1$, that is, $\Lambda_t$ is maximally non-Markovian. Interestingly, if there are at most two decoherence channels, then there is no difference between CP- and P-divisibility. Note, that random unitary dynamics (\ref{rud}) is unital, i.e. $\Lambda_t(\mathbb{I}) = \mathbb{I}$ and hence during the evolution the entropy never decreases $S(\Lambda_t(\rho)) \geq S(\rho)$ for any initial qubit state $\rho$. One easily shows that
P-divisibility is equivalent to
\begin{equation}\label{S1}
\frac{d}{dt} S(\Lambda_t(\rho)) \geq 0 \ ,
\end{equation}
for any qubit state $\rho$. Hence, for any weakly non-Markovian random unitary dynamics the von Neumann entropy monotonically increases. Violation of (\ref{S1}) proves that $\Lambda_t$ is essentially non-Markovian.
\end{Example}
\begin{Example}
Consider a qubit dynamics governed by the following local generator
\begin{equation}\label{}
L_t = \gamma_+(t) L_+ + \gamma_-(t) L_-\ ,
\end{equation}
where
\begin{eqnarray*}
L_+(\rho) &=& \frac 12 ([\sigma_+,\rho \sigma_-] + [\sigma_+\rho, \sigma_-]) \ ,\\
L_-(\rho) &=& \frac 12 ( [\sigma_-,\rho \sigma_+] + [\sigma_-\rho, \sigma_+]) \ ,
\end{eqnarray*}
{with} $\sigma_+ = |2\rangle\langle1|$ and $\sigma_- = |1\rangle\langle2|$. $L_+$ generates pumping from the ground state $|1\rangle$ to an excited state $|2\rangle$ and $L_-$ generates a decay from $|2\rangle$ to $|1\rangle$. One shows that $L_t$ generates legitimate dynamical map iff
\begin{equation}\label{gg}
0\leq \int_0^t \gamma_\pm(s) e^{\Gamma(s)}ds \leq e^{\Gamma(t)} - 1\ ,
\end{equation}
where $\Gamma(t) = \int_0^t [ \gamma_-(\tau) + \gamma_+(\tau)]d\tau$. In particular it follows from (\ref{gg}) that $\Gamma(t) \geq 0$.
Now, $\Lambda_t$ is CP-divisible iff
\begin{equation}\label{-+1}
\gamma_-(t)\geq 0 \ , \ \ \ \gamma_+(t) \geq 0\ ,
\end{equation}
and it is P-divisible iff
\begin{equation}\label{-+2}
\gamma_-(t) + \gamma_+(t) \geq 0\ .
\end{equation}
Note, that (\ref{-+1}) implies (\ref{gg}). However, it is not true for (\ref{-+2}), i.e. P-divisibility requires both (\ref{gg}) -- it {guarantees} that $\Lambda_t$ is completely positive -- and (\ref{-+2}).
\end{Example}
{\em Bloch equations and P-divisibility} --- The above examples illustrating qubit dynamics may be easily rewritten in terms of the Bloch vector $x_k(t) = {\rm Tr}[\sigma_k \Lambda_t(\rho)]$. Example 2 gives rise to
\begin{equation}\label{}
\frac{d}{dt} x_k(t) = - \frac{1}{T_k(t)} x_k(t)\ , \ \ k=1,2,3\ ,
\end{equation}
where $T_1(t) = [\gamma_2(t) + \gamma_3(t)]^{-1}$, and similarly for $T_2(t)$ and $T_3(t)$. Quantities $T_k(t)$ correspond to local relaxation times. It is therefore clear that P-divisibility is equivalent to $T_k(t) \geq 0$ for $k=1,2,3$. Hence, CP-divisibility requires that all local decoherence rates satisfy $\gamma_k(t) \geq 0$, whereas P-divisibility requires only $T_k(t) \geq 0$. Note, that CP-divisibility is equivalent to P-divisibility plus three extra conditions
\begin{equation*}\label{}
\frac{1}{T_1} + \frac{1}{T_2} \geq \frac{1}{T_3}\ ,\ \ \frac{1}{T_1} + \frac{1}{T_3} \geq \frac{1}{T_2}\ ,\ \ \frac{1}{T_2} + \frac{1}{T_3} \geq \frac{1}{T_1}\ .
\end{equation*}
Finally, let us observe that the initial volume of the
Bloch ball shrinks during the evolution according to
$$V(t) = e^{-[\Gamma_1(t) + \Gamma_2(t) + \Gamma_3(t)]} V(0)\ , $$
where $V(t)$ denotes a volume of the set of accessible states at time $t$. Authors of \cite{Pater} characterized non-Markovian evolution as a departure from $\frac{d}{dt} V(t) \leq 0$. One has $\frac{d}{dt} V(t) = - [\gamma_1(t) + \gamma_2(t) + \gamma_3(t)] V(t)$ and hence $\frac{d}{dt} V(t) \leq 0$ iff
\begin{equation}\label{3gamma}
\gamma_1(t) + \gamma_2(t) + \gamma_3(t) \geq 0\ .
\end{equation}
This condition is much weaker than (\ref{2gamma}). To violate (\ref{3gamma}) the evolution has to be essentially non-Markovian (i.e. $\Lambda_t$ can not be even P-divisible).
A similar conclusion may be { drawn} from Example 3: the corresponding Bloch equations read
\begin{eqnarray}\label{}
\frac{d}{dt}\, x_1(t) &=& - \frac{1}{T_\perp(t)}\, x_1(t)\ , \nonumber \\
\frac{d}{dt}\, x_2(t) &=& - \frac{1}{T_\perp(t)}\, x_2(t)\ , \\
\frac{d}{dt}\, x_3(t) &=& - \frac{1}{T_{||}(t)}\, x_3(t) + \Delta(t)\ ,\nonumber
\end{eqnarray}
where $\Delta(t) = [\gamma_+(t) - \gamma_-(t)]$, and $T_\perp(t) = 2/[\gamma_-(t) + \gamma_+(t)]$ and $T_{||}(t) = T_\perp(t)/2$ are transverse and longitudinal local relaxation times, respectively. Again, P-divisibility is equivalent to $T_\perp, T_{||}(t) \geq 0$ provided that the Bloch vector stays within a Bloch ball.
{\em Conclusions} --- In this Letter we provided further characterization of non-Markovian evolution in terms of non-Markovianity degree. This simple concept, being an analog of the Schmidt number in the entanglement theory, enables one to compare quantum evolutions and finally to define a maximally non-Markovian evolution being an analog of a maximally entangled state.
{\em Acknowledgements} --- Our research was partially completed while the authors were visiting the Institute for Mathematical Sciences, National University of Singapore { in the framework of the programme Mathematical Horizons for Quantum Physics 2} in 2013.
D.C. was partially supported by the National Science Center project DEC-2011/03/B/ST2/00136.
\section*{Supplementary material}
Let $\Lambda_t$ be an invertible map. We prove that $\Lambda_k$ is $k$-divisible if and only if the formula (\ref{contr-k}) holds for all Hermitian $X \in M_k {\,\otimes\,} \mathcal{B}(\mathcal{H}$).
We use the following \cite{Paulsen}
\begin{Lemma}
If $\Phi : \mathcal{B}(\mathcal{H}) \rightarrow \mathcal{B}(\mathcal{H})$ is trace-preserving, then $\Phi$ is positive if and only if
\begin{equation*}\label{}
||\Phi(X)||_1 \leq ||X||_1\ ,
\end{equation*}
for all Hermitian $X \in \mathcal{B}(\mathcal{H})$.
\end{Lemma}
Assuming that $\Lambda_t$ is $k$-divisible one has
\begin{eqnarray*}
& & \frac{d}{dt}\,||[\oper_k {\,\otimes\,} \Lambda_t](X)||_1 \\
& & = \lim_{\epsilon\rightarrow\, 0+} \frac{1}{\epsilon}\, \Big[ ||[\oper_k {\,\otimes\,} \Lambda_{t+\epsilon}](X)||_1 - ||[\oper_k {\,\otimes\,} \Lambda_t](X)||_1 \Big] \\ & & =
\lim_{\epsilon\rightarrow\, 0+} \frac{1}{\epsilon}\,\Big[ ||[\oper_k {\,\otimes\,} V_{t+\epsilon,t}\, \Lambda_{t}](X)||_1 - ||[\oper_k {\,\otimes\,} \Lambda_t](X)||_1 \Big]\\
& & \leq \lim_{\epsilon\rightarrow\, 0+} \frac{1}{\epsilon}\,\Big[ || [\oper_k {\,\otimes\,} \Lambda_{t}](X)||_1 - ||[\oper_k {\,\otimes\,} \Lambda_t](X)||_1 \Big] = 0 \ ,
\end{eqnarray*}
where we have used $k$-divisibility of $\Lambda_t$, i.e.
$$\Lambda_{t+\epsilon} = V_{t+\epsilon,t} \Lambda_t \ , $$
with $V_{t+\epsilon,t}$ being $k$-positive and the Lemma 1 for a positive and trace-preserving map $\oper_k {\,\otimes\,} V_{t,t+\epsilon}$
\begin{eqnarray*}
||[\oper_k {\,\otimes\,} V_{t+\epsilon,t}\, \Lambda_{t}](X)||_1 &=& ||[\oper_k {\,\otimes\,} V_{t+\epsilon,t}] [(\oper_k {\,\otimes\,} \Lambda_{t})(X)] ||_1 \\
&\leq& ||[\oper_k {\,\otimes\,} \Lambda_{t}](X) ||_1 \ .
\end{eqnarray*}
To prove the converse result let $ Y = [\oper_k {\,\otimes\,} \Lambda_{t}](X)$. Now, if $\frac{d}{dt}\,||[\oper_k {\,\otimes\,} \Lambda_t](X)||_1 \leq 0$, then
$$ ||[\oper_k {\,\otimes\,} V_{t+\epsilon,t}](Y)||_1 \leq ||Y||_1 \ . $$
Assuming that $\Lambda_t$ is invertible we proved that $\oper_k {\,\otimes\,} V_{t+\epsilon,t}$ is a contraction on all Hermitian elements $Y \in M_k {\,\otimes\,} \mathcal{B}(\mathcal{H})$ and hence, due to the Lemma 1, $\oper_k {\,\otimes\,} V_{t+\epsilon,t}$ is positive or, equivalently, $V_{t+\epsilon,t}$ is $k$-positive.
\end{document} | math | 24,684 |
\begin{document}
\title{A new example of limit variety of aperiodic monoids}
\thanks{This research was supported by the National
Natural Science Foundation of China (No.~10971086, 11371177)}
\author[W. T. Zhang]{Wen Ting Zhang$^\star$}\thanks{$^\star$Corresponding author}
\author[Y. F. Luo]{Yan Feng Luo}
\address{School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People's Republic of China; Key Laboratory of Applied Mathematics and Complex Systems, Gansu Province}
\email{[email protected]}
\email{[email protected]}
\subjclass[2000]{20M07, 03C05, 08B15}
\keywords{Monoid, finitely based, variety, limit variety, subvariety lattice}
\begin{abstract}
A limit variety is a variety that is minimal with respect to being non-finitely based. The two limit varieties of Marcel Jackson are the only
known examples of limit varieties of aperiodic monoids. Our previous work had shown that there exists a limit subvariety of aperiodic monoids that is different from Marcel Jackson's limit varieties. In this paper, we introduce a new limit variety of aperiodic monoids.
\end{abstract}
\maketitle
\section{Introduction} \label{sec intro}
A variety of algebras is \textit{finitely based} if there is a finite subset of its identities
from which all of its identities may be deduced, otherwise, the variety is
said to be \textit{non-finitely based}. An
algebra is \textit{finitely based} if it generates a finitely based
variety, otherwise, the algebra is said to be \textit{non-finitely based}. There are many finitely based and many non-finitely based
finite semigroups, and consequently the finite basis property for finite semigroups, and for finite algebras in general has
been one of the most extensively studied in facets of varieties. Refer to the surveys of Volkov \cite{Vol01} for a great deal of information on varieties, identities, and the finite basis problem for semigroups.
A variety is \textit{hereditarily finitely
based} if all its subvarieties are finitely based. A variety is called \textit{limit variety} if
it is non-finitely based but every proper subvariety is finitely based; in other words,
limit varieties are precisely minimal non-finitely based varieties. Zorn's lemma
implies that each non-finitely based variety contains some limit subvariety; thus, a
variety is hereditarily finitely based if and only if it contains no limit subvarieties.
Therefore classifying hereditarily finitely based varieties in a certain sense reduces
to classifying limit varieties whence the latter task appears to be very hard in
general. Moreover, finding any concrete limit variety turns out to be nontrivial.
For example, no concrete limit variety of groups is known so far even though a
recent result by Kozhevnikov \cite{Kozhevnikov} shows that there are uncountably many of them.
Amongst (locally finite) groups, there are known to be infinitely many limit varieties, however the explicit construction of such an example remains one of the
foremost unsolved problems in group variety theory \cite{Kra03}
In contrast, for inverse semigroup varieties, a complete classification of non-group
limit varieties (and hence, a characterization of hereditarily finitely based varieties
modulo groups) has been found by Kleiman \cite{Kleiman}.
Recall that a monoid is \textit{aperiodic} if all its subgroups are
trivial. This article is concerned with the class $\mathfrak{A}$ of
aperiodic monoids and its subvarieties. A result of Kozhevnikov
implies the existence of continuum many limit varieties of monoids
consisting of groups~\cite{Kozhevnikov}. This makes classification of
limit varieties of monoids unfeasible unless restrictions are placed
on the groups lying in the variety. The class $\mathfrak{A}$ is
arguably the most obvious natural candidate for attention. In the
early 2000s, Jackson proved that the variety $\mathsf{var} \{J_1\}$
generated by the monoid
\[
J_1 = \left\langle a,b,s,t \left|\, \text{$xy=0$ if $xy$ is not a factor of $asabtb$} \right. \right\rangle \cup \{1\}
\]
of order 21 and the variety $\mathsf{var} \{J_2\}$ generated by the monoid
\[
J_2 = \left\langle a,b,s,t \left|\, \text{$xy=0$ if $xy$ is not a factor of either $absatb$ or $asbtab$} \right. \right\rangle \cup \{1\}
\]
of order 35 are limit subvarieties of
$\mathfrak{A}$~\cite[Proposition~5.1]{Jac05fin}. As commented by
Jackson, no other similar examples of limit varieties could be found
\cite[Section~5]{Jac05fin}. This led him to pose the question~{\cite[Question~1]{Jac05fin}}:
Is there any finitely generated, non-finitely based
subvariety of $\mathfrak{A}$ that contains neither $\mathsf{var} \{J_1\}$
nor $\mathsf{var} \{J_2\}$?
In \cite{LeeZhang}, we show that the $\mathcal{J}$-trivial semigroup
\[
L=\left\langle\,a,b\left|\,a^2=a,\,b^2=b,\,aba=0\right.\right\rangle
\]
of order six is one of minimal non-finitely semigroups. Let $L^1$ denote the monoid obtained by adjoining an identity element to $L$ and let $\mathsf{var} \{L^1\}$ denote the variety generated by $L^1$. It is easy to see that $\mathsf{var} \{L^1\}$ is a subvariety of $\mathfrak{A}$ that contains neither $\mathsf{var} \{J_1\}$ nor
$\mathsf{var} \{J_2\}$. In \cite{zhang13}, we show that $\mathsf{var} \{L^1\}$
is non-finitely based, and so there exists a limit subvariety of $\mathfrak{A}$ that is different from $\mathsf{var} \{J_1\}$ and $\mathsf{var} \{J_2\}.$ Consequently, identify all limit subvarieties of $\mathsf{var} \{L^1\}$ is an unavoidable step in the classification of limit varieties of aperiodic monoids.
The main goal of this paper and its prequel is to give an explicit example of a limit variety of $\mathfrak{A}$.
Let $A^1$ denote the monoid obtained by adjoining an identity element to the semigroup $A = \{ 0,a,b,c,d,e\}$ given by the following multiplication table:
\[
\begin{array}
[c]{c|cccccc}
A & 0 & a & b & c & d & e \\ \hline
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
a & 0 & 0 & 0 & 0 & 0 & a \\
b & 0 & 0 & 0 & 0 & 0 & b \\
c & 0 & 0 & a & 0 & c & 0 \\
d & 0 & 0 & b & 0 & d & 0 \\
e & 0 & a & a & c & c & e
\end{array}
\]
The semigroup $A$ was first investigated by Lee and Zhang~\cite[Section~19]{LeeZhang}, where it was shown to be finitely based. Let $B^1$ be the semigroup that is dual to $A^1$.
In \cite{zhang16}, by using a sufficient condition, we have shown that the semigroup $A^1 \times B^1$ is non-finitely based. In this paper, all of proper monoid subvarieties of the variety generated by $A^1 \times B^1$ are shown to be finitely based, and the monoid subvariety lattice of $\mathsf{var}_{\mathbb{M}} \{A^1 \times B^1\}$ will be completely described. Hence the monoid variety $\mathsf{var}_{\mathbb{M}} \{A^1 \times B^1\}$ is a limit monoid variety. Also,
an identity basis $A^1 \times B^1$ will be given, the finite membership problem for the variety generated by $A^1 \times B^1$ admits a polynomial algorithm.
In section \ref{sec basis}, an identity basis for $\mathsf{var} \{A^1 \times B^1\}$ will be given.
In section \ref{sec lattice}, all monoid subvarieties of $\mathsf{var} \{A^1 \times B^1\}$ will be characterized and each of them is finitely based. Furthermore, the monoid subvariety lattice of $\mathsf{var}_{\mathbb{M}} \{A^1 \times B^1\}$ will be completely described. Hence $\mathsf{var}_{\mathbb{M}} \{A^1 \times B^1\}$ is a limit monoid variety.
Recall that a variety is \textit{small} if it has finitely many subvarieties, a small variety
is \textit{cross} if it is finitely based and finitely generated, and a non-cross variety is
\textit{almost cross} if all its proper subvarieties are Cross.
Hence $\mathsf{var} \{A^1 \times B^1\})$ is a small almost cross variety.
\section{Preliminaries} \label{sec prelim}
Most of the notation and background material of this article are
given in this section. Refer to the monograph~\cite{BurSan81} for
any undefined terminology.
Let $\mathcal{X}$ be a fixed countably infinite alphabet
throughout. For any subset $\mathcal{A}$ of $\mathcal{X}$, denote
by $\mathcal{A}^{\ast}$ the free monoid over $\mathcal{A}$.
Elements of $\mathcal{X}$ and $\mathcal{X}^{\ast}$ are referred to
as \textit{letters} and \textit{words} respectively.
The \textit{content} of a
word $\mathbf{w}$, denoted by $\mathsf{con} (\mathbf{w})$, is the
set of letters occurring in $\mathbf{w}$; The \textit{multiplicity} of a letter $x$ in $\mathbf{w}$, denoted by
$\mathsf{m} (x, \mathbf{w})$, is the number of times $x$ occurs in
$\mathbf{w}$; A letter $x$ is
\textit{simple} in a word $\mathbf{w}$ if $\mathsf{m} (x,
\mathbf{w}) = 1$; otherwise, $x$ is \textit{non-simple} in
$\mathbf{w}$. The set of simple letters of a word $\mathbf{w}$ is
denoted by $\mathsf{sim} (\mathbf{w})$ and the set of non-simple
letters of $\mathbf{w}$ is denoted by $\mathsf{non} (\mathbf{w})$.
Note that $\mathsf{con} (\mathbf{w}) = \mathsf{sim} (\mathbf{w})
\cup \mathsf{non} (\mathbf{w})$ and $\mathsf{sim} (\mathbf{w}) \cap
\mathsf{non} (\mathbf{w}) = \emptyset$.
An identity is typically written as $\mathbf{w} \approx
\mathbf{w}'$ where $\mathbf{w}$ and $\mathbf{w}'$ are nonempty
words. Let $\Pi$ be any set of identities. The deducibility of an
identity $\mathbf{w} \approx \mathbf{w}'$ from $\Pi$ is indicated
by $\Pi \vdash \mathbf{w} \approx \mathbf{w}'$ or $\mathbf{w}
\stackrel{\Pi}{\approx} \mathbf{w}'$. The variety \textit{defined}
by $\Pi$, denoted by $\mathbb{V}(\Pi)$, is the class of all semigroups that satisfy all
identities in $\Pi$; in this case, $\Pi$ is said to be a
\textit{basis} for the variety.
For any word $\mathbf{w}$ and any set $\mathcal{B}$ of letters in
$\mathbf{w}$, let $\mathbf{w}_\mathcal{B}$ denote the word obtained
from $\mathbf{u}$ by retaining the letters from $\mathcal{B}$ (but
removing all others). It is easy to see that if the identity
$\mathbf{u} \approx \mathbf{v}$ is satisfied by a monoid $M$, then
any identity of the form $\mathbf{u}_\mathcal{B} \approx
\mathbf{v}_\mathcal{B}$ is also satisfied by $M$.
For any class $\mathfrak{C}$ of semigroups or monoids, let $\mathsf{var} \{\mathfrak{C}\}$ denote the semigroup variety generated by $\mathfrak{C}$. For any class $\mathfrak{C}$ of monoids, let $\mathsf{var}_{\mathbb{M}} \{\mathfrak{C}\}$ denote the monoid variety generated by $\mathfrak{C}$.
It is easy to see that a monoid $S$ is contained in the semigroup variety
of a monoid $H$ if and only if it is contained in the monoid variety of $H$.
The following small semigroups are required throughout the article:
\begin{align*}
J & =\Big\langle a,b\,\Big|\,ab=0,\,ba=a,\,b^2=b\Big\rangle, \\
A_0 & = \Big\langle \, a,b \,\Big|\, a^{2} = a, \, b^2 = b, \, ba = 0 \Big\rangle, \\%
B_0 & = \Big\langle \, a,b,c \,\Big|\, a^{2} = a, \, b^2= b, \, ab = ba =0, \, ac=cb=c \Big\rangle, \\%
L & = \Big\langle \, a,b \,\Big|\, a^{2} = ba = 0, \, ab = a, \, b^2 = b \Big\rangle, \\%
R & = \Big\langle \, a,b \,\Big|\, a^{2} = ab = 0, \, ba=a, \, b^{2} = b\, \Big\rangle, \\%
M & = \Big\langle \, a,b,c \,\Big|\, cb=a, \, \mbox{and all other products equal to $0$} \Big\rangle, \\%
N & = \Big\langle \, a \,\Big|\, a^{2} = 0 \Big\rangle. \\%
\end{align*}
For any non-unital semigroup $S$, let $S^1$ denote the monoid
obtained by adjoining a unit element to $S$.
\begin{proposition}
\begin{enumerate}
\item $\mathbf {A}_{0}^{1}= \mathsf{var}\{x^3\approx x^2, x^{2}yx\approx xyx \approx xyx^{2}, xyhxty \approx yxhxty, xhytxy \approx xhytyx\};$
\item $\mathbf {B}_{0}^{1}= \mathsf{var}\{x^3\approx x^2, x^{2}yx\approx xyx \approx xyx^{2}, xyhxty \approx yxhxty, xhytxy \approx xhytyx, x^2y^2\approx y^2x^2\};$
\item $\mathbf {L}^{1}= \mathsf{var}\{x^3\approx x^2, xyx\approx x^2y, x^2y^2 \approx y^2x^2 \};$
\item $\mathbf {R}^{1}= \mathsf{var}\{x^3\approx x^2, xyx\approx yx^2, x^2y^2 \approx y^2x^2 \};$
\item $\mathbf {M}^{1}= \mathsf{var}\{x^3\approx x^2, xyx\approx x^2y, xyx \approx yx^2 \};$
\item $\mathbf {N}^{1}= \mathsf{var}\{x^3\approx x^2, xy\approx yx \};$
\end{enumerate}
\end{proposition}
For any letters $x$ and $y$ of a word $\mathbf{w}$, write $x
\prec_\mathbf{w} y$ to indicate that within $\mathbf{w}$, each
occurrence of $x$ precedes every occurrence of $y$. In other words,
if $x \prec_\mathbf{w} y$ with $p=\mathsf{m}(x,\mathbf{w})$ and
$q=\mathsf{m}(y,\mathbf{w})$, then retaining only the letters $x$ and $y$
in $\mathbf{w}$ results in the word $x^p y^q$.
\begin{lemma}[{\cite[Lemma~1.3]{LeeZhang}}] \label{prelim LEM J1 words}
Suppose that $\mathbf{w} \approx \mathbf{w}'$ is any identity
satisfied by the semigroup $J^1.$ Then
\begin{enumerate}
\item $\operatorname{\mathsf{con}} (\mathbf{w}) = \operatorname{\mathsf{con}} (\mathbf{w}')$ and $\operatorname{\mathsf{sim}} (\mathbf{w}) = \operatorname{\mathsf{sim}} (\mathbf{w}');$
\item for any $x \in \operatorname{\mathsf{con}} (\mathbf{w}) = \operatorname{\mathsf{con}} (\mathbf{w}')$ and $y \in \operatorname{\mathsf{sim}} (\mathbf{w}) = \operatorname{\mathsf{sim}} (\mathbf{w}'),$ the conditions $x \prec_\mathbf{w} y$ and $x \prec_{\mathbf{w}'} y$ are equivalent\emph{;}
\item $\mathbf{w}_{\operatorname{\mathsf{sim}}} = \mathbf{w}_{\operatorname{\mathsf{sim}}}'.$
\end{enumerate}
\end{lemma}
\section{An identity basis for $\mathsf{var} \{A^1 \times B^1\}$} \label{sec basis}
The present section establishes an identities basis for the variety $\mathsf{var} \{A^1 \times B^1\}.$
\begin{theorem}
\label{S basis theorem} The variety $\mathsf{var} \{A^1 \times B^1\}$ is defined
by the identities
\begin{gather}
x^{2} \approx x^{3} , \ \ xyx \approx x^{2}yx \approx xyx^{2}, \label{basis S xx} \\
xy^{2}x \approx (xy)^2 \approx (xy)^2x \approx yx^{2}y \approx (yx)^2, \label{basis S xy2x} \\
xytxsy \approx (xy)^{2}txsy, \ \ xt(yx)^{2}sy \approx xtyxsy, \ \ xtysxy \approx xtys(xy)^{2}, \label{basis S txsy} \\%
xy_{1}^{2} y_{2}^{2} \cdots y_{n}^{2} x \approx x y_{1}^{2} x y_{2}^{2} x \cdots y_{n}^{2} x, \qquad n=2,3,\ldots, \label{basis S xy2z2x} \\%
xytxz_{1}^{2} \cdots z_{n}^{2} y \approx yxtx z_{1}^{2} \cdots z_{n}^{2} y, \qquad n=0,1,\ldots, \label{basis S txaay} \\%
xz_{1}^{2} \cdots z_{n}^{2} ytxy \approx xz_{1}^{2} \cdots z_{n}^{2} y tyx, \qquad n=0,1,\ldots. \label{basis S xaayt}
\end{gather}
\end{theorem}
\noindent The proof of Theorem~\ref{S basis theorem} is given at
the end of the section.
Most of the equational deductions in this article are deductions
within the equational theory of $\mathsf{var} \{A^1 \times B^1\}$. Therefore, it
will be convenient to refer to the identities in Theorem~\ref{S
basis theorem} collectively by $\circledS$, that is,
\[
\circledS = \{ (\ref{basis S xx}), (\ref{basis S xy2x}), (\ref{basis S txsy}), (\ref{basis S xy2z2x}),
(\ref{basis S txaay}), (\ref{basis S xaayt}) \}.
\]
For any sets $\Pi_{1}$ and $\Pi_{2}$ of identities, the deduction
$\circledS \cup \Pi_{1} \vdash \Pi_{2}$ is abbreviated to $\Pi_{1}
\Vdash \Pi_{2}$.
For any nonempty set $Z = \{ z_1,\ldots,z_r \}$ of letters, the word of the form
\[
(z_1 \cdots z_r)^2,
\]
is said to be the \textit{$Z$-square},
in particular, if $z_1, \ldots, z_r$ are in alphabetical order, then it said to be the
\textit{perfect $Z$-square}.
More generally, a \textit{square} (resp. \textit{perfect square}) is a $Z$-square (resp. perfect $Z$-square)
for some nonempty set $Z$ of letters.
\begin{lemma}\label{lem: perfect square}
Let $\mathbf{z}$ be any square. Then the identities~$\circledS$ imply the identity
\begin{equation}
\mathbf{z} \approx \overline{\mathbf{z}}, \label{perfect square}
\end{equation}
where $\overline{\mathbf{z}}$ is the perfect $\mathsf{con}(\mathbf{z})$-square.
\end{lemma}
\begin{proof}
Without loss of generality, we may assume that $\mathbf{z}=(z_1 \cdots z_r)^2$.
Then
\begin{align*}
\mathbf{z} & \makebox[0.42in]{$=$} z_1 \cdots z_iz_{i+1} \cdots z_r z_1 \cdots z_{i}z_{i+1} \cdots z_r \\
& \makebox[0.42in]{$\stackrel{\eqref{basis S txaay}}{\approx}$} z_1 \cdots z_{i+1}z_{i} \cdots z_r z_1 \cdots z_{i}z_{i+1} \cdots z_r \\
& \makebox[0.42in]{$\stackrel{\eqref{basis S xaayt}}{\approx}$} z_1 \cdots z_{i+1}z_{i} \cdots z_r z_1 \cdots z_{i+1}z_{i} \cdots z_r \\
& \makebox[0.42in]{$=$} (z_1 \cdots z_{i+1}z_{i} \cdots z_r)^{2} .
\end{align*}
Hence the identities ~$\circledS$ can be used to permute any letters within $\mathbf{z}$ in any manner. Specifically, the identities ~$\circledS$ can be used to permute any letters within $\mathbf{z}$ into alphabetical order, whence $\mathbf{z} \stackrel{\circledS}{\approx} \overline{\mathbf{z}}$.
\end{proof}
\begin{lemma}\label{lem: perfect square reduce}
Let $\mathbf{z}$ and $\mathbf{z}'$ be any squares with $\mathsf{con}(\mathbf{z}') \subseteq \mathsf{con}(\mathbf{z})$. Then the identities~$\circledS$ imply the identity
\begin{equation}
\mathbf{z}'\mathbf{z} \approx \mathbf{z} \approx \mathbf{z}\mathbf{z}'. \label{perfect square reduce}
\end{equation}
\end{lemma}
\begin{proof}
By symmetry$,$ it suffices to prove that $\mathbf{z}'\mathbf{z} \approx \mathbf{z}.$ Without loss of generality, we may assume that $\mathsf{t}(\mathbf{z}')=z$, and $\mathbf{z}=(z_1 \cdots z_i z z_{i+1} \cdots z_{r})^2$.
Then
\begin{align*}
z\mathbf{z} & \makebox[0.42in]{$=$} zz_1 \cdots z_iz z_{i+1} \cdots z_r z_1 \cdots z_iz z_{i+1} \cdots z_r \\
& \makebox[0.42in]{$\stackrel{\eqref{basis S xx}}{\approx}$} z(z_1 \cdots z_i)^2z z_{i+1} \cdots z_r z_1 \cdots z_iz z_{i+1} \cdots z_r \\
& \makebox[0.42in]{$\stackrel{\eqref{basis S xy2x}}{\approx}$} (z_1 \cdots z_iz)^2 z_{i+1} \cdots z_r z_1 \cdots z_iz z_{i+1} \cdots z_r \\
& \makebox[0.42in]{$\stackrel{\eqref{basis S xx}}{\approx}$} z_1 \cdots z_iz z_{i+1} \cdots z_r z_1 \cdots z_iz z_{i+1} \cdots z_r \\
& \makebox[0.42in]{$=$} \mathbf{z} .
\end{align*}
It is easily seen how this procedure can be repeated so that the word $\mathbf{z}'\mathbf{z}$ can be converted to $\mathbf{z}.$
\end{proof}
\begin{lemma}\label{lem: perfect square product}
Let $\mathbf{w}$ be any non-simple word such that $\mathsf{sim}(\mathbf{w}) = \emptyset$. Then the identities~$\circledS$ imply the identity $\mathbf{w} \approx \overline{\mathbf{w}},$ where
\[
\overline{\mathbf{w}}=\mathbf{z}_1 \cdots \mathbf{z}_{p_k}
\]
where
\begin{enumerate}
\item the words $\mathbf{z}_1^{(k)}, \ldots, \mathbf{z}_{p_k}^{(k)}\in \mathcal{X}^{+}$ are perfect squares;
\item if $\operatorname{\mathsf{con}}(\mathbf{z}_\ell^{(k)}) \cap \operatorname{\mathsf{con}}(\mathbf{z}_g^{(k)}) \ne \emptyset$ for some $\ell<g$ and $\ell, g\in \{1, \ldots, p_k\}$, then $\operatorname{\mathsf{con}}(\mathbf{z}_\ell^{(k)}) \cap \operatorname{\mathsf{con}}(\mathbf{z}_g^{(k)}) \subseteq \operatorname{\mathsf{con}}(\mathbf{z}_j^{(k)})$ for each $\ell\leq j\leq g$;
\item $\operatorname{\mathsf{con}}(\mathbf{z}_\ell^{(k)}) \nsubseteq \operatorname{\mathsf{con}}(\mathbf{z}_g^{(k)})$ for each $\ell \ne g$ and $\ell, g\in \{1, \ldots, p_k\}$;
\end{enumerate}
\end{lemma}
\begin{proof}
Since each non-simple letter $x$ in $\mathbf{w}$ can be replaced by its square $x^2$ by applying the identities \eqref{basis S xx},
the word $\mathbf{w}$ can be written into the form of
\[
\mathbf{w}_k=\mathbf{z}_1 \cdots \mathbf{z}_p,
\]
where the words $\mathbf{z}_1, \ldots, \mathbf{z}_p \in \mathcal{X}^{+}$ are squares. Then by Lemma~\ref{lem: perfect square} we may assume that the words $\mathbf{z}_1, \ldots, \mathbf{z}_p$
are perfect squares. Hence the condition (CF2a) is satisfied.
If $z\in \operatorname{\mathsf{con}}(\mathbf{z}_\ell) \cap \operatorname{\mathsf{con}}(\mathbf{z}_g)$
for some $1 \leq \ell < g \leq p$, then
\begin{align*}
\mathbf{w} & \makebox[0.42in]{$=$} \cdots \mathbf{z}_\ell \mathbf{z}_{\ell+1} \cdots \mathbf{z}_g \cdots \\
& \makebox[0.42in]{$\stackrel{\eqref{perfect square reduce}}{\approx}$} \cdots z^2\mathbf{z}_\ell \mathbf{z}_{\ell+1} \cdots \mathbf{z}_g z^2\cdots \\
& \makebox[0.42in]{$\stackrel{\eqref{basis S xy2z2x}}{\approx}$} \cdots z^2\mathbf{z}_\ell z \mathbf{z}_{\ell+1} z \cdots z \mathbf{z}_g z^2 \cdots \\
& \makebox[0.42in]{$\stackrel{\eqref{basis S xx}}{\approx}$} \cdots z^2(z\mathbf{z}_\ell z)(z\mathbf{z}_{\ell+1} z) \cdots (z \mathbf{z}_g z) z^2 \cdots \\
& \makebox[0.42in]{$\stackrel{\eqref{perfect square}}{\approx}$} \cdots z^2 \overline{\mathbf{z}}_\ell \overline{\mathbf{z}}_{\ell+1} \cdots \overline{\mathbf{z}}_g z^2 \cdots \\
& \makebox[0.42in]{$\stackrel{\eqref{perfect square reduce}}{\approx}$} \cdots \overline{\mathbf{z}}_\ell \overline{\mathbf{z}}_{\ell+1} \cdots \overline{\mathbf{z}}_g \cdots
\end{align*}
where $\overline{\mathbf{z}}_j$ is the perfect $\mathsf{con}(\mathbf{z}_j)\cup \{z\}$-square for each $j= \ell, \ldots, g$. Now if $\operatorname{\mathsf{con}}(\mathbf{z}_\ell) \cap \operatorname{\mathsf{con}}(\mathbf{z}_g) = \emptyset$, then by repeating the above processes, each letter in $\operatorname{\mathsf{con}}(\mathbf{z}_\ell) \cap \operatorname{\mathsf{con}}(\mathbf{z}_g)$ can be put into $\mathbf{z}_j$ for each $\ell \leq j \leq g$. Hence we may assume that $\operatorname{\mathsf{con}}(\mathbf{z}_\ell) \cap \operatorname{\mathsf{con}}(\mathbf{z}_g) \subseteq \operatorname{\mathsf{con}}(\mathbf{z}_j)$, and so the condition (CF2b) is satisfied.
Suppose that $\operatorname{\mathsf{con}}(\mathbf{z}_\ell^{(k)}) \subseteq \operatorname{\mathsf{con}}(\mathbf{z}_g^{(k)})$ for some $\ell < g$.
Then $\operatorname{\mathsf{con}}(\mathbf{z}_\ell^{(k)}) \subseteq \operatorname{\mathsf{con}}(\mathbf{z}_{\ell+1}^{(k)})$ by the condition (CF2b). Hence by applying the identities \eqref{basis S xx} and \eqref{basis S xy2x}, the identity $\mathbf{z}_{\ell}^{(k)}\mathbf{z}_{\ell+1}^{(k)} \approx \mathbf{z}_{\ell+1}^{(k)}$ is hold, and so $\mathbf{z}_{\ell}^{(k)}$ can be deleted from $\mathbf{w}_k$.
Hence we may assume that the condition (CF2c) is satisfied.
\end{proof}
A word $\mathbf{w}$ is
said to be in \textit{canonical form} if
\begin{equation}
\mathbf{w} = \mathbf{w}_0 \prod_{i=1}^n (\mathbf{s}_i \mathbf{w}_i) \label{canon form}
\end{equation}
for some $n \geq 0$ such that the following conditions are all satisfied:
\begin{enumerate}
\item[(I)] the letters of $\mathbf{s}_1,\ldots,\mathbf{s}_n \in \mathcal{X}^+$ are simple in $\mathbf{w}$;
\item[(II)] $\mathbf{w}_0, \mathbf{w}_n \in \mathcal{X}^{*}$ and $\mathbf{w}_1, \ldots, \mathbf{w}_{n-1} \in \mathcal{X}^{+}$ and for each $k=0, 1, \ldots, n$,
\[
\mathbf{w}_k=\mathbf{z}_1^{(k)} \cdots \mathbf{z}_{p_k}^{(k)}
\]
where
\begin{enumerate}
\item the words $\mathbf{z}_1^{(k)}, \ldots, \mathbf{z}_{p_k}^{(k)}\in \mathcal{X}^{+}$ are perfect squares;
\item if $\operatorname{\mathsf{con}}(\mathbf{z}_\ell^{(k)}) \cap \operatorname{\mathsf{con}}(\mathbf{z}_g^{(k)}) \ne \emptyset$ for some $\ell<g$ and $\ell, g\in \{1, \ldots, p_k\}$, then $\operatorname{\mathsf{con}}(\mathbf{z}_\ell^{(k)}) \cap \operatorname{\mathsf{con}}(\mathbf{z}_g^{(k)}) \subseteq \operatorname{\mathsf{con}}(\mathbf{z}_j^{(k)})$ for each $\ell\leq j\leq g$;
\item $\operatorname{\mathsf{con}}(\mathbf{z}_\ell^{(k)}) \nsubseteq \operatorname{\mathsf{con}}(\mathbf{z}_g^{(k)})$ for each $\ell \ne g$ and $\ell, g\in \{1, \ldots, p_k\}$;
\item if $x\in \operatorname{\mathsf{con}}(\mathbf{z}_{\ell-1}^{(k)}) \setminus \operatorname{\mathsf{con}}(\mathbf{z}_\ell^{(k)})$ and $y\in \operatorname{\mathsf{con}}(\mathbf{z}_\ell^{(k)}) \setminus \operatorname{\mathsf{con}}(\mathbf{z}_{\ell-1}^{(k)})$ for some $k\in\{0, 1, \ldots, n\}$ and $\ell\in \{1,\ldots, p_k\}$, then $x, y$ satisfy neither of the following conditions:
\begin{enumerate}
\item $x \in \operatorname{\mathsf{con}}(\mathbf{w}_g)$ and $y \in \operatorname{\mathsf{con}}(\mathbf{w}_h)$ for some $g<h<k$ or $k<g<h$ or $h<k<g $;
\item $x, y \in \operatorname{\mathsf{con}}(\mathbf{w}_g)$ for some $g \ne k$.
\end{enumerate}
\end{enumerate}
\end{enumerate}
An identity $\mathbf{u} \approx \mathbf{v}$ is \textit{canonical}
if the words $\mathbf{u}$ and $\mathbf{v}$ are in canonical form.
\begin{lemma} \label{canonical word}
Let $\mathbf{w}$ be any word\emph{.} Then there exists
some word $\overline{\mathbf{w}}$ in canonical form such that the
identities~$\circledS$ imply the identity $\mathbf{w}
\approx \overline{\mathbf{w}}.$
\end{lemma}
\begin{proof}
It suffices to convert the word $\mathbf{w}$, using the
identities~$\circledS$, into a word in canonical form.
It is easy to see that the word $\mathbf{w}$ can be written into the form of
\begin{equation}
\mathbf{w} = \mathbf{w}_0 \prod_{i=1}^n (\mathbf{s}_i \mathbf{w}_i)
\end{equation}
where $\mathbf{s}_1,\ldots, \mathbf{s}_n \in \mathcal{X}^{+}$, $\mathbf{w}_0, \mathbf{w}_n \in \mathcal{X}^{*}$,
$\mathbf{w}_1,\ldots, \mathbf{w}_{n-1} \in \mathcal{X}^{+}$, the letters of $\mathbf{s}_1,\ldots, \mathbf{s}_n$
are simple in $\mathbf{w}$ and the letters of $\mathbf{w}_0,\ldots, \mathbf{w}_{n}$ are non-simple in $\mathbf{w}$. Hence the condition (CF1) is satisfied.
Since each non-simple letter $x_j$ in $\mathbf{w}_k$ can be replaced by its square $x_j^2$ by applying the identities \eqref{basis S xx},
the word $\mathbf{w}_k$ can be written into the form of
\[
\mathbf{w}_k=\mathbf{z}_1^{(k)} \cdots \mathbf{z}_{p_k}^{(k)},
\]
where the words $\mathbf{z}_1^{(k)}, \ldots, \mathbf{z}_{p_k}^{(k)} \in \mathcal{X}^{+}$ are squares.
The identities \eqref{basis S txaay} and \eqref{basis S xaayt} can be applied to alphabetically order the letters in each of
$\mathbf{z}_1^{(k)}, \ldots, \mathbf{z}_{p_k}^{(k)}$, and so we may assume that the words $\mathbf{z}_1^{(k)}, \ldots, \mathbf{z}_{p_k}^{(k)}$
are perfect squares. Hence the condition (CF2a) is satisfied.
If $z\in \operatorname{\mathsf{con}}(\mathbf{z}_\ell^{(k)}) \cap \operatorname{\mathsf{con}}(\mathbf{z}_g^{(k)})$
for some $\ell < g$ and $\ell, g\in \{1, \ldots, p_k\}$, then
\begin{align*}
\mathbf{w}_k & \makebox[0.42in]{$=$} \cdots \mathbf{z}_\ell^{(k)} \mathbf{z}_{l+1}^{(k)} \cdots \mathbf{z}_g^{(k)} \cdots \\
& \makebox[0.42in]{$\stackrel{\eqref{basis S xy2x}}{\approx}$} \cdots (z\mathbf{z}_\ell^{(k)}) \mathbf{z}_{\ell+1}^{(k)} \cdots (\mathbf{z}_g^{(k)} z)\cdots \\
& \makebox[0.42in]{$\stackrel{\eqref{basis S xy2z2x}}{\approx}$} \cdots z\mathbf{z}_\ell^{(k)} z \mathbf{z}_{\ell+1}^{(k)} \cdots z \mathbf{z}_k^{(k)} z \cdots \\
& \makebox[0.42in]{$\stackrel{\eqref{basis S xx}}{\approx}$} \cdots (z\mathbf{z}_\ell^{(k)} z)(z\mathbf{z}_{\ell+1}^{(k)}z) \cdots (z \mathbf{z}_g^{(k)} z) \cdots \\
& \makebox[0.42in]{$\stackrel{\eqref{basis S xy2x}}{\approx}$} \cdots (z\mathbf{z}_\ell^{(k)})^2(z\mathbf{z}_{\ell+1}^{(k)})^{2} \cdots (z\mathbf{z}_g^{(k)})^2 \cdots .
\end{align*}
Hence for each $\ell \leq j \leq g$, the perfect square $\mathbf{z}_j^{(k)}$ can be replaced by the perfect square $\overline{\mathbf{z}}_j^{(k)}$ where $\operatorname{\mathsf{con}}(\mathbf{z}_j^{(k)})\cup \{z\}=\operatorname{\mathsf{con}}(\overline{\mathbf{z}}_j^{(k)})$. Now if $\operatorname{\mathsf{con}}(\mathbf{z}_\ell^{(k)}) \cap \operatorname{\mathsf{con}}(\mathbf{z}_g^{(k)}) = \emptyset$, then by repeating the above processes, each letter in $\operatorname{\mathsf{con}}(\mathbf{z}_\ell^{(k)}) \cap \operatorname{\mathsf{con}}(\mathbf{z}_g^{(k)})$ can be put into $\mathbf{z}_j^{(k)}$ for each $\ell \leq j \leq g$. Hence we may assume that $\operatorname{\mathsf{con}}(\mathbf{z}_\ell^{(k)}) \cap \operatorname{\mathsf{con}}(\mathbf{z}_g^{(k)}) \subseteq \operatorname{\mathsf{con}}(\mathbf{z}_j^{(k)})$, and so the condition (CF2b) is satisfied.
Suppose that $\operatorname{\mathsf{con}}(\mathbf{z}_\ell^{(k)}) \subseteq \operatorname{\mathsf{con}}(\mathbf{z}_g^{(k)})$ for some $\ell < g$.
Then $\operatorname{\mathsf{con}}(\mathbf{z}_\ell^{(k)}) \subseteq \operatorname{\mathsf{con}}(\mathbf{z}_{\ell+1}^{(k)})$ by the condition (CF2b). Hence by applying the identities \eqref{basis S xx} and \eqref{basis S xy2x}, the identity $\mathbf{z}_{\ell}^{(k)}\mathbf{z}_{\ell+1}^{(k)} \approx \mathbf{z}_{\ell+1}^{(k)}$ is hold, and so $\mathbf{z}_{\ell}^{(k)}$ can be deleted from $\mathbf{w}_k$.
Hence we may assume that the condition (CF2c) is satisfied.
Let $x\in \operatorname{\mathsf{con}}(\mathbf{z}_{\ell-1}^{(k)}) \setminus \operatorname{\mathsf{con}}(\mathbf{z}_\ell^{(k)})$ and $y\in \operatorname{\mathsf{con}}(\mathbf{z}_\ell^{(k)}) \setminus \operatorname{\mathsf{con}}(\mathbf{z}_{\ell-1}^{(k)})$ for some $k\in\{0, 1, \ldots, n\}$ and $\ell\in \{1,\ldots, p_k\}$. Suppose that
$x, y$ satisfy one of the conditions (a), (b) and (c) in (CF3). Then since
\[
\cdots \mathbf{z}_{\ell-1}^{(k)} \mathbf{z}_{l}^{(k)} \cdots \stackrel{\eqref{basis S xy2x}}{\approx} \cdots (\mathbf{z}_{\ell-1}^{(k)}x) (y\mathbf{z}_{l}^{(k)}) \cdots \stackrel{(\dag)}{\approx} \cdots \mathbf{z}_{\ell-1}^{(k)} (xy)^2\mathbf{z}_{l}^{(k)}) \cdots
\]
where $(\dag) = \eqref{basis S txsy}$ if $x, y$ satisfy the condition (a) or (b), and $(\dag) = \{\eqref{basis S txaay}, \eqref{basis S xaayt}\}$ if $x, y$ satisfy the condition (c). Hence in these cases, we may assume that there is a perfect square for the set $\{x, y\}$ between $\mathbf{z}_{\ell-1}^{(k)}$ and $\mathbf{z}_\ell^{(k)}$, and so the condition (CF3) is satisfied.
\end{proof}
\begin{lemma} \label{lem: non-id}
The variety $\mathsf{var} \{A^1 \times B^1\}$ does not satisfy the following identities
\begin{align}
xy^{2}tx & \approx xy^{2}xtx, \label{id xyyxt} \\[0.04in]
xty^{2}x & \approx xtxy^{2}x, \label{id txyyx} \\[0.04in]
xsxtx &\approx xstx, \label{id xtxsx}\\[0.04in]
xxyy &\approx xy^2x \label{id xxyy}.
\end{align}
\end{lemma}
\begin{proof}
Let $x=e$, $y=d$ and $t=b$ in $A^1$. Then the left side of \eqref{id xyyxt} is $ed^2be =
a$, but the right side of \eqref{id xyyxt} is $ed^2ebe =0$, and so $A^1$ does not satisfy the identity \eqref{id xyyxt}. By a dual argument we may show that $B^{1}$ does not satisfy the identity \eqref{id txyyx}.
Let $x=e$, $s=c$ and $t=b$ in $A^1$. Then the left side of \eqref{id xtxsx} is $ecebe =
0$, but the right side of \eqref{id xtxsx} is $ecbe =0$, and so $A^1$ does not satisfy the identity \eqref{id xtxsx}.
Let $x=e$ and $y=d$ in $A^1$. Then the left side of \eqref{id xxyy} is $e^2d^2 =
c$, but the right side of \eqref{id xxyy} is $d^2e^2 =0$, and so $A^1$ does not satisfy the identity \eqref{id xxyy}.
\end{proof}
For any word $\mathbf{w}$, let $\mathsf{F_{SS}} (\mathbf{w})$ denote
the set of factors of $\mathbf{w}$ of length two that are formed by
simple letters:
\[
\mathsf{F_{SS}} (\mathbf{w}) = \{ xy \in \mathcal{X}^2 \mid \mathbf{w} \in \mathcal{X}^* xy \mathcal{X}^*, \, x,y \in \mathsf{sim} (\mathbf{w}) \}.
\]
For example, if $\mathbf{w} = x^3 abcyxdy^2 efx$, then
$\mathsf{F_{SS}} (\mathbf{w}) = \{ ab, bc, ef \}$.
\begin{lemma} \label{sim}
Suppose that $\mathbf{w} \approx \mathbf{w}'$ is any identity
satisfied by the semigroup $S.$ Then
\begin{enumerate}
\item $\operatorname{\mathsf{con}} (\mathbf{w}) = \operatorname{\mathsf{con}} (\mathbf{w}')$ and $\operatorname{\mathsf{sim}} (\mathbf{w}) = \operatorname{\mathsf{sim}} (\mathbf{w}');$
\item $\mathsf{F_{SS}} (\mathbf{w}) = \mathsf{F_{SS}} (\mathbf{w}');$
\item $\mathbf{w}_{\operatorname{\mathsf{sim}}} = \mathbf{w}_{\operatorname{\mathsf{sim}}}'.$
\end{enumerate}
\end{lemma}
\begin{proof}
(1) and (3) follow from Lemma~\ref{prelim LEM J1 words} since the subsemigroup $\{0, 1, b, d\}$ of $A^1$ is isomorphic to $J^1$.
(2) follows from Lemma~1.10 of \cite{LeeZhang} since the variety $\mathsf{var} \{A^1 \times B^1\}$ does not satisfy the identity \eqref{id xtxsx}.
\end{proof}
For the remainder of this section, suppose that $\mathbf{w}
\approx \mathbf{w}'$ is any identity satisfied by the variety $\mathsf{var} \{A^1 \times B^1\}$, where the words
\begin{align}\label{w=w'}
\mathbf{w} = \mathbf{w}_0 \prod_{i=1}^n (\mathbf{s}_i \mathbf{w}_i) \quad \text{and} \quad \mathbf{w}' = \mathbf{w}_0' \prod_{i=1}^{n'} (\mathbf{s}_i' \mathbf{w}_i')
\end{align}
are in canonical form. It follows from Lemma~\ref{sim} that $n=n'$ and $\mathbf{s}_k =\mathbf{s}_k'$ for each $k=0, \ldots, n$. The remainder of this section is devoted to the verification $\mathbf{w}_k = \mathbf{w}_k'$ for each $k=0, \ldots, n$.
\begin{lemma}\label{z con}
$\operatorname{\mathsf{con}}(\mathbf{w}_k)= \operatorname{\mathsf{con}}(\mathbf{w}_k')$.
\end{lemma}
\begin{proof}
Suppose that $x\in \operatorname{\mathsf{con}}(\mathbf{w}_k)\setminus \operatorname{\mathsf{con}}(\mathbf{w}_k')$. Since the variety $\mathsf{var} \{A^1 \times B^1\}$ satisfies the identity $\mathbf{w}
\approx \mathbf{w}'$, it is easy to see that the variety $\mathsf{var} \{A^1 \times B^1\}$ satisfies the identity $a\mathbf{w}b
\approx a\mathbf{w}'b$ where $a \ne b$ and $a, b \not\in \operatorname{\mathsf{con}}(\mathbf{w})=\operatorname{\mathsf{con}}(\mathbf{w}')$.
Then
\[
xsxtx \stackrel{\eqref{basis S xx}} \approx x(a\mathbf{w}b)_{\{s, t, x\}}x \approx x(a\mathbf{w}'b)_{\{s, t, x\}}x \stackrel{\eqref{basis S xx}} \approx xstx,
\]
where $s\in \operatorname{\mathsf{con}}(\mathbf{s}_k)$ if $k\geq 1$ and $s=a$ if $k=1$, and $t\in \operatorname{\mathsf{con}}(\mathbf{s}_{k+1})$ if $k< n$ and $t = b$ if $k=n$.
But this implies that the semigroup $S$ satisfies the identity \eqref{id xtxsx}, contradicting Lemma~\ref{lem: non-id}. Hence $\operatorname{\mathsf{con}}(\mathbf{w}_k)= \operatorname{\mathsf{con}}(\mathbf{w}_k')$.
\end{proof}
\begin{lemma}\label{z equivalent}
If for each $x,y \in \operatorname{\mathsf{con}}(\mathbf{w}_k)= \operatorname{\mathsf{con}}(\mathbf{w}_k')$, the condition
\begin{align}\label{equivalent}
x \prec_{\mathbf{w}_k} y \quad \mbox{if and only if}\quad x \prec_{\mathbf{w}_k'} y
\end{align}
is satisfied by the identity $\mathbf{w} \approx \mathbf{w}'$, then $\mathbf{w}_k=\mathbf{w}_k'$.
\end{lemma}
\begin{proof}
Without loss of generality, we may assume that
\[
\mathbf{w}_k= \mathbf{z}_1 \cdots \mathbf{z}_p \quad \mbox{and} \quad \mathbf{w}_k'= \mathbf{z}_1' \cdots \mathbf{z}_{p'}'.
\]
First we may show that
\begin{enumerate}
\item[(\dag)] if $\mathbf{z}_i$ is a perfect square factor of $\mathbf{w}_k$, then
$\mathbf{z}_i$ also is a perfect square factor of $\mathbf{w}_k'$.
\end{enumerate}
Suppose that $\operatorname{\mathsf{con}}(\mathbf{z}_i)$ is not any subset of $\operatorname{\mathsf{con}}(\mathbf{z}_g')$ for $g=1, \ldots, p'$. Since $\operatorname{\mathsf{con}}(\mathbf{w}_k)= \operatorname{\mathsf{con}}(\mathbf{w}_k')$ by Lemma~\ref{z con}, without loss of generality, we may assume that
$\operatorname{\mathsf{con}}(\mathbf{z}_l') \cup \operatorname{\mathsf{con}}(\mathbf{z}_g') \subseteq \operatorname{\mathsf{con}}(\mathbf{z}_i)$ for some $1 \leq \ell < g \leq p'$. By the condition (CF2c), let $x\in \operatorname{\mathsf{con}}(\mathbf{z}_l') \setminus \operatorname{\mathsf{con}}(\mathbf{z}_{l+1}')$ and $y\in \operatorname{\mathsf{con}}(\mathbf{z}_g') \setminus \operatorname{\mathsf{con}}(\mathbf{z}_l')$. It follows from (CF2b) that $x\not \in \operatorname{\mathsf{con}}(\mathbf{z}_{l+1}' \cdots \mathbf{z}_g'\cdots \mathbf{z}_p')$ and $y \not \in \operatorname{\mathsf{con}}(\mathbf{z}_1'\cdots \mathbf{z}_l')$, that is $x \prec_{\mathbf{w}_k'} y$. But $x \not\prec_{\mathbf{w}_k} y$ since $x,y \in \operatorname{\mathsf{con}}(\mathbf{z}_i)$, which contradicts the assumption. Therefore, we may assume that $\operatorname{\mathsf{con}}(\mathbf{z}_i) \subseteq \operatorname{\mathsf{con}}(\mathbf{z}_g')$ for some $g=1, \ldots, p'$.
Suppose that $\operatorname{\mathsf{con}}(\mathbf{z}_i) \subset \operatorname{\mathsf{con}}(\mathbf{z}_g')$. Without loss of generality, we may assume that $ z \in \operatorname{\mathsf{con}}(\mathbf{z}_g')\setminus \operatorname{\mathsf{con}}(\mathbf{z}_i)$. Then $z\in \operatorname{\mathsf{con}}(\mathbf{z}_1 \cdots \mathbf{z}_{i-1}\mathbf{z}_{i+1} \cdots \mathbf{z}_p)$ by Lemma~\ref{z con} and $z\not\in \operatorname{\mathsf{con}}(\mathbf{z}_1 \cdots \mathbf{z}_{i-1})\cap \operatorname{\mathsf{con}}(\mathbf{z}_{i+1} \cdots \mathbf{z}_p)$ by the condition (CF2b). Hence by symmetry, we may assume that $z\in \operatorname{\mathsf{con}}(\mathbf{z}_1 \cdots \mathbf{z}_{i-1})\setminus \operatorname{\mathsf{con}}(\mathbf{z}_{i+1} \cdots \mathbf{z}_p)$, in particular, say $z\in \operatorname{\mathsf{con}}(\mathbf{z}_\ell)\setminus \operatorname{\mathsf{con}}(\mathsf{z}_{\ell+1} \cdots \mathbf{z}_{i-1})$ for some $\ell < i$. Since $z \not\prec_{\mathbf{w}_k'} x$ for each $x \in \operatorname{\mathsf{con}}(\mathbf{z}_i)\subseteq \operatorname{\mathsf{con}}(\mathbf{z}_g')$, it follows from the assumption that $x \in \operatorname{\mathsf{con}}(\mathbf{z}_1 \cdots \mathbf{z}_{\ell})$. Hence $\operatorname{\mathsf{con}}(\mathbf{z}_i) \subseteq \operatorname{\mathsf{con}}(\mathbf{z}_1 \cdots \mathbf{z}_{\ell})$. Now it follows from the condition (CF2b) that $\operatorname{\mathsf{con}}(\mathbf{z}_i) \subseteq \operatorname{\mathsf{con}}(\mathbf{z}_{\ell})$, which contradicts the condition (CF2c). Hence $\operatorname{\mathsf{con}}(\mathbf{z}_i) =\operatorname{\mathsf{con}}(\mathbf{z}_g')$. Now by the definition of perfect square it is easy to see that $\mathbf{z}_i= \mathbf{z}_g'$ and so (\dag) holds.
The converse of (\dag) also holds by symmetry. It then follows that $\mathbf{z}_i$ is a perfect square factor of $\mathbf{w}_k$ if and only if $\mathbf{z}_i$ also is a perfect square factor of $\mathbf{w}_k'$. Hence it follows from the conditions (CF2) that
$p=p'$ and $\{\mathbf{z}_1, \ldots, \mathbf{z}_{p}\} = \{\mathbf{z}_1', \ldots, \mathbf{z}_{p}'\}$.
Suppose that the occurrence of $\mathbf{z}_{i+1}$ precedes the occurrence of $\mathbf{z}_{i}$ in $\mathbf{w}_k'$. By the condition (CF2c), let $x\in \operatorname{\mathsf{con}}(\mathbf{z}_i) \setminus \operatorname{\mathsf{con}}(\mathbf{z}_{i+1})$ and $y\in \operatorname{\mathsf{con}}(\mathbf{z}_{i+1}) \setminus \operatorname{\mathsf{con}}(\mathbf{z}_{i})$. Then $x \prec_{\mathbf{w}_k} y$, but $x \not\prec_{\mathbf{w}_k'} y$, which contradicts the assumption. Hence the order of occurrence of $\{\mathbf{z}_1, \ldots, \mathbf{z}_{p}\}$ in $\mathbf{w}_k$ is the same as the order of occurrence of $\{\mathbf{z}_1, \ldots, \mathbf{z}_{p}\}$ in $\mathbf{w}_k'$. Therefore $\mathbf{w}_k=\mathbf{w}_k'$.
\end{proof}
\begin{lemma} \label{zi=zi}
For each $x,y\in \operatorname{\mathsf{con}}(\mathbf{w}_k)=\operatorname{\mathsf{con}}(\mathbf{w}_k')$, if
$x \prec_{\mathbf{w}_k} y$, then $x \prec_{\mathbf{w}_k'} y$.
\end{lemma}
\begin{proof} Let
\[
\mathbf{w}_k= \mathbf{z}_1 \cdots \mathbf{z}_p.
\]
Seeking a contradiction, we may assume that
\begin{enumerate}
\item[(a)] $x\in \operatorname{\mathsf{con}}(\mathbf{z}_{\ell})\setminus \operatorname{\mathsf{con}}(\mathbf{z}_{\ell+1} \cdots \mathbf{z}_{p})$ and $y \in \operatorname{\mathsf{con}}(\mathbf{z}_{g})\setminus \operatorname{\mathsf{con}}(\mathbf{z}_{1} \cdots \mathbf{z}_{g-1})$ for some $1\leq \ell<g \leq p$;
\item[(b)] for each $z\in \operatorname{\mathsf{con}}(\mathbf{z}_{\ell+1}\cdots \mathbf{z}_{g-1})$, if $x \prec_{\mathbf{w}_k} z$, then $x \prec_{\mathbf{w}_k'} z$;
\item[(c)] for each $z\in \operatorname{\mathsf{con}}(\mathbf{z}_{\ell+1}\cdots \mathbf{z}_{g-1})$, if $z \prec_{\mathbf{w}_k} y$, then $z \prec_{\mathbf{w}_k'} y$;
\item[(d)] $x \not\prec_{\mathbf{w}_k'} y$, that is, there exist some $x$ occur after some $y$ in $\mathbf{w}_k'$.
\end{enumerate}
There are three cases to consider.
\noindent{\bf Case~1.} $y\not\in \operatorname{\mathsf{con}}(\mathbf{w}_0 \cdots \mathbf{w}_{k-1})$ and $x \not\in \operatorname{\mathsf{con}}(\mathbf{w}_{k+1} \cdots \mathbf{w}_{n})$.
Then
\[
x^2y^2\stackrel{\eqref{basis S xx}} \approx x\mathbf{w}_{\{x, y\}} \approx x\mathbf{w}'_{\{x, y\}} \stackrel{\eqref{basis S xy2x}} \approx xy^2x,
\]
but this implies that the variety $\mathsf{var} \{A^1 \times B^1\}$ satisfies the identity \eqref{id xxyy}, contradicting Lemma~\ref{lem: non-id}.
\noindent{\bf Case~2.} $x \in \operatorname{\mathsf{con}}(\mathbf{w}_{k+1} \cdots \mathbf{w}_{n})$, say $x \in \operatorname{\mathsf{con}}(\mathbf{w}_{h}) \setminus \operatorname{\mathsf{con}}(\mathbf{w}_{k+1}\cdots \mathbf{w}_{h-1})$ for some $h>k$ and let $t$ be a simple letter in $\mathbf{s}_{h}$.
\noindent{\bf 2.1.} $g=l+1$. Then $y \not\in \operatorname{\mathsf{con}} (\mathbf{z}_{1}\cdots \mathbf{z}_{\ell})$ by (CF2b) and (a), and so by (CF3), $y \not\in \operatorname{\mathsf{con}} (\mathbf{w}_{0}\cdots \mathbf{w}_{k-1}\mathbf{w}_{h}\cdots \mathbf{w}_{n})$. Hence it follows from Lemma~\ref{z con} that
\[
xy^2tx\stackrel{\eqref{basis S xx}} \approx x\mathbf{w}_{\{x, y, t\}} \approx x\mathbf{w}'_{\{x, y, t\}} \stackrel{\eqref{basis S xx}, \eqref{basis S xy2x}} \approx xy^2xtx,
\]
but this implies that the variety $\mathsf{var} \{A^1 \times B^1\}$ satisfies the identity \eqref{id xyyxt}, contradicting Lemma~\ref{lem: non-id}.
\noindent{\bf 2.2.} $g>l+1$. Then by (CF2c), there exist a letter $z$ such that $z \in \operatorname{\mathsf{con}}(\mathbf{z}_{\ell+1}) \setminus \operatorname{\mathsf{con}}(\mathbf{z}_{\ell})$ , and so $z \not\in \operatorname{\mathsf{con}}(\mathbf{z}_{1} \cdots \mathbf{z}_{\ell})$ by (CF2b). Suppose that $z\not\in \operatorname{\mathsf{con}}(\mathbf{z}_g)$. Then $z \not\in \operatorname{\mathsf{con}}(\mathbf{z}_{g} \cdots \mathbf{z}_{p})$ by (CF2b). Hence it is easy to see that $x \prec_{\mathbf{w}_{k}} z \prec_{\mathbf{w}_{k}} y$. Since $x \prec_{\mathbf{w}_{k}} z$, it follows from the assumption (b) that $x \prec_{\mathbf{w}_{k}'} z$. Since some $x$ occur after some $y$ in $\mathbf{w}_k'$ by (d),
\[
\mathbf{w}_k'= \cdots y\cdots x \cdots z\cdots,
\]
that is some $y$ occur before some $z$ in $\mathbf{w}_k'$. Hence $z \not\prec_{\mathbf{w}_{k}'} y$, which contradicts the assumption (c). Therefore $z\in \operatorname{\mathsf{con}}(\mathbf{z}_g)$. It follows from the condition (CF2c) that there exists a letter $s \ne z$ such that $s \in \operatorname{\mathsf{con}}(\mathbf{z}_{g-1}) \setminus \operatorname{\mathsf{con}}(\mathbf{z}_{g})$, and so $s\not\in \operatorname{\mathsf{con}}(\mathbf{z}_{g} \cdots \mathbf{z}_{p})$ by (CF2b). Hence it is easy to show that $x \prec_{\mathbf{w}_{k}} z$ and $s \prec_{\mathbf{w}_{k}} y$. It follows from the assumptions (b) and (c) that $x \prec_{\mathbf{w}_{k}'} z$ and $s \prec_{\mathbf{w}_{k}'} y$. Since some $x$ occur after some $y$ in $\mathbf{w}_k'$ by (d), it follows that $s \prec_{\mathbf{w}_{k}'} z$.
If $s\in \operatorname{\mathsf{con}}(\mathbf{w}_{k+1} \cdots \mathbf{w}_{h-1})$, say $s \in \operatorname{\mathsf{con}}(\mathbf{w}_{q}) \setminus \operatorname{\mathsf{con}}(\mathbf{w}_{k+1}\cdots \mathbf{w}_{q-1})$ for some $k<q<h$, then since
$s \in \operatorname{\mathsf{con}}(\mathbf{z}_{g-1}) \cap \operatorname{\mathsf{con}}(\mathbf{w}_{q})$ and $y\in \operatorname{\mathsf{con}}(\mathbf{z}_{g})\setminus \operatorname{\mathsf{con}}(\mathbf{z}_{1} \cdots \mathbf{z}_{g-1})$, it follows from (CF3) that $y \not\in \operatorname{\mathsf{con}} (\mathbf{w}_{0}\cdots \mathbf{w}_{k-1}\mathbf{w}_{q}\cdots\mathbf{w}_{h}\cdots \mathbf{w}_{n})$. Hence it follows from Lemma~\ref{z con} that
\[
xy^2tx\stackrel{\eqref{basis S xx}} \approx x\mathbf{w}_{\{x, y, t\}} \approx x\mathbf{w}'_{\{x, y, t\}} \stackrel{\eqref{basis S xx}, \eqref{basis S xy2x}} \approx xy^2xtx
\]
but this implies that the variety $\mathsf{var} \{A^1 \times B^1\}$ satisfies the identity \eqref{id xyyxt}, contradicting Lemma~\ref{lem: non-id}.
If $s\not\in \operatorname{\mathsf{con}}(\mathbf{w}_{k+1} \cdots \mathbf{w}_{h-1})$, then $s\not\in \operatorname{\mathsf{con}}(\mathbf{w}_{k+1}' \cdots \mathbf{w}_{h-1}')$ by Lemma~\ref{z con}. Since $x\in \operatorname{\mathsf{con}}(\mathbf{z}_\ell) \cap \operatorname{\mathsf{con}}(\mathbf{w}_{h})$ and $z\in \operatorname{\mathsf{con}}(\mathbf{z}_{\ell+1}) \setminus \operatorname{\mathsf{con}}(\mathbf{z}_{1}\cdots \mathbf{z}_{\ell})$, it follows from (CF3) that $z \not\in \operatorname{\mathsf{con}} (\mathbf{w}_{0}\cdots \mathbf{w}_{k-1}\mathbf{w}_{h}\cdots \mathbf{w}_{n})$, and so $z \not\in \operatorname{\mathsf{con}} (\mathbf{w}_{0}'\cdots \mathbf{w}_{k-1}'\mathbf{w}_{h}'\cdots \mathbf{w}_{n}')$ by Lemma~\ref{z con}.
It follows that
\[
sz^2sts\stackrel{\eqref{basis S xx}} \approx \mathbf{w}_{\{z, t, s\}}s \approx\mathbf{w}'_{\{z, t, s\}}s \stackrel{\eqref{basis S xx}, \eqref{basis S xy2x}} \approx s^2z^2ts,
\]
but this implies that the variety $\mathsf{var} \{A^1 \times B^1\}$ satisfies the identity \eqref{id xyyxt}, contradicting Lemma~\ref{lem: non-id}.
\noindent{\bf Case~3.} $y\in \operatorname{\mathsf{con}}(\mathbf{w}_0 \cdots \mathbf{w}_{k-1})$. By arguments that are dual to Case 2 we may show that the variety $\mathsf{var} \{A^1 \times B^1\}$ satisfies either the identity \eqref{id xxyy} or the identity \eqref{id txyyx}, contradicting Lemma~\ref{lem: non-id}.
\end{proof}
\begin{proof}[\bf Proof of Theorem~\ref{S basis theorem}]
It is routine to verify that the identities $\circledS$ hold in semigroups $A^{1}$ and $A^{1}$
so that the variety $\mathsf{var} \{A^1 \times B^1\}$
satisfies the identities $\circledS$. It remains to show that any
identity $\mathbf{w} \approx \mathbf{w}'$ of the
variety $\mathsf{var} \{A^1 \times B^1\}$ is a consequence of the identities $\circledS$.
In the presence of Lemma~\ref{canonical word}, it suffices to
assume that the identity $\mathbf{w} \approx \mathbf{w}'$ is
canonical. Without loss of generality, we may assume that $\mathbf{w}$ and $\mathbf{w}'$ are in the form of \eqref{w=w'},
and $n=n'$ and $\mathbf{s}_k =\mathbf{s}_k'$ for each $k=0, \ldots, n$.
By Lemma~\ref{zi=zi} and its dual, it is easy to see that the condition \eqref{equivalent} is satisfied by the identity $\mathbf{w} \approx \mathbf{w}'$. Hence it follows from Lemma~\ref{z equivalent} that $\mathbf{w}_k = \mathbf{w}_k'$ for each $k=0, \ldots, n$.
Thus the identity $\mathbf{w} \approx \mathbf{w}'$ is
trivial and so is vacuously a consequence of the identities $\circledS$.
\end{proof}
\section{Monoid subvarieties of $\mathsf{var} \{A^1 \times B^1\}$}\label{sec lattice}
In this section, all monoid subvarieties of $\mathsf{var} \{A^1 \times B^1\}$ will be characterized and the monoid subvariety lattice of $\mathsf{var} \{A^1 \times B^1\}$ will be completely described. For convenience,
the monoid subvariety of $\mathsf{var} \{A^1 \times B^1\}$
defined by $\Pi$ is denoted by $\mathsf{var}_{\mathbb{M}} \{ \Pi \}$.
\begin{lemma}
\label{lem zi=zi}
Let $\mathbf{w} \approx \mathbf{w}'$ be any identity in canonical form where
\[
\mathbf{w} = \mathbf{w}_0 \prod_{i=1}^{k-1}(\mathbf{s}_i \mathbf{w}_i)\ \ \mathbf{s}_k \underbrace{\mathbf{z}_{1} \cdots \mathbf{z}_{p}}_{\mathbf{w}_k}\prod_{i=k+1}^{n} (\mathbf{s}_i \mathbf{w}_i) \quad \text{and} \quad \mathbf{w}' = \mathbf{w}_0' \prod_{i=1}^{n} (\mathbf{s}_i \mathbf{w}_i')
\]
and $\operatorname{\mathsf{con}}(\mathbf{w}_i)= \operatorname{\mathsf{con}}(\mathbf{w}_i')$ for each $i=0, 1, \ldots, n$.
Suppose that $x$
and $y$ are non-simple letters of $\operatorname{\mathsf{con}}(\mathbf{w}_k) = \operatorname{\mathsf{con}}(\mathbf{w}_k')$ such that
\begin{enumerate}
\item[(a)] $x \prec_{\mathbf{w}_k} y$, say $x\in \operatorname{\mathsf{con}}(\mathbf{z}_{\ell})\setminus \operatorname{\mathsf{con}}(\mathbf{z}_{\ell+1} \cdots \mathbf{z}_{p})$ and $y \in \operatorname{\mathsf{con}}(\mathbf{z}_{g})\setminus \operatorname{\mathsf{con}}(\mathbf{z}_{1} \cdots \mathbf{z}_{g-1})$ for some $1\leq \ell<g \leq p$;
\item[(b)] for each $z\in \operatorname{\mathsf{con}}(\mathbf{z}_{\ell+1}\cdots \mathbf{z}_{g-1})$, if $x \prec_{\mathbf{w}_k} z$, then $x \prec_{\mathbf{w}_k'} z$;
\item[(c)] for each $z\in \operatorname{\mathsf{con}}(\mathbf{z}_{\ell+1}\cdots \mathbf{z}_{g-1})$, if $z \prec_{\mathbf{w}_k} y$, then $z \prec_{\mathbf{w}_k'} y$;
\item[(d)] $x \not\prec_{\mathbf{w}_k'} y$.
\end{enumerate}
Then
\[
\mathsf{var}_{\mathbb{M}} \{ \mathbf{w} \approx \mathbf{w}' \} =
\mathsf{var}_{\mathbb{M}} \{\mathbf{w}^{*} \approx \mathbf{w}', \Lambda \}
\]
where $\mathbf{w}^{*}$ equal either
\[
\mathbf{w}_0 \prod_{i=1}^{k-1}(\mathbf{s}_i \mathbf{w}_i)\ \ \mathbf{s}_k \underbrace{\mathbf{z}_{1} \cdots \mathbf{z}_{\ell}\cdots \mathbf{z}_{g-1}(xy)^2\mathbf{z}_{g}\cdots \mathbf{z}_{p}}_{\mathbf{w}_k}\prod_{i=k+1}^{n} (\mathbf{s}_i \mathbf{w}_i)
\]
or
\[
\mathbf{w}_0 \prod_{i=1}^{k-1}(\mathbf{s}_i \mathbf{w}_i)\ \ \mathbf{s}_k \underbrace{\mathbf{z}_{1} \cdots \mathbf{z}_{\ell}(xy)^2 \mathbf{z}_{\ell+1}\cdots \mathbf{z}_{g}\cdots \mathbf{z}_{p}}_{\mathbf{w}_k}\prod_{i=k+1}^{n} (\mathbf{s}_i \mathbf{w}_i)
\]
and $\Lambda$ is some subset of $\{ (\ref{id xyyxt}), (\ref{id txyyx}), (\ref{id xtxsx})\}$.
\end{lemma}
\begin{proof}
There are three cases to consider.
\noindent{\bf Case~1.} $y\not\in \operatorname{\mathsf{con}}(\mathbf{w}_0 \cdots \mathbf{w}_{k-1})$ and $x \not\in \operatorname{\mathsf{con}}(\mathbf{w}_{k+1} \cdots \mathbf{w}_{n})$. Then by Case 1 of Lemma~\ref{zi=zi} that
\[
\mathbf{w} \approx \mathbf{w}' \Vdash \eqref{id xxyy}.
\]
Therefore
\begin{equation}
\label{z lemma display1}
\mathsf{var}_{\mathbb{M}} \{ \mathbf{w} \approx \mathbf{w}' \} = \mathsf{var}_{\mathbb{M}} \{ \eqref{id xxyy}, \mathbf{w} \approx \mathbf{w}' \}.
\end{equation}
Now the deduction $\eqref{id xxyy} \Vdash \mathbf{w} \approx
\mathbf{w}^{*}$ holds since
\[
\begin{array}[c]{rcl}
\mathbf{w} \!\!\!\! & = & \!\!\!\! \mathbf{w}_0 \prod_{i=1}^{k-1} (\mathbf{s}_i \mathbf{w}_i)\ \ \mathbf{s}_k \mathbf{z}_{1} \cdots \mathbf{z}_{\ell} \mathbf{z}_{\ell+1} \cdots \mathbf{z}_{g-1}\mathbf{z}_{g} \cdots \mathbf{z}_{p} \prod_{i=k+1}^{n} \ (\mathbf{s}_i \mathbf{z}_i) \\[0.04in]
& \stackrel{\eqref{basis S xy2x}}{\approx} & \!\!\!\! \mathbf{w}_0 \prod_{i=1}^{k-1} (\mathbf{s}_i \mathbf{w}_i)\ \ \mathbf{s}_k \mathbf{z}_{1} \cdots (\mathbf{z}_{\ell}x^2) \mathbf{z}_{\ell+1} \cdots \mathbf{z}_{g-1}(y^2\mathbf{z}_{g}) \cdots \mathbf{z}_{p} \prod_{i=k+1}^{n} \ (\mathbf{s}_i \mathbf{z}_i) \\[0.04in]
& \stackrel{\eqref{id xxyy}}{\approx} & \!\!\!\! \mathbf{w}_0 \prod_{i=1}^{k-1} (\mathbf{s}_i \mathbf{w}_i)\ \ \mathbf{s}_k \mathbf{z}_{1} \cdots \mathbf{z}_{\ell}x \mathbf{z}_{\ell+1}x \cdots x \mathbf{z}_{g-1} x y^2 x\mathbf{z}_{g} \cdots \mathbf{z}_{p} \prod_{i=k+1}^{n} \ (\mathbf{s}_i \mathbf{z}_i) \\[0.04in]
& \stackrel{\eqref{basis S xy2x}}{\approx} & \!\!\!\! \mathbf{w}_0 \prod_{i=1}^{k-1} (\mathbf{s}_i \mathbf{w}_i)\ \ \mathbf{s}_k \mathbf{z}_{1} \cdots \mathbf{z}_{\ell}x \mathbf{z}_{\ell+1}x \cdots x \mathbf{z}_{g-1} (xy)^2\mathbf{z}_{g} \cdots \mathbf{z}_{p} \prod_{i=k+1}^{n} \ (\mathbf{s}_i \mathbf{z}_i) \\[0.04in]
& \stackrel{\eqref{id xxyy}}{\approx} & \!\!\!\! \mathbf{w}_0 \prod_{i=1}^{k-1} (\mathbf{s}_i \mathbf{w}_i)\ \ \mathbf{s}_k \mathbf{z}_{1} \cdots \mathbf{z}_{\ell}x^2 \mathbf{z}_{\ell+1} \cdots \mathbf{z}_{g-1} (xy)^2\mathbf{z}_{g} \cdots \mathbf{z}_{p} \prod_{i=k+1}^{n} \ (\mathbf{s}_i \mathbf{z}_i) \\[0.04in]
& \stackrel{\eqref{basis S xy2x}}{\approx} & \!\!\!\! \mathbf{w}_0 \prod_{i=1}^{k-1} (\mathbf{s}_i \mathbf{w}_i)\ \ \mathbf{s}_k \mathbf{z}_{1} \cdots \mathbf{z}_{\ell} \mathbf{z}_{\ell+1} \cdots \mathbf{z}_{g-1} (xy)^2\mathbf{z}_{g} \cdots \mathbf{z}_{p} \prod_{i=k+1}^{n} \ (\mathbf{s}_i \mathbf{z}_i) \\[0.04in]
& = & \!\!\!\! \mathbf{w}^{*}.
\end{array}
\]
It follows from $(\ref{z lemma display1})$ that
$\mathsf{var}_{\mathbb{M}} \{ \mathbf{w} \approx \mathbf{w}' \} = \mathsf{var}_{\mathbb{M}} \{ (\ref{id xxyy}), \mathbf{w}^{*} \approx
\mathbf{w}'\}$.
\noindent{\bf Case~2.} $x \in \operatorname{\mathsf{con}}(\mathbf{w}_{k+1} \cdots \mathbf{w}_{n})$.
Then by Case 2 of Lemma~\ref{zi=zi} that
\[
\mathbf{w} \approx \mathbf{w}' \Vdash \eqref{id xyyxt}.
\]
Therefore
\begin{equation}
\label{z lemma display2}
\mathsf{var}_{\mathbb{M}} \{ \mathbf{w} \approx \mathbf{w}' \} = \mathsf{var}_{\mathbb{M}} \{ \eqref{id xyyxt}, \mathbf{w} \approx \mathbf{w}' \}.
\end{equation}
Now the deduction $\eqref{id xyyxt} \Vdash \mathbf{w} \approx
\mathbf{w}^{*}$ holds since
\[
\begin{array}[c]{rcl}
\mathbf{w} \!\!\!\! & = & \!\!\!\! \mathbf{w}_0 \prod_{i=1}^{k-1} (\mathbf{s}_i \mathbf{w}_i)\ \ \mathbf{s}_k \mathbf{z}_{1} \cdots \mathbf{z}_{\ell} \mathbf{z}_{\ell+1} \cdots \mathbf{z}_{g-1}\mathbf{z}_{g} \cdots \mathbf{z}_{p} \prod_{i=k+1}^{n} \ (\mathbf{s}_i \mathbf{z}_i) \\[0.04in]
& \stackrel{\eqref{basis S xy2x}}{\approx} & \!\!\!\! \mathbf{w}_0 \prod_{i=1}^{k-1} (\mathbf{s}_i \mathbf{w}_i)\ \ \mathbf{s}_k \mathbf{z}_{1} \cdots (\mathbf{z}_{\ell}x^2) \mathbf{z}_{\ell+1} \cdots \mathbf{z}_{g-1}(y^2\mathbf{z}_{g}) \cdots \mathbf{z}_{p} \prod_{i=k+1}^{n} \ (\mathbf{s}_i \mathbf{z}_i) \\[0.04in]
& \stackrel{\eqref{id xxyy}}{\approx} & \!\!\!\! \mathbf{w}_0 \prod_{i=1}^{k-1} (\mathbf{s}_i \mathbf{w}_i)\ \ \mathbf{s}_k \mathbf{z}_{1} \cdots \mathbf{z}_{\ell}x \mathbf{z}_{\ell+1}x \cdots x \mathbf{z}_{g-1} x y^2 x\mathbf{z}_{g} \cdots \mathbf{z}_{p} \prod_{i=k+1}^{n} \ (\mathbf{s}_i \mathbf{z}_i) \\[0.04in]
& \stackrel{\eqref{basis S xy2x}}{\approx} & \!\!\!\! \mathbf{w}_0 \prod_{i=1}^{k-1} (\mathbf{s}_i \mathbf{w}_i)\ \ \mathbf{s}_k \mathbf{z}_{1} \cdots \mathbf{z}_{\ell}x \mathbf{z}_{\ell+1}x \cdots x \mathbf{z}_{g-1} (xy)^2\mathbf{z}_{g} \cdots \mathbf{z}_{p} \prod_{i=k+1}^{n} \ (\mathbf{s}_i \mathbf{z}_i) \\[0.04in]
& \stackrel{\eqref{id xxyy}}{\approx} & \!\!\!\! \mathbf{w}_0 \prod_{i=1}^{k-1} (\mathbf{s}_i \mathbf{w}_i)\ \ \mathbf{s}_k \mathbf{z}_{1} \cdots \mathbf{z}_{\ell}x^2 \mathbf{z}_{\ell+1} \cdots \mathbf{z}_{g-1} (xy)^2\mathbf{z}_{g} \cdots \mathbf{z}_{p} \prod_{i=k+1}^{n} \ (\mathbf{s}_i \mathbf{z}_i) \\[0.04in]
& \stackrel{\eqref{basis S xy2x}}{\approx} & \!\!\!\! \mathbf{w}_0 \prod_{i=1}^{k-1} (\mathbf{s}_i \mathbf{w}_i)\ \ \mathbf{s}_k \mathbf{z}_{1} \cdots \mathbf{z}_{\ell} \mathbf{z}_{\ell+1} \cdots (xy)^2\mathbf{z}_{g} \cdots \mathbf{z}_{p} \prod_{i=k+1}^{n} \ (\mathbf{s}_i \mathbf{z}_i) \\[0.04in]
& = & \!\!\!\! \mathbf{w}^{*}.
\end{array}
\]
It follows from $(\ref{z lemma display2})$ that
$\mathsf{var}_{\mathbb{M}} \{ \mathbf{w} \approx \mathbf{w}' \} =
\mathsf{var}_{\mathbb{M}} \{(\ref{id xyyxt}), \mathbf{w}^{*} \approx
\mathbf{w}'\}$.
\noindent{\bf Case~3.} $y\in \operatorname{\mathsf{con}}(\mathbf{w}_0 \cdots \mathbf{w}_{k-1})$. Then by Case 3 of Lemma~\ref{zi=zi} that
\[
\mathbf{w} \approx \mathbf{w}' \Vdash \eqref{id txyyx}.
\]
Therefore
\begin{equation}
\label{z lemma display3}
\mathsf{var}_{\mathbb{M}} \{ \mathbf{w} \approx \mathbf{w}' \} = \mathsf{var}_{\mathbb{M}} \{ \eqref{id txyyx}, \mathbf{w} \approx \mathbf{w}' \}.
\end{equation}
Now the deduction $\eqref{id txyyx} \Vdash \mathbf{w} \approx
\mathbf{w}^{*}$ holds since
\[
\begin{array}[c]{rcl}
\mathbf{w} \!\!\!\! & = & \!\!\!\! \mathbf{w}_0 \prod_{i=1}^{k-1} (\mathbf{s}_i \mathbf{w}_i)\ \ \mathbf{s}_k \mathbf{z}_{1} \cdots \mathbf{z}_{\ell} \mathbf{z}_{\ell+1} \cdots \mathbf{z}_{g-1}\mathbf{z}_{g} \cdots \mathbf{z}_{p} \prod_{i=k+1}^{n} \ (\mathbf{s}_i \mathbf{z}_i) \\[0.04in]
& \stackrel{\eqref{basis S xy2x}}{\approx} & \!\!\!\! \mathbf{w}_0 \prod_{i=1}^{k-1} (\mathbf{s}_i \mathbf{w}_i)\ \ \mathbf{s}_k \mathbf{z}_{1} \cdots (\mathbf{z}_{\ell}x^2) \mathbf{z}_{\ell+1} \cdots \mathbf{z}_{g-1}(y^2\mathbf{z}_{g}) \cdots \mathbf{z}_{p} \prod_{i=k+1}^{n} \ (\mathbf{s}_i \mathbf{z}_i) \\[0.04in]
& \stackrel{\eqref{id xxyy}}{\approx} & \!\!\!\! \mathbf{w}_0 \prod_{i=1}^{k-1} (\mathbf{s}_i \mathbf{w}_i)\ \ \mathbf{s}_k \mathbf{z}_{1} \cdots \mathbf{z}_{\ell} yx^2 y\mathbf{z}_{\ell+1}y \cdots y \mathbf{z}_{g-1} y \mathbf{z}_{g} \cdots \mathbf{z}_{p} \prod_{i=k+1}^{n} \ (\mathbf{s}_i \mathbf{z}_i) \\[0.04in]
& \stackrel{\eqref{basis S xy2x}}{\approx} & \!\!\!\! \mathbf{w}_0 \prod_{i=1}^{k-1} (\mathbf{s}_i \mathbf{w}_i)\ \ \mathbf{s}_k \mathbf{z}_{1} \cdots \mathbf{z}_{\ell} (xy)^2 \mathbf{z}_{\ell+1}y \cdots y \mathbf{z}_{g-1} y \mathbf{z}_{g} \cdots \mathbf{z}_{p} \prod_{i=k+1}^{n} \ (\mathbf{s}_i \mathbf{z}_i) \\[0.04in]
& \stackrel{\eqref{id xxyy}}{\approx} & \!\!\!\! \mathbf{w}_0 \prod_{i=1}^{k-1} (\mathbf{s}_i \mathbf{w}_i)\ \ \mathbf{s}_k \mathbf{z}_{1} \cdots \mathbf{z}_{\ell}(xy)^2 \mathbf{z}_{\ell+1} \cdots \mathbf{z}_{g-1} y^2\mathbf{z}_{g} \cdots \mathbf{z}_{p} \prod_{i=k+1}^{n} \ (\mathbf{s}_i \mathbf{z}_i) \\[0.04in]
& \stackrel{\eqref{basis S xy2x}}{\approx} & \!\!\!\! \mathbf{w}_0 \prod_{i=1}^{k-1} (\mathbf{s}_i \mathbf{w}_i)\ \ \mathbf{s}_k \mathbf{z}_{1} \cdots \mathbf{z}_{\ell} (xy)^2 \mathbf{z}_{\ell+1} \cdots \mathbf{z}_{g-1}\mathbf{z}_{g} \cdots \mathbf{z}_{p} \prod_{i=k+1}^{n} \ (\mathbf{s}_i \mathbf{z}_i) \\[0.04in]
& = & \!\!\!\! \mathbf{w}^{*}.
\end{array}
\]
It follows from $(\ref{z lemma display3})$ that
$\mathsf{var}_{\mathbb{M}} \{ \mathbf{w} \approx \mathbf{w}' \} =
\mathsf{var}_{\mathbb{M}} \{ (\ref{id txyyx}), \mathbf{w}^{*} \approx
\mathbf{w}'\}$.
\end{proof}
\begin{lemma}
\label{*zi=zi}
Let $\mathbf{w} \approx \mathbf{w}'$ be any identity in canonical form where
\[
\mathbf{w} = \mathbf{w}_0 \prod_{i=1}^{n}(\mathbf{s}_i \mathbf{w}_i) \quad \text{and} \quad \mathbf{w}' = \mathbf{w}_0' \prod_{i=1}^{n} (\mathbf{s}_i \mathbf{w}_i')
\]
and $\operatorname{\mathsf{con}}(\mathbf{w}_i)= \operatorname{\mathsf{con}}(\mathbf{w}_i')$ for each $i=0, 1, \ldots, n$.
Suppose that $\mathbf{w} \approx \mathbf{w}'$ does not satisfy the condition \eqref{equivalent}.
Then
\[
\mathsf{var}_{\mathbb{M}} \{ \mathbf{w} \approx \mathbf{w}' \} =
\mathsf{var}_{\mathbb{M}} \{\Lambda \}
\]
for some set $\Lambda$ of identities from $\{ (\ref{id xyyxt}), (\ref{id txyyx}), (\ref{id xtxsx})\}$.
\end{lemma}
\begin{proof}
Since $\mathbf{w} \approx \mathbf{w}'$ does not satisfy the condition \eqref{equivalent}, there must exist letters $x,y\in \operatorname{\mathsf{con}}(\mathbf{w}_k)= \operatorname{\mathsf{con}}(\mathbf{w}_k')$ for some $k$ such that $x$ and $y$ satisfy the conditions (1)-(4) of Lemma~\ref{zi=zi} or its dual conditions. First if there exist some $x_1$ and $y_1$ satisfy the conditions (1)-(4) in $\mathbf{w}$. Then by Lemma~\ref{zi=zi},
the identities $\{ (\ref{id xyyxt}), (\ref{id txyyx}), (\ref{id xtxsx})\}$ can be used to convert $\mathbf{w} \approx \mathbf{w}'$ into $\mathbf{w}^{(1)} \approx \mathbf{w}'$ such that
\begin{enumerate}
\item $x_1 \prec_{\mathbf{w}_k} y_1$ and $x_1 \not \prec_{\mathbf{w}_k^{(1)}} y_1$;
\item if $x_1 \not\prec_{\mathbf{w}_k} z$ (resp. $z \not\prec_{\mathbf{w}_k} y_1$) for any $z \in \operatorname{\mathsf{con}}(\mathbf{w})$, then $x_1 \not\prec_{\mathbf{w}_k^{(1)}} z$ (resp. $z \not\prec_{\mathbf{w}_k^{(1)}} y_1$);
\item $z \not\prec_{\mathbf{w}_k} t$ if and only if $z \not\prec_{\mathbf{w}_k^{(1)}} t$ for each $z , t \in \operatorname{\mathsf{con}}(\mathbf{w}) \setminus \{x_1, y_1\}$.
\end{enumerate}
whence $\mathsf{var}_{\mathbb{M}} \{ \mathbf{w} \approx \mathbf{w}' \} =
\mathsf{var}_{\mathbb{M}}\{\mathbf{w}^{(1)} \approx \mathbf{w}', \Lambda^{(1)} \}$ for some set $\Lambda^{(1)}$ of identities from $\{ (\ref{id xyyxt}), (\ref{id txyyx}), (\ref{id xtxsx})\}$.
Now if there still exist letters $x_2, y_2$ in $\mathbf{w}^{(1)} \approx \mathbf{w}'$ such that $x_2, y_2$ does not satisfy the conditions (1)-(4) in $\mathbf{w}^{(1)}$, then the above procedure can be repeated to
construct an identity $\mathbf{w}^{(2)} \approx
\mathbf{w}'$ and some subset $\Lambda^{(2)} \subseteq \{ (\ref{id xyyxt}), (\ref{id txyyx}), (\ref{id xtxsx})\}$.
The construction of $\mathbf{w}^{(1)} \approx \mathbf{w}',
\mathbf{w}^{(2)} \approx \mathbf{w}', \ldots$ and
$\Lambda^{(1)}, \Lambda^{(2)}, \ldots$ cannot continue
indefinitely since that is bounded above by $C_{|\operatorname{\mathsf{con}}(\mathbf{w})|}^{2}$.
This
procedure can be repeated to obtain an identity $\mathbf{w}^{*}
\approx \mathbf{w}'$ that satisfies the property
\begin{align}\label{left equ}
\mbox{for any $k=0,1,\ldots, n$, $x,y \in \operatorname{\mathsf{con}}(\mathbf{w}_k')$, if $x \not\prec_{\mathbf{w}_k'} y$, then $x \not\prec_{\mathbf{w}_k^{*}} y$.}
\end{align}
Hence $\mathsf{var}_{\mathbb{M}} \{ \mathbf{w} \approx \mathbf{w}' \} =
\mathsf{var}_{\mathbb{M}}\{\mathbf{w}^{*} \approx \mathbf{w}', \Lambda \}$.
Now if there exist letters $x$ and $y$ satisfy the conditions (1)-(4) in $\mathbf{w}'$ of Lemma~\ref{zi=zi}, then by the same arguments to $\mathbf{w}'$, we can construct an identity $\mathbf{w}^{*} \approx \mathbf{w}^{'*}$ that satisfies the property
\begin{align}\label{right equ}
\mbox{for any $k=0,1,\ldots, n$, $x,y \in \operatorname{\mathsf{con}}(\mathbf{w}_k^{*})$, if $x \not\prec_{\mathbf{w}_k^{*}} y$, then $x \not\prec_{\mathbf{w}_k^{'*}} y$.}
\end{align}
Hence $\mathbf{S} \{ \mathbf{w}^{*} \approx \mathbf{w}'\} =
\mathbf{S}\{\mathbf{w}^{*} \approx \mathbf{w}^{'*}, \Lambda \}$.
Since $\mathbf{w}^{*} \approx \mathbf{w}^{'*}$ satisfies the conditions \eqref{left equ} and \eqref{right equ}, by Lemma~\ref{z equivalent} that $\mathbf{w}^{*} \approx \mathbf{w}^{'*}$ is the trivial identity, and so $\mathsf{var}_{\mathbb{M}} \{ \mathbf{w}^{*} \approx \mathbf{w}'\} =
\mathsf{var}_{\mathbb{M}}\{\Lambda \}$.
\end{proof}
\begin{lemma}
\label{cont-w}
Let $\mathbf{w} \approx \mathbf{w}'$ be any identity in canonical form where
\[
\mathbf{w} = \mathbf{w}_0 \prod_{i=1}^{n}(\mathbf{s}_i \mathbf{w}_i) \quad \text{and} \quad \mathbf{w}' = \mathbf{w}_0' \prod_{i=1}^{n} (\mathbf{s}_i \mathbf{w}_i').
\]
If $\operatorname{\mathsf{con}}(\mathbf{w}_i)= \operatorname{\mathsf{con}}(\mathbf{w}_i')$ for some $i=1, \ldots, n$, then
$\mathbf{w} \approx \mathbf{w}' \Vdash \eqref{id xtxsx}$.
\end{lemma}
\begin{proof}
It follows from the proof of Lemma~\ref{z con}.
\end{proof}
\begin{lemma}
Let $\mathbf{w} \approx \mathbf{w}'$ be any identity.
\begin{enumerate}
\item If either $\operatorname{\mathsf{con}} (\mathbf{w}) \ne \operatorname{\mathsf{con}} (\mathbf{w}')$ or $\operatorname{\mathsf{sim}} (\mathbf{w}) \ne \operatorname{\mathsf{sim}} (\mathbf{w}')$, then $\mathbf{w} \approx \mathbf{w}' \Vdash x^2 \approx x \Vdash \eqref{id xtxsx}$;
\item If $\mathsf{F_{SS}} (\mathbf{w}) \ne \mathsf{F_{SS}} (\mathbf{w}')$, then $\mathbf{w} \approx \mathbf{w}' \Vdash \eqref{id xtxsx}$;
\item If $\mathbf{w}_{\operatorname{\mathsf{sim}}} \ne \mathbf{w}_{\operatorname{\mathsf{sim}}}',$ then $\mathbf{w} \approx \mathbf{w}' \Vdash \eqref{id xtxsx}$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) If $\mathbf{w} \approx \mathbf{w}'$ satisfy either $\operatorname{\mathsf{con}} (\mathbf{w}) \ne \operatorname{\mathsf{con}} (\mathbf{w}')$ or $\operatorname{\mathsf{sim}} (\mathbf{w}) = \operatorname{\mathsf{sim}} (\mathbf{w}')$, then monoid $N_2^1$ does not in the variety $\mathsf{var}_{\mathbb{M}}\{A^1 \times B^1\}$. It follows from Lemma~1.9 of \cite{LeeZhang}, that
\[
\mathbf{w} \approx \mathbf{w}' \Vdash y^2xy^2xy^2xy^2\approx y^2xy^2 \Vdash x^2 \approx x \Vdash \eqref{id xtxsx}
\]
(2) It follows from the proof of (2) of Lemma~\ref{sim}.
(3) If $\mathbf{w}_{\operatorname{\mathsf{sim}}} \ne \mathbf{w}_{\operatorname{\mathsf{sim}}}',$ then we can derive some identity $\overline{\mathbf{w}} \approx \overline{\mathbf{w}'}$ from $\mathbf{w} \approx \mathbf{w}'$ by deleting some letters in $\operatorname{\mathsf{con}}(\mathbf{w})\cup \operatorname{\mathsf{con}}(\mathbf{w}')$, such that
$\mathsf{F_{SS}} (\overline{\mathbf{w}}) \ne \mathsf{F_{SS}} (\overline{\mathbf{w}'})$. Then it follows from (2) that
$\mathbf{w} \approx \mathbf{w}' \Vdash \eqref{id xtxsx}$.
\end{proof}
\begin{theorem}
The subvariety lattice of $\mathsf{var}_{\mathbb{M}} \{A^1 \times B^1\}$ is as shown in Figure \ref{lattice}.
\end{theorem}
\begin{proof}
Let $\mathbf{w} \approx \mathbf{w}'$ be any nontrivial identity in canonical form where
\[
\mathbf{w} = \mathbf{w}_0 \prod_{i=1}^{n}(\mathbf{s}_i \mathbf{w}_i) \quad \text{and} \quad \mathbf{w}' = \mathbf{w}_0' \prod_{i=1}^{n'} (\mathbf{s}_i' \mathbf{w}_i').
\]
If $\mathbf{w} \approx \mathbf{w}'$ does not satisfy
one of the following conditions:
\begin{enumerate}
\item[(a)] $\operatorname{\mathsf{con}} (\mathbf{w}) = \operatorname{\mathsf{con}} (\mathbf{w}')$ and $\operatorname{\mathsf{sim}} (\mathbf{w}) = \operatorname{\mathsf{sim}} (\mathbf{w}')$;
\item[(b)] $\mathsf{F_{SS}} (\mathbf{w}) = \mathsf{F_{SS}}$;
\item[(c)] $\mathbf{w}_{\operatorname{\mathsf{sim}}} = \mathbf{w}_{\operatorname{\mathsf{sim}}}'$;
\item[(d)] $\operatorname{\mathsf{con}}(\mathbf{w}_i)= \operatorname{\mathsf{con}}(\mathbf{w}_i')$,
\end{enumerate}
then $\mathbf{w} \approx \mathbf{w}' \Vdash \eqref{id xtxsx}$. Hence each subvariety that does not satisfy one of the conditions (a)-(d) is contained in the variety $\mathsf{var}_{\mathbb{M}}\circledS \cup \{\eqref{id xtxsx}\})$. It is easy to see that the variety $\mathsf{var}_{\mathbb{M}}\circledS \cup \{\eqref{id xtxsx}\}$ is just the variety $\mathsf{var}_{\mathbb{M}}\{A_0^1\}$ and its subvariety lattice can be found in \cite{Lee08}.
If $\mathbf{w} \approx \mathbf{w}'$ satisfy all of the conditions (a)-(d), then since $\mathbf{w} \approx \mathbf{w}'$ is nontrivial, it follows that $\mathbf{w}_k \ne \mathbf{w}_k'$ for some $k=0,\ldots,n$. Now it follows from Lemma~\ref{*zi=zi} that
\[
\mathsf{var}_{\mathbb{M}} \{ \mathbf{w} \approx \mathbf{w}' \} =
\mathsf{var}_{\mathbb{M}} \{\Lambda \}
\]
for some set $\Lambda$ of identities from $\{ (\ref{id xyyxt}), (\ref{id txyyx}), (\ref{id xtxsx})\}$.
It is easy to show that
\[
\eqref{id xxyy} \Vdash \eqref{id xyyxt},\quad \eqref{id xxyy} \Vdash \eqref{id txyyx},\quad \eqref{id xtxsx} \Vdash \eqref{id xyyxt},\quad \eqref{id xtxsx} \Vdash \eqref{id txyyx}.
\]
It follows from Proposition 4.3 of \cite{LeeLi11} that $\mathsf{var}_{\mathbb{M}}\{\circledS \cup \{\eqref{id xxyy}\}\}={\mathbb Q}^1$.
Clearly, the variety $\mathsf{var}_{\mathbb{M}} \circledS \cup \{\eqref{id xtxsx}\}=\mathsf{var}_{\mathbb{M}} \{A_0^1\}$ and $\mathsf{var}_{\mathbb{M}} \circledS \cup \{\eqref{id xxyy}\}=\mathsf{var}_{\mathbb{M}} \{Q^1\}$ are incomparable. It is routine to verify that $\mathsf{var}_{\mathbb{M}}\{A^{1}\}$ satisfy the identity $\eqref{id txyyx}$ but not $\eqref{id xyyxt}$, and $\mathsf{var}_{\mathbb{M}}\{A^{1})$ satisfy the identity $\eqref{id xyyxt}$ but not $\eqref{id txyyx}$, and the variety $\mathsf{var}_{\mathbb{M}}\circledS \cup \{\eqref{id xyyxt}\}$ and $\mathsf{var}_{\mathbb{M}}\circledS \cup \{\eqref{id txyyx}\}$ are incomparable. Hence $\mathsf{var}_{\mathbb{M}}\circledS \cup \{\eqref{id txyyx}\}=\mathsf{var}_{\mathbb{M}}\{A^{1}\}$ and $\mathsf{var}_{\mathbb{M}}\circledS \cup \{\eqref{id xyyxt}\}=\mathsf{var}_{\mathbb{M}}\{B^1\}$, which are the maximal proper monoid subvarieties of $\mathsf{var}_{\mathbb{M}}\{A^1 \times B^1\}$. Hence
the monoid subvariety lattice of $\mathsf{var}_{\mathbb{M}}\{A^1 \times B^1\}$ is as shown in Figure \ref{lattice}.
\end{proof}
\begin{figure}
\caption{The monoid subvariety lattice $\mathsf{var}
\label{lattice}
\end{figure}
{\bf Acknowledgments} The authors would like to express their
gratitude to Dr E. W. H. Lee for his comments and helps.
\end{document} | math | 66,683 |
\begin{document}
\begin{frontmatter}
\title{The solution of a generalized secretary problem via analytic expressions}
\author[rvt]{Adam Woryna}
\ead{[email protected]}
\address[rvt]{Silesian University of Technology, Institute of Mathematics, ul. Kaszubska 23, 44-100 Gliwice, Poland}
\begin{abstract}
Given integers $1\leq k<n$, the Gusein-Zade version of a generalized secretary problem is to choose one of the $k$ best of $n$ candidates for a secretary, which are interviewing in random order. The stopping rule in the selection is based only on the relative ranks of the successive arrivals. It is known that the best policy can be described by a non--decreasing sequence $(s_1, \ldots, s_k)$ of integers with $l\leq s_l<n$ for every $1\leq l\leq k$, and conversely, any such a sequence determines the general structure of the best policy. We found a finite analytic expression for the probability of success when using the optimal policy with a sequence $(s_1, \ldots, s_k)$. We also study the problem of the construction of the optimal sequence, i.e. a sequence which maximizes the corresponding probability of success. We discovered finite analytic expressions which
enable to calculate the elements $s_l$ of an optimal sequence one by one, from $l=k$ to $l=1$. Until now, such expressions were derived separately, and only for the values $k\leq 3$.
\end{abstract}
\begin{keyword}
Secretary problem \sep optimal stopping \sep optimal sequence \sep analytic expression \sep combinatorial identity
\MSC[2010]60C05\sep 62P25\sep 05A19\sep 05A05\sep 33C90\sep 90C27
\end{keyword}
\end{frontmatter}
\section{Introduction and the main results}
In the paper we study the Gusein-Zade version of a generalized secretary problem (see~\cite{1}). There are many ways for presenting this optimal stopping problem. In the romantic version, instead of interviewing the candidates for a secretary, we have a bachelor who has an occasion to meet a certain number of girls during his bachelorhood and who found out about this number (denoted further by $n$) in some miraculous way. The bachelor wants to marry one of the $k$ best girls, where $k$ is fixed and less than $n$. He can not be sure of success as he follows in his live the following principles: in every time he gets to know only one of the girls and after some time he must decide to marry her or to split up. In the latter case he starts to meet the next girl, but later on he can not go back to any girl he decided to split up. The order in which he gets to know the girls is random, thus there are $n!$ equally likely orderings.
The bachelor is able to judge only the present girl or the girls he met previously, and he has no idea about the attraction of the future girls. However, we assume that no two girls will turn out equally attractive for him. The problem is to find the best policy for the bachelor, i.e. the policy which maximizes the probability of the marriage to the girl that is one of the $k$ best. In the paper~\cite{1} it was proved the following general structure of such a policy:
\begin{proposition}[\cite{1}]\label{p1}
The best policy for a bachelor who wants to marry one of the $k$ best girls is described by a certain non--decreasing sequence $(s_1, \ldots, s_k)$ of integers with $l\leq s_l<n$ for every $1\leq l\leq k$ in the following way: marry the $i_0$-th girl, where $1\leq i_0\leq n$ is the smallest integer such that there is $1\leq l\leq k$ which satisfies:
\begin{itemize}
\item[(i)] $s_l<i_0\leq s_{l+1}$ (assume $s_{k+1}:=n-1$),
\item[(ii)] the $i_0$-th girl is one of the $l$ best of $i_0$ girls met so far.
\end{itemize}
If such a number $i_0$ does not exist, then marry the $n$-th girl.
\end{proposition}
Given an arbitrary non--decreasing sequence $(s_1, \ldots, s_k)$ of integers such that $l\leq s_l<n$ for every $1\leq l\leq k$, it is natural to ask about the probability of success when using the above described policy with the sequence $(s_1, \ldots, s_k)$. Namely, we would like to know how this probability depends on the elements of this sequence and how to construct a sequence which maximizes this probability.
\begin{definition}\label{ddddef}
We call a sequence $(s_1, \ldots, s_k)$ which maximizes the probability of success for the policy described in Proposition~\ref{p1} as an {\it optimal sequence}.
\end{definition}
The classical version, i.e. the case $k=1$, was solved by Lindley (\cite{5}) by using equations arising from the principle of dynamic programming. He solved these equations by simple backward recursion and obtained that the only element of an optimal sequence is equal to the smallest integer $1\leq x\leq n-1$ such that $H(n-1)-H(x)\leq 1$, where $H(x):=\sum_{j=1}^{x}1/j$ is the $x$-th harmonic number, as well as that the probability of success when using the optimal policy with an element $s_1$ is equal to $s_1(H(n-1)-H(s_1-1))/n$ (see also~\cite{13} for the survey paper). The cases $k=2,3$ were solved by using backward induction and exploiting the existence of an imbedded Markov chain. In the case $k=2$ the corresponding analytic expressions were stated by Gilbert and Mosteller (\cite{4}) and the proof was outlined by Dynkin and Yushkevich (\cite{14}). The case $k=3$ was derived by Quine and Law (\cite{2}).
In the present paper, we extended to an arbitrary value of $k$ the formula for an optimal sequence in the following way.
\begin{theorem}\label{mt}
Let $(s_1, \ldots, s_k)$ be a sequence such that $s_l=t_l+l-1$ for every $1\leq l\leq k$, where each $t_l$ is defined as the smallest integer $1\leq x\leq n-l$ satisfying the inequality
\begin{equation}\label{ineqqq}
d_l(x)\leq \frac{1}{l}\sum_{i=l}^{k-1}\left(\prod_{j=l+1}^i t_j\right)d_i(t_{i+1})+\frac{1}{l}\left(\prod_{j=l+1}^k t_j\right)\delta_{k,n},
\end{equation}
where
\begin{equation}\label{ajj}
\delta_{k,n}:=\left\{
\begin{array}{ll}
-\frac{1}{n}\cdot\sum_{j=1}^{n-1}\frac{1}{j},&if\;\;k=1,\\
\\
\frac{(n-k)!}{n!},&if\;\;k>1,
\end{array}
\right.
\end{equation}
and the map $d_l\colon \{1,\ldots, n-l\}\to\mathbb{R}$ is defined for each $0\leq l\leq k$ as follows:
\begin{equation}\label{aj}
d_l(x):=\left\{
\begin{array}{ll}
-1+\frac{k\cdot x}{n}+\frac{{n-x-1\choose k}}{{n\choose k}}+\frac{x}{{n\choose k}}\sum\limits_{j=1}^{x}\frac{{n-j-1\choose k-1}}{j},& if\;\;l=0,\\
\\
-\frac{k}{n}-\frac{1}{{n\choose k}}\sum\limits_{j=1}^{x}\frac{{n-j-1\choose k-1}}{j},&if\;\;l=1,\\
\\
\frac{k\cdot x!\cdot (n-l-x)!}{l\cdot (l-1)\cdot n!}\sum\limits_{j=0}^{l-2}{k-1\choose j}{n-k\choose x+l-j-1},&if\;\;1<l\leq k.
\end{array}
\right.
\end{equation}
Then $(s_1, \ldots, s_k)$ is an optimal sequence.
\end{theorem}
In the right side of (\ref{ineqqq}) we use the standard conventions for the empty sum and the empty product and evaluate them to 0 and to 1, respectively. In particular, for $l=k$ the right side of (\ref{ineqqq}) just equals $\delta_{k,n}/k$. Hence, the above formula allows to calculate the elements $s_l$ ($1\leq l\leq k$) one by one, from $l=k$ down to $l=1$.
In the present paper we also proved the following
\begin{theorem}\label{mtt}
The probability of success for the policy described in Proposition~\ref{p1} is equal to
\begin{equation}\label{expr}
-\sum\limits_{i=0}^{k-1}\left(\prod_{j=1}^i t_j\right)d_i(t_{i+1})-\left(\prod_{j=1}^k t_j\right)\delta_{k,n},
\end{equation}
where $t_l:=s_l-l+1$ for $1\leq l\leq k$.
\end{theorem}
The known constructions of the optimal sequence via analytic expressions were presented in the similar form as in Theorem~\ref{mt} but, as we have mentioned above, only in the cases $k=1,2,3$. For higher values of $k$, as well as for some other versions of this problem, the algorithms computing the probability of success and the elements of an optimal sequence can be found in various of papers (see~\cite{15,8,7,10,11,16,3,1,9,17}). However, in contrast to the analytic solution, these methods apply mechanisms via dynamic or linear programming, and hence only numerically allow to determine the elements of an optimal sequence.
\section{The strategy of the proofs}
Our proofs are purely elementary and only combinatorial arguments are used. At first, since no two girls are equally attractive for the bachelor, we assign the rank to each girl, which is an integer from 1 to $n$, i.e. the rank 1 to the best girl, the rank 2 to the next best girl and so on. Then each of the possible $n!$ orderings of the girls defines uniquely a permutation $\pi$ of the set $\{1,\ldots, n\}$ such that $\pi(i)$ is the rank of the $i$-th girl for every $i\in\{1,\ldots, n\}$, and conversely, any permutation $\pi$ of the set $\{1,\ldots, n\}$ defines in the obvious way the possible ordering of the girls.
Let $w=(s_1, \ldots, s_k)$ be a non-decreasing sequence such that $l\leq s_l<n$ for every $l\in\{1,\ldots, k\}$ and let us assume that the policy from Proposition~\ref{p1} is used with the sequence $w$. Let $\pi$ be a permutation defining the ordering of the girls. If the policy successfully chooses a candidate of top $k$, then we say that $\pi$ is a {\it lucky permutation corresponding to $w$}. We distinguish the case when there exists $l\in\{1,\ldots, k\}$ such that for some $i\in\{1,\ldots, n-1\}$ the following two conditions hold:
\begin{itemize}
\item $s_l<i\leq s_{l+1}$,
\item among the first $i$ girls, there are at most $l-1$ girls which are more attractive than the $i$-th girl (equivalently, the set $\{\pi(1), \ldots, \pi(i)\}$ contains at most $l$ elements which are not greater than $\pi(i)$).
\end{itemize}
We call such a number $l$ a {\it $w$-threshold} of the permutation $\pi$, and the corresponding number $i$ we call a {\it $(\pi, l, w)$-element} (see also Definition~\ref{deee1} in Section~\ref{s3}). Let now assume that the permutation $\pi$ has a $w$-threshold. If $l_0\in\{1,\ldots, k\}$ is the smallest $w$-threshold of $\pi$, then the bachelor using the policy will marry to the $i_0$-th girl, where $i_0$ is the smallest $(\pi,l_0,w)$-element. Otherwise (i.e. when $\pi$ has no $w$-thresholds), the bachelor will marry to the $n$-th girl. In particular, the set $\Pi_{w}$ of all lucky permutations corresponding to the sequence $w$ naturally splits into two subsets: the subset $\Pi_{w,1}$ of permutations having a $w$-threshold and the subset $\Pi_{w,2}$ of permutations without $w$-thresholds. Obviously, the probability of success when using the policy is equal to the ratio
$$
\frac{|\Pi_{w}|}{n!}=\frac{|\Pi_{w,1}|}{n!}+\frac{|\Pi_{w,2}|}{n!}.
$$
For every $l\in\{1,\ldots, k\}$ we define the following sets:
$$
X_l:=\{l,\ldots,n-1\},\;\;\;X^{(l)}:=X_{k-l+1}\times\ldots\times X_k,
$$
and the set $X^{(0)}:=\{\epsilon\}$, where $\epsilon$ is the empty sequence. Further, we refer to the elements of the sets $X^{(l)}$ as {\it words} and to the elements of the sets $X_l$ as {\it letters}.
In Theorem~\ref{tt1} (Section~\ref{s3}), we provide for every non-decreasing sequence $w=(x_1, \ldots, x_k)\in X^{(k)}$ the analytic formulae for the cardinalities of the sets $\Pi_{w,1}$ and $\Pi_{w,2}$. To derive the formula for $|\Pi_{w,1}|$, we consider for each $1\leq l\leq k$ and $x_l<i\leq x_{l+1}$ the subset $S(l,i)\subseteq \Pi_{w,1}$ of all permutations $\pi$ such that the number $l$ is the smallest $w$-threshold of $\pi$ and the number $i$ is the smallest $(\pi,l,w)$-element. In particular, we can write
$$
|\Pi_{w,1}|=\sum\limits_{l=1}^k\sum_{i=x_{l}+1}^{x_{l+1}}|S(l, i)|.
$$
Further, for every $\pi\in S(l,i)$, we divide the set $\{\pi(1), \ldots, \pi(i)\}$ into two subsets: the subset of those elements which are not greater than $k$ and the subset of those elements which are greater than $k$. Conversely, given arbitrarily the sets $Y,Y'$ satisfying
$$
Y\subseteq \{1,\ldots, k\},\;\;\;\;Y'\subseteq \{k+1, \ldots, n\},\;\;\;|Y|+|Y'|=i,
$$
we consider the subset $S(Y,Y', l,i)\subseteq S(l,i)$ of those permutations $\pi$ for which
$$
\{1,\ldots, k\}\cap \{\pi(1), \ldots, \pi(i)\}=Y,\;\;\;\{k+1, \ldots, n\}\cap \{\pi(1), \ldots, \pi(i)\}=Y'.
$$
Then, we have:
$$
|S(l, i)|=\sum\limits_{j=1}^k\sum_{(Y, Y')\in M_j}|S(Y, Y', l, i)|,
$$
where $M_j$ ($j\in\{1,\ldots, k\}$) is the set of those pairs $(Y,Y')$ for which $|Y|=j$. In Proposition~\ref{pomol1}, we characterize the elements of the set $S(Y,Y', l,i)$, which allows to find the following formula for its cardinality:
$$
|S(Y,Y', l,i)|=\min\{|Y|, l\}\cdot (i-l-1)!\cdot (n-i)!\cdot \prod_{j=1}^{l} (x_j-j+1).
$$
We use the above formula to find the cardinality of the set $S(l,i)$ and, consequently, the following formula for $|\Pi_{w,1}|$:
$$
|\Pi_{w,1}|=
n!\cdot\sum\limits_{l=1}^k\left((r_{l-1}(x_l)-r_{l-1}(x_{l+1}))\cdot \prod\limits_{j=1}^l(x_j-j+1)\right),
$$
where the map $r_l\colon \{l+1,\ldots,n\}\to\mathbb{R}$ is defined for every integer $l\leq k$ as follows:
\begin{equation}\label{rere}
r_l(x):=\left\{
\begin{array}{ll}
0,&if\;\;l<0,\\
\\
\frac{1}{x}-\frac{k}{n}-\frac{{n-x-1\choose k}}{x {n\choose k}}-\frac{1}{{n\choose k}}\sum\limits_{j=1}^x\frac{{n-j-1\choose k-1}}{j},&if\;\;l=0,\\
\\
\frac{(x-l-1)!}{x!}\left(1-\frac{1}{l{n\choose x}}\sum\limits_{j=0}^l(l-j){k\choose j}{n-k\choose x-j}\right),&if\;\;1\leq l\leq k.
\end{array}
\right.
\end{equation}
By using a similar idea as in Proposition~\ref{pomol1}, we also characterize the elements of the set $\Pi_{w,2}$ (see Proposition~\ref{pomol3}), which gives the following formula for $|\Pi_{w,2}|$:
$$
|\Pi_{w,2}|=k(n-k-1)!\cdot\prod\limits_{j=1}^k(x_j-j+1).
$$
In Section~\ref{s4}, we derive the formula for the elements of an optimal sequence. To this aim, we introduce the notion of an {\it optimal point} (see Definition~\ref{jjdef}) of an arbitrary map $f\colon Z\to \mathbb{R}$, where $Z\subseteq X^{(l)}$ or $Z\subseteq X_l$ for some $l\in\{1,\ldots, k\}$. Next, we define for every $l\in\{0,1\ldots, k\}$ a map $T_l\colon X^{(k-l)}\to\mathbb{R}$ (see formula (\ref{uogt})), which constitutes a natural generalization of the map
$$
T\colon X^{(k)}\to\mathbb{R},\;\;\;T(w)=\frac{|\Pi_{w,1}|}{n!}+\frac{|\Pi_{w,2}|}{n!}.
$$
We study the maps $T_l$ in relation to the maps $c_l$, $D_l$ ($l\in\{0,\ldots, k\}$) and $F_{l, w}$ ($l\in\{1,\ldots, k\}$, $w\in X^{(k-l)}$) defined as follows:
\begin{eqnarray}
c_l(x)&:=&d_l(x-l),\;\;\;\;x\in \{l+1,\ldots,n\},\label{cccd}\\
D_l(w)&:=&r_{l-1}(x_1)-T_l(w),\;\;\;\;w\in X^{(k-l)},\label{dddd}\\
F_{l,w}(x)&:=&-c_{l-1}(x)-x\cdot D_{l}(w),\;\;\;\;x\in \{l,\ldots,n-1\}\label{fffd},
\end{eqnarray}
where $x_1$ in (\ref{dddd}) denotes the first letter of a word $w\in X^{(k-l)}$ or $x_1:=n-1$ depending on whether $l<k$ or $l=k$. In particular, we obtain
$$
D_0=-T_0=-T.
$$
In Proposition~\ref{proo8}, we show how to describe the maps $D_l$ in terms of the maps $d_l$. As a result, we obtain for every $l\in\{1,\ldots, k\}$ that the right side of (\ref{ineqqq}) is equal to
$$
\frac{D_l\left(w^{(l)}\right)}{l},
$$
where $w^{(l)}\in X^{(k-l)}$ arises from the sequence $w:=(t_1, t_2+1, \ldots, t_k+k-1)$ by deleting the first $l$ letters.
In Proposition~\ref{prop10}, for any $l\in\{1,\ldots, k\}$ and $w\in X^{(k-l)}$, we show that if $t$ is the smallest element $x\in\{1,\ldots, n-l\}$ such that $d_l(x)\leq D_l(w)/l$, then the number $t+l-1$ is an optimal point of the map $F_{l,w}$. Next, we show (Proposition~\ref{p8}) that an arbitrary sequence $w\in X^{(k)}$ is an optimal point of the map $T$ if and only if for every $l\in\{1,\ldots, k\}$ the $l$-th letter of $w$ is an optimal point of the map $F_{l, w^{(l)}}$. Finally, in Proposition~\ref{trud}, we show that if $w\in X^{(k)}$ is an optimal point of $T$, then $w$ must be a non-decreasing sequence. As a simple consequence of Theorem~\ref{tt1} and the above propositions, we obtain our main results (see Section~\ref{secpr}). The proofs of Propositions~\ref{proo8}-\ref{trud} are based on various combinatorial identities and on some auxiliary properties of the maps $r_l, c_l, D_l$ and $F_{l,w}$. We derive them in Section~\ref{sqa1}.
Further, we use for all $i,j\in\mathbb{Z}$ the following notations:
\begin{itemize}
\item $[i]:=\{t\in\mathbb{Z}\colon 1\leq t\leq i\}$,
\item $[i]_0:=[i]\cup\{0\}$,
\item $[i,j]:=\{t\in\mathbb{Z}\colon i<t\leq j\}$,
\item $[\leq i]:=\{t\in\mathbb{Z}\colon t\leq i\}$.
\end{itemize}
\section{The formula for the probability of success}\label{s3}
Let $Sym(n)$ be the set of all permutations of the set $[n]$. For every $\pi\in Sym (n)$ and every $i\in[n]$ we call the image $\pi(i)$ the {\it $\pi$-rank} of the element $i$. We call the element $i$ a {\it $\pi$-candidate} if $\pi(i)\in[k]$. In particular, if the ordering of the girls is defined by a permutation $\pi\in Sym(n)$, then the bachelor's win is to marry to the $i_0$-th girl, where $i_0\in[n]$ is an arbitrary $\pi$-candidate.
For every $\pi\in Sym(n)$ and every $i\in[n]$ we also consider the {\it relative $\pi$-rank} of the element $i$, i.e. the number of the elements from the set $[i]$ such that their $\pi$-ranks are not greater than $\pi(i)$; we denote this number by $\rho_\pi(i)$. In other words, $\rho_{\pi}(i)$ is the number of those elements from the set $\pi([i])$ which are not greater than $\pi(i)$.
\begin{definition}\label{deee1}
Let $\pi\in Sym(n)$, $l\in[k]$ and let $w=(s_1, \ldots, s_{k})\in X^{(k)}$ be a non-decreasing sequence. We call an arbitrary element $i\in[s_{l}, s_{l+1}]$ satisfying the inequality $\rho_\pi(i)\leq l$ a $(\pi,l,w)$-{\it element}. If the set $[s_{l}, s_{l+1}]$ contains at least one $(\pi,l,w)$-element, then we call the number $l$ a {\it $w$-threshold} of the permutation $\pi$. In other words, the element $l\in[k]$ is a $w$-threshold of $\pi$ if there is $i\in[s_l, s_{l+1}]$ such that $\rho_\pi(i)\leq l$ (as before, we assume $s_{k+1}:=n-1$).
\end{definition}
Let now assume that the bachelor uses the policy from Proposition~\ref{p1} with a sequence $w=(s_1, \ldots, s_k)$ and that the ordering of the girls is defined by a permutation $\pi\in Sym(n)$. Then $\pi$ is a lucky permutation corresponding to $w$ if and only if one of the following conditions holds:
\begin{itemize}
\item $\pi$ has a $w$-threshold and if $l_0\in[k]$ is the smallest $w$-threshold of $\pi$, then the smallest $(\pi,l_0,w)$-element is a $\pi$-candidate,
\item $\pi$ has no $w$-thresholds and the element $n$ is a $\pi$-candidate.
\end{itemize}
\begin{theorem}\label{tt1}
Let $w_0=(x_1, \ldots, x_k)\in X^{(k)}$ be a non-decreasing sequence. Then the number of lucky permutations corresponding to $w_0$ and having a $w_0$-threshold is equal to
$$
n!\cdot\sum\limits_{l=1}^k\left((r_{l-1}(x_l)-r_{l-1}(x_{l+1}))\cdot \prod\limits_{j=1}^l(x_j-j+1)\right),
$$
where the maps $r_l\colon [l,n]\to\mathbb{R}$ ($l\in[\leq k]$) are defined as in (\ref{rere}). The number of lucky permutations corresponding to $w_0$ and having no $w_0$-thresholds is equal to
$$
k(n-k-1)!\cdot\prod\limits_{j=1}^k(x_j-j+1).
$$
\end{theorem}
\begin{proof}
Let us fix the integers $l_0,i_0$ such that $l_0\in[k]$ and $i_0\in[x_{l_0},x_{l_0+1}]$. At first, we determine the number of lucky permutations $\pi\in \Pi_{w_0}$ such that $l_0$ is the smallest $w_0$-threshold of $\pi$ and $i_0$ is the smallest $(\pi,l_0,w_0)$-element. Let us denote by $S(l_0, i_0)$ the set of all such permutations.
For every permutation $\pi\in Sym(n)$, we denote
$$
\mathcal{X}_\pi:=\pi([i_0])\cap [k],\;\;\;\mathcal{X}'_\pi:=\pi([i_0])\cap [k,n].
$$
Obviously, we have
$$
\pi([i_0])=\mathcal{X}_\pi\cup \mathcal{X}'_\pi,\;\;\;\mathcal{X}_\pi\subseteq [k],\;\;\;\mathcal{X}'_\pi\subseteq [k,n],\;\;\;|\mathcal{X}_\pi|+|\mathcal{X}'_\pi|=i_0.
$$
Moreover, if $\pi\in S(l_0, i_0)$, then $\pi$ is a lucky permutation, and hence $i_0$ is a $\pi$-candidate, which implies $\pi(i_0)\in[k]$ and consequently $\pi(i_0)\in \mathcal{X}_\pi$. Further, since $i_0$ is a $(\pi, l_0, w_0)$-element, we obtain $\rho_\pi(i_0)\leq l_0$, which means that the set $\pi([i_0])$ contains at most $l_0$ elements which are not greater than $\pi(i_0)$. Since $\pi(i_0)\in \mathcal{X}_\pi\subseteq [k]$ and $\mathcal{X}'_\pi\subseteq [k,n]$, all these elements must belong to the set $\mathcal{X}_\pi$. In particular, if we denote $\mathcal{X}_\pi:=\{y_1, \ldots, y_{j_0}\}$ for some $j_0\in [k]$, where $y_1<y_2<\ldots<y_{j_0}$, then we obtain: $\pi(i_0)=y_\iota$ for some $1\leq \iota\leq \min\{j_0, l_0\}$. Note that $\iota$ is the relative $\pi$-rank of the element $i_0$.
Let $Y, Y'\subseteq [n]$ be arbitrary subsets which satisfy the following conditions
\begin{equation}\label{1}
Y\subseteq[k],\;\;\;Y'\subseteq [k,n],\;\;\;|Y|+|Y'|=i_0.
\end{equation}
Let us denote
\begin{eqnarray*}
S(Y, Y', l_0, i_0)&:=&\{\pi\in S(l_0, i_0)\colon \mathcal{X}_\pi=Y,\;\mathcal{X}'_\pi=Y'\},\\
\mu_{j, l_0}&:=&\min\{j, l_0\},\;\;\;j\in[k].
\end{eqnarray*}
\begin{proposition}\label{pomol1}
For every permutation $\pi\in S(Y, Y', l_0, i_0)$ the following three conditions hold:
\begin{itemize}
\item [(i)] $\pi([i_0])=Y\cup Y'$,
\item [(ii)] if $Y=\{y_1,y_2,\ldots,y_{j_0}\}$ for some $j_0\in[k]$ and $y_1<y_2<\ldots< y_{j_0}$, then there is $\iota\in[\mu_{j_0, l_0}]$ such that $\pi(i_0)=y_{\iota}$,
\item [(iii)] if $\pi([i_0-1])=\{y_1',y_2',\ldots,y'_{i_0-1}\}$ and $y_1'<y_2'<\ldots<y'_{i_0-1}$, then $y'_j\in \pi([x_j])$ for every $j\in [l_0]$.
\end{itemize}
Conversely, if the sets $Y$, $Y'$ satisfy (\ref{1}) and a permutation $\pi\in Sym(n)$ satisfies (i)-(iii), then $\pi\in S(Y, Y', l_0, i_0)$.
\end{proposition}
\begin{proof}[of Proposition~\ref{pomol1}]
Let $\pi\in S(Y, Y', l_0, i_0)$ be arbitrary. By the above reasoning, the conditions (i)-(ii) directly follow from the equalities $Y=\mathcal{X}_\pi$ and $Y'=\mathcal{X}'_\pi$. To justify (iii) let us assume contrary that there is $j_1\in[l_0]$ such that $i_1:=\pi^{-1}(y'_{j_1}){\mathbb{N}}tin [x_{j_1}]$. Since $\pi(i_1)=y'_{j_1}\in \pi([i_0-1])$, we have $i_1<i_0$ and $\pi([i_1])\subseteq \pi([i_0-1])=\{y_1',y_2',\ldots,y'_{i_0-1}\}$. Thus the set $\pi([i_1])$ contains at most $j_1$ elements which are not greater than $y'_{j_1}=\pi(i_1)$. Hence the relative $\pi$-rank of the element $i_1$ is not greater than $j_1$, i.e. $\rho_{\pi}(i_1)\leq j_1$. Since the sequence $(x_{j_1}, \ldots, x_{l_0+1})$ is non--decreasing, we have
$$
[x_{l_0+1}]=[x_{j_1}]\cup\bigcup_{j_1\leq j\leq l_0}[x_j, x_{j+1}].
$$
Since $i_1\in [i_0]\setminus [x_{j_1}]\subseteq [x_{l_0+1}]\setminus [x_{j_1}]$, there is $j_1\leq j_2\leq l_0$ such that $i_1\in [x_{j_2}, x_{j_2+1}]$. But $\rho_{\pi}(i_1)\leq j_1\leq j_2$, and hence $i_1$ is a $(\pi, j_2, w_0)$-element. Consequently $j_2$ is a $w_0$-threshold of $\pi$. Since $j_2\leq l_0$ and $l_0$ is the smallest $w_0$-threshold of $\pi$, we obtain: $j_2=l_0$. Consequently, the element $i_1$ is a $(\pi, l_0, w_0)$-element. Since $i_1<i_0$, we obtain the contradiction with the assumption that $i_0$ is the smallest $(\pi, l_0, w_0)$-element. This justifies the first part of Proposition~\ref{pomol1}.
Conversely, let $\pi\in Sym(n)$ be an arbitrary permutation which satisfies (i)-(iii). We show that $\pi\in S(Y, Y', l_0, i_0)$. By (ii), we have $\pi(i_0)\in Y\subseteq [k]$, and hence $i_0$ is a $\pi$-candidate. The equalities $Y=\mathcal{X}_\pi$ and $Y'=\mathcal{X}'_{\pi}$ directly follows from the definition of the sets $\mathcal{X}_\pi$, $\mathcal{X}'_\pi$ as well as from the conditions~(\ref{1}) and from (i). Next, we have $\pi(i_0)=y_\iota$ for some $\iota\in [\mu_{j_0, l_0}]$. Since $\pi([i_0])=Y\cup Y'$ and every element in $Y'$ is greater than every element in $Y$, we see by (ii) that $\{y_1, \ldots, y_\iota\}$ is the set of all elements from $\pi([i_0])$ which are not greater than $y_\iota=\pi(i_0)$. Thus the relative $\pi$-rank of $i_0$ is equal to $\iota$. But $\iota\leq l_0$ and hence $\rho_{\pi}(i_0)\leq l_0$. Since $i_0\in [x_{l_0}, x_{l_0+1}]$, we see that $i_0$ is a $(\pi, l_0, w_0)$-element. Thus $l_0$ is a $w_0$-threshold of $\pi$. To show that $l_0$ is the smallest $w_0$-threshold of $\pi$, suppose contrary that there is $l_1<l_0$ such that $l_1$ is a $w_0$-threshold of $\pi$. Then there is $i_1\in [x_{l_1}, x_{l_1+1}]$ such that
\begin{equation}\label{mmmmm}
\rho_{\pi}(i_1)\leq l_1.
\end{equation}
By (iii) we have $y'_j\in \pi([x_j])$ for every $j\in[l_1]$. But for every $j\in[l_1]$ we have $[x_j]\subseteq [x_{l_1}]\subseteq [i_1]$. Thus we have
\begin{equation}\label{mmm}
\{y_1', \ldots, y_{l_1}', \pi(i_1)\}\subseteq \pi([i_1]).
\end{equation}
Since $i_0\in[x_{l_0}, x_{l_0+1}]$, $i_1\in [x_{l_1}, x_{l_1+1}]$ and $l_1<l_0$, we have $i_1<i_0$ and hence, by (iii), we have
$\pi([i_1])\subseteq \{y_1', \ldots, y_{i_0-1}'\}$. Thus there is $j_1\in[i_0-1]$ such that
\begin{equation}\label{mmmm}
\pi(i_1)=y'_{j_1}.
\end{equation}
Assuming $j_1\in[l_1]$, we would have by (iii): $\pi(i_1)=y'_{j_1}\in \pi([x_{j_1}])$, and hence $i_1\in [x_{j_1}]\subseteq [x_{l_1}]$. But, this is impossible as $i_1\in [x_{l_1}, x_{l_1+1}]$. Thus it must be $j_1>l_1$, and consequently, we see by (\ref{mmm})--(\ref{mmmm}) that the set $\pi([i_1])$ contains at least $l_1+1$ elements which are not grater than $\pi(i_1)$. Thus $\rho_{\pi}(i_1)\geq l_1+1$ and we have a contradiction with~ (\ref{mmmmm}). Hence $l_0$ is indeed the smallest $w_0$-threshold of $\pi$. To show that $\pi\in S(Y, Y', l_0, i_0)$, we now need to show that $i_0$ is the smallest $(\pi, l_0, w_0)$-element. Suppose contrary that there is $i_1<i_0$ such that $i_1$ is a ($\pi, l_0, w_0)$-element. We have $i_1\in [x_{l_0}, x_{l_0+1}]$ and $\rho_{\pi}(i_1)\leq l_0$. Similarly as above, we obtain by (iii) the inclusion $\{y'_1, \ldots, y'_{l_0}, \pi(i_1)\}\subseteq \pi([i_1])$. Since $i_1<i_0$, there is $j_1\in[i_0-1]$ such that $\pi(i_1)=y'_{j_1}$. Similarly as above, we show that $j_1>l_0$. This implies $\rho_{\pi}(i_1)\geq l_0+1$ and we have a contradiction. Consequently $i_0$ is indeed the smallest $(\pi, l_0, w_0)$-element, and hence $\pi\in S(Y, Y', l_0, i_0)$. \qed
\end{proof}
\begin{proposition}\label{lolip}
The number of elements in the set $S(Y, Y', l_0, i_0)$ is equal to
$$
\mu_{j_0, l_0}\cdot (i_0-l_0-1)!\cdot (n-i_0)!\cdot \prod_{j=1}^{l_0} (x_j-j+1),
$$
where $j_0:=|Y|$.
\end{proposition}
\begin{proof}[of Proposition~\ref{lolip}]
We use the characterization of the set $S(Y, Y', l_0, i_0)$ from Proposition~\ref{pomol1}. By the conditions (i)--(iii), we see that every permutation $\pi\in S(Y, Y', l_0, i_0)$ can be constructed as follows. At first, we choose arbitrarily an element $\iota\in[\mu_{j_0, l_0}]$ and define: $\pi(i_0):=y_\iota$. We can do that in $\mu_{j_0, l_0}$ ways. Next, for every $j\in [l_0]$ we choose an element $i_j\in [x_j]$ and define $\pi(i_j):=y'_j$. We can do that in
$\prod_{j=1}^{l_0} (x_j-j+1)$ ways. Further, we define the $\pi$-ranks from the set
$$
\widetilde{Y}:=(Y\cup Y')\setminus\{y_\iota, y'_1, \ldots, y'_{l_0}\}=\{y'_{l_0+1}, \ldots, y'_{i_0-1}\}.
$$
Namely, for every $y\in \widetilde{Y}$ we choose an element $i_y\in [i_0-1]\setminus \{i_1, \ldots, i_{l_0}\}$ and define $\pi(i_y):=y$. We can do that in $(i_0-l_0-1)!$ ways. Finally, we define the $\pi$-ranks from the set $[n]\setminus (Y\cup Y')$, i.e. for every $i\in [n]\setminus [i_0]$ we choose an element $y_i\in [n]\setminus (Y\cup Y')$ and define $\pi(i):=y_i$. We can do that in $(n-i_0)!$ ways. Hence, the claim directly follows from the above construction.\qed
\end{proof}
\begin{proposition}\label{lolip2}
The number of elements in the set $S(l_0, i_0)$ is equal to
$$
n!\cdot l_0\cdot r_{l_0}(i_0)\cdot \prod_{j=1}^{l_0} (x_j-j+1).
$$
\end{proposition}
\begin{proof}[of Proposition~\ref{lolip2}]
We have:
$$
|S(l_0, i_0)|=\sum_{j\in[k]}\sum_{(Y, Y')\in M_j}|S(Y, Y', l_0, i_0)|,
$$
where for every $j\in[k]$ we define
$$
M_j:=\{(Y, Y')\colon Y\subseteq[k],\;\;Y'\subseteq [n]\setminus[k],\;\;|Y|=j,\;\;|Y'|=i_0-j\}.
$$
Since $|M_j|={k\choose j}{n-k\choose i_0-j}$, we obtain from Proposition~\ref{lolip}:
\begin{equation}\label{eq3}
|S(l_0, i_0)|=(i_0-l_0-1)!\cdot (n-i_0)!\cdot \lambda(i_0, l_0)\cdot \prod_{j=1}^{l_0} (x_j-j+1),
\end{equation}
where
\begin{eqnarray*}
\lambda(i_0, l_0)&=&\sum_{j=1}^k{k\choose j}{n-k\choose i_0-j}\mu_{j, l_0}=\\
&=&\sum_{j=1}^{l_0}{k\choose j}{n-k\choose i_0-j} j+\sum_{j=l_0+1}^k{k\choose j}{n-k\choose i_0-j}l_0=\\
&=&\sum_{j=0}^{l_0}{k\choose j}{n-k\choose i_0-j} j+\sum_{j=0}^k{k\choose j}{n-k\choose i_0-j}l_0-\sum_{j=0}^{l_0}{k\choose j}{n-k\choose i_0-j}l_0=\\
&=&\sum_{j=0}^k{k\choose j}{n-k\choose i_0-j}l_0-\sum_{j=0}^{l_0}(l_0-j){k\choose j}{n-k\choose i_0-j}.
\end{eqnarray*}
From the Vandermonde's identity, we obtain
$$
\lambda(i_0, l_0)=l_0{n\choose i_0}-\sum_{j=0}^{l_0}(l_0-j){k\choose j}{n-k\choose i_0-j}=\frac{l_0\cdot n!}{(n-i_0)!\cdot (i_0-l_0-1)!}\cdot r_{l_0}(i_0).
$$
The claim now follows from~(\ref{eq3}).\qed
\end{proof}
Obviously, the number of all lucky permutations from $\Pi_{w_0}$ which have a $w_0$-threshold is equal to
$\sum_{l=1}^k\sum_{i=x_{l}+1}^{x_{l+1}}|S(l, i)|$. We see by Proposition~\ref{lolip2} that this double sum can be written as follows
$$
n!\cdot\sum_{l=1}^k\left(\left(\sum\limits_{i=x_{l}+1}^{x_{l+1}}lr_l(i)\right)\cdot \prod_{j=1}^{l} (x_j-j+1)\right).
$$
The crucial point for the further study, which also finishes the proof of the first part of Theorem~\ref{tt1}, is the observation that the sum $\sum_{i=x_{l}+1}^{x_{l+1}}lr_l(i)$ in the above expression can be written in a closed form as follows:
$$
\sum_{i=x_{l}+1}^{x_{l+1}}lr_l(i)=r_{l-1}(x_l)-r_{l-1}(x_{l+1}).
$$
The last equality follows from the identity $lr_l(x)=r_{l-1}(x)-r_{l-1}(x-1)$ for all $l\in[\leq k]$ and $x\in [l,n]$, which we derive in
Section~\ref{sqa1} (see Lemma~\ref{propppp}~(iii) therein).
To show the second part of Theorem~\ref{tt1}, we provide the following characterization of all lucky permutations from $\Pi_{w_0}$ which have no $w_0$-thresholds.
\begin{proposition}\label{pomol3}
Let $\pi\in \Pi_{w_0}$ be an arbitrary lucky permutation without $w_0$-thresholds. Then the following two conditions hold:
\begin{itemize}
\item[(i)] there is $\iota\in[k]$ such that $\pi(n)=\iota$,
\item[(ii)] if $[k+1]\setminus\{\iota\}=\{y_1, \ldots, y_k\}$ and $y_1<y_2<\ldots<y_{k}$, then $y_j\in \pi([x_j])$ for every $j\in[k]$.
\end{itemize}
Conversely, if $\pi\in Sym(n)$ is an arbitrary permutation which satisfies (i)--(ii), then $\pi$ is a lucky permutation corresponding to $w_0$ and $\pi$ has no $w_0$-thresholds.
\end{proposition}
\begin{proof}[of Proposition~\ref{pomol3}]
The condition (i) directly follows from the definition of a lucky permutation. To show (ii), we proceed in the similar way as in the proof of Proposition~\ref{pomol1}. Namely, suppose contrary that there is $j_0\in[k]$ such that $i_0:=\pi^{-1}(y_{j_0}){\mathbb{N}}tin [x_{j_0}]$. By~(i), we have $i_0\neq n$. Thus $i_0\in [n-1]$, and since
$$
[n-1]:=[x_{j_0}]\cup \bigcup_{j_0\leq j\leq k}[x_j, x_{j+1}],
$$
we obtain that there is $j_0\leq j_1\leq k$ such that $i_0\in [x_{j_1}, x_{j_1+1}]$. Since $y_{j_0}\in [k+1]\setminus\{\iota\}$ and $\iota{\mathbb{N}}tin \pi([i_0])$, every element in $\pi([i_0])$ which is not grater than $y_{j_0}$ belongs to $[k+1]\setminus\{\iota\}$. Since $\pi(i_0)=y_{j_0}$ and
$[k+1]\setminus \{\iota\}=\{y_1, \ldots, y_k\}$, we see that the set of all elements from $\pi([i_0])$ which are not grater than $\pi(i_0)$ is contained in the set $\{y_1, \ldots, y_{j_0}\}$. Thus $\rho_{\pi}(i_0)\leq j_0\leq j_1$. Consequently the element $i_0$ is a $(\pi, j_1, w_0)$-element. Thus the set $[x_{j_1}, x_{j_1+1}]$ contains a $(\pi, j_1, w_0)$-element, which means that $j_1$ is a $w_0$-threshold of $\pi$, contrary to our assumption.
Conversely, let $\pi\in Sym(n)$ be an arbitrary permutation which satisfies (i)--(ii). We show that $\pi$ is a lucky permutation corresponding to $w_0$ and $\pi$ has no $w_0$-thresholds. By (i), it is enough to show that $\pi$ has no $w_0$-thresholds. We proceed in the similar way as in the proof of Proposition~\ref{pomol1}. Namely, suppose contrary that $\pi$ has a $w_0$-threshold. Then there are $j_0\in [k]$ and $i_0\in [x_{j_0}, x_{j_0+1}]$ such that $\rho_{\pi}(i_0)\leq j_0$. By (ii), we have $y_j\in \pi([x_j])$ for every $j\in [j_0]$. But $[x_j]\subseteq [x_{j_0}]\subseteq [i_0]$ for every $j\in [j_0]$. Hence
$\{y_1, \ldots, y_{j_0}, \pi(i_0)\}\subseteq \pi([i_0])$. If $\pi(i_0){\mathbb{N}}tin [k+1]$, then $\pi(i_0)>y_{j_0}$ and consequently $\rho_{\pi}(i_0)\geq j_0+1$. Thus, it must be $\pi(i_0)\in [k+1]\setminus\{\iota\}$ (note that $i_0\neq n$ and hence $\pi(i_0)\neq \iota$). By (ii), there is $j_1\in [k]$ such that $\pi(i_0)=y_{j_1}$. Assuming $j_1\in[j_0]$, we would have $\pi(i_0)=y_{j_1}\in \pi([x_{j_1}])$. Hence $i_0\in [x_{j_1}]\subseteq [x_{j_0}]$, which is impossible as $i_0\in [x_{j_0}, x_{j_0+1}]$. Thus it must be $j_1>j_0$ and consequently $\rho_{\pi}(i_0)\geq j_0+1$ contrary to our assumption. This finishes the proof of Proposition~\ref{pomol3}. \qed
\end{proof}
By using the conditions (i)-(ii) from Proposition~\ref{pomol3}, we see that every lucky permutation $\pi\in \Pi_{w_0}$ without $w_0$-thresholds can be constructed as follows. At first, we choose arbitrarily an element $\iota\in[k]$ and we define: $\pi(n):=\iota$. We can do that in $k$ ways. Next, we choose for every $j\in [k]$ an element $i_j\in [x_j]$ and define $\pi(i_j):= y_j$. This can be done in $\prod_{j=1}^{k} (x_j-j+1)$ ways. Finally, we define the $\pi$-ranks from the set $[n]\setminus\{\iota, y_1, \ldots, y_{k}\}$, which can be done in $(n-k-1)!$ ways. As a result of this construction, we obtain the required formula. This completes the proof of Theorem~\ref{tt1}.
\qed
\end{proof}
\section{The formula for an optimal sequence}\label{s4}
Let $T\colon X^{(k)}\to\mathbb{R}$ be the map defined for every $w=(x_1, \ldots, x_k)\in X^{(k)}$ as follows: $T(w):=P_1(w)+P_2(w)$, where
\begin{eqnarray*}
P_1(w)&:=&\frac{|\Pi_{w,1}|}{n!}=\sum\limits_{l=1}^k\left((r_{l-1}(x_l)-r_{l-1}(x_{l+1}))\cdot \prod\limits_{j=1}^l(x_j-j+1)\right),\\
P_2(w)&:=&\frac{|\Pi_{w,2}|}{n!}=\xi_{k,n}\cdot\prod\limits_{j=1}^k(x_j-j+1),
\end{eqnarray*}
and $\xi_{k,n}:=k(n-k-1)!/n!$. We see by Theorem~\ref{tt1} that the probability of success for the policy described in Proposition~\ref{p1} is equal to $T(w_0)$, where $w_0=(s_1, \ldots, s_k)$.
\begin{definition}\label{jjdef}
If $f\colon Z\to\mathbb{R}$ is a map with $Z\subseteq X^{(l)}$ or $Z\subseteq X_l$ ($l\in[k]$), then we call an element $w_0\in Z$ such that $f(w_0)\geq f(w)$ for every $w\in Z$ an {\it optimal point} of this map.
\end{definition}
In particular, we see that if $w_0\in X^{(k)}$ is an optimal point of the map $T$, then $w_0$ is an optimal sequence (see Definition~\ref{ddddef}) if and only if it is a non--decreasing sequence. In this section, we show (see Proposition~\ref{trud}) that every optimal point of the map $T$ is indeed a non--decreasing sequence, which implies that every optimal point of $T$ is simultaneously an optimal sequence. We also derive the formula for an optimal point of $T$ (see Propositions~\ref{prop10},\ref{p8}). For this aim, we introduce certain natural generalizations of this map. Namely, for every $l\in [k]_0$ we define the map $T_l\colon X^{(k-l)}\to\mathbb{R}$ as follows:
\begin{equation}\label{uogt}
T_l(w):=\sum\limits_{j=1}^{k-l}R_{l,j}(w)\cdot\Gamma_{l,1,j}(w)+\xi_{k,n}\cdot\Gamma_{l,1,k-l}(w),
\end{equation}
where the maps $R_{l,j},\;\Gamma_{l,j,j'}\colon X^{(k-l)}\to\mathbb{R}$ ($l\in[k]_0$, $j,j'\in[n]_0$) are defined as follows (in the formula for $R_{l,j}$ below we assume $x_{k-l+1}:=n-1$):
\begin{eqnarray*}
R_{l,j}((x_1, \ldots, x_{k-l}))&:=&\left\{
\begin{array}{ll}
r_{l+j-1}(x_j)-r_{l+j-1}(x_{j+1}),&{\rm if}\;\;1\leq j\leq k-l,\\
0,&{\rm otherwise},
\end{array}
\right.\\
\Gamma_{l,j,j'}((x_1, \ldots, x_{k-l}))&:=&\left\{
\begin{array}{ll}
\prod\limits_{t=j}^{j'}(x_t-l-t+1),&{\rm if}\;\;1\leq j\leq j'\leq k-l,\\
1,&{\rm otherwise}.
\end{array}
\right.
\end{eqnarray*}
In particular $T_0=T$ and $T_k=\xi_{k,n}$.
Let us consider the maps $D_l$ ($l\in[k]_0$) defined by (\ref{dddd}). In Section~\ref{sqa1}, we derive some properties of these maps (see Lemma~\ref{prop8} therein), which allows to describe them in terms of the maps $d_l$ defined by (\ref{aj}) in the following way.
\begin{proposition}\label{proo8}
For every $w=(x_1, \ldots x_k)\in X^{(k)}$ and every $l\in[k]_0$ we have
$$
D_l(\sigma^l(w))=\sum_{i=l}^{k-1}\left(\prod_{j=l+1}^i \widetilde{x}_j\right)d_i(\widetilde{x}_{i+1})+\left(\prod_{j=l+1}^k \widetilde{x}_j\right)\delta_{k,n},
$$
where $\sigma\colon X^{(k-l)}\to X^{(k-l-1)}$ is a left-shift operator removing the first letter from a non-empty word and $\widetilde{x}_i:=x_i-i+1$ for every $i\in [k]$.
\end{proposition}
\begin{proof}
The case $l=k$ directly follows from the equality $D_k(\epsilon)=\delta_{k,n}$ (see Lemma~\ref{prop8}~(i)). In the case $l\in[k-1]_0$, we have by Lemma~\ref{prop8}~(ii)
$$
D_l(\sigma^l(w))=c_l(x_{l+1})+(x_{l+1}-l)D_{l+1}(\sigma^{l+1}(w)),
$$
where the map $c_l$ is defined by (\ref{cccd}). Hence, since $c_l(x_{l+1})=d_l(\widetilde{x}_{l+1})$, we can write
$$
D_l(\sigma^l(w))=d_l(\widetilde{x}_{l+1})+\widetilde{x}_{l+1}D_{l+1}(\sigma^{l+1}(w)).
$$
By easy induction on $m$, we can extend the last formula as follows:
\begin{equation}\label{erex}
D_l(\sigma^l(w))=\sum_{i=l}^m\left(\prod_{j=l+1}^i\widetilde{x}_j\right)d_i(\widetilde{x}_{i+1})+\left(\prod_{j=l+1}^{m+1}\widetilde{x}_j\right)D_{m+1}(\sigma^{m+1}(w))
\end{equation}
for every $l\in[k-1]_0$ and $m\in[l-1, k-1]$. The claim now follows by taking $m:=k-1$ in~(\ref{erex}).\qed
\end{proof}
Let $F_{l, w}$ ($l\in[k]$, $w\in X^{(k-l)}$) be the maps defined by (\ref{fffd}). In the proof of the next proposition, we use some properties of the maps $c_l$, which we derive in Section~\ref{sqa1}.
\begin{proposition}\label{prop10}
Let $l\in[k]$ and $w\in X^{(k-l)}$. If $s$ is the smallest number $x\in[l-1, n-1]$ such that $c_l(x+1)\leq D_l(w)/l$, then $s$ is an optimal point of the map $F_{l,w}$. Consequently, if $t$ is the smallest number $x\in[n-l]$ such that $d_l(x)\leq D_l(w)/l$, then $t+l-1$ is an optimal point of $F_{l,w}$.
\end{proposition}
\begin{proof}
For every $x\in [l-1,n-2]$ the following equality holds:
\begin{equation}\label{uss}
F_{l,w}(x+1)-F_{l,w}(x)=l\cdot c_{l}(x+1)-D_l(w).
\end{equation}
Indeed, directly by the definition of the map $F_{l,w}$, the left side of (\ref{uss}) is equal to
$c_{l-1}(x)-c_{l-1}(x+1)-D_l(w)$, which, by the identity $lc_l(x+1)=c_{l-1}(x)-c_{l-1}(x+1)$ for all $x\in [l-1,n-1]$ (see Lemma~\ref{prop7}~(iii) in Section~\ref{sqa1}), is equal to the right side of (\ref{uss}). Hence, the first part follows from~(\ref{uss}) and from the fact that the map $c_l$ is non--increasing (see Lemma~\ref{prop7}~(iv)). The second part follows now from the equalities $d_l(x)=c_l(x+l)$ for all $x\in [n-l]$.
\qed
\end{proof}
The below proposition is based on the observation that for every word $w=(x_1, \ldots, x_k)\in X^{(k)}$ and every $l\in[k]$ there are $A\in \mathbb{R}_+$, $B\in \mathbb{R}$ which do not depend on the letter $x_l$ and such that $T(w)=A\cdot F_{l, \sigma^l(w)}(x_l)+B$ (for the proof see Lemma~\ref{prop9} in Section~\ref{sqa1}).
\begin{proposition}\label{p8}
A sequence $w=(x_1, \ldots, x_k)\in X^{(k)}$ is an optimal point of the map $T$ if and only if for every $l\in[k]$ the letter $x_l$ is an optimal point of the map $F_{l, \sigma^l(w)}$.
\end{proposition}
\begin{proof}
Suppose, contrary, that the sequence $w=(x_1, \ldots, x_k)$ is an optimal point of $T$ and there is $l\in[k]$ such that the letter $x_l$ is not an optimal point of $F_{l, \sigma^l(w)}$. Let $x'_l\in X_l$ be an optimal point of $F_{l, \sigma^l(w)}$ and let $w'\in X^{(k)}$ be the word arising from $w$ by replacing the $l$-th coordinate with $x'_l$. Since $\sigma^l(w')=\sigma^l(w)$, we have:
$$
F_{l, \sigma^l(w')}(x'_l)=F_{l, \sigma^l(w)}(x'_l)>F_{l, \sigma^l(w)}(x_l).
$$
By Lemma~\ref{prop9}, there are $A\in \mathbb{R}_+$, $B\in\mathbb{R}$ such that
$$
T(w')=A\cdot F_{l, \sigma^l(w')}(x'_l)+B,\;\;\;T(w)=A\cdot F_{l, \sigma^l(w)}(x_l)+B.
$$
Consequently $T(w')>T(w)$, which contradicts with the assumption that $w$ is an optimal point of $T$.
Conversely, let $w=(x_1,\ldots, x_k)\in X^{(k)}$ be such that for every $l\in[k]$ the letter $x_l$ is an optimal point of the map $F_{l, \sigma^l(w)}$. We show that $w$ is an optimal point of $T$. Let $v=(y_1, \ldots, y_k)\in X^{(k)}$ be an arbitrary optimal point of the map $T$. Let us define the words $w_l\in X^{(k)}$ ($0\leq l\leq k$) as follows: $w_0:=w$, $w_k:=v$ and $w_l:=(y_1, \ldots, y_l, x_{l+1},\ldots, x_k)$ for every $l\in [k-1]$. In particular, for each $l\in[k]$ the two words $w_{l-1}$ and $w_l$ differ only in the $l$-th position, which is equal to $x_l$ in $w_{l-1}$ and to $y_l$ in $w_l$. Hence, by Lemma~\ref{prop9}, there are $A\in\mathbb{R}_+$ and $B\in\mathbb{R}$ such that
$$
T(w_{l-1})=A\cdot F_{l, \sigma^l(w_{l-1})}(x_l)+B,\;\;\;T(w_l)=A\cdot F_{l, \sigma^l(w_l)}(y_l)+B.
$$
Since the letter $x_l$ is an optimal point of $F_{l, \sigma^l(w)}$ and $\sigma^l(w_{l-1})=\sigma^l(w_l)=\sigma^l(w)$, we have $T(w_{l-1})\geq T(w_l)$. Consequently, we obtain the inequalities:
$$
T(w)=T(w_0)\geq T(w_1)\geq \ldots\geq T(w_k)=T(v).
$$
Since $v$ is an optimal point of $T$, we have $T(w)=T(v)$. Thus $w$ is an optimal point of $T$.
\qed
\end{proof}
The proof of the next proposition is based on various properties of the maps $D_l$, $c_l$ ($l\in[k]_0$), which we derive in Section~\ref{sqa1} (see
Lemmas~\ref{prop7},~\ref{prop8}).
\begin{proposition}\label{trud}
If $w=(x_1, \ldots, x_k)\in X^{(k)}$ is an optimal point of the map $T$, then $w$ is a non--decreasing sequence.
\end{proposition}
\begin{proof}
Let $w=(x_1, \ldots, x_k)$ be an optimal point of the map $T$. By Proposition~\ref{p8}, we see that for every $l\in[k]$ the letter $x_l$ is an optimal point of the map $F_{l, \sigma^l(w)}$. Let us fix $l\in[k]\setminus\{1\}$ and let us denote $x:=x_{l-1}$, $y:=x_l$, $v:=\sigma^l(w)$. We have to show that $x\leq y$. We can assume that $x\neq l-1$ and $y\neq n-1$. Since $x$ and $y$ are optimal points of $F_{l-1 yv}$ and $F_{l, v}$, respectively, we obtain:
$$
F_{l,v}(y+1)-F_{l,v}(y)\leq 0, \;\;\;F_{l-1, yv}(x)-F_{l-1,yv}(x-1)\geq 0.
$$
By~(\ref{uss}), we have
\begin{equation}\label{w1}
l\cdot c_l(y+1)\leq D_l(v),\;\;\;(l-1)\cdot c_{l-1}(x)\geq D_{l-1}(yv).
\end{equation}
Since $y-l+1>0$, we obtain from the first of the inequalities in~(\ref{w1}):
\begin{equation}\label{wwww1}
(y-l+1)\cdot l\cdot c_l(y+1)\leq (y-l+1)\cdot D_l(v).
\end{equation}
But from Lemma~\ref{prop8}~(ii), we have
\begin{equation}\label{w2}
(y-l+1)\cdot D_l(v)=D_{l-1}(yv)-c_{l-1}(y).
\end{equation}
The second inequality in~(\ref{w1}) gives:
\begin{equation}\label{www1}
D_{l-1}(yv)-c_{l-1}(y)\leq (l-1)\cdot c_{l-1}(x)-c_{l-1}(y).
\end{equation}
From~(\ref{wwww1})--(\ref{www1}), we obtain:
\begin{equation}\label{ww1}
(y-l+1)\cdot l\cdot c_l(y+1)\leq (l-1)\cdot c_{l-1}(x)-c_{l-1}(y).
\end{equation}
Since $l\cdot c_{l}(y+1)=c_{l-1}(y)-c_{l-1}(y+1)$ (see Lemma~\ref{prop7}~(iii)), we obtain
$$
(y-l+1)\cdot (c_{l-1}(y)-c_{l-1}(y+1))\leq (l-1)\cdot c_{l-1}(x)-c_{l-1}(y),
$$
or equivalently:
\begin{equation}\label{lin}
(l-1)\cdot (c_{l-1}(x)-c_{l-1}(y+1))\geq (y-l+2)\cdot c_{l-1}(y)-y\cdot c_{l-1}(y+1).
\end{equation}
But the right side of~(\ref{lin}) is equal to $\frac{{n-y-1\choose k-l+1}}{(l-1)!\cdot {n\choose k}}$ (see Lemma~\ref{prop7}~(v)), which in the case $y\leq n-k+l-2$ is a positive number.
Consequently, we have $c_{l-1}(x)-c_{l-1}(y+1)>0$ in this case. Since the map $c_{l-1}$ is non--increasing (see Lemma~\ref{prop7}~(iv)), we have $x\leq y$. So, we can assume $y\geq n-k+l-1$. In the case $l\geq 3$ we have $D_{l-1}(yv)>0$ by Lemma~\ref{prop8}~(iii), and hence, by~(\ref{w1}), we obtain in this case: $c_{l-1}(x)>0$. But then, directly from the definition of the map $c_{l-1}$, we have
$$
c_{l-1}(x)=\frac{k\cdot (x-l+1)!\cdot (n-x)!}{(l-1)\cdot (l-2)\cdot n!}\sum\limits_{j=0}^{l-3}{k-1\choose j}{n-k\choose x-j-1}>0.
$$
Thus there must be $j\in[l-3]_0$ such that $n-k\geq x-1-j$. Consequently $x\leq n-k+1+j\leq n-k+l-2<y$. Hence, we can assume $l=2$. Then by~(\ref{w2}), we have $D_1(yv)-c_1(y)=(y-1)\cdot D_2(v)$ and by the second inequality in~(\ref{w1}), we have $c_1(x)\geq D_1(yv)$. Hence
$c_1(x)-c_1(y)\geq (y-1)\cdot D_2(v)>0$, where the last inequality follows from Lemma~\ref{prop8}~(iii). Since the map $c_1$ is non--increasing, we obtain $x<y$, which finishes the proof.\qed
\end{proof}
\section{The proofs of Theorems~\ref{mt}-\ref{mtt}}\label{secpr}
The main results are a straightforward consequence of Theorem~\ref{tt1} and Propositions~\ref{proo8}-\ref{trud}.
\begin{proof}[of Theorem~\ref{mt}]
Let $w_0:=(s_1, \ldots, s_k)$ be a sequence constructed as in Theorem~\ref{mt}. By Proposition~\ref{proo8}, we see that for every $l\in[k]$ the right side of~(\ref{ineqqq}) is equal to $D_l(\sigma^l(w_0))/l$. Thus for every $l\in[k]$ the number $t_l=s_l-l+1$ is the smallest number $x\in[n-l]$ which satisfies $d_l(x)\leq D_l(\sigma^l(w_0))/l$, and hence, by Proposition~\ref{prop10}, the number $s_l$ is an optimal point of the map $F_{l, \sigma^l(w_0)}$. By Proposition~\ref{p8}, we obtain that $w_0$ is an optimal point of the map $T$. Moreover, the sequence $w_0$ is non-decreasing by Proposition~\ref{trud}. Hence, we see by the definition of the map $T$ and by Theorem~\ref{tt1} that $w_0$ is an optimal sequence. \qed
\end{proof}
\begin{proof}[of Theorem~\ref{mtt}]
Let us denote $w_0:=(s_1, \ldots, s_k)$. By Proposition~\ref{proo8}, the expression~(\ref{expr}) is equal to $-D_0(w_0)$, which, by the definition of the map $D_0$, is equal to $T_0(w_0)=T(w_0)$. Hence, by the definition of the map $T$ and by Theorem~\ref{tt1}, this expression is equal to the probability of success for the policy described in Proposition~\ref{p1}.\qed
\end{proof}
\section{The auxiliary properties of the maps $r_l$, $c_l$, $D_l$ and $F_{l, w}$}\label{sqa1}
Let us define for every $l\in[k]$ the maps $a_l,b_l\colon [n]\to\mathbb{R}$ as follows:
$$
a_l(x):=\frac{1}{{n \choose x}}\sum\limits_{j=0}^l{k\choose j}{n-k\choose x-j},\;\;\;b_l(x):=\frac{1}{l{n \choose x}}\sum\limits_{j=0}^lj{k\choose j}{n-k\choose x-j}.
$$
\begin{lemma}\label{prop1}
For all $x\in[n]$, $l\in[k-1]$ we have:
\begin{eqnarray}
a_{l+1}(x)-a_l(x)&=&\gamma(x,l+1),\label{ajj1}\\
(l+1)\cdot b_{l+1}(x)-l\cdot b_l(x)&=&(l+1)\cdot\gamma(x,l+1)\label{ajj2},
\end{eqnarray}
and for all $x\in[n-1]$ and $l\in[k]$ we have:
\begin{eqnarray}
a_l(x)-a_{l}(x+1)&=&\frac{k-l}{n-x}\cdot\gamma(x,l),\label{ajj3}\\
(x+1)\cdot b_l(x)-x\cdot b_l(x+1)&=&\frac{(x+1)\cdot(k-l)}{n-x}\cdot\gamma(x,l)\label{bi},
\end{eqnarray}
where $\gamma(x,l):={k\choose l}{n-k\choose x-l}/{n\choose x}$. In particular, for all $x\in[1,n]$ and $l\in[1,k]$ we have
\begin{eqnarray}
x\cdot a_{l-1}(x-1)-(x-l)\cdot a_{l-1}(x)&=&l\cdot a_{l}(x),\label{acor}\\
x\cdot b_{l-1}(x-1)-(x-l)\cdot b_{l-1}(x)&=&l\cdot b_{l}(x)\label{bcor}.
\end{eqnarray}
\end{lemma}
\begin{proof}
The identities~(\ref{ajj1})-(\ref{ajj2}) directly follow from the definitions of the maps $a_l$, $b_l$; the identities~(\ref{ajj3})-(\ref{bi}) can be easily proved by induction on $l$. These four identities together with the definitions of the maps $a_l$, $b_l$ imply the identities (\ref{acor})--(\ref{bcor}).\qed
\end{proof}
\begin{lemma}\label{propppp}
The maps $r_l$ have the following properties:
\begin{itemize}
\item[(i)] $r_l(x)=\frac{(x-l-1)!}{x!}\cdot\left(b_l(x)-a_l(x)+1\right)$ for $l\in[k]$ and $x\in [l,n]$,
\item[(ii)] $r_0(x)=-\sum_{j=2}^x r_1(j)$ for $x\in[n]$,
\item[(iii)] $l\cdot r_l(x)=r_{l-1}(x-1)-r_{l-1}(x)$ for $l\in[\leq k]$ and $x\in [l,n]$.
\end{itemize}
\end{lemma}
\begin{proof}
The item (i) follows directly from the definition of the map $r_l$. To show (ii), we can write by the definition of $r_1$:
$$
r_1(j)=\left(\frac{1}{j-1}-\frac{1}{j}\right)-\frac{(n-j)!\cdot (j-2)!}{n!}\cdot{n-k\choose j},\;\;j\in[1,n].
$$
Hence, by the following easily verifiable identity
$$
\frac{(n-j)!\cdot (j-2)!}{n!}\cdot{n-k\choose j}=\frac{1}{{n\choose k}}
\left(
\frac{{n-j\choose k}}{j-1}-\frac{{n-(j+1)\choose k}}{j}\right)-\frac{{n-j-1\choose k-1}}{j{n\choose k}},
$$
we obtain:
$$
r_1(j)=\left(\frac{1}{j-1}-\frac{1}{j}\right)+\frac{{n-j-1\choose k-1}}{j{n\choose k}}-
\frac{1}{{n\choose k}}
\left(
\frac{{n-j\choose k}}{j-1}-\frac{{n-(j+1)\choose k}}{j}\right),\;\;j\in[1,n].
$$
Hence, the item (ii) in the case $x\in[1,n]$ simply follows from the above equalities. The case $x=1$ can be directly verified. The item~(iii) in the case $l\leq 0$ simply follows from the definition of the map $r_l$. In the case $l=1$ it follows from the item~(ii).
If $l>1$, then for every $x\in[l,n]$ we can write by the item (i):
\begin{eqnarray*}
r_{l-1}(x-1)&=&\frac{(x-l-1)!}{(x-1)!}(b_{l-1}(x-1)-a_{l-1}(x-1)+1),\\
r_{l-1}(x)&=&\frac{(x-l)!}{x!}\left(b_{l-1}(x)-a_{l-1}(x)+1\right).
\end{eqnarray*}
Now, we can use the equalities~(\ref{acor})--(\ref{bcor}) from Lemma~\ref{prop1} and obtain that the difference $r_{l-1}(x-1)-r_{l-1}(x)$ is equal to
$$
\frac{(x-l-1)!}{x!}\left(l\cdot b_{l}(x)-l\cdot a_{l}(x)+l\right)=l\cdot r_{l}(x),
$$
which finishes the proof of Lemma~\ref{propppp}.
\qed
\end{proof}
\begin{lemma}\label{prop7}
The maps $c_l$ ($l\in[k]_0$) have the following properties:
\begin{itemize}
\item[(i)] $c_{l}(x)=\frac{b_{l-1}(x)}{l\cdot l! \cdot{x\choose l}}$ for $l\in[1,k]$, $x\in[l,n]$,
\item[(ii)] $c_l(x)=r_{l-1}(x)-(x-l)\cdot r_l(x)$ for $l\in[k]_0$, $x\in[l,n]$,
\item[(iii)] $l\cdot c_{l}(x)=c_{l-1}(x-1)-c_{l-1}(x)$ for $l\in[k]$, $x\in [l,n]$,
\item[(iv)] for every $l\in[k]$ the map $c_l$ is non--increasing,
\item[(v)] $(x+1-l)\cdot c_l(x)-x\cdot c_l(x+1)=\frac{{n-x-1\choose k-l}}{l!\cdot {n\choose k}}$ for $l\in[k]$ and $x\in [l,n-1]$.
\end{itemize}
\end{lemma}
\begin{proof}
By the definition of the maps $c_l$, we can write
$$
c_l(x)=\left\{
\begin{array}{ll}
-1+\frac{k\cdot x}{n}+\frac{{n-x-1\choose k}}{{n\choose k}}+\frac{x}{{n\choose k}}\sum\limits_{j=1}^{x}\frac{{n-j-1\choose k-1}}{j},&{\rm if}\;\;l=0,\\
\\
-\frac{k}{n}-\frac{1}{{n\choose k}}\sum\limits_{j=1}^{x-1}\frac{{n-j-1\choose k-1}}{j},&{\rm if}\;\;l=1,\\
\\
\frac{k\cdot (x-l)!\cdot (n-x)!}{l\cdot (l-1)\cdot n!}\sum\limits_{j=0}^{l-2}{k-1\choose j}{n-k\choose x-j-1},&{\rm if}\;\;1<l\leq k.
\end{array}
\right.
$$
Hence, the item (i) easily follows from the definition of the map $b_{l-1}$ and from the identity $j{k\choose j}=k{k-1\choose j-1}$ for $j\in\mathbb{Z}$. The item~(ii) in the case $l\in\{0,1\}$ directly follows from the definitions of the maps $c_l$, $r_{l-1}$, $r_l$. In the case $l\in[1,k]$, we can use the item (i) for the left side and Lemma~\ref{propppp}~(i) for the right side, and then the claim easily follows from the identities~(\ref{ajj1})-(\ref{ajj2}) from Lemma~\ref{prop1}. To show the item~(iii), we see by (ii) that the difference $c_{l-1}(x-1)-c_{l-1}(x)$ is equal to:
$$
(r_{l-2}(x-1)-r_{l-2}(x))-(x-l)\cdot r_{l-1}(x-1)+(x-l+1)\cdot r_{l-1}(x).
$$
By Lemma~\ref{propppp}~(iii), we have $r_{l-2}(x-1)-r_{l-2}(x)=(l-1)\cdot r_{l-1}(x)$. Hence
$$
c_{l-1}(x-1)-c_{l-1}(x)=x\cdot r_{l-1}(x)-(x-l)\cdot r_{l-1}(x-1).
$$
Again, by Lemma~\ref{propppp}~(iii), we have $r_{l-1}(x-1)=r_{l-1}(x)+l\cdot r_l(x)$, and hence
$$
c_{l-1}(x-1)-c_{l-1}(x)=l(r_{l-1}(x)-(x-l)\cdot r_l(x)).
$$
The claim now follows from the item (ii). To show~(iv), we obtain by the item~(iii) that for every $l\in[k-1]$ and $x\in [l+1,n]$ the difference $c_l(x-1)-c_l(x)$ is equal to $(l+1)c_{l+1}(x)$, which is a nonnegative number by the definition of the map $c_{l+1}$. Hence the map $c_l$ is non--increasing for every $l\in[k-1]$. If $l=k=1$, then the item~(iv) follows directly from the definition of the map $c_l$. If $l=k>1$, then by the definition of the map $c_l$, we obtain
$$
c_k(x)=\frac{(x-k)!}{n(k-1)(x-1)!}-\frac{(n-k)!}{(k-1)n!}.
$$
Hence, we see that also in this case the map $c_l$ is non-increasing. As for the item~(v), the case $l=1$ follows directly from the item~(iii) and from the definition of the map $c_2$. In the case $l\in[1,k]$, by the item~(i), we obtain that the difference $(x+1-l)c_l(x)-x c_l(x+1)$ is equal to
$$
\frac{(x-l+1)!}{l (x+1)!}\left((x+1)b_{l-1}(x)-x b_{l-1}(x+1)\right),
$$
which is equal to $\frac{{n-x-1\choose k-l}}{l!\cdot {n\choose k}}$ by the equality~(\ref{bi}) from Lemma~\ref{prop1}.
\qed
\end{proof}
\begin{lemma}\label{prop8}
The maps $D_l$ ($l\in[k]_0$) have the following properties:
\begin{itemize}
\item[(i)] $D_k(\epsilon)=\delta_{k,n}$,
\item[(ii)] $D_l(w)=(x_1-l)\cdot D_{l+1}(\sigma(w))+c_{l}(x_1)$ for each $l\in[k-1]_0$ and $w\in X^{(k-l)}$, where $x_1$ denotes the first letter of $w$,
\item[(iii)] $D_l(w)>0$ for each $l\in[1,k]$ and $w\in X^{(k-l)}$.
\end{itemize}
\end{lemma}
\begin{proof}
The item~(i) directly follows from the definition of the map $D_k$ and from the formulae~(\ref{ajj}) and~(\ref{rere}) defining, respectively, the number $\delta_{k,n}$ and the map $r_{k-1}$. As for the item~(ii), for every $l\in[k-1]_0$ we obtain by Lemma~\ref{prop7}~(ii) and by the definitions of the maps $R_{l,1}$, $\Gamma_{l,1,1}$:
\begin{eqnarray*}
D_l(w)=r_{l-1}(x_1)-T_l(w)=\\
=r_{l-1}(x_1)-R_{l,1}(w)\cdot\Gamma_{l,1,1}(w)-\left(\sum\limits_{j=2}^{k-l}R_{l,j}(w)\cdot\Gamma_{l,1,j}(w)+\xi_{k,n}\cdot\Gamma_{l,1,k-l}(w)\right)=\\
=c_l(x_1)+r_l(x_2)\cdot (x_1-l)-\left(\sum\limits_{j=2}^{k-l}R_{l,j}(w)\cdot\Gamma_{l,1,j}(w)+\xi_{k,n}\cdot\Gamma_{l,1,k-l}(w)\right),
\end{eqnarray*}
where $x_2$ denotes the second letter of $w$ in the case $l<k-1$ and $x_2:=n-1$ in the case $l=k-1$. In particular, if $l=k-1$, then $\Gamma_{l,1,k-l}(w)=\Gamma_{k-1,1,1}(w)=x_1-l$, and hence
\begin{eqnarray*}
D_l(w)&=&c_l(x_1)+(x_1-l)\cdot (r_{k-1}(n-1)-\xi_{k,n})=\\
&=&c_l(x_1)+(x_1-l)\cdot D_k(\epsilon)=c_l(x_1)+(x_1-l)\cdot D_{l+1}(\sigma(w)).
\end{eqnarray*}
If $l<k-1$, then $\Gamma_{l,1,j}(w)=(x_1-l)\cdot \Gamma_{l,2,j}(w)$ for every $j\in[k-l]$, and hence
$$
D_l(w)=c_l(x_1)+r_l(x_2)\cdot (x_1-l)-(x_1-l)\cdot \Lambda,
$$
where
$$
\Lambda:=\sum\limits_{j=2}^{k-l}R_{l,j}(w)\cdot\Gamma_{l,2,j}(w)+\xi_{k,n}\cdot\Gamma_{l,2,k-l}(w).
$$
Since $R_{l,j}(w)=R_{l+1,j-1}(\sigma(w))$ and $\Gamma_{l,2,j}(w)=\Gamma_{l+1,1,j-1}(\sigma(w))$ for every $j\in[1,k-l]$, we obtain
\begin{eqnarray*}
\Lambda=\sum\limits_{j=2}^{k-l}R_{l+1,j-1}(\sigma(w))\cdot\Gamma_{l+1,1,j-1}(\sigma(w))+\xi_{k,n}\cdot\Gamma_{l+1,1,k-l-1}(\sigma(w))=\\
=\sum\limits_{j=1}^{k-l-1}R_{l+1,j}(\sigma(w))\cdot\Gamma_{l+1,1,j}(\sigma(w))+\xi_{k,n}\cdot\Gamma_{l+1,1,k-l-1}(\sigma(w))=T_{l+1}(\sigma(w)).
\end{eqnarray*}
Consequently, we have:
\begin{eqnarray*}
D_l(w)=c_l(x_1)+(x_1-l)\cdot r_l(x_2)-(x_1-l)\cdot T_{l+1}(\sigma(w))=\\
=c_l(x_1)+(x_1-l)(r_l(x_2)-T_{l+1}(\sigma(w)))=c_l(x_1)+(x_1-l)\cdot D_{l+1}(\sigma(w)),
\end{eqnarray*}
which finishes the proof of the item~(ii). The item~(iii) directly follows from the items~(i)--(ii) and from the inequalities $c_l(x)\geq 0$ for all $l\in[1,k]$, $x\in[l,n]$.
\qed
\end{proof}
\begin{lemma}\label{prop9}
Let $w=(x_1, \ldots, x_k)\in X^{(k)}$ be arbitrary. Then for every $l\in[k]$ there exist $A\in\mathbb{R}_+$, $B\in\mathbb{R}$ which do not depend on the letter $x_l$ and such that $T(w)=A\cdot F_{l, \sigma^l(w)}(x_l)+B$.
\end{lemma}
\begin{proof}
Let us fix $l\in[k]$ and let us define $A:=\Gamma_{0,1,l-1}(w)$. Then we see by the definition of the map $\Gamma_{0,1,l-1}$ that $A$ does not depend on $x_l$ and $A>0$. By the definition of the map $T=T_0$, we have $T(w)=B_1+B_2+B_3$, where
\begin{eqnarray*}
B_1&:=&\sum_{j=1}^{l-2}R_{0,j}(w)\cdot\Gamma_{0,1,j}(w),\\
B_2&:=&\sum_{j=l-1}^lR_{0,j}(w)\Gamma_{0,1,j}(w)=AR_{0,l-1}(w)+R_{0,l}(w)\Gamma_{0,1,l}(w),\\
B_3&:=&\sum_{j=l+1}^{k}R_{0,j}(w)\Gamma_{0,1,j}(w)+\xi_{k,n}\Gamma_{0,1,k}(w).
\end{eqnarray*}
By the definitions of the maps $R_{0,j}$, $\Gamma_{0,1,j}$ ($j\in[l-2]$), we see that $B_1$ does not depend on $x_l$. Since $\Gamma_{0,1,j}(w)=A(x_l-l+1)\Gamma_{0,l+1,j}(w)$ for every $j\in[l-1,k]$, we obtain $T(w)=B_1+AB_2'+AB_3'$, where
\begin{eqnarray*}
B_2'&:=&R_{0, l-1}(w)+(x_l-l+1)\cdot R_{0,l}(w),\\
B_3'&:=&(x_l-l+1)\cdot B_4,\\
B_4&:=&\sum\limits_{j=l+1}^k R_{0,j}(w)\cdot\Gamma_{0,l+1,j}(w)+\xi_{k,n}\cdot \Gamma_{0,l+1,k}(w).
\end{eqnarray*}
Next, by the definitions of the maps $R_{0,l-1}$, $R_{0,l}$ and by Lemma~\ref{prop7}~(ii), we obtain
$$
B_2'=B_2''-c_{l-1}(x_l)-x_l\cdot r_{l-1}(x_{l+1}),
$$
where $B_2'':=r_{l-2}(x_{l-1})+(l-1)\cdot r_{l-1}(x_{l+1})$ does not depend on $x_l$. Further, since $R_{0,j}(w)=R_{l, j-l}(\sigma^l(w))$ and $\Gamma_{0,l+1,j}(w)=\Gamma_{l,1, j-l}(\sigma^l(w))$ for every $j\in[l-1,k]$, we obtain
\begin{eqnarray*}
B_4&=&\sum\limits_{j=l+1}^k R_{l,j-l}(\sigma^l(w))\cdot\Gamma_{l,1,j-l}(\sigma^l(w))+\xi_{k,n}\cdot \Gamma_{l,1,k-l}(\sigma^l(w))=\\
&=&\sum\limits_{j=1}^{k-l} R_{l,j}(\sigma^l(w))\cdot\Gamma_{l,1,j}(\sigma^l(w))+\xi_{k,n}\cdot \Gamma_{l,1,k-l}(\sigma^l(w))=T_l(\sigma^l(w)).
\end{eqnarray*}
Hence
$$
B_3'=(x_l-l+1)\cdot B_4=x_l\cdot T_l(\sigma^l(w))-B_3'',
$$
where $B_3'':=(l-1)\cdot T_l(\sigma^l(w))$ does not depend on $x_l$. We can now write
\begin{eqnarray*}
T(w)&=&B_1+A\cdot (B_2'+B_3')=\\
&=&B_1+A\cdot (B_2''-c_{l-1}(x_l)-x_l\cdot r_{l-1}(x_{l+1})+x_l\cdot T_l(\sigma^l(w))-B_3'')=\\
&=&B_1+A\cdot (B_2''-c_{l-1}(x_l)-x_l\cdot D_l(\sigma^l(w))-B_3'')=\\
&=&B_1+A\cdot(B_2''-B_3'')+A\cdot F_{l,\sigma^l(w)}(x_l)=B+A\cdot F_{l,\sigma^l(w)}(x_l),
\end{eqnarray*}
where $B:=B_1+A\cdot(B_2''-B_3'')$ does not depend on $x_l$.\qed
\end{proof}
\section{Conclusion}
In the present paper, we obtained the analytic formulae for an optimal sequence $(s_1, \ldots, s_k)$ in the Gusein-Zade version (\cite{1}) of a generalized secretary problem. In this problem, the interviewer would like to choose one of the $k$ best of $n$ candidates arriving in random order and the stopping rule is based on the relative ranks of the successive arrivals. For any sequence $(s_1, \ldots, s_k)$ describing the optimal policy, we also found the analytic formula for the probability of success when using the policy with this sequence. Our original approach is purely elementary and bases on the combinatorial analysis of the problem. The obtained formulae reveal the possibility of an extension to an arbitrary value of $k$ for closed expressions describing the elements of an optimal sequence. Until now such expressions were derived only for $k\leq 3$. Since the maps $d_l$ in the inequalities (\ref{ineqqq}) describing the optimal sequence are all non-increasing, our formula reduces the determination of elements in this sequence to solving a system of $k$ equations. In other words, we need to solve a recurrence with the number of steps bounded by $k$, which is substantially more advantageous than the implicit solution via computing the optimum from the known mechanism of dynamic or linear programming. On the other hands, in recent years, the linear programming approach was discovered to analyze a broader class of secretary problems. For example, in \cite{7} the authors consider a so-called $J$-choice $K$-best secretary problem (the case $J=1$ was the subject of the present paper), where finding of an optimal solution reduces to solving the corresponding linear program. In~\cite{10} the authors use linear programming but to the so-called continuous and infinite models of the secretary problem (see also \cite{8,9}). In~\cite{17} even a more general problem is studied via this technique -- a so called shared $Q$-queue $J$-choice $K$-best secretary problem. Therefore, it seems natural to analyze and develop our combinatorial approach also for wider classes of secretary problems, which might result in finding some simplifications in the corresponding formulae. The construction of the optimal sequence from Theorem~\ref{mt} could also be applied in the study of the limits $\tau_l(k):=\lim_{n\to\infty}s_l/n$ ($1\leq l\leq k$) and their behaviour. This could help in solving some (according to our knowledge) open questions concerning these limits, such as (see also \cite{3,13}): Is it true that $\tau_1(k)$ monotonically decreases with $k$?
\end{document} | math | 60,221 |
\begin{document}
\title{Unconditional quantum cloning of coherent states with linear optics}
\author{Ulrik L. Andersen, Vincent Josse and Gerd Leuchs}
\email{[email protected]}
\affiliation{Institut f\"{u}r Optik, Information und Photonik, Max-Planck Forschungsgruppe, Universit\"{a}t Erlangen-N\"{u}rnberg, G\"{u}nther-Scharowsky str. 1, 91058, Erlangen, Germany}
\date{\today}
\begin{abstract}
A scheme for optimal Gaussian cloning of optical coherent states is proposed and experimentally demonstrated. Its optical realization is based entirely on simple linear optical elements and homodyne detection. The optimality of the presented scheme is only limited by detection inefficiencies. Experimentally we achieved a cloning fidelity of about 65\%, which almost touches the optimal value of 2/3.
\end{abstract}
\pacs{03.67.Hk, 03.65.Ta, 42.50.Lc}
\maketitle
According to the basic laws of quantum mechanics, an unknown nonorthogonal quantum state cannot be copied exactly \cite{wooters82.nat,dieks82.pla}. In other words, it would be an impossible task to devise a process that produces perfect clones of an arbitrary quantum state. However, a physical realization of a quantum cloning machine with less restrictive requirements to the quality of the clones is possible. Such a quantum cloning machine was first considered in a seminal paper by Buzek and Hillery \cite{buzek96.pra} where they went beyond the no-cloning theorem by considering the possibility of producing approximate clones for qubits. These considerations were later extended to the finite-dimensional regime \cite{buzek98.prl} and finally to the continuous variable (CV) regime \cite{cerf00.prl}.
This extension is stimulated by the relative ease in preparing and manipulating quantum states in the CV regime as well as the unconditionalness: Every prepared state is used in the protocols. Governed by these motivations many quantum protocols have been experimentally realized in this regime \cite{vanloock04.xxx}.
Studies on quantum cloning were initially motivated by the apparent implications on quantum information processing but also because they opened an avenue for a clearer understanding of the fundamental concepts of quantum mechanics and measurement theory. Recently, however it has been shown that quantum cloning might improve the performance of some quantum computational task \cite{galvao00.pra} and it is believed to be the optimal eaveasdropping attack for a certain class of quantum key distribution protocols employing coherent states and CV detection \cite{grosshans04.prl}. Furthermore, quantum cloning also provides a means of partial covariant distribution of quantum information between two (or more) parties in a quantum network \cite{braunstein01.pra}.
To date, convincing experimental realizations of quantum cloning have been restricted to the two-dimensional qubit regime where the polarization state of single photons has been conditionally cloned~\cite{linares02.sci,fasel02.prl}. In parallel there have been some theoretical proposals for the experimental implementations of quantum cloning of CV Gaussian states of light~\cite{dariano01.prl,fiurasek01.prl,braunstein01.prl}. These protocols have been shown to be optimal for the Gaussian N$\rightarrow$M cloner where M identical copies are produced from N originals~\cite{cerf00.pra} - a special case being the Gaussian 1$\rightarrow$2 cloner~\cite{nongaussian}. However all these proposals are based on at least one parametric amplifier rendering the practical realization quite difficult. In this Letter we propose a new simple scheme of an optimal Gaussian cloning machine, which does not rely on any non linear interaction but is only based on simple unitary beam splitter transformations and homodyne detection. Furthermore we implemented this idea experimentally for the 1$\rightarrow$2 cloner.
\begin{figure}
\caption{\it Schematic drawing showing the principle of the continuous variable cloning scheme using only linear optics and homodyne detection. $v_1$, $v_2$ and $v_3$ are vacuum inputs, D is a displacement governed by the measurement $\alpha$ scaled with the gain $\lambda$.}
\label{fig1}
\end{figure}
In this Letter we will consider coherent states of light, for which two canonical conjugate quadratures characterizing the state - e.g. the amplitude, $\hat{x}$, and phase, $\hat{p}$ - have Gaussian statistics. The unknown coherent state to be cloned is then uniquely described by $|\alpha_{in}\rangle=|\frac{1}{2}(x_{in}+ip_{in})\rangle$ where $x_{in}$ and $p_{in}$ are the expectation values of $\hat{x}_{in}$ and $\hat{p}_{in}$. The outputs of the cloning machine are Gaussian mixed states with the expectation values $x_{clone}$ and $p_{clone}$ and characterized by the density operator $\rho_{clone}$. The efficiency of the cloning machine is typically quantified by the fidelity, which gauges the similarity between an input state and an output state. It is defined by $F=\langle \alpha_{in} |\rho_{clone} |\alpha_{in}\rangle$~\cite{furusawa98.sci} and for the particular case of unity cloning gains (corresponding to $x_{clone}=x_{in}$ and $p_{clone}=p_{in}$) it reads
\begin{equation}
F= \frac{2}{\sqrt{(1+\Delta^2x_{clone})(1+\Delta^2p_{clone})}}
\label{fid}
\end{equation}
where $\Delta^2x_{clone}$ and $\Delta^2p_{clone}$ denote the variances.
A straight forward way to produce approximate clones uses a measure-and-prepare strategy~\cite{braunstein00.mod,grosshans01.pra}. In such a "classical" scenario, the best approach to cloning an arbitrary coherent state is to measure simultaneously both quadratures $\hat{x}_{in}$ and $\hat{p}_{in}$~\cite{arthur,hammer04.xxx} and subsequently, based on the outcomes of this measurement, clones of the input state are constructed. However, using this procedure two additional units of quantum noise are added to the clones partly due to the attempt to measure two non-commuting variables simultaneously and partly due to the construction of the clones. Although this method enables the production of an infinite number of clones (1$\rightarrow \infty$ cloner), the optimal fidelity is limited to 1/2~\cite{furusawa98.sci,braunstein00.mod,grosshans01.pra,hammer04.xxx}.
In contrast our quantum approach to cloning uses intrinsic correlations, and runs as follows (see Fig. 1). At the input side of the cloning machine the unknown quantum state is divided by a 50/50 beam splitter. At one output we perform an optimal estimation of the coherent state: the state is split at another 50/50 beam splitter and the amplitude and the phase quadratures are measured simultaneously using ideal homodyne detection~\cite{arthur,hammer04.xxx}.
According to the measurement outcomes the other half of the input state is displaced with a scaling factor, $\lambda$~\cite{lam97.prl}. Using the Heisenberg representation, the displaced field can be expressed as
\begin{eqnarray}
\hat{a}_{disp}=(\frac{1}{\sqrt{2}}+\frac{\lambda}{2})\hat{a}_{in}+(\frac{1}{\sqrt{2}}-\frac{\lambda}{2})\hat{v}_1-\frac{\lambda}{\sqrt{2}}\hat{v}_2^{\dagger}
\end{eqnarray}
where $\hat{v}_1$ and $\hat{v}_2$ refer to the annihilation operators associated with the uncorrelated vacuum modes entering the two beam splitters (see Fig. \ref{fig1}), and $\hat{a}_{in}$ and $\hat{a}_{disp}$ are the annihilation operators for the input and displaced states.
In a final step the displaced state is separated in two clones by a 50/50 beamsplitter:
\begin{eqnarray}
\hat{a}_{clone1}&=&\hat{a}_{in}+\frac{1}{\sqrt{2}}(\hat{v}_3-\hat{v}_2^{\dagger})\nonumber\\
\hat{a}_{clone2}&=&\hat{a}_{in}-\frac{1}{\sqrt{2}}(\hat{v}_3+\hat{v}_2^{\dagger})
\label{clone_transfer}
\end{eqnarray}
where $\hat{v}_3$ is uncorrelated vaccum noise entering the last beam splitter and $\lambda$ has been taken to be $\sqrt{2}$ to assure unity gain.
The transformations in Eq. (\ref{clone_transfer}) are known to describe an optimal Gaussian cloning machine \cite{fiurasek01.prl,braunstein01.prl}. In particular we see that it is invariant with respect to rotation and displacement in phase space as required by a phase independent or covariant cloner.
Normalizing the variance of the vacuum state to unity, the variances of the clones for the amplitude and phase quadratures are $\Delta^2x_{clone}=\Delta^2x_{in}+1$ and $\Delta^2p_{clone}=\Delta^2p_{in}+1$, respectively. Note that using the quantum approach only one unit of quantum noise is added in contrast to the classical approach where two units are added.
Using Eq.(\ref{fid}) the fidelity is found to be 2/3 which corresponds to the optimal fidelity for a Gaussian cloning machine~\cite{cerf00.pra}.
\begin{figure}
\caption{\it Schematic of the CV cloning setup divided into three boxes defining the preparation stage (where an arbitrary input coherent state can be generated), the cloning stage (where two clones are produced) and the verification stage (where the quality of the cloning process is quantified). BS: Beam splitter, $\lambda$: Electronic gain, LO: Local oscillator, AM: Amplitude modulator, PM: Phase modulator and AUX: Auxiliary beam.}
\label{fig2}
\end{figure}
We now proceed by discussing the experimental demonstration of the proposed scheme. First we present the experimental setup shown in Fig.~2. The laser source for our experiment was a monolithic Nd:YAG(yttrium aluminum garnat) nonplanar ring laser at 1064nm delivering 500mW of power in a single transverse mode. A small part of the power was used to create an input signal to the cloning machine whereas the rest served as local oscillator beams and auxiliary beams. The setup comprises three parts; a preparation stage, a cloning stage and finally a verification stage.
{\it Preparation:}
In our experiment, we define the quantum state to be frequency sidebands at $\pm$14.3~MHz (with a bandwidth of 100kHz) of a bright electro-magnetic field (similar to previous realisations of CV quantum protocols~\cite{furusawa98.sci}). At this frequency the laser was found to be shot noise limited, ensuring a pure coherent input state. An arbitrary input state is then easily generated by independently controlling the modulations of the amplitude quadrature ($x_{in}$) and the phase quadrature ($p_{in}$), using two electro-optical modulators.
{\it Cloning:} The prepared state is then directed to the cloning machine where it is divided into two halves by the first beam splitter (BS1). One of the halves was combined with an auxiliary beam (AUX1) at the second beam splitter (BS2) with a $\pi/2$ relative phase shift and balanced intensities. The two beam splitter outputs are detecteted by high quantum efficiency photodiodes so that the sum (difference) of the photo currents provide a measure of the amplitude (phase) quadrature of the two beam splitter outputs. This corresponds to an optimal coherent state measurement and therefore a simpler alternative to the one shown in Fig.\ref{fig1}~\cite{leuchs99.mod}. The added and subtracted photocurrents are scaled appropriately with electronic gains $\lambda_x$ and $\lambda_p$ to ensure unity cloning gains, and used to modulate the amplitude and phase of an auxiliary beam (AUX2) via two independent modulators. This beam is then combined at a 99/1 beam splitter with the other half of the signal beam, hereby displacing this part according to measurement outcomes \cite{furusawa98.sci}.
In a final step, the clones are generated at the output of the third beam splitter (BS3).
{\it Verification:} To characterize the performance of the cloning machine, the spectral noise properties of the two clones are measured by two homodyne detectors with strong local oscillator beams (LO1 and LO2). Since the statistics of the involved light fields are Gaussian we need only measure two conjugate quadratures to fully characterise the states. Therefore the homodyne detectors were set to measure stably - employing electronic servo feedback loops - either the amplitude or the phase quadrature. We note that the input state is also measured by the same homodyne detectors, to ensure a consistent comparison between the input state and the clones.
\begin{figure}
\caption{\it Spectral densities for the input state (traces (ii)) and the output clones (traces (iii)), 1 and 2, of the Gaussian cloning machine, measured by the two homodyne detection systems in a 2 MHz span mode and a center frequency of 14.3MHz corresponding to the location of the sidebands. The traces are normalised to the quantum noise limit (i). Resolution bandwidth is 100~kHz and video band width is 30~Hz. (a) and (b) show the spectra for the amplitude and the phase quadratures of clone 1 and similarily (c) and (d) the spectra for clone 2. Since the input signal was measured with the same homodyne detectors as the output clones, the measurement of the input signal is degraded by the two 50/50 beam splitters BS1 and BS3 and unity cloning gain is ensured by a 6~dB difference between the input signal and the clones.}
\label{fig3}
\end{figure}
An example of a cloning run is reported in Fig.~3. The spectral densities of the amplitude and phase quadratures are here shown over a 1 MHz frequency span for the input state (ii) and the two clones (iii).
From these traces the coherent amplitude of the various fields, $x_{in,out}$ and $p_{in,out}$, are measured by the heights of the peaks at 14.3~MHz relative to the quantum noise level (i). Using these signal powers we estimate an average photon number of 62 per unit bandwidth per unit time~\cite{photonno}. As evident from the figure, the electronic gains of the feed forward loops are adjusted such that the cloning gains are close to unity (which corresponds to a 6~dB difference between the measured input signal and the output signals due to the degradation of the input signal by BS1 and BS3). In order to simplify the following analysis of the measurement data we will assume unity gains and will later consider the consequences of small deviations from unity which is the case for real cloning machines.
From Fig.~3 it is also evident that additional noise has been added to the clones relative to the input state which is a result of the cloning action.
In order to quantify accurately the performance of the cloning machine, we estimated precisely this amount of added noise at 14.3 MHz (in a 100kHz bandwidth). To do so, we switched off the modulations of the input beam, and recorded the noise in a zero span measurement over 2 seconds. These results are displayed in Fig.~4 where the added noise in amplitude and phase are reported for both clones. To avoid an erroneously underestimation of the noise power, the traces are corrected to account for the detection efficiencies of the two homodyne stations (which amount to 78.5\% and 77.5\%). From these data, the fidelities of the two copies can be easily determined using Eq.~\ref{fid} and are found to be 64.3$\pm 1\%$ (clone 1) and 65.2$\pm 1\%$ (clone 2), assuming unity cloning gains. These values clearly demonstrate successfull operation of our cloning machine since they significantly surpass the maximum classical fidelity of 50\% and approach the optimal value of 2/3$\approx$66.7\%.
The performance of our system is limited solely by imperfections of the in-line feedforward loop, which include non-unity quantum efficiency of the diodes, electronic noise of the detector circuit and non-perfect interference contrast at the beam splitter BS2 in Fig.~2. The electronic noise was completely overcome by using newly designed ultra low noise detectors (with electronic noise 25dB below the shot noise level)~\cite{bruno}, and the detection efficiencies were maximized by optimizing the mode matching at the beam splitter (99\%) and by using high quantum efficiency photo diodes (95\%).
Based on these efficiencies we calculate an expected fidelity of 65\% which is in nice agreement with our experimental results. Note that despite of the imperfect detection system, the fidelity is still close to the optimum of 2/3, proving the robustness of the cloning scheme.
\begin{figure}
\caption{\it Spectral noise densities of the clones relative to the quantum noise level (black trace) recorded by the homodyne detectors both for the amplitude quadrature (red trace) and for the phase quadrature (blue trace). The added noise contributions are 3.28$\pm 0.13$~dB (3.16$\pm 0.13$~dB) and 3.20$\pm 0.11$~dB (3.15$\pm 0.13$~dB) in the amplitude quadrature and phase quadrature of clone 1(2). The optimal cloning limit as well as the classical limit are shown by solid lines.
The measurement frequency is 14.3~MHz, the sweep time 2 seconds, the resolution bandwidth 100~kHz and the video bandwidth 300~Hz. On the right hand side we plot the associated noise contours of the Wigner functions corresponding to the input state (green contour), the experimentally achieved clones (light blue contour) and the classical clones (dashed line).}
\label{fig4}
\end{figure}
In the discussion above we assumed unity gains. However, experimental imperfections lead to a small deviation from unity, and the gains were accurately determined to be $g_{x1}=0.96\pm0.01$ and $g_{p1}=1.00\pm0.01$ for clone 1 and $g_{x2}=1.03\pm0.01$ and $g_{p2}=1.03\pm0.01$ for clone 2 for the amplitude and phase quadratures, respectively.
As a result of the deviations from unity gain, the fidelity depends on the photon number of the input coherent state and the figure of merit for cloning is an average of the "single-shot" fidelities~\cite{cochrane04.pra}. E.g. considering a Gaussian distributed set of input coherent states with a spread in photon number of 50 (which is a huge number in quantum information science) the average fidelities equal 62.7\% and 63.3\%, which still significantly exceed the classical cloning boundary of 50.2\% for the same span of input states.
Despite the fact that the gains are not exactly unity, the obtained fidelities are far above the classical limits and approach the optimal limits for a large set of input states, demonstrating the suitability of this cloning machine for realistic experimental quantum information tasks.
In conclusion, we have proposed a simple Gaussian cloning protocol based on linear optics and homodyne detection which is optimal for coherent state inputs, and we have experimentally demonstrated the idea and obtained near optimal unconditional quantum cloning of coherent states.
Finally, let us stress that it is straightforward to extend the presented scheme for 1$\rightarrow$2 cloning (using linear optics) to a large variety of different copying functions such as optimal N$\rightarrow$M Gaussian cloning function which takes N originals and produces M clones~\cite{cerf00.pra,fiurasek01.prl,braunstein01.prl} and an asymmetric cloning function which produces output clones of different quality~\cite{fiurasek01.prl}, a procedure which is crucial in controlled partial information transfer between different parties in a network
We thank T.C. Ralph, R. Filip, N. Treps, W. Bowen, R. Schnabel and J. Sherson for stimulating discussions and B. Menegozzi for the construction of the photodetectors. This work has been supported by DFG (the Schwerpunkt programm 1078), the network of competence QIP (A8), and EU projects COVAQIAL (project no. FP6-511004) and SECOQC. ULA acknowledges an Alexander von Humboldt fellowship.
\end{document} | math | 19,380 |
\begin{document}
\title{Minimum Dominating Set for a Point Set in $\IR^2$}
\begin{abstract}
In this article, we consider the problem of computing minimum dominating set for a given set $S$
of $n$ points in $\mathbb{R}^2$. Here the objective is to find a minimum cardinality
subset $S'$ of $S$ such that the union of the unit radius disks centered at the points in $S'$ covers
all the points in $S$. We first propose a simple 4-factor and 3-factor approximation algorithms in $O(n^6 \log n)$
and $O(n^{11} \log n)$ time respectively improving time complexities by a factor of $O(n^2)$ and
$O(n^4)$ respectively over the best known result available in the literature
[M. De, G.K. Das, P. Carmi and S.C. Nandy, {\it Approximation
algorithms for a variant of discrete piercing set problem for unit disk}, Int. J. of Comp. Geom. and Appl.,
to appear]. Finally, we propose a very important shifting lemma, which is of independent
interest and using this lemma we propose a $\frac{5}{2}$-factor approximation algorithm and a PTAS for
the minimum dominating set problem.
\end{abstract}
{\bf Keywords:} minimum dominating set, unit disk graph, approximation algorithm.
\section{Introduction}
A minimum dominating set $S'$ for a set $S$ of $n$ points in $\mathbb{R}^2$ is defined as follows: (i) $S' \subseteq S$
(ii) each point $s \in S$ is covered by at least one unit radius disk centered at a point in $S'$, and (iii) size of
$S'$ is minimum. The {\it minimum dominating set} (MDS) problem for a point set $S$ of size $n$ in $\mathbb{R}^2$
involves finding a minimum dominating set $S'$ for the set $S$. We call this problem as a geometric version of
MDS problem. The MDS problem for a point set can be modeled
as an MDS problem in unit disk graph (UDG) as follows: A unit disk graph $G = (V,E)$ for a set ${\cal U}$ of $n$
unit diameter disks in $\mathbb{R}^2$ is the intersection graph of the family of disks in ${\cal U}$ i.e., the vertex
set $V$ corresponds to the set ${\cal U}$ and two vertices are connected by an edge if the corresponding disks have
common intersection. The minimum dominating set for the graph $G$ is a minimum size subset $V'$ of $V$ such
that for each of the vertex $v \in V$ is either in $V'$ or adjacent to to a node in $V'$ in $G$. Several people have done
research on MDS problem because of its wide applications such as wireless networking,
facility location problem, to name a few. Our interest in this problem arose from the following reason:
suppose in a city we have a set $S$ of $n$ important locations (houses, etc.); the objective is to provide
some emergency services (ambulance, fire station, etc.) to each of the locations in $S$ so that
each location is within a predefined distance of at least one service center. Note that positions of the
emergency service centers are from the predefined set $S$ of locations only.
\subsection{Related Work}
The MDS problem can be viewed as a general set cover problem, but it is an NP-hard problem \cite{GJ79,J82}
and not approximable within $c \log n$ for some constant $c$ unless P = NP \cite{RS97}. Therefore
$O(\log n)$-factor approximation algorithm is possible for MDS problem by applying the algorithm for
general set cover problem \cite{chvatal79}. Some exciting results for the geometric version of MDS problem
are available in the literature.
In the {\it discrete unit disk cover} (DUDC) problem, two sets $P$ and $Q$ of points in $\mathbb{R}^2$ are given, the
objective is to choose minimum number of unit disks $D'$ centered at the points in $Q$ such that
the union of the disks in $D'$ covers all the points in $P$. Johnson \cite{J82} proved that
the DUDC problem is NP-hard. Mustafa and Ray in 2010 \cite{MR10} proposed a $(1+\delta)$-approximation
algorithm for $0 < \delta \leq 2$ (PTAS) for the DUDC problem using $\epsilon$-net based local improvement approach.
The fastest algorithm is obtained by setting $\delta = 2$ for a 3-factor approximation algorithm, which runs
in $O(m^{65}n$) time, where $m$ and $n$ are number of unit radius disks and number of points respectively \cite{DFLN12}.
The high complexity of the PTAS leads to further research on constant factor
approximation algorithms for the DUDC problem. A series of constant factor approximation algorithms for DUDC
problem are available in the literature:
\begin{itemize}
\item 108-approximation algorithm [C\u{a}linescu et al., 2004 \cite{CMWZ04}]
\item 72-approximation algorithm [Narayanappa and Voytechovsky, 2006 \cite{NV06}]
\item 38-approximation algorithm in O($m^{2}n^{4}$) time [Carmi et al., 2007 \cite{CKL07}]
\item 22-approximation algorithm in O($m^{2}n^{4}$) time [Claude et al., 2010 \cite{CDDDFLNS10}]
\item 18-approximation algorithm in O($mn+n\log n+m\log m$) time [Das et al., 2012 \cite{DFLN12}]
\item 15-approximation algorithm in O($m^{6}n$) time [Fraser and L\'{o}pez-Ortiz, 2012 \cite{FL12}]
\item $(9+\epsilon)$-approximation algorithm in $O(m^{3(1+\frac{6}{\epsilon})} n \log n)$ time
[Acharyya et al., 2013 \cite{ABD13}]
\end{itemize}
The DUDC problem is a geometric version of MDS problem for $P=Q$. Therefore all results for the DUDC problem
are applicable to MDS problem.
The geometric version of MDS problem is known to be NP-hard \cite{CCJ90}.
Nieberg and Hurink \cite{NH06} proposed $(1+\epsilon)$-factor approximation algorithm
for $0 < \epsilon \leq 1$. The fastest algorithm is obtained by setting $\epsilon = 1$ for a
$2$-approximation result, which runs in $O(n^{81})$ time \cite{DDCN13}, which is not
practical even for $n = 2$. Another PTAS for dominating set of arbitrary size disk graph is available
in the literature proposed by Gibson and Pirwani \cite{GP10}. The running time of this PTAS is
$n^{O(\frac{1}{\epsilon^2})}$.
Marathe et al. \cite{MBIRR95} proposed a 5-factor approximation algorithm for the MDS problem.
Amb{\"u}hl et al. \cite{AEMN06} proposed 72-factor approximation algorithm for weighted dominating
set (WDS) problem. In the WDS problem, each node has a positive weight and the objective is to find
the minimum weight dominating set of the nodes in the graph. Huang et al. \cite{HGZW08}, Dai and Yu
\cite{DY09}, and Zou et al. \cite{ZWXLDWW11} improved the approximation factor for WDS problem to
$6+\epsilon$, $5+\epsilon$, and $4+\epsilon$ respectively. First, they proposed $\gamma$-factor
($\gamma = 6, 5, 4$ in \cite{HGZW08}, \cite{DY09}, and \cite{ZWXLDWW11} respectively) approximation
algorithm for a subproblem and using the result of their corresponding sub-problems they proposed
$(\gamma+\epsilon)$-factor approximation algorithms. The time complexity of their algorithms
are $O(\alpha(n) \times \beta(n))$, where $O(\alpha(n))$ is the time complexity of the algorithm
for the sub-problem and $O(\beta(n)) = O(n^{4 (\lceil\frac{84}{\epsilon} \rceil)^2})$ is the number
of times the sub-problem needs to be invoked to solve the original problem. The $(\gamma + 1)$-factor
approximation algorithm can be obtained by setting $\epsilon = 1$, but the time complexity becomes a
very high degree polynomial function in $n$. Carmi et al. \cite{CKL08} proposed a 5-factor approximation
algorithm of the MDS problem for arbitrary size disk graph. Fonseca et al. \cite{FFSM12} proposed a
$\frac{44}{9}$-factor approximation algorithm for the MDS problem in UDG which can be achieved in $O(n+m)$
time, when the input is a graph with $n$ vertices and $m$ edges, and in $O(n \log n)$ time, in the geometric
version of the problem. The same set of authors also proposed a $\frac{43}{9}$-factor approximation algorithm
for the MDS problem in UDG which runs in $O(n^2 m)$ time \cite{FFSM12-1}. Recently, De at al. \cite{DDCN13}
considered the geometric version of MDS problem and proposed 12-factor, 4-factor, and 3-factor approximation
algorithms with running time $O(n \log n)$, $O(n^8 \log n)$, and $O(n^{15}\log n)$ respectively. They also
proposed a PTAS with high degree polynomial running time.
\subsection{Our Contribution}
In this paper, we consider the geometric version of MDS problem and propose a series of constant factor
approximation algorithms. We first propose 4-factor and 3-factor approximation algorithms with running time
$O(n^6 \log n)$ and $O(n^{11}\log n)$ respectively improving the time complexities by a factor of $O(n^2)$ and
$O(n^4)$ respectively over the best known result in the literature \cite{DDCN13}. Finally, we propose a new
shifting strategy lemma. Using our shifting strategy lemma we propose $\frac{5}{2}$-factor and
$(1+\frac{1}{k})^2$-factor (i.e., PTAS) approximation algorithms for the MDS problem. The running time of
proposed $\frac{5}{2}$-factor and $(1+\frac{1}{k})^2$-factor approximation algorithms are $O(n^{20} \log n)$
and $n^{O(k)}$ respectively. Though the time complexity of the proposed PTAS is same as the PTAS proposed by
De et al. \cite{DDCN13} in terms of $O$ notation, but the constant involved in our PTAS is smaller than the
same in \cite{DDCN13}.
\section{4-Factor Approximation Algorithm for the MDS Problem}\label{4factor}
In this section, a set $S$ of $n$ points in $\mathbb{R}^2$ is given inside a rectangular region ${\cal R}$. The
objective is to find an MDS for $S$. Here we propose a simple 4-factor approximation algorithm. The running time
of our algorithm is $O(n^6 \log n)$, which is an improvement by a factor of $O(n^2)$ over the best known existing
result \cite{DDCN13}. In order to obtain a 4-factor approximation algorithm, we consider a partition of
${\cal R}$ into regular hexagons of side length $\frac{1}{2}$ (see Figure \ref{figure-2}(a)). We use {\it cell}
to denote a regular hexagon of side length $\frac{1}{2}$.
\begin{lemma} \label{lemma-1x}
All points inside a single cell can be covered by an unit radius disk centered at any point inside that cell.
\end{lemma}
\begin{proof}
The lemma follows from the fact that the distance between any two points inside a regular hexagon of side length
$\frac{1}{2}$ is at most 1 (for demonstration see the Figure \ref{figure-2}(b)).
\end{proof}
\begin{figure}
\caption{(a) Regular hexagonal partition (b) single regular hexagon of side length $\frac{1}
\label{figure-2}
\end{figure}
\begin{definition}
A {\it septa-hexagon} is a combination of 7 adjacent cells such that one cell is inscribed by six other
cells as shown in Figure \ref{figure-2}(c).
For a point set $U$, we use $\Delta(U)$ to denote the set of unit radius disks centered at the points in $U$.
Let $U_1$ and $U_2$ be two point sets such that $U_1 \subseteq U_2$. We use $\chi(U_1, U_2)$ to denote the set
of points such that $\chi(U_1, U_2) \subseteq U_2$ and an unit radius disk centered at any point in
$\chi(U_1, U_2)$ covers at least one point of $U_1$.
\end{definition}
\subsection{Algorithm overview}
Let us consider a septa-hexagon ${\cal C}$. Recall that ${\cal C}$ is a combination of 7 cells (regular
hexagon of side length $\frac{1}{2}$). Let $S_1 = S \cap {\cal C}$ and $S_2 = \chi(S_1, S)$. For the
4-factor approximation algorithm, we first find minimum size subset $S' \subseteq S_2$ such that
$S_1 \subseteq \bigcup_{d \in \Delta(S')}d$. Call this problem as {\it single septa-hexagon MDS} problem.
Using the optimum (minimum size) solution of single septa-hexagon MDS problem, we present our main 4-factor
approximation algorithm. The Lemma \ref{lemma-2x} gives an important feature to design optimum algorithm for
single septa-hexagon MDS problem.
\begin{lemma} \label{lemma-2x}
If $OPT_{\cal C}$is a minimum cardinality subset of $S_2$ such that
$S_1 \subseteq \bigcup_{d \in \Delta(OPT_{\cal C})}d$, then $|OPT_{\cal C}| \leq 7$.
\end{lemma}
\begin{proof}
The septa-hexagon ${\cal C}$ has at most 7 non-empty cells. From Lemma \ref{lemma-1x}, we know that
an unit radius disk centered at a point in a cell covers all points in that cell. Therefore one point
from each of the non-empty cells is sufficient to cover all the points in ${\cal C}$. Thus the
Lemma follows.
\end{proof}
\begin{algorithm}[!ht]
\caption{Algorithm\_4\_Factor($S, {\cal C}, n$)}
\begin{algorithmic}[1]
\STATE {\bf Input:} A set $S$ of $n$ points and a septa-hexagon ${\cal C}$
\STATE {\bf Output:} A set $S' (\subseteq S)$ such that $(S \cap {\cal C}) \subseteq \bigcup_{d \in \Delta(S')}d$.
\STATE $S' \leftarrow \emptyset$
\IF{($S \cap {\cal C} \neq \emptyset$)}
\STATE Choose one arbitrary point from each non-empty cell of ${\cal C}$ and add to $S'$.
\STATE $m \leftarrow |S'|$ /* $m$ is at most 7 */
\STATE Let $S_1 = S \cap {\cal C}$ and $S_2 = \chi(S_1, S)$.
\FOR{($i = m-1, m-2, \ldots, 1$)}
\IF{($i=6$)}
\FOR {(Each possible combination of 5 points $X = \{p_1, p_2, \ldots, p_5\}$ of $S_2$)}
\STATE Find $Y \subseteq S_1$ such that no point in $Y$ is covered by $\bigcup_{d \in \Delta(X)}d$.
\STATE Compute the farthest point Voronoi diagram of $Y$ \cite{BCKO08}
\STATE Find a point $p$ (if any) from $S_2\setminus X$ (using planar point location algorithm \cite{PS09})
such that the farthest point in $Y$ from $p$ is less than or equal to 1. If such $p$ exists, then
set $S' \leftarrow X \cup \{p\}$ and exit {\bf for} loop.
\ENDFOR
\ELSE
\FOR {(Each possible combination of $i$ points $X = \{p_1, p_2, \ldots, p_i\}$ of $S_2$)}
\IF{($S_1 \subseteq \bigcup_{d \in \Delta(X)}d$)}
\STATE Set $S' \leftarrow X$ and exit from {\bf for} loop
\ENDIF
\ENDFOR
\ENDIF
\ENDFOR
\ENDIF
\STATE Return $S'$
\end{algorithmic}
\label{algo-4factor}
\end{algorithm}
\begin{lemma} \label{lemma-3x}
For a given set $S$ of $n$ points and a septa-hexagon ${\cal C}$, the Algorithm \ref{algo-4factor} computes an
MDS for $S \cap {\cal C}$ using the points of $S$ in $O(n^6 \log n)$ time.
\end{lemma}
\begin{proof}
The optimality of the Algorithm \ref{algo-4factor} follows from the fact that Algorithm \ref{algo-4factor}
considers all possible set of sizes $0, 1, \ldots, 7$ (see Lemma \ref{lemma-2x}) as its solution and
reports minimum size solution.
The line number 7 of the algorithm can be computed in $O(n \log n)$ time as follows: (i) computation of the set
$S_1$ takes $O(n)$ time, (ii) computation of $S_2$ can be done in $O(n \log n)$ time using nearest
point Voronoi diagram of $S_1$ in $O(n \log n)$ time and for each point $p \in S$ apply planar point
location algorithm to find the nearest point in $S_1$ in $O(\log n)$ time.
The running time of the {\bf else} part in the line number 15 of the algorithm is at most $O(n^6)$ time.
The worst case running time of the algorithm comes from line numbers 9-14. The complexity of line numbers 11-13
is $O(n \log n)$ time. Therefore the running time of the line numbers 9-14 is $O(n^6 \log n)$ time.
Thus the overall worst case running time of the proposed Algorithm \ref{algo-4factor} is $O(n^6 \log n)$.
\end{proof}
Let us consider a septa-hexagonal partition of ${\cal R}$ such that no point of $S$ is on the boundary of
any septa-hexagon and a 4 coloring scheme of it (see Figure \ref{fig:fig10}). Consider an unicolor
septa-hexagon of color A (say). Its adjacent septa-hexagons are assigned colors B, C and D (say) such
that opposite septa-hexagons are assigned the same color (see Figure \ref{fig:fig10}).
\begin{figure}
\caption{A septa-hexagonal partition and 4-coloring scheme}
\label{fig:fig10}
\end{figure}
\begin{lemma} \label{lemma-4x}
If ${\cal C}'$ and ${\cal C}''$ are two same colored septa-hexagons, then
$({\cal C}' \cup {\cal C}'') \cap S \cap d = \emptyset$ for any unit radius disk $d$.
\end{lemma}
\begin{proof}
According to the 4-coloring scheme, size of the septa-hexagons, and no point of $S$ is on the boundary
of ${\cal C}'$ and ${\cal C}''$ the minimum distance between two points $s_1 \in {\cal C}' \cap S$
and $s_2 \in {\cal C}''\cap S$) is greater than 2 (see Figure \ref{fig:fig10}). Thus the lemma follows.
\end{proof}
\begin{theorem} \label{theorem-1y}
The 4-coloring scheme gives a 4-factor approximation algorithm for the MDS problem in $O(n^{6}\log n)$ time,
where $n$ is the input size.
\end{theorem}
\begin{proof}
Let $N_1, N_2, N_3$, and $N_4$ be the sets of septa-hexagons of colors $A, B, C$, and $D$ respectively.
Let $S_1^i = S \cap \bigcup_{{\cal C} \in N_i} {\cal C}$ and $S_2^i = \chi(S_1^i, S)$
for $1 \leq i \leq 4$. By Lemma
\ref{lemma-4x}, the pair ($S_1^i, S_2^i$) can be partitioned into $|N_i|$ pairs ($S_{1j}^i, S_{2j}^i$)
such that for each pair Algorithm \ref{algo-4factor} is applicable for solving the covering problem optimally
to cover $S_1^i$ using $S_2^i$, where $1 \leq j \leq |N_i|$. Let $N_i'$ be the optimum solution for the set
$S_1^i$ ($1 \leq i \leq 4$) using the Algorithm \ref{algo-4factor}. If $OPT$ is the optimum solution for the
set $S$, then $|N_i'| \leq |OPT|$. Therefore $\Sigma_{i=1}^4 |N_i'| \leq 4 \times |OPT|$. Thus the
approximation factor of the algorithm follows.
The time complexity result of the theorem follows from Lemma \ref{lemma-3x} and the fact that each point in
$S$ can participate in the Algorithm \ref{algo-4factor} at most constant number of times.
\end{proof}
\section{3-Factor Approximation Algorithm for the MDS Problem}\label{3factor}
Given a set $S$ of $n$ points in a rectangular region ${\cal R}$, we wish to find an MDS for $S$.
Here we present a 3-factor approximation algorithm in $O(n^{11} \log n)$ time for the MDS
problem, which is an improvement by a factor of $O(n^4)$ over the best known result available in
the literature \cite{DDCN13}.
\begin{definition}
A {\it super-cell} is a combination of 15 regular hexagons of side length $\frac{1}{2}$ arranged
in three consecutive rows as shown in Figure \ref{fig:fig14}.
\end{definition}
\begin{figure}
\caption{An example of a super-cell}
\label{fig:fig14}
\end{figure}
\subsection{Algorithm overview}
Let us consider a {\it super-cell} ${\cal D}$. Let $S_1 = S \cap {\cal D}$ and $S_2 = \chi(S_1, S)$.
In order to obtain 3-factor approximation algorithm for the MDS problem, we first find a minimum
size subset $S' \subseteq S_2$ such that $S_1 \subseteq \bigcup_{d \in \Delta(S')}d$. Call this problem
as a {\it single super-cell MDS} problem. Using the optimum solution of single super-cell MDS problem, we
present our main 3-factor approximation algorithm.
\begin{lemma} \label{lemma-5x}
If $OPT_{\cal D}$ is the minimum cardinality subset of $S_2$ such that
$S_1 \subseteq \bigcup_{d \in \Delta(OPT_{\cal D})} d$, then $|OPT_{\cal D}| \leq 15$.
\end{lemma}
\begin{proof}
The lemma follows from the Lemma \ref{lemma-1x} and the fact that the super-cell ${\cal D}$
has at most 15 non-empty cells.
\end{proof}
We decompose a super-cell ${\cal D}$ into 3 regions namely $G_{\cal D}^1, G_{\cal D}^2$, and $G_{\cal D}^3$
(see Figure \ref{fig:fig16}, where $G_{\cal D}^1, G_{\cal D}^2$, and $G_{\cal D}^3$ correspond to unshaded,
light shaded, and dark shaded regions respectively).
\begin{figure}
\caption{Decomposition of a super-cell}
\label{fig:fig16}
\end{figure}
\begin{lemma} \label{lemma-6x}
For any unit radius disk $d$ and a super-cell ${\cal D}$, $(G_{\cal D}^1 \cup G_{\cal D}^3) \cap d = \emptyset$.
\end{lemma}
\begin{proof}
The lemma follows from the fact that if $s$ and $t$ are two arbitrary points of $G_{\cal D}^1$ and $G_{\cal D}^3$
respectively, then the Euclidean distance between $s$ and $t$ is greater than 2.
\end{proof}
Let $S_1 = S \cap {\cal D}$ and $S_2 = \chi(S_1, S)$, where ${\cal D}$ is a super-cell. Our objective
is to find a minimum cardinality set $S' (\subseteq S_2)$ such that $S_1 \subseteq \bigcup_{d \in \Delta(S')}d$.
Let $S_1^1 = S_1 \cap G_{\cal D}^1$, $S_1^2 = S_1 \cap G_{\cal D}^2$, and $S_1^3 = S_1 \cap G_{\cal D}^3$.
A point on a boundary can be assigned to any set associated with that boundary. Let $S_2^1 = \chi(S_1^1, S_2)$,
$S_2^2 = \chi(S_1^2, S_2)$, and $S_2^3 = \chi(S_1^3, S_2)$. The Lemma \ref{lemma-6x} says that
$S_2^1 \cap S_2^3 = \emptyset$.
\begin{algorithm}
\caption{Algorithm\_3\_Factor($S, {\cal D}, n$)}
\begin{algorithmic}[1]
\STATE {\bf Input:} A set $S$ of $n$ points and a super-cell ${\cal D}$
\STATE {\bf Output:} A set $S' (\subseteq S)$ such that $(S \cap {\cal D}) \subseteq \bigcup_{d \in \Delta(S')}d$
\STATE $S' \leftarrow S$.
\STATE Find the sets $S_1^1, S_1^2,S_1^3, S_2^1, S_2^2$,and $S_2^3$ as defined above.
\FOR {(Each possible combination $X = \{p_1, p_2, \ldots, p_j\}$ of $j (0 \leq j \leq 9)$ points in $S_2^2$)}
\IF {($S_1^2 \subseteq \bigcup_{d \in \Delta(X)}d$)}
\STATE Let $U$ and $V$ be the subsets of $S_1^1$ and $S_1^3$ respectively such that no point in
$U \cup V$ is covered by $\bigcup_{d \in \Delta(X)}d$.
\STATE Let $Y$ be the minimum size subset of $S_2^1$ such that $U \subseteq \bigcup_{d \in \Delta(Y)}d$.
\STATE Let $Z$ be the minimum size subset of $S_2^3$ such that $V \subseteq \bigcup_{d \in \Delta(Z)}d$.
\IF{($|S'| > |X| + |Y| + |Z|$)}
\STATE Set $S' \leftarrow X \cup Y \cup Z$
\ENDIF
\ENDIF
\ENDFOR
\STATE Return $S'$
\end{algorithmic}
\label{algo-3factor}
\end{algorithm}
\begin{lemma} \label{lemma-7x}
For a given set $S$ of $n$ points and a super-cell ${\cal D}$, the Algorithm \ref{algo-3factor} computes an
MDS for $S \cap {\cal D}$ using the points of $S$ in $O(n^{11} \log n)$ time.
\end{lemma}
\begin{proof}
In the case of selecting 3 points in $S_2^1$ in line number 8 of the algorithm, we can choose one point from
each of the non-empty cells of $G_{\cal D}^1$. Therefore, the worst case of line number 8 appears for the
case of choosing all possible combinations of two points in $S_2^1$. This can be done in $O(n^2 \log n)$ using
the technique of the Algorithm \ref{algo-4factor} (line numbers 12-13). Similar analysis is applicable to line
number 9. Line numbers 6-7 and 10-12 can be implemented in $O(n)$ time.
The worst case running time of the algorithm depends on the {\bf for} loop in the line number 5. In this
{\bf for} loop, we are choosing all possible 9 points from a set of $n$ points in worst case. Therefore the
time complexity of the Algorithm \ref{algo-3factor} is $O(n^{11} \log n)$.
The optimality of the algorithm follows from the Lemma \ref{lemma-6x} and fact that Algorithm \ref{algo-3factor}
considers all possible combinations as its solution and reports minimum size solution.
Note that Algorithm \ref{algo-3factor} checks {\bf if} condition in line number 6 because of the definition
of $S_2^1, S_2^2$, and $S_2^3$.
\end{proof}
Let us consider a super-cell partition of ${\cal R}$ such that no point of $S$ lies on the boundary and a
3-coloring scheme (see Figure \ref{fig:fig15}). Consider an unicolor super-cell which has been assigned
color A (say). Its adjacent super-cells are assigned colors B, and C alternately (see Figure~\ref{fig:fig15}).
\begin{figure}
\caption{A super-cell partition and 3-coloring scheme}
\label{fig:fig15}
\end{figure}
\begin{lemma} \label{lemma-8x}
If ${\cal D}'$ and ${\cal D}''$ are two same colored super-cells, then
$({\cal D}' \cup {\cal D}'') \cap S \cap d = \emptyset$ for any unit radius disk $d$.
\end{lemma}
\begin{proof}
The lemma follows from the following facts: (i) size of the super-cells ${\cal D}'$ and ${\cal D}''$
(ii) no point of $S$ on the boundary of ${\cal D}'$ and ${\cal D}''$, and (iii) the 3-coloring scheme.
\end{proof}
\begin{theorem} \label{theorem-2y}
The 3-coloring scheme gives a 3-factor approximation algorithm for the MDS problem in
$O(n^{11} \log n)$ time, where $n$ is the input size.
\end{theorem}
\begin{proof}
The follows by the similar argument of Theorem \ref{theorem-1y}.
\end{proof}
\section{Shifting Strategy and its Application to the MDS Problem} \label{ShiftingStrategy}
In this section, we first propose a shifting strategy for the MDS problem, which is
a generalization of the shifting strategy proposed by Hochbaum and Maass \cite{HM85}. Next we propose
$\frac{5}{2}$-factor approximation algorithm and a PTAS algorithm for MDS problem using our shifting strategy.
\subsection{The Shifting Strategy} \label{shifting-strategy}
Our shifting strategy is very similar to the shifting strategy in \cite{HM85}. We include a brief
discussion here for completeness. Let a set $S$ of $n$ points be distributed inside an axis aligned
rectangular region ${\cal R}$. Our objective is to find an MDS for $S$.
\begin{definition}
A {\it monotone chain} $c$ with respect to line $L$ is a chain of line segments
such that any line perpendicular to $L$ intersect it only once. We define the distance between two
monotone chains $c'$ and $c''$ as the minimum Euclidean distance between any two points $p'$ and $p''$
on the chains $c'$ and $c''$ respectively. A {\it monotone strip} denoted by $M_s$ and is defined by the
area bounded by any two monotone chains $c'$ and $c''$ such that the area is left closed and
right open.
\end{definition}
Consider a set $c_1, c_2, \ldots, c_r$ of $r$ monotone chains with respect to the line parallel to $y$-axis
from left to right dividing the region ${\cal R}$ such that distance between each pair of monotone chains is
at least $D (>0)$, where $c_1$ and $c_r$ are the left and right boundary of ${\cal R}$ respectively
(see Figure \ref{shifting}). Let ${\cal A}$ be an $\alpha$-factor approximation algorithm, which provides a
solution of any $\ell$ consecutive monotone strips for the MDS problem.
\begin{figure*}
\caption{Demonstration of shifting strategy}
\label{shifting}
\end{figure*}
\begin{theorem} \label{shift-factor}
We can design an $\alpha(1+\frac{1}{\ell})$-factor approximation algorithm for finding an MDS for $S$.
\end{theorem}
\begin{proof}
The algorithm is exactly same as the algorithm proposed by Hochbaum and Maass \cite{HM85}. The approximation factor
follows from exactly the same argument proved in the shifting lemma \cite{HM85}.
\end{proof}
\subsection{$\frac{5}{2}$-Factor Approximation Algorithm for the MDS Problem}
Here we propose a $\frac{5}{2}$-factor approximation algorithm for MDS problem for a given
set $S$ of $n$ points in $\mathbb{R}^2$ using shifting strategy discussed in Subsection \ref{shifting-strategy}.
\begin{definition}
A {\it duper-cell} is a combination of 30 cells (regular hexagon of side length $\frac{1}{2}$) as shown in
Figure \ref{figure-8}. A duper-cell ${\cal E}$ generates four monotone chains with respect to
vertical and horizontal lines along its boundary. See Figure \ref{figure-8}, where $uv, vw, wx$, and $xu$
are the monotone chains. We rename them as {\bf left, bottom, right}, and {\bf top} monotone chains.
\end{definition}
The basic idea is as follows: first optimally solve the subproblem {\it duper-cell} i.e.,
find an MDS for the set $S \cap {\cal E}$, where ${\cal E}$ is a duper-cell and then apply shifting
strategy in both horizontal and vertical directions separately. The Lemma \ref{lemma-1x} leads
to restriction on the size of the MDS, which is at most 30. Therefore an easy optimum solution
for MDS can be obtained in $O(n^{30})$ time. Here we propose a different technique for the MDS problem
leading to lower time complexity as follows:
\begin{figure*}
\caption{Demonstration of $\frac{5}
\label{figure-8}
\end{figure*}
We divide the duper-cell ${\cal E}$ into 2 groups unshaded region ($U_R$) and shaded region ($S_R$) as shown in
Figure \ref{figure-8}. Let $\mu$ be the common boundary of the regions and two extended lines
(see Figure \ref{figure-8}). Let $Q_1$ and $Q_2$ be two sets of points in the left (resp. right) of $\mu$ such
that each disk in $\Delta(Q_1)$ and $\Delta(Q_2)$ intersects $\mu$.
\begin{algorithm}[!ht]
\caption{MDS\_for\_duper-cell($S, {\cal E}, n$)}
\begin{algorithmic}[1]
\STATE {\bf Input:} A set $S$ of $n$ points and a duper-cell ${\cal E}$.
\STATE {\bf Output:} A set $S' (\subseteq S)$ for an MDS of $S \cap {\cal E}$.
\STATE Find $Q_1$ and $Q_2$ as described above.
\STATE Let $S_2^L$ and $S_2^R$ be the set of points in $S\setminus (Q_1 \cup Q_2)$ such that each
disk in $\Delta(S_2^L)$ and $\Delta(S_2^R)$ covers at least one point in $S \cap U_R$ and
$S \cap S_R$ respectively.
\STATE $S' \leftarrow \emptyset, X \leftarrow \emptyset$
\FOR {($i=0, 1, \ldots, 9$)}
\STATE choose all possible $i$ disks in $\Delta(Q_1)$ (resp. $\Delta(Q_2)$) and for each combination
of $i$ disks find $S_1^L$ and $S_1^R$ such that $S_1^L \subseteq (S \cap U_R)$ and uncovered by that $i$ disks,
and $S_1^R \subseteq (S \cap S_R)$ and uncovered by that $i$ disks.
\STATE Call Algorithm \ref{algo-3factor} for finding an MDS for the sets $S_1^L$ and $S_1^R$ separately.
\ENDFOR
\STATE Return $S'$
\end{algorithmic}
\label{algo-shifting}
\end{algorithm}
\begin{lemma} \label{lemma-9x}
An MDS for the set of points inside a duper-cell ${\cal E}$ can be computed optimally
in $O(n^{20} \log n)$ time, where $n$ is the input size.
\end{lemma}
\begin{proof}
The time complexity of line number 8 of the Algorithm \ref{algo-shifting} is $O(n^{11} \log n)$
(see Lemma \ref{lemma-7x}). The line number 8 executes at most $O(n^9)$ time by the {\bf for} loop
in line number 6. Therefore the time complexity of the lemma follows.
In the {\bf for} loop (line number 6 of the algorithm), we considered
all possible $i$ ($0 \leq i \leq 9$) disks in $\Delta(Q_1)$ and $\Delta(Q_2)$ separately. Since
the number of cells that can intersect with such $i$ disks is at most 9, therefore the range of $i$ is correct.
For each combination of $i$ disks, we considered all possible combinations to solve the problem
for $S_1^L$ and $S_1^R$ separately. Therefore the correctness of the algorithm follows.
\end{proof}
\begin{theorem} \label{theorem-3y}
The shifting strategy discussed in Subsection \ref{shifting-strategy} gives a $\frac{5}{2}$-factor
approximation algorithm, which runs in $O(n^{20} \log n)$ time for the MDS problem, where $n$ is the
input size.
\end{theorem}
\begin{proof}
The distance between the monotone chains {\bf left} and {\bf right} of ${\cal E}$ is greater than 8,
the distance between the monotone chains {\bf bottom} and {\bf top} is 2, and the diameter $(D)$ of the
disks is 2. Now, if we apply shifting
strategy in horizontal and vertical directions separately, then we get $(1+\frac{1}{4})(1+\frac{1}{1})$-factor
i.e. $\frac{5}{2}$-factor approximation algorithm in $O(n^{20} \log n)$ time (see Lemma \ref{lemma-9x})
for the MDS problem.
\end{proof}
\subsection{A PTAS for MDS Problem}
In this section, we present a $(1+\frac{1}{k})^2$-factor approximation algorithm in $n^{O(k)}$ time for
a positive integer $k$. Suppose a set $S$ of $n$ points within a rectangular region ${\cal R}$ is given.
Consider a partition of ${\cal R}$ into regular hexagonal cells of side length $\frac{1}{2}$. The idea
of our algorithm is to solve the MDS problem optimally for the points inside regular hexagons (say ${\cal F}$)
such that the distance between {\bf left} and {\bf right} (resp. {\bf bottom} and {\bf top}) monotone chains
is $2k$ (see Figure \ref{figure-9}) and using our proposed shifting strategy carefully
(see Subsection \ref{shifting-strategy}).
\begin{figure*}
\caption{Demonstration of PTAS}
\label{figure-9}
\end{figure*}
To solve the MDS problem in $S \cap {\cal F}$ we further decompose ${\cal F}$ into four parts using
the monotone chains $L_1$ and $L_2$ as shown in Figure \ref{figure-9}. The number of disks in the
optimum solution intersecting the chain $L_1$ with centers {\bf left} (resp. {\bf right}) side of
$L_1$ is at most $\lceil 3 \times 3 \times \frac{2k}{2} \rceil$ which is less than $10k$ and the
number of disks in the optimum solution intersecting the chain $L_2$ with centers {\bf bottom}
(resp. {\bf top}) side of $L_2$ is at most $\lceil 5 \times 3 \times \frac{2k}{4} \rceil$ which
is less than $8k$. Next we apply recursive procedure to solve four independent sub-problems of
size $k \times k$. If $T(n,2k)$ is the running time of the recursive algorithm for the MDS problem
for $S \cap {\cal F}$, then using the technique of \cite{DDCN13} we have the following recurrence
relation: $T(n, 2k) = 4 \times T(n, k) \times n ^{10k + 8k}$, which leads to the following theorem.
\begin{theorem}\label{theorem-4y}
For a given set $S$ of $n$ points in $\mathbb{R}^2$, the proposed algorithm produces an MDS of
$S$ in $n^{O(k)}$ time, whose size is at most $(1+\frac{1}{k})^2 \times |OPT|$, where $k$ is a
positive integer and $OPT$ is the optimum solution.
\end{theorem}
\section{Conclusion}
In this paper, we proposed a series of constant factor approximation algorithms for the MDS
problem for a given set $S$ of $n$ points. Here we used hexagonal partition very carefully.
We first presented a simple 4-factor and 3-factor approximation algorithms in $O(n^6 \log n)$
and $O(n^{11} \log n)$ time respectively, which improved the time complexities of best known
result by a factor of $O(n^2)$ and $O(n^4)$ respectively \cite{DDCN13}. Finally, we proposed a
very important shifting lemma and using this lemma we presented a $\frac{5}{2}$-factor approximation
algorithm and a PTAS for the MDS problem. Though the complexity of the proposed PTAS is same as that
of the PTAS proposed by De et al. \cite{DDCN13} in terms of $O$ notation, but the constant involved
in our PTAS is smaller than the same in \cite{DDCN13}.
\end{document} | math | 33,478 |
\begin{document}
\title{Optomechanical simulation of a parametric oscillator}
\author{F.~E. Onah}
\email[e-mail: ]{[email protected]}
\affiliation{Tecnol\'ogico de Monterrey, Escuela de Ingenier\'ia y Ciencias, Ave. Eugenio Garza Sada 2501, Monterrey, N.L., Mexico, 64849}
\affiliation{The Division of Theoretical Physics, Physics and Astronomy, University of Nigeria Nsukka, Nsukka Campus, Enugu State, Nigeria}
\author{C. Ventura-Vel\'azquez}
\email[e-mail: ]{[email protected]}
\affiliation{Deparment of Optics, Palack\'y University, 17.Listopadu 12, 77146 Olomouc, Czech Republic}
\author{F.~H. Maldonado-Villamizar}
\email[e-mail: ]{[email protected]}
\affiliation{CONACYT - Instituto Nacional de Astrof\'isica, \'Optica y Electr\'onica, Calle Luis Enrique Erro No. 1. Sta. Ma. Tonantzintla, Pue. C.P. 72840, Mexico}
\author{B.~R. Jaramillo-\'Avila}
\email[e-mail: ]{[email protected]}
\affiliation{CONACYT - Instituto Nacional de Astrof\'isica, \'Optica y Electr\'onica, Calle Luis Enrique Erro No. 1. Sta. Ma. Tonantzintla, Pue. C.P. 72840, Mexico}
\author{B.~M. Rodr\'iguez-Lara}
\email[e-mail: ]{[email protected]}
\affiliation{Tecnol\'ogico de Monterrey, Escuela de Ingenier\'ia y Ciencias, Ave. Eugenio Garza Sada 2501, Monterrey, N.L., Mexico, 64849}
\date{\today}
\begin{abstract}
We study an optomechhanical device supporting at least three optical modes in the infrared telecommunication band and three mechanical vibration modes.
We model the coherent driving of each optical mode, independently of each other, to obtain an effective Hamiltonian showing the different types of parametric processes allowed in the device.
We propose a bichromatic driving scheme, in the lossy optical cavity regime, under a mean field approximation, that provides the quantum simulation of a parametric oscillator with optical control of its parameters.
\end{abstract}
\maketitle
\section{Introduction}
Cavity optomechanics explores the interaction between optical and mechanical excitation modes in the quantum regime \cite{Aspelmeyer2014}.
In an optomechanical device, the interaction arises from the radiation pressure that optical modes exert on a material structure and, in consequence, its mechanical vibration modes.
The structure is typically a solid and its quantum control is in itself a milestone in physics research \cite{Aspelmeyer2014}.
Further motivation for research in the field comes from quantum technologies, where photon-to-photon transducers \cite{Guha2021,Balram2022} and quantum memories \cite{Wallucks2020} are necessary for quantum communication and information processing, and quantum sensing, where optomechanical devices take the role of high precision sensors \cite{Qiao2018,Li2021}, for example.
There exists a plethora of experimental platforms to realize cavity optomechanics; for example, suspended micromirrors \cite{Groeblacher2009}, microtoroids \cite{Riviere2011}, microsphere resonators \cite{Ma2007}, micromechanical membranes in a superconducting microwave cavity \cite{Teufel2011}, one-dimensional photonic crystal nanobeams \cite{Chan2011}, and levitated nanoparticles \cite{Delic2020}.
These devices vary in size and shape, covering a wide range of frequencies and coupling strength values.
However, they are accurately modeled by a relatively simple Hamiltonian,
\begin{eqnarray}
\frac{\hat{H}}{\hbar} = \omega \hat{a}^{\dagger} \hat{a} + \omega_{m} \hat{b}^{\dagger} \hat{b} - g_{0} \hat{a}^{\dagger} \hat{a} \left( \hat{b}^{\dagger} + \hat{b} \right) + \Omega \cos \left( \omega_{d} t\right) \left( \hat{a}^{\dagger} + \hat{a} \right),
\end{eqnarray}
describing a single mode of an optical oscillator, frequency $\omega$ and creation (annihilation) operators $\hat{a}^{\dagger}$ ($\hat{a}$), interacting with a single mode of a mechanical oscillator, frequency $\omega_{m}$ and creation (annihilation) operators $\hat{b}^{\dagger}$ ($\hat{b}$), with optomechanical coupling strength $g_{0}$, and external driving of the optical mode with strength $\Omega$ and frequency $\omega_{d}$.
For the sake of simplicity, we assume all parameters real.
This Hamiltonian model allows the description of experimentally observed optomechanical effects such as strong coupling \cite{Groeblacher2009}, mechanical squeezing \cite{Wollman2015}, entanglement \cite{Palomaki2013,Riedinger2018}, nonreciprocal behaviour \cite{Bernier2017,Barzanjeh2017}, and non-clasical mechanical states \cite{Riedinger2016,Hong2017}, among others \cite{Weis2010,Lecocq2015,Marinkovic2018}.
From a theoretical point of view, it is possible to recast the standard driven optomechanical model into an effective Hamiltonian,
\begin{eqnarray}
\frac{\hat{H}_{\mathrm{eff}}}{\hbar} &=& - \frac{g_{0}^{2}}{\omega_{m }} \left( \hat{a}^{\dagger} \hat{a} \right)^{2} + \frac{\Omega}{2} e^{- \frac{\alpha^{2}}{2}} \left\{ \hat{a}^{\dagger} \left[ \sum_{p=0}^{\infty}\frac{1}{p!} (-\alpha \hat{b}^{\dagger})^{p} ~_{1}F_{1}(- \hat{b}^{\dagger} \hat{b}; p+1; \alpha^{2} ) e^{i ( \Delta + p \omega_{m } ) t } + \right. \right. \nonumber \\
&& \left. \left. +\sum_{p=1}^{\infty}\frac{1}{p!} ~_{1}F_{1}(- \hat{b}^{\dagger} \hat{b}; p+1; \alpha^{2} ) (\alpha \hat{b})^{p} e^{i ( \Delta - p \omega_{m } ) t } \right] + \hat{a} \left[ \sum_{p=0}^{\infty}\frac{1}{p!} ~_{1}F_{1}(- \hat{b}^{\dagger} \hat{b}; p+1; \alpha^{2} ) \right. \right. \nonumber \\
&& \left. \left. (-\alpha \hat{b})^{p} e^{-i ( \Delta + p \omega_{m } ) t } + \sum_{p=1}^{\infty}\frac{1}{p!} (\alpha \hat{b}^{\dagger})^{p} ~_{1}F_{1}(- \hat{b}^{\dagger} \hat{b}; p+1; \alpha^{2} ) e^{-i ( \Delta - p \omega_{m } ) t } \right] \right\},
\end{eqnarray}
where we use the confluent hypergeometric function $_{1}F_{1}(a;b;z)$, define a detuning between the optical mode and driving frequencies $\Delta = \omega - \omega_{d}$, and introduce an auxiliary dimensionless parameter $\alpha = g_{0} / \omega_{m }$.
The first term of this effective Hamiltonian is a Kerr term for the optical oscillator mode, the second term vanishes in the absence of driving and we recover the well-known fact that the standard optomechanical Hamiltonian is feasible of diagonalization \cite{Restrepo2017}.
In this form, the second term allows us to realize that the driving frequency selects the type of dominant processes in the dynamics \cite{Ventura2015}.
Driving the system with a detuning proportional to an integer of the mechanical mode frequency, $\Delta = p \omega_{m }$ with $p = 0, 1, 2, \ldots$, yields an effective Hamiltonian for parametric conversion of order $p$ between the optical and mechanical modes with Kerr nonlinearity in the optical mode,
\begin{eqnarray}
\left. \frac{\hat{H}_{\mathrm{eff}}}{\hbar} \right\vert_{\Delta = p \omega_{m }} &\approx& - \frac{g_{0}^{2}}{\omega_{m }} \left( \hat{a}^{\dagger} \hat{a} \right)^{2} + \frac{\Omega}{2} e^{- \frac{\alpha^{2}}{2}} \left[ \frac{1}{p!} ~_{1}F_{1}(- \hat{b}^{\dagger} \hat{b}; p+1; \alpha^{2} ) \hat{a}^{\dagger} (\alpha \hat{b} )^{p} \right. \nonumber \\
&& \left. + \frac{1}{p!} ~ \hat{a} ( \alpha \hat{b}^{\dagger} )^{p} ~_{1}F_{1}(- \hat{b}^{\dagger} \hat{b}; p+1; \alpha^{2} ) \right],
\end{eqnarray}
informing us that resonant driving of the optical oscillator $p=0$ produces an effective optical driven Kerr system where the drive is modified by the optomechanical coupling strength and the excitation number of the mechanical mode \cite{Gong2009,Aldana2013,Xiong2016}.
For detuning equal to the mechanical mode frequency, $\Delta = \omega_{m }$ with $p=1$, the effective Hamiltonian takes the form of an optomechanical beam-splitter with Kerr nonlinearity in the optical mode and suggests its use for coherent state transfer between the optical and mechanical modes \cite{Ventura2019}.
For the next order, $p=2$, it becomes a second order parametric down-converter that suggest its use to generate mechanical squeezed states \cite{Wollman2015}.
Choosing a driving frequency that delivers a negative detuning $\Delta = - p \omega_{m }$ with $p=1,2,3,\ldots$ yields an effective Hamiltonian,
\begin{eqnarray}
\left. \frac{\hat{H}_{\mathrm{eff}}}{\hbar} \right\vert_{\Delta = -p \omega_{m }} &\approx& - \frac{g_{0}^{2}}{\omega_{m }} \left( \hat{a}^{\dagger} \hat{a} \right)^{2} + \frac{\Omega}{2} e^{- \frac{\alpha^{2}}{2}} \left[ \frac{1}{p!} \hat{a}^{\dagger} (-\alpha \hat{b}^{\dagger})^{p} ~_{1}F_{1}(- \hat{b}^{\dagger} \hat{b}; p+1; \alpha^{2} ) + \right. \nonumber \\
&& \left. \frac{1}{p!} ~_{1}F_{1}(- \hat{b}^{\dagger} \hat{b}; p+1; \alpha^{2} ) ~\hat{a} (-\alpha \hat{b})^{p} \right],
\end{eqnarray}
that, for the case of detuning equal to the negative mechanical mode frequency $\Delta = - \omega_{m }$ with $p=1$ suggests its use for two-mode squeezing.
Of course, we may think of processes involving switching from resonant to off-resonant driving to implement more complex processes like sideband cooling \cite{Riviere2011}.
Our interest focuses on one-dimensional photonic crystal cavities etched on nanobeams whose structure may be optimized to enhance the optomechanical coupling or decrease losses \cite{Chan2012}.
These devices localize optical and mechanical modes around a defect in a large periodic array and sustain multiple optical and mechanical modes \cite{Chan2012,Eichenfield2009,Yu2018}; the former may be addressed separately.
In Section \ref{sec:S2}, we introduce a finite element model for such a device, the Hamiltonian describing the individual interaction of multiple optical modes with one mechanical mode, and the effective Hamiltonian form that allows us to create insight of its dynamics.
In Section \ref{sec:S3}, we propose a driving scheme that allows us to simulate a quantum parametric process where the annihilation (creation) of an optical excitation produces the first and second order creation (annihilation) of a mechanical excitation.
This parametric process in the strong driving regime, mean optical field approximation, produces an effective Hamiltonian of a parametric oscillator in the mechanical mode.
In principle, the full optical control of the parameter values in the parametric oscillator should allow probing the controlled squeezing of the mechanical mode.
We close our contribution with a brief conclusion in Section \ref{sec:S4}.
\section{Driven polychrome optomechanical model} \label{sec:S2}
Nanobeams with engraved one-dimensional photonic crystal cavities, Fig.~\ref{Fig1}, are a standard experimental realization for current optomechanical setups \cite{Chan2012, Eichenfield2009}.
A typical device is made of a silica nanobeam where a one-dimensional photonic cavity is created by introducing a defect in an otherwise periodic structure to allow for localized optical and mechanical modes in it.
We focus on the structure proposed in Ref.~\cite{Eichenfield2009}.
It consists of 75 rectangular cells in a periodic array along the $x$ axis.
Quadratic reduction of the size along the $x$-axis of 15 cells in the middle of the array introduces a defect.
The resulting photonic defect cavity supports optical modes that approximately follow the eigenstates of the one-dimensional quantum mechanical harmonic oscillator along the $x$ axis.
Each regular cell has length $360~\textrm{nm}$ ($x$-axis), width $1400~\textrm{nm}$ ($y$-axis), and thickness $220~\textrm{nm}$ ($z$-axis).
These type of structures allow for many optical modes \cite{Buckley2014} which may even be coupled among themselves \cite{Yu2018}.
We use Finite Element Modeling to find three optical modes where the electric field is mostly polarized along the $y$-axis.
These modes have frequencies $204~\mathrm{THz}$, $195~\mathrm{THz}$ and $188~\mathrm{THz}$, Fig.~\ref{Fig1}(a), Fig.~\ref{Fig1}(b) and Fig.~\ref{Fig1}(c), in that order.
We also find three mechanical modes localized around the defect.
These are breathing, accordion, and pinch modes with frequency $2.23~\mathrm{GHz}$, $1.55~\mathrm{GHz}$ and $0.885~\mathrm{GHz}$, Fig.~\ref{Fig1}(d), Fig.~\ref{Fig1}(e) and Fig.~\ref{Fig1}(f), in that order.
Our results, fundamental frequencies and mode shapes, are in agreement with those of Ref.~\cite{Eichenfield2009}.
While this structure is designed to increase the total optomechanical coupling, there is one optomechanical coupling between each pair of optical and mechanical modes and also between pairs of optical modes. All of these couplings, of course, depend on the particular details and symmetries of the structure. Finite Element techniques offer a powerful tool to study and optimize these structures \cite{Chan2012,Eichenfield2009,Buckley2014,Deotare2009}.
Due to their difference in frequency, some of these optical modes may be separately addressed and, therefore, it becomes worthwhile to study the quantum dynamics of driven polychrome optomechanical systems.
\begin{figure}
\caption{(a)--(c) Optical and (d)--(f) mechanical modes of a nanobeam with their corresponding frequencies. The optical modes are in the infrared telecommunication band.}
\label{Fig1}
\end{figure}
In the following, we assume that only the optical modes are driven by an external coherent source such that the effective Hamiltonian becomes,
\begin{eqnarray}
\frac{\hat{H}}{\hbar} = \sum_{j} \omega_{j} \hat{a}_{j}^{\dagger} \hat{a}_{j} + \omega_{m } \hat{b}^{\dagger} \hat{b} - \sum_{j} g_{0j} \hat{a}_{j}^{\dagger} \hat{a}_{j} \left( \hat{b}^{\dagger} + \hat{b} \right) + \sum_{j} \Omega_{j} \cos \left( \omega_{dj} t\right) \left( \hat{a}_{j}^{\dagger} + \hat{a}_{j} \right),
\end{eqnarray}
where the optical modes, frequency $\omega_{j}$ and creation (annihilation) operators $\hat{a}_{j}^{\dagger}$ ($\hat{a}_{j}$), interact with a single mechanical mode, frequency $\omega_{m}$ and creation (annihilation) operators $\hat{b}^{\dagger}$ ($\hat{b}$), under coupling strength $g_{0j}$, external optical driving strength $\Omega_{j}$, and driving frequency $\omega_{dj}$.
We follow an approach similar to that in Ref. \cite{Ventura2015}.
First, we move into a frame defined by the driving frequencies and the excitation number of each optical mode.
Then, we perform an optical rotating wave approximation.
At this point, we move into a new frame defined by the displacement of the mechanical mode by a factor proportional to the optomechanical coupling and the excitation number of the corresponding optical mode divided by the mechanical frequency.
Then, we move into a frame defined by the free optical and mechanical oscillators to obtain an effective Hamiltonian,
\begin{eqnarray}
\frac{\hat{H}_{\mathrm{eff}}}{\hbar} &=& -\sum_{j,k} \frac{g_{0j} g_{0k}}{\omega_{m }} \hat{a}_{j}^{\dagger} \hat{a}_{j} \hat{a}_{k}^{\dagger} \hat{a}_{k} + \sum_{j} \frac{ \Omega_{j}}{2} e^{- \frac{\alpha_{j}^{2}}{2}} \left\{ \hat{a}_{j}^{\dagger} \left[ \sum_{p_{j}=0}^{\infty} \frac{1}{p_{j}!} (-\alpha_{j} \hat{b}^{\dagger})^{p_{j}} ~_{1}F_{1}(- \hat{b}^{\dagger} \hat{b}; p_{j}+1; \alpha_{j}^{2} ) \right. \right. \nonumber \\
&& \left. \left. e^{i ( \Delta_{j} + p_{j} \omega_{m } ) t } + \sum_{p_{j}=1}^{\infty}\frac{1}{p_{j}!} ~_{1}F_{1}(- \hat{b}^{\dagger} \hat{b}; p_{j}+1; \alpha_{j}^{2} ) (\alpha_{j} \hat{b})^{p_{j}} e^{i ( \Delta_{j} - p_{j} \omega_{m } ) t } \right] + \mathrm{h.c.} \right\}
\end{eqnarray}
written in terms of the optical detunings $\Delta_{j}=\omega_{j}-\omega_{dj}$ and the displacement parameters $\alpha_{j} = g_{0j} / \omega_{m }$.
Again, we are assuming all parameters real for the sake of simplicity.
The first term of this effective Hamiltonian is a collection of self- and cross-Kerr interactions, $\left( \hat{a}_{j}^{\dagger} \hat{a}_{j}\right)^{2}$ and $\hat{a}_{j}^{\dagger} \hat{a}_{j} \hat{a}_{k}^{\dagger} \hat{a}_{k}$, in that order.
The second term is identical to that in the driven standard optomechanical model for each optical mode and informs us of the ability to select effective dynamics depending on the optical detuning between optical oscillators and driving fields.
In the following, we will explore a particular driving scheme that may help us realize the uses of this polychrome optomechanical platform as an analog quantum simulator.
\section{Analog simulation of a parametric oscillator} \label{sec:S3}
We may choose to drive just two optical modes, for example, with a driving strength that adiabatically changes with time $\Omega_{j}(t)$, close to the first and second sideband transition of the mechanical mode by a small factor that changes adiabatically in time, $p_{1} = -\omega_{m } -\epsilon(t)$ and $p_{2} = - 2 \omega_{m } - 2 \epsilon(t)$ with $\epsilon(t) \ll \omega_{m }$, such that we may approximate the effective dynamics,
\begin{eqnarray}
\frac{\hat{H}_{\mathrm{eff}}}{\hbar} &\approx& -\sum_{j,k} \frac{g_{0j} g_{0k}}{\omega_{m }} \hat{a}_{j}^{\dagger} \hat{a}_{j} \hat{a}_{k}^{\dagger} \hat{a}_{k} + \epsilon(t) \hat{b}^{\dagger} \hat{b} + \frac{1}{2} \sum_{j=1}^{2} \frac{\Omega_{j}(t)}{j!} e^{- \frac{\alpha_{j}^{2}}{2}} \left[ \hat{a}_{j}^{\dagger} ~_{1}F_{1}(- \hat{b}^{\dagger} \hat{b}; j+1; \alpha_{j}^{2} ) (\alpha_{j} \hat{b})^{j} \right. \nonumber \\
&& \left. + \hat{a}_{j} (\alpha_{j} \hat{b}^{\dagger})^{j} ~_{1}F_{1}(- \hat{b}^{\dagger} \hat{b}; j+1; \alpha_{j}^{2} ) \right] ,
\end{eqnarray}
as long as the driving strength and the frequency change slowly in time, which is experimentally plausible in principle.
We now turn to the fact that one-dimensional photonic cavities on nanobeams have optical losses.
This produces dynamics that reaches a coherent state with large photon population as a steady state of the optical modes.
Experimental values for the losses in these devices lead to optical steady states with up to a few hundred photons; for example, the optomechanical device in Ref.~\cite{Chan2012} has losses of about $1.34~\mathrm{GHz}$ and a steady state with up to a mean optical excitation number of $|\beta|^{2}=120$ photons.
Thus, it is possible to work out an effective mean-field approximation on this driven system \cite{Ventura2019,Jaramillo2020}, such that we obtain an effective mean-field Hamiltonian,
\begin{eqnarray}
\frac{\hat{H}_{\mathrm{mf}}}{\hbar} &\approx& \epsilon(t) \hat{b}^{\dagger} \hat{b} + g_{1}(t) \left( \hat{b}^{\dagger} + \hat{b} \right) + g_{2}(t) \left( \hat{b}^{\dagger2} + \hat{b}^{2} \right),
\end{eqnarray}
that is identical in shape to that of a parametric oscillator, up to a constant phase, with time-dependent parameters controlled by the optical driving fields,
\begin{eqnarray}
g_{j}(t) &\approx& \frac{\Omega_{j}(t)}{j!} e^{- \frac{\alpha_{j}^{2}}{2}} \alpha_{j} \beta_{j},
\end{eqnarray}
where we safely assume that $ ~_{1}F_{1}(- \hat{b}^{\dagger} \hat{b}; j+1; \alpha_{j}^{2} ) \approx 1$ for typical values of the optomechanical coupling.
For the sake of providing an example, let us work out the simplest scenario where the driving strength and frequency are constant.
This allows us to move into a displaced reference frame $\hat{D}(\xi)$ with displacement $\xi=-g_{1}/(\epsilon+2g_{2})$ where the effective displaced Hamiltonian,
\begin{eqnarray}
\frac{\hat{H}_{\mathrm{mf}}}{\hbar} &\approx& \frac{1}{2}\left( \epsilon + 2g_{2} \right) \hat{x}^{2} +\frac{1}{2} \left( \epsilon - 2g_{2} \right) \hat{y}^{2},
\end{eqnarray}
up to a constant, in terms of the scaled canonical variables ($\hat{x},\hat{y}$) defined by the relation $\hat{b}= \left(\hat{x} + i\hat{y} \right) / \sqrt{2}$ and $\hat{b}^{\dagger}= \left(\hat{x} - i\hat{y}\right)/\sqrt{2}$.
The effective Hamiltonian looks like that of a particle under the action of a quadratic potential $\frac{1}{2} \left( \epsilon - 2g_{2} \right)\hat{y}^{2}$ controlled by the second-order parametric driving strength.
It takes the form of a harmonic oscillator Hamiltonian for $ \epsilon > 2g_{2}$, a free particle for $ \epsilon = 2g_{2}$, and an inverted quadratic potential for $ \epsilon < 2g_{2}$.
In principle, we may probe the model near the transition from an harmonic oscillator to a free particle, $2g_{2} \gtrapprox \epsilon $.
For example, we may observe the deformation of the ground state from a coherent state, Fig. \ref{Fig2}(a), to a squeezed state, Fig. \ref{Fig2}(b), and to a squeezed state that starts resembling an eigenstate of the scaled canonical variable $\hat{x}$, Fig. \ref{Fig2}(c).
\begin{figure}
\caption{ Hussimi Q-function for the ground state of a parametric oscillator with parameter values $\epsilon = 10 g_{1}
\label{Fig2}
\end{figure}
\section{Conclusion} \label{sec:S4}
We presented a finite element simulation of a common experimental realization of the standard optomechanical model in the form of a one-dimensional photonic cavity etched on a silica nanobeam.
The model shows at least three optical modes in the infrared telecommunication band and three mechanical vibration modes in the GHz range localized near the photonic cavity.
In principle, each one of the optical modes may be coherently driven on its own.
This motivated us to explore a Hamiltonian model describing the interaction of multiple optical modes with a single mechanical mode to obtain an effective Hamiltonian producing insight on the parametric processes that may be supported by the system.
We use this insight to propose a bichromatic driving scheme that provides us with an analog simulation of the parametric oscillator in the lossy optical cavity regime under a mean field approximation.
The parametric oscillator is optically controlled and requires adiabatic time-dependent driving strength and frequency in its most general realization.
As a simple example, we studied the ground state of the time-independent parametric oscillator and show that it should be possible to explore the transition from a coherent state to a eigenstate of a canonical variable in the ground state of the system.
\section*{Acknowledgments}
F.~E.~O., F~.H~.M.-V. and B.~R.~J.-A. thank CONACYT for their financial support.
F.~E.~O. thanks U.~N.~N. for study leave support.
B.~M.~R.-L. thanks the Quantum Fest 2021 organizers for their hospitality.
\begin{thebibliography}{34}
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\citenamefont {Kippenberg}}]{Riviere2011}
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oscillator close to the quantum ground state},\ }\href
{https://doi.org/10.1103/PhysRevA.83.063835} {\bibfield {journal} {\bibinfo
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{NoStop}
\bibitem [{\citenamefont {Ma}\ \emph {et~al.}(2007)\citenamefont {Ma},
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{https://doi.org/10.1364/OL.32.002200} {\bibfield {journal} {\bibinfo
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{arXiv:0702252 [physics.optics]} \BibitemShut {NoStop}
\bibitem [{\citenamefont {Teufel}\ \emph {et~al.}(2011)\citenamefont {Teufel},
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\citenamefont {Allman}, \citenamefont {Cicak}, \citenamefont {Sirois},
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{arXiv:1103.2144 [quant-ph]} \BibitemShut {NoStop}
\bibitem [{\citenamefont {Chan}\ \emph {et~al.}(2011)\citenamefont {Chan},
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{\bibfield {journal} {\bibinfo {journal} {Nature}\ }\textbf {\bibinfo
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\bibitem [{\citenamefont {Deli\'c}\ \emph {et~al.}(2020)\citenamefont
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quantum ground state},\ }\href {https://doi.org/10.1126/science.aba3993}
{\bibfield {journal} {\bibinfo {journal} {Science}\ }\textbf {\bibinfo
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\bibitem [{\citenamefont {Wollman}\ \emph {et~al.}(2015)\citenamefont
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}\href {https://doi.org/10.1126/science.aac5138} {\bibfield {journal}
{\bibinfo {journal} {Science}\ }\textbf {\bibinfo {volume} {349}},\ \bibinfo
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\bibitem [{\citenamefont {Palomaki}\ \emph {et~al.}(2013)\citenamefont
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\citenamefont {Lehnert}}]{Palomaki2013}
\BibitemOpen
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{Lehnert}},\ }\bibfield {title} {\bibinfo {title} {Entangling mechanical
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{https://doi.org/10.1126/science.1244563} {\bibfield {journal} {\bibinfo
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{710} (\bibinfo {year} {2013})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Riedinger}\ \emph {et~al.}(2018)\citenamefont
{Riedinger}, \citenamefont {Wallucks}, \citenamefont {Marinkovi\'c},
\citenamefont {L\"oschnauer}, \citenamefont {Aspelmeyer}, \citenamefont
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{\bibfnamefont {S.}~\bibnamefont {Gr\"oblacher}},\ }\bibfield {title}
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oscillators},\ }\href {https://doi.org/10.1038/s41586-018-0036-z} {\bibfield
{journal} {\bibinfo {journal} {Nature}\ }\textbf {\bibinfo {volume} {556}},\
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\bibitem [{\citenamefont {Bernier}\ \emph {et~al.}(2017)\citenamefont
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\citenamefont {Ioannou}, \citenamefont {Malz}, \citenamefont {Nunnenkamp},
\citenamefont {Feofanov},\ and\ \citenamefont {Kippenberg}}]{Bernier2017}
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\bibnamefont {Kippenberg}},\ }\bibfield {title} {\bibinfo {title}
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{8}},\ \href {https://doi.org/10.1038/s41467-017-00447-1}
{10.1038/s41467-017-00447-1} (\bibinfo {year} {2017}),\ \Eprint
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\bibitem [{\citenamefont {Barzanjeh}\ \emph {et~al.}(2017)\citenamefont
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\citenamefont {Fink}}]{Barzanjeh2017}
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{doi:10.1038/s41467-017-01304-x} (\bibinfo {year} {2017}),\ \Eprint
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\bibitem [{\citenamefont {Riedinger}\ \emph {et~al.}(2016)\citenamefont
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{\bibfield {journal} {\bibinfo {journal} {Nature}\ }\textbf {\bibinfo
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\bibitem [{\citenamefont {Hong}\ \emph {et~al.}(2017)\citenamefont {Hong},
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{https://doi.org/10.1126/science.aan7939} {\bibfield {journal} {\bibinfo
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{arXiv:1706.03777 [quant-ph]} \BibitemShut {NoStop}
\bibitem [{\citenamefont {Weis}\ \emph {et~al.}(2010)\citenamefont {Weis},
\citenamefont {Rivi\`ere}, \citenamefont {Del\'eglise}, \citenamefont
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{\bibfnamefont {O.}~\bibnamefont {Arcizet}}, \bibinfo {author} {\bibfnamefont
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{https://doi.org/10.1126/science.1195596} {\bibfield {journal} {\bibinfo
{journal} {Science}\ }\textbf {\bibinfo {volume} {330}},\ \bibinfo {pages}
{1520} (\bibinfo {year} {2010})},\ \Eprint {https://arxiv.org/abs/1007.0565}
{arXiv:1007.0565 [quant-ph]} \BibitemShut {NoStop}
\bibitem [{\citenamefont {Lecocq}\ \emph {et~al.}(2015)\citenamefont {Lecocq},
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\bibinfo {author} {\bibfnamefont {R.~W.}\ \bibnamefont {Simmonds}},\
}\bibfield {title} {\bibinfo {title} {Resolving the vacuum fluctuations of
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{https://doi.org/doi:10.1038/nphys3365} {\bibfield {journal} {\bibinfo
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{arXiv:1409.0872 [quant-ph]} \BibitemShut {NoStop}
\bibitem [{\citenamefont {Marinkovi\'c}\ \emph {et~al.}(2018)\citenamefont
{Marinkovi\'c}, \citenamefont {Wallucks}, \citenamefont {Riedinger},
\citenamefont {Hong}, \citenamefont {Aspelmeyer},\ and\ \citenamefont
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{\bibfnamefont {S.}~\bibnamefont {Gr\"oblacher}},\ }\bibfield {title}
{\bibinfo {title} {Optomechanical {B}ell test},\ }\href
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{\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {121}},\
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\bibitem [{\citenamefont {Restrepo}\ \emph {et~al.}(2017)\citenamefont
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}\bibfield {title} {\bibinfo {title} {Fully coupled hybrid cavity
optomechanics: Quantum interferences and correlations},\ }\href
{https://doi.org/10.1103/PhysRevA.95.023832} {\bibfield {journal} {\bibinfo
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\bibitem [{\citenamefont {Ventura-Vel\'azquez}\ \emph
{et~al.}(2015)\citenamefont {Ventura-Vel\'azquez}, \citenamefont
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{H.~M.}\ \bibnamefont {Moya-Cessa}},\ }\bibfield {title} {\bibinfo {title}
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{https://doi.org/10.1088/0031-8949/90/6/068010} {\bibfield {journal}
{\bibinfo {journal} {Phys. Scr.}\ }\textbf {\bibinfo {volume} {90}},\
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\bibitem [{\citenamefont {Gong}\ \emph {et~al.}(2009)\citenamefont {Gong},
\citenamefont {Ian}, \citenamefont {Liu}, \citenamefont {Sun},\ and\
\citenamefont {Nori}}]{Gong2009}
\BibitemOpen
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{\bibfnamefont {C.~P.}\ \bibnamefont {Sun}},\ and\ \bibinfo {author}
{\bibfnamefont {F.}~\bibnamefont {Nori}},\ }\bibfield {title} {\bibinfo
{title} {Effective {H}amiltonian approach to the {K}err nonlinearity in an
optomechanical system},\ }\href {https://doi.org/10.1103/PhysRevA.80.065801}
{\bibfield {journal} {\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo
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\BibitemShut {NoStop}
\bibitem [{\citenamefont {Aldana}\ \emph {et~al.}(2013)\citenamefont {Aldana},
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\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
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}\bibfield {title} {\bibinfo {title} {Equivalence between an optomechanical
system and a {K}err medium},\ }\href
{https://doi.org/10.1103/PhysRevA.88.043826} {\bibfield {journal} {\bibinfo
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{pages} {043826} (\bibinfo {year} {2013})},\ \Eprint
{https://arxiv.org/abs/1306.0415} {arXiv:1306.0415 [quant-ph]} \BibitemShut
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\bibitem [{\citenamefont {Xiong}\ \emph {et~al.}(2016)\citenamefont {Xiong},
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\end{document} | math | 58,804 |
\begin{document}
\title[Twists of hyperelliptic curves]{Twists of hyperelliptic curves by integers in progressions modulo $p$}
\author[D. Krumm]{David Krumm}
\address{Mathematics Department\\
Reed College}
\email{[email protected]}
\urladdr{http://maths.dk}
\author[P. Pollack]{Paul Pollack}
\address{Department of Mathematics\\
University of Georgia}
\email{[email protected]}
\urladdr{http://pollack.uga.edu}
\thanks{The second author was supported by NSF award DMS-1402268.}
\keywords{Hyperelliptic curve, quadratic twist, abc conjecture}
\subjclass[2010]{Primary: 11N32. Secondary: 11N36, 11G30}
\maketitle
\begin{abstract}
Let $f(x)$ be a nonconstant polynomial with integer coefficients and nonzero discriminant. We study the distribution modulo primes of the set of squarefree integers $d$ such that the curve $dy^2=f(x)$ has a nontrivial rational or integral point.
\end{abstract}
\section{Introduction}
Let $f(x)\in\mathbb{Z}[x]$ be a nonconstant polynomial with nonzero discriminant, and let $C$ be the hyperelliptic curve over $\mathbb{Q}$ defined by $y^2=f(x)$. For every squarefree integer $d$, let $C_d$ denote the quadratic twist $dy^2=f(x)$. The main object of interest in this article is the set $S_{\mathbb{Q}}(f)$ consisting of all squarefree integers $d$ such that $C_d$ has a nontrivial rational point, i.e., an affine rational point $(x_0,y_0)$ with $y_0\ne 0$. Specifically, we are interested in the following conjecture, which was proposed by the first author in \cite{krumm}.
\begin{conj}\label{krumm_conjecture}
For every large enough prime $p$, and every integer $r$ not divisible by $p$, there exist infinitely many $d\in S_{\mathbb{Q}}(f)$ such that $d\equiv r\mathfrak pmod p$.
\end{conj}
This conjecture is proved in \cite{krumm} in the case where $\deg f\le 2$. Furthermore, when $\deg f=3$, or when $\deg f=4$ and $f(x)$ has a rational root, the conjecture is shown to follow from the Parity Conjecture for elliptic curves over $\mathbb{Q}$. In this paper we explain how to leverage known results on squarefree values of polynomials and binary forms to prove the following two theorems.
First, using work of Granville \cite{granville} we show that Conjecture \ref{krumm_conjecture} follows from the abc conjecture; in fact, the latter can be used to prove a stronger statement. Let us denote by $S_{\mathbb{Z}}(f)$ the set of all squarefree integers $d$ such that $C_d$ has a nontrivial \emph{integral} point.
\begin{thm}\label{abc_thm}
The abc conjecture implies that for every large enough prime $p$, and every integer $r$ not divisible by $p$, there exist infinitely many $d\in S_{\mathbb{Z}}(f)$ such that $d\equiv r\mathfrak pmod p$.
\end{thm}
Second, we prove an unconditional result by using work of Greaves \cite{greaves}.
\begin{thm}\label{unconditional_thm}
Conjecture \ref{krumm_conjecture} holds if every irreducible factor of $f(x)$ over $\mathbb{Q}$ has degree at most six.
\end{thm}
In addition, we consider the distribution of elements of $S_{\mathbb{Z}}(f)$ modulo $p$ when $p$ is a ``small" prime, by which we mean that at least one of the conditions in \eqref{p_sufficient_conditions} is not satisfied.
\section{Assuming abc: Proof of Theorem \ref{abc_thm}}
We will need the following special case of Theorem 1 in \cite{granville}.
\begin{prop}[Granville]\label{granville_prop} Assume the abc conjecture is true. Let $g(x)$ be a nonconstant polynomial with integer coefficients and nonzero discriminant, and suppose that there is no prime $p$ such that $p|g(n)$ for all integers $n$. Then there exist infinitely many integers $n$ such that $g(n)$ is squarefree.
\end{prop}
Recall that an integer $k$ is called a \emph{fixed divisor} of $f(x)$ if $k|f(n)$ for every integer $n$. The set of all fixed divisors of $f(x)$ is finite, and therefore has a largest element, which we denote by $D$. It is a simple exercise to show that $D$ is maximal also in the sense that every fixed divisor of $f(x)$ divides $D$.
Let $p$ be a prime number, let $\operatorname{ord}_p$ denote the $p$-adic valuation on $\mathbb{Z}$, and let $\varepsilon\in\{0,1\}$ be the parity of $\operatorname{ord}_p(D)$. For every integer $r\not\equiv 0\mathfrak pmod p$ and for every integer $v\ge 0$ we define a statement $S(r,v)$ as follows:
\begin{equation}\label{S_statement}
S(r,v)\;
\begin{cases}
\text{There exist $h,x_0,y_0\in\mathbb{Z}$ satisfying}\\
\;\;\bullet\;\;hy_0^2\equiv f(x_0)\mathfrak pmod*{p^{2(v+\varepsilon) + 1}},\\
\;\;\bullet\;\;\operatorname{ord}_p(y_0)=v+\varepsilon,\text{ and}\\
\;\;\bullet\;\;h\equiv r\mathfrak pmod* p.
\end{cases}
\end{equation}
The proof of the following proposition establishes the key ideas to be used throughout this article.
\begin{prop}[Assuming abc]\label{S_reverse_direction} Let $r$ be an integer not divisible by $p$.
Suppose that $S(r,v)$ holds true for some $v\ge 0$, and that $f(x)$ has an irreducible factor whose discriminant is not divisible by $p$. Then there exist infinitely many integers $d\in S_{\mathbb{Z}}(f)$ such that $d\equiv r\mathfrak pmod p$.
\end{prop}
\begin{proof}
For every nonzero rational number $x$ we denote by $\operatorname{sqf}(x)$ the squarefree part of $x$, i.e., the unique squarefree integer representing the coset of $x$ in $(\mathbb{Q}^{\ast})/(\mathbb{Q}^{\ast})^2$. By definition of $\varepsilon$, we have $\operatorname{ord}_p(D)=2k+\varepsilon$ for some nonnegative integer $k$. Writing $D=\operatorname{sqf}(D)t^2$, it is necessarily the case that $\operatorname{ord}_p(t)=k$ and $\operatorname{ord}_p(\operatorname{sqf}(D))=\varepsilon$; thus, we may write $t=p^ku$, where $p\nmid u$, and $\operatorname{sqf}(D)=p^{\varepsilon}\delta$ for some squarefree integer $\delta$ not divisible by $p$.
Since $S(r,v)$ holds true, there exist integers $h, x_0,$ and $y_0$ satisfying the properties listed in \eqref{S_statement}. In particular, $\operatorname{ord}_p(y_0)=v+\varepsilon$, so we may write $y_0=p^{v+\varepsilon}z_0$, where $p\nmid z_0$.
By the Chebotarev density theorem\footnote{See Lemma 4.4 in \cite{krumm} for details. The crucial fact we use here is that if $h(x)$ is an irreducible factor of $f(x)$ such that $p\nmid\operatorname{disc} h(x)$, then the intersection of the splitting field of $h(x)$ and the cyclotomic field $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}$.}, there exists a prime $q\nmid D$ such that $qu\equiv z_0\mathfrak pmod p$ and $f(x)$ has a simple root modulo $q$. The latter property ensures, via Hensel's lemma\footnote{Let $\alpha$ be a simple root of $f(x)$ modulo $q$. Hensel lifting allows us to find an integer $\beta$ such that $\beta\equiv\alpha\mathfrak pmod q$ and $f(\beta)\equiv 0\mathfrak pmod{q^3}$. Then $m=\beta+q^2$ satisfies $q^2\|f(m)$.}, that there exists $m\in\mathbb{Z}$ such that $q^2\|f(m)$.
For every prime $s\ne p$ dividing $D$, let $e_s=\operatorname{ord}_s(D)$ and let $n_s$ be an integer such that $f(n_s)\not\equiv 0\mathfrak pmod{s^{e_s+1}}$. (Such an integer must exist, for otherwise $\operatorname{lc}m(s^{e_s+1},D)=sD$ would be a fixed divisor of $f(x)$, contradicting the maximality of $D$.)
Choose $b\in\mathbb{Z}$ satisfying
\begin{align*}
b&\equiv x_0\mathfrak pmod*{p^{2(v+\varepsilon)+1}},\\
b&\equiv m\mathfrak pmod*{q^3},\text{ and}\\
b&\equiv n_s\mathfrak pmod*{s^{e_s+1}}\text{ for every prime } s|D, s\ne p.
\end{align*}
Let $a=q^2p^{2(v+\varepsilon)+1}\mathfrak prod_{s}s^{e_s+1}$, and define a polynomial $g(x)$ by the equation
\[\Delta\cdot g(x)=f(ax+b),\text{ where } \Delta=Dq^2p^{2(v-k)+\varepsilon}.\]
Note that $v\ge k$, so that $\Delta\in\mathbb{Z}$. Indeed, the properties in \eqref{S_statement} imply that $\operatorname{ord}_p(f(x_0))=2(v+\varepsilon)$. Since $D|f(x_0)$, then $\operatorname{ord}_p(D)\le\operatorname{ord}_p(f(x_0))$, so $2k+\varepsilon\le 2v+2\varepsilon$, and therefore $k\le v$.
We claim that $g(x)$ satisfies all the hypotheses of Proposition \ref{granville_prop}. A Taylor expansion shows that $f(ax+b)=f(b)+a\cdot P(x)$ for some polynomial $P(x)\in\mathbb{Z}[x]$. Thus, in order to show that $g(x)\in\mathbb{Z}[x]$ it suffices to show that $\Delta$ divides both $f(b)$ and $a$. From the definitions it follows easily that $\operatorname{ord}_{\ell}(a)\ge\operatorname{ord}_{\ell}(\Delta)$ for every prime $\ell$ dividing $\Delta$, so $\Delta|a$. Similarly, the definition of $b$ implies that $\Delta|f(b)$. Hence $g(x)\in\mathbb{Z}[x]$. Now suppose that $\ell$ is a fixed prime divisor of $g(x)$. We claim that $\ell\nmid a$. If $\ell=q$, then $q|g(q)$, so $q^3|f(aq+b)$. However, $f(aq+b)\equiv f(b)\equiv f(m)\not\equiv 0\mathfrak pmod{q^3}$. Thus $\ell\ne q$. Suppose now that $\ell$ is one of the primes $s$, and let $n\in\mathbb{Z}$. Then $s|g(n)$, so $s^{e_s+1}|f(an+b)$. However, $f(an+b)\equiv f(b)\equiv f(n_s)\not\equiv 0\mathfrak pmod{s^{e_s+1}}$. Thus $\ell\ne s$. Similarly, we can show that $p$ does not divide $g(n)$ for any integer $n$. For if $p|g(n)$, then $f(an+b)\equiv 0\mathfrak pmod{p^{2(v+\varepsilon)+1}}$. However, $f(an+b)\equiv f(b)\equiv f(x_0)\not\equiv 0\mathfrak pmod*{p^{2(v+\varepsilon)+1}}$.
This proves that $\ell\nmid a$. Now, since the map $x\mapsto (ax+b)$ is invertible modulo $\ell$, the assumption that $\ell$ is a fixed divisor $g(x)$ implies that it is also a fixed divisor of $f(x)$. It follows that $\ell| D$, but this has already been ruled out above. Therefore, $g(x)$ has no fixed prime divisor. Finally, $\operatorname{disc} g(x)\ne 0$ since $\operatorname{disc} f(x)\ne 0$ by assumption.
As shown above, neither $p$ nor any of the primes $s$ can divide $g(n)$ for any integer $n$. Thus,\begin{equation}\label{coprime_values}
\gcd(g(n), pD)=1\text{ for every integer }n.
\end{equation}
The last step in the proof is to show that there is a well-defined map
\[\mathfrak psi:\{n\in\mathbb{Z}\;\vert\;g(n)\text{ is squarefree}\}\rightarrow\{d\in S_{\mathbb{Z}}(f)\;\vert\; d\equiv r\mathfrak pmod* p\}\]
given by $n\mapsto \delta g(n)$. Note that the domain of $\mathfrak psi$ is infinite by Proposition \ref{granville_prop}. Let $n\in\mathbb{Z}$ be such that $g(n)$ is squarefree. Tracking through the definitions we find that
\begin{equation}\label{f_value}
f(ax+b)=\delta g(x)(qu)^2p^{2v+2\varepsilon}.
\end{equation}
By \eqref{coprime_values} we have $\gcd(g(n), \delta)=1$, so \eqref{f_value} implies that \[d:=\operatorname{sqf}(f(an+b))=\delta g(n).\]
Reducing \eqref{f_value} modulo $p^{2(v+\varepsilon)+1}$ and recalling that $y_0=p^{v+\varepsilon}z_0$, we obtain
\[d(qu)^2p^{2v+2\varepsilon}\equiv f(b)\equiv f(x_0)\equiv hy_0^2\equiv hp^{2v+2\varepsilon}z_0^2\mathfrak pmod*{p^{2(v+\varepsilon)+1}}.\]
It follows that $d(qu)^2\equiv hz_0^2\mathfrak pmod* p$. Since $qu\equiv z_0\mathfrak pmod p$ by construction and $h\equiv r\mathfrak pmod p$ by the assumptions in \eqref{S_statement}, this implies that $d\equiv r\mathfrak pmod p$. Moreover, it is clear from the definitions that $d\in S_{\mathbb{Z}}(f)$. Thus, we have shown that the map $\mathfrak psi$ is well defined.
The equation $g(x)=g(y)$ has only finitely many integral solutions with $x\ne y$, since the polynomial map $x\mapsto g(x)$ is injective outside some bounded interval in the real line. Hence, the fact that the domain of $\mathfrak psi$ is infinite implies that its image is infinite as well. This completes the proof.
\end{proof}
\begin{rems}\mbox{}
\begin{enumerate}[leftmargin=7mm]
\item[(i)] Proposition \ref{granville_prop} is known to hold unconditionally if every irreducible factor of $f(x)$ has degree at most three (see \cite[Chap. 4]{hooley}). Our arguments show that Proposition \ref{S_reverse_direction} also holds unconditionally in this case.
\item[(ii)] The version of Proposition \ref{granville_prop} given in \cite{granville} states that the number of positive integers $n\le B$ such that $g(n)$ is squarefree is asymptotic to $\kappa B$ (as $B\rightarrow\infty$) for some positive constant $\kappa$. Modifying the proof of Proposition \ref{S_reverse_direction} appropriately to take advantage of this, one can show that
\[\#\{d\in S_{\mathbb{Z}}(f): |d|\le B \text{ and }d\equiv r\mathfrak pmod* p\}\gg B^{1/\deg f}.\]
\end{enumerate}
\end{rems}
\begin{cor}[Assuming abc]\label{large_p_cor}
Let $r$ be an integer not divisible by $p$. Suppose that $p\nmid D$, $p\nmid\operatorname{disc} f(x)$, and $ry_0^2\equiv f(x_0)\mathfrak pmod p$ for some integers $x_0,y_0$ with $p\nmid y_0$. Then there exist infinitely many integers $d\in S_{\mathbb{Z}}(f)$ such that $d\equiv r\mathfrak pmod p$.
\end{cor}
\begin{proof}
The hypotheses imply that the statement $S(r,0)$ holds true. The result then follows immediately from Proposition \ref{S_reverse_direction}.
\end{proof}
\subsection*{Proof of Theorem \ref{abc_thm}}
Assuming the abc conjecture, we must show that for every large enough prime $p$, and every integer $r$ not divisible by $p$, there exist infinitely many $d\in S_{\mathbb{Z}}(f)$ such that $d\equiv r\mathfrak pmod p$. Let $\operatorname{lc}(f)$ be the leading coefficient of $f(x)$, and let $g$ be the genus of the curve $y^2=f(x)$. Suppose that $p$ is a prime satisfying
\begin{equation}\label{p_sufficient_conditions}
p\nmid\operatorname{lc}(f),\;p\nmid D,\;p\nmid\operatorname{disc} f(x),\;\text{and }p>4g^2+6g+4.
\end{equation}
Let $r$ be an integer not divisible by $p$. The Hasse-Weil bound implies that every smooth projective curve of genus $g$ over $\mathbb{F}_{p}$ has at least $2g+5$ points defined over $\mathbb{F}_p$; in particular, this applies to the hyperelliptic curve over $\mathbb{F}_p$ defined by $ry^2=f(x)$. This curve can have at most $2g+4$ trivial points defined over $\mathbb{F}_p$, so it must have a nontrivial point. Applying Corollary \ref{large_p_cor} we obtain the desired result.
\mathfrak qed
\section{The case of small primes $p$}
Let $R(p)\subseteq\mathbb{F}_p^{\ast}$ be the set consisting of all the nonzero residue classes modulo $p$ which are represented in the set $S_{\mathbb{Z}}(f)$. We have shown that if $p$ is large enough, then $R(p)=\mathbb{F}_p^{\ast}$. In this section we discuss the problem of determining $R(p)$ when $p$ is a ``small" prime, meaning that the conditions \eqref{p_sufficient_conditions} are not all satisfied.
\begin{lem}\label{square_classes_lem}
Let $r$ be an integer not divisible by $p$, and let $v$ be a nonnegative integer. Suppose that $S(r,v)$ holds. Then $S(a,v)$ holds for every integer $a$ in the same square class as $r$ modulo $p$.
\end{lem}
\begin{proof}
Let $h, x_0,$ and $y_0$ be integers satisfying the conditions in \eqref{S_statement}. Let $g$ be a primitive root modulo $p$, and let $z$ be a multiplicative inverse of $g$ modulo $p^{2(v+\varepsilon)+1}$. By hypothesis, $a\equiv rg^{2k}\mathfrak pmod p$ for some positive integer $k$. From the definitions it follows that
\begin{itemize}[itemsep=2mm]
\item $hg^{2k}(z^ky_0)^2\equiv hy_0^2\equiv f(x_0)\mathfrak pmod{p^{2(v+\varepsilon)+1}}$,
\item $\operatorname{ord}_p(z^ky_0)=\operatorname{ord}_p(y_0)=v+\varepsilon$, and
\item $hg^{2k}\equiv rg^{2k}\equiv a\mathfrak pmod p$.
\end{itemize}
Hence, $S(a,v)$ holds.
\end{proof}
\begin{prop}[Assuming abc]\label{reduction_possibilities}
Suppose that $f(x)$ has an irreducible factor whose discriminant is not divisible by $p$. Then $R(p)$ is either empty or equal to one of the sets $\mathbb{F}_p^{\ast}$, $(\mathbb{F}_p^{\ast})^2$, or $\mathbb{F}_p^{\ast}\setminus(\mathbb{F}_p^{\ast})^2$.
\end{prop}
\begin{proof}
We claim that if $R(p)$ contains a square, then $R(p)\supseteq(\mathbb{F}_p^{\ast})^2$. Let $a$ and $r$ be nonzero quadratic residues modulo $p$, and suppose that there exists $d\in S_{\mathbb{Z}}(f)$ such that $d\equiv r\mathfrak pmod p$. Then we have $dy_0^2=f(x_0)$ for some integers $x_0,y_0$ with $y_0\ne 0$. Letting $v=\operatorname{ord}_p(y_0)-\varepsilon$, it is easy to verify that $v\ge 0$ and $S(r,v)$ holds. By Lemma \ref{square_classes_lem}, $S(a,v)$ also holds. Hence, by Proposition \ref{S_reverse_direction}, there exists $d'\in S_{\mathbb{Z}}(f)$ such that $d'\equiv a\mathfrak pmod p$. This proves the claim. A similar argument shows that if $R(p)$ contains a nonsquare, then $R(p)\supseteq\mathbb{F}_p^{\ast}\setminus(\mathbb{F}_p^{\ast})^2$.
Suppose that $R(p)$ is nonempty. If $R(p)$ contains only squares, then the above argument implies that $R(p)=(\mathbb{F}_p^{\ast})^2$; similarly, if $R(p)$ contains only nonsquares, then $R(p)=\mathbb{F}_p^{\ast}\setminus(\mathbb{F}_p^{\ast})^2$. Finally, if $R(p)$ contains both a square and a nonsquare, then $R(p)=\mathbb{F}_p^{\ast}$.
\end{proof}
We now provide examples in which the various possibilities of Proposition \ref{reduction_possibilities} occur with small primes $p$.
\begin{ex}
Let $p$ be any prime such that $p\equiv 3\mathfrak pmod 4$, and consider the polynomial $f(x)=(x^2+1)((x^p-x)^2+p)$. Note that $f(x)$ has a repeated root modulo $p$, so that $p|\operatorname{disc} f(x)$, and $p$ is a small prime for $f(x)$. We have $\operatorname{ord}_p(f(n))=1$ for every integer $n$, which implies that $p|\operatorname{sqf}(f(n))$ for all $n$. Hence, every element of $S_{\mathbb{Z}}(f)$ is divisible by $p$, and $R(p)=\emptyset$.
\end{ex}
\begin{ex}
Let $p$ be an arbitrary prime, and consider the polynomial $f(x)=x^p-x+1$. Note that $p$ is small for $f(x)$. We claim that $R(p)=(\mathbb{F}_p^{\ast})^2$. Let $r$ be an integer not divisible by $p$, and suppose that $d\in S_{\mathbb{Z}}(f)$ satisfies $d\equiv r\mathfrak pmod p$. Then $dy_0^2=x_0^p-x_0+1$ for some integers $x_0,y_0$. Reducing modulo $p$ we obtain $ry_0^2\equiv 1\mathfrak pmod p$, from which it follows that $r$ is a square modulo $p$. Thus, $R(p)\subseteq(\mathbb{F}_p^{\ast})^2$. Conversely, if $r$ is a nonzero square modulo $p$, then $ry_0^2\equiv 1\equiv f(x_0)\mathfrak pmod p$ for some integer $y_0$ and for every integer $x_0$. Since $p\nmid D=1$ and $p\nmid\operatorname{disc} f(x)$, Corollary \ref{large_p_cor} implies that there exists $d\in S_{\mathbb{Z}}(f)$ such that $d\equiv r\mathfrak pmod p$. Hence, $R(p)=(\mathbb{F}_p^{\ast})^2$, as claimed. A similar argument shows that if we define $f(x)=x^p-x+a$, where $a$ is a quadratic nonresidue modulo $p$, then $R(p)=\mathbb{F}_p^{\ast}\setminus(\mathbb{F}_p^{\ast})^2$.
\end{ex}
\begin{ex}
Let $p$ be prime, let $v$ be a nonnegative integer, and consider
\[f(x)=x(x^p-x)^{2v+2}+p^{2v+1}x.\]
We will show that $R(p)=\mathbb{F}_p^{\ast}$. Note that $p|\operatorname{disc} f(x)$, so $p$ is small for $f(x)$. Clearly, $p^{2v+1}$ is a fixed divisor of $f(x)$, so $p^{2v+1}|D$; in fact $p^{2v+1}\|D$ since $p^{2v+2}\nmid f(1)$. In particular, the parity of $\operatorname{ord}_p(D)$ is $\varepsilon=1$. The statement $S(r,v)$ can now be seen to hold for every integer $r\not\equiv 0\mathfrak pmod p$: indeed,
\[r(p^{v+1})^2\equiv f(rp)\mathfrak pmod*{p^{2v+3}}.\]
Moreover, $f(x)$ has an irreducible factor (namely, $x$) whose discriminant is not divisible by $p$. Thus, by Proposition \ref{S_reverse_direction}, there exists $d\in S_{\mathbb{Z}}(f)$ such that $d\equiv r\mathfrak pmod p$. We conclude that $R(p)=\mathbb{F}_p^{\ast}$.
\end{ex}
In the last example we show that when the discriminant condition in Proposition \ref{reduction_possibilities} is not satisfied, the conclusion may not hold.
\begin{ex}
Let $p$ be an odd prime, and let $f(x)$ be the $p$-th cyclotomic polynomial. Then $f(x)$ is irreducible and $p|\operatorname{disc} f(x)$. We will show that $R(p)=\{1\}$. Clearly $1\in R(p)$ because the curve $y^2=f(x)$ has a nontrivial integral point, namely $(0,1)$. Now suppose that $d\in S_f(\mathbb{Z})$ is not divisible by $p$. We have $d>0$ because $f(x)$ only takes positive values for $x\in\mathbb{R}$. If $q$ is any prime dividing $d$, then $f(x)$ has a simple root modulo $q$. Let $K$ denote the cyclotomic field $\mathbb{Q}[x]/(f(x))$. By the Dedekind-Kummer theorem in algebraic number theory (Proposition 8.3 in \cite{neukirch}), some prime (and therefore every prime) of $\mathcal O_K$ lying over $(q)$ has ramification index and residue degree equal to 1. Hence, $q$ splits completely in $K$. It follows that $q\equiv 1\mathfrak pmod p$ (see Corollary 10.4 in \cite{neukirch}, for instance). Since $d>0$ and every prime divisor of $d$ is congruent to 1 modulo $p$, then $d\equiv 1\mathfrak pmod p$. Therefore, $R(p)=\{1\}$.
\end{ex}
\section{An unconditional result: proof of Theorem \ref{unconditional_thm}}
We will need the following special case of the main theorem in \cite{greaves}.
\begin{prop}[Greaves]\label{greaves_prop}
Let $F(x,y)\in\mathbb{Z}[x,y]$ be a binary form of degree $d$ with nonzero discriminant, and suppose that the coefficient of $y^d$ in $F(x,y)$ is nonzero. Let $A,B,M$ be integers with $M>0$. Assume that for every prime $\ell$ there exist integers $\alpha$ and $\beta$ such that
\begin{equation}\label{greaves_congruences}
\alpha\equiv A\mathfrak pmod* M,\; \beta\equiv B\mathfrak pmod* M,\text{ and }\;\ell^2\nmid F(\alpha,\beta).
\end{equation}
If every irreducible factor of $F(x,y)$ has degree at most six, then there exist infinitely many pairs of integers $\alpha,\beta$ such that $\alpha\equiv A\mathfrak pmod M$, $\beta\equiv B\mathfrak pmod M$, and $F(\alpha,\beta)$ is squarefree.
\end{prop}
\begin{rem}
The result in \cite{greaves} assumes that $F(x,y)$ has nonzero terms in both $x^d$ and $y^d$. To obtain Proposition \ref{greaves_prop}, one should apply the result of \cite{greaves} with $F(x,y)$ replaced with $F(x,kx+y)$ for an integer $k$ chosen so that the coefficient of $x^d$ is nonzero.
\end{rem}
\begin{prop}\label{unconditional_prop}
Let $r$ be an integer not divisible by $p$.
Suppose that $S(r,v)$ holds true for some $v\ge 0$, and that $f(x)$ has an irreducible factor whose discriminant is not divisible by $p$. Moreover, suppose that $\deg f\ge 3$ and that every irreducible factor of $f(x)$ has degree at most six. Then there exist infinitely many integers $d\in S_{\mathbb{Q}}(f)$ such that $d\equiv r\mathfrak pmod p$.
\end{prop}
\begin{proof}
The hypotheses allow us to define a polynomial $g(x)$ as in the proof of Proposition \ref{S_reverse_direction}; we will use here the notation introduced in that proof. Let $G(x,y)$ be the homogenization of $g(x)$, $\mathfrak partial=\deg g$, and $F(x,y)=y^{\sigma}G(x,y)$, where $\sigma\in\{0,1\}$ is the parity of $\mathfrak partial$. We have $\operatorname{disc} F\ne 0$ since $\operatorname{disc} g(x)\ne 0$. Note that $g(0)=f(b)/\Delta$, and $f(b)\ne 0$ because $f(b)\equiv f(m)\not\equiv 0\mathfrak pmod{q^3}$. It follows that the coefficient of $y^{\mathfrak partial+\sigma}$ in $F(x,y)$ is nonzero.
We will apply Proposition \ref{greaves_prop} with $A=q, B=1, M=pD$. We must show that for every prime $\ell$ there exist $\alpha, \beta\in\mathbb{Z}$ satisfying \eqref{greaves_congruences}. By \eqref{coprime_values} we have $\gcd(g(q),pD)=1$. Thus, if $\ell|pD$, then $\ell\nmid g(q)=F(q,1)$, so we may take $\alpha=q,\beta=1$. For $\ell=q$, we have $q\nmid g(q)=F(q,1)$, as shown in the proof of Proposition \ref{S_reverse_direction}. Suppose now that $\ell\nmid pqD$, so that $\ell\nmid a$. We claim that there exists $\alpha\equiv q\mathfrak pmod {pD}$ such that $\ell\nmid F(\alpha,1)$. If not, then $\ell|f(a\alpha+b)$ for every such $\alpha$. Since $a$ is invertible modulo $\ell$, this implies that $\ell$ is a fixed divisor of $f(x)$, and hence divides $D$, which is a contradiction.
Let $P$ be the set of all pairs of integers $(\alpha, \beta)$ such that $\alpha\equiv q\mathfrak pmod{pD}$, $\beta\equiv 1\mathfrak pmod{pD}$, and $F(\alpha,\beta)$ is squarefree. By Proposition \ref{greaves_prop}, $P$ is an infinite set. We claim that there is a well-defined map
\[\mathfrak psi:P\rightarrow\{d\in S_{\mathbb{Q}}(f)\;|\;d\equiv r\mathfrak pmod* p\},\;(\alpha,\beta)\mapsto F(\alpha,\beta)\delta.\]
Given $(\alpha,\beta)\in P$, let $\lambda=\alpha/\beta$ and $d=F(\alpha,\beta)\delta$. Then $\beta^{\mathfrak partial+\sigma} g(\lambda)=F(\alpha,\beta)$, so $\operatorname{sqf}(g(\lambda))=F(\alpha,\beta)$. Note that $F(\alpha,\beta)$ is relatively prime to $D$: if $\ell$ is a prime dividing $D$, then $\ell\nmid g(q)=F(q,1)$, and therefore $\ell\nmid F(\alpha,\beta)$ since $F(\alpha,\beta)\equiv F(q,1)\mathfrak pmod\ell$. Thus $d$ is squarefree. Using \eqref{f_value} we obtain
\[\beta^{\mathfrak partial+\sigma}f(a\lambda+b)=d(qu)^2p^{2v+2\varepsilon},\] from which it follows that $\operatorname{sqf}(f(a\lambda+b))=d$, and therefore $d\in S_{\mathbb{Q}}(f)$. We claim that $d\equiv r\mathfrak pmod p$. Since $\beta$ is a unit modulo $p$, $\lambda$ belongs to the local ring $\mathbb{Z}_{(p)}$. In this ring we have the congruence $a\lambda+b\equiv b\mathfrak pmod{p^{2(v+\varepsilon)+1}}$; hence, by the displayed equation above,
\[d(qu)^2p^{2v+2\varepsilon}\equiv \beta^{\mathfrak partial+\sigma}f(b)\mathfrak pmod*{p^{2(v+\varepsilon)+1}}.\]
The definition of $b$ implies that $f(b)\equiv hz_0^2p^{2(v+\varepsilon)}\mathfrak pmod{p^{2(v+\varepsilon)+1}}$. It follows that $d(qu)^2\equiv\beta^{\mathfrak partial+\sigma}hz_0^2\equiv\beta^{\mathfrak partial+\sigma}rz_0^2\mathfrak pmod p$. Since $qu\equiv z_0\mathfrak pmod p$ and $\beta\equiv 1\mathfrak pmod p$, we obtain $d\equiv r\mathfrak pmod p$, as claimed. This proves that the map $\mathfrak psi$ is well defined.
We end by showing that $\mathfrak psi$ has finite fibers. For this purpose it suffices to show that $F$ can represent a given nonzero integer only finitely many times. If $F$ is irreducible, then, since $\deg F\ge\deg f\ge 3$, this follows from a well-known theorem of Thue. If $F$ is reducible, the proof of this finiteness statement is a straightforward exercise.
\end{proof}
\subsection*{Proof of Theorem \ref{unconditional_thm}}
Assuming that every irreducible factor of $f(x)$ has degree at most six, we must show that for every large enough prime $p$, and every integer $r$ not divisible by $p$, there exist infinitely many $d\in S_{\mathbb{Q}}(f)$ such that $d\equiv r\mathfrak pmod p$.
By the results of \cite{krumm} mentioned in the introduction, we may assume that $\deg f\ge 3$. As seen in the proof of Theorem \ref{abc_thm}, if $p$ satisfies the conditions \eqref{p_sufficient_conditions}, then $S(r,0)$ holds for every integer $r\not\equiv 0\mathfrak pmod p$. Applying Proposition \ref{unconditional_prop} we obtain the desired result.
\mathfrak qed
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\end{document} | math | 27,586 |
\begin{document}
\title[Volume preserving Gauss curvature flow]{Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space}
\author[Y. Wei]{Yong Wei}
\address{School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, P.R. China}
\email{\href{mailto:[email protected]}{[email protected]}}
\author[B. Yang]{Bo Yang}
\address{Institute of Mathematics, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences,
Beijing, 100190, P. R. China}
\email{\href{mailto:[email protected]}{[email protected]}}
\author[T. Zhou]{Tailong Zhou}
\address{School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, P.R. China}
\email{\href{mailto:[email protected]}{[email protected]}}
\subjclass[2010]{53C44; 53C42}
\keywords{Volume preserving Gauss Curvature flow, Hyperbolic space, Convex, Curvature measures, Alexandrov reflection.}
\begin{abstract}
We consider the volume preserving flow of smooth, closed and convex hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1} (n\geq 2)$ with the speed given by arbitrary positive power $\alpha$ of the Gauss curvature. We prove that if the initial hypersurface is convex, then the smooth solution of the flow remains convex and exists for all positive time $t\in [0,\infty)$. Moreover, we apply a result of Kohlmann which characterises the geodesic ball using the hyperbolic curvature measures and an argument of Alexandrov reflection to prove that the flow converges to a geodesic sphere exponentially in the smooth topology. This can be viewed as the first result for non-local type volume preserving curvature flows for hypersurfaces in the hyperbolic space with only convexity required on the initial data.
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction}
Let $X_0:M^n\to\mathbb{H}^{n+1} (n\geq 2)$ be a smooth embedding such that $M_0=X_0(M)$ is a closed and convex hypersurface in the hyperbolic space $\mathbb{H}^{n+1}$. We consider the smooth family of embeddings $X:M^n\times [0,T)\to \mathbb{H}^{n+1}$ satisfying
\begin{equation}\label{flow-VMCF}
\left\{\begin{aligned}
\frac{\partial}{\partial t}X(x,t)=&~(\phi(t)-K^{\alpha})\nu(x,t),\\
X(\cdot,0)=&~X_0(\cdot),
\end{aligned}\right.
\end{equation}
where $\alpha>0$, $\nu$ is the unit outer normal of $M_t=X(M,t)$, $K$ is the Gauss curvature of $M_t$ and
\begin{equation}\label{eqphi}
\phi(t)=\frac{1}{|M_t|}\int_{M_t}K^{\alpha}\,\mathrm{d}\mu_t
\end{equation}
such that the domain $\Omega_t$ enclosed by $M_t$ has a fixed volume $|\Omega_t|=|\Omega_0|$ along the flow \eqref{flow-VMCF}.
\begin{defn}
Let $M^n$ be a smooth closed hypersurface in the hyperbolic space $\mathbb{H}^{n+1}$. Denote the principal curvatures of $M$ by $\kappa=(\kappa_1,\cdots,\kappa_n)$.
(i). $M$ is said to be horospherically convex (or simply called $h$-convex), if its principal curvatures satisfy $\kappa_i\geq 1$ for all $i=1,\cdots,n$ everywhere on $M$. Equivalently, $M$ is h-convex if for any point $p\in M$ there exists a horosphere enclosing $M$ and touching $M$ at $p$. We also say that a bounded domain $\Omega\subset \mathbb{H}^{n+1}$ is $h$-convex if its boundary $\partial\Omega$ is $h$-convex.
(ii). $M$ is said to be positively curved (or called having positive sectional curvatures), if its principal curvatures satisfy
\begin{equation*}
\kappa_i\kappa_j>1,\qquad \forall~i\neq j
\end{equation*}
everywhere on $M$.
(iii). $M$ is said to be convex if its principal curvatures satisfy $\kappa_i>0$ for all $i=1,\cdots,n$ everywhere on $M$. We also say that a bounded domain $\Omega\subset \mathbb{H}^{n+1}$ is convex if its boundary $\partial\Omega$ is convex.
\end{defn}
As the main result of this paper, we prove the following convergence result for the flow \eqref{flow-VMCF} with convex initial hypersurfaces:
\begin{thm}\label{theo}
Let $X_0:M^n\to \mathbb{H}^{n+1} (n\geq 2)$ be a smooth embedding such that $M_0=X_0(M)$ is a closed and convex hypersurface in $\mathbb{H}^{n+1}$. Then for any $\alpha>0$, the volume preserving flow \eqref{flow-VMCF} has a unique smooth convex solution $M_t$ for all time $t\in[0,\infty)$, and the solution $M_t$ converges smoothly and exponentially as $t\to \infty$ to a geodesic sphere of radius $\rho_{\infty}$ which encloses the same volume as $M_0$.
\end{thm}
The volume preserving mean curvature flow
\begin{equation}\label{eqH}
\frac{\partial}{\partial t}X(x,t)=(\phi(t)-H)\nu(x,t)
\end{equation}
was introduced by Huisken \cite{Hui87} in 1987 for convex hypersurfaces in the Euclidean space $\mathbb{R}^{n+1}$, and it has been proved that for any smooth convex initial hypersurface, the solution of the flow \eqref{eqH} converges smoothly to a round sphere. The analogue of the flow \eqref{eqH} in the hyperbolic space $\mathbb{H}^{n+1}$ was studied by Cabezas-Rivas and Miquel \cite{Cab-Miq2007} in 2007 assuming a stronger condition that the initial hypersurface is $h$-convex. There are further generalizations of the flow \eqref{eqH} in $\mathbb{H}^{n+1}$ with the mean curvature $H$ replaced by more general curvature functions including powers of the $k$th mean curvature $\sigma_k(\kappa),k=1,\cdots,n$ (see \cite{BenWei,Be-Pip2016,GLW-CAG,Mak2012,WX} for instance). In all cases, the $h$-convexity is assumed on the initial hypersurfaces. This is mainly because that the $h$-convexity is convenient for the analysis of the curvature evolution equations and so that the tensor maximum principle or the constant rank theorem can be applied to derive that the $h$-convexity is preserved along the flow. Moreover, the $h$-convexity is strong enough geometrically such that the outer radius of the enclosed domain is uniformly controlled by its inner radius (see \cite{BM99}), and then the a prior $C^0$ estimate of the flow can be proved easily. In a recent work \cite{BenChenWei}, the first author with Andrews and Chen proved the smooth convergence of volume preserving $k$th mean curvature flows in $\mathbb{H}^{n+1}$ for initial hypersurfaces with positive sectional curvatures. This condition is weaker than $h$-convexity but still stronger than the convexity $\kappa_i>0.$
An open question in the field is whether the convexity ($\kappa_i>0$, $i=1,\cdots,n$) is sufficient to guarantee the smooth convergence of the volume preserving curvature flow in the hyperbolic space. Our result in Theorem \ref{theo} provides the first affirmative answer to this question. The curve case ($n=1$) of Theorem \ref{theo} was also treated recently by the first and second authors in \cite{WY2022}, where the idea is that in this case there is only one curvature and we can calculate the curvature evolution explicitly to derive the convexity preserving immediately. However, the higher dimensional case ($n\geq 2$) as stated in Theorem \ref{theo} requires on some new ideas. The key step is again to show the convexity preserving, but the tensor maximum principle no longer works when applying to the evolution of the second fundamental form directly. Instead, we shall use the projection method via the Klein model of the hyperbolic space that was described earlier by the first author and Andrews in \cite{BenWei}. Based on this, we can treat the flow \eqref{flow-VMCF} as an equivalent flow in the Euclidean space and this allows us to derive a time-dependent positive lower bound on the principal curvatures, and a time-dependent upper bound on the Gauss curvature $K$. A continuation argument then implies that the flow exists for all positive time $t>0$.
To study the asymptotical behavior of the flow as time $t\to\infty$, we shall use some machinery from the theory of convex bodies in $\mathbb{H}^{n+1}$. In particular, we use the Blaschke selection theorem to show that for a subsequence of times $t_i\to\infty$, the enclosed domain $\Omega_{t_i}$ of $M_{t_i}$ converges in Hausdorff sense to a limit convex domain $\hat{\Omega}$. We then apply the monotonicity of a certain quermassintegral to show that $\hat{\Omega}$ satisfies
\begin{equation}\label{s1-cur}
\Phi_0(\hat{\Omega},\beta)=c\Phi_n(\hat{\Omega},\beta)
\end{equation}
for any Borel set $\beta$ in $\mathbb{H}^{n+1}$, where $\Phi_0$ and $\Phi_n$ are the hyperbolic curvature measures of $\hat{\Omega}$, and $c$ is a constant. A theorem of Kohlmann \cite{Koh1998} which characterises the geodesic ball in the hyperbolic space $\mathbb{H}^{n+1} (n\geq 2)$ using the equation \eqref{s1-cur} for curvature measures can be used to conclude that $\hat{\Omega}$ is a geodesic ball. Moreover, if we denote the center of the inner ball of $\Omega_t$ as $p_t$, then using the Alexandrov reflection argument and the subsequential Hausdorff convergence, we can deduce that for all time $t\to\infty$, the points $p_t$ converge to a fixed point $p$, which is the center of the ball $\hat{\Omega}$. With the help of this, we can prove the uniform positive bounds for the principal curvatures and then obtain the smooth convergence of the flow to a geodesic sphere.
The convergence result in Theorem \ref{theo} has an application in the Alexandrov-Fenchel inequalities for quermassintegrals in the hyperbolic space (see \S \ref{sec2} for the definitions). Along the flow \eqref{flow-VMCF}, we find that the $(n-1)$th quermassintegral $\mathcal{A}_{n-1}(\Omega_t)$ is monotone decreasing in time $t$. As the volume $|\Omega_t|$ is preserved, the smooth convergence proved in Theorem \ref{theo} yields the following Alexandrov-Fenchel inequality of quermassintegrals for convex domains in the hyperbolic space:
\begin{cor}\label{coro}
Suppose that $\Omega\subset\mathbb{H}^{n+1}$ is a bounded weakly convex domain with smooth boundary $\partial\Omega$. Then the following inequality holds:
\begin{equation}\label{eqA-F}
\mathcal{A}_{n-1}(\Omega)\geq \psi_n\left(|\Omega|\right),
\end{equation}
where $\psi_n:[0,\infty)\to \mathbb{R}$ is a strictly increasing function such that the equality is achieved for geodesic balls. Moreover, the equality holds in \eqref{eqA-F} if and only if $\Omega$ is a geodesic ball.
\end{cor}
Here we say that a hypersurface is weakly convex if all principal curvatures $\kappa_i\geq 0$. We remark that the inequality \eqref{eqA-F} for convex domains in the hyperbolic space was proved also by Brendle, Guan and Li in a preprint \cite{BGL} using a different method. The argument we present here provides a new proof of \eqref{eqA-F}.
The paper is organized as follows: In $\S$\ref{sec2}, we collect some preliminaries including the geometry of hypersufaces in the hyperbolic space, the evolution equations along the flow \eqref{flow-VMCF}, the quermassintegrals and curvature measures in the hyperbolic space. In \S \ref{subsec}, we project the flow to the Euclidean space via the Klein model. This projection has the advantage that the second fundamental forms of the corresponding solutions have a simple relation. In order to show that convexity is preserving along the flow \eqref{flow-VMCF}, it suffices to show that the corresponding flow \eqref{projected flow eq} in the Euclidean space preserves the convexity. In $\S$\ref{sec3}, we give the a priori $C^0$ and $C^1$ estimates of the flow \eqref{flow-VMCF} and the projected flow \eqref{projected flow eq}, which follows by a similar argument as in previous work \cite{BenWei,BenChenWei}. In $\S$\ref{sec4}, we prove the positive lower bound for the principal curvatures along the flow \eqref{flow-VMCF}, by proving the corresponding estimate along the flow \eqref{projected flow eq} in the Euclidean space. In \S \ref{sec.upK}, we adapt Tso's\cite{Tso85} technique to prove an upper bound on the Gauss curvature on any finite time interval. Since we only assumed convexity, the terms involving global term $\phi(t)$ need to be carefully treated. This implies two-sided curvature bounds of the solution on any finite time interval, and then we obtain the long time existence of the flow \eqref{flow-VMCF} in $\S$\ref{sec5}. In \S \ref{sec.hau}, we show the subsequential Hausdorff convergence of $M_t$ and the convergence of the center of the inner ball of $\Omega_t$ to a fixed point. Finally, in \S \ref{final}, we complete the proofs of Theorem \ref{theo} and Corollary \ref{coro}.
\end{ack}
\section{Preliminaries}\label{sec2}
In this section, we collect some preliminary results concerning the geometry of hypersurfaces in the hyperbolic space, the evolution equations for geometric quantities along the flow \eqref{flow-VMCF}, the quermassintegrals and curvature measures in the hyperbolic space.
\subsection{Hyperbolic space}
The hyperbolic space $\mathbb{H}^{n+1}$ can be viewed as a warped product manifold $(\mathbb{R}_{+}\times \mathbb{S}^n,g_{\mathbb{H}^{n+1}})$ with
\begin{equation*}
g_{\mathbb{H}^{n+1}}=d\rho^2+\sinh^2\rho g_{\mathbb{S}^n},
\end{equation*}
where $g_{\mathbb{S}^n}$ is the round metric on unit sphere $\mathbb{S}^n$. Let $D$ be the Levi-Civita connection on $\mathbb{H}^{n+1}$. The vector field $V=\sinh \rho\partial_\rho$ is a conformal Killing field satisfying $DV=\cosh \rho g_{\mathbb{H}^{n+1}}.$
Let $\Omega$ be a convex domain in $\mathbb{H}^{n+1}$ with a smooth boundary $M=\partial\Omega$. Then $M$ is a smooth convex hypersurface in $\mathbb{H}^{n+1}$. We denote by $g_{ij}, h_{ij}$ and $\nu$ the induced metric, the second fundamental form and the unit outward normal vector of $M$ respectively. As $M$ is convex, there exists a point $p_0\in\Omega$ such that $M$ is star-shaped with respect to $p_0$ and can be written as a radial graph $M=\{(\rho(\theta),\theta),~\theta\in \mathbb{S}^n\}$ with respect to $p_0$ for a smooth function $\rho\in C^\infty(\mathbb{S}^n)$. Equivalently, the support function of $M$ with respect to $p_0\in \Omega$ defined by
\begin{equation*}
u=\langle V,\nu\rangle=\langle \sinh \rho\partial_\rho,\nu\rangle
\end{equation*}
is positive everywhere on $M$. It is well known that (see e.g.\cite{GL15})
\begin{align*}
g_{ij}&=\rho_{i}\rho_{j}+\sinh^2\rho\sigma_{ij},\\
h_{ij}&=\frac{1}{\sqrt{\sinh^2\rho+|\bar{\nabla} \rho|^2}}(-(\sinh \rho)\rho_{ij}+2(\cosh \rho)\rho_i \rho_j+\sinh^2\rho\cosh \rho\sigma_{ij}),\\
\nu=&\frac{1}{\sqrt{1+|\bar{\nabla}\rho|^2/\sinh^2\rho}}\left(1,-\frac{\rho_1}{\sinh^2\rho},\cdots,-\frac{\rho_n}{\sinh^2\rho}\right),
\end{align*}
where $\bar{\nabla}$ is the covariant derivative on $\mathbb{S}^n$ with respect to the round metric $g_{\mathbb{S}^n}=(\sigma_{ij})$ and $\rho_i=\bar{\nabla}_i\rho, \rho_{ij}=\bar{\nabla}_i\bar{\nabla}_j\rho$. It follows that the Gauss curvature $K$ of $M$ can be expressed as a function of $\rho$ and its first and second derivatives:
\begin{equation}\label{eq-Gauss}
K=\frac{\det h_{ij}}{\det g_{ij}}=\frac{\det(-\sinh \rho\rho_{ij}+2\cosh \rho\rho_i \rho_j+\sinh^2\rho\cosh \rho\sigma_{ij})}{(\sinh^2\rho+|\bar{\nabla} \rho|^2)^{\frac{n+2}{2}}(\sinh \rho)^{2n-2}}.
\end{equation}
\subsection{Evolution equations}
Let $M_t$ be a smooth solution to the curvature flow \eqref{flow-VMCF} in the hyperbolic space $\mathbb{H}^{n+1}$. We have the following evolution equations (see \cite{BenWei}) for the induced metric $g_{ij}$, the area element $d{\mu_t}$ and the speed function $K^{\alpha}$:
\begin{align}
\frac{\partial}{\partial t}g_{ij}&=2\left(\phi(t)-K^{\alpha}\right)h_{ij},\label{eq-g}\\
\frac{\partial}{\partial t}d\mu_t&=H\left(\phi(t)-K^{\alpha}\right)d\mu_t,\label{eq-dmu}\\
\frac{\partial}{\partial t}K^{\alpha}&=\alpha K^{\alpha-1}\dot{K}^{ij}\left(\nabla_i\nabla_j{K^{\alpha}}+(K^{\alpha}-\phi(t))(h_i^k h_{kj}-\delta_{ij})\right),\label{eq-KK}
\end{align}
where $\dot{K}^{ij}$ denote the derivatives of $K$ with respect to the components of the second fundamental form, and $\nabla$ denotes the Levi-Civita connection on $M_t$ with respect to the induced metric $g_{ij}$. On the time interval when $M_t$ is star-shaped with respect to some point $p_0$, the support function $u(x,t)=\langle{\sinh \rho_{p_0}(x)\partial_{\rho_{p_0}},\nu}\rangle$ of $M_t$ with respect to $p_0$ evolves by (see \cite[Lemma 4.3]{BenWei}):
\begin{equation}\label{equeq}
\frac{\partial}{\partial t}{u}=\alpha K^{\alpha-1}\dot{K}^{ij}\nabla_i\nabla_j{u}+\cosh{\rho_{p_0}}(x)\left(\phi(t)-(n\alpha+1)K^{\alpha}\right)+\alpha K^{\alpha}Hu.
\end{equation}
\subsection{Quermassintegrals}
Let $\mathcal{K}(\mathbb{H}^{n+1})$ be the set of compact convex sets in $\mathbb{H}^{n+1}$ with nonempty interior. For any $\Omega\in \mathcal{K}(\mathbb{H}^{n+1})$ , the quermassintegrals of $\Omega$ are defined as follows (see \cite[Definition 2.1]{Sol05} \footnote{Note that the definition for $\mathcal{A}_k$ given here is the same as that for $W_{k+1}$ given in \cite{Sol05} up to a constant. In fact, we have $\mathcal{A}_k=(n+1)\binom{n}{k}W_{k+1}$.}):
\begin{equation}\label{Wk}
\mathcal{A}_k(\Omega)=(n-k)\binom{n}{k}\frac{\omega_k\cdots\omega_0}{\omega_{n-1}\cdots\omega_{n-k-1}}\int_{\mathcal{L}_{k+1}}{\chi(L_{k+1}\cap\Omega)dL_{k+1}}
\end{equation}
for $k=0,1,\dots,n-1$, where $\omega_k=|\mathbb{S}^k|$ denotes the area of $k$-dimensional unit sphere, $\mathcal{L}_{k+1}$ is the space of $(k+1)$-dimensional totally geodesic subspaces $L_{k+1}$ in $\mathbb{H}^{n+1}$ and $\binom{n}{k}=\frac{n!}{k!(n-k)!}$. The function $\chi$ is defined to be 1 if $L_{k+1}\cap\Omega\neq\emptyset$ and to be 0 otherwise. In particular, we have
\begin{equation*}
\mathcal{A}_{-1}(\Omega)=|\Omega|,\qquad \mathcal{A}_0(\Omega)=|\partial\Omega|,\qquad \mathcal{A}_{n}(\Omega)=\frac{\omega_{n}}{n+1}.
\end{equation*}
If the boundary $M=\partial\Omega$ is smooth (or at least of class $C^2$), we can define the principal curvatures $\kappa=(\kappa_1,\dots,\kappa_n)$ as the eigenvalues of the Weingarten matrix $\mathcal{W}$ of $M$. For each $k\in\{1,\dots,n\}$, the $k$th mean curvature $\sigma_k$ of $M$ is then defined as the $k$th elementary symmetric functions of the principal curvatures of $M$:
\begin{equation*}
\sigma_k=\sum_{1\leq i_1<\cdots<i_k\leq n}{\kappa_{i_1}\cdots\kappa_{i_k}}.
\end{equation*}
These include the mean curvature $H=\sigma_1$ and Gauss curvature $K=\sigma_n$ as special cases.
In the smooth case, the quermassintegrals and the curvature integrals of a smooth convex domain $\Omega$ in $\mathbb{H}^{n+1}$ are related as follows:
\begin{align}
\mathcal{A}_1(\Omega)=&\int_{\partial\Omega}{\sigma_1}d\mu-n\mathcal{A}_{-1}(\Omega),\label{eq-V1}\\
\mathcal{A}_k(\Omega)=&\int_{\partial\Omega}{\sigma_k}d\mu-\frac{n-k+1}{k-1}\mathcal{A}_{k-2}(\Omega),\quad k=2,\cdots,n \label{eq-VW}.
\end{align}
The quermassintegrals for smooth domains satisfy a nice variational property (see \cite{BA97}):
\begin{equation}\label{eqWk}
\frac{d}{dt}\mathcal{A}_k(\Omega_t)=(k+1)\int_{M_t}\eta \sigma_{k+1}d\mu_t,\quad k=0,\cdots,n-1
\end{equation}
along any normal variation with velocity $\eta$.
The quermassintegrals defined by \eqref{Wk} are monotone with respect to inclusion of convex sets. That is, if $E,F\in \mathcal{K}(\mathbb{H}^{n+1})$ satisfy $E\subset F$, we have
\begin{equation}\label{s2.Akmo}
\mathcal{A}_k(E)\leq \mathcal{A}_k(F)
\end{equation}
for all $k=0,1,\cdots,n$. Moreover, they are continuous with respect to the Hausdoff distance. Recall that the Hausdorff distance between two convex sets $\Omega, L\in \mathcal{K}(\mathbb{H}^{n+1})$ is defined as
\begin{equation*}
\mathrm{dist}_{\mathcal{H}}(\Omega,L):=\mathrm{inf}\{\lambda>0:\Omega\subset B_{\lambda}(L)\, \text{and}\, L\subset B_{\lambda}(\Omega)\},
\end{equation*}
where $ B_{\lambda}(L):=\{q\in \mathbb{H}^{n+1}|~\mathrm{d}_{\mathbb{H}^{n+1}}(q,L)<\lambda\}$.
\begin{lem}\label{inquer}
Suppose that $\{\Omega_i\}_{i=1}^\infty, \Omega\in\mathcal{K}(\mathbb{H}^{n+1})$ and $\Omega_i$ converges to $\Omega$ in the Hausdorff sense. Then we have $\lim_{i\to\infty}{\mathcal{A}_k(\Omega_i)}=\mathcal{A}_k(\Omega)$ for all $k=-1,0,\cdots, n-1$.
\end{lem}
To see this, we use an expression of the measure $dL_{k+1}$. Every totally geodesic subspace $L_{k+1}$ in $\mathbb{H}^{n+1}$ is determined by its orthogonal subspace $L_{n-k}[o]$, which passes through the origin $o\in\mathbb{H}^{n+1}$, and by the intersection point $m=L_{k+1}\cap L_{n-k}[o]$. In this way, $\mathcal{L}_{k+1}$ can be identified as a bundle over the Grassmannian manifolds $G_{n+1,n-k}$ which consists of all subspaces $L_{n-k}[o]$, and then $dL_{k+1}$ is written as (see \cite{San04})
\begin{equation}
dL_{k+1}=\cosh^{k+1}(\rho) d\mu_{n-k} dL_{n-k}[o],
\end{equation}
where $d\mu_{n-k}$ is the volume element on $L_{n-k}[o]$, $dL_{n-k}[o]$ is the volume element on $G_{n+1,n-k}$ and $\rho=d_{\mathbb{H}^{n+1}}(x,o)$ denotes the distance of a point $x$ in $L_{n-k}[o]$ to the origin. Then we can rewrite the integral in \eqref{Wk} equivalently as
\begin{equation}\label{s2.Ak2}
\int_{\mathcal{L}_{k+1}}{\chi(L_{k+1}\cap\Omega)dL_{k+1}}=\int_{G_{n+1,n-k}}\left(\int_{\Pi_{L_{n-k}[o]}(\Omega)}\cosh^{k+1}(\rho) d\mu_{n-k}\right)dL_{n-k}[o],
\end{equation}
where $\Pi_{L_{n-k}[o]}:\mathbb{H}^{n+1}\to\mathbb{H}^{n+1}$ is the orthogonal projection over $L_{n-k}[o]\in G_{n+1,n-k}$ along geodesics. Then the conclusion in Lemma \ref{inquer} follows from \eqref{s2.Ak2} immediately.
\subsection{Curvature measures in $\mathbb{H}^{n+1}$}For any $\Omega
\in \mathcal{K}(\mathbb{H}^{n+1})$ and $\varepsilon>0$, define the parallel set
\begin{equation*}
\Omega_{\varepsilon}=\{x\in\mathbb{H}^{n+1}|~d_{\mathbb{H}^{n+1}}(x,\Omega)\leq\varepsilon\}.
\end{equation*}
The maps $f_\Omega:\mathbb{H}^{n+1}\setminus \Omega\to\partial \Omega$ and $F_\Omega:\mathbb{H}^{n+1}\setminus \Omega\to T_{\partial \Omega}\mathbb{H}^{n+1}$:
\begin{equation*}
d_{\mathbb{H}^{n+1}}(f_\Omega(x),x)=d_{\mathbb{H}^{n+1}}(x,\Omega) \quad \text{and} \quad x=\mathrm{exp}_{f_\Omega(x)}^{\mathbb{H}^{n+1}}(d(\Omega,x)F_{\Omega}(x))
\end{equation*}
are well-defined. For any Borel set $\beta\subset\mathbb{H}^{n+1}$, define the local parallel set
\begin{equation*}
M_{\varepsilon}(\Omega,\beta)=f_{\Omega}^{-1}(\beta\cap \partial \Omega)\cap(\Omega_{\varepsilon}\setminus \Omega).
\end{equation*}
\begin{figure}
\caption{Local parallel set $M_{\varepsilon}
\end{figure}
Following \cite{Koh1991}, given a convex set $\Omega$ and $\varepsilon>0$, let us define a Radon measure $\mu_\varepsilon$ on the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{H}^{n+1})$ of the hyperbolic space by
\begin{equation*}
\mu_{\varepsilon}(\Omega,\beta)=\mathrm{Vol}_{\mathbb{H}^{n+1}}(M_{\varepsilon}(\Omega,\beta)).
\end{equation*}
Set
\begin{equation*}
l_{k+1}(t)=\int_0^{t}{\sinh^{k}(s)\cosh^{n-k}(s) ds},\quad k=0,1,\cdots,n
\end{equation*}
and $l_0\equiv 1$. Then there holds the following Steiner-type formula:
\begin{equation*}
\mu_{\varepsilon}(\Omega,\beta)=\sum_{k=0}^{n}l_{n+1-k}(\varepsilon)\Phi_{k}(\Omega,\beta),\quad \forall\beta\in\mathcal{B}(\mathbb{H}^{n+1}).
\end{equation*}
The coefficients $\Phi_0(\Omega,\cdot),\cdots,\Phi_n(\Omega,\cdot)$ are called the curvature measures of the convex body $\Omega$, which are Borel measures on $\mathbb{H}^{n+1}$.
When the boundary $\partial \Omega$ is smooth, $\Phi_k(\Omega,\cdot)$ has a nice expression:
\begin{equation*}
\Phi_k(\Omega,\beta)=\int_{\partial \Omega\cap\beta}\sigma_{n-k}d\mu,
\end{equation*}
where $\sigma_{n-k}$ is the $(n-k)$th mean curvature of $\partial \Omega$. We refer \cite{Koh1991} to the readers for a general integral representation for $\Phi_k(\Omega,\cdot)$.
As in the classical setting of convex bodies in the Euclidean space, the curvature measures introduced by Kohlmann are solid enough to be weakly continuous with respect to the topology induced on $\mathcal{K}(\mathbb{H}^{n+1})$.
\begin{thm}[\cite{VG-2019}]\label{s2.thmcurv}
Let $\{\Omega_j\}_{j=1}^{\infty}\subset \mathcal{K}(\mathbb{H}^{n+1})$ be a sequence of convex sets such that $\Omega_j\to \Omega$ as $j\to\infty$ in the Hausdorff topology. Then for every $k=0,\cdots,n$ we have
\begin{equation*}
\Phi_k(\Omega_j,\cdot)\to\Phi_k(\Omega,\cdot)
\end{equation*}
as $j\to\infty$, weakly in the sense of measure.
\end{thm}
The following characterization of geodesic balls via the hyperbolic curvature measures was proved by Kohlmann \cite{Koh1998} and can be viewed as a generalization of the classical Alexandrov Theorem in differential geometry.
\begin{thm}[\cite{Koh1998}]\label{gAT}
Let $n\geq 2$ and $k\in\{0,\dots,n-2\}$. Assume that $\Omega$ is a compact, connected, locally convex subset of $\mathbb{H}^{n+1}$ with nonempty interior satisfying
\begin{equation}\label{Ath}
\Phi_{k}(\Omega,\beta)=c\Phi_n(\Omega,\beta)
\end{equation}
for any Borel set $\beta\in\mathcal{B}(\mathbb{H}^{n+1})$, where $c>0$ is a constant, then $\Omega$ is a geodesic ball.
\end{thm}
Note that there is no smoothness assumption on the domain $\Omega$ in Theorem \ref{gAT}. When the boundary $\partial\Omega$ is smooth, the equation \eqref{Ath} means that the $(n-k)$th mean curvature $\sigma_{n-k}$ is a constant on $\partial\Omega$, and the conclusion of Theorem \ref{gAT} in this case reduces to the classical result of Montiel and Ros \cite{MS91}.
\section{Klein model and Projection}\label{subsec}
To investigate the flow \eqref{flow-VMCF} for convex hypersurfaces, it is convenient to project it to the Euclidean space $\mathbb{R}^{n+1}$ and apply the Gauss map parametrization as in the work \cite[\S 5]{BenWei} by the first author and Andrews. We briefly review the argument here and refer the readers to \cite{BenWei} for details.
Let us denote by $\mathbb{R}^{1,n+1}$ the Minkowski spacetime, that is the vector space $\mathbb{R}^{n+2}$ endowed with the Minkowski spacetime metric $\langle\cdot,\cdot\rangle$ given by
\begin{equation*}
\langle X,X\rangle~=~-X_0^2+\sum_{i=1}^{n+1}X_i^2
\end{equation*}
for any vector $X=(X_0,X_1,\cdots,X_{n+1})\in\mathbb{R}^{n+2}$. The hyperbolic space $\mathbb{H}^{n+1}$ is then
\begin{equation*}
\mathbb{H}^{n+1}=\{X\in\mathbb{R}^{1,n+1}~|~\langle X,X\rangle=-1,\ X_0>0\}.
\end{equation*}
The Klein model parametrizes the hyperbolic space using the unit disc, which induces a projection from an embedding $X:M^n\rightarrow\mathbb{H}^{n+1}$ to an embedding $Y:M^n\rightarrow B_1(0)\subset\mathbb{R}^{n+1}$ by
\begin{equation}\label{s3.proj}
X~=~\frac{(1,Y)}{\sqrt{1-|Y|^2}}.
\end{equation}
Let $\nu\in T\mathbb{H}^{n+1}, g^X_{ij}, h^X_{ij}, K^X$ and $N\in\mathbb{R}^{n+1}, g^Y_{ij}, h^Y_{ij}, K^Y$ denote the unit normal vectors, induced metrics, the second fundamental forms and Gauss curvatures of $X(M^n)\subset\mathbb{H}^{n+1}$ and $Y(M^n)\subset \mathbb{R}^{n+1}$ respectively. We have the following relations:
\begin{align}
h^X_{ij}&=\frac{h^Y_{ij}}{\sqrt{\left(1-|Y|^2\right)\left(1-\langle N,Y\rangle^2\right)}},\label{eq-hij}\\
g^X_{ij}&=\frac{1}{1-|Y|^2}\left(g^Y_{ij}+\frac{\langle Y,\partial_i Y\rangle\langle Y,\partial_j Y\rangle}{1-|Y|^2}\right).\label{eq-metric}
\end{align}
It follows from \eqref{eq-hij} that $X(M^n)$ is convex in $\mathbb{H}^{n+1}$ is equivalent to that $Y(M^n)$ is convex in $\mathbb{R}^{n+1}$. If we take the determinant on both sides of \eqref{eq-hij} and \eqref{eq-metric}, the Gauss curvatures satisfy
\begin{equation}\label{s3.K}
K^X=\left(\frac{1-|Y|^2}{1-\langle N,Y\rangle^2}\right)^{\frac{n+2}{2}}K^Y.
\end{equation}
Suppose that $X: M^n\times [0,T)\to \mathbb{H}^{n+1}$ is a smooth convex solution to the flow \eqref{flow-VMCF}. Then the corresponding solution $Y:M^n\times [0,T)\to B_1(0)\subset\mathbb{R}^{n+1}$ satisfies the evolution equation:
\begin{equation}\label{s3.evlY}
\frac{\partial}{\partial t}Y=\sqrt{(1-|Y|^2)(1-\langle N,Y\rangle^2)}\Big(\phi(t)-(K^X)^{\alpha}\Big)N.
\end{equation}
Since each $Y_t(M^n)=Y(M^n,t)\subset B_1(0)\subset \mathbb{R}^{n+1}$ is convex, we can parametrize $Y_t(M^n)$ using the Gauss map and the support function $s(z):=\langle Y_t(N^{-1}(z)),z\rangle$, where $N^{-1}:\mathbb{S}^n\to M^n$ is the inverse map of the Gauss map. Then $Y_t(M^n)$ is given by the embedding $Y:\mathbb{S}^n\to \mathbb{R}^{n+1}$ with (see \cite[\S 2]{Ur90})
\begin{equation}\label{s3.Y-Gas}
Y(z)=~s(z)z+\bar{\nabla} s
\end{equation}
where $\bar{\nabla}$ is the gradient with respect to the round metric $g_{\mathbb{S}^n}$ on $\mathbb{S}^n$. The derivative of this map is given by
\begin{equation*}
\partial_iY=~\mathfrak{r}_{ik}\sigma^{kl}\partial_lz
\end{equation*}
in local coordinates, where $ \mathfrak{r}_{ij}$ is given as follows
\begin{equation}\label{s2.r-def}
\mathfrak{r}_{ij}=~\bar{\nabla}_i\bar{\nabla}_js+s\sigma_{ij}.
\end{equation}
The eigenvalues $ \mathfrak{r}_i$ of $ \mathfrak{r}_{ij}$ with respect to the round metric $g_{\mathbb{S}^n}$ are the inverse of the principal curvatures $\kappa_i^Y$, i.e., $ \mathfrak{r}_i=1/{\kappa_i^Y}$, and are called the principal radii of curvature. The determinant of \eqref{s2.r-def} gives the Gauss curvature of $Y_t(M^n)$:
\begin{equation}\label{s3.Ky}
K^Y=\frac 1{\mathrm{det}(\mathfrak{r}_{ij})}.
\end{equation}
By the identity \eqref{s3.K}, we obtain
\begin{equation}\label{eq-K}
K^X=\left(\frac{1-(s^2+|\bar{\nabla} s|^2)}{1-s^2}\right)^\frac{n+2}{2}K^Y.
\end{equation}
The evolution equation \eqref{s3.evlY} is equivalent to the following scalar parabolic equation:
\begin{equation*}
\frac{\partial}{\partial t}s=~\sqrt{(1-s^2-|\bar{\nabla}s|^2)(1-s^2)}\Big(\phi(t)-(K^X)^{\alpha}\Big)
\end{equation*}
on the round sphere $\mathbb{S}^n$ for the support function $s(z,t)$ of $Y(M^n,t)$.
We summarize the results as the following lemma:
\begin{lem}
Assume that $X(M^n,t)\subset \mathbb{H}^{n+1}, t\in [0,T)$ is a smooth convex solution to the flow \eqref{flow-VMCF}. Then up to a tangential diffeomorphism, the corresponding solution $Y(M^n,t)\subset B_1(0)\subset \mathbb{R}^{n+1}$ and its support function $s(z,t)$ satisfy the following equations:
\begin{align}
\frac{\partial}{\partial t}Y=&~\sqrt{(1-|Y|^2)(1-\langle N,Y\rangle^2)}\Big(\phi(t)-(K^X)^{\alpha}\Big)N,\label{projected flow eq}\\
\frac{\partial}{\partial t}s=&~\sqrt{(1-s^2-|\bar{\nabla}s|^2)(1-s^2)}\Big(\phi(t)-(K^X)^{\alpha}\Big),\label{eq-sptf0}
\end{align}
where the Gauss curvature $K^X$ is related to the Gauss curvature $K^Y$ via the identities \eqref{s3.K} and \eqref{eq-K}.
\end{lem}
For simplicity of the notation, we denote $A$ and $B$ as the following functions:
\begin{equation}\label{eqAB}
\left\{\begin{aligned}
A=A(s,\bar{\nabla} s)&:=\sqrt{(1-s^2-|\bar{\nabla} s|^2)(1-s^2)},\\
B=B(s,\bar{\nabla} s)&:=(1-s^2-|\bar{\nabla} s|^2)^{\frac{n+2}{2}\alpha+\frac{1}{2}}(1-s^2)^{-\frac{n+2}{2}\alpha+\frac{1}{2}}.
\end{aligned}\right.
\end{equation}
Then \eqref{eq-sptf0} can be simplified as:
\begin{equation}
\frac{\partial}{\partial t}s=A\phi(t)-B(K^{Y})^{\alpha}.\label{eq-sptf}
\end{equation}
In \S \ref{sec4}, we will estimate the upper bound of the largest eigenvalues of $\mathfrak{r}_{ij}$ along the equation \eqref{eq-sptf} and this implies a positive lower bound on the principal curvatures of the solution $M_t$ to the original flow \eqref{flow-VMCF}. In the following lemma, we derive the evolution equation of $\mathfrak{r}_{ij}$ along the flow \eqref{eq-sptf}:
\begin{lem}\label{s3.lem2}
Let $s(z,t)$ be a smooth solution of the flow \eqref{eq-sptf}. Choose a local orthonormal frame $e_1,\cdots, e_n$ around the point $z\in \mathbb{S}^n$ such that $(\mathfrak{r}_{ij})$ is diagonal at $(z,t)$ and $\mathfrak{r}_{11}$ corresponds to the largest eigenvalue of $(\mathfrak{r}_{ij})$. Along the equation \eqref{eq-sptf}, we have the following evolution equation of $\mathfrak{r}_{11}$:
\begin{align}\label{s2.tau11}
\frac{\partial}{\partial t}\mathfrak{r}_{11}-F^{k\ell}\bar{\nabla}_k\bar{\nabla}_\ell\mathfrak{r}_{11}=&\phi(t)\Big(A_{s_1s_1}{\mathfrak{r}_{11}}^2+2A_{ss_1}s_1\mathfrak{r}_{11}-2A_{s_1s_1}\mathfrak{r}_{11}s+A_s\mathfrak{r}_{11}+A_{s_k}\bar{\nabla}_k{\mathfrak{r}_{11}}\notag\\
&+A+A_{ss}{s_1}^2-2A_{ss_1}ss_1-A_ss+A_{s_1s_1}s^2-A_{s_1}s_1\Big)\notag\\
&-(K^Y)^{\alpha}\Big(B_{s_1s_1}{\mathfrak{r}_{11}}^2+2B_{ss_1}s_1\mathfrak{r}_{11}-2B_{s_1s_1}\mathfrak{r}_{11}s+B_s\mathfrak{r}_{11}\notag\\
&+B+B_{ss}{s_1}^2-2B_{ss_1}ss_1-B_ss+B_{s_1s_1}s^2-B_{s_1}s_1\notag\\
&+B_{s_k}\bar{\nabla}_k{\mathfrak{r}_{11}}+\alpha B\mathfrak{r}^{kk}\mathfrak{r}^{\ell\ell}(\bar{\nabla}_1{\mathfrak{r}_{k\ell}})^2+{\alpha}^2 B(\sum_k\mathfrak{r}^{kk}\bar{\nabla}_1\mathfrak{r}_{kk})^2\notag\\
&-2\alpha \mathfrak{r}^{kk}\bar{\nabla}_1\mathfrak{r}_{kk}(B_{s_1}\mathfrak{r}_{11}+B_ss_1-B_{s_1}s)+\alpha B\mathfrak{r}^{kk}(\mathfrak{r}_{11}-\mathfrak{r}_{kk})\Big),
\end{align}
where $F^{k\ell}=\alpha B(K^Y)^{\alpha}\mathfrak{r}^{k\ell}$.
The notations such as $A_s, A_{s_k}$ are partial derivatives of $A$ with respect to its first and second arguments.
\end{lem}
\proof
We first recall that $\mathfrak{r}_{ij}$ satisfies a Codazzi-type identity
\begin{equation}\label{s2.codz}
\bar{\nabla}_k\mathfrak{r}_{ij}=\bar{\nabla}_i\mathfrak{r}_{kj}.
\end{equation}
This follows from the Ricci identity for commuting covariant derivatives on $\mathbb{S}^n$. Differentiating \eqref{s2.codz} and commuting derivatives, we have the Simons' identity for the second derivatives
\begin{equation}\label{s2.Sim}
\bar{\nabla}_{(i}\bar{\nabla}_{j)}\mathfrak{r}_{k\ell}=\bar{\nabla}_{(k}\bar{\nabla}_{\ell)}\mathfrak{r}_{ij}+\mathfrak{r}_{k\ell}\sigma_{ij}-\mathfrak{r}_{ij}\sigma_{k\ell},
\end{equation}
where the brackets denote symmetrisation. Let $(\mathfrak{r}^{ij})=(\mathfrak{r}_{ij})^{-1}$ be the inverse of $(\mathfrak{r}_{ij})$. The spatial derivatives of $K^Y$ satisfy
\begin{align}
\bar{\nabla}_iK^Y= & -\frac{1}{(\mathrm{det} (\mathfrak{r}_{ij}))^2} \bar{\nabla}_i\mathrm{det} (\mathfrak{r}_{ij})\nonumber\\
=& -\frac{1}{\mathrm{det} (\mathfrak{r}_{ij})}\mathfrak{r}^{k\ell}\bar{\nabla}_i\mathfrak{r}_{k\ell}\nonumber\\
= & -K^Y \mathfrak{r}^{k\ell}\bar{\nabla}_i\mathfrak{r}_{k\ell},\label{s2.dK}
\end{align}
and
\begin{align}
\bar{\nabla}_j\bar{\nabla}_iK^Y=&-K^Y \mathfrak{r}^{k\ell}\bar{\nabla}_j\bar{\nabla}_i\mathfrak{r}_{k\ell}-\bar{\nabla}_jK^Y \mathfrak{r}^{k\ell}\bar{\nabla}_i\mathfrak{r}_{k\ell}-K^Y \bar{\nabla_j}\mathfrak{r}^{k\ell}\bar{\nabla}_i\mathfrak{r}_{k\ell}\nonumber\\
=&-K^Y \mathfrak{r}^{k\ell}\bar{\nabla}_j\bar{\nabla}_i\mathfrak{r}_{k\ell}+K^Y\left(\mathfrak{r}^{k\ell}\bar{\nabla}_i\mathfrak{r}_{k\ell}\right)\left(\mathfrak{r}^{pq}\bar{\nabla}_j\mathfrak{r}_{pq}\right)\nonumber\\
&\quad +K^Y\mathfrak{r}^{kq}\mathfrak{r}^{p\ell}\bar{\nabla}_j\mathfrak{r}_{pq}\bar{\nabla}_i\mathfrak{r}_{k\ell}. \label{s2.d2K}
\end{align}
On the other hand, the spatial derivatives of $A(s,\bar{\nabla}s)$ can be calculated as
\begin{align}
\bar{\nabla}_iA= & A_ss_i+A_{s_k}s_{ki} \nonumber\\
=&A_{s_k}\mathfrak{r}_{ki}+A_ss_i-sA_{s_k}\sigma_{ki},\label{s2.dA}
\end{align}
and
\begin{align}
\bar{\nabla}_j\bar{\nabla}_iA =& A_ss_{ij}+A_{ss}s_is_j+A_{ss_\ell}s_{\ell j}s_i+A_{s_k s}s_{j}s_{ki}+A_{s_ks_\ell}s_{ki}s_{\ell j}+A_{s_k}s_{kij}\nonumber\\
=& A_s(\mathfrak{r}_{ij}-s\sigma_{ij})+A_{ss}s_is_j+A_{ss_\ell}(\mathfrak{r}_{\ell j}-s\sigma_{\ell j})s_i\nonumber\\
&+A_{s_k s}s_{j}(\mathfrak{r}_{ki}-s\sigma_{ki})+A_{s_ks_\ell}(\mathfrak{r}_{ki}-s\sigma_{ki})(\mathfrak{r}_{\ell j}-s\sigma_{\ell j})\nonumber\\
&+A_{s_k}\left(\bar{\nabla}_j\mathfrak{r}_{ki}-s_j\sigma_{ki}\right)\nonumber\\
=&A_{s_k}\bar{\nabla}_k\mathfrak{r}_{ij}+A_s\mathfrak{r}_{ij}+A_{ss_\ell}s_i\mathfrak{r}_{\ell j}+A_{s_k s}s_{j}\mathfrak{r}_{ki}+A_{s_ks_\ell}\mathfrak{r}_{ki}\mathfrak{r}_{\ell j}\nonumber\\
&-A_{s_k s_l}\mathfrak{r}_{ki}\sigma_{\ell j}s-A_{s_k s_\ell}\mathfrak{r}_{\ell j}\sigma_{ki}s-A_ss\sigma_{ij}+A_{ss}s_is_j-A_{ss_{\ell}}\sigma_{\ell j}ss_i\nonumber\\
&-A_{s_k s}\sigma_{ki}s_{j}s+A_{s_k s_\ell}\sigma_{ki}\sigma_{lj}s^2-A_{s_k}\sigma_{ki}s_j,\label{s2.d2A}
\end{align}
where $A_s, A_{s_k}$ are partial derivatives of $A$ with respect to its first and second arguments, and we used \eqref{s2.r-def} and the Codazzi-type identity \eqref{s2.codz}. The calculation for $B(s,\bar{\nabla}s)$ is similar.
Taking the spatial derivatives to the equation \eqref{eq-sptf} and using \eqref{s2.dK}$-$\eqref{s2.d2A}, we compute that
\begin{align}\label{s2.d2S}
\bar{\nabla}_1\bar{\nabla}_1(\partial_ts)
=&\phi(t)\bar{\nabla}_1\bar{\nabla}_1A-(K^{Y})^{\alpha}\bar{\nabla}_1\bar{\nabla}_1B-2\alpha (K^Y)^{\alpha-1}\bar{\nabla}_1K^Y\bar{\nabla}_1B\nonumber\\
&-\alpha (K^Y)^{\alpha-1}\bar{\nabla}_1\bar{\nabla}_1K^Y B-\alpha(\alpha-1)(K^Y)^{\alpha-2}(\bar{\nabla}_1K^Y)^2B\notag\\
=&\phi(t)\bar{\nabla}_1\bar{\nabla}_1A-(K^{Y})^{\alpha}\Big(\bar{\nabla}_1\bar{\nabla}_1B-2\alpha \mathfrak{r}^{k\ell}\bar{\nabla}_1\mathfrak{r}_{k\ell}\bar{\nabla}_1B\nonumber\\
&- \alpha B \mathfrak{r}^{k\ell}\bar{\nabla}_1\bar{\nabla}_1\mathfrak{r}_{k\ell}+\alpha^2 B \big(\sum_{k,\ell}\mathfrak{r}^{k\ell}\bar{\nabla}_1\mathfrak{r}_{k\ell}\big)^2+\alpha B\mathfrak{r}^{kq}\mathfrak{r}^{p\ell}\bar{\nabla}_1\mathfrak{r}_{pq}\bar{\nabla}_1\mathfrak{r}_{k\ell}\Big)\nonumber\\
=&\phi(t)\Big(A_{s_k}\bar{\nabla}_k{\mathfrak{r}_{11}}+A_s\mathfrak{r}_{11}+2A_{ss_1}s_1\mathfrak{r}_{11}+A_{s_1s_1}{\mathfrak{r}_{11}^2}-2A_{s_1s_1}\mathfrak{r}_{11}s\nonumber\\
&-A_ss+A_{ss}{s_1}^2-2A_{ss_1}ss_1+A_{s_1s_1}s^2-A_{s_1}s_1\Big)\notag\\ &-(K^Y)^{\alpha}\Big(B_{s_k}\bar{\nabla}_k{\mathfrak{r}_{11}}+B_s\mathfrak{r}_{11}+2B_{ss_1}s_1\mathfrak{r}_{11}+B_{s_1s_1}{\mathfrak{r}_{11}^2}-2B_{s_1s_1}\mathfrak{r}_{11}s\nonumber\\
&-B_ss+B_{ss}{s_1}^2-2B_{ss_1}ss_1+B_{s_1s_1}s^2-B_{s_1}s_1\notag\\
&-2\alpha \mathfrak{r}^{k\ell}\bar{\nabla}_1\mathfrak{r}_{k\ell}(B_{s_1}\mathfrak{r}_{11}+B_ss_1-B_{s_1}s)-\alpha B\mathfrak{r}^{k\ell}\bar{\nabla}_1\bar{\nabla}_1{\mathfrak{r}_{k\ell}}\notag\\
&+{\alpha}^2 B(\sum_{k,\ell}\mathfrak{r}^{k\ell}\bar{\nabla}_1\mathfrak{r}_{k\ell})^2+\alpha B\mathfrak{r}^{ik}\mathfrak{r}^{j\ell}\bar{\nabla}_1{\mathfrak{r}_{ij}}\bar{\nabla}_1{\mathfrak{r}_{k\ell}}\Big).
\end{align}
Let $F^{k\ell}=\alpha B(K^Y)^{\alpha}\mathfrak{r}^{k\ell}$. Taking derivatives of \eqref{s2.r-def},
\begin{align}\label{s3.1}
\frac{\partial}{\partial t}\mathfrak{r}_{11}-F^{k\ell}\bar{\nabla}_k\bar{\nabla}_\ell\mathfrak{r}_{11}=&\bar{\nabla}_1\bar{\nabla}_1(\partial_ts)+\partial_ts-F^{k\ell}\bar{\nabla}_k\bar{\nabla}_\ell\mathfrak{r}_{11}.
\end{align}
Substituting \eqref{eq-sptf}, \eqref{s2.Sim}, \eqref{s2.d2S} into \eqref{s3.1}, using the fact that $(\mathfrak{r}_{k\ell})$ is diagonal at the point we are considering and rearranging the terms, we obtain the equation \eqref{s2.tau11}.
\endproof
We also calculated the evolution equation of $r^2=|Y|^2=s^2+|\bar{\nabla}s|^2$.
\begin{lem}\label{s3.lem3}
Let $s(z,t)$ be a smooth solution of the flow \eqref{eq-sptf}. If we choose a local orthonormal frame $e_1,\cdots, e_n$ around the point $z\in \mathbb{S}^n$, then we have:
\begin{align}\label{s4.dr2}
&\frac{\partial}{\partial t}r^2-F^{k\ell}\bar{\nabla}_k\bar{\nabla}_\ell r^2\nonumber\\
=&2\phi(t)\Big(sA+A_s|\bar{\nabla} s|^2+A_{s_k}s_i\mathfrak{r}_{ki}-A_{s_k}s_ks\Big)\notag\\
-&2(K^Y)^{\alpha}\Big((1-n\alpha)sB+B_s|\bar{\nabla} s|^2+B_{s_k}s_i\mathfrak{r}_{ki}-B_{s_k}s_ks+\alpha B\sum_{k}{\mathfrak{r}_{kk}}\Big),
\end{align}
where $F^{k\ell}=\alpha B(K^Y)^{\alpha}\mathfrak{r}^{k\ell}$.
\end{lem}
\proof
Since $r^2=s^2+|\bar{\nabla}s|^2$, using \eqref{eq-sptf}, \eqref{s2.dK} and \eqref{s2.dA}, we have
\begin{align*}
\partial_tr^2=&2ss_t+2s_is_{it}\notag\\
=&2s(A\phi(t)-B(K^Y)^{\alpha})+2s_i\Big(\phi(t)\bar{\nabla}_iA-(K^Y)^{\alpha}\bar{\nabla}_iB+\alpha B(K^Y)^{\alpha}\mathfrak{r}^{k\ell}\bar{\nabla}_i\mathfrak{r}_{k\ell}\Big)\nonumber\\
=&2\phi(t)\left(sA+A_s|\bar{\nabla} s|^2+A_{s_k}s_i\mathfrak{r}_{ki}-A_{s_k}s_i\sigma_{ki}s\right)\nonumber\\
&-2(K^Y)^{\alpha}\left(sB+B_s|\bar{\nabla} s|^2+B_{s_k}s_i\mathfrak{r}_{ki}-B_{s_k}s_i\sigma_{ki}s-\alpha B\mathfrak{r}^{k\ell}\bar{\nabla}_i\mathfrak{r}_{k\ell}s_i\right),
\end{align*}
and
\begin{align*}
\bar{\nabla}_k\bar{\nabla}_\ell r^2=&2 \bar{\nabla}_k\left(ss_\ell+s_is_{i\ell}\right)\nonumber\\
=&2s_ks_\ell+2ss_{k\ell}+2s_{ik}s_{i\ell}+2s_is_{ik\ell}\nonumber\\
=&2\mathfrak{r}_{ik}\mathfrak{r}_{i\ell}-2s\mathfrak{r}_{k\ell}+2s_i\bar{\nabla}_i\mathfrak{r}_{k\ell}.
\end{align*}
Combining the above two equations gives the equation \eqref{s4.dr2}.
\endproof
We conclude the section with following remarks.
\begin{rem} (i). The projection method via Klein model has also been used recently by Chen and Huang \cite{ChenHuang} to study the contracting Gauss curvature flows in the hyperbolic space.
(ii). Denote $\mathcal{W}^X$ the Weingarten matrix of $X(M^n)\subset \mathbb{H}^{n+1}$. Then \eqref{eq-hij} and \eqref{eq-metric} imply that the inverse matrix $\mathcal{W}_X^{-1}$ of $\mathcal{W}^X$ satisfies
\begin{align}\label{s5:W-inv}
(\mathcal{W}_X^{-1})_{ij} =& (h^{-1}_X)^{jk}g_{ki}^X \nonumber\\
=&(h^{-1}_Y)^{kj}\left(g_{ki}^Y+\frac{\langle Y,\partial_iY\rangle\langle Y,\partial_kY\rangle }{(1-|Y|^2)}\right)\sqrt{\frac{1-\langle N,Y\rangle^2}{1-|Y|^2}}.
\end{align}
We can also express \eqref{s5:W-inv} using $s,\bar{\nabla}s$ and $\mathfrak{r}_{ij}$ as follows:
\begin{align}\label{s5:W-inv-2}
(\mathcal{W}_X^{-1})_{ij} =&\left(\sigma^{jq}+\frac{\langle \sigma^{ja}\bar{\nabla}_as,\sigma^{qb}\bar{\nabla}_bs\rangle}{1-s^2-|\bar{\nabla}s|^2}\right) \mathfrak{r}_{qi}\sqrt{\frac{1-s^2}{1-s^2-|\bar{\nabla}s|^2}}.
\end{align}
Since the eigenvalues of $(\mathcal{W}_X^{-1})_{ij}$ correspond to the reciprocal of the principal curvatures $\kappa_1,\cdots,\kappa_n$ of $X(M^n,t)\subset \mathbb{H}^{n+1}$, the above equation means that in order to estimate the lower bound of $\kappa_i$, it suffices to estimate the upper bound of $\mathfrak{r}_{ij}$ together with the $C^0, C^1$ estimates of $s$. This observation will be used in the next section. Moreover, we believe that \eqref{s5:W-inv-2} would also be useful for studying general fully nonlinear curvature flows in the hyperbolic space via the projection method described in this section.
\end{rem}
\section{A priori estimate}
In this section, we first review the $C^0$ and $C^1$ estimates of the solution $M_t$ to the flow \eqref{flow-VMCF}, which can be proved using a similar argument as in \cite{BenChenWei,BenWei}. Then we derive a positive lower bound on the principal curvatures $\kappa_i$ of the solution $M_t$, and prove the upper bound of the Gauss curvature $K$. Both of them may depend on the time $t$.
\subsection{$C^0$ and $C^1$ estimates}\label{sec3}
Let $M_0$ be a smooth, closed and convex hypersurface in the hyperbolic space. The flow \eqref{flow-VMCF} has a unique smooth solution $M_t$ for at least a short time and $M_t$ is convex on a possibly shorter time interval. Without loss of generality, we suppose that $M_t$ a smooth convex solution to the flow \eqref{flow-VMCF} starting from $M_0$ on a maximum time interval $[0,T)$, where $T\leq \infty$.
Denote by $\Omega_t$ as the domain enclosed by $M_t$ such that $\partial\Omega_t=M_t$. As the velocity of the flow \eqref{flow-VMCF} only depends on the curvature which is invariant under reflection with respect to a totally geodesic hyperplane, we can argue as in \cite[\S 4]{BenChenWei} using the Alexandrov reflection method to show that the inner radius and outer radius of $\Omega_t$ are uniformly bounded.
\begin{lem}\label{in-out}
Let $M_t$ be the smooth convex solution to the flow \eqref{flow-VMCF} on the time interval $t\in [0,T)$. Denote $\rho_-(t)$, $\rho_+(t)$ be the inner radius and outer radius of $\Omega_t$. Then there exist positive constants $c_1,\ c_2$ depending only on $n$, $\alpha$ and $M_0$ such that
\begin{equation}\label{in-out-radius}
0<c_1\leq\rho_-(t)\leq\rho_+(t)\leq c_2
\end{equation}
for all time $t\in[0,T)$.
\end{lem}
By \eqref{in-out-radius}, the inner radius of $\Omega_t$ is bounded below by a positive constant $c_1$. This implies that for each $t\in[0,T)$ there exists a geodesic ball $B_{c_1}(p_t)$ of radius $c_1$ centered at some point $p_t$ such that $B_{c_1}(p_t)\subset\Omega_t$. A similar argument as in \cite[Lemma 4.2]{BenWei} yields the existence of a geodesic ball with fixed center enclosed by the flow hypersurface on a suitable fixed time interval.
\begin{lem}\label{lem3.2}
Let $M_t$ be the smooth convex solution to the flow \eqref{flow-VMCF} on the time interval $[0,T)$. For any $t_0\in[0,T)$, let $B_{\rho_0}(p_0)$ be the inball of $\Omega_{t_0}$, where $\rho_0=\rho_-(t_0)$. Then
\begin{equation}\label{inball-t0}
B_{\rho_0/2}(p_0)\subset\Omega_t,\ \ t\in[t_0,\min\{T,t_0+\tau\})
\end{equation}
for some $\tau$ depending only on $n$, $\alpha$ and $M_0$.
\end{lem}
Since the projection \eqref{s3.proj} is a diffeomorphism, by the equation \eqref{eq-hij} we know that if the flow \eqref{flow-VMCF} has a smooth convex solution on the maximal time interval $[0,T)$, then so does the projected flow \eqref{projected flow eq}. Given $t_0\in[0,T)$ and let $B_{\rho_0}(p_0)$ be the inball of $\Omega_{t_0}$, where $\rho_0=\rho_-(t_0)$.
Consider the support function $u(x,t)=\sinh \rho_{p_0}\langle\partial_{\rho_{p_0}},\nu\rangle$ of $M_t$ with respect to the point $p_0$, where $\rho_{p_0}$ is the distance function in $\mathbb{H}^{n+1}$ from the point $p_0$. Since $M_t$ is convex, by \eqref{in-out-radius} and \eqref{inball-t0}, we see
\begin{equation}\label{u-bound}
\begin{split}
u(x,t)&\geq\sinh\left(\frac{\rho_0}{2}\right)\geq\sinh\left(\frac{c_1}{2}\right)=:2c,\\
u(x,t)&\leq\sinh(2c_2),
\end{split}
\end{equation}
and
\begin{equation}
0<\frac{c_1}{2}\leq\rho_{p_0}(t)\leq 2c_2<\infty
\end{equation}
for any $t\in[t_0,\min\{T,t_0+\tau\})$. Assume that $p_0$ is the origin of $\mathbb{H}^{n+1}$, we project the flow \eqref{flow-VMCF} in $\mathbb{H}^{n+1}$ onto $B_1(0)\subset \mathbb{R}^{n+1}$ with respect to the original point $p_0$. Then on the time interval $[t_0,\min\{T,t_0+\tau\})$, the Euclidean distance $r$ from the origin satisfies $r=\tanh{\rho_{p_0}}$ and is uniformly bounded from below and above. Under the Gauss map parametrization \eqref{s3.Y-Gas}, the support function $s$ of $Y_t(M^n)$ satisfies
\begin{equation*}
r^2=s^2+|\bar{\nabla} s|^2.
\end{equation*}
We conclude with the following $C^0$ and $C^1$ estimates:
\begin{lem}\label{t0-bound on u,du}
Let $M_t$ be the smooth convex solution of the flow \eqref{flow-VMCF} on the time interval $[0,T)$. Given $t_0\in[0,T)$ and $\tau$ be defined as in Lemma \ref{lem3.2}. Let $B_{\rho_0}(p_0)$ be the inball of $\Omega_{t_0}$ with $\rho_0=\rho_-(t_0)$ and we project $\mathbb{H}^{n+1}$ onto $B_1(0)\subset \mathbb{R}^{n+1}$ with respect to the original point $p_0$. We have the following estimates of the projected flow \eqref{projected flow eq}:
\begin{equation}
0<c_3\leq 1-r^2\leq 1-s^2\leq c_4<1,\qquad |s|\leq c_5<1,\qquad |\bar{\nabla} s|\leq c_6<1
\end{equation}
for any $t\in[t_0,\min\{T,t_0+\tau\})$, where $\{c_i\}_{i=3,4,5,6}$ are positive constants depending only on $n$, $\alpha$ and $M_0$.
\end{lem}
\subsection{Preserving convexity}\label{sec4}
In this subsection, we prove the lower bound on the principal curvatures of the solution $M_t$ of the flow \eqref{flow-VMCF}.
\begin{prop}\label{preserve convex}
Let $M_0$ be a smooth, closed and convex hypersurface in $\mathbb{H}^{n+1}$ and $M_t, t\in [0,T)$, be the smooth solution of the flow \eqref{flow-VMCF} starting from $M_0$. Then there exist constants $\Lambda_1$ and $\Lambda_3$ depending only on $n$, $\alpha$ and $M_0$ such that the principal curvatures $\kappa_i$ of $M_t$ satisfy
\begin{equation}\label{s4.2-0}
\kappa_i\geq \Lambda_3^{-1}\Lambda_1^{-\frac{2t}{\tau}}
\end{equation}
for all $i\in\{1,\dots,n\}$ and $t\in[0,T)$, where $\tau$ is the constant in Lemma \ref{lem3.2}.
\end{prop}
\proof
Since $M_0$ is convex, by continuity the solution $M_t$ is convex for at least a short time. Without loss of generality, we assume that the solution is convex on the time interval $[0,T)$ and aim to derive the estimate \eqref{s4.2-0} on this interval $[0,T)$. In fact, if $T_1<T$ is the largest time such that $M_t$ is convex for all $0\leq t<T_1$, then the estimate \eqref{s4.2-0} implies that the principal curvatures of $M_{T_1}$ has a positive lower bound and this contradicts with the maximality of $T_1$. Therefore, $M_t$ remains convex for all time $t\in [0,T)$ and satisfies the estimate \eqref{s4.2-0}.
So in the following, we will derive the estimate \eqref{s4.2-0} on the interval $[0,T)$ where the solution $M_t$ is convex. Given any fixed time $t_0\in[0,T)$ and $\tau$ be defined as in Lemma \ref{lem3.2}. Let $B_{\rho_0}(p_0)$ be the inball of $\Omega_{t_0}$, where $\rho_0=\rho_-(t_0)$. We consider the solution $Y_t(M^n)$ of the projected flow \eqref{projected flow eq} on $\mathbb{R}^{n+1}$ with respect to the original point $p_0$. Let $\mathfrak{r}_{ij}=s_{ij}+s\sigma_{ij}$ be the inverse of the Weingarten matrix of the projected hypersurface $Y_t$ with respect to an orthonormal frame on $\mathbb{S}^n$, whose eigenvalues $(\mathfrak{r}_1,\dots,\mathfrak{r}_n)$ are the principal radii of curvature of $Y_t$. As the principal curvatures $\kappa_i$ of $M_t=X(M^n,t)$ are the eigenvalues of the Weingarten matrix $\mathcal{W}_X$, by the expression \eqref{s5:W-inv-2} and the $C^0, C^1$ estimates in Lemma \ref{t0-bound on u,du}, in order to prove the lower bound of $\kappa_i$, it suffices to prove the upper bound of $\mathfrak{r}(z,t)=\max_{i=1,\dots,n}{\mathfrak{r}_i(z,t)}$.
For this purpose, we consider the function
\begin{equation*}
{G}(z,t):=\log{\mathfrak{r}}+\frac{L}{2}r^2,\qquad t\in [t_0,\min\{T,t_0+\tau\}),
\end{equation*}
where $L>0$ is a large constant to be determined. Suppose that the maximum of ${G}$ on $\mathbb{S}^n\times [t_0,\min\{T,t_0+\tau\})$ is attained at $(\bar{z},\bar{t})$. We choose a local orthonormal frame $e_1,\dots,e_n$ around $\bar{z}$ such $\{\mathfrak{r}_{ij}(z,t)\}$ is diagonal at $(\bar{z},\bar{t})$ and $\mathfrak{r}_{11}(\bar{z},\bar{t})=\mathfrak{r}_1(\bar{z},\bar{t})=\mathfrak{r}(\bar{z},\bar{t})$. Then without loss of generality, we can view the function $G$ as the following form
\begin{equation}\label{eqG}
G(z,t)=\log{\mathfrak{r}_{11}(z,t)}+\frac{L}{2}r^2.
\end{equation}
If $\bar{t}=t_0$, we have
\begin{equation}\label{s4.G1}
G(z,t)\leq \max_{z\in\mathbb{S}^n}G(z,t_0)~\leq \log\max_{z\in\mathbb{S}^n}\mathfrak{r}_{11}(z,t_0)+\frac{L}{2}(1-c_3)
\end{equation}
for $(z,t)\in \mathbb{S}^n\times [t_0,\min\{T,t_0+\tau\})$. In the following, we assume $\bar{t}>t_0$. We shall apply the maximum principle to the evolution equation of $G$ to derive the upper bound of $\mathfrak{r}_{11}$.
We have calculated the evolution equations of $\mathfrak{r}_{11}$ and of $r^2$ in Lemma \ref{s3.lem2} and Lemma \ref{s3.lem3}. Since $(\bar{z},\bar{t})$ is a maximum point of $G$, at $(\bar{z},\bar{t})$ there hold
\begin{align}\label{s4.dQ}
0=\bar{\nabla}_iG=&\frac{\bar{\nabla}_i\mathfrak{r}_{11}}{\mathfrak{r}_{11}}+L(ss_i+s_ks_{ki}) =\frac{\bar{\nabla}_i\mathfrak{r}_{11}}{\mathfrak{r}_{11}}+Ls_k\mathfrak{r}_{ki}
\end{align}
and
\begin{align}\label{s4.dtQ}
0&\leq \partial_tG-F^{k\ell}\bar{\nabla}_k\bar{\nabla}_\ell G\nonumber\\
&=\frac{1}{\mathfrak{r}_{11}}(\partial_t \mathfrak{r}_{11}-F^{k\ell}\bar{\nabla}_k\bar{\nabla}_\ell \mathfrak{r}_{11})+\frac{L}{2}(\partial_tr^2-F^{k\ell}\bar{\nabla}_k\bar{\nabla}_\ell r^2)+F^{k\ell}\frac{\bar{\nabla}_k\mathfrak{r}_{11}\bar{\nabla}_\ell \mathfrak{r}_{11}}{(\mathfrak{r}_{11})^2}\nonumber\\
&=:Q_1+Q_2,
\end{align}
where $Q_1$ involves the term $\phi(t)$:
\begin{align}\label{s4.Q1-0} Q_1=&\frac{\phi(t)}{\mathfrak{r}_{11}}\Big(A_{s_1s_1}{\mathfrak{r}_{11}}^2+2A_{ss_1}s_1\mathfrak{r}_{11}-2A_{s_1s_1}\mathfrak{r}_{11}s+A_s\mathfrak{r}_{11}+A_{s_k}\bar{\nabla}_k{\mathfrak{r}_{11}}\notag\\
&+A+A_{ss}{s_1}^2-2A_{ss_1}ss_1-A_ss+A_{s_1s_1}s^2-A_{s_1}s_1\Big)\notag\\
&+\phi(t)L\Big(sA+A_s|\bar{\nabla} s|^2+A_{s_k}s_i\mathfrak{r}_{ki}-A_{s_k}s_i\sigma_{ki}s\Big),
\end{align}
and $Q_2$ involves the term $(K^Y)^{\alpha}$:
\begin{align}\label{s4.Q2-0}
Q_2=&-\frac{(K^Y)^{\alpha}}{{\mathfrak{r}}_{11}}\Big(B_{s_1s_1}{\mathfrak{r}_{11}}^2+2B_{ss_1}s_1\mathfrak{r}_{11}-2B_{s_1s_1}\mathfrak{r}_{11}s+B_s\mathfrak{r}_{11}\notag\\
&+B+B_{ss}{s_1}^2-2B_{ss_1}ss_1-B_ss+B_{s_1s_1}s^2-B_{s_1}s_1\notag\\
&+B_{s_k}\bar{\nabla}_k{\mathfrak{r}_{11}}+\alpha B\mathfrak{r}^{kk}\mathfrak{r}^{\ell\ell}(\bar{\nabla}_1{\mathfrak{r}_{k\ell}})^2+{\alpha}^2 B(\sum_k\mathfrak{r}^{kk}\bar{\nabla}_1\mathfrak{r}_{kk})^2\notag\\
&-2\alpha \mathfrak{r}^{kk}\bar{\nabla}_1\mathfrak{r}_{kk}(B_{s_1}\mathfrak{r}_{11}+B_ss_1-B_{s_1}s)+\alpha B\mathfrak{r}^{kk}(\mathfrak{r}_{11}-\mathfrak{r}_{kk})\Big)\nonumber\\
&-(K^Y)^{\alpha}L\Big((1-n\alpha)sB+B_s|\bar{\nabla} s|^2+B_{s_k}s_i\mathfrak{r}_{ki}-B_{s_i}s_is+\alpha B\sum_{k}{\mathfrak{r}_{kk}}\Big)\nonumber\\
&+F^{k\ell}\frac{\bar{\nabla}_k\mathfrak{r}_{11}\bar{\nabla}_\ell \mathfrak{r}_{11}}{(\mathfrak{r}_{11})^2}.
\end{align}
In the following, we estimate the terms $Q_1$ and $Q_2$ separately. First, we use \eqref{s4.dQ} to cancel the term on $\bar{\nabla}_k{\mathfrak{r}_{11}}$ in \eqref{s4.Q1-0} and get
\begin{align}\label{s4.Q1}
Q_1=&\frac{\phi(t)}{\mathfrak{r}_{11}}\Big(A_{s_1s_1}{\mathfrak{r}_{11}}^2+2A_{ss_1}s_1\mathfrak{r}_{11}-2A_{s_1s_1}\mathfrak{r}_{11}s+A_s\mathfrak{r}_{11}\notag\\
&+A+A_{ss}{s_1}^2-2A_{ss_1}ss_1-A_ss+A_{s_1s_1}s^2-A_{s_1}s_1\Big)\notag\\
&+\phi(t)L\Big(sA+A_s|\bar{\nabla} s|^2-A_{s_i}s_is\Big).
\end{align}
Since
\begin{equation*}
A_{s_1s_1}=-\frac{(1-s^2)^{\frac{1}{2}}}{(1-{s}^2-|\bar{\nabla}{s}|^2)^{\frac{3}{2}}}\left(1-({s}^2+|\bar{\nabla}{s}|^2)+s_1^2\right)<0
\end{equation*}
and the coefficients such as $A, A_s, A_{ss_1}, \cdots$ depend only on $s$ and $\bar{\nabla}s$ which are uniformly bounded on the interval $[t_0, \min\{T,t_0+\tau\})$, we obtain the estimate:
\begin{align}\label{s4.Q1'}
Q_1 \leq&~ \frac{\phi(t)}{{\mathfrak{r}}_{11}}\left(a_1+(a_2+a_3L){\mathfrak{r}}_{11}-a_4 {\mathfrak{r}}_{11}^2\right),
\end{align}
where $a_i>0,i=1,2,3,4$ depend only on $n,\alpha$ and $M_0$.
To estimate $Q_2$, we use \eqref{s4.dQ} to kill the terms involving $\bar{\nabla}_k\mathfrak{r}_{11}$ and get
\begin{align*}
& -\frac{(K^Y)^{\alpha}}{{\mathfrak{r}}_{11}} \alpha B\mathfrak{r}^{kk}\mathfrak{r}^{\ell\ell}(\bar{\nabla}_1{\mathfrak{r}_{k\ell}})^2 +F^{k\ell}\frac{\bar{\nabla}_k\mathfrak{r}_{11}\bar{\nabla}_\ell \mathfrak{r}_{11}}{(\mathfrak{r}_{11})^2} \\
\leq & -\frac{(K^Y)^{\alpha}}{{\mathfrak{r}}_{11}} \alpha B\mathfrak{r}^{kk}\mathfrak{r}^{11}(\bar{\nabla}_k{\mathfrak{r}_{11}})^2 +\alpha B (K^Y)^\alpha \mathfrak{r}^{kk}\frac{(\bar{\nabla}_k\mathfrak{r}_{11})^2}{(\mathfrak{r}_{11})^2}=0.
\end{align*}
Then
\begin{align}\label{s4.Q2}
Q_2\leq &-\frac{(K^Y)^{\alpha}}{{\mathfrak{r}}_{11}}\Big(B_{s_1s_1}{\mathfrak{r}_{11}}^2+\left(2B_{ss_1}s_1-2B_{s_1s_1}s+B_s\right)\mathfrak{r}_{11}\notag\\
&+B+B_{ss}{s_1}^2-2B_{ss_1}ss_1-B_ss+B_{s_1s_1}s^2-B_{s_1}s_1\notag\\
&+{\alpha}^2 B(\sum_k\mathfrak{r}^{kk}\bar{\nabla}_1\mathfrak{r}_{kk})^2-2\alpha \mathfrak{r}^{kk}\bar{\nabla}_1\mathfrak{r}_{kk}(B_{s_1}\mathfrak{r}_{11}+B_ss_1-B_{s_1}s)\Big)\nonumber\\
&-(K^Y)^{\alpha}L\Big((1-n\alpha)sB+B_s|\bar{\nabla} s|^2-B_{s_i}s_is+\alpha B\mathfrak{r}_{11}\Big).
\end{align}
The third line on the right hand side of \eqref{s4.Q2} can be estimated using Cauchy-Schwarz inequality:
\begin{align*}
&- {\alpha}^2 B(\sum_k\mathfrak{r}^{kk}\bar{\nabla}_1\mathfrak{r}_{kk})^2+2\alpha \mathfrak{r}^{kk}\bar{\nabla}_1\mathfrak{r}_{kk}(B_{s_1}\mathfrak{r}_{11}+B_ss_1-B_{s_1}s) \\
\leq & \frac{1}{B}\left(B_{s_1}\mathfrak{r}_{11}+B_ss_1-B_{s_1}s\right)^2.
\end{align*}
Since the coefficients such as $B, B_{s_i}, B_s, B_{ss_i}$ are bounded and $B>0$, we obtain the estimate
\begin{align}\label{s4.Q2'}
Q_2 \leq&\frac{(K^Y)^{\alpha}}{{\mathfrak{r}}_{11}}\bigg(b_1+(b_2+b_3L){\mathfrak{r}}_{11}+(b_4-b_5L){\mathfrak{r}}_{11}^2\bigg),
\end{align}
where $b_i>0,i=1,2,3,4,5$ depend only on $n,\alpha$ and $M_0$.
Combining the two estimates \eqref{s4.Q1'} and \eqref{s4.Q2'} and choosing $L={2b_4}/{b_5}$, we can get an uniform upper bound ${\mathfrak{r}}_{11}(\bar{z},\bar{t})\leq \Lambda$, where $\Lambda$ depends only on $n$, $\alpha$ and $M_0$. It follows from
\begin{align*}
G(z,t)&\leq \max\left\{G(\bar{z},\bar{t}),\max_{z\in\mathbb{S}^n}G(z,t_0)\right\}
\end{align*}
that
\begin{align}\label{tau1}
\mathfrak{r}_{11}(z,t)&\leq \max\left\{\mathfrak{r}_{11}(\bar z,\bar t),\max_{z\in\mathbb{S}^n}\mathfrak{r}_{11}(z,t_0)\right\}\exp\Big(\frac{L}{2}\big(1-c_3-r^2(z,t)\big)\Big)\notag\\
&\leq \max\left\{\Lambda,\max_{z\in\mathbb{S}^n}\mathfrak{r}_{11}(z,t_0)\right\}\exp\Big(\frac{L}{2}(c_4-c_3)\Big)\notag\\
&=\Lambda_1\max\left\{\Lambda,\max_{z\in\mathbb{S}^n}\mathfrak{r}_{11}(z,t_0)\right\}
\end{align}
for all $(z,t)\in \mathbb{S}^n\times [t_0, \min\{T,t_0+\tau\})$, where $\Lambda_1=\exp\big(\frac{L}{2}(c_4-c_3)\big)\geq 1$ and $c_3,c_4$ are constants in Lemma \ref{t0-bound on u,du}.
Note that $t_0\in [t_0-\frac{\tau}{2},\min\{T,t_0+\frac{\tau}{2}\})$. Applying the above argument for the time interval $[t_0-\frac{\tau}{2},\min\{T,t_0+\frac{\tau}{2}\})$ gives
\begin{equation}\label{tau2}
\max_{z\in\mathbb{S}^n}\mathfrak{r}_{11}(z,t_0)\leq \Lambda_1\max\left\{\Lambda,\max_{z\in\mathbb{S}^n}\mathfrak{r}_{11}(z,t_0-\frac{\tau}{2})\right\}.
\end{equation}
Combining \eqref{tau1}, \eqref{tau2} and the fact that $\Lambda_1\geq 1$, we have
\begin{equation*}
\mathfrak{r}_{11}(z,t)\leq \Lambda_1^2\max\left\{\Lambda,\max_{z\in\mathbb{S}^n}\mathfrak{r}_{11}(z,t_0-\frac{\tau}{2})\right\}.
\end{equation*}
By backward induction on the time interval, we finally get
\begin{align}\label{tauup}
\mathfrak{r}_{11}(z,t)&\leq \Lambda_1^{\left[\frac{2t_0}{\tau}\right]+2}\max\{\Lambda,\max_{z\in\mathbb{S}^n}\mathfrak{r}_{11}(z,0)\}\notag\\
&\leq \Lambda_1^{\frac{2t}{\tau}+2}\max\{\Lambda,\max_{z\in\mathbb{S}^n}\mathfrak{r}_{11}(z,0)\}~=:\Lambda_2\Lambda_1^{\frac{2t}{\tau}}
\end{align}
for all $(z,t)\in \mathbb{S}^n\times [t_0, \min\{T,t_0+\tau\})$, where $[\cdot]$ denotes the integer part of a real constant, and $\Lambda_1$, $\Lambda_2=\Lambda_1^{2}\max\{\Lambda,\max_{z\in\mathbb{S}^n}\mathfrak{r}_{11}(z,0)\}$ are constants depending only on $n$, $\alpha$ and $M_0$. Since $t_0$ is arbitrary, by \eqref{s5:W-inv-2} and the $C^0,C^1$ estimates, we conclude that the principal curvatures $\kappa_i$ of the solution $M_t$ of the flow \eqref{flow-VMCF} satisfy
\begin{equation*}
\kappa_i\geq \Lambda_3^{-1}\Lambda_1^{-\frac{2t}{\tau}},\qquad i=1,\dots,n,
\end{equation*}
for all time $t\in [0,T)$, where $\Lambda_3=C\Lambda_2$ for a constant $C$ which estimates the bound on the coefficient of \eqref{s5:W-inv-2} involving $s$ and $\bar{\nabla}s$.
\endproof
\subsection{Upper bound of Gauss curvature} \label{sec.upK}
Now we use the technique of Tso \cite{Tso85} to prove the upper bound of the Gauss curvature of the solution $M_t$ along the flow \eqref{flow-VMCF}.
\begin{prop}\label{propKupp}
Let $M_t,\ t\in[0,T)$ be the smooth solution of the flow \eqref{flow-VMCF} starting from a smooth closed convex hypersurface $M_0$. If $T<\infty$, then there is a constant $C$ depending on $n, \alpha, M_0$ and $T$ such that the Gauss curvature $K$ of $M_t$ satisfies
\begin{equation*}
\max_{M_t} K\leq C
\end{equation*}
for any $t\in [0,T)$.
\end{prop}
\proof
For any given $t_0\in[0,T)$, let $B_{\rho_0}(p_0)$ be the inball of $\Omega_{t_0}$, where $\rho_0=\rho_{-}(t_0)$. Consider the support function $u(x,t)=\sinh \rho_{p_0}(x)\langle{\partial_{\rho_{p_0}},\nu}\rangle$ of $M_t$ with respect to point $p_0$, where $\rho_{p_0}(x)$ is the distance function in $\mathbb{H}^{n+1}$ from the point $p_0$. Since $M_t$ is convex for all $t\in [0,T)$, by \eqref{u-bound} we have
\begin{equation}\label{equbound}
2c\leq u\leq \sinh(2c_2)
\end{equation}
on $M_t$ for any $t\in \left[t_0,\min\{T,t_0+\tau\}\right)$. We define the auxiliary function
\begin{equation*}
W=\frac{K^\alpha}{u-c},
\end{equation*}
which is well-defined for time $t\in \left[t_0,\min\{T,t_0+\tau\}\right)$. We shall apply the maximum principle to the evolution equation of $W$ to derive the upper bound of $K$.
Combining \eqref{eq-KK} and \eqref{equeq}, we compute that along the flow \eqref{flow-VMCF} $W$ evolves as
\begin{align}
\frac{\partial}{\partial t}{W}
=&\alpha K^{\alpha-1}\dot{K}^{ij}(W_{ij}+\frac{2}{u-c}u_iW_j)\nonumber\\
&\quad -\frac{\phi(t)}{u-c}\bigg(\alpha K^{\alpha-1}(HK-\sigma_{n-1}(\kappa))+W\cosh\rho_{{p_0}}(x)\bigg)\notag\\
&\quad +(1+n\alpha)\frac{K^{2\alpha}}{(u-c)^2}\cosh\rho_{{p_0}}(x)-\frac{c\alpha K^{2\alpha}}{(u-c)^2}H-\alpha WK^{\alpha-1}\sigma_{n-1}(\kappa)\notag\\
\leq&~\alpha K^{\alpha-1}\dot{K}^{ij}(W_{ij}+\frac{2}{u-c}u_iW_j)+\underbrace{\frac{\phi(t)}{u-c}\alpha K^{\alpha-1}\left(\sigma_{n-1}(\kappa)-HK\right)}_{(I)}\notag\\
&\qquad +(1+n\alpha)W^2\cosh\rho_{{p_0}}(x)-\alpha cHW^2.\label{eqW}
\end{align}
In the previous work \cite{BenWei,BenChenWei}, $M_t$ is assumed to be either $h$-convex ($\kappa_i>1, ~i=1,\cdots,n$) or positively curved ($\kappa_i\kappa_j>1$,~$\forall~i\neq j$), so the terms $(I)$ involving $\phi(t)$ can be thrown away when we estimate the upper bound of $W$. But in our case we only have convexity, so we still need to estimate the terms $(I)$ in \eqref{eqW} carefully.
Let $\tilde{W}(t)=\max_{M_t}W(x,t)$. Noting that $K^\alpha=(u-c)W$, by the definition \eqref{eqphi} of $\phi(t)$ and the upper bound \eqref{equbound} of $u$, we have
\begin{equation*}
\phi(t)=\frac{1}{|M_t|}\int_{M_t}K^{\alpha}\,\mathrm{d}t\leq \max_{M_t}{K^{\alpha}(\cdot,t)}\leq (\sinh(2c_2)-c)\tilde{W}.
\end{equation*}
By the lower bound on the principal curvatures in Lemma \ref{preserve convex}, we also have
\begin{align*}
\sigma_{n-1}(\kappa)&=K(\frac{1}{\kappa_1}+\cdots\frac{1}{\kappa_n})\leq nK(\min_{1\leq i\leq n} \kappa_i)^{-1}\leq nK\Lambda_3\Lambda_1^{{2T}/{\tau}}.
\end{align*}
It follows that the terms $(I)$ can be estimated as
\begin{equation}\label{s4.2-1}
(I)\leq ~n\alpha (\sinh(2c_2)-c)\Lambda_3\Lambda_1^{{2T}/{\tau}}\tilde{W}^2~=:~\beta_1(T)\tilde{W}^2,
\end{equation}
where for simplicity of the notations we denote $\beta_1(T)=n\alpha (\sinh(2c_2)-c)\Lambda_3\Lambda_1^{{2T}/{\tau}}$, which depends on the maximal existence time $T<\infty$.
The first term on the last line of \eqref{eqW} can be simply estimated from above by
\begin{equation}\label{s4.2-2}
(1+n\alpha)W^2\cosh\rho_{{p_0}}(x)~\leq (1+n\alpha)\cosh(2c_2)\tilde{W}^2.
\end{equation}
The last term of \eqref{eqW} provides sufficient negative term, since by $H\geq nK^{1/n}$ there holds:
\begin{align}\label{s4.2-3}
-\alpha cHW^2\leq & -\alpha nc K^{1/n}W^2\nonumber\\
= & -\alpha nc(u-c)^{\frac{1}{n\alpha}}W^{2+\frac{1}{n\alpha}}\nonumber\\
\leq &-\alpha nc^{1+\frac{1}{n\alpha}}W^{2+\frac{1}{n\alpha}}.
\end{align}
Combining \eqref{s4.2-1} - \eqref{s4.2-3}, we arrive at
\begin{align}\label{s4.2-4}
\frac{d}{dt}\tilde{W} \leq & \bigg(\beta_1(T)+(1+n\alpha)\cosh(2c_2)-\alpha nc^{1+\frac{1}{n\alpha}}\tilde{W}^{\frac{1}{n\alpha}}\biggr)\tilde{W}^2.
\end{align}
The coefficient of the hightest order term on the right hand side of \eqref{s4.2-4} is negative, so the comparison principle implies that $\tilde{W}$ is bounded above by a positive constant. In fact, whenever
\begin{equation*}
\tilde{W}\geq \bigg(\frac{2}{\alpha n}\left(\beta_1(T)+(1+n\alpha)\cosh(2c_2)\right)\bigg)^{n\alpha}c^{-n\alpha-1}~=:\beta_2(T),
\end{equation*}
we have
\begin{align*}
\frac{d}{dt}\tilde{W} \leq & -\frac{\alpha n}{2}c^{1+\frac{1}{n\alpha}}\tilde{W}^{2+\frac{1}{n\alpha}}.
\end{align*}
Therefore,
\begin{equation}\label{s4.2-5}
\tilde{W}(t)\leq \max\left\{\bigg(W^{-1-\frac{1}{n\alpha}}(t_0)+\frac{n\alpha+1}{2}c^{1+\frac{1}{n\alpha}}(t-t_0)\bigg)^{-\frac{n\alpha}{n\alpha+1}},\beta_2(T)\right\}
\end{equation}
for all time $t\in \left[t_0,\min\{T,t_0+\tau\}\right)$.
For $t_0=0$, we obtain from \eqref{s4.2-5} the upper bound
\begin{equation*}
\tilde{W}(t)\leq\max\left\{\tilde{W}(0),\beta_2(T)\right\},\quad \forall t\in[0,\min\{\tau,T\})
\end{equation*}
and so
\begin{equation}\label{eqKalpha}
K^{\alpha}\leq \sinh(2c_2)\max\left\{\tilde{W}(0),\beta_2(T)\right\},\quad \forall t\in[0,\min\{\tau,T\}).
\end{equation}
Next, for $t_0=\tau/2$, the estimate \eqref{s4.2-5} implies
\begin{align*}
\tilde{W}(t)\leq &\max\left\{(\frac{n\alpha+1}{2}(t-t_0))^{-\frac{n\alpha}{n\alpha+1}}c^{-1},\beta_2(T)\right\}\nonumber\\
\leq&~\max\left\{(\frac{(n\alpha+1)\tau}{4})^{-\frac{n\alpha}{n\alpha+1}}c^{-1},\beta_2(T)\right\}
\end{align*}
for $t\in [\tau, \min\{3\tau/2,T\})$, and so
\begin{equation}\label{s4.tdW3}
K^\alpha\leq \sinh(2c_2)\max\left\{(\frac{(n\alpha+1)\tau}{4})^{-\frac{n\alpha}{n\alpha+1}}c^{-1},\beta_2(T)\right\}
\end{equation}
for $t\in [\tau, \min\{3\tau/2,T\})$. Repeating the above argument for $t_0=m\tau/2$ ($m\geq 2$), we can get the estimate \eqref{s4.tdW3} for $t\in [\frac{(m+1)\tau}2, \min\{\frac{(m+2)\tau}2,T\})$, which covers the whole time interval $[0,T)$. Combining \eqref{eqKalpha} and \eqref{s4.tdW3}, we complete the proof of Proposition \ref{propKupp}.
\endproof
\section{Long time existence}\label{sec5}
In this section, we prove the long time existence of the flow \eqref{flow-VMCF}.
\begin{thm}\label{long}
Let $M_0$ be a smooth closed convex hypersurface in $\mathbb{H}^{n+1}$ and $M_t$ be the smooth solution of the flow \eqref{flow-VMCF} starting from $M_0$. Then $M_t$ remains convex and exists for all time $t\in[0,\infty)$.
\end{thm}
\proof
We will argue by contradiction. Let $[0,T)$ be the maximum interval such that the solution of the flow \eqref{flow-VMCF} exists with $T<\infty$. Then combining Proposition \ref{preserve convex} and Proposition \ref{propKupp} yields that the principal curvatures $\kappa=(\kappa_1,\dots,\kappa_n)$ of the solution $M_t$ satisfy
\begin{equation}\label{s5.1}
0<\underline{\kappa}_0\leq \kappa_i\leq \overline{\kappa}_0,\quad i=1,\dots,n
\end{equation}
for all time $t\in [0,T)$, where the constants $\underline{\kappa}_0, \overline{\kappa}_0$ depend on $n,\alpha,M_0$ and $T$.
To prove the long time existence of the solution $M_t$ of the flow \eqref{flow-VMCF}, we need to derive the higher order regularity estimates. Recall that in $\S$\ref{subsec}, up to a tangential diffeomorphism, the flow equation \eqref{flow-VMCF} is equivalent the following scalar parabolic equation
\begin{align}\label{eqqs}
\frac{\partial}{\partial t}s=&A\phi(t)-B(K^{Y})^{\alpha}=~A\phi(t)-B(\det \mathfrak{r})^{-\alpha}
\end{align}
on the sphere $\mathbb{S}^n$, where $\mathfrak{r}_{ij}=s_{ij}+s\sigma_{ij}$ and $A$, $B$ are functions defined in \eqref{eqAB} which depend only on $s$ and $\bar{\nabla}s$. Combing Lemma \ref{t0-bound on u,du} and the curvature estimate \eqref{s5.1} gives the $C^2$ estimates of $s(z,t)$. Denote the right hand side of \eqref{eqqs} by $G(\bar{\nabla}^2 s,\bar{\nabla} s,s,z,t)$. Then the derivatives of $G$ with respect to the first argument are given by
\begin{equation}
\dot{G}^{ij}=n\alpha B(\det \mathfrak{r})^{-\alpha-\frac{1}{n}}\frac{\partial(\det \mathfrak{r})^{\frac{1}{n}}}{\partial s_{ij}}
\end{equation}
and
\begin{equation}
\begin{split}
\ddot{G}^{ij,k\ell}&=n\alpha B(\det \mathfrak{r})^{-\alpha-\frac{1}{n}}\frac{\partial^2(\det \mathfrak{r})^{\frac{1}{n}}}{\partial s_{ij}\partial s_{k\ell}}\\
&-n\alpha(n\alpha+1)B(\det \mathfrak{r})^{-\alpha-\frac{2}{n}}\frac{\partial(\det \mathfrak{r})^{\frac{1}{n}}}{\partial s_{ij}}\frac{\partial(\det \mathfrak{r})^{\frac{1}{n}}}{\partial s_{k\ell}}.
\end{split}
\end{equation}
The estimates we have established imply the existence of a constant $C=C(T)>0$, such that $1/C I\leq (\dot{G^{ij}})\leq CI$, that is, the operator $G$ is uniformly elliptic on the finite time interval $[0,T)$. Since $(\det \mathfrak{r})^{\frac{1}{n}}$ is concave with respect to $s_{ij}$, we see that $G$ is also a concave operator. We can apply Theorem 1.1 in \cite{TW13} to obtain a $C^{2,\gamma}$ estimate on $s$, for a suitable $\gamma\in(0,1)$. See also the arguments in \cite{CS10,Mc05} for the $C^{2,\gamma}$ estimate of the solutions to volume preserving curvature flows. Then by the parabolic Schauder theory (see \cite{Lie96}), we can deduce all higher order regularity estimates of $s$ on $[0,T)$ and a standard continuation argument then shows that $T=+\infty$.
\endproof
\begin{rem}
Note that the curvature estimate \eqref{s5.1} of the solution $M_t$ of the flow \eqref{flow-VMCF} depends on time $t$ and may degenerate as time $t\to\infty$. To study the asymptotical behavior of $M_t$ as $t\to\infty$, we still need to get an uniform curvature estimate which does not depend on time. This will be obtained in the next two sections.
\end{rem}
\section{Monotonicity and Hausdorff convergence}\label{sec.hau}
In this section, we prove the monotonicity of $\mathcal{A}_{n-1}(\Omega_t)$, the subsequential Hausdorff convergence of the solution $M_t$ of \eqref{flow-VMCF} and the convergence of the center of the inner ball of $\Omega_t$ to a fixed point.
Denote the average integral of the Gauss curvature by
\begin{equation}\label{s6.0}
\bar{K}=\frac{1}{|M_t|}\int_{M_t}{K d\mu_t}=\frac{\mathcal{A}_n(\Omega_t)+\frac{1}{n-1}\mathcal{A}_{n-2}(\Omega_t)}{\mathcal{A}_0(\Omega_t)}.
\end{equation}
It follows from the monotonicity \eqref{s2.Akmo} of quermassintegrals with respect to inclusion of convex sets and the estimates on inner radius and outer radius in Lemma \ref{in-out} that $\bar{K}$ is uniformly bounded from above and below by positive constants depending only on $n$, $\alpha$ and $M_0$.
\subsection{Monotonicity for $\mathcal{A}_{n-1}$}
We first show the following monotonicity of $\mathcal{A}_{n-1}(\Omega_t)$ along the flow \eqref{flow-VMCF}, which will be useful in proving the subsequential Hausdorff convergence of $M_t$.
\begin{lem}\label{lemmono}
Let $M_t$ be a smooth convex solution of the volume preserving flow \eqref{flow-VMCF}. Denote by $\Omega_t$ the domain enclosed by $M_t$. Then $\mathcal{A}_{n-1}(\Omega_t)$ is monotone decreasing in time $t$, and is strictly decreasing unless $\Omega_t$ is a geodesic ball.
\end{lem}
\proof
From the evolution equation \eqref{eqWk} for the quermassintegrals of $\Omega_t$, we have
\begin{equation*}
\frac{d}{dt}\mathcal{A}_{n-1}(\Omega_t)=n\int_{M_t}{K(\phi(t)-K^{\alpha})d\mu_t}.
\end{equation*}
Since $\phi(t)$ is defined as in \eqref{eqphi}, we have
\begin{align}\label{eqWmo}
\frac{d}{dt}\mathcal{A}_{n-1}(\Omega_t)&=\frac{n}{|M_t|}\left(\int_{M_t}K d\mu_t\int_{M_t}K^{\alpha} d\mu_t-|M_t|\int_{M_t}K^{\alpha+1} d\mu_t\right)\notag\\
&=-n\int_{M_t}{(K-\bar{K})(K^{\alpha}-{\bar{K}}^{\alpha})d\mu_t}\leq 0.
\end{align}
Note that equality holds in \eqref{eqWmo} if and only if $K$ is a constant on $M_t$, which means $M_t$ is a geodesic sphere by the Alexandrov type theorem for hypersurfaces with constant Gauss curvature in the hyperbolic space (see \cite{MS91}).
\endproof
\subsection{Subsequential Hausdorff convergence}
In this subsection, we prove that there exists a sequence of times $t_i\to \infty$ such that the solution $M_{t_i}$ converges in Hausdorff sense to a geodesic sphere. We first prove the following estimate:
\begin{lem}
Let $M_0$ be a smooth closed and convex hypersurface in $\mathbb{H}^{n+1}$ and $M_t$ be the smooth solution of the flow \eqref{flow-VMCF} starting from $M_0$. Then there exists a sequence of times $\{t_i\},t_i\to\infty$, such that
\begin{equation}\label{s6.2-1}
\int_{M_{t_i}}{|K-\bar{K}|d\mu_{t_i}}\to 0,\quad \text{as}\,\, t_i\to\infty.
\end{equation}
\end{lem}
\proof
Applying the monotonicity in Lemma \ref{lemmono} and the long time existence of the flow \eqref{flow-VMCF}, we have
\begin{equation*}
n\int_0^{\infty}\int_{M_t}{(K-\bar{K})(K^{\alpha}-{\bar{K}}^{\alpha})d\mu_t}dt\leq {\mathcal{A}_{n-1}}(\Omega_0)<\infty.
\end{equation*}
Therefore there exists a sequence of times $t_i\to\infty$ such that
\begin{equation}\label{s6.2-2}
\int_{M_{t_i}}{(K-\bar{K})(K^{\alpha}-{\bar{K}}^{\alpha})d\mu_{t_i}}\to 0.
\end{equation}
If $\alpha\geq 1$, we have
\begin{align*}
(K-\bar{K})(K^{\alpha}-{\bar{K}}^{\alpha})&=\alpha\int_0^1{((1-s)\bar{K}+sK)^{\alpha-1}ds}\cdot (K-\bar{K})^2\\
&\geq \alpha\int_0^1{(1-s)^{\alpha-1}ds}\bar{K}^{\alpha-1}(K-\bar{K})^2\\
&\geq C(K-\bar{K})^2,
\end{align*}
where we used the fact that $\bar{K}$ is uniformly bounded and $\alpha\geq 1$. Therefore, \eqref{s6.2-2} implies that
\begin{equation*}
\int_{M_{t_i}}{(K-\bar{K})^2d\mu_{t_i}}\to 0,\quad \text{as}\,\, i\to\infty
\end{equation*}
for a sequence of times $t_i\to\infty$ and the estimate \eqref{s6.2-1} follows by the H\"{o}lder inequality.
We next prove the estimate \eqref{s6.2-1} for $0<\alpha<1$. In this case, we have
\begin{align*}
&\int_{M_{t_i}}|K-\bar{K}|d\mu_{t_i} \\
=&\int_{M_{t_i}}|K-\bar{K}|^{1/2}|K-\bar{K}|^{1/2}d\mu_{t_i} \\
=& \int_{M_{t_i}}\Big(\frac 1{\alpha} \int_0^1{((1-s)\bar{K}^\alpha+sK^\alpha)^{\frac{1}{\alpha}-1}ds}\Big)^{1/2}\cdot \Big|(K^\alpha-\bar{K}^\alpha)(K-\bar{K})\Big|^{1/2}d\mu_{t_i} \\
\leq & \Big(\underbrace{\int_{M_{t_i}}\frac 1{\alpha} \int_0^1{((1-s)\bar{K}^\alpha+sK^\alpha)^{\frac{1}{\alpha}-1}ds}d\mu_{t_i}}_{(I)}\Big)^{1/2}\Big(\underbrace{\int_{M_{t_i}}(K-\bar{K})(K^{\alpha}-{\bar{K}}^{\alpha})d\mu_{t_i}}_{(II)}\Big)^{1/2}.
\end{align*}
The second term $(II)$ tends to zero as $t_i\to\infty$ by \eqref{s6.2-2}. We show that the first term $(I)$ is bounded for any $0<\alpha<1$. In fact, if $\alpha\in [\frac 12,1)$, we have $1<1/\alpha\leq 2$. This implies that
\begin{align*}
(I)= & \frac 1{\alpha} \int_{M_{t_i}}\int_0^1{((1-s)\bar{K}^\alpha+sK^\alpha)^{\frac{1}{\alpha}-1}ds}d\mu_{t_i} \\
\leq & \frac 1{\alpha} \int_{M_{t_i}}(\bar{K}^\alpha+K^\alpha)^{\frac{1}{\alpha}-1}d\mu_{t_i}\\
=& \frac 1{\alpha} \int_{M_{t_i}}\frac{\bar{K}^\alpha+K^\alpha}{(\bar{K}^\alpha+K^\alpha)^{2-\frac{1}{\alpha}}}d\mu_{t_i}\\
\leq &\frac 1{\alpha\bar{K}^{2\alpha-1}} \int_{M_{t_i}}(\bar{K}^\alpha+K^\alpha)d\mu_{t_i}\\
\leq & \frac 2{\alpha} \bar{K}^{1-\alpha}\mathcal{A}_0(\Omega_{t_i})\leq ~C
\end{align*}
is uniformly bounded from above, where we used the H\"{o}lder inequality in the fourth inequality to get the estimate
\begin{equation*}
\int_{M_{t_i}}K^\alpha d\mu_{t_i}\leq \left(\int_{M_{t_i}}Kd\mu_{t_i}\right)^\alpha \mathcal{A}_0(\Omega_{t_i})^{1-\alpha}=\bar{K}^{\alpha}\mathcal{A}_0(\Omega_{t_i})
\end{equation*}
for $\alpha<1$ and that $\mathcal{A}_0(\Omega_{t_i})\leq \mathcal{A}_0(B_{c_2}(0))\leq C$ which is due to that the outer radius of $\Omega_{t_i}$ is bounded from above by $c_2$. We repeat the argument for $\alpha\in [\frac{1}{k+1},\frac{1}{k})$ with $k=2,3,\cdots$, where we have $(k+1)\alpha\geq 1$ and $0<k\alpha<1$. Then
\begin{align*}
(I) \leq & \frac 1{\alpha} \int_{M_{t_i}}(\bar{K}^\alpha+K^\alpha)^{\frac{1}{\alpha}-1}d\mu_{t_i}\\
=& \frac 1{\alpha} \int_{M_{t_i}}\frac{(\bar{K}^\alpha+K^\alpha)^k}{(\bar{K}^\alpha+K^\alpha)^{k+1-\frac{1}{\alpha}}}d\mu_{t_i}\\
\leq &\frac 1{\alpha\bar{K}^{(k+1)\alpha-1}} \int_{M_{t_i}}2^{k-1}(\bar{K}^{k\alpha}+K^{k\alpha})d\mu_{t_i}\\
\leq & \frac {2^k}{\alpha} \bar{K}^{1-\alpha}\mathcal{A}_0(\Omega_{t_i})\leq ~C
\end{align*}
is uniformly bounded, where in the third inequality we used the H\"{o}lder inequality again as $k\alpha<1$. Therefore for any $0<\alpha<1$, the term $(I)$ is uniformly bounded from above and thus the estimate \eqref{s6.2-1} follows in this case.
\endproof
Note that the curvature estimate \eqref{s5.1} of the solution $M_t$ of the flow \eqref{flow-VMCF} depends on time $t$ and may degenerate as time $t\to\infty$. So we can not conclude from \eqref{s6.2-1} that the solution $M_{t_i}$ converges to a geodesic sphere as $t_i\to\infty$, as we do not have an uniform regularity estimate to guarantee the existence of a smooth limit of $M_{t_i}$. However, we can apply the similar idea in the work \cite[\S 6]{AW21} by the first author and Andrews in the Euclidean space to prove the following subsequential Hausdorff convergence result:
\begin{lem}\label{subcon}
Let $M_0$ be a smooth, closed convex hypersurface in $\mathbb{H}^{n+1}$ and $M_t$ be the smooth solution of the flow \eqref{flow-VMCF} starting from $M_0$. Then there exists a sequence of times $\{t_i\},t_i\to\infty$, such that $M_{t_i}$ converges to a geodesic sphere $S_{\rho_{\infty}}(p)$ in Hausdorff sense as $t_i\to\infty$, where $p$ is the center of the sphere and the radius $\rho_{\infty}$ is determined by the fact that $S_{\rho_{\infty}}(p)$ encloses the same volume of $M_0$.
\end{lem}
\proof
Let $p_t$ be the center of the inball of $\Omega_t$ and let $\varphi_t:\mathbb{H}^{n+1}\to\mathbb{H}^{n+1}$ be an isometry carrying $p_t$ to the origin $o\in \mathbb{H}^{n+1}$. Clearly, each $\varphi_t(M_t)$ is a closed convex hypersurface with an inball centered at the origin and having inner radius $\rho_{-}(t)\geq c_1$. For simplicity of the notations, we still denote the transformed solution $\varphi_t(M_t)$ as $M_t$.
As in \S \ref{subsec}, we project $\Omega_t\subset \mathbb{H}^{n+1}$ onto $B_1(0)\subset\mathbb{R}^{n+1}$ and get the corresponding $\tilde{\Omega}_t\subset B_1(0)\subset \mathbb{R}^{n+1}$. Since the outer radius of $\Omega_t$ is uniformly bounded above and so does $\tilde{\Omega}_t$, the Blaschke selection theorem (see Theorem 1.8.7 of \cite{RS2014}) implies that there exists a sequence of times $t_i$ and a convex body $\tilde{\Omega}$ such that $\tilde{\Omega}_{t_i}$ converges to $\tilde{\Omega}$ in Hausdorff sense as $t_i\to\infty$. Note that the projection yields an one-to-one correspondence between the convex bodies in $\mathbb{H}^{n+1}$ and the convex bodies in $B_1(0)\subset \mathbb{R}^{n+1}$, then there exists a convex set $\hat{\Omega}\in\mathcal{K}(\mathbb{H}^{n+1})$ such that $\Omega_{t_i}$ converges to $\hat{\Omega}$ in Hausdorff sense as $t_i\to\infty$. As each $\Omega_{t_i}$ has inner radius $\rho_{-}(\Omega_{t_i})\geq c_1$, the limit convex set $\hat{\Omega}$ has positive inner radius. Without loss of generality, we may assume that the sequence $t_i$ is the same sequence such that \eqref{s6.2-1} holds.
By the continuity of quermassintegrals with respect to the Hausdorff distance in Lemma \ref{inquer}, the Hausdorff convergence of $\Omega_{t_i}$ to $\hat{\Omega}$ implies that
\begin{equation}\label{s6.1-0}
\bar{K}=\frac{\mathcal{A}_n(\Omega_{t_i})+\frac{1}{n-1}\mathcal{A}_{n-2}(\Omega_{t_i})}{\mathcal{A}_0(\Omega_{t_i})}~\to~\frac{\mathcal{A}_n(\hat{\Omega})+\frac{1}{n-1}\mathcal{A}_{n-2}(\hat{\Omega})}{\mathcal{A}_0(\hat{\Omega})}~=:c
\end{equation}
as $t_i\to \infty$. We will show that $\hat{\Omega}$ satisfies the equation $\Phi_0(\hat{\Omega},\cdot)=c\Phi_n(\hat{\Omega},\cdot)$ for the curvature measures $\Phi_0$ and $\Phi_n$. In fact, by the weak continuity of the curvature measures in Theorem \ref{s2.thmcurv}, for any bounded continuous function $f$ on $\mathbb{H}^{n+1}$ with compact support, we have that $\int{f d\Phi_0(\Omega_{t_i})}$ converges to $\int{f d\Phi_0(\hat{\Omega})}$, and $\int{f d\Phi_n(\Omega_{t_i})}$ converges to $\int{f d\Phi_n(\hat{\Omega})}$, as $i\to\infty$. Then
\begin{align}\label{s6.1-1}
&\left|\int{f d\Phi_0(\Omega_{t_i})}-c\int{f d\Phi_n(\Omega_{t_i})}\right|\nonumber\\
=&\left|\int_{M_{t_i}}{fK d\mathcal{H}^n}-\int_{M_{t_i}}{cf d\mathcal{H}^n}\right|\nonumber\\
\leq&\sup|f|\int_{M_{t_i}}{|K-c| d\mathcal{H}^n}\nonumber\\
\leq&\sup|f|\int_{M_{t_i}}{|K-\bar{K}| d\mathcal{H}^n}+\sup|f|\int_{M_{t_i}}{|\bar{K}-c| d\mathcal{H}^n}~\to~0
\end{align}
by the estimates \eqref{s6.2-1} and \eqref{s6.1-0}.
It follows that $\int{f d\Phi_0(\hat{\Omega})}=c\int{f d\Phi_n(\hat{\Omega})}$ for all bounded continuous functions $f$, and therefore $\Phi_0(\hat{\Omega},\cdot)=c\Phi_n(\hat{\Omega},\cdot)$ as claimed. Then by Theorem \ref{gAT}, $\hat{\Omega}$ is a geodesic ball. This means that $\varphi_{t_i}(\Omega_{t_i})$ converges to a geodesic ball of some radius $\rho_\infty$ centered at the origin. The radius is uniquely determined by the preserving of the volume. Moreover, by an argument of Alexandrov reflection, the evolving domains $\Omega_t$ cannot leave away from the initial domain $\Omega_0$, that is, there exists a geodesic ball $B_R\subset \mathbb{H}^{n+1}$ of radius $R>0$ containing $\Omega_0$ such that $\Omega_t\cap B_R\neq \emptyset$ for all time $t\in [0,\infty)$, see Lemma 4.1 in \cite{BenChenWei}. In particular, this implies that the centers $p_{t_i}$ of inner ball of $\Omega_{t_i}$ are located in a compact subset of $\mathbb{H}^{n+1}$, and then there exists a subsequence of $t_i$ (still denoted by $t_i$) such that $p_{t_i}$ converges to a limit point $p\in \mathbb{H}^{n+1}$. This concludes that for this subsequence of times $t_i$, $\Omega_{t_i}$ converges to a geodesic ball $B_{\rho_\infty}(p)$ centered at the point $p$ without correction of ambient isometry $\varphi_{t_i}$. This completes the proof of Lemma \ref{subcon}.
\endproof
\subsection{Convergence of the center of the inner ball}
The argument in \cite{AW21} is not sufficient for us to deduce the Hausdorff convergence of $M_t$ to the geodesic sphere for all time $t\to \infty$, as we do not have the analogue stability estimate as in \cite[Eq.(7.124)]{RS2014} for the hyperbolic case. However, if we denote $p_t$ as the center of the inner ball of $\Omega_t$, we can prove that $p_t$ converges to the fixed point $p\in \mathbb{H}^{n+1}$ for all time $t\to\infty$ using the Alexandrov reflection and the subsequential Hausdorff convergence of $M_t$ proved in Lemma \ref{subcon}.
Recall that $p\in \mathbb{H}^{n+1}$ is the center of the limit geodesic sphere $S_{\rho_{\infty}}(p)$ in Lemma \ref{subcon}. Take an arbitrary direction $z\in T_p\mathbb{H}^{n+1}$. Let $\gamma_z$ be the normal geodesic line (i.e. $|\gamma'|=1$) through the point $p$ with $\gamma_z(0)=p$ and $\gamma'_z(0)=z$, and let $H_{z,s}$ be the totally geodesic hyperplane in $\mathbb{H}^{n+1}$ that is perpendicular to $\gamma_z$ at $\gamma_z(s), s\in\mathbb{R}$. We use the notation $H_{z,s}^{+}$ and $H_{z,s}^{-}$ for the half-spaces in $\mathbb{H}^{n+1}$ determined by $H_{z,s}$ as follows:
\begin{equation*}
H_{z,s}^{+}:=\bigcup_{s'\geq s}H_{z,s'},\qquad H_{z,s}^{-}:=\bigcup_{s'\leq s}H_{z,s'}.
\end{equation*}
For a bounded domain $\Omega$ in $\mathbb{H}^{n+1}$, denote
\begin{equation*}
\Omega_z^{+}(s)=\Omega\cap H_{z,s}^{+},\qquad \Omega_z^{-}(s)=\Omega\cap H_{z,s}^{-}.
\end{equation*}
The reflection map across $H_{z,s}$ is denoted by $R_{\gamma_z,s}$. We define
\begin{align*}
S_{\gamma_z}^{+}(\Omega)&:=\inf\{s\in\mathbb{R}~|~R_{\gamma_z,s}(\Omega_z^{+}(s))\subset\Omega_z^{-}(s)\},\\
S_{\gamma_z}^{-}(\Omega)&:=\sup\{s\in\mathbb{R}~|~R_{\gamma_z,s}(\Omega_z^{-}(s))\subset\Omega_z^{+}(s)\}.
\end{align*}
The Alexandrov reflection argument implies that $S_{\gamma_z}^{+}(\Omega_t)$ is non-increasing in $t$ for each $z$ (see \cite[Lemma 6.1]{BenWei}). By the definitions of $S_{\gamma_z}^{+}(\Omega_t)$ and $S_{\gamma_z}^{-}(\Omega_t)$, we have $S_{\gamma_z}^{-}(\Omega_t)\leq S_{\gamma_z}^{+}(\Omega_t)$. Since $S_{\gamma_z}^{-}(\Omega_t)=-S_{\gamma_{-z}}^{+}(\Omega_t)$, we also have that $S_{\gamma_z}^{-}(\Omega_t)$ is non-decreasing in $t$ for each $z$. Note that the paper \cite{BenWei} deals with the flow with $h$-convex initial hypersurfaces, the argument in Lemma 6.1 of \cite{BenWei} works for convex solutions as well. The readers may refer to \cite{Chow97,CG96} for more details on the Alexandrov reflection method.
We need the following lemma in proving the convergence of the center of the inner ball.
\begin{lem}\label{center}
Let $\Omega$ be a bounded convex domain in $\mathbb{H}^{n+1}$, and $\gamma_z, H_{z,s}, S_{\gamma_z}^{+}(\Omega)$ and $S_{\gamma_z}^{-}(\Omega)$ be defined as above. Denote $p_0$ as the center of an inner ball of $\Omega$ and assume that $p_0\in H_{z,s_0}$. Then we have $S_{\gamma_z}^{-}(\Omega)\leq s_0\leq S_{\gamma_z}^{+}(\Omega)$.
\end{lem}
\proof
Denote $S_{\gamma_z}^{+}(\Omega)$ by $\bar{s}$, then we have $R_{\gamma_z,\bar{s}}(\Omega_z^{+}(\bar{s}))\subset\Omega_z^{-}(\bar{s})$. Assume that $B_{r_0}(p_0)$ is the inner ball of $\Omega$ centered at $p_0$. First we prove that $s_0\leq \bar{s}$. If not (i.e. $s_0>\bar{s}$), then $B_{r_0}(p'_0):=R_{\gamma_z,\bar{s}}(B_{r_0}(p_0))\subset\Omega$ by $R_{\gamma_z,\bar{s}}(\Omega_z^{+}(\bar{s}))\subset\Omega_z^{-}(\bar{s})$ and the assumption that $s_0>\bar{s}$. See Figure \ref{figure center}. Here $p'_0=R_{\gamma_z,\bar{s}}(p_0)\subset H_{z,s'_0}$ with $s'_0<\bar{s}<s_0$. Then we can deduce that the reflection ball $B_{r_0}(p'_0)\cap \partial\Omega=\emptyset$. It follows that $B_{r_0+\varepsilon}(p'_0)\subset\Omega$ for sufficiently small $\varepsilon$ and hence $p_0$ cannot be the center of an inner ball, which leads to a contradiction.
\begin{figure}
\caption{Center of the inner ball.}
\label{figure center}
\end{figure}
The inequality $S_{\gamma_z}^{-}(\Omega)\leq s_0$ can be checked similarily as above. This completes the proof of Lemma \ref{center}.
\endproof
\begin{rem}
For a bounded domain $\Omega$, the center of the inner ball may not be unique. Lemma \ref{center} says that all of them must satisfy the two-sided bounds.
\end{rem}
Then we can prove the following convergence result:
\begin{lem}\label{innercon}
Let $M_0$ be a smooth, closed convex hypersurface in $\mathbb{H}^{n+1}$ and $M_t$ be the smooth solution of the flow \eqref{flow-VMCF} starting from $M_0$. Denote the enclosed domain of $M_t$ by $\Omega_t$. Take an arbitrary direction $z\in T_p{\mathbb{H}^{n+1}}$ and let $\gamma_z, S_{\gamma_z}^{-}(\Omega_t), S_{\gamma_z}^{+}(\Omega_t)$ be defined as above. Then along the flow \eqref{flow-VMCF}, we have
\begin{equation}\label{reflection}
\lim_{t\to\infty}S_{\gamma_z}^{-}(\Omega_t)=\lim_{t\to\infty}S_{\gamma_z}^{+}(\Omega_t)=0.
\end{equation}
As a consequence, if we set $p_t$ as the center of an inner ball of $\Omega_t$, then we have $d(p_t,p)\to 0$ as $t\to\infty$.
\end{lem}
\begin{proof}
Recall that in Lemma \ref{subcon}, we have proved that there exists a sequence of times $t_i\to\infty$, such that $M_{t_i}$ converges to a geodesic sphere $S_{\rho_{\infty}}(p)=\partial B_{\rho_{\infty}}(p)$ in Hausdorff sense as $t_i\to\infty$, where $\rho_{\infty}$ is the radius determined by $\Omega_0$ and $p$ is the center of the ball. Then there exists a sequence of $d_i\rightarrow 0$ with $d_i<\rho_{\infty}$ such that
\begin{equation*}
M_{t_i}\subset B_{\rho_{\infty}+d_i}(p)/ B_{\rho_{\infty}-d_i}(p).
\end{equation*}
Hence by the monotonicity of $S_{\gamma_z}^{+}(\Omega_t)$, if we can show that $S_{\gamma_z}^{+}(\Omega_{t_i})\leq C\sqrt{d_i}$ for some $C=C(\rho_{\infty})>0$, then we have $S_{\gamma_z}^{+}(\Omega_{t})\leq C\sqrt{d_i},\ \forall t\geq t_i$.
Denote the geodesic segment starting from $A_1$ to $A_2$ by $\overline{A_1A_2}$, and the length by $|\overline{A_1A_2}|$. Let $P,\ P'$ be two totally geodesic hyperplane which is perpendicular to $\gamma_{z}$ and passes through $p$ and some $p'$ respectively on $\gamma_{z}$. Let $E=P\cap\partial B_{\rho_\infty-d_i}(p)$ and $E'=P'\cap\partial B_{\rho_\infty+d_i}(p)$. By a simple continuity argument, there exists $P'$ such that $\forall e'\in E'$, the geodesic line starting form $e'$ and is perpendicular to $P'$ would pass through some $e\in E$, see figure \ref{fig2}.
\begin{figure}
\caption{$R_{\gamma_z,s}
\label{fig2}
\end{figure}
Furthermore, we can pick $P'$ such that
\begin{equation}
d(P,P')=|\overline{pp'}|\leq O(\sqrt{d_i}).
\end{equation}
To see this, by the law of sines and cosines in hyperbolic space, denote $\angle A_1A_2A_3$ as the angle between $\overline{A_1A_2}$ and $\overline{A_2A_3}$, we have
\begin{equation}\label{pythargorean}
\left\{\begin{aligned}
&\cosh|\overline{pp'}| \cosh|\overline{e'p'}|=\cosh(\rho_\infty+d_i),\\
&\cosh|\overline{e'p'}| \cosh|\overline{ee'}|=\cosh(\rho_\infty-d_i) \cosh|\overline{pp'}|\left(=\cosh|\overline{ep'}|\right)\\
&\frac{\sinh|\overline{e'p'}|}{\sinh(\rho_\infty+d_i)}=\sin\angle e'pp'=\cos\angle epe'=\frac{\cosh(\rho_\infty-d_i)\cosh(\rho_\infty+d_i)-\cosh|\overline{ee'}|}{\sinh(\rho_\infty-d_i)\sinh(\rho_\infty+d_i)}.
\end{aligned}\right.
\end{equation}
Solving \eqref{pythargorean}, we get
\begin{equation*}
\sinh(\rho_{\infty}-d_i)\left(1+\sinh^2|\overline{e'p'}|\right)=\cosh(\rho_\infty-d_i)\cosh(\rho_\infty+d_i)\sinh|\overline{e'p'}|.
\end{equation*}
We see that there is a solution with $\sinh|\overline{e'p'}|=\sinh\rho_\infty+O(d_i)$ and hence $\cosh|\overline{pp'}|=1+O(d_i),\ |\overline{pp'}|=O(\sqrt{d_i})$.
Next, we claim that $S_{\gamma_z}^+(\Omega_{t_i})\leq |\overline{pp'}|$, which yields
$\lim_{t\to\infty}S_{\gamma_z}^+(\Omega_{t})\leq 0$. To see this, we have to prove that for $s:=|\overline{pp'}|$, $R_{\gamma_{z},s}(\Omega^+_{t_i}(s))\subset\Omega^-_{t_i}(s)$.
Note that $\Omega^+_{t_i}(s)$ is contained in the cylinder region whose boundary corresponds to the union of normal geodesic lines through $e'\in E'$ in direction perpendicular to $P'$. Hence $\forall A\in\Omega_{t_i,z}^+(s)$, the geodesic line which goes through $A$ and is perpendicular to $P'$, would intersect $\partial B_{\rho_{\infty}+d_i}(p)\cap H_{z,s}^+$, $P'$ and $\partial B_{\rho_{\infty}-d_i}(p)\cap H_{z,s}^-$ at some points $r$, $b$, $l$ respectively. Let $a$ be the closest point on $\overline{rl}$ to $p$, see figure \ref{fig2}. By the convexity of $\Omega_{t_i}$ and that $B_{\rho_{\infty}-d_i}(p)\subset\Omega_{t_i}\subset B_{\rho_{\infty}+d_i}(p)$, we have
\begin{equation*}
\overline{lb}\subset \Omega_{t_i,z}^-(s)
\end{equation*}
and $|\overline{bA}|\leq |\overline{br}|$. For our purpose, if we can show
\begin{equation*}
|\overline{lb}|>|\overline{br}|,
\end{equation*}
then the reflection argument follows.
Denote $\beta=|\overline{bp'}|$. Again by the law of sines and cosines in hyperbolic space, one can calculate
\begin{equation*}
\tanh|\overline{pp'}|=\tanh|\overline{ab}|\cosh\beta,
\end{equation*}
hence $|\overline{ab}|$ is monotonously decreasing in $\beta$. It's obvious that $|\overline{ap}|$ is monotonously increasing in $\beta$, thus
\begin{equation*}
|\overline{la}|-|\overline{ar}|=\text{arccosh}\left(\frac{\cosh(\rho_\infty-d_i)}{\cosh|\overline{ap}|}\right)-\text{arccosh}\left(\frac{\cosh(\rho_\infty+d_i)}{\cosh|\overline{ap}|}\right)
\end{equation*}
is monotonously decreasing in $\beta$. Then
\begin{equation}
|\overline{lb}|-|\overline{br}|=(|\overline{la}|-|\overline{ar}|)+2|\overline{ab}|\geq\left(|\overline{lb}|-|\overline{br}|\right)\bigg|_{\beta=|\overline{p'e'}|}=|\overline{ee'}|>0.
\end{equation}
The same argument can also show $\lim_{t\to\infty}S_{\gamma_z}^-(\Omega_{t})\geq 0$. Then
\begin{equation*}
0\leq \lim_{t\to\infty}S_{\gamma_z}^-(\Omega_{t_i})\leq\lim_{t\to\infty}S_{\gamma_z}^+(\Omega_{t_i})\leq 0,
\end{equation*}
which finishes our proof.
\end{proof}
\section{Proofs of Theorem \ref{theo} and Corollary \ref{coro}}\label{final}
In this section, we complete the proofs of Theorem \ref{theo} and Corollary \ref{coro}.
\subsection{Proof of Theorem \ref{theo}}
Firstly, we prove the following uniform estimate for the principal curvatures of $M_t$ along the flow \eqref{flow-VMCF}.
\begin{lem}\label{uni}
Let $M_0$ be a smooth, closed and convex hypersurface in $\mathbb{H}^{n+1} (n\geq 2)$, and $M_t$ be the smooth solution of the flow \eqref{flow-VMCF} starting from $M_0$. Then there exists constants $\underline{\kappa}$, $\overline{\kappa}$ depending only on $n$, $\alpha$ and $M_0$ such that the principal curvatures $\kappa_i$ of $M_t$ satisfy:
\begin{equation}
\underline{\kappa}\leq \kappa_i\leq \overline{\kappa},\qquad i=1,\dots,n
\end{equation}
for all time $t\in [0,+\infty)$.
\end{lem}
\proof
By Lemma \ref{innercon}, the center $p_t$ of an inner ball of $\Omega_t$ converges to a fixed point $p$ as $t\to\infty$. Since the inner radius of $\Omega_t$ has a positive lower bound $\rho_-(t)>c_1$, there exists a sufficiently large time $t^{*}$, depending on $c_1$ and hence depending only on $n$, $\alpha$ and $M_0$, such that $d(p_t,p)<{c_1}/{4}$ for $t\geq t^{*}$. Then we have:
\begin{equation}\label{s7.0}
B_{c_1/4}(p)\subset \Omega_t,\qquad \forall~t\geq t^{*}.
\end{equation}
Applying Proposition \ref{preserve convex} to the time interval $[0,t^*)$ and $[t^*,\infty)$ respectively gives a uniform lower bound for the principal curvatures of $M_t$ for all time $t>0$. In fact, on the time interval $[0,t^*)$, the estimate \eqref{s4.2-0} implies that the principal curvatures $\kappa_i$ of $M_t$ satisfy
\begin{equation}\label{s7.1}
k_i\geq \Lambda_3^{-1}\Lambda_1^{-\frac{2t^*}{\tau}},\qquad t\in[0,t^*).
\end{equation}
While for time $t\in [t^*,\infty)$, since $B_{c_1/4}(p)\subset \Omega_t$ for all time $t\in [t^*,\infty)$, without loss of generality, we can view the point $p$ as the origin and project $\Omega_t$ onto $B_1(0)\subset\mathbb{R}^{n+1}$ with respect to the point $p$ for all time $t\geq t^{*}$. By the estimate \eqref{tau1} in the proof of Proposition \ref{preserve convex}, we have
\begin{equation*}
\mathfrak{r}_{11}(z,t)\leq \Lambda_1\max\left\{\Lambda,\max_{z\in \mathbb{S}^n}\mathfrak{r}_{11}(z,t^*)\right\}
\end{equation*}
for all $t\in [t^*,\infty)$. This together with \eqref{s7.1} and \eqref{s5:W-inv-2} implies that the principal curvatures of $M_t$ are uniformly bounded from below by a positive constant $\underline{\kappa}$ which depends only on $n$, $\alpha$ and $M_0$.
Since we get the uniform lower bound for the principal curvatures, the uniform upper bound for the Gauss curvature $K$ follows easily from the proof of Proposition \ref{propKupp}. In fact, the upper bound \eqref{s4.2-1} for the terms $(I)$ in the proof of Proposition \ref{propKupp} now has a uniform coefficient $\beta_1$, which is independent of time $t$. Therefore there exists a constant $\overline{\kappa}$, such that $\kappa_i\leq \overline{\kappa}$ for all $i=1,\dots,n$. This completes the proof of Lemma \ref{uni}.
\endproof
It follows from Lemma \ref{uni} that the flow \eqref{flow-VMCF} is uniformly parabolic for all time $t>0$. Then an argument similar to that in the proof of Theorem \ref{long} can be applied to show that all derivatives of curvatures are uniformly bounded on $M_t$ for all $t>0$. This together with Lemma \ref{subcon} implies there exists a sequence of times $t_i\to\infty$, such that $M_{t_i}$ converges smoothly to a geodesic sphere $S_{\rho_{\infty}}(p)$ as $t_i\to\infty$.
The full convergence and the exponential convergence can be obtained by studying the linearization of the flow \eqref{flow-VMCF}. Fix a sufficiently large time $t_i$, we write $M_{t}$ for $t\geq t_i$ as the graph of the radial function $\rho(\cdot,t)$ over $\mathbb{S}^n$ centered at the fixed point $p\in \mathbb{H}^{n+1}$ and the flow equation \eqref{flow-VMCF} is equivalent to the following scalar parabolic PDE
\begin{equation}\label{eq-rho}
\left\{\begin{aligned}
\frac{\partial}{\partial t}\rho&=\left(\phi(t)-K^{\alpha}\right)\sqrt{1+{|\bar{\nabla}\rho|^2}/{\sinh^2\rho}},\quad t>t_i,\\
\rho(\cdot&,t_i)=\rho_{t_i}(\cdot),
\end{aligned}\right.
\end{equation}
for the radial function $\rho$ over the sphere $\mathbb{S}^n$, where $K$ is expressed as a function of $\rho, \bar{\nabla}\rho$ and $\bar{\nabla}^2\rho$ via the equation \eqref{eq-Gauss}. Since $M_{t_i}$ converges to a geodesic sphere $S_{\rho_\infty}(p)$ as $t_i\to \infty$, we can assume that the oscillation of $\rho(\cdot,{t_i})-\rho_{\infty}$ is sufficiently small by choosing $t_i$ large enough. By a direct computation, the linearized equation of the flow \eqref{eq-rho} about the geodesic sphere $S_{\rho_\infty}(p)$ is given by
\begin{equation}\label{eqdeta2}
\frac{\partial }{\partial t}\eta=\frac{ \alpha\coth^{\alpha-1}{\rho_{\infty}}}{n\sinh^2\rho_{\infty}}\left(\bar{\Delta}\eta+n\eta-\frac{n}{|\mathbb{S}^n|}\int_{\mathbb{S}^n}{\eta\, d\sigma}\right).
\end{equation}
Since the oscillation of $\rho(\cdot,{t_i})-\rho_{\infty}$ is already sufficiently small, it follows exactly as in \cite{Cab-Miq2007}, using \cite{Esc98}, that the solution $\rho(\cdot,t)$ of \eqref{eq-rho} starting at $\rho(\cdot,{t_i})$ exists for all time and converges exponentially to a constant $\rho_{\infty}$. This means that the hypersurface $\overline{M}_t=$ graph $\rho(\cdot,t)$ solves \eqref{flow-VMCF} with initial condition $M_{t_i}$ and by uniqueness $\overline{M}_t$ coincides with $M_t$ for $t\geq t_i$, and hence the solution $M_t$ of \eqref{flow-VMCF} with initial condition $M_0$ converges exponentially as $t\to\infty$ to the geodesic sphere $S_{\rho_\infty}(p)$. This completes the proof of Theorem \ref{theo}.
\subsection{Proof of Corollary \ref{coro}}\label{sec6}
Finally, we give the proof of Corollary \ref{coro} using the monotonicity of $\mathcal{A}_{n-1}(\Omega_t)$ and the convergence result of the flow \eqref{flow-VMCF}. Firstly, if $\Omega$ is convex, we evolve the boundary $M=\partial\Omega$ by the flow $\eqref{flow-VMCF}$. Then the inequality \eqref{eqA-F} follows from Theorem \ref{theo} and the monotonicity in Lemma \ref{lemmono} immediately. If equality holds in \eqref{eqA-F} for such $\Omega$, then equality also holds in \eqref{eqWmo} for all time $t$ which means that $\Omega
$ and $\Omega_t$ are all geodesic balls.
In general, for weakly convex $\Omega$ we can approximate $\Omega$ by a family of convex domains $\Omega_\varepsilon$ as $\varepsilon\to 0$. In fact, if we project the domain $\Omega$ into $B_1(0)\subset \mathbb{R}^{n+1}$ as in $\S$\ref{sec2}, the equation \eqref{eq-hij} implies that the image $\hat{\Omega}$ of the projection is also weakly convex in $\mathbb{R}^{n+1}$. Hence we can approximate $\hat{\Omega}$ by convex domains (e.g., by using mean curvature flow). Since the projection is a diffeomorphism, we find a family of convex domains $\Omega_{\varepsilon}$ that approximate $\Omega$ as $\varepsilon\to 0$. Then the inequality \eqref{eqA-F} for $\Omega$ follows from the one for $\Omega_\varepsilon$ by letting $\varepsilon\to 0$.
To prove the equality case for weakly convex $\Omega$, we employ an argument previously used in \cite{G-L2009}. Suppose that $\Omega$ is a weakly convex domain which attains the equality of \eqref{eqA-F}. Let $M_{+}=\{x\in M=\partial\Omega, K>0\}$. Since there exists at least one point $p$ on a closed hypersurface in $\mathbb{H}^{n+1}$ such that all the principal curvatures are strictly larger than 1 at $p$, the subset $M_{+}$ is open and nonempty. We claim that $M_{+}$ is closed as well. In fact, pick any $\eta\in C_0^2(M_{+})$ compactly supported in $M_{+}$, let $M_\varepsilon$ be a smooth family of variational hypersurfaces generated by the vector field $V=\eta\nu$. Let $\Omega_\varepsilon$ be the domain enclosed by $M_\varepsilon$. It is easy to show that $M_\varepsilon$ is weakly convex when $|\varepsilon|$ is small enough. Hence
\begin{equation*}
\mathcal{A}_{n-1}(\Omega_\varepsilon)\geq \psi_n\left(|\Omega_\varepsilon|\right)
\end{equation*}
holds for sufficiently small $|\varepsilon|$ and with equality holding at $\varepsilon=0$. Thus
\begin{align*}
0=&\frac{d}{d\varepsilon}\bigg|_{\varepsilon=0}\Big(\mathcal{A}_{n-1}(\Omega_\varepsilon)- \psi_n\left(|\Omega_\varepsilon|\right)\Big)=n\int_{M}{\left(K-\psi'_n(|\Omega|)\right)\eta}d\mu.
\end{align*}
Since $\eta\in C_0^2(M_{+})$ is arbitrary, we have $K=\psi'_n(|\Omega|)$ everywhere on $M_{+}$. As this is a closed condition, we conclude that $M_{+}$ is closed. Therefore $M=M_{+}$ and so $\Omega$ is a convex domain. Then by the equality case for convex domain, we conclude that $\Omega$ is a geodesic ball.
\end{document} | math | 99,449 |
\begin{document}
\author[J.~P.~Gollin]{J.~Pascal Gollin}
\address{J. Pascal Gollin, Discrete Mathematics Group, Institute for Basic Science (IBS), 55 Expo-ro, Yuseong-gu, Daejeon,
Korea, 34126}
\email{\tt [email protected]}
\thanks{The first author was supported by the Institute for Basic Science (IBS-R029-Y3).}
\author[A.~Jo\'{o}]{Attila Jo\'{o}}
\address{Attila Jo\'{o}, Department of Mathematics, University of Hamburg, Bundesstra{\ss}e 55 (Geomatikum), 20146
Hamburg, Germany and Set theory and general topology research division,
Alfr\'{e}d R\'{e}nyi Institute of Mathematics, 13-15 Re\'{a}ltanoda St.,
Budapest, Hungary}
\email{\tt [email protected]}
\thanks{Funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation)-513023562 and partially by NKFIH OTKA-129211}
\title[Matching variables to equations in infinite linear equation systems]{Matching variables to equations in infinite linear equation systems}
\date{November 23}
\keywords{linear equation system, matching, thin sum}
\subjclass[2020]{Primary: 15A06, 05C50 Secondary: 05C63 }
\begin{abstract}
A fundamental result in linear algebra states that if a homogenous linear equation system has only the trivial solution, then there are at most as many variables as equations.
We prove the following generalisation of this phenomenon.
If a possibly infinite homogenous linear equation system with finitely many variables in each equation has only the trivial solution, then there exists an injection from the variables to the equations that maps each variable to an equation in which it appears.
\end{abstract}
\maketitle
\@ifstar{\origsection*}{\@startsection{section}{1}\z@{.7\ifmmode\ell\else\polishlcross\fiinespacing\@plus\ifmmode\ell\else\polishlcross\fiinespacing}{.5\ifmmode\ell\else\polishlcross\fiinespacing}{\normalfont\scshape\centering\S}}{Introduction}
Infinite linear equation systems appear in the most diverse areas of mathematics.
They have a key role in boundary value problems for linear partial differential equations.
Banach devoted them in his book~\cite{banach1987book} a whole section entitled ``Systems of linear equations in infinitely
many unknowns''.
In this setting, there are countably many variables and infinite sums are defined via convergence.
Another possible approach deals with sums that are ``thin'', i.e.~that there are only finitely many non-zero summands in each.
Such sums appear for example in horizon planning programs (see \cite{romeijn1998shadowprices}).
More recently, these sums turned out to be fruitful in the representation theory of infinite matroids.
For a set~$I$ and field~$\mathbb{F}$, a family~${\mathcal{F} = ( f_j \in \mathbb{F}^I \,\colon\, j \in J )}$ is called \emph{thin} if for each~${i \in I}$ there are only finitely many~${j \in J}$ with~${f_j(i) \neq 0}$.
Infinite linear combinations of the functions~$f_j$ can be defined in a natural way.
Indeed, if~${\ifmmode\ell\else\polishlcross\fiambda_j \in \mathbb{F}}$ for~${j \in J}$, then for each~${i \in I}$ the sum~${\sum_{j \in J} \ifmmode\ell\else\polishlcross\fiambda_j f_j(x)}$ is a well-defined element of~${\mathbb{F}}$, therefore~${\sum_{j \in J} \ifmmode\ell\else\polishlcross\fiambda_j f_j}$ can be considered as an element of~$\mathbb{F}^{I}$.
If the constant~$0$ function on~$I$ is obtained only if~${\ifmmode\ell\else\polishlcross\fiambda_j = 0}$ for every~${j \in J}$, then~$\mathcal{F}$ is said to be \emph{thinly independent}.
In other words, $\mathcal{F}$ is thinly independent if the (possibly infinite) homogenous linear equation system~${\sum_{j \in J} f_j(i) x_j = 0\ (i \in I)}$ has only the trivial solution.
Investigation of the concept of thin dependence in the context of matroid theory was initiated by Bruhn and Diestel~\cite{bruhn2011infmatrgraph} and became a relatively well-understood subject after the discoveries of Afzali and Bowler~\cite{borujeni2016finitary}.
Our main result states that for a thinly independent~$\mathcal{F}$, there is always a system of distinct representatives for the family~${\{ \mathsf{supp}(f_j) \, \colon \, j \in J \}}$ where~${\mathsf{supp}(f_j)}$ denotes the \emph{support} of~$f_j$, that is the set of those elements of~$X$ on which~$f_j$ is non-zero.
Note that by considering arbitrary families with the the usual linear independence in~$\mathbb{F}^I$, the analogous statement fails.
Indeed, for example the dimension of the vector space~${\mathsf{GF}(2)^{\mathbb{N}}}$ is continuum and therefore no base of it can be injectively mapped into~$\mathbb{N}$.
Considering thin families but still the usual independence does not fix this issue.
To demonstrate this, let us take in~$\mathsf{GF}(2)^{\mathbb{N}}$ the unit vectors together with their thin sum, the constant~$1$ vector.
Then, no desired injection exists, although this family is linearly independent.
Let us rephrase our main result in a more elementary way.
A basic fact in linear algebra states that if a finite homogeneous linear equation system has only the trivial solution, then there are at most as many variables as equations.
Naively lifting observations such as this to an infinite setting often loses some interesting structural information about the problem, since a pure comparison between cardinalities of sets is a rough measure.
Instead, let us strengthen the fact to include more structural information.
If a finite homogeneous linear equation system has only the trivial solution, then it is not too hard to show (using standard techniques from matching theory) that there exists an injection from the variables to the equations where each variable is mapped to an equation in which it has non-zero coefficient.
Our main result states that this remains true for every homogeneous thin linear equation system.
\begin{thm-intro}\ifmmode\ell\else\polishlcross\fiabel{thm: main}
Let~$I$ and~$J$ be sets, let~$\mathbb{F}$ be a field, and let~${a_{i,j} \in \mathbb{F}}$ for~${i \in I}$ and~${j \in J}$ such that for each~${i \in I}$ there are only finitely many~${j \in J}$ with~${a_{i,j} \neq 0}$.
Suppose that the (possibly infinite) homogeneous linear equation system
\begin{equation}
\ifmmode\ell\else\polishlcross\fiabel{eq: thinLES}\tag{$\ast$}
\sum_{j \in J} a_{i,j} x_j = 0 \quad (i \in I)
\end{equation}
has only the trivial solution.
Then there is an injection~${\varphi \colon J \to I}$ such that~${a_{\varphi(j),j} \neq 0}$ for every~${j \in J}$.
\end{thm-intro}
In a regular matrix, one can rearrange its rows to obtain a matrix in which every entry of the diagonal is non-zero. Using the terminology of thinly independent families as before and using the Cantor-Bernstein Theorem (see Theorem~\ref{thm: cantor-bernstein}), we obtain the following generalisation of this fact as a corollary of Theorem~\ref{thm: main}.
\begin{cor-intro}
\ifmmode\ell\else\polishlcross\fiabel{cor: diagonal}
Let~$I$ and~$J$ be sets, let~$\mathbb{F}$ be a field, and let~${a_{i,j} \in \mathbb{F}}$ for~${i \in I}$ and~${j \in J}$ such that the families ${( ( i \mapsto a_{i,j} ) \in \mathbb{F}^I \, \colon \, j \in J)}$ and ${( ( j \mapsto a_{i,j} ) \in \mathbb{F}^J \, \colon \, i \in I )}$ are both thinly independent.
Then there is a bijection~${\psi \colon J \to I}$ such that~${a_{\psi(j),j} \neq 0}$ for every~${j \in J}$.
\end{cor-intro}
\@ifstar{\origsection*}{\@startsection{section}{1}\z@{.7\ifmmode\ell\else\polishlcross\fiinespacing\@plus\ifmmode\ell\else\polishlcross\fiinespacing}{.5\ifmmode\ell\else\polishlcross\fiinespacing}{\normalfont\scshape\centering\S}}{Notation and Preliminaries}
\ifmmode\ell\else\polishlcross\fiabel{sec: prelims}
For the domain and range of a function~$f$ we write~${\mathsf{dom}(f)}$ and~${\mathsf{ran}(f)}$ respectively. We write ${\mathsf{ran}_I(f)}$ as an abbreviation of ${\mathsf{ran}(f)}\cap I$.
For a subset~${S \subseteq \mathsf{dom}(f)}$, we denote by~$f {\upharpoonright} S$ the \emph{restriction} of~$f$ to~$S$.
A \emph{bipartite graph}~$G$ is a triple~$(S,T,E)$, where~$S$ and~$T$ are disjoint sets \ifmmode\ell\else\polishlcross\fiinebreak and~${E \subseteq \{\{ s,t \} \, \colon \, s \in S \textnormal{ and } t \in T\}}$.
The elements of~${S \cup T}$ are the \emph{vertices} of~$G$ and the elements of~$E$ are the \emph{edges} of~$G$.
The set containing all~${w \in S \cup T}$ for which ~${\{v,w\} \in E}$ is the \emph{neighbourhood}~${N_G(v)}$ of~$v$, and the cardinal~$\abs{N_G(v)}$ is the \emph{degree} of~$v$.
A \emph{matching}~$M$ in~$G$ is a set of edges no two of which share a vertex.
We say a matching~$M$ \emph{covers} a set~${X \subseteq S \cup T}$ if each vertex in~$X$ is contained in some edge in~$M$.
A matching is \emph{perfect} if it covers~${S \cup T}$.
Let~$I$ and~$J$ be sets and let~$\mathbb{F}$ be a field.
We denote by~$\mathbb{F}^I$ the vector space of functions from~$I$ to~$\mathbb{F}$.
Given an element~${b \in \mathbb{F}^I}$ and~${i \in I}$ we write~$b_i$ instead of~${b(i)}$.
A \emph{matrix} in this paper is a function~${A \colon I \times J \to \mathbb{F}}$.
For~${i \in I}$ and~${j \in J}$, we write~$a_{i,j}$ instead of~${A(i,j)}$.
For a fixed~${i \in I}$, the map~${j \mapsto a_{i,j}}$ is the \emph{row} of~$A$ corresponding to~$i$ while columns are defined analogously.
The \emph{rank}~${r(A)}$ of a finite matrix~$A$ is the dimension of the subspace of~${\mathbb{F}^{J}}$ spanned by its rows (equivalently the dimension of the subspace of~${\mathbb{F}^{I}}$ spanned by its columns).
Let~${A \colon I \times J \to \mathbb{F}}$ be a matrix, and let~${b \in \mathbb{F}^I}$.
We say that~$A$ is \emph{row-thin} if the support of each row of~$A$ is finite.
If~$A$ is row-thin, then we denote by
\[
\sum_{j \in J} a_{i,j} x_j = b_i \quad (i \in I)
\]
a \emph{thin system of linear equations} with \emph{variables}~${x_j \ (j \in J)}$.
We may also denote this system by~${Ax = b}$.
If~${b_i = 0}$ for all~${i \in I}$, we call the system \emph{homogeneous}.
Note that given an element~${s \in \mathbb{F}^J}$, the sum~$\sum_{j \in J} a_{i,j} s_j$ is a well-defined element of~$\mathbb{F}$.
A \emph{solution} for~${Ax = b}$ is an element~${\ifmmode\ell\else\polishlcross\fiambda \in \mathbb{F}^J}$ such that~$\sum_{j \in J} a_{i,j} \ifmmode\ell\else\polishlcross\fiambda_j = b_i$ for each~${i \in I}$.
We say that~${Ax = b}$ is \emph{solvable} if it has a solution.
If the field~$\mathbb{F}$ is finite, then standard compactness arguments show that the solvability of a thin linear equation system is equivalent with the solvability of all its finite subsystems.
Maybe surprisingly, this remains true without any restriction on~$\mathbb{F}$.
\begin{theorem}[Compactness of thin linear equation systems, Cowen and Emerson \cite{cowen1996complin}\footnotemark]
\ifmmode\ell\else\polishlcross\fiabel{thm: compact LES}
If every finite subset of the equations of a thin linear equation system is solvable, then the whole system is solvable.
\end{theorem}
\footnotetext{This theorem was rediscovered independently by Bruhn and Georgakopoulos~\cite{bruhn2011bases}.
Their proof was later simplified by Afzali and Bowler {\cite[Lemma 4.2]{borujeni2015thin}}.}
To prove our main theorem, we also need a tool from infinite matching theory developed by Wojciechowski~\cite{wojciechowski1997criteria}.
Let~${G = (S,T,E)}$ be a bipartite graph.
A \emph{string} corresponding to~$G$ is an injective function~$f$ defined on an ordinal number with range~${\mathsf{ran}(f) \subseteq S \cup T}$.
A string~$f$ is called \emph{saturated} if whenever~$f(\alpha) = v \in S$, then~${N_G(v) \subseteq \mathsf{ran}(f {\upharpoonright} \alpha})$.
In other words, a vertex~${v \in S}$ can only appear in the transfinite sequence~$f$ after all of its neighbours already appeared.
For a saturated string~$f$ with~${\mathsf{dom}(f) = \alpha}$, the quantity~${\mu_G(f) \in \mathbb{Z} \cup \{ -\infty, +\infty \}}$ is defined by transfinite recursion on~$\alpha$ as follows.
\[
\mu_G(f) :=
\begin{cases} 0 &\mbox{if } \alpha=0, \\
\ifmmode\ell\else\polishlcross\fiiminf \{ \mu_G(f {\upharpoonright} \beta) \, \colon \, \beta<\alpha \} & \mbox{if } \alpha \text{ is a limit ordinal},\\
\mu_G(f {\upharpoonright} \beta)-1 & \mbox{if } \alpha=\beta+1 \text{ and } f(\beta)\in S,\\
\mu_G(f {\upharpoonright} \beta)+1 & \mbox{if } \alpha=\beta+1 \text{ and } f(\beta)\in T,
\end{cases}
\]
where we use the convention that~${\pm \infty + k = \pm \infty}$ for~${k \in \mathbb{Z}}$.
It is not too hard to prove that if~$G$ admits a matching that covers~$S$, then we must have~${\mu_G(f) \geq 0}$ for every saturated string~$f$.
Under some assumption the reverse is also true.
\begin{thm}[Wojciechowski {\cite[Theorem 1]{wojciechowski1997criteria}}\footnotemark]
\ifmmode\ell\else\polishlcross\fiabel{thm: jerzy}
Let~${G = (S,T,E)}$ be a bipartite graph in which each vertex in~$T$ has countable degree.
Then there is a matching in~$G$ covering~$S$ if and only if~${\mu_G(f) \geq 0}$ for every saturated string~$f$ corresponding to~$G$
\end{thm}
\footnotetext{The criterion given in Theorem~\ref{thm: jerzy} is called $\mu$\nobreakdash-admissibility and was inspired by the $q$\nobreakdash-admissibility criterion of Nash-Williams (see \cite{nash1978another}).
A characterisation of matchability for arbitrary bipartite graphs was discovered by Aharoni, Nash-Williams and Shelah~\cite{aharoni1983general}.
For a survey on infinite matching theory (including the non-bipartite case) we refer to~\cite{aharoni1991infinite}.}
To obtain Corollary~\ref{cor: diagonal}, we now state the well-known theorem of Cantor and Bernstein in a stronger, graph-theoretic form.
\begin{thm}[Cantor-Bernstein \cite{cantor1987}]
\ifmmode\ell\else\polishlcross\fiabel{thm: cantor-bernstein}
If ${G = (S,T,E)}$ is a bipartite graph and there exist a matching~$M_S$ that covers~$S$ as well as a matching~$M_T$ that covers~$T$, then~$G$ admits a perfect matching.
\end{thm}
\@ifstar{\origsection*}{\@startsection{section}{1}\z@{.7\ifmmode\ell\else\polishlcross\fiinespacing\@plus\ifmmode\ell\else\polishlcross\fiinespacing}{.5\ifmmode\ell\else\polishlcross\fiinespacing}{\normalfont\scshape\centering\S}}{Proof of the main results}
Let us fix a homogeneous thin linear equation system~${Ax = 0}$, where~${A \colon I \times J \to \mathbb{F}}$,
that admits only the trivial solution.
Without loss of generality, we may assume that~$I$ and~$J$ are disjoint.
We define a bipartite graph~$G_A = (J,I,E)$ where~${\{i,j\} \in E}$ for~${i \in I}$ and~${j \in J}$ if and only if~$a_{i,j}$ is non-zero.
We will simply write~$\mu$ instead of~$\mu_{G_A}$.
Moreover, when we refer to a saturated string we will always mean a saturated string with respect to~$G_A$.
\begin{obs}
\ifmmode\ell\else\polishlcross\fiabel{obs: saturated}
If a string~$f$ is saturated, then for~${i \in I \setminus (\mathsf{ran}_I(f)})$ and~${j \in \mathsf{ran}_J(f)}$, we have~${a_{i,j} = 0}$.
In other words, the matrix~${A {\upharpoonright} (I \times (\mathsf{ran}_J(f)))}$ is obtained from the matrix~${A {\upharpoonright} ((\mathsf{ran}_I(f)) \times (\mathsf{ran}_J(f)))}$ by extending the columns by zeroes.
\end{obs}
\begin{lemma}
\ifmmode\ell\else\polishlcross\fiabel{lem: main}
If~$f$ is a saturated string such that~$\mu$ takes non-negative finite values on all proper initial segments of~$f$,
then for every finite~${I_0 \subseteq \mathsf{ran}_I(f)}$, there is a finite~${I' \subseteq \mathsf{ran}_I(f)}$ extending~${I_0}$ and a finite~${J' \subseteq \mathsf{ran}_J(f)}$ such that~${\mu(f) = \abs{I'} - r(A {\upharpoonright} (I' \times J'))}$.
\end{lemma}
\begin{proof}
We apply transfinite induction on~${\mathsf{dom}(f) =: \alpha}$.
If~${\alpha = 0}$, we must have~${I_0 = \emptyset}$ and we can only take~${I' := J' := \emptyset}$.
This is appropriate because~${\mu(\emptyset) = \abs{\emptyset} = r(\emptyset) = 0}$.
If~$\alpha$ is a limit ordinal, then~${U := \{ \beta < \alpha \, \colon \, \mu(f {\upharpoonright} \beta) = \mu(f) \}}$ is unbounded in~$\alpha$ by the definition of~$\mu$.
Let a finite set~${I_0 \subseteq \mathsf{ran}_I(f)}$ be given and let~${\beta \in U}$ large enough to satisfy~${I_0 \subseteq \mathsf{ran}(f {\upharpoonright} \beta)}$.
By induction,
we obtain a finite set~${I' \subseteq \mathsf{ran}_{I}(f {\upharpoonright} \beta)}$ extending~$I_0$ and a finite set~${J' \subseteq \mathsf{ran}_J(f {\upharpoonright} \beta)}$ such that~${\mu(f {\upharpoonright} \beta) = \abs{I'} - r(A {\upharpoonright} (I' \times J'))}$.
Since ${\mathsf{ran}(f {\upharpoonright} \beta) \subseteq \mathsf{ran}(f)}$,
these~$I'$ and~$J'$ are as desired.
Finally, assume that~${\alpha = \beta+1}$ and let a finite~${I_0 \subseteq \mathsf{ran}_I(f)}$ be given.
Suppose first that~${i := f(\beta) \in I}$.
We may assume without loss of generality that~${i \in I_0}$.
By applying the induction hypotheses for~${f {\upharpoonright} \beta}$ and~${I_0 \setminus \{ i \}}$,
we can pick finite
sets~${I^{*} \subseteq \mathsf{ran}_I(f {\upharpoonright} \beta)}$
and~${J^{*} \subseteq \mathsf{ran}_J(f {\upharpoonright} \beta)}$
with~${I^{*} \supseteq I_0 \setminus \{ i \}}$
and~${\mu(f {\upharpoonright} \beta) = \abs{I^{*}} - r(A {\upharpoonright} (I^{*} \times J^{*}))}$.
On the one hand, ${\abs{I^{*} \cup \{ i \}} = \abs{I^{*}} + 1}$ since~${i \notin I^{*}}$ and~${\mu(f) = \mu(f {\upharpoonright} \beta) + 1}$ since~${f(\beta) = i \in I}$.
On the other hand, ${r(A {\upharpoonright} (I^{*} \times J^{*})) = r(A {\upharpoonright} ((I^{*} \cup \{ i \})\times J^{*}))}$
because Observation~\ref{obs: saturated} ensures that each column is extended only by a new~$0$ coordinate.
By combining these, we conclude that~${\mu(f) = \abs{I^{*} \cup \{ i \}} - r(A {\upharpoonright} ((I^{*} \cup \{ i \}) \times J^{*}))}$, thus~${I' := I^{*} \cup \{ i \}}$ and~${J' := J^{*}}$ are appropriate.
Now we suppose that~${j_0 := f(\beta) \in J}$.
The thin linear equation system
\[
\sum_{j \in \mathsf{ran}_J(f {\upharpoonright} \beta)} a_{i,j} x_j = a_{i, j_0} \quad (i \in I)
\]
has no solution since a solution would yield a non-trivial solution of~(\ref{eq: thinLES}).
Note that for~${i \in I \setminus \mathsf{ran}(f {\upharpoonright} \beta)}$, the equations above are trivial (i.e.~all coefficients and the right side are zeroes) by Observation~\ref{obs: saturated}.
Therefore the subsystem of the equations corresponding the indices in~${\mathsf{ran}_I(f {\upharpoonright} \beta)}$ is unsolvable.
By Theorem~\ref{thm: compact LES}, there is already a finite~${I_1 \subseteq \mathsf{ran}_I(f {\upharpoonright} \beta)}$ such that the corresponding subsystem is unsolvable.
Now we apply the induction hypothesis for~${f {\upharpoonright} \beta}$ and~${I_0 \cup I_1}$ to pick finite sets~${I^{*} \subseteq \mathsf{ran}_I(f {\upharpoonright} \beta)}$ and~${J^{*} \subseteq \mathsf{ran}_J(f {\upharpoonright} \beta)}$ with~${I^{*} \supseteq
I_0 \cup I_1}$ and~${\mu(f {\upharpoonright} \beta) = \abs{I^{*}} - r(A {\upharpoonright} (I^{*} \times J^{*}))}$.
On one hand,~${\mu(f) = \mu(f {\upharpoonright} \beta) - 1}$ since~${f(\beta) = j_0 \in J}$.
On the other hand, ${r(A {\upharpoonright} (I^{*} \times (J^{*} \cup \{ j_0 \}))) = r(A {\upharpoonright} (I^{*} \times J^{*})) + 1}$ because the new column is not spanned by the old ones because~${I_1 \subseteq I^{*}}$.
By combining these, we conclude that~${\mu(f) = \abs{I^{*}} - r(A {\upharpoonright} (I^{*} \times (J^{*} \cup \{ j_0 \})))}$, thus~${I' := I^{*}}$ and~${J' := J^{*} \cup \{ j_0 \}}$ are appropriate, which completes the proof.
\end{proof}
Let us restate our main theorem using the notation from Section~\ref{sec: prelims}.
\setcounter{thm-intro}{0}
\begin{thm-intro}
Let~$I$ and~$J$ be sets, let~$\mathbb{F}$ be a field, and let~$A \colon I \times J \to \mathbb{F}$ be a row-thin matrix.
If the homogeneous thin linear equation system~${Ax = 0}$ has only the trivial solution,
then there is an injection~${\varphi \colon J \to I}$ such that~${a_{\varphi(j),j} \neq 0}$ for every~${j \in J}$.
\end{thm-intro}
\begin{proof}
Suppose for a contradiction that the desired injection does not exist, i.e.~there is no matching in~$G_A$ that covers~$J$.
By Theorem~\ref{thm: jerzy}, we can pick a saturated string~$f$ with~${\mu(f) < 0}$.
By replacing~$f$ with an initial segment of itself if necessary, we can assume that~$\mu$ takes
only non-negative finite values on the proper initial segments of~$f$.
According to Lemma~\ref{lem: main}, we have~${\mu(f) = \abs{I'} - r(A {\upharpoonright} (I'\times J'))}$ for some finite~${I' \subseteq I}$ and~${J' \subseteq J}$.
Since the rank of a matrix is at most the number of rows this leads to~${\mu(f) \geq 0}$, contradicting the choice of~$f$.
\end{proof}
Let us now prove Corollary~\ref{cor: diagonal}, which we restate using the terminology from Section~\ref{sec: prelims}.
\begin{cor-intro}
Let~$I$ and~$J$ be sets, let~$\mathbb{F}$ be a field, and let~${A \colon I \times J \to \mathbb{F}}$ such that the family of rows and the family of columns are both thinly independent.
Then there is a bijection~${\psi \colon J \to I}$ such that~${a_{\psi(j),j} \neq 0}$ for every~${j \in J}$.
\end{cor-intro}
\begin{proof}
Consider the graph~$G_A$ as above.
By Theorem~\ref{thm: main},
since the family of rows is thinly independent there is matching that covers~$I$, and since the family of columns is thinly independent, there is a matching that covers~$J$.
By Theorem~\ref{thm: cantor-bernstein}, $G_A$ admits a perfect matching~$M$.
Setting~$\psi(j)$ to be the unique~${i \in I}$ for which~${\{i,j\} \in M}$ completes the proof.
\end{proof}
\printbibliography
\end{document} | math | 22,086 |
\begin{document}
\begin{abstract}
We introduce a new perfect sampling technique that can be applied to general Gibbs distributions and runs in linear time if the correlation decays faster than the neighborhood growth.
In particular, in graphs with sub-exponential neighborhood growth like $\mathbb{Z}^d$, our algorithm achieves linear running time as long as Gibbs sampling is rapidly mixing.
As concrete applications, we obtain the currently best perfect samplers for colorings and for monomer-dimer models in such graphs.
\end{abstract}
\title{Perfect sampling from spatial mixing}
\section{Introduction}
Spin systems model nearest neighbor interactions of complex systems.
These models originated from statistical physics, and have found a wide range of applications in probability theory, machine learning, and theoretical computer science,
often under different names such as \emph{Markov random fields}.
Given an underlying graph $G=(V,E)$,
a \emph{configuration} $\sigma$ is an assignment from vertices to a finite set of spins, usually denoted by $[q]$.
The \emph{weight} of a configuration is specified by the $q$-dimensional vector $b_v$ assigned to each vertex $v \in V$ and the $q$-by-$q$ symmetric interaction matrix $A_e$ assigned to each edge $e \in E$, namely,
\begin{align}\label{eqn:spin-system}
w(\sigma) = \prod_{v \in V}b_v(\sigma_v)\prod_{e = \{u,v\}\in E} A_e(\sigma_u, \sigma_v).
\end{align}
The equilibrium state of the system is described by the \emph{Gibbs distribution} $\mu$,
where the probability of a configuration is proportional to its weight.
A central algorithmic problem related to spin systems is to sample from the Gibbs distribution.
A canonical Markov chain for sampling approximately from the Gibbs distribution is the \emph{Gibbs sampler} (a.k.a.~\emph{heat bath} or \emph{Glauber dynamics}).
One (conjectured) general criterion for the rapid mixing of this chain is the \emph{spatial mixing} property,
which roughly states that correlation decays rapidly in the system as distance increases.
It is widely believed that spatial mixing (in some form) implies the rapid mixing of the Gibbs sampler.
However, rigorous implications have only been established for special classes of graphs or systems,
such as for lattice graphs \cite{Mar99,DSVW04}, neighborhood amenable graphs \cite{GMP05}, or for monotone systems \cite{mossel2013exact}.
One main drawback for Gibbs samplers or Markov chains in general,
is that one needs to know the mixing time in advance to be able to implement them to make the error provably small.
The so-called \emph{perfect} samplers are thus more desirable,
which run in Las Vegas fashion and return exact samples upon halting.
There have been a number of techniques available to design perfect samplers,
such as Coupling From The Past (CFTP) \cite{propp1996exact} including the bounding chains \cite{huber2004perfect,bhandari2019perfect},
Randomness Recycler (RR) \cite{fill2000randomness}, and Partial Rejection Sampling (PRS) \cite{GJL19,FVY19}.
Nevertheless, none of these previous techniques addresses general spin systems or relates to the important spatial mixing properties of the system.
In this paper, we introduce a new technique to perfectly sample from Gibbs distributions of spin systems.
The correctness of our algorithm relies on only the conditional independence property of Gibbs distributions.
Moreover, the expected running time is linear in the size of the system, when the correlation decays more rapidly than the growth of the neighborhood.
\begin{theorem}[{informal}] \label{thm:main-informal}
For any spin system with bounded maximum degree,\footnote{We remark that our algorithm remains in polynomial-time (but not in linear time) for graphs with unbounded degrees,
as long as the degree does not grow too quickly.
This requirement on the degree comes from the cost of updating a block of certain radius,
similar to the cost of block dynamics,
and the exact upper bound needed varies from problem to problem.
See \Cref{theorem-general} and the discussion thereafter.
For the most part, we state our results for bounded degree cases to keep the statements clean.}
if strong spatial mixing holds with a rate faster than the neighborhood growth of the underlying graph,
then there exists a perfect sampler with running time $O(n)$ in expectation, where $n$ represents the number of vertices of the graph.
\end{theorem}
More details and undefined terms are explained in \Cref{section-results}.
Formal statements of our results are given in \Cref{theorem-sub-graph} for spin systems on sub-exponential neighborhood growth graphs,
and in \Cref{theorem-general} for spin systems on general graphs.
Applications on list colorings and monomer-dimer models are given in \Cref{corollary-main-coloring} and \Cref{corollary-main-matching}.
Lattice graphs, such as $\mathbb{Z}^d$, are of special interests in statistical physics and combinatorics.
These graphs have sub-exponential neighborhood growth,
which implies that temporal mixing is equivalent to spatial mixing on them~\cite{DSVW04}.
Therefore our sampler runs in linear time as long as the standard Glauber dynamics has $O(n\log n)$ mixing time.
This is a direct strengthening of aforementioned results~\cite{Mar99,DSVW04,GMP05} from approximate to perfect sampling, with an improved running time.
\begin{corollary}[{informal}] \label{cor:main-informal}
For spin systems on graphs with sub-exponential neighborhood growth,
if the Gibbs sampler has $O(n\log n)$ mixing time, where $n$ is the number of vertices, or the system shows strong spatial mixing,
then there exists a perfect sampler with running time $O(n)$ in expectation.
\end{corollary}
It is worth noting that many traditional perfect sampling algorithms,
especially those rely on CFTP \cite{propp1996exact,huber2004perfect,bhandari2019perfect},
suffer from ``non-interruptibility''.
That is, early termination of the algorithm induces a bias on the sample.
In contrast, our algorithm is interruptible in the following sense:
conditioned on its termination at any particular step, the algorithm guarantees to return a correct sample.
Therefore, had the algorithm been running for too long, one can simply stop it and restart.
One can also run many independent copies in parallel and output the earliest returned sample without biasing the output distribution.
In addition, our algorithm can be used to solve the recently introduced dynamic sampling problem~\cite{FVY19,feng2019dynamicMCMC},
where the Gibbs distribution itself changes dynamically and the algorithm needs to efficiently maintain a sample from the current Gibbs distribution.
The detail of this part is given in Section~\ref{section-dynamic}.
Our perfect sampler also generalizes straightforwardly to Gibbs distributions with multi-body interactions
(namely spin systems on hypergraphs / constraint satisfaction problems),
and similar efficiency can be achieved when some appropriate variant of spatial mixing holds.
Morally, we believe that efficient perfect sampling algorithms exist whenever efficient approximate samplers exist.
Our result verifies this belief for spin systems on lattice graphs.
However, in general the gap between approximate and perfect sampling persists (e.g.~for sampling proper colorings \cite{bhandari2019perfect}),
and designing efficient perfect sampling algorithms matching their approximate counterparts remain an interesting research direction.
\mathsf{Sub}section{Algorithm overview}\label{sec:alg-overview}
In this section, we give a simplified single-site version of our algorithm to illustrate the main ideas.
Recall that the Gibbs sampler is a Markov chain on state space $\Omega=[q]^V$.
The chain starts from an arbitrary initial configuration $\boldsymbol{X}\in[q]^V$. At each step, a vertex $u\in V$ is picked uniformly at random and $\boldsymbol{X}$ is updated as following:
\begin{itemize}
\item the spin $X_u$ is redrawn according to the marginal distribution $\mu_u^{X_{\Gamma(u)}}$ induced at vertex $u$ by Gibbs distribution $\mu$ conditioned on the current spins of the neighborhood $\Gamma(u)$,
\end{itemize}
where $\Gamma(u)\triangleq\{v\in V\mid \{u,v\}\in E\}$ denotes the neighborhood of $u$ in graph $G=(V,E)$.
It is a basic fact that this chain converges to the desired Gibbs distribution $\mu$.
Our perfect sampler makes use of the same update rule.
For ease of exposition, here we consider only soft constraints (where all $A_e$ and $b_v$ are positive).
The single-site version of our perfect sampler is described in Algorithm~\ref{alg:perfect-sampler-ss}.
{
\let\oldnl\nl
\newcommand{\nonl}{\renewcommand{\nl}{\let\nl\oldnl}}
\begin{algorithm}[htbp]
\SetKwInOut{Input}{Input}
\SetKwComment{Comment}{\quad$\triangleright$\ }{}
\SetKwIF{withprob}{}{}{with probability}{do}{}{}{}
Start from an arbitrary initial configuration $\boldsymbol{X} \in [q]^V$ and $\mathcal{R} \gets V$\;
\While{$\mathcal{R} \neq \emptyset$}{
pick a $u\in\mathcal{R}$ uniformly at random\;
let $\mu_{\min}$ be the minimum value of $\mu_{u}^{\sigma}(X_u)$ over all $\sigma\in[q]^{\Gamma(u)}$ that $\sigma_{\mathcal{R}\cap\Gamma(u)}=X_{\mathcal{R}\cap\Gamma(u)}$\;
\withprob{\,\,${\mu_{\min}}/{\mu_{u}^{X_{\Gamma(u)}}(X_u)}$\,\,}{
\nonl
\hspace{206.6pt}\texttt{$\triangleright$ Bayes filter}
update $\boldsymbol{X}$ by redrawing $X_u\sim\mu_{u}^{{X_{\Gamma(u)}}}$;
\hspace{44.3pt}\texttt{$\triangleright$ Gibbs sampler update}\\
$\mathcal{R} \gets\mathcal{R} \setminus \{u\}$\;
}
\Else{
$\mathcal{R} \gets\mathcal{R} \cup \Gamma(u)$\;
}
}
\mathcal{R}eturn{$\boldsymbol{X}$}\;
\caption{Perfect Gibbs sampler (single-site version)}\label{alg:perfect-sampler-ss}
\end{algorithm}
}
The algorithm starts from an arbitrary initial configuration $\boldsymbol{X}\in [q]^V$,
and gradually ``repairs'' $\boldsymbol{X}$ to a perfect sample of the Gibbs distribution $\mu$.
We maintain a set $\mathcal{R}\mathsf{Sub}seteq V$ of vertices that currently ``incorrect'', initially set as $\mathcal{R}=V$.
At each step, a random vertex $u$ is picked from $\mathcal{R}$, and we try to remove $u$ from $\mathcal{R}$ while maintaining the following invariant,
known as the \emph{conditional Gibbs property}:
\begin{align}
\begin{array}{l}
X_{\overline{\mathcal{R}}}\text{ always follows the law } \mu_{\overline{\mathcal{R}}}^{X_\mathcal{R}},\\
\text{which is the marginal distribution induced by $\mu$ on $\overline{\mathcal{R}}\triangleq V\setminus\mathcal{R}$ conditioned on $X_\mathcal{R}$}.
\end{array}\label{eq:invariant-conditional-gibbs}
\end{align}
This property ensures that the configuration on $\overline{\+R}$ follows the correct distribution conditioned on the configuration on $\+R$.
In particular, when $\mathcal{R}=\emptyset$, $\overline{\+R}=V$, $\mu_{\overline{\mathcal{R}}}^{X_\mathcal{R}}=\mu$,
and the sample $\boldsymbol{X}$ follows precisely the distribution $\mu$.
This is the goal of our algorithm: to reduce $\+R$ to the empty set.
Our first attempt is simply to update the spin $X_u$ according to its marginal distribution $\mu_u^{{X_{\Gamma(u)}}}$
as in the Gibbs sampler and then remove $u$ from $\mathcal{R}$.
This gives the transition $(\boldsymbol{X},\mathcal{R})\to(\boldsymbol{X}',\mathcal{R}')$,
where $\mathcal{R}'=\mathcal{R}\setminus\{u\}$, and $\boldsymbol{X}=\boldsymbol{X}'$ except at vertex $u$, where $X'_u\sim\mu_u^{{X_{\Gamma(u)}}}$.
Ideally, if we had $X_{\overline{\mathcal{R}}}\sim\mu_{\overline{\mathcal{R}}}^{X_{\mathcal{R}'}}$, since $\mathcal{R}'=\mathcal{R}\setminus\{u\}$,
after the spin of $u$ being updated as $X'_u\sim\mu_u^{{X_{\Gamma(u)}}}$, the partial configuration $X_{\overline{\mathcal{R}}}$ would be extended to a {$X'_{\overline{\mathcal{R}'}}\sim \mu_{\overline{\mathcal{R}'}}^{X_{\mathcal{R}'}}\equiv\mu_{\overline{\mathcal{R}'}}^{X'_{\mathcal{R}'}}$} as $X_{\mathcal{R}'}=X'_{\mathcal{R}'}$, giving us the invariant~\eqref{eq:invariant-conditional-gibbs} on the new pair $(\boldsymbol{X}',\mathcal{R}')$.
However, the invariant~\eqref{eq:invariant-conditional-gibbs} on the original $(\boldsymbol{X},\mathcal{R})$ only guarantees $X_{\overline{\mathcal{R}}}\sim\mu_{\overline{\mathcal{R}}}^{X_{\mathcal{R}}}$ rather than $X_{\overline{\mathcal{R}}}\sim\mu_{\overline{\mathcal{R}}}^{X_{\mathcal{R}'}}$.
To remedy this, we construct a filter $\mathcal{F}=\mathcal{F}(u,\boldsymbol{X})$ that corrects $\mu_{\overline{\mathcal{R}}}^{X_{\mathcal{R}}}$ to $\mu_{\overline{\mathcal{R}}}^{X_{\mathcal{R}'}}$.
We call $\+F$ the \emph{Bayes filter}.
Specifically, $\mathcal{F}$
is determined by a biased coin depending on only part of $\boldsymbol{X}$ so that
\begin{align}
\Pr[\text{\,$\mathcal{F}$ succeeds\,}]
&\propto
\frac{\mu_{\overline{\mathcal{R}}}^{X_{\mathcal{R}'}}(X_{\overline{\mathcal{R}}})}{\mu_{\overline{\mathcal{R}}}^{X_{\mathcal{R}}}(X_{\overline{\mathcal{R}}})}
=
\frac{\mu_u^{X_{\mathcal{R}'}}(X_u)}{\mu_{u}^{X_{\Gamma(u)}}(X_u)},\label{eq:filter-property-1}
\end{align}
where $\propto$ is taken over all ${X_{\overline{\mathcal{R}}}}$ and
the equality is due to Bayes' theorem:
\begin{align*}
\mu_{\overline{\mathcal{R}}}^{X_{\mathcal{R}}}(X_{\overline{\mathcal{R}}})
&=
\mu_{\overline{\mathcal{R}}}^{X_{\mathcal{R}'}\land X_u}(X_{\overline{\mathcal{R}}})
=
{\mu_{u}^{X_{\overline{\mathcal{R}}}\land X_{\mathcal{R}'}}(X_u)\cdot\mu_{\overline{\mathcal{R}}}^{X_{\mathcal{R}'}}(X_{\overline{\mathcal{R}}})}/{\mu_u^{X_{\mathcal{R}'}}(X_u)},
\end{align*}
together with the \emph{conditional independence property} (formally, \Cref{property-cond-ind}) of Gibbs distribution which guarantees that
$\mu_{u}^{X_{\overline{\mathcal{R}}}\land X_{\mathcal{R}'}}(X_u)=\mu_{u}^{X_{\Gamma(u)}}(X_u)$.
We observe that despite that the exact value of the marginal probability $\mu_{u}^{{X_{\mathcal{R}'}}}(X_u)$ in~\eqref{eq:filter-property-1} is hard to compute, it does not depend on $X_{\overline{\mathcal{R}}}$.
Therefore, $\mu_{u}^{X_{\mathcal{R}'}}(X_u)$ can be treated as a constant and \eqref{eq:filter-property-1} holds as long as
$\Pr[\text{\,$\mathcal{F}$ succeeds\,}]\propto{1}/{\mu_{u}^{X_{\Gamma(u)}}(X_u)}$,
which is satisfied by the Bayes filter in Algorithm~\ref{alg:perfect-sampler-ss}.
Now by~\eqref{eq:filter-property-1}, conditioned on the success of the filter $\mathcal{F}$, the new $(\boldsymbol{X}',\mathcal{R}')$ satisfies the invariant~\eqref{eq:invariant-conditional-gibbs}.
Meanwhile, since the filter only reveals the neighborhood spins $X_{\Gamma(u)}$, upon failure of $\mathcal{F}$, the invariant \eqref{eq:invariant-conditional-gibbs} remains to hold as long as the revealed sites $\Gamma(u)$ are included into $\mathcal{R}$ and $\boldsymbol{X}$ is unchanged.
This sampler is valid for general Gibbs distributions,
since the only property we require is conditional independence.
However for efficiency purposes our general algorithm, \Cref{alg:perfect-sampler-gen},
uses \emph{block} updates rather than the single-site updates in \Cref{alg:perfect-sampler-ss}.
Block updates guarantee that the filter $\mathcal{F}$ always has a positive chance to succeed for Gibbs distributions with hard constraints,
and a suitable block length (chosen according to spatial mixing) is the key to efficiency of our analysis.
Moreover, to make sure that marginal distributions are well-defined,
we restrict our attention to \emph{permissive} systems, which contain all soft constraint systems as well as all hard constraint systems of interest.
The details are given in Section~\ref{section-results} and~\ref{sec:algorithm}.
The algorithm is efficient as long as the size of $\mathcal{R}$ shrinks in expectation in every step.
For the more general \Cref{alg:perfect-sampler-gen}, this is implied by correlation decay faster than neighborhood growth.
The details are in Section~\ref{section-running-time}.
Let us remark that it is surprising to us that the Gibbs sampler, studied for decades as the go-to approximate sampling algorithm,
can be turned into a perfect sampler by simply adding a filter that accesses only local information.
\mathsf{Sub}section{Related work and open problems}
The conditional Gibbs property have been used implicitly in previous works such as partial rejection sampling~\cite{GJL19,GJ18,GJ19a,GJ19b}, dynamic sampling~\cite{FVY19}, and randomness recycler~\cite{fill2000randomness}. Furthermore, the invariant of the conditional Gibbs property ensures that the sampling algorithm is correct even when the input spin system is dynamically changing over time~\cite{FVY19}.
Before our work, all previous resampling algorithms~\cite{fill2000randomness,GJL19,GJ18,GJ19a,GJ19b,FVY19} fall into the paradigm of rejection sampling: a new sample is generated, usually from modifying the old sample, and (part of) the new sample is rejected independently with some probability determined by the new sample. In our algorithm, the filtration is executed \emph{before} the generation of the new sample, with a bias independent of the new sample.
While our algorithm works for spin systems in general,
it still requires a rather strong notion of spatial mixing to be efficient.
Weaker forms of spatial mixing are known to imply rapid mixing of the Gibbs sampler for some particular systems. See for example \cite{MS13}.
If by ``efficient'' we allow algorithms whose running times are high degree polynomials and may depend on the degree of the graph,
then indeed even weak spatial mixing corresponds to the optimal threshold for efficient samplability in anti-ferromagnetic 2-spin systems \cite{Wei06,LLY13,SST14,SS14,GSV16}.
It remains to be an interesting open problem whether spatial mixing implies the existence of efficient samplers in general,
and whether these samplers can be perfect.
Our algorithm handles the failure of the filter in a very pessimistic way --- all revealed variables are considered ``incorrect'' and are added to $\mathcal{R}$.
Perhaps better algorithms can be obtained by handling the failure case more efficiently.
\section{Our results}
\label{section-results}
\mathsf{Sub}section{Model and definitions}
Let $G=(V, E)$ be an undirected graph, and $[q]=\{1,2,\ldots,q\}$ a finite domain of $q\ge2$ spins.
An instance of $q$-state spin system is specified by a tuple $\mathcal{I}=(G, [q], \boldsymbol{b}, \boldsymbol{A})$, where $\boldsymbol{b} = (b_v)_{v \in V}$ assigns every vertex $v\in V$ a vector $b_v \in \mathbb{R}_{\geq 0}^q$ and
$\boldsymbol{A} = (A_e)_{e \in E}$ assigns every edge $e\in E$ a symmetric matrix $A_e \in \mathbb{R}_{\geq 0}^{q\times q}$.
The \emph{Gibbs distribution} $\mu_{\mathcal{I}}$ over $[q]^V$ is defined as
\begin{align}
\label{eq-def-Gibbs}
\forall \sigma \in [q]^V: \quad \mu_{\mathcal{I}}(\sigma) \triangleq \frac{w_{\mathcal{I}}(\sigma)}{Z_{\mathcal{I}}} = \frac{1}{Z_{\mathcal{I}}}\prod_{v \in V}b_v(\sigma_v)\prod_{e = \{u,v\}\in E} A_e(\sigma_u, \sigma_v),
\end{align}
where $w_{\mathcal{I}}(\sigma)$ is the \emph{weight} defined in~\eqref{eqn:spin-system} and $Z_{\mathcal{I}} \triangleq \sum_{\sigma \in [q]^V} w_{\mathcal{I}}(\sigma)$ is the \emph{partition function}.
We restrict our attention to the so-called ``{permissive}'' spin systems, where the marginal distributions are always well-defined.
Let $\mathcal{I}=(G, [q], \boldsymbol{b}, \boldsymbol{A})$ be an instance of spin system.
A configuration on $V$ is called \emph{feasible} if its weight is positive, and a partial configuration is \emph{feasible} if it can be extended to a feasible configuration.
For any (possibly empty) subset $\Lambda \mathsf{Sub}seteq V$ and any (not necessarily feasible) partial configuration $\sigma \in [q]^\Lambda$, we use $w_{\mathcal{I}}^{\sigma}\left(\tau\right)$ to denote the weight of $\tau\in[q]^{V \setminus \Lambda}$ conditional on $\sigma$:
\begin{align}
\label{eq-def-proper}
w_{\mathcal{I}}^{\sigma}\left(\tau\right)
= \prod_{v \in V \setminus \Lambda}b_v(\tau_v)
\prod_{\mathsf{Sub}stack{e=\{u,v\} \in E \\ u, v \in V \setminus \Lambda}}A_e(\tau_u, \tau_v)
\prod_{\mathsf{Sub}stack{e=\{u,v\} \in E \\ u \in \Lambda, v\in V \setminus \Lambda}}A_e(\sigma_u, \tau_v).
\end{align}
Define the partition function $Z_{\mathcal{I}}^{\sigma}$ conditional on $\sigma$ as
$Z_{\mathcal{I}}^{\sigma} \triangleq \sum_{\tau \in [q]^{V \setminus \Lambda}}w_{\mathcal{I}}^{\sigma}\left( \tau\right)$.
\begin{definition}[permissive]\label{definition-locally-admissible}
A spin system $\mathcal{I}=(G, [q], \boldsymbol{b}, \boldsymbol{A})$, where $G=(V,E)$, is called \emph{permissive} if $Z_{\mathcal{I}}^{\sigma} > 0$ for any partial configuration $\sigma \in [q]^{\Lambda}$ specified on any subset $\Lambda \mathsf{Sub}seteq V$.
\end{definition}
Permissive systems are very common,
including, for examples,
uniform proper $q$-coloring when $q \geq \mathcal{D}elta + 1$, where $\mathcal{D}elta$ is the maximum degree,
and spin systems with soft constraints, e.g.~the Ising model, or with a ``permissive'' state that is compatible with all other states, e.g.~the hardcore model.
For permissive systems, a feasible configuration is always easy to construct by greedy algorithm.
More importantly, with permissiveness, marginal probabilities are always well defined, which is crucial for Gibbs sampler and spatial mixing property.
Formally, we use $\mu^\sigma_{\mathcal{I}}$ to denote the conditional distribution over $[q]^{V \setminus \Lambda}$ given $\sigma \in [q]^{\Lambda}$, that is,
\begin{align}
\label{eq-def-conditional-Gibbs}
\forall \tau \in [q]^{V \setminus \Lambda},\quad \mu^\sigma_{\mathcal{I}}(\tau) \triangleq \frac{w_{\mathcal{I}}^{\sigma}\left( \tau \right)}{Z_{\mathcal{I}}^{\sigma}}.
\end{align}
And for any $v \in V \setminus \Lambda$, we use $\mu_{v,\mathcal{I}}^\sigma$ to denote the marginal distribution at $v$ projected from $\mu^\sigma_{\mathcal{I}}$.
For any $u,v \in V$, we use $\mathrm{dist}_G(u,v)$ to denote the shortest-path distance between $v$ and $u$ in $G$.
\begin{definition}[strong spatial mixing~\cite{weitz2006counting,weitz2004mixing}]
\label{definition-standard-SSM}
Let $\delta:\mathbb{N}\to\mathbb{R}^+$.
A class $\mathfrak{I}$ of permissive spin systems is said to exhibit \emph{strong spatial mixing} with rate $\delta(\cdot)$ if for every instance $\mathcal{I}=(G, [q], \boldsymbol{b}, \boldsymbol{A})\in\mathfrak{I}$, where $G=(V,E)$, for every $v \in V$, $\Lambda \mathsf{Sub}seteq V$, and any two partial configurations $\sigma, \tau \in [q]^\Lambda$,
\begin{align}
\label{eq:condition-standard-SSM}
\mathcal{D}TV{\mu_{v, \mathcal{I}}^\sigma}{\mu_{v, \mathcal{I}}^\tau} \le \delta(\ell),
\end{align}
where $\ell = \min\{\mathrm{dist}_G(v, u) \mid u \in \Lambda,\ \sigma_u\neq\tau_u\}$, and $\mathcal{D}TV{\mu_{v, \mathcal{I}}^\sigma}{\mu_{v, \mathcal{I}}^\tau}\triangleq \frac{1}{2}\sum_{a \in [q]}\left\vert\mu_{v, \mathcal{I}}^\sigma(a)-\mu_{v, \mathcal{I}}^\tau(a) \right\vert$
denotes the total variation distance between $\mu_{v, \mathcal{I}}^\sigma$ and $\mu_{v, \mathcal{I}}^\tau$.
In particular, $\mathfrak{I}$ exhibits \emph{strong spatial mixing with exponential decay} if \eqref{eq:condition-standard-SSM} is satisfied for $\delta(\ell)= \alpha\exp(-\beta\ell)$ for some constants $\alpha,\beta>0$.
\end{definition}
Our first result holds for spin systems on graphs with bounded neighborhood growth.
\begin{definition}[sub-exponential neighborhood growth]\label{definition-sub-exp-graph}
A class $\mathfrak{G}$ of graphs is said to have \emph{sub-exponential neighborhood growth} if there is a function $s:\mathbb{N}\to\mathbb{N}$ such that $s(\ell)=\exp(o(\ell))$ and for every graph $G=(V,E)\in \mathfrak{G}$,
\begin{align*}
\forall v \in V, \forall \ell \geq 0, \quad |S_\ell(v)| \leq s(\ell),
\end{align*}
where $S_{\ell}(v)\triangleq \{u \in V \mid \mathrm{dist}_G(v, u) = \ell\}$ denotes the {sphere} of radius $\ell$ centered at $v$ in $G$.
\end{definition}
Note that graphs with sub-exponential neighborhood growth necessarily have bounded maximum degree because we can set $\ell=1$ and get $s(1)=O(1)$.
\mathsf{Sub}section{Main results}
Our first result shows that for spin systems on graphs with sub-exponential neighborhood growth, strong spatial mixing implies the existence of linear-time perfect sampler.
\begin{theorem}[main theorem: bounded-growth graphs]
\label{theorem-sub-graph}
Let $q>1$ be a finite integer and $\mathfrak{I}$ a class of permissive $q$-state spin systems on graphs with sub-exponential neighborhood growth.
If $\mathfrak{I}$ exhibits strong spatial mixing with exponential decay, then there exists an algorithm such that given any instance $\mathcal{I}=(G,[q],\boldsymbol{b},\boldsymbol{A})\in \mathfrak{I}$, the algorithm outputs a perfect sample from $\mu_{\mathcal{I}}$ within $O\left( n \right)$ time in expectation, where $n$ is the number of vertices in $G$.
\end{theorem}
\noindent
The factor in $O(\cdot)$ is the cost for a block update in (block) Gibbs sampler with $\ell_0$-radius blocks, where $\ell_0=O(1)$ is determined by both rates of correlation decay and neighborhood growth (by~\eqref{eq-set-ell}).
It is already known that for spin systems on sub-exponential neighborhood growth graphs, the strong spatial mixing with exponential decay implies $O(n \log n)$ mixing time for block Gibbs sampler \cite{DSVW04}, which only generates approximate samples. We give a perfect sampler with $O(n)$ expected running time under the same condition.
The notion of sub-exponential neighborhood growth is related to, but should be distinguished from neighborhood-amenability (see e.g.~\cite{goldberg2005strong}),
which says that in an infinite graph, for any constant real $c > 0$, there is an $\ell$ such that $\frac{|S_{\ell+1}(v)|}{|B_{\ell}(v)|} \leq c$ holds everywhere.
Our main result on general graphs assumes the following strong spatial mixing condition.
\begin{condition}
\label{condition-lower-bound}
Let $\mathcal{I}=(G, [q], \boldsymbol{b}, \boldsymbol{A})$ be a permissive spin system where $G=(V,E)$.
There is an integer $\ell=\ell(q)\ge 2$ such that the following holds:
for every $v \in V$, $\Lambda \mathsf{Sub}seteq V$,
for any two partial configurations $\sigma, \tau \in [q]^\Lambda$ satisfying $\min\left\{\mathrm{dist}_G(v, u) \mid u \in \Lambda,\ \sigma(u)\neq\tau(u)\right\} = \ell$,
\begin{align}
\label{eq:condition-stronger-SSM-general}
\mathcal{D}TV{\mu_{v, \mathcal{I}}^\sigma}{\mu_{v, \mathcal{I}}^\tau}
\leq \frac{\gamma}{ 5|S_{\ell}(v)|},
\end{align}
where
$S_{\ell}(v)$ denotes the sphere of radius $\ell$ centered at $v$ in $G$, and
\begin{align}
\label{eq:condition-stronger-lower}
\gamma=\gamma(v,\Lambda)
\triangleq
\min\left\{\mu_{v,\mathcal{I}}^{\rho}(a)\mid \rho \in [q]^{\Lambda},
a\in[q]\text{ that }\mu_{v,\mathcal{I}}^{\rho}(a)>0\right\}
\end{align}
denotes the lower bound of positive marginal probabilities at $v$.
\end{condition}
The above condition basically says that the spin systems exhibit strong spatial mixing with a decay rate faster than that of neighborhood growth, given that the marginal probabilities are appropriately lower bounded (which holds with $\gamma=\Omega(1)$ when entries of $\boldsymbol{A}$ and $\boldsymbol{b}$ are of finite precision and the maximum degree $\mathcal{D}elta$ is finitely bounded).
Our result on general graphs is stated as the following theorem.
\begin{theorem}[main theorem: general graphs]
\label{theorem-general}
Let $\mathfrak{I}$ be a class of permissive spin systems satisfying Condition~\ref{condition-lower-bound}.
There exists an algorithm which given any instance $\mathcal{I}=(G,[q],\boldsymbol{b},\boldsymbol{A})\in\mathfrak{I}$, outputs a perfect sample from $\mu_{\mathcal{I}}$ within $ n\cdot q^{O\left(\mathcal{D}elta^\ell \right)} $ time in expectation,
where $n$ is the number of vertices in $G$, $\mathcal{D}elta$ is the maximum degree of $G$, and $\ell=\ell(q)$ is determined by Condition~\ref{condition-lower-bound}.
\end{theorem}
The $q^{O(\mathcal{D}elta^\ell)}$ factor in the time cost is contributed by the block Gibbs sampler update on $\ell$-radius blocks.
This extra cost could remain polynomial if $q=\omega(1)$, but the upper bound on $q$ for that will vary from problem to problem.
See e.g.~the discussion after \Cref{corollary-main-coloring}.
The conditions of both Theorem~\ref{theorem-sub-graph}
and Theorem~\ref{theorem-general}
are special cases of a more technical condition (\Cref{condition-SSM-ratio}),
formally stated in Section~\ref{section-running-time}.
\mathsf{Sub}section{Applications on specific systems}
Our results can be applied on various spin systems.
We consider two important examples:
\begin{itemize}
\item{\bf Uniform list coloring:}
A list coloring instance is specified by $\mathcal{I}=(G, [q],\mathcal{L})$, where $\mathcal{L} \triangleq \{L_v\mathsf{Sub}seteq [q] \mid v\in V \}$ assigns each vertex $v\in V$ a list of colors $L_v\mathsf{Sub}seteq [q]$.
A proper list coloring of instance $\mathcal{I}$ is a $\sigma \in [q]^V$ that $\sigma_v \in L_v$ for all $v \in V$ and $\sigma_u \neq \sigma_v$ for all $\{u,v\} \in E$.
Let $\mu_{\mathcal{I}}$ denote the uniform distribution over all proper list colorings of $\mathcal{I}$.
\item{\bf The monomer-dimer model:}
The monomer-dimer model defines a distribution over graph matchings.
An instance is specified by $\mathcal{I} = (G, \lambda)$, where $G = (V, E)$ is a graph and $\lambda> 0$.
Each matching $M\mathsf{Sub}seteq E$ in $G$ is assigned a weight $w_{\mathcal{I}}(M) = \lambda^{|M|}$.
Let $\mu_{\mathcal{I}}$ be the distribution over all matchings in $G$ such that $\mu_{\mathcal{I}}(M) \propto w_{\mathcal{I}}(M)$.
\end{itemize}
We use $\deg_G(v)$ to denote the degree of $v$ in $G$ and $\mathcal{D}elta=\max_{v\in V}\deg_G(v)$ the maximum degree.
First, for the list coloring problem, we define the following two conditions for instance $\mathcal{I}=(G,[q],\+L)$.
Let $\deg_G(v)$ denote the degree of $v \in V$ in graph $G$, and $\mathcal{D}elta\triangleq\max_{v\in V}\deg_G(v)$ the maximum degree.
\begin{condition}
\label{condition-subexp}
For every $v \in V$,
$|L(v)| \geq \alpha \deg_G(v) + \beta$,
where either one of the followings holds:
\begin{itemize}
\item $\alpha = 2$ and $\beta = 0$;
\item $G$ is triangle-free, $\alpha > \alpha^*$ where $\alpha^* = 1.763\cdots$ is the positive root of $x^x = \mathrm{e}$,
and $\beta \geq \frac{\sqrt{2}}{\sqrt{2}-1}$ satisfies $(1-1/\beta) \alpha \mathrm{e}^{\frac{1}{\alpha}(1-1/\beta)} > 1$.
\end{itemize}
\end{condition}
\begin{condition}
\label{condition-list-coloring}
For every $v \in V$,
$|L(v)| \geq \mathcal{D}elta^2 - \mathcal{D}elta + 2$.
\end{condition}
Our perfect sampler for list coloring runs in linear time in either above condition.
\begin{theorem}
\label{corollary-main-coloring}
Let $\mathfrak{L}$ be a class of list coloring instances with at most $q$ colors for a finite $q>0$.
If either of the two followings holds for all instances $\mathcal{I}=(G,[q],\mathcal{L})\in\mathfrak{L}$:
\begin{itemize}
\item \Cref{condition-subexp}
and $G$ has sub-exponential neighborhood growth; or
\item \Cref{condition-list-coloring},
\end{itemize}
then there exists a perfect sampler for $\mu_{\mathcal{I}}$ that runs in expected $O\left( n \right)$ time, where $n$ is the number of vertices.
\end{theorem}
\noindent
The constant factor in $O(\cdot)$, can be determined in the same way as in Theorem~\ref{theorem-sub-graph} in the case of \Cref{condition-subexp},
but is much higher in the case of \Cref{condition-list-coloring} due to the arbitrary neighborhood growth.
Nevertheless, even in this costly case, $\ell=O(q^2\log q)$ and the overall overhead can by upper bounded by a rough estimate $\exp(\exp(\mathrm{poly}(q)))$.
Sampling proper $q$-colorings (where the lists are identical for all vertices) has been extensively studied, especially using Markov chains.
Approximate sampling
received considerable attention \cite{jerrum1995very,dyer2003randomly,hayes2003non, molloy2004glauber,dyer2013randomly}.
The current best result~\cite{vigoda2000improved,chen2019improved} is the $O(n\log n)$ mixing time of the flip chain if $q \geq (\frac{11}{6}-\epsilon_0)\mathcal{D}elta$ for some constant $\epsilon_0 > 0$.
For perfect sampling $q$-colorings,
Huber introduced a bounding chain \cite{huber2004perfect} based on CFTP~\cite{wilson1996generating},
which terminates within $O(n\log n)$ steps in expectation if $q\ge\mathcal{D}elta^2+2\mathcal{D}elta$.
In a very recent breakthrough, Bhandari and Chakraborty~\cite{bhandari2019perfect} introduced a novel bounding chain that has expected running time $O(n \log^2n)$ in a substantially broader regime $q > 3\mathcal{D}elta$.
Another way to obtain perfect samplers is to use standard reductions between counting and sampling~\cite{jerrum1986random}.
Using this technique, any FPTAS for the number of colorings can be turned into a polynomial-time {perfect} sampler.
(It is important that the approximate counting algorithm is deterministic,
or at least with errors that can be detected.)
Currently, the FPTAS with the best regime for general graphs is due to Liu et al.\ \cite{liu2019deterministic}, which requires $q\ge 2\mathcal{D}elta$,
and it runs in time $n^{\mathsf{EXP}(\mathcal{D}elta)}$.
Comparing to the results above, our algorithm draws perfect samples and achieves the $O(n)$ expected running time.
In case of sub-exponential growth graphs such as $\mathbb{Z}^d$, it improves the result of \cite{bhandari2019perfect} by requiring only $q > \alpha\mathcal{D}elta+O(1)$, where $\alpha>\alpha^*=1.763\cdots$.
Next we consider the monomer-dimer model.
It was proved in~\cite{bayati2007simple,song2019counting} that instances $\mathcal{I}=(G,\lambda)$ on graphs $G$ with maximum degree $\mathcal{D}elta$ exhibits strong spatial mixing with rate $(1-\Omega(1/\sqrt{1+\lambda\mathcal{D}elta}))^{-\ell}$.
Applying our algorithm yields the following perfect sampling result.
\begin{theorem}
\label{corollary-main-matching}
Let $\mathfrak{M}$ be a class of monomer-dimer instances $\mathcal{I}=(G,\lambda)$ on graphs $G$ with sub-exponential neighborhood growth and constant $\lambda$.
There exists an algorithm which given any instance $\mathcal{I}=(G,\lambda)\in\mathfrak{M}$, outputs a perfect sample from $\mu_{\mathcal{I}}$ within expected $O(n)$ time.
\end{theorem}
Previously, Markov chains were the most successful techniques for sampling weighted matchings.
The Jerrum-Sinclair chain~\cite{jerrum1989approximating} on a monomer-dimer model $\mathcal{I}=(G,\lambda)$,
generates {approximate samples} from $\mu_{\mathcal{I}}$ within $\widetilde{O}(n^2m)$ steps, where $m = |E|$.
This chain also mixes in $O(n\log^2 n)$ time for finite subgraphs of the 2D lattice $\mathbb{Z}^2$ \cite{vdBB00}.
It is difficult to convert the Jerrum-Sinclair chain to perfect samplers.
Before our work, the only perfect sampler for the monomer-dimer model we are aware of
is the one obtained via standard reductions from sampling to counting~\cite{jerrum1986random} (similar to the case of colorings),
together with deterministic approximate counting algorithms~\cite{bayati2007simple}.
This is a perfect sampler with running time $n^{\mathrm{Poly}(\mathcal{D}elta,\lambda)}$.
Our algorithm is the first linear-time perfect sampler for the monomer-dimer model on graphs with sub-exponential neighborhood growth,
such as finite subgraphs of lattices $\mathbb{Z}^d$ for any constant $d$ and constant weight $\lambda$.
\section{The Algorithm}
\label{sec:algorithm}
We now describe our general perfect Gibbs sampler.
It generalizes the single-site version (Algorithm~\ref{alg:perfect-sampler-ss}) by allowing block updates.
This generalization allows us to bypass some pathological situations,
and to greatly improve the efficiency of the algorithm.
The pseudocode is given in Algorithm~\ref{alg:perfect-sampler-gen}.
Let $\mathcal{I}=(G, [q], \boldsymbol{b}, \boldsymbol{A})$ be a permissive spin system instance and $G=(V,E)$.
For any $u\in V$ and integer $\ell\ge 0$, recall that $B_{\ell}(u) \triangleq \left\{v \in V \mid \mathrm{dist}_{G}(u,v) \le \ell\right\}$ denotes the $\ell$-ball centered at $u$ in $G$.
And for any $B\mathsf{Sub}seteq V$, we use $\partial B\triangleq\{v\in V\setminus B\mid \{u,v\}\in E\}$ to denote the vertex boundary of $B$ in $G$.
{
\let\oldnl\nl
\newcommand{\nonl}{\renewcommand{\nl}{\let\nl\oldnl}}
\begin{algorithm}[htbp]
\SetKwInOut{Input}{Input}
\SetKwInOut{Input}{Parameter}
\SetKwComment{Comment}{\quad$\triangleright$\ }{}
\SetKwIF{withprob}{}{}{with probability}{do}{}{}{}
\mathcal{I}nput{an integer $\ell \geq 0$;}
Start from an arbitrary feasible configuration $\boldsymbol{X} \in [q]^V$, i.e.~$w_{\mathcal{I}}(\boldsymbol{X})>0$\;
$\mathcal{R} \gets V$\;
\While{$\mathcal{R} \neq \emptyset$}{
pick a $u\in\mathcal{R}$ uniformly at random and let $B\gets (B_{\ell}(u)\setminus \mathcal{R})\cup\{u\}$\label{line-pick}\;
let $\mu_{\min}$ be the minimum value of $\mu_{u}^{\sigma}(X_u)$ over all $\sigma\in[q]^{\partial B}$ that $\sigma_{\mathcal{R}\cap\partial B}=X_{\mathcal{R}\cap\partial B}$\label{line-F}\;
\withprob{$\frac{\mu_{\min}}{\mu_{u}^{X_{\partial B}}(X_u)}$}{
\nonl
\hspace{204.5pt}\texttt{$\triangleright$ Bayes filter}
update $\boldsymbol{X}$ by redrawing $X_B\sim\mu_{B}^{X_{\partial B}}$;\label{line-sample}
\hspace{44.3pt}\texttt{$\triangleright$ block Gibbs sampler update}\\
$\mathcal{R} \gets\mathcal{R} \setminus \{u\}$\;
}
\Else{
$\mathcal{R} \gets\mathcal{R} \cup \partial B$\;
}
}
\mathcal{R}eturn{$\boldsymbol{X}$}\;
\caption{Perfect Gibbs sampler (general version)}\label{alg:perfect-sampler-gen}
\end{algorithm}
}
The algorithm is parameterized by an integer $\ell\ge 1$, which is set rigorously in~\Cref{section-running-time}.
The initial $\boldsymbol{X} \in [q]^V$ is an arbitrary feasible configuration, which is easy to construct by greedy algorithm since $\mathcal{I}$ is permissive (\Cref{definition-locally-admissible}).
After each iteration of the \textbf{while} loop,
either $\boldsymbol{X}$ is unchanged or $X_B$ is redrawn from $\mu_B^{X_{\partial B}}$, which is the marginal distribution of $\mu=\mu_{\mathcal{I}}$, conditioned on the current $\boldsymbol{X}_B$.
Thus, we have the following observation.
\begin{observation}
\label{observation-positive}
In \Cref{alg:perfect-sampler-gen}, the configuration $\boldsymbol{X}$ is always feasible,~i.e. $w_{\mathcal{I}}(\boldsymbol{X})>0$.
\end{observation}
\noindent
The observation implies that $\mu_{u}^{X_{\partial B}}(X_u) > 0$ all along.
The {Bayes filter} in Line~\ref{line-F} is always well-defined.
If the filer succeeds, $X_B$ is resampled according to the correct marginal distribution $\mu_B^{X_{\partial B}}$ and $u$ is removed from $\mathcal{R}$ (that is, $u$ has been successfully ``fixed'');
otherwise, $\boldsymbol{X}$ is unchanged and $\mathcal{R}$ is enlarged by $\partial B$ (because variables in $\partial B$ are revealed and no longer random).
The key to the correctness of \Cref{alg:perfect-sampler-gen} is the conditional Gibbs property~\eqref{eq:invariant-conditional-gibbs}: the law of $\boldsymbol{X}$ over $\overline{R}\triangleq V \setminus \mathcal{R}$ is always the conditional distribution $\mu^{X_\mathcal{R}}=\mu^{X_\mathcal{R}}_{\mathcal{I}}$.
By similar argument as in Section~\ref{sec:alg-overview}, just redrawing $X_B$ from $\mu_{B}^{X_{\partial B}}$ will introduce a bias $\propto\mu_{u}^{X_{\partial B}}(X_u)$ to the sample $\boldsymbol{X}$, relative to its target distribution $\mu^{X_{\mathcal{R}\setminus\{u\}}}=\mu_{\mathcal{I}}^{X_{\mathcal{R}\setminus\{u\}}}$.
In the algorithm, we use the Bayes filter that succeeds with probability $\propto1/\mu_{u}^{X_{\partial B}}(X_u)$ to cancel this bias,
with the risk of enlarging $\mathcal{R}$ by $\partial B$ upon failure.
Balancing the success probability and the size of $\partial B$ is the key to getting an efficient algorithm,
and this depends on choosing an appropriate $\ell$ according to the spatial mixing rate.
The correctness and efficiency of the algorithm are then analyzed in next two sections.
\section{Correctness of the perfect sampling}
\label{section-correctness}
In this section, we prove the correctness of \Cref{alg:perfect-sampler-gen},
stated by the following theorem.
\begin{theorem}[correctness theorem]
\label{theorem-correctness}
Given any permissive spin system $\mathcal{I}=(G,[q],\boldsymbol{b},\boldsymbol{A})$,
\Cref{alg:perfect-sampler-gen} with any parameter $\ell\ge 1$ terminates with probability 1,
and outputs $\boldsymbol{X}$ that follows the law of $\mu_{\mathcal{I}}$.
\end{theorem}
The theorem is implied by two key properties of the Gibbs distribution $\mu_\mathcal{I}$.
\mathsf{Sub}section{Key properties of Gibbs distributions}
Note that the $\mu_{\min}$ in \Cref{alg:perfect-sampler-gen} is determined by the set $\+R$, the vertex $u \in \+R$, and the partial feasible configuration $X_{\+R}$. Formally, fixing the parameter $\ell \geq 0$ in \Cref{alg:perfect-sampler-gen},
\begin{align*}
\mu_{\min}(\+R, u, X_{\mathcal{R}}) \triangleq \min\left\{ \mu^{\sigma}_{u,\mathcal{I}}(X_u) \mid \sigma \in [q]^{\partial B} \text{ s.t. } \sigma_{\+R \cap \partial B } = X_{\+R \cap \partial B}, \text{ where } B \triangleq (B_{\ell}(u) \setminus \+R) \cup \{u\} \right\}.
\end{align*}
\begin{property}[positive lower bound of $\mu_{\min}$]
\label{property-lower-bound}
The lower bound $\gamma_{\mathcal{I}}$ of $\mu_{\min}$ is positive:
\begin{align}
\label{eq-lower-bound-suss}
\gamma_{\mathcal{I}} \triangleq \min\left\{ \mu_{\min}(\mathcal{R}, u, X_{\mathcal{R}}) \mid \mathcal{R} \mathsf{Sub}seteq V, u \in \mathcal{R}, X_{\mathcal{R}} \in [q]^{\mathcal{R}} \text{ s.t. } X_{\mathcal{R}} \text{ is feasible} \right\} > 0.
\end{align}
\end{property}
To state the next property, we introduce some notations:
For any $\Lambda \mathsf{Sub}seteq V$, $\sigma \in [q]^\Lambda$ and $S \mathsf{Sub}seteq V \setminus \Lambda$, we use $\mu^{\sigma}_{S,\mathcal{I}}(\cdot)$ to denote the marginal distribution on $S$ projected from $\mu^{\sigma}_{\mathcal{I}}$. For any disjoint sets $\Lambda,\Lambda' \mathsf{Sub}seteq V$, $\sigma \in [q]^{\Lambda}$ and $\sigma' \in [q]^{\Lambda'}$, we use $\sigma \uplus \sigma'$ to denote the configuration on $\Lambda \uplus \Lambda'$ that is consistent with $\sigma$ on $\Lambda$ and consistent with $\sigma'$ on $\Lambda'$.
\begin{property}[conditional independence]
\label{property-cond-ind}
Suppose $A,B,C\mathsf{Sub}set V$ are three disjoint non-empty subsets such that the removal of $C$ disconnects $A$ and $B$ in $G$.
For any $\sigma_A \in [q]^A, \sigma_B \in [q]^B$ and $\sigma_C \in [q]^C$,
\begin{align*}
\mu^{\sigma_A \uplus \sigma_C}_{B,\mathcal{I}}(\sigma_B) = \mu^{\sigma_C}_{B, \mathcal{I}}(\sigma_B).
\end{align*}
\end{property}
Theorem~\ref{theorem-correctness} is proved using only these two properties,
thus \Cref{alg:perfect-sampler-gen} is correct for general permissive Gibbs distributions satisfying these two properties.
In particular, we verify that all permissive spin systems satisfy these two properties.
First, the conditional independence (\Cref{property-cond-ind}) holds generally for Gibbs distributions~\cite{mezard2009information}.
Next, for the positive lower bound of $\mu_{\min}$ (\Cref{property-lower-bound}): for spin systems with soft constraints, clearly \Cref{property-lower-bound} holds for all $\ell \geq 0$;
and for general permissive spin systems $\mathcal{I}$, we need to verify that \Cref{property-lower-bound} holds if $\ell \geq 1$.
Fix a tuple $(\mathcal{R},u,X_{\mathcal{R}})$ in~\eqref{eq-lower-bound-suss}.
The following fact follows from the definition of set $B$.
\begin{fact}
\label{fact-0}
$\partial B \mathsf{Sub}seteq S_{\ell+1}(u) \cup \+R$, where $B = (B_{\ell}(u) \setminus \+R) \cup \{u\}$.
\end{fact}
\noindent
The fact implies $\partial B \setminus \+R \mathsf{Sub}seteq S_{\ell +1}(u)$.
Since $\ell \geq 1$, $u$ is not adjacent to any vertex in $\partial B \setminus \+R$.
Since $\mathcal{I}$ is permissive and $X_{\mathcal{R}}$ is feasible, $\mu_{u,\mathcal{I}}^{\sigma}(X_u) > 0$ for all $\sigma \in [q]^{ \partial B }$ such that $\sigma_{\+R \cap \partial B } = X_{\+R \cap \partial B}$.
This implies $\mu_{\min}(\mathcal{R},u,X_{\mathcal{R}})$ is positive, thus the \Cref{property-lower-bound} holds.
We then prove \Cref{theorem-correctness} assuming only \Cref{property-lower-bound} and \Cref{property-cond-ind}.
More specifically, termination of the algorithm is guaranteed by \Cref{property-lower-bound}, and correctness of the output upon termination is guaranteed by \Cref{property-cond-ind}.
\mathsf{Sub}section{Termination of the algorithm}
Denote by $T$ the number of iterations of the \textbf{while} loop in \Cref{alg:perfect-sampler-gen}.
To prove that the algorithm terminates with probability 1, we show that $T$ is stochastically dominated by a geometric distribution.
We use $\+F$ to denote the Bayes filter in \Cref{alg:perfect-sampler-gen}. Then,
\begin{align*}
\Pr[\mathcal{F}\text{ succeeds}] = \frac{\mu_{\min}(\+R, u, X_{\mathcal{R}}) }{\mu_{u,\mathcal{I}}^{X_{\partial B}}(X_u)} \geq \mu_{\min}(\+R, u, X_{\mathcal{R}}).
\end{align*}
If $\+F$ succeeds for $n = |V|$ consecutive iterations of the \textbf{while} loop, then the set $\mathcal{R}$ must become empty and the algorithm terminates.
By \Cref{property-lower-bound},
we have
\begin{align}
\label{eq-dominate}
\forall k \geq 0, \quad \Pr[ T \geq k n] \leq \gamma_{\mathcal{I}}^{kn}.
\end{align}
This implies $T$ is stochastically dominated by a geometric distribution.
Each iteration of the \textbf{while}{} loop terminates within finite number of steps.
Thus, the algorithm terminates with probability~1.
\mathsf{Sub}section{Correctness upon termination}
We show that upon termination, the output follows the correct distribution.
Let $(\boldsymbol{X}, \mathcal{R}) \in [q]^V \times 2^V$ be the random pair maintained by the algorithm.
The following condition is the ``loop invariant'' of the random pair $(\boldsymbol{X}, \mathcal{R})$.
\begin{condition}[conditional Gibbs property]
\label{condition-invariant}
For any $R \mathsf{Sub}seteq V$ and $\sigma \in [q]^R$,
conditioned on $\mathcal{R} = R$ and $X_R = \sigma$,
the random configuration $X_{V \setminus R}$ follows the law $\mu_{\mathcal{I}}^\sigma$.
\end{condition}
\Cref{condition-invariant} is satisfied initially by the initial pair $(\boldsymbol{X}, \mathcal{R}) = (\boldsymbol{X}, V)$.
Furthermore, consider the \textbf{while}{} loop that transforms
\begin{align*}
(\boldsymbol{X},\mathcal{R}) \rightarrow (\boldsymbol{X}', \mathcal{R}').
\end{align*}
Then next lemma shows that \Cref{condition-invariant} holds inductively assuming \Cref{property-cond-ind}.
\begin{lemma}
\label{lemma-detailed-invariant}
Suppose that $(\boldsymbol{X}, \mathcal{R}) \in [q]^V \times 2^V$ is a random pair such that $\boldsymbol{X}$ is feasible and the pair $(\boldsymbol{X}, \mathcal{R})$ satisfies Condition~\ref{condition-invariant}.
Then, the random pair $(\boldsymbol{X}',\mathcal{R}')$ satisfies Condition~\ref{condition-invariant}.
\end{lemma}
\noindent
By \Cref{observation-positive}, the random configuration $\boldsymbol{X} \in [q]^V$ maintained by algorithm is always feasible.
\Cref{lemma-detailed-invariant} guarantees that \Cref{condition-invariant} holds all along for the random pair $(\boldsymbol{X}, \mathcal{R})$ maintained by \Cref{alg:perfect-sampler-gen}.
In particular, when the algorithm terminates, $\mathcal{R} = \emptyset$ and the output $\boldsymbol{X}$ follows the correct distribution $\mu_{\mathcal{I}}$.
This proves Theorem~\ref{theorem-correctness}.
\begin{proof}[Proof of \Cref{lemma-detailed-invariant}]
It is sufficient to prove that for any $R \mathsf{Sub}seteq V$, any feasible partial configuration $\rho \in [q]^R$ and any vertex $u \in R$,
conditioned on $\mathcal{R} = R $, $X_R = \rho$, and the vertex picked in \Cref{line-pick} being $u$, the new random pair $(\boldsymbol{X}',\mathcal{R}')$ after one iteration of the \textbf{while}{} loop satisfies Condition~\ref{condition-invariant}.
Fix $R \mathsf{Sub}seteq V$ and a feasible partial configuration $\rho \in [q]^R$.
Let $\*u \in R$ denote the uniform random vertex picked in \Cref{line-pick}.
Fix a vertex $u \in R$.
Let $\mathcal{E}$ denote the event
\begin{align*}
\mathcal{E}: \quad X_R = \rho \land \mathcal{R} = R \land \*u = u.
\end{align*}
Since $(\boldsymbol{X}, \mathcal{R})$ satisfies \Cref{condition-invariant} and given the set $R$, $\*u$ is independent from $\boldsymbol{X}$, we have
\begin{align}
\label{eq-X-V-setminus-R}
\forall \tau \in [q]^{V \setminus R}:\quad
\Pr\left[ X_{V \setminus R} = \tau \mid \mathcal{E} \right] = \mu_{\mathcal{I}}^\rho(\tau).
\end{align}
Recall that $\+F$ is the Bayes filter.
Depending on whether $\+F$ succeeds or not, we have two cases.
The easier case is when $\+F$ fails.
Recall that the set $B$ is fixed by $R$ and $u$. In this case, $\mathcal{R}'= R \cup \partial B$ and $\boldsymbol{X}' = \boldsymbol{X}$.
Conditioned on $\+E$, we know that $X_u = \rho_u$ and $X_{\partial B \cap R} = \rho_{\partial B \cap R}$,
the filter $\+F$ depends only on the partial configuration $X_{\partial B \setminus R}$.
For any configuration $\sigma \in [q]^{\partial B \setminus R}$,
conditioned on $\mathcal{E}$ and $X_{\partial B \setminus R} = \sigma$,
the failure of $\mathcal{F}$ is independent from $X_{V \setminus (R \cup \partial B)}=X_{V\setminus \mathcal{R}'}$.
Thus, by~\eqref{eq-X-V-setminus-R}, conditioned on $\mathcal{E}$, $X_{\partial B \setminus R} = \sigma$ and the failure of $\+F$,
we have that $X'_{V \setminus \mathcal{R}'} = X_{V\setminus \mathcal{R}'} \sim \mu_{\mathcal{I}}^{\rho \uplus \sigma}$, i.e.~$(\boldsymbol{X}', \mathcal{R}')$ satisfies \Cref{condition-invariant}.
Now we analyze the main case that $\+F$ succeeds.
If this case does occur, we must have
\begin{align}
\label{eq-proof-assume-mu-min}
\mu_{\min}(R,u,\rho)\triangleq \min\{ \mu^{\sigma}_{u,\mathcal{I}}(\rho_u) \mid \sigma \in [q]^{\partial B} \text{ s.t. } \sigma_{R \cap \partial B } = \rho_{R \cap \partial B}\} > 0.
\end{align}
Define $R_u \triangleq R \setminus \{u\}$.
The fact that $\+F$ succeeds means $\mathcal{R}' = R \setminus \{u\} = R_u$ and $X'_{R_u}=X_{R_u} = \rho_{R_u}$.
Hence, we only need to show that
\begin{align}
\label{eq-X-V-setminus-Ru}
\forall \tau \in [q]^{V \setminus R_u}:\quad
\Pr\left[ X'_{V \setminus R_u} = \tau \mid \mathcal{E} \land \mathcal{F}\text{ succeeds} \right] = \mu_{\mathcal{I}}^{\rho(R_u)}(\tau).
\end{align}
Recall $ B = (B_{\ell}(u) \setminus R) \cup \{u\} = B_{\ell}(u) \setminus R_u$.
We define the following set:
\begin{align*}
H \triangleq V \setminus \left\{ R_u \cup B \right\}.
\end{align*}
Namely, $B$ is the set whose configuration is resampled, and $H$ is the set whose configuration is untouched, i.e.~$X'_H=X_H$.
Note that $B \uplus H \uplus R_u = V$. By the chain rule:
\begin{align}\label{eq:chain-rule-success}
&\Pr\left[X'_{V \setminus R_u}=\tau \land \mathcal{F}\text{ succeeds} \mid \mathcal{E}\right]
= \Pr\left[ X'_H = \tau_H\land X'_B = \tau_B\land \mathcal{F}\text{ succeeds} \mid \+E \right]\\
=\,& \Pr[X'_H = \tau_H \mid \mathcal{E}]\cdot \Pr[\mathcal{F}\text{ succeeds} \mid \mathcal{E}\land X'_H = \tau_H]\cdot
\Pr[X'_B = \tau_B \mid \mathcal{E}\land X'_H = \tau_H\land \mathcal{F}\text{ succeeds}].\notag
\end{align}
As $X'_H=X_H$, \eqref{eq-X-V-setminus-R} implies that
\begin{align*}
\Pr[X'_H = \tau_H \mid \mathcal{E}] = \mu_{H,\mathcal{I}}^\rho(\tau_H).
\end{align*}
By Line~\ref{line-sample} of Algorithm~\ref{alg:perfect-sampler-gen},
$X'_B$ is redrawn from the distribution
$\mu_{B,\mathcal{I}}^{X_{\partial B}}(\cdot)$.
By conditional independence property (\Cref{property-cond-ind}), we have $\mu_{B,\mathcal{I}}^{X_{\partial B}}(\cdot) = \mu_{\mathcal{I}}^{X_{V \setminus B}}(\cdot)$.
Note that $V \setminus B = R_u \uplus H$.
Conditioned on $\+E\land X'_H = \tau_H$, $X_{R_u} = \rho_{R_u}$ and $X_{H}=X'_H=\tau_H$, thus $\mu_{B, \mathcal{I}}^{X_{\partial B}}(\cdot) = \mu_{\mathcal{I}}^{\rho(R_u)\uplus \tau(H)}(\cdot)$.
Hence,
\begin{align}
\eqref{eq:chain-rule-success}
=\mu_{H,\mathcal{I}}^\rho(\tau_H)\cdot \mu_{\mathcal{I}}^{\rho(R_u)\uplus \tau(H)}(\tau_B)
\cdot \Pr[\mathcal{F}\text{ succeeds} \mid \mathcal{E}\land X'_H = \tau_H]. \label{eq-chain-rule}
\end{align}
To finish the proof, we need to calculate $\Pr[\mathcal{F}\text{ succeeds}\mid \mathcal{E}\land X'_H = \tau_H]$.
This is done by the following claim, whose proof is deferred to the end of the section.
\begin{claim}
\label{claim-correctness}
Assume~\eqref{eq-proof-assume-mu-min}. It holds that
\begin{align}
\label{eq-claim-1}
\mu_{H, \mathcal{I}}^{\rho}(\tau_H) > 0 \quad \Longleftrightarrow \quad \mu_{H, \mathcal{I}}^{\rho(R_u)}(\tau_H) > 0.
\end{align}
Furthermore, for $\tau_H$ such that $\Pr[X'_H = \tau_H \mid \mathcal{E}] = \mu_{H, \mathcal{I}}^{\rho}(\tau_H) > 0$,
\begin{align}
\label{eq-claim-2}
\Pr[\mathcal{F}\text{ succeeds}\mid \mathcal{E}\land X'_H = \tau_H] = C\cdot \frac{\mu_{H, \mathcal{I}}^{\rho(R_u)}(\tau_H)}{\mu_{H,\mathcal{I}}^\rho(\tau_H)},
\end{align}
where $C = C(R, u, \rho) > 0$ is a constant depending only on $R, u, \rho$ but not on $\tau$.
\end{claim}
Combining~\eqref{eq-chain-rule} and \Cref{claim-correctness}, we have
\begin{align}
\label{eq-final-prob}
\forall \tau \in [q]^{V \setminus R_u}, \quad \Pr\left[X'_{V \setminus R_u}=\tau \land \mathcal{F}\text{ succeeds}\mid \mathcal{E}\right] = C\cdot\mu^{\rho(R_u)}_{\mathcal{I}}(\tau).
\end{align}
This equation can be verified in two cases:
\begin{itemize}
\item If $\mu_{H,\mathcal{I}}^\rho(\tau_H) = 0$, then by~\eqref{eq-claim-1}, $\mu_{H, \mathcal{I}}^{\rho(R_u)}(\tau_H) = 0$, thus $\text{LHS} = \text{RHS} = 0$.
\item If $\mu_{H,\mathcal{I}}^\rho(\tau_H) > 0$, by~\eqref{eq-chain-rule}~and~\eqref{eq-claim-2}, we have $\text{LHS} = C \cdot \mu_{H, \mathcal{I}}^{\rho(R_u)}(\tau_H)\cdot \mu_{\mathcal{I}}^{\rho(R_u)\uplus \tau(H)}(\tau_B) = \text{RHS}$, where the last equation holds because $\tau \in [q]^{V \setminus R_u}$ and $V \setminus R_u= {H \uplus B}$.
\end{itemize}
Thus, the probability that $\mathcal{F}\text{ succeeds}$ is
\begin{align}
\label{eq-succeeds-prob}
\Pr\left[ \mathcal{F}\text{ succeeds}\mid \mathcal{E}\right] = \sum_{\sigma \in [q]^{V \setminus R_u }} \Pr\left[X'_{V \setminus R_u}=\sigma \land \mathcal{F}\text{ succeeds}\mid \mathcal{E}\right] = \sum_{\sigma \in [q]^{V \setminus R_u }}C\cdot\mu^{\rho(R_u)}_{\mathcal{I}}(\sigma) =C,
\end{align}
where the last equation holds because $\sum_{\sigma \in [q]^{V \setminus R_u }}\mu^{\rho(R_u)}_{\mathcal{I}}(\sigma) = 1$ and $C = C(R, u, \rho) > 0$ is a constant depending only on $R, u, \rho$.
Thus, for any $\tau \in [q]^{V \setminus R_u}$, combining~\eqref{eq-final-prob} and~\eqref{eq-succeeds-prob}, we have
\begin{align*}
\Pr\left[X'_{V \setminus R_u}=\tau \mid \mathcal{F}\text{ succeeds} \land \mathcal{E}\right]=\frac{\Pr\left[X'_{V \setminus R_u}=\tau \land \mathcal{F}\text{ succeeds}\mid \mathcal{E}\right]}{\Pr\left[ \mathcal{F}\text{ succeeds}\mid \mathcal{E}\right]}
=\frac{C \cdot \mu^{\rho(R_u)}_{\mathcal{I}}(\tau)}{C} = \mu^{\rho(R_u)}_{\mathcal{I}}(\tau),
\end{align*}
where the last equation holds due to $C = C(R,u,\rho) > 0$. This proves~\eqref{eq-X-V-setminus-Ru}.
\end{proof}
\begin{proof}[Proof of Claim~\ref{claim-correctness}]
We first introduce the following definitions.
Recall that $R_u \uplus B \uplus H = V$. We further partition $\partial B$ into two disjoint sets $\partial B \cap R_u$ and $\partial B \setminus R_u$. Define
\begin{equation}
\label{eq-def-part-B}
\begin{split}
S &\triangleq \partial B \setminus R_u = \partial B \cap H,\\
\Psi &\triangleq \partial B \cap R_u = \partial B \cap R.
\end{split}
\end{equation}
We now prove~\eqref{eq-claim-1}.
Since $\rho= \rho(R_u) \uplus \rho(u)$, by the Bayes law, we have the following relation between $\mu_{H, \mathcal{I}}^{\rho(R_u)}(\tau_{H})$ and $\mu_{H, \mathcal{I}}^{\rho}(\tau_{H})$:
\begin{align}
\label{eq-proof-bayes}
\mu_{H,\mathcal{I}}^\rho(\tau_{H}) = \mu_{H,\mathcal{I}}^{\rho(R_u)\uplus \rho(u)}(\tau_{H}) = \frac{\mu_{u,\mathcal{I}}^{\rho(R_u) \uplus \tau(H)}(\rho_u) }{ \mu^{\rho(R_u)}_{u,\mathcal{I}}(\rho_u)} \cdot \mu^{\rho(R_u)}_{H,\mathcal{I}}(\tau_{H}).
\end{align}
Note that $\rho \in [q]^R$ is a feasible configuration, thus $\mu^{\rho(R_u)}_{u,\mathcal{I}}(\rho_u) > 0$ and the above ratio is well-defined. Note that $u \in B$ and $R_u \uplus B \uplus H = V$. The set $\partial B$ separates $u$ from $(R_u \uplus H) \setminus \partial B$. Note that $\partial B = S \uplus \Psi$, where $S$ and $\Psi$ is defined in~\eqref{eq-def-part-B}. By the conditional independence property (\Cref{property-cond-ind}), we have
\begin{align}
\label{eq-proof-cond-ind}
\mu_{u,\mathcal{I}}^{\rho(R_u) \uplus \tau(H)}(\rho_u) = \mu_{u,\mathcal{I}}^{\rho(\Psi) \uplus \tau(S)}(\rho_u) \geq \mu_{\min}(R,u,\rho) > 0,
\end{align}
where the first inequality is because $\mu_{\min}(R,u,\rho)$ in~\eqref{eq-proof-assume-mu-min} can be rewritten as $\min_{\eta \in [q]^{ S }}\mu_{u,\mathcal{I}}^{\rho(\Psi) \uplus \eta}(\rho_u)$, and the second inequality is because $\mu_{\min}(R,u,\rho) > 0$ due to the lower bound in~\eqref{eq-proof-assume-mu-min}.
Next, we prove~\eqref{eq-claim-2}. Suppose $\mu_{H, \mathcal{I}}^{\rho}(\tau_H) > 0$.
Combining~\eqref{eq-proof-bayes} and~\eqref{eq-proof-cond-ind}, it remains to prove that
\begin{align}
\label{eq-proof-new-target}
\Pr[\mathcal{F}\text{ succeeds}\mid \mathcal{E}\land X'_H = \tau_H] = C\cdot \frac{\mu_{H, \mathcal{I}}^{\rho(R_u)}(\tau_H)}{\mu_{H,\mathcal{I}}^\rho(\tau_H)} = C\cdot \frac{\mu^{\rho(R_u)}_{u,\mathcal{I}}(\rho_u) }{\mu_{u,\mathcal{I}}^{\rho(\Psi) \uplus \tau(S)}(\rho_u)}.
\end{align}
Conditional on $\+E$, we have $X_{\Psi} = \rho_{\Psi}$ and $X_u = \rho_u$.
Recall that $X'_H=X_H$, $S \mathsf{Sub}seteq H$ and $S \uplus \Psi =\partial B$.
Conditional on $X'_H = \tau_H$, it holds that $X_S = \tau_S$.
By the definition of the filter $\mathcal{F}$ in Line~\ref{line-F} of \Cref{alg:perfect-sampler-gen}, we have that
\begin{align}
\label{eq-original-P}
\Pr\left[ \mathcal{F}\text{ succeeds} \mid \mathcal{E} \land X'_H = \tau_H\right]
& = \frac{\mu_{\min}(\+R,u,X_{\+R})}{\mu_{u,\mathcal{I}}^{X_{\partial B}}(X_u)} = \frac{\mu_{\min}(R,u,\rho)}{\mu_{u,\mathcal{I}}^{\rho(\Psi) \uplus \tau(S)}(\rho_u)}.
\end{align}
Combining~\eqref{eq-original-P} and~\eqref{eq-proof-new-target}, we can set $ C = C(R,u,\rho)$ in~\eqref{eq-proof-new-target} as
\begin{align}
\label{eq-set-C}
C = C(R,u,\rho) \triangleq \frac{\mu_{\min}( R,u,\rho) }{ \mu_{u,\mathcal{I}}^{\rho(R_u)} (\rho_u) } = \frac{1}{\mu_{u,\mathcal{I}}^{\rho(R_u)} (\rho_u)}\cdot \min_{\eta \in [q]^{ S }}\mu_{u,\mathcal{I}}^{\rho(\Psi) \uplus \eta}(\rho_u).
\end{align}
Note that $\mu^{\rho(R_u)}_{u,\mathcal{I}}(\rho_u) > 0$ because $\rho$ is feasible, and $\mu_{\min}( R,u,\rho) > 0$ due to the lower bound in~\eqref{eq-proof-assume-mu-min}. This implies $C(R,u,\rho) > 0$.
Remark the sets $S$ and $\Psi$ are determined by $R$ and $u$.
This implies that the $C(R,u,\rho)$ defined as above depends only on $R,u,\rho$.
This proves \eqref{eq-claim-2}.
\end{proof}
\section{Efficiency under strong spatial mixing}
\label{section-running-time}
In this section,
we prove the following result.
\begin{condition}
\label{condition-SSM-ratio}
Let $\mathcal{I}=(G, [q], \boldsymbol{b}, \boldsymbol{A})$ be a permissive spin system where $G=(V,E)$.
There is an integer $\ell=\ell(q)\ge 2$ such that the following holds:
for every $v \in V$, $\Lambda \mathsf{Sub}seteq V$, and any two partial configurations $\sigma, \tau \in [q]^\Lambda$ satisfying $\min\{\mathrm{dist}_G(v, u) \mid u \in \Lambda,\ \sigma_u\neq\tau_u\}=\ell$,
\begin{align}
\label{eq:condition-SSM}
\forall a \in [q]: \quad \left\vert \frac{\mu_{v, \mathcal{I}}^\sigma(a)}{ \mu_{v,\mathcal{I}}^\tau(a) } - 1 \right\vert \leq \frac{ 1 }{5 \left\vert S_{\ell}(v) \right\vert}
\quad(\text{with the convention $0/0 = 1$}),
\end{align}
where $S_{\ell}(v)\triangleq \{u \in V \mid \mathrm{dist}_G(v, u) = \ell\}$ denotes the {sphere} of radius $\ell$ centered at $v$ in $G$.
\end{condition}
\begin{theorem}
\label{theorem-general-ratio}
Let $\mathfrak{I}$ be a class of permissive spin systems satisfying Condition~\ref{condition-SSM-ratio}.
Given any instance $\mathcal{I}=(G,[q],\boldsymbol{b},\boldsymbol{A})\in\mathfrak{I}$, the \Cref{alg:perfect-sampler-gen} with parameter $\ell = \ell^* - 1$ outputs a perfect sample from $\mu_{\mathcal{I}}$ within $ O\left(n\cdot q^{2\mathcal{D}elta^{\ell^*}}\right) $ time in expectation,
where $n$ is the number of vertices in $G$, $\mathcal{D}elta$ is the maximum degree of $G$, $\ell^* = \ell^*(q) \geq 2$ is determined by \Cref{condition-SSM-ratio}, and $O(\cdot)$ hides only absolute constants.
\end{theorem}
The correctness part of \Cref{theorem-general-ratio} follows from \Cref{theorem-correctness}, we focus on the efficiency part.
The proof sketch is that if $\mathfrak{I}$ satisfies \Cref{condition-SSM-ratio} with parameter $\ell^*$, we set the parameter $\ell$ in \Cref{alg:perfect-sampler-gen} as $\ell = \ell^* - 1$. We prove that after each iteration of the \textbf{while}{} loop, the size of $\+R$ decreases by at least $\frac{1}{5}$ in expectation.
Note that the initial $\+R = V$, thus the initial size of $\+R$ is $n$.
By the optional stopping theorem, the number of iterations of the \textbf{while}{} loop is $O(n)$ in expectation.
One can verify that the time complexity of the \textbf{while}{} loop is $O(q^{2\mathcal{D}elta^{\ell+1}}) = O(q^{2\mathcal{D}elta^{\ell^*}})$.
This proves the running time result.
To analyze the expected size of $\+R$ after each iteration of the \textbf{while} loop, we analyze the Bayes filter $\+F$ in \Cref{line-F}. The probability that $\+F$ fails is $1 - \mu_{\min}/ \mu_{u}^{X_{\partial B}}(X_u)$. Suppose $\sigma \in [q]^{\partial B}$ achieves $\mu_{\min} = \mu_{u}^\sigma(X_u)$. By \Cref{fact-0}, we can verify that $\sigma$ and $X_{\partial B}$ can be differ only at $\partial B \setminus \+R \mathsf{Sub}seteq S_{\ell+1}(u) = S_{\ell^*}(u)$. By \Cref{condition-SSM-ratio}, we know that $\Pr[\+F \text{ fails}] \leq \frac{1}{5\abs{S_{\ell^*}(u)}}$. In addition, we have
\begin{itemize}
\item if $\mathcal{F}\text{ succeeds}$, the size of $\+R$ decreases by 1;
\item if $\+F$ fails, the size of $\+R$ increases by $\abs{\partial B \setminus \+R}$, it easy to verify $\partial B \setminus \+R \mathsf{Sub}seteq S_{\ell^*}(u)$ by \Cref{fact-0}, thus, the size of $\+R$ increases by at most $\abs{S_{\ell^*}(u)}$.
\end{itemize}
Thus, after each iteration of the \textbf{while}{} loop, the size of $\+R$ decreases by at least $\frac{1}{5}$ in expectation.
In the formal proof, we actually prove a stronger result. We first introduce the following condition.
\begin{condition}
\label{condition-SSM-weak}
Let $\mathcal{I}=(G, [q], \boldsymbol{b}, \boldsymbol{A})$ be a permissive spin system where $G=(V,E)$.
There is an integer $\ell=\ell(q)\ge 2$ such that the following holds:
for every $v \in V$, any two disjoint sets $A, B \mathsf{Sub}seteq V$ with $\mathrm{dist}_G(v,B) = \min\{\mathrm{dist}_G(v,u) \mid u \in B\} = \ell$, and any partial configuration $\sigma \in [q]^A$,
\begin{align}
\label{eq:condition-SSM-weak}
\forall a \in [q],\quad 1 - \frac{\min_{\tau \in [q]^B} \mu_{v, \mathcal{I}}^{\sigma \uplus \tau}(a)}{ \mu_{v,\mathcal{I}}^\sigma(a) } \leq \frac{ 1 }{5 \left\vert S_{\ell}(v) \right\vert}
\quad(\text{with the convention $0/0 = 1$}),
\end{align}
where $S_{\ell}(v)\triangleq \{u \in V \mid \mathrm{dist}_G(v, u) = \ell\}$ denotes the {sphere} of radius $\ell$ centered at $v$ in $G$.
\end{condition}
It is straightforward to verify that \Cref{condition-SSM-ratio} implies \Cref{condition-SSM-weak}. In the rest of this section, we prove that the efficiency result in \Cref{theorem-general-ratio} holds under \Cref{condition-SSM-weak}.
Let $\mathcal{I} = (G, [q], \boldsymbol{b}, \boldsymbol{A}) \in \mathfrak{I}$ be the input instance
satisfying \Cref{condition-SSM-weak} with some $\ell^* \geq 2$.
Set the parameter $\ell$ in Algorithm~\ref{alg:perfect-sampler-gen} as $\ell = \ell^* - 1$.
Denote by $T$ the number of iterations of the \textbf{while}{} loop in \Cref{alg:perfect-sampler-gen}.
To prove the efficiency result in Theorem~\ref{theorem-general-ratio},
we bound the maximum running time of the \textbf{while}{} loop and the expectation of $T$ in the following two lemmas.
\begin{lemma}
\label{lemma-running-time-alg2}
The running time of each \textbf{while}{} loop is at most \ensuremath{O(q^{2\Delta^{\ell+1}})} $= O(q^{2\mathcal{D}elta^{\ell^*}})$.
\end{lemma}
\begin{lemma}
\label{lemma-ET}
$\E{T} \leq 5n$.
\end{lemma}
Since the input instance $\mathcal{I}$ is permissive (\Cref{definition-locally-admissible})
, the initial feasible configuration can be constructed by a simple greedy algorithm.
The running time of the first two lines in Algorithm~\ref{alg:perfect-sampler-gen} is $O(n\mathcal{D}elta )$.
Combining this with Lemma~\ref{lemma-running-time-alg2} and Lemma~\ref{lemma-ET} proves the efficiency result in Theorem~\ref{theorem-general-ratio}.
\begin{proof}[Proof of Lemma~\ref{lemma-running-time-alg2}]
We first show that $\mu_{u,\mathcal{I}}^{X_{\partial B}}(X_u)$ can be computed in time \ensuremath{O(q^{\Delta^{\ell+1}})},
where $O(\cdot)$ hides an absolute constant.
Let $\widetilde{G}=G[B \cup \partial B]$ and $\widetilde{\mathcal{I}}$ be the instance restricted to $\widetilde{G}$.
By the conditional independence property (\Cref{property-cond-ind}), it is straightforward to verify
\begin{align}
\label{eq-K-reduced}
\mu_{u,\mathcal{I}}^{X_{\partial B}}(X_u) = \mu_{u,\widetilde{\mathcal{I}}}^{X_{\partial B}}(X_u)
\end{align}
since $\partial B$ separates $B$ from $V\setminus (B\cup \partial B)$ and $u \in B$.
Since $\abs{B}\leq\abs{B_{\ell}(u)}\le \frac{\mathcal{D}elta^{\ell+1}-1}{\mathcal{D}elta-1}\le \mathcal{D}elta^{\ell+1}$,
it takes at most \ensuremath{O(q^{\Delta^{\ell+1}})}\ to enumerate all possibilities and compute $ \mu_{u,\mathcal{I}}^{X_{\partial B}}(X_u)$ using~\eqref{eq-K-reduced}.
By \Cref{fact-0}, $\partial B \mathsf{Sub}seteq S_{\ell+1}(u) \cup R$.
Since $\abs{\partial B \setminus R } \le \abs{S_{\ell+1}(u)}\le \mathcal{D}elta^{\ell+1}$,
we can enumerate all $[q]^{\partial B \setminus R}$ to compute $\mu_{\min}$ in time \ensuremath{O(q^{2\Delta^{\ell+1}})}.
The total running time for the first three lines of the \textbf{while}{} loop is at most \ensuremath{O(q^{2\Delta^{\ell+1}})}.
Another non-trivial computation is to sample $X(B)$ from $\mu_{\mathcal{I}}^{X_{\partial B}}$.
Similar to \eqref{eq-K-reduced},
conditional independence implies that this can be done by straightforward enumeration in time \ensuremath{O(q^{\Delta^{\ell+1}})}.
The total running time of the \textbf{while}{} loop is thus \ensuremath{O(q^{2\Delta^{\ell+1}})}$= O(q^{2\mathcal{D}elta^{\ell^*}})$.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lemma-ET}]
Define a sequence of random pairs $(\boldsymbol{X}_0,\mathcal{R}_0),(\boldsymbol{X}_1,\mathcal{R}_1),\ldots,(\boldsymbol{X}_T,\mathcal{R}_T)$, where each $(\boldsymbol{X}_t,\mathcal{R}_t) \in [q]^V \times 2^V$.
The initial $(\boldsymbol{X}_0,\mathcal{R}_0)$ is constructed by the first two lines of Algorithm~\ref{alg:perfect-sampler-gen}.
In $t$-th \textbf{while}{} loop, Algorithm~\ref{alg:perfect-sampler-gen} updates $(\boldsymbol{X}_{t-1},\mathcal{R}_{t-1}) $ to $(\boldsymbol{X}_t,\mathcal{R}_t)$.
For any $t \geq 0$, we use a random variable $Y_t \triangleq \abs{\mathcal{R}_t}$ to denote the size of $\mathcal{R}_t$.
The stopping time $T$ is the smallest integer such that $Y_t=0$.
Define the \emph{execution log} of \Cref{alg:perfect-sampler-gen} up to time $t$ as
\begin{align*}
\+L_t \triangleq (X_0(\mathcal{R}_0), \mathcal{R}_0), (X_1(\mathcal{R}_1), \mathcal{R}_1),\ldots, (X_t(\mathcal{R}_t), \mathcal{R}_t).
\end{align*}
Note that the algorithm terminates at time $T$ if and only if $\mathcal{R}_T = \emptyset$.
In the $t$-th iteration of the \textbf{while}{} loop, we use $\+F_t$ to denote the Bayes filter and $\*u_t$ to denote the random vertex picked in~\Cref{line-pick}.
We have the following claim.
\begin{claim}
\label{claim-lower-bound}
Given any execution log $\+L_{t-1}$ created by \Cref{alg:perfect-sampler-gen} such that $\mathcal{R}_{t-1}\neq \emptyset$, for any $u \in \mathcal{R}_{t-1}$,
\begin{align*}
\Pr[\+F_t \text{ succeeds} \mid \+L_{t-1}, \*u_t =u] \geq \begin{cases}
1 &\text{if } |S_t| = \emptyset;\\
1- \frac{2}{5|S_t|}&\text{if }|S_t| \neq \emptyset,
\end{cases}
\end{align*}
where $S_t = \partial B_t \setminus \+R_{t-1}$ and $B_t = (B_{\ell}(u) \setminus \mathcal{R}_{t-1})\cup\{u\}$ is the set $B$ in the $t$-th iteration the \textbf{while}{} loop.
\end{claim}
Note that given $\+L_{t-1}$, the vertex $\*u_t \in \mathcal{R}_{t-1}$ is sampled uniformly and independently.
For any $1\leq t \leq T$ and any execution log $\+L_{t-1}$ created by \Cref{alg:perfect-sampler-gen}, if $S_t = \emptyset$, by \Cref{claim-lower-bound}, we have
\begin{align*}
\E{Y_t \mid \+L_{t-1}, S_t = \emptyset} = Y_{t-1} - 1;
\end{align*}
Suppose $S_t \neq \emptyset$. If $\+F_t$ fails, then $\mathcal{R}_{t} = \mathcal{R}_{t-1} \cup \partial B_t = \mathcal{R}_{t-1} \cup (\partial B_t \setminus \mathcal{R}_{t-1})$.
In other words, $|S_t|$ new vertices will be added into $\mathcal{R}_{t-1}$ if $\+F_t$ fails.
We have
\begin{align*}
\E{Y_t \mid \+L_{t-1}, S_t \neq \emptyset} &\leq Y_{t-1} -\Pr[\mathcal{F}_t \text{ succeeds} \mid \+L_{t-1},S_t \neq \emptyset] + \Pr[\mathcal{F}_t \text{ fails} \mid \+L_{t-1},S_t \neq \emptyset] \cdot |S_t|\\
&\leq Y_{t-1} + \frac{2}{5|S_t|} - \frac{3}{5} \tag{by \Cref{claim-lower-bound}}\\
&\leq Y_{t-1} - \frac{1}{5}.
\end{align*}
Combining two cases together implies
\begin{align*}
\E{Y_t \mid \+L_{t-1}} = \E{Y_t\mid (X_0(\mathcal{R}_0), \mathcal{R}_0), (X_1(\mathcal{R}_1), \mathcal{R}_1),\ldots, (X_{t-1}(\mathcal{R}_{t-1}), \mathcal{R}_{t-1}) } \leq Y_{t-1} - \frac{1}{5}.
\end{align*}
We now define a sequence $Y'_0,Y'_1,\ldots,Y'_T$ where each $Y'_t = Y_t + \frac{t}{5}$. Thus, $Y'_0,Y'_1,\ldots,Y'_T$ is a super-martingale with respect to $(X_0(\mathcal{R}_0), \mathcal{R}_0), (X_1(\mathcal{R}_1), \mathcal{R}_1),\ldots, (X_T(\mathcal{R}_T), \mathcal{R}_T)$ and $T$ is a stopping time. Note that each $|Y'_t - Y'_{t-1}| \leq n + 1$ is bounded and $\E{T}$ is finite due to~\eqref{eq-dominate}. Due to the optional stopping theorem~\cite[Chapter 5]{durrett2019probability} , we have $\E{Y'_T} \leq \E{Y'_0} = \E{Y_0}$. Hence
\begin{align*}
\E{T} \leq 5\E{Y_0} = 5n,
\end{align*}
where the last eqaution is because $\E{Y_0} = \E{|\mathcal{R}_0|} = n$.
\end{proof}
\begin{proof}[Proof of \Cref{claim-lower-bound}]
Suppose $S_t = \partial B_t \setminus \+R_{t-1} = \emptyset$.
This implies $\partial B_t \mathsf{Sub}seteq \mathcal{R}_{t-1}$.
By the definition of $\mu_{\min}$, we have $\mu^{X_{t-1}(\partial B_t)}_{u,\mathcal{I}}(X_{t-1}(u)) = \mu_{\min}$ and $\Pr[\+F_t \text{ succeeds}] = 1$.
In the following proof, we assume $S_t \neq \emptyset$.
We need the following property to prove the claim.
Fix an execution log $L_{t}$ up to time $t \geq 0$:
\begin{align*}
L_{t} = (\rho_0, R_0), (\rho_1,R_1),\ldots, (\rho_{t}, R_{t}),
\end{align*}
where each $R_i \mathsf{Sub}seteq V$, each $\rho_i \in [q]^{R_i}$. Assume $R_{t} \neq \emptyset$ and $L_{t}$ is a feasible execution log, i.e. $\Pr[\+L_{t} = L_{t}] > 0$.
We claim that given the log $L_t$, the random $\boldsymbol{X}_t \in [q]^V$ satisfies $X_t(R_t) = \rho_t$ and
\begin{align}
\label{eq-martingale}
\forall \tau \in [q]^{V \setminus R_{t}}, \quad \Pr[X_{t}(V \setminus R_{t}) = \tau \mid \+L_{t}=L_{t}] = \mu_{\mathcal{I}}^{\rho_{t}}(\tau).
\end{align}
\Cref{eq-martingale} is proved by the induction on $t$. If $t = 0$, we have $R_0 = V$, \Cref{eq-martingale} holds trivially. Assume \Cref{eq-martingale} holds for all $t < k$. Fix any feasible execution log $L_{k} = (\rho_0, R_0), (\rho_1,R_1),\ldots, (\rho_{k}, R_{k})$ such that $R_k \neq \emptyset$.
Since $L_k$ is feasible, we have $R_{k-1} \neq \emptyset$. Consider the $k$-th iteration of the \textbf{while}{} loop. The $k$-th iteration exists because $R_{k-1} \neq \emptyset$. By induction hypothesis, conditioning on the execution log $\+L_{k-1} = (\rho_0, R_0), (\rho_1,R_1),\ldots, (\rho_{k-1}, R_{k-1})$, the random pair $(\boldsymbol{X}_{k-1},\mathcal{R}_{k-1})$ satisfies the \Cref{condition-invariant} and $\boldsymbol{X}_{k-1}$ is a feasible configuration (since $L_k$ is a feasible execution log, thus $\rho_{k-1}$ is feasible). By \Cref{lemma-detailed-invariant}, conditioning on the execution log $\+L_{k-1} = (\rho_0, R_0), (\rho_1,R_1),\ldots, (\rho_{k-1}, R_{k-1})$, the random pair $(\boldsymbol{X}_{k},\mathcal{R}_{k})$ satisfies the \Cref{condition-invariant}. By \Cref{condition-invariant}, assuming the further condition $\mathcal{R}_{k} = R_k$ and $X_k(R_k) = \rho_k$, \Cref{eq-martingale} holds for $t = k$. This proves \Cref{eq-martingale}.
Consider a feasible execution log up to time $t - 1 \geq 0$:
\begin{align*}
L_{t-1} = (\rho_0, R_0), (\rho_1,R_1),\ldots, (\rho_{t-1}, R_{t-1})
\end{align*}
satisfying $R_{t-1} \neq \emptyset$, where each $R_i \mathsf{Sub}seteq V$ and each $\rho_i \in [q]^{R_i}$.
Given the execution log $\+L_{t-1} = L_{t-1}$, we fix a vertex $u \in R_{t-1}$ and assume $\*u_t = u$.
We analyze the $t$-th iteration of the \textbf{while}{} loop.
To simplify the notation, we drop the index and denote
\begin{align*}
\boldsymbol{X} = \boldsymbol{X}_{t-1},\quad R= R_{t-1},\quad\rho = \rho_{t-1},\quad B = B_t = (B_{\ell}(u) \setminus R )\cup \{u\} ,\quad S = S_t = \partial B\setminus R.
\end{align*}
Note that the vertex $\*u_t$ is sampled from $R$ uniformly and independently.
By~\eqref{eq-martingale}, given $\+L_{t-1}=L_{t-1}$ and $\*u_t = u$, it holds that $X(R) = \rho$ and $(\boldsymbol{X}, R)$ satisfies \Cref{condition-invariant}.
By \Cref{property-lower-bound}, we know $\mu_{\min}(R,u,\rho) > 0$, thus the lower bound in~\eqref{eq-proof-assume-mu-min} holds.
According to the proof of \Cref{lemma-detailed-invariant}, combining~\eqref{eq-succeeds-prob} and~\eqref{eq-set-C}, we have
\begin{align*}
\Pr[\+F_t \text{ succeeds} \mid \+L_{t-1} = L_{t-1}, \*u_t = u] = \frac{1}{\mu_{u,\mathcal{I}}^{\rho(R_u)} (\rho_u)}\cdot \min_{\eta \in [q]^{ S }}\mu_{u,\mathcal{I}}^{\rho(\Psi) \uplus \eta}(\rho_u),
\end{align*}
where $R_u = R \setminus \{u\}$, $S = \partial B \setminus R$ and $\Psi = \partial B \cap R$.
Note that $\partial B = S \uplus \Psi$ and $u \in B$, the set $\partial B$ separates $u$ from $V \setminus (B \cup \partial B)$.
Since $\Psi \mathsf{Sub}seteq R_u$ and two sets $R_u$ and $B$ are disjoint,
by the conditional independence property (\Cref{property-cond-ind}), we have
$\mu_{u,\mathcal{I}}^{\rho(\Psi) \uplus \eta}(\rho_u) = \mu_{u,\mathcal{I}}^{\rho(R_u) \uplus \eta}(\rho_u)$.
This implies
\begin{align*}
\Pr[\+F_t \text{ succeeds} \mid \+L_{t-1} = L_{t-1}, \*u_t = u] = \frac{1}{\mu_{u,\mathcal{I}}^{\rho(R_u)} (\rho_u)}\cdot \min_{\eta \in [q]^{ S }}\mu_{u,\mathcal{I}}^{\rho(R_u) \uplus \eta}(\rho_u).
\end{align*}
By \Cref{fact-0}, we have $S = \partial B \setminus R \mathsf{Sub}seteq S_{\ell + 1}(u)$.
We take $A = R_u, B = S, v= u, \sigma = \rho(R_u)$ and $a = \rho_u$ in \Cref{condition-SSM-weak}, since $\mathrm{dist}_G(u, S) = \ell + 1 = \ell^*$ and $|S| \leq |S_{\ell^*}(u)|$, this proves that
\begin{align*}
\Pr[\+F_t \text{ succeeds} \mid \+L_{t-1} = L_{t-1}, \*u_t = u] \geq 1 - \frac{1}{5|S_{\ell^*}(u)|} \geq 1 - \frac{1}{5|S|} \geq 1 - \frac{2}{5|S|}.
\end{align*}
\end{proof}
\begin{remark}
\label{remark-improve}
Suppose the input instance from the class $\mathfrak{I}$ satisfies~\cref{condition-SSM-ratio} with some $\ell^* \geq 2$ and take $\ell = \ell^* - 1$ in \Cref{alg:perfect-sampler-gen}.
We could tweak \Cref{alg:perfect-sampler-gen} to reduce its running time to $O\left( n\cdot q^{ \mathcal{D}elta^{\ell^*} } \right)$.
Let $\Psi \triangleq \partial B \cap \mathcal{R}$ and $S \triangleq \partial B \setminus \mathcal{R}$.
Note that $S \mathsf{Sub}seteq S_{\ell^*}(u)$ by \Cref{fact-0}.
The idea is that instead of calculating $\mu_{\min}$, we may simply compute $\mu_{u,\mathcal{I}}^{X(\Psi) \uplus \sigma}(X_u)$ where $\sigma = \boldsymbol{1} \in [q]^{S}$ is a one-vector, then use~\eqref{eq:condition-SSM} to get a lower bound $\mu_{\mathrm{low}}$ of $\mu_{\min}$ as
\begin{align*}
\mu_{\mathrm{low}} \triangleq \left(1 - \frac{ 1 }{5 \left\vert S_{\ell^*}(u) \right\vert}\right)\mu_{u,\mathcal{I}}^{X(\Psi) \uplus \sigma}(X_u).
\end{align*}
\noindent
By \Cref{condition-SSM-ratio}, $\mu_{\mathrm{low}} \leq \mu_{u,\mathcal{I}}^{X(\partial B)}(X_u)$.
Then we use $\mu_{\mathrm{low}}$ instead of $\mu_{\min}$ in the definition of $\+F$.
It is straightforward to check that \Cref{alg:perfect-sampler-gen} is still correct with this tweak, and for each iteration of the \textbf{while}{} loop, given any $\boldsymbol{X}, \mathcal{R}$, it holds that
\begin{align*}
\Pr[\mathcal{F}\text{ succeeds} \mid \boldsymbol{X}, \mathcal{R}] &= \frac{\mu_{\mathrm{low}}}{ \mu_{u,\mathcal{I}}^{X(\partial B)}(X_u)} = \left(1 - \frac{1}{5|S_{\ell^*}(v)|}\right) \frac{\mu_{u,\mathcal{I}}^{X(\Psi) \uplus \sigma}(X_u)}{ \mu_{u,\mathcal{I}}^{X(\partial B)}(X_u)}= \left(1 - \frac{1}{5|S_{\ell^*}(v)|}\right) \frac{\mu_{u,\mathcal{I}}^{X(\Psi) \uplus \sigma}(X_u)}{ \mu_{u,\mathcal{I}}^{X(\Psi) \uplus X(S)}(X_u)}\\
&\geq \left(1 - \frac{1}{5|S_{\ell^*}(v)|}\right)^2\qquad (\text{by } S \mathsf{Sub}seteq S_{\ell^*}(u) \text{ and \Cref{condition-SSM-ratio}})\\
&\geq 1 - \frac{2}{5|S|}.
\end{align*}
This proves \Cref{claim-lower-bound}. Besides, we do not need to enumerate all configurations in $[q]^S$ to compute $\mu_{\min}$, the expected running time of \Cref{alg:perfect-sampler-gen} can be reduced to $ O\left( n\cdot q^{ \mathcal{D}elta^{\ell^*} } \right)$.
\end{remark}
\section{Analysis of strong spatial mixing}
In this section, we use \Cref{theorem-general-ratio} to prove other results mentioned in \Cref{section-results}.
We analyze the strong spatial mixing properties for the classes of spin systems mentioned in \Cref{section-results}, so that we can use~\Cref{theorem-general-ratio} to prove the existences of perfect samplers.
\mathsf{Sub}section{Spin systems on sub-exponential neighborhood growth graphs}
In this section, we prove \Cref{theorem-sub-graph} using \Cref{theorem-general-ratio}.
We need the following proposition, which explains the relation between the multiplicative form of decay in~\eqref{eq:condition-SSM} and the
additive form of decay in~\eqref{eq:condition-standard-SSM}. Similar results appeared in~\cite{K98, feng2018local,spinka2018finitary,alexander2004mixing}.
\begin{proposition}
\label{proposition-SSM}
Let $\delta: \mathbb{N} \to \mathbb{N}$ be a non-increasing function.
Let $\mathfrak{I}$ be a class of permissive spin system instances exhibiting strong spatial mixing with decay rate $\delta$.
For every instance $\mathcal{I}=(G, [q], \boldsymbol{b}, \boldsymbol{A}) \in \mathfrak{I}$, where $G=(V,E)$,
for every $v \in V$, $\Lambda \mathsf{Sub}seteq V$, and any two partial configurations $\sigma, \tau \in [q]^\Lambda$ with $\ell \geq 2$,
\begin{align}
\label{eq-implied-SSM}
\forall a \in [q],\quad
\min\left(\left\vert \frac{\mu_{v, \mathcal{I}}^\sigma(a)}{ \mu_{v,\mathcal{I}}^\tau(a) } - 1 \right\vert, 1\right) \leq 10 q\cdot|S_{\lfloor \ell / 2 \rfloor }(v)|\cdot\delta(\lfloor \ell / 2 \rfloor) \quad(\text{with the convention $0/0 = 1$}),
\end{align}
where $\ell \triangleq \min\{\mathrm{dist}_G(v, u) \mid u \in \Lambda,\ \sigma(u)\neq\tau(u)\}\geq 2$ and $S_{\lfloor \ell / 2 \rfloor}(v) \triangleq \{u \in V \mid \mathrm{dist}_G(v, u) = \lfloor \ell / 2 \rfloor\}$ denotes the sphere of radius $\lfloor \ell / 2 \rfloor$ centered at $v$ in $G$.
\end{proposition}
The proof of \Cref{proposition-SSM} is deferred to the end of this section.
We first use \Cref{proposition-SSM} to prove \Cref{theorem-sub-graph}.
\begin{proof}[Proof of \Cref{theorem-sub-graph}]
Let $q$ be a finite integer.
Suppose $\mathfrak{I}$ is a class of $q$-state spin systems that is defined on a class of graphs that have sub-exponential neighborhood growth in \Cref{definition-sub-exp-graph} with function $s:\mathbb{N}\to\mathbb{N}$.
Suppose $\mathfrak{I}$ exhibits strong spatial mixing with exponential decay with some constants $\alpha > 0, \beta > 0$. Let $\delta$ be the function $\delta(x) = \alpha \exp(-\beta x)$.
Fix a instance $\mathcal{I}=(G, [q], \boldsymbol{b}, \boldsymbol{A}) \in \mathfrak{I}$, where $G=(V,E)$.
By \Cref{proposition-SSM}, we have
for any $\Lambda \mathsf{Sub}seteq V$, any $v \in V$, and any two partial configurations $\sigma, \tau \in [q]^\Lambda$
satisfying $\ell \triangleq \min\{\mathrm{dist}_G(v, u) \mid u \in \Lambda,\ \sigma(u)\neq\tau(u)\}\geq 2$,
\begin{align}
\label{eq-proof-app-1}
\forall a \in [q],\quad
\min\left(\left\vert \frac{\mu_{v, \mathcal{I}}^\sigma(a)}{ \mu_{v,\mathcal{I}}^\tau(a) } - 1 \right\vert, 1\right) &\leq 10 q \cdot |S_{\lfloor \ell / 2 \rfloor }(v)| \cdot \delta(\lfloor \ell / 2 \rfloor ) \notag\\
(\text{by }|S_r(v)| \leq s(r))\quad &\leq 10 q \cdot s(\lfloor \ell / 2 \rfloor) \cdot \delta(\lfloor \ell / 2 \rfloor )
\end{align}
Note that $\delta(\ell) = \alpha \exp(-\beta \ell)$.
We take $\ell_0 = \ell_0(q, \alpha, \beta, s)$ sufficiently large such that $\ell_0 \geq 2$ and
\begin{align}
\label{eq-proof-app-2}
10\alpha q \cdot s(\lfloor\ell_0 / 2\rfloor ) \exp(-\beta \lfloor\ell_0 / 2\rfloor) \leq \frac{1}{5s( \ell_0 )} \leq \frac{1}{5 \left\vert S_{\ell_0}(v) \right\vert}.
\end{align}
Note that~\eqref{eq-proof-app-2} is equivalent to
\begin{align}
\label{eq-set-ell}
\alpha\exp(-\beta \lfloor\ell_0 / 2\rfloor) \leq \frac{1}{50q \cdot s(\lfloor\ell_0 / 2\rfloor) \cdot s(\ell_0)}.
\end{align}
Such $\ell_0 = \ell_0(q, \alpha, \beta, s)$ must exist because $s(r)=\exp(o(r))$ for all $r \geq 0$.
Combining~\eqref{eq-proof-app-1} and~\eqref{eq-proof-app-2} implies that $\mathcal{I}$ satisfies \Cref{condition-SSM-ratio} with $\ell_0 \geq 2$.
By \Cref{theorem-general-ratio}, if the parameter $\ell$ in Algorithm~\ref{alg:perfect-sampler-gen} is set so that $\ell = \ell_0-1$, given $\mathcal{I}$, the expected running time of \Cref{alg:perfect-sampler-gen} is
$
n\cdot q^{O\left( \mathcal{D}elta^{\ell_0} \right)}.
$
Since $\ell_0 = O(1)$, $q=O(1)$ and $\mathcal{D}elta \leq s(1) = O(1)$,
the expected running time of \Cref{alg:perfect-sampler-gen} is $O(n)$.
\end{proof}
We now prove \Cref{proposition-SSM}. Similar results are proved in~\cite{K98,feng2018local,spinka2018finitary,alexander2004mixing}.
\begin{proof}[Proof of \Cref{proposition-SSM}]
Fix a instance $\mathcal{I}=(G, [q], \boldsymbol{b}, \boldsymbol{A}) \in \mathfrak{I}$, where $G=(V,E)$.
Fix two partial configurations $\sigma, \tau \in [q]^\Lambda$
with $\ell \triangleq \min\{\mathrm{dist}_G(v, u) \mid u \in \Lambda,\ \sigma(u)\neq\tau(u)\}$. We use $D \triangleq \{v \in \Lambda \mid \sigma(v) \neq \tau(v) \}$ to denote the set at which $\sigma$ and $\tau$ disagree. Fix a spin $a \in [q]$. Since $\mathcal{I}$ is permissive (\Cref{definition-locally-admissible}), we have
\begin{align*}
\mu^{\sigma}_{v,\mathcal{I}}(a) = 0 \quad\Longleftrightarrow\quad b_v(a)\prod_{u \in \Gamma_G(v) \cap \Lambda}A_{uv}(a,\sigma(v)) = 0,\\
\mu^{\tau}_{v,\mathcal{I}}(a) = 0 \quad\Longleftrightarrow\quad b_v(a)\prod_{u \in \Gamma_G(v) \cap \Lambda}A_{uv}(a,\tau(v)) = 0,
\end{align*}
where $\Gamma_G(v)$ is the neighborhood of $v$ in $G$.
Since $\ell \geq 2$, there is no edge between $v$ and $D$, we have $\Gamma_G(v) \cap D = \emptyset$.
This implies $\mu^{\sigma}_{v,\mathcal{I}}(a) = 0$ if and only if $\mu^{\tau}_{v,\mathcal{I}}(a) = 0$. If $\mu^{\sigma}_{v,\mathcal{I}}(a) = \mu^{\tau}_{v,\mathcal{I}}(a) = 0$, the proposition holds trivially. We assume
\begin{align}
\label{eq-proof-assume}
\mu^{\sigma}_{v,\mathcal{I}}(a) >0 \land \mu^{\tau}_{v,\mathcal{I}}(a) > 0.
\end{align}
Define the set of vertices
$H \triangleq S_{\lfloor \ell/2 \rfloor}(v) \setminus \Lambda$,
where $S_{\lfloor \ell /2 \rfloor}(v) = \{u \in V \mid \mathrm{dist}(u,v) = \lfloor \ell/2 \rfloor \}$ is the sphere of radius $\lfloor \ell/2 \rfloor$ centered at $v$ in graph $G$.
By the definitions, we have $H \cap D = \emptyset$.
If $H = \emptyset$, then $S_{\lfloor \ell/2 \rfloor}(v) \mathsf{Sub}seteq \Lambda$,
the proposition holds due to the conditional independence property. In the rest of the proof, we assume $H \neq \emptyset$.
For any two disjoint sets $S,S' \mathsf{Sub}seteq V$ and any partial configurations $\eta \in [q]^S, \eta' \in [q]^{S'}$, we use $\mu_{\mathcal{I}}^{S\gets \eta, S' \gets \eta'}$ to denote the distribution $\mu_{\mathcal{I}}^{\eta \uplus \eta'}$.
For any $\rho \in [q]^H$ satisfying $\mu^{\Lambda \gets \sigma,v \gets a}_{H,\mathcal{I}}(\rho) > 0$ and $\mu^{\Lambda \gets \tau,v \gets a}_{H,\mathcal{I}}(\rho) > 0$
we have
\begin{align*}
\mu^{\sigma}_{v,\mathcal{I}}(a) = \frac{ \mu_{H,\mathcal{I}}^{\sigma}(\rho) \cdot \mu^{\sigma \uplus \rho}_{v,\mathcal{I}}(a) }{\mu^{\Lambda \gets \sigma,v \gets a}_{H,\mathcal{I}}(\rho)},\quad
\mu^{ \tau}_{v,\mathcal{I}}(a)= \frac{ \mu_{H,\mathcal{I}}^{\tau}(\rho) \cdot \mu^{\tau \uplus \rho}_{v,\mathcal{I}}(a) }{\mu^{\Lambda \gets \tau,v \gets a}_{H,\mathcal{I}}(\rho)}.
\end{align*}
The first equation holds since $\mu_{H,\mathcal{I}}^{\sigma}(\rho) \cdot \mu^{\sigma \uplus \rho}_{v,\mathcal{I}}(a) =\mu^{\sigma}_{v,\mathcal{I}}(a) \cdot \mu^{\Lambda \gets \sigma,v \gets a}_{H,\mathcal{I}}(\rho)$ and $\mu^{\Lambda \gets \sigma,v \gets a}_{H,\mathcal{I}}(\rho) > 0$;
the second equation holds similarly.
Note that $\mu^{\sigma}_{v,\mathcal{I}}(a) >0$ and $\mu^{\tau}_{v,\mathcal{I}}(a) > 0$.
We have
\begin{align*}
\frac{\mu^{\sigma}_{v,\mathcal{I}}(a)}{\mu^{ \tau}_{v,\mathcal{I}}(a)} = \left(\frac{\mu^{ \sigma \uplus \rho}_{v,\mathcal{I}}(a)}{\mu^{\tau \uplus \rho}_{v,\mathcal{I}}(a)}\right) \left(\frac{\mu_{H,\mathcal{I}}^{ \sigma}(\rho) }{\mu_{H,\mathcal{I}}^{\tau}(\rho) }\right) \left( \frac{\mu^{\Lambda \gets \tau,v \gets a}_{H,\mathcal{I}}(\rho)}{\mu^{\Lambda \gets \sigma,v \gets a}_{H,\mathcal{I}}(\rho)} \right).
\end{align*}
Note that $(\Lambda \setminus D ) \cup H$ separates $v$ from $D$ in graph $G$,
and the two configurations $\sigma \uplus \rho$ and $\tau \uplus \rho$ disagree only at $D$.
By the conditional independence property, we have $\mu^{\sigma \uplus \rho}_{v,\mathcal{I}}(a) = \mu^{\tau \uplus \rho}_{v,\mathcal{I}}(a)$. Hence, we have
\begin{align}
\label{eq-proof-ratios}
\frac{\mu^{\sigma}_{v,\mathcal{I}}(a)}{\mu^{ \tau}_{v,\mathcal{I}}(a)} = \left(\frac{\mu_{H,\mathcal{I}}^{\sigma}(\rho) }{\mu_{H,\mathcal{I}}^{ \tau}(\rho) }\right) \left( \frac{\mu^{\Lambda \gets \tau,v \gets a}_{H,\mathcal{I}}(\rho)}{\mu^{\Lambda \gets \sigma,v \gets a}_{H,\mathcal{I}}(\rho)} \right).
\end{align}
Note that~\eqref{eq-proof-ratios} holds for any $\rho \in [q]^H$ satisfying $\mu^{\Lambda \gets \sigma,v \gets a}_{H,\mathcal{I}}(\rho) > 0$ and $\mu^{\Lambda \gets \tau,v \gets a}_{H,\mathcal{I}}(\rho) > 0$. Our goal is to choose a suitable $\rho$ and bound the RHS.
Let
\begin{align*}
\epsilon \triangleq \delta(\lfloor \ell / 2 \rfloor).
\end{align*}
Without loss of generality,
we assume
\begin{align}
\label{eq-proof-assume2}
10 q\cdot|S_{\lfloor \ell / 2 \rfloor }(v)|\cdot\delta(\lfloor \ell / 2 \rfloor) = 10 q \epsilon\cdot|S_{\lfloor \ell / 2 \rfloor }(v)| < 1.
\end{align}
If~\eqref{eq-proof-assume2} does not hold, then the inequality~\eqref{eq-implied-SSM} holds trivially.
We have the following claim.
\begin{claim}
\label{claim-exist-rho}
Assume~\eqref{eq-proof-assume2}.
There exists a configuration $\rho \in [q]^H$ satisfying $\mu^{\Lambda \gets \sigma,v \gets a}_{H,\mathcal{I}}(\rho) > 0$ and $\mu^{\Lambda \gets \tau,v \gets a}_{H,\mathcal{I}}(\rho) > 0$ such that
\begin{align*}
\left( 1 - \frac{2q\epsilon}{1+q\epsilon} \right)^{2m} \leq \left(\frac{\mu_{H,\mathcal{I}}^{\sigma}(\rho) }{\mu_{H,\mathcal{I}}^{ \tau}(\rho) }\right) \left( \frac{\mu^{\Lambda \gets \tau,v \gets a}_{H,\mathcal{I}}(\rho)}{\mu^{\Lambda \gets \sigma,v \gets a}_{H,\mathcal{I}}(\rho)} \right) \leq \left( 1 + \frac{2q\epsilon}{1-q\epsilon} \right)^{2m},
\end{align*}
where $ m \triangleq |S_{\lfloor \ell / 2 \rfloor }(v)|$ and $\epsilon \triangleq \delta(\lfloor \ell / 2 \rfloor)$.
\end{claim}
The inequality \eqref{eq-proof-assume2} implies that
\begin{align}
q\epsilon m\le \frac{1}{10}.
\label{eq-proof-assume2p}
\end{align}
Combining~\Cref{claim-exist-rho} with the above, we have
\begin{align*}
\left(\frac{\mu_{H,\mathcal{I}}^{\sigma}(\rho) }{\mu_{H,\mathcal{I}}^{ \tau}(\rho) }\right) \left( \frac{\mu^{\Lambda \gets \tau,v \gets a}_{H,\mathcal{I}}(\rho)}{\mu^{\Lambda \gets \sigma,v \gets a}_{H,\mathcal{I}}(\rho)} \right) &\leq \exp \left( \frac{4q\epsilon m}{1-q\epsilon} \right)\\
&\leq \exp\left( 5q\epsilon m \right) \tag{by~\eqref{eq-proof-assume2p}}\\
&\leq 1 + 10q\epsilon m. \tag{by~\eqref{eq-proof-assume2p}}
\end{align*}
Similarly, we have
\begin{align*}
\left(\frac{\mu_{H,\mathcal{I}}^{\sigma}(\rho) }{\mu_{H,\mathcal{I}}^{ \tau}(\rho) }\right) \left( \frac{\mu^{\Lambda \gets \tau,v \gets a}_{H,\mathcal{I}}(\rho)}{\mu^{\Lambda \gets \sigma,v \gets a}_{H,\mathcal{I}}(\rho)} \right) &\geq \left( 1 - 2q\epsilon \right)^{ 2m}\\
&\geq \exp\left( -8q\epsilon m \right) \tag{by~\eqref{eq-proof-assume2p}}\\
&\geq 1 - 10q\epsilon m.
\end{align*}
Recall $ m \triangleq |S_{\lfloor \ell / 2 \rfloor }(v)|$ and $\epsilon \triangleq \delta(\lfloor \ell / 2 \rfloor)$. This proves the proposition.
\end{proof}
\begin{proof}[Proof of Claim~\ref{claim-exist-rho}]
Suppose $|H| = h\geq1$.
Let $H = \{v_1,v_2,\ldots,v_h\}$. Define a sequence of subsets $H_0,H_1,\ldots,H_h$ as $H_i \triangleq \{v_j \mid 1\leq j\leq i\}$. Note that $H_0 = \emptyset$ and $H_h=H$. We now construct the configuration $\rho \in [q]^H$ by the following $h$ steps.
\begin{itemize}
\item initially, $\rho = \emptyset$ is an empty configuration;
\item in $i$-th step, note that $\rho \in [q]^{H_{i-1}}$, choose $c_i \in [q]$ that maximizes $\mu_{v_i,\mathcal{I}}^{\Lambda\gets \sigma,v \gets a,H_{i-1} \gets \rho}(c_i)$ (break tie arbitrarily), extend $\rho$ further at position $v_i$ and set $\rho(v_i) = c_i$, thus $\rho \in [q]^{H_{i}}$ after the $i$-th step.
\end{itemize}
By the construction, we have
\begin{align*}
\forall 1\leq i \leq h,\quad \mu_{v_i,\mathcal{I}}^{\Lambda\gets \sigma,v \gets a,H_{i-1} \gets \rho(H_{i-1})}(\rho(v_i)) \geq \frac{1}{q} > 0.
\end{align*}
Recall $H \triangleq S_{\lfloor \ell/2 \rfloor}(v) \setminus \Lambda$. We have $\mathrm{dist}_G(H, D) \geq \ell - \lfloor \ell / 2 \rfloor \geq \lfloor\ell / 2\rfloor$, where $D$ is the set at which $\sigma$ and $\tau$ disagree, and $\mathrm{dist}_G(H, D) \triangleq \min\{ \mathrm{dist}_G(u_1,u_2) \mid u_1 \in H \land u_2 \in D \}$.
Recall $\epsilon \triangleq \delta(\lfloor \ell / 2 \rfloor)$ and $\delta$ is a non-increasing function.
By the strong spatial mixing property in~\Cref{definition-standard-SSM}, we have
\begin{align*}
\forall 1\leq i \leq h,\quad \mu_{v_i,\mathcal{I}}^{\Lambda\gets \tau,v \gets a,H_{i-1} \gets \rho(H_{i-1})}(\rho(v_i)) \geq \frac{1}{q} -\epsilon > 0,
\end{align*}
where $\frac{1}{q} -\epsilon > 0$ holds due to~\eqref{eq-proof-assume2}. By the chain rule, we have $\mu_{H,\mathcal{I}}^{\Lambda \gets \sigma, v \gets a}(\rho) > 0$ and $\mu_{H,\mathcal{I}}^{\Lambda \gets \tau, v \gets a}(\rho) > 0$.
We now prove that $\rho$ satisfies the inequalities in \Cref{claim-exist-rho}.
For any $1\leq i \leq h$, define
\begin{align}
\label{eq-def-pi}
p_i \triangleq \mu_{v_i,\mathcal{I}}^{\Lambda\gets \sigma,v \gets a,H_{i-1} \gets \rho(H_{i-1})}(\rho(v_i)).
\end{align}
Recall that $\mathrm{dist}_G(H,v) \geq \lfloor\ell / 2\rfloor$ and $\mathrm{dist}_G(H, D) \geq \lfloor\ell / 2\rfloor$. Recall $\epsilon \triangleq \delta(\lfloor \ell / 2 \rfloor)$ and $\delta$ is a non-increasing function. By the strong spatial mixing property in~\Cref{definition-standard-SSM}, we have for any $c \in [q]$ and any $1\leq i \leq h$,
\begin{align}
0 <p_i -\epsilon \leq \mu_{v_i,\mathcal{I}}^{\Lambda\gets \sigma,v \gets c,H_{i-1} \gets \rho(H_{i-1})}(\rho(v_i)) \leq p_i + \epsilon,\label{eq-lu-1}\\
0< p_i -\epsilon \leq \mu_{v_i,\mathcal{I}}^{\Lambda\gets \tau,v \gets c,H_{i-1} \gets \rho(H_{i-1})}(\rho(v_i)) \leq p_i + \epsilon.\label{eq-lu-2}
\end{align}
Note that $p_i - \epsilon \geq \frac{1}{q} - \epsilon > 0$ due to~\eqref{eq-proof-assume2}. Combining ~\eqref{eq-lu-1}, ~\eqref{eq-lu-2} and the chain rule implies
\begin{align*}
\forall c,c'\in[q],\quad
\prod_{i=1}^h\left(\frac{p_i - \epsilon}{p_i + \epsilon}\right) \leq \frac{\mu^{\Lambda \gets \tau,v \gets c}_{H,\mathcal{I}}(\rho)}{\mu^{\Lambda \gets \sigma,v \gets c'}_{H,\mathcal{I}}(\rho)} \leq \prod_{i=1}^h \left(\frac{p_i + \epsilon}{p_i- \epsilon}\right)
\end{align*}
Note that $p_i \geq \frac{1}{q}$ for all $1\leq i \leq h$ due the construction of $\rho$, and $q\epsilon < 1$ due to~\eqref{eq-proof-assume2}. We have
\begin{align}
\label{eq-proof-lower-up-1}
\forall c,c'\in[q],\quad
\left( 1 - \frac{2q\epsilon}{1+q\epsilon} \right)^h \leq \frac{\mu^{\Lambda \gets \tau,v \gets c}_{H,\mathcal{I}}(\rho)}{\mu^{\Lambda \gets \sigma,v \gets c'}_{H,\mathcal{I}}(\rho)} \leq \left( 1 + \frac{2q\epsilon}{1-q\epsilon} \right)^h,
\end{align}
and
\begin{align}
\label{eq-proof-lower-up-2}
\forall c,c'\in[q],\quad
\left( 1 - \frac{2q\epsilon}{1+q\epsilon} \right)^h \leq \frac{\mu^{\Lambda \gets \sigma,v \gets c}_{H,\mathcal{I}}(\rho)}{\mu^{\Lambda \gets \tau,v \gets c'}_{H,\mathcal{I}}(\rho)} \leq \left( 1 + \frac{2q\epsilon}{1-q\epsilon} \right)^h.
\end{align}
Note that
\begin{align*}
\mu^{\sigma}_{H,\mathcal{I}}(\rho) = \sum_{c \in [q]} \mu^{\sigma}_{v,\mathcal{I}}(c) \mu^{\Lambda \gets \sigma,v \gets c}_{H,\mathcal{I}}(\rho),\quad \mu^{\tau}_{H,\mathcal{I}}(\rho) = \sum_{c \in [q]} \mu^{\tau}_{v,\mathcal{I}}(c) \mu^{\Lambda \gets \tau,v \gets c}_{H,\mathcal{I}}(\rho).
\end{align*}
$\mu^{\sigma}_{H,\mathcal{I}}(\rho)$ is a convex combination of $\mu^{\Lambda \gets \sigma,v \gets c}_{H,\mathcal{I}}(\rho)$, and $\mu^{\tau}_{H,\mathcal{I}}(\rho)$ is a convex combination of $\mu^{\Lambda \gets \tau,v \gets c}_{H,\mathcal{I}}(\rho)$. By~\eqref{eq-proof-lower-up-1} and~\eqref{eq-proof-lower-up-2}, it holds that
\begin{align*}
\left( 1 - \frac{2q\epsilon}{1+q\epsilon} \right)^{2h } \leq \left(\frac{\mu_{H,\mathcal{I}}^{\sigma}(\rho) }{\mu_{H,\mathcal{I}}^{ \tau}(\rho) }\right) \left( \frac{\mu^{\Lambda \gets \tau,v \gets a}_{H,\mathcal{I}}(\rho)}{\mu^{\Lambda \gets \sigma,v \gets a}_{H,\mathcal{I}}(\rho)} \right) \leq \left( 1 + \frac{2q\epsilon}{1-q\epsilon} \right)^{2h }.
\end{align*}
Note that $h = |H|$ and $H \mathsf{Sub}seteq S_{\lfloor \ell /2 \rfloor}(v)$, then $m = |S_{\lfloor \ell /2 \rfloor}(v)| \geq h$.
This proves the claim.
\end{proof}
\mathsf{Sub}section{Spin systems on general graphs}
In this section, we prove \Cref{theorem-general} by showing that \Cref{condition-lower-bound} implies \Cref{condition-SSM-ratio}.
\begin{proof}[Proof of \Cref{theorem-general}]
Fix a instance $\mathcal{I} = (G,[q], \boldsymbol{A},\boldsymbol{b}) \in \mathfrak{I}$
satisfying \Cref{condition-lower-bound} with parameter $\ell = \ell(q) \geq 2$.
Fix subset $\Lambda \mathsf{Sub}seteq V$ and vertex $v \in V \setminus \Lambda$.
For any two partial configurations $\sigma, \tau \in [q]^\Lambda$ satisfying $\min\{\mathrm{dist}_G(v, u) \mid u \in \Lambda,\ \sigma(u)\neq\tau(u)\} =\ell \geq 2$, we claim
\begin{align}
\label{eq-claim-application}
\forall a \in [q], \quad \mu^{\sigma}_{v,\mathcal{I}}(a) = 0 \quad \Longleftrightarrow \quad \mu^{\tau}_{v,\mathcal{I}}(a) = 0.
\end{align}
Let $D \triangleq \{u \in \Lambda \mid \sigma(u) \neq \tau(u) \}$, $H\triangleq \Lambda \setminus D$ and $\rho \triangleq \sigma_H = \tau_H$.
Since $\ell \geq 2$, $\Gamma_G(v) \cap \Lambda = \Gamma_G(v) \cap H$, where $\Gamma_G(v)$ is the neighborhood of $v$ in $G$.
Since $\mathcal{I}$ is a permissive spin system (\Cref{definition-locally-admissible}), $\mu^{\sigma}_{v,\mathcal{I}}(a) = 0$ if and only if $b_v(a)\prod_{u \in \Gamma_G(v) \cap H}A_{uv}(a,\rho_u) = 0$; similarly, $\mu^{\tau}_{v,\mathcal{I}}(a) = 0$ if and only if $b_v(a)\prod_{u \in \Gamma_G(v) \cap H}A_{uv}(a,\rho_u) = 0$. This proves~\eqref{eq-claim-application}.
If $\mu^{\sigma}_{v,\mathcal{I}}(a) = \mu^{\tau}_{v,\mathcal{I}}(a) = 0$, then~\eqref{eq:condition-SSM} holds trivially. Otherwise, by \Cref{condition-lower-bound}, $\mu^{\sigma}_{v,\mathcal{I}}(a) \geq \gamma$ and $\mu^{\tau}_{v,\mathcal{I}}(a) \geq \gamma$, where $\gamma = \gamma(\Lambda, v)> 0$ is positive and depends only on $\Lambda$ and $v$.
By~\eqref{eq:condition-stronger-lower} and~\eqref{eq:condition-stronger-SSM-general}, we have
\begin{align*}
\left\vert \frac{\mu_{v, \mathcal{I}}^\sigma(a)}{ \mu_{v,\mathcal{I}}^\tau(a) } - 1 \right\vert \leq \frac{\gamma+ \mathcal{D}TV{\mu^{\sigma}_{v,\mathcal{I}}}{\mu^{\tau}_{v,\mathcal{I}}} }{\gamma} - 1 \leq \frac{1}{5|S_{\ell}(v)|}.
\end{align*}
This implies that any instance $\mathcal{I} = (G,[q], \boldsymbol{A},\boldsymbol{b}) \in \mathfrak{I}$ satisfies \Cref{condition-SSM-ratio} with parameter $\ell=\ell(q) \geq 2$.
\Cref{theorem-general} is a corollary of \Cref{theorem-general-ratio}.
\end{proof}
\mathsf{Sub}section{Uniform list coloring}
\label{section-list-coloring}
We now prove \Cref{corollary-main-coloring}.
Let $\mathfrak{L}$ be a class of list coloring instances with at most $q$ colors for a finite $q>0$.
Let $\alpha^* \approx 1.763\ldots$ be the positive root of the equation $x^x =\mathrm{e}$.
Suppose there exist $\alpha > \alpha^*$ and $\beta \geq \frac{\sqrt{2}}{\sqrt{2}-1}$ satisfying $(1-1/\beta) \alpha \mathrm{e}^{\frac{1}{\alpha}(1-1/\beta)} > 1$ such that for all $\mathcal{I}=(G=(V,E),[q],\+L) \in \mathfrak{L}$, the graph $G$ is triangle-free and
\begin{align*}
\forall v \in V, \quad |L(v)| \geq \alpha \deg_G(v) + \beta.
\end{align*}
Gamarnik, Katz, and Misra~\cite{GKM15} proved that $\mathfrak{L}$ exhibits the strong spatial mixing with exponential decay. If $\mathfrak{L}$ is defined on sub-exponential neighborhood growth graphs, then by \Cref{theorem-sub-graph}, the linear time perfect sampler exists for every instance in $\mathfrak{L}$.
There are two remaining cases in \Cref{corollary-main-coloring}.
We now assume that the class $\mathfrak{L}$ of list coloring instances satisfies one of the following two conditions.
\begin{enumerate}[label=(\mathcal{R}oman*)]
\item \label{condition-color-1}
there is an $s:\mathbb{N}\to \mathbb{N}$ with $s(\ell) = \exp(o(\ell))$ such that
for any $\mathcal{I}=(G = (V,E),[q],\+L) \in \mathfrak{L}$,
\begin{align*}
\forall v \in V, \ell \geq 0, \quad & |S_{\ell}(v)|\leq s(\ell), \\
\forall v \in V, \quad &|L(v)| \geq 2\deg_G(v);
\end{align*}
\item \label{condition-color-2} for any $\mathcal{I}=(G=(V,E),[q],\+L) \in \mathfrak{L}$,
\begin{align*}
\forall v \in V, \quad |L(v)| \geq \mathcal{D}elta^2 - \mathcal{D}elta + 2.
\end{align*}
\end{enumerate}
\begin{lemma}
\label{lemma-coloring-to-real}
Let $\mathfrak{L}$ be a class of list coloring instances with at most $q$ colors for a finite $q>0$. Suppose $\mathfrak{L}$ satisfies \ref{condition-color-1} or \ref{condition-color-2}. There exist finite $A > 0$ and $\theta > 0$ such that for every $\mathcal{I} =(G,[q],\mathcal{L}) \in \mathfrak{L}$, where $G=(V,E)$,
for any $v \in V$, any $\Lambda \mathsf{Sub}seteq V$, and any $\sigma, \tau \in [q]^\Lambda$
with $\ell \triangleq \min\{\mathrm{dist}_G(v, u) \mid u \in \Lambda,\ \sigma(u)\neq\tau(u)\} = \Omega(q\log q)$, it holds that
\begin{align*}
\forall a \in [q]: \quad \left\vert \frac{\mu_{v, \mathcal{I}}^\sigma(a)}{ \mu_{v,\mathcal{I}}^\tau(a) } - 1 \right\vert \leq \frac{ A \mathrm{e}^{-\theta \ell } }{\left\vert S_{\ell}(v) \right\vert}
\quad(\text{with the convention $0/0 = 1$}),
\end{align*}
where $A = A(q, s) > 0$ and $\theta= \frac{1}{2q}> 0$ if $\mathfrak{L}$ satisfies~\ref{condition-color-1} with the function $s: \mathbb{N}\rightarrow\mathbb{N}$;
or $A = \mathrm{poly}(q)$ and $\theta = \frac{1}{2q^2} > 0$ if $\mathfrak{L}$ satisfies~\ref{condition-color-2}.
\end{lemma}
\Cref{theorem-general-ratio} together with Lemma~\ref{lemma-coloring-to-real} proves the remaining two cases in \Cref{corollary-main-coloring}.
We take a sufficiently large $\ell^*$ such that $\ell^* = \Omega(q \log q)$ and $A\mathrm{e}^{-\theta \ell^*} \leq \frac{1}{5}$.
By Lemma~\ref{lemma-coloring-to-real}, instances of $\mathfrak{L}$ satisfy Condition~\ref{condition-SSM-ratio} with this $\ell^* \geq 2$.
Thus the perfect sampler exists due to \Cref{theorem-general-ratio}.
Note that $\ell^*$ depends only on $q$ and the function $s$, and for any instance $\mathcal{I} \in \mathfrak{L}$, the maximum degree $\mathcal{D}elta \leq q$.
Thus, the expected running time of our algorithm is $n \cdot q^{O(q^{\ell^*})} = O(n)$.
Furthermore, if $\mathfrak{L}$ satisfies~\ref{condition-color-2}, then $\ell^* = \Theta(q^2\log q)$, thus the expected running time is $n \cdot \exp(\exp(\mathrm{poly}(q)))$.
\mathsf{Sub}subsection{The multiplicative SSM of list coloring (proof of \texorpdfstring{\Cref{lemma-coloring-to-real}}{\texttwoinferior})}
In \cite[Theorem~3]{GKM15}, Gamarnik, Katz, and Misra established the best known strong spatial mixing result for list colorings in bounded degree graphs.
This is almost what we need, except that we want to control the decay rate under conditions~\ref{condition-color-1} and ~\ref{condition-color-2}.
Going through the proof of \cite[Theorem~3]{GKM15} and keeping track of the decay rate,
we have the proposition below.
The similar analysis technique are also used in~\cite{liu2019deterministic}.
\begin{proposition}[\cite{GKM15}]
\label{proposition-single-instance}
Let $\mathcal{I}=(G, [q], \mathcal{L})$ be a list coloring instance, where $G=(V, E)$. Assume that $\mathcal{I}$ satisfies $|L(v)| \geq \deg_G(v)+ 1$ for all $v \in V$. Suppose
\begin{align*}
\max_{u \in V} \frac{\deg_G(u) - 1 }{|L(u)| - \deg_G(u)} \leq \chi < 1.
\end{align*}
Then for any $\Lambda \mathsf{Sub}seteq V$, any vertex $v \in V \setminus \Lambda$, and any two partial colorings $\sigma, \tau \in [q]^\Lambda$
satisfying $\ell \triangleq \min\{\mathrm{dist}_G(v, u) \mid u \in \Lambda,\ \sigma(u)\neq\tau(u)\} = \Omega(\frac{\log q}{\log(1/\chi)})$, it holds that
\begin{align*}
\forall a \in [q]:\quad
\left\vert \frac{\mu^\sigma_{v,\mathcal{I}}(a)}{\mu^\tau_{v, \mathcal{I}}(a)} - 1 \right\vert \leq B\chi^{\ell}, \quad\text{(with convention $0/0 = 1$)}
\end{align*}
where $B = \mathrm{poly}(q/\chi)$ depends only on $q$ and $\chi$.
\end{proposition}
\begin{proof}[Proof of Lemma~\ref{lemma-coloring-to-real}]
Fix a instance $\mathcal{I} = (G, [q],\mathcal{L}) \in \mathfrak{L}$, where $G=(V,E)$.
Suppose $\mathfrak{L}$ satisfies Condition in~\ref{condition-color-1}.
We have
\begin{align*}
\max_{u \in V} \frac{\deg_G(u) - 1 }{|L(u)| - \deg_G(u)} \leq \max_{u \in V} \frac{\deg_G(u) - 1 }{ \deg_G(u)} = \frac{\mathcal{D}elta-1}{\mathcal{D}elta} \leq \frac{q-1}{q}.
\end{align*}
The $\chi$ and $B$ in \Cref{proposition-single-instance} can be set as $\chi = \frac{q-1}{q}$ and $B = \mathrm{poly}(q/\chi) \leq B_{\max} = \mathrm{poly}(q)$.
Then
for any subset $\Lambda \mathsf{Sub}seteq V$, any vertex $v \in V \setminus \Lambda$, any two colorings $\sigma,\tau \in [q]^\Lambda$ that disagree on $D \mathsf{Sub}seteq \Lambda$ satisfying $\ell \triangleq \min\{\mathrm{dist}_G(u,v)\mid u \in D\} = \Omega(\frac{\log q}{\log 1/\chi}) = \Omega(q\log q)$, it holds that
\begin{align*}
\forall a \in [q]:\quad
\left\vert \frac{\mu^\sigma_{v,\mathcal{I}}(a)}{\mu_{v,\mathcal{I}}^\tau(a)}-1 \right\vert
\leq B_{\max}\chi^{\ell}
\leq B_{\max}\cdot\frac{ \left\vert S_\ell(v)\right\vert}{\left\vert S_\ell(v)\right\vert}\cdot\chi^{\ell}.
\end{align*}
Since $G$ has sub-exponential growth,
we have that $\left\vert S_\ell(v)\right\vert \leq s(\ell) = \exp(o(\ell))$.
Thus,
\begin{align*}
\forall a \in [q]:\quad
\left\vert \frac{\mu^\sigma_{v,\mathcal{I}}(a)}{\mu_{v,\mathcal{I}}^\tau(a)}-1 \right\vert
\leq B_{\max}\cdot\frac{s(\ell) }{\left\vert S_\ell(v)\right\vert}\cdot\left(\frac{q-1}{q} \right)^{\ell}
\leq \frac{s(\ell)B_{\max} }{\left\vert S_\ell(v)\right\vert}\cdot\mathrm{e}^{-\ell/q}
\leq \frac{ A\mathrm{e}^{-\theta \ell}}{\left\vert S_\ell(v)\right\vert},
\end{align*}
for some $A=A(q, s) > 0$ and $\theta =\frac{1}{2q} > 0$.
Suppose $\mathfrak{L}$ satisfies~\ref{condition-color-2}.
Recall that $\mathcal{D}elta$ is the maximum degree of graph $G$.
we have
\begin{align*}
\max_{u \in V} \frac{\deg_G(u) - 1 }{|L(u)| - \deg_G(u)} \leq \frac{\mathcal{D}elta-1}{(\mathcal{D}elta-1)^2 + 1}.
\end{align*}
The $\chi$ and $B$ in \Cref{proposition-single-instance} can be set as $\chi = \frac{\mathcal{D}elta-1}{(\mathcal{D}elta-1)^2 + 1}$ and $B = \mathrm{poly}(q/\chi)$. Thus $1/\chi \leq \mathcal{D}elta^2 \leq q^2$. We have $B = \mathrm{poly}(q/\chi) \leq B_{\max}=\mathrm{poly}(q)$.
For any subset $\Lambda \mathsf{Sub}seteq V$, any vertex $v \in V \setminus \Lambda$, any two colorings $\sigma,\tau \in [q]^\Lambda$ that disagree on $D \mathsf{Sub}seteq \Lambda$ satisfying $\ell \triangleq \min\{\mathrm{dist}_G(u,v)\mid u \in D\} = \Omega(\frac{\log q}{\log 1/\chi}) = \Omega(\log q)$, it holds that
\begin{align*}
\forall a \in [q]:\quad
\left\vert \frac{\mu^\sigma_{v,\mathcal{I}}(a)}{\mu_{v,\mathcal{I}}^\tau(a)}-1 \right\vert
\leq B_{\max}\chi^{\ell}
\leq B_{\max}\cdot\frac{ \mathcal{D}elta (\mathcal{D}elta - 1)^{\ell-1}}{\left\vert S_\ell(v)\right\vert}\cdot\chi ^{\ell},
\end{align*}
where the last inequality due to $|S_{\ell}(v)| \leq \mathcal{D}elta (\mathcal{D}elta - 1)^{\ell-1}$.
Since $\chi = \frac{\mathcal{D}elta-1}{(\mathcal{D}elta-1)^2 + 1}$, we have
\begin{align*}
\forall a \in [q]:\quad
\left\vert \frac{\mu^\sigma_{v,\mathcal{I}}(a)}{\mu_{v,\mathcal{I}}^\tau(a)}-1 \right\vert
&\leq \frac{B_{\max}\mathcal{D}elta}{\mathcal{D}elta-1}\cdot\frac{1}{\left\vert S_\ell(v)\right\vert} \cdot\left(\frac{(\mathcal{D}elta-1)^2}{(\mathcal{D}elta-1)^2 + 1} \right)^{\ell}\\
(\text{by } \mathcal{D}elta \leq q)\quad&\leq\frac{2B_{\max}}{\left\vert S_\ell(v)\right\vert} \cdot\left(\frac{(q-1)^2}{(q-1)^2 + 1} \right)^{\ell}\\
&\leq \frac{2B_{\max}}{\left\vert S_\ell(v)\right\vert} \cdot \mathrm{e}^{-\frac{\ell}{2q^2}} = \frac{A\mathrm{e}^{-\theta \ell}}{\left\vert S_\ell(v)\right\vert},
\end{align*}
where $A = 2B_{\max} = \mathrm{poly}(q)$ and $\theta = \frac{1}{2q^2} > 0$.
\end{proof}
\mathsf{Sub}section{The monomer-dimer model}
We now prove \Cref{corollary-main-matching}.
We first present the monomer-dimer model instance as a spin system instance, then we use \Cref{theorem-sub-graph} to prove~\Cref{corollary-main-matching}.
Given a graph $G = (V, E)$, we use $G^{*}=(V^*,E^*) = \mathrm{Lin}(G)$ to denote the \emph{line graph} of $G$. Each vertex $v_e \in V^*$ in line graph $G^*$ represents an edge $e \in E$ in the original graph $G$, and two vertices $v_e,v_{e'}$ in $G^*$ are adjacent if and only if $e$ and $e'$ share a vertex in $G$. We call $S \mathsf{Sub}seteq V^*$ an \emph{independent set} in $G^*$ if no two vertices in $S$ are adjacent in $G^*$. It is easy to verify that there is a one-to-one correspondence between the matchings in $G$ and the independent sets in $G^*$.
Given a monomer-dimer model instance $\mathcal{I} = (G, \lambda)$, we define a \emph{hardcore model} instance $\mathcal{I}^*=(G^*,\lambda)$ in the line graph $G^* = \mathrm{Lin}(G)$. Each independent set $S$ in $G^*$ is assigned a weight $w_{\mathcal{I}^*}(S) = \lambda^{|S|}$. Let $\mu_{\mathcal{I}^*}$ be a distribution over all independent sets in $G^*$ such that $\mu_{\mathcal{I}^*}(S) \propto w_{\mathcal{I}^*}(S)$. Hence, $\mathcal{I}^*$ is a spin system instance and $\mathcal{I}^*$ is permissive.
Besides, if we can sample independent sets from $\mu_{\mathcal{I}^*}$, then we can sample matchings from $\mu_{\mathcal{I}}$.
Suppose the class of monomer-dimer model instances $\mathfrak{M}$ satisfies the condition in \Cref{corollary-main-matching}.
Then, there exist a constant $C$ and a function $s:\mathbb{N}\to \mathbb{N}$ with $s(\ell) = \exp(o(\ell))$ such that for all $\mathcal{I} = (G,\lambda) \in \mathfrak{M}$, $\lambda \leq C = O(1)$, $|S_\ell(v)| \leq s(\ell) = \exp(o(\ell))$ for all $v \in V$ and $\ell \geq 0$, and $\mathcal{D}elta_G \leq s(1) = O(1)$.
Thus, $\mathfrak{M}$ exhibits strong spatial mixing with exponential decay with constants $\alpha = \alpha(C,s)>0$ and $\beta = \beta(C,s)>0$~\cite{bayati2007simple,song2016counting}.
Observe that if $e_1,e_2,\ldots,e_\ell$ is a path of edges in $G$, then $v_{e_1},v_{e_2},\ldots,v_{e_\ell}$ is a path of vertices in $G^*$, and vice versa.
Hence, the following results hold for the class of hardcore instances $\mathfrak{H} = \{\mathcal{I}^*=(G^*,\lambda) \mid \mathcal{I} \in \mathfrak{M}\}$.
\begin{itemize}
\item The class of hardcore instances $\mathfrak{H}$ exhibits strong spatial mixing with exponential decay with constants $\alpha' = \alpha'(C,s)>0$ and $\beta = \beta(C,s)>0$.
\item for any instance $(G^*,\lambda) \in \mathfrak{H}$, the graph $G^*$ has sub-exponential growth. Suppose $G^* = (V^*,E^*)$ is the line graph of $G=(V,E)$. For all $ e = \{u,v\} \in E$, $\ell \geq 1$, it holds that $|S^*_{\ell}(v_e)| \leq \mathcal{D}elta_G(|S_{\ell - 1}(u)| + |S_{\ell - 1}(v)|) \leq 2s(1)s(\ell - 1) = \exp(o(\ell))$, where $v_e \in V^*$ represents the edge $e$, $S^*_{\ell}(v_e)$ is the sphere of radius $\ell$ centered at $v_e$ in $G^*$, $S_{\ell-1}(v)$ is the sphere of radius $\ell-1$ centered at $v$ in $G$.
\end{itemize}
Note that the number of vertices in $G^*$ is at most $n\mathcal{D}elta_G = O(n)$, where $n$ is number of vertices in $G$.
\Cref{corollary-main-matching} is a corollary of \Cref{theorem-sub-graph}.
\section{Dynamic sampling}
\label{section-dynamic}
In this section, we use our algorithm to solve the dynamic sampling problem~\cite{FVY19,feng2019dynamicMCMC}.
In this problem, the Gibbs distribution itself changes dynamically and the algorithm needs to maintain a random sample efficiently with respect to the current Gibbs distribution.
We first define the update for the spin system instance.
Let $\mathcal{I}=(G,[q],\boldsymbol{b},\boldsymbol{A})$ be a spin system instance, where $G = (V, E)$.
\begin{itemize}
\item updates for vertices: modifying the vector $b_v$ of vertex $v \in V$;
\item updates for edges: modifying the matrix $A_e$ of edge $e \in E$; or adding new edge $e \notin E$.
\end{itemize}
We use $(D_V, D_E, \+C)$ to denote the update for instance $\mathcal{I}$, where $D_V \mathsf{Sub}seteq V$, $D_E \mathsf{Sub}seteq \{\{u,v\}\mid u,v \in V, u\neq v\}$, and $\+C = (b_v)_{v \in D_V} \cup (A_e)_{e \in D_E}$. For each $v \in D_V$, we modify its vector to $b_v \in \+C$, and for each $e \in D_E$, we either add the new edge $e$ with matrix $A_e \in \+C$ (if $e \notin E$), or modify its matrix to $A_e \in \+C$ (if $e \in E$).
\begin{definition}[dynamic sampling problem]
\label{definition-dynamic}
Given a spin system instance $\mathcal{I}=(G,[q],\boldsymbol{b},\boldsymbol{A})$ where $G=(V,E)$, a random sample $\boldsymbol{X} \in [q]^V$ such that $\boldsymbol{X} \sim \mu_{\mathcal{I}}$, and an update $(D_V, D_E, \+C)$ that modifies the instance $\mathcal{I}$ to an updated instance $\mathcal{I}'= (G',[q],\boldsymbol{b}',\boldsymbol{A}')$ where $G'=(V, E')$, the algorithm updates $\boldsymbol{X}$ to a new sample $\boldsymbol{X}' \in [q]^V$ such that $\boldsymbol{X}' \sim \mu_{\mathcal{I}'}$.
\end{definition}
\begin{theorem}
\label{theorem-dynamic}
Let $\mathfrak{I}$ be a class of permissive spin systems satisfying Condition~\ref{condition-SSM-ratio}.
There exists an algorithm such that if the updated instance $\mathcal{I}'= (G',[q],\boldsymbol{b}',\boldsymbol{A}') \in \mathfrak{I}$, then the algorithm solves the dynamic sampling problem within $\mathcal{D}elta(|D_V| + |D_E|) q^{O(\mathcal{D}elta^{\ell})}$ time in expectation,
where $\mathcal{D}elta$ is the maximum degree of $G'$ and $\ell = \ell(q) \geq 2$ is determined by \Cref{condition-SSM-ratio}.
\end{theorem}
Suppose $q,\mathcal{D}elta,\ell = O(1)$. By \Cref{theorem-dynamic},
the running time of our algorithm is linear in the size of the update.
Hence, the efficient dynamic sampling algorithm exists if strong spatial mixing holds with a rate faster than the neighborhood growth.
The relation between the spatial mixing property and the static sampling is well studied, we extend such relation further to the dynamic setting.
The dynamic sampling algorithm is given in~\Cref{dynamic-perfect-sampler}.
\begin{algorithm}[ht]
\SetKwInOut{Input}{Input}
\mathcal{I}nput{a spin system instance $\mathcal{I}=(G=(V,E), [q], \boldsymbol{b}, \boldsymbol{A})$, a random sample $\boldsymbol{X} \sim \mu_{\mathcal{I}}$, and an update $(D_V, D_E, \+C)$ that modifies $\mathcal{I}$ to $\mathcal{I}' = (G'=(V,E'), [q], \boldsymbol{b}', \boldsymbol{A}')$.}
$\+D \gets D_V \cup \left( \bigcup_{e \in D_E} e\right)$ and $\partial \+D \gets \{v \in V \setminus \+D \mid \exists u \in \+D \text{ s.t. } \{u,v\}\in E' \}$\;
based on $X_{\partial \mathcal{D}}$, modify the partial configuration $X_{\mathcal{D}}$ so that $w_{\mathcal{I}'}(\boldsymbol{X}) > 0$\label{line-fix}\;
$\mathcal{R} \gets \+D \cup \partial \+D$ \;
\While{$\mathcal{R} \neq \emptyset$}{
$(\boldsymbol{X}, \mathcal{R}) \gets \mathcal{R}eSample(\mathcal{I}', \boldsymbol{X}, \mathcal{R})$\label{line-dynamic-resample}\;
}
\mathcal{R}eturn{$\boldsymbol{X}$\;}
\caption{Dynamic perfect Gibbs sampler}\label{dynamic-perfect-sampler}
\end{algorithm}
\begin{algorithm}[htbp]
\SetKwInOut{Input}{Input}
\SetKwIF{withprob}{}{}{with probability}{do}{}{}{}
\mathcal{I}nput{a spin system instance $\mathcal{I}=(G=(V,E), [q], \boldsymbol{b}, \boldsymbol{A})$, a configuration $\boldsymbol{X} \in [q]^V$, a non-empty subset $\+R \mathsf{Sub}seteq V$, and an integer parameter $\ell \geq 0$;}
pick a $u\in\mathcal{R}$ uniformly at random and let $B\gets (B_{\ell}(u)\setminus \mathcal{R})\cup\{u\}$\;
let $\mu_{\min}$ be the minimum value of $\mu_{u,\mathcal{I}}^{\sigma}(X_u)$ over all $\sigma\in[q]^{\partial B}$ that $\sigma_{\mathcal{R}\cap\partial B}=X_{\mathcal{R}\cap\partial B}$\;
\withprob{$\frac{\mu_{\min}}{\mu_{u,\mathcal{I}}^{X_{\partial B}}(X_u)}$}{
update $\boldsymbol{X}$ by redrawing $X_B\sim\mu_{B,\mathcal{I}}^{X_{\partial B}}$\;
$\mathcal{R} \gets\mathcal{R} \setminus \{u\}$\;
}
\Else{
$\mathcal{R} \gets\mathcal{R} \cup \partial B$\;
}
\mathcal{R}eturn{$(\boldsymbol{X},\mathcal{R})$}
\caption{$\mathcal{R}eSample(\mathcal{I}, \boldsymbol{X}, \mathcal{R})$}\label{alg:dynamic-resample}
\end{algorithm}
In \Cref{dynamic-perfect-sampler},
the set $\+D \mathsf{Sub}seteq V$ contains all the vertices incident to the update.
Note that the input $\boldsymbol{X}$ can be an infeasible configuration for $\mathcal{I}'$, i.e.~$w_{\mathcal{I}'}(\boldsymbol{X}) = 0$, because the configuration $X_{\mathcal{D}}$ may violate the new constraints in $\+C$. Hence, in Line~\ref{line-fix}, we modify the configuration $X_{\+D}$ so that $w_{\mathcal{I}'}(\boldsymbol{X}) > 0$. Given the $X_{\partial \+D}$, this step can be achieved by a simple greedy algorithm since $\mathcal{I}'$ is permissive. Then, we construct the initial $\+R$ as $\+D \cup \partial \+D$. In Line~\ref{line-dynamic-resample}, we call the subroutine $\mathcal{R}eSample$ on the updated instance $\mathcal{I}'$.
Note that $\+R = \+D \cup \partial\+D$ and $\partial \+D$ separates $\+D$ from $V \setminus \mathcal{R}$ in both $G$ and $G'$.
In Line~\ref{line-fix}, we only modify the partial configuration $X_{\+D}$.
Such modification only reviews the information of $\boldsymbol{X}$ in $\+D \cup \partial \+D$.
Thus, after the modification, the $X_{V \setminus \+R}$ follows the distribution $\mu_{\mathcal{I}}^{X_{\+R}} = \mu_{V \setminus \+R,\mathcal{I}}^{X_{\partial \+D}}$ due to the conditional independence property. Since two instances $\mathcal{I}$ and $\mathcal{I}'$ differ only at the subset $\+D$, due to the conditional independence property, two distributions $\mu_{\mathcal{I}}^{X_{\+R}}$ and $\mu_{\mathcal{I}'}^{X_{\+R}}$ are identical. Thus, the initial $\boldsymbol{X}, \+R$ satisfies \Cref{condition-invariant} with respect to $\mathcal{I}'$, and $\boldsymbol{X}$ is a feasible configuration for $\mathcal{I}'$. In each iteration of the \textbf{while}{} loop, we call the subroutine $\mathcal{R}eSample$ on $\mathcal{I}'$.
By the identical proof in \Cref{section-correctness}, the output $\boldsymbol{X} \sim \mu_{\mathcal{I}'}$.
Let $\mathcal{D}elta$ denote the maximum degree of graph $G'$.
Note that $|\+D| = O(|D_V| + |D_E|)$. The time complexity of the first three lines of \Cref{dynamic-perfect-sampler} is $O(\mathcal{D}elta |\+D|)$. Note that the size of the initial $\+R$ is $O(\mathcal{D}elta |\+D|)$. The efficiency result in \Cref{theorem-dynamic} follows from the identical proof in \Cref{section-running-time}.
\end{document} | math | 117,759 |
\begin{document}
\title{The Stochastic Wave Equation
with Fractional Noise: a random field approach}
\author{Raluca M. Balan\footnote{Corresponding author. Department of Mathematics and Statistics, University of Ottawa,
585 King Edward Avenue, Ottawa, ON, K1N 6N5, Canada. E-mail address:
[email protected]} \ \footnote{Research supported by a grant from
the Natural Sciences and Engineering Research Council of Canada.} \
and Ciprian A Tudor \footnote{Laboratoire Paul Painlev\'e,
Universit\'e de Lille 1,
F-59655 Villeneuve d'Ascq, France. Email address: [email protected]. Associate member: SAMOS/MATISSE, Centre d'Economie de La
Sorbonne, Universit\'e de Panth\'eon-Sorbonne Paris 1, 90 rue de
Tolbiac, 75634 Paris Cedex 13, France. }}
\date{December 17, 2009}
\maketitle
\begin{abstract}
\noindent We consider the linear stochastic wave equation with
spatially homogenous Gaussian noise, which is fractional in time
with index $H>1/2$. We show that the necessary and sufficient
condition for the existence of the solution is a relaxation of the
condition obtained in \cite{dalang99}, when the noise is white in
time. Under this condition, we show that the solution is
$L^2(\Omega)$-continuous. Similar results are obtained for the heat
equation. Unlike the white noise case, the necessary and sufficient
condition for the existence of the solution in the case of the heat
equation is {\em different} (and more general) than the one obtained
for the wave equation.
\end{abstract}
{\em MSC 2000 subject classification:} Primary 60H15; secondary
60H05
{\em Keywords and phrases:} stochastic wave equation, random field
solution, spatially homogenous Gaussian noise, fractional Brownian
motion
\section{Introduction}
The random field approach to s.p.d.e.'s initiated in \cite{walsh86},
has become increasingly popular in the past few decades, as an
alternative to the semigroup approach developed in
\cite{daprato-zabczyk92}, or the analytic approach of
\cite{krylov99}.
Generally speaking, a random field solution of the (non-linear)
equation:
\begin{equation}
\label{spde}Lu(t,x)=\alpha(u(t,x))\dot W(t,x)+\beta(u(t,x)), \quad
t>0,x \in \mathbb{R}^d
\end{equation} (with vanishing initial conditions) is a
collection $\{u(t,x), t \geq 0, x \in \mathbb{R}^d\}$ of square integrable
random variables, which satisfy the following integral equation:
$$u(t,x)=\int_0^t
\int_{\mathbb{R}^d}G(t-s,x-y)\alpha(u(s,y))W(ds,dy)+\int_0^t
\int_{\mathbb{R}^d}G(t-s,x-y)\beta(u(s,y))dy ds,$$ provided that both
integrals above are well-defined (the first being a stochastic
integral). In this context, $L$ is a second-order partial
differential operator with constant coefficients, $G$ is the
fundamental solution of $Lu=0$, and $\dot W$ is a formal way of
denoting the random noise perturbing the equation.
When the equation is driven by a space-time white noise (i.e. a
Gaussian noise which has the covariance structure of a Brownian
motion in space-time), the random field solution exists only if the
spatial dimension is $d=1$. In this case, the stochastic integral
above is defined with respect to a martingale measure, and the
solution is well-understood for most operators $L$, in particular
for the heat and wave operators (see \cite{walsh86},
\cite{carmona-nualart88} or \cite{sanz-sole05}).
To obtain a random field solution in higher dimensions, one needs to
consider a different type of noise, which can be either Gaussian,
but with a spatially homogenous covariance structure given formally
by:
$$E[\dot W(t,x) \dot W(s,y)]=\delta(t-s)f(x-y),$$
or of Poisson type. Historically, the two approaches have been
initiated at about the same time (see \cite{mueller97},
\cite{dalang-frangos98}, \cite{millet-sanzsole99} for the wave
equation with Gaussian noise in dimension $d=2$, and
\cite{dalang-hou97}, \cite{saint-loubert98} for the Poisson case).
After the ingenious extension of the martingale measure stochastic
integral due to \cite{dalang99}, it became clear that the random
field approach can be pursued for the study of general s.p.d.e.'s
with spatially homogenous Gaussian noise. Since this extension
allows for integrands which are non-negative measures (in space),
the theory developed in \cite{dalang99} covers instantly the case of
the (non-linear) wave equation in dimensions $d \in \{1,2,3\}$, and
the case of the heat equation in any dimensions $d$. In the
non-linear case, the existence of the solution is obtained by a
Picard's iteration scheme, under the usual Lipschitz assumptions on
$\alpha, \beta$, and the following condition, linking the operator
$L$ and the spatial covariance function $f$:
\begin{equation}
\label{Dalang-cond}\int_{\mathbb{R}^d} \int_0^t |\mathcal{F} G(u,\cdot)(\xi)|^2 du
\mu(d\xi)<\infty. \end{equation} (Here $\mu$ is a non-negative
tempered measure, whose Fourier transform in $f$.)
Moreover, (\ref{Dalang-cond}) is the necessary any sufficient
condition for the stochastic integral $\int_0^t
\int_{\mathbb{R}^d}G(t-s,x-y)W(ds,dy)$ to be well-defined, and hence the
necessary any sufficient condition for the existence of the solution
in the linear case, when $\alpha \equiv 1$ and $\beta \equiv 0$.
Since for both heat and wave operators,
\begin{equation}
\label{estimates} c_t^{(1)}\frac{1}{1+|\xi|^2} \leq \int_0^t |\mathcal{F}
G(u,\cdot)(\xi)|^2 du \leq c_t^{(2)}\frac{1}{1+|\xi|^2}, \quad
\mbox{for all} \ \xi \in \mathbb{R}^d,
\end{equation} for some constants
$c_t^{(1)},c_t^{(2)}>0$, condition (\ref{Dalang-cond}) is equivalent
to:
$$\int_{\mathbb{R}^d} \frac{1}{1+|\xi|^2}\mu(d\xi)<\infty.$$
Subsequently, using the Malliavin calculus techniques, it was shown
that the random variable $u(t,x)$ has an absolutely continuous law
with respect to the Lebesgue measure on $\mathbb{R}$, and this density is
infinitely differentiable. These results are valid for the heat
equation in any dimension $d$, and for the wave equation in
dimension $d \in \{1,2,3\}$ (see \cite{QS-SanzSole04a},
\cite{QS-SanzSole04b}, \cite{sanz-sole05}), under the additional
assumption (which was removed in \cite{Nualart-QS07}):
\begin{equation}
\label{Dalang-cond-alpha} \int_{\mathbb{R}^d} \left(\frac{1}{1+|\xi|^2}
\right)^{\alpha}\mu(d\xi)<\infty, \quad \mbox{for some} \ \alpha \in
(0,1).
\end{equation}
Under (\ref{Dalang-cond-alpha}), one also obtains the H\"{o}lder
continuity of the solution for the heat equation in any dimension
$d$ and the wave equation in dimensions $d \in \{1,2,3\}$. This is
done using Kolmogorov's criterion and some estimates for the $p$-th
moments of the increments of the solution (see
\cite{sanzsole-sarra00}, \cite{sanzsole-sarra02},
\cite{dalang-sanzsole08}).
The case of the wave equation in dimension $d \geq 4$ was solved in
the recent article \cite{conus-dalang09}, using an extension of the
integral developed in \cite{dalang99}. The existence of a
random-field solution is obtained under condition
(\ref{Dalang-cond}). In the the affine case (i.e. $\alpha(u)=au+b,
a,b \in \mathbb{R}$ and $\beta \equiv 0$), and under the additional
assumption (\ref{Dalang-cond-alpha}), the solution is shown to be
H\"older continuous.
In parallel with these developments, a new process began to be used
intensively in stochastic analysis: the {\em fractional Brownian
motion} (fBm) with index $H \in (0,1)$, a zero-mean Gaussian process
$(B_t)_{t \geq 0}$ with covariance:
$$R_{H}(t,s)=\frac{1}{2}(t^{2H}+s^{2H}-|t-s|^{2H}).$$
The case $H=1/2$ corresponds to the classical Brownian motion, while
the cases $H>1/2$ and $H<1/2$ have many contrasting properties. We
refer the reader to the survey article \cite{nualart03} and the
monographs \cite{BHOZ08} and \cite{mishura08} for more details. Most
importantly, in the case $H>1/2$,
\begin{equation}
\label{formula-RH} R_{H}(t,s)=\alpha_H\int_0^t
\int_0^s|u-v|^{2H-2}dudv,
\end{equation}
where $\alpha_H=H(2H-1)$. This shows that $(B_t)_{t \geq 0}$ has a
homogenous covariance structure, similar to the spatial structure of
the noise $\dot W$ considered above.
Returning to our discussion about s.p.d.e.'s with a Gaussian noise,
it seems natural to consider equation (\ref{spde}), when the
covariance of the noise $\dot W$ is given formally by:
\begin{equation}
\label{noise} E[\dot W(t,x) \dot W(s,y)]=\alpha_H|t-s|^{2H-2}f(x-y).
\end{equation}
However, this simple modification changes the problem drastically,
since unless $H=1/2$, the fBm is {\em not} a semimartingale, and
therefore the previous method, based on martingale measure
stochastic integrals, cannot be applied.
Several methods have been proposed for developing a stochastic
calculus with respect to fBm: (i) the Malliavin calculus (see
\cite{decreusefond-ustunel99}, \cite{AMN01}, \cite{alos-nualart03},
\cite{nualart06}), which exploits the fact that the fBm is Gaussian;
(ii) the method of generalized Lebesgue-Stieltjes integration (see
\cite{zahle98}), which uses the H\"older continuity of the fBm
trajectories; (iii) the rough path analysis (see \cite{lyons98},
\cite{lyons-qian02}), which uses the fact that the paths of the fBm
have bounded $p$-variation, for $p>1/H$; (iv) the stochastic
calculus via regularization based also in general on the properties
of the paths of the fBm (see \cite{GRV}).
These methods have been applied to s.p.d.e.'s (see
\cite{maslowski-nualart03}, \cite{nualart-vuillermont06},
\cite{sanzsole-vuillermont07}, \cite{GLT06}), \cite{QS-tindel07}),
but not using the random field approach
A notable exception is the heat equation. The linear equation with
noise (\ref{noise}) and $H>1/2$ was examined in
\cite{balan-tudor08}, for particular functions $f$ (e.g.
$f(x)=|x|^{-(d-\alpha)}$ with $\alpha \in (0,d)$). We also mention
the works \cite{EV} and \cite{TTV} for the case of the space
variable belonging to the unit circle. The quasi-linear equation
(i.e. $\alpha \equiv 0$) was treated in \cite{oksendal-zhang01}, and
the equation with multiplicative noise (i.e. $\alpha(u)=u, \beta
\equiv 0)$ was studied in \cite{hu01}; in these two references, the
covariance structure of the noise is a particular case of
(\ref{noise}): for $H,H_i>1/2$
$$E[\dot W(t,x) \dot
W(s,y)]=\alpha_H|t-s|^{2H-2}\prod_{i=1}^{d}(\alpha_{H_i}|x_i-y_i|^{2H_i-2}).$$
(This type of noise is called {\em fractional Brownian field}.)
The heat equation with multiplicative noise (\ref{noise}) was
studied in \cite{balan-tudor09} (for particular functions $f$ and
$H>1/2$) and \cite{hu-nualart09} (in the case $H \in (0,1)$ and
$f=\delta_0$). In the case when the spatial dimension is $d=1$, the
non-linear equation has been treated in \cite{QS-tindel07} using a
two-parameter Young integral based on the H\"older continuity of
fBm.
To the best of our knowledge, there is no study of the wave equation
driven by a noise $\dot W$, whose covariance is given by
(\ref{noise}). The goal of the present article is to start filling
this gap, by identifying the necessary and sufficient conditions for
the existence of a random field solution of the linear wave equation
with noise (\ref{noise}) and $H>1/2$. We also treat the heat
equation.
When $H>1/2$, it turns out that under relatively mild assumptions on
the fundamental solution $G$ of the operator $L$, the necessary and
sufficient condition for the existence of the random-field solution
of the linear equation $Lu=\dot W$ is:
\begin{equation}
\label{general-cond} \int_{\mathbb{R}^d}\int_0^t \int_0^t \mathcal{F}
G(u,\cdot)(\xi)
\overline{\mathcal{F} G(v,\cdot)(\xi)} |u-v|^{2H-2} dudv \mu(d\xi)<\infty,
\end{equation}
which is more general than (\ref{Dalang-cond}). Note that the
integrand of the $\mu(d\xi)$ integral in (\ref{general-cond}) is the
$\mathcal{H}(0,t)$-norm of the function $u \mapsto \mathcal{F} G(u,\cdot)(\xi)$.
Quite surprisingly, and in contrast with (\ref{estimates}), the
estimates that we obtain for this norm are {\em different} in the
case of the wave and heat operators: in the case of the wave
equation, (\ref{general-cond}) is equivalent to
\begin{equation}
\label{wave-cond} \int_{\mathbb{R}^d}
\left(\frac{1}{1+|\xi|^2}\right)^{H+1/2}\mu(d\xi)<\infty,
\end{equation}
whereas in the case of the heat equation, (\ref{general-cond}) is
equivalent to:
\begin{equation}
\label{heat-cond} \int_{\mathbb{R}^d}
\left(\frac{1}{1+|\xi|^2}\right)^{2H}\mu(d\xi)<\infty,
\end{equation}
The amazing fact is that for the wave operator, these estimates can
be deduced using only the estimates of the $L^2(0,t)$-norm (given by
(\ref{estimates})), the trick being to pass to the spectral
representation of the $\mathcal{H}(0,t)$-norm of $u \mapsto \mathcal{F}
G(u,\cdot)(\xi)$. In the case of the heat operator, there is no need
for this machinery, since $u \mapsto \mathcal{F} G(u,\cdot)(\xi)$ is a
non-negative function, and its $\mathcal{H}(0,t)$-norm can be bounded
directly by the $L^{1/H}(0,t)$-norm, which is easily computable.
This article is organized as follows. Section 2 contains some
preliminaries, and a basic result which ensures that under
(\ref{general-cond}), the stochastic integral of the fundamental
solution $G$ of the wave operator is well defined. In Section 3, we
show that the solution of the wave equation exists if and only if
(\ref{wave-cond}) holds (Theorem \ref{wave-th}). Moreover, the
solution is $L^2(\Omega)$-continuous. Similar results are obtained
in Section 4 for the heat equation, using (\ref{heat-cond}).
Appendix A contains some useful identities, which are needed in the
sequel. Appendix B gives the spectral representation of the
$\mathcal{H}(0,t)$-norm of the function $\sin$.
\section{The Basics}
We denote by $C_0^{\infty}(\mathbb{R}^{d+1})$ the space of infinitely
differentiable functions on $\mathbb{R}^{d+1}$ with compact support, and
$\mathcal{S}(\mathbb{R}^d)$ the Schwartz space of rapidly decreasing $C^{\infty}$
functions in $\mathbb{R}^d$. For $\varphi \in L^1(\mathbb{R}^d)$, we let $\mathcal{F}
\varphi$ be the Fourier transform of $\varphi$:
$$\mathcal{F} \varphi (\xi)=\int_{\mathbb{R}^d} e^{-i \xi \cdot x}\varphi (x)dx.$$
We begin by introducing the framework of \cite{dalang99}.
Let $\mu$ be a non-negative tempered measure on $\mathbb{R}^d$, i.e. a
non-negative measure which satisfies: $$\int_{\mathbb{R}^d}
\left(\frac{1}{1+|\xi|^2} \right)^l \mu(d\xi)<\infty, \quad
\mbox{for some} \ l >0.$$
Since the integrand is non-increasing in $l$, we may assume that $l
\geq 1$ is an integer. Note that $1+|\xi|^2$ behaves as a constant
around $0$, and as $|\xi|^2$ at $\infty$, and hence
(\ref{mu-tempered}) is equivalent to:
\begin{equation}
\label{mu-tempered} \int_{|\xi| \leq 1}\mu(d\xi)<\infty, \quad
\mbox{and} \quad \int_{|\xi| \geq 1}\frac{1}{|\xi|^{2l}}<\infty,
\quad
\mbox{for some integer} \ l \geq 1.
\end{equation}
Let $f: \mathbb{R}^d \to \mathbb{R}_{+}$ be the Fourier transform of $\mu$ in
$\mathcal{S}'(\mathbb{R}^d)$, i.e.
$$\int_{\mathbb{R}^d}f(x)\varphi(x)dx=\int_{\mathbb{R}^d}\mathcal{F}
\varphi(\xi)\mu(d\xi), \quad \forall \varphi \in \mathcal{S}(\mathbb{R}^d).$$
Simple properties of the Fourier transform show that for any
$\varphi, \psi \in \mathcal{S}(\mathbb{R}^d)$,
$$\int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \varphi(x)f(x-y)\psi(y)dx dy=
\int_{\mathbb{R}^d}\mathcal{F} \varphi(\xi) \overline{\mathcal{F} \psi(\xi)}\mu(d\xi).$$
An approximation argument shows that the previous equality also
holds for indicator functions $\varphi=1_{A},\psi=1_{B}$, with $A,B
\in \mathcal{B}_{b}(\mathbb{R}^d)$, where $\mathcal{B}_b(\mathbb{R}^d)$ is the class of bounded
Borel sets of $\mathbb{R}^d$:
\begin{equation}
\label{Fourier-indicator} \int_A \int_B f(x-y)dx dy=\int_{\mathbb{R}^d}\mathcal{F}
1_{A}(\xi) \overline{\mathcal{F} 1_{B}(\xi)} \mu(d\xi).
\end{equation}
As in \cite{balan-tudor08}, \cite{balan-tudor09}, on a complete
probability space $(\Omega,\mathcal{F},P)$, we consider a zero-mean Gaussian
process $W=\{W_t(A); t \geq 0, A \in \mathcal{B}_{b}(\mathbb{R}^d)\}$ with
covariance:
$$E(W_t(A)W_s(B))=R_{H}(t,s) \int_{A} \int_{B} f(x-y)dx dy=:
\langle 1_{[0,t] \times A}, 1_{[0,s] \times B} \rangle_{\mathcal{H} \mathcal{P}}.$$
Let $\mathcal{E}$ be the set of linear combinations of elementary functions
$1_{[0,t] \times A}$, $t \geq 0, A \in \mathcal{B}_b(\mathbb{R}^d)$, and $\mathcal{H} \mathcal{P}$
be the Hilbert space defined as the closure of $\mathcal{E}$ with respect to
the inner product $\langle \cdot , \cdot \rangle_{\mathcal{H} \mathcal{P}}$.
(Alternatively, $\mathcal{H} \mathcal{P}$ can be defined as the completion of
$C_0^{\infty}(\mathbb{R}^{d+1})$, with respect to the inner product
$\langle \cdot, \cdot \rangle_{\mathcal{H} \mathcal{P}}$.)
The map $1_{[0,t] \times A} \mapsto W_t(A)$ is an isometry between
$\mathcal{E}$ and the Gaussian space $H^{W}$ of $W$, which can be extended
to $\mathcal{H} \mathcal{P}$. We denote this extension by: $$\varphi \mapsto
W(\varphi)=\int_0^{\infty}\int_{\mathbb{R}^d} \varphi(t,x)W(dt,dx).$$
In the present work, we assume that $H>1/2$. Hence,
(\ref{formula-RH}) holds. From (\ref{Fourier-indicator}) and
(\ref{formula-RH}), it follows that for any $\varphi,\psi \in \mathcal{E}$,
\begin{eqnarray*}
\langle \varphi, \psi \rangle_{\mathcal{H} \mathcal{P}}&=& \alpha_H
\int_{0}^{\infty} \int_0^{\infty}\int_{\mathbb{R}^d}\int_{\mathbb{R}^d}
\varphi(u,x)\psi(v,y)f(x-y)|u-v|^{2H-2} dx
dy du dv \\
&=& \alpha_H \int_{0}^{\infty} \int_0^{\infty} \int_{\mathbb{R}^d} \mathcal{F}
\varphi(u,\cdot)(\xi)\overline{\mathcal{F}\psi(v,\cdot)(\xi)} |u-v|^{2H-2}
\mu(d\xi) du dv.
\end{eqnarray*}
Moreover, we can interchange the order of the integrals $dudv$ and
$\mu(d\xi)$, since for indicator functions $\varphi$ and $\psi$, the
integrand is a product of a function of $(u,v)$ and a function of
$\xi$. Hence, for $\varphi,\psi \in \mathcal{E}$, we have:
\begin{equation}
\label{norm-HP-2} \langle \varphi, \psi \rangle_{\mathcal{H} \mathcal{P}}= \alpha_H
\int_{\mathbb{R}^d} \int_{0}^{\infty} \int_0^{\infty} \mathcal{F}
\varphi(u,\cdot)(\xi)\overline{\mathcal{F}\psi(v,\cdot)(\xi)} |u-v|^{2H-2}
du dv \mu(d\xi).
\end{equation}
The space $\mathcal{H} \mathcal{P}$ may contain distributions, but contains the
space $|\mathcal{H} \mathcal{P}|$ of measurable functions $\varphi: \mathbb{R}_{+} \times
\mathbb{R}^d \to \mathbb{R}$ such that
$$\|\varphi \|_{|\mathcal{H} \mathcal{P}|}^2:=\alpha_H
\int_{0}^{\infty} \int_0^{\infty}\int_{\mathbb{R}^d}\int_{\mathbb{R}^d}
|\varphi(u,x)||\varphi(v,y)|f(x-y)|u-v|^{2H-2} dx dy du dv<\infty.$$
We recall now several facts related to the fBm (see e.g.
\cite{nualart03}).
Let $B=(B_t)_{t \geq 0}$ be a fBm of index $H>1/2$. For a fixed
$T>0$, let $\mathcal{H}(0,T)$ be the Hilbert space defined as the closure of
$\mathcal{E}(0,T)$ (the set of step functions on $[0,T]$), with respect to
the inner product:
$$\langle 1_{[0,t]}, 1_{[0,s]} \rangle_{\mathcal{H}(0,T)}=R_{H}(t,s).$$
One can prove that
$$R_{H}(t,s)=\int_{0}^{t \wedge s} K_{H}(t,r)K_H(s,r)dr,$$
where $K_H(t,r)= c_H^*\int_r^t (u-r)^{H-3/2} u^{H-1/2}du$ and
$c_H^*=\left(\frac{\alpha_H}{\beta(H-1/2,2-2H)}\right)^{1/2}$. (Here
$\beta$ denotes the Beta function.) Therefore, the map $K_H^*$
defined by:
$$(K_H^* 1_{[0,t]}) (s)=K_{H}(t,s)1_{[0,t]}(s)$$
is an isometry between $\mathcal{E}(0,T)$ and $L^2(0,T)$. This isometry can
be extended to $\mathcal{H}(0,T)$, and is denoted by $\phi \mapsto
B(\phi)=\int_0^T \phi(s)dB_s$.
The transfer operator $K_H^*$ can be expressed in terms of
fractional integrals, as follows: for any $\phi \in \mathcal{E}(0,T)$,
$$(K_H^* \phi)(s)=c_{H}^{*}\Gamma(H-1/2)
s^{1/2-H}I_{T-}^{H-1/2}(u^{H-1/2}\phi(u))(s),$$ where
$$I_{T-}^{\alpha}f (s)=
\frac{1}{\Gamma(\alpha)}\int_{s}^{T}(u-s)^{\alpha-1}f(u)du$$ denotes
the fractional integral of $f \in L^1(0,T)$, of order $\alpha \in
(0,1)$.
$K_H^*$ can be extended to complex-valued functions, as follows. Let
$\mathcal{E}_{\mathbb{C}}(0,T)$ be the set of all complex linear combinations of
functions $1_{[0,t]}, t \in [0,T]$, and $\mathcal{H}_{\mathbb{C}}(0,T)$ be the
closure of $\mathcal{E}_{\mathbb{C}}(0,T)$ with respect to the inner product:
$$\langle \varphi,\psi \rangle_{\mathcal{H}_{\mathbb{C}}(0,T)}=
\alpha_H \int_0^T \int_0^T \varphi(u)
\overline{\psi(v)}|u-v|^{2H-2}du dv.$$
The operator $K_H^*$ is an isometry which maps $\mathcal{H}_{\mathbb{C}}(0,T)$ onto
$L_{\mathbb{C}}^2(0,T)$ (the space of functions $\varphi:[0,T] \to \mathbb{C}$,
with $\int_0^T |\varphi(t)|^2 dt<\infty$): for any $\phi \in
\mathcal{H}_{\mathbb{C}}(0,T)$,
\begin{equation}
\label{isometry} \alpha_H \int_0^T \int_0^T \phi(u)
\overline{\phi(v)}|u-v|^{2H-2}du dv=d_{H} \int_0^T
|I_{T-}^{H-1/2}(u^{H-1/2}\phi(u))(s)|^2 \lambda_H(ds),
\end{equation} where $d_H=(c_{H}^{*})^2\Gamma(H-1/2)^2$ and
$\lambda_H(ds)=s^{1-2H}ds$.
Let $\mathcal{E}_T$ be the class of elementary functions on $[0,T] \times
\mathbb{R}^d$. Note that for any $\varphi \in \mathcal{E}_T$, the function $t
\mapsto \mathcal{F} \varphi (t,\cdot)(\xi)$ belongs to $\mathcal{H}_{\mathbb{C}}(0,T)$, for
all $\xi \in \mathbb{R}^d$. Using (\ref{norm-HP-2}) and (\ref{isometry}),
we obtain that for any $\varphi \in \mathcal{E}_T$,
\begin{equation}
\label{new-def-norm-HP} \|\varphi \|_{\mathcal{H} \mathcal{P}}^2=d_H \int_{\mathbb{R}^d}
\int_0^T |I_{T-}^{H-1/2}(u^{H-1/2} \mathcal{F} \varphi (u,\cdot)(\xi))(s)|^2
\lambda_H(ds) \mu(d\xi)=:\|\varphi\|_{0}^2.
\end{equation}
We are now ready to state our result. Note that, although the
conclusion of this result resembles that of Theorem 3 of
\cite{dalang99} (for deterministic integrands), the hypothesis are
different, since the proof uses techniques specific to the fBm.
\begin{theorem}
\label{theorem-about-HP} Let $[0,T] \ni t \mapsto \varphi(t,\cdot)
\in \mathcal{S}'(\mathbb{R}^d)$ be a deterministic function such that $\mathcal{F}
\varphi(t,\cdot)$ is a function for all $t \in [0,T]$. Suppose that:\\
(i) the function $t \mapsto \mathcal{F} \varphi(t,
\cdot)(\xi)$ belongs to
$\mathcal{H}_{\mathbb{C}}(0,T)$ for all $\xi \in \mathbb{R}^d$; \\
(ii) the function $(t,\xi) \mapsto \mathcal{F} \varphi(t, \cdot)(\xi)$ is
measurable on $(0,T) \times \mathbb{R}^d$; \\
(iii) $\int_s^T u^{H-1/2} (u-s)^{H-3/2} |\mathcal{F}
\varphi(u,\cdot)(\xi)|du<\infty$ for all $(s,\xi) \in (0,T) \times
\mathbb{R}^d$ (or $\mathcal{F} \varphi (s,\cdot)(\xi) \geq 0$ for all $(s,\xi) \in
(0,T) \times \mathbb{R}^d$).
If
\begin{equation}
\label{HP-norm-finite}I_T:=\alpha_H \int_{\mathbb{R}^d} \int_0^T \int_0^T
\mathcal{F} \varphi(u,\cdot)(\xi) \overline{\mathcal{F}
\varphi(v,\cdot)(\xi)}|u-v|^{2H-2}du dv \mu(d\xi)<\infty,
\end{equation}
then $\varphi \in \mathcal{H} \mathcal{P}$ and $\|\varphi \|_{\mathcal{H} \mathcal{P}}^2 =I_T$. (By
convention, we set $\varphi(t,\cdot)=0$ for $t>T$.)
\end{theorem}
\begin{remark}
{\rm Conditions (i)-(iii) are satisfied by the fundamental solution
$G$ of the wave (or heat) equation. In this case, $|\mathcal{F}
G(t,\cdot)(\xi)| \leq 1$ for all $t \geq 0$, and hence the map $t
\mapsto \mathcal{F} G(t,\cdot)(\xi)$ belongs to $L_{\mathbb{C}}^2(0,T)$, which is
included in $\mathcal{H}_{\mathbb{C}}(0,T)$.}
\end{remark}
\noindent {\bf Proof:} The argument is a modified version of the
proof of Theorem 3.8 of \cite{balan-tudor08}. For any $\xi \in
\mathbb{R}^d$ fixed, we apply (\ref{isometry}) to the function
$\phi_{\xi}(t)=\mathcal{F} \varphi(t,\cdot)(\xi)$. We get:
\begin{eqnarray}
\label{equality-xi} \lefteqn{ \alpha_H \int_0^T \int_0^T \mathcal{F}
\varphi(u,\cdot)(\xi) \overline{\mathcal{F}
\varphi(v,\cdot)(\xi)}|u-v|^{2H-2}du dv=} \\
\nonumber & & d_H \int_0^T|I_{T-}^{H-1/2}(u^{H-1/2}\mathcal{F}
\varphi(u,\cdot)(\xi))(s)|^2 \lambda_H(ds).
\end{eqnarray}
It will be shown later that:
\begin{equation}
\label{a-measurable} (s,\xi) \mapsto
a(s,\xi):=I_{T-}^{H-1/2}(u^{H-1/2}\mathcal{F}\varphi(u,\cdot)(\xi))(s)\
\mbox{is measurable on} \ (0,T) \times \mathbb{R}^d.
\end{equation}
\noindent Hence, we can integrate with respect to $\mu(d\xi)$ in
(\ref{equality-xi}). Using (\ref{HP-norm-finite}), we obtain:
\begin{equation}
\label{a-in-L2} I_T=d_{H}\int_{\mathbb{R}^d} \int_0^T
|I_{T-}^{H-1/2}(u^{H-1/2}\mathcal{F} \varphi(u,\cdot)(\xi))(s)|^2
\lambda_H(ds) \mu(d\xi)=:\|\varphi \|_{0}^{2}<\infty.
\end{equation}
By the definition of $\mathcal{H} \mathcal{P}$ and (\ref{new-def-norm-HP}), it
suffices to show that for any $\varepsilon>0$, there exists a
function $l=l_{\varepsilon} \in \mathcal{E}_T$ such that:
\begin{equation}
\label{E-is-dense} \|\varphi -l \|_{0} <\varepsilon.
\end{equation}
Let $\varepsilon>0$ be arbitrary. By (\ref{a-measurable}) and
(\ref{a-in-L2}), it follows that $a \in L^2((0,T) \times \mathbb{R}^d,
\lambda_H(ds) \times \mu(d\xi) )$. Hence, there exists a simple
function $h(s,\xi)$ such that
\begin{equation}
\label{approx-a-h} \int_{\mathbb{R}^d}
\int_{0}^{T}|a(s,\xi)-h(s,\xi)|^{2}\lambda_{H}(ds)\mu(d\xi)
<\varepsilon.
\end{equation}
Without loss of generality, we assume that
$h(s,\xi)=1_{(c,d]}(s)1_{A}(\xi)$, with $c,d \in [0,T],c < d$ and $A
\in \mathcal{B}_{b}(\mathbb{R}^d)$. By relation (8.1) of \cite{pipiras-taqqu01}, we
approximate the function $1_{(c,d]}(s)$ in
$L^2((0,T),\lambda_{H}(ds))$ by $I_{T-}^{H-1/2}(u^{H-1/2}l_0(u))(s)$
with $l_0 \in \mathcal{E}(0,T)$, i.e.
\begin{equation}
\label{approx-l0} \int_0^T
|1_{(c,d]}(s)-I_{T-}^{H-1/2}(u^{H-1/2}l_0(u))(s)|^2\lambda_{H}(ds)
<\varepsilon.
\end{equation}
By Lemma 3.7 of \cite{balan-tudor08}, we approximate the function
$1_{A}(\xi)$ in $L^2(\mathbb{R}^d,\mu(d\xi))$ by $\mathcal{F} l_1(\xi)$ with $l_1
\in \mathcal{E}(\mathbb{R}^d)$, i.e.
\begin{equation}
\label{approx-l1} \int_{\mathbb{R}^d}|1_{A}(\xi)-\mathcal{F}
l_1(\xi)|^2\mu(d\xi)<\varepsilon.
\end{equation}
We define $l(u,x)=l_0(u)l_1(x)$. Clearly $l \in \mathcal{E}_T$ and $\mathcal{F}
l(u,\cdot)(\xi)=l_0(u)\mathcal{F} l_{1}(\xi)$. Let
$$b(s,\xi):=I_{T-}^{H-1/2}(u^{H-1/2} \mathcal{F} l (u,\cdot)(\xi))(s)=
I_{T-}^{H-1/2}(u^{H-1/2} l_0(u))(s) \cdot \mathcal{F} l_1(\xi).$$
\noindent Using (\ref{approx-l0}) and (\ref{approx-l1}), we obtain
that:
\begin{eqnarray}
\nonumber \lefteqn{\int_{\mathbb{R}^d}\int_0^T |h(s,\xi)-b(s,\xi)|^2
\lambda_{H}(ds)\mu(d\xi) }
\\
\nonumber & \leq & 2 \left\{ \int_{\mathbb{R}^d}\int_0^T
|1_{(c,d]}(s)-I_{T-}^{H-1/2}(u^{H-1/2}
l_0(u))(s)|^2 1_{A}(\xi) \lambda_{H}(ds)\mu(d\xi) + \right. \\
\nonumber & & \left. \int_{\mathbb{R}^d}\int_0^T |I_{T-}^{H-1/2}(u^{H-1/2}
l_0(u))(s)|^2
|1_{A}(\xi)-\mathcal{F} l_1(\xi)|^2 \lambda_{H}(ds)\mu(d\xi) \right\}\\
\label{approx-b-h} & \leq & 2
\{\varepsilon\mu(A)+\varepsilon\|l_0\|_{\mathcal{H}
(0,T)}^2/d_H\}:=C_1\varepsilon.
\end{eqnarray}
From (\ref{approx-a-h}) and (\ref{approx-b-h}), it follows that
$$\|\varphi-l
\|_{0}^{2}=d_H\int_{\mathbb{R}^d}
\int_{0}^{T}|a(s,\xi)-b(s,\xi)|^{2}\lambda_{H}(ds)\mu(d\xi) <2d_H
(\varepsilon+C_1\varepsilon):=C_2 \varepsilon.$$ This concludes the
proof of (\ref{E-is-dense}).
We now return to the proof of (\ref{a-measurable}), which uses
assumptions (ii) and (iii). If $\mathcal{F} (u,\cdot)(\xi) \geq 0$, then
$(u,s,\xi) \mapsto \phi(u,s,\xi) =1_{\{s \leq
u\}}u^{H-1/2}(u-s)^{H-3/2}\mathcal{F} \varphi(u,\cdot)(\xi)$ is measurable
and non-negative, and $a(s,\xi)=\int_0^T \phi(u,s,\xi)du$ is
measurable, by Fubini's theorem.
Suppose next that $\int_s^T u^{H-1/2}(u-s)^{H-3/2}|\mathcal{F} \varphi
(u,\cdot)(\xi)|du<\infty$. If $l(s,\xi)=1_{(c,d]}(s)1_{A}(\xi)$ is
an elementary function with $c,d \in [0,T],A \in \mathcal{B}_b(\mathbb{R}^d)$, then
$$a_{l}(s,\xi)=I_{T-}^{H-1/2}(u^{H-1/2}l(u,\xi))(s)=1_{A}(\xi)\int_s^T
u^{H-1/2}1_{(c,d]}(u)(u-s)^{H-3/2}du$$ is clearly measurable.
In general, since $(u,\xi) \mapsto \mathcal{F} \varphi(u,\cdot)(\xi)$ is
measurable, there exists a sequence $(l_n)_n$ of simple functions
such that $l_n(u,\xi) \to \mathcal{F} \varphi(u,\cdot)(\xi)$ for all
$(u,\xi)$ and $|l_n(u,\xi)| \leq |\mathcal{F} \varphi(u,\cdot)(\xi)|$ for
all $(u,\xi),n$ (see e.g. Theorem 13.5 of \cite{billingsley95}). By
the dominated convergence theorem, for every $(s,\xi)$
$$|a_{l_n}(s,\xi)-a(s,\xi)| \leq \int_s^T u^{H-1/2} (u-s)^{H-3/2}|l_n(s,\xi)-
\mathcal{F} \varphi (u,\cdot)(\xi)|du \to 0.$$ Since $a_{l_n}(s,\xi)$ is
measurable for every $n$, it follows that $a(s,\xi)$ is measurable.
$\Box$
\section{The wave equation}
We consider the linear wave equation:
\begin{eqnarray}
\label{wave} \frac{\partial^2 u}{\partial t^2}(t,x)&=& \Delta u
(t,x) +\dot
W(t,x), \quad t>0, x \in \mathbb{R}^d \\
\nonumber
u(0, x)&=& 0, \quad x \in \mathbb{R}^d \\
\nonumber \frac{\partial u}{\partial t}(0,x) &=& 0, \quad x \in
\mathbb{R}^d.
\end{eqnarray}
Let $G_1$ be the fundamental solution of $u_{tt}-\Delta u=0$. It is
known that $G_1(t, \cdot)$ is a distribution in $\mathcal{S}'(\mathbb{R}^d)$ with
rapid decrease, and
\begin{equation}
\label{Fourier-G-wave} \mathcal{F}
G_1(t,\cdot)(\xi)=\frac{\sin(t|\xi|)}{|\xi|},
\end{equation}
for any $\xi \in \mathbb{R}^d,t>0,d \geq 1$ (see e.g. \cite{treves75}). In
particular,
\begin{eqnarray*}
G_1(t,x)&=&\frac{1}{2}1_{\{|x|<t\}}, \quad \mbox{if} \ d=1 \\
G_1(t,x)&=&\frac{1}{2 \pi}\frac{1}{\sqrt{t^2-|x|^2}}1_{\{|x|<t\}},
\quad \mbox{if} \ d=2 \\
G_1(t,x)&=&c_{d}\frac{1}{t}\sigma_t, \quad \mbox{if} \ d=3,
\end{eqnarray*}
where $\sigma_t$ denotes the surface measure on the 3-dimensional
sphere of radius $t$.
The solution of (\ref{wave}) is a square-integrable process
$u=\{u(t,x); t \geq 0, x \in \mathbb{R}^d\}$ defined by:
$$u(t,x)=\int_{0}^{t} \int_{\mathbb{R}^d}G_1(t-s,x-y)W(ds,dy).$$
By definition, $u(t,x)$ exists if and only if the stochastic
integral above is well-defined, i.e. $g_{tx}:=G_1(t-\cdot,x-\cdot)
\in \mathcal{H} \mathcal{P}$. In this case, $E|u(t,x)|^2 = \|g_{tx}\|_{\mathcal{H} \mathcal{P}}^2$.
The following theorem is the main result of this article.
\begin{theorem}
\label{wave-th} The solution $u=\{u(t,x); t \geq 0,x \in \mathbb{R}^d\}$ of
(\ref{wave}) exists if and only if the measure $\mu$ satisfies
(\ref{wave-cond}). In this case, for all $p \geq 2$ and $T>0$
\begin{equation}\label{sup-L2-norm} \sup_{t \in [0,T]} \sup_{x \in
\mathbb{R}^d} E|u(t,x)|^p<\infty,
\end{equation}
and the map $(t,x) \mapsto u(t,x)$ is continuous from $\mathbb{R}_{+}
\times \mathbb{R}^d$ into $L^2(\Omega)$.
\end{theorem}
\begin{example}
{\rm Let $f(x)=\gamma_{\alpha,d}|x|^{-(d-\alpha)}$ be the Riesz
kernel of order $\alpha \in (0,d)$. Then
$\mu(d\xi)=|\xi|^{-\alpha}d\xi$ and (\ref{wave-cond}) is equivalent
to $\alpha>d-2H-1$.}
\end{example}
\begin{example}
{\rm Let $f(x)=\gamma_{\alpha}\int_0^{\infty}w^{(\alpha-d)/2-1}
e^{-w}e^{-|x|^2/(4w)}dw$ be the Bessel kernel of order $\alpha>0$.
Then $\mu(d\xi)=(1+|\xi|^2)^{-\alpha/2}$ and (\ref{wave-cond}) is
equivalent to $\alpha>d-2H-1$. }
\end{example}
\begin{example}
{\rm Let $f(x)=\prod_{i=1}^{d}(\alpha_{H_i}|x_i|^{2H_i-2})$ be the covariance function of a fractional Brownian field with $H_i>1/2$ for all $i=1,\ldots,d$.
Then $\mu(d\xi)=\prod_{i=1}^{d}(c_{H_i}|\xi_i|^{-(2H_i-1)})$ and (\ref{wave-cond}) is
equivalent to $\sum_{i=1}^{d}(2H_i-1)>d-2H-1$. (This can be seen using the change of variables to the polar coordinates.)}
\end{example}
\begin{remark}
\label{rem-AB-equiv} {\rm Condition (\ref{wave-cond}) is equivalent
to
$$\int_{|\xi| \leq 1}\mu(d\xi)<\infty \quad \mbox{and} \quad \int_{|\xi| \geq
1}\frac{1}{|\xi|^{2H+1}}\mu(d\xi)<\infty.$$ }
\end{remark}
\noindent {\bf Proof of Theorem \ref{wave-th}:} Note that
$g_{tx}=G_1(t-\cdot,x-\cdot)$ satisfies conditions (i)-(iii) of
Theorem \ref{theorem-about-HP}. Hence, $g_{tx} \in \mathcal{H} \mathcal{P}$ (i.e.
the solution $u$ of (\ref{wave}) exists) if and only if $I_t<\infty$
for all $t>0$, where
$$I_t:=\alpha_H \int_{\mathbb{R}^d} \int_0^t \int_0^t \mathcal{F}
g_{tx}(u,\cdot)(\xi) \overline{\mathcal{F}
g_{tx}(v,\cdot)(\xi)}|u-v|^{2H-2}dudv \mu(d\xi),$$ and
$E|u(t,x)|^2=\|g_{tx} \|_{\mathcal{H} \mathcal{P}}^2 =I_t$. Since $\mathcal{F}
g_{tx}(u,\cdot)(\xi)=e^{-i\xi \cdot x} \overline{\mathcal{F}
G_1(t-u,\cdot)(\xi)}$,
$$ I_t= \alpha_H
\int_{\mathbb{R}^d} \int_0^t \int_0^t \mathcal{F} G_1(u,\cdot)(\xi) \overline{\mathcal{F}
G_1(v,\cdot)(\xi)}|u-v|^{2H-2}dudv \mu(d\xi).$$
\noindent Using (\ref{Fourier-G-wave}), we obtain:
$$I_t=\alpha_H \int_{\mathbb{R}^d} \frac{\mu(d\xi)}{|\xi|^2}\int_0^t \int_0^t
\sin(u|\xi|)\sin(v|\xi|) |u-v|^{2H-2}dudv.$$
We split the integral $\mu(d\xi)$ into two parts, which correspond
to the regions $\{|\xi| \leq 1\}$ and $\{|\xi| \geq 1\}$. We denote
the respective integrals by $I_t^{(1)}$ and $I_t^{(2)}$. Since the
integrand is non-negative
$I_t<\infty$ if and only if $I_t^{(1)}<\infty$ and
$I_t^{(2)}<\infty$.
The fact that condition (\ref{wave-cond}) is sufficient for
$I_t<\infty$ follows by Proposition \ref{wave-prop-1} below. The
necessity follows by Proposition \ref{wave-prop-2} (using Remark
\ref{rem-AB-equiv}).
Relation (\ref{sup-L2-norm}) with $p=2$ follows from the estimates
obtained for $I_t=E|u(t,x)|^2$, using Proposition \ref{wave-prop-1}.
For arbitrary $p \geq 2$, we use the fact that $E|u(t,x)|^p \leq C_p
(E|u(t,x)|^2)^{p/2}$, since $u(t,x)$ is a Gaussian random variable.
The $L^2(\Omega)$-continuity is proved in Proposition
\ref{sol-cont-L2}. $\Box$
We begin with an auxiliary result. To simplify the notation, we
introduce the following functions: for $\lambda>0,\tau>0$, let
\begin{equation}
\label{def-ft-gt} f_t(\lambda,\tau)=\sin \tau \lambda t -\tau \sin
\lambda t, \quad g_t(\lambda,\tau)=\cos \tau \lambda t -\cos \lambda
t.
\end{equation}
\begin{lemma}
\label{lemma-sin-cos} For any $\lambda>0$ and $t>0$,
$$c_t^{(1)} \frac{\lambda^3}{1+\lambda^2} \leq \int_{\mathbb{R}}\frac{1}{(\tau^2-1)^2}
[f_t^2(\lambda,\tau) +g_t^2(\lambda,\tau)]d\tau \leq c_t^{(2)}
\frac{\lambda^3}{1+\lambda^2},$$ where $c_t^{(1)}=c_1 (t \wedge
t^3)$ and $c_t^{(2)}=c_2(t+t^3)$, for some positive constants
$c_1,c_2$.
\end{lemma}
\noindent {\bf Proof:} From the proof of Lemma \ref{H-norm-sin}, we
see that:
$$\frac{1}{(\tau^2-1)^2}[f_t^2(\lambda,\tau) +g_t^2(\lambda,\tau)]= |\mathcal{F}_{0,\lambda t} \varphi(\tau)|^2,$$ where
$\varphi(x)=\sin x$. Using the Plancharel's identity
(\ref{Plancharel-lemma}), we obtain:
\begin{eqnarray*}
\lefteqn{\int_{\mathbb{R}}\frac{1}{(\tau^2-1)^2}[f_t^2(\lambda,\tau)
+g_t^2(\lambda,\tau)]d\tau= \int_{\mathbb{R}} |\mathcal{F}_{0,\lambda t}
\varphi(\tau)|^2d\tau = }\\
& & 2\pi \int_0^{\lambda t}|\sin x|^2 dx=2\pi \lambda \int_0^t
|\sin \lambda s|^2 ds = 2\pi \lambda^3 \int_0^t \frac{|\sin \lambda
s|^2}{\lambda^2} ds
\end{eqnarray*}
\noindent The result follows using (\ref{estimates}): (see e.g.
Lemma 6.1.2) of \cite{sanz-sole05})
$$c_t^{(1)}\frac{1}{1+\lambda^2} \leq \int_0^t
\frac{|\sin \lambda s|^2}{\lambda^2}ds \leq
c_t^{(2)}\frac{1}{1+\lambda^2}.$$ $\Box$
We denote by $N_t(\xi)$ the $\mathcal{H}(0,t)$-norm of $u \mapsto \mathcal{F} G_1(u,
\cdot)(\xi)$, i.e.
$$N_t(\xi)=\frac{\alpha_H}{|\xi|^2}\int_0^t \int_0^t \sin(u|\xi|)
\sin(v|\xi|) |u-v|^{2H-2}dudv.$$
\begin{proposition}
\label{wave-prop-1} For any $t>0, \xi \in \mathbb{R}^d$
\begin{eqnarray*}
N_t(\xi) & \leq & C_{H}t^{2H+2} \left(\frac{1}{1+|\xi|^2}
\right)^{H+1/2}, \quad \mbox{if} \quad |\xi| \leq 1 \\
N_t(\xi) & \leq & c_{t,H}^{(3)} \left(\frac{1}{1+|\xi|^2}
\right)^{H+1/2}, \quad \mbox{if} \quad |\xi| \geq 1
\end{eqnarray*}
where $C_H= b_H^2 2^{H+1/2}/3$ and $c_{t,H}^{(3)}=c_H (
\frac{C}{1-H} + c_t^{(2)}) 2^{3H-1/2}$. Here $c_t^{(2)}$ is the
constant given by Lemma \ref{lemma-sin-cos}.
\end{proposition}
\noindent {\bf Proof:} a) Suppose that $|\xi| \leq 1$. We use the
fact that $\|\varphi \|_{\mathcal{H}(0,t)}^2 \leq b_H^2 \|\varphi
\|_{L^{1/H}(0,t)}^2 \leq b_H^2 t^{2H-1} \|\varphi \|_{L^2(0,t)}^2$
for any $\varphi \in L^2(0,t)$, and $|\sin x| \leq x$ for any $x>0$.
Hence,
\begin{eqnarray*}
N_t(\xi) & \leq & b_H^2 t^{2H-1} \frac{1}{|\xi|^2}\int_0^t \sin^2
(u|\xi|)du \leq
b_H^2 t^{2H-1} \int_0^t u^2 du \\
&=& b_H^2 t^{2H-1} \frac{t^3}{3} \leq \frac{1}{3} b_H^2 t^{2H+2}
2^{H+1/2}
\left(\frac{1}{1+|\xi|^2} \right)^{H+1/2},
\end{eqnarray*}
where for the last inequality we used the fact that $\frac{1}{2}
\leq \frac{1}{1+|\xi|^2}$ if $|\xi| \leq 1$.
b) Suppose that $|\xi| \geq 1$. Using the change of variable
$u'=u|\xi|$, $v'=v|\xi|$,
\begin{eqnarray*}
N_t(\xi) &=& \frac{\alpha_H }{|\xi|^{2H+2}}
\int_0^{t|\xi|}\int_0^{t|\xi|} \sin(u')\sin(v')|u'-v'|^{2H-2}dudv
\\
&=& \frac{1}{|\xi|^{2H+2}} \|\sin(\cdot)\|_{\mathcal{H}(0,t|\xi|)}^2.
\end{eqnarray*}
Using the expression of the $\mathcal{H}(0,t|\xi|)$-norm of $\sin(\cdot)$
given by Lemma \ref{H-norm-sin},
we obtain:
\begin{equation}
\label{expression-N} N_t(\xi)=\frac{c_{H} }{|\xi|^{2H+2}} \int_{\mathbb{R}}
\frac{|\tau|^{-(2H-1)}}{(\tau^2-1)^2} [f_t^2(|\xi|,\tau)+
g_t^2(|\xi|,\tau)]d\tau.
\end{equation}
We split the integral into the regions $|\tau| \leq 1/2$ and $|\tau|
\geq 1/2$, and we denote the two integrals by $N_{t}^{(1)}(\xi)$ and
$N_{t}^{(2)}(\xi)$.
Since $|f_t(\lambda,\tau)|\leq 1+|\tau|$ and $|g_t(\lambda,\tau)|
\leq 2$ for any $\lambda>0,\tau>0$, we have:
\begin{eqnarray*}
N_{t}^{(1)}(\xi) & \leq & c_H \frac{1}{|\xi|^{2H+2}} \int_{|\tau|
\leq 1/2} \frac{|\tau|^{-(2H-1)}}{(1-\tau^2)^2} [(1+|\tau|)^2+4]
d\tau \\
& \leq & c_H \frac{1}{|\xi|^{2H+1}}
\int_{|\tau| \leq 1/2} C|\tau|^{-(2H-1)} d\tau\\
&=& C
\frac{c_H}{1-H}\left(\frac{1}{2}\right)^{2-2H}\frac{1}{|\xi|^{2H+1}}.
\end{eqnarray*}
We used the fact that $|\xi|^{2H+2} \geq |\xi|^{2H+1}$ if $|\xi|
\geq 1$, and $\frac{1}{(1-\tau^2)^2}[(1+|\tau|)^2+4] \leq
\frac{1}{(3/4)^2}[(3/2)^2+4]=C$ if $|\tau| \leq 1/2$.
Using the fact that $|\tau|^{-(2H-1)} \leq (\frac{1}{2})^{-(2H-1)}$
if $|\tau| \geq \frac{1}{2}$, Lemma \ref{lemma-sin-cos}, and the
fact that $|\xi|^2/(1+|\xi|^2) \leq 1$, we obtain:
\begin{eqnarray*}
N_{t}^{(2)}(\xi) & \leq & \frac{c_H}{2^{-(2H-1)}}
\frac{1}{|\xi|^{2H+2}}\int_{|\tau| \geq 1/2}
\frac{1}{(\tau^2-1)^2}[f_{t}^2(|\xi|,\tau)+g_{t}^2(|\xi|,\tau)]d\tau \\
& \leq & \frac{c_H}{2^{-(2H-1)}} \frac{1}{|\xi|^{2H+2}}\int_{\mathbb{R}}
\frac{1}{(\tau^2-1)^2}[f_{t}^2(|\xi|,\tau)+g_{t}^2(|\xi|,\tau)]d\tau\\
& \leq & \frac{c_H}{2^{-(2H-1)}} c_{t}^{(2)}
\frac{1}{|\xi|^{2H+2}} \cdot |\xi| \frac{|\xi|^2}{1+|\xi|^2} \\
& \leq & \frac{c_H}{2^{-(2H-1)}} c_{t}^{(2)} \frac{1}{|\xi|^{2H+1}}.
\end{eqnarray*}
$\Box$
\begin{proposition}
\label{wave-prop-2} a) If $I_t^{(1)}<\infty$ for $t=1$, then
$\int_{|\xi| \leq 1}\mu(d\xi)<\infty$.
b) Let $l \geq 1$ be the integer from (\ref{mu-tempered}) and
$m=2l-2$. For any $t>0$,
\begin{equation}
\label{lower-estimate-It2} \int_{|\xi| \geq 1}
\frac{\mu(d\xi)}{|\xi|^{2H+1}} \leq a_{H,t}
(\sum_{i=0}^{m}b_t^{i})I_t^{(2)}+b_t^{m+1} \int_{|\xi| \geq
1}\frac{\mu(d\xi)}{|\xi|^{2H+2+m}},
\end{equation}
where
$a_{H,t}=2^{2H} /(c_H c_{t}^{(1)})$, $b_t=2C /c_{t}^{(1)}$ and
$c_t^{(1)}$ is the constant of Lemma \ref{lemma-sin-cos}.
In particular, if $I_t^{(2)}<\infty$ for some $t>0$, then
$\int_{|\xi| \geq 1} |\xi|^{-(2H+1)} \mu(d\xi)<\infty$.
\end{proposition}
\noindent {\bf Proof:} a) Using the fact that $\sin x/x \geq \sin1$
for all $x \in [0,1]$, we have:
\begin{eqnarray*}
I_1^{(1)}&=&\int_{|\xi| \leq 1}\frac{\mu(d\xi)}{|\xi|^2} \int_0^1
\int_0^1 \sin (u|\xi|)\sin(v|\xi|)|u-v|^{2H-2}du dv \\
& \geq & \sin^2 1 \int_{|\xi| \leq 1}\mu(d\xi)\int_0^1 \int_0^1
uv|u-v|^{2H-2}du dv.
\end{eqnarray*}
b) According to (\ref{expression-N}),
\begin{equation}
\label{new-expr-It2} I_t^{(2)}=c_{H} \int_{|\xi| \geq 1}
\frac{\mu(d\xi)}{|\xi|^{2H+2}} \int_{\mathbb{R}}
\frac{|\tau|^{-(2H-1)}}{(\tau^2-1)^2} [f_t^2(|\xi|,\tau)+
g_t^2(|\xi|,\tau)]d\tau.
\end{equation}
For any $k \in \{-1,0, \ldots,m\}$, let
$$I(k):=\int_{|\xi| \geq 1}\frac{1}{|\xi|^{2H+2+k}} \mu(d\xi).$$ By
(\ref{mu-tempered}), $I(m)= \int_{|\xi| \geq 1}|\xi|^{-(2H+2+m)}
\mu(d\xi)\leq \int_{|\xi| \geq 1}|\xi|^{-2l}\mu(d\xi)<\infty$.
We will prove that the integrals $I(k)$ satisfy a certain recursive
relation. By reverse induction, this will imply that all integrals
$I(k)$ with $k \in \{-1,0,\ldots,m\}$ are finite. For this, for $k
\in \{0,1 \ldots,m\}$, we let
\begin{equation}
\label{claim2}
A_t(k):=\int_{|\xi| \geq
1}\frac{\mu(d\xi)}{|\xi|^{2H+2+k}}\int_{\mathbb{R}}
\frac{1}{(\tau^2-1)^2}[f_t^2(|\xi|,\tau)+g_t^2(|\xi|,\tau)]d\tau.
\end{equation}
We consider separately the regions $\{|\tau| \leq 2\}$ and $\{|\tau|
\geq 2\}$. For the region $\{|\tau| \leq 2\}$, we use the expression
(\ref{new-expr-It2}) of $I_t^{(2)}$. Using the fact that
$|\xi|^{2H+2+k} \geq |\xi|^{2H+2}$ ({\em since $k \geq 0$}), and
$|\tau|^{-(2H-1)} \geq 2^{-(2H-1)}$ if $|\tau| \leq 2$, we obtain:
\begin{eqnarray*}
A_t'(k)& := &\int_{|\xi| \geq 1}\frac{\mu(d\xi)}{|\xi|^{2H+2+k}}
\int_{|\tau| \leq 2}\frac{1}{(\tau^2-1)^2}
[f_t^2(|\xi|,\tau)+g_t^2(|\xi|,\tau)]d\tau \\
& \leq & 2^{2H-1} \int_{|\xi| \geq 1}\frac{\mu(d\xi)}{|\xi|^{2H+2}}
\int_{|\tau| \leq 2}\frac{|\tau|^{-(2H-1)}}{(\tau^2-1)^2}
[f_t^2(|\xi|,\tau)+g_t^2(|\xi|,\tau)]d\tau \\
& \leq & 2^{2H-1}\frac{1}{c_{H}}I_{t}^{(2)}, \quad \mbox{by}
(\ref{new-expr-It2}).
\end{eqnarray*}
For the region $\{|\tau| \geq 2\}$, we use the fact
$|f_t(\lambda,\tau)|\leq 1+|\tau|$ and $|g_t(\lambda,\tau)|\leq 2$
for all $\lambda>0,\tau>0$. Hence,
\begin{eqnarray*}
A_t''(k) &:=& \int_{|\xi| \geq 1}\frac{\mu(d\xi)}{|\xi|^{2H+2+k}}
\int_{|\tau| \geq 2}
\frac{1}{(\tau^2-1)^2}[f_t^2(|\xi|,\tau)+g_t^2(|\xi|,\tau)]d\tau
\\
& \leq & \int_{|\xi| \geq 1}\frac{\mu(d\xi)}{|\xi|^{2H+2+k}}
\int_{|\tau| \geq 2} \frac{1}{(\tau^2-1)^2}[(1+|\tau|)^2+4]d\tau
=CI(k).
\end{eqnarray*}
Hence, for any $k \in \{0,1,\ldots,m\}$
$$A_t(k) \leq 2^{2H-1}\frac{1}{c_{H}}I_{t}^{(2)}+CI(k).$$
Using Lemma \ref{lemma-sin-cos}, and the fact that
$\frac{|\xi|^2}{1+|\xi|^2} \geq \frac{1}{2}$ if $|\xi| \geq 1$, we
obtain: $$A_t(k) \geq c_t^{(1)} \int_{|\xi| \geq
1}\frac{\mu(d\xi)}{|\xi|^{2H+2+k}} \cdot \frac{|\xi|^3}{1+|\xi|^2}
\geq \frac{1}{2} c_t^{(1)} I(k-1),$$ for all $k \in
\{0,1,\ldots,m\}$. From the last two relations, we conclude that:
\begin{equation}
\label{recursion} \frac{1}{2} c_t^{(1)} I(k-1) \leq
2^{2H-1}\frac{1}{c_{H}}I_{t}^{(2)}+CI(k), \quad \forall k \in
\{0,1,\ldots,m\},
\end{equation}
or equivalently, $I(k-1) \leq a_{H,t}I_t^{(2)}+b_t I(k)$, for all $k
\in \{0,1,\ldots,m\}$. Relation (\ref{lower-estimate-It2}) follows
by recursion. $\Box$
\begin{remark}
{\rm In the previous argument, the recursion relation
(\ref{recursion}) uses the fact that $k$ is {\em non-negative} (see
the estimate of $A_t'(k)$). Therefore, the ``last'' index $k$ for
which this relation remains true (counting downwards from $m$) is
$k=0$, leading us to the conclusion that $\int_{|\xi| \geq
1}|\xi|^{-(2H+1)} \mu(d\xi)<\infty$, if $I_t^{(2)}<\infty$. }
\end{remark}
The next result shows that the map $(t,x) \to u(t,x)$ from $\mathbb{R}_{+}
\times \mathbb{R}^d$ into $L^2(\Omega)$ is continuous.
\begin{proposition}
\label{sol-cont-L2} Suppose that (\ref{wave-cond}) holds, and let
$u=\{u(t,x), t \geq 0, x \in \mathbb{R}^d\}$ be the solution of
(\ref{wave}). For any $t \geq 0$,
\begin{equation} \label{cont-t} E|u(t+h,x)-u(t,x)|^2 \to 0 \quad
\mbox{as} \ |h| \to 0, \quad \mbox{uniformly in} \ x \in
\mathbb{R}^d\end{equation} and
\begin{equation}\label{cont-x}E|u(t,x)-u(t,y)|^2 \to 0 \quad
\mbox{as} \quad |x-y| \to 0.
\end{equation}
\end{proposition}
\noindent {\bf Proof:} We use the same argument as in Lemma 19 of
\cite{dalang99} (see also the erratum to \cite{dalang99}). We first
show (\ref{cont-t}).
Suppose that $h>0$. Splitting the interval $[0,t+h]$ into the
intervals $[0,t]$ and $[t,t+h]$, and using the inequality $|a+b|^2
\leq 2(a^2+b^2)$, we obtain:
\begin{eqnarray*}
E|u(t+h,x)-u(t,x)|^2 & \leq & 2 \{\|(g_{t+h,x}-g_{tx})1_{[0,t]} \|_{\mathcal{H}
\mathcal{P}}^2 +
\|g_{t+h,x}1_{[t,t+h]} \|_{\mathcal{H} \mathcal{P}}^2\}\\
&=:& 2[E_{1,t}(h)+E_2(h)].
\end{eqnarray*}
Since $\mathcal{F} (g_{t+h,x}-g_{tx})(u,\cdot)(\xi)=e^{-i \xi \cdot x}
\overline{\mathcal{F} G_1(t+h-u,\cdot)(\xi)-\mathcal{F} G_1(t-u,\cdot)(\xi)}$,
\begin{eqnarray*}
E_{1,t}(h) &= & \alpha_H \int_{\mathbb{R}^d} \mu(d\xi )\int_0^t \int_0^t dv dv
|u-v|^{2H-2}\mathcal{F} (g_{t+h,x}-g_{tx})(u,\cdot)(\xi) \\
& & \overline{\mathcal{F}
(g_{t+h,x}-g_{tx})(v,\cdot)(\xi)}\\
&=& \alpha_H \int_{\mathbb{R}^d} \mu(d\xi)\int_0^t \int_0^t du dv |u-v|^{2H-2}
[\mathcal{F} G_1(u+h,\cdot)(\xi)-\mathcal{F} G_1(u,\cdot)(\xi)] \\
& & \overline{\mathcal{F} G_1(v+h,\cdot)(\xi)-\mathcal{F} G_1(v,\cdot)(\xi)} \\
&=& \int_{\mathbb{R}^d}\frac{\mu(d\xi)}{|\xi|^2} k_t(h,|\xi|),
\end{eqnarray*}
where
\begin{eqnarray*}
k_t(h,|\xi|)&=&\alpha_H \int_0^t \int_0^t (\sin ((u+h)|\xi|) -\sin
(u|\xi|)) (\sin ((v+h)|\xi|) -\sin (v|\xi|)) \\
& & |u-v|^{2H-2}du dv =\| \sin ((\cdot \ +h)|\xi|) -\sin (\cdot \
|\xi|)\|_{\mathcal{H}(0,t)}^2.
\end{eqnarray*}
\noindent By the Bounded Convergence Theorem, $\lim_{h \downarrow
0}k_t(h,|\xi|)=0$, for any $\xi \in \mathbb{R}^d$.
The fact that $E_{1,t}(h) \to 0$ as $h \downarrow 0$ will follow from
the Dominated Convergence Theorem, once we prove that:
\begin{equation}
\label{bound-of-k} k_t(h,|\xi|) \leq k_t(|\xi|), \quad \forall h \in
[0,1], \forall \xi \in \mathbb{R}^d, \quad \mbox{and} \quad
\int_{\mathbb{R}^d}\frac{\mu(d\xi)}{|\xi|^2} k_t(|\xi|)<\infty.
\end{equation}
When $|\xi| \leq 1$, using the same argument as in Proposition
\ref{wave-prop-1}, we get:
\begin{eqnarray*}
k_t(h,|\xi|) & \leq & b_H^2 t^{2H-1} \| \sin ((\cdot \ +h)|\xi|)
-\sin
(\cdot \ |\xi|)\|_{L^2(0,t)}^2 \\
& \leq & 2b_H^2 t^{2H-1} \left(\int_0^t \sin^2
((u+h)|\xi|)du+\int_0^t \sin^2 (u|\xi|)du \right) \\
& \leq & 2 b_H^2 t^{2H-1}|\xi|^2\left(\int_0^t 2(u^2+1)du+\int_0^t
u^2 du \right)=:k_t(|\xi|).
\end{eqnarray*}
Suppose that $|\xi| \geq 1$. We use the fact that:
$$k_t(h,|\xi|) \leq 2 (\| \sin ((\cdot \ +h)|\xi|)\|_{\mathcal{H}(0,t)}^2+
\|\sin (\cdot \ |\xi|)\|_{\mathcal{H}(0,t)}^2).$$
Using the change of variables $u'=(u+h)|\xi|, v'=(v+h)|\xi|$, and
(\ref{lemmaA1}) (Appendix A) we obtain:
\begin{eqnarray*}
\| \sin ((\cdot \ +h)|\xi|)\|_{\mathcal{H}(0,t)}^2 &=& \alpha_H \int_0^t
\int_0^t \sin ((u+h)|\xi|) \ \sin ((v+h)|\xi|) \ |u-v|^{2H-2}du dv \\
&=& \frac{\alpha_H}{|\xi|^{2H}} \int_{h|\xi|}^{(t+h)|\xi|}
\int_{h|\xi|}^{(t+h)|\xi|} \sin (u') \sin (v') |u'-v'|^{2H-2}du' dv'
\\
&=& \frac{c_H}{|\xi|^{2H}} \int_{\mathbb{R}} |\mathcal{F}_{h|\xi|, (t+h)|\xi|}
\varphi (\tau)|^2 |\tau|^{-(2H-1)} d\tau,
\end{eqnarray*}
where $\varphi(t)=\sin t$. Note that the square of the real part of
$\mathcal{F}_{h|\xi|, (t+h)|\xi|} \varphi(\tau)$ is:
$$\left|\int_{h|\xi|}^{(t+h)|\xi|} \cos \tau t \sin t dt\right|^2 \leq
2 \left|\int_{0}^{(t+h)|\xi|} \cos \tau t \sin t dt\right|^2 +2
\left|\int_{0}^{h|\xi|} \cos \tau t \sin t dt\right|^2,$$ and the
square of the imaginary part of $\mathcal{F}_{h|\xi|, (t+h)|\xi|}
\varphi(\tau)$ is:
$$\left|\int_{h|\xi|}^{(t+h)|\xi|} \sin \tau t \sin t dt\right|^2 \leq
2 \left|\int_{0}^{(t+h)|\xi|} \sin \tau t \sin t dt\right|^2 +2
\left|\int_{0}^{h|\xi|} \sin \tau t \sin t dt\right|^2.$$
We now use the following fact (see Appendix B): for any $T>0$
$$\left| \int_0^T \cos \tau t \sin t dt \right|^2+
\left| \int_0^T \sin \tau t \sin t dt
\right|^2=\frac{1}{(\tau^2-1)^2}[(\sin \tau T-\tau \sin T)^2+(\cos
\tau T-\cos T)^2].$$
From here, it follows that $k_t(h,|\xi|)$ is bounded by:
$$\frac{2c_H}{|\xi|^{2H}}
\int_{\mathbb{R}}\frac{|\tau|^{-(2H-1)}}{(\tau^2-1)^2}[f_{t+h}^2(|\xi|,\tau)+
g_{t+h}^2(|\xi|,\tau)+f_{h}^2(|\xi|,\tau)+g_{h}^2(|\xi|,\tau)+
f_{t}^2(|\xi|,\tau)+g_{t}^2(|\xi|,\tau)]d\tau,$$ where
$f_t(\lambda,\tau)$ and $g_{t}(\lambda,\tau)$ are defined by
(\ref{def-ft-gt}). The argument of Proposition \ref{wave-prop-1}
shows that for any $t>0$ and $|\xi| \geq 1$
$$\int_{\mathbb{R}}
\frac{|\tau|^{-(2H-1)}
}{(\tau^2-1)^2}[f_t^2(|\xi|,\tau)+g_t^2(|\xi|,\tau)] d\tau \leq
c_{t,H}^{(4)}\frac{|\xi|^3}{1+|\xi|^2} \leq c_{t,H}^{(4)}|\xi|,$$
where $c_{t,H}^{(4)}=\frac{2C}{1-H}\left(\frac{1}{2} \right)^{2-2H}
+ \left(\frac{1}{2}\right)^{-(2H-1)}c_t^{(2)}$. Since
$c_{t,H}^{(4)}$ is non-decreasing in $t$ and $h \in [0,1]$,
$k_t(h,|\xi|)$ is bounded by
$$\frac{2c_H}{|\xi|^{2H-1}} (c_{t+h,H}^{(4)}+c_{h,H}^{(4)}+c_{t,H}^{(4)})
\leq \frac{2c_H}{|\xi|^{2H-1}}
(c_{t+1,H}^{(4)}+c_{1,H}^{(4)}+c_{t,H}^{(4)}):=k_t(|\xi|).$$ This
concludes the proof of (\ref{bound-of-k}).
A similar argument shows that $E_2(h) \to 0$ as $h \downarrow 0$,
since
\begin{eqnarray*}
E_2(h) &=& \alpha_H \int_{\mathbb{R}^d} \int_{t}^{t+h} \int_t^{t+h} \mathcal{F} G_1
(t+h-u, \cdot)(\xi) \overline{\mathcal{F} G_1 (t+h-v,
\cdot)(\xi)}|u-v|^{2H-2}du dv \mu(d\xi) \\
&=& \alpha_H \int_{\mathbb{R}^d}\frac{\mu(d\xi)}{|\xi|^2} \int_0^h \int_0^h
\sin (u|\xi|) \sin (v|\xi|) |u-v|^{2H-2}du dv.
\end{eqnarray*}
The case $h<0$ is treated similarly. Using the same argument as
above, it follows that for any $h>0$, $E|u(t-h,x)-u(t,x)|^2 \leq 2
(E_{1,t}'(h)+E_2(h))$, where
$$E_{1,t}'(h)=\int_{\mathbb{R}^{d}} \frac{\mu(d\xi)}{|\xi|^2} k_t'(h,|\xi|), \
\mbox{and} \ k_t'(h,|\xi|)=\| \sin (\cdot \ |\xi|)-\sin ( (\cdot \ -h) |\xi|) \|_{\mathcal{H}(h,t)}^2.$$
To prove (\ref{cont-x}), note that
\begin{eqnarray*}
\lefteqn{E|u(t,x)-u(t,y)|^2=\|g_{tx}-g_{ty}\|_{\mathcal{H} \mathcal{P}}^2 = }\\
& & \alpha_H \int_{\mathbb{R}^d} \int_{0}^{t} \int_{0}^{t} \mathcal{F}
(g_{tx}-g_{ty}) (u, \cdot)(\xi) \overline{\mathcal{F} (g_{tx}-g_{ty})(v,
\cdot)(\xi)}|u-v|^{2H-2}du dv \mu(d\xi)= \\
& & \alpha_H \int_{\mathbb{R}^d} |e^{-i \xi \cdot x}-e^{-i \xi \cdot
y}|^2\int_{0}^{t} \int_{0}^{t} \mathcal{F} G_1(u, \cdot)(\xi) \overline{\mathcal{F}
G_1(v, \cdot)(\xi)}|u-v|^{2H-2}du dv \mu(d\xi),
\end{eqnarray*}
which converges to $0$ as $|x-y| \to 0$, by the Dominated
Convergence Theorem. $\Box$
\begin{example}
{\rm There exists an interesting connection between the solution of
the wave equation with fractional noise in time and Riesz covariance
in space and the odd and even parts of the fBm. Indeed, if $f$ be
the Riesz kernel of order $\alpha \in (0,d)$, then
\begin{eqnarray*}
I_t&=& \alpha_H \int_{\mathbb{R}^d}d\xi |\xi|^{-\alpha-2H-2}\int_{\mathbb{R}}
\frac{|\tau|^{-(2H-1)}}{(\tau^2-1)^2}[f_t^2(|\xi|,\tau)+g_t^2(|\xi|,\tau)]
d\tau \\
&=& 2\alpha_H c_d \int_{\mathbb{R}} \frac{|\tau|^{-(2H-1)}}{(\tau^2-1)^2}
\left(\int_0^{\infty} \frac{(\sin \tau \lambda t -\tau \sin \lambda
t)^2}{\lambda^2}\lambda^{-\theta}d\lambda+\int_0^{\infty}
\frac{(\cos \tau \lambda t -\cos \lambda
t)^2}{\lambda^2}\lambda^{-\theta}d\lambda \right),
\end{eqnarray*}
where $\theta=\alpha+1-d+2H >0$ under (\ref{wave-cond}). If
$\theta<1$, the two integrals $d\lambda$ can be expressed in terms
of the covariance functions of the odd and even parts of the fBm
(see \cite{DZ}).}
\end{example}
\section{The heat equation}
In this section, we consider the the heat equation with additive
noise:
\begin{eqnarray}
\label{heat} \frac{\partial u}{\partial t}(t,x)&=&\frac{1}{2} \Delta
u(t,x)+\dot
W(t,x), \quad t>0, x \in \mathbb{R}^d \\
\nonumber u(0,x)&=& 0, \quad x \in \mathbb{R}^d.
\end{eqnarray}
Equation (\ref{heat}) was treated in \cite{balan-tudor08}, in the
case of particular covariance kernels $f$. We give here an unitary
approach which covers the case of any covariance kernel $f$, which
satisfies (\ref{heat-cond}).
The case of the heat equation is actually much simpler than the case of the
wave equation, since both the fundamental solution $G$ and its
Fourier transform are non-negative functions.
More precisely, let $G_2$ be the fundamental solution of
$u_t-\frac{1}{2}\Delta u=0$. Then
$$G_2(t,x)=\frac{1}{(2\pi t)^{d/2}} \exp \left(-\frac{|x|^2}{2t}\right), \quad t >0,x \in
\mathbb{R}^d$$ and
\begin{equation}
\label{Fourier-heat} \mathcal{F} G_2(t,\cdot)(\xi)=\exp
\left(-\frac{t|\xi|^2}{2} \right), \quad t>0,\xi \in \mathbb{R}^d.
\end{equation}
We will prove the following result.
\begin{theorem}
\label{heat-th} The solution $u=\{u(t,x),t \geq 0, x \in \mathbb{R}^d\}$ of
(\ref{heat}) exists if and only if the measure $\mu$ satisfies
(\ref{heat-cond}). In this case, (\ref{sup-L2-norm}) holds for all
$p \geq 2$ and $T>0$, and the solution is $L^2(\Omega)$-continuous.
\end{theorem}
\begin{remark}
{\rm (i) When $f$ is the Riesz kernel of order $\alpha$, or the
Bessel kernel of order $\alpha$, condition (\ref{heat-cond}) is
equivalent to $\alpha>d-4H$. When $f$ is the covariance function of the fractional Brownian field with
$H_i>1/2$ for all $i=1,\ldots,d$, condition (\ref{heat-cond}) is
equivalent to $\sum_{i=1}^{d}(2H_i-1)>d-4H$. Note
that this condition is weaker than the condition given in \cite{oksendal-zhang01}.
(ii) In Theorem 2.1 of the Erratum to \cite{balan-tudor08} it has
been proven that condition (\ref{heat-cond}) implies that
$\|g_{tx}\|_{\mathcal{H} \mathcal{P}}<\infty$ for any $t\geq 0$ and $x\in \mathbb{R} ^{d}$.
}
\end{remark}
\noindent {\bf Proof of Theorem \ref{heat-th}:} Note that
$g_{tx}=G_2(t-\cdot,x-\cdot)$ is non-negative. Hence, $g_{tx} \in
\mathcal{H} \mathcal{P}$
if and only if $g_{tx} \in |\mathcal{H} \mathcal{P}|$. This is equivalent to saying
that $J_t:=\|g_{tx}\|_{|\mathcal{H} \mathcal{P}|}^2<\infty$ for all $t>0$. Note that
\begin{eqnarray*}
J_t&=& \alpha_H \int_0^t \int_0^t \int_{\mathbb{R}^d} \int_{\mathbb{R}^d}
g_{tx}(u,y)
g_{tx}(v,z)f(y-z)|u-v|^{2H-2}dy dz dudv \\
&=& \alpha_H \int_0^t \int_0^t \int_{\mathbb{R}^d} \mathcal{F}
g_{tx}(u,\cdot)(\xi) \overline{\mathcal{F} g_{tx}(v,\cdot)(\xi)}|u-v|^{2H-2}
\mu(d\xi) dudv
\\
&=& \alpha_H \int_0^t \int_0^t \int_{\mathbb{R}^d} \mathcal{F}
G_2(t-u,\cdot)(\xi) \overline{\mathcal{F} G_2(t-v,\cdot)(\xi)}|u-v|^{2H-2}
\mu(d\xi) dudv.
\end{eqnarray*}
Using (\ref{Fourier-heat}) and Fubini's theorem (whose application
is justified since the integrand is non-negative), we obtain:
$$J_t=\alpha_H \int_{\mathbb{R}^d} \int_0^t \int_0^t \exp
\left(-\frac{u|\xi|^2}{2} \right) \exp \left(-\frac{v|\xi|^2}{2}
\right) |u-v|^{2H-2}dudv \mu(d\xi).$$
The existence of the solution follows from Proposition
\ref{estimate-heat} below, which also gives estimates for
$J_t=E|u(t,x)|^2$ (and hence for $E|u(t,x)|^p$). The
$L^2(\Omega)$-continuity is given by Proposition \ref{cont-L2-heat}.
$\Box$
Let
$$A_t(\xi)=\alpha_H\int_0^t \int_0^t \exp
\left(-\frac{u|\xi|^2}{2} \right) \exp \left(-\frac{v|\xi|^2}{2}
\right) |u-v|^{2H-2}dudv $$
The next result is similar to Lemma 6.1.1) of \cite{sanz-sole05}.
\begin{proposition}
\label{estimate-heat} For any $t>0, \xi \in \mathbb{R}^d$,
$$\frac{1}{4} (t^{2H} \wedge 1)\left(\frac{1}{1+|\xi|^2}\right)^{2H}
\leq A_t(\xi) \leq C_H (t^{2H}+1)
\left(\frac{1}{1+|\xi|^2}\right)^{2H},$$ where $C_H=b_H^2
(4H)^{2H}$.
\end{proposition}
\noindent {\bf Proof:} Suppose that $|\xi| \leq 1$. Using the fact
that $\|\varphi \|_{\mathcal{H}(0,t)}^2 \leq b_H^2 t^{2H-1}
\|\varphi\|_{L^2(0,t)}^2$ for all $\varphi \in L^2(0,t)$, $e^{-x}
\leq 1$ for any $x>0$, and $\frac{1}{2} \leq \frac{1}{1+|\xi|^2}$ if
$|\xi| \leq 1$,
$$A_t(\xi) \leq b_H^2 t^{2H-1} \int_0^t
\exp(-u|\xi|^2)du \leq b_H^2 t^{2H} \leq b_H^2 2^{2H} t^{2H}
\left(\frac{1}{1+|\xi|^2}\right)^{2H}.$$
Suppose that $|\xi| \geq 1$. Using the fact that $\|\varphi
\|_{\mathcal{H}(0,t)}^2 \leq b_H^2 \|\varphi\|_{L^{1/H}(0,t)}^2$ for any
$\varphi \in L^{1/H}(0,t)$, $1-e^{-x} \leq 1$ for all $x>0$, and
$\frac{1}{|\xi|^2} \leq \frac{2}{1+|\xi|^2}$, we obtain:
\begin{eqnarray*}
A_t(\xi) & \leq & b_{H}^2 \left[\int_0^t
\exp\left(-\frac{u|\xi|^2}{2H} \right) du \right]^{2H} = b_{H}^2
\left(\frac{2H}{|\xi|^2}\right)^{2H}
\left[1-\exp\left(-\frac{t|\xi|^2}{2H}\right) \right]^{2H} \\
& \leq & b_{H}^2 (4H)^{2H}
\left(\frac{1}{1+|\xi|^{2}}\right)^{2H}.
\end{eqnarray*}
This proves the upper bound.
Next, we show the lower bound. Suppose that $t|\xi|^2 \leq 1$. For
any $u \in [0,t]$, $\frac{u|\xi|^2}{2} \leq \frac{t|\xi|^2}{2} \leq
\frac{1}{2}$. Using the fact that $e^{-x} \geq 1-x$ for all $x>0$,
we conclude that:
$$\exp\left(-\frac{u|\xi|^2}{2}\right) \geq 1-\frac{u|\xi|^2}{2}
\geq \frac{1}{2}, \quad \forall u \in [0,t].$$ Hence
$$A_t(\xi) \geq \alpha_H \left(\frac{1}{2} \right)^2
\int_0^t \int_0^t |u-v|^{2H-2}dudv=\frac{1}{4}t^{2H} \geq
\frac{1}{4}t^{2H} \left( \frac{1}{1+|\xi|^2}\right)^{2H}.$$ For the
last inequality, we used the fact that $1 \geq \frac{1}{1+|\xi|^2}$.
Suppose that $t|\xi|^2 \geq 1$. Using the change of variables
$u'=u|\xi|^2/2, v'=v|\xi|^2/2$, we obtain:
$$A_{t}(\xi)=\alpha_H \frac{2^{2H}}{|\xi|^{4H}} \int_0^{t|\xi|^2/2}
\int_{0}^{t|\xi|^2/2} e^{-u'}e^{-v'}|u'-v'|^{2H-2}du' dv'.$$ Since
the integrand is
non-negative,
\begin{eqnarray*}
A_t(\xi) & \geq & \alpha_H \frac{2^{2H}}{|\xi|^{4H}} \int_0^{1/2}
\int_{0}^{1/2} e^{-u}e^{-v}|u-v|^{2H-2}dudv \\
& = & 2^{2H}\|e^{-u} \|_{\mathcal{H}(0,1/2)}^2 \frac{1}{|\xi|^{4H}} \geq
2^{2H} \left(\frac{1}{2} \right)^{2H+2} \left(\frac{1}{1+|\xi|^2}
\right)^{2H},
\end{eqnarray*}
where for the last inequality we used the fact that
$\frac{1}{|\xi|^2} \geq \frac{1}{1+|\xi|^2}$, and $\|e^{-u}
\|_{\mathcal{H}(0,1/2)}^2 \geq \left(\frac{1}{2} \right)^{2H+2}$. (This
follows since $e^{-u} \geq 1-u \geq \frac{1}{2}$ for all $u \in
[0,\frac{1}{2}]$.)
$\Box$
\begin{proposition}
\label{cont-L2-heat} Suppose that (\ref{heat-cond}) holds, and let
$u=\{u(t,x), t \geq 0, x \in \mathbb{R}^d\}$ be the mild-sense solution of
(\ref{heat}). Then the map $(t,x) \to u(t,x)$ from $\mathbb{R}_{+} \times
\mathbb{R}^d$ into $L^2(\Omega)$ is continuous.
\end{proposition}
\noindent {\bf Proof:} The argument is similar to that of
Proposition \ref{sol-cont-L2}. In this case, if $h>0$,
$$E_{1,t}(h)=\int_{\mathbb{R}^d}\mu(d\xi)k_t(h,|\xi|),$$
where
$$k_t(h,|\xi|)=\left\| \exp \left(-\frac{(\cdot +h)|\xi|^2}{2} \right)-\exp
\left(-\frac{ \cdot \ |\xi|^2}{2} \right) \right\|_{\mathcal{H}(0,t)}^2,$$
and
$$E_2(h)=\alpha_H\int_{\mathbb{R}^d}\mu(d\xi)\int_0^h \int_0^h \exp
\left(-\frac{u|\xi|^2}{2} \right) \exp \left(-\frac{v|\xi|^2}{2}
\right)|u-v|^{2H-2}du dv .$$ We omit the details. $\Box$
\begin{remark}
{\rm We consider the operator $Lu=\partial_t
u-\sum_{i,j=1}^{d}a_{ij}\partial^2_{x_i x_j}u-\sum_{i=1}^{d}b_i
\partial_{x_i}u$. Let $G_3(t,x;s,y)$ be the fundamental
solution of $Lu=0$. We assume that:
(i) The functions $a_{ij},b_i:[0,T] \times \mathbb{R}^d \to \mathbb{R}$,
$i,j=1,\ldots,d$ are $\alpha/2$-H\"older continuous in $t$ and
$\alpha$-H\"older continuous in $x$, for some $\alpha \in (0,1)$.
(ii) There exist some $k,K>0$ such that for all $(t,x) \in [0,T]
\times \mathbb{R}^d, \xi \in \mathbb{R}^d$,
$$k|\xi|^2 \leq \sum_{i,j=1}^{d}a_{ij}(t,x)\xi_i \xi_j \leq
K|\xi|^2.$$
Under these assumptions, $G_3$ is a positive function defined on
$[0,T] \times \mathbb{R}^d \times [0,T] \times \mathbb{R}^d \cap \{(s,t); 0 \leq s
\leq t \leq T\}$, which satisfies: (see p. 376 of \cite{LSU68})
\begin{equation}
\label{ineq-G3} G_3(t,x;s,y) \leq c_1 (t-s)^{-d/2} \exp \left(-c_2
\frac{|x-y|^2}{t-s} \right):=G_2'(t-s,x-y).
\end{equation}
Since $G_2'(t,x)$ is essentially the same as the heat kernel
$G_2(t,x)$, the solution of $Lu(t,x)=\dot W(t,x)$ (with vanishing
initial conditions) exists, if the measure $\mu$ satisfies condition
(\ref{heat-cond}). }
\end{remark}
\appendix
\section{Some useful identities}
Recall that the Fourier transform of a function $\varphi \in
L^1(\mathbb{R})$ is defined by: $$\mathcal{F} \varphi(\tau)=\int_{\mathbb{R}}e^{-i\tau
x}\varphi(x)dx.$$
\noindent For an interval $(a,b) \subset \mathbb{R}$, we define the
restricted Fourier transform of a function $\varphi \in L^1(a,b)$:
$$\mathcal{F}_{a,b} \varphi(\tau):=\int_{a}^{b} e^{-i\tau x}\varphi(x)dx=
\mathcal{F} (\varphi 1_{[a,b]})(\tau).$$
One can prove that $\mathcal{F} \varphi \in L^2(\mathbb{R})$, for any $\varphi \in
L^1(\mathbb{R}) \cap L^2(\mathbb{R}).$ By the Plancharel's identity, for any
$\varphi,\psi \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$, we have:
$$ \int_{\mathbb{R}} \varphi(x)\psi(x)dx =(2\pi)^{-1}\int_{\mathbb{R}}\mathcal{F} \varphi(\tau)
\overline{\mathcal{F} \psi(\tau)}d\xi.$$
In particular, for any $\varphi,\psi \in L^2(a,b)$, we have:
\begin{equation}
\label{Plancharel-lemma} \int_{a}^{b} \varphi(x)\psi(x)dx
=(2\pi)^{-1}\int_{\mathbb{R}}\mathcal{F}_{a,b} \varphi(\tau) \overline{\mathcal{F}_{a,b}
\psi(\tau)}d\xi.
\end{equation}
\noindent (Consider $\tilde \varphi=\varphi 1_{[a,b]}$. Then $\tilde
\varphi \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$ and $\mathcal{F} \tilde
\varphi(\xi)=\mathcal{F}_{a,b} \varphi(\xi)$.)
The proof of Theorem \ref{wave-th} uses, in an essential way, a
formula for the $\mathcal{H}(0,T)$-norm of $\sin$ (developed in Appendix B),
which is in turn based on the following result. (This result can be
derived using for instance, the results of \cite{PT00}.)
\begin{lemma}
Let $H \in (\frac{1}{2},1)$. For any $\varphi,\psi \in L^1(\mathbb{R}) \cap
L^2(\mathbb{R})$,
$$\alpha_H \int_{\mathbb{R}} \int_{\mathbb{R}} \varphi(u)
\psi(v)|u-v|^{2H-2}du dv=c_{H} \int_{\mathbb{R}} \mathcal{F} \varphi
(\tau)\overline{\mathcal{F} \psi(\tau)}|\tau|^{-(2H-1)}d\tau,$$ where
$\alpha_H=H(2H-1)$ and $c_H=\Gamma(2H+1)\sin(\pi H)/(2\pi)$.
In particular, for any $\varphi,\psi \in L^2(a,b)$,
\begin{equation}
\label{lemmaA1} \alpha_H \int_a^b \int_a^b \varphi(u)
\psi(v)|u-v|^{2H-2}du dv=c_{H} \int_{\mathbb{R}} \mathcal{F}_{a,b}\varphi
(\tau)\overline{\mathcal{F}_{a,b}\psi(\tau)}|\tau|^{-(2H-1)}d\tau.
\end{equation}
\end{lemma}
\section{The $\mathcal{H}(0,T)$-norm of $\sin$}
\begin{lemma}
\label{H-norm-sin} Let $\varphi(t)=\sin t$, $t \in [0,T]$. Then
$$\|\varphi\|_{\mathcal{H}(0,T)}^2= c_H \int_{\mathbb{R}} \frac{(\sin \tau T-\tau \sin T)^2+(\cos\tau
T-\cos T)^2}{(\tau^2-1)^2} \ |\tau|^{-(2H-1)} d\tau,$$ where
$c_H=\Gamma(2H+1)\sin(\pi H)/(2\pi)$.
\end{lemma}
\noindent {\bf Proof}: By (\ref{lemmaA1}),
$$\|\varphi \|_{\mathcal{H}(0,T)}^2=\alpha_H \int_0^T \int_0^T \varphi(u)
\varphi(v) |u-v|^{2H-2}du dv=c_H \int_{\mathbb{R}}|\mathcal{F}_{0,T} \varphi
(\tau)|^2 |\tau|^{-(2H-1)}d\tau.$$ Note that
$|\mathcal{F}_{0,T}\varphi(\tau)|^2=\left|\int_{0}^{T}e^{-i\tau
t}\varphi(t)dt\right|^2=I_1^2+ J_1^2$, where
$$I_1={\rm Re}[\mathcal{F}_{0,T} \varphi (\tau)]=\int_0^T \cos \tau t \sin t dt,
\quad J_1={\rm Im}[\mathcal{F}_{0,T} \varphi (\tau)]=\int_0^T \sin \tau t
\sin t dt.$$
\noindent We calculate $I_1$ first. Using integration by parts, we
obtain: $$ I_1=1-\cos \tau T\cos T-\tau I_2,$$ where $I_2=\int_0^T
\sin \tau t \cos t dt$. On the other hand,
$$I_1+I_2=\int_0^T \sin [(\tau+1 )t]dt=\frac{1-\cos[(\tau+1)T]}{\tau+1}.$$
\noindent Solving for $I_1$ and $I_2$, we obtain:
$$I_1=\frac{1}{1-\tau^2}(1-\cos\tau T \cos T- \tau \sin\tau T \sin T).$$
\noindent Similarly, letting $J_2=\int_0^T \cos\tau t \cos t dt$,
obtain:
$$\tau J_2-J_1=\sin\tau T \cos T \quad \mbox{and} \quad
J_2-J_1=\frac{1}{\tau+1}\sin[(\tau+1)T].$$ Solving for $J_1$, we
obtain:
$$J_1=\frac{1}{1-\tau^2}(\tau \cos\tau T \sin T-\sin\tau T \cos T).$$
\noindent An elementary calculation shows that:
$$I_1^2+J_1^2 =\frac{1}{(1-\tau^2)^2} [(\sin \tau T-\tau \sin
T)^2+(\cos\tau T-\cos T)^2].$$ $\Box$
\begin{remark}
{\rm Let $B=(B_t)_{t \in \mathbb{R}}$ a fBm of index $H$ (on the whole real
line). Let $B^o=(B_t^o)_{t \in \mathbb{R}}$ and $B^e=(B_t^e)_{t \in \mathbb{R}}$
be the odd and even parts of $B$ (see \cite{DZ}). $B^o$ and $B^e$
are independent centered Gaussian processes with
$B_t=B_t^{o}+B_{t}^{e}$, and
\begin{eqnarray*} E(B_t^o
B_s^o)&=&\langle 1_{(0,t)}, 1_{(0,s)} \rangle_{o}:= c_{H}\int_{\mathbb{R}}
\frac{\sin \tau t \sin \tau s}{\tau^2} |\tau|^{-(2H-1)}d\tau \\
E(B_t^e B_s^e)&= &\langle 1_{(0,t)}, 1_{(0,s)} \rangle_{e}:=
c_{H}\int_{\mathbb{R}} \frac{(1-\cos \tau t)(1- \cos \tau s)}{\tau^2}
|\tau|^{-(2H-1)}d\tau.
\end{eqnarray*}
In general, for $\varphi \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$, we have:
$$E[B^o(\varphi)^2]=\|\varphi \|_{o}^2:=c_{H} \int_{\mathbb{R}}
|{\rm Re}[\mathcal{F} \varphi (\tau)]|^2 |\tau|^{-(2H-1)}d\tau$$
$$E[B^e(\varphi)^2]=\|\varphi \|_{e}^2:=c_{H} \int_{\mathbb{R}}
|{\rm Im}[\mathcal{F} \varphi (\tau)]|^2 |\tau|^{-(2H-1)}d\tau$$
In the proof of Lemma \ref{H-norm-sin}, $I_1=I_1(\tau)$ and
$J_1=J_1(\tau)$ are the real and imaginary parts of $\mathcal{F} (\varphi
1_{[0,T]})(\tau)$, where $\varphi (t)=\sin t$. Hence
\begin{eqnarray*}
E \left(\int_0^T \sin t dB_t^o \right)^2=\|\sin (\cdot)
1_{[0,T]}\|_{o}^2 &=& c_H \int_{\mathbb{R}} |I_1(\tau)|^2
|\tau|^{-(2H-1)}d\tau \\
E \left(\int_0^T \sin t dB_t^e \right)^2=\|\sin (\cdot)
1_{[0,T]}\|_{e}^2 &=& c_H \int_{\mathbb{R}} |J_1(\tau)|^2
|\tau|^{-(2H-1)}d\tau.
\end{eqnarray*}
}
\end{remark}
\footnotesize{{\em Acknowledgement.} The authors would like to thank
Professor Robert Dalang for the invitations to visit EPFL and the
useful discussions.
\normalsize{
\end{document} | math | 64,265 |
\begin{document}
\begin{center}
{\bf Some 2-adic conjectures concerning \\
polyomino tilings of Aztec diamonds} \\
\ \\
James Propp, UMass Lowell \\
August 10, 2022 \\
\ \\
{\it Dedicated to Michael Larsen on the occasion of his 60th birthday}
\end{center}
\begin{abstract}
\noindent
For various sets of tiles, we count the ways to tile
an Aztec diamond of order $n$ using tiles from that set.
The resulting function $f(n)$ often has interesting behavior
when one looks at $n$ and $f(n)$ modulo powers of 2.
\end{abstract}
\begin{section}{Introduction}
\label{sec:intro}
I had a great time working on domino tilings of Aztec diamonds
with Noam Elkies, Greg Kuperberg, and Michael Larsen back in the late 1980s,
and the paper we wrote together~\cite{EKLP}
had a huge impact on my career.
So I’d like to honor Michael by proposing
some new problems about tilings of Aztec diamonds
(and other regions), many of which are more challenging
than the one I shared with him thirty-something years ago
and have a number-theoretic slant that I think he will enjoy.
Ideally the solutions to these problems will involve
interesting applications of algebra to combinatorics.
\begin{figure}
\caption{A domino tiling of the Aztec diamond of order 4.}
\label{fig:dominos}
\end{figure}
Here is some general background.
The main result of~\cite{EKLP} was that the number of domino-tilings
of an Aztec diamond of order $n$ is $2^{n(n+1)/2}$
(\href{https://oeis.org/A006125}{A006125}),
where a domino is a rectangle in $\mathds{R}^2$ of the form
$[i,i+1] \times [j,j+2]$ or $[i,i+2] \times [j,j+1]$ (with $i,j \in \mathds{Z}$)
and the Aztec diamond of order $n$
is the union of the squares $[i,i+1] \times [j,j+1]$
contained within the region $\{(x,y): \ |x|+|y| \leq n+1\}$.
Figure~\ref{fig:dominos} shows one of the $2^{(4)(5)/2}$ domino tilings
of the Aztec diamond of order 4.
Mihai Ciucu~\cite{Ci} proved combinatorially
that the number of domino tilings of the $2n$-by-$2n$ square
(\href{https://oeis.org/A004003}{A004003})
can be written in the form $2^n f(n)^2$
where $f(n)$ is the number of domino tilings of the region
exemplified for $n=4$ in Figure~\ref{fig:ciucu}
(\href{https://oeis.org/A065072}{A065072}).
\begin{figure}
\caption{Ciucu's way of halving the 8-by-8 square.}
\label{fig:ciucu}
\end{figure}
Henry Cohn~\cite{Co} proved that the function sending $n$ to $f(n)$
is uniformly continuous under the 2-adic metric
and thus extends to a function defined on all of $\mathds{Z}$
and indeed all of $\mathds{Z}_2$;
moreover, he showed that this extension satisfies
\begin{equation}
f(-1-n) = \left\{\begin{array}{rl}
f(n) & \mbox{when $n$ is congruent to 0 or 3 (mod 4)}, \\
-f(n) & \mbox{when $n$ is congruent to 1 or 2 (mod 4)}.
\end{array} \right.
\end{equation}
Barkley and Liu~\cite{BL} have recently proved results
about 2-divisibility for the number of perfect matchings of a graph,
including as a special case the number of domino tilings of a rectangle,
but there is more refined work still to be done along the lines of Cohn's paper.
For instance, the mod 8 residue of
the number of domino tilings of the $2n$-by-$(2n+2)$ rectangle
appears to depend only on the mod 4 residue of $n$;
the same goes for the number of domino tilings of the $2n$-by-$4n$ rectangle.
In this article we extend the discussion
to other sorts of tiles, specifically, tetrominos.
A {\em tetromino} is a connected subset of the grid
that is a union of four grid-squares,
just as a domino is a union of two grid-squares.
Up to symmetry, there are five kinds of tetrominos: straight tetrominos,
skew tetrominos, L-tetrominos, square tetrominos, and T-tetrominos.
They are shown in Figure~\ref{fig:the-six}, preceded by the domino.
These six tiles can be placed on a square grid in
2, 2, 4, 8, 1, and 4 translationally-inequivalent ways, respectively
(where rotations and reflections are permitted).
These are the sorts of tiles considered in this article.
({\em Trominos} -- unions of three grid-squares --
will be considered elsewhere.)
\begin{figure}
\caption{A domino, a straight tetromino, a skew tetromino,
an L-tetromino, a square tetromino, and a T-tetromino.}
\label{fig:the-six}
\end{figure}
\end{section}
\begin{section}{Skew and straight tetrominos}
\label{sec:skewstraight}
\begin{figure}
\caption{Tiling the Aztec diamond of order 3 with skew and straight tetrominos.}
\label{fig:olympiad}
\end{figure}
I'll start with a warm-up puzzle that's
roughly at the level of a math olympiad:
Prove that an Aztec diamond of order $n$ can be tiled
by skew and straight tetrominos
(as shown in Figure~\ref{fig:olympiad} for $n=3$)
only if $n$ is congruent to 0 or 3 (mod 4).
The puzzle can be solved using a valuation argument
(sometimes called a generalized coloring argument):
one can construct a mapping from the grid-cells to an appropriate abelian group
(a ``weight function'') and show that when $n$ is 1 or 2 (mod 4),
the sum of the weights of the tiles
can't equal the sum of the weights of the region being tiled,
where the weight of a tile or a region being tiled
is the sum of the weights of the constituent cells.
Readers who are already familiar with this technique
might enjoy the challenge of attempting to solve the problem purely mentally.
\end{section}
\begin{section}{Dominos and square tetrominos}
\label{sec:domsquare}
In this section we use dominos and square tetrominos.
Thus an Aztec diamond of order 1
(better known as the 2-by-2 square) can be tiled in 3 ways:
with two horizontal dominos, two vertical dominos, or a single square tetromino.
The Aztec diamond of order 2 can be tiled in $2^{(2)(3)/2} = 8$ ways
using dominos, and can be tiled in an additional 11 ways
if one or more square tetrominos are included,
as shown in Figure~\ref{fig:eleven}.
Thus there are a total of $8+11=19$ ways
to tile an Aztec diamond of order 2
using dominos and square tetrominos.
\begin{figure}
\caption{Tiling the Aztec diamond of order 2
with dominos and at least one square tetromino.}
\label{fig:eleven}
\end{figure}
Define $M(n)$ (with $n \geq 0$) as
the number of tilings of the Aztec diamond of order $n$
using dominos and square tetrominos.
This is \href{https://oeis.org/A356512}{A356512}.
Trivially we have $M(0) = 1$ (since the Aztec diamond of order 0 is empty)
and we have already seen that $M(1) = 3$ and $M(2) = 19$.
Figure~\ref{fig:Mtable} shows the terms of the sequence $M(n)$
for $n$ ranging from 0 to 12,
computed using a program written by David desJardins.
The reader may wish to pause here to consider the problem of showing that $M(n)$ is always odd;
a solution will be given in section~\ref{sec:cong}.
\begin{figure}
\caption{Enumeration of tilings of Aztec diamonds
using dominos and square tetrominos.}
\label{fig:Mtable}
\end{figure}
These numbers grow quadratic-exponentially as a function of $n$,
and I have no conjectural formula for the $n$th term,
nor a conjectural recurrence relation for the sequence,
nor any efficient method of computing terms.
Nonetheless, something systematic is going on.
I have already mentioned that all the terms are odd.
Taking this observation further, one notices that
the numbers’ residues mod 4 are
$$1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1,$$
the residues mod 8 are
$$1, 3, 3, 5, 5, 7, 7, 1, 1, 3, 3, 5, 5,$$
and the residues mod 16 are
$$1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13.$$
{\bf Conjecture 1}: For all $k \geq 1$,
the mod $2^k$ residue of $M(n)$ is periodic with period dividing $2^k$.
That is, $2^k$ divides $M(n+2^k) - M(n)$ for all $k,n$.
I tried to prove this conjecture by reducing it to an assertion
about alternating-sign matrices but I was unsuccessful.
Note that if the conjecture is true
then $M(n) \equiv n + 1 + (1 + (-1)^{n + 1})/2$ (mod 8).
This congruence might also hold mod 16
but it certainly cannot hold mod $2^k$ for all $k$,
since that would require that $M(n)$ actually equals
$n + 1 + (1 + (-1)^{n + 1})/2$, which is clearly not the case for $n \geq 2$.
And indeed $M(2) = 19 \not\equiv 3$ (mod 32).
A deeper consequence of Conjecture 1 is that
the function sending $n$ to $M(n)$ is 2-adically continuous.
Moreover, the function appears to satisfy a kind of symmetry
analogous to the functional equation (1)
mentioned at the end of section~\ref{sec:intro}.
{\bf Conjecture 2}: For all $k \geq 1$,
if $n + n' \equiv -3$ (mod $2^k$) then $M(n) + M(n') \equiv 0$ (mod $2^k$).
That is, if one extends $M: \mathds{N} \rightarrow \mathds{N}$
to the 2-adic function $\widehat{M}: \mathds{Z}_2 \rightarrow \mathds{Z}_2$,
one has $\widehat{M}(-3-n) = - \widehat{M}(n)$.
Although in this article I am limiting myself
to discussion of tilings of Aztec diamonds,
I have also looked at tilings of other regions
using dominos and square tetrominos,
and the same phenomenon of 2-adic continuity arises fairly broadly there.
For instance, for the $2n$-by-$2n$ square, the $2n$-by-$(2n+2)$ rectangle, and the $2n$-by-$4n$ rectangle,
the number of tilings with dominos and square tetrominos
always seems to be congruent to $2n+1$ mod 8.
\end{section}
\begin{section}{Skew tetrominos and square tetrominos}
\label{sec:skewsquare}
In~\cite{Pr} I considered tilings of Aztec diamonds
by skew tetrominos and square tetrominos.
If we require that all skew tetrominos be horizontal,
interesting numerical patterns appear.
(Of course we would get the same result
if we required that all skew tetrominos be vertical.)
In this section we allow horizontal skew tetrominos and square tetrominos
as seen in Figure~\ref{fig:tetra}, which depicts
all six tilings of the Aztec diamond of order 3
using square tetrominos and horizontal skew tetrominos.
\begin{figure}
\caption{Tiling the Aztec diamond of order 3
with horizontal skew tetrominos and square tetrominos.}
\label{fig:tetra}
\end{figure}
\begin{figure}
\caption{Enumeration of tilings of Aztec diamonds
using horizontal skew tetrominos and square tetrominos.}
\label{fig:Ltable}
\end{figure}
Define $L(n)$ (with $n \geq 0$) as
the number of tilings of the Aztec diamond of order $n$
using horizontal skew tetrominos and square tetrominos.
This is \href{https://oeis.org/A356513}{A356513}.
Trivially we have $L(0) = 1$ and $L(1) = 1$.
Figure~\ref{fig:Ltable} shows the terms
of the sequence $L(n)$ for $n$ ranging from 0 to 15,
again computed using the program written by David desJardins.
The sequence grows quadratic-exponentially,
and once again, I have no conjectural formula,
but as before there are patterns that call out for explanation.
Noticing that all but the first two terms are even,
one might naturally think to look at the multiplicity of 2
in the prime factorization of $L(n)$, obtaining the sequence
0,0,1,1,3,2,5,3,7,4,9,5,11,6,13,7,\dots,
which (once we throw out the initial 0)
we recognize as an interspersal of the arithmetic progressions
0,1,2,3,4,5,6,7,\dots and 1,3,5,7,9,11,13,\dots.
{\bf Conjecture 3}:
For $n \geq 1$,
the multiplicity of 2 in the prime factorization of $L(n)$
is $n-1$ if $n$ is even and $(n-1)/2$ if $n$ is odd.
Going further,
let $L_0(m) = L(2m) / 2^{2m-1}$ and $L_1(m) = L(2m-1) / 2^{m-1}$,
so that (if Conjecture 3 holds)
$L_0(m)$ and $L_1(m)$ are odd integers for all $m$. These two new sequences
are shown in Figure~\ref{fig:JKtable}.
\begin{figure}
\caption{Values of $L_0(m)$ and $L_1(m)$.}
\label{fig:JKtable}
\end{figure}
The mod 4 residues of the $L_0$ sequence go $1, 1, 3, 3, 1, 1, 3$
while those of the $L_1$ sequence go $1, 3, 3, 1, 1, 3, 3, 1$.
That's not much evidence to go on,
so perhaps it would be prudent not to make a conjecture,
but I choose to be hopeful.
{\bf Conjecture 4}: For all $k \geq 0$, the mod $2^k$ residue of $L_0(m)$
is periodic with period dividing $2^k$. Likewise for $L_1(m)$.
We do not observe such patterns in the numbers of tilings
when both horizontal and vertical skew tetrominos are allowed
along with square tetrominos as in~\cite{Pr}.
More specifically, if we count tilings of Aztec diamonds
in which we are permitted to use all four kinds of skew tetrominos
as well as square tetrominos, the resulting sequence, taken mod 4, goes
1, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, \dots;
if there is a period here, and it is a power of 2,
it must be at least 16.
The prime $p=2$ appears to be special
for the enumerative problems I described above;
looking at the $M$ and $L$ sequences
mod 3 or mod 5 yields no discernible patterns.
\end{section}
\begin{section}{Assorted congruential problems}
\label{sec:cong}
For each of the sixty-three nonempty subsets of
the set of six tiles shown in Figure~\ref{fig:the-six},
we can ask in how many ways
it is possible to tile the Aztec diamond of order $n$
using only tiles from that set,
allowing translations, rotations, and reflections of tiles.
These are the enumerative problems considered in this section.
(One could expand the set of tiling problems
by distinguishing between different orientations of the tiles,
as was done in the preceding section
where we permitted horizontal skew tetrominos
but forbade vertical skew tetrominos;
since there are $2+2+4+8+1+4=21$ different tiles up to translation,
we would obtain over two million different problems,
and even if we mod out the $2^{21}-1$ problems by a dihedral action of order 8,
that is still too many problems to consider exhaustively.
One that appears to be interesting is discussed at the end of this section.)
In each of the sixty-three cases, I used the aforementioned program
to count the tilings of the Aztec diamond of order $n$,
with $n$ going from 1 to 8, using the allowed tiles.
Although no 2-adic continuity phenomena arose from these experiments,
there were definite patterns in the parity,
and in a few cases there were congruence patterns modulo higher powers of 2.
Here I will adopt a six-bit code to represent
the sixty-three tiling problems,
in which the six successive bits (from left to right) equal 1 or 0
according to whether or not dominos, straight tetrominos, skew tetrominos,
L-tetrominos, square tetrominos, and T-tetrominos are allowed.
For instance, the case treated in section~\ref{sec:skewstraight},
in which only straight tetrominos and skew tetrominos are allowed
(see Figure~\ref{fig:olympiad}), would be assigned the code 011000;
the case treated in section~\ref{sec:domsquare},
in which only dominos and square tetrominos are allowed
(see Figure~\ref{fig:eleven}), would be assigned the code 100010;
and the case of unconstrained skew and square tetrominos
(briefly discussed in section~\ref{sec:skewsquare})
would be assigned the code 001010.
In one-third of the 63 cases,
I observed that for all $n$ between 1 and 8,
the number of tilings of the Aztec diamond of order $n$ is even.
These were the cases associated with the six-bit codes
001001, 001100, 001101, 011001, 011100, 011101, 100001,
100100, 100101, 101000, 101001, 101100, 101101, 110000,
110001, 110100, 110101, 111000, 111001, 111100, and 111101.
Presumably some (perhaps all) of these examples can be resolved
by showing that there are no tilings that are invariant
under the full dihedral group, since in that case all orbits
would contain an even number of tilings.
Three of the 21 cases were especially interesting.
In case 011100, all terms were divisible by 8;
in case 100001, all terms after the first were divisible by 8;
and in case 110001, all terms were congruent to 2 (mod 4).
There were also four cases in which I observed that
the number of tilings of the Aztec diamond of order $n$ is even
for all $n$ between 2 and 8
(with the number of tilings being the odd number 1 in the case $n=1$).
These were the cases associated with the six-bit codes
001010, 001110, 011010, and 011110.
In the cases 001101, 100001, 100011, and 111000
it appears that the exponent of 2 in the number of tilings
may be going to infinity with $n$,
though with such scant evidence it would be rash
to place too much faith in this guess.
Additionally, there is one case in which the number of tilings
of the Aztec diamond of order $n$ is always odd,
namely, tilings using only dominos and square tetrominos.
Indeed, if we assign each tiling weight $(-1)^s$ where $s$ is the number of square tetrominos,
I claim that the sum of the weights is 1.
We can prove this using a sign-reversing involution
that scans through the tiling in some fashion in search of a 2-by-2 block
that is tiled either with a square tetromino or with two vertical dominos
and switches between the two possibilities.
The fixed points of this involution are tilings that use only horizontal dominos,
and there is just one of those.
Finally, leaving the small world of the $2^6-1$ problems
and dipping our toe into the big world of the $2^{21}-1$ problems,
we consider tilings of the Aztec diamond of order $n$
by dominos and horizontal straight tetrominos.
This is \href{https://oeis.org/A356523}{A356523}, and begins
1, 2, 11, 209, 12748, 2432209, 1473519065, \dots.
It appears that the number of tilings is
even when $n \equiv 1$ (mod 3) and odd otherwise;
this has been verified for $1 \leq n \leq 16$.
\end{section}
\begin{section}{Reduction to perfect matchings}
\label{sec:matchings}
The $L$ sequence from section~\ref{sec:skewsquare}
has an interpretation in terms of perfect matchings.
To see why, suppose we have a tiling of the Aztec diamond of order $n$
using horizontal skew tetrominos and square tetrominos.
Dividing each tetromino into two horizontal dominos
gives us a tiling of the Aztec diamond by horizontal dominos,
but it is easy to see that there is exactly one such tiling (call it $T$).
Hence each tetromino is obtained by gluing together two dominos in $T$.
That is, the tetromino tilings correspond to perfect matchings in the graph
whose vertices correspond to the dominos in $T$
with an edge joining two vertices if the corresponding dominos
form a horizontal skew tetromino or square tetromino.
It is not hard to see that this graph is similar
to the $n$-by-$n$ square except that the diagonal has been ``doubled'';
for instance, the right panel of Figure~\ref{fig:double-double}
shows the graph for $n=4$.
\begin{figure}
\caption{Deriving the square graph with doubled diagonal.}
\label{fig:double-double}
\end{figure}
A similar analysis can be applied to tilings of Aztec diamonds
using horizontal skew tetrominos and horizontal straight tetrominos.
In this case the Aztec diamond splits
into two non-interacting halves (top half and bottom half),
each of which can be tiled independently of the other,
and the tilings of either half correspond
to perfect matchings of a triangle graph
as shown in Figure~\ref{fig:triangles}.
Thus the number of such tetromino tilings of the Aztec diamond of order $n$
is equal to the square of the $n$th term of sequence
\href{https://oeis.org/A071093}{A071093}.
Studying the first 25 terms, I find that
the sequence seems to have 2-adic properties of its own.
The largest power of 2 dividing the $n$th term of the sequence A071093
appears to be $\lfloor n/2 \rfloor$,
and the 2-free part appears to satisfies 2-adic continuity:
for instance, its value mod 16 seems to be determined by $n$ mod 16.
\begin{figure}
\caption{Deriving the double triangle graph.}
\label{fig:triangles}
\end{figure}
What if we superimpose the two graphs,
obtaining the graph shown at the right half of
Figure~\ref{fig:superimpose}?
This is equivalent to tiling an Aztec diamond using
horizontal skew tetrominos, horizontal straight tetrominos, and square tetrominos.
Then, counting the tilings, we obtain the integer sequence
1, 2, 10, 116, 3212, 209152, 32133552, 11631456480, 9922509270288,
19946786274879008, 94492874103638971552, 1054865198752147761744448,
\dots.
This is \href{https://oeis.org/A356514}{A356514}.
It appears that the number of tilings is divisible
by $2^{\lfloor n/2 \rfloor}$.
\begin{figure}
\caption{Superimposing Figures~\ref{fig:double-double}
\label{fig:superimpose}
\end{figure}
\end{section}
\begin{section}{Some thoughts}
\label{sec:thoughts}
The articles of Lovasz~\cite{Lo}, Ciucu~\cite{Ci},
Pachter~\cite{Pa}, and Barkley and Liu~\cite{BL}
give ways to find the largest power of 2
that divides the number of perfect matchings of a graph.
This should provide traction for Conjecture 3,
since we saw in section~\ref{sec:matchings} that
the $L$ sequence has an interpretation in terms of perfect matchings
of certain graphs.
Graphs of this kind appear in the paper of Ciucu~\cite{Ci};
in particular, his Lemma 1.1 shows that
the number of perfect matchings is divisible by $2^{\lfloor n/2 \rfloor}$.
By bringing ideas from Pachter~\cite{Pa}, one might be able to prove Conjecture 3,
as well as some of the other 2-divisibility conjectures from this article.
The only work I know of that provides detailed 2-adic information
about the 2-free part of numbers that count tilings
is the work of Cohn~\cite{Co}.
Cohn's approach presupposes the existence of an exact formula
(in Cohn's case, an explicit product of algebraic integers);
perhaps something similar can be done for perfect matchings
of the square graph with doubled diagonal,
yielding a proof of Conjecture 4.
Conjectures 1 and 2 seem harder.
The product formula exploited by Cohn
was discovered by Temperley and Fischer~\cite{TF}
and independently by Kasteleyn~\cite{Ka} at about the same time;
those researchers made use of the fact that,
just as determinants and Pfaffians of matrices can be expressed as sums of terms
associated with perfect matchings of the set of rows and columns,
one can conversely express the number of perfect matchings of a planar graph
in terms of the determinant or Pfaffian of an associated matrix.
I know of no way of recast the $M$ sequence
as enumerating perfect matchings of graphs.
However, it is easy to recast the $M$ sequence as enumerating
perfect matchings of certain hypergraphs.
Can any of the existing notions of hyperdeterminants be brought to bear?
Perhaps a reading of~\cite{GKZ} would suggest possible approaches.
Kuperberg's elegant solution~\cite{Ku}
to the alternating sign matrix conjecture
exploits the power of the Yang-Baxter equation in statistical mechanics.
It's possible that tools for analyzing the new problems described in this article
will be found in the existing literature at
the interface between algebra and statistical mechanics.
In any case, inasmuch as Conjectures 1 and 2 are reminiscent of Cohn's work,
and inasmuch as Cohn's argument hinges on an exact product formula,
one might hope that an exact formula of some kind
can be found for the $M$ sequence. Such an exact formula would have other uses.
In~\cite{CEP} and~\cite{CLP}, Henry Cohn, Noam Elkies, Michael Larsen and I
used exact enumeration results to prove
concentration theorems for random tilings.
One might hope that the curious 2-adic phenomena discussed in this article
hint at the existence of algebraic machinery that
could be applied to the task of showing us
what random tilings associated with Conjecture 1
look like in the limit as size goes to infinity.
Preliminary experiments suggest that
there is a ``frozen region'' near the boundary,
but I have no idea how far into the interior it extends.
\end{section}
\noindent
I thank David desJardins for the software that made this research possible,
and Noam Elkies for helpful comments.
Above all I thank Michael Larsen for his many years
of friendship and mathematical camaraderie.
\label{sec:biblio}
\end{document} | math | 23,474 |
\begin{equation}gin{document}
\tildeitle{\bf\normalsize CONSTRAINED LARGE SOLUTIONS TO LERAY'S PROBLEM IN A DISTORTED STRIP WITH THE NAVIER-SLIP BOUNDARY CONDITION}
\author{\normalsize\sqrtc Zijin Li, Xinghong Pan and Jiaqi Yang}
\date{}
\title{f
ormalsize CONSTRAINED LARGE SOLUTIONS TO LERAY'S PROBLEM IN A DISTORTED STRIP WITH THE NAVIER-SLIP BOUNDARY CONDITION}
\begin{equation}gin{abstract}
In this paper, we solve the Leray's problem for the stationary Navier-Stokes system in a 2D infinite distorted strip with the Navier-slip boundary condition. The existence, uniqueness, regularity and asymptotic behavior of the solution are investigated. Moreover, we discuss how the friction coefficient affects the well-posedness of the solution. Due to the validity of the Korn's inequality, all constants in each a priori estimate are independent of the friction coefficient. Thus our method is also valid for the total-slip and no-slip cases. The main novelty is that the total flux of the velocity can be relatively large (proportional to the {\inftyt slip length}) when the friction coefficient of the Navier-slip boundary condition is small, which is essentially different from the 3D case.
{\sqrtc Keywords:} Stationary Navier-Stokes system, Navier-slip boundary condition, Leray's problem, Distorted strip.
{\sqrtc Mathematical Subject Classification 2020:} 35Q35, 76D05
\epsilonnd{abstract}
\tildeableofcontents
\sqrtection{Introduction}\lambdabel{SEC1}
Consider the Navier-Stokes equations
\begin{equation}\lambdabel{NS}
\left\{
\begin{equation}gin{aligned}
&\boldsymbol{u}\cdot\nabla \boldsymbol{u}+\nabla p-\Delta \boldsymbol{u}=0,\\
&\nabla\cdot \boldsymbol{u}=0,
\epsilonnd{aligned}
\right.\quad \tildeext{in}\quad \mathcal{S}\sqrtubset {\mathbb R}^2,
\epsilonnd{equation}
subject to the Navier-slip boundary condition:
\begin{equation}\lambdabel{NBC}
\left\{
\begin{equation}gin{aligned}
&2(\mathbb{S}\boldsymbol{u}\cdot\boldsymbol{n})_{\mathrm{tan}}+\alphapha \boldsymbol{u}_{\mathrm{tan}}=0,\\
&\boldsymbol{u}\cdot\boldsymbol{n}=0,\\
\epsilonnd{aligned}
\right.\quad\tildeext{on}\quad\partial\mathcal{S}.
\epsilonnd{equation}
Here $\mathbb{S}\boldsymbol{u}=\fracrac{1}{2}\left(\nablabla \boldsymbol{u}+\nablabla^T \boldsymbol{u}\right)$ is the stress tensor, where $\nablabla^T \boldsymbol{u}$ represent the transpose of $\nablabla \boldsymbol{u}$, and $\boldsymbol{n}$ is the unit outer normal vector of $\partial \mathcal{S}$. For a vector field $\boldsymbol{v}$, we denote $\boldsymbol{v}_{\mathrm{tan}}$ its tangential part:
\[
\boldsymbol{v}_{\mathrm{tan}}:=\boldsymbol{v}-(\boldsymbol{v}\cdot\boldsymbol{n})\boldsymbol{n},
\]
and $\alpha\gammaeq0$ in \epsilonqref{NBC} stands for the friction coefficient which may depend on various elements, such as the property of the boundary and the viscosity of the fluid. When $\alpha\tildeo0_+$, the boundary condition \epsilonqref{NBC} turns to be the total Navier-slip boundary condition, while when $\alpha\tildeo\inftynfty$, the boundary condition \epsilonqref{NBC} degenerates into the no-slip boundary condition $u\epsilonquiv 0$. In this paper, we consider the general case, which assumes $0\leq\alpha\leq +\infty$.
The domain $\mathcal{S}$ is a two dimensional infinitely distorted smooth strip as follows.
\begin{equation}gin{figure}[H]\lambdabel{FIG0}
\centering
\inftyncludegraphics[scale=0.3]{P1.pdf}
\caption{The infinitely distorted strip $\mathcal{S}$.}
\epsilonnd{figure}
Here, $\mathcal{S}_R$ and $\mathcal{S}_L$ are two semi-infinitely long straight strips. In the cartesian coordinate system $x_1Ox_2$, the strip
\[
\mathcal{S}_R:=\{x\inftyn\mathbb{R}^2\,:\,(x_1,x_2)\inftyn(0,\infty)\tildeimes(0,1)\},
\]
while in the cartesian coordinate system $y_1O'y_2$, the stripe
\[
\mathcal{S}_L:=\{y\inftyn\mathbb{R}^2\,:\,(y_1,y_2)\inftyn(-\infty,0)\tildeimes(0,c_0)\}.
\]
Here $c_0$ is a fixed constant. They are smoothly connected by the compact distorted part $\mathcal{S}_0$ in the middle. We do not insist the domain $\mathcal{S}$ to be simply connected, but all obstacles, with their boundaries are smooth Jordan curves, must lie in $\mathcal{S}_0$, and keep away from upper and lower boundaries of $\mathcal{S}$, i.e. $\partial\mathcal{S}_+$ and $\partial\mathcal{S}_-$, respectively.
Before stating our main results, we give some notations for later convenience.
\sqrtubsection*{Notations}
\quad\ Throughout this paper, $C_{a,b,c,...}$ denotes a positive constant depending on $a,\,b,\, c,\,...$, which may be different from line to line. For simplicity, a constant $C_\mathcal{S}$, depending on geometry properties of $\mathcal{S}$, is usually abbreviated by $C$. Dependence on $\mathcal{S}$ is default unless independence of $\mathcal{S}$ is particularly stated. We use $\boldsymbol{e}_1,\,\boldsymbol{e}_2$ to denote the unit standard basis in the cartesian coordinate system $x_1Ox_2$, and $\boldsymbol{e}_1',\,\boldsymbol{e}_2'$ to denote the unit standard basis in the cartesian coordinate system $y_1O'y_2$. Meanwhile, for any $\zeta>1$, $\mathcal{S}_{R,\zeta}=(0,\zeta)\tildeimes(0,1)$ in the cartesian coordinate system $x_1Ox_2$, and $\mathcal{S}_{L,\zeta}=(-\zeta,0)\tildeimes(0,c_0)$ in the cartesian coordinate system $y_1O'y_2$. Then the truncated strip is defined by:
\begin{equation}\lambdabel{Szeta}
\mathcal{S}_\zeta:=\mathcal{S}_{L,\zeta}\cup\mathcal{S}_0\cup\mathcal{S}_{R,\zeta}.
\epsilonnd{equation}
We use $\Upsilon^{\partialm}_\zeta$ to denote the right and left part of $\mathcal{S}_\zeta\backslash\mathcal{S}_{\zeta-1}$ as follows:
\[
\Upsilon^{+}_\zeta:=\mathcal{S}_{R,\zeta}\backslash\mathcal{S}_{R,\zeta-1},\quad\quad\Upsilon^{-}_\zeta:=\mathcal{S}_{L,\zeta}\backslash\mathcal{S}_{L,\zeta-1}.
\]
We also apply $A\lesssim B$ to state $A\leq CB$. Moreover, $A\sqrtigmameq B$ means both $A\lesssim B$ and $B\lesssim A$.
For $1\leq p\leq\inftynfty$ and $k\inftyn\mathbb{N}$, $L^p$ denotes the usual Lebesgue space with norm
\[
\|f\|_{L^p(D)}:=
\left\{
\begin{equation}gin{aligned}
&\left(\inftynt_{D}|f(x)|^pdx\right)^{1/p},\quaduad &1\leq p<\inftynfty,\\[3mm]
&\mathrm{ess sup}_{x\inftyn D}|f(x)|,\quaduad &p=\inftynfty,\\
\epsilonnd{aligned}
\right.
\]
while $W^{k,p}$ denotes the usual Sobolev space with its norm
\[
\begin{equation}gin{split}
\|f\|_{W^{k,p}(D)}:=&\sqrtum_{0\leq|L|\leq k}\|\nablabla^L f\|_{L^p(D)},\\
\epsilonnd{split}
\]
where $L=(l_1,l_2)$ is a multi-index. We also simply denote $W^{k,p}$ by $H^k$ provided $p=2$. Finally, $\omegaverline{D}$ denote the closure of a domain $D$. A function $g\inftyn W^{k,p}_{\mathrm{loc}}(D)$ or $W^{k,p}_{\mathrm{loc}}(\omegaverline{D})$ function means $g\inftyn W^{k,p}(\tildeilde{D})$, for any $\tildeilde{D}$ compactly contained in $D$ or $\omegaverline{D}$.
For the 2D vector-valued function, we define
\[
\begin{equation}gin{split}
\mathcal{H}(\mathcal{S})&:=\left\{\boldsymbol{\varphi}\inftyn H^1({\mathcal{S}};{\mathbb R}^2):\,\boldsymbol{\varphi}\cdot\boldsymbol{n}\big|_{\partial \mathcal{S}}=0 \right\},\\
\mathcal{H}_\sqrtigmagma({\mathcal{S}})&:=\left\{\boldsymbol{\varphi}\inftyn H^1({\mathcal{S}};\,\mathbb{R}^2):\,\nabla\cdot \boldsymbol{\varphi}=0,\ \boldsymbol{\varphi}\cdot \boldsymbol{n}\big|_{\partial \mathcal{S}}=0\right\},
\epsilonnd{split}
\]
and
\[
\mathcal{H}_{\sqrtigmagma,{\mathrm{loc}}}(\omegaverline{\mathcal{S}}):=\left\{\boldsymbol{\varphi}\inftyn H^1_{{\mathrm{loc}}}(\omegaverline{\mathcal{S}};\,\mathbb{R}^2):\,\nabla\cdot \boldsymbol{\varphi}=0,\ \boldsymbol{\varphi}\cdot \boldsymbol{n}\big|_{\partial \mathcal{S}}=0\right\}.
\]
We also denote
\[
\boldsymbol{X}:=\big\{\boldsymbol{\varphi}\inftyn C_c^\inftynfty(\omegaverline{\mathcal{S}}\,;\,\mathbb{R}^2):\ \nabla\cdot \boldsymbol{\varphi}=0,\ \boldsymbol{\varphi}\cdot\boldsymbol{n}\big|_{\partial\mathcal{S}}=0\big\}.
\]
Clearly, $\boldsymbol{X}$ is dense in $\mathcal{H}_\sqrtigmagma$ in $H^1(\mathcal{S})$ norm. For matrices $\boldsymbol{\Gammaamma}=(\gammaamma_{ij})_{1\leq i,j\leq 2}$ and $\boldsymbol{K}=(\kappa_{ij})_{1\leq i,j\leq 2}$, we denote
\[
\boldsymbol{\Gammaamma}:\boldsymbol{K}=\sqrtum^{2}_{i,j=1}\gammaamma_{ij}\kappa_{ij}.
\]
\sqrtubsection{The generalized Leray's problem in the distorted strip}
\quad\ For a given flux $\Phi$ which is supposed to be nonnegative without loss of generality, we consider Poiseuille-type flows, $\boldsymbol{P}^R_{\Phi}=:{P}^R_{\Phi} (x_2)\boldsymbol{e}_1$ and $\boldsymbol{P}^L_{\Phi}=:{P}^L_{\Phi} (y_2)\boldsymbol{e}_1'$, of \epsilonqref{NS} with the boundary condition \epsilonqref{NBC} in $\mathcal{S}_i$ ($i$ denotes $R$ or $L$), then we will find that
\begin{equation}\lambdabel{POSS2}
\left\{
\begin{equation}gin{aligned}
&-\left(P^R_\Phi(x_2)\right)''=C_R,\quad\quad\quad\tildeext{in }\quad (0,1),\\
&\left(P^R_\Phi(0)\right)'=\alpha P^R_\Phi(0),\quad \left(P^R_\Phi(1)\right)'=-\alpha P^R_\Phi(1),\\
&\inftynt_{0}^{1}P^R_\Phi(x_2)dx_2=\Phi, \\
\epsilonnd{aligned}
\right.
\epsilonnd{equation}
and
\begin{equation}\lambdabel{POSS1}
\left\{
\begin{equation}gin{aligned}
&-\left(P^L_\Phi(y_2)\right)''=C_L,\quad\quad\quad\tildeext{in }\quad (0,c_0),\\
&\left(P^L_\Phi(0)\right)'=\alpha P^L_\Phi(0),\quad \left(P^L_\Phi(c_0)\right)'=-\alpha P^L_\Phi(c_0),\\
&\inftynt_{0}^{c_0}P^L_\Phi(y_2)dy_2=\Phi, \\
\epsilonnd{aligned}
\right.
\epsilonnd{equation}
where the constants $C_i$ are uniquely related to $\Phi$. Direct computation shows that
\begin{equation}\lambdabel{PF}
\left\{
\begin{equation}gin{split}
P_{\Phi}^R(x_2)&=\frac{6\Phi}{6+\alpha}\left[\alpha(-x^2_2+x_2)+1\right];\\
P_{\Phi}^L(y_2)&=\frac{6\Phi}{c_0^2\left(6+c_0\alpha\right)}\left[\alpha(-y^2_2+c_0y_2)+c_0\right].\\
\epsilonnd{split}
\right.
\epsilonnd{equation}
And
\begin{equation}\lambdabel{EHP}
\begin{aligned}
|(P^i_\Phi)'|\leq C\frac{\alpha\Phi}{1+\alpha},\quad\fracorall i\inftyn\{L,R\},
\epsilonnd{aligned}
\epsilonnd{equation}
where $C>0$ is a constant independent of both $\alpha$ and $\Phi$.
The main objective of this paper is to study the solvability of the following generalized Leray's problem: For a given flux $\Phi$, to find a pair $(\boldsymbol{u},p)$ such that
\begin{equation}\lambdabel{GL1}
\left\{
\begin{aligned}
&\boldsymbol{u}\cdot\nabla \boldsymbol{u}+\nabla p-\Delta \boldsymbol{u}=0,\quad \nabla\cdot \boldsymbol{u}=0,\quad \hskip .1cm\tildeext{in}\quad \mathcal{S}, \\
&2(\mathbb{S}\boldsymbol{u}\cdot\boldsymbol{n})_{\mathrm{tan}}+\alphapha \boldsymbol{u}_{\mathrm{tan}}=0,\quad \boldsymbol{u}\cdot\boldsymbol{n}=0, \quad \tildeext{on}\quad \partial\mathcal{S},\\
\epsilonnd{aligned}
\right.
\epsilonnd{equation}
with
\begin{equation}\lambdabel{GL2}
\inftynt^{1}_{0}u_1(x_1, x_2) d x_2=\Phi,\quad\tildeext{for any}\quad x\inftyn\mathcal{S}_R
\epsilonnd{equation}
and
\begin{equation}\lambdabel{GL3}
\boldsymbol{u}\rightarrow \boldsymbol{P}^i_{\Phi},\quad \tildeext{as}\quad |x|\rightarrow \infty \ \tildeext{ in }\ \mathcal{S}_i.
\epsilonnd{equation}
To prove the existence of the above generalized Leray's problem, we first introduce a weak formulation. Multiplying \epsilonqref{GL1}$_1$ with $\boldsymbol{\varphi}\inftyn\boldsymbol{X}$ and integration by parts, by using the boundary condition \epsilonqref{GL1}$_2$, we can obtain
\begin{equation}\lambdabel{weaksd}
\begin{equation}gin{split}
&2\inftynt_{\mathcal{S}}\mathbb{S}\boldsymbol{u}:\mathbb{S}\boldsymbol{\varphi} dx+\alpha\inftynt_{\partial\mathcal{S}}\boldsymbol{u}_{\mathrm{tan}}\cdot\boldsymbol{\varphi}_{\mathrm{tan}}dS+\inftynt_{\mathcal{S}}\boldsymbol{u}\cdot\nablabla \boldsymbol{\varphi} \cdot \boldsymbol{u} dx=0,\quad \tildeext{for all }\ \boldsymbol{\varphi}\inftyn \boldsymbol{X}.
\epsilonnd{split}
\epsilonnd{equation}
Now we define the weak solution of the generalized Leary's problem:
\begin{equation}gin{definition}\lambdabel{weaksd1}
A vector $\boldsymbol{u}:\mathcal{S}\tildeo {\mathbb R}^2$ is called a {\inftyt weak} solution of the generalized Leray problem \epsilonqref{GL1} to \epsilonqref{GL3} if and only if
\begin{equation}gin{itemize}
\sqrtetlength{\inftytemsep}{-2 pt}
\inftytem[(i).] $\boldsymbol{u}\inftyn \mathcal{H}_{\sqrtigmagma,{\mathrm{loc}}}(\omegaverline{\mathcal{S}})$;
\inftytem[(ii).] $\boldsymbol{u}$ satisfies \epsilonqref{weaksd};
\inftytem[(iii).] $\boldsymbol{u}$ satisfies \epsilonqref{GL2} in the trace sense;
\inftytem[(iv).] $\boldsymbol{u}-\boldsymbol{P}^i_{\Phi}\inftyn {H}^1(\mathcal{S}_i)$, for $i=L,R$.
\epsilonnd{itemize}
\epsilonnd{definition}
\quaded
\begin{equation}gin{remark}
The weak solution also satisfies a generalized version of \epsilonqref{GL3}. Actually, it follows from the trace inequality (\cite[Theorem II.4.1]{Galdi2011}) that for any $x\inftyn\mathcal{S}_R$:
\begin{equation}\lambdabel{WK111}
\begin{aligned}
&\inftynt^1_0|\boldsymbol{u}(x_1,x_2)-\boldsymbol{P}^R_{\Phi}(x_2)|^2dx_2\leq C\|\boldsymbol{u}-\boldsymbol{P}^R_{\Phi}\|^2_{H^1([x_1,+\infty)\tildeimes[0,1])},\quad\fracorall x_1>0,
\epsilonnd{aligned}
\epsilonnd{equation}
where the constant $C$ is independent of $x_1$. Using $(iv)$ in Definition \ref{weaksd1}, the estimate \epsilonqref{WK111} implies that
\[
\inftynt^1_0|\boldsymbol{u}(x_1,x_2)-\boldsymbol{P}^R_{\Phi}(x_2)|^2dx_2 \tildeo0,\quaduad\tildeext{as $x_1\tildeo\inftynfty$}.
\]
The case in $\mathcal{S}_L$ is similar.
\epsilonnd{remark}
\quaded
The following result shows that for each weak solution we can associate a corresponding pressure field. See the proof in Section \ref{SEC3627} below.
\begin{equation}gin{lemma}\lambdabel{pressure}
Let $\boldsymbol{u}$ be a weak solution to the generalized Leray's problem defined above. Then there exists a scalar function $p\inftyn L^2_{{\mathrm{loc}}}(\omegaverline{\mathcal{S}})$ such that
\[
\inftynt_{\mathcal{S}}\nabla\boldsymbol{u}:\nabla\boldsymbol{\partialsi} dx+\inftynt_{\mathcal{S}}\boldsymbol{u}\cdot\nablabla \boldsymbol{u}\cdot \boldsymbol{\partialsi}dx=\inftynt_{\mathcal{S}} p\nabla\cdot\boldsymbol{\partialsi} dx
\]
holds for any $\boldsymbol{\partialsi}\inftyn C^\infty_c(\mathcal{S}; {\mathbb R}^2)$.
\epsilonnd{lemma}
\quaded
\sqrtubsection{Main results}
\quad\ Now we are ready to state the main theorems of this paper. The first one is existence and uniqueness of the weak solution, the second one addresses the regularity and decay estimates of the weak solution.
\begin{equation}gin{theorem}\lambdabel{PRO1.2}
Let $0\leq \alpha\leq +\infty$ be the friction coefficient given in \epsilonqref{NBC}. Assume that $\mathcal{S}$ is the aforementioned smooth strip. Then there exists $C_0>0$, independent of $\alpha$, such that
\begin{equation}gin{itemize}
\inftytem[(i)] if
\begin{equation}\lambdabel{COND1}
\frac{\alpha\Phi}{1+\alpha}\leq C_0,
\epsilonnd{equation}
then the 2D generalized Leray's problem \epsilonqref{GL1}-\epsilonqref{GL2}-\epsilonqref{GL3} has a weak solution $(\boldsymbol{u},p)\inftyn \mathcal{H}^1_{\sqrtigmagma,{\mathrm{loc}}}(\omegaverline{\mathcal{S}})\tildeimes L^2_{{\mathrm{loc}}}(\omegaverline{\mathcal{S}})$ satisfying
\begin{equation}\lambdabel{sesti}
\sqrtum_{i=L,R}\|\boldsymbol{u}-\boldsymbol{P}^i_\Phi\|_{H^1(\mathcal{S}_i)}\leq \Phi e^{C \Phi},
\epsilonnd{equation}
for some constant $C$ independent of $\alpha$.
\inftytem[(ii)] Moreover, if $\tilde{\boldsymbol{u}}$ is another weak solution of \epsilonqref{GL1}-\epsilonqref{GL2}-\epsilonqref{GL3} with the flux $\Phi\leq C_0$, and satisfies that for $\zeta\tildeo\infty$,
\begin{equation}\lambdabel{GROWC}
\|\nabla \tilde{\boldsymbol{u}}\|_{L^2(\mathcal{S}_\zeta)}=o\left(\zeta^{3/2}\right).
\epsilonnd{equation}
then $\tilde{\boldsymbol{u}}\epsilonquiv \boldsymbol{u}$.
\epsilonnd{itemize}
\epsilonnd{theorem}
\quaded
\begin{equation}gin{remark} Here we give several remarks.
\begin{equation}gin{itemize}
\inftytem In the existence result (i), noticing that the flux at the cross section $\Phi$ can be relatively large when $\alpha<1$ is small, since one only needs $\Phi\leq 2C_0\alpha^{-1}$. Here $\alpha=0$ means the flux $\Phi$ can be arbitrarily large.
\inftytem The limiting case $\alpha=0$ (i.e., the total slip situation) has already been considered in \cite{Mucha2003}, where an extra geometry restriction on the shape of the strip was imposed and the uniqueness was not considered there.
\inftytem The limiting case $\alpha=+\infty$ corresponds to the famous Leray's problem with the no-slip boundary condition which has been investigated for a long period of time. See a systematic review and study in \cite[Chapter XIII]{Galdi2011}.
\inftytem From the uniqueness result in Theorem \ref{PRO1.2}, we see that uniqueness can be only guaranteed by assuming that $\Phi$ is small enough, independent of the scale of $\alpha$. Actually uniqueness of the weak solution is a more complicated problem than existence. See some discussion and non-uniqueness results in \cite[Chapter XII.2]{Galdi2011} for the stationary 2D exterior problem.
\epsilonnd{itemize}
\epsilonnd{remark}
\quaded
The following Theorem gives the smoothness and the asymptotic behavior of a weak solution, which decays exponentially to the Poiseuille flow $\boldsymbol{P}^i_{\Phi}$ at each $\mathcal{S}_i$ as $|x|$ tends to infinity. Only the partial smallness condition \epsilonqref{COND1} is imposed.
\begin{equation}gin{theorem}\lambdabel{THMSA}
Let $\boldsymbol{u}$ be a weak solution stated in the item $(i)$ of Theorem \ref{PRO1.2}. Then
\[
\boldsymbol{u}\inftyn C^\infty(\omegaverline{\mathcal{S}})
\]
such that: For any $m\gammaeq 0$,
\begin{equation}\lambdabel{HODEST}
\sqrtum_{i=L,R}\|\nabla^m(\boldsymbol{u}-\boldsymbol{P}^i_\Phi)\|_{L^2(\mathcal{S}_i)}+\|\nabla^m \boldsymbol{u}\|_{L^2(\mathcal{S}_0)}\leq C_{\Phi,m}.
\epsilonnd{equation}
Meanwhile, the following pointwise decay estimates hold for sufficiently large $|x|$:
\begin{equation}\lambdabel{pointdecay}
\begin{equation}gin{split}
|\nabla^m(\boldsymbol{u}-\boldsymbol{P}^L_{\Phi})(y)|&\leq C_{\Phi,m} e^{\sqrtigmagma_{\Phi,m}y_1},\quad\tildeext{for all}\quad y_1<<-1;\\
|\nabla^m(\boldsymbol{u}-\boldsymbol{P}^R_{\Phi})(x)|&\leq C_{\Phi,m} e^{-\sqrtigmagma_{\Phi,m}x_1},\quad\tildeext{for all}\quad x_1>>1.
\epsilonnd{split}
\epsilonnd{equation}
Here $C_{\Phi,m}$ and $\sqrtigmagma_{\Phi,m}$ are positive constants depending only on $\Phi$ and $m$.
\epsilonnd{theorem}
\quaded
Also the corresponding pressure $p$ enjoys estimates akin to \epsilonqref{HODEST} and \epsilonqref{pointdecay}. See Remark \ref{RMMKK43}.
\begin{equation}gin{remark}
After our paper was posted on arXiv, we were informed by Professor Chunjing Xie that their group are also considering 2D Leray's problem with Navier-slip boundary and two manuscripts on this topic are finished. We are grateful for their kindness of sending us their manuscripts. After checking their manuscripts, though there are partial overlaps of results, the proof between theirs and ours differs in many aspects. Since our two groups' manuscripts are nearly posted at the same time, we believe that we independently solve this 2D Leray's problem at almost the same time. Reader can refer their works in \cite{SWX:2022ARXIV1, SWX:2022ARXIV2} for more details.
\epsilonnd{remark}
\sqrtubsection{Influence of the friction coefficient for the well-posedness}
\quad\ Unlike the 3D generalized Leray's problem with the Navier-slip boundary condition in our recent work \cite{LPY:2022ARXIV}, the friction coefficient $\alpha$ plays an important role for the well-posedness in the 2D problem. Some interesting results different from the 3D problem are presented as follows:
\begin{equation}gin{itemize}
\sqrtetlength{\inftytemsep}{-3 pt}
\inftytem[(i).]Largeness of the flux $\Phi$ when we show the existence, regularity and asymptotic behavior of the constructed $H^1$ weak solution.
\inftytem[(ii).] The $\alpha-$independence of all the estimates in Theorem \ref{PRO1.2} and Theorem \ref{THMSA} indicates that our results are uniform with the friction coefficient $\alpha$ and can be applied to the limiting cases $\alpha=0$ (total slip case) and $\alpha=\infty$ (classical Leray's problem).
\epsilonnd{itemize}
The main reason behind the above improvements is the validity of the Korn-type inequality ($L^2$ norm equivalence between $\nabla\boldsymbol{v}$ and $\mathbb{S}\boldsymbol{v}$) in the 2D strip domain $\mathcal{S}$ as displayed in Lemma \ref{lem-korn} and Corollary \ref{cor-korn}, which fails in the 3D pipe as Remark \ref{3korn}.
\sqrtubsection{Difficulties, outline of the proof and related works}
\sqrtubsubsection*{Difficulties and corresponding strategies}
\quad\ In two dimensional case, compared with the no-slip boundary condition, the main difficulties of the problem with Navier-slip boundary condition lie in the following:
\begin{equation}gin{itemize}
\sqrtetlength{\inftytemsep}{-3 pt}
\inftytem[(i).] For a given flux, construction of a smooth solenoidal flux carrier, satisfying the Navier-slip boundary condition, and equalling to the Poiseuille flow at a large distance;
\inftytem[(ii).] Achieving Poincar\'e-type inequalities and Korn-type inequalities in the distorted strip $\mathcal{S}$.
\epsilonnd{itemize}
In order to overcome difficulties listed above, our main strategies are as follows:
\begin{equation}gin{itemize}
\sqrtetlength{\inftytemsep}{-3 pt}
\inftytem[(i).] In order to construct the flux carrier, we first introduce a uniform curvilinear coordinates transform near the boundary: $T:\, x\rightarrow (s,t)$ to smoothly connect the two semi-infinite long straight strips and the middle distorted part. Under this curvilinear coordinates, a thin strip near the boundary of the middle distorted part is straightened. Under the curvilinear coordinates $(s,t)$, the flux carrier is constructed to smoothly connect the Poiseuille flows $\boldsymbol{P}_\Phi^L$ and $\boldsymbol{P}_\Phi^R$ at far field with a compact supported divergence-free vector $\sqrtigmagma'_\varepsilon(t)\boldsymbol{e}_s$ in $\mathcal{S}_0$. In the intermediate parts, they can be glued smoothly, and the divergence-free property together with the Navier-slip boundary condition keep valid.
\inftytem[(ii).] The Poincar\'e inequality and Korn's inequality play important roles during the proof. For the no-slip boundary condition, Poincar\'e inequality can be applied directly by using zero boundary condition. However, in the case of the Navier-slip boundary condition, the Poincar\'e inequality is not obvious in the middle distorted part $\mathcal{S}_0$. First, in $\mathcal{S}_L$ or $\mathcal{S}_R$, after subtracting the constant flux, $\boldsymbol{v}\cdot\boldsymbol{e_1}$ (or $\boldsymbol{v}\cdot\boldsymbol{e_1}'$) has zero mean value in any cross line, and then combining the impermeable boundary condition, which indicates that $\boldsymbol{v}\cdot\boldsymbol{e_2}=0$ (or $\boldsymbol{v}\cdot\boldsymbol{e_2}'=0$) on the boundary, we can achieve Poincar\'e inequality in the straight strips. Based on the result in the straight strip, we derive the Poincar\'e inequality in $\mathcal{S}_0$ by the trace theorem and a 2D Payne's identity \epsilonqref{Payne3D}. See Lemma \ref{TORPOIN}. The $\alpha$-independence of constants during the proof of main theorems is creditable to Korn's inequalities in 2D strips. The Korn's inequality is proved via a contradiction argument, which is given in Section \ref{PRE}, and is highly dependent on the compactness of the curvature of the boundary. It is not valid in the 3D case. See a counterexample in Remark \ref{3korn}.
\epsilonnd{itemize}
\sqrtubsubsection*{Outline of the proof}
\quad\ The existence and uniqueness of the solution will be given in Section \ref{SEU}. Before proving the existence theorem, a smooth solenoidal flux carrier $\boldsymbol{a}$ will be carefully constructed under the help of the uniformly curvilinear coordinates near the boundary $\partial\mathcal{S}_{-}$. By subtracting this smooth flux carrier, the existence problem of \epsilonqref{GL1}-\epsilonqref{GL2}-\epsilonqref{GL3} is reduced to a related one that the solution approaches zero at spacial infinity, which can be handled by the standard Galerkin method.
The main idea of proving the uniqueness is applying Lemma \ref{LEM2.3}, which was originally announced in reference \cite{Lady-Sol1980} and used to prove the uniqueness of the Leray's problem with no-slip boundary. Although our idea originates from \cite{Lady-Sol1980}, there are many differences from the previous literature. Some estimates of the present manuscript is much more complicated, involving the Poincar\'e inequality in a distorted strip as shown in Lemma \ref{TORPOIN} and Korn's inequality in Lemma \ref{lem-korn}.
Proofs of the asymptotic behavior and smoothness of weak solutions are given in Section \ref{SECH}. The main idea in deriving the exponential decay of $H^1$ weak solutions is to derive a first order ordinary differential inequality for the $L^2$ gradient integration in domain $\mathcal{S}\backslash\mathcal{S}_\zeta$. For the global estimates of higher-order norms, by using a ``decomposing-summarizing" technique, $H^1$ estimate of the vorticity in $\mathcal{S}$ will be obtained, and then Biot-Savart law indicates $H^2$ global estimate of the solution. Using the bootstrapping argument, higher-order global estimates then follow. This also leads to the higher-order exponential decay estimates, by utilizing the $H^1$-decay estimate and interpolation inequalities.
\sqrtubsubsection*{Some related works}
\quad\ The well-posedness study of the stationary Navier-Stokes equations in an infinite long pipe (or an infinite strip in the 2D case) with no-slip boundary condition and toward the Poiseuille flow laid down by Ladyzhenskaya in 1950s \cite{Ladyzhenskaya:1959UMN,Ladyzhenskaya:1959SPD}, in which the problem was called \epsilonmph{Leray's problem}. Later by reducing the problem to the resolution of a variational problem, Amick \cite{Amick:1977ASN, Amick:1978NATMA} obtained the existence result of the Leray's problem with small flux, but the uniqueness was left open. For the planar flow, Amick--Fraenkel \cite{Amick-Fraenkel1980} studied the Leray's problem in various types of stripes distinguished by their properties at infinity. An approach to solving the uniqueness of small-flux solution via energy estimate was built by Ladyzhenskaya-Solonnikov \cite{Lady-Sol1980}, in which authors also addressed the existence and asymptotic behavior results. See \cite{AP:1989SIAM, HW:1978SIAM, Pileckas:2002MB} for more related conclusion, also \cite[Chapter XIII]{Galdi2011} for a systematic review to the Leray's problem with the no-slip boundary condition. Recently Yang-Yin \cite{YY:2018SIAM} studied the well-posedness of weak solutions to the steady non-Newtonian fluids in pipe-like domains. Wang-Xie in \cite{WX:2019ARXIV,Wang-Xie2022ARMA} studied the existence, uniqueness and uniform structural stability of Poiseuille flows for the 3D axially symmetric inhomogeneous Navier-Stokes equations in the 3D regular cylinder, with a force term appearing on the right hand of the equations.
The Navier-slip boundary condition was initialed by Navier \cite{Navier}. It allows fluid slip on the boundary with a scale proportional to its stress tensor. Different from the no-slip boundary, the Leray's problem with the Navier-slip boundary condition requires much more complicated mathematical strategies. \cite{Mucha2003, Mucha:2003STUDMATH, Konie:2006COLLMATH} studied the solvability of the steady Navier-Stokes equations with the perfect Navier-slip condition ($\alpha=0$). In this case, the solution approaches to a constant vector at the spatial infinity. Authors in \cite{Amrouche2014,Ghosh:2018PHD,AACG:2021JDE} studied the properties of solutions to the steady Navier-Stokes equations with the Navier-slip boundary in bounded domains. Wang and Xie \cite{WX:2021ARXIV2} showed the uniqueness and uniform structural stability of Poiseuille flows in an infinite straight long pipe with the Navier-slip boundary condition. Authors of the present paper studied the related 3D Leray's problem with the Naiver-slip boundary condition \cite{LPY:2022ARXIV} under more strict smallness of the flux than the recent paper on 2D case. They also proved the characterization of bounded smooth solutions for the 3D axially symmetric Navier-Stokes equations with the perfect Navier-slip boundary condition in the infinitely long cylinder \cite{LP:2021ARXIV}.
This paper is arranged as follows: In Section \ref{PRE}, some preliminary work are contained, in which a uniform curvilinear coordinate near the boundary will be introduced and the Navier-slip boundary condition will be written under this curvilinear coordinate frame, and some useful lemmas will be presented. We will concern the existence and uniqueness results in Section \ref{SEU}. Finally, we focus on the higher-order regularity and exponential decay properties of the solution in Section \ref{SECH}.
\sqrtection{Preliminary}\lambdabel{PRE}
First, we introduce a uniformly curvilinear coordinate near the boundary, which will help to construct the flux carrier. This curvilinear coordinates can be viewed as the straightening of the boundary in the distorted part $\mathcal{S}_0$. Under this curvilinear coordinates, the Navier-slip boundary condition in \epsilonqref{NBC} on the boundary of $\mathcal{S}_0$ will share almost the same form as that in the semi-infinite straight part of $\mathcal{S}_L$ and $\mathcal{S}_R$. See \epsilonqref{NBCM} below.
\sqrtubsection{On the uniformly curvilinear coordinates near the boundary}
To investigate the delicate feature of the Navier-slip boundary condition, also to construct the flux carrier in the distorted part of the strip, one needs to parameterize the boundary of $\mathcal{S}$. Recalling that
\[
\partial\mathcal{S}=\partial\mathcal{S}_+\cup\partial\mathcal{S}_-\cup\partial\mathcal{S}_{Ob},
\]
where $\partial\mathcal{S}_{\partialm}$ are upper and lower boundary portions of $\mathcal{S}$, while $\partial\mathcal{S}_{Ob}$ denotes the union of boundaries of obstacles in the middle of the strip. For convenience, we only parameterize $\partial\mathcal{S}_-$ since the others are similar. Besides, under this parameterised curvilinear coordinates, we will construct the divergence-free flux carrier, which is supported in
\[
\mathcal{S}_-(\delta):=\{x\inftyn\omegaverline{\mathcal{S}}:\,\mathrm{dist }(x,\partial\mathcal{S}_-)<\delta\},
\]
for some suitably small $\delta$ in the next section.
\begin{equation}gin{figure}[H]\lambdabel{FIG1}
\centering
\inftyncludegraphics[scale=0.3]{P1NB.pdf}\\
\caption{A curvilinear coordinate system $(s,t)$ near the boundary portion $\partial\mathcal{S}_-$.}\lambdabel{FIG1}
\epsilonnd{figure}
Denoting \begin{equation}\lambdabel{PS-}
\partial\mathcal{S}_-=\{\boldsymbol{\mathfrak{b}}(s)=\left(\mathfrak{b}_1(s),\mathfrak{b}_2(s)\right)\inftyn\mathbb{R}^2:\,s\inftyn\mathbb{R}\},
\epsilonnd{equation}
where $\mathfrak{b}_1(s),\mathfrak{b}_2(s)$ are smooth functions of $s$. Without loss of generality, we suppose the parameter $s\inftyn\mathbb{R}$ being the \epsilonmph{arc length parameter} of $\partial\mathcal{S}_-$, so that
\[
|\boldsymbol{\mathfrak{b}}(s)|=\sqrtqrt{(\mathfrak{b}_1'(s))^2+(\mathfrak{b}_2'(s))^2}\epsilonquiv 1,\quad\fracorall s\inftyn\mathbb{R}.
\]
By the definition of $\mathcal{S}$ given in Section \ref{SEC1}, $\partial\mathcal{S}_-$ lies on part of straight lines $\{y_2=0\}$ or $\{x_2=0\}$ except a compact distorted part in the middle, and there exists $s_0>0$ such that
\begin{equation}\lambdabel{para}
\partial\mathcal{S}_-\cap\omegaverline{\mathcal{S}_0}=\{\boldsymbol{\mathfrak{b}}(s)=\left(\mathfrak{b}_1(s),\mathfrak{b}_2(s)\right)\inftyn\mathbb{R}^2:\,s\inftyn[-s_0,s_0]\}.
\epsilonnd{equation}
This indicates $\boldsymbol{\mathfrak{b}}(s_0)=O$ and $\boldsymbol{\mathfrak{b}}(-s_0)=O'$. Meanwhile, all ``obstacles" inside $\mathcal{S}$ are away from it.
Because of the compact distortion, $\partial\mathcal{S}_-$ must satisfy the following condition:
\begin{equation}gin{condition}[Uniform interior sphere]\lambdabel{COND1.1}
For any point $z\inftyn\partial\mathcal{S}_-$, there exists a disk $K_{z}$, with its radius being $R_{z}$, such that
\[
\omegaverline{K_{z}}\cap(\mathbb{R}^2-\mathcal{S})=\{z\}.
\]
Meanwhile, there exists $\delta>0$ such that
\begin{equation}\lambdabel{UISC}
R_{z}\gammaeq2\delta,\quad\fracorall z\inftyn\partial\mathcal{S}_-.
\epsilonnd{equation}
\epsilonnd{condition}
\quaded
Due to the uniform interior sphere condition, for any $x\inftyn\mathcal{S}_-(\delta)$, there exists a unique point $z\inftyn\partial\mathcal{S}_-$ such that $|x-z|=\mathrm{dist }(x,\partial\mathcal{S}_-)$. Recalling \epsilonqref{PS-}, there exists a unique pair $(s,t)\inftyn\mathbb{R}\tildeimes[0,\delta)$ such that $y=\boldsymbol{\mathfrak{b}}(s)$ and $t=\mathrm{dist }(x,z)$. In this way, the following mapping is one-to-one and well-defined:
\begin{equation}\lambdabel{MAPP}
x\tildeo(s,t),\quad\fracorall x\inftyn\mathcal{S}_-(\delta).
\epsilonnd{equation}
Meanwhile, one has
\begin{equation}gin{lemma}\lambdabel{LEM1.2}
The mapping defined in \epsilonqref{MAPP} is smooth.
\epsilonnd{lemma}
\begin{equation}gin{proof}
By the construction of this mapping, one deduces
\[
x=y-\mathrm{dist }(x,z)\boldsymbol{n}_z,
\]
where $\boldsymbol{n}_y$ is the unit outer normal of $y\inftyn\partial\mathcal{S}_-\,$. Since $y=\boldsymbol{\mathfrak{b}}(s)$, we define
\[
\boldsymbol{F}(s,t):=\boldsymbol{\mathfrak{b}}(s)-t\boldsymbol{n}_{\boldsymbol{\mathfrak{b}}(s)}\quad\fracorall(s,t)\inftyn\mathbb{R}\tildeimes[0,\delta)\,\,.
\]
Clearly $\boldsymbol{F}$ is well-defined and smooth in $\mathbb{R}\tildeimes[0,\delta)$, and its Jacobian matrix writes
\[
J\boldsymbol{F}=\left(
\begin{equation}gin{array}{cc}
\mathfrak{b}_1'(s)-t\left(\frac{d}{d s}\boldsymbol{n}_{\boldsymbol{\mathfrak{b}}(s)}\right)_1&\quad-(\boldsymbol{n}_{\boldsymbol{\mathfrak{b}}(s)})_1\\[2mm]
\mathfrak{b}_2'(s)-t\left(\frac{d}{d s}\boldsymbol{n}_{\boldsymbol{\mathfrak{b}}(s)}\right)_2&\quad-(\boldsymbol{n}_{\boldsymbol{\mathfrak{b}}(s)})_2\\
\epsilonnd{array}
\right).
\]
Here and below, $(\boldsymbol{X})_i$ with $i=1,2$ means the $i$-th component of the vector $\boldsymbol{X}$. Direct calculation shows
\[
\mathrm{det }(J\boldsymbol{F})=\left(\boldsymbol{\mathfrak{b}}'(s)-t\left(\frac{d}{d s}\boldsymbol{n}_{\boldsymbol{\mathfrak{b}}(s)}\right)\right)\cdot\boldsymbol{n}_{\boldsymbol{\mathfrak{b}}(s)}^{\partialerp}=1-t\left(\frac{d}{d s}\boldsymbol{n}_{\boldsymbol{\mathfrak{b}}(s)}\right)\cdot\boldsymbol{\mathfrak{b}}'(s),
\]
where
\[
\boldsymbol{n}_{\boldsymbol{\mathfrak{b}}(s)}^{\partialerp}=(-(\boldsymbol{n}_{\boldsymbol{\mathfrak{b}}(s)})_2,(\boldsymbol{n}_{\boldsymbol{\mathfrak{b}}(s)})_1)=\boldsymbol{\mathfrak{b}}'(s).
\]
This indicates that
\[
\mathrm{det }(J\boldsymbol{F})\gammaeq1-t\frac{1}{2\delta}>\frac{1}{2},\quad\fracorall(s,t)\inftyn\mathbb{R}\tildeimes[0,\delta)
\]
due to \epsilonqref{UISC} so that $\frac{1}{2\delta}$ can bound the curvature of $\partial\mathcal{S}_-\,$. Recalling the compactness of the distorted part, the lemma is claimed by the inverse mapping theorem.
\epsilonnd{proof}
For any $x=(x_1,x_2)\inftyn\mathcal{S}_-(\delta)$, Condition \ref{COND1.1} and Lemma \ref{LEM1.2} above guarantee the following well-defined curvilinear coordinate system
\[
(s,t)=(s(x),t(x))\inftyn\mathbb{R}\tildeimes[0,\delta).
\]
Geometrically, $t(x)$ is the distance of the given point $x\inftyn\mathcal{S}_-(\delta)$ to the boundary $\partial\mathcal{S}_-$, while $s(x)$ denotes the parameter coordinate of the unique point $y\inftyn\partial\mathcal{S}_-$ such that $|x-y|=\mathrm{dist }(x,\partial\mathcal{S}_-)$. As it is shown in Figure \ref{FIG1}, we denote
\begin{equation}\lambdabel{ESET}
\boldsymbol{e_s}=(e_{s1}\,,\,e_{s2})\quad\tildeext{and}\quad\boldsymbol{e_t}=(e_{t1}\,,\,e_{t2})
\epsilonnd{equation}
are unit tangent vector of $s-$curves and $t-$curves, respectively. Meanwhile, they are all independent with variable $t\inftyn[0,\delta)$. Clearly
\begin{equation}\lambdabel{ESET}
\left\{
\begin{equation}gin{aligned}
\boldsymbol{e_s}&\epsilonquiv\boldsymbol{e_1};\\
\boldsymbol{e_t}&\epsilonquiv\boldsymbol{e_2},\\
\epsilonnd{aligned}
\right.\quad\quad\fracorall x\inftyn\omegaverline{\mathcal{S}_R};\quad\quad\left\{
\begin{equation}gin{aligned}
\boldsymbol{e_s}&\epsilonquiv\boldsymbol{e_1}';\\
\boldsymbol{e_t}&\epsilonquiv\boldsymbol{e_2}',\\
\epsilonnd{aligned}
\right.\quad\quad\fracorall x\inftyn\omegaverline{\mathcal{S}_L}.
\epsilonnd{equation}
Moreover,
\begin{equation}\lambdabel{ENTS}
\nablabla_x t=\left(\frac{\partial t}{\partial x_1}\,,\,\frac{\partial t}{\partial x_2}\right)\epsilonquiv\boldsymbol{e_t},
\epsilonnd{equation}
and there exists a smooth function $\gammaamma(s,t)>\gammaamma_0>0$ that
\begin{equation}\lambdabel{ENTS1}
\nablabla_x s=\left(\frac{\partial s}{\partial x_1}\,,\,\frac{\partial s}{\partial x_2}\right)=\gammaamma\boldsymbol{e_s}.
\epsilonnd{equation}
Thus by denoting
\[
\boldsymbol{D}:=\left(
\begin{equation}gin{array}{cc}
\frac{\partial s}{\partial x_1}&\frac{\partial s}{\partial x_2}\\[2mm]
\frac{\partial t}{\partial x_1}&\frac{\partial t}{\partial x_2}\\
\epsilonnd{array}
\right),
\]
one derives
\[
\mathrm{det }\,\boldsymbol{D}=-\gammaamma\boldsymbol{e_s}\cdot\boldsymbol{e_t}^{\partialerp}=\gammaamma.
\]
by \epsilonqref{ENTS} and \epsilonqref{ENTS1}. Moreover, by calculating the inverse matrix of $\boldsymbol{D}$, one deduces
\[
\left(
\begin{equation}gin{array}{cc}
\frac{\partial x_1}{\partial s}&\frac{\partial x_1}{\partial t}\\[2mm]
\frac{\partial x_2}{\partial s}&\frac{\partial x_2}{\partial t}\\
\epsilonnd{array}
\right)=\left(\begin{equation}gin{array}{cc}
\frac{1}{\gammaamma}e_{t2} & -e_{s2}\\[2mm]
-\frac{1}{\gammaamma}e_{t1} & e_{s1}\\
\epsilonnd{array}
\right),
\]
which indicates
\[
\left\{
\begin{equation}gin{aligned}
&\frac{\partial x}{\partial s}=\frac{\boldsymbol{e_s}}{\gammaamma};\\
&\frac{\partial x}{\partial t}=\boldsymbol{e_t}.\\
\epsilonnd{aligned}
\right.
\]
Since $|\boldsymbol{e_s}|=|\boldsymbol{e_t}|\epsilonquiv 1$ and $\boldsymbol{e_s}\cdot\boldsymbol{e_t}\epsilonquiv0$, there exists a bounded smooth function $\kappa=\kappa(s,t)\inftyn\mathbb{R}$, which denotes the curvature of the boundary, such that
\[
\left\{
\begin{equation}gin{split}
\frac{d\boldsymbol{e_t}}{ds}&=-\frac{\kappa\boldsymbol{e_s}}{\gammaamma};\\[2mm]
\frac{d\boldsymbol{e_s}}{ds}&=\frac{\kappa\boldsymbol{e_t}}{\gammaamma},\\
\epsilonnd{split}
\right.\quad\quad\fracorall(s,t)\inftyn\mathbb{R}\tildeimes[0,\delta).
\]
By direct calculation, the divergence and curl of a vector field $\boldsymbol{w}=w_s(s,t)\boldsymbol{e_s}+w_t(s,t)\boldsymbol{e_t}$ writes
\begin{equation}\lambdabel{DIV-CURL}
\begin{equation}gin{split}
\mathrm{div }\,\boldsymbol{w}&=\gammaamma\partial_sw_s+\partial_tw_t-\kappa w_t;\\
\mathrm{curl }\,\boldsymbol{w}&=\partial_{x_2}w_1-\partial_{x_1}w_2=\partial_tw_s-\gammaamma\partial_sw_t-\kappa w_s,
\epsilonnd{split}
\epsilonnd{equation}
under this curvilinear coordinates.
To finish this subsection, let us focus on the Navier-slip boundary condition under the curvilinear coordinates. Writing
\[
\boldsymbol{u}=u_{s}\boldsymbol{e_s}+u_t\boldsymbol{e_t}.
\]
Then \epsilonqref{NBC} enjoys the following simplified expression:
\begin{equation}\lambdabel{NBCM}
\left\{
\begin{equation}gin{aligned}
&\frac{\partial u_s}{\partial\boldsymbol{n}}=-\partial_{t}u_{s}=\left(\kappa-\alpha\right)u_{s},\\
&u_t=0,\\
\epsilonnd{aligned}
\right.\quad\tildeext{on}\quad\partial\mathcal{S}_-.
\epsilonnd{equation}
See \cite[Proposition 2.1 and Corollary 2.2]{Watanabe2003} for a detailed calculation. Moreover, denoting $\mathfrak{w}=\partial_{x_2}u_1-\partial_{x_1}u_2$ and applying \epsilonqref{DIV-CURL}$_2$, one has \epsilonqref{NBCM}$_1$ is equivalent to
\[
\mathfrak{w}=\left(-2\kappa+\alpha\right)u_{s},\quad\tildeext{on}\quad\partial\mathcal{S}_-.
\]
\sqrtubsection{The Poincar\'e inequality and the Korn's inequality}
The following Poincar\'e inequalities and Korn's inequality will play crucial role in the existence and uniqueness results when the no-slip boundary is replaced by the Navier-Slip boundary.
\begin{equation}gin{lemma}[Poincar\'e inequality in a straight strip]\lambdabel{POIN}
Let $\boldsymbol{g}=g_1\boldsymbol{e_1}+g_2\boldsymbol{e_2}$ be a $H^1$ vector field in the box domain $S:=[a,b]\tildeimes[c,d]$, and satisfies that
\begin{equation}s
\inftynt^d_c g_1(x_1,x_2)dx_2=g_2(x_1,x_2)\big|_{x_2=c,d}=0, \quad \fracorall\ x_1\inftyn [a,b],
\epsilonnd{equation}s
then we have the following
\begin{equation}\lambdabel{poinfull}
\|\boldsymbol{g}\|_{L^2(S)}\leq C\|\partial_{x_2} \boldsymbol{g}\|_{L^2(S)}.
\epsilonnd{equation}
where $C\lesssim |c-d|$ is a constant depending on the width of the strip.
\epsilonnd{lemma}
\noindent {\bf Proof. \hspace{2mm}} Since $g_1$ has zero mean and $g_2$ has zero boundary in the $x_2$ direction. The classical one dimensional Poincar\'e inequality leads to
\begin{equation}s
\|\boldsymbol{g}(x_1)\|^2_{L^2_{x_2}([c,d])}\lesssim_{|d-c|} \|\partial_{x_2} \boldsymbol{g}(x_1)\|^2_{L^2_{x_2}([c,d])}, \quad \fracorall\ x_1\inftyn [a,b]
\epsilonnd{equation}s
Integration on $[a,b]$ with respect to $x_1$ variable indicates \epsilonqref{poinfull}.
\quaded
\begin{equation}gin{lemma}[Poincar\'e inequality in the torsion part]\lambdabel{TORPOIN}
Let $\zeta>1$ and $\boldsymbol{h}=h_1\boldsymbol{e_1}+h_2\boldsymbol{e_2}=\tildeilde{h}_1\boldsymbol{e_1}'+\tildeilde{h}_2\boldsymbol{e_2}'\inftyn H^1(\mathcal{S}_\zeta)$ be a divergence free vector with zero flux, that is
\[
\inftynt_{0}^1{h}_1(x_1,x_2)dx_2=0\quad \tildeext{for any}\ x\inftyn\mathcal{S}_\zeta\cap\mathcal{S}_R.
\]
If we suppose $\boldsymbol{h}\cdot\boldsymbol{n}\epsilonquiv 0$ on $\partial\mathcal{S}\cap\partial\mathcal{S}_\zeta$, where $\boldsymbol{n}$ is the unit outer normal vector on $\partial\mathcal{S}$, then the following Poincar\'e inequality holds:
\begin{equation}\lambdabel{TORPIPEPOIN}
\|\boldsymbol{h}\|_{L^2(\mathcal{S}_\zeta)}\leq C\|\nabla\boldsymbol{h}\|_{L^2(\mathcal{S}_\zeta)}.
\epsilonnd{equation}
Here $C>0$ is a uniform constant, independent of $\zeta$.
\epsilonnd{lemma}
\noindent {\bf Proof. \hspace{2mm}} Integrating the following identity on $\mathcal{S}_{1}$,
\begin{equation}\lambdabel{Payne3D}
\sqrtum^2_{i,j=1}\left[\partial_{x_i}(h_i x_j h_j)-\partial_{x_i} h_ix_jh_j-|\boldsymbol{h}|^2-h_ix_j\partial_{x_i} h_j\right]=0,
\epsilonnd{equation}
one deduces
\begin{equation}\lambdabel{ZJ1}
\begin{aligned}
\inftynt_{\mathcal{S}_{1}} |\boldsymbol{h}|^2dx=&\underbracederbrace{\sqrtum^2_{i,j=1}\inftynt_{\mathcal{S}_{1}}\partial_i(h_ix_j h_j)dx}_{J_1}-\sqrtum^2_{i,j=1}\inftynt_{\mathcal{S}_{1}}\partial_i h_ix_jh_jdx-\sqrtum^2_{i,j=1}\inftynt_{\mathcal{S}_{1}} h_ix_j\partial_i h_jdx.\\
\epsilonnd{aligned}
\epsilonnd{equation}
Using the divergence theorem and the boundary condition $\boldsymbol{h}\cdot{\boldsymbol{n}}=0$ on $\partial\mathcal{S}_{1}\cap\partial\mathcal{S}$, we can obtain
\[
J_1=\sqrtum^2_{j=1} \inftynt_{\partial\mathcal{S}_1}({\boldsymbol{n}}\cdot \boldsymbol{h})x_jh_jdS=\inftynt_{0}^1\left((x\cdot\boldsymbol{h})h_1\right)(1,x_2)dx_2-\inftynt_{0}^{c_0}\left((x\cdot\boldsymbol{h})\tildeilde{h}_1\right)(-1,y_2)dy_2.
\]
Thus by \epsilonqref{ZJ1} and the Cauchy-Schwarz inequality, we arrive at
\[
\begin{equation}gin{split}
\inftynt_{\mathcal{S}_{1}}|\boldsymbol{h}|^2dx\leq& \frac{1}{2}\inftynt_{\mathcal{S}_{1}}|\boldsymbol{h}|^2dx+C\inftynt_{\mathcal{S}_{1}}|\nabla\boldsymbol{h}|^2dx\\
&+\left|\inftynt_{0}^1\left((x\cdot\boldsymbol{h})h_1\right)(1,x_2)dx_2\right|+\left|\inftynt_{0}^{c_0}\left((x\cdot\boldsymbol{h})\tildeilde{h}_1\right)(-1,y_2)dy_2\right|,
\epsilonnd{split}
\]
which indicates
\begin{equation}\lambdabel{PCC1}
\inftynt_{\mathcal{S}_{1}}|\boldsymbol{h}|^2dx\leq C\left(\inftynt_{\mathcal{S}_{1}}|\nabla\boldsymbol{h}|^2dx+\inftynt_{0}^1\left|\boldsymbol{h}(1,x_2)\right|^2dx_2+\inftynt_{0}^{c_0}\left|\boldsymbol{h}(-1,y_2)\right|^2dy_2\right).
\epsilonnd{equation}
Meanwhile, using the trace theorem in $\mathcal{S}_{L,1}$, and Lemma \ref{POIN}, one derives
\begin{equation}\lambdabel{PCC2}
\begin{equation}gin{split}
\inftynt_{0}^{c_0}\left|\boldsymbol{h}(-1,y_2)\right|^2dy_2&\leq C\left(\inftynt_{(-1,0)\tildeimes(0,c_0)}\left|\boldsymbol{h}\right|^2dy_1dy_2+\inftynt_{(-1,0)\tildeimes(0,c_0)}\left|\nabla\boldsymbol{h}\right|^2dy_1dy_2\right)\\
&\leq C\inftynt_{\mathcal{S}_{L,1}}\left|\nabla\boldsymbol{h}\right|^2dx.
\epsilonnd{split}
\epsilonnd{equation}
Similarly, one deduces that
\begin{equation}\lambdabel{PCC3}
\inftynt_{0}^{1}\left|\boldsymbol{h}(1,x_2)\right|^2dx_2\leq C\inftynt_{\mathcal{S}_{R,1}}\left|\nabla\boldsymbol{h}\right|^2dx.
\epsilonnd{equation}
Substituting \epsilonqref{PCC2}--\epsilonqref{PCC3} in the right hand side of \epsilonqref{PCC1}, one deduces
\[
\inftynt_{\mathcal{S}_{1}}|\boldsymbol{h}|^2dx\leq C\inftynt_{\mathcal{S}_{1}}|\nabla\boldsymbol{h}|^2dx.
\]
Using Lemma \ref{POIN}, it is easy to see that
\[
\inftynt_{\mathcal{S}_\zeta\backslash\mathcal{S}_{1}}|\boldsymbol{h}|^2dx\leq C\inftynt_{\mathcal{S}_\zeta\backslash\mathcal{S}_{1}}|\nabla\boldsymbol{h}|^2dx.
\]
Combining the above two inequalities, we finish the proof of \epsilonqref{TORPIPEPOIN}.
\quaded
After showing Lemma \ref{POIN} and Lemma \ref{TORPOIN}, one concludes the following Poincar\'e inequality in the whole infinite strip:
\begin{equation}gin{corollary}[Poincar\'e inequality in $\mathcal{S}$]\lambdabel{CorPoin}
Let
\[
\boldsymbol{g}\inftyn\mathcal{V}:=\left\{\boldsymbol{f}=(f_1,f_2)\inftyn H^1(\mathcal{S}):\,\left(\boldsymbol{f}\cdot\boldsymbol{n}\right)\big|_{\partial\mathcal{S}}=0,\,\mathrm{div }\,\boldsymbol{f}=0\right\},
\]
then the following Poincar\'e inequality holds:
\begin{equation}\lambdabel{TORPIPEPOIN}
\|\boldsymbol{g}\|_{L^2(\mathcal{S})}\leq C\|\nabla\boldsymbol{g}\|_{L^2(\mathcal{S})}.
\epsilonnd{equation}
Here $C>0$ is a uniform constant.
\epsilonnd{corollary}
\noindent {\bf Proof. \hspace{2mm}} This is a direct conclusion by gluing results in Lemma \ref{POIN} and Lemma \ref{TORPOIN} together, after we have shown
\[
\inftynt_{\mathcal{S}\cap\{x_1=0\}}g_1dx_2=0
\]
holds unconditionally for $\boldsymbol{g}\inftyn\mathcal{V}$. By the divergence theorem,
\[
\inftynt_{\mathcal{S}\cap\{x_1=\mathfrak{s}\}}g_1dx_2-\inftynt_{\mathcal{S}\cap\{x_1=0\}}g_1dx_2=\inftynt_{\mathcal{S}\cap\{0<x_1<\mathfrak{s}\}}\mathrm{div }\,\boldsymbol{g}dx-\inftynt_{\partial\mathcal{S}\cap\{0<x_1<\mathfrak{s}\}}\left(\boldsymbol{g}\cdot\boldsymbol{n}\right)dS=0
\]
for any $\mathfrak{s}>0$. Thus if
\[
\left|\inftynt_{\mathcal{S}\cap\{x_1=0\}}g_1dx_2\right|=c_0>0,
\]
one deduces
\[
c_0\leq\inftynt_{\mathcal{S}\cap\{x_1=\mathfrak{s}\}}|g_1|dx_2\leq\left|\mathcal{S}\cap\{x_1=\mathfrak{s}\}\right|^{1/2}\left(\inftynt_{\mathcal{S}\cap\{x_1=\mathfrak{s}\}}|g_1|^2dx_2\right)^{1/2}.
\]
This implies that, for any $\mathfrak{s}>Z_0$
\[
\inftynt_{\mathcal{S}\cap\{x_1=\mathfrak{s}\}}|g_1|^2dx_2\gammaeq c_0^2,
\]
which results in a paradox with $\boldsymbol{g}\inftyn H^1(\mathcal{S})$ .
\quaded
Here goes the Korn's inequality in the truncated strip:
\begin{equation}gin{lemma}[Korn's inequality]\lambdabel{lem-korn}
Let $\mathcal{S}_{\zeta}$ with $\zeta>0$ be the finite truncated strip given in \epsilonqref{Szeta}. For any
\[
\boldsymbol{g}\inftyn\mathcal{V}_\zeta:=\left\{\boldsymbol{f}=(f_1,f_2)\inftyn H^1(\mathcal{S}_{\zeta};\,\mathbb{R}^2):\,\left(\boldsymbol{f}\cdot\boldsymbol{n}\right)\big|_{\partial\mathcal{S}\cap \partial\mathcal{S}_{\zeta}}=0,\,\mathrm{div }\,\boldsymbol{f}=0,\,\inftynt_{\mathcal{S}\cap\{x_1=0\}}f_1dx_2=0\right\},
\]
there exists $C>0$, which is independent of $\boldsymbol{g}$ or $\zeta$, such that
\begin{equation}\lambdabel{KN1.1}
\|\boldsymbol{g}\|^2_{H^{1}(\mathcal{S}_{\zeta})}\leq C\|\mathbb{S}\boldsymbol{g}\|^2_{L^2(\mathcal{S}_{\zeta})}+2\inftynt_{\{x_1=\zeta\}}|\boldsymbol{g}||\nabla \boldsymbol{g}|dx_2+2\inftynt_{\{y_1=-\zeta\}}|\boldsymbol{g}||\nabla \boldsymbol{g}|dy_2.
\epsilonnd{equation}
\epsilonnd{lemma}
\noindent {\bf Proof. \hspace{2mm}} Noting that
\begin{equation}\lambdabel{KN655}
\begin{equation}gin{aligned}
\inftynt_{\mathcal{S}_{\zeta}}|\mathbb{S}\boldsymbol{g}|^{2} dx &=\fracrac{1}{2} \inftynt_{\mathcal{S}_{\zeta}} \sqrtum_{i, j=1}^{2}\left(\partial_{x_j}g_{ i}+\partial_{x_i}g_{j}\right)^{2} dx \\
&=\inftynt_{\mathcal{S}_{\zeta}} \sqrtum_{i, j=1}^{2}\left(\left(\partial_{x_j}g_{i}\right)^{2}+\partial_{x_j}g_{i}\partial_{x_i}g_{j}\right) dx \\
&=\|\nablabla \boldsymbol{g}\|_{L^{2}(\mathcal{S}_{\zeta})}^{2}+\underbrace{\inftynt_{\mathcal{S}_{\zeta}} \sqrtum_{i, j=1}^{2}\partial_{x_j}g_{i}\partial_{x_i}g_{j}dx}_{K_1}.
\epsilonnd{aligned}
\epsilonnd{equation}
For the last term of \epsilonqref{KN655}, we can assume that $\boldsymbol{g}\inftyn C^{2}(\mathcal{S}_{\zeta})$ without loss of generality. By integration by parts, we get
\begin{equation}\lambdabel{KN111}
\begin{equation}gin{aligned}
K_1=&-\inftynt_{\mathcal{S}_{\zeta}} g_{i}\partial^2_{x_ix_j}g_jdx+\inftynt_{\partialartial \mathcal{S}_{\zeta}\cap\partial\mathcal{S} } \sqrtum_{i, j=1}^{2} n_{j}g_{i} \partial_{x_i}g_{j} dS+\inftynt_{\partialartial \mathcal{S}_{\zeta}\cap \{x_1=\partialm \zeta\} } \sqrtum_{i, j=1}^{2} n_{j}g_{i} \partial_{x_i}g_{j} dx_2 \\
=& \inftynt_{\mathcal{S}} (\mathrm{div}\,\boldsymbol{g})^2dx-\inftynt_{\partialartial \mathcal{S}_{\zeta}\cap\partial\mathcal{S}} \sqrtum_{i, j=1}^{2} n_{i}g_{i}\partial_{x_j}g_{j} dS+\inftynt_{\partialartial \mathcal{S}_{\zeta}\cap\partial\mathcal{S}} \sqrtum_{i, j=1}^{2} n_{j}g_{i} \partial_{x_i}g_{j} dS\\
&-\inftynt_{\{x_1=\zeta\}} \sqrtum_{i, j=1}^{2} n_{i}g_{i}\partial_{x_j}g_{j} dx_2-\inftynt_{\{y_1=-\zeta\}} \sqrtum_{i, j=1}^{2} n_{i}g_{i}\partial_{x_j}g_{j} dy_2\\
&+\inftynt_{\{x_1=\zeta\}} \sqrtum_{i, j=1}^{2} n_{j}g_{i} \partial_{x_i}g_{j} dx_2+\inftynt_{\{y_1=-\zeta\}} \sqrtum_{i, j=1}^{2} n_{j}g_{i} \partial_{x_i}g_{j} dy_2.
\epsilonnd{aligned}
\epsilonnd{equation}
The first, fourth and fifth terms on the far right of above equation vanish owing to the divergence-free property of $\boldsymbol{g}$, and the second one also vanishes because $\left.\boldsymbol{g} \cdot \boldsymbol{n}\right|_{\partialartial \mathcal{S}}=0$. Meanwhile, the condition $\left.\boldsymbol{g} \cdot \boldsymbol{n}\right|_{\partialartial \mathcal{S}}=0$ also implies that, at the boundary,
\begin{equation}\lambdabel{KN222}
\sqrtum_{i=1}^{2}g_i\partial_{x_i}(\boldsymbol{g}\cdot \boldsymbol{n})=0 \quaduad \tildeext { or } \quaduad \sqrtum_{i,j=1}^{2} g_i\partial_{x_i}g_{j} n_{j}=-\sqrtum_{i,j=1}^{2} g_ig_{j} \partial_{x_i}n_{j}.
\epsilonnd{equation}
Using \epsilonqref{KN111} to \epsilonqref{KN222}, we get
\[
\begin{aligned}
K_1=&-\inftynt_{\partial\mathcal{S}\cap\partial\mathcal{S}_\zeta }\kappa(x)|\boldsymbol{g}|^2dS+\inftynt_{\{x_1=\zeta\}} \sqrtum_{i, j=1}^{2} n_{j}g_{i} \partial_{x_i}g_{j} dx_2+\inftynt_{\{y_1=-\zeta\}} \sqrtum_{i, j=1}^{2} n_{j}g_{i} \partial_{x_i}g_{j} dy_2.
\epsilonnd{aligned}
\]
where $\kappa(x)$ is the curvature of the boundary $\partial\mathcal{S}$. By definition of $\mathcal{S}$, we have $\kappa(x) \epsilonquiv 0$ on $\partial\mathcal{S}\backslash\partial\mathcal{S}_0$, and $\boldsymbol{n}=\boldsymbol{e_1}$ on $\partial\mathcal{S}_{\zeta} \cap \{x_1=\zeta\}$, while $\boldsymbol{n}=-\boldsymbol{e_1}'$ on $\partial\mathcal{S}_{\zeta}\cap\{y_1=-\zeta\}$. This guarantees that
\begin{equation}\lambdabel{KN699}
|K_1|\leq\inftynt_{\partial\mathcal{S}\cap\partial\mathcal{S}_0}|\kappa(x)||\boldsymbol{g}|^2dS+\inftynt_{\{x_1=\zeta\}} |\boldsymbol{g}| |\nabla \boldsymbol{g}| dx_2+\inftynt_{\{y_1=-\zeta\}} |\boldsymbol{g}| |\nabla \boldsymbol{g}| dy_2.
\epsilonnd{equation}
Substituting \epsilonqref{KN111}--\epsilonqref{KN699} in \epsilonqref{KN655}, one concludes
\[
\|\nablabla \boldsymbol{g}\|_{L^{2}(\mathcal{S}_\zeta)}^{2}\leq \inftynt_{\mathcal{S}_\zeta}|\mathbb{S}\boldsymbol{g}|^{2} dx+\inftynt_{\partial\mathcal{S}\cap\partial\mathcal{S}_0}|\kappa(x)||\boldsymbol{g}|^2dS+\inftynt_{\{x_1=\zeta\}} |\boldsymbol{g}| |\nabla \boldsymbol{g}| dx_2+\inftynt_{\{y_1=-\zeta\}} |\boldsymbol{g}| |\nabla \boldsymbol{g}| dy_2.
\]
Noting that $\|\kappa(x)\|_{L^\infty(\partial\mathcal{S})}$ is uniformly bounded due to the smoothness of $\partial\mathcal{S}$ and combining with the Poincar\'e inequalities in Lemma \ref{POIN} and Lemma \ref{TORPOIN}, one deduces that there exists $C>0$ that
$$
\|\boldsymbol{g}\|_{H^1(\mathcal{S}_\zeta)}^{2} \leq C\left(\inftynt_{\mathcal{S}_\zeta}|\mathbb{S}\boldsymbol{g}|^{2} dx+\|\boldsymbol{g}\|_{L^{2}(\partial\mathcal{S}\cap\partial\mathcal{S}_0)}^{2}\right)+\inftynt_{\{x_1=\zeta\}} |\boldsymbol{g}| |\nabla \boldsymbol{g}| dx_2+\inftynt_{\{y_1=-\zeta\}} |\boldsymbol{g}| |\nabla \boldsymbol{g}| dy_2.
$$
To finish the proof, one only needs to show that there exists $C>0$ such that:
\begin{equation}\lambdabel{KN611}
\|\boldsymbol{g}\|_{L^{2}(\partial\mathcal{S}\cap\partial\mathcal{S}_0)}^{2} \leq\fracrac{1}{2 C}\|\boldsymbol{g}\|_{H^1(\mathcal{S}_{\zeta})}^{2}+C\inftynt_{\mathcal{S}_{\zeta}}|\mathbb{S}\boldsymbol{g}|^{2} dx .
\epsilonnd{equation}
We prove this by the method of contradiction. If a number the above $C$ does not exist, then there exists a bounded sequence $\left\{\boldsymbol{g}_{m}\right\}_{m=0}^{\inftynfty}\sqrtubset\mathcal{V}_{\zeta}$ such that
\[
\left\|\boldsymbol{g}_{m}\right\|_{L_{2}(\partial\mathcal{S}\cap\partial\mathcal{S}_0)}^{2} \gammaeq \fracrac{1}{2 C_{1}}\left\|\boldsymbol{g}_{m}\right\|_{H^1(\mathcal{S}_{\zeta})}^{2}+m \inftynt_{\mathcal{S}_{\zeta}}|\mathbb{S}\boldsymbol{g}_{m}|^{2} dx.
\]
Denoting $\boldsymbol{h}_{m}=\boldsymbol{g}_{m} /\left\|\boldsymbol{g}_{m}\right\|_{L^{2}(\partial\mathcal{S}\cap\partial\mathcal{S}_0)}$, one deduces that
\begin{equation}\lambdabel{KN613}
\left\|\boldsymbol{h}_{m}\right\|_{L^{2}(\partial\mathcal{S}\cap\partial\mathcal{S}_0)}=1 \quaduad \tildeext { and } \quaduad m \inftynt_{\mathcal{S}_{\zeta}}|\mathbb{S}\boldsymbol{h}_{m}|^{2} dx \leq 1.
\epsilonnd{equation}
Since the sequence $\left\{\boldsymbol{g}_{m}\right\}$ is bounded in $\mathcal{V}_{\zeta}$, we can choose a subsequence $\left\{\boldsymbol{h}_{m_{k}}\right\}_{k=0}^{\inftynfty}$ which is weakly convergent in $H^1(\mathcal{S}_{\zeta})$ and strongly in $L^{2}(\partial\mathcal{S}\cap\partial\mathcal{S}_0)$ to a vector $\boldsymbol{h}_{*} \inftyn \mathcal{V}_{\zeta}$. Particularly,
\[
\mathbb{S}\boldsymbol{h}_{m_{k}}\tildeo\mathbb{S}\boldsymbol{h}_*,\quad\tildeext{weakly in }L^2(\mathcal{S}_{\zeta}).
\]
By \epsilonqref{KN613}, one knows
$$
\inftynt_{\mathcal{S}_{\zeta}}|\mathbb{S}\boldsymbol{h}_{m_{k}}|^{2} dx \leq \fracrac{1}{m_{k}} \rightarrow 0,\quad\tildeext{as}\quad k\tildeo\infty.
$$
Thus one deduces
\[
\inftynt_{\mathcal{S}_{\zeta}}|\mathbb{S}\boldsymbol{h}_{*}|^{2} \mathrm{~d} x\leq\liminf_{k\tildeo\infty}\inftynt_{\mathcal{S}_{\zeta}}|\mathbb{S}\boldsymbol{h}_{m_{k}}|^{2} dx=0,
\]
by the Fatou's lemma for weakly convergent sequences. This concludes $\mathbb{S}\boldsymbol{h}_{*}\epsilonquiv0$ in $\mathcal{S}_{\zeta}$. It is well known that $\boldsymbol{h}_*$ has the form $\boldsymbol{h}_*=Ax+B$ (see \cite[\S 6]{KO:1988RMS}), where $A$ is a constant skew-symmetric matrix with constant entries and $B$ is a constant vector, that is,
\begin{equation}gin{equation*}
\boldsymbol{h}_*=
\begin{equation}gin{pmatrix}
0 & -a\\
a & 0
\epsilonnd{pmatrix}
\begin{equation}gin{pmatrix}
x_1 \\
x_2
\epsilonnd{pmatrix}
+\begin{equation}gin{pmatrix}
b_1 \\
b_2
\epsilonnd{pmatrix}=
\begin{equation}gin{pmatrix}
-a\,x_2+b_1 \\
a\,x_1+b_2
\epsilonnd{pmatrix}\,,
\epsilonnd{equation*}
where $a\,,b_i$ ($i=1\,,2$) are some constants. However, by the boundary condition $\boldsymbol{h}_*\cdot\boldsymbol{n}=0$ holds everywhere on $\partial\mathcal{S}\cap\partial\mathcal{S}_{\zeta}$, one has
\[
(h_*)_2=ax_1+b_2\epsilonquiv0,\quad\tildeext{for all}\quad 0<x_1<\zeta,
\]
which indicates $a=b_2\epsilonquiv 0$. This indicates $(h_*)_1=b_1$ and thus $b_1=0$ due to
\[
\inftynt_0^1(h_*)_1(x_1,x_2)dx_2=0,\quad\tildeext{for all}\quad 0<x_1<\zeta.
\]
Therefore one concludes $\boldsymbol{h}_*\epsilonquiv0$ in $\mathcal{S}_{\zeta}$. However, this creates a paradox to the fact
\[
\|\boldsymbol{h}_*\|_{L^2(\partial\mathcal{S}\cap\partial\mathcal{S}_0)}=1
\]
coming from \epsilonqref{KN613}. This indicates the validity of \epsilonqref{KN611} and therefore one concludes \epsilonqref{KN1.1}.
\quaded
If we replace the truncated strip with the infinite strip $\mathcal{S}$, the result in Lemma \ref{lem-korn} will be simpler with boundary term integrations on the segments $\{x_1=\zeta\}$ and $\{y_1=-\zeta\}$ disappearing. We have the following Corollary.
\begin{equation}gin{corollary}\lambdabel{cor-korn}
Let $\mathcal{S}$ be the infinite strip given in the previous section. For any $\boldsymbol{g}\inftyn \mathcal{V}$, there exists $C>0$, which is independent of $\boldsymbol{g}$, such that
\begin{equation}\lambdabel{KN1.1plus}
\|\boldsymbol{g}\|_{H^{1}(\mathcal{S})}\leq C\|\mathbb{S}\boldsymbol{g}\|_{L^2(\mathcal{S})}.
\epsilonnd{equation}
\epsilonnd{corollary}
\quaded
\begin{equation}gin{remark}\lambdabel{3korn}
Here let us give a brief explanation why this Korn's inequality fails to be valid in a 3D infinite pipe. Consider the vector
\[
\boldsymbol{w}=(-\xi(x_3)x_2\,,\,\xi(x_3)x_1\,,\,0)
\]
given in the cylindrical pipe $\mathcal{D}=B\tildeimes\mathbb{R}$, where $B$ is the unit disk in $\mathbb{R}^2$, and $\xi$ is a smooth cut-off function that:
\[
\xi(x_3)=\left\{
\begin{equation}gin{aligned}
1\,,&\quad\quad x_3\inftyn[-R,R];\\
0\,,&\quad\quad x_3\inftyn\mathbb{R}\backslash(-R-1,R+1),
\epsilonnd{aligned}
\right.
\]
with
\[
|\xi'(x_3)|\leq 2,\quad\tildeext{for any}\quad x_3\inftyn(-R-1,R)\cup(R,R+1).
\]
One notices that $\boldsymbol{w}$ is divergence-free and it satisfies $\boldsymbol{w}\cdot\boldsymbol{n}\epsilonquiv0$ on $\partial B\tildeimes\mathbb{R}$, also its flux in the cross section $B\tildeimes\{x_3=0\}$ is zero.
For the convenience of calculation, we introduce the cylindrical coordinates:
\[
\boldsymbol{e_r}=(\fracrac{x_1}{r},\fracrac{x_2}{r},0),\quaduad \boldsymbol{e_\tildeh}=(-\fracrac{x_2}{r},\fracrac{x_1}{r},0),\quaduad \boldsymbol{e_z}=(0,0,1),
\]
and we find
\[
\boldsymbol{w}=\xi(z)r\boldsymbol{e_\tildeh}.
\]
Using equation (A.4) in \cite{LP:2021ARXIV}, one finds
\[
\mathbb{S}\boldsymbol{w}=\frac{1}{2}\xi'(z)r\left(\boldsymbol{e_\tildeh}\omegatimes\boldsymbol{e_z}+\boldsymbol{e_z}\omegatimes\boldsymbol{e_\tildeh}\right).
\]
This indicates
\[
\inftynt_{\mathcal{D}}|\mathbb{S}\boldsymbol{w}|^2dx=O(1)
\]
which is independent with $R$. On the other hand
\[
\inftynt_{\mathcal{D}}|\nablabla\boldsymbol{w}|^2dx\gammaeq\inftynt_{\mathcal{D}}|\partial_2w_1|^2dx\gammaeq 2\partiali R.
\]
Noting that $R>0$ is arbitrary, one could not find a uniform constant $C>0$ such that a ``3D version" \epsilonqref{KN1.1} or \epsilonqref{KN1.1plus} holds.
\epsilonnd{remark}
\quaded
\begin{equation}gin{remark}
In the 3-dimensional case, the curvature of the domain boundary $\kappa(x)$ no longer has compact support. In this case one cannot find a subsequence $\left\{\boldsymbol{h}_{m_{k}}\right\}_{k=0}^{\inftynfty}$ which is strongly convergent in $L^{2}(\partial\mathcal{D})$ to a vector $\boldsymbol{h}_{*}$. That is why our method in the proof of Lemma \ref{lem-korn} fails in the 3-dimensional case.
\epsilonnd{remark}
\quaded
\sqrtubsection{Other useful lemmas}
The following Brouwer's fixed point theorem is crucial to establish the existence. See \cite{Lions1969} or \cite[Lemma IX.3.1]{Galdi2011}.
\begin{equation}gin{lemma}\lambdabel{FUNC}
Let $P$ be a continuous operator which maps $\mathbb{R}^N$ into itself, such that for some $\rho>0$
\[
P({\xi})\cdot{\xi}\gammaeq 0 \quaduad \tildeext { for all } {\xi} \inftyn\mathbb{R}^N \tildeext { with }\,|{\xi}|=\rho.
\]
Then there exists ${\xi}_{0} \inftyn\mathbb{R}^N$ with $|{\xi}_{0}| \leq \rho$ such that ${P}({\xi}_{0})=0$.
\epsilonnd{lemma}
\quaded
The following asymptotic estimate of a function that satisfies an ordinary differential inequality will be useful in our further proof. To the best of the authors' knowledge, it was originally derived by Ladyzhenskaya-Solonnikov in \cite{Lady-Sol1980}. We also refer readers to \cite[Lemma 2.7]{LPY:2022ARXIV} for a proof written in a relatively recent format.
\begin{equation}gin{lemma}\lambdabel{LEM2.3}
Let $Y(\zeta)\nequiv 0$ be a nondecreasing nonnegative differentiable function satisfying
\[
Y(\zeta)\leq\Psi(Y'(\zeta)),\quad\fracorall\zeta>0.
\]
Here $\Psi:\,[0,\inftynfty)\tildeo[0,\inftynfty)$ is a monotonically increasing function with $\Psi(0)=0$ and there exists $C,\,\tildeau_1>0$, $m>1$, such that
\[
\Psi(\tildeau)\leq C\tildeau^m,\quad\fracorall\tildeau>\tildeau_1.
\]
Then
\[
\liminf_{\zeta\tildeo+\inftynfty}\zeta^{-\frac{m}{m-1}}Y(\zeta)>0.
\]
\epsilonnd{lemma}
\quaded
The following two lemmas are essential in creating the pressure field for a weak solution to the Navier-Stokes equations. The first one is a special case of \cite[Theorem 17]{DeRham1960} by De Rham. See also \cite[Proposition 1.1]{Temam1984}.
\begin{equation}gin{lemma}\lambdabel{DeRham}
For a given open set $\Omega\sqrtubset\mathbb{R}^2$, let $\boldsymbol{\mathcal{F}}$ be a distribution in $\left(C_c^\infty(\Omega)\right)'$ which satisfies:
\[
\lambdangle \boldsymbol{\mathcal{F}},\boldsymbol{\partialhi}\rangle=0,\quad\tildeext{for all}\quad \boldsymbol{\partialhi}\inftyn\{\boldsymbol{g}\inftyn C_c^\infty(\Omega;\mathbb{R}^2):\,\tildeext{div }\boldsymbol{g}=0\}.
\]
Then there exists a distribution $q\inftyn\left(C_c^\infty(\Omega;\mathbb{R})\right)'$ such that
\[
\boldsymbol{\mathcal{F}}=\nabla q.
\]
\epsilonnd{lemma}
\quaded
The second one states the regularity of the aforementioned field $q$:
\begin{equation}gin{lemma}[See \cite{Temam1984}, Proposition 1.2]\lambdabel{LEM312}
Let $\Omegamega$ be a bounded Lipschitz open set in $\mathbb{R}^{2}$. If a distribution $q$ has all its first derivatives $\partial_{x_i} q$, $1\leq i \leq2$, in $H^{-1}(\Omegamega)$, then $q\inftyn L^{2}(\Omegamega)$ and
\begin{equation}\lambdabel{EEEE0}
\left\|q-\bar{q}_\Omega\right\|_{L^{2}(\Omegamega)}\leq C_\Omega\|\nabla q\|_{H^{-1}(\Omegamega)},
\epsilonnd{equation}
where $\bar{q}_\Omega=\frac{1}{|\Omega|}\inftynt_{\Omega}qdx$. Moreover, if $\Omegamega$ is any Lipschitz open set in $\mathbb{R}^{2}$, then $q\inftyn L_{\mathrm{loc}}^{2}(\omegaverline{\Omegamega})$.
\epsilonnd{lemma}
\quaded
Finally, we state the following lemma, which shows the existence of the solution to problem $\nablabla\cdot \boldsymbol{V}=f$ in a truncated regular stripe.
\begin{equation}gin{lemma}\lambdabel{LEM2.1}
For a boxed domain $S:= [a,b]\tildeimes [c,d]$, if $f\inftyn L^2(S)$ with $\inftynt_S fdx=0$, then there exists a vector valued function $\boldsymbol{V}:\,S\tildeo\mathbb{R}^2$ belongs to $H^1_0(S)$ such that
\begin{equation}\lambdabel{LEM2.11}
\nablabla\cdot \boldsymbol{V}=f,\quad\tildeext{and}\quad\|\nablabla \boldsymbol{V}\|_{L^2(S)}\leq C\|f\|_{L^2(S)}.
\epsilonnd{equation}
Here $C>0$ is an absolute constant.
\epsilonnd{lemma}
See \cite{Bme1, Bme2}, also \cite[Chapter III]{Galdi2011} for detailed proof of this lemma.
\quaded
\sqrtection{Existence and uniqueness of the weak solution}\lambdabel{SEU}
\sqrtubsection{Construction of the flux carrier}\lambdabel{SEC32}
\quad\ In this subsection, we are devoted to the construction of a flux carrier $\boldsymbol{a}$, which is divergence free, satisfying the Navier-slip boundary condition \epsilonqref{NBC}, and connects two Poiseuille flows in $\mathcal{S}_L$ and $\mathcal{S}_R$ smoothly. Meanwhile, the vector $\boldsymbol{a}$ will satisfy the following:
\begin{equation}gin{proposition}\lambdabel{Prop}
There exists a smooth vector field $\boldsymbol{a}(x)$ which enjoys the following properties
\begin{equation}gin{itemize}
\sqrtetlength{\inftytemsep}{-2 pt}
\inftytem[(i).] $\boldsymbol{a}\inftyn C^\infty(\omegaverline{\mathcal{S}})$, and $\nabla\cdot \boldsymbol{a}=0$ in $\mathcal{S}$;
\inftytem[(ii).] $2(\mathbb{S}\boldsymbol{a}\cdot\boldsymbol{n})_{\mathrm{tan}}+\alpha \boldsymbol{a}_{\mathrm{tan}}=0$, and $\boldsymbol{a}\cdot\boldsymbol{n}=0$ on $\partial\mathcal{S}$;
\inftytem[(iii).] For a fixed $\varepsilon\inftyn (0,1)$ ,
\begin{equation}\lambdabel{EADF}
\boldsymbol{a}=\left\{
\begin{equation}gin{aligned}
&\boldsymbol{P}^L_{\Phi}(y)\quad\tildeext{ in }\quad\mathcal{S}\cap \{y_1\leq -e^{\frac{2}{\varepsilon}}\},\\
&\boldsymbol{P}^R_{\Phi}(x)\quad\tildeext{ in }\quad\mathcal{S}\cap\{x_1\gammaeq e^{\frac{2}{\varepsilon}}\}.
\epsilonnd{aligned}\right.
\epsilonnd{equation}
Moreover, for any vector filed $\boldsymbol{v}\inftyn \mathcal{V}$ with
\begin{equation}\lambdabel{FUNCVV}
\mathcal{V}:=\left\{\boldsymbol{v}\inftyn H^1(\mathcal{S}):\,\omegaperatorname{div}\boldsymbol{v}=0,\ (\boldsymbol{v}\cdot\boldsymbol{n})\big|_{\partial\mathcal{S}}=0\right\},
\epsilonnd{equation}
there exists a constant $C$, independent of $\varepsilon$ and $\alpha$, such that
\begin{equation}
\left|\inftynt_{\mathcal{S}} \boldsymbol{v}\cdot \nabla\boldsymbol{a}\cdot \boldsymbol{v} dx\right|\leq C_\mathcal{S}\Phi\left(\varepsilon+\frac{\alpha}{1+\alpha}\right)\|\nabla \boldsymbol{v}\|^2_{L^2(\mathcal{S})}. \lambdabel{estisqu}
\epsilonnd{equation}
\epsilonnd{itemize}
\epsilonnd{proposition}
\quaded
The following lemma is useful in the construction of $\boldsymbol{a}$.
\begin{equation}gin{lemma}
There exists a smooth non-decreasing function $\sqrtigmagma_\varepsilon: [0,\delta)\tildeo[-\Phi,0]$, where $0<\varepsilon<<1$, such that
\[
\sqrtigmagma_\varepsilon(t)=\left\{
\begin{equation}gin{aligned}
0,&\quad\tildeext{for}\quad t\gammaeq \varepsilon;\\
-\Phi,&\quad\tildeext{for}\quad 0\leq t\leq \epsilon.
\epsilonnd{aligned}
\right.
\]
Here $\epsilon:=\varepsilon e^{-1/\varepsilon}/3$, and
\[
\delta:=\mathrm{dist}(\partial\mathcal{S}_-,\partial\mathcal{S}_{ob}\cup\partial\mathcal{S}_+)=\inftynf\left\{|x-z|:\,x\inftyn\partial\mathcal{S}_-,\,z\inftyn\partial\mathcal{S}_{ob}\cup\partial\mathcal{S}_+\right\}>2\varepsilon.
\]
Meanwhile, when $t\inftyn\left[0,\varepsilon\right]$, there exists constant $C$ such that
\begin{equation}\lambdabel{VEEST}
\left\{\begin{equation}gin{aligned}
&0\leq\sqrtigmagma_\varepsilon'(t)\leq \min\left\{\Phi e^{1/\varepsilon},\frac{2\Phi\varepsilon}{t}\right\},\\[1mm]
&\left|\sqrtigmagma_\varepsilon^{(k)}(t)\right|\leq C\Phi e^{1/\varepsilon}(\varepsilon^{-1}e^{1/\varepsilon})^{k-1},\quad\tildeext{for }k=2,3.\\
\epsilonnd{aligned}\right.
\epsilonnd{equation}
\epsilonnd{lemma}
\begin{equation}gin{proof}
We start with the piecewise smooth function
\begin{equation}\lambdabel{tauve}
\tildeau_\varepsilon(t)=\left\{
\begin{equation}gin{aligned}
0,&\quad\tildeext{for}\quad t\gammaeq\varepsilon;\\
\frac{\varepsilon}{t},&\quad\tildeext{for}\quad \varepsilon e^{-1/\varepsilon}<t<\varepsilon;\\
0,&\quad\tildeext{for}\quad 0\leq t\leq \varepsilon e^{-1/\varepsilon}.
\epsilonnd{aligned}
\right.
\epsilonnd{equation}
Then denoting $\varsigma$ the classical mollifier with radius equals $\epsilon$, the function $\sqrtigmagma_\varepsilon$ is given by:
\begin{equation}\lambdabel{PRESIG}
\sqrtigmagma_\varepsilon(t):=\left\{
\begin{equation}gin{array}{ll}
-\Phi,&\quad\tildeext{for}\quad 0\leq t<\epsilon;\\[2mm]
-\Phi+\tildeilde{C}\Phi\inftynt_0^t(\varsigma*\tildeau_\varepsilon)(s)ds,&\quad\tildeext{for}\quad \epsilon<t<\delta-\epsilon;\\[2mm]
0,&\quad\tildeext{for}\quad \delta-\epsilon\leq t<\delta.\\
\epsilonnd{array}\right.
\epsilonnd{equation}
Here $\tildeilde{C}>0$ is chosen such that
\[
\tildeilde{C}\inftynt_0^{\delta-\epsilon}(\varsigma*\tildeau_\varepsilon)(s)ds=1.
\]
Noting that $\tildeilde{C}$ must be sufficiently close to $1$, since $\inftynt_0^\delta\tildeau_\varepsilon(s)ds=1$ by \epsilonqref{tauve}. Finally, \epsilonqref{VEEST}$_1$ follows directly from \epsilonqref{tauve} and \epsilonqref{PRESIG}, while the validity of \epsilonqref{VEEST}$_2$ follows that
\[
\left\|\sqrtigmagma_\varepsilon^{(k)}\right\|_{L^\infty}=\tildeilde{C}\Phi\left\|\varsigma^{(k-1)}*\tildeau_\varepsilon\right\|_{L^\infty}\leq C\Phi\left\|\varsigma^{(k-1)}\right\|_{L^1}\left\|\tildeau_\varepsilon\right\|_{L^\infty}\leq C\Phi e^{1/\varepsilon}(\varepsilon^{-1}e^{1/\varepsilon})^{k-1},\quad\tildeext{for }k=2,3.
\]
\epsilonnd{proof}
{\noindent\bf Proof of Proposition \ref{Prop} : }Given $\varepsilon<\frac{\delta}{2}$, we define
\begin{equation}\lambdabel{Cons1}
\boldsymbol{a}=\sqrtigmagma'_\varepsilon(t)\boldsymbol{e_s},\quad\tildeext{in}\quad\mathcal{S}_{0},
\epsilonnd{equation}
where $\boldsymbol{e_s}$ is defined around \epsilonqref{ESET}, while
\begin{equation}\lambdabel{Cons}
\boldsymbol{a}=\left\{
\begin{equation}gin{array}{l} \left[\sqrtigmagma'_\varepsilon(x_2)(1-\epsilonta(x_1))+\epsilonta(x_1)P^R_\Phi(x_2)\right]\boldsymbol{e_1}-\left(\epsilonta'(x_1)\inftynt_0^{x_2}\left(P^R_\Phi(\xi)-\sqrtigmagma'_\varepsilon(\xi)\right)d\xi\right)\boldsymbol{e_2},\\
\hskip 13.5cm\tildeext{in}\quad\mathcal{S}_R;\\[3mm] \left[\sqrtigmagma'_\varepsilon(y_2)(1-\epsilonta(-y_1))+\epsilonta(-y_1)P^L_\Phi(y_2)\right]\boldsymbol{e_1}'+\left(\epsilonta'(-y_1)\inftynt_{0}^{y_2}\left(P^L_\Phi(\xi)-\sqrtigmagma'_\varepsilon(\xi)\right)d\xi\right)\boldsymbol{e_2}',\\
\hskip 13.5cm\tildeext{in}\quad\mathcal{S}_L.\\
\epsilonnd{array}
\right.
\epsilonnd{equation}
Here $\epsilonta=\epsilonta(s)$ be the smooth cut-off functions such that
\begin{equation}\lambdabel{ETA}
\epsilonta(s)=\left\{
\begin{equation}gin{aligned}
&1,\quad\tildeext{for}\quad s>e^{2/\varepsilon};\\
&0,\quad\tildeext{for}\quad s<0.\\
\epsilonnd{aligned}
\right.
\epsilonnd{equation}
and $\epsilonta$ satisfies
\[
|\epsilonta'|\leq 2e^{-2/\varepsilon}, \quad\tildeext{and}\quad|\epsilonta''|\leq 4e^{-4/\varepsilon}.
\]
$P^L_\Phi$ and $P^R_\Phi$, which are given in \epsilonqref{PF}, are $\boldsymbol{e_1}$-component and $\boldsymbol{e_1}'$-component of Poiseuille flows in pipes $\mathcal{S}_L$ and $\mathcal{S}_R$, respectively.
Using \epsilonqref{ESET} and \epsilonqref{DIV-CURL}$_1$, the flux carrier $\boldsymbol{a}$ constructed in \epsilonqref{Cons} is smooth and divergence-free. Meanwhile, since $\sqrtigmagma_\varepsilon(t)=0$ near $t=0$, one has $\boldsymbol{a}$ vanishes near $\partial\mathcal{S}\cap\partial\mathcal{S}_0$. This indicates $\boldsymbol{a}$ satisfies the homogeneous Navier-slip boundary condition on $\partial\mathcal{S}\cap\partial\mathcal{S}_0$.
Now we go to verify that $\boldsymbol{a}$ meets the Navier-slip boundary condition on $\partial\mathcal{S}\cap(\mathcal{S}_L\cup\mathcal{S}_R)$. Owing to cases in $\epsilonqref{Cons}_{1,2}$ are similar, we only consider $\epsilonqref{Cons}_{1}$ for simplicity. Since in this part, $\partial\mathcal{S}_R$ is straight, direct calculation of the Navier-slip boundary condition is to check
\[
\left\{
\begin{equation}gin{aligned}
&-\partial_{x_2}a_1(x_1,0)+\alpha a_1(x_1,0)=\partial_{x_2}a_1(x_1,1)+\alpha a_1(x_1,1)=0;\\[2mm]
&a_2(x_1,0)=a_2(x_1,1)=0,\\
\epsilonnd{aligned}
\right.\quad\quad\fracorall x_1\inftyn(0,\infty).
\]
This could be done by the definition of $P^R_\Phi$ in \epsilonqref{PF}, the construction of $\sqrtigmagma_\varepsilon$ above, and direct calculations.
Then items (i) and item (ii) in Proposition \ref{Prop} is proven and also it is easy to check that \epsilonqref{EADF} stands due to the choice of the cutoff function $\epsilonta(x_1)$. Now it remains to derive \epsilonqref{estisqu}, we define
\[
V:=\inftynt_{\mathcal{S}}\boldsymbol{v}\cdot\nablabla \boldsymbol{a}\cdot \boldsymbol{v} dx=\inftynt_{\mathcal{S}_L\cup\mathcal{S}_0\cup\mathcal{S}_R} \boldsymbol{v}\cdot\nablabla \boldsymbol{a}\cdot \boldsymbol{v}dx.
\]
{\noindent\bf Estimates of the $\mathcal{S}_0$-part integration:}\\[1mm]
In the distorted part $\mathcal{S}_0$, we denote that
\[
\boldsymbol{v}=v_s(s,t)\boldsymbol{e_s}+v_t(s,t)\boldsymbol{e_t},
\]
where the coordinates $(s,t)$ and vectors $\boldsymbol{e_s}$, $\boldsymbol{e_t}$ are defined in Section \ref{PRE}. Noting that
\[\boldsymbol{e_t}\cdot\nabla=\partial_t\,,\quaduad \boldsymbol{e_s}\cdot\nabla=\gammaamma(s,t)\partial_s\,,\quaduad \frac{d\boldsymbol{e_s}}{ds}=\frac{\kappa(s,t)}{\gammaamma(s,t)}\boldsymbol{e_t}\,,\]
in $\mathcal{S}_0$, one derives
\[
\begin{equation}gin{split}
\boldsymbol{v}\cdot\nablabla \boldsymbol{a}&=\gammaamma(s,t) v_s(s,t)\frac{\partial}{\partial s}(\sqrtigmagma_\varepsilon'(t)\boldsymbol{e_s})+v_t(s,t)\frac{\partial}{\partial t}(\sqrtigmagma_\varepsilon'(t)\boldsymbol{e_s})\\
&=\kappa(s,t) v_s(s,t)\sqrtigmagma_\varepsilon'(t)\boldsymbol{e_t}+v_t(s,t)\sqrtigmagma_\varepsilon''(t)\boldsymbol{e_s}\,.
\epsilonnd{split}
\]
Hence, we have
\[
\boldsymbol{v}\cdot\nablabla \boldsymbol{a}\cdot \boldsymbol{v}=\left(\sqrtigmagma_\varepsilon''(t)+\kappa(s,t)\sqrtigmagma_\varepsilon'(t)\right)v_s(s,t)v_t(s,t),\quad\tildeext{in}\quad\mathcal{S}_0.
\]
Recalling \epsilonqref{para}, we deduce
\[
\begin{equation}gin{split}
\inftynt_{\mathcal{S}_0}\boldsymbol{v}\cdot\nablabla \boldsymbol{a}\cdot \boldsymbol{v}dx=&\inftynt_0^\delta\inftynt_{-s_0}^{s_0}\left(\sqrtigmagma_\varepsilon''(t)+\kappa(s,t)\sqrtigmagma_\varepsilon'(t)\right)v_s(s,t)v_t(s,t)\frac{1}{\gammaamma(s,t)}ds dt\\[1mm]
=&\inftynt_0^\delta\inftynt_{-s_0}^{s_0}\sqrtigmagma_\varepsilon''(t)v_s(s,t)v_t(s,t)\frac{1}{\gammaamma(s,t)}dsdt+\inftynt_0^\delta\inftynt_{-s_0}^{s_0}\frac{\kappa(s,t)}{\gammaamma(s,t)}\sqrtigmagma_\varepsilon'(t)v_s(s,t)v_t(s,t)dsdt.
\epsilonnd{split}
\]
Noting that $\sqrtigmagma'_\varepsilon(t)$ vanishes near $\partial\mathcal{S}\cap\partial\mathcal{S}_0$, integration by parts for the first term of the right hand side of the above equality on $t$ indicate that
\begin{equation}\lambdabel{s0est}
\begin{equation}gin{split}
\inftynt_{\mathcal{S}_0}\boldsymbol{v}\cdot\nablabla \boldsymbol{a}\cdot \boldsymbol{v}dx=&-\inftynt_0^\delta\inftynt_{-s_0}^{s_0}\sqrtigmagma_\varepsilon'(t)\partial_t\left(\frac{v_s}{\gammaamma}\right)v_tdsdt-\inftynt_0^\delta\inftynt_{-s_0}^{s_0}\sqrtigmagma_\varepsilon'(t)v_s\partial_tv_t\frac{1}{\gammaamma(t,s)}dsdt\\
&+\inftynt_0^\delta\inftynt_{-s_0}^{s_0}\frac{\kappa(s,t)}{\gammaamma(s,t)}\sqrtigmagma_\varepsilon'(t)v_s(s,t)v_t(s,t)dsdt.
\epsilonnd{split}
\epsilonnd{equation}
Recalling \epsilonqref{DIV-CURL}$_1$, the divergence-free property of $\boldsymbol{v}$ in the curvilinear coordinates follows:
\[
\gammaamma\partial_sv_s+\partial_tv_t-\kappa v_t=0.
\]
Then inserting the divergence-free property into \epsilonqref{s0est}, we obtain that
\begin{equation}\lambdabel{s1est}
\begin{equation}gin{split}
\inftynt_{\mathcal{S}_0}\boldsymbol{v}\cdot\nablabla \boldsymbol{a}\cdot \boldsymbol{v}dx&=-\inftynt_0^\delta\inftynt_{-s_0}^{s_0}\sqrtigmagma_\varepsilon'(t)\partial_t\left(\frac{v_s}{\gammaamma}\right)v_tdsdt+\inftynt_0^\delta\inftynt_{-s_0}^{s_0}\sqrtigmagma_\varepsilon'(t)v_s\partial_sv_sdsdt\\
&:=V_1+V_2.
\epsilonnd{split}
\epsilonnd{equation}
First, noting that $\gammaamma(s,t)>\gammaamma_0>0$ is smooth, and $\sqrtigmagma_\varepsilon'(t)$, which is supported on $[0,\varepsilon]$, satisfies
\[
|\sqrtigmagma_\varepsilon'(t)|\leq\frac{2\Phi\varepsilon}{t},\quad\fracorall t\inftyn[0,\varepsilon],
\]
one bounds $V_1$ by using the Cauchy-Schwarz inequality and the Poincar\'e inequality in Lemma \ref{TORPOIN}
\begin{equation}\lambdabel{V210}
\begin{equation}gin{split}
|V_{1}|\leq& C \Phi\varepsilon\left(\inftynt_0^\delta\inftynt_{-s_0}^{s_0}|\nabla \boldsymbol{v}|^2dsdt\right)^{1/2}\underbrace{\left(\inftynt_0^\delta\inftynt_{-s_0}^{s_0}\frac{|v_t|^2}{t^2}dsdt\right)^{1/2}}_{V_{11}}.
\epsilonnd{split}
\epsilonnd{equation}
Due to $v_t=0$ on the $t=0$, the part $V_{11}$ can be estimated by the one dimensional Hardy inequality. In fact
\[
\begin{equation}gin{split}
\inftynt_0^\delta\frac{|v_t(s,t)|^2}{t^2}dt&=-\inftynt_0^\delta|v_t(s,t)|^2\left(\frac{1}{t}\right)'dt\\
&=-\frac{|v_t(s,\delta)|^2}{\delta}\inftynt_0^\delta\frac{|v_t(s,t)|^2}{t^2}dt+2\inftynt_0^\delta \frac{v_t(s,t)}{t}\partial_tv_t(s,t)dt\\
&\leq 2\left(\inftynt_0^\delta\frac{|v_t(s,t)|^2}{t^2}dt\right)^{1/2}\left(\inftynt_0^\delta|\partial_tv_t(s,t)|^2dt\right)^{1/2},
\epsilonnd{split}
\]
which indicates
\begin{equation}\lambdabel{V211}
\inftynt_0^\delta\frac{|v_t(s,t)|^2}{t^2}dt\leq 4\inftynt_0^\delta|\partial_tv_t(s,t)|^2dt.
\epsilonnd{equation}
Thus one concludes
\[
|V_{1}|\leq C\Phi\varepsilon\|\nablabla \boldsymbol{v}\|_{L^2(\mathcal{S}_0)}^2
\]
by combining \epsilonqref{V210} and \epsilonqref{V211}. For $V_{2}$ in \epsilonqref{s1est}, it follows that
\begin{equation}\lambdabel{V221E}
\begin{equation}gin{aligned}
V_{2}&=\frac{1}{2}\inftynt_0^\delta\inftynt_{-s_0}^{s_0}\sqrtigmagma_\varepsilon'(t)\partial_s(v_s)^2dsdt\\
&=\frac{1}{2}\inftynt_{0}^1\sqrtigmagma_\varepsilon'(x_2)(v_1)^2(0,x_2)dx_2-\frac{1}{2}\inftynt_{0}^{c_0}\sqrtigmagma_\varepsilon'(y_2)(v_1)^2(0,y_2)dy_2.
\epsilonnd{aligned}
\epsilonnd{equation}
The second equality above is established due to the Newton-Leibniz formula and fact that the curvilinear coordinates $(s,t)$ turns to be Euclidean in $\mathcal{S}\backslash\mathcal{S}_0$.\\[1mm]
{\noindent\bf Estimates in $\mathcal{S}_L$ and $\mathcal{S}_R$.}\\[1mm]
The cases in subsets $\mathcal{S}_L$ and $\mathcal{S}_R$ are similar, thus we only discuss the latter one for simplicity. At the beginning, we denote that
\[
\mathcal{S}_R:=\mathcal{S}_{R1}\cup\mathcal{S}_{R2},
\]
where
\[
\left\{
\begin{equation}gin{aligned}
&\mathcal{S}_{R1}=\mathcal{S}\cap\{x_1\inftyn[0,\,e^{2/\varepsilon}]\};\\
&\mathcal{S}_{R2}=\mathcal{S}\cap\{x_1\inftyn(e^{2/\varepsilon},\infty)\}.
\epsilonnd{aligned}
\right.
\]
Direct calculation shows
\[
\begin{equation}gin{aligned}
\inftynt_{\mathcal{S}_R}\boldsymbol{v}\cdot\nablabla \boldsymbol{a}\cdot \boldsymbol{v}dx=&\inftynt_{\mathcal{S}_{R1}}\epsilonta'(x_1)\left(P^R_\Phi(x_2)-\sqrtigmagma'_\varepsilon(x_2)\right)\left((v_1)^2-(v_2)^2\right)dx\\
&+\inftynt_{\mathcal{S}_{R1}}\epsilonta''(x_1)\left(\inftynt_0^{x_2}\left(\sqrtigmagma'_\varepsilon(\xi)-P^R_\Phi(\xi)\right)d\xi\right)v_1v_2dx\\
&+\inftynt_{\mathcal{S}_{R1}}\sqrtigmagma_\varepsilon''(x_2)\left(1-\epsilonta(x_1)\right)v_1v_2dx+\inftynt_{\mathcal{S}_{R2}}\epsilonta(x_1)(P_{\Phi}^R)'(x_2)v_1v_2dx\\
:=&J_1+J_2+J_3+J_4.
\epsilonnd{aligned}
\]
Here, noticing that $|\epsilonta'|\leq 2e^{-2/\varepsilon}$, $|\epsilonta''|\leq 4e^{-4\varepsilon}$ and
\[
|\sqrtigmagma_\varepsilon'(t)|\leq \Phi e^{1/\varepsilon}\quad\fracorall t\inftyn[0,\delta)
\]
which follows from \epsilonqref{VEEST}, one concludes that
\[
|J_1|+|J_2|\leq C\Phi e^{-1/\varepsilon}\inftynt_{\mathcal{S}_{R1}}|\boldsymbol{v}|^2dx\leq C\Phi\varepsilon\|\nablabla\boldsymbol{v}\|_{L^2(\mathcal{S})}^2
\]
by applying the Cauchy-Schwarz inequality and the Poincar\'e inequality in Corollary \ref{CorPoin}. Moreover, adopting integrating by parts, one deduces
\[
\begin{equation}gin{split}
J_3=&-\inftynt_{\mathcal{S}_{R1}}\sqrtigmagma_\varepsilon'(x_2)\left(1-\epsilonta(x_1)\right)\partial_{x_2}v_1v_2dx-\inftynt_{\mathcal{S}_{R1}}\sqrtigmagma_\varepsilon'(x_2)\left(1-\epsilonta(x_1)\right)v_1\partial_{x_2}v_2dx\\
:=&J_{31}+J_{32}.
\epsilonnd{split}
\]
Via an analogous route as we go through for $V_{1}$ in \epsilonqref{s1est} above, one deduces
\[
|J_{31}|\leq C \Phi\varepsilon\|\nablabla\boldsymbol{v}\|_{L^2(\mathcal{S})}^2.
\]
For the term $J_{32}$, applying the divergence-free property of $\boldsymbol{v}$ and using integration by parts, one arrives
\[
\begin{equation}gin{aligned}
J_{32}=&\frac{1}{2}\inftynt_{\mathcal{S}_{R1}}\sqrtigmagma_\varepsilon'(x_2)\left(1-\epsilonta(x_1)\right)\partial_{x_1}(v_1)^2dx\\ =&\frac{1}{2}\inftynt_{\mathcal{S}_{R1}}\sqrtigmagma_\varepsilon'(x_2)\epsilonta'(x_1)(v_1)^2dx-\frac{1}{2}\inftynt_{0}^1\sqrtigmagma_\varepsilon'(x_2)(v_1)^2(0,x_2)dx_2\\
:=&J_{321}+J_{322}.
\epsilonnd{aligned}
\]
Noting that $J_{321}$ can be estimated in the same way as we do on $J_1$ and $J_2$, that is
\[
|J_{321}|\leq C \Phi\varepsilon\|\nablabla\boldsymbol{v}\|_{L^2(\mathcal{S})}^2.
\]
Due to $J_{322}$ being cancelled out with the first term in \epsilonqref{V221E}$_2$, it remains only to estimate $J_4$. Recall the $L^\infty$ bound of $(P_\Phi^R)'$ in \epsilonqref{EHP}, one concludes that
\[
J_4\leq C\frac{\alpha\Phi}{1+\alpha}\|\nabla \boldsymbol{v}^N\|_{L^2(\mathcal{S})}^2.
\]
Collecting the above estimates and cancellations, we derive that
\[
|V|=\left|\inftynt_{\mathcal{S}}\boldsymbol{v}\cdot\nablabla \boldsymbol{a}\cdot \boldsymbol{v}dx\right|\leq C\Phi \left(\varepsilon+\frac{\alpha}{1+\alpha}\right)\|\nabla \boldsymbol{v}\|_{L^2(\mathcal{S})}^2,
\]
which concludes \epsilonqref{estisqu}. This completes the proof of Proposition \ref{Prop}.
\quaded
\sqrtubsection{Existence of the weak solution}\lambdabel{SEC3627}
We will look for a solution to \epsilonqref{NS}-\epsilonqref{GL3} of the form
\begin{equation}\lambdabel{EEPPSS}
\boldsymbol{u}=\boldsymbol{v}+\boldsymbol{a}.
\epsilonnd{equation}
Thus, our problem turns to the following equivalent form:
\begin{equation}gin{problem}\lambdabel{PP2}
Find $(\boldsymbol{v},p)$ such that
\[
\left\{
\begin{equation}gin{aligned}
&\boldsymbol{v}\cdot\nablabla \boldsymbol{v}+\boldsymbol{a}\cdot\nablabla \boldsymbol{v}+\boldsymbol{v}\cdot\nablabla \boldsymbol{a}+\nabla p-\Delta \boldsymbol{v}=\Delta \boldsymbol{a}-\boldsymbol{a}\cdot\nablabla \boldsymbol{a},\\
&\nabla\cdot \boldsymbol{v}=0,\\
\epsilonnd{aligned}
\right.\quad\tildeext{in }\quad\mathcal{S},
\]
subject to the Navier-slip boundary condition
\begin{equation}\lambdabel{BDR}
\left\{
\begin{equation}gin{aligned}
&2(\mathbb{S}\boldsymbol{v}\cdot\boldsymbol{n})_{\mathrm{tan}}+\alpha \boldsymbol{v}_{\mathrm{tan}}=0,\\
&\boldsymbol{v}\cdot\boldsymbol{n}=0,\\
\epsilonnd{aligned}
\right.\quad\tildeext{on }\quad\partial\mathcal{S},
\epsilonnd{equation}
with the asymptotic behavior as $|x|\tildeo\infty$
\[
\boldsymbol{v}(x)\tildeo \boldsymbol{0},\quad\tildeext{as}\quad |x|\tildeo \infty.
\]
\epsilonnd{problem}
\quaded
From the weak formulation \epsilonqref{weaksd}, we have that $\boldsymbol{v}$ satisfies the following weak formulation:
\begin{equation}gin{definition}\lambdabel{defws}
Let $\boldsymbol{a}$ be a smooth vector satisfying the properties stated in the above. We say that $\boldsymbol{v}\inftyn \mathcal{H}_\sqrtigmagma(\mathcal{S})$ is a weak solution of Problem \ref{PP2} if
\begin{equation}\lambdabel{EQU111}
\begin{equation}gin{split}
&2\inftynt_{\mathcal{S}}\mathbb{S}\boldsymbol{v}:\mathbb{S}\boldsymbol{\varphi} dx+\alpha\inftynt_{\partial\mathcal{S}}\boldsymbol{v}_{\mathrm{tan}}\cdot\boldsymbol{\varphi}_{\mathrm{tan}}dS+\inftynt_{\mathcal{S}}\boldsymbol{v}\cdot\nablabla \boldsymbol{v} \cdot \boldsymbol{\varphi} dx+\inftynt_{\mathcal{S}}\boldsymbol{v}\cdot\nablabla \boldsymbol{a}\cdot\boldsymbol{\varphi} dx\\
+&\inftynt_{\mathcal{S}}\boldsymbol{a}\cdot\nablabla \boldsymbol{v}\cdot\boldsymbol{\varphi} dx=\inftynt_{\mathcal{S}}\big(\Delta \boldsymbol{a}-\boldsymbol{a}\cdot\nablabla \boldsymbol{a}\big)\cdot\boldsymbol{\varphi} dx
\epsilonnd{split}
\epsilonnd{equation}
holds for any vector-valued function $\boldsymbol{\varphi}\inftyn \mathcal{H}_\sqrtigmagma(\mathcal{S})$.
\epsilonnd{definition}
\quaded
Now we state our main result of this part.
\begin{equation}gin{theorem}\lambdabel{THM3.8}
There is a constant $\Phi_0>0$ depending on the curvature of $\partial\mathcal{S}$ such that if $\frac{\alpha\Phi}{1+\alpha}<\Phi_0$, then Problem \ref{PP2} admits at least one weak solution $(\boldsymbol{v},p)\inftyn \mathcal{H}_\sqrtigmagma(\mathcal{S})\tildeimes L^2_{\mathrm{loc}}(\omegaverline{\mathcal{S}})$, with
\begin{equation}\lambdabel{EOFV627}
\|\boldsymbol{v}\|_{H^1(\mathcal{S})}\leq C(\|\boldsymbol{a}\cdot\nabla \boldsymbol{a}\|_{L^2(\mathcal{S}_{e^{C\Phi}})}+\|\Delta\boldsymbol{a}\|_{L^2(\mathcal{S}_{e^{C\Phi}})})\leq \Phi e^{C\Phi}\,.
\epsilonnd{equation}
\epsilonnd{theorem}
\quaded
\begin{equation}gin{proof}
Set
\[
\boldsymbol{X}:=C^\inftynfty_{\sqrtigmagma,c}(\omegaverline{\mathcal{S}};\,\mathbb{R}^2)=\big\{\boldsymbol{\varphi}\inftyn C^\inftynfty_c(\omegaverline{\mathcal{S}}\,;\,\mathbb{R}^2):\, \nabla\cdot \boldsymbol{\varphi}=0,\ \boldsymbol{\varphi}\cdot\boldsymbol{n}\big|_{\partial\mathcal{S}}=0\big\},
\]
and $\{\boldsymbol{\varphi}_k\}_{k=1}^\infty\sqrtubset\boldsymbol{X}$ be an unit orthonormal basis of $\mathcal{H}_\sqrtigmagma(\mathcal{S})$, that is:
\[
\lambdangle \boldsymbol{\varphi}_i,\boldsymbol{\varphi}_j\rangle_{H^1(\mathcal{S})}=\begin{equation}gin{cases}
1,&\quad\tildeext{if }\quad i=j;\\
0,&\quad\tildeext{if }\quad i\neq j,\\
\epsilonnd{cases}
\]
$\fracorall i,j\inftyn\mathbb{N}$. We look for an approximation of $\boldsymbol{v}$ of the form
\[
\boldsymbol{v}^N(x)=\sqrtum_{i=1}^Nc_i^N\boldsymbol{\varphi}_i(x).
\]
Testing the weak formulation \epsilonqref{EQU111} by $\boldsymbol{\varphi}_i$, with $i=1,2,...,N$, one has
\[
\begin{equation}gin{split}
&2\sqrtum_{i=1}^Nc_i^N\inftynt_{\mathcal{S}}\mathbb{S}\boldsymbol{\varphi}_i:\mathbb{S}\boldsymbol{\varphi}_j dx+\alpha\sqrtum_{i=1}^N c_i^N\inftynt_{\partial\mathcal{S}} (\boldsymbol{\varphi}_i)_{\mathrm{tan}}(\boldsymbol{\varphi}_j)_{\mathrm{tan}}dS+\sqrtum_{i,k=1}^Nc_i^Nc_k^N\inftynt_{\mathcal{S}}\boldsymbol{\varphi}_i\cdot\nablabla \boldsymbol{\varphi}_k\cdot \boldsymbol{\varphi}_jdx\\
+&\sqrtum_{i=1}^N\inftynt_{\mathcal{S}}\boldsymbol{\varphi}_i\cdot\nablabla \boldsymbol{a}\cdot \boldsymbol{\varphi}_jdx+\sqrtum_{i=1}^Nc_i^N\inftynt_{\mathcal{S}}\boldsymbol{a}\cdot\nablabla \boldsymbol{\varphi}_i\cdot \boldsymbol{\varphi}_jdx=\inftynt_{\mathcal{S}}\big(\Delta \boldsymbol{a}-\boldsymbol{a}\cdot\nablabla \boldsymbol{a}\big)\cdot \boldsymbol{\varphi}_jdx,\quad\fracorall j=1,2,...,N.
\epsilonnd{split}
\]
This is a system of nonlinear algebraic equations of $N$-dimensional vector
\[
\boldsymbol{c}^N:=(c_1^N,c_2^N,...,c_N^N).
\]
We denote ${P}:\,\mathbb{R}^N\tildeo\mathbb{R}^N$ such that
\begin{equation}s
\begin{equation}gin{split}
\big({P}(\boldsymbol{c}^N)\big)_j=&2\sqrtum_{i=1}^Nc_i^N\inftynt_{\mathcal{S}}\mathbb{S}\boldsymbol{\varphi}_i:\mathbb{S}\boldsymbol{\varphi}_j dx+\alpha\sqrtum_{i=1}^N c_i^N\inftynt_{\partial\mathcal{S}} (\boldsymbol{\varphi}_i)_{\mathrm{tan}}\cdot(\boldsymbol{\varphi}_j)_{\mathrm{tan}}dS+\sqrtum_{i,k=1}^Nc_i^Nc_k^N\inftynt_{\mathcal{S}}\boldsymbol{\varphi}_i\cdot\nablabla \boldsymbol{\varphi}_k\cdot \boldsymbol{\varphi}_jdx\\
&+\sqrtum_{i=1}^N\inftynt_{\mathcal{S}}\boldsymbol{\varphi}_i\cdot\nablabla \boldsymbol{a}\cdot \boldsymbol{\varphi}_jdx+\sqrtum_{i=1}^Nc_i^N\inftynt_{\mathcal{S}}\boldsymbol{a}\cdot\nablabla \boldsymbol{\varphi}_i\cdot \boldsymbol{\varphi}_jdx-\inftynt_{\mathcal{S}}\big(\Delta \boldsymbol{a}-\boldsymbol{a}\cdot\nablabla \boldsymbol{a}\big)\cdot \boldsymbol{\varphi}_jdx,\\
&\hskip 10cm\quad\fracorall j=1,2,...,N.
\epsilonnd{split}
\epsilonnd{equation}s
It is easy to check that
\[
{P}(\boldsymbol{c}^N)\cdot\boldsymbol{c}^N=\underbrace{2\inftynt_{\mathcal{S}}|\mathbb{S} \boldsymbol{v}^N|^2dx+\alpha\inftynt_{\partial\mathcal{S}}|(\boldsymbol{v}^N)_{\mathrm{tan}}|^2dS}_{A_1}+\underbrace{\inftynt_{\mathcal{S}}\left((\boldsymbol{v}^N+\boldsymbol{a})\cdot\nablabla(\boldsymbol{v}^N+\boldsymbol{a})\right)\cdot \boldsymbol{v}^Ndx}_{A_2}-\underbrace{\inftynt_{\mathcal{S}}\boldsymbol{v}^N\cdot\Delta \boldsymbol{a}dx}_{A_3}.
\]
By Lemma \ref{lem-korn}, we have
\[
A_1\gammaeq C_0\inftynt_{\mathcal{S}}|\nablabla \boldsymbol{v}^N|^2dx.
\]
Next, by using integration by parts, together with the divergence-free property of $\boldsymbol{v}^N$ and $\boldsymbol{a}$, one knows that
\[
A_2=\underbrace{\inftynt_{\mathcal{S}}\boldsymbol{v}^N\cdot\nablabla \boldsymbol{a}\cdot \boldsymbol{v}^Ndx}_{V}+\underbrace{\inftynt_{\mathcal{S}}\boldsymbol{a}\cdot\nablabla \boldsymbol{a}\cdot \boldsymbol{v}^Ndx}_{K}.
\]
We now focus on the term $V$. Applying \epsilonmph{(iii)} in Proposition \ref{Prop} to $\boldsymbol{v}^N$, one deduces
\begin{equation}\lambdabel{ESTA21}
|V|\leq C_1\Phi\left(\varepsilon+\frac{\alpha}{1+\alpha}\right)\|\nabla\boldsymbol{v}^N\|_{L^2(\mathcal{S})}^2,
\epsilonnd{equation}
where the constant $C_1$ is independent with $N$. For the term $K$, since $\boldsymbol{a}$ equals to the Poiseuille flow $\boldsymbol{P}^L_{\Phi}$ or $\boldsymbol{P}^R_{\Phi}$ in $\mathcal{S}-\mathcal{S}_{e^{2/\varepsilon}}$, we have $\boldsymbol{a}\cdot\nablabla \boldsymbol{a}\epsilonquiv 0$ in $\mathcal{S}-\mathcal{S}_{e^{2/\varepsilon}}$. Using the Cauchy-Schwarz inequality and the Poincar\'e inequality, one arrives at
\begin{equation}\lambdabel{ESTA22}
\begin{equation}gin{split}
|K|=\left|\inftynt_{\mathcal{S}}\boldsymbol{a}\cdot\nablabla \boldsymbol{a}\cdot \boldsymbol{v}^Ndx\right|
\leq C\|\boldsymbol{a}\cdot\nabla \boldsymbol{a}\|_{L^2(\mathcal{S}_{e^{2/\varepsilon}})}\|\nabla\boldsymbol{v}^N\|_{L^2(\mathcal{S})}\,.
\epsilonnd{split}
\epsilonnd{equation}
Thus, by combining \epsilonqref{ESTA21} and \epsilonqref{ESTA22}, one deduces
\[
|A_2|\leq C_1\Phi\left(\varepsilon+\frac{\alpha}{1+\alpha}\right)\|\nabla\boldsymbol{v}^N\|_{L^2(\mathcal{S})}^2+C\|\boldsymbol{a}\cdot\nabla \boldsymbol{a}\|_{L^2(\mathcal{S}_{e^{2/\varepsilon}})}\|\nabla\boldsymbol{v}^N\|_{L^2(\mathcal{S})}.
\]
Finally, by the construction of the Poiseuille flow $\boldsymbol{P}_{\Phi}^{R}$ and $\boldsymbol{P}_{\Phi}^{L}$, we have
\[
\inftynt_{(-\infty,-e^{2/\varepsilon})\tildeimes(0,c_0)}\boldsymbol{v}^N\cdot\Delta \boldsymbol{a}dy=-C_L\inftynt^{-e^{2/\varepsilon}}_{-\infty}\inftynt_{0}^{c_0}\boldsymbol{v}^N\cdot\boldsymbol{e_1}'dy=0
\]
and
\[
\inftynt_{(e^{2/\varepsilon},+\inftynfty)\tildeimes(0,1)}\boldsymbol{v}^N\cdot\Delta \boldsymbol{a}dx=-C_R\inftynt^{\infty}_{e^{2/\varepsilon}}\inftynt_{0}^{1}\boldsymbol{v}^N\cdot\boldsymbol{e_1}dx=0\,.
\]
Thus, by the Cauchy-Schwarz inequality and the Poincar\'e inequality, we deduce that
\[
|A_3|=\left|\inftynt_{\mathcal{S}_{e^{2/\varepsilon}}}\boldsymbol{v}^N\cdot\Delta\boldsymbol{a}dx\right|\leq C\|\Delta\boldsymbol{a}\|_{L^2(\mathcal{S}_{e^{2/\varepsilon}})}\|\nabla\boldsymbol{v}^N\|_{L^2(\mathcal{S})}.
\]
Substituting the above estimates for $A_1$--$A_3$, and choosing $\varepsilon>0$ being sufficiently small such that $C_1\varepsilon\Phi<\frac{C_0}{2}$, one derives
\[
{P}(\boldsymbol{c}^N)\cdot\boldsymbol{c}^N\gammaeq\|\boldsymbol{v}^N\|_{H^1(\mathcal{S})}\left(\left(\frac{C_0}{2}-C_1\frac{\alpha\Phi}{1+\alpha}\right)\|\boldsymbol{v}^N\|_{H^1(\mathcal{S})}-C(\|\boldsymbol{a}\cdot\nabla \boldsymbol{a}\|_{L^2(\mathcal{S}_{e^{2/\varepsilon}})}+\|\Delta\boldsymbol{a}\|_{L^2(\mathcal{S}_{e^{2/\varepsilon}})})\right),
\]
which guarantees
\[
{P}(\boldsymbol{c}^N)\cdot\boldsymbol{c}^N\gammaeq0,
\]
provided
\[
\frac{\alpha\Phi}{1+\alpha}< \Phi_0:=\frac12C_1^{-1}C_0\quad\tildeext{and}\quad|\boldsymbol{c}^N|=\|\boldsymbol{v}^N\|_{H^1(\mathcal{S})}\gammaeq\frac{C\left(\|\boldsymbol{a}\cdot\nabla \boldsymbol{a}\|_{L^2(\mathcal{S}_{e^{2/\varepsilon}})}+\|\Delta\boldsymbol{a}\|_{L^2(\mathcal{S}_{e^{2/\varepsilon}})}\right)}{C_0/2-C_1\alpha\Phi/(1+\alpha)}:=\rho.
\]
Using Lemma \ref{FUNC}, there exists
\begin{equation}\lambdabel{ubound}
(\boldsymbol{v}^N)^*\inftyn\tildeext{span}\left\{\boldsymbol{\varphi}_1,\boldsymbol{\varphi}_2,...,\boldsymbol{\varphi}_N\right\},\quad\tildeext{and}\quad\|(\boldsymbol{v}^N)^*\|_{H^1(\mathcal{S})}\leq\frac{C\left(\|\boldsymbol{a}\cdot\nabla \boldsymbol{a}\|_{L^2(\mathcal{S}_{e^{2/\varepsilon}})}+\|\Delta\boldsymbol{a}\|_{L^2(\mathcal{S}_{e^{2/\varepsilon}})}\right)}{C_0/2-C_1\alpha\Phi/(1+\alpha)},
\epsilonnd{equation}
such that
\begin{equation}gin{equation}\lambdabel{app-N}
\begin{equation}gin{split}
&2\inftynt_{\mathcal{S}}\mathbb{S}(\boldsymbol{v}^N)^*:\mathbb{S}\boldsymbol{\partialhi}_Ndx+\alpha\inftynt_{\partial\mathcal{S}}(\boldsymbol{v}^N)^*_{\mathrm{tan}}\cdot(\boldsymbol{\partialhi}_N)_{\mathrm{tan}}dS+\inftynt_{\mathcal{S}}(\boldsymbol{v}^N)^*\cdot\nablabla (\boldsymbol{v}^N)^*\cdot \boldsymbol{\partialhi}_Ndx+\inftynt_{\mathcal{S}}(\boldsymbol{v}^N)^*\cdot\nablabla \boldsymbol{a}\cdot\boldsymbol{\partialhi}_Ndx\\
+&\inftynt_{\mathcal{S}}\boldsymbol{a}\cdot\nablabla(\boldsymbol{v}^N)^*\cdot\boldsymbol{\partialhi}_N dx=\inftynt_{\mathcal{S}}\big(\Delta \boldsymbol{a}-\boldsymbol{a}\cdot\nablabla \boldsymbol{a}\big)\cdot\boldsymbol{\partialhi}_Ndx,\quad\fracorall\boldsymbol{\partialhi}_N\inftyn\tildeext{span}\left\{\boldsymbol{\varphi}_1,\boldsymbol{\varphi}_2,...,\boldsymbol{\varphi}_N\right\}.
\epsilonnd{split}
\epsilonnd{equation}
The above bound \epsilonqref{ubound} and Rellich-Kondrachov embedding theorem imply the existence of a field $\boldsymbol{v}\inftyn \mathcal{H}_{\sqrtigmagma}(\mathcal{S})$ and a subsequence, which we will always denote by $(\boldsymbol{v}^N)^*$, such that
\begin{equation}gin{equation*}
(\boldsymbol{v}^N)^*\tildeo \boldsymbol{v}\quaduad \tildeext{weakly in $\mathcal{H}_{\sqrtigmagma}(\mathcal{S})$}
\epsilonnd{equation*}
and
\begin{equation}gin{equation*}
(\boldsymbol{v}^N)^*\tildeo \boldsymbol{v}\quaduad \tildeext{strongly in $L^2(\mathcal{S}')$, for all bounded $\mathcal{S}'\sqrtubset\mathcal{S}$}\,.
\epsilonnd{equation*}
By passing to the limit in \epsilonqref{app-N}, one obtains
\begin{equation}\lambdabel{V}
\begin{equation}gin{split}
&2\inftynt_{\mathcal{S}}\mathbb{S}\boldsymbol{v}:\mathbb{S}\boldsymbol{\varphi} dx+\alpha\inftynt_{\partial\mathcal{S}}\boldsymbol{v}_{\mathrm{tan}}\cdot\boldsymbol{\varphi}_{\mathrm{tan}}dS+\inftynt_{\mathcal{S}}\boldsymbol{v}\cdot\nablabla \boldsymbol{v}\cdot\boldsymbol{\varphi} dx+\inftynt_{\mathcal{S}}\boldsymbol{v}\cdot\nablabla \boldsymbol{a}\cdot\boldsymbol{\varphi} dx\\
+&\inftynt_{\mathcal{S}}\boldsymbol{a}\cdot\nablabla \boldsymbol{v}\cdot\boldsymbol{\varphi} dx=\inftynt_{\mathcal{S}}\big(\Delta \boldsymbol{a}-\boldsymbol{a}\cdot\nablabla \boldsymbol{a}\big)\cdot\boldsymbol{\varphi} dx,\quad\tildeext{for any}\quad \boldsymbol{\varphi}\inftyn \mathcal{H}_\sqrtigmagma(\mathcal{S}).
\epsilonnd{split}
\epsilonnd{equation}
It follows from \epsilonqref{ubound} and the Fatou lemma for weakly convergent sequences that
\begin{equation}\lambdabel{EEEEST1}
\|\boldsymbol{v}\|_{H^1(\mathcal{S})}\leq C\left(\|\boldsymbol{a}\cdot\nabla \boldsymbol{a}\|_{L^2(\mathcal{S}_{e^{2/\varepsilon}})}+\|\Delta\boldsymbol{a}\|_{L^2(\mathcal{S}_{e^{2/\varepsilon}})}\right)\,.
\epsilonnd{equation}
Now it remains to verify \epsilonqref{EOFV627}. From the construction of $\boldsymbol{a}$ in \epsilonqref{Cons1}--\epsilonqref{Cons} and the estimate of $\sqrtigmagma_\varepsilon$ in \epsilonqref{VEEST}, we have
\[
\left|\nabla^k\boldsymbol{a}\right|\leq C\Phi e^{\frac1{\varepsilon}}(\varepsilon^{-1}e^{\frac1{\varepsilon}})^k,\quad\tildeext{for }k=0,1,2.
\]
According to the construction of $\boldsymbol{v}$ given before, it is legal to choose $\varepsilon=\min\left\{\frac{C_0}{4C_1\Phi},\frac{\delta}{2}\right\}$. This indicates that
\[
\|\boldsymbol{a}\cdot\nabla \boldsymbol{a}\|_{L^2(\mathcal{S}_{e^{2/\varepsilon}})}+\|\Delta\boldsymbol{a}\|_{L^2(\mathcal{S}_{e^{2/\varepsilon}})}\leq Ce^{1/\varepsilon}\left(\Phi^2\varepsilon^{-1}e^{3/\varepsilon}+\Phi\varepsilon^{-2}e^{3/\varepsilon}\right)\leq C\Phi\left(1+\Phi^2e^{C\Phi}\right),
\]
which gives
\[
\|\boldsymbol{v}\|_{H^1(\mathcal{S})}\leq \Phi e^{C\Phi}\,.
\]
Now we focus on the pressure. Let $\boldsymbol{v}$ be a weak solution of \epsilonqref{EQU111} constructed in the above. Using \epsilonqref{V}, one has $\boldsymbol{u}=\boldsymbol{v}+\boldsymbol{a}$ satisfies
\[
\inftynt_{\mathcal{S}}\nablabla \boldsymbol{u}\cdot\nablabla\boldsymbol{\partialhi}\,dx+\inftynt_{\mathcal{S}}\boldsymbol{u}\cdot\nablabla \boldsymbol{u}\cdot\boldsymbol{\partialhi}\,dx=0,\quad\tildeext{for all}\quad \boldsymbol{\partialhi}\inftyn\{\boldsymbol{g}\inftyn C_c^\infty(\mathcal{S};\mathbb{R}^2):\,\tildeext{div }\boldsymbol{g}=0\}.
\]
Thus by Lemma \ref{DeRham}, there exists $p\inftyn\left(C_c^\infty(\mathcal{S};\mathbb{R})\right)'$, such that
\begin{equation}\lambdabel{NS11}
\Delta\boldsymbol{u}-\boldsymbol{u}\cdot\nablabla\boldsymbol{u}=\nablabla p
\epsilonnd{equation}
in the sense of distribution. Furthermore, we have that \epsilonqref{NS11} is equivalent to
\begin{equation}\lambdabel{DEFPi}
\tildeext{div}\big(\nablabla\boldsymbol{v}-\boldsymbol{v}\omegatimes\boldsymbol{v}-\boldsymbol{a}\omegatimes\boldsymbol{v}-\boldsymbol{v}\omegatimes\boldsymbol{a}\big)+\Delta \boldsymbol{a}+{C_{R}}{\epsilonta(x_1)}\boldsymbol{e_1}+{C_{L}}{\epsilonta(-y_1)}\boldsymbol{e_1}'-\boldsymbol{a}\cdot\nablabla \boldsymbol{a}=\nabla\Pi,
\epsilonnd{equation}
with
\begin{equation}\lambdabel{DEFPI}
\Pi=p+{C_{R}}{\inftynt_{-\infty}^{x_1}\epsilonta(s)ds}-{C_{L}}{\inftynt_{-\infty}^{-y_1}\epsilonta(s)ds}\,,
\epsilonnd{equation}
where $C_L$ and $C_R$ are Poiseuille constants defined in \epsilonqref{POSS1}$_1$ and \epsilonqref{POSS2}$_1$, respectively. By the definition of $\boldsymbol{a}$, one has both
\[
\Delta \boldsymbol{a}+{C_{R}}{\epsilonta(x_1)}\boldsymbol{e_1}+{C_{L}}{\epsilonta(-y_1)}\boldsymbol{e_1}'
\]
and $\boldsymbol{a}\cdot\nablabla\boldsymbol{a}$ are smooth and have compact support. Since $\boldsymbol{v}\inftyn H^1(\mathcal{S})$ and $\boldsymbol{a}$ is uniformly bounded, one deduces
\[
\nablabla\boldsymbol{v}-\boldsymbol{v}\omegatimes\boldsymbol{v}-\boldsymbol{a}\omegatimes\boldsymbol{v}-\boldsymbol{v}\omegatimes\boldsymbol{a}\inftyn L^2(\mathcal{S}),
\]
directly by the Sobolev embedding and H\"older's inequality. Therefore one concludes the left hand side of \epsilonqref{DEFPi} belongs to $H^{-1}(\mathcal{S})$. Then applying Lemma \ref{LEM312}, we have $\Pi\inftyn L^2_{\mathrm{loc}}(\omegaverline{\mathcal{S}})$, which leads to $p\inftyn L^2_{\mathrm{loc}}(\omegaverline{\mathcal{S}})$ by \epsilonqref{DEFPI}.
\epsilonnd{proof}
\sqrtubsection{Uniqueness result}
The rest part of this section is devoted to the proof of uniqueness. We will show that the solution $(\boldsymbol{u},{p})$ constructed earlier in this section with its flux being $\Phi$ is unique for $\Phi$ being sufficiently small and independent of $\alpha$.
\sqrtubsubsection{Estimate of the pressure}
Below, we give a proposition to show that an integration estimate related to the pressure in the truncated strip $\Upsilon_Z^{+}:=\left(\mathcal{S}_Z\backslash\mathcal{S}_{Z-1}\right)\cap \{x_1>0\}$ or $\Upsilon_Z^{-}:=\left(\mathcal{S}_Z\backslash\mathcal{S}_{Z-1}\right)\cap \{y_1<0\}$.
\begin{equation}gin{proposition}\lambdabel{P2.4}
Let $(\tildeilde{\boldsymbol{u}},\tildeilde{p})$ be an alternative weak solution of \epsilonqref{NS} in the strip $\mathcal{S}$, subject to the Navier-slip boundary condition \epsilonqref{NBC}. If the total flux
\[
\inftynt_{\mathcal{S}\cap\{x_1=s\}}\tildeilde{\boldsymbol{u}}(s,x_2)\cdot\boldsymbol{e_1}dx_2=\Phi=\inftynt_{\mathcal{S}\cap\{x_1=s\}}{\boldsymbol{u}}(s,x_2)\cdot\boldsymbol{e_1}dx_2,\quad\tildeext{for any } s\gammaeq 1,
\]
then the following estimate of $\boldsymbol{w}:=\tildeilde{\boldsymbol{u}}-{\boldsymbol{u}}$ and the pressure holds
\begin{equation}\lambdabel{ZJZJ}
\left|\inftynt_{\Upsilon^{\partialm}_K}(\tildeilde{p}-p)w_1dx\right|\leq C\left(\|\boldsymbol{u}\|_{L^4(\Upsilon^{\partialm}_K)}\|\nablabla \boldsymbol{w}\|^2_{L^2(\Upsilon^{\partialm}_K)}+\|\nablabla \boldsymbol{w}\|_{L^2(\Upsilon^{\partialm}_K)}^2+\|\nablabla \boldsymbol{w}\|_{L^2(\Upsilon^{\partialm}_K)}^3\right),\fracorall K\gammaeq 2,
\epsilonnd{equation}
where $C>0$ is a constant independent of $K$.
\epsilonnd{proposition}
\noindent {\bf Proof. \hspace{2mm}} We only show \epsilonqref{ZJZJ} on the $\Upsilon^+$ since the rest part is similar. During the proof, we cancel the upper index ``$+$" of the domain for simplicity. Noticing
\[
\inftynt_{\mathcal{S}\cap\{x_1=s\}}w_1(s,x_2)dx_2\epsilonquiv0,\quad\fracorall s\gammaeq 1,
\]
by integrating the above equality for variable $s$ from $K-1$ to $K$, we deduce that
\[
\inftynt_{\Upsilon_K}w_1 dx=0,\quad\fracorall K\gammaeq 2.
\]
Using Lemma \ref{LEM2.1}, one derives the existence of a vector field $V$ satisfying \epsilonqref{LEM2.11} with $f=w_1$. Applying equation \epsilonqref{NS}$_1$, one arrives
\[
\begin{aligned}
\inftynt_{\Upsilon_K}(\tildeilde{p}-{p})w_1dx=&\inftynt_{\Upsilon_K}(\tildeilde{p}-{p})\nabla\cdot \boldsymbol{V}dx\\
=&-\inftynt_{\Upsilon_K}\nablabla(\tildeilde{p}-{p})\cdot \boldsymbol{V}dx=\inftynt_{\Upsilon_K}\left(\boldsymbol{w}\cdot\nabla \boldsymbol{w}+\boldsymbol{u}\cdot\nablabla \boldsymbol{w}+\boldsymbol{w}\cdot\nablabla \boldsymbol{u}-\Delta \boldsymbol{w}\right)\cdot \boldsymbol{V}dx.
\epsilonnd{aligned}
\]
Using integration by parts, one deduces
\[
\inftynt_{\Upsilon_K}(\tildeilde{p}-p)w_1dx=\sqrtum_{i,j=1}^2\inftynt_{\Upsilon_K}(\partial_iw_j-w_iw_j-u_iw_j-u_jw_i)\partial_iV_jdx.
\]
By applying H\"older's inequality and \epsilonqref{LEM2.11} in Lemma \ref{LEM2.1}, one deduces that
\begin{equation}\lambdabel{EP1}
\left|\inftynt_{\Upsilon_K}(\tildeilde{p}-p)w_1dx\right|\leq C\left(\|\nabla \boldsymbol{w}\|_{L^2(\Upsilon_K)}+\|\boldsymbol{w}\|_{L^4(\Upsilon_K)}^2+\|\boldsymbol{u}\|_{L^4(\Upsilon_K)}\|\boldsymbol{w}\|_{L^4(\Upsilon_K)}\right)\|w_1\|_{L^2(\Upsilon_K)}.
\epsilonnd{equation}
Since $w_1$ has a zero mean value on each cross section $\{x_1=s\}$ for $s\gammaeq 1$ and $w_2$ has zero boundary on in the $x_2$ direction, then Poincar\'e inequality in $x_2$ direction implies that
\begin{equation}\lambdabel{PON3}
\|\boldsymbol{w}\|_{L^2(\Upsilon_K)}\leq C \|\partial_{x_2} \boldsymbol{w}\|_{L^2(\Upsilon_K)}.
\epsilonnd{equation}
Substituting \epsilonqref{PON3} in \epsilonqref{EP1}, also noting the Gagliardo-Nirenberg inequality
\[
\|\boldsymbol{w}\|_{L^4(\Upsilon_K)}^2\leq C\left(\|\boldsymbol{w}\|_{L^2(\Upsilon_K)}\|\nabla \boldsymbol{w}\|_{L^2(\Upsilon_K)}+\|\boldsymbol{w}\|_{L^2(\Upsilon_K)}^{2}\right),
\]
one concludes
\[
\left|\inftynt_{\Upsilon_K}(\tildeilde{p}-p)w_1dx\right|\leq C\left(\|\boldsymbol{u}\|_{L^4(\Upsilon_K)}\|\nablabla \boldsymbol{w}\|^2_{L^2(\Upsilon_K)}+\|\nablabla \boldsymbol{w}\|_{L^2(\Upsilon_K)}^2+\|\nablabla \boldsymbol{w}\|_{L^2(\Upsilon_K)}^3\right).
\]
\quaded
\sqrtubsubsection{Main estimates of the uniqueness result}
\quad\ Subtracting the equation of $\boldsymbol{u}$ from the equation of $\tildeilde{\boldsymbol{u}}$, one finds
\begin{equation}\lambdabel{SUBT}
\boldsymbol{w}\cdot\nablabla \boldsymbol{w}+\boldsymbol{u}\cdot\nablabla \boldsymbol{w}+\boldsymbol{w}\cdot\nablabla \boldsymbol{u}+\nablabla(\tildeilde{p}-p)-\Delta \boldsymbol{w}=0.
\epsilonnd{equation}
Multiplying $\boldsymbol{w}$ on both sides of \epsilonqref{SUBT}, and integrating on $\mathcal{S}_\zeta$, one derives
\begin{equation}\lambdabel{Maint0}
-\inftynt_{\mathcal{S}_\zeta}\boldsymbol{w}\cdot\Delta \boldsymbol{w}dx=-\inftynt_{\mathcal{S}_\zeta}\boldsymbol{w}\big(\boldsymbol{w}\cdot\nablabla \boldsymbol{w}+\boldsymbol{u}\cdot\nablabla \boldsymbol{w}+\boldsymbol{w}\cdot\nablabla \boldsymbol{u}+\nablabla (\tildeilde{p}-p)\big)dx.
\epsilonnd{equation}
Using the divergence-free property and the Navier-slip boundary condition of $\boldsymbol{u}$ and $\tildeilde{\boldsymbol{u}}$, one deduces
\[
\begin{equation}gin{split}
-\inftynt_{\mathcal{S}_\zeta}\boldsymbol{w}\cdot\Delta \boldsymbol{w}dx&=-\inftynt_{\mathcal{S}_\zeta}w_i\partial_{x_j}(\partial_{x_j}w_i+\partial_{x_i}w_j)dx\\
&=\sqrtum_{i,j=1}^2\inftynt_{\mathcal{S}_\zeta}\partial_{x_j}w_i(\partial_{x_j}w_i+\partial_{x_i}w_j)dx-\sqrtum_{i,j=1}^2\inftynt_{\partial\mathcal{S}_\zeta}w_in_j(\partial_{x_j}w_i+\partial_{x_i}w_j)dx\\
&=2\inftynt_{\mathcal{S}_\zeta}|\mathbb{S} \boldsymbol{w}|^2dx+\alpha\inftynt_{\partial\mathcal{S}_\zeta\cap\partial\mathcal{S}}|w_{\tildeau}|^2dS\\
&\hskip 1cm-\sqrtum_{i=1}^2\inftynt_{\{x_1=\zeta\}}w_i(\partial_{x_1}w_i+\partial_{x_i} w_1)dx_2+\sqrtum_{i=1}^2\inftynt_{\{y_1=-\zeta\}}w_i(\partial_{x_1}w_i+\partial_{x_i}w_1)dy_2.
\epsilonnd{split}
\]
Here $\boldsymbol{n}=(n_1,n_2)$ is the unit outer normal vector on $\partial\mathcal{S}$. Then one concludes that
\[
-\inftynt_{\mathcal{S}_\zeta}\boldsymbol{w}\cdot\Delta \boldsymbol{w}dx+\inftynt_{\{x_1=\zeta\}}|\boldsymbol{w}||\nablabla \boldsymbol{w}|dx_2+\inftynt_{\{y_1=-\zeta\}}|\boldsymbol{w}||\nablabla \boldsymbol{w}|dy_2\gammaeq 2\inftynt_{\mathcal{S}_\zeta}|\mathbb{S} \boldsymbol{w}|^2dx+\alpha\inftynt_{\partial\mathcal{S}_\zeta\cap\partial\mathcal{S}}|\boldsymbol{w}_{\mathrm{tan}}|^2dS.
\]
Then using the Korn inequality \epsilonqref{KN1.1} in Lemma \ref{lem-korn}, we can achieve that
\begin{equation}\lambdabel{Maint1}
\inftynt_{\mathcal{S}_\zeta}|\nabla \boldsymbol{w}|^2dx\leq C\left(-\inftynt_{\mathcal{S}_\zeta}\boldsymbol{w}\cdot\Delta \boldsymbol{w}dx+\inftynt_{\{x_1=\zeta\}}|\boldsymbol{w}||\nablabla \boldsymbol{w}|dx_2+\inftynt_{\{y_1=-\zeta\}}|\boldsymbol{w}||\nablabla \boldsymbol{w}|dy_2\right).
\epsilonnd{equation}
Now we focus on the right hand side of \epsilonqref{Maint0}. Applying integration by parts, one derives
\begin{equation}\lambdabel{Maint3}
\begin{equation}gin{split}
-\inftynt_{\mathcal{S}_\zeta}\boldsymbol{w}\big(\boldsymbol{w}\cdot\nablabla \boldsymbol{w}+\nablabla (\tildeilde{p}-p)\big)dx=&-\inftynt_{\mathcal{S}\cap\{x_1=\zeta\}}\boldsymbol{w}\cdot\boldsymbol{e_1}\left(\frac{1}{2}|\boldsymbol{w}|^2+(\tildeilde{p}-p)\right)dx_2\\
&+\inftynt_{\mathcal{S}\cap\{y_1=-\zeta\}}\boldsymbol{w}\cdot\boldsymbol{e_1}'\left(\frac{1}{2}|\boldsymbol{w}|^2+(\tildeilde{p}-p)\right)dy_2.
\epsilonnd{split}
\epsilonnd{equation}
Applying H\"older's inequality, noting that $\boldsymbol{u}=\boldsymbol{v}+\boldsymbol{a}$, where $\boldsymbol{a}$ is the flux carrier constructed in Proposition \ref{Prop}, while $\boldsymbol{v}$ is the $H^1$-weak solution given in Section \ref{SEC3627}, one has
\begin{equation}\lambdabel{Maint5}
\begin{equation}gin{split}
\left|-\inftynt_{\mathcal{S}_\zeta}\big(\boldsymbol{w}\cdot\nablabla \boldsymbol{u}\cdot \boldsymbol{w}+\boldsymbol{u}\cdot\nabla\boldsymbol{w}\cdot\boldsymbol{w}\big)dx\right|\leq&\, \|\nablabla\boldsymbol{v}\|_{L^2(\mathcal{S}_\zeta)}\|\boldsymbol{w}\|_{L^4(\mathcal{S}_\zeta)}^2+\|\boldsymbol{v}\|_{L^4(\mathcal{S}_\zeta)}\|\nabla\boldsymbol{w}\|_{L^2(\mathcal{S}_\zeta)}\|\boldsymbol{w}\|_{L^4(\mathcal{S}_\zeta)}\\
&+\|\nabla\boldsymbol{a}\|_{L^\infty(\mathcal{S}_\zeta)}\|\boldsymbol{w}\|_{L^2(\mathcal{S}_\zeta)}^2+\|\boldsymbol{a}\|_{L^\infty(\mathcal{S}_\zeta)}\|\nabla\boldsymbol{w}\|_{L^2(\mathcal{S}_\zeta)}\|\boldsymbol{w}\|_{L^2(\mathcal{S}_\zeta)}\\
\leq&\, C\left(\|\boldsymbol{v}\|_{H^{1}(\mathcal{S}_\zeta)}+\|\boldsymbol{a}\|_{W^{1,\infty}(\mathcal{S}_\zeta)}\right)\inftynt_{\mathcal{S}_\zeta}|\nablabla \boldsymbol{w}|^2dx\\
\leq&\,\Phi e^{C \Phi}\inftynt_{\mathcal{S}_\zeta}|\nablabla \boldsymbol{w}|^2dx.
\epsilonnd{split}
\epsilonnd{equation}
Here in the second inequality, we have applied the Gagliardo-Nirenberg inequality and the Poincar\'e inequality \epsilonqref{TORPIPEPOIN} in Lemma \ref{TORPOIN}, which indicate
\[
\|\boldsymbol{w}\|_{L^4(\mathcal{S}_\zeta)}\leq C\left(\|\boldsymbol{w}\|^{1/2}_{L^2(\mathcal{S}_\zeta)}\|\nabla\boldsymbol{w}\|^{1/2}_{L^2(\mathcal{S}_\zeta)}+\|\boldsymbol{w}\|_{L^2(\mathcal{S}_\zeta)}\right)\leq C\left(\inftynt_{\mathcal{S}_\zeta}|\nablabla \boldsymbol{w}|^2dx\right)^{1/2}.
\]
Meanwhile, the third inequality in \epsilonqref{Maint5} is guaranteed by \epsilonqref{EOFV627} and Estimates for $\boldsymbol{a}$. Substituting \epsilonqref{Maint1}, \epsilonqref{Maint3} and \epsilonqref{Maint5} in \epsilonqref{Maint0}, one arrives
\[
\begin{equation}gin{split}
\inftynt_{\mathcal{S}_\zeta}|\nablabla \boldsymbol{w}|^2dx\leq& C\left(\inftynt_{\{x_1=\zeta\}}|\boldsymbol{w}|(|\nabla \boldsymbol{w}|+|\boldsymbol{w}|^2)dx_2+\inftynt_{\{y_1=-\zeta\}}|\boldsymbol{w}|(|\nabla \boldsymbol{w}|+|\boldsymbol{w}|^2)dy_2+\Phi e^{C\Phi}\inftynt_{\mathcal{S}_\zeta}|\nablabla \boldsymbol{w}|^2dx\right.\\
&\left.-\inftynt_{\{x_1=\zeta\}}\boldsymbol{w}\cdot\boldsymbol{e_1}\left(\tildeilde{p}-p\right)dx_2+\inftynt_{\{y_1=-\zeta\}}\boldsymbol{w}\cdot\boldsymbol{e_1}'\left(\tildeilde{p}-p\right)dy_2\right).
\epsilonnd{split}
\]
Now one concludes that if $\Phi<<1$ being small enough such that
\begin{equation}s
\Phi e^{C\Phi}<\frac{1}{2},
\epsilonnd{equation}s
then we achieve
\[
\begin{equation}gin{split}
\inftynt_{\mathcal{S}_\zeta}|\nablabla \boldsymbol{w}|^2dx\leq C&\left(\inftynt_{\{x_1=\zeta\}}|\boldsymbol{w}|(|\nabla \boldsymbol{w}|+|\boldsymbol{w}|^2)dx_2+\inftynt_{\{y_1=-\zeta\}}|\boldsymbol{w}|(|\nabla \boldsymbol{w}|+|\boldsymbol{w}|^2)dy_2\right.\\
&\left.-\inftynt_{\{x_1=\zeta\}}\boldsymbol{w}\cdot\boldsymbol{e_1}\left(\tildeilde{p}-p\right)dx_2+\inftynt_{\{y_1=-\zeta\}}\boldsymbol{w}\cdot\boldsymbol{e_1}'\left(\tildeilde{p}-p\right)dy_2\right).
\epsilonnd{split}
\]
Therefore, one derives the following estimate by integrating with $\zeta$ on $[K-1,K]$, where $K\gammaeq 2$:
\begin{equation}\lambdabel{ET+0}
\begin{equation}gin{split}
\inftynt_{K-1}^K\inftynt_{\mathcal{S}_\zeta}|\nablabla \boldsymbol{w}|^2dxd\zeta\leq C&\left(\inftynt_{\Upsilon_K^+}|\boldsymbol{w}|(|\nabla \boldsymbol{w}|+|\boldsymbol{w}|^2)dx+\inftynt_{\Upsilon_K^-}|\boldsymbol{w}|(|\nabla \boldsymbol{w}|+|\boldsymbol{w}|^2)dy\right.\\
&\left.+\Big|\inftynt_{\Upsilon_K^+}\boldsymbol{w}\cdot\boldsymbol{e_1}\left(\tildeilde{p}-p\right)dx\Big|+\Big|\inftynt_{\Upsilon_K^-}\boldsymbol{w}\cdot\boldsymbol{e_1}'\left(\tildeilde{p}-p\right)dy\Big|\right).
\epsilonnd{split}
\epsilonnd{equation}
Now we only handle integrations on $\Upsilon_K^+$ since the cases of $\Upsilon_K^-$ are similar. Using the Cauchy-Schwarz inequality and the Poincar\'e inequality Lemma \ref{POIN}, one has
\begin{equation}\lambdabel{ET+1}
\inftynt_{\Upsilon_K^+}|\boldsymbol{w}||\nabla \boldsymbol{w}|dx\leq\|\boldsymbol{w}\|_{L^2(\Upsilon_K^+)}\|\nablabla \boldsymbol{w}\|_{L^2(\Upsilon_K^+)}\leq C\|\nablabla \boldsymbol{w}\|^2_{L^2(\Upsilon_K^+)}.
\epsilonnd{equation}
Moreover, by H\"older's inequality and the Gagliardo-Nirenberg inequality, one writes
\[
\inftynt_{\Upsilon_K^+}|\boldsymbol{w}|^3dx\leq C\left(\|\boldsymbol{w}\|_{L^2(\Upsilon_K^+)}^{2}\|\nabla \boldsymbol{w}\|_{L^2(\Upsilon_K^+)}+\|\boldsymbol{w}\|_{L^2(\Upsilon_K^+)}^{3}\right),
\]
which follows by the Poincar\'e inequality that
\[
\inftynt_{\Upsilon_K^+}|\boldsymbol{w}|^3dx\leq C\|\nabla \boldsymbol{w}\|_{L^2(\Upsilon_K^+)}^{3}.
\]
Recalling Proposition \ref{P2.4}, one arrives at
\begin{equation}\lambdabel{ET+2}
\left|\inftynt_{\Upsilon_K^+}w_3\left(\tildeilde{p}-p\right)dx\right|\leq C\left(\|\boldsymbol{u}\|_{L^4(\Upsilon_K^+)}\|\nablabla \boldsymbol{w}\|^2_{L^2(\Upsilon_K^+)}+\|\nablabla \boldsymbol{w}\|_{L^2(\Upsilon_K^+)}^2+\|\nablabla \boldsymbol{w}\|_{L^2(\Upsilon_K^+)}^3\right).
\epsilonnd{equation}
Substituting \epsilonqref{ET+1}--\epsilonqref{ET+2}, together with their related inequality on domain $\Upsilon_K^{-}$, in \epsilonqref{ET+0}, one concludes
\begin{equation}\lambdabel{FEST}
\inftynt_{K-1}^K\inftynt_{\mathcal{S}_\zeta}|\nablabla \boldsymbol{w}|^2dxd\zeta\leq C\left(\|\nablabla \boldsymbol{w}\|_{L^2(\Upsilon_K^+\cup\Upsilon_K^-)}^2+\|\nablabla \boldsymbol{w}\|_{L^2(\Upsilon_K^+\cup\Upsilon_K^-)}^3\right).
\epsilonnd{equation}
\sqrtubsubsection{End of proof}
\quad\ Finally, by defining
\[
Y(K):=\inftynt_{K-1}^K\inftynt_{\mathcal{S}_\zeta}|\nablabla \boldsymbol{w}|^2dxd\zeta,
\]
\epsilonqref{FEST} indicates
\[
Y(K)\leq C\left(Y'(K)+\left(Y'(K)\right)^{3/2}\right),\quad\fracorall K\gammaeq 1.
\]
By Lemma \ref{LEM2.3}, we derive
\[
\liminf_{\zeta\tildeo\inftynfty}K^{-3}Y(K)>0,
\]
that is, there exists $C_0>0$ such that
\[
\inftynt_{K-1}^K\inftynt_{\mathcal{S}_\zeta}|\nablabla \boldsymbol{w}|^2dxd\zeta\gammaeq C_0K^3.
\]
However, this leads to a paradox with the condition \epsilonqref{sesti}. Thus $Y(K)\epsilonquiv0$ for all $K\gammaeq 1$, which proves $\boldsymbol{u}\epsilonquiv \tildeilde{\boldsymbol{u}}$. This concludes the uniqueness.
\quaded
\sqrtection{Asymptotic and regularity of the weak solution}\lambdabel{SECH}
\sqrtubsection{Decay estimate of the weak solution}
In this subsection we will show the weak solution constructed in the previous section decays exponentially to Poiseuille flows \epsilonqref{PF} as $|x|\tildeo\infty$. Our proof is also valid for stationary Navier-Stokes problem on domains which is less regular, say an infinite pipe only with a $C^{1,1}$ boundary.
For the convenience of our further statement, we localize the problem in the following way: Denoting
\begin{equation}\lambdabel{CUTFRAK}
\mathcal{S}=\bigcup_{k\inftyn\mathbb{Z}}\mathfrak{S}_k,
\epsilonnd{equation}
where
\[
\mathfrak{S}_k:=\left\{
\begin{equation}gin{array}{ll}
\mathcal{S}\cap\left\{x\inftyn\mathbb{R}^2:\,\left(\frac{3k}{2}-1\right)Z_\Phi\leq x_1\leq\left(\frac{3k}{2}+1\right)Z_\Phi\right\},
&\quad k>0;\\[1.5mm]
\mathcal{S}_{Z_\Phi},&\quad k=0;\\[1.5mm]
\mathcal{S}\cap\left\{x\inftyn\mathbb{R}^2:\,\left(\frac{3k}{2}-1\right)Z_\Phi\leq y_1\leq\left(\frac{3k}{2}+1\right)Z_\Phi\right\},
&\quad k<0,\\
\epsilonnd{array}
\right.
\]
where $Z_{\Phi}=e^{2/\varepsilon}\leq e^{C\Phi}$, while $\varepsilon>0$ is a fixed small constant given in the construction of $\boldsymbol{a}$. Here is the main result of this subsection:
\begin{equation}gin{proposition}\lambdabel{PROP5.1}
Let the conditions of the item (ii) in Theorem \ref{PRO1.2} be satisfied and $(\boldsymbol{v},\,\Pi)$ is given in \epsilonqref{EEPPSS} and \epsilonqref{DEFPI}. Then there exist positive constants $C$, $\sqrtigmagma$, depending only on $\Phi$, such that
\begin{equation}\lambdabel{ASYP}
\begin{equation}gin{split}
\left\|\boldsymbol{u}-\boldsymbol{P}^L_{\Phi}\right\|_{H^1(\mathcal{S}_L\backslash\mathcal{S}_\zeta)}+\left\|\boldsymbol{u}-\boldsymbol{P}^R_{\Phi}\right\|_{H^1(\mathcal{S}_R\backslash\mathcal{S}_\zeta)}&\leq C\|\boldsymbol{v}\|_{H^1(\mathcal{S})}\epsilonxp(-\sqrtigmagma\zeta),\\
\epsilonnd{split}
\epsilonnd{equation}
for any $\zeta$ being large enough.
\epsilonnd{proposition}
\quaded
During the proof of Proposition \ref{PROP5.1}, we need the following refined estimate of the pressure field:
\begin{equation}gin{lemma}\lambdabel{RMKK54}
The reformulated pressure field $\Pi$ given in \epsilonqref{DEFPI} enjoys the following uniform estimate:
\[
\sqrtum_{k\inftyn\mathbb{Z}}\|\Pi-\omegaverline{\Pi}_{\mathfrak{S}_k}\|^2_{L^2(\mathfrak{S}_k)}\leq \Phi^2 e^{C\Phi}<\infty.
\]
\epsilonnd{lemma}
\noindent {\bf Proof. \hspace{2mm}} Applying \epsilonqref{EEEE0} in Lemma \ref{LEM312}, one deduces
\begin{equation}\lambdabel{EEEE1}
\|\Pi-\omegaverline{\Pi}_{\mathfrak{S}_k}\|_{L^2(\mathfrak{S}_k)}\leq C_k\|\nabla\Pi\|_{H^{-1}(\mathfrak{S}_k)}.
\epsilonnd{equation}
Notice that, each $\mathfrak{S}_k$ ($k\inftyn\mathbb{Z}$) is congruent to an element in $\{\mathfrak{S}_{-1},\,\mathfrak{S}_0,\,\mathfrak{S}_1\}$. This indicates constants $C_k$ in estimates \epsilonqref{EEEE1} above could be chosen uniformly with respect to $k\inftyn\mathbb{Z}$. By equation
\[
\nablabla\Pi=\tildeext{div}\big(\nablabla\boldsymbol{v}-\boldsymbol{v}\omegatimes\boldsymbol{v}-\boldsymbol{a}\omegatimes\boldsymbol{v}-\boldsymbol{v}\omegatimes\boldsymbol{a}\big)+\Delta \boldsymbol{a}+{C_{R}}{\epsilonta(x_1)}\boldsymbol{e_1}+{C_{L}}{\epsilonta(-y_1)}\boldsymbol{e_1}'-\boldsymbol{a}\cdot\nablabla \boldsymbol{a},
\]
with both $\Delta \boldsymbol{a}+{C_{R}}{\epsilonta(x_1)}\boldsymbol{e_1}+{C_{L}}{\epsilonta(-y_1)}\boldsymbol{e_1}'$ and $\boldsymbol{a}\cdot\nablabla \boldsymbol{a}$ vanish in $\mathfrak{S}_k$ with $|k|\gammaeq 2$, one concludes from \epsilonqref{EEEE1} that
\[
\begin{equation}gin{split}
\|\Pi-\omegaverline{\Pi}_{\mathfrak{S}_k}\|_{L^2(\mathfrak{S}_k)}&\leq C\left(\|\nabla\boldsymbol{v}\|_{L^2(\mathfrak{S}_k)}+\|\boldsymbol{v}\|_{L^4(\mathfrak{S}_k)}^2+\Phi e^{C\Phi}\|\boldsymbol{v}\|_{L^2(\mathfrak{S}_k)}\right)+\Phi e^{C\Phi}\chi_{|k|\leq1}\\
&\leq C\|\boldsymbol{v}\|_{H^1(\mathfrak{S}_k)}\left(1+\Phi e^{C\Phi}+\|\boldsymbol{v}\|_{H^1(\mathfrak{S}_k)}\right)+\Phi e^{C\Phi}\chi_{|k|\leq1}.
\epsilonnd{split}
\]
Here we have applied the Sobolev imbedding theorem and interpolations of $L^p$ spaces. This completes the proof of Lemma \ref{RMKK54}.
\quaded
{\noindent\bf Proof of Proposition \ref{PROP5.1}: }
We only prove the estimate of term $\|\boldsymbol{u}-\boldsymbol{P}^R_{\Phi}\|_{H^1(\mathcal{S}_R\backslash\mathcal{S}_\zeta)}$ since the rest term is essentially identical. For $\zeta>Z_\Phi$, in $\mathcal{S}_R\backslash\mathcal{S}_\zeta$, the equation of $\boldsymbol{v}=\boldsymbol{u}-\boldsymbol{a}$ reads
\begin{equation}\lambdabel{EW}
\boldsymbol{v}\cdot\nablabla \boldsymbol{v}+\boldsymbol{a}\cdot\nablabla\boldsymbol{v}+\boldsymbol{v}\cdot\nabla\boldsymbol{a}+\nablabla\Pi-\Delta\boldsymbol{v}=0.
\epsilonnd{equation}
This is because
\[
\Delta \boldsymbol{a}+{C_{R}}{\epsilonta(x_3)}\boldsymbol{e_1}+{C_{L}}{\epsilonta(-y_1)}\boldsymbol{e_1}'-\boldsymbol{a}\cdot\nablabla\boldsymbol{a}=\left(\Delta {P}^R_{\Phi}+{C_{L}}\right)\boldsymbol{e_1}=0,\quad\tildeext{in}\quad\mathcal{S}_R\backslash\mathcal{S}_\zeta.
\]
In the following proof, we will drop (upper or lower) indexes ``$R$" for convenience. For any $Z_\Phi<\zeta\leq\zeta'<\zeta_1$, taking inner product with $\boldsymbol{v}$ on both sides of \epsilonqref{EW} and integrating on $\mathcal{S}_R\cap(\mathcal{S}_{\zeta_1}\backslash\mathcal{S}_{\zeta'})$, one has
\begin{equation}\lambdabel{Maint000}
\underbrace{\inftynt_0^1\inftynt_{\zeta'}^{\zeta_1}\boldsymbol{v}\cdot\Delta \boldsymbol{v}dx_1dx_2}_{LHS}=\underbrace{\inftynt_0^1\inftynt_{\zeta'}^{\zeta_1}\big(\boldsymbol{v}\cdot\nablabla \boldsymbol{v}+\boldsymbol{a}\cdot\nablabla \boldsymbol{v}+\boldsymbol{v}\cdot\nablabla \boldsymbol{a}+\nablabla\Pi\big)\cdot\boldsymbol{v}dx_1dx_2}_{RHS}.
\epsilonnd{equation}
To handle the left hand side of \epsilonqref{Maint000}, one first recalls the derivation of \epsilonqref{Maint1} that
\[
\begin{equation}gin{split}
\inftynt_0^1\inftynt_{\zeta'}^{\zeta_1}\boldsymbol{v}\cdot\Delta\boldsymbol{v}dx_1dx_2=&-2\inftynt_0^1\inftynt_{\zeta'}^{\zeta_1}|\mathbb{S} \boldsymbol{v}|^2dx_1dx_2-\alpha\inftynt_{\zeta'}^{\zeta_1}|\boldsymbol{v}_{tan}|^2\Big|_{x_2=1}dx_1-\alpha\inftynt_{\zeta'}^{\zeta_1}|\boldsymbol{v}_{tan}|^2\Big|_{x_2=0}dx_1\\
&-\sqrtum_{i=1}^2\inftynt_0^1v_i(\partial_{x_1}v_i+\partial_{x_i}v_1)\Big|_{x_1=\zeta'}dx_2+\sqrtum_{i=1}^2\inftynt_0^1v_i(\partial_{x_1}v_i+\partial_{x_i}v_1)\Big|_{x_1=\zeta_1}dx_2.
\epsilonnd{split}
\]
Applying Lemma \ref{lem-korn}, the Korn's inequality in a truncated stripe, one deduces the left hand side of \epsilonqref{Maint000} satisfies
\begin{equation}\lambdabel{LH}
\begin{equation}gin{split}
LHS\leq&C\left(-\inftynt_0^1\inftynt_{\zeta'}^{\zeta_1}|\nabla \boldsymbol{v}|^2dx_1dx_2+\inftynt_0^1|\boldsymbol{v}||\nabla\boldsymbol{v}|\Big|_{x_1=\zeta'}dx_2+\inftynt_0^1|\boldsymbol{v}||\nabla\boldsymbol{v}|\Big|_{x_1=\zeta_1}dx_2\right)
\epsilonnd{split}
\epsilonnd{equation}
Using integration by parts for the right hand side of \epsilonqref{Maint000}, one arrives
\begin{equation}\lambdabel{RH}
\begin{equation}gin{split}
RHS=&\inftynt_0^1\left(\frac{1}{2}\left(v_1+P_\Phi\right)|\boldsymbol{v}|^2+v_1\Pi+P_\Phi(v_1)^2\right)\Big|_{x_1=\zeta_1}dx_2\\
&-\inftynt_0^1\left(\frac{1}{2}\left(v_1+P_\Phi\right)|\boldsymbol{v}|^2+v_1\Pi+P_\Phi(v_1)^2\right)\Big|_{x_1=\zeta'}dx_2\\
&-\inftynt_0^1\inftynt_{\zeta'}^{\zeta_1}\boldsymbol{v}\cdot\nabla \boldsymbol{v}\cdot \boldsymbol{a} dx_1dx_2.
\epsilonnd{split}
\epsilonnd{equation}
Now we are ready to perform $\zeta_1\tildeo\infty$. To do this, one must be careful with the integrations on $\{x_1=\zeta_1\}\tildeimes(0,1)$ in both \epsilonqref{LH} and \epsilonqref{RH}. Recalling estimates of $(\boldsymbol{v},\Pi)$ in Theorem \ref{THM3.8} and Lemma \ref{RMKK54}, one derives
\begin{equation}\lambdabel{WESTTT}
\|\boldsymbol{v}\|^2_{H^1(\mathcal{S})}+\|\boldsymbol{v}\|^4_{L^4(\mathcal{S})}+\sqrtum_{k\inftyn\mathbb{Z}}\|\Pi-\omegaverline{\Pi}_{\mathfrak{S}_k}\|^2_{L^2(\mathfrak{S}_k)}\leq \Phi^2 e^{C\Phi}<\infty.
\epsilonnd{equation}
Choosing $M:=\frac{\Phi^2e^{C\Phi}}{Z_\Phi}$, one concludes that for any $k>1$, there exists a slice $\{x_1=\zeta_{1,k}\}\tildeimes(0,1)$ which satisfies
\[
\{x_1=\zeta_{1,k}\}\tildeimes(0,1)\sqrtubset\mathcal{S}\cap\left\{x\inftyn\mathbb{R}^2:\,\left(\frac{3k}{2}-\frac{1}{2}\right)Z_\Phi\leq x_1\leq\left(\frac{3k}{2}+\frac{1}{2}\right)Z_\Phi\right\}\sqrtubset\mathfrak{S}_k,
\]
and it holds that
\[
\inftynt_0^1\left(|\nabla\boldsymbol{v}|^2+|\boldsymbol{v}|^4+|\Pi-\omegaverline{\Pi}_{\mathfrak{S}_k}|^2\right)\Big|_{{x_1}=\zeta_{1,k}}dx_2\leq M.
\]
Otherwise, one has
\[
\|\boldsymbol{v}\|^2_{H^1(\mathfrak{S}_k)}+\|\boldsymbol{v}\|^4_{L^4(\mathfrak{S}_k)}+\|\Pi-\omegaverline{\Pi}_{\mathfrak{S}_k}\|^2_{L^2(\mathfrak{S}_k)}{>Z_\Phi M=\Phi ^2e^{C\Phi}},
\]
which creates a paradox to \epsilonqref{WESTTT}. Choosing $k_0>0$ being sufficiently large such that the sequence $\{\zeta_{1,k}\}_{k=k_0}^\infty\sqrtubset [\zeta',\infty)$, clearly one has $\zeta_{1,k}\nearrow\infty$ as $k\tildeo\infty$. Moreover, using the trace theorem of functions in the Sobolev space $H^1$, one has
\[
\inftynt_0^1|\boldsymbol{v}(x_1,x_2)|^2dx_2\leq C\inftynt_{z>x_1}\inftynt_0^1(|\boldsymbol{v}|^2+|\nablabla \boldsymbol{v}|^2)(z,x_2)dx_2dz\tildeo 0,\quad\tildeext{as}\quad x_1\tildeo\infty.
\]
Noting that $\inftynt_0^1v_1(\zeta_{1,k},x_2)dx_2=0$ for $k\gammaeq k_0$, we deduce the following by the Poincar\'e inequality:
\[
\begin{equation}gin{split}
\left|\inftynt_0^1v_3\Pi\Big|_{x_1=\zeta_{1,k}} dx_2\right|&=\left|\inftynt_0^1v_3\left(\Pi-\omegaverline{\Pi}_{\mathfrak{S}_k}\right)\Big|_{x_1=\zeta_{1,k}} dx_2\right|\\
&\leq \left(\inftynt_0^1|\boldsymbol{v}|^2\Big|_{x_1=\zeta_{1,k}}dx_2\right)^{1/2}\left(\inftynt_0^1|\Pi-\omegaverline{\Pi}_{\mathfrak{S}_k}|^2\Big|_{x_1=\zeta_{1,k}}dx_2\right)^{1/2}\tildeo 0,\quad\tildeext{as}\quad k\tildeo\infty.
\epsilonnd{split}
\]
Meanwhile, one finds
\[
\begin{equation}gin{split}
&\inftynt_{\Sigmagma\tildeimes\{x_3=\zeta_{1,k}\}}|\boldsymbol{v}|\left(|\nablabla\boldsymbol{v}|+|\boldsymbol{v}|^2\right)\Big|_{x_1=\zeta_{1,k}}dx_2\\
&\leq \left(\inftynt_0^1\left(|\nabla\boldsymbol{v}|^2+|\boldsymbol{v}|^4\right)\Big|_{x_1=\zeta_{1,k}}dx_2\right)^{1/2}\left(\inftynt_0^1|\boldsymbol{v}|^2\Big|_{x_1=\zeta_{1,k}}dx_2\right)^{1/2}\tildeo 0,\quad\tildeext{as}\quad k\tildeo\infty;
\epsilonnd{split}
\]
and
\[
\inftynt_0^1|P_\Phi||\boldsymbol{v}|^2\Big|_{x_1=\zeta_{1,k}}dx_2\leq\|P_\Phi\|_{L^\infty(\mathcal{S}_R)}\inftynt_0^1|\boldsymbol{v}|^2\Big|_{x_1=\zeta_{1,k}}dx_2\tildeo0,\quad\tildeext{as}\quad k\tildeo\infty.
\]
Choosing $\zeta_1=\zeta_{1,k}$ ($k\gammaeq k_0$) in \epsilonqref{LH} and \epsilonqref{RH}, respectively, and performing $k\tildeo\infty$, one can deduce that
\[
\begin{equation}gin{split}
\inftynt_0^1\inftynt_{\zeta'}^\infty|\nabla \boldsymbol{v}|^2dx\leq&\,\underbrace{C\inftynt_0^1\inftynt_{\zeta'}^\infty\boldsymbol{v}\cdot\nabla \boldsymbol{v}\cdot \boldsymbol{a} dx}_{R_1}\\
&+C\inftynt_0^1\Big(|\boldsymbol{v}|\left(|\boldsymbol{v}|^2+|P_\Phi||\boldsymbol{v}|+|\nablabla\boldsymbol{v}|\right)+v_1\Pi\Big)\Big|_{x_1=\zeta'}dx_2.
\epsilonnd{split}
\]
Using the Cauchy-Schwarz inequality, the Poincar\'e inequality in Lemma \ref{POIN}, and the construction of profile vector $\boldsymbol{a}$, one derives
\[
R_1\leq C\|P_\Phi\|_{L^\infty(\mathcal{S}_R)}\left(\inftynt_0^1\inftynt_{\zeta'}^\infty|\nabla \boldsymbol{v}|^2dx\right)^{1/2}\left(\inftynt_0^1\inftynt_{\zeta'}^\infty|\boldsymbol{v}|^2dx\right)^{1/2}\leq \frac{C\alpha\Phi}{1+\alpha}\inftynt_0^1\inftynt_{\zeta'}^\infty|\nabla \boldsymbol{v}|^2dx,
\]
which indicates the following estimate provided $\alpha\Phi$ is small enough such that $\frac{C\alpha\Phi}{1+\alpha}<1$:
\begin{equation}\lambdabel{MEST}
\inftynt_0^1\inftynt_{\zeta'}^\infty|\nabla \boldsymbol{v}|^2dx\leq C\inftynt_0^1\Big(|\boldsymbol{v}|\left(|\boldsymbol{v}|^2+|P_\Phi||\boldsymbol{v}|+|\nablabla\boldsymbol{v}|\right)+v_3\Pi\Big)\Big|_{x_1=\zeta'}dx_2.
\epsilonnd{equation}
Denoting
\begin{equation}\lambdabel{DEFG}
\mathcal{G}(\zeta'):=\inftynt_0^1\inftynt_{\zeta'}^\infty|\nabla \boldsymbol{v}|^2dx,
\epsilonnd{equation}
and integrating \epsilonqref{MEST} with $\zeta'$ on $(\zeta,\infty)$, one arrives
\begin{equation}\lambdabel{EEP0}
\inftynt_\zeta^\infty\mathcal{G}(\zeta')d\zeta'\leq C\left(\inftynt_0^1\inftynt_\zeta^\infty\Big(|\boldsymbol{v}|\left(|\boldsymbol{v}|^2+|P_\Phi||\boldsymbol{v}|+|\nablabla\boldsymbol{v}|\right)\Big)dx+\left|\inftynt_0^1\inftynt_\zeta^\infty v_1\Pi dx\right|\right).
\epsilonnd{equation}
Applying the Poincar\'e inequality in Lemma \ref{POIN}, one deduces
\begin{equation}\lambdabel{EEP1}
\inftynt_0^1\inftynt_\zeta^\infty|\boldsymbol{v}|\left(|\boldsymbol{v}|^2+|P_\Phi||\boldsymbol{v}|+|\nablabla\boldsymbol{v}|\right)dx\leq C\inftynt_0^1\inftynt_\zeta^\infty|\nablabla\boldsymbol{v}|^2dx.
\epsilonnd{equation}
Moreover, using a similar approach as in the proof of Proposition \ref{P2.4}, one notices that
\begin{equation}\lambdabel{EEP2}
\begin{equation}gin{split}
\left|\inftynt_0^1\inftynt_\zeta^\infty v_1\Pi dx\right|&\leq\sqrtum_{m=1}^\infty\left|\inftynt_{\Upsilon_{\zeta+m}^+}v_1\Pi dx\right|\\
&\leq C\sqrtum_{m=1}^\infty\left(\|P_\Phi\|_{L^\inftynfty(\Upsilon^{+}_{\zeta+m})}\|\nablabla \boldsymbol{v}\|^2_{L^2(\Upsilon^{+}_{\zeta+m})}+\|\nablabla \boldsymbol{v}\|_{L^2(\Upsilon^{+}_{\zeta+m})}^2+\|\nablabla \boldsymbol{v}\|_{L^2(\Upsilon^{+}_{\zeta+m})}^3\right)\\
&\leq C\inftynt_0^1\inftynt_\zeta^\infty|\nablabla\boldsymbol{v}|^2dx.
\epsilonnd{split}
\epsilonnd{equation}
Substituting \epsilonqref{EEP1} and \epsilonqref{EEP2} in \epsilonqref{EEP0}, one arrives at
\[
\inftynt_\zeta^\infty\mathcal{G}(\zeta')d\zeta'\leq C\mathcal{G}(\zeta),\quad\tildeext{for any}\quad\zeta>Z_\Phi.
\]
This implies
\[
\mathcal{N}(\zeta):=\inftynt_\zeta^\infty\mathcal{G}(\zeta')d\zeta'
\]
is well-defined for all $\zeta>Z_\Phi$, and
\begin{equation}\lambdabel{EEP3}
\mathcal{N}(\zeta)\leq-C\mathcal{N}'(\zeta),\quad\tildeext{for any}\quad\zeta>Z_\Phi.
\epsilonnd{equation}
Multiplying the factor $e^{C^{-1}\zeta}$ on both sides of \epsilonqref{EEP3} and integrating on $[Z_\Phi,\zeta]$, one deduces
\[
\mathcal{N}(\zeta)\leq C\epsilonxp\left(-C^{-1}\zeta\right),\quad\tildeext{for any}\quad\zeta>Z_\Phi.
\]
According to the definition \epsilonqref{DEFG}, one has $\mathcal{G}$ is both non-negative and non-increasing. Thus
\[
\mathcal{G}(\zeta)\leq\inftynt_{\zeta-1}^\zeta\mathcal{G}(\zeta')d\zeta'\leq\mathcal{N}(\zeta-1)\leq C\epsilonxp\left(-C^{-1}\zeta\right),\quad\tildeext{for any}\quad\zeta>Z_\Phi+1.
\]
This completes the proof of the \epsilonqref{ASYP} by choosing $\sqrtigmagma=C^{-1}$.
\quaded
\sqrtubsection{Higher-order regularity of weak solutions}
\sqrtubsubsection{$\boldsymbol{H^m}$-estimates of weak solutions}\lambdabel{SEC521}
Given an arbitrary $\partialhi\inftyn C_c^\infty(\omegaverline{\mathcal{S}}\,,\,\mathbb{R})$, with $\partialhi=0$ on $\partial\mathcal{S}$, direct calculation shows $\boldsymbol{\varphi}:=(-\partial_{x_2}\partialhi\,,\,\partial_{x_1}\partialhi)$ defines a well-defined test function in Definition \ref{defws}. By replacing $\boldsymbol{\varphi}$ with $(-\partial_{x_2}\partialhi\,,\,\partial_{x_1}\partialhi)$ in \epsilonqref{EQU111}, and denoting $\omega=\partial_{x_2}v_1-\partial_{x_1}v_2$, one deduces
\[
\begin{equation}gin{split}
-\inftynt_{\mathcal{S}}\omega\Delta\partialhi dx+\inftynt_{\partial\mathcal{S}}(\alpha-2\kappa)v_{\mathrm{tan}}\frac{\partial\partialhi}{\partial\boldsymbol{n}}dS+\inftynt_{\mathcal{S}}\boldsymbol{v}\cdot\nablabla \omega\cdot \partialhi dx=&\inftynt_{\mathcal{S}}\big(\Delta b-\boldsymbol{a}\cdot\nablabla b\big)\cdot\partialhi dx\\
&+\inftynt_{\mathcal{S}}\left(\boldsymbol{v}\cdot\nabla\boldsymbol{a}+\boldsymbol{a}\cdot\nablabla\boldsymbol{v}\right)^{\partialerp}\cdot\nabla\partialhi dx,
\epsilonnd{split}
\]
where $b=\partial_{x_2}a_1-\partial_{x_1}a_2$. This implies $\omega$ solves the following linear elliptic problem weakly:
\begin{equation}\lambdabel{PCURL}
\left\{
\begin{equation}gin{array}{ll}
-\Delta\omega+\boldsymbol{v}\cdot\nablabla\omega=(\Delta b-\boldsymbol{a}\cdot\nablabla b)-\nabla\cdot\left(\boldsymbol{v}\cdot\nabla\boldsymbol{a}+\boldsymbol{a}\cdot\nablabla\boldsymbol{v}\right)^{\partialerp}, & \tildeext{in}\quad\mathcal{S};\\[2mm]
\omega=\left(-2\kappa+\alpha\right)v_{\mathrm{tan}}, & \tildeext{on}\quad\partial\mathcal{S}.
\epsilonnd{array}
\right.
\epsilonnd{equation}
Here $\boldsymbol{v}\inftyn H^1(\mathcal{S})$ is treated as a known function solved in Section \ref{SEC3627}, while $\boldsymbol{a}$ is the smooth divergence-free flux carrier constructed in Section \ref{SEC32}.
To study bounds of higher-order norms of the solution, we split the problem \epsilonqref{PCURL} into a sequence of problems on bounded smooth domains. Recall the definition of $\mathfrak{S}_k$ in \epsilonqref{CUTFRAK}, we denote the related cut-off function
\[
\partialsi_k=\left\{
\begin{equation}gin{array}{ll}
\partialsi\left(x_1-\frac{3kZ_\Phi}{2}\right),&\quad\tildeext{for}\quad k>0;\\[1mm]
\partialsi\left(y_1-\frac{3kZ_\Phi}{2}\right),&\quad\tildeext{for}\quad k<0,
\epsilonnd{array}
\right.
\]
where $\partialsi$ is a smooth 1D cut-off function that satisfies:
\[
\left\{
\begin{equation}gin{array}{*{2}{ll}}
\mathrm{supp}\,\partialsi\sqrtubset\left[-9Z_\Phi/10\,,9Z_\Phi/10\right];\\
\partialsi\epsilonquiv 1,&\quad\tildeext{in }\left[-{4Z_\Phi}/{5}\,,{4Z_\Phi}/{5}\right];\\
0\leq\partialsi\leq1,&\quad\tildeext{in }\left[-Z_\Phi\,,Z_\Phi\right];\\
|\partialsi^{(m)}|\leq C/Z_\Phi^m\leq C,&\quad\tildeext{for }m=1,2. \\
\epsilonnd{array}
\right.
\]
Meanwhile, $\partialsi_0$ is a 2D smooth cut-off function that enjoys
\[
\left\{
\begin{equation}gin{array}{*{2}{ll}}
\mathrm{supp}\,\partialsi_0\sqrtubset\mathcal{S}_{9Z_\Phi/10};\\
\partialsi\epsilonquiv 1,&\quad\tildeext{in }\mathcal{S}_{4Z_\Phi/5};\\
0\leq\partialsi\leq1,&\quad\tildeext{in }\mathcal{S}_{Z_\Phi};\\
|\partialsi^{(m)}|\leq C/Z_\Phi^m\leq C,&\quad\tildeext{for }m=1,2. \\
\epsilonnd{array}
\right.
\]
However, domains $\mathfrak{S}_k$ given in previous subsection are only Lipschitzian, which may cause unnecessary difficulty in deriving higher-order regularity of $\omega$. To this end, we introduce $\tildeilde{\mathfrak{S}}_k$, a bounded smooth domain which contains $\mathfrak{S}_k$, with its boundary $\partial\tildeilde{\mathfrak{S}}_k\sqrtupset\partial\mathfrak{S}_k\cap\partial\mathcal{S}$. In order to make the constants of specific inequalities (i.e. imbedding inequalities, trace inequalities, Biot-Savart law) on each $\tildeilde{\mathfrak{S}}_k$ ($k\inftyn\mathbb{Z}$) being uniform, one chooses every $\tildeilde{\mathfrak{S}}_k$ with $k>0$ to be congruent to $\tildeilde{\mathfrak{S}}_1$, and every $\tildeilde{\mathfrak{S}}_k$ with $k<0$ to be congruent to $\tildeilde{\mathfrak{S}}_{-1}$. This can be guaranteed by the definition of $\mathfrak{S}_k$. By the splitting and constructions above, the ``distorted part" in the middle of the stripe is totally contained in $\mathfrak{S}_0\sqrtubset\tildeilde{\mathfrak{S}}_0$, and $\nabla\partialsi_k$ are totally supported away from this``distorted part" for each $k\inftyn\mathbb{Z}$.
Multiplying \epsilonqref{PCURL}$_1$ by $\partialsi_k$, we can convert the problem \epsilonqref{PCURL} to related problem in domain $\tildeilde{\mathfrak{S}}_k$, with $k\inftyn\mathbb{Z}$:
\[
\left\{
\begin{equation}gin{array}{ll}
-\Delta\omega_k+\boldsymbol{v}\cdot\nablabla\omega_k=\nablabla\cdot\boldsymbol{F}_k+\boldsymbol{f}_k, & \tildeext{in}\quad\tildeilde{\mathfrak{S}}_k;\\[2mm]
\omega_k=g_k, & \tildeext{on}\quad\partial\tildeilde{\mathfrak{S}}_k.
\epsilonnd{array}
\right.
\]
Here $\omega_k=\partialsi_k\omega$, while
\[
\begin{equation}gin{split}
\boldsymbol{F}_k=&-\partialsi_k\left(\boldsymbol{v}\cdot\nablabla\boldsymbol{a}+\boldsymbol{a}\cdot\nablabla\boldsymbol{v}\right)^{\partialerp}-2\omega\nablabla\partialsi_k;\\[2mm]
\boldsymbol{f}_k=&\,\,\partialsi_k\left(\Delta b-\boldsymbol{a}\cdot\nablabla b\right)+\nablabla\partialsi_k\cdot\left(\boldsymbol{v}\cdot\nabla \boldsymbol{a}+\boldsymbol{a}\cdot\nabla \boldsymbol{v}\right)^{\partialerp}+\omega\left(\Delta\partialsi_k+\boldsymbol{v}\cdot\nablabla\partialsi_k\right);\\[2mm]
g_k=&\left(-2\kappa+\alpha\right)\partialsi_k \boldsymbol{v}_{\mathrm{tan}}.
\epsilonnd{split}
\]
Using Gagliardo-Nirenberg interpolation together with the trace theorem, it is not hard to derive
\begin{equation}\lambdabel{EST0908}
\|\boldsymbol{F}_k\|_{L^2(\tildeilde{\mathfrak{S}}_k)}+\|\boldsymbol{f}_k\|_{L^2(\tildeilde{\mathfrak{S}}_k)}+\|g_k\|_{H^{1/2}(\partial\tildeilde{\mathfrak{S}}_k)}\leq C\|\boldsymbol{v}\|_{H^1(\mathfrak{S}_k)}\left(1+\Phi e^{C\Phi}+\|\boldsymbol{v}\|_{H^1(\mathfrak{S}_k)}\right),\quad\fracorall k\inftyn\mathbb{Z}.
\epsilonnd{equation}
Noting that the constant $C$ above is independent with $k$, due to congruent property of domains $\{\tildeilde{\mathfrak{S}}_k\}_{k\inftyn\mathbb{Z}}$. Therefore, using the classical theory of elliptic equations and \epsilonqref{EST0908}, one derives
\[
\|\omega_k\|_{H^1(\tildeilde{\mathfrak{S}}_k)}\leq C\|\boldsymbol{v}\|_{H^1(\mathfrak{S}_k)}\left(1+\Phi e^{C\Phi}+\|\boldsymbol{v}\|_{H^1(\mathfrak{S}_k)}\right),\quad\fracorall k\inftyn\mathbb{Z}.
\]
Applying the Biot-Savart law, one derives
\[
\|\boldsymbol{v}\|_{H^2(\mathfrak{S}_k')}\leq C\left(\|\omega_k\|_{H^1(\tildeilde{\mathfrak{S}}_k)}+\|\boldsymbol{v}\|_{L^2(\mathfrak{S}_k')}\right)\leq C\|\boldsymbol{v}\|_{H^1(\mathfrak{S}_k)}\left(1+\Phi e^{C\Phi}+\|\boldsymbol{v}\|_{H^1(\mathfrak{S}_k)}\right),\quad\fracorall k\inftyn\mathbb{Z},
\]
where
\[
\mathfrak{S}_k'=\{x\inftyn\mathfrak{S}_k:\,\partialsi_k=1\}.
\]
This implies, by summing over $k\inftyn\mathbb{Z}$, that
\[
\begin{equation}gin{split}
\|\boldsymbol{v}\|^2_{H^2(\mathcal{S})}\leq& \sqrtum_{k\inftyn\mathbb{Z}}\|\boldsymbol{v}\|^2_{H^2(\mathfrak{S}_k')}\\
\leq&C\sqrtum_{k\inftyn\mathbb{Z}}\|\boldsymbol{v}\|^2_{H^1(\mathfrak{S}_k)}\left(1+\Phi e^{C\Phi}+\|\boldsymbol{v}\|_{H^1(\mathfrak{S}_k)}\right)^2\\
\leq&C\|\boldsymbol{v}\|^2_{H^1(\mathcal{S})}\left(1+\Phi^2 e^{C\Phi}+\|\boldsymbol{v}\|^2_{H^1(\mathcal{S})}\right).\\
\epsilonnd{split}
\]
This concludes the global $H^2$-regularity estimate of $\boldsymbol{v}$. From this, similarly as we derive \epsilonqref{EST0908}, one achieves a ``one-order upper" regularity of $\boldsymbol{F}_k$, $\boldsymbol{f}_k$ and $g_k$, for any $k\inftyn\mathbb{Z}$, that is
\[
\|\boldsymbol{F}_k\|_{H^1(\tildeilde{\mathfrak{S}}_k)}+\|\boldsymbol{f}_k\|_{H^1(\tildeilde{\mathfrak{S}}_k)}+\|g_k\|_{H^{3/2}(\partial\tildeilde{\mathfrak{S}}_k)}\leq C_{\Phi},
\]
which indicates the $H^3$-regularity of $\boldsymbol{v}$. Following this bootstrapping argument, one deduces $\boldsymbol{v}$ is smooth and
\[
\|\boldsymbol{v}\|_{H^m(\mathcal{S})}\leq C_{\Phi,m},\quad\fracorall m\inftyn\mathbb{N}.
\]
This finished the proof of the regularity part of Theorem \ref{THMSA}.
\quaded
\sqrtubsubsection{Exponential decay of higher-order norms}
Finally, the higher-order regularity and the $H^1$-exponential decay estimate in previous subsection, indicates the higher-order exponential decay. In fact, using Sobolev imbedding, we first need to show the following decay of the solution in $H^m$ norms, with $m\gammaeq 2$:
\[
\begin{equation}gin{split}
&\|\boldsymbol{v}\|_{H^m(\mathcal{S}_L\backslash\mathcal{S}_\zeta))}+\|\boldsymbol{v}\|_{H^m(\mathcal{S}_R\backslash\mathcal{S}_\zeta)}\leq C_{\Phi,m}\left(\|\boldsymbol{v}\|_{H^1(\mathcal{S}_L\backslash\mathcal{S}_{\zeta-Z_\Phi})}+\|\boldsymbol{v}\|_{H^1(\mathcal{S}_R\backslash\mathcal{S}_{\zeta-Z_\Phi})}\right),\\[1mm]
\epsilonnd{split}
\]
for all $\zeta>2Z_\Phi$. This is derived by using the method in the proof of Section \ref{SEC521}, but summing over $k\inftyn\mathbb{Z}$ such that
\[
\mathrm{supp}\,\partialsi_k\cap\left(\mathcal{S}\backslash\mathcal{S}_\zeta\right)\neq\varnothing.
\]
Then, the proof is completed by the $H^1$ decay estimate \epsilonqref{ASYP}. This finishes the proof of Theorem \ref{THMSA}.
\quaded
\begin{equation}gin{remark}\lambdabel{RMMKK43}
For the pressure $p$, there exists two constants $C_{L},\,C_{R}>0$ (See \epsilonqref{POSS1} and \epsilonqref{POSS2}) , and a smooth cut-off function $\epsilonta$ given in \epsilonqref{ETA} such that: For any $m\gammaeq 0$,
\[
\left\|\nabla^m\nabla\left(p+{C_R\inftynt_{-\infty}^{x_1}\epsilonta(s)ds}-{C_L\inftynt_{-\infty}^{-y_1}\epsilonta(s)ds}\right) \right\|_{L^2(\mathcal{S})}\leq C_{\Phi,m}.
\]
Meanwhile, the following pointwise decay estimate holds: for all $|x|>>1$,
\[
\left|\nabla^m\nabla \left(p+{C_R\inftynt_{-\infty}^{x_1}\epsilonta(s)ds}-{C_L\inftynt_{-\infty}^{-y_1}\epsilonta(s)ds}\right)(x)\right|\leq C_{\Phi,m}\epsilonxp\left\{-\sqrtigmagma_{\Phi,m}|x|\right\},
\]
where $C_{\Phi,m}$ and $\sqrtigmagma_{\Phi,m}$ are positive constants depending on $\Phi$ and $m$. The subtracted term
\[
\partiali_{\boldsymbol{P}}:=-{C_R\inftynt_{-\infty}^{x_1}\epsilonta(s)ds}+{C_L\inftynt_{-\infty}^{-y_1}\epsilonta(s)ds}
\]
is set to balance the pressure of the Poiseuille flows.
\epsilonnd{remark}
\quaded
\sqrtection*{Data availability statement}
\addcontentsline{toc}{section}{Data availability statement}
\quad\ Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.
\sqrtection*{Conflict of interest statement}
\addcontentsline{toc}{section}{Conflict of interest statement}
\quad\ The authors declare that they have no conflict of interest.
\sqrtection*{Acknowledgments}
\addcontentsline{toc}{section}{Acknowledgments}
\quad\ Z. Li is supported by Natural Science Foundation of Jiangsu Province (No. BK20200803) and National Natural Science Foundation of China (No. 12001285). X. Pan is supported by National Natural Science Foundation of China (No. 11801268, 12031006). J. Yang is supported by National Natural Science Foundation of China (No. 12001429).
\begin{equation}gin{thebibliography}{}
\addcontentsline{toc}{section}{References}
\sqrtmall
\sqrtetlength{\inftytemsep}{-3 pt}
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\epsilonnd{thebibliography}
{\fracootnotesize
{\sqrtc Z. Li: School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China}
{\inftyt E-mail address:} [email protected]
{\sqrtc X. Pan: College of Mathematics and Key Laboratory of MIIT, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China}
{\inftyt E-mail address:} xinghong\[email protected]
{\sqrtc J. Yang: School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710129, China}
{\inftyt E-mail address:} [email protected]
}
\epsilonnd{document} | math | 145,234 |
\begin{document}
\title[Mean curvature flow with surgery]{Mean curvature flow with surgery of mean convex surfaces in $\mathbb{R}^3$}
\author{Simon Brendle and Gerhard Huisken}
\address{Department of Mathematics \\ Stanford University \\ Stanford, CA 94305}
\address{Mathematisches Institut \\ Universit\"at T\"ubingen \\ 72076 T\"ubingen \\ Germany}
\begin{abstract}
We define a notion of mean curvature flow with surgery for two-dimensional surfaces in $\mathbb{R}^3$ with positive mean curvature. Our construction relies on the earlier work of Huisken and Sinestrari in the higher dimensional case. One of the main ingredients in the proof is a new estimate for the inscribed radius established by the first author \cite{Brendle2}.
\end{abstract}
\maketitle
\section{Introduction}
The formation of singularities in geometric flows is a central problem in geometric analysis. In the 1990s, Hamilton \cite{Hamilton1} started a program aimed at understanding the singularities of the Ricci flow in dimensions $3$ and $4$. In particular, for Ricci flow on four-manifolds with positive isotropic curvature, Hamilton \cite{Hamilton2} showed that the flow can be extended beyond singularities by means of surgery procedure. In 2002, Perelman \cite{Perelman1}, \cite{Perelman2}, \cite{Perelman3} successfully carried out a surgery construction for the Ricci flow in dimension $3$, and used it to prove the Poincar\'e and Geometrization Conjectures.
It this paper, we focus on the mean curvature flow. In \cite{Huisken-Sinestrari3}, Huisken and Sinestrari defined a notion of mean curvature flow with surgery for two-convex hypersurfaces in $\mathbb{R}^{n+1}$, where $n \geq 3$. We assume that the reader is familiar with that paper. Our goal in this paper is to extend the construction in \cite{Huisken-Sinestrari3} to the case $n=2$:
\begin{theorem}
\label{main.theorem}
Let $M_0$ be a closed, embedded surface in $\mathbb{R}^3$ with positive mean curvature. Then there exists a mean curvature flow with surgeries starting from $M_0$ which terminates after finitely many steps.
\end{theorem}
As in \cite{Huisken-Sinestrari3}, the surgery construction involves three curvature thresholds $H_1,H_2,H_3$, where $H_3 = 10H_2 \gg H_1$. The basic idea is that we let the flow evolve smoothly until the maximum curvature reaches the threshold $H_3$. When that happens, we perform surgeries on necks with a curvature scale comparable to $H_1$. As a result of that, the maximum curvature drops below $H_2$ right after surgery. We then let the flow evolve smoothly until the maximum curvature reaches the threshold $H_3$ again, and repeat the process until the flow becomes extinct.
If we send the curvature thresholds $H_1,H_2,H_3$ to infinity, the flow with surgery will converge to the level set solution; this follows from results of Head \cite{Head} and Lauer \cite{Lauer}.
Our argument broadly follows the one in \cite{Huisken-Sinestrari3}. However, there are several major differences. One important difference is that the cylindrical estimate in Section 5 of \cite{Huisken-Sinestrari3} fails for $n=2$. To replace the cylindrical estimate, we use an estimate for the inscribed radius established in \cite{Brendle2} (see also \cite{Brendle3} for a recent survey). Given an embedded oriented surface $M$ in $\mathbb{R}^3$ and a point $p \in M$, the inscribed radius at $p$ is defined as the radius of the largest open ball in $\mathbb{R}^3$ which is disjoint from $M$ and touches $M$ at $p$ from the inside. Similarly, the outer radius at $p$ is defined as the radius of the largest open ball in $\mathbb{R}^3$ which is disjoint from $M$ and touches $M$ at $p$ from the outside. Following Sheng and Wang \cite{Sheng-Wang}, an embedded mean convex surface $M$ will be called $\alpha$-noncollapsed if the inscribed radius at each point $p \in M$ is bounded from below by $\frac{\alpha}{H}$, where $H$ denotes the mean curvature at the point $p$. It follows from general results of Brian White \cite{White2},\cite{White3} that every embedded solution of the mean curvature flow with positive mean curvature is $\alpha$-noncollapsed for some uniform constant $\alpha>0$ which is independent of $t$. An alternative proof of that fact was given by Sheng and Wang \cite{Sheng-Wang}. Andrews recently showed that the $\alpha$-noncollapsing condition is preserved by the flow; this uses a maximum principle argument similar in spirit to \cite{Huisken3}.
The main theorem in \cite{Brendle2} asserts that, for any smooth solution of the mean curvature flow with positive mean curvature, we have a pointwise estimate of the form $\mu \leq (1+\delta) \, H + C(\delta)$. Here, $\mu$ denotes the reciprocal of the inscribed radius, $\delta$ is a given positive number, and $C(\delta)$ is a constant that depends on $\delta$ and the initial data. We show that this estimate still holds in the presence of surgeries, at least for a suitable choice of surgery parameters. This is a subtle issue, as the ratio $\frac{\mu}{H}$ might deteriorate slightly under surgery. To overcome this obstacle, we show that the ratio $\frac{\mu}{H}$ improves immediately prior to surgery. By a suitable choice of the surgery parameters, we can ensure that this improvement in the noncollapsing constant prior to surgery is strong enough to absorb the error terms that arise during each surgery procedure.
Another problem is that the proof of the gradient estimate in Section 6 of \cite{Huisken-Sinestrari3} does not directly carry over to the case $n=2$. To get around this issue, we use a new interior gradient estimate due to Haslhofer and Kleiner \cite{Haslhofer-Kleiner}. The estimate of Haslhofer and Kleiner implies that $|\nabla A| \leq C \, H^2$, provided that the flow is $\alpha$-noncollapsed and has evolved smoothly for a long enough time (cf. Theorem \ref{interior.derivative.estimate} below). On the other hand, we can use the pseudolocality principle to control $|\nabla A|$ shortly after surgery (cf. Proposition \ref{consequence.of.pseudolocality}). By combining these two results, we obtain an estimate for $|\nabla A|$ which is valid at all points in space-time, even in the presence of surgeries (see Proposition \ref{gradient.estimate}).
In Section \ref{overview}, we state a number of auxiliary results. In Section \ref{neck.contin}, we use these auxiliary results to establish an analogue of the crucial Neck Continuation Theorem in \cite{Huisken-Sinestrari3}. We then implement the surgery construction from \cite{Huisken-Sinestrari3}, and complete the proof of Theorem \ref{main.theorem}. Finally, in Sections \ref{proof.of.pseudoloc} -- \ref{proof.of.7.19} we give the proofs of the auxiliary results stated in Section \ref{overview}.
Finally, let us mention some related results. Brian White has obtained several breakthroughs in the analysis of the singularities of mean convex mean curvature flow; see \cite{White1}, \cite{White2}, \cite{White3}, \cite{White4}, \cite{White5}, \cite{White6}. A different approach to the singularity analysis for mean curvature flow in the two-dimensional case was suggested by Colding and Kleiner in \cite{Colding-Kleiner}. Moreover, Wang \cite{Wang} has obtained a classification of translating solutions to the mean curvature flow in dimension $2$. These solutions arise as models for Type II singularities. Finally, the first author has recently obtained a classification of self-similar solutions to the Ricci flow in dimension $3$ under a noncollapsing assumption (see \cite{Brendle1}).
We are grateful Brian White for discussions concerning the pseudolocality property for the mean curvature flow. We thank the referees for their careful reading of the original manuscript, and for valuable comments.
\section{Overview of some auxiliary results}
\label{overview}
In this section, we collect a number of auxiliary results which are needed in order to prove the Neck Continuation Theorem and implement the surgery algorithm. The order has been arranged so as to make the consecutive choice of curvature thresholds and surgery parameters apparent. The proofs of these auxiliary results will be given in Sections \ref{proof.of.pseudoloc} -- \ref{proof.of.7.19}.
We first establish a pseudolocality principle for the mean curvature flow. We begin with a definition.
\begin{definition}
\label{regular.flow}
Consider a ball $B$ in $\mathbb{R}^3$ and a one-parameter family of smooth surfaces $M_t \subset B$ such that $\partial M_t \subset \partial B$. Moreover, suppose that each surface $M_t$ bounds a domain $\Omega_t \subset B$. We say that the surfaces $M_t$ form a regular mean curvature flow if the surfaces $M_t$ form a smooth solution to mean curvature flow, except at finitely many times where one or more connected components of $\Omega_t$ may be removed.
\end{definition}
\begin{theorem}[Pseudolocality Principle]
\label{pseudolocality}
There exist positive constants $\beta_0$ and $C$ such that the following holds. Suppose that $M_t$, $t \in [0,T]$, is a regular mean curvature flow in $B_4(0)$ in the sense of Definition \ref{regular.flow}. Moreover, we assume that the initial surface $M_0$ can be expressed as the graph of a (single-valued) function $u$ over a plane. If $\|u\|_{C^4} \leq \beta_0$, then
\[|A|+|\nabla A|+|\nabla^2 A| \leq C\]
for all $t \in [0,\beta_0] \cap [0,T]$ and all $x \in M_t \cap B_1(0)$.
\end{theorem}
Another important ingredient is the following curvature derivative estimate due to Haslhofer and Kleiner:
\begin{theorem}[cf. Haslhofer-Kleiner \cite{Haslhofer-Kleiner}, Theorem 1.8']
\label{interior.derivative.estimate}
Given any $\alpha \in (0,\frac{1}{1000}]$, there exists a constant $C(\alpha)$ with the following property. Suppose that $M_t$, $t \in [-1,0]$, is a regular mean curvature flow in the ball $B_4(0)$. Moreover, suppose that each surface $M_t$ is outward-minimizing within the ball $B_4(0)$. We further assume that the inscribed radius and the outer radius are at least $\frac{\alpha}{H}$ at each point on $M_t$. Finally, we assume that $M_0$ passes through the origin, and $H(0,0) \leq 1$. Then the surface $M_0$ satisfies $|\nabla A| \leq C(\alpha)$ and $|\nabla^2 A| \leq C(\alpha)$ at the origin.
\end{theorem}
In the following, we will fix an initial surface $M_0$ in $\mathbb{R}^3$. We assume that $M_0$ is closed, embedded, and has positive mean curvature. Moreover, let us fix a constant $\alpha \in (0,\frac{1}{1000}]$ such that the inscribed radius and the outer radius of the initial surface $M_0$ are at least $\frac{\alpha}{H}$.
We next describe the necks on which we will perform surgery.
\begin{definition}
Let $M$ be a mean convex surface in $\mathbb{R}^3$, and let $N$ be a region in $M$. As usual, we denote by $\nu$ the outward pointing unit normal vector field. We say that $N$ is an $(\hat{\alpha},\hat{\delta},\varepsilon,L)$-neck of size $r$ if (in a suitable coordinate system in $\mathbb{R}^3$) the following holds:
\begin{itemize}
\item There exists a simple closed, convex curve $\Gamma \subset \mathbb{R}^2$ with the property that $\text{\rm dist}_{C^{20}}(r^{-1} \, N,\Gamma \times [-L,L]) \leq \varepsilon$.
\item At each point on $\Gamma$, the inscribed radius is at least $\frac{1}{(1+\hat{\delta}) \, \kappa}$, where $\kappa$ denotes the geodesic curvature of $\Gamma$.
\item We have $\sum_{l=1}^{18} |\nabla^l \kappa| \leq \frac{1}{100}$ at each point on $\Gamma$.
\item There exists a point on $\Gamma$ where the geodesic curvature $\kappa$ is equal to $1$.
\item The region $\{x + a \, \nu(x): x \in N, \, a \in (0,2\hat{\alpha} \, r)\}$ is disjoint from $M$.
\end{itemize}
\end{definition}
The last assumption is needed to ensure that, immediately after performing surgery, the resulting surface has outer radius at least $\frac{\alpha}{H}$ everywhere (cf. Proposition \ref{noncollapsing.preserved.under.surgery}). It turns out that the necks obtained via the Neck Detection Lemma satisfy this condition; see Theorem \ref{neck.detection.a} below.
Given an $(\hat{\alpha},\hat{\delta},\varepsilon,L)$-neck, we can perform surgery on $N$. The procedure depends on a parameter $\Lambda$, and will be explained in detail in Section \ref{construction.of.cap}. We will refer to this as $\Lambda$-surgery. The exact choice of $\Lambda$ will be specified later.
\begin{theorem}[Properties of Surgery]
\label{properties.of.surgery}
Given any number $\hat{\alpha} > \alpha$, there exists a real number $\delta_0$ with the following significance. Suppose that we are given a pair of real numbers $\delta$ and $\hat{\delta}$ such that $\hat{\delta} < \delta < \delta_0$. Then we can find numbers $\bar{\varepsilon}$ and $\Lambda$, depending only on $\delta$ and $\hat{\delta}$, such that the following holds. Suppose that $N$ is an $(\hat{\alpha},\hat{\delta},\varepsilon,L)$-neck of size $r$ sitting in a mean convex surface in $\mathbb{R}^3$, where $\varepsilon \leq \bar{\varepsilon}$ and $\frac{L}{1000} \geq \Lambda$. If we perform a $\Lambda$-surgery on $N$, then the resulting surface $\tilde{N}$ will be $\frac{1}{1+\delta}$-noncollapsed. Furthermore, the outer radius is at least $\frac{\alpha}{H}$ at each point on $\tilde{N}$. Finally, if $\tilde{p} \in \tilde{N} \setminus N$ is a point in the surgically modified region, then either $\lambda_1(\tilde{p}) \geq 0$, or else there exists a point $p \in N$ such that $\lambda_1(\tilde{p}) \geq \lambda_1(p)$ and $H(\tilde{p}) \geq H(p)$.
\end{theorem}
A key point is that the deterioration in the noncollapsing constant can be made arbitrarily small by choosing $\varepsilon$ small and $\Lambda$ large.
\begin{assumption}
\label{a.priori.assumptions}
In the following, we will assume that $M_t$ is a solution of the mean curvature flow which is interrupted by finitely many surgeries as in \cite{Huisken-Sinestrari3}, p.~145. We will assume that this flow satisfies the following assumptions:
\begin{itemize}
\item The flow $M_t$ is smooth for $t \in [0,(100 \, \sup_{M_0} |A|)^{-2}]$.
\item Each surgery procedure involves performing a $\Lambda$-surgery on an $(\hat{\alpha},\hat{\delta},\varepsilon,L)$-neck of size $r \in [\frac{1}{2H_1},\frac{2}{H_1}]$, where $\hat{\alpha} > \alpha$, $\hat{\delta} \leq \frac{1}{10}$, $\frac{L}{1000} \geq \Lambda$, and $H_1 \geq \frac{(1000 \, \sup_{M_0} |A|)^2}{\inf_{M_0} H}$.
\item The region $\Omega_t$ enclosed by $M_t$ shrinks as $t$ increases.
\item For each $t$, the surface $M_t$ is outward-minimizing within the region $\Omega_0$.
\item For each $t$, the inscribed radius and the outer radius of $M_t$ are at least $\frac{\alpha}{H}$.
\end{itemize}
\end{assumption}
The exact values of the parameters $\hat{\alpha}$, $\hat{\delta}$, $\Lambda$, $\varepsilon$, $L$, and $H_1$ will be specified later.
In the first step, we want to apply the Pseudolocality Theorem to obtain derivative bounds shortly after a surgery. We begin by showing that surgeries are seperated in space:
\begin{proposition}[Separation of Surgery Regions]
\label{separation.of.surgery.regions}
Let $M_t$ be a mean curvature flow with surgery satisfying \ref{a.priori.assumptions}. Suppose that $t_0<t_1$ are two surgery times, and $x_0 \in M_{t_0+}$ and $x_1 \in M_{t_1+}$ are two points in the surgically modified regions. Then $|x_1-x_0| > \frac{\alpha}{1000} \, H_1^{-1}$.
\end{proposition}
Thus, if $t_0$ is a surgery time and $x_0$ is a point in the surgically modified region at time $t_0+$, then the flow $M_t \cap B_{\frac{\alpha}{1000} \, H_1^{-1}}(x_0)$, $t > t_0$, is a regular flow in the sense of Definition \ref{regular.flow}. Using the Pseudolocality Theorem, we can draw the following conclusion:
\begin{proposition}
\label{consequence.of.pseudolocality}
There exist positive constants $\beta_* \in (0,\frac{\alpha}{1000})$ and $C_*$ with the following property. Let $M_t$ be a mean curvature flow with surgery satisfying Assumption \ref{a.priori.assumptions}. Suppose that $t_0$ is a surgery time and $x_0$ is a point in the surgically modified region at time $t_0+$. Then we have
\[H_1^{-1} \, |A| + H_1^{-2} \, |\nabla A| + H_1^{-3} \, |\nabla^2 A| \leq C_*\]
for all times $t \in (t_0,t_0+\beta_* \, H_1^{-2}]$ and all points $x \in M_t \cap B_{\beta_* \, H_1^{-1}}(x_0)$. The constants $\beta_*$ and $C_*$ may depend on the noncollapsing constant $\alpha$, but they do not depend on the surgery parameters $\hat{\alpha}$, $\hat{\delta}$, $\Lambda$, $\varepsilon$, $L$, and $H_1$.
\end{proposition}
The exact values of the surgery parameters will depend on the value of the constant in the derivative estimate, which in turn depends on $\beta_*$ and $C_*$. It is therefore critically important that the constants $\beta_*$ and $C_*$ do not depend on the exact choice of the surgery parameters $\hat{\alpha}$, $\hat{\delta}$, $\Lambda$, $\varepsilon$, $L$, and $H_1$.
Combining Proposition \ref{consequence.of.pseudolocality} with the interior gradient estimate of Haslhofer and Kleiner \cite{Haslhofer-Kleiner}, we obtain pointwise bounds for the first and second derivatives of the second fundamental form which hold even in the presence of surgeries.
\begin{proposition}[Pointwise Derivative Estimate]
\label{gradient.estimate}
There exists a constant $C_\#$ with the following significance. Suppose that $M_t$ is a mean curvature flow with surgery satisfying Assumption \ref{a.priori.assumptions}. Then $|\nabla A| \leq C_\# \, (H+H_1)^2$ and $|\nabla^2 A| \leq C_\# \, (H+H_1)^3$ for all times $t \geq (1000 \, \sup_{M_0} |A|)^{-2}$ and all points $x \in M_t$. The constant $C_\#$ may depend on the initial noncollapsing constant $\alpha$, but is independent of the surgery parameters $\hat{\alpha}$, $\hat{\delta}$, $\Lambda$, $\varepsilon$, $L$, and $H_1$.
\end{proposition}
Having fixed the constant $C_\#$ in the derivative estimate, we next define $\Theta = \frac{400}{\alpha}$, $\theta_0 = 10^{-6} \, \min\{\alpha,\frac{1}{C_\# \, \Theta^3}\}$, and $\hat{\alpha} = \frac{\alpha}{1-\frac{\theta_0}{8}}$. Hence, if we start at a point $(p_0,t_0)$ with $H(p_0,t_0) \geq \frac{H_1}{\Theta}$ and follow this point back in time, then the mean curvature at the resulting point will be between $\frac{1}{2} \, H(p_0,t_0)$ and $2 \, H(p_0,t_0)$, provided that $t \in (t_0-2\theta_0 \, H(p_0,t_0)^{-2},t_0]$.
We next establish two auxiliary results concerning curves in the plane. It is here that we fix our choice of $\delta$ and $\hat{\delta}$. By applying these results to a blow-up limit that splits off line, we will show that the noncollapsing constants of a neck improve prior to surgery; this improvement offsets the deterioration of the noncollapsing constants under surgery (see Theorem \ref{properties.of.surgery} above).
\begin{proposition}
\label{choice.of.delta}
We can find a real number $\delta > 0$ such that the following holds:
\begin{itemize}
\item Suppose that $\Gamma$ is a (possibly non-closed) embedded curve in the plane with the property that $\kappa > 0$, $|\frac{d\kappa}{ds}| \leq C_\# \, (\kappa+2\Theta)^2$, and $|\frac{d^2\kappa}{ds^2}| \leq C_\# \, (\kappa+2\Theta)^3$. Moreover, suppose that the inscribed radius is at least $\frac{1}{(1+\delta) \, \kappa}$ at each point on $\Gamma$, and the outer radius is at least $\frac{\alpha}{\kappa}$ at each point on $\Gamma$. Finally, we assume that $\kappa(p)=1$ for some point $p \in \Gamma$. Then $L(\Gamma) \leq 3\pi$ and $\sup_\Gamma |\kappa-1| \leq \frac{1}{100}$.
\item Suppose that $\Gamma_t$, $t \in (-2\theta_0,0]$, is a family of simple closed, convex curves in the plane which evolve by curve shortening flow. Assume that, for each $t \in (-2\theta_0,0]$, the curve $\Gamma_t$ satisfies the derivative estimates $|\frac{d\kappa}{ds}| \leq C_\# \, (\kappa+2\Theta)^2$ and $|\frac{d^2\kappa}{ds^2}| \leq C_\# \, (\kappa+2\Theta)^3$. Moreover, we assume that the inscribed radius is at least $\frac{1}{(1+\delta) \, \kappa}$ at each point on $\Gamma_t$, and the outer radius is at least $\frac{\alpha}{\kappa}$ at each point on $\Gamma_t$. Finally, we assume that the geodesic curvature of $\Gamma_0$ is equal to $1$ somewhere. Then the curve $\Gamma_0$ satisfies $\sum_{l=1}^{18} |\nabla^l \kappa| \leq \frac{1}{1000}$. Moreover, we have $\sup_{\Gamma_{-\theta_0}} \kappa \leq 1-\frac{\theta_0}{4}$.
\end{itemize}
\end{proposition}
We assume that $\delta$ is chosen sufficiently small so that $\delta < \delta_0$, where $\delta_0$ is the constant in Theorem \ref{properties.of.surgery}. In the next step, we choose $\hat{\delta}$ such that the following holds:
\begin{proposition}
\label{choice.of.hat.delta}
Given $\theta_0 > 0$ and $\delta > 0$, we can find a real number $\hat{\delta} \in (0,\delta)$ with the following property: Consider a simple closed, convex solution $\Gamma_t$, $t \in (-2\theta_0,0]$, of the curve shortening flow in the plane which satisfies the derivative estimates $|\frac{d\kappa}{ds}| \leq C_\# \, (\kappa+2\Theta)^2$ and $|\frac{d^2\kappa}{ds^2}| \leq C_\# \, (\kappa+2\Theta)^3$. Moreover, we assume that the inscribed radius is at least $\frac{1}{(1+\delta) \, \kappa}$ at each point on $\Gamma_t$, and the outer radius is at least $\frac{\alpha}{\kappa}$ at each point on $\Gamma_t$. Finally, we assume that the geodesic curvature of $\Gamma_0$ is equal to $1$ somewhere. Then $\Gamma_0$ is $\frac{1}{1+\hat{\delta}}$-noncollapsed.
\end{proposition}
Having fixed the values of $\alpha$, $\hat{\alpha}$, $\delta$, $\hat{\delta}$, we will choose $\bar{\varepsilon}$ and $\Lambda$ such that the conclusion of Theorem \ref{properties.of.surgery} holds.
We next observe that the convexity estimates of Huisken and Sinestrari (cf. \cite{Huisken-Sinestrari1}, \cite{Huisken-Sinestrari2}) still hold for mean curvature flow with surgery.
\begin{proposition}[Huisken-Sinestrari \cite{Huisken-Sinestrari3}, Section 4]
\label{huisken.sinestrari.convexity}
Suppose that $\bar{\varepsilon}$ and $\Lambda$ are chosen such that the conclusion of Theorem \ref{properties.of.surgery} holds. Moreover, let $M_t$ be a mean curvature flow with surgery satisfying Assumption \ref{a.priori.assumptions}, where $\varepsilon \leq \bar{\varepsilon}$ and $L \geq 1000 \, \Lambda$. Given any $\eta > 0$, there exists a constant $C_1(\eta)$ such that $\lambda_1 \geq -\eta \, H - C_1(\eta)$. The constant $C_1(\eta)$ depends only on $\eta$ and the initial data, but is independent of the remaining surgery parameters $\varepsilon$, $L$, and $H_1$.
\end{proposition}
Theorem \ref{properties.of.surgery} implies that performing $\Lambda$-surgery on an $(\hat{\alpha},\hat{\delta},\varepsilon,L)$-neck will produce a surface which is $\frac{1}{1+\delta}$-noncollapsed, provided that $\varepsilon \leq \bar{\varepsilon}$ and $L \geq 1000 \, \Lambda$. This allows us to show that the cylindrical estimate from \cite{Brendle2} holds in the presence of surgeries:
\begin{proposition}[Cylindrical Estimate]
\label{cylindrical.estimate}
Let $\delta$ and $\hat{\delta}$ be chosen as above. Moreover, suppose that $\bar{\varepsilon}$ and $\Lambda$ are chosen such that the conclusion of Theorem \ref{properties.of.surgery} holds. Finally, let $M_t$ be a mean curvature flow with surgery satisfying Assumption \ref{a.priori.assumptions}, where $\varepsilon \leq \bar{\varepsilon}$ and $L \geq 1000 \, \Lambda$. Then $\mu \leq (1+\delta) \, H + C \, H^{1-\sigma}$ for $0 \leq t \leq T$, where $\mu$ denotes the reciprocal of the inscribed radius. Here, $\sigma$ and $C$ may depend on $\delta$ and the initial data, but they are independent of the exact choice of $\varepsilon$, $L$, and $H_1$.
\end{proposition}
Using the convexity estimate and the cylindrical estimate, we are able to prove an analogue of the Neck Detection Lemma in \cite{Huisken-Sinestrari3}. In fact, we will need two different versions.
\begin{theorem}[Neck Detection Lemma, Version A]
\label{neck.detection.a}
Let $\delta$ and $\hat{\delta}$ be chosen as above, and let $\bar{\varepsilon}$ and $\Lambda$ be chosen so that the conclusion of Theorem \ref{properties.of.surgery} holds. Let $M_t$ be a mean curvature flow with surgery satisfying Assumption \ref{a.priori.assumptions}, where $\varepsilon \leq \bar{\varepsilon}$ and $L \geq 1000 \, \Lambda$. Then, given $\varepsilon_0>0$ and $L_0 \geq 100$, we can find $\eta_0 > 0$ and $K_0$ with the following significance: Suppose that $t_0$ and $p_0 \in M_{t_0}$ satisfy
\begin{itemize}
\item $H(p_0,t_0) \geq \max\{K_0,\frac{H_1}{\Theta}\}$, $\frac{\lambda_1(p_0,t_0)}{H(p_0,t_0)} \leq \eta_0$,
\item the parabolic neighborhood $\hat{\mathcal{P}}(p_0,t_0,L_0+4,2\theta_0)$ does not contain surgeries.\footnote{See \cite{Huisken-Sinestrari3}, pp.~189--190, for the definition of $\hat{\mathcal{P}}(p_0,t_0,L_0+4,2\theta_0)$.}
\end{itemize}
Then $(p_0,t_0)$ lies at the center of an $(\hat{\alpha},\hat{\delta},\varepsilon_0,L_0)$-neck of size $H(p_0,t_0)^{-1}$. Finally, the constants $\eta_0$ and $K_0$ may depend on $\varepsilon_0$, $L_0$, $\delta$, $\hat{\delta}$, and the initial data, but they are independent of the remaining surgery parameters $\varepsilon$, $L$, and $H_1$.
\end{theorem}
\begin{theorem}[Neck Detection Lemma, Version B]
\label{neck.detection.b}
Let $\delta$ and $\hat{\delta}$ be chosen as above, and let $\bar{\varepsilon}$ and $\Lambda$ be chosen so that the conclusion of Theorem \ref{properties.of.surgery} holds. Let $M_t$ be a mean curvature flow with surgery satisfying Assumption \ref{a.priori.assumptions}, where $\varepsilon \leq \bar{\varepsilon}$ and $L \geq 1000 \, \Lambda$. Then, given $\theta$, $\varepsilon_0>0$ and $L_0 \geq 100$, we can find positive numbers $\eta_0$ and $K_0$ with the following significance: Suppose that $t_0$ and $p_0 \in M_{t_0}$ satisfy
\begin{itemize}
\item $H(p_0,t_0) \geq \max\{K_0,\frac{H_1}{\Theta}\}$, $\frac{\lambda_1(p_0,t_0)}{H(p_0,t_0)} \leq \eta_0$,
\item the parabolic neighborhood $\hat{\mathcal{P}}(p_0,t_0,L_0+4,\theta)$ does not contain surgeries.
\end{itemize}
Let us dilate the surface $\{x \in M_{t_0}: d_{g(t_0)}(p_0,x) \leq L_0 \, H(p_0,t_0)^{-1}\}$ by the factor $H(p_0,t_0)$. Then the resulting surface is $\varepsilon_0$-close to a product $\Gamma \times [-L_0,L_0]$ in the $C^3$-norm. Here, $\Gamma$ is a closed, convex curve satisfying $L(\Gamma) \leq 3\pi$ and $\sup_\Gamma |\kappa-1| \leq \frac{1}{100}$. The constant $K_0$ may depend on $\theta$, $\varepsilon_0$, $L_0$, $\delta$, $\hat{\delta}$, and the initial data, but they are independent of the remaining surgery parameters $\varepsilon$, $L$, and $H_1$.
\end{theorem}
The proof of the Neck Continuation Theorem in Section \ref{neck.contin} will require both versions of the Neck Detection Lemma. We will describe the proof of Version A in Section \ref{proof.of.neck.detection.lemma.version.a}. (The proof of Version B is analogous.) The main difference between the two versions is that Version A requires the assumption that $\hat{\mathcal{P}}(p_0,t_0,L_0+4,2\theta_0)$ does not contain surgeries, whereas Version B only requires that the parabolic neighborhood $\hat{\mathcal{P}}(p_0,t_0,L_0+4,\theta)$ is free of surgeries. (Note that $\theta$ can be much smaller than $\theta_0$.)
The following next result serves as a replacement for Lemma 7.12 in \cite{Huisken-Sinestrari3}:
\begin{proposition}[Replacement for Lemma 7.12 in \cite{Huisken-Sinestrari3}]
\label{7.12}
Let $M_t$ be a mean curvature flow with surgery satisfying Assumption \ref{a.priori.assumptions}. Suppose that $(p_1,t_1)$ is a point in spacetime such that $H(p_1,t_1) \geq H_1$ and the parabolic neighborhood $\hat{\mathcal{P}}(p_1,t_1,\tilde{L}+4,2\theta_0)$ contains at least one point belonging to a surgery region. Then there exists a point $q_1 \in M_{t_1}$ and an open set $V \subset M_{t_1}$ such that
$d_{g(t_1)}(p_1,q_1) \leq (\tilde{L}+4) \, H(p_1,t_1)^{-1}$, $\{x \in M_{t_1}: d_{g(t_1)}(q_1,x) \leq 500 \, H_1^{-1}\} \subset V$, and $V$ is diffeomorphic to a disk. Moreover, the mean curvature is at most $40 \, H_1$ at each point in $V$.
\end{proposition}
To construct $V$, we consider a surgical cap that was inserted shortly before time $t_1$. We then follow this cap forward in time (see Section \ref{proof.of.7.12} below).
Since we have a bound for the gradient of the mean curvature, we can apply Theorem 7.14 in \cite{Huisken-Sinestrari3}. This gives the following result:
\begin{proposition}[Huisken-Sinestrari \cite{Huisken-Sinestrari3}, Theorem 7.14]
\label{7.14}
Consider a closed surface in $\mathbb{R}^3$ which satisfies the estimate $|\nabla A| \leq C_\# \, (H+H_1)^2$ for suitable constants $C_\#$ and $H_1$. Then, given any $\eta>0$, we can find large numbers $\rho$ and $\gamma_0$ (depending only on $C_\#$ and $\eta$) with the following significance. Suppose that $p$ is a point on the surface with $\lambda_1(p) > \eta \, H(p)$ and $H(p) \geq \gamma_0 \, H_1$. Then either $\lambda_1 > \eta \, H > 0$ everywhere on the surface, or else there exists a point $q$ such that $\lambda_1(q) \leq \eta \, H(q)$; $d(p,q) \leq \frac{\rho}{H(p)}$; and $H(q') \geq \frac{H(p)}{\gamma_0} \geq H_1$ for all points $q'$ satisfying $d(p,q') \leq \frac{\rho}{H(p)}$. In particular, $H(q) \geq \frac{H(p)}{\gamma_0} \geq H_1$.
\end{proposition}
Moreover, using the noncollapsing property we can prove an analogue of Lemma 7.19 in \cite{Huisken-Sinestrari3}. This result will be needed for the proof of the Neck Continuation Theorem.
\begin{proposition}[Replacement for Lemma 7.19 in \cite{Huisken-Sinestrari3}]
\label{7.19}
Let $\Sigma$ be an embedded surface in $\mathbb{R}^3$ which is $\alpha$-noncollapsed, and let $y_1<y_2$ be two real numbers. We assume that the surface $\Sigma$ is contained in the cylinder $\{(x_1,x_2,x_3) \in \mathbb{R}^3: x_1^2+x_2^2 \leq 100, \, y_1 \leq x_3 \leq y_2\}$. Moreover, we assume that $\partial \Sigma = \Gamma_1 \cup \Gamma_2$, where $\Gamma_1 \subset \{x \in \mathbb{R}^3: x_3=y_1\}$ and $\Gamma_2 \subset \{x \in \mathbb{R}^3: x_3=y_2\}$. Then we have $H(x) \geq \frac{4}{\Theta}$ for all points $x \in \Sigma$ satisfying $x_3 \in [y_1+1,y_2]$ and $\langle \nu(x),e_3 \rangle \geq 0$. Here, $\nu$ denotes the outward-pointing unit normal to $\Sigma$ and $\Theta = \frac{400}{\alpha}$.
\end{proposition}
\section{The Neck Continuation Theorem and the proof of Theorem \ref{main.theorem}}
\label{neck.contin}
In this section, we use the auxiliary results collected in Section \ref{overview} to establish an analogue of the Neck Continuation Theorem of Huisken and Sinestrari \cite{Huisken-Sinestrari3}.
We begin by finalizing our choice of the surgery parameters. This step is similar to the discussion on pp.~208--209 in \cite{Huisken-Sinestrari3}. Recall that the parameters $\delta$, $\hat{\delta}$, $\hat{\alpha}$ and the constants $C_\#$, $\theta_0$, $\Theta$ have already been chosen at this stage. Moreover, we have chosen $\bar{\varepsilon}$ and $\Lambda$ so that the conclusion of Theorem \ref{properties.of.surgery} holds.
In the next step, we choose numbers $\varepsilon_0$ and $L_0$ so that $\varepsilon_0 < \bar{\varepsilon}$ and $L_0 > 1000 \, \Lambda$. In addition, we require that the mean curvature on an $(\hat{\alpha},\hat{\delta},\varepsilon_0,L_0)$-neck varies by at most a factor of $1+L_0^{-1}$. (This can always be achieved by choosing $\varepsilon_0$ very small.) We then choose real numbers $\eta_0 > 0$ and $K_0 > 1000 \, \sup_{M_0} |A|$ so that the conclusion of Version A of the Neck Detection Lemma can be applied for each $\tilde{L} \in [100,L_0]$. In other words, if $(p_0,t_0)$ satisfies $H(p_0,t_0) \geq \max \{K_0,\frac{H_1}{\Theta}\}$, $\lambda_1(p_0,t_0) \leq \eta_0 \, H(p_0,t_0)$, and if the parabolic neighborhood $\hat{\mathcal{P}}(p_0,t_0,\tilde{L}+4,2\theta_0)$ is free of surgeries for some $\tilde{L} \in [100,L_0]$, then $(p_0,t_0)$ lies at the center of a $(\hat{\alpha},\hat{\delta},\varepsilon_0,\tilde{L})$-neck in $M_{t_0}$.
In the next step, we put $\varepsilon_1 = \frac{\eta_0}{10}$. By Version A of the Neck Detection Lemma, we can find constants $\eta_1<\eta_0$ and $K_1>K_0$ such that the following holds: if $(p_0,t_0)$ satisfies $H(p_0,t_0) \geq \max \{K_1,\frac{H_1}{\Theta}\}$, $\lambda_1(p_0,t_0) \leq \eta_1 \, H(p_0,t_0)$, and if the parabolic neighborhood $\hat{\mathcal{P}}(p_0,t_0,104,2\theta_0)$ is free of surgeries, then $(p_0,t_0)$ lies at the center of a $(\hat{\alpha},\hat{\delta},\varepsilon_1,100)$-neck in $M_{t_0}$.
Having chosen $\eta_1$, we next choose $\gamma_0$ and $\rho$ so that the conclusion of Proposition \ref{7.14} holds with $\eta=\eta_1$.
By Version B of the Neck Detection Lemma, we can find a number $K_2>K_1$ such that the following holds: Suppose that $(p_0,t_0)$ satisfies $H(p_0,t_0) \geq \max\{K_2,\frac{H_1}{\Theta}\}$ and $\lambda_1(p_0,t_0) \leq 0$, and that the parabolic neighborhood $\hat{\mathcal{P}}(p_0,t_0,104,10^{-6} \, \Theta^{-2} \, \gamma_0^{-2})$ does not contain surgeries. Then, if we dilate the surface $\{x \in M_{t_0}: d_{g(t_0)}(p_0,x) \leq 100 \, H(p_0,t_0)^{-1}\}$ by the factor $H(p_0,t_0)$, the resulting surface is $\frac{\varepsilon_1}{10}$-close to a product $\Gamma \times [-100,100]$ in the $C^3$-norm. Here, $\Gamma$ is a closed, convex curve satisfying $L(\Gamma) \leq 3\pi$ and $\sup_\Gamma |\kappa-1| \leq \frac{1}{100}$.
Finally, we choose $H_1 \geq 1000 \, \Theta \, K_2$, and define $H_2 = 1000 \, \gamma_0 \, H_1$, and $H_3 = 10 \, H_2$.
Recall that the Neck Detection Lemma requires that a certain parabolic neighborhood is free of surgeries. It turns out that this assumption is not needed when the curvature is at least $1000 \, H_1$:
\begin{proposition}
\label{7.10}
Suppose that $M_t$ is a mean curvature flow with surgeries satisfying Assumption \ref{a.priori.assumptions}, where $\varepsilon \leq \bar{\varepsilon}$ and $L \geq 1000 \, \Lambda$. Moreover, suppose that $(p_0,t_0)$ satisfies $H(p_0,t_0) \geq 1000 \, H_1$ and $\lambda_1(p_0,t_0) \leq \eta_0 \, H(p_0,t_0)$, where $\eta_0$ and $H_1$ are defined as above. Then $p_0$ lies at the center of an $(\hat{\alpha},\hat{\delta},\varepsilon_0,L_0)$-neck.
\end{proposition}
\textbf{Proof.}
We distinguish two cases:
\textit{Case 1:} Suppose first that the parabolic neighborhood $\hat{\mathcal{P}}(p_0,t_0,104,2\theta_0)$ contains a point modified by surgery. By Proposition \ref{7.12}, we can find a point $q \in M_{t_0}$ and an open set $V \subset \{x \in M_{t_0}: H(x,t_0) \leq 40 \, H_1\}$ such that $d_{g(t_0)}(p_0,q) \leq 104 \, H(p_0,t_0)^{-1}$ and $\{x \in M_{t_0}: d_{g(t_0)}(q,x) \leq 500 \, H_1^{-1}\} \subset V$. Clearly, $p_0 \in V$. Consequently, $H(p_0,t_0) \leq 40 \, H_1$, contrary to our assumption.
\textit{Case 2:} We now assume that the parabolic neighborhood $\hat{\mathcal{P}}(p_0,t_0,104,2\theta_0)$ is free of surgeries. Let $\tilde{L} \in [100,L_0]$ be the largest number with the property that $\hat{\mathcal{P}}(p_0,t_0,\tilde{L}+4,2\theta_0)$ is free of surgeries. By Version A of the Neck Detection Lemma, the point $(p_0,t_0)$ lies at the center of an $(\hat{\alpha},\hat{\delta},\varepsilon_0,\tilde{L})$-neck $N$. If $\tilde{L}=L_0$, we are done. Hence, it remains to consider the case when $\tilde{L}<L_0$. In this case, the parabolic neighborhood $\hat{\mathcal{P}}(p_0,t_0,\tilde{L}+5,2\theta_0)$ must contain a point modified by surgery. By Proposition \ref{7.12}, we can find a point $q \in M_{t_0}$ and an open set $V \subset \{x \in M_{t_0}: H(x,t_0) \leq 40 \, H_1\}$ such that $d_{g(t_0)}(p_0,q) \leq (\tilde{L}+5) \, H(p_0,t_0)^{-1}$ and $\{x \in M_{t_0}: d_{g(t_0)}(q,x) \leq 500 \, H_1^{-1}\} \subset V$. Since the set $\{x \in M_{t_0}: d_{g(t_0)}(p_0,x) \leq (\tilde{L}-1) \, H(p_0,t_0)^{-1}\}$ is contained in $N$, we conclude that $\text{\rm dist}_{g(t_0)}(q,N) \leq 6 \, H(p_0,t_0)^{-1} \leq 6 \, H_1^{-1}$. Consequently, we have $N \cap V \neq \emptyset$. On the other hand, we have $H \geq \frac{1}{2} \, H(p_0,t_0) \geq 500 \, H_1$ at each point on $N$ and $H \leq 40 \, H_1$ at each point on $V$. This is a contradiction. This completes the proof of Proposition \ref{7.10}. \\
The following result is the analogue of the Neck Continuation Theorem in \cite{Huisken-Sinestrari3}:
\begin{theorem}[Neck Continuation Theorem]
\label{neck.continuation}
Suppose that $M_t$ is a mean curvature flow with surgery satisfying Assumption \ref{a.priori.assumptions}, where $\varepsilon \leq \bar{\varepsilon}$ and $L \geq 1000 \, \Lambda$. Suppose that $(p_0,t_0)$ satisfies $H(p_0,t_0) \geq 1000 \, H_1$ and $\lambda_1(p_0,t_0) \leq \eta_1 \, H(p_0,t_0)$, where $\eta_1$ and $H_1$ are defined as above. Then there exists a finite collection of points $p_1,\hdots,p_l$ with the following properties:
\begin{itemize}
\item For each $i=0,1,\hdots,l$, the point $p_i$ lies at the center of an $(\hat{\alpha},\hat{\delta},\varepsilon_0,L_0)$-neck $N^{(i)} \subset M_{t_0}$, and we have $H(p_i,t_0) \geq H_1$.
\item For each $i=1,\hdots,l-1$, the point $p_{i+1}$ lies on the neck $N^{(i)}$, and we have $\text{\rm dist}_{g(t_0)}(p_{i+1},\partial N^{(i)} \setminus N^{(i-1)}) \in [(L_0-100) \, H(p_i,t_0)^{-1},(L_0-50) \, H(p_i,t_0)^{-1}]$.
\item Finally, at least one of the following four statements holds: either the union $\mathcal{N} = \bigcup_{i=1}^l N^{(i)}$ covers the entire surface; or $H(p_l,t_0) \in [H_1,2H_1]$; or there exists a closed curve in $\mathcal{N} \cap \{x \in M_{t_0}: H(x,t_0) \leq 40 \, H_1\}$ which is homotopically non-trivial in $\mathcal{N}$ and bounds a disk in $\{x \in M_{t_0}: H(x,t_0) \leq 40 \, H_1\}$; or the outer boundary $\partial N^{(k)} \setminus N^{(k-1)}$ bounds a convex cap.
\end{itemize}
\end{theorem}
We now describe the proof of the Neck Continuation Theorem. Most of the arguments in \cite{Huisken-Sinestrari3} carry over to our situation. However, the proof of Lemma 7.19 does not work in our setting. The reason is that the gradient estimate in \cite{Huisken-Sinestrari3} works on all scales, whereas the gradient estimate in Proposition \ref{gradient.estimate} becomes weaker when the curvature is much smaller than $H_1$. We will use Proposition \ref{7.19} to overcome this problem.
\textbf{Proof.} By Proposition \ref{7.10}, the point $p_0$ lies at the center of an $(\hat{\alpha},\hat{\delta},\varepsilon_0,L_0)$-neck $N^{(0)} \subset M_{t_0}$. The construction of the points $p_1,p_2,\hdots$ is by induction. Suppose that we have constructed points $p_1,\hdots,p_k$ and necks $N^{(1)},\hdots,N^{(k)}$ with the following properties:
\begin{itemize}
\item For each $i=0,1,\hdots,k$, the point $p_i$ lies at the center of an $(\hat{\alpha},\hat{\delta},\varepsilon_0,L_0)$-neck $N^{(i)} \subset M_{t_0}$, and we have $H(p_i,t_0) \geq H_1$.
\item For each $i=1,\hdots,k-1$, the point $p_{i+1}$ lies on the neck $N^{(i)}$, and we have $\text{\rm dist}_{g(t_0)}(p_{i+1},\partial N^{(i)} \setminus N^{(i-1)}) \in [(L_0-100) \, H(p_i,t_0)^{-1},(L_0-50) \, H(p_i,t_0)^{-1}]$.
\end{itemize}
If $H(p_k,t_0) \in [H_1,2H_1]$, then we are done. Hence, for the remainder of the proof, we will assume that $H(p_k,t_0) \geq 2H_1$. We break the discussion into several cases:
\textit{Case 1:} Suppose that the there exists a point $p \in N^{(k)}$ such that $\text{\rm dist}_{g(t_0)}(p,\partial N^{(k)} \setminus N^{(k-1)}) \in [(L_0-100) \, H(p_k,t_0)^{-1},(L_0-50) \, H(p_k,t_0)^{-1}]$, and the parabolic neighborhood $\hat{\mathcal{P}}(p,t_0,L_0+4,2\theta_0)$ contains a point modified by surgery. In this case, Proposition \ref{7.12} implies that there exists a point $q \in M_{t_0}$ and an open set $V \subset \{x \in M_{t_0}: H(x,t_0) \leq 40 \, H_1\}$ such that $d_{g(t_0)}(p,q) \leq (L_0+4) \, H(p,t_0)^{-1}$, $\{x \in M_{t_0}: d_{g(t_0)}(q,x) \leq 500 \, H_1^{-1}\} \subset V$, and $V$ is diffeomorphic to a disk.
By our choice of $\varepsilon_0$ and $L_0$, the mean curvature on $N^{(k)}$ varies at most by a factor $1+L_0^{-1}$. Hence, $H(p_k,t_0) \leq (1+L_0^{-1}) \, H(p,t_0)$. Since the set $\{x \in M_{t_0}: d_{g(t_0)}(p,x) \leq (L_0-100) \, H(p_k,t_0)^{-1}\}$ is contained in $N^{(k)}$, we conclude that
\begin{align*}
\text{\rm dist}_{g(t_0)}(q,N^{(k)})
&\leq (L_0+4) \, H(p,t_0)^{-1}-(L_0-100) \, H(p_k,t_0)^{-1} \\
&\leq (L_0+4) \, (1+L_0^{-1}) \, H(p_k,t_0)^{-1}-(L_0-100) \, H(p_k,t_0)^{-1} \\
&\leq 200 \, H(p_k,t_0)^{-1} \\
&\leq 100 \, H_1^{-1}.
\end{align*}
Consequently, there exists a closed curve which is contained in $N^{(k)} \cap V$ and is homotopically non-trivial in $N^{(k)}$. Since $V$ is diffeomorphic to a disk, this curve bounds a disk in $V$, and we are done.
\textit{Case 2:} We now assume that the parabolic neighborhood $\hat{\mathcal{P}}(p,t_0,L_0+4,2\theta_0)$ is free of surgeries for all points $p \in N^{(k)}$ satisfying $\text{\rm dist}_{g(t_0)}(p,\partial N^{(k)} \setminus N^{(k-1)}) \in [(L_0-100) \, H(p_k,t_0)^{-1},(L_0-50) \, H(p_k,t_0)^{-1}]$. There are two possibilities now:
\textit{Subcase 2.1:} Suppose that there exists a point $p \in N^{(k)}$ with the property that $\text{\rm dist}_{g(t_0)}(p,\partial N^{(k)} \setminus N^{(k-1)}) \in [(L_0-100) \, H(p_k,t_0)^{-1},(L_0-50) \, H(p_k,t_0)^{-1}]$ and $\lambda_1(p,t_0) \leq \eta_0 \, H(p,t_0)$. By Version A of the Neck Detection Lemma, the point $p$ lies at the center of an $(\hat{\alpha},\hat{\delta},\varepsilon_0,L_0)$-neck $N$. Moreover, since $p \in N^{(k)}$ and $H(p_k,t_0) \geq 2H_1$, we have $H(p,t_0) \geq H_1$. Hence, we can put $p^{(k+1)} := p$ and $N^{(k+1)} := N$ and continue the process.
\textit{Subcase 2.2:} Suppose that $\lambda_1(p,t_0) > \eta_0 \, H(p,t_0)$ for all points $p \in N^{(k)}$ satisfying $\text{\rm dist}_{g(t_0)}(p,\partial N^{(k)} \setminus N^{(k-1)}) \in [(L_0-100) \, H(p_k,t_0)^{-1},(L_0-50) \, H(p_k,t_0)^{-1}]$. Let $\mathcal{N} = \bigcup_{i=0}^k N^{(i)}$, and let $\mathcal{A}$ be the set of all points $x \in \mathcal{N}$ satisfying $\text{\rm dist}_{g(t_0)}(p,\partial N^{(k)} \setminus N^{(k-1)}) \geq (L_0-50) \, H(p_k,t_0)^{-1}$ and $\lambda_1(x,t_0) \leq \eta_1 \, H(x,t_0)$. The assumptions of Theorem \ref{neck.continuation} imply that the initial point $p_0$ belongs to $\mathcal{A}$, so $\mathcal{A}$ is non-empty. Let us consider a point $p^*$ which has maximal intrinsic distance from $p_0$ among all points in $\mathcal{A}$.
\textit{Subcase 2.2.1:} Suppose that the parabolic neighborhood $\hat{\mathcal{P}}(p^*,t_0,104,2\theta_0)$ contains a point modified by surgery. In this case, Proposition \ref{7.12} implies that there exists a point $q \in M_{t_0}$ and an open set $V \subset \{x \in M_{t_0}: H(x,t_0) \leq 40 \, H_1\}$ such that $d_{g(t_0)}(p^*,q) \leq 104 \, H(p^*,t_0)^{-1}$, $\{x \in M_{t_0}: d_{g(t_0)}(q,x) \leq 500 \, H_1^{-1}\} \subset V$, and $V$ is diffeomorphic to a disk. Since $H(p^*,t_0) \geq \frac{H_1}{2}$, this implies
\begin{align*}
&\{x \in M_{t_0}: d_{g(t_0)}(p^*,x) \leq 100 \, H(p^*,t_0)^{-1}\} \\
&\subset \{x \in M_{t_0}: d_{g(t_0)}(q,x) \leq 204 \, H(p^*,t_0)^{-1}\} \\
&\subset \{x \in M_{t_0}: d_{g(t_0)}(q,x) \leq 500 \, H_1^{-1}\} \\
&\subset V.
\end{align*}
Consequently, there exists a closed curve in $\mathcal{N} \cap V$ which is homotopically non-trivial in $\mathcal{N}$. This curve bounds a disk which is contained in $V$. Hence, we can again terminate the process.
\textit{Subcase 2.2.2:} Suppose, finally, that the parabolic neighborhood $\hat{\mathcal{P}}(p^*,t_0,104,2\theta_0)$ is free of surgeries. In this case, Version A of the Neck Detection Lemma implies that the point $p^*$ lies at the center of an $(\hat{\alpha},\hat{\delta},\varepsilon_1,100)$-neck $N^*$. Clearly, $\lambda_1 \leq \varepsilon_1 \, H$ at each point on $N^*$. Consequently, the set $N^*$ is disjoint from the set $\{p \in N^{(k)}: \text{\rm dist}_{g(t_0)}(p,\partial N^{(k)} \setminus N^{(k-1)}) \in [(L_0-100) \, H(p_k,t_0)^{-1},(L_0-50) \, H(p_k,t_0)^{-1}]\}$. Furthermore, since $p^*$ has maximal distance from $p_0$ among all points in $\mathcal{A}$, we conclude that the part of $\mathcal{N}$ that lies between the neck $N^*$ and the set $\{p \in N^{(k)}: \text{\rm dist}_{g(t_0)}(p,\partial N^{(k)} \setminus N^{(k-1)}) \in [(L_0-100) \, H(p_k,t_0)^{-1},(L_0-50) \, H(p_k,t_0)^{-1}]\}$ is strictly convex.
Let $\omega$ be a unit vector in $\mathbb{R}^3$ which is parallel to the axis of the neck $N^*$. The arguments on p.~214 of \cite{Huisken-Sinestrari3} imply that $\langle \nu,\omega \rangle \geq -\varepsilon_1$ for all points $p \in N^{(k)}$ satisfying $\text{\rm dist}_{g(t_0)}(p,\partial N^{(k)} \setminus N^{(k-1)}) \in [(L_0-100) \, H(p_k,t_0)^{-1},(L_0-50) \, H(p_k,t_0)^{-1}]$. Moreover, we have $\lambda_1(p,t_0) > \eta_0 \, H(p,t_0)$ for all points $p \in N^{(k)}$ satisfying $\text{\rm dist}_{g(t_0)}(p,\partial N^{(k)} \setminus N^{(k-1)}) \in [(L_0-100) \, H(p_k,t_0)^{-1},(L_0-50) \, H(p_k,t_0)^{-1}]$. Putting these facts together (and using the fact that $\eta_0 \geq 10\varepsilon_1$), we conclude that $\langle \nu,\omega \rangle \geq 4\varepsilon_1$ for all points $p \in N^{(k)}$ satisfying $\text{\rm dist}_{g(t_0)}(p,\partial N^{(k)} \setminus N^{(k-1)}) \in [(L_0-100) \, H(p_k,t_0)^{-1},(L_0-75) \, H(p_k,t_0)^{-1}]$.
We claim that the boundary curve $\partial N^{(k)} \setminus N^{(k-1)}$ bounds a convex cap. To prove this, we follow the argument on pp.~215-216 of \cite{Huisken-Sinestrari3}. Let us choose a curve $\Gamma_0$ such that $\Gamma_0 \subset \{p \in N^{(k)}: \text{\rm dist}_{g(t_0)}(p,\partial N^{(k)} \setminus N^{(k-1)}) \in [(L_0-100) \, H(p_k,t_0)^{-1},(L_0-75) \, H(p_k,t_0)^{-1}]\}$ and $\Gamma_0$ is contained in a plane orthogonal to $\omega$. For each point on $\Gamma_0$, we solve the ODE $\dot{\gamma} = \frac{\omega^T(\gamma)}{|\omega^T(\gamma)|^2}$, where $\omega^T(\gamma)$ denotes the projection of $\omega$ to the tangent plane to $M_{t_0}$ at the point $\gamma$. This gives a family of curves $\Gamma_y \subset M_{t_0}$, each of which is contained in a plane orthogonal to $\omega$. The curves $\Gamma_y$ are well-defined for $y \in [0,y_{\text{\rm max}})$. Moreover, there exists a point $p \in \Gamma_0$ such that $\nu(\gamma(y,p)) \to \omega$ as $y \to y_{\text{\rm max}}$.
Following the arguments on p.~215 in \cite{Huisken-Sinestrari3}, we can show that the inequalities
\begin{equation}
\tag{$\star$}
\langle \nu,\omega \rangle < 1, \qquad \lambda_1 > 0, \qquad H > \frac{2H_1}{\Theta}, \qquad \langle \nu,\omega \rangle > \varepsilon_1
\end{equation}
hold for all $y \in [0,y_{\text{\rm max}})$. Indeed, the inequalities in ($\star$) are clearly satisfied for $y=0$. If one of the inequalities in ($\star$) fails for some $y>0$, we consider the smallest value of $y$ for which that happens. The first inequality in ($\star$) cannot fail first by definition of $y_{\text{\rm max}}$. If the second inequality in ($\star$) is the first one to fail, then we have $\lambda_1=0$. Since $H \geq \frac{2H_1}{\Theta}$, we may apply Version B of the Neck Detection Lemma to conclude that we are $\frac{\varepsilon_1}{10}$-close to a Cartesian product, but this is ruled out by the fourth inequality in ($\star$). If the third inequality in ($\star$) is the first one to fail, we obtain a contradiction with Proposition \ref{7.19}. Finally, $\langle \nu,\omega \rangle$ is montone increasing in $y$ as long as $\lambda_1$ remains nonnegative; this implies that the fourth inequality in ($\star$) cannot fail first. Thus, the inequalities in ($\star$) hold for all $y \in [0,y_{\text{\rm max}})$. Consequently, the union of the curves $\Gamma_y$ is a convex cap, and we can terminate the process. This completes the construction of the sequence $p_1,p_2,\hdots$.
If the sequence $p_1,p_2,\hdots$ terminates after finitely many steps, then the theorem is proved. On the other hand, if the sequence $p_1,p_2,\hdots$ never terminates, then the necks $N^{(1)},N^{(2)},\hdots$ will eventually cover the entire surface. This completes the proof of Theorem \ref{neck.continuation}. \\
Having established the Neck Continuation Theorem, we can now implement the surgery algorithm of Huisken and Sinestrari \cite{Huisken-Sinestrari3}, and complete the proof of Theorem \ref{main.theorem}. Starting from the given initial surface $M_0$, we run the mean curvature flow until the maximum of the mean curvature reaches the threshold $H_3$ for the first time. Let us denote this time by $T_1$. By a result of Andrews \cite{Andrews}, the inscribed radius and the outer radius are bounded from below by $\frac{\alpha}{H}$ for $0 \leq t \leq T_1$. Moreover, it is easy to see that the surfaces $M_t$ are outward-minimizing for $0 \leq t \leq T_1$. Therefore, Assumption \ref{a.priori.assumptions} is satisfied for $0 \leq t \leq T_1$. Consequently, we may apply the Neck Detection Lemma and the Neck Continuation Theorem for $0 \leq t \leq T_1$. By performing surgery on suitably chosen $(\hat{\alpha},\hat{\delta},\varepsilon_0,L_0)$-necks at time $T_1$, we can remove all regions where the mean curvature is between $H_2$ and $H_3$. Hence, immediately after surgery, the maximum of the mean curvature drops to a level below $H_2$. We then run the flow again until the maximum of the mean curvature reaches $H_3$ for the second time. Let us denote this time by $T_2$. We claim that, for $0 \leq t \leq T_2$, the flow satisfies Assumption \ref{a.priori.assumptions} with $\varepsilon=\varepsilon_0$ and $L=L_0$. Indeed, Theorem \ref{properties.of.surgery} implies that the inscribed radius and the outer radius of the surface $M_{T_1+}$ are bounded from below by $\frac{\alpha}{H}$, and this property continues to hold for all $T_1 < t \leq T_2$ by a result of Andrews \cite{Andrews}. Furthermore, the outward-minimizing property follows from work of Head (see \cite{Head}, Lemma 5.2). Therefore, Assumption \ref{a.priori.assumptions} is satisfied for $0 \leq t \leq T_2$ with $\varepsilon=\varepsilon_0$ and $L=L_0$. Hence, we can apply the Neck Detection Lemma and the Neck Continuation Theorem for $0 \leq t \leq T_2$. By performing surgery on suitably chosen $(\hat{\alpha},\hat{\delta},\varepsilon,L)$-necks, we can push the maxmimum of the mean curvature below $H_2$. We then restart the flow again. This process can be repeated until the solution becomes extinct.
\section{Proof of the pseudolocality principle (Theorem \ref{pseudolocality})}
\label{proof.of.pseudoloc}
We first recall the following analogue of Shi's local derivative estimate for the Ricci flow. The argument given here is standard and follows the proof in Ecker-Huisken \cite{Ecker-Huisken2}; see also \cite{Ecker}, Proposition 3.22.
\begin{lemma}
\label{shi.1}
Suppose that $M_t$, $t \in [0,T]$, is a regular mean curvature flow in $B_4(0)$ in the sense of Definition \ref{regular.flow}. Moreover, we assume that $|A| \leq 1$ for all $t \in [0,T]$ and all $x \in M_t \cap B_4(0)$. Finally, we assume that $|\nabla A| \leq 1$ for all $x \in M_0 \cap B_4(0)$. Then $|\nabla A| \leq C$ for all $t \in [0,1] \cap [0,T]$ and all $x \in M_t \cap B_2(0)$.
\end{lemma}
\textbf{Proof.}
Consider the cutoff function $\psi(x) = 1-\frac{|x|^2}{16}$. A straightforward calculation gives
\[\frac{\partial}{\partial t} (\psi^2 \, |\nabla A|^2) \leq \Delta (\psi^2 \, |\nabla A|^2) + C_0 \, |\nabla A|^2\]
for all $t \in [0,T]$ and all $x \in M_t \cap B_4(0)$. This implies that
\[\frac{\partial}{\partial t} (\psi^2 \, |\nabla A|^2 + C_0 \, |A|^2) \leq \Delta (\psi^2 \, |\nabla A|^2 + C_0 \, |A|^2) + C_1\]
for all $t \in [0,T]$ and all $x \in M_t \cap B_4(0)$. Applying the maximum principle to the function $\psi^2 \, |\nabla A|^2 + C_0 \, |A|^2 - C_1 \, t$, we obtain
\begin{align*}
&\sup_{t \in [0,T]} \sup_{x \in M_t \cap B_4(0)} (\psi^2 \, |\nabla A|^2 + C_0 \, |A|^2 - C_1 \, t) \\
&\leq \max \Big \{ \sup_{x \in M_0 \cap B_4(0)} (\psi^2 \, |\nabla A|^2 + C_0 \, |A|^2),\sup_{t \in [0,T]} \sup_{x \in M_t \cap \partial B_4(0)} (C_0 \, |A|^2 - C_1 \, t) \Big \} \\
&\leq 1+C_0.
\end{align*} From this, the assertion follows. \\
A similar estimate holds for the second derivatives of the second fundamental form:
\begin{lemma}
\label{shi.2}
Suppose that $M_t$, $t \in [0,T]$, is a regular mean curvature flow in $B_4(0)$ in the sense of Definition \ref{regular.flow}. Moreover, we assume that $|A| \leq 1$ for all $t \in [0,T]$ and all $x \in M_t \cap B_4(0)$. Finally, we assume that $|\nabla A|+|\nabla^2 A| \leq 1$ for all $x \in M_0 \cap B_4(0)$. Then $|\nabla A|+|\nabla^2 A| \leq C$ for all $t \in [0,1] \cap [0,T]$ and all $x \in M_t \cap B_1(0)$.
\end{lemma}
\textbf{Proof.}
By Lemma \ref{shi.1}, we have $|\nabla A| \leq C$ for all $t \in [0,1] \cap [0,T]$ and all $x \in M_t \cap B_2(0)$. To get a bound for $|\nabla^2 A|$, we apply the maximum principle to the function $\psi^2 \, |\nabla^2 A|^2 + C_0 \, |\nabla A|^2$, where $\psi = 1-\frac{|x|^2}{4}$ and $C_0$ is a large constant. \\
Our next result will require the monotonicity formula for mean curvature flow (cf. \cite{Huisken2}). We will need a local version of this result. Specifically, we consider the modified Gaussian density
\[\Theta(x_0,t_0;r) = \int_{M_{t_0}-r^2} \frac{1}{4\pi r^2} \, e^{-\frac{|x-x_0|^2}{4r^2}} \, (1 - |x-x_0|^2 + 4 \, r^2)_+^3.\]
The local monotonicity formula asserts that the function $r \mapsto \Theta(x_0,t_0;r)$ is monotone increasing. A proof of this fact can be found in \cite{Ecker}, pp.~64--65 (see also \cite{Ecker-Huisken1}).
\begin{proposition}
\label{pseudolocality.2}
There exist positive constants $\beta_0 \in (0,1)$ and $C$ such that the following holds. Suppose that $M_t$, $t \in [0,T]$, is a regular mean curvature flow in $B_4(0)$. Moreover, we assume that the initial surface $M_0$ can be expressed as the graph of a (single-valued) function $u$ over a plane. If $\|u\|_{C^4} \leq \beta_0$, then $|A(x,t)| \leq C$ for all $t \in [0,\beta_0] \cap [0,T]$ and all $x \in M_t \cap B_1(0)$.
\end{proposition}
\textbf{Proof.}
Our argument is inspired in part by the proof of Theorem C.1 in \cite{Haslhofer-Kleiner}. Suppose that the assertion is false. Then we can find a sequence of regular mean curvature flows $\mathcal{M}_j$ in $B_4(0)$ with the following properties:
\begin{itemize}
\item The initial surface $M_{0,j} \cap B_4(0)$ is the graph of a (single-valued) function $u_j$ over a plane, and $u_j$ satisfies $\|u_j\|_{C^4} \leq \frac{1}{j}$.
\item There exists a sequence of times $t_j \in [0,\frac{1}{j}] \cap [0,T_j]$ and a sequence of points $x_j \in M_{t_j,j} \cap B_1(0)$ such that $|A(x_j,t_j)| \geq j$.
\end{itemize}
Using a point picking argument as in Appendix C of \cite{Haslhofer-Kleiner}, we can find a pair $(\tilde{x}_j,\tilde{t}_j)$ such that $\tilde{t}_j \in [0,\frac{1}{j}] \cap [0,T_j]$, $\tilde{x}_j \in M_{\tilde{t}_j,j} \cap B_2(0)$, $Q_j := |A(\tilde{x}_j,\tilde{t}_j)| \geq j$, and
\[\sup_{t \in [0,\tilde{t}_j]} \sup_{x \in M_{t,j} \cap B_{\frac{j}{2} \, Q_j^{-1}}(\tilde{x}_j)} |A(x,t)| \leq 2 \, Q_j.\]
At this point, we distinguish two cases:
\textit{Case 1:} Suppose that $\limsup_{j \to \infty} \tilde{t}_j \, Q_j^2 = 0$. By assumption, we have $|\nabla A| \leq 1$ and $|\nabla^2 A| \leq 1$ on the initial surface $M_{0,j} \cap B_4(0)$. Hence, it follows from Lemma \ref{shi.1} and Lemma \ref{shi.2} that
\[\sup_{t \in [0,\tilde{t}_j]} \sup_{x \in M_{t,j} \cap B_{Q_j^{-1}}(\tilde{x}_j)} |\nabla A(x,t)| \leq C_0 \, Q_j^2\]
and
\[\sup_{t \in [0,\tilde{t}_j]} \sup_{x \in M_{t,j} \cap B_{Q_j^{-1}}(\tilde{x}_j)} |\nabla^2 A(x,t)| \leq C_0 \, Q_j^3,\]
where $C_0$ is a uniform constant independent of $j$. In the next step, we follow the point $\tilde{x}_j$ back in time. More precisely, we consider a path $\sigma_j: [0,\tilde{t}_j] \to \mathbb{R}^3$ such that $\sigma_j(t) \in M_{t,j}$, $\sigma_j'(t)$ equals the mean curvature vector of $M_{t,j}$ at the point $\sigma_j(t)$, and $\sigma_j(\tilde{t}_j) = \tilde{x}_j$. Then $|\sigma_j'(t)| \leq 4 \, Q_j$ as long as $\sigma_j(t) \in B_{Q_j^{-1}}(\tilde{x}_j)$. Hence, if $j$ is sufficiently large, then the curve $\sigma_j(t)$ will remain in the ball $B_{Q_j^{-1}}(\tilde{x}_j)$ for all $t \in [0,\tilde{t}_j]$. In particular, if $j$ is sufficiently large, then we have $|\nabla^2 A(\sigma_j(t),t)| \leq C_0 \, Q_j^3$ for all $t \in [0,\tilde{t}_j]$. This implies
\[\frac{d}{dt} |A(\sigma_j(t),t)| \leq C_1 \, Q_j^3\]
for all $t \in [0,\tilde{t}_j]$, provided that $j$ is sufficiently large. Integrating this inequality from $0$ to $\tilde{t}_j$ gives
\[Q_j = |A(\tilde{x}_j,\tilde{t}_j)| \leq |A(\sigma_j(0),0)| + C_1 \, \tilde{t}_j \, Q_j^3 \leq 1 + C_1 \, \tilde{t}_j \, Q_j^3\]
if $j$ is sufficiently large. Since $Q_j \to \infty$ and $\tilde{t}_j \, Q_j^2 \to 0$, we arrive at a contradiction.
\textit{Case 2:} We now assume that $\tau := \limsup_{j \to \infty} \tilde{t}_j \, Q_j^2 \in (0,\infty]$. Let us define a family of surfaces $M_{t,j}' \subset M_{t,j}$ in the following way: The surface $M_{\tilde{t}_j,j}'$ is defined as the intersection of $M_{\tilde{t}_j,j}$ with the ball $B_{\frac{j}{4} \, Q_j^{-1}}(\tilde{x}_j)$. Moreover, for each $t \in [0,\tilde{t}_j]$, the surface $M_{t,j}'$ is obtained by following each point on the surface $M_{\tilde{t}_j,j}$ back in time. It is clear that the surfaces $M_{t,j}'$, $t \in [0,\tilde{t}_j]$, form a solution of the mean curvature flow in the classical sense. Moreover, we have $\partial M_{t,j}' \cap B_{\frac{j}{8} \, Q_j^{-1}}(\tilde{x}_j) = \emptyset$ for all $t \in [0,\tilde{t}_j] \cap [\tilde{t}_j-\frac{j}{64} \, Q_j^{-2},\tilde{t}_j]$.
We next consider the rescaled surfaces $\tilde{M}_{s,j} := Q_j \, (M_{\tilde{t}_j+Q_j^{-2} \, s,j}' - \tilde{x}_j)$, $s \in [-\tilde{t}_j \, Q_j^2,0]$. These surfaces again form a solution of mean curvature flow in the classical sense. Moreover, we have $\partial \tilde{M}_{s,j} \cap B_{\frac{j}{40}}(0) = \emptyset$ for all $s \in [-\tilde{t}_j \, Q_j^2,0] \cap [-\frac{j}{64},0]$. Finally, the norm of the second fundamental form of $\tilde{M}_{s,j} \cap B_{\frac{j}{8}}(0)$ is bounded from above by $2$.
Taking the limit as $j \to \infty$, we obtain a complete, smooth, non-flat solution to the mean curvature flow which is defined on the time interval $(-\tau,0]$. The limiting solution has bounded curvature and nonnegative mean curvature. We claim that the (standard) Gaussian density of the limit flow is at most $1$. To see this, let us denote the limit flow by $\hat{M}_s$, $s \in (-\tau,0]$.
Moreover, let us consider an arbitrary point $(y_0,s_0) \in \mathbb{R}^3 \times (-\tau,0]$ and a number $r \in (0,\sqrt{\tau+s_0})$. Clearly, $Q_j^{-1} \, r < \sqrt{\tilde{t}_j+Q_j^{-2} \, s_0}$ for $j$ large. Using Ecker's monotonicity formula for the modified Gaussian density $\Theta_{\mathcal{M}_j}$, we obtain
\begin{align*}
&\int_{\hat{M}_{s_0-r^2}} \frac{1}{4\pi r^2} \, e^{-\frac{|y-y_0|^2}{4r^2}} \\
&\leq \limsup_{j \to \infty} \Theta_{\mathcal{M}_j}(\tilde{x}_j+Q_j^{-1} \, y_0,\tilde{t}_j+Q_j^{-2} \, s_0;Q_j^{-1} \, r) \\
&\leq \limsup_{j \to \infty} \Theta_{\mathcal{M}_j}(\tilde{x}_j+Q_j^{-1} \, y_0,\tilde{t}_j+Q_j^{-2} \, s_0;\sqrt{\tilde{t}_j+Q_j^{-2} \, s_0}) \\
&= \limsup_{j \to \infty} \int_{M_{0,j}} \frac{1}{4\pi \bar{r}_j^2} \, e^{-\frac{|x-\bar{x}_j|^2}{4\bar{r}_j^2}} \, (1-|x-\bar{x}_j|^2+4\bar{r}_j^2)_+^3,
\end{align*}
where $\bar{x}_j := \tilde{x}_j+Q_j^{-1} \, y_0$ and $\bar{r}_j := \sqrt{\tilde{t}_j+Q_j^{-2} \, s_0}$. Using our assumption on $M_{0,j}$, we obtain
\[\int_{\hat{M}_{s_0-r^2}} \frac{1}{4\pi r^2} \, e^{-\frac{|y-y_0|^2}{4r^2}} \leq 1\]
for all $(y_0,s_0) \in \mathbb{R}^3 \times (-\tau,0]$ and all $r \in (0,\sqrt{\tau+s_0})$. This easily implies that the limiting solution is a flat plane of multiplicity $1$. This is a contradiction. This completes the proof of Proposition \ref{pseudolocality.2}. \\
Theorem \ref{pseudolocality} follows by combining Proposition \ref{pseudolocality.2} with Lemma \ref{shi.1} and Lemma \ref{shi.2}.
\section{The gradient estimate of Haslhofer and Kleiner (Theorem \ref{interior.derivative.estimate}
)}
\label{haslhofer}
The curvature derivative estimate of Haslhofer and Kleiner is a consequence of the following result:
\begin{theorem}[cf. Haslhofer-Kleiner \cite{Haslhofer-Kleiner}]
\label{int.estimate}
Given any $\alpha \in (0,\frac{1}{100}]$, there exist constants $C = C(\alpha)$ and $\rho = \rho(\alpha)$ with the following property. Suppose that $M_t$, $t \in [-1,0]$, is a regular mean curvature flow in the ball $B_4(0)$. Moreover, suppose that each surface $M_t$ is outward-minimizing within the ball $B_4(0)$. We further assume that the inscribed radius and the outer radius are at least $\frac{\alpha}{H}$ at each point on $M_t$. Finally, we assume that $M_0$ passes through the origin, and $H(0,0) \leq 1$. Then $|A(x,t)| \leq C$ for all $t \in [-\rho^2,0]$ and all points $x \in M_t \cap B_\rho(0)$.
\end{theorem}
We sketch the proof of Haslhofer and Kleiner for the convenience of the reader. Suppose that there exists a sequence of regular flows $\mathcal{M}_j$ in $B_4(0)$ satisfying the assumptions of Theorem \ref{int.estimate}, and a sequence of pairs $(x_j,t_j)$ such that $t_j \in [-\frac{1}{j^2},0]$, $x_j \in M_{t_j,j} \cap B_{\frac{1}{j}}(0)$ and $|A(x_j,t_j)| \geq j^2$. By the point selection argument of Haslhofer-Kleiner, there exists a pair $(\tilde{x}_j,\tilde{t}_j)$ such that $\tilde{t}_j \in [-\frac{2}{j^2},0]$, $\tilde{x}_j \in M_{\tilde{t}_j,j} \cap B_{\frac{2}{j}}(0)$, $Q_j := |A(\tilde{x}_j,\tilde{t}_j)| \geq j^2$, and
\[\sup_{t \in [\tilde{t}_j-\frac{j^2}{4} \, Q_j^{-2},\tilde{t}_j]} \sup_{x \in M_{t,j} \cap B_{\frac{j}{2} \, Q_j^{-1}}(\tilde{x}_j)} |A(x,t)| \leq 2 \, Q_j.\]
We can define a family of surfaces $M_{t,j}' \subset M_{t,j}$ in the following way: The surface $M_{\tilde{t}_j,j}'$ is defined as the intersection of $M_{\tilde{t}_j,j}$ with the ball $B_{\frac{j}{4} \, Q_j^{-1}}(\tilde{x}_j)$. Moreover, for each $t \in [\tilde{t}_j-\frac{j}{64} \, Q_j^{-2},\tilde{t}_j]$, the surface $M_{t,j}'$ is obtained by following each point on the surface $M_{\tilde{t}_j,j}$ back in time. Clearly, the surfaces $M_{t,j}'$, $t \in [\tilde{t}_j-\frac{j}{64} \, Q_j^{-2},\tilde{t}_j]$, form a solution of the mean curvature flow in the classical sense. Moreover, we have $\partial M_{t,j}' \cap B_{\frac{j}{8} \, Q_j^{-1}}(\tilde{x}_j) = \emptyset$ for all $t \in [\tilde{t}_j-\frac{j}{64} \, Q_j^{-2},\tilde{t}_j]$.
Consider the rescaled surfaces $\tilde{M}_{s,j} := Q_j \, (M_{\tilde{t}_j+Q_j^{-2} \, s,j}' - \tilde{x}_j)$, $s \in [-\frac{j}{64},0]$. These surfaces again form a solution of mean curvature flow in the classical sense. Moreover, $\partial \tilde{M}_{s,j} \cap B_{\frac{j}{8}}(0) = \emptyset$ for all $s \in [-\frac{j}{64},0]$. Finally, the norm of the second fundamental form of $\tilde{M}_{s,j}$ is bounded from above by $2$. After passing to the limit as $j \to \infty$, one obtains a complete, non-flat, ancient solution to the mean curvature flow with bounded curvature. Let us denote this limit solution by $\hat{M}_s$, $s \in (-\infty,0]$.
We claim that the Gaussian density of the limit solution is at most $1$ everywhere. Let us fix a point $(y_0,s_0) \in \mathbb{R}^3 \times (-\infty,0]$ and a number $r > 0$. Using Ecker's local monotonicity formula, we obtain
\begin{align*}
&\int_{\hat{M}_{s_0-r^2}} \frac{1}{4\pi r^2} \, e^{-\frac{|y-y_0|^2}{4r^2}} \\
&\leq \limsup_{j \to \infty} \Theta_{\mathcal{M}_j}(\tilde{x}_j+Q_j^{-1} \, y_0,\tilde{t}_j+Q_j^{-2} \, s_0;Q_j^{-1} \, r) \\
&\leq \limsup_{j \to \infty} \Theta_{\mathcal{M}_j}(\tilde{x}_j+Q_j^{-1} \, y_0,\tilde{t}_j+Q_j^{-2} \, s_0;\sqrt{\tilde{t}_j+Q_j^{-2} \, s_0 + \frac{1}{j}}) \\
&= \limsup_{j \to \infty} \int_{M_{-\frac{1}{j},j}} \frac{1}{4\pi \bar{r}_j^2} \, e^{-\frac{|x-\bar{x}_j|^2}{4\bar{r}_j^2}} \, (1 - |x - \bar{x}_j|^2 + 4\bar{r}_j^2)_+^3,
\end{align*}
where $\bar{x}_j := \tilde{x}_j+Q_j^{-1} \, y_0$ and $\bar{r}_j := \sqrt{\tilde{t}_j+Q_j^{-2} \, s_0 + \frac{1}{j}}$.
By assumption, $\mathcal{M}_j$ satisfies $H(0,0) \leq 1$. Let $v_j$ denote the outward-pointing unit normal vector to the surface $M_{0,j}$ at the origin. Moreover, let $\Omega_{t,j}$ denote the region enclosed by $M_{t,j}$. The noncollapsing property implies that $B_\alpha(-\alpha \, v_j) \subset \Omega_{0,j} \subset \Omega_{-\frac{1}{j},j}$. On the other hand, since the surface $M_{0,j}$ passes through the origin, we must have $M_{-\frac{1}{j},j} \cap B_{\sqrt{\frac{\alpha^2}{4}+\frac{8}{j}}}(-\frac{\alpha}{2} \, v_j) \neq \emptyset$. For each $j$, we pick a point $z_j \in M_{-\frac{1}{j},j}$ which has minimal distance from $-\frac{\alpha}{2} \, v_j$ among all points on $M_{-\frac{1}{j},j}$. Then $|z_j + \frac{\alpha}{2} \, v_j| \leq \sqrt{\frac{\alpha^2}{4}+\frac{8}{j}}$ and $|z_j+\alpha \, v_j| \geq \alpha$. This implies $|z_j| \leq C \, j^{-\frac{1}{2}}$. Moreover, we have $H(z_j,-\frac{1}{j}) \leq \frac{4}{\alpha}$. In view of the noncollapsing assumption, we can find two open balls of radius $\frac{\alpha^2}{4}$ such that one of them is contained in $\Omega_{-\frac{1}{j},j}$; the other one is disjoint from $\Omega_{-\frac{1}{j},j}$; and the two balls touch each other at the point $z_j$. Consequently, the rescaled domains $j^{\frac{1}{2}} \, \Omega_{-\frac{1}{j},j}$ converge to a halfspace in the Hausdorff sense. Using the fact that $M_{-\frac{1}{j},j}$ is outward-minimizing, we conclude that the rescaled surfaces $j^{\frac{1}{2}} \, M_{-\frac{1}{j},j}$ converge, in the sense of geometric measure theory, to a plane of multiplicity at most $1$.
Since $|\bar{x}_j| \leq O(j^{-1})$ and $\bar{r}_j = (1+o(1)) \, j^{-\frac{1}{2}}$ for $j$ large, we conclude that
\[\limsup_{j \to \infty} \int_{M_{-\frac{1}{j},j}} \frac{1}{4\pi \bar{r}_j^2} \, e^{-\frac{|x-\bar{x}_j|^2}{4\bar{r}_j^2}} \, (1 - |x - \bar{x}_j|^2 + 4\bar{r}_j^2)_+^3 \leq 1.\]
Therefore, the limiting flow $\hat{M}_s$ has Gaussian density at most $1$. This contradicts the fact that the limit flow is non-flat.
\section{Proof of Theorem \ref{properties.of.surgery}}
\label{construction.of.cap}
In this section, we explain our procedure for capping off a neck. We begin by constructing an axially symmetric model surface.
\begin{lemma}
\label{model.surface}
The surface
\[\Sigma = \Big \{ \Big ( \sqrt{\frac{s}{1+s}} \, \cos(2\pi t),\sqrt{\frac{s}{1+s}} \, \sin(2\pi t),s \Big ) : s \in [0,\infty), \, t \in [0,1] \Big \}\]
closes up smoothly at $s=0$. Moreover, we have $0 < \lambda_1 < \lambda_2 = \mu$ whenever $s>0$. Here, $\mu$ denotes the reciprocal of the inscribed radius of $\Sigma$.
\end{lemma}
\textbf{Proof.}
The smoothness of $\Sigma$ is obvious. A straightforward calculation shows that the principal curvatures of $\Sigma$ are given by
\[\lambda_1 = 2 \, (1+4s \, (1+s)^3)^{-\frac{3}{2}} \, (1+s)^2 \, (1+4s)\]
and
\[\lambda_2 = 2 \, (1+4s \, (1+s)^3)^{-\frac{1}{2}} \, (1+s)^2.\]
Clearly, $0 < \lambda_1 < \lambda_2$ for $s > 0$. Hence, it remains to estimate the inscribed radius of $\Sigma$. To that end, let $U = \{x \in \mathbb{R}^3: x_3 > 0, \, x_1^2+x_2^2 < \frac{x_3}{1+x_3}\}$ be the region enclosed by $\Sigma$. For each $s>0$, we denote by $W_s$ the open ball of radius $\frac{1}{2} \, (1+4s \, (s+1)^3)^{1/2} \, (1+s)^{-2}$ centered at the point $(0,0,s+\frac{1}{2} \, (s+1)^{-2})$. For each $s>0$, the circle
\[C_s := \Big \{ \Big ( \sqrt{\frac{s}{1+s}} \, \cos(2\pi t),\sqrt{\frac{s}{1+s}} \, \sin(2\pi t),s \Big ) : t \in [0,1] \Big \}\]
is contained in $\Sigma \cap \partial W_s$. Moreover, the surfaces $\Sigma$ and $\partial W_s$ have the same tangent plane at each point on the circle $C_s$.
It is easy to see that $W_s \subset U$ if $s$ is sufficiently large. We claim that $W_s \subset U$ for all $s > 0$. Suppose this is false. Let $\bar{s} = \sup \{s > 0: W_s \not\subset U\}$. Then $W_{\bar{s}} \subset U$. Moreover, we can find a sequence of numbers $s_j \nearrow \bar{s}$ and a sequence of points $p_j \in W_{s_j} \setminus U$. After passing to a subsequence if necessary, the points $p_j$ converge to some point $p \in \bar{W}_{\bar{s}} \setminus U$. Since $W_{\bar{s}} \subset U$, we conclude that $p \in \Sigma \cap \partial W_{\bar{s}}$, and the surfaces $\Sigma$ and $\partial W_{\bar{s}}$ have the same tangent plane at the point $p$. On the other hand, since $\lambda_1 < \lambda_2$, we must have $\liminf_{j \to \infty} \text{\rm dist}(p_j,C_{s_j}) > 0$. Consequently, we have $p \in C_{\tilde{s}}$ for some $\tilde{s} \neq \bar{s}$. This implies that $p \in \Sigma \cap \partial W_{\tilde{s}}$, and the surfaces $\Sigma$ and $\partial W_{\tilde{s}}$ have the same tangent plane at the point $p$. Thus, the spheres $\partial W_{\bar{s}}$ and $\partial W_{\tilde{s}}$ touch each other at the point $p$, but this is impossible if $\bar{s} \neq \tilde{s}$. This shows that $W_s \subset U$ for all $s > 0$. Consequently, the inscribed radius is given by $\frac{1}{2} \, (1+4s \, (s+1)^3)^{1/2} \, (1+s)^{-2}$. This completes the proof of Lemma \ref{model.surface}. \\
In the remainder of this section, we consider an $(\hat{\alpha},\hat{\delta},\varepsilon,L)$-neck $N$ of size $1$, which is contained in a closed, embedded, mean convex surface $M \subset \mathbb{R}^3$. It is understood that $\varepsilon$ is much smaller than $\hat{\delta}$. By definition, we can find a simple closed, convex curve $\Gamma$ with the property that $\text{\rm dist}_{C^{20}}(N,\Gamma \times [-L,L]) \leq \varepsilon$. Moreover, the curve $\Gamma$ is $\frac{1}{1+\hat{\delta}}$-noncollapsed, and the derivatives of the geodesic curvature of $\Gamma$ satisfy $\sum_{l=1}^{18} |\nabla^l \kappa| \leq \frac{1}{100}$ at each point on $\Gamma$. Furthermore, there exists a point on $\Gamma$ where the geodesic curvature $\kappa$ is equal to $1$.
Since $\text{\rm dist}_{C^{20}}(N,\Gamma \times [-L,L]) \leq \varepsilon$, we can find a collection of curves $\Gamma_s$ such that
\[\{(\gamma_s(t),s): s \in [-(L-1),L-1], \, t \in [0,1]\} \subset N\]
and
\[\sum_{k+l \leq 20} \Big | \frac{\partial^k}{\partial s^k} \, \frac{\partial^l}{\partial t^l} (\gamma_s(t) - \gamma(t)) \Big | \leq O(\varepsilon).\]
Here, we have used the notation $\Gamma = \{\gamma(t): t \in [0,1]\}$ and $\Gamma_s = \{\gamma_s(t): t \in [0,1]\}$.
The following lemma is analogous to Proposition 3.17 in \cite{Huisken-Sinestrari3}:
\begin{lemma}[cf. Huisken-Sinestrari \cite{Huisken-Sinestrari3}, Proposition 3.17]
\label{aux}
Consider a bended surface of the form
\[\tilde{N} = \{((1-u(s)) \, \gamma_s(t),s): s \in (0,\Lambda^{\frac{1}{4}}], \, t \in [0,1]\},\]
where $|u|+|u'|+|u''| \leq \frac{1}{10}$ everywhere.
Then we have the pointwise estimates
\[\tilde{\lambda}_1(s,t) \geq \lambda_1(s,t) + c_0 \, u''(s) - \frac{1}{c_0} \, (|u(s)|+|u'(s)|)\]
and
\[\tilde{H}(s,t) \geq H(s,t) + c_0 \, u''(s) - \frac{1}{c_0} \, (|u(s)|+|u'(s)|),\]
where $c_0>0$ is a universal constant.
\end{lemma}
It will be convenient to translate the neck $N$ in space so that the center of mass of $\Gamma$ is at the origin. Using the curve shortening flow, we can construct a homotopy $\tilde{\gamma}_r(t)$, $(r,t) \in [0,1] \times [0,1]$, with the following properties:
\begin{itemize}
\item $\tilde{\gamma}_r(t)=\gamma(t)$ for $r \in [0,\frac{1}{4}]$.
\item $\tilde{\gamma}_r(t)=(\cos (2\pi t),\sin(2\pi t))$ for $r \in [\frac{1}{2},1]$.
\item For each $r \in [0,1]$, the curve $\tilde{\Gamma}_r$ is $\frac{1}{1+\hat{\delta}}$-noncollapsed.
\item We have $\sup_{(r,t) \in [0,1] \times [0,1]} |\frac{\partial}{\partial r} \tilde{\gamma}_r(t)| + |\frac{\partial^2}{\partial r \, \partial t} \tilde{\gamma}_r(t)| + |\frac{\partial^2}{\partial r^2} \tilde{\gamma}_r(t)| \leq \omega(\hat{\delta})$, where $\omega(\hat{\delta}) \to 0$ as $\hat{\delta} \to 0$.
\end{itemize}
Moreover, let $\chi: \mathbb{R} \to \mathbb{R}$ be a smooth cutoff function such that $\chi = 1$ on $(-\infty,1]$ and $\chi=0$ on $[2,\infty)$. We next define a surface $\tilde{F}_\Lambda: [-L,\Lambda] \times [0,1] \to \mathbb{R}^3$ by
\[\tilde{F}_\Lambda(s,t) = \begin{cases} (\gamma_s(t),s) & \text{\rm for $s \in [-(L-1),0]$} \\ ((1-e^{-\frac{4\Lambda}{s}}) \, \gamma_s(t),s) & \text{\rm for $s \in (0,\Lambda^{\frac{1}{4}}]$} \\ ((1-e^{-\frac{4\Lambda}{s}}) \, (\chi(s/\Lambda^{\frac{1}{4}}) \, \gamma_s(t)+(1-\chi(s/\Lambda^{\frac{1}{4}})) \, \gamma(t)),s) & \text{\rm for $s \in (\Lambda^{\frac{1}{4}},2 \, \Lambda^{\frac{1}{4}}]$} \\ ((1-e^{-\frac{4\Lambda}{s}}) \, \tilde{\gamma}_{s/\Lambda}(t),s) & \text{\rm for $s \in (2 \, \Lambda^{\frac{1}{4}},\Lambda]$.} \end{cases}\]
It is clear that $\tilde{F}_\Lambda$ is smooth. Moreover, $\tilde{F}_\Lambda$ is axially symmetric for $s \geq \frac{\Lambda}{2}$. \\
\begin{lemma}
\label{convexity.of.cap}
We can find real numbers numbers $\delta_1$, $\Lambda_1$, and a function $E(\Lambda)$ such that the following statements hold:
\begin{itemize}
\item Suppose that $\hat{\delta} < \delta_1$, $\frac{L}{1000} \geq \Lambda \geq \Lambda_1$, and $\varepsilon \leq E(\Lambda)$. Then, for each point $(s,t) \in (0,\Lambda^{\frac{1}{4}}] \times [0,1]$, the mean curvature of $\tilde{F}_\Lambda$ at $(s,t)$ is greater than the mean curvature of the original neck at $(s,t)$, and the smallest curvature eigenvalue of $\tilde{F}_\Lambda$ is greater than the smallest curvature eigenvalue of the original neck at $(s,t)$.
\item Suppose that $\hat{\delta} < \delta_1$, $\frac{L}{1000} \geq \Lambda \geq \Lambda_1$, and $\varepsilon \leq E(\Lambda)$. Then, for each point $(s,t) \in (\Lambda^{\frac{1}{4}},2 \, \Lambda^{\frac{1}{4}}] \times [0,1]$, the surface $\tilde{F}_\Lambda$ is strictly convex at $(s,t)$.
\item Suppose that $\hat{\delta} < \delta_1$, $\frac{L}{1000} \geq \Lambda \geq \Lambda_1$, and $\varepsilon \leq E(\Lambda)$. Then, for each point $(s,t) \in (2 \, \Lambda^{\frac{1}{4}},\Lambda] \times [0,1]$, the surface $\tilde{F}_\Lambda$ is strictly convex at $(s,t)$.
\end{itemize}
\end{lemma}
\textbf{Proof.}
We begin with the first statement. Let $\tilde{\lambda}_1$ denote the smallest curvature eigenvalue of the bended surface $\tilde{F}_\Lambda$ and let $\lambda_1$ be the smallest curvature eigenvalue of the original neck. Similarly, we denote by $\tilde{H}$ the mean curvature of the bended surface and by $H$ the mean curvature of the original neck. By choosing $\Lambda$ sufficiently large, we can arrange that the function $u(s) = e^{-\frac{4\Lambda}{s}}$ satisfies
\[u''(s) \geq \frac{1}{c_0^2} \, (|u(s)|+|u'(s)|)\]
for all $s \in (0,\Lambda^{\frac{1}{4}}]$, where $c_0$ is the constant from Lemma \ref{aux}. Using Lemma \ref{aux}, we conclude that $\tilde{\lambda}_1 \geq \lambda_1$ and $\tilde{H} \geq H$ for all points $(s,t) \in (0,\Lambda^{\frac{1}{4}}] \times [0,1]$. This proves the first statement.
To verify the second statement, we consider a point $(s,t) \in (\Lambda^{\frac{1}{4}},2 \, \Lambda^{\frac{1}{4}}] \times [0,1]$. It is easy to see that $\tilde{h}_{tt} \geq \frac{1}{2}$ and $\langle (\gamma(t),0),\tilde{\nu}(s,t) \rangle \geq \frac{1}{2}$. We next compute
\[\frac{\partial^2}{\partial s^2} \tilde{F}_\Lambda(s,t) = -\Big ( \frac{16\Lambda^2}{s^4} - \frac{8\Lambda}{s^3} \Big ) \, e^{-\frac{4\Lambda}{s}} \, (\gamma(t),0) + O(\varepsilon)\]
and
\begin{align*}
\frac{\partial^2}{\partial s \, \partial t} \tilde{F}_\Lambda(s,t)
&= -\frac{4\Lambda}{s^2} \, e^{-\frac{4\Lambda}{s}} \, \Big ( \chi(s/\Lambda^{\frac{1}{4}}) \, \frac{\partial}{\partial t} \gamma_s(t) + (1-\chi(s/\Lambda^{\frac{1}{4}})) \, \frac{\partial}{\partial t} \gamma(t),0 \Big ) + O(\varepsilon) \\
&= -\frac{4\Lambda}{s^2} \, \frac{e^{-\frac{4\Lambda}{s}}}{1-e^{-\frac{4\Lambda}{s}}} \, \frac{\partial \tilde{F}_\Lambda}{\partial t}(s,t) + O(\varepsilon)
\end{align*}
for $(s,t) \in (\Lambda^{\frac{1}{4}},2 \, \Lambda^{\frac{1}{4}}] \times [0,1]$. From this, we deduce that
\begin{align*}
\tilde{h}_{ss}
&= -\Big \langle \frac{\partial^2}{\partial s^2} \tilde{F}_\Lambda(s,t),\tilde{\nu}(s,t) \Big \rangle \\
&= \Big ( \frac{16\Lambda^2}{s^4} - \frac{8\Lambda}{s^3} \Big ) \, e^{-\frac{4\Lambda}{s}} \, \langle (\gamma(t),0),\tilde{\nu}(s,t) \rangle + O(\varepsilon) \\
&\geq \Big ( \frac{8\Lambda^2}{s^4} - \frac{4\Lambda}{s^3} \Big ) \, e^{-\frac{4\Lambda}{s}} + O(\varepsilon)
\end{align*}
and
\[\tilde{h}_{st} = -\Big \langle \frac{\partial^2}{\partial s \, \partial t} \tilde{F}_\Lambda(s,t),\tilde{\nu}(s,t) \Big \rangle = O(\varepsilon)\]
for $(s,t) \in (\Lambda^{\frac{1}{4}},2 \, \Lambda^{\frac{1}{4}}] \times [0,1]$. Hence, if $\varepsilon$ is small enough (depending on $\Lambda$), then $\tilde{h}_{ss} \, \tilde{h}_{tt} - \tilde{h}_{st}^2 > 0$, and the surface $\tilde{F}_\Lambda$ is strictly convex at $(s,t)$. This completes the proof of the second statement.
To prove the third statement, we consider a point $(s,t) \in (2 \, \Lambda^{\frac{1}{4}},\Lambda] \times [0,1]$. We clearly have $\tilde{h}_{tt} \geq \frac{1}{2}$ and $\langle (\tilde{\gamma}_{s/\Lambda}(t),0),\tilde{\nu}(s,t) \rangle \geq \frac{1}{2}$. We next compute
\[\frac{\partial^2}{\partial s^2} \tilde{F}_\Lambda(s,t) = -\Big ( \frac{16\Lambda^2}{s^4} - \frac{8\Lambda}{s^3} \Big ) \, e^{-\frac{4\Lambda}{s}} \, (\tilde{\gamma}_{s/\Lambda}(t),0) + O(\Lambda^{-2} \, \omega(\hat{\delta}) \, 1_{\{\frac{\Lambda}{4} \leq s \leq \frac{\Lambda}{2}\}})\]
and
\begin{align*}
\frac{\partial^2}{\partial s \, \partial t} \tilde{F}_\Lambda(s,t)
&= -\frac{4\Lambda}{s^2} \, e^{-\frac{4\Lambda}{s}} \, \Big ( \frac{\partial}{\partial t} \tilde{\gamma}_{s/\Lambda}(t),0 \Big ) + O(\Lambda^{-1} \, \omega(\hat{\delta}) \, 1_{\{\frac{\Lambda}{4} \leq s \leq \frac{\Lambda}{2}\}}) \\
&= -\frac{4\Lambda}{s^2} \, \frac{e^{-\frac{4\Lambda}{s}}}{1-e^{-\frac{4\Lambda}{s}}} \, \frac{\partial \tilde{F}_\Lambda}{\partial t}(s,t) + O(\Lambda^{-1} \, \omega(\hat{\delta}) \, 1_{\{\frac{\Lambda}{4} \leq s \leq \frac{\Lambda}{2}\}})
\end{align*}
for $(s,t) \in (2 \, \Lambda^{\frac{1}{4}},\Lambda] \times [0,1]$. From this, we deduce that
\begin{align*}
\tilde{h}_{ss}
&= -\Big \langle \frac{\partial^2}{\partial s^2} \tilde{F}_\Lambda(s,t),\tilde{\nu}(s,t) \Big \rangle \\
&= \Big ( \frac{16\Lambda^2}{s^4} - \frac{8\Lambda}{s^3} \Big ) \, e^{-\frac{4\Lambda}{s}} \, \langle (\tilde{\gamma}_{s/\Lambda}(t),0),\tilde{\nu}(s,t) \rangle + O(\Lambda^{-2} \, \omega(\hat{\delta}) \, 1_{\{\frac{\Lambda}{4} \leq s \leq \frac{\Lambda}{2}\}}) \\
&\geq \Big ( \frac{8\Lambda^2}{s^4} - \frac{4\Lambda}{s^3} \Big ) \, e^{-\frac{4\Lambda}{s}} + O(\Lambda^{-2} \, \omega(\hat{\delta}) \, 1_{\{\frac{\Lambda}{4} \leq s \leq \frac{\Lambda}{2}\}})
\end{align*}
and
\[\tilde{h}_{st} = -\Big \langle \frac{\partial^2}{\partial s \, \partial t} \tilde{F}_\Lambda(s,t),\tilde{\nu}(s,t) \Big \rangle = O(\Lambda^{-1} \, \omega(\hat{\delta}) \, 1_{\{\frac{\Lambda}{4} \leq s \leq \frac{\Lambda}{2}\}})\]
for $(s,t) \in (2 \, \Lambda^{\frac{1}{4}},\Lambda] \times [0,1]$. Hence, if $\hat{\delta}$ is sufficiently small, then $\tilde{h}_{ss} \, \tilde{h}_{tt} - \tilde{h}_{st}^2 > 0$, and the surface $\tilde{F}_\Lambda$ is strictly convex at $(s,t)$. This completes the proof of Lemma \ref{convexity.of.cap}. \\
Since the surface $\tilde{F}_\Lambda$ is axially symmetric in the region $\{\frac{\Lambda}{2} \leq s \leq \Lambda\}$, we may glue $\tilde{F}_\Lambda$ to a scaled copy of the axially symmetric cap constructed in Lemma \ref{model.surface}. We briefly sketch how this can be done. Let us fix a smooth, convex, even function $\Phi: \mathbb{R} \to \mathbb{R}$ such that $\Phi(z) = |z|$ for $|z| \geq \frac{1}{100}$. For $\Lambda$ very large, we define
\[a = 1-e^{-4}+\frac{1}{3} \, (1-e^{-4})^2 \, \Lambda^{-\frac{1}{4}}\]
and
\begin{align*}
v_\Lambda(s)
&= 1-e^{-\frac{4\Lambda}{s}} + a \, \sqrt{\frac{\Lambda+2\,\Lambda^{\frac{1}{4}}-s}{a+\Lambda+2\,\Lambda^{\frac{1}{4}}-s}} \\
&- \Lambda^{-\frac{1}{4}} \, \Phi \bigg ( \Lambda^{\frac{1}{4}} \, \Big ( 1-e^{-\frac{4\Lambda}{s}} - a \, \sqrt{\frac{\Lambda+2\,\Lambda^{\frac{1}{4}}-s}{a+\Lambda+2\,\Lambda^{\frac{1}{4}}-s}} \Big ) \bigg )
\end{align*}
for $s \in [\Lambda,\Lambda+\Lambda^{\frac{1}{4}}]$. Since the functions $s \mapsto 1-e^{-\frac{4\Lambda}{s}}$ and $s \mapsto a \, \sqrt{\frac{\Lambda+2 \, \Lambda^{\frac{1}{4}}-s}{a+\Lambda+2\,\Lambda^{\frac{1}{4}}-s}}$ are concave, the function $v_\Lambda$ is concave as well. Moreover, if $\Lambda$ is sufficiently large, then we have $v_\Lambda(s)=2 \, (1-e^{-\frac{4\Lambda}{s}})$ in a neighborhood of the point $s=\Lambda$, and $v_\Lambda(s)=2a \, \sqrt{\frac{\Lambda+2\,\Lambda^{\frac{1}{4}}-s}{a+\Lambda+2\,\Lambda^{\frac{1}{4}}-s}}$ in a neighborhood of the point $s=\Lambda+\Lambda^{\frac{1}{4}}$. We now extend $\tilde{F}_\Lambda$ to the region $[-(L-1),\Lambda+2\Lambda^{\frac{1}{4}}] \times [0,1]$ by putting
\[\tilde{F}_\Lambda(s,t) = \Big ( \frac{1}{2} \, v_\Lambda(s) \, \cos(2\pi t),\frac{1}{2} \, v_\Lambda(s) \, \sin(2\pi t),s \Big )\]
for $s \in (\Lambda,\Lambda+\Lambda^{\frac{1}{4}}]$ and
\[\tilde{F}_\Lambda(s,t) = \Big ( a \, \sqrt{\frac{\Lambda+2 \, \Lambda^{\frac{1}{4}} - s}{a+\Lambda+2\,\Lambda^{\frac{1}{4}}-s}} \, \cos(2\pi t),a \, \sqrt{\frac{\Lambda+2 \, \Lambda^{\frac{1}{4}} - s}{a+\Lambda+2\,\Lambda^{\frac{1}{4}}-s}} \, \sin(2\pi t),s \Big )\]
for $s \in (\Lambda+\Lambda^{\frac{1}{4}},\Lambda+2\Lambda^{\frac{1}{4}}]$. It is straightforward to verify that the resulting surface is smooth satisfies the curvature bounds $\frac{1}{2} \leq H \leq 10$.
\begin{lemma}
The surface $\tilde{F}_\Lambda$ is convex in the region $(\Lambda,\Lambda+2\Lambda^{\frac{1}{4}}] \times [0,1]$.
\end{lemma}
\textbf{Proof.}
Since the function $v_\Lambda$ is concave on the interval $[\Lambda,\Lambda+\Lambda^{\frac{1}{4}}]$, we conclude that $\tilde{F}_\Lambda$ is a convex surface for $(s,t) \in (\Lambda,\Lambda+\Lambda^{\frac{1}{4}}] \times [0,1]$. Moreover, it follows from Lemma \ref{model.surface} that the surface $\tilde{F}_\Lambda$ is convex for $(s,t) \in (\Lambda+\Lambda^{\frac{1}{4}},\Lambda+2\Lambda^{\frac{1}{4}}]$. \\
In the next step, we show that in the surgically modified region the inscribed radius is at least $\frac{1}{(1+\delta) \, H}$ and the outer radius is at least $\frac{\alpha}{H}$.
\begin{proposition}
\label{noncollapsing.preserved.under.surgery}
Given any number $\hat{\alpha} > \alpha$, we can find a number $\delta_2$ with the following property. Suppose that we are given a pair of real numbers $\delta$ and $\hat{\delta}$ such that $\hat{\delta} < \delta < \delta_2$. Then there exist real numbers $\bar{\varepsilon}$ and $\Lambda_2$ such that the surface $\tilde{F}_\Lambda$ is $\frac{1}{1+\delta}$-noncollapsed whenever $\varepsilon \leq \bar{\varepsilon}$ and $\frac{L}{1000} \geq \Lambda \geq \Lambda_2$. Furthermore, if $\varepsilon \leq \bar{\varepsilon}$ and $\frac{L}{1000} \geq \Lambda \geq \Lambda_2$, then the outer radius is at least $\frac{\alpha}{H}$ at each point on $\tilde{F}_\Lambda$.
\end{proposition}
\textbf{Proof.}
We first establish the bound for the inscribed radius. It follows from Lemma \ref{model.surface} that the inscribed radius of $\tilde{F}_\Lambda$ is at least $\frac{1}{(1+\delta) \, H}$ at each point in the region $(\Lambda+\Lambda^{\frac{1}{4}},\Lambda+2\Lambda^{\frac{1}{4}}]$. Consider now a point $(s_0,t_0) \in [-(L-1),\Lambda+\Lambda^{\frac{1}{4}}] \times [0,1]$. If $\Lambda$ is large, then we can approximate the map $\tilde{F}_\Lambda$ near the point $(s_0,t_0)$ by a cylinder whose cross-section is a simple closed, convex curve. Furthermore, the noncollapsing constant of the cross section is $\frac{1}{1+\hat{\delta}}-O(\varepsilon)$ or better. Since $\hat{\delta} < \delta$, we conclude that the surface $\tilde{F}_\Lambda$ is $\frac{1}{1+\delta}$-noncollapsed if $\Lambda$ is sufficiently large and $\varepsilon$ is sufficiently small.
It remains to prove the bound for the outer radius. Let $\tilde{N}$ denote the image of the map $\tilde{F}_\Lambda: (-(,\Lambda+2\Lambda^{\frac{1}{4}}] \times [0,1] \to \mathbb{R}^3$. By assumption, the region $\{x + a \, \nu(x): x \in N, \, a \in (0,2\hat{\alpha})\}$ is disjoint from $M \setminus N$. Since the surface $\tilde{N} \setminus N$ lies inside the original neck $N$, it follows that the region $\{x + a \, \tilde{\nu}(x): x \in \tilde{N} \setminus N, \, a \in (0,2\hat{\alpha})\}$ is disjoint from $M \setminus N$. Consequently, for each point $x \in \tilde{N} \setminus N$, we can find a ball of radius $\hat{\alpha}$ which touches $\tilde{N}$ at the point $x$ from the outside, and which is disjoint from $M \setminus N$. On the other hand, if $\hat{\delta}$ and $\varepsilon$ are sufficiently small and $\Lambda$ is sufficiently large, then the mean curvature of the surface $\tilde{N} \setminus N$ is greater than $\frac{\alpha}{\hat{\alpha}}$ everywhere. Putting these facts together, we conclude that the outer radius is at least $\hat{\alpha} > \frac{\alpha}{H}$ at each point in $\tilde{N} \setminus N$. This completes the proof of Proposition \ref{noncollapsing.preserved.under.surgery}. \\
We note that our surgery procedure always produces an embedded surface. Finally, it is clear from the construction that the resulting cap is at least of class $C^5$ with uniform bounds independent of the surgery parameters $\hat{\alpha}$, $\hat{\delta}$, $\varepsilon$, $L$, and $\Lambda$.
\section{Proof of Proposition \ref{separation.of.surgery.regions}}
By assumption, the point $x_0$ lies in the surgically modified region of $M_{t_0+}$. Hence, the surface $M_{t_0-}$ must have contained an $(\hat{\alpha},\hat{\delta},\varepsilon,L)$-neck of size $r \in [\frac{1}{2H_1},\frac{2}{H_1}]$. Let us denote this neck by $N$. At time $t_0$ the neck $N$ is replaced by a capped-off neck $\tilde{N}$. More precisely, suppose that the original neck $N$ satisfies
\[\{(\gamma_s(t),s): s \in [-(L-1),L-1], \, t \in [0,1]\} \subset r^{-1} \, N.\]
Then the surface $\tilde{N}$ satisfies
\[N \cap \{x \in \mathbb{R}^3: \langle x,e_3 \rangle \leq 0\} \subset \tilde{N}\]
and
\[\tilde{N} \subset \{x \in \mathbb{R}^3: \langle x,e_3 \rangle \leq 4\Lambda \, H_1^{-1}\}.\]
Since the point $x_0$ lies in the surgically modified part of $\tilde{N}$, we have $\langle x_0,e_3 \rangle \geq 0$.
By assumption, the outer radius of $M_{t_0+}$ is at least $\frac{\alpha}{H}$ everywhere. Moreover, we have $H \leq 100 \, H_1$ at each point on $\tilde{N}$. Therefore, for each point $x \in \tilde{N}$, the outer radius is at least $\frac{\alpha}{100} \, H_1^{-1}$. Hence, if we denote by $\nu$ the outward-pointing unit normal vector field to $\tilde{N}$, then the set
\[E = \{x + a \, \nu(x): x \in \tilde{N}, \, a \in (0,\frac{\alpha}{100} \, H_1^{-1})\}\]
is disjoint from the region $\Omega_{t_0+}$.
By assumption, the point $x_1 \in M_{t_1+}$ lies in the surgically modified region at time $t_1$. This region was created by performing surgery on an $(\hat{\alpha},\hat{\delta},\varepsilon,L)$-neck in $M_{t_1-}$. This neck has length at least $\frac{L}{2} \, H_1^{-1} \geq 50 \, \Lambda \, H_1^{-1}$. Hence, we can find two points $y,z \in \mathbb{R}^3$ such that $|y-z| = 40 \, \Lambda \, H_1^{-1}$, $|\frac{y+z}{2}-x_1| \leq \frac{\alpha}{1000} \, H_1^{-1}$, and the line segment joining $y$ and $z$ is contained in the region $\Omega_{t_1-}$. Since $\Omega_{t_1-} \subset \Omega_{t_0+}$ is disjoint from $E$, the line segment joining $y$ and $z$ cannot intersect the set $E$.
Now, if $|\frac{y+z}{2}-x_0| \leq \frac{\alpha}{500} \, H_1^{-1}$, then it is easy to see that the line segment joining $y$ and $z$ must intersect the set $E$. Therefore, we have $|\frac{y+z}{2}-x_0| > \frac{\alpha}{500} \, H_1^{-1}$, hence $|x_1-x_0| > \frac{\alpha}{1000} \, H_1^{-1}$. This completes the proof of Proposition \ref{separation.of.surgery.regions}.
\section{Proof of Proposition \ref{consequence.of.pseudolocality}}
Let $\beta_0$ be chosen as in Theorem \ref{pseudolocality}. Let us fix a positive number $\beta_1 \in (0,\frac{\alpha}{8000})$ such that the following holds: Suppose that $N$ is an $(\hat{\alpha},\hat{\delta},\varepsilon,L)$-neck of size $r \in [\frac{1}{2H_1},\frac{2}{H_1}]$. Moreover, let $\tilde{N}$ denote the capped-off neck obtained by performing a $\Lambda$-surgery on $N$. Then, for each point $x_0$ in the surgically modified region, the dilated surface $(\beta_1^{-1} \, H_1 \, (\tilde{N} - x_0)) \cap B_4(0)$ can be expressed as the graph of function which has $C^4$-norm less than $\beta_0$. In view of the construction of the cap in Section \ref{construction.of.cap}, we can choose the constant $\beta_1$ in such a way that $\beta_1$ depends only on the noncollapsing constant $\alpha$, but not on the exact choice of the surgery parameters $\hat{\alpha}$, $\hat{\delta}$, $\varepsilon$, $L$, and $H_1$.
After these preparations, we now complete the proof of Proposition \ref{consequence.of.pseudolocality}. Suppose that $t_0$ is a surgery time and $x_0$ lies in the surgically modified region. By Proposition \ref{separation.of.surgery.regions}, the flow $M_t \cap B_{4\beta_1 \, H_1^{-1}}(x_0)$ is smooth for all times $t > t_0$. Moreover, the surface $(\beta_1^{-1} \, H_1 \, (M_{t_0+} - x_0)) \cap B_4(0)$ is a graph of a function with $C^4$-norm less than $\beta_0$. Hence, Theorem \ref{pseudolocality} implies that
\[\beta_1 \, H_1^{-1} \, |A| + \beta_1^2 \, H_1^{-2} \, |\nabla A| + \beta_1^3 \, H_1^{-3} \, |\nabla^2 A| \leq C\]
for all $t \in (t_0,t_0+\beta_0 \, \beta_1^2 \, H_1^{-2}]$ and all $x \in M_t \cap B_{\beta_1 \, H_1^{-1}}(x_0)$. From this, the assertion follows.
\section{Proof of Proposition \ref{gradient.estimate}}
Let us consider an arbitrary time $t_1 \geq (1000 \, \sup_{M_0} |A|)^{-2}$ and an arbitrary point $x_1 \in M_{t_1}$ for which we want to verify the estimate. To avoid confusion (and without any loss of generality), we will assume that $t_1$ is not itself a surgery time. There are two cases:
\textit{Case 1:} There exists a surgery time $t_0$ and a point $x_0$ such that $|x_1-x_0| \leq \beta_* \, H_1^{-1}$, $0 < t_1-t_0 \leq \beta_* \, H_1^{-2}$, and $x_0$ lies in the surgically modified region at time $t_0+$. Applying Proposition \ref{consequence.of.pseudolocality}, we conclude that
\[H_1^{-1} \, |A| + H_1^{-2} \, |\nabla A| + H_1^{-3} \, |\nabla^2 A| \leq C_*\]
at the point $(x_1,t_1)$. Hence, $|\nabla A| \leq C_* \, (H+H_1)^2$ and $|\nabla^2 A| \leq C_* \, (H+H_1)^3$ at the point $(x_1,t_1)$.
\textit{Case 2:} There does not exist a surgery time $t_0$ and a point $x_0$ such that $|x_1-x_0| \leq \beta_* \, H_1^{-1}$, $0 < t_1-t_0 \leq \beta_* \, H_1^{-2}$, and $x_0$ lies in the surgically modified region at time $t_0+$. In this case, the surfaces $M_t \cap B_{\beta_* \, H_1^{-1}}(x_1)$, $t \in (t_1-\beta_* \, H_1^{-2},t_1]$, form a regular mean curvature flow in the sense of Definition \ref{regular.flow}. Note that, since $t_1 \geq (1000 \, \sup_{M_0} |A|)^{-2}$, we have $t_1-\beta_* \, H_1^{-2} > 0$. Moreover, the ball $B_{\beta_* \, H_1^{-1}}(x_1)$ is contained in the region $\Omega_0$, so the surfaces $M_t$ are outward-minimizing within the ball $B_{\beta_* \, H_1^{-1}}(x_1)$. Hence, Theorem \ref{interior.derivative.estimate} implies that $|\nabla A| \leq B \, (H+H_1)^2$ and $|\nabla^2 A| \leq B \, (H+H_1)^3$ at the point $(x_1,t_1)$. Here, $B$ is a positive constant that depends only on $\beta_*$ and the noncollapsing constant $\alpha$. This completes the proof.
\section{Proof of Proposition \ref{choice.of.delta}}
We next prove some auxiliary results about curves. In the following, we assume that $C_\#$ is the constant in Proposition \ref{gradient.estimate}.
\begin{lemma}
\label{const.curvature}
Let $\Gamma$ be a (possibly non-closed) embedded curve in the plane of class $C^3$ with geodesic curvature $\kappa>0$. Moreover, suppose that the inscribed radius is at least $\frac{1}{\kappa}$ at each point on $\Gamma$. Then $\kappa$ is constant.
\end{lemma}
\textbf{Proof.}
The assumption implies that the function
\[Z(s,t) := \frac{1}{2} \, \kappa(s) \, |\gamma(s)-\gamma(t)|^2 - \langle \gamma(s)-\gamma(t),\nu(s) \rangle\]
is nonnegative for all $s,t$. A straightforward calculation gives
\[\frac{\partial Z}{\partial t}(s,t) \Big |_{s=t} = 0, \quad \frac{\partial^2 Z}{\partial t^2}(s,t) \Big |_{s=t} = 0, \quad \frac{\partial^3 Z}{\partial t^3}(s,t) \Big |_{s=t} = -\frac{d\kappa}{ds}(s).\]
Since $Z$ is nonnegative everywhere, we conclude that $\frac{d\kappa}{ds}(s) = 0$ at each point on $\Gamma$.
\begin{lemma}
\label{sup}
Let $\Gamma_j$ be a sequence of (possibly non-closed) embedded curves in the plane with the property that $\kappa > 0$, $|\frac{d\kappa}{ds}| \leq C_\# \, (\kappa+2\Theta)^2$, and $|\frac{d^2\kappa}{ds^2}| \leq C_\# \, (\kappa+2\Theta)^3$. Moreover, suppose that the inscribed radius is at least $\frac{1}{(1+\frac{1}{j}) \, \kappa}$ at each point on $\Gamma_j$. Finally, we assume that $L(\Gamma_j) \leq 4\pi$ and $\kappa(p_j)=1$ for some point $p_j \in \Gamma_j$. Then $\sup_{\Gamma_j} \kappa \to 1$ as $j \to \infty$.
\end{lemma}
\textbf{Proof.}
Suppose that there exists a real number $a>0$ such that $\sup_{\Gamma_j} \kappa \geq 1+2a$ for $j$ large. We can find a segment $\tilde{\Gamma}_j \subset \Gamma_j$ such that the geodesic curvature increases from $1+a$ to $1+2a$ along $\tilde{\Gamma}_j$. Using our assumptions, we obtain $\limsup_{j \to \infty} \sup_{\tilde{\Gamma}_j} |\frac{d\kappa}{ds}| < \infty$ and $\limsup_{j \to \infty} \sup_{\tilde{\Gamma}_j} |\frac{d^2\kappa}{ds^2}| < \infty$. Since $\kappa$ varies between $1+a$ and $1+2a$ along $\tilde{\Gamma}_j$, we must have $\liminf_{j \to \infty} L(\tilde{\Gamma}_j) > 0$. On the other hand, we have $L(\tilde{\Gamma}_j) \leq L(\Gamma_j) \leq 4\pi$. Hence, after passing to a subsequence, the curves $\tilde{\Gamma}_j$ converge in $C^3$ to a curve $\hat{\Gamma}$. The geodesic curvature of the limiting curve $\hat{\Gamma}$ increases from $1+a$ to $1+2a$ as we travel along the curve $\hat{\Gamma}$. Finally, at each point on $\hat{\Gamma}$, the inscribed radius is at least $\frac{1}{\hat{\kappa}}$, where $\hat{\kappa}$ denotes the geodesic curvature of $\hat{\Gamma}$. By Lemma \ref{const.curvature}, $\hat{\kappa}$ is constant. This contradicts the fact that $\hat{\kappa}$ varies between $1+a$ and $1+2a$. \\
\begin{lemma}
\label{inf}
Let $\Gamma_j$ be a sequence of (possibly non-closed) embedded curves in the plane with the property that $\kappa > 0$, $|\frac{d\kappa}{ds}| \leq C_\# \, (\kappa+2\Theta)^2$, and $|\frac{d^2\kappa}{ds^2}| \leq C_\# \, (\kappa+2\Theta)^3$. Moreover, suppose that the inscribed radius is at least $\frac{1}{(1+\frac{1}{j}) \, \kappa}$ at each point on $\Gamma_j$. Finally, we assume that $L(\Gamma_j) \leq 4\pi$ and $\kappa(p_j)=1$ for some point $p_j \in \Gamma_j$. Then $\inf_{\Gamma_j} \kappa \to 1$ as $j \to \infty$.
\end{lemma}
\textbf{Proof.}
Suppose that there exists a real number $a>0$ such that $\inf_{\Gamma_j} \kappa \leq 1-2a$ for $j$ large. We can find a segment $\tilde{\Gamma}_j \subset \Gamma_j$ such that the geodesic curvature decreases from $1-a$ to $1-2a$ along $\tilde{\Gamma}_j$. Using our assumptions, we obtain $\limsup_{j \to \infty} \sup_{\tilde{\Gamma}_j} |\frac{d\kappa}{ds}| < \infty$ and $\limsup_{j \to \infty} \sup_{\tilde{\Gamma}_j} |\frac{d^2\kappa}{ds^2}| < \infty$. Since $\kappa$ varies between $1-a$ and $1-2a$ along $\tilde{\Gamma}_j$, we must have $\liminf_{j \to \infty} L(\tilde{\Gamma}_j) > 0$. On the other hand, we have $L(\tilde{\Gamma}_j) \leq L(\Gamma_j) \leq 4\pi$. Hence, after passing to a subsequence, the curves $\tilde{\Gamma}_j$ converge in $C^3$ to a curve $\hat{\Gamma}$. The geodesic curvature of the limiting curve $\hat{\Gamma}$ decreases from $1-a$ and $1-2a$ as we travel along the curve $\hat{\Gamma}$. Finally, at each point on $\hat{\Gamma}$, the inscribed radius is at least $\frac{1}{\hat{\kappa}}$, where $\hat{\kappa}$ denotes the geodesic curvature of $\hat{\Gamma}$. By Lemma \ref{const.curvature}, $\hat{\kappa}$ is constant. This contradicts the fact that $\hat{\kappa}$ varies between $1-a$ and $1-2a$. \\
\begin{proposition}
\label{closing.up}
Let $\Gamma_j$ be a sequence of (possibly non-closed) embedded curves in the plane with the property that $\kappa > 0$, $|\frac{d\kappa}{ds}| \leq C_\# \, (\kappa+2\Theta)^2$, and $|\frac{d^2\kappa}{ds^2}| \leq C_\# \, (\kappa+2\Theta)^3$. Moreover, suppose that the inscribed radius is at least $\frac{1}{(1+\frac{1}{j}) \, \kappa}$ at each point on $\Gamma_j$, and the outer radius is at least $\frac{\alpha}{\kappa}$ at each point on $\Gamma_j$. Finally, we assume that $\kappa(p_j)=1$ for some point $p_j \in \Gamma_j$. Then $L(\Gamma_j) < 3\pi$ for $j$ large, and we have $\lim_{j \to \infty} \sup_{\Gamma_j} |\kappa-1| = 0$.
\end{proposition}
\textbf{Proof.}
We first show that $L(\Gamma_j) < 3\pi$ for $j$ large. Suppose by contradiction that $L(\Gamma_j) \geq 3\pi$ for all $j$. By shortening $\Gamma_j$ if necessary, we can arrange that $L(\Gamma_j) = 3\pi$ for all $j$. Let $\gamma_j: [0,3\pi] \to \mathbb{R}^2$ be a parametrization of $\Gamma_j$ by arclength. It follows from Lemma \ref{sup} and Lemma \ref{inf} that the geodesic curvature of $\Gamma_j$ is close to $1$ when $j$ is sufficiently large. This implies that $\gamma_j(2\pi) - \gamma_j(0) \to 0$ as $j \to \infty$. Let us pick a sequence of numbers $s_j \in [0,3\pi]$ such that $s_j \to 2\pi$ as $j \to \infty$ and the function $s \mapsto |\gamma_j(s) - \gamma_j(0)|^2$ has a local minimum at $s_j$. Then the vector $\gamma_j(s_j)-\gamma_j(0)$ is parallel to $\nu_j(s_j)$. Consequently, we have
\[|\langle \gamma_j(s_j)-\gamma_j(0),\nu_j(s_j) \rangle| = |\gamma_j(s_j)-\gamma_j(0)|.\]
On the other hand, we know that the inscribed radius and the outer radius of $\Gamma_j$ are at least $\frac{\alpha}{\kappa}$. This implies
\[\frac{1}{2} \, \kappa_j(s_j) \, |\gamma_j(s_j)-\gamma_j(0)|^2 \geq \alpha \, |\langle \gamma_j(s_j)-\gamma_j(0),\nu_j(s_j) \rangle|.\]
Putting these facts together, we obtain
\[\frac{1}{2} \, \kappa_j(s_j) \, |\gamma_j(s_j)-\gamma_j(0)| \geq \alpha.\]
But $\kappa_j(s_j) \to 1$ and $|\gamma_j(s_j)-\gamma_j(0)| \to 0$ as $j \to \infty$, so we arrive at a contradiction. Consequently, we must have $L(\Gamma_j) < 3\pi$ when $j$ is sufficiently large. Using Lemma \ref{sup} and Lemma \ref{inf}, we obtain $\lim_{j \to \infty} \sup_{\Gamma_j} \kappa = 1$ and $\lim_{j \to \infty} \inf_{\Gamma_j} \kappa = 1$. This completes the proof. \\
\begin{corollary}
\label{closing.up.2}
We can find a number $\delta>0$ with the following property: Suppose that $\Gamma$ is a
(possibly non-closed) embedded curve in the plane with the property that $\kappa > 0$, $|\frac{d\kappa}{ds}| \leq C_\# \, (\kappa+2\Theta)^2$, and $|\frac{d^2\kappa}{ds^2}| \leq C_\# \, (\kappa+2\Theta)^3$. Moreover, suppose that the inscribed radius is at least $\frac{1}{(1+\delta) \, \kappa}$ at each point on $\Gamma$, and the outer radius is at least $\frac{\alpha}{\kappa}$ at each point on $\Gamma$. Finally, we assume that $\kappa=1$ at some point $p \in \Gamma$. Then $L(\Gamma) < 3\pi$ and $\sup_\Gamma |\kappa-1| \leq \frac{1}{100}$.
\end{corollary}
Note that the constant $\delta$ will depend only on the constants $\alpha$ and $C_\#$, which have already been chosen.
In the following, we define $\theta_0 = 10^{-6} \, \min\{\alpha,\frac{1}{C_\# \, \Theta^3}\}$.
\begin{proposition}
\label{smoothing}
We can choose $\delta$ small enough so that the following holds: Consider a family of simple closed, convex curves $\Gamma_t$, $t \in (-2\theta_0,0]$, in the plane which evolve by curve shortening flow. Assume that, for each $t \in (-2\theta_0,0]$, the curve $\Gamma_t$ satisfies the derivative estimates $|\frac{d\kappa}{ds}| \leq C_\# \, (\kappa+2\Theta)^2$ and $|\frac{d^2\kappa}{ds^2}| \leq C_\# \, (\kappa+2\Theta)^3$. Moreover, we assume that the inscribed radius is at least $\frac{1}{(1+\delta) \, \kappa}$ at each point on $\Gamma_t$, and the outer radius is at least $\frac{\alpha}{\kappa}$ at each point on $\Gamma_t$. Finally, we assume that the geodesic curvature of $\Gamma_0$ is equal to $1$ somewhere. Then the curve $\Gamma_0$ satisfies $\sum_{l=1}^{18} |\nabla^l \kappa| \leq \frac{1}{1000}$. Moreover, we have $\sup_{\Gamma_{-\theta_0}} \kappa \leq 1-\frac{\theta_0}{4}$.
\end{proposition}
\textbf{Proof.}
Suppose that the assertion is false, and consider a sequence of counterexamples. These counterexamples converge to a smooth solution of the curve shortening flow which is defined for $t \in (-2\theta_0,0]$. The limiting solution is a family of homothetically shrinking circles. This gives a contradiction. \\
Proposition \ref{choice.of.delta} follows by combining Corollary \ref{closing.up} and Proposition \ref{smoothing}.
\section{Proof of Proposition \ref{choice.of.hat.delta}}
We again argue by contradiction. Let us fix $\theta_0$ and $\delta$ as above, and suppose that there is no real number $\hat{\delta} \in (0,\delta)$ for which the conclusion of Proposition \ref{choice.of.delta} holds. By taking a sequence of counterexamples and passing to the limit, we obtain a smooth solution $\Gamma_t$, $t \in (-2\theta_0,0]$, to the curve shortening flow with the property that $\sup_{\Gamma_t} \frac{\mu}{\kappa} \leq 1+\delta$ for each $t \in (-2\theta_0,0]$ and $\sup_{\Gamma_0} \frac{\mu}{\kappa} = 1+\delta$. (As usual, $\mu$ denotes the reciprocal of the inscribed radius and $\kappa$ denotes the geodesic curvature.) The geodesic curvature satisfies the evolution equation
\[\frac{\partial}{\partial t} \kappa = \Delta \kappa + \kappa^3.\]
Moreover, $\mu$ satisfies the inequality
\[\frac{\partial}{\partial t} \mu \leq \Delta \mu + \kappa^2 \, \mu - \frac{2}{\mu-\kappa} \, |\nabla \mu|^2\]
on the set $\{\mu>\kappa\}$, where $\Delta \mu$ is interpreted in the sense of distributions (see \cite{Brendle2}, Proposition 2.3). In particular, the function $(1+\delta) \, \kappa - \mu$ is nonnegative and satisfies the inequality
\[\frac{\partial}{\partial t} ((1+\delta) \, \kappa - \mu) \geq \Delta ((1+\delta) \, \kappa - \mu) + \kappa^2 \, ((1+\delta) \, \kappa - \mu) + \frac{2}{\mu-\kappa} \, |\nabla \mu|^2\]
on the set $\{\mu>\kappa\}$. Since $\inf_{\Gamma_0} ((1+\delta) \, \kappa - \mu) = 0$, the function $(1+\delta) \, \kappa - \mu$ vanishes identically by the strict maximum principle. This implies $\nabla \mu = 0$, hence $\nabla \kappa = 0$. Therefore, our solution is a family of shrinking circles. In that case, we have $\mu = \kappa$, which contradicts the fact that $(1+\delta) \, \kappa - \mu = 0$.
\section{Proof of Proposition \ref{cylindrical.estimate}}
Let $\delta$ and $\hat{\delta}$ be chosen such that Theorem \ref{choice.of.delta} holds. In the following, we put
\[f_{\delta,\sigma} = H^{\sigma-1} \, (\mu-(1+\delta) \, H) - C_1(\delta/2),\]
where $\mu$ denotes the reciprocal of the inscribed radius and $C_1(\delta)$ is the constant in the convexity estimate of Huisken and Sinestrari (see Proposition \ref{huisken.sinestrari.convexity} above). The following result was established in \cite{Brendle2}:
\begin{proposition}[cf. \cite{Brendle2}]
\label{Lp.bound.for.f}
We can find a constant $c_0$, depending only on $\delta$ and the initial data, with the following property: if $p \geq \frac{1}{c_0}$ and $\sigma \leq c_0 \, p^{-\frac{1}{2}}$, then we have
\[\frac{d}{dt} \bigg ( \int_{M_t} f_{\delta,\sigma,+}^p \bigg ) \leq C \, \sigma \, p \int_{M_t} f_{\delta,\sigma,+}^p + \sigma \, p \, K_0^p \int_{M_t} |A|^2\]
except if $t$ is a surgery time. Here, $C$ and $K_0$ depend only on $\delta$ and the initial data, but not on $\sigma$ and $p$.
\end{proposition}
We next analyze the behavior of $f_{\delta,\sigma}$ under surgery.
\begin{lemma}
\label{behavior.under.surgery}
The integral $\int_{M_t} f_{\delta,\sigma,+}^p$ does not increase under surgery.
\end{lemma}
\textbf{Proof.}
Consider a surgery time $t_0$. By assumption, each surgery is being performed on an $(\hat{\alpha},\hat{\delta},\varepsilon,L)$-neck with $\varepsilon \leq \bar{\varepsilon}$ and $L \geq 1000 \, \Lambda$. Hence, Theorem \ref{properties.of.surgery} implies that the inscribed radius of $M_{t_0+}$ is at least $\frac{1}{(1+\delta) \, H}$ in the surgically modified region. In other words, we have $f_{\delta,\sigma} \leq 0$ in the surgically modified region of $M_{t_0+}$. Consequently, we have $\int_{M_{t_0+}} f_{\delta,\sigma,+}^p \leq \int_{M_{t_0-}} f_{\delta,\sigma,+}^p$, as claimed. \\
Combining Proposition \ref{Lp.bound.for.f} and Lemma \ref{behavior.under.surgery}, we can draw the following conclusion:
\begin{proposition}
We can find a constant $c_0$, depending only on $\delta$ and the initial data, with the following property: if $p \geq \frac{1}{c_0}$ and $\sigma \leq c_0 \, p^{-\frac{1}{2}}$, then we have
\[\int_{M_t} f_{\delta,\sigma,+}^p \leq C\]
for all $t$. Here, $C$ is a constant that depends on $\delta$, $\sigma$, $p$, and the initial data.
\end{proposition}
We can now use Stampacchia iteration to show that $f_{\delta,\sigma} \leq C$, where $\sigma$ and $C$ depend only on $\delta$ and the initial data. This completes the proof of Proposition \ref{cylindrical.estimate}.
\section{Proof of the Neck Detection Lemma (Version A)}
\label{proof.of.neck.detection.lemma.version.a}
The proof is by contradiction. Suppose that the assertion is false. Then there exists a sequence of flows $\mathcal{M}_j$ and a sequence of points $(p_j,t_j)$ with the following properties:
\begin{itemize}
\item For each $j$, $\mathcal{M}_j$ is a mean curvature flow with surgery satisfying Assumption \ref{a.priori.assumptions}.
\item $H_{\mathcal{M}_j}(p_j,t_j) \geq \max \{j,\frac{H_{1,j}}{\Theta}\}$ and $\frac{\lambda_{1,\mathcal{M}_j}(p_j,t_j)}{H_{\mathcal{M}_j}(p_j,t_j)} \leq \frac{1}{j}$.
\item The neighborhood $\hat{\mathcal{P}}_{\mathcal{M}_j}(p_j,t_j,L_0+4,2\theta_0)$ does not contain surgeries.
\item The point $p_j$ does not lie at the center of an $(\hat{\alpha},\hat{\delta},\varepsilon_0,L_0)$-neck in the surface $M_{t_j,j}$.
\end{itemize}
For each $j$, we put
\begin{align*}
\rho_j
&= \min \Big \{ \inf \{d_{g(t_j)}(p_j,x) \, H_{\mathcal{M}_j}(p_j,t_j): \\
&\hspace{30mm} x \in M_{t_j,j}, \, H_{\mathcal{M}_j}(x,t_j) > 4 \, H_{\mathcal{M}_j}(p_j,t_j)\},L_0+2 \Big \}.
\end{align*}
Using Proposition \ref{gradient.estimate}, we obtain
\[\liminf_{j \to \infty} \rho_j > 0.\]
By definition of $\rho_j$, we have
\[H_{\mathcal{M}_j}(x,t_j) \leq 4 \, H_{\mathcal{M}_j}(p_j,t_j)\]
for all points $x \in M_{t_j,j}$ satisfying $d_{g(t_j)}(p_j,x) < \rho_j \, H_{\mathcal{M}_j}(p_j,t_j)^{-1}$. Using Proposition \ref{gradient.estimate}, we obtain
\[H_{\mathcal{M}_j}(x,t) \leq 8 \, H_{\mathcal{M}_j}(p_j,t_j)\]
for all points $(x,t) \in \hat{\mathcal{P}}_{\mathcal{M}_j}(p_j,t_j,\rho_j,2\theta_0)$.
We next consider the restriction of the flow $\mathcal{M}_j$ to the parabolic region $\hat{\mathcal{P}}_{\mathcal{M}_j}(p_j,t_j,\rho_j,2\theta_0)$. Let us shift $(p_j,t_j)$ to $(0,0)$ and dilate the surface by the factor $H_{\mathcal{M}_j}(p_j,t_j)$. As a result, we obtain a flow $\tilde{\mathcal{M}_j}$ which is defined in the parabolic region $\mathcal{P}_{\tilde{\mathcal{M}}_j}(0,0,\rho_j,2\theta_0)$ and satisfies $H_{\tilde{M}_j}(0,0)=1$ and $\lambda_{1,\tilde{\mathcal{M}}_j}(0,0) \leq \frac{1}{j}$. Furthermore, the mean curvature of $\tilde{\mathcal{M}}_j$ is at most $8$ everywhere in the parabolic region $\mathcal{P}_{\tilde{\mathcal{M}}_j}(0,0,\rho_j,2\theta_0)$.
After passing to a subsequence, the flows $\tilde{\mathcal{M}_j}$ converge smoothly to a limit flow $\hat{\mathcal{M}}$. The limit flow is defined in a parabolic region $\mathcal{P}(0,0,\rho,2\theta_0)$, where $\rho = \lim_{j \to \infty} \rho_j > 0$. Moreover, the limit flow $\hat{\mathcal{M}}$ satisfies $H(0,0)=1$ and $\lambda_1(0,0) = 0$. Finally, the mean curvature of $\hat{\mathcal{M}}$ is at most $8$ everywhere in the parabolic region $\mathcal{P}(0,0,\rho,2\theta_0)$.
By the strict maximum principle, the limit flow $\hat{\mathcal{M}}$ splits as a product. In other words, we can find a one-parameter family of curves $\Gamma_t$, $t \in (-2\theta_0,0]$, such that $\hat{M}_t \subset \Gamma_t \times \mathbb{R}$. We may assume that the curve $\Gamma_t$ coincides with the image of $\hat{M}_t$ under the projection from $\mathbb{R}^3$ to $\mathbb{R}^2$. Note that the curves $\Gamma_t$ need not be closed.
It follows from the Huisken-Sinestrari convexity estimate that the second fundamental form of the limiting solution $\hat{\mathcal{M}}$ is nonnegative. Hence, the curve $\Gamma_t$ has positive geodesic curvature. Since the original flow $\mathcal{M}_j$ satisfies the gradient estimate in Proposition \ref{gradient.estimate}, the curve $\Gamma_t$ satisfies the derivative estimates $|\frac{d\kappa}{ds}| \leq C_\# \, (\kappa+\Theta)^2$ and $|\frac{d^2\kappa}{ds^2}| \leq C_\# \, (\kappa+\Theta)^3$. Furthermore, since the original flow $\mathcal{M}_j$ satisfies the cylindrical estimate in Proposition \ref{cylindrical.estimate}, the limit flow $\hat{M}_t$ is $\frac{1}{1+\delta}$-noncollapsed. Hence, the inscribed radius of $\Gamma_t$ is at least $\frac{1}{(1+\delta) \, \kappa}$, where $\kappa$ denotes the geodesic curvature of $\Gamma_t$. Furthermore, the outer radius of $\Gamma_t$ is at least $\frac{\alpha}{\kappa}$ at each point on $\Gamma_t$.
Finally, the curve $\Gamma_0$ passes through the origin, and the geodesic curvature of $\Gamma_0$ is equal to $1$ at the origin. Hence, Proposition \ref{choice.of.delta} implies that $\Gamma_0$ has length at most $3\pi$, and $\sup_{\Gamma_0} |\kappa-1| \leq \frac{1}{100}$. Moreover, Proposition \ref{gradient.estimate} implies that each curve $\Gamma_t$ contains a point where the geodesic curvature is between $\frac{1}{2}$ and $2$. Applying Proposition \ref{choice.of.delta} to a scaled copy of $\Gamma_t$, we conclude that each curve $\Gamma_t$ has length at most $6\pi$.
At this point, we distinguish two cases:
\textit{Case 1:} Suppose first that $0 < \rho < L_0+2$. Then $\rho_j < L_0+2$ for $j$ large. From this, we deduce that $\sup_{\tilde{M}_{0,j}} H \geq 4$ for $j$ large. Using this fact and the gradient estimate, we obtain $\sup_{\hat{M}_0} H \geq 2$. Consequently, $\sup_{\Gamma_0} \kappa \geq 2$, where $\kappa$ denotes the geodesic curvature of $\Gamma_0$. On the other hand, we have established earlier that $\sup_{\Gamma_0} |\kappa-1| \leq \frac{1}{100}$. This is a contradiction.
\textit{Case 2:} We now assume that $\rho = L_0+2 > 100$. Since $\Gamma_t$ has length at most $6\pi$, we conclude that $\Gamma_t$ must be a closed curve. Hence, the curves $\Gamma_t$ are simple closed, convex curves in the plane, which evolve by curve shortening flow.
By Proposition \ref{choice.of.delta}, the curve $\Gamma_0$ satisfies $\sum_{l=1}^{18} |\nabla^l \kappa| \leq \frac{1}{1000}$. Moreover, we have $\sup_{\Gamma_{-\theta_0}} \kappa \leq 1-\frac{\theta_0}{4}$. Finally, Proposition \ref{choice.of.hat.delta} implies that, for each point on $\Gamma_0$, the inscribed radius is at least $\frac{1}{(1+\hat{\delta}) \, \kappa}$.
If $j$ is sufficiently large, we can find a region $N_j \subset \{x \in M_{t_j,j}: d_{g(t_j)}(p_j,t_j) \leq (L_0+1) \, H(p_j,t_j)^{-1}\}$ such that $\text{\rm dist}_{C^{20}}(H(p_j,t_j) \, (N_j-p_j),\Gamma_0 \times [-L_0,L_0]) < \varepsilon_0$. We again divide the discussion into two subcases:
\textit{Subcase 2.1:} Suppose that, for $j$ large, the region
\[\{x + a \, \nu(x): x \in M_{t_j,j}, \, d_{g(t_j)}(p_j,x) \leq (L_0+1) \, H(p_j,t_j)^{-1}, \, a \in (0,2\hat{\alpha} \, H(p_j,t_j)^{-1})\}\]
is disjoint from $M_{t_j,j}$. Consequently, the point $p_j$ lies at the center of an $(\hat{\alpha},\hat{\delta},\varepsilon_0,L_0)$-neck in $M_{t_j,j}$ if $j$ is sufficiently large. This contradicts our assumption.
\textit{Subcase 2.2:} Suppose that, for $j$ large, the region
\[\{x + a \, \nu(x): x \in M_{t_j,j}, \, d_{g(t_j)}(p_j,x) \leq (L_0+1) \, H(p_j,t_j)^{-1}, \, a \in (0,2\hat{\alpha} \, H(p_j,t_j)^{-1})\}\]
does intersect $M_{t_j,j}$. In this case, we can find a sequence of points $x_j \in M_{t_j,j}$ and a sequence of numbers $a_j \in (0,2\hat{\alpha} \, H(p_j,t_j)^{-1})$ such that $d_{g(t_j)}(p_j,x_j) \leq (L_0+1) \, H(p_j,t_j)^{-1}$ and $z_j := x_j + a_j \, \nu(x_j) \in M_{t_j,j}$. We next observe that $H(x_j,t_j) \leq 4 \, H(p_j,t_j)$ by definition of $\rho_j$. Hence, the outer radius of the surface $M_{t_j,j}$ at the point $x_j$ is at least $\alpha \, H(x_j,t_j)^{-1} \geq \frac{\alpha}{4} \, H(p_j,t_j)^{-1}$. Consequently, $a_j \geq \frac{\alpha}{2} \, H(p_j,t_j)^{-1}$.
We now let $\tau_j = t_j - \theta_0 \, H(p_j,t_j)^{-2}$. If $j$ is sufficiently large, we may write $z_j = y_j + b_j \, \nu(y_j)$, where $(y_j,\tau_j) \in \hat{\mathcal{P}}_{\mathcal{M}_j}(p_j,t_j,L_0+1,2\theta_0)$ and $0 \leq b_j \leq a_j < 2\hat{\alpha} \, H(p_j,t_j)^{-1}$. On the other hand, since $\sup_{\Gamma_{-\theta_0}} \kappa \leq 1-\frac{\theta_0}{4}$, we have $H(y_j,\tau_j) \leq (1-\frac{\theta_0}{8}) \, H(p_j,t_j)$. This implies that the outer radius of the surface $M_{\tau_j,j}$ at the point $y_j$ is at least
\[\alpha \, H(y_j,\tau_j)^{-1} \geq \frac{\alpha}{1-\frac{\theta_0}{8}} \, H(p_j,t_j)^{-1} = \hat{\alpha} \, H(p_j,t_j)^{-1} > \frac{b_j}{2}.\]
Consequently, the point $z_j = y_j + b_j \, \nu(y_j)$ does not lie in the region enclosed by $M_{\tau_j,j}$. This contradicts the fact that $z_j \in M_{t_j,j}$. This completes the proof.
\section{Proof of Proposition \ref{7.12}}
\label{proof.of.7.12}
By assumption, the parabolic neighborhood $\hat{\mathcal{P}}(p_1,t_1,\tilde{L}+4,2\theta_0)$ contains a point which belongs to a surgery region. Consequently, we can find a surgery time $t_0 \in [t_1-2\theta_0 \, H(p_1,t_1)^{-2},t_1)$ and a point $q_1 \in M_{t_1}$ such that the following holds:
\begin{itemize}
\item $d_{g(t_1)}(p_1,q_1) \leq (\tilde{L}+4) \, H(p_1,t_1)^{-1}$.
\item If we follow the point $q_1 \in M_{t_1}$ back in time, then the corresponding point $q_0 \in M_{t_0+}$ lies in the region modified by surgery at time $t_0$.
\end{itemize}
Let us consider the region modified by surgery at time $t_0$, and let $U_0$ denote the connected component of this set that contains the point $q_0$. In other words, $U_0 \subset M_{t_0+}$ is a cap that was inserted at time $t_0$. We next define $V_0 = \{x \in M_{t_0+}: \text{\rm dist}_{g(t_0+)}(U_0,x) \leq 1000 \, H_1^{-1}\}$. Clearly, $V_0$ is diffeomorphic to a disk. Let
\[D = \{y \in \mathbb{R}^3: \text{\rm there exists a point $x \in V_0$ such that $|y - x| < \frac{\alpha}{1000} \, H_1^{-1}$}\}.\]
Arguing as in Proposition \ref{separation.of.surgery.regions} above, we can show that, for every surgery time $t>t_0$, the set $D$ is disjoint from the region modified by surgery at time $t$. Consequently, the surfaces $M_t \cap D$ form a regular mean curvature flow for $t > t_0$. In other words, the surfaces $M_t \cap D$ evolve smoothly for $t > t_0$, but we allow the possibility that some components of $M_t \cap D$ may disappear as a result of surgeries in other regions.
At each point on $V_0 \subset M_{t_0+}$, the mean curvature is at most $20 \, H_1$. We now follow the surface $V_0 \subset M_{t_0+}$ forward in time. This gives a one-parameter family of surfaces which are all diffeomorphic to a disk. It follows from Proposition \ref{gradient.estimate} that, for $t \in (t_0,t_0+2\theta_0 \, H_1^{-2}]$, the resulting surfaces remain inside the region $D$ and have mean curvature at most $40 \, H_1$. Moreover, since $q_1 \in M_{t_1}$, the resulting surfaces cannot disappear before time $t_1$.
Let $V_1 \subset M_{t_1}$ denote the region in $M_{t_1}$ which is obtained by following the region $V_0 \subset M_{t_0+}$ forward in time. Clearly, $V_1$ is diffeomorphic to a disk, and the mean curvature is at most $40 \, H_1$ at each point in $V_1$. Since $q_0 \in V_0$, we have $q_1 \in V_1$. Furthermore, since $\text{\rm dist}_{g(t_0+)}(q_0,\partial V_0) \geq 1000 \, H_1^{-1}$, we obtain $\text{\rm dist}_{g(t_1)}(q_1,\partial V_1) \geq 500 \, H_1^{-1}$. From this, we deduce that
\[\{x \in M_{t_1}: d_{g(t_1)}(q_1,x) \leq 500 \, H_1^{-1}\} \subset V_1.\]
Hence, if we put $V := V_1$, then $V$ has the required properties.
\section{Proof of Proposition \ref{7.19}}
\label{proof.of.7.19}
To fix notation, let $U \subset \{x \in \mathbb{R}^3: y_1 \leq x_3 \leq y_2\}$ denote the region enclosed by $\Sigma$. Moreover, let $\nu$ denote the outward-pointing unit normal to $U$.
Suppose by contradiction that there exists a point $\bar{x} \in \Sigma$ such that $\bar{x}_3 \in [y_1+1,y_2]$, $\langle \nu(\bar{x}),e_3 \rangle \geq 0$, and $H(\bar{x}) \leq \frac{\alpha}{100}$. The noncollapsing assumption implies that there exists a ball $B \subset \mathbb{R}^3$ of radius $100$ such that
\[B \cap \{x \in \mathbb{R}^3: y_1 \leq x_3 \leq y_2\} \subset U.\]
This implies
\[B \cap \{x \in \mathbb{R}^3: x_3=\bar{x}_3-1\} \subset U \cap \{x \in \mathbb{R}^3: x_3=\bar{x}_3-1\}.\]
Since $\langle \nu(\bar{x}),e_3 \rangle \geq 0$, the set
\[B \cap \{x \in \mathbb{R}^3: x_3=\bar{x}_3-1\}\]
is a disk of radius at least $\sqrt{100^2-99^2} > 10$. On the other hand, our assumptions imply that the set $U \cap \{x \in \mathbb{R}^3: x_3=\bar{x}_3-1\}$ is contained in a disk of radius $10$. This is a contradiction.
\end{document} | math | 117,140 |
\begin{document}
\begin{abstract}
We prove that the classical Coifman-Meyer theorem holds on any polydisc ${\bf T}^d$ of arbitrary dimension $d\geq 1$.
\end{abstract}
\title{Multi-parameter paraproducts}
\section{Introduction}
This article is a continuation of our previous paper \cite{cjtc}. For $n\geq 1$ let $m (=m(\tau))$ in $L^{\infty}({\mbox{\rm I{\bf k}ern-.22em R}}^n)$ be a bounded function,
smooth away from the origin and satisfying
\begin{equation}{\bf l}abel{m1}
|\partial^{\alpha} m(\tau)|{\bf l}esssim \frac{1}{|\tau|^{|\alpha|}}
\end{equation}
for sufficiently many multi-indices $\alpha$ \footnote{ $A{\bf l}esssim B$ means that there exists an universal constant $C>0$ so that $A{\bf l}eq CB$.}. Denote by $T_m^{(1)}$ the $n$-linear operator defined by
\begin{equation}{\bf l}abel{tm1}
T_m^{(1)}(f_1,...,f_n)(x) = \int_{{\mbox{\rm I{\bf k}ern-.22em R}}^n} m(\tau)
\widehat{f_1}(\tau_1)...
\widehat{f_n}(\tau_n) e^{2\pi i x (\tau_1+...+\tau_n)}\, d\tau
\end{equation}
where $f_1,...,f_n$ are Schwartz functions on the real line ${\mbox{\rm I{\bf k}ern-.22em R}}$. The following statement of Coifman and Meyer is a classical theorem in Analysis \cite{cm},
\cite{ks}, \cite{gt}.
\begin{theorem}{\bf l}abel{cm1}
$T_m^{(1)}$ maps $L^{p_1}\times...\times L^{p_n}\rightarrow L^p$ boundedly, as long as
$1<p_1,...,p_n{\bf l}eq \infty$, $\frac{1}{p_1}+...+\frac{1}{p_n} = \frac{1}{p}$ and $0<p<\infty$.
\end{theorem}
In \cite{cjtc} we considered the bi-parameter analogue of $T_m^{(1)}$ defined as follows. Let $m( = m(\gamma,\eta))$ in $L^{\infty}({\mbox{\rm I{\bf k}ern-.22em R}}^{2n})$ be a bounded function,
smooth away from the subspaces $\{\gamma=0\} \cup \{\eta = 0\}$ and satisfying
\begin{equation}{\bf l}abel{m2}
|\partial^{\alpha}_{\gamma}
\partial^{\beta}_{\eta}
m(\gamma,\eta)|{\bf l}esssim
\frac{1}{|\gamma|^{|\alpha|}}
\frac{1}{|\eta|^{|\beta|}}
\end{equation}
for sufficiently many multi-indices $\alpha$ and $\beta$.
Denote by $T_m^{(2)}$ the $n$-linear operator defined by
\begin{equation}{\bf l}abel{tm2}
T_m^{(2)}(f_1,...,f_n)(x) = \int_{{\mbox{\rm I{\bf k}ern-.22em R}}^{2n}} m(\gamma,\eta)
\widehat{f_1}(\gamma_1,\eta_1)...
\widehat{f_n}(\gamma_n,\eta_n) e^{2\pi i x [(\gamma_1,\eta_1)+...+(\gamma_n,\eta_n)]}\, d\gamma d\eta
\end{equation}
where $f_1,...,f_n$ are Schwartz functions on the plane ${\mbox{\rm I{\bf k}ern-.22em R}}^2$. The following theorem has been proven in \cite{cjtc}.
\begin{theorem}{\bf l}abel{cm2}
$T_m^{(2)}$ maps $L^{p_1}\times...\times L^{p_n}\rightarrow L^p$ boundedly, as long as
$1<p_1,...,p_n{\bf l}eq \infty$, $\frac{1}{p_1}+...+\frac{1}{p_n} = \frac{1}{p}$ and $0<p<\infty$.
\end{theorem}
The main goal of the present paper is to generalize Theorem \ref{cm2} to the $d$-parameter setting, for any $d\geq 1$.
In general, if $\xi_1= (\xi_1^i)_{i=1}^d,...,\xi_n= (\xi_n^i)_{i=1}^d$ are $n$ generic vectors in ${\mbox{\rm I{\bf k}ern-.22em R}}^d$, they naturally generate the following
$d$ vectors in ${\mbox{\rm I{\bf k}ern-.22em R}}^n$ which we will denote by
$\overline{\xi_1}= (\xi_j^1)_{j=1}^n,...,\overline{\xi_d}= (\xi_j^d)_{j=1}^n$. As before, let $m(=m(\xi)=m(\overline{\xi}))$ in
$L^{\infty}({\mbox{\rm I{\bf k}ern-.22em R}}^{dn})$ be a bounded symbol, smooth away from the subspaces $\{\overline{\xi_1}=0\}\cup...\cup\{\overline{\xi_d}=0\}$ and satisfying
\begin{equation}{\bf l}abel{m3}
|\partial^{\alpha_1}_{\overline{\xi_1}}...
\partial^{\alpha_d}_{\overline{\xi_d}} m(\overline{\xi})|
{\bf l}esssim
\prod_{i=1}^d \frac{1}{|\overline{\xi_i}|^{|\alpha_i|}}
\end{equation}
for sufficiently many multi-indices $\alpha_1,...,\alpha_d$.
Denote by $T_m^{(d)}$ the $n$-linear operator defined by
\begin{equation}{\bf l}abel{tm3}
T_m^{(d)}(f_1,...,f_n)(x) = \int_{{\mbox{\rm I{\bf k}ern-.22em R}}^{dn}} m(\xi)
\widehat{f_1}(\xi_1)...
\widehat{f_n}(\xi_n) e^{2\pi i x (\xi_1+...+\xi_n)}\, d\xi
\end{equation}
where $f_1,...,f_n$ are Schwartz functions on ${\mbox{\rm I{\bf k}ern-.22em R}}^d$. The main theorem of the article is the following.
\begin{theorem}{\bf l}abel{cm3}
$T_m^{(d)}$ maps $L^{p_1}\times...\times L^{p_n}\rightarrow L^p$ boundedly, as long as
$1<p_1,...,p_n{\bf l}eq \infty$, $\frac{1}{p_1}+...+\frac{1}{p_n} = \frac{1}{p}$ and $0<p<\infty$.
\end{theorem}
Classically, \cite{cm}, \cite{ks}, \cite{gt} an estimate as the one in Theorem \ref{cm1} is proved
by using the $T(1)$ theorem of David and Journ\'{e} \cite{stein} together with the Calder\'{o}n-Zygmund decomposition. In particular, the theory
of BMO functions and Carleson measures is involved.
On the other hand, it is well known \cite{cf}, \cite{journe} that in the multi-parameter setting all these results and concepts are much more delicate
(BMO, John-Nirenberg inequality, Calder\'{o}n-Zygmund decomposition). To overcome these difficulties, in \cite{cjtc} we had to develop a completely
new approach to prove Theorem \ref{cm2}. This approach relied on the one dimensional BMO theory and also on Journ\'{e}'s lemma \cite{journe} \cite{fl},
but did not extend to prove the general $d$-parameter case.
The novelty of the present paper is that it simplifies the method introduced in \cite{cjtc} and this simplification works equally well in all dimensions.
Surprisingly, it turned out that one doesn't need to rely on any knowledge of BMO, Carleson measures or Journ\'{e}'s lemma in order to prove the estimates in
Theorem \ref{cm3}.
We shall rely on our previous paper \cite{cjtc} and for the reader's convenience we chose to present the argument in the same bi-linear bi-parameter setting
(so both $n$ and $d$ will be equal to $2$). However, it will be clear from the proof that its extension to the $n$-linear $d$-parameter case is straightforward.
The paper is organized as follows. In Section 2 we recall the discretization procedure from \cite{cjtc} which reduces the study of our operator to the study of some
general multi-parameter paraproducts. In Section 3 we present the proof of our main theorem, Theorem \ref{cm3} and in the Appendix we give a proof of Lemma \ref{desc}
which plays
an important role in our simplified construction.
$\bf{Acknowledgements}$ C.Muscalu and J.Pipher were partially supported by NSF Grants. T.Tao was partially supported by a Packard Foundation Grant.
C.Thiele was partially supported by the NSF Grants 9985572 and DMS 9970469. Two of the authors (C.Muscalu and C.Thiele) are particularly grateful
to Centro di Ricerca Matematica Ennio de Giorgi in Pisa for their warm hospitality during their visit in June 2004.
\section{Discrete paraproducts}
As we promised, assume throughout the paper that $n=d=2$. In this case, our operator $T_m^{(d)}$ can be written as
\begin{equation}{\bf l}abel{last}
T_m^{(2)}(f,g)(x)= \int_{{\mbox{\rm I{\bf k}ern-.22em R}}^4} m(\gamma, \eta)
\widehat{f}(\gamma_1, \eta_1)
\widehat{g}(\gamma_2, \eta_2) e^{2\pi i x[ (\gamma_1, \eta_1)+(\gamma_2, \eta_2)]}\, d\gamma d\eta.
\end{equation}
In \cite{cjtc} Section 1, we decomposed the operator $T_m^{(2)}$ into smaller pieces, well adapted to its bi-parameter structure. This allowed us to reduce its analysis
to the analysis of some simpler discretized dyadic paraproducts. We will recall their definitions below.
An interval $I$ on the real line ${\mbox{\rm I{\bf k}ern-.22em R}}$ is called dyadic if it is of the form $I=2^k[n,n+1]$ for some $k,n\in{\bf Z}$. If ${\bf l}ambda, t\in [0,1]$ are two parameters
and $I$ is as above, we denote by $I_{{\bf l}ambda,t}$ the interval $I_{{\bf l}ambda,t} = 2^{k+{\bf l}ambda}[n+t, n+t+1]$.
\begin{definition}{\bf l}abel{bump}
For $J\subseteq{\mbox{\rm I{\bf k}ern-.22em R}}$ an arbitrary interval, we say that a smooth function ${\bf P}hi_J$ is a bump adapted to $J$, if and only if the following inequalities hold
\begin{equation}
|{\bf P}hi_J^{(l)}(x)|{\bf l}eq C_{l,\alpha} \frac{1}{|J|^{l}}\frac{1}{{\bf l}eft(1+\frac{{\rm dist}(x,J)}{|J|}\right)^{\alpha}},
\end{equation}
for every integer $\alpha\in {{\mbox{\rm I{\bf k}ern-.2em N}}}$ and for sufficiently many derivatives $l\in{{\mbox{\rm I{\bf k}ern-.2em N}}}$. If ${\bf P}hi_J$ is a bump adapted to $J$, we say that $|J|^{-1/2}{\bf P}hi_J$ is an
$L^2$ - normalized bump adapted to $J$.
\end{definition}
For ${\bf l}ambda, t_1, t_2, t_3\in [0,1]$ and $j\in\{1,2,3\}$ we define the discretized dyadic paraproduct ${\bf P}i^j_{{\bf l}ambda, t_1, t_2, t_3}$ of ``type $j$'' by
\begin{equation}{\bf l}abel{para}
{\bf P}i^j_{{\bf l}ambda, t_1, t_2, t_3}(f,g)=
\sum_{I\in\cal{D}}\frac{1}{|I|^{1/2}}
{\bf l}angle f, {\bf P}hi^1_{I_{{\bf l}ambda, t_1}}\rangle
{\bf l}angle g, {\bf P}hi^2_{I_{{\bf l}ambda, t_2}}\rangle
{\bf P}hi^3_{I_{{\bf l}ambda,t_3}},
\end{equation}
where $f,g$ are complex-valued measurable functions on ${\mbox{\rm I{\bf k}ern-.22em R}}$ and ${\bf P}hi^i_{I_{{\bf l}ambda, t_i}}$ are $L^2$-normalized bumps adapted to $I_{{\bf l}ambda, t_i}$
with the additional property that $\int_{{\mbox{\rm I{\bf k}ern-.22em R}}}{\bf P}hi^i_{I_{{\bf l}ambda, t_i}}(x) dx = 0$ for $i{\bf n}eq j$, $i=1,2,3$.
$\cal{D}$ is an arbitrary finite set of dyadic intervals and by ${\bf l}angle\cdot, \cdot\rangle$ we denoted the complex scalar product.
Similarly, for $\vec{{\bf l}ambda}, \vec{t_1}, \vec{t_2}, \vec{t_3}\in [0,1]^2$ and $\vec{j}\in\{1,2,3\}^2$, we define the discretized dyadic bi-parameter
paraproduct of ``type $\vec{j}$''
$${\bf P}i^{\vec{j}}_{\vec{{\bf l}ambda}, \vec{t_1}, \vec{t_2}, \vec{t_3}}= {\bf P}i^{j'}_{{\bf l}ambda', t'_1, t'_2, t'_3}\otimes
{\bf P}i^{j''}_{{\bf l}ambda'', t''_1, t''_2, t''_3}$$
by
\begin{equation}{\bf l}abel{para2}
{\bf P}i^{\vec{j}}_{\vec{{\bf l}ambda}, \vec{t_1}, \vec{t_2}, \vec{t_3}}(f,g) =
\sum_{R\in\vec{\cal{D}}}\frac{1}{|R|^{1/2}}
{\bf l}angle f, {\bf P}hi^1_{R_{\vec{{\bf l}ambda}, \vec{t_1}}}\rangle
{\bf l}angle g, {\bf P}hi^2_{R_{\vec{{\bf l}ambda}, \vec{t_2}}}\rangle
{\bf P}hi^3_{R_{\vec{{\bf l}ambda},\vec{t_3}}},
\end{equation}
where this time $f,g$ are complex-valued measurable functions on ${\mbox{\rm I{\bf k}ern-.22em R}}^2$, $R=I\times J$ are dyadic rectangles and ${\bf P}hi^i_{R_{\vec{{\bf l}ambda}, \vec{t_i}}}$
are given by
$${\bf P}hi^i_{R_{\vec{{\bf l}ambda}, \vec{t_i}}} = {\bf P}hi^i_{I_{{\bf l}ambda', t'_i}}\otimes{\bf P}hi^i_{J_{{\bf l}ambda'', t''_i}}$$
for $i=1,2,3$. In particular, if $i{\bf n}eq j'$ then $\int_{{\mbox{\rm I{\bf k}ern-.22em R}}}{\bf P}hi^i_{I_{{\bf l}ambda', t'_i}}(x) dx = 0$ and if $i{\bf n}eq j''$ then
$\int_{{\mbox{\rm I{\bf k}ern-.22em R}}}{\bf P}hi^i_{J_{{\bf l}ambda'', t''_i}}(x) dx = 0$. $\vec{\cal{D}}$ is an arbitrary finite collection of dyadic rectangles.
We will also denote by ${\cal{L}}ambda^{\vec{j}}_{\vec{{\bf l}ambda}, \vec{t_1}, \vec{t_2}, \vec{t_3}}(f,g,h)$ the trilinear form given by
\begin{equation}
{\cal{L}}ambda^{\vec{j}}_{\vec{{\bf l}ambda}, \vec{t_1}, \vec{t_2}, \vec{t_3}}(f,g,h)=
\int_{{\mbox{\rm I{\bf k}ern-.22em R}}^2}{\bf P}i^{\vec{j}}_{\vec{{\bf l}ambda}, \vec{t_1}, \vec{t_2}, \vec{t_3}}(f,g)(x,y) h(x, y) dx dy.
\end{equation}
In \cite{cjtc} we showed that Theorem \ref{cm2} can be reduced to the following Proposition.
\begin{proposition}{\bf l}abel{reduction}
Fix $\vec{j}\in \{1,2,3\}^2$ and let $1<p,q<\infty$ be two numbers arbitrarily close to $1$. Let also $f\in L^p$, $\|f\|_p=1$,
$g\in L^q$, $\|g\|_q=1$ and $E\subseteq {\mbox{\rm I{\bf k}ern-.22em R}}^2$, $|E|=1$. Then, there exists a subset $E'\subseteq E$ with
$|E'|\sim 1$ such that \footnote{$A\sim B$ means that $A{\bf l}esssim B$ and $B{\bf l}esssim A$}
\begin{equation}{\bf l}abel{inegalitatea}
{\bf l}eft|{\cal{L}}ambda^{\vec{j}}_{\vec{{\bf l}ambda}, \vec{t_1}, \vec{t_2}, \vec{t_3}}(f,g,h)\right|{\bf l}esssim1
\end{equation}
uniformly in the parameters $\vec{{\bf l}ambda}, \vec{t_1}, \vec{t_2}, \vec{t_3}\in [0,1]^2$, where $h:=\chi_{E'}$.
\end{proposition}
It is therefore enough to prove the above Proposition \ref{reduction}, in order to complete the proof of our main Theorem \ref{cm2}. Since all the cases are similar, we assume as in
\cite{cjtc} that $\vec{j} = (1,2)$.
To construct the desired set $E'$, we need to recall the ``maximal-square'', ``square-maximal'' and ``square-square'' functions considered in \cite{cjtc}.
For $(x,y)\in{\mbox{\rm I{\bf k}ern-.22em R}}^2$ define
\begin{equation}{\bf l}abel{ms}
MS(f)(x,y) = \sup_I\frac{1}{|I|^{1/2}}
{\bf l}eft(\sum_{J:R=I\times J\in\vec{\cal{D}}}
\sup_{\vec{{\bf l}ambda},\vec{t_1}}
\frac{|{\bf l}angle f, {\bf P}hi^1_{R_{\vec{{\bf l}ambda}, \vec{t_1}}}\rangle|^2}{|J|}
\chi_J(y)
\right)
\chi_I(x),
\end{equation}
\begin{equation}{\bf l}abel{sm}
SM(g)(x,y)=
{\bf l}eft(
\sum_I
\frac{\sup_{J:R=I\times J\in\vec{\cal{D}}} \sup_{\vec{{\bf l}ambda},\vec{t_2}}\frac{|{\bf l}angle g, {\bf P}hi^2_{R_{\vec{{\bf l}ambda}, \vec{t_2}}}\rangle|^2 }{|J|}\chi_J(y) }{|I|}
\chi_I(x)\right)^{1/2}
\end{equation}
and
\begin{equation}{\bf l}abel{ss}
SS(h)(x,y) =
{\bf l}eft(
\sum_{R\in\vec{\cal{D}}}\sup_{\vec{{\bf l}ambda}, \vec{t_3}}
\frac{|{\bf l}angle h, {\bf P}hi^3_{R_{\vec{{\bf l}ambda}, \vec{t_3}}}\rangle|^2 }{|R|}
\chi_R(x,y)
\right)^{1/2}.
\end{equation}
Then, we also recall (see \cite{stein}) the bi-parameter Hardy-Littlewood maximal function
\begin{equation}{\bf l}abel{mm}
MM(F)(x,y) = \sup_{(x,y)\in I\times J}
\frac{1}{|I| |J|}
\int_{I\times J}
|F(x',y')| dx' dy'.
\end{equation}
The following simple estimates explain the appearance of these functions. In particular, we will see that our desired bounds in Theorem \ref{cm2}
can be easily obtained as long as all the indices involved are strictly between $1$ and $\infty$.
We start by recalling the following basic inequality, \cite{cjtc}. If ${\bf P}i^1$ ia a one-parameter paraproduct of ``type $1$'' given by
\begin{equation}
{\bf P}i^1(f_1,f_2) =
\sum_I
\frac{1}{|I|^{1/2}}
{\bf l}angle f_1, {\bf P}hi^1_I \rangle
{\bf l}angle f_2, {\bf P}hi^2_I \rangle
{\bf P}hi^3_I
\end{equation}
then we can write
$${\bf l}eft|{\cal{L}}ambda^1(f_1,f_2,f_3)\right|=
{\bf l}eft|\int_{{\mbox{\rm I{\bf k}ern-.22em R}}}{\bf P}i^1(f_1, f_2)(x) f_3(x) dx\right|$$
$${\bf l}esssim
\sum_I
\frac{1}{|I|^{1/2}}
|{\bf l}angle f_1, {\bf P}hi^1_I \rangle|
|{\bf l}angle f_2, {\bf P}hi^2_I \rangle|
|{\bf l}angle f_3, {\bf P}hi^3_I \rangle|$$
$$=
\int_{{\mbox{\rm I{\bf k}ern-.22em R}}}
{\bf l}eft(
\sum_I
\frac{|{\bf l}angle f_1, {\bf P}hi^1_I \rangle|}{|I|^{1/2}}
\frac{|{\bf l}angle f_2, {\bf P}hi^2_I \rangle|}{|I|^{1/2}}
\frac{|{\bf l}angle f_3, {\bf P}hi^3_I \rangle|}{|I|^{1/2}}
\chi_I(x)
\right)
dx$$
\begin{equation}{\bf l}abel{unu}
{\bf l}esssim \int_{{\mbox{\rm I{\bf k}ern-.22em R}}} M(f_1)(x) S(f_2)(x) S(f_3)(x) dx
\end{equation}
where $M$ denotes the Hardy-Littlewood maximal function and $S$ is the square function of Littlewood and Paley. In particular, we easily see that
${\bf P}i^1: L^p\times L^q\rightarrow L^r$ for any $1<p,q,r<\infty$ satisfying $1/p+1/q=1/r$.
Analogous estimates hold for any other type of paraproducts ${\bf P}i^j$ for $j=1,2,3$.
Similarly, for the bi-parameter paraproduct ${\bf P}i^{(1,2)}$ of ``type $(1,2)$'' formally defined by ${\bf P}i^{(1,2)} = {\bf P}i^1\otimes {\bf P}i^2$ one obtains the inequalities
$${\bf l}eft|{\cal{L}}ambda^{(1,2)}(f_1,f_2,f_3)\right|=
{\bf l}eft|\int_{{\mbox{\rm I{\bf k}ern-.22em R}}}{\bf P}i^{(1,2)}(f_1, f_2)(x,y) f_3(x,y) dx dy\right|$$
\begin{equation}{\bf l}abel{doi}
{\bf l}esssim \cdots {\bf l}esssim
\int_{{\mbox{\rm I{\bf k}ern-.22em R}}^2} MS(f_1)(x,y) SM(f_2)(x,y) SS(f_3)(x,y) dx dy,
\end{equation}
and analogous estimates hold for any other type of paraproducts ${\bf P}i^{\vec{j}}$ for $\vec{j}\in \{1,2,3\}^2$.
It is important that all these $MS$, $SM$ and $SS$ functions are bounded on $L^p$ for any $1<p<\infty$. We recall the proof of this fact here (see \cite{cjtc}).
We start with $SM(f_2)(x,y)$. It can be written as
\begin{equation}
SM(f_2)(x,y)=
{\bf l}eft(
\sum_{\tilde{I}}
\frac{\sup_{\tilde{J}} \frac{|{\bf l}angle f_2, {\bf P}hi^2_{\tilde{I}}\otimes {\bf P}hi^2_{\tilde{J}}\rangle|^2 }{|\tilde{J}|}\chi_{\tilde{J}}(y) }{|\tilde{I}|}
\chi_{\tilde{I}}(x)\right)^{1/2}
\end{equation}
$${\bf l}esssim
{\bf l}eft(
\sum_{\tilde{I}}
M(\frac{{\bf l}angle f_2, {\bf P}hi^2_{\tilde{I}}\rangle }{|\tilde{I}|^{1/2}})^2(y)
\chi_{\tilde{I}}(x)\right)^{1/2}
$$
where $\tilde{I}$ and $\tilde{J}$ are the intervals where the corresponding supremums over $\vec{{\bf l}ambda}, \vec{t_2}\in [0,1]^2$ in (\ref{sm}) are attained.
In particular, by using Fefferman-Stein \cite{fs} and Littlewood-Paley \cite{stein} inequalities, we have
\begin{equation}
\|SM(f_2)\|_p {\bf l}esssim
\|
{\bf l}eft(
\sum_{\tilde{I}}
M(\frac{{\bf l}angle f_2, {\bf P}hi^2_{\tilde{I}}\rangle }{|\tilde{I}|^{1/2}})^2(y)
\chi_{\tilde{I}}(x)\right)^{1/2}
\|_p
\end{equation}
$${\bf l}esssim
\|
{\bf l}eft(
\sum_{\tilde{I}}
\frac{|{\bf l}angle f_2, {\bf P}hi^2_{\tilde{I}}\rangle|^2 }{|\tilde{I}|}(y)
\chi_{\tilde{I}}(x)\right)^{1/2}
\|_p{\bf l}esssim \|f_2\|_p
$$
for any $1<p<\infty$. Then, we observe that the $MS$ function is pointwise smaller than a certain $SM$ type function and hence bounded on $L^p$, while
the $SS$ function is a classical double square function and its boundedness on $L^p$ spaces is well known, \cite{cf}. As a consequence, it follows as before
that ${\bf P}i^{(1,2)}: L^p\times L^q\rightarrow L^r$ as long as $1<p,q,r<\infty$ with $1/p+1/q=1/r$.
\section{Proof of Proposition \ref{reduction}}
It remains to prove Proposition \ref{reduction}. First, we state the following Lemma.
\begin{lemma}{\bf l}abel{desc}
Let $J\subseteq {\mbox{\rm I{\bf k}ern-.22em R}}$ be an arbitrary interval. Then, every bump function $\phi_J$ adapted to $J$ can be written as
\begin{equation}
\phi_J = \sum_{k\in{{\mbox{\rm I{\bf k}ern-.2em N}}}} 2^{-1000 k} \phi^k_J
\end{equation}
where for each $k\in{{\mbox{\rm I{\bf k}ern-.2em N}}}$, $\phi^k_J$ is also a bump adapted to $J$ but with the additional property that ${\rm supp} (\phi^k_J)\subseteq 2^k J$
\footnote{$2^k J$ is the interval having the same center as $J$ and whose length is $2^k |J|$.}.
Moreover, if we assume $\int_{R}\phi_J(x) dx = 0$ then all the functions $\phi^k_J$ can be chosen so that $\int_{{\mbox{\rm I{\bf k}ern-.22em R}}}\phi^k_J(x) dx = 0$
for every $k\in {{\mbox{\rm I{\bf k}ern-.2em N}}}$.
\end{lemma}
The proof of this Lemma will be presented later on in the Appendix. It is the main new ingredient which allows us to simplify our previous argument in \cite{cjtc}.
Using it, we can decompose our trilinear form in (\ref{para2}) as
\begin{equation}{\bf l}abel{split}
{\cal{L}}ambda^{\vec{j}}_{\vec{{\bf l}ambda}, \vec{t_1}, \vec{t_2}, \vec{t_3}}(f,g,h) = \sum_{\vec{k}\in{{\mbox{\rm I{\bf k}ern-.2em N}}}^2} 2^{-1000|\vec{k}|}
\sum_{R\in\vec{\cal{D}}}\frac{1}{|R|^{1/2}}
{\bf l}angle f, {\bf P}hi^1_{R_{\vec{{\bf l}ambda}, \vec{t_1}}}\rangle
{\bf l}angle g, {\bf P}hi^2_{R_{\vec{{\bf l}ambda}, \vec{t_2}}}\rangle
{\bf l}angle h, {\bf P}hi^{3,\vec{k}}_{R_{\vec{{\bf l}ambda},\vec{t_3}}}\rangle,
\end{equation}
where the new functions ${\bf P}hi^{3,\vec{k}}_{R_{\vec{{\bf l}ambda},\vec{t_3}}}$ have basically the same structure as the old ${\bf P}hi^{3}_{R_{\vec{{\bf l}ambda},\vec{t_3}}}$
but they also have the additional property that ${\rm supp} ({\bf P}hi^{3,\vec{k}}_{R_{\vec{{\bf l}ambda},\vec{t_3}}})\subseteq 2^{\vec{k}}R_{\vec{{\bf l}ambda},\vec{t_3}}$.
We denoted by $2^{\vec{k}}R_{\vec{{\bf l}ambda},\vec{t_3}}:= 2^{k_1}I_{{\bf l}ambda', t'_3}\times 2^{k_2}J_{{\bf l}ambda'', t''_3}$, $\vec{k}=(k_1, k_3)$ and
$|\vec{k}|=k_1+k_2$.
Fix now $f,g, E, p, q$ as in Proposition \ref{reduction}. For each $\vec{k}\in{{\mbox{\rm I{\bf k}ern-.2em N}}}^2$ define
\begin{equation}
\Omega_{-5|\vec{k}|}=
\{ (x,y)\in{\mbox{\rm I{\bf k}ern-.22em R}}^2 : MS(f)(x,y)> C2^{5|\vec{k}|} \} \cup \{ (x,y)\in{\mbox{\rm I{\bf k}ern-.22em R}}^2 : SM(g)(x,y)> C2^{5|\vec{k}|} \}.
\end{equation}
Also, define
\begin{equation}
\tilde{\Omega}_{-5|\vec{k}|}= \{ (x,y)\in{\mbox{\rm I{\bf k}ern-.22em R}}^2 : MM(\chi_{\Omega_{-5|\vec{k}|}})(x,y) >\frac{1}{100} \}
\end{equation}
and then
\begin{equation}
\tilde{\tilde{\Omega}}_{-5|\vec{k}|}= \{ (x,y)\in{\mbox{\rm I{\bf k}ern-.22em R}}^2 : MM(\chi_{\tilde{\Omega}_{-5|\vec{k}|}})(x,y) >\frac{1}{2^{|\vec{k}|}} \}.
\end{equation}
Finally, we denote by
$$\Omega = \begin{itemize}gcup_{\vec{k}\in{{\mbox{\rm I{\bf k}ern-.2em N}}}^2}\tilde{\tilde{\Omega}}_{-5|\vec{k}|}.$$
It is clear that $|\Omega|< 1/2$ if $C$ is a big enough constant, which we fix from now on. Then, define $E':= E\setminus\Omega$ and observe that
$|E'|\sim 1$. We now want to show that the corresponding expression in (\ref{inegalitatea}) is $O(1)$ uniformly in the parameters
$\vec{{\bf l}ambda}, \vec{t_1}, \vec{t_2}, \vec{t_3}\in [0,1]^2$. Since our argument will not depend on these parameters, we can assume for simplicity that they are all zero
and in this case we will write ${\bf P}hi^i_R$ instead of ${\bf P}hi^i_{R_{\vec{{\bf l}ambda},\vec{t_i}}}$ for $i=1,2$ and ${\bf P}hi^{3,\vec{k}}_R$ instead of
${\bf P}hi^{3,\vec{k}}_{R_{\vec{{\bf l}ambda},\vec{t_3}}}$.
Fix then $\vec{k}\in{{\mbox{\rm I{\bf k}ern-.2em N}}}^2$ and look at the corresponding inner sum in (\ref{split}). We split it into two parts as follows.
Part I sums over those rectangles $R$ with the property that
\begin{equation}
R\cap\tilde{\Omega}_{-5|\vec{k}|}^c {\bf n}eq \emptyset
\end{equation}
while Part II sums over those rectangles with the property that
\begin{equation}
R\cap\tilde{\Omega}_{-5|\vec{k}|}^c =\emptyset.
\end{equation}
We observe that Part II is identically equal to zero, because if $R\cap\tilde{\Omega}_{-5|\vec{k}|}^c {\bf n}eq \emptyset$ then $R\subseteq\tilde{\Omega}_{-5|\vec{k}|}$
and in particular this implies that $2^{\vec{k}}R\subseteq \tilde{\tilde{\Omega}}_{-5|\vec{k}|}$ which is a set disjoint from $E'$. It is therefore enough to estimate Part I
only. This can be done by using the technique developed in \cite{cjtc}.
Since $R\cap\tilde{\Omega}_{-5|\vec{k}|}^c{\bf n}eq\emptyset$, it follows that
$\frac{|R\cap\Omega_{-5|\vec{k}|}|}{|R|}< \frac{1}{100}$ or equivalently,
$|R\cap\Omega^c_{-5|\vec{k}|}|> \frac{99}{100}|R|$.
We are now going to describe three decomposition procedures, one for each function $f, g, h$. Later on, we will
combine them, in order to handle our sum.
First, define
$$\Omega_{-5|\vec{k}|+1}= \{ (x,y)\in{\mbox{\rm I{\bf k}ern-.22em R}}^2 : MS(f)(x,y)> \frac{C 2^{5|\vec{k}|} }{2^1} \}$$
and set
$${\bf T}_{-5|\vec{k}| +1}= \{ R\in \vec{\cal{D}} : |R\cap\Omega_{-5|\vec{k}| +1}|>\frac{1}{100} |R| \},$$
then define
$$\Omega_{-5|\vec{k}| +2}= \{ (x,y)\in{\mbox{\rm I{\bf k}ern-.22em R}}^2 : MS(f)(x,y)> \frac{C 2^{5|\vec{k}|} }{2^2} \}$$
and set
$${\bf T}_{-5|\vec{k}| +2}= \{ R\in \vec{\cal{D}}\setminus{\bf T}_{-5|\vec{k}| +1} : |R\cap\Omega_{-5|\vec{k}| +2}|>\frac{1}{100} |R| \},$$
and so on. The constant $C>0$ is the one in the definition of the set $E'$ above.
Since there are finitely many rectangles, this algorithm ends after a while, producing the sets $\{\Omega_n\}$
and $\{{\bf T}_n\}$ such that $\vec{\cal{D}}=\cup_n{\bf T}_n$.
Independently, define
$$\Omega'_{-5|\vec{k}|+1}= \{ (x,y)\in{\mbox{\rm I{\bf k}ern-.22em R}}^2 : SM(g)(x,y)> \frac{C 2^{5|\vec{k}|} }{2^1} \}$$
and set
$${\bf T}'_{-5|\vec{k}| +1}= \{ R\in \vec{\cal{D}} : |R\cap\Omega'_{-5|\vec{k}| +1}|>\frac{1}{100} |R| \},$$
then define
$$\Omega'_{-5|\vec{k}| +2}= \{ (x,y)\in{\mbox{\rm I{\bf k}ern-.22em R}}^2 : SM(g)(x,y)> \frac{C 2^{5|\vec{k}|} }{2^2} \}$$
and set
$${\bf T}'_{-5|\vec{k}| +2}= \{ R\in \vec{\cal{D}}\setminus{\bf T}'_{-5|\vec{k}| +1} : |R\cap\Omega'_{-5|\vec{k}| +2}|>\frac{1}{100} |R| \},$$
and so on, producing the sets $\{\Omega'_n\}$ and $\{{\bf T}'_n\}$ such that $\vec{\cal{D}}=\cup_n{\bf T}'_n$.
We would like to have such a decomposition available for the function $h$ also. To do this, we first need to
construct the analogue of the set $\Omega_{-5|\vec{k}|}$, for it. Pick $N>0$ a big enough integer such that for every
$R\in\vec{\cal{D}}$ we have $|R\cap\Omega^{''c}_{-N}|> \frac{99}{100} |R|$ where we defined
$$\Omega''_{-N}= \{ (x,y)\in{\mbox{\rm I{\bf k}ern-.22em R}}^2 : SS^{\vec{k}}(h)(x,y)> C 2^N \}.$$
Here $SS^{\vec{k}}$ denotes the same ``square-square'' function defined in (\ref{ss}) but with the functions ${\bf P}hi^{3,\vec{k}}_{R_{\vec{{\bf l}ambda},\vec{t_3}}}$
instead of ${\bf P}hi^{3}_{R_{\vec{{\bf l}ambda},\vec{t_3}}}$
Then, similarly to the previous algorithms, we define
$$\Omega''_{-N+1}= \{ (x,y)\in{\mbox{\rm I{\bf k}ern-.22em R}}^2 : SS^{\vec{k}}(h)(x)> \frac{C 2^N}{2^1} \}$$
and set
$${\bf T}''_{-N+1}= \{ R\in \vec{\cal{D}} : |R\cap\Omega''_{-N+1}|>\frac{1}{100} |R| \},$$
then define
$$\Omega''_{-N+2}= \{ x\in{\mbox{\rm I{\bf k}ern-.22em R}}^2 : SS^{\vec{k}}(h)(x)> \frac{C 2^N}{2^2} \}$$
and set
$${\bf T}''_{-N+2}= \{ R\in \vec{\cal{D}}\setminus{\bf T}''_{-N+1} : |R\cap\Omega''_{-N+2}|>\frac{1}{100} |R| \},$$
and so on, constructing the sets $\{\Omega''_n\}$ and $\{{\bf T}''_n\}$ such that $\vec{\cal{D}}=\cup_n{\bf T}''_n$.
Then we write Part I as
\begin{equation}{\bf l}abel{in5}
\sum_{n_1, n_2>-5|\vec{k}|, n_3>-N} \sum_{R\in {\bf T}_{n_1, n_2, n_3}}
\frac{1}{|R|^{3/2}}
|{\bf l}angle f, {\bf P}hi^1_{R}\rangle|
|{\bf l}angle g, {\bf P}hi^2_{R}\rangle|
|{\bf l}angle h, {\bf P}hi^{3,\vec{k}}_{R}\rangle| |R|,
\end{equation}
where ${\bf T}_{n_1, n_2, n_3}:= {\bf T}_{n_1}\cap {\bf T}'_{n_2}\cap {\bf T}''_{n_3}$. Now, if $R$ belongs to
${\bf T}_{n_1, n_2, n_3}$ this means in particular that $R$ has not been selected at the previous $n_1 -1$, $n_2 -1$ and
$n_3 -1$ steps respectively, which means that $|R\cap\Omega_{n_1-1}|<\frac{1}{100} |R|$,
$|R\cap\Omega'_{n_2-1}|<\frac{1}{100} |R|$ and $|R\cap\Omega''_{n_3-1}|<\frac{1}{100} |R|$ or equivalently,
$|R\cap\Omega^c_{n_1-1}|>\frac{99}{100} |R|$,
$|R\cap\Omega^{'c}_{n_2-1}|>\frac{99}{100} |R|$ and
$|R\cap\Omega^{''c}_{n_3-1}|>\frac{99}{100} |R|$. But this implies that
\begin{equation}{\bf l}abel{in6}
|R\cap\Omega^c_{n_1-1}\cap\Omega^{'c}_{n_2-1}\cap\Omega^{''c}_{n_3-1}|> \frac{97}{100}|R|.
\end{equation}
In particular, using (\ref{in6}), the term in (\ref{in5}) is smaller than
$$\sum_{n_1, n_2>-5|\vec{k}| , n_3>-N} \sum_{R\in {\bf T}_{n_1, n_2, n_3}}
\frac{1}{|R|^{3/2}}
|{\bf l}angle f, {\bf P}hi^1_{R}\rangle|
|{\bf l}angle g, {\bf P}hi^2_{R}\rangle|
|{\bf l}angle h, {\bf P}hi^{3,\vec{k}}_{R}\rangle| |R\cap\Omega^c_{n_1-1}\cap\Omega^{'c}_{n_2-1}\cap\Omega^{''c}_{n_3-1}|=$$
$$\sum_{n_1, n_2> -5|\vec{k}|, n_3>-N} \int_{\Omega^c_{n_1-1}\cap\Omega^{'c}_{n_2-1}\cap\Omega^{''c}_{n_3-1}}
\sum_{R\in {\bf T}_{n_1, n_2, n_3}}
\frac{1}{|R|^{3/2}}
|{\bf l}angle f, {\bf P}hi^1_{R}\rangle|
|{\bf l}angle g, {\bf P}hi^2_{R}\rangle|
|{\bf l}angle h, {\bf P}hi^{3,\vec{k}}_{R}\rangle| \chi_{R}(x,y)\, dx dy$$
$${\bf l}esssim \sum_{n_1, n_2> -5|\vec{k}|, n_3>-N} \int_{\Omega^c_{n_1-1}\cap\Omega^{'c}_{n_2-1}\cap\Omega^{''c}_{n_3-1}\cap
\Omega_{{\bf T}_{n_1, n_2, n_3}}}
MS(f)(x,y) SM(g)(x,y) SS^{\vec{k}}(h)(x,y)\, dx dy$$
\begin{equation}{\bf l}abel{in7}
{\bf l}esssim \sum_{n_1, n_2> -5|\vec{k}|, n_3>-N} 2^{-n_1} 2^{-n_2} 2^{-n_3} |\Omega_{{\bf T}_{n_1, n_2, n_3}}|,
\end{equation}
where
$$\Omega_{{\bf T}_{n_1, n_2, n_3}}:= \begin{itemize}gcup_{R\in{\bf T}_{n_1, n_2, n_3}}R .$$
On the other hand we can write
$$|\Omega_{{\bf T}_{n_1, n_2, n_3}}|{\bf l}eq |\Omega_{{\bf T}_{n_1}}|{\bf l}eq
|\{ (x,y)\in{\mbox{\rm I{\bf k}ern-.22em R}}^2 : MM(\chi_{\Omega_{n_1}})(x,y)> \frac{1}{100} \}|$$
$${\bf l}esssim |\Omega_{n_1}|= |\{ (x,y)\in{\mbox{\rm I{\bf k}ern-.22em R}}^2 : MS(f)(x,y)>\frac{C}{2^{n_1}} \}|{\bf l}esssim 2^{n_1 p}.$$
Similarly, we have
$$|\Omega_{{\bf T}_{n_1, n_2, n_3}}|{\bf l}esssim 2^{n_2 q}$$
and also
$$|\Omega_{{\bf T}_{n_1, n_2, n_3}}|{\bf l}esssim 2^{n_2 \alpha},$$
for every $\alpha >1$. Here we used the fact that all the operators $SM$, $MS$, $SS^{\vec{k}}$, $MM$ are bounded
on $L^s$ (independently of $\vec{k}$) as long as $1<s< \infty$ and also that $|E'|\sim 1$.
In particular, it follows that
\begin{equation}{\bf l}abel{*}
|\Omega_{{\bf T}_{n_1, n_2, n_3}}|{\bf l}esssim 2^{n_1 p \theta_1}
2^{n_2 q \theta_2} 2^{n_3 \alpha \theta_3}
\end{equation}
for any $0{\bf l}eq \theta_1, \theta_2, \theta_3 < 1$, such that $\theta_1+ \theta_2 +\theta_3= 1$.
Now we split the sum in (\ref{in7}) into
\begin{equation}{\bf l}abel{last}
\sum_{n_1, n_2> -5|\vec{k}|, n_3>0} 2^{-n_1} 2^{-n_2} 2^{-n_3} |\Omega_{{\bf T}_{n_1, n_2, n_3}}|+
\sum_{n_1, n_2> -5|\vec{k}|, 0>n_3>-N} 2^{-n_1} 2^{-n_2} 2^{-n_3} |\Omega_{{\bf T}_{n_1, n_2, n_3}}|.
\end{equation}
To estimate the first term in (\ref{last}) we use the inequality (\ref{*}) in the particular case
$\theta_1=\theta_2=1/2$, $\theta_3=0$, while to estimate the second term we use (\ref{*}) for $\theta_j$, $j=1,2,3$
such that $1-p\theta_1>0$, $1-q\theta_2>0$ and $\alpha\theta_3 -1>0$. With these choices, the sum in (\ref{last})
is $O(2^{10|\vec{k}|})$ and this makes the expression in (\ref{split}) to be $O(1)$, after summing over $\vec{k}\in{{\mbox{\rm I{\bf k}ern-.2em N}}}^2$.
This completes our proof.
It is now clear that our argument works equally well in all dimensions. In the general case, exactly as in \cite{cjtc} Section 1, one first reduces
the study of the operator $T_m^{(d)}$ to the study of generic $d$-parameter dyadic paraproducts ${\bf P}i^{\vec{j}}$ for $\vec{j}=(j_1,...,j_d)\in \{1,2,3\}^d$ formally defined by
${\bf P}i^{\vec{j}} = {\bf P}i^{j_1}\otimes \cdots \otimes {\bf P}i^{j_d}$. Then, one observes as before, by using the linear theory and Fefferman-Stein inequality,
that all the corresponding ``square and maximal'' type functions which naturally appear in
inequalities analogous to (\ref{unu}), (\ref{doi}) are bounded in $L^p$ for $1<p<\infty$ (in fact, as before, it is enough to observe this in the $SS...SMM...M$ case, because
all the other expressions are pointwise smaller quantities).
Having all these ingredients, the argument used in Section 3 works similarly.
Finally, the $n$-linear case follows in the same way. The details are left to the reader.
\section{Appendix: proof of Lemma \ref{desc}}
In this section we prove Lemma \ref{desc}. Fix $J\subseteq{\mbox{\rm I{\bf k}ern-.22em R}}$ an interval and let $\phi_J$ be a bump function adapted to $J$.
Consider $\psi$ a smooth function such that ${\rm supp} (\psi) \subseteq [-1/2, 1/2]$ and $\psi = 1$ on $[-1/4, 1/4]$. If $I\subseteq{\mbox{\rm I{\bf k}ern-.22em R}}$ is a generic interval with center $x_I$,
we denote by $\psi_I$ the function defined by
\begin{equation}
\psi_I(x) = \psi(\frac{x-x_I}{|I|}).
\end{equation}
Since
$$1=\psi_J + (\psi_{2J}-\psi_J) + (\psi_{2^2J}-\psi_{2J})+...$$
it follows that
$$\phi_J = \phi_J\cdot\psi_J + \sum_{k=1}^{\infty}\phi_J\cdot (\psi_{2^kJ}-\psi_{2^{k-1}J})
$$
$$=\phi_J\cdot\psi_J + \sum_{k=1}^{\infty}2^{-1000 k}\cdot[2^{1000 k}\phi_J\cdot (\psi_{2^kJ}-\psi_{2^{k-1}J})]
$$
$$:=\sum_{k=0}^{\infty} 2^{-1000 k} \phi^k_J$$
and it is easy to see that all the $\phi^k_J$ functions are bumps adapted to $J$, having the property that ${\rm supp} (\phi^k_J)\subseteq 2^k J$.
Suppose now that in addition we have $\int_{{\mbox{\rm I{\bf k}ern-.22em R}}}\phi_J(x) dx = 0$. This time, we write
$$\phi_J = \phi_J\cdot\psi_J + \phi_J \cdot(1-\psi_J)$$
$$={\bf l}eft[\phi_J\cdot\psi_J - {\bf l}eft(\frac{1}{\int_{{\mbox{\rm I{\bf k}ern-.22em R}}}\psi_J(x) dx}\cdot\int_{{\mbox{\rm I{\bf k}ern-.22em R}}}\phi_J(x)\psi_J(x) dx\right)\cdot \psi_J\right]$$
$$+ {\bf l}eft[{\bf l}eft(\frac{1}{\int_{{\mbox{\rm I{\bf k}ern-.22em R}}}\psi_J(x) dx}\cdot\int_{{\mbox{\rm I{\bf k}ern-.22em R}}}\phi_J(x)\psi_J(x) dx\right)\cdot \psi_J + \phi_J(1-\psi_J)\right]$$
$$:=\phi^0_J + R^0_J.$$
Clearly, by construction we have that $\int_{{\mbox{\rm I{\bf k}ern-.22em R}}}\phi^0_J(x) dx = 0$ and therefore $\int_{{\mbox{\rm I{\bf k}ern-.22em R}}}R^0_J(x) dx =0$. Moreover, $\phi^0_J$ is a bump adapted to the interval $J$
having the property that ${\rm supp} (\phi^0_J)\subseteq J$. On the other hand, since
\begin{equation}
{\bf l}eft|\frac{1}{\int_{{\mbox{\rm I{\bf k}ern-.22em R}}}\psi_J(x) dx}\cdot\int_{{\mbox{\rm I{\bf k}ern-.22em R}}}\phi_J(x)\psi_J(x) dx\right|=
{\bf l}eft|\frac{1}{\int_{{\mbox{\rm I{\bf k}ern-.22em R}}}\psi_J(x) dx}\cdot\int_{{\mbox{\rm I{\bf k}ern-.22em R}}}\phi_J(x)(1-\psi_J(x)) dx\right|
\end{equation}
$${\bf l}esssim 2^{-1000}$$
it follows that $\|R^0_J\|_{\infty}{\bf l}esssim 2^{-1000}$.
Then, we perform a similar decomposition for the ``rest function'' $R^0_J$, but this time we localize it on the larger interval $2J$.
We have
$$R^0_J = R^0_J\cdot\psi_{2J} + R^0_J\cdot(1-\psi_{2J})$$
$$={\bf l}eft[ R^0_J\cdot\psi_{2J}-{\bf l}eft(\frac{1}{\int_{{\mbox{\rm I{\bf k}ern-.22em R}}}\psi_{2J}(x) dx}\cdot\int_{{\mbox{\rm I{\bf k}ern-.22em R}}}R^0_J(x)\psi_{2J}(x) dx\right)\cdot \psi_{2J}\right]$$
$$+ {\bf l}eft[{\bf l}eft(\frac{1}{\int_{{\mbox{\rm I{\bf k}ern-.22em R}}}\psi_{2J}(x) dx}\cdot\int_{{\mbox{\rm I{\bf k}ern-.22em R}}}R^0_J(x)\psi_{2J}(x) dx\right)\cdot \psi_{2J} + R^0_J\cdot(1-\psi_{2J})\right]$$
$$:= 2^{-1000}\phi^1_J + R^1_J.$$
As before, we observe that $\int_{{\mbox{\rm I{\bf k}ern-.22em R}}}\phi^1_J(x) dx = 0$ and also $\int_{{\mbox{\rm I{\bf k}ern-.22em R}}}R^1_J(x) dx=0$. Moreover, $\phi^1_J$ is a bump adapted to $J$ whose support lies in $2J$
and $\|R^1_J\|_{\infty}{\bf l}esssim 2^{-1000\cdot 2}$. Iterating this procedure $N$ times, we obtain the decomposition
\begin{equation}
\phi_J = \sum_{k=0}^N 2^{-1000 k} \phi^k_J + R^N_J
\end{equation}
where all the functions $\phi^k_J$ are bumps adapted to $J$ with $\int_{{\mbox{\rm I{\bf k}ern-.22em R}}}\phi^k_J(x) dx = 0$ and ${\rm supp} (\phi^k_J)\subseteq 2^k J$, while
$\|R^N_J\|_{\infty}{\bf l}esssim 2^{-1000 N}$.
This completes the proof of the Lemma.
\end{document} | math | 33,665 |
\begin{document}
\begin{abstract}
We define and study an equivariant version of Farber's topological complexity for spaces with a given compact group action. This is a special case of the equivariant sectional category of an equivariant map, also defined in this paper. The relationship of these invariants with the equivariant Lusternik-Schnirelmann category is given. Several examples and computations serve to highlight the similarities and differences with the non-equivariant case. We also indicate how the equivariant topological complexity can be used to give estimates of the non-equivariant topological complexity.
\end{abstract}
\maketitle
\section{Introduction}
The sectional category of a map $p\colon\thinspace E\to B$, denoted $\mbox{\rm secat}\,(p)$, is the minimum number of open sets needed to cover $B$, on each of which $p$ admits a homotopy section. It was first studied extensively by \v Svarc \cite{S} for fibrations (under the name genus) and later by Berstein and Ganea \cite{BG} for arbitrary maps. The notion of sectional category generalizes the classical (Lusternik-Schnirelmann) category, since $\mbox{\rm secat}\,(p)=\mbox{\rm cat}(B)$ whenever $E$ is contractible and $p$ is surjective. For a general overview of these and other category-type notions, we refer the reader to the survey article of James \cite{Jam} and the book of Cornea-Lupton-Oprea-Tanr\' e \cite{CLOT}.
Further to the classical applications of category to critical point theory, the concept of sectional category has been applied in a variety of settings.
We mention the work of Smale \cite{Sma} and Vassiliev \cite{V} on the complexity of algorithms for solving polynomial equations,
and applications to the theory of embeddings \cite{S}. More recently, Farber has applied these ideas to the motion planning problem in robotics \cite{Far03,Far04}.
He defines the {\em topological complexity} of a space $X$, denoted $\mathbf{TC}(X)$, to be the sectional category of the free path fibration on $X$.
The topological complexity is a numerical homotopy invariant which measures the `navigational complexity' of $X$, when viewed as the configuration space of a mechanical system. Along with various related invariants, it has enjoyed much attention in the recent literature (see \cite{GL,BGRT,G} for example).
In this paper we begin a systematic study of the equivariant versions of these notions. For simplicity, we restrict to compact group actions (although most of our results remain true for proper actions). Let $G$ be a compact Hausdorff topological group, and let $p\colon\thinspace E\to B$ be a $G$-map. Then the equivariant sectional category of $p$, denoted $\mbox{\rm secat}_G\,(p)$, is the minimum number of invariant open sets needed to cover $B$, on each of which $p$ admits a $G$-homotopy section. If $p$ is a $G$-fibration, this is equivalent to asking for a $G$-equivariant section on each open set in the cover. In the case when the actions are trivial, $\mbox{\rm secat}_G\,(p)$ reduces to the ordinary (non-equivariant) sectional category $\mbox{\rm secat}\,(p)$.
The equivariant sectional category does not seem to have appeared in the literature until now, although we note below (Corollary \ref{contractible}) that it generalizes the equivariant category, or $G$-category, in many cases of interest. This latter invariant has been extensively studied (see for example \cite{F,Mar,HC}), and gives a lower bound for the number of critical orbits of a $G$-invariant functional on a $G$-manifold. We include a review of some of its properties in Section 3 below, where we also prove product inequalities for equivariant category.
The equivariant topological complexity of a $G$-space $X$, denoted $\mathbf{TC}G(X)$, is defined in Section 5 to be the equivariant sectional category of the free path fibration $\pi\colon\thinspace X^I\to X\times X$, where $G$ acts diagonally on the product and in the obvious way on paths in $X$. After proving that $\mathbf{TC}G(X)$ is a $G$-homotopy invariant (Theorem \ref{Ghinv}), we give several inequalities relating $\mathbf{TC}G(X)$ to the equivariant and non-equivariant categories and topological complexities of the various fixed point sets. We also show by examples that $\mathbf{TC}G(X)$ can be equal to $\mathbf{TC}(X)$, or at the other extreme, one can be finite and the other infinite (this always happens for example if $X$ is a $G$-manifold which is connected but not $G$-connected). For a group acting on itself by left translations, we show that $\mathbf{TC}G(G)=\mbox{\rm cat}(G)$, so that category of Lie groups is obtained as a special case of equivariant topological complexity (Theorem \ref{catgrp}). Various other results are given, including a lower bound in terms of equivariant cohomology (Theorem \ref{eqcohom}) and an inequality which bounds the ordinary topological complexity of the fibre space with fibre $X$ associated to a numerable principal $G$-bundle by the product of $\mathbf{TC}G(X)$ and the topological complexity of the base space (Theorem \ref{fred}).
The invariant $\mathbf{TC}G(X)$ has an interpretation in terms of the motion planning problem, when $X$ is viewed as the configuration space of a mechanical system which exhibits $G$ as a group of symmetries. Namely, it is the minimum number of domains of continuity of motion planners in $X$ which preserve the symmetry. Whilst we do not pursue this viewpoint here, it is conceivable that the invariant $\mathbf{TC}G(X)$ may find applications in practical problems of engineering. For more background on the topological approach to motion planning, we refer the reader to \cite{Far06}.
The computation of category and topological complexity in the non-equivariant case are difficult problems which continue to inspire a great deal of research in homotopy theory, and serve to gauge the power of new topological techniques. We believe that the equivariant counterparts of these problems can fill a similar niche in equivariant homotopy theory.
\section{Topological Complexity}
We begin by recalling some definitions and fixing some notation. The term {\em fibration} will always refer to a Hurewicz fibration. The sectional category of a fibration was introduced by \v{S}varc (under the name {\em genus}) and generalized to any map by Berstein and Ganea.
\begin{defn} \cite{BG} The {\em sectional category} of a map $p\colon\thinspace E\to B$, denoted $\mbox{\rm secat}\,(p)$, is the least integer $k$ such that $B$ may be covered by $k$ open sets $\{ U_1,\ldots, U_k\}$ on each of which there exists a map $s\colon\thinspace U_i\to E$ such that $ps\colon\thinspace U_i\to B$ is homotopic to the inclusion $i_{U_i}\colon\thinspace U_i\hookrightarrow B$. If no such integer exists we set $\mbox{\rm secat}\,(p)=\infty$.
\end{defn}
The sets $U_i\subseteq B$ in the above definition are called {\em sectional categorical} for $p$.
\begin{remark} \v{S}varc's original definition \cite{S} of the genus of a fibration $p\colon\thinspace E\to B$ was as the least integer $k$ such that $B$ may be covered by $k$ open sets $\{ U_1,\ldots, U_k\}$ on each of which there exists a local section of $p$, that is to say a map $s\colon\thinspace U_i\to E$ such that $ps=i_{U_i}$. It is easy to see (using the HLP) that this coincides with $\mbox{\rm secat}\,(p)$ defined above.
\end{remark}
\begin{remark} In this paper, all our category-type invariants are un-normalized (they are equal to the number of open sets in the cover). For instance, $\mbox{\rm secat}\,(p)=1$ if and only if $p$ admits a homotopy section. Thus our definitions exceed by one those in the book \cite{CLOT}, but are in agreement with those used in previous works of the authors, such as \cite{HC,G}.
\end{remark}
Recall that the {\em (Lusternik-Schnirelmann) category} of a space $X$, denoted $\mbox{\rm cat}(X)$, is the least $k$ such that $X$ may be covered by $k$ open sets $\{ U_1,\ldots , U_k\}$ such that each inclusion $i_{U_i}\colon\thinspace U_i\hookrightarrow X$ is null-homotopic. The $U_i$ are called {\em categorical} sets.
The sectional category of a surjective fibration $p\colon\thinspace E\to B$ is bounded above by the category of the base, $\mbox{\rm secat}\,(p)\le \mbox{\rm cat}(B)$, and they coincide if the space $E$ is contractible \cite{S}.
Lower bounds for the sectional category can be found using cohomology. If $I$ is an ideal in the commutative ring $R$, the {\em nilpotency} of $I$, denoted $\mbox{\rm nil}\, I$, is the maximum number of factors in a nonzero product of elements from $I$. Let $H^*$ denote cohomology with coefficients in an arbitrary commutative ring.
\begin{prop}\cite{S}
Let $p\colon\thinspace E\to B$ be a fibration and $p^*\colon\thinspace H^*(B)\to H^*(E)$ be the induced homomorphism. Then $\mbox{\rm secat}\,(p)> \mbox{\rm nil}\ker p^*$.
\end{prop}
The topological complexity of a space $X$, denoted $\mathbf{TC}(X)$, is a homotopy invariant defined by Farber \cite{Far03} in order to study the motion planning problem in robotics. We recall now some of its important properties. For more detail we refer the reader to the original papers of Farber \cite{Far03,Far04,Far06}. For any space $X$, let $X^I$ denote the space of paths in $X$ endowed with the compact-open topology. The {\em free path fibration} is the map $\pi\colon\thinspace X^I\to X\times X$ given by $\pi({\mathfrak{genus}}_Gamma)=({\mathfrak{genus}}_Gamma(0), {\mathfrak{genus}}_Gamma(1))$. It is surjective if $X$ is path-connected.
\begin{defn} The {\em topological complexity} of a space $X$ is
\[
\mathbf{TC}(X)=\mbox{\rm secat}\,(\pi),
\]
the sectional category of the free path fibration $\pi\colon\thinspace X^I\to X\times X$.
\end{defn}
\begin{prop}
If $X$ dominates $Y$, then $\mathbf{TC}(X){\mathfrak{genus}}_Geq \mathbf{TC}(Y)$. In particular, if $X\simeq Y$ then $\mathbf{TC}(X)=\mathbf{TC}(Y)$.
\end{prop}
\begin{prop} For a path-connected space $X$,
$
\mbox{\rm cat}(X)\le\mathbf{TC}(X)\le\mbox{\rm cat}(X\times X).
$
\end{prop}
\begin{prop} If $X$ is path-connected and paracompact then
$
\mathbf{TC}(X)\le 2\dim X +1,
$
where $\dim$ denotes the covering dimension.
\end{prop}
\begin{defn} Let $\Bbbk$ be a field. Then cup product defines a homomorphism of rings
\[
\xymatrix{
H^*(X;\Bbbk)\otimes_\Bbbk H^*(X;\Bbbk)\ar[r]^-{\cup} & H^*(X;\Bbbk).
}
\]
The {\em ideal of zero-divisors} $\mathcal{Z}_\Bbbk\subseteq H^*(X;\Bbbk)\otimes_\Bbbk H^*(X;\Bbbk)$ is the kernel of this homomorphism.
\end{defn}
\begin{prop}[Cohomological lower bound] $\mathbf{TC}(X)>\mbox{\rm nil} \,\mathcal{Z}_\Bbbk$ for any field $\Bbbk$.
\end{prop}
\begin{exam}\label{TCspheres}
The topological complexity of the standard $n$-sphere is
\[
\mathbf{TC}(S^n) = \left\{ \begin{array}{ll} \infty & (n=0) \\
2 & (n{\mathfrak{genus}}_Ge 1\mbox{ odd}) \\
3 & (n{\mathfrak{genus}}_Ge 2\mbox{ even}). \end{array} \right.
\]
\end{exam}
\section{Equivariant category}
In this section we recall some definitions and results related to the equivariant (Lusternik-Schnirelmann) category of a $G$-space. We also prove a product inequality for a diagonal action with fixed points, and state the analogous inequality for product actions.
{\em For the remainder of the paper, $G$ will denote a compact Hausdorff topological group acting continuously on a Hausdorff space $X$ on the left}. In this case, we say that $X$ is a {\em $G$-space}. For each $x\in X$ the {\em isotropy group} $G_x=\{h\in G\mid hx=x\}$ is a closed subgroup of $G$. The set $Gx=\{gx\mid g\in G\}\subseteq X$ is called the {\em orbit} of $x$, and also denoted ${\mathcal O} (x)$. There is a homeomorphism from the coset space $G/G_x$ to $Gx$, which sends $gG_x$ to $gx$ for each $g\in G$.
The {\em orbit space} $X/G$ is the set of equivalence classes determined by the action, endowed with the quotient topology. Since $G$ is compact and $X$ is Hausdorff, $X/G$ is also Hausdorff, and the {\em orbit map} $p\colon\thinspace X\to X/G$ sending a point to its orbit is both open and closed \cite[Chapter I.3]{tD}.
If $H$ is a closed subgroup of $G$, then $X^H=\{x\in X|\; hx=x \mbox{ for all }h\in H\}$ is called the {\em $H$-fixed point set} of $X$.
Let $X$ and $Y$ be $G$-spaces.
Two $G$-maps $\phi, \psi\colon\thinspace X\to Y$ are {\em $G$-homotopic}, written $\phi\simeq_G \psi$, if there is a $G$-map $F\colon\thinspace X \times I \to Y$ with $F_0=\phi$ and $F_1=\psi$, where $G$ acts trivially on $I$ and diagonally on $X\times I$.
We now begin to discuss the equivariant category of a $G$-space $X$, as studied for instance in \cite{HC,F,Mar}. An open set $U\subseteq X$ is described as {\em invariant} if $gU\subseteq U$ for all $g\in G$.
\begin{defn} An invariant set $U$ in a $G$-space $X$ is called {\em $G$-categorical} if the inclusion $i_U\colon\thinspace U\to X$ is $G$-homotopic to a map with values in a single orbit.
\end{defn}
\begin{defn} The {\em equivariant category} of a $G$-space $X$, denoted $\mbox{\rm cat}_G(X)$, is the least integer $k$ such that $X$ may be covered by $k$ open sets $\{ U_1,\ldots , U_k\}$, each of which is $G$-categorical.
\end{defn}
\begin{defn} A $G$-space $X$ is said to be {\em $G$-contractible} if $\mbox{\rm cat}_G(X)=1$.
\end{defn}
\begin{exam}\label{circle} Let $G=S^1$ acting freely on $X=S^1$ by rotations. Since the action is transitive we have $\mbox{\rm cat}_G(X)=1$, whilst $\mbox{\rm cat}(X)=2$. Note that $X$ is $G$-contractible but not contractible.
\end{exam}
\begin{prop}[\cite{F,Mar}]\label{X/G}
When $X$ is a free metrizable $G$-space we have
\[
\mbox{\rm cat}_G(X)=\mbox{\rm cat}(X/G),
\]
the non-equivariant category of the orbit space. In general, $\mbox{\rm cat}_G(X){\mathfrak{genus}}_Geq\mbox{\rm cat}(X/G)$.
\end{prop}
The equivariant category of a $G$-space is independent from the category of the space, as the following family of examples illustrates.
\begin{exam} For $n{\mathfrak{genus}}_Ge 1$, let $G=S^1\subset\mathbb{C}$ act on the unit sphere $S^{2n-1}\subset\mathbb{C}^{n}$ by complex multiplication in each co-ordinate. Then $\mbox{\rm cat}_G(S^{2n-1}) = \mbox{\rm cat}(\mathbb{C} P^{n-1}) = n$, whilst $\mbox{\rm cat}(S^{2n-1})=2$.
\end{exam}
Just like its non-equivariant counterpart, the $G$-category finds applications in critical point theory.
\begin{thm}[\cite{F,Mar}] \label{crit}
Let $M$ be a compact $G$-manifold, and let $f\colon\thinspace M\to \mathbb{R}$ be a smooth $G$-invariant function on $M$. Then $f$ has at least $\mbox{\rm cat}_G(M)$ critical orbits.
\end{thm}
\begin{exam} \label{U(n)}
Let $U(n)$ denote the compact Lie group of $n\times n$-unitary matrices. Then $U(n)$ acts smoothly on itself by conjugation, $A\cdot B = ABA^{-1}$. We can apply Theorem \ref{crit} to obtain an upper bound for $\mbox{\rm cat}_{U(n)}(U(n))$, as follows.
By diagonalization, two unitary matrices are conjugate if and only if they have the same set of eigenvalues (all of which lie on the unit circle in $\mathbb{C}$). Thus we can define an invariant functional
\[
f\colon\thinspace U(n)\to \mathbb{R},\qquad f(A) = \sum_{i=1}^n |\lambda_i - 1|,
\]
where $\{\lambda_1, \ldots , \lambda_n\}$ is the set of eigenvalues of $A$. It is easy to see that $f$ is smooth, and that the critical orbits are the conjugacy classes of the matrices $\operatorname{diag}(-1,\ldots , -1,1,\ldots, 1)$. There are precisely $n+1$ such orbits. We therefore have $\mbox{\rm cat}_{U(n)}(U(n))\le n+1$ by Theorem \ref{crit}.
In Example \ref{conjugation} below we will see that $\mbox{\rm cat}_{U(n)}(U(n))= n+1$, by relating it to the notion of equivariant topological complexity. \qed
\end{exam}
We now give an equivariant version of the product inequality for category. Our treatment is based on \cite[Theorem 1.37]{CLOT}, which in turn is based on that of Fox \cite[Theorem 9]{Fox}. In particular, our proof relies on a notion of categorical sequence.
\begin{defn} A {\em $G$-categorical sequence} in $X$ of length $k$ is a nested sequence $A_0\subseteq A_1\subseteq\cdots\subseteq A_k=X$ with $A_0=\emptyset$ and the property that each difference $A_i-A_{i-1}$ is invariant and is contained in some $G$-categorical open set $U_i$. (Note that we do not require the $A_i$ to themselves be invariant.)
\end{defn}
\begin{lemma} A $G$-space $X$ has a $G$-categorical sequence of length $k$ if and only if $\mbox{\rm cat}_G(X)\leq k$. \qed
\end{lemma}
Just as in the non-equivariant case, we also need some separation and connectedness conditions. Recall that a space $X$ is called {\em completely normal} if whenever $A,B\subseteq X$ such that $\overline{A}\cap B = \emptyset = A\cap\overline{B}$, then $A$ and $B$ have disjoint open neighbourhoods in $X$. For example, metric spaces and CW-complexes are completely normal.
\begin{defn}
A $G$-space $X$ is called {\em $G$-completely normal} if whenever $A,B\subseteq X$ are invariant sets such that $\overline{A}\cap B = \emptyset = A\cap\overline{B}$, then $A$ and $B$ have disjoint open invariant neighbourhoods in $X$.
\end{defn}
\begin{lemma}
If $X$ is a completely normal $G$-space, then $X$ is $G$-completely normal.
\end{lemma}
\begin{proof}
It is well known that if $X$ is completely normal and $G$ is compact, then $X/G$ is completely normal. Thus it suffices to prove that complete normality of $X/G$ implies $G$-complete normality of $X$. This is an exercise in general topology, using the orbit map $p\colon\thinspace X\to X/G$, and is left to the reader.
\end{proof}
\begin{defn}
A $G$-space $X$ is said to be {\em $G$-connected} if
the $H$-fixed point set $X^H$ is path-connected for every closed subgroup $H$ of
$G$.
\end{defn}
\begin{lemma}[Conservation of isotropy]\label{4}
Let $X$ be a $G$-connected $G$-space, and let $x, y \in X$ such that $G_x\subseteq G_y$. Then there exists a $G$-homotopy $F\colon\thinspace {\mathcal O}(x)\times I\to X$ such that $F_0=i_{{\mathcal O}(x)}$ and $F_1({\mathcal O}(x))\subset {\mathcal O}(y)$.
\end{lemma}
\begin{proof}
Let $H=G_x$. Then $x, y\in X^H$ since $G_{x}\subseteq G_{y}$. Consider a path $\alpha\colon\thinspace I\to X^H$ joining $x$ and $y$. Then $H\subseteq G_{\alpha(t)}$ for all $t\in I$. Define a homotopy $F\colon\thinspace {G/{G_x}}\times I \to X$ given by $F(gG_x, t)=g\alpha(t)$. We have that $F$ is well defined, is equivariant and is a homotopy of the inclusion into the orbit ${\mathcal O}(y)$.
\end{proof}
\begin{thm}\label{productI}
Let $X$ and $Y$ be $G$-connected $G$-spaces such that $X\times Y$ is completely normal. If $X^G\neq \emptyset$ or $Y^G\neq\emptyset$, then
\[
\mbox{\rm cat}_G(X\times Y)\leq\mbox{\rm cat}_G(X)+\mbox{\rm cat}_G(Y)-1,
\]
where $X\times Y$ is given the diagonal $G$-action.
\end{thm}
\begin{proof}
Suppose that $\mbox{\rm cat}_G(X)=n$ with $G$-categorical sequence $\{A_0,A_1,\ldots , A_n\}$ and $\mbox{\rm cat}_G(Y)=m$ with $G$-categorical sequence $\{B_0,B_1,\ldots , B_m\}$. Denote by $U_i\subseteq X$ the open $G$-categorical set containing $A_i-A_{i-1}$, and by $W_j\subseteq Y$ the open $G$-categorical set containing $B_j-B_{j-1}$. Suppose for concreteness that $X^G\neq\emptyset$. By Lemma \ref{4} we may assume that the inclusions $i_{U_i}\colon\thinspace U_i\to X$ are all $G$-homotopic into ${\mathcal O} (x_0) = \{x_0\}$, where $x_0\in X^G$ is some fixed point. Each inclusion $i_{W_j}\colon\thinspace W_j\to Y$ is $G$-homotopic into ${\mathcal O}(y_j)$ for some $y_j\in Y$. It follows that $U_i\times W_j\subseteq X\times Y$ is $G$-homotopic into ${\mathcal O}(x_0,y_j) = \{x_0\}\times{\mathcal O}(y_j)$, and hence these products are all $G$-categorical in $X\times Y$.
Define subsets of $X\times Y$ by
\[
C_0=\emptyset,\qquad C_k = \bigcup_{r=1}^{k} A_r\times B_{k+1-r}\quad(1\leq k\leq n+m-1),
\]
where we set $A_i=\emptyset$ for $i>n$ and $B_j=\emptyset$ for $j>m$. We claim that $\{C_0,\ldots, C_{n+m-1}\}$ is a $G$-categorical sequence for $X\times Y$.
The proof of this claim proceeds by analogy with the non-equivariant case \cite[Theorem 1.37]{CLOT}, using the $G$-complete normality of $X\times Y$. Therefore we omit the details.
\end{proof}
We remark that a similar result (with a similar proof) was given by Cicorta\c s \cite[Proposition 3.2]{Cic}. There the assumption on the existence of fixed points was omitted, however, leading to counter-examples. For example, let $G=S^1$ acting on $X=Y=S^1$ by rotations as in Example \ref{circle}. Then $\mbox{\rm cat}_G(X\times Y) = \mbox{\rm cat}((S^1\times S^1)/S^1) = \mbox{\rm cat}(S^1) = 2$, whilst $\mbox{\rm cat}_G(X) + \mbox{\rm cat}_G(Y) - 1 = 1$. The problem is that an orbit of the diagonal action is not necessarily a product of orbits.
The hypothesis on fixed point sets can be dropped when considering more general product actions. Let $K$ be another compact Hausdorff group. Then the product of a $G$-space $X$ and a $K$-space $Y$ becomes a $G\times K$-space in an obvious way. The orbits of this action are the products of orbits, and one easily obtains the following result.
\begin{thm}\label{productII}
Let $X$ be a path-connected $G$-space and $Y$ be a path-connected $K$-space, such that $X\times Y$ is completely normal. Then
\[
\mbox{\rm cat}_{G\times K}(X\times Y)\leq \mbox{\rm cat}_G(X) + \mbox{\rm cat}_K(Y)-1.
\]
\end{thm}
\section{Equivariant sectional category}
In this section we generalize the notion of sectional category to the equivariant setting.
\begin{defn} The {\em equivariant sectional category} of a $G$-map $p\colon\thinspace E\to B$, denoted $\mbox{\rm secat}_G\,(p)$, is the least integer $k$ such that $B$ may be covered by $k$ invariant open sets $\{ U_1,\ldots, U_k\}$ on each of which there exists a $G$-map $s\colon\thinspace U_i\to E$ such that $ps\simeq_G i_{U_i}\colon\thinspace {U_i}\hookrightarrow B$.
\end{defn}
The sets $U_i\subseteq B$ in the above will be called {\em $G$-sectional categorical} for $p$.
\begin{prop}\label{secatfibr} If $p\colon\thinspace E\to B$ is a $G$-fibration, then $\mbox{\rm secat}_G\,(p)\leq k$ if and only if $B$ may be covered by $k$ invariant open sets $\{ U_1,\ldots, U_k\}$ on each of which there exists a {\em local $G$-section}, that is a $G$-map $\sigma_i\colon\thinspace U_i\to E$ such that $p\sigma_i = i_{U_i}\colon\thinspace {U_i}\hookrightarrow B$.
\end{prop}
\begin{proof}
This is analogous to the non-equivariant case, using the $G$-HLP \cite[page 53]{tD}.
\end{proof}
Next we observe that equivariant sectional category of $G$-fibrations cannot increase under taking pullbacks (compare \cite[Proposition 7]{S}).
\begin{prop}\label{pullback}
Let $p\colon\thinspace E\to B$ be a $G$-fibration and $f\colon\thinspace A\to B$ be a $G$-map. The pullback $q\colon\thinspace A\times_B E \to A$ of $p$ along $f$ satisfies
$\mbox{\rm secat}_G\,(q)\leq\mbox{\rm secat}_G\,(p)$.
\end{prop}
\begin{proof}
Given an invariant open set $U\subseteq B$ with $G$-section $s\colon\thinspace U\to E$ of $p$, one obtains a $G$-section $\sigma\colon\thinspace f^{-1}(U)\to A\times_B E$ of $q$ by setting $\sigma(a) = (a,sf(a))$.
\end{proof}
We now study conditions under which the equivariant category of a $G$-space $B$ is an upper bound for the equivariant sectional category of a $G$-map $p\colon\thinspace E\to B$.
\begin{prop}[Equivariant version of the connectivity condition]\label{5}
Let $p\colon\thinspace E\to B$ be a $G$-map.
If $B$ is $G$-connected and $E^G\not = \emptyset$, then $\mbox{\rm secat}_G\,(p)\le \mbox{\rm cat}_G(B)$.
\end{prop}
\begin{proof}
Let $U$ be a $G$-categorical set for $B$ and let $F\colon\thinspace U\times I\to B$ be the $G$-homotopy such that $F_0=i_U$ and $F_1=c$ with $c(U)\subseteq {\mathcal O}(x_0)$. Choose an $e\in E^G$, and let $b=p(e)$. By Lemma \ref{4}, since $B$ is $G$-connected and $G_{x_0}\subseteq G_b=G$ we have that there exists a $G$-homotopy $\Phi\colon\thinspace {\mathcal O}(x_0)\times I\to B$ such that $\Phi_0=i_{{\mathcal O}(x_0)}$ and $\Phi_1({\mathcal O}(x_0))\subseteq {\mathcal O}(b)=\{b\}$.
Consider $s\colon\thinspace U\to E$ given by $s(x)=e$ for all $x\in U$. The map $s$ is equivariant and the composition of the homotopies $F$ and $\Phi$ provides a homotopy from $i_U$ to $ps$.
\end{proof}
\begin{prop}[Equivariant version of the surjectivity condition]\label{6}
Let $p\colon\thinspace E\to B$ be a $G$-map.
If $p(E^H)=B^H$ for all closed subgroups $H$ of $G$, then $\mbox{\rm secat}_G\,(p)\le \mbox{\rm cat}_G(B)$.
\end{prop}
\begin{proof}
Let $U$ be a $G$-categorical set for $B$ and let $F\colon\thinspace U\times I\to B$ be the $G$-homotopy such that $F_0=i_U$ and $F_1=c$ with $c(U)\subseteq {\mathcal O}(x_0)$.
Let $H=G_{x_0}$, then $x_0\in B^H=p(E^H)$. Therefore there exists $z_0\in E^H$ such that $p(z_0)=x_0$. Define $s\colon\thinspace U\to E$ by $s(x)=g z_0$ if $c(x)=g x_0$.
The proof that $s$ is equivariant follows from the fact that $c$ is equivariant. Moreover, $ps$ is $G$-homotopic to the inclusion since $ps(x)=p(g z_0)=g p(z_0)=g x_0=c(x)$ and $c\simeq_G i_U$.
\end{proof}
Finally in this section, we investigate equivariant sectional category in the case when the total space is $G$-contractible.
\begin{prop}\label{Gcontract} Let $p\colon\thinspace E\to B$ be a $G$-map.
If $E$ is a $G$-contractible space then $\mbox{\rm cat}_G (B) \le \mbox{\rm secat}_G\, (p)$.
\end{prop}
\proof
Let $U$ be a $G$-sectional categorical set for $p$ and $s\colon\thinspace U\to E$ be a $G$-map such that $ps\simeq_G i_U$. We have that the identity map $\operatorname{id}_E$ is $G$-homotopic to a $G$-map $c$ whose image is contained in a single orbit ${\mathcal O}(x)$.
Then $pcs\simeq_G ps\simeq_Gi_U$ and $pcs(U)$ is contained in ${\mathcal O}(p(x))$. Thus $U$ is $G$-categorical in $B$.
\qed
\begin{cor} \label{contractible} Let $p\colon\thinspace E\to B$ be a $G$-map and $E$ be a $G$-contractible space. If $B$ is $G$-connected and $E^G\not = \emptyset$ or if $p(E^H)=B^H$ for all closed subgroups $H$ of $G$, then $\mbox{\rm secat}_G\, (p)= \mbox{\rm cat}_G (B)$.
\end{cor}
In particular, if $X$ is a $G$-connected space with a fixed point $x\in X^G$, then the inclusion $p\colon\thinspace \{x\}\to X$ is a $G$-map with $\mbox{\rm secat}_G\, (p)= \mbox{\rm cat}_G (X)$.
\section{Equivariant topological complexity}
When $X$ is a $G$-space, the free path fibration $\pi\colon\thinspace X^I\to X\times X$ is a $G$-fibration with respect to the actions
$$G\times X^I\to X^I, \qquad G\times X\times X\to X\times X,$$
$$g({\mathfrak{genus}}_Gamma)(t)=g({\mathfrak{genus}}_Gamma(t)),\qquad g(x,y)=(gx,gy).$$
The verification of this fact is straightforward; one may use a $G$-equivariant version of \cite[Theorem 2.8.2]{Spa}.
\begin{defn} The {\em equivariant topological complexity} of the
$G$-space $X$, denoted $\mathbf{TC}G(X)$, is defined as the equivariant sectional category of the free path fibration $\pi\colon\thinspace X^I\to X\times X$. That is, $$\mathbf{TC}G(X)=\mbox{\rm secat}_G\,(\pi\colon\thinspace X^I\to X\times X).$$
\end{defn}
In other words, the equivariant topological complexity is the least integer $k$ such that
$X\times X$ may be covered by $k$ invariant open sets
$\{U_1,\ldots ,U_k\}$, on each of which there is a $G$-equivariant
map $s_i\colon\thinspace U_i\to X^I$ such that $\pi s_i\simeq_G i_{U_i}\colon\thinspace U_i\hookrightarrow X\times X$. (Since $\pi$ is a $G$-fibration, this is equivalent to requiring $\sigma_i\colon\thinspace U_i\to X^I$ such that $\pi \sigma_i = i_{U_i}$.) If no such integer exists then we set $\mathbf{TC}G(X)=\infty$.
We first show that equivariant topological complexity is a $G$-homotopy invariant. Let $X$ and $Y$ be $G$-spaces. We say that $X$ {\em $G$-dominates} $Y$ if there exist $G$-maps $\phi\colon\thinspace X\to Y$ and $\psi\colon\thinspace Y\to X$ such that $\phi\psi\simeq_G \operatorname{id}_Y$. If in addition $\psi\phi\simeq_G \operatorname{id}_X$, then $\phi$ and $\psi$ are {\em $G$-homotopy equivalences}, and $X$ and $Y$ are {\em $G$-homotopy equivalent}, written $X\simeq_G Y$.
\begin{thm}\label{Ghinv}
If $X$ $G$-dominates $Y$ then $\mathbf{TC}G(X){\mathfrak{genus}}_Ge \mathbf{TC}G(Y)$. In particular, if $X\simeq_G Y$ then $\mathbf{TC}G(X)=\mathbf{TC}G(Y)$.
\end{thm}
\begin{proof}
Let $\phi\colon\thinspace X\to Y$ and $\psi\colon\thinspace Y\to X$ be $G$-maps such that $\phi\psi\simeq_G \operatorname{id}_Y$, and let $U\subseteq X\times X$ be $G$-sectional categorical for $\pi_X\colon\thinspace X^I\to X\times X$. Hence there exists a $G$-map $s\colon\thinspace U\to X^I$ such that $\pi_X s\simeq_G i_U\colon\thinspace U\hookrightarrow X\times X$.
We will show that $V=(\psi\times \psi)^{-1}U\subseteq Y\times Y$ is $G$-sectional categorical for $\pi_Y\colon\thinspace Y^I\to Y\times Y$. Denote by $\overline{(\psi\times\psi)}\colon\thinspace V\to U$ the map obtained by restricting the domain and range of $(\psi\times\psi)$. Let $\sigma\colon\thinspace V\to Y^I$ be the composition $\sigma = \widetilde\phi \circ s \circ \overline{(\psi\times\psi)}$, where $\widetilde\phi\colon\thinspace X^I\to Y^I$ is the map induced by $\phi$. Then
$$
\pi_Y\sigma
= (\phi\times\phi)\pi_X s \overline{(\psi\times\psi)}
\simeq_G (\phi\times\phi) i_U \overline{(\psi\times\psi)}
= (\phi\times\phi)(\psi\times \psi) i_V
\simeq_G i_V,
$$
hence $V$ is $G$-sectional categorical for $\pi_Y$.
Now if $\{U_1,\ldots , U_k\}$ is a $G$-sectional categorical cover of $X\times X$, then $\{V_1,\ldots, V_k\}$ defined as above is a $G$-sectional categorical cover of $Y\times Y$. This proves the first statement, and the second follows immediately.
\end{proof}
It is obvious that $\mathbf{TC}(X)\leq\mathbf{TC}G(X)$ for any $G$-space $X$. More generally we have the following.
\begin{prop}\label{fixed}
Let $X$ be a $G$-space, and let $H$ and $K$ be closed subgroups of $G$ such that $X^H$ is $K$-invariant. Then $\mathbf{TC}K(X^H)\le\mathbf{TC}G (X)$.
\end{prop}
\begin{proof}
Let $U\subseteq X\times X$ be $G$-sectional categorical for $\pi\colon\thinspace X^I\to X\times X$, and let $\sigma\colon\thinspace U\to X^I$ be a $G$-map such that $\pi \sigma = i_{U}\colon\thinspace U\hookrightarrow X\times X$.
Define $V=U\cap (X^H\times X^H)\subseteq X^H\times X^H$, and note that $V$ is $K$-invariant.
Since $\sigma$ is $G$-equivariant it takes $H$-fixed points to $H$-fixed points, and so restricts to a $K$-equivariant map $\sigma_V\colon\thinspace V\to (X^H)^I$. It is clear that $\pi\sigma_V=i_V$ and therefore $V$ is $K$-sectional categorical for $\pi|_{X^H}\colon\thinspace (X^H)^I\to X^H\times X^H$.
\end{proof}
\begin{cor}\label{subgroups} Let $X$ be a $G$-space. Then
\begin{enumerate}
\item $\mathbf{TC}(X^H)\le\mathbf{TC}G (X)$ for all closed subgroups $H$ of $G$.
\item $\mathbf{TC}K(X)\le\mathbf{TC}G (X)$ for all closed subgroups $K$ of $G$.
\end{enumerate}
\end{cor}
\begin{cor} If $X$ is not $G$-connected, then $\mathbf{TC}G(X)=\infty$.
\end{cor}
\begin{proof} Let $H$ be a closed subgroup of $G$ such that $X^H$ is not path-connected. Then $\mathbf{TC}(X^H)=\infty$ and the result follows from Corollary \ref{subgroups} (1).
\end{proof}
The next three results describe the basic relationship of equivariant topological complexity with equivariant category.
\begin{prop}\label{upper} If $X$ is $G$-connected, then $\mathbf{TC}G(X)\le \mbox{\rm cat}_G(X\times X)$.
\end{prop}
\begin{proof} Given a closed subgroup $H$ of $G$, $(X\times X)^H=X^H\times X^H$ is path-connected since $X$ is $G$-connected. Then the map
$\pi|_{X^H}\colon\thinspace (X^H)^I \to X^H\times X^H$ is surjective and the result follows from Proposition \ref{6}.
\end{proof}
\begin{prop}\label{lower} If $X$ is $G$-connected, and $H=G_z\subseteq G$ is the isotropy group of some $z\in X$, then
$\mbox{\rm cat}_H(X)\le\mathbf{TC}G(X)$.
\end{prop}
\begin{proof}
We have $\mathbf{TC}_H(X)\le \mathbf{TC}G(X)$ by Corollary \ref{subgroups} (2). When we pull back $\pi\colon\thinspace X^I\to X\times X$ along the $H$-equivariant map
\[
j\colon\thinspace X\to X\times X,\qquad j(x)=(z,x)
\]
we obtain an $H$-fibration $p\colon\thinspace PX\to X$ whose total space $PX=\{ {\mathfrak{genus}}_Gamma\colon\thinspace I\to X \mid {\mathfrak{genus}}_Gamma(0)=z\}$ is $H$-contractible. Therefore $\mbox{\rm cat}_H(X)\le \mbox{\rm secat}\,_H(p)\le \mathbf{TC}_H(X)$ by Propositions \ref{Gcontract} and \ref{pullback}.
\end{proof}
\begin{cor}\label{catfixed} Let $X$ be a $G$-connected $G$-space with $X^G\not=\emptyset$. Then
\begin{enumerate}
\item $\mbox{\rm cat}_G(X)\le\mathbf{TC}G(X)\le 2\mbox{\rm cat}_G(X)-1$.
\item $\mathbf{TC}G(X)=1$ if and only if $X$ is $G$-contractible.
\end{enumerate}
\end{cor}
\begin{proof}
Part (1) follows directly from Propositions \ref{upper} and \ref{lower}, and Theorem \ref{productI}. Part (2) follows from part (1), since by definition $X$ is $G$-contractible if and only if $\mbox{\rm cat}_G(X)=1$.
\end{proof}
We now turn to examples.
\begin{exam}[Spheres under reflection] \label{spheres} For $n{\mathfrak{genus}}_Geq 1$, let $X=S^n\subseteq \mathbb{R}^{n+1}$ with the group $G=\mathbb{Z}_2$ acting by the reflection given by multiplication by $-1$ in the last co-ordinate.
When $n=1$ the fixed point set $X^G = \{(1,0),(-1,0)\}$ is disconnected, and so $\mathbf{TC}G(X) = \infty$ in this case.
When $n{\mathfrak{genus}}_Geq 2$ the fixed point set $X^G = S^{n-1}$ is the equatorial sphere, hence is connected. Note that $\mbox{\rm cat}_G(X)=2$ in this case ($X$ is clearly not $G$-contractible since the orbits are discrete; we leave it to the reader to construct a cover by two $G$-categorical open sets). Therefore we have
\[
\mathbf{TC}G(X)\le 2\mbox{\rm cat}_G(X)-1 = 3
\]
by Corollary \ref{catfixed} (1). When $n$ is even we have $3=\mathbf{TC}(X)\le \mathbf{TC}G(X)$, and when $n$ is odd we have $3=\mathbf{TC}(X^G)\le \mathbf{TC}G(X)$ by Corollary \ref{subgroups} (1). We have therefore shown that
\[
\mathbf{TC}G(S^n) = \left\{ \begin{array}{rl} \infty & (n=1)\\
3 & (n{\mathfrak{genus}}_Ge 2). \end{array}\right.
\]
\end{exam}
\begin{exam}
If $X=S^1$ is the circle with $G=S^1$ acting on $X$ by
rotations, then $\mathbf{TC}G(X)=2$.
For the usual motion planner on $S^1$ with two local rules is equivariant
with respect to rotations. So $\mathbf{TC}G(X)\leq 2$, and $\mathbf{TC}G(X){\mathfrak{genus}}_Geq
\mathbf{TC}(X)=2$.
\end{exam}
Generalizing the previous example, we have the following result which shows that the category of a connected group is a particular instance of its equivariant topological complexity.
\begin{thm} \label{catgrp} Let $G$ be a connected metrizable group acting on itself by left translation. Then $\mathbf{TC}G(G)=\mbox{\rm cat}(G)$.
\end{thm}
\begin{proof} Since the diagonal action of $G$ on $G\times G$ is free, we have
\begin{align*}
\mbox{\rm cat}(G) & \leq \mathbf{TC}G(G) \qquad\mbox{(Proposition \ref{lower})}\\
& \leq\mbox{\rm cat}_G(G\times G) \qquad\mbox{(Proposition \ref{upper})} \\
& = \mbox{\rm cat}\left((G\times G)/G\right) \qquad\mbox{(Proposition \ref{X/G})}\\
& = \mbox{\rm cat}(G),
\end{align*}
where at the last step we have made use of the fact that there is a homeomorphism $(G\times G)/G$ to $G$ sending $[g,h]$ to $g^{-1}h$.
\end{proof}
This illustrates the importance of the fixed point set in determining equivariant topological complexity. In particular it shows that $\mathbf{TC}G(X)$ can be arbitrarily large even when $X$ is $G$-contractible, as long as $X$ has no fixed points. By contrast, a $G$-contractible space with fixed points has $\mathbf{TC}G(X)=1$, by Corollary \ref{catfixed} (2).
Next we give an equivariant version of a result of Farber \cite[Lemma 8.2]{Far04} which states that the topological complexity of a connected topological group equals its category.
\begin{prop}\label{topgroups}
Let $X$ be a topological group. Assume that $G$ acts on $X$ by topological group homomorphisms, and that $X$ is $G$-connected. Then $\mathbf{TC}G(X)=\mbox{\rm cat}_G(X)$.
\end{prop}
\begin{proof}
We first note that the identity element $e\in X$ is a fixed point, since $G$ acts by group homomorphisms. It follows from Proposition \ref{lower} that $\mbox{\rm cat}_G(X)\le \mathbf{TC}G(X)$.
Now suppose that $\mbox{\rm cat}_G(X)=k$. Let $\{ U_1,\ldots , U_k\}$ be a $G$-categorical open cover of $X$. By Lemma \ref{4}, for each $i$ we can find a $G$-homotopy $F\colon\thinspace U_i\times I\to X$ such that $F_0=i_{U_i}$ and $F_1(U_i)\subseteq {\mathcal O}(e)=\{e\}$.
Set $V_i = \{ (a,b)\mid ab^{-1}\in U_i\}\subseteq X\times X$. Since multiplication and inversion are continuous and $G$-equivariant, the $V_i$ are open and $G$-invariant. Define a $G$-section on $V_i$ by setting $s_i(a,b)(t) = F(ab^{-1},t)b$; this is easily checked as being $G$-equivariant. Hence $\{V_1,\ldots , V_k\}$ forms a $G$-sectional categorical open cover, and $\mathbf{TC}G(X)\le k = \mbox{\rm cat}_G(X)$. This completes the proof.
\end{proof}
\begin{exam} \label{conjugation} Let $G$ be a connected Lie group, acting on itself by conjugation $g\cdot a = gag^{-1}$. Note that $G$ acts by homomorphisms. The fixed point set $G^H$ of a closed subgroup $H\subseteq G$ is the centralizer $C_G(H) = \{ g\in G\mid ga=ag\mbox{ for all } a\in H\}$. Hence $G$ is $G$-connected if and only if the centralizer of every closed subgroup is connected (this holds for example if $G$ is $U(n)$ for $n{\mathfrak{genus}}_Geq 1$, or a product of such). In this case, Proposition \ref{topgroups} applies and gives
\[
\mathbf{TC}G(G)=\mbox{\rm cat}_G(G).
\]
We now look at the case $G=U(n)$ in more detail. By results of Farber \cite[Lemma 8.2]{Far04} and Singhof \cite[Theorem 1(b)]{Si}, we have
\[
\mathbf{TC}_{U(n)}(U(n)){\mathfrak{genus}}_Ge \mathbf{TC}(U(n)) = \mbox{\rm cat}(U(n)) = n+1.
\]
On the other hand, Example \ref{U(n)} gives $\mathbf{TC}_{U(n)}(U(n))=\mbox{\rm cat}_{U(n)}(U(n))\le n+1$. Therefore,
\[
\mathbf{TC}_{U(n)}(U(n)) =\mbox{\rm cat}_{U(n)}(U(n)) = n+1.
\]
\end{exam}
Next we give a cohomological lower bound for $\mathbf{TC}G(X)$, using equivariant cohomology theory. We use the following equivariant generalization of \cite[Lemma 18.1]{Far06}, whose proof is routine.
\begin{lemma}
An invariant open subset $U\subseteq
X\times X$ is $G$-sectional categorical with respect to $\pi\colon\thinspace X^I\to X\times X$ if and only if the inclusion $i_U\colon\thinspace U\hookrightarrow X\times X$
is $G$-homotopic to a map with values in the diagonal
$\triangle X\subseteq X\times X$. \qed
\end{lemma}
Let $EG\to BG$ denote a universal principal $G$-bundle, and $X^h_G=EG\times_G X$ the corresponding homotopy orbit space of $X$. Denote by
$H^*_G(X)=H^*(X^h_G)$ the Borel $G$-equivariant cohomology of $X$, with coefficients in an arbitrary commutative ring. Note that
the diagonal map $\triangle\colon\thinspace X\to X\times X$ is equivariant, and
hence induces a map $\triangle_G\colon\thinspace X^h_G\to (X\times X)^h_G$.
\begin{thm}[Cohomological lower bound] \label{eqcohom}
Suppose that there are cohomology classes $z_1,\ldots ,z_k\in
H^*_G(X\times X)$ such that $0= \triangle_G^*(z_i)\in H^*_G(X)$ for all $i$ and the
product $z_1\cdots z_k$ is non-zero. Then $\mathbf{TC}G(X)>k$.
\end{thm}
\begin{proof}
Suppose $\mathbf{TC}G(X)\leq k$, and let $\{U_1,\ldots , U_k\}$ be a $G$-sectional categorical open cover. Since the inclusion $i_{U_i}\colon\thinspace U_i\hookrightarrow X\times X$ factors through $\triangle\colon\thinspace X\to X\times X$ up to $G$-homotopy, it follows that the restriction $(i_{U_i})^*\colon\thinspace H^*_G(X\times X)\to H^*_G(U_i)$ maps $z_i$ to zero, and hence
$z_i$ is in the image of $H^*_G(X\times X,U_i)\to H^*_G(X\times X)$
for each $i=1,\ldots , k$. By naturality of cup products, it then follows that the product $z_1\cdots z_k$ is
zero.
\end{proof}
We do not currently know of any examples of $G$-connected spaces $X$ where the lower bound for $\mathbf{TC}G(X)$ given by Theorem \ref{eqcohom} improves on the non-equivariant lower bound $\mbox{\rm nil}\ker (\cup)<\mathbf{TC}(X)\le\mathbf{TC}G(X)$ given by the zero-divisors cup-length.
Finally in this section, we prove a result which relates equivariant and non-equivariant topological complexity, and give an example indicating that the former may be useful in estimating the latter.
\begin{thm}\label{fred} Let $X$ be a $G$-space, and let $E\to B=E/G$ be a numerable principal $G$-bundle. Then
$$\mathbf{TC}(X_G)\leq\mathbf{TC}G(X)\mathbf{TC}(B),$$
where $X_G=E\times_G X$ denotes the associated fibre space with fibre $X$.
\end{thm}
\begin{proof}
Suppose that $\mathbf{TC}(B)=k$ by a cover $B\times
B=U_1\cup\cdots\cup U_k$ and $\mathbf{TC}G(X)=\ell$ by a cover $X\times
X=W_1\cup\cdots\cup W_\ell$ by $G$-invariant open sets admitting $G$-sections $s_j\colon\thinspace W_j\to X^I$. Our aim is to cover $X_G\times X_G$ by $k\ell$ open sets on which the map $\Pi\colon\thinspace (X_G)^I\to X_G\times X_G$ admits a homotopy section. We have a strictly commuting diagram
\[
\xymatrix{
E\times_G(X^I)\ar[r] \ar[d]_{E\times_G\pi} \ar[dr]^p & (X_G)^I \ar[d]^{\Pi} \\
E\times_G(X\times X) \ar@{^{(}->}[r]^-\iota \ar[d] & X_G\times X_G \ar[d]^{\rho\times\rho} \\
B \ar@{^{(}->}[r]^\triangle & B\times B
}
\]
whose bottom square is a pullback. Here $\iota$ is the map sending $[e,x,x']$ to $([e,x],[e,x'])$ and $\rho\colon\thinspace X_G\to B$ is the fibration sending $[e,x]$ to $[e]$.
To prove the proposition it suffices to cover $X_G\times X_G$ by $k\ell$ open sets on which the map $p$ given by $p[e,{\mathfrak{genus}}_Gamma] = ([e,{\mathfrak{genus}}_Gamma(0)],[e,{\mathfrak{genus}}_Gamma(1)])$ admits a homotopy section.
By Lemma 18.1 of \cite{Far06} each
inclusion $U_i\hookrightarrow B\times B$ is homotopic to a map with
values in the diagonal $\triangle B\subseteq B\times B$. Setting $V_i=(\rho\times \rho)^{-1}(U_i)\subseteq X_G\times X_G$ and applying the homotopy lifting property of the map $\rho\times \rho$, we obtain a homotopy from the inclusion $V_i\hookrightarrow X_G\times X_G$ to a map $H_i\colon\thinspace V_i\to X_G\times X_G$ with values in $E\times_G(X\times X)$, and hence by restricting the range a map $h_i\colon\thinspace V_i\to E\times_G(X\times X)$.
Since the sets $W_j\subseteq X\times X$ form an invariant open cover, the sets $E\times_G W_j\subseteq E\times_G(X\times X)$ form an open cover. The equivariant sections $s_j\colon\thinspace W_j\to X^I$ give rise to sections $\sigma_j:=E\times_G s_j\colon\thinspace E\times_G W_j\to E\times_G (X^I)$ such that $(E\times_G \pi)\sigma_j = i_{E\times_G W_j}\colon\thinspace E\times_G W_j\hookrightarrow E\times_G(X\times X)$.
We now have a cover of $X_G\times X_G$ by the $k\ell$ open sets
\[
\Omega_{ij} = h_i^{-1}(E\times_G W_j),\qquad i=1,\ldots, k,\qquad j=1,\ldots,\ell,
\]
and candidate sections $\zeta_{ij}=\sigma_j\circ h_i|_{\Omega_{ij}}\colon\thinspace \Omega_{ij}\to E\times_G(X^I)$ of the map $p$. These are in fact homotopy sections, since \[
p\zeta_{ij} = p\sigma_j h_i|_{\Omega_{ij}} = \iota(E\times_G \pi)\sigma_j h_i|_{\Omega_{ij}} = \iota i_{E\times_G W_j} h_i|_{\Omega_{ij}}= H_i|_{\Omega_{ij}} \simeq i_{\Omega_{ij}}.
\]
\end{proof}
\begin{exam}[{Compare \cite[Example 5.8]{G}}]
Let $K^{n+1}$ be the `$(n+1)$-dimensional Klein bottle'. This is the mapping torus of the involution $S^n\to S^n$ given by reflection in the last co-ordinate. Note that $K^2$ is the usual Klein bottle.
Letting $E=S^1$ with free $\mathbb{Z}_2$-action given by the antipodal map, we see that $K^{n+1}=E\times_{\mathbb{Z}_2} S^n$. If $n{\mathfrak{genus}}_Ge 2$ we have
$\mathbf{TC}_{\mathbb{Z}_2}(S^n)=3$ by Example \ref{spheres}, and $\mathbf{TC}(B)=\mathbf{TC}(S^1)=2$. Then Theorem \ref{fred} gives $\mathbf{TC}(K)\leq
3\cdot 2=6$. However Theorem \ref{fred} says nothing in the $n=1$ case, since $\mathbf{TC}_{\mathbb{Z}_2}(S^1)=\infty$.
\end{exam}
\end{document} | math | 43,876 |
\begin{document}
\title[On the Adams Spectral Sequence for $R$-modules]{On the Adams
Spectral Sequence for \boldmath$R$-modules}
\covertitle{On the Adams Spectral Sequence for $R$-modules}
\asciititle{On the Adams Spectral Sequence for R-modules}
\authors{Andrew Baker\\Andrey Lazarev}
\address{Mathematics Department, Glasgow University,
Glasgow G12 8QW, UK. \\
Mathematics Department, Bristol University,
Bristol BS8 1TW, UK.}
\asciiemail{[email protected], [email protected]}
\email{[email protected]\qua{\rm and}\qua [email protected]}
\url{www.maths.gla.ac.uk/$\sim$ajb\qua{\rm and}\qua
www.maths.bris.ac.uk/$\sim$pure/staff/maxal/maxal}
\begin{abstract}
We discuss the Adams Spectral Sequence for $R$-modules based on commutative
localized regular quotient ring spectra over a commutative $S$-algebra $R$
in the sense of Elmendorf, Kriz, Mandell, May and Strickland. The formulation
of this spectral sequence is similar to the classical case and the calculation
of its $\mathrm{E}_2$-term involves the cohomology of certain `brave new Hopf algebroids'
$E^R_*E$. In working out the details we resurrect Adams' original approach to
Universal Coefficient Spectral Sequences for modules over an $R$ ring spectrum.
We show that the Adams Spectral Sequence for $S_R$ based on a commutative localized
regular quotient $R$ ring spectrum $E=R/I[X^{-1}]$ converges to the homotopy of
the $E$-nilpotent completion
\[
\pi_*\hat{\LOC}^R_ES_R=R_*[X^{-1}]\sphat_{I_*}.
\]
We also show that when the generating regular sequence of $I_*$ is finite,
$\hat\LOC^R_ES_R$ is equivalent to $\LOC^R_ES_R$, the Bousfield localization
of $S_R$ with respect to $E$-theory.
The spectral sequence here collapses at its $\mathrm{E}_2$-term but it does not have
a vanishing line because of the presence of polynomial generators of positive
cohomological degree. Thus only one of Bousfield's two standard convergence
criteria applies here even though we have this equivalence. The details involve
the construction of an $I$-adic tower
\[
R/I\longleftarrow R/I^2\longleftarrow\cdots\longleftarrow R/I^s\longleftarrow R/I^{s+1}\longleftarrow\cdots
\]
whose homotopy limit is $\hat\LOC^R_ES_R$. We describe some examples for the
motivating case $R={MU}$.
\end{abstract}
\asciiabstract{ We discuss the Adams Spectral Sequence for R-modules
based on commutative localized regular quotient ring spectra over a
commutative S-algebra R in the sense of Elmendorf, Kriz, Mandell, May
and Strickland. The formulation of this spectral sequence is similar
to the classical case and the calculation of its E_2-term involves the
cohomology of certain `brave new Hopf algebroids' E^R_*E. In working
out the details we resurrect Adams' original approach to Universal
Coefficient Spectral Sequences for modules over an R ring spectrum.
We show that the Adams Spectral Sequence for S_R based on a
commutative localized regular quotient R ring spectrum E=R/I[X^{-1}]
converges to the homotopy of the E-nilpotent completion
pi_*hat{L}^R_ES_R=R_*[X^{-1}]^hat_{I_*}. We also show that
when the generating regular sequence of I_* is finite,
hatL^R_ES_R is equivalent to L^R_ES_R, the Bousfield localization of
S_R with respect to E-theory. The spectral sequence here collapses at
its E_2-term but it does not have a vanishing line because of the presence
of polynomial generators of positive cohomological degree. Thus only one
of Bousfield's two standard convergence criteria applies here even though
we have this equivalence. The details involve the construction of an I-adic
tower
R/I <-- R/I^2 <-- ... <-- R/I^s <-- R/I^{s+1} <-- ...
whose homotopy limit is hatL^R_ES_R. We describe some examples for the
motivating case R=MU.}
\primaryclass{55P42, 55P43, 55T15; 55N20}
\Bbbkeywords{$S$-algebra, $R$-module, $R$ ring spectrum, Adams Spectral
Sequence, regular quotient}
\asciikeywords{S-algebra, R-module, R ring spectrum, Adams Spectral
Sequence, regular quotient}
\maketitle
\vglue12pt
\section*{Erratum}
While this paper was in e-press, the authors discovered
that the original versions of Theorems~\ref{thm:LQMU-nilcomp}
and~\ref{thm:LQ-R} were incorrect since they did not assume
that the regular sequence $u_j$ was finite. With the agreement
of the Editors, we have revised this version to include the
appropriate finiteness assumptions. We have also modified the
Abstract and Introduction to reflect this and in
Section~\ref{sec:LRQ-MU} have replaced Bousfield localizations
$\LOC^R_EX$ by $E$-nilpotent completions $\hat{\LOC}^R_EX$. As
far as we are aware, there are no further problems arising from
this mistake.
\par\vglue 5pt
\rightline{\it
Andrew Baker and Andrey Lazarev\qua 9 May 2001}
\section*{Introduction}
We consider the Adams Spectral Sequence for $R$-modules based on localized
regular quotient ring spectra over a commutative $S$-algebra $R$ in the sense
of~\cite{EKMM,Strickland:MU}, making systematic use of ideas and notation from
those two sources. This work grew out of a preprint~\cite{AB:BraveNewASS} and
the work of~\cite{AB+AJ:Brave-MU}; it is also related to ongoing collaboration
with Alain Jeanneret on Bockstein operations in cohomology theories defined on
$R$-modules~\cite{AB+AJ:Brave-BocksteinOps}.
One slightly surprising phenomenon we uncover concerns the convergence of the
Adams Spectral Sequence based on $E=R/I[X^{-1}]$, a commutative localized regular
quotient of a commutative $S$-algebra $R$. We show that the spectral sequence
for $\pi_*S_R$ collapses at $\mathrm{E}_2$, however for $r\geqslant2$, $\mathrm{E}_r$ has no vanishing
line because of the presence of polynomial generators of positive cohomological
degree which are infinite cycles. Thus only one of Bousfield's two convergence
criteria~\cite{Bousfield:ASS} (see Theorems~\ref{thm:BousfieldCgceASS}
and~\ref{thm:BousfieldCgceASS-Comp=Loc} below) apply here.
Despite this, when the generating regular sequence of $I_*$ is finite,
the spectral sequence converges to $\pi_*\LOC^R_ES_R$, where $\LOC^R_E$ is
the Bousfield localization functor with respect to $E$-theory on the
category of $R$-modules and
\[
\pi_*\LOC^R_ES_R=R_*[X^{-1}]\sphat_{I_*},
\]
the $I_*$-adic completion of $R_*[X^{-1}]$; we also show that
in this case
$\LOC^R_ES_R\simeq\hat\LOC^R_ES_R$, the $E$-nilpotent completion of $S_R$.
In the final section we describe some examples for the important case
of $R={MU}$, leaving more delicate calculations for future work.
To date there seems to have been very little attention paid to the detailed
homotopy theory associated with the category of $R$-modules, apart from general
results on Bousfield localizations and Wolbert's work on $K$-theoretic localizations
in~\cite{EKMM,Wolbert}. We hope this paper leads to further work in this area.
\section*{Background assumptions, terminology and technology}
We work in a setting based on a good category of spectra ${\mathcal S}$ such
as the category of $\mathbb{L}$-spectra of~\cite{EKMM}. Associated to this
is the subcategory of $S$-modules $\mathcal M_S$ and its derived homotopy
category $\mathcal D_S$.
Throughout, $R$ will denote a commutative $S$-algebra in the sense of~\cite{EKMM}.
There is an associated subcategory $\mathcal M_R$ of $\mathcal M_S$ consisting of
the $R$-modules, and its derived homotopy category $\mathcal D_R$ and our homotopy
theoretic work is located in the latter. Because we are working in $\mathcal D_R$,
we frequently make constructions using cell $R$-modules in place of non-cell
modules (such as $R$ itself).
For $R$-modules $M$ and $N$, we set
\[
M^R_*N=\pi_*M\SSmash{R}N,\quad N_R^*M=\mathcal{D}_R(M,N)^*,
\]
where $\mathcal{D}_R(M,N)^n=\mathcal{D}_R(M,\Sigma^nN)$.
We will use the following terminology of Strickland~\cite{Strickland:MU}.
If the homotopy ring $R_*=\pi_*R$ is concentrated in even degrees, a
\emph{localized quotient} of $R$ will be an $R$ ring spectrum of the form
$R/I[X^{-1}]$. A localized quotient is \emph{commutative} if it is a commutative
$R$ ring spectrum. A localized quotient $R/I[X^{-1}]$ is \emph{regular} if the
ideal $I_*\triangleleft R_*$ is generated by a regular sequence $u_1,u_2,\ldots$ say.
The ideal $I_*\triangleleft R_*$ extends to an ideal of $R_*[X^{-1}]$ which we will
again denote by $I_*$; then as $R$-modules, $R/I[X^{-1}]\simeq R[X^{-1}]/I$.
We will make use of the language and ideas of algebraic derived categories
of modules over a commutative ring, mildly extended to deal with evenly graded
rings and their modules. In particular, this means that chain complexes are
often bigraded (or even multigraded) objects with their first grading being
homological and the second and higher ones being internal.
\section{Brave new Hopf algebroids and their cohomology}
\leftarrowbel{sec:Brave-HA}
If $E$ is a commutative $R$-ring spectrum, the smash product $E\SSmash{R}E$
is also a commutative $R$-ring spectrum. More precisely, it is naturally
an $E$-algebra spectrum in two ways induced from the left and right units
\[
E\xrightarrow{\;\cong\;}E\SSmash{R}R\longrightarrow E\SSmash{R}E
\longleftarrow E\SSmash{R}R\xleftarrow{\;\cong\;}E.
\]
\begin{thm}\leftarrowbel{thm:ERE-HA}
Let $E^R_*E$ be flat as a left or equivalently right $E_*$-module. Then
the following are true. \\
{\rm i)}
$(E_*,E^R_*E)$ is a Hopf algebroid over $R_*$. \\
{\rm ii)} for any $R$-module $M$, $E^R_*M$ is a left $E^R_*E$-comodule.
\end{thm}
\begin{proof}
This is proved using essentially the same argument as
in~\cite{Adams:Chicago,Ravenel:book}. The natural map
\[
E\SSmash{R}M\xrightarrow{\;\cong\;}E\SSmash{R}R\SSmash{R}M
\longrightarrow E\SSmash{R}E\SSmash{R}M
\]
induces the coaction
\[
\psi\colonE^R_*M\longrightarrow\pi_*E\SSmash{R}E\SSmash{R}M
\xrightarrow{\;\cong\;}E^R_*E\circTimes{E_*}E^R_*M,
\]
which uses an isomorphism
\[
\pi_*E\SSmash{R}E\SSmash{R}M\cong E^R_*E\circTimes{E_*}E^R_*M.
\]
that follows from the flatness condition.
\end{proof}
For later use we record a general result on the Hopf algebroids associated
with commutative regular quotients. A number of examples for the case $R={MU}$
are discussed in Section~\ref{sec:LRQ-MU}.
\begin{prop}\leftarrowbel{prop:E*R/I}
Let $E=R/I$ be a commutative regular quotient where $I_*$ is generated by
the regular sequence $u_1,u_2,\ldots$. Then as an $E_*$-algebra,
\[
E^R_*E=\LOCambda_{E_*}(\tau_i:i\geqslant1),
\]
where $\mathbb{D}ERIVeg\tau_i=\mathbb{D}ERIVeg u_i+1$. Moreover, the generators $\tau_i$ are
primitive with respect to the coaction, and $E^R_*E$ is a primitively
generated Hopf algebra over $E_*$.
Dually, as an $E_*$-algebra,
\[
E_R^*E=\hat\LOCambda_{E_*}(Q^i:i\geqslant1),
\]
where $Q^i$ is the Bockstein operation dual to $\tau_i$ with
$\mathbb{D}ERIVeg Q^i=\mathbb{D}ERIVeg u_i+1$ and $\hat\LOCambda_{E_*}(\ )$ indicates the
completed exterior algebra generated by the anti-commuting $Q^i$
elements.
\end{prop}
The proof requires the K\"unneth Spectral Sequence for $R$-modules
of~\cite{EKMM},
\[
\mathrm{E}^2_{p,q}=\Tor^{R_*}_{p,q}(E_*,E_*)\LOCra E^R_{p+q}E.
\]
This spectral sequence is multiplicative, however there seems to be no
published proof in the literature. At the suggestion of the referee, we
indicate a proof of this due to M.~Mandell and which originally appeared
in a preprint version of~\cite{Lazarev}.
\begin{lem}\leftarrowbel{lem:KSS-Mult}
If $A$ and $B$ are $R$ ring spectra then the K\"unneth Spectral Sequence
\[
\Tor^{R_*}(A_*,B_*)\LOCra A^R_*B=\pi_*A\SSmash{R}B
\]
is a spectral sequence of differential graded $R_*$-algebras.
\end{lem}
\begin{proof}[Sketch proof]
To deal with the multiplicative structure we need to modify the
original construction given in Part~IV section~5 of~\cite{EKMM}.
We remind the reader that we are working in the derived homotopy
category $\mathcal D_R$.
Let
\[
\cdots\longrightarrow F_{p,*}\xrightarrow{f_p}F_{p-1,*}\longrightarrow\cdots
\xrightarrow{f_1}F_{0,*}\xrightarrow{f_0}A_*\rightarrow0
\]
be an free $R_*$-resolution of $A_*$. Using freeness, we can choose
a map of complexes
\[
\mu\colonF_{*,*}\circTimes{R_*}F_{*,*}\longrightarrow F_{*,*}
\]
which lifts the multiplication on $A_*$.
For each $p\geqslant 0$ let $\mathbf F_p$ be a wedge of sphere $R$-modules
satisfying $\pi_*\mathbf F_p=F_{p,*}$. Set $A'_0=\mathbf F_0$ and choose
a map $\varphi_0\colonA'_0\longrightarrow A$ inducing $f_0$ in homotopy. If $\mathbf Q_0$
is the homotopy fibre of $\varphi_0$ then
\[
\pi_*\mathbf Q_0=\Bbbker f_0
\]
and we can choose a map $\mathbf F_1\longrightarrow\mathbf Q_0$ for which the composition
$\varphi'_1\colon\mathbf F_1\longrightarrow\mathbf Q_0\longrightarrow\mathbf F_0$ induces $f_1$ in homotopy.
Next take $A'_1$ to be the cofibre of $\varphi'_1$. The map $\varphi_0$ has a canonical
extension to a map $\varphi_1\colonA'_1\longrightarrow A$. If $\mathbf Q_1$ is the homotopy fibre
of $\varphi_1$ then
\[
\pi_*\Sigma^{-1}\mathbf Q_1=\Bbbker f_1,
\]
and we can find a map $\mathbf F_2\longrightarrow\Sigma\mathbf Q_1$ for which the composite
map $\varphi'_2\colon\mathbf F_2\longrightarrow\mathbf Q_1\longrightarrow\mathbf F_1$ induces $f_2$ in homotopy.
We take $A'_2$ to be the cofibre of $\varphi'_2$ and find that there is a canonical
extension of $\varphi_1$ to a map $\varphi_2\colonA'_2\longrightarrow A$.
Continuing in this way we construct a directed system
\begin{equation}\leftarrowbel{eqn:fil}
A'_0\longrightarrow A'_1\longrightarrow\cdots\longrightarrow A'_p\longrightarrow\cdots
\end{equation}
whose telescope $A'$ is equivalent to $A$. Since we can assume that all consecutive
maps are inclusions of cell subcomplexes, there is an associated filtration on $A'$.
Smashing this with $B$ we get a filtration on $A'\SSmash{R}B$ and an associated
spectral sequence converging to $A_*^RB$. The identification of the $\mathrm{E}_2$-term is
routine.
Recall that $A$ and therefore $A'$ are $R$ ring spectra. Smashing the directed
system of~\eqref{eqn:fil} with itself we obtain a filtration on $A'\SSmash{R}A'$,
\begin{equation}\leftarrowbel{eqn:mul}
A'_0\SSmash{R}A'_0\longrightarrow\cdots\longrightarrow\bigcup_{i+j=k}A'_i\SSmash{R}A'_j
\longrightarrow\bigcup_{i+j=k+1}A'_i\SSmash{R}A'_j\longrightarrow\cdots,
\end{equation}
where the filtrations terms are unions of the subspectra $A_i'\SSmash{R}A_j'$.
Proceeding by induction, we can realize the multiplication map $A'\SSmash{R}A'\longrightarrow A'$
as a map of filtered $R$-modules so that on the cofibres of the filtration terms
of~\eqref{eqn:mul} it agrees with the pairing $\mu$.
We have constructed a collection of maps $A'_i\SSmash{R}A'_j\longrightarrow A'_{i+j}$.
Using these maps and the multiplication on $B$ we can now construct maps
\[
A'_i\SSmash{R}B\SSmash{R}A'_j\SSmash{R}B\longrightarrow A_{i+j}\SSmash{R}B
\]
which induce the required pairing of spectral sequences.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:E*R/I}]
As in the discussion preceding Proposition~\ref{prop:Tor(R/I,R/I)},
making use of a Koszul resolution we obtain
\[
\mathrm{E}^2_{*,*}=\LOCambda_{E_*}(e_i:i\geqslant1).
\]
The generators have bidegree $\bideg e_i=(1,|u_i|)$, so the differentials
\[
\mathbb{D}ERIV^r\colon\mathrm{E}^r_{p,q}\longrightarrow\mathrm{E}^r_{p-r,q+r-1}
\]
are trivial on the generators $e_i$ for dimensional reasons. Together with
multiplicativity, this shows that spectral sequence collapses, giving
\[
E^R_*E=\LOCambda_{E_*}(\tau_i:i\geqslant1),
\]
where the generator $\tau_i$ has degree $\mathbb{D}ERIVeg\tau_i=\mathbb{D}ERIVeg u_i+1$ and is
represented by $e_i$.
For each $i$,
\[
(R/u_i)^R_*(R/u_i)=\LOCambda_{R_*/(u_i)}(\tau'_i)
\]
with $\mathbb{D}ERIVeg\tau'_i=|u_i|+1$. Under the coproduct, $\tau'_i$ is primitive
for degree reasons. By comparing the two K\"unneth Spectral Sequences we
find that $\tau_i\in E^R_*E$ can be chosen to be the image of $\tau'_i$
under the evident ring homomorphism $(R/u_i)^R_*(R/u_i)\longrightarrow E^R_*E$, which
is actually a morphism of Hopf algebroids over $R_*$. Hence $\tau_i$ is
coaction primitive in $E^R_*E$.
For $E_R^*E$, we construct the Bockstein operation $Q^i$ using the composition
\[
R/u_i\longrightarrow\Sigma^{|u_i|+1}R\longrightarrow\Sigma^{|u_i|+1}R/u_i
\]
to induce a map $E\longrightarrow\Sigma^{|u_i|+1}E$, then use the Koszul resolution
to determine the Universal Coefficient Spectral sequence
\[
\mathrm{E}_2^{p,q}=\mathrm{E}xt_{R_*}^{p,q}(E_*,E_*)\LOCra E_R^{p+q}E
\]
which collapses at its $\mathrm{E}_2$-term. Further details on the construction
of these operations appear in~\cite{Strickland:MU,AB+AJ:Brave-BocksteinOps}.
\end{proof}
\begin{cor}\leftarrowbel{cor:E*R/I}
{\rm i)} The natural map $E_*=E^R_*R\longrightarrow E^R_*E$ induced by the unit
$R\longrightarrow R/I$ is a split monomorphism of $E_*$-modules. \\
{\rm ii)} $E^R_*E$ is a free $E_*$-module.
\end{cor}
\begin{proof}
An explicit splitting as in (i) is obtained using the multiplication
map $E\SSmash{R}E\longrightarrow E$ which induces a homomorphism of $E_*$-modules
$E^R_*E\longrightarrow E_*$.
\end{proof}
We will use $\mathbb{C}oext$ to denote the cohomology of such Hopf algebroids
rather than $\mathrm{E}xt$ since we will also make heavy use of $\mathrm{E}xt$ groups
for modules over rings; more details of the definition and calculations
can be found in~\cite{Adams:Chicago,Ravenel:book}. Recall that for
$E^R_*E$-comodules $L_*$ and $M_*$ where $L_*$ is $E_*$-projective,
$\mathbb{C}oext_{E^R_*E}^{s,t}(L_*,M_*)$ can be calculated as follows. Consider
a resolution
\[
0\rightarrow M_*\longrightarrow J_{0,*}\longrightarrow J_{1,*}\longrightarrow\cdots\longrightarrow J_{s,*}\longrightarrow\cdots
\]
in which each $J_{s,*}$ is a summand of an extended comodule
\[
E^R_*E\Square{E_*}N_{s,*},
\]
for some $E_*$-module $N_{s,*}$. Then the complex
\begin{multline*}
0\rightarrow\Hom_{E^R_*E}^*(L_*,J_{0,*})\longrightarrow\Hom_{E^R_*E}^*(L_*,J_{1,*})
\\
\longrightarrow\cdots\longrightarrow\Hom_{E^R_*E}^*(L_*,J_{s,*})
\longrightarrow\cdots
\end{multline*}
has cohomology
\[
\mathrm{H}^s(\Hom_{E^R_*E}^*(L_*,J_{*,*}))=\mathbb{C}oext_{E^R_*E}^{s,*}(L_*,M_*).
\]
The functors $\mathbb{C}oext_{E^R_*E}^{s,*}(L_*,\ )$ are the right derived functors
of the left exact functor
\[
M_*\rightsquigarrow\Hom_{E^R_*E}^*(L_*,M_*)
\]
on the category of left $E^R_*E$-comodules. By analogy with~\cite{Ravenel:book},
when $L_*=E_*$ we have
\[
\mathbb{C}oext_{E^R_*E}^{s,*}(E_*,M_*)=\mathbb{C}otor_{E^R_*E}^{s,*}(E_*,M_*).
\]
\section{The Adams Spectral Sequence for $R$-modules}\leftarrowbel{sec:ASS}
We will describe the $E$-theory Adams Spectral Sequence in the homotopy
category of $R$-module spectra. As in the classical case of sphere spectrum
$R=S$, it turns out that the $\mathrm{E}_2$-term is can be described in terms of the
functor $\mathbb{C}oext_{E^R_*E}$.
Let $L,M$ be $R$-modules and $E$ a commutative $R$-ring spectrum with
$E^R_*E$ flat as a left (or right) $E_*$-module.
\begin{thm}\leftarrowbel{thm:ASS}
If $E^R_*L$ is projective as an $E_*$-module, there is an
Adams Spectral Sequence with
\[
\mathrm{E}_2^{s,t}(L,M)=\mathbb{C}oext_{E^R_*E}^{s,t}(E^R_*L,E^R_*M).
\]
\end{thm}
\begin{proof}
Working throughout in the derived category $\mathcal{D}_R$, the proof
follows that of Adams~\cite{Adams:Chicago}, with $S_R\simeq R$ replacing
the sphere spectrum $S$. The canonical Adams resolution of $M$ is built
up in the usual way by splicing together the cofibre triangles in the
following diagram.
\[
\begin{xy}
\[email protected]{
\ar[dr]M&&\bar{E}\SSmash{R}\ar[ll]M
\ar[dr]&&\bar{E}\SSmash{R}\bar{E}\SSmash{R}\ar[ll]M
\ar[dr]&\ar[l]&\ar@{.}[l]
&&
\\
&E\SSmash{R}M\ar[ur]&
&E\SSmash{R}\bar{E}\SSmash{R}M\ar[ur]&
&&
&&
}
\end{xy}
\]
The algebraic identification of the $\mathrm{E}_2$-term proceeds as
in~\cite{Adams:Chicago}.
\end{proof}
In the rest of this paper we will have $L=S_R\simeq R$, and set
\[
\mathrm{E}_2^{s,t}(M)=\mathbb{C}oext_{E^R_*E}^{s,t}(E_*,E^R_*M).
\]
We will refer to this spectral sequence as the Adams Spectral
Sequence based on $E$ for the $R$-module $M$.
To understand convergence of such a spectral sequence we use
a criterion of Bousfield~\cite{Bousfield:ASS,Ravenel:LocnPaper}.
For an $R$-module $M$, let $D_sM$ ($s\geqslant0$) be the $R$-modules
defined by $D_0M=M$ and taking $D_sM$ to be the fibre of the
natural map
\[
D_{s-1}M\cong R\SSmash{R}D_{s-1}M\longrightarrow E\SSmash{R}D_{s-1}M.
\]
Also for each $s\geqslant0$ let $K_sM$ be the cofibre of the natural map
$D_sM\longrightarrow M$. Then the \emph{$E$-nilpotent completion} of $M$ is the
homotopy limit
\[
\hat{\LOC}^R_EM=\holim_sK_sM.
\]
\begin{rem}\leftarrowbel{rem:Ehat-Invce}
It is easy to see that if $M\longrightarrow N$ is a map of $R$-modules which
is an $E$-equivalence, then for each $s$, there is an equivalence
$K_sM\longrightarrow K_sN$, hence
\[
\hat{\mathrm{L}}^R_EM\simeq\hat{\mathrm{L}}^R_EN.
\]
\end{rem}
\begin{thm}\leftarrowbel{thm:BousfieldCgceASS}
If for each pair $(s,t)$ there is an $r_0$ for which
$\mathrm{E}_r^{s,t}(M)=\mathrm{E}_\infty^{s,t}(M)$ whenever $r\geqslant r_0$,
then the Adams Spectral Sequence for $M$ based on $E$
converges to $\pi_*\hat{\mathrm{L}}^R_EM$.
\end{thm}
Although there is a natural map $\mathrm{L}^R_EM\longrightarrow\hat{\mathrm{L}}^R_EM$,
it is not in general a weak equivalence; this equivalence is guaranteed
by another result of Bousfield~\cite{Bousfield:ASS}.
\begin{thm}\leftarrowbel{thm:BousfieldCgceASS-Comp=Loc}
Suppose that there is an $r_1$ such that for every $R$-module $N$ there
is an $s_1$ for which $\mathrm{E}_{r}^{s,t}(N)=0$ whenever $r\geqslant r_1$ and
$s\geqslant s_1$. Then for every $R$-module $M$ the Adams Spectral Sequence
for $M$ based on $E$ converges to $\pi_*\mathrm{L}^R_EM$ and
\[
\mathrm{L}^R_EM\simeq\hat{\mathrm{L}}^R_EM.
\]
\end{thm}
\section{The Universal Coefficient Spectral Sequence for regular quotients}
\leftarrowbel{sec:UCKSS-LQR}
Let $R$ be a commutative $S$-algebra and $E=R/I$ a commutative regular
quotient of $R$, where $u_1,u_2,\ldots$ is a regular sequence generating
$I_*\triangleleft R_*$.
We will discuss the existence of the Universal Coefficient Spectral Sequence
\begin{equation}\leftarrowbel{eqn:E*-UCSS}
\mathrm{E}^2_{r,s}=\mathrm{E}xt^{r,s}_{E_*}(E^R_*M,N_*)\LOCra N_R^*M,
\end{equation}
where $M$ and $N$ are $R$-modules and $N$ is also an $E$-module spectrum in
$\mathcal M_R$. The classical prototype of this was described by
Adams~\cite{Adams:Chicago} (who generalized a construction of Atiyah~\cite{Atiyah}
for the K\"unneth Theorem in $K$-theory) and used in setting up the $E$-theory
Adams Spectral Sequence. It is routine to verify that Adams' approach can be
followed in $\mathcal D_R$. We remark that if $E$ were a commutative $R$-algebra
then the Universal Coefficient Spectral Sequence of~\cite{EKMM} would be applicable
but that condition does not hold in the generality we require.
The existence of such a spectral sequence depends on the following
conditions
being satisfied.
\begin{cond}\leftarrowbel{cond:UCSS}
$E$ is a homotopy colimit of finite cell $R$-modules $E_\alpha$ whose
$R$-Spanier Whitehead duals $\mathrm{D}_RE_\alpha=\mathbb{F}unc_R(E_\alpha,R)$
satisfy the two conditions \\
(A) $E^R_*\mathrm{D}_RE_\alpha$ is $E_*$-projective; \\
(B) the natural map
\[
N_R^*M\longrightarrow\Hom_{E_*}(E^R_*M,N_*)
\]
is an isomorphism.
\end{cond}
\begin{thm}\leftarrowbel{thm:UCSS-RQ-Conditions}
For a commutative regular quotient $E=R/I$ of $R$, $E$ can be expressed as
a homotopy colimit of finite cell $R$-modules satisfying the conditions of
\emph{Condition~\ref{cond:UCSS}}. In fact we can take
$E^R_*\mathrm{D}_RE_\alpha$ to be $E_*$-free.
\end{thm}
The proof will use the following Lemma.
\begin{lem}\leftarrowbel{lem:UCSS-lemma}
Let $u\in R_{2d}$ be non-zero divisor in $R_*$. Suppose that $P$ is an
$R$-module for which $E^R_*P$ is $E_*$-projective and for an $E$-module
$R$-spectrum $N$,
\[
N_R^*P\cong\Hom_{E_*}(E^R_*P,N_*).
\]
Then $E^R_*P\SSmash{R}R/u$ is $E_*$-projective and
\[
N_R^*P\SSmash{R}R/u\cong\Hom_{E_*}(E^R_*P\SSmash{R}R/u,N_*).
\]
\end{lem}
\begin{proof}
Smashing $E\SSmash{R}P$ with the cofibre sequence~\eqref{eqn:UCSS-CofibSeq}
and taking homotopy, we obtain an exact triangle
\[
\begin{xy}
\xymatrix{
E^R_*P\ar[rr]^{\mathbb{D}ERIVs u}&&E^R_*P\ar[dl]\\
&E^R_*P\SSmash{R}R/u\ar[ul]&
}
\end{xy}
\]
As multiplication by $u$ induces the trivial map in $E^R$-homology,
this is actually a short exact sequence of $E_*$-modules,
\[
0\rightarrow E^R_*P\longrightarrow E^R_*P\SSmash{R}R/u\longrightarrow E^R_*P\rightarrow0
\]
which clearly splits, so $E^R_*P\SSmash{R}R/u$ is $E_*$-projective.
In the evident diagram of exact triangles
\[
\begin{xy}
\xymatrix@-1pc{
N_R^*P\ar[rr]\ar[dd]&&N_R^*P\ar[dl]\ar[dd] \\
&N_R^*P\SSmash{R}R/u\ar[ul]\ar[dd]& \\
\Hom_{E_*}(E^R_*P,N_*)\ar'[r][rr]&&\Hom_{E_*}(E^R_*P,N_*)\ar[dl] \\
&\Hom_{E_*}(E^R_*P\SSmash{R}R/u,N_*)\ar[ul]&
}
\end{xy}
\]
the map $N_R^*P\longrightarrow\Hom_{E_*}(E^R_*P,N_*)$ is an isomorphism, so
\[
N_R^*P\SSmash{R}R/u\longrightarrow\Hom_{E_*}(E^R_*P\SSmash{R}R/u,N_*)
\]
is also an isomorphism by the Five Lemma.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:UCSS-RQ-Conditions}]
Let $u_1,u_2,\ldots$ be a regular sequence generating $I_*\triangleleft R_*$.
Using the notation $R/u=R/(u)$, we recall from~\cite{Strickland:MU}
that
\[
E=\hocolim_k R/u_1\SSmash{R}R/u_2\SSmash{R}\cdots\SSmash{R}R/u_k.
\]
For $u\in R_{2d}$ a non-zero divisor, the $R_*$-free resolution
\[
0\rightarrow R_*\xrightarrow{\ph{\;u\;}}R_*\xrightarrow{\;u\;}R_*/(u)\rightarrow0
\]
corresponds to an $R$-cell structure on $R/u$ with one cell in each of
the dimensions $0$ and $2d+1$. There is an associated cofibre sequence
\begin{equation}\leftarrowbel{eqn:UCSS-CofibSeq}
\cdots\longrightarrow
\Sigma^{2d}R\xrightarrow{u}R\longrightarrow R/u\longrightarrow\Sigma^{2d+1}R
\longrightarrow\cdots,
\end{equation}
for which the induced long exact sequence in $E^R$-homology shows that
$E^R_*R/u$ is $E_*$-free. The dual $\mathrm{D}_RR/u$ is equivalent to
$\Sigma^{-(2d+1)}R/u$, hence $R/u$ is essentially self dual.
For an $E$-module spectrum $N$ in $\mathcal D_R$, there are two exact
triangles and morphisms between them,
\[
\begin{xy}
\xymatrix@-1pc{
N_R^*R\ar[rr]\ar[dd]&&N_R^*R\ar[dl]\ar[dd] \\
&N_R^*R/u\ar[ul]\ar[dd]& \\
\Hom_{E_*}(E_*,N_*)\ar'[r][rr]&&\Hom_{E_*}(E_*,N_*)\ar[dl] \\
& \Hom_{E_*}(E^R_*R/u,N_*)\ar[ul]&
}
\end{xy}
\]
The identifications
\[
N_*\cong N_R^*R\cong\Hom_{E_*}(E_*,N_*),
\]
and the Five Lemma imply that
\[
N_R^*R/u\cong\Hom_{E_*}(E^R_*R/u,N_*).
\]
Lemma~\ref{lem:UCSS-lemma} now implies that each of the spectra
$R/u_1\SSmash{R}R/u_2\SSmash{R}\cdots\SSmash{R}R/u_k$ satisfies
conditions (A) and (B).
\end{proof}
\section{The Adams Spectral Sequence based on a regular quotient}
\leftarrowbel{sec:ASS-RQ}
For an $R$-module $M$, let $M^{(s)}$ denote the $s$-fold $R$-smash
power of $M$,
\[
M^{(s)}=M\SSmash{R}M\SSmash{R}\cdots\SSmash{R}M.
\]
If $M$ is an $R[X^{-1}]$-module, then
\[
M^{(s)}=
M\SSmash{R[X^{-1}]}M\SSmash{R[X^{-1}]}\cdots\SSmash{R[X^{-1}]}M.
\]
Let $E=R/I[X^{-1}]$ be a localized regular quotient and $u_1,u_2,\ldots$
a regular sequence generating $I_*$. We will discuss the Adams Spectral
Sequence based on $E$. By Remark~\ref{rem:Ehat-Invce}, we can work in the
category of $R[X^{-1}]$-modules and replace the Adams Spectral Sequence of
$S_R$ by that of $S_{R[X^{-1}]}$. To simplify notation, from now on we will
replace $R$ by $R[X^{-1}]$ and therefore assume that $E=R/I$ is a regular
quotient of $R$.
First we identify the canonical Adams resolution giving rise to the Adams
Spectral Sequence based on the regular quotient $E=R/I$. We will relate this
to a tower described by the second author~\cite{Lazarev}, but the reader
should beware that his notation for $I^{(s)}$ is $I^s$ which we will use for
a different spectrum.
There is a fibre sequence $I\longrightarrow R\longrightarrow R/I$ and a tower of maps of $R$-modules
\[
R\longleftarrow I\longleftarrow I^{(2)}\longleftarrow\cdots\longleftarrow I^{(s)}\longleftarrow I^{(s+1)}\longleftarrow\cdots
\]
in which $I^{(s+1)}\longrightarrow I^{(s)}$ is the evident composite
\[
I^{(s+1)}\longrightarrow R\SSmash{R}I^{(s)}=I^{(s)}.
\]
Setting $R/I^{(s)}=\cofibre(I^{(s)}\longrightarrow R)$, we obtain a tower
\[
R/I\longleftarrow R/I^{(2)}\longleftarrow\cdots\longleftarrow R/I^{(s)}\longleftarrow R/I^{(s+1)}\longleftarrow\cdots
\]
which we will refer to as the \emph{external $I$-adic tower}. The
next result is immediate from the definitions.
\begin{prop}\leftarrowbel{prop:ASS-LRQ-Tower}
We have
\[
D_0S_{R}=R,
\qquad
D_sS_{R}=I^{(s)},
\quad(s\geqslant1),
\]
and
\[
K_sS_R=R/I^{(s+1)}
\quad(s\geqslant0).
\]
\end{prop}
It is not immediately clear how to determine the limit
\[
\hat{\mathrm{L}}^R_ES_R=\holim_{s}R/I^{(s)}.
\]
Instead of doing this directly, we will adopt an approach suggested by
Bousfield~\cite{Bousfield:ASS}, making use of another $E$-nilpotent
resolution, associated with the \emph{internal $I$-adic tower} to be
described below.
In order to carry this out, we first need to understand convergence.
We will see that the condition of Theorem~\ref{thm:BousfieldCgceASS}
is satisfied for a commutative regular quotient $E=R/I$.
\begin{prop}\leftarrowbel{prop:LRQ-ASS-E2}
The $\mathrm{E}_2$-term of the $E$-theory Adams Spectral Sequence for $\pi_*S_R$
is
\[
\mathrm{E}_2^{s,t}(S_R)=\mathbb{C}oext_{E^R_*E}^{s,t}(E_*,E_*)
=E_*[U_i:i\geqslant1],
\]
where $\bideg U_i=(1,|u_i|+1)$. Hence this spectral sequence
collapses at its $\mathrm{E}_2$-term
\[
\mathrm{E}_2^{*,*}(S_R)=\mathrm{E}_\infty^{*,*}(S_R)
\]
and converges to $\pi_*\hat{\mathrm{L}}^R_ES_R$.
\end{prop}
\begin{proof}
By Proposition~\ref{prop:E*R/I},
\[
E^R_*E=\LOCambda_{R_*}(\tau_i:i\geqslant1),
\]
with generators $\tau_i$ which are primitive with respect to the
coproduct of this Hopf algebroid. The determination of
\[
\mathbb{C}oext_{E^R_*E}^{*,*}(E_*,E_*)
\]
is now standard and the differentials are trivial for degree reasons.
\end{proof}
Induction on the number of cells now gives
\begin{cor}\leftarrowbel{cor:LRQ-ASS-E2}
For a finite cell $R$-module $M$, the $E$-theory Adams Spectral Sequence
for $\pi_*M$ converges to $\pi_*\hat{\LOC}^R_EM$.
\end{cor}
\section{The internal $I$-adic tower}\leftarrowbel{sec:I-adictower}
Suppose that $I_*\triangleleft R_*$ is generated by a regular sequence
$u_1,u_2,\ldots$. We will often indicate a monomial in the $u_i$ by writing
$u_{(i_1,\ldots,i_k)}=u_{i_1}\cdots u_{i_k}$. We will write $E=R/I$ and make
use of algebraic results from~\cite{AB:HomRegQuot} which we now recall in detail.
For $s\geqslant0$, we define the $R$-module $I^s/I^{s+1}$ to be the wedge of copies
of $E$ indexed on the distinct monomials of degree $s$ in the generators $u_i$.
For an explanation of this, see Corollary~\ref{cor:I*s/I*(s+1)-R/Ifree}.
We will show that there is an (\emph{internal}) \emph{$I$-adic tower}
of $R$-modules
\[
R/I\longleftarrow R/I^2\longleftarrow\cdots\longleftarrow R/I^s\longleftarrow R/I^{s+1}\longleftarrow\cdots
\]
so that for each $s\geqslant0$ the fibre sequence
\[
R/I^s\longleftarrow R/I^{s+1}\longleftarrow I^s/I^{s+1}
\]
corresponds to a certain element of
\[
\mathrm{E}xt_{R_*}^{1}(R_*/I_*^s,I_*^s/I_*^{s+1})
\]
in $\mathrm{E}_2$-term of the Universal Coefficient Spectral Sequence
of~\cite{EKMM} converging to $\mathcal{D}_R(R/I^s,I^s/I^{s+1})^*$.
On setting $I^s=\fibre(R\longrightarrow R/I^s)$ we obtain another tower
\[
R\longleftarrow I\longleftarrow I^2\longleftarrow\cdots\longleftarrow I^s\longleftarrow I^{s+1}\longleftarrow\cdots
\]
which is analogous to the external version of~\cite{Lazarev}.
A related construction appeared in~\cite{AB:Ainfty,AB-UW:Bockop}
for the case of $R=\hat{E(n)}$ (which was shown to admit a not
necessarily commutative $S$-algebra structure) and $I=I_n$.
Underlying our work is the classical \emph{Koszul resolution}
\[
\mathbf{K}_{*,*}\longrightarrow R_*/I_*\rightarrow0,
\]
where
\[
\mathbf{K}_{*,*}=\LOCambda_{R_*}(e_i:i\geqslant1),
\]
which has grading given by $\mathbb{D}ERIVeg e_i=|u_i|+1$ and differential
\begin{align*}
\mathbb{D}ERIV e_i&=u_i, \\
\mathbb{D}ERIV(xy)&=(\mathbb{D}ERIV x)y+(-1)^{r}x\mathbb{D}ERIV y
\quad(x\in\mathbf{K}_{r,*},\;y\in\mathbf{K}_{s,*}).
\end{align*}
Hence $(\mathbf{K}_{*,*},\mathbb{D}ERIV)$ is an $R_*$-free resolution of $R_*/I_*$
which is a differential graded $R_*$-algebra. Tensoring with $R_*/I_*$
and taking homology leads to a well known result.
\begin{prop}\leftarrowbel{prop:Tor(R/I,R/I)}
As an $R_*/I_*$-algebra,
\[
\Tor^{R_*}_{*,*}(R_*/I_*,R_*/I_*)=\LOCambda_{R_*/I_*}(e_i:i\geqslant1).
\]
\end{prop}
\begin{cor}\leftarrowbel{cor:Tor(R/I,R/I)}
$\Tor^{R_*}_{*,*}(R_*/I_*,R_*/I_*)$ is a free $R_*/I_*$-module.
\end{cor}
This is of course closely related to the topological result
Proposition~\ref{prop:E*R/I}.
Now returning to our algebraic discussion, we recall the following
standard result.
\begin{lem}[\cite{Matsumura}, Theorem 16.2]\leftarrowbel{lem:I*s/I*(s+1)-R/Ifree}
For $s\geqslant0$, $I_*^s/I_*^{s+1}$ is a free $R_*/I_*$-module with a
basis consisting of residue classes of the distinct monomials
$u_{(i_1,\ldots,i_s)}$ of degree $s$.
\end{lem}
\begin{cor}\leftarrowbel{cor:I*s/I*(s+1)-R/Ifree}
For $s\geqslant0$, there is an isomorphism of $R_*$-modules
\[
\pi_*I^s/I^{s+1}=I_*^s/I_*^{s+1}.
\]
Hence $\pi_*I^s/I^{s+1}$ is a free $R_*/I_*$-module with a basis indexed
on the distinct monomials $u_{(i_1,\ldots,i_s)}$ of degree $s$.
\end{cor}
Let $\mathrm{U}^{(s)}_*$ be the free $R_*$-module on a basis indexed on
the distinct monomials of degree $s$ in the $u_i$. For $s\geqslant0$, set
\[
\mathbf{Q}^{(s)}_{*,*}=\mathbf{K}_{*,*}\circTimes{R_*}\mathrm{U}^{(s)}_*,
\quad
\mathbb{D}ERIV_{\mathbf{Q}}^{(s)}=\mathbb{D}ERIV\circtimes1,
\]
and also for $x\in\mathbf{K}_{*,*}$ write
\[
x\tilde u_{(i_1,\ldots,i_s)}=x\circtimes u_{(i_1,\ldots,i_s)}.
\]
There is an obvious augmentation
\[
\mathbf{Q}^{(s)}_{0,*}\longrightarrow I_*^s/I_*^{s+1}.
\]
\begin{lem}\leftarrowbel{lem:I*s/I*(s+1)}
For $s\geqslant1$,
\[
\mathbf{Q}^{(s)}_{*,*}\xrightarrow{\varepsilon^{(s)}}I_*^s/I_*^{s+1}\rightarrow0
\]
is a resolution by free $R_*$-modules.
\end{lem}
Given a complex $(\mathbf{C}_{*,*},\mathbb{D}ERIV_{\mathbf{C}})$, the $k$-shifted
complex $(\mathbf{C}[k]_{*,*},\mathbb{D}ERIV_{\mathbf{C}[k]})$ is defined by
\[
\mathbf{C}[k]_{n,*}=\mathbf{C}_{n+k,*},
\quad
\mathbb{D}ERIV_{\mathbf{C}[k]}=(-1)^k\mathbb{D}ERIV_{\mathbf{C}}.
\]
There is a morphism of chain complexes
\begin{align*}
\mathbb{D}ERIVel^{(s+1)}\colon&\mathbf{Q}^{(s)}_{*,*}\longrightarrow\mathbf{Q}^{(s+1)}[-1]_{*,*}; \\
\mathbb{D}ERIVel^{(s+1)}e_{i_1}\cdots e_{i_r}\tilde u_{(j_1,\ldots,j_s)}&=
\sum_{k=1}^r(-1)^k
e_{i_1}\cdots\hat e_{i_k}\cdots e_{i_r}\tilde u_{(j_1,\ldots,j_s)}.
\end{align*}
Using the identification
$\mathbf{Q}^{(s+1)}[-1]_{n,*}=\mathbf{Q}^{(s+1)}_{n-1,*}$,
we will often view $\mathbb{D}ERIVel^{(s+1)}$ as a homomorphism
\[
\mathbb{D}ERIVel^{(s+1)}\colon\mathbf{Q}^{(s)}_{*,*}\longrightarrow\mathbf{Q}^{(s+1)}_{*,*}
\]
of bigraded $R_*$-modules of degree $-1$.
There are also external pairings
\begin{align*}
\mathbf{Q}^{(r)}_{*,*}\circTimes{R_*}\mathbf{Q}^{(s)}_{*,*}
& \longrightarrow\mathbf{Q}^{(r+s)}_{*,*}; \\
x\tilde u_{(i_1,\ldots,i_s)}\circtimes y\tilde u_{(j_1,\ldots,j_s)}
&\longmapsto
xy\tilde u_{(i_1,\ldots,i_s,j_1,\ldots,j_s)}
\quad(x,y\in\mathbf{K}_{*,*}).
\end{align*}
In particular, each $\mathbf{Q}^{(r)}_{*,*}$ is a differential module
over the differential graded $R_*$-algebra $\mathbf{K}^{(0)}_{*,*}$
and $\mathbb{D}ERIVel^{(s+1)}$ is a $\mathbf{K}^{(0)}_{*,*}$-derivation.
\begin{thm}\leftarrowbel{thm:R/Is-Resn}
For $s\geqslant1$, there is a resolution
\[
\mathbf{K}^{(s-1)}_{*,*}\xrightarrow{\varepsilon^{(s-1)}}R_*/I_*^s\rightarrow0,
\]
by free $R_*$-modules, where
\[
\mathbf{K}^{(s-1)}_{*,*}=
\mathbf{Q}^{(0)}_{*,*}\circplus\mathbf{Q}^{(1)}_{*,*}
\circplus\cdots\circplus\mathbf{Q}^{(s-1)}_{*,*},
\]
and the differential is
\[
\mathbb{D}ERIV^{(s-1)}=
(\mathbb{D}ERIV_{\mathbf{Q}}^{(0)},\mathbb{D}ERIVel^{(1)}+\mathbb{D}ERIV_{\mathbf{Q}}^{(1)},
\mathbb{D}ERIVel^{(2)}+\mathbb{D}ERIV_{\mathbf{Q}}^{(2)},\ldots,
\mathbb{D}ERIVel^{(s-1)}+\mathbb{D}ERIV_{\mathbf{Q}}^{(s-1)}).
\]
In fact $(\mathbf{K}^{(s-1)}_{*,*},\mathbb{D}ERIV^{(s-1)})$ is a differential
graded $R_*$-algebra which provides a multiplicative resolution
of $R_*/I^s$, with the augmentation given by
\[
\varepsilon^{(s-1)}
(x_0,x_1\tilde u_{\mathbf{i}_1},\ldots,x_{s-1}\tilde u_{\mathbf{i}_{s-1}})
=
x_0+x_1u_{\mathbf{i}_1}+\cdots+x_{s-1}u_{\mathbf{i}_{s-1}}.
\]
\end{thm}
The algebraic extension of $R_*$-modules
\[
0\leftarrow R_*/I_*^s\longleftarrow R_*/I_*^{s+1}\longleftarrow I_*^s/I_*^{s+1}\leftarrow0
\]
is classified by an element of
\[
\mathrm{E}xt_{R_*}^{1}(R_*/I_*^s,I_*^s/I_*^{s+1})=
\Hom_{\mathcal{D}_{R_*}}(R_*/I_*^s,I_*^s/I_*^{s+1}[-1]),
\]
where $\Hom_{\mathcal{D}_{R_*}}$ denotes morphisms in the derived category
$\mathcal{D}_{R_*}$ of the ring $R_*$~\cite{Weibel}. This element is
represented by the composite
\begin{equation}\leftarrowbel{eqn:R/Is-R/I(s+1)-Ext1}
\tilde\mathbb{D}ERIVel^{(s)}_*\colon
\mathbf{K}^{(s-1)}_{*,*}\xrightarrow{\text{proj}}
\mathbf{Q}^{(s-1)}_{*,*}\xrightarrow{\mathbb{D}ERIVel^{(s)}}\mathbf{Q}^{(s)}[-1]_{*,*}.
\end{equation}
The analogue of the next result for ungraded rings was proved
in~\cite{AB:HomRegQuot}; the proof is easily adapted to the
graded case.
\begin{prop}\leftarrowbel{prop:QComp-Exactness}
For each $s\geqslant2$, the following complex is exact:
\begin{multline*}
\Tor^{R_*}_{*,*}(R_*/I_*,R_*/I_*)\xrightarrow{\mathbb{D}ERIVel^{(1)}_*}
\Tor^{R_*}_{*,*}(R_*/I_*,I_*/I_*^2) \\
\xrightarrow{\mathbb{D}ERIVel^{(2)}_*}
\cdots\xrightarrow{\mathbb{D}ERIVel^{(s-1)}_*}
\Tor^{R_*}_{*,*}(R_*/I_*,I_*^{s-1}/I_*^{s}).
\end{multline*}
\end{prop}
\begin{thm}\leftarrowbel{thm:Tor(R/I,R/Is)}
For $s\geqslant2$,
\[
\Tor^{R_*}_{*,*}(R_*/I_*,R_*/I_*^s)=R_*/I_*\circplus\coker\mathbb{D}ERIVel^{(s-1)}_*.
\]
This is a free $R_*/I_*$-module and with its natural $R_*/I_*$-algebra
structure, $\Tor^{R_*}_{*,*}(R_*/I_*,R_*/I_*^s)$ has trivial products.
\end{thm}
Given this algebraic background, we can now construct the $I$-adic
tower.
\begin{thm}\leftarrowbel{thm:R/I^s}
There is a tower of $R$-modules
\[
R/I\longleftarrow R/I^2\longleftarrow\cdots\longleftarrow R/I^s\longleftarrow R/I^{s+1}\longleftarrow\cdots
\]
whose maps define fibre sequences
\[
R/I^s\longleftarrow R/I^{s+1}\longleftarrow I^s/I^{s+1}
\]
which in homotopy realise the exact sequences of $R_*$-modules
\[
0\leftarrow R_*/I_*^s\longleftarrow R_*/I_*^{s+1}\longleftarrow I_*^s/I_*^{s+1}\leftarrow0.
\]
Furthermore, the following conditions are satisfied for each $s\geqslant1$.\\
{\rm(i)} $E^R_*R/I^s$ is a free $E_*$-module and the unit induces
a splitting
\[
E^R_*R/I^s=E_*\circplus(\Bbbker\colonE^R_*R/I^s\longrightarrow E_*);
\]
{\rm(ii)} the projection map $R/I^{s+1}\longrightarrow R/I^s$ induces
the zero map
\[
(\Bbbker\colonE^R_*R/I^{s+1}\longrightarrow E_*)\longrightarrow(\Bbbker\colonE^R_*R/I^s\longrightarrow E_*);
\]
{\rm(iii)} the inclusion map $j_s\colonI^s/I^{s+1}\longrightarrow R/I^{s+1}$
induces an exact sequence
\[
E^R_*I^{s-1}/I^s\xrightarrow{\mathbb{D}ERIVel^{(s)}_*}E^R_*I^s/I^{s+1}
\xrightarrow{{j_s}_*}(\Bbbker\colonE^R_*R/I^{s+1}\longrightarrow E_*)\rightarrow0.
\]
\end{thm}
\begin{proof}
The proof is by induction on $s$. Assuming that $R/I^s$ exists
with the asserted properties, we will define a suitable map
$\mathbb{D}ERIVelta_s\colonR/I^s\longrightarrow\Sigma I^s/I^{s+1}$ which induces a fibre
sequence of the form
\begin{equation}\leftarrowbel{eqn:FibSeq(s+1)}
R/I^s\longleftarrow X^{(s+1)}\longleftarrow I^s/I^{s+1},
\end{equation}
for which $\pi_*X^{(s+1)}=R_*/I_*^{s+1}$ as an $R_*$-module.
If $M$ is an $R$-module which is an $E$ module spectrum,
Theorem~\ref{thm:UCSS-RQ-Conditions} provides a Universal
Coefficient Spectral Sequence
\[
\mathrm{E}_2^{*,*}=
\mathrm{E}xt^{p,q}_{E_*}(E^R_*R/I^s,M_*)\LOCra\mathcal{D}_R(R/I^s,M)^{p+q}.
\]
Since $E^R_*R/I^s$ is $E_*$-free, this spectral sequence collapses
to give
\[
\mathcal{D}_R(R/I^s,M)^*=\Hom^*_{E_*}(E^R_*R/I^s,M_*).
\]
In particular, for $M=I^s/I^{s+1}$,
\[
\mathcal{D}_R(R/I^s,I^s/I^{s+1})^n=\Hom^n_{E_*}(E^R_*R/I^s,I_*^s/I_*^{s+1}).
\]
By~\eqref{eqn:R/Is-R/I(s+1)-Ext1} and Theorem~\ref{thm:R/Is-Resn}, there is
an element
\[
\tilde\mathbb{D}ERIVel^{(s)}_*\in
\Hom_{E_*}^0(E^R_*R/I^s,I_*^s/I_*^{s+1}[-1])
=
\Hom_{E_*}^1(E^R_*R/I^s,I_*^s/I_*^{s+1}),
\]
corresponding to an element $\mathbb{D}ERIVelta_s\colonR/I^s\longrightarrow\Sigma I^s/I^{s+1}$ inducing
a fibre sequence as in \eqref{eqn:FibSeq(s+1)}. It still remains to verify that
$\pi_*X^{(s+1)}=R_*/I_*^{s+1}$ as an $R_*$-module.
For this, we will use the resolutions $\mathbf{K}^{(s-1)}_{*,*}\longrightarrow R_*/I_*^s\rightarrow0$
and $\mathbf{K}_{*,*}\longrightarrow R_*/I_*\rightarrow0$. These free resolutions give rise to
cell $R$-module structures on $R/I^s$ and $E$. By~\cite{EKMM}, the $R$-module
$E\SSmash{R}R/I^s$ admits a cell structure with cells in one-one correspondence
with the elements of the obvious tensor product basis of
$\mathbf{K}_{*,*}\circTimes{R_*}\mathbf{K}^{(s-1)}_{*,*}$. Hence there is a resolution
by free $R_*$-modules
\[
\mathbf{K}_{*,*}\circTimes{R_*}\mathbf{K}^{(s-1)}_{*,*}\longrightarrow E^R_*R/I^s\rightarrow0.
\]
There are morphisms of chain complexes
\[
\mathbf{K}^{(s-1)}_{*,*}
\xrightarrow{\rho_s}
\mathbf{K}_{*,*}\circTimes{R_*}\mathbf{K}^{(s-1)}_{*,*}
\xrightarrow{\tilde\mathbb{D}ERIVelta_s}
\mathbf{Q}^{(s)}_{*,*}[-1],
\]
where $\rho_s$ is the obvious inclusion and $\tilde\mathbb{D}ERIVelta_s$ is
a chain map lifting $\tilde\mathbb{D}ERIVel^{(s)}_*$ which can be chosen so
that
\[
\tilde\mathbb{D}ERIVelta_s(e_i\circtimes x)=0.
\]
The effect of the composite $\tilde\mathbb{D}ERIVelta_s\rho_s$ on the generator
$e_i\tilde u_{(j_1,\ldots,j_{s-1})}\in\mathbf{K}^{(s-1)}_{1,*}$ turns
out to be
\[
\tilde\mathbb{D}ERIVel^{(s)}_*e_i\tilde u_{(j_1,\ldots,j_{s-1})}=
\tilde u_{(i,j_1,\ldots,j_{s-1})},
\]
while the elements of form $e_i\circtimes\tilde u_{(j_1,\ldots,j_{k-1})}$ with
$k<s$ are annihilated. The composite homomorphism
\[
\mathbf{K}^{(s-1)}_{1,*}\xrightarrow{\tilde\mathbb{D}ERIVelta_s\rho_s}
\mathbf{Q}^{(s)}_{0,*}[-1]\xrightarrow{\varepsilon_1} I_*^s/I_*^{s+1}[-1]
\]
is a cocycle. There is a morphism of exact sequences
\[
\begin{CD}
0@<<<R_*/I_*^s @<<<\mathbf{K}^{(s-1)}_{0,*}@<<<\mathbf{K}^{(s-1)}_{1,*}
@<<<\mathbf{K}^{(s-1)}_{2,*}\\
@. @| @V\alpha_0 VV @V\alpha_1 VV @VVV \\
0@<<<R_*/I_*^s @<<< R_*/I_*^{s+1} @<<<I_*^s/I_*^{s+1} @<<< 0 @.
\end{CD}
\]
where the cohomology class
\[
[\alpha_1]\in\mathrm{E}xt_{R_*}^{1,*}(R_*/I_*^s,I_*^s/I_*^{s+1})
\]
represents the extension of $R_*$-modules on the bottom row. It is
easy to see that $[\alpha_1]=[\varepsilon_1\tilde\mathbb{D}ERIVelta_s\rho_s]$,
hence this class also represents the extension of $R_*$-modules
\[
0\leftarrow R_*/I_*^s \longleftarrow \pi_*X^{s+1}\longleftarrow I_*^s/I_*^{s+1}\leftarrow 0.
\]
There is a diagram of cofibre triangles
\[
\begin{xy}
\xymatrix{
R/I^{s+1}\ar[d]&I/I^{s+1}\ar[d]\ar[l]
&\ar[l]
&\cdots
&&I^{s-1}/I^{s+1}\ar[l]\ar[d]&I^s/I^{s+1}\ar[l]\ar[d]^{=}\\
R/I\ar[ur]&I/I^2\ar[ur]
&& &\ar[ur]&I^{s-1}/I^{s}\ar[ur]&I^s/I^{s+1}
}
\end{xy}
\]
and applying $E^R_*(\ )$ we obtain a spectral sequence converging to
$E^R_*R/I^{s+1}$ whose $\mathrm{E}_2$-term is the homology of the complex
\[
0\rightarrow E^R_*R/I\xrightarrow{\mathbb{D}ERIVel^{(1)}_*}E^R_*I/I^2\xrightarrow{\mathbb{D}ERIVel^{(2)}_*}
E^R_*I^2/I^3\longrightarrow\cdots\xrightarrow{\mathbb{D}ERIVel^{(s)}_*}E^R_*I^s/I^{s+1}\rightarrow0,
\]
where the $\mathbb{D}ERIVel^{(k)}_*$ are essentially the maps used to compute
$\Tor^{R_*}_*(R_*/I_*,R_*/I_*^{s+1})$ in~\cite{AB:HomRegQuot}. By
Proposition~\ref{prop:QComp-Exactness} and Theorem~\ref{thm:Tor(R/I,R/Is)},
this complex is exact except at the ends, where we have $\Bbbker\mathbb{D}ERIVel^{(1)}_*=E_*$.
As a result, this spectral sequence collapses at $\mathrm{E}_3$ giving the desired form
for $E^R_*R/I^{s+1}$.
\end{proof}
\begin{cor}\leftarrowbel{cor:R/Is-E*Proj}
For any $E$-module spectrum $N$ and $s\geqslant1$,
\[
N_R^*R/I^s\cong\Hom_{E_*}(E^R_*R/I^s,N_*).
\]
\end{cor}
\begin{proof}
This follows from Theorem~\ref{thm:R/I^s}(i).
\end{proof}
We will also use the following result.
\begin{cor}\leftarrowbel{cor:ER*R/Is-TowerSurj}
For $s\geqslant1$, the natural map
\begin{align*}
E^R_*R/I^{s+1}\longrightarrow E^R_*R/I^s,
\end{align*}
has image equal to $E_*=E^R_*R$.
\end{cor}
\begin{proof}
This follows from Theorem~\ref{thm:R/I^s}(ii).
\end{proof}
\begin{cor}\leftarrowbel{cor:R/Is-TowerSurj}
For any $E$-module spectrum $N$ and $s\geqslant1$,
\[
\colim_sN_R^*R/I^s\cong N_R^*R\cong N_*.
\]
\end{cor}
\begin{proof}
This is immediate from Corollaries~\ref{cor:R/Is-E*Proj}
and \ref{cor:ER*R/Is-TowerSurj} since
\[
\colim_s\Hom_{E_*}(E^R_*R/I^s,N_*)\cong\Hom_{E_*}(E_*,N_*).
\]
\end{proof}
\section{The $I$-adic tower and Adams Spectral Sequence}
\leftarrowbel{sec:I-adictower-ASS}
Continuing with the notation of Section~\ref{sec:I-adictower},
the first substantial result of this section is
\begin{thm}\leftarrowbel{thm:I-adic-holim}
The $I$-adic tower
\[
R/I\longleftarrow R/I^2\longleftarrow\cdots\longleftarrow R/I^s\longleftarrow R/I^{s+1}\longleftarrow\cdots
\]
has homotopy limit
\[
\holim_sR/I^s\simeq\hat{\mathrm{L}}^R_ES_R.
\]
\end{thm}
Our approach follows ideas of Bousfield~\cite{Bousfield:ASS}
where it is shown that the following Lemma implies
Theorem~\ref{thm:I-adic-holim}.
\begin{lem}\leftarrowbel{lem:E-nilpotentTow-Conditions}
Let $E=R/I$. Then the following are true. \\
{\rm i)} Each $R/I^s$ is $E$-nilpotent. \\
{\rm ii)} For each $E$-nilpotent $R$-module $M$,
\[
\colim_s\mathcal{D}_R(R/I^s,M)^*=M_{-*}.
\]
\end{lem}
\begin{proof}
(i) is proved by an easy induction on $s\geqslant1$. \\
(ii) is a consequence of Corollary~\ref{cor:R/Is-TowerSurj}.
\end{proof}
Since the maps $R_*/I_*^{s+1}\longrightarrow R_*/I_*^s$ are surjective,
from the standard exact sequence for $\pi_*(\ )$ of a homotopy
limit we have
\begin{equation}\leftarrowbel{eqn:LQMU-nilcomp-pi*}
\pi_*\hat{\mathrm{L}}^R_ES_R=\lim_sR_*/I_*^s.
\end{equation}
We can generalize this to the case where $E$ is a commutative
localized regular quotient.
\begin{thm}\leftarrowbel{thm:LQMU-nilcomp}
Let $E=R/I[X^{-1}]$ be a commutative localized regular quotient
of $R$. Then
\[
\pi_*\hat{\mathrm{L}}^R_ES_R=R_*[X^{-1}]\sphat_{I_*}
=\invlim_sR_*[X^{-1}]/I_*^s.
\]
If the regular sequence generating $I_*$ is finite, then the
natural map $S_R\longrightarrow\hat{\mathrm{L}}^R_ES_R$ is an
$E$-equivalence,
hence
\begin{align*}
\mathrm{L}^R_ES_R&\simeq\hat{\mathrm{L}}^R_ES_R, \\
\pi_*\mathrm{L}^R_ES_R&=R_*[X^{-1}]\sphat_{I_*}.
\end{align*}
\end{thm}
\begin{proof}
The first statement is easy to verify.
By Remark~\ref{rem:Ehat-Invce}, to simplify notation we may as well
replace $R$ by $R[X^{-1}]$ and so assume that $E=R/I$ is a commutative
regular quotient of $R$.
Using the Koszul complex $(\LOCambda_{R_*}(e_j:j),\mathbb{D}ERIV)$, we see that
$\Tor^{R_*}_{*,*}(E_*,(R_*)\sphat_{I_*})$ is the homology of the
complex
\[
\LOCambda_{R_*}(e_j:j)\circTimes{R_*}(R_*)\sphat_{I_*}
=\LOCambda_{(R_*)\sphat_{I_*}}(e_j:j)
\]
with differential $\mathbb{D}ERIV'=\mathbb{D}ERIV\circtimes1$.
Since the sequence $u_j$ remains regular in $(R_*)\sphat_{I_*}$,
this complex provides a free resolution of $E_*=R_*/I_*$ as an
$(R_*)\sphat_{I_*}$-module (this is \emph{false} if the sequence
$u_j$ is not finite).
Hence we have
\[
\Tor^{R_*}_{*,*}(E_*,(R_*)\sphat_{I_*})
=
\Tor^{(R_*)\sphat_{I_*}}_{*,*}(E_*,(R_*)\sphat_{I_*})
=E_*.
\]
To calculate $E_*^R\hat{\mathrm{L}}^R_ES_R$ we may use the K\"unneth
Spectral Sequence of~\cite{EKMM},
\[
\mathrm{E}_2^{s,t}=\Tor^{R_*}_{s,t}(E_*,\hat{\mathrm{L}}^R_ES_R)
\LOCra E^R_{s+t}\hat{\mathrm{L}}^R_ES_R.
\]
By the first part,
the $\mathrm{E}_2$-term is
\[
\Tor^{R_*}_{*,*}(E_*,(R_*)\sphat_{I_*})=E_*=E^R_*R.
\]
Hence the natural homomorphism
\[
E^R_*S_R\longrightarrow E^R_*\hat{\mathrm{L}}^R_ES_R
\]
is an isomorphism.
\end{proof}
If the sequence $u_j$ is infinite, the calculation of this proof
shows that
\[
E^R_*\hat{\mathrm{L}}^R_ES_R=(R_*)\sphat_{I_*}/I_*\neq
R_*/I_*=E_*S_R
\]
and the Adams Spectral Sequence does not converge to the homotopy
of the $E$-localization.
An induction on the number of cells of $M$ proves a generalization
of Theorem~\ref{thm:LQMU-nilcomp}.
\begin{thm}\leftarrowbel{thm:LQ-R}
Let $E$ be a commutative localized regular quotient of $R$ and $M$
a finite cell $R$-module. Then
\[
\pi_*\hat{\mathrm{L}}^R_EM=M_*[X^{-1}]\sphat_{I_*}
=R_*[X^{-1}]\sphat_{I_*}\circTimes{R_*}M_*.
\]
If the regular sequence generating $I_*$ is finite, then the
natural map $M\longrightarrow\hat{\mathrm{L}}^R_EM$ is an $E$-equivalence,
hence
\begin{align*}
\LOC^R_EM&\simeq\hat{\mathrm{L}}^R_EM, \\
\pi_*\LOC^R_EM&=M_*[X^{-1}]\sphat_{I_*}
=R_*[X^{-1}]\sphat_{I_*}\circTimes{R_*}M_*.
\end{align*}
\end{thm}
The reader may wonder if the following conjecture is true, the algebraic
issue being that it does not appear to be true that for a commutative ring
$A$, the extension $A\longrightarrow A\sphat_J$ is always flat for an ideal $J\triangleleft A$,
a Noetherian condition normally being required to establish such a result.
\begin{conj}\leftarrowbel{conj:LQ}
The conclusion of Theorem~\ref{thm:LQ-R} holds when $E$ is any commutative
localized quotient of $R$.
\end{conj}
\section{Some examples associated with ${MU}$}
\leftarrowbel{sec:LRQ-MU}
An obvious source of commutative localized regular quotients is the
commutative $S$-algebra $R={MU}$ and we will describe some important
examples. It would appear to be algebraically simpler to work with
${BP}$ at a prime $p$ in place of ${MU}$, but at the time of writing,
it seems not to be known whether ${BP}$ admits a commutative $S$-algebra
structure.
\subsection*{Example A: ${MU}\longrightarrow H\mathbb{F}_p$.}
Let $p$ be a prime. By considering the Eil\-en\-berg\--Mac~Lane
spectrum $H\mathbb{F}_p$ as a commutative ${MU}$-algebra~\cite{EKMM},
we can form $H\mathbb{F}_p\SSmash{{MU}}H\mathbb{F}_p$. The K\"unneth Spectral
Sequence gives
\[
\mathrm{E}^2_{s,t}=\Tor_{s,t}^{{MU}_*}(\mathbb{F}_p,\mathbb{F}_p)
\LOCra
{H\mathbb{F}_p\,}^{{MU}}_{s+t}H\mathbb{F}_p.
\]
Using a Koszul complex over ${MU}_*$, it is straightforward to see that
\[
\mathrm{E}^2_{*,*}=\LOCambda_{\mathbb{F}_p}(\tau_j:j\geqslant0),
\]
the exterior algebra over $\mathbb{F}_p$ with generators
$\tau_j\in\mathrm{E}^2_{1,2j}$.
Taking $R={MU}$ and $E=H\mathbb{F}_p$, we obtain a spectral sequence
\[
\mathrm{E}_2^{s,t}({MU})=
\mathbb{C}oext_{\LOCambda_{\mathbb{F}_p}(\tau_j:j\geqslant0)}^{s,t}(\mathbb{F}_p,\mathbb{F}_p)
\LOCra
\pi_{s+t}
\hat{\LOC}
^{MU}_{H\mathbb{F}_p}S_{MU},
\]
where $I_\infty\triangleleft{MU}_*$ is generated by $p$ together with
all positive degree elements, so ${MU}_*/I_\infty=\mathbb{F}_p$. Also,
\[
\pi_*
\hat{\LOC}
^{MU}_{H\mathbb{F}_p}S_{MU}=({MU}_*)\sphat_{I_\infty}.
\]
More generally, for a finite cell ${MU}$-module $M$, the Adams
Spectral Sequence has the form
\[
\mathrm{E}_2^{s,t}(M)=
\mathbb{C}oext_{\LOCambda_{\mathbb{F}_p}(\tau_j:j\geqslant0)}^{s,t}(\mathbb{F}_p,{H\mathbb{F}_p}^{MU}_*M)
\LOCra
\pi_{s+t}
\hat{\LOC}
^{MU}_{H\mathbb{F}_p}M,
\]
where
\[
\pi_*
\hat{\LOC}
^{MU}_{H\mathbb{F}_p}M=(M_*)\sphat_{I_\infty}.
\]
\subsection*{Example B: ${MU}\longrightarrow\mathrm{E}n$.}
By~\cite{EKMM,Strickland:MU}, the Johnson-Wilson spectrum $\mathrm{E}n$ at an
\emph{odd} prime $p$ is a commutative ${MU}$-ring spectrum. According
to proposition~2.10 of~\cite{Strickland:MU}, at the prime $2$ a
certain modification of the usual construction also yields a commutative
${MU}$-ring spectrum which we will still denote by $\mathrm{E}n$ rather than
Strickland's $\mathrm{E}n'$. In all cases we can form the commutative ${MU}$-ring
spectrum $\mathrm{E}n\SSmash{{MU}}\mathrm{E}n$ and there is a K\"unneth Spectral Sequence
\[
\mathrm{E}^2_{s,t}=\Tor_{s,t}^{{MU}_*}(\mathrm{E}n_*,\mathrm{E}n_*)
\LOCra
\mathrm{E}n^{{MU}}_{s+t}\mathrm{E}n.
\]
By using a Koszul complex for ${MU}n_*$ over ${MU}_*$ and localizing
at $v_n$, we find that
\[
\mathrm{E}^2_{*,*}=
\LOCambda_{\mathrm{E}n_*}(\tau_j:\text{$j\geqslant1$ and $j\neq p^k-1$ with $1\leqslant k\leqslant n$}),
\]
where $\LOCambda$ denotes an exterior algebra and $\tau_j\in\mathrm{E}^2_{1,2j}$.
So
\[
\mathrm{E}n^{{MU}}_*\mathrm{E}n=
\LOCambda_{\mathrm{E}n_*}(\tau_j:
\text{$j\geqslant1$ and $j\neq p^k-1$ with $1\leqslant k\leqslant n$})
\]
as an $\mathrm{E}n_*$-algebra.
When $R={MU}$ and $E=\mathrm{E}n$, we obtain a spectral sequence
\[
\mathrm{E}_2^{s,t}({MU})=
\mathbb{C}oext_{\LOCambda_{\mathrm{E}n_*}(\tau_j:j\geqslant n+1)}^{s,t}(\mathrm{E}n_*,\mathrm{E}n_*)
\LOCra
\pi_{s+t}
\hat{\LOC}
^{MU}_{\mathrm{E}n}{MU},
\]
where
\[
\pi_*
\hat{\LOC}
^{MU}_{\mathrm{E}n}{MU}=({MU}_*)_{(p)}[v_n^{-1}]\sphat_{J_{n+1}}
\]
and
\[
J_{n+1}=(\Bbbker\colon({MU}_*)_{(p)}[v_n^{-1}]\longrightarrow\mathrm{E}n_*)\triangleleft{MU}_*[v_n^{-1}].
\]
In the $\mathrm{E}_2$-term we have
\[
\mathrm{E}_2^{s,t}({MU})=
\mathrm{E}n_*[U_j:\text{$0\leqslant j\neq p^k-1$ for $0\leqslant k\leqslant n$}],
\]
with generator $U_j\in\mathrm{E}_2^{1,2j+1}({MU})$ corresponding to
an exterior generator in $\mathrm{E}n^{{MU}}_*\mathrm{E}n$ associated with
a polynomial generator of ${MU}_*$ in degree $2j$ lying in
$\Bbbker{MU}_*\longrightarrow\mathrm{E}n_*$.
More generally, for a finite cell ${MU}$-module $M$,
\[
\mathrm{E}_2^{s,t}(M)=
\mathbb{C}oext_{\LOCambda_{\mathrm{E}n_*}(\tau_j:j\geqslant n+1)}^{s,t}(\mathrm{E}n_*,\mathrm{E}n^{{MU}}_{\ *}M),
\LOCra
\pi_{s+t}
\hat{\LOC}
^{MU}_{\mathrm{E}n}M,
\]
where
\[
\pi_*
\hat{\LOC}
^{MU}_{\mathrm{E}n}M=M\sphat_{J_{n+1}}
=({MU}_*)_{(p)}[v_n^{-1}]\sphat_{J_{n+1}}\circTimes{{MU}}M.
\]
\subsection*{Example C: ${MU}\longrightarrowK(n)$.}
We know from~\cite{EKMM,Strickland:MU} that for an odd prime $p$, the
spectrum $K(n)$ representing the $n$\,th Morava $K$-theory $K(n)^*(\ )$
is a commutative ${MU}$ ring spectrum. There is a K\"unneth Spectral
Sequence
\[
\mathrm{E}^2_{s,t}=\Tor_{s,t}^{{MU}_*}(K(n)_*,K(n)_*)
\LOCra
K(n)^{{MU}}_{s+t}K(n),
\]
and we have
\[
\mathrm{E}^2_{*,*}=\LOCambda_{K(n)_*}(\tau_j:0\leqslant j\neq p^n-1).
\]
Taking $R={MU}$ and $E=K(n)$, we obtain a spectral sequence
\[
\mathrm{E}_2^{s,t}({MU})=
\mathbb{C}oext_{\LOCambda_{K(n)_*}(\tau_j:\text{$0\leqslant j\neq n$})}^{s,t}(K(n)_*,K(n)_*)
\LOCra\pi_{s+t}
\hat{\LOC}
^{{MU}}_{K(n)}{MU},
\]
where
\[
\pi_*
\hat{\LOC}
^{{MU}}_{K(n)}{MU}=({MU}_*)\sphat_{I_{n,\infty}}
\]
with $I_{n,\infty}=\Bbbker{MU}_*\longrightarrowK(n)_*$. In the $\mathrm{E}_2$-term
we have
\[
\mathrm{E}_2^{s,t}({MU})=
\mathrm{E}n_*[U_j:\text{$0\leqslant j\neq p^n-1$}],
\]
with generator $U_j\in\mathrm{E}_2^{1,2j+1}({MU})$ corresponding to
an exterior generator in $\mathrm{E}n^{{MU}}_*\mathrm{E}n$ associated with
a polynomial generator of ${MU}_*$ in degree $2k$ lying in
$\Bbbker{MU}_*\longrightarrow\mathrm{E}n_*$ (or when $j=0$, associated with $p$).
More generally, for a finite cell ${MU}$-module $M$,
\[
\mathrm{E}_2^{s,t}(M)=
\mathbb{C}oext_{\LOCambda_{K(n)_*}(\tau_j:\text{$0\leqslant j\neq n$})}^{s,t}(K(n)_*,K(n)^{{MU}}_*M)
\LOCra\pi_{s+t}
\hat{\LOC}
^{{MU}}_{K(n)}M,
\]
where
\[
\pi_*
\hat{\LOC}
^{{MU}}_{K(n)}M=(M_*)_{I_n,\infty}\sphat
=({MU}_*)\sphat_{I_{n,\infty}}\circTimes{{MU}_*}M_*.
\]
\section*{Concluding remarks}
There are several outstanding issues raised by our work.
Apart from the question of whether it is possible to weaken the assumptions
from (commutative) regular quotients to a more general class, it seems reasonable
to ask whether the internal $I$-adic tower is one of $R$ ring spectra.
Since $\LOC^R_ER=\mathbb{D}ERIVs\holim_s R/I^s$ (at least when $I_*$ is finitely
generated),
the localization theory of~\cite{EKMM,Wolbert} shows that this can be
realized as a commutative $R$-algebra. However, showing that each $R/I^s$
is an $R$ ring spectrum or even an $R$-algebra seem to involve far more
intricate calculations. We expect that this will turn out to be true and even
that the tower is one of $R$-algebras. This should involve techniques similar
to those of~\cite{Lazarev,AB+AJ:Brave-MU}. It is also worth noting that our
proofs make no distinction between the cases where $I_*\triangleleft R_*$ is infinitely
or finitely generated. There are a number of algebraic simplifications possible
in the latter case, however we have avoided using them since the most interesting
examples we know are associated with infinitely generated regular ideals in
${MU}_*$. The spectra $E_n$ of Hopkins, Miller \emph{et al.} have Noetherian
homotopy rings and there are towers based on powers of their maximal ideals
similar to those in the first author's previous work~\cite{AB:Ainfty,AB-UW:Bockop}.
We also hope that our preliminary exploration of Adams Spectral Sequences for
$R$-modules will lead to further work on this topic, particularly in the case
$R={MU}$ and related examples. A more ambitious project would be to investigate
the commutative $S$-algebra $MSp$ from this point of view, perhaps reworking
the results of Vershinin, Gorbounov and Botvinnik in the context of
$MSp$-modules~\cite{Botvinnik,Vershinin}.
\Addresses\recd
\end{document} | math | 57,102 |
\begin{equation}gin{document}
\noindent
Contribution to ``Bohmian Mechanics and Quantum Theory:
An Appraisal,''\\
edited by J.T.\ Cushing, A.\ Fine, and S.\ Goldstein
\begin{itemize}gskip
\begin{itemize}ggi
{\huge \bf Scattering Theory}
\noindent
{\huge \bf from a Bohmian Perspective}
\begin{itemize}gskip
\begin{itemize}ggi
Martin Daumer\\
Mathematisches Institut der Universit\"{a}t M\"{u}nchen,\\
Theresienstra{\ss}e 39,
80333 M\"{u}nchen, Germany
\begin{itemize}ggi
\today
Quantum mechanical scattering theory is a subject with a long and winding
history. We shall pick out some of the most important concepts and ideas
of scattering theory and look at them
{}from the perspective of Bohmian mechanics: Bohmian mechanics , having real particle
trajectories,
provides an excellent basis for analyzing scattering phenomena.
Schr\"odinger equationction{A very brief historical sketch}
We begin with a quote taken from Born 1926, shortly after Heisenberg 1925
had invented matrix mechanics and Schr\"odinger 1926 his wave mechanics:
\begin{itemize}gskip
{\footnotesize
Neither of these two conceptions appear satisfactory to me. I should like
to
attempt here to give a third interpretation and to test its utility on
collision processes.
In this attempt, I adhere to an observation of Einstein on the
relationship of wave field and light quanta; he said, for example, that
the waves are
present
only to show the corpuscular light quanta the way, and he spoke in the
sense of
a ``ghost field''. This determines the probability that
a light quantum \dots takes a certain path; \dots
And here it is obvious to regard the de Broglie-Schr\"odinger waves as
the ghost
field or, better, ``guiding field''.
I should therefore like to investigate experimentally the following
idea: the guiding
field, represented by a scalar function $\psi$ of the coordinates of
all
the particles
involved and the time, propagates in accordance with Schr\"odinger's
differential
equation. \dots
The paths of these corpuscules are determined only to the extent that
the laws
of energy and momentum restrict them; otherwise, only a probability
for a certain path is found.}
\begin{itemize}gskip
and the
\begin{itemize}gskip
{\footnotesize
Closing remarks:
On the basis of the above discussion, I should like to put forward
the opinion
that quantum mechanics permits not only the formulation and solution
of the
problem of stationary states, but also that of transition processes.
In these
circumstances Schr\"odinger's version appears to do justice to the
facts in
by far the easiest manner; moreover, it permits the retention of the
conventional ideas of space and time in which events take place in a
completely
normal manner. On the other hand, the proposed theory does not
correspond to the
requirement of the causal determinacy of the individual event. In
my preliminary
communication I stressed this indeterminacy quite particularly,
since it appears
to me in best agreement with the practice of the experimenter.
But it is natural for him
who will not be satisfied with this to remain unconverted and to
assume that
there are other parameters, not given in the theory, that
determine the individual
event. In classical mechanics these are the ``phases'' of the
motion, i.e.\
the coordinates of the particles at a given instant. It appears
to me {\it a priori}\/
improbable that quantities corresponding to these phases can
easily be introduced into
the new theory, but Mr.~Frenkel has told me that this may
perhaps be the case.
However this may be, this possibility would not alter anything
relating the
practical indeterminacy
of collision processes, since it is in fact impossible to
give the values of the
phases; it must in fact lead to the same formulae as
the ``phaseless'' theory
proposed here.
{
Born 1926, translated in Ludwig 1968, p.224}
}
\begin{itemize}gskip
Schr\"odinger 1926 had shown with his wave mechanics
how to obtain the discrete
energy levels of the hydrogen atom, by seeking stationary
square integrable solutions
of the Schr\"odinger equation\ (in units $\hbar=m=1$)
\begin{equation}
\label{SCHREQ}
i \frac{\partial \psi_t({\bf x})}{\partial t} =
-\frac{1}{2}\Delta \psi_t({\bf x}) +
V({\bf x}) \psi_t({\bf x}).
\end{equation}
What are called stationary states are solutions
of the form $\psi_t({\bf x}) = e^{-iEt}
\psi({\bf x})$, where
$\psi({\bf x})$ obeys the stationary Schr\"odinger equation
\begin{equation}
\label{STATSCH}
-\frac{1}{2} \Delta \psi({\bf x}) + V({\bf x})\psi({\bf x}) =
E\psi({\bf x}),
\end{equation}
which has square integrable solutions only for certain discrete energies
$E_n < 0$, in agreement with the experimental results.
The physical meaning of the wave function was however unclear.
A description of a time-dependent scattering process
was soon given by
Max Born 1926, who explored the hypothesis that the wave function
might be a ``guiding field'' for the motion of the electron. As a
consequence
of this hypothesis, Born is led in his paper to the ``statistical
interpretation'' of the wave function: $\rho_t({\bf x}) = |\psi_t({\bf x})|^2$
is the
probability density for a particle to be at point ${\bf x}$ at time $t$.
It follows from (\ref{SCHREQ}) that there is a conserved flux
corresponding
to this density, the quantum flux $\bj^{\psi_t}({\bf x}) :=
{\rm Im} \begin{itemize}gl(\psi_t^*({\bf x}) \nabla \psi_t({\bf x})\begin{itemize}gr)$, which
obeys the continuity
equation
\begin{equation}
\label{CONTEQ}
\frac{\partial |\psi_t({\bf x})|^2}{\partial t} + \nabla \cdot \bj^{\psi_t}({\bf x})
= 0.
\end{equation}
Born interpreted the quantum flux as a probability current of particles.
His basic ansatz, sometimes called ``naive'' scattering theory
(Reed, Simon 1979, p.\ 355),
is to seek non-normalized solutions of the stationary Schr\"odinger equation\ (\ref{STATSCH})
for {\it positive} energies $E=k^2/2$, which have the long-distance
behavior
\begin{equation}
\label{LONGDIS}
\psi({\bf x}) \stackrel{x\to\infty}{\sim} e^{i{\bf k}\cdot{\bf x}} + f(\theta,\phi)
\frac{e^{ikx}}{x}.
\end{equation}
$e^{i{\bf k}\cdot{\bf x}}$ is interpreted as the incoming plane wave and
$f(\theta,\phi) \frac{e^{ikx}}{x}$ as the outgoing spherical wave with
angular dependent density.
The flux corresponding to the incoming wave is ${\bf k}$, such that the
number
of crossings per unit time and a unit surface orthogonal to ${\bf k}$ is $k :=
|{\bf k}|$. The flux
corresponding to the spherical wave is $\frac{{\bf x}}{x^3} k|f(\theta,\phi)|^2$
and is obviously
purely radial. The number
of crossings of the surface element $x^2d\Omega$ of a distant sphere
in a direction
specified by the angles $\theta,\phi$ per unit time divided by the number
of incoming particles per unit time and unit surface
is called ``differential cross section,'' and one finds
\begin{equation}
\label{BORNFORM}
\frac{d\sigma}{d\Omega} = |f(\theta,\phi)|^2.
\end{equation}
This description of a scattering process is however not convincing for
simple reasons: there is no hint in the equations what
the particles do which are responsible for
the flux and how their motion is related
to the wave function such that $\rho_t({\bf x}) = |\psi_t({\bf x})|^2$
holds for all times.
Moreover, the picture is entirely time-independent although a scattering
process is certainly a process in space and time; stuff moves.
The arguments leading to
the formula (\ref{BORNFORM}) for the cross section
``wouldn't convince an educated first grader'' (Goldberger, cited in
Simon 1971, p 97).
{}From a physical point of view it might have seemed natural
that a time-dependent justification of Born's time-independent method
was developed, involving a detailed analysis of the behavior
of the wave function\ and its corresponding flux, but scattering theory proceeded
along a
different direction. (For the reasons why Born later abandoned the idea
of a guiding field see e.g.\ Beller 1990.)
Heisenberg aimed at casting all in-principle-measurable quantities,
such as
energy levels and the cross section, into a single abstract object: the
unitary $S$-matrix, or better, writing $S=e^{i\eta}$, into the self-adjoint\
``phase matrix''
$\eta$ (Heisenberg 1946).
$S$ should be the map from freely evolving
``in''-states, which are controllable
by an experimenter, to the freely evolving ``out''-states, whose
properties can
be measured.
Motivated by this idea M{\o}ller
introduced the concept of ``wave operators'' (M{\o}ller 1946),
the building blocks of the $S$-matrix: one expects that
in a scattering process any wave function has a simple asymptotic
behavior for large negative and positive times, where it should evolve
almost freely with
$e^{-iH_0t}$ ($H_0 := -\frac{1}{2}\Delta$).
Hence there should be states $\phi_+$ and $\phi_-$ such that
\begin{equation}
\label{WAVEOPDER}
\lim_{t\to\pm\infty} \|e^{-iHt}\psi - e^{-iH_0t} \phi_{\pm}\|_2 = 0,
\end{equation}
where $\|\cdot\|_2$ is the norm in $L^2$.
If the wave operators
\begin{equation}
\label{WAVEOPDEF}
\Omega_{\mp} = {\mbox{\rm s-}} \lim_{t\to\pm\infty} e^{iHt}e^{-iH_0t}
\end{equation}
(``$ {\mbox{\rm s-}} \lim$'' denotes the strong limit) exist, then,
if we now imagine that $\phi_\pm$ are given,
obviously $\psi:=\Omega_\mp \phi_\pm$ obeys (\ref{WAVEOPDER}).
(The strange sign convention is a tradition, see Reed, Simon 1979, p.17.)
The operators $\Omega_\pm$ should be unitary (we assume here that there
are no bound states) such that $\Omega_\pm^\dagger = \Omega_\pm^{-1}$.
The $S$-matrix can now be defined as $S= \Omega_-^\dagger\Omega_+$,
because it maps freely evolving ``in''-states onto ``out''-states.
There has been a lot of work on the precise definition and
properties of wave operators.
We
should mention here the
concept of modified wave operators which was introduced (Dollard 1964)
in order to bypass the problem, that the usual wave operators don't
exist for long range potentials such as the Coulomb potential.
The program of ``asymptotic completeness,'' which aims at proving
certain additional properties of the wave operators and the spectrum
of the Hamiltonian, kept mathematicians and physicists busy for
a long time, until recently this problem could be solved in
great generality (Derezi\'{n}ski, Gerard 1993).
Interestingly enough, these achievements
would not have been possible without
the ``renaissance'' of geometrical ideas (Ruelle 1969, Enss 1978), namely
by realizing that the wave packets evolve in space and time and are
far away from the scattering center most of the time,
instead of working with very abstract
and complicated methods in momentum space (Faddeev 1965), where this
simple
geometrical picture easily gets lost.
Most of this work on the $S$-matrix, wave operators and asymptotic
completeness
is however not much concerned with the original question of scattering
theory,
the justification of the formula for the cross section (\ref{BORNFORM})
of the ``naive''
scattering theory, which was
commonly used to do the actual numerical calculations.
It is true that formulas for the differential
cross section have been suggested from an analysis of
$\langle {\bf k}'|S|{\bf k}\mbox{\rm ran} gle$,
which was interpreted as the ``probability density''
to find the momentum eigenstate
$|{\bf k}'\mbox{\rm ran} gle$ in the final state $S|{\bf k}\mbox{\rm ran} gle$
(Lippmann and Schwinger 1950
used
``adiabatic switching,'' Gell-Mann and Goldberger 1953
Abelian limits), but these arguments
were not much more convincing than the original argument to arrive at
(\ref{BORNFORM}) (see also Reed, Simon 1979, p.356).
An exception is the work of Ikebe 1960, who
rigorously established the
physicist's notation of expansions in
continuum eigenfunctions (for ``Ikebe''-potentials)
and linked wave operators with solutions of the
``Lippmann-Schwinger equation.''
The time-dependent wave function\ (in physicists notation formally
$\langle{\bf x}|e^{-iHt}|\psi\mbox{\rm ran} gle =
\int d^3k e^{-i\kkt} \langle{\bf x}\widetilde{|{\bf k}\mbox{\rm ran} gle}
\widetilde{\langle {\bf k}|}\psi\mbox{\rm ran} gle)$
may be written as
\begin{equation}
\label{PSIT}
\psi_t({\bf x}) = (e^{-iHt}\psi)({\bf x}) =
(2\pi)^{-3/2}\int d^3k e^{-i\kkt}\phi({\bf x},{\bf k}) \psi^{\#}({\bf k}),
\end{equation}
where the ``generalized eigenfunctions'' $\phi({\bf x},{\bf k})$
are solutions of the
Lippmann-Schwinger equation
\begin{equation}
\label{LS1}
\phi({\bf x},{\bf k}) = e^{i{\bf k}\cdot{\bf x}} - \frac{1}{2\pi} \int d^3y
\frac{e^{-i{k|{\bf x}-{\bf y}|}}}{|{\bf x}-{\bf y}|}
V({\bf y}) \phi({\bf y},{\bf k}).
\end{equation}
Here $\psi^{\#}({\bf k}) := (2\pi)^{-3/2} \int d^3x \phi^*({\bf x},{\bf k}) \psi({\bf x})$
is the ``generalized Fourier transform'' of $\psi$
and is connected with the wave operators by
$(\widehat{\Omega_-^\dagger \psi})({\bf k}) = \psi^{\#}({\bf k})$,
where $\widehat{\ }$ denotes the usual Fourier transform.
(There is another set of eigenfunctions corresponding to $\Omega_+$
which are in fact the ones most frequently used.)
The solutions of the Lippmann-Schwinger equation
are also solutions of the stationary Schr\"odinger equation\
with the asymptotic behavior as in (\ref{LONGDIS})
(the ones for $\Omega_+$) such that
the differential cross section
may be read off their long distance asymptotics as in (\ref{BORNFORM}).
At least one could now find the formula (\ref{BORNFORM}) of the ``naive''
scattering theory
somewhere in an appropriate expansion of the
time-dependent wave function.
Another exception is the work of Dollard 1969.
Dollard suggested to use the probability to find a particle in the far
future
in a given cone $C\subset\IR^3$ as
a natural time-dependent {\it definition} of the
cross section.
Dollard's ``scattering-into-cones-theorem '' relates this probability to the wave operators:
\begin{equation}
\label{SICT}
\lim_{t \to \infty} \int_C d^3x |\psi_t({\bf x})|^2 = \int_C d^3v
|\widehat{\Omega_-^\dagger\psi}({\bf v})|^2.
\end{equation}
The scattering-into-cones-theorem\ has come to be regarded as the fundamental result from
which the differential
cross section ought to be derived (e.g.\ Reed, Simon 1979, p.356, and
Enss, Simon 1980).
Dollard's approach was however criticized
by Combes, Newton and Shtokhamer 1975. They observe that
the experimental relevance of the scattering-into-cones-theorem\ rests on the connection of
the probability of
finding the particle in the far future in a cone with the probability
that the particle
has, at some time, crossed a given distant surface subtended by the cone.
Heuristically,
the last
probability should be given by integrating the
quantum mechanical flux over the total time interval and this surface.
(The flux is often used that way in textbooks.) Combes, Newton and
Shtokhamer hence
conjecture the ``{\cal F }AS''
\begin{equation}
\label{FAST}
\lim_{R \to \infty} \int_0^\infty dt \int_{C\cap{\partial B_R}} {\bf j}^{\psi_t} \cdot {\bf n} d\sigma=
\int_C d^3v |\widehat{\Omega_-^\dagger\psi}({\bf v})|^2,
\end{equation}
where $B_R$ is the ball with radius $R$ and outward normal ${\bf n}$.
There exists no proof of this theorem. Even the
``free {\cal F }AS\,'' for freely evolving $\psi_t$,
\begin{equation}
\label{FASFREET}
\lim_{R \to \infty} \int_0^\infty dt \int_{C\cap{\partial B_R}} {\bf j}^{\psi_t} \cdot {\bf n} d\sigma=
\int_C d^3v |\hat{\psi}({\bf v})|^2
\end{equation}
which should be physically good enough, because the scattered wave packet
is expected to move almost freely
after the scattering is essentially completed,
has not been proven.
This is certainly strange because the physical
importance of
(\ref{FAST}) and (\ref{FASFREET}) for scattering theory
is obvious and the mathematical
problem does not
seem to be too hard.
Perhaps it was the vagueness in the meaning attached to the flux which
is responsible for this matter of fact. For example,
the authors try to
reformulate the problem in operator language and are faced with the
problem
that for general $L^2$-functions the current across a given surface
may well be
infinite. Instead of using smooth functions they use ``smeared-out''
surfaces and
therefore fail to find a proof of the original theorem.
Or, for example, they argue that
{\footnotesize
At large distances the
scattering part of the wave function contains outgoing particles only.
Therefore the particles cannot describe loops there and the flux can
be measured by the
interposition of counters on ${\partial B_R}$.}\\
and seem to have in mind a picture very
similar to Born's original proposal (cf.\ the Born quote), also without
giving a precise guiding law for the trajectories which would allow,
for example,
to check the
``no-loop-conjecture.''
Next we want to show how the ideas of Combes, Newton and Shtokhamer
arise naturally
by analyzing a scattering process in the framework of Bohmian mechanics
(Bell 1987,
Bohm 1952, Bohm and Hiley 1993, D\"urr, Goldstein and Zangh\a'{\i} 1992
and their
essay in this volume, Holland 1993):
in this theory
{\it particles move} along trajectories determined by the quantum flux,
controlling the expected number of particles crossings of surfaces.
We will sketch the main ideas of the proof of (\ref{FASFREET})
(for the complete proof see Daumer 1995)
and indicate the extension to the interacting case.
We shall see
that the {\cal F }AS\
in Bohmian mechanics\ is a relation between the flux across a distant surface and the
asymptotic probability of outward crossings of the trajectories
of this surface---obviously the quantity of interest for the scattering
analysis
of {\it any} mechanical theory of point particles.
Schr\"odinger equationction{Bohmian Mechanics}
Bohmian mechanics\ does what not only Born found ``a priori improbable,'' namely it
shows that the
introduction of additional parameters into the theory, represented by
the Schr\"odinger equation\
(\ref{SCHREQ}),
which ``determine the
individual event'' is easily possible (see the
essay of D\"urr, Goldstein and Zangh\a'{\i}):
the integral curves of the velocity field
\begin{equation}
{\bf v}^{\psi_t}({\bf x}) = \frac{\bj^{\psi_t}}{\rho_t}({\bf x}) =
{\rm Im}\frac{\nabla \psi_t}{\psi_t}({\bf x})
\end{equation}
which are solutions of
\begin{equation}gin{equation}
\label{BM1}
\frac{d}{dt} {{\bf x}}(t) = {\bf v}^{\psi_t}({{\bf x}}(t)),
\end{equation}
together with an initial position ${\bf x}_0$ determine
the trajectory of the particle.
(For a proof of the global existence of the solutions
for general $N$-particle systems
and a large class of potentials see the essay of Berndl.)
The initial position is distributed according to the quantum equilibrium
probability
$\IP^\psi$ ($\psi$ is normalized) with density $\rho=|\psi|^2$ (for
a justification
of ``quantum equilibrium'' see
D\"urr, Goldstein and Zangh\a'{\i} 1992).
Thus, in Bohmian mechanics\ a particle moves along a trajectory
guided by the particle's wave function .
Hence, given $\psi_t$, the solutions ${{\bf x}}(t,{{\bf x}}_0)$ of equation
(\ref{BM1})
are random trajectories,
where the randomness comes from the $\IP^\psi$-distributed random initial
position ${{\bf x}}_0$,
$\psi$ being the initial wave function .
Consider now a region $G\subset \IR^3$ and let $N^{\Sigma,\Delta}$ be
the number of crossings of ${\bf x}(t)$ of
subsets $\Sigma \subset \partial G$ in
time intervals $\Delta \subset [0,\infty)$.
Splitting $N^{\Sigma,\Delta} =: N_+^{\Sigma,\Delta} + N_-^{\Sigma,\Delta}$,
where $N_+^{\Sigma,\Delta}$
denotes the number of outward crossings
and $N_-^{\Sigma,\Delta}$ the number of backward crossings of $\Sigma$ in
$\Delta$,
we define for the number of ``signed crossings''
$N_s^{\Sigma,\Delta} := N_+^{\Sigma,\Delta} - N_-^{\Sigma,\Delta}.$
By the very meaning of the probability flux it is rather
clear (and it can easily be computed, Berndl 1995),
that the expectated value of these numbers of
crossings in quantum equilibrium is given by integrals of the current,
namely
\begin{eqnarray}
\label{EXPECT}
\IE^\psi(N^{\Sigma,\Delta }) &=&
\int_\Delta dt\int_\Sigma |\bj^{\psi_t} \cdot {\bf n}| d\sigma \/ \Bigl( =
\int d^3x |\psi({\bf x})|^2
N^{\Sigma,\Delta}({\bf x})\Bigr)\\
\label{EXPECT1}
\IE^\psi(N_s^{\Sigma,\Delta}) &=& \int_\Delta dt\int_\Sigma \bj^{\psi_t}
\cdot {\bf n} d\sigma.
\end{eqnarray}
(This relation between the current and the expected number of crossings is
also one of the fundamental insights used in the proof of global existence
of solutions.)
Schr\"odinger equationction{Scattering analysis of Bohmian mechanics}
We want to analyze the {\it scattering regime} of Bohmian mechanics,
i.e.\ the
asymptotic behavior of the
distribution of crossings of the trajectories traversing some distant
surface surrounding the
scattering center (see also Daumer 1995).
As surfaces we chose, for the sake of simplicity, spheres and we fix
the notation
illustrated in figure 1.
(We may imagine the surface surrounding the scattering center or,
more generally,
simply an area in which the particle happens to be.)
We consider the random variables
(functions of the paths) {\it first exit time} from $B_R$
\begin{equation}
\label{ESCAPETIME}
t_e := \inf \{t\ge 0| {\bf x}(t) \notin B_R\}
\end{equation}
and the corresponding {\it exit position}
\begin{equation}
\label{ESCAPEPOSITION}
{\bf x}_e = {\bf x}(t_e).
\end{equation}
Upon solving (\ref{BM1}) and (\ref{SCHREQ}) the statistical
distributions for $t_e$ and ${\bf x}_e$
can of course be calculated (see e.g.\ Leavens 1990 and his
essay on the
related problem of tunneling times).
In general we should expect this to be a very hard task
but it turns out that if ${\partial B_R}$ is at most crossed once by every
trajectory---this is
what we expect to happen asymptotically in the scattering
regime---a very simple formula
involving the current obtains.
The probability of the exit positions $\IP^\psi({\bf x}_e \in R\Sigma)$
should become ``independent''
of $R$ for large $R$ such that we may focus on the map
$\sigma^\psi: {\cal B}(S^2) \to \IR^+$ defined by
\begin{equation}
\label{SIGMA0}
\sigma^\psi(\Sigma) := \lim_{R \to \infty} \IP^\psi({\bf x}_e\in R\Sigma) =
\lim_{R \to \infty} \IP^\psi(\frac{{\bf x}_e}{x_e} \in \Sigma),
\end{equation}
where $\frac{{\bf x}_e}{x_e}$ is the exit direction, which we expect
to be a probability
measure on
the unit sphere (if eventually all trajectories go off to infinity,
as they should.)
This measure
gives us the asymptotic probability of outward crossings of a distant
surface, certainly the quantity of interest for the scattering analysis of
any mechanical theory of point particles and it seems
appropriate to {\it define} $\sigma^\psi$ in (\ref{SIGMA0})
as the cross section measure (see also the last section).
How can we find a handy expression for this probability?
With formula (\ref{EXPECT}) we have already a
formula for the expected number of crossings
and the expected number of
signed crossings
of the surface $R\Sigma$ in the time interval $\Delta$.
For large $R$ the sphere ${\partial B_R}$ should be crossed at most once, from
the inside
to the outside, such that the number of crossings equals the number of
signed crossings, both being either 0 or 1, such that furthermore their
expectation value equals the probability, that the particle has crossed
the surface $R\Sigma$ at some time.
Hence, if there are asymptotically no backward crossings, i.e.\
if $ \lim_{R \to \infty} \IE^\psi(N_-^{{\partial B_R},[0,\infty)} ) = 0$,
we find for the asymptotic {\it probability}
that a trajectory crosses the surface $R\Sigma$ from the inside to the
outside
\begin{eqnarray}
\label{SIGMA1}
\sigma(\Sigma ) =
\lim_{R \to \infty} \IP^\psi({\bf x}_e\in R\Sigma) &=& \lim_{R \to \infty} \int_0^\infty dt \int_{R\Sigma}
{\bf j}^{\psi_t} \cdot {\bf n} d\sigma.
\end{eqnarray}
This is a very nice result, because it connects our (natural)
definition of the cross section (\ref{SIGMA0}) with the quantity
considered in the {\cal F }AS\ (\ref{FAST}).
Up to now the discussion leading to formula
(\ref{SIGMA1}) has been completely general concerning
the time evolution.
Let us now process (\ref{SIGMA1}) further, taking the
simplest case, namely free evolution. Our goal is to find a formula
where
the limit is taken.
The flux will contribute to the integral in (\ref{SIGMA1}) only for large
times,
because the packet has to travel a long time until it reaches the distant
sphere ${\partial B_R}$
such that we may use the long-time asymptotics of the free evolution.
We use the well-known formula (Reed, Simon 1975, p.59)
\begin{equation}
\psi_t({\bf x}) = (e^{-iH_0t} \psi)({\bf x}) =
\int d^3y \frac{ e^{i\frac{|{\bf x}-{\bf y}|^2}{2t}} }{(2\pi it)^{3/2}} \psi({\bf y})
\end{equation}
and obtain with the splitting
\begin{equation}
\psi_t({\bf x}) = \frac{e^{i\xxt}}{(it)^{3/2}} \hat{\psi}(\xt)
+ \frac{e^{i\xxt}}{(it)^{3/2}} \int \frac{d^3y}{(2\pi)^{3/2}}e^{-i\xyt}
(e^{i\yyt} - 1) \psi({\bf y}),
\end{equation}
neglecting the second term, as $t\to\infty$
\begin{equation}
\label{ASYMP}
\psi_t({\bf x}) \sim (it)^{-3/2}e^{i\xxt} \hat{\psi}(\xt).
\end{equation}
This asymptotics
for scattering theory has since long been realized as important (e.g.\
Brenig and Haag 1959 and
Dollard 1969, who proved that the asymptotics (\ref{ASYMP})
holds in the $L_2$ sense).
{}From
(\ref{ASYMP}) we find for $t\to\infty$
\begin{equation}
\label{JASYMPHEU}
\bj^{\psi_t}({\bf x}) = {\mbox{\rm Im}} \/ \psi_t^*({\bf x}) \nabla \psi_t({\bf x}) \approx
\xt t^{-3} |\hat{\psi}(\xt)|^2,
\end{equation}
and note, that the current and hence also the velocity
is strictly radial for large times, i.e.\
parallel to the outward normal ${\bf n}$ of ${\partial B_R}$, reflecting our
expectation that the
expected value of backward
crossings of ${\partial B_R}$ vanishes as $R\to\infty$.
Back to (\ref{SIGMA1}). Using the approximation
(\ref{JASYMPHEU}) and substituting ${\bf v} := \xt$ we arrive
at
\begin{eqnarray}
\label{NICE}
\int_0^\infty dt \int_{R\Sigma} \bj^{\psi_t} \cdot {\bf n} d\sigma
&\approx&
\int_0^\infty dt \int_{R\Sigma} t^{-3}|\hat{\psi}(\xt)|^2 \xt \cdot
{\bf n}({\bf x}) d\sigma \nonumber \\
&=& \int_0^\infty dv v^2 \int_C d\Omega |\hat{\psi}({\bf v})|^2 =
\int_C d^3v |\hat{\psi}({\bf v})|^2.
\end{eqnarray}
This heuristic argument for the free {\cal F }AS\ (\ref{FASFREET})
is so simple and intuitive that it is indeed
strange that it does not appear in any primer on scattering theory!
Now let us turn to the interacting case. Using Ikebe's eigenfunction
expansion
(\ref{PSIT}) and the relation of the generalized eigenfunctions with
the wave operators we
find that
\begin{equation}
\psi_t({\bf x}) = (2\pi)^{-3/2}\int d^3k e^{-i\kkt} \phi({\bf x},{\bf k})
\widehat{\Omega_-^\dagger\psi}({\bf k}).
\end{equation}
The Lippmann-Schwinger equation (\ref{LS1}) for $\phi({\bf x},{\bf k})$
allows us
to split off the free evolution of $\Omega_-^\dagger\psi$ and we
obtain
\begin{eqnarray}
\psi_t({\bf x}) &=& e^{-iH_0t}\Omega_-^\dagger\psi \nonumber \\
&-&
(2\pi)^{-3/2}\int d^3k e^{-i\kkt} \widehat{\Omega_-^\dagger\psi}({\bf k})
\Bigl(\frac{1}{2\pi} \int d^3y \frac{e^{-i{k|{\bf x}-{\bf y}|}}}{|{\bf x}-{\bf y}|}
V({\bf y}) \phi({\bf y},{\bf k})\Bigr).
\end{eqnarray}
Now observe that the first term immediately gives with (\ref{NICE})
the desired result
\begin{equation}
\label{INTFORM}
\lim_{R \to \infty} \int_0^\infty dt \int_{C\cap{\partial B_R}} {\bf j}^{\psi_t} \cdot {\bf n} d\sigma=
\int_C d^3v |\widehat{\Omega_-^\dagger\psi}({\bf v})|^2.
\end{equation}
The other terms which appear in the flux
should not contribute for the following reason: they all contain the
phase factor $e^{-i\kkt-ikx}$, where we used $e^{-ik|{\bf x}-{\bf y}|}
\approx e^{-ikx}$
for $x$ far away from the range of the potential.
This factor is rapidly oscillating
for large
$x$ and $t$ such that the $k$-integrals
should decay fast enough in $x$
and $t$ to give no contribution to the flux across surfaces.
Schr\"odinger equationction{Morals}
Let us recollect what we have achieved from our Bohmian perspective.
The analysis of the scattering regime of Bohmian mechanics\ suggests a natural
{\it definition}
of the cross section measure as the asymptotic probability distribution
of the
exit positions, formula (\ref{SIGMA0}).
This
is very similar to the definition of the cross section in classical
mechanics
(e.g.\ Reed, Simon 1979, p.15). The random distribution of impact
parameters
used in classical mechanics to define the cross section measure
corresponds to
the $|\psi|^2$ distribution of the initial positions of the particle.
The ``individual event'' (cf.\ the Born quote at the beginning),
that is the deflection of one particle in a certain direction,
is indeed determined by the initial position. However, the
initial positions are randomly distributed according to $|\psi|^2$
such that
there is no way to control the individual event. What is relevant
for
a scattering experiment with a given wave function
is the statistical distribution of these individual
events and the {\cal F }AS\ provides us with a formula for the asymptotic
distribution.
Let us now examine how this formula may be connected with the usual
operator formalism of quantum mechanics.
The first step has already been done, by writing formula (\ref{SIGMA0})
in terms of wave operators instead of using the
generalized eigenfunctions directly. (Existence of the wave operators
and asymptotic
completeness may
appear as a by-product, once an eigenfunction expansion has been
established
(Green and Lanford 1960).)
Note that formula (\ref{SIGMA0}) may be rewritten as
\begin{equation}
\sigma(\Sigma) = \int_C d^3v |\widehat{\Omega_-^\dagger\psi}({\bf v})|^2
= (\psi, \Omega_-{\cal F}^{-1}P_C{\cal F} \Omega_-^\dagger \psi),
\end{equation}
where $P_C$ denotes the projection operator on the cone $C$ and
${\cal F}$ is
the unitary operator of the fourier transformation. The map
$\Sigma \mapsto
\Omega_-{\cal F}^{-1}P_C{\cal F} \Omega_-^\dagger$ is an
explicit example
of what is called a projection operator valued measure which
corresponds to a unique
self-adjoint\ operator by the spectral theorem. This particular example
of escape statistics
exemplifies the general situation, namely that operators as
``observables'' appear
merely as computational tools in the phenomenology of certain
types of experiments---those
for which the statistics of the result are governed by a projection
operator valued measure.
Now we come to the end of the closing remarks of the Born quote
at the
beginning. Is it true that any deterministic completion of
quantum mechanics must lead
to the
same formulae for the cross section as Born's formula?
Certainly not!
Further assumptions would be required, e.g.\
the initial packet must be close to a plane wave, initially far away
from the scattering
center, in order to have a chance to arrive from formula (\ref{FAST})
(or from Dollard's formula (\ref{SICT}))
at Born's formula (\ref{BORNFORM}).
This is however not the strongest
point we can make here, because
Born's formula is not really taken seriously as the fundamental formula
for the cross section nowadays; but Dollard's formula certainly is.
But what is the quantum mechanical prediction if we go further and
do not want to make
such strong idealizations, e.g.\ if we happen to
place detectors around the scattering center
which are {\it not}\/ that
far away? What are the
predictions for the times and positions at which the detectors click?
Dollard's formula doesn't apply here, because it is an asymptotic formula.
Maybe---after some reflection---one comes up with what seems a very natural
candidate for the joint density of exit position and exit time, namely
$\bj^{\psi_t}({\bf x}) \cdot {\bf n}({{\bf x}})
d\sigma dt$.
But further
scrutinizing this answer within quantum mechanics should leave one uneasy.
After all, this is a formula---a prediction---which is not at all
a quantum mechanical prediction of the common type:
$\bj^{\psi_t}({\bf x}) \cdot {\bf n}({{\bf x}})$ can well be negative and thus
the formula makes sense only for (very) particular wave functions,
for which
$\bj^{\psi_t}({\bf x}) \cdot {\bf n}({{\bf x}})$ is positive.
But what is then the right formula for
all the other wave functions? Why is there such a simple formula for some
wave functions?
In Bohmian mechanics\ there is no need to feel uneasy. $\IP^\psi(({{\bf x}}_e,t_e) \in
(d\sigma,dt))$ is the
probability
that the particle exits at ${\bf x}_e$ at time $t_e$---and that
can be calculated in the usual way for {\it any} wave function .
For some wave function s it turns out (Daumer, D\"urr, Goldstein and Zangh\a'{\i}
1994) to be
indeed given by
\begin{equation}
\label{FUNDAMENT}
\IP^\psi(({{\bf x}}_e,t_e) \in (d\sigma,dt)) = \bj^{\psi_t}({\bf x}) \cdot {\bf n}({{\bf x}})
d\sigma dt,
\end{equation}
for others it isn't.
That's alright, isn't it?
\begin{equation}gin{figure}[b]
\label{STREUNG}
\begin{equation}gin{center}
\leavevmode
\epsfxsize=13cm
\epsffile{cone1.eps}
\end{center}
\caption{Possible trajectories of a particle
starting in the wave packet localized in the ball $B_R$.}
\end{figure}
Schr\"odinger equationction*{References}
Schr\"odinger equationtlength{\parindent}{0pt}
\parskip 3ex plus 0.1ex minus 0.1ex
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Mech.~Anal.} {\bf 5}, 1--34.
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\end{document} | math | 37,785 |
\begin{document}
\begin{abstract}
The training process of neural networks usually optimize weights and bias parameters of linear transformations, while nonlinear activation functions are pre-specified and fixed. This work develops a systematic approach to constructing matrix activation functions whose entries are generalized from ReLU. The activation is based on matrix-vector multiplications using only scalar multiplications and comparisons. The proposed activation functions depend on parameters that are trained along with the weights and bias vectors. Neural networks based on this approach are simple and efficient and are shown to be robust in numerical experiments.
\end{abstract}
\title{Neural networks with trainable matrix activation functions}
\section{Introduction}
In recent decades, deep neural networks (DNNs) have achieved significant successes in many fields such as computer vision and natural language processing~\cite{vision}, \cite{DBLP:journals/corr/abs-1807-10854}. The DNN surrogate model is constructed using recursive composition of linear transformations and nonlinear activation functions. To achieve good performance, it is essential to choose activation functions suitable for specific applications.
In practice, Rectified Linear Unit (ReLU) is one of the most popular activation functions for its simplicity and efficiency.
A drawback of ReLU is the presence of vanishing gradient in the training process, known as dying ReLU problem~\cite{Lu_2020}. Several relatively new activation approaches are proposed to overcome this problem, e.g., the simple Leaky ReLU, and
Piecewise Linear Unit (PLU)~\cite{plu}, Softplus~\cite{softplus}, Exponential Linear Unit (ELU)~\cite{elu}, Scaled Exponential Linear Unit (SELU)~\cite{selu}, Gaussian Error Linear Unit (GELU)~\cite{gelu}.
Although the aforementioned activation functions are shown to be competitive in benchmark tests, they are still fixed nonlinear functions. In a DNN structure, it is often hard to determine a priori the optimal activation function for a specific application. In this paper, we shall generalize ReLU and introduce arbitrary trainable matrix activation functions. The effectiveness of the proposed approach is validated using function approximation examples and well-known benchmark datasets such as \texttt{CIFAR-10} and \texttt{CIFAR-100}. There are a few classical works on adaptively tuning of parameters in the training process, e.g., the parametric ReLU~\cite{para_relu}. However, our adaptive matrix activation functions are shown to be competitive and more robust in those experiments.
\subsection{Preliminaries}
We consider the general learning process based on a given training set
$\{(x_n,f_n)\}_{n=1}^N$, where the inputs $\{x_n\}_{n=1}^N\subset\mathbb{R}^d$ and outputs $\{f_n\}_{n=1}^N\subset\mathbb{R}^J$ are implicitly related via an unknown target function $f:\mathbb{R}^d\to\mathbb{R}^J$ with $f_n=f(x_n)$.
The ReLU activation function is a piecewise linear function given by
\begin{equation}\label{ReLU}
\sigma(t)=\max\left\{t,0\right\},
\quad \mbox{for} \quad t\in \mathbb{R}.
\end{equation}
In the literature $\sigma$ is acting component-wise on an input vector. In a DNN, let $L$ be the
number of layers and $n_\ell$ denote the number of neurons at the $\ell$-th layer for $0\leq\ell\leq L$ with $n_0=d$ and $n_L=J$. Let
$\mathcal{W}=(W_1,W_2,\ldots,W_L)\in\prod_{\ell=1}^L\mathbb{R}^{n_{\ell}\times n_{\ell-1}}$ denote the tuple of admissible weight matrices and $\mathcal{B}=(b_1,b_2,\ldots,b_L)\in\prod_{\ell=1}^{L}\mathbb{R}^{n_\ell}$ the tuple of admissible bias vectors. The ReLU DNN approximation to $f$ at the $\ell$-th layer is recursively defined as
\begin{equation}\label{DNN}
\eta_{L,\mathcal{W},\mathcal{B}}:=h_{W_L,b_L}\circ\sigma\circ\cdots\circ h_{W_2,b_2}\circ\sigma\circ h_{W_1,b_1},
\end{equation}
where $h_{\mathcal{W}_\ell,b_\ell}(x)=W_\ell x+b_\ell$ is an affine linear transformation at the $\ell$-th layer.
The traditional training process for such a DNN is to find optimal
$\mathcal{W}_*\in\prod_{\ell=1}^L\mathbb{R}^{n_{\ell}\times n_{\ell-1}}$, $\mathcal{B}_*\in\prod_{\ell=1}^{L}\mathbb{R}^{n_\ell}$, (and thus optimal $\eta_{L,\mathcal{W}_*,\mathcal{B}_*}$) such that
\begin{equation}\label{bestfit}
(\mathcal{W}_*,\mathcal{B}_*)=\arg\min_{\mathcal{W}, \mathcal{B}}
E(\mathcal{W},\mathcal{B}), \quad\mbox{where}\quad
E(\mathcal{W},\mathcal{B})=\frac1N\sum_{n=1}^N \left|f_n-\eta_{L,\mathcal{W},\mathcal{B}}(x_n)\right|^2.
\end{equation}
In other words, $\eta_{L,\mathcal{W}_*,\mathcal{B}_*}$ best fits the data with respect to the discrete $\ell^2$ norm within the function class $\{\eta_{L,\mathcal{W},\mathcal{B}}\}$. In practice, the sum of squares norm in $E$ could be replaced with more convenient norms.
\section{Trainable matrix activation function}
Having a closer look at ReLU $\sigma,$ we observe that the activation $\sigma\circ\eta_\ell(x)=\sigma(\eta_\ell(x))$ could be realized as a matrix-vector multiplication $\sigma\circ\eta_\ell(x)=D_\ell(\eta_\ell(x))\eta_\ell(x)$ or equivalently $\sigma\circ\eta_\ell=(D_\ell\circ\eta_\ell) \eta_\ell$,
where $D_\ell$ is a \emph{diagonal} matrix-valued function mapping from $\mathbb{R}^{n_\ell}$ to $\mathbb{R}^{n_\ell\times n_\ell}$ with entries from the discrete set $\{0,1\}$. This is a simple but quite useful observation. There is no reason to restrict on $\{0,1\}$ and we thus look for a larger set of values over which the diagonal entries of $D_\ell$ are running or sampled. With slight abuse of notation, our new DNN approximation to $f$ is calculated using the following recurrence relation
\begin{equation}\label{newDNN}
\eta_1=h_{W_1,b_1},\quad\eta_{\ell+1}=h_{W_{\ell+1},b_{\ell+1}}\circ \big[(D_\ell\circ\eta_{\ell})\eta_{\ell}\big],\quad
\ell=1,\ldots,L-1.
\end{equation}
Here each $D_\ell$ is diagonal and is of the form \begin{equation}\label{Dell}
D_\ell(y) = \operatorname{diag}(\alpha_{\ell,1}(y_1),\alpha_{\ell,2}(y_2),\ldots,\alpha_{\ell,n_\ell}(y_{n_\ell})),\quad y\in\mathbb{R}^{n_\ell},
\end{equation}
where $\alpha_{\ell,i}(y_i)$ is a nonlinear function to be determined. Since piecewise constant functions can approximate a continuous function within arbitrarily high accuracy, we choose $\alpha_{\ell,i}$ with $1\leq i\leq n_\ell$ to be the following step function
\begin{equation}\label{alphali}
\alpha_{\ell,i}(s) = \left\{
\begin{aligned}
t_{\ell,i,0}, && s \in (-\infty, s_{\ell,i,1}], \\
t_{\ell,i,1}, && s \in (s_{\ell,i,1}, s_{\ell,i,2}],\\
\vdots \\
t_{\ell,i,m_{\ell,i}-1}, && s \in (s_{\ell,i,m_{\ell,i}-1}, s_{\ell,i,m_{\ell,i}}], \\
t_{\ell,i,m_{\ell,i}}, && s \in (s_{\ell,i,m_{\ell,i}}, \infty),
\end{aligned}
\right.
\end{equation}
where $m_{\ell,i}$ is a positive integer and $\{t_{\ell,i,j}\}_{j=0}^{m_{\ell,i}}$ and $\{s_{\ell,i,j}\}_{j=1}^{m_{\ell,i}}$ are constants. We may suppress the indices $\ell,i$ in $\alpha_{\ell,i}$, $m_{\ell,i}$, $t_{\ell,i,j}$, $s_{\ell,i,j}$ and write them as $\alpha$, $m$, $t_j$, $s_j$ when those quantities are uniform across layers and neurons.
If $m=1$, $s_1=0$, $t_0=0$, $t_1=1,$ then the DNN in \eqref{newDNN} is exactly ReLU DNN.
If $m=1$, $s_1=0$, $t_1=1$ and $t_0$ is a fixed small negative number, \eqref{newDNN} reduces to a DNN based on Leaky ReLU. If $m=2$, $s_1=0$, $s_2=1$, $t_0=t_2=0$, $t_1=1$, then $\alpha=\alpha_{\ell,i}$ actually represents the action of a discontinuous activation function.
In our case, we may train the parameters $\cup_{\ell=1}^L\cup_{i=1}^{n_\ell}\{t_{\ell,i,j}\}_{j=0}^{m_{\ell,i}}$ and $\cup_{\ell=1}^L\cup_{i=1}^{n_\ell}\{s_{\ell,i,j}\}_{j=1}^{m_{\ell,i}}$. In such a way, the resulting DNN may use different activation functions for different neurons and layers, and these activation functions are naturally adapted to the target function $f$ (or the target dataset). Since ReLU and Leaky ReLU are included by our DNN as special cases, the proposed DNN is clearly not worse than the traditional ones in practice. In the following, we call the neural network in \eqref{newDNN}, with the activation approach given in \eqref{Dell} and \eqref{alphali}, a DNN based on the ``Trainable Matrix Activation Function (TMAF)".
\subsection*{remark}
\emph{The activation functions used in TMAF neural network are not piecewise constants. Instead, TMAF activation is realized using matrix-vector multiplication, where entries of those matrices are trainable piecewise constants.}
Starting from the diagonal activation $D_\ell$, we can go
one step further to construct more general activation matrices. First we note that $D_\ell$ could be viewed as a nonlinear operator $T_\ell: [C(\mathbb{R}^d)]^{n_\ell}\rightarrow [C(\mathbb{R}^d)]^{n_\ell}$, where
\begin{equation*}
[T_\ell(g)](x)=D_\ell(g(x))g(x),\quad g\in [C(\mathbb{R}^d)]^{n_\ell},\quad x\in\mathbb{R}^d.
\end{equation*}
There seems to be no convincing reason to consider only diagonal operators. A more ambitious possibility is to consider a trainable nonlinear activation \emph{operator} determined by more general matrices, for example, by following tri-diagonal operator \begin{equation}\label{Tell}
[T_\ell(g)](x)=\begin{pmatrix}\alpha_{\ell,1}&\beta_{\ell,2}&0&\cdots&0\\
\gamma_{\ell,1}&\alpha_{\ell,2}&\beta_{\ell,3}&\cdots&0\\
\vdots&\ddots&\ddots&\ddots&\vdots\\
0&0&\cdots&\alpha_{\ell,n_\ell-1}&\beta_{\ell,n_\ell}\\
0&0&\cdots&\gamma_{\ell,n_\ell-1}&\alpha_{\ell,n_\ell}\end{pmatrix}g(x),\quad x\in\mathbb{R}^d.
\end{equation}
The diagonal $\{\alpha_{\ell,i}\}$ is given in \eqref{alphali} while the off-diagonals $\beta_{\ell,i}$, $\gamma_{\ell,i}$ are piecewise constant functions in the $i$-th coordinate $y_i$ of $y\in\mathbb{R}^{n_\ell}$ defined in a fashion similar to $\alpha_{\ell,i}$.
Theoretically speaking, even trainable full matrix activation is possible despite of potentially huge training cost. In summary,
a DNN based on trainable nonlinear activation operators $\{T_\ell\}_{\ell=1}^L$ reads
\begin{equation}\label{newDNN2}
\eta_1=h_{W_1,b_1},\quad\eta_{\ell+1}=h_{W_{\ell+1},b_{\ell+1}}\circ T_\ell(\eta_{\ell}),\quad
\ell=1,\ldots,L-1.
\end{equation}
The evaluation of $D_\ell$ and $T_\ell$ are cheap because they require only scalar multiplications and comparisons. When calling a general-purpose packages such as PyTorch or TensorFlow in the training process, it is observed that the computational time of $D_\ell$ and $T_\ell$ is comparable to the classical ReLU.
\subsubsection*{Remark}
\emph{Our observation also applies to an activation function $\sigma$ other than ReLU. For example, we may rescale $\sigma(x)$ to obtain $\sigma(\omega_{i,\ell}x)$ using a set of constants $\{\omega_{i,\ell}\}$ varying layer by layer and neuron by neuron. Then $\sigma(\omega_{i,\ell}x)$ are used to form a matrix activation function and a TMAF DNN, where $\{\omega_{i,\ell}\}$ are trained according to given data and are adapted to the target function. }
\section{Numerical results}
In this section, we demonstrate the feasibility and efficiency of TMAF by comparing it with traditional ReLU-type activation functions. Recall that neurons in the $\ell$-th layer will be activated by the matrix $D_\ell$. In principle, all parameters in \eqref{alphali} are allowed to be trained. To ensure practical efficiency, each diagonal entry $\alpha_\ell=\alpha_{\ell,i}$ of $D_\ell$ remains the same for all $i$. We shall only let function values $\{t_{\ell,i,j}\}$ in \eqref{alphali} be trained in the following. In each experiment, we use the same learning rates, stochastic optimization methods, and number NE of epochs (optimization iterations). In particular, the learning rate is 1e-4 from epoch $1$ to $\frac{\rm NE}{2}$ and 1e-5 is used from epoch $\frac{\rm NE}{2}+1$ to ${\rm NE}$. The optimization method is ADAM (\cite{adam}) based on mini-batches of size 500. Numerical experiments are performed in PyTorch (\cite{pytorch}). We provide two sets of numerical examples:
\begin{itemize}
\item Function approximations by TMAF networks and ReLU-type networks;
\item Classification problems for \texttt{MNIST} and \texttt{CIFAR} set solved by TMAF and ReLU networks.
\end{itemize}
For the first class of examples we use the $\ell^2$-loss function as defined in~\eqref{bestfit}. For the classification problems we consider the \emph{cross-entropy} that is widely used as a loss function in classification models. The cross entropy is defined using a training set having $p$ images, each with $N$ pixels. Thus, the training dataset is equivalent to the vector set $\{z_j\}_{j=1}^p\subset{R}^{N}$ with each $z_j$ being an image. The $j$-th image belongs to a class $c_j\in \{1,\ldots,M\}$. The neural network maps $z_j$ to $x_j\in \mathbb{R}^{M}$,
\begin{equation*}
x_j := \eta_{L,\mathcal{W},\mathcal{B}}(z_j)\in \mathbb{R}^M, \quad z_j\in \mathbb{R}^N, \quad j=1,\ldots,p.
\end{equation*}
The cross entropy loss function of $\eta_{L,\mathcal{W},\mathcal{B}}$ then is defined by
\begin{equation*}
\mathcal{E}(\mathcal{W},\mathcal{B}) = -\frac{1}{p}\sum_{k=1}^p \log\left(\frac{\exp(x_{c_k,k})}{\sum_{j=1}^M\exp(x_{j,k})}\right).
\end{equation*}
\subsection{Approximation of a smooth function}\label{subsecsin}
As our first example, we use neural networks to approximate
\begin{equation*}
f(x_1,\cdots,x_n) = \sin(\pi x_1+ \cdots+\pi x_n),
\quad x_k \in [-2,2], \quad k=1,\ldots,n.
\end{equation*}
The training datasets consist of 20000 input-output data pairs where the input data are randomly sampled from the hypercube $[-2,2]^n$ based on uniform distribution.
The neural networks have single or double hidden layers. Each layer (except input and output layers) has $20$ neurons. For TMAF $D_\ell$ in \eqref{Dell}, the function $\alpha=\alpha_{\ell,i}$ uses intervals $(-\infty,-1.4)$, $(-1.4,-0.92]$, $(-0.92,-0.56]$, $(-0.56,-0.26]$, $(-0.26,0]$, $(0,0.26]$, $(0.26,0.56]$, $(0.56,0.92]$, $(0.92,1.4]$,
$(1.4,\infty)$ such that probability over each of the ten intervals is 0.1 with respect to Gaussian distribution. Moreover, we apply \texttt{BatchNorm1d} in PyTorch to the linear output of neural networks in each hidden layer.
The approximation results are shown in
Table~\ref{tab:2} and Figures~\ref{fig:u1}--\ref{fig:u56}, where Para-ReLU stands for the parametric ReLU neural network. It is observed that TMAF is the most accurate activation approach in these examples.
\begin{table}\centering
\begin{tabular}{|l|l|l|l|l|l|}
\hline
& \multicolumn{4}{|c|}{Approximation error} \\ \cline{2-5}
& \multicolumn{2}{|c|}{Single hiden layer} & \multicolumn{2}{|c|}{Two hiden layers}\\ \hline
$n$ & 1 & 2 & 5 & 6 \\ \hline
ReLU & 0.09 & 0.34
& 0.14 & 0.48 \\
\hline
Para ReLU & 0.04 & 0.11
& 0.09 & 0.47 \\
\hline
TMAF & 0.01 & 0.05
& 0.02 & 0.13 \\
\hline\hline
\end{tabular}
\caption{Approximation errors for $\sin(\pi x_1+\cdots+\pi x_n)$ by neural networks\label{tab:2}}
\end{table}
\begin{figure}
\caption{Training errors for $\sin(\pi x_{1}
\label{fig:u1-1}
\label{fig:u1-2}
\label{fig:u1}
\end{figure}
\subsection{Approximation of an oscillatory function}
The next example is on approximating the following function having high, medium and low frequency components
\begin{equation}\label{func}
f(x) = \sin (100 \pi x) + \cos(50 \pi x) + \sin (\pi x),
\end{equation}
see Figure~\ref{exact100fig} for an illustration. In fact, the function in \eqref{func} is rather difficult to capture by traditional approximation methods as it is highly oscillatory. We consider ReLU, parametric ReLU, and diagonal TMAF neural networks with 3 hidden layers and 400 neurons per layer (except the first and last layers). We also consider the tri-diagonal TMAF (denoted by Tri-diag TMAF, see \eqref{Tell}) with 3 hidden layers and 300 neurons per layer. The training datasets are 20000 input-output data pairs where the input data are randomly sampled from the interval $[-1,1]$ based on uniform distribution.
The diagonal TMAF uses $\alpha=\alpha_{\ell,i}$ in \eqref{alphali} with intervals $(-\infty,-1]$, $(-1+kh,-1+(k+1)h]$, $(1,\infty)$ for $h=0.02,$ $0\leq k\leq99$. The tri-diagonal TMAF is given in \eqref{Tell}, where $\{\alpha_{\ell,i}\}$ is the same as the diagonal TMAF, while $\{\beta_{\ell,i}\}$ are piecewise constants with respect to intervals $(-\infty,-2.01+\underline{h}]$, $\big\{(-2.01+kh,-2.01+(k+1)h]\big\}_{k=0}^{99}$, $(-0.01,\infty)$, and $\{\gamma_{\ell,i}\}$ are piecewise constants based on $(-\infty,0.01]$, $\big\{(0.01+kh,0.01+(k+1)h]\big\}_{k=0}^{99}$, $(2.01,\infty)$, respectively. Numerical results could be found in Figures~\ref{comparison100fig}, \ref{TMAF100fig}, \ref{ReLUpReLU100fig} and Table \ref{comparison100tab}.
\begin{figure}
\caption{Training errors for $\sin(\pi x_{1}
\label{fig:u1-5}
\label{fig:u1-6}
\label{fig:u56}
\end{figure}
For this challenging problem, we note that the diagonal TMAF and tri-diagonal TMAF produce high-quality approximations (see Figures \ref{TMAF100fig} and \ref{comparison100fig}) while ReLU and parametric ReLU are not able to approximate the highly oscillating function within reasonable accuracy, see Figure \ref{comparison100fig} and Table \ref{comparison100tab}. Moreover, it is observed from Figure \ref{ReLUpReLU100fig} that ReLU and parametric ReLU actually approximate the low frequency part of the target function. To capture the high frequency, ReLU-type neural networks are clearly required to use much more neurons, introducing significantly amount of weight and bias parameters. On the other hand, increasing the number of intervals in TMAF only lead to a few more training parameters.
\begin{figure}
\caption{Plot of $\sin(100 \pi x) + \cos(50 \pi x) + \sin(\pi x)$ and training loss comparison}
\label{exact100fig}
\label{comparison100fig}
\end{figure}
\begin{figure}
\caption{Approximations to $\sin(100 \pi x) + \cos (50 \pi x) + \sin(\pi x)$, TMAF-type}
\label{TMAF100fig}
\end{figure}
\begin{figure}
\caption{Approximations to $\sin(100 \pi x) + \cos (50 \pi x) + \sin(\pi x)$, ReLU-type}
\label{ReLUpReLU100fig}
\end{figure}
\begin{table}\centering
\begin{tabular}{|l|l|l|}
\hline
& final loss\\ \hline
ReLU & 1.00 \\ \hline
Para ReLU & 1.00 \\ \hline
Diag TMAF & 6.45e-2 \\ \hline
Tri-diag TMAF & 5.81e-2\\ \hline\hline
\end{tabular}
\caption{Error comparison for $\sin(100 \pi x) + \cos (50 \pi x) + \sin(\pi x)$}
\label{comparison100tab}
\end{table}
\subsection{Classification of \texttt{MNIST} and \texttt{CIFAR} datasets}
We now test TMAF by classifying images in the \texttt{MNIST}, \texttt{CIFAR-10} and \texttt{CIFAR-100} dataset.
For the \texttt{MNIST} set, we implement double layer fully connected networks defined as in \eqref{DNN} and \eqref{newDNN} with $10$ neurons per layer (except at the first layer $n_0=764$), and we use ReLU or diagonal TMAF as described in Subsection \ref{subsecsin}. Numerical results are shown in Figures~\ref{u15}, \ref{u16} and Table \ref{evaluation-cifar}. We observe that performance of the TMAF and the ReLU networks are similar. Such a behavior clearly should be expected for a simple dataset such as \texttt{MNIST}.
For the \texttt{CIFAR-10} dataset, we use the \verb|ResNet18| network structure provided by~\cite{DBLP:journals/corr/HeZRS15}. The activation functions are still ReLU and the diagonal TMAF used in Subsection \ref{subsecsin}. Numerical results are presented in
Figures~\ref{u16-1}, ~\ref{u17} and Table~\ref{evaluation-cifar}. Those parameters given in \cite{pytorch} are already tuned well with respect to ReLU. Nevertheless, TMAF still produces smaller errors in the training process and returns better classification results in the evaluation stage, see Table \ref{evaluation-cifar}.
For the \texttt{CIFAR-100} dataset, we use the \verb|ResNet34| network structure provided by ~\cite{DBLP:journals/corr/HeZRS15} with the ReLU and TMAF activation functions in Subsection \ref{subsecsin}. Numerical results are presented in
Figures~\ref{u19} and \ref{u20}. In the training process, TMAF again outperforms the classical ReLU network.
It is possible to improve the performance of TMAF applied to those benchmark datasets. The key point is to select suitable intervals in $\alpha_{\ell,i}$ to optimize the performance. A simple strategy is to let those intervals in \eqref{alphali} be varying and adjusted in the training process, which will be investigated in our future research.
\begin{figure}
\caption{\texttt{MNIST}
\label{u15}
\label{u16}
\end{figure}
\begin{table}\centering
\begin{tabular}{|l|l|l|}
\hline
\multirow{2}{*}{Dataset} & \multicolumn{2}{|c|}{Evaluation Accuracy} \\ \cline{2-3}
& ReLU & TMAF\\ \hline
\texttt{MNIST} (2 hidden layers) & 91.8\% & 92.2\% \\ \hline
\texttt{CIFAR-10} (Resnet18) & 77.5\% & 80.2\% \\
\hline \hline
\end{tabular}
\caption{Evaluation accuracy for \texttt{CIFAR-10}\label{evaluation-cifar}}
\end{table}
\begin{figure}
\caption{Comparison between ReLU and TMAF for \texttt{CIFAR-10}
\label{u16-1}
\label{u17}
\end{figure}
\begin{figure}
\caption{Comparison between ReLU and TMAF for \texttt{CIFAR-100}
\label{u19}
\label{u20}
\end{figure}
\end{document} | math | 21,304 |
\begin{document}
\title{\Large\textbf{Generating set for general multipartite entangled states} \thispagestyle{empty}
\begin{abstract}
We propose a entanglement generating set for a general multipartite
state based on the of concurrence. In particular, we show that
concurrence for general multipartite states can be constructed by
different classes of local operators which are defined by complement
of positive operator valued measure on quantum phases. The
entanglement generating set consists of different classes of
entanglement that are detected by these classes of operators and
contributes to the degree of entanglement for a general multipartite
state.
\end{abstract}
\section{Introduction}
Quantification and classification of multipartite quantum entangled
states is an ongoing research activity in the emerging field of
quantum information and quantum computation
\cite{Lewen00,Hor00,Bennett96a,Dur00,Eisert01,Pan,Verst,Miyake}.
There are several well-known entanglement measures for bipartite
states and among these entanglement measures the concurrence
\cite{Wootters98} is widely known. Recently, there have been some
proposals to generalize this measure to general bipartite and
multipartite states \cite{Uhlmann00,Gerjuoy,Albeverio,Mintert}. We
have also defined concurrence classes for multi-qubit mixed states
and for general pure multipartite states based on an orthogonal
complement of a positive operator valued measure ( POVM) on quantum
phase \cite{Hosh5, Hosh6}.
In this paper, we propose a minimal
entanglement generating set (MEGS) for a general multipartite state
based on our concurrence and it's generating operators
\cite{Hosh7}. In particular, in section \ref{povm} we give short
introduction our POVM and the construction of the operators that
generate the concurrence for general multipartite states. In section
\ref{megs} we will define a minimal entanglement generating set for
multipartite states and we will also give some example our
construction for bi-, three-, and four-partite states.
We will denote a general, composite quantum system with $m$
subsystems as $\mathcal{Q}=
\mathcal{Q}_{1}\mathcal{Q}_{2}\cdots\mathcal{Q}_{m}$ with th pure
state
\begin{equation}
\ket{\Psi}=\sum^{N_{1}}_{i_{1}=1}\cdots\sum^{N_{m}}_{i_{m}=1}
\alpha_{i_{1},i_{2},\ldots,i_{m}} \ket{i_{1},i_{2},\ldots,i_{m}}
\end{equation}
and corresponding Hilbert space $
\mathcal{H}_{\mathcal{Q}}=\mathcal{H}_{\mathcal{Q}_{1}}\otimes
\mathcal{H}_{\mathcal{Q}_{2}}\otimes\cdots\otimes\mathcal{H}_{\mathcal{Q}_{m}},
$ where the dimension of the $j$th Hilbert space is given by
$N_{j}=\dim(\mathcal{H}_{\mathcal{Q}_{j}})$. In order to simplify
our presentation, we will use $\Lambda_{m}=k_{1},l_{1};$
$\ldots;k_{m},l_{m}$ as an abstract
multi-index notation.
\section{Construction of concurrence for general multipartite states}\label{povm}
In this section we give a short review of our POVM and the
construction of concurrence for general multipartite states based on
orthogonal complement of these operators. A general and symmetric
POVM in a single $N_{j}$-dimensional Hilbert space
$\mathcal{H}_{\mathcal{Q}_{j}}$ is given by
\begin{eqnarray}
&&\Delta(\varphi_{1_j,2_j},\ldots,\varphi_{1_j,N_j},
\varphi_{2_j,3_j},\ldots,\varphi_{N_{j}-1,N_{j}})=
\sum^{N_{j}}_{l_{j},k_{j}=1}
e^{i\varphi_{k_{j},l_{j}}}\ket{k_{j}}\bra{l_{j}}\\\nonumber&&=
\left(
\begin{array}{ccccc}
1 &e^{i\varphi_{1,2}} & \cdots
& e^{i\varphi_{1,N_{j}-1}} &e^{i\varphi_{1,N_{j}}}\\
e^{-i\varphi_{1,2}} & 1 & \cdots
& e^{i\varphi_{2,N_{j}-1}} &e^{i\varphi_{2,N_{j}}}\\
\vdots& \vdots&\ddots &\vdots& \vdots\\
e^{-i\varphi_{1,N_{j}-1}} & e^{-i\varphi_{2,N_{j}-1}} &\cdots&1&e^{i\varphi_{N_{j}-1,N_{j}}}\\
e^{-i\varphi_{1,N_{j}}} & e^{-i\varphi_{2,N_{j}}} &\cdots&e^{-i\varphi_{N_{j}-1,N_{j}}}&1\\
\end{array}
\right),
\end{eqnarray}
where $\ket{k_{j}}$ and $\ket{l_{j}}$ are the basis vectors in
$\mathcal{H}_{\mathcal{Q}_j}$ and the quantum phases satisfy the
following relation $ \varphi_{k_{j},l_{j}}=
-\varphi_{l_{j},k_{j}}(1-\delta_{k_{j} l_{j}})$. The POVM is a
function of the $N_{j}(N_{j}-1)/2$ phases
$(\varphi_{1_j,2_j},\ldots,\varphi_{1_j,N_j},\varphi_{2_j,3_j},\ldots,\varphi_{N_{j}-1,N_{j}})$.
Moreover, our POVM is self-adjoint, positive, and normalized. In
following text we will use the short notation
$\widetilde{\Delta}(\varphi_{k_{j},l_{j}})$ for our POVM. It is now
possible to form a POVM of a multipartite system by simply forming
the tensor product
\begin{eqnarray}\label{POVM}
\Delta_\mathcal{Q}(\varphi_{k_{1},l_{1}},\ldots,
\varphi_{k_{m},l_{m}})&=&
\Delta_{\mathcal{Q}_{1}}(\varphi_{k_{1},l_{1}}) \otimes\cdots
\otimes\Delta_{\mathcal{Q}_{m}}(\varphi_{k_{m},l_{m}}),
\end{eqnarray}
where, e.g., $\varphi_{k_{1},l_{1}}$ is the set of POVMs phase
associated with subsystems $\mathcal{Q}_{1}$, for all
$k_{1},l_{1}=1,2,\ldots,N_{1}$.
In the $m$-partite case, the off-diagonal elements of the matrix
corresponding to
\begin{equation}\widetilde{\Delta}_\mathcal{Q}(\varphi_{k_{1},l_{1}},\ldots,
\varphi_{k_{m},l_{m}})=
\widetilde{\Delta}_{\mathcal{Q}_{1}}(\varphi_{k_{1},l_{1}})
\otimes\cdots
\otimes\widetilde{\Delta}_{\mathcal{Q}_{m}}(\varphi_{k_{m},l_{m}}),
\end{equation}
where the orthogonal complement of our POVM is
defined by
$\widetilde{\Delta}(\varphi_{k_{j},l_{j}})=\mathcal{I}_{N_{j}}-
\Delta_{\mathcal{Q}_{j}}(\varphi_{k_{j},l_{j}})$.
$\mathcal{I}_{N_{j}}$ is the $N_{j}$-by-$N_{j}$ identity matrix for
subsystem $j$.
$\widetilde{\Delta}_\mathcal{Q}(\varphi_{k_{1},l_{1}},\ldots,
\varphi_{k_{m},l_{m}})$ has phases that are sums or differences of
phases originating from $2,3,\ldots,m$ subsystems. That is, in the
latter case the phases of
$\widetilde{\Delta}_\mathcal{Q}(\varphi_{k_{1},l_{1}},\ldots,
\varphi_{\mathcal{Q}_{m};k_{m},l_{m}})$ take the form
$(\varphi_{k_{1},l_{1}}\pm\varphi_{k_{2},l_{2}}
\pm\ldots\pm\varphi_{k_{m},l_{m}})$ and identification of these
joint phases makes our distinguishing possible. Thus, we can define
linear operators for the $\mathrm{EPR}^{m}$ class which are sums and
differences of phases of two subsystems, i.e.,
$(\varphi_{k_{r_{1}},l_{r_{1}}} \pm\varphi_{k_{r_{2}},l_{r_{2}}})$.
That is, for the $\mathrm{EPR}^{m}$ class we have
\begin{equation}
\widetilde{\Delta}^{
\mathrm{EPR}^{m}_{\Lambda_{m}}}_{\mathcal{Q}_{r_{1},r_{2}}(N_{r_{1}},N_{r_{2}})}
=\mathcal{I}_{N_{1}} \otimes\cdots
\otimes\widetilde{\Delta}_{\mathcal{Q}_{r_{1}}}
(\varphi^{\frac{\pi}{2}}_{k_{r_{1}},l_{r_{1}}}) \otimes\cdots\otimes
\widetilde{\Delta}_{\mathcal{Q}_{r_{2}}}
(\varphi^{\frac{\pi}{2}}_{k_{r_{2}},l_{r_{2}}})\otimes\cdots\otimes\mathcal{I}_{N_{m}},
\end{equation}
where $\varphi^{\frac{\pi}{2}}_{k_{j},l_{j}}=\frac{\pi}{2}$ for all
$k_{j}<l_{j}, ~j=1,2,\ldots,m$.
Next, we rewrite the linear operator
$\widetilde{\Delta}^{\mathrm{EPR}^{m}_{\Lambda_{m}}}_{\mathcal{Q}_{r_{1},r_{2}}(N_{r_{1}},N_{r_{2}})}$
as a direct sum of the upper and lower anti-diagonal
\begin{equation}
\widetilde{\Delta}^{
\mathrm{EPR}^{m}_{\Lambda_{m}}}_{\mathcal{Q}_{r_{1},r_{2}}(N_{r_{1}},N_{r_{2}})}
=U\widetilde{\Delta}^{
\mathrm{EPR}^{m}_{\Lambda_{m}}}_{\mathcal{Q}_{r_{1},r_{2}}(N_{r_{1}},N_{r_{2}})}+L\widetilde{\Delta}^{
\mathrm{EPR}^{m}_{\Lambda_{m}}}_{\mathcal{Q}_{r_{1},r_{2}}(N_{r_{1}},N_{r_{2}})}.
\end{equation}
The set of linear operators for the $\mathrm{EPR}^{m}$ classes
gives
the $\mathrm{W}^{m}$ class concurrence.
The next class we will consider is what we call the
$\mathrm{GHZ}^{m}$ class which given by
\begin{eqnarray}\nonumber
\widetilde{\Delta}^{
\mathrm{GHZ}^{m}_{\Lambda_{m}}}_{\mathcal{Q}_{r_{1},r_{2}}(N_{r_{1}},N_{r_{2}})}
&=&\widetilde{\Delta}_{\mathcal{Q}_{1}}
(\varphi^{\pi}_{k_{1},l_{1}})\otimes\cdots
\otimes\widetilde{\Delta}_{\mathcal{Q}_{r_{1}}}
(\varphi^{\frac{\pi}{2}}_{k_{r_{1}},l_{r_{1}}}) \otimes\cdots\otimes
\widetilde{\Delta}_{\mathcal{Q}_{r_{2}}}
(\varphi^{\frac{\pi}{2}}_{k_{r_{2}},l_{r_{2}}})\\&&\otimes\cdots\otimes
\widetilde{\Delta}_{\mathcal{Q}_{m}} (\varphi^{\pi}_{k_{m},l_{m}}),
\end{eqnarray}
where by choosing $\varphi^{\pi}_{k_{j},l_{j}}=\pi$ for all
$k_{j}<l_{j}, ~j=1,2,\ldots,m$, we get an operator which has the
structure of the Pauli operator $\sigma_{x}$ embedded in a
higher-dimensional Hilbert space and coincides with $\sigma_{x}$ for
a single-qubit.
Next, we write the linear operators for the $\mathrm{GHZ}^{m}$ class
as
\begin{equation}
\widetilde{\Delta}^{
\mathrm{GHZ}^{m}_{\Lambda_{m}}}_{\mathcal{Q}_{r_{1},r_{2}}(N_{r_{1}},N_{r_{2}})}
=P_{1}\widetilde{\Delta}^{
\mathrm{GHZ}^{m}_{\Lambda_{m}}}_{\mathcal{Q}_{r_{1},r_{2}}(N_{r_{1}},N_{r_{2}})}+P_{2}\widetilde{\Delta}^{
\mathrm{GHZ}^{m}_{\Lambda_{m}}}_{\mathcal{Q}_{r_{1},r_{2}}(N_{r_{1}},N_{r_{2}})}+\ldots,
\end{equation}
where the
operators $P_{i}\widetilde{\Delta}^{
\mathrm{GHZ}^{m}_{\Lambda_{m}}}_{\mathcal{Q}_{r_{1},r_{2}}(N_{r_{1}},N_{r_{2}})}$
are constructed by pairing of elements of the POVM with sums and
differences of quantum phases.
We can also define the $\mathrm{GHZ}^{m-1}$ class operator
\begin{eqnarray} \widetilde{\Delta}^{
\mathrm{GHZ}^{m-1}_{\Lambda_{m}}}_{\mathcal{Q}_{r_{1}r_{2},r_{3}}(N_{r_{1}},N_{r_{2}})}
&=&\widetilde{\Delta}_{\mathcal{Q}_{r_{1}}}
(\varphi^{\frac{\pi}{2}}_{k_{r_{1}},l_{r_{1}}})
\otimes\widetilde{\Delta}_{\mathcal{Q}_{r_{2}}}
(\varphi^{\frac{\pi}{2}}_{k_{r_{2}},l_{r_{2}}})
\otimes\widetilde{\Delta}_{\mathcal{Q}_{r_{3}}}
(\varphi^{\pi}_{k_{r_{3}},l_{r_{3}}}) \\\nonumber&&\otimes\cdots
\otimes\widetilde{\Delta}_{\mathcal{Q}_{m-1}}
(\varphi^{\pi}_{k_{r_{m-1}},l_{r_{m-1}}})\otimes\mathcal{I}_{N_{m}}
,
\end{eqnarray}
where $1\leq r_{1}<r_{2}<\cdots<r_{m-1}<m$. In the same way we can
construct all other operators in these classes. Now, by taking the
expectation value of each of these classes of operators, we are able
to construct the concurrence for general multipartite states
\cite{Hosh7}. In the next section we will propose a minimal
entanglement generating set for general multipartite states based
on our construction of the concurrence.
\section{Minimal entanglement generating set for multipartite
states}\label{megs} As we have shown there are different classes of
operators that generates concurrence for general multipartite
states. So, the construction of a measures of entanglement for
general multipartite states suggests that there exists different
classes of entanglement that contributes to the degree entanglement.
These different classes of entanglement are detected by
$\mathrm{EPR}^{m}$ and $\mathrm{GHZ}^{m}$ classes of operators.
Thus, we can propose a $\mathrm{MEGS}$ for general multipartite
states based on this construction of the concurrence as follows
\begin{eqnarray}\label{MEGS}\nonumber
\mathcal{E}^{m}_{MEGS}&=&\{\mathrm{EPR}_{\mathcal{Q}_{1}\mathcal{Q}_{2}}
,\ldots,\mathrm{EPR}_{\mathcal{Q}_{1}\mathcal{Q}_{m}},\ldots,
\mathrm{EPR}_{\mathcal{Q}_{m-2}\mathcal{Q}_{m-1}},
\ldots,\mathrm{EPR}_{\mathcal{Q}_{m-1}\mathcal{Q}_{m}},\\\nonumber&&
\mathrm{GHZ}^{3}_{\mathcal{Q}_{1}\mathcal{Q}_{2}\mathcal{Q}_{3}},\ldots
\mathrm{GHZ}^{3}_{\mathcal{Q}_{m-2}\mathcal{Q}_{m-1}\mathcal{Q}_{m}},\ldots,
,
\mathrm{GHZ}^{m-1}_{\mathcal{Q}_{1}\mathcal{Q}_{2}\cdots\mathcal{Q}_{m-1}},\ldots,\\&&
\mathrm{GHZ}^{m-1}_{\mathcal{Q}_{2}\mathcal{Q}_{3}\cdots\mathcal{Q}_{m}},
\mathrm{GHZ}^{m}_{\mathcal{Q}_{1}\mathcal{Q}_{2}\cdots\mathcal{Q}_{m}}\},
\end{eqnarray}
where for $m$-partite states we have $C(m,2)=\frac{m(m-1)}{2}$
$\mathrm{EPR}_{\mathcal{Q}_{r_{1}}\mathcal{Q}_{r_{2}}}$ classes and
$C(m,k)$
~$\mathrm{GHZ}^{k}_{\mathcal{Q}_{r_{1}}\mathcal{Q}_{r_{2}}\cdots\mathcal{Q}_{r_{k}}}$
classes, for $2<k\leq m$.
For example the minimal entanglement generating set for bipartite
states has only one element, that is
$\mathcal{E}^{2}_{MEGS}=\{\mathrm{EPR}_{\mathcal{Q}_{1}\mathcal{Q}_{2}}
\} $. This is exactly what we have expect to see. For three-partite states
the MEGS has $C(3,2)=3$ EPR elements and $C(3,3)=1$ $\mathrm{GHZ}^{3}$ element
\begin{equation}
\mathcal{E}^{3}_{MEGS}=\{\mathrm{EPR}_{\mathcal{Q}_{1}\mathcal{Q}_{2}}
,\mathrm{EPR}_{\mathcal{Q}_{1}\mathcal{Q}_{3}},\mathrm{EPR}_{\mathcal{Q}_{2}\mathcal{Q}_{3}},
\mathrm{EPR}_{\mathcal{Q}_{1}\mathcal{Q}_{2}\mathcal{Q}_{3}} \}.
\end{equation}
Note also that the combinations of EPR elements gives the W class
entanglement. Our last example is the MEGS for four-partite states
which is given by
\begin{eqnarray}
\mathcal{E}^{4}_{MEGS}&=&\{\mathrm{EPR}_{\mathcal{Q}_{1}\mathcal{Q}_{2}}
,\mathrm{EPR}_{\mathcal{Q}_{1}\mathcal{Q}_{3}},\mathrm{EPR}_{\mathcal{Q}_{1}\mathcal{Q}_{4}},
\mathrm{EPR}_{\mathcal{Q}_{2}\mathcal{Q}_{3}},
\mathrm{EPR}_{\mathcal{Q}_{2}\mathcal{Q}_{4}}
,\\\nonumber&&\mathrm{EPR}_{\mathcal{Q}_{3}\mathcal{Q}_{4}},
\mathrm{GHZ}^{3}_{\mathcal{Q}_{1}\mathcal{Q}_{2}\mathcal{Q}_{3}},
\mathrm{GHZ}^{3}_{\mathcal{Q}_{1}\mathcal{Q}_{2}\mathcal{Q}_{4}},
\mathrm{GHZ}^{3}_{\mathcal{Q}_{1}\mathcal{Q}_{3}\mathcal{Q}_{4}},
\mathrm{GHZ}^{3}_{\mathcal{Q}_{2}\mathcal{Q}_{3}\mathcal{Q}_{4}},\\\nonumber&&
\mathrm{GHZ}^{4}_{\mathcal{Q}_{1}\mathcal{Q}_{2}\mathcal{Q}_{3}\mathcal{Q}_{4}}\}.
\end{eqnarray}
This set has $C(4,2)=6$ EPR elements, $C(4,3)=4$ $\mathrm{GHZ}^{3}$
elements, and $C(4,4)=1$ $\mathrm{GHZ}^{4}$ element. Note that, for
each element of EPR class there is only one operator that correspond
to a given element of the MEGS. But for each element of GHZ classes
there are a set of operators that correspond to a given element of
the MEGS. The elements of MEGS are inequivalent under local quantum
operations and classical communication by construction an in the
case of three-partite states and in general between
$\mathrm{EPR}^{m}$ and $\mathrm{GHZ}^{m}$ classes.
The MEGS set is not equivalent to minimal reversible
entanglement generating set, but there is some similarity between
these two sets.
Thus the MEGS set
gives
information
about the nature of multipartite entangled states in a time where
there is no well known and accepted classification of multipartite
states available.
\begin{flushleft}
\textbf{Acknowledgments:} The author acknowledges the financial
support of the Japan Society for the Promotion of Science (JSPS).
\end{flushleft}
\end{document} | math | 14,115 |
\begin{document}
\begin{frontmatter}
\title{Existence and uniqueness of global solutions of fractional functional differential equations with bounded delay}
\author[Firstaddress]{Chung-Sik Sin\corref{mycorrespondingauthor}}
\cortext[mycorrespondingauthor]{Corresponding author}
\ead{[email protected]}
\address[Firstaddress]{Faculty of Mathematics, \textit{\textbf {Kim Il Sung}} University, Kumsong Street, Taesong District, Pyongyang, D.P.R.KOREA}
\begin{abstract}
This paper deals with initial value problems for fractional functional differential equations with bounded delay.
The fractional derivative is defined in the Caputo sense.
By using the Schauder fixed point theorem and the properties of the Mittag-Leffler function, new existence and uniqueness results for global solutions of the initial value problems are established.
In particular, the unique existence of global solution is proved under the condition close to the Nagumo-type condition.
\end{abstract}
\begin{keyword}
Caputo derivative, fractional functional differential equation with delay, existence and uniqueness of solutions, initial value problem
\end{keyword}
\end{frontmatter}
\section{Introduction}\label{Sec:1}
The present paper considers initial value problems for fractional functional differential equations of the form
\begin{equation}
\label{governing_equation}
^CD^\alpha u(t) =f(t,u_t),
\end{equation}
subject to initial conditions
\begin{equation}
\label{initial_condition}
u^{(i)}(t)=\phi^{(i)}(t) \text{ on } [-h,0], i=0,1,...,\lceil \alpha \rceil-1,
\end{equation}
where $ \alpha,h>0 $, $ \phi \in C^{\lceil \alpha \rceil-1}[-h,0] $, the symbol $ ^cD^\alpha $ denotes the Caputo fractional differential operator and $ u_t $ is defined by $ u_t(\theta)=u(t+\theta) \text{ for } \theta \in [-h,0] $.
Here $\lceil \alpha \rceil$ is the smallest integer $ \geq \alpha $.
Fractional calculus has been widely used to describe the complex nonlinear phenomena in continuum mechanics, thermodynamics, quantum mechanics, plasma dynamics, electrodynamics and so on \cite{Uchaikin}. The constitutive relation of the viscoelastic media is successfully modeled by fractional differential equations \cite{Mainardi_book}. Wang et al. \cite{WangHuang} proposed a fractional order financial system with time delay and considered the dynamical behaviors of such a system.
Existence and uniqueness of solutions of initial value problems for fractional ordinary differential equations have been investigated by many authors \cite{DieFor,DieBoo,DieNag,KilSri,Lin,LakLee,Sin}.
Diethelm and Ford \cite{DieFor} used the Schauder fixed point theorem to prove that the initial value problem for the fractional differential equation with Caputo derivative has a local solution under appropriate assumptions.
Lakshmikantham \cite{LakLee} first extended the Nagumo-type existence and uniqueness result for fractional differential equations involving Riemann-Liouville fractional derivative.
In \cite{DieNag} the classical Nagumo-type theorem is generalized to Caputo-type fractional differential equations.
Lin \cite{Lin} obtained existence results for solutions of the initial value problems under more general assumptions.
Recently, Sin et al. \cite{Sin} improved sufficient conditions for existence and uniqueness of global solutions of the initial value problem by proving a new property of the two parameter Mittag-Leffler function.
With the development of mathematical theory on fractional ordinary differential equations, fractional functional differential equations with delay have also been intensively studied \cite{Benchohra,Lakshmikantham,ZhouJiao,AgarwalZhou,Zhou,YangCao,WangXiao}.
Lakshmikantham \cite{Lakshmikantham} established existence results for solutions of initial value problems for fractional functional differential equations with bounded delay.
Agarwal et al. \cite{AgarwalZhou} used Krasnoselskii¡Çsfixed point theorem to prove sufficient conditions for existence of solutions of fractional neutral functional differential equations with bounded delay.
Zhou et al. \cite{ZhouJiao} obtained various criteria on existence and uniqueness of solutions of fractional neutral functional differential equations with infinite delay.
Yang et al. \cite{YangCao} considered existence and uniqueness of local and global solutions of the initial value problem (\ref{governing_equation})-(\ref{initial_condition}).
In order to obtain the existence result for global solutions, they supposed the following condition:
there exist constants $ c_1, c_2 \geq 0 $ and $ 0<\lambda<1 $ such that
\begin{equation}
\label{previous_condition}
|f(x,u_t)|\leq c_1+c_2 \|u_t\|_{C[-h,0]}^{\lambda}.
\end{equation}
As pointed in \cite{YangCao}, the condition is violated even by some very elementary equations like linear equation.
They proved the unique existence of solution when $ f $ is continuous and satisfies the Lipschitz condition with respect to the second variable.
In this paper, by using Schauder fixed point theorem and properties of two parameter Mittag-Leffler function, existence and uniqueness results for global solutions of the equation (\ref{governing_equation})-(\ref{initial_condition}) are improved. Firstly, the condition $ 0<\lambda<1 $ for global solutions is replaced with the more general one $ 0<\lambda \leq 1 $. Then the unique existence of global solutions is established when $ f $ satisfies the condition close to the Nagumo-type condition weaker than the Lipschitz condition.
The rest of the paper is organized as follows.
Section \ref{Sec:2} introduces the preliminary results which are used in deriving the main results of this paper.
In Section \ref{Sec:3}, the existence of solutions of the initial value problem (\ref{governing_equation})-(\ref{initial_condition}) is discussed.
Section \ref{Sec:4} deals with the uniqueness of global solutions of the equation.
\section{Preliminaries}\label{Sec:2}
In this section we give definitions and preliminary results which are needed in our investigation.
Firstly, we recall the concepts of Riemann-Liouville fractional integral operator and Caputo fractional differential operator.
\begin{definition}[\cite{DieBoo}]
Let $ \beta,l \geq 0 $ and $ u \in L^{1}[0,l] $.
The Riemann-Liouville integral of order $ \beta $ of $ u $ is defined by
\begin{equation}
\nonumber
I^\beta u(t)={\frac{1}{\Gamma (\beta) } \int^{t}_{0}{(t-s)}^{\beta-1} u(s) ds}.
\end{equation}
\end{definition}
\begin{definition}[\cite{DieBoo}]
Let $ \beta,l\geq 0 $ and $ D^{\lceil \beta \rceil}u \in L^{1}[0,l] $. The Caputo fractional derivative of order $ \beta $ of $ u $ is defined by
\begin{equation}
\nonumber
^CD^\beta u(t)=I^{\lceil \beta \rceil-\beta}D^{\lceil \beta \rceil}u(t).
\end{equation}
\end{definition}
Then the Caputo fractional derivative of order $ \beta $ is defined for $ u $ belonging to the space $ AC^{\lceil \beta \rceil}[0,l] $ of absolutely continuous functions of order $\lceil \beta \rceil$.
\begin{definition}
Let $ T>0 $. A function $ u:[-h,T] \rightarrow R $ is called a solution of the initial value problem (\ref{governing_equation})-(\ref{initial_condition}) on $ [0,T]$ if
$ u|_{[-h,0]}=\phi$, $ u|_{[0,T]} \in AC^{\lceil \alpha \rceil}[0,T] $ and $ u $ satisfies the equation (\ref{governing_equation}) for $ t \in [0,T] $.
\end{definition}
Here $ u|_{[0,T]} $ means the restriction of the function $ u $ to $[0,T]$.
For the next lemmas, we make the following assumption:\\
(H2-1) $ T>0 $ and the function $f :[0,T]\times C[-h,0] \rightarrow R $ is continuous.
\begin{lemma}
Suppose that (H2-1) holds and $ u:[-h,T] \rightarrow R $ satisfies the conditions such that $ u|_{[-h,0]}=\phi$ and $ u|_{[0,T]} \in AC^{\lceil \alpha \rceil}[0,T] $. Then the function $ u $ is a solution of the initial value problem (\ref{governing_equation})-(\ref{initial_condition}) on $ [0,T]$ if and only if $ u|_{[0,T]} $ is a solution of the integral equation
\begin{equation}\label{equivalent_integral_equation}
u(t)=\sum^{\lceil \alpha \rceil-1}_{i=0}\frac{t^i}{i!} \phi^{(i)}(0)+\frac{1}{\Gamma (\alpha) }\int_0^t (t-s)^{\alpha-1}f(s,u_s)ds .
\end{equation}
\end{lemma}
\begin{proof}
See Lemma 2.8 in \cite{YangCao}.
\end{proof}
We can easily see that if $f$ is continuous and $ u $ is the solution of the integral equation (\ref{equivalent_integral_equation}) on $ [0,T]$, then $ u|_{[0,T]} \in AC^{\lceil \alpha \rceil}[0,T] $.
Thus, in order to obtain the solution of the initial value problem (\ref{governing_equation})-(\ref{initial_condition}), we will find the solution of the integral equation (\ref{equivalent_integral_equation}) in $ Y $, where
$ Y $ means the Banach space consisting of all continuous functions belonging in $C[0,T]$ with the Chebyshev norm $ \|\cdot\|_{C[0,T]} $.
Then the integral equation (\ref{equivalent_integral_equation}) is transformed into the fixed point problem with the operator $ J:Y \rightarrow Y $ defined by
\begin{equation}\label{fixed point problem}
Ju(t)=\sum^{\lceil \alpha \rceil-1}_{i=0}\frac{t^i}{i!} \phi^{(i)}(0)+\frac{1}{\Gamma (\alpha) }\int_0^t (t-s)^{\alpha-1}f(s,u_s)ds.
\end{equation}
\begin{lemma}\label{operator_continuity}
Let $B \subset Y $ be a bounded set. If (H2-1) holds, then the operator $ J $ is continuous in $ B $.
\end{lemma}
\begin{proof}
See Theorem 3.1 in \cite{YangCao}.
\end{proof}
\begin{lemma}\label{equicontinuity}
Let $B \subset Y $ be a bounded set. If (H2-1) holds, then the set $ J(B) $ is equicontinuous.
\end{lemma}
\begin{proof}
See Theorem 3.1 in \cite{YangCao}.
\end{proof}
\begin{lemma}\label{fixed point}
Let $B \subset Y $ be a bounded and convex set. If (H2-1) holds and $J(B)\subset B$, then $ J $ has a fixed point in $B$.
\end{lemma}
\begin{proof}
By Lemma \ref{operator_continuity}, Lemma \ref{equicontinuity} and Arzela-Ascoli theorem, the operator $ J:B \rightarrow B $ is compact.
Then by Schauder fixed point theorem, $ J$ has at least one fixed point in $B$.
\end{proof}
The Mittag-Leffler functions play a crucial role in our investigation.
\begin{definition}[\cite{DieBoo}]
Let $ c,d>0 $. A two-parameter function of the Mittag-Leffler type is defined by the series expansion
\begin{equation}\label{eq2-1}
E_{c,d}(t)={\sum^\infty_{i=0}{\frac{t^i}{\Gamma (ci+d)}}}.
\end{equation}
\end{definition}
\begin{lemma}[\cite{Sin}]
\label{Mittag_function_property}
Let $ c, l, r, \beta>0$. If $d<min\{\beta,1\}$, then there exists a real number $ \lambda>0 $ such that
\begin{equation}\label{eq2-6}
\frac{1}{\Gamma (\beta)}\int^{t}_{0}{(t-s)}^{\beta-1}{\frac {E_{c,1-d}({\lambda s^c})}{s^d}ds }<rE_{c,1-d}({\lambda t^c}), t \in [0,l].
\end{equation}
\end{lemma}
\section{Existence of global solutions}\label{Sec:3}
In this section the existence of global solutions of the initial value problem for the fractional functional differential equation (\ref{governing_equation})-(\ref{initial_condition}) is investigated.
For the next theorem, we make the following assumption:\\
(H3-1) there exist $T_1 \in (0,T]$, $ \eta \in (T_1,T) $, $a_1,a_2,q_1,q_2 \in [0,\alpha)$, $ p_1, p_2 \in (0,1] $, $ b_1, b_2>0 $, $ m_1(t) \in L^{\frac{1}{q_1}} [0,T_1] $,
$ m_2(t) \in L^{\frac{1}{q_2}} [T_1,T] $ such that
\begin{equation}
\nonumber
|f(t,u)| \leq \left\{
\begin{aligned}
& \frac{b_1}{t^{a_1}}\|u\|_{C[-h,0]}^{p_1}+m_1(t) && \text{for $ t \in (0, T_1] $ and $ u \in C[-h,0] $} \\
& \frac{b_2}{|t-\eta|^{a_2}}\|u\|_{C[-h,0]}^{p_2}+ m_2(t) && \text{for $ t \in [T_1,\eta) \cup(\eta, T] $ and $ u \in C[-h,0] $}.\\
\end{aligned}
\right.
\end{equation}
\begin{theorem}
\label{existence_theorem}
Suppose that (H2-1) and (H3-1) hold. Then the initial value problem (\ref{governing_equation})-(\ref{initial_condition}) has a solution on $ [0,T]$.
\end{theorem}
\begin{proof}
By Lemma \ref{Mittag_function_property}, there exist $ \lambda_1,\lambda_2,\lambda_3>0 $ such that
for $ t \in [0,T] $,
\begin{align}
\nonumber
&\{I^\alpha[E_{1,1-a_1}(\lambda_1 s)s^{-a_1}]\}(t)<\frac{1}{2b_1}E_{1,1-a_1}(\lambda_1 t),\\
\nonumber
&\{I^{\alpha-a_2}[E_{1,1}(\lambda_2 s)]\}(t)<\frac{\Gamma(\alpha)}{2b_2\Gamma(\alpha-a_2)}E_{1,1}(\lambda_2 t),\\
\nonumber
&\{I^\alpha[E_{1,1-a_2}(\lambda_3 s)s^{-a_2}]\}(t)<\frac{1}{2b_2}E_{1,1-a_2}(\lambda_3 t).
\end{align}
We define a convex bounded closed subset $ G $ of $ Y $ as follows\\
\begin{equation}
\nonumber
G=\Biggl \{ x \in C[0,T] :|x(t)| \leq \left\{
\begin{aligned}
& 2D_1E_{1,1-a_1}(\lambda_1 t) && \text{for $ t \in [0, T_1] $} \\
& 2D_2E_{1,1}(\lambda_2 t) && \text{for $ t \in [T_1, \eta] $}\\
& 2D_3E_{1,1-a_2}(\lambda_3 (t-\eta)) && \text{for $ t \in [\eta, T] $} \\
\end{aligned}
\right.
\Biggr \},
\end{equation}
where
\begin{align}
\nonumber
&D_1= \Gamma (1-a_1) max \Bigg\{1,\|\phi\|_{C[-h,0]}, \sum^{\lceil \alpha \rceil-1}_{i=0}\frac{T^i}{i!} |\phi^{(i)}(0)|
+\frac{{T_1}^{\alpha-q_1} }{\Gamma(\alpha)}\bigg(\frac{1-q_1}{\alpha-q_1}\bigg)^{1-q_1}\\
\nonumber
&\|m_1\|_{L^{\frac{1}{q_1}}[0,T_1]}\Bigg\},\\
\nonumber
&D_2= 2D_1E_{1,1-a_1}(\lambda_1 {T_1})+\frac{1}{\Gamma(\alpha)}\bigg(\frac{1-q_2}{\alpha-q_2}\bigg)^{1-q_2}
{(\eta-T_1)}^{\alpha-q_2} \|m_2\|_{L^{\frac{1}{q_2}}[T_1,\eta]},\\
\nonumber
&D_3=\Gamma (1-a_2)\Bigg[2D_2E_{1,1}(\lambda_2 \eta) +\frac{1}{\Gamma(\alpha)}\bigg(\frac{1-q_2}{\alpha-q_2}\bigg)^{1-q_2}
(T-\eta)^{\alpha-q_2} \|m_2\|_{L^{\frac{1}{q_2}}[\eta,T]}\Bigg].
\end{align}
By Holder inequality, we have that for any $ x \in G $ and $ t \in [0,T_1] $,
\begin{align}
\nonumber
&|Jx(t)|\leq \sum^{\lceil \alpha \rceil-1}_{i=0}\frac{t^i}{i!} |\phi^{(i)}(0)|+\frac{1}{\Gamma(\alpha)}\int_0^t (t-s)^{\alpha-1}|f(s,x_s)|ds\\
\nonumber
&\leq\sum^{\lceil \alpha \rceil-1}_{i=0}\frac{t^i}{i!} |\phi^{(i)}(0)|+\frac{1}{\Gamma(\alpha)}\int_0^t (t-s)^{\alpha-1} \bigg(\frac{b_1}{s^{a_1}}\|x_s\|_{C[-h,0]}^{p_1}+m_1(s)\bigg)ds\\
\nonumber
&\leq \sum^{\lceil \alpha \rceil-1}_{i=0}\frac{t^i}{i!} |\phi^{(i)}(0)|+\frac{2b_1 D_1}{\Gamma(\alpha)}\int_0^t (t-s)^{\alpha-1} \frac{E_{1,1-a_1}(\lambda_1 s)}{s^{a_1}}ds\\
\nonumber
&+\frac{1}{\Gamma(\alpha)}\bigg(\int_0^t (t-s)^{\frac{\alpha-1}{1-q_1}}ds\bigg)^{1-q_1}\|m_1\|_{L^{\frac{1}{q_1}}[0,T_1]} \\
\nonumber
&\leq \frac{D_1}{\Gamma(1-a_1)}+D_1E_{1,1-a_1}(\lambda_1 t)< 2D_1E_{1,1-a_1}(\lambda_1 t).
\end{align}
We have that for any $ x \in G $ and $ t \in [T_1,\eta] $,
\begin{align}
\nonumber
&|Jx(t)|\leq \sum^{\lceil \alpha \rceil-1}_{i=0}\frac{t^i}{i!} |\phi^{(i)}(0)|+\int_0^{T_1} \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}|f(s,x_s)|ds +\int_{T_1}^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}|f(s,x_s)|ds
\end{align}
\begin{align}
\nonumber
&\leq 2D_1E_{1,1-a_1}(\lambda_1 {T_1})+\frac{1}{\Gamma(\alpha)}\int_{T_1}^t (t-s)^{\alpha-1}\bigg(\frac{b_2}{(\eta-s)^{a_2}}\|x_s\|^{p_2}+ m_2(s)\bigg)ds\\
\nonumber
&\leq 2D_1E_{1,1-a_1}(\lambda_1 {T_1})+\int_{T_1}^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}m_2(s)ds+\int_{T_1}^t \frac{b_2(t-s)^{\alpha-a_2-1}}{\Gamma(\alpha)}\|x_s\|_{C[-h,0]}^{p_2}ds\\
\nonumber
&\leq 2D_1E_{1,1-a_1}(\lambda_1 {T_1})+\frac{1}{\Gamma(\alpha)}\bigg(\int_{T_1}^t (t-s)^{\frac{\alpha-1}{1-q_2}}ds \bigg)^{1-q_2}\|m_2\|_{L^{\frac{1}{q_2}}[T_1,\eta]}\\
\nonumber
&+\frac{2b_2\Gamma(\alpha-a_2)}{\Gamma(\alpha)}D_2\{I^{\alpha-a_2}[E_{1,1}(\lambda_2 s)]\}(t)
\leq D_2+D_2E_{1,1}(\lambda_2 t)\leq 2D_2E_{1,1}(\lambda_2 t).
\end{align}
We have that for any $ x \in G $ and $ t \in [\eta,T] $,
\begin{align}
\nonumber
&|Jx(t)|\leq \sum^{\lceil \alpha \rceil-1}_{i=0}\frac{t^i}{i!} |\phi^{(i)}(0)|+\int_0^{\eta} \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}|f(s,x_s)|ds +\int_{\eta}^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}|f(s,x_s)|ds\\
\nonumber
&\leq 2D_2E_{1,1}(\lambda_2 \eta)+\frac{1}{\Gamma(\alpha)}\int_\eta^t (t-s)^{\alpha-1}\bigg(\frac{b_2}{(s-\eta)^{a_2}}\|x_s\|_{C[-h,0]}^{p_2}
+ m_2(s)\bigg)ds\\
\nonumber
&\leq 2D_2E_{1,1}(\lambda_2 \eta)+\int_\eta^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}m_2(s)ds+ \int_0^{t-\eta}\frac{(t-\eta-s)^{\alpha-1}}{\Gamma(\alpha)} \frac{b_2}{s^{a_2}}\|x_{s+\eta}\|_{C[-h,0]}^{p_2}ds\\
\nonumber
&\leq 2D_2E_{1,1}(\lambda_2 \eta)+\frac{1}{\Gamma(\alpha)}\bigg(\int_\eta^t (t-s)^{\frac{\alpha-1}{1-q_2}}ds\bigg)^{1-q_2}
\|m_2\|_{L^{\frac{1}{q_2}}[\eta,T]}\\
\nonumber
&+2b_2D_3\{I^\alpha[E_{1,1-a_2}(\lambda_3 s)]\}(t-\eta)\leq \frac{D_3}{\Gamma(1-a_2)}+D_3E_{1,1-a_2}(\lambda_3 (t-\eta))\\
\nonumber
&\leq 2D_3E_{1,1-a_2}(\lambda_3 (t-\eta)).
\end{align}
Thus $ JG \subset G $.
By Lemma \ref{fixed point}, $ J $ has at least one fixed point in $ G $.
\end{proof}
\begin{remark}
The condition (H3-1) of Theorem \ref{existence_theorem} can be replaced by the following condition.
There exist $ n\in N $, $0=T_0<T_1<\cdots<T_n=T $, $ \eta_i \in (T_{i-1},T_i)$ for $i=2,\cdots,n $, $a_j,q_j \in [0,\alpha)$, $ p_j \in (0,1] $, $ b_j>0 $, $ m_j(t) \in L^{\frac{1}{q_j}} [T_{j-1},T_j] $ for $ j=1,\cdots,n $ such that
\begin{equation}
\nonumber
|f(t,u)| \leq \left\{
\begin{aligned}
& \frac{b_1}{t^{a_1}}\|u\|_{C[-h,0]}^{p_1}+m_1(t) && \text{for $ t \in (0, T_1] $ and $ u \in C[-h,0] $} \\
& \frac{b_i}{|t-\eta_i|^{a_i}}\|u\|_{C[-h,0]}^{p_i}+ m_i(t) && \text{for
$ t \in [T_{i-1},\eta_i) \cup(\eta_i, T_i] $ and $ u \in C[-h,0] $},\\
\end{aligned}
\right.
\end{equation}
where $ i=2,\cdots,n $.
\end{remark}
\begin{remark}
Theorem \ref{existence_theorem} is an improvement on the result of Corollary 3.1 in \cite{YangCao}.
\end{remark}
\begin{remark}
Theorem \ref{existence_theorem} can be easily extended to vector-valued functions.
\end{remark}
\begin{corollary}
\label{global_existence}
Suppose that (H2-1) and (H3-1) hold, except that the number $ T $ is taken to be $ \infty $.
Then the initial value problem (\ref{governing_equation})-(\ref{initial_condition}) has a solution on $ [0, \infty)$.
\end{corollary}
\begin{proof}
By Theorem \ref{existence_theorem}, for any $ T \in R $, the fractional functional differential equation (\ref{governing_equation})-(\ref{initial_condition}) has a solution.
Since $ T $ can be chosen arbitrarily large, the equation (\ref{governing_equation})-(\ref{initial_condition}) has at least one global solution on $ [0, \infty)$.
\end{proof}
\section{Uniqueness of global solutions}\label{Sec:4}
In this section the uniqueness of global solutions of the initial value problem for the fractional functional differential equation (\ref{governing_equation})-(\ref{initial_condition}) is studied.
For the uniqueness theorem, we make the following hypotheses:\\
(H4-1) there exist constants $ a_1,a_2 \in [0,\alpha), b_1, b_2>0 $, $ T_1 \in (0,T]$,
$ \eta \in (T_1,T)$ such that
\vskip -11pt
\begin{equation}
\nonumber
|f(t,u)-f(t,v)|\leq \left\{ \begin{aligned}
& \frac {b_1}{t^{a_1}}\|u-v\|_{C[-h,0]} && \text {for $t \in (0,T_1]$, $ u,v \in C[-h,0] $}\\
& \frac {b_2}{|t-\eta|^{a_2}}\|u-v\|_{C[-h,0]} && \text { for $ t \in [T_1,\eta)\cup (\eta,T]$, $ u,v \in C[-h,0] $}.
\end{aligned}
\right.
\end{equation}
\begin{theorem}\label{uniqueness_theorem}
Suppose that (H2-1) and (H4-1) hold. Then the initial value problem (\ref{governing_equation})-(\ref{initial_condition}) has a unique solution on $ [0,T]$.
\end{theorem}
\begin{proof}
It follows from the condition (H4-1) that (H3-1) holds.
Thus, by Theorem \ref{existence_theorem}, the fractional functional differential equation (\ref{governing_equation})-(\ref{initial_condition}) has at least one solution on $ [0, T]$.
By using the method of proof by contradiction, we will obtain a uniqueness result for solutions of the initial value problem (\ref{governing_equation})-(\ref{initial_condition}).
Assume that the equation (\ref{governing_equation})-(\ref{initial_condition}) has two solutions on $ [0, T]$.
Then the operator $ J $ has also two fixed points $x,y \in C[0,T]$ such that $\|x-y\|_{C[0,T]}>0$.
By Lemma \ref{Mittag_function_property}, there exists a real number $ \lambda_1>0 $
such that for $ t \in [0,T] $,
\begin{equation}
\nonumber
\{I^\alpha[E_{1,1-a_1}(\lambda_1 s)s^{-a_1}]\}(t)<\frac{1}{b_1}E_{1,1-a_1}(\lambda_1 t).
\end{equation}
We define $W_1$ and $Q_1$ as follows
\begin{align}
\nonumber
W_1&=\inf\{w:|x(t)-y(t)|\leq wE_{1,1-a_1}(\lambda_1t),t\in [0,T_1]\},\\
\nonumber
Q_1&=\inf\{t\in[0,T_1]:|x(t)-y(t)|=W_1E_{1,1-a_1}(\lambda_1t)\}.
\end{align}
If $ W_1 \neq 0 $, then we have
\begin{align}
\nonumber
W_1E_{1,1-a_1}&(\lambda_1{Q_1})=|x(Q_1)-y(Q_1)|\\
\nonumber
&\leq \frac{1}{\Gamma(\alpha)}\int_0^{Q_1} (Q_1-s)^{\alpha-1}|f(s,x_s)-f(s,y_s)|ds\\
\nonumber
&\leq \frac{1}{\Gamma(\alpha)}\int_0^{Q_1} (Q_1-s)^{\alpha-1}\frac{b_1}{s^{a_1}}\|x_s-y_s\|_{C[-h,0]}ds\\
\nonumber
&\leq \frac{b_1}{\Gamma(\alpha)}\int_0^{Q_1} (Q_1-s)^{\alpha-1} \frac{1}{s^{a_1}}W_1E_{1,1-a_1}(\lambda_1s)ds\\
\nonumber
&< W_1E_{1,1-a_1}(\lambda_1{Q_1}),
\end{align}
which implies that $ W_1=0 $. Therefore $ x(t)=y(t), t\in [0,T_1] $.
By Lemma \ref{Mittag_function_property}, there exists a real number $ \lambda_2>0 $
such that for $ t \in [0,T] $,
\begin{equation}
\nonumber
\{I^{\alpha-a_2}[E_{1,1}(\lambda_2 s)]\}(t)<\frac{\Gamma(\alpha)}{b_2\Gamma(\alpha-a_2)}E_{1,1}(\lambda_2 t).
\end{equation}
We define $W_2$ and $Q_2$ as follows
\begin{align}
\nonumber
W_2&=\inf\{w:|x(t)-y(t)|\leq wE_{1,1}(\lambda_2t),t\in [T_1,\eta]\},\\
\nonumber
Q_2&=\inf\{t\in[T_1,\eta]:|x(t)-y(t)|=W_2E_{1,1}(\lambda_2t)\}.
\end{align}
If $ W_2 \neq 0 $, then we have
\begin{align}
\nonumber
W_2&E_{1,1}(\lambda_2{Q_2})=|x(Q_2)-y(Q_2)|\\
\nonumber
&\leq \frac{1}{\Gamma(\alpha)}\int_0^{Q_2} (Q_2-s)^{\alpha-1}|f(s,x_s)-f(s,y_s)|ds\\
\nonumber
&\leq \frac{1}{\Gamma(\alpha)}\int_{T_1}^{Q_2} (Q_2-s)^{\alpha-1}\frac {b_2}{(\eta-s)^{a_2}}\|x_s-y_s\|_{C[-h,0]}ds\\
\nonumber
&\leq \frac{b_2}{\Gamma(\alpha)}\int_0^{Q_2} (Q_2-s)^{\alpha-a_2-1}W_2E_{1,1}(\lambda_2s)ds
<W_2E_{1,1}(\lambda_2{Q_2}),
\end{align}
which implies that $ W_2=0 $. Therefore $ x(t)=y(t), t\in [0,\eta] $.
By Lemma \ref{Mittag_function_property}, there exists a real number $ \lambda_3>0 $
such that for $ t \in [0,T] $,
\begin{equation}
\nonumber
\{I^q[E_{1,1-a_2}(\lambda_3 s)s^{-a_2}]\}(t)<\frac{1}{b_2}E_{1,1-a_2}(\lambda_3 t).
\end{equation}
We define $W_3$ and $Q_3$ as follows
\begin{align}
\nonumber
W_3&=\inf\{w:|x(t)-y(t)|\leq wE_{1,1-a_2}(\lambda_3(t-\eta)),t\in [\eta,T]\},\\
\nonumber
Q_3&=\inf\{t\in[\eta,T]:|x(t)-y(t)|=W_3E_{1,1-a_2}(\lambda_3(t-\eta))\}.
\end{align}
From the assumption $\|x-y\|_{C[0,T]}>0$, it is clear that $ W_3 \neq 0 $.
Then we have
\begin{align}
\nonumber
W_3&E_{1,1-a_2}(\lambda_3 (Q_3-\eta))=|x(Q_3)-y(Q_3)|\\
\nonumber
&\leq \frac{1}{\Gamma(\alpha)}\int_\eta^{Q_3} (Q_3-s)^{\alpha-1}|f(s,x_s)-f(s,y_s)|ds\\
\nonumber
&\leq \frac{1}{\Gamma(\alpha)}\int_\eta^{Q_3} (Q_3-s)^{\alpha-1}\frac{b_2}{(s-\eta)^{a_2}}\|x_s-y_s\|_{C[-h,0]}ds\\
\nonumber
&\leq \frac{b_2}{\Gamma(\alpha)}\int_0^{Q_3-l} (Q_3-l-s)^{\alpha-1} \frac{1}{s^{a_2}}W_3E_{1,1-a_2}(\lambda_3s)ds\\
\nonumber
&< W_3E_{1,1-a_2}(\lambda_3(Q_3-\eta)).
\end{align}
This contradiction shows that the equation (\ref{governing_equation})-(\ref{initial_condition}) has a unique solution.
\end{proof}
\begin{remark}
The condition (H4-1) of Theorem \ref{uniqueness_theorem} can be replaced by the following condition.
There exist $ n\in N $, $0=T_0<T_1<\cdots<T_n=T $, $ \eta_i \in (T_{i-1},T_i)$ for $i=2,\cdots,n $, $a_j \in [0,\alpha)$, $ b_j>0 $ for $ j=1,\cdots,n $ such that
\begin{equation}
\nonumber
|f(t,u)-f(t,v)| \leq \left\{
\begin{aligned}
& \frac{b_1}{t^{a_1}}\|u-v\|_{C[-h,0]} && \text{for $ t \in (0, T_1] $ and $ u,v \in C[-h,0] $} \\
& \frac{b_i}{|t-\eta_i|^{a_i}}\|u-v\|_{C[-h,0]} && \text{for $ t \in [T_{i-1},\eta_i) \cup(\eta_i, T_i] $}\\
& && \text{ and $ u,v \in C[-h,0] $},\\
\end{aligned}
\right.
\end{equation}
where $ i=2,\cdots,n $.
\end{remark}
\begin{remark}
Theorem \ref{existence_theorem} is an improvement on the result of Theorem 5.1 in \cite{YangCao}.
\end{remark}
\begin{corollary}
\label{global_uniqueness}
Suppose that (H2-1) and (H4-1) hold, except that the number $ T $ is taken to be $ \infty $.
Then the initial value problem (\ref{governing_equation})-(\ref{initial_condition}) has a unique solution on $ [0, \infty)$.
\end{corollary}
\begin{proof}
Similar to Corollary \ref{global_existence}, we can prove this result.
\end{proof}
\section*{Acknowledgement(s)}
The authors would like to thank refrees for their valuable advices for the improvement of this article.
\section*{References}
\end{document} | math | 24,802 |
\begin{document}
\title{Quantum nonlocality: How does Nature perform the trick?\cite{Bellprize}
Since our early childhood we know in our bones that in order to interact with an object we have either to go to it or to throw something at it. Yet, contrary to all our daily experience, Nature is nonlocal: there are spatially separated systems that exhibit nonlocal correlations. In recent years this led to new experiments, deeper understanding of the tension between quantum physics and relativity and to proposals for disruptive technologies.
Consider two spatially separated quantum systems, one controlled by Alice, the other by Bob, in a pure state $\psi$. Alice and Bob may perform some measurements $x$ and $y$ on their systems and collect the results $a$ and $b$, respectively. This situation is described by a conditional probability distribution $p_\psi (a,b|x,y)$. In general this correlation doesn't factorize: $p_\psi (a,b|x,y)\neq p_\psi (a|x)\cdot p_\psi (b|y)$, i.e. the two systems are correlated. At first, this is no surprise, correlations are everywhere. For example, consider two cups of the same color, either both red or both green, one in Alice's and one in Bob's hands. If they looks at the color of their cups, Alice and Bob's results are correlated. In this example the origin of the correlation is obvious, Alice and Bob had only partial information: they knew that both have the same color, but they ignored which color. This differs deeply from the quantum situation, as quantum theory claims that a pure state provides a complete description of the two systems. This led EPR\cite{EPR} to believe that quantum theory is incomplete in the same sense as the description "of the same color" provides only an incomplete description of the color state of the cups.
Let us now consider the situation described by any possible future theory. Define $\leftarrowambda$ as the state that this future theory ascribes to the two spatially separated systems and assume that: $p_\leftarrowambda (a,b|x,y)= p_\leftarrowambda (a|x)\cdot p_\leftarrowambda (b|y)$. A priori it seems hard to make any prediction from this assumption since we do not know this future theory. But John Bell noticed that the experiments we perform today necessarily correspond to a statistical mixture of the more refined states of this future theory: $p_\psi(a,b|x,y) = \int d\leftarrowambda \rightarrowho(\leftarrowambda) p_\leftarrowambda(a,b|x,y)$, from which he derived his famous inequality satisfied by all local correlations. Let us emphasize that a violation of Bell's inequality not only tells us something about quantum physics, but - more impressively - tells us that in all possible future theories some spatially separated systems exhibit nonlocal correlations. Consequently, it is Nature herself that is nonlocal.
Many physicists feel uneasy with nonlocality\cite{nonrealism}. A part of the uneasiness comes from a confusion between nonlocal correlations and nonlocal signalling. The latter means the possibility to signal at arbitrarily fast speeds, a clear contradiction to relativity. However, the nonlocal correlations of quantum physics are nonsignalling. This should remove some of the uneasiness. Furthermore, note that in a nonsignalling world, correlations can be nonlocal only if the measurement results were not pre-determined. Indeed, if the results were pre-determined (and accessible with future theories and technologies), then one could exploit nonlocal correlations to signal. This fact has recently been used to produce bit strings with proven randomness \cite{StefanoRandom}. This is fascinating because it places quantum nonlocality no longer at the center of a debate full of susceptibilities and prejudice, but as a resource for future quantum technologies. We'll come back to this, but beforehand let us present a few recent experimental tests of quantum nonlocality.
The pioneering experiment by Clauser\cite{Clauser} suffered from a few loopholes, but these have since been separately closed\cite{locality, DetLoophole}. Still, correlations cry out for explanations, as emphasized by Bell\cite{BellCorrCry}. Everyone confronted with nonlocal correlations feels that the two systems somehow influence each other (e.g. Einstein's famous {\it spooky action at a distance}). This is also the way textbooks describe the process: a first measurement triggers a collapse of the entire state vector, hence modifying the state at the distant side. In recent years these intuitions have been taken seriously, leading to new experimental tests. If there is an influence from Alice to Bob, this influence has to propagate faster than light, as existing experiments have already demonstrated violation of Bell's inequality between space-like separated regions\cite{FinishMeasrmt}. But a faster than light speed can only be defined with respect to a hypothetical universal privileged reference frame, as the one in which the cosmic background radiation is isotropic. The basic idea is that if correlations are due to some "hidden influence" that propagates at finite speed, then, if the two measurements are sufficiently well synchronized in the hypothetical privileged frame, the influence doesn't arrive on time and one shouldn't observe nonlocal correlations. Remains the problem that one doesn't know {\it a priori} the privileged frame in which one should synchronize the measurements. This difficulty was recently circumvented by taking advantage of the Earth's 24 hours rotation, setting thus stringent lower bounds on the speed of these hypothetical influences\cite{SalartNature}. Hence, nonlocal correlations happen without one system influencing the oter. In another set of experiments the two observers, Alice and Bob, were set in motion in opposite directions in such a way that each in its own inertial reference frame felt he performed his measurement first and could thus not be influenced by his partner\cite{SuarezScarani97,MovingObs}. Hence, quantum correlations happen without any time-ordering.
All of today's experimental evidence points to one conclusion: Nature is nonlocal. As for any truly deep finding, this one has implications both for our world view and for future technologies. Let us first give an example of the second implication. Quantum Key Distribution (QKD) is the most advanced application of quantum information science. Today's commercial QKD systems rely on sound principles, but their implementation has to be thoroughly tested in order to check for unwanted side channels that an adversary could exploit. For example, the photons emitted by Alice could, in addition to carrying a quantum bit encoded in its polarization state, also carry redundant information unwittingly encoded in the timing of the photons, or in their spectra. This is possible because today's QKD systems do not rely on nonlocal correlations. If they would, the mere fact that the correlations between the data collected by Alice and Bob violate Bell's inequality would suffice to guarantee the absence of any side channel. This was the intuition of Ekert in 1991 \cite{Ekert91}, but was proven only in 2007 \cite{QKDDI}. Note the amazing consequence\cite{EkertPW}: in future, it will be possible to buy cryptography systems from ones adversary as the observation of nonlocal correlations will guarantee the proper functioning of the system!
To conclude let us come to the conceptual implications. In modern quantum physics entanglement is fundamental; furthermore, space is irrelevant - at least in quantum information science space plays no central role and time is a mere discrete clock parameter. In relativity space-time is fundamental and there is no place for nonlocal correlations. To put the tension in other words: no story in space-time can tell us how nonlocal correlations happen, hence nonlocal quantum correlations seem to emerge, somehow, from outside space-time.
\end{document} | math | 7,932 |
\begin{document}
\title{Bloch sphere like construction of SU(3) Hamiltonians using Unitary
Integration}
\author{Sai Vinjanampathy and A R P Rau}
\address{
Dept of Physics and Astronomy, Louisiana State University, Baton Rouge,
Louisiana 70803-4001,USA\\
}
\ead{[email protected] and [email protected]}
\begin{abstract}
The Bloch sphere is a familiar and useful geometrical picture of the time
evolution of a single spin or a quantal two-level system. The analogous
geometrical picture for three-level systems is presented, with several
applications. The relevant SU(3) group and su(3) algebra are eight-dimensional
objects and are realized in our picture as two four-dimensional manifolds that
describe the time evolution operator. The first, called the base manifold, is
the counterpart of the S$^2$ Bloch sphere, whereas the second, called the fiber,
generalizes the single U(1) phase of a single spin. Now four-dimensional, it
breaks down further into smaller objects depending on alternative
representations that we discuss. Geometrical phases are also developed and
presented for specific applications. Arbitrary time-dependent couplings between
three levels or between two spins (qubits) with SU(3) Hamiltonians can be
conveniently handled through these geometrical objects.
\end{abstract}
\pacs{02.40.Yy, 02.20.Qs, 03.67.Lx, 03.65.Vf, 03.65.Fd}
\submitto{\JPA}
\maketitle
\section{\label{intro}Introduction}
Three-level systems are of fundamental importance to many branches of
physics. While two levels give the simplest model for
the dynamics of discrete systems, three levels illustrate the role that an
intermediate state can play in inducing
transitions between the other two.
Canonical examples of this include applications in quantum optics
that use three-level atoms to control quantum state evolution
\cite{eit}. Such laser control is used, for instance, to
transfer population between two states using stimulated Raman adiabatic
passage (STIRAP)
\cite{stirap,stirap2} and chirped adiabatic passage (CARP) \cite{Chelkowski}. In
some of these systems, the interaction of the radiation with the atom is
represented as a time-dependent Hamiltonian inducing an energy
separation between the two states that varies with time. For a non-zero sweep
rate, it can be shown that there is finite transition probability between the
states \cite{LZ,LZ2,LZ3}. The study of Landau-Zener transitions in multilevel
systems is
of interest to understand the interplay between various level
crossings \cite{vitanov}. Particle physics represents another example where
three-level
systems play a central role as, for example, the oscillations of neutrino flavor
eigenstates \cite{neutrino}.
The general Hamiltonian of a three-level system involves 8 independent
operators. Such a set can also naturally arise as a subgroup of higher level
systems where there is some degeneracy involved. Thus, several important
two-qubit
problems in quantum computing and quantum information can be so written in
terms of eight operators that
form a subalgebra of the full fifteen operators that describe two spins. The
Hamiltonian describing anisotropic spin
exchange is an example of one such important physical problem. While
isotropic spin exchange has been explored to design two-qubit gates
in quantum computing, anisotropic spin exchange has been studied as
a possible impediment to two-qubit gate operations
\cite{divincenzo,divincenz2}. Such a SU(3) Hamiltonian is given by
\begin{eqnarray}\label{DV}
\mathbf{H}(t)=J(t)(\vec{\bm{\sigma}}.\vec{\bm{\tau}} +
\vec{\beta}(t).(\vec{\bm{\sigma}}
\times \vec{\bm{\tau}}) + \vec{\bm{\sigma}}.\mathbf{\Gamma}(t).\vec{\bm{\tau}}),
\end{eqnarray}
when written in terms of a scalar, a vector and a symmetric tensor operator
expressed in terms of two Pauli spins. Here, $\vec{\beta}(t)$ is the
Dzyaloshinksii-Moriya
vector \cite{dz1,dz2} and $\mathbf{\Gamma}(t)$ is the (traceless) symmetric
interaction term. While the first term is the familiar Ising interaction
Hamiltonian \cite{Chandler}, the last two terms are due to spin-orbit coupling.
Given this wide applicability, a geometrical picture of the
dynamics of three-level systems can be useful. For a two-level system, the
geometry of the evolution
operator is well
known. Any density matrix can be written
as $\bm{\rho}=(I^{(2)}+\vec{n}.\vec{\bm{\sigma}})/2$, where $\vec{\bm{\sigma}}$
are the Pauli matrices. Unitary evolution of $\bm{\rho}$ is represented as the
vector $\vec{n}$ rotating on the surface of a three dimensional unit
sphere called the Bloch sphere \cite{poincare}. This vector, along with a
phase, accounts for
the three parameters describing the time evolution operator of a two-level
system. The vector $\vec{n}$, along
with the phase factor, is shown in Fig. (\ref{Fig1}). The vector $\vec{n}$
shown traces
out the ``base manifold'' and together with the global phase factor or ``fiber''
at each point on that manifold is referred to as a ``fiber
bundle'' \cite{Bengtsson}.
While the
density matrix is independent of
it, the complete description of the system requires this phase as well. The aim
of this paper is to provide an analogous geometrical picture for a three-level
system with appropriate
generalizations of the base and fiber.
Some work already exists regarding the geometry of SU(3). Following Wei
and Norman \cite{wei}, Dattoli and Torre have constructed the ``Rabi matrix"
for a general SU(3) unitary evolution in \cite{dattoli}. Mosseri and Dandoloff
in \cite{remy} described the generalization of the Bloch sphere
construction of single qubits to two qubits via the Hopf fibration
description. This method relies upon the homomorphism between the SU(2) and
SO(3) groups and likewise between the SU(4) and SO(6) groups. In \cite{tilma},
the authors propose a generalized Euler angle parameterization for SU(4). This
decomposition is similar to the work in
\cite{oldui,oldui2,oldui3,oldui4,restui,restui2,restui3} into which fits our
treatment of SU(3) in this paper.
Another well known choice of the $(N^{2}-1)$ generators $\mathbf{s}_{j}$ of the
SU(N) group was studied in \cite{hioe,dattoli2}. Consider $\mathbf{s}_{j}$,
chosen to be traceless and Hermitian such that
$[\mathbf{s}_{i},\mathbf{s}_{j}]=2i f_{ijk}\mathbf{s}_{k}$ and $Tr \{
\mathbf{s}_{i}\mathbf{s}_{j} \} = 2 \delta_{jk} $. Here, $f_{ijk}$ is the
completely antisymmetric symbol which for a two-level system is the Levi-Civita
symbol $\epsilon_{ijk}$, and a repeated index is summed over. In this basis, the
Hamiltonian is written as $
\mathbf{H}(t) = \Gamma_{i} \mathbf{s}_{i} $. With this choice, the Liouville-Von
Neumann equation for the density matrix
$\bm{\rho}=\mathbf{I}/N+S_{j}\mathbf{s}_{j}/2$ becomes
$\dot{S}_{i}=f_{ijk}\Gamma_{j}S_{k}$. Note that for the N=2 case, this is the
familiar Bloch sphere representation. But, for SU(3), this representation
differs from the one we
present in two aspects. Firstly, the ``coherence vector'', whose elements
are
real and are given by $S_{j}$, experiences rotations in a $(N^{2}-1)$
dimensional
space. For instance, for SU(3), the coherence vector undergoes rotations in an
eight-dimensional space. Arbitrary rotations in eight
dimensions are characterized by 28 parameters. But since a three-level
Hamiltonian is only characterized by 8 real quantities, this means that the
coherence vector is not permitted arbitrary rotations and is instead
constrained.
Secondly, the coherence vector representation does not differentiate between
local and non-local operations. Our decomposition of the time evolution operator
into a diagonal and an off-diagonal term in this paper is more suited for this
differentiation. Such a parameterization of the time evolution operator
in terms of local and non-local operations can be
useful in understanding entanglement. The aim of this paper is to
discuss the geometry of two-qubit time evolution operators in terms of such a
decomposition. The authors in \cite{englert} discuss an alternative
decomposition of two-qubit states in terms of two three-vectors and a $3 \times
3$ dyadic to discuss entanglement.
A series of papers presented a systematic approach to
studying N-level systems using a program of unitary integration
\cite{oldui,oldui2,oldui3,oldui4,newui,newui2,restui,restui2,restui3}.
Continuing this program, we
present a complete analytical solution to the three-level problem that
generalizes the Bloch sphere approach to three levels.
Below, we define the fiber bundle via two different decompositions which allows
us to extract the geometric
phases associated with a three-level system (for a discussion on the
quantum phases of three-level systems, see
\cite{gpqutrit,berry2and3}). These fiber bundles are $\{$SU(3)/SU(2)$\times$
U(1)$\}\times\{$SU(2)$\times$U(1)$\}$ and $\{$SU(4)/[SU(2)$\times$SU(2)]$\}
\times\{$SU(2)$\times$SU(2)$\}$.
\begin{figure}
\caption{Bloch or Poincare sphere representation for SU(2). The base
manifold is the $S^{2}
\label{Fig1}
\end{figure}
The structure of this paper is as follows: Section \ref{ui} outlines
the unitary integration program to solve time-dependent operator
equations. Section \ref{su3} uses this technique for the
solution of a general time-dependent SU(3) Hamiltonian
completely analytically. Section
\ref{discussion} presents the geometry of the time
evolution operator for SU(3) with some applications. Section
\ref{GP} presents a coordinate description
that is useful to define the geometric phase for three-level systems, and
Section \ref{conclusions} presents the conclusions. The appendix will
present an alternative analytical solution to the three-level problem by
exploiting the natural embedding of SU(3) in SU(4).
\section{\label{ui}Unitary Integration}
Many important applications in physics involve time dependence in
the Hamiltonian. For such systems, the time evolution operator is
not given by the simple exponentiation of the Hamiltonian
\cite{sakurai}. To handle the time evolution for such Hamiltonians
iteratively, ``Unitary Integration" was proposed in
\cite{oldui,oldui2,oldui3,oldui4}.
Earlier work with this technique is presented in \cite{wei,dattoli2}. Later,
the technique was
presented as generalizing the SU(2) example
to solve iteratively for the time evolution operator $\mathbf{U}^{(N)}(t)$ of
N-level
systems \cite{newui,newui2}. Consider the N-dimensional Hamiltonian
$\mathbf{H}^{(N)}$
given by
\begin{equation}\label{Hamiltonian}
\mathbf{H}^{(N)}=\left(\begin{array}{cc}\mathbf{H}^{(N-n)}&\mathbf{V
}\\ \mathbf{V}^{\dagger}&
\mathbf{H}^{(n)}\end{array} \right).
\end{equation}
The diagonal blocks are (N$\--$n)- and (n)-dimensional square
matrices, respectively, while $\mathbf{V}$ is an $(N-n) \times (n)$-dimensional
matrix.
The evolution operator $\mathbf{U}^{(N)}(t)$ for such a
$\mathbf{H}^{(N)}$ is written as a product of
two operators
$\mathbf{U}^{(N)}(t)=\widetilde{U}_{1}\widetilde{U}_{2}$, where
\begin{eqnarray}\label{nonunit_u}
\widetilde{U}_{1}=\left(\begin{array}{cc}\mathbf{I}^{(N-n)}& \mathbf{z}(t)\\
\mathbf{0}^{\dagger}&
\mathbf{I}^{(n)} \end{array} \right)\left(\begin{array}{cc}\mathbf{I}^{(N-n)}&
\mathbf{0}\\
\mathbf{w}^{\dagger}(t)& \mathbf{I}^{(n)} \end{array} \right)
,\\
\widetilde{U}_{2}=\left(\begin{array}{cc}\widetilde{\mathbf{U}}^{(N-n)}&
\mathbf{0}\\ \mathbf{0}^{\dagger}&
\widetilde{\mathbf{U}}^{n} \end{array} \right).\nonumber
\end{eqnarray}
For any $N$, $n$ is arbitrary with $1 \leq n < N$, and tilde denotes
that the matrices need not be unitary. The product of three factors
parallels the product of exponentials in three Pauli matrices. Equations
defining
the rectangular matrices $\mathbf{z}(t)$ and $\mathbf{w}^{\dagger}(t)$ are
developed and the problem is reduced to the two residual $(N-n)$- and
$(n)$ dimensional evolution problems sitting as diagonal blocks of
$\widetilde{U}_{2}$. $\mathbf{z}(t)$ and
$\mathbf{w}^{\dagger}(t)$ are
related to each other through the unitarity of
$\mathbf{U}^{(N)}(t)$ \cite{newui,newui2}:
\begin{eqnarray}
\mathbf{z}=-\bm{\gamma}_{1}\mathbf{w}=-\mathbf{w}\bm{\gamma}_{2},
\end{eqnarray}
with $\bm{\gamma}_{1}=\hat{\mathbf{I}}^{(N-n)}+\mathbf{z}.\mathbf{z}^{\dagger}$
and
$\bm{\gamma}_{2}=\hat{\mathbf{I}}^{(n)}+\mathbf{z}^{\dagger}.\mathbf{z}$.
With $\mathbf{U}^{(N)}(t)$ in such a product form, the
Schr\"{o}dinger equation is written as
\begin{eqnarray}\label{effeqn}
i \dot{\widetilde{U}}_{2}(t)=\mathbf{H}_{\mathtt{eff}}\widetilde{U}_{2},\\
\nonumber
\mathbf{H}_{\mathtt{eff}}=\widetilde{U}^{-1}_{1}\mathbf{H}^{(N)}\widetilde{U}_{1
} \-- i \widetilde{U}^{-1}_{1}\dot{\widetilde{U}}_{1}.
\end{eqnarray}
Here, overdot denotes differentiation with respect to time. Since
$\widetilde{U}_{2}$ is block diagonal, the off-diagonal blocks of
equation~(\ref{effeqn}) define the equation satisfied by $\mathbf{z}$
\begin{equation}\label{zdot}
i\dot{\mathbf{z}}=\mathbf{H}^{(N-n)}\mathbf{z} + \mathbf{V}
\--\mathbf{z}(\mathbf{V}^{\dagger}\mathbf{z}+\mathbf{H}^{(n)}).
\end{equation}
Note that the initial condition $U^{N}(0)=\mathbf{I}^{N}$ implies
that $\widetilde{U}_{1}(0)=\mathbf{I}^{(N-n)}$,
$\widetilde{U}_{2}(0)=\mathbf{I}^{(n)}$
and $\mathbf{z}(0)=\mathbf{0}^{(N-n)}$. equation~(\ref{zdot}), along with
the initial condition can be solved to determine $\mathbf{z}$ and thereby
$\widetilde{U}_{1}$ and $\mathbf{H}_{\mathtt{eff}}$ for subsequent solution of
equation (\ref{effeqn}) for $\widetilde{U}_{2}$. In this manner, the
procedure iteratively determines $U^{(N)}(t)$.
Before discussing the geometry of the time evolution operators for
this unitary case, we briefly mention the procedure to deal with non-Hermitian
Hamiltonians. For such a non-Hermitian Hamiltonian,
\begin{equation}\label{Hamiltonian_non_Hermitian}
\mathbf{H}^{(N)}=\left(\begin{array}{cc}\widetilde{\mathbf{H}}^{(N-n)}&\mathbf{V
}\\ \mathbf{Y}^{\dagger}&
\widetilde{\mathbf{H}}^{(n)}\end{array} \right),
\end{equation}
where tilde denotes possibly non-Hermitian character, and the off-diagonal
components $\mathbf{V}$ and $\mathbf{Y}$ are independent.
In this case, equation (\ref{zdot}) is replaced by
\begin{equation}\label{zdot_non_Hermitian}
i\dot{\mathbf{z}}=\widetilde{\mathbf{H}}^{(N-n)}\mathbf{z} + \mathbf{V}
\--\mathbf{z}(\mathbf{Y}^{\dagger}\mathbf{z}+\widetilde{\mathbf{H}}^{(n)}),
\end{equation}
and there is a separate equation governing the evolution of $\mathbf{w}$ given
by
\begin{equation}\label{wdot_non_Hermitian}
i\dot{\mathbf{w}}^{\dagger}=\mathbf{w}^{\dagger}(\mathbf{z}\mathbf{Y}^{\dagger}
-\widetilde{\mathbf{H}}^{(N-n)})+(\widetilde{\mathbf{H}}^{(n)} +
\mathbf{Y}^{\dagger}\mathbf{z})\mathbf{w}^{\dagger} +\mathbf{Y}^{\dagger}.
\end{equation}
The diagonal terms of the time-evolution operators are governed by
\begin{eqnarray}\label{effeqn_non_Hermitian}
i \dot{\widetilde{U}}_{2}(t)=\left(\begin{array}{cc}
\widetilde{\mathbf{H}}^{(N-n)}-\mathbf{z}\mathbf{Y}^{\dagger}&0\\
0&\widetilde{\mathbf{H}}^{(n)}+\mathbf{Y}^{\dagger}\mathbf{z}
\end{array}
\right)\widetilde{U}_{2}.
\end{eqnarray}
Returning to the case where the Hamiltonian is Hermitian, it is convenient to
render the two matrices $\widetilde{U}_{1}$ and
$\widetilde{U}_{2}$ themselves unitary \cite{newui,newui2}. For this purpose, a
``gauge factor" $b$ is chosen such that the unitary counterparts of
$\widetilde{U}_{1}$ and $\widetilde{U}_{2}$ are defined via
$U_{1}=\widetilde{U}_{1}b$ and $U_{2}=b^{-1}\widetilde{U}_{1}$. Since
$\widetilde{U}^{\dagger}_{1}\widetilde{U}_{1}=diag(\gamma^{(-1)}_{1},\gamma_{2}
)$,
this would imply that b is the ``Hermitian square-root" of
$diag(\gamma^{(-1)}_{1},\gamma_{2})$. This ``Hermitian square-root"
is defined by the relation
$(b^{(-1)})^{\dagger}b^{(-1)}=diag(\gamma^{(-1)}_{1},\gamma_{2})$.
Inspection of the power series expansion of
$\gamma_{1}^{(\pm\frac{1}{2})}=(\hat{\mathbf{I}}+\mathbf{z}.\mathbf{z}^{\dagger}
)^{(\pm\frac{1}{2})}$
and
$\gamma_{2}^{(\pm\frac{1}{2})}=(\hat{\mathbf{I}}+\mathbf{z}^{\dagger}.\mathbf{z}
)^{(\pm\frac{1}{2})}$
show that since each term in the expansion is Hermitian, matrices
$\gamma^{\pm\frac{1}{2}}_{1}$ and $\gamma^{\pm\frac{1}{2}}_{2}$ are Hermitian
and
have non-negative eigenvalues. Because of this, it is
sufficient to define $b$ as the inverse square root via
$b^{(-2)}=diag(\gamma^{(-1)}_{1},\gamma_{2})$.
Furthermore, $H_{\mathtt{eff}}$ in equation (\ref{effeqn}) is Hermitian for
the
unitary
counterpart $U_{1}$. The upper diagonal block of this Hermitian
Hamiltonian accompanying the decomposition $U=U_{1}U_{2}$ is given
by
\begin{equation}\label{heffup}
\frac{i}{2}[\frac{d(\gamma_{1}^{-\frac{1}{2}})}{dt},\gamma_{1}^{\frac{1}{2}}]
+
\frac{1}{2}\left(\gamma_{1}^{-\frac{1}{2}}(\widetilde{\mathbf{H}}^{(N-n)}
-\mathbf { z }
\mathbf{V}^{\dagger})\gamma_{1}^{\frac{1}{2}}
+ H.c.\right),
\end{equation}
where [,] represents the commutator and \textit{H.c.} stands for the Hermitian
conjugate. The lower diagonal block is similarly given by
\begin{equation}\label{heffdown}
\frac{i}{2}[\frac{d(\gamma_{2}^{-\frac{1}{2}})}{dt},\gamma_{2}^{\frac{1}{2}}]
+
\frac{1}{2}\left(\gamma_{2}^{-\frac{1}{2}}(\widetilde{\mathbf{H}}^{(N-n)}
+\mathbf { z } ^
{\dagger}\mathbf{V})\gamma_{2}^{\frac{1}{2}}
+ H.c.\right).
\end{equation}
For $N=3$, $n=1$, these diagonal blocks define an SU(2)- and a U(1) Hamiltonian
and $\mathbf{z}$ is a pair of complex numbers.
The SU(2) Hamiltonian is in turn rendered in terms of its fiber bundle in Fig.
(\ref{Fig1}) and the U(1)
Hamiltonian corresponds to a phase. Together, they describe a four-dimensional
fiber for SU(3) over the base manifold, also four dimensional, of
$\mathbf{z}$.
Alternatively, $N=3$ SU(3) problems may be conveniently seen as a part of $N=4$
SU(4) problems, making contact with two qubit systems that are extensively
studied. In this case, for $N=4$, $n=2$, these diagonal blocks define two SU(2)
Hamiltonians and $\mathbf{z}$ is a $2\times2$ matrix representable in terms of
Pauli spinors. Generally, it is 8-dimensional while the fiber has seven
dimensions (two SU(2) and a mutual phase) but for the SU(3) subgroup of
SU(4),both the base and manifold again reduce to four dimensions each. With
$\textbf{z}$ a pair of complex numbers, the non-trivial part of geometrizing
SU(3) is thereby reduced to describing this four-dimensional manifold. Exploring
this for the $N=3$, $n=1$ decomposition will be the content of the next
section whereas the Appendix gives the alternative SU(4) rendering.
\section{\label{su3}Geometry of general SU(3) time evolution operator}
A general time-dependent three-level Hamiltonian may be written in terms of
eight linearly independent operators of a three-level system. Such a Hamiltonian
can also be
written in terms of a subgroup of 15 operators of a four-level system. Before
the time evolution operator is presented in the SU(3) basis in terms of a $N=3$,
$n=1$ decomposition, we will note that it can be rendered in a few alternative
ways.
First, a general time-dependent four-level Hamiltonian may be written as
$H(t)=\sum_i c_{i}\mathbf{O}_{i}$. Here $c_{i}$ are time-dependent and
$\mathbf{O}_{i}$ are the unit matrix and 15 linearly independent
operators of a 4-level system that may be chosen in a variety of matrix
representations. One choice used in particle physics
are the so called Greiner matrices \cite{greiner,oldui,oldui2,oldui3,oldui4}.
Another choice
consists of using $\vec{\bm{\sigma}}$, $\vec{\bm{\tau}}$,
$\vec{\bm{\sigma}}\otimes\vec{\bm{\tau}}$ and the $4\times4$ unit matrix. Such
a choice was discussed in \cite{restui,restui2} and will be used throughout
this paper. As it stands, the above Hamiltonian describes a general
four-level atom with 4 energies and 6 complex couplings. Note that
only the three differences in energies are important. Restricting
the 15 coefficients $c_{i}$ to a smaller number allows this
Hamiltonian to describe various physical Hamiltonians, forming different
subalgebras of the su(4) algebra \cite{restui}. For
example, if two of the six complex couplings are zero (levels 1 and 4 and levels
2 and 3
of a four-level atom not coupled),
then the Hamiltonian may be recast such that the operators involved
belong to an so(5) subalgebra \cite{restui}. On the other hand, if levels 2
and 3
are degenerate and level 4 is uncoupled from the rest, then
the problem may be recast in terms of only eight operators belonging
to the su(3) subalgebra of su(4). This is illustrated in Fig.
\ref{Fig_energylevels} and is one of the systems of interest in this paper.
\begin{figure}
\caption{Levels $\vert2\rangle$ and $\vert3\rangle$ couple equally to
$\vert1\rangle$ and to $\vert4\rangle$, which are themselves coupled. The three
complex coupling matrix elements and two energy positions define such an SU(3)
system.}
\label{Fig_energylevels}
\end{figure}
Alternatively, after one arrives at the linear equation for the
$N=4$, $n=2$ decomposition, one can represent the resulting vector in terms of
six homogeneous coordinates. This is the so-called ``Pl\"{u}cker
coordinate'' representation for the SU(3) Hamiltonian. These coordinates as
well as the alternative
derivation are presented in the appendix. The $N=3$, $n=1$ decomposition will
be the content of the rest of this section.
Consider the Hamiltonian in the basis of the Gell-Mann lambda
matrices \cite{georgi} $H(t)=\sum_{i}a_{i}\bm{\lambda}_{i}$. The $N=3$, $n=1$
decomposition consists of writing the time evolution operator in terms of a
product of two matrices $U=\widetilde{U}_{1} \widetilde{U}_{2}$ where
$\widetilde{U}_{1}$ is composed of a (2$\times$1)-dimensional \textbf{z}, as
explained in Sec. II. The equation that governs the evolution of \textbf{z},
equation
(\ref{zdot}), can be written in this case as
\begin{equation}\label{su3_z_eqn}
\dot{z}_{\mu}=-iV_{\mu}-iF_{\mu\nu}z_{\nu}+iV^{*}_{\nu}z_{\nu}z_{\mu}
;\; \mu ,\nu=1,2.
\end{equation}
Here, the symbols used in defining $\dot{\mathbf{z}}$ are defined as
$V=(a_{4}-ia_{5},a_{6}-ia_{7})$,
and
\begin{equation*}
F=\left(\begin{array}{cc}
a_{3}+\sqrt{3} a_{8}&a_{1}-i a_{2}\\
a_{1}+i a_{2}&-a_{3}+\sqrt{3} a_{8}\\
\end{array} \right).
\end{equation*}
Using the transformation equations $m_{1,2}=-z_{1,2}(De^{i\phi})^{-1}$,
$m_{3}=(De^{i\phi})^{-1}$ and $\vert m_{1} \vert^{2}+\vert m_{2}\vert^{2}+\vert
m_{3} \vert^{2}=1$ leads to the evolution equation for
$\vec{m}=(m_{1r},m_{2r},m_{3r},m_{1i},m_{2i},m_{3i})^{T}$:
\begin{equation}\label{su3_Rotation}
\fl
\dot{\vec{m}}=\left(\begin{array}{cccccc}
0&-a_{2}&a_{5}&a_{3}+\sqrt{3}a_{8}&a_{1}&-a_{4}\\
a_{2}&0&a_{7}&a_{1}&-a_{3}+\sqrt{3}a_{8}&-a_{6}\\
-a_{5}&-a_{7}&0&-a_{4}&-a_{6}&0\\
-a_{3}-\sqrt{3}a_{8}&-a_{1}&a_{4}&0&-a_{2}&a_{5}\\
-a_{1}&a_{3}-\sqrt{3}a_{8}&a_{6}&a_{2}&0&a_{7}\\
a_{4}&a_{6}&0&-a_{5}&-a_{7}&0\\
\end{array} \right)
\vec{m},
\end{equation}
which describes the rotation of a unit vector in a six dimensional space of the
real and imaginary parts of $\vec{m}$ defined by $m_{\mu}=m_{\mu r}+im_{\mu
i}$. In the
above equations, $D=(1+\vert z_{1}\vert^{2}+\vert z_{2}\vert^{2})^{1/2}$
and $i\dot{\phi}=i(V^{*}_{\nu}z_{\nu}+V_{\nu}z^{*}_{\nu})$. The phase $\phi$ is
real and determined only up to a constant factor.
Since the real and imaginary parts of $m_{3}$ are not
independently defined, the geometrical
description of the base manifold for the
$N=3$, $n=1$ decomposition may be
thought of as a point on the surface of a constrained six-dimensional
unit sphere.
The two constraints, namely $\vert m_{1}
\vert^{2}+\vert m_{2}
\vert^{2}+\vert m_{3} \vert^{2}=1$ and the ``phase arbitrariness" of
$\phi$, reduce the 6-dimensional manifold of the three-dimensional
complex vector $\vec{m}$ to a four-dimensional manifold in agreement with there
being only four independent parameters in $\textbf{z}$.The first condition
defines the
base as a vector on an $S^{5}$ sphere while the phase arbitrariness serves as
an additional constraint. The fiber, on the other
hand, is an SU(2) block, evolving as a vector on $S^{2}$
Poincare-like sphere with a phase at each point, and a U(1) block that amounts
to an extra phase.This is presented schematically in
Fig.(\ref{Bloch3_SU3}), as the product of three matrices of the evolution
operator.
\begin{figure}
\caption{The base and fiber for the SU(3) group. The first two factors give the
base manifold,
an $S^{5}
\label{Bloch3_SU3}
\end{figure}
The alternative $N=4$, $n=2$ decomposition in the appendix yields the
equation of motion for
$m_{\mu}=-z_{\mu}/De^{i \phi}$ in equation (\ref{rotation_SU4}). Following
equation (\ref{heffup}) and equation (\ref{heffdown}), we see that for this
case, the two
remaining blocks of the time evolution operator, namely
$\widetilde{U}^{(4-2)}$ and $\widetilde{U}^{(2)}$, can be transformed
into unitary matrices for SU(2). The fiber evolves as vectors on two identical
$S^{2}$
Bloch-like spheres with a mutual phase, whose evolution is coupled to the base
that evolves as a
vector on an $S^{5}$ sphere. This is illustrated in Fig. (\ref{Bloch3_SU4}).
\begin{figure}
\caption{The base and fiber for the SU(3) group via the $N=4$, $n=2$
decomposition. The base again is given by
an $S^{5}
\label{Bloch3_SU4}
\end{figure}
Either decomposition can be used to study various physical processes as will
be discussed in the next section.
\section{\label{discussion}Applications}
It is often desirable to control the time evolution of quantum states to
manipulate an input state into a desirable output state. In \cite{mitra,mitra2},
the
authors considered a Hamiltonian of the form $\mathbf{H}_{0}-\mu
\mathcal{E}(t)$, where $\mathbf{H}_{0}$ is a free-field Hamiltonian and $\mu
\mathcal{E}(t)$ is a control field. To illustrate the ``Hamiltonian encoding''
scheme to control quantum systems, the authors considered a three-level system
and studied stimulated Raman adiabatic passage (STIRAP), an atomic
coherence effect that employs
interference between quantum states to transfer population completely from a
given initial state to a specific final state. This is done through a
``counterintuitive'' pulse sequence. Consider the Hamiltonian
\begin{eqnarray}\label{Ham_mitra}
\mathbf{H}(t)=\left(\begin{array}{ccc}
0&G_{1}(t)&0\\
G_{1}(t)&2 \Delta&G_{2}(t)\\
0&G_{2}(t)&0
\end{array}\right).
\end{eqnarray}
Here $G_{1,2}(t)=2.5$exp$[-(t-t_{1,2})^{2}/\tau^{2}]$ and $\Delta=0.1$. The
initial
population is in the upper state. For $t_{1}=\tau$, $t_{2}=0$ and $\tau=3$, it
is seen that the two empty states are coupled first via $G_{2}(t)$ and then the
levels $\vert 1 \rangle$ and $\vert 2 \rangle$ are coupled through $G_{1}$. The
dynamics of the
populations reveal complete population transfer. A complete solution as per
Section \ref{su3} was constructed for this model and the results are presented
in Fig. \ref{Fig_Mitra} in total agreement with the results of \cite{mitra}.
\begin{figure}
\caption{Population $P_{1j}
\label{Fig_Mitra}
\end{figure}
Quantum control can also be achieved by understanding the nature of tunneling.
The famous Landau-Zener formula \cite{LZ,LZ2,LZ3} predicts the transition
probability of
the ground state of a two-level system when the energy levels adiabatically
undergo a crossing. The study of level crossings has since been extended to
multi-level systems. For example, in \cite{agarwal}, the authors considered a
three-level atom to study population trapping by manipulating the phase acquired
as a three-level system evolves under the influence of frequency modulated
fields \cite{agarwal2}. Such a frequency modulated field is given by
\begin{eqnarray}
\mathbf{E}(t)=
\mathbf{E}_{1}e^{-i[\omega_{1}t+\varphi_{1}(t)]}+\mathbf{E}_{2}e^{-i[\omega_{2}
t+\varphi_{2}(t)]}+c.c.\\
\varphi_{i}(t)=M_{i}\sin{\Omega_{i}t}.
\end{eqnarray}
Here, $c.c.$ stands for complex conjugation. The phase $\varphi_{i}(t)$ in the
exponent can be written in terms of Bessel functions as \cite{Abramowitz}
\begin{equation}
e^{M_{j}\sin{\Omega_{j}t}}=\sum^{\infty}_{k=-\infty}J_{k}(M_{j})e^{ik\Omega_{j}
t}.
\end{equation}
For large values of $\Omega_{j}$, the leading contribution for slow time
scales would come from $J_{0}(M_{j})$. Hence, for large $\Omega_{j}$, the
interaction Hamiltonian can be written as
\begin{eqnarray}
\mathbf{H}_{int}(t)=
-\mathbf{d}.(\mathbf{E}_{1}J_{0}(M_{1})+\mathbf{E}_{2}J_{0}(M_{2}).
\end{eqnarray}
Hence, for values of $M_{1,2}$ that are zeros of the zeroth-order Bessel
functions, the interaction Hamiltonian is zero and population trapping is
observed.
Under this assumption, consider the full Hamiltonian under the rotating wave
approximation,
\begin{eqnarray*}
\mathbf{H}(t)=\left(\begin{array}{ccc}
E_{1}(t)&G_{1}(t)&0\\
G^{*}_{1}(t)&0&G_{2}(t)\\
0&G^{*}_{2}(t)&E_{3}(t)
\end{array}\right).
\end{eqnarray*}
Here, $E_{1}(t)=\Delta_{1}-M_{1}\Omega_{1}\cos(\Omega_{1} t+\theta)$ and
$E_{3}(t)=-\Delta_{2}+M_{2}\Omega_{2}\cos(\Omega_{2}t)$. Results are
presented in Fig. \ref{Fig_Agarwal}, and for the parameter
values $\Omega_{1,2}=1$, $\Delta_{1}=-\Delta_{2}=10$, $\theta=0$ and
$G_{1,2}=6$,
demonstrate the phenomenon of population localization discussed in
\cite{agarwal}.
\begin{figure}
\caption{(a) For $M_{1,2}
\label{Fig_Agarwal}
\end{figure}
As a final illustration of the unitary integration technique applied to
three-level systems, let us consider the example discussed in \cite{kancheva}.
Here, a three-level system is subject to strong fields and the correlation
between the scattered light spectrum and the atom dynamics is discussed. The
authors consider the Hamiltonian
\begin{eqnarray}\label{Ham_kancheva}
\mathbf{H}(t)=\left(\begin{array}{ccc}
0&0&G_{1}(t)\\
0&0&G_{2}(t)\\
G^{*}_{1}(t)&G^{*}_{2}(t)&0
\end{array}\right).
\end{eqnarray}
Here, $G_{1,2}(t)=-V_{1,2}e^{-i \delta t}$. The time evolution of the states
calculated as per our procedure in Section \ref{su3} is plotted in Fig.
\ref{Fig_kancheva} for different values of the parameters.
All of these results agree with those given in \cite{kancheva}. Further
features of the base and fiber will be presented at the end of the next section.
\begin{figure}
\caption{(a) Populations $P_{1j}
\label{Fig_kancheva}
\end{figure}
\section{\label{GP}Geometric phase for SU(3) group}
Many physical systems give rise to a measurable phase that does not depend
directly on the dynamical equations that govern the evolution of the system, but
depends only on the geometry of the path traversed by vectors characterizing the
state of the system. This geometric phase is denoted by $\gamma_{g}$ and is
given by the integral \cite{berry},
\begin{eqnarray}
\gamma_{g}=\int d\mathbf{R}\;.\langle n(\mathbf{R}(t)) \vert
i\nabla_{\mathbf{R}} \vert n(\mathbf{R}(t)) \rangle,
\end{eqnarray}
where the state evolution is governed by a set of internal coordinates that
parameterize the Hamiltonian $\mathbf{R}(t)$, and $\nabla_{\mathbf{R}}$ is the
gradient in the space of these internal coordinates. This phase has been
generalized to non-cyclic non-adiabatic evolution of quantum
systems \cite{wilczek,wilczek2,wilczek3}. The purpose of this section is to
present
this phase in terms of coordinates on the Bloch sphere for two-level systems
and extend it to three-level systems.
In two-level systems, the time evolution operator is described by three
parameters as described in Section \ref{intro}. Two of these parameters describe
a point on the Bloch sphere. Traversing closed loops on this Bloch sphere
returns the
quantum system to its initial state as described
by the two parameters on the Bloch sphere but not the third parameter of an
overall phase. Hence,
general closed loops on the Bloch sphere do not correspond to closed loops in
the space of the full unitary operator. This discrepancy in the phase between
the initial and final state corresponds to the geometric phase given above and
amounts to changes along the fiber at each point on the sphere. To
formalize this, consider $U_{1}$, given by equation (\ref{nonunit_u}) as
unitarized
through the matrix $b$ in Section \ref{ui}, which for $N=2$, $n=1$ takes the
form
\begin{eqnarray}
U_{1}=\frac{1}{\sqrt{1+\vert
\mathbf{z}\vert^{2}}}\left(\begin{array}{cc}1&\mathbf{z}\\ -\mathbf{z}^{*}&
1 \end{array} \right).
\end{eqnarray}
By identifying $\cos{\frac{\theta}{2}}=(1+\vert
\mathbf{z}\vert^{2})^{-\frac{1}{2}}$ and
$\sin{\frac{\theta}{2}}e^{-i \epsilon}=-\mathbf{z} (1+\vert
\mathbf{z}\vert^{2})^{-\frac{1}{2}}$, we get the usual description of the base
manifold in
terms of the angles $0\leq\theta<\pi$ and $0\leq\epsilon<2\pi$ that are
associated with the Bloch sphere, namely,
\begin{eqnarray}
U_{1}=\left(\begin{array}{cc}\cos{\frac{\theta}{2}}&-\sin{\frac{\theta}{2}}e^{
-i\epsilon}\\\sin{\frac{\theta}{2}}e^{i\epsilon}&
\cos{\frac{\theta}{2}} \end{array} \right).
\end{eqnarray}
In terms of the parameters $\theta$ and $\epsilon$, the Hamiltonian
$H(t)=-\vec{a}.\vec{\bm{\sigma}}$ is given by
\begin{eqnarray}
H(t)=\left(\begin{array}{cc}-\cos{\theta}&-\sin{\theta}e^{-i\epsilon}\\-\sin{
\theta}e^{i\epsilon}&
\cos{\theta} \end{array} \right).
\end{eqnarray}
equation (\ref{effeqn}) governing the evolution of the fiber $U_{2}$ has two
terms. The first term is evaluated as
\begin{eqnarray}
U^{\dagger}_{1}H(t)U_{1}=\left(\begin{array}{cc}-1&0\\0&
1 \end{array} \right),
\end{eqnarray}
which corresponds to the eigenvalues of the Hamiltonian. To evaluate the second
term, consider the case whereby the vector on the Bloch sphere traverses a
closed path defined by a constant $\theta$. The second term is then given by
\begin{eqnarray}
U^{\dagger}_{1}\frac{\partial{U}_{1}}{\partial(-i\epsilon)}=\left(\begin{array}{
cc}-\sin^{2}{\frac{\theta}{2}}&-\frac{1}{2}\sin{\theta}e^{-i\epsilon}\\-\frac{1}
{
2}\sin{\theta}e^{i\epsilon}&\sin^{2}{\frac{\theta}{2}}
\end{array} \right).
\end{eqnarray}
Integrating $\epsilon$ from 0 to $2\pi$ yields
\begin{eqnarray*}
\int^{2\pi}_{0}{d\epsilon}U^{\dagger}_{1}\frac{\partial{U}_{1}}{
\partial(-i\epsilon)}=\left(\begin{array}{cc}\pi(1-\cos{\theta}
)&0\\0&-\pi(1-\cos
{\theta})
\end{array} \right),
\end{eqnarray*}
which is the correct formula for the geometric phase of a two-level system
\cite{berry}.
To extend this analysis to three-level systems, we consider the $N=3$, $n=1$
decomposition. The matrix $U_{1}=\widetilde{U}_{1}.b$ is now given by
\begin{eqnarray}
U_{1}=\left(\begin{array}{cc}I^{(2)}-\frac{1}{D(D+1)}\mathbf{z}\mathbf{z}^{
\dagger}&\frac{\mathbf{z}}{D}\\-\frac{\mathbf{z}^{\dagger}}{D}&
\frac{1}{D} \end{array} \right),
\end{eqnarray}
where $\mathbf{z}$ is a complex column vector $(z_{1},z_{2})^{T}$ and
$D=\sqrt{1+\vert\mathbf{z}\vert^{2}}$. Care has to be taken in assigning
angles to elements of this matrix such that the transformation satisfies two
conditions: the $U_{1}$ matrix should not depend on $\phi$ and the
transformation must be
commensurate with the definition of $\vec{m}$. To this effect, we transform
$\mathbf{z}$ into polar coordinates:
$z_{1}=-\tan{\frac{\theta_{1}}{2}}\cos{\frac{\theta_{2}}{2}}e^{i\epsilon_{1}}$,
$z_{2}=-\tan{\frac{\theta_{1}}{2}}\sin{\frac{\theta_{2}}{2}}e^{i\epsilon_{2}}$.
These transformation equations imply that
$D=\sqrt{1+\vert\mathbf{z}\vert^{2}}=\sec{\frac{\theta_{1}}{2}}$,
$m_{1}=\sin{\frac{\theta_{1}}{2}}\cos{\frac{\theta_{2}}{2}}e^{i(\epsilon_{1}
-\phi)}$,
$m_{2}=\sin{\frac{\theta_{1}}{2}}\sin{\frac{\theta_{2}}{2}}e^{i(\epsilon_{2}
-\phi)}$ and $m_{3}=\cos{\frac{\theta_{2}}{2}}e^{-i\phi}$. The $U_{1}$
matrix is given by
\begin{eqnarray}
\fl
U_{1}=\left(\begin{array}{ccc}
1-2\sin^{2}{\frac{\theta_{1}}{4}}\cos^{2}{\frac{\theta_{2}}{2}}
&-\sin^{2}{\frac{\theta_{1}}{4}}\sin{\theta_{2}}e^{i(\epsilon_{1}-\epsilon_{2})}
&-\sin{\frac{\theta_{1}}{2}}\cos{\frac{\theta_{2}}{2}}e^{i\epsilon_{1}} \\
-\sin^{2}{\frac{\theta_{1}}{4}}\sin{\theta_{2}}e^{-i(\epsilon_{1}-\epsilon_{2})}
&1-2\sin^{2}{\frac{\theta_{1}}{4}}\sin^{2}{\frac{\theta_{2}}{2}}&
-\sin{\frac{\theta_{1}}{2}}\sin{\frac{\theta_{2}}{2}}e^{i\epsilon_{2}}\\
\sin{\frac{\theta_{1}}{2}}\cos{\frac{\theta_{2}}{2}}e^{-i\epsilon_{1}}&\sin{
\frac{\theta_{1}}{2}}\sin{\frac{\theta_{2}}{2}}e^{-i\epsilon_{2}}&\cos{\frac{
\theta_{1}}{2}}\\
\end{array}\right).
\end{eqnarray}
In the above equation, the range on the angles $0\leq\theta_{i}<\pi$ and
$0\leq\epsilon_{i}<2\pi$ are chosen so that the absolute value of each
element of the time-evolution operator is positive \cite{aravind}. Hence $U_{1}$
can be represented as two vectors on a sphere, at angles
$(\theta_{1},\epsilon_{1})$ and $(\theta_{2},\epsilon_{2})$ respectively. This
is represented in Fig. (\ref{SU3_GP_Blochfig}).
\begin{figure}
\caption{The base manifold $U_{1}
\label{SU3_GP_Blochfig}
\end{figure}
Since the columns of a unitary operator correspond to normalized eigenvectors,
we can consider the last column of the matrix above,
$\vert\psi\rangle=(-\sin{\frac{\theta_{1}}{2}}\cos{\frac{\theta_{2}}{2}}e^{
i\epsilon_{1}},-\sin{\frac{\theta_{1}}{2}}\sin{\frac{\theta_{2}}{2}}e^{
i\epsilon_{2}},\cos{\frac{\theta_{1}}{2}})^{T}$, and evaluate the so-called
connection 1-form given by \cite{arno bohms book}
\begin{eqnarray}
\mathcal{A}=-i\langle\psi\vert d\vert\psi\rangle.
\end{eqnarray}
The Abelian geometric phase, given by $\gamma_{g}=\int{\mathcal{A}}$ is
evaluated to be
\begin{eqnarray}
\gamma_{g}=-\frac{1}{2}\int{\sin^{2}{\frac{\theta_{1}}{2}}\left((d\epsilon_{1}
+d\epsilon_{2})+\cos{\theta_{2}}(d\epsilon_{1}-d\epsilon_{2})\right)}.
\end{eqnarray}
If the various angles are relabelled $\epsilon_{1}\rightarrow-\gamma-\alpha$,
$\epsilon_{2}\rightarrow-\gamma+\alpha$, $\theta_{1}\rightarrow2\theta$ and
$\theta_{2}\rightarrow2\beta$, the formula above agrees with
\cite{byrd} and \cite{aravind}.
The time-evolution operator above can now be used as in the case of SU(2) to
evaluate the dynamic contribution $\int U^{\dagger}_{1}H(t)U_{1}$ and the
geometric contribution to the time evolution operator which is given by $-i\int
U^{\dagger}_{1}dU_{1}$, where $dU_{1}=\frac{dU_{1}}{d\theta_{i}}
d\theta_{i}+\frac{dU_{1}}{d\epsilon_{i}}d\epsilon_{i}$, $i=1,2$.
This description of the base manifold in terms of $(\theta_{i},\epsilon_{i})$
can now be used to describe the dynamics of various physical processes. Fig.
(\ref{Kancheva2vec}) represents the base manifold corresponding to the results
in Fig. (\ref{Fig_kancheva}). $(\theta_{1},\epsilon_{1})$ depend on all the
parameters that define the
system while $(\theta_{2},\epsilon_{2})$ depend only on the ratio $V_{1}/V_{2}$.
Also note that the maximum value of $\epsilon_{2}$, corresponding to the
maximum latitude traversed by the black curve, is inversely proportional to
$\delta$. Such observations can be used to control the dynamics of this
system.
\begin{figure}
\caption{ The base manifold corresponding to the results in
Fig. (\ref{Fig_kancheva}
\label{Kancheva2vec}
\end{figure}
\section{\label{conclusions}Conclusions}
The ability to decouple the time dependence of operator equations
from the non-commuting nature of the operators is the central
feature of unitary integration and also characterizes the Bloch sphere
representation for the evolution of a single spin.
By doing so, the quantum mechanical evolution is rendered a
``classical" picture of a rotating unit vector. For a two-level atom,
the Bloch sphere representation along with a phase completely
determines the time evolution operator. In this paper, we have
extended this program to deal with the time evolution operator belonging to
the SU(3) group. This complements the work in \cite{newui} for SU(4)
Hamiltonians of two qubit systems. We have also extended the
analysis of geometric phase to three-level systems by providing an explicit
coordinate representation for the SU(3) time evolution operator.
\appendix
\section{\label{appendix}Alternative derivations for a general SU(3)
Hamiltonian.}
Consider a three-level Hamiltonian written in terms of the Gell-Mann
matrices \cite{georgi} as $H(t)=\sum_{i=1}^{8}a_{i}\bm{\lambda}_{i}$. To
exploit the fact that this Hamiltonian is a subgroup
of four-level problems, it is represented in terms of the O
matrices \cite{restui} as
\begin{eqnarray}\label{SU(3) derivation}
\fl
2\frac{a_{8}}{\sqrt{3}}\mathbf{O}_{2}+(a_{3}-\frac{a_{8}}{\sqrt{3}})\mathbf
{O}_{3}+(2a_{3}+2\frac{a_{8}}{\sqrt{3}})\mathbf{O}_{4}
+a_{4}\mathbf{O}_{5}+a_{5}\mathbf{O}_{6}+2a_{4}\mathbf{O}_{7}+2a_{5}\mathbf{O}_{
8}+\nonumber\\
\fl\qquad
a_{1}\mathbf{O}_{9}+
a_{2}\mathbf{O}_{10}+2a_{1}\mathbf{O}_{11}+2a_{2}\mathbf{O}_{12}+2a_{6}\mathbf{O
}_{13}+2a_{6}\mathbf{O}_{14}-2a_{7}\mathbf{O}_{15}+2a_{7}\mathbf{O}_{16}.
\end{eqnarray}
This embeds the Hamiltonian $H(t)=\sum_{i}a_{i}\bm{\lambda}_{i}$ as a
4$\times$4 matrix with zeros along the last row and column. In such a
representation,
the various entries of the Hamiltonian equation (\ref{Hamiltonian}) are given
by
\begin{eqnarray}
H^{(4-2)}=\frac{1}{\sqrt{3}}a_{8}\mathbf{I}^{(2)}+a_{1}\bm{\sigma}_{1}+a_{2}
\bm{\sigma}_{2}+a_{3}\bm{\sigma}_{3},\\
H^{(2)}=-\frac{1}{\sqrt{3}}a_{8}\mathbf{I}^{(2)}-\frac{1}{\sqrt{3}}a_{8}\bm{
\sigma}_{1},\\
\mathbf{V}=\frac{1}{2}(a_{4}-ia_{5})\mathbf{I}^{(2)}+\frac{1}{2}(a_{6}-ia_{7}
)\bm{\sigma}_{1}\\\nonumber
-i\frac{1}{2}(a_{6}-ia_{7})\bm{\sigma}_{2}+\frac{1}{2}(a_{4}-ia_{5})\bm{
\sigma}_{3}.
\end{eqnarray}
Writing $\mathbf{z}$ in the standard Clifford basis as
$\mathbf{z}=\frac{1}{2}z_{4}\bm{I}^{(2)}-\frac{i}{2}\sum_{i}z_{i}\bm{\sigma}_{i}
$, it follows from equation~(\ref{zdot}) that $z_{1}=iz_{2}$ and $z_{3}=iz_{4}$
and
the equation reduces precisely to equation (\ref{su3_z_eqn}).
The geometry described in Section \ref{su3} can thus be derived from either of
these decompositions of the time evolution operator.
The SU(3) subgroup in equation (\ref{SU(3) derivation}) is one among many SU(3)
subgroups embedded in SU(4). Another choice corresponds to the
Dzyaloshinskii-Moriya interaction Hamiltonian \cite{dz1,dz2} and is also of
interest because the 4$\times$4 matrices now do not have a trivial row and
column of zeros. In the
two-spin basis, this Hamiltonian is given by
\begin{eqnarray}\label{expandedDV}
\fl
H(t)=\sum_{i}c_{i}\mathbf{O}_{i}=a_{1}(\mathbf{O}_{2}+\mathbf{O}_{3})+2
a_{2}(\mathbf{O}_{15}+\mathbf{O}_{16})+2 a_{3}(\mathbf{O}_{14}\--
\mathbf{O}_{13})+2 a_{4}(\mathbf{O}_{7}+\mathbf{O}_{11})\nonumber\\
\fl\qquad
+a_{5}(\mathbf{O}_{6}+\mathbf{O}_{10})+a_{6}(\mathbf{O}_{5}+\mathbf{O}_{9})+2
a_{7}(\mathbf{O}_{8}+\mathbf{O}_{12})+\frac{2 a_{8}}{\sqrt{3}}(2
\mathbf{O}_{4}\--\mathbf{O}_{13}\--\mathbf{O}_{14}).
\end{eqnarray}
The correspondence between the coefficients in terms of $\mathbf{O}$ and in
terms of the $\bm{\lambda}$ matrices is : $c_{1}=0$, $c_{2}=a_{1}$,
$c_{3}=a_{1}$,
$c_{4}=4a_{8}/\sqrt{3}$, $c_{5}=a_{6}$, $c_{6}=a_{5}$, $c_{7}=2a_{4}$,
$c_{8}=2a_{7}$, $c_{9}=a_{6}$, $c_{10}=a_{5}$, $c_{11}=2a_{4}$,
$c_{12}=2a_{7}$, $c_{13}=-2a_{3}-2a_{8}/\sqrt{3}$,
$c_{14}=2a_{3}-2a_{8}/\sqrt{3}$, $c_{15}=2a_{2}$ and $c_{16}=2a_{2}$. Relabeling
of the states $1\rightarrow2$, $2\rightarrow3$,
$3\rightarrow4$ and $4\rightarrow1$ expresses the Hamiltonian as
\begin{eqnarray}
H^{(4-2)}=\frac{1}{\sqrt{3}}a_{8}\mathbf{I}^{(2)}-a_{3}\bm{\sigma}_{1}-a_{2}
\bm{\sigma}_{2}-a_{1}\bm{\sigma}_{3},\\
H^{(2)}=-\frac{1}{\sqrt{3}}a_{8}\mathbf{I}^{(2)}-\frac{1}{\sqrt{3}}a_{8}\bm{
\sigma}_{1},\\
\mathbf{V}=\frac{1}{2}(a_{6}-ia_{7})\mathbf{I}^{(2)}+\frac{1}{2}(a_{6}-ia_{7}
)\bm{\sigma}_{1}\\\nonumber
-\frac{1}{2}(a_{5}+ia_{4})\bm{\sigma}_{2}-\frac{1}{2}(a_{4}-ia_{5})\bm{
\sigma}_{3}.
\end{eqnarray}
If $\mathbf{z}$ is written in terms of the standard Clifford basis
$(\hat{\mathbf{I}},-i\vec{\bm{\sigma}})$ as
$\mathbf{z}=\frac{1}{2}z_{4}\mathbf{I}^{(2)}-\frac{i}{2}\sum_{i=1}^{3}z_{i}\bm{
\sigma}_{i} $ ,
it follows from equation (\ref{zdot}) that
$z_{1}=i z_{4}$ and
$z_{2}=i z_{3}$. This is consistent with the
parameter count that since the inhomogeneity $\mathbf{V}$ has only
two free complex parameters (namely $V_{1}=a_{6}-ia_{7}$ and
$V_{2}=a_{4}-ia_{5}$), the complex $\mathbf{z}$ matrix
should be composed only of two independent complex parameters,
$z_{1}$ and $z_{2}$. With the above analysis, equation~(\ref{zdot}) becomes for
the
pair of complex numbers
\begin{equation}
\frac{1}{2}\dot{z}_{\mu}=\frac{1}{2}X_{\mu}-iF_{\mu\nu}z_{\nu}+2G_{\nu}z_{\nu}z_
{ \mu}
;\;\mu,\nu=1,2.
\end{equation}
Here $X=(V_{1}/2,-i V_{2}/2)$,
$G=(2V_{1}^{*},2i V_{2}^{*})$ and
\begin{equation*}
-i F=\left(\begin{array}{cc}
i a_{3}-\sqrt{3} i a_{8}&a_{1}+i a_{2}\\
-a_{1}+i a_{2}&-i a_{3}-\sqrt{3}i a_{8}\\
\end{array} \right).
\end{equation*}
Paralleling the technique employed to solve an
SO(5) Hamiltonian in \cite{newui,newui2}, we transform $\mathbf{z}$ into a
complex
vector $\vec{m}$: $m_{\mu}=\frac{-2z_{\mu}e^{i\phi}}{D}$ and
$m_{3}=\frac{e^{i\phi}}{D}$ such that $\vert m_{1} \vert^{2}+\vert
m_{2} \vert^{2}+\vert m_{3} \vert^{2}=1$,with
$D=(1+4(\vert z_{1} \vert^{2}+\vert z_{2}
\vert^{2}))^{1/2}$. This leads to the new set of
evolution equations
\begin{small}
\begin{equation}
\dot{\vec{m}}=\left(\begin{array}{ccc}
i a_{3}-\sqrt{3} i a_{8}&a_{1}+i a_{2}&-a_{6}+i a_{7}\\
-a_{1}+i a_{2}&-i a_{3}-\sqrt{3}i a_{8}&a_{5}+i a_{4}\\
a_{6}+i a_{7}&-a_{5}+i a_{4}&0
\end{array} \right)\vec{m}.
\end{equation}
\end{small}
This can be written as an equation describing the rotation of the real and
imaginary components of the vector
$\vec{m}=(m_{1r},m_{2r},m_{3r},m_{1i},m_{2i},m_{3i})^{T}$,
\begin{eqnarray}\label{rotation_SU4}
\fl
\dot{\vec{m}}=\left(\begin{array}{cccccc}
0&a_{1}&-a_{6}&-a_{3}+\sqrt{3}a_{8}&-a_{2}&-a_{7}\\
-a_{1}&0&a_{5}&-a_{2}&a_{3}+\sqrt{3}a_{8}&-a_{4}\\
a_{6}&-a_{5}&0&-a_{7}&-a_{4}&0\\
a_{3}-\sqrt{3}a_{8}&a_{2}&a_{7}&0&a_{1}&-a_{6}\\
a_{2}&-a_{3}-\sqrt{3}a_{8}&a_{4}&-a_{1}&0&a_{5}\\
a_{7}&a_{4}&0&a_{6}&-a_{5}&0\\
\end{array} \right)\vec{m}.
\end{eqnarray}
Here, the coefficients $c_{i}$ are written in terms of the
coefficients $a_{i}$, whose correspondence was given earlier in
this section. Also note that $m_{\mu}=m_{\mu
r}+im_{\mu i}$, $D=(1+\vert z_{1}\vert^{2}+\vert
z_{2}\vert^{2})^{\frac{1}{2}}$ and
$\dot{\phi}=(V^{*}_{\nu}z_{\nu}+V_{\nu}z^{*}_{\nu})$.
Simplifying this leads to the equation
$i\dot{\phi}=-2(X_{\mu}z_{\mu}^{*}-X_{\mu}^{*}z_{\mu})$
for the evolution of $\phi$ which is clearly real but determined only to within
a constant.
A little algebra yields for the effective
Hamiltonian given by equation~(\ref{heffup}),
\begin{small}
\begin{eqnarray*}
H^{(4-2)}-\frac{1}{(D+1)}(\mathbf{z}\mathbf{V}^{\dagger}+\mathbf{V}\mathbf{z}^{
\dagger})-
\frac{1}{2(D+1)^{2}}(\mathbf{z}\mathbf{V}^{\dagger}\mathbf{z}\mathbf{z}^{\dagger
} \- +\mathbf{z}\mathbf{z}^{\dagger}\mathbf{V}\mathbf{z}^{\dagger}),
\end{eqnarray*}
\end{small}
and for the effective Hamiltonian given by equation~(\ref{heffdown}), the
expression
$H^{(2)}+(\mathbf{z}^{\dagger}\mathbf{V}+\mathbf{V}^{\dagger}\mathbf{z})/2.$
Another representation of the SU(3) subgroup of SU(4) Hamiltonians is given by
the so called ``Pl\"{u}cker coordinate'' representation of the SU(4)
group discussed
in
\cite{newui,newui2}. For an arbitrary SU(4) matrix, the Pl\"{u}cker coordinates
are
defined as a set of six parameters $(P_{12},P_{13},P_{14},P_{23},P_{24},P_{34})$
such that $P_{12}P_{34}-P_{13}P_{24}+P_{14}P_{23}=0$ and $\sum \vert
P_{ij}\vert^{2}=1$. They can be written in terms of the unit vector $\vec{m}$
and are given by
\begin{equation}
\left(\begin{array}{c}
P_{12}\\
P_{13}\\
P_{14}\\
P_{23}\\
P_{24}\\
P_{34}\\
\end{array} \right)=\frac{1}{2}
\left(\begin{array}{c}
im_{6}-m_{5}\\
im_{1}+m_{2}\\
-im_{3}+m_{4}\\
-im_{3}-m_{4}\\
-im_{1}+m_{2}\\
im_{6}+m_{5}\\
\end{array} \right).
\end{equation}
The linear equation of motion for $\vec{m}$ translates into an evolution
equation for $\mathbf{P}=(P_{12},-P_{13},P_{14},P_{23},P_{24},P_{34})$ of the
form $i\dot{\mathbf{P}}=\mathbf{H}_{P}\mathbf{P}$. Here, $\mathbf{H}_{P}$ is
given by
\begin{equation}
\mathbf{H}_{P}=\left(\begin{array}{cc}
\mathbf{H}_{P1}&\mathbf{V}_{P}\\
\mathbf{V}^{\dagger}_{P}&\mathbf{H}_{P2}\\
\end{array}\right),
\end{equation}
where
\begin{eqnarray*}
\fl
\mathbf{H}_{P1}=\left(\begin{array}{ccc}
2a_{8}/\sqrt{3}&a_{64-}+ia_{75-}&a_{64-}+ia_{75-}\\
a_{64-}-ia_{75-}&-a_{1}&a_{8}/\sqrt{3}\\
a_{64-}-ia_{75-}&a_{8}/\sqrt{3}&-a_{1}
\end{array}
\right),\\
\fl
\mathbf{H}_{P2}=\left(\begin{array}{ccc}
a_{1}&-a_{8}/\sqrt{3}&-a_{64-}-ia_{75-}\\
-a_{8}/\sqrt{3}&a_{1}&-a_{64-}-ia_{75-}\\
-a_{64-}+ia_{75-}&-a_{64-}+ia_{75-}&-2a_{8}/\sqrt{3}
\end{array}
\right),\\
\fl
\mathbf{V}_{P}=\left(\begin{array}{ccc}
-a_{64+}-ia_{75+}&a_{64+}+ia_{75+}&0\\
a_{32-}&0&-a_{64+}-ia_{75+}\\
0&-a_{32-}&a_{64+}-ia_{75+}
\end{array}
\right).
\end{eqnarray*}
In the above equation, $a_{ij\pm}$ denotes $a_{i}\pm a_{j}$.
\section*{References}
\end{document} | math | 48,556 |
\begin{document}
\title{Factorizations of Analytic Self-Maps of the Upper Half-Plane}
\author{Hari Bercovici and Dan Timotin}
\subjclass[2010]{Primary:30H15}
\thanks{HB was supported in part by grants from the National Science Foundation.
DT was supported in part by a grant of the Romanian National Authority for Scientific
Research, CNCS--UEFISCDI, project number PN-II-ID-PCE-2011-3-0119.}
\address{HB: Department of Mathematics, Indiana University, Bloomington, IN
47405, USA}
\email{[email protected]}
\address{DT: Simion Stoilow Institute of Mathematics of the Romanian Academy,
PO Box 1-764, Bucharest 014700, Romania}
\email{[email protected]}
\begin{abstract}
We extend a factorization due to Kre\u\i n to arbitrary analytic
functions from the upper half-plane to itself. The factorization represents
every such function as a product of fractional linear factors times
a function which, generally, has fewer zeros and singularities than
the original one. The reult is used to construct functions with given zeros and poles on the real line.
\end{abstract}
\maketitle
\section{Introduction}
Denote by $\mathbb{C}^{+}=\{x+iy:x\in\mathbb{R},y>0\}$ the upper
half of the complex plane. We consider the multiplicative structure
of the set $\mathcal{H}$ of analytic functions $f\colon\mathbb{C}^{+}\to\overline{\mathbb{C}^{+}}$.
We start with a classical result of M.
G. Kre\u\i n \cite[Theorem 27.2.1]{levin}. Assume that $k\in\mathcal{H}$
is the restriction of a meromorphic function defined on $\mathbb{C}$
with the property that $k(\overline{z})=\overline{k(z)}$ and, in
addition, $k$ has infinitely many positive and infinitely many negative
zeros. Denote by $\{a_{n})_{n=1}^{\infty}$ the zeros of $k$, all
of which are real and simple, and by $\{b_{n}\}_{n=1}^{\infty}$ all
the poles of $k$, all of which are also real and simple, arranged
so that the interval $(b_{n},a_{n})$ contains no zeros or poles.
Then Kre\u\i n proved that there exist positive constants $\{c_{n}\}_{n=1}^{\infty}$
such that
\begin{equation}
k(z)=\prod_{n=1}^{\infty}c_{n}\frac{z-a_{n}}{z-b_{n}}.\label{eq:k-product}
\end{equation}
The constants $c_{n}$ are needed for convergence of the product,
and they can be taken to be equal to $b_{n}/a_{n}$ for all but one
or two values of $n$. This result was studied in further detail by
Chalendar, Gorkin, and Partington \cite{chal-et-al}, including the
cases where there are, for instance, only finitely many negative zeros.
Furthermore, \cite{chal-et-al} addresses the more general situation
where $f$ is not meromorphic, but is allowed a finite number of essential
singularities on the real line. However the results are not as complete
in this more general situation.
Our main result states that any function $f\in\mathcal{H}$ can be
written as
\[
f=kg,
\]
where $k$ is a product of the form (\ref{eq:k-product}), and $g\in\mathcal{H}$
has the following additional property: if $I\subset\mathbb{R}$ is
any interval such that $g$ extends continuously to a real-valued
function on $I$, then the extension is positive on $I$. This factorization
is of interest only for functions $f$ which do extend to a real function
on a nonempty open subset $\Omega$ of $\mathbb{R}$. When the complement
of this set $\Omega$ has linear measure equal to zero, the factor
$g$ is a positive constant, and this extends Kre\u\i n's theorem
as well as the factorization results in \cite{chal-et-al}. In particular, we se in Section 4 aunder what conditions we can prescribe the real zeros and poles of a function in $\mathcal{H}$.
Our factorization is closely related to results of Aronszajn and Donoghue
\cite{aron-don}. These authors also consider the multiplicative
structure of $\mathcal{H}$ starting with the observation that $\log f\in\mathcal{H}$
provided that $f\in\mathcal{H}\setminus\{0\}$. A result essentially
equivalent to Kre\u\i n's theorem is stated in \cite[page 331]{aron-don}.
The emphasis in these works is however not on factorization into linear
fractional factors. We also refer to Donoghue's book \cite{donog}
for information about the class $\mathcal{H}$, and Section 2
of \cite{gest-tsek} for a brisk review of the basic results concerning
this class.
\section{Kre\u\i n Products}
We now describe in more detail the products $k$ needed in our results.
Given a proper open interval $J\subset\mathbb{R}\cup\{\infty\}$, there exists a unique conformal automorphism $p_{J}\in\mathcal{H}$
such that $p_{J}$ is negative precisely on $J$, and $|p_{J}(i)|=1$.
Explicitly, if $J=(b,a)$ with $b<a$ is a finite interval, then
\[
p_{J}(z)=\frac{|i-b|}{|i-a|}\cdot\frac{z-a}{z-b}.
\]
If $J=(-\infty,a),$ then
\[
p_{J}(z)=\frac{z-a}{|i-a|}.
\]
If $J$ is the complement of a closed interval $\overline{J'}$, where
$J'$ is of the preceding two types, then
\[
p_{J}=-\frac{1}{p_{J'}}.
\]
We also agree that $p_{\varnothing}(z)=1$ and $p_{\mathbb{R}\cup\{\infty\}}(z)=-1$.
In order to simplify the notation of intervals in $\mathbb R\cup\{\infty\}$, we set
\[
(b,a)=(b,\infty)\cup\{\infty\}\cup(-\infty,a)
\]
when $b>a$.
Consider now an arbitrary open subset $O\subset\mathbb{R}\cup\{\infty\}$,
and write it as a union of pairwise disjoint open intervals
\[
O=\bigcup_{0\le n<N}J_{n},\quad J_n=(b_n,a_n),\quad0\le n<N,
\]
where $N\in\{0,1,\dots,\infty\}$. Denote by $X$ the closure of the set $\{b_n:0\le n<N\}$ in $\mathbb{R}\cup\{\infty\}$. The function
\[
k_{O}(z)=\prod_{0\le n<N}p_{J_{n}}
\]
is called the \emph{Kre\u\i n product }associated with the set $O$.
We collect in the following statement the basic properties of Kre\u\i n products. The proofs are similar to the ones available in the classical case where $\sup_n b_n=+\infty$ and $\inf_n b_n=-\infty$. We sketch these arguments for the reader's convenience.
\begin{prop}
\label{prop:krein-prod-properties}Consider an open set $O=\bigcup_{0\le n<N}(b_n,a_n)\subset\mathbb R\cup\{\infty\}$, where the intervals $(b_n,a_n)$ are pairwise disjoint. Denote by $X$ the closure of the set $\{b_n:0\le n<N\}$ in $\mathbb R\cup\{\infty\}$.
\begin{enumerate}
\item The product $k_O$ converges uniformly on compact subsets of $\mathbb{C}^+$, and it defines a function in $\mathcal H$.
\item The function $k_O$ continues analytically to $(\mathbb{C}\cup\{\infty\})\setminus X$.
\item The function $k_O|( \mathbb{R}\cup\{\infty\})\setminus X$ is real-valued. More precisely, $k_O(x)<0$ for $x\in O$ and $k_O(x)>0$ for $x\in(\mathbb{R}\cup\{\infty\})\setminus\overline{O}$.
\item If $a_n\notin X$ for some $n$, then $a_n$ is a simple zero of $k_O$.
\item If $b_n$ is an isolated point of $X$ for some $n$, then $b_n$ is a simple pole of $k_O$.
\item Let $O_{1},O_{2}\subset\mathbb{R}\cup\{\infty\}$ be two open sets such that the
symmetric difference $O_{1}\triangle O_{2}$ has linear Lebesgue measure
equal to zero. Then $k_{O_{1}}=k_{O_{2}}$.
\item If $\varphi$ is a
conformal automorphism of $\mathbb{C}^{+}$, then $k_{\varphi^{-1}(O)}=ck_{O}\circ\varphi$,
where $c=1/|k_{O}(\varphi(i))|>0$.
\end{enumerate}
\end{prop}\begin{proof}
The convergence in (1) only needs to be discussed when $N=\infty$.
For $z\in\mathbb{C}^{+}$,
the argument $\arg p_{J_{n}}(z)$ is equal to the angle at $z$ subtended
by the interval $(b_n,a_n)$, and therefore the sum of these arguments converges
to a number $\le\pi$. Since $|p_{(b_n,a_n)}|=1$, we conclude that the infinite product $k_O(i)$ converges and $|k_O(i)|=1$.
Convergence
elsewhere in $\mathbb{C}^{+}$ can be deduced via a normal family
argument. Indeed, the partial products $p_{(b_0,a_0)}p_{(b_1,a_1)}\cdots p_{(b_n,a_n)}$
form a normal family in $\mathbb{C}^{+}$, and any limit point $f$
of this family is such that $|f(i)|=1$ and $\arg f(z)$ equals the
angle at $z$ subtended by the set $O$. This determines the
function $f$ completely, hence $f$ is the limit (uniform on compact
subsets of $\mathbb{C}^{+}$) of these partial products.
To verify (2) and (3), we use the alternative formula $k_{O}=e^{v}$, where
\[
v(z)=\int_{O}\frac{1+tz}{t-z}\cdot\frac{dt}{1+t^{2}}.
\]
Indeed, $\Re v(i)=0$, while
\[
\Im v(z)=\Im z\int_{O}\frac{dt}{|t-z|^{2}},\quad z\notin\overline{O}.
\]
When $\Im z>0$, this integral equals precisely the angle at $z$
subtended by $O$. The function $v$ is analytic
on $(\mathbb{C}\cup\{\infty\})\setminus\overline{O}$, and it is real-valued on $(\mathbb{R}\cup\{\infty\})\setminus\overline{O}$. Thus $k_O(x)$ is positive on this last set. The same argument shows that $k_O/p_{(a_n,b_n)}$ can be continued analytically across the interval $(b_n,a_n)$, and the continuation is positive on this interval. We deduce that $k_O$ can also be continued across $(b_n,a_n)$, and the continuation is negative on this interval.
Properties (4) and (5) follow from the fact that $k_O/p_(b_n,a_n)$ is analytic and positive at $a_n$ if $a_n\notin X$, and it is analytic and positive at $b_n$ if $b_n\notin X$.
To prove (6), observe that $\arg k_{O_{1}}(z)=\arg k_{O_{2}}(z)$
for $z\in\mathbb{C}^{+}$. It follows that $k_{O_{1}}/k_{O_{2}}$
is constant, and the constant is 1, as can be seen by evaluating the
quotient at $z=i$.
For (7), observe that $p_{\varphi^{-1}(J)}/p_{J}\circ\varphi$
is a positive constant (depending on $J$) for every interval $J$.
It follows that $k_{\varphi^{-1}(J)}/k_{J}\circ\varphi$ is a positive
constant as well.
\end{proof}
Part (6) of the preceding proposition indicates how a Kre\u\i n
product can be modified to minimize the number of factors. To make this precise, we define the \emph{Lebesgue regularization} of an open set $O$ to consist of
all points $x\in\mbox{\ensuremath{\mathbb{R}}}$ for which there is
$\varepsilon>0$ such that $(x-\varepsilon,x+\varepsilon)\setminus O$
has Lebesgue measure equal to zero. Thus, the Lebesgue regularization of $O$ is the largest open set $O_1\supset O$ such that the Lebesgue measure of $O_1\setminus O$ is equal to zero, and therefore we have $k_O=k_{O_1}$. For instance,
\[
k_{(1,2)\cup(2,3)}=p_{(1,3)}.
\]
More generally, the Lebesgue regularization of a union $\bigcup_{n}(b_n,a_n)$ of subintervals of $(b,a)$ is equal to $(b,a)$ if $a-b=\sum_{n}(a_n-b_n)$. Thus, $k_{\mathbb{R}\setminus C}=-1$ if $C$ denotes the usual Cantor
ternary set.
An open set $O$ is said to be \emph{Lebesgue regular} if it equals its Lebesgue regularization.
\section{The Nevanlinna Representation}
The additive structure of the class $\mathcal{H}$ is described by
the Nevanlinna representation. Given $\alpha\ge0,\beta\in\mathbb{R}$,
and a finite, positive Borel measure $\rho$ on $\mathbb{R}$, the
function
\begin{equation}
f(z)=\alpha z+\beta+\int_{\mathbb{R}}\frac{1+zt}{t-z}\, d\rho(t),\quad z\in\mathbb{C}^{+},\label{eq:nev-formule}
\end{equation}
belongs to $\mathcal{H}$. Conversely, every function $f\in\mathcal{H}$
can be represented under this form. The constants $\alpha,\beta$
and the measure $\rho$ are uniquely determined by the function $f$.
For instance,
\[
\alpha=\lim_{y\to+\infty}\frac{f(iy)}{iy},
\]
while $\rho$ is the weak limit as $\varepsilon\downarrow0$ of the
measures
\[
d\rho_{\varepsilon}(t)=\frac{\Im f(t+i\varepsilon)}{\pi(1+t^{2})}\, dt.
\]
We denote by $\sigma(f)$ the closed support of the measure $\rho$,
to which we add $\infty$ if $\alpha>0$ or the support is unbounded. If the set $\Omega(f)=(\mathbb{R}\cup\{\infty\})\setminus\sigma(f)$
is not empty, then the formula (\ref{eq:nev-formule}) defines an
analytic function in $(\mathbb{C}\cup\{\infty\})\setminus\sigma(f)$
which takes real values on $\Omega(f)$. Conversely, if $J\subset\mathbb{R}$
is an interval such that $f$ can be extended to a continuous function
on $J$, then $J\subset\Omega(f)$. The isolated points of $\sigma(f)$
are simple poles of $f$. Observe that
\[f'(z)=\alpha+\int_{\mathbb{R}}\frac{1+t^2}{(t-z)^2}\,d\rho(t),\]
and this shows that $f$ is an increasing function on any interval in $\mathbb R$ disjoint from $\sigma(f)$. In particular, the zeros of $f$ in such an interval must be simple.
A few simple illustrations will help clarify these notions. The function
$f(z)=z+i$ belongs to $\mathcal{H}$, and it is the restriction to
$\mathbb{C}^{+}$ of an entire function. However, this entire function
is not real-valued at any point of $\mathbb{R}$, and therefore $\sigma(f)=\mathbb{R}\cup\{\infty\}.$
The measure $\rho$ corresponding to this function is the Cauchy distribution
\[
d\rho(t)=\frac{dt}{\pi(1+t^{2})}.
\]
A more interesting example is the function
\[
f(z)=z+\sqrt{z^{2}-1},
\]
where the square root is taken to be positive for $z>1$, for which
we have $\sigma(f)=[-1,1]\cup\{\infty\}$. For this example we have
\[
-\frac{1}{f(z)}=\sqrt{z^{2}-1}-z,
\]
and $\sigma(-1/f)=[-1,1]$. For a Kre\u\i n factor $p_J$, where $J=(b,a)$ is a proper interval, we have $\sigma(p_J)=\{b\}$.
For any function $f\in\mathcal{H}$, the limit
\[
f(t)=\lim_{\varepsilon\downarrow0}f(t+i\varepsilon)
\]
exists for almost every $t\in\mathbb{R}$ (relative to Lebesgue measure).
Moreover, the absolutely continuous part of the measure $\rho$ in
the representation (\ref{eq:nev-formule}) is equal to
\[
\frac{\Im f(t)\, dt}{\pi(1+t^{2})}.
\]
In particular, the measure $\rho$ is singular relative to Lebesgue
measure if and only if $f(x)$ is real for almost every $x\in\mathbb{R}$.
The following result was proved by G. Letac \cite{letac-proc}; the
case of measures with finite support is due to Boole \cite{boole}.
\begin{thm}
\label{thm:measure-preserving}Assume that the function $f\in\mathcal{H}$
is given by
\[
f(z)=z+\beta+\int_{\mathbb{R}}\frac{1+zt}{t-z}\, d\rho(t),\quad z\in\mathbb{C}^{+},
\]
where $\rho$ is singular relative to Lebesgue measure. Then the map
$f$ preserves Lebesgue measure, that is, the Lebesgue
measure of the set
\[
\{x\in\mathbb{R}:f(x)\in(c,d)\}
\]
equals $d-c$ when $-\infty<c<d<\infty$.
\end{thm}
We will actually require an equivalent result, first proved by Hru\v s\v c\"ev
and Vinogradov \cite{hru-vin}. We include the simple derivation from
Theorem \ref{thm:measure-preserving}. Given a finite, positive, measure
$\mu$ on $\mathbb{R}$, singular with respect to Lebesgue measure,
the Cauchy transform
\[
G_{\mu}(z)=\int_{\mathbb{R}}\frac{d\mu(t)}{z-t},\quad z\in\mathbb{C}^{+},
\]
has real limit values
\[
G_{\mu}(x)=\lim_{\varepsilon\downarrow0}G_{\mu}(x+i\varepsilon)
\]
for almost every $x\in\mathbb{R}$. Here is the result of \cite{hru-vin};
the special case of measures with finite support is due again to Boole
\cite{boole}.
\begin{thm}
\label{thm:hruscev-vino}Let $\mu$ be a finite, positive measure
on $\mathbb{R}$, singular with respect to Lebesgue measure. For every
$y>0$, the Lebesgue measure of the sets
\[
\{x\in\mathbb{R}\colon G_{\mu}(x)>y\},\quad\{x\in\mathbb{R}\colon G_{\mu}(x)<-y\}
\]
is equal to $\mu(\mathbb{R})/y$.\end{thm}
\begin{proof}
It suffices to consider the case when $\mu$ is a probability measure.
The function $f(z)=1/G_{\mu}(z)$ is easily seen to belong to $\mathcal{H}$,
and $f(x)$ is real for almost every $x\in\mathbb R$. Moreover, since
\[
\lim_{y\to\infty}iyG_{\mu}(iy)=\mu(\mathbb{R})=1,
\]
we deduce that Theorem \ref{thm:measure-preserving} applies to $f$.
The result follows now immediately because, for instance,
\[
\{x\in\mathbb{R}\colon G_{\mu}(x)>y\}=\{x\in\mathbb{R}:f(x)\in(0,1/y)\}.\qedhere
\]
\end{proof}
We need an easy consequence of this result.
\begin{cor}
\label{cor:unbounded_f}Assume that the function $f\in\mathcal{H}$
is given by \emph{(\ref{eq:nev-formule}),} and the restriction of
the measure $\rho$ to an interval $J$ is nonzero but singular relative
to Lebesgue measure. Then the function $f$ is not essentially
bounded below or above on the interval $J$.\end{cor}
\begin{proof}
Denote by $\mu$ the restriction of the measure $(1+t^{2})\, d\rho(t)$
to the interval $J$. Then the function $f(z)+G_{\mu}(z)$ can be
continued analytically across the interval $J$, thus $f(x)+G_{\mu}(x)$
is bounded on any compact subset of $J$. The corollary follows now
immediately from Theorem \ref{thm:hruscev-vino}.\end{proof}
\begin{cor}
\label{cor:lebesgur-regularity-of_f>0}For every $f\in\mathcal{H}$,
the set
\[
\Gamma(f)=\{x\in\Omega(f):f(x)<0\}
\]
is Lebesgue regular.\end{cor}
\begin{proof}
Assume to the contrary that there exists $x\in\mathbb{R}\setminus\Gamma(f)$
and $\varepsilon>0$ such that the interval $J=(x-\varepsilon,x+\varepsilon)$
is contained almost everywhere in $\Gamma(f)$, but $J\cap\sigma(f)\ne\varnothing.$
If the function $f\in\mathcal{H}$ is given by (\ref{eq:nev-formule}),
it follows that the restriction of $\rho$ to $J$ is nonzero, and
this restriction is singular as well since its support is a closed
set of Lebesgue measure equal to zero. Corollary \ref{cor:unbounded_f}
implies that $f$ is not essentially bounded above on $J$, contrary
to the assumption that $f<0$ almost everywhere on $J$.
\end{proof}
Proposition \ref{prop:krein-prod-properties} describes the sets
$\sigma(f),\Omega(f),$ and $\Gamma(f)$ in
the case of a Kre\u\i n product. We state the result for further use.
\begin{prop}
Consider a Lebesgue-regular open set $O=\bigcup_{0\le n<N}(b_n,a_n)\subset\mathbb R\cup\{\infty\}$, where the intervals $(b_n,a_n)$ are pairwise disjoint. Denote by $X$ the closure of the set $\{b_n:0\le n<N\}$ in $\mathbb R\cup\{\infty\}$. Then
\begin{enumerate}
\item $\sigma(k_O)=X$;
\item $\Gamma(k_O)=O$;
\item the real poles of $k_O$ are precisely the points $b_n$ which are isolated in $X$;
\item a point $a_n$ is in $\sigma(k_O)$ precisely when $a_n\in X$, otherwise $a_n$ is a simple zero.
\end{enumerate}
\end{prop}
\section{Factorization}
We begin by considering factors of the form $p_{J}$.
\begin{lem}
\label{lem:factor_a_single_p_J}Consider a nonzero function $f\in\mathcal{H}$,
and an interval $J\subset\Omega(f)$ such that $f(x)<0$ for all $x\in J$.
Then there exists a function $g\in\mathcal{H}$ such that $f=p_{J}g$.\end{lem}
\begin{proof}
Replacing the function $f$ by $f\circ\varphi$ for some conformal
automorphism of $\mathbb{C}^{+}$, we may assume that $J=(-\infty,0)$,
so that we need to prove that the function $f(z)/z$ belongs to $\mathcal{H}$.
Since
\[
\frac{f(z)}{z}=\lim_{\varepsilon\downarrow0}\frac{f(z-\varepsilon)}{z},\quad z\in\mathbb{C}^{+},
\]
and $\mathcal{H}$ is closed under pointwise limits, we may also assume
that $\sigma(f)\subset[\varepsilon,+\infty]$ for some $\varepsilon>0$,
and $f(0)<0$. Thus $f$ can be written as
\[
f(z)=\alpha z+\beta+\int_{[\varepsilon,+\infty)}\frac{1+zt}{t-z}\, d\rho(t),\quad z\in\mathbb{C}^{+},
\]
for some $\alpha>0,\beta\in\mathbb{R}$, and some finite, positive,
Borel measure $\rho$ on $[\varepsilon,+\infty).$ We have then
\[
f(0)=\beta+\int_{[\varepsilon,+\infty)}\frac{1}{t}\, d\rho(t)<0,
\]
so that
\begin{eqnarray*}
\frac{f(z)}{z} & = & \alpha+\left[\beta+\int_{[\varepsilon,+\infty)}\frac{1}{t}\, d\rho(t)\right]\frac{1}{z}+\int_{[\varepsilon,+\infty)}\left[\frac{1+zt}{z(t-z)}-\frac{1}{zt}\right]\, d\rho(t)\\
& = & \alpha+\left[\beta+\int_{[\varepsilon,+\infty)}\frac{1}{t}\, d\rho(t)\right]\frac{1}{z}+\int_{[\varepsilon,+\infty)}\frac{1+t^{2}}{t(t-z)}\, d\rho(t),
\end{eqnarray*}
and it is clear that this function has nonnegative imaginary part
for $z\in\mathbb{C}^{+}$.
\end{proof}
Our main factorization result follows. Recall that $\Omega(f)=(\mathbb{R}\cup\{\infty\})\setminus\sigma(f)$
and $\Gamma(f)=\{x\in\Omega(f)\colon f(x)<0\}$.
\begin{thm}
\label{thm:main-factorization}For every nonzero function $f\in\mathcal{H}$
there exists $g\in\mathcal{H}$ such that $f=k_{\Gamma(f)}g$. The
function $g$ has the following properties.
\begin{enumerate}
\item $\sigma(g)\subset\sigma(f)$.
\item $g(x)>0$ for every $x\in\Omega(g)$.
\item The set $\Omega(g)$ is Lebesgue regular.
\item $\Omega(f)=\Omega(k_{\Gamma(f)})\cap\Omega(g)$.
\end{enumerate}
\end{thm}
\begin{proof}
Denote by $(J_{n})_{0\le n<N}$ the connected components of $\Gamma(f)$,
so that
\[
k_{\Gamma(f)}=\prod_{0\le n<N}p_{J_{n}}.
\]
We first verify by induction that the function
\[
g_{k}=f/\prod_{0\le n<k}p_{J_{n}}
\]
belongs to $\mathcal{H}$ for every finite $k\le N$. The case $k=0$
is vacuously verified since $g_{0}=f$. The function $g_{k}$ has
the property that $J_{k+1}\subset\Gamma(g_{k})$, and thus the fact
that $g_{k+1}\in\mathcal{H}$ follows by an appplication of Lemma
\ref{lem:factor_a_single_p_J}. If $N$ is finite, we are done showing
that $g\in\mathcal{H}$. If $N$ is infinite, we have $g(z)=\lim_{n\to\infty}g_{n}(z)$
for all $z\in\mathbb{C}^{+}$, and we conclude again that $g\in\mathcal{H}$.
Assume next that $x\in\Omega(f)$. If $f(x)>0$ then $x$ is at positive
distance from $\Gamma(f)$, and therefore the product $k_{\Gamma(f)}$
converges on a neighborhood of $x$, and it takes positive values
at real points close to $x$. (This argument also works for $x=\infty$
with the usual interpretation of the word `close'.) It follows that
$x\in\Omega(g)$. If $f(x)<0$ then $k_{\Gamma(f)}$ is again analytic
in a neighborhood of $x$, and it takes negative values at real points
close to $x$. We conclude again that $x\in\Omega(g)$. If $f(x)=0$,
it follows that, for some $\varepsilon>0$, $(x-\varepsilon,x)\subset\Gamma(f)$
and $(x,x+\varepsilon)\cap\Gamma(f)=\varnothing$. In particular,
$f$ has a simple zero at $x$. In this case $k_{\Gamma(f)}$ also
has a simple zero at $x$, and the ratio $g=f/k_{\Gamma(f)}$ is positive
in $(x-\varepsilon,x+\varepsilon)$ and analytic near $x$. In particular,
$x\in\Omega(g)$. This verifies property (1).
The above argument also shows that $g(x)>0$ for every $x\in\Omega(f)$.
Assume now that $x\in\Omega(g)\setminus\Omega(f)$ and $g(x)<0$.
Since $k_{\Gamma(f)}=f/g$, we deduce that we have $x\in\sigma(k_{\Gamma(f)})$,
and therefore $x$ is an accumulation point of a sequence of endpoints
of some intervals $J_{n}$. Since $g$ is positive on each $J_{n}$,
we deduce $g(x)\ge0,$ a contradiction. Thus we must have $g(x)\ge0$
for $x\in\Omega(g)$. However, when a function $g$ in $\mathcal{H}$
vanishes at some point $x\in\Omega(g)$, the function $g$ must change
sign in a neighborhood of that point. This proves (2).
Property (3) follows from Corollary \ref{cor:lebesgur-regularity-of_f>0}
applied to the function $-1/g$.
The inclusion $\Omega(f)\supset\Omega(k_{\Gamma(f)})\cap\Omega(g)$ is obvious. Conversely, observe that any point $a\in\Omega(f)$ such that $f(a)=0$ is a simple zero of $f$ and of $k_{\Gamma(f)}$. Indeed, there is an interval $(b,a)\subset\Gamma(f)$, and $a$ is isolated in $\partial\Gamma(f)$. Moreover, $k_{\Gamma(f)}$ is analytic and real in a neighborhood of $a$. It follows that the quotient $g=f/k_{\Gamma(f)}$ is analytic and real on a neighborhood of $a$. It is also clear that $f$ and $k_{\Gamma(f)}$ are analytic and nonzero in the neighborhood of any point $a\in\Omega(f)$ such that $f(a)\ne0$. Thus both $k_{\Gamma(f)}$ and $g$ are analytic and real on $\Omega(f)$, thus verifying the opposite inclusion.
\end{proof}
The preceding result is most effective when $\sigma(f)$ is small.
\begin{cor}
\label{cor:sigma-of-measure-zero}Assume that $f\in\mathcal{H}$ and
$\sigma(f)$ has Lebesgue measure equal to zero. Then we have $f=ck_{\Gamma(f)}$
for some positive constant $c$.\end{cor}
\begin{proof}
Consider the factorization $f=k_{\Gamma(f)}g$ provided by the preceding
theorem. The set $\Omega(g)$ is Lebesgue regular, and hence $\Omega(g)=\mathbb{R}\cup\{\infty\}$
because its complement has measure zero. The Nevanlinna representation
of $g$ shows then that $g$ is a constant function, and the constant
is positive by Theorem \ref{thm:main-factorization}(2).
\end{proof}
The preceding corollary recovers Kre\u\i n's original factorization
result, as well as the extensions considered in \cite{chal-et-al}. Note that
the measure $\rho$ in the Nevanlinna representations of the functions
considered in \cite{chal-et-al} is a discrete measure whose support has only
finitely many accumulation points. Our corollary covers functions
for which the support of $\rho$ is an arbitrary closed set of Lebesgue
measure zero, for instance the ternary Cantor set.
\section{Functions $g$ Positive on $\Omega(g)$}
In this section we analyze more carefully the factor $g$ in the decomposition
$f=k_{\Gamma(f)}g$. The characteristic properties of these functions
are that $\Omega(g)$ is Lebesgue regular, and $g(x)>0$ for $x\in\Omega(g)$.
Such a function can then be written as $g=e^{h}$ for some function
$f\in\mathcal{H}$ such that
\[
0<\Im h(z)<\pi
\]
for all $z\in\mathbb{C}^{+}$. Moreover, the function $h$ can be
continued analytically across $\Omega(g)$, and $h(x)\in\mathbb{R}$
for $x\in\Omega(g)$. Conversely, $g$ can be continued analytically
across $\Omega(h)$, and $g(x)>0$ for $x\in\Omega(h)$. We conclude
that $\Omega(h)=\Omega(g)$ and $\sigma(h)=\sigma(g)$. These facts
imply that the Nevanlinna representation of the function $h$ is of
the form
\[
h(z)=\gamma+\int_{\sigma(g)}\frac{1+zt}{t-z}\cdot\frac{\psi(t)}{1+t^{2}}\, dt,\quad z\in\mathbb{C}^{+},
\]
where $\gamma\in\mathbb R$, $\psi:\sigma(g)\to(0,1]$ is a measurable function, and there is no open interval $J$ such that $\psi(t)=1$ almost everywhere on $J$. The fact that $\sigma(h)=\sigma(g)$ amounts
to saying that the support of the measure $\psi(t)\, dt$ is equal
to $\sigma(g)$ or, equivalently since $\Omega(g)$ is Lebesgue regular,
$\psi(t)>0$ for almost every $t\in\sigma(g)$.
Conversely, assume that $\sigma\subset\mathbb{R}$ is a closed set
such that $\mathbb{R}\setminus\sigma$ is Lebesgue regular, and $\varphi:\sigma\to(0,1]$
is measurable. We can then define a function $v\in\mathcal{H}$ by
setting
\[
v(z)=\int_{\sigma}\frac{1+zt}{t-z}\cdot\frac{\varphi(t)}{1+t^{2}}\, dt,\quad z\in\mathbb{C}^{+}.
\]
This function satisfies
\[
\Im v(z)=\Im z\int_{\sigma}\frac{\varphi(t)}{|t-z|^{2}}\, dt\le\Im z\int_{\sigma}\frac{dt}{|t-z|^{2}}\le\Im z\int_{-\infty}^{\infty}\frac{dt}{|t-z|^{2}}=\pi.
\]
Moreover, $\Gamma(g)=\varnothing$ provided that $\varphi$ is not equal to $1$ almost everywhere on any open interval.
We summarize these observations below.
\begin{thm}
\label{thm:additional-info-about-g}Assume that $f\in\mathcal{H}$.
There exist a constant $\gamma\in\mathbb{R}$ and a measurable function
$\psi:\sigma(f)\to[0,1]$ such that
\[
f(z)=k_{\Gamma(f)}(z)e^{h(z)},\quad z\in\mathbb{C}^{+},
\]
where $h\in\mathcal{H}$ is defined by
\[
h(z)=\gamma+\int_{\sigma(f)}\frac{1+zt}{t-z}\cdot\frac{\psi(t)}{1+t^{2}}\, dt,\quad z\in\mathbb{C}^{+}.
\]
Conversely, given a closed set $\sigma\subset\mathbb{R}$, an open
subset $O\subset(\mathbb{R}\cup\{\infty\})\setminus\sigma$, and a
measurable function $\varphi:\sigma\to[0,1]$ such that $\phi$ is not equal to $1$ almost everywhere on any open interval, the function
\[
k_{O}(z)e^{v(z)},\quad z\in\mathbb{C}^{+},
\]
where $v$ is defined by
\[
v(z)=\int_{\sigma}\frac{1+zt}{t-z}\cdot\frac{\varphi(t)}{1+t^{2}}\, dt,\quad z\in\mathbb{C}^{+},
\]
belongs to the class $\mathcal{H}$, and $e^{v(t)}>0$ for every $t\in\Gamma(e^v)$.\end{thm}
\begin{proof}
We only need to verify the last assertion. Recall that the argument
of $k_{O}(z)$ equals the angle subtended by the set $\Gamma$ at
$z$, that is
\[
\Im z\int_{O}\frac{dt}{|t-z|^{2}}.
\]
As shown above, the argment of $e^{v(z)}$ is at most
\[
\Im z\int_{\sigma}\frac{dt}{|t-z|^{2}},
\]
and the sum of these two numbers is at most $\pi$ because $\sigma\cap O=\varnothing$.
It follows that the argument of $k_{\Gamma}e^{v}$ is at most equal
to $\pi$, and therefore this function belongs to $\mathcal{H}$.
\end{proof}
\section{Interpolation}
A natural question arises as to which pairs $(\Omega,O)$ of open
subsets of $\mathbb{R}\cup\{\infty\}$ are of the form $(\Omega(f),\Gamma(f))$.
\begin{prop}
\label{prop:which-omega-and-gamma-occur}Consider open sets $\Omega$
and $O=\bigcup_{0\le n<N}(b_n,a_n)$ in $\mathbb{R}\cup\{\infty\}$. Denote by $X$ the closure in $\mathbb{R}\cup\{\infty\}$ of the set $\{b_n:0\le n<N\}$, and by $\Omega_1$ the Lebesgue regularization of $\Omega$. The following conditions are equivalent.
\begin{enumerate}
\item There exists a function $f\in\mathcal{H}$ such that $\Omega(f)=\Omega$
and $\Gamma(f)=O$.
\item The sets $\Omega$ and $O$ satisfy the following three requirements:
\begin{enumerate}
\item $O\subset\Omega$;
\item $O$ is Lebesgue regular;
\item $\Omega=\Omega_1\setminus X$.
\end{enumerate}
\end{enumerate}
\end{prop}
\begin{proof}
Assume first $f\in\mathcal{H}$ satisfies (1), and factor $f=k_Og$ for some $g\in\mathcal{H}$. Condition (a) is obviously satisfied. The sets $O$ and $\Omega(g)$ are Lebesgue regular by Corollary \ref{cor:lebesgur-regularity-of_f>0} and Theorem \ref{thm:main-factorization}(3), and
\[
\Omega=\Omega(f)=\Omega(g)\cap\Omega(k_O)=\Omega(g)\setminus X.
\]
Condition (c) now follows because $\Omega(g)\supset\Omega_1$ by regularity, while $\Omega\cap X=\varnothing$.
Conversely, assume that conditions (a--c) are satisfied, and set $f=k_Og$, where $g=e^v$ and $v$ is defined as
\[v(z)=\frac12\int_{\mathbb{R}\setminus\Omega_1}\frac{1-zt}{t-z}\cdot\frac{dt}{1+t^2}\quad z\in\mathbb{C}^+.\]
We have $\Gamma(f)=O$ and $\Omega(g)=\Omega_1$, while $\Omega(k_O)=\mathbb R\setminus X$. It is now easy to conclude using Theorem \ref{thm:main-factorization}(4) that \[\Omega(f)=\Omega(g)\cap\Omega(k_O)=\Omega_1\cap\Omega(k_O)=\Omega_1\setminus X=\Omega.\]
We conclude that $f$ satisfies (1).
\end{proof}
It is easy to construct examples of sets $\Omega$ and $O$ satisfying conditions (a-c) of the preceding theorem. Consider, for instance the ternary Cantor set $C$, obtained by removing $2^{k-1}$ intervals of length $3^{-k}$ from the interval $[0,1]$ for $k\ge1$, and set $\Omega=\mathbb{R}\setminus C$. Denote by $(b_n,c_n)$ these intervals, and select for each $n$ a point $a_n\in(b_n,c_n)$. Then $O=\bigcup_n(b_n,a_n)$ is a Lebesgue regular open set and $k_O$ has zeros at the points $a_n$ and essential singularities at all the points of $C$. If we consider instead the union $O$ of all intervals $(b_n,c_n)$ of length $3^{-k}$ with $k$ even, then $k_O$ has no zeros or poles and it is still singular at all points in $C$. In general, if $\Omega=\Omega(f),O=\Gamma(f)$, and $(b,c)$ is a connected component of $\Omega$, the intersection $(b,c)\cap O$ can be any interval of the form $(b,a)$ for some $a\in[b,c]$.
The following result can be viewed as an interpolation result which yields, in the special case of a finite set $Y$, the results of \cite{chal-et-al}. The set $A$ in the statement is the proposed set of zeros of a function in $\mathcal H$, $B$ is the proposed set of poles, and $Y$ is a set where the function is allowed to be essentially singular. The conditions in assertion (3) could be summarized by saying that the points of $A$ and $B$ are \emph{interlaced} in each component of the complement of $Y$.
\begin{thm}
Consider pairwise disjoint sets $A,B,Y\subset\mathbb R$ such that $Y$ is closed, and the limit points of $A\cup B$ belong to $Y$. The following conditions are equivalent.
\begin{enumerate}
\item There exists a function $f\in\mathcal{H}$ such that $A\subset\{x\in\Omega(f):f(x)=0\}\subset A\cup Y$ and $B\subset\sigma(f)\subset B\cup Y$.
\item There exists a Lebesgue regular open set $O=\bigcup_{0\le n<N}(b_n,a_n)$ such that the intervals $(b_n,a_n)$ are pairwise disjoint, $A\subset\{a_n:0\le n<N\}\subset A\cup Y$, and $B\subset\{b_n:0\le n<N\}\subset B\cup Y$.
\item For every connected component $J$ of $(\mathbb{R}\cup\{\infty\})\setminus Y$, the following two conditions are satisfied.\begin{enumerate}
\item If $a$ and $a'$ are two distinct points in $A\cap J$, there is a point $b\in B$ between $a$ and $a'$.
\item If $b$ and $b'$ are two distinct points in $B\cap J$, there is a point $a\in A$ between $b$ and $b'$.
\end{enumerate}
\end{enumerate}
If these equivalent conditions are satisfied and $Y$ is of Lebesgue measure equal to zero, then every function $f$ satisfying \emph{(1)} is of the form $f=ck_O$, where $c>0$ and $O$ is one of the open sets satisfying \emph{(2)}.
\end{thm}
\begin{proof}
Assume first that $f$ satisfies the conditions in (1), set $O=\Gamma(f)$, and write $O=\bigcup_{0\le n<N}(b_n,a_n)$ with pairwise disjoint intervals $(b_n,a_n)$. Then $f$ has a zero at each point $a\in A$, and therefore $a=a_n$ for some $n$. On the other hand, if $a\notin A\cup Y$ then $f$ is (at worst) meromorphic in a neighborhood of $a$ and $f(a)\ne0$, so that $a\ne a_n$ for all $n$. Similarly, if $b\in B$ then $b$ is isolated in $B\cup Y$, in particular $b$ is isolated in $\sigma(f)$. It follows that $b$ is a pole of $f$, and therefore $b=b_n$ for some $n$. Finally, if $b\notin B\cup Y$ then $b\notin\sigma(f)$ so that $f$ is analytic at $b$. We conclude that $b\ne b_n$ for all $n$. We have then verified that $O$ satisfies the conditions in (2).
Conversely, assume that $O$ is a Lebesgue regular open set satisfying the conditions in (2), and set $f=k_O$. The set $\sigma(f)$ is the closure of $\{b_n:0\le n<N\}$ and it therefore contained in $B\cup Y$. Every $a\in A$ is of the form $a=a_n$ for some $n$, and the hypothesis on the set $A$ implies that $a_n\notin B\cup Y$. We conclude that $a_n\in\Omega(f)$ and $f(a_n)=0$. On the other hand, if $a\notin A\cup Y$ then $a\ne a_n$ for all $n$ and therefore $f(a)\ne0$ when $a\in\Omega(f)$. This proves the first two inclusions in (1). Similarly, each $b\in B$ is equal to some $b_n$ and it is an isolated point in $\sigma(f)$, thus a pole. Points $b\notin B\cup Y$ belong to $\Omega(f)$, and this verifies the last two inclusions in (1).
We have established the equivalence of (1) and (2). The equivalence of (2) and (3) is easily verified. Indeed, assume that (2) is satisfied, $J$ is a component of the complement of $Y$, and $a,a'\in A\cap Y$. There are then integers $n,m$ so that $a=a_n$ and $a'=a_m$. Then we have $b_n,b_m\in B\cap Y$, and therefore one of these points is between $a$ and $a'$. The second condition in (3) is verified similarly.
Conversely, assume that (3) holds. For each connected component $(c,d)$ of the complement of $Y$ there exists then a family $\{(b_i,a_i):i\in I\}$
of pairwise disjoint intervals contained in $(c,d)$ such that $A\cap (c,d)\supset\{a_i :i\in I\}$, $B\cap (c,d)\supset\{b_i :i\in I\}$, and the sets $(A\cap (c,d))\setminus\{a_i:i\in I\}$ and $(B\cap (c,d))\setminus\{b_i:i\in I\}$ contain at most one point each. If $(A\cap (c,d))\setminus\{a_i:i\in I\}=\{a\}$, then $(c,a)$ is disjoint from $(b_i,a_i)$ for all $i$. Analogously, if $(B\cap (c,d))\setminus\{b_i:i\in I\}={b}$ then $(b,d)$ is disjoint from all the intervals $(b_i,a_i)$. The set $O$ is now defined to be the smallest Lebesgue regular open set containing all the intervals $(b_i,a_i)$ as well as the intervals $(c,a)$ and $(b,d)$ when needed.
The last assertion in the statement follows immediately from Corollary \ref{cor:sigma-of-measure-zero}.
\end{proof}
The special points $a,b$ which appear in the proof of $(3)\Rightarrow(2)$ are the `loners' of \cite{chal-et-al}. When a loner $a$ exists, the function $f=k_O$ has a pole at $c\in Y$ unless there is also a loner on the other side of $c$. Similarly, the function $f$ may have a zero at $d$.
When a component $(c,d)$ of $(\mathbb{R}\cup\{\infty\})\setminus Y$ contains no points in $A\cup B$, the entire interval $(c,d)$ can be added to the set $O$. This is the extent of nonuniqueness allowed in the selection of the set $O$, and thus in the choice of the function $f$, when $Y$ has zero Lebesgue measure.
The preceding result can be reformulated as an interpolation result for self maps of the unit disk using the conformal map $z\mapsto (z-\zeta)/(z+\overline{\zeta})$ of $\mathbb{C}^+$ onto the disk $\mathbb D$ for some $\zeta\in\mathbb{C}^+$. We state a particular case which may be useful in other contexts. We denote by $\mathbb{T}=\partial\mathbb{D}$ the unit circle.
\begin{cor}
Assume that $A,B$, and $Z$ are three pairwise disjoint subsets of $\mathbb T$ such that $Z$ is closed, the limit points of $A$ and $B$ are contained in $Z$, and $A$ and $B$ are interlaced on every component of $\mathbb{T}\setminus Z$. Fix $\alpha,\beta\in\mathbb T$ with $\alpha\ne\beta$. Then there exists an analytic function $\vartheta:\mathbb D\to\mathbb D$ such that
\begin{enumerate}
\item $\vartheta$ can be continued analytically across $\mathbb{T}\setminus Z$;
\item $|\vartheta(z)|=1$ for all $z\in\mathbb{T}\setminus Z$;
\item $A=\{a\in\mathbb{T}\setminus Z:\vartheta(a)=\alpha\}$ and $B=\{b\in\mathbb{T}\setminus Z:\vartheta(b)=\beta\}$.
\end{enumerate}
When $Z$ has zero linear Lebesgue measure, the function $\theta$ can be chosen to be inner.
\end{cor}
The case of a finite set $Z$ is considered in \cite{chal-et-al}, while $Z=\varnothing$, corresponding to finite Blaschke products, was studied in \cite{go}.
\end{document} | math | 36,713 |
\begin{document}
\title[Rate of convergence]
{\textsc{Remarks on the vanishing viscosity process of state-constraint Hamilton--Jacobi equations}}
\thanks{The authors contributed equally to this work. The authors are supported in part by NSF grant DMS-1664424 and NSF CAREER grant DMS-1843320. The work of Son N. T. Tu is supported in part by the GSSC Fellowship, University of Wisconsin--Madison.}
\begin{abstract}
We investigate the convergence rate in the vanishing viscosity process of the solutions to the subquadratic state-constraint Hamilton-Jacobi equations. We give two different proofs of the fact that, for nonnegative Lipschitz data that vanish on the boundary, the rate of convergence is $\mathcal{O}(\sqrt{\varepsilon})$ in the interior. Moreover, the one-sided rate can be improved to $\mathcal{O}(\varepsilon)$ for nonnegative compactly supported data and $\mathcal{O}(\varepsilon^{1/p})$ (where $1<p<2$ is the exponent of the gradient term) for nonnegative data $f\in \mathrm{C}^2(\overline{\Omegaega})$ such that $f = 0$ and $Df = 0$ on the boundary. Our approach relies on deep understanding of the blow-up behavior near the boundary and semiconcavity of the solutions.
\end{abstract}
\author{Yuxi Han}
\address[Y. Han]
{
Department of Mathematics,
University of Wisconsin Madison, 480 Lincoln Drive, Madison, WI 53706, USA}
\email{[email protected]}
\author{Son N. T. Tu}
\address[S. N.T. Tu]
{
Department of Mathematics,
University of Wisconsin Madison, 480 Lincoln Drive, Madison, WI 53706, USA}
\email{[email protected]}
\date{\today}
\keywords{first-order Hamilton--Jacobi equations; second-order Hamilton--Jacobi equations; state-constraint problems; optimal control theory; rate of convergence; viscosity solutions; semiconcavity; boundary layer.}
\subjclass[2010]{
35B40,
35D40,
49J20,
49L25,
70H20
}
\maketitle
\setcounter{tocdepth}{1}
\section{Introduction}\label{sec:intro}
\subsection{Settings} Let $\Omegaega$ be an open, bounded and connected domain in $\mathbb{R}^n$ with $\mathrm{C}^2$ boundary, $f\in \mathrm{C}(\overline{\Omegaega})\cap W^{1,\infty}(\Omegaega)$. For $\varepsilon>0$, let $u^\varepsilon\in \mathrm{C}^2(\Omegaega)$ (see \cite{Lasry1989} for the existence and the uniqueness) be the solution to
\begin{equation}\label{eq:PDEepsa}
\begin{cases}
u^\varepsilon(x) + H(Du^\varepsilon(x)) - f(x) - \varepsilon \Deltata u^\varepsilon(x) = 0 \qquad
\text{in}\;\Omegaega,
\\
\displaystyle \lim_{\mathrm{dist}(x,\partial \Omegaega)\to 0} u^\varepsilon(x) = +\infty,
\end{cases}
\end{equation}
where $H:\mathbb{R}^n\to\mathbb{R}^n$ is a given continuous Hamiltonian. The solution that blows up uniformly on the boundary is also called a \emph{large solution}. A typical Hamiltonian that has been considered in the literature is $H(\xi) = \left\lvert \xi\right\rvert ^p$ for $\xi\in \mathbb{R}^n$ where $1<p\leq 2$, and equation \eqref{eq:PDEepsa} becomes
\begin{equation}\label{eq:PDEeps}
\begin{cases}
u^\varepsilon(x) + \left\lvert Du^\varepsilon(x)\right\rvert ^p - f(x) - \varepsilon \Deltata u^\varepsilon(x) = 0 \qquad
\text{in}\;\Omegaega,
\\
\displaystyle \lim_{\mathrm{dist}(x,\partial \Omegaega)\to 0} u^\varepsilon(x) = +\infty.
\end{cases} \tag{PDE$_\varepsilon$}
\end{equation}
It turns out that for this specific subquadratic Hamiltonian, $u^\varepsilonsilon$ is also the unique solution to the second-order state-constrait problem (see \cite{Lasry1989})
\begin{equation}\label{eq:HJB}
\begin{cases}
u^\varepsilon(x) + \left\lvert Du^\varepsilon(x)\right\rvert ^p - f(x) - \varepsilon \Deltata u^\varepsilon(x) \leq 0 \qquad\text{in}\;\Omegaega,\\
u^\varepsilon(x) + \left\lvert Du^\varepsilon(x)\right\rvert ^p - f(x) - \varepsilon \Deltata u^\varepsilon(x) \geq 0 \qquad\text{on}\;\overline{\Omegaega}.
\end{cases}
\end{equation}
We focus on this Hamiltonian in our paper, which follows the setting of \cite{Lasry1989}, where the specific structure of the Hamiltonian enables more explicit estimates for the solution of \eqref{eq:PDEeps}.
In fact, for $1<p\leq 2$, the solution to equation \eqref{eq:PDEeps} is the value function associated with a minimization problem in stochastic optimal control theory with state constraints (\cite{fabbri_stochastic_2017,Lasry1989}). We briefly recall the setting and all the domains and target spaces are omitted for simplicity. For a given stochastic control $\alphapha(\cdot)$, we can solve for a solution (a state process) of the feedback control system
\begin{equation}\label{eq:u2}
\begin{cases}
dX_t = \alphapha\left(X_t\right)dt + \sqrt{2\varepsilon}\,d\mathbb{B}_t \qquad \text { for } t >0,\\
\;\; X_0 = x.
\end{cases}
\end{equation}
Here, $\mathbb{B}_t\sim \mathcal{N}(0,t)$ is the Brownian motion with mean zero and variance $t$. To constrain the state $X_t$ inside $\overline{\Omegaega}$, we define
\begin{equation*}
\widehat{\mathcal{A}}_x = \Big\lbrace\alphapha(\cdot)\in \mathrm{C}(\Omegaega): \mathbb{P}(X_t\in \Omegaega) = 1\;\text{for all}\;t\geq 0\Big\rbrace
\end{equation*}
and hope to minimize a cost function in expectation to get the value function
\begin{equation}\label{eq:valueeps}
u^\varepsilon(x) = \inf_{\alphapha\in \widehat{\mathcal{A}}_x} \mathbb{E}\left[\int_0^\infty e^{-t}L\big(X_t,\alphapha(X_t) \big)\;dt\right],
\end{equation}
where $L(x,v):\overline{\Omegaega}\times \mathbb{R}^n \to \mathbb{R}$ is the running cost. More specifically, $L(x,v) = c\left\lvert v\right\rvert ^q+f(x)$ is the Legendre transform of $H(x,\xi): = \left\lvert \xi\right\rvert ^p - f(x)$ with $q>1$, $f\in \mathrm{C}(\overline{\Omegaega})$ nonnegative, and some constant $c$. Using the Dynamic Programming Principle (see
\cite{Lasry1989}), we expect the value function \eqref{eq:valueeps} to solve \eqref{eq:HJB}, which means that $u^\varepsilon$ is a subsolution in $\Omegaega$ and a supersolution on $\overline{\Omegaega}$.
We are interested in studying the asymptotic behavior of $\{u^\varepsilon\}_{\varepsilon>0}$ as $\varepsilon\rightarrow 0^+$. Heuristically, the solution of the second-order state-constraint equation converges to that of a first-order state-constraint equation associated with the deterministic optimal control problem, namely,
\begin{equation}\label{eq:PDE0}
\begin{cases}
u(x) + \left\lvert Du(x)\right\rvert ^p - f(x) \leq 0\;\qquad\text{in}\;\Omegaega,\\
u(x) + \left\lvert Du(x)\right\rvert ^p - f(x) \geq 0\;\qquad\text{on}\;\overline{\Omegaega}.
\end{cases} \tag{PDE$_0$}
\end{equation}
and indeed equation \eqref{eq:PDE0} admits a unique viscosity solution $u \in \mathrm{C}(\overline{\Omegaega})$(see \cite{Capuzzo-Dolcetta1990,Soner1986}). From the viewpoint of optimal control theory, as $\varepsilon \to 0^+$, the stochastic control system \eqref{eq:u2} becomes a deterministic control system. In particular, let $\mathcal{A}_x = \{\zeta\in \mathrm{AC}([0,\infty);\overline{\Omegaega}): \zeta(0)=x\}$ and we have
\begin{equation*}
u(x) = \inf_{\zeta\in \mathcal{A}_x} \int_0^\infty e^{-t}L\big(\zeta(t),\dot{\zeta}(t)\big)dt
\end{equation*}
where $L(x,v)$ is again the Legendre transform of $H(x,\xi) := \left\lvert \xi\right\rvert ^p - f(x)$.
The problem is interesting since in the limit there is no blowing up behavior near the boundary, as $u\in \mathrm{C}(\overline{\Omegaega})$. In this paper, we investigate the rate of convergence of $u^\varepsilon \to u$ as $\varepsilon\to 0^+$. What is intriguing and delicate here is the blow-up behavior of $u^\varepsilon$ in a narrow strip near $\partial \Omegaega$ as $\varepsilon\to 0^+$. This is often called the boundary layer theory in the literature.
Note that a comparison principle holds for \eqref{eq:PDE0} since we always assume $\Omegaega$ is an open, bounded and connected domain in $\mathbb{R}^n$ with $\mathrm{C}^2$ boundary (\cite{Capuzzo-Dolcetta1990,Soner1986}).
\subsection{Relevant literature} There is a
vast amount of work in the literature on viscosity solutions with state constraints and large solutions. We would like to first mention that the problem \eqref{eq:PDE0} with general Hamiltonian is a huge subject of research interest, started with the pioneer work \cite{Soner1986} (see also \cite{ishii_new_1996,ishii_class_2002}). Some of the recent work related to the asymptotic behavior of solutions of \eqref{eq:PDE0} can be found in \cite{ishii_vanishing_2017,kim_state-constraint_2020,mitake_asymptotic_2008,tu2021vanishing}. The problem \eqref{eq:PDEeps}
was first studied in \cite{Lasry1989} and subsequently many works have been done in understanding deeper the properties of solutions (see \cite{Porretta_a,alessio_asymptotic_2006,Bandle_1994,marcus_existence_2003} and the references therein). The time-dependent version of \eqref{eq:PDEepsa} was also studied by many works, for instance, \cite{barles_generalized_2004,barles_large_2010,leonori_local_2011,moll_large_2012} and the references therein.
In terms of rate of convergence, that is, the convergence rate of $u^\varepsilon\to u$ as $\varepsilon\to 0^+$, to the best of our knowledge, such a question has not been studied in the literature. For the case where \eqref{eq:PDEeps} is equipped with the Dirichlet boundary condition, a rate $\mathcal{O}(\sqrt{\varepsilon})$ is well known with multiple proofs (see \cite{Bardi1997,crandall1984,tran_hamilton-jacobi_2021}).
\subsection{Main results}
For $1 <p \leq 2$, define
\begin{equation*}
\alphapha = \frac{2-p}{p-1} \in [0,\infty).
\end{equation*}
Let $\Omegaega$ be an open, bounded and connected subset of $\mathbb{R}^n$ with boundary $\partial\Omegaega$ of class $C^2$. For small $\deltata>0$, denote $\Omegaega_\deltata = \{x\in \Omegaega: \mathrm{dist}(x, \partial \Omegaega) > \deltata\}$ and $\Omegaega^\deltata = \{x\in \mathbb{R}^n: \mathrm{dist}(x,\overline{\Omegaega}) < \deltata\}$.
\begin{defn} Define
\begin{equation}\label{def:delta_0}
\deltata_{0,\Omegaega} =\frac{1}{2}\sup \big\{ \deltata > 0: x\mapsto\mathrm{dist}(x,\partial\Omegaega)\;\text{is}\;\mathrm{C}^2\;\text{in}\;\Omegaega^\deltata \backslash\overline{\Omegaega}_{\deltata} \big\}.
\end{equation}
We will write $\deltata_0$ instead of $\deltata_{0,\Omegaega}$ when the underlying domain is understood.
\end{defn}
The reader is referred to \cite{gilbarg_elliptic_2001} for the regularity of the distance function defined in $\Omegaega^{\deltata_0} \backslash \Omegaega_{\deltata_0}$. We then extend $\mathrm{dist}(x,\partial\Omegaega)$ to a function $d(x)\in \mathrm{C}^2(\mathbb{R}^n)$ such that
\begin{equation}\label{e:distance_def}
\begin{cases}
d(x)\geq 0\;\text{for}\;x\in\Omegaega\;\text{with}\;d(x) = +\mathrm{dist}(x,\partial\Omegaega)\;\text{for}\;x\in \Omegaega\backslash \Omegaega_{\deltata_0},\\
d(x)\leq 0\;\text{for}\;x\notin \Omegaega\;\text{with}\;d(x) = -\mathrm{dist}(x,\partial\Omegaega)\;\text{for}\;x\in \Omegaega^{\deltata_0}\backslash \Omegaega.
\end{cases}
\end{equation}
\paragraph{Assumption on $f$.}
We assume that $f\in \mathrm{C}(\overline{\Omegaega}) \cap W^{1, \infty}(\Omegaega)$ with $f = \min_{\overline{\Omegaega}}$ on $\partial\Omegaega$.
By replacing $f$ by $f - \min_{\overline{\Omegaega}}$, without loss of generality, we can assume $ \min_{\overline{\Omegaega}} f = 0$ and $f = 0$ on $\partial\Omegaega$. The reason why this assumption is needed is elaborated in Remark 1. The main results of the paper are the following theorems.
\begin{theorem}\label{main_thm1} Let $\Omegaega$ be an open, bounded and connected subset of $\mathbb{R}^n$ with $\mathrm{C}^2$ boundary. Assume that $1 < p\leq 2$ and $f$ is nonnegative and Lipschitz with $f = 0$ on $\partial\Omegaega$. Let $u^\varepsilon$ be the unique solution to \eqref{eq:PDEeps} and $u$ be the unique solution to \eqref{eq:PDE0}. Then there exists a constant $C$ independent of $\varepsilon\in (0,1)$ such that for $x\in \Omegaega$,
\begin{align*}
&-C\sqrt{\varepsilon}\leq u^\varepsilon(x) - u(x)\leq C\left(\sqrt{\varepsilon} + \frac{\varepsilon^{\alphapha+1}}{d(x)^\alphapha}\right), \qquad\; 1<p < 2,\\
&-C\sqrt{\varepsilon}\leq u^\varepsilon(x) - u(x)\leq C\left(\sqrt{\varepsilon} + \varepsilon\left\lvert \log(d(x))\right\rvert \right), \qquad \; p = 2.
\end{align*}
\end{theorem}
\begin{remark} To the best of our knowledge, this theorem is new in the literature. The precise boundary behavior is very delicate and deserves further investigation. The condition $f = 0$ on $\partial\Omegaega$ is a little bit restrictive but is naturally needed in the proof. As is illustrated in the proof of Theorem \ref{main_thm1}, we will first show the result for $f$ that is compactly supported and nonnegative. Then, to further generalize the main result, if we make the assumption that $f = 0$ on $\partial\Omegaega$ and $f$ is nonnegative, we
can approximate $f$ \emph{uniformly} in $L^\infty(\Omegaega)$ by a sequence of compactly supported Lipschitz functions with uniformly bounded Lipschitz constants. Using the previous result obtained for the case where $f$ is compactly supported and nonnegative, we can pass to the limit and prove Theorem \ref{main_thm1} for nonnegative $f$ with $f=0$ on $\partial \Omegaega$, which is more general than the compactly supported case. At the current moment, we do not yet know how to extend the result to general f
where f does not vanish or is not equal to its minimum on the boundary.
To prove the result for the case where $f$ is compactly supported and nonnegative, it is natural to consider the doubling variable method. Indeed, for instance, if $1<p<2$, one would consider constructing an auxiliary function with
\begin{equation}\label{heur1}
\psi^\varepsilon(x) := u^\varepsilon(x) - \frac{C_\alphapha\varepsilon^{\alphapha+1}}{d(x)^{\alphapha}}
\end{equation}
and $u(x)$, where $C_\alphapha\varepsilon^{\alphapha+1}d(x)^{-\alphapha}$ is the leading order term in the asymptotic expansion of $u^\varepsilon(x)$ as $d(x) \to 0^{+}$ with $C_\alphapha:= \alphapha^{-1}(\alphapha+1)^{\alphapha+1}$. If we take the derivative of \eqref{heur1} formally, it becomes
\begin{equation}\label{heur2}
D\psi^\varepsilon(x) = Du^\varepsilon(x) + C_\alphapha\alphapha \left(\frac{\varepsilon}{d(x)}\right)^{\alphapha+1}Dd(x).
\end{equation}
We will see that $D\psi^\varepsilon(x)$ is uniformly bounded if $d(x)\geq \varepsilon$ (Lemma \ref{lem:boundDu^eps}). Indeed,
\begin{equation*}
-C_\alphapha \alphapha \left(\frac{\varepsilon}{d(x)}\right)^{\alphapha+1}Dd(x)
\end{equation*}
is more or less the leading order term in the asymptotic expansion of $Du^\varepsilon$ near $\partial\Omegaega$. Heuristically, this means that the boundary layer is $\mathcal{O}(\varepsilon)$ from the boundary.
However, to get a useful estimate by the doubling variable method, at the maximum point $x_0$ of $\psi^\varepsilon(x) - u(x)$, we need to have $d(x_0)\geq \varepsilon^{\gammama}$ for $\gammama<1$ so that the latter term in \eqref{heur2} vanishes as $\varepsilon\to 0^+$. Otherwise, we cannot obtain a convergence rate via the doubling variable method as there are still nonvanishing constant terms. In the other case where $d(x_0) < \varepsilon^{\gammama}$, we introduce a new localization idea, that is, we construct a blow-up solution in the ball of radius $\varepsilon^\gammama$ from the boundary. Finally, a technical (and common for the doubling variable method) computation leads to $\gammama = 1/2$.
\end{remark}
As a different approach, the convexity of $\left\lvert \xi\right\rvert ^p$ and the semiconcavity of the solution to \eqref{eq:PDE0} give us a better one-sided $\mathcal{O}(\varepsilon)$ estimate for nonnegative compactly supported $f$ which is semiconcave in its support (see Theorem \ref{thm:rate_doubling2}). Such an one-sided $\mathcal{O}(\varepsilon)$ rate is well known for the Dirichlet boundary problem (see \cite{Bardi1997,tran_hamilton-jacobi_2021}). Moreover, the result in Theorem \ref{thm:rate_doubling2} further provides us with a better one-sided estimate $\mathcal{O}(\varepsilon^{1/p})$ than that in Theorem \ref{main_thm1}, as in Corollary \ref{cor:key}. We recall that $f$ is (uniformly) semiconcave in $\overline{\Omegaega}$ with linear modulus (or semiconcavity constant) $c>0$ if
\begin{equation*}
f(x+h)-2f(x)+f(x-h)\leq c\left\lvert h\right\rvert ^2, \quad \forall x, h \in \mathbb{R}^n \,\, \text{such that} \, \, x+h, x, \text{and}\, \, x-h \in \overline{\Omegaega}.
\end{equation*}
Note that any $f\in \mathrm{C}^2_c(\Omegaega)$ is semiconcave on its support with the constant
$$c = \max \left\{ D^2f(x) \xi \cdot \xi: \left\lvert \xi\right\rvert =1, x\in \Omegaega \right\}$$
in the above definition. It is well known that the solution $u$ to \eqref{eq:PDE0} is \emph{locally} semiconcave given $f$ is uniformly semiconcave in $\overline{\Omegaega}$. Using tools from the optimal control theory, we provide the explicit blow-up rate of the semiconcavity modulus of $u(x)$ when $x$ approaches $\partial\Omegaega$. As an application, we can improve the rate of convergence as follows.
\begin{theorem}[One-sided $\mathcal{O}(\varepsilon)$ rate for nonnegative compactly supported data]\label{thm:rate_doubling2}
Under the conditions of Theorem \ref{main_thm1}, suppose $f$ also satisfies the following conditions:
\begin{itemize}
\item $f$ is semiconcave in its support;
\item $f$ has a compact support in $\Omegaega_\kappapa := \{x\in \Omegaega: \mathrm{dist}(x,\Omegaega) > \kappapa\}$ for some $\kappapa\in(0,\deltata_0)$\,, $0 < \deltata_0 <1$ defined in \eqref{def:delta_0}.
\end{itemize}
Then there exist two constants $\nu > 1$ and $C$ independent of $\varepsilon$ and $\kappapa$ such that $\forall x \in \Omegaega$,
\begin{equation*}
\begin{aligned}
u^\varepsilon(x) - u(x) &\leq\frac{\nu C_\alphapha \varepsilon^{\alphapha+1}}{d(x)^\alphapha} + C \left(\left(\frac{\varepsilon}{\kappapa}\right)^{\alphapha+1}+\left(\frac{\varepsilon}{\kappapa}\right)^{\alphapha+2}\right) + \frac{Cn\varepsilon}{\kappapa}, &\quad \text{if } p <2, \\
u^\varepsilon(x) - u(x) & \leq \nu \varepsilon \log\left( \frac{1}{d(x)}\right)+C \left(\left(\frac{\varepsilon}{\kappapa}\right)+\left(\frac{\varepsilon}{\kappapa}\right)^2\right)+ \frac{Cn\varepsilon}{\kappapa} , &\quad \text{if } p=2.
\end{aligned}
\end{equation*}
\end{theorem}
\begin{remark}\label{rem:C} If $f\in \mathrm{C}^2_c(\Omegaega)$, then the last term $\left(Cn\varepsilon\right)\kappapa^{-1}$ in the equations above can be improved to $nc\varepsilon$, where $c$ is the semiconcavity constant of $f$. This improvement is due to the fact that we can prove $u$ is uniformly semiconcave with a semiconcavity constant that only depends on the semiconcavity constant $c$ of $f$ (see Theorem \ref{convex} in the Appendix). Hence, in the proof of Theorem \ref{thm:rate_doubling2}, in equation \eqref{mybound}, instead of $C \kappapa^{-1}$, we can bound $c(x_0)$ by the semiconcavity constant $c$ of $f$, independent of $\kappapa$. Similarly, see Remark \ref{rem:nice} for this improvement on the last term. It turns out that in general, if $f$ can be extended to a semiconcave function $\tilde{f}:\mathbb{R}^n \to \mathbb{R}$ by setting $f=0$ on $\Omegaega^c$, then $u$ is uniformly semiconcave, and hence this improvement happens. See Fig. \ref{fig:Domains2} for two examples where $f$ can and cannot be extended to a semiconcave function in the whole space by setting $f=0$ outside $\Omegaega$.
\end{remark}
\noindent
\begin{figure}
\caption{The one on the right corresponds to a general $f$ in Theorem
\ref{thm:rate_doubling2}
\label{fig:Domains2}
\end{figure}
\noindent
\begin{cor}[One-sided $\mathcal{O}(\varepsilon^{1/p})$ rate]\label{cor:key} Let $1<p<2$. If $f \in \mathrm{C}^2(\overline{\Omegaega})$ is nonnegative, $f = 0$ and $Df = 0$ on $\partial\Omegaega$, then there exists a constant $C$ independent of $\varepsilon\in (0,1)$ such that
\begin{align*}
&-C\varepsilon^{1/2}\leq u^\varepsilon(x) - u(x)\leq C\left(\varepsilon^{1/p}+ \frac{\varepsilon^{\alphapha+1}}{d(x)^\alphapha}\right)
\end{align*}
for all $x\in \Omegaega$.
\end{cor}
\begin{remark} While the second approach looks more powerful, we need the gradient bound of $u^\varepsilon$ (Lemma \ref{lem:boundDu^eps}), the blow-up rate of the semiconcavity constant of $u$ (Theorem \ref{thm:newsemi}), and higher regularity on $f$. On the other hand, the first approach by doubling variable is relatively simple and does not require any explicit asymptotic behavior of $Du^\varepsilon$, except the fact that it is locally bounded.
\end{remark}
\subsection*{Organization of the paper} Section 2 contains some preliminary results. The proof of Theorem \ref{main_thm1} is given in Section 3. Then in Section 4, we give the proof of Theorem \ref{thm:rate_doubling2} and Corollary \ref{cor:key}. Finally, the proofs of some useful lemmas are presented in Appendix.
\section{Preliminaries}\label{sec:prelim}
Let $K_0:= \max_{x\in \overline{\Omegaega}}\left\lvert d(x)\right\rvert$, $K_1 := \max_{x\in \overline{\Omegaega}} \left\lvert D d(x)\right\rvert$, and $K_2 := \max_{x\in \overline{\Omegaega}} \left\lvert \Deltata d(x)\right\rvert$. Note that $d(x) = \mathrm{dist}(x,\partial\Omegaega)$ for $x \in \Omegaega$ and $\left\lvert D d(x)\right\rvert = 1$ in the classical sense in $\Omegaega^{\deltata_0}\backslash \Omegaega_{\deltata_0}$. Denote by $\mathcal{L}^\varepsilon:\mathrm{C}^2(\Omegaega)\to \mathrm{C}(\Omegaega)$ the operator
\begin{equation*}
\mathcal{L}^\varepsilon[u](x) := u(x) + \left\lvert Du(x)\right\rvert ^p - f(x) - \varepsilon \Deltata u(x), \qquad x\in \Omegaega.
\end{equation*}
\subsection{Local gradient estimate} For $\varepsilon \in (0,1)$ and $p>1$, we state an a priori estimate for $\mathrm{C}^2$ solutions to \eqref{eq:PDEeps} (\cite[Appendix]{Lasry1989}). Since we are working with smooth solutions, the proof is relatively simple by the classical Bernstein method (\cite{bernstein_sur_1910}), which is provided in Appendix for the reader's convenience.
\begin{theorem}\label{thm:grad_1} Let $f\in \mathrm{C}(\overline{\Omegaega})\cap W^{1,\infty}(\Omegaega)$ and $u^\varepsilon \in \mathrm{C}^2(\Omegaega)$ be a solution to $\mathcal{L}^\varepsilon[u^\varepsilon] = 0$ in $\Omegaega$ with $1 < p \leq 2$. Let $m:= \max_{\overline{\Omegaega}}f(x)$. Then for $\deltata>0$, there exists $C_\deltata = C(m,p,\deltata, \Vert D f\Vert_{L^\infty(\Omegaega)})$ such that
\begin{equation*}
\sup_{x\in \overline{\Omegaega}_\deltata} \Big(\left\lvert u^\varepsilon(x)\right\rvert +\left\lvert Du^\varepsilon(x)\right\rvert \Big) \leq C_\deltata
\end{equation*}
for $\varepsilon$ small enough.
\end{theorem}
\subsection{Well-posedness of \eqref{eq:PDEeps}} In this section, we recall the existence and the uniqueness of solutions to \eqref{eq:PDEeps} for $1<p\leq 2$ and $f\in \mathrm{C}(\overline{\Omegaega})\cap W^{1,\infty}(\Omegaega)$. In fact, the assumption of $f$ can be relaxed to $f\in L^\infty(\Omegaega)$ (\cite{Lasry1989}).
\begin{theorem}\label{thm:wellposed1<p<2} Let $f\in \mathrm{C}(\overline{\Omegaega})\cap W^{1,\infty}(\Omegaega)$. There exists a unique solution $u^\varepsilon\in \mathrm{C}^2(\Omegaega)$ of \eqref{eq:PDEeps} such that:
\begin{itemize}
\item[(i)] If $1<p< 2$, then
\begin{equation}\label{rate_p<2}
\lim_{d(x)\to 0}\left( u^\varepsilon(x) \,d(x)^\alphapha \right)= C_\alphapha \varepsilon^{\alphapha+1},
\end{equation}
where $\alphapha = (p-1)^{-1}(2-p)$ and $C_\alphapha = \alphapha^{-1}(\alphapha+1)^{\alphapha+1}$.
\item[(ii)] If $p=2$, then
\begin{equation}\label{rate_p=2}
\lim_{d(x)\to 0} \left(-\frac{u^\varepsilon(x)}{\log(d(x))}\right) = \varepsilon.
\end{equation}
\end{itemize}
Furthermore, $u^\varepsilon$ is the maximal subsolution among all the subsolutions $v\in W^{2,r}_{\mathrm{loc}}(\Omegaega)$ for all $r\in [1,\infty)$ of \eqref{eq:PDEeps}.
\end{theorem}
This is Theorem I.1 in \cite{Lasry1989} with an explicit dependence on $\varepsilon$. The proof of this theorem is carried out explicitly in Appendix for later use. Also, it is useful to note that $\alphapha+1 = (p-1)^{-1}$. More results on the behavior of the gradient of $u^\varepsilon$ can be found in \cite{alessio_asymptotic_2006} and Lemma \ref{lem:boundDu^eps}, where we show $\left\lvert Du^\varepsilon\right\rvert \leq C + C\left(\frac{\varepsilon}{d(x)}\right)^{\alphapha+1}$. We believe Lemma \ref{lem:boundDu^eps} is new in the literature.
\subsection{Convergence results} We first state the following lemma (\cite{Capuzzo-Dolcetta1990}), which characterizes the solution to the first-order state-constraint equation \eqref{eq:PDE0}.
\begin{lem}\label{lem:max} Let $u\in \mathrm{C}(\overline{\Omegaega})$ be a viscosity subsolution of \eqref{eq:PDE0} such that, for any viscosity subsolution $v\in \mathrm{C}(\overline{\Omegaega})$ of \eqref{eq:PDE0}, one has $v\leq u$ on $\overline{\Omegaega}$. Then $u$ is a viscosity supersolution of \eqref{eq:PDE0} on $\overline{\Omegaega}$.
\end{lem}
Again, the proof of Lemma \ref{lem:max} is given in Appendix for the reader's convenience.
\begin{lem}\label{lem:lower-bound} Assume $1<p\leq 2$. Let $u^\varepsilon\in \mathrm{C}^2(\Omegaega)$ be the solution to \eqref{eq:PDEeps} and $u \in \mathrm{C}(\overline{\Omegaega})$ be the solution to \eqref{eq:PDE0}. We have $\{ u^\varepsilon\}_{\varepsilon>0}$ is uniformly bounded from below by a constant independent of $\varepsilon$. More precisely, $ u^\varepsilon \geq \min_\Omegaega f$ and $ u\geq \min_\Omegaega f$.
\end{lem}
\begin{proof} For $m\in \mathbb{N}$, let $u^{\varepsilon}_m\in \mathrm{C}^2(\Omegaega)\cap \mathrm{C}(\overline{\Omegaega})$ solve the Dirichlet problem
\begin{equation}\label{e:uepsm}
\begin{cases}
u^{\varepsilon}_m(x) + \left\lvert Du^{\varepsilon}_m(x)\right\rvert ^p - f(x) - \varepsilon \Deltata u^{\varepsilon}_m(x) = 0 &\qquad
\text{in}\;\Omegaega,
\\
\quad \qquad\quad\qquad\qquad\qquad\qquad u^{\varepsilon}_m(x) = m &\qquad
\text{on}\;\partial\Omegaega.
\end{cases} \tag{PDE$_{\varepsilon,m}$}
\end{equation}
We have $u^{\varepsilon}_m(x) \to u^\varepsilon(x)$ in $\Omegaega$ as $m\to \infty$. Let $\varphi(x) \equiv \inf_{\Omegaega} f$ for $x\in \overline{\Omegaega}$. Then $\varphi(x)$ is a classical subsolution of \eqref{e:uepsm} in $\Omegaega$ with
\begin{equation*}
\varphi(x) = \inf_\Omegaega f \leq m = u^\varepsilon_m(x) \qquad \text{on } \partial \Omegaega
\end{equation*}
for $m$ large enough.
By the comparison principle of the uniformly elliptic equation \eqref{e:uepsm},
\begin{equation*}
\inf_\Omegaega f \leq u^{\varepsilon}_m(x) \qquad\text{for all}\;x\in \Omegaega.
\end{equation*}
As $m\to \infty$, we obtain $ u^\varepsilon \geq \min_\Omegaega f$. The inequality $ u\geq \min_{\Omegaega}f$ follows from the comparison principle of \eqref{eq:PDE0} applied to the supersolution $u$ on $\overline{\Omegaega}$ and the subsolution $\varphi$ in $\Omegaega$.
\end{proof}
We present here a simple proof of the convergence $u^\varepsilon \to u$ using Lemma \ref{lem:max}. See also \cite[Theorem VII.3]{Capuzzo-Dolcetta1990}.
\begin{theorem}[Vanishing viscosity]\label{thm:qual} Let $u^\varepsilon$ be the solution to \eqref{eq:PDEeps}. Then there exists $u \in \mathrm{C}(\overline{\Omegaega})$ such that $u^\varepsilon \rightarrow u$ locally uniformly in $\Omegaega$ as $\varepsilon\rightarrow 0$ and $u$ solves \eqref{eq:PDE0}.
\end{theorem}
\begin{proof} By the a priori estimate (Theorem \ref{thm:grad_1}),
\begin{equation}\label{e:priorie_eps}
\left\lvert u^\varepsilon(x)\right\rvert + \left\lvert Du^\varepsilon(x)\right\rvert \leq C_\deltata \qquad\text{for}\;x\in \overline{\Omegaega}_\deltata.
\end{equation}
By the Arzel\`a--Ascoli theorem, there exists a subsequence $\varepsilon_j\to 0$ and a function $u\in \mathrm{C}(\Omegaega)$ such that $u^{\varepsilon_j}\to u$ locally uniformly in $\Omegaega$. From the stability of viscosity solutions, we easily deduce that
\begin{equation}\label{eq:u0int}
u(x) + \left\lvert Du(x)\right\rvert ^p - f(x) = 0 \qquad\text{in}\;\Omegaega.
\end{equation}
From Lemma \ref{lem:lower-bound}, $ u^\varepsilon(x)\geq \min_{\Omegaega} f$ and $ u(x)\geq \min_{\Omegaega} f$ for all $x\in \Omegaega$. Together with \eqref{eq:u0int}, we obtain $\left\lvert \xi\right\rvert ^p \leq \max_\Omegaega f - \min_\Omegaega f$ for all $\xi\in D^+u(x)$ and $x\in \Omegaega$. This implies there exists a constant $C_0$ such that
\begin{equation}\label{e:C0}
\left\lvert u(x) - u(y)\right\rvert \leq C_0\left\lvert x-y\right\rvert \qquad\text{for all}\;x,y\in \Omegaega.
\end{equation}
Thus, we can extend $u$ uniquely to $u\in \mathrm{C}(\overline{\Omegaega})$. We use Lemma \ref{lem:max} to show that $u$ is a supersolution of \eqref{eq:PDE0} on $\overline{\Omegaega}$.
It suffices to show that $u\geq w$ on $\overline{\Omegaega}$, where $w\in \mathrm{C}(\overline{\Omegaega})$ is the unique solution to \eqref{eq:PDE0}. For $\deltata>0$, let $u_\deltata\in\mathrm{C}(\overline{\Omegaega}_\deltata)$ be the unique viscosity solution to
\begin{equation}\label{e:v_v}
\begin{cases}
u_\deltata(x) + \left\lvert Du_\deltata(x)\right\rvert ^p-f(x) \leq 0 &\qquad\text{in}\;\Omegaega_\deltata,\\
u_\deltata(x) + \left\lvert Du_\deltata(x)\right\rvert ^p - f(x) \geq 0 &\qquad\text{on}\;\overline{\Omegaega}_\deltata.
\end{cases}
\end{equation}
Since $u_\deltata\rightarrow w$ locally uniformly as $\deltata\rightarrow 0^+$ (see \cite{kim_state-constraint_2020}) and $w$ is bounded, $\{u_\deltata\}_{\deltata>0}$ is uniformly bounded. Let $v^\varepsilon_\deltata\in \mathrm{C}^2(\Omegaega_\deltata)\cap \mathrm{C}(\overline{\Omegaega}_\deltata)$ be the unique solution to the Dirichlet problem
\begin{equation}\label{eq:vv_eps}
\begin{cases}
v_\deltata^\varepsilon(x) + \left\lvert Dv_\deltata^\varepsilon(x)\right\rvert ^p - f(x) = \varepsilon \Deltata v_\deltata^\varepsilon(x) &\qquad\text{in}\;\Omegaega_\deltata,\\
\;\quad\qquad\qquad\qquad\qquad v_\deltata^\varepsilon = u_\deltata &\qquad \text{on}\;\partial\Omegaega_\deltata.
\end{cases}
\end{equation}
It is well known that $v^\varepsilon_\deltata\to u_\deltata$ uniformly on $\overline{\Omegaega}_\deltata$ as $\varepsilon\to 0$ (\cite{crandall1984,fleming_convergence_1986,Tran2011}).
For $\deltata$ small enough, $u_\deltata\leq u^\varepsilon$ on $\partial \Omegaega_\deltata$. Hence, by the maximum principle, $v^\varepsilon_\deltata \leq u^\varepsilon$ on $\overline{\Omegaega}_\deltata$. Now we first let $\varepsilon\to 0$ to obtain $u_\deltata \leq u$ on $\overline{\Omegaega}_\deltata$.
Then let $\deltata\rightarrow 0$ to get $w\leq u$ in $\Omegaega$, which implies $w\leq u$ on $\overline{\Omegaega}$ since both $w,u$ belong to $\mathrm{C}(\overline{\Omegaega})$.
\end{proof}
\section{Rate of convergence}
In this section, we focus on the rate of convergence for the case where $f\in \mathrm{W}^{1,\infty}(\Omegaega)\cap\mathrm{C}(\overline{\Omegaega})$ is nonnegative. As a consequence, $u^\varepsilon(x),u(x)\geq 0$ for $x\in \Omegaega$ by Lemma \ref{lem:lower-bound}. In our main results, we have an additional assumption that $f=0$ on $\partial \Omegaega$.
Before we show any result about the rate of convergence, we would like to mention a lower bound of $u^\varepsilon - u$ and some properties of $u$ from its optimal control formulation.
\begin{theorem}\label{thm:lowerbound} Let $u^\varepsilon$ be the unique solution to \eqref{eq:PDEeps} and $u$ be the unique solution to \eqref{eq:PDE0}. Then there exists a constant $C$ independent of $\varepsilon$ such that
\begin{equation}\label{e:lower1}
-C\sqrt{\varepsilon} \leq u^\varepsilon(x) - u(x) \qquad\text{for all}\;x\in \Omegaega.
\end{equation}
\end{theorem}
\begin{proof} The proof relies on a well-known rate of convergence for vanishing viscosity of the viscous Hamilton--Jacobi equation with the Dirichlet boundary condition (see \cite{crandall1984,evans_adjoint_2010,fleming_convergence_1961,Tran2011}). Let $g(x) = u(x)$ for $x\in \partial\Omegaega$. Let $v^\varepsilon\in \mathrm{C}^2(\Omegaega)\cap \mathrm{C}(\overline{\Omegaega})$ be the unique viscosity solution to
\begin{equation*}
\left\{
\begin{aligned}
v^\varepsilon(x) + \left\lvert Dv^\varepsilon(x)\right\rvert ^p - f(x) - \varepsilon \Deltata v^\varepsilon(x) &= 0 \,\qquad\text{in}\;\Omegaega,\\
v^\varepsilon(x) &= g(x) \ \ \ \text{on}\;\partial\Omegaega.
\end{aligned}\right.
\end{equation*}
It is well known that $v^\varepsilon \to u$ (\cite{crandall1984,fleming_convergence_1986,Tran2011}). Furthermore, there exists a positive constant $C$ independent of $\varepsilon\in (0,1)$ such that
\begin{equation}\label{e:cp1}
\left\lvert v^\varepsilon(x) - u(x)\right\rvert \leq C\sqrt{\varepsilon} \qquad\text{for}\;x\in \overline{\Omegaega}.
\end{equation}
By the comparison principle for \eqref{eq:PDEeps}, we have
\begin{equation}\label{e:cp2}
v^\varepsilon(x)\leq u^\varepsilon(x) \qquad\text{for}\;x\in \Omegaega.
\end{equation}
From \eqref{e:cp1} and \eqref{e:cp2}, we obtain the lower bound \eqref{e:lower1}.
\end{proof}
\begin{lem}\label{lem:f=0} Assume $f\geq 0$ in $\Omegaega$. Then $u(x) = 0$ if and only if $f(x) = 0$. In particular, $f \equiv 0$ implies $u \equiv 0$.
\end{lem}
\begin{proof} It is clear to see that $f\equiv 0$ implies $u\equiv 0$ by the uniqueness of \eqref{eq:PDE0}.
It is not hard to prove the converse by contradiction. Suppose $u \equiv 0$ and $f(x_0)>0$. Then there exists $\varepsilon$, $\deltata >0$ such that $f(x) > \varepsilon$ for all $x \in B_{\deltata} (x_0)$. Let $\eta \in AC([0,\infty);\overline{\Omegaega})$ such that $\eta(0)=x_0$ and t be the time that $\eta$ first hits $\partial B_{\deltata}(x_0)$. Note that $t$ could be $+\infty$. Then
\begin{equation*}
\begin{aligned}
\int_0^\infty e^{-s} \left(\left\lvert \dot{\eta}(s)\right\rvert ^q + f\left(\eta(s) \right)\right) ds &\geq \int_0^t e^{-s} \left(\left\lvert \dot{\eta}(s)\right\rvert ^q + f \left(\eta (s) \right)\right) ds\\
&\geq \frac{1}{e^t t^{q-1}}\left\lvert \int_0^t \dot{\eta} (s) ds\right\rvert ^q+ \varepsilon \left(1-e^{-t} \right)\\
&\geq \frac{\deltata^q}{e^t t^{q-1}}+\varepsilon \left(1-e^{-t} \right),
\end{aligned}
\end{equation*}
where we used Jensen's inequality in the second line.
This implies $u(x_0)>0$ since $q \geq 2$, which is a contradiction.
\end{proof}
The following lemma is about a crucial estimate that will be used. It is a refined construction of a supersolution for \eqref{eq:PDEeps}.
\begin{lem}\label{lem:super_refined} Let $\deltata_0$ be defined as in \eqref{def:delta_0}. There exist positive constants $\nu = \nu(\deltata_0)> 1$ and $C_{\deltata_0} =\mathcal{O}\left(\deltata_0^{-(\alphapha+2)}\right)$ such that
\begin{equation}\label{e:superwa}
w(x) = \begin{cases}
\displaystyle\frac{\nu C_\alphapha \varepsilon^{\alphapha+1}}{d(x)^\alphapha} + \max f + C_{\deltata_0} \varepsilon^{\alphapha+2}, \qquad\;\;\, p<2,
\\
\displaystyle\nu \varepsilon \log\left(\frac{1}{d(x)}\right) + \max f+ C_{\deltata_0} \varepsilon^2, \qquad p=2,
\end{cases}
\end{equation}
is a supersolution of \eqref{eq:PDEeps} in $\Omegaega$.
\end{lem}
\begin{proof} Let us first consider $1<p<2$. Recall from Theorem \ref{thm:wellposed1<p<2} that $C_\alphapha^p \alphapha^p = C_\alphapha \alphapha (\alphapha+1)$ and $p(\alphapha+1) = \alphapha+2$. Compute
\begin{equation*}
\left\lvert D w(x)\right\rvert ^p = \nu^p\frac{(C_\alphapha\alphapha)^p\varepsilon^{p(\alphapha+1)}}{d(x)^{p(\alphapha+1)}} \left\lvert Dd(x) \right\rvert ^p= \nu^p\frac{C_\alphapha \alphapha(\alphapha+1)\varepsilon^{\alphapha+2}}{d(x)^{\alphapha+2}} \left\lvert Dd(x) \right\rvert ^p
\end{equation*}
and
\begin{equation*}
\varepsilon\Deltata w(x) = \nu\frac{C_\alphapha\alphapha(\alphapha+1)\varepsilon^{\alphapha+2}}{d(x)^{\alphapha+2}} \left\lvert Dd(x)\right\rvert ^2- \nu\frac{C_\alphapha\alphapha\varepsilon^{\alphapha+2}\Deltata d(x)}{d(x)^{\alphapha+1}}.
\end{equation*}
We have
\begin{equation*}
\begin{aligned}
\mathcal{L}^\varepsilon\left[ w \right] = &\frac{\nu C_\alphapha\varepsilon^{\alphapha+1}}{d(x)^\alphapha} + \max f - f(x) + C_{\deltata_0} \varepsilon^{\alphapha+2} \\&+ \frac{C_\alphapha \alphapha (\alphapha+1)\varepsilon^{\alphapha+2}}{d(x)^{\alphapha+2}}\left[\nu^p\left\lvert Dd(x) \right\rvert ^p-\nu \left\lvert Dd(x)\right\rvert ^2+\nu\frac{d(x)\Deltata d(x)}{\alphapha+1}\right] .
\end{aligned}
\end{equation*}
\paragraph{\textbf{Case 1.}} If $0< d(x)\leq \deltata_0$, we have $\left\lvert D d(x)\right\rvert = 1$. Recall that $K_2= \Vert \Deltata d\Vert_{L^\infty}$ and observe
\begin{equation*}
\left\lvert \frac{d(x)\Deltata d(x)}{\alphapha+1}\right\rvert \leq \frac{\deltata_0\Vert \Deltata d\Vert_{L^\infty}}{\alphapha+1} \leq \frac{K_2\deltata_0}{\alphapha+1}\leq K_2\deltata_0.
\end{equation*}
Therefore,
\begin{equation}\label{e:choose_nu}
\begin{split}
\nu^p-\nu +\nu\frac{d(x) \Deltata d(x)}{(\alphapha+1)} \geq \nu^p - \nu - \nu K_2\deltata_0 = \nu\Big(\nu^{p-1} - (1+K_2\deltata_0)\Big).
\end{split}
\end{equation}
We will choose $\nu$ as follows. For $\gammama>1$, we have the inequality
\begin{equation}\label{e:ineq}
\left\lvert \left\lvert x+y\right\rvert ^\gammama - \left\lvert x\right\rvert ^\gammama\right\rvert \leq \gammama\left(\left\lvert x\right\rvert +\left\lvert y\right\rvert \right)^{\gammama-1}\left\lvert y\right\rvert
\end{equation}
for $x,y\in \mathbb{R}$, which implies that
\begin{equation*}
0 \leq (1+K_2\deltata_0)^{\alphapha+1} - 1 \leq\underbrace{(\alphapha+1)\left(1+K_2\deltata_0\right)^\alphapha K_2}_{C_2}\deltata_0.
\end{equation*}
Hence, $(1+K_2\deltata_0)^{\alphapha+1} \leq 1 + C_2\deltata_0$. Since $\alphapha+1 = \frac{1}{p-1}$,
\begin{equation}\label{e:cru1}
(1+K_2\deltata_0) \leq (1+C_2\deltata_0)^{\frac{1}{\alphapha+1}} = (1+C_2\deltata_0)^{p-1}.
\end{equation}
Choose $\nu = 1+C_2\deltata_0$ in \eqref{e:choose_nu} and we obtain $\mathcal{L}[w]\geq 0$ in $\{x\in \Omegaega_\deltata: \deltata <d(x)\leq \deltata_0\}$. \\
\paragraph{\textbf{Case 2.}} If $d(x)\geq \deltata_0$, recall that $K_0 = \Vert d\Vert_{L^\infty}$ and $K_1 = \Vert D d\Vert_{L^\infty}$. And we have
\begin{align*}
\mathcal{L}[w] =& \frac{\nu C_\alphapha \varepsilon^{\alphapha+1}}{d(x)^\alphapha} + \max_{\Omegaega} f - f(x)\\
&+ \nu^p\frac{C_\alphapha\alphapha(\alphapha+1)\varepsilon^{\alphapha+2}}{d(x)^{\alphapha+2}}\left\lvert D d(x)\right\rvert ^p - \nu \frac{C_\alphapha \alphapha(\alphapha+1)\varepsilon^{\alphapha+2}}{d(x)^{\alphapha+2}}\left\lvert D d(x)\right\rvert ^2\\
&+ \nu \frac{C_\alphapha \alphapha \varepsilon^{\alphapha+2}\Deltata d(x)}{d(x)^{\alphapha+1}} + C_{\deltata_0} \varepsilon^{\alphapha+2}\\
\geq &\frac{C_\alphapha \alphapha(\alphapha+1)\varepsilon^{\alphapha+2}}{d(x)^{\alphapha+2}}\left(\nu^p\left\lvert D d(x)\right\rvert ^p - \nu \left\lvert D d(x)\right\rvert ^2 + \nu \frac{d(x) \Deltata d(x)}{\alphapha+1}\right) + C_{\deltata_0} \varepsilon^{\alphapha+2}\\
\geq& \left[C_{\deltata_0} - C_3\left(\frac{1}{\deltata_0}\right)^{\alphapha+2}\right]\varepsilon^{\alphapha+2},
\end{align*}
where
\begin{equation*}
C_3 = C_\alphapha\alphapha(\alphapha+1) \left(\nu^pK_1^p + \nu K_1^2 + \nu \frac{K_0K_2}{\alphapha+1}\right).
\end{equation*}
We can choose $C_{\deltata_0} = C_3\deltata_0^{-(\alphapha+2)}$ to obtain $\mathcal{L}[w]\geq 0$ in $\{x\in \Omegaega_\deltata:d(x)\geq \deltata_0\}$.
If $p=2$, then $\alphapha = 0$. We can easily see that the similar calculation holds true with $\nu := 1+K_2\deltata_0$ and $C_{\deltata_0} := \deltata_0^{-2}\nu (\nu K_1^2 + K_1^2+ K_0K_2)$.
\end{proof}
Now we begin to present the rate of convergence for the special case where $f = C_f$ in $\Omegaega$ for some constant $C_f$.
\begin{theorem}[Constant data]\label{thm:rate_doubling0} Assume $f\equiv C_f$ in $\Omegaega$. Let $u^\varepsilon$ be the unique solution to \eqref{eq:PDEeps} and $u \equiv C_f$ be the unique solution to \eqref{eq:PDE0}. Then there exists a constant $C$ independent of $\varepsilon\in(0,1)$ such that
\begin{equation*}
\begin{split}
&0\leq u^\varepsilon(x) - u(x)\leq C \left(\frac{ \varepsilon^{\alphapha+1}}{d(x)^\alphapha} + \frac{\varepsilon^{\alphapha+2}}{\deltata_{0,\Omegaega}^{\alphapha+2}}\right), \qquad\qquad \;\;\; \text{if}\; 1<p<2,\\
&0\leq u^\varepsilon(x) - u(x)\leq C \left(\varepsilon \mathrm{log}\left(\frac{1}{d(x)}\right) + \frac{\varepsilon^{2}}{\deltata_{0,\Omegaega}^{2}}\right), \qquad \text{if}\; p=2,
\end{split}
\end{equation*}
for $x\in \Omegaega$, where $\deltata_{0,\Omegaega}$ is defined as in \eqref{def:delta_0}. In particular,
\begin{itemize}
\item[(i)] if $1<p<2$, we have $C_f\leq u^\varepsilon(x)\leq C_f + C\varepsilon$ for $x\in \Omegaega_\varepsilon $, and
\item[(ii)] for any $K\subset\subset \Omegaega$, there holds $\Vert u^\varepsilon - u\Vert_{L^\infty(K)} \leq C\varepsilon^{\alphapha+1}$ .
\end{itemize}
\end{theorem}
\begin{proof} Lemma \ref{lem:f=0} implies $u \equiv C_f$ in $\Omegaega$. And Lemma \ref{lem:lower-bound} tells us $u^\varepsilon-u=u^\varepsilon-C_f \geq 0$. By the comparison principle of \eqref{eq:PDEeps} and Lemma \ref{lem:super_refined}, the conclusion follows.
\end{proof}
\begin{remark} The conclusion of Theorem \ref{thm:rate_doubling0} also holds if $f = C_f+ \mathcal{O}(\varepsilon^{\beta})$ for $\beta \geq \alphapha+1$.
\end{remark}
Even this special case (Theorem \ref{thm:rate_doubling0}) is new in the literature. As an immediate consequence, we obtain the rate of convergence on any compact subset that is disjoint from the support of $f$.
\begin{cor} Assume $f$ is Lipschitz with compact support and $K$ is a connected compact subset of $\Omegaega$ that is disjoint from $\mathrm{supp}(f)$. Then there exists a constant $C = C(K)$ independent of $\varepsilon\in (0,1)$ such that
\begin{equation*}
\Vert u^\varepsilon - u\Vert_{L^\infty(K)} \leq C\varepsilon^{\alphapha+1}.
\end{equation*}
\end{cor}
\begin{proof} We choose an open, bounded and connected set $U$ such that $\partial U$ is $\mathrm{C}^2$ and $K\subset\subset U \subset\subset \Omegaega$. Let $w^\varepsilon$ be the solution to \eqref{eq:PDEeps} with $\Omegaega$ replaced by $U$. Then by Theorem \ref{thm:rate_doubling0}, we have
\begin{equation*}
0\leq w^\varepsilon(x)\leq C\left(\varepsilon^{\alphapha+1} + \varepsilon^{\alphapha+2}\right), \qquad x\in K,
\end{equation*}
where $C$ depends on $\mathrm{dist}(K, \partial U)$ and $U$. Recall that $u = 0$ outside the support of $f$. By the comparison principle in $U$, we see that $u^\varepsilon \leq w^\varepsilon$ and thus the conclusion follows.
\end{proof}
For the general result of nonnegative compactly supported data, we have the following theorem.
\begin{theorem}[Nonnegative compactly supported data]\label{thm:rate_doubling1} Assume that $f$ is nonnegative and Lipschitz with compact support in $\Omegaega_{\kappapa}$ for some $\kappapa >0$. Let $u^\varepsilon$ be the unique solution to \eqref{eq:PDEeps} and $u$ be the unique solution to \eqref{eq:PDE0}. Then there exists a constant $C$ independent of $\varepsilon\in(0,1)$ and $\kappapa$ such that
\begin{align}
&-C \sqrt{\varepsilon}\leq u^\varepsilon(x) - u(x) \leq C\left(\sqrt{\varepsilon}+\left(\frac{\varepsilon}{\kappapa}\right)^{\alphapha+2}\right) + \frac{\nu C_\alphapha \varepsilon^{\alphapha+1}}{d(x)^\alphapha}, \quad\; \qquad p<2,\label{eq:cp1}\\
&-C\sqrt{\varepsilon}\leq u^\varepsilon(x) - u(x) \leq C \left(\sqrt{\varepsilon}+ \left(\frac{\varepsilon}{\kappapa}\right)^2\right) + \nu \varepsilon \log\left(\frac{1}{d(x)}\right), \qquad p=2, \label{eq:cp2}
\end{align}
for any $x\in \Omegaega$. As a consequence, $\left\lvert u^\varepsilon(x)-u(x)\right\rvert \leq C\sqrt{\varepsilon}$ for all $x\in \Omegaega_\varepsilon$.
\end{theorem}
We state the following lemma as a preparation.
\begin{lem} Let $0<\kappapa < \deltata_0$ and $U_\kappapa = \big\{x\in \Omegaega: 0<\mathrm{dist}(x,\partial\Omegaega) < \kappapa\big\} = \Omegaega\backslash \overline{\Omegaega}_\kappapa$. There holds
\begin{equation*}
\mathrm{dist}(x,\partial\Omegaega_\kappapa) = \kappapa - \mathrm{dist}(x,\partial\Omegaega) \qquad\text{for all}\;x\in U_\kappapa.
\end{equation*}
As a consequence, $x\mapsto \mathrm{dist}(x,\partial U_\kappapa) = \min\big\{\mathrm{dist}(x,\partial \Omegaega_k),\mathrm{dist}(x,\partial \Omegaega)\big\}$ is twice continuously differentiable for $x\in \Omegaega\backslash \overline{\Omegaega}_{\kappapa/2}$. Hence, we can choose
\begin{equation}\label{e:delta_kappa}
\deltata_{0,U_\kappapa} \geq \frac{\kappapa}{4}
\end{equation}
where $\deltata_{0,\Omegaega}$ is defined as in \eqref{def:delta_0}.
\end{lem}
\begin{proof} By the definition of $\deltata_0 = \deltata_{0,\Omegaega}$, we have $d(x) = \mathrm{dist}(x,\partial\Omegaega)$ is twice continuously differentiable in the region $U_{\deltata_0} = \Omegaega\backslash \overline{\Omegaega}_{\deltata_0}$. The proof follows from \cite[p. 355]{gilbarg_elliptic_2001}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:rate_doubling1}] Without loss of generality, assume that $f$ is supported in $\Omegaega_\kappapa$ where $0<\kappapa < \deltata_{0}$.
Let $g_\kappapa = u^\varepsilon$ on $\partial\Omegaega_{\kappapa}$. Then the solution $u^\varepsilon$ of \eqref{eq:PDEeps} also solves
\begin{equation*}
\left\{
\begin{aligned}
u^\varepsilon(x) + \left\lvert Du^\varepsilon(x)\right\rvert ^p-\varepsilon \Deltata u^\varepsilon(x) &=0 \;\qquad \text{in } U_\kappapa ,\\
u^\varepsilon(x) &= +\infty \quad \text{on } \partial \Omegaega,\\
u^\varepsilon(x) &= g_\kappapa \;\;\quad \text{on } \partial \Omegaega_{\kappapa},
\end{aligned}
\right.
\end{equation*}
in $U_\kappapa= \Omegaega \setminus \overline{\Omegaega}_{\kappapa} = \{x\in \Omegaega: 0< d(x) < \kappapa\}$. Let $\tilde{u}^\varepsilon\in \mathrm{C}^2(U_\kappapa)$ be the solution to the following problem
\begin{equation*}
\left\{
\begin{aligned}
\tilde{u}^\varepsilon(x) + \left\lvert D\tilde{u}^\varepsilon(x)\right\rvert ^p-\varepsilon \Deltata \tilde{u}^\varepsilon(x) &=0 \;\qquad \text{in } U_\kappapa ,\\
\tilde{u}^\varepsilon(x) &= +\infty \quad \text{on } \partial U_\kappapa = \partial \Omegaega\cup \partial \Omegaega_{\kappapa},
\end{aligned}
\right.
\end{equation*}
whose existence is guaranteed by Theorem \ref{thm:wellposed1<p<2}.
Here the boundary condition is understood in the sense that $\tilde{u}^\varepsilon(x)\to \infty$ as $d_\kappapa(x)\to 0$, where $d_\kappapa(\cdot)$ is the distance function from the boundary of $U_\kappapa$, i.e.,
\begin{equation*}
d_\kappapa(x) = \min \big\lbrace \mathrm{dist}(x,\partial \Omegaega_\kappapa),\mathrm{dist}(x,\partial\Omegaega) \big\rbrace \leq d(x) \qquad\text{for}\;x\in U_\kappapa.
\end{equation*}
Since $f = 0$ in $\overline{U}_\kappapa$, by Lemma \ref{lem:f=0}, $u=0$ in $\overline{U}_\kappapa$. Hence, $u$ is also the unique state-constraint solution to
\begin{equation*}
\left\{
\begin{aligned}
u(x)+ \left\lvert Du(x)\right\rvert ^p &=0 \quad \text{in } U_\kappapa ,\\
u(x)+ \left\lvert Du(x)\right\rvert ^p &\geq 0 \quad \text{on } \partial U_\kappapa = \partial \Omegaega \cup\partial \Omegaega_{\kappapa}.
\end{aligned}
\right.
\end{equation*}
The vanishing viscosity of $\tilde{u}^\varepsilon \to 0$ in $U_\kappapa$ can be quantified by Theorem \ref{thm:rate_doubling0}, which gives us
\begin{equation*}
\begin{split}
&0\leq \tilde{u}^\varepsilon(x) \leq \frac{\nu C_\alphapha \varepsilon^{\alphapha+1}}{d_\kappapa(x)^\alphapha}+C_3\left(\frac{\varepsilon}{\deltata_{0,U_\kappapa}}\right)^{\alphapha+2}\qquad\quad\;\text{for}\;p<2,\\
&0\leq \tilde{u}^\varepsilon(x) \leq \nu \varepsilon \log\left(\frac{1}{d_\kappapa(x)}\right)+C\left(\frac{\varepsilon}{\deltata_{0,U_\kappapa}}\right)^{2}\qquad\text{for}\;p=2,
\end{split}
\end{equation*}
for $x\in U_\kappapa$. From \eqref{e:delta_kappa} and the comparison principle in $U_\kappapa$, we have
\begin{align}
&0\leq u^\varepsilon(x) \leq \tilde{u}^\varepsilon(x) \leq \frac{\nu C_\alphapha\varepsilon^{\alphapha+1}}{d_\kappapa(x)^{\alphapha}} + C_3\left(\frac{4\varepsilon}{\kappapa}\right)^{\alphapha+2} \qquad\quad\;\text{for}\;p<2, \qquad \label{annulus2}\\
&0\leq u^\varepsilon(x)\leq \tilde{u}^\varepsilon(x) \leq \nu \varepsilon \log\left(\frac{1}{d_\kappapa(x)}\right)+C\left(\frac{4\varepsilon}{\kappapa}\right)^{2}\qquad\text{for}\;p=2,\label{annulus2p=2}
\end{align}
for $x\in U_\kappapa$.
We proceed with the doubling variable method. For $p<2$, consider the auxiliary functional
\begin{equation*}
\Phi(x,y)= u^\varepsilon(x) - u(y) -\frac{C_0\left\lvert x-y\right\rvert ^2}{\sigmama} - \frac{\nu C_\alphapha \varepsilon^{\alphapha +1}}{d(x)^\alphapha}, \qquad (x,y)\in \overline{\Omegaega}\times \overline{\Omegaega},
\end{equation*}
where $C_0$ is the Lipschitz constant of $u$ from \eqref{e:C0}, $\sigmama\in (0,1)$. The fact that $\displaystyle d(x)^\alphapha u^\varepsilon(x) \to C_\alphapha \varepsilon^{\alphapha+1}$ as $d(x) \to 0^+$ implies
\begin{equation*}
\max_{(x,y) \in \overline{\Omegaega} \times \overline{\Omegaega}} \Phi(x,y) = \Phi(x_\sigmama, y_\sigmama) \qquad\text{for some}\;(x_\sigmama,y_\sigmama) \in \Omegaega \times \overline{\Omegaega}.
\end{equation*}
From $\Phi(x_\sigmama, y_\sigmama) \geq \Phi(x_\sigmama, x_\sigmama)$, we can deduce that
\begin{equation}\label{e:sigma}
\left\lvert x_\sigmama - y_\sigmama \right\rvert \leq \sigmama.
\end{equation}
If $ d(x_\sigmama) \geq \frac{1}{2}\kappapa$, since $x\mapsto \Phi(x,y_\sigmama)$ has a maximum over $\Omegaega$ at $x=x_\sigmama$, the subsolution test for $u^\varepsilon(x)$ gives us
\begin{align}\label{e:subsln}
&u^\varepsilon(x_\sigmama) + \left\lvert \frac{2C_0(x_\sigmama - y_\sigmama)}{\sigmama} - \frac{\nu C_\alphapha\alphapha \varepsilon^{\alphapha+1} D d(x_\sigmama)}{d(x_\sigmama)^{\alphapha+1}}\right\rvert ^p - f(x_\sigmama)\nonumber\\
&\qquad -\varepsilon\left(\frac{2nC_0}{\sigmama}+ \frac{\nu C_\alphapha\alphapha(\alphapha+1) \varepsilon^{\alphapha+1}\left\lvert D d(x_\sigmama)\right\rvert ^2}{d(x_\sigmama)^{\alphapha+2}} - \frac{\nu C_\alphapha\alphapha \varepsilon^{\alphapha+1}\Deltata d(x_\sigmama)}{d(x_\sigmama)^{\alphapha+1}}\right) \leq 0.
\end{align}
Since $y\mapsto \Phi(x_\sigmama,y)$ has a maximum over $\overline{\Omegaega}$ at $y = y_\sigmama$, the supersolution test for $u(y)$ gives us
\begin{align}\label{e:supersln}
u(y_\sigmama) + \left\lvert \frac{2C_0(x_\sigmama - y_\sigmama)}{\sigmama}\right\rvert ^p - f(y_\sigmama) \geq 0.
\end{align}
For simplicity, define
\begin{equation*}
\xi_\sigmama := \frac{2C_0(x_\sigmama - y_\sigmama)}{\sigmama} \qquad\text{and}\qquad \zeta_\sigmama :=- \frac{\nu C_\alphapha\alphapha \varepsilon^{\alphapha+1} D d(x_\sigmama)}{d(x_\sigmama)^{\alphapha+1}}.
\end{equation*}
From \eqref{e:sigma} and $d(x_\sigmama) \geq \frac{1}{2}\kappapa$,
\begin{equation*}
\left\lvert \xi_\sigmama\right\rvert \leq 2C_0, \qquad\text{and}\qquad \left\lvert \zeta_\sigmama\right\rvert \leq \nu K_1 C_\alphapha\alphapha \left(\frac{\varepsilon}{d(x_\sigmama)}\right)^{\alphapha+1} \leq \nu K_1 C_\alphapha \alphapha \left(\frac{2\varepsilon}{\kappapa}\right)^{\alphapha+1}.
\end{equation*}
Using the inequality \eqref{e:ineq} with $\gammama = p > 1$, we deduce that
\begin{align}\label{e:estia}
\left\lvert \left\lvert \xi_\sigmama +\zeta_\sigmama\right\rvert ^p - \left\lvert \xi_\sigmama\right\rvert ^p \right\rvert &\leq p\Big(\left\lvert \xi_\sigmama\right\rvert +\left\lvert \zeta_\sigmama\right\rvert \Big)^{p-1}\left\lvert \zeta_\sigmama\right\rvert \nonumber\\
&\leq p\left[2C_0+\nu K_1 C_\alphapha\alphapha \left(\frac{2\varepsilon}{\kappapa}\right)^{\alphapha+1}\right]^{p-1}\nu K_1 C_\alphapha\alphapha \left(\frac{2\varepsilon}{\kappapa}\right)^{\alphapha+1}.
\end{align}
Combine \eqref{e:estia} together with \eqref{e:subsln}, \eqref{e:supersln} and $\left\lvert f(x_\sigmama) - f(y_\sigmama)\right\rvert \leq C\left\lvert x_\sigmama - y_\sigmama\right\rvert \leq C\sigmama$ to obtain
\begin{align*}
u^\varepsilon(x_\sigmama) - u(y_\sigmama) \leq & p\left(2C_0+\nu K_1 C_\alphapha \alphapha\left( \frac{2\varepsilon}{\kappapa}\right)^{\alphapha+1}\right)^{p-1}\nu K_1 C_\alphapha \alphapha \left(\frac{2\varepsilon}{\kappapa}\right)^{\alphapha+1} + C\sigmama\nonumber\\
&+2nC_0\left(\frac{\varepsilon}{\sigmama}\right) + \nu K_1^2C_\alphapha \alphapha(\alphapha+1)\left(\frac{2\varepsilon}{\kappapa}\right)^{\alphapha+2} + \nu K_2 C_\alphapha \alphapha \left(\frac{2\varepsilon}{\kappapa}\right)^{\alphapha+1}\varepsilon\nonumber\\
\leq & C\left[\sigmama + \frac{\varepsilon}{\sigmama} + \left(1+\left(\frac{\varepsilon}{\kappapa}\right)^{\alphapha+1}\right)^{p-1}\left(\frac{\varepsilon}{\kappapa}\right)^{\alphapha+1} + \left(\frac{\varepsilon}{\kappapa}\right)^{\alphapha+2} \right].
\end{align*}
By the fact that $(1+x)^\gammama \leq 1+x^\gammama$ for $x\in [0,1]$ and $\gammama\in [0,1]$, we know
\begin{equation*}
\left(1+\left(\frac{\varepsilon}{\kappapa}\right)^{\alphapha+1}\right)^{p-1} \leq 1+ \left(\frac{\varepsilon}{\kappapa}\right),
\end{equation*}
as $0<p-1\leq 1$.
Therefore,
\begin{equation*}
u^\varepsilon(x_\sigmama) - u(y_\sigmama) \leq C\left[\sigmama + \frac{\varepsilon}{\sigmama} + \left(\frac{\varepsilon}{\kappapa}\right)^{\alphapha+1} + \left(\frac{\varepsilon}{\kappapa}\right)^{\alphapha+2}\right],
\end{equation*}
where $C$ is independent of $\kappapa$ and $\varepsilon$. Now choose $\sigmama = \sqrt{\varepsilon}$ to get (with $\kappapa$ fixed)
\begin{equation}\label{e:final1}
\Phi(x_\sigmama,y_\sigmama) \leq u^\varepsilon(x_\sigmama) - u(y_\sigmama) \leq C\sqrt{\varepsilon}.
\end{equation}
If $d(x_\sigmama) < \frac{1}{2}\kappapa$, then $x_\sigmama \in U_\kappapa$ and furthermore $\mathrm{dist}(x_\sigmama,\partial \Omegaega_\kappapa) > \frac{1}{2}\kappapa$. Indeed, for any $y\in\partial\Omegaega$ and $z\in \partial\Omegaega_k$, we have $\left\lvert x_\sigmama - z\right\rvert + \left\lvert x_\sigmama - y\right\rvert \geq \left\lvert y-z\right\rvert $. Taking the infimum over all $y\in \partial\Omegaega$, we deduce that
\begin{equation*}
\left\lvert x_\sigmama - z\right\rvert + d(x_\sigmama) \geq \inf_{y\in \partial \Omegaega}\left\lvert y-z\right\rvert = d(z) = \kappapa
\end{equation*}
since $z\in \partial\Omegaega_k = \{x \in \Omegaega: d(x) = \kappapa\}$. Thus, $\left\lvert x_\sigmama - z\right\rvert \geq \kappapa - d(x_\sigmama) > \frac{1}{2}\kappapa$ for all $z\in \partial\Omegaega_k$, which implies that $\mathrm{dist}(x_\sigmama,\partial\Omegaega_k)>\frac{1}{2}\kappapa$ and hence $d_\kappapa(x_\sigmama) = d(x_\sigmama)$. By \eqref{annulus2} and the fact that $u \geq 0$, we have
\begin{align}
\Phi(x_\sigmama, y_\sigmama)\leq u^\varepsilon(x_\sigmama) - \frac{\nu C_\alphapha \varepsilon^{\alphapha+1}}{d(x_\sigmama)^\alphapha} \leq C_3\left(\frac{4\varepsilon}{\kappapa}\right)^{\alphapha+2} \label{e:final2}.
\end{align}
Since $\Phi(x,x) \leq \Phi(x_\sigmama,y_\sigmama)$ for all $x\in \Omegaega$, we obtain from \eqref{e:final1} and \eqref{e:final2} that
\begin{equation*}
u^\varepsilon(x)-u(x)-\frac{\nu C_\alphapha \varepsilon^{\alphapha+1}}{d(x)^\alphapha} \leq C\sqrt{\varepsilon} +C_3\left(\frac{4\varepsilon}{\kappapa}\right)^{\alphapha+2}
\end{equation*}
and thus \eqref{eq:cp1} follows.
For $p=2$, we consider instead the functional
\begin{equation*}
\Phi(x,y) = u^\varepsilon(x) - u(y) - \frac{C_0\left\lvert x-y\right\rvert ^2}{\sigmama} - \nu \varepsilon \mathrm{log}\left(\frac{1}{d(x)}\right), \qquad (x,y)\in \overline{\Omegaega}\times \overline{\Omegaega}.
\end{equation*}
Similar to the previous case where $1<p<2$, the maximum of $\Phi$ occurs at some point $(x_\sigmama,y_\sigmama)\in \Omegaega\times\overline{\Omegaega}$ and $\left\lvert x_\sigmama - y_\sigmama\right\rvert \leq \sigmama$. If $d(x_\sigmama)\geq \frac{1}{2}\kappapa$, by the subsolution test for $u^\varepsilon(x)$, we have
\begin{align}\label{e:subslnp=2}
u^\varepsilon(x_\sigmama)&+ \left\lvert \frac{2C_0(x_\sigmama - y_\sigmama)}{\sigmama} - \nu \varepsilon \frac{D d(x_\sigmama)}{d(x_\sigmama)}\right\rvert ^2 -f(x_\sigmama) \nonumber\\
&- 2nC_0\left(\frac{\varepsilon}{\sigmama}\right) - \nu \left\lvert D d(x_\sigmama)\right\rvert ^2 \left(\frac{\varepsilon}{d(x_\sigmama)}\right)^2 + \nu \Deltata d(x_\sigmama)\left(\frac{\varepsilon^2}{d(x_\sigmama)}\right) \leq 0.
\end{align}
By the supersolution test for $u(y)$, we have
\begin{equation}\label{e:superslnp=2}
u(y_\sigmama) + \left\lvert \frac{2C_0(x_\sigmama - y_\sigmama)}{\sigmama}\right\rvert ^2 - f(y_\sigmama) \geq 0.
\end{equation}
Subtract \eqref{e:superslnp=2} from \eqref{e:subslnp=2} to get
\begin{align*}
u^\varepsilon(x_\sigmama) - u(y_\sigmama) \leq &\left(4C_0+ \nu\varepsilon\frac{D d(x_\sigmama)}{d(x_\sigmama)}\right)\left(\nu \varepsilon \frac{ D d(x_\sigmama)}{d(x_\sigmama)}\right) \\
&+ C\sigmama + 2nC_0 \left(\frac{\varepsilon}{\sigmama}\right)+ \nu \left\lvert D d(x_\sigmama)\right\rvert ^2 \left(\frac{\varepsilon}{d(x_\sigmama)}\right)^2 + \nu\left\lvert \Deltata d(x_\sigmama)\right\rvert \frac{\varepsilon^2}{d(x_\sigmama)}.
\end{align*}
Using $d(x_\sigmama)\geq \frac{1}{2}\kappapa$ and bounds on $d(x)$, we see that
\begin{align}\label{p=2a}
\Phi(x_\sigmama,y_\sigmama)&\leq u^\varepsilon(x_\sigmama) - u(y_\sigmama)\nonumber \\
&\leq 4K_1^2\nu(1+\nu)\left(\frac{\varepsilon}{\kappapa}\right)^2 + C\sigmama + 2nC_0\left( \frac{\varepsilon}{\sigmama}\right) + 2\nu(K_2\varepsilon+4C_0K_1)\left(\frac{\varepsilon}{\kappapa}\right)\nonumber\\
&\leq C\left(\sigmama+\frac{\varepsilon}{\sigmama} + \frac{\varepsilon}{\kappapa} + \left(\frac{\varepsilon}{\kappapa}\right)^2\right) \leq C\sqrt{\varepsilon}
\end{align}
if we choose $\sigmama = \sqrt{\varepsilon}$.
If $d(x_\sigmama)<\frac{1}{2}\kappapa$, then $x_\sigmama\in U_\kappapa$. Again, we have $d_\kappapa(x_\sigmama) = d(x_\sigmama)$ and from \eqref{annulus2p=2}
\begin{equation}\label{p=2b}
\Phi(x_\sigmama,y_\sigmama) \leq u^\varepsilon(x_\sigmama) - \nu \varepsilon \log\left(\frac{1}{d(x_\sigmama)}\right) \leq C\left(\frac{4\varepsilon}{\kappapa}\right)^2.
\end{equation}
Since $\Phi(x,x)\leq \Phi(x_\sigmama,y_\sigmama)$ for $x\in \Omegaega$, we obtain from \eqref{p=2a} and \eqref{p=2b} that
\begin{equation*}
u^\varepsilon(x) - u(x) - \nu\varepsilon\log\left(\frac{1}{d(x)}\right) \leq C\sqrt{\varepsilon} +C\left(\frac{4\varepsilon}{\kappapa}\right)^2
\end{equation*}
and thus \eqref{eq:cp2} follows.
\end{proof}
\begin{remark} For general nonnegative Lipschitz data $f\in \mathrm{C}(\overline{\Omegaega})$, it is natural to try a cutoff function argument. Let $\chi_{{\kappapa}}\in \mathrm{C}_c^\infty(\Omegaega)$ such that $0\leq \chi_{\kappapa}\leq 1$, $\chi_\kappapa = 1$ in $\Omegaega_{2\kappapa}$ and $\mathrm{supp}\;\chi_\kappapa\subset\Omegaega_\kappapa$. Let $u^\varepsilon_\kappapa\in \mathrm{C}^2(\Omegaega)\cap\mathrm{C}(\overline{\Omegaega})$ solve \eqref{eq:PDEeps} with data $f\chi_{\kappapa}$. Then $u^\varepsilon_\kappapa\to u^\varepsilon$ as $\kappapa\to 0$ (since $f\chi_\kappapa\to f$ in the weak$^*$ topology of $L^\infty(\Omegaega)$ and we have the continuity of the solution to \eqref{eq:PDEeps} with respect to data in this topology \cite[Remark II.1]{Lasry1989}). However, it is not clear at the moment how to quantify this rate of convergence, since $f\chi_\kappapa$ does not converge to $f$ in the uniform norm, unless $f = 0$ on $\partial\Omegaega$.
\end{remark}
\subsection{A rate for nonnegative zero boundary data}
We prove the rate of convergence for the case where $f$ is nonnegative with $f=0$ on $\partial \Omegaega$.
\begin{proof}[Proof of Theorem \ref{main_thm1}]
Let $L = \Vert D f\Vert_{L^\infty(\Omegaega)}$ be the Lipschitz constant of $f$. For $\kappapa>0$ small such that $0<\kappapa<\deltata_0$ and $x\in \Omegaega\backslash \Omegaega_\kappapa$, let $x_0$ be the projection of $x$ onto $\partial\Omegaega$. We observe that
\begin{equation}\label{e:bound_aa}
f(x) = f(x) - f(x_0) \leq L\left\lvert x-x_0\right\rvert = L\kappapa.
\end{equation}
Define
\begin{equation*}
g_\kappapa(x) =
\begin{cases}
0 &\qquad\text{if}\;0\leq d(x) \leq \kappapa/2,\\
2L\left(d(x)-\kappapa/2\right) &\qquad\text{if}\;\kappapa/2\leq d(x) \leq \kappapa.
\end{cases}
\end{equation*}
It is clear that for $x\in \partial\Omegaega_\kappapa$, $g_\kappapa(x) = L\kappapa \geq f(x)$ since \eqref{e:bound_aa}. Therefore, we can define the following continuous function
\begin{equation}\label{def_f_kappa}
f_\kappapa(x) =
\begin{cases}
0 &\qquad\text{if}\;0\leq d(x) \leq \kappapa/2,\\
\min \left\lbrace g_\kappapa(x), f(x) \right\rbrace &\qquad\text{if}\;\kappapa/2\leq d(x) \leq \kappapa,\\
f(x) &\qquad\text{if}\;\kappapa \leq d(x).
\end{cases}
\end{equation}
A graph of $f_\kappapa$ is given in Figure \ref{fig:f_kappa}.
\begin{figure}
\caption{Graph of the function $f_\kappapa$.}
\label{fig:f_kappa}
\end{figure}
The continuity at $x\in \partial\Omegaega_\kappapa$ comes from the fact that when $d(x) =\kappapa$, we have $g_k(x) = L\kappapa \geq f(x)$ by \eqref{e:bound_aa}. It is clear that $f_\kappapa$ is Lipschitz with $\Vert f_\kappapa\Vert_{L^\infty(\Omegaega)}\leq L$ as well and $f_\kappapa\to f$ uniformly as $\kappapa\to 0$. Indeed, we have $0\leq f_\kappapa \leq f$ and
\begin{equation*}
0\leq \max_{x\in \overline{\Omegaega}} (f(x) - f_\kappapa(x)) \leq \max_{x\in \overline{\Omegaega}\backslash \overline{\Omegaega}_\kappapa} (f(x) - f_\kappapa(x)) = \max_{x\in \overline{\Omegaega}\backslash \overline{\Omegaega}_\kappapa} f(x) \leq L\kappapa.
\end{equation*}
Let $u^\varepsilon_\kappapa\in \mathrm{C}^2(\Omegaega)\cap\mathrm{C}(\overline{\Omegaega})$ be the solution to \eqref{eq:PDEeps} with data $f\chi_{\kappapa}$ and $u_k\in \mathrm{C}(\overline{\Omegaega})$ be the corresponding solution to \eqref{eq:PDE0} with data $f{\chi_\kappapa}$. By the comparison principle (\cite[Corollary II.1]{Lasry1989}), we have
\begin{equation}\label{e:bound_2aa}
0\leq u^\varepsilon(x) - u^\varepsilon_\kappapa(x) \leq L\kappapa \qquad\text{for}\;x\in \Omegaega.
\end{equation}
By the comparison principle for \eqref{eq:PDE0}, we also have
\begin{equation}\label{e:bound_2ab}
0\leq u(x) - u_\kappapa(x) \leq L\kappapa \qquad\text{for}\;x\in \Omegaega.
\end{equation}
If $1<p<2$, by Theorem \ref{thm:rate_doubling1}, there exists a constant $C$ independent of $\kappapa$ such that
\begin{equation}\label{e:bound_2ac}
-C\sqrt{\varepsilon}\leq u^\varepsilon_\kappapa(x) - u_\kappapa(x)\leq C\left[\sqrt{\varepsilon} + \left(\frac{\varepsilon}{\kappapa}\right)^{\alphapha+2} + \frac{\varepsilon^{\alphapha+1}}{d(x)^\alphapha}\right], \qquad x\in \Omegaega.
\end{equation}
Combining \eqref{e:bound_2aa}, \eqref{e:bound_2ab} and \eqref{e:bound_2ac}, we obtain
\begin{equation*}
\begin{split}
-C\sqrt{\varepsilon}\leq u^\varepsilon(x) - u(x) &= \Big(u^\varepsilon(x) - u^\varepsilon_\kappapa(x)\Big) + \Big(u^\varepsilon_\kappapa(x) - u_\kappapa(x)\Big) + \Big(u_\kappapa(x) - u(x)\Big) \\
&\leq L\kappapa + C\left[\sqrt{\varepsilon} + \left(\frac{\varepsilon}{\kappapa}\right)^{\alphapha+2} + \frac{\varepsilon^{\alphapha+1}}{d(x)^\alphapha}\right], \qquad x\in \Omegaega.
\end{split}
\end{equation*}
Choose $\kappapa = \sqrt{\varepsilon}$ and we deduce that
\begin{equation*}
-C\sqrt{\varepsilon}\leq u^\varepsilon(x) - u(x) \leq C\sqrt{\varepsilon} + \frac{C\varepsilon^{\alphapha+1}}{d(x)^\alphapha}
\end{equation*}
for $x\in \Omegaega$. Thus, the conclusion follows. \\
If $p=2$, by Theorem \ref{thm:rate_doubling1}, there exists a constant $C$ independent of $\kappapa$ such that
\begin{equation}\label{e:bound_2acp=2}
-C\sqrt{\varepsilon}\leq u^\varepsilon_\kappapa(x) - u_\kappapa(x) \leq C\left[\sqrt{\varepsilon} + \left(\frac{\varepsilon}{\kappapa}\right)^2 + \varepsilon\log\left(\frac{1}{d(x)}\right)\right], \qquad x\in \Omegaega.
\end{equation}
Combining \eqref{e:bound_2aa}, \eqref{e:bound_2ab} and \eqref{e:bound_2acp=2}, we obtain
\begin{equation*}
\begin{split}
-C\sqrt{\varepsilon}\leq u^\varepsilon(x) - u(x) &= \Big(u^\varepsilon(x) - u^\varepsilon_\kappapa(x)\Big) + \Big(u^\varepsilon_\kappapa(x) - u_\kappapa(x)\Big) + \Big(u_\kappapa(x) - u(x)\Big) \\
&\leq L\kappapa + C\left[\sqrt{\varepsilon} +\; \left(\frac{\varepsilon}{\kappapa}\right)^{2} + \varepsilon\log\left(\frac{1}{d(x)}\right) \right], \qquad x\in \Omegaega.
\end{split}
\end{equation*}
Choose $\kappapa = \varepsilon$ and we deduce that
\begin{equation*}
-C\sqrt{\varepsilon}\leq u^\varepsilon(x) - u(x) \leq C\sqrt{\varepsilon} +\varepsilon\log\left(\frac{1}{d(x)}\right)
\end{equation*}
for $x\in \Omegaega$. Thus, the conclusion follows.
\end{proof}
\section{Improved one-sided rate of convergence}
In this section, we assume $f\in \mathrm{C}^2(\overline{\Omegaega})$ (or uniformly semiconcave in $\overline{\Omegaega}$) such that $f = 0$ on $\partial\Omegaega$ and $f\geq 0$. It is known that for the problem on $\mathbb{R}^n$, namely,
\begin{equation*}
u(x) + \left\lvert Du\right\rvert ^p - f(x) = 0 \qquad\text{in}\;\mathbb{R}^n,
\end{equation*}
if $f$ is semiconcave in the whole space $\mathbb{R}^n$, then the solution $u$ is also semiconcave (Theorem \ref{convex}, see also \cite{Calder2021}).
\begin{remark}\label{rem:heuristic} The heuristic idea that we will use in this section is the following. Assume that $u^\varepsilon(x) - u(x)$ has a maximum over $\overline{\Omegaega}$ at some interior point $x_0\in \Omegaega$. Then by the equation \eqref{eq:PDEeps} at $x_0$ and the supersolution test for \eqref{eq:PDE0} at $x_0$, we obtain
\begin{equation*}
\max_{x\in \overline{\Omegaega}}\Big( u^\varepsilon(x) - u(x)\Big) \leq u^\varepsilon(x_0) - u(x_0) \leq \varepsilon \Deltata u^\varepsilon(x_0).
\end{equation*}
If $u$ is uniformly semiconcave in $\overline{\Omegaega}$, then $\Deltata u^\varepsilon(x_0)\leq \Deltata u(x_0) \leq C$. Thus, we obtain a better one-sided rate $\mathcal{O}(\varepsilon)$ for $u^\varepsilon-u$. However, there are a couple of problems with this argument. Firstly, as $u^\varepsilon = +\infty$ on $\partial\Omegaega$, we need to subtract an appropriate term from $u^\varepsilon$ to make a maximum over $\overline{\Omegaega}$ happen in the interior. Secondly, unless $f\in \mathrm{C}^2_c(\Omegaega)$, in general, $u$ is not uniformly semiconcave but only \emph{locally semiconcave}. In this section, we provide estimates on the local semiconcavity constant of $u$ and rigorously show how the upper bound of $u^\varepsilon-u$ can be obtained.
\end{remark}
From Lemma \ref{lem:f=0}, we have $u = 0$ on $\partial\Omegaega$. It is clear that the solution $u$ to \eqref{eq:PDE0} is also the unique solution to the following Dirichlet boundary problem
\begin{equation}\label{eq:Dirichlet}
\begin{cases}
u(x) + \left\lvert Du(x)\right\rvert ^p = f(x) &\qquad\text{in}\;\Omegaega,\\
\quad\quad \quad\quad\;\;\; u(x) = 0 &\qquad\text{on}\;\partial\Omegaega.
\end{cases}
\end{equation}
Since $H(x,\xi) = \left\lvert \xi\right\rvert ^p - f(x)$, the corresponding Legendre transform is
\begin{equation*}
L(x,v) = C_p\left\lvert v\right\rvert ^q + f(x)
\end{equation*}
where $p^{-1} + q^{-1} = 1$ and $C_p$ is defined in Lemma \ref{lem:f=0}. Let us extend $f$ to a function $\tilde{f}:\mathbb{R}^n\to \mathbb{R}$ by setting $\tilde{f}(x) = 0$ for $x\notin \Omegaega$.
\begin{defn} Define
\begin{equation*}
C^k_0(\overline{\Omegaega}) = \Big\{\varphi\in \mathrm{C}^k(\overline{\Omegaega}): D^\beta\varphi(x) = 0\;\text{on}\;\partial\Omegaega\;\,\text{with}\ |\beta| \in [0, k] \Big\}\,,
\end{equation*}
where $\beta$ is a multiindex and $|\beta|$ is its order.
\end{defn}
We summarize the results about the semiconcavity of $u$ as follows.
\begin{theorem}[Semiconcavity]\label{thm:newsemi} Assume $f \geq 0$, $f = 0$ on $\partial\Omegaega$ and $f$ is uniformly semiconcave in $\overline{\Omegaega}$ with semiconcavity constant $c$. Let $u$ be the solution to \eqref{eq:PDE0}.
\begin{itemize}
\item[(i)] If $\tilde{f}$ is uniformly semiconcave in $\mathbb{R}^n$, then $u$ is uniformly semiconcave in $\overline{\Omegaega}$.
\item[(ii)] In general, $u$ is locally semiconcave. More specifically, there exists a constant $C > 0$ independent of $x \in \Omegaega$ such that $\forall x \in \Omegaega$,
\begin{equation}
u(x+h)-2u(x)+u(x-h)\leq \frac{C}{d(x)}\left\lvert h\right\rvert ^2,
\end{equation}
$\forall h \in \mathbb{R}^n$ with $\left\lvert h\right\rvert \leq M_x$ for some constant $M_x$ that depends on $x$.
\end{itemize}
\end{theorem}
The proof of Theorem \ref{thm:newsemi} is given at the end of this section.
\begin{remark} If $f\in \mathrm{C}_c^2(\mathbb{R}^n)$ (or $\mathrm{C}_0^2(\overline{\Omegaega})$), then $f$ is uniformly semiconcave with semiconcavity constant
\begin{equation}\label{def_c}
c = \max \big\lbrace D^2f(x)\xi \cdot \xi: \left\lvert \xi\right\rvert =1, x\in \mathbb{R}^n \big\rbrace\geq 0.
\end{equation}
Also, the condition that $\tilde{f}$ is semiconcave in $\mathbb{R}^n$ holds for $\mathrm{C}_c^2(\Omegaega)$ and $\mathrm{C}_0^2(\overline{\Omegaega})$.
\end{remark}
The following lemma is a refined version of the local gradient bound in Theorem \ref{thm:grad_1}. We follow \cite[Theorem 3.1]{Armstrong2015a} where the authors use Bernstein's method inside a doubling variable argument and explicitly keep track of all the dependencies. We refer the reader to \cite{barles_weak_1991,capuzzo_dolcetta_holder_2010} and the references therein for related versions of the gradient bound. We believe this result is new in the literature since it is uniform in $\varepsilon$, namely, we give the explicit dependence of the gradient bound on $d(x)$. It also indicates that the boundary layer is a strip of size $\mathcal{O}(\varepsilon)$ from the boundary.
\begin{lem}\label{lem:boundDu^eps} For all $\varepsilon$ small enough, there exists a constant $C$ independent of $\varepsilon$ such that
\begin{equation}\label{eq:es_final}
\left\lvert Du^\varepsilon(x)\right\rvert \leq C\left( 1 + \left(\frac{\varepsilon}{d(x)}\right)^{\alphapha+1}\right) \qquad\text{for}\;x\in \Omegaega.
\end{equation}
\end{lem}
\begin{proof}[Proof of Lemma \ref{lem:boundDu^eps}]
Fix $x_0 \in \Omegaega\backslash \Omegaega_{\deltata_0}$. Let $\deltata := \frac{1}{4}d(x_0)$ and
\begin{equation*}
v(x) := \frac{1}{\deltata}u^\varepsilon(x_0+\deltata x), \qquad x\in B(0,2).
\end{equation*}
Then $v$ solves
\begin{equation}\label{eq:eqn_of_v}
\deltata v(x) + \left\lvert Dv(x)\right\rvert ^p - \tilde{f}(x) - \frac{\varepsilon}{\deltata}\Deltata v(x) = 0 \qquad\text{in}\;B(0,2),
\end{equation}
where $\tilde{f}(x): = f(x_0+\deltata x)$ on $\overline{B(0,2)}$. Note that $\Vert \tilde{f}\Vert_{L^\infty}\leq \Vert f\Vert_{L^\infty}$ and
\begin{equation*}
B(x_0,2\deltata)\subset \Omegaega_{2\deltata}\subset\subset\Omegaega.
\end{equation*}
By Lemma \ref{lem:super_refined}, there is a constant $C$ independent of $\deltata,\varepsilon$ such that
\begin{equation*}
\deltata\Vert v\Vert_{L^\infty\left(B\left(0,\frac{3}{2}\right)\right)} \leq \Vert u^\varepsilon\Vert_{L^\infty(\Omegaega_{2\deltata}))} \leq C\left(1 + \frac{\varepsilon^{\alphapha+1}}{\deltata^\alphapha}\right).
\end{equation*}
Apply Theorem 3.1 in \cite{Armstrong2015a} to obtain
\begin{equation*}
\begin{split}
\sup_{x\in B(0,1)}\left\lvert Dv(x)\right\rvert &\leq C\left[\left(\frac{\varepsilon}{\deltata}\right)^{\frac{1}{p-1}} + \left(\Vert f\Vert_{L^\infty}+\deltata \Vert v\Vert_{L^\infty\left(B\left(0,\frac{3}{2}\right)\right)}\right)^{\frac{1}{p}}\right] \\
&\leq C\left[\left(\frac{\varepsilon}{\deltata}\right)^{\alphapha+1} + \left(1+\frac{\varepsilon^{\alphapha+1}}{\deltata^\alphapha}\right)^{\frac{\alphapha+1}{\alphapha+2}}\right]\leq C \left(1+\left(\frac{\varepsilon}{\deltata}\right)^{\alphapha+1}\right),
\end{split}
\end{equation*}
where $p = \frac{\alphapha+2}{\alphapha+1}$ and $\alphapha+1 = \frac{1}{p-1}$.
Plugging in $\deltata = \frac{1}{4}d(x_0)$, we obtain
\begin{equation*}
\left\lvert Du^\varepsilon(x_0)\right\rvert = \left\lvert Dv(0)\right\rvert \leq C\left(1+ \left(\frac{\varepsilon}{d(x_0)}\right)^{\alphapha+1}\right).
\end{equation*}
In other words, we have \eqref{eq:es_final} for all $x\in \Omegaega\backslash\Omegaega_{\deltata_0}$. On the other hand, from Theorem \ref{thm:grad_1}, there exists a constant $C$ independent of $\varepsilon$ such that $\left\lvert Du^\varepsilon(x)\right\rvert \leq C$ for all $x\in \Omegaega_{\deltata_0}$. Thus, the proof is complete.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:rate_doubling2}] For $1<p<2$, we proceed as in the proof of Theorem \ref{thm:rate_doubling1} to obtain
\begin{align}
&0\leq u^\varepsilon(x) \leq \tilde{u}^\varepsilon(x) \leq \frac{\nu C_\alphapha\varepsilon^{\alphapha+1}}{d_\kappapa(x)^{\alphapha}} + C_3\left(\frac{4\varepsilon}{\kappapa}\right)^{\alphapha+2} \label{annulus2a}
\end{align}
for $x\in U_\kappapa$. Let
\begin{equation*}
\psi^\varepsilon(x) := u^\varepsilon(x) - \frac{\nu C_\alphapha \varepsilon^{\alphapha+1}}{d(x)^\alphapha}, \qquad x\in \Omegaega,
\end{equation*}
where $\nu > 1$ is chosen as in Lemma \ref{lem:super_refined}. It is clear that $u-\psi^\varepsilon$ has a local minimum at some point $x_0\in \Omegaega$ since $\psi^\varepsilon(x)\to -\infty$ as $x\to \partial\Omegaega$. The normal derivative test gives us
\begin{equation*}
D\psi^\varepsilon(x_0) = Du^\varepsilon(x_0) + \nu C_\alphapha\alphapha \left(\frac{\varepsilon}{d(x_0)}\right)^{\alphapha+1} D d(x_0) \in D^-u(x_0).
\end{equation*}
There are two cases to consider:
\begin{itemize}
\item If $\displaystyle d(x_0)< \frac{1}{2}\kappapa$, then as in the proof of Theorem \ref{thm:rate_doubling1}, $x_0\in U_\kappapa$ and $d_\kappapa(x_0) = d(x_0)$. By the definition of $x_0$, for any $x\in \Omegaega$, there holds
\begin{align*}
u(x) - \left(u^\varepsilon(x) - \frac{\nu C_\alphapha \varepsilon^{\alphapha+1}}{d(x)^\alphapha}\right) \geq u(x_0) - \left(u^\varepsilon(x_0) - \frac{\nu C_\alphapha \varepsilon^{\alphapha+1}}{d(x_0)^\alphapha}\right).
\end{align*}
Therefore,
\begin{equation*}
\begin{split}
u^\varepsilon(x) - u(x) - \frac{\nu C_\alphapha \varepsilon^{\alphapha+1}}{d(x)^\alphapha} &\leq \left(u^\varepsilon(x_0) - \frac{\nu C_\alphapha \varepsilon^{\alphapha+1}}{d(x_0)^\alphapha}\right) - u(x_0) \leq C_3 \left(\frac{4\varepsilon}{\kappapa}\right)^{\alphapha+2}
\end{split}
\end{equation*}
thanks to \eqref{annulus2a}. Thus, in this case
\begin{equation*}
u^\varepsilon(x) - u(x) \leq \frac{\nu C_\alphapha \varepsilon^{\alphapha+1}}{d(x)^\alphapha} + C_3 \left(\frac{4\varepsilon}{\kappapa}\right)^{\alphapha+2}, \qquad x\in \Omegaega.
\end{equation*}
\item If $\displaystyle d(x_0) \geq \frac{1}{2}\kappapa$, from the fact that $u$ is semiconcave in $\Omegaega$ with a linear modulus $c(x)$ as in Theorem \ref{thm:newsemi}, we have
\begin{equation*}
D^2\psi^\varepsilon(x_0) \leq c(x_0)\;\mathbb{I}_n,
\end{equation*}
where $\mathbb{I}_n$ denotes the identity matrix of size n. This implies that
\begin{equation}\label{mybound}
\Deltata \psi^\varepsilon(x_0) \leq nc(x_0) \leq \frac{Cn}{d(x_0)} \leq \frac{Cn}{\kappapa}.
\end{equation}
In other words, we have
\begin{equation*}
\varepsilon\Deltata u^\varepsilon (x_0) - \frac{\nu C_\alphapha \alphapha(\alphapha+1)\varepsilon^{\alphapha+2}}{d(x_0)^{\alphapha+2}}\left\lvert Dd(x_0)\right\rvert ^2 + \frac{\nu C_\alphapha \alphapha \varepsilon^{\alphapha+2}}{d(x_0)^{\alphapha+1}}\Deltata d(x_0) \leq \frac{Cn\varepsilon}{\kappapa}.
\end{equation*}
Since $d(x_0)\geq \frac{1}{2}\kappapa$, we can further deduce that
\begin{equation}\label{delta_psi}
\varepsilon\Deltata u^\varepsilon(x_0) \leq \frac{Cn\varepsilon}{\kappapa} + \frac{C\varepsilon^{\alphapha+2}}{d(x_0)^{\alphapha+2}} \leq \frac{Cn\varepsilon}{\kappapa} + C\left(\frac{\varepsilon}{\kappapa}\right)^{\alphapha+2},
\end{equation}
where $C$ is independent of $\varepsilon$. Since $\psi^\varepsilon\in \mathrm{C}^2(\Omegaega)$, the viscosity supersolution test for $u$ gives us
\begin{equation}\label{convex1}
u(x_0) + \left\lvert Du^\varepsilon(x_0) + \frac{\nu C_\alphapha \alphapha \varepsilon^{\alphapha+1}}{d(x_0)^{\alphapha+1}}D d(x_0)\right\rvert ^p - f(x_0) \geq 0.
\end{equation}
On the other hand, since $u^\varepsilon$ solves \eqref{eq:PDEeps}, we have
\begin{equation}\label{convex2}
u^\varepsilon(x_0) + \left\lvert Du^\varepsilon(x_0)\right\rvert ^p - f(x_0) - \varepsilon \Deltata u^\varepsilon(x_0) = 0.
\end{equation}
Combine \eqref{convex1} and \eqref{convex2} to obtain that
\begin{equation}\label{convex3}
u^\varepsilon(x_0) - u(x_0) \leq \left\lvert Du^\varepsilon(x_0) + \frac{\nu C_\alphapha \alphapha \varepsilon^{\alphapha+1}}{d(x_0)^{\alphapha+1}}D d(x_0)\right\rvert ^p - \left\lvert Du^\varepsilon(x_0)\right\rvert ^p + \varepsilon\Deltata u^\varepsilon(x_0).
\end{equation}
By Lemma \ref{lem:boundDu^eps}, we can bound $Du^\varepsilon(x_0)$ as
\begin{equation}\label{eq:bound_Du^eps}
\left\lvert Du^\varepsilon(x_0)\right\rvert \leq C + C \left(\frac{\varepsilon}{d(x_0)}\right)^{\alphapha+1} \leq C +C\left(\frac{\varepsilon}{\kappapa}\right)^{\alphapha+1}
\end{equation}
since $d(x_0)\geq \frac{1}{2}\kappapa$. We estimate the gradient terms on the right hand side of \eqref{convex3} using \eqref{eq:bound_Du^eps} as follows.
\begin{equation}\label{convex4}
\begin{aligned}
&\left\lvert Du^\varepsilon(x_0) + \frac{\nu C_\alphapha \alphapha \varepsilon^{\alphapha+1}}{d(x_0)^{\alphapha+1}}D d(x_0)\right\rvert ^p - \left\lvert Du^\varepsilon(x_0)\right\rvert ^p\nonumber \\
&\qquad \qquad \leq p\left(\left\lvert Du^\varepsilon(x_0)\right\rvert + \frac{\nu C_\alphapha \alphapha \varepsilon^{\alphapha+1}}{d(x_0)^{\alphapha+1}}\left\lvert D d(x_0)\right\rvert \right)^{p-1} \frac{\nu C_\alphapha \alphapha \varepsilon^{\alphapha+1}}{d(x_0)^{\alphapha+1}}\left\lvert D d(x_0)\right\rvert \nonumber\\
&\qquad \qquad \leq p\left(C + C\left(\frac{\varepsilon}{\kappapa}\right)^{\alphapha+1}\right)^{p-1} C\left(\frac{\varepsilon}{\kappapa}\right)^{\alphapha+1}\leq C\left(\frac{\varepsilon}{\kappapa}\right)^{\alphapha+1}\left(1 + \left(\frac{\varepsilon}{\kappapa}\right)\right),
\end{aligned}
\end{equation}
where $C$ is a constant depending only on $\nu$, $\alphapha$, and $d$. Plugging \eqref{delta_psi} and \eqref{convex4} in the right hand side of \eqref{convex3}, we get
\begin{equation*}
u^\varepsilon(x_0) - u(x_0) \leq \frac{Cn\varepsilon}{\kappapa} + C\left(\frac{\varepsilon}{\kappapa}\right)^{\alphapha+1}\left(1 + \left(\frac{\varepsilon}{\kappapa}\right)\right).
\end{equation*}
Therefore,
\begin{equation*}
u^\varepsilon(x) - u(x) \leq \frac{\nu C_\alphapha \varepsilon^{\alphapha+1}}{d(x)^\alphapha} +C\left( \left(\frac{\varepsilon}{\kappapa}\right)^{\alphapha+1} + \left(\frac{\varepsilon}{\kappapa}\right)^{\alphapha+2}\right) + \frac{Cn\varepsilon}{\kappapa}, \qquad x\in \Omegaega.
\end{equation*}
\end{itemize}
For $p = 2$, the argument is similar. We take $\psi^\varepsilon(x):=u^\varepsilon(x)-\nu \varepsilon \log \left(\frac{1}{d(x)}\right)$ instead and still $u-\psi^\varepsilon$ attains a local minimum at some point $x_0 \in \Omegaega$. Carrying out the similar computations as in the case of $1 < p < 2$, we have:
\begin{itemize}
\item If $\displaystyle d(x_0) < \frac{1}{2} \kappapa$, then
\begin{equation*}
u^\varepsilon(x)-u(x) \leq \nu \varepsilon \log \left( \frac{1}{d(x)} \right) + C\left( \frac{4\varepsilon}{\kappapa}\right)^2, \quad x \in \Omegaega.
\end{equation*}
\item If $\displaystyle d(x_0) \geq \frac{1}{2} \kappapa$, then
\begin{equation*}
u^\varepsilon(x)-u(x) \leq \nu \varepsilon \log\left(\frac{1}{d(x)}\right) + C\left(\left( \frac{\varepsilon}{\kappapa} \right)+ \left( \frac{\varepsilon}{\kappapa} \right)^2\right) + \frac{Cn\varepsilon}{\kappapa}, \quad x \in \Omegaega.
\end{equation*}
\end{itemize}
From these two cases, the conclusion for $p=2$ follows.
\end{proof}
\begin{remark}\label{rem:nice} If $f\in \mathrm{C}^2(\overline{\Omegaega})$ with $f = 0,Df = 0$ and $D^2f = 0$ on $\partial\Omegaega$, then \eqref{mybound} can be improved to $\Deltata \psi^\varepsilon(x_0) \leq nc$ where $c$ is the semiconcavity constant of $f$, and thus the final estimate becomes
\begin{equation*}
u^\varepsilon(x) - u(x) \leq \frac{\nu C_\alphapha \varepsilon^{\alphapha+1}}{d(x)^\alphapha} +C\left( \left(\frac{\varepsilon}{\kappapa}\right)^{\alphapha+1} + \left(\frac{\varepsilon}{\kappapa}\right)^{\alphapha+2}\right) + nc\varepsilon, \qquad x\in \Omegaega.
\end{equation*}
\end{remark}
\begin{remark} \quad
\begin{itemize}
\item We only need the local gradient bound in Theorem \ref{thm:grad_1} to obtain the local rate of convergence $\mathcal{O}(\varepsilon)$ in \eqref{convex4}. However, to make the dependence on $\kappapa$ explicit, we need to bound $Du^\varepsilon(x_0)$ as in \eqref{convex4}.
\item Another way to get \eqref{eq:bound_Du^eps} without using Lemma \ref{lem:boundDu^eps} (which is true for all $x\in \Omegaega$) is using the fact that $D\psi^\varepsilon(x_0)\in D^-u(x_0)$, which implies
\begin{equation*}
\left\lvert D\psi^\varepsilon(x_0)\right\rvert = \left\lvert Du^\varepsilon(x_0) + \nu C_\alphapha\alphapha \left(\frac{\varepsilon}{d(x_0)}\right)^{\alphapha+1} D d(x_0) \right\rvert \leq C_0
\end{equation*}
since $u$ is Lipschitz with constant $C_0$.
\end{itemize}
\end{remark}
Before giving the proof of Corollary \ref{cor:key}, we need to modify the construction of the cutoff function in the proof of Theorem \ref{main_thm1}.
\begin{lem}\label{conca} Assume $f\in \mathrm{C}^2(\overline{\Omegaega})$ such that $f=0$ and $Df = 0$ on $\partial\Omegaega$. For all $\kappapa>0$ small enough, there exists $f_\kappapa\in \mathrm{C}^2_c(\Omegaega)$ such that
\begin{equation*}
\Vert f_\kappapa - f\Vert_{L^\infty(\Omegaega)}\leq C\kappapa \qquad\text{and}\qquad \Vert D^2f_\kappapa\Vert_{L^\infty(\Omegaega)}\leq C
\end{equation*}
where $C$ is independent of $\kappapa$.
\end{lem}
\begin{proof} Choose a smooth function $\chi \in \mathrm{C}^\infty(\mathbb{R})$ such that $\chi\geq 0$, $\chi = 0$ if $x\leq 1$, $\chi = 1$ if $x\geq 2$ and $0 \leq \chi' \leq 2$ in $\mathbb{R}$.
For $\kappapa>0$ such that $0<2\kappapa<\deltata_0$ and $x\in \Omegaega\backslash \Omegaega_{2\kappapa}$, let $x_0$ be the projection of $x$ onto $\partial\Omegaega$ and denote by $\nu(x_0)$ the outward unit normal vector at $x_0$. Write $x = x_0 - d(x)\nu(x_0)$ where $d(x)\leq 2\kappapa$. We have
\begin{equation*}
f(x) = f(x_0)-Df(x_0)\cdot \nu(x_0) d(x) + \int_0^{d(x)}(d(x)-s)\nu(x_0)\cdot D^2f(x_0-s\nu(x_0))\cdot \nu(x_0)ds.
\end{equation*}
Since $f = 0$ and $Df = 0$ on $\partial\Omegaega$, we deduce that
\begin{equation}\label{bound_near_bdr}
\left\lvert f(x)\right\rvert \leq \left( \left\Vert\frac{1}{2} D^2f \right\Vert_{L^\infty(\overline{\Omegaega})}\right)d(x)^2 \leq C\kappapa^2 \qquad\text{and}\qquad
\left\lvert Df(x)\right\rvert \leq C\kappapa
\end{equation}
for all $d(x)\leq 2\kappapa$.
Define
\begin{equation*}
f_\kappapa(x) = f(x)\chi\left(\frac{d(x)}{\kappapa}\right) \qquad\text{for}\;x\in \overline{\Omegaega}.
\end{equation*}
It is clear that $0\leq f_\kappapa(x)\leq f(x)$ for all $x\in \overline{\Omegaega}$ and $f_\kappapa(x) = f(x)$ if $d(x)\geq 2\kappapa$. Furthermore, we observe that
\begin{equation*}
0\leq \max_{x\in \overline{\Omegaega}} \big(f(x) - f_\kappapa(x)\big) \leq \max_{0\leq d(x) \leq 2\kappapa} \big(f(x) - f_\kappapa(x)\big)\leq \max_{0\leq d(x) \leq 2\kappapa} f(x) \leq C\kappapa^2.
\end{equation*}
We have
\begin{equation*}
Df_\kappapa(x) = Df(x)\chi\left(\frac{d(x)}{\kappapa}\right)+ f(x)\chi'\left(\frac{d(x)}{\kappapa}\right)\frac{Dd(x)}{\kappapa}
\end{equation*}
and
\begin{equation*}
\begin{split}
D^2f_\kappapa(x) = &D^2f(x)\chi\left(\frac{d(x)}{\kappapa}\right) + 2\chi'\left(\frac{d(x)}{\kappapa}\right)\frac{Df(x)\otimes Dd(x)}{\kappapa} \\
&+ f(x)\left(\chi''\left(\frac{d(x)}{\kappapa}\right)\frac{Dd(x)\otimes Dd(x)}{\kappapa^2} + \chi'\left(\frac{d(x)}{\kappapa}\right)\frac{D^2d(x)}{\kappapa}\right)
\end{split}
\end{equation*}
is uniformly bounded thanks to \eqref{bound_near_bdr}.
\end{proof}
\begin{proof}[Proof of Corollary \ref{cor:key}] Let $u^\varepsilon_\kappapa\in \mathrm{C}^2(\Omegaega)\cap \mathrm{C}(\overline{\Omegaega})$ be the solution to \eqref{eq:PDEeps} and $u_k$ be the solution to \eqref{eq:PDE0} with $f$ replaced by $f_k$, respectively. It is clear that
\begin{equation*}
0\leq u^\varepsilon(x) - u^\varepsilon_\kappapa(x)\leq C\kappapa \qquad\text{for}\;x\in \Omegaega
\end{equation*}
and
\begin{equation*}
0\leq u(x) - u_\kappapa(x)\leq C\kappapa \qquad\text{for}\;x\in \Omegaega.
\end{equation*}
Therefore,
\begin{equation}\label{cool}
u^\varepsilon(x)-u(x)\leq 2C\kappapa + \Big(u^\varepsilon_\kappapa(x) - u_\kappapa(x)\Big).
\end{equation}
By Theorem \ref{thm:rate_doubling2} and Remark \ref{rem:nice}, as $f_\kappapa\in \mathrm{C}_c^2(\Omegaega)$ with a uniform bound on $D^2f_\kappapa$, we have
\begin{equation*}
\begin{aligned}
u_\kappapa^\varepsilon(x) - u_\kappapa(x) &\leq\frac{\nu C_\alphapha \varepsilon^{\alphapha+1}}{d(x)^\alphapha} + C\left(\left(\frac{\varepsilon}{\kappapa}\right)^{\alphapha+1} +\left(\frac{\varepsilon}{\kappapa}\right)^{\alphapha+2}\right) + 4nC\varepsilon, &\quad p <2, \\
u_\kappapa^\varepsilon(x) - u_\kappapa(x) & \leq \nu \varepsilon \log\left( \frac{1}{d(x)}\right)+C \left(\left(\frac{\varepsilon}{\kappapa}\right)+\left(\frac{\varepsilon}{\kappapa}\right)^2\right)+ 4nC\varepsilon, &\quad p=2
\end{aligned}
\end{equation*}
for some constant $C$ independent of $\kappapa$. Choose $\kappapa = \varepsilon^{\gammama}$ with $\gammama \in (0,1)$. Then \eqref{cool} becomes
\begin{align*}
&u^\varepsilon(x) - u(x) \leq C\varepsilon^{\gammama} + C\varepsilon + \frac{C\varepsilon^{\alphapha+1}}{d(x)^\alphapha} + C\varepsilon^{(1-\gammama)(\alphapha+1)}, &\qquad p < 2,\\
&u^\varepsilon(x) - u(x) \leq C\varepsilon^{\gammama} + C\varepsilon + C\varepsilon \left\lvert \log d(x)\right\rvert + C\varepsilon^{1-\gammama}, &\qquad p = 2.
\end{align*}
If $p = 2$, then $\gammama = 1/2$ is the best value to choose, which implies the $\mathcal{O}(\sqrt{\varepsilon})$ estimate in Theorem \ref{main_thm1}. If $p<2$, by setting $\gammama = (1-\gammama)(\alphapha+1)$, we can get the best value of $\gammama$, that is,
\begin{equation*}
\gammama = \frac{\alphapha+1}{\alphapha+2} = \frac{1}{p} > \frac{1}{2},
\end{equation*}
and we obtain a better estimate $\mathcal{O}(\varepsilon^{1/p})$.
\end{proof}
\begin{remark} If we do not assume $Df = 0$ on $\partial\Omegaega$, then the best we can get from the above argument is
\begin{equation*}
u^\varepsilon(x) - u(x) \leq C\varepsilon^{\gammama} + C\varepsilon^{1-\gammama} + \frac{C\varepsilon^{\alphapha+1}}{d(x)^\alphapha} + C\varepsilon^{(1-\gammama)(\alphapha+1)}, \qquad p < 2
\end{equation*}
and we obtain the rate $\mathcal{O}(\varepsilon^{1/2})$ again.
\end{remark}
\begin{proof}[Proof of Theorem \ref{thm:newsemi}]\quad
\begin{itemize}
\item[(i)] It is clear that
\begin{equation*}
\tilde{u}(x) = \begin{cases}
u(x) &\qquad\text{if}\;x\in \overline{\Omegaega},\\
0 &\qquad\text{if}\;x\notin \overline{\Omegaega},
\end{cases}
\end{equation*}
solves the equation $\tilde{u}(x)+\left\lvert D\tilde{u}(x)\right\rvert ^p - \tilde{f}(x) = 0$ in $\mathbb{R}^n$. Now we can use a classical doubling variable argument to show that
$-D^2u \geq -c\;\mathbb{I}_n$ in $\mathbb{R}^n$ where
\begin{equation*}
c = \max \big\lbrace D^2f(x)\xi \cdot \xi: \left\lvert \xi\right\rvert =1, x\in \mathbb{R}^n \big\rbrace\geq 0.
\end{equation*}
We give the proof of this fact in Appendix for the reader's convenience (see also \cite{Calder2021}).
\item[(ii)] Fix $x \in \Omegaega$ and let $\eta$ be a minimizing curve for $u(x)$. Then
$$u(x)=\int^\infty_0 e^{-s} \left(C_q\left\lvert \dot{\eta}(s)\right\rvert ^q +f(\eta(s)) \right) ds .$$
Since $\eta(0)=x \in \Omegaega$, then there exists $T > 0$ such that $\eta(s) \in \Omegaega, \forall 0 \leq s \leq T$. In fact, we can choose $\displaystyle T \geq \frac{d(x)}{C_0}$ for some constant $C_0$ independent of $x$, since $\left\|\dot{\eta}\right\|_\infty \leq C$ where $C$ is independent of $x$.
Note that
\begin{equation}\label{x}
u(x)=\int_0^{T} e^{-s}\left( C_q\left\lvert \dot{\eta}(s)\right\rvert ^q +f(\eta(s)) \right)ds + e^{-T} u(\eta(T)).
\end{equation}
Define $\mathbb{T}ilde{\eta}:[0, +\infty)\to \mathbb{R}^n$ by
\begin{equation*}
\mathbb{T}ilde{\eta}(s):=\left\{
\begin{aligned}
&\eta(s)+\left( 1-\frac{s}{T} \right) h, \quad \text{if } 0\leq s\leq T,\\
&\eta(s),\qquad \qquad \qquad \quad \text{if } s \geq T.
\end{aligned}
\right.
\end{equation*}
Choose $h$ small enough so that $\mathbb{T}ilde{\eta}(s) \in \Omegaega, \forall s\geq 0$.
(This can be done because there exists $r>0$ such that $B(\eta(s),r) \subset \Omegaega$, for all $0 \leq s \leq T$.)
By the optimal control formula of $u(x+h)$ and $u(x-h)$, we have
\begin{equation} \label{h+}
u(x+h)\leq \int_0^{T} e^{-s}\left( C_q \left\lvert \dot{\eta}(s)-\frac{h}{T}\right\rvert ^q +f \left(\eta(s)+\left(1-\frac{s}{T}\right)h\right) \right)ds + e^{-T} u(\eta(T)),
\end{equation}
and
\begin{equation}\label{h-}
u(x-h)\leq \int_0^{T} e^{-s}\left( C_q \left\lvert \dot{\eta}(s)+\frac{h}{T} \right\rvert ^q +f \left(\eta(s)-\left(1-\frac{s}{T} \right)h\right) \right)ds + e^{-T} u(\eta(T)).
\end{equation}
Hence, from \eqref{x}, \eqref{h+}, and \eqref{h-}, for $h$ small enough,
\begin{equation}\label{diff}
\begin{aligned}
& u(x+h)+u(x-h)-2u(x) \\
\leq &\int_0^T e^{-s} C_q \left( \left\lvert \dot{\eta}(s)-\frac{h}{T}\right\rvert ^q + \left\lvert \dot{\eta}(s)+\frac{h}{T} \right\rvert ^q -2\left\lvert \dot{\eta}(s) \right\rvert ^q\right) ds\\
&+\int_0^Te^{-s}\left(f \left(\eta(s)+\left(1-\frac{s}{T}\right)h\right) + f \left(\eta(s)-\left(1-\frac{s}{T} \right)h\right) -2f\left(\eta(s)\right) \right)ds\\
\leq &\int_0^T e^{-s} C_q \left( \left\lvert \dot{\eta}(s)-\frac{h}{T}\right\rvert ^q + \left\lvert \dot{\eta}(s)+\frac{h}{T} \right\rvert ^q -2 \left\lvert \dot{\eta}(s)\right\rvert ^q\right) ds\\
&+C \left\lvert h\right\rvert ^2 \int_0^Te^{-s} \left( 1-\frac{s}{T}\right)^2 ds,
\end{aligned}
\end{equation}
where the second inequality follows from the semiconcavity of $f$.
By Taylor's theorem, for any $y \in \mathbb{R}^n$,
\begin{equation}
\begin{aligned}
\left\lvert y+\frac{1}{T}h\right\rvert ^q =&\left\lvert y\right\rvert ^q+q\left\lvert y\right\rvert ^{q-2}y \cdot \frac{1}{T} h \\
&+\int^1_0 q(q-2) \left\lvert y +\frac{t}{T} h \right\rvert ^{q-4} \left( \left(y+\frac{t}{T}h\right) \cdot \frac{h}{T}\right)^2(1-t)dt\\
&+\int^1_0q\left\lvert y+\frac{t}{T}h\right\rvert ^{q-2} \left\lvert \frac{h}{T}\right\rvert ^2(1-t)dt\\
\leq &\left\lvert y\right\rvert ^q+q\left\lvert y\right\rvert ^{q-2}y \cdot \frac{1}{T} h + C\int^1_0 \left\lvert y +\frac{t}{T} h \right\rvert ^{q-2} \left\lvert \frac{h}{T}\right\rvert ^2dt\\
\leq &\left\lvert y\right\rvert ^q+q\left\lvert y\right\rvert ^{q-2}y \cdot \frac{1}{T} h + C\left(\left\lvert y\right\rvert ^{q-2}\left\lvert \frac{h}{T}\right\rvert ^2+\left\lvert \frac{h}{T}\right\rvert ^q \right)
\end{aligned}
\end{equation}
and similarly
\begin{equation}
\begin{aligned}
\left\lvert y-\frac{1}{T}h\right\rvert ^q \leq \left\lvert y\right\rvert ^q-q\left\lvert y\right\rvert ^{q-2}y \cdot \frac{1}{T} h + C\left(\left\lvert y\right\rvert ^{q-2}\left\lvert \frac{h}{T}\right\rvert ^2+\left\lvert \frac{h}{T}\right\rvert ^q \right),
\end{aligned}
\end{equation}
which implies
\begin{equation}\label{eta}
\left\lvert \dot{\eta}(s)-\frac{h}{T}\right\rvert ^q + \left\lvert \dot{\eta}(s)+\frac{h}{T} \right\rvert ^q -2\left\lvert \dot{\eta}\right\rvert ^q\leq C \left(\left\lvert \frac{h}{T}\right\rvert ^2 +\left\lvert \frac{h}{T}\right\rvert ^q \right)\leq C\left\lvert \frac{h}{T}\right\rvert ^2
\end{equation}
where $q\geq 2$, $C=C(q, \|\dot{\eta}\|_\infty)$, and $h$ is chosen to be small enough so that $\displaystyle \left\lvert \frac{h}{T}\right\rvert \leq 1$. Plugging \eqref{eta} into \eqref{diff}, we get
\begin{equation}
\begin{aligned}
&u(x+h)+u(x-h)-2u(x) \leq C\left\lvert h\right\rvert ^2 \int_0^T \frac{e^{-s}}{T^2}ds +C \left\lvert h\right\rvert ^2 \int_0^Te^{-s} \left( 1-\frac{s}{T}\right)^2 ds \\
&\qquad\qquad\qquad \leq C\frac{\left\lvert h\right\rvert ^2}{T} \int_0^1e^{-sT}ds+C\left\lvert h\right\rvert ^2 \int_0^T e^{-s}ds\\
& \qquad\qquad\qquad \leq C\left(1+\frac{1}{T}\right)\left\lvert h\right\rvert ^2 \leq C\left(1 + \frac{1}{d(x)}\right)\left\lvert h\right\rvert ^2\leq \frac{C}{d(x)}\left\lvert h\right\rvert ^2
\end{aligned}
\end{equation}
since $\displaystyle T \geq \frac{d(x)}{C_0}$.
\end{itemize}
\end{proof}
\subsection{Future work}
In the end, we would like to mention some questions that are worth investigating in the future.
\begin{itemize}
\item \textsc{General $f$.} As is mentioned earlier, one interesting question is to figure out the rate of convergence for the case of general $f$ where $f$ is not equal to its minimum on the boundary.
\item \textsc{General $H$.} In our proof, an explicit estimate of the asymptotic behavior of the solution $u^\varepsilon$ near the boundary is obtained due to the specific form of Hamiltonian $H(\xi) = | \xi|^p$. We believe that a similar but more technical computation can be done to establish such an estimate of the asymptotic behavior of the solution for Hamiltonian that satisfies
\begin{equation*}
\deltata^{\frac{p}{p-1}}H\left(\deltata^{\frac{-1}{p-1}}\xi\right) = | \xi |^p
\end{equation*}
locally uniformly in $\xi$ as $\deltata\to 0$. This condition is mentioned in \cite{sardarli_ergodic_2021}. For more general Hamiltonian, the question is still open.
\item \textsc{The case $p>2$.} In this case, the solution to the second order state-constraint equation is no longer blowing up near the boundary and we do not know any explicit boundary information, which becomes a main difficulty. In fact, loss of boundary data can happen in this case, that is, the Dirichlet boundary problem may not be solvable for any boundary condition in the classical sense. (\cite{barles_generalized_2004})
\end{itemize}
\appendix
\section{Appendix}
\subsection{Estimates on solutions}
We present here a proof for the gradient bound of the solution to \eqref{eq:PDEeps} using Bernstein's method (see also \cite{Lasry1989,lions_quelques_1985}). Another proof using Berstein's method inside a doubling variable argument is given in \cite{Armstrong2015a}.
\begin{proof}[Proof of Theorem \ref{thm:grad_1}] Let $\theta\in (0,1)$ be chosen later, $\varphi\in \mathrm{C}_c^\infty(\Omegaega)$, $0\leq \varphi\leq 1$, $\mathrm{supp}\;\varphi\subset \Omegaega$ and $\varphi = 1$ on $\Omegaega_\deltata$ such that
\begin{equation}\label{e:ass_power}
\left\lvert \Deltata \varphi(x)\right\rvert \leq C\varphi^\theta \qquad\text{and}\qquad \left\lvert D \varphi(x)\right\rvert ^2 \leq C\varphi^{1+\theta},\quad \forall x\in\Omegaega,
\end{equation}
where $C = C(\deltata,\theta)$ is a constant depending on $\deltata,\theta$.
Define $w(x) := \left\lvert Du^\varepsilon(x)\right\rvert ^2$ for $x \in \Omegaega$. The equation for $w$ is given by
\begin{equation*}
-\varepsilon \Deltata w + 2 p\left\lvert D u^\varepsilon\right\rvert ^{p-2}D u^\varepsilon \cdot D w + 2 w - 2 D f\cdot D u^\varepsilon + 2 \varepsilon \left\lvert D^2u^\varepsilon\right\rvert ^2 = 0 \qquad\text{in}\;\Omegaega.
\end{equation*}
Then an equation for $(\varphi w)$ can be derived as follows.
\begin{align*}
&-\varepsilon \Deltata (\varphi w) + 2 p\left\lvert D u^\varepsilon\right\rvert ^{p-2}D u^\varepsilon \cdot D (\varphi w) + 2 (\varphi w) + 2 \varepsilon \varphi\left\lvert D^2u^\varepsilon\right\rvert ^2 + 2\varepsilon \frac{D \varphi}{\varphi}\cdot D (\varphi w) \\
= &\varphi(D f\cdot D u^\varepsilon) + 2 p\left\lvert D u^\varepsilon\right\rvert ^{p-2}(D u^\varepsilon\cdot D \varphi)w -\varepsilon w \Deltata \varphi + 2\varepsilon \frac{\left\lvert D \varphi\right\rvert ^2}{\varphi}w
\qquad\text{in}\;\mathrm{supp}\;\varphi.
\end{align*}
Assume that $\varphi w$ achieves its maximum over $\overline{\Omegaega}$ at $x_0\in \Omegaega$. And we can further assume that $x_0\in \mathrm{supp}\;\varphi$, since otherwise the maximum of $\varphi w$ over $\overline{\Omegaega}$ is zero. By the classical maximum principle,
\begin{equation*}
-\varepsilon \Deltata(\varphi w)(x_0)\geq 0 \qquad\text{and}\qquad \left\lvert D(\varphi w)(x_0)\right\rvert = 0.
\end{equation*}
Use this in the equation of $\varphi w$ above to obtain
\begin{equation*}
\varepsilon \varphi\left\lvert D^2u^\varepsilon \right\rvert ^2 \leq \varphi (Df\cdot Du^\varepsilon)+ 2 p\left\lvert Du^\varepsilon\right\rvert ^{p-1} \left\lvert D\varphi\right\rvert w + \varepsilon w \left\lvert \Deltata\varphi\right\rvert + 2\varepsilon w\frac{\left\lvert D\varphi\right\rvert ^2}{\varphi},
\end{equation*}
where all terms are evaluated at $x_0$. From \eqref{e:ass_power}, we have
\begin{equation}\label{e:est_for_D^2u}
\varepsilon \varphi\left\lvert D^2u^\varepsilon\right\rvert ^2 \leq \varphi \left\lvert Df\right\rvert w^{\frac{1}{2}}+ 2 Cp w^{\frac{p-1}{2}+1} \varphi^{\frac{1+\theta}{2}} + C\varepsilon w \varphi^{\theta} + 2C\varepsilon w\varphi^\theta.
\end{equation}
By Cauchy-Schwartz inequality, $n\left\lvert D^2u^\varepsilon\right\rvert ^2\geq (\Deltata u^\varepsilon)^2$. Thus, if $n\varepsilon < 1$, then
\begin{equation}\label{e:est_for_D^2u_2}
\begin{aligned}
\varepsilon \left\lvert D^2u^\varepsilon\right\rvert ^2 &\geq \frac{(\varepsilon \Deltata u^\varepsilon)^2}{n\varepsilon} \geq (\varepsilon \Deltata u^\varepsilon)^2 = \left( u^\varepsilon + \left\lvert Du^\varepsilon\right\rvert ^p - f\right)^2\\ &\geq \left\lvert Du^\varepsilon\right\rvert ^{2p} - 2C\left\lvert Du^\varepsilon\right\rvert ^p \geq \frac{\left\lvert Du^\varepsilon\right\rvert ^{2p}}{2} - 2C,
\end{aligned}
\end{equation}
where $C$ depends on $\max_{\overline{\Omegaega}}f$ only. Using \eqref{e:est_for_D^2u_2} in \eqref{e:est_for_D^2u}, we obtain that
\begin{equation*}
\varphi\left(\frac{1}{2}w^p - 2C\right) \leq \varphi \left\lvert Df\right\rvert w^{\frac{1}{2}}+ 2 Cp w^{\frac{p-1}{2}+1} \varphi^{\frac{1+\theta}{2}} + 3C\varepsilon w \varphi^{\theta}.
\end{equation*}
Multiply both sides by $\varphi^{p-1}$ to deduce that
\begin{align*}
(\varphi w)^p \leq 4C\varphi^{p-1} + 2\Vert Df\Vert_{L^\infty}\varphi^p w^{\frac{1}{2}} + 4Cp \varphi^{\frac{2p+\theta - 1}{2}}w^{\frac{p+1}{2}} + 6C\varepsilon \varphi^{p+\theta - 1}w.
\end{align*}
Choose $2p+\theta -1 \geq p+1$, i.e., $p+\theta\geq 2$. This is always possible with the requirement $\theta \in (0,1)$, as $1<p <\infty$. Then we get
\begin{equation}\label{e:est_for_D^2u_3}
(\varphi w)^p \leq C\left(1+ (\varphi w)^\frac{1}{2} + (\varphi w)^\frac{p+1}{2} +(\varphi w)\right).
\end{equation}
As a polynomial in $z = (\varphi w)(x_0)$, this implies that $(\varphi w)(x_0)\leq C$ where $C$ depends on coefficients of the right hand side of \eqref{e:est_for_D^2u_3}, which gives our desired gradient bound since $w(x)=(\varphi w)(x) \leq (\varphi w)(x_0)$ for $x \in \overline{\Omegaega}_\deltata\subset \mathrm{supp}\;\varphi$.
\end{proof}
\subsection{Well-posedness of \eqref{eq:PDEeps}}
\begin{proof} [Proof of Theorem \ref{thm:wellposed1<p<2}] If $p\in (1,2)$, we use the ansatz $ u(x) = C_\varepsilon d(x)^{-\alphapha}$ to find a solution to \eqref{eq:PDEeps}. Plug the ansatz into \eqref{eq:PDEeps} and compute
\begin{equation*}
\begin{split}
\left\lvert Du (x)\right\rvert ^p &= \frac{(\alphapha C_\varepsilon)^p }{d(x)^{p(\alphapha+1)}}\left\lvert D d(x)\right\rvert ^p,\\
\varepsilon\Deltata u(x) &= \frac{\varepsilon C_\varepsilon\alphapha(\alphapha+1)}{d(x)^{\alphapha+2}}\left\lvert D d(x)\right\rvert ^2 - \frac{\varepsilon C_\varepsilon\alphapha}{d(x)^{\alphapha+1}}\Deltata d(x).
\end{split}
\end{equation*}
Since $\left\lvert D d(x)\right\rvert = 1$ for $x$ near $\partial\Omegaega$, as $x\to \partial \Omegaega$, the explosive terms of the highest order are
\begin{equation*}
C_\varepsilon^p \alphapha^p d^{-(\alphapha+1)p} -\varepsilon C_\varepsilon \alphapha(\alphapha+1)d^{-(\alphapha+2)}.
\end{equation*}
Set the above to be zero to obtain that
\begin{equation}\label{e:relation}
\displaystyle\alphapha = \frac{2-p}{p-1} \qquad\text{and}\qquad C_\varepsilon = \left(\frac{1}{\alphapha}(\alphapha+1)^\frac{1}{p-1}\right) \varepsilon^{\frac{1}{p-1}} = \frac{1}{\alphapha}(\alphapha+1)^{\alphapha+1}\varepsilon^{\alphapha+1}.
\end{equation}
For $0<\deltata < \frac{1}{2}\deltata_0$ and $\eta$ small, define
\begin{equation*}
\begin{split}
\overline{w}_{\eta,\deltata}(x) &:= \frac{(C_\alphapha+\eta)\varepsilon^{\alphapha+1}}{(d(x)-\deltata)^\alphapha} + M_\eta, \qquad x\in \Omegaega_\deltata,\\
\underline{w}_{\eta,\deltata}(x) &:= \frac{(C_\alphapha-\eta)\varepsilon^{\alphapha+1}}{(d(x)+\deltata)^\alphapha} - M_\eta, \qquad x\in \Omegaega^\deltata,
\end{split}
\end{equation*}
where $C_\alphapha := \frac{1}{\alphapha} (\alphapha+1)^{\alphapha+1} $, $M_\eta$ to be chosen. Next, we show that $\overline{w}_{\eta,\deltata}$ is a supersolution of \eqref{eq:PDEeps} in $\Omegaega_\deltata$, while $\underline{w}_{\eta,\deltata}$ is a subsolution of \eqref{eq:PDEeps} in $\Omegaega^\deltata$. Compute
\begin{align*}
\mathcal{L}^\varepsilon\left[\overline{w}_{\eta,\deltata}\right](x) = &\frac{ (C_\alphapha + \eta)\varepsilon^{\alphapha+1}}{(d(x)-\deltata)^\alphapha} + M_\eta + \frac{(C_\alphapha+\eta)^p \alphapha^p\varepsilon^{\alphapha+2}}{(d(x)-\deltata)^{\alphapha+2}}\left\lvert D d(x)\right\rvert ^p - f(x) \\
&- \frac{(C_\alphapha+\eta)\alphapha(\alphapha+1)\varepsilon^{\alphapha+2}}{(d(x)-\deltata)^{\alphapha+2}}\left\lvert D d(x)\right\rvert ^2 + \frac{(C_\alphapha+\eta)\alphapha \varepsilon^{\alphapha+2}}{(d(x)-\deltata)^{\alphapha+1}}\Deltata d(x)\\
\geq &M_\eta - f(x) \\
&+ \underbrace{\frac{\nu C_\alphapha\alphapha(\alphapha+1)\varepsilon^{\alphapha+2}}{(d(x)-\deltata)^{\alphapha+2}}\left[\nu^{p-1}\left\lvert Dd(x)\right\rvert ^{p} - \left\lvert Dd(x)\right\rvert ^2 + \frac{(d(x)-\deltata)\Deltata d(x)}{\alphapha+1}\right]}_{I},
\end{align*}
where we use $(C_\alphapha\alphapha)^p = C_\alphapha\alphapha(\alphapha+1)$ and $\nu = \frac{C_\alphapha+\eta}{C_\alphapha} \in (1,2)$ for small $\eta$. Let
\begin{equation*}
\deltata_\eta : = \frac{\alphapha+1}{K_2}\left[\nu^{p-1}-1\right]
\end{equation*}
and $\deltata_\eta \to 0$ as $\eta \to 0$.
To get $\mathcal{L}^\varepsilon\left[\overline{w}_{\eta,\deltata}\right]\geq 0$, there are two cases to consider, depending on how large $d(x)-\deltata$ is.
\begin{itemize}
\item If $0< d(x)-\deltata <\deltata_\eta < \deltata_0$ for $\eta$ small and fixed, then $\left\lvert Dd(x)\right\rvert = 1$, and thus $I\geq 0$. Hence, $\mathcal{L}^\varepsilon\left[\overline{w}_{\eta,\deltata}\right]\geq 0$ if we choose $M_\eta \geq \max_{\overline{\Omegaega}} f$.
\item If $d(x)-\deltata\geq \deltata_\eta$, then
\begin{equation*}
I \leq \left(\frac{1}{\deltata_\eta}\right)^{\alphapha+2}\nu C_\alphapha \alphapha(\alphapha+1)\left[\nu^{p-1}K_1^{p}+K_1^2+K_2K_0\right]\varepsilon^{\alphapha+2}.
\end{equation*}
Thus, we can choose $M_\eta = \max_{\overline{\Omegaega}} f + C\varepsilon^{\alphapha+2}$ for $C$ large enough (depending on $\eta$) so that $\mathcal{L}^\varepsilon\left[\overline{w}_{\eta,\deltata}\right]\geq 0$.
\end{itemize}
Therefore, $\overline{w}_{\eta,\deltata}$ is a supersolution in $\Omegaega_\deltata$.
Similarly, we have
\begin{align*}
&\mathcal{L}_\varepsilon\left[\underline{w}_{\eta,\deltata}\right](x) \\
= &\frac{ (C_\alphapha - \eta)\varepsilon^{\alphapha+1}}{(d(x)+\deltata)^\alphapha} - M_\eta + \frac{(C_\alphapha-\eta)^p \alphapha^p\varepsilon^{\alphapha+2}}{(d(x)+\deltata)^{\alphapha+2}}\left\lvert D d(x)\right\rvert ^p - f(x) \\
& - \frac{(C_\alphapha-\eta)\alphapha(\alphapha+1)\varepsilon^{\alphapha+2}}{(d(x)+\deltata)^{\alphapha+2}}\left\lvert D d(x)\right\rvert ^2 + \frac{(C_\alphapha-\eta)\alphapha \varepsilon^{\alphapha+2}}{(d(x)+\deltata)^{\alphapha+1}}\Deltata d(x)\\
= &- M_\eta - f(x) \\
&+\underbrace{\frac{\nu C_\alphapha\alphapha(\alphapha+1)\varepsilon^{\alphapha+2}}{(d(x)+\deltata)^{\alphapha+2}}\left[\nu^{p-1}\left\lvert Dd(x)\right\rvert ^{p} - \left\lvert Dd(x)\right\rvert ^2 + \frac{(d(x)+\deltata)\Deltata d(x)}{\alphapha+1}+\frac{(d(x)+\deltata)^2}{\alphapha(\alphapha+1)\varepsilon}\right]}_{J},
\end{align*}
where $\nu = \frac{C_\alphapha-\eta}{C_\alphapha}\in (0,1)$ for small $\eta$. Let
\begin{equation*}
\deltata_\eta := \left(1-\nu^{p-1}\right)\left(\frac{\alphapha(\alphapha+1)\varepsilon}{1+K_2\alphapha\varepsilon}\right)
\end{equation*}
and $\deltata_\eta \to 0$ as $ \eta \to 0$.
To obtain $\mathcal{L}^\varepsilon\left[\underline{w}_{\eta,\deltata}\right]\leq 0$, there are two cases to consider depending on how large $d(x)+\deltata$ is.
\begin{itemize}
\item If $0<d(x)+\deltata<\deltata_\eta < \deltata_0$ for $\eta$ small and fixed, then $\left\lvert Dd(x)\right\rvert = 1$, and thus $J\leq 0$. Hence, $\mathcal{L}^\varepsilon\left[\underline{w}_{\eta,\deltata}\right]\leq 0$ if we choose $M_\eta \geq -\max_{\Omegaega}f$.
\item If $d(x)+\deltata\geq \deltata_\eta$, then
\begin{equation*}
\left\lvert J\right\rvert \leq \left(\frac{1}{\deltata_\eta}\right)^{\alphapha+2} \nu C_\alphapha\alphapha(\alphapha+1)\left[\nu^{p-1}K_1^{p}+K_1^2 + \frac{(K_0+1)K_2}{\alphapha+1} + \frac{(K_0+1)^2}{\alphapha(\alphapha+1)\varepsilon}\right]\varepsilon^{\alphapha+2}
\end{equation*}
Thus, we can choose $M_\eta = -\max_{\overline{\Omegaega}} f - C\varepsilon^{\alphapha+2}$ for $C$ large enough (depending on $\eta$) so that $\mathcal{L}^\varepsilon\left[\underline{w}_{\eta,\deltata}\right]\leq 0$.
\end{itemize}
Therefore, $\underline{w}_{\eta,\deltata}$ is a subsolution in $\Omegaega^\deltata$.
For $p=2$, we use the ansatz $u(x) = -C_\varepsilon \log(d(x))$ instead. Similar to the previous case, one can find $u(x) = -\varepsilon\log(d(x))$. For $0<\deltata<\frac{1}{2}\deltata_0$, define
\begin{equation*}
\begin{split}
&\overline{w}_{\eta,\deltata}(x) = -(1+\eta)\varepsilon\log(d(x)-\deltata) + M_\eta, \qquad x\in \Omegaega_\deltata,\\
&\underline{w}_{\eta,\deltata}(x) = -(1-\eta)\varepsilon\log(d(x)+\deltata) - M_\eta, \qquad x\in \Omegaega^\deltata,
\end{split}
\end{equation*}
where $M_\eta$ is to be chosen so that $\overline{w}_{\eta,\deltata}(x)$ is a supersolution in $\Omegaega_\deltata$ and $\underline{w}_{\eta,\deltata}$ is a subsolution in $\Omegaega^\deltata$. The computations are omitted here, as they are similar to the previous case.
We divide the rest of the proof into 3 steps. We first construct a minimal solution, then a maximal solution to \eqref{eq:PDEeps}, and finally show that they are equal to conclude the existence and the uniqueness of the solution to \eqref{eq:PDEeps}.
\paragraph{\textbf{Step 1.}} There exists a minimal solution $\underline{u}\in \mathrm{C}^2(\Omegaega)$ of \eqref{eq:PDEeps} such that $v\geq \underline{u}$ for any other solution $v\in \mathrm{C}^2(\Omegaega)$ solving \eqref{eq:PDEeps}.
\begin{proof} Let $w_{\eta,\deltata}\in \mathrm{C}^2(\Omegaega)$ solve
\begin{equation}\label{e:w_def}
\begin{cases}
\mathcal{L}^\varepsilon\left[w_{\eta,\deltata}\right] = 0 &\qquad\text{in}\;\Omegaega,\\
\qquad w_{\eta,\deltata} = \underline{w}_{\eta,\deltata} &\qquad\text{on}\;\partial\Omegaega.
\end{cases}
\end{equation}
\begin{itemize}
\item Fix $\eta>0$. As $\deltata\to 0^+$, the value of $\underline{w}_{\eta,\deltata}$ blows up on the boundary. Therefore, by the standard comparison principle for the second-order elliptic equation with the Dirichlet boundary, $\deltata_1 \leq \deltata_2$ implies $w_{\eta,\deltata_1}\geq w_{\eta,\deltata_2}$ on $\overline{\Omegaega}$.
\item For $\deltata'>0$, since $\underline{w}_{\eta,\deltata'}$ is a subsolution in $\overline{\Omegaega}$ with finite boundary,
\begin{equation}\label{e:cp_delta1}
0<\deltata \leq \deltata'\qquad\Longrightarrow\qquad \underline{w}_{\eta,\deltata'} \leq w_{\eta_,\deltata'}\leq w_{\eta,\deltata} \qquad\text{on}\;\overline{\Omegaega}.
\end{equation}
\item Similarly, since $\overline{w}_{\eta,\deltata'}$ is a supersolution on $\Omegaega_{\deltata'}$ with infinity value on the boundary $\partial\Omegaega_{\deltata'}$, by the comparison principle,
\begin{equation}\label{e:cp_delta2}
w_{\eta,\deltata} \leq \overline{w}_{\eta, \deltata'} \qquad\text{in}\;\Omegaega_{\deltata'} \qquad\Longrightarrow\qquad w_{\eta,\deltata} \leq \overline{w}_{\eta,0} \qquad\text{in}\;\Omegaega.
\end{equation}
\end{itemize}
From \eqref{e:cp_delta1} and \eqref{e:cp_delta2}, we have
\begin{equation}\label{e:cp_delta3}
0<\deltata \leq \deltata'\qquad\Longrightarrow\qquad \underline{w}_{\eta,\deltata'} \leq w_{\eta_,\deltata'}\leq w_{\eta,\deltata} \leq \overline{w}_{\eta,0} \qquad\text{in}\;\Omegaega.
\end{equation}
Thus, $\{w_{\eta,\deltata}\}_{\deltata>0}$ is locally bounded in $L^{\infty}_{\mathrm{loc}}(\Omegaega)$ ($\{w_{\eta,\deltata}\}_{\deltata>0}$ is uniformly bounded from below). Using the local gradient estimate for $w_{\eta,\deltata}$ solving \eqref{e:w_def}, we deduce that $\{w_{\eta,\deltata}\}_{\deltata>0}$ is locally bounded in $W^{1,\infty}_{\mathrm{loc}}(\Omegaega)$. Since $w_{\eta,\deltata}$ solves \eqref{e:w_def}, we further have that $\{w_{\eta,\deltata}\}_{\deltata>0}$ is locally bounded in $W^{2,r}_{\mathrm{loc}}(\Omegaega)$ for all $r<\infty$ by Calderon-Zygmund estimates.
Local boundedness of $\{w_{\eta,\deltata}\}_{\deltata>0}$ in $W^{2,r}_{\mathrm{loc}}(\Omegaega)$ implies weak$^*$ compactness, that is, there exists a function $u\in W^{2,r}_{\mathrm{loc}}(\Omegaega)$ such that (via subsequence and monotonicity)
\begin{equation*}
w_{\eta,\deltata} \rightharpoonup u \qquad\text{weakly in}\;W^{2,r}_{\mathrm{loc}}(\Omegaega),\qquad \text{and}\qquad
w_{\eta,\deltata} \to u \qquad\text{strongly in}\;W^{1,r}_{\mathrm{loc}}(\Omegaega).
\end{equation*}
In particular, $w_{\eta,\deltata}\to u$ in $\mathrm{C}^1_{\mathrm{loc}}(\Omegaega)$ thanks to Sobolev compact embedding. Let us rewrite the equation $\mathcal{L}^\varepsilon\left[w_{\eta,\deltata}\right] = 0$ as $\varepsilon\Deltata w_{\eta,\deltata}(x) = F[w_{\eta,\deltata}](x)$ for $x \in U\subset\subset \Omegaega$, where
\begin{equation*}
F[w_{\eta,\deltata}](x) = w_{\eta,\deltata}(x) + H(x,Dw_{\eta,\deltata}(x)).
\end{equation*}
Since $w_{\eta,\deltata}\to u$ in $\mathrm{C}^1(U)$ as $\deltata \to 0$, we have $F[w_{\eta,\deltata}](x) \to F(x)$ uniformly in $U$ as $\deltata \to 0$, where
\begin{equation*}
F(x) = u(x) + H(x,Du(x)).
\end{equation*}
In the limit, we obtain that $u\in L^2(U)$ is a weak solution of $\varepsilon\Deltata u = F$ in $U$ where $F$ is continuous. Thus, $u\in \mathrm{C}^2(\Omegaega)$ and by stability, $u$ solves $\mathcal{L}^\varepsilon[u] = 0$ in $\Omegaega$. From \eqref{e:cp_delta3}, we also have
\begin{equation*}
\underline{w}_{\eta,0} \leq u \leq \overline{w}_{\eta,0} \qquad\text{in}\;\Omegaega.
\end{equation*}
Moreover, $u(x)\to \infty$ as $\mathrm{dist}(x,\partial\Omegaega)\to 0$ with the precise rate like \eqref{rate_p<2} or \eqref{rate_p=2}. Note that by construction, $u$ may depend on $\eta$. But next, we will show that $u$ is independent of $\eta$, by proving $u$ is the unique minimal solution of $\mathcal{L}^\varepsilon[u] = 0$ in $\Omegaega$ with $u = +\infty$ on $\partial\Omegaega$.
Let $v\in \mathrm{C}^{2}(\Omegaega)$ be a solution to \eqref{eq:PDEeps}. Fix $\deltata>0$. Since $v(x)\to \infty$ as $x\to \partial\Omegaega$ while $w_{\eta,\deltata}$ remains bounded on $\partial \Omegaega$, the comparison principle yields
\begin{equation*}
v\geq w_{\eta,\deltata} \qquad\text{in} \; \Omegaega.
\end{equation*}
Let $\deltata\to 0$ and we deduce that $v\geq u$ in $\Omegaega$. This concludes that $u$ is the minimal solution in $\mathrm{C}^2(\Omegaega)(\forall\,r<\infty)$ and thus $u$ is independent of $\eta$.
\end{proof}
\paragraph{\textbf{Step 2.}} There exists a maximal solution $\overline{u}\in \mathrm{C}^2(\Omegaega)$ of \eqref{eq:PDEeps} such that $v\leq \overline{u}$ for any other solution $v\in \mathrm{C}^2(\Omegaega)$ solving \eqref{eq:PDEeps}.
\begin{proof} For each $\deltata>0$, let $u_\deltata\in \mathrm{C}^2(\Omegaega_\deltata)$ be the minimal solution to $\mathcal{L}^\varepsilon[u_\deltata] = 0$ in $\Omegaega_\deltata$ with $u_\deltata = +\infty$ on $\partial\Omegaega_\deltata$. By the comparison principle, for every $\eta>0$, there holds
\begin{equation*}
\underline{w}_{\eta,\deltata} \leq u_\deltata \leq \overline{w}_{\eta,\deltata} \qquad\text{in}\;\Omegaega_\deltata,
\end{equation*}
and
\begin{equation*}
0<\deltata<\deltata' \qquad \Longrightarrow\qquad u_\deltata \leq u_\deltata' \qquad\text{in}\;\Omegaega_{\deltata'}\,.
\end{equation*}
The monotoniciy, together with the local boundedness of $\{u_\deltata\}_{\deltata>0}$ in $W^{2,r}_{\mathrm{loc}}(\Omegaega)$, implies that there exists $u\in W^{2,r}_{\mathrm{loc}}(\Omegaega)$ for all $r<\infty$ such that $u_\deltata\to u$ strongly in $\mathrm{C}^1_{\mathrm{loc}}(\Omegaega)$. Using the equation $\mathcal{L}^\varepsilon [u_\deltata] = 0$ in $\Omegaega_\deltata$ and the regularity of Laplace's equation, we can further deduce that $u\in \mathrm{C}^2(\Omegaega)$ solves \eqref{eq:PDEeps} and
\begin{equation*}
\underline{w}_{\eta,0} \leq u\leq \overline{w}_{\eta,0} \qquad\text{in}\;\Omegaega
\end{equation*}
for all $\eta>0$. As $u_\deltata$ is independent of $\eta$ by the previous argument in Step 1, it is clear that $u$ is also independent of $\eta$. Now we show that $u$ is the maximal solution of \eqref{eq:PDEeps}. Let $v\in\mathrm{C}^2(\Omegaega)$ solve \eqref{eq:PDEeps}. Clearly $v\leq u_\deltata$ on $\Omegaega_\deltata$. Therefore, as $\deltata \to 0$, we have $v\leq u$.
\end{proof}
In conclusion, we have found a minimal solution $\underline{u}$ and a maximal solution $\overline{u}$ in $\mathrm{C}^2(\Omegaega)$ such that
\begin{equation}\label{e:chain}
\underline{w}_{\eta,0} \leq \underline{u}\leq \overline{u}\leq \overline{w}_{\eta,0} \qquad\text{in}\;\Omegaega
\end{equation}
for any $\eta>0$. This extra parameter $\eta$ now enables us to show that $\overline{u} = \underline{u}$ in $\Omegaega$. The key ingredient here is the convexity in the gradient slot of the operator.
\paragraph{\textbf{Step 3.}} We have $\overline{u}\equiv \underline{u}$ in $\Omegaega$. Therefore, the solution to \eqref{eq:PDEeps} in $\mathrm{C}^2(\Omegaega)$ is unique.
\begin{proof} Let $\theta\in (0,1)$. Define $w_\theta = \theta \overline{u} + (1-\theta) \inf_{\Omegaega} f$. It can be verified that $w_\theta$ is a subsolution to \eqref{eq:PDEeps}. Then one may argue that by the comparison principle,
\begin{equation*}
w_\theta = \theta \overline{u} + (1-\theta)\inf_{\Omegaega} f\leq \underline{u} \qquad\text{in}\;\Omegaega,
\end{equation*}
and conclude that $\overline{u} \leq \underline{u}$ by letting $\theta\to 1$. But we have to be careful here. As they are both explosive solutions, to use the comparison principle, we need to show that $w_\theta \leq \underline{u}$ in a neighborhood of $\partial\Omegaega$. From \eqref{e:chain}, we see that
\begin{align*}
&1\leq \frac{\overline{u}(x)}{\underline{u}(x)} \leq \frac{\overline{w}_{\eta,0}(x)}{\underline{w}_{\eta,0}(x)} = \frac{(C_\alphapha+\eta)+ M_\eta d(x)^\alphapha}{(C_\alphapha-\eta)- M_\eta d(x)^\alphapha},& 1<p<2,\\
&1 \leq \frac{\overline{u}(x)}{\underline{u}(x)} \leq \frac{\overline{w}_{\eta,0}(x)}{\underline{w}_{\eta,0}(x)} = \frac{-(1+\eta)\log(d(x)) + M_\eta}{-(1-\eta)\log(d(x))-M_\eta}, & p=2,
\end{align*}
for $x\in \Omegaega$. Hence,
\begin{align*}
&1\leq \lim_{d(x)\to 0} \left(\frac{\overline{u}(x)}{\underline{u}(x)}\right) \leq \frac{C_\alphapha+\eta}{C_\alphapha-\eta}, & 1< p < 2,\\
&1\leq \lim_{d(x)\to 0} \left(\frac{\overline{u}(x)}{\underline{u}(x)}\right) \leq \frac{-(1+\eta)}{-(1-\eta)}, & p = 2.
\end{align*}
Since $\eta>0$ is chosen arbitrary, we obtain
\begin{equation*}
\lim_{d(x)\to 0} \left(\frac{\overline{u}(x)}{\underline{u}(x)}\right) = 1.
\end{equation*}
This means for any $\varsigma\in(0,1)$, there exists ${\deltata_1}(\varsigma)>0$ small such that
\begin{equation*}
\frac{\overline{u}(x)}{\underline{u}(x)}\leq (1+\varsigma)\Longrightarrow \left(\frac{1}{1+\varsigma}\right)\overline{u}(x) \leq \underline{u}(x) \qquad\text{in}\; \Omegaega\backslash \Omegaega_{\deltata_1}.
\end{equation*}
For a fixed $\theta\in (0,1)$, one can always choose $\varsigma$ small enough so that $\displaystyle (1+\varsigma)^{-1} \geq \frac{1+\theta}{2}$. Since $\overline{u}(x) \to +\infty$ as $d(x) \to 0$, there exists $\deltata_2 > 0$ such that $\overline{u}(x) \geq 2 \inf_\Omegaega f$ for all $x \in \Omegaega \setminus \Omegaega_{\deltata_2}$.
Now we have
\begin{equation*}
\underline{u}(x)\geq \left(\frac{1}{1+\varsigma}\right)\overline{u}(x) \geq \theta \overline{u}(x)+ \left( \frac{1-\theta}{2}\right)\overline{u}(x)\geq \theta \overline{u}(x) + (1-\theta)\left(\inf_\Omegaega f\right)
\end{equation*}
for all $x \in \Omegaega \setminus \Omegaega_\deltata$ where $\deltata: = \min \{\deltata_1, \deltata_2\}$. This implies for any fixed $\theta\in(0,1)$, $w_\theta \leq \underline{u}$ in a neighborhood of $\partial\Omegaega$. Hence, by the comparison principle,
\begin{equation*}
w_\theta = \theta \overline{u} + (1-\theta)\inf_{\Omegaega} f\leq \underline{u} \qquad\text{in}\;\Omegaega,
\end{equation*}
for any $\theta \in (0,1)$. Then let $\theta \to 1$ to get the conclusion.
\end{proof}
This finishes the proof of the well-posedness of \eqref{eq:PDEeps} for $1<p\leq2$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:max}] The proof is a variation of Perron's method (see \cite{Capuzzo-Dolcetta1990}) and we proceed by contradiction. Let $\varphi\in \mathrm{C}(\overline{\Omegaega})$ and $x_0\in \overline{\Omegaega}$ such that $u(x_0) = \varphi(x_0)$ and $u-\varphi$ has a global strict minimum over $\overline{\Omegaega}$ at $x_0$ with
\begin{equation}\label{eq:max_a1}
\varphi(x_0) + H(x_0,D\varphi(x_0)) < 0.
\end{equation}
Let $\varphi^\varepsilon(x) = \varphi(x) - \left\lvert x-x_0\right\rvert ^2 + \varepsilon$ for $x\in \overline{\Omegaega}$. Let $\deltata > 0$. We see that for $x\in \partial B(x_0,\deltata)\cap \overline{\Omegaega}$,
\begin{equation*}
\varphi^\varepsilon(x) = \varphi(x) - \deltata^2 +\varepsilon \leq \varphi(x) - \varepsilon
\end{equation*}
if $2\varepsilon \leq \deltata^2$. We observe that
\begin{equation*}
\begin{split}
\varphi^\varepsilon(x) - \varphi(x_0) &= \varphi(x)-\varphi(x_0) + \varepsilon - \left\lvert x-x_0\right\rvert ^2 \\
D\varphi^\varepsilon(x) - D\varphi(x_0) &= D\varphi(x) - D\varphi(x_0) - 2(x-x_0)
\end{split}
\end{equation*}
for $x\in B(x_0,\deltata)\cap \overline{\Omegaega}$. By the continuity of $H(x,p)$ near $(x_0,D\varphi(x_0))$ and the fact that $\varphi\in \mathrm{C}^1(\overline{\Omegaega})$, we can deduce from \eqref{eq:max_a1} that if $\deltata$ is small enough and $0<2\varepsilon < \deltata^2$, then
\begin{equation}\label{eq:max_a2}
\varphi^\varepsilon(x)+H(x,D\varphi^\varepsilon(x)) < 0 \qquad\text{for}\;x\in B(x_0,\deltata)\cap \overline{\Omegaega}.
\end{equation}
We have found $\varphi^\varepsilon\in \mathrm{C}^1(\overline{\Omegaega})$ such that $\varphi^\varepsilon(x_0)>u(x_0)$, $\varphi^\varepsilon<u$ on $\partial B(x_0,\deltata)\cap \overline{\Omegaega}$ and \eqref{eq:max_a2}. Let
\begin{equation*}
\tilde{u}(x) = \begin{cases}
\max \big\lbrace u(x),\varphi^\varepsilon(x) \big\rbrace &x\in B(x_0,\deltata)\cap \overline{\Omegaega},\\
u(x)&x\notin B(x_0,\deltata)\cap \overline{\Omegaega}.\\
\end{cases}
\end{equation*}
We see that $\tilde{u}\in \mathrm{C}(\overline{\Omegaega})$ is a subsolution of \eqref{eq:PDE0} in $\Omegaega$ with $\tilde{u}(x_0) > u(x_0)$, which is a contradiction. Thus, $u$ is a supersolution of \eqref{eq:PDE0} on $\overline{\Omegaega}$.
\end{proof}
\subsection{Semiconcavity}
We present a proof for the semiconcavity of solution to first-order Hamilton--Jacobi equation using the doubling variable method (see also \cite{Calder2021}).
\begin{theorem}\label{convex} Let $H(x,p) = G(p)-f(x)$ where $G\geq 0$ with $G(0) = 0$ is a convex function from $\mathbb{R}^n\to\mathbb{R}^n$ and $f\in \mathrm{C}^2_c(\mathbb{R}^n)$. Let $u\in \mathrm{C}_c(\mathbb{R}^n)$ be a viscosity solution to $u+H(x,Du) = 0$ in $\mathbb{R}^n$. Then $u$ is semiconcave, i.e., $u$ is a viscosity solution of $-D^2u \geq -c\;\mathbb{I}_n$ in $\mathbb{R}^n$ where
\begin{equation*}
c = \max \big\lbrace D_{\xi\xi}f(x): \left\lvert \xi\right\rvert =1, x\in \mathbb{R}^n \big\rbrace\geq 0.
\end{equation*}
\end{theorem}
\begin{proof} Consider the auxiliary functional
\begin{equation*}
\Phi(x,y,z) = u(x)-2u(y)+u(z) - \frac{\alphapha}{2}\left\lvert x-2y+z\right\rvert ^2 - \frac{c}{2}\left\lvert y-x\right\rvert ^2-\frac{c}{2}\left\lvert y-z\right\rvert ^2
\end{equation*}
for $(x,y,z)\in \mathbb{R}^n\times\mathbb{R}^n\times\mathbb{R}^n$. By the a priori estimate, $u$ is bounded and Lipschitz. Thus, we can assume $\Phi$ achieves its maximum over $\mathbb{R}^n\times\mathbb{R}^n\times \mathbb{R}^n$ at $(x_\alphapha,y_\alphapha,z_\alphapha)$. The viscosity solution tests give us
\begin{align*}
&u(x_\alphapha) + G\big(p_\alphapha+c(x_\alphapha-y_\alphapha)\big) \leq f(x_\alphapha)\\
&u(z_\alphapha) + G\big(p_\alphapha+c(z_\alphapha-y_\alphapha)\big) \leq f(z_\alphapha)\\
&u(y_\alphapha) + G\left(p_\alphapha+\frac{c}{2}(x_\alphapha-y_\alphapha) + \frac{c}{2}(z_\alphapha-y_\alphapha)\right) \geq f(y_\alphapha),
\end{align*}
where $p_\alphapha = \alphapha(x_\alphapha-2y_\alphapha+z_\alphapha)$. By the convexity of $G$, we have
\begin{equation*}
2G\left(p_\alphapha+\frac{c}{2}(x_\alphapha-y_\alphapha) + \frac{c}{2}(z_\alphapha-y_\alphapha)\right) \leq G\big(p_\alphapha+c(x_\alphapha-y_\alphapha)\big) + G\big(p_\alphapha+c(z_\alphapha-y_\alphapha)\big)
\end{equation*}
Therefore,
\begin{equation*}
u(x_\alphapha) - 2u(y_\alphapha) + u(z_\alphapha) \leq f(x_\alphapha) - 2f(y_\alphapha) + f(z_\alphapha).
\end{equation*}
\begin{itemize}
\item $\Phi(x_\alphapha,y_\alphapha,z_\alphapha)\geq \Phi(0,0,0)$ gives us
\begin{equation*}
\frac{\alphapha}{2}\left\lvert x_\alphapha-2y_\alphapha+z_\alphapha\right\rvert ^2 + \frac{c}{2}\left\lvert y_\alphapha-x_\alphapha\right\rvert ^2+\frac{c}{2}\left\lvert y_\alphapha-z_\alphapha\right\rvert ^2 \leq C.
\end{equation*}
Thus, $(x_\alphapha-y_\alphapha)\to h_0$ and $(y_\alphapha-z_\alphapha)\to h_0$ as $\alphapha\to \infty$ for some $h_0 \in \mathbb{R}^n$.
\item $\Phi(x_\alphapha,y_\alphapha,z_\alphapha )\geq \Phi(y_\alphapha+h_0,y_\alphapha,y_\alphapha-h_0)$ gives us
\begin{align*}
u(x_\alphapha)-2u(y_\alphapha) + u(z_\alphapha) - \frac{\alphapha}{2}\left\lvert x_\alphapha - 2y_\alphapha+z_\alphapha\right\rvert ^2 - \frac{c}{2}\left\lvert x_\alphapha-y_\alphapha\right\rvert ^2 - \frac{c}{2}\left\lvert y_\alphapha-z_\alphapha\right\rvert ^2 \\
\geq u(y_\alphapha+h_0) - 2u(y_\alphapha) + u(y_\alphapha-h_0) - c\left\lvert h_0\right\rvert ^2.
\end{align*}
Therefore, by the fact that $u$ is Lipschitz, we have
\begin{equation*}
\begin{split}
\frac{\alphapha}{2}\left\lvert x_\alphapha - 2y_\alphapha+z_\alphapha\right\rvert ^2
\leq &c\left(\frac{2\left\lvert h_0\right\rvert ^2 - \left\lvert x_\alphapha-y_\alphapha\right\rvert ^2 - \left\lvert y_\alphapha-z_\alphapha\right\rvert ^2}{2}\right) \\
&+ C\Big(\left\lvert (x_\alphapha-y_\alphapha) - h_0\right\rvert + \left\lvert (z_\alphapha - y_\alphapha) + h_0\right\rvert \Big) \to 0
\end{split}
\end{equation*}
as $\alphapha\to \infty$.
\end{itemize}
For any $x\in \mathbb{R}^n$, we have $\Phi(x_\alphapha,y_\alphapha,z_\alphapha) \geq \Phi(x+h,x,x-h)$, i.e.,
\begin{align*}
&u(x+h)-2u(x)+u(x-h)-c\left\lvert h\right\rvert ^2\\
\leq &f(x_\alphapha)-2f(y_\alphapha) + f(z_\alphapha) \\
&- \frac{\alphapha}{2}\left\lvert x_\alphapha - 2y_\alphapha+z_\alphapha\right\rvert ^2-\frac{c}{2}\left\lvert y_\alphapha-x_\alphapha\right\rvert ^2-\frac{c}{2}\left\lvert y_\alphapha-z_\alphapha\right\rvert ^2.
\end{align*}
If $\{y_\alphapha\}$ is unbounded, then since $f\in \mathrm{C}_c^2(\mathbb{R}^n)$, we have $f(y_\alphapha)\to 0$ as $\alphapha\to \infty$. As a consequence, $x_\alphapha,z_\alphapha \to \infty$ as well and thus $f(x_\alphapha)-2f(y_\alphapha) + f(z_\alphapha)\to 0$ as $\alphapha\to \infty$. Therefore,
\begin{equation*}
u(x+h)-2u(x)+u(x-h)-c\left\lvert h\right\rvert ^2 \leq 0.
\end{equation*}
If $\{y_\alphapha\}$ is bounded, then $y_\alphapha\to y_0$ for some $y_0 \in \mathbb{R}^n$ as $\alphapha\to \infty$. Thus,
\begin{equation*}
u(x+h)-2u(x)+u(x-h)-c\left\lvert h\right\rvert ^2 \leq f(y_0+h_0) - 2f(y_0) + f(y_0-h_0) -c\left\lvert h_0\right\rvert ^2.
\end{equation*}
Let $\xi = h_0$ and we have
\begin{equation*}
\begin{cases}
f(y_0+h_0) - f(y_0) = \displaystyle\int_0^1 D_x f(y_0 + t\xi)\cdot \xi dt,
\\
f(y_0) - f(y_0-h_0) = \displaystyle\int_0^1 D_x f(y_0 - \xi + t\xi)\cdot \xi dt.
\end{cases}
\end{equation*}
Therefore,
\begin{equation*}
\begin{split}
f(y_0+h_0) - 2f(y_0) + f(y_0-h_0) &= \int_0^1 \Big(D_x f(y_0 + t\xi) - D_x f(y_0 - \xi + t\xi) \Big)\cdot \xi dt\\
&= \int_0^1 \int_0^1 \xi^\mathsf{T} D^2 f(y_0-\xi +t\xi+s \xi) \xi\;dsdt.
\end{split}
\end{equation*}
which implies
\begin{equation*}
\left\lvert f(y_0+h_0) - 2f(y_0) + f(y_0-h_0)\right\rvert \leq \left(\max_{\left\lvert \xi\right\rvert =1}D_{\xi\xi} f\right)\left\lvert \xi\right\rvert ^2.
\end{equation*}
Hence,
\begin{equation*}
u(x+h)-2u(x)+u(x-h)-c\left\lvert h\right\rvert ^2 \leq 0
\end{equation*}
and thus $u$ is semiconcave. It is easy to see that if $\varphi$ is smooth and $u-\varphi$ has a local min at $x$, then $D^2\varphi(x) \leq c\;\mathbb{I}$, i.e., $-D^2\varphi(x)\geq -c\;\mathbb{I}$.
\end{proof}
\end{document} | math | 127,908 |
\begin{document}
\title[Fractional Jumps]{Full Orbit Sequences in Affine Spaces via Fractional Jumps and Pseudorandom Number Generation}
\author{Federico Amadio Guidi}
\address{Mathematical Institute, University of Oxford, Oxford, UK}
\email{[email protected]}
\author{Sofia Lindqvist}
\address{Mathematical Institute, University of Oxford, Oxford, UK}
\email{[email protected]}
\author[Giacomo Micheli]{Giacomo Micheli$^*$}
\thanks{$^*$ Corresponding author.}
\address{Mathematical Institute, University of Oxford, Oxford, UK}
\email{[email protected]}
\maketitle
\begin{abstract}
Let $n$ be a positive integer. In this paper we provide a general theory to produce full orbit sequences in the affine $n$-dimensional space over a finite field. For $n=1$ our construction covers the case of the Inversive Congruential Generators (ICG).
In addition, for $n>1$ we show that the sequences produced using our construction are easier to compute than ICG sequences. Furthermore, we prove that they have the same discrepancy bounds as the ones constructed using the ICG.
\end{abstract}
{\footnotesize\noindent\textbf{Keywords}: full orbit sequences, pseudorandom number generators, inversive congruential generators, discrepancy.}\\
{\footnotesize\noindent\textbf{MSC2010 subject classification}: 11B37, 15B33, 11T06, 11K38, 11K45, 11T23, 65C10, 37P25.}
\section{Introduction}
In recent years there has been a great interest in the construction of discrete dynamical systems with given properties (see for example \cite{bib:eich93,bib:EHHW98,bib:HBM17,bib:ost10,bib:OPS10,bib:OS10degree,bib:OS10length}) both for applications (see for example \cite{bib:BW05,bib:chou95,bib:eich91,bib:eich92,bib:GPOS14, bib:NS02, bib:NS03, bib:TW06,bib:winterhof10}) and for the purely mathematical interest that these objects have (see for example \cite{bib:eich91,bib:EMG09,bib:ferraguti2016existence,bib:FMS16,bib:FMS17,bib:GSW03}).
This paper deals with the problem of finding discrete dynamical systems which can be new candidates for pseudorandom number generation.
Let us denote the set of natural numbers by $\mathbb{N}$. Given a finite set $S$, a sequence $\{a_m\}_{m\in \mathbb{N}}$ of elements in $S$ is said to have \emph{full orbit} if for any $s\in S$ there exists $m\in \mathbb{N}$ such that $a_m=s$.
Let $q$ be a prime power, $\mathbb{F}_q$ be the finite field of cardinality $q$, and $n$ be a positive integer. In this paper we produce maps $\psi:\mathbb{F}_q^n \rightarrow \mathbb{F}_q^n$ such that
\begin{itemize}
\item the sequences $\{\psi^m(0)\}_{m\in \mathbb{N}}$ have full orbit (whenever this property is verified, we say that the map $\psi$ is \emph{transitive}),
\item the sequences constructed from $\psi$ have nice discrepancy bounds, analogous to those constructed from an Inversive Congruential Generator (ICG),
\item they are very inexpensive to iterate: if $n>1$ they are asymptotically less expensive than an ICG for the same bitrate.
\end{itemize}
In addition, such maps can be described using quotients of degree one polynomials.
From a purely theoretical point of view related to the full orbit property, one of the reasons why such constructions are interesting is that one cannot build transitive affine maps (i.e. of the form $x\mapsto Ax+b$, with $A$ an invertible $n\times n$ matrix and $b$ an $n$-dimensional vector) unless either $n=1$ and $q$ is prime, or $n=2$ and $q=2$ (see Theorem \ref{affine_transitivity_theorem}).
For $n=1$ our construction covers the well-studied case of the ICG, for which we obtain easy proofs of classical facts (see for example Remark \ref{remarkICGfullorbit}). In fact, we fit the theory of full orbit sequences in a much wider context, where tools from projective geometry can be used to establish properties of the sequences produced with our method (see for example Proposition \ref{theorem_uniformity}).
Let us now summarise the results of the paper. The main tool we use to construct full orbit sequences is the notion of fractional jump of projective maps, which is described in Section \ref{affine_jumps}.
With such a notion we are able to produce maps in the affine space which can be guaranteed to be transitive when they are fractional jumps of transitive projective maps. In Section \ref{transitivity_projective} we characterise transitive projective maps using the notion of projective primitivity for polynomials (see Definition \ref{projectively_primitive_polynomial}). In Section \ref{uniformity} we show that whenever our sequences come from the iterations of transitive projective automorphisms, they behave quite uniformly with respect to proper projective subspaces (i.e. not many consecutive element in the sequence can lie in a proper subspace of the projective space). This fact (and in particular Proposition \ref{theorem_uniformity}) will allow us in Section \ref{explicit} to give an explicit description of the fractional jump of a transitive projective map, finally leading to the new explicit constructions of full orbit sequences promised earlier. In turn, such a description and the theory developed in Section \ref{transitivity_projective} allow us to prove the discrepancy bounds of Theorem \ref{thm:discrepancy} in Section \ref{discrepancy}. In Section \ref{computation} we show the computational advantage of our approach compared to the classical ICG one.
Finally, we include some conclusions which summarise the results of the paper.
\subsection*{Notation}
Let us denote the set of natural numbers by $\mathbb{N}$, and the ring of integers by $\mathbb{Z}$. For a commutative ring with unity $R$, let us denote by $R^*$ the group of invertible elements of $R$.
We denote by $\mathbb{F}_q$ the finite field of cardinality $q$, which will be fixed throughout the paper, and by $\overline{\mathbb{F}}_q$ an algebraic closure of $\mathbb{F}_q$. Given an integer $n \geq 1$, we often denote the $n$-dimensional affine space $\mathbb{F}_q^n$ by $\mathbb{A}^n$. The $n$-dimensional projective space over the finite field $\mathbb{F}_q$ is denoted by $\mathbb{P}^n$. Also, we denote by $\mathbb{G}\mathrm{r} (d, n)$ the set of $d$-dimensional projective subspaces of $\mathbb{P}^n$.
We denote by $\mathbb{F}_q[x_1,...,x_n]$ the ring of polynomials in $n$ variables with coefficients in $\mathbb{F}_q$. For a polynomial $a \in \mathbb{F}_q[x_1,...,x_n]$ we denote by $\deg a$ its total degree, which we will simply call its degree. Also, for $b\in\mathbb{F}_q[x_1,\dots,x_n]$ we let $V(b)$ denote the set of points $x\in \mathbb{A}^n$ such that $b(x)=0$.
We denote by $\mathrm{GL}_n (\mathbb{F}_q)$ the general linear group over the field $\mathbb{F}_q$, i.e. the group of $n \times n$ invertible matrices with entries in $\mathbb{F}_q$, and by $\mathrm{PGL}_{n+1} (\mathbb{F}_q)$ the group of automorphisms of $\mathbb{P}^{n}$. Recall that $\mathrm{PGL}_{n+1} (\mathbb{F}_q)$ can be identified with the quotient group $\mathrm{GL}_{n+1} (\mathbb{F}_q) / \mathbb{F}_q^*\mathrm{Id}$, where $\mathbb{F}_q^*\mathrm{Id}$ is just the subgroup of nonzero scalar multiples of the identity matrix $\mathrm{Id}$. Given a matrix $M \in \mathrm{GL}_{n+1} (\mathbb{F}_q)$, we denote by $[M]$ its class in $\mathrm{PGL}_{n+1} (\mathbb{F}_q)$.
Let $X$ be either $\mathbb{A}^n$ or $\mathbb{P}^n$. We will say that a map $f : X \rightarrow X$ is \emph{transitive}, or equivalently that it \emph{acts transitively on $X$}, if for any $x, y \in X$ there exists an integer $i \geq 0$ such that $y = f^i (x)$. Equivalently, $f$ is transitive if and only if for any $x \in X$ the sequence $\{ f^m(x)\}_{m \in \mathbb{N}}$ has full orbit, that is $\{ f^m (x) \, : \, m \in \mathbb{N} \} = X$. A map $f:\mathbb{A}^n\rightarrow \mathbb{A}^n$ is said to be affine if there exist $A\in \mathrm{GL}_n(\mathbb{F}_q)$ and $b\in \mathbb{F}_q^n$ such that $f(x)=Ax+b$ for any $x\in \mathbb{A}^n$.
Let $G$ be a group acting on a set $S$. The orbit of an element $s\in S$ will be denoted by $\mathcal O(s)$. For any element $g\in G$, let us denote by $o(g)$ the order of $g$ in $G$.
We write $f\ll g$ or $f=O(g)$ to mean that for some positive constant $C$ it holds that $|f|\le Cg$. The notation $f\ll_\delta g$ or $f=O_\delta(g)$ means the same, but now the constant $C$ may depend on the parameter $\delta$.
For any real vector $\mathbf{h}=(h_1,\dots, h_n)$, we write
$\|\mathbf{h}\|_{\infty}=
\max\{|h_j|\, : \, j\in \{1,\dots, n\}\}$.
Finally, for any prime $p$ and any $z \in \mathbb{Z}$ we write $e_p (z) = \exp (2 \pi i z / p)$.
\section{Fractional jumps} \label{affine_jumps}
Fix the standard projective coordinates $X_0, \ldots, X_n$ on $\mathbb{P}^n$, and the canonical decomposition
\begin{equation} \label{decomposition}
\mathbb{P}^n = U \cup H,
\end{equation}
where
\begin{align*}
U &= \set{[X_0: \ldots: X_n] \in \mathbb{P}^n \, : \, X_n \neq 0}, \\
H &= \set{[X_0: \ldots: X_n] \in \mathbb{P}^n \, : \, X_n = 0}.
\end{align*}
There is a natural isomorphism of the affine $n$-dimensional space into $U$ given by
\begin{equation} \label{pi_definition}
\pi : \mathbb{A}^n \xrightarrow{\sim} U, \quad (x_1, \ldots, x_n) \mapsto [x_1: \ldots: x_n: 1],
\end{equation}
Let now $\Psi$ be an automorphism of $\mathbb{P}^n$. We give the following definitions:
\begin{definition}
For $P \in U$, the \emph{fractional jump index of $\Psi$ at $P$} is
\begin{equation*}
\mathfrak{J}_P = \min \set{k \geq 1 \, : \, \Psi^k (P) \in U}.
\end{equation*}
\end{definition}
\begin{remark}
The fractional jump index $\mathfrak{J}_P$ is always finite, as it is bounded by the order of $\Psi$ in $\mathrm{PGL}_{n+1} (\mathbb{F}_q)$.
\end{remark}
\begin{definition}
The \emph{fractional jump of $\Psi$} is the map
\begin{equation*}
\psi : \mathbb{A}^n \rightarrow \mathbb{A}^n, \quad x \mapsto \pi^{-1} \Psi^{\mathfrak{J}_{\pi(x)}} \pi (x).
\end{equation*}
\end{definition}
Roughly speaking, the purpose of defining this new map is to avoid the points which are mapped outside $U$ via $\Psi$. This is done simply by iterating $\Psi$ until $\Psi(\pi (x))$ ends up again in $U$. In this definition, $\pi$ is simply used to obtain the final map defined over $\mathbb{A}^n$ instead of $U$. A priori, one of the issues here is that a global description of the map might be difficult to compute, as in principle it depends on each of the $x\in \mathbb{A}^n$. It is interesting to see that this does not happen in the case in which $\Psi$ is transitive on $\mathbb{P}^n$:
in fact, we will show in Section \ref{explicit} that there always exists a set of indices $I$, a disjoint covering $\set{U_i}_{i \in I}$ of $\mathbb{A}^n$, and a family $\set{f^{(i)}}_{i \in I}$ of rational maps of degree $1$ on $\mathbb{A}^n$ such that
\begin{enumerate}
\item[i)] $|I| \leq n+1$,
\item[ii)] $f^{(i)}$ is well-defined on $U_i$ for every $i \in I$,
\item[iii)] $\psi (x) = f^{(i)} (x)$ if $x \in U_i$.
\end{enumerate}
That is, $\psi$ can be written as a multivariate linear fractional transformation on each $U_i$. In addition, for any fixed $i\in \{1,\dots n+1\}$, all the denominator of the $f^{(i)}$'s will be equal.
\begin{example} \label{inversive}
Let $n=1$. For $a\in \mathbb{F}_q^*$ and $b\in \mathbb{F}_q$ and
\begin{equation*}
\Psi ([X_0: X_1]) = [b X_0 + a X_1: X_0]
\end{equation*}
we get the case of the inversive congruential generator. In fact, the fractional jump index of $\Psi$ is given by
\begin{equation*}
\mathfrak{J}_P = \begin{cases}
1, & \text{if } P \neq [0, 1], \\
2, & \text{if } P = [0, 1],
\end{cases}
\end{equation*}
and $\Psi^2 ([0, 1]) = [b, 1]$. Therefore, the fractional jump $\psi$ of $\Psi$ is defined on the covering $\set{U_1, U_2}$, where $U_1 = \mathbb{A}^1 \setminus \set{0}$ and $U_2 = \set{0}$, by
\begin{equation*}
\psi (x) = \begin{cases}
\frac{a}{x} + b, & \text{if } x \neq 0, \\
b, & \text{if } x = 0.
\end{cases}
\end{equation*}
The inversive sequence is then given by $\set{\psi^m (0)}_{m \in \mathbb{N}}$, which has full orbit under suitable assumptions on $a$ and $b$ (see for example \cite[Lemma FN]{bib:chou95}).
\end{example}
\begin{remark} \label{remark_transitivity_affine_jump}
Let $\Psi$ be an automorphism of $\mathbb{P}^n$ and let $\psi$ be its fractional jump. It is immediate to see that if $\Psi$ acts transitively on $\mathbb{P}^n$ then $\psi$ acts transitively on $\mathbb{A}^n$.
\end{remark}
For the case of $n = 1$, the next proposition shows that the notion of transitivity for $\Psi$ and its fractional jump $\psi$ are actually equivalent, under the additional assumption that $\Psi$ sends a point of $U$ to a point of $H$ (which is equivalent to ask that the induced map on $\mathbb{A}^1$ is not affine).
\begin{proposition} \label{transitivity_affine_jump_P1}
Let $\Psi$ be an automorphism of $\mathbb{P}^1$ and let $\psi$ be its fractional jump. Assume that $\Psi$ sends a point of $U$ to the point at infinity. Then, $\Psi$ acts transitively on $\mathbb{P}^1$ if and only if $\psi$ acts transitively on $\mathbb{A}^1$.
\end{proposition}
\begin{proof}
As already stated in Remark \ref{remark_transitivity_affine_jump}, if $\Psi$ is transitive on $\mathbb{P}^1$ then $\psi$ is obviously transitive on $\mathbb{A}^1$. Conversely, assume that $\psi$ is transitive on $\mathbb{A}^1$. Consider the decomposition $\mathbb{P}^1 = U \cup H$ of $\mathbb{P}^1$ as in \eqref{decomposition}. Since $n = 1$, we have $H = \set{P_0}$, for $P_0 = [1 : 0]$. Since there exists $P_1 \in U$ such that $\Psi (P_1) = P_0$, we have that $\Psi^2 (P_1) = \Psi (P_0) \in U$, as otherwise the point $P_0$ would have two preimages under $\Psi$, which is not possible as $\Psi$ is an automorphism, and so in particular a bijection. We have to prove that given $P, Q \in \mathbb{P}^1$ there exists an integer $i\geq 0$ such that $Q = \Psi^i (P)$. Assume that $P$ and $Q$ are distinct, as otherwise we can simply set $i = 0$. We distinguish two cases: either $P, Q \in U$, or one of the two, say $P$, is equal to $P_0$ and $Q \in U$. In the first case, the claim follows by transitivity of $\psi$. In the second case, reduce to the previous case by considering $\Psi (P_0), Q \in U$.
\end{proof}
One can actually prove that affine transformations of $\mathbb{A}^n$ are never transitive, unless restrictive conditions on $q$ and $n$ apply. Actually, the result that follows will not be used in the rest of the paper but provides additional motivation for the study of fractional jumps of projective maps and for completeness we include its proof.
\begin{theorem} \label{affine_transitivity_theorem}
There is no affine transitive transformation of $\mathbb{A}^n$ unless $n = 1$ and $q$ is prime, or $q = 2$ and $n = 2$, with explicit examples in both cases.
\end{theorem}
\begin{proof}
For convenience of notation, in this proof we will identify the points of $\mathbb{A}^n$ with columns vectors in $\mathbb{F}_q^n$.
Let us first deal with the pathological cases.
For $n=1$ it is trivial to observe that $x\mapsto x+1$ has full orbit if and only if $q$ is prime.
For $n=2$ and $q=2$, we get by direct check that the map
\begin{equation*}
\varphi \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} + \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \quad \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \in \mathbb{F}_2^2,
\end{equation*}
has full orbit.
Let $\varphi$ be an affine transformation of the $n$-dimensional affine space over $\mathbb{F}_q$. Then, by definition there exist $A \in \mathrm{GL}_n (\mathbb{F}_q)$ and $b \in \mathbb{F}_q^n$ such that
\begin{equation*}
\varphi (x) = A x + b, \quad x \in \mathbb{F}_q^n.
\end{equation*}
Assume by contradiction that $\varphi$ is transitive, so that the order $o (\varphi)$ of $\varphi$ is $q^n$. Denote by $p$ the characteristic of $\mathbb{F}_q$. We firstly prove that the order $o (A)$ of $A$ in $\mathrm{GL}_n (\mathbb{F}_q)$ is $q^n / p$. Then, we will show how this will lead to a contradiction.
Let $j$ be the smallest integer such that
\begin{equation*}
\varphi^j (x) = x + c, \quad \text{for all }x \in \mathbb{F}_q^n,
\end{equation*}
for some $c \in \mathbb{F}_q^n$. As
\begin{equation} \label{explicit_affine}
\varphi^j (x) = A^j x + \sum_{i = 0}^{j-1} A^i b, \quad x \in \mathbb{F}_q^n,
\end{equation}
we get $o(A) = j$. If $c = 0$, then $o(\varphi) = j = o (A) \leq q^n - 1$, so that $\varphi$ cannot be transitive. We then have $c \neq 0$. By \eqref{explicit_affine}, we get $\varphi^{j p} = \mathrm{Id}$, therefore $o(\varphi)\mid jp$. We now prove that $o(\varphi) = jp$. Write $o(\varphi) = j s + r$, with $r < j$. Then, we have
\begin{align*}
\varphi^{j s + r} (x) &= \varphi^r (x) + s c \\
&= A^r x + v, \quad x \in \mathbb{F}_q^n,
\end{align*}
for a suitable $v \in \mathbb{F}_q^n$. Since $\varphi^{j s + r} = \mathrm{Id}$, we get that $A^r x + v = x$ for all $x \in \mathbb{F}_q^n$, and so we must have $r = 0$ and $v=0$. It follows that $\varphi^{j s} (x) = x + s c = x$ for all $x \in \mathbb{F}_q^n$, which gives $p \mid s$, as $c \neq 0$, so that we get $p \leq s$, and then $o(\varphi) = j s \geq jp$. Therefore we conclude that $o (\varphi) = jp$. As $\varphi$ is assumed to be transitive, we have that $j p = q^n$, and so $o (A) = j = q^n / p$.
Essentially, what we have proved up to now is that, if such a transitive affine map $\varphi(x)=Ax+b$ exists, then it must have the property that $o (A) = q^n / p$.
Let $\mu_A (T) \in \mathbb{F}_q [T]$ be the minimal polynomial of $A$. By the fact that $o (A) = q^n / p$ we get
\begin{equation*}
\mu_A (T) \mid T^{q^n / p} - 1 = (T-1)^{q^n / p}.
\end{equation*}
Then, $\mu_A (T) = (T - 1)^d$, for some $d \leq n$, as the degree of the minimal polynomial is less than or equal to the degree of the characteristic polynomial by Cayley-Hamilton. From basic ring theory, one gets that the order of $A$ in $\mathrm{GL}_n (\mathbb{F}_q)$ is equal to the order of the class $\overline{T}$ of $T$ in the quotient ring $(\mathbb{F}_q[T] / (\mu_A (T)))^* = (\mathbb{F}_q[T] / ((T - 1)^d))^*$.
Let us now assume $q^n / p^2 \geq n$. In this case we have
\begin{equation*}
\overline{T}^{q^n / p^2} = (\overline{T}-1)^{q^n / p^2} + 1 = 1,
\end{equation*}
as $q^n / p^2 \geq n \geq d$. Therefore, $o (A)=o(\overline T)\leq q^n/p^2 < q^n / p$ from which the contradiction follows.
Therefore we can restrict to the case $q^n / p^2 < n$. It is easy to see that this inequality forces $q=p$: in fact if $q=p^k$ and $k\geq 2$, then $q^n/p^2=p^{kn-2}\geq p^{2n-2}\geq 4^{n-1}\geq n$. Therefore, the only uncovered cases are in correspondence with the solutions of $p^{n-2}<n$, which consist only of the following: $n=3$ and $p=2$, or $n=1$ and $p$ any prime, or $n=2$ and $p$ any prime.
For $n=3$ and $p=2$ an exhaustive computation shows that there is no transitive affine map.
Also, we already know that in the case $n=1$ and $p$ any prime we have such a transitive map, as this is one of the pathological cases.
For the case $n=2$ we argue as follows. Let
\[\varphi(x)=Ax+b\]
be such a transitive affine map.
Clearly $A\in \mathrm{GL}_2(\mathbb{F}_p)$ must be different from the identity matrix, as otherwise $\varphi$ cannot have full orbit. So the minimal polynomial of $A$ is different from $T-1$. On the other hand, the minimal polynomial of $A$ must divide $(T-1)^d$. Since $n=2$, we have that $d=2$. In $\mathrm{GL}_2(\mathbb{F}_p)$ having minimal polynomial $(T-1)^2$ forces a matrix to be conjugate to a single Jordan block of size $2$ with eigenvalue $1$, hence there exists $C\in \mathrm{GL}_2(\mathbb{F}_p)$ such that
\[CAC^{-1}= \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \]
Let us now consider again the map $\varphi$. Clearly, $\varphi$ is transitive if and only
if the map $\widetilde \varphi=C\varphi C^{-1}$ is. For any $x\in \mathbb{F}_p^2$ we have that $\widetilde \varphi(x)=C(AC^{-1}x+b)$. Therefore
the map $\widetilde \varphi$ can be written as
\[\widetilde \varphi\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}=\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} + \begin{pmatrix} r \\ s \end{pmatrix}.\]
for some $r,s\in \mathbb{F}_p$.
We will now prove that $\widetilde \varphi^p\begin{pmatrix} r \\ s\end{pmatrix}=\begin{pmatrix} r \\ s \end{pmatrix}$ so that $\varphi$ cannot be transitive, as starting form $c:=\begin{pmatrix} r \\ s\end{pmatrix}$ only visits $p$ points.
\begin{align*}
\widetilde \varphi^p\begin{pmatrix} r \\ s\end{pmatrix}&=\sum^{p}_{i=0} \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}^ic \\
&= c+\sum^p_{i=1}\begin{pmatrix} 1 & i \\ 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} r \\ s \end{pmatrix} \\
&= c+\sum^p_{i=1}\begin{pmatrix} r+is \\ s \end{pmatrix}=c+ \begin{pmatrix} s\sum^p_{i=1}i \\ 0 \end{pmatrix}
\end{align*}
But now the sum $\sum^p_{i=1}i$ is different from zero if and only if $p= 2$. Therefore, such a transitive map could exist only for $p=n=2$. Since we already provided such an example of transitive map, the proof of the theorem is now concluded.
\end{proof}
\section{Transitive actions via projective primitivity} \label{transitivity_projective}
In this section we characterise transitive projective automorphisms.
\begin{definition} \label{projectively_primitive_polynomial}
A polynomial $\chi(T) \in \mathbb{F}_q [T]$ of degree $m$ is said to be \emph{projectively primitive} if the two following conditions are satisfied:
\begin{enumerate}
\item[i)] $\chi(T)$ is irreducible over $\mathbb{F}_q$,
\item[ii)] for any root $\alpha$ of $\chi(T)$ in $\mathbb{F}_{q^m} \cong \mathbb{F}_q[T] / (\chi(T))$ the class $\overline{\alpha}$ of $\alpha$ in the quotient group $G = \mathbb{F}_{q^m}^* / \mathbb{F}_q^*$ generates $G$.
\end{enumerate}
\end{definition}
\begin{remark}
Note that if a polynomial $\chi(T)\in \mathbb{F}_q[T]$ of degree $m$ is primitive, i.e. it is irreducible and any of its roots in $\mathbb{F}_{q^m} \cong \mathbb{F}_q[T] / (\chi(T))$ generates the multiplicative group $\mathbb{F}_{q^m}^*$, then it is obviously projectively primitive. The class of projectively primitive polynomials is in general larger than then the class of primitive polynomials: for example take the polynomial $\chi(T)=T^3+T+1\in \mathbb{F}_5[T]$. One can check that this polynomial is irreducible but is not primitive. In fact, let $\alpha$ be a root of $\chi(T)$. Since $\alpha^{62}=1\in \mathbb{F}_5[T] / (\chi(T))\cong \mathbb{F}_{5^3}$, and therefore $o(\alpha)\neq 5^3-1=124$, we have that $\chi(T)$ is not primitive. On the other hand $G=\mathbb{F}_{5^3}^* / \mathbb{F}_5^*$ has prime cardinality equal to $|G|=(5^3-1)/(5-1)=31$ and $\overline{\alpha}\neq 1\in G$. It follows immediately that $\overline \alpha$ has to be a generator of $G$.
\end{remark}
\begin{remark}
Let $M, M' \in \mathrm{GL}_{n+1} (\mathbb{F}_q)$ be such that $[M] = [M']$ in $\mathrm{PGL}_{n+1} (\mathbb{F}_q)$, and let $\chi_M (T), \chi_{M'} (T) \in \mathbb{F}_q [T]$ be their characteristic polynomials. It is immediate to see that $\chi_M (T)$ is projectively primitive if and only if $\chi_{M'} (T)$ is projectively primitive.
\end{remark}
We are now ready to give a full characterisation of transitive projective automorphisms on $\mathbb{P}^n$.
\begin{theorem} \label{transitivity_characterisation}
Let $\Psi$ be an automorphism of $\mathbb{P}^n$. Write $\Psi = [M] \in \mathrm{PGL}_{n+1} (\mathbb{F}_q)$ for some $M \in \mathrm{GL}_{n+1} (\mathbb{F}_q)$. Then, $\Psi$ acts transitively on $\mathbb{P}^n$ if and only if the characteristic polynomial $\chi_M (T) \in \mathbb{F}_q [T]$ of $M$ is projectively primitive.
\end{theorem}
\begin{proof}
For simplicity of notation, set $N = |\mathbb{P}^n| = (q^{n+1}-1) / (q-1)$. Assume that $\chi_M (T)$ is projectively primitive. Now we prove that for any $P \in \mathbb{P}^n$ we have that $\Psi^k (P) \neq P$ for $k \in \set{1, \ldots, N -1}$. Suppose by contradiction that there exists $P_0 \in \mathbb{P}^n$ such that $\Psi^k (P_0) = P_0$ for some $k \in \set{1, \ldots, N-1}$. Let $v_0 \in \mathbb{F}_q^{n+1} \setminus \set{0}$ be a representative of $P_0$. Then, there exists $\lambda \in \mathbb{F}_q^*$ such that
\begin{equation*}
M^k v_0 = \lambda v_0.
\end{equation*}
This means that $v_0$ is an eigenvector for the eigenvalue $\lambda$ of $M^k$, which implies that $\lambda = \alpha^k$ for some root $\alpha$ of $\chi_M (T)$ in $\mathbb{F}_{q^{n+1}}$. But now, the class $\overline{\alpha}^k$ of $\alpha^k$ in $G = \mathbb{F}_{q^{n+1}}^* / \mathbb{F}_q^*$ is $\overline{\lambda} = \overline{1}$, contradicting the hypothesis that $\overline{\alpha}$ generates $G$.
Conversely, assume that $\Psi$ is transitive, so that for any $P \in \mathbb{P}^n$ we have that $\Psi^k (P) \neq P$ for $k \in \set{1, \ldots, N -1}$. Let $\alpha$ be a root of $\chi_M(x)$ in its splitting field and $h$ be a positive integer such that $\mathbb{F}_{q^h}\cong\mathbb{F}_q(\alpha)$. Clearly $1\leq h\leq n+1$. We also have that $\alpha \neq 0$ as $\det M \neq 0$, and $\alpha \notin \mathbb{F}_q^*$ as otherwise $M v_0 = \alpha v_0$ for some eigenvector $v_0 \in \mathbb{F}_q^{n+1} \setminus \set{0}$ for the eigenvalue $\alpha$, so that $\Psi (P_0) = P_0$ for $P_0$ the class of $v_0$ in $\mathbb{P}^n$, in contradiction with the fact that $\Psi$ is transitive. Let $d$ be the order of the class $\overline{\alpha}$ of $\alpha$ in $\mathbb{F}_{q^h}^* / \mathbb{F}_q^*$. Then, there exists $\lambda \in \mathbb{F}_q^*$ such that $\alpha^d = \lambda$. Now, $\alpha^d$ is an eigenvalue of $M^d$, and so $M^d v_1 = \lambda v_1$ for some eigenvector $v_1\in \mathbb{F}_q^{n+1} \setminus \set{0}$ for the eigenvalue $\alpha^d$. Thus, $\Psi^d (P_1) = P_1$ for $P_1$ the class of $v_1$ in $\mathbb{P}^n$, and so $d=N$ by the transitivity of $\Psi$. Therefore we have $(q^{n+1}-1)/(q-1)=N=d\leq (q^{h}-1)/(q-1)$. Now, since $h\leq n+1$, this forces $h = n+1$, so that $\chi_M$ is irreducible, which together with $d=N$ gives projective primitivity for $\chi_M$, as we wanted.
\end{proof}
\begin{remark}\label{remarkICGfullorbit}
When $n = 1$, our approach gives immediately the criterion to get maximal period for inversive congruential generators, see for example \cite[Lemma FN]{bib:chou95}. To see this, set $\Psi$ and $\psi$ as in Example \ref{inversive}, so that $\set{\psi^m (0)}_{m \in \mathbb{N}}$ is an inversive sequence. By Proposition \ref{transitivity_affine_jump_P1}, transitivity of $\Psi$ and $\psi$ are equivalent. If $\chi (T) = T^2 - a T - b$ is irreducible, then $\psi$ acts transitively on $\mathbb{A}^1$ if and only if the class $\overline{\alpha}$ of a root $\alpha$ of $\chi(T)$ in $G = \mathbb{F}_{q^{2}}^* / \mathbb{F}_q^*$ generates $G$, which is itself equivalent to the fact that $\alpha^{q-1}$ has order $q+1$ in $\mathbb{F}_{q^2}^*$ (which is in fact the condition given in \cite[Lemma FN]{bib:chou95}).
\end{remark}
\section{Subspace Uniformity} \label{uniformity}
In this section we show that sequences associated to iterations of transitive projective maps behave ``uniformly'' with respect to subspaces, i.e. not too many consecutive points can lie in the same projective subspace of $\mathbb{P}^n$.
More precisely, we have the following:
\begin{proposition} \label{theorem_uniformity}
Let $\Psi$ be a transitive automorphism of $\mathbb{P}^n$. For any $P \in \mathbb{P}^n$ and any $d \in \set{1, \ldots, n-1}$ there is no $W \in \mathbb{G}\mathrm{r} (d, n)$ such that $\Psi^i (P) \in W$ for all $i \in \set{0, \ldots, d+1}$.
\end{proposition}
\begin{proof}
Suppose by contradiction that there exists a projective subspace $W$ of dimension $d$ such that there exists $P\in \mathbb{P}^n$ such that $\Psi^i (P) \in W$ for all $i\in \set{0, \ldots, d+1}$. Let $W'$ be the subspace of $\mathbb{F}_q^{n+1}$ whose projectification is $W$, and let $v \in \mathbb{F}_q^{n+1} \setminus \set{0}$ be a representative for $P$. Let also $M\in \mathrm{GL}_{n+1}(\mathbb{F}_q)$ be a representative for $\Psi$. Consider now the smallest integer $h$ such that $M^h v$ is linearly dependent on $\{M^i v \, : \, i\in \{0,\dots h-1\}\}$ over $\mathbb{F}_q$. Since $M^iv$ is contained in $W'$ for any $i\in \{0,\dots d+1\}$, and $W'$ has dimension $d+1$, then we have that $h$ is at most $d+1$. Therefore, $M^hv$ can be rewritten in terms of lower powers of $M$, which in turn forces the span of $\{ M^iv \, : \, i\in \{0,\dots, h-1\}\}$ over $\mathbb{F}_q$ to be an invariant space for $M$. It follows that the characteristic polynomial of $M$ has a non-trivial factor of degree less than or equal to $d$. Since $d\leq n$, we have the claim, as by Theorem \ref{transitivity_characterisation} the characteristic polynomial of $M$ has to be irreducible.
\end{proof}
\begin{remark}
This result is optimal with respect to $d$, as for any set $S$ of $d+2$ points of $\mathbb{P}^n$ there always exists a $W\in \mathbb{G}\mathrm{r} (d+1, n)$ containing $S$.
\end{remark}
Fix the canonical decomposition $\mathbb{P}^n = U \cup H$ as in \eqref{decomposition}.
\begin{corollary} \label{bound_affine_jump_index}
Let $\Psi$ be a transitive automorphism of $\mathbb{P}^n$. For any $P \in U$ the fractional jump index $\mathfrak{J}_P$ of $\Psi$ at $P$ is bounded by $n+1$.
\end{corollary}
\begin{proof}
Assume by contradiction $\mathfrak{J}_P \geq n+2$. Then, setting $P' = \Psi (P)$ we get $\Psi^i (P') \in H$ for all $i \in \set{0, \ldots, n}$. But we have $H \in \mathbb{G}\mathrm{r} (n-1, n)$, and so this violates Proposition \ref{theorem_uniformity}.
\end{proof}
\section{Explicit description of fractional jumps} \label{explicit}
Let $\Psi$ be an automorphism of $\mathbb{P}^n$. In this section we will give an explicit description of the fractional jump $\psi$ of $\Psi$.
First of all, fix homogeneous coordinates $X_0, \ldots, X_n$ on $\mathbb{P}^n$, fix the canonical decomposition $\mathbb{P}^n = U \cup H$ as in \eqref{decomposition} and the map $\pi$ as in \eqref{pi_definition}, and write $\Psi\in \mathrm{PGL}_{n+1} (\mathbb{F}_q)$ as
\begin{equation*}
\Psi = [F_0: \ldots: F_n],
\end{equation*}
where each $F_j$ is an homogeneous polynomial of degree $1$ in $\mathbb{F}_q [X_0, \ldots, X_n]$.
Fix now affine coordinates $x_1, \ldots, x_n$ on $\mathbb{A}^n$, and for each $j \in \set{1, \ldots, n}$ set
\begin{equation} \label{rational_functions}
f_j (x_1, \ldots, x_n) = \frac{F_{j-1} (x_1, \ldots, x_n, 1)}{F_n (x_1, \ldots, x_n, 1)}.
\end{equation}
Let $K = \mathbb{F}_q (x_1, \ldots, x_n)$ be the field of rational functions on $\mathbb{A}^n$. Then, \eqref{rational_functions} defines elements $f_j \in K$ for $j \in \set{1, \ldots, n}$, and $f_\Psi = (f_1, \ldots, f_n) \in K^n$.
In turn this process defines a map
\begin{equation} \label{map_pgl}
\imath : \mathrm{PGL}_{n+1} (\mathbb{F}_q) \rightarrow K^n, \quad \Psi \mapsto f_\Psi.
\end{equation}
It is easy to see that this map is well defined and for any element $f=(f_1,\dots,f_n)$ in the image of $\imath$ all the denominators of the $f_j$'s are equal. It also holds that $\imath(\Psi\circ \Phi)=\imath(\Psi)\circ \imath(\Phi)$, where the composition in $K^n$ is defined in the obvious way, i.e. just plugging in the components of $\imath (\Phi)$ in the variables of $\imath(\Psi)$.
Let us go back to $f_\Psi = (f_1, \ldots, f_n) \in K^n$ for a fixed automorphism $\Psi$. For any $i \geq 1$, let us define $f^{(i)} = \imath(\Psi^{i})$. For each $f^{(i)}$ and for each $j\in \{1,\dots,n\}$, write the $j$-th component of $f^{(i)}$ as
\begin{equation*}
f^{(i)}_j = \frac{a^{(i)}_j}{b^{(i)}_j}, \quad \text{for } a^{(i)}_j, b^{(i)}_j \in \mathbb{F}_q [x_1, \ldots, x_n].
\end{equation*}
As we already observed, for fixed $i\geq 1$ all the $b^{(i)}_j$'s are equal, so we can set $b^{(i)}=b^{(i)}_1$. Define now
\begin{align*}
V_0 &= \mathbb{A}^n, \\
V_i &= \bigcap_{k = 1}^i V(b^{(k)}), \quad \text{for } i \geq 1.
\end{align*}
These sets will be the main ingredient in the definition of the covering mentioned in Section \ref{affine_jumps}. The following result characterises the $V_i$'s in terms of the position of a bunch of iterates of $\Psi$.
\begin{lemma} \label{characterisation_vanishing_loci}
Let $x \in \mathbb{A}^n$, and $P = \pi (x) \in U$. Then, $x \in V_i$ if and only if $\Psi^k (P) \in H$ for $k \in \set{1, \ldots, i}$.
\end{lemma}
\begin{proof}
By definition, $x \in V_i$ if and only if $x \in V(b^{(k)})$ for every $k \in \set{1, \ldots, i}$, which means $b^{(k)} (x) = 0$ for every $k \in \set{1, \ldots, i}$. Now, $b^{(k)} (x) = 0$ if and only if the last component of $\Psi^k (P)$ is zero, which is equivalent to the condition $\Psi^k (P) \in H$.
\end{proof}
\begin{definition}
Define the \emph{absolute fractional jump index $\mathfrak{J}$ of $\Psi$} to be the quantity
\begin{equation*}
\mathfrak{J} = \max \set{\mathfrak{J}_P \, : \, P \in U}.
\end{equation*}
\end{definition}
When $\Psi$ is transitive, Corollary \ref{bound_affine_jump_index} ensures that $\mathfrak{J} \leq n+1$. We will now show that the absolute jump index equals the number of non empty $V_i$'s.
\begin{proposition}
We have that
\begin{equation*} \label{absolute_jump_index}
\min \set{i \in \mathbb{N} \, : \, V_i = \emptyset} = \mathfrak{J}.
\end{equation*}
\end{proposition}
\begin{proof}
Set $i_0 = \min \set{i \in \mathbb{N} \, : \, V_i = \emptyset}$. In order to show that $i_0 \leq \mathfrak{J}$, it is enough to prove that $V_{\mathfrak{J}} = \emptyset$. Assume that there exists $x \in V_{\mathfrak{J}}$. Then, if $P = \pi (x)$, we have by Lemma \ref{characterisation_vanishing_loci} that $\Psi^j (P) \in H$ for $j \in \set{1, \ldots, \mathfrak{J}}$, and so the jump index $\mathfrak{J}_P$ must be strictly greater than $\mathfrak{J} $, a contradiction.
Conversely, in order to show that $\mathfrak J \leq i_0$, it is enough to prove that $V_{\mathfrak J -1}\neq \emptyset $. To do so, take $P_0 \in U$ for which $\mathfrak{J}_{P_0} = \mathfrak{J}$. Then $\Psi^k(P_0)\in H$ for any $k\in \{1,\dots \mathfrak{J}-1\}$. Let $x_0 = \pi^{-1} (P_0)$. Then, by Lemma \ref{characterisation_vanishing_loci} we have $x_0 \in V_{\mathfrak{J}-1}$.
\end{proof}
Define now
\begin{equation*}
U_i = V_{i-1} \setminus V_i, \quad \text{for } i \in \set{1, \ldots, \mathfrak{J}}.
\end{equation*}
Thus, for $I = \set{1, \ldots, \mathfrak{J}}$, the family $\set{U_i}_{i \in I}$ is a disjoint covering of $\mathbb{A}^n$ and each $f^{(i)}$ is a rational map of degree $1$ on $\mathbb{A}^n$. Also, we observe that by construction $f^{(i)}$ is well defined on $U_i$, so that the fractional jump is defined as
\begin{equation*}
\psi (x) = f^{(i)} (x), \quad \text{if } x \in U_i.
\end{equation*}
To clarify this contruction, we now give an explicit description of a fractional jump over $\mathbb{A}^2$.
\begin{example}
Let $q = 101$ and $n = 2$. Consider the automorphism of $\mathbb{P}^2$ defined by
\begin{align*}
\Psi([X_0: X_1: X_2]) &= [F_0 : F_1 : F_2] \\
&= [X_0 + 2 X_2: 3 X_1+4 X_2: 4X_0 + 2 X_1 + 3 X_2].
\end{align*}
Notice that
\begin{equation*}
M = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 3 & 4 \\ 4 & 2 & 3 \end{pmatrix}
\end{equation*}
is a representative of $\Psi$ in $\mathrm{GL}_3 (\mathbb{F}_{101})$. The characteristic polynomial $\chi_M (T) \in \mathbb{F}_{101} [T]$ of $M$ is given by
\begin{equation*}
\chi_M (T) = T^3 - 7 T^2 - T + 23,
\end{equation*}
which is irreducible over $\mathbb{F}_{101}$. Now, as
\begin{align*}
\frac{q^{n+1}-1}{q-1} &= \frac{101^3-1}{101-1} \\
&= 10303
\end{align*}
is prime, any irreducible polynomial of degree $3$ in $\mathbb{F}_{101}[T]$ is projectively primitive. By Theorem \ref{transitivity_characterisation} we have that $\Psi$ is transitive on $\mathbb{P}^n$. Since $n=2$ and $\Psi$ is transitive, by Proposition \ref{absolute_jump_index} and the definition of the $U_i$'s we know that the
fractional jump of $\Psi$ will be defined using at most $U_1,U_2,U_3$.
As in \eqref{rational_functions}, we consider rational functions
\begin{align*}
f_1(x_1, x_2) &= \frac{x_1+2}{4x_1+2 x_2 +3}, \\
f_2(x_1, x_2) &= \frac{3x_2+4}{4x_1+2 x_2 +3}.
\end{align*}
in $\mathbb{F}_{101} (x_1, x_2)$, and set $f = (f_1, f_2) \in \mathbb{F}_{101}(x_1, x_2)^2$. Given the definition of $f$, we have
\begin{align*}
V_1 &= V(4x_1+2 x_2 +3), \\
U_1 &= \mathbb{A}^2 \setminus V_1.
\end{align*}
Let now $f^{(1)} = f$ and $f^{(2)} = f \circ f = (f_1^{(2)}, f_2^{(2)}) \in \mathbb{F}_{101}(x_1, x_2)^2$, where
\begin{align*}
f_1^{(2)}(x_1, x_2) &= f_1 (f_1 (x_1, x_2), f_2 (x_1, x_2)) \\
&= \frac{9x_1 + 4x_2 + 8}{16 x_1 + 12 x_2 +25}, \\
f_2^{(2)}(x_1, x_2) &= f_2 (f_1 (x_1, x_2), f_2 (x_1, x_2)) \\
&= \frac{16 x_1 + 17 x_2 + 24}{16 x_1 + 12 x_2 +25}.
\end{align*}
Define
\begin{align*}
V_2 &= V_1 \cap V(16 x_1 + 12 x_2 +25) = \set{(64, 22)}, \\
U_2 &= V_1 \setminus V_2.
\end{align*}
Finally, let $f^{(3)} = f \circ f \circ f = (f_1^{(3)}, f_2^{(3)}) \in \mathbb{F}_{101}(x_1, x_2)^2$, where
\begin{align*}
f_1^{(3)}(x_1, x_2) &= f_1 (f_1^{(2)} (x_1, x_2), f_2^{(2)} (x_1, x_2)) \\
&= \frac{41 x_1 + 28 x_2 - 43}{15 x_1 - 15 x_2 - 47}, \\
f_2^{(3)}(x_1, x_2) &= f_2 (f_1^{(2)} (x_1, x_2), f_2^{(2)} (x_1, x_2)) \\
&= \frac{11 x_1 - 2 x_2 - 30}{15 x_1 - 15 x_2 - 47},
\end{align*}
and $U_3 =V_2= \set{(64, 22)}$, since $V_3=V_2 \cap V(15x_1-15x_2-47)=\emptyset$.
By construction, $\mathbb{A}^2 = U_1 \cup U_2 \cup U_3$, and therefore we are ready to describe the fractional jump $\psi$ of $\Psi$ as
\begin{equation*}
\psi (x_1, x_2) = \begin{cases}
f^{(1)} (x_1, x_2), & \text{if }(x_1, x_2) \in U_1, \\
f^{(2)} (x_1, x_2), & \text{if }(x_1, x_2) \in U_2, \\
f^{(3)} (x_1, x_2), & \text{if }(x_1, x_2) \in U_3.
\end{cases}
\end{equation*}
Notice that $f^{(3)} (64, 22) = (63, 78)$, and so $\psi (x_1, x_2) = (63, 78)$ if $(x_1, x_2) \in U_3 = \set{(64, 22)}$.
\end{example}
\section{The discrepancy of fractional jump sequences}\label{discrepancy}
In the context of pseudorandom number generation, it is of interest to say something about the distribution of a sequence. A statistic that is of particular interest is the discrepancy of a sequence, of which we recall the definition below. The goal of this section is to show that for sequences generated by fractional jumps one can prove the same discrepancy bounds as for the sequences generated by the ICG.
For simplicity, we let $q=p$ be prime. We assume the set $ \mathbb{F}_p\cong \mathbb{Z}/p\mathbb{Z}$ to be represented by $\{0,1,\dots, p-1\}\subseteq \mathbb{Z}$ as in \cite{shparlinski10}. For $x\in \mathbb{F}_p$ we then write $\frac{x}{p}$ for the corresponding element in $\frac{1}{p}\mathbb{Z}\subseteq \mathbb{R}$.
For a sequence
\begin{equation*}
\Gamma = \{(\gamma_{m,0},\dots,\gamma_{m,s-1})\}_{m=0}^{N-1}
\end{equation*}
of $N$ points in $[0,1)^s$, for $s\in \mathbb{N}$, the \emph{discrepancy} of $\Gamma$ is defined by
\begin{equation*}
D_\Gamma = \sup_{B\subseteq [0,1)^s} \bigg|\frac{T_\Gamma(B)}{N}-|B|\bigg|,
\end{equation*}
where the supremum is taken over boxes $B$ of the form
\begin{equation*}
B = [\alpha_1,\beta_1)\times \dots \times [\alpha_s,\beta_s) \subseteq [0,1)^s,
\end{equation*}
and $T_\Gamma(B)$ denotes the number of points of $\Gamma$ which lie inside $B$.
For a sequence $\{u_m\}_{m \in \mathbb{N}}$ of points in $\mathbb{F}_p$ the main interest lies in bounding the discrepancy of the sequence
\[ \Big(\frac{u_{m}}{p},\frac{u_{m+1}}{p},\dots,\frac{u_{m+s-1}}{p}\Big)_{m=0}^{N-1}\]
for $s\ge 1$. In the case of a sequence generated by an ICG such a bound was given in \cite{shparlinski10}. The goal of this section is to extend the results in \cite{shparlinski10} to give discrepancy bounds for full orbit sequences generated by fractional jumps also in the case where the dimension $n$ satisfies $n>1$.
Given a fractional jump $\psi:\mathbb{F}_p^n\to \mathbb{F}_p^n$ and an initial value $x\in \mathbb{F}_p^n$ we define the sequence $\{ \mathbf{u}_m (x) \}_{m \in \mathbb{N}}$ of points in $\mathbb{F}_p^n$ by setting $\mathbf{u}_0(x) = x$ and
\begin{equation*}
\mathbf{u}_m(x) = \psi^{m}(x), \quad \text{for } m \geq 1.
\end{equation*}
We also define the \emph{snake sequence} $\{ v_m (x) \}_{m \ge 1}$ of points in $\mathbb{F}_p$ by setting
\[ (v_{kn+1}(x),v_{kn+2}(x),\dots,v_{(k+1)n}(x)) = \mathbf{u}_{k}(x), \quad \text{for } k \in \mathbb{N}.\]
Let $D_{s,\psi}(N;x)$ denote the discrepancy of the sequence
\[ \Big(\frac{v_{m+1}}{p},\frac{v_{m+2}}{p},\dots,\frac{v_{m+s}}{p}\Big)_{m=0}^{N-1}\]
and let $D^n_{s,\psi}(N;x)$ denote the discrepancy of the sequence
\[ \Big(\frac{\mathbf{u}_m}{p},\frac{\mathbf{u}_{m+1}}{p},\dots,\frac{\mathbf{u}_{m+s-1}}{p}\Big)_{m=0}^{N-1}.\]
Note that in the first case the individual points of the sequence lie in $\mathbb{F}_p^s$, while in the second case the points of the sequence lie in $\mathbb{F}_p^{ns}$.
Our main result for the discrepancy $D_{s,\psi}(N;x)$ is a direct generalization of \cite[Theorem 4]{shparlinski10}, which deals with the discrepancy of a sequence generated by an ICG. We also provide the analogous bounds for the $n$-dimensional discrepancy $D_{s,\psi}^n(N;x)$.
\begin{theorem}\label{thm:discrepancy}
Let $\Psi$ be a transitive automorphism of $\mathbb{P}^n$ and let $\psi$ be its fractional jump. Then for any integer $s\ge 1$ and any real $\Delta >0$, for all but $O(\Delta p^n)$ initial values $x\in \mathbb{F}_p^n$ it holds that
\[ D_{s,\psi}(N;x) \ll_{s,n} (\Delta^{-2/3}N^{-1/3}+ p^{-1/4}\Delta^{-1})(\log N)^s \log p\]
and
\[ D_{s,\psi}^n(N;x) \ll_{s,n} (\Delta^{-2/3}N^{-1/3}+ p^{-1/4}\Delta^{-1})(\log N)^{sn} \log p\]
for all $N$ with $1\le N\le p^n$.
\end{theorem}
The proof of Theorem \ref{thm:discrepancy} follows the same lines as the proof of \cite[Theorem 4]{shparlinski10}, but with Lemma \ref{lem:2nd-moment} below extending \cite[Lemma 1]{shparlinski10} to $n>1$.
In the proofs we will make use of the Koksma--Sz\"{u}sz inequality as well as the Bombieri--Weil bound.
\begin{theorem}[{\cite[Theorem 1.21]{drmota}}]\label{thm:koksma}
For any integer $H\ge 1$, the discrepancy $D_\Gamma$ of the sequence $\Gamma = (\gamma_{m,0},\dots,\gamma_{m,s-1})_{m=0}^{N-1}$ satisfies
\[ D_\Gamma \ll \frac{1}{H} + \frac{1}{N}\sum_{0<\| \mathbf{h} \|_\infty \le H} \frac{1}{\rho(\mathbf{h})} \bigg| \sum_{m=0}^{N-1} \exp\Big( 2\pi i \sum_{j=0}^{s-1} h_j \gamma_{m,j} \Big)\bigg|, \]
where $\rho(\mathbf{h}) = \prod_{j=0}^{s-1} \max\{|h_j|,1\}$ for $\mathbf{h}=(h_0,\dots, h_{s-1})\in \mathbb{Z}^s$.
\end{theorem}
\begin{theorem}[{\cite[Theorem 2]{moreno}}]\label{thm:weil}
Let $f/g$ be a rational function over $\mathbb{F}_p$ with $\deg(f)>\deg(g)$. Suppose that $f/g$ is not of the form $h^p-h$, where $h$ is a rational function over $\overline{\mathbb{F}}_p$.
Then
\[ \bigg|\sum_{\substack{x\in \mathbb{F}_p:\\g(x)\ne 0}} e_p\left( \frac{f(x)}{g(x)} \right) \bigg| \le (\deg(f) + v-1)p^{1/2},\]
where $v$ is the number of distinct roots of $g$ in $\overline{\mathbb{F}}_p$.
\end{theorem}
We will also need to use the explicit description of $\psi$ given in Section \ref{explicit} to describe powers of $\psi$, which is done in the next lemma.
\begin{lemma}\label{lem:explicit}
Let $\Psi$ be a transitive automorphism of $\mathbb{P}^n$ and let $\psi$ be its
fractional jump. Then there are polynomials $a^{(i)}_j, b^{(i)} \in \mathbb{F}_p [x_1, \ldots, x_n]$ of degree less or equal than $1$, for $i\in\{1,\dots,p-1\}$ and $j\in\{1,\dots,n\}$, with $b^{(i)}$ not
identically a constant, and such that
\[ \psi^i_j(x) = \frac{a^{(i)}_j(x)}{b^{(i)}(x)}, \quad \text{for } x\not\in \bigcup_{k=1}^i V(b^{(k)}),\]
where $\psi^i_j (x)$ denotes the $j$-th component of $\psi^i (x)$.
\end{lemma}
\begin{proof}
The functions $a_j^{(i)}, b^{(i)}$ are defined as in Section \ref{explicit}. Indeed, recall that there is a set $U_1$ and there is a rational map
\[ f^{(1)}=\Big(\frac{a^{(1)}_1}{b^{(1)}},\dots,\frac{a^{(1)}_n}{b^{(1)}}\Big) \]
of degree $1$ such that
\[ \psi(x) = f^{(1)}(x),\quad \text{for } x\in U_1 = \mathbb{F}_p^n\setminus V(b^{(1)}).\]
For $i\in \{1,\dots,p-1\}$ and $j\in\{1,\dots,n\}$, define the maps $a_j^{(i)},b^{(i)}$ by iterating the map $f^{(1)}$, that is
\[ f^{(i)} = (f^{(1)})^i = \left(\frac{a^{(i)}_1}{b^{(i)}},\dots,\frac{a^{(i)}_n}{b^{(i)}}\right),\quad i\in\{1,\dots,p-1\}.\]
Let us notice that in Section \ref{explicit} the function $f^{(i)}$ was used to describe the map $\psi$ on the set $U_i$. In this section we are instead using $f^{(i)}$ to describe the $i$'th iterate of $\psi$ on the set $U_1\cap \psi^{-1}(U_1) \cap \cdots \cap \psi^{-i+1}(U_1)$. In particular, Section \ref{explicit} made use of $f^{(i)}$ for $i\in\{1,\dots,n\}$, but here we instead use $f^{(i)}$ on the range $i\in \{1,\dots,p-1\}$.
To see that $b^{(i)}$ isn't identically a constant for $i \in \set{1, \ldots, p-1}$ we need to show that $f^{(i)} = \imath(\Psi^i)$, where $\imath$ is the map in \eqref{map_pgl}, is not affine. Let $M\in \mathrm{GL}_{n+1}(\mathbb{F}_p)$ be such that $\Psi = [M] \in \mathrm{PGL}_{n+1}(\mathbb{F}_p)$.
Suppose by contradiction that $\imath(\Psi^i)$ is affine for some $i \in \set{1, \ldots, p-1}$. Since $M$ has irreducible characteristic polynomial by Theorem \ref{transitivity_characterisation}, we have that $\mathbb{F}_p[M]$ is a field and in turn that $\mathbb{F}_p[M^i]$ is a proper subfield, therefore the minimal polynomial of $M^i$ is irreducible. Now, since $\imath(\Psi^i)$ is assumed to be affine, we have that $\Psi^i(H)=H$, which in turn forces $M^i$ to fix a proper subspace of $\mathbb{F}_p^{n+1}$. This directly implies that the (irreducible) minimal polynomial of $M^i$ cannot be equal to the characteristic polynomial of $M^i$, and therefore it must have degree $d<n+1$.
We know, again by Theorem \ref{transitivity_characterisation}, that $[M]$ is a generator for the quotient group $\mathbb{F}_p[M]^*/ \mathbb{F}_p^*$ as $\Psi$ is transitive, but $[1]=[M^i]^{(p^d-1)/(p-1)}= [M]^{i (p^d-1)/(p-1)}$, so $(p^{n+1}-1)/(p-1)\mid i (p^d-1)/(p-1)$ which forces $i\geq
(p^{n+1}-1)/(p^d-1)\geq p$, a contradiction.
\end{proof}
We are now ready to prove the technical heart of the argument.
\begin{lemma}\label{lem:technical}
Let $\Psi$ be a transitive automorphism of $\mathbb{P}^n$ and let $\psi$ be its fractional jump. Then for any integers $j_0,
s\ge 1$, $d\le (p-1)n-s$ and $\mathbf{h} \in \mathbb{F}_p^s\setminus \{0\}$ it holds that
\[ \Big|\sum_{x \in \mathbb{F}_p^n} e_p\Big( \sum_{j=0}^{s-1}h_j(v_{j_0+d+j}(x)-v_{j_0+j}(x))\Big)\Big| \le 3\Big(\frac{s+d}{n}+1\Big)p^{n-1}+4\Big(\frac{s}{n}+1\Big)p^{n-1/2}.
\]
\end{lemma}
\begin{proof}
Observe first that the result is trivial for $s \ge p^{1/2}n$, so assume that $s\le p^{1/2}n$.
Let $r = \min\{j: h_j\ne 0\}$, $s'=s-r$ and $h_j'=h_{j+r}$ for $j\in\{0,\dots,s'-1\}$. Let $m= \floor{(j_0+r)/n}$. Since $\psi$ is a bijection, we can make the substitution $x'=\psi^m (x)$ and sum over $x'\in\mathbb{F}_p^n$ in place of summing over $x\in \mathbb{F}_p^n$. Then, we get $v_{i}(x')=v_{mn+i}(x) $, and so
\begin{align*}
\sum_{x\in\mathbb{F}_p^n} e_p\Big(\sum_{j=0}^{s-1} h_j(v_{j_0+d+j}(x)-v_{j_0+j}(x))\Big) &= \sum_{x'\in\mathbb{F}_p^n} e_p\Big(\sum_{j=0}^{s'-1} h_{j}'(v_{j_1+d+j}(x')-v_{j_1+j}(x'))\Big)\\
&=\sum_{x\in\mathbb{F}_p^n} e_p\Big(\sum_{j=0}^{s'-1} h_{j}'(v_{j_1+d+j}(x)-v_{j_1+j}(x))\Big),
\end{align*}
for some $j_1$ with $1\le j_1 \le n$, where in the last equality we have simply relabeled the summation index $x'$ to $x$, which we do for simplicity of notation. Notice that in this way we have that $h_0' \ne 0$, which was the entire point of shifting the sum.
As $d\le (p-1)n-s$ we have $j_1+j \le j_1+d+j \le
pn-1 < pn$. This means that $v_{j_1+d+j} = \psi_k^i(x)$ for some $i< p$ and some
$k\in\{1,\dots, n\}$. An analogous statement also holds for $v_{j_1+j}$, i.e. $v_{j_1+j} = \psi_{k'}^{i'}(x)$ for some $i'< p$ and some $k'\in\{1,\dots, n\}$.
We can therefore apply Lemma \ref{lem:explicit} to write
\[v_{in+j}(x) = \psi_j^i(x) = \frac{a_j^{(i)}(x)}{b^{(i)}(x)},\quad x\not\in \bigcup_{k=1}^i V(b^{(k)}),\]
for $i\in\{1,\dots,\floor{\frac{n+d+s-1}{n}}\}$, $j\in \{1,\dots,n\}$, and for $i=0$ we clearly have
\[ v_{j}(x) = x_j,\]
for $j \in \{1,\dots,n\}$.
Since we want to estimate the sum
\[\Big|\sum_{x \in \mathbb{F}_p^n} e_p\Big( \sum_{j=0}^{s'-1}h_j'(v_{j_0+d+j}(x)-v_{j_0+j}(x))\Big)\Big|,\]
we consider for fixed $\tilde{x} = (x_1,\dots,x_{j_1-1},x_{j_1+1},\dots,x_n)\in \mathbb{F}_q^{n-1}$ the inner sum
\[G(x_{j_1};\tilde{x})= \sum_{j=0}^{s'-1} h_j'(v_{j_1+d+j}(x)-v_{j_1+j}(x))\]
as a function of the variable $x_{j_1}$. Since we want to apply Theorem \ref{thm:weil} to $G$ for fixed $\tilde x$ (and considered as a univariate polynomial in $x_{j_1}$) we first need to give a nice description of $G$ outside a certain set. We can do that outside of the set
\[E=\bigcup_{i=1}^{\floor{(n+d+s-1)/n}}V(b^{(i)}).\]
In fact, one may write
\[G(x_{j_1};\tilde{x})= \frac{a(x_{j_1};\tilde{x})}{b(x_{j_1};\tilde{x})} = \frac{\tilde{a}(x_{j_1};\tilde{x})}{b(x_{j_1};\tilde{x})} - h_0' x_{j_1} + c(\tilde{x}),\]
where $a,\tilde{a},b$ are polynomials and $c(\tilde{x})$ is constant with respect to $x_{j_1}$.
In order to apply Theorem \ref{thm:weil} to the sum over $x_{j_1}$, we need to check that the conditions of the theorem are verified apart from a small set $F$ of $\tilde x$'s, whose size we can estimate. To begin with we check that $\deg(a) = \deg(b)+1$. This follows immediately if either $\deg(\tilde{a})\le \deg(b)$, or if $\deg(\tilde{a})=\deg(b)+1$ and the leading coefficient in $\tilde{a}/b$ doesn't cancel the term $-h_0'x_{j_1}$.
By considering the possible powers of $\psi$ that can appear in the definition of $G$, we see that
\[b(x_{j_1};\tilde{x}) = \prod_{i\in I} b^{(i)}(x),\]
with the product taken over a set $I$ of $i$ satisfying
\begin{equation}
i \in \left[ \frac{j_1}{n}-1,\frac{j_1+s'-1}{n} \right)\cup\left[ \frac{j_1+d}{n}-1,\frac{j_1+d+s'-1}{n} \right) \subseteq [0, p)
\label{eq:ugly}
\end{equation}
and such that the coefficient of $x_{j_1}$ is nonzero in $b^{(i)}$.
Since $\tilde{a}/b$ was defined by a linear sum of rational functions of degree $1$, it follows that $\deg(\tilde{a}) \le \deg(b) + 1$. If $\deg(\tilde{a}) = \deg(b)+1$, the coefficient of the highest order term in $a$ is of the form a constant times $\prod_{i\in J} b^{(i)}(x)$ for some set $J$ of $i$ satisfying \eqref{eq:ugly} and such that $b^{(i)}$ doesn't depend on $x_{j_1}$. Think of this coefficient as a polynomial in $\tilde{x}$. In particular, there are no more than $2 \Big(\frac{s'}{n}+1 \Big)$ values of $\tilde{x}$ satisfying \eqref{eq:ugly}, and therefore the coefficient is equal to $h_0'$ for at most $2 \Big( \frac{s'}{n}+1 \Big) p^{n-2}$ values of $\tilde{x}$. We can therefore define a set $F \subseteq \mathbb{F}_p^{n-1}$ with
\[ |F| \le 2\Big(\frac{s'}{n}+1\Big)p^{n-2},\]
and such that $\deg(a) = \deg(b) + 1$~for $\tilde{x} \not \in F$.
Finally, for $\tilde{x}\not\in F$ we want to check that $G$ is not of the form $h^p-h$ for some rational function $h$ over $\overline{\mathbb{F}}_p$. Assume therefore that in fact $a/b = h^p-h$ for some rational function $h = h_1/h_2$, where $h_1$ and $h_2$ are coprime. Then ${h_2^pa = h_1^p b - h_1bh_2^{p-1}}$, and so in particular $h_2^p | b$. Note that
\[ \deg(b) \le 2(s'/n+1) < p \]
since we initially assumed that $s\le p^{1/2}n$, and so $h_2$ must be constant. This gives $a = b(c_1h_1^{p} - c_2h_1)$ for some constants $c_1,c_2$. But then $\deg(a) -\deg(b)$ is a multiple of $p$, contradicting $\deg(a) = \deg(b)+1$.
Combining all of this we may apply Theorem \ref{thm:weil} to the sum over $x_{j_1}$ to conclude that whenever $\tilde{x}\not\in F$ it holds that
\[\bigg|\sum_{x_{j_1}} e_p\Big(\frac{a(x_{j_1};\tilde{x})}{b(x_{j_1};\tilde{x})}\Big)\bigg| \le 4\Big(\frac{s}{n}+1\Big)p^{1/2},\]
where the sum is taken over values $x_{j_1}$ where $b\ne 0$. For $\tilde{x}\in F$ we have the trivial bound
\[\bigg|\sum_{x_{j_1}} e_p\Big(\frac{a(x_{j_1};\tilde{x})}{b(x_{j_1};\tilde{x})}\Big)\bigg| \le p.\]
Finally, these bounds together with the union bound
\[ |E| \le \sum_{i=0}^{\floor{(n+d+s-1)/n}} |V(b^{(i)})| \le \left(\frac{s+d}{n}+1\right)p^{n-1}\]
and the triangle inequality give
\begin{align*}
\Big|\sum_{x\in\mathbb{F}_p^n} e_p(G(x_{j_1};\tilde{x}))\Big| &\le |E| + \Big|\sum_{x\not\in E} e_p\Big(\frac{a(x_{j_1};\tilde{x})}{b(x_{j_1};\tilde{x})}\Big)\Big| \\
&\le |E| + p|F| + \Big|\sum_{\tilde{x}\not\in F}\sum_{\substack{x_{j_1}:\\ x\not\in E}} e_p\Big(\frac{a(x_{j_1};\tilde{x})}{b(x_{j_1};\tilde{x})}\Big)\Big|\\
&\le \Big(\frac{s+d}{n}+1\Big)p^{n-1} + 2p\Big(\frac{s}{n}+1\Big)p^{n-2} + 4\Big(\frac{s}{n}+1\Big)p^{1/2}p^{n-1}\\
&\le 3\Big(\frac{s+d}{n}+1\Big)p^{n-1}+4\Big(\frac{s}{n}+1\Big)p^{n-1/2}.
\end{align*}
\end{proof}
We now need an additional ancillary result, which will be used in the proof of the main theorem.
\begin{lemma}\label{lem:2nd-moment}
Let $\Psi$ be a transitive automorphism of $\mathbb{P}^n$ and let $\psi$ be its fractional jump. Then for any integers $j_0, s \geq 1$ and $K$ with $1\le K \le p^n$, and any $\mathbf{h}\in \mathbb{F}_p^s\setminus \{0\}$ one has
\[ \sum_{x\in \mathbb{F}_p^n} \bigg| \sum_{k=0}^{K-1} e_p\Big(\sum_{j=0}^{s-1}h_jv_{j_0+j+k}(x)\Big)\bigg|^2 \ll_{s,n} Kp^n + K^2p^{n-1/2}. \]
\end{lemma}
\begin{proof}
We divide into the two cases $K\le p^{1/2}$ and $K> p^{1/2}$. In the first case we have
\begin{align*}
\sum_{x\in \mathbb{F}_p^n} \bigg| \sum_{k=0}^{K-1} e_p\Big(\sum_{j=0}^{s-1}h_jv_{j_0+j+k}(x)\Big)\bigg|^2 = \sum_{x\in\mathbb{F}_p^n} \sum_{m,l=0}^{K-1}e_p\Big(\sum_{j=0}^{s-1}h_j (v_{j_0+j+m}(x)-v_{j_0+j+l}(x))\Big)\\
\le Kp^n + 2\sum_{d=1}^{K-1}\sum_{m=0}^{K-1-d}\bigg|\sum_{x\in\mathbb{F}_p^n} e_p\Big(\sum_{j=0}^{s-1} h_j(v_{j_0+m+j+d}(x)-v_{j_0+m+j}(x))\Big)\bigg| ,
\end{align*}
where we have split into the cases $m=l$ and $m\ne l$. Applying Lemma \ref{lem:technical} to the innermost sum when $d\le (p-1)n-s$, and applying the trivial bound for the $O_{s,n}(1)$ remaining values of $d$ then gives that this is
\begin{align*}
&\ll_{s,n} K p^n + \sum_{d=1}^{K-1} (K-d)\left(\frac{d}{n}p^{n-1} +(s/n+1)p^{n-1/2}\right) + p^n\\
&\ll_{s,n} Kp^n + K^3p^{n-1} + K^2p^{n-1/2}.
\end{align*}
As the middle term never dominates for the considered range of $K$, we are done in this case.
In the second case, split the sum over $k$ into at most $K/M+1$ intervals of length $M=p^{1/2}$. On each interval we bound the sum as in the first case, and so by Cauchy--Schwarz it follows that
\begin{align*}
\sum_{x\in \mathbb{F}_p^n} \bigg|\sum_{k=0}^{K-1} e_p\Big(\sum_{j=0}^{s-1}h_jv_{j_0+j+k}(x)\Big)\bigg|^2 &\ll_{s,n} \left(\frac{K^2}{M^2}+1\right)(Mp^n+M^3p^{n-1}+M^2p^{n-1/2})\\
&\ll_{s,n} K^2p^{n-1/2}.
\end{align*}
\end{proof}
We are now ready to prove the main theorem.
\begin{proof}[Proof of Theorem \ref{thm:discrepancy}]
Apply Theorem \ref{thm:koksma} with $H = \floor{N/2}$ to get
\begin{equation}
D_{s,\psi}(N;x) \ll \frac{1}{N}+\frac{1}{N}\sum_{0<\| \mathbf{h} \|_\infty \le N/2} \frac{1}{\rho(\mathbf{h})}\bigg| \sum_{m=0}^{N-1} e_p\Big(\sum_{j=0}^{s-1}h_j v_{m+j}(x)\Big)\bigg|.
\label{eq:bound1}
\end{equation}
Let $k\geq 1$ be an integer. Observe that if $k > N-1$, we have that
\[\bigg|\sum_{m=0}^{N-1}e_p\Big(\sum_{j=0}^{s-1}h_j v_{m+j}(x)\Big)-\sum_{m=0}^{N-1}e_p\Big(\sum_{j=0}^{s-1} h_j v_{m+j+k}(x)\Big)\bigg| \leq 2N \leq 2k,\]
if $k \leq N-1$, since the two sums in $m$ overlap in all but $2k$ terms, we have that
\begin{align*}
&\bigg|\sum_{m=0}^{N-1}e_p\Big(\sum_{j=0}^{s-1}h_j v_{m+j}(x)\Big)-\sum_{m=0}^{N-1}e_p\Big(\sum_{j=0}^{s-1} h_j v_{m+j+k}(x)\Big)\bigg| \\
& = \bigg|\sum_{m=0}^{k-1}e_p\Big(\sum_{j=0}^{s-1}h_j v_{m+j}(x)\Big)-\sum_{m=N-k}^{N-1}e_p\Big(\sum_{j=0}^{s-1} h_j v_{m+j+k}(x)\Big)\bigg| \\
&\leq 2k.
\end{align*}
Therefore, for any integer $K \geq 1$ it holds that
\begin{equation*}
\begin{split}
K\bigg|\sum_{m=0}^{N-1}e_p\Big(\sum_{j=0}^{s-1}h_j v_{m+j}(x)\Big) \bigg| &\le \bigg|\sum_{k=0}^{K-1} \sum_{m=0}^{N-1} e_p\Big(\sum_{j=0}^{s-1}h_j v_{m+j+k}(x)\Big) \bigg|+\sum_{k=0}^K 2k\\
&\le \sum_{m=0}^{N-1} \bigg|\sum_{k=0}^{K-1} e_p\Big(\sum_{j=0}^{s-1}h_j v_{m+j+k}(x)\Big)\bigg| + O(K^2).
\end{split}
\end{equation*}
Combining this with \eqref{eq:bound1}, and noting that $\sum_{0<\| \mathbf{h} \|_\infty \le H} \frac{1}{\rho(\mathbf{h})}\ll (\log H)^s$, then gives
\begin{equation}
D_{s,\psi}(N;x) \ll \frac{K}{N}(\log N)^s + \frac{1}{N}R(N,K,x)
\label{eq:bound4}
\end{equation}
where
\begin{equation}
R(N,K,x) = \frac{1}{K}\sum_{0<\| \mathbf{h} \|_\infty \le N/2} \frac{1}{\rho(\mathbf{h})}\sum_{m=0}^{N-1}\bigg|\sum_{k=0}^{K-1}e_p\Big(\sum_{j=0}^{s-1}h_jv_{m+j+k}(x)\Big)\bigg|.
\label{eq:bound2}
\end{equation}
We now average over initial values $x$. By Cauchy--Schwarz one has
\[\left( \sum_{x\in\mathbb{F}_p^n}\bigg|\sum_{k=0}^{K-1}e_p\Big(\sum_{j=0}^{s-1}h_jv_{m+j+k}(x)\Big)\bigg| \right)^2 \le p^n \sum_{x\in\mathbb{F}_p^n} \bigg|\sum_{k=0}^{K-1} e_p\Big(\sum_{j=0}^{s-1}h_j v_{m+j+k}(x)\Big)\bigg|^2.
\]
Inserting this and the bound from Lemma \ref{lem:2nd-moment} into \eqref{eq:bound2} gives
\begin{equation}
\sum_{x \in \mathbb{F}_p^n} R(N,K,x) \ll_{s,n} Np^n (K^{-1/2}+p^{-1/4})(\log N)^s.
\label{eq:bound3}
\end{equation}
Now, let $N_j = 2^j$ and $K_j = \ceil{\Delta^{-2/3}N_j^{2/3}}$ for $j \in \set{ 0,1,\dots,\ceil{\log_2 p^n}}$. Let $\Omega_j\subseteq \mathbb{F}_p^n$ be the set of $x$ for which
\[ R(N_j,K_j,x) \ge C_{s,n}\Delta^{-1}N_j(K_j^{-1/2}+p^{-1/4}) (\log N_j)^s \log p^n,\]
where $C_{s,n}$ is the implied constant in \eqref{eq:bound3}. By \eqref{eq:bound3} we must have that $|\Omega_j| \le \Delta p^n/\log p^n$. Setting $\Omega = \cup_j \Omega_j$ we then have $|\Omega| \le \Delta p^n$, and for $x \not\in \Omega$ it holds that
\begin{equation}
R(N_j,K_j,x) \le C_{s,n} \Delta^{-1}N_j(K_j^{-1/2}+p^{-1/4}) (\log N_j)^s \log p^n
\label{eq:bound5}
\end{equation}
for all $j \le \ceil{\log_2 p^n}$.
Given $N$ such that $1\le N \le p^n$, take $\nu\in\mathbb{N}$ such that $N_{\nu-1}\le N < N_{\nu}$. By \eqref{eq:bound4} we have
\[ D_{s,\psi}(N;x) \ll \frac{K_\nu}{N_\nu}(\log N_\nu)^s + \frac{1}{N_\nu}R(N_\nu,K_\nu,x),\]
and so for $x \not\in \Omega$ it holds that
\[ D_{s,\psi}(N;x) \ll (\Delta^{-2/3}N^{-1/3} + p^{-1/4}\Delta^{-1})(\log N)^s \log p^n\]
by \eqref{eq:bound5}, completing the first bound in the theorem.
For $D_{s,\psi}^n(N;x)$ we also apply Theorem \ref{thm:koksma} with $H = \ceil{N/2}$ to get
\begin{equation*}
D_{s,\psi}^n(N;x) \ll \frac{1}{N}+\frac{1}{N}\sum_{0< \| \mathbf{h} \|_\infty \le N/2} \frac{1}{\rho(\mathbf{h})}\Big| \sum_{m=0}^{N-1} e_p\Big(\sum_{j=0}^{s-1}\mathbf{h}_j\cdot \mathbf{u}_{m+j}(x)\Big)\Big|,
\end{equation*}
where now $\mathbf{h} = (\mathbf{h}_0,\dots,\mathbf{h}_{s-1})$ and $\mathbf{h}_j = (h_{j,1},\dots,h_{j,n})$ for $j \in \set{0,\dots,s-1}$. Observe that
\[ \sum_{j=0}^{s-1} \mathbf{h}_j\cdot \mathbf{u}_{m+j}(x) = \sum_{j=0}^{s-1}\sum_{i=1}^{n} h_{j,i}v_{(m+j)n+i}(x) = \sum_{k=0}^{sn-1}h_j' v_{mn+k}(x), \]
where $h_k' = h_{j,i}$ if $k = nj+i$, $0\le i \le s-1$. We may therefore bound all sums exactly as before, with the only difference being that $sn$~replaces $s$.
\end{proof}
\section{The computational complexity of fractional jump sequences} \label{computation}
Let $\Psi$ be a transitive automorphism of $\mathbb{P}^n$, and let $\psi$ be its fractional jump. We now want to establish the computational complexity of computing the $m$-th term of the sequence $\{ \psi^m (0) \}_{m \in \mathbb{N}}$. In particular in this section we will show that computing a term in our sequence is less expensive than computing a term of a classical inversive sequence of the same bit size.
Fix notations as in Section \ref{explicit}. For simplicity, let us restrict to the case in which $q$ is prime.
Let us first deal with the regime in which $q$ is large (which is the regime in which we got the discrepancy bounds in Section \ref{discrepancy}), so that most of the computations will be performed for points in $U_1$.
If one chooses $\Psi$ in such a way that the coefficients of the $F_j$'s are small (this is possible for example by taking $\Psi$ as the companion matrix of a projectively primitive polynomial with small coefficients), so that also the coefficients of the $f_j^{(1)}$'s are small, the multiplications for such coefficients cost essentially the same as sums. Therefore the computational cost of computing the $m$-th term of the sequence (given the $(m-1)$-th term) is reduced to the cost of computing $n$ multiplications in $\mathbb{F}_q$ and one inversion in
$\mathbb{F}_q$ (as all the denominators of the $f_j^{(1)}$'s are equal). Let $M(q)$ (resp. $I(q)$) denote the cost of one multiplication (resp. inversion) in
$\mathbb{F}_q$. The total cost of bit operations involved to compute a single term in the sequence is then \[C^{\text{new}}(q,n)=n M(q)+ I(q).\]
Using the fast Fourier transform for multiplications \cite{bib:SchStr71} and the
extended Euclidean algorithm for inversion \cite{bib:schonhage71}
one gets
\begin{align*}
M(q) &= O(\log(q)\log\log(q) \log \log \log (q)), \\
I(q) &= O(M(q)\log\log(q)).
\end{align*}
Let us compare this complexity with the complexity of computing the $m$-th term of an inversive sequence of the form $x_{m}=a/x_{m-1}+b$ over $\mathbb{F}_p$. The correct analogue is obtained when $q^n$ has roughly the same bit size as $p$. If one chooses $a,b$ small, one obtains that the complexity of computing $x_m$ is essentially the complexity of computing only one inversion modulo $p$, which is
\[C^{\text{old}}(p)=O(\log(p)[\log \log(p)]^2 \log \log \log(p)).\]
Now, since $q,n$ are chosen in such a way that $q^n$ has roughly the same bit size as $p$, we can write $C^{\text{old}}(q,n)=C^{\text{old}}(p)$.
It is easy to see that up to a positive constant we have
\[\frac{C^{\text{new}}(q,n)}{C^{\text{old}}(q,n)}\leq \frac{1}{\log \log q}+\frac{1}{n},\] which goes to zero as $n$ and $q$ grow.
It is also interesting to see that with our construction we have the freedom to choose $q$ relatively small and $n$ large (so that again one gets $q^n\sim p$).
In this case one can see that ${C^{\text{new}}(q,n)}/{C^{\text{old}}(q,n)}$ goes to zero as $([\log(n)]^2\log \log(n))^{-1}$.
If one tries to do something similar with an ICG (i.e. reducing the characteristic but keeping the size of the field large), one would anyway have to compute an inversion in $\mathbb{F}_{q^n}$ which costs $O(n^2)$ $\mathbb{F}_q$-operations, see \cite[Table 2.8]{bib:MVOV96}, while in our case we would only need to invert one element and multiply $n$ elements in $\mathbb{F}_q$, which costs $(n+1)$ $\mathbb{F}_q$-operations.
\section{Conclusion and further research}
Using the theory of projective maps, we provided a general construction for full orbit sequences over $\mathbb A^n$. Our theory generalises the standard construction for the inversive congruential generators.
Let us summarise the properties of fractional jump sequences obtained in this paper:
\begin{itemize}
\item We completely characterise the full orbit condition for such sequences (Theorem \ref{transitivity_characterisation}).
\item In dimension $1$ they cover the theory of ICG sequences.
\item In dimension greater than $1$, they are automatically full orbit whenever $(q^{n+1}-1)/(q-1)$ is a prime number, which is something that can never occur in the case of ICG sequences, as $2$ always divides $q+1$ when $q$ is odd.
\item In any dimension, they enjoy the same discrepancy bound as the one of the ICG, so they appear to be a good source of pseudorandomness both when one desires a one dimensional sequence of pseudorandom elements or if one desires a stream of $n$-dimensional pseudorandom points (Theorem \ref{thm:discrepancy}).
\item They are very inexpensive to compute: for $n>1$ computations are asymptotically quicker than the ones of an ICG sequence, as described in Section \ref{computation}. The moral reason for this is that at each step the ICG generates a $1$-dimensional pseudorandom point with exactly one inversion in $\mathbb{F}_q$, on the other hand at each step our construction generates an $n$-dimensional point of $\mathbb{F}_q$ (again, using only one inversion).
\end{itemize}
Some research questions arising are the following.
\begin{enumerate}
\item
As our bound on the discrepancy holds for any transitive non-affine fractional jump sequence, can one build special fractional jump sequences having strictly better discrepancy bounds with respect to the one of the ICG?
\item What happens if one replaces the finite field with a finite ring? Can we extend the fractional jump construction to this case?
\item Can the notion of fractional jump be extended to more general objects such as quasi-projective varieties and produce competitive behaviours as in the projective space setting?
\end{enumerate}
\section*{Acknowledgments}
The authors would like to thank Violetta Weger for checking the preliminary version of this manuscript.
The third author would like to thank the Swiss National Science Foundation grant number 171248.
\end{document} | math | 64,849 |
\begin{document}
\title{\bf Uniform Energy Decay for Wave Equations with Unbounded Damping Coefficients}
\author{Ryo IKEHATA\thanks{Corresponding author: [email protected]} \\ {\small Department of Mathematics, Graduate School of Education} \\{\small Hiroshima University} \\ {\small Higashi-Hiroshima 739-8524, Japan}\\ Hiroshi TAKEDA
\thanks{[email protected]} \\{\small Department of Intelligent Mechanical Engineering, Faculty of Engineering} \\{\small Fukuoka Institute of Technology} \\{\small Fukuoka 811-0295, Japan}}
\maketitle
\begin{abstract}
We consider the Cauchy problem for wave equations with unbounded damping coefficients in ${\bf R}^{n}$. For a general class of unbounded damping coefficients, we derive uniform total energy decay estimates together with a unique existence result of a weak solution. In this case we never impose strong assumptions such as compactness of the support of the initial data. This means that we never rely on the finite propagation speed property of the solutions, and we try to deal with an essential unbounded coefficient case. One of our methods comes from an idea developed in \cite{IM}.
\end{abstract}
\section{Introduction}
\footnote[0]{Keywords and Phrases: Unbounded damping; Wave equation; Cauchy problem; Weighted initial data; Multiplier method; Fourier analysis; Total energy decay, Weak solutions.}
\footnote[0]{2010 Mathematics Subject Classification. Primary 35L05; Secondary 35B35, 35B40.}
We consider the mixed problem for wave equations with a localized damping in ${\bf R}^{n}$ ($n \geq 1$)
\begin{equation}
u_{tt}(t,x) -{\cal D}elta u(t,x) + a(x)u_{t}(t,x) = 0,\ \ \ (t,x)\in (0,\infty)\times {\bf R}^{n},\label{eqn}
\end{equation}
\begin{equation}
u(0,x)= u_{0}(x),\ \ u_{t}(0,x)= u_{1}(x),\ \ \ x\in{\bf R}^{n} ,\label{initial}
\end{equation}
where $(u_{0},u_{1})$ are initial data chosen as:
\[u_{0} \in H^{2}({\bf R}^{n}),\quad u_{1} \in H^{1}({\bf R}^{n}),\]
and
\[u_{t}=\frac{\partial u}{\partial t},\quad u_{tt}=\frac{\partial^2 u}{\partial t^2}, \quad {\cal D}elta = \sum_{j=1}^{n}\frac{\partial^{2}}{\partial x_{j}^{2}}, \quad x = (x_1,\cdots,x_n).\]
Note that solutions and/or functions considered in this paper are all real valued except for several parts concerning the Fourier transform.
Concerning the decay or non-decay property of the total or local energy to problem (1.1)-(1.2) with $x$-dependent variable damping coefficients, many research manuscripts are already published by Alloui-Ibrahim-Khenissi \cite{AIK}, Bouclet-Royer \cite{BR}, Daoulatli \cite{D}, Ikehata \cite{Ik-1}, Ikehata-Todorova-Yordanov \cite{ITY}, Joly-Royer \cite{JR}, Kawashita \cite{K}, Khader \cite{Kha}, Matsumura \cite{Ma}, Mochizuki \cite{M}, Mochizuki-Nakazawa \cite{MN}, Nakao \cite{Na}, Nishiyama \cite{Nishiyama}, Nishihara \cite{Ni}, Radu-Todorova-Yordanov \cite{RTY}, Sobajima-Wakasugi \cite{SW-0}, Todorova-Yordanov \cite{TY}, Uesaka \cite{U} and Wakasugi \cite{W}, Zhang \cite{Z} and the references therein. However, we should emphasize that those cases are quite restricted to the bounded damping coefficient case, i.e., $a \in L^{\infty}({\bf R}^{n})$. This condition seems to be essential to get the unique existence of mild or weak solutions to problem (1.1)-(1.2). So, when we do not assume the boundedness of the coefficient $a(x)$, a natural question arises whether one can construct a unique weak or mild solution $u(t,x)$ to problem (1.1)-(1.2) together with some decay property of the total energy or not. Quite recently Sobajima-Wakasugi \cite{SW} have announced an interesting result from the viewpoint of the diffusion phenomenon of the solution to the equation (1.1) together with a unique global existence result. The mixed problem to the equation (1.1) in \cite{SW} is considered in the exterior domain of a bounded obstacle, and the Dirichlet null boundary condition is treated. They treated typically $a(x) = a_{0}\vert x\vert^{\alpha}$ with $a_{0} > 0$ and $\alpha > 0$. But, their results require a stronger assumption such as the compactness of the support of the initial data. This implies that they have to rely on the finite speed of propagation property (FSPP for short) of the solution, so that in an essential meaning, their framework seems to be still restricted to the bounded damping coefficient case for each $t \geq 0$. In the treatment of the unbounded coefficient $a(x)$, it seems important and interesting not to assume such FSPP. Additionally, they essentially used rather stronger regularity condition on the initial data such that $[u_{0},u_{1}] \in (H^{2}\cap H_{0}^{1})\times H_{0}^{1}$. In connection with this topic, D'Abbicco \cite{Da} (and for a more general class, see D'Abbicco-Ebert \cite{DE} and Reissig \cite{R}) and Wirth \cite{Wi} have ever studied $t$-dependent unbounded damping coefficient case:
\[u_{tt}(t,x) -{\cal D}elta u(t,x) + b(1+t)^{\alpha}u_{t}(t,x) = f(u),\]
where $\alpha \in [0,1]$ and $b > 0$. Therefore, in the $x$-dependent unbounded coefficient case, a unique existence of the solution itself together with some decay property of the total energy are completely open in the framework of non-compactly supported initial data class. When Sobajima-Wakasugi \cite{SW} constructs a unique solution, they have used directly the well-known result due to Ikawa \cite{I}. That means their result is still under the already known framework. In our result to be announced, we have to discuss how we should construct a weak solution itself because no any well-known theories can be applied directly. This is a main difficulty in our result.
In this connection, originally Komatsu \cite{Ko} first proposed this open question in his Master thesis in January 2016 such that for the unbounded damping coefficient case $a \notin L^{\infty}({\bf R}^{n})$, can one construct a global in time solution?\, Unfortunately, Komatsu \cite{Ko} could not solve his problem before finishing Master course. This paper gives an answer to his problem.
Now, let us start with introducing our new result. Before stating our result, we shall give the following only one assumption ({\bf A}) on the damping coefficient $a(x)$, and the definition of the solution to be constructed.\\
\noindent
({\bf A})\,$a \in C({\bf R}^{n})$, and there exists a constant $V_{0} > 0$ such that $0 < V_{0} \leq a(x)$ ($\forall x \in {\bf R}^{n}$).\\
\noindent
{\bf Definition.} {\rm A function $u:\,[0,\infty)\times{\bf R}^{n} \to {\bf R}$ is called as the weak solution if it satisfies
\[\int_{0}^{\infty}\int_{{\bf R}^{n}}u(t,x)(\phi_{tt}(t,x)-{\cal D}elta\phi(t,x) - a(x)\phi_{t}(t,x))dx dt\]
\[= \int_{{\bf R}^{n}}u_{1}(x)\phi(0,x)dx - \int_{{\bf R}^{n}}u_{0}(x)\phi_{t}(0,x)dx+ \int_{{\bf R}^{n}}a(x)u_{0}(x)\phi(0,x)dx\]
for any $\phi \in C_{0}^{\infty}([0,\infty)\times{\bf R}^{n})$.}\\
Our new result reads as follows.
\begin{theo} Let $n \geq 3$ and assume {\rm ({\bf A})}. If the initial data $[u_{0},u_{1}] \in (H^{2}({\bf R}^{n})\cap L^{1}({\bf R}^{n}))\times (H^{1}({\bf R}^{n})\cap L^{1}({\bf R}^{n}))$ further satisfies $a(\cdot)u_{0} \in L^{1}({\bf R}^{n})\cap L^{2}({\bf R}^{n})$, then there exists a unique weak solution $u \in L^{\infty}(0,\infty;H^{1}({\bf R}^{n}))\cap W^{1,\infty}(0,\infty;L^{2}({\bf R}^{n}))$ to problem {\rm (1.1)}-{\rm (1.2)} satisfying
\[\Vert u(t,\cdot)\Vert^{2} \leq CI_{00}^{2}(1+t)^{-1},\quad E_{u}(t) \leq CI_{00}^{2}(1+t)^{-2},\]
with some constant $C > 0$, where
\[I_{00} := \left(\Vert u_{0}\Vert^{2} + \Vert\nabla u_{0}\Vert^{2} + \Vert a(\cdot)u_{0}\Vert_{1}^{2} + \Vert a(\cdot)u_{0}\Vert^{2} + \Vert u_{1}\Vert^{2} +\Vert u_{1}\Vert_{1}^{2}\right)^{1/2}.\]
\end{theo}
\begin{rem}{\rm If one considers the mixed problem (1.1)-(1.2) with the Dirichlet null boundary condition on the smooth exterior domain, one can treat the two dimensional case. In that case, we need to assume the logarithmic type weight condition such that $\Vert\log(B\vert x\vert)(a(\cdot)u_{0} + u_{1})\Vert < +\infty$ on the initial data with some constant $B > 0$. This has a close relation to the place where one obtains Lemma 2.1 below in the two dimensional exterior domain case. For more detail, one can refer the reader to \cite{IM}.}
\end{rem}
\begin{rem}{\rm The assumption $[u_{0},u_{1}] \in H^{2}({\bf R}^{n})\times H^{1}({\bf R}^{n})$ is not essential. It will be used only to justify the integration by parts in the course of the proof. That condition seems not to be so rare (cf. \cite{MN}). One may be able to generalize to the more weak case such that $[u_{0},u_{1}] \in H^{1}({\bf R}^{n})\times L^{2}({\bf R}^{n})$, however, for simplicity we do not deal with such case.}
\end{rem}
\noindent
{\bf Example.} {\rm As the typical unbounded example for $a(x)$, we can choose $a(x) := (1+\vert x\vert^{2})^{\frac{\alpha}{2}}$ with $\alpha \in [0,\infty)$, $a(x) := e^{\vert x\vert}$, and so on.}
\par
This paper is organized as follows. In section 2 we shall prove Theorem 1.1 by relying on a multiplier method which was introduced in \cite{IM}. In section 3 we give several remarks and open problems. Section 4 is devoted to the appendix to check the known result.\\
{\bf Notation.} {\small Throughout this paper, $\| \cdot\|_q$ stands for the usual $L^q({\bf R}^{n})$-norm. For simplicity of notation, in particular, we use $\| \cdot\|$ instead of $\| \cdot\|_2$. Furthermore, we denote $\Vert\cdot\Vert_{H^{1}}$ as the usual $H^{1}$-norm. The $L^{2}$-inner product is denoted by $(f,g) := \displaystyle{\int_{{\bf R}^{n}}}f(x)g(x)dx$ for $f,g \in L^{2}({\bf R}^{n})$. The total energy $E_{u}(t)$ corresponding to the solution $u(t,x)$ of (1.1) is defined by
\[E_{u}(t):=\frac{1}{2}(\| u_t(t,\cdot)\|^2+\|\nabla u(t,\cdot)\|^2),\]
where
\[\vert\nabla f(x)\vert^{2} := \sum_{j=1}^{n}\vert\frac{\partial f(x)}{\partial x_{j}}\vert^{2}.\]
The weighted $L^{1}$-space and its norm $\Vert \cdot\Vert_{1,m}$ can be defined as
\[f \in L^{1,m}({\bf R}^{n}) \Leftrightarrow f \in L^{1}({\bf R}^{n}),\quad \Vert f\Vert_{1,m} := \int_{{\bf R}^{n}}(1+\vert x\vert)^{m}\vert f(x)\vert dx < +\infty.\]
The subspace $X_{m}^{n}$ of $L^{1,m}$ is defined by
\[X_{m}^{n} := \{f \in L^{1,m}({\bf R}^{n}):\,\int_{{\bf R}^{n}}f(x)dx = 0\}.\]
And also, for the Hilbert space $X$ we define a class of vector valued continuous functions $C_{0}([0,\infty);X)$ as follows: $f \in C_{0}([0,\infty);X)$ if and only if $f \in C([0,\infty);X)$ and the closure of the set $\{t \in [0,\infty);\,\Vert f(t)\Vert_{X} \ne 0\}$ is compact in $[0,\infty)$.
On the other hand, we denote the Fourier transform of $f(x)$ by $\hat{f}(\xi) := (\displaystyle{\frac{1}{2\pi}})^{\frac{n}{2}}\displaystyle{\int_{{\bf R}^{n}}}e^{-ix\cdot\xi}f(x)dx$ as usual with $i := \sqrt{-1}$, and we define the usual convolution by
\[(f*g)(x) := \int_{{\bf R}^{n}}f(x-y)g(y)dy.\]
}\\
\section{Proof of Theorem 1.1.}
In the course of the proof, the next inequality concerning the Fourier image of the Riesz potential plays an alternative role for the Hardy inequality. This comes from \cite[Proposition 2.1]{Ike-0}. It should be emphasized that this inequality holds even in the low dimensional case if we control the weight parameter $\gamma$.
\begin{pro}\,Let $n \geq 1$ and $\gamma \in [0,1]$.\\
{\rm (1)}\, If $f \in L^{2}({\bf R}^{n})\cap L^{1,\gamma}({\bf R}^{n})$, and $\theta \in [0,\displaystyle{\frac{n}{2}})$, then there exists a constant $C = C_{n,\theta,\gamma} > 0$ such that\\
\[\displaystyle{\int_{{\bf R}^{n}}}\displaystyle{\frac{\vert \hat{f}(\xi)\vert^{2}}{\vert\xi\vert^{2\theta}}}d\xi \leq C\left(\Vert f\Vert_{1,\gamma}^{2} + \vert\displaystyle{\int_{{\bf R}^{n}}f(x)dx}\vert^{2}+ \Vert f\Vert^{2}\right) .\]
{\rm (2)}\, If $f \in L^{2}({\bf R}^{n})\cap X_{\gamma}^{n}$, and $\theta \in [0,\gamma + \displaystyle{\frac{n}{2}})$, then it is true that
\[\displaystyle{\int_{{\bf R}^{n}}}\displaystyle{\frac{\vert \hat{f}(\xi)\vert^{2}}{\vert\xi\vert^{2\theta}}}d\xi \leq C(\Vert f\Vert_{1,\gamma}^{2} + \Vert f\Vert^{2}) \]
with some constant $C = C_{n,\theta,\gamma} > 0$.
\end{pro}
To construct a global weak solution we first define a sequence of the weak solutions $\{u^{(m)}(t,x)\}$ ($m \in {\bf N}$) to the approximated problem below:
\begin{equation}
u_{tt}^{(m)}(t,x) -{\cal D}elta u^{(m)}(t,x) + a_{m}(x)u_{t}^{(m)}(t,x) = 0,\ \ \ (t,x)\in (0,\infty)\times {\bf R}^{n},\label{eqn2}
\end{equation}
\begin{equation}
u^{(m)}(0,x)= u_{0}(x),\ \ u_{t}^{(m)}(0,x)= u_{1}(x),\ \ \ x\in{\bf R}^{n},\label{initial2}
\end{equation}
where $a_{m} \in C({\bf R}^{n})$ can be chosen to satisfy
\begin{eqnarray}a_{m}(x) = \left\{
\begin{array}{ll}
\displaystyle{a(x)}&
\qquad (\vert x\vert \leq m) \\[0.2cm]
\displaystyle{V_{0}}& \qquad (\vert x\vert > m+1),
\end{array} \right. \end{eqnarray}
and
\begin{equation}
V_{0} \leq a_{m}(x) \leq a(x), \quad a_{m}(x) \to a(x) \quad \textstyle{as}\quad m \to \infty \quad\textstyle{(pointwise)}\quad x \in {\bf R}^{n}
\end{equation}
for each $x \in {\bf R}^{n}$.\\
Now let us consider the problem (2.1)-(2.2) with initial data $[u_{0}, u_{1}] \in H^{2}({\bf R}^{n})\times H^{1}({\bf R}^{n})$. Then, for each $m \in {\bf N}$ since $a_{m} \in C({\bf R}^{n})\cap L^{\infty}({\bf R}^{n})$ it is well known that the Cauchy problem (2.1)-(2.2) has a unique strong solution $u^{(m)} \in C([0,\infty);H^{2}({\bf R}^{n}))\cap C^{1}([0,\infty);H^{1}({\bf R}^{n}))\cap C^{2}([0,\infty);L^{2}({\bf R}^{n}))$ satisfying the energy identity:
\begin{equation}
E_{u^{(m)}}(t) + \int_{0}^{t}\int_{{\bf R}^{n}}a_{m}(x)\vert u_{s}^{(m)}(s,x)\vert^{2}dxds = E_{}(0),
\end{equation}
where
\[E(0) := \frac{1}{2}(\Vert u_{1}\Vert^{2} + \Vert\nabla u_{0}\Vert^{2}).\]
To begin with we prove the following crucial estimate. The lemma below is a combination of the method introduced in \cite{IM} ($=$ the modified Morawetz method \cite{Mora}) and Proposition 2.1.
\begin{lem}\,Let $n \geq 3$. Under the same assumptions as in Theorem {\rm 1.1}, the {\rm (}unique{\rm )} solution $u^{(m)}(t,x)$ to problem {\rm (2.1)-(2.2)} satisfies
\[\Vert u^{(m)}(t,\cdot)\Vert^{2} + \int_{0}^{t}\int_{{\bf R}^{n}}a_{m}(x)\vert u^{(m)}(s,x)\vert^{2}dxds\]
\[\leq C\left(\Vert a(\cdot)u_{0}\Vert_{1}^{2} + \Vert a(\cdot)u_{0}\Vert^{2} + \Vert u_{1}\Vert_{1}^{2} + \Vert u_{1}\Vert^{2}\right) =:CI_{0}^{2}\]
with a constant $C > 0$, where $C$ is independent of $m$.
\end{lem}
{\it Proof.}\, The original idea comes from \cite{IM}. For the solution $u^{(m)}(t,x)$ to problem (2.1)-(2.2), one introduces an auxiliary function
$$W(t,x) := \int^{t}_{0}u^{(m)}(s,x)ds.$$
Then $W(t,x)$ satisfies
\begin{equation}
W_{tt} - {\cal D}elta W + a_{m}(x)W_{t} = a_{m}(x)u_{0} + u_{1},\ \ \ \ (t,x) \in (0,\infty) \times {\bf R}^{n},
\end{equation}
\begin{equation}
W(0,x) = 0,\quad W_{t}(0,x) = u_{0}(x),\,\,\,\, x \in {\bf R}^{n}.
\end{equation}
Multiplying $(2.6)$ by $W_{t}$ and integrating over $[0,t]\times {\bf R}^{n}$ we get
$$\frac{1}{2}(\|W_{t}(t,\cdot)\|^{2} + \|\nabla W(t,\cdot)\|^{2}) + \int^{t}_{0}\|\sqrt{a_{m}(\cdot)}W_{s}(s,\cdot)\|^{2}ds$$
\begin{equation}
= \frac{1}{2}\|u_{0}\|^{2} + \int^{t}_{0}(a_{m}(\cdot)u_{0} + u_{1}, W_{s}(s,\cdot))ds.
\end{equation}
Next one uses (1) of Proposition 2.1 with $\theta = 1$ and $\gamma = 0$, the Plancherel theorem and the Cauchy-Schwarz inequality to obtain a series of inequalities below:\\
$$\left\vert\int^{t}_{0}(a_{m}(\cdot)u_{0} + u_{1},W_{s}(s,\cdot))ds\right\vert = \left\vert\int^{t}_{0}\frac{d}{ds}(a_{m}(\cdot)u_{0} + u_{1},W(s,\cdot))ds\right\vert$$
$$= \left\vert\int_{{\bf R}^{n}}(a_{m}(x)u_{0}(x) + u_{1}(x))W(t,x)dx\right\vert$$
$$= \left\vert\int_{{\bf R}_{\xi}^{n}}(\widehat{(a_{m}u_{0})}(\xi) + \hat{u}_{1}(\xi))\overline{\hat{W}(t,\xi)}d\xi\right\vert$$
$$\leq \int_{{\bf R}_{\xi}^{n}}\frac{\vert\widehat{(a_{m}u_{0})}(\xi) + \hat{u}_{1}(\xi)\vert}{\vert\xi\vert}(\vert\xi\vert\vert\hat{W}(t,\xi)\vert)d\xi$$
$$\leq \left(\int_{{\bf R}_{\xi}^{n}}\frac{\vert\widehat{(a_{m}u_{0})}(\xi) + \hat{u}_{1}(\xi)\vert^{2}}{\vert\xi\vert^{2}}d\xi\right)^{1/2}\left(\int_{{\bf R}_{\xi}^{n}}\vert\xi\vert^{2}\vert\hat{W}(t,\xi)\vert^{2}d\xi\right)^{1/2}$$
\[\leq \int_{{\bf R}_{\xi}^{n}}\frac{\vert\widehat{(a_{m}u_{0})}(\xi) + \hat{u}_{1}(\xi)\vert^{2}}{\vert\xi\vert^{2}}d\xi + \frac{1}{4}\int_{{\bf R}_{\xi}^{n}}\vert\xi\vert^{2}\vert\hat{W}(t,\xi)\vert^{2}d\xi\]
\[\leq C\Vert a_{m}u_{0}+u_{1}\Vert_{1}^{2} + \Vert a_{m}u_{0}+u_{1}\Vert^{2} + C\left\vert\int_{{\bf R}^{n}}(a_{m}(x)u_{0}(x)+u_{1}(x))dx\right\vert^{2} + \frac{1}{4}\Vert\nabla W(t,\cdot)\Vert^{2}\]
\begin{equation}
\leq C\left(\Vert a_{m}(\cdot)u_{0}\Vert_{1}^{2} +\Vert u_{1}\Vert_{1}^{2} + \Vert a_{m}(\cdot)u_{0}\Vert^{2} +\Vert u_{1}\Vert^{2} + \vert\int_{{\bf R}^{n}}(a_{m}(x)u_{0}(x)+u_{1}(x))dx\vert^{2}\right) + \frac{1}{4}\Vert\nabla W(t,\cdot)\Vert^{2}
\end{equation}
with some constant $C > 0$.
Combining $(2.8)$ with $(2.9)$ we can derive
$$\frac{1}{2}\|W_{t}(t,\cdot)\|^{2} + \frac{1}{4}\|\nabla W(t,\cdot)\|^{2} + \int^{t}_{0}\int_{{\bf R}^{n}}a_{m}(x)\left\vert W_{s}(s,x)\right\vert^{2}dxds$$
$$\leq C\left(\Vert a_{m}(\cdot)u_{0}\Vert_{1}^{2} +\Vert u_{1}\Vert_{1}^{2} + \Vert a_{m}(\cdot)u_{0}\Vert^{2} +\Vert u_{1}\Vert^{2}+ \vert \int_{{\bf R}^{n}}(a_{m}(x)u_{0}(x)+u_{1}(x))dx\vert^{2}\right)$$
with some constant $C > 0$. Since $a_{m}(x) \leq a(x)$, in the case when $n \geq 3$ one has
$$\frac{1}{2}\|W_{t}(t,\cdot)\|^{2} + \frac{1}{4}\|\nabla W(t,\cdot)\|^{2} + \int^{t}_{0}\int_{{\bf R}^{n}}a_{m}(x)\left\vert W_{s}(s,x)\right\vert^{2}dxds$$
$$\leq C\left(\Vert a(\cdot)u_{0}\Vert_{1}^{2} +\Vert u_{1}\Vert_{1}^{2} + \Vert a(\cdot)u_{0}\Vert^{2} +\Vert u_{1}\Vert^{2} + \vert\int_{{\bf R}^{n}}(a(x)\vert u_{0}(x)\vert + \vert u_{1}(x)\vert)dx\vert^{2}\right).$$
We easily see that the constant $C$ in the above estimates is independent of $m$.
Thus, one has the desired estimate because of the fact $W_{t}=u^{(m)}$.
$
\Box$
\begin{lem}Under the same assumptions as in Theorem {\rm 1.1}, the {\rm (}unique{\rm )} solution $u^{(m)}(t,x)$ to problem {\rm (2.1)-(2.2)} satisfies
\[(1+t)E_{u^{(m)}}(t) \leq E(0)(1+\frac{1}{V_{0}} + \frac{1}{2\varepsilon}) + \frac{1}{2}(u_{1},u_{0}) =: I_{1}^{2},\quad (t \geq 0),\]
\[\int_{0}^{t}E_{u^{(m)}}(s)ds \leq I_{1}^{2},\quad (t \geq 0),\]
with some small constant $\varepsilon > 0$, where $C$ is independent of $m$.
\end{lem}
{\it Proof.}\,For simplicity of notation, we use $w(t,x)$ in place of $u^{(m)}(t,x)$, i.e., $w(t,x)$ satisfies the equation below:
\begin{equation}
w_{tt}(t,x) -{\cal D}elta w(t,x) + a_{m}(x)w_{t}(t,x) = 0,\ \ \ (t,x)\in (0,\infty)\times {\bf R}^{n},
\end{equation}
\begin{equation}
w(0,x)= u_{0}(x),\ \ w_{t}(0,x)= u_{1}(x),\ \ \ x\in{\bf R}^{n}.
\end{equation}
Note that $E_{w}(t) = E_{u^{(m)}}(t)$ satisfies (2.5). Then, since
\[\frac{d}{dt}\{(1+t)E_{w}(t)\} \leq E_{w}(t),\]
it follows from (2.5) that
\[(1+t)E_{w}(t) \leq E(0) + \frac{1}{2}\int_{0}^{t}\Vert w_{s}(s,\cdot)\Vert^{2}ds + \frac{1}{2}\int_{0}^{t}\Vert\nabla w(s,\cdot)\Vert^{2}ds\]
\[\leq E(0) + \frac{1}{2V_{0}}\int_{0}^{t}\int_{{\bf R}^{n}}a_{m}(x)\vert w_{s}(s,x)\vert^{2}dxds + \frac{1}{2}\int_{0}^{t}\Vert\nabla w(s,\cdot)\Vert^{2}ds\]
\begin{equation}
\leq E(0) + \frac{1}{2V_{0}}E(0) + \frac{1}{2}\int_{0}^{t}\Vert\nabla w(s,\cdot)\Vert^{2}ds.
\end{equation}
On the other hand, by multiplying both sides of (2.10) by $w(t,x)$ it follows that
\begin{equation}
\frac{d}{dt}(w_{t}(t,\cdot),w(t,\cdot)) + \Vert\nabla w(t,\cdot)\Vert^{2} + \frac{1}{2}\frac{d}{dt}\int_{{\bf R}^{n}}a_{m}(x)\vert w(t,x)\vert^{2}dx = \Vert w_{t}(t,\cdot)\Vert^{2},
\end{equation}
so that by integrating both sides over $[0,t]$ one has
\[\int_{0}^{t}\Vert\nabla w(s,\cdot)\Vert^{2}ds + \frac{1}{2}\int_{{\bf R}^{n}}a_{m}(x)\vert w(t,x)\vert^{2}dx\]
\[= \int_{0}^{t}\Vert w_{s}(s,\cdot)\Vert^{2}ds - (w_{t}(t,\cdot),w(t,\cdot)) + (u_{1},u_{0})\]
\[\leq \frac{1}{V_{0}}\int_{0}^{t}\int_{{\bf R}^{n}}a_{m}(x)\vert w_{s}(s,x)\vert^{2}dxds +\frac{1}{2\varepsilon}\Vert w_{t}(t,\cdot)\Vert^{2} + \frac{\varepsilon}{2}\Vert w(t,\cdot)\Vert^{2} + (u_{1},u_{0})\]
\[\leq \frac{1}{V_{0}}\int_{0}^{t}\int_{{\bf R}^{n}}a_{m}(x)\vert w_{s}(s,x)\vert^{2}dxds +\frac{1}{\varepsilon}E_{w}(t) + \frac{\varepsilon}{2V_{0}}\int_{{\bf R}^{n}}a_{m}(x)\vert w(t,x)\vert^{2}dx + (u_{1},u_{0})\]
\[\leq \frac{1}{V_{0}}E(0) + \frac{1}{\varepsilon}E(0) + \frac{\varepsilon}{2V_{0}}\int_{{\bf R}^{n}}a_{m}(x)\vert w(t,x)\vert^{2}dx + (u_{1},u_{0}),\]
where we have just used (2.5) and the Cauchy-Schwarz inequality with some positive parameter $\varepsilon > 0$. This implies
\[\int_{0}^{t}\Vert\nabla w(s,\cdot)\Vert^{2}ds + \frac{1}{2}(1-\frac{\varepsilon}{V_{0}})\int_{{\bf R}^{n}}a_{m}(x)\vert w(t,x)\vert^{2}dx\]
\begin{equation}
\leq E(0)(\frac{1}{V_{0}} + \frac{1}{\varepsilon}) + (u_{1},u_{0}).
\end{equation}
By choosing $\varepsilon > 0$ sufficiently small, from (2.13) and (2.14) one can get the desired two estimates. In this final check, because of (2.6) we have to make the following estimate once more:
\[\int_{0}^{t}\Vert w_{t}(s,\cdot)\Vert^{2}ds \leq \frac{1}{V_{0}}\int_{{\bf R}^{n}}a_{m}(x)\vert w_{s}(s,x)\vert^{2}dxds \leq \frac{1}{V_{0}}E(0).\]
$
\Box$
\begin{lem}Under the same assumptions as in Theorem {\rm 1.1}, the {\rm (}unique{\rm )} solution $u^{(m)}(t,x)$ to problem {\rm (2.1)-(2.2)} satisfies
\[(1+t)^{2}E_{u^{(m)}}(t) \leq E(0) + \frac{1}{V_{0}}(E(0) + I_{1}^{2}) + (u_{1},u_{0}) + \frac{1}{2}\int_{{\bf R}^{n}}a(x)\vert u_{0}(x)\vert^{2}dx + \frac{C}{2}I_{0}^{2} +\frac{I_{1}^{2}}{\varepsilon}=: I_{2}^{2},\]
\[(1+t)\Vert u^{(m)}(t,\cdot)\Vert^{2} \leq 2(V_{0}-\epsilon)^{-1}\{(u_{1},u_{0}) + \frac{1}{2}\int_{{\bf R}^{n}}a(x)\vert u_{0}(x)\vert^{2}dx + \frac{C}{2}I_{0}^{2} + \frac{1}{V_{0}}(E(0) + I_{1}^{2})+\frac{I_{1}^{2}}{\varepsilon} \} =: I_{3}^{2}\]
with some constant $C > 0$, where $C$ is independent of $m$.
\end{lem}
{\it Proof.}\,We use the same notation as in the proof of Lemma 2.2. Then, by multiplying both sides of (2.11) by $(1+t)w$, and integrating it over $[0,t] \times {\bf{R}}^{n}$ one can arrive at
the important identity:
\[\frac{1}{2}\Vert u_{0}\Vert^{2} + \int_{0}^{t}(1+s)\Vert\nabla w(s,\cdot)\Vert^{2}ds + \frac{(1+t)}{2}\int_{{\bf R}^{n}}a_{m}(x)\vert w(t,x)\vert^{2}dx\]
\[=-(1+t)(w_{t}(t,\cdot),w(t,\cdot)) + (u_{1},u_{0}) + \frac{1}{2}\Vert w(t,\cdot)\Vert^{2} + \frac{1}{2}\int_{0}^{t}\int_{{\bf R}^{n}}a_{m}(x)\vert w(s,x)\vert^{2}dxds\]
\begin{equation}
+ \frac{1}{2}\int_{{\bf R}^{n}}a_{m}(x)\vert u_{0}(x)\vert^{2}dx + \int_{0}^{t}(1+s)\Vert w_{s}(s,\cdot)\Vert^{2}ds.
\end{equation}
Now, by using the Cauchy-Schwarz inequality and Lemma 2.2 we can first estimate
\[-(1+t)(w_{t}(t,\cdot),w(t,\cdot)) \leq \frac{(1+t)}{2\varepsilon}\Vert w_{t}(t,\cdot)\Vert^{2} + \frac{\varepsilon}{2}(1+t)\Vert w(t,\cdot)\Vert^{2}\]
\[\leq \frac{1+t}{\varepsilon}E_{w}(t) + \frac{\varepsilon}{2V_{0}}(1+t)\int_{{\bf R}^{n}}a_{m}(x)\vert w(t,x)\vert^{2}dx\]
\begin{equation}
\leq \frac{I_{1}^{2}}{\varepsilon} + \frac{\varepsilon}{2V_{0}}(1+t)\int_{{\bf R}^{n}}a_{m}(x)\vert w(t,x)\vert^{2}dx.
\end{equation}
(2.16) and (2.17) imply
\[\frac{1}{2}\Vert u_{0}\Vert^{2} + \int_{0}^{t}(1+s)\Vert\nabla w(s,\cdot)\Vert^{2}ds + \frac{(1+t)}{2}(1- \frac{\varepsilon}{V_{0}})\int_{{\bf R}^{n}}a_{m}(x)\vert w(s,x)\vert^{2}dx\]
\[\leq (u_{1},u_{0}) + \frac{1}{2}\int_{{\bf R}^{n}}a(x)\vert u_{0}(x)\vert^{2}dx + \frac{1}{2}\Vert w(t,\cdot)\Vert^{2} + \frac{1}{2}\int_{0}^{t}\int_{{\bf R}^{n}}a_{m}(x)\vert w(s,x)\vert^{2}dxds
\]
\begin{equation}
+ \frac{I_{1}^{2}}{\varepsilon} + \int_{0}^{t}(1+s)\Vert w_{s}(s,\cdot)\Vert^{2}ds.
\end{equation}
On the other hand, it follows from (2.5) and Lemma 2.2 that
\[\int_{0}^{t}(1+s)\Vert w_{s}(s,\cdot)\Vert^{2}ds \leq \frac{1}{V_{0}}\int_{0}^{t}(1+s)\int_{{\bf R}^{n}}a_{m}(x)\vert w_{s}(s,x)\vert^{2}dxds\]
\[= -\frac{1}{V_{0}}\int_{0}^{t}(1+s)E'_{w}(s)ds = -\frac{1}{V_{0}}(1+t)E_{w}(t) + \frac{1}{V_{0}}E(0) + \frac{1}{V_{0}}\int_{0}^{t}E_{w}(s)ds\]
\begin{equation}
\leq \frac{1}{V_{0}}(E(0) + I_{1}^{2}).
\end{equation}
Let us finalize the proof of Lemma 2.3. To do so, we rely on the inequality:
\[\frac{d}{dt}\{(1+t)^{2}E_{w}(t)\} \leq 2(1+t)E_{w}(t),\]
so that
\begin{equation}
(1+t)^{2}E_{w}(t) \leq E(0) + \int_{0}^{t}(1+s)\Vert w_{s}(s,\cdot)\Vert^{2}ds + \int_{0}^{t}(1+s)\Vert \nabla w(s,\cdot)\Vert^{2}ds
\end{equation}
Because of Lemma 2.1, (2.17) with small $\varepsilon > 0$ and (2.18) one has
\[(1+t)^{2}E_{w}(t) \leq E(0) + \frac{2}{V_{0}}(E(0) + I_{1}^{2}) + (u_{1},u_{0}) + \frac{1}{2}\int_{{\bf R}^{n}}a(x)\vert u_{0}(x)\vert^{2} dx+ \frac{C}{2}I_{0}^{2} +\frac{I_{1}^{2}}{\varepsilon} =: I_{2}^{2} .\]
Concerning the fast $L^{2}$-decay estimate, we use (2.5) and (2.17) with small $\varepsilon > 0$ to have
\[\frac{(1+t)}{2}(1- \frac{\varepsilon}{V_{0}})V_{0}\int_{{\bf R}^{n}}\vert w(t,x)\vert^{2}dx \leq \frac{(1+t)}{2}(1- \frac{\varepsilon}{V_{0}})\int_{{\bf R}^{n}}a_{m}(x)\vert w(t,x)\vert^{2}dx\]
\[\leq (u_{1},u_{0}) + \frac{1}{2}\int_{{\bf R}^{n}}a(x)\vert u_{0}(x)\vert^{2}dx + \frac{1}{2}\Vert w(t,\cdot)\Vert^{2} + \frac{1}{2}\int_{0}^{t}\int_{{\bf R}^{n}}a_{m}(x)\vert w(s,x)\vert^{2}dxds
\]
\begin{equation}
+\frac{I_{1}^{2}}{\varepsilon}+ \int_{0}^{t}(1+s)\Vert w_{s}(s,\cdot)\Vert^{2}ds.
\end{equation}
The result follows from (2.20) and (2.18) and Lemma 2.1 by choosing $\varepsilon > 0$ small enough.
$
\Box$
\par
{\it Proof of Theorem 1.1.} From Lemma 2.3 we first notice that $\{u^{(m)}\}$ is a bounded sequence in $L^{\infty}(0,\infty;H^{1}({\bf R}^{n}))$ and in $L^{\infty}(0,\infty;L^{2}({\bf R}^{n}))$. Furthermore, $\{u^{(m)}_{t}\}$ is also a bounded sequence in $L^{\infty}(0,\infty;L^{2}({\bf R}^{n}))$. Therefore, there exist a subsequence $\{u^{(\mu)}\}$ of the original one $\{u^{(m)}\}$, and a function $u = u(t,x) \in L^{\infty}(0,\infty;H^{1}({\bf R}^{n}))$ satisfying $u_{t} \in L^{\infty}(0,\infty;L^{2}({\bf R}^{n}))$ such that
\begin{equation}
u^{(\mu)} \to u\quad (\textstyle{weakly*}) \quad \textstyle{in} \quad L^{\infty}(0,\infty;H^{1}({\bf R}^{n})) \quad (\mu \to \infty),
\end{equation}
\begin{equation}
u_{t}^{(\mu)} \to u_{t} \quad (\textstyle{weakly*}) \quad \textstyle{in} \quad L^{\infty}(0,\infty;L^{2}({\bf R}^{n})) \quad (\mu \to \infty),
\end{equation}
\begin{equation}
u^{(\mu)} \to u \quad (\textstyle{weakly*}) \quad \textstyle{in} \quad L^{\infty}(0,\infty;L^{2}({\bf R}^{n})) \quad (\mu \to \infty).
\end{equation}
By multiplying both sides of the approximated equation (2.1) with $m$ replaced by $\mu$ the test function $\phi \in C_{0}^{\infty}([0,\infty)\times{\bf R}^{n})$, one can get the following weak form of the problem (2.1)-(2.2) with the help of the integration by parts:
\[\int_{0}^{\infty}\int_{{\bf R}^{n}}u^{(\mu)}(t,x)(\phi_{tt}(t,x)-{\cal D}elta\phi(t,x) - a_{\mu}(x)\phi_{t}(t,x))dx dt\]
\begin{equation}
= \int_{{\bf R}^{n}}u_{1}(x)\phi(0,x)dx - \int_{{\bf R}^{n}}u_{0}(x)\phi_{t}(0,x)dx+ \int_{{\bf R}^{n}}a_{\mu}(x)u_{0}(x)\phi(0,x)dx.
\end{equation}
Now, it follows from (2.23) that as $\mu \to \infty$,
\begin{equation}
\int_{0}^{\infty}\int_{{\bf R}^{n}}u^{(\mu)}(t,x)(\phi_{tt}(t,x)-{\cal D}elta\phi(t,x))dx dt \to \int_{0}^{\infty}\int_{{\bf R}^{n}}u(t,x)(\phi_{tt}(t,x)-{\cal D}elta\phi(t,x))dx dt.
\end{equation}
Furthermore, because of the Lebesgue dominated convergence theorem one can get
\begin{equation}
\int_{{\bf R}^{n}}a_{\mu}(x)u_{0}(x)\phi(0,x)dx \to \int_{{\bf R}^{n}}a(x)u_{0}(x)\phi(0,x)dx \quad (\mu \to \infty).
\end{equation}
On the other hand, for each fixed $\phi \in C_{0}^{\infty}([0,\infty)\times{\bf R}^{n})$, if we take $\mu \in {\bf N}$ large enough, then it follows from the compact support condition on $\phi(t,x)$ that
\[\int_{0}^{\infty}\int_{{\bf R}^{n}}u^{(\mu)}(t,x)a_{\mu}(x)\phi_{t}(t,x)dx dt = \int_{0}^{\infty}\int_{{\bf R}^{n}}u^{(\mu)}(t,x)a(x)\phi_{t}(t,x)dx dt.\]
So, since $a(\cdot)\phi_{t}(t,\cdot) \in L^{1}([0,\infty);L^{2}({\bf R}^{n}))$, it follows from (2.23) that
\begin{equation}
\int_{0}^{\infty}\int_{{\bf R}^{n}}u^{(\mu)}(t,x)a_{\mu}(x)\phi_{t}(t,x)dx dt \to \int_{0}^{\infty}\int_{{\bf R}^{n}}u(t,x)a(x)\phi_{t}(t,x)dx dt \quad (\mu \to \infty).
\end{equation}
Therefore, by taking $\mu \to \infty$ in (2.24), by means of (2.25)-(2.27) one can check that the limit function $u(t,x)$ is the weak solution to problem (1.1)-(1.2).\\
Finally, let us check decay estimates of the energy and $L^{2}$-norm of solutions such that
\begin{equation}
E_{u}(t) \leq C(1+t)^{-2}, \quad \Vert u(t,\cdot)\Vert^{2} \leq C(1+t)^{-1}
\end{equation}
with some constant $C > 0$. \\
For this end, one first remarks that for each $\phi \in L^{1}(0,\infty; C_{0}^{\infty}({\bf R}^{n}))$, it follows from the integration by parts that
\[\int_{0}^{\infty}(\frac{\partial u^{(\mu)}}{\partial x_{j}}(t,\cdot),\phi(t,\cdot))dt = -\int_{0}^{\infty}(u^{(\mu)}(t,\cdot), \frac{\partial \phi}{\partial x_{j}}(t,\cdot))dt,\]
so that we can have
\[\lim_{\mu \to \infty}\int_{0}^{\infty}(\frac{\partial u^{(\mu)}}{\partial x_{j}}(t,\cdot),\phi(t,\cdot))dt = -\int_{0}^{\infty}(u(t,\cdot), \frac{\partial \phi}{\partial x_{j}}(t,\cdot))dt.\]
Since $u \in L^{\infty}([0,\infty);H^{1}({\bf R}^{n}))$, it follows from the integration by parts again one can get
\[\lim_{\mu \to \infty}\int_{0}^{\infty}(\frac{\partial u^{(\mu)}}{\partial x_{j}}(t,\cdot),\phi(t,\cdot))dt = \int_{0}^{\infty}(\frac{\partial u}{\partial x_{j}}(t,\cdot),\phi(t,\cdot))dt.\]
By density of $L^{1}(0,\infty; C_{0}^{\infty}({\bf R}^{n}))$ into $L^{1}(0,\infty;L^{2}({\bf R}^{n}))$, for each $j = 1,2,\cdots, n$ it is true that
\begin{equation}
\frac{\partial u^{(\mu)}}{\partial x_{j}} \to \frac{\partial u}{\partial x_{j}} \quad (\textstyle{weakly*}) \quad \textstyle{in} \quad L^{\infty}(0,\infty;L^{2}({\bf R}^{n})) \quad (\mu \to \infty).
\end{equation}
Now, let us finalize the proof of Theorem 1.1. First of all, we prepare the following basic lemma.
\begin{lem}\,Assume that a sequence $\{v_{m}\} \subset L^{\infty}(0,\infty;L^{2}({\bf R}^{n})$ satisfies
\[v_{m} \to v \quad (\textstyle{weakly*}) \quad \textstyle{in} \quad L^{\infty}(0,\infty;L^{2}({\bf R}^{n})) \quad (m \to \infty),\]
for some $v \in L^{\infty}(0,\infty;L^{2}({\bf R}^{n}))$, and
\[\Vert v_{m}(t,\cdot)\Vert \leq C(1+t)^{-\gamma}\]
with some constants $C > 0$, and $\gamma > 0$. Then, it is also true that
\[\Vert v(t,\cdot)\Vert \leq C(1+t)^{-\gamma}.\]
\end{lem}
{\it Proof.}\,Take $\psi \in C_{0}([0,\infty);L^{2}({\bf R}^{n}))$, and set $w_{m}(t,x) := (1+t)^{\gamma}v_{m}(t,x)$. Then, since\\
$(1+t)^{\gamma}\psi(t,x) \in L^{1}(0,\infty;L^{2}({\bf R}^{n}))$, by assumption it follows that
\[\lim_{m \to \infty}\int_{0}^{\infty}\int_{{\bf R}^{n}}w_{m}(t,x)\psi(t,x)dxdt = \lim_{m \to \infty}\int_{0}^{\infty}\int_{{\bf R}^{n}}v_{m}(t,x)\left((1+t)^{\gamma}\psi(t,x)\right)dxdt\]
\[= \int_{0}^{\infty}\int_{{\bf R}^{n}}v(t,x)\left((1+t)^{\gamma}\psi(t,x)\right)dxdt = \int_{0}^{\infty}\int_{{\bf R}^{n}}\left((1+t)^{\gamma}v(t,x)\right)\psi(t,x)dxdt.\]
Because of the density of $C_{0}([0,\infty);L^{2}({\bf R}^{n}))$ into $L^{1}(0,\infty;L^{2}({\bf R}^{n}))$ again (cf. Miyadera \cite[Theorem 15.3]{Miya}), we have
\[w_{m} = (1+t)^{\gamma}v_{m} \to (1+t)^{\gamma}v\quad (\textstyle{weakly*}) \quad \textstyle{in} \quad L^{\infty}(0,\infty;L^{2}({\bf R}^{n})) \quad (m \to \infty),\]
so that it follows that
\[\Vert (1+t)^{\gamma}v(t,\cdot)\Vert \leq \liminf_{m \to \infty}\Vert (1+t)^{\gamma}v_{m}\Vert_{L^{\infty}(0,\infty;L^{2}({\bf R}^{n}))} \leq C\]
for each $t \geq 0$. This implies the desired statement.
$
\Box$
{\it Proof of Theorem 1.1 completed.}\, It follows from Lemma 2.3 that
\[
\Vert\frac{\partial u^{(\mu)}(t,\cdot)}{\partial x_{j}}\Vert \leq \sqrt{2}I_{2}(1+t)^{-1}, \quad \Vert u^{(\mu)}_{t}(t,\cdot)\Vert \leq \sqrt{2}I_{2}(1+t)^{-1}.
\]
Thus, it follows from (2.29) and (2.22) and Lemma 2.4 that
\begin{equation}
\Vert\frac{\partial u(t,\cdot)}{\partial x_{j}}\Vert \leq \sqrt{2}I_{2}(1+t)^{-1},
\end{equation}
\begin{equation}
\Vert u_{t}(t,\cdot)\Vert \leq \sqrt{2}I_{2}(1+t)^{-1}.
\end{equation}
Furthermore, from Lemma 2.3 one also has
\[\Vert u^{(\mu)}(t,\cdot)\Vert \leq I_{3}(1+t)^{-1/2}\quad (t \geq 0).\]
Therefore, (2.23) and Lemma 2.4 imply
\begin{equation}
\Vert u(t,\cdot)\Vert \leq I_{3}(1+t)^{-1/2},\quad (t \geq 0).
\end{equation}
(2.30)-(2.32) imply the desired estimates (2.28) of Theorem 1.1. Note that all quantities $I_{j}$ ($j = 0,1,2,3$) can be absorbed into $CI_{00}$ defined in Theorem 1.1 with some constant $C > 0$. In this connection, we should remark the following relation because of the Cauchy-Schwarz inequality:
\[\int_{{\bf R}^{n}}a(x)\vert u_{0}(x)\vert^{2}dx \leq \sqrt{\int_{{\bf R}^{n}}a(x)^{2}\vert u_{0}(x)\vert^{2}dx}\sqrt{\int_{{\bf R}^{n}}\vert u_{0}(x)\vert^{2}dx}.\]
A uniqueness argument is standard, so we shall omit its detail.
$
\Box$
\section{Remark on the low dimensional case.}
Let us give some remarks on the low dimensional case and several open problems.\\
{\rm (I)}\,When one gets the result for $n = 2$, we may use (2) of Proposition 2.1 with $\theta = 1$ and $\gamma \in (0,1]$ to get the similar estimate to (2.9), which is an essential part of proof. But, in this case we have to assume a stronger assumption such that
\begin{equation}
\int_{{\bf R}^{2}}(a(x)u_{0}(x) + u_{1}(x))dx = 0.
\end{equation}
Unfortunately, this assumption (3.1) is not hereditary to the approximate solution, i.e.,
$$\displaystyle{\int_{{\bf R}^{2}}}(a_{m}(x)u_{0}(x) + u_{1}(x))dx = 0$$
is not necessarily true. So, it is completely open to get the similar result in the low dimensional case (i.e., $n = 1,2$). \\
{\rm (II)}\,One can not treat the associated nonlinear equation such that
\begin{equation}
u_{tt}(t,x) -{\cal D}elta u(t,x) + a(x)u_{t}(t,x) = f(u(t,x)).\ \ \
\end{equation}
This is because one encounters lack of some compactness argument when we want to get the limit such that (see Lions \cite[(1.42)]{L})
\[f(u^{(m)}) \to f(u)\quad \textstyle{weakly*}\quad \textstyle{in}\quad L^{\infty}([0,\infty);X)\]
(as $m \to \infty$), where $X$ is a Banach space.\\
In the unbounded coefficient case there are still many obstacles to be overcome. In this sense, the damped wave equation with unbounded coefficient and non-compactly supported initial data seem to be quite difficult to be treated.\\
{\rm (III)}\,We have another idea to deal with the two dimensional case, that is, the idea to rely on the well-known inequality:
\begin{equation}
\int_{{\bf R}^{2}}\frac{\vert\hat{f}(\xi)\vert^{2}}{\vert\xi\vert^{2}}d\xi \leq C\Vert f\Vert_{{\cal H}^{1}}^{2},
\end{equation}
in place of (2) of Proposition 2.1, where ${\cal H}^{1}({\bf R}^{2})$ is the so called Hardy space. Unfortunately, in this case we also encounter the same problem as the heredity of the property
$a(\cdot)u_{0} \in {\cal H}^{1}({\bf R}^{2})$ to $a_{m}(\cdot)u_{0} \in {\cal H}^{1}({\bf R}^{2})$.
\section{Appendix.}
In this section, let us check the density of $L^{1}(0,\infty; C_{0}^{\infty}({\bf R}^{n}))$ into $L^{1}(0,\infty;L^{2}({\bf R}^{n}))$.\\
Take $f \in L^{1}(0,\infty;L^{2}({\bf R}^{n}))$ and for each $L > 0$ set
\[ f_{L}(t,x) = \left\{
\begin{array}{ll}
\displaystyle{f(t,x)}&
\qquad (\vert x\vert \leq L) \\[0.2cm]
\displaystyle{0}& \qquad (\vert x\vert > L).
\end{array} \right. \]
Next set
\[ h(t) = \left\{
\begin{array}{ll}
\displaystyle{e^{-\frac{1}{1-t^{2}}}}&
\qquad (\vert t\vert < 1) \\[0.2cm]
\displaystyle{0}& \qquad (\vert t\vert \geq 1),
\end{array} \right. \]
and $\rho(x) := h(\vert x\vert^{2})\left(\displaystyle{\int_{{\bf R}^{n}}}h(\vert x\vert^{2})dx\right)^{-1}$, and for $\varepsilon > 0$ define
\[\rho_{\varepsilon}(x) := \frac{1}{\varepsilon^{n}}\rho(\frac{x}{\varepsilon}).\]
Under these preparations, we define an approximating sequence of the original function $f$ by
\[f_{\varepsilon,L}(t,x) := (f_{L}(t,\cdot)*\rho_{\varepsilon})(x).\]
Then it is standard to check that $f_{\varepsilon,L}(t,\cdot) \in C_{0}^{\infty}({\bf R}^{n})$ for each $t \geq 0$ and $L > 0$, and
\[\Vert f_{\varepsilon,L}(t,\cdot)\Vert \leq \Vert f_{L}(t,\cdot)\Vert \leq \Vert f(t,\cdot)\Vert,\]
so that $f_{\varepsilon,L} \in L^{1}(0,\infty;L^{2}({\bf R}^{n}))$ for each $L > 0$ and $\varepsilon > 0$. Furthermore, for each $L > 0$ and $t \geq 0$ it is known that
\[\Vert f_{\varepsilon,L}(t,\cdot) - f_{L}(t,\cdot)\Vert \to 0\quad (\varepsilon \to 0).\]
Now, for an arbitrary fixed $\eta > 0$, choose $M > 0$ so large such that
\begin{equation}
\Vert f_{M} - f\Vert_{X} < \frac{\eta}{2},
\end{equation}
where $X := L^{1}(0,\infty;L^{2}({\bf R}^{n}))$, and $\Vert\cdot\Vert_{X}$ is the standard norm of $X$:
\[\Vert g\Vert_{X} := \int_{0}^{\infty}\Vert g(t,\cdot)\Vert dt.\]
Next, for such a fixed $M > 0$, by taking $\varepsilon > 0$ sufficiently small, if one applies the Lebesgue convergence theorem, one can get
\begin{equation}
\Vert f_{\varepsilon,M}-f_{M}\Vert_{X} < \frac{\eta}{2}.
\end{equation}
Therefore, it follows from (4.1) and (4.2) with such large $M > 0$ and small $\varepsilon > 0$ that
\[\Vert f_{\varepsilon,M} - f\Vert_{X} \leq \Vert f_{\varepsilon,M} - f_{M}\Vert_{X} + \Vert f_{M} - f\Vert_{X} < \frac{\eta}{2} + \frac{\eta}{2} = \eta,\]
which implies the density of $L^{1}(0,\infty; C_{0}^{\infty}({\bf R}^{n}))$ into $L^{1}(0,\infty;L^{2}({\bf R}^{n}))$.
$
\Box$
\par
\noindent{\em Acknowledgement.}
The work of the first author (R. IKEHATA) was supported in part by Grant-in-Aid for Scientific Research 15K04958 of JSPS.
The work of the second author (H. TAKEDA) was supported in part by Grant-in-Aid for Young Scientists (B)15K17581 of JSPS.
\end{document} | math | 37,520 |
\begin{document}
\title{Stable ODE-type blowup for some quasilinear wave equations
with derivative-quadratic nonlinearities
}
\author[JS]{Jared Speck$^{* \dagger}$}
\thanks{$^{\dagger}$JS gratefully acknowledges support from NSF grant \# DMS-1162211,
from NSF CAREER grant \# DMS-1454419,
from a Sloan Research Fellowship provided by the Alfred P. Sloan foundation,
and from a Solomon Buchsbaum grant administered by the Massachusetts Institute of Technology.
}
\thanks{$^{*}$Massachusetts Institute of Technology, Cambridge, MA, USA.
\texttt{[email protected]}}
\begin{abstract}
We prove a constructive stable ODE-type blowup result for
open sets of solutions to a family of quasilinear wave equations
in three spatial dimensions
featuring a Riccati-type derivative-quadratic semilinear term.
The singularity is more severe than a shock
in that the solution itself blows up like the log
of the distance to the blowup-time. We assume that the quasilinear terms
satisfy certain structural assumptions, which in particular ensure
that the ``elliptic part'' of the wave operator vanishes precisely at the singular points.
The initial data are compactly supported and
can be small or large in $L^{\infty}$, but
the spatial derivatives must initially satisfy a nonlinear
smallness condition compared to the time derivative.
The first main idea of the proof is to construct a quasilinear integrating factor,
which allows us to reformulate the wave equation as a
system whose solutions remain regular,
all the way up to the singularity. This is equivalent to constructing
quasilinear vectorfields adapted to the nonlinear flow.
The second main idea is to exploit some
crucial monotonic terms in various estimates, especially the energy estimates,
that feature the integrating factor.
The availability of the monotonicity is tied to our assumptions on the data
and on the structure of the quasilinear terms.
The third main idea is to propagate the relative smallness of
the spatial derivatives all the way up to the singularity
so that the solution behaves, in many ways, like
an ODE solution. As a corollary of our main results, we show that there are quasilinear wave equations
that exhibit two distinct kinds of blowup: the formation of shocks for one non-trivial set of data,
and ODE-type blowup for another non-trivial set.
\noindent \textbf{Keywords}: blowup, ODE blowup, shocks, singularities, stable blowup
\noindent \textbf{Mathematics Subject Classification (2010)} Primary: 35L67; Secondary: 35L05, 35L45, 35L52, 35L72
\end{abstract}
\maketitle
\centerline{\today}
\tableofcontents
\setcounter{tocdepth}{1}
\section{Introduction}
\label{S:INTRO}
A fundamental issue surrounding the study of nonlinear hyperbolic PDEs
is that singularities can form in finite time, starting from smooth initial data.
For a given singularity-forming solution, perhaps the most basic question one can ask is
whether or not it is stable under perturbations
of its initial data. Our main result provides an affirmative answer to this question
for some solutions to a class of quasilinear wave equations. Specifically, in three spatial dimensions,
we provide a sharp, constructive proof of stable ODE-type blowup for solutions corresponding to an \underline{open}
set (in a suitable Sobolev topology in which there are no radial weights in the norms)
of initial data for a class of quasilinear wave equations
that are well-modeled by
\begin{align} \label{E:FIRSTMODELWAVE}
-\partial_t^2 \Phi + \frac{1}{1 + \partial_t \Phi} \Delta \Phi = - (\partial_t \Phi)^2.
\end{align}
For the solutions from our main results, $\Phi$ itself blows up.
This is a much more drastic singularity compared to the case of the formation of shocks,
which for equations of type \eqref{E:FIRSTMODELWAVE} would correspond to the blowup
of $\partial^2 \Phi$ with $\Phi$ and $\partial \Phi$ remaining bounded;
see Subsect.\ \ref{SS:PRIORBREAKDOWNRESULTS} for further discussion.
As a corollary of our main results, we show
(see Subsect.\ \ref{SS:DIFFERENTKINDSOFSINGULARITYFORMATION})
that there are quasilinear wave equations
that exhibit \emph{two distinct kinds of blowup}:
ODE-type blowup for one non-trivial (but not necessarily open) set of initial data,
and the formation of a shock for a different non-trivial set of data.
We view this as a parable highlighting two key phenomena
that would have to be accounted for in any sufficiently broad theory of
singularity formation in solutions to quasilinear wave equations;
i.e., in principle, a quasilinear\footnote{For certain \emph{semilinear} wave equations,
it is well-known that
different kinds of blowup can occur: blowup of the $L^{\infty}$ norm of the solution itself (known as Type I blowup)
and a different kind of blowup in which the solution remains bounded in an appropriate Sobolev norm
(known as Type II blowup); see Subsect.\ \ref{SS:PRIORBREAKDOWNRESULTS}.}
wave equation can admit radically
different types of singularity-forming solutions.\footnote{These phenomena can also be exhibited in the
much simpler setting of quasilinear transport equations. For example, the inhomogeneous Burgers equation
$\partial_t \Psi + \Psi \partial_x \Psi = \Psi^2$ admits the $T$-parameterized family of
spatially homogeneous singularity-forming solutions $\Psi_{(ODE);T} := (T-t)^{-1}$ as well as solutions that form shocks,
i.e., $|\partial_x \Psi|$ blows up but $|\Psi|$ remains bounded.}
It is only for concreteness that we restrict our attention to three spatial dimensions;
our approach can be applied to any number of spatial dimensions, with only slight modifications needed.
See Subsects.\ \ref{SS:STATEMENTOFWAVEEQUATION} and \ref{SS:WEIGHTASSUMPTIONS}
for a precise description of the class of equations that we treat,
Subsubsect.\ \ref{SSS:SUMMARYOFRESULTS} for a summary of our main results,
and Sect.\ \ref{S:MAINTHM} for the detailed statement of our main theorem.
Obtaining a sharp description of the blowup is
particularly important if one aims to weakly continue the solution
past the singularity, as is sometimes possible; one expects to need sharp information
in order to even properly set up the problem of weakly continuing.\footnote{The most significant weak continuation result in more than one spatial dimension
is Christodoulou's recent solution \cite{dC2017} of the restricted
shock development
problem in compressible fluid mechanics, which, roughly speaking,
is a local well-posedness result for weak solutions and their corresponding
hypersurfaces of discontinuity, starting from the first shock, whose formation
from smooth initial conditions was described in detail in his breakthrough work
for relativistic fluids \cite{dC2007} and in his follow-up work with Miao \cite{dCsM2014}
on non-relativistic compressible fluids.
The term ``restricted'' means that the jump in entropy across the shock hypersurface was ignored.}
Our work shows that a standard type of weak continuation
is not possible for the solutions that we study,
since $\Phi$ itself blows up.
The precise algebraic details of the weight
$
\frac{1}{1 + \partial_t \Phi}
$
in front of the Laplacian term in equation \eqref{E:FIRSTMODELWAVE}
are not important for our proof.
What is important is that the weight decays at an appropriate rate
as $\partial_t \Phi \to \infty$, that is, as the singularity forms;
see Subsect.\ \ref{SS:WEIGHTASSUMPTIONS} for our assumptions on the weight.
As we will explain, this decay yields
a friction-type spacetime integral that is important for closing the energy estimates,
and it also helps us to prove that spatial derivative terms remain small relative to the time derivative terms,
up to the singularity. The problem of providing a sharp description of blowup for solutions to
derivative-quadratic semilinear wave equations, such as
$-\partial_t^2 \Phi + \Delta \Phi = - (\partial_t \Phi)^2$,
remains open, even though John \cite{fJ1981} showed, via proof by contradiction,
that all non-trivial, smooth, compactly supported solutions to the equation
$-\partial_t^2 \Phi + \Delta \Phi = - (\partial_t \Phi)^2$
in three spatial dimensions blow up in finite time.
Our results show, in part due to the weight in front of the Laplacian,
that the spatial-derivative-involving nonlinearities in equation \eqref{E:FIRSTMODELWAVE} (and the other equations that we study)
exhibit a subcritical\footnote{In contrast,
for the semilinear equation $-\partial_t^2 \Phi + \Delta \Phi = - (\partial_t \Phi)^2$,
our approach suggests, but does not prove, that the blowup-rate for the
Laplacian term $\Delta \Phi$ might be critical with respect to the expected blowup-rate
for the other two terms in the equation, i.e., that all terms might blow up at the same rate.}
blowup-rate relative to the pure time derivative terms.
However, as we explain below, this
\emph{subcritical behavior does not seem detectable relative to the standard partial derivatives $\partial_{\alpha}$};
to detect the behavior, we will use a combination of ``quasilinear vectorfield derivatives'' $\mathcal{I} \partial_{\alpha}$
and standard derivatives $\partial_{\alpha}$,
where $\mathcal{I}$ is a ``quasilinear integrating factor'' that we describe below. In fact, we will show that
$\mathcal{I} \partial_{\alpha} \Phi$ remains bounded up to the singularity and that the singularity formation
coincides with the vanishing of $\mathcal{I}$.
In total, our approach allows us to treat the equations under study
as quasilinear perturbations of the Riccati ODE
$\frac{d^2}{dt^2} \Phi = (\frac{d}{dt} \Phi)^2$.
By ``perturbation of the Riccati ODE'',
we mean in particular that the singularity formation
is similar to the one that occurs in the following $T$-parameterized family of ODE solutions to \eqref{E:FIRSTMODELWAVE}:
\begin{align} \label{E:ODEBLOWWINGUPSOLUTION}
\Phi_{(ODE);T}(t) := \ln \left((T-t)^{-1}\right),
\end{align}
where $T \in \mathbb{R}$ is the blowup-time.
Our methods are tailored to the quadratic term on RHS~\eqref{E:FIRSTMODELWAVE}
in that they do not apply, at least in their current form, to
semilinear terms of type $(\partial_t \Phi)^p$ for $p \neq 2$.
However, derivative-quadratic terms are of particular interest in view of the fact
that they commonly arise in nonlinear field theories
(though the derivative-quadratic terms in such theories
are often not Riccati-type, like the one featured on RHS~\eqref{E:FIRSTMODELWAVE}).
There are many results on stable breakdown for wave equations, some of which
we review in Subsect.\ \ref{SS:PRIORBREAKDOWNRESULTS}.
The ``theory'' of stable breakdown is quite fragmented
in that the techniques that have been employed
vary wildly between different classes of equations.
In particular, the techniques that have been developed
do not seem to apply to the equations under study here.
This will become clear after we describe the main ideas
of our proof (see Subsect.\ \ref{SS:IDEASBEHINDPROOF})
and review prior works on stable
breakdown. Although ODE-type blowup is arguably the simplest blowup scenario,
there do not seem to be any prior constructive stable blowup results
of this type for scalar wave equations with derivative-quadratic nonlinearities,
in any number of spatial dimensions.
We mention, however, that in \cite{iRjS2014b},
we proved, using rather different techniques specialized to Einstein's equations,
a singularity formation result for Einstein's equations
that can be interpreted as a stable ODE-type blowup result
for the first derivatives\footnote{Relative to a geometrically defined coordinate system,
the second fundamental form of the metric blows up, though the metric
components do not; this can be viewed as the blowup of the first derivatives of the metric.}
of a solution
to a quasilinear system with derivative-quadratic nonlinearities.
\subsection{Paper outline}
\label{SS:PAPEROUTLINE}
\begin{itemize}
\item In the remainder of Sect.\ \ref{S:INTRO},
we summarize our results, outline their proofs,
place our work in context by discussing prior works,
discuss (see Subsect.\ \ref{SS:DIFFERENTKINDSOFSINGULARITYFORMATION})
a corollary (which we described just below equation \eqref{E:FIRSTMODELWAVE}) of our main results,
and summarize our notation.
\item In Sect.\ \ref{S:MATHEMATICALSETUP},
we define the quantities play a role in our analysis
and derive various evolution equations.
\item In Sect.\ \ref{S:DATAANDBOOTSTRAP}, we state our assumptions on the
initial data and state bootstrap assumptions that are useful for studying the solution.
\item In Sect.\ \ref{S:ENERGY}, we derive energy identities.
\item In Sect.\ \ref{S:ESTIMATES}, which is the main section of the article,
we derive a priori estimates that in particular yield strict improvements
of the bootstrap assumptions.
\item In Sect.\ \ref{S:WELLPOSEDNESS}, we state a standard local well-posedness result
and continuation criteria for the equations under study.
\item In Sect.\ \ref{S:MAINTHM}, we prove the main theorem.
\end{itemize}
\subsection{The class of wave equations under study}
\label{SS:STATEMENTOFWAVEEQUATION}
Our main theorem concerns solutions to the Cauchy problem for
quasilinear wave equations in three spatial dimensions
of the following form:
\begin{subequations}
\begin{align} \label{E:WAVE}
-
\partial_t^2 \Phi
+
\mathscr{W}(\partial_t \Phi)
\Delta \Phi
& = - (\partial_t \Phi)^2,
\\
(\partial_t \Phi|_{\Sigma_0},\partial_1 \Phi|_{\Sigma_0}, \partial_2 \Phi|_{\Sigma_0}, \partial_3 \Phi|_{\Sigma_0})
& = (\mathring{\Psi}_0,\mathring{\Psi}_1,\mathring{\Psi}_2,\mathring{\Psi}_3),
\label{E:DATAWAVE}
\end{align}
\end{subequations}
where throughout, $\Sigma_t$ denotes the hypersurface of constant time $t$.
Our use of the notation ``$\mathring{\Psi}_{\alpha}$'' for the data functions
is tied to our use of the renormalized solutions variables $\Psi_{\alpha}$ that
we will use in studying solutions; see Def.~\ref{D:RENORMALIZEDSOLUTION}.
\begin{remark}[\textbf{Viewing} \eqref{E:WAVE} \textbf{as an equation in} $\partial \Phi$]
\label{R:NOPHIINWAVEEQUATION}
Since $\Phi$ itself is not featured in equation \eqref{E:WAVE}
(only its derivatives appear),
we only need to prescribe the derivatives of $\Phi$ along $\Sigma_0$
in order to solve for $\lbrace \partial_{\alpha} \Phi \rbrace_{\alpha = 0,1,2,3}$.
This is relevant in that we do not bother to derive estimates for
$\Phi$ itself (see, however, Remark~\ref{R:BLOWUPOFPHI}).
\end{remark}
In \eqref{E:WAVE},
$\Delta := \sum_{a=1}^3 \partial_a^2$ is the standard Euclidean Laplacian on $\mathbb{R}^3$
and $\mathscr{W} = \mathscr{W}(\partial_t \Phi)$ is a nonlinear ``weight function''
verifying certain technical conditions stated below,
specifically \eqref{E:WEIGHTISPOSITIVE}-\eqref{E:WEIGHTVSWEIGHTDERIVATIVECOMPARISON}.
Prototypical examples of
weights verifying \eqref{E:WEIGHTISPOSITIVE}-\eqref{E:WEIGHTVSWEIGHTDERIVATIVECOMPARISON}
are the functions
\begin{align} \label{E:POWERLAWWEIGHT}
\mathscr{W}(y)
& = \frac{1}{1 + y^M}
&
\mbox{or }
\mathscr{W}(y)
& = \frac{1}{(1 + y)^M},
\end{align}
where $M \geq 1$ is an integer,
and the function
\begin{align} \label{E:EXPWEIGHT}
\mathscr{W}(y)
& = \exp(-y).
\end{align}
\subsection{Rough summary of the results and discussion of the proof}
\label{SS:IDEASBEHINDPROOF}
\subsubsection{Rough summary of the results}
\label{SSS:SUMMARYOFRESULTS}
We now briefly summarize the main results;
see Theorem~\ref{T:STABILITYOFODEBLOWUP} for precise statements.
\begin{theorem}[\textbf{Stable ODE-type blowup} (rough version)]
\label{T:ROUGHMAINTHM}
Under suitable assumptions (stated in Subsect.\ \ref{SS:WEIGHTASSUMPTIONS})
on the weight $\mathscr{W}(\partial_t \Phi)$,
there exists an open set of compactly supported initial data for equation \eqref{E:WAVE},
with $\mathring{\Psi}_{\alpha} \in H^5(\mathbb{R}^3)$,
such that the solution blows up in finite time in a manner similar
to the solutions $\Phi_{(ODE);T}$ from \eqref{E:ODEBLOWWINGUPSOLUTION}.
In particular, there exists a time $0 < T_{(Lifespan)} < \infty$
such that\footnote{More precisely, one can conclude that $\| \Phi\|_{L^{\infty}(\Sigma_t)}$ blows up at
$t = T_{(Lifespan)}$ if initial data for $\Phi$ itself is prescribed; Remark~\ref{R:NOPHIINWAVEEQUATION}.}
$
\| \partial_t \Phi \|_{L^{\infty}(\Sigma_t)}
$
and
$\| \Phi\|_{L^{\infty}(\Sigma_t)}$ blow up as
$t \uparrow T_{(Lifespan)}$.
The data functions $\lbrace \mathring{\Psi}_{\alpha} \rbrace_{\alpha=0,1,2,3}$
are allowed to be large or small as measured
by a Sobolev norm without radial weights, but
$\lbrace \mathring{\Psi}_a \rbrace_{a=1,2,3}$,
$\nabla \mathring{\Psi}_0$,
and their spatial derivatives up to top order must satisfy
a nonlinear smallness condition
relative to $\max_{\Sigma_0} [\mathring{\Psi_0}]_+$.
Moreover, let the integrating factor $\mathcal{I}$ be the solution to
\begin{align} \label{E:INTROINTEGRATINFACTORODEANDIC}
\partial_t \mathcal{I}
& =
- \mathcal{I} \partial_t \Phi,
&& \mathcal{I}|_{\Sigma_0} = 1.
\end{align}
Then $\mathcal{I}$, the variables
\begin{align} \label{E:RENORMALIZEDRICCATIBLOWUPVARIABLES}
\Psi_{\alpha} := \mathcal{I} \partial_{\alpha} \Phi,
\end{align}
and their partial derivatives with respect to the Cartesian coordinates
\textbf{remain regular all the way up to time $T_{(Lifespan)}$},
except possibly at the top derivative level due to
the vanishing of the weight $\mathscr{W}(\partial_t \Phi)$
(which appears in the energy estimates)
as $\partial_t \Phi \uparrow \infty$.
\end{theorem}
\begin{remark}[\textbf{Maximal development}]
\label{R:MAXIMALDEVELOPMENT}
We anticipate that the sharp results of Theorem~\ref{T:ROUGHMAINTHM} should be useful
for obtaining detailed information about the solution not just up to the first singular time,
but also up to the boundary
of the maximal development.\footnote{The maximal development of the data is, roughly,
the largest possible classical solution that is uniquely determined by the data.
Readers may consult \cites{jSb2016,wW2013} for further discussion.
\label{FN:MAXIMALDEVELOPMENT}}
In the context of shock formation for fluids, Christodoulou \cite{dC2007}*{Chapter 15}
used similar sharp estimates to follow the solution up to boundary.
Broadly similar results were obtained by Merle--Zaag in \cite{fMzH2012a},
in which, in the case of one spatial dimension,
they gave a sharp description of the
boundary of the maximal development for \emph{any} singularity-forming solution
to the semilinear focusing wave equation $-\partial_t^2 \Psi + \partial_x^2 \Psi = - |\Psi|^{p-1} \Psi$
with $p > 1$ and showed in particular that characteristic points on the boundary are isolated.
\end{remark}
\begin{remark}[\textbf{The blowup of} $\Phi$]
\label{R:BLOWUPOFPHI}
We now make some remarks on the blowup of $\Phi$ itself since,
as we highlighted in Remark~\ref{R:NOPHIINWAVEEQUATION}, one does
not need to prescribe the initial data for $\Phi$ itself
(and since in the rest of the paper we do not assume that initial data for $\Phi$ itself are prescribed).
If one does prescribe its initial data,
then the results of Theorem~\ref{T:STABILITYOFODEBLOWUP} can easily be used to show
that $\Phi$ itself blows up at time $T_{(Lifespan)}$ (such a result is not stated in Theorem~\ref{T:STABILITYOFODEBLOWUP}).
This is philosophically important in that it dashes
any hope of weakly continuing the solution past the singularity, at least in a standard sense.
To deduce the blowup for $\Phi$, one can first use
\eqref{E:INTROINTEGRATINFACTORODEANDIC} and the fundamental theorem of calculus
to deduce that
$\ln \mathcal{I}(t,\underline{x}) + \Phi(t,\underline{x}) = \Phi(0,\underline{x})$,
where $\Phi(0,\cdot)$ is a regular function that by assumption verifies
$\| \Phi(0,\cdot) \|_{L^{\infty}} < \infty$.
Since the singularity formation for $\partial_t \Phi$ yielded by Theorem~\ref{T:STABILITYOFODEBLOWUP}
coincides with the vanishing of $\mathcal{I}$ for the first time at $t = T_{(Lifespan)}$,
it follows that $\lim_{t \uparrow T_{(Lifespan)}} \sup_{s \in [0,t)} \| \Phi\|_{L^{\infty}(\Sigma_s)} = \infty$,
as is claimed in Theorem~\ref{T:ROUGHMAINTHM}.
\end{remark}
\subsubsection{The main ideas behind the proof}
\label{SSS:DISCUSSIONOFTHEPROOF}
The initial data that we consider are such that the spatial derivatives
of $\Phi$ up to top order are initially small relative to $\partial_t \Phi$.
We also assume that the spatial derivatives of $\partial_t \Phi$ up to top order are initially small.
The smallness assumptions that we need to close the proof are nonlinear in nature,\footnote{In particular, our smallness assumptions
on the data \eqref{E:DATAWAVE}
are \emph{not} generally invariant under rescalings of the form
$(\mathring{\Psi}_0,\mathring{\Psi}_1,\mathring{\Psi}_2,\mathring{\Psi}_3)
\rightarrow
\uplambda^{-1}
(\mathring{\Psi}_0,\mathring{\Psi}_1,\mathring{\Psi}_2,\mathring{\Psi}_3)
$ if $\uplambda$ is too large.}
for reasons described just below \eqref{E:CARICATURENERGYAPRIORI};
see Subsect.\ \ref{SS:SMALLNESSASSUMPTIONS}
for our precise smallness assumptions
and Subsect.\ \ref{SS:EXISTENCEOFDATA} for a simple proof that such data exist.
In our analysis, we propagate certain aspects of this smallness all the way up to the singularity.
As we mentioned earlier, this allows us to effectively treat equation \eqref{E:WAVE} as a perturbation of
the Riccati ODE $\frac{d^2}{dt^2} \Phi = (\frac{d}{dt} \Phi)^2$.
We again stress that the vanishing of the coefficient $\mathscr{W}(\partial_t \Phi)$ of the Laplacian term in \eqref{E:WAVE}
as the singularity forms is important
for our estimates, in particular for showing that spatial derivative terms
remain relatively small.
A key point is that it does not seem possible to follow the solution all the way to the singularity
by studying the wave equation in the form
\eqref{E:WAVE}. To caricature the situation, let us pretend that the singularity occurs at $t = 1$.
Our proof shows, roughly, that $\partial^k \Phi$ blows up like $(1-t)^{-k}$,
where $\partial^k$ denotes $k^{th}$-order Cartesian coordinate partial derivatives.
This means, in particular, that commuting equation \eqref{E:WAVE} with more and more spatial derivatives makes the
singularity strength of the nonlinear terms worse and worse,
which is a serious obstacle to closing nonlinear estimates.
For this reason, as our statement of Theorem~\ref{T:ROUGHMAINTHM} already makes clear,
our proof is fundamentally based on the solution to \eqref{E:INTROINTEGRATINFACTORODEANDIC},
that is, the integrating factor
$\mathcal{I}$ solving the transport equation
$\partial_t \mathcal{I} = - \mathcal{I} \partial_t \Phi$
with initial conditions $\mathcal{I}|_{\Sigma_0} = 1$.
Note that the finite-time blowup $\partial_t \Phi \uparrow \infty$
would follow from the finite-time vanishing of $\mathcal{I}$.
Indeed, to show that a singularity forms,
\emph{we will show that $\mathcal{I}$ vanishes in finite time}.
Using $\mathcal{I}$, we are able to transform the wave equation into
a ``regularized'' system, equivalent to equation \eqref{E:WAVE} up to the singularity,
that we analyze to show that the weighted derivatives
$\lbrace \Psi_{\alpha} := \mathcal{I} \partial_{\alpha} \Phi \rbrace_{\alpha=0,1,2,3}$,
$\mathcal{I}$, and their \emph{Cartesian} spatial partial derivatives remain
bounded, in appropriate norms (some with $\mathcal{I}$ weights),
all the way up to the singularity.
In particular, our proof relies on a combination of the derivatives
$\lbrace \mathcal{I} \partial_{\alpha} \rbrace_{\alpha=0,1,2,3}$ and
$\lbrace \partial_{\alpha} \rbrace_{\alpha=0,1,2,3}$, where the weighted derivatives
$ \mathcal{I} \partial_{\alpha}$ act first.
Here and throughout,
$\partial_0 = \partial_t$ and
$\lbrace \partial_i \rbrace_{i=1,2,3}$
are the standard Cartesian coordinate spatial partial derivatives.
In Prop.~\ref{P:RENORMALIZEDEVOLUTOINEQUATIONS}, we derive
the ``regularized'' system of equations verified by
$\lbrace \Psi_{\alpha} \rbrace_{\alpha=0,1,2,3}$.
Here we only note that the system is first-order hyperbolic
and that a \emph{seemingly dangerous factor of $\mathcal{I}^{-1}$ appears in the equations}
(recall that $\mathcal{I}$ vanishes at the singularity).
However, the factor $\mathcal{I}^{-1}$ is multiplied by the weight
$\mathscr{W} = \mathscr{W}(\mathcal{I}^{-1} \Psi_0)$
from equation \eqref{E:WAVE},
and due to our assumptions on $\mathscr{W}$,
we are able to show that the product
$\mathcal{I}^{-1} \mathscr{W}(\mathcal{I}^{-1} \Psi_0)$
remains uniformly bounded up to the singularity.
Moreover, the spatial derivatives of the product
$\mathcal{I}^{-1} \mathscr{W}(\mathcal{I}^{-1} \Psi_0)$
also are controllable up to the singularity;
it is in this sense that the equations verified by $\lbrace \Psi_{\alpha} \rbrace_{\alpha=0,1,2,3}$
can be viewed as a ``regularizing'' of the original problem.
The proof (see Lemma~\ref{L:ESTIMATESINVOLVINGWEIGHT}) of these bounds for the product $\mathcal{I}^{-1} \mathscr{W}(\mathcal{I}^{-1} \Psi_0)$
constitutes the most technical analysis of the article
and is based on separately treating regions
where $\mathcal{I}$ is large and $\mathcal{I}$ is small.
To prove that $\partial_t \Phi$ blows up,
we derive, in an appropriate \emph{localized} region of spacetime,
a pointwise bound for $\Psi_0$ of the form $\Psi_0 \gtrsim 1$.
In view of the evolution equation for $\mathcal{I}$, we see that
such a bound is strong enough to drive $\mathcal{I}$ to $0$ in finite time.
To prove that $\Psi_0 \gtrsim 1$, we of course rely on the size assumptions described
in the first paragraph of this subsusbsection,
which in particular include the assumption that $\Psi_0|_{\Sigma_0} \gtrsim 1$ (in a localized region).
If we caricature the situation by assuming the estimate\footnote{Here we use the notation ``$A \sim B$''
to imprecisely indicated that $A$ is well-approximated by $B$.}
$\Psi_0 \sim \delta$
for some $\delta > 0$, then it follows
from the evolution equation for $\mathcal{I}$ that
$\mathcal{I} \sim 1 - \delta t$,
$\partial_t \Phi \sim (1- \delta t)^{-1}$,
$\ln \mathcal{I} + \Phi \sim data$,
and thus $\Phi \sim \ln (1- \delta t)^{-1} + data$,
where $data$ is a smooth function determined by the initial data.
Note that $\ln (1- \delta t)^{-1}$ is one of the ODE blowup solutions \eqref{E:ODEBLOWWINGUPSOLUTION}.
It is in this sense that our results yield the stability of ODE-type blowup.
In reality, to close the proof sketch described above,
we must overcome several major difficulties. The first
is that the blowup time is not known in advance.
However, we are able to make a good approximate guess
for it, which is sufficient for closing a bootstrap argument.
We now describe what we mean by this.
The discussion in the previous paragraph suggests
that the (future) blowup time is approximately $\frac{1}{\mathring{A}_*}$,
where $\mathring{A}_* := \max_{\Sigma_0} [\mathring{\Psi}_0]_+$ (where $\mathring{A}_* > 0$ by assumption).
Indeed, if we set all spatial derivative terms equal to zero in equation \eqref{E:WAVE},
then the time of first blowup is precisely $\frac{1}{\mathring{A}_*}$.
Our main theorem confirms that for data with small spatial derivatives,
the blowup time is a small perturbation of $\frac{1}{\mathring{A}_*}$.
This is conceptually important in that it enables us to use a bootstrap argument
in which we only aim to control the solution for times less than
$\frac{2}{\mathring{A}_*}$; the factor of $2$ gives us a sufficient margin of error
to show that the singularity does form, and it allows us, in most cases,
to soak factors of $\frac{1}{\mathring{A}_*}$ into the constants ``$C$''
in our estimates; see Subsect.\ \ref{SS:CONVENTIONSFORCONSTANTS} for further
discussion on our conventions for constants.
The second and main difficulty that we encounter in our proof
is that we need to derive energy estimates
for $\lbrace \Psi_{\alpha} \rbrace_{\alpha=0,1,2,3}$
that hold up to the singularity and,
at the same time, to control the integrating factor $\mathcal{I}$;
most of our work in this paper is towards this goal.
Our energies are roughly of the following form,
where $V = (V_0,V_1,V_2,V_3)$ should be thought of as
some $k^{th}$ Cartesian spatial derivative of $(\Psi_0,\Psi_1,\Psi_2,\Psi_3)$:
\begin{align} \label{E:INTROBASICENERGY}
\mathbb{E}[V]
& = \mathbb{E}[V](t)
:= \int_{\Sigma_t}
\left\lbrace
V_0^2
+
\sum_{a=1}^3
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
V_a^2
\right\rbrace
\, d \underline{x}.
\end{align}
For the data under study, $\mathbb{E}[V](0)$ is small.
Since $\mathcal{I}$ is small near the singularity and $\Psi_0$ is order-unity,
our assumptions on $\mathscr{W}$ imply that the factor $\mathscr{W}(\mathcal{I}^{-1} \Psi_0)$
on RHS~\eqref{E:INTROBASICENERGY} is small near the singularity, i.e.,
the energy provides only weak control over $V_a^2$.
This makes it difficult to control certain terms in the energy identities,
which arise from commutator error terms (that are generated upon commuting the evolution equations for
$\lbrace \Psi_{\alpha} \rbrace_{\alpha = 0,1,2,3}$ with spatial derivatives)
and from the basic integration by parts argument that we use to derive the energy
identities.
To control the most difficult error integrals,
we exploit the following spacetime integral, which also appears in the energy identities
(roughly it is generated when $\partial_t$ falls on the weight $\mathscr{W}(\mathcal{I}^{-1} \Psi_0)$ on RHS~\eqref{E:INTROBASICENERGY}):
\begin{align} \label{E:INTROKEYSPACETIMEINTEGRAL}
\mathfrak{I}[V](t) :=
\sum_{a=1}^3
\int_{s=0}^t
\int_{\Sigma_s}
(\mathcal{I}^{-1} \Psi_0)^2 \mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
(V_a)^2
\, d \underline{x}
\, ds,
\end{align}
where $\mathscr{W}'(y) = \frac{d}{dy} \mathscr{W}(y)$.
A good model scenario to keep in mind is the case $\mathscr{W} = \frac{1}{1 + \partial_t \Phi}$
in regions where $\partial_t \Phi$ is large (and thus the energy \eqref{E:INTROBASICENERGY} is weak),
in which case $\mathscr{W}' = - \frac{1}{(1 + \partial_t \Phi)^2}$,
and the factor
$
(\mathcal{I}^{-1} \Psi_0)^2 \mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
$
on RHS~\eqref{E:INTROKEYSPACETIMEINTEGRAL}
can be expressed as
$
- \frac{(\partial_t \Phi)^2}{(1 + \partial_t \Phi)^2}
$.
In view of our assumptions on $\mathscr{W}$, the
term $\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)$ has a \emph{quantitatively negative} sign
in the difficult regions where $\mathcal{I}$ is small (which is equivalent to the largeness of $\partial_t \Phi$).
More precisely,
\eqref{E:INTROKEYSPACETIMEINTEGRAL} has a \emph{friction-type} sign.
This is important because the difficult error integrals mentioned above
can be bounded by $\lesssim \varepsilon \mathfrak{I}[V](t)$,
where, roughly, $\varepsilon$ is the small $L^{\infty}$ size of the spatial derivatives.
For this reason, the integral \eqref{E:INTROKEYSPACETIMEINTEGRAL} can be used
to absorb the difficult error integrals.
In total, this allows us to prove a priori energy estimates, roughly of the following form:
\begin{align} \label{E:CARICATURENERGYAPRIORI}
\mathbb{E}[V](t)
+
\mathfrak{I}[V](t)
& \leq data \times C \exp(C t),
\end{align}
where ``$data$'' is, roughly, the small size of the spatial derivatives of the data.
For our proof to close, RHS~\eqref{E:CARICATURENERGYAPRIORI} must be sufficiently small.
Thanks to our bootstrap assumption that $t < \frac{2}{\mathring{A}_*}$,
it suffices to choose that initial data so that
$
data \times C \exp(\frac{C}{\mathring{A}_*})
$
is sufficiently small. This is one example of the \emph{nonlinear smallness} of the spatial derivatives,
relative to $\mathring{A}_*$, that we impose to close the proof.
In reality, to make this procedure work,
we must separately treat regions where $\mathcal{I}$ is small and $\mathcal{I}$ is large; see
Prop.~\ref{P:APRIORIESTIMATES} and its proof for the details.
We stress that absorbing the difficult error integrals into the friction integral \eqref{E:INTROKEYSPACETIMEINTEGRAL}
\emph{is crucial for showing that the energies remain bounded up to the vanishing of $\mathcal{I}$},
which is in turn central for our approach.
In the model case $\mathscr{W} = \frac{1}{1 + \partial_t \Phi}$,
if we had instead tried to directly control the difficult error integrals by the energy,
then we would have obtained the inequality
$\mathbb{E}[V](t) \leq
C
\int_{s=0}^t
\left\|
\partial_t \Phi
\right\|_{L^{\infty}(\Sigma_s)}
\mathbb{E}[V](s)
\, ds
+
\cdots
$.
Since
$\left\|
\partial_t \Phi
\right\|_{L^{\infty}(\Sigma_s)}$
goes to infinity at a non-integrable rate\footnote{With a bit of additional effort,
Theorem~\ref{T:STABILITYOFODEBLOWUP} could be sharpened to show that
$
\displaystyle
\left\|
\partial_t \Phi
\right\|_{L^{\infty}(\Sigma_t)}$
blows up like
$\frac{c}{T_{(Lifespan)} - t}$,
where $c$ is a positive data-dependent constant.}
as the blowup-time is approached,
this would have led to a priori energy estimates
allowing for the possibility that the energies blow up at the singularity,
which would have completely invalidated our philosophy of obtaining non-singular estimates
for the $\lbrace \Psi_{\alpha} \rbrace_{\alpha=0,1,2,3}$.
We also highlight that the regularity theory of $\mathcal{I}$ is somewhat subtle at top order:
our proof requires that we show that $\mathcal{I}$ and $\Phi$ have the same degree of differentiability
and that the estimates for $\mathcal{I}$ do not involve any dangerous factors of $\mathcal{I}^{-1}$;
these features are not immediately apparent from equation \eqref{E:INTROINTEGRATINFACTORODEANDIC}.
The circle of ideas tied to the ``regularization approach'' that we have taken
here seems to be new in the context of proving the stability of ODE-type blowup
for a quasilinear wave equation.
However, our approach has some parallels with
the known proofs of stable shock formation in multiple spatial dimensions, which we describe
in Subsect.\ \ref{SS:PRIORBREAKDOWNRESULTS}. In those problems, the crux of the proofs also involve
quasilinear integrating factors that ``hide'' the singularity.
In shock formation problems, the integrating factor is
tied to nonlinear geometric optics,\footnote{In the shock formation problems
described in Subsect.\ \ref{SS:PRIORBREAKDOWNRESULTS},
the integrating factor is the inverse foliation density
of a family of characteristic hypersurfaces, which are the level sets of an eikonal function.}
and its top-order regularity theory
is very difficult (much more so than the top-order regularity theory of the integrating factor employed in the present article).
The proofs also crucially rely on friction-type spacetime integrals,
in analogy with \eqref{E:INTROKEYSPACETIMEINTEGRAL},
that are available because the integrating factor has a negative derivative (in an appropriate direction)
in regions near the singularity.
However, in shock formation problems, the top-order energy identities feature dangerous terms,
analogous to terms of strength $\mathcal{I}^{-1}$, which leads to a priori energy estimates
allowing for the possibility that the high-order energies might blow up like $\mathcal{I}^{-P}$
for some large universal constant $P$. This makes it difficult to derive the non-singular estimates
at the lower derivative levels, which are central for closing the proof. In contrast, in our work here,
the difficult factors of $\mathcal{I}^{-1}$ are always multiplied by the term $\mathscr{W}$,
which effectively ameliorates them, making it easier to close the energy estimates.
On the other hand, the singularities that form in the solutions from our main results are much
more severe in that $\Phi$ and $\partial_t \Phi$ blow up;
in contrast, in the shock formation results
(see Subsect.\ \ref{SS:PRIORBREAKDOWNRESULTS})
for equations whose principal part is similar
to that of \eqref{E:FIRSTMODELWAVE},
$|\Phi|$ and $\lbrace |\partial_{\alpha} \Phi| \rbrace_{\alpha=0,1,2,3}$ remain
bounded up to the singularity, while
$\max_{\alpha, \beta = 0,1,2,3}|\partial_{\alpha} \partial_{\beta} \Phi|$
blows up in finite time. Our approach to proving Theorem~\ref{T:ROUGHMAINTHM}
also has some parallels with Kichenassamy's stable blowup results
\cite{sKi1996a} for semilinear wave equations with exponential nonlinearities,
but we defer further discussion of this point until the next subsection.
\subsection{Our results in the context of prior breakdown results}
\label{SS:PRIORBREAKDOWNRESULTS}
There are many prior breakdown results for solutions to various hyperbolic equations,
especially of wave type. Here we give a non-exhaustive account of some of these works,
which is meant to give the reader some feel
for the kinds of results that are known
and how they compare with/contrast against our main results.
In particular, we aim to expose how the
proof techniques vary wildly between different types of blowup results.
We separate the results into seven classes.
\begin{enumerate}
\item (\textbf{Proofs of blowup by contradiction})
For various hyperbolic systems, there are proofs of blowup by contradiction,
based on showing that for smooth solutions,
certain spatially averaged quantities satisfy ordinary differential
inequalities that force them to blow up in finite time,
contradicting the assumed smoothness.
Notable contributions of this type are
John's works \cites{fJ1979,fJ1981} on several classes of nonlinear wave equations
with signed nonlinearities and Sideris' proof \cite{tS1984} of blowup for various hyperbolic systems,
for semilinear wave equations in higher dimensions \cite{tS1984b} (which improved upon Kato's result \cite{tK1980}),
and for the compressible Euler equations in three spatial dimensions \cite{tS1985}.
See also \cite{yGsTZ1998} for similar results in the case of the relativistic Euler and Euler--Maxwell equations.
None of these results yield constructive information about the nature of the blowup, nor do they
apply to the wave equations under study here.
\item (\textbf{Blowup for semilinear wave equations with power-law nonlinearities})
There are many interesting constructive blowup results,
in various spatial dimensions,
for focusing semilinear wave equations of the form
$\square_m \Phi = - |\Phi|^{p-1} \Phi$,
where $\square_m := - \partial_t^2 + \Delta$ is the wave operator of
the Minkowski metric $m$.
A notable difference between these works
and our work here is that these works relied on a careful analysis
of the spectrum of a linearized operator.
We now discuss some specific examples.
In \cite{rD2010}, in three spatial dimensions and under the assumption of radial symmetry,
Donninger proved
the nonlinear stability of the ODE blowup solutions
$\Phi_{(ODE);T} := c_p(T-t)^{- \frac{2}{p-1}}$,
where\footnote{Actually, for convenience, Donninger considered the semilinear term
$- \Phi^p$ in \cite{rD2010}. However, as he noted there, his work could be extended to apply to
the term $- |\Phi|^{p-1} \Phi$ for $p > 1$.}
$p=3,5,7,\cdots$.
In three spatial dimensions, in the subcritical cases $p \in (1,3]$,
Donninger--Sch\"{o}rkhuber proved \cite{rDbS2012} an asymptotic stability result
for $\Phi_{(ODE);T}$
under radially symmetric perturbations of the data
in the energy space, thereby sharpening (in the near-ODE case) the works \cites{fMhZ2003,fMhZ2005},
which yielded the \emph{all} solutions that blow up
do so at the rate
$
(T-t)^{- \frac{2}{p-1}}
$,
but which did not yield the asymptotic profile.
In \cite{rDbS2014}, Donninger--Sch\"{o}rkhuber
extended their stability results (still within radial symmetry)
to the supercritical cases $p > 3$, but
they assumed additional regularity on the initial data
(which they believed to be essential for closing the proof).
In three spatial dimensions, in the critical case $p=5$,
there are many blowup results tied to the ground
state solution $W(r) := (1 + r^2/3)^{-1/2}$.
In \cite{ceKfM2008}, for solutions with (conserved) energy below that of the ground state,
Kenig--Merle established a sharp dichotomy showing that solutions blow up
in finite time to the past and future
if $\| \Phi \|_{\dot{H}^1(\Sigma_0)} > \| W \|_{\dot{H}^1(\Sigma_0)}$, while they
exist globally and scatter if $\| \Phi \|_{\dot{H}^1(\Sigma_0)} < \| W \|_{\dot{H}^1(\Sigma_0)}$.
For the same equation, the authors of \cite{jKwSdT2009} proved the existence of
radially symmetric ``slow'' Type II blowup solutions $\Phi(t,r) = \lambda^{1/2}(t) W(\lambda(t) r) + w(t,r)$,
where $w$ is a small error term,
$\lambda(t) := t^{-1-\nu}$, $\nu > 1/2$, and the singularity occurs at $t=0$.
In this context, a Type II singularity is such that the solution remains uniformly bounded in the energy space
(which is critical) up to the time of first blowup.
The results were extended to $\nu > 0$ in \cite{jKwS2014}.
In \cite{rDmHjKwS2014}, the results were extended
to cases in which $\lambda(t)$ does not behave like a power law.
In \cite{mHpR2012}, Hillairet--Rapha\"{e}l constructed
Type II blowup solutions for the critical focusing wave equation in four spatial dimensions.
Jendrej treated the case of five spatial dimensions in \cite{jJ2017}.
For the radial critical focusing wave equation in three spatial dimensions,
the work \cite{tDcKfM2011} yielded that if the blowup-time
$T$ is finite
and if the quantitative type II condition
$\sup_{t \in [0,T)}
\left\lbrace
\| \partial_t \Phi \|_{L^2(\Sigma_t)}^2
+
\| \nabla \Phi \|_{L^2(\Sigma_t)}^2
\right\rbrace
\leq \| \nabla W \|_{L^2}^2 + \eta_0
$
holds, where $W$ is the ground state and $\eta_0 > 0$ is a small constant,
then the blowup asymptotics are of the type exhibited by the solutions constructed in \cite{jKwSdT2009}.
The results were extended to the non-radial case in three and five spatial dimensions in
\cite{tDcKfM2012}. Similar results were obtained in the case of four
spatial dimensions in \cite{rCcKaLwS2014} in the radial case.
In \cite{tDcKfM2013}, the authors gave a detailed description
of the possible large-time behaviors of all finite-energy
radial solutions to the focusing critical wave equation in three spatial
dimensions, extending the work \cite{tDcKfM2012b}, where information
along a sequence of times was obtained.
In \cite{jJ2016}, for $n \in \lbrace 3,4,5 \rbrace$ spatial dimensions,
Jendrej proved an upper bound for
the blowup rate $\lambda(t)$
for Type II blowup solutions whose asymptotics are
$\Phi(t,r) = [\lambda(t)]^{(n-2)/2} W(\lambda(t) r) + w(t,r)$,
with $w$ sufficiently regular.
\item (\textbf{Constructive blowup results for wave maps})
There are similar blowup results for some wave maps whose targets
admit a non-trivial harmonic map.
For example, for the critical case of the wave maps equation
$\square_m \Phi = \Phi(|\partial_t \Phi|^2 - |\nabla \Phi|^2)$,
where $\Phi:\mathbb{R}^{1+2} \rightarrow \mathbb{S}^2$,
under the equivariant symmetry assumption
$\Phi(t,r) = (k \theta,\phi(t,r))$, where
the first and second entries on the RHS are Euler angles parameterizing $\mathbb{S}^2$
and $k \in \mathbb{Z}_+$,
there are blowup results tied to the
ground state $Q(r) := 2 \arctan(r^k)$.
In \cite{iRjS2010}, Rodnianski--Sterbenz
gave a sharp description of \emph{stable} blowup
when $k \geq 4$. They showed that (under the symmetry assumptions),
there is an open set of data with energy slightly larger than the ground state
whose solutions blow up at a time $T < \infty$.
Moreover, the asymptotics can be described as
$\phi(t,r) = Q(t,r/\lambda(t)) + q(t,r)$,
where $\lambda(t) \to 0$ as $t \uparrow T$,
$\lambda(t) \geq \frac{T-t}{|\ln(T-t)|^{1/4}}$,
and
$(q,\partial_t q)$ is small in $\dot{H}^1 \times L^2$.
In particular, $Q$ is the universal blowup profile.
A key point of the proof is to derive and analyze an appropriate \emph{modulation equation},
that is, the ODE (which is coupled to the PDE) that governs the evolution of $\lambda(t)$.
The function $\lambda(t)$ is somewhat analogous to the integrating factor $\mathcal{I}$ that we use in our work here.
In \cite{pRiR2012},
Rodnianski--Rapha\"{e}l extended the results to all cases $k \geq 1$,
proving \emph{stable} blowup with
$\lambda(t) = c_k \left(1 + o(1) \right) \frac{T-t}{\left| \ln(T-t) \right|^{\frac{1}{2k-2}}}$
as $t \uparrow T$ in the cases $k \geq 2$,
and, in the case $k=1$,
$\lambda(t) = (T-t) \exp \left(- \sqrt{\ln|T-t|} + O(1) \right)$
as $t \uparrow T$.
In \cite{jKwSdD2008}, in the case $k=1$,
the authors proved the existence
of a continuum of related solutions
(believed to be non-generic)
exhibiting the blowup rates
$\lambda(t) = (T-t)^{\nu}$, where $\nu > 3/2$.
The results were extended to $\nu > 1$ in \cite{cGjK2015}.
In \cites{rCcKaLwS2015a}, in the equivariance class $k=1$,
the authors proved that within the sub-class of degree $0$ maps (i.e., in radial coordinates $(t,r)$, one assumes
$\Phi(0,0) = \Phi(0,\infty) = 0$),
there exist solutions with energy above but arbitrarily close to twice the energy of the ground
state that blow up in finite time.
Within the sub-class of degree $1$ maps (i.e., $\Phi(0,0) = 0$ and $\Phi(0,\infty) = \pi$),
for maps with energy bigger than that of the ground state but less than three times
the energy of the ground state, the authors show that if a singularity forms,
then the solution has asymptotics whose blowup profiles
are the same as those from the works \cites{jKwSdD2008,iRjS2010,pRiR2012}.
For equivariant wave maps $\Phi : \mathbb{R} \times \mathbb{S}^2 \rightarrow \mathbb{S}^2$,
in the class $k=1$,
Shahshahani proved \cite{sS2016} the existence of a continuum of blowup solutions.
In \cites{rD2011,rDbSpA2012},
in the supercritical context of equivariant wave maps from $\mathbb{R}^{1+3}$ into $\mathbb{S}^3$,
the authors proved the stability of self-similar blowup solutions
$\Phi_T(t,r) := 2 \arctan \left(\frac{r}{T-r} \right)$.
More precisely, the results were conditional and relied
on some mode stability results for which there is strong evidence
in the form of analysis and numerics.
\item (\textbf{Blowup for semilinear wave equations with exponential nonlinearities})
In \cite{sKi1996a}, for the focusing semilinear wave equation $\square_m \Phi = - e^{\Phi}$
in three spatial dimensions,
Kichenassamy proved that the singular solution $\ln(2/t^2)$
is stable under perturbations of the data along the constant-time hypersurface
$\lbrace t = -1 \rbrace$.
Moreover, he showed that the blowup-hypersurface is
of the form $\lbrace t = f(x) \rbrace$, where $f(x)$ loses Sobolev regularity
compared to the initial data.
It would be interesting to see if our main results could be extended to show a similar result
for the equations under study here. More precisely, we conjecture that
a portion of the blowup surface is $\lbrace \mathcal{I} = 0 \rbrace$ for the solutions under study here.
Kichenassamy's work has one key feature in common with ours:
he devises a reformulation of the wave equation in which no singularity is visible, in his
case by constructing a singular change of coordinates adapted to the singularity;
this is broadly similar to the approach taken by authors who have proved shock formation results,
as we describe just below.
However, unlike the ``forwards approach'' that we take in this article,
Kichenassamy used a ``backwards approach''
in which he first solved problems in which the singularity was \emph{prescribed}
along blowup-hypersurfaces and then showed that the map from the singularity to the Cauchy data along
$\lbrace t = -1 \rbrace$ is invertible. His proof of the invertibility of the
map from the singularity data to the Cauchy data
was based on studying appropriately linearized versions of
the equations and on using Fuchsian techniques.
The linearized equations exhibited derivative loss, and
Kichenassamy used a Nash--Moser approach to overcome the loss.
\item (\textbf{Shock formation for quasilinear equations})
Roughly speaking, a shock singularity is such that the solution remains bounded but
one of its derivatives blows up. There are many classical shock formation results
in one spatial dimension, based on the method of characteristics,
with important contributions coming from
Riemann \cite{bR1860},
Lax \cites{pL1964,pL1972,pL1973},
and John \cite{fJ1974}, among others. Even in one spatial dimension, the field is still active,
as is evidenced by the recent work of \cite{dCdRP2016} of Christodoulou--Perez,
in which they significantly sharpened John's work \cite{fJ1974}, giving a complete
description of the singularity.
Alinhac \cites{sA1999a,sA1999b,sA2001b} obtained the first results
on shock formation without symmetry assumptions in more than one spatial
dimension. The main new difficulty in more than one spatial dimension
is that the method of characteristics must
be supplemented with energy estimates, which leads to enormous technical complications.
Alinhac's work applied
to small-data solutions to a class of
scalar quasilinear wave equations
of the form
\begin{align} \label{E:ALINHACWAVE}
(g^{-1})^{\alpha \beta}(\partial \Phi) \partial_{\alpha} \partial_{\beta} \Phi = 0
\end{align}
that fail to satisfy the null condition.
He showed that for a set of ``non-degenerate'' small data,
$\Phi$ and $\partial \Phi$ remain bounded, while $\partial^2 \Phi$ blows up in finite time
due to the intersection of the characteristics.
Alinhac's proof fundamentally relied on nonlinear geometric optics,
that is, on an eikonal function, which is a solution
to the eikonal equation
\begin{align} \label{E:WAVEEIKONAL}
(g^{-1})^{\alpha \beta}(\partial \Phi) \partial_{\alpha} u \partial_{\beta} u
& = 0,
\end{align}
supplemented with appropriate initial data.
The level sets of $u$ are characteristic hypersurfaces for equation \eqref{E:ALINHACWAVE}.
As it turns out, the intersection of the level sets of $u$ is tied to the formation of a singularity
in the solution to \eqref{E:ALINHACWAVE}.
Much like in the present work, the main estimates in Alinhac's proof did not concern singularities;
the crux of his proof was to construct a system of geometric coordinates, one of which is $u$,
and to prove that relative to them,
the solution remains very smooth, except possibly at the very high derivative levels.
He then showed that a singularity forms
in the standard second-order derivatives $\partial^2 \Phi$
as a consequence of a finite-time degeneracy
between the geometric coordinates and the standard coordinates; roughly, the level sets of $u$
intersect and cause the blowup, much like in the classical example of the blowup of solutions to Burgers' equation.
The main challenge in the proof is that
to derive energy estimates relative to the geometric coordinates, one must control the eikonal function,
whose top-order regularity properties are difficult to obtain; naive estimates lead to the loss of a derivative.
The regularity properties of eikonal functions had previously been understood
in certain problems for quasilinear wave equations in which singularities did not form.
For example, the first quasilinear wave problem in which the regularity properties of eikonal functions were fully exploited was the
celebrated proof \cites{dCsK1993} of the stability of Minkowski spacetime.
Eikonal functions have also played a central role
in proofs of low-regularity well-posedness for quasilinear wave equations \cites{sKiR2003,sKiR2005d,hSdT2005,sKiRjS2015}.
However, unlike in these problems,
in the problem of shock formation, the top-order geometric energy estimates feature a degenerate weight
that vanishes as the shock forms, which leads to a priori estimates allowing for the possibility that the high-order
energies might blow up; note that this possible geometric energy blowup is distinct from the formation of the shock,
which happens at the low derivative levels relative to the standard coordinates.
The ``degenerate weight'' is the inverse foliation density of the level sets of $u$.
It vanishes when the characteristics intersect,
and it is in some ways analogous to the integrating factor $\mathcal{I}$ that we use in our work here.
Alinhac closed the singular top-order energy estimates with a Nash--Moser iteration scheme
that was adapted to the singularity and that handled the issue of the regularity theory of $u$ in a different
way than \cites{dCsK1993,sKiR2003,sKiR2005d,hSdT2005,sKiRjS2015}.
He then used a ``descent scheme'' to show that the top-order geometric energy blowup
does not propagate down too far to the lower derivative levels. Consequently,
the solution remains highly differentiable relative to the geometric coordinates.
Due to his reliance on the Nash--Moser iteration scheme, Alinhac's proof applied only to
``non-degenerate'' initial data such that the first
singularity is isolated in the constant-time hypersurface of first blowup, and his framework breaks down
precisely at the time of first blowup.
For this reason, his approach is inadequate
for following the solution to the boundary of the maximal development
of the data (see Footnote~\ref{FN:MAXIMALDEVELOPMENT}),
which intersects the future of the first singular time.
In his breakthrough work \cite{dC2007}, Christodoulou overcame this drawback and significantly sharpened
Alinhac's results for the quasilinear wave equations of irrotational relativistic fluid mechanics.
More precisely, Christodoulou's proof yielded a sharp description of the solution up to the boundary of the maximal development.
This information was essential even for setting up the shock development problem, which, roughly speaking,
is the problem of uniquely extending the solution past the singularity in a weak sense, subject to
appropriate jump conditions.
We note that the shock development problem
in relativistic fluid mechanics was solved in spherical symmetry by Christodoulou--Lisibach in \cite{dCaL2016}
and, by Christodoulou in a recent breakthrough work \cite{dC2017},
for the non-relativistic compressible Euler equations without symmetry assumptions
in a restricted case
(known as the restricted shock development problem)
such that the jump in entropy across the shock hypersurface is ignored.
The wave equations studied by Christodoulou in \cite{dC2007} form a sub-class of the ones \eqref{E:ALINHACWAVE}
studied by Alinhac. They enjoy special
properties that Christodoulou used in his proofs, notably
an Euler-Lagrange formulation whose Lagrangian is invariant under various symmetry groups.
The main technical improvement afforded by Christodoulou's framework is that
in closing the energy estimates, he avoided using a Nash--Moser iteration scheme
and instead used an approach similar to the one employed in the aforementioned works
\cites{dCsK1993,sKiR2003}. This approach is more robust and is capable of accommodating solutions such that
the blowup occurs along a hypersurface, which, in the problem of shock formation,
is what typically occurs along a portion
of the boundary of the maximal development.
Christodoulou's results have since been extended in many directions, including
to apply to other wave equations \cites{dCsM2014,jS2016b},
different sets of initial data \cites{jSgHjLwW2016,sMpY2017,sM2016},
the compressible Euler equations with non-zero vorticity \cites{jLjS2016a,jLjS2016b,jS2017a},
systems of wave equations with multiple speeds \cite{jS2017b},
and quasilinear systems in which a solution to a transport equation forms a shock
\cite{jS2017c}.
Some of the earlier extensions are explained in detail in the survey article \cite{gHsKjSwW2016}.
\item (\textbf{Breakdown results for Einstein's equations})
The Einstein equations of general relativity have many remarkable properties and as such,
it is not surprising that there are stable breakdown results that are specialized to
these equations. Here we simply highlight the following constructive
results in three spatial dimensions without symmetry assumptions:
Christodoulou's breakthrough results \cite{dC2009} on the formation of trapped surfaces
and the stable singularity formation results \cites{iRjS2014a,iRjS2014b,jL2013}.
The work \cite{iRjS2014b} can be viewed as a stable ODE-type blowup result
for Einstein's equations in which the wave speed became infinite at the singularity.
Note that in contrast, for equation \eqref{E:WAVE}, the wave speed vanishes when $\partial_t \Phi$
blows up.
\item (\textbf{Finite time degeneration of hyperbolicity})
In \cite{jS2016}, we studied the wave equations
$- \partial_t^2 \Psi + (1 + \Psi)^P \Delta \Psi = 0$
in three spatial dimensions, for $P = 1,2$. We showed that there
exists an open set of data such that $\Psi$ is initially small but
$1 + \Psi$ vanishes in finite time,
corresponding to a breakdown in the hyperbolicity of the equation,
but without any singularity forming. The difficult part of the proof is
closing the energy estimates in regions where $1 + \Psi$ is small.
The proof has some features in common with the proof of the main results of this paper.
For example, the proof relies on monotonicity tied to the sign of $\partial_t \Psi$
and the small size of $\nabla \Psi$. In particular, this leads to the availability
of a friction-type integral in the energy identities,
analogous to the one \eqref{E:INTROKEYSPACETIMEINTEGRAL},
which is crucially important for controlling certain error terms.
\end{enumerate}
\subsection{Different kinds of singularity formation within the same quasilinear hyperbolic system}
\label{SS:DIFFERENTKINDSOFSINGULARITYFORMATION}
In this subsection, we show that there are quasilinear wave equations,
closely related to the wave equation \eqref{E:WAVE},
that can exhibit two distinct kinds of blowup:
ODE-type blowup for one set of data, and
the formation of shocks for another set.
The ODE-type blowup is provided by our main
results, so in this subsection,
most of the discussion is centered on
shock formation. Our discussion is based on ideas and
techniques found in the works \cites{dC2007,jS2016b}.
To initiate the discussion, we define
\begin{align} \label{E:PHIALPHA}
\Phi_0
& := \partial_t \Phi.
\end{align}
For convenience, we will restrict our discussion to the specific
weight $\mathscr{W} = \frac{1}{1 + \partial_t \Phi} = \frac{1}{1 + \Phi_0}$,
though similar results hold for any weight that verifies the assumptions
of Subsect.\ \ref{SS:WEIGHTASSUMPTIONS}.
To proceed, we differentiate equation \eqref{E:WAVE} with $\partial_t$
to obtain the following closed equation in $\Phi_0$:
\begin{align} \label{E:PHI0CLOSEDEQUATION}
\partial_t^2 \Phi_0
-
\frac{1}{1 + \Phi_0}
\Delta \Phi_0
=
-
\frac{1}{1 + \Phi_0}
(\partial_t \Phi_0)^2
+
\frac{2 \Phi_0}{1 + \Phi_0} \partial_t \Phi_0
+
\frac{3 \Phi_0^2}{1 + \Phi_0} \partial_t \Phi_0.
\end{align}
In the remainder of our discussion of shock formation,
we will only consider plane symmetric solutions,
that is, solutions that depend only on $t$ and $x^1$.
Note that in equation \eqref{E:PHI0CLOSEDEQUATION},
$\Delta = \partial_1^2$ for plane symmetric solutions.
To study plane symmetric solutions to \eqref{E:PHI0CLOSEDEQUATION}, we will use the characteristic vectorfields
\begin{align}
L := \partial_t + \frac{1}{\sqrt{1 + \Phi_0}} \partial_1,
&& \underline{L} := \partial_t - \frac{1}{\sqrt{1 + \Phi_0}} \partial_1.
\end{align}
We next define the characteristic coordinate $u$ to be the solution to the following transport equation:
\begin{align} \label{E:EIKONALDEF}
L u
& = 0,
&
u|_{\Sigma_0}
& = 1 - x^1.
\end{align}
We view $u$ as a new coordinate adapted to the characteristics,
and we will use the ``geometric'' coordinate system $(t,u)$ when analyzing solutions.
In particular, the level sets of $u$ are characteristic for equation \eqref{E:PHI0CLOSEDEQUATION}.
We define $\Sigma_t^{u'}$,
relative to the geometric coordinates,
to be the following subset:
$\Sigma_t^{u'} := \lbrace (t,u) \ | \ 0 \leq u \leq u' \rbrace$.
Note that $\Sigma_0^1$ can be identified with an orientation reversed version of the unit $x^1$ interval $[0,1]$.
Associated to $u$, we define the \emph{inverse foliation density} $\upmu > 0$
by
\begin{align} \label{E:MUDEF}
\upmu & := \frac{1}{\partial_t u}.
\end{align}
$1/\upmu$ is a measure of the density of the level sets of $u$.
$\upmu = 0$ corresponds to the intersection of the characteristics,
that is, the formation of a shock.
From \eqref{E:EIKONALDEF}, it follows that $\upmu|_{\Sigma_0} = \sqrt{1 + \Phi_0} = 1 + \mathcal{O}(\Phi_0)$
(for $\Phi_0$ small).
One can check that from the above definitions, we have
$L u = 0$, $L t = 1$, $\upmu \underline{L} t = \upmu $,
and $\upmu \underline{L} u = 2 $.
In particular,
$L = \frac{d}{dt}$ along the integral curves of $L$
and $\upmu \underline{L} = 2 \frac{d}{du}$ along the integral curves of
$\upmu \underline{L}$.
Next, we differentiate equation \eqref{E:WAVE}
with $\partial_t$ and carry out tedious but straightforward calculations
to obtain the following system in $\Phi_0$ and $\upmu$:
\begin{subequations}
\begin{align}
L (\upmu \underline{L} \Phi_0)
& =
-
\frac{1}{2(1 + \Phi_0)} (L \Phi_0) (\upmu \underline{L} \Phi_0)
\label{E:LONOUTSIDE} \\
& \ \
+
\upmu
\frac{\Phi_0}{1 + \Phi_0}
\left\lbrace
1 + \frac{3}{2} \Phi_0
\right\rbrace
L \Phi_0
+
\frac{\Phi_0}{1 + \Phi_0}
\left\lbrace
1 + \frac{3}{2} \Phi_0
\right\rbrace
(\upmu \underline{L} \Phi_0),
\notag \\
\upmu \underline{L} L \Phi_0
& =
-
\frac{\upmu}{4(1 + \Phi_0)}
(L \Phi_0)^2
-
\frac{3}{4(1 + \Phi_0)} (L \Phi_0) (\upmu \underline{L} \Phi_0)
\label{E:LBARONOUTSIDE} \\
& \ \
+
\upmu
\frac{\Phi_0}{1 + \Phi_0}
\left\lbrace
1 + \frac{3}{2} \Phi_0
\right\rbrace
L \Phi_0
+
\frac{\Phi_0}{1 + \Phi_0}
\left\lbrace
1 + \frac{3}{2} \Phi_0
\right\rbrace
(\upmu \underline{L} \Phi_0),
\notag
\\
L \upmu
& = \frac{1}{4(1 + \Phi_0)} \upmu L \Phi_0
+
\frac{1}{4(1 + \Phi_0)} (\upmu \underline{L} \Phi_0).
\label{E:UPMUEVOLUTION}
\end{align}
\end{subequations}
For convenience, we will prove shock formation only for a restricted class of
initial data supported in $\Sigma_0^1$;
as can easily be inferred from our proof,
the shock-forming solutions that we will construct are stable under plane symmetric perturbations,
and our approach could be applied to a much larger set of plane symmetric initial data.
Specifically, we will prove shock formation for solutions corresponding to initial data such that
\begin{align} \label{E:SHOCKDATA}
\sup_{\Sigma_0^1} |\Phi_0|
& \leq \epsilon,
&&
L \Phi_0|_{\Sigma_0} = 0,
&
\sup_{\Sigma_0^1} |\underline{L} \Phi_0|
& = 4,
\end{align}
such that $\underline{L} \Phi_0|_{\Sigma_0^1}$ is \emph{negative} at some maximum of $|\underline{L} \Phi_0|$ on $\Sigma_0^1$,
and such that $\epsilon$ is small. The negativity of $\underline{L} \Phi_0$ will drive the vanishing of $\upmu$.
To show the existence of such data, it is convenient to refer to the Cartesian coordinate $x^1$.
Specifically, we fix a smooth non-trivial function $f = f(x^1)$ supported in $\Sigma_0^1$
and set
$\Phi_0|_{\Sigma_0^1}(x^1) := \upkappa f(\uplambda x^1)$,
where $\upkappa$ and $\uplambda$ are real parameters.
Note that $\partial_1 \Phi_0|_{\Sigma_0}(x^1) = \upkappa \uplambda f'(\uplambda x^1)$.
We then set
$\partial_t \Phi_0|_{\Sigma_0^1}
:=
- \frac{1}{\sqrt{1 + \Phi_0|_{\Sigma_0^1}}} \partial_1 \Phi_0|_{\Sigma_0^1}
$, which implies that
$L \Phi_0|_{\Sigma_0^1} = 0$
and
$\underline{L} \Phi_0|_{\Sigma_0^1}(x^1) = - 2 \upkappa \uplambda \frac{1}{\sqrt{1 + \upkappa f(\uplambda x^1)}} f'(\uplambda x^1)$.
We now choose $|\upkappa|$ sufficiently small and $\uplambda$
sufficiently large, which allows us to achieve \eqref{E:SHOCKDATA} with $\epsilon > 0$ arbitrarily small.
Moreover, by adjusting the sign of $\upkappa$, we can ensure that
$\underline{L} \Phi_0|_{\Sigma_0^1}$ is negative at some maximum of $|\underline{L} \Phi_0|$ on $\Sigma_0^1$.
We also note that from domain of dependence considerations,
it follows that in terms of the geometric coordinates,
solutions with data supported in $\Sigma_0^1$
vanish when $u \leq 0$, and that the level set
$\lbrace u = 0 \rbrace$ can be described in Cartesian coordinates
as $\lbrace (t,x^1) \ | \ 1 - x^1 + t = 0 \rbrace$.
To derive estimates, we make the following
bootstrap assumptions on any region of classical existence such that
$0 \leq t \leq 2$ and $0 \leq u \leq 1$:
\begin{align} \label{E:INTROSHOCKBA}
0 < \upmu \leq 3,
\qquad
|\Phi_0| \leq \sqrt{\epsilon},
\qquad
|L \Phi_0| \leq \sqrt{\epsilon},
\qquad
|\underline{L} \Phi_0| \leq 5.
\end{align}
Note also that the solution verifies $\Phi_0(t,u=0) = 0$
and that the assumptions \eqref{E:INTROSHOCKBA} are consistent
with the initial data when $\epsilon$ is small.
We now derive estimates. We define
\begin{align} \label{E:INTROSHOCKGOODDERIVATIVES}
Q(t,u)
& := \sup_{(t',u') \in [0,t] \times [0,u]} \left\lbrace |\Phi_0|(t',u') + |L \Phi_0|(t',u') \right\rbrace.
\end{align}
Note that $Q(0,u) \lesssim \epsilon$,
while our data support assumptions imply that $Q(t,0) = 0$.
Using the evolution equation \eqref{E:LBARONOUTSIDE}, the bootstrap assumptions,
the fact that $L = \frac{d}{dt}$ along the integral curves of $L$,
and the fact that $\upmu \underline{L} = 2 \frac{d}{du}$ along the integral curves of
$\upmu \underline{L}$,
we deduce
$Q(t,u)\leq C Q(0,u) + c \int_{t'=0}^t Q(t',u) \, dt' + c \int_{u'=0}^u Q(t,u') \, du'$.
From this estimate and Gronwall's inequality (in two variables), we deduce that there are constants $C > 0$ and $c' > c$
such that for $0 \leq t \leq 2$ and $0 \leq u \leq 1$, we have
\begin{align} \label{E:APRIORIESTIMATEINTROSHOCKGOODDERIVATIVES}
Q(t,u) \leq C Q(0,u) e^{c't} e^{c'u} \leq C Q(0,u) e^{3c'} \leq C e^{3c'} \epsilon \lesssim \epsilon.
\end{align}
Using the estimate \eqref{E:APRIORIESTIMATEINTROSHOCKGOODDERIVATIVES}
and the bootstrap assumptions for
$\upmu$
and $\upmu \underline{L} \Phi_0$
to control the terms on RHS~\eqref{E:LONOUTSIDE},
we deduce
$|L (\upmu \underline{L} \Phi_0)| \lesssim \epsilon$.
Integrating this estimate along the integral curves of $L$
and using that $\upmu(0,u) = 1 + \mathcal{O}(\epsilon)$,
we find that for $0 \leq t \leq 2$ and $0 \leq u \leq 1$, we have
$\upmu \underline{L} \Phi_0(t,u) = \upmu \underline{L} \Phi_0(0,u) + \mathcal{O}(\epsilon)
= \underline{L} \Phi_0(0,u) + \mathcal{O}(\epsilon)$.
Inserting this information into \eqref{E:UPMUEVOLUTION},
we deduce
$L \upmu = \frac{1}{4} \underline{L} \Phi_0(0,u) + \mathcal{O}(\epsilon)$.
Integrating in time and using the initial condition
$\upmu(0,u) = 1 + \mathcal{O}(\epsilon)$, we deduce that
$\upmu(t,u) = 1 + \frac{1}{4} \underline{L} \Phi_0(0,u) t + \mathcal{O}(\epsilon)
= 1 + \frac{1}{4} \upmu \underline{L} \Phi_0(t,u) t + \mathcal{O}(\epsilon)
$.
We now note that if $\epsilon$ is sufficiently small, then the above estimates
yield strict improvements of the bootstrap assumptions \eqref{E:INTROSHOCKBA}.
By a standard continuity argument in $t$ and $u$,
this justifies the bootstrap assumptions
and shows that the solution exists
on regions of the form
$0 \leq t \leq 2$ and $0 \leq u \leq 1$,
as long as $\upmu$ remains positive;
the positivity of $\upmu$ and the above estimates guarantee that
$|\Phi_0| + \max_{\alpha = 0,1} |\partial_{\alpha} \Phi_0|$ is finite.
Moreover, since (by construction)
$\sup_{\Sigma_0^1} |\underline{L} \Phi_0| = 4$
and since there is a value $u_* \in (0,1)$ such that
$\underline{L} \Phi_0(0,u_*) = - 4$,
the above estimates for
$\upmu \underline{L} \Phi_0$
and
$\upmu$
guarantee that $\min_{\Sigma_t^u} \upmu = 1 + \mathcal{O}(\epsilon) - t$
and that $\upmu \underline{L} \Phi_0(t,u) \leq - 1$ at points $(t,u)$ with $\upmu(t,u) \leq 1/4$
and $0 \leq t \leq 2$.
It follows that
$\min_{\Sigma_t^1} \upmu$ cannot remain positive for times larger than
$1 + \mathcal{O}(\epsilon)$
and that $\min_{\Sigma_t^1} \upmu \leq 1/4 \implies \sup_{\Sigma_t^1} |\underline{L} \Phi_0| \geq \frac{1}{\min_{\Sigma_t^1} \upmu}$.
This implies that $\sup_{\Sigma_t^1} |\underline{L} \Phi_0|$ blows up
at some time $t_{(Shock)} = 1 + \mathcal{O}(\epsilon)$
while $|\Phi_0|$ and $|L \Phi_0|$ remain uniformly bounded by $\lesssim \epsilon$.
We have thus shown that a shock forms.
We now revisit the solutions from our main results under the weight $\mathscr{W} = \frac{1}{1 + \partial_t \Phi}$.
Notice that for such solutions,
$\Phi_0$ also solves equation \eqref{E:PHI0CLOSEDEQUATION}
but is such that such that $|\Phi_0|$ blows up at the singularity.
This is \emph{different blowup behavior} compared to the shock-forming solutions
to equation \eqref{E:PHI0CLOSEDEQUATION}
constructed above, in which
$|\Phi_0|$ remained bounded.
Notice also that our main theorem requires, roughly, that $\Phi_0|_{\Sigma_0}$
should not be too small, which is in contrast to the initial data
for the shock-forming formation solutions described above.
To close this subsection, we clarify that it could be, in principle,
that the ODE-type blowup solutions that we have constructed
are \emph{unstable} when viewed as solutions to equation \eqref{E:PHI0CLOSEDEQUATION},
even though they are stable solutions of the
original wave equation \eqref{E:WAVE}. The key point is that
to solve \eqref{E:PHI0CLOSEDEQUATION} (viewed as a wave equation for $\Phi_0$),
we need to prescribe the data functions
$\Phi_0|_{\Sigma_0}$
and
$\partial_t \Phi_0|_{\Sigma_0}$,
whereas for the ODE-type blowup solutions we have constructed,
we can freely prescribe (in plane symmetry) only $\Phi_0|_{\Sigma_0}$;
the quantity $\partial_t \Phi_0|_{\Sigma_0}$ is not ``free,'' but rather
is uniquely determined from $\Phi_0|_{\Sigma_0}$
via the wave equation \eqref{E:WAVE}.
Put differently,
the ODE-type blowup solutions that we have constructed yield
``special'' solutions to equation \eqref{E:PHI0CLOSEDEQUATION}
that are constrained by the fact that
$\Phi_0$ is the time derivative of a solution
to the original wave equation \eqref{E:WAVE}.
In contrast, we note that we expect that the methods of \cite{jSgHjLwW2016} could be used
to show that the plane symmetric shock-forming
solutions to \eqref{E:PHI0CLOSEDEQUATION} that we constructed above
are stable under perturbations that break the plane symmetry.
\subsection{Notation}
\label{SS:NOTATION}
In this subsection, we summarize some notation that we use throughout.
\begin{itemize}
\item $\lbrace x^{\alpha} \rbrace_{\alpha=0,1,2,3}$ denotes
the standard Cartesian coordinates on $\mathbb{R}^{1+3} = \mathbb{R} \times \mathbb{R}^3$
and
$
\displaystyle
\partial_{\alpha}
:=
\frac{\partial}{\partial x^{\alpha}}
$
denotes the corresponding coordinate partial derivative vectorfields.
$x^0 \in \mathbb{R}$ is the time coordinate and $\underline{x} := (x^1,x^2,x^3) \in \mathbb{R}^3$
are the spatial coordinates.
\item We often use the alternate notation $x^0 = t$ and $\partial_0 = \partial_t$.
\item $\Sigma_t := \lbrace (t,\underline{x}) \ | \ \underline{x} \in \mathbb{R}^3 \rbrace$
is the standard flat hypersurface of constant time.
\item Greek ``spacetime'' indices such as $\alpha$ vary over $0,1,2,3$ and
Latin ``spatial'' indices such as $a$ vary over $1,2,3$.
We use primed indices, such as $a'$, in the same way that we use their non-primed counterparts.
We use Einstein's summation convention in that repeated indices are summed
over their respective ranges.
\item We sometimes omit the arguments of functions appearing in pointwise inequalities. For example,
we sometimes write $|f| \leq C \mathring{\upepsilon}$
instead of $|f(t,\underline{x})| \leq C \mathring{\upepsilon}$.
\item $\nabla^k \Psi$ denotes the array comprising all $k^{th}-$order
derivatives of $\Psi$ with respect to the Cartesian spatial coordinate vector fields.
We often use the alternate notation $\nabla \Psi$ in place of $\nabla^1 \Psi$.
For example, $\nabla^1 \Psi = \nabla \Psi := (\partial_1 \Psi, \partial_2 \Psi, \partial_3 \Psi)$.
\item
$|\nabla^{\leq k} \Psi|
:= \sum_{k'=0}^k |\nabla^{k'} \Psi|
$.
\item
$|\nabla^{[a,b]} \Psi|
:= \sum_{k'=a}^b |\nabla^{k'} \Psi|
$.
\item $H^N(\Sigma_t)$ denotes the standard Sobolev space of functions on $\Sigma_t$
with corresponding norm
\[
\displaystyle
\| f \|_{H^N(\Sigma_t)}
:=
\left\lbrace
\sum_{a_1 + a_2 + a_3 \leq N}
\int_{\underline{x} \in \mathbb{R}^3}
|\partial_1^{a_1} \partial_2^{a_2} \partial_3^{a_3}f(t,\underline{x})|^2
\, d \underline{x}
\right\rbrace^{1/2}.
\]
In the case $N=0$, we use the notation ``$L^2$'' in place of ``$H^0$.''
\item
$L^{\infty}(\Sigma_t)$ denotes the standard Lebesgue space of functions on $\Sigma_t$
with corresponding norm
$
\displaystyle
\| f \|_{L^{\infty}(\Sigma_t)}
:=
\mbox{\upshape ess sup}_{\underline{x} \in \mathbb{R}^3}
|f(t,\underline{x})|
$.
\item Above and throughout, $d \underline{x} = d x^1 dx^2 dx^3$ is the standard Euclidean integration measure on $\Sigma_t$.
\item If $A$ and $B$ are two quantities, then we often write
$A \lesssim B$
to indicate that ``there exists a constant $C > 0$ such that $A \leq C B$.''
\item We sometimes write $\mathcal{O}(B)$ to denote a quantity $A$
with the following property: there exists a constant $C > 0$ such that $|A| \leq C |B|$.
\item Explicit and implicit constants
are allowed to depend on the data-size parameters
$\mathring{A}$
and
$\mathring{A}_*^{-1}$
from Subsect.\ \ref{SS:DATAASSUMPTIONS},
in a manner that we more fully explain in Subsect.\ \ref{SS:CONVENTIONSFORCONSTANTS}.
\end{itemize}
\section{Mathematical setup and the evolution equations}
\label{S:MATHEMATICALSETUP}
In this section, we state our assumptions on the nonlinearities,
define the quantities that we will study in the rest of the paper,
and derive evolution equations.
\subsection{Assumptions on the weight}
\label{SS:WEIGHTASSUMPTIONS}
Let $\mathscr{W}$ be the scalar function from equation \eqref{E:WAVE}.
We assume that there are constants $C_k > 0$ such that
\begin{align}
\mathscr{W}(y)
& > 0,
&& y \in (-1/2,\infty),
\label{E:WEIGHTISPOSITIVE} \\
\mathscr{W}(0)
& = 1,
&&
\label{E:WOFZEROISONE} \\
\mathscr{W}'(y)
& \leq 0,
&& y \in [0,\infty),
\label{E:WEIGHTNEGATIVEDERIVATIVE} \\
\left|
\left\lbrace
(1 + y)^2
\frac{d}{dy}
\right\rbrace^k
\left[
(1 + y) \mathscr{W}(y)
\right]
\right|
& \leq C_k,
\qquad
0 \leq k \leq 5,
&& y \in (-1/2,\infty).
\label{E:ESTIMATEFORDERIVATIVESOFWEIGHT}
\end{align}
We also assume that there is a constant $\upalpha > 0$ such that
\begin{align} \label{E:WEIGHTVSWEIGHTDERIVATIVECOMPARISON}
\mathscr{W}(y)
& \leq \upalpha
\left|
\mathscr{W}'(y)
\right|^{1/2},
&& y \in [1,\infty).
\end{align}
Note that
\eqref{E:WEIGHTISPOSITIVE},
\eqref{E:WEIGHTNEGATIVEDERIVATIVE},
and \eqref{E:WEIGHTVSWEIGHTDERIVATIVECOMPARISON} imply in particular that
\begin{align} \label{E:WEIGHTPRIMEISNEGATIVE}
\mathscr{W}'(y)
& < 0,
&& y \in [1,\infty).
\end{align}
\subsection{The integrating factor and the renormalized solution variables}
\label{SS:INTEGRATINGFACTOR}
\subsubsection{Definitions}
\label{SSS:DEFINITIONSOFINTEGRATINGFACTORANDRENORMALIZEDVARIABLES}
As we described in Subsubsect.\ \ref{SSS:DISCUSSIONOFTHEPROOF},
our analysis fundamentally relies on the following integrating factor.
\begin{definition}[\textbf{The integrating factor}]
\label{D:INTEGRATINGFACTOR}
Let $\Phi$ be the solution to the wave equation \eqref{E:WAVE}.
We define $\mathcal{I} = \mathcal{I}(t,\underline{x})$
to be the solution to the following transport equation:
\begin{align} \label{E:INTEGRATINFACTORODEANDIC}
\partial_t \mathcal{I}
& =
- \mathcal{I} \partial_t \Phi,
&& \mathcal{I}|_{\Sigma_0} = 1.
\end{align}
Moreover, we define
\begin{align} \label{E:IFACTMIN}
\mathcal{I}_{\star}(t)
& := \min_{\Sigma_t} \mathcal{I}.
\end{align}
\end{definition}
\begin{remark}[\textbf{The vanishing of} $\mathcal{I}$ \textbf{implies singularity formation}]
\label{R:VANISHINGOFINTEGRATINGFACTORISSINGULARITY}
It is straightforward to see from \eqref{E:INTEGRATINFACTORODEANDIC}
that if $\mathcal{I}(T,\underline{x}) = 0$ for some $T > 0$ and for one or more $\underline{x} \in \mathbb{R}^3$, then
at such values of $\underline{x}$, we have
$\lim_{t \uparrow T} \sup_{s \in [0,t)} \partial_t \Phi(s,\underline{x}) = \infty$.
In fact, it follows that $\int_{s=0}^t |\partial_t \Phi(s,\underline{x})| \, ds = \infty$.
\end{remark}
Most of our effort will go towards analyzing the following ``renormalized''
solution variables. We will show that they remain regular up to the singularity.
\begin{definition}[\textbf{Renormalized solution variables}]
\label{D:RENORMALIZEDSOLUTION}
Let $\Phi$ be the solution to the wave equation \eqref{E:WAVE}
and let $\mathcal{I}$ be as in Def.~\ref{D:INTEGRATINGFACTOR}.
For $\alpha = 0,1,2,3$, we define
\begin{align} \label{E:PSIDEF}
\Psi_{\alpha}
& := \mathcal{I} \partial_{\alpha} \Phi.
\end{align}
\end{definition}
\subsubsection{A crucial identity for $\mathcal{I}$ and the $\mathcal{I}$-weighted evolution equations}
\label{SSS:EVOLUTIONEQUATIONS}
Our main goal in this subsubsection is to derive evolution equations
for the renormalized solution variables; see Prop.~\ref{P:RENORMALIZEDEVOLUTOINEQUATIONS}.
As a preliminary step, we first provide a lemma
that shows that
$\partial_i \mathcal{I}$ can be controlled in terms $\Psi_i$ and the initial data,
and that no singular factors of $\mathcal{I}^{-1}$ appear in the relationship.
Though simple, the lemma is crucial for the top-order regularity theory of $\mathcal{I}$.
\begin{lemma}[\textbf{Identity for the spatial derivatives of the integrating factor}]
The following identity holds for $i=1,2,3$:
\begin{align} \label{E:FORMULAFORDERIVATIVESOFINTEGRATINGFACTOR}
\partial_i \mathcal{I}
& = - \Psi_i
+
\mathcal{I} \mathring{\Psi}_i.
\end{align}
\end{lemma}
\begin{proof}
Dividing equation \eqref{E:INTEGRATINFACTORODEANDIC}
by $\mathcal{I}$ and then applying $\partial_i$,
we compute that
\begin{align} \label{E:CRUCIALEVOLUTIONEQUATIONFORINTEGRATINGFACTORSPATIALDERIVATIVES}
\partial_t
\left\lbrace
\frac{\partial_i \mathcal{I} + \Psi_i}{\mathcal{I}}
\right\rbrace
& = 0.
\end{align}
Integrating \eqref{E:CRUCIALEVOLUTIONEQUATIONFORINTEGRATINGFACTORSPATIALDERIVATIVES} with respect
to time and using the initial conditions $\mathcal{I}|_{\Sigma_0} = 1$ and $\Psi_i|_{\Sigma_0} = \mathring{\Psi}_i$,
we arrive at \eqref{E:FORMULAFORDERIVATIVESOFINTEGRATINGFACTOR}.
\end{proof}
We now derive the main evolution equations that we will study in the remainder of the paper.
\begin{proposition}[\textbf{$\mathcal{I}$-weighted evolution equations}]
\label{P:RENORMALIZEDEVOLUTOINEQUATIONS}
For solutions to the wave equation \eqref{E:WAVE},
the renormalized solution variables of Def.~\ref{D:RENORMALIZEDSOLUTION}
verify the following system:
\begin{subequations}
\begin{align}
\partial_t \Psi_0
& =
\mathscr{W}(\mathcal{I}^{-1} \Psi_0) \sum_{a=1}^3 \partial_a \Psi_a
+
\mathcal{I}^{-1} \mathscr{W}(\mathcal{I}^{-1} \Psi_0) \sum_{a=1}^3 \Psi_a \Psi_a
-
\mathscr{W}(\mathcal{I}^{-1} \Psi_0) \sum_{a=1}^3 \mathring{\Psi}_a \Psi_a,
\label{E:PARTALTPSI0EVOLUTION} \\
\partial_t \Psi_i
&
= \partial_i \Psi_0
- \mathring{\Psi}_i \Psi_0.
\label{E:PARTALTPSIIEVOLUTION}
\end{align}
\end{subequations}
\end{proposition}
\begin{proof}
We first prove \eqref{E:PARTALTPSI0EVOLUTION}.
From equations \eqref{E:WAVE} and \eqref{E:INTEGRATINFACTORODEANDIC},
we deduce
$\partial_t(\mathcal{I} \partial_t \Phi)
= \mathcal{I} \mathscr{W}(\partial_t \Phi) \Delta \Phi
= \mathscr{W}(\partial_t \Phi) \sum_{a=1}^3 \partial_a (\mathcal{I} \partial_a \Phi)
-
\mathscr{W}(\partial_t \Phi) \sum_{a=1}^3 (\partial_a \mathcal{I}) \partial_a \Phi
$.
Using equation \eqref{E:FORMULAFORDERIVATIVESOFINTEGRATINGFACTOR} to substitute for
$\partial_a \mathcal{I}$ and appealing to Def.~\eqref{D:RENORMALIZEDSOLUTION},
we arrive at the desired equation \eqref{E:PARTALTPSI0EVOLUTION}.
To prove \eqref{E:PARTALTPSIIEVOLUTION}, we first use
Def.~\eqref{D:RENORMALIZEDSOLUTION} and the symmetry property $\partial_t \partial_i \Phi = \partial_i \partial_t \Phi$
to obtain
$\partial_t \Psi_i = (\partial_t \ln \mathcal{I}) \Psi_i + \partial_i \Psi_0 - (\partial_i \ln \mathcal{I}) \Psi_0$.
Using \eqref{E:INTEGRATINFACTORODEANDIC} to replace
$\partial_t \ln \mathcal{I}$ with $- \mathcal{I}^{-1} \Psi_0$
and equation \eqref{E:FORMULAFORDERIVATIVESOFINTEGRATINGFACTOR}
to replace
$
- \partial_i \ln \mathcal{I}
$
with
$\mathcal{I}^{-1} \Psi_i - \mathring{\Psi}_i$,
we conclude \eqref{E:PARTALTPSIIEVOLUTION}.
\end{proof}
\section{Assumptions on the initial data and bootstrap assumptions}
\label{S:DATAANDBOOTSTRAP}
In this section, we state our size assumptions on the data
$
(\partial_t \Phi|_{\Sigma_0},\partial_1 \Phi|_{\Sigma_0}, \partial_2 \Phi|_{\Sigma_0}, \partial_3 \Phi|_{\Sigma_0})
= (\mathring{\Psi}_0,\mathring{\Psi}_1,\mathring{\Psi}_2,\mathring{\Psi}_3),
$
for the wave equation \eqref{E:WAVE}
and formulate bootstrap assumptions that are convenient
for studying the solution. We also precisely describe the smallness assumptions
that we need to close our estimates and show the existence of initial data
that verify the smallness assumptions.
\subsection{Assumptions on the data}
\label{SS:DATAASSUMPTIONS}
We assume that the initial data
are compactly supported
and verify the following size assumptions
for $i=1,2,3$:
\begin{subequations}
\begin{align} \label{E:DATASIZE}
&
\| \nabla^{\leq 2} \mathring{\Psi}_i \|_{L^{\infty}(\Sigma_0)}
+
\| \nabla^{[1,3]} \mathring{\Psi}_0 \|_{L^{\infty}(\Sigma_0)}
\\
& \ \
+
\| \mathring{\Psi}_i \|_{H^5(\Sigma_0)}
+
\mathring{\upepsilon}^{3/2} \| \nabla \mathring{\Psi}_0 \|_{L^2(\Sigma_0)}
+
\| \nabla^2 \mathring{\Psi}_0 \|_{H^3(\Sigma_0)}
\leq \mathring{\upepsilon},
\notag
\end{align}
\begin{align}
\| \mathring{\Psi}_0 \|_{L^{\infty}(\Sigma_0)}
& \leq \mathring{A},
\label{E:LARGEDATASIZE}
\end{align}
\begin{align}
- \frac{1}{4}
& \leq \min_{\Sigma_0} \mathring{\Psi}_0,
\label{E:PSI0NOTTOONEGATIVE}
\end{align}
\end{subequations}
where $\mathring{\upepsilon} > 0$
and $\mathring{A} > 0$
are two data-size parameters that
we will discuss below
(roughly, $\mathring{\upepsilon}$ will have to be small for our proofs to close).
Roughly, in our analysis, we will propagate the above
size assumptions during the solution's classical lifespan.
A possible exception can occur for the top-order spatial derivatives of $\Psi_i$,
which we are are not able to control uniformly in the norm $\| \cdot \|_{L^2(\Sigma_t)}$
due to the presence of the weight $\mathscr{W}$ in our energy,
which can go to $0$ as the singularity forms
(see Def.~\ref{D:L2CONTROLLINGQUANTITY}).
We now introduce the
crucial parameter $\mathring{A}_*$ that
controls the time of first blowup;
our analysis shows that for
$\mathring{\upepsilon}$ sufficiently small, the time
of first blowup is
$\left\lbrace
1 + \mathcal{O}(\mathring{\upepsilon})
\right\rbrace
\mathring{A}_*^{-1}$;
see also Remark~\ref{R:CRUCIALDELTAPARAMETER}.
\begin{definition}[\textbf{The parameter that controls the time of first blowup}]
\label{D:CRUCIALDATASIZEPARAMETER}
We define the data-dependent parameter $\mathring{A}_*$ as follows:
\begin{align} \label{E:CRUCIALDATASIZEPARAMETER}
\mathring{A}_*
& :=
\max_{\Sigma_0} [\mathring{\Psi}_0]_+,
\end{align}
where $[\mathring{\Psi}_0]_+ := \max \lbrace \mathring{\Psi}_0,0 \rbrace$.
\end{definition}
Our main results concern solutions such that
$\mathring{A}_* > 0$, so we will assume in the
rest of the article that this is the case.
\begin{remark}[\textbf{The relevance of $\mathring{A}_*$}]
\label{R:CRUCIALDELTAPARAMETER}
The solutions that we study are such that\footnote{Here ``$A \sim B$'' imprecisely indicates
that $A$ is well-approximated by $B$.}
$
\partial_t \mathcal{I}
= - \Psi_0
$
and
$
\partial_t \Psi_0 \sim 0
$
(throughout the evolution).
Hence, by the fundamental theorem of calculus,
we have
$\Psi_0(t,\underline{x}) \sim \mathring{\Psi}_0(\underline{x})$
and $\mathcal{I}(t,\underline{x}) \sim 1 - t \mathring{\Psi}_0(\underline{x})$.
From this last expression, we see that
$\mathcal{I}$ is expected to vanish for the first time at approximately
$t = \mathring{A}_*^{-1}$ which, since $\partial_t \mathcal{I} = - \mathcal{I} \partial_t \Phi$,
implies the blowup of $\partial_t \Phi$ (see Remark~\ref{R:VANISHINGOFINTEGRATINGFACTORISSINGULARITY}).
See Lemmas~\ref{L:POINTWISEFORPSIANDPARTIALTPSI}
and \ref{L:INTEGRATINGFACTORCRUCIALESTIMATES}
for the precise statements.
\end{remark}
\subsection{Bootstrap assumptions}
\label{SS:BOOTSTRAP}
To prove our main results, we find it convenient to rely on
a set of bootstrap assumptions,
which we provide in this subsection.
\noindent \underline{\textbf{The size of} $T_{(Boot)}$}.
We assume that $T_{(Boot)}$
is a bootstrap time with
\begin{align} \label{E:BOOTSTRAPTIME}
0 < T_{(Boot)} \leq 2 \mathring{A}_*^{-1},
\end{align}
where $\mathring{A} > 0$ is the data-size parameter from Def.~\ref{D:CRUCIALDATASIZEPARAMETER}.
The assumption \eqref{E:BOOTSTRAPTIME}
gives us a sufficient margin of error
to prove that finite-time blowup occurs
(see Remark~\ref{R:CRUCIALDELTAPARAMETER}).
\noindent \underline{\textbf{Blowup has not yet occurred}}.
Recall that for the solutions under study, the vanishing of $\mathcal{I}$
will coincide with the formation of a singularity in $\partial_t \Phi$.
For this reason,
we assume that for $t \in [0,T_{(Boot)})$,
we have
\begin{align} \label{E:HYPERBOLICBOOTSTRAP}
\mathcal{I}_{\star}(t) > 0,
\end{align}
where $\mathcal{I}_{\star}$ is defined in \eqref{E:IFACTMIN}.
\underline{\textbf{The solution is contained in the regime of hyperbolicity}}.\footnote{In particular, the assumptions of
Subsect.\ \ref{SS:WEIGHTASSUMPTIONS} guarantee that $\mathscr{W}(\mathcal{I}^{-1} \Psi_0) > 0$
whenever \eqref{E:BOOTSTRAPRATIO} holds.}
We assume that for $(t,\underline{x}) \in [0,T_{(Boot)}) \times \mathbb{R}^3$, we have
\begin{align} \label{E:BOOTSTRAPRATIO}
\frac{\Psi_0(t,\underline{x})}{\mathcal{I}(t,\underline{x})}
> - \frac{1}{2}.
\end{align}
\noindent \underline{\textbf{Smallness of} $\mathcal{I}$ \textbf{implies largeness of} $\Psi_0$}.
We assume that for $(t,\underline{x}) \in [0,T_{(Boot)}) \times \mathbb{R}^3$,
\begin{align} \label{E:BOOTSTRAPSMALLINTFACTIMPMLIESPSI0ISLARGE}
\mathcal{I}(t,\underline{x})
\leq \frac{1}{8}
\implies
\Psi_0(t,\underline{x})
\geq \frac{1}{8} \mathring{A}_*.
\end{align}
\noindent \underline{$L^{\infty}$ \textbf{bootstrap assumptions}}.
We assume that for $t \in [0,T_{(Boot)})$, we have
\begin{subequations}
\begin{align} \label{E:PSI0ITSELFBOOTSTRAP}
\| \Psi_0 \|_{L^{\infty}(\Sigma_t)}
& \leq
\mathring{A}
+
\varepsilon,
\\
\| \nabla^{[1,3]} \Psi_0 \|_{L^{\infty}(\Sigma_t)}
& \leq \varepsilon,
\label{E:PSI0DERIVATVESBOOTSTRAP} \\
\| \nabla^{\leq 2} \Psi_i \|_{L^{\infty}(\Sigma_t)}
& \leq \varepsilon,
\label{E:PSIIANDDERIVATIVESBOOTSTRAP}
\\
\| \mathcal{I} \|_{L^{\infty}(\Sigma_t)}
& \leq
1
+
2 \mathring{A}_*^{-1} \mathring{A}
+
\varepsilon,
\label{E:IFACTITSELFBOOTSTRAP}
\end{align}
\end{subequations}
where $\varepsilon > 0$ is a small bootstrap parameter;
we describe our smallness assumptions in the next subsection.
\begin{remark}[\textbf{The solution remains compactly supported in space}]
\label{R:BOUNDEDWAVESPEED}
From the bootstrap assumptions
and the assumptions of Subsect.\ \ref{SS:WEIGHTASSUMPTIONS} on $\mathscr{W}$,
we see that the wave speed
$\left\lbrace
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\right\rbrace^{1/2}$
associated to equation~\eqref{E:WAVE}
remains uniformly bounded for $(t,\underline{x}) \in [0,T_{(Boot)}) \times \mathbb{R}^3$.
It follows that there exists a large, data-dependent ball $B \subset \mathbb{R}^3$
such that $\Psi_{\alpha}(t,\underline{x})$ and $\mathcal{I} - 1$
vanish for $(t,\underline{x}) \in [0,T_{(Boot)}) \times B^c$.
\end{remark}
\subsection{Smallness assumptions}
\label{SS:SMALLNESSASSUMPTIONS}
For the rest of the article,
when we say that ``$A$ is small relative to $B$,''
we mean that $B > 0$ and that there exists a continuous increasing function
$f :(0,\infty) \rightarrow (0,\infty)$
such that
$
\displaystyle
A < f(B)
$.
For brevity, we typically do not
specify the form of $f$.
In the rest of the article, we make the following
relative smallness assumptions. We
continually adjust the required smallness
in order to close the estimates.
\begin{itemize}
\item The bootstrap parameter $\varepsilon$ from Subsect.\ \ref{SS:BOOTSTRAP}
is small relative to $1$ (i.e., in an absolute sense, without regard for the other parameters).
\item $\varepsilon$ is small relative to $\mathring{A}^{-1}$,
where $\mathring{A}$ is the data-size parameter
from \eqref{E:LARGEDATASIZE}.
\item $\varepsilon$ is small relative to
the data-size parameter $\mathring{A}_*$
from \eqref{E:CRUCIALDATASIZEPARAMETER}.
\item We assume that
\begin{align} \label{E:DATAEPSILONVSBOOTSTRAPEPSILON}
\varepsilon^{4/3}
& \leq
\mathring{\upepsilon}
\leq \varepsilon,
\end{align}
where $\mathring{\upepsilon}$ is the data smallness parameter from \eqref{E:DATASIZE}.
\end{itemize}
The first two assumptions will allow us to control error terms that,
roughly speaking, are of size $\varepsilon \mathring{A}^k$
for some integer $k \geq 0$. The third assumption
is relevant because the expected blowup-time is
approximately $\mathring{A}_*^{-1}$
(see Remark~\ref{R:CRUCIALDELTAPARAMETER});
the assumption will allow us to show that various
error products featuring a small factor $\varepsilon$
remain small for $t \leq 2 \mathring{A}_*^{-1}$,
which is plenty of time for us to show that $\mathcal{I}$ vanishes
and $\partial_t \Phi$ blows up.
\eqref{E:DATAEPSILONVSBOOTSTRAPEPSILON} is convenient
for closing our bootstrap argument.
\subsection{Existence of initial data verifying the smallness assumptions}
\label{SS:EXISTENCEOFDATA}
It is easy to construct initial data
such that the parameters
$\mathring{\upepsilon}$,
$\mathring{A}$,
and
$\mathring{A}_*$
satisfy the size assumptions stated in Subsect.\ \ref{SS:SMALLNESSASSUMPTIONS}.
For example, we can start with \emph{any} smooth compactly supported data
$(\mathring{\Psi}_0,\mathring{\Psi}_1,\mathring{\Psi}_2,\mathring{\Psi}_3)$
such that $\max_{\Sigma_0} \mathring{\Psi}_0 > 0$
and $-\frac{1}{4} \leq \min_{\Sigma_0} \mathring{\Psi}_0$.
We then consider the one-parameter family (for $i=1,2,3$)
\[
\left(
\leftexp{(\uplambda)}{\mathring{\Psi}_0}(\underline{x}),
\leftexp{(\uplambda)}{\mathring{\Psi}_i}(\underline{x})
\right)
:=
\left(
\mathring{\Psi}_0(\uplambda^{-1} \underline{x}),
\uplambda^{-1} \mathring{\Psi}_i(\underline{x})
\right).
\]
It is straightforward to check that for $\uplambda > 0$ sufficiently large,
all of the size assumptions of
Subsect.\ \ref{SS:SMALLNESSASSUMPTIONS} are satisfied by the rescaled data
(where, roughly speaking, the role of $\mathring{\upepsilon}$ is played by $\uplambda^{-1}$),
as is \eqref{E:PSI0NOTTOONEGATIVE}.
The proof relies on the simple scaling identities
$\nabla^k \leftexp{(\uplambda)}{\mathring{\Psi}_0}(\underline{x})
= \uplambda^{-k} (\nabla^k \mathring{\Psi}_0)(\uplambda^{-1} \underline{x})
$,
$\nabla^k \leftexp{(\uplambda)}{\mathring{\Psi}_i}(\underline{x})
= \uplambda^{-1} (\nabla^k \mathring{\Psi}_i)(\underline{x})
$,
$
\left\|
\nabla^k \leftexp{(\uplambda)}{\mathring{\Psi}_0}
\right\|_{L^2(\Sigma_0)}
= \uplambda^{3/2 - k}
\| \mathring{\Psi}_0 \|_{L^2(\Sigma_0)}
$,
and
$
\left\|
\nabla^k \leftexp{(\uplambda)}{\mathring{\Psi}_i}
\right\|_{L^2(\Sigma_0)}
= \uplambda^{-1} \| \mathring{\Psi}_i \|_{L^2(\Sigma_0)}
$.
\begin{remark}[\textbf{Blowup generically occurs for appropriately
rescaled non-trivial data}]
\label{R:GENERICDEGENERACY}
The discussion in Subsect.\ \ref{SS:EXISTENCEOFDATA}
can easily be extended to show that
if $\max_{\Sigma_0} \mathring{\Psi}_0 > 0$
and
$-\frac{1}{4} \leq \min_{\Sigma_0} \mathring{\Psi}_0$,
then
one \emph{always} generates data to which
our results apply
by considering the rescaled data
$
\left(
\leftexp{(\uplambda)}{\mathring{\Psi}_0}(\underline{x}),
\leftexp{(\uplambda)}{\mathring{\Psi}_i}(\underline{x})
\right)
$
with $\uplambda$ sufficiently large.
\end{remark}
\section{Energy identities}
\label{S:ENERGY}
In this section, we define the energies that we use to control the solution in $L^2$
up to top order. We then derive energy identities.
\subsection{Definitions}
\label{SS:ENERGYDEFINITIONS}
The following energy functional serves as a building block for our energies.
\begin{definition}[\textbf{Basic energy functional}]
\label{D:BASICENERGY}
To any array-valued function
$V = V(t,\underline{x}) := (V_0,V_1,V_2,V_3)$,
we associated the following energy:
\begin{align} \label{E:BASICENERGY}
\mathbb{E}[V]
& = \mathbb{E}[V](t)
:= \int_{\Sigma_t}
\left\lbrace
V_0^2
+
\sum_{a=1}^3
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
V_a^2
\right\rbrace
\, d \underline{x}.
\end{align}
\end{definition}
We now define $\mathbb{Q}_{(\mathring{\upepsilon})}(t)$, which is the main $L^2$-type
quantity that we use to control the solution up to top order.
\begin{definition} [\textbf{The} $L^2$-\textbf{controlling quantity}]
\label{D:L2CONTROLLINGQUANTITY}
Let $\mathring{\upepsilon} > 0$ be the data-size parameter from Subsect.\ \ref{SS:DATAASSUMPTIONS}.
We define the $L^2$-controlling quantity $\mathbb{Q}_{(\mathring{\upepsilon})}$ as follows:
\begin{align} \label{E:ENERGYTOCONTROLSOLNS}
\mathbb{Q}_{(\mathring{\upepsilon})}(t)
& :=
\sum_{k=2}^5
\int_{\Sigma_t}
\left\lbrace
|\nabla^{k} \Psi_0|^2
+
\sum_{a=1}^3
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
|\nabla^{k} \Psi_a|^2
\right\rbrace
\, d \underline{x}
\\
& \ \
+
\sum_{k=1}^4
\int_{\Sigma_t}
|\nabla^{k} \Psi_a|^2
\, d \underline{x}
+
\mathring{\upepsilon}^3
\int_{\Sigma_t}
\left\lbrace
|\nabla \Psi_0|^2
+
\sum_{a=1}^3
|\Psi_a|^2
\right\rbrace
\, d \underline{x}.
\notag
\end{align}
\end{definition}
\begin{remark}[\textbf{The} $\mathring{\upepsilon}$ \textbf{weight in the definition of} $\mathbb{Q}_{(\mathring{\upepsilon})}$]
\label{R:EPSILONWEIGHTSINCONTROLLINGQUANTITY}
Our main a priori energy estimate
shows that $\mathbb{Q}_{(\mathring{\upepsilon})}(t) \lesssim \mathring{\upepsilon}^2$
up to the singularity.
The small coefficient of $\mathring{\upepsilon}^3$
in front of the last integral on RHS~\eqref{E:ENERGYTOCONTROLSOLNS}
is needed to ensure the $\mathcal{O}(\mathring{\upepsilon}^2)$
smallness of $\mathbb{Q}_{(\mathring{\upepsilon})}$.
However, the small coefficient of $\mathring{\upepsilon}^3$
implies that $\mathbb{Q}_{(\mathring{\upepsilon})}(t)$
provides only weak $L^2$ control of
$\nabla \Psi_0$
and $\Psi_a$,
i.e., their $L^2$ norms can be as large as
$\mathcal{O}(\mathring{\upepsilon}^{-1/2})$.
We clarify that the possible $\mathcal{O}(\mathring{\upepsilon}^{-1/2})$ size of $\nabla \Psi_0$
is consistent with the construction of initial data described in
Subsect.\ \ref{SS:EXISTENCEOFDATA}.
Despite the possible $\mathcal{O}(\mathring{\upepsilon}^{-1/2})$ largeness,
we will nonetheless be able to show,
through a separate argument,
the following crucial bounds:
$\nabla \Psi_0$
and $\Psi_a$
are bounded in the norm $\| \cdot \|_{L^{\infty}(\Sigma_t)}$ by $\lesssim \mathring{\upepsilon}$,
up to the singularity; see Prop.~\ref{P:APRIORIESTIMATES}.
\end{remark}
\subsection{Basic energy identity}
\label{SS:BASICENERGYIDENTITY}
We aim to derive an energy identity for
the controlling quantity $\mathbb{Q}_{(\mathring{\upepsilon})}$
defined in \eqref{E:ENERGYTOCONTROLSOLNS}.
As a preliminary step, in this subsection,
we derive a standard energy identity for the building block energy from \eqref{E:BASICENERGY}.
\begin{lemma}[\textbf{Basic energy identity}]
\label{L:BASICENERGYID}
Let $\mathbb{E}[V](t)$ be the building block energy defined in \eqref{E:BASICENERGY}.
Solutions $V := (V_0,V_1,V_2,V_3)$ to the inhomogeneous linear system
\begin{subequations}
\begin{align}
\partial_t V_0
& =
\sum_{a=1}^3
\mathscr{W}(\mathcal{I}^{-1} \Psi_0) \partial_a V_a
+
F_0,
\label{E:LINEARPSI0EVOLUTION} \\
\partial_t V_i
&
= \partial_i V_0
+
F_i
\label{E:LINEARPSIIEVOLUTION}
\end{align}
\end{subequations}
verify the following energy identity:
\begin{align} \label{E:BASICENERGYID}
\frac{d}{dt}
\mathbb{E}[V](t)
& =
\sum_{a=1}^3
\int_{\Sigma_t}
(\mathcal{I}^{-1} \Psi_0)^2 \mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
(V_a)^2
\, d \underline{x}
\\
& \ \
+
\sum_{a=1}^3
\sum_{b=1}^3
\int_{\Sigma_t}
\mathcal{I}^{-1} \mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\partial_a \Psi_a
(V_b)^2
\, d \underline{x}
\notag \\
& \ \
+ \sum_{a=1}^3
\sum_{b=1}^3
\int_{\Sigma_t}
\mathcal{I}^{-2}
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
(\Psi_a)^2
(V_b)^2
\, d \underline{x}
\notag \\
& \ \
-
\sum_{a=1}^3
\sum_{b=1}^3
\int_{\Sigma_t}
\mathcal{I}^{-1}
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\mathring{\Psi}_a \Psi_a
(V_b)^2
\, d \underline{x}
\notag \\
& \ \
-
2
\sum_{a=1}^3
\int_{\Sigma_t}
\mathcal{I}^{-1} \mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
(\partial_a \Psi_0) V_a V_0
\, d \underline{x}
\notag
\\
& \ \
-
2
\sum_{a=1}^3
\int_{\Sigma_t}
\mathcal{I}^{-2} \Psi_0
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\Psi_a V_a V_0
\, d \underline{x}
\notag
\\
& \ \
+
2
\sum_{a=1}^3
\int_{\Sigma_t}
\mathcal{I}^{-1} \Psi_0
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\mathring{\Psi}_a V_a V_0
\, d \underline{x}
\notag
\\
& \ \
+
2
\int_{\Sigma_t}
V_0 F_0
\, d \underline{x}
+
2
\sum_{a=1}^3
\int_{\Sigma_s}
\mathscr{W}(\mathcal{I}^{-1} \Psi_0) V_a F_a
\, d \underline{x}.
\notag
\end{align}
\end{lemma}
\begin{proof}
First, using equations \eqref{E:INTEGRATINFACTORODEANDIC} and \eqref{E:PARTALTPSI0EVOLUTION},
we compute that
\begin{align} \label{E:TIMEDERIVATIVEOFWEIGHT}
\partial_t
\left\lbrace
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\right\rbrace
&
=
\mathcal{I}^{-1} \mathscr{W}'(\mathcal{I}^{-1} \Psi_0) (\partial_t \Psi_0)
+
(\mathcal{I}^{-1} \Psi_0)^2 \mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\\
& =
\sum_{a=1}^3
\mathcal{I}^{-1} \mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
(\partial_a \Psi_a)
\notag \\
& \ \
+
\sum_{a=1}^3
\mathcal{I}^{-2}
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
(\Psi_a)^2
\notag \\
& \ \
-
\sum_{a=1}^3
\mathcal{I}^{-1}
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\mathring{\Psi}_a \Psi_a
\notag \\
& \ \
+
(\mathcal{I}^{-1} \Psi_0)^2
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0).
\notag
\end{align}
Taking the time derivative of \eqref{E:BASICENERGY}, using \eqref{E:TIMEDERIVATIVEOFWEIGHT},
and using \eqref{E:LINEARPSI0EVOLUTION}-\eqref{E:LINEARPSIIEVOLUTION}
for substitution, we obtain
\begin{align} \label{E:TIMEDERIVATIVEOFBASICENERGY}
\frac{d}{dt} \mathbb{E}[V](t)
& =
2
\sum_{a=1}^3
\int_{\Sigma_t}
\left\lbrace
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
V_0 \partial_a V_a
+
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
V_a \partial_a V_0
\right\rbrace
\, d \underline{x}
\\
& \ \
+
2
\int_{\Sigma_t}
\left\lbrace
V_0 F_0
+
\sum_{a=1}^3
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
V_a F_a
\right\rbrace
\, d \underline{x}
\notag \\
& \ \
+
\sum_{a=1}^3
\sum_{b=1}^3
\int_{\Sigma_t}
\mathcal{I}^{-1} \mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
(\partial_a \Psi_a)
(V_b)^2
\, d \underline{x}
\notag
\\
& \ \
+
\sum_{a=1}^3
\sum_{b=1}^3
\int_{\Sigma_t}
\mathcal{I}^{-2}
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
(\Psi_a)^2
(V_b)^2
\, d \underline{x}
\notag
\\
& \ \
-
\sum_{a=1}^3
\sum_{b=1}^3
\int_{\Sigma_t}
\mathcal{I}^{-1}
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\mathring{\Psi}_a \Psi_a
(V_b)^2
\, d \underline{x}
\notag
\\
& \ \
+
\sum_{a=1}^3
\int_{\Sigma_t}
(\mathcal{I}^{-1} \Psi_0)^2 \mathscr{W}'(\mathcal{I}^{-1} \Psi_0) (V_a)^2
\, d \underline{x}.
\notag
\end{align}
Integrating by parts in the first integral on RHS~\eqref{E:TIMEDERIVATIVEOFBASICENERGY}
and using the identity \eqref{E:FORMULAFORDERIVATIVESOFINTEGRATINGFACTOR},
we obtain
\begin{align} \label{E:IBPFIRSTINTEGRALINTIMEDERIVATIVEOFBASICENERGY}
&
2
\sum_{a=1}^3
\int_{\Sigma_t}
\left\lbrace
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
V_0 \partial_a V_a
+
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
V_a \partial_a V_0
\right\rbrace
\, d \underline{x}
\\
& =
- 2
\sum_{a=1}^3
\int_{\Sigma_t}
\mathcal{I}^{-1} \mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
(\partial_a \Psi_0) V_a V_0
\, d \underline{x}
\notag \\
& \ \
-
2
\sum_{a=1}^3
\int_{\Sigma_t}
\mathcal{I}^{-2} \Psi_0
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\Psi_a V_a V_0
\, d \underline{x}
+
2
\sum_{a=1}^3
\int_{\Sigma_t}
\mathcal{I}^{-1} \Psi_0
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\mathring{\Psi}_a V_a V_0
\, d \underline{x}.
\notag
\end{align}
Using \eqref{E:IBPFIRSTINTEGRALINTIMEDERIVATIVEOFBASICENERGY} to substitute
for the first integral on RHS~\eqref{E:TIMEDERIVATIVEOFBASICENERGY},
we arrive at \eqref{E:BASICENERGYID}.
\end{proof}
\subsection{Integral identity for the fundamental $L^2$-controlling quantity}
We now derive an energy identity for
the controlling quantity $\mathbb{Q}_{(\mathring{\upepsilon})}$.
\begin{lemma}[\textbf{Integral identity for the $L^2$-controlling quantity}]
\label{L:INTEGRALIDENTITYFORENERGY}
Consider the following inhomogeneous system,
obtained by commuting \eqref{E:PARTALTPSI0EVOLUTION}-\eqref{E:PARTALTPSIIEVOLUTION}
with $\nabla^k$:
\begin{subequations}
\begin{align}
\partial_t \nabla^k \Psi_0
& =
\mathscr{W}(\mathcal{I}^{-1} \Psi_0) \sum_{a=1}^3 \partial_a \nabla^k \Psi_a
+
F_0^{(k)},
\label{E:ENERGYIDCOMMUTEDPARTALTPSI0EVOLUTION} \\
\partial_t \nabla^k \Psi_i
&
= \partial_i \nabla^k \Psi_0
+ F_i^{(k)}.
\label{E:ENERGYIDCOMMUTEDPARTALTPSIIEVOLUTION}
\end{align}
\end{subequations}
For solutions,
the $L^2$-controlling quantity $\mathbb{Q}_{(\mathring{\upepsilon})}$ of Def.~\ref{D:L2CONTROLLINGQUANTITY}
satisfies the following integral identity:
\begin{align} \label{E:INTEGRALIDENTITYFORENERGY}
\mathbb{Q}_{(\mathring{\upepsilon})}(t)
& =
\mathbb{Q}_{(\mathring{\upepsilon})}(0)
+
\sum_{k=2}^5
\sum_{a=1}^3
\int_{s=0}^t
\int_{\Sigma_s}
(\mathcal{I}^{-1} \Psi_0)^2
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
|\nabla^{k} \Psi_a|^2
\, d \underline{x}
\, ds
\\
& \ \
+
\sum_{k=2}^5
\sum_{a=1}^3
\sum_{b=1}^3
\int_{s=0}^t
\int_{\Sigma_s}
\mathcal{I}^{-1} \mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
(\partial_a \Psi_a)
|\nabla^{k} \Psi_b|^2
\, d \underline{x}
\, ds
\notag \\
& \ \
+ \sum_{k=2}^5
\sum_{a=1}^3
\sum_{b=1}^3
\int_{s=0}^t
\int_{\Sigma_s}
\mathcal{I}^{-2}
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
|\Psi_a|^2
|\nabla^{k} \Psi_b|^2
\, d \underline{x}
\, ds
\notag \\
& \ \
-
\sum_{k=2}^5
\sum_{a=1}^3
\sum_{b=1}^3
\int_{s=0}^t
\int_{\Sigma_s}
\mathcal{I}^{-1}
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\mathring{\Psi}_a \Psi_a
|\nabla^{k} \Psi_b|^2
\, d \underline{x}
\, ds
\notag \\
& \ \
-
2
\sum_{k=2}^5
\sum_{a=1}^3
\int_{s=0}^t
\int_{\Sigma_s}
\mathcal{I}^{-1} \mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
(\partial_a \Psi_0) \nabla^{k} \Psi_a \cdot \nabla^{k} \Psi_0
\, d \underline{x}
\, ds
\notag
\\
& \ \
-
2
\sum_{k=2}^5
\sum_{a=1}^3
\int_{s=0}^t
\int_{\Sigma_s}
\mathcal{I}^{-2} \Psi_0
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\Psi_a \nabla^{k} \Psi_a \cdot \nabla^{k} \Psi_0
\, d \underline{x}
\, ds
\notag \\
& \ \
+
2
\sum_{k=2}^5
\sum_{a=1}^3
\int_{s=0}^t
\int_{\Sigma_s}
\mathcal{I}^{-1} \Psi_0
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\mathring{\Psi}_a \nabla^{k} \Psi_a \cdot \nabla^{k} \Psi_0
\, d \underline{x}
\, ds
\notag
\\
& \ \
+
2
\sum_{a=1}^3
\sum_{k=1}^4
\int_{s=0}^t
\int_{\Sigma_s}
\nabla^{k} \Psi_a \cdot \partial_a \nabla^{k} \Psi_0
\, d \underline{x}
\, ds
\notag \\
& \ \
+
2
\sum_{k=2}^5
\int_{s=0}^t
\int_{\Sigma_s}
\nabla^{k} \Psi_0 \cdot F_0^{(k)}
\, d \underline{x}
\, ds
\notag \\
& \ \
+
2
\sum_{k=2}^5
\sum_{a=1}^3
\int_{s=0}^t
\int_{\Sigma_s}
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\nabla^{k} \Psi_a \cdot F_a^{(k)}
\, d \underline{x}
\, ds
\notag \\
& \ \
+
2
\sum_{k=1}^4
\sum_{a=1}^3
\int_{s=0}^t
\int_{\Sigma_s}
\nabla^{k} \Psi_a \cdot F_a^{(k)}
\, d \underline{x}
\, ds
\notag \\
& \ \
+
2
\mathring{\upepsilon}^3
\sum_{a=1}^3
\int_{s=0}^t
\int_{\Sigma_s}
\mathscr{W}(\mathcal{I}^{-1} \Psi_0) \nabla \Psi_0 \cdot \partial_a \nabla \Psi_a
\, d \underline{x}
\, ds
\notag
\\
& \ \
+
2
\mathring{\upepsilon}^3
\int_{s=0}^t
\int_{\Sigma_s}
\nabla \Psi_0 \cdot F_0^{(1)}
\, d \underline{x}
\, ds
\notag \\
& \ \
+
2
\mathring{\upepsilon}^3
\sum_{a=1}^3
\int_{s=0}^t
\int_{\Sigma_s}
\Psi_a \partial_a \Psi_0
\, ds
\notag \\
& \ \
-
2
\mathring{\upepsilon}^3
\sum_{a=1}^3
\int_{s=0}^t
\int_{\Sigma_s}
\Psi_0
\Psi_a \mathring{\Psi}_a
\, d \underline{x}
\, ds.
\notag
\end{align}
\end{lemma}
\begin{proof}
We take the time derivative of both sides of \eqref{E:ENERGYTOCONTROLSOLNS}.
The time derivative of the first line of RHS~\eqref{E:ENERGYTOCONTROLSOLNS}
is given by \eqref{E:BASICENERGYID}, where the role of
$(V_0,V_1,V_2,V_3)$
in \eqref{E:BASICENERGYID}
is played by
$(\nabla^{k} \Psi_0,\nabla^{k} \Psi_1,\nabla^{k} \Psi_2,\nabla^{k} \Psi_3)$
and the role of the inhomogeneous terms
$F_{\alpha}$ on RHS~\eqref{E:BASICENERGYID}
is played by the terms $F_{\alpha}^{(k)}$
from \eqref{E:ENERGYIDCOMMUTEDPARTALTPSI0EVOLUTION}-\eqref{E:ENERGYIDCOMMUTEDPARTALTPSIIEVOLUTION}.
Moreover, with the help of
\eqref{E:PARTALTPSIIEVOLUTION}
and
\eqref{E:ENERGYIDCOMMUTEDPARTALTPSI0EVOLUTION}-\eqref{E:ENERGYIDCOMMUTEDPARTALTPSIIEVOLUTION},
we compute that the time derivatives of the terms on the second line of RHS~\eqref{E:ENERGYTOCONTROLSOLNS}
are equal to
\begin{align} \label{E:ENERGYTIMEDERIVATIVESOFLOWERORDERTERMS}
&
2
\sum_{k=1}^4
\sum_{a=1}^3
\int_{\Sigma_t}
\nabla^{k} \Psi_a \cdot \partial_a \nabla^{k} \Psi_0
\, d \underline{x}
+
2
\sum_{k=1}^4
\sum_{a=1}^3
\int_{\Sigma_t}
\nabla^{k} \Psi_a \cdot F_a^{(k)}
\, d \underline{x}
\\
& \ \
+
2 \mathring{\upepsilon}^3
\sum_{a=1}^3
\int_{\Sigma_t}
\left\lbrace
\mathscr{W}(\mathcal{I}^{-1} \Psi_0) \nabla \Psi_0 \cdot \partial_a \nabla \Psi_a
+
\nabla \Psi_0 \cdot F_0^{(1)}
+
\Psi_a \partial_a \Psi_0
-
\Psi_0 \Psi_a \mathring{\Psi}_a
\right\rbrace
\, d \underline{x}.
\notag
\end{align}
Combining these calculations, we deduce
\begin{align} \label{E:DIFFERENTIALFORMENERGYID}
\frac{d}{dt}
\mathbb{Q}_{(\mathring{\upepsilon})}(t)
&
=
\sum_{k=2}^5
\sum_{a=1}^3
\int_{\Sigma_t}
(\mathcal{I}^{-1} \Psi_0)^2 \mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
|\nabla^{k} \Psi_a|^2
\, d \underline{x}
\\
& \ \
+
\sum_{k=2}^5
\sum_{a=1}^3
\sum_{b=1}^3
\int_{\Sigma_t}
\mathcal{I}^{-1} \mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\partial_a \Psi_a
|\nabla^{k} \Psi_b|^2
\, d \underline{x}
\notag \\
& \ \
+ \sum_{k=2}^5
\sum_{a=1}^3
\sum_{b=1}^3
\int_{\Sigma_t}
\mathcal{I}^{-2}
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
|\Psi_a|^2
|\nabla^{k} \Psi_b|^2
\, d \underline{x}
\notag \\
& \ \
-
\sum_{k=2}^5
\sum_{a=1}^3
\sum_{b=1}^3
\int_{\Sigma_t}
\mathcal{I}^{-1}
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\mathring{\Psi}_a \Psi_a
|\nabla^{k} \Psi_b|^2
\, d \underline{x}
\notag \\
& \ \
-
2
\sum_{k=2}^5
\sum_{a=1}^3
\int_{\Sigma_t}
\mathcal{I}^{-1} \mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
(\partial_a \Psi_0) \nabla^{k} \Psi_a \cdot \nabla^{k} \Psi_0
\, d \underline{x}
\notag
\\
& \ \
-
2
\sum_{k=2}^5
\sum_{a=1}^3
\int_{\Sigma_t}
\mathcal{I}^{-2} \Psi_0
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\Psi_a \nabla^{k} \Psi_a \cdot \nabla^{k} \Psi_0
\, d \underline{x}
\notag
\\
& \ \
+
2
\sum_{k=2}^5
\sum_{a=1}^3
\int_{\Sigma_t}
\mathcal{I}^{-1} \Psi_0
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\mathring{\Psi}_a \nabla^{k} \Psi_a \cdot \nabla^{k} \Psi_0
\, d \underline{x}
\notag
\\
& \ \
+
2
\sum_{k=2}^5
\int_{\Sigma_t}
\nabla^{k} \Psi_0 \cdot F_0^{(k)}
\, d \underline{x}
+
2
\sum_{a=1}^3
\int_{\Sigma_t}
\mathscr{W}(\mathcal{I}^{-1} \Psi_0) \nabla^{k} \Psi_a \cdot F_a^{(k)}
\, d \underline{x}
+
\mbox{\eqref{E:ENERGYTIMEDERIVATIVESOFLOWERORDERTERMS}}.
\notag
\end{align}
Integrating \eqref{E:DIFFERENTIALFORMENERGYID}
from time $0$ to time $t$,
we arrive at the desired identity \eqref{E:INTEGRALIDENTITYFORENERGY}.
\end{proof}
\section{A priori estimates}
\label{S:ESTIMATES}
In this section, we use the data-size and bootstrap assumptions of Sect.\ \ref{S:DATAANDBOOTSTRAP}
and the energy identities of Sect.\ \ref{S:ENERGY} to derive a priori estimates.
\subsection{Conventions for constants}
\label{SS:CONVENTIONSFORCONSTANTS}
In our estimates, the explicit constants $C > 0$ and $c > 0$ are free to vary from line to line.
\textbf{These explicit constants, and implicit ones as well, are allowed to depend on the data-size parameters
$\mathring{A}$
and
$\mathring{A}_*^{-1}$
from Subsect.\ \ref{SS:DATAASSUMPTIONS}}.
However, the constants can be chosen to be
independent of the parameters $\mathring{\upepsilon}$
and $\varepsilon$ whenever $\mathring{\upepsilon}$ and $\varepsilon$
are sufficiently small relative to
$\mathring{A}^{-1}$
and $\mathring{A}_*$
in the sense described in Subsect.\ \ref{SS:SMALLNESSASSUMPTIONS}.
For example, under our conventions, we have that $\mathring{A}_*^{-2} \varepsilon = \mathcal{O}(\varepsilon)$.
\subsection{Pointwise estimates tied to the integrating factor}
\label{SS:POINTWISEESTIMATES}
In this subsection, we derive pointwise estimates that are important for
analyzing $\mathcal{I}$.
We start by deriving sharp estimates for $\Psi_0$.
The proof is based on separately considering regions where
$\mathcal{I}$ is small and $\mathcal{I}$ is large.
In Lemma~\ref{L:INTEGRATINGFACTORCRUCIALESTIMATES},
we will use these estimates
to derive further information about the behavior of $\Psi_0$ in regions where
$\mathcal{I}$ is small (i.e., near the singularity),
which is crucial for closing the energy estimates.
\begin{lemma}[\textbf{Pointwise estimates for} $\Psi_0$]
\label{L:POINTWISEFORPSIANDPARTIALTPSI}
Under the data-size assumptions of Subsect.\ \ref{SS:DATAASSUMPTIONS},
the bootstrap assumptions of Subsect.\ \ref{SS:BOOTSTRAP},
and the smallness assumptions of Subsect.\ \ref{SS:SMALLNESSASSUMPTIONS},
the following pointwise estimates hold for
$(t,\underline{x}) \in [0, T_{(Boot)}) \times \mathbb{R}^3$:
\begin{align}
\Psi_0(t,\underline{x})
& =
\mathring{\Psi}_0(\underline{x})
+
\mathcal{O}(\varepsilon),
\label{E:PSI0WELLAPPROXIMATED}
\end{align}
where $\mathring{\Psi}_0(\underline{x}) = \Psi_0(0,\underline{x})$.
In addition,
\begin{align} \label{E:PSI0BIGGERTHANMINUSONEHALF}
-5/16 & \leq \min_{\Sigma_t} \Psi_0.
\end{align}
\end{lemma}
\begin{proof}
We first prove \eqref{E:PSI0WELLAPPROXIMATED}.
We will show that
$\left| \partial_t \Psi_0(t,\underline{x}) \right| \lesssim \varepsilon$.
Then from this estimate and the fundamental theorem of calculus,
we obtain the desired bound \eqref{E:PSI0WELLAPPROXIMATED}.
It remains for us to prove the bound
$\left| \partial_t \Psi_0(t,\underline{x}) \right| \lesssim \varepsilon$.
We first consider points $(t,\underline{x})$ such that $\mathcal{I}(t,\underline{x}) > 1/8$.
Then all factors of $\mathcal{I}^{-1}$ in the evolution equation \eqref{E:PARTALTPSI0EVOLUTION} can be bounded
by $\lesssim 1$. For this reason,
the desired bound follows as a straightforward consequence of
equation \eqref{E:PARTALTPSI0EVOLUTION},
the bootstrap assumptions,
the data-size assumptions \eqref{E:DATASIZE},
and the assumptions of Subsect.\ \ref{SS:WEIGHTASSUMPTIONS} on $\mathscr{W}$.
We now prove the desired bound at points $(t,\underline{x})$ such that
$0 < \mathcal{I}(t,\underline{x}) \leq 1/8$.
From the bootstrap assumption \eqref{E:BOOTSTRAPSMALLINTFACTIMPMLIESPSI0ISLARGE},
we deduce that $1 \lesssim \Psi_0(t,\underline{x})$ at such points.
From this bound, the bootstrap assumptions,
the data-size assumptions \eqref{E:DATASIZE},
and the assumptions of Subsect.\ \ref{SS:WEIGHTASSUMPTIONS} on $\mathscr{W}$,
we deduce the following bound for some factors on RHS~\eqref{E:PARTALTPSI0EVOLUTION}
at the spacetime points under consideration:
\[
\left|\mathcal{I}^{-1} \mathscr{W}(\mathcal{I}^{-1} \Psi_0)\right|
=
\Psi_0^{-1}
\left|(\mathcal{I}^{-1} \Psi_0) \mathscr{W}(\mathcal{I}^{-1} \Psi_0)\right|
\lesssim
\left|(\mathcal{I}^{-1} \Psi_0) \mathscr{W}(\mathcal{I}^{-1} \Psi_0)\right|
\lesssim 1.
\]
With the help of this bound,
the desired estimate
$\left| \partial_t \Psi_0(t,\underline{x}) \right| \lesssim \varepsilon$
follows as a straightforward consequence of
equation \eqref{E:PARTALTPSI0EVOLUTION},
the bootstrap assumptions,
the data-size assumptions \eqref{E:DATASIZE},
and the assumptions of Subsect.\ \ref{SS:WEIGHTASSUMPTIONS} on $\mathscr{W}$.
We have therefore proved \eqref{E:PSI0WELLAPPROXIMATED}.
The bound \eqref{E:PSI0BIGGERTHANMINUSONEHALF} then follows
from \eqref{E:PSI0NOTTOONEGATIVE}
and
\eqref{E:PSI0WELLAPPROXIMATED}.
\end{proof}
In the next lemma, we derive sharp estimates for $\mathcal{I}$.
The estimates are important for closing the energy estimates up to the singularity
and for precisely tying the vanishing of $\mathcal{I}$ to the blowup of $\partial_t \Phi$.
\begin{lemma}[\textbf{Crucial estimates for the integrating factor}]
\label{L:INTEGRATINGFACTORCRUCIALESTIMATES}
Under the data-size assumptions of Subsect.\ \ref{SS:DATAASSUMPTIONS},
the bootstrap assumptions of Subsect.\ \ref{SS:BOOTSTRAP},
and the smallness assumptions of Subsect.\ \ref{SS:SMALLNESSASSUMPTIONS},
the following estimates hold for $(t,\underline{x}) \in [0,T_{(Boot)}) \times \mathbb{R}^3$:
\begin{subequations}
\begin{align} \label{E:IFACTCRUCIALPOINTWISE}
\mathcal{I}(t,\underline{x})
& = 1 - t \mathring{\Psi}_0(\underline{x}) + \mathcal{O}(\varepsilon),
\\
\mathcal{I}_{\star}(t)
& = 1 - t \mathring{A}_* + \mathcal{O}(\varepsilon),
\label{E:IFACTSTARCRUCIALPOINTWISE}
\end{align}
\end{subequations}
where $\mathring{\Psi}_0(\underline{x}) = \Psi_0(0,\underline{x})$
and $\mathring{A}_* > 0$ is the data-size parameter from Def.~\ref{D:CRUCIALDATASIZEPARAMETER}.
Moreover, the following implications hold for $(t,\underline{x}) \in [0,T_{(Boot)}) \times \mathbb{R}^3$:
\begin{subequations}
\begin{align} \label{E:SMALLINTFACTIMPMLIESRATIOISLARGE}
\mathcal{I}(t,\underline{x})
\leq \frac{1}{4} \min \lbrace 1, \mathring{A}_* \rbrace
\implies
\frac{\Psi_0(t,\underline{x})}{\mathcal{I}(t,\underline{x})}
\geq 1,
\\
\mathcal{I}(t,\underline{x})
\leq \frac{1}{4}
\implies
\Psi_0(t,\underline{x})
\geq \frac{1}{4} \mathring{A}_*.
\label{E:SMALLINTFACTIMPMLIESPSI0ISLARGE}
\end{align}
\end{subequations}
Finally, the following implications hold for $(t,\underline{x}) \in [0,T_{(Boot)}) \times \mathbb{R}^3$:
\begin{align} \label{E:PSI0NEGATIVEIMPMLIESINHYPERBOLICREGIME}
\Psi(t,\underline{x})
& \leq 0
\implies
\mathcal{I}(t,\underline{x})
\geq 1 - \mathcal{O}(\varepsilon)
&&
\mbox{and }
\Psi(t,\underline{x})
\leq 0
\implies
\frac{\Psi_0(t,\underline{x})}{\mathcal{I}(t,\underline{x})}
\geq - \frac{3}{8}.
\end{align}
\end{lemma}
\begin{remark}[\textbf{Improvement of a bootstrap assumption}]
\label{R:IMPROVEMENTOFANUNUSUALBOOTSTRAPASSUMPTION}
Note in particular that
the estimate \eqref{E:SMALLINTFACTIMPMLIESPSI0ISLARGE}
provides a strict improvement of the bootstrap assumption
\eqref{E:BOOTSTRAPSMALLINTFACTIMPMLIESPSI0ISLARGE}.
\end{remark}
\begin{remark}[\textbf{The significance of} \eqref{E:PSI0NEGATIVEIMPMLIESINHYPERBOLICREGIME}]
\label{R:SOLUTIONREMAINSINSIDEREGIMEOFHYPERBOLICITITY}
Note that
\eqref{E:PSI0NEGATIVEIMPMLIESINHYPERBOLICREGIME}
is a strict improvement of the bootstrap assumption
\eqref{E:BOOTSTRAPRATIO}
and implies that $-3/8 \leq \partial_t \Phi(t,\underline{x})$
for
$(t,\underline{x}) \in [0,T_{(Boot)}) \times \mathbb{R}^3$.
In view of the assumption \eqref{E:WEIGHTISPOSITIVE} for $\mathscr{W}$,
we conclude that the solution never escapes the set of state-space
values for which the wave equation \eqref{E:WAVE} is hyperbolic.
In the rest of article, we often silently use this fact.
\end{remark}
\begin{proof}
From equation \eqref{E:INTEGRATINFACTORODEANDIC} and the estimate \eqref{E:PSI0WELLAPPROXIMATED},
we deduce
$\partial_t \mathcal{I}(t,\underline{x}) = - \mathring{\Psi}_0(\underline{x}) + \mathcal{O}(\varepsilon)$.
Integrating in time and using the initial condition \eqref{E:INTEGRATINFACTORODEANDIC}, we
find that $\mathcal{I}(t,\underline{x}) = 1 - t \mathring{\Psi}_0(\underline{x}) + \mathcal{O}(\varepsilon)$,
which is \eqref{E:IFACTCRUCIALPOINTWISE}.
\eqref{E:IFACTSTARCRUCIALPOINTWISE} follows a simple consequence of \eqref{E:IFACTCRUCIALPOINTWISE}
and Def.~\ref{D:CRUCIALDATASIZEPARAMETER}.
To prove \eqref{E:SMALLINTFACTIMPMLIESRATIOISLARGE}, we
first consider the case $\mathring{A}_* \geq 1$.
From
\eqref{E:IFACTCRUCIALPOINTWISE} and
\eqref{E:PSI0WELLAPPROXIMATED},
we deduce that
$\mathcal{I}(t,\underline{x}) = 1 - t \Psi_0(t,\underline{x}) + \mathcal{O}(\varepsilon)$.
It follows that if $\mathcal{I}(t,\underline{x}) \leq 1/4$,
then $t \Psi_0(t,\underline{x}) \geq 1/2$.
Since $0 \leq t \leq 2 \mathring{A}_*^{-1} \leq 2$,
we deduce that
$
\frac{\Psi_0(t,\underline{x})}{\mathcal{I}(t,\underline{x})}
\geq 1
$,
which is the desired conclusion.
Next, we consider the case $\mathring{A}_* < 1$.
Using \eqref{E:IFACTCRUCIALPOINTWISE} and
\eqref{E:PSI0WELLAPPROXIMATED},
we deduce that
$\mathcal{I}(t,\underline{x}) = 1 - t \Psi_0(t,\underline{x}) + \mathcal{O}(\varepsilon)$.
It follows that if $\mathcal{I}(t,\underline{x}) \leq (1/4) \mathring{A}_*$,
then $t \Psi_0(t,\underline{x}) \geq 1 - (1/2) \mathring{A}_*$.
Since $0 \leq t \leq 2 \mathring{A}_*^{-1}$,
we deduce that
$
\frac{\Psi_0(t,\underline{x})}{\mathcal{I}(t,\underline{x})}
\geq 2 \left\lbrace 1 - (1/2) \mathring{A}_* \right\rbrace
= 2 - \mathring{A}_*
$,
which, in view of our assumption $\mathring{A}_* < 1$,
is
$
> 1
$.
This completes our proof of \eqref{E:SMALLINTFACTIMPMLIESRATIOISLARGE}.
The implication \eqref{E:SMALLINTFACTIMPMLIESPSI0ISLARGE}
can be proved using arguments similar to the ones that we used to prove
\eqref{E:SMALLINTFACTIMPMLIESRATIOISLARGE}, and we therefore omit the details.
Next, we note that when $\Psi_0(t,\underline{x}) \leq 0$,
the estimate
$\mathcal{I}(t,\underline{x}) = 1 - t \Psi_0(t,\underline{x}) + \mathcal{O}(\varepsilon)$
proved above implies that $\mathcal{I}(t,\underline{x}) \geq 1 - \mathcal{O}(\varepsilon)$,
which yields the first implication stated in
\eqref{E:PSI0NEGATIVEIMPMLIESINHYPERBOLICREGIME}.
To obtain the second implication stated in
\eqref{E:PSI0NEGATIVEIMPMLIESINHYPERBOLICREGIME},
we use the first implication and the estimate
\eqref{E:PSI0BIGGERTHANMINUSONEHALF}.
\end{proof}
In the next lemma, we derive some simple pointwise estimates showing
the spatial derivatives of $\mathcal{I}$ up to top order can be controlled in terms
of the spatial derivatives of $\lbrace \Psi_a \rbrace_{a=1,2,3}$.
\begin{lemma}[\textbf{Estimates for the derivatives of the integrating factor}]
\label{L:POINTWISEESTIMATEFORDERIVATIVESOFIFACT}
Under the data-size assumptions of Subsect.\ \ref{SS:DATAASSUMPTIONS},
the bootstrap assumptions of Subsect.\ \ref{SS:BOOTSTRAP},
and the smallness assumptions of Subsect.\ \ref{SS:SMALLNESSASSUMPTIONS},
the following pointwise estimates hold
for $(t,\underline{x}) \in [0,T_{(Boot)}) \times \mathbb{R}^3$:
\begin{subequations}
\begin{align}
\left|
\nabla \mathcal{I}
\right|
& \lesssim
\sum_{a=1}^3
\left|
\Psi_a
\right|
+
\sum_{a=1}^3
\left|
\mathring{\Psi}_a
\right|.
\label{E:LOWESTLEVELPOINTWISEESTIMATEFORDERIVATIVESOFIFACT}
\end{align}
Moreover, for $2 \leq k \leq 6$, the following estimate holds:
\begin{align}
\left|
\nabla^k \mathcal{I}
\right|
& \lesssim
\sum_{a=1}^3
\left|
\nabla^{[1,k-1]} \Psi_a
\right|
+
\sum_{a=1}^3
\left|
\nabla^{[1,k-1]} \mathring{\Psi}_a
\right|
+
\varepsilon
\sum_{a=1}^3
\left|
\mathring{\Psi}_a
\right|.
\label{E:POINTWISEESTIMATEFORDERIVATIVESOFIFACT}
\end{align}
\end{subequations}
Finally, the following estimate holds for $t \in [0,T_{(Boot)})$:
\begin{align} \label{E:LINFINITYFORDERIVATIVESOFIFACT}
\left\|
\nabla^{[1,3]} \mathcal{I}
\right\|_{L^{\infty}(\Sigma_t)}
& \lesssim
\varepsilon.
\end{align}
\end{lemma}
\begin{proof}
The estimate \eqref{E:LOWESTLEVELPOINTWISEESTIMATEFORDERIVATIVESOFIFACT}
is straightforward consequence
of equation \eqref{E:FORMULAFORDERIVATIVESOFINTEGRATINGFACTOR}
and the bootstrap assumptions.
Similarly, the estimate \eqref{E:POINTWISEESTIMATEFORDERIVATIVESOFIFACT}
is straightforward to derive via induction in $k$ with the help
of equation \eqref{E:FORMULAFORDERIVATIVESOFINTEGRATINGFACTOR},
the bootstrap assumptions, the data-size assumptions \eqref{E:DATASIZE},
and \eqref{E:DATAEPSILONVSBOOTSTRAPEPSILON}.
\eqref{E:LINFINITYFORDERIVATIVESOFIFACT} then follows from
\eqref{E:LOWESTLEVELPOINTWISEESTIMATEFORDERIVATIVESOFIFACT}-\eqref{E:POINTWISEESTIMATEFORDERIVATIVESOFIFACT},
the bootstrap assumptions, the data-size assumptions \eqref{E:DATASIZE},
and \eqref{E:DATAEPSILONVSBOOTSTRAPEPSILON}.
\end{proof}
\subsection{Pointwise estimates involving the weight}
\label{SS:POINTWISEWEIGHT}
In the next lemma, we derive precise pointwise estimates
for quantities that involve the weight function
$\mathscr{W}$. The detailed information is important for
closing the energy estimates
and for showing that the spatial derivatives of $\mathscr{W} = \mathscr{W}(\partial_t \Phi)$
are controllable. Some of the analysis is delicate in that
$\partial_t \Phi$ and its derivatives
are allowed to be arbitrarily large
(i.e., the estimates hold uniformly, arbitrarily close to the expected singularity).
\begin{lemma}[\textbf{Pointwise estimates involving the weight} $\mathscr{W}$]
\label{L:ESTIMATESINVOLVINGWEIGHT}
Let $\mathbf{1}_{\left\lbrace 0 < \mathcal{I} \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}$
be the characteristic function of the spacetime subset
$
\lbrace
(t,\underline{x})
\ | \
0 < \mathcal{I} (t,\underline{x})
\leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace
\rbrace
$.
Under the data-size assumptions of Subsect.\ \ref{SS:DATAASSUMPTIONS},
the bootstrap assumptions of Subsect.\ \ref{SS:BOOTSTRAP},
and the smallness assumptions of Subsect.\ \ref{SS:SMALLNESSASSUMPTIONS},
the following pointwise estimates hold
for $(t,\underline{x}) \in [0,T_{(Boot)}) \times \mathbb{R}^3$:
\begin{subequations}
\begin{align}
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
& \lesssim 1,
\label{E:WEIGHTBOUNDEDBYONE}
\\
\left|
\nabla
\left\lbrace
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\right\rbrace
\right|
& \lesssim
\varepsilon
\mathbf{1}_{\left\lbrace 0 < \mathcal{I} \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}
\left\lbrace
\mathcal{I}^{-2}
\left|
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|
\right\rbrace^{1/2}
+
\varepsilon
\left\lbrace
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\right\rbrace^{1/2}
\label{E:WEIGHTEXACTLYONEDERIVATIVEPOINTWISE} \\
& \lesssim
\varepsilon.
\label{E:WEIGHTEXACTLYONEDERIVATIVEPOINTWISESMALL}
\end{align}
\end{subequations}
In addition, for $2 \leq k \leq 5$, the following estimates hold:
\begin{align}
\left|
\nabla^k
\left\lbrace
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\right\rbrace
\right|
& \lesssim
\left|
\nabla^{[1,k]} \Psi_0
\right|
+
\sum_{a=1}^3
\left|
\nabla^{\leq k-1} \Psi_a
\right|
+
\sum_{a=1}^3
\left|
\nabla^{\leq k-1} \mathring{\Psi}_a
\right|.
\label{E:WEIGHTATLEASTONEDERIVATIVEPOINTWISE}
\end{align}
Furthermore, the following estimates hold:
\begin{subequations}
\begin{align}
\mathcal{I}^{-1}
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
& \lesssim
\mathbf{1}_{\left\lbrace 0 < \mathcal{I} \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}
\left\lbrace
\mathcal{I}^{-2}
\left|
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|
\right\rbrace^{1/2}
+
\left\lbrace
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\right\rbrace^{1/2}
\label{E:SINGULARFACTORTIMESWEIGHTPOINTWISE}
\\
& \lesssim 1.
\label{E:SINGULARFACTORTIMESWEIGHTBOUNDEDBYONE}
\end{align}
\end{subequations}
Moreover,
for $1 \leq k \leq 5$,
the following estimates hold:
\begin{align}
\left|
\nabla^k
\left\lbrace
\mathcal{I}^{-1}
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\right\rbrace
\right|
& \lesssim
\left|
\nabla^{[1,k]} \Psi_0
\right|
+
\sum_{a=1}^3
\left|
\nabla^{\leq k-1} \Psi_a
\right|
+
\sum_{a=1}^3
\left|
\nabla^{\leq k-1} \mathring{\Psi}_a
\right|.
\label{E:ATLEASTONEDERIVATIVESINGULARFACTORTIMESWEIGHTPOINTWISE}
\end{align}
Finally, for $P \in [0,2]$, the following estimates hold:
\begin{subequations}
\begin{align}
\left|
\mathcal{I}^{-2}
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
+
\mathbf{1}_{\left\lbrace 0 < \mathcal{I} \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}
\mathcal{I}^{-2}
\left|
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|
\right|
&
\lesssim
\mathscr{W}(\mathcal{I}^{-1} \Psi_0),
\label{E:POINTWISEESTIMATESIFACTMINUSTWOTIMESWEIGHTDERIVATIVE} \\
\left|
\mathcal{I}^{-P}
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|
& \lesssim 1.
\label{E:BOUNDEDBYONEPOINTWISEESTIMATESIFACTMINUSTWOTIMESWEIGHTDERIVATIVE}
\end{align}
\end{subequations}
\end{lemma}
\begin{proof}
Throughout this proof, we denote
$
\displaystyle
y = y(t,\underline{x}):= \frac{\Psi_0(t,\underline{x})}{\mathcal{I}(t,\underline{x})}
$.
Also, we silently use the observations of Remark~\ref{R:SOLUTIONREMAINSINSIDEREGIMEOFHYPERBOLICITITY}.
\noindent \textbf{Proof of \eqref{E:WEIGHTBOUNDEDBYONE}}:
This bound is a trivial consequence of our assumption \eqref{E:ESTIMATEFORDERIVATIVESOFWEIGHT} on $\mathscr{W}$.
\noindent \textbf{Proof of \eqref{E:WEIGHTEXACTLYONEDERIVATIVEPOINTWISE} and \eqref{E:WEIGHTEXACTLYONEDERIVATIVEPOINTWISESMALL}}:
We first prove \eqref{E:WEIGHTEXACTLYONEDERIVATIVEPOINTWISE}
at spacetime points $(t,\underline{x})$ such that
$\mathcal{I}(t,\underline{x}) > (1/4) \min \lbrace 1, \mathring{A}_* \rbrace$.
This is the easy case because $\mathcal{I}^{-1} < 4 \max \lbrace 1, \mathring{A}^{-1} \rbrace \leq C$,
and we therefore do not have to concern ourselves
with the possibility of small denominators. Specifically,
using the identity \eqref{E:FORMULAFORDERIVATIVESOFINTEGRATINGFACTOR},
the bootstrap assumptions,
the data-size assumptions \eqref{E:DATASIZE},
and the assumptions of Subsect.\ \ref{SS:WEIGHTASSUMPTIONS},
we deduce that when $\mathcal{I}(t,\underline{x}) > (1/4) \min \lbrace 1, \mathring{A}_* \rbrace$,
we have
\begin{align} \label{E:EASYCASEWEIGHTEXACTLYONEDERIVATIVEPOINTWISE}
\left|
\nabla
\left\lbrace
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\right\rbrace
\right|
& \lesssim
\left|
\nabla \Psi_0
\right|
+
\sum_{a=1}^3
\left|
\Psi_a
\right|
+
\sum_{a=1}^3
\left|
\mathring{\Psi}_a
\right|
\lesssim \varepsilon.
\end{align}
Next, we use the bootstrap assumptions
and the assumptions of Subsect.\ \ref{SS:WEIGHTASSUMPTIONS} on $\mathscr{W}$
(specifically the uniform positivity of $\mathscr{W}(y)$ for $y \in [-3/8,C]$)
to obtain
\[
\mathbf{1}_{\left\lbrace \mathcal{I} > (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}
\lesssim
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\lesssim
\left\lbrace
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\right\rbrace^{1/2}.
\]
It follows that
$\mbox{RHS~\eqref{E:EASYCASEWEIGHTEXACTLYONEDERIVATIVEPOINTWISE}}$
is
$\lesssim$ the second term on RHS~\eqref{E:WEIGHTEXACTLYONEDERIVATIVEPOINTWISE}
as desired.
We now prove \eqref{E:WEIGHTEXACTLYONEDERIVATIVEPOINTWISE}
at points $(t,\underline{x})$ such that $0 < \mathcal{I}(t,\underline{x}) \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace$.
Along the way, we will prove some additional estimates that we will use later on.
We start by defining the following weighted differential operator, which acts on
functions $f = f(y)$:
$
D_Y f := y^2 \frac{d}{dy} f
$.
Note that the chain rule implies that
\begin{align}
\nabla \mathscr{W}(y)
= - D_Y \mathscr{W}(y) \nabla (y^{-1}).
\end{align}
We therefore inductively deduce that for $1 \leq k \leq 5$, we have
\begin{align} \label{E:FIRSTPOINTWISEESTIMATECHAINRULEWITHWEIGHT}
\left|
\nabla^k \mathscr{W}(y)
\right|
& \lesssim
\sum_{n=1}^k
\left|
D_Y^n \mathscr{W}(y)
\right|
\left\lbrace
\mathop{\mathop{\sum}_{\sum_{i=1}^n k_i = k}}_{k_i \geq 1}
\prod_{i=1}^n
\left|
\nabla^{k_i} (y^{-1})
\right|
\right\rbrace.
\end{align}
The case $k=1$ in \eqref{E:FIRSTPOINTWISEESTIMATECHAINRULEWITHWEIGHT} yields
$
\left|
\nabla \mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\right|
\lesssim
(\mathcal{I}^{-1} \Psi_0)^2
\left|
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|
\left|
\nabla
\left\lbrace
\mathcal{I} \Psi_0^{-1}
\right\rbrace
\right|
$.
Also using the identity \eqref{E:FORMULAFORDERIVATIVESOFINTEGRATINGFACTOR},
the bootstrap assumptions,
the data-size assumptions \eqref{E:DATASIZE},
\eqref{E:DATAEPSILONVSBOOTSTRAPEPSILON},
the assumptions of Subsect.\ \ref{SS:WEIGHTASSUMPTIONS},
and the crucially important estimate \eqref{E:SMALLINTFACTIMPMLIESPSI0ISLARGE}
(which implies that $\Psi_0^{-1} \lesssim 1$),
we deduce that when $\mathcal{I}(t,\underline{x}) \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace$,
we have
$
\left|
\nabla
\left\lbrace
\mathcal{I} \Psi_0^{-1}
\right\rbrace
\right|
\lesssim \varepsilon
$
and thus
\begin{align} \label{E:INEQUALITYCHAIN}
\left|
\nabla \mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\right|
& \lesssim
\varepsilon
(\mathcal{I}^{-1} \Psi_0)^2
\left|
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|
\lesssim
\varepsilon
\left\lbrace
(\mathcal{I}^{-1} \Psi_0)^2
\left|
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|
\right\rbrace^{1/2}
\\
& \lesssim
\varepsilon
\left\lbrace
\mathcal{I}^{-2}
\left|
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|
\right\rbrace^{1/2},
\notag
\end{align}
which is $\lesssim$ the first term on RHS~\eqref{E:WEIGHTEXACTLYONEDERIVATIVEPOINTWISE} as desired.
This finishes the proof of \eqref{E:WEIGHTEXACTLYONEDERIVATIVEPOINTWISE}.
We clarify that to derive the next-to-last inequality in \eqref{E:INEQUALITYCHAIN},
in which we bounded
$
(\mathcal{I}^{-1} \Psi_0)^2
\left|
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|
$
by its square root,
we used \eqref{E:ESTIMATEFORDERIVATIVESOFWEIGHT}
to deduce
$y^2 \left| \mathscr{W}'(y) \right| \lesssim 1$.
We now prove \eqref{E:WEIGHTEXACTLYONEDERIVATIVEPOINTWISESMALL}.
From Remark~\ref{R:SOLUTIONREMAINSINSIDEREGIMEOFHYPERBOLICITITY},
the assumptions of Subsect.\ \ref{SS:WEIGHTASSUMPTIONS} on $\mathscr{W}$,
and \eqref{E:SMALLINTFACTIMPMLIESPSI0ISLARGE},
we deduce that
\begin{align} \label{E:MAINTERMWEIGHTEXACTLYONEDERIVATIVEPOINTWISESMALL}
\mathbf{1}_{\left\lbrace 0 < \mathcal{I} \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}
\left\lbrace
\mathcal{I}^{-2}
\left|
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|
\right\rbrace^{1/2}
& \lesssim
\mathbf{1}_{\left\lbrace 0 < \mathcal{I} \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}
\left\lbrace
(\mathcal{I}^{-2} \Psi_0^2)
\left|
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|
\right\rbrace^{1/2}
\\
& \lesssim 1
\notag
\end{align}
and that
$
\left\lbrace
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\right\rbrace^{1/2}
\lesssim 1
$.
That is, the non-$\varepsilon$ factors on RHS~\eqref{E:WEIGHTEXACTLYONEDERIVATIVEPOINTWISE}
are $\lesssim 1$. This yields \eqref{E:WEIGHTEXACTLYONEDERIVATIVEPOINTWISESMALL}.
\noindent \textbf{Proof of \eqref{E:WEIGHTATLEASTONEDERIVATIVEPOINTWISE}}:
The proof is similar to that of \eqref{E:WEIGHTEXACTLYONEDERIVATIVEPOINTWISE} but slightly simpler.
Note that $k \in [2,5]$ by assumption in this estimate.
We first prove the estimate at points $(t,\underline{x})$ such that
$\mathcal{I}(t,\underline{x}) > (1/4) \min \lbrace 1, \mathring{A}_* \rbrace$.
This is the easy case because $\mathcal{I}^{-1} < 4 \max \lbrace 1, \mathring{A}^{-1} \rbrace \leq C$,
and we therefore do not have to concern ourselves
with the possibility of small denominators. Specifically,
using the identity \eqref{E:FORMULAFORDERIVATIVESOFINTEGRATINGFACTOR},
the bootstrap assumptions,
the data-size assumptions \eqref{E:DATASIZE},
\eqref{E:DATAEPSILONVSBOOTSTRAPEPSILON},
and the assumptions of Subsect.\ \ref{SS:WEIGHTASSUMPTIONS},
we deduce that when $\mathcal{I}(t,\underline{x}) > (1/4) \min \lbrace 1, \mathring{A}_* \rbrace$,
we have
\begin{align} \label{E:EASYCASEWEIGHTATLEASTONEDERIVATIVEPOINTWISE}
\left|
\nabla^k
\left\lbrace
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\right\rbrace
\right|
& \lesssim
\left|
\nabla^{[1,k]} \Psi_0
\right|
+
\sum_{a=1}^3
\left|
\nabla^{\leq k-1} \Psi_a
\right|
+
\sum_{a=1}^3
\left|
\nabla^{\leq k-1} \mathring{\Psi}_a
\right|,
\end{align}
which is $\lesssim \mbox{RHS~\eqref{E:WEIGHTATLEASTONEDERIVATIVEPOINTWISE}}$ as desired.
It remains for us to prove \eqref{E:WEIGHTATLEASTONEDERIVATIVEPOINTWISE} at points $(t,\underline{x})$
such that $0 < \mathcal{I}(t,\underline{x}) \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace$.
Note that the estimate \eqref{E:FIRSTPOINTWISEESTIMATECHAINRULEWITHWEIGHT} holds
and that by \eqref{E:ESTIMATEFORDERIVATIVESOFWEIGHT} and Remark~\ref{R:SOLUTIONREMAINSINSIDEREGIMEOFHYPERBOLICITITY},
we have the following bound\footnote{In obtaining this bound, it is helpful to note that
$D_Y f = - \frac{d}{dz} f$, where $z := 1/y$.
\label{FN:DBIGYVSDDZ}}
for the factors of $D_Y^n \mathscr{W}(y)$ on RHS~\eqref{E:FIRSTPOINTWISEESTIMATECHAINRULEWITHWEIGHT}:
$
\left|
D_Y^n \mathscr{W}(y)
\right|
\lesssim 1
$.
From this bound,
\eqref{E:FIRSTPOINTWISEESTIMATECHAINRULEWITHWEIGHT},
the bootstrap assumptions,
and the data-size assumptions \eqref{E:DATASIZE},
we see that the desired bound
\eqref{E:WEIGHTATLEASTONEDERIVATIVEPOINTWISE} will follow once we show that the following bound holds
when $2 \leq k \leq 5$ and $\mathcal{I}(t,\underline{x}) \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace$:
\begin{align} \label{E:POINTWISEBOUNDFORDERIVATIVESOFIFACTOVERPSI0}
\left|
\nabla^k (y^{-1})
\right|
& \lesssim
\left|
\nabla^{[1,k]} \Psi_0
\right|
+
\sum_{a=1}^3
\left|
\nabla^{\leq k-1} \Psi_a
\right|
+
\sum_{a=1}^3
\left|
\nabla^{\leq k-1} \mathring{\Psi}_a
\right|.
\end{align}
To prove \eqref{E:POINTWISEBOUNDFORDERIVATIVESOFIFACTOVERPSI0},
we first note that \eqref{E:SMALLINTFACTIMPMLIESPSI0ISLARGE} implies that
$1 \lesssim \Psi_0(t,\underline{x})$ in the present context.
Thus,
$
\mbox{LHS~\eqref{E:EASYCASEWEIGHTATLEASTONEDERIVATIVEPOINTWISE}}
=
\left|
\nabla^k
\left(
\frac{\mathcal{I}}{\Psi_0}
\right)
\right|
$
is the $k^{th}$ derivative of a ratio with a denominator uniformly bounded from below away from $0$,
and the desired estimate \eqref{E:WEIGHTATLEASTONEDERIVATIVEPOINTWISE}
follows as a straightforward consequence of
the identity \eqref{E:FORMULAFORDERIVATIVESOFINTEGRATINGFACTOR},
the data-size assumptions \eqref{E:DATASIZE},
and the bootstrap assumptions.
\noindent \textbf{Proof of \eqref{E:SINGULARFACTORTIMESWEIGHTPOINTWISE},
\eqref{E:SINGULARFACTORTIMESWEIGHTBOUNDEDBYONE},
and \eqref{E:ATLEASTONEDERIVATIVESINGULARFACTORTIMESWEIGHTPOINTWISE}}:
These estimates can be proved using arguments similar to the ones
we used to prove
\eqref{E:WEIGHTEXACTLYONEDERIVATIVEPOINTWISE}
and
\eqref{E:WEIGHTATLEASTONEDERIVATIVEPOINTWISE},
based on separately considering the cases
$\mathcal{I}(t,\underline{x}) > (1/4) \min \lbrace 1, \mathring{A}_* \rbrace$
and
$0 < \mathcal{I}(t,\underline{x}) \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace$
and using the assumptions of Subsect.\ \ref{SS:WEIGHTASSUMPTIONS}.
We omit the details, noting only that we can write
$
\mathcal{I}^{-1} \mathscr{W}(\mathcal{I}^{-1} \Psi_0)
= \Psi_0^{-1} y \mathscr{W}(y)
$
and that
the assumptions of Subsect.\ \ref{SS:WEIGHTASSUMPTIONS}
(especially \eqref{E:WEIGHTVSWEIGHTDERIVATIVECOMPARISON}),
\eqref{E:SMALLINTFACTIMPMLIESRATIOISLARGE},
\eqref{E:SMALLINTFACTIMPMLIESPSI0ISLARGE},
and Remark~\ref{R:SOLUTIONREMAINSINSIDEREGIMEOFHYPERBOLICITITY}
imply that we have the following key estimates,
relevant for the more difficult case $0 < \mathcal{I}(t,\underline{x}) \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace$:
\begin{align} \label{E:USEOFWEIGHTVSWEIGHTPRIMEASSUMPTION}
\mathbf{1}_{\left\lbrace 0 < \mathcal{I} \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}
\left\lbrace
\Psi_0^{-1} y \mathscr{W}(y)
\right\rbrace
& \lesssim
\mathbf{1}_{\left\lbrace 0 < \mathcal{I} \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}
\left\lbrace
y^2
\left|
\mathscr{W}'(y)
\right|
\right\rbrace^{1/2}
\\
& \lesssim
\mathbf{1}_{\left\lbrace 0 < \mathcal{I} \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}
\left\lbrace
\mathcal{I}^{-2}
\left|
\mathscr{W}'(y)
\right|
\right\rbrace^{1/2}
\notag
\end{align}
and,
for $n \leq 5$:
$\left|
D_Y^n (y \mathscr{W}(y))
\right|
\lesssim 1
$
(Footnote~\ref{FN:DBIGYVSDDZ} is also relevant for obtaining this latter bound).
\noindent \textbf{Proof of \eqref{E:POINTWISEESTIMATESIFACTMINUSTWOTIMESWEIGHTDERIVATIVE}}:
We first note that by \eqref{E:WEIGHTPRIMEISNEGATIVE}
and
\eqref{E:SMALLINTFACTIMPMLIESRATIOISLARGE},
we have $\mathscr{W}'(\mathcal{I}^{-1} \Psi_0) < 0$
at points $(t,\underline{x})$ such that
$\mathcal{I}(t,\underline{x}) \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace$.
From this fact and the identity
$
1
=
\mathbf{1}_{\left\lbrace \mathcal{I} > (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}
+
\mathbf{1}_{\left\lbrace 0 < \mathcal{I} \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}
$,
it follows that
\[
\mbox{LHS~\eqref{E:POINTWISEESTIMATESIFACTMINUSTWOTIMESWEIGHTDERIVATIVE}}
=
\left|
\mathbf{1}_{\left\lbrace \mathcal{I} > (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}
\mathcal{I}^{-2}
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|
\lesssim
\mathbf{1}_{\left\lbrace \mathcal{I} > (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}
\left|
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|.
\]
Also using the bound $\left|\mathscr{W}'(y) \right| \lesssim 1$,
which is a simple consequence of \eqref{E:ESTIMATEFORDERIVATIVESOFWEIGHT},
we find that
$
\mbox{LHS~\eqref{E:POINTWISEESTIMATESIFACTMINUSTWOTIMESWEIGHTDERIVATIVE}}
\lesssim
\mathbf{1}_{\left\lbrace \mathcal{I} > (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}
$.
Next, we recall the estimate
$\mathbf{1}_{\left\lbrace \mathcal{I} > (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}
\lesssim \mathscr{W}(\mathcal{I}^{-1} \Psi_0)$
that we derived in our proof of
\eqref{E:WEIGHTEXACTLYONEDERIVATIVEPOINTWISE}.
Combining the above estimates, we conclude the desired bound \eqref{E:POINTWISEESTIMATESIFACTMINUSTWOTIMESWEIGHTDERIVATIVE}.
\noindent \textbf{Proof of \eqref{E:BOUNDEDBYONEPOINTWISEESTIMATESIFACTMINUSTWOTIMESWEIGHTDERIVATIVE}}:
We first prove \eqref{E:BOUNDEDBYONEPOINTWISEESTIMATESIFACTMINUSTWOTIMESWEIGHTDERIVATIVE} at points $(t,\underline{x})$
such that $\mathcal{I}(t,\underline{x}) > (1/4) \min \lbrace 1, \mathring{A}_* \rbrace$.
Using the bootstrap assumptions
and the assumptions of Subsect.\ \ref{SS:WEIGHTASSUMPTIONS} on $\mathscr{W}$, we deduce,
in view of Remark~\ref{R:SOLUTIONREMAINSINSIDEREGIMEOFHYPERBOLICITITY},
that
$
\left|
\mathcal{I}^{-P}
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|
\lesssim
\left|
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|
\lesssim 1
$
as desired.
It remains for us to prove \eqref{E:BOUNDEDBYONEPOINTWISEESTIMATESIFACTMINUSTWOTIMESWEIGHTDERIVATIVE} at points $(t,\underline{x})$
such that $0 < \mathcal{I}(t,\underline{x}) \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace$.
Using \eqref{E:SMALLINTFACTIMPMLIESPSI0ISLARGE}, we see
that $1 \lesssim \Psi_0(t,\underline{x})$ at such points,
and it follows that
$
\left|
\mathcal{I}^{-P}
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|
\lesssim
\left|
\left\lbrace
\mathcal{I}^{-1} \Psi_0
\right\rbrace^P
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|
$.
Using the assumptions of Subsect.\ \ref{SS:WEIGHTASSUMPTIONS} on $\mathscr{W}$ and the assumption $P \in [0,2]$,
we deduce,
in view of Remark~\ref{R:SOLUTIONREMAINSINSIDEREGIMEOFHYPERBOLICITITY},
that the RHS of the previous expression is $\lesssim 1$ as desired.
This finishes the proof of \eqref{E:BOUNDEDBYONEPOINTWISEESTIMATESIFACTMINUSTWOTIMESWEIGHTDERIVATIVE}
and completes the proof of the lemma.
\end{proof}
\subsection{Pointwise estimates for the inhomogeneous terms in the commuted evolution equations}
\label{SS:POINTWISEINHOMOGENEOUSEVOLUTIONEQUATIONS}
With the estimates of Lemma~\ref{L:ESTIMATESINVOLVINGWEIGHT} in hand, we are now ready to derive
pointwise estimates for the inhomogeneous terms in the $\nabla^k$-commuted evolution equations.
\begin{lemma}[\textbf{Pointwise estimates for the inhomogeneous terms}]
\label{L:POINTWISEESTIMATES}
Let $\mathcal{I}$ be a solution to \eqref{E:INTEGRATINFACTORODEANDIC} and let
$\lbrace \Psi_{\alpha} \rbrace_{\alpha =0,1,2,3}$ be a solution to the system \eqref{E:PARTALTPSI0EVOLUTION}-\eqref{E:PARTALTPSIIEVOLUTION}.
Consider the following system,\footnote{We do not bother to state the precise form of $F^{(k)}$ here.}
obtained by commuting \eqref{E:PARTALTPSI0EVOLUTION}-\eqref{E:PARTALTPSIIEVOLUTION}
with $\nabla^k$:
\begin{subequations}
\begin{align}
\partial_t \nabla^k \Psi_0
& =
\mathscr{W}(\mathcal{I}^{-1} \Psi_0) \sum_{a=1}^3 \partial_a \nabla^k \Psi_a
+
F_0^{(k)},
\label{E:COMMUTEDPARTALTPSI0EVOLUTION} \\
\partial_t \nabla^k \Psi_i
&
= \partial_i \nabla^k \Psi_0
+ F_i^{(k)}.
\label{E:COMMUTEDPARTALTPSIIEVOLUTION}
\end{align}
\end{subequations}
Under the data-size assumptions of Subsect.\ \ref{SS:DATAASSUMPTIONS},
the bootstrap assumptions of Subsect.\ \ref{SS:BOOTSTRAP},
and the smallness assumptions of Subsect.\ \ref{SS:SMALLNESSASSUMPTIONS},
for
$k=2,3,4,5$
and
$(t,\underline{x}) \in [0,T_{(Boot)}) \times \mathbb{R}^3$,
the following estimate holds:
\begin{align}
\left|
F_0^{(k)}
\right|
& \lesssim
\varepsilon
\left|
\nabla^{[2,k]} \Psi_0
\right|
+
\varepsilon
\mathbf{1}_{\left\lbrace 0 < \mathcal{I} \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}
\sum_{a=1}^3
\left\lbrace
\mathcal{I}^{-2}
\left|
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|
\right\rbrace^{1/2}
\left|
\nabla^k \Psi_a
\right|
\label{E:PSI0INHOMOGENEOUSTERMPOINTWISEBOUND} \\
& \ \
+
\varepsilon
\sum_{a=1}^3
\left\lbrace
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\right\rbrace^{1/2}
\left|
\nabla^k \Psi_a
\right|
\notag \\
& \ \
+
\sum_{a=1}^3
\left|
\nabla^{[1,k-1]} \Psi_a
\right|
+
\varepsilon^2
\sum_{a=1}^3
\left|
\Psi_a
\right|
+
\sum_{a=1}^3
\left|
\nabla^{\leq k} \mathring{\Psi}_a
\right|.
\notag
\end{align}
Moreover, for $k=0,1,2,3,4$, the following estimate holds:
\begin{align} \label{E:ANNOYINGLOWORDERPSI0INHOMOGENEOUSTERMPOINTWISEBOUND}
\left|
F_0^{(k)}
\right|
& \lesssim
\underbrace{
\varepsilon
\left|
\nabla^{[1,k]} \Psi_0
\right|
}_{\mbox{\upshape Absent if $k=0$}}
+
\underbrace{
\sum_{a=1}^3
\left|
\nabla^{[1,k]} \Psi_a
\right|
}_{\mbox{\upshape Absent if $k=0$}}
+
\varepsilon
\sum_{a=1}^3
\left|
\Psi_a
\right|
+
\sum_{a=1}^3
\left|
\nabla^{\leq k} \mathring{\Psi}_a
\right|.
\end{align}
Finally, for $k=0,1,2,3,4,5$, the following estimate holds:
\begin{align}
\sum_{a=1}^3
\left|
F_a^{(k)}
\right|
& \lesssim
\underbrace{
\varepsilon
\left|
\nabla^{[2,k]} \Psi_0
\right|
}_{\mbox{\upshape Absent if $k=0,1$}}
+
\sum_{a=1}^3
\left|
\nabla^{\leq k} \mathring{\Psi}_a
\right|.
\label{E:PSIIINHOMOGENEOUSTERMPOINTWISEBOUND}
\end{align}
\end{lemma}
\begin{proof}
The estimate \eqref{E:PSIIINHOMOGENEOUSTERMPOINTWISEBOUND}
follows in a straightforward fashion from commuting
equation \eqref{E:PARTALTPSIIEVOLUTION}
with $\nabla^k$ and using the bootstrap assumptions,
the data-size assumptions \eqref{E:DATASIZE},
and \eqref{E:DATAEPSILONVSBOOTSTRAPEPSILON}.
To prove \eqref{E:PSI0INHOMOGENEOUSTERMPOINTWISEBOUND},
we first commute equation \eqref{E:PARTALTPSI0EVOLUTION}
with $\nabla^k$ to obtain equation \eqref{E:COMMUTEDPARTALTPSI0EVOLUTION}.
The only products in $F_0^{(k)}$ that are difficult to bound are those
that feature a factor in which $k$ derivatives fall on $\Psi_a$,
specifically the products
$
\sum_{a=1}^3
\left\lbrace
\nabla
\left[
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\right]
\right\rbrace
\partial_a \nabla^{k-1} \Psi_a
$,
$
\sum_{a=1}^3
\mathcal{I}^{-1} \mathscr{W}(\mathcal{I}^{-1} \Psi_0) \Psi_a \nabla^k \Psi_a
$,
and
$
\sum_{a=1}^3
\mathscr{W}(\mathcal{I}^{-1} \Psi_0) \mathring{\Psi}_a \nabla^k \Psi_a
$.
To bound the first of these, we use
the estimate \eqref{E:WEIGHTEXACTLYONEDERIVATIVEPOINTWISE},
which implies that the product is bounded by the second and third terms on RHS~\eqref{E:PSI0INHOMOGENEOUSTERMPOINTWISEBOUND}
as desired.
To handle the second and third products,
we use
\eqref{E:WEIGHTBOUNDEDBYONE},
\eqref{E:SINGULARFACTORTIMESWEIGHTPOINTWISE},
the bootstrap assumptions,
the data-size assumptions \eqref{E:DATASIZE},
and \eqref{E:DATAEPSILONVSBOOTSTRAPEPSILON}
to bound them in magnitude by
\begin{align*}
&\lesssim
\varepsilon
\mathbf{1}_{\left\lbrace 0 < \mathcal{I} \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}
\sum_{a=1}^3
\left\lbrace
\mathcal{I}^{-2}
\left|
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|
\right\rbrace^{1/2}
\left|
\nabla^k \Psi_a
\right|
\\
& \ \
+
\varepsilon
\sum_{a=1}^3
\left\lbrace
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\right\rbrace^{1/2}
\left|
\nabla^k \Psi_a
\right|,
\end{align*}
which is in turn bounded by the second and third terms on RHS~\eqref{E:PSI0INHOMOGENEOUSTERMPOINTWISEBOUND}
as desired.
The remaining terms in $F_0^{(k)}$ feature $\leq k-1$ derivatives of $\Psi_a$.
These terms are easily seen to be $\lesssim \mbox{RHS}~\eqref{E:PSI0INHOMOGENEOUSTERMPOINTWISEBOUND}$
with the help of the estimates
\eqref{E:WEIGHTATLEASTONEDERIVATIVEPOINTWISE},
\eqref{E:SINGULARFACTORTIMESWEIGHTBOUNDEDBYONE},
and
\eqref{E:ATLEASTONEDERIVATIVESINGULARFACTORTIMESWEIGHTPOINTWISE},
the bootstrap assumptions,
the data-size assumptions \eqref{E:DATASIZE},
and \eqref{E:DATAEPSILONVSBOOTSTRAPEPSILON}.
The estimate \eqref{E:ANNOYINGLOWORDERPSI0INHOMOGENEOUSTERMPOINTWISEBOUND}
is easier to prove and can be obtained in a similar fashion
with the help of the estimates
\eqref{E:WEIGHTBOUNDEDBYONE},
\eqref{E:WEIGHTEXACTLYONEDERIVATIVEPOINTWISESMALL},
\eqref{E:WEIGHTATLEASTONEDERIVATIVEPOINTWISE},
\eqref{E:SINGULARFACTORTIMESWEIGHTBOUNDEDBYONE},
\eqref{E:ATLEASTONEDERIVATIVESINGULARFACTORTIMESWEIGHTPOINTWISE},
the bootstrap assumptions,
the data-size assumptions \eqref{E:DATASIZE},
and \eqref{E:DATAEPSILONVSBOOTSTRAPEPSILON}.
\end{proof}
\subsection{The main a priori estimates}
\label{SS:MAINAPRIORI}
We now derive the main result of this section: a priori estimates
that hold up to top order and that in particular yield a strict
improvement of the bootstrap assumptions. These are the main
ingredients in the proof of our main theorem.
\begin{proposition}[\textbf{The main a priori estimates}]
\label{P:APRIORIESTIMATES}
Let $\mathbf{1}_{\left\lbrace 0 < \mathcal{I} \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}$
be the characteristic function of the spacetime subset
$
\lbrace
(t,\underline{x})
\ | \
0 < \mathcal{I} (t,\underline{x})
\leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace
\rbrace
$.
There exists a constant $C > 0$ such that
under the data-size assumptions of Subsect.\ \ref{SS:DATAASSUMPTIONS},
the bootstrap assumptions of Subsect.\ \ref{SS:BOOTSTRAP},
and the smallness assumptions of Subsect.\ \ref{SS:SMALLNESSASSUMPTIONS},
for solutions to the system
\eqref{E:INTEGRATINFACTORODEANDIC}
+
\eqref{E:PARTALTPSI0EVOLUTION}-\eqref{E:PARTALTPSIIEVOLUTION},
the $L^2$-controlling quantity $\mathbb{Q}_{(\mathring{\upepsilon})}$ of Def.~\ref{D:L2CONTROLLINGQUANTITY}
verifies the following estimate for $t \in [0,T_{(Boot)})$:
\begin{align} \label{E:MAINAPRIORIENERGYESTIMATES}
&
\mathbb{Q}_{(\mathring{\upepsilon})}(t)
+
\frac{1}{20} \mathring{A}_*^2
\sum_{k=2}^5
\sum_{a=1}^3
\int_{s=0}^t
\int_{\Sigma_s}
\mathbf{1}_{\left\lbrace 0 < \mathcal{I} \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}
\mathcal{I}^{-2}
\left|
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|
|\nabla^{k} \Psi_a|^2
\, d \underline{x}
\, ds
\\
&
\leq C \mathring{\upepsilon}^2.
\notag
\end{align}
In addition the following estimates hold for $t \in [0,T_{(Boot)})$ and $i=1,2,3$:
\begin{subequations}
\begin{align} \label{E:PARTIALTPSI0H4ESTIMATE}
\mathring{\upepsilon}
\| \partial_t \Psi_0 \|_{L^2(\Sigma_t)}^2
+
\| \nabla \partial_t \Psi_0 \|_{H^3(\Sigma_t)}^2
& \leq C \mathring{\upepsilon}^2,
\\
\mathring{\upepsilon}^3
\| \partial_t \Psi_i \|_{L^2(\Sigma_t)}^2
+
\| \nabla \partial_t \Psi_i \|_{H^3(\Sigma_t)}^2
& \leq C \mathring{\upepsilon}^2.
\label{E:PARTIALTPSIIH4ESTIMATE}
\end{align}
\end{subequations}
Moreover, the integrating factor $\mathcal{I}$ from Def.~\ref{D:INTEGRATINGFACTOR}
verifies the following estimate for $t \in [0,T_{(Boot)})$:
\begin{align} \label{E:IFACTAPRIORIENERGYESTIMATE}
&
\mathring{\upepsilon}^3
\| \nabla \mathcal{I} \|_{L^2(\Sigma_t)}^2
+
\| \nabla^{[2,5]} \mathcal{I} \|_{L^2(\Sigma_t)}^2
\\
&
+
\int_{s=0}^t
\int_{\Sigma_s}
\mathbf{1}_{\left\lbrace 0 < \mathcal{I} \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}
\mathcal{I}^{-2}
\left|
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|
\left|
\nabla^6 \mathcal{I}
\right|^2
\, d \underline{x}
\, ds
\leq C \mathring{\upepsilon}^2.
\notag
\end{align}
Finally, we have the following estimates for $t \in [0,T_{(Boot)})$,
which in particular yield strict improvements of the bootstrap assumptions
\eqref{E:PSI0ITSELFBOOTSTRAP}-\eqref{E:IFACTITSELFBOOTSTRAP}
whenever $C \mathring{\upepsilon} < \varepsilon$:
\begin{subequations}
\begin{align} \label{E:PSI0ITSELFIMPROVED}
\| \Psi_0 \|_{L^{\infty}(\Sigma_t)}
& \leq
\mathring{A}
+
C \mathring{\upepsilon},
\\
\| \nabla^{[1,3]} \Psi_0 \|_{L^{\infty}(\Sigma_t)}
& \leq C \mathring{\upepsilon},
\label{E:PSI0DERIVATVESIMPROVED} \\
\| \nabla^{\leq 2} \Psi_i \|_{L^{\infty}(\Sigma_t)}
& \leq C \mathring{\upepsilon},
\label{E:PSIIANDDERIVATIVESIMPROVED}
\\
\| \mathcal{I} \|_{L^{\infty}(\Sigma_t)}
& \leq
1
+
2 \mathring{A}_*^{-1} \mathring{A}
+
C \mathring{\upepsilon},
\label{E:IFACTITSELFIMPROVED}
\\
\| \nabla^{[1,3]} \mathcal{I} \|_{L^{\infty}(\Sigma_t)}
& \leq C \mathring{\upepsilon}.
\label{E:IFACTDERIVATIVESLIFTYIMPROVED}
\end{align}
\end{subequations}
\end{proposition}
\begin{proof}
\noindent \textbf{Proof of \eqref{E:MAINAPRIORIENERGYESTIMATES}}:
The main step is to derive the following estimate:
\begin{align} \label{E:CONTROLLINGQUANTITYGRONWALLREADY}
&
\mathbb{Q}_{(\mathring{\upepsilon})}(t)
+
\frac{1}{16} \mathring{A}_*^2
\sum_{k=2}^5
\sum_{a=1}^3
\int_{s=0}^t
\int_{\Sigma_s}
\mathbf{1}_{\left\lbrace 0 < \mathcal{I} \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}
\mathcal{I}^{-2}
\left|
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|
|\nabla^{k} \Psi_a|^2
\, d \underline{x}
\, ds
\\
&
\leq C \mathring{\upepsilon}^2
+
C \varepsilon
\sum_{k=2}^5
\sum_{a=1}^3
\int_{s=0}^t
\int_{\Sigma_s}
\mathbf{1}_{\left\lbrace 0 < \mathcal{I} \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}
\mathcal{I}^{-2}
\left|
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|
|\nabla^{k} \Psi_a|^2
\, d \underline{x}
\notag \\
& \ \
+
C
\int_{s=0}^t
\mathbb{Q}_{(\mathring{\upepsilon})}(s)
\, ds.
\notag
\end{align}
Once we have shown
\eqref{E:CONTROLLINGQUANTITYGRONWALLREADY},
we can absorb the second term on RHS~\eqref{E:CONTROLLINGQUANTITYGRONWALLREADY}
into the second term on LHS~\eqref{E:CONTROLLINGQUANTITYGRONWALLREADY}, which,
for $\varepsilon$ sufficiently small,
at most reduces the coefficient of $\frac{1}{16} \mathring{A}_*^2$ in front of the second term on the left to the value of
$\frac{1}{20} \mathring{A}_*^2$,
as is stated on LHS~\eqref{E:MAINAPRIORIENERGYESTIMATES}.
We then use
Gronwall's inequality and the assumption $0 < t < T_{(Boot)} \leq 2 \mathring{A}_*^{-1}$
to conclude that
$\mbox{LHS}~\eqref{E:MAINAPRIORIENERGYESTIMATES}
\leq C \exp(C t) \mathring{\upepsilon}^2
\leq C \exp(C \mathring{A}_*^{-1}) \mathring{\upepsilon}^2
\leq C \mathring{\upepsilon}^2$
as desired.
To prove \eqref{E:CONTROLLINGQUANTITYGRONWALLREADY},
we must bound the terms on
RHS~\eqref{E:INTEGRALIDENTITYFORENERGY}.
First, we note the following bound for the first term on the RHS:
$\mathbb{Q}_{(\mathring{\upepsilon})}(0) \leq C \mathring{\upepsilon}^2$,
an estimate that follows as a straightforward consequence of
definition \eqref{E:ENERGYTOCONTROLSOLNS},
the data-size assumptions \eqref{E:DATASIZE}-\eqref{E:PSI0NOTTOONEGATIVE},
the initial condition $\mathcal{I}|_{\Sigma_0} = 1$ stated in \eqref{E:INTEGRATINFACTORODEANDIC},
and the assumptions of Subsect.\ \ref{SS:WEIGHTASSUMPTIONS} on $\mathscr{W}$.
Next, we treat the spacetime integral
on the first line of RHS~\eqref{E:INTEGRALIDENTITYFORENERGY}.
Using
\eqref{E:SMALLINTFACTIMPMLIESPSI0ISLARGE},
\eqref{E:POINTWISEESTIMATESIFACTMINUSTWOTIMESWEIGHTDERIVATIVE},
and the bootstrap assumption \eqref{E:PSI0DERIVATVESBOOTSTRAP} for
$\| \Psi_0 \|_{L^{\infty}(\Sigma_t)}$,
we can express the integral as the negative integral
\[
- \sum_{k=2}^5
\sum_{a=1}^3
\int_{s=0}^t
\int_{\Sigma_s}
\mathbf{1}_{\left\lbrace 0 < \mathcal{I} \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}
(\mathcal{I}^{-1} \Psi_0)^2
\left|
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|
|\nabla^{k} \Psi_a|^2
\, d \underline{x}
\, ds,
\]
which is bounded from above by the negative ``favorable integral''
\[
-
\frac{1}{16} \mathring{A}_*^2
\sum_{k=2}^5
\sum_{a=1}^3
\int_{s=0}^t
\int_{\Sigma_s}
\mathbf{1}_{\left\lbrace 0 < \mathcal{I} \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}
\mathcal{I}^{-2}
\left|
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|
|\nabla^{k} \Psi_a|^2
\, d \underline{x}
\, ds,
\]
plus an error integral that is bounded in magnitude by
\[
\lesssim
\sum_{k=2}^5
\sum_{a=1}^3
\int_{s=0}^t
\int_{\Sigma_s}
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
|\nabla^{k} \Psi_a|^2
\, d \underline{x}
\, ds.
\]
We can therefore bring the favorable integral over to LHS~\eqref{E:CONTROLLINGQUANTITYGRONWALLREADY},
where it appears with a ``$+$'' sign.
Moreover, from Def.~\ref{D:L2CONTROLLINGQUANTITY}, we deduce that the error integral
is bounded by the last term on RHS~\eqref{E:CONTROLLINGQUANTITYGRONWALLREADY} as desired.
We now bound the spacetime integrals on lines two to four of
RHS~\eqref{E:INTEGRALIDENTITYFORENERGY}.
Using
the estimates
\eqref{E:WEIGHTBOUNDEDBYONE}
and
\eqref{E:BOUNDEDBYONEPOINTWISEESTIMATESIFACTMINUSTWOTIMESWEIGHTDERIVATIVE},
the bootstrap assumptions,
the data-size assumptions \eqref{E:DATASIZE},
and \eqref{E:DATAEPSILONVSBOOTSTRAPEPSILON},
we deduce that all three integrands are bounded in magnitude
by
$
\lesssim
\sum_{k=2}^5
\sum_{a=1}^3
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
|\nabla^{k} \Psi_a|^2.
$
From Def.~\ref{D:L2CONTROLLINGQUANTITY},
we conclude that the corresponding error integrals
are bounded by the last term on RHS~\eqref{E:CONTROLLINGQUANTITYGRONWALLREADY} as desired.
Using similar reasoning, we bound the last two spacetime integrals on RHS~\eqref{E:INTEGRALIDENTITYFORENERGY}
by $\leq \mbox{RHS~\eqref{E:CONTROLLINGQUANTITYGRONWALLREADY}}$.
We now bound the spacetime integrals on lines five to seven of RHS~\eqref{E:INTEGRALIDENTITYFORENERGY}.
Using
the estimates
\eqref{E:POINTWISEESTIMATESIFACTMINUSTWOTIMESWEIGHTDERIVATIVE}
and
\eqref{E:BOUNDEDBYONEPOINTWISEESTIMATESIFACTMINUSTWOTIMESWEIGHTDERIVATIVE},
the bootstrap assumptions,
the data-size assumptions \eqref{E:DATASIZE},
\eqref{E:DATAEPSILONVSBOOTSTRAPEPSILON},
and Young's inequality,
we deduce that all three integrands are bounded in magnitude
by
\begin{align*}
& \lesssim
\varepsilon
\mathbf{1}_{\left\lbrace 0 < \mathcal{I} \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}
\sum_{k=2}^5
\sum_{a=1}^3
\mathcal{I}^{-2}
\left|
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|
|\nabla^{k} \Psi_a|^2
\\
&
+
\varepsilon
\sum_{k=2}^5
\sum_{a=1}^3
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
|\nabla^{k} \Psi_a|^2
+
\sum_{k=2}^5
|\nabla^{k} \Psi_0|^2.
\end{align*}
Appealing to Def.~\ref{D:L2CONTROLLINGQUANTITY},
we conclude that the corresponding error integrals
are bounded in magnitude by
$\lesssim \mbox{RHS~\eqref{E:CONTROLLINGQUANTITYGRONWALLREADY}}$ as desired.
We now bound the spacetime integral on line eight of
RHS~\eqref{E:INTEGRALIDENTITYFORENERGY},
in which the integrand is
$
2
\sum_{k=1}^4
\sum_{a=1}^3
\nabla^{k} \Psi_a \cdot \partial_a \nabla^{k} \Psi_0
$.
Using Young's inequality, we bound this integrand by
$
\lesssim
\left|
\nabla^{[2,5]} \Psi_0
\right|^2
+
\sum_{a=1}^3
\left|
\nabla^{[1,4]} \Psi_a
\right|^2
$.
From Def.~\ref{D:L2CONTROLLINGQUANTITY},
we conclude that the integral of the RHS of this expression
over the spacetime domain $(s,\underline{x}) \in [0,t] \times \mathbb{R}^3$
is bounded by the last term on RHS~\eqref{E:CONTROLLINGQUANTITYGRONWALLREADY} as desired.
We now bound the spacetime integral on line nine of
RHS~\eqref{E:INTEGRALIDENTITYFORENERGY},
in which the integrand is
$
2 \sum_{k=2}^5
\nabla^{k} \Psi_0 \cdot F_0^{(k)}
$.
Using Young's inequality,
\eqref{E:PSI0INHOMOGENEOUSTERMPOINTWISEBOUND},
and \eqref{E:DATAEPSILONVSBOOTSTRAPEPSILON},
we pointwise bound this integrand in magnitude by
\begin{align} \label{E:MOSTDIFFICULTERRORINTEGRANDPOINTWISEBOUND}
& \lesssim
\left|
\nabla^{[2,5]} \Psi_0
\right|^2
+
\varepsilon
\mathbf{1}_{\left\lbrace 0 < \mathcal{I} \leq (1/4) \min \lbrace 1, \mathring{A}_* \rbrace \right\rbrace}
\sum_{a=1}^3
\mathcal{I}^{-2}
\left|
\mathscr{W}'(\mathcal{I}^{-1} \Psi_0)
\right|
\left|
\nabla^{[2,5]} \Psi_a
\right|^2
\\
& \ \
+
\sum_{a=1}^3
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\left|
\nabla^{[2,5]} \Psi_a
\right|^2
\notag \\
& \ \
+
\sum_{a=1}^3
\left|
\nabla^{[1,4]} \Psi_a
\right|^2
+
\mathring{\upepsilon}^3
\sum_{a=1}^3
\left|
\Psi_a
\right|^2
+
\sum_{a=1}^3
\left|
\nabla^{\leq 5} \mathring{\Psi}_a
\right|^2.
\notag
\end{align}
From Def.~\ref{D:L2CONTROLLINGQUANTITY}
and the data-size assumptions \eqref{E:DATASIZE},
we conclude that the integral of RHS~\eqref{E:MOSTDIFFICULTERRORINTEGRANDPOINTWISEBOUND}
over the spacetime domain $(s,\underline{x}) \in [0,t] \times \mathbb{R}^3$
is $\lesssim \mbox{RHS~\eqref{E:CONTROLLINGQUANTITYGRONWALLREADY}}$ as desired.
We now bound the spacetime integral on line ten of RHS~\eqref{E:INTEGRALIDENTITYFORENERGY},
in which the integrand is
$2 \sum_{k=2}^5
\sum_{a=1}^3
\mathscr{W}(\mathcal{I}^{-1} \Psi_0) \nabla^{k} \Psi_a \cdot F_a^{(k)}$.
Using Young's inequality,
\eqref{E:WEIGHTBOUNDEDBYONE},
and
\eqref{E:PSIIINHOMOGENEOUSTERMPOINTWISEBOUND},
we pointwise bound this integrand in magnitude by
$\lesssim
\left|
\nabla^{[2,5]} \Psi_0
\right|^2
+
\sum_{a=1}^3
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\left|
\nabla^{[2,5]} \Psi_a
\right|^2
+
\sum_{a=1}^3
\left|
\nabla^{\leq 5} \mathring{\Psi}_a
\right|^2
$.
From Def.~\ref{D:L2CONTROLLINGQUANTITY}
and the data-size assumptions \eqref{E:DATASIZE},
we conclude that the integral of the RHS of this expression
over the spacetime domain $(s,\underline{x}) \in [0,t] \times \mathbb{R}^3$
is $\lesssim \mbox{RHS~\eqref{E:CONTROLLINGQUANTITYGRONWALLREADY}}$ as desired.
We now bound the spacetime integral on line eleven of RHS~\eqref{E:INTEGRALIDENTITYFORENERGY},
in which the integrand is
$
2
\sum_{k=1}^4
\sum_{a=1}^3
\nabla^{k} \Psi_a \cdot F_a^{(k)}
$.
Using Young's inequality
and
\eqref{E:PSIIINHOMOGENEOUSTERMPOINTWISEBOUND},
we pointwise bound this integrand in magnitude by
$\lesssim
\left|
\nabla^{[2,4]} \Psi_0
\right|^2
+
\sum_{a=1}^3
\left|
\nabla^{[1,4]} \Psi_a
\right|^2
+
\sum_{a=1}^3
\left|
\nabla^{\leq 4} \mathring{\Psi}_a
\right|^2
$.
From Def.~\ref{D:L2CONTROLLINGQUANTITY}
and the data-size assumptions \eqref{E:DATASIZE},
we conclude that the integral of the RHS of this expression
over the spacetime domain $(s,\underline{x}) \in [0,t] \times \mathbb{R}^3$
is $\lesssim \mbox{RHS~\eqref{E:CONTROLLINGQUANTITYGRONWALLREADY}}$ as desired.
We now bound the spacetime integral on line twelve of RHS~\eqref{E:INTEGRALIDENTITYFORENERGY},
in which the integrand is
$
2
\mathring{\upepsilon}^3
\mathscr{W}(\mathcal{I}^{-1} \Psi_0) \nabla \Psi_0 \cdot \sum_{a=1}^3 \partial_a \nabla \Psi_a
$.
Using the estimate \eqref{E:WEIGHTBOUNDEDBYONE} and Young's inequality,
we bound this integrand by
$
\lesssim
\mathring{\upepsilon}^3
\left|
\nabla \Psi_0
\right|^2
+
\sum_{a=1}^3
\left|
\nabla^2 \Psi_a
\right|^2
$.
From Def.~\ref{D:L2CONTROLLINGQUANTITY},
we conclude that the integral of the RHS of this expression
over the spacetime domain $(s,\underline{x}) \in [0,t] \times \mathbb{R}^3$
is bounded by the last term on RHS~\eqref{E:CONTROLLINGQUANTITYGRONWALLREADY} as desired.
Finally, we bound the spacetime integral on line thirteen of RHS~\eqref{E:INTEGRALIDENTITYFORENERGY},
in which the integrand is
$
2
\mathring{\upepsilon}^3
\nabla \Psi_0 \cdot F_0^{(1)}
$.
Using Young's inequality
and \eqref{E:ANNOYINGLOWORDERPSI0INHOMOGENEOUSTERMPOINTWISEBOUND},
we pointwise bound this integrand in magnitude by
\begin{align} \label{E:LOWORDERANNOYINGERRORINTEGRANDPOINTWISEBOUND}
& \lesssim
\mathring{\upepsilon}^3
\left|
\nabla \Psi_0
\right|^2
+
\sum_{a=1}^3
\left|
\nabla \Psi_a
\right|^2
+
\mathring{\upepsilon}^3
\sum_{a=1}^3
\left|
\Psi_a
\right|^2
+
\sum_{a=1}^3
\left|
\nabla^{\leq 1} \mathring{\Psi}_a
\right|^2.
\end{align}
From Def.~\ref{D:L2CONTROLLINGQUANTITY}
and the data-size assumptions \eqref{E:DATASIZE},
we conclude that the integral of RHS~\eqref{E:LOWORDERANNOYINGERRORINTEGRANDPOINTWISEBOUND}
over the spacetime domain $(s,\underline{x}) \in [0,t] \times \mathbb{R}^3$
is $\lesssim \mbox{RHS~\eqref{E:CONTROLLINGQUANTITYGRONWALLREADY}}$ as desired.
This completes our proof of \eqref{E:CONTROLLINGQUANTITYGRONWALLREADY}
and therefore finishes the proof of \eqref{E:MAINAPRIORIENERGYESTIMATES}.
\noindent \textbf{Proof of \eqref{E:PSI0DERIVATVESIMPROVED} and \eqref{E:PSIIANDDERIVATIVESIMPROVED}}:
In view of Def.~\ref{D:L2CONTROLLINGQUANTITY},
we see that the estimates
$
\| \nabla^{[2,3]} \Psi_0 \|_{L^{\infty}(\Sigma_t)} \lesssim \mathring{\upepsilon}
$
and
$\| \nabla^{[1,2]} \Psi_i \|_{L^{\infty}(\Sigma_t)} \lesssim \mathring{\upepsilon}$
follow from \eqref{E:MAINAPRIORIENERGYESTIMATES} and Sobolev embedding $H^2(\mathbb{R}^3) \hookrightarrow L^{\infty}(\mathbb{R}^3)$.
To bound $\| \nabla \Psi_0 \|_{L^{\infty}(\Sigma_t)}$,
we first use equation \eqref{E:COMMUTEDPARTALTPSI0EVOLUTION},
the bootstrap assumptions,
the data-size assumptions \eqref{E:DATASIZE},
the estimates \eqref{E:WEIGHTBOUNDEDBYONE} and \eqref{E:ANNOYINGLOWORDERPSI0INHOMOGENEOUSTERMPOINTWISEBOUND},
inequality \eqref{E:DATAEPSILONVSBOOTSTRAPEPSILON},
and the already proven bound
$\| \nabla^{[1,2]} \Psi_i \|_{L^{\infty}(\Sigma_t)} \lesssim \mathring{\upepsilon}$
to obtain
$\left|
\partial_t \nabla \Psi_0
\right|
\lesssim
\varepsilon^2
+
\mathring{\upepsilon}
+
\sum_{a=1}^3
\left|
\nabla^{[1,2]} \Psi_a
\right|
\lesssim
\mathring{\upepsilon}
$.
From this bound, the fundamental theorem of calculus, and the data-size assumptions \eqref{E:DATASIZE},
we find that
$\left|
\nabla \Psi_0
\right|
\lesssim
\mathring{\upepsilon}
+
\int_{s=0}^t
\mathring{\upepsilon}
\, ds
\lesssim \mathring{\upepsilon}
$.
This implies that
$
\| \nabla \Psi_0 \|_{L^{\infty}(\Sigma_t)} \lesssim \mathring{\upepsilon}
$, which completes the proof of \eqref{E:PSI0DERIVATVESIMPROVED}.
Similarly, from equation \eqref{E:PARTALTPSIIEVOLUTION},
the bootstrap assumptions,
the data-size assumptions \eqref{E:DATASIZE},
and the already proven bound
$\| \nabla \Psi_0 \|_{L^{\infty}(\Sigma_t)} \lesssim \mathring{\upepsilon}$,
we deduce
$
\sum_{a=1}^3
\left|
\partial_t \Psi_a
\right|
\lesssim
\mathring{\upepsilon}
$.
From this bound, the fundamental theorem of calculus, and the data-size assumption \eqref{E:DATASIZE},
we find that
$\sum_{a=1}^3
\left|
\Psi_a
\right|
\lesssim
\mathring{\upepsilon}
$,
which implies that
$
\sum_{a=1}^3 \| \Psi_a \|_{L^{\infty}(\Sigma_t)} \lesssim \mathring{\upepsilon}
$,
thereby completing the proof of \eqref{E:PSIIANDDERIVATIVESIMPROVED}.
\noindent \textbf{Proof of \eqref{E:PSI0ITSELFIMPROVED}}:
We first use equation \eqref{E:PARTALTPSI0EVOLUTION},
the estimates \eqref{E:WEIGHTBOUNDEDBYONE} and \eqref{E:SINGULARFACTORTIMESWEIGHTBOUNDEDBYONE},
the bootstrap assumptions,
the data-size assumptions \eqref{E:DATASIZE},
and the already proven bound
$\| \nabla^{\leq 1} \Psi_i \|_{L^{\infty}(\Sigma_t)} \lesssim \mathring{\upepsilon}$
to obtain
$\left|
\partial_t \Psi_0
\right|
\lesssim
\mathring{\upepsilon}
$.
From this bound, the fundamental theorem of calculus, the data-size assumption \eqref{E:LARGEDATASIZE},
and the fact that $0 < t \leq 2 \mathring{A}_*^{-1}$,
we find that
$\|
\Psi_0
\|_{L^{\infty}(\Sigma_t)}
\leq
\|
\mathring{\Psi}_0
\|_{L^{\infty}(\Sigma_0)}
+
C \mathring{\upepsilon}
\leq
\mathring{A}
+
C \mathring{\upepsilon}
$,
which is the desired bound \eqref{E:PSI0ITSELFIMPROVED}.
\noindent \textbf{Proof of \eqref{E:IFACTITSELFIMPROVED} and \eqref{E:IFACTDERIVATIVESLIFTYIMPROVED}}:
We repeat the proof of
\eqref{E:IFACTCRUCIALPOINTWISE},
but using the bootstrap assumption
\eqref{E:IFACTITSELFBOOTSTRAP} and the
estimates \eqref{E:PSI0ITSELFIMPROVED}-\eqref{E:PSIIANDDERIVATIVESIMPROVED}
instead of using the full set of bootstrap assumptions.
We find that
$\mathcal{I}(t,\underline{x}) = 1 - t \mathring{\Psi}_0(\underline{x}) + \mathcal{O}(\mathring{\upepsilon})$.
From this estimate, the fact that $0 < t < 2 \mathring{A}_*^{-1}$,
and the data-size assumption \eqref{E:LARGEDATASIZE},
we conclude the desired bound \eqref{E:IFACTITSELFIMPROVED}.
Similarly, to prove \eqref{E:IFACTDERIVATIVESLIFTYIMPROVED},
we repeat the proof of \eqref{E:LINFINITYFORDERIVATIVESOFIFACT},
but using
the estimates \eqref{E:PSI0ITSELFIMPROVED}-\eqref{E:IFACTITSELFIMPROVED}
instead of the bootstrap assumptions.
\noindent \textbf{Proof of \eqref{E:IFACTAPRIORIENERGYESTIMATE}}:
The estimate \eqref{E:IFACTAPRIORIENERGYESTIMATE}
follows as a straightforward consequence of
the pointwise estimates
\eqref{E:LOWESTLEVELPOINTWISEESTIMATEFORDERIVATIVESOFIFACT}-\eqref{E:POINTWISEESTIMATEFORDERIVATIVESOFIFACT},
the weight estimate \eqref{E:BOUNDEDBYONEPOINTWISEESTIMATESIFACTMINUSTWOTIMESWEIGHTDERIVATIVE},
the energy estimate \eqref{E:MAINAPRIORIENERGYESTIMATES},
and the data-size assumptions \eqref{E:DATASIZE}.
\noindent \textbf{Proof of \eqref{E:PARTIALTPSI0H4ESTIMATE} and \eqref{E:PARTIALTPSIIH4ESTIMATE}}:
To prove \eqref{E:PARTIALTPSI0H4ESTIMATE},
we first use equation \eqref{E:COMMUTEDPARTALTPSI0EVOLUTION}
and the estimate \eqref{E:WEIGHTBOUNDEDBYONE}
to deduce that
\begin{align} \label{E:PARTIATPSI0FIRSTH4BOUND}
\mathring{\upepsilon}
\| \partial_t \Psi_0 \|_{L^2(\Sigma_t)}^2
+
\| \nabla \partial_t \Psi_0 \|_{H^3(\Sigma_t)}^2
& \lesssim
\sum_{k=2}^5
\sum_{a=1}^3
\left\|
\left\lbrace
\mathscr{W}(\mathcal{I}^{-1} \Psi_0)
\right\rbrace^{1/2}
\nabla^{k} \Psi_a
\right\|_{L^2(\Sigma_t)}^2
+
\sum_{a=1}^3
\left\|
\nabla \Psi_a
\right\|_{L^2(\Sigma_t)}^2
\\
& \ \
+
\mathring{\upepsilon} \left\| F_0^{(0)} \right\|_{L^2(\Sigma_t)}^2
+
\sum_{k=1}^4
\left\| F_0^{(k)} \right\|_{L^2(\Sigma_t)}^2.
\notag
\end{align}
Next, we recall that
the already proven estimates
\eqref{E:PSI0ITSELFIMPROVED}-\eqref{E:IFACTITSELFIMPROVED}
imply the bootstrap assumptions
\eqref{E:PSI0ITSELFBOOTSTRAP}-\eqref{E:IFACTITSELFBOOTSTRAP}
hold with $C \mathring{\upepsilon}$ in place of $\varepsilon$.
It follows that the pointwise estimate \eqref{E:ANNOYINGLOWORDERPSI0INHOMOGENEOUSTERMPOINTWISEBOUND}
holds with $C \mathring{\upepsilon}$ in place of $\varepsilon$.
From this fact,
Def.~\ref{D:L2CONTROLLINGQUANTITY},
the energy estimate \eqref{E:MAINAPRIORIENERGYESTIMATES},
and the data-size assumptions \eqref{E:DATASIZE},
we deduce that $\mbox{RHS~\eqref{E:PARTIATPSI0FIRSTH4BOUND}} \lesssim \mathring{\upepsilon}^2$,
which is the desired bound \eqref{E:PARTIALTPSI0H4ESTIMATE}.
The estimate \eqref{E:PARTIALTPSIIH4ESTIMATE} can be proved using similar arguments
based on the evolution equation \eqref{E:COMMUTEDPARTALTPSIIEVOLUTION}
and the pointwise estimate \eqref{E:PSIIINHOMOGENEOUSTERMPOINTWISEBOUND},
and we omit the details.
\end{proof}
\section{Local well-posedness and continuation criteria}
\label{S:WELLPOSEDNESS}
In this section, we provide a proposition that yields
standard well-posedness results and continuation criteria
pertaining to the quantities
$\lbrace \partial_{\alpha} \Phi \rbrace_{\alpha = 0,1,2,3}$, $\mathcal{I}$, and $\lbrace \Psi_{\alpha} \rbrace_{\alpha = 0,1,2,3}$.
\begin{proposition}
\label{P:LOCALWELLPOSEDNESSANDCONTINUATIONCRITERIA}
Let $N \geq 3$ be an integer and let
$(\partial_t \Phi|_{\Sigma_0},\partial_1 \Phi|_{\Sigma_0}, \partial_2 \Phi|_{\Sigma_0}, \partial_3 \Phi|_{\Sigma_0})
= (\mathring{\Psi}_0,\mathring{\Psi}_1,\mathring{\Psi}_2,\mathring{\Psi}_3)$
be initial data (see Remark~\ref{R:NOPHIINWAVEEQUATION})
for the equation \eqref{E:WAVE} verifying
$\mathring{\Psi}_{\alpha} \in H^N(\mathbb{R}^3)$, $(\alpha = 0,1,2,3)$.
Let $\mathcal{H} := (-1/2,\infty)$,
and note that the following holds:
equation \eqref{E:WAVE} is a non-degenerate\footnote{By non-degenerate,
we mean that relative to the Cartesian coordinates, the
$4 \times 4$ matrix of components $g_{\alpha \beta}$ has signature $(-,+,+,+)$,
where $g := - dt^2 + \frac{1}{\mathscr{W}(\partial_t \Phi)}\sum_{a=1}^3 (dx^a)^2$
is the metric corresponding to equation \eqref{E:WAVE}.}
wave equation
at points $(t,\underline{x})$ such that
$\partial_t \Phi(t,\underline{x}) \in \mathcal{H}$
(see \eqref{E:WEIGHTISPOSITIVE} for justification of this assertion).
Assume that $\mathring{\Psi}_0(\mathbb{R}^3)$ is contained in a compact subset $\mathfrak{K}$ of $\mathcal{H}$.
Let $\mathcal{I}$, $\mathcal{I}_{\star}$, and $\lbrace \Psi_{\alpha} \rbrace_{\alpha=0,1,2,3}$ be the quantities defined in
Defs.\ \ref{D:INTEGRATINGFACTOR} and \ref{D:RENORMALIZEDSOLUTION}.
Then there exist a compact set $\mathfrak{K}'$ of $\mathcal{H}$ containing $\mathfrak{K}$ in its interior
and a time $T > 0$ depending on $\mathfrak{K}$
and
$\sum_{\alpha = 0}^3 \| \mathring{\Psi}_{\alpha} \|_{H^N}$,
such that a unique classical solution to equation \eqref{E:WAVE} exists on
$[0,T) \times \mathbb{R}^3$, such that
$\partial_t \Phi([0,T) \times \mathbb{R}^3) \subset \mathfrak{K}'$,
and such that the following regularity properties hold for $\alpha = 0,1,2,3$:
\begin{align} \label{E:LWPREGULARITY}
\partial_{\alpha} \Phi \in C\left([0,T),H^N \right).
\end{align}
In addition, the solution depends continuously on the data.
Let $T_{(Lifespan)}$ be the supremum of all times $T > 0$ such that the
classical solution to \eqref{E:WAVE} exists on $[0,T) \times \mathbb{R}^3$ and verifies the above properties.
Then either $T_{(Lifespan)} = \infty$, or $T_{(Lifespan)} < \infty$ and one of the following two breakdown scenarios must occur:
\begin{enumerate}
\item There exists a sequence of points
$\lbrace (t_n,\underline{x}_n) \rbrace_{n=1}^{\infty} \subset [0,T_{(Lifespan)}) \times \mathbb{R}^3$
such that $\partial_t \Phi(t_n,\underline{x}_n)$ escapes every compact subset of $\mathcal{H}$ as $n \to \infty$.
\item
$
\displaystyle
\lim_{t \uparrow T_{(Lifespan)}}
\sup_{s \in [0,t)}
\left\|
\nabla^{\leq 1} \partial_t \Phi
\right\|_{L^{\infty}(\Sigma_s)}
=
\infty
$.
\end{enumerate}
Moreover,
on $[0,T_{(Lifespan)}) \times \mathbb{R}^3$,
$\mathcal{I}$ and $\lbrace \Psi_{\alpha} \rbrace_{\alpha=0,1,2,3}$
are classical solutions to
equations \eqref{E:INTEGRATINFACTORODEANDIC} and
\eqref{E:PARTALTPSI0EVOLUTION}-\eqref{E:PARTALTPSIIEVOLUTION}
such that
\begin{align} \label{E:PSILWPREGULARITY}
& \mathcal{I} - 1 \in C\left([0,T_{(Lifespan)}),H^{N+1}(\mathbb{R}^3) \right),
&& \Psi_{\alpha} \in C\left([0,T_{(Lifespan)}),H^N(\mathbb{R}^3) \right).
\end{align}
Finally, $\mathcal{I}_{\star}$ satisfies the following estimates:
\begin{align} \label{E:IFACTSTRICTLYPOSITIVEDURINGCLASSICALLIFESPAN}
& 0 < \mathcal{I}_{\star}(t) < \infty,
&& \mbox{for } t \in [0,T_{(Lifespan)}).
\end{align}
\end{proposition}
\begin{proof}
The statements concerning $\Phi$ are standard
and can be proved using the ideas found, for example, in \cite{jS2008c}.
Next, we note that the evolution equation + initial condition for $\mathcal{I}$
stated in \eqref{E:INTEGRATINFACTORODEANDIC},
the fact that $\mathcal{I}(t,\cdot) - 1$ is compactly supported in space (see Remark~\ref{R:BOUNDEDWAVESPEED}),
and the fact that
$\partial_t \Phi \in C\left([0,T_{(Lifespan)}),H^N(\mathbb{R}^3) \right) \subset C\left([0,T_{(Lifespan)}),C^1(\mathbb{R}^3) \right)$
(i.e., \eqref{E:LWPREGULARITY})
can be used to deduce \eqref{E:IFACTSTRICTLYPOSITIVEDURINGCLASSICALLIFESPAN}.
Similarly, from
\eqref{E:INTEGRATINFACTORODEANDIC},
the identity \eqref{E:FORMULAFORDERIVATIVESOFINTEGRATINGFACTOR},
the definition $\Psi_{\alpha} := \mathcal{I} \partial_{\alpha} \Phi$ (see Def.\ \ref{D:RENORMALIZEDSOLUTION}),
\eqref{E:LWPREGULARITY},
and the standard Sobolev--Moser calculus,
it is straightforward to deduce \eqref{E:PSILWPREGULARITY}.
\end{proof}
\section{The main theorem}
\label{S:MAINTHM}
In this section, we state and prove our main stable blowup result.
\begin{theorem}[\textbf{Stable ODE-type blowup}]
\label{T:STABILITYOFODEBLOWUP}
Assume that the weight function $\mathscr{W}$ verifies the assumptions
stated in Subsect.\ \ref{SS:WEIGHTASSUMPTIONS}.
Consider compactly supported initial data
$(\partial_t \Phi|_{\Sigma_0},\partial_1 \Phi|_{\Sigma_0}, \partial_2 \Phi|_{\Sigma_0}, \partial_3 \Phi|_{\Sigma_0})
= (\mathring{\Psi}_0,\mathring{\Psi}_1,\mathring{\Psi}_2,\mathring{\Psi}_3)$
for the wave equation \eqref{E:WAVE}
(see Remark~\ref{R:NOPHIINWAVEEQUATION} concerning the data)
that verify the data-size assumptions \eqref{E:DATASIZE}-\eqref{E:PSI0NOTTOONEGATIVE}
involving the parameters $\mathring{\upepsilon}$ and $\mathring{A}$,
and let $\mathring{A}_*$ be the data-size parameter defined in \eqref{E:CRUCIALDATASIZEPARAMETER}.
Let $\mathcal{I}$, $\mathcal{I}_{\star}$, and $\lbrace \Psi_{\alpha} \rbrace_{\alpha=0,1,2,3}$ be the quantities defined in
Defs.\ \ref{D:INTEGRATINGFACTOR} and \ref{D:RENORMALIZEDSOLUTION}.
We define
\begin{align} \label{E:TLIFESPAN}
T_{(Lifespan)}
& := \sup
\left\lbrace
t > 0 \ | \ \lbrace \partial_{\alpha} \Phi \rbrace_{\alpha=0,1,2,3} \mbox{ exist classically on } [0,t) \times \mathbb{R}^3
\right\rbrace.
\end{align}
If
$\mathring{\upepsilon} > 0$,
$\mathring{A} > 0$,
and
$\mathring{A}_* > 0$,
and if $\mathring{\upepsilon}$ is small relative to
$\mathring{A}^{-1}$
and
$\mathring{A}_*$
in the sense explained in Subsect.\ \ref{SS:SMALLNESSASSUMPTIONS}, then
the following conclusions hold.
\noindent \underline{\textbf{Characterization of the solution's classical lifespan}}:
The solution's classical lifespan is characterized by $\mathcal{I}_{\star}$ as follows:
\begin{align} \label{E:LIFESPANCHARACTERIZATION}
T_{(Lifespan)}
= \sup\left\lbrace t > 0 \ | \ \inf_{s \in [0,t)} \mathcal{I}_{\star}(s) > 0 \right\rbrace.
\end{align}
Moreover,
\begin{subequations}
\begin{align}
& \mathcal{I}(t,\underline{x}) > 0 \mbox{ for } (t,\underline{x}) \in [0,T_{(Lifespan)}) \times \mathbb{R}^3,
\label{E:IFACTPOSITIVEDURINGCLASSICALLIFESPAN} \\
& \lim_{t \uparrow T_{(Lifespan)}} \mathcal{I}_{\star}(t) = 0.
\label{E:IFACTMINVANISHESATLIFESPAN}
\end{align}
\end{subequations}
In addition, the following estimate holds:
\begin{align} \label{E:MAINTHMLIFESPANESTIMATE}
T_{(Lifespan)}
& = \mathring{A}_*^{-1}
\left\lbrace
1
+
\mathcal{O}(\mathring{\upepsilon})
\right\rbrace.
\end{align}
\noindent \underline{\textbf{Regularity properties of}
$\Psi_{\alpha}$ \textbf{and } $\mathcal{I}$ \textbf{on } $[0,T_{(Lifespan)}) \times \mathbb{R}^3$}:
On the slab
$[0,T_{(Lifespan)}) \times \mathbb{R}^3$,
the solution verifies the energy bounds
\eqref{E:MAINAPRIORIENERGYESTIMATES}-\eqref{E:IFACTAPRIORIENERGYESTIMATE},
the $L^{\infty}$ estimates \eqref{E:PSI0ITSELFIMPROVED}-\eqref{E:IFACTDERIVATIVESLIFTYIMPROVED},
\eqref{E:PSI0WELLAPPROXIMATED}-\eqref{E:PSI0BIGGERTHANMINUSONEHALF},
and
\eqref{E:IFACTCRUCIALPOINTWISE}-\eqref{E:SMALLINTFACTIMPMLIESRATIOISLARGE}
(with $C \mathring{\upepsilon}$ on the RHS in place of $\varepsilon$ in these equations).
Moreover, $\lbrace \Psi_{\alpha} \rbrace_{\alpha=0,1,2,3}$ and $\mathcal{I}$
enjoy the following regularity:
\begin{subequations}
\begin{align}
\label{E:MAINTHEOREMPSI0REGULARITY}
\Psi_0
& \in
C\left([0,T_{(Lifespan)}), H^5(\mathbb{R}^3) \right)
\cap
L^{\infty}\left([0,T_{(Lifespan)}), H^5(\mathbb{R}^3) \right),
\\
\Psi_i
& \in C\left([0,T_{(Lifespan)}), H^5(\mathbb{R}^3) \right)
\cap
L^{\infty}\left([0,T_{(Lifespan)}), H^4(\mathbb{R}^3) \right),
\label{E:MAINTHEOREMPSIIREGULARITY}
\\
\mathcal{I} - 1
& \in
C\left([0,T_{(Lifespan)}), H^6(\mathbb{R}^3) \right)
\cap
L^{\infty}\left([0,T_{(Lifespan)}), H^5(\mathbb{R}^3) \right).
\label{E:MAINTHEOREMIFACTREGULARITY}
\end{align}
\end{subequations}
\noindent \underline{\textbf{Regularity properties of}
$\Psi_{\alpha}$ \textbf{and } $\mathcal{I}$ \textbf{on } $[0,T_{(Lifespan)}] \times \mathbb{R}^3$}:
$\Psi_{\alpha}$ and $\mathcal{I}$
\textbf{do not blow up} at time $T_{(Lifespan)}$, but rather
continuously extend to $[0,T_{(Lifespan)}] \times \mathbb{R}^3$
as functions that enjoy the following regularity for any $N < 5$:
\begin{subequations}
\begin{align}
\label{E:EXTENDEDMAINTHEOREMPSI0REGULARITY}
\Psi_0
& \in
L^{\infty}\left([0,T_{(Lifespan)}], H^5(\mathbb{R}^3) \right)
\cap
C\left([0,T_{(Lifespan)}], H^N(\mathbb{R}^3) \right),
\\
\Psi_i
& \in C\left([0,T_{(Lifespan)}], H^4(\mathbb{R}^3) \right),
\label{E:EXTENDEDMAINTHEOREMPSIIREGULARITY}
\\
\mathcal{I} - 1
& \in C\left([0,T_{(Lifespan)}], H^5(\mathbb{R}^3) \right).
\label{E:EXTENDEDMAINTHEOREMIFACTREGULARITY}
\end{align}
\end{subequations}
\noindent \underline{\textbf{Description of the vanishing of } $\mathcal{I}$ \textbf{and the blowup of} $\partial_t \Phi$}:
For $(t,\underline{x}) \in [0,T_{(Lifespan)}) \times \mathbb{R}^3$,
we have
\begin{align} \label{E:MAINTHMSMALLINTFACTIMPMLIESPSI0ISLARGE}
\mathcal{I}(t,\underline{x})
\leq \frac{1}{2}
\implies
\partial_t \Phi(t,\underline{x})
\geq \frac{1}{4 \mathcal{I}(t,\underline{x})} \mathring{A}_*.
\end{align}
Let
\begin{align} \label{E:BLOWUPSET}
\Sigma_{T_{(Lifespan)}}^{Blowup}
& := \lbrace (T_{(Lifespan)}, \underline{x}) \ | \ \mathcal{I}(T_{(Lifespan)},\underline{x}) = 0 \rbrace.
\end{align}
Then if $(T_{(Lifespan)}, \underline{x}) \in \Sigma_{T_{(Lifespan)}}^{Blowup}$,
we have\footnote{See also Remark~\ref{R:BLOWUPOFPHI} concerning the blowup of $\Phi$ itself, if initial data for
$\Phi$ itself are prescribed.}
\begin{align} \label{E:BLOWUPBEHAVIOR}
\lim_{t \uparrow T_{(Lifespan)}} \partial_t \Phi(t,\underline{x}) = \infty.
\end{align}
Finally, if $(T_{(Lifespan)}, \underline{x}) \notin \Sigma_{T_{(Lifespan)}}^{Blowup}$,
then there exists an open ball $B_{\underline{x}} \subset \mathbb{R}^3$ centered at $\underline{x}$
such that for $\alpha = 0,1,2,3$, we have
$\partial_{\alpha} \Phi \in C\left([0,T_{(Lifespan)}], H^5(B_{\underline{x}}) \right)$.
\end{theorem}
\begin{proof}
Let $C_* > 1$ be a constant; we will enlarge $C_*$ as needed throughout the proof.
Let $T_{(Max)}$ be the supremum of times $0 \leq T \leq 2 \mathring{A}_*^{-1}$ such that
the following properties hold:
\begin{itemize}
\item $\lbrace \partial_{\alpha} \Phi \rbrace_{\alpha=0,1,2,3}$ is a classical solution to \eqref{E:WAVE} on $[0,T) \times \mathbb{R}^3$
(see Remark~\ref{R:NOPHIINWAVEEQUATION})
verifying the properties stated in Prop.~\ref{P:LOCALWELLPOSEDNESSANDCONTINUATIONCRITERIA}
(with $N=5$ in the proposition).
\item $\mathcal{I}$ is a classical solution to \eqref{E:INTEGRATINFACTORODEANDIC} on $[0,T) \times \mathbb{R}^3$
verifying the properties stated in Prop.~\ref{P:LOCALWELLPOSEDNESSANDCONTINUATIONCRITERIA}.
\item $\lbrace \Psi_{\alpha} \rbrace_{\alpha = 0,1,2,3}$ are classical solutions to
\eqref{E:PARTALTPSI0EVOLUTION}-\eqref{E:PARTALTPSIIEVOLUTION}
for $(t,\underline{x}) \in [0,T) \times \mathbb{R}^3$
such that the properties stated in Prop.~\ref{P:LOCALWELLPOSEDNESSANDCONTINUATIONCRITERIA}
hold.
\item The bootstrap assumptions \eqref{E:BOOTSTRAPRATIO} and \eqref{E:BOOTSTRAPSMALLINTFACTIMPMLIESPSI0ISLARGE}
hold for $(t,\underline{x}) \in [0,T) \times \mathbb{R}^3$.
\item The $L^{\infty}$ bootstrap assumptions \eqref{E:PSI0ITSELFBOOTSTRAP}-\eqref{E:IFACTITSELFBOOTSTRAP}
hold for $t \in [0,T)$ with $\varepsilon := C_* \mathring{\upepsilon}$.
\item $\inf \left\lbrace \mathcal{I}_{\star}(t) \ | \ t \in [0,T) \right\rbrace > 0$,
where $\mathcal{I}_{\star}$ is defined in \eqref{E:IFACTMIN}.
Note that this implies that the bootstrap assumption \eqref{E:HYPERBOLICBOOTSTRAP} holds on $[0,T)$.
\end{itemize}
Throughout the rest of the proof, we will assume that
$\mathring{\upepsilon}$ is sufficiently small and that $C_*$ is sufficiently large
without explicitly mentioning it every time.
Next, we note that the hypotheses of
Prop.~\ref{P:LOCALWELLPOSEDNESSANDCONTINUATIONCRITERIA} hold with $N=5$.
Hence, by Prop.~\ref{P:LOCALWELLPOSEDNESSANDCONTINUATIONCRITERIA} and Sobolev embedding,
we have $T_{(Max)} > 0$.
We will now show that
$T_{(Max)} = T_{(Lifespan)}$.
Clearly $T_{(Max)} \leq T_{(Lifespan)}$ and thus it suffices to show that $T_{(Lifespan)} \leq T_{(Max)}$.
To proceed, we assume for the sake of deriving a contradiction that
\[
\inf_{s \in [0,T_{(Max)})} \mathcal{I}_{\star}(s) > 0.
\]
Then, in view of Defs.~\ref{D:INTEGRATINGFACTOR} and ~\ref{D:RENORMALIZEDSOLUTION} and the bootstrap assumptions,
we see that this assumption implies that
\[
\lim_{t \uparrow T_{(Max)}}
\sup_{s \in [0,t)}
\left\lbrace
\left\|
\partial_t \Phi
\right\|_{L^{\infty}(\Sigma_s)}
+
\sum_{\alpha = 0}^3
\left\|
\partial_{\alpha}
\partial_t \Phi
\right\|_{L^{\infty}(\Sigma_s)}
\right\rbrace
<
\infty.
\]
It follows that neither of the two breakdown scenarios of
Prop.~\ref{P:LOCALWELLPOSEDNESSANDCONTINUATIONCRITERIA} occur on $[0,T_{(Max)}) \times \mathbb{R}^3$.
Moreover, by Prop.~\ref{P:APRIORIESTIMATES}, if $C_*$ is large enough,
then the bootstrap assumption inequalities
\eqref{E:PSI0ITSELFBOOTSTRAP}-\eqref{E:IFACTITSELFBOOTSTRAP}
hold in a strict sense (that is, with ``$\leq$'' replaced by ``$<$'') on $[0,T_{(Max)}) \times \mathbb{R}^3$.
Moreover, all estimates proved prior to Prop.~\ref{P:APRIORIESTIMATES} hold with
$\varepsilon$ replaced by $C \mathring{\upepsilon}$, and we will use this fact in the rest of the proof
without mentioning it again.
Furthermore,
\eqref{E:PSI0NEGATIVEIMPMLIESINHYPERBOLICREGIME}
and
\eqref{E:SMALLINTFACTIMPMLIESPSI0ISLARGE}
respectively yield strict improvements of the bootstrap assumptions
\eqref{E:BOOTSTRAPRATIO} and \eqref{E:BOOTSTRAPSMALLINTFACTIMPMLIESPSI0ISLARGE}
for $(t,\underline{x}) \in [0,T_{(Max)}) \times \mathbb{R}^3$.
Next, we note that the estimate \eqref{E:IFACTSTARCRUCIALPOINTWISE}
implies that $\mathcal{I}_{\star}(t)$ cannot remain positive for $t$ larger than
$
\mathring{A}_*^{-1}
\left\lbrace
1
+
\mathcal{O}(\mathring{\upepsilon})
\right\rbrace
$.
From this fact, it follows that $T_{(Max)} < 2 \mathring{A}_*^{-1}$.
Combining these facts and appealing to Prop.~\ref{P:LOCALWELLPOSEDNESSANDCONTINUATIONCRITERIA},
we deduce that
$\lbrace \partial_{\alpha} \Phi \rbrace_{\alpha=0,1,2,3}$,
$\lbrace \Psi_{\alpha} \rbrace_{\alpha=0,1,2,3}$,
and $\mathcal{I}$ extend as classical solutions to
a region of the form $[0,T_{(Max)} + \Delta) \times \mathbb{R}^3$ for some $\Delta > 0$ with $T_{(Max)} + \Delta < 2 \mathring{A}_*^{-1}$
on which the solution has the same Sobolev regularity as the data,
such that
$\inf_{s \in [0,T_{(Max)} + \Delta]} \mathcal{I}_{\star}(s) > 0$,
and such that the bootstrap assumptions
\eqref{E:BOOTSTRAPRATIO}-\eqref{E:IFACTITSELFBOOTSTRAP}
hold for $(t,\underline{x}) \in [0,T_{(Max)} + \Delta] \times \mathbb{R}^3$.
In total, this contradicts the definition of $T_{(Max)}$.
We therefore conclude that
\begin{align} \label{E:IFACTVANISHESONTMAX}
T_{(Max)}
= \sup
\left\lbrace
t > 0 \ | \ \inf_{s \in [0,t)} \mathcal{I}_{\star}(s) > 0
\right\rbrace
\end{align}
and that the estimates \eqref{E:MAINAPRIORIENERGYESTIMATES}-\eqref{E:IFACTAPRIORIENERGYESTIMATE}
and
\eqref{E:PSI0ITSELFIMPROVED}-\eqref{E:IFACTDERIVATIVESLIFTYIMPROVED}
hold for $t \in [0,T_{(Max)})$.
Next, we note that the estimate
\eqref{E:MAINTHMSMALLINTFACTIMPMLIESPSI0ISLARGE} follows from \eqref{E:SMALLINTFACTIMPMLIESPSI0ISLARGE}.
In particular, it follows from
\eqref{E:MAINTHMSMALLINTFACTIMPMLIESPSI0ISLARGE} and \eqref{E:IFACTVANISHESONTMAX} and that
$
\displaystyle
\lim_{t \uparrow T_{(Max)}} \sup_{s \in [0,t)}
\left\|
\partial_t \Phi
\right\|_{L^{\infty}(\Sigma_s)}
=
\infty
$,
that is, that $\partial_t \Phi$ blows up at time $T_{(Max)}$.
We have therefore shown that
$T_{(Max)} = T_{(Lifespan)}$
and that $T_{(Lifespan)}$ is characterized by \eqref{E:LIFESPANCHARACTERIZATION}.
Moreover, the arguments given in the previous paragraph imply that
$\mathcal{I}_{\star}$ vanishes for the first time at
$
T_{(Lifespan)}
=
\mathring{A}_*^{-1}
\left\lbrace
1
+
\mathcal{O}(\mathring{\upepsilon})
\right\rbrace$,
which in total yields
\eqref{E:IFACTPOSITIVEDURINGCLASSICALLIFESPAN}
and
\eqref{E:MAINTHMLIFESPANESTIMATE}.
In the rest of this proof,
we sometimes silently use that
$
\Psi_0 \in L^{\infty}\left([0,T_{(Lifespan)}),L^2(\mathbb{R}^3)\right)
$
and
$
\mathcal{I} - 1 \in L^{\infty}\left([0,T_{(Lifespan)}),L^2(\mathbb{R}^3)\right)
$.
These facts do not follow from
the energy estimates \eqref{E:MAINAPRIORIENERGYESTIMATES} and \eqref{E:IFACTAPRIORIENERGYESTIMATE},
but instead follow from
\eqref{E:PSI0ITSELFIMPROVED}, \eqref{E:IFACTITSELFIMPROVED},
and the compactly supported (in space) nature of $\Psi_0$ and $\mathcal{I} - 1$.
Next, we easily conclude from the definition \eqref{E:ENERGYTOCONTROLSOLNS}
of $\mathbb{Q}_{(\mathring{\upepsilon})}(t)$
and the fact that the estimate \eqref{E:MAINAPRIORIENERGYESTIMATES}
holds on $[0,T_{(Lifespan)})$
that
$
\Psi_0 \in L^{\infty}\left([0,T_{(Lifespan)}), H^5(\mathbb{R}^3) \right)
$
(as is stated in \eqref{E:MAINTHEOREMPSI0REGULARITY})
and that
$
\Psi_i
\in
L^{\infty} \left([0,T_{(Lifespan)}), H^4(\mathbb{R}^3) \right)
$
(as is stated in \eqref{E:MAINTHEOREMPSIIREGULARITY}).
The same reasoning yields that
$
\Psi_0 \in L^{\infty}\left([0,T_{(Lifespan)}], H^5(\mathbb{R}^3) \right)
$
(as is stated in \eqref{E:EXTENDEDMAINTHEOREMPSI0REGULARITY}),
where the open time interval is replaced with $[0,T_{(Lifespan)}]$.
The fact that
$
\Psi_{\alpha} \in C\left([0,T_{(Lifespan)}), H^5(\mathbb{R}^3) \right)
$
(as is stated in \eqref{E:MAINTHEOREMPSI0REGULARITY})
is a standard result that can be proved using
energy estimate arguments (similar to the ones we used to prove \eqref{E:MAINAPRIORIENERGYESTIMATES})
and standard facts from functional analysis.
We omit the details and instead refer the reader to
\cite{jS2008c}*{Section 2.7.5}.
We clarify that in proving this ``soft result,''
it is important that for fixed $t \in [0,T_{(Lifespan)})$,
we have
$\min_{[0,t] \times \mathbb{R}^3} \mathcal{I} > 0$,
which implies in particular that the weight $\mathscr{W}(\mathcal{I}^{-1} \Psi_0)$
on the right-hand side of \eqref{E:ENERGYTOCONTROLSOLNS}
is bounded from above and from below away from $0$ on $[0,t] \times \mathbb{R}^3$
(and thus the energy estimates are non-degenerate away from $\Sigma_{T_{(Lifespan)}}$).
Through similar reasoning based on
equation \eqref{E:INTEGRATINFACTORODEANDIC} (which states that $\partial_t \mathcal{I} = - \Psi_0$),
the identity \eqref{E:FORMULAFORDERIVATIVESOFINTEGRATINGFACTOR},
and the estimate \eqref{E:IFACTAPRIORIENERGYESTIMATE},
we deduce that
$
\mathcal{I} - 1
\in
L^{\infty} \left([0,T_{(Lifespan)}), H^5(\mathbb{R}^3) \right)
\cap
C\left([0,T_{(Lifespan)}), H^6(\mathbb{R}^3) \right)
$.
We have therefore proved \eqref{E:MAINTHEOREMPSI0REGULARITY}-\eqref{E:MAINTHEOREMIFACTREGULARITY}.
We will now prove \eqref{E:EXTENDEDMAINTHEOREMPSI0REGULARITY}-\eqref{E:EXTENDEDMAINTHEOREMIFACTREGULARITY}.
We first note that the estimates
\eqref{E:MAINAPRIORIENERGYESTIMATES} and \eqref{E:PARTIALTPSI0H4ESTIMATE}-\eqref{E:PARTIALTPSIIH4ESTIMATE}
and equation \eqref{E:INTEGRATINFACTORODEANDIC}
imply that
$
\partial_t
\Psi_{\alpha}
\in
L^{\infty}\left([0,T_{(Lifespan)}), H^4(\mathbb{R}^3) \right)
$
and
$
\partial_t
\mathcal{I}
\in
L^{\infty}\left([0,T_{(Lifespan)}), H^5(\mathbb{R}^3) \right)
$.
Hence, from the fundamental theorem of calculus,
the initial conditions \eqref{E:INTEGRATINFACTORODEANDIC} and \eqref{E:DATASIZE}-\eqref{E:LARGEDATASIZE},
and the completeness
of the Sobolev spaces $H^M(\mathbb{R}^3)$, we obtain
$
\Psi_{\alpha}
\in
C\left([0,T_{(Lifespan)}],H^4(\mathbb{R}^3) \right)
$
and
$
\mathcal{I} - 1
\in
C\left([0,T_{(Lifespan)}],H^5(\mathbb{R}^3) \right)
$.
In particular, we have shown
\eqref{E:EXTENDEDMAINTHEOREMPSIIREGULARITY}-\eqref{E:EXTENDEDMAINTHEOREMIFACTREGULARITY}.
Moreover, \eqref{E:EXTENDEDMAINTHEOREMIFACTREGULARITY} and Sobolev embedding together yield that
$\mathcal{I} \in C\left([0,T_{(Lifespan)}],C(\mathbb{R}^3) \right)$
and thus $\mathcal{I}_{\star} \in C\left([0,T_{(Lifespan)}],C(\mathbb{R}^3) \right)$.
Since we have already shown that $T_{(Lifespan)} = \mbox{RHS~\eqref{E:IFACTVANISHESONTMAX}}$,
it follows that $\mathcal{I}_{\star}(T_{(Lifespan)}) = 0$ and that
$\lim_{t \uparrow T_{(Lifespan)}} \mathcal{I}_{\star}(t) = 0$,
that is, that \eqref{E:IFACTMINVANISHESATLIFESPAN} holds.
To obtain that for $N < 5$, we have
$\Psi_0
\in
C\left([0,T_{(Lifespan)}],H^N(\mathbb{R}^3) \right)
$
(as is stated in \eqref{E:EXTENDEDMAINTHEOREMPSI0REGULARITY}),
we interpolate between\footnote{Here, we mean the following standard inequality:
if $f \in H^5(\Sigma_t)$ and $0 \leq N \leq 5$,
then there exists a constant $C_N > 0$ such that
$\| f \|_{H^N(\Sigma_t)} \leq C_N \| f \|_{L^2(\Sigma_t)}^{1-N/5} \| f \|_{H^5(\Sigma_t)}^{N/5}$. \label{FN:INTERPOLATION}}
$L^2$ and $H^5$
and
use the already shown facts
$
\Psi_0 \in L^{\infty}\left([0,T_{(Lifespan)}], H^5(\mathbb{R}^3) \right)
\cap
C\left([0,T_{(Lifespan)}],H^4(\mathbb{R}^3) \right)
$.
We have therefore proved \eqref{E:EXTENDEDMAINTHEOREMPSI0REGULARITY}.
The desired localized blowup result \eqref{E:BLOWUPBEHAVIOR}
for points in $\Sigma_{T_{(Lifespan)}}^{Blowup}$
(where $\Sigma_{T_{(Lifespan)}}^{Blowup}$ is defined in \eqref{E:BLOWUPSET})
now follows from
\eqref{E:MAINTHMSMALLINTFACTIMPMLIESPSI0ISLARGE}
and the continuous extension property $\mathcal{I} \in C\left([0,T_{(Lifespan)}],C(\mathbb{R}^3) \right)$
mentioned in the previous paragraph.
Finally, we will show that
if $(T_{(Lifespan)}, \underline{x}) \notin \Sigma_{T_{(Lifespan)}}^{Blowup}$,
then there exists an open ball $B_{\underline{x}} \subset \mathbb{R}^3$ centered at $\underline{x}$
such that for $\alpha = 0,1,2,3$, we have
$\partial_{\alpha} \Phi \in C\left([0,T_{(Lifespan)}], H^5(B_{\underline{x}}) \right)$.
To proceed, we first note that if $(T_{(Lifespan)}, \underline{x}) \notin \Sigma_{T_{(Lifespan)}}^{Blowup}$,
then the results proved above imply that there
exist a $\delta > 0$ and a radius $r_{\underline{x}} > 0$
such that,
with $B_{\underline{x};r_{\underline{x}}} \subset \mathbb{R}^3$ denoting
the open ball of radius $r_{\underline{x}}$ centered at $\underline{x}$,
we have $\mathcal{I}(t,\underline{y}) > 0$ for
$(t,\underline{y}) \in [T_{(Lifespan)} - \delta, T_{(Lifespan)}] \times \bar{B}_{\underline{x};r_{\underline{x}}}$
(where $\bar{B}_{\underline{x};r_{\underline{x}}}$ denotes the closure of $B_{\underline{x};r_{\underline{x}}}$)
and such that for $t \in [T_{(Lifespan)} - \delta, T_{(Lifespan)})$,
we have
$\| \Psi_{\alpha} \|_{H^5(\lbrace t \rbrace \times B_{\underline{x};r_{\underline{x}}})} < \infty$
and
$\| \mathcal{I} - 1 \|_{H^6(\lbrace t \rbrace \times B_{\underline{x};r_{\underline{x}}})} < \infty$.
Hence, since the wave speed of the system is uniformly bounded on
$[T_{(Lifespan)} - \delta, T_{(Lifespan)}] \times \bar{B}_{\underline{x};r_{\underline{x}}}$
(see Remark~\ref{R:BOUNDEDWAVESPEED}),
since
$\mathcal{I}$ is uniformly bounded from above and from below \emph{strictly away from $0$}
on $[T_{(Lifespan)} - \delta, T_{(Lifespan)}] \times \bar{B}_{\underline{x};r_{\underline{x}}}$,
and since the estimates
\eqref{E:MAINTHEOREMPSI0REGULARITY}-\eqref{E:MAINTHEOREMIFACTREGULARITY}
\eqref{E:EXTENDEDMAINTHEOREMPSI0REGULARITY}-\eqref{E:EXTENDEDMAINTHEOREMIFACTREGULARITY} hold,
we can derive Sobolev estimates
(based on energy arguments)
similar to the ones that we derived three paragraphs above,
but localized in space,\footnote{For example, for $\upsigma > 0$ chosen sufficiently large,
for $t$ near $T_{(Lifespan)}$,
and for $s \in [t,T_{(Lifespan)}]$,
one can consider the state of the solution on
$\lbrace t \rbrace \times B_{\underline{x};r_{\underline{x}}}$
as an initial condition and use energy identities to obtain
Sobolev estimates on
$\lbrace s \rbrace \times B_{\underline{x};r_{\underline{x}} - \upsigma s} \subset \Sigma_s$.}
for equations
\eqref{E:INTEGRATINFACTORODEANDIC}
and
\eqref{E:PARTALTPSI0EVOLUTION}-\eqref{E:PARTALTPSIIEVOLUTION},
starting from initial conditions on
$\lbrace t \rbrace \times B_{\underline{x};r_{\underline{x}}}$
for some $t$ sufficiently close
(in a manner that depends on $\underline{x}$)
to $T_{(Lifespan)}$.
This yields the existence of an open ball
$B_{\underline{x}} \subset B_{\underline{x};r_{\underline{x}}}$
centered at $\underline{x}$
such that the following regularity properties hold:
$\Psi_{\alpha} \in C\left([0,T_{(Lifespan)}], H^5(B_{\underline{x}}) \right)$
and
$\mathcal{I} - 1 \in C\left([0,T_{(Lifespan)}], H^6(B_{\underline{x}}) \right)$.
We clarify that to derive the localized energy estimates on the closed time interval $[0,T_{(Lifespan)}]$,
it is crucially important that the bounds noted above imply
that the spatial derivative weight $\mathscr{W}(\mathcal{I}^{-1} \Psi_0)$
(which appears, for example, on the right-hand side of \eqref{E:ENERGYTOCONTROLSOLNS})
is strictly positive on the domain
$[T_{(Lifespan)} - \delta, T_{(Lifespan)}] \times \bar{B}_{\underline{x}}$.
From the above regularity properties of $\lbrace \Psi_{\alpha} \rbrace_{\alpha = 0,1,2,3}$ and $\mathcal{I}$,
the positivity of $\mathcal{I}$ on $[T_{(Lifespan)} - \delta, T_{(Lifespan)}] \times \bar{B}_{\underline{x}}$,
and the standard Sobolev--Moser calculus,
we conclude,
in view of Def.~\ref{D:RENORMALIZEDSOLUTION},
the desired result
$\partial_{\alpha} \Phi \in C\left([0,T_{(Lifespan)}], H^5(B_{\underline{x}}) \right)$.
We have therefore proved the theorem.
\end{proof}
\end{document} | math | 200,371 |
\begin{document}
{\scriptscriptstyle\mathsf{T}}itle{Fast and reliable entanglement distribution with quantum repeaters: principles for improving protocols using reinforcement learning}
\author{Stav Haldar}\mathrm{e}mail{[email protected]}
\affiliation{Hearne Institute for Theoretical Physics, Department of Physics and Astronomy,
Louisiana State University, Baton Rouge, Louisiana 70803, USA}
\author{Pratik J. Barge}\mathrm{e}mail{[email protected]}
\affiliation{Hearne Institute for Theoretical Physics, Department of Physics and Astronomy,
Louisiana State University, Baton Rouge, Louisiana 70803, USA}
\author{Sumeet Khatri}\mathrm{e}mail{[email protected]}
\affiliation{Dahlem Center for Complex Quantum Systems, Freie Universit\"{a}t Berlin, 14195 Berlin, Germany}
\author{Hwang Lee}\mathrm{e}mail{[email protected]}
\affiliation{Hearne Institute for Theoretical Physics, Department of Physics and Astronomy,
Louisiana State University, Baton Rouge, Louisiana 70803, USA}
\date{{\scriptscriptstyle\mathsf{T}}oday}
\begin{abstract}
Future quantum technologies such as quantum communication, quantum sensing, and distributed quantum computation, will rely on networks of shared entanglement between spatially separated nodes. Distributing entanglement between these nodes, especially over long distances, currently remains a challenge, due to limitations resulting from the fragility of quantum systems, such as photon losses, non-ideal measurements, and quantum memories with short coherence times. In the absence of full-scale fault-tolerant quantum error correction, which can in principle overcome these limitations, we should understand the extent to which we can circumvent these limitations. In this work, we provide improved protocols/policies for entanglement distribution along a linear chain of nodes, both homogeneous and inhomogeneous, that take practical limitations into account. For a wide range of parameters, our policies improve upon previously known policies, such as the ``swap-as-soon-as-possible'' policy, with respect to both the waiting time and the fidelity of the end-to-end entanglement. This improvement is greatest for the most practically relevant cases, namely, for short coherence times, high link losses, and highly asymmetric links. To obtain our results, we model entanglement distribution using a Markov decision process, and then we use the Q-learning reinforcement learning (RL) algorithm to discover new policies. These new policies are characterized by dynamic, state-dependent memory cutoffs and collaboration between the nodes. In particular, we quantify this collaboration between the nodes. Our quantifiers tell us how much ``global'' knowledge of the network every node has, specifically, how much knowledge two distant nodes have of each other's states. In addition to the usual figures of merit, these quantifiers add an extra important dimension to the performance analysis and practical implementation of quantum repeaters. Finally, our understanding of the performance of large quantum networks is currently limited by the computational inefficiency of simulating them using RL or other optimization methods. The other main contribution of our work is to address this limitation. We present a method for nesting policies in order to obtain policies for large repeater chains. By nesting our RL-based policies for small repeater chains, we obtain policies for large repeater chains that improve upon the swap-as-soon-as-possible policy, and thus we pave the way for a scalable method for obtaining policies for long-distance entanglement distribution under practical constraints.
\mathrm{e}nd{abstract}
\title{Fast and reliable entanglement distribution with quantum repeaters: principles for improving protocols using reinforcement learning}
\section{Introduction}
The development of advanced and practical quantum technologies is a hallmark of the {\scriptscriptstyle\mathsf{T}}extit{second quantum revolution}~\cite{dowling2003quantum}. One of the frontiers of this revolution is a global-scale quantum internet~\cite{VanMeter_book,Dowling_book2,kimble2008quantuminternet,simon2017towards,WEH18,CCT+20,ICM+22,munro2022tomorrowquantuminternet}, which promises the realization of a plethora of quantum technologies, such as distributed quantum computing~\cite{cirac1999distributed,barz2012demonstration}, distributed quantum sensing~\cite{ge2018distributed, proctor2018multiparameter, komar2014quantum, simon2017towards, gottesman2012longer}, and quantum key distribution~\cite{bennett2020quantum, ekert1991quantum,xu2020QKDrealistic,pirandola2020advancescrypto}. A critical milestone on the road to a quantum internet is the ability to distribute quantum entanglement over long distances.
One of the main obstacles to achieving long-distance entanglement distribution, and consequently many of the above promises of quantum technologies, is noise, i.e., errors caused by the difficulty of maintaining good control over qubits and their environment. Noise arises in entanglement distribution due to loss in the quantum channels used to send qubits between spatially-separated nodes, and at every node noise arises due to short-lived quantum memories and imperfect entanglement swapping~\cite{heshami2016quantum,awschalom2021interconnects}. These sources of noise ultimately limit the rate, the distance, and the quality of entanglement distribution. Quantum error correction~\cite{Shor95errorcorrection,shor96faulttolerant,gottesman2009qecreview,roffe2019qecreview}, which includes entanglement distillation~\cite{bennett1996concentrating,bennett1996purification,bennett1996distillQEC}, has been understood for almost two decades to be the primary method to combat noise in order to achieve long-distance entanglement distribution, as well as full-scale, fault-tolerant quantum computation more generally. However, building devices with several thousands of qubits and then implementing error correction is currently a major technological and engineering challenge.
Motivated by this current state of affairs, our work is inspired by the following ensuing idea: instead of viewing noise as something that should be fought, let us take noise as a {\scriptscriptstyle\mathsf{T}}extit{given} and then see what protocols we can design, and what performance and potential advantages we can achieve.
This idea lies at the intersection of theory and experiment, and our goal is to prove theoretical statements that provide a guide to researchers in the lab on how to design their devices in order to achieve the best performance.
We note that this type of question, on making the best use of noisy quantum resources, has already been the focus of recent theoretical and experimental research on noisy intermediate-scale quantum computing~\cite{Preskill2018quantumcomputingin,bharti2021NISQreview,cerezo2021VQAreview}, particularly in the context of noise resilience~\cite{sharma2020noiseresilience,gentini2020noiseresilient,fontana2021noiseresilience}, quantum error mitigation~\cite{cai2022errormitigationreview}, and quantum advantage~\cite{bravyi2020quantumadvantagenoisy}.
Long-distance entanglement distribution typically proceeds by breaking up a quantum communication channel between a sender and a receiver into segments with intermediate nodes called ``quantum repeaters''~\cite{briegel1998quantum, duer1999repeaterspurification,muralidharan2016optimal}. The quantum repeaters perform entanglement distillation and entanglement swapping~\cite{bennett1993teleporting,zukowski93swapping} in order to increase the quality and distance, respectively, of an entangled quantum state; see Fig.~\ref{fig:q_net}. Once entanglement between the sender and the receiver has been established, they can use the quantum teleportation protocol~\cite{bennett1993teleporting} in order to send arbitrary quantum information to each other. The advantage of using quantum repeaters is that the rates for the end-to-end quantum information transmission can be improved compared to directly sending the quantum information. Entanglement distribution has been experimentally demonstrated on numerous platforms for neighboring nodes~\cite{olmschenk2009quantum, nolleke2013efficient, langenfeld2021quantum, humphreys2018deterministic, pfaff2014unconditional}, and recently for non-neighboring nodes in a quantum network of solid-state spin qubits~\cite{hermans2022qubit}.
Understanding what parameter regimes of today's noisy devices correspond to high performance (high rates and high fidelities)---or, conversely, what performance can be achieved with such devices---is crucial in order to realize long-distance entanglement distribution with currently available quantum devices. One challenge is to determine what protocols achieve {\scriptscriptstyle\mathsf{T}}extit{optimal} performance in the presence of device imperfections. For example, a simple strategy one can always do is perform entanglement swapping as soon as both elementary links are active at a particular node~\cite{coopmans2021thesis,kamin2022exactrateswapasap}, the so-called ``swap-as-soon-as-possible'' ({\scriptscriptstyle\mathsf{T}}extsc{swap-asap}) policy---but is this the {\scriptscriptstyle\mathsf{T}}extit{best} strategy, especially in the presence of device imperfections, and particularly probabilistic elementary link generation and entanglement swapping? This is the primary question we are concerned with in this work. In more detail, we are concerned with the following questions:
\begin{itemize}
\item Can we find optimal entanglement distribution policies, or at least policies better than {\scriptscriptstyle\mathsf{T}}extsc{swap-asap}, for small, inhomogeneous, resource-constrained quantum networks, which provide faster connectivity as well as high end-to-end fidelities?
\item Can general unifying features be identified for such improved policies?
\item Can these unifying features help in devising improved policies for large networks with many nodes, for which a direct optimization is not practical?
\mathrm{e}nd{itemize}
Under certain conditions, which we outline below, we answer {\scriptscriptstyle\mathsf{T}}extit{``yes''} to all of the above questions.
\subsection{Summary of our work}
In this work, we provide a theoretical framework for both modeling protocols/policies for entanglement distribution and for optimizing policies with respect to the performance parameters of average waiting time and fidelity. We do so while taking specific aspects of physical implementations into account, but the framework is abstract enough that our results are agnostic to any particular implementation. We consider specifically the task of distributing entanglement between two distant nodes separated by a linear chain of quantum repeaters; see Fig.~\ref{fig:q_net}. We consider probabilistic elementary link generation, probabilistic entanglement swapping, and quantum memories with finite coherence time at the nodes. The policies that we obtain are thus functions of these high-level parameters that characterize the noise and imperfections of the quantum devices that are available today. Our theoretical framework is based on Markov decision processes (MDPs), which we use to model states of the repeater chain and the actions that can be taken at the nodes; see Sec.~\ref{sec:MDP_model}
In principle, MDPs can be solved exactly using dynamic and linear programming~\cite{Puterman2014book}. However, because the state and action space grows exponentially with the input size (in our case, the input is the number of nodes), other techniques are needed. In this work, we use reinforcement learning (RL)~\cite{SuttonBarto2018book}, and we use it to obtain policies that, while not necessarily globally optimal, are better in certain parameter regimes than human-developed policies such as the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy referred to above.
\begin{figure*}
\centering
\includegraphics[width=0.95{\scriptscriptstyle\mathsf{T}}extwidth]{figures/Fig1_alt.pdf}
\caption{(a) A schematic of a general quantum network. The intermediate nodes represent quantum repeaters, and the symbols at the end nodes indicate possible applications. (b) A closer look at the elementary links and nodes of a linear network topology. Every node has two quantum memories, and every pair of neighboring quantum memories has an associated entanglement source. (c) Once both memories at a node are active (each storing one-half of a two-qubit entangled state), entanglement swapping can be performed to extend the entanglement to non-neighboring nodes, with success probability $p_{sw}$. (d) A linear quantum repeater chain. In a homogeneous chain, all elementary links have the same elementary link success probability $p_{\mathrm{e}ll}$. For an inhomogeneous chain, these success probabilities can be different. We use $p_{\mathrm{e}ll}^{i,j}$ to denote the elementary link success probability for nodes $i$ and $j$.}
\label{fig:q_net}
\mathrm{e}nd{figure*}
The main results of our work are as follows.
\begin{itemize}
\item Using Q-learning, a model-free RL technique, we find improved policies for entanglement distribution in a linear repeater chain with up to five nodes. Our policies yield faster communication (i.e., lower waiting time) (Sec.~\ref{sec:opt_pol_waiting_time}) and higher fidelity (Sec.~\ref{sec-opt_pol_fidelity}) compared to the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy. Furthermore, our fidelity-based results are independent of the specific noise model for the quantum memories. The improvement of our policies over the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy is the largest in the most non-ideal cases, and thus the most practically relevant cases: when the memory coherence times are low, the elementary link success probabilities are low, and the asymmetries are high. Furthermore, the policies obtained using Q-learning are known to be optimal, in principle, as long as enough training is done. In this sense, our policies are near-optimal. We refer to Appendix~\ref{sec-Q_learning} for more information about the Q-learning algorithm and the specifics of our implementation.
\item Another important practical consideration is inhomogeneous device parameters. In this work, we also find improved policies for small inhomogeneous repeater chains of up to five nodes; see Sec.~\ref{sec-waiting_time_inhomogeneous}.
\item Apart from discovering policies that outperform {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} over a wide range of parameters, in Sec.~\ref{sec:advantage} we also quantitatively address the question: \mathrm{e}mph{Where do improved policies derive their advantage from?} We find that our improved policies have the following key features:
\begin{enumerate}
\item Using global knowledge of the network, i.e., using the state of the entire repeater chain when executing actions at a particular node.
\item Collaboration between nodes of the network, i.e., correlations between the states and actions of links and nodes.
\item Dynamic, state-dependent cutoffs for the links. This means that links are kept only for a limited amount of time before being discarded due to too much decoherence. Compared to the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy, these cutoffs improve the end-to-end fidelity, as one would expect, but surprisingly they also improve the expected waiting time.
\mathrm{e}nd{enumerate}
We provide in this work concrete quantifiers for the aforementioned ``global knowledge'', ``collaboration'', and ``dynamic cutoffs''; see Sec.~\ref{sec:advantage}. These results, and in particular the quantifiers that we use, provide new conceptual insights into the functioning of quantum repeaters. In particular, because our policies provide the most improvement in highly non-ideal parameter regimes, as described above, we see that collaboration between the nodes, one of the central features of our policies, is crucial when dealing with realistic, small-to-medium scale quantum networks with noisy and imperfect quantum devices.
\item Finally, practical quantum networks require policies for distributing entanglement over very long distances. Finding an optimal policy directly for large repeater chains under general assumptions is neither analytically tractable nor computationally efficient. Global policies for such repeater chains, i.e., those in which the actions are based on the entire state of the repeater chain, would also not necessarily be practically useful, because classical communication times would adversely effect the average waiting time and end-to-end fidelity. Another main contribution of this work is to address this challenge. We introduce a nesting method that takes our improved RL-based policies for small sections of a long repeater chain and combines them with a local policy for connecting the sections amongst themselves; see Sec.~\ref{sec:nested}. We demonstrate this method for up to 13 nodes with two levels of nesting. In this case, we show that the nested policies substantially outperform fully local policies, and thus we demonstrate the utility of our RL-based approach even for very large quantum repeater chains.
\mathrm{e}nd{itemize}
It is worth mentioning that the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy is a ``system-agnostic'' policy, not depending explicitly on the device parameters of the repeater chain. In contrast, our RL-based policies do in general depend on the input device parameters and on the entanglement swapping success probability. The advantage of obtaining these ``system-dependent'' policies is that it allows practitioners to use policies that are tailored to their specific device parameters. Furthermore, while it might not be surprising that we would obtain waiting time and fidelity improvements by using RL (or any other optimization technique that takes information about the system into account), understanding {\scriptscriptstyle\mathsf{T}}extit{how} these improvements come about is certainly of interest, even for small repeater chains, and it is this question that we primarily address in this work. Notably, just by extracting the general principles of improved policies for small repeater chains in Sec.~\ref{sec-discussion} and Sec.~\ref{sec:advantage}, we are able to construct improved policies for large repeater chains in Sec.~\ref{sec:nested}.
\subsection{Summary of prior work}
Due to the complexity of the problem that we consider in this work, analytical treatments of performance and policy optimization~\cite{khatri2019practical, vinay2019statistical, shchukin2019waiting, VGNT20, PGGT20, DT21, Khatri2021policieselementary,coopmans2022improved,Kha22} are limited to simple geometries (e.g., linear or star networks) of only a handful of nodes, often with the simplifying assumption of infinite quantum memory coherence time. Hence, numerical optimization seems to be a more accessible route to analyzing realistic quantum networks. For example, Ref.~\cite{li2020efficient} numerically optimizes memory cutoffs, Ref.~\cite{jiang2007optimal} uses dynamic programming to optimize quantum repeater protocols, and Ref.~\cite{daSilva2021genetic} uses genetic algorithms. There also exist several numerical simulation platforms for quantum networks~\cite{MMG+15,DW18,Bart18,Mat19,DNZB20,WKC+20,CKD+20,wallnofer2022ReQuSim}.
Recently, in Ref.~\cite{wallnofer2020MLQComm}, the idea of using machine learning, in particular reinforcement learning, for quantum communication tasks was introduced. Then, in Ref.~\cite{Khatri2021policieselementary} (see also Ref.~\cite{Kha22}), a particular kind of MDP for elementary links in a quantum network was introduced, taking memory decoherence into account, with an exact solution using dynamic~\cite{Khatri2021policieselementary} and linear~\cite{Kha22} programming. Thereafter, in Refs.~\cite{shchukin2022optimal,inesta2022optimal,reiss2022deep}, an analogous MDP formulation for repeater chains was presented, with exact solutions based on linear programming in Ref.~\cite{shchukin2022optimal} and RL-based solutions in Refs.~\cite{inesta2022optimal,reiss2022deep}. This work is most similar in spirit to Refs.~\cite{inesta2022optimal,reiss2022deep}; however, there are notable differences in the modeling of the repeater chain using MDPs between those two works and ours, and we defer a detailed outline of these and other differences between our work and Refs.~\cite{inesta2022optimal,reiss2022deep} to Sec.~\ref{sec-compare_prior_work}.
Finally, it is important to emphasize that this line of work departs from the usual information-theoretic analysis of entanglement distribution and quantum communication~\cite{Hol12_book,NC00_book,KW20_book}, which is concerned with what rates, in principle, can be achieved when constrained by the laws of quantum mechanics alone. (See, in particular, Refs.~\cite{BCHW15,Pir16,AML16,LP17,CM17,AK17,BA17,RKB+18,Pir19,Pir19b,DBWH19,BAKE20,HP22} for information-theoretic analyses for quantum communication networks.) From a practical perspective, the fundamental rate limitations derived in those works should be thought of as informing us about what performance rates {\scriptscriptstyle\mathsf{T}}extit{cannot at all} be achieved in practice. On the other hand, while this work still uses tools from quantum information theory, it also dives deeper into the physical implementation, thus resulting in a more realistic performance analysis, telling us what rates and performance criteria {\scriptscriptstyle\mathsf{T}}extit{can} be achieved with quantum devices available to us today, and provides us with a mathematical framework for obtaining optimal protocols in realistic settings. Prior works along similar lines include Refs.~\cite{VLMN09,AME11,JKR+16,MDV19,rozpedek2018practicalentdistill,krastanov2019optentpurif,CCvM20,DPW20,CERW20,GEW20,bugalho2021multipartitenoisy,ramiro2021optimized,PGGT20,VGNT20,DT21,VanMeter_book,daSilva2021genetic,azuma2021tools}. We emphasize that while the framework and results presented in this work take the specifics of physical implementations into account, they are agnostic to any particular implementation.
\section{Theoretical Model} \label{sec:MDP_model}
In this section, we outline our theoretical model for entanglement distribution in a quantum repeater chain. We start with the model for the creation of elementary and virtual links, and then proceed to the definition of the Markov decision process that we use to define policies for creating end-to-end entanglement in the quantum repeater chain. Further mathematical details on the model presented here can be found in Ref.~\cite{Kha22}.
In summary, our model consists of the following elements.
\begin{itemize}
\item The quantum repeater chain consists of $n$ nodes. Every node has at most two quantum memories; see Fig.~\ref{fig:q_net}. All of these memories have a maximum cutoff time $m^{\star}\in\{0,1,2,\dotsc\}$, which can be thought of as the coherence time of the quantum memory. If a pair of memories has been holding an entangled state for $m^{\star}$ time steps, then the state is discarded and the entanglement is regenerated.
\item Elementary link generation succeeds with probability $p_{\mathrm{e}ll}\in[0,1]$. Different elementary links can have different success probabilities, in which case we refer to the repeater chain as ``inhomogeneous''. Otherwise, when all of the elementary links have the same value of $p_{\mathrm{e}ll}$, we refer to the repeater chain as ``homogeneous''.
\item Entanglement swapping is assumed to be error-free but non-deterministic in general, succeeding with probability $p_{sw}\in[0,1]$.
\item We consider discrete time steps. In every time step, nodes, or pairs of nodes, can perform the following actions:
\begin{itemize}
\item A pair of neighboring nodes can attempt to generate an elementary link between them, discard the link that might already exist between them, or wait (do nothing).
\item A node can attempt entanglement swapping on its two quantum memories.
\mathrm{e}nd{itemize}
\mathrm{e}nd{itemize}
It is important to note that, in this work, we consider the abstract setting in which the size of the repeater chain is given simply by the total number $n$ of nodes, and neither the physical distance between the end points of the chain nor the distances between the intermediate nodes is explicitly specified. Also, classical communication---specifically, local operations and classical communications (LOCC)---is assumed to be free in our framework. This abstract setting is entirely analogous to the information-theoretic setting of quantum communication, in which only the number of channel uses (or the number of ``network uses''---see, e.g., Refs.~\cite{Pir16,Pir19}) is the resourceful quantity used to determine the waiting time/rate, and LOCC is assumed to be free. Accordingly, in this work, a time step is synonymous with a single use of the repeater chain, and thus entanglement swapping operations (because they are LOCC operations) do not add a time step, and thus do not contribute to the waiting time; see Sec.~\ref{sec:opt_pol_waiting_time} below for more information. A detailed calculation of waiting times, taking into account classical communication in both the uses of the repeater chain and of the entanglement swapping operations, is left for future work.
We now describe our model for the creation of elementary and virtual links.
\paragraph*{Creation of elementary links.} As shown in Fig.~\ref{fig:q_net}(b), an \mathrm{e}mph{elementary link} exists between two neighboring nodes when one part of a bipartite entangled state is successfully delivered to each of them from a common source. Elementary link creation succeeds with probability $p_{\mathrm{e}ll} \in [0,1]$. This success probability includes the effects of channel loss, detector inefficiencies, noise, finite absorption cross-sections of quantum memories, etc.; see Refs.~\cite{Khatri2021policieselementary,Kha22} for details and examples. We call the elementary link ``active'' if an entangled state is shared between the two nodes.
Assume that the quantum state shared by the quantum memories of two nodes, call them $R_1$ and $R_2$, after successful creation of the elementary link, is the two-qubit state $\sigma_{R_1R_2}^0$. Let us also assume that the qubit memories $R_1$ and $R_2$ undergo decoherence according to quantum channels $\mathcal{N}^1$ and $\mathcal{N}^2$, respectively. Then, let
\begin{equation}\label{eq-elem_link_state}
\sigma_{R_1R_1}(m)\coloneqq ((\mathcal{N}_{R_1}^{1})^{\circ m}\otimes(\mathcal{N}_{R_2}^2)^{\circ m})(\sigma_{R_1R_2}^0)
\mathrm{e}nd{equation}
be the quantum state of the two memories after $m\in\{0,1,2,\dotsc\}$ discrete time steps, where $(\mathcal{N}^1)^{\circ m}\mathrm{e}quiv \mathcal{N}^1\circ\mathcal{N}^1\circ\dotsb\circ\mathcal{N}^1$ ($m$ times), and similarly for $(\mathcal{N}^2)^{\circ m}$. We refer to $m$ as the {\scriptscriptstyle\mathsf{T}}extit{age} of the elementary link.
As time progresses, the fidelity of the quantum state in \mathrm{e}qref{eq-elem_link_state} deteriorates until it is considered too noisy to be useful after a given number $m^{\star}$ of time steps, at which point the state is discarded and the elementary link is created again. We let
\begin{equation}\label{eq-elem_link_fidelity}
f(m)\coloneqq \bra{\Phi^+}\sigma(m)\ket{\Phi^+}
\mathrm{e}nd{equation}
be the fidelity of the state of the elementary link after $m$ time steps, where $\ket{\Phi^+}=\frac{1}{\sqrt{2}}(\ket{0,0}+\ket{1,1})$ is the two-qubit maximally-entangled Bell state.
\paragraph*{Creation of virtual links.} As shown in Fig.~\ref{fig:q_net}(c), a \mathrm{e}mph{virtual link} can be established between two distant nodes with intermediate nodes acting as quantum repeaters, which execute the entanglement swapping protocol with success probability $p_{sw} \in [0,1]$. With reference to Fig.~\ref{fig:q_net}(c), consider three nodes, corresponding to two elementary links, such that the quantum state of the two elementary links is $\sigma_{AR_1}(m_1)\otimes\sigma_{R_2B}(m_2)$. By performing the entanglement swapping protocol on the two qubits $R_1,R_2$, the resulting quantum state shared by $A$ and $B$ is $\rho_{AB}(m_1,m_2)$; we refer to Ref.~\cite{Kha22} for an explicit expression for this state, which is not required for our purposes in what follows. Now, we would like to determine the fidelity
\begin{equation}\label{eq-virtual_link_fidelity}
f_{{\scriptscriptstyle\mathsf{T}}extsc{swap}}(m_1,m_2)\coloneqq\bra{\Phi^+}\rho(m_1,m_2)\ket{\Phi^+}
\mathrm{e}nd{equation}
of the swapped state without keeping track of the full quantum state of the elementary links. Depending on the noise model of the individual quantum memories, there exists an $m'\in\{0,1,\dotsc\}$ such that $f_{{\scriptscriptstyle\mathsf{T}}extsc{swap}}(m_1,m_2)=f(m')$. The age $m'$ of the swapped state can thus be calculated as $m'=f^{-1}(f_{{\scriptscriptstyle\mathsf{T}}extsc{swap}}(m_1,m_2))$, as long as $f$ is invertible. For example, for Pauli noise, because it commutes with the entanglement swapping operation~\cite{bowen2001teleportation,schmidt2020memoryassisted}, we have that $f_{{\scriptscriptstyle\mathsf{T}}extsc{swap}}(m_1,m_2)=f(m_1+m_2)$, so that $m'=m_1+m_2$. In our simulations, we take the age of the oldest link as the age of the swapped state, i.e., we let $m'=\max\{m_1,m_2\}$ be the age of the swapped state. In the case of the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy, this rule is exact regardless of the noise model, because in that case one of the two links has age 0 and because $f_{{\scriptscriptstyle\mathsf{T}}extsc{swap}}(0,m)=f(m)$~\cite{Kha22}. For other policies, and for general noise models, this rule is not exact, but we use it in order to obtain results that are independent of the underlying noise model. We leave the analysis of a noise-specific update rule to future work.
With the basic physical elements of our model in place, we now show how to model the evolution in time of a quantum repeater chain in terms of a Markov decision process.
\subsection{Definition of the MDP}\label{sec-MDP}
A Markov decision process (MDP)~\cite{Puterman2014book} (see also Ref.~\cite[Appendix~A]{Kha22}) is a mathematical model of an agent that has the ability to interact with a system by taking different actions on it. The system consists of a set $\mathbf{S}$ of (classical) states, and similarly the actions form a set $\mathbf{A}$. The transition probabilities between states are given by transition matrices $\{T_A\}_{A\in\mathbf{A}}$, such that the matrix element $T_A(S';S)$ is equal to the probability of going to the state $S' \in \mathbf{S}$, when the current state is $S \in \mathbf{S}$ and the action taken is $A \in \mathbf{A}$. We now present these defining elements of an MDP within the context of quantum repeater chains; see Fig.~\ref{fig:mdp} for a summary. We note that our definitions of the states and actions in a quantum repeater chain are similar to, but not exactly the same as, those in Refs.~\cite{inesta2022optimal,reiss2022deep}; see Sec.~\ref{sec-compare_prior_work} for details on the differences.
\begin{figure}
\centering
\includegraphics[width=0.75\columnwidth]{figures/mdp.pdf}
\caption{Illustration of the states, actions, and transition rules of our MDP for entanglement distribution in a quantum repeater chain. (a)~The effect of the elementary link request action ($R_{\mathrm{e}ll}$) between two nodes is to discard the entangled state, if there is one stored in their respective quantum memories, and attempt to generate an elementary link between the two nodes, which succeeds with probability $p_{\mathrm{e}ll}$. We note that the agent can choose to discard a link at any time $t^{\star}\leq m^{\star}$, thereby allowing it to choose a cutoff that is different from the maximum cutoff $m^{\star}$. (b)~The effect of the wait action ($W$) is to increase the age $m$ of a link by one time unit, if $m$ is strictly less than the maximum cutoff time $m^{\star}$; otherwise, if $m=m^{\star}$, then the link is discarded. (c)~Given two elementary links with ages $m_1$ and $m_2$, the entanglement swapping action ($R_{sw}$) succeeds with probability $p_{sw}$, in which case the age of the new virtual link is given by the age of the oldest elementary link, i.e., $\max\{m_1,m_2\}$.}
\label{fig:mdp}
\mathrm{e}nd{figure}
\paragraph*{States.} The state space consists of all possible configurations of the repeater chain. In particular, we represent the state of the repeater chain at any given time $t$ by an $n{\scriptscriptstyle\mathsf{T}}imes n$ matrix $S^t$, where $n$ is the number of nodes in the chain. The state of the link between two nodes $i,j\in\{1,2,\dotsc,n\}$ is represented by the entry $S^t_{i,j}$, which can lie between $-1$ and $m^{\star}$, with $S^t_{i,j}\geq 0$ indicating that the link is active and $S_{i,j}^{t}=-1$ indicating that the link is inactive.
For example, consider a four-node repeater chain, with an elementary link between node 1 and 2 that is active for one time step and a virtual link between 2 and 4 that is active for two time steps. Such a state is represented as
\begin{equation}
S^t = \begin{bmatrix}
\ast & 1 & \ast & \ast\\
1 & \ast & \ast & 2\\
\ast & \ast & \ast & \ast\\
\ast & 2 & \ast & \ast
\mathrm{e}nd{bmatrix},
\mathrm{e}nd{equation}
where the asterisks `$\ast$' indicate that the corresponding entries could be arbitrary elements in the set $\{-1,0,1,\dotsc,m^{\star}\}$. The {\scriptscriptstyle\mathsf{T}}extit{terminal/absorbing states} are states in which the first and the last node are connected by a virtual link. Such states have the form $S^t_{1,n}=m$, for $m\in\{0,1,\dotsc,m^{\star}\}$.
We remark that not all $n{\scriptscriptstyle\mathsf{T}}imes n$ matrices are possible states of the repeater chain. First, the states are symmetric and nodes cannot link to themselves, so that diagonal elements of the form $S^t_{i,i}$ are not allowed. Furthermore, because we are only dealing with a linear repeater chain in which every node has at most two quantum memories, every node can be connected to not more than two more nodes, one ahead it and one behind it. Therefore, every row and column of $S$ can have only two elements that are not equal to $-1$. These constraints mean that the number of possible states scales as $\approx (m^{\star}+1)^{n}$. For example, for a repeater chain with $n=5$ nodes and maximum cutoff $m^{\star} = 2$, the number of allowed states is 562.
\paragraph*{Actions.} The action space consists of the following.
\begin{itemize}
\item Request ($R_\mathrm{e}ll$):
This action is a request to create an elementary link between two adjacent nodes. The success probability of elementary link creation is denoted by $p_{\mathrm{e}ll}$. If the elementary link is already active between the two nodes, then this action first discards the link and then attempts to create a new one.
\item Wait ($W$): This action keeps a link, either a virtual link or an elementary link, stored in the quantum memories. It increases the age of active links by one; however, if the current age is equal to the maximum cutoff $m^{\star}$, then the link becomes inactive.
\item Request Swapping ($R_{sw}$) : This action can take place at any node when it is linked to two other nodes via elementary or virtual links. We assume that the entanglement swapping is error free but in general non-deterministic, succeeding with probability $p_{sw}\in[0,1]$. If it succeeds, a virtual link is established between the corresponding end nodes. In case of failure, both links become inactive.
\mathrm{e}nd{itemize}
Mathematically, similar to the states, we describe the action taken on a state at any given time $t$ by an $n{\scriptscriptstyle\mathsf{T}}imes n$ matrix $A^t$. Now, $A^t_{i,j}$ represents the action taken by the nodes $i,j$ together. This is also a symmetric matrix. An elementary link request ($R_{\mathrm{e}ll}$) is given by $A^t_{i,j}=1$, with $j=i+1$. The wait action ($W$) is given by $A^t_{i,j}=0$ for $i\neq j$. Entanglement swapping requests ($R_{sw}$) are given by diagonal elements of the matrix, so that $A_{i,i}^t=1$ means a request for entanglement swapping at the $i^{{\scriptscriptstyle\mathsf{T}}ext{th}}$ node. For example, consider a four-node repeater chain. The action of requesting an elementary link between node 1 and node 2 along with a swapping request at node 3 is represented as
\begin{equation*}
A^t = \begin{bmatrix}
0 & 1 & 0 & 0\\
1 & 0 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 0
\mathrm{e}nd{bmatrix}.
\mathrm{e}nd{equation*}
As with the states, the number of allowed actions is limited. Again, the actions are symmetric, but now the elements $A^t_{i,i}$ can be non-zero. Also, no $R_\mathrm{e}ll$ action is allowed between non-adjacent nodes, so the only non-diagonal non-zero elements are of the form $A_{i,i+1}^t$. These constraints bring down the number of possible actions to the order of $\approx 2^{(2n-3)}$ for a repeater chain with $n$ nodes. Further, if a swap action $R_{sw}$ is taken at a node, then the same node is not involved in an elementary link request $R_\mathrm{e}ll$; therefore, if $A_{i,i}^t = 1$, then the $i^{{\scriptscriptstyle\mathsf{T}}ext{th}}$ row and column can have no other non-zero entry. These constraints reduce the number of allowed actions even more. For example, for a repeater chain with $n=5$ nodes, the number of allowed actions is only 34 (out of the $2^{(2n-3)} = 128$ possible actions taking all combinations of elementary link generation and entanglement swap requests).
\paragraph*{Transition rules.} Let us now look at the effect of various actions on the states of the repeater chain.
\begin{itemize}
\item Effect of request ($R_\mathrm{e}ll$): If $A^t_{i,i+1} = 1$, then with probability $p_{\mathrm{e}ll}$ we obtain $S^{t+1}_{i,i+1} = 1$, and with probability $1 - p_{\mathrm{e}ll}$ we obtain $S^{t+1}_{i,i+1} = 0$.
\item Effect of wait ($W$):
If $A^t_{i,j} = 0$ and $S^t_{i,j} \geq 0$, then
\begin{equation}\label{eq-state_update_wait}
S^{t+1}_{i,j} = -1 + ((S^t_{i,j} + 2){\scriptscriptstyle\mathsf{T}}ext{ mod } (m^\star+2)).
\mathrm{e}nd{equation}
If the link is already inactive, i.e., if $S^t_{i,j} = -1$, then $S^{t+1}_{i,j} = S^t_{i,j}$. Observe that the action $W$ automatically discards any link (elementary or virtual) that has an age equal to the maximum cutoff time $m^\star$.
\item Effect of request for entanglement swapping ($R_{sw}$):
If $A^t_{i,i} =1 $, then check if $S^t_{i,k}\geq 0$ and $S^t_{j,i}\geq 0$, i.e., check if the node $i$ has an active link with some nodes $j$ and $k$. If so, then with probability $p_{sw}$, $S^{t+1}_{j,k} = \max\{S_{i,k}^t,S_{j,i}^t\}$, and with probability $1 - p_{sw}$, $S^{t+1}_{j,k} = -1$. For both outcomes, the parent links become inactive, i.e., $S^{t+1}_{i,k} = -1$ and $S^{t+1}_{j,i} = -1$. Note that if an entanglement swap is requested at a node that is connected to only one other node, then that link becomes inactive.
\mathrm{e}nd{itemize}
The request and wait actions effect individual links independently. The swap request, on the other hand, depends on and effects the states of two links.
The transition rules described above form the transition matrices $\{T^A\}_{A\in\mathbf{A}}$ of our MDP. As a result of having absorbing states, we can partition every transition matrix $T_A$, $A\in\mathbf{A}$, into a block matrix as follows:
\begin{equation}
T_A=\begin{pmatrix} Q_A & 0 \\ R_A & \mathbbm{1} \mathrm{e}nd{pmatrix},
\mathrm{e}nd{equation}
where the top-left block $Q_A$ corresponds to transitions between transient (non-terminal) states, the bottom-left block $R_A$ corresponds to transitions from transient states to terminal states, and the bottom-right block corresponds to transitions between terminal states, which is the identity matrix $\mathbbm{1}$ by our definition of a terminal state.
\subsection{Optimization of waiting time and fidelity}
A policy is defined by the function $\pi:\mathbf{A}{\scriptscriptstyle\mathsf{T}}imes\mathbf{S}{\scriptscriptstyle\mathsf{T}}o[0,1]$, such that $\pi(A|S)$ is defined as the probability of taking action $A\in\mathbf{A}$ given that the state of the repeater chain is $S\in\mathbf{S}$. In other words, the policy is a collection of conditional probability distributions, one for every state.
For every policy $\pi$, we define the transition matrix $P_{\pi}$ such that, for all $S,S'\in\mathbf{S}$,
\begin{equation}
P_{\pi}(S';S)=\sum_{A\in\mathbf{A}}\pi(A|S)T_A(S';S).
\mathrm{e}nd{equation}
Based on the block structure of the transition matrices $T_A$, as described above, we have that for every policy $\pi$ the transition matrix $P_{\pi}$ has the following block structure:
\begin{equation}
P_{\pi}=\begin{pmatrix} Q_{\pi} & 0 \\ R_{\pi} & \mathbbm{1} \mathrm{e}nd{pmatrix},
\mathrm{e}nd{equation}
where
\begin{equation}
Q_{\pi}(S';S)\coloneqq\sum_{A\in\mathbf{A}}\pi(A|S)Q_A(S';S)
\mathrm{e}nd{equation}
for all transient states $S,S'$, and
\begin{equation}
R_{\pi}(S';S)\coloneqq\sum_{A\in\mathbf{A}}\pi(A|S)R_A(S';S)
\mathrm{e}nd{equation}
for all transient states $S$ and terminal states $S'$.
\paragraph*{Optimization of waiting time.} For a given policy $\pi$, let
\begin{equation}
N_{\pi}\coloneqq (\mathbbm{1}-Q_{\pi})^{-1}.
\mathrm{e}nd{equation}
Then, it follows from the theory of Markov chains (see, e.g., \cite[Theorem~9.6.1]{Stewart09_book}) that the expected time to reach a terminal state (i.e., the expected waiting time for an end-to-end link) is given by
\begin{equation}
W_{\pi}\coloneqq \sum_{S,S'\in\mathbf{S}} N_{\pi}(S';S)p_1(S),
\mathrm{e}nd{equation}
where $p_1(S)\mathrm{e}quiv\Pr[S^1=S]$ is the initial probability distribution of (transient) states at time $t=1$. The minimum expected waiting time is then given by the following optimization problem:
\begin{equation}\label{eq-waiting_time_opt_problem}
\begin{array}{l l} {\scriptscriptstyle\mathsf{T}}ext{minimize} & W_{\pi} \\[1ex] {\scriptscriptstyle\mathsf{T}}ext{subject to} & \pi:\mathbf{A}{\scriptscriptstyle\mathsf{T}}imes\mathbf{S}{\scriptscriptstyle\mathsf{T}}o[0,1], \\[1ex] & \displaystyle \sum_{A\in\mathbf{A}}\pi(A|S)=1~~\forall~S\in\mathbf{S}.\mathrm{e}nd{array}
\mathrm{e}nd{equation}
\paragraph*{Optimization of fidelity.} For every state $S\in\mathbf{S}$ of the repeater chain, we define
\begin{equation}
\widetilde{f}(S)\coloneqq\left\{\begin{array}{l l} f(S_{1,n}) & {\scriptscriptstyle\mathsf{T}}ext{if } S {\scriptscriptstyle\mathsf{T}}ext{ is a terminal state}, \\ 0 & {\scriptscriptstyle\mathsf{T}}ext{otherwise}, \mathrm{e}nd{array}\right.
\mathrm{e}nd{equation}
where we recall the function $f$ defined in \mathrm{e}qref{eq-elem_link_fidelity}. In other words, $\widetilde{f}(S)$ gives the fidelity of the end-to-end entangled state, when it is active. We are then interested in the expected value of this quantity with respect to a policy $\pi$ in the steady-state limit:
\begin{align}
\widetilde{F}_{\pi}&\coloneqq \lim_{t{\scriptscriptstyle\mathsf{T}}o\infty}\widetilde{F}_{\pi}(t),\\
\widetilde{F}_{\pi}(t)&\coloneqq\langle\widetilde{f}(S^t)\rangle_{\pi},\\
&=\sum_{S,S'\in\mathbf{S}} \widetilde{f}(S')P_{\pi}^{t-1}(S';S)p_1(S),
\mathrm{e}nd{align}
where in the last line $p_1(S)\mathrm{e}quiv\Pr[S^1=S]$ is the initial probability distribution of (transient) states at time $t=1$. The maximum fidelity is then given by the following optimization problem:
\begin{equation}\label{eq-fidelity_opt_problem}
\begin{array}{l l} {\scriptscriptstyle\mathsf{T}}ext{maximize} & \widetilde{F}_{\pi} \\[1ex] {\scriptscriptstyle\mathsf{T}}ext{subject to} & \pi:\mathbf{A}{\scriptscriptstyle\mathsf{T}}imes\mathbf{S}{\scriptscriptstyle\mathsf{T}}o[0,1], \\[1ex] & \displaystyle \sum_{A\in\mathbf{A}}\pi(A|S)=1~~\forall~S\in\mathbf{S}. \mathrm{e}nd{array}
\mathrm{e}nd{equation}
\subsection{Policies via reinforcement learning}
We are interested in solving the optimization problems in \mathrm{e}qref{eq-waiting_time_opt_problem} and \mathrm{e}qref{eq-fidelity_opt_problem}. To do this, in this work we resort to using reinforcement learning, specifically, the Q-learning reinforcement learning algorithm. Q-learning is a model-free reinforcement learning algorithm, in the sense that the learning agent is not given a description of the transition probability matrices $T_A$. In principle, the Q-learning algorithm provides us with an optimal policy, in the limit that all states and actions are visited by the agent sufficiently often. In practice, therefore, the policies obtained will be near optimal, providing us with bounds on the optimal quantities defined by \mathrm{e}qref{eq-waiting_time_opt_problem} and \mathrm{e}qref{eq-fidelity_opt_problem}. We refer to Appendix~\ref{sec-Q_learning} for a brief description of the Q-learning algorithm and the details of our implementation.
The Q-learning algorithm, like any other reinforcement learning algorithm, requires the learning agent to receive a reward in a manner such that maximizing the expected total reward over time corresponds to optimizing either the waiting time or the fidelity according to the optimization problems in \mathrm{e}qref{eq-waiting_time_opt_problem} and \mathrm{e}qref{eq-fidelity_opt_problem}, respectively. We now provide our definitions of these rewards.
\paragraph*{Reward for the waiting time.} When optimizing for the expected waiting time, we want the agent to receive high rewards for actions that bring it to the terminal state, namely, the state in which the two end nodes are connected. Hence, we define the reward $R$ as follows:
\begin{equation}\label{eq-opt_waiting_time_reward}
R(S,A) = \left\{
\begin{array}{l l}
100 & \quad {\scriptscriptstyle\mathsf{T}}ext{if } S {\scriptscriptstyle\mathsf{T}}ext{ is a terminal state}, \\
-1 & \quad {\scriptscriptstyle\mathsf{T}}ext{otherwise}, \\
\mathrm{e}nd{array}\right.
\mathrm{e}nd{equation}
for every state $S\in\mathbf{S}$ and action $A\in\mathbf{A}$. Note that with this definition, the agent, in order to achieve the highest expected total reward, is incentivized to take the least number of steps to reach the terminal state, and this is why it can be used to minimize the expected waiting time for an end-to-end link.
\paragraph*{Reward for the fidelity.} When maximizing the fidelity, we want to give rewards to the Q-learning agent based on minimizing the age of the end-to-end entangled state. Note that when optimizing for the waiting time, the reward in \mathrm{e}qref{eq-opt_waiting_time_reward} only depends on whether the state is a terminal state. Thus, in that case, the total reward collected is the highest if the number of intermediate states required to reach the terminal states from a fully disconnected initial state is minimized (least number of negative rewards collected). Now, we are not only interested in taking a small number of steps, but also at the end we want to reach a terminal state with a small value of the age $S_{1,n}$ of the end-to-end entangled state. The reward should therefore incentivize the agent to not only reduce the waiting time, but also reduce the age of the end-to-end link. We therefore define the reward as follows:
\begin{equation}\label{eq-reward_Q_learning_fidelity}
R(S,A) = \left\{
\begin{array}{l l}
\frac{100}{S_{1,n}} & \quad {\scriptscriptstyle\mathsf{T}}ext{if } S {\scriptscriptstyle\mathsf{T}}ext{ is a terminal state}, \\
-1 &\quad {\scriptscriptstyle\mathsf{T}}ext{otherwise}, \\
\mathrm{e}nd{array}\right.
\mathrm{e}nd{equation}
for every state $S\in\mathbf{S}$ and action $A\in\mathbf{A}$. In other words, higher positive rewards are awarded to the agent if the age of the end-to-end link is low. We also emphasize that by defining the reward in terms of the age of the entangled state, we circumvent the need to keep track of quantum states during our simulations.
\section{Improved policies for the waiting time}\label{sec:opt_pol_waiting_time}
In this section, we apply the Q-learning reinforcement learning algorithm to find policies that reduce the expected waiting time for end-to-end entanglement in a linear quantum repeater chain. We consider four-node and five-node repeater chains, and we compare the policy obtained via Q-learning to the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} (swap-as-soon-as-possible) policy~\cite{coopmans2021thesis,kamin2022exactrateswapasap}. We refer to Appendix~\ref{sec-Q_learning} for a brief description of the Q-learning algorithm and the details of our implementation of it.
\subsection{Evolution of the state and calculation of the waiting time}
As described at the beginning of Sec.~\ref{sec:MDP_model}, the state of the repeater chain evolves in discrete time steps. Initially, all elementary links in the repeater chain are inactive. In every time step, we have a combination of elementary link requests and requests for entanglement swapping. Specifically, we have the following:
\begin{enumerate}
\item Check if the action matrix $A^t$ contains any elementary link requests, i.e., if there are off-diagonal elements of $A^t$ that are equal to 1. If so, then increase the age of all existing elementary and virtual links according to the rule in \mathrm{e}qref{eq-state_update_wait}. Otherwise, leave the state unchanged.
\item Perform all requests for elementary links.
\item Perform all entanglement swapping requests.
\mathrm{e}nd{enumerate}
Since we consider local operations and classical communication (LOCC) to be free, any time step in which the action consists only of entanglement swapping requests does not increase the age of existing links and also does not count towards the waiting time. Time steps in which only elementary links are requested or both swaps and elementary links are requested contribute one time step towards the waiting time and also to the age of all links. This is a reasonable assumption for short chains, in which the classical communication time between different nodes of the chain is small compared to the time required to create the elementary links by storing the shared entangled qubits in quantum memories. For longer chains/larger networks, classical communication times become more relevant. Even though we neglect the time taken for entanglement swapping, some reasonable restrictions are put on the allowed actions at any given time step in order to mimic the finite-time requirements of entanglement generation, processing, and storage in memories. Specifically, if $A_{i,i}^t = 1$, then $A_{i-1,i}^t = 0$ and $A_{i,i+1}^t = 0$, i.e., nodes that request entanglement swapping cannot request elementary links in the same time step, and vice versa. Another restriction is that virtual links cannot cross. For example, consider a seven-link repeater chain. If nodes 1 and 5 are connected, then nodes 2 and 6 cannot be connected; similarly, nodes 3 and 7 cannot be connected, and so on. However, nodes 2, 3, and 4 can form links amongst themselves.
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{figures/swap_asap.pdf}
\caption{Fixed and dynamic versions of the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy. Consider a five-node repeater chain in which all elementary links are active at the same time and the entanglement swapping is non-deterministic. In the fixed {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy, all Bell measurements are performed at once, leading to two possibilities: either one of the swaps fails and consequently all elementary links become inactive, or all of them succeed and we obtain end-to-end entanglement. On the other hand, for the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy, there are four possibilities illustrated on the right-hand side. These possibilities are allowed within the framework of {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} when local operations and classical communication (LOCC) are considered free.}
\label{fig:swap-asap}
\mathrm{e}nd{figure}
\subsection{Fixed and dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policies} \label{subsec:swap_asap}
In this work, we consider two versions of the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy, one that we refer to as the {\scriptscriptstyle\mathsf{T}}extit{dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap}} policy (which has been considered before in Ref.~\cite{shchukin2022optimal}) and the other that we refer to as the {\scriptscriptstyle\mathsf{T}}extit{fixed {\scriptscriptstyle\mathsf{T}}extsc{swap-asap}} policy (which has been considered before in Ref.~\cite{inesta2022optimal}). See Fig.~\ref{fig:swap-asap} for an illustration of the difference between the two versions.
The dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy is defined as follows.
\begin{itemize}
\item The action $R_\mathrm{e}ll$ of requesting an elementary link between two neighboring nodes is performed whenever an elementary link is inactive.
\item An elementary link is discarded if and only if its age is equal to the maximum cutoff value $m^\star$.
\item A virtual link is discarded when its age is equal to $m^\star$.
\item Entanglement swaps are performed ``as soon as possible'', i.e., as soon as two elementary links sharing a common node are active.
\item If more than one entanglement swap can be performed sequentially, then we are free to choose the order of the swaps, in a \mathrm{e}mph{dynamic} way; see Fig.~\ref{fig:swap-asap} for an example. When LOCC is considered free, all such different orderings correspond to a single time step.
\mathrm{e}nd{itemize}
The policy described above is called the ``dynamic'' {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy because the entanglement swapping is performed sequentially when multiple entanglement swapping operations can be done at the same time. On the other hand, the \mathrm{e}mph{fixed {\scriptscriptstyle\mathsf{T}}extsc{swap-asap}} policy is a strictly local policy in which the entanglement swapping operations are done in parallel rather than sequentially. Doing the entanglement swapping operations in parallel means that they can only succeed together or fail together. For example, in Fig.~\ref{fig:swap-asap}, we see that if one of the entanglement swapping operations fails, then this causes all of the other elementary links to become inactive, even if the swapping succeeds at the other nodes. Instead, in the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy, because the swapping operations are done sequentially, it is possible to communicate the failure of a swapping operation to the adjacent nodes, such the adjacent nodes can avoid doing the entanglement swapping. In fact, a better strategy in the four-segment case is thus to first try to perform the swaps at nodes 2 and 4 and only if they succeed attempt swapping at node 3 (see Fig.~\ref{fig:swap-asap}~(right)). The extra classical communication involved in the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy does not add to the waiting time, because in our model LOCC is considered free. As a result of this freedom in choosing the order of the entanglement swapping, and because LOCC is assumed free, the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy can, in principle, outperform the fixed {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy.
Despite their differences, from the point of view of actions on elementary links (namely, waiting and requesting on an elementary link), both the fixed and dynamic versions of the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy are considered to be local, because in these policies the elementary link actions of waiting and requesting depend only on the state of the nodes performing those actions, and not on the state of the other elementary links or of the network as a whole.
\subsection{Homogeneous repeater chains}
We now analyse the performance of policies obtained via Q-learning, henceforth referred to as ``Q-learning policies''. Specifically, we compare our Q-learning policies to the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy using the improvement factor~\cite{inesta2022optimal}, which is defined as $(T_{RL} - T_{SA})/T_{RL}$, where $T_{RL}$ is the average waiting time using the Q-learning policy and $T_{SA}$ is the average waiting time using the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy.
\begin{figure*}
\centering
\includegraphics[width=0.45{\scriptscriptstyle\mathsf{T}}extwidth]{figures/waiting_time_t_star.pdf}\quad
\includegraphics[width=0.45{\scriptscriptstyle\mathsf{T}}extwidth]{figures/imp_fac_wt_t_star.pdf}
\caption{(Left) Average waiting time as a function of the maximum memory cutoff $m^{\star}$ for a repeater chain with $n=4$ nodes. The elementary link success probability is $p_{\mathrm{e}ll}=0.5$, and the entanglement swapping success probability is $p_{sw}=0.5$. (Right) Improvement factor of our Q-learning policy compared to the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy, as a function of the maximum memory cutoff $m^{\star}$. For large values of $m^{\star}$, we find that the Q-learning policy performs slightly worse compared to the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy, essentially due to the large state space of our MDP for values of $m^{\star}$.}
\label{fig:t_star_benchmark}
\mathrm{e}nd{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.45{\scriptscriptstyle\mathsf{T}}extwidth]{figures/avg_wt_deterministic.pdf}\quad
\includegraphics[width=0.45{\scriptscriptstyle\mathsf{T}}extwidth]{figures/imp_fac_deterministic.pdf}
\caption{(Left) Average waiting time as a function of the elementary link success probability $p_{\mathrm{e}ll}$ when entanglement swapping is deterministic, $p_{sw} = 1$, for a repeater chain with $n=5$ nodes. (Right) Improvement factor of our Q-learning policy compared to the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy as a function of $p_{\mathrm{e}ll}$.}
\label{fig:deterministic}
\mathrm{e}nd{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.45{\scriptscriptstyle\mathsf{T}}extwidth]{figures/wt_non_determ.pdf}\quad
\includegraphics[width=0.45{\scriptscriptstyle\mathsf{T}}extwidth]{figures/imp_fact_non_determ.pdf}
\caption{(Left) Average waiting time as a function of the elementary link success probability $p_{\mathrm{e}ll}$ when entanglement swapping is not deterministic ($p_{sw} = 0.5$), for a repeater chain with $n=5$ nodes. (Right) Improvement factor compared to the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy as a function of $p_{\mathrm{e}ll}$.}
\label{fig:non-deterministic}
\mathrm{e}nd{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.45{\scriptscriptstyle\mathsf{T}}extwidth]{figures/imp_fact_non_determ_compare.pdf}\quad
\includegraphics[width=0.45{\scriptscriptstyle\mathsf{T}}extwidth]{figures/imp_fac_wt_t_star_compare.pdf}
\caption{Improvement factor of our Q-learning policy compared to the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy (fixed or dynamic) as a function of the elementary link success probability $p_{\mathrm{e}ll}$ and the maximum cutoff $m^{\star}$ for non-deterministic entanglement swapping. Our Q-learning policies provide an advantage for all values of $p_{\mathrm{e}ll}$. (Left) The dependence of the improvement factor on $p_{\mathrm{e}ll}$ is non-monotonic when comparing with the fixed {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy. On the other hand, compared to the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy, the advantage decreases monotonically with $p_{\mathrm{e}ll}$ beyond $p_{\mathrm{e}ll}=0.4$. (Also see Fig.~\ref{fig:asymmetry_comparison} below.) (Right) The dependence of the improvement factor on $m^{\star}$ is monotonic when compared to both the fixed and dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policies.}
\label{fig:comparison}
\mathrm{e}nd{figure*}
Our main results are as follows.
\begin{enumerate}
\item For a fixed elementary link and entanglement swapping success probability, the Q-learning policy has a larger improvement factor for smaller values of the maximum memory cutoff $m^{\star}$. In Fig.~\ref{fig:t_star_benchmark}, we show this trend for $p_{\mathrm{e}ll} = 0.5$ and $p_{sw} = 0.5$. As $m^{\star}$ increases, the Q-learning policy approaches the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy, and the improvement factor falls. In fact, for large $m^{\star}$, we find that the Q-learning policy performs slightly worse compared to the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy. This worse performance could be due to the large state space of the MDP for large $m^{\star}$, which means that the training performed is not adequate to precisely learn the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy. This shows that the Q-learning approach is also limited in terms of scaling with $n$ and $m^{\star}$, like other policy optimization techniques such as linear programming~\cite{shchukin2022optimal}, policy and value iteration~\cite{inesta2022optimal} and even deep reinforcement learning~\cite{reiss2022deep}.
Also, it is interesting to note that the waiting time for the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy approaches a fixed value for large $m^{\star}$.
\item For deterministic entanglement swapping, the improvement factor monotonically falls for increasing elementary link success probability; see Fig.~\ref{fig:deterministic}. We note that, in this case, both the fixed and dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policies have the same waiting times. In particular, we see that our Q-learning policies provide greater advantage for low values of the elementary link success probability $p_{\mathrm{e}ll}$. For larger values of $p_{\mathrm{e}ll}$, the waiting times of the Q-learning policy and the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy coincide.
\item If entanglement swapping is non-deterministic, then the improvement factor decreases monotonically with increasing $p_{\mathrm{e}ll}$ when compared to the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy. On the other hand, when compared to the fixed {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy, the trend is non-monotonic with $p_{\mathrm{e}ll}$; see Fig.~\ref{fig:non-deterministic}. For larger values of $m^{\star}$, the trends remain the same, although the improvement factor reduces for all values of $p_{\mathrm{e}ll}$ when compared to a smaller value of $m^{\star}$.
\mathrm{e}nd{enumerate}
It is also worth understanding how our Q-learning policies compare to the fixed {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy. In Fig.~\ref{fig:comparison}, we plot the improvement factor of our Q-learning policy with respect to both the fixed and dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policies. When comparing to the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy, the advantage offered by the Q-learning policy decreases monotonically with increasing elementary link success probability, which is not the case if we compare with the fixed {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy. In contrast, this change in trends is not seen for the dependence of the waiting time on $m^{\star}$. In this case, when the Q-learning policy is compared with the fixed and dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policies, the trends remain the same: the improvement factor decreases monotonically with increasing $m^{\star}$.
The trends we obtain for the improvement in waiting time in the deterministic swapping case are qualitatively and quantitatively in close agreement with the results in Ref.~\cite{inesta2022optimal}, in which the comparison of their policies was done against the fixed {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy. This agreement is to be expected, because for deterministic entanglement swapping both the fixed and dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policies yield the same waiting time. However, in the non-deterministic case, our trends do not agree with Ref.~\cite{inesta2022optimal}, in which the improvement factor increases monotonically with increasing $p_\mathrm{e}ll, m^\star$, while in our case the improvement factor decreases. This difference arises because of some of the differences in our assumptions. First, unlike Ref.~\cite{inesta2022optimal}, our MDP formulation allows the memory cutoff to be dynamic, which leads to greater improvement over the policies found in Ref.~\cite{inesta2022optimal}. Furthermore, since we do not allow requests and swaps at the same nodes to take place in the same time step, our waiting times are slightly different from those found in Ref.~\cite{inesta2022optimal}. The improvement factors for a five-node repeater chain in Ref.~\cite{inesta2022optimal} vary between $\approx$ 6-13\% for $p_{sw} = 0.5$, $p_\mathrm{e}ll\in(0.3,1.0)$, and $m^\star\in(2,6)$. In our work, for $p_{sw} = 0.5$ and $m^\star = 2$, the improvement factors against the fixed {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} and dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policies vary between $\approx$ 35-50\% and $\approx$ 10-50\%, respectively. We have thus shown that by using our more general MDP formulation, a greater improvement in waiting times, even over the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy (which is itself better than the fixed {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy), can be obtained.
\subsection{Inhomogeneous repeater chains}\label{sec-waiting_time_inhomogeneous}
\begin{figure}
\centering
\includegraphics[width=0.5\columnwidth]{figures/inhomo3.pdf}\\
\includegraphics[width=0.9\columnwidth]{figures/wt_symmetry_factor_3link.pdf}
\caption{Average waiting time of our Q-learning policy for a four-node inhomogeneous repeater chain compared to the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy as a function of the inhomogeneity $\delta p$ in the elementary link success probabilities, where $\delta p = p_{\mathrm{e}ll}^{2,3} - p_{\mathrm{e}ll}^{1,2} = p_{\mathrm{e}ll}^{3,4} - p_{\mathrm{e}ll}^{2,3}$. Here, $p_{\mathrm{e}ll}^{i,j}$ is the elementary link success probability between nodes $i$ and $j$. Our Q-learning policies provide an advantage over the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy even for short inhomogeneous chains.} \label{fig:asymmetry_comparison}
\mathrm{e}nd{figure}
\begin{figure}
\centering
\includegraphics[width=0.6\columnwidth]{figures/inhomo4.pdf}\\
\includegraphics[width=0.9\columnwidth]{figures/wt_symmetry_factor_dumbell.pdf}
\caption{Average waiting time of our Q-learning policy compared to the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy as a function of the inhomogeneity $\delta p$ in the elementary link success probabilities for a five-node inhomogeneous repeater chain. The repeater chain has longer central elementary links ($p_{\mathrm{e}ll}^{2,3} = p_{\mathrm{e}ll}^{3,4} = 0.6-\delta p$) and shorter outer elementary links ($p_{\mathrm{e}ll}^{1,2} = p_{\mathrm{e}ll}^{4,5} = 0.6+\delta p$), where $p_{\mathrm{e}ll}^{i,j}$ is the elementary link success probability for nodes $i$ and $j$. Such a repeater chain is a toy model for a multi-layered quantum network in which small-scale networks between nearby nodes are connected via long links to a central node.}
\label{fig:dumbell_comparison}
\mathrm{e}nd{figure}
We now consider inhomogeneous quantum repeater chains, in which the elementary link success probabilities are different. We provide the following two sets of results in this setting.
\begin{enumerate}
\item A four-node inhomogeneous repeater chain with all elementary links having different success probabilities $p_{\mathrm{e}ll}^{i,j}$, where $p_{\mathrm{e}ll}^{i,j}$ is the elementary link success probability for pair $(i,j)$ of adjacent nodes; see Fig.~\ref{fig:asymmetry_comparison}. We fix the central elementary link (between nodes 2 and 3) to have $p_{\mathrm{e}ll}^{2,3} = 0.6$, and we vary the other elementary link success probabilities $p_{\mathrm{e}ll}^{1,2}$ and $p_{\mathrm{e}ll}^{3,4}$. We define the inhomogeneity of the chain as $\delta p = p_{\mathrm{e}ll}^{2,3} - p_{\mathrm{e}ll}^{1,2} = p_{\mathrm{e}ll}^{3,4} - p_{\mathrm{e}ll}^{2,3}$. In Fig.~\ref{fig:asymmetry_comparison}, we show that our Q-learning policies can improve the average waiting times compared to the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy. The dependence of the improvement on $\delta p$ does not have a monotonic trend, but overall, more advantage can be gained in the more asymmetric case. The improvement factor varies between 2\% and 8\% in this specific case.
\item Next, we consider a five-node inhomogeneous repeater chain that includes central links (elementary links emanating from the central node of the chain) that have low success probability, and outer elementary links that have high success probability; see Fig.~\ref{fig:dumbell_comparison}. In particular, the first and fourth elementary links have high success probability compared to the interior and central links. In particular, $p_{\mathrm{e}ll}^{1,2} = p_{\mathrm{e}ll}^{4,5} = 0.6 +\delta p$ and $p_{\mathrm{e}ll}^{2,3} = p_{\mathrm{e}ll}^{3,4} = 0.6 - \delta p$, where as before $\delta p$ quantifies the inhomogeneity of the chain. This configuration mimics a quantum network built out of short (high $p_{\mathrm{e}ll}$) local elementary links amongst client nodes and long (low $p_{\mathrm{e}ll}$) elementary links to a central hub that connects to several clusters of nodes or small local networks. We consider non-deterministic entanglement swapping, with $p_{sw} = 0.5$. In Fig.~\ref{fig:dumbell_comparison}, we show that even in this inhomogeneous setting our Q-learning policies improve upon the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy. The average waiting time improvement factor varies between 26\% and 38\% for this case. Therefore, as in the case of homogeneous repeater chains, a greater advantage can be gained for longer chains by using our Q-learning policies.
\mathrm{e}nd{enumerate}
\section{Improved policies for the fidelity}\label{sec-opt_pol_fidelity}
We now look at another figure of merit for the performance of quantum repeater chains, namely, the average age of the end-to-end entangled state, which allows us to determine the fidelity of the end-to-end entangled state with respect to the desired maximally entangled state via \mathrm{e}qref{eq-elem_link_fidelity} and \mathrm{e}qref{eq-virtual_link_fidelity}. Fidelity-based policy optimization using MDPs was previously presented in Ref.~\cite{Khatri2021policieselementary} for elementary links (see also Ref.~\cite[Sec.~II]{Kha22}).
We present the results of using the Q-learning algorithm, with the reward as defined in \mathrm{e}qref{eq-reward_Q_learning_fidelity}. The average age of the end-to-end link, $\langle S_{1,n} \rangle$, is calculated by simulating the learned policy, and the average is performed over multiple runs, each run starting from the state of no active elementary links and letting the network evolve until the end-nodes are connected, just as in the case of average waiting time. We emphasize that the policies obtained in this section, using the reward in \mathrm{e}qref{eq-reward_Q_learning_fidelity}, will in general be different from the policies obtained in Sec.~\ref{sec:opt_pol_waiting_time} based on the waiting time reward in \mathrm{e}qref{eq-opt_waiting_time_reward}, because when using \mathrm{e}qref{eq-reward_Q_learning_fidelity} as the reward the agent, while learning, considers not only the total time taken to reach the terminal state (end-to-end entangled state) but also the age/fidelity of the terminal state, once it is reached. This allows us to explore the trade-off between the waiting time and the fidelity, namely, that links that are stored longer will decrease the waiting time for an end-to-end link, at the cost of increasing the average age of the end-to-end entangled state (decreasing its fidelity); conversely, storing the links for less time will decrease the average age (increase the fidelity), but it will increase the waiting time.
Our results are shown in Fig.~\ref{fig:avg_age_improvement}, Fig.~\ref{fig:trade-off}, and Fig.~\ref{fig:trade-off_asymmetry}. Similar to the case of waiting time, we compare our Q-learning policies to the fixed and dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policies by considering the improvement factor.
\begin{figure*}
\centering
\includegraphics[width=0.9\columnwidth]{figures/avg_age_pbm75.pdf}
\includegraphics[width=0.9\columnwidth]{figures/avg_age_imp_fac.pdf}
\caption{Average age of the end-to-end entangled state as a function of the elementary link success probability $p_{\mathrm{e}ll}$ for a four-node repeater chain with maximum cutoff $m^{\star}=2$ and various entanglement swapping success probabilities $p_{sw}$. (Left) Average age of the end-to-end entangled state when performing either the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy, the policy based on the waiting time reward in \mathrm{e}qref{eq-opt_waiting_time_reward}, as presented in Sec.~\ref{sec:opt_pol_waiting_time} (``RL: wt''), and the policy based on the fidelity-based reward in \mathrm{e}qref{eq-reward_Q_learning_fidelity} (``RL: wt+fid''). Here, $p_{sw}=0.75$. (Right) Average age improvement factor, as a function of $p_{\mathrm{e}ll}$ and different choices of $p_{sw}$, for the Q-learning policy based on the reward in \mathrm{e}qref{eq-reward_Q_learning_fidelity}, when compared to the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy.}
\label{fig:avg_age_improvement}
\mathrm{e}nd{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.9\columnwidth]{figures/imp_fac_trade_off.pdf}
\includegraphics[width=0.9\columnwidth]{figures/imp_factor_tradeoff_4link.pdf}
\caption{Improvement factor for the average waiting time and the end-to-end link fidelity (average age) when our Q-learning policy is based on the fidelity-based reward defined in \mathrm{e}qref{eq-reward_Q_learning_fidelity}. (Left) Four-node homogeneous repeater chain. Interestingly, a positive improvement can be obtained simultaneously for both quantities using fidelity-based Q-learning policies. (Right) Five-node homogeneous chain. The trends are similar to the four-node case. In fact, the advantage in the low-probability regime is on average much higher for the five-node case compared to the four-node case, showing that our Q-learning policies become more advantageous for longer chains.}
\label{fig:trade-off}
\mathrm{e}nd{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.9\columnwidth]{figures/average_age_comparison_asymmetric.pdf}
\includegraphics[width=0.9\columnwidth]{figures/imp_factor_tradeoff_asymmetric.pdf}
\caption{(Left) Average age of the end-to-end link for a four-node inhomogeneous repeater chain. (Right) Improvement factors for the average waiting time and the end-to-end link fidelity (average age). Just as in the homogeneous case (Fig.~\ref{fig:trade-off}), both the fidelity and waiting time can be simultaneously improved for most parameter choices; however, with increasing non-homogeneity, as the improvement in waiting time increases the improvement in fidelity falls, indicating a trade-off between the two quantities.}
\label{fig:trade-off_asymmetry}
\mathrm{e}nd{figure*}
\begin{enumerate}
\item In homogeneous repeater chains, the average age $\langle S_{1,n} \rangle$ of the end-to-end entangled state falls with increasing elementary link success probability $p_{\mathrm{e}ll}$ and increasing swapping probability $p_{sw}$, as expected. The Q-learning policy reduces the average age of the end-to-end entangled state when compared to both the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy and when compared to Q-learning policies based on the waiting time reward in \mathrm{e}qref{eq-opt_waiting_time_reward}; see Fig.~\ref{fig:avg_age_improvement}~(left). Furthermore, the improvement factor of the average age (compared to the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy), calculated analogously to the waiting time improvement factor, decreases with increasing $p_{\mathrm{e}ll}$ for deterministic entanglement swapping, whereas it increases with increasing $p_{\mathrm{e}ll}$ for non-deterministic entanglement swapping; see Fig.~\ref{fig:avg_age_improvement}~(right). It is interesting to see that this trend is different from what we obtain when considering the waiting time reward (see Fig.~\ref{fig:deterministic} and \ref{fig:non-deterministic}). In particular, as shown in Fig.~\ref{fig:trade-off}, even though there exists a trade-off between the average waiting time and the average age of the end-to-end link, as described at the beginning of this section, both can be simultaneously improved (in certain parameter regimes) when the Q-learning agent is trained according to the fidelity-based reward in \mathrm{e}qref{eq-reward_Q_learning_fidelity}.
\item Fig.~\ref{fig:trade-off_asymmetry} shows that for inhomogeneous repeater chains, a similar improvement in fidelity as in homogeneous repeater chains can be obtained from our Q-learning policies. In this case, for some parameter regimes, the fidelity is enhanced at the expense of larger waiting time (negative improvement), unlike the homogeneous case (Fig.~\ref{fig:trade-off}) in which both the fidelity and waiting time could be simultaneously improved. While the waiting time improvement increases with increasing inhomogeneity, the fidelity improvements reduce. This result reinforces the trade-off between maximizing fidelity and minimizing waiting time.
\mathrm{e}nd{enumerate}
In Appendix~\ref{sec-opt_pol_example}, we provide an explicit example of a Q-learning policy obtained from our fidelity-based reward in \mathrm{e}qref{eq-reward_Q_learning_fidelity}, providing the action for every possible state. We compare this policy to the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy and a Q-learning policy obtained from the waiting time reward in \mathrm{e}qref{eq-opt_waiting_time_reward}.
\section{Discussion of results}\label{sec-discussion}
\begin{figure*}
\centering
\includegraphics[width=0.95{\scriptscriptstyle\mathsf{T}}extwidth]{figures/swap_asap_vs_Q_learning.pdf}
\caption{Some examples of differences in the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy and improved Q-learning policies obtained using waiting time-based rewards. We consider probabilistic elementary link generation ($p_{\mathrm{e}ll}$) and $m^\star = 2$ for these illustrations. The ages of active links are written on top of the link. (a)~Homogeneous chains. (Top)~Deterministic swaps when links 1 and 2 are active. The dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy (left) always requests swaps for active adjacent links, whereas the Q-learning policy determines the action based on the ages of links and also the status of links in other parts of the network. (Bottom) Non-deterministic swaps. Here, the Q-learning policy sometimes chooses to discard active links, even before they have reached the maximum cutoff time, in order to provide more chances for elementary link generation, if the network is still far away from the terminal state. This \mathrm{e}mph{dynamic} cutoff becomes even more pronounced for inhomogeneous chains. (b)~Inhomogeneous chains. (Top) The Q-learning policy chooses to discard the link between nodes 1 and 2 before it reaches the maximum cutoff $m^{\star}$, because it has a success probability of $p_{\mathrm{e}ll}^{1,2}=0.9$ and is therefore easy to produce. (Bottom) Similarly, the Q-learning policy can sometimes delay swapping depending on the relative ages of the involved links in order to preserve elementary links that are more difficult to produce.}
\label{fig:swap-asap_vs_optimal}
\mathrm{e}nd{figure*}
In this work, we have considered entanglement distribution in a linear chain of quantum repeaters in the setting of non-deterministic elementary link creation and entanglement swapping. We have done so by modeling the repeater chain using an MDP in Sec.~\ref{sec-MDP}. Then, using the Q-learning algorithm, we have obtained policies that improve upon the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy, both in terms of the average waiting time and the average age of the end-to-end link, in certain parameter regimes. In addition, by using the Q-learning algorithm, we can explicitly identify the best action for every state, and we provide an example of this in Appendix~\ref{sec-opt_pol_example}. In this section, we provide some details on the high-level insights about our results. In Sec.~\ref{sec-swap_asap_vs_Q_learning}, we identify some of the key differences between our Q-learning policies and the fixed and dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policies, which lead to the improvements in waiting time and fidelity shown in Sec.~\ref{sec:opt_pol_waiting_time} and Sec.~\ref{sec-opt_pol_fidelity}. Then, in Sec.~\ref{sec-compare_prior_work}, we outline the differences between our work and prior related work on reinforcement learning for entanglement distribution in quantum repeater chains.
\subsection{Differences between the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} and Q-learning policies}\label{sec-swap_asap_vs_Q_learning}
We outline below of the some key differences that we observed between our Q-learning policies and the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policies. These differences are also illustrated in Fig.~\ref{fig:swap-asap_vs_optimal}.
\begin{enumerate}
\item For homogeneous repeater chains with deterministic entanglement swapping, it is clear that most of the advantage comes from adopting a ``global'' policy, in which all of the nodes have knowledge of each other's states. Specifically, this means that entanglement swapping is not always performed right away, but instead some links will wait until a large number of adjacent nodes are all active. (We quantitatively explore this concept of global knowledge in Sec.~\ref{sec:advantage} below.) For an illustration, see Fig.~\ref{fig:swap-asap_vs_optimal}(a)(top), where two different age configurations are shown for a five-node chain with maximum cutoff $m^{\star}=2$. If both links are fresh, a swap is attempted, but if one of the links is at the maximum cutoff, a swap is not attempted; instead, a new link is requested, because the agent knows that links 3 and 4 are still inactive and will need at least one more time step to become active, by which time the swapped link would have reached its maximum cutoff time of $2$, according to the rule $m'=\max\{m_1,m_2\}$ (see Sec.~\ref{sec:MDP_model}). Thus, the second link is saved for one more time step by avoiding an unnecessary swap attempt. More generally, waiting can be advantageous when both the elementary link success probability $p_{\mathrm{e}ll}$ and the maximum cutoff $m^{\star}$ are low. Indeed, when $p_{\mathrm{e}ll}$ is low, discarding a link becomes ``risky'', because the new elementary link attempt is not likely to succeed. Similarly, when $m^{\star}$ is low, then due to our update rule for the age of a virtual link (see Sec.~\ref{sec:MDP_model}), intermediate swapped states will not survive for too long if $m^{\star}$ is low. This also tells us why the advantage of our Q-learning policy is reduced for increasing elementary link success probability and increasing memory cutoff (see Fig.~\ref{fig:deterministic}).
\item For homogeneous repeater chains with non-deterministic entanglement swapping, as in Fig.~\ref{fig:swap-asap_vs_optimal}(a)(bottom), the Q-learning policy often chooses to discard links even before they have reached the maximum cutoff time, in contrast to the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy. For example, in Fig.~\ref{fig:swap-asap_vs_optimal}(a)(bottom), segments (2, 3) and (3, 4) are active and (1, 2) and (4, 5) are inactive. The {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy would attempt a swap at node 3 and request links at (1, 2) and (4, 5). A doubling policy (see Ref.~\cite{shchukin2019waiting} for details) would not perform the swap and just request at (1, 2) and (4, 5). On the other hand, the Q-learning policy chooses several different strategies depending on the absolute and relative ages of the two links and also the values $p_{\mathrm{e}ll}$, $p_{sw}$, $m^{\star}$, two of which are shown in the figure.
\item For inhomogeneous repeater chains, the key differences in policy occur with respect to the swapping of links with different elementary link success probabilities. This also explains why the improvement factor falls with increasing symmetry of the chain (see Fig.~\ref{fig:trade-off_asymmetry}). For example, in Fig.~\ref{fig:swap-asap_vs_optimal}(b)(top), because the elementary link success probability between the outer nodes 1 and 2 is high, whenever a request for a link between central nodes 2 and 3 is made, the adjacent outer link is also requested even when it is already active. In contrast, in the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy (either fixed or dynamic), unless the link is at the cutoff time, it would not be requested. Our Q-learning policy adopts this strategy because it helps keep the outer link fresh, so that whenever the central link is created and a successful swap occurs between it and the adjacent outer link, the age of the virtual link remains low. Otherwise, if the created virtual link has a high age, because the easy-to-produce outer link was old, the precious central link's lifetime would be effectively shortened. A similar choice of holding off swaps if the effective age of central links would increase due to swapping with an old but easy-to-produce outer link can be seen in Fig.~\ref{fig:swap-asap_vs_optimal}(b)(bottom).
\mathrm{e}nd{enumerate}
\subsection{Comparison to prior work}\label{sec-compare_prior_work}
The work presented here is most similar to Refs.~\cite{inesta2022optimal,reiss2022deep}, in which linear, homogeneous repeater chains are studied. One key difference between those works and ours is that our MDP is different from those in Refs.~\cite{inesta2022optimal,reiss2022deep}. In particular, in Ref.~\cite{inesta2022optimal}, the elementary link policy was fixed to the memory-cutoff policy, and only the entanglement swapping policy was optimized. Then, using policy/value iteration, the authors of Ref.~\cite{inesta2022optimal} obtain policies that improve the average waiting time. On the other hand, although optimization of the elementary link policy is considered in Ref.~\cite{reiss2022deep}, the authors of that work fix the swapping policy to {\scriptscriptstyle\mathsf{T}}extsc{swap-asap}, and they also assume that the entanglement swapping is perfect and deterministic. In this work, we optimize both the policy of the elementary links as well as the policy for the entanglement swapping, and we take as our figure of merit both the average waiting time as well as the fidelity of the end-to-end entangled state. One advantage of our more general MDP is that cutoffs less than the maximum cutoff $m^{\star}$ can be chosen. This then allows implicitly for an optimization of the cutoffs simultaneously with an optimization of the entanglement swapping strategy, in contrast to prior works~\cite{li2020efficient,reiss2022deep} that perform an optimization of memory cutoffs while keeping the entanglement swapping strategy fixed.
Furthermore, in reality, quantum networks will in all likelihood resemble inhomogeneous repeater chains, with elementary links of different lengths, or using different physical platforms/channels, and hence the elementary link would have different success probabilities; see, e.g., Refs.~\cite{khatri2021spooky,wallnofer2022sats}. Thus, in this work we also consider examples of four- and five-node inhomogeneous repeater chains, and we find improved policies for such networks. Finally, the use of the Q-learning algorithm allows us to explicitly extract the learned policy, i.e., a list containing the best action for every possible state (see Appendix~\ref{sec-opt_pol_example} for an example), while for deep reinforcement learning techniques, as used in Ref.~\cite{reiss2022deep}, extracting the policy is generally difficult.
\section{How do improved policies derive their advantage?}\label{sec:advantage}
In this section, we quantify the various sources of advantage of our Q-learning policies, particularly the advantages that we outlined in the previous section. Broadly, there are three key differences in our improved policies when compared to a fixed-cutoff local {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy, which are: different cutoffs for the elementary links, global knowledge and coordination between non-adjacent nodes, and decision making with foresight.
\subsection{Non-uniform and state-dependent cutoffs}
By analyzing the policies obtained using the Q-learning algorithm, we find that these improved policies have a dynamic and global-state-dependent cutoff for the different elementary links of the network. These cutoffs are, in general, less than the maximum cutoff $m^{\star}$, which we recall is allowed within our MDP framework; see Sec.~\ref{sec-MDP}.
In order to see that the cutoffs are not only state dependent but that different links can have different cutoffs, we calculate an average cutoff $\langle t^\star \rangle_{i,j}$ for every node pair $(i,j)$. We do this by averaging the age of a link over instances when it is discarded, i.e., when a link request $R_\mathrm{e}ll$ is performed on that link whenever it is active. Results for four-node and five-node repeater chains are shown in Table~\ref{tab:dynamic_cut-off}. We find that, even for homogeneous repeater chains, interior links (closer to the center of the chain) have a higher average cutoff than outer links (closer to the ends of the chain). It appears that the Q-learning agent views the interior links as more ``precious'' than the outer links. Intuitively, this could be because interior links have the ability to become joined with two links, one on each side, which is not the case for outer links. For inhomogeneous repeater chains, the effect of the difference in the elementary link success probabilities on the choice of the cutoffs can also be seen by looking at the quantities $\langle t^\star \rangle_{i,j}$. For a fixed $m^\star$, the $t^\star$ choice made by the Q-learning agent is on average smaller for elementary links with higher probability, and vice versa.
\begin{table}
\renewcommand{1.2}{1.2}
\begin{center}
\begin{tabular}{|p{1.5cm}|p{1.5cm}|p{1.5cm}|p{1.5cm}|}
\hline
\multicolumn{4}{|c|}{$n=4, m^\star=2, p_{sw}=0.5$} \\
\hline\hline
& link 1 & link 2 & link 3\\
\hline
$p_{\mathrm{e}ll}$ & 0.6 & 0.6 & 0.6 \\
$\langle t^\star \rangle_{ij}$ & 1.5 & 2.0 & 1.6\\
\hline
$p_{\mathrm{e}ll}$ & 0.3 & 0.6 & 0.9\\
$\langle t^\star \rangle_{i,j}$ & 2.0 & 1.0 & 0.0 \\
\hline
\mathrm{e}nd{tabular}\\[0.4cm]
\begin{tabular}{|p{1.5cm}|p{1.5cm}|p{1.5cm}|p{1.5cm}|p{1.5cm}|}
\hline
\multicolumn{5}{|c|}{$n=5, m^\star=2, p_{sw}=0.5$} \\
\hline\hline
& link 1 & link 2 & link 3 & link 4\\
\hline
$p_{\mathrm{e}ll}$ & 0.6 & 0.6 & 0.6 & 0.6\\
$\langle t^\star \rangle_{i,j}$ & 0.3 & 1.1 & 1.0 & 0.3\\
\hline
$p_{\mathrm{e}ll}$ & 0.9 & 0.3 & 0.3 & 0.9\\
$\langle t^\star \rangle_{i,j}$ & 0.3 & 2.0 & 2.0 & 0.1\\
\hline
\mathrm{e}nd{tabular}
\caption{Average values $\langle t^\star \rangle_{i,j}$ of the cutoffs for the links between nodes pairs $(i,j)$ in homogeneous and inhomogeneous repeater chains. In general, interior links are retained for longer compared to outer links, and low probability links are retained for longer than high probability links. In contrast, for a fixed memory cutoff policy, the maximum possible cutoff $m^\star$ is chosen for all links irrespective of its position in the chain or its elementary link success probability.}
\label{tab:dynamic_cut-off}
\mathrm{e}nd{center}
\mathrm{e}nd{table}
\begin{table*}
\renewcommand{1.2}{1.2}
\begin{center}
\begin{tabular}{|p{4cm}|p{2cm}|p{2cm}|p{2cm}|p{2cm}|}
\hline
\multicolumn{5}{|c|}{$n=5, m^\star=2, p_{sw}=0.5, p_{\mathrm{e}ll}=0.6$} \\
\hline\hline
Correlator & $\vert i-k \vert = 0$ & $\vert i-k \vert = 1$ & $\vert i-k \vert = 2$ & $\vert i-k \vert = 3$ \\
\hline
$r[A_{i,i}^t,A_{k,k}^t]$ (SA) & N/A & $0.48 \pm 0.02$ & $0.05 \pm 0.02$ & N/A \\
$r[A_{i,i}^t,A_{k,k}^t]$ (RL) & N/A & $0.39 \pm 0.01$ & $0.23 \pm 0.01$ & N/A \\
\hline
$r[A_{i,i+1}^t,A_{k,k+1}^t]$ (SA) & N/A & $0.37 \pm 0.02$ & $0.13 \pm 0.01$ & $0.02 \pm 0.01$\\
$r[A_{i,i+1}^t,A_{k,k+1}^t]$ (RL) & N/A & $0.42 \pm 0.01$ & $0.42 \pm 0.01$ & $0.40 \pm 0.01$\\
\hline
$r[A_{i,i}^t,X_{k,k+1}^t]$ (SA) & $0.70 \pm 0.02$ & $0.09 \pm 0.01$ & $0.02 \pm 0.01$ & N/A \\
$r[A_{i,i}^t,X_{k,k+1}^t]$ (RL) & $0.57 \pm 0.01$ & $0.15 \pm 0.01$ & $0.12 \pm 0.01$ & N/A \\
\hline
$r[A_{i,i+1}^t,X_{k,k+1}^t]$ (SA) & $0.81 \pm 0.01$ & $0.25 \pm 0.01$ & $0.10 \pm 0.01$ & $0.01 \pm 0.01$ \\
$r[A_{i,i+1}^t,X_{k,k+1}^t]$ (RL) & $0.61 \pm 0.01$ & $0.17 \pm 0.01$ & $0.09 \pm 0.01$ & $0.06 \pm 0.01$ \\
\hline
\mathrm{e}nd{tabular}
\caption{Equal-time correlation coefficients of states and actions (see \mathrm{e}qref{eq:pearson_coeff}). For this table, we chose $t = 5$. The values range between 0 and 1 (0 represents no correlation and 1 represents maximum correlation or anti-correlation). For the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} (SA) policy the correlation between actions fall off very quickly with increasing distance $\vert i-k \vert$, whereas the correlations for our Q-learning policy (RL) survive even between the two ends of the chain. In the case of {\scriptscriptstyle\mathsf{T}}extsc{swap-asap}, all the correlation is concentrated between the states and actions at the same site, i.e., only local knowledge of the state is being used. For our Q-learning policies, the correlations are well distributed. In particular, the actions at a node/link are not only influenced by the state at that link but by the state of other links. This indicates global knowledge of the state.}
\label{tab:equal_time_corr}
\mathrm{e}nd{center}
\mathrm{e}nd{table*}
\begin{figure*}
\centering
\includegraphics[height=5cm]{figures/correlators_0.pdf}
\includegraphics[height=5cm]{figures/correlators_2.pdf}
\caption{(Left) Unequal-time correlation coefficients for states and actions at the same link. For these plots we chose $t = 5$. We see that for the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy, actions at a time $t$ are influenced only by the states at the next time step, because the correlator falls off exponentially with increasing ${\scriptscriptstyle\mathsf{T}}au$. This indicates that the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy maximizes only short-term rewards. On the other hand, our Q-learning policy maximizes both short- and long-term rewards through discounting. The unequal-time correlators fall with increasing ${\scriptscriptstyle\mathsf{T}}au$, but are sustained for much longer time scales. (Right) Unequal-time correlation functions at different links. For the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy, these correlations are close to $0$ to begin with (equal-time), and stay so with increasing ${\scriptscriptstyle\mathsf{T}}au$. In stark contrast, for our Q-learning policies, the non-zero equal-time correlations between states and actions at faraway sites are sustained, even for large ${\scriptscriptstyle\mathsf{T}}au$.}
\label{fig:unequal_correlators}
\mathrm{e}nd{figure*}
\subsection{Global knowledge and coordination between nodes}
Next, we come to global knowledge and the coordination between non-adjacent nodes. It was noted in Sec.~\ref{subsec:swap_asap} that {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policies use local knowledge of the network to decide on what actions to take. Therefore, only adjacent nodes can coordinate on elementary link requests, waiting, and entanglement swapping requests. On the other hand, in the Q-learning framework, the agent is all-knowing: it can decide on actions based on complete global knowledge of the state of the network. This is one of the main reasons for the improvement over {\scriptscriptstyle\mathsf{T}}extsc{swap-asap}, as we demonstrate quantitatively in this section. It is worth mentioning that this source of improvement is already present even within the framework of {\scriptscriptstyle\mathsf{T}}extsc{swap-asap}: the fixed version, which is completely local and does not allow for any collaboration, even among adjacent nodes, performs worse than the dynamic version, which allows for some collaboration/flexibility in terms of the decision to do Bell measurements and the order in which swaps are performed.
The extent of global knowledge used in our Q-learning policies can be quantified using correlation functions between actions and states at different links and nodes of the network. Specifically, we use the absolute value of the Pearson correlation coefficient. For two random variables $X$ and $Y$, it is defined as follows:
\begin{equation}
r[X,Y] = \frac{\vert \langle XY \rangle - \langle X \rangle\langle Y \rangle \vert}{\sigma_X \sigma_Y},
\label{eq:pearson_coeff}
\mathrm{e}nd{equation}
where, $\vert \cdot \vert$ represents the absolute value, $\langle \cdot \rangle$ stands for the mean value, and $\sigma_X$, $\sigma_Y$ are the standard deviations of $X$ and $Y$, respectively. This quantity varies between 0 and 1, with 0 indicating no correlation between the random variables and 1 representing maximum correlation. We look at the following cases:
\begin{enumerate}
\item If the actions $A_{i,j}^t$ and $A_{k,l}^t$ at some fixed time $t$ at two faraway links/nodes are coordinated, then the covariance between them should be strictly greater than zero, i.e.,
\begin{equation}
r[A_{i,j}^t,A_{k,l}^t] > 0 \quad {\scriptscriptstyle\mathsf{T}}ext{(collaboration).}
\mathrm{e}nd{equation}
From the first two rows of Table~\ref{tab:equal_time_corr}, it is clear that for the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy the correlation between action pairs $A_{i,i}^t,\, A_{k,k}^t$ and $A_{i,i+1}^t,\, A_{k,k+1}^t$ falls off very quickly with increasing distance $\vert i-k \vert$, whereas the correlations for our Q-learning policy survive even between the two ends of the chain.
\item Similarly, if the action $A_{i,j}^t$ at fixed time $t$ at a link or node is correlated with the status $X_{i,j}^t$ of a faraway link, i.e.,
\begin{equation}
r[A_{i,j}^t,X_{k,l}^t] > 0\quad{\scriptscriptstyle\mathsf{T}}ext{(global knowledge)},
\mathrm{e}nd{equation}
then this would indicate the use of global knowledge of the state by the policy. Here, the status $X_{i,j}^t$ is defined as $X_{i,j}^t=1$ if $S_{i,j}^t\in\{0,1,\dotsc,m^{\star}\}$, and $X_{i,j}^t=0$ if $S_{i,j}^t=-1$. From the final two rows of Table~\ref{tab:equal_time_corr}, we see that in the case of {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} all the correlation is concentrated between the states and actions at the same site, i.e., only local knowledge of the state is being used, whereas, for our Q-learning policy the correlations are well distributed. The actions at a node/link are not only influenced by the state at that like but by state of other links. This indicates global knowledge of the state.
\mathrm{e}nd{enumerate}
\subsection{Decision making with foresight}
Our Q-learning policies also take future rewards into account, unlike local policies such as {\scriptscriptstyle\mathsf{T}}extsc{swap-asap}, since they sometimes decide to take actions that might lead to a worse state in the very next time step, but over a longer time scale, and on average, lead to more frequent and larger network connectivity. This maximization of future rewards on larger time scales, discounting immediate rewards, is quantified in the Bellman equations (see Appendix~\ref{sec-Q_learning}) as the discount factor $\gamma$.
The discounting for future rewards can be examined by looking at unequal-time correlators at a given site or between different sites. In particular, if
\begin{equation}\label{eq-correlator_3}
r[A_{i,j}^t,X_{k,l}^{t+{\scriptscriptstyle\mathsf{T}}au}] > 0
\mathrm{e}nd{equation}
for large ${\scriptscriptstyle\mathsf{T}}au$, then this indicates that actions at time $t$ are correlated with states at time $t+{\scriptscriptstyle\mathsf{T}}au$. Similar conclusions can be drawn if
\begin{equation}
r[A_{i,j}^t,A_{k,l}^{t+{\scriptscriptstyle\mathsf{T}}au}] > 0.
\mathrm{e}nd{equation}
In Fig.~\ref{fig:unequal_correlators}, we see that for the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy, actions at a time $t$ are influenced only by the state outcomes at the next time step, and the correlator falls off exponentially with increasing ${\scriptscriptstyle\mathsf{T}}au$. This indicates that it maximizes only short-term rewards. On the other hand, our Q-learning policy maximizes both short- and long-term rewards through discounting. The unequal-time correlators fall with increasing ${\scriptscriptstyle\mathsf{T}}au$ but are sustained for much longer time scales. The same is true for correlations between different sites. For {\scriptscriptstyle\mathsf{T}}extsc{swap-asap}, these correlations are close to zero to begin with (equal-time), and stay so with increasing ${\scriptscriptstyle\mathsf{T}}au$. In stark contrast, for our Q-learning policies, the non-zero equal-time correlations between states and actions at faraway sites are sustained, even for large ${\scriptscriptstyle\mathsf{T}}au$.
As a side remark, we note that the correlation measure in \mathrm{e}qref{eq-correlator_3} is similar in spirit to the ``empowerment'' measure presented in Ref.~\cite{klyubin05empowerment}, which considers, instead of the covariance, the mutual information between the random variables $A_{i,j}^t$ and $S_{k,l}^{t+{\scriptscriptstyle\mathsf{T}}au}$ in order to assess a learning agent's ability to influence its environment.
\section{Improved policies for large repeater chains via nesting} \label{sec:nested}
\begin{figure*}
\centering
\includegraphics[width=0.7{\scriptscriptstyle\mathsf{T}}extwidth]{figures/nesting.pdf}
\caption{An example of the policy nesting strategy presented in Sec.~\ref{sec:nested}. In this example, the policy for distributing entanglement to the end nodes of a quantum repeater chain with 13 nodes consists of performing a fixed policy for the three five-node segments indicated by the dashed boxes in the ``first'' nesting level. Then, in the ``second'' nesting level, a policy for four nodes is executed.}
\label{fig:nesting}
\mathrm{e}nd{figure*}
So far, we have considered relatively small repeater chains, up to five nodes, with small values of the maximum cutoff $m^{\star}$. Remarkably, this has been enough for us to extract general features of policies that outperform the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy in certain parameter regimes. Now, we would of course like to obtain policies for larger repeater chains. Doing so, however, becomes quickly intractable, even using reinforcement learning, because the number of states and actions in our MDP grows exponentially with the number of nodes. In this section, we present a method for using policies for small repeater chains in order to construct policies for large repeater chains. This allows us to extend our Q-learning policies to larger repeater chains without having to train over the exponentially large state-action space of such chains. To illustrate the method, we present a specific example with two nesting levels. This makes it clear how to generalize the method to an arbitrary number of nesting levels.
\begin{figure*}
\centering
\includegraphics[width=0.45{\scriptscriptstyle\mathsf{T}}extwidth]{figures/concat_3x3.pdf}\quad
\includegraphics[width=0.45{\scriptscriptstyle\mathsf{T}}extwidth]{figures/concat_3x4.pdf}
\caption{Examples of our nesting strategy in Sec.~\ref{sec:nested}, which uses our Q-learning policies in combination with the local {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy in order to improve waiting times for large quantum repeater chains consisting of 10 nodes (left) and 13 nodes (right). In particular, nested policies such as the ``{\scriptscriptstyle\mathsf{T}}extsc{swap-asap} -- Q-learning'' policy (see the description in the main text) are ``quasi-local'', in the sense that only nearby nodes need to coordinate and share knowledge of their states.
For comparison, we also show the waiting times obtained by nesting Q-learning policies (``Q-learning -- Q-learning nesting'') and nesting the fixed {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy with itself (``{\scriptscriptstyle\mathsf{T}}extsc{swap-asap} -- {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} nesting'').}
\label{fig:concat}
\mathrm{e}nd{figure*}
Consider the repeater chain shown in Fig.~\ref{fig:nesting}, containing a large number of nodes, and let us split it up into $k$ smaller chains, each with $n$ nodes. (In Fig.~\ref{fig:nesting}, $k=3$ and $n=5$.) Therefore, the full repeater chain has $k(n-1) + 1$ nodes. Now, our ``nesting'' policy refers to the use of a given policy for the smaller chains with $n$ nodes---this is the ``first'' nesting level---and then for the ``second'' nesting level using a policy for $k+1$ nodes. More specifically, the following rules are followed for the two nesting levels.
\begin{enumerate}
\item An elementary link request $R_\mathrm{e}ll$ at the second nesting level is considered as a request to create an end-to-end link for the smaller $n$-node chain in the first nesting level. For example, referring to Fig.~\ref{fig:nesting}, we regard the chain at the second nesting level to be a four-node chain among the nodes 1, 5, 9, and 13, such that a request for an ``elementary link'' between nodes 1 and 5 is interpreted as a request for the end-to-end link in the five-node chain, with end nodes 1 and 5, at the first nesting level.
\item All non-terminal states of the first nesting level are considered inactive states for the second nesting level, and once the terminal state in the first/lower level is achieved, its age becomes the age of the elementary link of the second nesting level.
\item Once a request is performed for elementary link generation at the second level, another request is not allowed for that link until it becomes active. The waiting time keeps on increasing in such a case, and any other active links at the second level keep getting older. Lower level links keep evolving as per the policy at that level.
\item Also, like before, an active link at both levels can be discarded anytime as determined by the policy.
\item A new action is performed at the second level only when the state of at least one link at this level changes.
\mathrm{e}nd{enumerate}
For the budgeting of classical communication time, it is intuitive that any policy for the second nesting level should ideally be local, because nodes at the second nesting level will, in general, be far away from each other, and because global policies require long-distance classical communication, which increases the waiting time. We therefore consider three kinds of nesting of policies for two nesting levels, and we provide examples of the waiting times for these policies for large chains of 10 and 13 nodes in Fig.~\ref{fig:concat}.
\begin{enumerate}
\item {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} -- {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} nesting: This is a ``local-local'' nesting of policies, i.e., the fixed {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy is used at both levels. This is similar to the ``doubling'' policy studied in previous literature and is known to be better (in terms of the average waiting time) than a fully local {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy (without any nesting) for large chains with low elementary link generation success probability and low entanglement swapping success probability~\cite{shchukin2022optimal}.
\item {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} -- Q-learning nesting: This we refer to as a ``quasi-local'' or ``local-global'' policy, in which the fixed {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy is used at the highest (second) nesting level and a Q-learning policy is used at the lowest (first) nesting level. This type of nesting makes sense because, for many repeater chains and network scenarios more generally, it is possible that small sections of the chain (at lower nesting levels) can collaborate amongst themselves and thus use a global policy, without heavily impacting waiting times due to slow classical communication. At the higher (first) level, however, because the corresponding nodes are generally far away, a local policy such as {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} should be utilized in order to keep keep waiting times low. Overall, this type of nesting can reduce waiting times compared to using {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} for the full repeater chain. Intuitively, this can also be thought of as increasing the effective link probability of the elementary link of the next nesting level, in turn, reducing the total waiting time. In Fig.~\ref{fig:concat}, we can see that this policy offers a considerable advantage compared to the fully local {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} -- {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} nested policy.
\item Q-learning -- Q-learning nesting: This refers to the use of a Q-learning policy at both nesting levels. This is not necessarily a practical policy from the point of view of classical communication time, but it serves as a benchmark to see how advantageous the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} -- {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} and the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} -- Q-learning nested policies can be. Indeed, in Fig.~\ref{fig:concat}, we see that, as expected, this policy outperforms both the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} -- {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} and the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} -- Q-learning nested policies.
\mathrm{e}nd{enumerate}
We remark that ideas similar in spirit to our nesting idea can be found in Refs.~\cite{jiang2007optimal,wallnofer2020MLQComm}. Furthermore, from the perspective of Markov decision processes, our nesting idea can be thought of as an example of a multi-agent strategy (as opposed to the single-agent strategy that we have considered throughout this work so far), in which each agent is in charge of a portion of the full chain at the lowest nesting level and acts independently of the other agents. These agents coordinate via another agent at the highest level, which executes a policy that joins the individual portions of the full chain in order to obtain the full end-to-end entanglement.
While we have illustrated our nesting method only for a specific example with two nesting levels, the method can be straightforwardly generalized to an arbitrary number of nesting levels. Also, unlike the example shown here, the repeater chain segments at each nesting level do not have to have the same size. Examining various nesting scenarios for a given repeater chain, in order to determine optimal nesting strategies, is an interesting direction for future work.
\section{Summary and outlook}
In this work, we have presented policies for entanglement distribution for linear quantum repeater chains, both homogeneous (with equal elementary link probabilities throughout) and inhomogeneous (in which the elementary link probabilities can be different). We have shown that these policies improve upon the fixed and dynamic versions of the ``swap-as-soon-as-possible'' ({\scriptscriptstyle\mathsf{T}}extsc{swap-asap}) policy, both with respect to the waiting time and the fidelity of the end-to-end entanglement; see Sec.~\ref{sec:opt_pol_waiting_time} and Sec.~\ref{sec-opt_pol_fidelity}. While it might not be surprising that we would obtain improvements, understanding {\scriptscriptstyle\mathsf{T}}extit{how} this improvement comes about is certainly of interest, and it is this question that we have aimed to address in this work. Specifically, we have identified some of the key differences between our policies and the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policies that lead to these improvements; see Sec.~\ref{sec-discussion}. We have also quantified three of these key features, namely: state-dependent cutoff times for the elementary and virtual links, collaboration between the nodes, and knowledge of how current actions will affect the state of the repeater chain far in to the future; see Sec.~\ref{sec:advantage}. Although we have considered relatively small chains of up to five nodes, and limited values of the maximum cutoff time $m^{\star}$, this was already enough for us to extract general principles about how the improvement in waiting time and fidelity comes about. We then applied these principles in Sec.~\ref{sec:nested} in order to obtain waiting time improvements for larger repeater chains with 10 and 13 nodes.
We used the Q-learning reinforcement learning algorithm to obtain our improved policies. Q-learning is a model-free reinforcement learning algorithm, unlike model-dependent methods such as value and policy iteration. Drawing an analogy to a game of chess, the agent in Q-learning not only learns the best strategy to win, but does so while learning the rules of the game on the go. This method is thus well suited to situations in which the exact transition probabilities between the different states of a quantum network are either unknown or very difficult to establish, for example in the case of large networks. Furthermore, unlike deep reinforcement learning techniques, the Q-learning algorithm automatically provides us explicitly with the learned policy.
One noteworthy aspect of our results is that the improvement due to our policies is the largest in the most non-ideal cases, that is, when the maximum memory cutoff is small, the elementary link success probabilities are low, and the asymmetries in the repeater chain are the highest. This shows that collaboration between the nodes, one of the central features of our policies, is crucial when dealing with realistic, small-to-medium-scale networks with noisy, imperfect quantum devices.
Our work opens up several interesting avenues for future work. One direction for future work is to add the possibility of doing entanglement distillation, which is a form of quantum error correction, to the MDP formalism presented here. As entanglement distillation is now within the reach of experiments~\cite{PSBZ01,Kalb+17,RWC+21}, this question is especially pertinent, and much remains to be explored about what policies are optimal in this scenario, building on prior works that have considered specific policies~\cite{duer1999repeaterspurification,RPL09,VLMN09,AME11,bratzik2013repeatersdistill,PWD18,PD19,MDV19}; see also Refs.~\cite{CKD+20,wallnofer2022ReQuSim}, which numerically investigate several of these scenarios. We therefore expect our insights on the features of improved policies to be a valuable starting point when designing better protocols for quantum repeater chains with multiple memories and entanglement distillation. In particular, it would be interesting to compare a policy that incorporates entanglement distillation to one without entanglement distillation that simply does the memory-cutoff policy with a single memory~\cite{wallnofer2022ReQuSim}. It is not obvious that entanglement distillation will always give a better overall fidelity, and understanding in what parameter regimes this is the case will help guide us towards a concept of a noise and loss threshold for quantum communication, similar to the fault-tolerance thresholds for quantum computation that are defined by the point at which the logical error rate of an error-corrected quantum computation is less than the raw, physical error rate.
The analysis in this work considers only the number of nodes in the chain as the input to the problem, and not the physical distances between the nodes. Consequently, the policies we obtain do not explicitly take the actual classical communication time (in physical units of time) into account. While our work still provides an important first step towards understanding what types of policies improve waiting times and fidelities, ultimately, in order for these policies to be useful in practice, we have to take classical communication explicitly into account. One way to do this, albeit indirectly, would be to incorporate the quantifiers considered in Sec.~\ref{sec:advantage} into the reward, such that the policies obtained have locality constraints based on the fact that classical communication times between distant nodes can reduce the end-to-end entanglement distribution rate. It would then be interesting to see whether any global-knowledge policy can outperform the local {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy.
Finally, the ideas in Sec.~\ref{sec:nested} provide an example of using a multi-agent strategy for discovering improved policies in quantum networks. It would be interesting to explore how this nesting strategy, and multi-agent strategies more generally, could be extended to networks with arbitrary topologies, not just linear topologies as we have considered here. Doing so would help guide efforts to develop good policies for arbitrary networks, with the ultimate goal of providing policies for large-scale entanglement distribution, towards a global-scale quantum internet.
\begin{acknowledgments}
The authors thank Paras Regmi and Roy Pace for helpful discussions. This work was supported by the Army Research Office Multidisciplinary University Research Initiative (ARO MURI) through the grant number W911NF2120214. PB and HL also acknowledge the support of the U.S. Air Force Office of Scientific Research as well as the US-Israel Binational Science Foundation. In addition, SK is supported by the German BMBF (Hybrid).
\mathrm{e}nd{acknowledgments}
\onecolumngrid
\appendix
\section{The Q-learning algorithm}\label{sec-Q_learning}
In this work, we used the Q-learning algorithm~\cite{SuttonBarto2018book} to train the learning agent. Q-learning employs the following update rule:
\begin{equation}
Q'(S^t,A^t) = Q(S^t,A^t) +\alpha[R^{t+1}+\gamma \max_a Q(S^{t+1},a)-Q(S^t, A^t)],
\label{eqn:bellman}
\mathrm{e}nd{equation}
where $Q$ is the expected return of the state-action pair ($S^t$ and $A^t$ at time $t$, respectively). The Q-learning algorithm is known to converge to an optimal policy in the limit that all state-action pairs are visited infinitely often~\cite{watkins89_thesis,tsitsiklis94Qlearning,jaakkola94Qlearning}. Convergence rates and sampling complexities for Q-learning have been studied in Refs.~\cite{kearns98Qlearningconvergence,dar03Qlearningconvergence}.
The Q-learning algorithm initializes a Q-matrix of dimension $|\mathbf{S}| {\scriptscriptstyle\mathsf{T}}imes |\mathbf{A}$|. The initial state is chosen randomly from all possible non-terminal states. Then, the initial action is chosen using an $\mathrm{e}psilon$-greedy algorithm, in which the reward is maximized with probability $\mathrm{e}psilon$, otherwise a random action is taken. The new Q-value for the state-action pair is updated in the Q-matrix according to a pre-established reward. The algorithm repeats this cycle until a terminal state is reached; see Fig.~\ref{fig:q_learn}. This sequence starting from an initial state and reaching the terminal state is defined as an episode. After the algorithm has run over an adequate number of episodes, the column corresponding to the maximum value in the $i^{{\scriptscriptstyle\mathsf{T}}ext{th}}$ row of the Q-matrix represents the best possible action in the state ``$i$''. Hence, the Q-matrix explicitly determines the improved policy.
\begin{figure*}
\centering
\includegraphics[width=0.95{\scriptscriptstyle\mathsf{T}}extwidth]{figures/flowchart_QL.pdf}
\caption{Schematic of the Q-learning algorithm, which is a model-free reinforcement learning technique. The Q-table is the central object in the algorithm and its update through numerous episodes starting from a disconnected state to a terminal state (end-to-end connected via a virtual link) constitutes the learning process. The final Q-table is an instruction on what are the best actions to take in a particular state, with the aim to reach the terminal state and at the same time optimize the desired figure of merit, namely the waiting time and the end-to-end link fidelity. The update of the Q-table is done according to the update rule in \mathrm{e}qref{eqn:bellman}.}
\label{fig:q_learn}
\mathrm{e}nd{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.35{\scriptscriptstyle\mathsf{T}}extwidth]{figures/single_link_time.pdf}\qquad\qquad
\includegraphics[width=0.35{\scriptscriptstyle\mathsf{T}}extwidth]{figures/single_link_p_l.pdf}\\
\includegraphics[width=0.35{\scriptscriptstyle\mathsf{T}}extwidth]{figures/single_link_time_analytic.pdf}\qquad\qquad
\includegraphics[width=0.35{\scriptscriptstyle\mathsf{T}}extwidth]{figures/single_link_p_l_analytic.pdf}
\caption{Probability that an elementary link is active for the memory-cutoff policy with cutoff $m^{\star}$. (Top) Q-learning results. (Bottom) Analytical results taken from Ref.~\cite[Corollary~4.3]{Khatri2021policieselementary}.}
\label{fig:elem_link_Q_learning}
\mathrm{e}nd{figure*}
\paragraph*{Role of training hyperparameters in learning.} Four training hyperparameters, namely $\alpha, \gamma, \mathrm{e}psilon$, and the number of training episodes, need to be tuned in order to find improved policies.
\begin{enumerate}
\item The role of $\alpha$ is to determine the learning rate. A very low $\alpha$ makes the training too slow; a very large $\alpha$ makes us ``overshoot'' the optimal policy. A value of $\alpha \approx 0.01$ was found to be ideal for training in this work.
\item The discount factor $\gamma$ determines the discounting for future rewards. This parameter also needs to be tuned in order to find a good balance between prioritizing immediate (next state) rewards versus long-term rewards. We found $\gamma \approx 0.8$ to be ideal.
\item The hyperparameter $\mathrm{e}psilon$, as mentioned above, controls the trade-off between the exploration of the agent (choosing random actions) and its exploitation (using learned actions). We found $\mathrm{e}psilon$ to be the most relevant hyperparameter, and the one that requires the most fine tuning for finding improved policies. We found $\mathrm{e}psilon \approx 0.15$ to be optimal.
\item The figures of merit improve with an increasing number of training episodes (at the expense of a higher training time) until they saturate and more training does not help. Furthermore, the larger the state-action space (i.e., the higher the number of nodes and/or the maximum cutoff $m^\star$), and the smaller the probability of creating elementary and virtual links (small $p_\mathrm{e}ll$, $p_{sw}$), the larger the number of training episodes required for getting an improved policy. The number of training episodes needed for saturating the figures of merit varied between a few hundred thousand to approximately one million.
\mathrm{e}nd{enumerate}
\paragraph*{Verification of the Q-learning algorithm for a single elementary link.} At the elementary link level, if one assumes a finite memory cutoff, the improved policy is to request ($R_\mathrm{e}ll$) until the link is created and then wait for $m^{\star}$ time steps before requesting again. For the single link, analytical expressions for figures of merit such as expected state of the link or average wait times can be obtained (see Corollary 4.3 in Ref.~\cite{Khatri2021policieselementary}) and show that the aforementioned policy is optimal. As a sanity check, we see in Fig.~\ref{fig:elem_link_Q_learning} that the Q-learning algorithm also rediscovers this policy and gives qualitative and quantitative agreement with the aforementioned analytical results.
\section{Comparing the {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy with Q-learning policies}\label{sec-opt_pol_example}
In Sec.~\ref{sec-discussion}, we compared our Q-learning policies with the fixed and dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policies. Our Q-learning policies are based on two different rewards. The first reward is based on optimizing the average waiting time alone (see \mathrm{e}qref{eq-opt_waiting_time_reward}), and the second reward is based on optimizing both the average waiting time and the fidelity of the end-to-end entangled state (see \mathrm{e}qref{eq-reward_Q_learning_fidelity}). In Table~\ref{Q_table}, we compare these improved policies with each other and also with the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy. Since the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy allows freedom in the order of swaps (see Sec.~\ref{subsec:swap_asap}) when an action consisting of two swap requests is taken, these swaps are performed sequentially, i.e., if the first swap fails the second one is not performed. On the other hand, for the Q-learning policies, such freedom is not allowed. All swaps that are requested by the policy are performed in one go. This is in keeping with our assumptions that Q-learning is free to find the optimal order of actions, which includes the order of swaps. In the case as shown in the table below the Q-learning policies do in fact choose to perform the swaps in one go.
{\renewcommand{1.2}{1.2}
\begin{longtable}[c]{| p{2.5cm} | p{3cm} | p{3cm} | p{3cm}|}
\hline
\centering State & \centering Q-learning action (waiting time) & \centering Q-learning action (fidelity \& waiting time) & \centering\arraybackslash {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} action \\
\hline\hline
($-1$, $-1$, $-1$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) \\
($-1$, $-1$, $0$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $W$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $W$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $W$) \\
($-1$, $-1$, $1$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $W$) \\
($-1$, $-1$, $2$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) \\
($-1$, $0$) & ($R_\mathrm{e}ll$, $W$) & ($R_\mathrm{e}ll$, $W$) & ($R_\mathrm{e}ll$, $W$) \\
($-1$, $1$) & ($R_\mathrm{e}ll$, $W$) & ($R_\mathrm{e}ll$, $W$) & ($R_\mathrm{e}ll$, $W$) \\
($-1$, $2$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$ ) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) \\
($-1$, 0, $-1$) & ($R_\mathrm{e}ll$, $W$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $W$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $W$, $R_\mathrm{e}ll$) \\
($-1$, 0, 0) & ($R_\mathrm{e}ll$, $R_{sw}$) & ($R_\mathrm{e}ll$, $R_{sw}$) & ($R_\mathrm{e}ll$, $R_{sw}$) \\
($-1$, 0, 1) & ($R_\mathrm{e}ll$, $R_{sw}$) & ($R_\mathrm{e}ll$, $R_{sw}$) & ($R_\mathrm{e}ll$, $R_{sw}$) \\
($-1$, 0, 2) & ($R_\mathrm{e}ll$, $W$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $W$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_{sw}$) \\
($-1$, 1, $-1$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $W$, $R_\mathrm{e}ll$) \\
($-1$, 1, 0) & ($R_\mathrm{e}ll$, $R_{sw}$) & ($R_\mathrm{e}ll$, $R_{sw}$) & ($R_\mathrm{e}ll$, $R_{sw}$) \\
($-1$, 1, 1) & ($R_\mathrm{e}ll$, $R_{sw}$) & ($R_\mathrm{e}ll$, $R_{sw}$) & ($R_\mathrm{e}ll$, $R_{sw}$) \\
($-1$, 1, 2) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_{sw}$) \\
($-1$, 2, $-1$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) \\
($-1$, 2, 0) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $W$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_{sw}$) \\
($-1$, 2, 1) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_{sw}$) \\
($-1$, 2, 2) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_{sw}$) \\
0 - ($-1$, $-1$, $-1$) & N/A & N/A & N/A \\
0 - ($-1$, 0, $-1$) & N/A & N/A & N/A \\
0 - ($-1$, 1, $-1$) & N/A & N/A & N/A \\
0 - ($-1$, 2, $-1$) & N/A & N/A & N/A \\
1 - ($-1$, $-1$, $-1$) & N/A & N/A & N/A \\
1 - ($-1$, 0, $-1$) & N/A & N/A & N/A \\
1 - ($-1$, 1, $-1$) & N/A & N/A & N/A \\
1 - ($-1$, 2, $-1$) & N/A & N/A & N/A \\
2 - ($-1$, $-1$, $-1$) & N/A & N/A & N/A \\
2 - ($-1$, 0, $-1$) & N/A & N/A & N/A \\
2 - ($-1$, 1, $-1$) & N/A & N/A & N/A \\
2 - ($-1$, 2, $-1$) & N/A & N/A & N/A \\
(0, $-1$) & ($W$, $R_\mathrm{e}ll$) & ($W$, $R_\mathrm{e}ll$) & ($W$, $R_\mathrm{e}ll)$ \\
(0, 0) & $R_{sw}$ & $R_{sw}$ & $R_{sw}$ \\
(0, 1) & $R_{sw}$ & $R_{sw}$ & $R_{sw}$ \\
(0, 2) & $R_{sw}$ & $R_{sw}$ & $R_{sw}$ \\
(1, $-1$) & ($W$, $R_\mathrm{e}ll$) & ($W$, $R_\mathrm{e}ll$) & ($W$, $R_\mathrm{e}ll$) \\
(1, 0) & $R_{sw}$ & $R_{sw}$ & $R_{sw}$ \\
(1, 1) & $R_{sw}$ & $R_{sw}$ & $R_{sw}$ \\
(1, 2) & $R_{sw}$ & $R_{sw}$ & $R_{sw}$ \\
(2, $-1$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) \\
(2, 0) & $R_{sw}$ & $R_{sw}$ & $R_{sw}$ \\
(2, 1) & $R_{sw}$ & $R_{sw}$ & $R_{sw}$ \\
(2, 2) & $R_{sw}$ & $R_{sw}$ & $R_{sw}$ \\
(0, $-1$, $-1$) & ($W$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($W$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) \\
(0, $-1$, 0) & ($W$, $R_\mathrm{e}ll$, $W$) & ($W$, $R_\mathrm{e}ll$, $W$) & ($W$, $R_\mathrm{e}ll$, $W$) \\
(0, $-1$, 1) & ($W$, $R_\mathrm{e}ll$, $W$) & ($W$, $R_\mathrm{e}ll$, $W$) & ($W$, $R_\mathrm{e}ll$, $W$) \\
(0, $-1$, 2) & ($W$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($W$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) \\
(0, 0, $-1$) & ($R_{sw}$, $R_\mathrm{e}ll$) & ($R_{sw}$, $R_\mathrm{e}ll$) & ($R_{sw}$, $R_\mathrm{e}ll$) \\
(0, 0, 0) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) \\
(0, 0, 1) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) \\
(0, 0, 2) & ($R_{sw}$, $R_{sw}$ ) & ($R_{sw}$, $R_{sw}$ ) & ($R_{sw}$, $R_{sw}$) \\
(0, 1, $-1$) & ($R_{sw}$, $R_\mathrm{e}ll$) & ($R_{sw}$, $R_\mathrm{e}ll$ ) & ($R_{sw}$, $R_\mathrm{e}ll$) \\
(0, 1, 0) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) \\
(0, 1, 1) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) \\
(0, 1, 2 ) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) \\
(0, 2, $-1$) & ($W$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($W$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_{sw}$, $R_\mathrm{e}ll$) \\
(0, 2, 0) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) \\
(0, 2, 1) & ($R_{sw}$, $R_{sw}$ ) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) \\
(0, 2, 2) & ($R_{sw}$, $R_{sw}$ ) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) \\
(1, $-1$, $-1$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($W$, $R_\mathrm{e}ll$ $R_\mathrm{e}ll$) \\
(1, $-1$, 0) & ($W$, $R_\mathrm{e}ll$, $W$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($W$, $R_\mathrm{e}ll$, $W$) \\
(1, $-1$, 1) & ($W$, $R_\mathrm{e}ll$, $W$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($W$, $R_\mathrm{e}ll$, $W$) \\
(1, $-1$, 2) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($W$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) \\
(1, 0, $-1$) & ($R_{sw}$, $R_\mathrm{e}ll$) & ($R_{sw}$, $R_\mathrm{e}ll$) & ($R_{sw}$, $R_\mathrm{e}ll$) \\
(1, 0, 0) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) \\
(1, 0, 1) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) \\
(1, 0, 2) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) \\
(1, 1, $-1$) & ($R_{sw}$, $R_\mathrm{e}ll$) & ($R_{sw}$, $R_\mathrm{e}ll$) & ($R_{sw}$, $R_\mathrm{e}ll$) \\
(1, 1, 0) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) \\
(1, 1, 1) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) \\
(1, 1, 2) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) \\
(1, 2, $-1$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_{sw}$, $R_\mathrm{e}ll$) \\
(1, 2, 0) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) \\
(1, 2, 1) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) \\
(1, 2, 2) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) \\
(2, $-1$, $-1$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) \\
(2, $-1$, 0) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $W$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $W$) \\
(2, $-1$, 1) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $W$) \\
(2, 0, $-1$) & ($R_\mathrm{e}ll$, $W$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $W$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $W$, $R_\mathrm{e}ll$) \\
(2, 0, 0) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) \\
(2, 0, 1) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) \\
(2, 0, 2) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) \\
(2, 1, $-1$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_{sw}$, $R_\mathrm{e}ll$) \\
(2, 1, 0) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) \\
(2, 1, 1) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) \\
(2, 1, 2) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) \\
(2, 2, $-1$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$, $R_\mathrm{e}ll$) & ($R_\mathrm{e}ll$, $R_\mathrm{e}ll$ $R_\mathrm{e}ll$) & ($R_{sw}$, $R_\mathrm{e}ll$) \\
(2, 2, 0) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) \\
(2, 2, 0) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) \\
(2, 2, 2) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) & ($R_{sw}$, $R_{sw}$) \\
\hline
\caption{Comparison of the dynamic {\scriptscriptstyle\mathsf{T}}extsc{swap-asap} policy with our Q-learning policies for a four-node repeater chain with $p_{\mathrm{e}ll} = 0.6$, $p_{sw} = 0.5$, and $m^\star = 2$. See Sec.~\ref{sec-MDP} for a description of the states and actions. Here, we write the states as $(m_1,m_2,m_3)$, with $m_i$ indicating the age of the $i^{{\scriptscriptstyle\mathsf{T}}ext{th}}$ elementary link. States like $(m_1, m_2)$ indicate an elementary link with age $m_1$ between nodes $1$ and $2$ and a virtual link of age $m_2$ between nodes $2$ and $4$. It also represents the mirror image state, i.e., a state with a virtual link with age $m_2$ between nodes $1$ and $3$ and an elementary link with age $m_1$ between nodes $3$ and $4$, because this state of the repeater chain results in identical optimal actions for the homogeneous chain under consideration. Terminal states are written as $m_1-(-1,m_2,-1)$, indicating that the end-to-end link has age $m_1$ and that, in addition, there exists an elementary link between nodes $2$ and $3$ with age $m_2$. The action N/A indicates that no action needs to be taken, because the state is terminal.}\label{Q_table}
\mathrm{e}nd{longtable}
}
\mathrm{e}nd{document} | math | 144,165 |
\begin{document}
\title
{Proving a conjecture on chromatic polynomials
by counting the number of acyclic orientations\thanks{This article is partially supported by
NTU AcRf Project (RP 3/16 DFM) of Singapore and
NSFC grants (No. 11701401, 11961070 and 11971346).}}
\date{}
\def \bg {\hspace{0.3 cm}}
\author{Fengming Dong\thanks{Corresponding author. Email: [email protected] and [email protected]. },\bg Jun Ge,\bg Helin Gong
\\
Bo Ning,\bg Zhangdong Ouyang\bg and\bg Eng Guan Tay}
\maketitle
\begin{abstract}
The chromatic polynomial $P(G,x)$ of a graph $G$ of order $n$ can be expressed as $\sum\limits_{i=1}^n(-1)^{n-i}a_{i}x^i$, where
$a_i$ is interpreted as the number of broken-cycle free spanning subgraphs of $G$ with exactly $i$ components. The parameter
$\epsilon(G)=\sum\limits_{i=1}^n (n-i)a_i/\sum\limits_{i=1}^n a_i$
is the mean size of a broken-cycle-free spanning subgraph of $G$.
In this article, we confirm and strengthen a conjecture proposed by Lundow and Markstr\"{o}m in 2006
that $\epsilon(T_n)< \epsilon(G)<\epsilon(K_n)$ holds for any
connected graph $G$ of order $n$ which is neither the complete graph $K_n$ nor a tree $T_n$ of order $n$.
The most crucial step of our proof is to obtain the
interpretation of all $a_i$'s by the number of acyclic
orientations of $G$.
\end{abstract}
\noindent {\bf Keywords:} chromatic polynomial;
graph; acyclic orientation; combinatorial interpretation
\noindent {\bf Mathematics Subject Classification (2010): 05C31, 05C20}
\section{Introduction}
All graphs considered in this paper are simple graphs.
For any graph $G=(V, E)$ and any positive integer $k$,
a {\it proper $k$-coloring} $f$ of $G$ is a mapping
$f: V\rightarrow \{1, 2, \ldots, k\}$
such that $f(u)\neq f(v)$ holds
whenever $uv\in E$.
The chromatic polynomial of $G$
is the function $P(G, x)$ such that $P(G, k)$
counts the number of proper $k$-colorings of
$G$ for any positive integer $k$.
In this article, the variable $x$ in $P(G,x)$
is a real number.
The study of chromatic polynomials is one of the
most active areas in graph theory.
For basic concepts and properties on chromatic polynomials,
we refer the reader to the monograph~\cite{DKT2005}.
For the most celebrated results on this topic,
we recommend surveys~\cite{Dong2020,Jackson2015,Royle2009, RT1988}.
The first interpretation of the coefficients
of $P(G,x)$ was provided by Whitney~\cite{Whitney1932}:
for any simple graph $G$ of order $n$ and size $m$,
\begin{align}\relabel{int1}
P(G,x)=\sum_{i=1}^{n}\left( \sum_{r=0}^m (-1)^r N(i,r) \right )x^i,
\end{align}
where $N(i,r)$ is the number of spanning subgraphs of $G$
with exactly $i$ components and $r$ edges.
Whitney further simplified (\ref{int1})
by introducing the notion of broken cycles.
Let $\eta:E\rightarrow \{1,2,\ldots,|E|\}$
be a bijection.
For any cycle $C$ in $G$, the path $C-e$ is called a
{\it broken cycle} of $G$ with respect to
$\eta$, where $e$ is the edge on $C$
with $\eta(e)\le \eta(e')$ for
every edge $e'$ on $C$.
When there is no confusion, a broken cycle of $G$
is always assumed to be with respect to
a bijection $\eta:E\rightarrow \{1,2,\ldots,|E|\}$.
\begin{theorem}[\cite{Whitney1932}]\relabel{brokencycle}
Let $G=(V,E)$ be a graph of order $n$
and $\eta:E\rightarrow \{1,2,\ldots,|E|\}$ be a bijection.
Then,
\begin{align}\relabel{int2}
P(G,x)=\sum_{i=1}^{n} (-1)^{n-i}a_i(G)x^i,
\end{align}
where $a_i(G)$ is the number of
spanning subgraphs of $G$
with $n-i$ edges and $i$ components
which do not contain broken cycles.
\end{theorem}
Let $G$ be a simple graph of order $n$.
When there is no confusion,
$a_i(G)$ is written as $a_i$ for short.
Clearly, by Theorem~\ref{brokencycle},
$P(G,x)$ is indeed a polynomial in $x$
in which the constant term is $0$,
the leading coefficient $a_n$ is $1$
and all coefficients are integers
alternating in signs.
Thus, $(-1)^nP(G,x)>0$ holds for all $x<0$.
The concept of broken cycles has the following connection
with Tutte's work of expressing the Tutte polynomial
${\bf T}_G(x,y)$ of a connected graph $G$
in terms of spanning trees \cite{Crapo1969, Tutte1954}:
\begin{align}\label{Tuute-ex1}
{\bf T}_G(x,y)=\sum_{T}x^{ia_{\omega}(T)}y^{ea_{\omega}(T)},
\end{align}
where the sum runs over all spanning trees of $G$
and $ia_{\omega}(T)$ and $ea_{\omega}(T)$
are respectively the internal and external activities
of $T$ with respect to a bijection
$\omega: E\rightarrow \{1,2,\ldots,|E|\}$.
If we take $\omega$ to be $\eta$, then
$ea_{\eta}(T)$ is exactly the number of edges $e\in E(G)\setminus E(T)$
such that $\eta(e)\le \eta(e')$ holds for all edges $e'$
on the unique cycle $C$ of $T\cup e$.
As $G$ is a simple graph, $ea_{\eta}(T)$ equals the
number of broken cycles contained in $T$ with respect to $\eta$.
In particular, $ea_{\eta}(T)=0$ if and only if
$T$ does not contain broken cycles with respect to $\eta$.
By Theorem~\ref{brokencycle}, $a_1(G)$ is
the number of spanning trees $T$ of $G$ with $ea_{\eta}(T)=0$.
If
\begin{align}\label{Tuute-ex2}
{\bf T}_G(x,y)=\sum\limits_{i\ge 0, j\ge 0} c_{i,j}x^iy^j,
\end{align}
then $a_1(G)=\sum_{i\ge 0}c_{i,0}={\bf T}_G(1,0)$.
As in \cite{LM2006},
for $i=0,1,2,\ldots,n-1$,
we define $b_i(G)$ (or simply $b_i$)
as the probability
that a randomly chosen broken-cycle-free spanning subgraph
of $G$ has size $i$.
Then
\begin{align}\relabel{prob-bi}
b_i=\frac{a_{n-i}}{a_1+a_2+\cdots+a_n},
\quad \forall i=0, 1, \ldots, n-1.
\end{align}
Let $\epsilon(G)$ denote the mean size of a
broken-cycle-free spanning subgraph of $G$. Then
\begin{align}\relabel{meansize}
\epsilon(G)=
\sum_{i=0}^{n-1} ib_i
=\frac{(n-1)a_1+(n-2)a_2+\cdots+a_{n-1}}{a_1+a_2+\cdots+a_n}.
\end{align}
An elementary property of $\epsilon(G)$
is given below.
\begin{prop}[\cite{LM2006}]\relabel{prop-eps}
For any graph $G$ of order $n$,
$\epsilon(G)=n+\frac{P'(G, -1)}{P(G, -1)}$.
\end{prop}
Let $T_n$ denote a tree of order $n$
and $K_n$ denote the complete graph of order $n$.
By Proposition~\ref{prop-eps},
$\epsilon(T_n)=\frac{n-1}{2}$, ,
while
\begin{align}\label{constant}
\epsilon(K_n)=n-\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)\sim n-\log n-\gamma
\end{align}
as $n\rightarrow \infty$, where $\gamma\approx 0.577216$
is the Euler-Mascheroni constant.
Lundow and Markstr\"{o}m~\cite{LM2006}
proposed the following conjecture on $\epsilon(G)$.
\begin{conjecture}
[\cite{LM2006}]\relabel{mainconj}
For any connected graph $G$ of order $n$,
where $n\ge 4$,
if $G$ is neither $K_n$ nor a $T_n$,
then $\epsilon(T_n)<\epsilon(G)<\epsilon(K_n)$.
\end{conjecture}
In this paper, we aim to prove and strengthen
Conjecture~\ref{mainconj}.
For any graph $G$, define the function $\epsilon(G,x)$
as follows:
\begin{align}\relabel{epsi-G}
\epsilon(G,x)=\frac{P'(G, x)}{P(G, x)}.
\end{align}
By Proposition~\ref{prop-eps},
$\epsilon(G)=n+\epsilon(G,-1)$ holds for every graph
$G$ of order $n$.
Thus, for any graphs $G$ and $H$ of the same order, $\epsilon(G)<\epsilon(H)$
if and only if $\epsilon(G,-1)<\epsilon(H,-1)$.
Conjecture~\ref{mainconj} is equivalent
to the statement that
$\epsilon(T_n,-1)<\epsilon(G,-1)<\epsilon(K_n,-1)$
holds for any connected graph $G$ of order $n$
which is
neither $K_n$ nor a $T_n$.
A graph $Q$ is said to be {\it chordal}
if $Q[V(C)]\not\cong C$
for every cycle $C$ of $Q$ with $|V(C)|\ge 4$,
where $Q[V']$ is the subgraph of $Q$ induced by $V'$
for $V'\subseteq V(G)$.
In Section~\ref{firstIn},
we will establish the following result.
\begin{theorem}\relabel{compare-Q}
For any graph $G$,
if $Q$ is a chordal and proper spanning subgraph
of $G$, then
$\epsilon(G,x)>\epsilon(Q,x)$ holds for all $x<0$.
\end{theorem}
Note that any tree is a chordal graph
and any connected graph contains a spanning tree.
Thus, we have the following corollary
which obviously implies the first part of
Conjecture~\ref{mainconj}.
\begin{corollary}\relabel{compare-T}
For any connected graph $G$ of order $n$
which is not a tree,
$\epsilon(G, x)>\epsilon(T_n, x)$
holds for all $x<0$.
\end{corollary}
The second part of Conjecture~\ref{mainconj}
is extended to the inequality
$\epsilon(K_n,x)>\epsilon(G,x)$
for any non-complete graph $G$ of order $n$ and all $x<0$.
In order to prove this inequality,
we will show in Section~\ref{SecondIn} that
it suffices to establish the following result.
\begin{theorem}\relabel{average-th}
For any non-complete graph $G=(V,E)$ of order $n$,
\begin{align}
(-1)^{n}(x-n+1)\sum_{u\in V}P(G-u, x)
+(-1)^{n+1}nP(G, x)> 0
\relabel{right-2}
\end{align}
holds for all $x<0$.
\end{theorem}
Note that the left-hand side of
(\ref{right-2}) vanishes when $G\cong K_n$.
Theorem~\ref{average-th} will be proved in
Section~\ref{finalproof},
based on Greene $\&$ Zaslavsky's interpretation in \cite{GZ1983}
for coefficients $a_i(G)$'s
of $P(G,x)$ by acyclic orientations
introduced in Section~\ref{interp}.
By applying Theorem~\ref{average-th} and
two lemmas in Section~\ref{SecondIn},
we will finally prove the second main result in this article.
\begin{theorem}\relabel{compare-K}
For any non-complete graph $G$ of order $n$,
$\epsilon(G,x)<\epsilon(K_n,x)$ holds for
all $x<0$.
\end{theorem}
\section{Proof of Theorem~\ref{compare-Q}
\relabel{firstIn}}
A vertex $u$ in a graph $G$ is called a {\it simplicial
vertex} if
$\{u\}\cup N_G(u)$ is a clique of $G$,
where $N_G(u)$ is the set of vertices in
$G$ which are adjacent to $u$.
For a simplicial vertex $u$ of $G$,
$P(G,x)$ has the following property
(see~\cite{DKT2005, Read1968, RT1988}):
\begin{align}\relabel{sim-ch}
P(G,x)=(x-d(u))P(G-u,x),
\end{align}
where $G-u$ is the subgraph of $G$ induced by $V-\{u\}$
and $d(u)$ is the degree of $u$ in $G$.
By (\ref{sim-ch}), it is not difficult to show the following.
\begin{prop}\relabel{sim-epsi}
If $u$ is a simplicial vertex of a graph $G$, then
\begin{align}\relabel{sim-ch1}
\epsilon(G,x)=\frac{1}{x-d(u)}+\epsilon(G-u,x).
\end{align}
\end{prop}
It has been shown that a graph $Q$ of order $n$
is chordal if and only if
$Q$ has an ordering $u_1,u_2,\ldots,u_n$
of its vertices such that
$u_i$ is a simplicial vertex in
$Q[\{u_1,u_2,\ldots,u_i\}]$
for all $i=1,2,\ldots,n$
(see \cite{Dirac1961, FG1965}).
Such an ordering of vertices in $Q$ is
called a {\it perfect elimination ordering} of $Q$.
For any perfect
elimination ordering $u_1,u_2,\ldots,u_n$ of
a chordal graph $Q$,
by Proposition~\ref{sim-epsi},
\begin{align}\relabel{sim-ch2-1}
\epsilon(Q,x)=\sum_{i=1}^n \frac 1{x-d_{Q_i}(u_i)},
\end{align}
where $Q_i$ is the subgraph $Q[\{u_1,u_2,\ldots,u_i\}]$.
Now we are ready to prove Theorem~\ref{compare-Q}.
\noindent {\it Proof of Theorem~\ref{compare-Q}}:
Let $G$ be any graph of order $n$
and $Q$ be any chordal
and proper spanning subgraph of $G$.
When $n\le 3$, it is not difficult to verify that
$\epsilon(G,x)>\epsilon(Q,x)$ holds for all $x<0$.
Suppose that Theorem~\ref{compare-Q} fails
and $G=(V,E)$ is a counter-example to this result
such that $|V|+|E|$ has the minimum value among all
counter-examples. Thus the result holds
for any graph $H$ with $|V(H)|+|E(H)|<|V|+|E|$
and any chordal and proper spanning subgraph $Q'$ of $H$,
but $G$ has a chordal and proper spanning subgraph $Q$ such that $\epsilon(G,x)\le \epsilon(Q,x)$ holds for some $x<0$.
We will establish the following claims.
Let $u_1,u_2,\ldots,u_n$ be a perfect elimination ordering
of $Q$ and $Q_i=Q[\{u_1,\ldots,u_i\}]$ for all
$i=1,2,\ldots,n$.
So $u_i$ is a simplicial vertex of $Q_i$ for
all $i=1,2,\ldots,n$.
\noindent {\bf Claim 1}:
$u_n$ is not a simplicial vertex of $G$.
Note that $Q-u_n$ is chordal and a spanning
subgraph of $G-u_n$.
By the assumption on the minimality of $|V|+|E|$,
$\epsilon(G-u_n,x)\ge \epsilon(Q-u_n,x)$ holds
for all $x<0$,
where the inequality is strict whenever
$Q-u_n\not\cong G-u_n$.
Clearly $d_G(u_n)\ge d_Q(u_n)$.
As $Q$ is a proper subgraph of $G$,
$d_G(u_n)>d_Q(u_n)$
in the case that $G-u_n\cong Q-u_n$.
If $u_n$ is also a simplicial vertex of $G$,
then by Proposition~\ref{sim-epsi},
\begin{align}\relabel{comp-GQ}
\epsilon(G,x)=\frac 1{x-d_G(u_n)}+\epsilon(G-u_n,x),
\quad
\epsilon(Q,x)=\frac 1{x-d_Q(u_n)}+\epsilon(Q-u_n,x),
\end{align}
implying that $\epsilon(G,x)>\epsilon(Q,x)$ holds for all
$x<0$, a contradiction.
Hence Claim 1 holds.
\noindent {\bf Claim 2}: $d_G(u_n)>d_Q(u_n)$.
Clearly $d_G(u_n)\ge d_Q(u_n)$.
Since $u_n$ is a simplicial vertex of $Q$
and $Q$ is a subgraph of $G$,
$d_G(u_n)=d_Q(u_n)$ implies that $u_n$ is a simplicial
vertex of $G$,
contradicting Claim 1.
Thus Claim 2 holds.
For any edge $e$ in $G$, let $G-e$ be the graph
obtained from $G$ by deleting $e$. Let $G/e$ be
the graph obtained from $G$ by contracting $e$
and replacing multiple edges, if any arise,
by single edges.
\noindent {\bf Claim 3}:
For any $e=u_nv\in E-E(Q)$,
both $\epsilon(G-e,x)\ge \epsilon(Q,x)$
and $\epsilon(G/e,x)\ge \epsilon(Q-u_n,x)$
hold for all $x<0$.
As $e=u_nv\in E-E(Q)$,
$Q$ is a spanning subgraph of $G-e$ and $Q-u_n$ is a spanning subgraph of $G/e$.
As both $Q$ and $Q-u_n$ are chordal, by the assumption on
the minimality of $|V|+|E|$, the theorem holds for both $G-e$ and $G/e$.
Thus this claim holds.
\noindent {\bf Claim 4}:
$\epsilon(G,x)>\epsilon(Q,x)$ holds for all
$x<0$.
By Claim 2, there exists $e=u_nv\in E-E(Q)$.
By Claim 3, $\epsilon(G-e,x)\ge \epsilon(Q,x)$
and $\epsilon(G/e,x)\ge \epsilon(Q-u_n,x)$
hold for all $x<0$.
By (\ref{epsi-G}) and (\ref{sim-ch2-1}),
\begin{eqnarray}\relabel{G-1-eq1}
\ & & (\epsilon(G-e,x)-\epsilon(Q,x))
\times (-1)^nP(G-e,x)\nonumber \\
& = & (-1)^nP'(G-e,x)+(-1)^{n+1}P(G-e,x)\sum_{i=1}^n
\frac 1{x-d_{Q_i}(u_i)}.
\end{eqnarray}
As $(-1)^nP(G-e,x)>0$ and
$\epsilon(G-e,x)\ge \epsilon(Q,x)$ for all $x<0$,
the left-hand side of (\ref{G-1-eq1}) is non-negative for $x<0$,
implying that
the right-hand side of (\ref{G-1-eq1}) is also non-negative
for $x<0$,
i.e.,
\begin{eqnarray}\relabel{G-1-eq1-1}
(-1)^nP'(G-e,x)+(-1)^{n+1}P(G-e,x)\sum_{i=1}^n
\frac 1{x-d_{Q_i}(u_i)}\ge 0,\quad \forall x<0.
\end{eqnarray}
As $u_1,\ldots,u_{n-1}$
is a perfect elimination ordering of $Q-u_n$
and $\epsilon(G/e,x)\ge \epsilon(Q-u_n,x)$ holds
for all $x<0$, similarly we have:
\begin{align}\relabel{G-1-eq1-2}
(-1)^{n-1}P'(G/e,x)+(-1)^{n}P(G/e,x)\sum_{i=1}^{n-1}
\frac 1{x-d_{Q_i}(u_i)}\ge 0,\quad \forall x<0.
\end{align}
As $(-1)^{n-1}P(G/e,x)>0$ holds for all $x<0$,
(\ref{G-1-eq1-2}) implies that
\begin{eqnarray}\relabel{G-1-eq1-3}
\ & &
(-1)^{n-1}P'(G/e,x)+(-1)^{n}P(G/e,x)\sum_{i=1}^{n}
\frac 1{x-d_{Q_i}(u_i)}
\nonumber \\
& \ge &
\frac {(-1)^{n}P(G/e,x)}{x-d_{Q_n}(u_n)} >0,
\qquad \forall x<0.
\end{eqnarray}
By the deletion-contraction formula for chromatic
polynomials,
\begin{align}\label{del-con}
P(G, x)=P(G-e, x)-P(G/e, x),\quad
P'(G, x)=P'(G-e, x)-P'(G/e, x).
\end{align}
Then (\ref{G-1-eq1-1}), (\ref{G-1-eq1-3})
and (\ref{del-con}) imply that
\begin{align}\relabel{G-1-eq1-4}
(-1)^nP'(G,x)+(-1)^{n+1}P(G,x)\sum_{i=1}^n
\frac 1{x-d_{Q_i}(u_i)}> 0,\quad \forall x<0.
\end{align}
By (\ref{epsi-G}) and (\ref{sim-ch2-1}),
inequality (\ref{G-1-eq1-4}) implies that
\begin{align}\relabel{G-1-eq1-5}
\left (\epsilon(G,x)-\epsilon(Q,x)\right )(-1)^nP(G,x)
> 0,\quad \forall x<0.
\end{align}
Since $(-1)^nP(G,x)>0$ holds for all $x<0$,
inequality (\ref{G-1-eq1-5}) implies
Claim 4.
As Claim 4 contradicts the assumption of $G$,
there are no counter-examples to this result and
the theorem is proved.
\qed
\section{An approach for proving
Theorem~\ref{compare-K} \relabel{SecondIn}}
In this section, we will mainly show that, in order
to prove Theorem~\ref{compare-K},
it suffices to prove Theorem \ref{average-th}.
By (\ref{sim-ch2-1}),
we have
\begin{align}\relabel{Kn-epsi}
\epsilon(K_n,x)=\sum\limits_{i=0}^{n-1}\frac{1}{x-i}.
\end{align}
Thus,
\begin{align}\relabel{Kn-epsi2}
\epsilon(K_n,x)-\epsilon(G,x)
=\frac{(-1)^n}{P(G,x)}
\left (
(-1)^{n}P(G, x)\sum_{i=0}^{n-1}\frac {1}{x-i}
+(-1)^{n+1}P'(G, x)\right ).
\end{align}
For any graph $G$ of order $n$, define
\begin{align}
\xi(G, x)=(-1)^{n}P(G, x)\sum_{i=0}^{n-1}\frac {1}{x-i}+(-1)^{n+1}P'(G, x).
\relabel{xi}
\end{align}
Note that $\xi(G,x)\equiv 0$ if $G$ is a complete graph.
For any non-complete graph $G$ and any $x<0$,
we have $(-1)^nP(G,x)>0$ and so (\ref{Kn-epsi2})
implies that
$\epsilon(K_n,x)-\epsilon(G,x)>0$
if and only if $\xi(G,x)> 0$.
\begin{prop}\relabel{compare-K-eq}
Theorem~\ref{compare-K} holds
if and only if
$\xi(G,x)> 0$ holds for every non-complete graph $G$
and all $x<0$.
\end{prop}
It can be easily verified that $\xi(G,x)>0$ holds
for all non-complete graphs $G$ of order at most $3$
and all $x<0$.
For the general case, we will prove it by induction.
In the rest of this section,
we will find a relation between $\xi(G,x)$ and $\xi(G-u,x)$
for a vertex $u$ in $G$ in two cases.
Lemma~\ref{ud0} is for the case when $u$ is a simplicial vertex and
Lemma~\ref{rec2} when $d(u) \ge 1$.
We then explain why
Theorem~\ref{average-th} implies
$\xi(G,x)>0$ for all non-complete graphs $G$
and all $x<0$.
\begin{lemma}\relabel{ud0}
Let $G$ be a graph of order $n$.
If $u$ is a simplicial vertex of $G$ with $d(u)=d$, then
\begin{align}\relabel{ud0-eq1}
\xi(G, x)=(d-x)\xi(G-u, x)
+\frac{(-1)^{n-1}(n-1-d)P(G-u,x)}{n-1-x}.
\end{align}
\end{lemma}
\begin{proof}
As $u$ is a simplicial vertex of $G$ with $d(u)=d$,
$P(G,x)=(x-d)P(G-u,x)$ by (\ref{sim-ch}).
Thus
$
P'(G, x)=P(G-u, x)+(x-d)P'(G-u, x).
$
By (\ref{xi}),
\begin{eqnarray}
\xi(G,x)
&=&(-1)^n (x-d)P(G-u,x)\sum_{i=0}^{n-1}\frac 1{x-i}
+(-1)^{n+1}(P(G-u, x)+(x-d)P'(G-u, x))\nonumber \\
&=&(d-x)\xi(G-u,x)+\frac{(-1)^n(x-d)P(G-u,x)}{x-n+1}
+(-1)^{n+1}P(G-u, x)\nonumber \\
&=&(d-x)\xi(G-u, x)+
\frac{(-1)^{n-1}(n-1-d)P(G-u,x)}{n-1-x}.
\end{eqnarray}
\end{proof}
Note that
$d\le n-1$ and $(-1)^{n-1} P(G-u,x)>0$ holds for all $x<0$,
implying that the second term in the right-hand side of
(\ref{ud0-eq1}) is non-negative.
Thus, if $u$ is a simplicial vertex of $G$ and $x<0$,
by Lemma~\ref{ud0},
$\xi(G-u,x)>0$ implies that $\xi(G,x)>0$.
Now consider the case that
$u$ is a vertex in $G$ with $d(u)=d\ge 1$.
Assume that $N(u)=\{u_1,u_2,\ldots,u_d\}$.
For any $i=1, 2, \ldots, d-1$,
let $G_i$ denote the graph obtained from $G-u$ by adding
edges joining $u_i$ to $u_j$ whenever $u_iu_j\notin E(G)$
for all $j$ with $i+1\le j\le d$.
Thus, $u_i$ is adjacent to $u_j$ in $G_i$ for all $j$
with $i+1\le j\le d$.
In the case that $u$ is a simplicial vertex of $G$,
$G_i\cong G-u$ for all $i=1,2,\cdots,d-1$.
By applying the deletion-contraction formula
for chromatic polynomials
(see \cite{DKT2005,Read1968}),
$P(G,x)$ can be expressed in terms of $P(G-u,x)$ and $P(G_i,x)$ for
$i=1,2,\cdots,d-1$.
\begin{lemma}\relabel{rec0}
Let $u$ be a vertex in $G$ with $d(x)=d\ge 1$
and for $i=1,2,\cdots,d-1$, let $G_i$ be the graph defined above. Then,
\begin{align}
P(G, x)=(x-1)P(G-u, x)-\sum_{i=1}^{d-1}P(G_i, x).
\relabel{rec1}
\end{align}
\end{lemma}
\begin{proof}
For $1\le i\le d$,
let $E_i$ denote the set of edges $uu_j$ in $G$ for $j=1,2,\cdots,i-1$.
So $|E_i|=i-1$ and $E_1=\emptyset$.
For any $i$ with $1\le i\le d-1$,
applying the deletion-contraction formula
for chromatic polynomials to edge $uu_i$ in $G-E_i$,
the graph obtained from $G$ by removing
all edges in $E_i$,
we have
\begin{align}
P(G-E_i, x)=P(G-E_{i+1}, x)-P((G-E_i)\slash uu_i, x)
=P(G-E_{i+1}, x)-P(G_i, x),
\relabel{rec0-1}
\end{align}
where the last equality follows from the fact that
$(G-E_i)\slash uu_i\cong G_i$
by the assumption of $G_i$.
Thus, by (\ref{rec0-1}),
\begin{align}
P(G,x)=P(G-E_1,x)=P(G-E_d,x)-\sum_{i=1}^{d-1}P(G_i,x).
\relabel{rec0-2}
\end{align}
As $u$ is of degree $1$ in $G-E_d$,
$P(G-E_d,x)=(x-1)P(G-u,x)$.
Hence (\ref{rec1}) follows.
\end{proof}
\begin{lemma}\relabel{rec2}
Let $G$ be a graph of order $n$ and let $u$ be
a vertex of $G$ with $d(u)=d\ge 1$. Then
\begin{align}\relabel{rec2-eq1}
\xi(G, x)=(1-x)\xi(G-u, x)+\sum_{i=1}^{d-1}\xi(G_i, x)
+\frac{(-1)^{n}\left[(x-n+1)P(G-u, x)-P(G, x)\right]}{n-x-1},
\end{align}
where $G_1,\ldots,G_{d-1}$ are graphs defined above.
\end{lemma}
\begin{proof}
By (\ref{rec1}), we have
\begin{align}\relabel{rec2-eq0}
P'(G, x)=P(G-u, x)+(x-1)P'(G-u, x)-\sum_{i=1}^{d-1}P'(G_i, x).
\end{align}
Thus
\begin{eqnarray}
\ \xi(G, x) & = & (-1)^{n}P(G, x)
\sum_{j=0}^{n-1}\frac {1}{x-j}+(-1)^{n+1}P'(G, x) \nonumber \\
& = & (-1)^{n}\left[(x-1)P(G-u, x)-\sum_{i=1}^{d-1}P(G_i, x)\right]\sum_{j=0}^{n-1}\frac {1}{x-j} \nonumber \\
& & +(-1)^{n+1}\left[P(G-u, x)+(x-1)P'(G-u, x)-\sum_{i=1}^{d-1}P'(G_i, x)\right] \nonumber \\
& = & (1-x)\left[(-1)^{n-1}P(G-u, x)
\sum_{j=0}^{n-2}\frac {1}{x-j}
+(-1)^{n}P'(G-u, x)\right] \nonumber \\
& & +\sum_{i=1}^{d-1}\left[(-1)^{n-1}P(G_i, x)
\sum_{j=0}^{n-2}\frac {1}{x-j}
+(-1)^{n}P'(G_i, x)\right] +(-1)^{n+1}P(G-u, x)
\nonumber \\
& & +(-1)^{n}\left[\frac{(x-1)P(G-u,x)}{x-(n-1)}
-\frac{1}{x-(n-1)}\sum_{i=1}^{d-1} P(G_i,x)\right]
\nonumber
\end{eqnarray}
\begin{eqnarray}
& = & (1-x)\xi(G-u, x)+\sum_{i=1}^{d-1}\xi(G_i, x)
\nonumber \\ & &
+\frac{(-1)^{n}\left[(x-n+1)P(G-u, x)
-P(G, x)\right]}{n-x-1},
\relabel{rec2-eq2}
\end{eqnarray}
where the last expression follows from (\ref{rec1})
and the definitions
of $\xi(G-u, x)$ and $\xi(G_i,x)$.
The result then follows.
\end{proof}
It is known that $\xi(G,x)>0$ holds for all
non-complete graphs $G$ of order at most $3$ and all $x<0$.
For any non-complete graph $G$ of order $n\ge 4$,
by Lemma~\ref{ud0},
$\xi(G-u,x)>0$ implies $\xi(G,x)>0$
for each simplicial vertex $u$ in $G$
and all $x<0$;
by Lemma~\ref{rec2},
for any $x<0$,
$\xi(G-u,x)>0$ implies $\xi(G,x)>0$
whenever $u$ is an non-isolated vertex in $G$
satisfying the following
inequality:
\begin{align} \relabel{ineq2}
(-1)^{n}((x-n+1)P(G-u, x)-P(G, x))> 0.
\end{align}
Note that the left-hand side of (\ref{ineq2})
vanishes when $G$ is $K_n$.
Also notice that there exist
non-complete graph $G$ and some vertex $u$ in $G$
such that inequality (\ref{ineq2}) does not hold
for some $x<0$.
For example, if $G$ is the complete bipartite graph $K_{2,3}$
and $u$ is a vertex of degree $3$ in $G$, then (\ref{ineq2}) fails for all real $x$ with $-2.3<x<0$.
However,
to prove that for any $x<0$, there exists some vertex $u$ in $G$
such that inequality (\ref{ineq2}) holds,
it suffices to prove the following inequality
(i.e., Theorem~\ref{average-th}):
\begin{align}
(-1)^n (x-n+1)\sum_{u\in V}P(G-u, x)
+(-1)^{n+1}nP(G, x)
> 0 \relabel{average}
\end{align}
for any non-complete graph $G=(V,E)$ of order $n$ and all $x<0$.
By Proposition~\ref{compare-K-eq} and
inequality (\ref{ineq2}),
to prove Theorem~\ref{compare-K},
we can now just focus on
proving inequality~(\ref{average})
(i.e., Theorem~\ref{average-th}).
The proof of Theorem~\ref{average-th}
will be given in Section~\ref{finalproof}
based on the interpretations for the coefficients
of chromatic polynomials introduced in
Section~\ref{interp}.
\section{Combinatorial interpretations for
coefficients of $P(G,x)$\relabel{interp}}
Let $G=(V,E)$ be any graph.
In this section, we will introduce
Greene $\&$ Zaslavsky's
combinatorial interpretation in \cite{GZ1983}
for the coefficients of
$P(G,x)$ in terms of acyclic orientations.
The result will be applied in the next section to prove
Theorem~\ref{average-th}.
An orientation $D$ of $G$ is called {\it acyclic}
if $D$ does not contain any directed cycle.
Let $\alpha (G)$ be the number of acyclic
orientations of a graph $G$.
In~\cite{Stanley1973}, Stanley
gave a nice combinatorial interpretation
of $(-1)^n P(G, -k)$ for any positive integer $k$
in terms of acyclic
orientations of $G$. In particular, he proved:
\begin{theorem}
[\cite{Stanley1973}]\relabel{Stanley}
For any graph $G$ of order $n$,
$(-1)^nP(G, -1)=\alpha(G)$,
i.e.,
\begin{align}\label{Stanley-eq1}
\sum\limits_{i=1}^{n}a_i(G)=\alpha(G).
\end{align}
\end{theorem}
In a digraph $D$, any vertex of $D$
with in-degree (resp. out-degree) zero is called a
{\it source} (resp. {\it sink}) of $D$.
It is well known that
any acyclic digraph has at least one
source and at least one sink.
If $v$ is an isolated vertex of $G$,
then $v$ is a source and also a sink in
any orientation of $G$.
For any $v\in V$,
let $\alpha(G, v)$ be the number of acyclic
orientations of $G$ with $v$ as its
unique source.
Clearly $\alpha(G, v)=0$ if and only if $G$ is
not connected.
In 1983, Greene and Zaslavsky \cite{GZ1983}
showed that $a_1(G)=\alpha(G, v)$.
\begin{theorem}[\cite{GZ1983}]\relabel{source}
For any graph $G=(V,E)$,
$a_1(G)=\alpha(G,v)$ holds for every $v\in V$.
\end{theorem}
This theorem was proved originally by using the
theory of hyperplane arrangements. See \cite{GS2000}
for three other nice proofs.
By Whitney's Broken-cycle Theorem
(i.e., Theorem~\ref{brokencycle}), $a_i(G)$
equals the number of spanning subgraphs of $G$ with
$i$ components and $n-i$ edges, containing no broken
cycles of $G$. In particular, $a_1(G)$ is the
number of spanning trees of $G$ containing no broken
cycles of $G$. Now we have two different combinatorial
interpretations for $a_1$.
For any $a_i(G)$, $2\leq i\leq n$,
its combinatorial interpretation can be obtained by
applying these two different combinatorial
interpretations for $a_1$.
Let $\mathcal{P}_i(V)$ be the set of
partitions $\{V_1,V_2,\ldots,V_i\}$ of $V$
such that $G[V_j]$ is connected for all $j=1,2,\ldots,i$
and let $\beta_i(G)$ be the number
of ordered pairs $(P_i, F)$, where
\begin{enumerate}
\item[(a)] $P_i=\{V_1,V_2,\ldots,V_i\}\in \mathcal{P}_i(V)$;
\item[(b)]
$F$ is a spanning forest of $G$ with exactly $i$ components
$T_1,T_2, \ldots, T_i$, where each $T_j$ is a spanning tree of
$G[V_j]$ containing no broken cycles of $G$.
\end{enumerate}
For any subgraph $H$ of $G$,
let $\widetilde{\tau}(H)$ be the number
of spanning trees of $H$ containing
no broken cycles of $G$.
By Theorem~\ref{brokencycle},
$\widetilde{\tau}(H)=a_1(H)$ holds
and the next result follows.
\begin{theorem}\relabel{interpre1}
For any graph $G$ and any $1\leq i\leq n$,
\begin{align}
a_i(G)=\beta_i(G)=\sum_{\{V_1,\ldots, V_i\}\in \mathcal{P}_i(V)}\prod_{j=1}^{i} \widetilde{\tau}(G[V_j]).
\end{align}
\end{theorem}
Now let $V=\{1, 2, \ldots, n\}$.
For any $i:1\le i\le n$
and any vertex $v\in V$, let
$\mathcal{OP}_{i, v}(V)$ be the family of
ordered partitions $(V_1,V_2,\ldots,V_i)$ of $V$
such that
\begin{enumerate}
\item[(a)] $\{V_1,V_2,\ldots,V_i\}\in \mathcal{P}_i(V)$,
where $v\in V_1$;
\item[(b)] for $j=2, \ldots, i$, the minimum number
in the set $\bigcup_{j\le s\le i} V_s$ is within $V_j$.
\end{enumerate}
Clearly, for any $v\in V$ and
any $\{V_1,V_2,\ldots,V_i\}\in \mathcal{P}_i(V)$,
there is exactly one permutation
$(\pi_1,\pi_2,\ldots,\pi_i)$ of $1,2,\ldots,i$
such that $(V_{\pi_1}, V_{\pi_2},\ldots,V_{\pi_i})\in
\mathcal{OP}_{i, v}(V)$.
By Theorem \ref{source},
$\widetilde{\tau}(G[V_j])=\alpha(G[V_j],u)$ holds
for any vertex $u$ in $G[V_j]$ and
Theorem \ref{interpre1} is equivalent to
a result in \cite{GZ1983}
which we illustrate differently below.
\begin{theorem}[\cite{GZ1983}, Theorem 7.4]\relabel{interpre2}
For any $v\in V$ and any $1\leq i\leq n$,
\begin{align}
a_i(G)=\sum_{(V_1,\ldots, V_i)\in \mathcal{OP}_{i, v}(V)}\alpha(G[V_1],v) \prod_{j=2}^{i} \alpha(G[V_j],m_j),
\relabel{interpre}
\end{align}
where $m_j$ is the minimum number in $V_j$ for
$j=2,\ldots,i$.
\end{theorem}
Note that the theorem above indicates that the
right hand side of (\ref{interpre}) is independent
of the choice of $v$. Thus, for any $1\le i\le n$,
\begin{align}
na_i(G)=\sum_{v\in V}
\sum_{(V_1,\ldots, V_i)\in \mathcal{OP}_{i, v}(V)}
\alpha(G[V_1],v) \prod_{j=2}^{i} \alpha(G[V_j],m_j).
\relabel{interpre-n}
\end{align}
Let $P^{(i)}(G, x)$ be the $i$-th derivative of $P(G,x)$.
Very recently,
Bernardi and Nadeau \cite{Bernardi2020} gave an interpretation of $P^{(i)}(G, -j)$
for any nonnegative integers $i$ and $j$
in terms of acyclic orientations.
When $i=0$, their result is exactly
Theorem \ref{Stanley} due to Stanley~\cite{Stanley1973};
and when $j=0$, it is Theorem \ref{interpre2} due to
Greene $\&$ Zaslavsky~\cite{GZ1983}.
\section{Proofs of Theorems \ref{average-th} and \ref{compare-K}
\relabel{finalproof}}
By the explanation in Section 3,
to prove Theorem~\ref{compare-K},
it suffices to prove Theorem~\ref{average-th}.
In this section, we will
prove Theorem~\ref{average-th} by
showing that
the coefficient of
$x^i$ in the expansion of the left-hand side of (\ref{right-2})
in Theorem~\ref{average-th}
is of the form $(-1)^i d_i$
with $d_i\ge 0$
for all $i=1,2,\ldots,n$.
Furthermore,
$d_i>0$ holds for some $i$ when $G$ is not complete.
We first establish the following result.
\begin{lemma}\relabel{le5-1}
Let $G=(V,E)$ be a non-complete graph of order
$n\geq 3$
and component number $c$.
\begin{enumerate}
\renewcommand{\rm (\alph{enumi})}{\rm (\alph{enumi})}
\item If $c=1$ and $G$ is not the $n$-cycle $C_n$,
then there exist non-adjacent vertices $u_1,u_2$
of $G$ such that $G-\{u_1,u_2\}$ is connected.
\item If $2\le c\le n-1$,
then for any integer $i$ with $c\le i\le n-1$,
there exists a partition $V_1,V_2,\ldots,V_i$ of $V$
such that $G[V_j]$ is connected for all $j=2,\ldots,i$
and $G[V_1]$ has exactly two components
one of which is an isolated vertex.
\end{enumerate}
\end{lemma}
\begin{proof} (a). As $c=1$, $G$ is connected.
As $G$ is non-complete, the result is trivial when
$G$ is 3-connected.
If $G$ is not $2$-connected, choose vertices
$u_1$ and $u_2$ from distinct blocks $B_1$ and $B_2$ of $G$
such that both $u_1$ and $u_2$ are not cut-vertices of $G$.
Then $u_1u_2\notin E(G)$ and $G-\{u_1,u_2\}$ is connected.
Now consider the case that $G$ is 2-connected
but not $3$-connected.
Since $G$ is not $C_n$, there exists a vertex $w$ such that
$d(w)\geq 3$.
If $d(w)=n-1$, then
$G-\{u_1,u_2\}$ is connected for any two non-adjacent
vertices $u_1$ and $u_2$ in $G$.
If $G-w$ is $2$-connected and $d(w)\leq n-2$,
then $G-\{w,u\}$ is connected
for any $u\in V-N_G(w)$.
If $G-w$ is not $2$-connected,
then $G-w$ contains two non-adjacent vertices $u_1,u_2$
such that $G-\{w,u_1,u_2\}$ is connected,
implying that $G-\{u_1,u_2\}$ is connected
as $d(w)\geq 3$.
(b).
Let $G_1, G_2,\ldots, G_c$ be the components of $G$ with
$|V(G_1)|\ge |V(G_j)|$ for all $j=1,2,\ldots,c$.
As $c\le n-1$, $|V(G_1)|\ge 2$.
Choose $u\in V(G_1)$ such that $G_1-u$ is connected. Then
$V(G_2)\cup \{u\}, V(G_1)-\{u\}, V(G_3),\ldots, V(G_c)$
is a partition of $V$ satisfying the condition in (b) for $i=c$.
Assume that (b) holds for $i=k$, where $c\le k<n-1$,
and $V_1,V_2,\ldots,V_k$ is a partition of $V$
satisfying the condition in (a).
Then $G[V_1]$ has an isolated vertex $u$
and $G[V'_1]$ is connected, where $V'_1=V_1-\{u\}$.
Since $k\le n-2$, either $|V'_1|\ge 2$
or $|V_j|\ge 2$ for some $j\ge 2$.
If $|V'_1|\ge 2$, then $V'_1$ has a partition $V'_{1,1}, V'_{1,2}$
such that both $G[V'_{1,1}]$ and $G[V'_{1,2}]$ are connected,
implying that
$V'_{1,1}\cup \{u\}, V'_{1,2}, V_2, V_3,\ldots, V_k$
is a partition of $V$ satisfying the condition in (b)
for $i=k+1$.
Similarly,
if $|V_j|\ge 2$ for some $j\ge 2$ (say $j=2$),
then $V_2$ has a partition $V_{2,1}, V_{2,2}$
such that both $G[V_{2,1}]$ and $G[V_{2,2}]$ are connected,
implying that $V_{1}, V_{2,1},V_{2,2}, V_3,\ldots, V_k$
is a partition of $V$ satisfying the condition in (b)
for $i=k+1$.
\end{proof}
For any graph $G=(V,E)$ of order $n$, write
\begin{align}\relabel{new-coe}
(-1)^n \left[(x-n+1)\sum_{u\in V(G)}P(G-u, x)-nP(G, x)\right]=\sum_{i=1}^{n}(-1)^i d_ix^i.
\end{align}
By comparing coefficients, it can be shown that
\begin{align}\relabel{ci-exp}
d_i=\sum_{u\in V(G)}\left
[a_{i-1}(G-u)+(n-1)a_i(G-u)\right ]-na_{i}(G),
\quad \forall i=1, 2, \ldots, n.
\end{align}
It is obvious that when $G$ is the
complete graph $K_n$, the left-hand side of
(\ref{new-coe}) vanishes and thus
$d_i=0$ for all $i=1,2,\ldots,n$.
Now we consider the case that $G$ is not complete.
\begin{prop}\relabel{pos-d}
Let $G=(V,E)$ be a non-complete graph of order $n$
and component number $c$.
Then, for any $i=1,2,\ldots,n$, $d_i\ge 0$
and equality holds
if and only if one of the following cases happens:
\begin{enumerate}
\renewcommand{\rm (\alph{enumi})}{\rm (\alph{enumi})}
\item $i=n$;
\item $1\le i\le c-2$;
\item $i=c-1$ and $G$ does not have isolated vertices;
\item $i=c=1$ and $G$ is $C_n$.
\end{enumerate}
\end{prop}
\begin{proof}
We first
show that $d_i=0$ in any one of the four cases above.
By (\ref{ci-exp}),
$d_n=\sum_{u\in V}\left[1+(n-1)\cdot 0\right]-n\cdot1=0$.
It is known that for $1\le i\le n$,
$a_i(G)=0$ if and only if $i<c$ (see~\cite{DKT2005,Read1968,RT1988}).
Similarly, $a_i(G-u)=0$ for all $i$ with $1\le i<c-1$
and all $u\in V$,
and $a_{c-1}(G-u)=0$ if $u$ is not an isolated
vertex of $G$.
By (\ref{ci-exp}), $d_i=0$ for all $i$ with
$1\le i\leq c-2$,
and $d_{c-1}=0$ when $G$ does not have isolated vertices.
If $G$ is $C_n$,
then $a_1(G)=n-1$, $a_0(G-u)=0$ and $a_1(G-u)=1$ for each
$u\in V$, implying that $d_1=0$ by (\ref{ci-exp}).
In the following, we will show that
$d_i>0$ when $i$ does not belong to any one
of the four cases.
If $G$ has isolated vertices, then
$a_{c-1}(G-u)>0$ for any isolated vertex $u$ of $G$ and
\begin{align}\label{isolated}
\sum_{u\in V}a_{c-1}(G-u)=
\sum_{u\in V\atop
u \text{ isolated}}a_{c-1}(G-u)>0.
\end{align}
As $a_{c-1}(G)=0$, by (\ref{ci-exp}), we have $d_{c-1}>0$
in this case.
Now it remains to show that
$d_i>0$ holds for all $i$ with $c\le i\le n-1$,
except when $i=c=1$ and $G$ is $C_n$.
For any $v\in V$, let
$\mathcal{OP}'_{i,v}(V)$ be the set of
ordered partitions $(V_1,\ldots,V_i)\in \mathcal{OP}_{i,v}(V)$
with $V_1=\{v\}$.
As $\alpha(G[V_1],v)=1$, for any $i$ with
$c\leq i\leq n$,
by Theorem~\ref{interpre2},
\begin{align}\relabel{G-u-i-1}
a_{i-1}(G-v)
=\sum_{(V_1,\ldots, V_i)\in \mathcal{OP}'_{i,v}(V)}
\alpha(G[V_1],v) \prod_{j=2}^{i} \alpha(G[V_j],m_j),
\end{align}
where $m_j$ is the minimum number in $V_j$ for all
$j=2,\ldots,i$.
Let $s$ and $v$ be distinct members in $V$.
For any $V_1\subseteq V-\{s\}$ with $v\in V_1$,
let $\alpha(G[V_1\cup \{s\}],v,s)$ be the
number of those acyclic orientations of $G[V_1\cup \{s\}]$
with $v$ as the unique source and $s$ as one sink.
Then $\alpha(G[V_1\cup \{s\}],v,s)\le
\alpha(G[V_1],v)$ holds,
where the inequality is strict if and only if
$G[V_1]$ is connected but
$G[V_1\cup \{s\}]$ is not.
Observe that
\begin{align}
a_i(G-s)
&=\sum_{(V_1,\ldots,V_i)\in \mathcal{OP}_{i,v}(V-\{s\})}
\alpha(G[V_1],v) \prod_{j=2}^{i} \alpha(G[V_j],m_j)
\nonumber \\
&\ge \sum_{(V_1,\ldots,V_i)\in \mathcal{OP}_{i,v}(V-\{s\})}
\alpha(G[V_1\cup \{s\}],v,s) \prod_{j=2}^{i} \alpha(G[V_j],m_j)
\relabel{G-u-i-0} \\
&=\sum_{(V_1',\ldots,V_i')\in \mathcal{OP}_{i,v,s}(V)}
\alpha(G[V_1'],v,s) \prod_{j=2}^{i} \alpha(G[V_j'],m_j),
\relabel{G-u-i}
\end{align}
where $\mathcal{OP}_{i,v,s}(V)$ is the set of ordered partitions
$(V_1',\ldots,V_i')\in \mathcal{OP}_{i,v}(V)$ with
$s,v\in V_1'$.
By the explanation above,
inequality (\ref{G-u-i-0}) is strict
whenever $V-\{s\}$ has a partition $V_1,V_2,\ldots,V_i$
with $v\in V_1$
such that each $G[V_j]$ is connected
for all $j=1,2,\ldots,i$ but
$G[V_1\cup \{s\}]$ is not connected.
By (\ref{interpre-n}), we have
\begin{eqnarray}
n a_i(G)
&=&\sum_{v\in V}
\sum_{(V_1,\ldots,V_i)\in \mathcal{OP}_{i,v}(V)}
\alpha(G[V_1],v) \prod_{j=2}^{i} \alpha(G[V_j],m_j)\nonumber \\
&=&\sum_{v\in V}
\sum_{(V_1,\ldots,V_i)\in \mathcal{OP}'_{i,v}(V)}
\alpha(G[V_1],v) \prod_{j=2}^{i} \alpha(G[V_j],m_j)
\nonumber \\
& &+\sum_{v\in V}
\sum_{(V_1,\ldots,V_i)\in
\mathcal{OP}_{i,v}(V)-\mathcal{OP}'_{i,v}(V)}
\alpha(G[V_1],v) \prod_{j=2}^{i} \alpha(G[V_j],m_j).
\relabel{ci-po-1}
\end{eqnarray}
By (\ref{G-u-i-1}),
\begin{eqnarray}
\sum_{v\in V}
\sum_{(V_1,\ldots,V_i)\in \mathcal{OP}'_{i,v}(V)}
\alpha(G[V_1],v) \prod_{j=2}^{i} \alpha(G[V_j],m_j)
=
\sum_{v\in V} a_{i-1}(G-v)
\relabel{ci-po-2},
\end{eqnarray}
and by (\ref{G-u-i}),
\begin{eqnarray}
& &\sum_{v\in V}
\sum_{(V_1,\ldots,V_i)\in \mathcal{OP}_{i,v}(V)-\mathcal{OP}'_{i,v}(V)}
\alpha(G[V_1],v) \prod_{j=2}^{i}
\alpha(G[V_j],m_j)\nonumber \\
&\le &
\sum_{v\in V}\sum_{s\in V-\{v\}}
\sum_{(V_1,\ldots,V_i)\in \mathcal{OP}_{i,v,s}(V)}
\alpha(G[V_1],v,s) \prod_{j=2}^{i} \alpha(G[V_j],m_j)
\relabel{ci-po-0} \\
&\leq &
\sum_{v\in V}\sum_{s\in V-\{v\}}
a_i(G-s)\relabel{ci-po-00} \\
&=&
(n-1)\sum_{v\in V}a_i(G-v),
\relabel{ci-po}
\end{eqnarray}
where inequality (\ref{ci-po-0}) is strict
if there exists $(V_1,\ldots,V_i)\in \mathcal{OP}_{i,v}(V)$
for some $v\in V$
such that $G[V_j]$ is connected for all $j=1,\ldots,i$
and $G[V_1]$ has acyclic orientations
with $v$ as the unique source
but with at least two sinks,
and by (\ref{G-u-i-0}) and (\ref{G-u-i}),
inequality (\ref{ci-po-00})
is strict if $V$ can be partitioned into $V_1,\ldots, V_i$
such that $G[V_j]$ is connected for all $j=2,\ldots,i$
but $G[V_1]$ has exactly two components,
one of which is an isolated vertex in $G[V_1]$.
As $G$ is not complete, by Lemma~\ref{le5-1}
and the above explanation,
the inequality of (\ref{ci-po})
is strict for all $i$ with $c\le i\le n-1$,
except when $i=c=1$ and $G$ is $C_n$.
Then, by (\ref{ci-po-1}), (\ref{ci-po-2}) and (\ref{ci-po}),
we conclude that
\begin{align}
d_i=\sum_{v\in V}\left [a_{i-1}(G-u)+(n-1)a_i(G-u)\right ]
-na_i(G)>0,\quad \forall c\le i\le n-1,
\end{align}
except that $i=c=1$ and $G$ is $C_n$.
Hence the proof is complete.
\end{proof}
Now everything is ready for proving Theorems~\ref{average-th} and \ref{compare-K}.
\noindent
{\it Proof of Theorem~\ref{average-th}}:
Let $G$ be a non-complete graph of order $n$.
Recall (\ref{new-coe}) that
\begin{align}\label{proof-th3}
(-1)^n \left[(x-n+1)\sum_{u\in V(G)}P(G-u, x)-nP(G, x)\right]=\sum_{i=1}^{n}(-1)^i d_ix^i.
\end{align}
By Proposition \ref{pos-d}, we know that $d_i\geq 0$ for
all $i$ with $1\leq i\leq n$ and $d_{n-1}>0$.
Thus
$\sum_{i=1}^{n}(-1)^i d_ix^i>0$ holds
for all $x<0$, which completes the proof of
Theorem~\ref{average-th}.
\qed
\begin{prop}\relabel{pro5-3}
For any non-complete graph $G$, $\xi(G,x)>0$ holds
for all $x<0$.
\end{prop}
\proof
We will prove this result by induction
on the order $n$ of $G$.
When $n=2$, the empty graph $N_2$ of order $2$
is the only non-complete graph of order $2$.
As $P(N_2,x)=x^2$, by (\ref{xi}),
we have
\begin{align}\label{proof-pro5}
\xi(N_2,x)=(-1)^2x^2\left ( \frac 1x +\frac 1{x-1}\right )
+(-1)^32x=\frac{x}{x-1}>0
\end{align}
for all $x<0$.
Assume that this result holds for any non-complete graph
$G$ of order less than $n$, where $n\ge 3$.
Now let $G$ be any non-complete graph of order $n$.
\noindent {\bf Case 1}:
$G$ contains an isolated vertex $u$.
By the inductive assumption,
$\xi(G-u,x)\ge 0$ holds for all $x<0$,
where equality holds when $G-u$ is a complete graph.
By Lemma~\ref{ud0}, $\xi(G,x)>0$ holds for all $x<0$.
\noindent {\bf Case 2}:
$G$ has no isolated vertex.
By Theorem~\ref{average-th},
(\ref{right-2}) holds for all $x<0$.
Thus, for any $x<0$, there exists some $u\in V(G)$ such that
$(-1)^n (x-n+1)P(G-u,x)+(-1)^{n+1}P(G,x)>0$
holds. Then, by Lemma~\ref{rec2} and by the inductive assumption,
$\xi(G,x)>0$ holds for any $x<0$.
Hence the result holds.
\endproof
\noindent
{\it Proof of Theorem~\ref{compare-K}}:
It follows directly from
Propositions~\ref{compare-K-eq} and~\ref{pro5-3}.
\qed
\section{Remarks and problems\relabel{further}}
First we give some remarks here.
\begin{enumerate}
\renewcommand{\rm (\alph{enumi})}{\rm (\alph{enumi})}
\item Theorem~\ref{compare-K} implies that
for any non-complete graph $G$ of order $n$,
$\frac{P(G, x)}{P(K_n, x)}$
is strictly decreasing when $x<0$.
\item Let $G$ be a non-complete graph of order $n$
and $P(G, x)=\sum\limits_{i=1}^n (-1)^{n-i}a_i x^i$.
Then $\epsilon(G)<\epsilon(K_n)$ implies that
\begin{align}\label{average0}
\frac{a_1+2a_2+\cdots+na_n}{a_1+a_2+\cdots+a_n}> 1+\frac{1}{2}+\cdots+\frac{1}{n}.
\end{align}
\item When $x=-1$, Theorem~\ref{average-th}
implies that for any graph $G$ of order $n$,
\begin{align}
(-1)^{n-1}\sum_{u\in V}P(G-u, -1)\ge (-1)^nP(G, -1),
\relabel{average1}
\end{align}
where the inequality holds if and only if $G$
is complete.
By Stanley's interpretation for $(-1)^nP(G,-1)$
in~\cite{Stanley1973},
the inequality above implies that
for any graph $G=(V,E)$,
the number of acyclic
orientations of $G$ is at most
the total number of acyclic
orientations of $G-u$ for all $u\in V$,
where the equality holds if and only if $G$ is complete.
\end{enumerate}
Now we raise some problems for further study.
It is clear that for any graph $G$ of order $n$,
\begin{align}\label{con2-ex}
\frac{d}{dx}\left (\ln[(-1)^nP(G,x)]\right )
=\frac{P'(G,x)}{P(G,x)}<0
\end{align}
holds for all $x<0$.
We surmise that this property holds for higher derivatives of
the function $\ln[(-1)^nP(G,x)]$ in the interval $(-\infty,0)$.
\begin{conjecture}\relabel{con6-1}
Let $G$ be a graph of order $n$.
Then $\frac{d^k}{dx^k}\left (\ln[(-1)^nP(G,x)]\right )<0$
holds for all $k\ge 2$ and
$x\in (-\infty,0)$.
\end{conjecture}
Observe that $\epsilon(G,x)=\frac{d}{dx}\left (\ln[(-1)^nP(G,x)]\right )$.
We believe that
Theorems~\ref{compare-Q} and~\ref{compare-K}
can be extended to higher derivatives of
the function $\ln[(-1)^nP(G,x)]$.
\begin{conjecture}\relabel{con6-2}
Let $G$ be any non-complete graph
of order $n$
and $Q$ be any chordal and proper spanning subgraph $Q$ of $G$.
Then
\begin{align}\label{con3-ex}
\frac{d^k}{dx^k}\left (\ln[(-1)^nP(Q,x)]\right )
< \frac{d^k}{dx^k}\left (\ln[(-1)^nP(G,x)]\right )
<\frac{d^k}{dx^k}\left (\ln[(-1)^nP(K_n,x)]\right )
\end{align}
holds for any integer $k\geq 2$ and all $x<0$.
\end{conjecture}
It is not difficult to show that Conjecture~\ref{con6-1} holds
for $G\cong K_n$.
Thus the second inequality of Conjecture~\ref{con6-2}
implies Conjecture~\ref{con6-1}.
It is natural to extend the second part
of Conjecture~\ref{mainconj} (i.e., $\epsilon(G)<\epsilon(K_n)$
for any non-complete graph $G$ of order $n$)
to the inequality
$\epsilon(G)\le \epsilon(G')$ for any graph $G'$
which contains $G$ as a subgraph.
However, this inequality is not always true.
Let $G_n$ denote the graph obtained from the complete bipartite graph
$K_{2,n}$ by adding a new edge joining
the two vertices in the partite set of size $2$.
Lundow and Markstr\"{o}m \cite{LM2006}
stated that
$\epsilon(K_{2,n})>\epsilon(G_n)$ holds for all $n\ge 3$.
In spite of this,
we believe that for any non-complete graph $G$,
we can add a new edge to $G$ to obtain a graph $G'$
with the property that
$\epsilon(G)<\epsilon(G')$, as stated below.
\begin{conjecture}\relabel{con6-4}
For any non-complete graph $G$, there exist non-adjacent vertices $u$ and $v$
in $G$ such that $\epsilon(G)<\epsilon(G+uv)$.
\end{conjecture}
Obviously, Conjecture~\ref{con6-4} implies
$\epsilon(G)<\epsilon(K_n)$ for any non-complete graph $G$
of order $n$ (i.e., Theorem~\ref{compare-K}).
Conjecture~\ref{con6-4} is similar to but may be not
equivalent to the following conjecture
due to Lundow and Markstr\"{o}m \cite{LM2006}.
\begin{conjecture}[\cite{LM2006}]\relabel{con6-3}
For any $2$-connected graph $G$,
there exists an edge $e$ in $G$ such that
$\epsilon(G-e)<\epsilon(G)$.
\end{conjecture}
(F. Dong and E. Tay) Mathematics and Mathematics Education,
National Institute of Education,
Nanyang Technological University,
Singapore.
Email (Tay): [email protected].
(J. Ge) School of Mathematical Sciences,
Sichuan Normal University, Chengdu, P. R. China.
Email: [email protected].
(H. Gong)
Department of Mathematics,
Shaoxing University, Shaoxing,
P. R. China.
Email: [email protected].
(B. Ning)
College of Computer Science, Nankai University,
Tianjin 300071, P.R. China.
Email: [email protected].
(Z. Ouyang)
Department of Mathematics,
Hunan First Normal University,
Changsha, P. R. China. Email:
[email protected].
\end{document} | math | 43,762 |
\begin{document}
\title[Propagation rules]{Propagation rules for $(u,m,{\bf e},s)$-nets and $(u,{\bf e},s)$-sequences}
\author{PETER KRITZER}
\address{Peter Kritzer,
Department of Financial Mathematics,
Johannes Kepler University Linz,
Altenbergerstr. 69,
A-4040 Linz,
AUSTRIA}
{\bf e}mail{[email protected] (P. Kritzer)}
\author{HARALD NIEDERREITER}
\address{
Harald Niederreiter,
Johann Radon Institute for Computational and Applied Mathematics,
Austrian Academy of Sciences,
Altenbergerstr. 69,
A-4040 Linz,
AUSTRIA,
and
Department of Mathematics, University of Salzburg, Hellbrunnerstr. 34,
A-5020 Salzburg, AUSTRIA
}
{\bf e}mail{[email protected] (H. Niederreiter)}
\thanks{P. Kritzer gratefully acknowledges the support of the Austrian Science Fund (FWF), Project P23389-N18}
\subjclass[2010]{11K31, 11K38, 65C05}
\date{\today}
\keywords{low-discrepancy point sets, digital construction methods, propagation rule, duality theory, global function field}
\begin{abstract}
The classes of $(u,m,{\bf e},s)$-nets and $(u,{\bf e},s)$-sequences were recently introduced by Tezuka, and in a slightly more restrictive
form by Hofer and Niederreiter. We study propagation rules for these point sets, which state how one can obtain $(u,m,{\bf e},s)$-nets
and $(u,{\bf e},s)$-sequences with new parameter configurations from existing ones. In this way,
we show generalizations and extensions of several well-known construction
methods that have previously been shown for $(t,m,s)$-nets and $(t,s)$-sequences. We also develop a duality theory for digital
$(u,m,{\bf e},s)$-nets and present a new construction of such nets based on global function fields.
{\bf e}nd{abstract}
\maketitle
\section{Introduction}\label{secintro}
Finite or infinite sequences of points with good equidistribution properties are frequently studied in number-theoretic questions,
and they also play an important role as the node sets of quadrature rules in numerical integration (see, e.g., \cite{DP} and \cite{N92}).
When studying the question of how to evenly distribute (a large number of) points in a certain domain, one frequently restricts oneself
to considering the $s$-dimensional unit cube $[0,1]^s$.
If we would like to distribute
a large number of points in the unit cube, a very popular and powerful method is to use $(t,m,s)$-nets
(for the case of finitely many points) and $(t,s)$-sequences (for the case of
infinitely many points), or, more generally, the recently introduced $(u,m,{\bf e},s)$-nets and $(u,{\bf e},s)$-sequences.
The former classes of point sets and sequences were introduced
in their nowadays most common form by Niederreiter in~\cite{N87}, see also~\cite{N92} for detailed information, while the latter
were introduced by Tezuka in~\cite{T13} and studied in a slightly
modified form by Hofer and Niederreiter~\cite{HN13} and Niederreiter and Yeo~\cite{NY}. The underlying idea of these nets and sequences is to guarantee fair
distribution of the points for certain subintervals
of the half-open unit cube $[0,1)^s$. To be more precise, let $s\ge 1$ be a given dimension and let $b\ge 2$ be an integer
(which we usually refer to as the base
in the following). An interval $J\subseteq [0,1)^s$ is
called an {\bf e}mph{elementary interval in base} $b$ if it is of the form
$$J=\prod_{i=1}^s \left[\frac{a_i}{b^{d_i}},\frac{a_i +1}{b^{d_i}}\right),$$
with integers $d_i\ge 0$ and $0\le a_i < b^{d_i}$ for $1\le i\le s$. These intervals play a crucial role in the subsequent
definition of a $(u,m,{\bf e},s)$-net, which we state below.
Here and in the following, we denote by ${\mathbb{N}}N$ the set of positive integers and by $\lambda_s$ the $s$-dimensional Lebesgue measure.
\begin{definition}\label{defumes} {\rm
Let $b\ge 2$, $s\ge 1$, and $0\le u\le m$ be integers and let ${\bf e}=(e_1,\ldots,e_s)\in{\mathbb{N}}N^s$. A point set $\mathcal{P}$ of $b^m$
points in $[0,1)^s$ is called a $(u,m,{\bf e},s)$-{\bf e}mph{net in base} $b$ if every elementary interval
$J\subseteq [0,1)^s$ in base $b$ of volume $\lambda_s (J)\ge b^{u-m}$ and of the form
$$J=\prod_{i=1}^s \left[\frac{a_i}{b^{d_i}},\frac{a_i +1}{b^{d_i}}\right),$$
with integers $d_i\ge 0$, $0\le a_i < b^{d_i}$, and $e_i |d_i$ for $1\le i\le s$, contains exactly $b^m \lambda_s (J)$ points of $\mathcal{P}$.}
{\bf e}nd{definition}
Note that the points of a $(u,m,{\bf e},s)$-net are particularly evenly distributed if $u$ is small. On the other hand, also the choice of $e_1,\ldots,e_s$
plays an important role, as larger values of the $e_i$ in general
mean less restrictions on the distribution of the points in the unit cube.
Definition~\ref{defumes} is the definition of a $(u,m,{\bf e},s)$-net in base $b$ in the sense of~\cite{HN13}. Previously, Tezuka~\cite{T13} introduced
a slightly more general definition, where the conditions
on the number of points in the elementary intervals only need to hold for those elementary intervals $J$ in base $b$ with $\lambda_s (J) = b^{u-m}$. The
narrower definition used in~\cite{HN13} guarantees, as stated in that
paper, that any $(u,m,{\bf e},s)$-net in base $b$ is also a $(v,m,{\bf e},s)$-net in base $b$ for any integer $v$ with $u\le v\le m$. The latter property
is a very useful property when working with such point sets (see also \cite{HN13} for further details);
hence, whenever we speak of a $(u,m,{\bf e},s)$-net here, we mean a $(u,m,{\bf e},s)$-net in the narrower sense of Definition~\ref{defumes}.
The definition of a $(u,{\bf e},s)$-sequence is based on $(u,m,{\bf e},s)$-nets and is given next. As usual, we write
$[{\bf x}]_{b,m}$ for the coordinatewise $m$-digit truncation in base $b$ of ${\bf x} \in [0,1]^s$.
\begin{definition}\label{defues} {\rm
Let $b\ge 2$, $s\ge 1$, and $u\ge 0$ be integers and let ${\bf e}\in{\mathbb{N}}N^s$. A sequence ${\bf x}_0,{\bf x}_1,\ldots$ of points in $[0,1]^s$ is a
$(u,{\bf e},s)$-{\bf e}mph{sequence in base} $b$ if for all integers $k\ge 0$ and $m>u$ the
points $[{\bf x}_n]_{b,m}$ with $kb^m \le n < (k+1)b^m$ form a $(u,m,{\bf e},s)$-net in base $b$. }
{\bf e}nd{definition}
Again, the points of a $(u,{\bf e},s)$-sequence are evenly distributed if $u$ is small, but also in this case the choice of
${\bf e}$ has influence on the conditions on how the points are spread over the elementary intervals
in the unit cube.
If we choose ${\bf e}=(1,\ldots,1) \in {\mathbb{N}}^s$ in Definitions~\ref{defumes} and~\ref{defues}, then the definitions coincide with those of a classical
$(t,m,s)$-net or a classical $(t,s)$-sequence with $t=u$, respectively,
as they were introduced in~\cite{N87}. The reasons why the more general
$(u,m,{\bf e},s)$-nets and $(u,{\bf e},s)$-sequences were introduced are twofold. On the one hand, as pointed out by Tezuka in~\cite{T13},
one can derive better bounds on the discrepancy for at least some examples of $(t,m,s)$-nets
and $(t,s)$-sequences by viewing them as special cases of $(u,m,{\bf e},s)$-nets and $(u,{\bf e},s)$-sequences (see also the recent paper~\cite{FL13} for related results).
Furthermore, by viewing, e.g., $(t,s)$-sequences as $(u,{\bf e},s)$-sequences, it is possible to deal with special types of $(t,s)$-sequences in a very natural way.
For example, as pointed out in~\cite{T13}
(see also~\cite{HN13}), a generalized Niederreiter sequence over the finite field
${\mathbb{F}}ield_q$ ($q$ a prime power) is a $(u,{\bf e},s)$-sequence in base $q$ with $u=0$, where ${\bf e}=(e_1,\ldots,e_s)\in{\mathbb{N}}N^s$ is such that
$e_i$, $1\le i\le s$, exactly corresponds to the degree of the $i$th base polynomial over ${\mathbb{F}}ield_q$ used
in the construction of the sequence. The interested reader is referred to~\cite{HN13} and~\cite{T13} for further information.
Most of the constructions of $(u,m,{\bf e},s)$-nets and $(u,{\bf e},s)$-sequences are based on the digital construction method
introduced by Niederreiter in~\cite{N87}. Let us first outline the digital construction method
for $(u,m,{\bf e},s)$-nets. Let $s\ge 1$ be a given dimension, let $q$ be a prime power, and consider the finite field ${\mathbb{F}}_q$ with $q$ elements.
Furthermore, put $Z_q:=\{0,1,\ldots,q-1\} \subset {\mathbb{Z}}$. Choose bijections ${\bf e}ta_r:Z_q\rightarrow{\mathbb{F}}ield_q$ for
all integers $0\le r\le m-1$ and bijections $\kappa_{i,j}:{\mathbb{F}}ield_q\rightarrow Z_q$ for $1\le i\le s$ and $1\le j\le m$. Furthermore, we choose
$m\times m$ {\bf e}mph{generating matrices} $C_1,\ldots,C_s$ over ${\mathbb{F}}ield_q$. For $n\in\{0,1,\ldots,q^m-1\}$, let
$n=\sum_{v=0}^{m-1}n_v q^v$ be the base $q$ expansion of $n$. For $1\le i\le s$, we compute the matrix-vector product
$$ C_i \cdot ({\bf e}ta_0 (n_0),{\bf e}ta_1 (n_1),\ldots,{\bf e}ta_{m-1}(n_{m-1}))^\top=:\left(y_{n,1}^{(i)},y_{n,2}^{(i)},\ldots,y_{n,m}^{(i)}\right)^\top$$ and then we put
$$x_n^{(i)}:=\sum_{j=1}^m \kappa_{i,j} \left(y_{n,j}^{(i)}\right) q^{-j}.$$
Finally, we put ${\bf x}_n:=(x_n^{(1)},\ldots,x_n^{(s)}) \in [0,1)^s$ for $0\le n\le q^m-1$. Then the point set consisting of the points ${\bf x}_0,{\bf x}_1,\ldots,{\bf x}_{q^m-1}$ is
called a {\bf e}mph{digital net over} ${\mathbb{F}}ield_q$.
Regarding the parameters of a digital net, we recall the following proposition from~\cite{HN13}.
\begin{proposition}\label{propdignet}
The matrices $C_1,\ldots,C_s\in{\mathbb{F}}ield_q^{m\times m}$ generate a digital $(u,m,{\bf e},s)$-net over ${\mathbb{F}}_q$ if and only if, for any nonnegative
integers $d_1,\ldots,d_s$ with $e_i | d_i$ for $1\le i\le s$ and
$d_1+\cdots + d_s\le m-u$, the collection of the $d_1+ \cdots +d_s$ vectors obtained by taking the first $d_i$ rows of
$C_i$ for $1\le i\le s$ is linearly independent over ${\mathbb{F}}ield_q$.
{\bf e}nd{proposition}
For the digital construction of a $(u,{\bf e},s)$-sequence, we choose bijections ${\bf e}ta_r:Z_q\rightarrow{\mathbb{F}}ield_q$ for
all integers $r\ge 0$, satisfying ${\bf e}ta_r(0)=0$ for all sufficiently large $r$, and bijections $\kappa_{i,j}:{\mathbb{F}}ield_q\rightarrow Z_q$ for $1\le i\le s$ and $j\ge 1$.
Furthermore, we choose $\infty \times \infty$ {\bf e}mph{generating matrices} $C_1,\ldots,C_s$ over ${\mathbb{F}}_q$ (by an $\infty \times \infty$ matrix over ${\mathbb{F}}_q$
we mean a matrix over ${\mathbb{F}}_q$ with denumerably many rows and columns).
For an integer $n\ge 0$, let $n=\sum_{v=0}^{\infty}n_v q^v$ be the base $q$ expansion of $n$.
For $1\le i\le s$, we compute the matrix-vector product
$$ C_i \cdot ({\bf e}ta_0 (n_0),{\bf e}ta_1 (n_1),\ldots)^\top=:\left(y_{n,1}^{(i)},y_{n,2}^{(i)},\ldots\right)^\top$$ and then we put
$$x_n^{(i)}:=\sum_{j=1}^\infty \kappa_{i,j} \left(y_{n,j}^{(i)}\right) q^{-j}.$$
Finally, we put ${\bf x}_n:=(x_n^{(1)},\ldots,x_n^{(s)}) \in [0,1]^s$ for $n=0,1,\ldots$ . Then the \se ${\bf x}_0,{\bf x}_1,\ldots$ is called a {\bf e}mph{digital
sequence over} ${\mathbb{F}}ield_q$. As for digital nets,
the properties of the matrices $C_1,\ldots,C_s$ are intimately related to the parameters of a digital sequence; the following proposition is also
due to~\cite{HN13}.
\begin{proposition}\label{propdigseq}
The $\infty \times \infty$ matrices $C_1,\ldots,C_s$ over ${\mathbb{F}}_q$ generate a digital $(u,{\bf e},s)$-sequence over ${\mathbb{F}}_q$ if and only if, for every integer
$m>u$, the left upper $m\times m$ submatrices $C_1^{(m)},\ldots,C_s^{(m)}$ of the
$C_i$ generate a digital $(u,m,{\bf e},s)$-net over ${\mathbb{F}}_q$.
{\bf e}nd{proposition}
The problem of how to find $(u,m,{\bf e},s)$-nets with good parameter configurations is non-trivial. One way to tackle this question is to
consider so-called propagation rules.
A propagation rule for (digital) nets is a rule that, from one (digital) net or several (digital) nets, produces a (digital) net with new parameters,
and similarly for (digital) sequences. The theory of propagation rules for classical $(t,m,s)$-nets
and $(t,s)$-sequences has attracted much interest in the past and there are many results to be found in, e.g.,
\cite{N05}, \cite{NP01}, \cite{NX98}, \cite{NX02} and related papers. We also refer to
the database MinT (\cite{MinT}), where information on all relevant propagation rules for $(t,m,s)$-nets and $(t,s)$-sequences
and the resulting parameter configurations can be found. We also remark that there
exist propagation rules for so-called higher-order nets and sequences as introduced by Dick (see, e.g., \cite{BDP11}, \cite{DK10}, and \cite{DP}).
In the present paper, we study to which extent it is possible to find propagation rules for the new concepts of $(u,m,{\bf e},s)$-nets and
$(u,{\bf e},s)$-sequences. We will first present propagation rules for (digital) $(u,m,{\bf e},s)$-nets in Section~\ref{secpr}. Then in Section~\ref{secbc},
we consider propagation rules for $(u,{\bf e},s)$-sequences that employ a change of the base. In Section~\ref{secduality},
we develop a duality theory for digital $(u,m,{\bf e},s)$-nets, and we show
in Section~\ref{secad} how this theory can be used for finding new digital $(u,m,{\bf e},s)$-nets.
\section{Propagation rules for (digital) $(u,m,{\bf e},s)$-nets} \label{secpr}
In this section, we derive propagation rules for $(u,m,{\bf e},s)$-nets. We first present results that are valid for arbitrary $(u,m,{\bf e},s)$-nets,
and then we move on to propagation rules that hold for digital nets.
Our first result generalizes the propagation rule that is called Propagation Rule~1 in~\cite{N05}. We write ${\mathbb{N}}_0$ for the set of nonnegative integers.
\begin{proposition} \label{prpr}
Let $b \ge 2$, $m \ge 0$, $s \ge 1$, and $u\ge 0$ be integers and let ${\bf e} =(e_1,\ldots,e_s) \in {\mathbb{N}}^s$. If a $(u,m,{\bf e},s)$-net in base $b$ is given, then
for every integer $k$ with $u \le k \le m$ such that $m-k$ is a linear combination of $e_1,\ldots,e_s$ with coefficients from ${\mathbb{N}}_0$, we can construct
a $(u,k,{\bf e},s)$-net in base $b$.
{\bf e}nd{proposition}
\begin{proof}
Let the point set $\mathcal{P}$ be a $(u,m,{\bf e},s)$-net in base $b$. By assumption, we can write $m-k=\sum_{i=1}^s f_ie_i$ with $f_i \in {\mathbb{N}}_0$ for $1 \le i \le s$.
Consider the interval
$$
E=\prod_{i=1}^s [0,b^{-f_ie_i}) \subseteq [0,1]^s.
$$
Then $\lambda_s(E)=b^{k-m} \ge b^{u-m}$, and so the definition of a $(u,m,{\bf e},s)$-net in base $b$ implies that $E$ contains exactly $b^k$ points of the point set
$\mathcal{P}$. Let these $b^k$ points be
$$
{\bf x}_n=(x_n^{(1)},\ldots,x_n^{(s)}) \in E \qquad \mbox{for } n=1,\ldots,b^k.
$$
Now we define the points
$$
{\bf y}_n=\left(b^{f_1e_1}x_n^{(1)},\ldots,b^{f_se_s}x_n^{(s)} \right) \in [0,1)^s \qquad \mbox{for } n=1,\ldots,b^k.
$$
We will show that the points ${\bf y}_n$, $n=1,\ldots,b^k$, form a $(u,k,{\bf e},s)$-net in base $b$. Let $J \subseteq [0,1]^s$ be an interval of the form
$$
J=\prod_{i=1}^s [a_ib^{-d_i},(a_i+1)b^{-d_i})
$$
with $a_i,d_i \in {\mathbb{N}}_0$, $a_i < b^{d_i}$, and $e_i|d_i$ for $1 \le i \le s$, and with $\lambda_s(J) \ge b^{u-k}$. Then ${\bf y}_n \in J$ if and only if
$$
{\bf x}_n \in J^{\prime} := \prod_{i=1}^s [a_ib^{-f_ie_i-d_i},(a_i+1)b^{-f_ie_i-d_i}) \subseteq E.
$$
Note that $\lambda_s(J^{\prime})=b^{k-m} \lambda_s(J) \ge b^{u-m}$. Again by the definition of a $(u,m,{\bf e},s)$-net in base $b$, the number of points ${\bf x}_n \in J^{\prime}$
is $b^m \lambda_s(J^{\prime})=b^k \lambda_s(J)$, and so the number of points ${\bf y}_n \in J$ is $b^k \lambda_s(J)$, which is the desired property.
{\bf e}nd{proof}
The following proposition generalizes a well-known result on $(t,m,s)$-nets and $(t,s)$-sequences from~\cite{N87} (see also \cite[Lemma~4.22]{N92}).
\begin{proposition} \label{prsn}
Let $b \ge 2$, $s \ge 1$, and $u \ge 0$ be integers and let ${\bf e} =(e_1,\ldots,e_s) \in {\mathbb{N}}^s$. If a $(u,{\bf e},s)$-sequence in base $b$ is given, then for every
integer $m \ge u$ we can construct a $(u,m,{\bf e}^{\prime},s+1)$-net in base $b$, where ${\bf e}^{\prime}=(1,e_1,\ldots,e_s) \in {\mathbb{N}}^{s+1}$.
{\bf e}nd{proposition}
\begin{proof}
Let ${\bf x}_0,{\bf x}_1,\ldots$ be a $(u,{\bf e},s)$-\se in base $b$. We fix an integer $m \ge u$ and define the points
$$
{\bf y}_n=(nb^{-m},[{\bf x}_n]_{b,m}) \in [0,1)^{s+1} \qquad \mbox{for } n=0,1,\ldots,b^m-1.
$$
We claim that these points form a $(u,m,{\bf e}^{\prime},s+1)$-net in base $b$. Consider an interval $J^{\prime} \subseteq [0,1]^{s+1}$ of the form
$$
J^{\prime}=\prod_{i=1}^{s+1} [a_ib^{-d_i},(a_i+1)b^{-d_i})
$$
with $a_i,d_i \in {\mathbb{N}}_0$ and $a_i < b^{d_i}$ for $1 \le i \le s+1$, $e_i|d_i$ for $2 \le i \le s+1$, and $\lambda_{s+1}(J^{\prime}) \ge b^{u-m}$, that is,
$\sum_{i=1}^{s+1} d_i \le m-u$. We have ${\bf y}_n \in J^{\prime}$ if and only if $a_1b^{m-d_1} \le n < (a_1+1)b^{m-d_1}$ and
$$
[{\bf x}_n]_{b,m} \in J := \prod_{i=2}^{s+1} [a_ib^{-d_i},(a_i+1)b^{-d_i}).
$$
Now $m-d_1 \ge u+\sum_{i=2}^{s+1} d_i \ge u$, and so the definition of a $(u,{\bf e},s)$-\se in base $b$ implies that the points $[{\bf x}_n]_{b,m}$ with $a_1b^{m-d_1}
\le n < (a_1+1)b^{m-d_1}$ form a $(u,m-d_1,{\bf e},s)$-net in base $b$. Since
$$
\lambda_s(J)=b^{-(d_2+ \cdots + d_{s+1})} \ge b^{u-m+d_1},
$$
it follows that the number of points ${\bf y}_n$, $0 \le n \le b^m-1$, contained in the interval $J^{\prime}$ is equal to $b^{m-d_1} \lambda_s(J)=b^m \lambda_{s+1}
(J^{\prime})$, and so we are done.
{\bf e}nd{proof}
Before we proceed to the formulation of propagation rules for digital $(u,m,{\bf e},s)$-nets, we give the following definition of a
$(d,m,{\bf e},s)$-system. For the corresponding definition for classical $(t,m,s)$-nets, see, e.g., \cite{NP01} and~\cite{NX98}.
\begin{definition} \label{defsystem} {\rm
Let $q$ be a prime power, let $m \ge 1$ and $s \ge 1$ be integers, let $d$ be an integer with $0\le d\le m$, and let
${\bf e}=(e_1,\ldots,e_s)\in {\mathbb{N}}^s$. The system
$$A=\left\{\boldsymbol{a}_j^{(i)}\in{\mathbb{F}}ield_q^m: 1\le i\le s, 1\le j\le m\right\}$$
of vectors is called a $(d,m,{\bf e},s)$-{\bf e}mph{system over} ${\mathbb{F}}ield_q$ if for any choice of
nonnegative integers $d_1,\ldots,d_s$ with $e_i |d_i$ for $1\le i\le s$ and $\sum_{i=1}^s d_i\le d$, the
vectors $\boldsymbol{a}_j^{(i)}$, $1\le j\le d_i$, $1\le i\le s$, are linearly independent over ${\mathbb{F}}ield_q$ (this property is
assumed to be trivially satisfied for $d=0$). }
{\bf e}nd{definition}
The next lemma is analogous to \cite[Lemma~3]{NX98}.
For an $m \times m$ matrix $C_i$ over ${\mathbb{F}}_q$ and $1 \le j \le m$, we write $\boldsymbol{c}_j^{(i)}$ for the $j$th
row vector of $C_i$.
\begin{lemma} \label{lemnetsystem}
Let $q$ be a prime power, let $m \ge 1$, $s \ge 1$, and $u \ge 0$ be integers, and let ${\bf e} \in {\mathbb{N}}^s$.
A digital net over ${\mathbb{F}}ield_q$ with $m \times m$ generating matrices $C_1,\ldots,C_s$ over ${\mathbb{F}}_q$
is a digital $(u,m,{\bf e},s)$-net over ${\mathbb{F}}_q$ if and only if the system $\{\boldsymbol{c}_j^{(i)} \in {\mathbb{F}}_q^m: 1 \le i \le s, 1 \le j \le m\}$
of row vectors of the matrices
$C_1,\ldots,C_s$ is an $(m-u,m,{\bf e},s)$-system over ${\mathbb{F}}ield_q$.
{\bf e}nd{lemma}
\begin{proof}
The result follows immediately from the definition of a $(d,m,{\bf e},s)$-system and Proposition~\ref{propdignet}.
{\bf e}nd{proof}
Lemma~\ref{lemnetsystem} enables us to show the following propagation rule, which is a generalization of \cite[Theorem~10]{NX98} (also
referred to as Direct Product Rule or Propagation Rule 4 in \cite{N05}).
\begin{theorem}\label{thmdirectproduct}
Let $q$ be a prime power, let $m_1,m_2 \ge 1$, $s_1,s_2 \ge 1$, and $u_1,u_2 \ge 0$ be integers, and let ${\bf e}= (e_1,\ldots,e_{s_1})\in {\mathbb{N}}^{s_1}$,
${\bf f}=(f_1,\ldots,f_{s_2})\in {\mathbb{N}}^{s_2}$. If a digital $(u_1,m_1,{\bf e},s_1)$-net over ${\mathbb{F}}_q$ and a digital
$(u_2,m_2,{\bf f},s_2)$-net over ${\mathbb{F}}ield_q$ are given,
then we can construct a digital $(u,m_1 + m_2,({\bf e},{\bf f}),s_1 + s_2)$-net over ${\mathbb{F}}ield_q$ with
$$u=\max\{m_1 + u_2, m_2 + u_1\}$$
and
$$({\bf e},{\bf f}) =(e_1,\ldots, e_{s_1}, f_1,\ldots, f_{s_2}) \in {\mathbb{N}}^{s_1+s_2}.$$
{\bf e}nd{theorem}
\begin{proof}
By Lemma~\ref{lemnetsystem} it suffices to show that a $(d_1,m_1,(e_1,\ldots,e_{s_1}),s_1)$-system over ${\mathbb{F}}_q$ and a
$(d_2,m_2,(f_1,\ldots,f_{s_2}),s_2)$-system over ${\mathbb{F}}ield_q$ yield a $(d,m_1 + m_2,(e_1,\ldots, e_{s_1},f_1,\ldots,f_{s_2}), s_1 + s_2)$-system
over ${\mathbb{F}}ield_q$ with $d=\min\{d_1,d_2\}$. Let
$$A=\left\{\boldsymbol{a}_j^{(i)}\in{\mathbb{F}}ield_q ^{m_1}: 1\le i\le s_1, 1\le j\le m_1\right\}$$
be a $(d_1,m_1,(e_1,\ldots,e_{s_1}),s_1)$-system
and
$$B=\left\{\boldsymbol{b}_j^{(i)}\in{\mathbb{F}}ield_q ^{m_2}: 1\le i\le s_2, 1\le j\le m_2\right\}$$
be a $(d_2,m_2,(f_1,\ldots,f_{s_2}),s_2)$-system over ${\mathbb{F}}ield_q$, where we assume, without loss of
generality, that $m_2\ge m_1$. Now we define the system
$$C=\left\{\boldsymbol{c}_j^{(i)}\in{\mathbb{F}}ield_q ^{m_1+m_2}: 1\le i\le s_1 + s_2, 1\le j\le m_1+m_2\right\}$$
with
$$\boldsymbol{c}_j^{(i)} :=\begin{cases}
(\boldsymbol{a}_j^{(i)},\boldsymbol{0}) & \mbox{for $1\le i\le s_1,\ 1\le j\le m_1$,}\\
(\boldsymbol{0},\boldsymbol{b}_j^{(i-s_1)}) & \mbox{for $s_1< i\le s_1 + s_2,\ 1\le j\le m_1$,}\\
\boldsymbol{0} & \mbox{for $1\le i\le s_1 + s_2,\ m_1 < j\le m_1 + m_2$.}
{\bf e}nd{cases}$$
We now show that $C$ is indeed a $(d,m_1 + m_2,(e_1,\ldots, e_{s_1},f_1,\ldots,f_{s_2}), s_1 + s_2)$-system over ${\mathbb{F}}ield_q$.
To this end, choose nonnegative integers $r_1,\ldots,r_{s_1},r_{s_1 +1},\ldots, r_{s_1+s_2}$ with $e_i| r_i$ for $1\le i\le s_1$ and $f_{i-s_1} | r_i$ for
$s_1 < i\le s_1 + s_2$, and $\sum_{i=1}^{s_1 + s_2} r_i \le d$. We need to check that the vectors
$$(\boldsymbol{a}_j^{(i)},\boldsymbol{0}),\ 1\le j\le r_i,\ 1\le i\le s_1,$$
and
$$(\boldsymbol{0},\boldsymbol{b}_j^{(i-s_1)}),\ 1\le j\le r_i,\ s_1< i\le s_1 + s_2,$$
are linearly independent over ${\mathbb{F}}ield_q$. But this property of $C$ follows immediately by the assumptions made on $A$ and $B$ and the fact
that $d$ does not exceed $d_1$ or $d_2$.
{\bf e}nd{proof}
The following corollary, which follows immediately from Theorem~\ref{thmdirectproduct} by induction, is a generalization of \cite[Corollary~3]{NX98}.
\begin{corollary}
Let $q$ be a prime power, let $m_1,\ldots,m_n \ge 1$, $s_1,\ldots,s_n \ge 1$, and $u_1,\ldots,u_n\ge 0$ be integers,
and let ${\bf e}_k= (e_1,\ldots,e_{s_k})\in {\mathbb{N}}^{s_k}$
for $1\le k\le n$. If digital $(u_k,m_k,{\bf e}_k,s_k)$-nets over ${\mathbb{F}}ield_q$ for $1\le k\le n$ are given, then there exists a digital
$$\left(u,\sum_{k=1}^{n} m_k,{\bf e},\sum_{k=1}^n s_k \right)\mbox{-net}$$
over ${\mathbb{F}}ield_q$ with
$$u=\sum_{k=1}^n m_k -\min_{1\le k\le n}(m_k-u_k)$$
and
$${\bf e}=(e_1^{(1)},\ldots, e_{s_1}^{(1)}, e_1^{(2)},\ldots,e_{s_2}^{(2)},\ldots\ldots, e_1^{(n)},
\ldots,e_{s_n}^{(n)}) \in {\mathbb{N}}^{s_1 + \cdots + s_n}.$$
{\bf e}nd{corollary}
Up to now, we have dealt only with propagation rules where one or several point sets or sequences in a certain base were given, and we constructed
a new point set in the same base. However, it is known that one can derive propagation rules by making a transition from one base to another.
These propagation rules are called base-change propagation rules.
We now show the following theorem, which is a generalization of Theorem~9 in \cite{NX98} (or Propagation Rule~7 in \cite{N05}).
\begin{theorem} \label{thmbasechange}
Let $q$ be a prime power, let $m\ge 1$, $s\ge 1$, $u \ge 0$, and $r\ge 1$ be integers, and let
${\bf e}=(e_1,\ldots,e_s)\in{\mathbb{N}}N^s$. If there exists a digital $(u,m,{\bf e},s)$-net over ${\mathbb{F}}ield_{q^r}$,
then there exists a digital $((r-1)m+u,rm,{\bf f},rs)$-net over ${\mathbb{F}}ield_q$, where
$${\bf f}=(\underbrace{e_1,\ldots,e_1}_{\mbox{$r$ times}},
\underbrace{e_2,\ldots,e_2}_{\mbox{$r$ times}},\ldots,\underbrace{e_s,\ldots,e_s}_{\mbox{$r$ times}}).$$
{\bf e}nd{theorem}
\begin{proof}
By Lemma~\ref{lemnetsystem} it suffices to show that we can obtain a $(d,rm,{\bf f},rs)$-system over ${\mathbb{F}}ield_q$ from a
$(d,m,{\bf e},s)$-system over ${\mathbb{F}}ield_{q^r}$. To this end, let
$$ A=\left\{\boldsymbol{a}_j^{(i)}\in{\mathbb{F}}ield_{q^r}^m: 1\le i\le s, 1\le j\le m\right\} $$
be a $(d,m,{\bf e},s)$-system over ${\mathbb{F}}ield_{q^r}$. Choose an ordered basis $\beta_1,\ldots,\beta_r$ of ${\mathbb{F}}ield_{q^r}$ over ${\mathbb{F}}ield_q$ and an
${\mathbb{F}}ield_q$-linear isomorphism $\varphi: {\mathbb{F}}ield_{q^r}^m\rightarrow{\mathbb{F}}ield_q^{rm}$. Define now a system
$$ B=\left\{\boldsymbol{b}_j^{(h)}\in{\mathbb{F}}ield_{q}^{rm}: 1\le h\le rs, 1\le j\le rm\right\}$$
by setting, for $1\le i\le s$ and $1\le k\le r$,
$$\boldsymbol{b}_j^{((i-1)r+k)}:=\begin{cases}
\varphi (\beta_k\boldsymbol{a}_j^{(i)}) & \mbox{for}\ 1\le j\le m,\\
\boldsymbol{0} & \mbox{for}\ m< j\le rm.
{\bf e}nd{cases}$$
We claim that $B$ is a $(d,rm,{\bf f},rs)$-system over ${\mathbb{F}}ield_q$. Indeed, choose nonnegative integers $d_{i,k}$ for $1 \le i \le s$, $1 \le k \le r$, with
the properties $e_i | d_{i,k}$ for $1\le i\le s$, $1\le k\le r$, and
$$ \sum_{i=1}^s \sum_{k=1}^r d_{i,k}\le d. $$
Suppose that
\begin{equation}\label{eqsupposezero}
\sum_{i=1}^s \sum_{k=1}^r\sum_{j=1}^{d_{i,k}} c_j^{(i,k)} \boldsymbol{b}_j^{((i-1)r +k)}=\boldsymbol{0}\in{\mathbb{F}}ield_q^{rm},
{\bf e}nd{equation}
where all $c_j^{(i,k)}\in{\mathbb{F}}ield_q$. For $1\le i\le s$ define
$ d_i:=\max_{1\le k\le r} d_{i,k}, $
and note that $d_i\le m$, as $d_{i,k}\le d\le m$, and that $e_i | d_i$.
For $1\le i\le s$ and $1\le k\le r$, let $h_j^{(i,k)}:=1\in{\mathbb{F}}ield_q$ for $1\le j\le d_{i,k}$ and
$h_j^{(i,k)}=0\in{\mathbb{F}}ield_q$ for $d_{i,k} < j\le d_i$. Then we can write~{\bf e}qref{eqsupposezero} as
$$\sum_{i=1}^s \sum_{k=1}^r\sum_{j=1}^{d_{i}} h_j^{(i,k)} c_j^{(i,k)} \varphi(\beta_k \boldsymbol{a}_j^{(i)})=\boldsymbol{0}\in{\mathbb{F}}ield_q^{rm},$$
from which we conclude, due to the properties of $\varphi$, that
$$ \sum_{i=1}^s \sum_{j=1}^{d_i} \gamma_j^{(i)} \boldsymbol{a}_j^{(i)}=\boldsymbol{0}\in{\mathbb{F}}ield_{q^r}^m$$
with
$$\gamma_j^{(i)}=\sum_{k=1}^r h_j^{(i,k)} c_j^{(i,k)} \beta_k \in{\mathbb{F}}ield_{q^r}. $$
Note now that
$$\sum_{i=1}^s d_i \le \sum_{i=1}^s \sum_{k=1}^r d_{i,k}\le d.$$
As we have $e_i | d_i$ for $1\le i\le s$, and since we assumed $A$ to be a $(d,m,{\bf e},s)$-system over ${\mathbb{F}}ield_{q^r}$,
we must have $\gamma_j^{(i)}=0$ for $1\le j\le d_i$, $1\le i\le s$, and therefore
also $h_j^{(i,k)}c_j^{(i,k)}=0$ for $1\le j\le d_i$, $1\le i\le s$, $1\le k\le r$. Therefore we see that all coefficients $c_j^{(i,k)}$
in~{\bf e}qref{eqsupposezero} must be equal to $0$.
{\bf e}nd{proof}
We also generalize the following propagation rule that is called ``base reduction for projective spaces'' in~\cite{MinT},
by proving the following result.
\begin{theorem} \label{thmprojective}
Let $q$ be a prime power, let $m\ge 1$, $s\ge 1$, $u \ge 0$, and $r\ge 2$ be integers, and let
${\bf e}=(e_1,\ldots,e_s)\in{\mathbb{N}}N^s$. If there exists a digital $(u,m,{\bf e},s)$-net
over ${\mathbb{F}}ield_{q^r}$, then there exists a digital $((r-1)m-(r-1)+u,rm-(r-1),{\bf e},s)$-net over ${\mathbb{F}}ield_q$.
{\bf e}nd{theorem}
\begin{proof}
The result can be shown by an adaptation of the proof of Theorem~\ref{thmbasechange}.
Let a digital $(u,m,{\bf e},s)$-net over ${\mathbb{F}}ield_{q^r}$ be given and let $C_1,\ldots,C_s$ be its generating matrices.
Note that the linear independence conditions in the definition of a digital $(u,m,{\bf e},s)$-net stay unchanged if we
multiply a row of a matrix $C_i$ with some nonzero element of ${\mathbb{F}}ield_{q^r}$. Doing so, we can obtain generating
matrices $A_1,\ldots,A_s$ over ${\mathbb{F}}ield_{q^r}$ which also generate a digital $(u,m,{\bf e},s)$-net over ${\mathbb{F}}_{q^r}$ and for which the first column
of each $A_i$, $1\le i\le s$, consists only of zeros and ones. Let $\boldsymbol{a}_{j}^{(i)}$, $1\le j\le m$, $1\le i\le s$, be the row vectors
of the matrices $A_i$. By Lemma~\ref{lemnetsystem}, the system
$$A=\{\boldsymbol{a}_j^{(i)}\in{\mathbb{F}}ield_{q^r}^m: 1\le i\le s, 1\le j\le m\}$$
is an $(m-u,m,{\bf e},s)$-system over ${\mathbb{F}}ield_{q^r}$. Let now $\psi$ be an ${\mathbb{F}}ield_q$-linear isomorphism from ${\mathbb{F}}ield_{q^r}$ to ${\mathbb{F}}ield_q^r$
such that $\psi (1)= (0,\ldots,0,1)\in{\mathbb{F}}ield_q^r$. For a vector $\boldsymbol{a}\in{\mathbb{F}}ield_{q^r}^m$ with $\boldsymbol{a}=(\alpha_1,\ldots,\alpha_m)$, put
$\varphi (\boldsymbol{a})=(\psi (\alpha_1),\ldots,\psi (\alpha_m))\in{\mathbb{F}}ield_q^{rm}$.
Define now a new system
$$B=\{\boldsymbol{b}_j^{(i)}\in{\mathbb{F}}ield_q^{rm}: 1\le i\le s, 1\le j\le rm\} $$
by setting, for $1\le i\le s$,
$$\boldsymbol{b}_j^{(i)}=\begin{cases}
\varphi (\boldsymbol{a}_j^{(i)}) & \mbox{for $1\le j\le m$,}\\
\boldsymbol{0} & \mbox{for $m< j\le rm$.}
{\bf e}nd{cases}$$
We are now going to show that $B$ is an $(m-u,rm,{\bf e},s)$-system over ${\mathbb{F}}ield_q$. Choose nonnegative integers $d_1,\ldots,d_s$ with
the properties $e_i | d_{i}$ for $1\le i\le s$ and $\sum_{i=1}^s d_{i}\le m-u$.
Suppose that
\begin{equation}\label{eqsupposezero2}
\sum_{i=1}^s \sum_{j=1}^{d_{i}} c_j^{(i)} \boldsymbol{b}_j^{(i)}=\boldsymbol{0}\in{\mathbb{F}}ield_q^{rm},
{\bf e}nd{equation}
where all $c_j^{(i)}\in{\mathbb{F}}ield_q$. Then we can write~{\bf e}qref{eqsupposezero2} as
$$\sum_{i=1}^s \sum_{j=1}^{d_{i}} c_j^{(i)} \varphi(\boldsymbol{a}_j^{(i)})=\boldsymbol{0}\in{\mathbb{F}}ield_q^{rm},$$
from which we conclude, due to the properties of $\varphi$, that
$$ \sum_{i=1}^s \sum_{j=1}^{d_i} c_j^{(i)} \boldsymbol{a}_j^{(i)}=\boldsymbol{0}\in{\mathbb{F}}ield_{q^r}^m.$$
As we have $\sum_{i=1}^s d_{i}\le m-u$ and $e_i | d_i$ for $1\le i\le s$, and since we assumed $A$ to be an $(m-u,m,{\bf e},s)$-system over ${\mathbb{F}}ield_{q^r}$,
we must have $c_j^{(i)}=0$ for $1\le j\le d_i$, $1\le i\le s$, in~{\bf e}qref{eqsupposezero2}. Consequently,
$B$ is indeed an $(m-u,rm,{\bf e},s)$-system over ${\mathbb{F}}ield_q$.
Note now that the first $r-1$ coordinates of each $\boldsymbol{b}_j^{(i)}$ are equal to zero, due to the choices of $\psi$ and $\varphi$ and due to the
fact that the first coordinates of the $\boldsymbol{a}_j^{(i)}$ are all either zero or one. Furthermore, note that
$\boldsymbol{b}_{rm-(r-2)}^{(i)},\boldsymbol{b}_{rm-(r-3)}^{(i)},\ldots,\boldsymbol{b}_{rm}^{(i)}$ are all $\boldsymbol{0}$, as we assumed $r\ge 2$.
We now remove $\boldsymbol{b}_{rm-(r-2)}^{(i)},\boldsymbol{b}_{rm-(r-3)}^{(i)},\ldots,\boldsymbol{b}_{rm}^{(i)}$ from $B$ and
discard for each of the remaining $\boldsymbol{b}_j^{(i)}\in B$ its first $r-1$ coordinates. In this way we end up with a system
$$D=\{\boldsymbol{d}_j^{(i)}\in{\mathbb{F}}ield_q^{rm - (r-1)}: 1\le i\le s, 1\le j\le rm - (r-1)\},$$
where the $\boldsymbol{d}_j^{(i)}$ are the projections of the original $\boldsymbol{b}_j^{(i)}$ onto their last $rm- (r-1)$ coordinates. However, the linear
independence properties of the vectors in $D$ are the same as those of the vectors in $B$, since we removed only zeros to derive $D$ from $B$.
Hence, $D$ is an $(m-u,rm- (r-1),{\bf e},s)$-system over ${\mathbb{F}}ield_q$. By Lemma~\ref{lemnetsystem}, the vectors in $D$ generate a
digital $((r-1)m-(r-1)+u,rm-(r-1),{\bf e},s)$-net over ${\mathbb{F}}ield_{q}$, as claimed.
{\bf e}nd{proof}
Now we establish the digital analog of the propagation rule in Proposition~\ref{prpr}. In the classical case of digital $(t,m,s)$-nets, this propagation rule
was shown in~\cite{SW} (see also \cite[Theorem~4.60]{DP}).
\begin{proposition} \label{prprd}
Let $q$ be a prime power, let $m \ge 1$, $s \ge 1$, and $u \ge 0$ be integers, and let ${\bf e} =(e_1,\ldots,e_s) \in {\mathbb{N}}^s$.
If a digital $(u,m,{\bf e},s)$-net over ${\mathbb{F}}_q$
is given, then for every integer $k$ with $\max (1,u) \le k \le m$ such that $m-k$ is a linear combination of $e_1,\ldots,e_s$
with coefficients from ${\mathbb{N}}_0$,
we can construct a digital $(u,k,{\bf e},s)$-net over ${\mathbb{F}}_q$.
{\bf e}nd{proposition}
\begin{proof}
It suffices to consider the case where $m-k$ is divisible by some $e_i$, since we can then proceed by induction. By using a permutation of the
coordinates, we can
assume that $e_s$ divides $m-k$. In the generating matrix $C_s$ of the given digital $(u,m,{\bf e},s)$-net over ${\mathbb{F}}_q$, the first $e_s \lfloor (m-u)/e_s \rfloor$ row vectors
are linearly independent over ${\mathbb{F}}_q$, whereas the remaining row vectors do not matter (compare with Lemma~\ref{lemnetsystem}).
Therefore we can assume that the row
vectors of $C_s$ form a basis of ${\mathbb{F}}_q^m$, and by changing the coordinate system in ${\mathbb{F}}_q^m$ we can take $C_s$ to be the antidiagonal matrix $E_m^{\prime}$ in the
proof of \cite[Theorem~4.60]{DP}. Now we can imitate that proof (but note that $n$ in that proof plays the role of our $k$ and that the condition $d_1 + \cdots + d_s
=n-t$ is now replaced by $d_1 + \cdots + d_s \le k-u$). The only point we need to observe is that the number $m-k$ of auxiliary unit vectors in the displayed scheme
of vectors in \cite[p.~157]{DP} (namely the unit vectors with the single coordinate $1$ between position $k+1$ and position $m$) must be divisible by $e_s$. But this
is guaranteed by our assumption.
{\bf e}nd{proof}
Next we show the digital analog of Proposition~\ref{prsn}. First we reformulate Proposition~\ref{propdigseq} in the language of $(d,m,{\bf e},s)$-systems.
\begin{lemma} \label{lemsnd}
Let $q$ be a prime power, let $s \ge 1$ and $u \ge 0$ be integers, and let ${\bf e} \in {\mathbb{N}}^s$. Then the $\infty \times \infty$ matrices $C_1,\ldots,C_s$ over ${\mathbb{F}}_q$ generate
a digital $(u,{\bf e},s)$-sequence over ${\mathbb{F}}_q$ if and only if, for every integer $m \ge \max (1,u)$, the system of row vectors of the matrices $C_1^{(m)},\ldots,C_s^{(m)}$
is an $(m-u,m,{\bf e},s)$-system over ${\mathbb{F}}_q$.
{\bf e}nd{lemma}
\begin{proposition} \label{prsnd}
Let $q$ be a prime power, let $s \ge 1$ and $u \ge 0$ be integers, and let ${\bf e} =(e_1,\ldots,e_s) \in {\mathbb{N}}^s$. If a digital $(u,{\bf e},s)$-sequence over ${\mathbb{F}}_q$ is given,
then for every integer $m \ge \max (1,u)$ we can construct a digital $(u,m,{\bf e}^{\prime},s+1)$-net over ${\mathbb{F}}_q$, where ${\bf e}^{\prime}=(1,e_1,\ldots,e_s) \in {\mathbb{N}}^{s+1}$.
{\bf e}nd{proposition}
\begin{proof}
Let $D_1,\ldots,D_s$ be $\infty \times \infty$ generating matrices over ${\mathbb{F}}_q$ of the given digital $(u,{\bf e},s)$-sequence over ${\mathbb{F}}_q$. Fix an integer $m \ge \max (1,u)$.
We define $m \times m$ generating matrices $C_1,\ldots,C_{s+1}$ over ${\mathbb{F}}_q$ by letting $C_1$ be a nonsingular right lower triangular matrix over ${\mathbb{F}}_q$ and by setting
$C_i=D_{i-1}^{(m)}$ for $2 \le i \le s+1$. By Lemma~\ref{lemnetsystem} it suffices to show that the system of row vectors of the matrices $C_1,\ldots,C_{s+1}$ is an
$(m-u,m,{\bf e}^{\prime},s+1)$-system over ${\mathbb{F}}_q$. To this end, we choose $d_1,\ldots,d_{s+1} \in {\mathbb{N}}_0$ with $e_i|d_{i+1}$ for $1 \le i \le s$ and $\sum_{i=1}^{s+1} d_i
\le m-u$, and we have to prove that the row vectors $\boldsymbol{c}_j^{(i)}$, $1 \le j \le d_i$, $1 \le i \le s+1$, of the matrices $C_1,\ldots,C_{s+1}$ are linearly independent
over ${\mathbb{F}}_q$. If $d_1=m-u$, then $d_i=0$ for $2 \le i \le s+1$, and we are done since $C_1$ is nonsingular. Hence we can assume that $d_1 < m-u$. Suppose that we have
\begin{equation} \label{eqla}
\sum_{i=1}^{s+1} \sum_{j=1}^{d_i} b_j^{(i)} \boldsymbol{c}_j^{(i)} = \boldsymbol{0} \in {\mathbb{F}}_q^m
{\bf e}nd{equation}
with all $b_j^{(i)} \in {\mathbb{F}}_q$. Let $\pi : {\mathbb{F}}_q^m \to {\mathbb{F}}_q^{m-d_1}$ be the projection to the first $m-d_1$ coordinates of a vector in ${\mathbb{F}}_q^m$. Then $\pi (\boldsymbol{c}_j^{(1)})=
\boldsymbol{0} \in {\mathbb{F}}_q^{m-d_1}$ for $1 \le j \le d_1$ since $C_1$ is a right lower triangular matrix, and so applying $\pi$ to~{\bf e}qref{eqla} we obtain
\begin{equation} \label{eqlb}
\sum_{i=2}^{s+1} \sum_{j=1}^{d_i} b_j^{(i)} \pi(\boldsymbol{c}_j^{(i)})=\boldsymbol{0} \in {\mathbb{F}}_q^{m-d_1}.
{\bf e}nd{equation}
Now the vectors $\pi(\boldsymbol{c}_j^{(i)})$, $1 \le j \le d_i$, $2 \le i \le s+1$, are row vectors of the matrices $D_1^{(m-d_1)},\ldots,D_s^{(m-d_1)}$. The system of all row vectors
of these matrices forms an $(m-d_1-u,m-d_1,{\bf e},s)$-system over ${\mathbb{F}}_q$ by Lemma~\ref{lemsnd}. By observing that $\sum_{i=2}^{s+1} d_i \le m-d_1-u$, we conclude from~{\bf e}qref{eqlb}
that $b_j^{(i)}=0$ for $1 \le j \le d_i$, $2 \le i \le s+1$, and returning to~{\bf e}qref{eqla} we see that $b_j^{(1)}=0$ for $1 \le j \le d_1$.
{\bf e}nd{proof}
\section{Base-change rules for $(u,{\bf e},s)$-sequences} \label{secbc}
Now we apply the idea of a base change to $(u,{\bf e},s)$-sequences. Note that the results in this section are not propagation rules
in the sense that they yield a new sequence, but they are statements on how we can view a given sequence with respect to different bases.
We first need the following auxiliary result.
\begin{lemma} \label{lebc}
Let $b \ge 2$, $g \ge 1$, $m \ge 1$, and $s \ge 1$ be integers. Let the subinterval $J$ of $[0,1]^s$ be of the form
\begin{equation} \label{eqj}
J=\prod_{i=1}^s [a_i b^{-gf_i}, (a_i+1) b^{-gf_i})
{\bf e}nd{equation}
with $a_i,f_i \in {\mathbb{N}}_0$, $a_i < b^{gf_i}$, and $f_i \le m$ for $1 \le i \le s$. Then for ${\bf x} \in [0,1]^s$ we have $[{\bf x}]_{b^g,m} \in J$ if and only if
$[{\bf x}]_{b,gm} \in J$.
{\bf e}nd{lemma}
\begin{proof}
It suffices to show the lemma for $s=1$. Thus, let $J=[a b^{-gf},(a+1) b^{-gf})$ with $a,f \in {\mathbb{N}}_0$, $a < b^{gf}$, and $f \le m$. Then $[x]_{b^g,m} \in J$
means that the first $f$ digits of $x \in [0,1]$ in base $b^g$ are prescribed. Now $f$ digits in base $b^g$ correspond to $gf$ digits in base $b$, and so
$[x]_{b^g,m} \in J$ is equivalent to $[x]_{b,gf} \in J$. Finally, $f \le m$ implies that we have $[x]_{b,gf} \in J$ if and only if $[x]_{b,gm} \in J$.
{\bf e}nd{proof}
\begin{theorem} \label{thbc}
Let $b \ge 2$, $s\ge 1$, and $u \ge 0$
be integers and let ${\bf e} =(e_1,\ldots,e_s) \in {\mathbb{N}}^s$. Then any $(u,{\bf e},s)$-\se in base $b$ is a $(\lceil u/g \rceil,
s)$-\se in base $b^g$, where $g$ is the least common multiple of $e_1,\ldots,e_s$.
{\bf e}nd{theorem}
\begin{proof}
It suffices to show that any $(gu,{\bf e},s)$-\se in base $b$ is a $(u,s)$-\se in base $b^g$. For then, with $v=\lceil u/g \rceil$, we have $gv \ge u$, hence a
given $(u,{\bf e},s)$-\se in base $b$ is also a $(gv,{\bf e},s)$-\se in base $b$ by \cite[Remark~1 and Definition~2]{HN13}, and so it is a $(v,s)$-\se in base $b^g$.
Thus, let ${\bf x}_0,{\bf x}_1,\ldots$ be a given $(gu,{\bf e},s)$-\se in base $b$, where $g$ is the least common multiple of the components of ${\bf e}$. We want to prove that ${\bf x}_0,{\bf x}_1,\ldots$ is a $(u,s)$-\se in base $b^g$. For given
integers $k \ge 0$ and $m > u$, we have to show that the points $[{\bf x}_n]_{b^g,m}$ with $kb^{gm} \le n < (k+1)b^{gm}$ form a $(u,m,s)$-net in base $b^g$. Take
an elementary interval $J$ in base $b^g$ as in~{\bf e}qref{eqj} with $\lambda_s(J)=(b^g)^{u-m}$, that is, with $\sum_{i=1}^s f_i=m-u$. Since ${\bf x}_0,{\bf x}_1,\ldots$
form a $(gu,{\bf e},s)$-\se in base $b$ and since $gm > gu$, it follows that the points $[{\bf x}_n]_{b,gm}$ with $kb^{gm} \le n < (k+1)b^{gm}$ form a $(gu,gm,{\bf e},s)$-net
in base $b$. Now $e_i | gf_i$ for $1 \le i \le s$ and $\lambda_s(J)=b^{gu-gm}$, and so the definition of a $(gu,gm,{\bf e},s)$-net in base $b$ implies that the
number of integers $n$ with $kb^{gm} \le n < (k+1)b^{gm}$ and $[{\bf x}_n]_{b,gm} \in J$ (or equivalently $[{\bf x}_n]_{b^g,m} \in J$ by Lemma~\ref{lebc}) is given by
$b^{gm} \lambda_s(J)=b^{gu}$. This shows that the points $[{\bf x}_n]_{b^g,m}$ with $kb^{gm} \le n < (k+1)b^{gm}$ form a $(u,m,s)$-net in base $b^g$.
{\bf e}nd{proof}
\begin{theorem} \label{thbc2}
Let $b \ge 2$, $s\ge 1$, and $u \ge 0$
be integers and let ${\bf e} =(e_1,\ldots,e_s) \in {\mathbb{N}}^s$. Furthermore, denote by $h$ the greatest common divisor of $e_1,\ldots,e_s$.
Then any $(u,{\bf e},s)$-\se in base $b$ is a $(\lceil u/h \rceil, {\bf e}/h,s)$-\se in base $b^h$, where we write ${\bf e} /h$ for $(e_1/h, e_2/h,\ldots, e_s/h)$.
{\bf e}nd{theorem}
\begin{proof}
Similarly to the proof of Theorem~\ref{thbc}, it is sufficient to show that any $(hu,{\bf e},s)$-\se in base $b$ is a $(u,{\bf e}/h,s)$-\se in base $b^h$, with $h$
being the greatest common divisor of the components of ${\bf e}$.
Let therefore ${\bf e}$ be given and let $h$ be defined as above. Let ${\bf x}_0,{\bf x}_1,\ldots$ be a given $(hu,{\bf e},s)$-\se in base $b$.
We want to prove that ${\bf x}_0,{\bf x}_1,\ldots$ is a $(u,{\bf e} /h,s)$-\se in base $b^h$. The argument works in an analogous way to the proof of Theorem~\ref{thbc}, the only
difference being that, for $m>u$, we consider an elementary interval $J$ in base $b^h$ with $\lambda_s (J)\ge (b^h)^{u-m}$ (that is, with $\sum_{i=1}^s f_i \le m-u$),
and with $(e_i/h)|f_i$ for $1\le i\le s$. We can then follow the argument in the proof of Theorem~\ref{thbc}, and we then have $\lambda_s (J)\ge b^{hu-hm}$ and
$e_i | hf_i$ for $1\le i\le s$, which yields, in exactly the same fashion as above, the desired result.
{\bf e}nd{proof}
\section{Duality theory} \label{secduality}
We generalize the classical duality theory for digital nets introduced in~\cite{NP01} by developing a duality theory for digital $(u,m,{\bf e},s)$-nets over the finite
field ${\mathbb{F}}_q$, where $q$ is an arbitrary prime power. Throughout this section, the prime power $q$ and the positive integer $m$ are fixed.
For $\boldsymbol{a}=(a_1,\ldots,a_m)\in{\mathbb{F}}ield_q^m$ and $e\in{\mathbb{N}}N$, we introduce the weight $v_e (\boldsymbol{a})$ by
$$ v_e(\boldsymbol{a})=\begin{cases}
0 &\mbox{if $\boldsymbol{a}=\boldsymbol{0}$,}\\
\min\left\{m,e\left\lceil\max\{j: a_j\neq 0\}/e\right\rceil\right\} & \mbox{otherwise.}
{\bf e}nd{cases}$$
This definition can be extended to ${\mathbb{F}}ield_q^{sm}$ by considering a vector $\boldsymbol{A}\in{\mathbb{F}}ield_q^{sm}$ as the concatenation of
$s$ vectors of length $m$ each, i.e.,
$$\boldsymbol{A}=(\boldsymbol{a}^{(1)},\ldots,\boldsymbol{a}^{(s)})\in{\mathbb{F}}ield_q^{sm}$$
with $\boldsymbol{a}^{(i)}\in{\mathbb{F}}ield_q^m$ for $1\le i\le s$,
and by putting, for ${\bf e}=(e_1,\ldots,e_s)\in{\mathbb{N}}N^s$,
$$ V_{m,{\bf e}}(\boldsymbol{A}):=\sum_{i=1}^s v_{e_i} (\boldsymbol{a}^{(i)}).$$
\begin{remark} \label{revm} {\rm
For ${\bf e}= {\bf 1}=(1,\ldots,1)$, $V_{m,{\bf e}}$ coincides with $V_m$ defined in~\cite{NP01}.}
{\bf e}nd{remark}
\begin{definition} \label{defmd} {\rm
Let $q$ be a prime power, let $s,m\ge 1$ be integers, and let $\mathcal{N}$ be a linear subspace of ${\mathbb{F}}ield_q^{sm}$.
Then we define the {\bf e}mph{minimum distance} $\delta_{m,{\bf e}}(\mathcal{N})$ of $\mathcal{N}$ (for ${\bf e}\in{\mathbb{N}}N^s$) as
$$\delta_{m,{\bf e}}(\mathcal{N}):=\begin{cases}
\min_{\boldsymbol{A} \in\mathcal{N}\setminus \{\boldsymbol{0}\}} V_{m,{\bf e}} (\boldsymbol{A}) & \mbox{if $\mathcal{N}\neq\{\boldsymbol{0}\}$},\\
sm+1 & \mbox{otherwise.}
{\bf e}nd{cases}
$$
}
{\bf e}nd{definition}
It is trivial that we have $\delta_{m,{\bf e}} (\mathcal{N})\ge 1$ for every linear subspace $\mathcal{N}$ of ${\mathbb{F}}ield_q^{sm}$. We now show the following result, which is a
generalization of \cite[Proposition~1]{NP01}.
\begin{proposition}
Let $q$ be a prime power and $s,m\ge 1$ be integers. For any linear subspace $\mathcal{N}$ of ${\mathbb{F}}ield_q^{sm}$ and for any ${\bf e}\in{\mathbb{N}}N^s$, we have
$$\delta_{m,{\bf e}} (\mathcal{N})\le sm -\dim (\mathcal{N}) + \min_{1\le i\le s} e_i. $$
{\bf e}nd{proposition}
\begin{proof}
If $\mathcal{N}=\{\boldsymbol{0}\}$, the result is trivial.
If $\mathcal{N}\neq \{\boldsymbol{0}\}$, let $h:=\dim(\mathcal{N})\ge 1$. We write $sm-h+1=km+r$ with integers $0 \le k \le s-1$ and $1 \le r \le m$. Without loss of generality, we can
use a permutation of the $e_i$ and the same permutation for the components $\boldsymbol{a}^{(i)}$ of each $\boldsymbol{A} \in {\mathbb{F}}_q^{sm}$ such that $\min_{1 \le i \le s} e_i=e_{k+1}$.
Let $\pi:\mathcal{N}\rightarrow{\mathbb{F}}ield_q^h$ be the linear transformation which maps
$\boldsymbol{A}\in\mathcal{N}$ to the $h$-tuple of the last $h$ coordinates of $\boldsymbol{A}$. If $\pi$ is surjective, then there exists a nonzero
$\boldsymbol{A}_1\in\mathcal{N}$ with
$$\pi (\boldsymbol{A}_1)=(1,0,\ldots,0)\in{\mathbb{F}}ield_q^h.$$
Let $\boldsymbol{A}_1=(\boldsymbol{b}^{(1)},\ldots,\boldsymbol{b}^{(s)})$ with $\boldsymbol{b}^{(i)} \in {\mathbb{F}}_q^m$ for $1 \le i \le s$. For $1 \le i \le k$ we have the trivial bound $v_{e_i}(\boldsymbol{b}^{(i)})
\le m$. For $i=k+1$ we have
$$
v_{e_{k+1}}(\boldsymbol{b}^{(k+1)}) \le e_{k+1} \lceil r/e_{k+1} \rceil \le r+e_{k+1}-1.
$$
For $k+2 \le i \le s$ we have $v_{e_i}(\boldsymbol{b}^{(i)}) =0$. Therefore
$$
V_{m,{\bf e}}(\boldsymbol{A}_1) \le km+r+e_{k+1}-1=sm-h+ \min_{1 \le i \le s} e_i.
$$
If $\pi$ is not surjective, then for any nonzero $\boldsymbol{A}_2$ in the kernel of $\pi$ we obtain by a similar argument,
$$V_{m,{\bf e}} (\boldsymbol{A}_2)\le sm-h-1 +\min_{1\le i\le s} e_i,$$
and so in both cases the result follows.
{\bf e}nd{proof}
We introduce some additional notation. With a given system of vectors
$$\left\{\boldsymbol{c}_j^{(i)}\in{\mathbb{F}}ield_q^m: 1\le j\le m, 1\le i\le s\right\},$$
we associate the matrices $C_i$, $1\le i\le s$, as the $m\times m$ matrices with column vectors
$\boldsymbol{c}_1^{(i)},\ldots,\boldsymbol{c}_m^{(i)}$. Combine these matrices into the $m \times sm$ matrix
$$C=(C_1 | C_2 | \cdots | C_s),$$
so that $C_1,\ldots,C_s$ are submatrices of $C$. By $\mathcal{C} \subseteq {\mathbb{F}}_q^{sm}$ we denote the row space of $C$ and by
$\mathcal{C}^\perp \subseteq {\mathbb{F}}_q^{sm}$ the dual space of $\mathcal{C}$.
We now show the following theorem.
\begin{theorem} \label{thmsystemdual}
Let $q$ be a prime power, let $s,m\ge 1$ be integers, and ${\bf e}\in{\mathbb{N}}N^s$. Furthermore, let the system
$$\left\{\boldsymbol{c}_j^{(i)}\in{\mathbb{F}}ield_q^m: 1\le j\le m, 1\le i\le s\right\}$$
be given. Then with the notation above, the following results hold for an integer $d\in\{0,\ldots,m\}$.
\begin{itemize}
\item[(a)] If the system is
a $(d,m,{\bf e},s)$-system over ${\mathbb{F}}ield_q$, then $\delta_{m,{\bf e}} (\mathcal{C}^\perp)\ge d$.
\item[(b)] If the matrices $C_1,\ldots,C_s$ are all nonsingular and if the system is
a $(d,m,{\bf e},s)$-system over ${\mathbb{F}}ield_q$, then $\delta_{m,{\bf e}} (\mathcal{C}^\perp)\ge d+1$.
\item[(c)] If $\delta_{m,{\bf e}} (\mathcal{C}^\perp)\ge d+1$, then it follows that the system is a
$(d,m,{\bf e},s)$-system over ${\mathbb{F}}ield_q$.
{\bf e}nd{itemize}
{\bf e}nd{theorem}
\begin{proof}
The result is trivial if $\mathcal{C}={\mathbb{F}}ield_q^{sm}$, so we can assume that $\mathcal{C}\subsetneq{\mathbb{F}}ield_q^{sm}$ in the following. Note that then $\dim(\mathcal{C}^{\perp}) \ge 1$.
For $\boldsymbol{A}=(\boldsymbol{a}^{(1)},\ldots,\boldsymbol{a}^{(s)})\in{\mathbb{F}}ield_q^{sm}$ with $\boldsymbol{a}^{(i)}=(a_1^{(i)},\ldots,a_m^{(i)})\in{\mathbb{F}}ield_q^m$
for $1\le i\le s$, we have
$$\sum_{i=1}^s \sum_{j=1}^m a_j^{(i)}\boldsymbol{c}_j^{(i)}=\boldsymbol{0}\in{\mathbb{F}}ield_q^m$$
if and only if $C \boldsymbol{A}^{\top}=\boldsymbol{0} \in {\mathbb{F}}_q^m$, i.e., if and only if $\boldsymbol{A} \in \mathcal{C}^\perp$.
Now let the given system be a $(d,m,{\bf e},s)$-system over ${\mathbb{F}}ield_q$ and consider any nonzero $\boldsymbol{A}\in\mathcal{C}^\perp$. Then from
the previous observation we obtain
\begin{equation}\label{eqlindependent}
\sum_{i=1}^s \sum_{j=1}^m a_j^{(i)} \boldsymbol{c}_j^{(i)}=\boldsymbol{0}\in{\mathbb{F}}ield_q^m.
{\bf e}nd{equation}
Suppose first that there exists an index $i_0\in\{1,\ldots,s\}$ with $v_{e_{i_0}}(\boldsymbol{a}^{(i_0)})=m$. In this case, we distinguish two subcases.
\begin{itemize}
\item If for some $i_1\in\{1,\ldots,s\}\setminus\{i_0\}$, we have $v_{e_{i_1}}(\boldsymbol{a}^{(i_1)})>0$, then
$$ V_{m,{\bf e}}(\boldsymbol{A})\ge v_{e_{i_0}}(\boldsymbol{a}^{(i_0)})+v_{e_{i_1}}(\boldsymbol{a}^{(i_1)})\ge m+1\ge d+1,$$
as $d$ cannot exceed $m$ by definition. In this case, the results in~(a) and~(b) are shown.
\item Suppose now that there is no index $i_1\in\{1,\ldots,s\}\setminus\{i_0\}$ for which $v_{e_{i_1}}(\boldsymbol{a}^{(i_1)})>0$. In this case, the result
in~(a) follows immediately as $V_{m,{\bf e}}(\boldsymbol{A})= v_{e_{i_0}}(\boldsymbol{a}^{(i_0)})=m\ge d$. To show the result in~(b), note that equation~{\bf e}qref{eqlindependent} is equivalent to
$$\sum_{j=1}^m a_j^{(i_0)}\boldsymbol{c}_j^{(i_0)}=\boldsymbol{0}\in{\mathbb{F}}ield_q^m,$$
where not all $a_j^{(i_0)}$ are zero. However, this is a contradiction to the assumption that $C_{i_0}$ is nonsingular,
so this situation cannot occur under the hypotheses in~(b).
{\bf e}nd{itemize}
In summary, we have shown the results in~(a) and~(b) for the case where there exists an index $i_0\in\{1,\ldots,s\}$ with $v_{e_{i_0}}(\boldsymbol{a}^{(i_0)})=m$.
Suppose now that $v_{e_i}(\boldsymbol{a}^{(i)})< m$ for all $i\in\{1,\ldots,s\}$. Note that this implies that $e_i| v_{e_i}(\boldsymbol{a}^{(i)})$ for all $i\in\{1,\ldots,s\}$.
In this case,~{\bf e}qref{eqlindependent} boils down to
$$\sum_{i=1}^s \sum_{j=1}^{v_{e_i}(\boldsymbol{a}^{(i)})} a_j^{(i)} \boldsymbol{c}_j^{(i)}=\boldsymbol{0}\in{\mathbb{F}}ield_q^m.$$
As $e_i | v_{e_i}(\boldsymbol{a}^{(i)})$ for all $i\in\{1,\ldots,s\}$, and in view of Definition~\ref{defsystem}, this linear dependence relation can hold only if
$$V_{m,{\bf e}}(\boldsymbol{A})=\sum_{i=1}^s v_{e_i} (\boldsymbol{a}^{(i)})\ge d+1,$$
and so we obtain the desired results in~(a) and~(b).
To show the assertion in~(c), assume now that $\delta_{m,{\bf e}}(\mathcal{C}^\perp)\ge d+1$. We have to show that any system
$\{\boldsymbol{c}_j^{(i)}\in{\mathbb{F}}ield_q^m: 1\le j\le d_i, 1\le i\le s\}$ with $0\le d_i\le m$, $e_i|d_i$, $1\le i\le s$, and $\sum_{i=1}^s d_i \le d$ is
linearly independent over ${\mathbb{F}}ield_q$. Suppose, on the contrary, that such a system were linearly dependent over ${\mathbb{F}}ield_q$, i.e.,
that there exist coefficients $a_j^{(i)}\in{\mathbb{F}}ield_q$, not all 0, such that
$$\sum_{i=1}^s \sum_{j=1}^{d_i} a_j^{(i)} \boldsymbol{c}_j^{(i)}=\boldsymbol{0}\in{\mathbb{F}}ield_q^m.$$
Define $a_j^{(i)}=0$ for $d_i < j\le m$, $1\le i\le s$, then
$$\sum_{i=1}^s \sum_{j=1}^m a_j^{(i)} \boldsymbol{c}_j^{(i)} =\boldsymbol{0}\in{\mathbb{F}}ield_q^m.$$
This implies that $\boldsymbol{A}\in\mathcal{C}^\perp$, and so $V_{m,{\bf e}}(\boldsymbol{A})\ge d+1$. On the other hand,
$$v_{e_i}(\boldsymbol{a}^{(i)})\le e_i \left\lceil \frac{d_i}{e_i}\right\rceil=d_i\ \mbox{for}\ 1\le i\le s,$$
and so
$$V_{m,{\bf e}} (\boldsymbol{A})=\sum_{i=1}^s v_{e_i}(\boldsymbol{a}^{(i)})\le \sum_{i=1}^s d_i \le d,$$
which is a contradiction. Hence the proof of~(c) is complete.
{\bf e}nd{proof}
We now have the following consequence.
\begin{theorem} \label{thmnetdual}
Let $q$ be a prime power, let $s \ge 1$, $m \ge 1$, and $0\le u\le m$ be integers, and let ${\bf e}\in{\mathbb{N}}N^s$. Let $C_1,\ldots,C_s$ be $m\times m$ matrices
over ${\mathbb{F}}ield_q$ and put
$$C:=(C_1^\top | C_2^\top |\cdots | C_s^\top) \in {\mathbb{F}}_q^{m \times sm}.$$
Let $\mathcal{C}^{\perp}$ be the dual space of the row space $\mathcal{C}$ of $C$.
\begin{itemize}
\item[(a)] If $C_1,\ldots,C_s$ are all nonsingular, then they generate a digital $(u,m,{\bf e},s)$-net over ${\mathbb{F}}ield_q$ if and only if
$\delta_{m,{\bf e}} (\mathcal{C}^\perp)\ge m-u+1$.
\item[(b)] If $C_1,\ldots,C_s$ are arbitrary, then the following assertions hold:
\begin{itemize}
\item If $\delta_{m,{\bf e}}(\mathcal{C}^\perp)\ge m-u+1$, this implies that $C_1,\ldots,C_s$ generate a digital $(u,m,{\bf e},s)$-net over ${\mathbb{F}}ield_q$.
\item If $C_1,\ldots,C_s$ generate a digital $(u,m,{\bf e},s)$-net over ${\mathbb{F}}ield_q$, it follows that $\delta_{m,{\bf e}}(\mathcal{C}^\perp)\ge m-u$.
{\bf e}nd{itemize}
{\bf e}nd{itemize}
{\bf e}nd{theorem}
\begin{proof}
The result follows by combining Lemma~\ref{lemnetsystem} and Theorem~\ref{thmsystemdual}.
{\bf e}nd{proof}
For the special case where ${\bf e}={\bf 1}=(1,\ldots,1)\in{\mathbb{N}}N^s$, there exists a stronger version of Theorem~\ref{thmnetdual}, which is the following Theorem~2
in~\cite{NP01}.
\begin{theorem} \label{thmnetdualNP}
Let $q$ be a prime power, let $s \ge 1$, $m \ge 1$, and $0\le t\le m$ be integers. Let $C_1,\ldots,C_s$ be $m\times m$ matrices over ${\mathbb{F}}ield_q$ and put
$$C:=(C_1^\top | C_2^\top | \cdots | C_s^\top) \in {\mathbb{F}}_q^{m \times sm}.$$
Let $\mathcal{C}^{\perp}$ be the dual space of the row space $\mathcal{C}$ of $C$.
Then $C_1,\ldots,C_s$ generate a digital $(t,m,s)$-net over ${\mathbb{F}}ield_q$ if and only if $\delta_m (\mathcal{C}^\perp)\ge m-t+1$, where
$\delta_m=\delta_{m,\boldsymbol{1}}$, as introduced in~\cite{NP01}.
{\bf e}nd{theorem}
We now establish a result similar to Lemma~1 in~\cite{NX02}.
\begin{proposition} \label{prNdual}
Let $q$ be a prime power, let $s \ge 2$ and $m \ge 1$ be integers, and let ${\bf e}\in{\mathbb{N}}N^s$. Then from any
${\mathbb{F}}ield_q$-linear subspace $\mathcal{N}$ of ${\mathbb{F}}ield_q^{sm}$ with $\dim (\mathcal{N})\ge sm-m$ we obtain a digital $(u,m,{\bf e},s)$-net over
${\mathbb{F}}ield_q$ with $u=\max\{0,m-\delta_{m,{\bf e}} (\mathcal{N})+1\}$.
{\bf e}nd{proposition}
\begin{proof}
For $\mathcal{C}:=\mathcal{N}^\perp$ we have $\dim (\mathcal{C})\le m$, and so $\mathcal{C}$ is the row space of a suitable digital net over ${\mathbb{F}}ield_q$.
Now let $u=\max\{0,m-\delta_{m,{\bf e}} (\mathcal{N})+1\}$. Then $\delta_{m,{\bf e}}(\mathcal{C}^\perp)=\delta_{m,{\bf e}}(\mathcal{N}) \ge m-u+1$, and so, by Theorem~\ref{thmnetdual},
$\mathcal{C}=\mathcal{N}^\perp$ generates a digital $(u,m,{\bf e},s)$-net over ${\mathbb{F}}ield_q$.
{\bf e}nd{proof}
\begin{remark} \label{redu} {\rm
We obviously have $v_e(\boldsymbol{a}) \le v_1(\boldsymbol{a}) +e-1$ for any $\boldsymbol{a} \in {\mathbb{F}}_q^m$, and so
$$
V_{m,{\bf e}}(\boldsymbol{A}) \le V_{m,{\bf 1}}(\boldsymbol{A})+\sum_{i=1}^s (e_i-1)=V_m(\boldsymbol{A})+\sum_{i=1}^s (e_i-1)
$$
for any $\boldsymbol{A} \in {\mathbb{F}}_q^{sm}$ (compare with Remark~\ref{revm}). It follows that
$$
\delta_{m,{\bf e}}(\mathcal{N}) \le \delta_{m,{\bf 1}}(\mathcal{N})+\sum_{i=1}^s (e_i-1) = \delta_m(\mathcal{N})+\sum_{i=1}^s (e_i-1)
$$
for any linear subspace $\mathcal{N}$ of ${\mathbb{F}}_q^{sm}$. This is in good accordance with \cite[Proposition~1]{HN13} where the term $\sum_{i=1}^s (e_i-1)$ governs
the transition from $(u,m,{\bf e},s)$-nets to $(t,m,s)$-nets.}
{\bf e}nd{remark}
\section{Applications of the duality theory} \label{secad}
In the following, we present an application of the duality theory in Section~\ref{secduality} and of the theory of global function fields. We use the same notation
and terminology for global function fields as in the monograph~\cite{NX09}. In particular, for any divisor $D$ of a global function field $F$, let
$$
\mathcal{L}(D) =\{f \in F^*: {\rm div}(f)+D \ge 0\} \cup \{0\}
$$
be the Riemann-Roch space associated with $D$, where ${\rm div}(f)$ denotes the principal divisor of $f \in F^* := F \setminus \{0\}$. Note that $\mathcal{L}(D)$ is a
finite-dimensional vector space over the full constant field of $F$ (compare with \cite[Section~3.4]{NX09}).
\begin{theorem} \label{thgf}
Let $F$ be a global function field with full constant field ${\mathbb{F}}_q$ and genus $g$. For an integer $s \ge 2$, let $P_1,\ldots,P_s$ be $s$ distinct places of $F$.
Put $e_i=\deg(P_i)$ for $1 \le i \le s$ and ${\bf e} =(e_1,\ldots,e_s) \in {\mathbb{N}}^s$. Then for every integer $m \ge \max (1,g)$ which is a multiple of ${\rm lcm} (e_1,\ldots,e_s)$,
we can construct a digital $(g,m,{\bf e},s)$-net over ${\mathbb{F}}_q$.
{\bf e}nd{theorem}
\begin{proof}
We fix integers $s$ and $m$ satisfying the hypotheses of the theorem. It follows from \cite[Corollary~5.2.10(c)]{St} that for a sufficiently large integer $d >
\max_{1 \le i \le s} e_i$, there exist places $Q_1$ and $Q_2$ of $F$ with $\deg(Q_1)=d+1$ and $\deg(Q_2)=d$. We define the divisor $G$ of $F$ by
$$
G=(sm-m+g-1)(Q_1-Q_2).
$$
Then
\begin{equation} \label{eqdg}
\deg(G)=sm-m+g-1.
{\bf e}nd{equation}
(More generally, we can take any divisor $G$ of $F$ such that $\deg(G)$ satisfies~{\bf e}qref{eqdg} and the support of $G$ is disjoint from the set $\{P_1,\ldots,P_s\}$
of places.) Take any $f \in \mathcal{L}(G)$. Fix $i \in \{1,\ldots,s\}$ for the moment. Let $\nu_{P_i}$ be the normalized discrete valuation of $F$ corresponding to the place
$P_i$. We have $\nu_{P_i}(f) \ge 0$ by the choices of $f$ and $G$, and so the local expansion of $f$ at $P_i$ has the form
$$
f=\sum_{j=0}^{\infty} a_j^{(i)}(f)z_i^j
$$
with all $a_j^{(i)}(f) \in {\mathbb{F}}_{q^{e_i}}$, where $z_i$ is a local parameter at $P_i$. Choose an ${\mathbb{F}}_q$-linear isomorphism $\psi_i: {\mathbb{F}}_{q^{e_i}} \to {\mathbb{F}}_q^{e_i}$ and for
convenience put $k_i=m/e_i \in {\mathbb{N}}$. Then we define the ${\mathbb{F}}_q$-linear map $\theta_i: \mathcal{L}(G) \to {\mathbb{F}}_q^m$ by
\begin{equation} \label{eqti}
\theta_i(f)=\left(\psi_i(a_{k_i-1}^{(i)}(f)),\psi_i(a_{k_i-2}^{(i)}(f)),\ldots,\psi_i(a_0^{(i)}(f)) \right) \qquad \mbox{for all } f \in \mathcal{L}(G).
{\bf e}nd{equation}
Furthermore, we define the ${\mathbb{F}}_q$-linear map $\theta : \mathcal{L}(G) \to {\mathbb{F}}_q^{sm}$ by
$$
\theta (f)=(\theta_1(f),\ldots,\theta_s(f)) \qquad \mbox{for all } f \in \mathcal{L}(G).
$$
By definition, let the ${\mathbb{F}}_q$-linear space $\mathcal{N} \subseteq {\mathbb{F}}_q^{sm}$ be the image of $\theta$.
Now we take any nonzero $f \in \mathcal{L}(G)$. We put
$$
w_i(f)=\min (k_i,\nu_{P_i}(f)) \qquad \mbox{for } 1 \le i \le s.
$$
We claim that for the weights $v_{e_i}(\theta_i(f))$ we have
\begin{equation} \label{eqth}
v_{e_i}(\theta_i(f))=m-e_iw_i(f) \qquad \mbox{for } 1 \le i \le s.
{\bf e}nd{equation}
If $\nu_{P_i}(f) \ge k_i$, then $a_j^{(i)}(f)=0$ for $0 \le j \le k_i-1$, and so $\theta_i(f)=\boldsymbol{0} \in {\mathbb{F}}_q^m$. Then $v_{e_i}(\theta_i(f))=0$, and so~{\bf e}qref{eqth} holds
in this case. In the remaining case, we have $0 \le h_i := \nu_{P_i}(f) < k_i$. Then $a_j^{(i)}(f)=0$ for $0 \le j < h_i$ and $a_{h_i}^{(i)}(f) \ne 0$. It follows that
$\psi_i(a_j^{(i)}(f))=\boldsymbol{0} \in {\mathbb{F}}_q^{e_i}$ for $0 \le j < h_i$ and $\psi_i(a_{h_i}^{(i)}(f)) \ne \boldsymbol{0}$. The definition of $\theta_i(f)$ in~{\bf e}qref{eqti} shows then that
$$
v_{e_i}(\theta_i(f))=(k_i-h_i)e_i=m-e_i \nu_{P_i}(f),
$$
and so~{\bf e}qref{eqth} holds again. Consequently, we obtain
$$
V_{m,{\bf e}}(\theta(f))=\sum_{i=1}^s v_{e_i}(\theta_i(f))=sm- \sum_{i=1}^s e_iw_i(f).
$$
We have $\nu_{P_i}(f) \ge w_i(f)$ for $1 \le i \le s$, and so $f \in \mathcal{L} \left(G-\sum_{i=1}^s w_i(f)P_i \right)$. Since $f \ne 0$, this means that
$$
0 \le {\rm div}(f)+G-\sum_{i=1}^s w_i(f)P_i.
$$
By applying the degree map for divisors and noting that $\deg({\rm div}(f))=0$ according to \cite[Corollary~3.4.3]{NX09}, we deduce that
$$
0 \le \deg(G)- \sum_{i=1}^s e_iw_i(f)=\deg(G)+V_{m,{\bf e}}(\theta(f))-sm.
$$
Therefore by~{\bf e}qref{eqdg},
$$
V_{m,{\bf e}}(\theta(f)) \ge sm-\deg(G)=m-g+1 > 0.
$$
This shows, in particular, that the map $\theta$ is injective, and also
$$
\delta_{m,{\bf e}}(\mathcal{N}) \ge m-g+1.
$$
Moreover,
$$
\dim(\mathcal{N})=\dim(\mathcal{L}(G)) \ge \deg(G)+1-g=sm-m,
$$
where we applied the Riemann-Roch theorem (see \cite[Theorem~3.6.14]{NX09}) in the second step. The rest follows from Proposition~\ref{prNdual}.
{\bf e}nd{proof}
\begin{remark} \label{regf} {\rm
If we combine Theorem~\ref{thgf} with Proposition~\ref{prprd}, then we usually get many more integers $m \ge \max (1,g)$ for which we can construct a digital
$(g,m,{\bf e},s)$-net over ${\mathbb{F}}_q$. For instance, if at least one $e_i=1$, then we obtain a digital $(g,m,{\bf e},s)$-net over ${\mathbb{F}}_q$ for any integer $m \ge \max (1,g)$. }
{\bf e}nd{remark}
\begin{example} \label{exgf} {\rm
Let $q$ be an arbitrary prime power and let $s=q+2$. Let $F$ be the rational function field over ${\mathbb{F}}_q$. Then $F$ has genus $g=0$ and exactly $q+1$ places of
degree $1$ (the infinite place corresponding to the degree valuation and the $q$ places corresponding to the distinct monic linear polynomials over ${\mathbb{F}}_q$).
Choose distinct places $P_1,\ldots,P_s$ of $F$ such that $\deg(P_i)=1$ for $1 \le i \le s-1=q+1$ and $\deg(P_s)=2$. Note that $P_s$ corresponds to a monic
irreducible quadratic polynomial over ${\mathbb{F}}_q$. Then Theorem~\ref{thgf} shows that for every even integer $m \ge 2$ we can obtain a digital $(0,m,{\bf e},q+2)$-net
over ${\mathbb{F}}_q$ with ${\bf e} =(1,\ldots,1,2) \in {\mathbb{N}}^{q+2}$. In combination with Proposition~\ref{prprd}, we get a digital $(0,m,{\bf e},q+2)$-net over ${\mathbb{F}}_q$ for any integer
$m \ge 1$. Note that for ${\bf 1}=(1,\ldots,1) \in {\mathbb{N}}^{q+2}$ and $m \ge 2$, there cannot exist a $(0,m,{\bf 1},q+2)$-net in base $q$, that is, a $(0,m,q+2)$-net
in base $q$, as this would violate a combinatorial bound for nets in~\cite{N87} (see also \cite[Corollary~4.19]{DP} and \cite[Corollary~4.21]{N92}). }
{\bf e}nd{example}
\begin{example} \label{exco} {\rm
Let $q=2$, $s=4$, $m=3$, and ${\bf e} =(1,1,1,2)$. The following is a concrete example of a digital $(0,3,{\bf e},4)$-net over ${\mathbb{F}}_2$. The four generating matrices over
${\mathbb{F}}_2$ are given by
$$
C_1=\left(
\begin{array}{ccc}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
{\bf e}nd{array}
\right), \ \
C_2=\left(
\begin{array}{ccc}
0 & 0 & 1\\
1 & 1 & 0\\
0 & 1 & 0
{\bf e}nd{array}
\right), \ \
C_3=\left(
\begin{array}{ccc}
0 & 1 & 1\\
1 & 0 & 1\\
0 & 0 & 1\\
{\bf e}nd{array}
\right), \ \
C_4=\left(
\begin{array}{ccc}
1 & 0 & 1\\
0 & 1 & 0\\
0 & 0 & 0
{\bf e}nd{array}
\right).
$$
It is easily verified that the row vectors of these matrices form a $(3,3,{\bf e},4)$-system over ${\mathbb{F}}_2$ (note that from $C_4$ we take either no row vectors or
the first two row vectors). Hence it follows from Lemma~\ref{lemnetsystem} that $C_1,C_2,C_3,C_4$ do indeed generate a digital $(0,3,{\bf e},4)$-net over ${\mathbb{F}}_2$. }
{\bf e}nd{example}
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{\bf e}nd{document} | math | 65,775 |
\begin{document}
\parindent=0cm
\title{Trees maximizing the number of almost-perfect matchings}
\begin{abstract}
We characterize the extremal trees that maximize the number of almost-perfect matchings, which are matchings covering all but one or two vertices, and those that maximize the number of strong almost-perfect matchings, which are matchings missing only one or two leaves. We also determine the trees that minimize the number of maximal matchings. We apply these results to extremal problems on the weighted Hosoya index for several choices of vertex-degree-based weight function.
\end{abstract}
\blfootnote{2020 Mathematics Subject Classification. Primary 05C05; Secondary 05C09, 05C70}
\blfootnote{Keywords: trees, matching, almost-perfect matching, maximal matching, weighted Hosoya index}
\section{Introduction}\label{sec:intro}
A matching in a graph $G$ is a set of edges such that no two edges share a vertex. Matching theory has a rich history with many applications. In particular, extremal problems and enumeration problems involving matchings have been studied extensively. A maximum matching can be the best solution to a variety of problems, such as scheduling and task assignment problems or matching donors with receivers~\cite{Gale-Shapley-1962,LP86}.
Computing the number of matchings in a graph, also called the Hosoya index, is \#P-complete in general. In this paper, we focus on enumeration of matchings in trees.
It is known that the number of matchings in the path graph $P_n$ is equal to $F_n$, the $n$th Fibonacci number, and that this is the maximum over all $n$-vertex trees~\cite{Gutman80}. In fact, the star and the path respectively minimize and maximize not only the number of matchings but also the number of induced matchings~\cite{KKKL17}, and the number of matchings of a fixed size~\cite[Thm.~4.7.3]{WH19}. In the last case, we show that the path is, in fact, the unique maximizer. For a more comprehensive background on matching enumeration problems, see e.g. the survey of Wagner and Gutman~\cite{WG10}, or the book by Wagner and Wang~\cite{WH19}.
To prove that the path uniquely maximizes the number of matchings of a fixed size, we require results on almost-perfect matchings which are of independent interest.
Perfect matchings are central to the study of matching theory---Hall's matching theorem and Tutte's theorem are perhaps the most notable examples \cite{Hall1935, Tutte47}. However, the notion of a matching that is ``close to perfect'' has been less prevalent. When the number of vertices in a graph is odd, an almost-perfect matching is a matching that avoids a single vertex. This is also called a near-perfect matching, and enumeration problems for other classes of graphs (such as factor-critical) have been considered~\cite{Liu02}.
When the number of vertices is even, an almost-perfect matching avoids two vertices. We characterize the extremal trees in both cases and prove the following:
\begin{theorem}\label{thr:odd_apm}
If $n$ is odd, a tree $T$ of order $n$ has at most $\frac{n+1}{2}$ almost-perfect matchings.
Equality holds if and only if $T$ is a $1$-subdivision of a tree of order $\frac{n+1}{2}.$
\end{theorem}
\begin{theorem}\label{thr:even_apm}
If $n$ is even, a tree $T$ of order $n$ has at most $\binom{ \frac{n}{2}+1}{2}=\frac{n(n+2)}{8}$ almost-perfect matchings.
\begin{itemize}
\item For $n > 4$, equality holds if and only if $T$ is the path $P_n$.
\item For $n=4$, equality holds for both $P_4$ and $S_4$.
\end{itemize}
\end{theorem}
We consider the same questions for strong almost-perfect matchings, which are matchings that cover all vertices except one or two leaves, and again find precise characterizations of the extremal graphs. When the order is odd, the number of strong almost-perfect matchings is maximized by the basic spider (a star with all but at most one edge subdivided once, as presented in Figure~\ref{fig:spiders}); this is the same construction that, for example, maximizes the number of maximal independent sets~\cite{Wilf86}.
\begin{theorem}\label{thr:sapmcombined}
Let $T$ be a tree of order $n\ge 28$.
Then the number of strong almost-perfect matchings in $T$ is at most $\frac{n-1}{2}$ if $n$ is odd and $\floorfrac{(n-4)^2}{12}$ if $n$ is even.
Equality holds if and only if $T$ is a spider (when $n$ is odd) or a balanced spider-trio (when $n$ is even).
\end{theorem}
In fact, we are able to give a complete picture of the bounds and extremal trees for all values of $n$, but postpone the precise statements until Section~\ref{sec:sapm}.
The trees that maximize the number of maximal matchings have been previously determined by G\'{o}rska and Skupie\'{n}~\cite{GS07}.
Surprisingly, the characterization of the trees that minimize the number of maximal matchings has not appeared in the literature.
We show that the extremal trees are precisely the most basic spiders again, or a special spider if $n$ is odd.
\begin{theorem} \label{thr:minmaxmatchings}
A tree $T$ of order $n$ has at least $\lceil \frac n2 \rceil $ maximal matchings.
Equality holds if and only if $T$ is a spider $\Sp_n$, or an odd special spider $\SSS_n$.
\end{theorem}
In Section~\ref{sec:HosoyaIndex}, we consider applications to the weighted Hosoya index $Z_\phi(T)$, where each edge is assigned a weight by $\phi$ and the weight of a matching is the product of its edge weights. Motivated by recent results by Cruz, Gutman, and Rada~\cite{CRUZ2022}, we consider certain choices of vertex-degree-based weight functions and prove that while the minimizer is the star in all cases that we consider, the maximizer can be the path, the double-broom, or a type of spider.
The outline of the paper is as follows: in Section~\ref{sec:ExtrTrees_mk}, we prove Theorems~\ref{thr:odd_apm} and~\ref{thr:even_apm}. In Section~\ref{sec:sapm}, we prove Theorem~\ref{thr:sapmcombined}. In Section~\ref{sec:minmaxmatching}, we prove Theorem~\ref{thr:minmaxmatchings}. In Section~\ref{sec:HosoyaIndex}, we prove results related to the weighted Hosoya index. We conclude by discussing some further questions to pursue in Section~\ref{sec:future}.
\subsection{Preliminaries}\label{subsec:def¬}
In this subsection, we introduce definitions and notation that will be used throughout the paper.
Given a graph $G=(V,E)$, a \emph{matching} $M$ in $G$
is a set of edges such that no two edges share a common
vertex. A matching is \emph{maximal} if it is not a subset of any other matching.
A matching is a \emph{maximum matching} if it has the largest possible cardinality.
A \emph{perfect} matching is a matching that includes all vertices.
The {\em diameter} of a graph $G$,
denoted $diam(G)$, is the maximum distance between any pair of vertices over all pairs in $G$. When we say ``a diameter of $G$,'' we mean a path with length equal to $diam(G)$. A {\em $1$-subdivision of an edge} $uv$ is obtained by adding a new vertex $x_{uv}$ and replacing $uv$ with the path $\{u, x_{uv}, v\}$. A {\em $1$-subdivision of a graph} is a graph all of whose edges are ($1$-)subdivided.
\begin{defi}
In a tree of order $n$, a matching $M$ is an {\em almost-perfect matching} (APM) if it covers all but one vertex when $n$ is odd and all but two vertices when $n$ is even. The vertices not covered by $M$ are called the {\em avoided vertices}. $M$ is a {\em strong almost-perfect matching} (SAPM) if the avoided vertices are leaves.
\end{defi}
Let $\mathcal M(G)$ be the set of all matchings of $G$, $\mathcal M_k(G)$ be the set of matchings in $G$ of size $k$, and $m_k(G) = |\mathcal M_k(G)|$ with $m_0(G) = 1$.
The \emph{Hosoya index} of $G$ is the number of matchings of $G$, which can be written as
$$Z(G)= |\mathcal M(G)| = \Sigma_{k\geq0} m_k(G).$$
The name of the index refers to Hosoya~\cite{HOSOYA71} who observed that if $G$ is a molecular graph, the Hosoya index is correlated with its chemical properties.
Let $(G,\omega)$ denote a graph with weight function
$\omega:E\to\mathbb{R}^+$. The weight of a matching $M$ is then the product of the weights of the edges in $M$,
and the \emph{weighted Hosoya index} is
$$Z(G, \omega) = \sum_{M \in \mathcal{M}}\left(\prod_{e \in M} \omega(e)\right)$$
As an aside, observe that this coincides with the definition of the (multivariate) monomer-dimer partition function from statistical physics when $\omega(e)$ is a positive constant $\lambda_e$ for each edge $e$.
Let $e=uv$ with $\deg(u)=i$
and $\deg(v)=j$. The weight function
is \emph{vertex-degree-based} if $\omega(e)=\phi(i,j)$
where $\phi$ is a function such that $\phi(i,j) = \phi(j,i)$.
In this case, we denote $Z(G,\omega)$ by $Z_{\phi}(G).$
We finish this section with some statements that we will use in our proofs throughout. We will refer to the fact $\sum_{v \in V(G)} \deg(v) = 2|E(G)|$ as the ``handshaking lemma.'' The following is also a well-known statement:
\begin{lemma}\label{lem:treepm}
Every tree has at most one perfect matching.
\end{lemma}
\begin{proof}
Suppose $M$ and $M'$ are two perfect matchings of $T$. In the graph $(V(T), M \cup M')$, every component must be either a single edge (contained in both $M$ and $M'$) or a cycle. As $T$ is a tree, it contains no cycles and so we must have $M = M'$.
\end{proof}
A sequence $(x_i)_{i=1}^k$ where $x_1 \geq x_2 \geq \cdots \geq x_k$ {\em majorizes} the sequence $(y_i)_{i=1}^k$, $y_1 \geq y_2 \geq \cdots \geq y_k$, if and only if $\sum_{i=1}^j x_i \geq \sum_{i=1}^j y_i$ for every $1 \le j \le k$ and equality holds for $j=k$.
Observe that an equivalent definition is the following: $(x_i)_{i=1}^k$ where $x_1 \leq x_2 \leq \cdots \leq x_k$ majorizes $(y_i)_{i=1}^k$, $y_1 \leq y_2 \leq \cdots \leq y_k$, if and only if $\sum_{i=1}^j x_i \leq \sum_{i=1}^j y_i$ for all $1 \leq j \leq k$ with equality when $j = k$.
\begin{lemma}[Karamata's inequality, \cite{Kar32}]\label{lem:karamata}
If $(x_i)_{i=1}^k$ is a sequence of real numbers that majorizes $(y_i)_{i=1}^k$, and $f$ is a real-valued convex function, then $\sum_{i=1}^k f(x_i) \geq \sum_{i=1}^k f(y_i)$. This inequality is strict if the sequences are not equal and $f$ is a strictly convex function. If $f$ is concave, the reverse inequality holds.
\end{lemma}
\section{Maximum number of matchings of a fixed size}\label{sec:ExtrTrees_mk}
In this section, we characterize the trees $T$ of order $n$ maximizing the number of almost-perfect matchings and use this to characterize those maximizing the number of matchings of a fixed size $k$.
When $n=2k$, any tree with a perfect matching is extremal since the perfect matching must be unique. While it is straightforward to check if a given tree has a perfect matching, there is not a clear characterization of the set of all trees with perfect matchings (there are exponentially many of them~\cite{Simion91}).
When $n=2k+1$, we prove that there are many extremal trees maximizing the number of almost-perfect matchings. For convenience, we recall the theorem statement:
\begin{customthm} {\bf \ref{thr:odd_apm}} If $n$ is odd, a tree $T$ of order $n$ has at most $\frac{n+1}{2}$ almost-perfect matchings.
Equality holds if and only if $T$ is a $1$-subdivision of a tree of order $\frac{n+1}{2}.$
\end{customthm}
\begin{proof}
Recall that every tree has a unique bipartition, so let $T=(X \cup Y,E)$. If $T$ has an almost-perfect matching, then the bipartition classes must differ by $1$ and the larger one, without loss of generality $X$, contains precisely $\frac{n+1}{2}$ vertices. This immediately implies the upper bound, since $T\setminus y$ for $y \in Y$ cannot have a perfect matching.
Next, we prove the characterization of the extremal trees.
Let $T'$ be an arbitrary tree of order $\frac{n+1}{2}$.
Let $T$ be a 1-subdivision of $T'$, with $Y$ the set of added vertices.
For every $v \in V(T')$, we can construct a perfect matching of $T \setminus v$: viewing $T$ as a tree rooted at $v$, match each $u \in V(T')\setminus v$ with its neighbor (in $Y$) on the unique path from $u$ to $v$.
Hence any subdivision of a tree of order $\frac{n+1}{2}$ contains $\frac{n+1}{2}$ almost-perfect matchings.
We prove the other direction by induction on $n$.
The cases $n \in \{1,3\}$ are straightforward, since $P_3$ is the only tree of order $3$ and is the $1$-subdivision of $P_2$.
Thus, assume the statement is true for $n-2,$ where $n \ge 5$.
Let $T=(X \cup Y,E)$ be a tree that attains equality and without loss of generality, let $|X| = \frac{n+1}{2}$. Then for every choice of $v \in X$, $T \setminus v$ has a perfect matching.
Observe that every leaf of $T$ must be in $X$. If there is a leaf in $Y$, then its unique neighbor in $X$ cannot be avoided by an almost-perfect matching, since any such matching would also avoid the leaf itself.
This implies that no two leaves can have the same neighbor. Indeed, if $x_1, x_2 \in X$ are leaves, then for any $v \in X$ such that $v \neq x_1, x_2$, the almost-perfect matching that avoids $v$ would have to cover both leaves $x_1$ and $x_2$, a contradiction.
Now let $x \in X$ be a leaf such that its unique neighbor $y$ has degree 2 (such a leaf must exist, e.g.~one of the ends of a diameter). Then $T':=T \setminus \{x, y\}$ is a tree. Furthermore, for any $v \in X \setminus \{x\}$, we know that $T \setminus \{v\}$ has a perfect matching, which must contain the edge $xy$. Thus, $T'$ contains $\frac{n-1}{2}$ almost-perfect matchings so we may apply the inductive hypothesis to conclude that $T'$ is a $1$-subdivision of a tree, and hence $T$ is as well.
\end{proof}
Next, we will show that when $n=2k+2$, the path is extremal in all non-trivial cases. We start with a lemma that will be applied in the proof of this and some further theorems.
\begin{lemma}\label{lem:leavesatdistance2=>n_bound}
If $n$ is even and $T$ is a tree of order $n$ with at least one pair of leaves sharing a common neighbor, then $T$ has at most $n-1$ almost-perfect matchings.
\end{lemma}
\begin{proof} Recall that for $n$ even, an almost-perfect matching $M$ covers all but two vertices, and we call these vertices the avoided pair of $M$.
Let $T$ be a tree of order $n$, and let $u$ and $u'$ be the two leaves with a common neighbor (siblings).
No matching can cover both leaves, so for every almost-perfect matching $M$ of $T$, at least one of $u$ and $u'$ is contained in the avoided pair of $M$.
By Theorem~\ref{thr:odd_apm}, $T\setminus u$ and $T\setminus u'$ have both at most $\frac{n}{2}$ almost-perfect matchings.
Equality is only possible if $T\setminus u$ and $T\setminus u'$ are $1$-subdivisions of a tree. Since the perfect matching of $T\setminus \{u, u'\}$ has been counted twice, $T$ has at most $n-1$ almost-perfect matchings.
Because $T\setminus u$ and $T\setminus u'$ are isomorphic, we also know that equality is attained if and only if $T$ is a $1$-subdivision of a tree of order $\frac{n}{2}$ where one copy of a leaf is added (i.e. a copy $u'$ of a leaf $u$ with the same neighbor).
\end{proof}
We use this to prove the case for $n$ even:
\begin{customthm} {\bf \ref{thr:even_apm}}
If $n$ is even, a tree $T$ of order $n$ has at most $\binom{ \frac{n}{2}+1}{2}=\frac{n(n+2)}{8}$ almost-perfect matchings.
\begin{itemize}
\item For $n > 4$, equality holds if and only if $T$ is the path $P_n$.
\item For $n=4$, equality holds for both $P_4$ and $S_4$.
\end{itemize}
\end{customthm}
\begin{proof}
One can check by hand that for $n = 4$ both $P_4$ and $S_4$ have three almost-perfect matchings, for $n=6$ the path $P_6$ has eight almost-perfect matchings, and in both cases these are the only examples attaining the maximum.
Now let $n \ge 8$ and suppose the claim holds for all even numbers less than or equal to $n-2$.
Since $n < \frac{n(n+2)}{8}$, by Lemma~\ref{lem:leavesatdistance2=>n_bound} we may assume there are no two leaves with a common neighbor. Taking a diameter of the tree, we note that there is an edge $uv$ with $u$ a leaf and $v$ its neighbor of degree $2$. Then either $uv$ is contained in the almost-perfect matching, or the avoided pair contains $u$. We count the number of matchings in each case and take the sum to get the total number of almost-perfect matchings in $T$.
For the first term, the number of almost-perfect matchings containing $uv$ is equal to the number of almost-perfect matchings in $T \setminus \{u,v\}$ which by the inductive hypothesis is at most $\frac{n(n-2)}{8}$.
For the second term, the number of avoided pairs containing $u$ is at most the number of almost-perfect matchings in $T \setminus u$, which is bounded by $\frac{(n-1)+1}{2}$ by Theorem~\ref{thr:odd_apm}.
Since $\frac{n(n-2)}{8}+\frac{(n-1)+1}{2}=\frac{n(n+2)}{8}$, the first statement of the theorem holds by induction.
To characterize equality, we see by the inductive hypothesis that equality holds for the first term if and only if $T \setminus \{u, v\} = P_{n-2}$. By Theorem~\ref{thr:odd_apm}, equality holds for the second term if and only if $T \setminus \{u\}$ is a 1-subdivision of a tree of order $\frac n2$. Since $v$ must have degree 2, these imply that equality holds overall if and only if $T = P_n$.
\end{proof}
Another way of stating Theorem~\ref{thr:even_apm} is that $P_n$ maximizes $m_k(T)$ when $n = 2k+2$ and $k \geq 2$. We can extend this to larger values of $n$. Note that $m_1(T)=n-1$ for every tree $T$ and so it has been omitted in the statement.
\begin{theorem}\label{thr:maxmatch}
If $k \geq 2$ and $T$ is a tree of order $n$ with $n \ge 2k+2$, then
$m_k(T) \le m_k(P_n)=\binom{n-k}{k}$. Equality holds if and only if $T=P_n.$
\end{theorem}
\begin{proof}
We will prove that the statement is true for forests of order $n$ using induction on $k$.
When $k=2$, for a tree $T$ we have $m_2(T)=\binom{n-1}{2}-\sum_v \binom{\deg(v)}{2}.$ Observe that the degree sequence of the path $P_n$ is majorized by the degree sequence of any other $n$-vertex tree. Indeed, the degree sequence of $P_n$ is $(2, 2, \ldots, 2, 1, 1)$. If $T \neq P_n$ has degree sequence $(d_1 \geq d_2 \geq \cdots \geq d_n)$ and $\sum_{i=1}^j d_i < 2j$ for some $j \leq n-2$, then we must have $d_i = 1$ for all $i \geq j$. Then $\sum_{i=1}^n d_i < 2j + (n-j) \leq 2n-2$, a contradiction since $\sum_{i=1}^n d_i = 2(n-1)$ by the handshaking lemma. Thus, we may apply Lemma~\ref{lem:karamata} (with strict inequality since the degree sequences of $T$ and $P_n$ must be different) to conclude that for any $T \neq P_n$, we have $m_2(T) < m_2(P_n) = {n-2 \choose 2}$.
If $F$ is a forest with at least two components, then $|E(F)| \leq n-2$ and as such, $m_2(F) \leq {n-2 \choose 2}$. Equality is not possible when $n > 4$.
Now assume the statement holds for $m_{k-1}(T)$ for any $T$ on $n \geq 2k$ vertices.
When $n=2k+2$, a matching of size $k$ is an almost-perfect matching and so $P_n$ is extremal among trees by Theorem~\ref{thr:even_apm}.
For a forest $F$ with multiple components, there are three cases.
{\bf Case 1:} All components of $F$ have even size. Let the components of $F$ be $T_1, T_2, \dots, T_r$ where $|T_i| = 2k_i$ for $1 \leq i \leq r$. An almost-perfect matching of $F$ consists of an almost-perfect matching of $T_i$ for some $i$ and a perfect matching of $T_j$ for every $j \neq i$, of which there is at most one by Lemma~\ref{lem:treepm}. Then
$$m_k(F) \le \sum_{i=1}^r m_{k_i-1}(T_i) \leq \sum_i m_{k_i - 1}(P_{2k_i})$$
We claim this is strictly less than $m_k(P_n)$. By Theorem~\ref{thr:even_apm}, it suffices to observe that
$$\sum_{i=1}^r \binom{k_i +1}{2} < \binom {\sum k_i + 1}2.$$
This is true by induction on $r$ since $\binom{a+b+1}{2} > \binom{a+1}{2}+\binom{b+1}2$ for every $a,b \ge 1$ (a straightforward computation).
{\bf Case 2:} At least four components are odd. Then $m_k(F)=0$ since at least one vertex from each of the four components must be avoided.
{\bf Case 3:} Exactly two components are odd, with order $a$ and $b$. In the last case, we apply Theorem~\ref{thr:odd_apm} and the inequality between the arithmetic and geometric mean (AM-GM) to conclude
$m_k(F) \le \frac{a+1}{2}\cdot \frac{b+1}{2} \le \left(\frac{k+2}{2}\right)^2 < \binom{k+2}{2}=m_k(P_n).$
Thus, the statement is true for forests when $n=2k+2.$
So for $n\ge 2k+3$, we may assume that the claim holds for $n-1$.
Let $F$ be a forest and $u$ a leaf of $F$, with unique neighbor $v$ (a forest without edges is clearly not extremal).
We observe that $m_k(F)=m_k(F \setminus u) + m_{k-1}(F \setminus uv)$, since either $u$ is covered by the matching or it is not.
By the inductive hypothesis, we have that $m_k(F \setminus u)$ is maximized when $F \setminus u = P_{n-1}$ and $m_{k-1}(F \setminus uv)$ is maximized when $F \setminus uv = P_{n-2}$. This shows that $m_k(F)$ is maximized if and only if $F = P_n$, completing the induction.
\end{proof}
\section{Maximum number of strong almost-perfect matchings}\label{sec:sapm}
Recall that a strong almost-perfect matching (SAPM) is a matching which avoids one leaf when $n$ is odd and two leaves when $n$ is even. In this section, we characterize the trees that maximize the number of {SAPM}s with three theorems which, combined, give a more complete version of Theorem~\ref{thr:sapmcombined}.
We first define the relevant families of trees. Often a spider can refer to any subdivision of a star, i.e. any tree which has at most one vertex of degree at least $3$. We will refer to a spider $\Sp_n$ as the unique spider of order $n$ whose legs all have length $2$, except possibly one leg of length $1$ when $n$ is even.
\begin{defi}
An {\em odd spider} is a $1$-subdivision of a star. An {\em even spider} is an odd spider with an additional pendant edge added to the unique vertex of degree $\geq 3$. We use the notation $\Sp_n$ for both. See Figure~\ref{fig:spiders}.
\end{defi}
\begin{figure}
\caption{Two spiders}
\label{fig:oddspider}
\label{fig:evenspider}
\label{fig:spiders}
\end{figure}
\begin{defi}
A {\em double broom} $\DB_{a,b}$ is a tree of diameter $3$ whose non-leaves have degree $a+1$ and $b+1$ respectively.
A {\em balanced double broom} on $n$ vertices is a tree of diameter $3$ whose non-leaves have degree $\lfloor \frac{n}{2} \rfloor$ on one side and $\lceil \frac{n}{2} \rceil$ on the other side. See Figure~\ref{fig:doublebroom}.
\end{defi}
\begin{figure}
\caption{A double broom and wide spider}
\label{fig:doublebroom}
\label{fig:widespider}
\label{fig:db+widespider}
\end{figure}
\begin{theorem}\label{thr:odd_sapm}
Let $n$ be odd. If $n \geq 5$, a tree $T$ of order $n$ has at most $\frac{n-1}{2}$ strong almost-perfect matchings.
For $n \geq 7$, equality holds if and only if $T$ is an odd spider.
For $n=5$, equality holds for both $\DB_{1,2}$ and $P_5.$
\end{theorem}
\begin{proof}
If $T$ is a tree that contains two leaves with a common neighbor, then $T$ has at most two {SAPM}s, since every matching must avoid at least one of these two leaves.
Otherwise, $T$ has at least as many internal vertices as leaves and thus at most $\frac{n-1}{2}$ leaves. It follows that $T$ has at most $\frac{n-1}{2}$ {SAPM}s. The $n$-vertex spider achieves this upper bound.
For $n=5$, we have $\frac{n-1}{2}=2$, and one can check by hand that the extremal graphs are the path $P_5$ (which is a spider) and the double broom $\DB_{1,2}$.
For $n \ge 7$, let $T$ be a tree of order $n$ with $\frac{n-1}{2}$ {SAPM}s. Then $T$ must have $\frac{n-1}{2}$ leaves $\{u_i : 1 \leq i \leq \frac{n-1}{2}\}$. For each $i$, let $v_i$ be the unique neighbor of $u_i$. This gives us $n-1$ vertices, so $T$ contains one additional vertex $x$. For each $i$, $T \setminus \{u_i\}$ has a perfect matching, which must contain $u_jv_j$ for all $j \neq i$. Thus, $x \sim v_i$ for all $i$, and so $T$ is indeed a spider.
\end{proof}
When $n$ is even, an almost-perfect matching $M$ of $T$ contains all but two vertices, so counting the number of almost-perfect matchings of $T$ is equivalent to counting the number of avoided pairs $P \subset V(T)$ for which $T \setminus P$ has a perfect matching. This is the strategy we employ to prove the next theorem, in which we have the additional assumption that $T$ contains a perfect matching.
\begin{defi}
A {\em wide spider} is constructed by subdividing every pendant edge of a double broom. For $n=2k$, the {\em balanced wide spider} $W_n$ on $n$ vertices is the tree one obtains from subdividing every pendant edge of a balanced double broom on $k+1$ vertices. See Figure~\ref{fig:widespider}.
\end{defi}
\begin{theorem}\label{thr:even_sapm_wpm}
Let $n$ be even. If $T$ is a tree of order $n$ that contains a perfect matching, then the number of strong almost-perfect matchings in $T$ is at most
$$\max \left\{1,\frac{n-2}{2}, \floorfrac{(n-2)^2}{16}\right\} = \begin{cases} 1 & n = 2, \\ \frac{n-2}{2} & 2 < n \leq 10, \\ \floorfrac{(n-2)^2}{16} & n \geq 10.\end{cases}$$
Equality holds if and only if either
\begin{itemize}
\item $n \leq 10$, and $T$ is obtained by attaching a leaf to each vertex of a tree of order $\frac{n}{2}$, or
\item $n \geq 10$, and $T$ is the balanced wide spider.
\end{itemize}
\end{theorem}
\begin{proof}
If $n = 2$, then there is precisely one SAPM, namely the empty matching, so assume that $n > 2$.
Let $T$ be an arbitrary tree of order $n$ with a perfect matching. Let $\{u_1, \dots, u_k\}$ be the leaves of $T$. Observe that no two leaves have the same neighbor; else, the perfect matching would match both with their common neighbor. It follows that $k \leq \frac{n}{2}$. Let $v_i$ be the unique neighbor of $u_i$, $1 \leq i \leq k$.
Recall that a bipartite graph can have a perfect matching only if the parts of the bipartition are the same size. Consider the (unique) bipartition $V(T) = X \cup Y$. An SAPM{} of $T$ that avoids $u_i$ and $u_j$ is a perfect matching of $T \setminus \{u_i, u_j\}$. This can occur only if $u_i$ and $u_j$ belong to different parts of the bipartition $X \cup Y$. Thus, there are at most $\floorfrac{k^2}{4}$ {SAPM}s of $T$.
If $k = \frac{n}{2}$, then $T[\{v_1, \dots, v_k\}]$ must be a tree of order $\frac{n}{2}$, and every SAPM{} in $T$ can only avoid $u_i$ and $u_j$ such that $v_i \sim v_j$. This gives $\frac{n}{2}-1 = \frac{n-2}{2}$ {SAPM}s in $T$, one for each edge of $T[\{v_1, \dots, v_k\}]$.
Otherwise, $k \leq \frac{n}{2} - 1 = \frac{n-2}{2}$, which means that there are at most $\floorfrac{(n-2)^2}{16}$ {SAPM}s. This is achieved by the wide spider $W_n$. Indeed, for every pair of leaves on opposite sides, there is an APM{} that avoids those leaves, so the number of {SAPM}s in $W_n$ is $\floorfrac{(n-2)^2}{16}$.
Thus, we have shown that the maximum is
$$\max \left\{1,\frac{n-2}{2}, \floorfrac{(n-2)^2}{16} \right\},$$
and it remains to discuss the cases of equality. Note first that
$\frac{n-2}{2} \geq \floorfrac{(n-2)^2}{16}$ if and only if $n \leq 10$ (with equality for $n=10$), so the case that there are $k = \frac{n}{2}$ leaves yields the maximum for $2 < n \leq 10$, while the case that there are $k = \frac{n}{2} - 1$ leaves and $\floorfrac{k^2}{4}$ {SAPM}s yields the maximum for $n \geq 10$.
It only remains to show that the wide spider $W_n$ is unique in the second case. Suppose $T$ has $k = \frac{n-2}2$ leaves and $\floorfrac{(n-2)^2}{16}$ almost-perfect matchings. Let $x$ and $y$ be the two vertices that are neither leaves nor neighbors of leaves. Their degree is at least $2$. The perfect matching of $T$ must contain the edge $u_iv_i$ for all $i$, so $x \sim y$. Suppose without loss of generality that $x \in X, y \in Y$. Then there is some $v_i \in X, v_j \in Y$ such that $x \sim v_j$ and $y \sim v_i$.
Let $v_\ell \in X$ for $\ell \neq i$. Suppose for the sake of contradiction that $v_\ell \not\sim y$. For each $m$ such that $v_m \in Y$, in order to have a perfect matching of $T \setminus \{u_\ell, u_m\}$, we must have $v_\ell \sim v_m$. This implies $v_m \not\sim x$ for all $m \neq j$; else, $\{x, v_j, v_\ell, v_m\}$ would form a cycle. Then there cannot be an almost-perfect matching that avoids $u_i$ and $u_m$ for $m \neq j$, since there is no disjoint set of edges to cover $v_m, v_\ell$, and $u_\ell$, which is a contradiction.
This tells us that $v_\ell \sim y$ for all $\ell$ such that $v_\ell \in X$, and a similar argument shows that $v_m \sim x$ for all $m$ such that $v_m \in Y$, and so $T$ is isomorphic to the balanced wide spider.
\end{proof}
When we remove the requirement that $T$ contains a perfect matching, we can still characterize the extremal examples as follows:
\begin{defi}
An {\em even special spider} $\SSS_n$ on $n$ vertices for $n$ even is an odd spider $\Sp_{n-1}$ with a sibling added for one of the leaves. See Figure~\ref{fig:evenspecialspider}.
A {\em spider trio} $\ST_{a,b,c}$ is a tree constructed from three spiders $\Sp_a$, $\Sp_b$, $\Sp_c$ whose center vertices are attached to a single new vertex. See Figure~\ref{fig:spider-trio}.
\end{defi}
\begin{figure}
\caption{Other variants of spiders}
\label{fig:evenspecialspider}
\label{fig:spider-trio}
\label{fig:adaptedspiders}
\end{figure}
\begin{theorem}\label{thr:even_sapm_gen}
Let $f(n)$ be the maximum number of SAPM's in a tree $T$ of order $n$. Then, for $n$ even,
$$f(n)=
\begin{cases}
\floorfrac{3n}{4} & \text{ if } 2 \le n \le 6,\\
n-3 & \text{ if } 8 \le n \le 14,\\
\floorfrac{n^2}{16}-1 & \text{ if } 16 \le n \le 28,\\
\floorfrac{(n-4)^2}{12} & \text{ if } n \ge 28.\\
\end{cases}$$
Equality holds if and only if $T$ is one of the following trees:
$$\begin{cases}
P_2, S_4, \& \DB_{2,2} &\text{ for } 2 \le n \le 6,\\
\SSS_n & \text{ if } 8 \le n \le 12,\\
\SSS_{14} \& \ST_{3,2,0} & \text{ if } n=14,\\
\ST_{\ceilfrac{n-4}{4}, \floorfrac{n-4}{4},0} & \text{ if } 16 \le n \le 26,\\
\ST_{6,6,0} \& \ST_{4,4,4} & \text{ if } n=28,\\
\ST_{\floorfrac{n}{6}, \floorfrac{n-2}{6},\floorfrac{n-4}{6}} & \text{ if } n \ge 28,\\
\end{cases}$$
\end{theorem}
\begin{proof}
For $n \leq 16$, the values of $f(n)$ have been checked with a computer, so we may assume $n\ge 18$.
Let $T$ be a tree on $n$ vertices. If two leaves of $T$ share a common neighbor, then by Lemma~\ref{lem:leavesatdistance2=>n_bound}, $T$ has at most $n$ {SAPM}s. Since $f(n)>n$ (as evidenced by the trees given in the claim), $T$ cannot be extremal in this case.
Thus, we may also assume that every leaf has a unique neighbor and so the number of leaves is at most $\frac n2$.
If $T$ has exactly $\frac n2$ leaves, then $T$ must also contain a perfect matching. By Theorem~\ref{thr:even_sapm_wpm}, $T$ has at most $\floorfrac{(n-2)^2}{16}$ {SAPM}s, which is smaller than both $\floorfrac{n^2}{16}-1$ and $\floorfrac{(n-4)^2}{12}$ when $n \geq 18$.
Now suppose the number of leaves in $T$ is at most $\frac n2-1.$ Let $V(T)=X \cup Y$ be the bipartition of $T$, and without loss of generality, suppose $\abs X \geq \abs Y$. Recall that $T$ containing an almost-perfect matching implies that either $\abs X = \abs Y$ or $\abs X = \abs Y + 2$.
If $\abs X = \abs Y,$ then every avoided pair of leaves must have one element in $X$ and one element in $Y$. There are at most $\floorfrac{(n-2)^2}{16}$ such pairs, which again is less than the stated upper bounds.
So now we assume that $\abs X = \abs Y+2=\frac n2+1.$
{\bf Case 1:} $Y$ contains no leaves. Then the degree of every vertex in $Y$ is at least $2$. As $\abs Y = \frac{n-2}{2}$, and $\sum_{v \in Y}\deg(v) = n-1$, this implies that all but one vertex in $Y$ has degree 2. Let $z$ be the unique vertex in $Y$ which has degree 3.
In that case $T \setminus z$ has three components---call them $A, B, C$. Each component is itself a tree whose vertices in $Y$ have degree 2. Thus each of them has one more vertex in $X$ than in $Y$. In particular, each component has an odd number of vertices in total.
Observe that if a component, say $C$, of $T \setminus z$ does not contain a member of the avoided pair, then since $\abs C$ is odd, the SAPM{} must match $z$ with its neighbor in $C$. Thus, an avoided pair of leaves cannot be contained in a single component.
Let $a,b,c$ be the number of leaves in $A, B$, and $C$ respectively, with $a \ge b \ge c$. Then the number of {SAPM}s is bounded above by $ab+bc+ac.$
If $B$ and $C$ are singletons, this gives an upper bound of $1+2a< n$.
If only $C$ is a singleton, then we have $ab+a+b=(a+1)(b+1)-1\le \floorfrac{n^2}{16}-1.$
Furthermore, we can see that equality is attained only for the tree $\ST_{a,b,0}$ when $a = b = \frac{n}{4}$. Indeed, each SAPM{} of $T$ is a union of {SAPM}s of $A$ and $B$, and the edge from $z$ to $C$. By Theorem~\ref{thr:odd_sapm}, $A$ and $B$ have the maximum number of {SAPM}s if and only if they are both spiders.
Else, each neighbor of $z$ is a non-leaf vertex in $X$, and as such $a+b+c \le \frac{n-4}{2}.$
By the AM-GM inequality, we know that $ab+bc+ac \le \frac{(a+b+c)^2}{3}$ and since the number of {SAPM}s is an integer, we conclude that $\floorfrac{(n-4)^2}{12}$ is an upper bound.
By a similar argument to the previous case, the extremal tree is precisely the balanced spider-trio $\ST_{a,b,c}$ where $a\ge b \ge c \ge a-1$. This has exactly $\floorfrac{(n-4)^2}{12}$ {SAPM}s.
It is now sufficient to note that $\floorfrac{n^2}{16}-1 \le \floorfrac{(n-4)^2}{12} $ if and only if $n \ge 28.$
{\bf Case 2:} There is a leaf $y\in Y$. Let $x$ be the unique neighbor of $Y$.
Since $\abs X = \abs Y+2$, every avoided pair must be contained in $X$ and so every almost-perfect matching must contain $xy$.
If $\deg(x)=2$, the number of {SAPM}s of $T$ equals the number of {SAPM}s of $T\setminus \{x, y\}$, which is at most $f(n-2)$. Since $f(n-2) < f(n)$ for every (even) positive integer $n$, the result follows from induction.
If $\deg(x)>2$ and $T\setminus \{x, y\}$ is a forest, the result also follows similarly by induction since $T\setminus \{x, y\}$ is a subgraph of a tree.
If the components of $T\setminus \{x, y\}$ are all even, with orders $n_1, n_2, \ldots n_k$, then there are at most $f(n_1)+f(n_2)+\ldots+f(n_k)<f(n)$ SAPM's.
If two components are odd, we have at most $\floorfrac{(n-2)^2}{16}$ pairs of leaves, one in each odd component.
\end{proof}
\section{Minimum number of maximal matchings}\label{sec:minmaxmatching}
In this section, we characterize trees with the minimum number of maximal matchings. This complements the result of G\'{o}rska and Skupie\'{n}~\cite{GS07} on the maximum number of maximal matchings.
\begin{defi}
An {\em odd special spider} $\SSS_n$ on $n$ vertices for $n$ odd is an even spider $\Sp_{n-1}$ with a pendant edge added to the center vertex (for $n \geq 7$, this is the unique vertex of degree $\geq 3$). See Figure~\ref{fig:oddSSSn}.
\end{defi}
\begin{figure}
\caption{An odd special spider $\SSS_n$}
\label{fig:oddSSSn}
\end{figure}
\begin{customthm} {\bf \ref{thr:minmaxmatchings}}
A tree $T$ of order $n$ has at least $\lceil \frac n2 \rceil $ maximal matchings.
Equality holds if and only if $T$ is a spider $\Sp_n$, or an odd special spider $\SSS_n$.
\end{customthm}
\begin{proof}
We proceed by induction on $n$. A small computer verification confirms the claim for $n\le 7$ (the paths $P_n$ are also spiders $\Sp_n$ when $n \le 5$).
Now let $n \ge 8$.
We consider two cases.
Either there are two leaves $u_1, u_2$ with a common neighbor $v$, or there is a leaf $u$ whose unique neighbor $v$ has degree $2$, i.e., has only $2$ neighbors $\ell_1$ and $u.$
In the first case, let $T' = T\setminus \{u_1, u_2\}$. Observe that every maximal matching $M$ of $T'$ can be extended to a maximal matching of $T$. If $M$ covers $v$, then $M$ is also a maximal matching of $T$. If $M$ does not contain $v$, then we can extend $M$ with $u_1v$ or with $u_2v$.
In the second case, let $T' = T \setminus \{u, v\}$ and let $x$ be the other neighbor of $v$. As before, every maximal matching $M$ of $T'$ can be extended to a maximal matching of $T$. If $M$ covers $x$, the unique extension is given by $M \cup \{uv\}$. If $M$ does not cover $x$, we can extend using either $uv$ or $vx$.
Since we can always find a maximal matching of $T$ that uses $u_2v$ ($uv$, respectively; in general any particular edge), we note that $T$ has strictly more maximal matchings than $T'$ and we conclude by induction.
For equality to be attained by $T$, equality need to be attained by $T'$ and thus $T'$ is a spider or a special spider (if $n$ is odd). Note that $T'$ has at least two legs of length $2$ and thus $T$ has at least one leg $L$ of length $2$.
By applying the inductive hypothesis to $T \setminus L$, we know that $T \setminus L$ has to be either $\Sp_{n-2}$, or $\SSS_{n-2}$ if $n$ is odd. We can then conclude that $T$ itself is $\Sp_{n}$ or $\SSS_{n}$, and these indeed attain equality.
\end{proof}
Initially, this question about determining the minimum number of maximal matching was motivated by the following observation about maximal matchings, which we will utilize in applications to the weighted Hosoya index.
\begin{lemma}\label{lem:sumdeg_maxmatching}
If $n \geq 2$, then the sum of degrees in a maximal matching $M$ in a tree $T$ of order $n$ is at least $n$.
Equality occurs only if $T$ is a star, or $T$ has diameter $3$ and $M$ is the central edge.
\end{lemma}
Note that the statement is trivially false for a $1$-vertex tree, so $n \geq 2$ is required.
\begin{proof}
Let $M$ be a maximal matching. If $v \in V$ is not covered by $M$, then there must exist some $u \in N(v)$ that is covered by $M$; else, $M \cup \{uv\}$ is a larger matching, contradicting the maximality of $M$. This implies that $\bigcup\{N(u) : u \text{ is covered by }M\} = V$, from which the statement follows.
Equality can occur only if every vertex has precisely one neighbor covered by the matching. By the argument above, for every $v$ not covered by $M$, we must have all of $N(v)$ covered by $M$. This implies that all non-leaf vertices of $T$ are covered by $M$. If the diameter of $T$ is at least $4$, then there exist internal vertices of a diameter path which have more than one neighbor covered by $M$, and so equality does not occur. If $T$ has diameter $3$, there are only two types of maximal matchings and we conclude easily.
\end{proof}
As a corollary of Theorem~\ref{thr:minmaxmatchings} and Lemma~\ref{lem:sumdeg_maxmatching}, we have the following result on the weighted Hosoya index. This is a special case of a more general theorem that we prove in the next section. Recall that $S_n$ denotes the $n$-vertex star.
\begin{corollary}\label{exponential}
Let $\phi(i,j) = c^{i+j}$ where $c \geq 2$. Then for every $n$-vertex tree $T$, we have $Z_\phi(S_n) \leq Z_\phi(T)$.
\end{corollary}
\begin{proof}
First note that $S_n$ has $n-1$ nonempty matchings, each with weight $c^n$, so $Z_{\phi}(S_n)=(n-1)c^n+1$.
For a tree $T$ that is not a star, by Theorem~\ref{thr:minmaxmatchings}, $T$ has at least $\lceil \frac{n}{2} \rceil$ maximal matchings, and by Lemma~\ref{lem:sumdeg_maxmatching}, each has degree sum strictly greater than $n$, with possibly one exception which has degree sum exactly equal to $n$. Thus we have
$Z_{\phi}(T)> \left(\ceilfrac{n}{2}-1\right) c^{n+1}+c^n+1 \ge (n-1)c^n+1=Z_{\phi}(S_n)$.
The first inequality is strict since there are pendant edges which are not maximal matchings and contribute in the sum as well. The second inequality holds by our assumption $c \geq 2$.
\end{proof}
\section{Applications to the weighted Hosoya index}\label{sec:HosoyaIndex}
Recall that the weighted Hosoya index is defined as
$$Z(T, \omega) = \sum_{M \in \mathcal{M}} \prod_{e \in M} \omega(e)$$
In this section, we characterize the extremal examples for some natural choices of weights that are vertex-degree-based, meaning $\omega(e) = \phi(i,j)$ where $i$ and $j$ are the degrees of the endpoints of $e$. We then write $Z(T, \omega)$ as $Z_\phi(T)$.
We consider the following weight functions for $c > 0$:
\begin{align*} \phi_1(i,j) &= c^{i+j}\\
\phi_2(i,j) &= c^{ij}\\
\phi_3(i,j) &= (i+j)^c\\
\phi_4(i,j) &= (ij)^c
\end{align*}
Note that these examples are all symmetric in $i$ and $j$. It was shown in~\cite{CRUZ2022} that for $c < 0$, the star minimizes the weighted Hosoya index for $\phi_3$ and $\phi_4$ (also referred to as the general Randi\'{c} index and general sum-connectivity index, respectively).
This can also be derived from the following proof that the star minimizes the weighted Hosoya index for $c>0$.
\begin{theorem}\label{thr:starminZ_phi}
Let $c > 0$. Then for every $n$-vertex tree $T$, we have $Z_{\phi_\ell}(T) \geq Z_{\phi_\ell}(S_n)$ for all $\ell \in [4] = \{1,2,3,4\}$. Equality holds in each case if and only if $T = S_n$.
\end{theorem}
\begin{proof}
Let $Z_\phi(T, v, k)$ be the weighted Hosoya index of $T$ where the weight of $v$ is increased by $k$. That is, if $uv = e \in E(T)$ for some $u$, then $\phi(e) = \phi(\deg(u), \deg(v)+k)$. We will use induction to prove a stronger statement:
\begin{claim}
Let $T$ be a tree on $n$ vertices, and let $v \in V(T)$ be a non-leaf vertex that has at least one leaf neighbor. For any nonnegative integer $k$ and $\ell \in [4]$, $Z_{\phi_\ell}(T, v, k) \geq Z_{\phi_\ell}(S_n, x, k)$ where $x$ is the center vertex of the star.
\end{claim}
Observe that taking $k = 0$ gives the theorem statement.
We proceed by induction on $n$. We can check the first non-trivial base case ($n=4$) directly to see that the claim holds true for all $k \geq 0$ and all $c > 0$.
Now suppose that for every $n' < n$, the claim holds for all $k \geq 0$.
Let $T$ be an $n$-vertex tree different from $S_n$, and let $k$ be a nonnegative integer.
Let $v \in T$ be a non-leaf vertex. Let $u_1, \dots, u_s$ be the
leaf neighbors of $v$ and let $x_1, \dots, x_t$ be the non-leaf neighbors.
Let $T_i$ be the component of $x_i$ in $T \setminus \{v, u_1, \dots, u_s\}$.
Then
$$Z_\phi(T, v, k) = Z_\phi(T\setminus \{\ell_1, \dots,\ell_s\}, v, s+k) + s \phi(1, deg(v)+k)\prod_{i=1}^t Z_\phi(T_i, x_i, 1)$$
Given a star graph $S_n$, let $z_n$ denote the center vertex.
First consider $\phi_1(i,j) = c^{i+j}$. We can calculate that $Z_{\phi_1}(S_n, z_n, k) = (n-1)c^{n+k}+1$. By the inductive hypothesis,
\begin{align*} Z_{\phi_1}(T, v, k) &\geq Z_{\phi_1}(S_{n-s},z_{n-s}, s+k) + s\phi_1(1, s+t+k)\prod_{i=1}^t Z_{\phi_1}(S_{|T_i|},z_{|T_i|}, 1)\\
&\geq (n-s-1)c^{(n-s)+(s+k)}+1+sc^{(s+t)+k+1}\prod_{i=1}^t ((|T_i|-1)c^{|T_i|+1}+1)
\end{align*}
Observe that $\sum_{i=1}^t |T_i| = n-s-1$, so we have
\begin{align*}
Z_{\phi_1}(T, v, k) &\geq (n-s-1)c^{n+k}+1+sc^{s+t+k+1}\left(c^{(n-s-1)+t}\prod_{i=1}^t (|T_i|-1)+1\right)\\
&= (n-s-1)c^{n+k}+1+sc^{n+2t+k}\prod_{i=1}^t (|T_i|-1)+sc^{s+t+k+1}\\
\end{align*}
We want the last expression to be at least $(n-1)c^{n+k}+1$. Recall that $s \geq 1$ by assumption. Dividing both sides of the desired inequality by $sc^{n+k}$ and rearranging, it suffices to show
$$c^{2t} \prod_{i=1}^t (|T_i|-1) +c^{s+t+1-n}\geq 1$$
If $T$ is not isomorphic to $S_n$, then $t \geq 1$ and $|T_i| \geq 2$ for all $i$, so when $c \geq 1$, the first term is at least 1 and the inequality holds. The second term is also positive, so the inequality must be strict in this case.
When $c < 1$, we use that $n = (s+1)+\sum_{i=1}^t |T_i| \geq s+1+2t$, so the second term is at least $1$, and as before, the inequality holds and is strict for $T \neq S_n$.
(Note that we may separately address the case $c < 1$ for $\phi_1$. Since $T$ is a tree, $\deg(u)+\deg(v) \leq n$ for every edge $uv$ in $T$. Thus, $Z_{\phi_1}(T) \geq (n-1)c^n + 1$ where equality holds if and only if $T$ is the star. However, this argument does not apply to the case of, for example, $\phi_2(i,j) = c^{ij}$.)
The calculations are almost identical for $\phi_2, \phi_3$, and $\phi_4$, so we omit most of the details here. For $\phi_2$, we arrive at
$$Z_{\phi_2}(T, v, k) \geq (n-s-1)c^{n+k-1}+1+sc^{n+t+k-1}\prod (|T_i|-1) + sc^{s+t+k}$$
and similarly show that this is at least $(n-1)c^{n+k-1}+1$ with equality if and only if $T = S_n$.
For $\phi_3$, we have
$$Z_{\phi_3}(T, v, k) \geq (n-s-1) (n+k)^c + 1 + s (s+t+k+1)^c \prod (|T_i|+1)^c$$
Since $n+k < n+k+t= (s+t+k)+\sum |T_i|+1$ and each of the terms $(s+t+k)$ and $\sum |T_i|$ is at least $2$,
we know that $n+k \le (s+t+k+1)\prod (|T_i|+1)$ and so the result follows.
Lastly, for $\phi_4$, we have
$$Z_{\phi_4}(T, v, k) \geq (n-s-1) (n+k-1)^c + 1 + s (s+t+k)^c \prod (|T_i|)^c$$
and similarly the result follows from
$n+k-1 \le (s+t+k)\prod |T_i|$.
\end{proof}
From this and previous work on the Hosoya index, one might be tempted to conjecture that the path and the star are always extremal, even if not uniquely so. It turns out this is not the case.
\begin{theorem}\label{thr:exponential_max}
For $c$ sufficiently large in terms of $n$, the tree that (uniquely) maximizes $Z_{\phi_\ell}(T)$ among all trees of order $n$ is
\begin{itemize}
\item\label{thm:exp_max1}$\ell=1$: $W_n$ if $n\ge 12$ is even and $\Sp_n$ if $n \geq 7$ is odd,
\item\label{thm:exp_max2}$\ell = 2$: $\DB_{\floorfrac{n}{2}-1,\ceilfrac n2 -1}$ if $n \ge 6,$
\item\label{thm:exp_max34}$\ell \in \{3,4\}$: $P_n$
\end{itemize}
\end{theorem}
\begin{proof}[Proof for {\hyperref[thm:exp_max1]{$\phi_1$}}]
Recall that $\phi_1(i,j) = c^{i+j}$. Suppose first that $n$ is even.
For a perfect matching $M$, we must have $$\sum_{e=uv \in M} \left( \deg(u)+\deg(v) \right) =2(n-1)$$
Thus, for $c \geq 1$, the leading term of $Z_{\phi_1}(T)$ is at most $c^{2n-2}$ (since every tree has at most one perfect matching). The second leading term corresponds to the weight of strong almost-perfect matchings in $T$, which must each have weight $c^{2n-4}$. Because we are taking $c$ sufficiently large, it is enough to recall from Theorem~\ref{thr:even_sapm_wpm} that the balanced wide spider uniquely maximizes the number of {SAPM}s for $n \geq 12$.
If $n$ is odd, the leading term of $Z_{\phi_1}(T)$ is given by {SAPM}s which each have weight $c^{2n-3}$. From Theorem~\ref{thr:odd_sapm}, we know that the odd spider has the maximum number of {SAPM}s when $n \geq 7$.
\end{proof}
\begin{proof}[Proof for {\hyperref[thm:exp_max2]{$\phi_2$}}]
To prove the result for $\phi_2(i,j) = c^{ij}$ we use the following claim.
\begin{claim}
Let $T$ be a tree of order $n \ge 6$ and $M$ a matching of $T$.
Then $\sum_{uv \in M} \deg(u)\cdot \deg(v) \le \floorfrac{n^2}{4}$, with equality if and only if $T$ is the balanced double-broom with diameter $3$ and $M$ consists of the central edge.
\end{claim}
\begin{claimproof}[Proof of Claim]
Let $M = \{e_1,\dots, e_k\}$ and for every $i$, let $d_i$ be the sum of the degrees of the endpoints of $e_i$. Assume the edges are ordered such that $d_1 \leq d_2 \leq \cdots \leq d_k$.
Since $n>2$, we know $d_i\ge 3$ for every $i$.
There are $n-2k$ vertices not covered by $M$, so by the handshaking lemma, we know that
$\sum_{i=1}^k d_i \le 2(n-1)-(n-2k)=n+2k-2.$
If $k=1$, then $M$ consists of a single edge $uv$ and by the AM-GM inequality,
$$\deg(u)\deg(v) \le \floorfrac{d_1^2}{4} \le \floorfrac{n^2}{4}$$
When $k \ge 2$, it suffices to prove that
$$\sum_i \floorfrac{d_i^2}{4} \le \floorfrac{n^2}{4}.$$
In this case, we apply Lemma~\ref{lem:karamata}. Consider the sequences
$A= (3,3,\dots, 3, \sum_{i=1}^k d_i - 3(k-1))$ with the first $k-1$ terms equal to 3, and $B = (d_1, \dots, d_k)$. Both sequences sum to $\sum_{i=1}^k d_i$ and since $d_i \geq 3$ for all $i$, we have that for any $\ell < k$, the sum of the first $\ell$ terms of $A$ is at most the sum of the first $\ell$ terms of $B$. This means that $A$ majorizes $B$, so we may apply Karamata's inequality with the convex function $f(x) = x^2$
to get that
$\sum_{i=1}^k d_i^2 \leq (k-1)3^2+ ((n+2k-2)-3(k-1))^2
= n^2 -(k-1)(2n-k-8)<n^2-1,$ where we use that $n \geq \max\{6, 2k\}$ and conclude.
Since the last inequality is strict, equality cannot be achieved by any matching of size at least two. In the case $k = 1$, the extremal tree must contain an edge whose endpoints each have degree $\frac{n}{2}$; the balanced double-broom is the unique example.
\end{claimproof}
Now let $T$ be a tree of order $n$ such that $T \not =\DB_{\floorfrac{n}{2}-1,\ceilfrac n2 -1}$. We know that $|\mathcal M(T)| \leq F_n$, the $n$th Fibonacci number. Thus,
$$Z_{\phi_2}(T) \le F_n c^{\floorfrac{n^2}{4}-1}<c^{\floorfrac{n^2}{4}} < Z_{\phi_2}\left(\DB_{\floorfrac{n}{2}-1,\ceilfrac n2 -1}\right).$$
\end{proof}
\begin{proof}[Proof for {\hyperref[thm:exp_max34]{$\phi_3$}}]
For $\phi_3(i,j) = (i+j)^c$, the leading term of $Z_{\phi_3}$ does not correspond with a unique graph. We first prove a general claim about maximizing the product of a set of integers, which follows from Karamata's inequality.
\begin{claim}\label{clm:maxprod_whenbalanced}
Assume integers $a_1,a_2, \ldots, a_k$ sum to $qk+r$, where $0 \le r <q$.
Then
$\prod_{i=1}^k a_i \le (q+1)^r q^{k-r}$ and equality occurs only if $r$ of the $a_i$ equal $q+1$ and $k-r$ of them equal $q.$
\end{claim}
\begin{claimproof}
Let the $a_i$ be ordered such that $a_1 \ge a_2 \ge \ldots \ge a_k.$
Then $\sum_{i=1}^{\ell} a_i \ge \ell(q+1)$ for every $\ell \le r.$
If not, $a_{i} \le q$ for every $i \ge \ell$ and thus
$\sum_{i=1}^{k} a_i \le \ell(q+1)-1 + (k-\ell)q<kq+\ell \le kq+r,$ a contradiction.
Similarly, the sum of the $\ell \le k-r$ smallest values among the $a_i$ is bounded by $\ell q.$
This implies that the sequence $(a_1,a_2 \ldots, a_k)$ majorizes the sequence
$$\left(\underbrace{q+1,q+1, \ldots, q+1}_{r },\underbrace{q,q, \ldots, q}_{k-r } \right).$$
By Lemma~\ref{lem:karamata} applied to the concave function $\log(x),$ we conclude that
$\sum_i \log(a_i) \le r\log(q+1)+(k-r) \log(q).$
\end{claimproof}
\begin{claim}\label{clm:maxprodsum}
Let $T$ be a tree of order $n \ge 4$ and $M$ a matching in $T$.
Then $$\prod_{uv \in M} \left(\deg(u)+\deg(v)\right) \le
\begin{cases}
3^2\cdot 4^{(n-4)/2} & \text{ if n is even,}\\
3\cdot 4^{(n-3)/2} & \text{ if n is odd.}\\
\end{cases}$$
Equality is possible only if $M$ is a perfect matching (for $n$ even) or strong almost-perfect matching (for $n$ odd).
Moreover, for all other matchings the upper bound can be reduced by a factor of $2$.
\end{claim}
\begin{claimproof}[Proof of Claim]
Let $V=\{v_1, \ldots, v_n\}$ be an arbitrary permutation of the vertices and $d_i = \deg(v_{2i-1})+\deg(v_{2i})$ (observe that $v_{2i-1}v_{2i}$ is not necessarily an edge of $T$).
If $n=2k$ is even, then $\sum_{i=1}^k d_i = 2n-2 = 4k-2$ by the handshaking lemma and by Claim~\ref{clm:maxprod_whenbalanced}, $\prod_{i=1}^k d_i \leq 3^2 \cdot 4^{k-2}.$
Equality happens only when all $d_i$ are $3$ or $4$. If $M$ is not a perfect matching and covers vertices $\{v_1, \dots, v_{2m}\}$ (without loss of generality), then $2m < 2k$ so we may divide both sides by $d_k \geq 2$ to get the final part of the claim.
If $n=2k+1$ is odd, then $\sum_{i=1}^k d_i\le 2(n-1)-1=4k-1$ and $\prod_{i=1}^k d_i \le 3\cdot 4^{k-1}$ by Claim~\ref{clm:maxprod_whenbalanced}.
Similar arguments hold for equality and when $M$ is not an SAPM as in the even case.
\end{claimproof}
We say $uv$ is an $(a,b)$-edge if $\deg(u) = a, \deg(v) = b$. To determine which tree achieves equality in the previous claim, a trivial but crucial observation is that $\deg(u)+\deg(v)=4$ if and only if $uv$ is a $(2,2)$- or $(3,1)$-edge.
The path $P_n$ is a tree that attains equality in Claim~\ref{clm:maxprodsum}, but not the unique one.
First, consider the case where $n=2k+1$ is odd.
The leading terms for $Z_{\phi_3}(P_n)$ are $2\cdot (3\cdot 4^{k-1})^c+(k-1)\cdot (3^2 \cdot 4^{k-2})^c.$
Let $T \not= P_n$ be a tree of order $n$.
The (strong) almost-perfect matching(s) for which equality is attained in Claim~\ref{clm:maxprodsum} consist of $(3,1)$- and $(2,2)$-edges with exactly one $(2,1)$-edge.
Observe that the extremal tree must then consist of a central path (the diameter) with pendant leaves connected to the internal vertices.
There are at most two choices for an SAPM\ that attains equality in Claim~\ref{clm:maxprodsum}.
If there is only one $(2,1)$-edge in $T$ (an example is presented in Figure~\ref{fig:oneconfigurationtoexplain} on the left), then there are two leaves with the same neighbor of degree $3$ and hence only $2$ APM's in total (since an APM\ must contain exactly one of the two neighboring leaves).
That implies whenever $c$ is sufficiently large that
$$Z_{\phi_3}(T)< 2\cdot (3\cdot 4^{k-1})^c+F_n \cdot (3\cdot 4^{k-1}/2)^c<Z_{\phi_3}(P_n)$$
If there are two $(2,1)$-edges in $T$, let $u$ and $v$ be the two leaves at the end of the diameter. Observe that we cannot have perfect matchings of both $T \setminus u$ and $T \setminus v$; this is because $T$ has a $(3,1)$-edge (since $T \neq P_n$) which must be contained in both matchings. Deleting the endpoints of the $(3,1)$-edge splits the tree into two components, both of which have odd order in one of $T \setminus u$ or $T \setminus v$.
So in this case there is at most one matching for which equality in Claim~\ref{clm:maxprodsum} holds. Thus the leading term in $Z_{\phi_3}(T)$ is smaller than $2\cdot (3\cdot 4^{k-1})^c$ and the result follows again.
\begin{figure}
\caption{Trees for which $Z_{\phi_3}
\label{fig:oneconfigurationtoexplain}
\end{figure}
Next, we consider the case where $n$ is even.
A tree $T$ that attains equality in Claim~\ref{clm:maxprodsum} has maximum degree at most $3$, every vertex of degree $3$ has a neighboring leaf, and $T$ has a perfect matching (see the right hand side of Figure~\ref{fig:oneconfigurationtoexplain} for an example).
As before, $T$ consists of a diameter with pendant leaves.
The second largest contribution to $Z_{\phi_3}(T)$ would be due to a strong almost-perfect matching $M$, for which $\prod_{uv \in M} \left(\deg(u)+\deg(v)\right) \le 2^{n-2}$ by Claim~\ref{clm:maxprodsum}, and equality occurs only if the avoided pair are the endpoints of the diameter, and the degree $3$ vertices in $M$ are paired with leaves.
Let the diameter have vertices $v_1, v_2, \dots, v_k$ where $\{v_1, v_k\}$ is the avoided pair, and let $i$ be minimum such that $\deg(v_i) = 3$. In order for $M$ to cover the vertices $v_2, \dots, v_{i-1}$, we must have $i$ even. But $T$ has a perfect matching, and in order for the perfect matching to cover $v_1, v_2, \dots, v_{i-1}$, we must have $i$ odd - a contradiction unless $T$ contains no vertices of degree 3, i.e. $T = P_n$.
\end{proof}
\begin{proof}[Proof for {\hyperref[thm:exp_max34]{$\phi_4$}}]
Finally, to prove the result for $\phi_4(i,j) = (ij)^c$ we use the following claim.
\begin{claim}
Let $T$ be a tree on $n \ge 2$ vertices.
Then the product of the degrees of all vertices is at most $2^{n-2}$, with equality if and only if $T$ is a path $P_n$.
\end{claim}
\begin{claimproof}
A tree has at least $2$ leaves, so by the handshaking lemma the other $n-2$ vertices have degree $2$ on average.
The claim follows from the AM-GM inequality and the fact that the path is the only tree with exactly two leaves.
\end{claimproof}
As such, the leading term of $Z_{\phi_4}(P_n)$ is $\left( 2^{n-2}\right)^c$, while for any other tree $T$ of order $n$, we have $Z_{\phi_4}(T) \le F_n \left( 3\cdot 2^{n-4}\right)^c,$ which is smaller than $\left( 2^{n-2}\right)^c$ once e.g. $c>2n.$
\end{proof}
\section{Further Directions}\label{sec:future}
We conclude with some further open problems.
\subsection{Maximum number of strong $k$-almost-perfect matchings for $k\ge 3$} \label{sec:k-sapm}
We can generalize the notion of an almost-perfect matching to a {\em $k$-almost-perfect matching} ($k$-APM), a matching which covers all but $k$ vertices. Similarly, a {\em strong $k$-almost-perfect matching} ($k$-SAPM) covers all but $k$ leaves. Observe that $k \in \{1, 2\}$ recovers our original notion of {APM} and SAPM. For $k \geq 3$, we can consider the question of which trees have the maximum number of {$k$-SAPM}s.
In Theorem~\ref{thr:even_sapm_gen}, for $n$ sufficiently large, we showed that the spider-trio maximizes the number of {SAPM}s.
For $n$ sufficiently large, one could hope that a generalisation of the spider-trio, presented in Figure~\ref{fig:GoodConstruction_k_sapm} for $k=7$, is extremal.
This might be true for small $k$ but is false for $k \ge 13$. When $k=13$
a tree consisting of spiders connected to a path has more {$k$-SAPM}s than the $(k+1)$-spider. Asymptotically, however, the generalized $(k+1)$-spider has the optimal order.
\begin{figure}
\caption{A construction $T$ with $k+1$ spiders.}
\label{fig:GoodConstruction_k_sapm}
\end{figure}
\begin{proposition}\label{lem:spider-leading-term}
The number of strong $k$-almost-perfect matchings in a tree of order $n$ is $\Theta_k(n^k).$
\end{proposition}
\begin{proof}
The number of {$k$-SAPM}s is bounded by
$\binom{n}{k}=O_k(n^k)$.
For the lower bound, let $T$ be the union of $k+1$ odd spiders
all of order $2a+1$, whose centers are connected with an additional vertex $v$
as in Figure~\ref{fig:GoodConstruction_k_sapm}.
Then $n=(k+1)(2a+1)+1$ and the number of {$k$-SAPM}s
in $T$ is $(k+1)a^k \sim \frac{n^k}{2^k(k+1)^{k-1}}=\Omega_k(n^k).$
\end{proof}
We can also determine some structural properties that the extremal tree(s) must have.
\begin{proposition}\label{lem:noleavesatdist2ingen}
If $T$ contains two leaves that share a common neighbor, then the number of strong $k$-almost-perfect matchings is at most $2\binom{n}{k-1}=O(n^{k-1}).$
\end{proposition}
\begin{proof}
If two leaves, $\ell_1$ and $\ell_2,$ have the same neighbor, then no matching can contain both of them. Thus, in any {$k$-APM}
at least one of the two is in the avoided set of $k$ leaves.
For both $\ell_1$ and $\ell_2$, the number of {$k$-APM}s is bounded
by the number of ways to choose the remaining $k-1$ leaves to avoid.
\end{proof}
As a consequence, for $n$ sufficiently large, we can partition the vertices
into three sets: the set of leaves $L$, the set of (unique) neighbors of the leaves
$N := N(L)$ and the set of vertices $X$ that are at distance at least two from any leaf.
Observe that $\lvert L\rvert = \lvert N \lvert$ and $\lvert X \rvert + 2\lvert L \rvert =n$.
\begin{proposition}\label{lem:noedgeinT[N]}
The number of strong $k$-almost-perfect matchings that contain an edge in $T[N]$ is $O_k(n^{k-1}).$
\end{proposition}
\begin{proof}
A {$k$-SAPM} $M$ is uniquely determined by the avoided set of $k$ leaves.
If there is an edge in $T[N] \cap M$, then the endpoints have neighbors in $L$ which must also be in the avoided set, together with $k-2$ other leaves.
There are fewer than $n-1$ edges in $T[N]$ and $O(n^{k-2})$ choices for the other $k-2$ leaves, which results in a total of $O(n^{k-1})$ {$k$-SAPM}s that contain an edge in $T[N]$.
\end{proof}
Thus, there are few {$k$-SAPM}s that contain an edge between two vertices in $N$.
We suspect the extremal trees are spiders connected to a relatively
small $X$, but the structure of $X$ seems difficult to characterize.
Surprisingly, when $T[X]$ is a path, the optimal number of
leaves in each spider is not monotone as the following example illustrates.
\begin{examp}
When $T$ is the union of $8$ spiders whose centers are connected with a path, such that $T$ is symmetric, the number of strong $4$-APM's is maximized when the ratios of their sizes (counted from outer to inner spiders) is approximately $\frac{27}{16} \colon 1 \colon \frac{9}{16} \colon \frac{3}{4}.$
\end{examp}
\begin{figure}
\caption{A construction where $T[X]=P_8$}
\label{fig:8spiders}
\end{figure}
To see this, let the number of leaves of the $8$ spiders be $a,b,c,d,d,c,b,a$ respectively, as presented in Figure~\ref{fig:8spiders}.
Then $2(a+b+c+d)=\frac{n-8}{2}.$
A strong $4$-APM\ can only be obtained when taking four leaves from different spiders in such a way that the centers of the other four form a matching in the path.
The total number of choices for these leaves is equal to $(b^2 + (2c + 2d)b + d^2)a^2 + 2c((c + 2d)b + d^2)a + c^2d^2.$
Using a small computer verification\footnote{\url{https://github.com/StijnCambie/MRC_trees_projects}, document OptimalDistributionBehaviour.}, we conclude that this homogenous multivariate polynomial is maximized when $(a \colon b \colon c \colon d) = \left(\frac{27}{16} \colon 1 \colon \frac{9}{16} \colon \frac{3}{4}\right).$
The preceding discussion may provide some intuition to tackle the following question:
\begin{question}
For $k\ge 3$, characterize the trees which maximize the number of strong $k$-almost-perfect matchings.
\end{question}
We can generalize the notion of {$k$-SAPM} even further to that of a {\em connected matching}, which is a matching $M$ in a tree $T$ such that $T[V(M)]$ is a connected graph.
The notion of connected matching (among some other variants) was defined in~\cite{GHHL05}.
Observe that every {$k$-SAPM} is a connected matching. Conversely, when $n$ is odd, a connected matching with $\frac{n-1}2$ edges is an SAPM.
One could ask similar questions about characterizing the extremal trees for connected matchings.
It may be interesting to consider these type of questions for connected matchings in general graphs as well, or classes such as $d$-regular graphs which are well-studied in matching theory.
\subsection{Weighted Hosoya index}
Regarding applications to the weighted Hosoya index, it would be interesting to determine the extremal trees for various other choices of degree-based weight function. In particular, can we characterize when the path and the star are the maximizer and minimizer, respectively?
For example, consider the function
$$\phi(i,j) = \varphi^{16\lfloor \frac{ij}{16}\rfloor}$$
where $\varphi$ is the golden ratio. While this choice of weight function may seem rather artificial, it is interesting to note that neither $P_n$ nor $S_n$ are extremal; one can check that $Z_\phi(P_n) = F_n \sim \varphi^n$ and $Z_\phi(S_n) = (n-1)\varphi^{n-1}$. Here $F_n$ is the $n^{th}$ Fibonacci number.
In fact, a weight of at least approximately $\varphi^{8n/3}$ can be achieved by the construction in Figure~\ref{fig:largeZphi} below.
\begin{figure}
\caption{Two trees indicating that $P_n$ and $S_n$ are not extremal}
\label{fig:largeZphi}
\label{fig:smallZphi}
\label{fig:2treesforZphi}
\end{figure}
On the other hand, a weight of at most $8^{n/7}< \varphi^{3n/4}$ is given by the construction in Figure~\ref{fig:smallZphi}.
One might try to generalize the results of Section~\ref{sec:HosoyaIndex} to answer the following:
\begin{question}
Can we fully characterize the class of functions $\phi$ for which $Z_\phi(T)$ is minimized by the star, or maximized by the path?
\end{question}
\section*{Acknowledgments}
The authors would like to express their gratitude to the American Mathematical Society for organizing the Mathematics Research Community workshops where this
work began, in the workshop ``Trees in Many Contexts,'' and to Wanda Payne for contributing to the initial discussions. This event was supported by the National Science Foundation under Grant Number DMS $1916439$.
The first author is supported by the Institute for Basic Science (IBS-R029-C4). The fourth author is supported by the Knut and Alice Wallenberg Foundation (KAW 2017.0112).
\paragraph{Open access statement.} For the purpose of open access,
a CC BY public copyright license is applied
to any Author Accepted Manuscript (AAM)
arising from this submission.
\end{document} | math | 64,257 |
\begin{document}
\title{A note on the Bures-Wasserstein metric}
\author{Shravan Mohan\\ 17-004, Mantri Residency, Bannerghatta Main Road, Bangalore.
}
\newgeometry{top=1in,bottom=0.75in,right=0.75in,left=0.75in}
\maketitle
\begin{abstract}
In this brief note, it is shown that the Bures-Wasserstein (BW) metric on the space positive definite matrices lends itself to convex optimization. In other words, the computation of the BW metric can be posed as a convex optimization problem. In turn, this leads to efficient computations of (i) the BW distance between convex subsets of positive definite matrices, (ii) the BW barycenter, and (iii) incorporating BW distance from a given matrix as a convex constraint. Computations are provided for corroboration.
\end{abstract}
\begin{IEEEkeywords}
Bures-Wasserstein Metric, Schur Complement, Semidefinite Programming.
\end{IEEEkeywords}
\section{Introduction}
\noindent Consider the set of positive definite matrices of dimension $n$ given by $\mathcal{P}(n)$. The Bures-Wasserstein (BW) metric between $A$ and $B$ in $\mathcal{P}(n)$ is given by the closed form \cite{bhatia2019bures}:
\begin{align}
\rho^2(A, B) = \mbox{Tr}(A) + \mbox{Tr}(B) - 2\mbox{Tr}\left(\sqrt{\sqrt{A}B\sqrt{A}}\right).
\end{align}
Here, $\sqrt{X}$ denotes the unique symmetric square root of a positive definite matrix $X$. That is:
\begin{align}
\sqrt{X} = U\Sigma^{\frac{1}{2}}U^\top,
\end{align}
where $X = U\Sigma U^\top$ is the singular value decomposition of $X$ and $\Sigma^{\frac{1}{2}}$ is the
element-wise square root of $\Sigma$.\\\\
The BW metric is defined in the following way. The set of square roots of a positive definite matrix $X$ is given by:
\begin{align}
\left\{ \sqrt{X}U: U \in O(n)\right\},
\end{align}
where $O(n)$ is the set of real unitary matrices of dimension $n$. Then, for two positive definite matrices $A$ and $B$, the Frobenius distance ($||.||_F$) between the sets of their respective square roots is defined as the BW metric. Mathematically, this gives:
\begin{align}
\rho(A, B) = \min_{U, V\in O(n)}\left|\left|\sqrt{A}U - \sqrt{B}V\right|\right|_F
\end{align}
\section{BW metric \& Semidefinite Programming}
Since the Frobenius norm is unitary invariant, the BW metric can also be written as:
\begin{align}
\rho(A, B) = \min_{V\in O(n)}\left|\left|\sqrt{A} - \sqrt{B}V\right|\right|_F,
\end{align}
Thus,
\begin{align}
\rho^2(A, B) = \min_{U\in O(n)}\mbox{Tr}(A) + \mbox{Tr}(B) - 2\mbox{Tr}\left(\sqrt{A}\sqrt{B}U\right).
\end{align}
Now, the following well-known lemma comes to the aid for solving the above optimization problem as a convex optimization problem \cite{boyd2004convex}.\\
\textbf{Lemma}: The linear SDP given by
\begin{align}
&\max_{U}~~~~\mbox{Tr}(KU)\\
&\mbox{subject to~~} \begin{bmatrix}
I & U^\top\\
U & I
\end{bmatrix} \succeq 0.
\end{align}
has an optimal solution $\tilde{U}$ such that $\tilde{U}^\top \tilde{U} = I$. \\
\textbf{\textit{Proof}}: Firstly, note that if a feasible point $U$ is such that some diagonal elements of $KU$ are negative, then the matrix $UD$, where $D$ is diagonal such that
$$
D_{i,i}=
\begin{cases}
1,& \text{if } (KU)_{i,i}\geq 0\\
-1, & \text{otherwise},
\end{cases}
$$
would yield a higher cost function value. Thus, at optimality, the diagonal elements of $KU$ are non-negative. Also note that $DU$ satisfies the semidefinite constraint if $U$ does. Secondly, suppose the optima $\tilde{U}$ was such that $I \succ \tilde{U}^\top \tilde{U}$. Let $\tilde{U} = PSQ^\top$ by SVD. By our assumption, some of the elements of $S$ have to be zero or less than 1. Now consider the unitary matrix $U = PD Q^\top $, where $D$ is a diagonal matrix (from the first observation) which makes all the diagonal elements of $KPD Q^\top$ non-negative.
Also note that
\begin{align}\small
\mbox{Tr}(KPDQ^\top ) = \mbox{Tr}(Q^\top KPD)\geq \mbox{Tr} (Q^\top KPS) \nonumber = \mbox{Tr}\left(KU\right)\nonumber,
\end{align}
which finally implies that the convex relaxation is tight. The result also applies to the case where the constraint is:
\begin{align}
\begin{bmatrix}
G & U^\top \\
U & I
\end{bmatrix} \succeq 0,
\end{align}
since this is equivalent to the constraint
\begin{align}
\begin{bmatrix}
I & \sqrt{G^{-1}}U^\top \\
U\sqrt{G^{-1}} & I
\end{bmatrix} \succeq 0,
\end{align}
With the above lemma, the computation of BW distance can also be written as:
\begin{align}
\rho^2(A, B) =~~& \min~~ \mbox{Tr}(A) + \mbox{Tr}(B) - 2\mbox{Tr}\left(\sqrt{A}U\right)\\
&\mbox{subject to~~} \begin{bmatrix}
B & U^\top\\
U & I
\end{bmatrix}\succeq 0.
\end{align}
\begin{figure*}
\caption{The Main Algorithms}
\label{fig:minmaxalgo}
\end{figure*}
\begin{table*}[h!]
\centering
\caption{Parameters and computational results.}
\resizebox{6.9in}{!}{\begin{tabular}{|c|c|c|}
\hline
Purpose & Parameters & Results \\ \hline
BW distance between convex subsets of PD matrices & $\mathcal{A}=\left\{X\in S_5^+|\mbox{Tr}(X)=1\right\}$, $\mathcal{B}=\left\{X\in S_5^+|\mbox{Tr}(X)=2\right\}$ & \textcolor{blue}{$A = \begin{bmatrix}
0.3209& -0.1364& -0.1069& -0.1686& 0.0726\\
-0.1364& 0.5256& 0.1634& -0.0637& -0.1171\\
-0.1069& 0.1634& 0.5295& 0.0262& -0.095 \\
-0.1686& -0.0637& 0.0262& 0.2931& 0.0048\\
0.0726& -0.1171& -0.095 & 0.0048& 0.3308
\end{bmatrix}$, $B = \begin{bmatrix}
0.1605& -0.0682& -0.0535& -0.0843& 0.0363\\
-0.0682& 0.2628& 0.0817& -0.0319& -0.0585\\
-0.0535& 0.0817& 0.2647& 0.0131& -0.0475\\
-0.0843& -0.0319& 0.0131& 0.1466& 0.0024\\
0.0363& -0.0585& -0.0475& 0.0024& 0.1654
\end{bmatrix}$} \\ \hline
& $w = \left[0.8766, 0.6682, 1.0852, 1.1009, 0.524\right]$ & \\
& $A1 = \begin{bmatrix}
2.7273& -1.3426& -1.4873& 1.1069& -0.5844\\
-1.3426& 5.6047& -0.7192& 0.3519& 1.0648\\
-1.4873& -0.7192& 4.6821& -0.9547& -1.6117\\
1.1069& 0.3519& -0.9547& 2.4089& -0.9744\\
-0.5844& 1.0648& -1.6117& -0.9744& 3.5771
\end{bmatrix}$, $A2 = \begin{bmatrix}
5.6143& -0.1039& 1.4161& 0.0105& 0.9256\\
-0.1039& 4.6277& 0.5304& 0.0571& 0.6138\\
1.4161& 0.5304& 7.017 & -0.4625& -0.3483\\
0.0105& 0.0571& -0.4625& 5.4935& 1.2015\\
0.9256& 0.6138& -0.3483& 1.2015& 8.2474
\end{bmatrix}$ &
\textcolor{blue}{$\textcolor{blue}{X_{\mbox{opt}} = \begin{bmatrix}
3.8514& -0.5993& -0.0722& 0.5644& -0.4899\\
-0.5993& 4.8924& 0.1755& 0.0716& 0.2198\\
-0.0722& 0.1755& 4.4109& -0.1818& -0.6419\\
0.5644& 0.0716& -0.1818& 3.9922& -0.3331\\
-0.4899& 0.2198& -0.6419& -0.3331& 4.5659
\end{bmatrix}}$}\\
BW barycenter & $A3 = \begin{bmatrix}
5.4601& -0.1268& -0.7682& -0.729 & -0.909 \\
-0.1268& 7.7425& 0.1735& 0.4499& -0.511 \\
-0.7682& 0.1735& 6.8627& -0.3396& -1.259 \\
-0.729 & 0.4499& -0.3396& 6.7328& 1.1921\\
-0.909 & -0.511 & -1.259 & 1.1921& 4.2019
\end{bmatrix}$, $A4 = \begin{bmatrix}
2.937 & -1.1282& 0.3996& 0.9282& -0.3372\\
-1.1282& 3.3586& -0.4808& -1.112 & 0.3812\\
0.3996& -0.4808& 2.1708& 0.4026& -0.1732\\
0.9282& -1.112 & 0.4026& 3.043 & -0.8748\\
-0.3372& 0.3812& -0.1732& -0.8748& 4.4907
\end{bmatrix}$ & $X_{\mbox{opt}}$ is computed using the optimization algorithm, while $X_{\mbox{fp}}$ is calculated using the fixed point equation \\
& $A5 = \begin{bmatrix}
4.5401& 1.2074& 1.3077& 1.6847& -1.2072\\
1.2074& 3.9336& 2.5037& 1.5876& -0.3888\\
1.3077& 2.5037& 3.8015& 0.5648& 0.9108\\
1.6847& 1.5876& 0.5648& 4.1194& -1.8946\\
-1.2072& -0.3888& 0.9108& -1.8946& 4.6055
\end{bmatrix}$ &
$\textcolor{blue}{X_{\mbox{fp}} = \begin{bmatrix}
3.8514& -0.5994& -0.0722& 0.5644& -0.4899\\
-0.5994& 4.8924& 0.1756& 0.0716& 0.2198\\
-0.0722& 0.1756& 4.4108& -0.1818& -0.6419\\
0.5644& 0.0716& -0.1818& 3.9921& -0.3331\\
-0.4899& 0.2198& -0.6419& -0.3331& 4.5658
\end{bmatrix}}$
\\ \hline
BW distance as convex constraint &
$f(X) = \left|\left|X\right|\right|_{F}$,~ $\rho^2(A,X)\leq 10$, ~$A = \begin{bmatrix}
6.5722& -0.4557& 0.018 & 0.0854& 0.1883\\
-0.4557& 6.3399& -0.0739& -0.1726& -0.2416\\
0.018 & -0.0739& 5.8477& -0.2659& -0.2295\\
0.0854& -0.1726& -0.2659& 5.5408& -0.3855\\
0.1883& -0.2416& -0.2295& -0.3855& 5.6995
\end{bmatrix}$
& \textcolor{blue}{$X = \begin{bmatrix}
1.1203& -0.036 & 0.0016& 0.0071& 0.0152\\
-0.036 & 1.1016& -0.0065& -0.0149& -0.0201\\
0.0016& -0.0065& 1.0617& -0.0234& -0.0202\\
0.0071& -0.0149& -0.0234& 1.0346& -0.0341\\
0.0152& -0.0201& -0.0202& -0.0341& 1.0482
\end{bmatrix}$}\\ \hline
\end{tabular}}
\label{tab:min_max}
\end{table*}
\noindent \textbf{\underline{BW distance between convex sets}}: The convex routine to compute the BW metric can be used to find the BW distance between two convex subsets of positive definite matrices using alternating projections \cite{bauschke1996projection}. The alternating projections method proceeds by finding the distance of a point from one set to the other alternatively, while latching onto the latest iterate. Recall that this method converges to the distance between convex sets (and the corresponding matrices in the two subsets), given a metric on the point set. More precisely, consider two convex sets of positive definite matrices $\mathcal{A}$ and $\mathcal{B}$. Then, the alternating steps would be:
\begin{align}
&\min_{K, A}~~ \mbox{Tr}(A) + \mbox{Tr}(B) - 2\mbox{Tr}(\sqrt{B}K)\\
&\mbox{subject to~~} \begin{bmatrix}
A & K^\top\\
K & I
\end{bmatrix} \succeq 0, ~ A\in \mathcal{A}.
\end{align}
and
\begin{align}
&\min_{K, B}~~ \mbox{Tr}(A) + \mbox{Tr}(B) - 2\mbox{Tr}( \sqrt{A}K)\\
&\mbox{subject to~~} \begin{bmatrix}
B & K^\top\\
K & I
\end{bmatrix} \succeq 0, ~ A\in \mathcal{A}.
\end{align}
\textbf{\underline{BW barycenter}}: The same idea can be applied towards finding a weighted BW barycenter. Conventionally, the BW barycenter is computed as a solution to a fixed point equation given by:
\begin{align}
X = \sqrt{X^{-1}}\left(\sum_{i=1}^N w_i \sqrt{\sqrt{X} A_i \sqrt{X}}\right)^2\sqrt{X^{-1}}.
\end{align}
Although, the fixed point iteration converges to the BW barycenter, a convex approach is easy to understand and allows inclusion of convex constraints on the barycenter. Consider the convex optimization problem:
\begin{align}
&\min_{X}~~ \sum_{i=1}^N w_i \left( \mbox{Tr}(A_i) + \mbox{Tr}(X) - 2\mbox{Tr}( \sqrt{A_i}K_i)\right)\\
&\mbox{subject to~~} \begin{bmatrix}
X & K_i^\top\\
K_i & I
\end{bmatrix} \succeq 0, ~ \forall i.
\end{align}
Note that the optima in this case too has to lie on the boundary of each constraint (by the aforementioned lemma), and hence the BW barycenter can be calculated this way. \\\\
\textbf{\underline{BW distance as convex constraint}}: Consider the constraint set given by:
\begin{align}
\left\{X\in \mbox{PD}(n) ~~|~~ \rho(A,X) \leq d \right\}.
\end{align}
This set can be represented as a convex constraint:
\begin{align}
\left\{X\in PD(n), ~K\in R^{n,n} ~~|~~
\begin{bmatrix}
X & K^\top \\
K & I
\end{bmatrix}\succeq 0 \right. ~\&~\\
\left. \mbox{Tr}(A) + \mbox{Tr}(X) -2\mbox{Tr}(\sqrt{A} K)\leq d^2
\right\}.
\end{align}
Note that if $(X, K)$ belongs to the constraint set, then obviously $$
\min_{X, K\in O(n)} ~\left(\mbox{Tr}(A) + \mbox{Tr}(X) -2\mbox{Tr}(\sqrt{A} K)\right) = \rho^2(A,X) \leq d^2.
$$
An example of convex optimization problem incorporating such a constraint would be:
\begin{align}
&\min_X ~~~~||X||_{{F}}\\
&\mbox{sub to~~} \rho(A,X) \leq d.
\end{align}
\section{Computations}
Computational results for the three use cases are shown in Table 1. For the first use case, the convex subsets of positive definite matrices are BW distance between two convex subsets of positive definite matrices are $\mathcal{A}=\left\{X\in P(n)|\mbox{Tr}(X)=1\right\}$ and $\mathcal{B}=\left\{X\in P(n)|\mbox{Tr}(X)=2\right\}$. For the second use case, the weight vector and the matrices of which the BW barycenter needs to be calculated are presented. The computation of the barycenter using the fixed point equation also yields essentially the same result, thereby corroborating this paper's claim. For the third case, the example optimization problem is chosen with the matrix $A$ given in the table and $d$ set as $\sqrt{10}$. All the computations were done using CVXPY \cite{diamond2016cvxpy}.
\section{Conclusion}
In this paper it was shown that the computation of the BW metric can be done using convex optimization. This resulted in numerically efficient routines for calculating the BW distance between convex subsets of matrices, the BW barycenter of a finite set of positive definite matrices and incorporating BW distance from a matrix as a convex constraint in a convex optimization problem. Computational examples were provided for corroboration.
\end{document} | math | 13,217 |
\begin{document}
\title{The provability logic of all provability predicates}
\begin{abstract}
We prove that the provability logic of all provability predicates is exactly Fitting, Marek, and Truszczy\'nski's pure logic of necessitation $\mathsf{N}$.
Moreover, we introduce three extensions $\mathsf{N}F$, $\mathsf{N}R$, and $\mathsf{N}RF$ of $\mathsf{N}$ and investigate the arithmetical semantics of these logics.
In fact, we prove that $\mathsf{N}F$, $\mathsf{N}R$, and $\mathsf{N}RF$ are the provability logics of all provability predicates satisfying the third condition $\D{3}$ of the derivabiity conditions, all Rosser's provability predicates, and all Rosser's provability predicates satisfying $\D{3}$, respectively.
\end{abstract}
\section{Introduction}
Let $T$ be a consistent primitive recursively axiomatized $\mathcal{L}_A$-theory containing Peano Arithmetic $\mathsf{PA}$, where $\mathcal{L}_A$ is the language of first-order arithmetic.
G\"odel's second incompleteness theorem states that if a provability predicate $\mathrm{Pr}_T(x)$ of $T$ satisfies the following two conditions $\D{2}$ and $\D{3}$, then the consistency statement $\neg \mathrm{Pr}_T(\gn{0=1})$ of $T$ cannot be proved in $T$: Let $\varphi$ and $\psi$ be any $\mathcal{L}_A$-formulas.
\begin{description}
\item [$\D{2}$:] $T \vdash \mathrm{Pr}_T(\gn{\varphi \to \psi}) \to \bigl(\mathrm{Pr}_T(\gn{\varphi}) \to \mathrm{Pr}_T(\gn{\psi}) \bigr)$.
\item [$\D{3}$:] $T \vdash \mathrm{Pr}_T(\gn{\varphi}) \to \mathrm{Pr}_T(\gn{\mathrm{Pr}_T(\gn{\varphi})})$.
\end{description}
In particular, a conventional provability predicate $\mathrm{Prov}_T(x)$ of $T$, which naturally expresses that $x$ is the G\"odel number of a $T$-provable formula, satisfies $\D{2}$ and $\D{3}$.
Therefore, $T \nvdash \neg \mathrm{Prov}_T(\gn{0=1})$ holds.
Every provability predicate $\mathrm{Pr}_T(x)$ is thought as a kind of modality, and modal logical study of provability predicates has been developed.
For each provability predicate $\mathrm{Pr}_T(x)$ of $T$, the set of all $T$-verifiable modal formulas under the interpretation that $\Box$ is interpreted by $\mathrm{Pr}_T$ is called the \textit{provability logic} of $\mathrm{Pr}_T(x)$.
The most striking result of this study is Solovay's arithmetical completeness theorem \cite{Sol} stating that if $T$ is $\Sigma_1$-sound, then the provability logic of $\mathrm{Prov}_T(x)$ is exactly the G\"odel--L\"ob modal logic $\mathsf{GL}$.
On the other hand, not all provability logics are exactly $\mathsf{GL}$.
In particular, there exist $\Sigma_1$ provability predicates for which the second incompleteness theorem does not hold.
A typical example of such a provability predicate is Rosser's one that was essentially introduced by Rosser \cite{Ros}.
Let $\mathrm{Pr}_T^{\mathrm{R}}(x)$ be a Rosser's provability predicate of $T$, then it is known that $\neg \mathrm{Pr}_T^{\mathrm{R}}(\gn{0=1})$ is provable in $\mathsf{PA}$.
Hence, the provability logic of $\mathrm{Pr}_T^{\mathrm{R}}(x)$ is completely different from $\mathsf{GL}$ because it contains the modal formula $\neg \Box \bot$ that is inconsistent with $\mathsf{GL}$.
Also, by the proof of the second incompleteness theorem, $\mathrm{Pr}_T^{\mathrm{R}}(x)$ does not satisfy at least one of the conditions $\D{2}$ and $\D{3}$.
And, it has been shown that whether $\mathrm{Pr}_T^{\mathrm{R}}(x)$ does not satisfy either $\D{2}$ or $\D{3}$ depends on the choice of $\mathrm{Pr}_T^{\mathrm{R}}(x)$.
In other words, whether the corresponding provability predicate contains either $\Box (A \to B) \to (\Box A \to \Box B)$ or $\Box A \to \Box \Box A$ depends on the choice of $\mathrm{Pr}_T^{\mathrm{R}}(x)$.
Indeed, Bernardi and Montagna \cite{BM} and Arai \cite{Ara} proved that there exists a Rosser's provability predicate $\mathrm{Pr}_T^{\mathrm{R}}(x)$ of $T$ satisfying $\D{2}$, and hence such a predicate does not satisfy $\D{3}$.
Arai also proved the existence of a Rosser's provability predicate satisfying $\D{3}$.
Especially, provability logics of Rosser's provability predicates satisfying $\D{2}$ contain the normal modal logic $\mathsf{K}D$.
The author proved in \cite{Kur20} that there exists a Rosser's provability logic whose provability logic is exactly $\mathsf{K}D$.
Provability predicates that are not $\Sigma_1$ whose provability logics are different from $\mathsf{GL}$ have also been studied.
For example, the author proved in \cite{Kur18_1} that there exists a $\Sigma_2$ provability predicate of $T$ whose provability logic is exactly the weakest normal modal logic $\mathsf{K}$.
Also, for several normal modal logics, the existence of corresponding $\Sigma_2$ provability predicates has been shown (cf.~\cite{Kur18_2,Mon,Sha94,Vis}).
In previous studies, all modal logics that have been considered as provability logics are normal, that is, containing the logic $\mathsf{K}$.
Provability predicates corresponding to such logics always satisfy the condition $\D{2}$.
In general, however, not all provability predicates satisfy $\D{2}$.
For example, Rosser's provability predicates satisfying $\D{3}$, whose existence was proved by Arai, do not satisfy $\D{2}$.
The provability logics corresponding to such predicates are non-normal.
In the present paper, we discuss non-normal provability logics through the following questions:
\begin{itemize}
\item [Q1] What is the intersection of all provability logics, that is, the provability logic of all provability predicates?
\item [Q2] What is the provability logic of all Rosser's provability predicates?
\end{itemize}
The property common to all provability predicates is ``$T \vdash \varphi \Rightarrow T \vdash \mathrm{Pr}_T(\gn{\varphi})$'' that corresponds to Necessitation $\dfrac{A}{\Box A}$, and presumably no other.
The non-normal modal logic $\mathsf{N}$, obtained by adding Necessitation $\dfrac{A}{\Box A}$ as an inference rule to classical propositional logic, was introduced by Fitting, Marek, and Truszczy\'nski \cite{FMT}.
In that paper, $\mathsf{N}$ is called the \textit{pure logic of necessitation}.
This logic $\mathsf{N}$ is our candidate for the answer to Q1, but a problem arises.
The usual proof of Solovay's theorem is to embed Kripke models into arithmetic, and similar techniques have been used in the proofs of the previous results for various normal modal logics.
On the other hand, since the logic $\mathsf{N}$ is not a normal modal logic, $\mathsf{N}$ does not have Kripke semantics.
However, Fitting, Marek, and Truszczy\'nski introduced a Kripkean relational semantic corresponding to $\mathsf{N}$, and the soundness, completeness, and finite frame property of $\mathsf{N}$ with respect to that semantics were proved.
Then, we can attempt to apply Solovay's method to that semantics.
Indeed, in Section \ref{Sec:N}, we prove that $\mathsf{N}$ is exactly the provability logic of all provability predicates.
This is the answer to Q1.
Moreover, we actually prove more: There exists a $\Sigma_1$ provability predicate of $T$ whose provability logic is exactly $\mathsf{N}$.
In Section \ref{Sec:Compl}, we introduce the logic $\mathsf{N}R$ that is obtained from $\mathsf{N}$ by adding the inference rule $\dfrac{\neg A}{\neg \Box A}$.
Then, we prove the finite frame property of $\mathsf{N}R$ with respect to Fitting, Marek, and Truszczy\'nski's semantics.
By using this result, in Section \ref{Sec:NR}, we prove that $\mathsf{N}R$ is exactly the provability logic of all Rosser's provability predicates.
This is the answer to Q2.
Furthermore, in the present paper, we deal with provability predicates satisfying the condition $\D{3}$.
In Section \ref{Sec:Compl}, we also introduce the logics $\mathsf{N}F$ and $\mathsf{N}RF$ that are obtained from $\mathsf{N}$ and $\mathsf{N}R$ by adding the axiom scheme $\Box A \to \Box \Box A$, respectively.
Then, we prove the finite frame property of $\mathsf{N}F$ and $\mathsf{N}RF$.
Also, in Sections \ref{Sec:NF} and \ref{Sec:NRF}, we prove that $\mathsf{N}F$ and $\mathsf{N}RF$ are exactly the provability logics of all provability predicates satisfying $\D{3}$ and all Rosser's provability predicates satisfying $\D{3}$, respectively.
In Appendix 1, as a related topic, we prove the existence of a $\Sigma_1$ provability predicate whose provability logic is exactly $\mathsf{K}$.
In Appendix 2, we prove the interchangeability of $\Box$ and $\Diamond$ in $\mathsf{N}R$.
As a continuation of the present paper, in \cite{KK}, non-normal provability logics having the rule $\dfrac{A \to B}{\Box A \to \Box B}$ are investigated.
\section{Preliminaries}\label{Sec:Pre}
Let $\mathcal{L}_A$ be the language of first-order arithmetic.
Also, let $\omega$ be the set of all natural numbers.
We fix a natural G\"odel numbering such that if $\psi$ is a proper subformula of $\varphi$, then the G\"odel number of $\psi$ is smaller than that of $\varphi$.
Let $\{\xi_t\}_{t \in \omega}$ be the reputation-free primitive recursive enumeration of all $\mathcal{L}_A$-formulas arranged in ascending order of G\"odel numbers.
That is, if $\xi_s$ is a proper subformula of $\xi_u$, then $s < u$.
For each $n \in \omega$, let $\overline{n}$ be the numeral for $n$.
For each $\mathcal{L}_A$-formula $\varphi$, let $\gn{\varphi}$ be the numeral for the G\"odel number of $\varphi$.
Throughout the present paper, $T$ always denotes a consistent primitive recursively axiomatized $\mathcal{L}_A$-theory containing Peano Arithmetic $\mathsf{PA}$.
We say that an $\mathcal{L}_A$-formula $\mathrm{Pr}_T(x)$ is a \textit{provability predicate} of $T$ if for any $\mathcal{L}_A$-formula $\varphi$, $T \vdash \varphi$ if and only if $\mathsf{PA} \vdash \mathrm{Pr}_T(\gn{\varphi})$.
In his proof of the incompleteness theorems, G\"odel constructed a primitive recursive proof predicate $\mathrm{Proof}_T(x, y)$ of $T$ naturally saying that $y$ is the G\"odel number of a $T$-proof of an $\mathcal{L}_A$-formula whose G\"odel number is $x$.
Then, the $\Sigma_1$ formula $\mathrm{Prov}_T(x)$ defined by $\exists y \mathrm{Proof}_T(x, y)$ is a provability predicate of $T$.
Then, it is shown that $\mathrm{Prov}_T(x)$ satisfies the conditions $\D{2}$ and $\D{3}$ given in the introduction.
We say that a $\Sigma_1$ formula $\mathrm{Pr}_T^{\mathrm{R}}(x)$ is a \textit{Rosser's provability predicate} of $T$ if there exists a primitive recursive formula $\mathrm{Prf}_T(x, y)$ satisfying the following three conditions:
\begin{enumerate}
\item For any $\mathcal{L}_A$-formula $\varphi$ and $n \in \omega$, $\mathsf{PA} \vdash \mathrm{Proof}_T(\gn{\varphi}, \overline{n}) \leftrightarrow \mathrm{Prf}_T(\gn{\varphi}, \overline{n})$.
\item $\mathsf{PA} \vdash \forall x \Bigl(\mathrm{Fml}_{\mathcal{L}_A}(x) \to \bigl(\mathrm{Prov}_T(x) \leftrightarrow \exists y \mathrm{Prf}_T(x, y) \bigr) \Bigr)$, where $\mathrm{Fml}_{\mathcal{L}_A}(x)$ is a primitive recursive formula naturally expressing that $x$ is the G\"odel number of an $\mathcal{L}_A$-formula.
\item $\mathrm{Pr}_T^{\mathrm{R}}(x)$ is of the form $\exists y \bigl(\mathrm{Fml}_{\mathcal{L}_A}(x) \land \mathrm{Prf}_T(x, y) \land \forall z < y \, \neg \mathrm{Prf}_T(\dot{\neg}(x), z) \bigr)$, where $\dot{\neg}(x)$ is a primitive recursive term corresponding to a primitive recursive function calculating the G\"odel number of $\neg \varphi$ from that of $\varphi$.
\end{enumerate}
It is shown that each Rosser's provability predicate of $T$ is in fact a $\Sigma_1$ provability predicate of $T$.
It is also shown that for any Rosser's provability predicate $\mathrm{Pr}_T^{\mathrm{R}}(x)$ of $T$ and any $\mathcal{L}_A$-formula $\varphi$, if $T \vdash \neg \varphi$, then $\mathsf{PA} \vdash \neg \mathrm{Pr}_T^{\mathrm{R}}(\gn{\varphi})$.
The language $\mathcal{L}(\Box)$ of modal propositional logic consists of propositional variables, the logical constant $\bot$, propositional connectives $\land, \lor, \to$, and the modal operator $\Box$.
Let $\mathsf{MF}$ be the set of all $\mathcal{L}(\Box)$-formulas.
The axioms of the modal logic $\mathsf{K}$ are all propositional tautologies in $\mathcal{L}(\Box)$ and the axiom scheme $\Box(A \to B) \to (\Box A \to \Box B)$.
The inference rules of $\mathsf{K}$ are Modus Ponens (\textsc{MP}) $\dfrac{A \quad A \to B}{B}$ and Necessitation (\textsc{Nec}) $\dfrac{A}{\Box A}$.
The modal logics $\mathsf{K}D$, $\mathsf{K4}$, and $\mathsf{GL}$ are obtained from $\mathsf{K}$ by adding the axiom schemata $\neg \Box \bot$, $\Box A \to \Box \Box A$, and $\Box(\Box A \to A) \to \Box A$, respectively.
When we interpret $\Box$ by a provability predicate $\mathrm{Pr}_T(x)$, then the axiom schemata $\Box(A \to B) \to (\Box A \to \Box B)$, $\Box A \to \Box \Box A$, and $\Box(\Box A \to A) \to \Box A$ correspond to $\D{2}$, $\D{3}$, and L\"ob's theorem, respectively.
To state these correspondences precisely, we introduce the notion of arithmetical interpretations.
For each provability predicate $\mathrm{Pr}_T(x)$ of $T$, a mapping $f$ from $\mathsf{MF}$ to a set of $\mathcal{L}_A$-sentences is called an \textit{arithmetical interpretation based on $\mathrm{Pr}_T(x)$} if it satisfies the following conditions:
\begin{enumerate}
\item $f(\bot)$ is $0=1$,
\item $f(\neg A)$ is $\neg f(A)$,
\item $f(A \circ B)$ is $f(A) \circ f(B)$ for $\circ \in \{\land, \lor, \to\}$,
\item $f(\Box A)$ is $\mathrm{Pr}_T(\gn{f(A)})$.
\end{enumerate}
Let $\mathsf{PL}(\mathrm{Pr}_T)$ be the set of all $\mathcal{L}(\Box)$-formulas $A$ satisfying that for any arithmetical interpretation $f$ based on $\mathrm{Pr}_T(x)$, $T \vdash f(A)$.
The set $\mathsf{PL}(\mathrm{Pr}_T)$ is called the \textit{provability logic} of $\mathrm{Pr}_T(x)$.
It is obvious that for any provability predicate $\mathrm{Pr}_T(x)$ of $T$, $\mathsf{PL}(\mathrm{Pr}_T)$ is closed under \textsc{Nec}.
If $\mathrm{Pr}_T(x)$ satisfies $\D{2}$, then $\mathsf{PL}(\mathrm{Pr}_T)$ contains the logic $\mathsf{K}$, that is, $\mathsf{PL}(\mathrm{Pr}_T)$ is a normal modal logic.
The study of provability logics can be approached from two directions corresponding to the following two problems, respectively.
\begin{prob}\label{MProb1}
For each provability predicate $\mathrm{Pr}_T(x)$ of $T$, how is $\mathsf{PL}(\mathrm{Pr}_T)$ axiomatized and what properties does it have?
\end{prob}
\begin{prob}\label{MProb2}
For which modal logic $L$ is there a provability predicate $\mathrm{Pr}_T(x)$ such that $L = \mathsf{PL}(\mathrm{Pr}_T)$?
\end{prob}
The most striking result concerning the first problem is Solovay's arithmetical completeness theorems \cite{Sol}.
It states that if $T$ is $\Sigma_1$-sound, then $\mathsf{PL}(\mathrm{Prov}_T)$ is exactly $\mathsf{GL}$.
Visser \cite{Vis} proved that if $T$ is not $\Sigma_1$-sound, then $\mathsf{PL}(\mathrm{Prov}_T)$ is either $\mathsf{GL}$ or $\mathsf{GL} + \Box^n \bot$ for some $n \geq 1$.
As an interesting example regarding the first problem, we present here Shavrukov's result \cite{Sha94}.
Let $\Pr_{\mathsf{PA}}^{\mathrm{Sh}}(x)$ be the $\Sigma_2$ provability predicate $\exists y \bigl(\mathrm{Pr}_{\mathbf{I\Sigma}_y}(x) \land \neg \mathrm{Prov}_{\mathbf{I\Sigma}_y}(\gn{0=1}) \bigr)$ of $\mathsf{PA}$.
Then, Shavrukov proved that $\mathsf{PL}(\mathrm{Pr}_{\mathsf{PA}}^{\mathrm{Sh}})$ is the logic $\mathsf{K}D + (\Box A \to \Box((\Box B \to B) \lor \Box A))$.
For the second problem, the following results have been obtained by previous studies.
\begin{itemize}
\item (Kurahashi \cite{Kur20}) There exists a Rosser's provability predicate $\mathrm{Pr}_T^{\mathrm{R}}(x)$ of $T$ such that $\mathsf{PL}(\mathrm{Pr}_T^{\mathrm{R}}) = \mathsf{K}D$.
\item (Kurahashi \cite{Kur18_1}) There exists a $\Sigma_2$ provability predicate $\mathrm{Pr}_T(x)$ of $T$ such that $\mathsf{PL}(\mathrm{Pr}_T) = \mathsf{K}$.
\item (Kurahashi \cite{Kur18_2}) For each $n \geq 2$, there exists a $\Sigma_2$ provability predicate $\mathrm{Pr}_T(x)$ of $T$ such that $\mathsf{PL}(\mathrm{Pr}_T) = \mathsf{K} + (\Box(\Box^n A \to A) \to \Box A)$.
\item (Montague \cite{Mon63}) For any provability predicate $\mathrm{Pr}_T(x)$ of $T$, $\mathsf{PL}(\mathrm{Pr}_T) \nsupseteq \mathsf{KT}$ ($= \mathsf{K} + (\Box A \to A)$).
\item (L\"ob \cite{Lob}) For any provability predicate $\mathrm{Pr}_T(x)$ of $T$, $\mathsf{PL}(\mathrm{Pr}_T) \neq \mathsf{K4}$.
\item (Kurahashi \cite{Kur18_2}) For any provability predicate $\mathrm{Pr}_T(x)$ of $T$, if $T$ does not prove $\mathrm{Pr}_T(\gn{0=1})$, then $\mathsf{PL}(\mathrm{Pr}_T) \nsupseteq \mathsf{KB}$ ($= \mathsf{K} + (A \to \Box \Diamond A)$) and $\mathsf{PL}(\mathrm{Pr}_T) \nsupseteq \mathsf{K5}$ ($= \mathsf{K} + (\Diamond A \to \Box \Diamond A)$).
\end{itemize}
All of the above results are for normal mode logics.
On the other hand, there is a result concerning a non-normal modal logic.
Shavrukov \cite{Sha91} introduced the bimodal logic $\mathsf{GR}$ of the usual and Rosser's provability predicates.
Let $\mathcal{L}(\Box, \blacksquare)$ be the language of modal propositional logic equipped with an additional modal operator $\blacksquare$.
The axiom schemata of $\mathsf{GR}$ are as follows:
\begin{enumerate}
\item Those of $\mathsf{GL}$ for $\Box$,
\item $\blacksquare A \to \Box A$,
\item $\Box A \to \Box \blacksquare A$,
\item $\Box A \to (\Box \bot \lor \blacksquare A)$,
\item $\Box \neg A \to \Box \neg \blacksquare A$.
\end{enumerate}
The inference rules of $\mathsf{GR}$ are \textsc{MP}, \textsc{Nec}, and $\dfrac{\Box A}{A}$.
A \textit{bimodal arithmetical interpretation $f$ based on $(\mathrm{Pr}_T, \mathrm{Pr}_T^{\mathrm{R}})$} is an arithmetical interpretation based on $\mathrm{Pr}_T(x)$ such that $f(\blacksquare A)$ is $\mathrm{Pr}_T^{\mathrm{R}}(\gn{f(A)})$.
Shavrukov proved the following arithmetical soundness and completeness theorems.
\begin{thm}[The arithmetical soundness theorem of $\mathsf{GR}$ {\cite[Lemma 2.5]{Sha91}}]
For any Rosser's provability predicate $\mathrm{Pr}_T^{\mathrm{R}}(x)$ of $T$, any bimodal arithmetical interpretation $f$ based on $(\mathrm{Prov}_T, \mathrm{Pr}_T^{\mathrm{R}})$, and any $\mathcal{L}(\Box, \blacksquare)$-formula $A$, if $\mathsf{GR} \vdash A$, then $\mathsf{PA} \vdash f(A)$.
\end{thm}
\begin{thm}[The uniform arithmetical completeness theorem of $\mathsf{GR}$ {\cite[Theorem 3.1]{Sha91}}]
Suppose that $T$ is $\Sigma_1$-sound.
Then, there exist a Rosser's provability predicate $\mathrm{Pr}_T^{\mathrm{R}}(x)$ of $T$ and a bimodal arithmetical interpretation $f$ based on $(\mathrm{Prov}_T, \mathrm{Pr}_T^{\mathrm{R}})$ such that for any $\mathcal{L}(\Box, \blacksquare)$-formula $A$, $\mathsf{GR} \vdash A$ if and only if $T \vdash f(A)$.
\end{thm}
Let $L^{\mathrm{R}}$ be the unimodal logic obtained by replacing all $\blacksquare$ in the $\Box$-free fragment of $\mathsf{GR}$ by $\Box$.
The following corollary follows from Shavrukov's theorems.
\begin{cor}\label{Cor:LR}
If $T$ is $\Sigma_1$-sound, then
\[
L^{\mathrm{R}} = \bigcap \{\mathsf{PL}(\mathrm{Pr}_T^{\mathrm{R}}) \mid \mathrm{Pr}_T^{\mathrm{R}}(x)\ \text{is a Rosser's provability predicate of}\ T\}.
\]
Furthermore, there exists a Rosser's provability predicate $\mathrm{Pr}_T^{\mathrm{R}}(x)$ of $T$ such that $L^{\mathrm{R}} = \mathsf{PL}(\mathrm{Pr}_T^{\mathrm{R}})$.
\end{cor}
Corollary \ref{Cor:LR} states that $L^{\mathrm{R}}$ is the provability logic of all Rosser's provability predicates.
The logic $L^{\mathrm{R}}$ is a non-normal modal logic because there are Rosser's provability predicates that do not satisfy $\D{2}$.
However, since no specific axiomatization for $L^{\mathrm{R}}$ is obtained, Corollary \ref{Cor:LR} is not sufficient for us in view of Problems \ref{MProb1}.
In this context, our purpose in the present paper is to axiomatize the following four logics:
\begin{enumerate}
\item $\bigcap \{\mathsf{PL}(\mathrm{Pr}_T) \mid \mathrm{Pr}_T(x)\ \text{is a provability predicate of}\ T\}$,
\item $\bigcap \{\mathsf{PL}(\mathrm{Pr}_T) \mid \mathrm{Pr}_T(x)\ \text{is a provability predicate of}\ T\ \text{satisfying}\ \D{3}\}$,
\item $\bigcap \{\mathsf{PL}(\mathrm{Pr}_T^{\mathrm{R}}) \mid \mathrm{Pr}_T^{\mathrm{R}}(x)\ \text{is a Rosser's provability predicate of}\ T\}$,
\item $\bigcap \{\mathsf{PL}(\mathrm{Pr}_T^{\mathrm{R}}) \mid \mathrm{Pr}_T^{\mathrm{R}}(x)\ \text{is a Rosser's provability predicate of}\ T\ \text{satisfying}\ \D{3}\}$.
\end{enumerate}
In the next section, we introduce the logics $\mathsf{N}$, $\mathsf{N}F$, $\mathsf{N}R$, and $\mathsf{N}RF$ which are candidates for axiomatizations of these logics.
We study these logics from the point of view of Problems \ref{MProb1} and \ref{MProb2}.
\section{The logic $\mathsf{N}$ and its extensions}\label{Sec:Compl}
For any provability predicate $\mathrm{Pr}_T(x)$, the provability logic $\mathsf{PL}(\mathrm{Pr}_T)$ is closed under \textsc{Nec}.
Thus, the provability logic
\[
\bigcap \{\mathsf{PL}(\mathrm{Pr}_T) \mid \mathrm{Pr}_T(x)\ \text{is a provability predicate of}\ T\}
\]
of all provability predicates is also closed under \textsc{Nec}.
On the other hand, there seems to be no other non-trivial modal logical principle that is common to all provability predicates.
Then, our candidate for the axiomatization of the provability logic of all provability predicates is the pure logic of necessitation $\mathsf{N}$ that was introduced by Fitting, Marek, and Truszczy\'nski \cite{FMT}.
The axioms of $\mathsf{N}$ are propositional tautologies in the language $\mathcal{L}(\Box)$ and the inference rules of $\mathsf{N}$ are \textsc{MP} and \textsc{Nec}.
Fitting, Marek, and Truszczy\'nski introduced the following natural relational semantics for $\mathsf{N}$.
\begin{defn}[$\mathsf{N}$-frames]\leavevmode
\begin{itemize}
\item We say that a tuple $(W, \{\prec_B\}_{B \in \mathsf{MF}})$ is an \textit{$\mathsf{N}$-frame} if $W$ is a non-empty set and for each $B \in \mathsf{MF}$, $\prec_B$ is a binary relation on $W$.
\item We say that a triple $(W, \{\prec_B\}_{B \in \mathsf{MF}}, \Vdash)$ is an \textit{$\mathsf{N}$-model} if $(W, \{\prec_B\}_{B \in \mathsf{MF}})$ is an $\mathsf{N}$-frame and $\Vdash$ is a satisfaction relation on $W \times \mathsf{MF}$ satisfying the usual conditions for propositional connectives and
\[
x \Vdash \Box B \iff \forall y \in W(x \prec_B y \Rightarrow y \Vdash B).
\]
\item A formula $A$ is \textit{valid} in an $\mathsf{N}$-model $(W, \{\prec_B\}_{B \in \mathsf{MF}}, \Vdash)$ if for any $x \in W$, $x \Vdash A$.
\item A formula $A$ is \textit{valid} in an $\mathsf{N}$-frame $(W, \{\prec_B\}_{B \in \mathsf{MF}})$ if $A$ is valid in any $\mathsf{N}$-model $(W, \{\prec_B\}_{B \in \mathsf{MF}}, \Vdash)$ based on $(W, \{\prec_B\}_{B \in \mathsf{MF}})$.
\end{itemize}
\end{defn}
Fitting, Marek, and Truszczy\'nski proved that $\mathsf{N}$ is sound and complete and has the finite frame property with respect to this semantics.
\begin{fact}[Fitting, Marek, and Truszczy\'nski {\cite[Theorems 3.6 and 4.10]{FMT}}]\label{Fact1}
For any $A \in \mathsf{MF}$, the following are equivalent:
\begin{enumerate}
\item $\mathsf{N} \vdash A$.
\item $A$ is valid in all $\mathsf{N}$-frames.
\item $A$ is valid in all finite $\mathsf{N}$-frames.
\end{enumerate}
\end{fact}
Each $\mathsf{N}$-model has infinitely many binary relations $\{\prec_B\}_{B \in \mathsf{MF}}$, but the truth of each $\mathcal{L}(\Box)$-formula in each world is determined by referring to only a finite number of those relations.
Let $\mathsf{Sub}(A)$ be the set of all subformulas of $A \in \mathsf{MF}$.
\begin{fact}[Fitting, Marek, and Truszczy\'nski {\cite[Theorem 4.11]{FMT}}]\label{Fact2}
Let $A \in \mathsf{MF}$.
Let $(W, \{\prec_B\}_{B \in \mathsf{MF}}, \Vdash)$ and $(W, \{\prec^*_B\}_{B \in \mathsf{MF}}, \Vdash^*)$ be any $\mathsf{N}$-models satisfying the following two conditions:
\begin{enumerate}
\item For each $x \in W$ and $p \in \mathsf{Sub}(A)$, $x \Vdash p \iff x \Vdash^* p$;
\item For each $\Box B \in \mathsf{Sub}(A)$, $\prec_B = \prec^*_B$.
\end{enumerate}
Then, for every $x \in W$, $x \Vdash A \iff x \Vdash^* A$.
\end{fact}
We introduce three extensions $\mathsf{N}R$, $\mathsf{N}F$, and $\mathsf{N}RF$ of $\mathsf{N}$.
\begin{defn}\leavevmode
\begin{itemize}
\item The logic $\mathsf{N}R$ is obtained from $\mathsf{N}$ by adding the inference rule $\dfrac{\neg B}{\neg \Box B}$.
\item The logics $\mathsf{N}F$ and $\mathsf{N}RF$ are obtained from $\mathsf{N}$ and $\mathsf{N}R$ by adding the axiom scheme $\Box B \to \Box \Box B$, respectively.
\end{itemize}
\end{defn}
We call the rule $\dfrac{\neg B}{\neg \Box B}$ the Rosser rule (\textsc{Ros}).
Before proving the completeness theorems of these logics, we show that the validity of these logics are related to some appropriate conditions of $\mathsf{N}$-frames.
\begin{defn}
Let $A \in \mathsf{MF}$ and $\Gamma \subseteq \mathsf{MF}$.
Let $\mathcal{F} = (W, \{\prec_B\}_{B \in \mathsf{MF}})$ be any $\mathsf{N}$-frame.
\begin{itemize}
\item $\mathcal{F}$ is called \textit{$A$-serial} if for every $x \in W$, there exists a $y \in W$ such that $x \prec_A y$.
\item $\mathcal{F}$ is said to be \textit{$\Gamma$-serial} if $\mathcal{F}$ is $A$-serial for every $\Box A \in \Gamma$.
\item $\mathcal{F}$ is called \textit{serial} if $\mathcal{F}$ is $\mathsf{MF}$-serial.
\end{itemize}
\end{defn}
\begin{prop}
Let $A \in \mathsf{MF}$ and $\mathcal{M} = (W, \{\prec_B\}_{B \in \mathsf{MF}}, \Vdash)$ be any $\mathsf{N}$-model.
Suppose that the $\mathsf{N}$-frame $\mathcal{F} = (W, \{\prec_B\}_{B \in \mathsf{MF}})$ is $A$-serial.
If $\neg A$ is valid in $\mathcal{M}$, then $\neg \Box A$ is also valid in $\mathcal{M}$.
\end{prop}
\begin{proof}
Suppose that $\mathcal{F}$ is $A$-serial and $\neg A$ is valid in $\mathcal{M}$.
Let $x \in W$ be any element.
Since $\mathcal{F}$ is $A$-serial, there exists a $y \in W$ such that $x \prec_A y$.
Since $\neg A$ is valid in $\mathcal{M}$, we have $y \Vdash \neg A$.
Thus, $x \Vdash \neg \Box A$.
Therefore, $\neg \Box A$ is valid in $\mathcal{M}$.
\end{proof}
\begin{cor}\label{Cor:NR}
Let $A \in \mathsf{MF}$.
If $\mathsf{N}R \vdash A$, then $A$ is valid in all serial $\mathsf{N}$-frames.
\end{cor}
\begin{defn}
Let $A \in \mathsf{MF}$ and $\Gamma \subseteq \mathsf{MF}$.
Let $\mathcal{F} = (W, \{\prec_B\}_{B \in \mathsf{MF}})$ be any $\mathsf{N}$-frame.
\begin{itemize}
\item $\mathcal{F}$ is called \textit{$A$-transitive} if for every $x, y, z \in W$, if $x \prec_{\Box A} y$ and $y \prec_A z$, then $x \prec_A z$.
\item $\mathcal{F}$ is said to be \textit{$\Gamma$-transitive} if $\mathcal{F}$ is $A$-transitive for every $\Box \Box A \in \Gamma$.
\item $\mathcal{F}$ is called \textit{transitive} if $\mathcal{F}$ is $\mathsf{MF}$-transitive.
\end{itemize}
\end{defn}
\begin{prop}
Let $A \in \mathsf{MF}$ and $\mathcal{F} = (W, \{\prec_B\}_{B \in \mathsf{MF}})$ be any $\mathsf{N}$-frame.
If $\mathcal{F}$ is $A$-transitive, then $\Box A \to \Box \Box A$ is valid in $\mathcal{F}$.
\end{prop}
\begin{proof}
Suppose that $\mathcal{F}$ is $A$-transitive.
Let $(\mathcal{F}, \Vdash)$ be any $\mathsf{N}$-model based on $\mathcal{F}$.
Let $x \in W$ be any element with $x \Vdash \Box A$.
Let $y, z \in W$ be such that $x \prec_{\Box A} y$ and $y \prec_A z$.
Since $\mathcal{F}$ is $A$-transitive, we have $x \prec_A z$.
Then, $z \Vdash A$.
Since $z$ is an arbitrary element with $y \prec_A z$, we have $y \Vdash \Box A$.
Also, we obtain $x \Vdash \Box \Box A$.
We conclude that $\Box A \to \Box \Box A$ is valid in $\mathcal{F}$.
\end{proof}
\begin{cor}\label{Cor:NF}
Let $A \in \mathsf{MF}$.
\begin{enumerate}
\item If $\mathsf{N}F \vdash A$, then $A$ is valid in all transitive $\mathsf{N}$-frames.
\item If $\mathsf{N}RF \vdash A$, then $A$ is valid in all transitive and serial $\mathsf{N}$-frames.
\end{enumerate}
\end{cor}
\begin{rem}
Unlike the case of Kripke frames, the validity of $\Box A \to \Box \Box A$ in an $\mathsf{N}$-frame is not equivalent to the $A$-transitivity in general.
Let $W = \{x, y, z\}$, $\prec_{\Box \Box A} = \{(x, y)\}$, $\prec_{\Box A} = \{(y, z)\}$, and $\prec_B = \emptyset$ for $B \notin \{\Box \Box A, \Box A\}$.
Then, it is shown that $\Box \Box A \to \Box \Box \Box A$ is valid in the $\mathsf{N}$-frame $\mathcal{F} = (W, \{\prec_B\}_{B \in \mathsf{MF}})$, but $\mathcal{F}$ is not $\Box A$-transitive\footnote{This example is due to Sohei Iwata.}.
\end{rem}
We prove the completeness and finite frame property of the logics $\mathsf{N}R$, $\mathsf{N}F$, and $\mathsf{N}RF$.
\begin{thm}[The completeness and finite frame property of $\mathsf{N}R$]\label{Thm:complNR}
For any $A \in \mathsf{MF}$, the following are equivalent:
\begin{enumerate}
\item $\mathsf{N}R \vdash A$.
\item $A$ is valid in all serial $\mathsf{N}$-frames.
\item $A$ is valid in all finite serial $\mathsf{N}$-frames.
\item $A$ is valid in all finite $\mathsf{Sub}(A)$-serial $\mathsf{N}$-frames.
\end{enumerate}
\end{thm}
\begin{thm}[The completeness and finite frame property of $\mathsf{N}F$]\label{Thm:complNF}
For any $A \in \mathsf{MF}$, the following are equivalent:
\begin{enumerate}
\item $\mathsf{N}F \vdash A$.
\item $A$ is valid in all transitive $\mathsf{N}$-frames.
\item $A$ is valid in all finite transitive $\mathsf{N}$-frames.
\item $A$ is valid in all finite $\mathsf{Sub}(A)$-transitive $\mathsf{N}$-frames.
\end{enumerate}
\end{thm}
\begin{thm}[The completeness and finite frame property of $\mathsf{N}RF$]\label{Thm:complNRF}
For any $A \in \mathsf{MF}$, the following are equivalent:
\begin{enumerate}
\item $\mathsf{N}RF \vdash A$.
\item $A$ is valid in all transitive and serial $\mathsf{N}$-frames.
\item $A$ is valid in all finite transitive and serial $\mathsf{N}$-frames.
\item $A$ is valid in all finite $\mathsf{Sub}(A)$-transitive and $\mathsf{Sub}(A)$-serial $\mathsf{N}$-frames.
\end{enumerate}
\end{thm}
\begin{proof}
We prove Theorems \ref{Thm:complNR}, \ref{Thm:complNF}, and \ref{Thm:complNRF} simultaneously.
Let $L$ be one of $\mathsf{N}R$, $\mathsf{N}F$, and $\mathsf{N}RF$.
$(1 \Rightarrow 2)$: This is already proved in Corollaries \ref{Cor:NR} and \ref{Cor:NF}.
$(2 \Rightarrow 3)$: Obvious.
$(3 \Rightarrow 4)$: Suppose that $A$ is valid in all finite $\mathsf{N}$-frames satisfying the corresponding conditions.
Let $\mathcal{M} = (W, \{\prec_B\}_{B \in \mathsf{MF}}, \Vdash)$ be any finite $\mathsf{N}$-model whose frame $\mathcal{F} = (W, \{\prec_B\}_{B \in \mathsf{MF}})$ satisfies the corresponding conditions restricted to $\mathsf{Sub}(A)$.
For example, if $L = \mathsf{N}F$, then $\mathcal{F}$ is $\mathsf{Sub}(A)$-transitive.
For each $B \in \mathsf{MF}$, let $\prec^*_B$ be the binary relation on $W$ defined as follows:
\[
\prec^*_B : = \begin{cases} \prec_B & \text{if}\ \Box B \in \mathsf{Sub}(A), \\
\{(x, x) \mid x \in W\} & \text{otherwise}.
\end{cases}
\]
Let $\mathcal{F}^* := (W, \{\prec^*_B\}_{B \in \mathsf{MF}})$.
\begin{cl}
If $L \in \{\mathsf{N}R, \mathsf{N}RF\}$, then $\mathcal{F}^*$ is serial.
\end{cl}
\begin{proof}
Let $x \in W$ and $B \in \mathsf{MF}$.
\begin{itemize}
\item If $\Box B \in \mathsf{Sub}(A)$, then there exists a $y \in W$ such that $x \prec_B y$ because $\mathcal{F}$ is $\mathsf{Sub}(A)$-serial.
Thus, $x \prec^*_B y$.
\item If $\Box B \notin \mathsf{Sub}(A)$, then $x \prec^*_B x$.
\end{itemize}
We have proved that $\mathcal{F}^*$ is $B$-serial.
\end{proof}
\begin{cl}
If $L \in \{\mathsf{N}F, \mathsf{N}RF\}$, then $\mathcal{F}^*$ is transitive.
\end{cl}
\begin{proof}
Let $x, y, z \in W$ and $B \in \mathsf{MF}$ be such that $x \prec^*_{\Box B} y$ and $y \prec^*_B z$.
\begin{itemize}
\item If $\Box \Box B \in \mathsf{Sub}(A)$, then $\Box B \in \mathsf{Sub}(A)$, and hence $x \prec_{\Box B} y$ and $y \prec_B z$.
Since $\mathcal{F}$ is $\mathsf{Sub}(A)$-transitive, we have $x \prec_B z$.
Thus, $x \prec^*_B z$.
\item If $\Box \Box B \notin \mathsf{Sub}(A)$, then $x = y$ by the definition of $\prec^*_{\Box B}$.
Since $y \prec^*_B z$, we obtain $x \prec^*_B z$.
\end{itemize}
We have proved that $\mathcal{F}^*$ is $B$-transitive.
\end{proof}
Therefore, $\mathcal{F}^*$ is a finite $\mathsf{N}$-frame satisfying the corresponding conditions.
Let $\Vdash^*$ be the satisfaction relation on $\mathcal{F}^*$ defined by $x \Vdash^* p : \iff x \Vdash p$.
By the supposition, $A$ is valid in $\mathcal{F}^*$.
In particular, $A$ is valid in $(\mathcal{F}^*, \Vdash^*)$.
Since $\prec^*_B = \prec_B$ for any $B \in \mathsf{MF}$ with $\Box B \in \mathsf{Sub}(A)$, by Fact \ref{Fact2}, $A$ is also valid in $\mathcal{M}$.
$(4 \Rightarrow 1)$: We prove the contrapositive.
Suppose $L \nvdash A$, and we would like to find a corresponding finite $\mathsf{N}$-frame in which $A$ is not valid.
For each formula $B \in \mathsf{MF}$, let ${\sim}B$ be $C$ if $B$ is of the form $\neg C$ and $\neg B$ othewise.
Let $\overline{\mathsf{Sub}(A)} : = \mathsf{Sub}(A) \cup \{{\sim}B \mid B \in \mathsf{Sub}(A)\}$.
We say that $X \subseteq \overline{\mathsf{Sub}(A)}$ is \textit{$L$-consistent} if $L \nvdash \neg \bigwedge X$ where $\bigwedge X$ is a conjunction of all elements of $X$.
Also, $X$ is called \textit{$A$-maximally $L$-consistent} if $X$ is $L$-consistent and for any $B \in \overline{\mathsf{Sub}(A)} \setminus X$, $X \cup \{B\}$ is $L$-inconsistent.
It is easily shown that every $L$-consistent subset $X$ of $\overline{\mathsf{Sub}(A)}$ is extended to an $A$-maximally $L$-consistent set.
We define the $\mathsf{N}$-model $\mathcal{M} = (W, \{\prec_B\}_{B \in \mathsf{MF}}, \Vdash)$ as follows:
\begin{itemize}
\item $W: = \{X \subseteq \overline{\mathsf{Sub}(A)} \mid X$ is $A$-maximally $L$-consistent$\}$;
\item For $X, Y \in W$, $X \prec_B Y : \iff \Box B \notin X$ or $B \in Y$;
\item For each propositional variable $p$ and $X \in W$, $X \Vdash p : \iff p \in X$.
\end{itemize}
Let $n$ be the number of elements of $\overline{\mathsf{Sub}(A)}$.
Then, the number of elements of $X$ is smaller than $2^n$.
Since $L \nvdash A$, $\{{\sim}A\}$ is $L$-consistent.
Then, we have $X_A \in W$ such that ${\sim}A \in X_A$.
\begin{cl}\label{TL}
For any $X \in W$ and $B \in \overline{\mathsf{Sub}(A)}$,
\[
X \Vdash B \iff B \in X.
\]
\end{cl}
\begin{proof}
We prove the claim by induction on the construction of $B$.
We only give a proof of the case that $B$ is of the form $\Box C$.
$(\Rightarrow)$: We prove the contrapositive.
Suppose $\Box C \notin X$.
Since $X$ is maximal, $\neg \Box C \in X$.
Assume, towards a contradiction, that $\{{\sim}C\}$ is $L$-inconsistent.
Then, $L \vdash C$.
By \textsc{Nec}, $L \vdash \Box C$.
This contradicts the $L$-consistency of $X$.
We proved that $\{{\sim}C\}$ is $L$-consistent.
Let $Y \in W$ be such that $\{{\sim}C\} \subseteq Y$.
Since $\Box C \notin X$, we have $X \prec_C Y$ by the definition of $\prec_C$.
Since ${\sim}C \in Y$, we have $C \notin Y$.
By the induction hypothesis, $Y \nVdash C$.
We conclude that $X \nVdash \Box C$.
$(\Leftarrow)$: Suppose $\Box C \in X$.
Let $Y \in W$ be such that $X \prec_C Y$.
By the definition of $\prec_C$, we have $C \in Y$.
By the induction hypothesis, $Y \Vdash C$.
Hence, $X \Vdash \Box C$.
\end{proof}
Since $A \notin X_A$, by Claim \ref{TL}, we obtain $X_A \nVdash A$.
Therefore, $A$ is not valid in $\mathcal{M}$.
\begin{cl}
If $L \in \{\mathsf{N}R, \mathsf{N}RF\}$, then $(W, \{\prec_B\}_{B \in \mathsf{MF}})$ is $\mathsf{Sub}(A)$-serial.
\end{cl}
\begin{proof}
Let $X \in W$ and $\Box B \in \mathsf{Sub}(A)$.
We distinguish the following two cases:
\begin{itemize}
\item Case 1: $\Box B \notin X$. \\
By Claim \ref{TL}, $X \nVdash \Box B$.
Then, there exists a $Y \in W$ such that $X \prec_B Y$ and $Y \nVdash B$.
\item Case 2: $\Box B \in X$. \\
Suppose, towards a contradiction, that $\{B\}$ is $L$-inconsistent.
Then, $L \vdash \neg B$.
By the rule \textsc{Ros}, we have $L \vdash \neg \Box B$.
This contradicts the $L$-consistency of $X$.
Hence, $\{B\}$ is $L$-consistent and there exists a $Y \in W$ such that $B \in Y$.
By the definition of $\prec_B$, $X \prec_B Y$.
\end{itemize}
In either case, we have a $Y \in W$ such that $X \prec_B Y$.
We conclude that $(W, \{\prec_B\}_{B \in \mathsf{MF}})$ is $\mathsf{Sub}(A)$-serial.
\end{proof}
\begin{cl}
If $L \in \{\mathsf{N}F, \mathsf{N}RF\}$, then $(W, \{\prec_B\}_{B \in \mathsf{MF}})$ is $\mathsf{Sub}(A)$-transitive.
\end{cl}
\begin{proof}
Let $X, Y, Z \in W$ and $\Box \Box B \in \mathsf{Sub}(A)$ be such that $X \prec_{\Box B} Y$ and $Y \prec_B Z$.
If $\Box B \in X$, then $\Box \Box B \in X$ because $L \vdash \Box B \to \Box \Box B$.
Since $X \prec_{\Box B} Y$, we have $\Box B \in Y$.
Also, since $Y \prec_B Z$, we have $B \in Z$.
By the definition of $\prec_B$, we obtain $X \prec_B Z$.
Therefore, $(W, \{\prec_B\}_{B \in \mathsf{MF}})$ is $\mathsf{Sub}(A)$-transitive.
\end{proof}
Our proof is finished.
\end{proof}
Furthermore, from the proofs of Theorems \ref{Thm:complNR}, \ref{Thm:complNF}, and \ref{Thm:complNRF}, we obtain that the sets of all theorems of $\mathsf{N}R$, $\mathsf{N}F$, and $\mathsf{N}RF$ are primitive recursive.
For example, to show that $A \in \mathsf{MF}$ is $\mathsf{N}F$-unprovable, it is sufficient to find a finite fragment $(W, \{\prec_B\}_{\Box B \in \mathsf{Sub}(A)}, \Vdash)$ of a $\mathsf{Sub}(A)$-transitive $\mathsf{N}$-model such that the cardinality of $W$ is smaller than $2^{2n}$ where $n$ is the number of subformulas of $A$.
Thus, an primitive recursive algorithm that simultaneously searches for such finite structures and $\mathsf{N}F$-proofs determines whether each $A \in \mathsf{MF}$ is provable in $\mathsf{N}F$ or not primitive recursively.
It is clear that this procudure is also applied to the logic $\mathsf{N}$.
\section{Arithmetical completeness of $\mathsf{N}$}\label{Sec:N}
It is easy to see that for any provability predicate $\mathrm{Pr}_T(x)$ of $T$, $\mathsf{N} \subseteq \mathsf{PL}(\mathrm{Pr}_T)$.
In this section, we prove that $\mathsf{N}$ is exactly the provability logic of all provability predicates.
Moreover, we prove that $\mathsf{N}$ is one of the logics considered in Problem \ref{MProb2}, namely, there exists a $\Sigma_1$ provability predicate $\mathrm{Pr}_T(x)$ of $T$ such that $\mathsf{N} = \mathsf{PL}(\mathrm{Pr}_T)$.
\begin{thm}[The uniform arithmetical completeness of $\mathsf{N}$]\label{Thm:N}
There exist a $\Sigma_1$ provability predicate $\mathrm{Pr}_T(x)$ of $T$ such that
\begin{enumerate}
\item for any $A \in \mathsf{MF}$ and any arithmetical interpretation $f$ based on $\mathrm{Pr}_T(x)$, if $\mathsf{N} \vdash A$, then $\mathsf{PA} \vdash f(A)$; and
\item there exists an arithmetical interpretation $f$ based on $\mathrm{Pr}_T(x)$ such that for any $A \in \mathsf{MF}$, $\mathsf{N} \vdash A$ if and only if $T \vdash f(A)$.
\end{enumerate}
\end{thm}
Before proving the theorem, we prepare a primitive recursive function $h$ which plays an important role in our proofs of the theorems in this paper.
The function $h$ was originally introduced in \cite{Kur20} to prove the existence of a Rosser's provability predicate whose provability logic is exactly the logic $\mathsf{KD}$.
We say that an $\mathcal{L}_A$-formula is \textit{propositionally atomic} if it is not a Boolean combination of its proper subformulas.
For each propositionally atomic formula $\varphi$, we prepare a propositional variable $p_\varphi$.
We define a primitive recursive mapping $I$ from $\mathcal{L}_A$-formulas to propositional formulas as follows:
\begin{enumerate}
\item For each propositionally atomic formula $\varphi$, $I(\varphi)$ is $p_\varphi$;
\item $I(\neg \varphi)$ is $\neg I(\varphi)$;
\item $I(\varphi \circ \psi)$ is $I(\varphi) \circ I(\psi)$ for $\circ \in \{\land, \lor, \to\}$.
\end{enumerate}
Then, it is shown that $I$ is an injection.
Let $\varphi$ be an $\mathcal{L}_A$-formula and $X$ be a finite set of $\mathcal{L}_A$-formulas.
We say that $\varphi$ is a \textit{tautological consequence} (\textit{t.c.}) of $X$ if $\bigwedge_{\psi \in X} I(\psi) \to I(\varphi)$ is a tautology.
For each natural number $n$, let $P_{T, n}$ be the set of all $\mathcal{L}_A$-formulas having a $T$-proof with the G\"odel number less than or equal to $n$.
Then, it is proved that the set $\{(n, \varphi) \mid \varphi$ is a t.c.~of $P_{T, n}\}$ is primitive recursive.
The above notions and sets are formalized in $\mathsf{PA}$.
In particular, we suppose that $P_{T, n}$ is formalized by using the proof predicate $\mathrm{Proof}_T(x, y)$.
The function $h$ is defined as follows by using the recursion theorem:
\begin{itemize}
\item $h(0) = 0$.
\item $h(m+1) = \begin{cases} i & \text{if}\ h(m) = 0\\
& \quad \&\ i = \min \{j \in \omega \setminus \{0\} \mid \neg S(\overline{j}) \ \text{is a t.c.~of}\ P_{T, m}\}, \\
h(m) & \text{otherwise}.
\end{cases}$
\end{itemize}
Here, $S(x)$ is the $\Sigma_1$ formula $\exists y(h(y) = x)$.
Then, it is shown that the following proposition holds.
\begin{prop}[Cf.~{\cite[Lemma 3.2.]{Kur20}}]\label{Prop:h}
\leavevmode
\begin{enumerate}
\item $\mathsf{PA} \vdash \forall x \forall y(0 < x < y \land S(x) \to \neg S(y))$.
\item $\mathsf{PA} \vdash \neg \mathrm{Con}_T \leftrightarrow \exists x(S(x) \land x \neq 0)$, where $\mathrm{Con}_T$ is the $\Pi_1$ consistency statement $\neg \mathrm{Prov}_T(\gn{0=1})$.
\item For each $i \in \omega \setminus \{0\}$, $T \nvdash \neg S(\overline{i})$.
\item For each $n \in \omega$, $\mathsf{PA} \vdash \forall x \forall y(h(x) = 0 \land h(x+1) = y \land y \neq 0 \to x > \overline{n})$.
\end{enumerate}
\end{prop}
We are ready to prove Theorem \ref{Thm:N}.
\begin{proof}[Proof of Theorem \ref{Thm:N}]
Let $\langle A_n \rangle_{n \in \omega}$ be a primitive recursive enumeration of all $\mathsf{N}$-unprovable $\mathcal{L}(\Box)$-formulas.
For each $n \in \omega$, let $(W_n, \{\prec_{n, B}\}_{B \in \mathsf{MF}}, \Vdash_n)$ be a primitive recursively constructed finite $\mathsf{N}$-model falsifying $A_n$.
We may assume that $\{W_n\}_{n \in \omega}$ is a pairwise disjoint family of subsets of $\omega$ and $\bigcup_{n \in \omega} W_n = \omega \setminus \{0\}$.
Let $\mathcal{M} = (W, \{\prec_B\}_{B \in \mathsf{MF}}, \Vdash)$ be an $\mathsf{N}$-model defined as follows:
\begin{itemize}
\item $W : = \bigcup_{n \in \omega} W_n = \omega \setminus \{0\}$.
\item $x \prec_B y : \iff x, y \in W_n$ and $x \prec_{n, B} y$ for some $n \in \omega$.
\item $x \Vdash p :\iff x \in W_n$ and $x \Vdash_n p$ for some $n \in \omega$.
\end{itemize}
We may assume that $\mathcal{M}$ is primitive recursively represented in $\mathsf{PA}$.
Moreover, we assume that $\mathsf{PA}$ proves basic properties of $\mathcal{M}$.
For each primitive recursive function $g$ enumerating all theorems of $T$, let $\mathrm{Pr}_g(x)$ be the $\Sigma_1$ formula $\exists y(g(y) = x \land \mathrm{Fml}_{\mathcal{L}_A}(x))$.
Then, $\mathrm{Pr}_g(x)$ is a provability predicate of $T$.
We define an arithmetical interpretation $f_g$ based on $\mathrm{Pr}_g(x)$ by $f_g(p): \equiv \exists x (S(x) \land x \Vdash p)$.
Then, from an index of such a function $g$ and $A \in \mathsf{MF}$, the $\mathcal{L}_A$-sentence $f_g(A)$ is primitive recursively computed.
Furthermore, it is shown that each $f_g$ is an injective mapping, and so from an index of $g$ and $f_g(A)$, the $\mathcal{L}(\Box)$-formula $A$ is primitive recursively computed.
Next, we define a primitive recursive function $g_0$ enumerating all theorems of $T$.
The definition of $g_0$ consists of two procedures.
The definition starts with Procedure 1.
The values of $g_0$ are defined step by step in the procedure by referring to $T$-proofs.
At the first time the value of the function $h$ is non-zero, the definition of $g_0$ switches to Procedure 2.
By using the recursion theorem, the arithmetical interpretation $f_{g_0}$ based on the provability predicate $\mathrm{Pr}_{g_0}(x)$ is used in the definition of $g_0$.
Also in the definition, each $\mathcal{L}_A$-formula $\varphi$ and its G\"odel number are identified.
\textsc{Procedure 1}\\
Stage $m$.
\begin{itemize}
\item If $h(m+1) = 0$, then
\[
g_0(m) = \begin{cases} \varphi & \text{if} \ m\ \text{is a}\ T\text{-proof of}\ \varphi, \\
0 & \text{otherwise.}
\end{cases}
\]
Go to stage $m+1$.
\item If $h(m+1) \neq 0$, then go to Procedure 2.
\end{itemize}
\textsc{Procedure 2}\\
Let $m$, $i \neq 0$ and $n$ be such that $h(m) = 0$, $h(m+1) = i$, and $i \in W_n$.
Let $\{\xi_t\}_{t \in \omega}$ be the primitive recursive enumeration of all $\mathcal{L}_A$-formulas introduced in Section \ref{Sec:Pre}.
Define
\[
g_0(m+t) = \begin{cases} \xi_t & \text{if}\ \xi_t \equiv f_{g_0}(B)\ \&\ i \Vdash_n \Box B\ \text{for some}\ \Box B \in \mathsf{Sub}(A_n), \\
0 & \text{otherwise.}
\end{cases}
\]
The definition of $g_0$ is finished.
\begin{cl}\label{NCL}
$\mathsf{PA} + \mathrm{Con}_T \vdash \forall x \forall y \Bigl(\mathrm{Fml}_{\mathcal{L}_A}(x) \to \bigl(\mathrm{Proof}_T(x, y) \leftrightarrow x = g_0(y) \bigr) \Bigr)$.
\end{cl}
\begin{proof}
We argue in $\mathsf{PA} + \mathrm{Con}_T$:
By Proposition \ref{Prop:h}.2, $h(x) = 0$ for all $x$.
Thus, the construction of $g_0$ never switches to Procedure 2.
Then, it is shown that for any $\mathcal{L}_A$-formula $\varphi$ and number $a$, $a$ is a $T$-proof of $\varphi$ if and only if $\varphi = g_0(a)$.
\end{proof}
Then, it is shown that for any $\mathcal{L}_A$-formula $\varphi$ and $n \in \omega$, $\mathsf{PA} \vdash \mathrm{Proof}_T(\gn{\varphi}, \overline{n})$ if and only if $\mathsf{PA} \vdash \gn{\varphi} = g_0(\overline{n})$.
It follows that $\mathrm{Pr}_{g_0}(x)$ is a $\Sigma_1$ provability predicate of $T$.
\begin{cl}\label{Cl:g_0}
Let $i \in W_n$ and $B \in \mathsf{Sub}(A_n)$.
\begin{enumerate}
\item If $i \Vdash_n B$, then $\mathsf{PA} \vdash S(\overline{i}) \to f_{g_0}(B)$.
\item If $i \nVdash_n B$, then $\mathsf{PA} \vdash S(\overline{i}) \to \neg f_{g_0}(B)$.
\end{enumerate}
\end{cl}
\begin{proof}
Clauses 1 and 2 are proved simultaneously by induction on the construction of $B \in \mathsf{Sub}(A_n)$.
We give only a proof of the case that $B$ is of the form $\Box C$.
1. Suppose $i \Vdash_n \Box C$.
We reason in $\mathsf{PA} + S(\overline{i})$:
Let $m$ be such that $h(m) = 0$ and $h(m+1) = i$.
Let $t$ be the number such that $\xi_t \equiv f_{g_0}(C)$.
Since $i \Vdash_n \Box C$ and $\Box C \in \mathsf{Sub}(A_n)$, we have $g_0(m+t) = f_{g_0}(C)$.
Thus, $\mathrm{Pr}_{g_0}(\gn{f_{g_0}(C)})$ holds.
This means that $f_{g_0}(\Box C)$ holds.
2. Suppose $i \nVdash_n \Box C$.
Then, there exists a $j \in W_n$ such that $i \prec_{n, C} j$ and $j \nVdash_n C$.
By the induction hypothesis, $\mathsf{PA} \vdash S(\overline{j}) \to \neg f_{g_0}(C)$.
Let $p$ be a $T$-proof of $S(\overline{j}) \to \neg f_{g_0}(C)$.
We argue in $\mathsf{PA} + S(\overline{i})$:
Let $m$ be such that $h(m) = 0$ and $h(m+1) = i$.
If $f_{g_0}(C)$ is output in Procedure 1, there exists a $T$-proof $q < m$ of $f_{g_0}(C)$.
It follows that $f_{g_0}(C) \in P_{T, m-1}$.
Since $m > p$ by Proposition \ref{Prop:h}.4, we have $S(\overline{j}) \to \neg f_{g_0}(C) \in P_{T, m-1}$.
Hence, $\neg S(\overline{j})$ is a t.c.~of $P_{T, m-1}$.
This contradicts $h(m) = 0$.
If $f_{g_0}(C)$ is output in Procedure 2, then $f_{g_0}(C) \equiv f_{g_0}(D)$ and $i \Vdash_n \Box D$ for some $\Box D \in \mathsf{Sub}(A_n)$.
Since $f_{g_0}$ is injective, we have $C \equiv D$.
Then, $i \Vdash_n \Box C$, this is a contradiction.
We have proved that $f_{g_0}(C)$ is not output by $g_0$.
Thus, $\neg \mathrm{Pr}_{g_0}(\gn{f_{g_0}(C)})$ holds, and hence $\neg f_{g_0}(\Box C)$ holds.
\end{proof}
We finish our proof of Theorem \ref{Thm:N}.
The first clause of the theorem follows from Claim \ref{NCL}.
We show the second clause.
The implication $\Rightarrow$ is obvious.
We prove the implication $\Leftarrow$.
Suppose that $\mathsf{N} \nvdash A$.
Then, $A \equiv A_n$ for some $n \in \omega$ and $i \nVdash_n A$ for some $i \in W_n$.
By Claim \ref{Cl:g_0}, $\mathsf{PA} \vdash S(\overline{i}) \to \neg f_{g_0}(A)$.
Since $T \nvdash \neg S(\overline{i})$ by Proposition \ref{Prop:h}.3, we obtain $T \nvdash f_{g_0}(A)$.
\end{proof}
\begin{cor}
\begin{align*}
\mathsf{N} & = \bigcap \{\mathsf{PL}(\mathrm{Pr}_T) \mid \mathrm{Pr}_T(x)\ \text{is a provability predicate of}\ T\}, \\
& = \bigcap \{\mathsf{PL}(\mathrm{Pr}_T) \mid \mathrm{Pr}_T(x)\ \text{is a}\ \Sigma_1\ \text{provability predicate of}\ T\}.
\end{align*}
Moreover, there exists a $\Sigma_1$ provability predicate $\mathrm{Pr}_T(x)$ of $T$ such that $\mathsf{N} = \mathsf{PL}(\mathrm{Pr}_T)$.
\end{cor}
\section{Arithmetical completeness of $\mathsf{N}F$}\label{Sec:NF}
In this section, we investigate provability predicates satisfying the condition $\D{3}$.
It is easy to show that for any provability predicate $\mathrm{Pr}_T(x)$ satisfying $\D{3}$, $\mathsf{N}F \subseteq \mathsf{PL}(\mathrm{Pr}_T)$.
In this section, we prove that $\mathsf{N}F$ is exactly the provability logic of all provability predicates satisfying $\D{3}$.
Moreover, we prove the following uniform version of arithmetical completeness.
\begin{thm}[The uniform arithmetical completeness of $\mathsf{N}F$]\label{Thm:NF}
There exist a $\Sigma_1$ provability predicate $\mathrm{Pr}_T(x)$ of $T$ such that
\begin{enumerate}
\item for any $A \in \mathsf{MF}$ and any arithmetical interpretation $f$ based on $\mathrm{Pr}_T(x)$, if $\mathsf{N}F \vdash A$, then $\mathsf{PA} \vdash f(A)$; and
\item there exists an arithmetical interpretation $f$ based on $\mathrm{Pr}_T(x)$ such that for any $A \in \mathsf{MF}$, $\mathsf{N}F \vdash A$ if and only if $T \vdash f(A)$.
\end{enumerate}
\end{thm}
\begin{proof}
Let $\langle A_n \rangle_{n \in \omega}$ be a primitive recursive enumeration of all $\mathsf{N}F$-unprovable $\mathcal{L}(\Box)$-formulas.
For each $n \in \omega$, let $(W_n, \{\prec_{n, B}\}_{B \in \mathsf{MF}}, \Vdash_n)$ be a primitive recursively constructed finite $\mathsf{Sub}(A_n)$-transitive $\mathsf{N}$-model falsifying $A_n$.
Let $\mathcal{M}$ be a primitive recursively representable $\mathsf{N}$-model defined as the disjoint union of these finite $\mathsf{N}$-models as in the proof of Theorem \ref{Thm:N}.
We define a primitive recursive function $g_1$ corresponding to this theorem.
By the recursion theorem, we use $\mathrm{Pr}_{g_1}$ and $f_{g_1}$ in the definition of $g_1$ where $\mathrm{Pr}_{g_1}(x)$ is the formula $\exists y(g_1(y) = x \land \mathrm{Fml}_{\mathcal{L}_A}(x))$ and $f_{g_1}$ is the arithmetical interpretation based on $\mathrm{Pr}_{g_1}(x)$ defined by $f_{g_1}(p) \equiv \exists x(S(x) \land x \Vdash p)$.
As in the definition of the function $g_0$, the definition of $g_1$ consists of Procedures 1 and 2.
Moreover, the definition of Procedure 1 is completely same as that of $g_0$, so here we only give the definition of Procedure 2.
\textsc{Procedure 2}\\
Let $m$, $i \neq 0$, and $n$ be such that $h(m) = 0$, $h(m+1) = i$, and $i \in W_n$.
Define
\[
g_1(m+t) = \begin{cases} \xi_t & \text{if}\ \xi_t \equiv f_{g_1}(B)\ \&\ i \Vdash_n \Box B\ \text{for some}\ \Box B \in \mathsf{Sub}(A_n) \\
& \quad\ \text{or}\ \xi_t \equiv \mathrm{Pr}_{g_1}(\gn{\varphi})\ \&\ g_1(l) = \varphi\ \text{for some}\ \varphi\ \text{and}\ l < m+t, \\
0 & \text{otherwise.}
\end{cases}
\]
Since Procedure 1 in the definition of $g_1$ is same as that of $g_0$, the following claim is proved as in the proof of Theorem \ref{Thm:N}.
\begin{cl}\label{Cl:g_1_eq}
$\mathsf{PA} + \mathrm{Con}_T \vdash \forall x \forall y \Bigl(\mathrm{Fml}_{\mathcal{L}_A}(x) \to \bigl(\mathrm{Proof}_T(x, y) \leftrightarrow x = g_1(y) \bigr) \Bigr)$.
\end{cl}
Hence, $\mathrm{Pr}_{g_1}(x)$ is a $\Sigma_1$ provability predicate of $T$.
We prove that $\mathrm{Pr}_{g_1}(x)$ satisfies the condition $\D{3}$.
\begin{cl}\label{NFCL}
For any $\mathcal{L}_A$-formula $\varphi$, $\mathsf{PA} \vdash \mathrm{Pr}_{g_1}(\gn{\varphi}) \to \mathrm{Pr}_{g_1}(\gn{\mathrm{Pr}_{g_1}(\gn{\varphi})})$.
\end{cl}
\begin{proof}
Since $\mathrm{Pr}_{g_1}(\gn{\varphi})$ is a $\Sigma_1$ sentence, $\mathsf{PA} \vdash \mathrm{Pr}_{g_1}(\gn{\varphi}) \to \mathrm{Prov}_T(\gn{\mathrm{Pr}_{g_1}(\gn{\varphi})})$.
By Claim \ref{Cl:g_1_eq}, $\mathsf{PA} + \mathrm{Con}_T \vdash \forall x \Bigl(\mathrm{Fml}_{\mathcal{L}_A}(x) \to \bigl(\mathrm{Prov}_T(x) \leftrightarrow \mathrm{Pr}_{g_1}(x) \bigr) \Bigr)$.
Thus, we have $\mathsf{PA} + \mathrm{Con}_T \vdash \mathrm{Pr}_{g_1}(\gn{\varphi}) \to \mathrm{Pr}_{g_1}(\gn{\mathrm{Pr}_{g_1}(\gn{\varphi})})$.
We reason in $\mathsf{PA} + \neg \mathrm{Con}_T + \mathrm{Pr}_{g_1}(\gn{\varphi})$:
By Proposition \ref{Prop:h}.2, there exists an $i \neq 0$ such that $S(i)$ holds.
Let $m$ and $n$ be such that $h(m) = 0$, $h(m+1) = i$, and $i \in W_n$.
Since $\mathrm{Pr}_{g_1}(\gn{\varphi})$ holds, $\varphi$ is output by $g_1$.
Let $s$ be such that $\xi_s \equiv \varphi$.
If $\varphi$ is output in Procedure 1, then $\varphi = g_1(k)$ for some $k < m$.
If $\varphi$ is output in Procedure 2, then $\varphi = g_1(m + s)$.
In either case, we have that $\varphi \in \{g_1(0), \ldots, g_1(m+s)\}$.
Let $u$ be such that $\xi_u \equiv \mathrm{Pr}_{g_1}(\gn{\varphi})$.
Since the G\"odel number of $\mathrm{Pr}_{g_1}(\gn{\varphi})$ is larger than that of $\varphi$, we have $s < u$ by the choice of the enumeration $\langle \xi_t \rangle_{t \in \omega}$.
Since there is an $l < m + u$ such that $g_1(l) = \varphi$, $g_1(m + u) = \mathrm{Pr}_{g_1}(\gn{\varphi})$.
Thus, $\mathrm{Pr}_{g_1}(\gn{\mathrm{Pr}_{g_1}(\gn{\varphi})})$ holds.
We have proved $\mathsf{PA} + \neg \mathrm{Con}_T \vdash \mathrm{Pr}_{g_1}(\gn{\varphi}) \to \mathrm{Pr}_{g_1}(\gn{\mathrm{Pr}_{g_1}(\gn{\varphi})})$.
By the law of excluded middle, we conclude $\mathsf{PA} \vdash \mathrm{Pr}_{g_1}(\gn{\varphi}) \to \mathrm{Pr}_{g_1}(\gn{\mathrm{Pr}_{g_1}(\gn{\varphi})})$.
\end{proof}
\begin{cl}\label{Cl:g_1}
Let $i \in W_n$ and $B \in \mathsf{Sub}(A_n)$.
\begin{enumerate}
\item If $i \Vdash_n B$, then $\mathsf{PA} \vdash S(\overline{i}) \to f_{g_1}(B)$.
\item If $i \nVdash_n B$, then $\mathsf{PA} \vdash S(\overline{i}) \to \neg f_{g_1}(B)$.
\end{enumerate}
\end{cl}
\begin{proof}
This is proved by induction on the construction of $B \in \mathsf{Sub}(A_n)$.
We only prove the case that $B$ is of the form $\Box C$.
Clause 1 is proved in the similar way as in the proof of Claim \ref{Cl:g_0}.
2. Suppose $i \nVdash_n \Box C$.
We prove that $f_{g_1}(C)$ is not output by $g_1$.
We distinguish the following two cases:
\begin{itemize}
\item Case 1: $C$ is not of the form $\Box D$. \\
There exists a $j \in W_n$ such that $i \prec_{n, C} j$ and $j \nVdash_n C$.
By the induction hypothesis, $\mathsf{PA} \vdash S(\overline{j}) \to \neg f_{g_1}(C)$.
Let $p$ be a $T$-proof of $S(\overline{j}) \to \neg f_{g_1}(C)$.
We proceed in $\mathsf{PA} + S(\overline{i})$:
Let $m$ be such that $h(m) = 0$ and $h(m+1) = i$.
By Proposition \ref{Prop:h}.4, we have $m > p$, and hence $S(\overline{j}) \to \neg f_{g_1}(C)$ is in $P_{T, m-1}$.
If $f_{g_1}(C)$ is output in Procedure 1, then $f_{g_1}(C) \in P_{T, m-1}$, and hence $\neg S(\overline{j})$ is a t.c.~of $P_{T, m-1}$.
This contradicts $h(m) = 0$.
If $f_{g_1}(C)$ is output in Procedure 2, then $\xi_t \equiv f_{g_1}(C)$ and $g_1(m+t) = f_{g_1}(C)$ for some $t$.
Since $C$ is not of the form $\Box D$, there is no $\varphi$ such that $f_{g_1}(C) \equiv \mathrm{Pr}_{g_1}(\gn{\varphi})$.
Then, by the definition of $g_1$, $f_{g_1}(C) \equiv f_{g_1}(D)$ and $i \Vdash_n \Box D$ for some $\Box D \in \mathsf{Sub}(A_n)$.
It follows $C \equiv D$ and this contradicts $i \nVdash_n \Box C$.
\item Case 2: $C$ is of the form $\Box D$. \\
Then, $\Box \Box D \in \mathsf{Sub}(A_n)$.
Since $(W_n, \{\prec_{n, B}\}_{B \in \mathsf{MF}}, \Vdash_n)$ is $\mathsf{Sub}(A_n)$-transitive, $\Box D \to \Box \Box D$ is valid in the model.
Since $i \nVdash_n \Box \Box D$, we have $i \nVdash_n \Box D$.
By the induction hypothesis,
\begin{equation}\label{eq1}
\mathsf{PA} \vdash S(\overline{i}) \to \neg f_{g_1}(\Box D).
\end{equation}
We reason in $\mathsf{PA} + S(\overline{i})$:
If $f_{g_1}(\Box D)$ is output in Procedure 1, then $f_{g_1}(\Box D) \in P_{T, m-1}$.
By (\ref{eq1}), $S(\overline{i}) \to \neg f_{g_1}(\Box D)$ is also in $P_{T, m-1}$.
Hence, $\neg S(\overline{i})$ is a t.c.~of $P_{T, m-1}$, a contradiction.
If $f_{g_1}(\Box D)$ is output in Procedure 2, then $\xi_t \equiv f_{g_1}(\Box D)$ and $g_1(m+t) = f_{g_1}(\Box D)$ for some $t$.
If $f_{g_1}(\Box D) \equiv f_{g_1}(E)$ and $i \Vdash_n \Box E$ for some $\Box E \in \mathsf{Sub}(A_n)$, then $\Box D \equiv E$ and $i \Vdash_n \Box \Box D$, a contradiction.
Thus, we have that $f_{g_1}(\Box D) \equiv \mathrm{Pr}_{g_1}(\gn{\varphi})$ and $g_1(l) = \varphi$ for some $\varphi$ and $l < m + t$.
Since $g_1(l) = \varphi$, $\mathrm{Pr}_{g_1}(\gn{\varphi})$ holds.
Since $\mathrm{Pr}_{g_1}(\gn{f_{g_1}(D)}) \equiv \mathrm{Pr}_{g_1}(\gn{\varphi})$, we have $f_{g_1}(D) \equiv \varphi$.
Thus $\mathrm{Pr}_{g_1}(\gn{f_{g_1}(D)})$ holds, and hence $f_{g_1}(\Box D)$ holds.
This contradicts (\ref{eq1}).
\end{itemize}
In either case, we proved that $f_{g_1}(C)$ is not output by $g_1$.
Therefore, $\neg f_{g_1}(\Box C)$ holds.
\end{proof}
The first clause of the theorem follows from Claims \ref{Cl:g_1_eq} and \ref{NFCL}.
The second clause follows from Proposition \ref{Prop:h}.3 and Claim \ref{Cl:g_1}.
\end{proof}
\begin{cor}
\begin{align*}
\mathsf{N}F & = \bigcap \{\mathsf{PL}(\mathrm{Pr}_T) \mid \mathrm{Pr}_T(x)\ \text{is a provability predicate of}\ T\ \text{satisfying}\ \D{3}\}, \\
& = \bigcap \{\mathsf{PL}(\mathrm{Pr}_T) \mid \mathrm{Pr}_T(x)\ \text{is a}\ \Sigma_1\ \text{provability predicate of}\ T\ \text{satisfying}\ \D{3}\}.
\end{align*}
Moreover, there exists a $\Sigma_1$ provability predicate $\mathrm{Pr}_T(x)$ of $T$ such that $\mathsf{N}F = \mathsf{PL}(\mathrm{Pr}_T)$.
\end{cor}
\section{Arithmetical completeness of $\mathsf{N}R$}\label{Sec:NR}
It is known that for any Rosser's provability predicate $\mathrm{Pr}_T^{\mathrm{R}}(x)$ of $T$ and any $\mathcal{L}_A$-formula $\varphi$, if $T \vdash \neg \varphi$, then $T \vdash \neg \mathrm{Pr}_T^{\mathrm{R}}(\gn{\varphi})$.
This fact corresponds to the rule \textsc{Ros} $\dfrac{\neg A}{\neg \Box A}$, and hence it is shown that $\mathsf{N}R \subseteq \mathsf{PL}(\mathrm{Pr}_T^{\mathrm{R}})$.
Therefore, our logic $\mathsf{N}R$ is a candidate for the axiomatization of the logic $L^{\mathrm{R}}$ introduced in Section \ref{Sec:Pre}.
In this section, we prove that this is the case.
Namely, we prove that $\mathsf{N}R$ is exactly the provability logic of all Rosser's provability predicates.
\begin{thm}[The uniform arithmetical completeness of $\mathsf{N}R$]\label{Thm:NR}
There exist a Rosser's provability predicate $\mathrm{Pr}_T^{\mathrm{R}}(x)$ of $T$ such that
\begin{enumerate}
\item for any $A \in \mathsf{MF}$ and any arithmetical interpretation $f$ based on $\mathrm{Pr}_T^{\mathrm{R}}(x)$, if $\mathsf{N}R \vdash A$, then $\mathsf{PA} \vdash f(A)$; and
\item there exists an arithmetical interpretation $f$ based on $\mathrm{Pr}_T^{\mathrm{R}}(x)$ such that for any $A \in \mathsf{MF}$, $\mathsf{N}R \vdash A$ if and only if $T \vdash f(A)$.
\end{enumerate}
\end{thm}
\begin{proof}
Let $\langle A_n \rangle_{n \in \omega}$ be a primitive recursive enumeration of all $\mathsf{N}R$-unprovable $\mathcal{L}(\Box)$-formulas.
For each $n \in \omega$, let $(W_n, \{\prec_{n, B}\}_{B \in \mathsf{MF}}, \Vdash_n)$ be a primitive recursively constructed finite $\mathsf{Sub}(A)$-serial $\mathsf{N}$-model in which $A_n$ is not valid.
Let $\mathcal{M} = (W, \{\prec_B\}_{B \in \mathsf{MF}}, \Vdash)$ be an $\mathsf{N}$-model defined as in the previous sections.
We define a corresponding primitive recursive function $g_2$ enumerating all theorems of $T$.
Let $\mathrm{Pr}_{g_2}^{\mathrm{R}}(x)$ be the formula $\exists y \bigl(\mathrm{Fml}_{\mathcal{L}_A}(x) \land x = g_2(y) \land \forall z < y\, \dot{\neg}(x) \neq g_2(z) \bigr)$.
In the definition of $g_2$, we use the arithmetical interpretation $f_{g_2}$ based on $\mathrm{Pr}_{g_2}^{\mathrm{R}}(x)$ defined as $f_{g_2}(p) \equiv \exists x (S(x) \land x \Vdash p)$.
Procedure 1 in the construction of $g_2$ is same as that of $g_0$, and so we only give the definition of Procedure 2.
\textsc{Procedure 2}\\
Let $m$, $i \neq 0$, and $n$ be such that $h(m) = 0$, $h(m+1) = i$, and $i \in W_n$.
We define a finite set $X$ of $\mathcal{L}_A$-formulas as follows:
\[
X : = \{\neg f_{g_2}(B) \mid i \nVdash_n \Box B\ \&\ \Box B \in \mathsf{Sub}(A_n)\}.
\]
Let $\chi_0, \ldots, \chi_{k-1}$ be the enumeration of all elements of $X$ arranged in descending order of G\"odel numbers.
For $l < k$, define
\[
g_2(m + l) = \chi_l.
\]
And define
\[
g_2(m + k + t) = \xi_t.
\]
The definition of $g_2$ is finished.
\begin{cl}\label{Cl:g_2_eq}\leavevmode
\begin{enumerate}
\item $\mathsf{PA} + \mathrm{Con}_T \vdash \forall x \forall y \Bigl(\mathrm{Fml}_{\mathcal{L}_A}(x) \to \bigl(\mathrm{Proof}_T(x, y) \leftrightarrow x = g_2(y) \bigr) \Bigr)$.
\item $\mathsf{PA} \vdash \forall x \Bigl(\mathrm{Fml}_{\mathcal{L}_A}(x) \to \bigl(\mathrm{Prov}_T(x) \leftrightarrow \mathrm{Pr}_{g_2}(x) \bigr) \Bigr)$.
\end{enumerate}
\end{cl}
\begin{proof}
Clause 1 is proved similarly as in the proof of Theorem \ref{Thm:N}.
2. By Clause 1, $\mathsf{PA} + \mathrm{Con}_T \vdash \forall x \Bigl(\mathrm{Fml}_{\mathcal{L}_A}(x) \to \bigl(\mathrm{Prov}_T(x) \leftrightarrow \mathrm{Pr}_{g_2}(x) \bigr) \Bigr)$.
Also, $\mathsf{PA} + \neg \mathrm{Con}_T \vdash \forall x \bigl(\mathrm{Fml}_{\mathcal{L}_A}(x) \to \mathrm{Prov}_T(x) \bigr)$.
Proposition \ref{Prop:h}.2 says that $\mathsf{PA}$ verifies that if $T$ is inconsistent, then the construction of $g_2$ eventually switches to Procedure 2.
Since $g_2$ outputs all $\mathcal{L}_A$-formulas in Procedure 2, we have $\mathsf{PA} + \neg \mathrm{Con}_T \vdash \forall x \bigl(\mathrm{Fml}_{\mathcal{L}_A}(x) \to \mathrm{Pr}_{g_2}(x) \bigr)$.
Hence, $\mathsf{PA} + \neg \mathrm{Con}_T \vdash \forall x \Bigl(\mathrm{Fml}_{\mathcal{L}_A}(x) \to \bigl(\mathrm{Prov}_T(x) \leftrightarrow \mathrm{Pr}_{g_2}(x) \bigr) \Bigr)$.
By the law of excluded middle, we conclude $\mathsf{PA} \vdash \forall x \Bigl(\mathrm{Fml}_{\mathcal{L}_A}(x) \to \bigl(\mathrm{Prov}_T(x) \leftrightarrow \mathrm{Pr}_{g_2}(x) \bigr) \Bigr)$.
\end{proof}
It follows from this claim, $\mathrm{Pr}_{g_2}^{\mathrm{R}}(x)$ is a Rosser's provability predicate of $T$.
\begin{cl}\label{Cl:g_2}
Let $i \in W_n$ and $B \in \mathsf{Sub}(A_n)$.
\begin{enumerate}
\item If $i \Vdash_n B$, then $\mathsf{PA} \vdash S(\overline{i}) \to f_{g_2}(B)$.
\item If $i \nVdash_n B$, then $\mathsf{PA} \vdash S(\overline{i}) \to \neg f_{g_2}(B)$.
\end{enumerate}
\end{cl}
\begin{proof}
This claim is proved by induction on the construction of $B \in \mathsf{Sub}(A_n)$.
We only give a proof of the case $B \equiv \Box C$.
1. Suppose that $i \Vdash_n \Box C$.
Since $\Box C \in \mathsf{Sub}(A_n)$ and $(W_n, \{\prec_{n, B}\}_{B \in \mathsf{MF}}, \Vdash_n)$ is $\mathsf{Sub}(A_n)$-serial, there is a $j \in W_n$ such that $i \prec_C j$.
Then, $j \Vdash_n C$.
By the induction hypothesis, $\mathsf{PA} \vdash S(\overline{j}) \to f_{g_2}(C)$.
Let $p$ be a $T$-proof of $S(\overline{j}) \to f_{g_2}(C)$.
We argue in $\mathsf{PA} + S(\overline{i})$:
Let $m$ be such that $h(m) = 0$ and $h(m+1) = i$.
Also, let $X$ be the finite set of $\mathcal{L}_A$-formulas as in Procedure 2 and let $k$ be the cardinality of $X$.
By Proposition \ref{Prop:h}.4, $m > p$, and hence $S(\overline{j}) \to f_{g_2}(C)$ is in $P_{T, m-1}$.
If $\neg f_{g_2}(C) \in P_{T, m-1}$, then $\neg S(\overline{j})$ is a t.c.~of $P_{T, m-1}$, a contradiction.
Hence, $\neg f_{g_2}(C) \notin P_{T, m-1}$, that is, $\neg f_{g_2}(C) \notin \{g_2(0), \ldots, g_2(m-1)\}$.
If $\neg f_{g_2}(C) \in X$, then there exists a $\Box D \in \mathsf{Sub}(A_n)$ such that $\neg f_{g_2}(C) \equiv \neg f_{g_2}(D)$ and $i \nVdash_n \Box D$.
Then, we have $C \equiv D$, and this contradicts $i \Vdash_n \Box C$.
Thus, we have $\neg f_{g_2}(C) \notin X$, that is, $\neg f_{g_2}(C) \notin \{g_2(m), \ldots, g_2(m + k -1)\}$.
Therefore, we obtain $\neg f_{g_2}(C) \notin \{g_2(0), \ldots, g_2(m + k -1)\}$.
Let $s$ and $u$ be such that $\xi_s \equiv f_{g_2}(C)$ and $\xi_u \equiv \neg f_{g_2}(C)$.
Then, $s < u$, $g_2(m + k + s) = f_{g_2}(C)$, and $g_2(m + k + u) = \neg f_{g_2}(C)$.
In particular, $g_2(m + k + u)$ is the first output of $\neg f_{g_2}(C)$ by $g_2$.
Therefore, $\mathrm{Pr}_{g_2}^{\mathrm{R}}(\gn{f_{g_2}(C)})$ holds.
That is, $f_{g_2}(\Box C)$ holds.
2. Suppose $i \nVdash_n \Box C$.
Then, there exists a $j \in W_n$ such that $i \prec_{n, C} j$ and $j \nVdash_n C$.
By the induction hypothesis, $\mathsf{PA} \vdash S(\overline{j}) \to \neg f_{g_2}(C)$.
We reason in $\mathsf{PA} + S(\overline{i})$:
Let $m$ be such that $h(m) = 0$ and $h(m+1) = i$.
Also, let $X$ and $k$ be as in the definition of $g_2$.
As above, $S(\overline{j}) \to \neg f_{g_2}(C)$ is in $P_{T, m-1}$.
If $f_{g_2}(C) \in P_{T, m-1}$, then $\neg S(\overline{j})$ is a t.c.~of $P_{T, m-1}$, and this is a contradiction.
Thus, $f_{g_2}(C) \notin P_{T, m-1}$, and hence $f_{g_2}(C) \notin \{g_2(0), \ldots, g_2(m-1)\}$.
On the other hand, since $\Box C \in \mathsf{Sub}(A_n)$ and $i \nVdash_n \Box C$, we obtain $\neg f_{g_2}(C) \in X$.
That is, $\neg f_{g_2}(C) \in \{g_2(m), \ldots, g_2(m+k-1)\}$.
Since the G\"odel number of $\neg f_{g_2}(C)$ is larger than that of $f_{g_2}(C)$, even if $f_{g_2}(C) \in X$, $\neg f_{g_2}(C)$ is listed earlier than $f_{g_2}(C)$ in the enumeration $\chi_0, \ldots, \chi_{k-1}$ of $X$.
Thus, $\neg f_{g_2}(C)$ is output by $g_2$ earlier than any output of $f_{g_2}(C)$.
Therefore, $\neg \mathrm{Pr}_{g_2}^{\mathrm{R}}(\gn{f_{g_2}(C)})$ holds.
This means that $\neg f_{g_2}(\Box C)$ holds.
\end{proof}
The first clause of the theorem follows from Claim \ref{Cl:g_2_eq}.
The second clause follows from Proposition \ref{Prop:h}.3 and Claim \ref{Cl:g_2}.
\end{proof}
\begin{cor}\label{Cor:PLRos}
\begin{align*}
\mathsf{N}R & = \bigcap \{\mathsf{PL}(\mathrm{Pr}_T^{\mathrm{R}}) \mid \mathrm{Pr}_T(x)\ \text{is a Rosser's provability predicate of}\ T\}.
\end{align*}
Moreover, there exists a Rosser's provability predicate $\mathrm{Pr}_T^{\mathrm{R}}(x)$ of $T$ such that $\mathsf{N}R = \mathsf{PL}(\mathrm{Pr}_T^{\mathrm{R}})$.
\end{cor}
Therefore, $\mathsf{N}R$ is exactly the $\Box$-free fragment $L^{\mathrm{R}}$ of Shavrukov's bimodal logic $\mathsf{GR}$ (cf.~Corollary \ref{Cor:LR}).
Furthermore, Corollary \ref{Cor:PLRos} states that the provability logic of all Rosser's provability predicates coincides with $\mathsf{N}R$ regardless of whether $T$ is $\Sigma_1$-sound or not.
We state the coincidence of $\mathsf{N}R$ and $L^{\mathrm{R}}$ more precisely.
For each $A \in \mathsf{MF}$, let $A^{\blacksquare}$ be the $\Box$-free $\mathcal{L}(\Box, \blacksquare)$-formula obtained from $A$ by replacing every $\Box$ in $A$ with $\blacksquare$.
\begin{cor}
For any $A \in \mathsf{MF}$, the following are equivalent:
\begin{enumerate}
\item $\mathsf{N}R \vdash A$.
\item $\mathsf{GR} \vdash A^\blacksquare$.
\end{enumerate}
\end{cor}
In fact, it is easy to show that the rule $\dfrac{\neg A}{\neg \blacksquare A}$ is admissible in $\mathsf{GR}$, and hence the implication ($1 \Rightarrow 2$) is also modal logically straightforward.
On the other hand, we do not know how to show the converse implication ($2 \Rightarrow 1$) in a modal logical way without going through arithmetic.
\section{Arithmetical completeness of $\mathsf{N}RF$}\label{Sec:NRF}
Arai \cite{Ara} proved the existence of Rosser's provability predicates satisfying the condition $\D{3}$.
For such Rosser's provability predicates $\mathrm{Pr}_T^{\mathrm{R}}(x)$, it is shown that $\mathsf{N}RF \subseteq \mathsf{PL}(\mathrm{Pr}_T^{\mathrm{R}})$.
In this section, we investigate the provability logic of Arai's predicates, and prove that $\mathsf{N}RF$ is exactly the provability logic of all Rosser's provability predicates satisfying $\D{3}$.
\begin{thm}[The uniform arithmetical completeness of $\mathsf{N}RF$]\label{Thm:NRF}
There exist a Rosser's provability predicate $\mathrm{Pr}_T^{\mathrm{R}}(x)$ of $T$ such that
\begin{enumerate}
\item for any $A \in \mathsf{MF}$ and any arithmetical interpretation $f$ based on $\mathrm{Pr}_T^{\mathrm{R}}(x)$, if $\mathsf{N}RF \vdash A$, then $\mathsf{PA} \vdash f(A)$; and
\item there exists an arithmetical interpretation $f$ based on $\mathrm{Pr}_T^{\mathrm{R}}(x)$ such that for any $A \in \mathsf{MF}$, $\mathsf{N}RF \vdash A$ if and only if $T \vdash f(A)$.
\end{enumerate}
\end{thm}
\begin{proof}
Let $\langle A_n \rangle_{n \in \omega}$ be a primitive recursive enumeration of all $\mathsf{N}RF$-unprovable $\mathcal{L}(\Box)$-formulas.
For each $n \in \omega$, let $(W_n, \{\prec_{n, B}\}_{B \in \mathsf{MF}}, \Vdash_n)$ be a primitive recursively constructed finite $\mathsf{Sub}(A_n)$-transitive and $\mathsf{Sub}(A_n)$-serial $\mathsf{N}$-model falsifying $A_n$.
Let $\mathcal{M} = (W, \{\prec_B\}_{B \in \mathsf{MF}}, \Vdash)$ be a primitive recursively representable infinite model constructed as the disjoint union of the family $\{(W_n, \{\prec_{n, B}\}_{B \in \mathsf{MF}}, \Vdash_n)\}_{n \in \omega}$.
Unlike the proofs in the previous sections, our proof of this theorem uses a different function $h'$ than the function $h$.
By using the double recursion theorem, we simultaneously define primitive recursive functions $h'$ and $g_3$.
Firstly, we define the function $h'$.
\begin{itemize}
\item $h'(0) = 0$.
\item $h'(m+1) = \begin{cases}
i & \text{if}\ h'(m) = 0\\
& \&\ i = \min \bigl\{j \in \omega \setminus \{0\} \mid \neg S'(\overline{j}) \ \text{is a t.c.~of}\ P_{T, m}\\
& \quad\ \text{or}\ \exists \varphi, \psi \bigl[\neg \varphi \notin P_{T, m} \cup X_{j, m} \ \&\ \psi \in P_{T, m}\\
& \quad \quad \quad \quad \&\ \psi \to (S'(\overline{j}) \to \neg \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi})) \in P_{T, m}\bigr]\bigr\}\\
h'(m) & \text{otherwise}.
\end{cases}$
\end{itemize}
Here, $S'(x)$ is the $\Sigma_1$ formula $\exists y(h'(y) = x)$.
Also, for each $j \in W_n$ and number $m$, $X_{j, m}$ is the set
\begin{equation*}
\left\{ \neg f_{g_3}(D) \left|
\begin{array}{l}
\Box D \in \mathsf{Sub}(A_n)\ \&\ \exists \psi \in P_{T, m}\ \exists l \in W_n \\
\quad \quad \text{s.t.}\ j \prec_{n, D} l\ \&\ \psi \to (S'(\overline{l}) \to \neg f_{g_3}(D)) \in P_{T, m}
\end{array}
\right.\right\},
\end{equation*}
where $f_{g_3}$ is the arithmetical interpretation based on $\mathrm{Pr}_{g_3}^{\mathrm{R}}(x)$ defined as $f_{g_3}(p) : \equiv \exists x (S'(x) \land x \Vdash p)$.
We show that if $h'(m) = 0$ and $h'(m+1) = i \neq 0$, then $i \leq m$.
It follows that $h'$ is actually a primitive recursive function.
Suppose $h'(m) = 0$ and $h'(m+1) = i \neq 0$.
If $P_{T, m}$ is propositionally unsatisfiable, then $\neg S'(\overline{1})$ is a t.c,~of $P_{T, m}$, and then $i = 1 \leq m$.
If $P_{T, m}$ is propositionally satisfiable and $\neg S'(\overline{i})$ is a t.c.~of $P_{T, m}$, then $S'(\overline{i})$ is a subformula of a formula contained in $P_{T, m}$.
Then, the G\"odel number of $S'(\overline{i})$ is smaller than $m$, and hence $i \leq m$.
If there exist $\varphi$ and $\psi$ such that $\neg \varphi \notin P_{T, m} \cup X_{j, m}$, $\psi \in P_{T, m}$, and $\psi \to (S'(\overline{j}) \to \neg \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi})) \in P_{T, m}$, then the G\"odel number of $S'(\overline{i})$ is also smaller than $m$, and thus $i \leq m$.
Secondly, we define the function $g_3$.
We only give the definition of Procedure 2 of the construction of $g_3$.
\textsc{Procedure 2}\\
Let $m$, $i \neq 0$, and $n$ be such that $h(m) = 0$, $h(m+1) = i$, and $i \in W_n$.
Let $\chi_0, \ldots, \chi_{k-1}$ be the enumeration of all elements of the set $X_{i, m-1}$ arranged in descending order of G\"odel numbers.
For $l < k$, define
\[
g_3(m + l) = \chi_l.
\]
And define
\[
g_3(m + k + t) = \xi_t.
\]
The definition of $g_3$ is finished.
The construction of $g_3$ is exactly the same as that of $g_2$ in the proof of Theorem \ref{Thm:NR}, except that it is based on the family of $\mathsf{N}$-models corresponding to the logic $\mathsf{N}RF$ and uses $X_{i, m-1}$ and $h'$ instead of $X$ and $h$, respectively.
Similar as Proposition \ref{Prop:h}, the following claim holds.
\begin{cl}\label{Cl:h}
\leavevmode
\begin{enumerate}
\item $\mathsf{PA} \vdash \forall x \forall y(0 < x < y \land S'(x) \to \neg S'(y))$.
\item $\mathsf{PA} \vdash \neg \mathrm{Con}_T \leftrightarrow \exists x(S'(x) \land x \neq 0)$.
\item For each $i \in \omega \setminus \{0\}$, $T \nvdash \neg S'(\overline{i})$.
\item For each $n \in \omega$, $\mathsf{PA} \vdash \forall x \forall y(h'(x) = 0 \land h'(x + 1) = y \land y \neq 0 \to x > \overline{n})$.
\end{enumerate}
\end{cl}
\begin{proof}
1. This is straightforward from the definition of $h'$.
2. The implication $\to$ is easy, and so we prove the implication $\leftarrow$.
Argue in $\mathsf{PA}$:
Suppose that $S'(i)$ holds for some $i \neq 0$.
Let $m$ and $n$ be such that $h'(m) = 0$, $h'(m+1) = i$, and $i \in W_n$.
Also, let $k$ be the cardinality of the set $X_{i, m-1}$.
We would like to show that $T$ is inconsistent.
We distinguish the following two cases:
\begin{itemize}
\item Case 1: $\neg S'(\overline{i})$ is a t.c.~of $P_{T, m}$. \\
Then, $\neg S'(\overline{i})$ is $T$-provable.
Since $S'(\overline{i})$ is a true $\Sigma_1$ sentence, it is provable in $T$.
Therefore, $T$ is inconsistent.
\item Case 2: There exist $\varphi$ and $\psi$ such that $\neg \varphi \notin P_{T, m} \cup X_{i, m}$, $\psi \in P_{T, m}$, and $\psi \to (S'(\overline{i}) \to \neg \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi})) \in P_{T, m}$. \\
Then, $\neg \varphi \notin P_{T, m-1} \cup X_{i, m-1}$.
That is, $\neg \varphi \notin \{g_3(0), \ldots, g_3(m + k -1)\}$.
Let $s$ and $u$ be such that $\xi_s \equiv \varphi$ and $\xi_u \equiv \neg \varphi$.
Then, $s < u$, $g_3(m + k + s) = \varphi$, and $g_3(m + k + u) = \neg \varphi$.
In particular, $g_3(m + k + u)$ is the first output of $\neg \varphi$ by $g_3$.
Hence, $\mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi})$ holds.
Then, $S'(\overline{i}) \land \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi})$ is a true $\Sigma_1$ sentence, and so it is provable in $T$.
On the other hand, since $\psi$ and $\psi \to (S'(\overline{i}) \to \neg \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi}))$ are $T$-provable, $S'(\overline{i}) \to \neg \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi})$ is also provable in $T$.
Therefore, $T$ is inconsistent.
\end{itemize}
3. Suppose $T \vdash \neg S'(\overline{i})$ for $i \neq 0$.
Let $p$ be a $T$-proof of $\neg S'(\overline{i})$.
Then, $\neg S'(\overline{i}) \in P_{T, p}$, and thus $h'(p+1) \neq 0$.
This means that $\exists x(S'(x) \land x \neq 0)$ is true.
By clause 2, $T$ is inconsistent, a contradiction.
4. This is because $h'(m+1) = 0$ for all $m \in \omega$.
\end{proof}
The following claim is proved as in the proof of Theorem \ref{Thm:NR}.
\begin{cl}\label{Cl:g_3_eq}\leavevmode
\begin{enumerate}
\item $\mathsf{PA} + \mathrm{Con}_T \vdash \forall x \forall y \Bigl(\mathrm{Fml}_{\mathcal{L}_A}(x) \to \bigl(\mathrm{Proof}_T(x, y) \leftrightarrow x = g_3(y) \bigr) \Bigr)$.
\item $\mathsf{PA} \vdash \forall x \Bigl(\mathrm{Fml}_{\mathcal{L}_A}(x) \to \bigl(\mathrm{Prov}_T(x, y) \leftrightarrow \mathrm{Pr}_{g_3}(x) \bigr) \Bigr)$.
\end{enumerate}
\end{cl}
Hence, $\mathrm{Pr}_{g_3}^{\mathrm{R}}(x)$ is a Rosser's provability predicate of $T$.
\begin{cl}\label{Cl:g_3}
Let $i \in W_n$ and $B \in \mathsf{Sub}(A_n)$.
\begin{enumerate}
\item If $i \Vdash_n B$, then $\mathsf{PA} \vdash S'(\overline{i}) \to f_{g_3}(B)$.
\item If $i \nVdash_n B$, then $\mathsf{PA} \vdash S'(\overline{i}) \to \neg f_{g_3}(B)$.
\end{enumerate}
\end{cl}
\begin{proof}
We prove the claim by induction on the construction of $B \in \mathsf{Sub}(A_n)$.
We only give a proof of the case $B \equiv \Box C$.
1. Suppose that $i \Vdash_n \Box C$.
Since $\Box C \in \mathsf{Sub}(A_n)$ and $(W_n, \{\prec_{n, B}\}_{B \in \mathsf{MF}}, \Vdash_n)$ is $\mathsf{Sub}(A_n)$-serial, there exists a $j \in W_n$ such that $i \prec_{n, C} j$.
Then, $j \Vdash_n C$.
By the induction hypothesis, $\mathsf{PA} \vdash S'(\overline{j}) \to f_{g_3}(C)$, and let $p$ be a $T$-proof of $S'(\overline{j}) \to f_{g_3}(C)$.
We argue in $\mathsf{PA} + S'(\overline{i})$:
Let $m$ be such that $h'(m) = 0$ and $h'(m+1) = i$.
If $\neg f_{g_3}(C) \in P_{T, m-1}$, then $\neg S'(\overline{j})$ is a t.c.~of $P_{T, m-1}$ because $m > p$ by Claim \ref{Cl:h}.4.
Then, $h'(m) \neq 0$ by the definition of $h'$, and this is a contradiction.
Hence, $\neg f_{g_3}(C) \notin P_{T, m-1}$.
If $\neg f_{g_3}(C) \in X_{i, m-1}$, then there exist $\Box D \in \mathsf{Sub}(A_n)$, $\psi \in P_{T, m-1}$, and $l \in W_n$ such that $\neg f_{g_3}(C) \equiv \neg f_{g_3}(D)$, $i \prec_{n, D} l$, and $\psi \to (S'(\overline{l}) \to \neg f_{g_3}(D)) \in P_{T, m-1}$.
Then, $C \equiv D$ and $S'(\overline{l}) \to \neg f_{g_3}(D)$ is a t.c.~of $P_{T, m-1}$.
Since $i \Vdash_n \Box C$ and $i \prec_{n, C} l$, we have $l \Vdash_n C$.
(Since $l$, $n$, and $C$ are standard, we may assume that $l \Vdash_n C$ actually holds.)
Then, $S'(\overline{l}) \to f_{g_3}(C)$ is a t.c.~of $P_{T, m-1}$ (by the induction hypothesis).
Since $S'(\overline{l}) \to \neg f_{g_3}(C)$ is also a t.c.~of $P_{T, m-1}$, $\neg S'(\overline{l})$ is a t.c.~of $P_{T, m-1}$, a contradiction.
Thus, we have $\neg f_{g_3}(C) \notin X_{i, m-1}$, that is, $\neg f_{g_3}(C) \notin \{g_3(m), \ldots, g_3(m + k -1)\}$.
Therefore, we obtain $\neg f_{g_3}(C) \notin \{g_3(0), \ldots, g_3(m + k -1)\}$.
Let $s$ and $u$ be such that $\xi_s \equiv f_{g_3}(C)$ and $\xi_u \equiv \neg f_{g_3}(C)$.
Then, $s < u$, $g_3(m + k + s) = f_{g_3}(C)$, $g_3(m + k + u) = \neg f_{g_3}(C)$, and this is the first output of $\neg f_{g_3}(C)$.
Therefore, $\mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{f_{g_3}(C)})$ holds.
That is, $f_{g_3}(\Box C)$ holds.
2. Suppose $i \nVdash_n \Box C$.
Then, there exists a $j \in W_n$ such that $i \prec_{n, C} j$ and $j \nVdash_n C$.
By the induction hypothesis, $\mathsf{PA} \vdash S'(\overline{j}) \to \neg f_{g_3}(C)$.
Let $p$ be a $T$-proof of $S'(\overline{j}) \to \neg f_{g_3}(C)$.
We reason in $\mathsf{PA} + S'(\overline{i})$:
Let $m$ be such that $h'(m) = 0$ and $h'(m+1) = i$.
Since $m > p$ by Claim \ref{Cl:h}.4, $0 = 0 \to (S'(\overline{j}) \to \neg f_{g_3}(C))$ is in $P_{T, m-1}$, and hence we have $\neg f_{g_3}(C) \in X_{i, m-1}$.
If $f_{g_3}(C) \in P_{T, m-1}$, then $\neg S'(\overline{j})$ is a t.c.~of $P_{T, m-1}$, a contradiction.
Hence, $f_{g_3}(C) \notin \{g_3(0), \ldots, g_3(m-1)\}$.
Then, even if $f_{g_3}(C) \in X_{i, m-1}$, $g_3$ outputs $\neg f_{g_3}(C)$ earlier than any output of $f_{g_3}(C)$.
Hence, $\neg \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{f_{g_3}(C)})$ holds.
That is, $\neg f_{g_3}(\Box C)$ holds.
\end{proof}
We prove that $\mathrm{Pr}_{g_3}^{\mathrm{R}}(x)$ satisfies $\D{3}$.
\begin{cl}\label{Cl:g_3_4}
For any $\mathcal{L}_A$-formula $\varphi$, $\mathsf{PA} \vdash \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi}) \to \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi})})$.
\end{cl}
\begin{proof}
Since $\mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi})$ is a $\Sigma_1$ sentence, $\mathsf{PA} \vdash \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi}) \to \mathrm{Prov}_T(\gn{\mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi})})$.
It follows from Claim \ref{Cl:g_3_eq}.1, we have $\mathsf{PA} + \mathrm{Con}_T \vdash \forall x \Bigl(\mathrm{Fml}_{\mathcal{L}_A}(x) \to \bigl(\mathrm{Prov}_T(x) \leftrightarrow \mathrm{Pr}_{g_3}^{\mathrm{R}}(x) \bigr) \Bigr)$.
Thus, $\mathsf{PA} + \mathrm{Con}_T \vdash \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi}) \to \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi})})$.
We reason in $\mathsf{PA} + \neg \mathrm{Con}_T + \neg \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi})})$:
By Claim \ref{Cl:h}.2, there exists an $i \neq 0$ such that $S'(i)$ holds.
Let $m$ and $n$ be such that $h'(m) = 0$, $h'(m+1) = i$, and $i \in W_n$.
If $\neg \varphi \in P_{T, m-1}$, then $\varphi \notin P_{T, m-1}$ because $\neg S(\overline{j})$ is not a t.c.~of $P_{T, m-1}$ for all $j \neq 0$.
In this case, $\neg \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi})$ holds.
Therefore, in the following, we assume that $\neg \varphi \notin P_{T, m-1}$.
Let $k$ be the cardinality of the set $X_{i, m-1}$.
Since $\neg \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi})})$ holds, $\neg \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi})$ is output by $g_3$ earlier than any output of $\mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi})$.
Let $s$ and $u$ be such that $\xi_s \equiv \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi})$ and $\xi_u \equiv \neg \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi})$.
Then, $g_3(m + k + u) = \neg \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi})$.
Since $s < u$ and $g_3(m + k + u) = \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi})$, $g_3(m + k + u)$ is not the first output of $\neg \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi})$.
It follows that $\neg \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi}) \in P_{T, m-1} \cup X_{i, m-1}$.
We would like to show that $\neg \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi})$ holds.
We distinguish the following two cases:
\begin{itemize}
\item Case 1: $\neg \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi}) \in P_{T, m-1}$. \\
If $\neg \varphi \notin P_{T, m-1} \cup X_{i, m-1}$, then $h'(m) = 1 \neq 0$ by the definition of $h'$ because $\neg \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi}) \to (\neg S'(\overline{1}) \to \neg \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi}))$ is in $P_{T, m-1}$ by Claim \ref{Cl:h}.
This is a contradiction.
Hence, $\neg \varphi \in P_{T, m-1} \cup X_{i, m-1}$.
Since $\neg \varphi \notin P_{T, m-1}$ by the assumption, we have $\neg \varphi \in X_{i, m-1}$.
Then, there exist $\Box D \in \mathsf{Sub}(A_n)$, $\psi \in P_{T, m-1}$, and $l \in W_n$ such that $\neg \varphi \equiv \neg f_{g_3}(D)$, $i \prec_{n, D} l$, and $\psi \to (S'(\overline{l}) \to \neg f_{g_3}(D)) \in P_{T, m-1}$.
It follows that $S'(\overline{l}) \to \neg \varphi$ is a t.c.~of $P_{T, m-1}$.
Since $\neg S'(\overline{l})$ is not a t.c.~of $P_{T, m-1}$, we have $\varphi \notin P_{T, m-1}$.
Thus, $\varphi \notin \{g_3(0), \ldots, g_3(m-1)\}$.
Then, even if $\varphi \in X_{i, m-1}$, $\neg \varphi$ is output by $g_3$ earlier than any output of $\varphi$.
Therefore, $\neg \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi})$ holds.
\item Case 2: $\neg \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi}) \in X_{i, m-1}$. \\
Then, there exist $\Box D \in \mathsf{Sub}(A_n)$, $\psi \in P_{T, m-1}$, and $j \in W_n$ such that $\neg \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi}) \equiv \neg f_{g_3}(D)$, $i \prec_{n, D} j$, and $\psi \to (S'(\overline{j}) \to \neg f_{g_3}(D)) \in P_{T, m-1}$.
It follows that $\mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi}) \equiv f_{g_3}(D)$.
By the definition of $f_{g_3}$, there exists a $\Box E \in \mathsf{Sub}(A_n)$ such that $D \equiv \Box E$ and $\mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi}) \equiv \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{f_{g_3}(E)})$.
Then, $\varphi \equiv f_{g_3}(E)$, $\Box \Box E \in \mathsf{Sub}(A_n)$, and $i \prec_{n, \Box E} j$.
If $\neg f_{g_3}(E) \notin X_{j, m-1}$, then $\neg f_{g_3}(E) \notin P_{T, m-1} \cup X_{j, m-1}$ by the assumption.
Since $\psi$ and $\psi \to (S(\overline{j}) \to \neg \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{f_{g_3}(E)}))$ are in $P_{T, m-1}$, we have $h'(m) \neq 0$.
This is a contradiction.
Hence, $\neg f_{g_3}(E) \in X_{j, m-1}$.
Then, there exist $\rho \in P_{T, m-1}$ and $l \in W_n$ such that $j \prec_{n, E} l$ and $\rho \to (S'(\overline{l}) \to \neg f_{g_3}(E)) \in P_{T, m-1}$.
Since $i \prec_{n, \Box E} j$ and $j \prec_{n, E} l$, we obtain $i \prec_{n, E} l$ because $(W_n, \{\prec_{n, B}\}_{B \in \mathsf{MF}}, \Vdash_n)$ is $\mathsf{Sub}(A_n)$-transitive.
Therefore, $\neg f_{g_3}(E) \in X_{i, m-1}$, and hence $\neg \varphi \in X_{i, m-1}$.
Since $S'(\overline{l}) \to \neg \varphi$ is a t.c.~of $P_{T, m-1}$, we have $\varphi \notin P_{T, m-1}$.
Hence, $\varphi \notin \{g_3(0), \ldots, g_3(m-1)\}$.
Then, even if $\varphi \in X_{i, m-1}$, $g_3$ outputs $\neg \varphi$ earlier than any output of $\varphi$.
Therefore, $\neg \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi})$ holds.
\end{itemize}
We have proved $\mathsf{PA} + \neg \mathrm{Con}_T \vdash \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi}) \to \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi})})$.
By the law of excluded middle, $\mathsf{PA} \vdash \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi}) \to \mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\mathrm{Pr}_{g_3}^{\mathrm{R}}(\gn{\varphi})})$.
\end{proof}
The first clause of the theorem follows from Claims \ref{Cl:g_3_eq} and \ref{Cl:g_3_4}.
The second clause follows from Claims \ref{Cl:h}.3 and \ref{Cl:g_3}.
\end{proof}
\begin{cor}
\begin{align*}
\mathsf{N}RF & = \bigcap \{\mathsf{PL}(\mathrm{Pr}_T^{\mathrm{R}}) \mid \mathrm{Pr}_T(x)\ \text{is a Rosser's provability predicate of}\ T\ \text{satisfying}\ \D{3}\}.
\end{align*}
Moreover, there exists a Rosser's provability predicate $\mathrm{Pr}_T^{\mathrm{R}}(x)$ of $T$ such that $\mathsf{N}RF = \mathsf{PL}(\mathrm{Pr}_T^{\mathrm{R}}R)$.
\end{cor}
\appendix
\section{Appendix: $\Sigma_1$ provability predicates corresponding to $\mathsf{K}$}\label{Sec:K}
In \cite{Kur18_1}, it is proved that there exists a $\Sigma_2$ provability predicate $\mathrm{Pr}_T(x)$ of $T$ such that $\mathsf{K} = \mathsf{PL}(\mathrm{Pr}_T)$.
It follows
\[
\mathsf{K} = \bigcap \{\mathsf{PL}(\mathrm{Pr}_T) \mid \mathrm{Pr}_T(x)\ \text{is a provability predicate of}\ T\ \text{satisfying}\ \D{2}\}.
\]
As in Theorems \ref{Thm:N} and \ref{Thm:NF}, we prove that the provability logic of all $\Sigma_1$ provability predicates satisfying $\D{2}$ is also $\mathsf{K}$.
\begin{thm}[The uniform arithmetical completeness of $\mathsf{K}$]\label{Thm:K}
There exist a $\Sigma_1$ provability predicate $\mathrm{Pr}_T(x)$ of $T$ such that
\begin{enumerate}
\item for any $A \in \mathsf{MF}$ and any arithmetical interpretation $f$ based on $\mathrm{Pr}_T(x)$, if $\mathsf{K} \vdash A$, then $\mathsf{PA} \vdash f(A)$; and
\item there exists an arithmetical interpretation $f$ based on $\mathrm{Pr}_T(x)$ such that for any $A \in \mathsf{MF}$, $\mathsf{K} \vdash A$ if and only if $T \vdash f(A)$.
\end{enumerate}
\end{thm}
\begin{proof}
Let $(W, \prec, \Vdash)$ be a primitive recursively representable Kripke model satisfying the following conditions:
\begin{itemize}
\item $W = \omega \setminus \{0\}$,
\item $(W, \prec, \Vdash)$ is a disjoint union of finite Kripke models, that is, for every $i \in W$, $\{j \in W \mid i \prec j\}$ is a finite set and may be empty,
\item for any $\mathsf{K}$-unprovable $\mathcal{L}(\Box)$-formula $A$, there exists an $i \in W$ such that $i \nVdash A$.
\end{itemize}
We define a primitive recursive function $g_4$ corresponding to this theorem.
We only describe Procedure 2.
\textsc{Procedure 2}\\
Let $m$ and $i \neq 0$ be such that $h(m) = 0$ and $h(m+1) = i$.
Define
\[
g_4(m + t) = \begin{cases}
\xi_t & \text{if}\ \xi_t\ \text{is a t.c.~of}\ P_{T, m-1} \cup \bigl\{\bigvee_{i \prec j} S(\overline{j}) \bigr\}, \\
0 & \text{otherwise}.
\end{cases}
\]
Notice that the empty disjunction represents $0=1$.
Our definition of $g_4$ is finished.
Let $f_{g_4}$ be the arithmetical interpretation based on $\mathrm{Pr}_{g_4}(x)$ defined as $\exists x(S(x) \land x \Vdash p)$.
The following claim is proved similarly as in the proof of Theorem \ref{Thm:N}.
\begin{cl}\label{Cl:g_4_eq}
$\mathsf{PA} + \mathrm{Con}_T \vdash \forall x \forall y \Bigl(\mathrm{Fml}_{\mathcal{L}_A}(x) \to \bigl(\mathrm{Proof}_T(x, y) \leftrightarrow x = g_4(y) \bigr) \Bigr)$.
\end{cl}
Thus, $\mathrm{Pr}_{g_4}(x)$ is a $\Sigma_1$ provability predicate of $T$.
\begin{cl}\label{Cl:g_4_eq2}
$\mathsf{PA}$ proves the following statement:
``Let $m$ and $i \neq 0$ be such that $h(m) = 0$ and $h(m+1) = i$.
Then, for any $\mathcal{L}_A$-formula $\varphi$,
\[
\mathrm{Pr}_{g_4}(\gn{\varphi})\ \text{holds}\ \iff \varphi\ \text{is a t.c.~of}\ P_{T, m-1} \cup \bigl\{\bigvee_{i \prec j} S(\overline{j}) \bigr\}\text{''}.
\]
\end{cl}
\begin{proof}
$(\Rightarrow)$: This is because if $\xi_t \in P_{T, m-1}$, then $\xi_t$ is a t.c.~of $P_{T, m-1} \cup \bigl\{\bigvee_{i \prec j} S(\overline{j}) \bigr\}$.
$(\Leftarrow)$: Immediate from the definition of $g_4$.
\end{proof}
\begin{cl}\label{Cl:g_4_2}
$\mathsf{PA} \vdash \forall x \forall y \bigl(\mathrm{Pr}_{g_4}(x \dot{\to} y) \land \mathrm{Pr}_{g_4}(x) \to \mathrm{Pr}_{g_4}(y) \bigr)$.
\end{cl}
\begin{proof}
Since $\mathsf{PA} \vdash \forall x \forall y \bigl(\mathrm{Prov}_{g_4}(x \dot{\to} y) \land \mathrm{Prov}_{g_4}(x) \to \mathrm{Prov}_{g_4}(y) \bigr)$.
We have $\mathsf{PA} + \mathrm{Con}_T \vdash \forall x \forall y \bigl(\mathrm{Pr}_{g_4}(x \dot{\to} y) \land \mathrm{Pr}_{g_4}(x) \to \mathrm{Pr}_{g_4}(y) \bigr)$ by Claim \ref{Cl:g_4_eq}.
We argue in $\mathsf{PA} + \neg \mathrm{Con}_T$:
By Proposition \ref{Prop:h}.2, there exists an $i \neq 0$ such that $S(\overline{i})$ holds.
Let $m$ be such that $h(m) = 0$ and $h(m+1) = i$.
Suppose $\mathrm{Pr}_{g_4}(\gn{\varphi \to \psi})$ and $\mathrm{Pr}_{g_4}(\gn{\varphi})$ hold.
By Claim \ref{Cl:g_4_eq2}, both $\varphi \to \psi$ and $\varphi$ are t.c.'s of $P_{T, m-1}$.
Then, $\psi$ is also a t.c.~of $P_{T, m-1}$.
By Claim \ref{Cl:g_4_eq2} again, we obtain that $\mathrm{Pr}_{g_4}(\gn{\psi})$ holds.
We have proved $\mathsf{PA} + \neg \mathrm{Con}_T \vdash \forall x \forall y \bigl(\mathrm{Pr}_{g_4}(x \dot{\to} y) \land \mathrm{Pr}_{g_4}(x) \to \mathrm{Pr}_{g_4}(y) \bigr)$.
By the law of excluded middle, we conclude $\mathsf{PA} \vdash \forall x \forall y \bigl(\mathrm{Pr}_{g_4}(x \dot{\to} y) \land \mathrm{Pr}_{g_4}(x) \to \mathrm{Pr}_{g_4}(y) \bigr)$.
\end{proof}
\begin{cl}\label{Cl:g_4_box}
Let $i, l \in W$.
\begin{enumerate}
\item $\mathsf{PA} \vdash S(\overline{i}) \to \mathrm{Pr}_{g_4} \Bigl( \gn{\bigvee_{i \prec j} S(\overline{j})} \Bigr)$.
\item If $i \prec l$, then $\mathsf{PA} \vdash S(\overline{i}) \to \neg \mathrm{Pr}_{g_4}(\gn{\neg S(\overline{l})})$.
\end{enumerate}
\end{cl}
\begin{proof}
We proceed in $\mathsf{PA} + S(\overline{i})$:
Let $m$ be such that $h(m) = 0$ and $h(m+1) = i$.
1. Since $\bigvee_{i \prec j} S(\overline{j})$ is a t.c.~of $P_{T, m-1} \cup \bigl\{\bigvee_{i \prec j} S(\overline{j}) \bigr\}$, it follows that $\mathrm{Pr}_{g_4} \Bigl( \gn{\bigvee_{i \prec j} S(\overline{j})} \Bigr)$ holds by Claim \ref{Cl:g_4_eq2}.
2. Suppose, towards a contradiction, that $\neg S(\overline{l})$ is a t.c.~of $P_{T, m-1} \cup \bigl\{\bigvee_{i \prec j} S(\overline{j}) \bigr\}$.
Then, $\bigvee_{i \prec j} S(\overline{j}) \to \neg S(\overline{l})$ is a t.c.~of $P_{T, m-1}$.
Since $S(\overline{l})$ is a disjunct of $\bigvee_{i \prec j} S(\overline{j})$, $S(\overline{l}) \to \neg S(\overline{l})$ is also a t.c.~of $P_{T, m-1}$.
Then, $\neg S(\overline{l})$ is a t.c.~of $P_{T, m-1}$.
This is a contradiction.
Hence, $\neg S(\overline{l})$ is not a t.c.~of $P_{T, m-1} \cup \bigl\{\bigvee_{i \prec j} S(\overline{j}) \bigr\}$.
By Claim \ref{Cl:g_4_eq2}, $\neg \mathrm{Pr}_{g_4}(\gn{\neg S(\overline{l})})$ holds.
\end{proof}
The following claim is proved in the same as in the usual proof of Solovay's arithmetical completeness theorem by using Claim \ref{Cl:g_4_box}.
\begin{cl}\label{Cl:g_4}
Let $i \in W$ and $B \in \mathsf{MF}$.
\begin{enumerate}
\item If $i \Vdash B$, then $\mathsf{PA} \vdash S(\overline{i}) \to f_{g_4}(B)$.
\item If $i \nVdash B$, then $\mathsf{PA} \vdash S'(\overline{i}) \to \neg f_{g_4}(B)$.
\end{enumerate}
\end{cl}
The first clause of the theorem follows from Claims \ref{Cl:g_4_eq} and \ref{Cl:g_4_2}.
The second clause follows from Proposition \ref{Prop:h}.3 and Claim \ref{Cl:g_4}.
\end{proof}
\begin{cor}
\[
\mathsf{K} = \bigcap \{\mathsf{PL}(\mathrm{Pr}_T) \mid \mathrm{Pr}_T(x)\ \text{is a}\ \Sigma_1\ \text{provability predicate of}\ T\ \text{satisfying}\ \D{2}\}.
\]
Moreover, there exists a $\Sigma_1$ provability predicate $\mathrm{Pr}_T(x)$ of $T$ such that $\mathsf{K} = \mathsf{PL}(\mathrm{Pr}_T)$.
\end{cor}
\section{Appendix: Interchangeability of $\Box$ and $\Diamond$ in $\mathsf{N}R$}\label{Sec:NR_Box}
The language of propositional modal logic does not have the symbol $\Diamond$ as a modal operator.
Then, we introduce the expression $\Diamond A$ as the abbreviation of $\neg \Box \neg A$.
However, in the logic $\mathsf{N}$, $\Box$ and $\Diamond$ are not dual operators, that is, it is shown that $\neg \Diamond \neg p \leftrightarrow \Box p$ is not provable in $\mathsf{N}$.
This is because the relations $\prec_{p}$ and $\prec_{\neg \neg p}$ in $\mathsf{N}$-frames may be irrelevant.
On the other hand, in $\mathsf{N}R$, the operators $\Box$ and $\Diamond$ have an interesting relationship.
That is, in a sense, $\Box$ and $\Diamond$ are interchangeable in $\mathsf{N}R$.
To state this fact precisely, we introduce the following translation $\chi$.
\begin{defn}
We define a translation $\chi$ of $\mathcal{L}(\Box)$-formulas recursively as follows:
\begin{enumerate}
\item $\chi(A)$ is $A$ if $A$ is a propositional variable or $\bot$,
\item $\chi(\neg A)$ is $\neg \chi(A)$,
\item $\chi(A \circ B)$ is $\chi(A) \circ \chi(B)$ for $\circ \in \{\land, \lor, \to\}$,
\item $\chi(\Box A)$ is $\Diamond \chi(A)$.
\end{enumerate}
\end{defn}
That is, $\chi(A)$ is obtained from $A$ by replacing every $\Box$ with $\Diamond$.
\begin{prop}
For any $A \in \mathsf{MF}$, $\mathsf{N}R \vdash A$ if and only if $\mathsf{N}R \vdash \chi(A)$.
\end{prop}
\begin{proof}
$(\Rightarrow)$: We prove this implication by induction on the length of proofs in $\mathsf{N}R$.
It suffices to prove that $\mathsf{N}R$ is closed under the rules $\dfrac{A}{\Diamond A}$ and $\dfrac{\neg A}{\neg \Diamond A}$.
Suppose $\mathsf{N}R \vdash A$.
Then, $\mathsf{N}R \vdash \neg \neg A$.
By the rule \textsc{Ros}, $\mathsf{N}R \vdash \neg \Box \neg A$, that is, $\mathsf{N}R \vdash \Diamond A$.
Suppose $\mathsf{N}R \vdash \neg A$.
By \textsc{Nec}, $\mathsf{N}R \vdash \Box \neg A$, and then $\mathsf{N}R \vdash \neg \neg \Box \neg A$.
This means $\mathsf{N}R \vdash \neg \Diamond A$.
$(\Leftarrow)$: We prove the contrapositive.
Suppose $\mathsf{N}R \nvdash A$.
Then, by Theorem \ref{Thm:complNR}, there exists a serial $\mathsf{N}$-model $\mathcal{M} = (W, \{\prec_B\}_{B \in \mathsf{MF}}, \Vdash)$ and $w \in W$ such that $w \nVdash A$.
For each $B \in \mathsf{MF}$, let $\prec_B^*$ be the binary relation on $W$ defined as follows:
\[
\prec_B^* : = \begin{cases} \prec_D & \text{if}\ B\ \text{is of the form}\ \neg \neg \chi(\chi(D)), \\
\prec_B & \text{otherwise}.
\end{cases}
\]
Let $\mathcal{M}^*$ be the $\mathsf{N}$-model $(W, \{\prec_B^*\}_{B \in \mathsf{MF}}, \Vdash^*)$ defined by $x \Vdash^* p : \iff x \Vdash p$.
It is easy to see that the frame of $\mathcal{M}^*$ is also serial.
\begin{cl}\label{Cl:NR}
For any $\mathcal{L}(\Box)$-formula $C$ and $x \in W$, $x \Vdash^* \chi(\chi(C))$ if and only if $x \Vdash C$.
\end{cl}
\begin{proof}
This claim is proved by induction on the construction of $C$.
If $C$ is a propositional variable, the claim is trivial because $\chi(\chi(p))$ is exactly $p$.
The cases of $\bot$ and propositional connectives are easy.
We prove the case that $C$ is of the form $\Box D$.
Notice $\prec_{\neg \neg \chi(\chi(D))}^* = \prec_D$.
By the induction hypothesis, for any $y \in W$, $y \Vdash^* \chi(\chi(D))$ if and only if $y \Vdash D$.
Then,
\begin{align*}
x \Vdash^* \chi(\chi(\Box D)) & \iff x \Vdash^* \Box \neg \neg \chi(\chi(D)), \\
& \iff \forall y \in W \bigl(x \prec_{\neg \neg \chi(\chi(D))}^* y \Rightarrow y \Vdash^* \neg \neg \chi(\chi(D)) \bigr), \\
& \iff \forall y \in W \bigl(x \prec_D y \Rightarrow y \Vdash^* \chi(\chi(D)) \bigr), \\
& \iff \forall y \in W \bigl(x \prec_D y \Rightarrow y \Vdash D \bigr), \\
& \iff x \Vdash \Box D.
\end{align*}
\end{proof}
Since $w \nVdash A$, we obtain $w \nVdash^* \chi(\chi(A))$ by Claim \ref{Cl:NR}.
By Theorem \ref{Thm:complNR} again, we obtain $\mathsf{N}R \nvdash \chi(\chi(A))$.
Since we have already proved the implication $(\Rightarrow)$ of the proposition, we conclude $\mathsf{N}R \nvdash \chi(A)$.
\end{proof}
\end{document} | math | 97,279 |
\begin{document}
\begin{abstract}
In this note, we prove the blow-up of solutions of the semilinear damped Klein-Gordon equation in a finite time for arbitrary positive initial energy on the Heisenberg group. This work complements the paper \cite{RT-18} by the first author and Tokmagambetov, where the global in time well-posedness was proved for the small energy solutions.
\end{abstract}
\maketitle
\section{Introduction}
\subsection{Setting of the problem} This note is devoted to study the blow up of solutions of the Cauchy problem for the semilinear damped Klein-Gordon equation for the sub-Laplacian $\mathcal{L}$ on the Heisenberg group $\mathbb{H}^n$:
\begin{align}\label{Wave-problem}
\begin{cases}
u_{tt}(t) - \mathcal{L} u(t) + bu_t(t)+ mu(t)= f(u), & t>0, \\
u(x,0) = u_0(x),\,\,\, & u_0 \in H^1_{\mathcal{L}}(\mathbb{H}^n),\\
u_t(x,0) = u_1(x),\,\,\, & u_1 \in L^2(\mathbb{H}^n),
\end{cases}
\end{align}
with the damping term determined by $b>0$ and the mass $m>0$. A total energy of problem \eqref{Wave-problem} is defined as
\begin{align*}
E(t)&=\frac{1}{2}|| u_t||^2_{L^2(\mathbb{H}^n)} + \frac{m}{2}||u||^2_{L^2(\mathbb{H}^n)}+ \frac{1}{2}||\nabla_{H} u||^2_{L^2(\mathbb{H}^n)} - \int_{\mathbb{H}^n}F(u) dx,
\end{align*}
where we assume the following
\begin{align}\label{Cond-F}
& g: [0,\infty] \rightarrow \mathbb{R},\nonumber\\
& F(z) = g(|z|) \,\, \text{ for } \,\, z \in \mathbb{C}^n, \\
& f(z) = \frac{g'(|z|)z}{|z|}. \nonumber
\end{align}
Then we have that
\begin{align*}
\frac{\partial}{\partial\varepsilon} F(z+ \varepsilon \xi) |_{\varepsilon=0} &= \frac{\partial}{\partial \varepsilon} g(|z + \varepsilon \xi|)|_{\varepsilon =0} \\
& = g'(|z+\varepsilon \xi| )\frac{\partial}{\partial \varepsilon}(|z+\varepsilon \xi|)|_{\varepsilon=0} \\
&= \frac{g'(|z|)}{|z|} \frac{1}{2}(\overline{z}\xi + z \overline{\xi}) \\
&={\rm Re } \left( f(z) \overline{\xi}\right),
\end{align*}
and
\begin{align*}
\frac{\partial}{\partial x_j}F(u(x)) &= g'(|u(x)|) \frac{\partial|u(x)| }{\partial x_j}\\
&= \frac{g'(|u(x)|)}{2|u(x)|} \left(u(x)\frac{\partial \overline{u}(x) }{\partial x_j} +\frac{\partial u(x) }{\partial x_j}\overline{u}(x)\right)\\
&= {\rm Re }\left( f(u(x)) \frac{\partial \overline{u}(x)}{\partial x_j}\right).
\end{align*}
The conservation of energy law follows from
\begin{align*}
\frac{\partial E(t)}{\partial t} & = \frac{\partial }{\partial t}\left[ \frac{1}{2} || u_t||^2_{L^2(\mathbb{H}^n)} +\frac{m}{2}||u||^2_{L^2(\mathbb{H}^n)}+ \frac{1}{2}||\nabla_{H} u||^2_{L^2(\mathbb{H}^n)} - \int_{\mathbb{H}^n} F(u) dx \right]\\
& = {\rm Re }\int_{\mathbb{H}^n } \overline{u}_t [u_{tt} + m u - \mathcal{L}u - f(u)] dx\\
& = - b \int_{\mathbb{H}^n } |u_t|^2 dx,
\end{align*}
this gives
\begin{equation}\label{eq-energy}
E(t) + b\int_{0}^t|| u_s(s) ||^2_{L^2(\mathbb{H}^n)}ds =E(0),
\end{equation}
where
\begin{equation*}
E(0):=\frac{1}{2} || u_1||^2_{L^2(\mathbb{H}^n)} +\frac{m}{2}||u_0||^2_{L^2(\mathbb{H}^n)}+ \frac{1}{2}||\nabla_{H} u_0||^2_{L^2(\mathbb{H}^n)} - \int_{\mathbb{H}^n} F(u_0) dx.
\end{equation*}
Also, let us define the Nehari functional
\begin{align*}
I(u) = m||u||^2_{L^2(\mathbb{H}^n)} + ||\nabla_{H} u||^2_{L^2(\mathbb{H}^n)} - {\rm Re } \int_{\mathbb{H}^n}f(u)\overline{u} dx.
\end{align*}
We assume that the nonlinear term $f(u)$ satisfies the condition
\begin{equation*}
f(0)=0, \,\,\, \text{ and } \,\, \alpha F(u) \leq {\rm Re } [ f(u)\overline{u}],
\end{equation*}
where $\alpha >2$. In particular, this includes the case
\begin{equation*}
f(u) = |u|^{p-1}u\,\,\,\text{ for } p>1.
\end{equation*}
\subsection{Literature overview }
The study of the damped wave equation on the Heisenberg group started in Bahouri-Gerrard-Xu \cite{BGX-00} to prove the dispersive and Strichartz inequalities based on the analysis in Besov-type spaces. Later, Greiner-Holcman-Kannai \cite{GHK-02} explicitly computed the wave kernel for the class of second-order subelliptic operators, where their class contains degenerate elliptic and hypoelliptic operators such as the sub-Laplacian and the Grushin operator. Also, M\"uller-Stein \cite{MS-99} established $L^p$-estimates for the wave equation on the Heisenberg group. Recently, M\"uller-Seeger \cite{MS-15} obtained the sharp version of $L^p$ estimates on the $H$-type groups.
The blow-up solutions of evolution equations on the Heisenberg group were considered by Georgiev-Palmieri \cite{GP-20} where they proved the global existence and nonexistence results of the Cauchy problem for the semilinear damped wave equation on the Heisenberg group with the power nonlinear term. The proof of blow-up solutions is based on the test function method. The first author and Yessirkegenov \cite{RY-22} established the existence and non-existence of global solutions for semilinear heat equations and inequalities on sub-Riemannian manifolds. In \cite{RY-22-1}, by using the comparison principle they obtain blow-up type results and global in $t$-boundedness of solutions of nonlinear equations for the heat $p$-sub-Laplacian on the stratified Lie groups. The global existence and nonexistence for the nonlinear porous medium equation were studied by the authors in \cite{RST-21} on the stratified Lie groups.
This work is motivated by the paper \cite{RT-18} of the first author and Tokmagambetov where the global existence of solutions for small data of problem \eqref{Wave-problem} was shown on the Heisenberg group and on general graded Lie groups. In the sense of the potential wells theory, we can understand this result in the sense that when the initial energy is less than the mountain pass level $E(0)<d$ and the Nehari functional is positive $I(u_0)>0$, there exists a global solution of the problem \eqref{Wave-problem}. A natural question arises when the solution of problem \eqref{Wave-problem} blows up in a finite time or $E(0)>0$ and $I(u_0)<0$.
The main aim of this paper is to obtain the blow-up solutions of problem \eqref{Wave-problem} in a finite time for arbitrary positive initial energy. Our proof is based on an adopted concavity method, which was introduced by Levine \cite{Levine-74} to establish the blow-up solutions of the abstract wave equation of the form $Pu_{tt}=-Au + F(u)$ (including the Klein-Gordon equation) for the negative initial energy. It was also used for parabolic type equations (see \cite{Levine-74-1, Levine73, Levine-90, LP-74, LP2-74}). Modifying the concavity method, Wang \cite{W-08} proved the nonexistence of global solutions to nonlinear damped Klein-Gordon equation for arbitrary positive initial energy under sufficient conditions. Later, Yang-Xu \cite{YX-18} extended this result by introducing a new auxiliary function and the adopted concavity method.
\subsection{Preliminaries on the Heisenberg group} Let us give a brief introduction of the Heisenberg group.
Let $\mathbb{H}^n$ be the Heisenberg group, that is, the set $\mathbb{R}^{2n+1}$ equipped with the group law
\begin{equation*}
\xi \circ \widetilde{\xi} := (x + \widetilde{x}, y + \widetilde{y}, t + \widetilde{t}+2 \sum_{i=1}^{n}(\widetilde{x}_i y_i - x_i \widetilde{y}_i )),
\end{equation*}
where $\xi:= (x,y,t) \in \mathbb{H}^n$, $x:=(x_1,\ldots,x_n)$, $y:=(y_1,\ldots,y_n)$, and $\xi^{-1}=-\xi$ is the inverse element of $\xi$ with respect to the group law. The dilation operation of the Heisenberg group with respect to the group law has the following form (see e.g. \cite{FR-book}, \cite{RS_book})
\begin{equation*}
\delta_{\lambda}(\xi) := (\lambda x, \lambda y, \lambda^2 t) \,\, \text{for}\,\, \lambda>0.
\end{equation*}
The Lie algebra $\mathfrak{h}$ of the left-invariant vector fields on the Heisenberg group $\mathbb{H}^n$ is spanned by
\begin{equation*}
X_i:= \frac{\partial }{\partial x_i} + 2y_i\frac{\partial }{\partial t} \,\, \text{for} \,\, 1\leq i \leq n,
\end{equation*}
\begin{equation*}
Y_i:= \frac{\partial }{\partial y_i} - 2x_i\frac{\partial }{\partial t} \,\, \text{for} \,\, 1\leq i \leq n,
\end{equation*}
and with their (non-zero) commutator
\begin{equation*}
[X_i,Y_i]= - 4 \frac{\partial}{\partial t}.
\end{equation*}
The horizontal gradient of $\mathbb{H}^n$ is given by
\begin{equation*}
\nabla_{H}:= (X_1,\ldots,X_n,Y_1,\ldots,Y_n),
\end{equation*}
so the sub-Laplacian on $\mathbb{H}^n$ is given by
\begin{equation*}
\mathcal{L}:= \sum_{i=1}^{n} \left(X_i^2 + Y_i^2\right).
\end{equation*}
\begin{defn}[Weak solution]
A function
\begin{align*}
&u \in C ([0,T_1);H^1_{\mathcal{L}}(\mathbb{H}^n)) \cap C^1 ([0,T_1);L^2(\mathbb{H}^n)), \\
&u_t \in L^2 ([0,T_1);H^1_{\mathcal{L}}(\mathbb{H}^n)),\\
&u_{tt} \in L^2 ([0,T_1);H^{-1}_{\mathcal{L}}(\mathbb{H}^n)),
\end{align*}
satisfying
\begin{equation}\label{eq-weak}
{\rm Re} \langle u_{tt}, v \rangle + {\rm Re}\int_{\mathbb{H}^n } \nabla_{H} u \cdot \nabla_{H} v dx +m{\rm Re} \int_{\mathbb{H}^n } uv dx + b {\rm Re}\int_{\mathbb{H}^n } u_tv dx\ = {\rm Re}\int_{\mathbb{H}^n } f(u)v dx,
\end{equation}
for all $v \in H^1_{\mathcal{L}}(\mathbb{H}^n)$ and a.e. $t \in [0,T_1)$ with $u(0)=u_0(x)$ and $u_t(0)=u_1(x) $ represents a weak solution of problem \eqref{Wave-problem}.
\end{defn}
Note that $T_1$ denotes the lifespan of the solution $u(x,t)$ and $\langle \cdot, \cdot \rangle$ is the duality between $H^{-1}_{\mathcal{L}}(\mathbb{H}^n)$ and $H^{1}_{\mathcal{L}}(\mathbb{H}^n)$. Here $H^1_{\mathcal{L}}(\mathbb{H}^n)$ denotes the sub-Laplacian Sobolev space, analysed by Folland \cite{Fol-75}, see also \cite{FR-book}.
\section{Main Result}
We now present the main result of this paper.
\begin{thm}\label{thm_main}
Let $b>0$, $m>0$ and $\mu= \max\{ b,m, \alpha \}$. Assume that nonlinearity $f(u)$ satisfies
\begin{equation}\label{eq-2.1}
\alpha F(u) \leq {\rm Re}[ f(u)\overline{u}]\,\,\, \text{ for }\,\, \alpha >2,
\end{equation}
where $F(u)$ is as in \eqref{Cond-F}.
Assume that the Cauchy data $u_0 \in H_{\mathcal{L}}^1(\mathbb{H}^n)$ and $u_1 \in L^2(\mathbb{H}^n)$ satisfy
\begin{equation}\label{eq-2.2}
I(u_0) = m||u_0||^2_{L^2(\mathbb{H}^n)} + ||\nabla_{H} u_0||^2_{L^2(\mathbb{H}^n)} - {\rm Re}\int_{\mathbb{H}^n}\overline{u}_0f(u_0) dx<0,
\end{equation}
and
\begin{equation}\label{eq-thm-cond}
{\rm Re}(u_0,u_1)_{L^2(\mathbb{H}^n)}\geq \frac{\alpha(\mu+1)}{m(\alpha-2)}E(0).
\end{equation}
Then the solution of equation \eqref{Wave-problem} blows up in finite time $T^*$ such that
\begin{equation*}
0< T^* \leq \frac{2(\mu+1)(bT_0 +1)}{(\alpha-2)(\mu+1-m)} \frac{||u_0||^2_{L^2(\mathbb{H}^n)}}{{\rm Re}(u_0,u_1)},
\end{equation*}
where the blow-up time $T^* \in (0,T_0)$ with $ T_0<+\infty$.
\end{thm}
\begin{rem} \begin{itemize}
\item[(i)] Note that we have times $T^*$, $T_0$ and $T_1$. The relationship between this times is the blow-up time $T^*\in (0,T_0) \subset (0,T_1)$ where $T_0<+\infty$ and $T_1 = +\infty$.
\item[(ii)] The local existence for the Klein-Gordon equation was shown in \cite{Caz-85} and \cite{CH-98}. The global in time well-posedness of problem \eqref{Wave-problem} was proved by the first author and Tokmagambetov \cite{RT-18} for the small energy solutions and the nonlinearity $f(u)$ satisfying
\begin{align*}
|f(u) - f(v)| \leq C (|u|^{p-1}+|v|^{p-1})|u-v|,
\end{align*}
with $1<p\leq 1+1/n$.
\end{itemize}
\end{rem}
\begin{proof}[Proof of Theorem \ref{thm_main} ]
First, recall the Nehari functional
\begin{align*}
I(u) = m||u||^2_{L^2(\mathbb{H}^n)} + ||\nabla_{H} u||^2_{L^2(\mathbb{H}^n)} - {\rm Re}\int_{\mathbb{H}^n}\overline{u}f(u) dx.
\end{align*}
Then the proof includes two steps.
\textbf{Step I.}
In this step, we claim that
\begin{equation*}
I(u(t))<0, \,\,\, \text{ and } \,\,\, A(t) > \frac{2\alpha(\mu+1)}{m(\alpha-2)}E(0),
\end{equation*}
for $0\leq t <T_1$ where $\mu= \max\{ b,m, \alpha \}$ and
\begin{align*}
A(t) = 2 {\rm Re}(u,u_t) + b ||u||^2_{L^2(\mathbb{H}^n)}.
\end{align*}
By using \eqref{eq-weak} along with $v =\overline{u}$ we get
\begin{align}\label{eq-*}
A'(t) &= 2 || u_t||^2_{L^2(\mathbb{H}^n)} + 2{\rm Re }\langle u_{tt},u \rangle + 2b {\rm Re}\int_{\mathbb{H}^n } \overline{u}u_t dx \nonumber\\
&= 2 || u_t||^2_{L^2(\mathbb{H}^n)} - 2 I(u), \,\,\, 0\leq t <T_1.
\end{align}
In the last line we have used that
\begin{align*}
{\rm Re} \langle u_{tt}, u \rangle &= {\rm Re}\int_{\mathbb{H}^n } f(u)\overline{u} dx- \int_{\mathbb{H}^n } |\nabla_{H} u|^2 dx - m\int_{\mathbb{H}^n } |u|^2 dx - b {\rm Re}\int_{\mathbb{H}^n } \overline{u} u_t dx\\
&= - I(u) - b {\rm Re}\int_{\mathbb{H}^n } \overline{u}u_t dx.
\end{align*}
Now let us suppose by contradiction that
\begin{equation*}
I(u(t))<0 \,\,\, \text{ for all } \,\, 0\leq t<t_0,
\end{equation*}
and
\begin{equation*}
I(u(t_0))=0.
\end{equation*}
Hereafter $0<t_0<T_1$. It is easy to see that $A'(t)>0$ over $[0,t_0)$ and
\begin{align}\label{eq-A}
A(t) > A(0) \geq 2 {\rm Re}(u_0,u_1) \geq \frac{2\alpha(\mu+1)}{m(\alpha-2)}E(0).
\end{align}
Since $u(t)$ and $u_t(t)$ are both continuous in $t$ that gives
\begin{align}\label{eq-2.6}
A(t_0) \geq \frac{2\alpha(\mu+1)}{m(\alpha-2)}E(0).
\end{align}
Next we need to show a contradiction to \eqref{eq-2.6}. Using \eqref{eq-energy} and \eqref{eq-2.1}, we have
\begin{align*}
E(0) &= E(t) + b \int_{0}^t||u_s||^2_{L^2(\mathbb{H}^n)}ds \\
& =\frac{1}{2} || u_t||^2_{L^2(\mathbb{H}^n)}+ \frac{m}{2}||u||^2_{L^2(\mathbb{H}^n)} + \frac{1}{2} ||\nabla_{H} u||^2_{L^2(\mathbb{H}^n)} \\
&- \int_{\mathbb{H}^n } F(u)dx + b \int_{0}^t||u_s||^2_{L^2(\mathbb{H}^n)}ds\\
&\geq \frac{1}{2} || u_t||^2_{L^2(\mathbb{H}^n)} + \frac{m}{2}||u||^2_{L^2(\mathbb{H}^n)} + \frac{1}{2} ||\nabla_{H} u||^2_{L^2(\mathbb{H}^n)} \\
&- \frac{1}{\alpha} {\rm Re}\int_{\mathbb{H}^n } \overline{u} f(u)dx + b \int_{0}^t||u_s||^2_{L^2(\mathbb{H}^n)}ds\\
& = \frac{1}{2} || u_t||^2_{L^2(\mathbb{H}^n)} + \frac{1}{\alpha}I(u) + \left( \frac{m}{2} -\frac{m}{\alpha} \right)||u||^2_{L^2(\mathbb{H}^n)}\\
&+\left(\frac{\alpha -2}{2\alpha} \right)||\nabla_{H} u||^2_{L^2(\mathbb{H}^n)} + b\int_{0}^t||u_s||^2_{L^2(\mathbb{H}^n)}ds.
\end{align*}
If we use $I(u(t_0))=0$ and $\frac{m(\alpha -2)}{\alpha(\mu+1)}<1$, then
\begin{align}\label{eq-EA}
E(0) & \geq \frac{1}{2} || u_t(t_0)||^2_{L^2(\mathbb{H}^n)} + \frac{m(\alpha -2)}{2\alpha}||u(t_0)||^2_{L^2(\mathbb{H}^n)}\nonumber\\
& \geq \frac{m(\alpha -2)}{2\alpha(\mu +1)} \left( || u_t(t_0)||^2_{L^2(\mathbb{H}^n)} + (\mu +1) ||u(t_0)||^2_{L^2(\mathbb{H}^n)} \right)\nonumber\\
& > \frac{m(\alpha -2)}{2\alpha(\mu +1)} \left( 2 {\rm Re}(u(t_0),u_t(t_0)) + \mu ||u(t_0)||^2_{L^2(\mathbb{H}^n)} \right)\nonumber \\
& \geq \frac{m(\alpha -2)}{2\alpha(\mu +1)} A(t_0).
\end{align}
Note that for the strict inequality above we use that the assumption \eqref{eq-2.2} implies that $||u_0||_{L^2(\mathbb{H}^n)} \neq 0$.
We have also used the fact $a^2 + b^2 -2ab \geq 0$, where $a = || u_t(t_0)||_{L^2(\mathbb{H}^n)}$ and $b= || u(t_0)||_{L^2(\mathbb{H}^n)}$. It gives the contradiction to \eqref{eq-2.6}. This proves our claim.
\textbf{Step II.} Define the functional
\begin{align*}
M(t) = ||u||^2_{L^2(\mathbb{H}^n)} + b \int_{0}^t ||u(s)||^2_{L^2(\mathbb{H}^n)} ds + b (T_0-t)||u_0||^2_{L^2(\mathbb{H}^n)},
\end{align*}
for $0\leq t \leq T_0$. Then
\begin{align*}
M'(t) &= 2 {\rm Re}(u,u_t) +b||u(t)||^2_{L^2(\mathbb{H}^n)}-b||u_0||^2_{L^2(\mathbb{H}^n)} \\
&=2 {\rm Re}(u,u_t) +2b \int_{0}^t {\rm Re}(u(s),u_s(s))ds,
\end{align*}
since
\begin{align*}
\int_{0}^t \frac{d}{ds} || u (s)||_{L^2(\mathbb{H}^n)}^2 ds = || u(t)||_{L^2(\mathbb{H}^n)}^2 - || u(0)||_{L^2(\mathbb{H}^n)}^2.
\end{align*}
We observe the following estimates
\begin{align*}
|{\rm Re}(u,u_t)|^2 &\leq ||u_t||^2_{L^2(\mathbb{H}^n)} || u||^2_{L^2(\mathbb{H}^n)}, \\
\left( \int_{0}^t |{\rm Re}(u(s),u_s(s))|ds \right)^2& \leq \left(\int_{0}^t || u(s)||^2_{L^2(\mathbb{H}^n)}ds \right)\left(\int_{0}^t || u_s(s)||^2_{L^2(\mathbb{H}^n)}ds\right),
\end{align*}
and
\begin{align*}
2 {\rm Re}(u,u_t)\int_{0}^t {\rm Re}(u(s),u_s(s))ds &\leq 2 ||u||_{L^2(\mathbb{H}^n)} ||u_t||_{L^2(\mathbb{H}^n)}\\
&\times \left(\int_{0}^t || u(s)||^2_{L^2(\mathbb{H}^n)}ds\right)^{1/2}\left( \int_{0}^t || u_s(s)||^2_{L^2(\mathbb{H}^n)}ds\right)^{1/2}\\
\leq ||u||^2_{L^2(\mathbb{H}^n)} &\int_{0}^t || u_s(s)||^2_{L^2(\mathbb{H}^n)}ds + ||u_t||^2_{L^2(\mathbb{H}^n)} \int_{0}^t || u(s)||^2_{L^2(\mathbb{H}^n)}ds.
\end{align*}
Using the above inequalities, we calculate
\begin{align*}
(M'(t))^2 &= 4 \left( |{\rm Re}(u,u_t)|^2 + 2b {\rm Re}(u,u_t) \int_0^t {\rm Re}(u(s),u_s(s))ds + b^2 \left(\int_0^t {\rm Re}(u(s),u_s(s))ds\right)^2 \right) \\
&\leq 4 \left( ||u||^2_{L^2(\mathbb{H}^n)} + b \int_{0}^t ||u(s)||^2_{L^2(\mathbb{H}^n)} ds \right)\left( ||u_t||^2_{L^2(\mathbb{H}^n)} + \int_{0}^t ||u_s(s)||^2_{L^2(\mathbb{H}^n)} ds \right),
\end{align*}
for all $0\leq t \leq T_0$. The second derivate with respect to time of $M(t)$ is
\begin{align*}
M''(t) = 2||u_t||^2_{L^2(\mathbb{H}^n)} - 2 I(u),
\end{align*}
for all $0\leq t\leq T_0$, where we used the equality from \eqref{eq-*}. Then we construct the differential inequality as follows
\begin{align*}
&M''(t) M(t) - \frac{\omega+3}{4} (M'(t))^2 \geq M(t)\left( M''(t) - (\omega +3)\left( ||u_t||^2 + b\int_{0}^t ||u_s(s)||^2_{L^2(\mathbb{H}^n)} ds \right) \right)\\
&=M(t) \left( -(\omega+1) ||u_t||^2_{L^2(\mathbb{H}^n)} - (\omega +3)b\int_{0}^t ||u_s(s)||^2_{L^2(\mathbb{H}^n)} ds -2I(u)\right),
\end{align*}
where we assume that $\omega >1$. We shall now show that the following term is nonnegative
\begin{align*}
\eta(t) &= -(\omega+1) ||u_t||^2_{L^2(\mathbb{H}^n)} - (\omega +3)b\int_{0}^t ||u_s(s)||^2_{L^2(\mathbb{H}^n)} ds -2I(u)\\
&\geq (\alpha -\omega -1) ||u_t||^2_{L^2(\mathbb{H}^n)} + b(2\alpha - \omega -3) \int_{0}^t ||u_s(s)||^2_{L^2(\mathbb{H}^n)} ds\\
& + m(\alpha -2)||u||^2_{L^2(\mathbb{H}^n)} + (\alpha -2)||\nabla_{H} u||^2_{L^2(\mathbb{H}^n)} -2\alpha E(0) \\
& = (\alpha -\omega -1) \left[||u_t||^2_{L^2(\mathbb{H}^n)} + (b+1) ||u||^2_{L^2(\mathbb{H}^n)}\right] + (\alpha -2)||\nabla_{H} u||^2_{L^2(\mathbb{H}^n)} -2\alpha E(0) \\
&+ b(2\alpha - \omega -3) \int_{0}^t ||u_s(s)||^2_{L^2(\mathbb{H}^n)} ds + (m(\alpha -2 )- (b+1) (\alpha - \omega -1)|)||u||^2_{L^2(\mathbb{H}^n)} \\
& \geq (\alpha -\omega -1) \left[2{\rm Re}(u,u_t) + b ||u||^2_{L^2(\mathbb{H}^n)}\right] + (\alpha -2)||\nabla_{H} u||^2 _{L^2(\mathbb{H}^n)}-2\alpha E(0) \\
&+ b(2\alpha - \omega -3) \int_{0}^t ||u_s(s)||^2_{L^2(\mathbb{H}^n)} ds + (m(\alpha -2) - (b+1) (\alpha - \omega -1)|)||u||^2_{L^2(\mathbb{H}^n)}.
\end{align*}
In the second line that we have used \eqref{eq-EA}.
By selecting $\omega = \alpha-1 - \frac{m(\alpha -2)}{\mu+1}$ which satisfies $\omega>1$ since $\mu +1>m$ and using the argument from Step I, we obtain
\begin{align*}
\eta(t)& > \frac{m(\alpha -2)}{\mu+1} (2 {\rm Re}(u,u_t) + b ||u||^2_{L^2(\mathbb{H}^n)}) -2\alpha E(0)\\
&>\frac{m(\alpha -2)}{\mu+1} (2 {\rm Re}(u_0,u_1) + b ||u_0||^2_{L^2(\mathbb{H}^n)}) -2\alpha E(0) \\
& > \left(\frac{m(\alpha -2)}{\mu+1}\right) 2{\rm Re}(u_0,u_1) -2\alpha E(0)\\
& \geq 0,
\end{align*}
Note that we have used the fact $A'(t)>0$ and the expression \eqref{eq-A} with $A(t)=2 {\rm Re}(u,u_t) + b ||u||^2_{L^2(\mathbb{H}^n)}$, and the condition \eqref{eq-thm-cond} in the last line, respectively. So we obtain the inequality
\begin{equation*}
M''(t) M(t) - \frac{\omega+3}{4} (M'(t))^2 > 0.
\end{equation*}
Then
\begin{equation*}
\frac{d}{dt} \left[ \frac{M'(t)}{M^{\frac{\omega+3}{4}}(t)} \right] > 0 \Rightarrow \begin{cases}
M'(t) \geq \left[ \frac{M'(0)}{M^{\frac{\omega+3}{4}}(0)} \right] M^{\frac{\omega+3}{4}}(t),\\
M(0)=(bT_0+1)|| u_0||_{L^2(\mathbb{H}^n)}^2.
\end{cases}
\end{equation*}
Let us denote $\sigma = \frac{\omega -1}{4}$. Then we have
\begin{equation*}
-\frac{1}{\sigma} \left[ M^{-\sigma}(t) - M^{-\sigma}(0) \right] \geq \frac{M'(0)}{M^{\sigma +1}(0)} t,
\end{equation*}
that gives
\begin{align*}
M(t) \geq \left( \frac{1}{M^{\sigma}(0)} - \frac{\sigma M'(0)}{M^{\sigma+1}(0)}t \right)^{-\frac{1}{\sigma}}.
\end{align*}
Then the blow-up time $T^*$ satisfies
\begin{equation*}
0<T^* \leq \frac{M(0)}{\sigma M'(0)},
\end{equation*}
where $M'(0)= 2{\rm Re} (u_0,u_1)$. This completes the proof.
\end{proof}
\end{document} | math | 20,288 |
\begin{eqnarray}gin{document}
\parindent 0cm
\title{Designing dose finding studies with an active control for exponential families}
\author{
{\small Holger Dette, Katrin Kettelhake } \\
{\small Ruhr-Universit\"at Bochum } \\
{\small Fakult\"at f\"ur Mathematik } \\
{\small 44780 Bochum, Germany } \\
{\small e-mail: [email protected] }\\
\and
{\small Frank Bretz } \\
{\small Statistical Methodology } \\
{\small Novartis Pharma AG } \\
{\small 4002 Basel, Switzerland } \\
{\small e-mail: [email protected] }\\
}
\maketitle
\begin{eqnarray}gin{abstract}
In a recent paper \cite{detkisbenbre2014} introduced optimal design problems for dose finding studies with an active control. These authors concentrated on regression models with normal distributed errors (with known variance) and the problem of determining optimal designs for estimating the smallest dose, which achieves the same treatment effect as the active control.
This paper discusses the problem of designing active-controlled dose finding studies from a broader perspective. In particular, we consider a general class of optimality criteria and models arising
from an exponential family, which are frequently used analyzing count data.
We investigate under which circumstances optimal designs for dose finding studies including a placebo can be used to obtain optimal designs for studies with an active control. Optimal designs are constructed for several situations and the
differences arising from different distributional assumptions are investigated in detail.
In particular, our results are applicable for constructing optimal experimental designs to analyze active-controlled dose finding studies with discrete data,
and we illustrate the efficiency of the new optimal designs with two recent examples from our consulting projects.
\end{abstract}
Keywords and Phrases: optimal designs, dose response, dose estimation,
active control\\
\section{Introduction}\label{sec1}
\def5.\arabic{equation}{1.\arabic{equation}}
\setcounter{equation}{0}
Dose finding studies are an important tool to investigate the effect of a compound on a response of interest and have numerous applications in various fields such as medicine, biology or toxicology. They are of particular importance in pharmaceutical drug development because marketed doses have to be safe and provide clinically relevant efficacy [see \cite{ruberg1995,ting2006}].
Most of the literature on statistical methodology for analyzing
dose response studies include placebo as a control group [see \cite{pinbrebra2006,bretzetal2008}, among others]. Numerous authors have worked on the problem of determining optimal designs for dose response experiments with a placebo group because the application of efficient designs can substantially increase the accuracy of statistical analysis [see \cite{zhuwong2000,fedleo2001,kresmyfung2002,wufedpro2005,drahsupad2007,milguidet2007,borbredet2011}, among many others].\\
However, dose response studies including a marketed drug as an active control are becoming more popular, especially in preparation for an
active-controlled confirmatory non-inferiority trial where the use of placebo may be unethical.
Thus, considerable interest on active-controlled studies
has emerged, as documented through the release of several related guidelines by regulatory agencies
[see \cite{iche4}, \cite{emea2006, emea2011}, \cite{emea2005}].
Recently \cite{helbenfri2014} investigated the finite sample properties of maximum likelihood estimates of the target dose in an active-controlled study, which achieves the same treatment effect as the active control and \cite{helbenzinknefri2014} studied nonparametric estimates for this quantity.
Despite of these important applications, to our best knowledge, optimal design problems for active-controlled dose finding studies
have only been considered in one paper so far [\cite{detkisbenbre2014}].
These authors investigated optimal designs
for estimating the target dose under the assumption of
a normal distribution with known variances. In particular, they demonstrated the superiority of the optimal designs compared to standard designs
used in pharmaceutical practice. However, this
work is restricted to normal distributed responses with known variances and a special $c$-optimality criterion and obviously the designs derived in their paper are not
necessarily useful for other applications. Therefore the goal of the present paper is to investigate optimal design problems for dose finding studies with an active control from a more general perspective. A first objective is to consider a general class of optimality criteria.
Second, as it will be pointed out in the following paragraph, in many dose finding trials with an active control the assumption of normal distributed responses is hard to justify, and we consider
exponential families for modeling the distribution of the responses of the new drug and the active control. This allows in particular to design experiments for controlled studies with discrete data as they have appeared
in the consulting projects described in the next paragraph. Third, even if the assumption of a normal distribution is justifiable, we will demonstrate that the estimation of the variances has a nontrivial effect on the optimal designs for an active-controlled study.
The research in the present paper is motivated by two clinical trial examples where the assumption of normal distributed responses made by \cite{detkisbenbre2014} is hard to justify.
The first example refers to a $24$-week, dose-ranging, Phase II study in patients with gouty
arthritis to determine the target dose of a compound in preventing signs and symptoms of flares in chronic gout patients starting allopurinol
therapy.
The study population consists of male and female patients (age $18-80$ years) diagnosed with
chronic gout as defined by the American College of Rheumatology preliminary criteria (ACR) and willing to either initiate allopurinol therapy
or having just initiated allopurinol therapy within less than one month. Approximately 500 patients are screened in order to randomize 440
patients in approximately $100$ centers worldwide. Patients who meet the entry criteria are randomized to receive either the active
comparator or a specific dose of the new compound. The primary endpoint is the number of
flares occurring per subject within 16 weeks of randomization, which are modeled using a negative binomial distribution for all treatment arms,
{where the corresponding probability is modeled by a
dose-response relationship between the (single) dose groups of the new compound and by a constant parameter for the comparator.} \\
The second example is a Phase IIb, multicenter, randomized, double-blind, active-controlled
dose-finding study in the treatment of acute migraine,
as measured by the percentage of patients reporting pain freedom at two hours post-dose.
Approximately $500$ patients are randomized worldwide.
Patients who meet the entry criteria are randomized to receive
either the active comparator or one dose of the new compound.
Once the dose of the new compound is selected,
Phase III studies are conducted to evaluate further the efficacy and safety of the
new compound in the targeted patient population.
In Section \ref{sec2} we give an introduction to optimal design theory for models with an active control under general distributional assumptions. In particular we present results, which relate optimal designs for dose finding studies with a placebo group
to optimal designs for models with an active control. This methodology is used in Section \ref{sec3} to construct $D$-optimal designs for dose finding studies with an active control. In Section \ref{sec4} we consider the optimal design problem for estimating the smallest dose,
which achieves the same treatment effect as the active control. In both sections we investigate the effect of the distributional assumption on the resulting optimal designs.
In particular, we show that different distributional assumptions (as for example a normal or Poisson distribution used to model continuous or discrete data) leads
to substantial changes in the structure of the optimal designs.
The Appendix contains the proofs of our main results.
For the sake of brevity this paper is restricted to locally optimal designs
which require a-priori information about the unknown model parameters
[see \cite{chernoff1953}, \cite{fortorwu1992}, \cite{fangheda2008}]. These designs can
be used as benchmarks for commonly used designs. Moreover,
locally optimal designs serve as basis for constructing optimal designs with respect to more sophisticated optimality
criteria, which
are robust against a misspecification of the unknown parameters [see \cite{pronwalt1985} or \cite{chaver1995}, \cite{dette1997},
\cite{imhof2001} among others]. Following this line of research the methodology introduced in the present paper can be further developed to adress uncertainty in the preliminary information for the unknown parameters.
\section{Modeling active-controlled dose finding studies using exponential families} \label{sec2}
\def5.\arabic{equation}{2.\arabic{equation}}
\setcounter{equation}{0}
Consider a clinical trial, where patients are treated either with
an active control (a standard treatment administered at a fixed dose level) or with
a new drug using different dose levels in order to investigate the corresponding
dose response relationship.
Given a total sample size $N$, we thus allocate $n_1$
and $n_2=N-n_1$ patients to the two treatments. In addition, we determine the optimal number of different dose levels
for the new drug, the dose levels themselves and the optimal number of patients allocated to each dose level to obtain
the design of the experiment.
More formally, we assume that $k$ different dose levels, say $d_1, \ldots , d_k$, are chosen in a dose range, say $\mathcal{D} \subset \mathbb{R}^+_0$,
for the new drug (the optimal number
$k$ and the dose levels will be determined
by the choice of the design) and that at each dose level $d_i$ the experimenter can investigate $n_{1i}$ patients ($i=1,\ldots ,k$), where $n_1= \sum_{i=1}^k n_{1i}$
denotes the number of patients treated with the new drug.
The optimal numbers $n_{1i}$, more precisely the optimal proportions $n_{1i}/n_1$, will be determined by the choice of the design.
The corresponding responses at dose level $d_i$ are modeled as realizations of independent real valued
random variables $Y_{ij}$ ($j=1,\ldots , n_{1i}$, $i=1,\ldots , k$). Similarly, the responses of patients treated with the active control are modeled as realizations
of independent real valued random variables $Z_1, \ldots , Z_{n_2}$, where the two samples corresponding to the new drug and active control are assumed to be independent.
For the statistical analysis we further assume
that the random variables $Z_j$ and $Y_{ij}$
have distributions from an exponential family, where the distributions of the latter depend on the corresponding dose levels $d_i$, that
is
\begin{eqnarray}gin{eqnarray}
\label{exp1} f_1(y|d_i, \theta_1) & := & \frac {\partial P_{\theta_1} ^{Y_{i1}}}{\partial \nu} (y) = \exp \{ c^T_1(d_i,\theta_1)T_1(y)-b_1(d_i,\theta_1) \} h_1 (y) ,\\
\label{exp2} f_2(z|\theta_2) & := & \frac {\partial P_{\theta_2}^{Z_1}}{\partial \nu} (z) = \exp \{ c^T_2 (\theta_2)T_2(z)-b_2(\theta_2)\} h_2(z) .
\end{eqnarray}
Here $\nu $ denotes a $\sigma$-finite measure on the real line, $\theta_1 \in\Theta_1 \subset \mathbb{R}^{s_1}, \ \theta_2 \in \Theta_2 \subset \mathbb{R}^{s_2}$ are unknown parameters
and we use common terminology for exponential families [see for example \cite{brown1986}]. In particular, the functions $c_1: \mathcal{D} \times \Theta_1 \to \mathbb{R}^{\ell_1}, \ b_1: \mathcal{D} \times \Theta_1 \to \mathbb{R} , \ c_2: \Theta_2 \to \mathbb{R}^{\ell_2}$ and $b_2: \Theta_2 \to \mathbb{R}$ are assumed to be twice continuously differentiable where $\tfrac{\partial c_1}{\partial \theta_1}, \tfrac{\partial c_2}{\partial \theta_2} \neq 0$ and $T_1$ and $T_2$ denote $\ell_1$- and $\ell_2$- dimensional statistics defined on the corresponding sample spaces. Additionally, the functions $h_1$ and $h_2$ are assumed to be positive (and measurable).
Throughout this paper let $\kappa$ be a variable indicating whether a patient receives the new drug $(\kappa=0)$ or
the active control ($\kappa=1$) and denote
\begin{eqnarray}gin{equation} \label{desspace}
\mathcal{X} = (\mathcal{D} \times \{ 0 \}) \cup \{ (C,1) \}
\end{equation}
as the design space of the experiment, where $\mathcal{D}$ is the dose
range for the new drug, $C$ the dose level of the active control and the second component of an experimental
condition $(d,\kappa) \in \mathcal{X}$ determines the treatment $(\kappa = 0, 1)$.
Straightforward calculation shows that the Fisher information at the point
$(d,\kappa)\in \mathcal{X}$ is given by the matrix
\begin{eqnarray}gin{equation} \label{finfo}
I( (d,\kappa) ,\theta) =
\begin{eqnarray}gin{pmatrix} I\{ \kappa =0\} I_1(d,\theta_1) & \mathbf{0} \\
\mathbf{0} & I\{ \kappa =1\} I_2(\theta_2)
\end{pmatrix},
\end{equation}
where $\mathbf{0 } $ denotes a matrix of appropriate dimension with all entries equal to $0$, $\theta=(\theta_1^T,\theta_2^T)^T \in \Theta_1 \times \Theta_2 \subset \mathbb{R}^{s_1+s_2}$ is the vector of all parameters, $I\{ \kappa =0\}$ is the indicator function of the event $\{ \kappa =0\} $ and the matrices $I_1$ and $I_2$ are the Fisher information matrices of the two models \eqref{exp1} and \eqref{exp2}, that is
\begin{eqnarray}gin{eqnarray} \label{i2}
I_1(d,\theta_1) &=& \mathbb{E} \mathcal{B}igl [ \mathcal{B}igl ( \frac {\partial}{\partial \theta_1} \log f_1 (Y | d,\theta_1) \mathcal{B}igl ) \mathcal{B}igl( \frac{\partial}{\partial \theta_1} \log f_1 (Y_{i1} | d_i,\theta_1) \mathcal{B}igr)^T \mathcal{B}igr] \nonumber \\
I_2(\theta_2) &=& \mathbb{E} \mathcal{B}igl [ \mathcal{B}igl ( \frac {\partial}{\partial \theta_2} \log f_2 (Z | \theta_2) \mathcal{B}igl ) \mathcal{B}igl( \frac{\partial}{\partial \theta_2} \log f_2 (Z_{1} | \theta_2) \mathcal{B}igr)^T \mathcal{B}igr].
\end{eqnarray}
where the random variables $Y$ and $Z$ have densities $f_1(y|d,\theta_1)$ and $f_2(z|\theta_2) $ defined by \eqref{exp1} and \eqref{exp2}, respectively.
{Note that the Fisher information in \eqref{finfo} is block diagonal because of the independence of the samples, as different
patients are either treated with the new drug or the active control. The following examples illustrate the general terminology.}
\begin{eqnarray}gin{Example} \label{examp1}{\rm In order to demonstrate the different structures of the Fisher information
arising from different distributions of the exponential family we consider several examples.
\begin{eqnarray}gin{itemize}
\item[(a)] \cite{detkisbenbre2014} investigated normal distributed responses with known variances
$\sigma_1^2$ and $\sigma_2^2$ for the new drug and the active control, respectively.
For the expectation of the response of the new drug at dose level $d$ they assumed a nonlinear regression model,
say $\eta (d, \vartheta)$, where $ \vartheta = (\vartheta_0,\dots,\vartheta_s)$,
while it is assumed to be equal to $\mu $ for the active control.
If the variances are not known and have to be estimated from the data,
we have $\theta_1=(\vartheta_0,\dots,\vartheta_s,\sigma^2_1), \ \theta_2 =(\mu, \sigma^2_2)$ for the parameters in models \eqref{exp1} and \eqref{exp2}, respectively.
Standard calculations show that the Fisher information at a point $(d, \kappa) \in \mathcal{X}$ is given by
\eqref{finfo}, where
\begin{eqnarray}gin{eqnarray} \label{finfnorm1}
I_1 (d,\theta_1 )
&=& \begin{eqnarray}gin{pmatrix} \tfrac{1}{\sigma_1^2}(\tfrac{\partial}{\partial \vartheta} \eta(d,\vartheta)) (\tfrac{\partial}{\partial \vartheta} \eta(d,\vartheta))^T
& \bold{0 }
\\\bold{0 } & \frac{1}{2\sigma_1^4}
\end{pmatrix}, ~
I_2 (\theta_2) = \begin{eqnarray}gin{pmatrix}
\frac{1 }{\sigma_2^2} & 0 \\
0 & \frac{ 1 }{2\sigma_2^4} \end{pmatrix}. ~~~~~~
\end{eqnarray}
\item[(b)] As motivated by the examples in Section~\ref{sec1}, it might be more reasonable to consider a different distribution than a normal distribution
in \eqref{exp1} and \eqref{exp2} to model discrete data.
Assume, for example, a
negative binomial distribution with parameter $r_1 \in \mathbb{N}$ for the number of failures and a function
$\pi(d,\theta_1) \in (0,1)$ for the probability of a success of the new drug (at dose level $d$) and
parameters $r_2 \in \mathbb{N}, \mu \in (0,1) $ for the active control. Then we have $\theta_2=\mu$ and the Fisher information matrix is given by \eqref{finfo},
where
\begin{eqnarray}gin{eqnarray} \label{finfnbin1}
I_1 (d,\theta_1 )
&=& \frac{ r_1 (\tfrac{\partial}{\partial \theta_1} \pi(d,\theta_1))(\tfrac{\partial}{\partial \theta_1} \pi(d,\theta_1))^T}{\pi^2(d,\theta_1) (1-\pi(d,\theta_1 ))} ~, ~
I_2 (\theta_2 ) =
\frac{ r_2}{ \mu^2(1- \mu) }. ~~~~
\end{eqnarray}
Here the parameters $r_1, r_2 \in \mathbb{N}$ for the number of failures are assumed to be known.
\item[(c)] Alternatively, in the case of a binary response we may use a Bernoulli distribution,
where $\pi(d,\theta_1) \in (0,1)$ and $ \mu \in (0,1)$ denote the probability of success for the new drug and the active control,
respectively. In this case we have $\theta_2=\mu$ and the Fisher information matrix is given by \eqref{finfo}, where
\begin{eqnarray}gin{eqnarray*}
I_1 (d,\theta_1) &=& \frac { ( \tfrac {\partial}{\partial \theta_1} \pi(d,\theta_1) ) ( \tfrac {\partial}{\partial \theta_1} \pi(d,\theta_1) )^T}{\pi(d,\theta_1)(1-\pi(d,\theta_1))} ~,~
I_2 (\theta_2) = \frac {1}{\mu(1-\mu)}.
\end{eqnarray*}
\item[(d)]
For a Poisson distribution, the parameters for the distribution of the responses corresponding to the new drug and the active control
are given by a function of the dose level, say $ \lambda (d, \theta_1) >0 $, and a parameter $\mu > 0 $, respectively.
In this case we have $\theta_2=\mu$ and the Fisher information matrix is given by \eqref{finfo}, where the two non-vanishing blocks are defined by
\begin{eqnarray}gin{eqnarray} \label{finfpoiss1}
I_1 (d,\theta_1 )
&=& \frac{ (\tfrac{\partial}{\partial \theta_1}
\lambda(d,\theta_1)(\tfrac{\partial}{\partial \theta_1} \lambda(d,\theta_1))^T }{\lambda(d,\theta)} ~,~~
I_2 (\theta_2 ) =
\frac {1 }{\mu} . \label{finfpoiss2}
\end{eqnarray}
\end{itemize}
}
\end{Example}
Throughout this paper we consider approximate designs in the sense
of \cite{kiefer1974}, which are defined as probability measures with finite support on the design space $\mathcal{X}$ in \eqref{desspace}. Therefore,
an experimental design is given by
\begin{eqnarray}gin{equation}\label{desall}
\xi=\begin{eqnarray}gin{pmatrix} (d_1,0)& \dots &(d_k,0) &(C,1)\\
w_1&\dots&w_k&w_{k+1}
\end{pmatrix},
\end{equation}
where $w_1,\dots,w_{k+1}$ are positive weights, such that $\sum^{k+1}_{i=1}w_i=1$.
Here, $w_i$ denotes the relative proportion of patients treated at dose level $d_i \ (i=1, \dots, k)$ or the active control $(i=k+1)$.
If $N$ observations can be taken, a rounding procedure
is applied
to obtain integers $n_{1i} $ ($i=1,\ldots,k)$ and $n_2$ from the not necessarily integer valued quantities
$w_iN$ ($i=1,\ldots, k+1$) [see \cite{pukrie1992}]. Thus, the experimenter assigns $n_{11}, \ldots, n_{1k}$ and $n_2$ patients to the dose levels $d_1, \ldots d_k$ of the new drug and the active control, respectively.
In the following discussion we will determine optimal designs, which also optimize the number $k$ of different dose levels. It turns out that for the models considered here the optimal designs usually allocate observations at less than $5$ dose levels. Note that in practice the number $k$ of different dose levels is in the range of $4-7$ and rarely larger than $10$.
The information matrix of an approximate design $\xi$ of the form \eqref{desall} is
defined by the $(s_1 + s_2) \times (s_1+s_2 )$ matrix
\begin{eqnarray}gin{align} \label{inf}
M(\xi,\theta)&=\int_{\mathcal{X}} I((d,\kappa) , \theta) d\xi(d,\kappa) = \begin{eqnarray}gin{pmatrix} (1-\omega_{k+1}) M_1(\tilde \xi, \theta_1) & \bold{0} \\ \bold{0} & \omega_{k+1}
I_2 (\theta_2). \end{pmatrix}
\end{align}
Here, the $s_1\times s_1$ matrix $ M_1(\tilde \xi, \theta_1)$ and the $s_2 \times s_2$ matrix $ I_2(\theta_2)$ are
given by
\begin{eqnarray}gin{eqnarray}\label{tildea}
M_1 (\tilde \xi,\theta_1) &=& \int_{\mathcal{D}} I_1(d,\theta_1) d \tilde \xi(d) ,
\end{eqnarray}
and \eqref{i2}, respectively, and
\begin{eqnarray}gin{equation}\label{tilde}
\tilde{\xi}=\begin{eqnarray}gin{pmatrix} d_1&\dots&d_k\\\tilde w_1 &\dots&\tilde w_k \end{pmatrix}
\end{equation}
denotes the design (on the design space $\mathcal{D}$) for the new drug, which is induced by the design $\xi$ in \eqref{desall} defining the weights $ \tilde w_i =\frac{w_i}{1-w_{k+1}}$, $i=1, \ldots, k$.
If observations are taken according to an approximate design it can be shown (assuming standard regularity conditions) that the maximum likelihood estimators $\hat \theta_1, \hat\theta_2$ in models \eqref{exp1} and \eqref{exp2} are asymptotically normal distributed, that is
$$
\sqrt{N} \big( (\hat \theta_1^T, \hat \theta_2^T)^T - (\theta^T_1, \theta^T_2) ^T \big) \stackrel{ \mathcal{D}}{\longrightarrow} \mathcal{N} (\bold{0}, M^{-1}(\xi,\theta))
$$
as $N {\longrightarrow}\infty$,
where the symbol $\stackrel{ \mathcal{D}}{\longrightarrow}$ denotes convergence in distribution. \cite{debrpepi2008} considered dose finding studies including a placebo group and
showed by means of a simulation study that the approximation of the variance of $\hat \theta = (\hat \theta_1^T, \hat \theta_2^T)^T $ by $\tfrac{1}{N}M^{-1}(\xi,\theta)$ is satisfactory for total sample sizes larger than $25$. As typical clinical dose finding trials have sample sizes in the range of
$200 - 300$ [see for example \cite{phrma:2007}], it is reasonable to use this approximation also for active-controlled studies. Consequently, optimal designs maximize an appropriate functional of the information matrix defined in \eqref{inf}.
In order to discriminate between competing designs we consider in this paper Kiefer's $\phi_p$-criteria [see \cite{kiefer1974} or \cite{pukelsheim2006}]. To be precise, let $ p \in [-\infty,1)$ and $ K \in \mathbb{R}^{(s_1 + s_2) \times t}$ denote a matrix of full column rank $t$. Then a design $\xi^*$ is called locally $\phi_p$-optimal for estimating the linear combination $K^T \theta$
in a dose response model with an active control, if $K^T \theta$ is estimable by the design $\xi^*$, that is, $K^T \theta \in$ Range$(M(\xi^*,\theta))$, and $\xi^*$ maximizes the functional
\begin{eqnarray}gin{equation} \label{crit}
\phi_p(\xi) = \mathcal{B}igl(\frac {1}{t} \mbox{tr} (K^T M^- (\xi,\theta)K)^{-p} \mathcal{B}igr)^{\frac {1}{p}}
\end{equation}
among all designs for which $K^T \theta$ is estimable, where tr$(A)$ and $A^-$ denote the trace and a generalized inverse of the matrix $A$, respectively.
Note that the cases $p=0$ and $p=-\infty $ correspond to the $D$- and $E$-optimality criterion, that is
$ \phi_0(\xi) = \det (K^T M^- (\xi,\theta)K )^{-\tfrac{1}{t}}$ and $ \phi_{-\infty} (\xi) = \lambda_{\min} ((K^T M^- (\xi,\theta)K) ^{-1})$.
An application of the general equivalence theorem [see \cite{pukelsheim2006}, chapter 7.19 and 7.21, respectively] to the situation considered in this paper yields immediately the following result.
\begin{eqnarray}gin{Lemma}\label{lem1} If $p \in (-\infty , 1)$, a
design $\xi^*$ with $K^T \theta \in$ Range$(M(\xi^*,\theta))$ is locally $\phi_p$-optimal for estimating the linear combination $K^T \theta$ in a dose response model with an active control if and only if there exists a generalized inverse $G$ of the information matrix $M(\xi^*,\theta)$, such that the inequality
{\small{
\begin{eqnarray}gin{equation}\label{aequ}
{ \rm tr} \bigl( I((d,\kappa),\theta)GK(K^TM^-(\xi^*,\theta)K)^{-p-1} K^TG^T\bigr) - { \rm tr} (K^TM^-(\xi^*,\theta)K)^{-p} \leq 0
\end{equation}
}}
holds for all $(d,\kappa ) \in \mathcal{X}$.
If $p = -\infty $, a design $\xi^*$ with $K^T \theta \in$ Range$(M(\xi^*,\theta))$ is
locally $\phi_{-\infty}$-optimal for estimating the linear combination $K^T \theta$
if and only if there exist a generalized inverse $G$ of the information matrix $M(\xi^*,\theta)$ and a nonnegative definite matrix $E \in \mathbb{R}^{t \times t}$ with $\mbox{\rm tr} (E) =1$,
such that the inequality
{\small{
\begin{eqnarray}gin{equation}\label{aequ1}
{ \rm tr} \bigl( I((d,\kappa),\theta)GK(K^TM^-(\xi^*,\theta)K)^{-1} E (K^TM^-(\xi^*,\theta)K)^{-1} K^TG^T\bigr) - \lambda_{\min}((K^TM^-(\xi^*,\theta)K)^{-1}) \leq 0
\end{equation}
}}
holds for all $(d,\kappa ) \in \mathcal{X}$.
Moreover, there is equality in \eqref{aequ} ($p> - \infty$) and \eqref{aequ1} ($p=-\infty$) for all support points of the design $\xi^*$.
\end{Lemma}
In the following discussion we assume that either $p=-1$ or that the matrix $K$ is a block matrix of the form
\begin{eqnarray}gin{equation}\label{kdiag}
K= \left( \begin{eqnarray}gin{array}{cc}
K_{11} & 0 \\ 0 & K_{22}
\end{array} \right) \in \mathbb{R}^{(s_1 + s_2) \times (t_1 + t_2)}
\end{equation}
with elements $K_{11} \in \mathbb{R}^{s_1 \times t_1} , \ K_{22} \in \mathbb{R}^{s_2 \times t_2}$, $t_1+t_2=t$. Roughly speaking, the choice $p=-1$ or a blockdiagonal structure of the matrix $K$ in \eqref{kdiag}
leads to a separation of the parameters from models \eqref{exp1} and \eqref{exp2} in the corresponding optimality criterion. As a consequence
optimal designs for dose finding studies with an active control can be obtained from
optimal designs for dose finding studies including a placebo group, which maximize the criterion
\begin{eqnarray}gin{equation} \label{crittilde}
\tilde \phi_p (\tilde \xi) = \mathcal{B}igl( \frac {1}{t_1} {\rm tr} (K^T_{11}M^-_1(\tilde \xi,\theta_1)K_{11})^{-p}\mathcal{B}igr)^{\frac {1}{p}}
\end{equation}
in the class of all designs $\tilde \xi$ for which $K^T_{11}\theta_1$ is estimable, i.e. $K_{11}^T \theta_1 \in \mbox{Range}( M_1(\tilde \xi,\theta_1))$. Throughout this paper these designs are called $\tilde \phi_p$-optimal for estimating the parameter $K_{11}^T\theta_1$ in the dose response model \eqref{exp1}. The proof can be found in the Appendix.
\begin{eqnarray}gin{Theorem} \label{thm1}
Assume $p \in [-\infty , 1)$, that the matrix $K$ is given by \eqref{kdiag} and that
\begin{eqnarray}gin{eqnarray} \label{tilde}
\tilde \xi^*_p= \left( \begin{eqnarray}gin{array}{ccc}
d^*_1 & \dots & d^*_k \\
\tilde w^*_1 & \dots & \tilde w^*_k
\end{array} \right)
\end{eqnarray}
is a locally $\tilde \phi_p$-optimal design for estimating $K^T_{11}\theta_1$ in the dose response model \eqref{exp1}. Then the design
\begin{eqnarray}gin{eqnarray*}
\xi^*_p = \left ( \begin{eqnarray}gin{array} {cccc}
(d^*_1,0) & \dots & (d^*_k,0) & (C,1) \\
w^*_1 & \dots & w^*_k & w^*_{k+1}
\end{array}\right )
\end{eqnarray*}
is locally $\phi_p$-optimal for estimating $K^T\theta$ in the dose response model with an active control, where the weights are given by
\begin{eqnarray}gin{equation} \label{weight}
w^*_{k+1} = \frac {1}{1+\rho_p} , \qquad w^*_i = \frac {\rho_p}{1+ \rho_p} \tilde w_i^* \qquad (i=1,\dots,k),
\end{equation}
and
\begin{eqnarray}gin{equation} \label{rho}
\rho_p = \frac {({\rm tr}((K^T_{22}I^-_2(\theta_2)K_{22})^{-p}))^{1/(p-1)}}
{({\rm tr}((K^T_{11}M^-_1(\tilde \xi^*_p, \theta_1)K_{11})^{-p}))^{1/(p-1)}}
\end{equation}
(the case $p=-\infty$ is interpreted as the corresponding limit).
\end{Theorem}
In the case $p=-1$ a more general statement is available without the restriction to block matrices of the form \eqref{kdiag}. The proof is obtained by similar arguments as presented in the proof of Theorem \ref{thm1} and therefore omitted.
\begin{eqnarray}gin{Theorem} \label{thm2}
Assume that $K^T=(K^T_{11}, K^T_{22}) \in \mathbb{R}^{t \times (s_1+s_2)}$ with $K^T_{11} \in \mathbb{R}^{t \times s_1}, K^T_{22} \in \mathbb{R}^{t \times s_2}$ and let $\tilde \xi_{-1}^*$ denote the $\tilde \phi_{-1}$-optimal design for estimating the parameter $K_{11}^T \theta_1$ in the dose response model \eqref{exp1}. Then the design $\xi^*_{-1}$ defined in Theorem \ref{thm1} is locally $\phi_{-1}$-optimal for estimating $K^T \theta$ in the dose response model with an active control.
\end{Theorem}
The final result of this section considers the special case $p=0$. The result is a direct consequence of Theorem \ref{thm1} considering the limit $p \to 0$ and observing that the quantity $\rho_p$ defined in $\eqref{rho}$ satisfies
$
\lim_{p \to 0} \rho_p = \tfrac{t_1}{t_2}.
$
\begin{eqnarray}gin{Corollary} \label{cor1}
Assume that the matrix $K$ is given by \eqref{kdiag} and let $\tilde \xi_0^*$ denote the locally $D$-optimal design of the form \eqref{tilde} for estimating the parameter $K^T_{11} \theta_1$ in the dose response model \eqref{exp1}, which maximizes $\det((K_{11}^T M_1^-(\tilde \xi,\theta_1)K_{11})^{-1})$ in the class of all designs for which $K_{11}^T \theta_1$ is estimable. Then the design
\begin{eqnarray}gin{equation}
\xi_\theta^* = \left ( \begin{eqnarray}gin{array}{cccc}
(d^*_1,0) & \dots & (d^*_k,0) & (C,1)
\\
\tfrac{t_1}{t_1+t_2} \tilde w^*_1 & \dots & \tfrac{t_1}{t_1+t_2} \tilde w_k^* & \tfrac{t_2}{t_1+t_2} \end{array} \right )
\end{equation}
is locally $D$-optimal for estimating the parameter $K^T \theta$ in the dose response model with an active control.
\end{Corollary}
\begin{eqnarray}gin{Remark} {\rm The assumption of a block matrix $K$ in Theorem \ref{thm1} and Corollary \ref{cor1} can not be omitted.
Consider for example the case of a binomial distribution and a Michaelis-Menten model $\pi(d,\theta_1)=\tfrac{\vartheta_1 d}{\vartheta_2 +d}$
for the dose response relationship of the new drug. Assume that one is interested in two functionals of the model parameters:
(i) The estimation of the distance between the effect of the active control, say $\theta_2$, and the effect of the new drug at a special dose level $d_0$, i.e. $\pi (d_0,\theta_1)$,
and (ii) the difference between $\theta_2$ and the maximum effect of the new drug, i.e. $\vartheta_1$.
In this case the matrix $K$ is given by
$$
K=\begin{eqnarray}gin{pmatrix} -\tfrac{d_0}{\vartheta_2+d_0} & \tfrac{\vartheta_1 d_0}{(\vartheta_2+d_0)^2}& 1 \\ -1 & 0 & 1 \end{pmatrix}^T.
$$
Consider exemplarily the choice $\mathcal{D}=[0,50]$, $\vartheta_1=0.5$, $\vartheta_2=2$, $\theta_2=0.4$ and $d_0=5$.
The locally $D$-optimal design for estimating
$K^T\theta$ in the dose response model with an active control
allocates $36\%$ and $32\%$ of the patients to the dose levels $0.93$ and $50$ of the new drug and $32\%$ of the patients to the active control, respectively. The corresponding function
\eqref{aequ}
of the equivalence theorem is shown in the left panel of Figure \ref{equAC} for the case $\kappa=0$.
In the case $\kappa=1$ this function reduces to the constant $0$ for all $d \in \mathcal{D}$.
\begin{eqnarray}gin{figure}
\tiny
\centering
\subfigure{\includegraphics[scale=0.4]{equi_geg_AC_kappa02.pdf}}
~~
\subfigure{\includegraphics[scale=0.4]{equi_geg_nd2.pdf}}
~
\subfigure{\includegraphics[scale=0.4]{equiv3}}
\caption{\it The inequality \eqref{aequ} of the equivalence theorem. Left panel: $D$-optimal design for estimating $K^T\theta$ in the dose response model with an active control. Middle
panel and right panel: optimal design for prediction and $D$-optimal design in the model \eqref{exp1}.}
\label{equAC}
\end{figure}
In the situation where no active control is available one could look at the
problem designing the experiment for a most efficient {estimation} of
$\pi (d_0,\theta_1)$. This corresponds to the
matrix $K_{11}=(\tfrac{d_0}{\vartheta_2+d_0} , -\tfrac{\vartheta_1 d_0}{(\vartheta_2+d_0)^2})^T$ and
the locally optimal design is a one point design which treats $100\%$ of the patients with the dose level $d_0=5$.
On the other hand the locally $D$-optimal design for estimating $\theta_1$ allocates
$50\%$ of the patients to each of the dose levels $1.15$ and $50$.
The corresponding inequalities of the equivalence theorem are shown in the middle and right panel of Figure \ref{equAC}.
Obviously the locally $D$-optimal design for estimating
$K^T\theta$ in the dose response model with an active control can not be derived from these designs and an assumption of the type \eqref{kdiag} is in fact necessary to obtain Theorem \ref{thm1}.
}
\end{Remark}
\section{$D$-optimal designs for the Michaelis-Menten and EMAX model}
\label{sec3}
\def5.\arabic{equation}{3.\arabic{equation}}
\setcounter{equation}{0}
In this section we determine some $D$-optimal designs for dose finding studies with an active control under different distributional assumptions. We assume that the dependence on the dose of the new drug is either described by the Michaelis-Menten model
$\frac {\vartheta_1d}{\vartheta_2+d},$
or the EMAX model
$\vartheta_0 + \frac {\vartheta_1d}{\vartheta_2+d},$
where the dose range is given by the interval $[L,R] \subset \mathbb{R}^+_0$. These models are widely used
when investigating the dose response relationship
of a new compound, such as a medicinal drug, a fertilizer, or an environmental toxin.
Note that in the case where the function describes a probability, one requires some restrictions on the parameters. For example, if $\pi(d,\theta_1)= \frac {\vartheta_1d}{\vartheta_2+d}$ is the probability of a success for the negative binomial distribution in Example \ref{examp1}(b), we implicitly assume $\frac {\vartheta_1R}{\vartheta_2+R}<1$ in the following discussion. In other models similar assumptions have to be made and we do not mention these restrictions explicitly for the sake of brevity. In the following, $x \vee y$ denotes the maximum of $x, y \in \mathbb{R}$.
\begin{eqnarray}gin{Theorem} (Michaelis-Menten model) \label{thm31a}
\begin{eqnarray}gin{itemize}
\item[(a)]
If the distributions of the responses corresponding to the new drug and active control are normal with parameters $(\frac {\vartheta_1d}{\vartheta_2+d}, \sigma^2_1)$ and $(\mu, \sigma^2_2)$, respectively, then the locally $D$-optimal design for the dose response model with an active control allocates $30\%$ of the patients to each of the dose levels $L \vee \frac {\vartheta_2R}{2\vartheta_2+R}$ and $R$ of the new drug and $40\%$ to the active control.
\item[(b)]
In the case of negative binomial distributions with probabilities $\pi(d,\theta)=\frac {\vartheta_1d}{ \vartheta_2+d}$ and $\mu$ the locally $D$-optimal design for the dose response model with an active control allocates $33.\overline{3}\%$ of the patients to each of the dose levels $L$ and $R$ of the new drug and $33.\overline{3}\%$ to the active control.
\item[(c)]
In the case of binomial distributions with probabilities $\pi(d,\theta)=\frac {\vartheta_1d}{ \vartheta_2+d}$ and $\mu$ the locally $D$-optimal design for the dose response model with an active control allocates $33.\overline{3}\%$ of the patients to each of the dose levels $L \vee \tfrac{\vartheta_2 R + 3 \vartheta_2^2-\vartheta_2 \sqrt{9 R^2 - 8R^2\vartheta_1+18R\vartheta_2-8R\vartheta_1\vartheta_2+9\vartheta_2^2}}{4\vartheta_1\vartheta_2-4R+4R\vartheta_1-6\vartheta_2}$ and $R$ of the new drug and $33.\overline{3}\%$ to the active control.
\item[(d)]
If Poisson distributions with parameters $\lambda(d,\theta_1)= \frac {\vartheta_1d}{\vartheta_2+d}$ and $\mu$ are used in \eqref{exp1} and \eqref{exp2}, the locally $D$-optimal design for the dose response model with an active control allocates $33.\overline{3}\%$ of the patients to each of the dose levels $L \vee \frac {\vartheta_2R}{3\vartheta_2+2R}$ and $R$ of the new drug and $33.\overline{3}\%$ to the active control.
\end{itemize}
\end{Theorem}
The proof of Theorem \ref{thm31a} is a direct consequence of Corollary \ref{cor1}, if the locally $D$-optimal designs for model \eqref{exp1} are known. For example, in the case of a normal distribution it follows from \cite{rasch1990} that the $D$-optimal design for the Michaelis-Menten model has equal masses at the points $L \vee \frac {\vartheta_2R}{2\vartheta_2+R}$ and $R$ and Corollary \ref{cor1} yields part $(a)$ of Theorem \ref{thm31a}. In the other cases the $D$-optimal designs for model \eqref{exp1} are not known and the proof can be found in the Appendix. \\
{ It is also worthwhile to note that the differences of the $D$-optimal designs
derived under different distributional assumptions can be substantial. For example if the design space is $[0,R]$ with a large
right boundary $R$, the non trivial dose level for the new drug is approximately $\vartheta_2$
and $\vartheta_2/2$ under the assumption of a normal and Poisson distribution, respectively.} We will now give the corresponding results for the EMAX model.
The proof follows by similar arguments as given in the proof of Theorem \ref{thm31a} and is therefore omitted.
\begin{eqnarray}gin{Theorem} (EMAX-model) \label{thm32a}
\begin{eqnarray}gin{itemize}
\item[(a)]
If the distribution of responses corresponding to the new drug and active control are normal distributions with parameters $(\vartheta_0 + \frac {\vartheta_1d}{\vartheta_2+d}, \sigma^2_1)$ and $(\mu, \sigma^2_2)$, respectively, then the locally $D$-optimal design for the dose response model with an active control allocates $22.\overline{2}\%$ of the patients to each of the dose levels $L$, $d^*=\tfrac{R(L+\vartheta_2) + L(R+\vartheta_2)}{(L+\vartheta_2)+(R+\vartheta_2)}$ and $R$ of the new drug and $33.\overline{3}\%$ to the active control.
\item[(b)]
In the case of negative binomial distributions with probabilities $\pi(d,\theta)=\vartheta_0 + \frac {\vartheta_1d}{\vartheta_2+d}$ and $\mu$, the locally $D$-optimal design for the dose response model with an active control allocates $25\%$ of the patients to each of the dose levels $L$, $d^*$ and $R$ of the new drug
and $25\%$ to the active control, where $d^*$ is the solution of the equation $$
\tfrac{2}{d-L}+\tfrac{2}{d-R}-\tfrac{\vartheta_0+\vartheta_1-1}{d (\vartheta_0+\vartheta_1-1)+(\vartheta_0-1) \vartheta_2}-\tfrac{2 (\vartheta_0+\vartheta_1)}{\vartheta_0 (\vartheta_2+d)+\vartheta_1 d}-\tfrac{1}{\vartheta_2+d}=0$$
\item[(c)]
In the case of binomial distributions with probabilities $\pi(d,\theta)=\vartheta_0 + \frac {\vartheta_1d}{\vartheta_2+d}$ and $\mu$, the locally $D$-optimal design is of the
same form as described in part (b), where $d^*$ is the solution of the equation
$$\tfrac{2}{d-L}+\tfrac{2}{d-R}-\tfrac{\vartheta_0+\vartheta_1-1}{d (\vartheta_0+\vartheta_1-1)+(\vartheta_0-1) \vartheta_2}-\tfrac{\vartheta_0+\vartheta_1}{\vartheta_0 (\vartheta_2+d)+\vartheta_1 d}-\tfrac{2}{\vartheta_2+d}=0.$$
\item[(d)]
If Poisson distributions with parameters $\lambda(d,\theta_1)= \vartheta_0 + \frac {\vartheta_1 d}{\vartheta_2+d}$ and $\mu$ are used in \eqref{exp1} and \eqref{exp2}, respectively, then the locally $D$-optimal is
of the same form as described in part (b), where
$$
d^*=\vartheta_2 \tfrac {4m(L)m(R)-\vartheta_1 (Lm(R)+Rm(L))-\vartheta_0 \sqrt{\kappa}}{-4m(L)m(R)- \vartheta_1 \vartheta_2 (m(R)+m(L))+(\vartheta_1 + \vartheta_0) \sqrt{\kappa}}$$
and $\kappa = ( (\vartheta_2+L)m(R)+(\vartheta_2+R)m(L))^2 + 12 (\vartheta_2+L)(\vartheta_2+R)m(R)m(L)$, $m(d)=\vartheta_0\vartheta_2+\vartheta_1d+\vartheta_0d$.
\end{itemize}
\end{Theorem}
\begin{eqnarray}gin{Example}{\rm
Under the assumption of a normal distribution, \cite{detkisbenbre2014} determined a $D$-optimal design for the EMAX dose response model ignoring the effect caused by estimating the variance. It follows from Theorem 4 in \cite{detkisbenbre2014} that the locally $D$-optimal design allocates $26.\overline{6}\%$ of the patients to the dose levels $L, d^*, R$ of the new drug and $20\%$ to the active control, respectively, where $d^*$ is defined in 3.2(a). Theorem \ref{thm32a} above shows that the design which accounts for the problem of estimating the variances uses the same
dose levels but allocates $ 13.3\%$ more patients to the active control. }
\end{Example}
\begin{eqnarray}gin{Example} \label{dataex}{\rm
In this example we discuss D-optimal designs for the two clinical trials considered in Section~\ref{sec1}.
\begin{eqnarray}gin{itemize}
\item[(a)]
We first consider the gouty arthritis example. The primary endpoint is modeled by a negative binomial distribution with parameters $r_1$ and $\pi(d,\theta_1) = \vartheta_0+\tfrac{\vartheta_1 d}{\vartheta_2 + d}$ for the new drug and parameters $r_2$ and $\theta_2$ for the comparator. The dose range is $[0,300]$mg
and we obtained from the clinical team the following preliminary information
for the unknown parameters: $\vartheta_0=0.26$, $\vartheta_1=0.73$, $\vartheta_2=10.5$, $\sigma_1=0.05$ and $\theta_2=0.9206$, $\sigma_2=0.05$. In addition, $r_1=r_2=10$ are fixed.
The $D$-optimal design is obtained from Theorem \ref{thm32a} and depicted in the upper part of Table \ref{tabDeffg}. It
allocates $25\%$ of all patients to the active control and $25\%$ of the patients to the dose levels $0, 8.23, 300$mg of the new drug, respectively.
The standard design actually used in this study allocates $14.3\%$ of the patients to the dose levels $25,50,100,200,300$mg of the new drug and $28.5\%$ of the patients to the active control.
To compare these designs we also show in the last column of Table \ref{tabDeffg}
the D-efficiency
\begin{eqnarray}gin{equation}
\mathrm{eff}_D(\xi,\theta)=\frac{\Phi_0 (\xi,\theta)}{\Phi_0(\xi_{D}^{*},\theta)} \in [0,1],
\end{equation}
where $\xi_{D}^{*}$ is the locally D-optimal design. We observe that in this example an optimal design improves the
standard design
substantially. We also observe that the differences between the $D$-optimal designs calculated under a different distributional assumption are rather small.
For this example, the $D$-optimal design calculated under the assumption of a normal distribution has efficiency $0.98$ in the model based on the negative binomial distribution.
\renewcommand{1.2}{1.2}
\begin{eqnarray}gin{table}[h]
\footnotesize
\centering{
\begin{eqnarray}gin{tabular}{|l|c|c|}
\hline
distribution & D-optimal design & $\mathrm{eff}_D$ \\
\hline
normal & \begin{eqnarray}gin{tabular}{cccc} (0,0) & (9.81,0) & (300,0) & (C,1) \\ \hline $22.\overline{2}\%$ & $22.\overline{2}\%$ & $22.\overline{2}\%$ & $33.\overline{3}\%$ \end{tabular} & 0.25 \\
\hline
negative binomial & \begin{eqnarray}gin{tabular}{cccc} (0,0) & (8.23,0) & (300,0) & (C,1) \\ \hline $25\%$ & $25\%$ & $25\%$ & $25\%$ \end{tabular} & 0.11 \\
\hline
\hline
normal & \begin{eqnarray}gin{tabular}{cccc} (0,0) & (10.95,0) & (200,0) & (C,1) \\ \hline $22.\overline{2}\%$ & $22.\overline{2}\%$ & $22.\overline{2}\%$ & $33.\overline{3}\%$ \end{tabular} & 0.84 \\
\hline
binomial & \begin{eqnarray}gin{tabular}{cccc} (0,0) & (9.05,0) & (200,0) & (C,1) \\ \hline $25\%$ & $25\%$ & $25\%$ & $25\%$ \end{tabular} & 0.86 \\
\hline
\end{tabular}
}
\caption{\small \it $D$-optimal designs in the two clinical trials discussed in Section~\ref{sec1} under different distributional assumptions. Upper part: gouty arthritis example;
lower part: acute migraine example. The last column shows the efficiencies of the designs, which were actually used in the study.}
\label{tabDeffg}
\end{table}
\item[(b)]
We now consider the acute migraine example, which measured the percentage of patients reporting pain freedom at two hours post-dose.
We assume a binomial distribution for this example. The probabilities of success are $\pi(d,\theta_1)= \vartheta_0+\tfrac{\vartheta_1 d}{\vartheta_2 + d}$
for the new compound (where the dose level varies in the interval $[0,200]$mg) and $\theta_2$ for the active control.
The sample sizes are $n_1=517$ and $n_2=100$ and the preliminary information obtained from the clinical team is given by
$\vartheta_0=0.098,$ $ \vartheta_1=0.2052,$ $\vartheta_2=12.3,$ $\sigma_1=0.05$ and $\theta_2=0.2505$, $\sigma_2=0.05$. The locally D-optimal designs under a normal and binomial distribution assumption are listed in the lower part of Table \ref{tabDeffg}. The design actually used for this study allocated $21, 5, 7, 10, 10, 11, 10, 10 \%$ of the patients to the dose levels $0,2.5,5,10,20,50,100,200$mg of the new drug and $16\%$ of the patients to the active control, respectively. The second column of Table \ref{tabDeffg} displays its efficiencies relative to the proposed designs and again a substantial improvement can be observed under both distributional assumptions.
{For this example, the $D$-optimal design calculated under the assumption of a normal distribution has also efficiency $0.98$ in the model based on the binomial distribution.}
\end{itemize}
}
\end{Example}
\section{Optimal designs for estimating the target dose}
\label{sec4}
\def5.\arabic{equation}{4.\arabic{equation}}
\setcounter{equation}{0}
In this section we investigate the problem of constructing locally optimal designs for estimating
the treatment effect of the active control and the target dose, that is
the smallest dose of the new compound which achieves the same treatment effect as the active control.
For this purpose we consider a dose range of the form $\mathcal{D}=[L,R]$ and introduce the notation
\begin{eqnarray}gin{eqnarray}\label{ex1}
\mathbb{E}_{\theta_1} [Y_{ij}|d_i] &=& \eta(d_i,\theta_1) \qquad \qquad (j=1,\dots,n_{1i}, \ i=1,\ldots,k) \\
\label{ex2}
\mathbb{E}_{\theta_2}[Z_i] &=& \Delta \qquad \qquad \qquad \ \ \ (i=1,\ldots,n_2)
\end{eqnarray}
for the expected values of responses corresponding to the new drug (for dose level $d_i$) and the active control, respectively. We assume
(for simplicity) that the function $\eta$ in \eqref{ex1} is strictly increasing in $d\in \mathcal{D}$ and that $d^*(\theta)=\eta^{-1}(\Delta,\theta_1)$, is an element of the dose range $\mathcal{D}=[L,R]$ for the new drug. Note that the expectation $\Delta$ in \eqref{ex2} is a function of the $s_2$-dimensional parameter $\theta_2$, say $\Delta =k(\theta_2)$. Consequently,
a natural estimate of $d^{\ast}$ is given by
$\hat{d}^{\ast} = d^*(\hat \theta) =\eta^{-1}(\hat{\Delta},\hat{\theta}_1)$, where $\hat{\Delta} = k (\hat \theta_2)$ and
$\hat{\theta}=( \hat{\theta}_1^T, \hat{\theta}_2^T)^T$ denotes the vector of the maximum likelihood estimates of
the parameter $\theta_1 $ and $\theta_2 $ in models \eqref{exp1} and \eqref{exp2}, respectively.
Standard calculations show that the variance of this estimator is approximately given by
\begin{eqnarray}gin{equation}\label{approx}
\mbox {Var}(d^*(\hat \theta)) \approx \tfrac{1}{N} \psi(\xi,\theta),
\end{equation}
where the function $\psi$ is defined by
\begin{eqnarray}gin{equation}\label{critpsi}
\psi(\xi,\theta) = \tfrac{1}{1-\omega_{k+1}} (\tfrac{\partial}{\partial \theta_1} d^*(\theta))^T M^-_1(\tilde \xi,\theta_1) (\tfrac{\partial}{\partial \theta_1}d^*(\theta)) + \tfrac{1}{\omega_{k+1}} (\tfrac{\partial}{\partial \theta_2} d^*(\theta))^T {I}_2 ^-(\theta_2) (\tfrac{\partial}{\partial \theta_2} d^*(\theta_2)),
\end{equation}
$\tilde \xi$ denotes the design for the new drug induced by the design $\xi$, see \eqref{tilde},
and $M^-_1(\tilde \xi,\theta_1)$ and $I_2^-(\theta_2)$
are generalized inverses of the information matrices $ M_1(\tilde \xi, \theta_1)$ and $I_2( \theta_2)$, respectively.
Following
\cite{detkisbenbre2014}, we call a design $\xi^*_{AC}$ locally AC-optimal design (for \underline{A}ctive \underline{C}ontrol) if $\tfrac{\partial}{\partial \theta_1}d^*(\theta) \in \mbox{Range}
(M_1(\tilde \xi,\theta_1))$, $\tfrac{\partial}{\partial \theta_2}d^*(\theta) \in \mbox{Range}
(I_2(\theta_2))$ and if $\xi^*_{AC}$ minimizes the function $\psi(\xi,{\theta})$ among all designs satisfying this
estimability condition. Note that the criterion \eqref{critpsi} corresponds to a $\phi_{-1}$-optimal design for estimating the parameter $K^T\theta$ in a dose response model with an active control, where the matrix $K$ is given by
$K = \big((\tfrac{\partial}{\partial \theta_1} d^* (\theta))^T, (\tfrac {\partial}{\partial \theta_2} d^*(\theta))^T \big)^T$.
In particular, Theorem \ref{thm2} is applicable and locally AC-optimal designs can be derived from the corresponding optimal designs for model \eqref{exp1}.
The following result provides an alternative representation of the criterion \eqref{critpsi} in the case $s_2=1$. As a consequence the design $\tilde \xi$ required
in Theorem \ref{thm2} is a locally $\tilde c$-optimal design in model \eqref{exp1} for a specific vector $\tilde c$, i.e. the design minimizing $\tilde c^T M_1^-(\tilde \xi, \theta_1) \tilde c$,
where $\tilde c = \tfrac{\partial}{\partial \theta_1} \eta(d^*,\theta_1)$.
\begin{eqnarray}gin{Theorem} \label{thm2A}
In the case $s_2=1$, the function in \eqref{critpsi} can be represented as
\begin{eqnarray}gin{equation*} \label{critalt}
\psi(\xi,\theta) = \frac {(\tfrac {\partial}{\partial \theta_2}d^*(\theta))^2}{(\tfrac {\partial}{\partial \theta_2}k(\theta_2))^2} \mathcal{B}igl \{ \tfrac {1}{1-w_{k+1}} (\tfrac {\partial}{\partial \theta_1}\eta(d^*, \theta_1))^T M^-_1 (\tilde \xi, \theta_1) (\tfrac {\partial}{\partial \theta_1} \eta (d^*,\theta_1)) + (\tfrac {\partial}{\partial \theta_2}k(\theta_2))^2 \tfrac {I_2^-(\theta_2)}{w_{k+1}} \mathcal{B}igr \}.
\end{equation*}
\end{Theorem}
In the following discussion we determine locally AC-optimal designs for several nonlinear
regression models accounting for an active control by minimizing the criterion
\eqref{critpsi}.
\subsection{Some explicit results for two-dimensional models}
In this section we present some examples illustrating different structures of
locally AC optimal designs. For this purpose we consider the situation where the Fisher information
matrix $I_1(d,\theta_1)$ defined in \eqref{i2} is of the form
\begin{eqnarray}gin{eqnarray}\label{geo1}
I_1 (d,\theta_1) = \left ( \begin{eqnarray}gin{array} {cc}
f(d,\theta_1)f^T(d,\theta_1) & \bm {0} \\
\bm{0} & \Sigma(\theta_1)
\end{array} \right ) \in \mathbb{R}^{s_1 \times s_1}
\end{eqnarray}
where $f(d,\theta_1)=(f_1(d,\theta_1), f_2(d,\theta_1))^T$ denotes a two-dimensional vector and $\Sigma(\theta_1)$ a $(s_1-2)\times(s_1-2)$ matrix, which does not depend on the dose level. By Theorem \ref{thm2} the locally AC-optimal design can be determined from the design $\tilde \xi^*$ which minimizes
the expression
\begin{eqnarray}gin{equation} \label{copt2}
\tilde c^T M^-_1 (\tilde \xi, \theta_1) \tilde c
\end{equation}
in the class of all designs defined on the dose range $\mathcal{D}$ for the new drug, where the vector is given by $\tilde c = (\tfrac{\partial}{\partial \theta_1} d^*(\theta))^T$. Because
of the block structure of the Fisher information in \eqref{geo1} (with a lower block not depending on the dose level) we may
assume without loss of generality that $s_1=2$, that is
\begin{eqnarray}gin{equation} \label{forf}
M_1 (\tilde \xi, \theta_1) = \int_{\mathcal{D}} f(d,\theta_1) f^T (d,\theta_1) d \tilde \xi (d).
\end{equation}
By Elfving's theorem [see \cite{elfving1952}] a design $\tilde \xi^*$ with weights $\tilde w_i^*$ at the points $d^*_i \ (i=1,\dots,k)$ minimizes
the expression
\eqref{copt2} if and only if there exists a constant $\gamma > 0$ and $\varepsilon_1,\dots,\varepsilon_k \in \{ -1,1 \}$, such that the
point $\gamma \tilde c$ is a boundary point of the Elfving set
\begin{eqnarray}gin{equation} \label{elfset}
\mathcal{R} = \mbox{conv} \mathcal{B}igl \{ \varepsilon f (d,\theta_1) \mid d \in \mathcal{D}, \ \varepsilon \in \{ -1,1 \} \mathcal{B}igr \}
\end{equation}
and the representation
$\gamma \tilde c = \sum^k_{i=1} \varepsilon_i \tilde w_i^* f (d^*_i,\theta_1)
$
is valid. Note that $\mathcal{R} = conv\{\mathcal{C} \cup (-\mathcal{C})\}$, where the curve $\mathcal{C}$ is defined by $\mathcal{C}=\{f(d,\theta_1) \mid d \in \mathcal{D}\}$.
The structure of the Elfving set $\mathcal{R}$ depends sensitively on the distributional assumptions and
we now consider several examples in the Michaelis-Menten model.
\begin{eqnarray}gin{Example} ~~~ \\
{\rm
{
Assume that the dependence on the dose
in model \eqref{exp1} is described by the Michaelis-Menten model, then the vector $f$ in \eqref{forf} has the form $v(d,\theta_1) (\tfrac{d}{\vartheta_2+d},-\tfrac{\vartheta_1 d}{(\vartheta_2+d)^2})^T$, where the function $v$ varies with the distributional assumption.
\begin{eqnarray}gin{itemize}
\item [(a)]
In the case of normal distributed responses we have $v(d,\theta_1)=1$ and it follows by an obvious generalization of Theorem \ref{thm2A},
that we have to consider a $\tilde c$-optimal design problem in model \eqref{exp1}, where the vector $\tilde c$ is now given by
$
\tilde c=\tfrac{\partial}{\partial \vartheta}\eta(d^*,\vartheta) = (\tfrac{d^*}{\vartheta_2 + d^*},-\tfrac{\vartheta_1d^*}{(\vartheta_2 + d^*)^2})^T.
$
From the left panel of Figure \ref{normandneg} we observe that the line $\{\gamma \tilde c | ~\gamma > 0\} $ intersects the boundary of the Elfving set
$\mathcal{R}$ at some point $\mathcal{C } \cup (- \mathcal{C}) $, whenever
$L \leq x^* \leq d^* < R$,
where
$$
x^*=L \lor \tfrac{\sqrt{2}R^2\vartheta_2+(\sqrt{2}-1)R\vartheta_2^2}{2R^2+4R\vartheta_2+\vartheta_2^2}.
$$
A typical situation is shown for the vector $\tilde c_2$ in the left panel of Figure \ref{normandneg} for $\vartheta_1=\vartheta_2=2, \mathcal{D}= [0.1,50]$. Consequently, Elfvings theorem shows that a one-point design minimizes \eqref{copt2} in this case. An application of Theorem \ref{thm2} yields that the locally AC-optimal design which allocates $\tfrac{\sigma_1}{\sigma_1+\sigma_2} 100\%$ of the patients to dose level $d^*=\eta^{-1}(\Delta,\vartheta)$ for the new drug and the remaining patients to the active control.
On the other hand, if $L < d^* \leq x^* < R$, the line $\{\gamma \tilde c | ~\gamma > 0\} $ does not intersect the set $\mathcal{C } \cup (- \mathcal{C})$
at the boundary of the Elfving set ${\cal R}$ and the situation is more complicated. A typical situation for this case is shown for the vector $\tilde c_1$ and the locally AC-optimal design allocates $\rho \tilde \omega_1 100\%$, $\rho \tilde \omega_2 100\%$ of the patients to dose levels $x^*$ and $R$ of the new drug, where $\rho=\tfrac{\sqrt{\delta}\sigma_1}{\sqrt{\delta}\sigma_1+\sigma_2}$ and the remaining patients to the active control,
where
\begin{eqnarray}gin{eqnarray} \label{gewform1}
\tilde \omega_1 &=& \tfrac{v(R,\theta_1) R(R-d^*)(\vartheta_2+x^*)^2}{v(R,\theta_1)R(R-d^*)(\vartheta_2+x^*)^2 + v(x^*,\theta_1)x^*(x^*-d^*)(\vartheta_2+R)^2},
\end{eqnarray}
$\tilde \omega_2= 1-\tilde \omega_1$,
$\delta =\tilde c^T M_1^{-1}(\tilde\xi^*,\theta_1) \tilde c$ and $ d^* = \eta^{-1}(\Delta,\theta_1)$.
\begin{eqnarray}gin{figure}[h]
\centering
\subfigure{\raisebox{0.5cm}{\includegraphics[scale=0.6]{NormalverteilungMMSchrift2.pdf}}}
\quad \quad
\subfigure{\includegraphics[scale=0.5]{negativebinomialverteilungMMSchrift2.pdf}}
\caption{\it The Elfving set \eqref{elfset} in model \eqref{exp1}, where the expected response is given by the Michaelis-Menten model. Left panel: normal distribution. Right panel: Negative-binomial distribution.}
\label{normandneg}
\end{figure}
\item [(b)] As a further example consider the Michaelis Menten model for the probability of a negative binomial distributed response. We have $s_1=2, s_2 =1, $
$\pi(d,\theta_1)=\tfrac{\vartheta_1 d}{\vartheta_2+d}$, $\tilde c = \tfrac {\partial}{\partial \theta_1} \eta(d^*,\theta_1) = \tfrac{r_1}{\vartheta_1 d^*}(-\tfrac{\vartheta_2+d^*}{\vartheta_1},1)^T$
and the function $v$ is given by $v(d,\theta_1)=\sqrt{\tfrac{r_1 (d+\vartheta_2)^3}{d^2 \vartheta_1^2 (d(1-\vartheta_1)+\vartheta_2)}}$.
A corresponding Elfving set is depicted in the right panel of Figure \ref{normandneg} for $\vartheta_1=1, \vartheta_2=0.5, \mathcal{D}= [0,10]$ and the locally AC-optimal design is always supported at three points. A straightforward calculation
shows that the locally AC-optimal design allocates $\rho \tilde \omega_1 100\%$, $\rho \tilde \omega_2 100\%$ of the patients to the dose levels $L$, $R$ for the new drug, where $\rho=\tfrac{\delta \theta_2^2 - \sqrt{(1-\theta_2)\delta \theta_2^2 r_2}}{\delta \theta_2^2- (1-\theta_2)r_2}$ and the remaining patients to the active control, where
\begin{eqnarray}gin{eqnarray}\label{gewform2}
\tilde \omega_1 &=& \tfrac{v(R,\theta_1)R (R-d^*) (\vartheta_2+L)^2 }{v(R,\theta_1)R (R-d^*) (\vartheta_2+L)^2 + v(L,\theta_1) L (d^*-L) (\vartheta_2+R)^2 },
\end{eqnarray}
$\tilde \omega_2 = 1-\tilde \omega_1$
$\delta=\tilde c^T M_1^{-1}(\tilde\xi^*,\theta_1) \tilde c$ and $d^* = \eta^{-1}(\Delta,\theta_1)$.
\item [(c)]
Consider now the Michaelis Menten model for binomial distributed responses. We have $s_1=2, s_2 =1, \pi(d,\theta_1)=\tfrac{\vartheta_1 d}{\vartheta_2+d}, \tilde c = \tfrac{\partial}{\partial \theta_1} \pi(d^*,\theta_1)$ and $v(d,\theta_1)=\sqrt{\tfrac{(d+\vartheta_2)^2}{d \vartheta_1 (d (1-\vartheta_1)+\vartheta_2)}}$.
The corresponding Elfving set is depicted in the left panel of Figure \ref{binandpois} for $\vartheta_1=1, \vartheta_2=0.1, \mathcal{D}= [0.02,2]$
and we have to distinguish three different cases. We observe that the line $\{\gamma \tilde c | ~\gamma > 0\} $ intersects the boundary of the Elfving set
$\mathcal{R}$ at some point $\mathcal{C } \cup (- \mathcal{C}) $
if and only if $L \leq x_1^* \leq d^* \leq x_2^* \leq R$,
where
$$
x_1^* = L \lor \tfrac{\vartheta_2(1-\sqrt{1-\pi(R,\theta_1)})}{2\vartheta_1 - 1 + \sqrt{1-\pi(R,\theta_1)}}, \quad x_2^*=
R \land \tfrac{\vartheta_2(1+\sqrt{1-\pi(R,\theta_1)})}{2\vartheta_1 - 1 - \sqrt{1-\pi(R,\theta_1)}} .
$$
A typical situation is shown for the vector $\tilde c_1$ in the left panel of Figure \ref{binandpois}.
Consequently, the same arguments as in the previous examples show that in this case the locally AC-optimal design allocates $\rho 100\%$ of the patients to the dose level $d^*$ of the new drug, where
$\rho=\frac{\delta-\sqrt{ \delta(1-\theta_2)\theta_2}}{\delta- (1-\theta_2) \theta_2}$ and the remaining patients to the active control,
where $\delta = \tilde c^T M_1^{-1}(\tilde\xi^*,\theta_1) \tilde c$ and $d^*=\eta^{-1}(\Delta,\theta_1)$. \\
On the other hand, if $L < d^* \leq x_1^*$, the locally AC-optimal design allocates $\rho \tilde \omega_{11} 100\%$, $\rho (1-\tilde \omega_{11}) 100\%$ of the patients to dose levels $x_1^*$ and $R$
of the new drug and the remaining patients to the active control,
where $\tilde \omega_{11}$ is of the form \eqref{gewform1} with $x^*=x_1^*$.
A typical situation is shown for the vector $\tilde c_2$.
The case $L \leq x_2^* \leq d^* \leq R$ corresponds to the vector $\tilde c_3$. Here
the locally AC-optimal design allocates $\rho \tilde \omega_{21} 100\%$, $\rho (1-\tilde \omega_{21}) 100\%$ of the patients to dose levels
$x_2^*$ and $R$ of the new drug and the remaining patients to the active control,
where with $L=x_2^*$ $\tilde \omega_{21}$ is of the form \eqref{gewform2} and $d^*=\eta^{-1}(\Delta,\theta_1)$.
\begin{eqnarray}gin{figure}[h]
\centering
\subfigure{\includegraphics[scale=0.4]{binomialverteilungMMSchrift2.pdf}}
\quad \quad \quad \quad \quad \quad
\subfigure{\raisebox{0.8cm}{\includegraphics[scale=0.4]{PoissonverteilungMMSchrift2.pdf}}}
\caption{\it The Elfving set \eqref{elfset} in model \eqref{exp1}, where the expected response is given by the Michaelis-Menten model. Left panel: binomial distribution. Right panel: Poisson distribution.}
\label{binandpois}
\end{figure}
\item [(d)] Finally we consider the case of Poisson distributed responses. We have $s_1=2, s_2 =1, \lambda(d,\theta_1)=\tfrac{\vartheta_1 d}{\vartheta_2+d}$, $v(d,\theta_1)
= {1} \big / {\sqrt{\tfrac{\vartheta_1 d}{\vartheta_2+d}}}$
and by Theorem \ref{thm2A} we have to solve a $\tilde c$-optimal design problem with $\tilde c = \tfrac {\partial}{\partial \theta_1} \lambda (d^*,\theta_1)= (\tfrac{d^*}{\vartheta_2+d^*},-\tfrac{\vartheta_1 d^{*}}{(\vartheta_2+d^*)^2})^T$. It is easy to see that the line $\{\gamma \tilde c | ~\gamma > 0\} $ intersects the boundary of the Elfving set
$\mathcal{R}$ at some point $\mathcal{C} \cup (- \mathcal{C}) $ if and only if $L \leq x^* \leq d^* < R$,
where $x^*=L \lor \tfrac{R \vartheta_2}{3R+4\vartheta_2}$ (see the right panel of Figure \ref{binandpois} for
$\vartheta_1=2.5, \vartheta_2=1.5, \mathcal{D}= [0.02,10]$ and the vector $\tilde c_2$).
Consequently, the same arguments as in the previous examples show that in this case the locally AC-optimal design allocates $\rho 100\%$ of the patients to dose levels $d^*$ of the new drug, where
$\rho = \tfrac{\sqrt{\delta}}{\sqrt{\delta}+\sqrt{\theta_2}}$ and the remaining patients to the active control,
where $\delta=\tfrac{d^* \vartheta_1}{\vartheta_2+d^*}$ and $d^*=\lambda^{-1}(\Delta,\theta_1)$.
On the other hand, if $L < d^* \leq x^* < R$, the locally AC-optimal design allocates $\rho \tilde \omega_1 100\%$, $\rho (1- \tilde \omega_1) 100\%$ of the patients to dose levels $x^*$ and $R$ of the new drug and the remaining patients to the active control,
where $\tilde \omega_1$ is of the form \eqref{gewform1} with $\delta=(\tfrac{\partial}{\partial \theta_1}\eta(d^*,\theta_1))^T M_1^-(\tilde\xi^*,\theta_1)(\tfrac{\partial}{\partial \theta_1}\eta(d^*,\theta_1))$.
A typical situation is shown for the vector $\tilde c_1$ in the right panel of Figure \ref{binandpois}.
\end{itemize}
}
}
\end{Example}
\subsection{Locally AC-optimal designs in the EMAX model}
Explicit expressions for the AC-optimal designs in the EMAX model are very complicated and for the sake of brevity and better illustration we conclude this paper discussing AC-optimal designs for the two data examples from Section~\ref{sec1}. \\
We begin with the gouty arthritis clinical trial where we use the same prior information as in Example \ref{dataex}. AC-optimal designs under the assumption
of a normal and negative binomial distribution can be found in the upper part of Table \ref{tabgout}. For example under the assumption of normal distributed endpoints,
the AC-optimal design allocates almost half of the patients to the dose level $101.06$mg and the rest to the active control. In order to compare the standard design introduced in Example \ref{dataex}
we display in the right column the efficiency
\begin{eqnarray}gin{equation} \label{eff}
\mathrm{eff}_{\text{AC}}(\xi,\theta)=\frac{\psi(\xi_{\text{AC}}^{*},\theta)}{\psi (\xi,\theta)} \in [0,1],
\end{equation}
where $\psi(\xi,\theta)$ is defined in \eqref{crit} and $\xi_{\text{AC}}^*$ is the locally AC-optimal design.
\renewcommand{1.2}{1.2}
\begin{eqnarray}gin{table}[h]
\footnotesize
\centering{
\begin{eqnarray}gin{tabular}{|l|c|c|}
\hline
distribution & AC-optimal design & $\mathrm{eff}_{AC}$ \\
\hline
normal & \begin{eqnarray}gin{tabular}{cc} $(101.06,0)$ & $(C,1)$ \\ \hline $49.99\%$ & $50.01\%$ \end{tabular} & 0.66 \\
\hline
negative binomial & \begin{eqnarray}gin{tabular}{ccc} $(5.44,0)$ & $(300,0)$ & $(C,1)$ \\ \hline $7.6\%$ & $35.6\%$ & $56.8\%$ \end{tabular} & 0.48 \\
\hline
\hline
normal & \begin{eqnarray}gin{tabular}{cc} $(35.739,0)$ & $(C,1)$ \\ \hline $49.99\%$ & $50.01\%$ \end{tabular} & 0.48 \\
\hline
binomial & \begin{eqnarray}gin{tabular}{ccc} $(0,0)$ & $(200,0)$ & $(C,1)$ \\ \hline $7.34\%$ & $41.95\%$ & $50.71\%$ \end{tabular} & 0.47
\\
\hline
\end{tabular}
\caption{ \small \it AC-optimal designs in the two examples from section 1 under different distributional assumptions. Upper part: gouty arthritis example, with target dose $d^*=100$mg;
lower part: acute migraine example, with target dose $d^*=35.6$mg. The last column shows the efficiencies of the designs, which were actually used in the study.}
\label{tabgout}
}
\end{table}
For example, the efficiency of the standard design for estimating the target dose under the assumption of a normal or negative binomial distribution is $66\%$ and $48\%$, respectively. \\
The second trial is the one in treating migraine and again we use the prior information from Example \ref{dataex}. AC-optimal designs for normal and binomial distributed responses can be found
in the lower part of Table \ref{tabgout}. The efficiencies of the standard design are given by $48\%$ and $47\%$ under the assumption of a normal and binomial distribution, respectively.
{\bf Acknowledgements} The authors would like to thank Martina
Stein, who typed parts of this manuscript with considerable
technical expertise.
This work has been supported in part by the
Collaborative Research Center "Statistical modeling of nonlinear
dynamic processes" (SFB 823) of the German Research Foundation
(DFG) and by the National Institute Of General Medical Sciences of the National
Institutes of Health under Award Number R01GM107639. The content is solely the responsibility of the authors and does not necessarily
represent the official views of the National
Institutes of Health.
\section{Conclusions}
In this paper the optimal design problem for active controlled dose finding studies is considered.
Sufficient conditions are provided such that the optimal design for a dose finding study with no active control can also be used
for the model with an active control. Our results apply to general optimality criteria and distributional assumptions. In particular they
are applicable in models with discrete responses, which appeared recently in two of our consulting projects. In several examples it is demonstrated that the optimal
designs may depend sensitively on the distributional assumptions. In the clinical trials under consideration these differences were less visible for
$D$-optimal designs. However, in the problem of estimating the target dose (i.e.
the smallest dose of the new compound which achieves the same treatment effect as the active control), the differences are more substantial, and an optimal design
calculated under a ''wrong'' distributional assumption (i.e. a normal distribution) might be inefficient, if it used in a different model (i.e. a Binomial model).
\setstretch{1.25}
\setlength{\bibsep}{1pt}
\begin{eqnarray}gin{small}
\end{small}
\normalsize
\section{Appendix: proofs} \label{appendix}
\def5.\arabic{equation}{5.\arabic{equation}}
\setcounter{equation}{0}
{\bf Proof of Theorem \ref{thm1}.}
Assume that the matrix $K$ is a blockdiagonal matrix of the form \eqref{kdiag}. Observing the representations \eqref{inf}, \eqref{kdiag}
and noting that the expression $K^TGK$ is independent of the choice of the generalized inverse of the matrix $M(\xi,\theta)$, we obtain
\begin{eqnarray}gin{eqnarray*}
K^T M^- (\xi,\theta)K= \left(
\begin{eqnarray}gin{array}{cc}
(1-w_{k+1})^{-1} K^T_{11} M^-_1 (\tilde\xi,\theta_1)K_{11} & 0 \\
0 & w^{-1}_{k+1} K^T_{22} I^-_2 (\theta_2)K_{22}
\end{array}
\right).
\end{eqnarray*}
In the case $p \neq 0, -\infty$ this gives for the criterion $\phi_p$ in \eqref{crit} the representation
\begin{eqnarray}gin{eqnarray} \label{crit0}
\phi_p (\xi) &=& \mathcal{B}igl( \frac {1}{t} \sum^{t}_{i=1} \lambda^{-p}_i (K^TM^-(\xi,\theta)K) \mathcal{B}igr)^{\frac {1}{p}} \\
&=& \mathcal{B}igl( \frac {1}{t} \bigl\{
(1-w_{k+1})^{p} \sum^{t_1}_{i=1} \lambda^{-p}_i (K^T_{11} M^-_1 (\tilde \xi, \theta_1)K_{11} \bigr) + w^{p}_{k+1}\sum^{t_2}_{i=1} \lambda^{-p}_i (K^T_{22}I_2^- (\theta_2)K_{22}) \bigr\} \mathcal{B}igr)^{\frac {1}{p}} \nonumber
\\ \nonumber
&=& \mathcal{B}igl( \frac {(1-w_{k+1})^{p}t_1 }{t} (\tilde \phi_p (\tilde \xi))^p + \frac {w^{p}_{k+1}}{t} tr( (K^T_{22}I_2^-(\theta_2)K_{22})^{-p} ) \mathcal{B}igr)^{\frac {1}{p}},
\end{eqnarray}
where $\lambda_1(A),\dots,\lambda_n(A)$ denote the eigenvalues of a matrix $A$, $t=t_1+t_2$ and the function $\tilde \phi_p$ is defined in \eqref{crittilde}.
Now it is easy to see that the function $\phi_p$ is an increasing function of $\tilde \phi_p(\tilde \xi)$. Consequently, the locally $\phi_p$-optimal design problem for the dose response model with an active control can be solved by determining a design $\tilde \xi^*_p$ which maximizes the criterion \eqref{crittilde} in a first step. If $\phi^* = \tilde \phi_p (\tilde \xi^*_p)= \max_{\tilde \xi} \tilde \phi_p(\tilde \xi)$ denotes the optimal value for this criterion, it remains to maximize the function $\phi_p$ in \eqref{crit0} with respect to the weight $w_{k+1}$ assigned to the active control, which gives the expression \eqref{weight} and proves the assertion for the case $p \neq 0, -\infty$. The remaining cases $p=0$ and $p=-\infty$ are proved similarly and the details are omitted for the sake of brevity.
$\mathcal{B}ox$
\textbf{Proof of Theorem \ref{thm31a} } The proof of part (a) has been given in Section \ref{sec3}. For the remaining cases we restrict ourselves to the case of the Poisson distribution for which the Fisher information in model \eqref{exp1} is given by
\begin{eqnarray}gin{equation} \label{31}
I_1 (d,\theta_1) = \frac {d}{\vartheta_1(\vartheta_2+d)} \left ( \begin{eqnarray}gin{array} {cc}
1 & -\frac { \vartheta_1}{\vartheta_2+d} \\
- \frac {\vartheta_1}{\vartheta_2+d} & \frac {\vartheta_1^2}{(\vartheta_2+d)^2}
\end{array}\right )
\end{equation}
[see equation \eqref{finfpoiss1}]. All other cases are treated similary. By Corollary \ref{cor1} the $D$-optimal design can be obtained from the $D$-optimal design $\tilde \xi^*$ in a regression model with Fisher information \eqref{31}. If $M_1(\tilde \xi, \theta_1) = \int_{\mathcal{D}}I_1(d,\theta_1) d \tilde \xi(d)$ denotes an information matrix of a design $\tilde \xi$ in this model, then $\tilde\xi^*$ is $D$-optimal if and only if the inequality
$
\mbox{tr} (I_1(d,\theta_1)M^{-1}_1(\tilde \xi^*,\theta_1)) \leq 2
$
holds for all $d \in \mathcal{D}$ (see Lemma \ref{lem1}). Moreover, there must be equality at the support points of the design $\tilde \xi^*$. It is easy to see that this inequality is equivalent to an inequality of the form $P_3(d) \leq 0$ where $P_3$ is a polynomial of degree $3$ with $P(0)<0$. A straightforward argument now shows that $\tilde \xi^*$ has exactly two support points $d^*_1 > 0$ and $d^*_2 = R$. Consequently, the $D$-optimal
design $\tilde \xi^*_1$ for the regression model with information matrix \eqref{31} has equal masses at the points
$d^*_1$ and $ R $,
where $d^*_1$ maximizes the function
$$
f(d) = \frac {R(R-d)^2d}{4(R+\vartheta_2)^3 (\vartheta_2+d)^3}
$$
in the interval $[L,R]$, that is $d^*_1= L \vee \frac {\vartheta_2R}{3\vartheta_2 + 2R}$.
The assertion now follows by an application of Corollary \ref{cor1}, observing that $t_1=2, t_2=2$ in the case under consideration.
\textbf{Proof of Thorem \ref{thm2A}} Note that $\Delta = k(\theta_2)$ and that the dose level $d^*(\theta) = \eta^{-1}(\Delta, \theta_1)$ can be defined as the (unique) solution of the equation
$
F(d,\theta) = k(\theta_2) - \eta(d,\theta_1) = 0
$
with respect to $d$. Consequently, the implicit function theorem shows that the function $\theta \to d^*(\theta)$ is differentiable with respect to $\theta$ with gradient given by
$$
\big( (\tfrac {\partial}{\partial \theta_1} d^*(\theta))^T, \tfrac {\partial}{\partial \theta_2} d^*(\theta) \big)^T = - \big( \tfrac{\partial}{\partial d} F(d,\theta) \mathcal{B}ig|_{d=d^* (\theta) }\big)^{-1}
\big( - (\tfrac {\partial}{\partial \theta_1} \eta(d,\theta_1))^T, \tfrac {\partial}{\partial \theta_2} k(\theta_2) \big)^T \mathcal{B}ig|_{d=d^*(\theta)},
$$
which implies (comparing the second components) $- \big( \tfrac{\partial}{\partial d} F(d,\theta) \mathcal{B}ig|_{d=d^* (\theta) }\big)^{-1} = {\tfrac {\partial}{\partial \theta_2} d^*(\theta)}
\big / {\tfrac {\partial}{\partial \theta_2} k(\theta_2)}. $
Altogether this gives for the first component
\begin{eqnarray}gin{eqnarray*}
(\tfrac {\partial}{\partial \theta_1} d^*(\theta))^T &=& \big( \tfrac{\partial}{\partial d} F(d,\theta) \mathcal{B}ig|_{d=d^* (\theta) }\big)^{-1}( \tfrac {\partial}{\partial \theta_1} \eta(d^*,\theta_1))^T = - \frac {\tfrac {\partial}{\partial \theta_2}d^*(\theta)}{\tfrac{\partial}{\partial \theta_2}k(\theta_2)} (\tfrac {\partial}{\partial \theta_1} \eta(d^*,\theta_1))^T
\end{eqnarray*}
and the result follows from the representation \eqref{crit}.
\end{document} | math | 72,491 |
\begin{document}
\iflineno \pagewiselinenumbers\fi
\renewcommand{\thesection.\Alph{subsection}}{\thesection.\Alph{subsection}}
\subjclass[2000]{Primary 03E35; secondary 03E17, 28E15}
\date{2011-12-27}
\title{Borel Conjecture and dual Borel Conjecture}
\author{Martin Goldstern}
\address{Institut f\"ur Diskrete Mathematik und Geometrie\\
Technische Universit\"at Wien\\
Wiedner Hauptstra{\ss}e 8--10/104\\
1040 Wien, Austria}
\email{[email protected]}
\tauladdr{http://www.tuwien.ac.at/goldstern/}
\author{Jakob Kellner}
\address{Kurt G\"odel Research Center for Mathematical Logic\\
Universit\"at Wien\\
W\"ahringer Stra\ss e 25\\
1090 Wien, Austria}
\email{[email protected]}
\tauladdr{http://www.logic.univie.ac.at/$\sim$kellner/}
\author{Saharon Shelah}
\address{Einstein Institute of Mathematics\\
Edmond J. Safra Campus, Givat Ram\\
The Hebrew University of Jerusalem\\
Jerusalem, 91904, Israel\\
and
Department of Mathematics\\
Rutgers University\\
New Brunswick, NJ 08854, USA}
\email{[email protected]}
\tauladdr{http://shelah.logic.at/}
\author{Wolfgang Wohofsky}
\address{Institut f\"ur Diskrete Mathematik und Geometrie\\
Technische Universit\"at Wien\\
Wiedner Hauptstra{\ss}e 8--10/104\\
1040 Wien, Austria}
\email{[email protected]}
\tauladdr{http://www.wohofsky.eu/math/}
\thanks{
We gratefully acknowledge the following partial support: US National Science
Foundation Grant No. 0600940 (all authors); US-Israel Binational Science
Foundation grant 2006108 (third author); Austrian Science Fund (FWF): P21651-N13 and P23875-N13 and EU FP7 Marie Curie grant PERG02-GA-2207-224747 (second and fourth author);
FWF grant P21968 (first and fourth author); \"OAW Doc fellowship (fourth author).
This is publication 969 of the third author.\\
We are grateful to the anonymous referee for pointing out several
unclarities in the original version.}
\dedicatory{Dedicated to the memory of Richard Laver}
\begin{abstract}
We show that it is consistent that the Borel Conjecture and the dual Borel
Conjecture hold simultaneously.
\end{abstract}
\maketitle
\section*{Introduction}
\subsection*{History}
A set $X$ of reals\footnote{In this paper, we use $2^\omega$ as
the set of reals. ($\omega=\{0,1,2,\ldots\}$.)
By well-known results both the definition and the theorem
also work for the unit interval~$ [0,1]$ or the torus $\mathbb R/\mathbb Z$.
Occasionally we also write ``$x$ is a real'' for ``$x\in \omega^\omega$''.}
is called ``strong measure zero'' (smz), if for
all functions $f:\omega\to\omega$ there are intervals $I_n$ of measure $\leq
1/f(n)$ covering $X$.
Obviously, a smz set is a null set (i.e., has Lebesgue measure zero), and it
is easy to see that the family of smz sets forms a $\sigma$-ideal and that
perfect sets (and therefore uncountable Borel or analytic sets) are not smz.
At the beginning of the 20th century, Borel~\cite[p.~123]{MR1504785}
conjectured:
\proofclaimnl{Every smz set is countable.}
This statement is known as the ``Borel Conjecture'' (BC). In the 1970s it was proved
that BC is \emph{independent}, i.e., neither provable nor refutable.
Let us very briefly comment on the notion of independence: A sentence $\varphi$
is called independent of a set $T$ of axioms, if neither $\varphi$ nor $\lnot\varphi$
follows from $T$. (As a trivial example, $(\forall x)(\forall y) x\cdot y= y\cdot x$ is
independent from the group axioms.) The set theoretic (first order) axiom system
ZFC (Zermelo Fraenkel with the axiom of choice) is considered to be the
standard axiomatization of all of mathematics: A mathematical proof is
generally accepted as valid iff it can be formalized in ZFC. Therefore we just
say ``$\varphi$ is independent'' if $\varphi$ is independent of ZFC.
Several mathematical statements are independent, the earliest and most
prominent example is Hilbert's first problem, the Continuum Hypothesis
(CH).
BC is independent as well: Sierpi\'nski~\cite{sierpinskiCH} showed
that CH implies $\lnot$BC (and, since G\"odel showed the consistency of CH, this
gives us the consistency of $\lnot$BC). Using the method of forcing,
Laver~\cite{MR0422027} showed that BC is consistent.
Galvin, Mycielski and Solovay~\cite{GMS} proved the following conjecture
of Prikry:
\proofclaimnl{$X \subseteq 2^\omega $ is smz
if and only if every comeager (dense $G_\delta$) set
contains a translate of $X$.}
Prikry also defined the following dual notion:
\proofclaimnl{$X \subseteq 2^\omega$ is called ``strongly meager'' (sm) if every set
of Lebesgue measure 1 contains a translate of $X$.}
The dual Borel Conjecture (dBC) states:
\proofclaimnl{Every sm set is countable.}
Prikry noted that CH implies $\lnot$dBC and conjectured dBC to be consistent
(and therefore independent), which was later proved by
Carlson~\cite{MR1139474}.
Numerous additional results regarding BC and dBC have been proved: The
consistency of variants of BC or of dBC, the consistency of BC or dBC together
with certain assumptions on cardinal characteristics, etc.
See~\cite[Ch.~8]{MR1350295} for several of these results. In this paper, we
prove the consistency (and therefore independence) of BC+dBC (i.e.,
consistently BC and dBC hold simultaneously).
\subsection*{The problem}
The obvious first attempt to force BC+dBC is to somehow combine Laver's and
Carlson's constructions. However, there are strong obstacles:
Laver's construction is a countable support iteration of Laver forcing. The
crucial points are:
\begin{itemize}
\item Adding a Laver real makes every old uncountable set $X$ non-smz.
\item And this set $X$ remains non-smz after another forcing $P$, provided
that $P$ has the ``Laver property''.
\end{itemize}
So we can start with CH and use a countable support iteration of Laver forcing of
length $\om2$. In the final model, every set $X$ of reals of size~$\al1$
already appeared at some stage $\alpha<\om2$ of the iteration; the next Laver real
makes $X$ non-smz, and the rest of the iteration (as it is a countable support
iteration of proper forcings with the Laver property) has the Laver property,
and therefore $X$ is still non-smz in the final model.
Carlson's construction on the other hand adds $\omega_2$ many
Cohen reals in a finite support iteration (or
equivalently: finite support product). The crucial points are:
\begin{itemize}
\item A Cohen real makes every old uncountable set $X$ non-sm.
\item And this set $X$ remains non-sm after another forcing $P$, provided
that $P$ has precaliber~$\al1$.
\end{itemize}
So we can start with CH, and use more or less the same argument as above:
Assume that $X$ appears at $\alpha<\om2$. Then the next Cohen makes $X$ non-sm. It
is enough to show that $X$ remains non-sm at all subsequent stages
$\beta<\om2$. This is guaranteed by the fact that a finite support iteration
of Cohen reals of length~$<\om2$ has precaliber~$\al1$.
So it is unclear how to combine the two proofs: A Cohen real makes all old sets
smz, and it is easy to see that whenever we add Cohen reals cofinally often in
an iteration of length, say, $\om2$, all sets of any intermediate extension
will be smz, thus violating BC. So we have to avoid Cohen
reals,\footnote{An iteration that forces dBC without adding Cohen reals was given in
\cite{MR2767969}, using non-Cohen oracle-cc.}
which also
implies that we cannot use finite support limits in our iterations.
So we have
a problem even if we find a replacement for Cohen forcing in Carlson's proof
that makes all old uncountable sets $X$ non-sm and that does not add Cohen
reals: Since we cannot use finite support, it seems hopeless to get
precaliber~$\aleph_1$,
an essential requirement to keep $X$ non-sm.
Note that it is the \emph{proofs} of BC and dBC that are seemingly
irreconcilable; this is not clear for the models. Of course Carlson's model,
i.e., the Cohen model, cannot satisfy BC, but it is not clear whether maybe
already the Laver model could satisfy dBC. (It is even still open whether a
single Laver forcing makes every old uncountable set non-sm.) Actually,
Bartoszy\'nski and Shelah~\cite{MR2020043} proved that the Laver model does
satisfy the following weaker variant of dBC (note that the continuum has size
$\al2$ in the Laver model):
\begin{quote}
Every sm set has size less than the continuum.
\end{quote}
In any case, it turns out that one \emph{can} reconcile Laver's and Carlson's proof,
by ``mixing'' them ``generically'', resulting in the following theorem:
\begin{Thmstar}
If ZFC is consistent, then ZFC+BC+dBC is consistent.
\end{Thmstar}
\subsection*{Prerequisites}
To understand anything of this paper, the reader
\begin{itemize}
\item should have some experience with finite and countable support
iteration, proper forcing, $\al2$-cc, $\sigma$-closed, etc.,
\item should know what a quotient forcing is,
\item should have seen some preservation theorem for proper countable support iteration,
\item should have seen some tree forcings (such as Laver forcing).
\end{itemize}
To understand everything, additionally the following is required:
\begin{itemize}
\item The ``case A'' preservation theorem from~\cite{MR1623206},
more specifically we build on
the proof
of~\cite{MR1234283} (or~\cite{MR2214624}).
\item In particular, some familiarity with the property ``preservation of randoms''
is recommended. We will use the fact that random and Laver forcing
have this property.
\item We make some claims about (a rather special case of)
ord-transitive models in Section~\ref{subsec:ordtrans}.
The readers can either believe these claims, or check them themselves
(by some rather straightforward proofs),
or look up the proofs (of more general
settings) in~\cite{MR2115943} or~\cite{kellnernep}.
\end{itemize}
{}From the theory of strong measure zero and strongly meager, we only need the
following two results (which are essential for our proofs of BC and dBC,
respectively):
\begin{itemize}
\item Pawlikowski's result from~\cite{MR1380640} (which we quote as Theorem~\ref{thm:pawlikowski} below), and
\item Theorem 8 of Bartoszy\'nski and Shelah's~\cite{MR2767969} (which we
quote as Lemma~\ref{lem:tomek}).
\end{itemize}
We do not need any other results of Bartoszy\'nski and Shelah's
paper~\cite{MR2767969}; in particular we do not use the notion of non-Cohen
oracle-cc (introduced in~\cite{MR2243849}); and the reader does not have to know the original proofs of Con(BC)
and Con(dBC), by Laver and Carlson, respectively.
The third author claims that our construction is more or less the same as a
non-Cohen oracle-cc construction, and that the
extended version presented in~\cite{MR2610747} is even closer to our
preparatory forcing.
\subsection*{Notation and some basic facts on forcing, strongly meager (sm) and
strong measure zero (smz) sets}
We call a lemma ``Fact'' if we think that no proof is necessary --- either because
it is trivial, or because it is well known (even without a reference),
or because we give an explicit reference to the literature.
Stronger conditions in forcing notions are smaller, i.e., $q\leq p$ means that
$q$ is stronger than $p$.
Let $P\subseteq Q$ be forcing notions. (As usual, we abuse notation by not
distinguishing between the underlying set and the quasiorder on it.)
\begin{itemize}
\item For $p_1,p_2\in P$ we write $p_1\perp_P p_2$ for
``$p_1$ and $p_2$ are incompatible''. Otherwise we write $p_1 \parallel_P p_2$.
(We may just write $\perp$ or $\parallel$ if $P$ is understood.)
\item\label{def:starorder} $q\leq^* p$ (or: $q\leq^*_P p$) means that $q$ forces that $p$ is in the generic
filter, or equivalently that
every $q'\leq q$ is compatible with $p$.
And $q=^* p$ means $q\leq^* p\ \wedge\ p\leq^* q$.
\item\label{def:separative} $P$ is separative, if $\leq$ is the same as $\leq^*$,
or equivalently, if for all $q\leq p$ with $q\neq p$ there is an $r\leq p$
incompatible with $q$. Given any $P$, we can define its ``separative
quotient'' $Q$ by first replacing (in $P$) $\leq$ by $\leq^*$ and then
identifying elements $p,q$ whenever $p=^*q$. Then $Q$ is separative
and forcing equivalent to $P$.
\item \qemph{$P$ is a subforcing of $Q$} means that the relation $\le_P$
is the restriction of $\le_Q$ to~$P$.
\item \qemph{$P$ is an incompatibility-preserving subforcing of $Q$}
means that $P$ is a subforcing of
$Q$ and that
$p_1\perp_P p_2$ iff $p_1\perp_Q p_2$ for all $p_1,p_2\in P$.
\end{itemize}
Let additionally $M$ be a countable
transitive\footnote{We will also use so-called
ord-transitive models, as defined in Section~\ref{subsec:ordtrans}.}
model (of a sufficiently large subset of ZFC) containing~$P$.
\begin{itemize}
\item ``$P$ is an $M$-complete subforcing of
$Q$'' (or: $P\lessdot_M Q$)
means that $P$ is a subforcing of $Q$ and: if $A\subseteq P$ is in
$M$ a maximal antichain, then it is a maximal antichain of~$Q$ as
well. (Or equivalently: $P$ is an incompatibility-preserving
subforcing of $Q$ and every predense subset of
$P$~in $M$ is predense in~$Q$.)
Note that this means that every $Q$-generic filter $G$ over~$V$
induces a $P$-generic filter over~$M$, namely $G^M\coloneqq G\cap P$
(i.e., every maximal antichain of
$P$ in~$M$ meets $G\cap P$ in exactly one point).
In particular, we can interpret a $P$-name $\tau$ in~$M$ as a $Q$-name.
More exactly, there is a $Q$-name $\tau'$ such that $\tau'[G]=\tau[G^M]$
for all $Q$-generic filters $G$. We will usually just identify $\tau$
and $\tau'$.
\item Analogously, if $P\in M$ and $i:P\to Q$ is a function, then
$i$ is called an $M$-complete embedding if it preserves $\leq$
(or at least $\leq^*$)
and $\perp$ and moreover: If $A\in M$ is predense in~$P$, then
$i[A]$ is predense in $Q$.
\end{itemize}
There are several possible characterizations of sm (``strongly
meager'') and smz (``strong measure zero'') sets; we will use the following
as definitions:
A set $X$ is not sm if there is a measure $1$ set into
which $X$ cannot be translated; i.e., if there is a null set $Z$
such that $(X+t)\cap Z\neq\emptyset$ for all reals $t$, or, in other words,
$Z+X=2^\omega$. To summarize:
\proofclaim{eq:notsm}{$X$ is {\em not} sm iff there is a
Lebesgue null set $Z$ such that $Z+X=2^\omega$.}
We will call such a $Z$ a ``witness'' for the fact that $X$ is not sm (or say
that $Z$ witnesses that $X$ is not sm).
The following theorem of Pawlikowski~\cite{MR1380640} is central for our
proof\footnote{We thank Tomek Bartoszy\'nski for pointing out Pawlikowski's
result to us, and for suggesting that it might be useful for our proof.}
that BC holds in our model:
\begin{Thm}\label{thm:pawlikowski}
$X\subseteq 2^\omega$ is smz iff $X+F$ is null for every closed null set $F$.
\\
Moreover, for every dense $G_\delta$ set $H$ we can \emph{construct} (in
an absolute way) a closed null set $F$ such that for every $X
\subseteq 2^\omega$ with $X+F$ null there is $t\in 2^\omega$ with
$t+X\subseteq H$.
\end{Thm}
In particular, we get:
\proofclaim{eq:notsmz}{$X$ is \emph{not} smz iff there is a closed null
set $F$ such that $X+F$ has positive outer Lebesgue measure.}
Again, we will say that the closed null set $F$ ``witnesses'' that
$X$ is not smz (or call $F$ a witness for this fact).
\subsection*{Annotated contents}
\begin{list}{}{\setlength{\leftmargin}{0.5cm}\addtolength{\leftmargin}{\labelwidth}}
\item[Section~\ref{sec:ultralaver}, p. \pageref{sec:ultralaver}:]
We introduce the family of ultralaver forcing notions
and prove some properties.
\item[Section~\ref{sec:janus}, p. \pageref{sec:janus}:]
We introduce the family of Janus forcing notions
and prove some properties.
\item[Section~\ref{sec:iterations}, p. \pageref{sec:iterations}:]
We define ord-transitive models and mention some basic properties.
We define the ``almost finite'' and ``almost countable'' support
iteration over a model.
We show that in many respects they behave like finite
and countable support, respectively.
\item[Section~\ref{sec:construction}, p. \pageref{sec:construction}:]
We introduce the
preparatory forcing notion $\mathbb{R}$ which adds a generic forcing
iteration~$\bar \mathbf{P}$.
\item[Section~\ref{sec:proof}, p. \pageref{sec:proof}:] Putting
everything together, we show that $\mathbb{R}*\mathbf{P}_{\om2}$
forces BC+dBC, i.e., that an uncountable $X$ is neither
smz nor sm. We show this under the assumption $X\in V$,
and then introduce a
factorization
of $\mathbb{R}*\bar \mathbf{P}$ that this assumption does not result in loss
of generality.
\item[Section~\ref{sec:alternativedefs}, p. \pageref{sec:alternativedefs}:] We briefly comment on alternative ways some notions could be defined.
\end{list}
An informal overview of the proof, including two illustrations, can be found
at~\taul{http://arxiv.org/abs/1112.4424/}.
\section{Ultralaver forcing}\label{sec:ultralaver}
In this section, we define the family of \emph{ultralaver forcings} $\mathbb{L}_{\bar
D}$, variants of Laver forcing which depend on a system $\bar D$ of
ultrafilters.
In the rest of the paper, we will use the following properties of $\mathbb{L}_{\bar
D}$.
(And we will use \emph{only} these properties. So readers who are willing to
take these properties for granted could skip to Section~\ref{sec:janus}.)
\begin{enumerate}
\item
$\mathbb{L}_{\bar D}$ is $\sigma$-centered, hence ccc.\label{item:sigmacentered}
\\
(This is Lemma~\ref{lem:newscentered}.)
\item $\mathbb{L}_{\bar D}$ is separative. \\
(This is Lemma~\ref{lem:LDMsep}.)
\item\label{item:absolutepositive}
\emph{Ultralaver kills smz:}
There is a canonical $\mathbb{L}_{\bar D}$-name $\bar{\n\ell}$ for a
fast growing real in~$\omega^\omega$ called the ultralaver real. From this
real, we can define (in an absolute way) a closed null set $F$
such that
$X+F$ is positive for all uncountable $X$ in~$V$ (and therefore
$F$ witnesses that $X$ is not smz, according to Theorem~\ref{thm:pawlikowski}).
\\
(This is Corollary~\ref{cor:absolutepositive}.)
\item
Whenever $X$ is uncountable, then $\mathbb{L}_{\bar D} $ forces that
$X$ is not ``thin''.
\\
(This is Corollary~\ref{cor:LDnotthin}.)
\item
If $(M,\in)$ is a countable model of ZFC$^*$ and if $\mathbb{L}_{\bar D^M}$ is an
ultralaver forcing in $M$,
then for any ultrafilter system $\bar D$ extending $\bar D^M$,
$\mathbb{L}_{\bar D^M} $ is an $M$-complete subforcing of the ultralaver forcing
$\mathbb{L}_{\bar D}$.
\\
(This is Lemma~\ref{lem:LDMcomplete}.)
\\
Moreover, the real $\bar{\n\ell}$ of
item~(\ref{item:absolutepositive}) is so ``canonical'' that we get:
If (in $M$) $\bar{\n\ell}^M$ is the $\mathbb{L}_{\bar D^M}$-name for the
$\mathbb{L}_{\bar D^M}$-generic real,
and if (in $V$) $\bar{\n\ell}$ is the $\mathbb{L}_{\bar D}$-name for the
$\mathbb{L}_{\bar D}$-generic real, and if $H$ is $\mathbb{L}_{\bar D}$-generic over
$V$ and thus $H^M\coloneqq H\cap \mathbb{L}_{\bar D^M}$ is the induced
$\mathbb{L}_{\bar D^M}$-generic filter over $M$, then
$\bar{\n\ell}[H]$ is equal to
$ \bar{\n \ell}^M[H^M]$.
\\
Since the closed null set $F$
is constructed from $\bar{\n\ell}$
in an absolute way, the same holds for $F$, i.e., the
Borel codes $F[H]$ and $F[H^M]$ are the same.
\item
Moreover, given $M$ and $\mathbb{L}_{\bar D^M}$ as above, and a random real
$r$ over~$M$, we can choose $\bar D$ extending $\bar D^M$
such that $\mathbb{L}_{\bar D}$
forces that randomness of~$r$ is
preserved (in a strong way that can be preserved in a
countable support iteration).
\\
(This is Lemma~\ref{lem:extendLDtopreserverandom}.)
\end{enumerate}
\subsection{Definition of ultralaver}
\begin{Notation} We use the following fairly standard notation:
A \emph{tree} is a nonempty
set $p \subseteq \omega^{<\omega}$ which is closed under initial segments
and has no maximal elements.\footnote{Except for the
proof of Lemma~\ref{lem:LDMcomplete},
where we also allow trees with maximal elements, and even empty trees.}
The elements (``nodes'') of a
tree are partially ordered by $\subseteq$.
For each sequence $s\in \omega^{<\omega}$ we write $\lh(s)$ for the
length of $s$.
For any tree $p \subseteq \omega^{<\omega}$ and any $s\in p$ we write
$\suc_p(s)$ for one of the following two sets:
\[ \{ k\in \omega: s^\frown k \in p \} \text{ \ \ or \ \ } \{ t\in p:
(\exists k\in \omega)\;\, t=s^\frown k \} \]
and we rely on the context to help the reader decide which set we
mean.
A \emph{branch} of $p$ is either of the following:
\begin{itemize}
\item A function $f:\omega\to \omega$ with $f\mathord\restriction n\in p$ for all
$n\in \omega$.
\item A maximal chain in the partial order $(p,\subseteq)$. (As
our trees do not have maximal elements, each
such chain $C$ determines a branch $\bigcup C$ in the first sense,
and conversely.)
\end{itemize}
We write $[p]$ for the set of all branches of~$p$.
For any tree $p\subseteq
\omega^{<\omega}$ and any $s\in p$ we write $p^{[s]}$ for the set
$\{t\in p: t \supseteq s \text{ or } t \subseteq s\}$, and we write
$[s]$ for either of the following sets:
\[ \{ t\in p: s \subseteq t \} \text{ \ \ or \ \ } \{ x \in [p]: s
\subseteq x \}. \]
The stem of a tree $p$ is the
shortest $s\in p $ with $|\suc_p(s)|>1$. (The trees we
consider will never be branches, i.e., will always have finite stems.)
\end{Notation}
\begin{Def}\label{def:LD}
\begin{itemize}
\item
For trees $q,p$ we write $q\le p$ if $q \subseteq p$ (``$q$ is stronger
than~$p$''), and
we say that \qemph{$q$ is a pure extension of~$p$} ($q\mathrel{\le_0} p$) if
$q\le p$ and $\stem(q)=\stem(p)$.
\item
A filter system $\bar D$ is a family $(D_s)_{s\in
\omega^{<\omega}}$ of filters on~$\omega$. (All our filters will
contain the Fr\'echet filter of cofinite sets.) We write $D_s^+$
for the collection of $D_s$-positive sets (i.e., sets whose complement
is not in $D_s$).
\item We define $\mathbb{L}_{\bar D} $
to be the set of all trees $p$ such that
$\suc_p(t)\in D_t^+$ for all $t\in p$ above the stem.
\item
The generic filter is determined by the generic branch $ \bar\ell
= (\ell_i)_{i\in \omega}\in \omega^\omega$, called the \emph{generic real}:
$\{\bar\ell\} = \bigcap_{p\in G} [p]$ or equivalently, $ \bar\ell =
\bigcup_{p\in G} \stem(p)$.
\item An ultrafilter system is a filter system consisting of
ultrafilters. (Since all our filters contain the Fr\'echet
filter, we only consider nonprincipal ultrafilters.)
\item
An \emph{ultralaver forcing} is a forcing $\mathbb{L}_{\bar D}$ defined from an
ultrafilter system. The generic real for an ultralaver forcing is
also called the \emph{ultralaver real}.
\end{itemize}
\end{Def}
Recall that a forcing notion $(P,\le)$ is \emph{$\sigma$-centered} if
$P = \bigcup_n P_n$, where for all $n,k\in \omega$ and for all
$p_1,\ldots, p_k\in P_n$ there is $q\le p_1,\ldots, p_k$.
\begin{Lem}\label{lem:newscentered}
All ultralaver forcings $\mathbb{L}_{\bar D}$ are $\sigma$-centered (hence ccc).
\end{Lem}
\begin{proof}
Every finite set of conditions sharing the same stem has a common lower bound.
\end{proof}
\begin{Lem}\label{lem:LDMsep}
$\mathbb{L}_{\bar D}$ is separative.\footnote{See page~\pageref{def:separative} for
the definition.}
\end{Lem}
\begin{proof} If $q\le p$, and $q\not=p$, then there is $s\in p\setminus q$.
Now $p^{[s]} \perp q$.
\end{proof}
If each $D_s$ is the Fr\'echet filter, then $\mathbb{L}_{\bar D}$ is Laver forcing
(often just written $\mathbb{L}$).
\subsection{$M$-complete embeddings}
Note that for all ultrafilter systems
$\bar D$ we have:
\proofclaim{eq:compatible}{
Two conditions in $\mathbb{L}_{\bar D}$ are compatible
if and only if their stems are comparable and moreover, the longer
stem is an element of the condition with the shorter stem.
}
\begin{Lem}\label{lem:LDMcomplete}
Let $M$ be countable.\footnote{Here,
we can assume that $M$ is a
countable transitive model of a sufficiently large finite
subset ZFC$^*$ of ZFC. Later, we will also use ord-transitive models
instead of transitive ones, which does not make any difference
as far as properties of $\mathbb{L}_{\bar D}$ are concerned, as our arguments
take place in transitive parts of such models.}
In~$M$, let $\mathbb{L}_{\bar D^M}$ be an ultralaver forcing. Let $\bar D$
be (in $V$) a filter system extending\footnote{I.e.,
$D_s^M \subseteq D_s$ for all $s\in \omega^{<\omega}$.}
$\bar D^M$.
Then $\mathbb{L}_{\bar D^M} $ is an $M$-complete subforcing of $\mathbb{L}_{\bar D}$.
\end{Lem}
\begin{proof}
For any tree\footnote{Here we also allow empty trees, and trees
with maximal nodes.}~$T$, any filter system
$\bar E = (E_s)_{s\in \omega^{<\omega}}$,
and any ${s_0}\in T$ we define a sequence
$(T_{\bar E,{s_0}}^\alpha)_{\alpha\in \omega_1}$
of ``derivatives''
(where we may abbreviate $T_{\bar E,{s_0}}^\alpha$ to $T^\alpha$)
as follows:
\begin{itemize}
\item $T^0\coloneqq T^{[{s_0}]}$.
\item Given $T^\alpha$, we let
$T^{\alpha+1}\coloneqq T^\alpha \setminus
\bigcup \{ [s] : s\in T^\alpha , {s_0}\subseteq s,
\suc_{T^\alpha}(s)\notin E_s^+ \}$,
where $[s]\coloneqq \{t: s\subseteq t\}$.
\item For limit ordinals $\delta>0$
we let $T^\delta\coloneqq \bigcap_{\alpha<\delta} T^\alpha$.
\end{itemize}
Then we have
\begin{itemize}
\item [(a)] Each $T^\alpha$ is closed under initial segments.
Also: $\alpha < \beta$ implies $ T^\alpha \supseteq T^\beta$.
\item [(b)] There is an $\alpha_0<\omega_1$ such that $T^{\alpha_0} =
T^{\alpha_0+1} = T^\beta$ for all $\beta>\alpha_0$. We write
$T^\infty$ or $T^\infty_{\bar E,{s_0}}$ for $T^{\alpha_0}$.
\item[(c)] If ${s_0}\in T_{\bar E,{s_0}}^\infty$, then $T_{\bar
E,{s_0}}^\infty\in \mathbb{L}_{\bar E}$ with stem~${s_0}$. \\
Conversely, if $\stem(T)={s_0}$, and $T\in \mathbb{L}_{\bar E}$, then
$T^\infty=T$.
\item[(d)] If $T$ contains a tree $q\in \mathbb{L}_{\bar E}$ with
$\stem(q)={s_0}$,
then $T^\infty$ contains $q^\infty=q$,
so in particular ${s_0}\in T^\infty$.
\item[(e)] Thus: $T$ contains a condition in $\mathbb{L}_{\bar E}$ with stem ${s_0}$
iff ${s_0}\in T^\infty_{\bar E,{s_0}}$.
\item[(f)] The computation of $T^\infty$ is absolute between any two models
containing $T$ and $\bar E$. (In particular, any transitive ZFC$^*$-model
containing $T$ and $\bar E$ will also contain
$\alpha_0$.)
\item[(g)] Moreover: Let $T\in M$, $\bar E\in M$, and let $\bar E'$ be a filter system
extending $\bar E$ such that for all ${s_0}$ and all
$A\in {\mathscr P}(\omega)\cap M$ we have: $A\in (E_{s_0})^+$ iff $A\in
(E_{s_0}')^+$.
(In particular, this will be true for any $\bar E'$ extending $\bar E$,
provided that each $E_{s_0}$ is an
$M$-ultrafilter.)
\\
Then
for each $\alpha\in M$ we have $T^\alpha_{\bar E,{s_0}}= T^\alpha_{\bar
E',{s_0}}$ (and hence $T^\alpha_{\bar E',{s_0}}\in M$).
(Proved by induction on~$\alpha$.)
\end{itemize}
Now let $A = (p_i:i\in I)\in M$ be a maximal antichain in
$\mathbb{L}_{\bar D^M}$, and assume (in $V$) that $q\in \mathbb{L}_{\bar D}$. Let
${s_0}\coloneqq \stem(q)$.
We will show that $q$ is compatible with some~$p_i$ (in $\mathbb{L}_{\bar D}$).
This is clear
if there is some $i$ with ${s_0}\in p_i$ and $\stem(p_i)\subseteq
{s_0}$, by~\eqref{eq:compatible}. (In this case, $p_i \cap q$
is a condition in $\mathbb{L}_{\bar D}$ with stem $s_0$.)
So for the rest of the proof
we assume that this is not the case, i.e.:
\proofclaim{eq:not.the.case}{
There is no $i$ with $s_0 \in p_i $ and $\stem(p_i)\subseteq s_0$.
}
Let $J\coloneqq \{ i\in I: {s_0}
\subseteq \stem(p_i)\}$. We claim that there is $j\in J$ with
$\stem(p_j)\in q$ (which as above implies that $q$ and $p_j$ are compatible).
Assume towards a contradiction that this is not the case.
Then $q$ is contained in the following tree $T$:
\begin{align}\label{def:T}
T \coloneqq
(\omega^{<\omega})^{[{{s_0}}]}\setminus
\bigcup _{j\in J} [\stem(p_j)].
\end{align}
Note that $T\in M$. In $V$ we have:
\proofclaim{eq:T.contains.q}{ The tree $T$ contains a condition
$q$ with stem ${s_0}$.}
So by (e) (applied in $V$), followed by (g), and again by (e) (now in $M$) we get:
\proofclaim{eq:T.contains.p}{
The tree $T$ also
contains a condition $p\in M$ with stem ${s_0}$.}
Now $p$ has to be
compatible with some~$p_i$. The sequences ${s_0}=\stem(p)$ and
$\stem(p_i)$ have to be comparable, so by~\eqref{eq:compatible}
there are two possibilities:
\begin{enumerate}
\item $\stem(p_i)\subseteq \stem(p) = s_0 \in p_i$. We have excluded this
case in our assumption \eqref{eq:not.the.case}.
\item $s_0 = \stem(p) \subseteq \stem(p_i)\in p$. So $i\in J$.
By construction of~$T$ (see~\eqref{def:T}),
we conclude $\stem(p_i)\notin T$, contradicting
$\stem(p_i)\in p\subseteq T$ (see~\ref{eq:T.contains.p}).
\qedhere
\end{enumerate}
\end{proof}
\subsection{Ultralaver kills strong measure zero}
The following lemma appears already in \cite[Theorem 9]{MR942525}. We will give a
proof below in Lemma~\ref{lem:pure}.
\begin{Lem}\label{lem:pure.finite}
If $A$ is a finite set, $\n \alpha$ an $\mathbb{L}_{\bar D}$-name, $p\in
\mathbb{L}_{\bar D}$, and $p\Vdash\n \alpha\in A$, then there is
$\beta\in A$ and a pure extension $q\mathrel{\le_0} p $ such that $q\Vdash
\n \alpha=\beta$.
\end{Lem}
\begin{Def}
Let $\bar\ell$ be an increasing sequence of natural numbers.
We say that $X\subseteq 2^\omega$ is
\emph{smz with respect to~$\bar\ell$}, if there
exists a sequence $(I_k)_{k\in\omega}$ of basic intervals
of $2^\omega$ of measure $\leq 2^{-\ell_k}$
(i.e., each $I_k$ is of the form $[s_k]$ for some
$s_k\in 2^{\ell_k }$) such that
$X\subseteq\bigcap_{m\in \omega} \bigcup_{k\ge m} I_k$.
\end{Def}
\begin{Rem}
It is well known and easy to see that the properties
\begin{itemize}
\item For all $\bar\ell$ there exists
exists a sequence $(I_k)_{k\in\omega}$ of basic intervals
of $2^\omega$ of measure $\leq 2^{-\ell_k}$
such that $X\subseteq \bigcup_{k\in\omega} I_k$.
\item For all $\bar\ell$ there exists
exists a sequence $(I_k)_{k\in\omega}$ of basic intervals
of $2^\omega$ of measure $\leq 2^{-\ell_k}$
such that $X\subseteq\bigcap_{m\in \omega} \bigcup_{k\ge m } I_k$.
\end{itemize}
are equivalent. Hence, a set $X$ is smz iff $X$ is smz with respect to all
$\bar\ell\in \omega^\omega$.
\end{Rem}
The following lemma is a variant of the corresponding lemma
(and proof)
for Laver forcing (see for example \cite[Lemma~28.20]{MR1940513}):
Ultralaver makes old uncountable sets non-smz.
\begin{Lem}\label{lem:LDdestroysSMZ}
Let $\bar D$ be a system of ultrafilters, and let $\bar{\n\ell}$ be the
$\mathbb{L}_{\bar D}$-name for the ultralaver real. Then each uncountable set $X \in V$ is forced to be non-smz (witnessed by the ultralaver real $\bar{\n\ell}$).
More precisely, the following holds:
\begin{equation}\label{eq:my_non_smz}
\Vdash_{\mathbb{L}_{\bar D}} \forall X \in V \cap [2^\omega]^{\aleph_1}\;\; \forall (x_k)_{ k \in \omega} \subseteq 2^\omega \;\; X \not\subseteq \bigcap_{m \in \omega} \bigcup_{k \geq m} [x_k \mathord\restriction \n\ell_k].
\end{equation}
\end{Lem}
We first give two technical lemmas:
\begin{Lem}\label{lem:first_technical}
Let $p \in \mathbb{L}_{\bar D}$ with stem $s \in \omega^{<\omega}$, and let $\n x$ be a $\mathbb{L}_{\bar D}$-name for a real in $2^\omega$. Then there exists a pure extension $q \leq_0 p$ and a real $\tau \in 2^\omega$ such that for every $n \in \omega$,
\begin{equation}\label{eq:first_technical}
\{ i \in\suc_q(s):\; q^{[s^\frown i]} \Vdash \n x \mathord\restriction n = \tau \mathord\restriction n \} \in D_s.
\end{equation}
\end{Lem}
\begin{proof}
For each $i \in \suc_p(s)$, let $q_i \leq_0 p^{[s^\frown i]}$ be such that $q_i$ decides
$\n x \mathord\restriction i$,
i.e.,
there is a $t_i$ of length $i$ such that
$q_i \Vdash \n x \mathord\restriction i = t_i$ (this is possible by Lemma~\ref{lem:pure.finite}).
Now we define the real $\tau \in 2^\omega$ as the $D_s$-limit of the $t_i$'s. In more detail: For each $n \in \omega$ there is a (unique) $\tau_n \in 2^n$ such that $\{ i:\; t_i \mathord\restriction n = \tau_n \} \in D_s$; since $D_s$ is a filter, there is a real $\tau \in 2^\omega$ with $\tau \mathord\restriction n = \tau_n$ for each $n$. Finally, let
$q \coloneqq \bigcup_i q_i$.
\end{proof}
\begin{Lem}\label{lem:second_technical}
Let $p \in \mathbb{L}_{\bar D}$ with stem $s$, and let $(\n x_k)_{ k \in \omega}$ be a
sequence of $\mathbb{L}_{\bar D}$-names for reals in $2^\omega$. Then there exists a
pure extension $q \leq_0 p$ and a family of reals $(\tau_\eta)_{ \eta \in q,\,
\eta \supseteq s} \subseteq 2^\omega$ such that for each $\eta \in q$
above~$s$, and every $n \in \omega$,
\begin{equation}\label{eq:second_technical}
\{ i \in \suc_q(\eta):\; q^{[\eta^\frown i]} \Vdash \n x_{|\eta|} \mathord\restriction n = \tau_\eta \mathord\restriction n \} \in D_\eta.
\end{equation}
\end{Lem}
\begin{proof}
We apply Lemma~\ref{lem:first_technical} to each node $\eta$ in $p$ above $s$ (and
to $\n x_{|\eta|}$) separately:
We first get a $p_1 \leq_0 p$ and a $\tau_s \in 2^\omega$;
for every immediate successor $\eta \in \suc_{p_1}(s)$, we get $q_\eta \leq_0
p_1^{[\eta]}$ and a $\tau_\eta \in 2^\omega$, and let $p_2 \coloneqq \bigcup_\eta
q_\eta$; in this way, we get a (fusion) sequence $(p,p_1,p_2,\ldots)$, and let
$q \coloneqq \bigcap_k p_k$.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:LDdestroysSMZ}]
We want to prove~\eqref{eq:my_non_smz}. Assume towards a contradiction that $X$ is an uncountable set in $V$,
and that $(\n x_k)_{ k \in \omega}$ is a sequence of names for reals in $2^\omega$ and $p
\in \mathbb{L}_{\bar D}$ such that
\begin{equation}\label{eq:towards_smz_contra}
p \Vdash X \subseteq \bigcap_{m \in \omega} \bigcup_{k \geq m} [\n x_k \mathord\restriction \n\ell_k].
\end{equation}
Let $s \in \omega^{<\omega}$ be the stem of $p$.
By Lemma~\ref{lem:second_technical}, we can fix a pure extension $q \leq_0 p$ and a family $(\tau_\eta)_{\eta \in q,\, \eta \supseteq s} \subseteq 2^\omega$ such that for each $\eta \in q$ above the stem $s$ and every $n \in \omega$, condition~\eqref{eq:second_technical} holds.
Since $X$ is (in $V$ and) uncountable, we can find a real $x^* \in X$ which is
different from each real in the countable family $(\tau_\eta)_{\eta \in q,\, \eta
\supseteq s}$; more specifically, we can pick a family of natural numbers
$(n_\eta)_{\eta \in q,\, \eta \supseteq s}$ such that $x^* \mathord\restriction n_\eta \neq \tau_\eta
\mathord\restriction n_\eta$ for any $\eta$.
We can now find $r\le_0 q$ such that:
\begin{itemize}
\item For all $\eta\in r$ above $s$ and all $i\in \suc_r(\eta)$
we have $i > n_\eta$.
\item For all $\eta\in r$ above $s$ and all $i\in \suc_r(\eta)$
we have $r^{[\eta^\frown i]} \Vdash \n x_{|\eta|} \mathord\restriction n_\eta =
\tau_\eta\mathord\restriction n_\eta \not= x^*\mathord\restriction n_\eta$.
\end{itemize}
So for all $\eta\in r$ above $s$ we have, writing $k$ for $|\eta|$, that
$r^{[\eta^\frown i]} $ forces $x^*\notin [ \n x_k \mathord\restriction n_\eta] \supseteq
[\n x_k \mathord\restriction \ell_k ] $.
We conclude that
$r$ forces $x^* \notin \bigcup_{k \ge |s|} [\n x_k \mathord\restriction \ell_k] $,
contradicting
\eqref{eq:towards_smz_contra}.
\end{proof}
\begin{Cor}\label{cor:LDdestroysSMZ}
Let $(t_k)_{k\in \omega}$ be a dense subset of $2^{\omega}$.
Let $\bar D$ be a system of ultrafilters, and let $\bar{\n\ell}$ be the
$\mathbb{L}_{\bar D}$-name for the ultralaver real. Then the set $$ \n H\coloneqq
\bigcap_{m\in\omega} \bigcup _{k\ge m} [ t_k \mathord\restriction {\n \ell_k}] $$
is forced to be a
comeager set with the property that $\n H$ does not contain any
translate of any old uncountable set.
\end{Cor}
Pawlikowski's theorem~\ref{thm:pawlikowski} gives us:
\begin{Cor}\label{cor:absolutepositive}
There is a canonical name $F$ for a closed null set such that
$X+F$ is positive for all uncountable $X$ in~$V$.
In particular, no uncountable ground model set is smz in the
ultralaver extension.
\end{Cor}
\subsection{Thin sets and strong measure zero}
\label{sec:thin}
For the notion of ``(very) thin'' set, we use an increasing
function $B^*(k) $ (the function we use will be described in
Corollary~\ref{cor:tomek}). We will
assume that $\bar\ell^*=(\ell^*_k)_{k\in\omega}$ is an
increasing sequence of natural numbers with $\ell^*_{k+1} \gg B^*(k)$.
(We will later use a subsequence
of the ultralaver real~$\bar\ell$ as~$\bar\ell^*$, see Lemma~\ref{lem:subsequence}).
\begin{Def}\label{def:thin}
For $X \subseteq 2^\omega$ and $k\in
\omega$ we write $X\mathord\restriction [\ell^*_k,\ell^*_{k+1}) $ for the set
$\{x\mathord\restriction [\ell^*_k,\ell^*_{k+1}) : x\in X\}$. We say that
\begin{itemize}
\item $X \subseteq 2^\omega$ is
\qemph{very thin with respect to $\bar
\ell^*$ and~$B^*$}, \ if there are infinitely many $k$ with $|X\mathord\restriction
[\ell^*_k,\ell^*_{k+1})|\le B^*(k) $.
\item $X\subseteq 2^\omega$ is \qemph{thin with respect to $\bar \ell^*$ and~$B^*$}, \ if $X$ is the union
of countably many very thin sets.
\end{itemize}
\end{Def}
Note that the family of thin sets is a $\sigma$-ideal, while the family of
very thin sets is not even an ideal. Also, every very thin set is covered by a
closed very thin (in particular nowhere dense) set. In particular, every thin
set is meager and the ideal of thin sets is a proper ideal.
\begin{Lem}\label{lem:subsequence}
Let $B^*$ be an increasing function.
Let $\bar\ell$ be an increasing sequence of natural numbers.
We define
a subsequence $\bar\ell^*$ of $\bar\ell$ in the following way:
$\ell^*_k=\ell_{n_k}$ where $n_{k+1}-n_k=B^*({k})\cdot 2^{\ell^*_k}$.
\\ Then we get: If $X$ is thin with respect to $\bar\ell^*$ and $B^*$,
then $X$ is smz with respect to~$\bar\ell$.
\end{Lem}
\begin{proof}
Assume that $X=\bigcup_{i\in\omega} Y_i$, each $Y_i$ very thin
with respect to~$\bar\ell^*$ and $B^*$.
Let $(X_j)_{j\in \omega}$ be an enumeration of $\{Y_i:i\in \omega\}$
where each $Y_i$ appears infinitely often. So $X \subseteq
\bigcap_{m\in \omega} \bigcup_{j\ge m} X_j$.
By induction on~$j\in\omega$, we find for all $j>0$ some $k_j>k_{j-1}$
such that
\[
|X_j\mathord\restriction [\ell^*_{k_j},\ell^*_{k_j+1}) |\leq B^*({k_j})
\quad\text{hence}\quad
|X_j\mathord\restriction [0,\ell^*_{k_j+1}) |\leq B^*({k_j})\cdot 2^{\ell^*_{k_j}}
= n_{k_j+1}-n_{k_j}.
\]
So we can enumerate $X_j\mathord\restriction [0,\ell^*_{k_j+1}) $ as $(s_i)_{n_{k_j}\leq
i<n_{k_{j}+1}}$. Hence $X_j$ is a subset of $\bigcup_{n_{k_j}\leq
i<n_{k_{j}+1}} [s_i]$;
and each $s_i $ has length $\ell^*_{k_j+1}\geq \ell_i$,
since $\ell^*_{k_j+1}=\ell_{n_{k_j+1}}$ and $i<n_{k_j+1}$.
This implies
\[ X \subseteq
\bigcap_{m\in \omega} \bigcup_{j\ge m} X_j \subseteq
\bigcap_{m\in \omega} \bigcup_{i\ge m} [s_i]. \]
Hence $X$ is smz with respect to~$\bar\ell$.
\end{proof}
Lemma~\ref{lem:LDdestroysSMZ}
and Lemma~\ref{lem:subsequence} yield:
\begin{Cor}\label{cor:LDnotthin}
Let $B^*$ be an increasing function.
Let $\bar D$ be a system
of ultrafilters, and $\n{\bar\ell}$ the name for the ultralaver real.
Let $\n{\bar\ell}^*$ be constructed from $B^*$ and $\n{\bar\ell}$ as in
Lemma~\ref{lem:subsequence}.
\\
Then $\mathbb{L}_{\bar D}$ forces that for every uncountable~$X\subseteq 2^\omega$:
\begin{itemize}
\item $X$ is not smz with respect to~$\n{\bar \ell}$.
\item $X$ is not thin with respect
to~$\n{\bar\ell}^*$ and~$B^*$.\label{item:LDnotthin}
\end{itemize}
\end{Cor}
\subsection{Ultralaver and preservation of Lebesgue positivity}\label{ss:ultralaverpositivity}
It is well known that both Laver forcing and random forcing preserve
Lebesgue positivity; in fact they satisfy a stronger property that is preserved
under countable support iterations.
(So in particular, a countable support iteration of Laver
and random also preserves positivity.)
Ultralaver forcing $\mathbb{L}_{\bar D}$ will in general not preserve
positivity. Indeed, if all ultrafilters $D_s$ are equal to the same
ultrafilter $D^*$,
then the range $L\coloneqq \{\ell_0, \ell_1, \ldots \} \subseteq \omega $
of the ultralaver real $\bar \ell$ will diagonalize $D^*$, so every
ground model real $x\in 2^\omega$ (viewed as a subset of $\omega$) will either
almost contain $L$ or be almost disjoint to $L$, which implies that
the set $2^\omega\cap V$
of old reals is covered by a null set in the
extension.
However, later in this paper it will become clear that if we choose the
ultrafilters $D_s$ in a sufficiently generic way, then many old positive sets
will stay positive.
More specifically, in this section we will show
(Lemma~\ref{lem:extendLDtopreserverandom}): If $\bar D^M$ is an ultrafilter
system in a countable model $M$ and $r$ a random real over $M$,
then we can find an extension $\bar D$ such that $\mathbb{L}_{\bar D}$ forces that
$r$ remains random over $M[H^M]$
(where $H^M$ denotes the $\mathbb{L}_{\bar D}$-name for the restriction of the
$\mathbb{L}_{\bar D}$-generic filter $H$ to $\mathbb{L}_{\bar D^M}\cap M$).
Additionally, some ``side conditions''
are met, which are necessary to preserve the property in forcing iterations.
In Section~\ref{subsec:almostCS} we will see how to use this property to
preserve randoms in limits.
The setup we use for preservation of randomness is basically the notation of
``Case A'' preservation introduced in~\cite[Ch.XVIII]{MR1623206}, see also
\cite{MR1234283,MR2214624} or the textbook~\cite[6.1.B]{MR1350295}:
\begin{Def}\label{def:nullset}
We write $\textsc{clopen}$ for the collection of clopen sets on $2^\omega$.
We say that the function $Z:\omega\to \textsc{clopen}$
is a code for a null set, if
the measure of $Z(n)$ is at most $ 2^{-n}$ for each~$n\in \omega $.
For such a code $Z$,
the set $\nullset(Z)$ coded by $Z$ is
\[
\nullset(Z)\coloneqq \bigcap_n \bigcup_{k\ge n} Z(k).
\]
\end{Def}
The set $\nullset(Z)$ obviously is a null set, and it is well known
that every null set is contained in such a set $\nullset(Z)$.
\begin{Def}\label{def:sqsubset}
For a real $r$ and any code $Z$, we define $Z \sqsubset_n
r$ by:
\[
(\forall k\geq n) \ r\notin Z(k).
\]
We write $Z \sqsubset r$ if $Z \sqsubset_n r$ holds for
some~$n$;
i.e., if $r\notin \nullset(Z)$.
\end{Def}
For later reference, we record the following trivial fact:
\proofclaim{eq:sq.n}{
$p \Vdash \n Z \sqsubset r$ iff
there is a name
$\n n $ for an element of $\omega$
such that $p\Vdash \n Z \sqsubset_{\n n} r$.
}
Let $P$ be a forcing notion, and $\n Z$ a $P$-name of a code for a null set. An
interpretation of $\n Z$ below $p$ is some code $Z^*$ such that there is a
sequence $p=p_0\geq p_1\geq p_2\geq \dots$ such that $p_m$ forces $\n Z \mathord\restriction m=
Z^*\mathord\restriction m$. Usually we demand (which allows a simpler proof of the
preservation theorem at limit stages) that the sequence $(p_0,p_1,\dots)$ is
inconsistent, i.e., $p$ forces that there is an $m$ such that $p_m\notin G$.
Note that whenever $P$ adds a new $\omega$-sequence of ordinals, we can find
such an interpretation for any~$\n Z$.
If $\n{\bar Z}=(\n Z_1,\ldots, \n Z_m)$ is a tuple of names of codes for null sets, then
an interpretation of $\bar{\n Z}$ below $p$ is some tuple $(Z_1^*,\ldots, Z_m^*)$ such that there is a
single sequence $p=p_0\geq p_1\geq p_2\geq \dots$ interpreting each $\n Z_i$ as $Z_i^*$.
We now turn to preservation of Lebesgue positivity:
\begin{Def} \label{def:random.random.random}
\begin{enumerate}
\item
A forcing notion $P$ \emph{preserves Borel outer measure}, if $P$
forces $\Leb^*(A^V)=\Leb(A^{V[G_P]})$ for every code $A$ for a Borel
set. ($\Leb^*$ denotes the outer Lebesgue measure, and for a
Borel code $A$ and a set-theoretic universe~$V$, $A^V$ denotes the
Borel set coded by $A$ in~$V$.)
\item
$P$ \emph{strongly preserves randoms}, if the following holds: Let
$N\prec H(\chi^*)$ be countable for a sufficiently large regular cardinal
$\chi^*$,
let $P,p, \bar {\n Z} = (\n Z_1,\ldots, \n Z_m)\in N$,
let $p\in P$ and let $r$ be random
over~$N$. Assume that in~$N$, $\bar Z^* $ is an interpretation of $\n
{\bar Z}$, and assume $Z_i^*\sqsubset_{k_i} r$ for each~$i$. Then there is an $N$-generic
$q\le p$ forcing that $r$ is still random over~$N[G]$ and
moreover, $\n Z_i\sqsubset_{k_i} r$ for each~$i$.
(In particular, $P$ has to be proper.)
\item Assume that $P$ is absolutely definable. $P$ \emph{strongly
preserves randoms over countable models} if (2) holds for all
countable (transitive\footnote{Later we will introduce
ord-transitive models, and it is easy to see that it does not make
any difference whether we demand transitive or not; this can be seen
using a transitive collapse.}) models $N$ of~ZFC$^*$.
\end{enumerate}
\end{Def}
It is easy to see that these properties are increasing in strength.
(Of course (3)$\Rightarrow$(2) works only if ZFC$^*$ is satisfied in~$H(\chi^*)$.)
In~\cite{MR2155272} it is shown that (1) implies (3), provided that $P$ is nep
(``non-elementary proper'',
i.e., nicely definable and proper with respect to countable models). In
particular, every Suslin ccc forcing notion such as random forcing, and also
many tree forcing notions including Laver forcing, are nep. However $\mathbb{L}_{\bar
D}$ is not nicely definable in this sense, as its definition uses ultrafilters
as parameters.
\begin{Lem}\label{lem:random.laver}
Both Laver forcing and random forcing strongly preserve randoms over
countable models.
\end{Lem}
\begin{proof}
For random forcing, this is easy and well known (see, e.g.,
\cite[6.3.12]{MR1350295}).
For Laver forcing: By the above, it is enough to show (1).
This was done by
Woodin (unpublished) and Judah-Shelah~\cite{MR1071305}. A nicer proof
(including a variant of (2)) is given by Pawlikowski~\cite{MR1367136}.
\end{proof}
Ultralaver will generally not preserve Lebesgue positivity, let alone
randomness. However, we get the following ``local'' variant of strong
preservation of randoms (which will be used in the preservation
theorem~\ref{lem:iterate.random}).
The rest of this section will be devoted to the proof of the following lemma.
\begin{Lem}\label{lem:extendLDtopreserverandom}
Assume that $M$ is a countable model,
$\bar D^M$ an
ultrafilter system in $M$ and
$r$ a random real over $M$. Then there is (in $V$) an
ultrafilter system $\bar D$ extending
\footnote{This implies, by
Lemma~\ref{lem:LDMcomplete}, that the $\mathbb{L}_{\bar D}$-generic
filter~$G$ induces an $\mathbb{L}_{\bar D^M}$-generic filter over~$M$,
which we call~$G^M$.}
$\bar D^M$, such that the following holds:
\\
\textbf{If}
\begin{itemize}
\item $p\in \mathbb{L}_{\bar D^M}$,
\item in $M$, $\n {\bar Z} = ( \n Z_1, \ldots , \n Z_m) $
is a sequence of $\mathbb{L}_{\bar D^M}$-names for
codes for null sets,\footnote{Recall that $\nullset(\n Z)= \bigcap_n
\bigcup_{k\ge n} \n Z(k)$ is a null set in the
extension.} and $Z_1^*,\dots , Z_m^*$ are interpretations under~$p$,
witnessed by a sequence $(p_n)_{n\in \omega}$ with
strictly increasing\footnote{It is enough to assume that the lengths of the
stems diverge to infinity; any thin enough subsequence
will then have strictly increasing stems and will still
interpret each $\n Z_i$ as $Z_i^*$.} stems,
\item $Z^*_i \sqsubset_{k_i} r$ for $i=1,\dots, m$,
\end{itemize}
\textbf{then} there is a $q\leq p$ in $\mathbb{L}_{\bar D}$ forcing that
\begin{itemize}
\item $r$ is random over $M[G^M]$,
\item $\n Z_i \sqsubset_{k_i} r$ for $i=1,\dots, m$.
\end{itemize}
\end {Lem}
For the proof of this lemma, we will use the following concepts:
\begin{Def}
Let $p\subseteq \omega^{< \omega} $ be a tree.
A \qemph{front name below $p$}
is
a function\footnote{Instead of $\textsc{clopen}$
we may also consider other ranges of front names,
such as the class of all ordinals, or the set $\omega$.}
$h:F\to \textsc{clopen}$, where $F\subseteq p$ is a front (a set that
meets every branch of~$p$ in a unique point). (For notational simplicity
we also allow $h$ to be defined on elements $\notin p$; this way,
every front name below $p$ is also a front name below $q$ whenever
$q\le p$.)
If $h$ is a front name and $\bar D$ is any filter system with $p\in \mathbb{L}_{\bar D}$,
we define the corresponding $\mathbb{L}_{\bar D}$-name (in the sense of forcing)
$\n z^h $ by
\begin{align}\label{def:n.alpha}
\n z^h\coloneqq \{ ( \check y, p^{[s]}): s\in F,\ y \in h(s)\}.
\end{align}
(This does not depend on the $\bar D$ we use, since we set
$\check y\coloneqq \{(\check x, \omega^{<\omega} ): x \in y \}$.)
Up to forced equality, the name $\n z^h$ is characterized
by the fact that $p ^{[s]} $ forces (in any ${\mathbb{L}_{\bar D}}$) that
$ \n z ^h = h(s)$, for every $s$ in the domain of $h$.
\end{Def}
Note that the same object~$h$ can be viewed as a front name below $p$ with respect
to different forcings $\mathbb{L}_{\bar D_1}$, $ \mathbb{L}_{\bar D_2}$, as long
as $p\in \mathbb{L}_{\bar D_1}\cap \mathbb{L}_{\bar D_2}$.
\begin{Def}
Let $p \subseteq \omega^{<\omega}$ be a tree.
A \qemph{continuous name below $p$}
is either of the following:
\begin{itemize}
\item An $\omega$-sequence of front names below $p$.
\item A $\subseteq$-increasing function $g:p\to \textsc{clopen}^{<\omega}$
such that $\lim_{n\to \infty } \lh(g(c\mathord\restriction n)) =\infty$
for every branch $c\in [p]$.
\end{itemize}
For each $n$, the set of minimal elements
in $\{ s\in p: \lh(g(s)) > n \}$ is a front, so each
continuous name in the second sense naturally
defines a name
in the first sense, and conversely.
Being a continuous name below $p$ does not involve the notion of $\Vdash$
nor does it depend on the filter system~$\bar D$.
If $g$ is a continuous name and $\bar D$ is any filter system,
we can again
define the corresponding $\mathbb{L}_{\bar D}$-name $\n Z^g $
(in the sense of forcing);
we leave a formal definition of $\n Z^g$ to the reader
and content ourselves with this characterization:
\begin{align}\label{def:z.g}
(\forall s\in p): p^{[s]} \Vdash_{\mathbb{L}_{\bar D}} g(s) \subseteq \n Z^g .
\end{align}
\end{Def}
Note that a continuous name below $p$ naturally corresponds
to a continuous function $F:[p] \to \textsc{clopen}^\omega$, and $ \n Z^g$ is forced
(by~$p$) to be the value of $F$ at the generic real $\n {\bar \ell}$.
\begin{Lem}\label{lem:pure}
$\mathbb{L}_{\bar D}$ has the following ``pure decision properties'':
\begin{enumerate}
\item\label{item:pure.one} Whenever ${\n y}$ is a name for an
element of $\textsc{clopen}$,
$p\in \mathbb{L}_{\bar D}$, then there is a pure extension $p_1\mathrel{\le_0} p$
such that $\n{y} = \n z^h $ (is forced)
for a front name $h$ below~$p_1$.
\item\label{item:pure.omega} Whenever ${\n Y }$ is a name for a sequence of
elements of $\textsc{clopen}$, $p\in \mathbb{L}_{\bar D}$, then there is a pure extension
$q\mathrel{\le_0} p$ such that ${\n Y } = \n Z ^g$ (is forced) for some
continuous name $g$ below~$q$.
\item\label{item:pure.finite} (This is Lemma~\ref{lem:pure.finite}.)
If $A$ is a finite set, $\n \alpha$ a name, $p\in
\mathbb{L}_{\bar D}$, and $p$ forces $\n \alpha\in A$, then there is
$\beta\in A$ and a pure extension $q\mathrel{\le_0} p $ such that $q\Vdash
\n \alpha=\beta$.
\end{enumerate}
\end{Lem}
\begin{proof}
Let $p\in \mathbb{L}_{\bar D}$, $s_0\coloneqq \stem(p)$, $\n y$ a name for an
element of $\textsc{clopen}$.
We call $t\in p$ a ``good node in $p$'' if $\n y$ is a front name
below~$p^{[t]}$ (more formally: forced to be equal to
$\n z^h$ for a front name $h$).
We can find $p_1\mathrel{\le_0} p$
such that for all $t\in p_1$ above $s_0$: If there is $q\mathrel{\le_0} p_1^{[t]}$
such that $t$ is good in~$q$, then $t$ is already good in~$p_1$.
We claim that $s_0$ is now good (in~$p_1$). Note that for any bad
node $s$ the set $\{\,t\in \suc_{p_1}(s): \ t \text{ bad}\,\}$ is
in~$D_s^+$. Hence, if $s_0$ is bad, we can inductively construct
$p_2\mathrel{\le_0} p_1$ such that all nodes of $p_2$ are bad nodes in~$p_1$.
Now let $q\le p_2$ decide $\n y$, $s\coloneqq \stem(q)$.
Then $q \mathrel{\le_0} p_1^{[s]}$, so $s$ is good in~$p_1$, contradiction.
This finishes the proof of (\ref{item:pure.one}).
To prove (\ref{item:pure.omega}), we first construct $p_1$ as in (\ref{item:pure.one}) with respect to $\n
y_0$. This gives a front $F_1\subseteq p_1$ deciding $\n
y_0$. Above each node in $F_1$ we now repeat the construction
from (\ref{item:pure.one}) with respect to $\n y_1$, yielding $p_2$, etc.
Finally, $q\coloneqq \bigcap_ n p_n$.
To prove (\ref{item:pure.finite}): Similar to (\ref{item:pure.one}), we can
find $p_1\mathrel{\le_0} p$ such that
for each $t\in p_1$: If there is a pure extension of $p_1^{[t]}$
deciding $\n\alpha$, then $p_1^{[t]}$ decides $\n \alpha$; in
this case we again call $t$ good. Since there are only finitely many
possibilities for the value of $\n \alpha$, any bad node $t$ has
$D_t^+$ many bad successors. So if the stem of $p_1$ is bad, we
can again reach a contradiction as in (\ref{item:pure.one}).
\end{proof}
\begin{Cor}\label{cor:obda.continuous}
Let $\bar D$ be a filter system, and let $G\subseteq \mathbb{L}_{\bar D}$
be generic. Then every $Y \in \textsc{clopen}^\omega$ in $V[G]$
is the evaluation of a continuous name $\n Z^g$ by $G$.
\end{Cor}
\begin{proof} In $V$, fix a $p\in \mathbb{L}_{\bar D}$ and a name $\n Y $ for an
element of $ \textsc{clopen}^\omega$.
We can find $q\le_0 p$
and a continuous name $g$ below $q$ such that $q \Vdash \n Y = \n Z^g$.
\end{proof}
We will need the following modification of the concept of ``continuous
names''.
\begin{Def}
Let $p \subseteq \omega^{<\omega}$ be a tree, $b\in [p]$ a branch.
An \qemph{almost continuous name below~$p$ (with respect to~$b$)}
is
a $\subseteq$-increasing function $g:p\to \textsc{clopen}^{<\omega}$
such that $\lim_{n\to \infty } \lh(g(c\mathord\restriction n)) =\infty$
for every branch $c\in [p]$, except possibly for $c=b$.
\end{Def}
Note that ``except possibly for $c=b$'' is the only difference between this definition and the definition of a continuous name.
Since for any $\bar D$ it is forced\footnote{
This follows from our assumption
that all our filters contain the Fr\'echet filter.}
that the generic real (for
$\mathbb{L}_{\bar D}$) is not equal to the exceptional branch $b$, we again get
a name $\n Z^g$ of a function in $\textsc{clopen}^\omega$ satisfying:
\[ (\forall s\in p): p^{[s]} \Vdash_{\mathbb{L}_{\bar D}} g(s) \subseteq \n Z^g. \]
An almost continuous name naturally corresponds to a
continuous function $F$ from $[p] \setminus \{b\}$ into
$\textsc{clopen}^\omega$.
Note that being an almost
continuous name is a very simple combinatorial
property of $g$ which does not depend on $\bar D$, nor does it
involve the notion $\Vdash$.
Thus, the same function $g$
can be viewed as an almost continuous name for two different
forcing notions $\mathbb{L}_{\bar D_1}$, $\mathbb{L}_{\bar D_2}$ simultaneously.
\begin{Lem} \label{lem:nicefy}
Let $\bar D$ be a system of filters
(not necessarily ultrafilters).
Assume that $\bar p = (p_n)_{n\in \omega}$ witnesses that $Y^*$
is an interpretation of~$\n Y$, and that the lengths of the stems of the $p_n$
are strictly increasing.\footnote{It is easy to see that for every
$\mathbb{L}_{\bar D}$-name $\n Y$ we can find such $\bar p$ and~$Y^*$:
First find $\bar p$ which interprets both $\n Y$ and $\bar{\n\ell}$,
and then thin out to get a strictly increasing sequence of stems.}
Then there exists a sequence $\bar q = (q_n)_{n\in \omega}$ such
that
\begin{enumerate}
\item $q_0\ge q_1\ge \cdots $.
\item $q_n\le p_n$ for all~$n$.
\item $\bar q$ also interprets $\n Y $ as~$Y^*$.
(This follows from the previous two statements.)
\item $\n Y$ is almost continuous below~$q_0$, i.e., there is
an almost continuous name $g$ such that $q_0$ forces
$\n Y = \n Z^g $.
\item $\n Y$ is almost continuous below~$q_n$, for all~$n$.
(This follows from the previous statement.)
\end{enumerate}
\end{Lem}
\begin{proof}
Let $b$ be the branch described by the stems of the conditions
$p_n$: \[b\coloneqq \{ s: (\exists n)\, s \subseteq \stem(p_n)\}.\]
We now construct
a condition~$q_0$.
For every $s\in b$ satisfying $\stem(p_n) \subseteq s \subsetneq
\stem(p_{n+1})$ we set $\suc _{q_0}(s) = \suc_{p_n}(s)$, and for all
$t\in \suc_{q_0}(s)$ except for the one in~$b$ we let $q_0^{[t]}
\mathrel{\le_0} p_n^{[t]} $ be such that $\n Y$ is continuous below
$q_0^{[t]}$. We can do this by Lemma~\ref{lem:pure}(\ref{item:pure.omega}).
Now we set
\[ q_n\coloneqq p_n \cap q_0 = q_0^{[\stem(p_n)]} \le p_n. \]
This takes care of~(1) and~(2). Now we show~(4):
Any branch $c$ of $q_0$ not equal to $b$ must contain a
node $s^\frown k\notin b$ with $s\in b$, so $c$ is a branch in
$q_0^{[s^\frown k]}$, below which $\n Y $ was continuous.
\end{proof}
The following lemmas and corollaries
are the motivation for considering continuous and
almost continuous names.
\begin{Lem} Let $\bar D$ be a system of filters
(not necessarily ultrafilters).
Let $p\in \mathbb{L}_{\bar D}$, let $b$ be a branch, and let
$g:p\to \textsc{clopen}^{<\omega}$ be an
almost continuous name
below~$p$ with respect to~$b$; write
$\n Z^g$ for the associated $\mathbb{L}_{\bar D}$-name.
Let $r\in 2^ \omega$ be a real, $n_0\in \omega$.
Then the following are equivalent:
\begin{enumerate}
\item $p\Vdash_{\mathbb{L}_{\bar D}} r \notin \bigcup_{n\ge n_0} \n Z^g(n)$,
i.e., $ \n Z^g \sqsubset_{n_0} r$.
\item For all $n\ge n_0$ and for all $s\in p $ for which
$g(s)$ has length $>n$ we have $r \notin g(s)(n)$.
\end{enumerate}
\end{Lem}
Note that (2) does not mention the notion $\Vdash$ and
does not depend on $\bar D$.
\begin{proof}
$\lnot$(2) $\Rightarrow$ $\lnot$(1):
Assume that there is $s\in p $ for which $g(s)= (C_0,\ldots,
C_n, \ldots, C_k)$ and $r\in C_n$. Then $p^{[s]}$ forces that the
generic sequence $\n Z^g = ( \n Z(0), \n Z(1), \ldots)$ starts with
$C_0,\ldots, C_n$, so $p^{[s]}$ forces $r\in \n Z^g(n)$.
$\lnot$(1) $\Rightarrow$ $\lnot$(2): Assume that $p$ does not force
$r \notin \bigcup_{n\ge n_0} \n Z^g(n)$. So there is a condition
$q\le p$ and some $n\ge n_0$ such that $q \Vdash r\in \n Z^g(n)$. By
increasing the stem of~$q$, if necessary, we may assume that
$s\coloneqq \stem(q)$ is not on $b$ (the ``exceptional'' branch), and
that $g(s)$ has already length~$>n$. Let $C_n\coloneqq g(s)(n)$ be the
$n$-th entry of~$g(s)$. So $p^{[s]}$ already forces $\n Z^g(n) =
C_n$; now $q^{[s]}\le p^{[s]}$, and $q^{[s]}$ forces the
following statements: $r\in \n Z^g(n) $, $\n Z^g(n) = C_n$. Hence $r\in
C_n$, so (2) fails.
\end{proof}
\begin{Cor}\label{cor:z.absolute}
Let $\bar D_1$ and $\bar D_2$ be systems of filters, and assume that
$p$ is in $\mathbb{L}_{\bar D_1} \cap \mathbb{L}_{\bar D_2}$. Let $g:p \to
\textsc{clopen}^{<\omega}$ be an
almost continuous name of a sequence of clopen sets, and
let $\n Z^g_1$ and $\n Z^g_2$ be the associated $\mathbb{L}_{\bar D_1}$-name
and $\mathbb{L}_{\bar D_2}$-name, respectively.
Then for any real $r$ and $n\in \omega$ we have
\[ p \Vdash_{\mathbb{L}_{\bar D_1}} \n Z^g_1 \sqsubset_n r
\ \ \Leftrightarrow \ \
p \Vdash_{\mathbb{L}_{\bar D_2}} \n Z^g_2 \sqsubset_n r.
\]
\end{Cor}
(We will use this corollary for the special case that $\mathbb{L}_{\bar D_1}$
is an ultralaver forcing, and $\mathbb{L}_{\bar D_2}$ is Laver forcing.)
\begin{Lem}
Let $\bar D_1$ and $\bar D_2$ be systems of filters, and assume that
$p$ is in $\mathbb{L}_{\bar D_1} \cap \mathbb{L}_{\bar D_2}$. Let $g:p \to
\textsc{clopen}^{<\omega}$ be a continuous
name of a
sequence of clopen sets, let $F \subseteq p$ be a front
and let $h:F\to \omega$ be a front name.
Again we will write $\n Z^g_1, \n Z^g_2$ for the associated names
of codes for null sets,
and we will write $\n n_1$ and $\n n_2$ for the associated
$\mathbb{L}_{\bar D_1}$- and
$\mathbb{L}_{\bar D_2}$-names, respectively, of natural numbers.
Then for any real $r$ we have:
\[p \Vdash_{\mathbb{L}_{\bar D_1}} \n Z^g_1 \sqsubset_{\n n_1} r
\ \ \Leftrightarrow \ \
p \Vdash_{\mathbb{L}_{\bar D_2}} \n Z^g_2 \sqsubset_{\n n_2} r.\]
\end{Lem}
\begin{proof}
Assume $p \Vdash_{\mathbb{L}_{\bar D_1}} \n Z^g_1 \sqsubset_{\n n_1} r$.
So for each $s\in F$ we have:
$p^{[s]}\Vdash_{\mathbb{L}_{\bar D_1}} \n Z^g_1 \sqsubset_{h(s) } r$.
By
Corollary~\ref{cor:z.absolute}, we also have
$p^{[s]}\Vdash_{\mathbb{L}_{\bar D_2}} \n Z^g_2 \sqsubset_{h(s)} r$.
So also
$p^{[s]}\Vdash_{\mathbb{L}_{\bar D_2}} \n Z^g_2 \sqsubset_{\n n_2} r$ for each $s\in F$.
Hence
$p\Vdash_{\mathbb{L}_{\bar D_2}} \n Z^g_2 \sqsubset_{\n n_2} r$.
\end{proof}
\begin{Cor}\label{cor:stays.random}
Assume $q\in \mathbb{L}$ forces in Laver forcing that
$ \n Z^{g_k} \sqsubset r$ for $k=1,2,\ldots$,
where each $g_k$ is a continuous name of a code for a
null set.
Then there is a Laver condition $q'\mathrel{\le_0} q$ such that for all
filter systems $\bar D$ we have:
\begin{quote} If $q'\in \mathbb{L}_{\bar D}$, then $q'$
forces (in ultralaver forcing ${\mathbb{L}_{\bar D}}$) that $
\n Z^{g_k} \sqsubset r$ for all $k$.
\end{quote}
\end{Cor}
\begin{proof} By \eqref{eq:sq.n} we can find a sequence $(\n n _k)_{k=1}^\infty$ of
$\mathbb{L}$-names such that $q\Vdash \n Z^{g_k} \sqsubset_{\n n_k} r$ for
each $k$. By Lemma~\ref{lem:pure}(\ref{item:pure.omega})
we can find $q'\mathrel{\le_0} q$ such that this sequence is continuous
below $q'$. Since each $\n n_k$ is now a front name below $q'$, we
can apply the previous lemma.
\end{proof}
\begin{Lem}\label{lem:continuous.is.enough}
Let $M$ be a countable model, $r\in 2^\omega$, $\bar D^M\in M$
an ultrafilter system, $\bar D $ a filter system extending $\bar D^M$, $q\in \mathbb{L}_{\bar D}$.
For any $V$-generic filter $G\subseteq \mathbb{L}_{\bar D}$ we write
$G^M$ for the ($M$-generic, by Lemma~\ref{lem:LDMcomplete}) filter on
$\mathbb{L}_{\bar D^M}$.
The following are equivalent:
\begin{enumerate}
\item $q\Vdash _{\mathbb{L}_{\bar D}} r $ is random over $M[G^M]$.
\item For all names $\n Z\in M$ of codes for null sets: $q\Vdash_{\mathbb{L}_{\bar D}} \n Z \sqsubset r $.
\item For all continuous names $g\in M$:
$q\Vdash_{\mathbb{L}_{\bar D}} \n Z^g \sqsubset r $.
\end{enumerate}
\end{Lem}
\begin{proof} (1)$\Leftrightarrow$(2) holds because every null set is contained in a set of the form $\nullset(Z)$, for some code $Z$.
(2)$\Leftrightarrow$(3): Every code for a null set in $M[G^M]$
is equal to~$\n Z^g[G^M]$, for some $g\in M$, by
Corollary~\ref{cor:obda.continuous}.
\end{proof}
The following lemma may be folklore. Nevertheless, we prove it for the convenience
of the reader.
\begin{Lem} \label{lem:random.over.mprime}
Let $r $ be random over a countable model $M$ and $A\in M$. Then there is a
countable model $M'\supseteq M$
such that $A$ is
countable in~$M'$, but $r$ is still random over~$M'$.
\end{Lem}
\begin{proof}
\def\namematrix{
\xymatrix@C=15mm{ M \ar[r]^C \ar[d]_{B_1} &
M^C \ar[d]^{\n B_2} \\
M^{B_1} \ar[r]_{\n P = C*\n B_2/ B_1} &
M^ {C*\n B_2} \\
}
}
\def\modelmatrix{
\xymatrix@C=15mm{ M \ar[r]^J \ar[d]_{r} &
M[J] \ar[d]^{K} \\
M[r] \ar[r]_H &
M[r][H] \\
}
}
We will need the following forcing notions, all defined in $M$:
\[\namematrix \]
\begin{itemize}
\item Let $C$ be the forcing that collapses the cardinality of~$A$ to
$\omega$ with finite conditions.
\item Let $B_1$ be random forcing (trees $T \subseteq 2^{<\omega}$
of positive measure).
\item Let $\n B_2$ be the $C$-name of random forcing.
\item Let $i:B_1\to C*\n B_2$ be the natural complete embedding $T\mapsto
(1_C,T)$.
\item Let $\n P$ be a $B_1$-name for the forcing $C*\n B_2/i[G_{B_1}]$, the
quotient of $C*\n B_2$ by the complete subforcing $i[B_1]$.
\end{itemize}
The random real $r$ is $B_1$-generic over~$M$. In $M[r]$ we let
$P\coloneqq \n P[r]$. Now let $H \subseteq P$ be generic over~$M[r]$.
Then $r*H \subseteq B_1*\n P \simeq C*\n B_2$ induces
an $M$-generic filter $J \subseteq C$ and an $M[J]$-generic filter
$K \subseteq \n B_2[J]$; it is easy to check
that $K$ interprets the $\n B_2$-name of the
canonical random real as the given random real~$r$.
Hence $r$ is random over the countable model
$M'\coloneqq M[J]$, and $A$ is countable
in~$M' $.
\[ \modelmatrix \]
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:extendLDtopreserverandom}]
We will first describe a construction that deals with a single
triple $ ( \bar p, \bar {\n Z}, \bar Z^ *)$ (where $\bar p$
is a sequence of conditions with strictly increasing stems which interprets
$ \bar {\n Z} $ as $ \bar Z^ *$); this construction will
yield a condition $q' = q'( \bar p, \bar {\n Z}, \bar Z^ *)$. We
will then show how to deal with all possible triples.
So let $p$ be a condition, and let
$\bar p = (p_k)_{k\in \omega}$ be a sequence interpreting
$\bar {\n Z}$ as $\bar Z^*$, where the lengths of the
stems of $p_n$ are strictly increasing and $p_0=p$.
It is easy to see that it is enough to deal with a single null set, i.e.,
$m=1$, and with $k_1=0$. We write $\n Z$ and $Z^*$ instead
of $\n Z_1$ and $Z_1^*$.
Using Lemma~\ref{lem:nicefy} we may (strengthening the conditions
in our interpretation) assume (in $M$) that the sequence
$(\n Z(k))_{k\in \omega}$ is almost continuous, witnessed by~$g:p\to
\textsc{clopen}^{<\omega}$.
By Lemma~\ref{lem:random.over.mprime}, we can find a model
$M'\supseteq M$ such that $(2^\omega)^M$ is
countable in~$M'$, but $r$ is still random over~$M'$.
We now work in~$M'$. Note that $g$ still defines an almost
continuous name, which we again call~$\n Z$.
Each filter in $D_s^M$ is now countably
generated; let $A_s$ be a pseudo-intersection of $D_s^M$
which additionally satisfies $A_s \subseteq \suc_p(s)$ for all $s\in p$
above the stem.
Let $D'_s$ be the Fr\'echet filter on $A_s$. Let $p'\in \mathbb{L}_{\bar D'}$
be the tree with the same stem as $p$ which satisfies
$\suc_{p'}(s)= A_s$ for all $s\in p'$ above the stem.
By Lemma~\ref{lem:LDMcomplete}, we know that $\mathbb{L}_{\bar D^M}$ is an
$M$-complete subforcing of $\mathbb{L}_{\bar D'}$ (in $M'$ as well as in $V$). We
write $G^M$ for
the induced filter on $\mathbb{L}_{\bar D^M}$.
We now work in $V$. Note that below the condition $p'$, the forcing
$\mathbb{L}_{\bar D'}$ is just Laver forcing $\mathbb{L}$, and that $p'\le_{\mathbb{L}} p$.
Using Lemma~\ref{lem:random.laver} we can find a condition $q\le p'$
(in Laver forcing $\mathbb{L}$) such that:
\begin{align}
& q \text{ is $M'$-generic}. \\
&q\Vdash_\mathbb{L} \text{ $r$ is random over $M'[G_{\mathbb{L}}]$
(hence also over $M[G^M]$)}\label{eq:r.random}.\\
& \text{Moreover, }q \Vdash_\mathbb{L} \n Z \sqsubset_0 r . \label{eq:z0r}
\end{align}
Enumerate all continuous
$\mathbb{L}_{\bar D^M}$-names of codes for null sets from $M$ as
$\n Z^{g_1}, \n Z^ {g_2}, \ldots $ \
Applying
Corollary~\ref{cor:stays.random} yields a condition $q'\le q$
such that
for all filter systems $\bar E$ satisfying $q'\in \mathbb{L}_{\bar E}$,
we have $q'\Vdash_{\mathbb{L}_{\bar E}} \n Z^{g_i} \sqsubset r$ for all $i$.
Corollary~\ref{cor:z.absolute} and Lemma~\ref{lem:continuous.is.enough} now imply:
\proofclaim{claim:p.prime}{ For every
filter system $\bar E$ satisfying $q'\in \mathbb{L}_{\bar E}$,
$q' $ forces in ${\mathbb{L}_{\bar E}}$ that $r$ is random over $M[G^M]$ and
that $\n Z \sqsubset_0 r$. }
By thinning out $q'$ we may assume that
\proofclaim{eq:basdf}{For each $\nu\in \omega^\omega\cap M$ there
is $k$ such that $\nu\mathord\restriction k\notin q'$. }
We have now described a construction of $q'= q'(\bar p, \n Z, Z^*)$.
Let $(\bar p^n , \n Z^n, Z^{*n})$ enumerate all triples
$(\bar p , \n Z, Z^{*})\in M$ where $\bar p$ interprets $\n Z$ as $Z^*$
(and consists of conditions with strictly increasing stems). For each
$n$ write $\nu^n$ for $\bigcup_k \stem (p^n_k)$,
the branch determined by the stems of the sequence $\bar p^ n$. We now
define by induction a sequence $q^n$ of conditions:
\begin{itemize}
\item $q^0 \coloneqq q'( \bar p^0 , \n Z^0, Z^{*0}) $.
\item Given $q^{n-1}$ and $(\bar p^n , \n Z^n, Z^{*n})$, we find $k_0$ such
that $\nu^n\mathord\restriction k_0 \notin q^0 \cup \cdots \cup q^{n-1}$
(using~\eqref{eq:basdf}). Let $k_1$ be such that
$\stem(p^n_{k_1})$ has length $>k_0$. We replace $\bar
p^n$ by $\bar p'\coloneqq (p^n_{k})_{k\ge k_1}$.
(Obviously, $\bar p'$
still interprets $\n Z^n$ as $Z^{*n}$.)
Now let $q^n\coloneqq q' (\bar p', \n Z^n, Z^{*n})$.
\end{itemize}
Note that the stem of $q^n$ is at least as long as
the stem of $p^n_{k_1}$, and is therefore not in $q^0 \cup
\cdots\cup q^{n-1}$, so $\stem(q^i)$ and $\stem(q^j)$ are
incompatible for all $i\not=j$. Therefore we can
choose for each $s$ an ultrafilter $D_s$ extending
$D^M_s$ such that
$\stem(q^i) \subseteq s $ implies $\suc_{q^i}(s)
\in D_s$.
Note that all $q^i$ are in $\mathbb{L}_{\bar D}$. Therefore,
we can use~\eqref{claim:p.prime}. Also, $q^i\le p^i_0$.
\end{proof}
Below, in Lemma~\ref{lem:iterate.random}, we will prove a preservation theorem
using the following ``local'' variant of ``random preservation'':
\begin{Def}\label{def:locally.random}
Fix a countable model $M$, a real $r\in 2^\omega$ and a
forcing notion $Q^M\in M$.
Let $Q^M$ be an $M$-complete subforcing of $Q$.
We say that \qemph{$Q$ locally preserves randomness of $r$ over $M$},
if
there is in $M$ a
sequence $(D^{Q^M}_n)_{n\in\omega}$ of
open dense subsets of $Q^M$ such that
the following holds:\\
{\bf Assume that }
\begin{itemize}
\item $M$ thinks that
$\bar p\coloneqq (p^n)_{n\in\omega}$ interprets
$(\n Z_1, \ldots, \n Z_m) $ as $(Z_1^*, \ldots, Z_m^*) $
(so each $\n Z_i$ is a $Q^M$-name of a code for a null set
and each $Z_i^*$ is a code for a null set, both in $M$);
\item moreover, each $p^n$ is in $D^{Q^M}_n$
(we call such a sequence $(p^n)_{n\in\omega}$, or the according interpretation, \qemph{quick});
\item $r$ is random over $M$;
\item $Z^*_i \sqsubset_{k_i} r$ for $i=1,\dots, m$.
\end{itemize}
{\bf Then}
there is a $q\leq_Q p^0$ forcing that
\begin{itemize}
\item
$r$ is random over $M[G^M]$;
\item $\n Z_i \sqsubset_{k_i} r$ for $i=1,\dots, m$.
\end{itemize}
\end{Def}
Note that this is trivially satisfied if $r$ is not random over $M$.
For a variant of this
definition, see Section~\ref{sec:alternativedefs}.
Setting
$D^{Q^M}_n$ to be the set of conditions with stem of length at least $n$,
Lemma~\ref{lem:extendLDtopreserverandom} gives us:
\begin{Cor}\label{cor:ultralaverlocalpreserving}
If $Q^M$ is an ultralaver forcing in $M$ and $r$ a real,
then there is an ultralaver forcing $Q$ over\footnote{``$Q$ over $Q^M$''
just means that $Q^M$ is an $M$-complete subforcing of $Q$.} $Q^M$ locally
preserving randomness of $r$ over~$M$.
\end{Cor}
\section{Janus forcing}\label{sec:janus}
In this section, we define a family of forcing notions that has two faces
(hence the name \qemph{Janus forcing}): Elements of this family may be countable (and therefore
equivalent to Cohen), and they may also be essentially random.
In the rest of the paper, we will use the following properties of Janus forcing
notions $\mathbb{J}$.
(And we will use \emph{only} these properties. So readers who are willing to
take these properties for granted could skip to Section~\ref{sec:iterations}.)
Throughout the whole paper we fix a function $B^*:\omega\to \omega$
given by Corollary~\ref{cor:tomek}.
The Janus forcings will depend on a real parameter
$\bar \ell^* = (\ell^*_m)_{m\in \omega}\in \omega^\omega$ which grows
fast with respect to~$B^*$. (In our application, $\bar \ell^*$
will be given by a subsequence of an ultralaver real.)
The sequence $\bar \ell^*$ and the function $B^*$ together define
a notion of a ``thin set'' (see Definition~\ref{def:thin}).
\begin{enumerate}
\item \label{item:canonical.null.set}
There is a canonical $\mathbb{J}$-name for a (code for a) null set~$\n
Z_\nabla$.
\\
Whenever $X \subseteq 2^\omega$ is not thin, and $\mathbb{J}$
is countable, then $\mathbb{J}$ forces that $X$ is not strongly meager,
witnessed\footnote{in the sense of~\eqref{eq:notsm}} by~$\nullset(\n Z_\nabla)$ (the set we get when we
evaluate the code $\n Z_\nabla$).
Moreover, for any $\mathbb{J}$-name~$\n Q$ of a $\sigma$-centered forcing,
also $\mathbb{J}*\n Q$ forces that $X$ is not strongly meager, again
witnessed by~$\nullset(\n Z_\nabla)$.
\\
(This is Lemma~\ref{lem:janusnotmeager}; ``thin'' is defined in Definition~\ref{def:thin}.)
\item
Let $M$ be a countable transitive model and $\mathbb{J}^M$ a Janus
forcing in~$M$. Then $\mathbb{J}^M$ is a Janus forcing in $V$ as well
(and of course countable in $V$). (Also note that trivially the forcing
$\mathbb{J}^M$ is an $M$-complete subforcing of itself.)
\\
(This is Fact~\ref{fact:janus.ctblunion}.)
\item
Whenever $M$ is a countable transitive model and $\mathbb{J}^M$ is a
Janus forcing in $M$,
then
there is a Janus forcing $\mathbb{J}$
such that
\begin{itemize}
\item
$\mathbb{J}^M$ is an $M$-complete subforcing of $\mathbb{J}$.
\item
$\mathbb{J}$ is (in $V$) equivalent to random forcing
(actually we just need that $\mathbb{J}$ preserves Lebesgue positivity
in a strong and iterable way).
\end{itemize}
(This is Lemma~\ref{lem:janusmayberandom} and Lemma~\ref{lem:janusrandompreservation}.)
\item
Moreover, the name $\n Z_\nabla$ referred to
in~(\ref{item:canonical.null.set}) is so ``canonical'' that
it evaluates to the same code in the $\mathbb{J}$-generic extension over $V$
as in the $\mathbb{J}^M$-generic extension over $M$.
\\
(This is Fact~\ref{fact:Znablaabsolute}.)
\end{enumerate}
\subsection{Definition of Janus}
A Janus forcing $\mathbb{J}$ will consist of:
\footnote{We thank Andreas Blass and Jind\v{r}ich Zapletal for their comments
that led to an improved presentation of Janus forcing.}
\begin{itemize}
\item
A countable ``core'' (or: backbone) $\nabla$ which is defined in a
combinatorial way from a parameter~$\bar\ell^*$. (In our
application, we will use a Janus forcing immediately after an
ultralaver forcing, and $\bar\ell^*$ will be a subsequence of the
ultralaver real.) This core is of course equivalent to Cohen
forcing.
\item
Some additional ``stuffing'' $\mathbb{J}\setminus \nabla$ (countable\footnote{Also
the trivial case $\mathbb{J}=\nabla $ is allowed.} or uncountable). We allow
great freedom for this, we just
require that the core $\nabla$ is a ``sufficiently'' complete subforcing (in a
specific combinatorial sense, see Definition~\ref{def:Janus}(\ref{item:fat})).
\end{itemize}
We will use the following
combinatorial theorem
from~\cite{MR2767969}:
\begin{Lem}[{\cite[Theorem 8]{MR2767969}\footnotemark}]
\footnotetext{The theorem
in~\cite{MR2767969} actually says ``for a sufficiently large
$I$'', but the proof shows that this should be read as ``for \emph{all}
sufficiently large $I$''. Also, the quoted theorem only claims that ${\mathcal A}_I$ will
be nonempty, but for $\varepsilon\le\frac12$ and $|I|> N_{\varepsilon,\delta}$
it is easy to see that ${\mathcal A}_I$ cannot be a singleton $\{A\}$: The set $X:= 2^I\setminus A$ has size $\ge 2^{|I|-1}\ge N_{\varepsilon,\delta}$
but satisfies $X+A\not=2^I$, as the constant sequence $\bar 0$ is not in $X+A$.}
\label{lem:tomek}
For every $\varepsilon,\delta>0$ there exists
$N_{\varepsilon,\delta}\in \omega$ such that for all sufficiently
large finite sets $I\subseteq \omega$ there is a family
${{\mathcal A}}_I $ with $|{\mathcal A}_I|\ge 2$ consisting of sets $A \subseteq 2^I$ with
$\dfrac{|A|}{2^{|I|}} \leq \varepsilon$ such that if $X \subseteq
2^I$, $|X| \geq N_{\varepsilon,\delta}$ then
\[
\frac{|\{ A \in {{\mathcal A}}_I: X+A=2^I\}|}{|{{\mathcal A}}_I|} \geq 1-\delta.
\]
(Recall that $X+A\coloneqq \{x+a: x\in X, a\in A\}$.)
\end{Lem}
Rephrasing and specializing to $\delta=\frac14$ and
$\varepsilon = \frac{1}{2^{i}}$ we get:
\begin{Cor}\label{cor:tomek}
For every
$i \in \omega$ there exists $B^*(i)$ such that for all finite
sets $I$
with $|I| \geq B^*(i)$
there is a nonempty
family ${{\mathcal A}}_I$
with $|{\mathcal A}_I| \geq 2$
satisfying the following:
\begin{itemize}
\item ${{\mathcal A}}_I$ consists of sets $A \subseteq 2^I$ with
$\dfrac{|A|}{2^{|I|}} \leq \dfrac{1}{2^{i}}$.
\item
For every $X \subseteq 2^I$ satisfying $|X| \geq B^*(i) $, the
set $\{ A \in {{{\mathcal A}}_I}: X+A=2^I \}$ has at least $\frac34 |{{\mathcal A}}_I|$ elements.
\end{itemize}
\end{Cor}
\begin{Asm}
We fix a sufficiently fast increasing sequence
$\bar\ell^*=(\ell^*_i)_{i\in\omega}$ of natural numbers; more precisely, the sequence $\bar\ell^*$ will be a subsequence of
an ultralaver real $\bar\ell$, defined as in Lemma~\ref{lem:subsequence} using the function $B^*$ from Corollary~\ref{cor:tomek}.
Note that in this case $\ell^*_{i+1}-\ell^*_i \geq B^*(i)$;
so we can fix for each $i$ a family ${\mathcal A}_i \subseteq
{\mathscr P}(2^{L_i})$ on
the interval $L_i \coloneqq [\ell^*_i,\ell^*_{i+1})$ according to Corollary~\ref{cor:tomek}.
\end{Asm}
\begin{Def}\label{def:Janus.nabla}
First we define the ``core'' $\nabla= \nabla_{\bar \ell^*}$ of our
forcing:
\[ \nabla = \bigcup_{i\in \omega} \prod_{j<i} {\mathcal A}_j .\]
In other words,
$\sigma\in \nabla$ iff
$\sigma= (A_0,\ldots, A_{i-1})$
for some $i\in \omega$, $A_0\in {\mathcal A}_0$, \dots, $A_{i-1}\in {\mathcal A}_{i-1}$.
We will denote the number $i$ by $\height(\sigma)$.
The forcing notion $\nabla$ is ordered by reverse inclusion (i.e., end extension): $\tau \leq \sigma$
if
$\tau \supseteq \sigma$.
\end{Def}
\begin{Def}\label{def:Janus}
Let $\bar \ell^* = (\ell^*_i)_{i\in \omega}$ be as in the assumption above.
We say that $\mathbb{J}$ is a Janus forcing based on $\bar \ell^*$ if:
\begin{enumerate}
\item\label{item:ic} $(\nabla, \supseteq)$ is an incompatibility-preserving
subforcing of $\mathbb{J}$.
\item\label{item:heightsarepredense} For each $i\in \omega$ the set $\{\sigma\in \nabla:\,
\height(\sigma)=i\}$ is predense in~$\mathbb{J}$.
So in particular,
$\mathbb{J}$ adds a
branch through $\nabla$. The union of this branch
is called $\n C^\nabla = (\n C^\nabla_0,\n C^\nabla_1,\n C^\nabla_2,\ldots)$, where $\n C^\nabla_i \subseteq 2^{L_i}$ with $\n C^\nabla_i \in {\mathcal A}_i$.
\item\label{item:fat} ``Fatness'':\footnote{This is the crucial combinatorial
property of Janus forcing. Actually, \eqref{item:fat}~implies~\eqref{item:heightsarepredense}.}
For all $p \in \mathbb{J}$ and all real numbers $\varepsilon>0$
there are arbitrarily large $i \in \omega$ such that there is a core condition $\sigma = (A_0,\ldots,A_{i-1}) \in \nabla$ (of length $i$) with
\[ \frac
{| \{ A \in {\mathcal A}_i: \, \sigma^\frown A \parallel_\mathbb{J} p\, \}|}
{| { {\mathcal A}_i } |}
\geq 1-\varepsilon.
\]
(Recall that $p \parallel_\mathbb{J} q$ means that $p$ and $q$ are compatible in $\mathbb{J}$.)
\item \label{item:janus.ccc} $\mathbb{J}$ is ccc.
\item \label{item:janus.sep} $\mathbb{J}$ is separative.\footnote{Separative is defined on page~\pageref{def:separative}.}
\item\label{item:janus.hc} (To simplify some technicalities:) $\mathbb{J} \subseteq H(\aleph_1)$.
\end{enumerate}
\end{Def}
We now define
$\n Z_\nabla$, which will be a canonical $\mathbb{J}$-name of (a code for) a null set. We will use the sequence $\n C^\nabla$ added by $\mathbb{J}$ (see Definition~\ref{def:Janus}(\ref{item:heightsarepredense})).
\begin{Def}\label{def:Znabla}
Each $\n C^\nabla_i$ defines a clopen set $\n Z^\nabla_i =
\{ x \in 2^\omega:\, x \mathord\restriction L_i \in \n C^\nabla_i \}$ of measure at most $\frac{1}{2^{i}}$.
The sequence
$\n Z_\nabla = (\n Z^\nabla_0,\n Z^\nabla_1,\n Z^\nabla_2,\ldots)$
is (a name for) a code for the null set
\[
\nullset(\n Z_\nabla) = \bigcap_{n < \omega} \bigcup_{i \geq n} \n Z^\nabla_i.
\]
\end{Def}
Since $\n C^\nabla$ is defined ``canonically'' (see in particular Definition~\ref{def:Janus}(\ref{item:ic}),(\ref{item:heightsarepredense})),
and $\n Z^\nabla$ is constructed in an absolute way from $\n C^\nabla$, we get:
\begin{Fact}\label{fact:Znablaabsolute}
If $\mathbb{J}$ is a Janus forcing, $M$ a countable model and $\mathbb{J}^M$ a Janus forcing in $M$ which is an $M$-complete subset of $\mathbb{J}$, if $H$ is $\mathbb{J}$-generic over $V$ and
$H^M$ the induced $\mathbb{J}^M$-generic filter over $M$, then $\n C^\nabla$
evaluates to the same real in $M[H^M]$ as in $V[H]$, and therefore
$\n Z^\nabla$ evaluates to the same code (but of course not to the same set of reals).
\end{Fact}
For later reference, we record the following trivial fact:
\begin{Fact} \label{fact:janus.ctblunion}
Being a Janus forcing is absolute. In particular, if $V\subseteq W$
are set theoretical universes and $\mathbb{J}$ is a Janus forcing in $V$,
then $\mathbb{J}$ is a Janus forcing in $W$. In particular, if $M$
is a countable model in $V$ and $\mathbb{J}\in M$ a Janus forcing in $M$, then $\mathbb{J}$ is also a Janus forcing in $V$.
\\
Let $(M^n)_{n\in \omega}$ be an increasing sequence of countable
models, and let $\mathbb{J}^n \in M^n$ be Janus forcings.
Assume that $\mathbb{J}^n$ is $M^n$-complete in~$\mathbb{J}^{n+1}$.
Then $\bigcup_n \mathbb{J}^n$ is a Janus forcing, and an
$M^n$-complete extension of $\mathbb{J}^n$ for all~$n$.
\end{Fact}
\subsection{Janus and strongly meager}
Carlson~\cite{MR1139474} showed that Cohen reals make every uncountable set $X$
of the ground model not strongly meager in the extension (and that not being
strongly meager is preserved in a subsequent forcing with precaliber~$\al1$).
We show that a {\em countable} Janus forcing $\mathbb{J}$ does the same
(for a subsequent forcing that is even $\sigma$-centered,
not just precaliber~$\al1$).
This sounds
trivial, since any (nontrivial) countable forcing is equivalent to Cohen
forcing anyway. However, we show (and will later use) that the canonical
null set $\n Z_\nabla$ defined above witnesses that $X$ is not strongly meager
(and not just some
null set that we get out of the isomorphism between $\mathbb{J}$ and Cohen forcing).
The point is that while $\nabla$ is
not a complete subforcing of~$\mathbb{J}$, the
condition~(\ref{item:fat})
of the Definition~\ref{def:Janus} guarantees that Carlson's argument still
works, if we assume that $X$ is non-thin (not just uncountable).
This is enough for us, since by Corollary~\ref{cor:LDnotthin} ultralaver forcing
makes any uncountable set non-thin.
Recall that we fixed the increasing sequence $\bar \ell^* = (\ell^*_i)_{i\in
\omega}$ and~$B^*$. In the following, whenever we say ``(very) thin'' we mean
``(very) thin with respect to $\bar \ell^*$ and $B^*$'' (see Definition~\ref{def:thin}).
\begin{Lem}\label{lem:janusnotmeager}
If $X$ is not thin, $\mathbb{J}$
is a countable Janus forcing based on $\bar \ell^*$,
and
$\n R$ is a $\mathbb{J}$-name
for a $\sigma$-centered forcing notion,
then
$\mathbb{J}*\n R$ forces that $X$ is not strongly meager witnessed by
the null set $\n Z_\nabla$.
\end{Lem}
\begin{proof}
Let $\n c$ be a $\mathbb{J}$-name for a function $\n c:\n R\to \omega$
witnessing that $\n R$ is $\sigma$-centered.
Recall that ``$\n Z_\nabla$ witnesses that $X$ is not strongly meager''
means that $X+\n Z_\nabla = 2^\omega$.
Assume towards a contradiction that $(p,r) \in \mathbb{J}*\n R$ forces that $X+\n Z_\nabla \neq 2^\omega$. Then we can fix a $(\mathbb{J}*\n R)$-name $\n \xi $ such that
$(p,r) \Vdash \n \xi \notin X + \n Z_\nabla$, i.e.,
$(p,r) \Vdash (\forall x \in X)\,\, \n \xi \notin x + \n Z_\nabla$.
By definition of $\n Z_\nabla$, we get
\[
(p,r) \Vdash (\forall x \in X)\, (\exists n \in \omega)\, (\forall i \geq n) \,\, \n \xi \mathord\restriction L_i \notin x \mathord\restriction L_i + \n C^\nabla_i.
\]
For each $x\in X$ we can find $(p_x,r_x) \leq (p,r)$ and
natural numbers
$n_x \in \omega$ and $m_x \in \omega$ such that
$p_x $ forces that $\n c(r_x) = m_x$
and
\[
(p_x,r_x) \Vdash (\forall i \geq n_x) \,\, \n \xi \mathord\restriction L_i \notin x \mathord\restriction L_i + \n C^\nabla_i.
\]
So $X = \bigcup_{p \in \mathbb{J}, m \in \omega, n \in \omega} X_{p,m,n}$,
where $X_{p,m,n}$ is the set of all $x$ with~$p_x=p$, $m_x=m$, $n_x=n$.
(Note that $\mathbb{J}$ is countable, so the union is countable.)
As $X$ is not thin, there is some $p^*, m^*, n^*$ such that $X^*\coloneqq X_{p^*,m^*,n^*}$ is not very thin.
So we get for all $x\in X^*$:
\begin{equation}\label{eq:prx}
(p^*,r_x) \Vdash (\forall i \geq n^*) \,\, \n \xi \mathord\restriction L_i \notin x \mathord\restriction L_i + \n C^\nabla_i.
\end{equation}
Since $X^*$ is not very thin,
there is some $i_0 \in \omega$ such that for all $i \geq i_0$
\begin{equation}\label{eq:star}
\textrm{the (finite) set } X^* \mathord\restriction L_i \textrm{ has more than } B^*(i)
\textrm{ elements.}
\end{equation}
Due to the fact that $\mathbb{J}$ is a Janus forcing (see Definition~\ref{def:Janus}~\eqref{item:fat}),
there are arbitrarily large $i \in \omega$ such that there is a core condition $\sigma = (A_0,\ldots,A_{i-1}) \in \nabla$ with
\begin{equation} \label{eq:sizeS}
\frac{ | \{ A \in {\mathcal A}_i: \, \sigma^\frown A \parallel_{\mathbb{J}} p^* \} | }
{ | {\mathcal A}_i | }
\geq \frac{2}{3}.
\end{equation}
Fix such an $i$
larger than both $i_0$ and $n^*$, and fix a condition $\sigma$ satisfying~\eqref{eq:sizeS}.
We now consider the following two subsets of ${\mathcal A}_i$:
\begin{equation}\label{eq:two_sets}
\{ A \in {\mathcal A}_i: \, \sigma^\frown A \parallel_{\mathbb{J}} p^* \}
\,\,
\textrm{ and }
\,\,
\{ A \in {\mathcal A}_i: \, X^* \mathord\restriction L_i + A = 2^{L_i} \}.
\end{equation}
By~\eqref{eq:sizeS}, the relative measure (in ${\mathcal A}_i$) of the left one is at least $\frac{2}{3}$;
due to~\eqref{eq:star} and the definition of ${\mathcal A}_i$ according to Corollary~\ref{cor:tomek}, the relative measure of the right one is at least $\frac{3}{4}$;
so the two sets in~\eqref{eq:two_sets} are not disjoint, and we can pick an $A$ belonging to both.
Clearly, $\sigma^\frown A$ forces (in $\mathbb{J}$) that
$\n C^\nabla_i$ is equal to~$A$. Fix $q \in \mathbb{J}$ witnessing $\sigma^\frown A \parallel_{\mathbb{J}} p^*$. Then
\begin{equation}\label{eq:wo_for_contradiction}
q \Vdash_\mathbb{J} X^* \mathord\restriction L_i + \n C^\nabla_i = X^* \mathord\restriction L_i + A = 2^{L_i}.
\end{equation}
Since $p^*$ forces that for each $x \in X^*$ the color $\n c(r_x) = m^*$, we can find an $r^*$
which is (forced by $q \leq p^*$ to be) a lower bound of the \emph{finite} set $\{r_x : \, x \in X^{**} \}$, where $X^{**} \subseteq X^*$ is any finite set with $X^{**} \mathord\restriction L_i = X^* \mathord\restriction L_i$.
By~\eqref{eq:prx},
\[
(q,r^*) \Vdash \n \xi \mathord\restriction L_i \notin X^{**} \mathord\restriction L_i + \n C^\nabla_i = X^* \mathord\restriction L_i + \n C^\nabla_i,
\]
contradicting~\eqref{eq:wo_for_contradiction}.
\end{proof}
Recall that by Corollary~\ref{cor:LDnotthin}, every uncountable set $X$ in $V$ will
not be thin in the $\mathbb{L}_{\bar D}$-extension. Hence we get:
\begin{Cor}\label{cor:ultraplusjanus}
Let $X$ be uncountable. If $\mathbb{L}_{\bar D}$ is any ultralaver forcing
adding an ultralaver real $\bar \ell$, and $\bar \ell^*$ is defined from
$\bar \ell$ as in Lemma~\ref{lem:subsequence}, and if $\n\mathbb{J}$ is a
countable Janus forcing based on $\bar \ell^*$, $\n Q$ is any $\sigma$-centered
forcing, then $\mathbb{L}_{\bar D}*\n\mathbb{J}*\n Q$ forces that $X$ is not strongly meager.
\end{Cor}
\subsection{Janus forcing and preservation of Lebesgue positivity}
We show that every Janus forcing in a countable model $M$ can be extended to
locally preserve a given random real over $M$. (We showed the same for ultralaver
forcing in Section~\ref{ss:ultralaverpositivity}.)
We start by proving that every countable Janus forcing can be embedded into a
Janus forcing which is equivalent to random forcing, preserving the maximality
of countably many maximal antichains. (In the following lemma, the letter
$M$ is just a label to distinguish $\mathbb{J}^M$ from $\mathbb{J}$,
and does not necessarily refer to a model.)
\newcommand{{\bJ^M}}{{\mathbb{J}^M}}
\newcommand{\bJ}{\mathbb{J}}
\begin{Lem}\label{lem:janusmayberandom}
Let $\mathbb{J}^M$ be a countable Janus
forcing (based on $\bar \ell^*$) and let
$\{D_k:\, k\in \omega\}$ be a countable family of open dense subsets
of $\mathbb{J}^M$.
Then there is a Janus forcing $\mathbb{J}$
(based on the same $\bar \ell^*$) such that
\begin{itemize}
\item $\mathbb{J}^M$ is an incompatibility-preserving subforcing of $\mathbb{J}$.
\item Each $D_k$ is still predense in $\mathbb{J}$.
\item $\mathbb{J}$ is forcing equivalent to random forcing.
\end{itemize}
\end{Lem}
\begin{proof}
Without loss of generality assume $D_0=\mathbb{J}^M$.
Recall that $\nabla = \nabla^{\mathbb{J}^M}$ was defined in
Definition~\ref{def:Janus.nabla}.
Note that for each $j$ the set
$\{ \sigma\in \nabla: \, \height(\sigma)=j\}$ is predense in ${\bJ^M}$,
so the set
\begin{align}\label{align:E.k}
E_j\coloneqq \{ p \in {\bJ^M}: \exists \sigma\in \nabla: \, \height(\sigma)=j, \ p \le \sigma\}
\end{align}
is dense open in ${\bJ^M}$; hence without loss of generality each $E_j$ appears in our list of $D_k$'s.
Let $\{r^n:\, n\in \omega\} $ be an enumeration of~$\mathbb{J}^M$.
We now fix $n$ for a while (up to \eqref{def:this.is.random}).
We will construct
a finitely splitting tree
$S^n \subseteq \omega^{<\omega}$ and a family $(\sigma^n_s,p^n_s,
\tau^{*n}_s)_{ s\in S^n}$
satisfying the following (suppressing the superscript~$n$):
\begin{enumerate}[(a)]
\item $\sigma_s\in \nabla$, $\sigma_{\langle\rangle} = \langle\rangle$,
$s\subseteq t$ implies $\sigma_s\subseteq \sigma_t$,
and $s\perp_{S^n} t$ implies $\sigma_s\perp_\nabla \sigma_t$.
\\ (So in particular the set $\{\sigma_t:\, t\in\suc_{S^n}(s)\}$
is a (finite) antichain above $\sigma_s$ in~$\nabla$.)
\item $p_s\in {\bJ^M}$, $p_{\langle\rangle} = r^n$;
if $s\subseteq t$ then
$p_t\leq_{\bJ^M} p_s$ (hence $p_t\leq r^n$);
$s\perp_{S^n} t$ implies $p_s\perp_{\bJ^M} p_t$.
\item $p_s\leq_{\bJ^M} \sigma_s$.
\item $\sigma_s \subseteq \tau^*_s \in \nabla$, and
$\{\sigma_t:\, t\in\suc_{S^n}(s)\}$
is the set of all $ \tau \in \suc_\nabla(\tau^*_s)$ which are compatible
with $p_s$.
\item The set $\{\sigma_t:\, t\in\suc_{S^n}(s)\}$
is a subset of $ \suc_\nabla(\tau^*_s)$
of relative size at least
$1-\frac 1 {\lh(s)+10}$.
\label{item:size.a}
\item Each $s\in {S^n}$ has at least 2 successors (in ${S^n}$).
\item If $k=\lh(s)$, then $p_s\in D_k$ (and therefore also
in all $D_l$ for $l<k$).
\end{enumerate}
Set $\sigma_{\langle\rangle}=\langle\rangle$ and $p_{\langle\rangle}=r^n$.
Given $s,\sigma_s$ and~$p_s$, we construct $\suc_{S^n}(s)$ and
$(\sigma_t,p_t)_{t\in\suc_{S^n}(s)}$:
We apply fatness~\ref{def:Janus}(\ref{item:fat}) to $p_s$ with
$\varepsilon=\frac 1 {\lh(s)+10}$. So we get some $\tau_s^*\in\nabla$
of height bigger than the height of $\sigma_s$ such that
the set $B$ of elements of $\suc_\nabla(\tau_s^*)$ which are compatible
with $p_s$ has relative size at least $1-\varepsilon$.
Since $p_s\leq_{\bJ^M} \sigma_s$ we get that $\tau^*_s$ is compatible
with (and therefore stronger than) $\sigma_s$.
Enumerate $B$ as $\{\tau_0,\dots,\tau_{l-1}\}$.
Set $\suc_{S^n}(s)=\{s^\frown i:\, i<l\}$ and $\sigma_{s^\frown i}=\tau_i$.
For $t\in\suc_{S^n}(s)$, choose $p_t\in {\bJ^M}$ stronger than both $\sigma_t$ and $p_s$
(which is obviously possible since $\sigma_t$ and $p_s$ are compatible),
and moreover $p_t\in D_{\lh(t)}$.
This concludes the construction of the family
$(\sigma^n_s,p^n_s, \tau^{*n}_s)_{ s\in S^n}$.
So $(S^n,\subseteq)$ is a finitely splitting nonempty
tree of height $\omega$ with no maximal nodes and no isolated
branches.
$[S^n]$ is the (compact) set of branches of~$S^n$.
The closed subsets of $[S^n]$ are exactly the sets of the form~$[T]$,
where $T\subseteq S^n$ is a subtree of $S^n$
with no maximal nodes. $[S^n]$
carries a natural (``uniform'') probability measure~$\mu_n$,
which is characterized by
\[
\mu_n((S^n)^{[t]}) = \frac{1}{|{\suc_{S^n}(s)}|}\cdot \mu_n((S^n)^{[s]})
\]
for all $s\in S^n$ and all $t\in \suc_{S^n}(s)$. (We just write $\mu_n(T)$
instead of $\mu_n([T])$ to increase readability.)
We call $T\subseteq S^n$ positive if $\mu_n(T)>0$, and we call $T$
pruned if $\mu_n(T^{[s]})>0$ for all $s\in T$.
(Clearly every positive tree $T$ contains a pruned tree $T'$ of the same
measure, which can be obtained from $T$ by removing all nodes $s$ with
$\mu_n(T^{[s]})=0$.)
Let $T\subseteq S^n$ be a positive pruned tree and $\varepsilon>0$.
Then on all but finitely many levels $k$ there is an $s\in T$ such that
\begin{equation}\label{eq:lebdense}
\suc_T(s)\subseteq \suc_{S^n}(s)\text{ has relative size }\geq
1-\varepsilon.
\end{equation}
(This follows from Lebesgue's density theorem, or can easily be seen directly:
Set $C_m=\bigcup_{t\in T,\,\lh(t)=m}{(S^n)}^{[t]}$. Then
$C_m$ is a decreasing sequence of closed sets, each containing~$[T]$.
If the claim fails, then $\mu_n(C_{m+1} ))\leq \mu_n(C_m)\cdot (1-\varepsilon)$
infinitely often; so $\mu_n(T) \le \mu_n( \bigcap_m C_m ) =0$.)
It is well known that the set of positive, pruned subtrees of~$S^n$, ordered
by inclusion, is forcing equivalent to random forcing (which can
be defined as the set of positive, pruned subtrees of $2^{<\omega}$).
We have now constructed $S^n$ for all $n$. Define
\begin{align}\label{def:this.is.random}
\bJ = {\bJ^M} \cup \bigcup_n \, \bigl\{\, (n,T) : \, T \subseteq S^n
\mbox{ is a positive pruned tree}\,\bigr \}
\end{align}
with the following partial order:
\begin{itemize}
\item The order on $\bJ$ extends the order on~${\bJ^M}$.
\item $(n',T')\le(n,T)$ if $n=n'$ and $T' \subseteq T$.
\item For $p\in {\bJ^M}$: $(n,T) \le p$ if there is a $k$ such that
$p^n_t\leq p$ for all $t\in T$ of length~$k$.
(Note that this will then be true for all bigger $k$ as well.)
\item $p \le (n,T)$ never holds (for $p\in {\bJ^M}$).
\end{itemize}
The lemma now easily follows from the following properties:
\begin{enumerate}
\item The order on $\bJ$ is transitive.
\item ${\bJ^M}$ is an incompatibility-preserving subforcing of $\bJ$.
\\
In particular, $\bJ$ satisfies item~\eqref{item:ic}
of Definition~\ref{def:Janus} of Janus forcing.
\item
For all $k$: the set $\{(n,T^{[t]}):\, t\in T,\ \lh(t)=k\}$
is a (finite) predense antichain below~$(n,T)$.
\item\label{item:compat} $(n,T^{[t]})$ is stronger than $p^n_t$ for each $t\in T$
(witnessed, e.g., by $k=\lh(t)$).
Of course, $(n,T^{[t]})$ is stronger than $(n,T)$ as well.
\item
Since $p^n_t\in D_k$ for $k=\lh(t)$, this implies that each
$D_k$ is predense below each
$(n,S^n)$ and therefore in~$\bJ$.
\\
Also, since each set $E_j$ appeared in our list of open dense subsets
(see \eqref{align:E.k}), the set
$\{\sigma\in \nabla: \, \height(\sigma)=j\}$ is still
predense in $\bJ$, i.e.,
item~\eqref{item:heightsarepredense}
of the Definition~\ref{def:Janus} of Janus forcing is satisfied.
\item The condition $(n,S^n)$ is stronger than~$r^n$, so
$\{(n,S^n):n\in \omega \}$ is predense in $\bJ$ and
$\bJ\setminus {\bJ^M}$ is dense in~$\bJ$.
\\
Below each $(n,S^n)$, the forcing $\bJ$ is isomorphic to random
forcing.
\\
Therefore, $\bJ$ itself is forcing equivalent to random
forcing. (In fact,
the complete Boolean algebra generated by $\bJ$ is
isomorphic to the standard random algebra, Borel sets modulo null sets.)
This proves in particular that $\mathbb{J}$ is ccc, i.e., satisfies
property \ref{def:Janus}(\ref{item:janus.ccc}).
\item
It is easy (but not even necessary) to check that $\bJ$ is separative, i.e.,
property~\ref{def:Janus}(\ref{item:janus.sep}). In any case,
we could replace $\le_\bJ$ by $\le^*_\bJ$, thus making $\bJ$ separative without changing
$\le_{\bJ^M}$, since ${\bJ^M}$ was already separative.
\item Property \ref{def:Janus}(\ref{item:janus.hc}), i.e., $\mathbb{J}\in H(\aleph_1)$,
is obvious.
\item \label{item:the.last}
The remaining item
of the definition
of Janus forcing, fatness~\ref{def:Janus}(\ref{item:fat}),
is satisfied.\\
I.e., given $(n,T)\in \bJ$ and $\varepsilon>0$ there is
an arbitrarily high $\tau^*\in\nabla$ such that the relative
size of the set $\{\tau\in\suc_\nabla(\tau^*):\, \tau\parallel (n,T)\}$
is at least $1-\varepsilon$. (We will show $\ge (1-\varepsilon)^2$ instead,
to simplify the notation.)
\end{enumerate}
We show~(\ref{item:the.last}): Given $(n,T)\in \bJ$ and $\varepsilon>0$,
we use~\eqref{eq:lebdense} to get an arbitrarily high
$s\in T$ such that $\suc_T(s)$ is of relative size
$\geq 1-\varepsilon$ in~$\suc_{S^n}(s)$. We may choose $s$ of length $>\frac 1 \varepsilon$.
We claim that $\tau^*_s$ is as required:
\begin{itemize}
\item Let
$B\coloneqq
\{\sigma_t:\, t\in\suc_{S^n}(s)\}
$.
Note that $B =
\{\tau\in\suc_\nabla(\tau^*_s):\, \tau\parallel p_s\}
$.
$B$ has relative size $\ge 1-\frac{1}{\lh(s)}\ge 1-\varepsilon$
in $\suc_\nabla(\tau^*_s)$
(according to property~(\ref{item:size.a}) of $S^n$).
\item $C\coloneqq \{\sigma_t:\, t\in\suc_T(s)\}$ is a subset of $B$
of relative size $\ge 1-\varepsilon$ according to our choice of~$s$.
\item So $C$ is of relative size $(1-\varepsilon)^2$
in~$\suc_\nabla(\tau^*_s)$.
\item Each $\sigma_t\in C$ is compatible with~$(n,T)$, as $(n,T^{[t]})
\le p_t \le \sigma_t$ (see~(\ref{item:compat})). \qedhere
\end{itemize}
\end{proof}
So in particular if $\mathbb{J}^M$ is a Janus forcing in a countable model $M$, then
we can extend it to a Janus forcing $\mathbb{J}$ which is in fact random forcing.
Since random forcing strongly preserves randoms over countable models (see
Lemma~\ref{lem:random.laver}), it is not surprising that we get local
preservation of randoms for Janus forcing, i.e., the analoga of
Lemma~\ref{lem:extendLDtopreserverandom} and
Corollary~\ref{cor:ultralaverlocalpreserving}. (Still, some additional argument
is needed, since the fact that~$\mathbb{J}$ (which is now random forcing) ``strongly preserves
randoms'' just means that a random real $r$ over $M$ is preserved with respect
to random forcing in $M$, not with respect to $\mathbb{J}^M$.)
\begin{Lem}\label{lem:janusrandompreservation}
If $\mathbb{J}^M$ is a Janus forcing in a countable model $M$ and
$r$ a random real over $M$, then there is a Janus forcing $\mathbb{J}$
such that $\mathbb{J}^M$ is an $M$-complete subforcing of $\mathbb{J}$ and the
following holds:
\\
\textbf{If}
\begin{itemize}
\item $p\in \mathbb{J}^M$,
\item in $M$, $\n {\bar Z} = ( \n Z_1, \ldots , \n Z_m) $
is a sequence of $\mathbb{J}^M$-names for
codes for null sets,
and $Z_1^*,\dots , Z_m^*$ are interpretations under~$p$,
witnessed by a sequence $(p_n)_{n\in \omega}$,
\item $Z^*_i \sqsubset_{k_i} r$ for $i=1,\dots, m$,
\end{itemize}
\textbf{then} there is a $q\leq p$ in $\mathbb{J}$ forcing that
\begin{itemize}
\item $r$ is random over $M[H^M]$,
\item $\n Z_i \sqsubset_{k_i} r$ for $i=1,\dots, m$.
\end{itemize}
\end{Lem}
\begin{Rem}
In the version for ultralaver forcings, i.e.,
Lemma~\ref{lem:extendLDtopreserverandom}, we had to assume that the stems of
the witnessing sequence are strictly increasing. In the Janus version, we do
not have any requirement of that kind.
\end{Rem}
\begin{proof}
Let $\mathcal D$ be the set of dense subsets of $\mathbb{J}^M$ in $M$.
According to Lemma~\ref{lem:random.over.mprime}, we
can first find some countable $M'$ such that $r$ is still random over $M'$
and such that in $M'$ both $\mathbb{J}^M$ and $\mathcal D$
are countable. According to Fact~\ref{fact:janus.ctblunion},
$\mathbb{J}^M$ is a (countable) Janus forcing in $M'$, so we can apply
Lemma~\ref{lem:janusmayberandom} to the set $\mathcal D$
to construct a Janus forcing $\mathbb{J}^{M'}$
which is equivalent to random forcing
such that (from the point of $V$) $\mathbb{J}^{M}\lessdot_M \mathbb{J}^{M'}$.
In $V$, let\footnote{More precisely:
Densely embed $\mathbb{J}^{M'}$ into
(Borel/null)$^{M'}$, the complete Boolean algebra associated with
random forcing in $M'$, and let $\mathbb{J}:=$ (Borel/null)$^V$. Using the
embedding, $\mathbb{J}^{M'}$ can now be viewed as an $M'$-complete
subset of $\mathbb{J}$.}
$\mathbb{J}$ be random forcing.
$\mathbb{J}^{M'}$ is an $M'$-complete subforcing of $\mathbb{J}$
and therefore $\mathbb{J}^{M}\lessdot_M \mathbb{J}$. Moreover,
as was noted in Lemma~\ref{lem:random.laver}, we even
know that random forcing strongly preserves randoms over $M'$
(see Definition~\ref{def:locally.random}). To show that
$\mathbb{J}$ is indeed a Janus forcing, we have to check the
fatness condition~\ref{def:Janus}(\ref{item:fat});
this follows easily from $\Pi^1_1$-absoluteness (recall that
incompatibility of random conditions is Borel).
So assume that (in $M$) the sequence $(p_n)_{n\in\omega}$
of $\mathbb{J}^M$-conditions interprets $\n {\bar Z}$ as $\bar Z^*$.
In $M'$, $\mathbb{J}^M$-names can be reinterpreted as $\mathbb{J}^{M'}$-names,
and the $\mathbb{J}^{M'}$-name $\n {\bar Z}$ is interpreted as $\bar Z^*$
by the same sequence $(p_n)_{n\in\omega}$.
Let $k_1,\ldots, k_m$ be such that $Z_i^*\sqsubset_{k_i} r$
for $i=1,\ldots, m$.
So by strong preservation of randoms, we can in $V$ find some
$q\leq p_0$ forcing that $r$ is random over $M'[H^{M'}]$ (and therefore
also over the subset $M[H^M]$), and that $\n Z_i\sqsubset_{k_i} r$ (where
$\n Z_i$ can be evaluated in
$M'[H^{M'}]$ or equivalently in $M[H^M]$).
\end{proof}
So Janus forcing is locally preserving randoms (just as ultralaver forcing):
\begin{Cor}\label{cor:januslocallypreserves}
If $Q^M$ is a Janus forcing in $M$ and $r$ a real, then there is a
Janus forcing $Q$ over~$Q^M$ (which is in fact equivalent to random
forcing) locally preserving randomness of~$r$ over~$M$.
\end{Cor}
\begin{proof}
In this case, the notion of ``quick'' interpretations is
trivial, i.e., $D^{Q^M}_k = Q^M$ for all~$k$, and the claim follows from
the previous lemma.
\end{proof}
\section{Almost finite and almost countable support iterations}\label{sec:iterations}
A main tool to construct the forcing for BC+dBC will be ``partial countable
support iterations'', more particularly ``almost finite support'' and ``almost
countable support'' iterations. A partial countable support iteration is a
forcing iteration $(P_\alpha,Q_\alpha)_{\alpha<\om2}$ such that for each limit
ordinal $\delta$ the forcing notion $P_\delta$ is a subset of the countable
support limit of $(P_\alpha,Q_\alpha)_{\alpha<\delta}$ which satisfies some
natural properties (see Definition~\ref{partial_CS}).
Instead of transitive models, we will use ord-transitive models (which are
transitive when ordinals are considered as urelements). Why do we do that? We
want to ``approximate'' the generic iteration $\bar\mathbf{P}$ of length $\omega_2$
with countable models; this can be done more naturally with ord-transitive
models (since obviously countable transitive models only see countable
ordinals). We call such an ord-transitive model a ``candidate'' (provided it
satisfies some nice properties, see Definition~\ref{def:candidate}). A basic
point is that forcing extensions work naturally with candidates.
In the next few paragraphs (and also in Section~\ref{sec:construction}),
$x=(M^x,\bar P^x)$ will denote a pair such that $M^x$ is a
candidate and $\bar P^x$ is (in $M^x$) a partial countable support iteration; similarly
we write, e.g., $y= (M^y, \bar P^ y) $ or $x_n=(M^{x_n},\bar P^{x_n})$.
We will need the following results to prove BC+dBC. (However, as opposed to the
case of the ultralaver and Janus section, the reader will probably have to read
this section to understand the construction in the next section, and not just
the following list of properties.)
Given $x=(M^x,\bar P^x)$, we can construct
by induction on $\alpha$ a
partial countable support iteration $\bar P = (P_\alpha, Q_\alpha)_{\alpha
< \om2}$ satisfying:
\begin{quote}
There is a canonical $M^x$-complete embedding from $\bar P^x$ to
$\bar P$.
\end{quote}
In this construction, we can use at each stage $\beta$
any desired $Q_\beta$, as long as
$P_\beta$ forces that $Q^x_\beta$ is (evaluated as)
an $M^x[H^x_\beta]$-complete subforcing of~$Q_\beta$ (where
$H^x_\beta\subseteq P^x_\beta$
is the $M^x$-generic filter induced by the generic filter~$H_\beta\subseteq P_\beta$).
\\
Moreover, we can demand either of
the following two additional properties\footnote{The $\sigma$-centered version is central for the proof of dBC; the random preserving version
for BC.}
of the limit of this iteration~$\bar P$:
\begin{enumerate}
\item
If all $Q_\beta$ are forced to be $\sigma$-centered,
and $Q_\beta$ is trivial for all $\beta\notin M^x$,
then $P_{\omega_2}$ is $\sigma$-centered.
\item
If $r$ is random over
$M^x$, and all $Q_\beta$ locally preserve randomness of $r$
over $M^x[H^x_\beta]$ (see
Definition~\ref{def:locally.random}), then also $P_{\om2}$
locally preserves the randomness of $r$.
\end{enumerate}
Actually, we need the following variant: Assume that we already
have $P_{\alpha_0}$
for some $\alpha_0\in M^x$, and that $P^x_{\alpha_0}$ canonically
embeds into~$P_{\alpha_0}$, and that
the respective assumption on $Q_\beta$ holds for all $\beta\ge
\alpha_0$. Then we get that $P_{\alpha_0}$ forces that the quotient
$P_{\omega_2}/P_{\alpha_0}$ satisfies the respective conclusion.
We also need:\footnote{This will give $\sigma$-closure
and $\al2$-cc for the preparatory forcing $\mathbb{R}$.}
\begin{enumerate}\setcounter{enumi}{2}
\item
If instead of a single $x$ we have a sequence $x_n$
such that each $P^{x_n}$ canonically (and $M^{x_n}$-completely)
embeds into~$P^{x_{n+1}}$, then
we can find a partial countable support
iteration $\bar P$ into which all $P^{x_n}$ embed canonically (and we can again use
any desired $Q_\beta$, assuming that $Q^{x_n}_\beta$
is an $M^{x_n}[H^{x_n}_\beta]$-complete subforcing of $Q_\beta$ for all $n\in\omega$).
\item
(A fact that is easy to prove but awkward to formulate.)
If a $\Delta$-system argument produces two $x_1$, $x_2$ as in Lemma~\ref{lem:prep.is.sigma.preparation}(\ref{item:karotte6}),
then we can find a partial countable support iteration $\bar P$
such that $\bar P^{x_i}$ canonically (and $M^{x_i}$-completely)
embeds into~$\bar P$ for $i=1,2$.
\end{enumerate}
\subsection{Ord-transitive models}\label{subsec:ordtrans}
We will use ``ord-transitive'' models, as introduced in~\cite{MR2115943} (see
also the presentation in~\cite{kellnernep}). We briefly summarize the basic
definitions and properties (restricted to the rather simple case needed in this
paper):
\begin{Def}\label{def:candidate} Fix a suitable finite subset ZFC$^*$ of ZFC (that is
satisfied by $H(\chi^*)$ for sufficiently large regular $\chi^*$).
\begin{enumerate}
\item A set $M$ is called a \emph{candidate}, if
\begin{itemize}
\item $M$ is countable,
\item $(M,\in)$ is a model of ZFC$^*$,
\item $M$ is ord-absolute:
$M \models \alpha\in \ON$ iff $\alpha\in \ON$,
for all $\alpha\in M$,
\item $M$ is ord-transitive: if $x\in M\setminus \ON$, then
$x\subseteq M$,
\item $\omega+1\subseteq M$.
\item ``$\alpha$ is a limit ordinal'' and ``$\alpha=\beta+1$''
are both
absolute between $M$ and $V$.
\end{itemize}
\item A candidate $M$ is called \emph{nice}, if
``$\alpha$ has countable cofinality''
and ``the countable set $A$ is cofinal in $\alpha$'' both are
absolute between $M$ and $V$. (So
if $\alpha\in M$ has countable cofinality, then
$\alpha\cap M$ is cofinal in $\alpha$.)
Moreover, we assume $\om1\in M$ (which implies
$\om1^M=\om1$) and $\om2\in M$ (but we do not require
$\om2^M= \om2$).
\item
Let $P^M$ be a forcing notion in a candidate $M$.
(To simplify notation, we can assume without loss of generality that
$P^M\cap \ON=\emptyset$ (or at least $\subseteq \omega$)
and that therefore $P^M\subseteq M$ and also $A\subseteq
M$ whenever $M$ thinks that $A$ is a subset of $P^M$.)
Recall that a subset $H^M$ of $P^M$ is $M$-generic (or: $P^M$-generic over $M$),
if $|A\cap H^M|=1$ for all maximal antichains $A$ in $M$.
\item Let $H^M$ be $P^M$-generic over $M$ and $\n\tau$ a $P^M$-name in $M$.
We define the evaluation $\n\tau[H^M]^M$ to be
$x$ if
$M$ thinks that $p\Vdash_{P^M} \n\tau=\std x$
for some $p\in H^M$ and $x\in M$ (or equivalently just for $x\in M\cap \ON$),
and $\{\n\sigma[H^M]^M:\, (\n\sigma,p)\in\n\tau,\,
p\in H^M\}$ otherwise.
Abusing notation we write $\n\tau[H^M]$ instead of $\n\tau[H^M]^M$,
and we write $M[H^M]$ for $\{\n\tau[H^M]:\, \n\tau\text{ is a $P^M$-name
in }M\}$.
\item For any set $N$ (typically, an elementary submodel of some $H(\chi)$),
the ord-collapse $k$ (or $k^N$) is a recursively defined function with domain $N$:
$k(x)=x$ if $x\in \ON$, and $k(x)=\{k(y):\,
y\in x\cap N\}$ otherwise.
\item We define $\ordclos(\alpha):=\emptyset$ for all ordinals $\alpha$.
The ord-transitive closure of a non-ordinal $x$ is defined
inductively on the rank:
\[
\ordclos(x)=x\cup\bigcup \{\ordclos(y):y\in x\setminus \ON\}
=x\cup\bigcup \{\ordclos(y):y\in x\}.
\]
So for $x\notin \ON$, the set
$\ordclos(x)$ is the smallest ord-transitive set
containing $x$ as a subset.
HCON is the collection of all sets $x$ such that
the ord-transitive closure of $x$ is countable.
$x$ is in HCON iff $x$ is element of some candidate.
In particular, all reals and all ordinals are HCON.
We write HCON$_\alpha$ for the family of all sets $x$ in HCON whose transitive
closure
only contains ordinals $<\alpha$.
\end{enumerate}
\end{Def}
The following facts can be found in~\cite{MR2115943} or~\cite{kellnernep} (they
can be proven by rather straightforward, if tedious, inductions on the ranks of
the according objects).
\begin{Fact}\label{fact:hcon}
\begin{enumerate}
\item The ord-collapse of a countable elementary submodel of $H(\chi^*)$ is a nice
candidate.
\item Unions, intersections etc.\ are generally not absolute for
candidates. For example, let $x\in M\setminus \ON$. In $M$ we can
construct a set $y$ such that $M\models y=\om1\cup\{x\}$.
Then $y$ is not an ordinal and therefore a subset of $M$, and
in particular $y$ is countable and $y\not=\om1\cup \{x\}$.
\item Let $j:M\to M'$ be the transitive collapse of a candidate $M$,
and $f:\om1\cap M'\to \ON$ the inverse (restricted to the ordinals).
Obviously $M'$ is a countable transitive model of ZFC$^*$;
moreover $M$ is characterized by the pair $(M',f)$
(we call such a pair a ``labeled transitive model'').
Note that $f$ satisfies $f(\alpha+1)=f(\alpha)+1$,
$f(\alpha)=\alpha$ for $\alpha\in \omega\cup\{\omega\}$.
$M\models (\alpha\text{ is a limit})$ iff $f(\alpha)$ is a limit.
$M\models \cf(\alpha)=\omega$ iff $\cf(f(\alpha))=\omega$, and
in that case $f[\alpha]$ is cofinal in $f(\alpha)$.
On the other hand, given a transitive countable model $M'$ of ZFC$^*$
and an $f$ as above, then we can construct a (unique) candidate $M$
corresponding to $(M',f)$.
\item All candidates $M$ with $M\cap \ON \subseteq \omega_1$ are hereditarily countable, so their number is at most $2^{\al0}$. Similarly, the cardinality of HCON$_\alpha$ is at most continuum
whenever $\alpha < \omega_2$.
\item If $M$ is a candidate, and if $H^M$ is $P^M$-generic over $M$, then $M[H^M]$ is a candidate
as well and an end-extension
of $M$ such that $M\cap\ON=M[H^M]\cap \ON$. If $M$ is nice and
($M$ thinks that) $P^M$ is proper, then
$M[H^M]$ is nice as well.
\item Forcing extensions commute with the transitive collapse $j$:
If $M$ corresponds to $(M',f)$, then
$H^M\subseteq P^M$ is $P^M$-generic over $M$ iff
$H'\coloneqq {j}[H^M]$ is $P'\coloneqq j(P^M)$-generic over $M'$,
and in that case $M[H^M]$ corresponds to $(M'[H'],f)$.
In particular, the forcing extension $M[H^M]$ of $M$ satisfies
the forcing theorem (everything that is forced is true, and
everything true is forced).
\item For elementary submodels, forcing extensions commute
with ord-collapses:
Let $N$ be a countable elementary submodel of $H(\chi^*)$,
$P\in N$, $k:N\to M$ the ord-collapse (so $M$ is a candidate),
and let $H$ be $P$-generic
over $V$.
Then $H$ is $P$-generic over $N$
iff $H^M\coloneqq k[H]$
is $P^M\coloneqq k(P)$-generic over $M$;
and in that case the ord-collapse of $N[H]$
is $M[H^M]$.
\end{enumerate}
\end{Fact}
Assume that a nice candidate $M$ thinks that $(\bar P^M,\bar Q^M)$ is a forcing
iteration of length $\om2^V$ (we will usually write $\om2$ for the length of
the iteration, by this we will always mean $\om2^V$ and not the possibly
different $\om2^M$). In this section, we will construct an iteration $(\bar
P,\bar Q)$ in $V$, also of length $\om2$, such that each $P^M_\alpha$
canonically and $M$-completely embeds into $P_\alpha$ for all $\alpha\in
\om2\cap M$. Once we know (by induction) that $P^M_\alpha$ $M$-completely
embeds into $P_\alpha$, we know that a $P_\alpha$-generic filter $H_\alpha$
induces a $P^M_\alpha$-generic (over $M$) filter which we call $H^M_\alpha$.
Then $M[H^M_\alpha]$ is a candidate, but nice only if $P^M_\alpha$ is proper.
We will not need that $M[H^M_\alpha]$ is nice, actually we will only
investigate sets of reals (or elements of $H(\al1)$) in $M[H^M_\alpha]$, so it
does not make any difference whether we use $M[H^M_\alpha]$ or its transitive
collapse.
\begin{Rem}\label{rem:fine.print}
In the discussion so far we omitted some details regarding the theory ZFC$^*$
(that a candidate has to satisfy). The following ``fine print'' hopefully
absolves us from any liability. (It is entirely irrelevant for the
understanding of the paper.)
We have to guarantee that each $M[H^M_\alpha]$ that we consider satisfies
enough of ZFC to make our arguments work (for example, the definitions and
basic properties of ultralaver and Janus forcings should work). This turns out
to be easy, since (as usual) we do not need the full power set axiom for these
arguments (just the existence of, say, $\beth_5$). So it is enough that each
$M[H^M_\alpha]$ satisfies some fixed finite subset of ZFC minus power set, which
we call ZFC$^*$.
Of course we can also find a bigger (still finite) set ZFC$^{**}$ that implies:
$\beth_{10}$ exists, and each forcing extension of the universe with a forcing
of size $\le \beth_4$ satisfies ZFC$^*$. And it is provable (in ZFC) that each
$H(\chi)$ satisfies ZFC$^{**}$ for sufficiently large regular $\chi$.
We define ``candidate'' using the weaker theory ZFC$^*$, and require that nice
candidates satisfy the stronger theory ZFC$^{**}$. This guarantees that all
forcing extensions (by small forcings) of nice candidates will be candidates
(in particular, satisfy enough of ZFC such that our arguments about Janus or
ultralaver forcings work). Also, every ord-collapse of a countable elementary
submodel $N$ of $H(\chi)$ will be a nice candidate.
\end{Rem}
\subsection{Partial countable support iterations}\label{subsec:partialCS}
We introduce the notion of ``partial countable support limit'': a
subset of the countable support (CS) limit containing the union (i.e.,
the direct limit) and satisfying some natural requirements.
Let us first describe what we mean by ``forcing iteration''. They have to satisfy the
following requirements:
\begin{itemize}
\item A \qemph{topless forcing iteration} $(P_\alpha,Q_\alpha)_{\alpha<\varepsilon}$
is a sequence of forcing notions $P_\alpha$ and $P_\alpha$-names
$Q_\alpha$ of quasiorders with a weakest element~$1_{Q_\alpha}$.
A \qemph{topped iteration} additionally has a final limit~$P_\varepsilon$.
Each $ P_\alpha$ is a set of partial functions
on $\alpha$
(as, e.g., in~\cite{MR1234283}). More specifically,
if $\alpha<\beta\le \varepsilon$ and $p\in P_\beta$, then
$p\mathord\restriction\alpha\in P_\alpha$. Also,
$p\mathord\restriction\beta\Vdash_{P_\beta} p(\beta)\in Q_\beta$ for all $\beta\in\dom(p)$.
The order on $P_\beta$ will always be the ``natural'' one:
$q\leq p$ iff
$q\mathord\restriction\alpha$ forces (in $P_\alpha$) that
$q^{\textrm{\textup{tot}}}(\alpha)\leq p^{\textrm{\textup{tot}}}(\alpha)$ for all $\alpha < \beta$,
where $r^{\textrm{\textup{tot}}}(\alpha)=r(\alpha)$
for all $\alpha\in\dom(r)$ and $1_{Q_\alpha}$ otherwise. $P_{\alpha+1}$ consists
of \emph{all} $p$ with $p\mathord\restriction\alpha\in P_\alpha$ and $p\mathord\restriction \alpha\Vdash p^{\textrm{\textup{tot}}}(\alpha)\in Q_\alpha$, so it is forcing equivalent to $P_\alpha*Q_\alpha$.
\item $P_\alpha \subseteq P_\beta$ whenever $\alpha < \beta\le
\varepsilon$. (In particular, the empty condition is an element of each
$P_\beta$.)
\item For any $p \in P_\varepsilon$ and any $q \in P_\alpha$ ($\alpha < \varepsilon$) with $q \leq p \mathord\restriction \alpha$, the partial function
$q \land p\coloneqq q\cup p\mathord\restriction[\alpha,\varepsilon)$ is a condition in $P_\varepsilon$ as well (so in particular, $p \mathord\restriction \alpha$ is a reduction of $p$, hence $P_\alpha$ is a complete subforcing of $P_\varepsilon$; and $q\land p$ is the weakest condition
in $P_\varepsilon$ stronger than both $q$ and $p$).
\item Abusing notation, we usually just write $\bar P$
for an iteration (be it topless or topped).
\item We usually write $H_\beta$ for the
generic filter on $P_\beta$ (which induces
$P_\alpha$-generic filters called $H_\alpha$ for $\alpha\le\beta$).
For topped iterations we call the filter on the final limit sometimes just $H$
instead of $H_\varepsilon$.
\end{itemize}
We use the following notation for quotients of iterations:
\begin{itemize}
\item For $\alpha<\beta$, in the $P_\alpha$-extension
$V[H_\alpha]$, we let $P_\beta/H_\alpha$ be the set
of all $p\in P_\beta$ with $p\mathord\restriction\alpha\in H_\alpha$
(ordered as in~$P_\beta$). We may occasionally write $P_\beta/P_\alpha$ for the $P_\alpha$-name of
$P_\beta/H_\alpha$.
\item Since $P_\alpha$ is a complete subforcing of $P_\beta$,
this is a quotient with the usual properties, in particular
$P_\beta$ is equivalent to $P_\alpha*(P_\beta/H_\alpha)$.
\end{itemize}
\begin{Rem}
It is well known that quotients of proper countable support iterations are
naturally equivalent to (names of) countable support iterations. In this paper,
we can restrict our attention to proper forcings, but we do not really have
countable support iterations. It turns out that it is not necessary to
investigate whether our quotients can naturally be seen as iterations of any
kind, so to avoid the subtle problems involved we will not consider the
quotient as an iteration by itself.
\end{Rem}
\begin{Def}\label{def:fullCS}
Let $\bar P$ be a (topless) iteration of limit length $\varepsilon$.
We define three limits of $\bar P$:
\begin{itemize}
\item The \qemph{direct limit} is the union of the $P_\alpha$ (for $\alpha<\varepsilon$).
So this is the smallest possible limit of the iteration.
\item The \qemph{inverse limit} consists of \emph{all} partial functions $p$
with domain $\subseteq \varepsilon$ such that $p\mathord\restriction\alpha\in P_\alpha$
for all $\alpha<\varepsilon$.
This is the largest possible limit of the iteration.
\item
The \qemph{full countable support limit $P^{\textrm{\textup{CS}}}_\varepsilon$} of $\bar P$
is the inverse limit if $\cf(\varepsilon)=\omega$ and the direct limit otherwise.
\end{itemize}
We say that $P_\varepsilon$ is a \qemph{partial CS limit}, if
$P_\varepsilon$ is a subset of the full CS limit and
the sequence $(P_\alpha)_{\alpha\le \varepsilon}$ is a topped iteration.
In particular, this means that $P_\varepsilon$
contains the direct limit,
and satisfies the following for each $\alpha<\varepsilon$:
$P_\varepsilon$ is closed under $p\mapsto p\mathord\restriction \alpha$, and
whenever $p\in P_\varepsilon$, $q\in P_\alpha$, $q\le p\mathord\restriction\alpha$,
then also the partial function $q\land p$ is in~$P_\varepsilon$.
\end{Def}
So for a given topless $\bar P$ there is a well-defined inverse, direct and full
CS limit. If $\cf(\varepsilon)>\omega$, then the direct and the full CS limit
coincide. If $\cf(\varepsilon)=\omega$,
then the direct limit and the full CS limit (=inverse limit) differ. Both of them are partial CS limits,
but there are many more
possibilities for partial CS limits. By definition, all of them will yield
iterations.
Note that the name ``CS limit'' is slightly inappropriate, as the size of
supports of conditions is not part of the definition. To give a
more specific example:
Consider a topped iteration $\bar P $ of length $\omega+\omega$ where
$P_\omega$ is the direct limit and $P_{\omega+\omega} $ is the full
CS limit. Let $p$ be any element of the full CS limit of $\bar P \mathord\restriction \omega$ which
is not in~$P_\omega$; then $p$ is not in $P_{\omega+\omega}$ either. So not
every countable subset of $\omega+\omega$ can appear as the support of a
condition.
\begin{Def}\label{partial_CS}
A forcing iteration $\bar P$ is called a \qemph{partial CS iteration}, if
\begin{itemize}
\item every limit is a partial CS limit, and
\item every $Q_\alpha$ is (forced to be) separative.\footnote{The reason for this requirement is briefly discussed in Section~\ref{sec:alternativedefs}. Separativity, as well as the relations $\leq^*$ and $=^*$, are defined on page~\pageref{def:separative}.}
\end{itemize}
\end{Def}
The following fact can easily be proved by transfinite induction:
\begin{Fact}\label{fact:eq.eqstar}
Let $\bar P$ be a partial CS iteration. Then for all $\alpha$ the forcing
notion $P_\alpha$ is separative.
\end{Fact}
{}From now on, all iterations we consider will be partial CS
iterations. In this paper, we will only be interested in proper
partial CS iterations, but properness is not part of the definition of
partial CS iteration. (The reader may safely assume that all
iterations are proper.)
Note that separativity of the $Q_\alpha $ implies that all partial CS
iterations satisfy the following (trivially equivalent) properties:
\begin{Fact}\label{fact:suitable.equivalent}
Let $\bar P$ be a topped partial CS iteration of length $\varepsilon$. Then:
\begin{enumerate}
\item Let $H$ be $ P_\varepsilon$-generic. Then $p\in H$ iff $p\mathord\restriction\alpha\in H_\alpha$ for all $\alpha < \varepsilon$.
\item For all $q,p \in P_\varepsilon$: If $q\mathord\restriction \alpha \leq^\ast p\mathord\restriction \alpha$ for each $\alpha < \varepsilon$, then $q \leq^\ast p$.
\item \label{item:3}
For all $q,p \in P_\varepsilon$: If $q\mathord\restriction \alpha \leq^\ast p\mathord\restriction \alpha$ for each $\alpha < \varepsilon$, then $q \parallel p$.
\end{enumerate}
\end{Fact}
We will be concerned with the following situation:
Assume that $M$ is a nice candidate, $\bar P^M$ is (in~$M$) a topped partial CS
iteration of length $\varepsilon$ (a limit ordinal in~$M$), and $\bar P$ is (in
$V$) a topless partial CS iteration of length
$\varepsilon'\coloneqq \sup(\varepsilon\cap M)$. (Recall that
``$\cf(\varepsilon)=\omega$'' is absolute between $M$ and $V$, and that
$\cf(\varepsilon)=\omega$ implies $\varepsilon'=\varepsilon$.) Moreover, assume
that we already have a system of $M$-complete coherent\footnote{I.e., they
commute with the restriction maps: $i_\alpha(p \mathord\restriction \alpha) = i_\beta(p) \mathord\restriction
\alpha$ for $\alpha < \beta$ and $p\in P^M_\beta$.} embeddings
$i_\beta:P^M_\beta\to P_\beta$ for $\beta\in \varepsilon'\cap M=\varepsilon\cap M$.
(Recall that any potential partial CS limit of $\bar P$ is a subforcing of
the full CS limit~$P^{\textrm{\textup{CS}}}_{\varepsilon'}$.)
It is easy to see that there is only one possibility for an embedding $j:
P^M_\varepsilon\to P^{\textrm{\textup{CS}}}_{\varepsilon'} $ (in fact, into any potential partial
CS limit of~$\bar P$) that extends the $i_\beta$'s naturally:
\begin{Def}\label{def:canonicalextension}
For a topped partial CS iteration $\bar P^M$ in $M$ of length $\varepsilon$
and a topless one $\bar P$ in $V$ of length $\varepsilon'\coloneqq \sup(\varepsilon\cap M)$
together with coherent embeddings $i_\beta$, we define
$j: P^M_\varepsilon\to P^{\textrm{\textup{CS}}}_{\varepsilon'} $, the \qemph{canonical extension},
in the obvious way:
Given $p \in P_\varepsilon^M$, take
the sequence of restrictions to $M$-ordinals,
apply the functions $i_\beta$, and let
$j(p)$ be the union of the resulting coherent sequence.
\end{Def}
We do not claim that $j: P^M_\varepsilon\to P^{\textrm{\textup{CS}}}_{\varepsilon'} $ is
$M$-complete.\footnote{\newcounter{myfootnote}\setcounter{myfootnote}{\value{footnote}}
\label{cs.fs.footnote}
For example, if $\varepsilon=\varepsilon'=\omega$ and if $P^M_\omega$ is the
finite support limit of a nontrivial iteration, then $j:P^M_\omega\to P^{\textrm{\textup{CS}}}_\omega$ is not
complete: For notational simplicity, assume that all $Q^M_n$ are (forced to be) Boolean
algebras. In $M$, let $c_n$ be (a $P^M_n$-name for) a nontrivial
element of $Q^M_n$ (so $\lnot c_n$, the Boolean complement, is also nontrivial).
Let $p_n$ be the $P^M_n$-condition $(c_0, \ldots, c_{n-1})$, i.e.,
the truth value of ``$c_m\in H(m)$ for all $m<n$''. Let $q_n$ be the
$P^M_{n+1}$-condition $(c_0, \ldots, c_{n-1}, \lnot c_n)$, i.e., the truth
value of
``$n$ is minimal with $c_n\notin H(n)$''. In $M$, the set $A=\{q_n:\, n\in\omega\}$
is a maximal antichain in $P^M_\omega$. Moreover, the sequence
$(p_n)_{n\in\omega}$ is a decreasing coherent sequence, therefore $i_n(p_n)$
defines an element $p_\omega$ in $P^{\textrm{\textup{CS}}}_\omega$, which is clearly incompatible
with all $j(q_n)$, hence $j[A] $ is not maximal.}
In the following, we will construct partial CS limits $P_{\varepsilon'}$ such
that $j:P^M_\varepsilon \to P_{\varepsilon'}$ is $M$-complete. (Obviously, one
requirement for such a limit is that $j[P^M_\varepsilon]\subseteq
P_{\varepsilon'}$.) We will actually define two versions: The almost FS (``almost
finite support'') and the
almost CS (``almost countable support'') limit.
Note that there is only one effect that the ``top'' of $\bar P^M$ (i.e., the
forcing $P^M_\varepsilon$) has on the canonical extension~$j$: It determines the
domain of $j$. In particular it will generally depend on $P^M_\varepsilon$
whether $j$ is complete or not. Apart from that, the value of any given $j(p)$
does not depend on $P^M_\varepsilon$.
Instead of arbitrary systems of embeddings $i_\alpha$, we will only be interested in
``canonical'' ones.
We assume for notational convenience that
$Q^M_\alpha$ is a subset of $Q_\alpha$ (this will naturally be the case in our application anyway).
\begin{Def}[The canonical embedding]\label{def:canonicalembedding}
Let $\bar P$ be a partial CS iteration in~$V$
and $\bar P^M$ a partial CS iteration in~$M$, both topped and of length~$\varepsilon\in M$.
We construct by induction on $\alpha\in (\varepsilon+1) \cap M$
the canonical $M$-complete embeddings $i_\alpha:P^M_\alpha\to P_\alpha$.
More precisely: We try to construct them, but it is possible that the
construction fails. If the construction succeeds, then
we say that \qemph{$\bar P^M$ (canonically) embeds into $\bar P$}, or
\qemph{the canonical embeddings work}, or
just:
\qemph{$\bar P$ is over $\bar P^M$}, or \qemph{over $P^M_\varepsilon$}.
\begin{itemize}
\item Let $\alpha=\beta+1$. By induction hypothesis,
$i_\beta$ is $M$-complete, so a $V$-generic
filter $H_\beta\subseteq P_\beta$ induces an $M$-generic filter $H^M_\beta
\coloneqq i_{\beta}^{-1}[H_\beta]\subseteq P^M_\beta$.
We require that (in the $H_\beta$ extension) the set $Q^M_\beta[H^M_\beta]$
is an $M[H^M_\beta]$-complete
subforcing of $Q_\beta[H_\beta]$. In this case, we define $i_\alpha$ in the
obvious way.
\item For $\alpha$ limit, let $i_\alpha$ be the canonical
extension of the family $(i_\beta)_{\beta\in \alpha\cap M}$.
We require that $P_\alpha$ contains the
range of $i_\alpha$, and that $i_\alpha$ is $M$-complete; otherwise the
construction fails.
(If $\alpha'\coloneqq \sup(\alpha\cap M) < \alpha$, then $i_\alpha $
will actually be an $M$-complete map into $P_{\alpha'}$, assuming that the
requirement is fulfilled.)
\end{itemize}
\end{Def}
In this section we try to construct a partial CS iteration $\bar P$
(over a given $\bar P^M$) satisfying additional properties.
\begin{Rem}
What is the role of $\varepsilon'\coloneqq \sup( \varepsilon\cap M)$? When our
inductive construction of $\bar P$ arrives at~$P_\varepsilon$ where $\varepsilon'<
\varepsilon$, it would be too late\footnote{
\label{fn:too.late}
For example:
Let $\varepsilon=\om1$ and $\varepsilon'=\om1\cap M$.
Assume that $P^M_{\om1}$ is (in $M$) a (or: the unique)
partial CS limit of a nontrivial iteration.
Assume that we have a topless iteration $\bar P$ of length $\varepsilon'$
in $V$ such that the canonical embeddings work for all $\alpha\in\om1\cap M$.
If we set $P_{\varepsilon'}$ to be the full CS limit, then
we cannot further extend it to any iteration of length $\om1$
such that the canonical embedding $i_{\om1}$ works: Let
$p_\alpha$ and $q_\alpha$ be as in footnote~\ref{cs.fs.footnote}. In $M$,
the set
$A=\{q_\alpha:\alpha\in\om1\}$ is a maximal antichain, and the sequence
$(p_\alpha)_{\alpha\in\om1}$ is a decreasing coherent
sequence. But in $V$ there is an element $p_{\varepsilon'}\in P^{\textrm{\textup{CS}}}_{\varepsilon'}$
with $p_{\varepsilon'}\mathord\restriction \alpha = j( p_\alpha)$ for all $\alpha\in \varepsilon\cap M$. This
condition $p_{\varepsilon'}$ is clearly incompatible
with all elements of $j[A] = \{ j(q_\alpha): \alpha \in \varepsilon\cap M\}$.
Hence $j[A] $ is not maximal.}
to take care of $M$-completeness of
$i_\varepsilon$ at this stage, even if all $i_\alpha$ work nicely for $\alpha\in
\varepsilon \cap M$. Note that $\varepsilon'<\varepsilon$ implies
that $\varepsilon$ is uncountable in $M$, and that therefore
$P^M_\varepsilon = \bigcup_{\alpha\in \varepsilon\cap M} P^M_\alpha$.
So the natural extension $j$ of the embeddings $(i_\alpha)_{\alpha\in
\varepsilon \cap M}$ has range in $ P_{\varepsilon'}$, which will be a complete
subforcing of $P_\varepsilon$. So we have to ensure
$M$-completeness already in the construction of $P_{\varepsilon'}$.
\end{Rem}
For now we just record:
\begin{Lem}\label{lem:wolfgang}
Assume that we have topped iterations
$\bar P^M$ (in~$M$) of length $\varepsilon$ and
$\bar P$ (in~$V$) of length $\varepsilon'\coloneqq \sup(\varepsilon\cap M)$,
and that for all $\alpha\in\varepsilon\cap M$
the canonical embedding $i_\alpha: P^M_\alpha\to P_\alpha$ works.
Let $i_\varepsilon: P^M_\varepsilon\to P^{\textrm{\textup{CS}}}_{\varepsilon'}$
be the canonical extension.
\begin{enumerate}
\item\label{item:pathetic099}
If $P^M_\varepsilon$ is (in $M$) a direct limit (which
is always the case if $\varepsilon$ has uncountable cofinality)
then $i_\varepsilon$
(might not work, but at least) has range in $P_{\varepsilon'}$
and preserves incompatibility.
\item\label{item:suitable_implies_filter}
If $i_\varepsilon$ has a range contained in $P_{\varepsilon'}$
and maps predense sets $D\subseteq P^M_\varepsilon$ in $M$
to predense sets $i_\varepsilon[D]\subseteq P_{\varepsilon'}$,
then $i_\varepsilon$ preserves incompatibility (and therefore works).
\end{enumerate}
\end{Lem}
\begin{proof} (1)
Since $P^M_\varepsilon$ is a direct limit, the canonical extension
$i_\varepsilon $ has range in $\bigcup_{\alpha<\varepsilon'}
P_\alpha$, which is subset of any partial CS limit $P_{\varepsilon'}$.
Incompatibility in $P^M_\varepsilon$ is the same as incompatibility in
$P^M_\alpha$ for sufficiently large $\alpha\in \varepsilon\cap M$, so
by assumption it is preserved by $i_\alpha$ and hence also by~$i_\varepsilon$.
(2)
Fix $p_1,p_2\in P^M_\varepsilon$, and assume that their images are
compatible in $P_{\varepsilon'}$; we have to show that they
are compatible in $P^M_\varepsilon$. So fix
a generic filter $H\subseteq P_{\varepsilon'} $ containing $i_\varepsilon(p_1)$
and $i_\varepsilon(p_2)$.
In $M$, we define the following set $D$:
\[
D \coloneqq \{ q \in P^M_\varepsilon:
(q \leq p_1 \land q \leq p_2) \textrm{ or }
(\exists \alpha < \varepsilon: q\mathord\restriction \alpha \perp_{P^M_\alpha} p_1\mathord\restriction \alpha) \textrm{ or }
(\exists \alpha < \varepsilon: q\mathord\restriction \alpha \perp_{P^M_\alpha} p_2\mathord\restriction \alpha)
\}.
\]
Using Fact~\ref{fact:suitable.equivalent}(\ref{item:3}) it is easy to check that $D$ is dense.
Since $i_\varepsilon$ preserves predensity,
there is $q\in D$ such that $i_\varepsilon(q)\in H$. We claim that
$q$ is stronger than $p_1$ and~$p_2$. Otherwise we would have without loss
of generality
$ q\mathord\restriction \alpha \perp_{P^M_\alpha} p_1\mathord\restriction \alpha$ for some
$\alpha<\varepsilon$. But the filter $H\mathord\restriction \alpha$ contains both $i_\alpha(q\mathord\restriction\alpha)$
and $i_\alpha(p_1\mathord\restriction\alpha)$, contradicting the assumption that $i_\alpha$
preserves incompatibility.
\end{proof}
\subsection{Almost finite support iterations}
Recall Definition~\ref{def:canonicalextension} (of the canonical extension)
and
the setup that was described there: We have to find a
subset $P_{\varepsilon'}$ of $P^{\textrm{\textup{CS}}}_{\varepsilon'}$ such that the canonical extension~$j:P^M_\varepsilon\to P_{\varepsilon'}$ is $M$-complete.
We now define the almost finite support limit. (The direct limit will in
general not do, as it may not contain the range $j[P^M_\varepsilon]$. The
almost finite support limit is the obvious modification of the direct limit,
and it is the smallest partial CS limit $P _{\varepsilon'}$ such that $j[P^M_\varepsilon]\subseteq
P_{\varepsilon'}$, and it indeed turns out to be $M$-complete as well.)
\begin{Def}\label{def:almostfs}
Let $\varepsilon$ be a limit ordinal in~$M$, and let
${\varepsilon'}\coloneqq \sup(\varepsilon\cap M)$.
Let $\bar P^M$ be a topped iteration in~$M$ of length $\varepsilon$,
and let $\bar P$ be a topless iteration in~$V$ of length $\varepsilon'$.
Assume that the canonical embeddings $i_\alpha$ work for all $\alpha
\in \varepsilon \cap M = \varepsilon' \cap M$. Let $i_\varepsilon$
be the canonical extension.
We define the \emph{almost finite support limit of $\bar P$ over
${\bar P^M}$} (or: almost FS limit) as the following subforcing
$P_{\varepsilon'}$ of $P^{\textrm{\textup{CS}}}_{\varepsilon'}$:
\[P_{\varepsilon'}\coloneqq \{\,
q \land i_\varepsilon(p) \in P^{\textrm{\textup{CS}}}_{\varepsilon'} :\ p\in P^M_\varepsilon \text{ and }
q\in P_\alpha\text{ for some } \alpha\in \varepsilon\cap M\text{ such that }
q\le _{P_\alpha} i_\alpha(p\mathord\restriction \alpha) \, \}.\]
\end{Def}
Note that for $\cf(\varepsilon)>\omega$, the almost FS limit
is equal to the direct limit, as each $p\in P^M_\varepsilon$ is in fact
in $P^M_\alpha$ for some $\alpha\in \varepsilon\cap M$, so $i_\varepsilon(p) =
i_\alpha(p)\in P_\alpha$.
\begin{Lem}\label{lem:afs.complete}
Assume that $\bar P$ and $\bar P^M$ are as above and let $P_{\varepsilon'}$ be
the almost FS limit. Then $\bar P^\frown P_{\varepsilon'}$ is a partial CS
iteration, and $i_\varepsilon$ works, i.e., $i_\varepsilon$ is an
$M$-complete embedding from $P^M_\varepsilon$ to $P_{\varepsilon'}$. (As
$P_{\varepsilon'} $ is a complete subforcing of~$P_\varepsilon$, this also
implies that $i_\varepsilon$ is $M$-complete from $P^M_\varepsilon$
to $P_\varepsilon$.)
\end{Lem}
\begin{proof}
It is easy to see that $P_{\varepsilon'}$ is a partial CS limit and contains
the range $i_\varepsilon[P^M_\varepsilon]$.
We now show preservation of
predensity; this implies $M$-completeness by Lemma~\ref{lem:wolfgang}.
Let $(p_j)_{j\in J} \in M$ be a maximal antichain in
$P^M_\varepsilon$. (Since $P^M_\varepsilon$ does not have to be ccc in $M$,
$J$ can have any cardinality in $M$.) Let $q\land {i_{\varepsilon}}(p)$ be a
condition in $P_{\varepsilon'} $. (If ${\varepsilon'}<\varepsilon$, i.e., if
$\cf(\varepsilon)>\omega$, then we can choose $p$ to be the empty condition.)
Fix $\alpha \in \varepsilon \cap M $ be such
that $q\in P_\alpha$.
Let $H_\alpha$ be $P_\alpha$-generic and contain $q$,
so $p\mathord\restriction\alpha$ is in $H^M_\alpha$.
Now in $M[H^M_\alpha]$ the set $\{p_j: j\in J, p_j\in
P_\varepsilon^M/H_\alpha^M\}$ is predense in
$P^M_\varepsilon/H^M_\alpha$ (since this is forced by the empty condition
in $P^M_\alpha$). In particular,
$p$ is compatible with some $p_j$, witnessed
by $p'\le p,p_j$ in $P^M_\varepsilon /H^M_\alpha$.
We can find $q'\le_{P_\alpha} q$ deciding $j$ and $p'$; since certainly
$q'\le^* {i_{\alpha}}(p'\mathord\restriction \alpha) $, we may assume even $\le$ without loss of generality. Now $q'\land {i_{\varepsilon}} (p') \le q\land
{i_\varepsilon}(p)$ (since $q'\leq q$ and $p'\leq p$),
and $q'\land {i_\varepsilon} (p') \le
{i_\varepsilon}(p_j)$ (since $p'\le p_j$).
\end{proof}
\begin{DefandClaim}\label{lem:kellnertheorem}
Let $\bar P^M$ be a topped partial CS iteration in $M$
of length $\varepsilon$. We can construct by induction on
$\beta\in\varepsilon+1$ an
\emph{almost finite support iteration $\bar P$ over $\bar P^M$}
(or: almost FS iteration) as follows:
\begin{enumerate}
\item As induction hypothesis we assume that the canonical
embedding $i_\alpha$
works for all $\alpha\in \beta \cap M$. (So the notation
$M [H^M_\alpha]$ makes sense.)
\item\label{item:qwrrqw} Let $\beta=\alpha+1$. If $\alpha\in M$, then we can use any
$Q_\alpha$ provided that (it is forced that) $Q^M_\alpha$ is an
$M[H^M_\alpha]$-complete subforcing of $Q_\alpha$.
(If $\alpha\notin M$, then there is no restriction on $Q_\alpha$.)
\item Let $\beta\in M$ and $\cf(\beta)=\omega$.
Then $P_\beta$ is the almost FS limit of
$(P_\alpha,Q_\alpha)_{\alpha<\beta}$ over $P^M_\beta$.
\item Let $\beta\in M$ and $\cf(\beta)>\omega$. Then $P_\beta$ is
again the almost FS limit of
$(P_\alpha,Q_\alpha)_{\alpha<\beta}$ over $P^M_\beta$ (which
also happens to be the direct limit).
\item For limit ordinals not in $M$,
$P_\beta$ is the direct limit.
\end{enumerate}
\end{DefandClaim}
So the claim includes that the resulting $\bar P$ is a (topped) partial CS
iteration of length $\varepsilon$ over $\bar P^M$ (i.e., the canonical embeddings
$i_\alpha$ work for all $\alpha\in (\varepsilon+1) \cap M$), where we only assume
that the $Q_\alpha$ satisfy the obvious requirement given in~(\ref{item:qwrrqw}).
(Note that we can always find some suitable $Q_\alpha$ for $\alpha\in M$,
for example we can just take $Q^M_\alpha$ itself.)
\begin{proof}
We have to show (by induction) that the resulting sequence $\bar P$
is a partial CS iteration, and that $\bar P^M$ embeds into $\bar P$.
For successor cases, there is nothing to do. So assume that $\alpha$
is a limit. If $P_\alpha$ is a direct limit, it is trivially a
partial CS limit; if $P_\alpha$ is an almost FS limit, then the
easy part of Lemma~\ref{lem:afs.complete} shows that it is a partial
CS limit.
So it remains to show that for a limit $\alpha\in M$, the (naturally defined)
embedding $i_\alpha:P^M_\alpha\to P_\alpha$ is $M$-complete.
This was the main claim in Lemma~\ref{lem:afs.complete}.
\end{proof}
The following lemma is natural and easy.
\begin{Lem}
Assume that we construct an almost FS
iteration $\bar P$ over $\bar P^M$ where
each $Q_\alpha$ is (forced to be) ccc.
Then $P_\varepsilon$ is ccc (and in particular proper).
\end{Lem}
\begin{proof}
We show that $P_\alpha$ is ccc by induction on
$\alpha\leq\varepsilon$. For successors, we use that $Q_\alpha$ is
ccc. For $\alpha$ of uncountable cofinality, we know that we took
the direct limit coboundedly often (and all $P_\beta$ are ccc for
$\beta<\alpha$), so by a result of Solovay $P_\alpha$ is again ccc.
For $\alpha$ a limit of countable cofinality not in $M$, just use
that all $P_\beta$ are ccc for $\beta<\alpha$, and the fact that
$P_\alpha$ is the direct limit. This leaves the case that
$\alpha\in M$ has countable cofinality, i.e., the $P_\alpha$ is the
almost FS limit. Let $A\subseteq P_\alpha$ be uncountable. Each
$a\in A$ has the form $q\land i_\alpha(p)$ for $p\in P^M_\alpha$ and
$q\in \bigcup_{\gamma<\alpha} P_\gamma$. We can thin out the set $A$ such that
$p$ are the same and all $q$ are in the same $P_\gamma$. So there have to be compatible
elements in $A$.
\end{proof}
All almost FS iterations that we consider in this paper will satisfy the
countable chain condition (and hence in particular be proper).
We will need a variant of this lemma for $\sigma$-centered forcing notions.
\begin{Lem}\label{lem:4.17}
Assume that we construct an almost FS
iteration $\bar P$ over $\bar P^M$
where only countably many $Q_\alpha$
are nontrivial (e.g., only those with $\alpha\in M$) and where
each $Q_\alpha$ is (forced to be) $\sigma$-centered.
Then $P_\varepsilon$ is $\sigma$-centered as well.
\end{Lem}
\begin{proof}
By induction: The direct limit of countably many
$\sigma$-centered forcings is $\sigma$-centered,
as is the almost FS limit of $\sigma$-centered forcings
(to color $q\land i_\alpha( p)$, use $p$ itself together with the color of $q$).
\end{proof}
We will actually need two variants of the almost FS construction:
Countably many models $M^n$; and starting the almost FS iteration with some $\alpha_0$.
Firstly, we can construct an almost FS iteration not just over one iteration
$\bar P^M$, but over an increasing chain of iterations.
Analogously to Definition~\ref{def:almostfs} and
Lemma~\ref{lem:afs.complete}, we can show:
\begin{Lem}\label{lem:418}
For each $n\in \omega$, let $M^n$ be a nice candidate, and let $\bar P^n$
be a topped partial CS iteration in $M^n$ of
length\footnote{Or only: $\varepsilon\in M^{n_0}$ for some $n_0$.} $\varepsilon\in M^0$ of countable
cofinality,
such that $M^m\in M^n $ and
$M^{n}$ thinks that $\bar P^m$ canonically embeds into $\bar P^{n}$,
for all $m<n$. Let $\bar P$ be a topless iteration of length $\varepsilon$ into
which all $\bar P^n$ canonically embed.
Then we can define the almost FS limit $P_\varepsilon$ over $(\bar
P^n)_{n\in \omega}$ as follows:
Conditions in $P_\varepsilon$ are of the form $q\land i^n_\varepsilon(p)$ where
$n\in\omega$, $p\in P^n_\varepsilon$,
and $q\in P_\alpha$ for some $\alpha\in M^n\cap\varepsilon$ with $q\le i^n_\alpha (p\mathord\restriction \alpha)$.
Then $P_\varepsilon$
is a partial CS limit over each $\bar P^n$.
\end{Lem}
As before, we get the following corollary:
\begin{Cor}\label{cor:ctblmanycandidates}
Given $M^n$ and $\bar P^n$ as above, we can construct a topped partial CS
iteration $\bar P$ such that each $\bar P^n$ embeds $M^n$-completely into it;
we can choose $Q_\alpha$ as we wish (subject to the obvious restriction that
each $Q^n_\alpha$ is an $M^n[H^n_\alpha]$-complete subforcing). If we always choose
$Q_\alpha$ to be ccc, then $\bar P$ is ccc; this is the case if we set
$Q_\alpha$ to be the union of the (countable) sets $Q^n_\alpha$.
\end{Cor}
\begin{proof}
We can define $P_\alpha$ by induction. If $\alpha\in \bigcup_{n\in
\omega} M^n$ has countable cofinality, then we use the almost
FS limit as in Lemma~\ref{lem:418}. Otherwise we use the direct
limit. If $\alpha\in M^n$ has uncountable cofinality, then
$\alpha'\coloneqq \sup(\alpha\cap M)$ is an element of $M^{n+1}$. In our
induction we have already considered $\alpha'$ and have defined
$P_{\alpha'}$ by Lemma~\ref{lem:418} (applied to the sequence
$(\bar P^{n+1}, \bar P^{n+2},\ldots)$).
This is sufficient to show that
$i^n_\alpha:P^n_\alpha\to P_{\alpha'} \lessdot P_\alpha$ is
$M^n$-complete.
\end{proof}
Secondly, we can start the almost FS iteration after some $\alpha_0$
(i.e., $\bar P$ is already given up to $\alpha_0$, and we can continue
it as an almost FS iteration up to $\varepsilon$), and get the same
properties that we previously showed for the almost FS iteration, but
this time for the quotient $P_\varepsilon/P_{\alpha_0}$. In
more detail:
\begin{Lem}\label{lem:almostfsstartatalpha}
Assume that $\bar P^M$ is in $M$ a (topped) partial CS iteration of length
$\varepsilon$, and that $\bar P$ is in $V$ a topped partial CS iteration
of length $\alpha_0$ over $\bar P^M\mathord\restriction {\alpha_0} $
for some $\alpha_0\in \varepsilon \cap M$.
Then we can extend $\bar P$ to a (topped) partial CS iteration of length
$\varepsilon$ over $\bar P^M$, as in the almost FS iteration (i.e.,
using the almost FS limit at limit points $\beta>\alpha_0$ with $\beta\in M$
of countable cofinality; and the direct limit everywhere else).
We can use
any $Q_\alpha$ for $\alpha\geq\alpha_0$ (provided $Q^M_\alpha$ is an
$M[H^M_\alpha]$-complete subforcing of $Q_\alpha$). If all $Q_\alpha$ are ccc,
then $P_{\alpha_0}$ forces that $P_{\varepsilon}/H_{\alpha_0}$ is ccc (in
particular proper); if moreover all $Q_\alpha$ are $\sigma$-centered and only
countably many are nontrivial, then $P_{\alpha_0}$ forces that
$P_{\varepsilon}/H_{\alpha_0}$ is $\sigma$-centered.
\end{Lem}
\subsection{Almost countable support iterations}\label{subsec:almostCS}
``Almost countable support iterations $\bar P$'' (over a given iteration $\bar P^M$ in a candidate~$M$)
will have the following two crucial properties: There is a canonical
$M$-complete embedding of~$\bar P^M $ into~$\bar P$,
and $\bar P$ preserves a given random real (similar to the usual countable support iterations).
\begin{DefandClaim}\label{def:almost_CS_iteration_wolfgang}
Let $\bar P^M$ be a topped partial CS iteration in $M$ of length $\varepsilon$.
We can construct by induction on $\beta\in\varepsilon+1$
the \emph{almost countable support iteration $\bar P$ over $\bar P^M$}
(or: almost CS iteration):
\begin{enumerate}
\item As induction hypothesis, we assume that the canonical embedding
$i_\alpha$ works for every $\alpha\in \beta\cap M$.
We set\footnote{
So for successors $\beta\in M$, we have $\delta'=\beta=\delta$.
For $\beta\in M$ limit, $\beta= \delta$ and $\delta'$ is as
in Definition~\ref{def:canonicalextension}.}
\begin{equation}
\label{eq:delta.prime}
\delta\coloneqq \min(M\setminus \beta),
\quad
\delta'\coloneqq \sup(\alpha+1:\, \alpha\in \delta \cap M ).
\end{equation}
Note that $\delta'\le \beta \le \delta$.
\item Let $\beta=\alpha+1$. We can choose any desired forcing $Q_\alpha$;
if $\beta\in M$ we of course require that
\begin{equation}\label{eq:jehrwetewt}
\text{$Q^M_\alpha$ is an $M[H^M_\alpha]$-complete subforcing of $Q_\alpha$.}
\end{equation}
This defines $P_\beta$.
\item Let $\cf(\beta)> \omega$. Then
$P_\beta$ is the direct limit.
\item Let $\cf(\beta) = \omega$ and assume that $\beta\in M$ (so $M\cap \beta$ is cofinal
in $\beta$ and $\delta' = \beta=\delta$).
We define $P_\beta=P_\delta$ as the union of the following two sets:
\begin{itemize}
\item The almost FS limit of $(P_\alpha,Q_\alpha)_{\alpha<\delta}$, see
Definition~\ref{def:almostfs}.
\item The set $P_{\delta}^\textup{\textrm{gen}}$ of $M$-generic conditions $q \in P^{\textrm{\textup{CS}}}_{\delta}$, i.e., those which satisfy
\begin{displaymath}
q \Vdash_{P^{\textrm{\textup{CS}}}_{\delta}} i^{-1}_\delta[H_{P^{\textrm{\textup{CS}}}_{\delta}}]\subseteq P^M_\delta \textrm{ is } M\textrm{-generic.}
\end{displaymath}
\end{itemize}
\item Let $\cf(\beta) = \omega$ and assume that $\beta\notin M$ but $M\cap \beta$ is cofinal
in $\beta$, so $\delta' = \beta< \delta$.
We define $P_\beta = P_{\delta'}$ as the union of the following two sets:
\begin{itemize}
\item The direct limit of $(P_\alpha,Q_\alpha)_{\alpha<\delta'}$.
\item The set $P_{\delta'}^\textup{\textrm{gen}}$ of $M$-generic conditions $q \in P^{\textrm{\textup{CS}}}_{\delta'}$, i.e., those which satisfy
\begin{displaymath}
q \Vdash_{P^{\textrm{\textup{CS}}}_{\delta'}} i^{-1}_\delta[H_{P^{\textrm{\textup{CS}}}_{\delta'}}]\subseteq
P^M_\delta
\textrm{ is } M\textrm{-generic.}
\end{displaymath}
\end{itemize}
(Note that the $M$-generic conditions form an open subset of $P^{\textrm{\textup{CS}}}_\beta=P^{\textrm{\textup{CS}}}_{\delta'}$.)
\item Let $\cf(\beta) = \omega$ and $M \cap \beta$
not cofinal in~$\beta$ (so $\beta\notin M$).
Then $P_\beta$ is the full CS limit of
$(P_\alpha,Q_\alpha)_{\alpha<\beta}$ (see Definition~\ref{def:fullCS}).
\end{enumerate}
\end{DefandClaim}
So the claim is that for every choice of $Q_\alpha$ (with the obvious
restriction~\eqref{eq:jehrwetewt}), this construction always results in a partial CS iteration
$\bar P$ over $\bar P^M$.
The proof is a bit cumbersome; it is a variant of the usual proof that
properness is preserved in countable support iterations (see
e.g.~\cite{MR1234283}).
We will use the following fact in $M$ (for the iteration $\bar P^M$):
\proofclaim{eq:prelim}{Let $\bar P$ be a topped iteration of length $\varepsilon$.
Let $\alpha_1\le\alpha_2\le\beta\le\varepsilon$. Let
$p_1$ be a $P_{\alpha_1}$-name for a condition in $P_\varepsilon$, and let
$D$ be an open dense set of $P_\beta$. Then there is a $P_{\alpha_2}$-name
$p_2$ for a condition in $D$ such that the empty condition of
$P_{\alpha_2}$ forces: $p_2\leq p_1\mathord\restriction\beta$ and: if $p_1$ is in
$P_\varepsilon/H_{\alpha_2}$, then the condition $p_2$ is as well.}
(Proof: Work in the $P_{\alpha_2}$-extension. We know that $p'\coloneqq
p_1\restriction \beta$ is a $P_\beta$-condition. We now define $p_2$ as
follows: If $p'\notin P_\beta/H_{\alpha_2}$ (which is equivalent to $p_1\notin
P_\epsilon/H_{\alpha_2}$), then we choose any $p_2\leq p'$ in $D$ (which is
dense in $P_\beta$). Otherwise (using that $D\cap P_\beta/H_{\alpha_2}$ is
dense in $P_\beta/H_{\alpha_2}$) we can choose $p_2\leq p'$ in $D\cap
P_\beta/H_{\alpha_2}$.)
The following easy fact will also be useful:
\proofclaim{eq:forces.forces}{Let $P$ be a subforcing of $Q$. We define
$P\mathord\restriction p\coloneqq \{ r\in P: r\le p\}$.
Assume that $p\in P$ and $P\mathord\restriction p = Q\mathord\restriction p$.
\\Then for any $P$-name $\n x$ and any formula $\varphi(x)$
we have:
$p\Vdash_{P} \varphi(\n x) $ iff $p\Vdash_{Q} \varphi(\n x)$.
}
We now prove by induction on $\beta\le\varepsilon$ the following statement
(which includes that the Definition and
Claim~\ref{def:almost_CS_iteration_wolfgang} works up to $\beta$). Let
$ \delta, \delta'$ be as in \eqref{eq:delta.prime}.
\begin{Lem}\label{lem:inductionA}
\begin{enumerate}[(a)]
\item
The topped iteration $\bar P$ of length $\beta$ is a partial CS iteration.
\item
The canonical embedding $i_\delta: P^M_\delta\to P_{\delta'}$ works, hence
also $i_\delta: P^M_\delta\to P_{\delta}$ works.
\item
Moreover, assume that
\begin{itemize}
\item $\alpha\in M\cap \delta $,
\item $\n p\in M$ is a $P^M_\alpha$-name
of a $P^M_\delta$-condition,
\item
$q\in P_\alpha$ forces (in $P_\alpha$)
that $\n p\mathord\restriction\alpha[H^M_\alpha]$ is in $H^M_\alpha$.
\end{itemize}
Then there is a $q^+\in P_{\delta'}$
(and therefore in $P_\beta$) extending $q$ and forcing that
$\n p[H^M_\alpha]$ is in $H^M_\delta$.
\end{enumerate}
\end{Lem}
\begin{proof}
First let us deal with the trivial cases.
It is clear that we always get a partial CS iteration.
\begin{itemize}
\item
Assume that $\beta=\beta_0+1\in M$, i.e., $\delta=\delta'=\beta$.
It is clear that $i_\beta$ works.
To get $q^+$, first extend $q$ to some $q'\in P_{\beta_0}$ (by induction
hypothesis), then define $q^+$ extending $q'$ by $q^+(\beta_0)\coloneqq \n
p(\beta_0)$.
\item
If $\beta=\beta_0+1\notin M$, there is nothing to do.
\item
Assume that $\cf(\beta)>\omega$ (whether $\beta\in M$ or not).
Then $\delta'<\beta$.
So $i_\delta: P^M_\delta\to P_{\delta'}$ works by induction, and
similarly (c) follows from the inductive assumption. (Use
the inductive assumption for $\beta=\delta'$; the $\delta$ that we got at that
stage is the same as the current~$\delta$, and the $q^+$ we obtained at
that stage will still satisfy all requirements at the current stage.)
\item
Assume that $\cf(\beta)=\omega$ and that $M\cap\beta$ is bounded in~$\beta$.
Then the proof is the same as in the previous case.
\end{itemize}
We are left with the cases corresponding to (4) and (5) of
Definition~\ref{def:almost_CS_iteration_wolfgang}: $\cf(\beta)=\omega$ and
$M\cap\beta$ is cofinal in~$\beta$. So either $\beta\in M$, then
$\delta'=\beta=\delta$, or $\beta\notin M$, then $\delta'=\beta<\delta$ and
$\cf(\delta)>\omega$.
We leave it to the reader to check that $P_\beta$ is indeed a partial CS limit.
The main point is to see that for all $p,q\in P_\beta$ the
condition $q\wedge p$ is in $P_\beta$ as well, provided $q\in P_\alpha$ and $q\le p\mathord\restriction \alpha$
for some $\alpha<\beta$.
If $p\in P^\textup{\textrm{gen}}_\beta$, then this follows because $P^\textup{\textrm{gen}}_\beta$ is open in $P^{\textrm{\textup{CS}}}_\beta$;
the other cases are immediate from the definition (by induction).
We now turn to claim (c). Assume $q\in P_\alpha$ and $\n p\in M$ are
given, $\alpha\in M\cap \delta$.
Let $(D_n)_{n \in \omega}$ enumerate all dense sets of~$P^M_\delta$
which lie in~$M$, and let $(\alpha_n)_{n \in \omega} $ be a sequence
of ordinals in $M$ which is cofinal in $\beta$, where $\alpha_0=\alpha$.
Using \eqref{eq:prelim} in~$M$, we can find a sequence $(\n p_n)_{n
\in \omega}$ satisfying the following in~$M$, for all $n>0$:
\begin{itemize}
\item
$\n p_0 = \n p$.
\item
$\n p_n\in M$ is a $P^M_{\alpha_n}$-name of a
$P^M_\delta$-condition in~$D_n$.
\item
$\Vdash_{P^M_{\alpha_{n}}} \n p_{n}\le_{P_\delta^M} \n p_{n-1}$.
\item
$\Vdash_{P^M_{\alpha_{n}}} $ If $
\n p_{n-1}\mathord\restriction\alpha_{n}\in H^M_{\alpha_{n}}$, then
$\n p_{n}\mathord\restriction\alpha_{n}\in H^M_{\alpha_{n}}$ as well.
\end{itemize}
Using the inductive assumption for the $\alpha_n$'s, we can now find
a sequence $(q_n)_{n \in \omega}$ of conditions satisfying the following:
\begin{itemize}
\item $q_0 = q$, $q_n\in P_{\alpha_n}$.
\item $q_{n}\mathord\restriction \alpha_{n-1} = q_{n-1}$.
\item $q_n \Vdash_{P_{\alpha_n}} \n p_{n-1}\mathord\restriction \alpha_n \in H^M_{\alpha_n}$,
so also $ \n p_{n }\mathord\restriction \alpha_n \in H^M_{\alpha_n}$.
\end{itemize}
Let $q^+\in P^{\textrm{\textup{CS}}}_\beta$ be the union of the $q_n$. Then for all $n$:
\begin{enumerate}
\item
$q_n \Vdash_{ P^{\textrm{\textup{CS}}}_\beta} \n p_n\mathord\restriction \alpha_n \in H^M_{\alpha_n}$, so also $q^+$ forces this.
\\(Using induction on $n$.)
\item For all $n$ and all $m\ge n$:
$q^+ \Vdash_{ P^{\textrm{\textup{CS}}}_\beta} \n p_m\mathord\restriction \alpha_m \in H^M_{\alpha_m}$,
so also $ \n p_n\mathord\restriction \alpha_m \in H^M_{\alpha_m}$.
\\(As $\n p_m \le \n p_n$.)
\item
$q^+ \Vdash_{ P^{\textrm{\textup{CS}}}_\beta} \n p_n\in H^M_{\delta}$.
\\ (Recall that $P^{\textrm{\textup{CS}}}_\beta$ is separative, see Fact~\ref{fact:eq.eqstar}.
So $i_\delta(\n p_n)\in H_\delta$ iff $i_{\alpha_n}(\n p\mathord\restriction \alpha_m)\in
H_{\alpha_m}$ for all large~$m$.)
\end{enumerate}
As $q^+\Vdash_{P^{\textrm{\textup{CS}}}_\beta } \n p_n \in D_n\cap H^M_\delta$, we conclude
that $q^+\in P^\textup{\textrm{gen}}_\beta$ (using Lemma~\ref{lem:wolfgang},
applied to ${P^{\textrm{\textup{CS}}}_\beta }$). In particular,
$P^\textup{\textrm{gen}}_\beta$ is dense in $P_\beta$: Let $q\wedge i_\delta(p)$
be an element of the almost FS limit; so $q\in P_\alpha$ for some
$\alpha < \beta$. Now find a generic $q^+$ extending $q$ and stronger than $i_\delta(p)$,
then $q^+\le q\wedge i_\delta(p)$.
It remains to show that $i_\delta$ is $M$-complete.
Let $A\in M$ be a maximal antichain of $P^M_\delta$, and $p\in P_\beta$.
Assume towards a contradiction that $p$ forces in $P_\beta$
that $ i^{-1}_{\delta}[H_\beta ]$ does not intersect $A$ in exactly one point.
Since $P^\textup{\textrm{gen}}_\beta$ is dense in $P_\beta$, we can find some $q\leq p$ in $P^\textup{\textrm{gen}}_\beta$.
Let \[P'\coloneqq \{r\in P^{\textrm{\textup{CS}}}_\beta: r\le
q\}=\{r\in P_\beta: r\le q\}, \]
where the equality holds because $P^\textup{\textrm{gen}}_\beta$ is open in $P^{\textrm{\textup{CS}}}_\beta$.
Let $\Gamma $ be the canonical name for a $P'$-generic filter,
i.e.:
$\Gamma\coloneqq \{(\check r, r): r\in P' \}$.
Let $R$ be either
$P^{\textrm{\textup{CS}}}_\beta$ or $ P_\beta$. We write
$\langle \Gamma\rangle _R$ for the filter generated
by~$\Gamma$ in~$R$, i.e., $\langle \Gamma\rangle _R \coloneqq \{r\in R: (\exists r'\in \Gamma ) \
r'\le r\}$. So
\begin{equation}\label{gehtsnochduemmer}
q\Vdash_R H_R =\langle \Gamma\rangle_R.
\end{equation}
We now see that the following hold:
\begin{enumerate}
\item[--] $ q\Vdash_{P_\beta} i^{-1}_{\delta}[ H _{P_\beta} ] $
does not intersect $A$ in exactly one point. (By
assumption.)
\item[--] $ q\Vdash_{P_\beta} i^{-1}_{\delta}[ \langle
\Gamma \rangle_{P_\beta}] $
does not intersect $A$ in exactly one point. (By
\eqref{gehtsnochduemmer}.)
\item[--] $ q\Vdash_{P_\beta^{\textrm{\textup{CS}}} } i^{-1}_{\delta}[ \langle \Gamma \rangle_{P_\beta}] $
does not intersect $A$ in exactly one point. (By \eqref{eq:forces.forces}.)
\item[--] $ q\Vdash_{P_\beta^{\textrm{\textup{CS}}} } i^{-1}_{\delta}[ \langle \Gamma
\rangle_{P_\beta^{\textrm{\textup{CS}}}}] $ does not intersect $A$ in exactly one point.
(Because $i_\delta$ maps $A$ into $P_\beta\subseteq P_\beta^{\textrm{\textup{CS}}}$, so
$A\cap i^{-1}_\delta[\langle Y\rangle_{P_\beta}] = A \cap i^{-1}_\delta[\langle
Y\rangle_{P_\beta^{\textrm{\textup{CS}}}}]$ for all~$Y$.)
\item[--] $ q\Vdash_{P_\beta^{\textrm{\textup{CS}}} } i^{-1}_{\delta}[ H_{P_\beta^{\textrm{\textup{CS}}}}]$
does not intersect $A$ in exactly one point. (Again by \eqref{gehtsnochduemmer}.)
\end{enumerate}
But this, according to
the definition of $P^\textup{\textrm{gen}}_\beta$, implies $q\notin
P^\textup{\textrm{gen}}_\beta$, a contradiction.
\end{proof}
We can also show that the almost CS iteration
of proper forcings $Q_\alpha$ is proper.
(We do not really need this fact, as we could allow non-proper iterations in
our preparatory forcing, see Section~\ref{sec:7a}(\ref{item:nonproper}). In some sense, $M$-completeness
replaces properness, so the proof of $M$-completeness was similar to the
``usual'' proof of properness.)
\begin{Lem}
Assume that in Definition~\ref{def:almost_CS_iteration_wolfgang},
every $Q_\alpha$ is (forced to be) proper. Then also each $P_\delta$
is proper.
\end{Lem}
\begin{proof}
By induction on $\delta\le \varepsilon$ we prove that for all $\alpha<\delta$
the quotient $P_\delta/H_ \alpha$ is (forced to be) proper. We use the following
facts about properness:
\proofclaim{claim:proper.successor}{
If $P$ is proper and $P$ forces that $Q$ is proper, then $P*Q$
is proper.}
\proofclaim{claim:proper.omega}{
If $\bar P$ is an iteration of length $\omega$
and if each $Q_n$ is forced to be proper, then
the inverse limit $P_\omega$ is proper, as are all quotients
$P_\omega/H_n$.}
\proofclaim{claim:proper.omega.1}{
If $\bar P$ is an iteration of length $\delta$ with
$\cf(\delta)>\omega$,
and if all quotients $P_\beta/H_\alpha$ (for $\alpha < \beta < \delta$)
are forced to be proper, then
the direct limit $P_\delta$ is proper, as are all quotients
$P_\delta/H_\alpha$.}
If $\delta$ is a successor, then our inductive claim easily follows from
the inductive assumption together with~\eqref{claim:proper.successor}.
Let $\delta$ be a limit of countable cofinality, say $\delta = \sup_n
\delta_n$. Define an iteration $\bar P'$ of length $\omega$
with $Q'_n\coloneqq P_{\delta_{n+1}} / H_{\delta_n}$. (Each $Q'_n$ is proper,
by inductive assumption.) There is a natural forcing
equivalence between $ P^{\textrm{\textup{CS}}}_\delta$ and $P^{\prime{\textrm{\textup{CS}}}}_\omega$, the full CS limit of $\bar
P'$.
Let $N \prec H(\chi^*)$ contain $\bar P, P_\delta, \bar P', M, \bar P^M$.
Let $p\in P_\delta\cap N$. Without loss of generality $p\in P_\delta^\textup{\textrm{gen}}$.
So below $p$ we can identify $P_\delta$ with $P^{\textrm{\textup{CS}}}_\delta$ and hence
with $P^{\prime{\textrm{\textup{CS}}}}_ \omega$; now apply~\eqref{claim:proper.omega}.
The case of uncountable cofinality is similar, using~\eqref{claim:proper.omega.1} instead.
\end{proof}
Recall the definition of $\sqsubset_n$ and $\sqsubset$ from Definition~\ref{def:sqsubset}, the notion of
(quick) interpretation $Z^*$ (of a name $\n Z$ of a code for a null set)
and the definition of local preservation
of randoms from Definition~\ref{def:locally.random}.
Recall that we have seen in Corollaries~\ref{cor:ultralaverlocalpreserving} and~\ref{cor:januslocallypreserves}:
\begin{Lem}\label{lem:4.28}
\begin{itemize}
\item If $Q^M$ is an ultralaver forcing in $M$ and $r$ a real,
then there is an ultralaver forcing $Q$ over $Q^M$ locally
preserving randomness of $r$ over~$M$.
\item If $Q^M$ is a Janus forcing in $M$ and $r$ a real, then there is a
Janus forcing $Q$ over~$Q^M$ locally preserving randomness
of~$r$ over~$M$.
\end{itemize}
\end{Lem}
We will prove the following preservation theorem:
\begin{Lem}\label{lem:iterate.random}
Let $\bar P$ be an almost CS iteration (of length $\varepsilon$) over~$\bar P^M$,
$r$ random over~$M$, and $p\in P^M_\varepsilon$. Assume
that each $P_\alpha$ forces
that $Q_\alpha$ locally preserves randomness of $r$ over
$M[H^M_\alpha]$.
Then there is some $q\leq p$ in $P_\varepsilon$ forcing that
$r$ is random over~$M[H^M_\varepsilon]$.
\end{Lem}
What we will actually need is the following variant:
\begin{Lem}\label{lem:preservation.variant}
Assume that $\bar P^M$ is in $M$ a topped partial CS iteration of length
$\varepsilon$, and we already have some topped partial CS iteration $\bar P$
over~$\bar P^M\mathord\restriction\alpha_0$ of length $\alpha_0\in M\cap\varepsilon$. Let $\n
r$ be a $ P_{\alpha_0}$-name of a random real over~$M[H^M_{\alpha_0}]$.
Assume that we extend $\bar P$ to length $\varepsilon$ as an almost CS
iteration\footnote{Of course our official definition of almost CS iteration assumes
that we start the construction at $0$, so we modify this definition in the
obvious way.} using forcings $Q_\alpha$ which
locally preserve the randomness of $\n r$ over~$M[H^M_\alpha]$,
witnessed by a sequence $(D_k^{Q_\alpha^M})_{k\in \omega}$.
Let $p\in P^M_\varepsilon$.
Then we can find a $q\leq p$ in $P_{\varepsilon}$
forcing that $\n r$ is random over
$M[H^M_\varepsilon]$.
\end{Lem}
Actually, we will only prove the two previous lemmas under the following
additional assumption (which is enough for our application, and saves some
unpleasant work). This additional assumption is not really necessary; without
it, we could use the method of~\cite{MR2214624} for the proof.
\begin{Asm}\label{asm:quick}
\begin{itemize}
\item For each $\alpha\in M\cap \varepsilon$,
($P^M_\alpha$ forces that) $Q^M_\alpha$ is either trivial\footnote{More specifically, $Q^M_\alpha=\{\emptyset\}$.}
or adds a new $\omega$-sequence of ordinals.
Note that in the latter case we can assume without loss of generality
that $\bigcap_{n\in\omega}D^{Q_\alpha^M}_n=\emptyset$ (and, of course, that
the $D^{Q_\alpha^M}_n$ are decreasing).
\item Moreover, we assume that already in $M$ there is a set $T\subseteq
\varepsilon$ such that $P_\alpha^M$ forces: $Q_\alpha^M$ is trivial iff
$\alpha\in T$. (So whether $Q_\alpha^M$ is trivial or not does not depend
on the generic filter below $\alpha$, it is already decided in the
ground model.)
\end{itemize}
\end{Asm}
The result will follow as a special case of the following lemma, which we prove
by induction on~$\beta$. (Note that this is a refined version of the proof of
Lemma~\ref{lem:inductionA} and similar to the proof of the preservation theorem
in~\cite[5.13]{MR1234283}.)
\begin{Def}\label{def:quick}
Under the assumptions of
Lemma~\ref{lem:preservation.variant} and
Assumption~\ref{asm:quick},
let $\n Z$ be a $P_\delta$-name, $\alpha_0\le \alpha <
\delta$, and let $\bar p = (p^k)_{k\in \omega}$ be a
sequence of $P_\alpha$-names of conditions in
$P_\delta/H_\alpha$. Let $Z^*$ be a $P_\alpha$-name.
We say that $(\bar p, Z^ *)$ is a \emph{quick} interpretation
of $\n Z$ if $\bar p$ interprets $\n Z$ as $Z^*$ (i.e.,
$P_\alpha$ forces that $p^k$ forces $\n Z \mathord\restriction k = Z^*\mathord\restriction
k$ for all $k$), and moreover:
\begin{quote}
Letting $\beta\ge \alpha$ be minimal with $Q^M_\beta$
nontrivial (if such $\beta$ exists):
$P_\beta$ forces that the sequence $(p^k(\beta))_{k\in \omega}$ is
quick in $Q^M_\beta$, i.e., $p^k(\beta)\in D^{Q_\beta^M}_k$ for all~$k$.
\end{quote}
\end{Def}
It is easy to see that:
\proofclaim{eq:find.quick}{For every name $\n Z$ there is a
quick interpretation $(\bar p, Z^*)$.}
\begin{Lem} \label{lem:induktion.wirklich}
Under the same assumptions as above,
let $\beta$, $\delta$, $\delta'$ be as in \eqref{eq:delta.prime}
(so in particular we have $\delta'\le\beta\le\delta\le\varepsilon$).
\\
{\bf Assume that }
\begin{itemize}
\item $\alpha\in M\cap \delta$ ($=M\cap \beta $) and $\alpha\ge \alpha_0$ (so $\alpha<\delta'$),
\item $ p\in M$ is a $P^M_\alpha$-name of a
$P^M_\delta$-condition,
\item $\n Z\in M$ is a $P^M_\delta$-name of a code for null set,
\item $Z^*\in M$ is a $P^M_\alpha$-name of a code for a null set,
\item $P^M_\alpha$ forces:
$\bar p = (p^k)_{k\in \omega}\in M$ is a quick
sequence in $P^M_\delta/H^M_\alpha$ interpreting $\n
Z $ as~$Z^*$ (as in Definition~\ref{def:quick}),
\item $P^M_\alpha$ forces:
if $p\mathord\restriction
\alpha \in H^M_\alpha$, then $p^0\le p$,
\item $q\in P_\alpha$ forces $p\mathord\restriction \alpha \in H^M_\alpha$,
\item
$q$ forces that $r$ is random over~$M[H^M_\alpha]$, so in particular
there is (in $V$) a $P_\alpha$-name $ \n c_0$ below $q$ for the minimal~$c$ with $Z^*\sqsubset_{c} r$.
\end{itemize}
{\bf Then} there is a condition $q^+\in P_{\delta'}$, extending
$q$, and forcing the following:
\begin{itemize}
\item
$p\in H^M_\delta$,
\item
$r$ is random over~$M[H^M_\delta]$,
\item
$\n Z\sqsubset_{\n c_0} r$.
\end{itemize}
\end{Lem}
We actually claim a slightly stronger version, where instead of
$Z^*$ and $\n Z$ we have finitely many codes for null sets and names of
codes for null sets, respectively. We will use this stronger claim as
inductive assumption, but for notational simplicity we only prove the weaker
version; it is easy to see that the weaker version implies the stronger
version.
\begin{proof}
\emph{\textbf{The nontrivial successor case:}} ${\beta=\gamma+1\in M}$.
If $Q^M_\gamma$ is trivial, there is nothing to do.
Now let $\gamma_0\ge \alpha $ be minimal with $Q^M_{\gamma_0}$ nontrivial. We
will distinguish two cases: $\gamma=\gamma_0$ and $\gamma>\gamma_0$.
Consider first the case that $\gamma=\gamma_0$.
Work in $V[H_\gamma]$ where $q\in H_\gamma$. Note that $M [H^M_\gamma] =
M[H^M_\alpha]$. So $r$ is random over~$ M [H^M_\gamma]$, and
$(p^k(\gamma))_{k\in \omega}$ quickly interprets $\n Z$ as $Z^*$ in $Q^M
_\gamma$.
Now let $q^+\mathord\restriction \gamma= q$, and use the fact that
$Q_\gamma$ locally preserves randomness to find $q^+(\gamma)\le p^0(\gamma)$.
Next consider the case that $Q^M_\gamma$ is nontrivial and $\gamma\ge \gamma_0+1$.
Again work in $V[H_\gamma]$.
Let $k^*$ be maximal with $p^{k^*}\mathord\restriction \gamma\in H^M_\gamma$. (This $k^*$
exists, since the sequence $(p^{k})_{k\in \omega}$ was quick, so there is even
a $k$ with $p^{k}\mathord\restriction ( {\gamma_0+1}) \notin H^M_{\gamma_0+1}$.)
Consider $\n Z$ as a $Q^M_\gamma$-name, and (using~\eqref{eq:find.quick})
find a quick interpretation $Z'$ of $\n Z$ witnessed by a
sequence starting with $p^{k^*}(\gamma)$. In $M[H^M_\alpha]$, $Z'$
is now a $P^M_\gamma/H^M_\alpha$-name. Clearly, the sequence
$(p^k\mathord\restriction \gamma)_{k\in \omega}$ is a quick sequence interpreting $Z'$ as
$Z^*$. (Use the fact that $p^k\mathord\restriction \gamma$ forces $k^*\ge k$.)
\\
Using the induction hypothesis, we can first extend $q$ to a condition $q'\in
P_\gamma$ and then (again by our assumption that $Q_\gamma$ locally preserves
randomness) to a condition $q ^+\in P_{\gamma+1}$.
\emph{\textbf{The nontrivial limit case:}}
${M\cap \beta}$ unbounded in ${\beta}$, i.e., $\delta'=\beta$.
(This deals with cases~(4) and~(5) in
Definition~\ref{def:almost_CS_iteration_wolfgang}.
In case (4) we have $\beta\in M$, i.e., $\beta=\delta$;
in case (5) we have $\beta\notin M$ and $\beta< \delta$.)
Let $\alpha=\delta_0 < \delta_1 < \cdots$ be a sequence of $M$-ordinals
cofinal in $M\cap \delta' = M\cap \delta$. We may assume\footnote{If from some $\gamma$ on
all $Q^M_\zeta$ are trivial, then $P^M_\delta=P^M_\gamma$, so by
induction there is nothing to do. If $Q^M_\alpha$ itself is
trivial, then we let $\delta_0\coloneqq \min\{\zeta: Q^M_\zeta \text{ nontrivial}\}$ instead.}
that each $Q^M_{\delta_n}$ is nontrivial.
Let $(\n Z_n)_{n\in\omega}$ be a list of all $P^ M_\delta$-names in $M$
of codes for null sets (starting with our given null set
$\n Z = \n Z_0$).
Let $(E_n)_{n\in\omega}$ enumerate all open dense sets of $P^M_\delta$ from $M$, without
loss of generality\footnote{well, if we just enumerate a basis of the open
sets instead of all of them\dots}
we can assume that:
\proofclaim{claim:E.n.decides}{
$E_n$ decides $\n Z_0\mathord\restriction n $, \dots, $\n Z_n\mathord\restriction n$.
}
We write $p^k_0$ for $p^k$, and $Z_{0,0}$ for $Z^*$; as mentioned
above, $\n Z=\n Z_0$.
By induction on $n$ we can now find a sequence $\bar p_n = (p^k_n)_{k\in
\omega}$ and $P^M_{\delta_n }$-names $Z_{i,n}$ for $i\in \{0,\dots, n\}$
satisfying the following:
\begin{enumerate}
\item $P^M_{\delta_n}$ forces that
$p^0_n \le p^{k}_{n-1}$ whenever
$p^k_{n-1}\in P^M_{\delta}/H^M_{\delta_n}$.
\item $P_{\delta_n}^M$ forces that $p^0_n\in E_n$. (Clearly $E_n\cap
P^M_\delta/H^M_{\delta_n}$ is a dense set.)
\item $\bar p_n\in M $ is a $P^M_{\delta_n}$-name for a quick sequence
interpreting $(\n Z_0,\ldots, \n Z_n)$ as
$(Z_{0,n},\ldots, Z_{n,n})$
(in $P^M_\delta/H^M_{\delta_n}$),
so $Z_{i,n}$ is a $P^M_{\delta_n}$-name of a code for a null set, for
$0\le i \le n$.
\end{enumerate}
Note that this implies that the sequence $(p^k_{n-1}\mathord\restriction \delta_{n})$ is
(forced to be) a quick sequence interpreting
$(Z_{0,{n}},\ldots, Z_{n-1, {n}})$ as
$(Z_{0,{n-1}},\ldots, Z_{n-1, {n-1}})$.
Using the induction hypothesis, we now
define a sequence $(q_n)_{n\in \omega}$ of conditions $q_n\in P_{\delta_n}$
and a sequence $(c_n)_{n\in\omega}$ (where $c_n$ is a $P_{\delta_n}$-name)
such that (for $n>0$) $q_n $ extends $q_{n-1}$ and forces the following:
\begin{itemize}
\item $ p_{n-1}^0\mathord\restriction \delta_n \in H^M_{\delta_n}$.
\item Therefore, $p_n^0 \le p_{n-1}^0$.
\item $r$ is random over~$M[H^M_{\delta_n}]$.
\item Let $c_n$ be the least $c$ such that $Z_{n,n}\sqsubset_c r$.
\item
$ Z_{i,n} \sqsubset_{c_i} r$ for $i=0,\ldots, n-1$.
\end{itemize}
Now let $q = \bigcup_n q_n\in P^{\textrm{\textup{CS}}}_{\delta'}$.
As in Lemma~\ref{lem:inductionA} it is easy to see that $q\in P^\textup{\textrm{gen}}_{\delta'}
\subseteq P_{\delta'}$.
Moreover, by~\eqref{claim:E.n.decides} we get that
$q$ forces that $\n Z_i = \lim_n Z_{i,n}$.
Since each set $C_{c,r}\coloneqq \{x:x\sqsubset_{c} r\}$ is closed, this implies that
$q $ forces
$\n Z_i \sqsubset_{c_i} r$, in particular $ \n Z= \n Z _0 \sqsubset_{c_0} r$.
\emph{\textbf{The trivial cases:}} In all other cases,
$M \cap \beta $ is bounded in $\beta$, so we already dealt with
everything at stage $ \beta_0\coloneqq \sup(\beta \cap M)$.
Note that $\delta_0'$ and $\delta_0$ used at stage $\beta_0$
are the same as the current $\delta'$ and $\delta$.
\end{proof}
\section{The forcing construction}\label{sec:construction}
In this section we describe a $\sigma$-closed ``preparatory'' forcing notion
$\mathbb{R}$; the generic filter will define a ``generic'' forcing iteration
$\bar \mathbf{P}$, so elements of $\mathbb{R}$ will be approximations to such an
iteration. In Section~\ref{sec:proof} we will show that the forcing
$\mathbb{R}*\mathbf{P}_\om2$ forces BC and dBC.
{} From now on, we assume CH in the ground model.
\subsection{Alternating iterations, canonical embeddings and the preparatory forcing $\mathbb{R}$}
The preparatory forcing $\mathbb{R}$ will consist of pairs $(M,\bar P)$, where $M$
is a countable model and $\bar P\in M$ is an iteration of ultralaver and
Janus forcings.
\begin{Def}\label{def:alternating}
An alternating iteration\footnote{See Section~\ref{sec:alternativedefs} for possible
variants of this definition.} is a topped partial CS iteration $\bar P$
of length $\om2$ satisfying the following:
\begin{itemize}
\item Each $P_\alpha$ is proper.\footnote{This does not seem to be
necessary, see Section~\ref{sec:alternativedefs}, but it is easy to ensure and
might be comforting to some of the readers and/or authors.}
\item For $\alpha$ even, either both $Q_{\alpha}$ and
$Q_{\alpha+1}$ are (forced by the empty condition to be) trivial,\footnote{
For definiteness, let us agree that the trivial forcing is the
singleton $\{\emptyset\}$.}
or $P_\alpha$ forces that $Q_\alpha$ is an ultralaver forcing adding the
generic real $\bar \ell_\alpha$, and $P_{\alpha+1}$ forces that
$Q_{\alpha+1}$ is a Janus forcing based on $\bar \ell^*_\alpha$ (where $ \bar
\ell^*$ is defined from $ \bar \ell $ as in
Lemma~\ref{lem:subsequence}).
\end{itemize}
\end{Def}
We will call an even index an ``ultralaver position'' and an odd one a ``Janus
position''.
As in any partial CS iteration, each $P_\delta$ for $\cf(\delta)>\omega$
(and in particular $P_\om2$) is a direct limit.
Recall that in Definition~\ref{def:canonicalembedding} we have defined the notion
``$\bar P^M$ canonically embeds into $\bar P$'' for nice candidates $M$ and
iterations $\bar P\in V$ and $\bar P^M\in M$.
Since our iterations now have length $\omega_2$, this
means that the canonical embedding works up to and
including\footnote{This is stronger than to require that the canonical
embedding works for every $\alpha\in\om2\cap M$, even though both $P_\om2$ and
$P^M_{\om2}$ are just direct limits; see footnote~\ref{fn:too.late}.}
$\om2$.
In the following, we will use pairs $x=(M^x,\bar P^x)$ as conditions in a
forcing, where $\bar P^x $ is an alternating iteration in the nice candidate $M^x$.
We will adapt our notation accordingly: Instead of writing $M$,
$\bar P^M$, $P^M_\alpha$
$H_\alpha^M$ (the induced filter), $Q_\alpha^M$, etc.,
we will write $M^x$, $\bar P^x$, $P^x_\alpha$, $H_\alpha^x$, $Q^x_\alpha$, etc.
Instead of ``$\bar P^x$ canonically embeds into $\bar P$'' we
will say
\footnote{Note the linguistic asymmetry here: A symmetric and more verbose variant
would say ``$x=(M^x,\bar P^x)$ canonically embeds into $(V,\bar P)$''.}
``$x$ canonically embeds into $\bar P$''
or ``$(M^x, \bar P^x)$ canonically embeds into $\bar P$''
(which is a more exact
notation anyway, since the test whether the embedding is $M^x$-complete uses
both $M^x$ and $\bar P^x$,
not just $\bar P^x$).
The following rephrases Definition~\ref{def:canonicalembedding} of
a canonical embedding in our new notation, taking into account that:
\begin{quote}
$\mathbb{L}_{{\bar D}^x}$ is an $M^x$-complete subforcing
of~$\mathbb{L}_{\bar D}$ \ \ iff \ \ $\bar D$
extends $\bar D^x$
\end{quote} (see Lemma~\ref{lem:LDMcomplete}).
\begin{Fact}\label{fact:canonical}
$x=(M^x,\bar P^x)$ canonically embeds into $\bar P$,
if (inductively) for all $\beta\in \om2\cap
M^x\cup\{\om2\}$ the following holds:
\begin{itemize}
\item Let $\beta=\alpha+1$ for $\alpha$ even (i.e., an ultralaver position).
Then either $Q^x_\alpha$ is trivial (and $Q_\alpha$ can be trivial or not),
or we require that
($P_\alpha$ forces that) the $V[H_\alpha]$-ultrafilter system $\bar D$ used for $Q_\alpha$
extends the $M^x[H^x_\alpha]$-ultrafilter system $\bar D^x$
used for $Q^x_\alpha$.
\item Let $\beta=\alpha+1$ for $\alpha$ odd (i.e., a Janus position).
Then either $Q^x_\alpha$ is trivial,
or we require that ($P_\alpha$ forces that)
the Janus forcing $Q^x_\alpha$ is an $M^x[H^x_\alpha]$-complete
subforcing of the Janus forcing $Q_\alpha$.
\item Let $\beta$ be a limit. Then the canonical extension
$i_\beta:P^x_\beta\to P_\beta$ is $M^x$-complete. (The canonical extension
was defined in Definition~\ref{def:canonicalextension}.)
\end{itemize}
\end{Fact}
Fix a sufficiently large regular cardinal $\chi^*$ (see Remark~\ref{rem:fine.print}).
\begin{Def}\label{def:prep}
The \qemph{preparatory forcing} $\mathbb{R}$ consists of
pairs $x=(M^x,\bar P^x)$ such that
$M^x\in H(\chi^*)$ is a nice candidate (containing $\om2$), and
$\bar P^x$ is in $M^x$ an alternating iteration (in particular topped and
of length $\om2$).
\\
We define $y$ to be stronger than $x$ (in symbols: $y\leq_{\mathbb{R}} x$),
if the following holds: either $x=y$, or:
\begin{itemize}
\item $M^x\in M^y$ and $M^x$ is countable in $M^y$.
\item $M^y$ thinks that $(M^x,\bar P^x)$ canonically embeds into $\bar P^y$.
\end{itemize}
\end{Def}
Note that this order on $\mathbb{R}$ is transitive.
We will sometimes write $i_{x,y}$ for
the canonical embedding (in $M^y$) from $P^x_{\om2}$ to $P^y_{\om2}$.
There are several variants of this definition which result in equivalent
forcing notions. We will briefly come back to this in
Section~\ref{sec:alternativedefs}.
The following is trivial by elementarity:
\begin{Fact}\label{fact:esmV}
Assume that $\bar P$ is an alternating iteration (in $V$), that $x=(M^x,\bar P^x) \in
\mathbb{R}$ canonically embeds into $\bar P$, and that $N \prec H( \chi^*)$
contains $x$ and $\bar P$. Let $y=(M^y, \bar P^y)$ be the ord-collapse of
$(N, \bar P)$. Then $y\in\mathbb{R}$ and $y\le x$.
\end{Fact}
This fact will be used, for example, to get from the following Lemma~\ref{lem:trivialexample}
to Corollary~\ref{cor:gurke3}.
\begin{Lem}\label{lem:trivialexample}
Given $x\in\mathbb{R}$, there is an alternating iteration $\bar P$
such that $x$ canonically embeds into $\bar P$.
\end{Lem}
\begin{proof}
For the proof, we use either of the partial CS constructions
introduced in the previous chapter (i.e., an almost CS iteration or an almost
FS iteration over $\bar P^x$). The only
thing we have to check is that we can indeed choose $Q_\alpha$ that satisfy the
definition of an alternating iteration (i.e., as ultralaver or Janus forcings) and
such that $Q^x_\alpha$ is $M^x$-complete in $Q_\alpha$.
In the ultralaver case we arbitrarily extend $\bar D^x$ to an ultrafilter
system $\bar D$, which is justified by Lemma~\ref{lem:LDMcomplete}.
In the Janus case, we take $Q_\alpha\coloneqq Q_\alpha^x$ (this works by Fact~\ref{fact:janus.ctblunion}). Alternatively, we could
extend $Q_\alpha^x$ to a random forcing (using Lemma~\ref{lem:janusrandompreservation}).
\end{proof}
\begin{Cor}\label{cor:gurke3}
Given $x\in\mathbb{R}$ and an HCON object $b\in H(\chi^*)$ (e.g., a real or an ordinal), there is a
$y\leq x$ such that $b\in M^y$.
\end{Cor}
What we will actually need are the following three variants:
\begin{Lem}\label{lem:prep.is.sigma.preparation}
\begin{enumerate}
\item
Given $x\in\mathbb{R}$ there is a $\sigma$-centered
alternating iteration $\bar P$ above $x$.
\item\label{item:karotte2}
Given a decreasing sequence $\bar x=(x_n)_{n\in \omega}$ in
$\mathbb{R}$, there is an alternating iteration $\bar P$
such that each $x_n$ embeds into $\bar P$. Moreover,
we can assume that for all Janus positions $\beta$, the
Janus \footnote{If all $Q^{x_n}_\beta$ are trivial, then we
may also set $Q_\beta$ to be the trivial forcing, which is formally
not a Janus forcing.}
forcing
$Q_\beta$ is (forced to be) the union of the
$Q^{x_n}_\beta$, and that for all limits $\alpha$, the forcing
$P_\alpha$ is the almost FS limit
over~$(x_n)_{n\in\omega}$
(as in Corollary~\ref{cor:ctblmanycandidates}).
\item\label{item:karotte6}
Let $x,y\in \mathbb{R}$. Let $j^x$ be the transitive collapse of
$M^x$, and define $j^y$ analogously.
Assume that $j^x{}[M^x]=j^y {}[ M^y]$, that $j^x(\bar P^x)=j^y(\bar P^y)$ and
that there are $\alpha_0\leq\alpha_1<\om2$ such that:
\begin{itemize}
\item $M^x\cap \alpha_0=M^y\cap \alpha_0$ (and thus $j^x\mathord\restriction \alpha_0=j^y\mathord\restriction\alpha_0$).
\item $M^x\cap [\alpha_0, \omega_2) \subseteq [\alpha_0, \alpha_1)$.
\item $M^y\cap [\alpha_0, \omega_2) \subseteq [\alpha_1, \om2)$.
\end{itemize}
Then there is an alternating iteration $\bar P$
such that both $x$ and $y$ canonically embed into it.
\end{enumerate}
\end{Lem}
\begin{proof}
For (1), use an almost FS iteration.
We only use the coordinates in $M^x$, and use the (countable!)
Janus forcings $Q_\alpha\coloneqq Q^x_\alpha$ for all Janus positions
$\alpha\in M^x$ (see Fact~\ref{fact:janus.ctblunion}).
Ultralaver forcings are $\sigma$-centered anyway, so
$P_\varepsilon$ will be $\sigma$-centered, by Lemma~\ref{lem:4.17}.
For (2), use the almost FS iteration over the sequence $(x_n)_{n\in \omega}$
as in Corollary~\ref{cor:ctblmanycandidates},
and at Janus positions $\alpha$ set $Q_\alpha$ to be the
union of the $Q^{x_n}_\alpha$. (By Fact~\ref{fact:janus.ctblunion},
$Q^{x_n}_\alpha$ is $M^{x_n}$-complete in $Q_\alpha$, so
Corollary~\ref{cor:ctblmanycandidates} can be applied here.)
For (3), we again use an almost FS construction. This
time we start with an almost FS construction over $x$ up
to $\alpha_1$, and then continue with an almost FS
construction over $y$.
\end{proof}
As above, Fact~\ref{fact:esmV} gives us the following consequences:
\begin{Cor}\label{cor:bigcor}
\begin{enumerate}
\item\label{item:gurke0}
$\mathbb{R}$ is $\sigma$-closed. Hence $\mathbb{R}$ does not add new HCON objects (and in particular: no new reals).
\item\label{item:gurke1} $\mathbb{R}$ forces that the generic filter $G\subseteq \mathbb{R}$ is
$\sigma$-directed, i.e., for every countable subset $B$
of $G$ there is a $y\in G$ stronger than each element of $B$.
\item\label{item:gurke2}
$\mathbb{R}$ forces CH. (Since we assume CH in $V$.)
\item\label{item:martin}
Given a decreasing sequence $\bar x=(x_n)_{n\in \omega}$ in
$\mathbb{R}$ and any HCON object $b\in H(\chi^*)$, there is a
$y\in \mathbb{R}$ such that
\begin{itemize}
\item $y\leq x_n$ for all $n$,
\item $M^y$ contains $b$ and the sequence $\bar x$,
\item for all Janus positions $\beta$, $M^y $ thinks that the
Janus forcing $Q^y_\beta$ is (forced to be) the union of
the~$Q^{x_n}_\beta$,
\item for all limits $\alpha$, $M^y $ thinks that $P^y_\alpha$
is the almost FS limit\footnote{constructed in Lemma~\ref{lem:418}}
over $(x_n)_{n\in\omega}$
(of~$(P^y_\beta)_{\beta<\alpha}$).
\end{itemize}
\end{enumerate}
\end{Cor}
\begin{proof}
Item~(\ref{item:martin}) directly follows from
Lemma~\ref{lem:prep.is.sigma.preparation}(\ref{item:karotte2}) and
Fact~\ref{fact:esmV}.
Item~(\ref{item:gurke0}) is a special case
of~(\ref{item:martin}), and~(\ref{item:gurke1}) and~(\ref{item:gurke2}) are
trivial consequences of~(\ref{item:gurke0}).
\end{proof}
Another consequence of Lemma~\ref{lem:prep.is.sigma.preparation} is:
\begin{Lem}\label{lem:al2cc}
The forcing notion $\mathbb{R}$ is $\al2$-cc.
\end{Lem}
\begin{proof}
Recall that we assume that $V$ (and hence $V[G])$
satisfies CH.
Assume towards a contradiction that $(x_i:i< \omega_2)$ is an antichain.
Using CH we may without loss of generality assume that for each
$i\in\omega_2$ the transitive collapse of $(M^{x_i},\bar P^{x_i})$ is the
same. Set $L_i\coloneqq M^{x_i}\cap\om2$. Using the $\Delta$-lemma we
find some uncountable $I\subseteq \om2$ such
that the $L_i$ for $i\in I$ form a $\Delta$-system with root~$L$.
Set $\alpha_0=\sup(L)+ 3$.
Moreover, we may assume $\sup(L_i)<\min(L_j\setminus \alpha_0)$ for all
$i<j$.
Now take any $i,j\in I$, set $x\coloneqq x_i$ and $y\coloneqq x_j$, and use
Lemma~\ref{lem:prep.is.sigma.preparation}(\ref{item:karotte6}).
Finally, use Fact~\ref{fact:esmV} to find $z\le x_i, x_j$.
\end{proof}
\subsection{The generic forcing $\mathbf{P}'$}
Let $G$ be $\mathbb{R}$-generic. Obviously $G$ is a $\le_{\mathbb{R}}$-directed system.
Using the canonical embeddings, we can construct in $V[G]$ a direct
limit $\mathbf{P}'_{\om2} $ of the directed system~$G$:
Formally, we set
\[\mathbf{P}'_{\om2}\coloneqq \{(x,p):\, x\in G \text{ and } p\in P^x_{\om2}\},\]
and we
set $(y,q)\leq (x,p)$ if $y\leq_{\mathbb{R}} x$ and $q$ is (in $y$) stronger than
$i_{x,y}(p)$ (where $i_{x,y}:P^x_{\om2} \to P^y_{\om2} $ is the canonical embedding).
Similarly, we define for each $\alpha$
\[\mathbf{P}'_{\alpha}\coloneqq \{(x,p):\, x\in G,\, \alpha\in
M^x\text{ and } p\in P^x_\alpha\}\] with the same order.
To summarize:
\begin{Def}\label{def:BPstrich} For $\alpha\leq\om2$,
the direct limit of the $P^x_\alpha$ with $x\in G$ is called
$\mathbf{P}'_{\alpha}$.
\end{Def}
Formally, elements of $\mathbf{P}'_{\om2}$ are defined as pairs $(x,p)$. However, the $x$
does not really contribute any information.
In particular:
\begin{Fact}\label{facts:trivial66}
\begin{enumerate}
\item Assume that $(x,p^x)$ and $(y,p^y)$ are in $\mathbf{P}'_{\om2}$,
that $y\leq x$, and that the canonical embedding $i_{x,y}$ witnessing~$y\le x$
maps $p^x$ to $p^y$. Then $(x,p^x)=^*(y,p^y)$.
\item $(y,q)$ is in $\mathbf{P}'_{\om2}$ stronger than $(x,p)$ iff
for some (or equivalently: for any)
$z\leq x,y$ in $G$ the canonically embedded $q$
is in $P^z_{\om2}$ stronger than the canonically embedded $p$.
The same holds if ``stronger than'' is replaced by ``compatible
with'' or by ``incompatible with''.
\item\label{item:bla3} If $(x,p)\in\mathbf{P}'_\alpha$, and if $y$ is such that
$M^y=M^x$ and $\bar P^y\mathord\restriction\alpha=\bar P^x\mathord\restriction\alpha$,
then $(y,p)=^*(x,p)$.
\end{enumerate}
\end{Fact}
In the following, we will therefore often abuse notation and just write $p$ instead of
$(x,p)$ for an element of~$\mathbf{P}'_\alpha$.
We can define a natural restriction map from $\mathbf{P}'_{\om2}$ to $\mathbf{P}'_\alpha$, by
mapping $(x,p)$ to $(x,p\mathord\restriction \alpha)$. Note that by the fact above,
we can assume without loss of generality that $\alpha\in M^x$.
More exactly: There is a $y\leq x$ in~$G$ such that $\alpha\in M^y$ (according
to Corollary~\ref{cor:gurke3}). Then in $\mathbf{P}'_{\om2}$ we have $(x,p)=^*(y,p)$.
\begin{Fact} \label{fact:5.12}
The following is forced by $\mathbb{R}$:
\begin{itemize}
\item $\mathbf{P}'_\beta$ is completely embedded into $ \mathbf{P}'_\alpha$
for $\beta< \alpha\le \om2$
(witnessed by the natural restriction map).
\item If $x\in G$, then $P^x_\alpha$ is $M^x$-completely embedded
into $\mathbf{P}'_\alpha$ for $\alpha\leq\om2$
(by the identity map
$p\mapsto (x,p)$).
\item If $\cf(\alpha)>\omega$, then $\mathbf{P}'_\alpha$ is the union of the
$\mathbf{P}'_\beta$ for $\beta<\alpha$.
\item By definition, $\mathbf{P}'_{\om2}$ is a subset of $V$.
\end{itemize}
\end{Fact}
$G$ will always denote an $\mathbb{R}$-generic filter, while the $\mathbf{P}'_{\om2}$-generic
filter over $V[G]$ will be denoted by $H'_{\om2}$ (and the induced
$\mathbf{P}'_\alpha$-generic by $H'_\alpha$). Recall that for each $x\in G$, the map
$p\mapsto (x,p)$ is an $M^x$-complete embedding of $P^x_{\om2}$ into $\mathbf{P}'_{\om2}$ (and of
$P^x_\alpha$ into $\mathbf{P}'_\alpha$). This way $H'_\alpha\subseteq \mathbf{P}'_\alpha$ induces an $M^x$-generic
filter $H^x_\alpha \subseteq P^x_\alpha$.
So $x\in \mathbb{R}$ forces that $\mathbf{P}'_\alpha$ is approximated by
$P^x_\alpha$. In particular we get:
\begin{Lem}\label{lem:pathetic0}
Assume that $x\in \mathbb{R}$, that $\alpha\leq \om2$ in $M^x$, that $p\in P^x_\alpha$, that $\varphi(t)$ is a first order formula of the language $\{\in\}$ with
one free variable $t$
and that $\dot \tau$ is a $P^x_\alpha$-name in $M^x$.
Then $M^x\models p\Vdash_{P^x_\alpha} \varphi(\dot\tau)$ iff
$x\Vdash_\mathbb{R} (x,p)\Vdash_{\mathbf{P}'_\alpha} M^x[H^x_\alpha]\models \varphi(\dot\tau[H^x_\alpha])$.
\end{Lem}
\begin{proof}
``$\Rightarrow$'' is clear. So assume that $\varphi(\dot\tau)$ is not forced
in $M^x$. Then some $q\leq_{P^x_\alpha} p$ forces the negation.
Now $x$ forces that $(x,q)\leq (x,p)$ in $\mathbf{P}'_\alpha$; but
the conditions $(x,p)$ and $(x,q)$ force contradictory statements.
\end{proof}
\subsection{The inductive proof of ccc} \label{sec:ccc}
We will now prove by induction on~$\alpha$
that $\mathbf{P}'_\alpha$ is (forced to be) ccc and
(equivalent to) an alternating iteration. Once we know this, we can prove
Lemma~\ref{lem:elemsub}, which easily implies all the lemmas in this section. So
in particular these lemmas will only be needed to prove
ccc and not for anything else (and they will probably not
aid the understanding of the construction).
In this section, we try to stick to the following notation: $\mathbb{R}$-names are
denoted with a tilde underneath (e.g., $\n \tau$), while $P^x_\alpha$-names or
$\mathbf{P}'_\alpha$-names (for any $\alpha\le\om2$) are denoted with a dot accent
(e.g., $\dot\tau$). We use both accents when we deal with $\mathbb{R}$-names for
$\mathbf{P}'_\alpha$-names (e.g., $\nd\tau$).
We first prove a few lemmas that are easy generalizations of the following
straightforward observation:
Assume that $x \Vdash_\mathbb{R}(\n z,\n p)\in\mathbf{P}'_\alpha$. In particular,
$x\Vdash \n z\in G$. We first strengthen $x$ to some $x_1$ that decides $\n
z$ and $\n p$ to be $z^*$ and $p^*$. Then $x_1\leq^* z^*$ (the order $\leq^*$
is defined on page~\pageref{def:starorder}), so we can further strengthen
$x_1$ to some $y\leq z^*$. By definition, this means that $z^*$ is canonically
embedded into $\bar P^y$; so (by Fact~\ref{facts:trivial66})
the $P^{z^*}_\alpha$-condition $p^*$ can be interpreted
as a $P^y_\alpha$-condition as well. So we end up with some $y\leq x$ and a
$P^y_\alpha$-condition $p^*$ such that $y\Vdash_\mathbb{R} (\n z,\n p)=^*(y, p^*)$.
Since $\mathbb{R}$ is $\sigma$-closed, we can immediately generalize this to countably
many ($\mathbb{R}$-names for) $\mathbf{P}'_{\alpha}$-conditions:
\begin{Fact}\label{fact:pathetic1}
Assume that $x\Vdash_{\mathbb{R}} \n p_n\in \mathbf{P}'_\alpha$ for all $n\in\omega$.
Then there is a $y\leq x$ and there are $p_n^*\in P^y_\alpha$ such that
$y\Vdash_{\mathbb{R}} \n p_n=^*p_n^*$ for all $n\in\omega$.
\end{Fact}
Recall that more formally we should write: $x\Vdash_{\mathbb{R}} (\n z_n,\n p_n)\in
\mathbf{P}'_\alpha$; and $y\Vdash_{\mathbb{R}} (\n z_n,\n p_n)=^*(y,p_n^*)$.
We will need a
variant of the previous fact:
\begin{Lem}\label{lem:pathetic2}
Assume that $\mathbf{P}'_\beta$ is forced to be ccc, and assume that
$x$ forces (in ${\mathbb{R}}$) that $\nd r_n$ is a $\mathbf{P}'_\beta$-name
for a real (or an HCON object) for every $n\in\omega$.
Then there is a $y\leq x$ and there are $P^y_\beta$-names $\dot{r}^*_n$
in $M^y$ such that
$y\Vdash_{\mathbb{R}} ( \Vdash _{\mathbf{P}'_\beta} \nd r_n=\dot{r}^*_n)$ for all $n$.
\end{Lem}
(Of course, we mean: $\nd r_n$ is evaluated by $G*H'_\beta$, while $\dot{r}^*_n$
is evaluated by $H_\beta^y$.)
\begin{proof}
The proof is an obvious consequence of the previous fact,
since names of reals in a
ccc forcing can be viewed as a countable sequence of conditions.
In more detail:
For notational simplicity assume all $\nd r_n$ are names for
elements
of $2^\omega$.
Working in $V$, we can find for each $n,m\in\omega$
names for a maximal antichain $\n A_{n,m}$ and for a function $\n f_{n,m}:\n A_{n,m}\to 2$
such that $x$ forces that ($\mathbf{P}'_\beta$ forces that) $\nd r_n(m)=\n f_{n,m}(a)$
for the unique $a\in \n A_{n,m}\cap H'_\beta$.
Since $\mathbf{P}'_\beta$ is ccc, each $\n A_{n,m}$ is countable, and
since ${\mathbb{R}}$ is $\sigma$-closed, it is forced
that the sequence $\n\Xi=(\n A_{n,m},\n
f_{n,m})_{n,m\in\omega}$ is in $V$.
In $V$, we strengthen $x$ to $x_1$ to decide $\n\Xi$
to be some $\Xi^*$. We can also assume that
$\Xi^*\in M^{x_1}$ (see Corollary~\ref{cor:gurke3}).
Each $A^*_{n,m}$ consists of countably many $a$ such that
$x_1$ forces $a\in\mathbf{P}'_\beta$. Using Fact~\ref{fact:pathetic1}
iteratively (and again the fact that ${\mathbb{R}}$ is $\sigma$-closed)
we get some $y\leq x_1$ such that each such $a$ is actually
an element of $P^y_\beta$. So in $M^y$, we can use
$( A^*_{n,m}, f^*_{n,m})_{n,m\in \omega}$ to construct $P^y_\beta$-names $\dot{r}^*_n$
in the obvious way.
Now assume that $y\in G$ and that $H'_\beta$ is $\mathbf{P}'_\beta$-generic
over $V[G]$. Fix any $a\in A^*_{n,m}=\n A_{n,m}$.
Since $a\in P^y_\beta$, we get $a \in H^y_\beta$ iff $a\in H'_\beta$.
So there is a unique element $a$ of $A^*_{n,m}\cap H^y_\beta$, and
$\dot{r}^*_n(m)=f^*_{n,m}(a)=\n f_{n,m}(a)=\nd r_n(m)$.
\end{proof}
We will also need the following modification:
\begin{Lem}\label{lem:pathetic3}
(Same assumptions as in the previous lemma.)
In $V[G][H'_\beta]$, let $\mathbf{Q}_\beta$ be the union of $Q^z_\beta[H^z_\beta]$
for all $z\in G$.
In $V$, assume that $x$ forces that each $\nd r_n$ is a name for
an element of $\mathbf{Q}_\beta$. Then there is a $y\leq x$ and there is in $M^y$
a sequence $(\dot r^*_n)_{n\in\omega}$ of
$P^y_\beta$-names for elements of $Q^y_\beta$ such that
$y$ forces $\nd r_n=\dot r^*_n$ for all $n$.
\end{Lem}
So the difference to the previous lemma is: We additionally assume that $\nd
r_n$ is in $\bigcup_{z\in G}Q^z_\beta$, and we additionally get that $\dot r^*_n$ is
a name for an element of $Q^y_\beta$.
\begin{proof}
Assume $x\in G$ and work in $V[G]$. Fix $n$.
$\mathbf{P}'_\beta$ forces that there is some
$y_n\in G$ and some
$P^{y_n}_\beta$-name $\tau_n\in M^{y_n}$ of an element of $Q^{y_n} _\beta$ such that
$\nd r_n$ (evaluated by $H'_\beta$) is the same as
$\tau_n$ (evaluated by $H^{y_n} _\beta$).
Since we assume that $\mathbf{P}'_\beta$ is ccc, we can
find a countable set $Y_n\subseteq G$ of the possible $y_n$,
i.e., the empty condition of $\mathbf{P}'_\beta$ forces $y_n\in Y_n$.
(As $\mathbb{R}$ is $\sigma$-closed and $Y_n\subseteq \mathbb{R} \subseteq V$,
we must have $Y_n\in V$.)
So in $V$, there is (for each $n$) an $\mathbb{R}$-name $\n Y_n$ for this countable
set. Since $\mathbb{R}$ is $\sigma$-closed, we can find some $z_0 \leq x$ deciding
each $\n Y_n$ to be some countable set $Y_n^* \subseteq \mathbb{R} $.
In particular,
for each $y\in Y_n^*$ we know that $z_0 \Vdash_\mathbb{R} y\in G$, i.e.,
$z_0 \leq^* y$;
so using once again that ${\mathbb{R}}$ is $\sigma$-closed we can find
some $z$ stronger
than $z_0 $ and all the $y\in \bigcup_{n\in\omega} Y^*_n$.
Let $X$ contain all $\tau\in M^y$ such that for some $y\in \bigcup_{n\in\omega} Y^*_n$,
$\tau$ is a $P^y_\beta$-name for a $Q^y_\beta$-element.
Since $z\leq y$,
each $\tau\in X$ is actually\footnote{
Here we use two consequences of
$z\leq y$: Every $P^y_\beta$-name in $M^y$ can be canonically interpreted
as a $P^z_\beta$-name in $M^z$, and $Q^y_\beta$ is (forced to be) a subset
of $Q^z_\beta$.}
a $P^z_\beta$-name
for an element of $Q^z_\beta$.
So $X$ is a set of $P^z_\beta$-names for $Q^z_\beta$-elements;
we can assume that $X\in M^z$.
Also, $z$ forces that $\nd r_n\in X$ for all~$n$.
Using Lemma~\ref{lem:pathetic2}, we can additionally assume that there
are names $P^z_\beta$-name $\dot r^*_n$ in $M^z$ such that
$z$ forces that $\nd r_n = \dot r^*_n$ is forced for each $n$.
By Lemma~\ref{lem:pathetic0}, we know that $M^z$
thinks that $P^z_\beta$ forces that $\dot r^*_n\in X$. Therefore
$\dot r^*_n$ is a $P^z_\beta$-name for a $Q^z_\beta$-element.
\end{proof}
We now prove by induction on $\alpha$ that $\mathbf{P}'_\alpha$ is equivalent to a ccc
alternating iteration:
\begin{Lem}\label{lem:halbfett}
The following holds in $V[G]$ for $\alpha<\om2$:
\begin{enumerate}
\item\label{item:iteration}
$\mathbf{P}'_\alpha$ is equivalent to
an alternating iteration. More formally:
There is an
iteration
$(\mathbf{P}_\beta,\mathbf{Q}_\beta)_{\beta<\alpha}$ with limit $\mathbf{P}_\alpha$
that satisfies the definition of alternating iteration
(up to $\alpha$), and there is
a naturally defined
dense embedding
$j_\alpha:\mathbf{P}'_\alpha\to \mathbf{P}_\alpha$,
such that
for $\beta < \alpha$ we have $j_\beta \subseteq j_\alpha$, and
the embeddings commute with the restrictions.\footnote{I.e.,
$j_\beta(x,p\mathord\restriction \beta) = j_\alpha(x,p\mathord\restriction \beta) = j_\alpha(x,p) \mathord\restriction
\beta$.}
Each $\mathbf{Q}_\alpha$ is the union of all $Q^x_\alpha$
with~$x\in G$.
For $x\in G$ with $\alpha\in M^x$,
the function $i_{x,\alpha}: P^x_\alpha\to \mathbf{P}_\alpha$
that maps $p$ to $j_\alpha(x,p)$ is the canonical $M^x$-complete embedding.
\item In particular, a $\mathbf{P}'_\alpha$-generic filter $H'_\alpha$
can be translated into a $\mathbf{P}_\alpha$-generic filter which we call
$H_\alpha$ (and vice versa).
\item\label{item:a1}
$\mathbf{P}_\alpha$ has a dense subset of size~$\al1$.
\item\label{item:ccc}
$\mathbf{P}_\alpha$ is ccc.
\item\label{item:ch}
$\mathbf{P}_\alpha$ forces CH.
\end{enumerate}
\end{Lem}
\begin{proof}
$\alpha=0$ is trivial (since $\mathbf{P}_0 $ and $\mathbf{P}'_0$ both
are trivial: $\mathbf{P}_0$ is a singleton, and $\mathbf{P}'_0$ consists of
pairwise compatible elements).
So assume that all items hold for all $\beta<\alpha$.
\proofsection{Proof of (\ref{item:iteration})}
\emph{\textbf{Ultralaver successor case:}} Let $\alpha=\beta+1$ with $\beta$
an ultralaver position.
Let $H_\beta$ be $\mathbf{P}_\beta$-generic over $V[G]$. Work in $V[G][H_\beta]$.
By induction, for every $x\in G$ the canonical embedding $i_{x,\beta}$
defines a $P^x_\beta$-generic filter over~$M^x$ called~$H^x_\beta$.
\emph{Definition of $\mathbf{Q}_\beta$ (and thus of $\mathbf{P}_{\alpha}$):}
In $M^x[H^x_\beta]$, the forcing notion $Q^x_\beta$ is defined as $\mathbb{L}_{\bar D^x}$
for some system of ultrafilters $\bar D^x$ in $M^x[H^x_\beta]$.
Fix some $s\in\omega^{{<}\omega}$.
If $y\leq x$ in $G$, then $D_s^y$ extends $D_s^x$.
Let $D_s$ be the union of all $D_s^x$ with $x\in G$.
So $D_s$ is a proper filter. It is even an ultrafilter:
Let $r$ be a $\mathbf{P}_\beta$-name for a real.
Using Lemma~\ref{lem:pathetic2}, we know that there is
some $y\in G$ and some
$P^y_\beta$-name $\n r^y\in M^y$
such that (in $V[G][H_\beta]$) we have
$\n r^y[H^y_\beta]=r$.
So $r\in M^y[H^y_\beta]$, hence
either $r$ or its complement is in $D_s^y$ and therefore in $D_s$.
So all filters in the family $\bar D = (D_s)_{s\in\omega^{{<}\omega}}$
are ultrafilters.
Now work again in $V[G]$. We set $\mathbf{Q}_\beta$ to be the
$\mathbf{P}_\beta$-name for $\mathbb{L}_{\bar D}$.
(Note that $\mathbf{P}_\beta$ forces that $\mathbf{Q}_\beta$ literally is the
union of the $Q^x_\beta[H^x_\beta]$ for $x\in G$,
again by Lemma~\ref{lem:pathetic2}.)
\emph{Definition of $j_\alpha$:}
Let $(x,p)$ be in $\mathbf{P}'_\alpha$.
If $p\in P^x_\beta$, then we set $j_\alpha(x,p)=j_\beta(x,p)$,
i.e., $j_\alpha$ will extend $j_\beta$. If $p=(p\mathord\restriction\beta,p(\beta))$
is in $P^x_\alpha$ but not in $P^x_\beta$, we set
$j_\alpha(x,p)=(r,s)\in \mathbf{P}_\beta*\mathbf{Q}_\beta$ where
$r=j_\beta(x,p\mathord\restriction\beta)$ and
$s$ is the ($\mathbf{P}_\alpha$-name for) $p(\beta)$ as evaluated in
$M^x[H^x_\beta]$. From $\mathbf{Q}_\beta = \bigcup_{x\in G} Q^x_\beta[H^x_\beta]$
we conclude that
this embedding is dense.
\emph{The canonical embedding:}
By induction we know that $i_{x,\beta}$ which
maps $p\in P^x_\beta$ to $j_\beta(x,p)$ is
(the restriction to $P^x_\beta$ of) the canonical
embedding of $x$ into $\mathbf{P}_{\om2}$. So we have to extend the
canonical embedding to $i_{x,\alpha}:P^x_\alpha\to \mathbf{P}_\alpha$.
By definition of ``canonical embedding'', $i_{x,\alpha}$ maps
$p\in P^x_\alpha$ to the pair
$(i_{x,\beta}(p\mathord\restriction\beta), p(\beta))$.
This is the same as $j_\alpha(x,p)$.
We already know that $D^x_s$ is (forced to be)
an $M^x[H^x_\beta]$-ultrafilter that is extended by~$D_s$.
\emph{\textbf{Janus successor case:}} This is similar, but simpler than
the previous case: Here, $\mathbf{Q}_\beta$ is just defined as the union of
all $Q^x_\beta[H^x_\beta]$ for~$x\in G$.
We will show below that this union satisfies the ccc;
just as in Fact~\ref{fact:janus.ctblunion},
it is then easy to see that this union is again a Janus forcing.
In particular, $\mathbf{Q}_\beta$ consists of hereditarily countable objects
(since it is the union of Janus forcings, which by definition
consist of hereditarily countable objects).
So since $\mathbf{P}_\beta$ forces CH,
$\mathbf{Q}_\beta$ is forced to have size~$\al1$.
Also note that since all Janus forcings involved are separative,
the union (which is a limit of an in\-com\-patibility-preserving
directed system)
is trivially separative as well.
\emph{\textbf{Limit case:}} Let $\alpha$ be a limit ordinal.
\emph{Definition of $\mathbf{P}_\alpha$ and $j_\alpha$:}
First we define $j_\alpha: \mathbf{P}_\alpha' \to \mathbf{P}^{\textrm{\textup{CS}}}_\alpha$: For each $(x,p)\in \mathbf{P}'_\alpha$,
let $j_\alpha(x,p)\in \mathbf{P}^{\textrm{\textup{CS}}}_\alpha$
be the union of all $j_\beta(x,p\mathord\restriction \beta)$ (for $\beta\in \alpha\cap M^x$).
(Note that $\beta_1<\beta_2$ implies that $j_{\beta_1}(x,p\mathord\restriction \beta_1)$ is
a restriction of $ j_{\beta_2}(x,p\mathord\restriction \beta_2)$, so this union is indeed
an element of $\mathbf{P}^{\textrm{\textup{CS}}}_\alpha$.)
$\mathbf{P}_\alpha$ is the set of all $q\wedge p$, where $p\in j_\alpha[\mathbf{P}'_\alpha]$,
$q\in \mathbf{P}_\beta$ for some $\beta < \alpha$, and $q \le p\mathord\restriction \beta$.
It is easy
to check that $\mathbf{P}_\alpha$ actually is a partial countable
support limit, and that $j_\alpha$ is dense.
We will show below that $\mathbf{P}_\alpha$ satisfies the ccc, so in
particular it is proper.
\emph{The canonical embedding:}
To see that $i_{x,\alpha}$ is the (restriction of the) canonical embedding,
we just have to check that $i_{x,\alpha}$ is $M^x$-complete.
This is the case since
$\mathbf{P}'_\alpha$ is the direct limit of all $P^y_\alpha$
for $y\in G$ (without loss of generality $y\le x$),
and each $i_{x,y}$ is $M^x$-complete (see Fact~\ref{fact:5.12}).
\proofsection{Proof of (\ref{item:a1})}
Recall that we assume CH in the ground model.
The successor case, $\alpha=\beta+1$, follows easily from
(\ref{item:a1})--(\ref{item:ch}) for $\mathbf{P}_\beta$
(since $\mathbf{P}_\beta$
forces that $\mathbf{Q}_\beta$ has size $2^{\al0}=\aleph_1 = \aleph_1^V$).
If $\cf(\alpha)>\omega$,
then $\mathbf{P}_\alpha=\bigcup_{\beta<\alpha} \mathbf{P}_\beta$, so the proof is easy.
So let $\cf(\alpha)= \omega $. The following straightforward argument works for
any ccc partial CS iteration where all iterands $\mathbf{Q}_\beta$ are of size $\le \aleph_1$.
For notational simplicity we assume $\Vdash_{\mathbf{P}_\beta } \mathbf{Q}_\beta \subseteq
\omega_1$ for all $\beta<\alpha$ (this is justified by inductive assumption~(\ref{item:ch})).
By induction, we can assume that for all $\beta<\alpha$ there is a dense
$\mathbf{P}^*_\beta\subseteq \mathbf{P}_\beta$ of size~$\al1$ and that every $\mathbf{P}^*_\beta$ is ccc.
For each $p\in \mathbf{P}_\alpha$ and
all $\beta\in \dom(p)$
we can find a maximal antichain
$A^p_\beta\subseteq \mathbf{P}_\beta^*$ such that each element $a\in A^p_\beta$
decides the value of $p(\beta)$, say $a \Vdash_{\mathbf{P}_\beta}
p(\beta)=\gamma^p_\beta(a)$. Writing\footnote{Since $\le $ is separative, $p\sim q$ iff $p=^*q$, but
this fact is not used here.} $p\sim q$ if $p\le q $ and $q\le p$,
the map $p\mapsto (A^p_\beta, \gamma^p_\beta)_{\beta\in\dom(p)}$ is 1-1
modulo $\sim$. Since each $A^p_\beta$ is countable,
there are only $\al1$ many possible values, therefore there are only $\al1$
many $\sim $-equivalence classes. Any set of representatives will be dense.
Alternatively, we can prove~(\ref{item:a1}) directly for $\mathbf{P}'_\alpha$.
I.e.,
we can find a $\le^*$-dense subset $\mathbf{P}'' \subseteq \mathbf{P}'_\alpha$ of
cardinality~$\aleph_1$. Note that a condition $(x,p)\in \mathbf{P}'_\alpha$
essentially depends only on $p$ (cf.~Fact~\ref{facts:trivial66}). More
specifically, given $(x,p)$ we can ``transitively\footnote{
In more detail: We define a function $f:M^x\to V$ by induction as follows: If
$\beta\in M^x\cap \alpha+1$ or if $\beta=\om2$, then $f(\beta)=\beta$.
Otherwise, if $\beta\in M^x\cap \ON$, then $f(\beta)$ is the smallest ordinal
above $f[\beta]$. If $a\in M^x\setminus\ON$, then $f(a)=\{f(b):\, b\in a\cap
M^x\}$. It is easy to see that $f$ is an isomorphism from $M^x$ to
$M^{x'}\coloneqq f[M^x]$ and that $M^{x'}$ is a candidate. Moreover,
the ordinals that occur in $M^{x'}$ are subsets of $\alpha+\om1$ together
with the interval $[\om2,\om2+\om1]$; i.e., there are $\al1$ many
ordinals that can possibly occur in $M^{x'}$, and therefore there are
$2^\al0$ many possible such candidates. Moreover, setting $p'\coloneqq f(p)$,
it is easy to check that $(x,p)=^*(x',p')$ (similarly to Fact~\ref{facts:trivial66}).
}
collapse $x$ above
$\alpha$'', resulting in a $=^*$-equivalent condition $(x',p')$. Since
$|\alpha|=\al1$, there are only $\al1^{\al0}=2^{\al0}$ many such candidates
$x'$ and since each $x'$ is countable and $p'\in x'$, there are only
$2^{\al0}$ many pairs $(x',p')$.
\proofsection{Proof of (\ref{item:ccc})}
\emph{\textbf{Ultralaver successor case:}} Let $\alpha=\beta+1$ with $\beta$ an ultralaver
position.
We already know that $\mathbf{P}_\alpha=\mathbf{P}_\beta*\mathbf{Q}_\beta$ where $\mathbf{Q}_\beta$ is an ultralaver
forcing, which in particular is ccc, so by induction $\mathbf{P}_\alpha$
is ccc.
{\bf\em Janus successor case:}
As above it suffices to show that $\mathbf{Q}_\beta$,
the union of the Janus forcings
$Q^x_\beta[H^x_\beta]$ for $x\in G$, is (forced to be) ccc.
Assume towards a contradiction that this is not the case, i.e., that we have
an uncountable antichain in~$\mathbf{Q}_\beta$. We already know that $\mathbf{Q}_\beta$
has size $\al1$ and therefore the uncountable antichain has size $\al1$. So,
working in $V$, we assume towards a contradiction that
\begin{equation}\label{eq:ijqprjqr0999}
x_0\Vdash_{\mathbb{R}} p_0\Vdash_{\mathbf{P}_\beta}
\{ \nd a_i:i\in \omega_1\}\text{ is a maximal (uncountable) antichain in }\mathbf{Q}_\beta.
\end{equation}
We construct by induction on $n\in\omega$ a
decreasing sequence of conditions such that $x_{n+1}$ satisfies the
following:
\begin{enumerate}
\item[(i)] For all $i\in\om1\cap M^{x_n}$ there is (in $M^{x_{n+1}}$)
a $P^{x_{n+1}}_\beta$-name $\dot{a}_i^*$ for
a $Q^{x_{n+1}}_\beta$-condition
such that
\[
x_{n+1}\Vdash_{\mathbb{R}} p_0\Vdash_{\mathbf{P}_\beta}\nd a_i=\dot{a}_i^*.
\]
Why can we get that? Just use Lemma~\ref{lem:pathetic3}.
\item[(ii)] If $\tau$ is in $M^{x_n}$ a $P^{x_n}_\beta$-name for
an element of $Q^{x_n}_\beta$, then there is $k^*(\tau)\in\om1$
such that
\[
x_{n+1}\Vdash_{\mathbb{R}} p_0 \Vdash_{\mathbf{P}_\beta}\,
(\exists i<k^*(\tau))\ \nd a_i \parallel_{\mathbf{Q}_\beta} \tau.
\]
Also, all these $k^*(\tau)$ are in $M^{x_{n+1}}$.
\\
Why can we get that?
First note that $x_n\Vdash p_0\Vdash (\exists i\in\om1) \ \nd a_i
\parallel \tau $. Since
$\mathbf{P}_\beta$ is ccc, $x_n$ forces that there is some bound $\n k(\tau)$
for $i$. So it suffices that $x_{n+1}$ determines $\n k(\tau)$ to be
$k^*(\tau)$ (for all the countably many $\tau$).
\end{enumerate}
Set $\delta^*\coloneqq \om1\cap\bigcup_{n\in\omega} M^{x_n}$.
By Corollary~\ref{cor:bigcor}(\ref{item:martin}), there is some $y$ such that
\begin{itemize}
\item $y \le x_n$ for all $n\in\omega$,
\item $(x_n)_{n\in\omega}$ and $(\dot{a}_i^*)_{i\in\delta^*}$ are in $M^y$,
\item ($M^y$ thinks that) $P^y_\beta$ forces that
$Q^y_\beta$ is the union of $Q^{x_n}_\beta$, i.e., as a formula:
$M^y\models P^y_\beta\Vdash Q^y_\beta=\bigcup_{n\in\omega} Q^{x_n}_\beta$.
\end{itemize}
Let $G$ be ${\mathbb{R}}$-generic (over $V$) containing $y$, and let $H_\beta$
be $\mathbf{P}_\beta$-generic (over $V[G]$) containing $p_0$.
Set $A^*\coloneqq \{\dot{a}^*_i[H^y_\beta]:\, i<\delta^*\}$.
Note that $A^*$ is in $M^y[H^y_\beta]$. We claim
\begin{equation}\label{eq:pijqr9}
A^*\subseteq Q^y_\beta[H^y_\beta]\text{ is predense.}
\end{equation}
Pick any $q_0\in Q^y_\beta$.
So there is some $n\in\omega$ and
some $\tau$ which is in
$M^{x_n}$ a $P^{x_n}_\beta$-name of a $Q^{x_n}_\beta$-condition,
such that $q_0=\tau[H^{x_n}_\beta]$. By (ii) above,
$x_{n+1}$ and therefore $y$ forces (in $\mathbb{R}$) that for some $i<k^*(\tau)$
(and therefore some $i< \delta^*$) the condition $p_0$ forces
the following (in $\mathbf{P}_\beta$):
\begin{quote}
The conditions
$\nd a_i$ and $\tau$ are compatible in~$\mathbf{Q}_\beta$. Also,
$\nd a_i=\dot a^*_i$ and $\tau $
both are in $Q^y_\beta$,
and
$Q^y_\beta$ is an incompatibility-preserving subforcing of $\mathbf{Q}_\beta$.
Therefore $M^y[H^y_\beta]$
thinks that $\dot a^*_i$ and $\tau$ are compatible.
\end{quote}
This proves~\eqref{eq:pijqr9}.
Since $Q^y_\beta[H^y_\beta]$ is $M^y[H^y_\beta]$-complete in $\mathbf{Q}_\beta[H_\beta]$,
and since $A^*\in M^y[H^y_\beta]$,
this implies
(as $\dot{a}^*_i[H^y_\beta]=\nd a_i[G*H_\beta]$ for all
$i<\delta^*$)
that $\{\nd a_i[G*H_\beta]:\, i<\delta^*\}$
already is predense, a contradiction to~\eqref{eq:ijqprjqr0999}.
{\bf\em Limit case:}
We work with $\mathbf{P}'_\alpha$, which by definition
only contains HCON objects.
Assume towards a contradiction that $\mathbf{P}'_\alpha$ has an
uncountable antichain. We already know that $\mathbf{P}'_\alpha$ has
a dense subset of size~$\al1$ (modulo $=^*$), so the antichain has size $\al1$.
Again, work in $V$. We assume towards a contradiction that
\begin{equation}\label{eq:lkjwtoi}
x_0\Vdash_{\mathbb{R}} \{\n a_i:\, i\in\om1\} \text{ is a maximal (uncountable)
antichain in }\mathbf{P}'_\alpha.
\end{equation}
So each $\n a_i$ is an ${\mathbb{R}}$-name for an HCON object $(x,p)$ in $V$.
To lighten the notation we will abbreviate elements $(x,p)\in \mathbf{P}'_\alpha$
by~$p$; this is justified by Fact~\ref{facts:trivial66}.
Fix any HCON object $p$ and $\beta<\alpha$.
We will now define
the $({\mathbb{R}}*\mathbf{P}' _\beta)$-names $\nd\iota(\beta,p)$ and $\nd r(\beta,p)$:
Let $G$ be ${\mathbb{R}}$-generic and containing $x_0$, and
$H'_\beta$ be $\mathbf{P}'_\beta$-generic.
Let $R$ be the quotient $\mathbf{P}'_\alpha / H'_\beta $.
If $p$ is not in $R$, set $\nd\iota(\beta, p)=\nd r(\beta,p)=0$.
Otherwise, let $\nd\iota(\beta, p)$ be the minimal $i$ such that
$\n a_i\in R$ and $\n a_i$ and $p$ are compatible (in $R$),
and set $\nd r(\beta, p)\in R$ to be a witness of this compatibility.
Since $\mathbf{P}'_\beta$ is (forced to be) ccc, we can find
(in~$V[G]$) a countable set $\n X^\iota(\beta, p)\subseteq \omega_1$
containing all possibilities for $\nd\iota(\beta, p)$
and similarly $\n X^r(\beta, p)$ consisting of HCON objects for $\nd r(\beta, p)$.
To summarize:
For every $\beta<\alpha$
and every HCON object $p$, we can define (in~$V$) the ${\mathbb{R}}$-names
$\n X^\iota(\beta, p)$ and $\n X^r(\beta, p)$ such that
\begin{equation}
x_0\Vdash_{\mathbb{R}} \ \Vdash_{\mathbf{P}'_\beta} \biggl(p\in \mathbf{P}'_\alpha/H'_\beta \ \rightarrow
\ (\exists i\in \n X^\iota(\beta,p))\,
(\exists r\in \n X^r(\beta, p))\ r\leq_{\mathbf{P}'_\alpha/H'_\beta}
p,\n a_i\biggr).
\end{equation}
Similarly to the Janus successor case, we define by induction on
$n\in\omega$ a decreasing sequence of conditions such that
$x_{n+1}$ satisfies the following:
For all $\beta\in\alpha\cap M^{x_n}$ and $p\in P^{x_n}_\alpha$,
$x_{n+1}$ decides $\n X^{\iota}(\beta,p)$
and $\n X^{r}(\beta,p)$ to be some $X^{\iota*}(\beta,p)$ and $X^{r*}(\beta,p)$.
For all $i\in\om1\cap M^{x_{n}}$,
$x_{n+1}$ decides $\n a_i$ to be some $a^*_i\in P^{x_{n+1}}_\alpha$.
Moreover, each
such $X^{\iota*}$ and $X^{r*}$ is in $M^{x_{n+1}}$,
and every $r\in X^{r*}(\beta,p)$ is in
$P^{x_{n+1}}_\alpha$.
(For this, we just use Fact~\ref{fact:pathetic1}
and Lemma~\ref{lem:pathetic2}.)
Set $\delta^*\coloneqq \om1\cap\bigcup_{n\in\omega} M^{x_{n}}$,
and set $A^*\coloneqq \{a^*_i:\, i\in\delta^*\}$.
By Corollary~\ref{cor:bigcor}(\ref{item:martin}), there is some $y$ such that
\begin{gather}
\text{$y\leq x_n$ for all $n\in \omega$},\\
\text{$\bar x\coloneqq (x_n)_{n\in\omega}$ and $A^*$ are in $M^y$},\\
\label{eq:gqetwet}\text{($M^y$ thinks that) $P^y_\alpha$ is defined as
the almost FS limit over $\bar x$}.
\end{gather}
We claim that $y$ forces
\begin{equation}\label{eq:khweqt}
A^*\text{ is predense in } P^y_\alpha.
\end{equation}
Since
$P^y_\alpha$ is $M^y$-completely embedded into $\mathbf{P}'_\alpha$,
and since $A^*\in M^y $ (and since
$\n a_i=a^*_i$ for all $i\in\delta^*$) we get that
$\{\n a_i:\, i\in\delta^*\}$ is predense, a contradiction
to~\eqref{eq:lkjwtoi}.
So it remains to show~\eqref{eq:khweqt}. Let $G$ be ${\mathbb{R}}$-generic containing $y$.
Let
$r$ be a condition in $P^y_\alpha$; we will find $i<\delta^*$ such that $r$
is compatible with $a^*_i$.
Since $P^y_\alpha$ is the almost FS limit over $\bar x$, there
is some $n\in\omega$ and $\beta\in \alpha\cap
M^{x_n}$ such that $r$
has the form $q\land p$ with
$p$ in $P^{x_n}_\alpha$, $q\in P^y_\beta$ and $q\le p\mathord\restriction \beta$.
Now let $H'_\beta$
be $\mathbf{P}'_\beta$-generic containing $q$.
Work in $V[G][H'_\beta]$. Since $q\leq p\mathord\restriction\beta$,
we get $p\in \mathbf{P}'_\alpha/H'_\beta$.
Let $\iota^*$ be the evaluation by $G*H'_\beta$ of $\nd\iota(\beta,p)$,
and let $r^*$ be the evaluation of $\nd r(\beta,p)$.
Note that $\iota^* < \delta^*$ and $r^*\in P^y_\alpha$.
So we know that
$a^*_{\iota^*}$ and $p$ are compatible in $\mathbf{P}'_\alpha/H'_\beta$
witnessed by $r^*$.
Find $q'\in H'_\beta$ forcing
$r^*\le_{\mathbf{P}'_\alpha/H'_\beta} p, a^*_{\iota^*}$. We may find $q'\le q$.
Now
$q'\land r^*$ witnesses that $q\land p$ and $
a^*_{\iota^*}$ are compatible in~$P^y_\alpha$.
To summarize: The crucial point in proving the ccc is that ``densely'' we
choose (a variant of) a finite support iteration, see~\eqref{eq:gqetwet}.
Still, it is a bit surprising that we get the ccc, since we can also argue
that densely we use (a variant of) a countable support iteration.
But this does not prevent the ccc, it only prevents
the generic iteration from having direct limits in
stages
of countable cofinality.
\footnote{Assume that $x$ forces that $\mathbf{P}'_\alpha$ is the union
of the $\mathbf{P}'_\beta$ for $\beta<\alpha$; then we can find a stronger
$y$ that uses an almost CS iteration over~$x$. This almost CS
iteration contains a condition $p$ with unbounded support. (Take
any condition in the generic part of the almost CS limit; if this
condition has bounded domain, we can extend it to have unbounded domain, see
Definition~\ref{def:almost_CS_iteration_wolfgang}.) Now $p$ will be
in $\mathbf{P}'_\alpha$ and have unbounded domain.}
\proofsection{Proof of (\ref{item:ch})}
This follows from (\ref{item:a1}) and (\ref{item:ccc}).
\end{proof}
\subsection{The generic alternating iteration $\mathaccent "7016{\mathbf{P}}$}
In Lemma~\ref{lem:halbfett} we have seen:
\begin{Cor}\label{cor:summary}
Let $G$ be ${\mathbb{R}}$-generic. Then we can construct\footnote{in an ``absolute
way'': Given $G$, we first define $\mathbf{P}'_\om2$ to be the direct limit of $G$,
and then inductively construct the $\mathbf{P}_\alpha$'s from $\mathbf{P}'_\om2$.}
(in $V[G]$)
an alternating iteration $\bar \mathbf{P}$ such that the following holds:
\begin{itemize}
\item $\bar \mathbf{P}$ is ccc.
\item If $x\in G$, then $x$ canonically
embeds into $\bar \mathbf{P}$.
(In particular, a $\mathbf{P}_\om2$-generic filter $H_\om2$
induces a $P^x_\om2$-generic filter over $M^x$, called $H^x_\om2$.)
\item Each $\mathbf{Q}_\alpha$ is the union of all $Q^x_\alpha[H^x_\alpha]$
with~$x\in G$.
\item $ \mathbf{P}_\om2$ is equivalent to the direct limit $\mathbf{P}'_\om2$ of $G$:
There is a dense embedding $j:\mathbf{P}'_\om2\to \mathbf{P}_\om2$, and
for each $x\in G$ the function
$p\mapsto j(x,p)$ is the canonical embedding.
\end{itemize}
\end{Cor}
\begin{Lem}\label{lem:weiothowet}
Let $x\in {\mathbb{R}}$. Then ${\mathbb{R}}$ forces the following:
$x\in G$ iff $x$ canonically embeds into $\bar \mathbf{P}$.
\end{Lem}
\begin{proof}
If $x\in G$, then we already know that $x$ canonically embeds into
$\bar \mathbf{P}$.
So assume (towards a contradiction)
that $y$ forces that $x$ embeds, but $y\Vdash x\notin G$.
Work in $V[G]$ where $y\in G$.
Both $x$ (by assumption) and $y\in G$ canonically embed into $\bar \mathbf{P}$.
Let $N$ be an elementary submodel
of $H^{V[G]}(\chi^*)$ containing $x,y,\bar \mathbf{P}$; let $z = (M^z, \bar P^z)$ be the
ord-collapse of $(N, \bar \mathbf{P})$. Then $z\in V$ (as $\mathbb{R}$ is $\sigma$-closed)
and $z\in \mathbb{R}$,
and (by elementarity) $z\leq x,y$. This shows that $x\parallel_\mathbb{R} y$,
i.e., $y$ cannot force $x\notin G$, a contradiction.
\end{proof}
Using ccc, we can now prove a lemma that is in fact stronger than
the lemmas in the previous Section~\ref{sec:ccc}:
\begin{Lem}\label{lem:elemsub}
The following is forced by ${\mathbb{R}}$: Let $N \prec H^{V[G]}(\chi^*)$ be
countable, and let $y$ be the ord-collapse of $(N,\bar\mathbf{P})$.
Then $y\in G$. Moreover, if $x\in G\cap N$, then $y \le x$.
\end{Lem}
\begin{proof}
Work in $V[G]$ with $x\in G$. Pick an elementary
submodel $N$ containing $x$ and $\bar \mathbf{P}$. Let $y$ be the
ord-collapse of $(N,\bar \mathbf{P})$ via a collapsing map $k$.
As above, it is clear that $y\in{\mathbb{R}}$
and $y\leq x$.
To show $y\in G$, it is (by the previous lemma)
enough to show that $y$ canonically embeds.
We claim that $k^{-1}$ is the canonical embedding of $y$ into $\bar\mathbf{P}$.
The crucial point is to show $M^y$-completeness. Let $B\in M^y$ be
a maximal antichain of $P^y_{\om2}$, say $B=k(A)$ where $A\in N$ is a maximal
antichain of $\mathbf{P}_{\om2}$. So (by ccc) $A$ is countable, hence $A\subseteq N$.
So not only $A=k^{-1}(B)$ but even
$A=k^{-1}[B]$.
Hence $k^{-1}$ is an $M^y$-complete embedding.
\end{proof}
\begin{Rem} We used the ccc of $\mathbf{P}_{\om2}$ to prove Lemma~\ref{lem:elemsub};
this use was essential in the sense that we can in turn easily prove the ccc of
$\mathbf{P}_{\om2}$ if we assume that Lemma~\ref{lem:elemsub} holds. In fact
Lemma~\ref{lem:elemsub} easily implies all other lemmas in
Section~\ref{sec:ccc} as well.
\end{Rem}
\section{The proof of \textup{BC+dBC}}\label{sec:proof}
We first
\footnote{Note that for this weak version, it would be enough to
produce a generic iteration of length 2 only, i.e., $\mathbf{Q}_0*\mathbf{Q}_1$, where
$\mathbf{Q}_0$ is an ultralaver forcing and $\mathbf{Q}_1$ a corresponding Janus forcing.}
prove that no uncountable $X$ in $V$ will be smz or sm in the
final extension $V[G*H]$.
Then we show how to modify the argument to work for all uncountable sets
in $V[G*H]$.
\subsection{\textup{BC+dBC} for ground model sets.}\label{sec:groundmodel}
\begin{Lem}\label{lem:6.1}
Let $X \in V$ be an uncountable set of reals.
Then $\mathbb{R}*\mathbf{P}_\om2$ forces that $X$ is not smz.
\end{Lem}
\begin{proof}\
\begin{enumerate}
\item
Fix any even $\alpha< \om2$ (i.e., an
ultralaver position) in our iteration.
The ultralaver forcing $\mathbf{Q}_\alpha$ adds a
(canonically defined code for a)
closed null set $\dot F$ constructed from the
ultralaver real $\bar \ell_\alpha$.
(Recall Corollary~\ref{cor:absolutepositive}.)
In the following, when
we consider various ultralaver forcings $\mathbf{Q}_\alpha$, $Q_\alpha$, $Q^x_\alpha$,
we treat
$\dot F$ not as an actual name, but rather as a definition
which depends on the forcing used.
\item
According to Theorem~\ref{thm:pawlikowski}, it is enough to show that $X+\dot
F$ is non-null in the $\mathbb{R}*\mathbf{P}_{\om2}$-extension, or equivalently, in every
$\mathbb{R}*\mathbf{P}_{\beta}$-extension ($\alpha < \beta<\om2$).
So assume towards a contradiction that there is a $\beta > \alpha$
and an $\mathbb{R}*\mathbf{P}_{\beta}$-name
$\nd Z$ of a (code for a) Borel null set such that some
$(x,p)\in \mathbb{R}*\mathbf{P}_\om2$ forces that
$X + \dot F \subseteq \nd Z$.
\item Using the dense embedding $j_\om2:\mathbf{P}'_\om2\to \mathbf{P}_\om2$, we may
replace $(x,p)$ by a condition $(x,p')\in \mathbb{R}*\mathbf{P}'_\om2$.
According to
Fact~\ref{fact:pathetic1} (recall that we now know that $\mathbf{P}_\om2$
satisfies ccc)
and Lemma~\ref{lem:pathetic2}
we can assume
that $p'$ is already a $P^x_\beta$-condition $p^x$
and that
$\nd Z $ is (forced by $x$ to be the same as)
a $P^x_\beta$-name $\dot Z^x$ in $M^x$.
\item
We construct (in $V$) an iteration $\bar P$ in the following way:
\begin{enumerate}
\item[(a)]
Up to $\alpha$, we take an arbitrary alternating iteration
into which $x$ embeds.
In particular, $P_\alpha$ will be proper and hence
force that $X$ is still uncountable.
\item[(b)] Let $Q_\alpha$ be any ultralaver forcing (over $Q^x_\alpha$
in case $\alpha\in M^x$).
So according
to Corollary~\ref{cor:absolutepositive}, we know that
$Q_\alpha$ forces that $X+\dot F$ is not null.
Therefore we can pick (in $V[H_{\alpha+1}]$)
some $\dot r$ in $X+\dot F$ which is random over
(the countable model)
$M^x[H^x_{\alpha+1}]$, where $H^x_{\alpha+1}$ is induced
by $H_{\alpha+1}$.
\item[(c)] In the rest of the construction, we preserve
randomness of $\dot r$ over $M^x[H^x_{\zeta}]$ for each $\zeta\le \om2$.
We can do this using an almost CS iteration
over~$x$
where at each Janus position we use a random version of Janus forcing and
at each ultralaver position we use a suitable ultralaver forcing;
this is possible by Lemma~\ref{lem:4.28}.
By Lemma~\ref{lem:preservation.variant}, this iteration
will preserve the randomness of $\dot r$.
\item[(d)] So we get $\bar P$ over $x$
(with canonical embedding $i_x$) and $q\leq_{P_\om2} i_x(p^x)$
such that $q\mathord\restriction\beta$ forces (in $P_\beta$)
that $\dot r$ is random over $M^x[H^x_{\beta}]$, in particular that
$\dot r\notin \dot Z^x$.
\end{enumerate}
We now pick a countable $N\prec H(\chi^*)$ containing
everything and ord-collapse $(N,\bar P)$ to $y\leq x$. (See Fact~\ref{fact:esmV}.)
Set $X^y\coloneqq X\cap M^y$ (the image of $X$ under the collapse).
By elementarity, $M^y$ thinks that (a)--(d) above holds for $\bar P^y$
and that $X^y$ is uncountable. Note that $X^y\subseteq X$.
\item
This gives a contradiction in the obvious way:
Let $G$ be $\mathbb{R}$-generic over $V$ and contain $y$,
and let $H_\beta$ be $\mathbf{P}_\beta$-generic over $V[G]$ and contain $q\mathord\restriction\beta$.
So $M^y[H^y_\beta]$ thinks that $r\notin \dot Z^x$ (which is
absolute) and that $r=x+f$ for some $x\in X^y\subseteq X$
and $f\in F$ (actually even in $F$ as evaluated in $M^y[H^y_{\alpha+1}]$).
So in $V[G][H_\beta]$, $r$ is the sum of an element of $X$
and an element of $F$. So $(y,q)\leq (x,p')$ forces that $\dot r\in (X+\dot
F)\setminus \nd Z$, a contradiction to~(2).
\qedhere
\end{enumerate}
\end{proof}
Of course, we need this result not just for ground model sets $X$, but for
$\mathbb{R}*\mathbf{P}_{\om2}$-names $\nd X=(\nd x_i:i\in\om1)$ of uncountable sets. It is
easy to see that it is enough to deal with $\mathbb{R}*\mathbf{P}_{\beta}$-names for (all)
$\beta<\om2$. So given $\nd X$, we can (in the proof) pick $\alpha$ such that
$\nd X$ is actually an $\mathbb{R}*\mathbf{P}_{\alpha}$-name. We can try to repeat the same
proof; however, the problem is the following: When constructing $\bar P$
in~(4), it is not clear how to simultaneously make all the uncountably many
names $(\nd x_i)$ into $\bar P$-names in a sufficiently ``absolute'' way. In
other words: It is not clear how to end up with some $M^y$ and $\dot X^y$
uncountable in $M^y$ such that it is guaranteed that $\dot X^y$ (evaluated in
$M^y[H^y_{\alpha}]$) will be a subset of $\nd X$ (evaluated in
$V[G][H_\alpha]$). We will solve this problem in the next section by factoring
$\mathbb{R}$.
Let us now give the proof of the corresponding weak version of dBC:
\begin{Lem}\label{lem:6.2}
Let $X \in V$ be an uncountable set of reals.
Then $\mathbb{R}*\mathbf{P}_\om2$ forces that $X$ is not strongly meager.
\end{Lem}
\begin{proof}
The proof is parallel to the previous one:
\begin{enumerate}
\item
Fix any even $\alpha< \om2$ (i.e., an
ultralaver position) in our iteration.
The Janus forcing $\mathbf{Q}_{\alpha+1}$
adds a (canonically defined code for a)
null set $\dot Z_\nabla$.
(See Definition~\ref{def:Znabla} and Fact~\ref{fact:Znablaabsolute}.)
\item
According to~\eqref{eq:notsm}, it is enough to show that
$X+\dot Z_\nabla=2^\omega$ in the $\mathbb{R}*\mathbf{P}_{\om2}$-extension, or equivalently, in every
$\mathbb{R}*\mathbf{P}_{\beta}$-extension ($\alpha<\beta<\om2$).
(For every real $r$, the statement
$r\in X+\dot Z_\nabla$, i.e., $(\exists x\in X)\ x+r\in\dot Z_\nabla$, is absolute.)
So assume towards a contradiction that there is a $\beta > \alpha$
and an $\mathbb{R}*\mathbf{P}_{\beta}$-name
$\nd r$ of a real such that some
$(x,p)\in \mathbb{R}*\mathbf{P}_\om2$ forces that
$\nd r\notin X + \dot Z_\nabla$.
\item Again, we can assume that $\nd r $ is a $P^x_\beta$-name $\dot r^x$ in $M^x$.
\item
We construct (in $V$) an iteration $\bar P$ in the following way:
\begin{enumerate}
\item[(a)]
Up to $\alpha$, we take an arbitrary alternating
iteration into which $x$ embeds.
In particular, $P_\alpha$ again forces that $X$ is still uncountable.
\item[(b1)] Let $Q_\alpha$ be any ultralaver forcing (over $Q_\alpha^x$). Then
$Q_\alpha$ forces that $X$ is not thin
(see Corollary~\ref{cor:LDnotthin}).
\item[(b2)] Let $Q_{\alpha+1}$ be a countable Janus forcing.
So $Q_{\alpha+1}$ forces $X+\dot Z_\nabla=2^\omega$. (See Lemma~\ref{lem:janusnotmeager}.)
\item[(c)] We continue the iteration in a $\sigma$-centered way.
I.e., we use an almost FS iteration over $x$ of
ultralaver forcings and countable Janus forcings,
using trivial $Q_\zeta$ for all $\zeta\notin M^x$; see
Lemma~\ref{lem:4.17}.
\item[(d)] So $P_\beta$ still
forces that $X+\dot Z_\nabla=2^\omega$, and in particular that $\dot
r^x\in X+\dot Z_\nabla$.
(Again by Lemma~\ref{lem:janusnotmeager}.)
\end{enumerate}
Again, by collapsing some $N$ as in the previous proof,
we get $y\le x $ and $X^y\subseteq X$.
\item
This again gives the obvious contradiction:
Let $G$ be $\mathbb{R}$-generic over $V$ and contain $y$,
and let $H_\beta$ be $\mathbf{P}_\beta$-generic over $V[G]$ and contain $p$.
So $M^y[H^y_\beta]$ thinks that
$r=x+ z$ for some $x\in X^y\subseteq X$
and $z \in Z_\nabla$ (this time, $\dot Z_\nabla$ is evaluated in $M^y[H^y_{\beta}]$),
contradicting~(2).
\qedhere
\end{enumerate}
\end{proof}
\subsection{A factor lemma}\label{sec:factor}
We can restrict $\mathbb{R}$ to any ${\alpha^*}<\om2$
in the obvious way: Conditions are pairs
$x=(M^x,\bar P^x)$ of nice candidates $M^x$ (containing ${\alpha^*}$) and alternating
iterations $\bar P^x$, but now $M^x$ thinks that $\bar P^x$ has length ${\alpha^*}$
(and not $\om2$). We call this variant $\mathbb{R}\mathord\restriction{\alpha^*}$.
Note that all results of Section~\ref{sec:construction} about $\mathbb{R}$ are still true for
$\mathbb{R}\mathord\restriction{\alpha^*}$. In particular, whenever $G\subseteq \mathbb{R}\mathord\restriction\alpha^*$ is generic,
it will define a direct limit (which we call $\mathbf{P}^{\prime*}$), and an
alternating iteration of length $\alpha^*$ (called $\bar \mathbf{P}^*$); again
we will have that $x\in G$ iff $x$ canonically embeds into $\bar
\mathbf{P}^*$.
There is a natural projection map from $\mathbb{R}$ (more exactly: from the dense
subset of those $x$ which satisfy ${\alpha^*}\in M^x$) into $\mathbb{R}\mathord\restriction\alpha^*$,
mapping $x=(M^x,\bar P^x)$ to
$x\mathord\restriction{\alpha^*}\coloneqq (M^x,\bar P^x\mathord\restriction{\alpha^*})$. (It is obvious that this projection is
dense and preserves $\leq$.)
There is also a natural embedding $\varphi$ from $\mathbb{R}\mathord\restriction{\alpha^*}$ to $\mathbb{R}$: We
can just continue an alternating iteration of length ${\alpha^*}$ by appending
trivial forcings.
$\varphi$ is complete: It preserves $\leq$ and $\perp$. (Assume that $z\leq
\varphi(x),\varphi(y)$. Then $z\mathord\restriction{\alpha^*}\leq x,y$.) Also, the projection is a
reduction: If $y\leq x\mathord\restriction{\alpha^*}$ in $\mathbb{R}\mathord\restriction{\alpha^*}$, then let $M^z$ be a model
containing both $x$ and $y$. In $M^z$, we can first construct
an alternating iteration of length $\alpha^*$ over $y$
(using almost FS over $y$, or almost CS ---
this does not matter here). We then continue this iteration $\bar P^z$
using almost FS or almost CS over $x$.
So $x$ and $y$ both embed into $\bar P^z$, hence
$z=(M^z,\bar P^z)\leq x,y$.
So according to the general factor lemma of forcing theory, we know that
$\mathbb{R}$ is forcing equivalent to $\mathbb{R}\mathord\restriction{\alpha^*} * (\mathbb{R}/\mathbb{R}\mathord\restriction{\alpha^*})$,
where $\mathbb{R}/\mathbb{R}\mathord\restriction{\alpha^*}$ is the quotient of $\mathbb{R}$ and $\mathbb{R}\mathord\restriction{\alpha^*}$,
i.e.,
the ($\mathbb{R}\mathord\restriction{\alpha^*}$-name for the)
set of $x\in\mathbb{R}$ which are compatible (in $\mathbb{R}$) with all $\varphi(y)$ for
$y\in G\mathord\restriction{\alpha^*}$ (the generic filter for $\mathbb{R}\mathord\restriction{\alpha^*}$),
or equivalently, the set of $x\in\mathbb{R}$ such that $x\mathord\restriction{\alpha^*}\in
G\mathord\restriction{\alpha^*}$. So Lemma~\ref{lem:weiothowet} (relativized to
$\mathbb{R}\mathord\restriction\alpha^*$) implies:
\proofclaim{eq:oetwji}{$\mathbb{R}/\mathbb{R}\mathord\restriction{\alpha^*}$ is the set of $x\in\mathbb{R}$ that
canonically embed (up to ${\alpha^*}$) into $\mathbf{P}_{\alpha^*}$.}
\begin{Setup}
Fix some $\alpha^*<\om2$ of uncountable cofinality.\footnote{Probably the cofinality is completely irrelevant, but the picture is clearer this way.}
Let $G\mathord\restriction{\alpha^*}$ be
$\mathbb{R}\mathord\restriction{\alpha^*}$-generic over $V$ and work in $V^*\coloneqq V[G\mathord\restriction{\alpha^*}]$.
Set $\bar\mathbf{P}^*=(\mathbf{P}^*_\beta)_{\beta<\alpha^*}$, the generic alternating
iteration added by $\mathbb{R}\mathord\restriction{\alpha^*}$.
Let $\mathbb{R}^*$ be the quotient
$\mathbb{R}/\mathbb{R}\mathord\restriction\alpha^*$.
\end{Setup}
We claim that $\mathbb{R}^*$ satisfies (in $V^*$) all the properties
that we proved in Section~\ref{sec:construction} for $\mathbb{R}$ (in $V$), with
the obvious modifications. In particular:
\begin{enumerate}[(A)$_{\alpha^*}$]
\item $\mathbb{R}^*$ is $ \al2$-cc, since it is
the quotient of an $\al2$-cc forcing.
\item $\mathbb{R}^*$ does not add new reals (and more generally, no new
HCON objects), since it is the quotient of a $\sigma$-closed forcing.\footnote{It is
easy to see that $\mathbb{R}^*$ is even $\sigma$-closed, by ``relativizing'' the
proof for $\mathbb{R}$, but we will not need this.}
\item
Let $G^*$ be $\mathbb{R}^*$-generic over $V^*$. Then $G^*$ is $\mathbb{R}$-generic
over $V$, and therefore
Corollary~\ref{cor:summary} holds for~$G^*$.
(Note that $\mathbf{P}'_\om2$ and then $\mathbf{P}_\om2$ is constructed from~$G^*$.)
Moreover, it is easy to see
\footnote{
For $\beta \le \alpha^*$, let
$\mathbf{P}^{\prime*}_\beta$ be the direct limit of $(G\mathord\restriction\alpha^*)\mathord\restriction \beta$
and $\mathbf{P}^{\prime}_\beta$ the direct limit of $G^*\mathord\restriction\beta$.
The function $k_\beta: \mathbf{P}^{\prime*}_\beta\to \mathbf{P}^{\prime}_\beta$
that maps $(x,p)$ to $(\varphi(x),p)$
preserves $\leq$ and $\perp$ and is surjective
modulo $=^*$,
see Fact~\ref{facts:trivial66}(\ref{item:bla3}).
So it is clear that
defining $\bar\mathbf{P}^*\mathord\restriction\beta$ by induction
from $\mathbf{P}^{\prime*}_\beta$ yields the same result as
defining $\bar\mathbf{P}\mathord\restriction\beta$ from $\mathbf{P}_{\beta}'$.
}
that $\bar\mathbf{P}$ starts with $\bar\mathbf{P}^*$.
\item In particular, we get a variant of Lemma~\ref{lem:elemsub}:
The following is forced by ${\mathbb{R}^*}$: Let $N \prec H^{V[G^*]}(\chi^*)$ be
countable, and let $y$ be the ord-collapse of $(N,\bar\mathbf{P})$.
Then $y\in G^*$. Moreover: If $x\in G^*\cap N$, then $y \le x$.
\end{enumerate}
We can use the last item to prove the $\mathbb{R}^*$-version of
Fact~\ref{fact:pathetic1}:
\begin{Cor}\label{cor:slkjte}
In $V^*$, the following holds:
\begin{enumerate}
\item \label{item:fangen.a}
Assume that $x\in\mathbb{R}^*$ forces that
$p\in \mathbf{P}_\om2$. Then there is a $y\leq x$
and a $p^y\in P^y_\om2$ such that $y$ forces
$p^y=^*p$.
\item \label{item:fangen.b}
Assume that $x\in\mathbb{R}^*$ forces that
$\nd r$ is a $\mathbf{P}_{\om2}$-name of a real. Then there is a $y\leq x$
and a $P^y_\om2$-name $\dot r^y$ such that $y$ forces
that $\dot r^y $ and $\nd r $ are equivalent as $\mathbf{P}_{\om2}$-names.
\end{enumerate}
\end{Cor}
\begin{proof} We only prove (\ref{item:fangen.a}), the proof of (\ref{item:fangen.b})
is similar.
Let $G^*$ contain $x$. In $V[G^*]$, pick an elementary
submodel $N$ containing $x,p,\bar\mathbf{P}$ and let $(M^{z},\bar P^{z},p^{z})$
be the ord-collapse of $(N,\bar\mathbf{P},p)$.
Then $z\in G^*$.
This whole situation is forced by some $y\leq z\leq x\in G^*$.
So $y$ and $p^y$ is as required, where
$p^y\in P^y_\om2$ is the canonical image of $p^z$.
\end{proof}
We also get the following analogue of Fact~\ref{fact:esmV}:
\proofclaim{claim:44}{
In $V^*$ we have: Let $x\in \mathbb{R}^*$.
Assume that $\bar P$ is an alternating iteration that extends
$ \bar \mathbf{P}\mathord\restriction \alpha^*$ and
that $x=(M^x,\bar P^x) \in
\mathbb{R}$ canonically embeds into $\bar P$, and that $N \prec H( \chi^*)$
contains $x$ and $\bar P$. Let $y=(M^y, \bar P^y)$ be the ord-collapse of
$(N, \bar P)$. Then $y\in\mathbb{R}^*$ and $y\le x$.
}
We now claim that $\mathbb{R}*\mathbf{P}_\om2$ forces BC+dBC.
We know that $\mathbb{R}$ is forcing equivalent to $\mathbb{R}\mathord\restriction{\alpha^*} *
\mathbb{R}^*$. Obviously we have
\[
\mathbb{R}*\mathbf{P}_\om2=\mathbb{R}\mathord\restriction{\alpha^*}* \mathbb{R}^**\mathbf{P}_{\alpha^*} * \mathbf{P}_{{\alpha^*},\,\om2}
\] (where $\mathbf{P}_{{\alpha^*},\,\om2}$ is the quotient of
$\mathbf{P}_\om2$ and $\mathbf{P}_{\alpha^*}$).
Note that $\mathbf{P}_{\alpha^*}$ is already determined by $\mathbb{R}\mathord\restriction{\alpha^*}$,
so $\mathbb{R}^**\mathbf{P}_{\alpha^*}$ is (forced by $\mathbb{R}\mathord\restriction{\alpha^*}$ to be)
a product $\mathbb{R}^*\times \mathbf{P}_{\alpha^*}=\mathbf{P}_{\alpha^*}\times \mathbb{R}^*$.
But note that this is not the same as $\mathbf{P}_{\alpha^*} * \mathbb{R}^*$, where we
evaluate the definition of~$\mathbb{R}^*$ in the $\mathbf{P}_{\alpha^*}$-extension of
$V[G\mathord\restriction{\alpha^*}]$: We would get new candidates and therefore new conditions
in~$\mathbb{R}^*$
after forcing with~$\mathbf{P}_{\alpha^*}$. In other words, we can
\emph{not} just argue as follows:
\begin{wrongproof}
$\mathbb{R}*\mathbf{P}_\om2$ is the same as
$(\mathbb{R}\mathord\restriction{\alpha^*}* \mathbf{P}_{\alpha^*})* (\mathbb{R}^**\mathbf{P}_{{\alpha^*},\om2})$;
so given an $\mathbb{R}*\mathbf{P}_\om2$-name $X$ of a set of reals of size~$\al1$,
we can choose $\alpha^*$ large enough so that
$X$ is an $(\mathbb{R}\mathord\restriction{\alpha^*}* \mathbf{P}_{\alpha^*})$-name. Then,
working in the $(\mathbb{R}\mathord\restriction{\alpha^*}* \mathbf{P}_{\alpha^*})$-extension,
we just apply Lemmas~\ref{lem:6.1} and~\ref{lem:6.2}.
\end{wrongproof}
So what do we do instead?
Assume that $\nd X=\{\nd\xi_i:\, i\in\om1\}$ is an $\mathbb{R}*\mathbf{P}_{\om2}$-name for
a set of reals of size~$\aleph_1$. So there is a $\beta<\om2$ such that
$\nd X$ is added by $\mathbb{R}*\mathbf{P}_\beta$.
In the $\mathbb{R}$-extension, $\mathbf{P}_{\beta}$ is ccc, therefore we can assume
that each $\nd \xi_i$ is a system of countably many countable
antichains $\n A^m_i$ of~$\mathbf{P}_\beta$,
together with functions $\n f^m_i:\n A^m_i\to\{0,1\}$.
For the following argument,
we prefer to work with the equivalent $\mathbf{P}_\beta'$ instead of~$\mathbf{P}_\beta$.
We can assume that each of the sequences $B_i\coloneqq (\n
A^m_i,\n f^m_i)_{m\in\omega}$ is an element of~$V$ (since $\mathbf{P}'_\beta$ is a
subset of~$V$ and since $\mathbb{R}$ is $\sigma$-closed).
So each $B_i$ is decided by a maximal antichain~$Z_i$ of~$\mathbb{R}$.
Since $\mathbb{R}$ is $\al2$-cc, these $\al1$ many antichains all
are contained in some $\mathbb{R}\mathord\restriction {\alpha^*}$ with ${\alpha^*}\geq \beta$.
So in the $\mathbb{R}\mathord\restriction {\alpha^*}$-extension $V^*$ we
have the following situation:
Each $\xi_i$ is a very ``absolute\footnote{or: ``nice'' in the sense of~\cite[5.11]{MR597342}}''
$\mathbb{R}^* * \mathbf{P}_{\alpha^*}$-name
(or equivalently, $\mathbb{R}^* \times \mathbf{P}_{\alpha^*}$-name),
in fact they are already determined by antichains that are in $\mathbf{P}_{\alpha^*}$
and do not depend on $\mathbb{R}^*$. So we can interpret them as
$\mathbf{P}_{\alpha^*}$-names.
Note that:
\proofclaim{claim:xi.i}{ The $\xi_i$ are forced
(by $\mathbb{R}^**\mathbf{P}_{\alpha^*}$)
to be pairwise different,
and therefore already by $\mathbf{P}_{\alpha^*}$.}
Now we are finally ready to prove that $\mathbb{R}* \mathbf{P}_{\om2}$ forces that every uncountable $X$
is neither smz nor sm. It is enough to show that for every name $\nd X$
of an uncountable set of reals of size $\al1$ the forcing $\mathbb{R}* \mathbf{P}_{\om2}$
forces that $\nd X$ is neither smz nor sm. For the rest of the proof
we fix such a name $\nd X$, the corresponding $\nd \xi_i$'s (for $i\in \omega_1$),
and the appropriate $\alpha^*$ as above. From now on, we
work in the $\mathbb{R}\mathord\restriction\alpha^*$-extension~$V^*$.
So we have to show that $\mathbb{R}^* * \mathbf{P}_{\om2}$ forces that $\nd X$
is neither smz nor sm.
After all our preparations, we can just repeat the proofs of BC (Lemma~\ref{lem:6.1})
and dBC (Lemma~\ref{lem:6.2})
of Section~\ref{sec:groundmodel}, with the
following modifications. The modifications are the same for both proofs;
for better readability we describe the results of the change only for the proof of dBC.
\begin{enumerate}
\item Change:
Instead of an arbitrary ultralaver position $\alpha<\om2$, we obviously
have to choose $\alpha\geq \alpha^*$.
\\ For the dBC: We choose $\alpha\ge \alpha^*$ an arbitrary ultralaver position.
The Janus forcing $\mathbf{Q}_{\alpha+1}$
adds a (canonically defined code for a)
null set $\dot Z_\nabla$.
\item
Change: No change here.
(Of course we now have an $\mathbb{R}^**\mathbf{P}_{\alpha^*}$-name $\nd X$ instead
of a ground model set.)\\
For the dBC:
It is enough to show that
$\nd X+\dot Z_\nabla=2^\omega$ in the $\mathbb{R}^**\mathbf{P}_{\om2}$-extension of~$V^*$,
or equivalently, in every
$\mathbb{R}^**\mathbf{P}_{\beta}$-extension ($\alpha<\beta<\om2$).
So assume towards a contradiction that there is a $\beta > \alpha$
and an $\mathbb{R}^**\mathbf{P}_{\beta}$-name
$\nd r$ of a real such that some
$(x,p)\in \mathbb{R}^**\mathbf{P}_\om2$ forces that
$\nd r\notin \nd X + \dot Z_\nabla$.
\item Change: no change. (But we use Corollary~\ref{cor:slkjte}
instead of Lemma~\ref{lem:pathetic2}.)\\
For dBC:
Using Corollary~\ref{cor:slkjte}(\ref{item:fangen.b}), without loss of generality $x$ forces $p^x=^* p $
and there is a $P^x_\beta$-name $\dot r^x$ in $M^x$ such that $\dot r^x=\nd r$ is forced.
\item Change:
The iteration obviously has to start with the $\mathbb{R}\mathord\restriction\alpha^*$-generic iteration
$\bar\mathbf{P}^*$ (which is ccc),
the rest is the same.
\\ For dBC:
In $V^*$ we construct an iteration $\bar P$ in the following way:
\begin{enumerate}
\item[(a1)]
Up to $\alpha^*$, we use the iteration $\bar\mathbf{P}^* $ (which
already lives in our current universe $V^*$). As explained
above in the paragraph preceding~\eqref{claim:xi.i},
$\nd X$ can be interpreted as a $\mathbf{P}_{\alpha^*}$-name $\dot X$, and by
\eqref{claim:xi.i}, $\dot X $ is forced to be uncountable.
\item[(a2)]
We continue the iteration from $\alpha^*$ to $\alpha$ in a way that
embeds $x$ and such that $P_\alpha$ is proper.
So $P_{\alpha}$ will force
that $\dot X$ is still uncountable.
\item[(b1)] Let $Q_\alpha$ be any ultralaver forcing (over $Q_\alpha^x$). Then
$Q_\alpha$ forces that $\dot X$ is not thin.
\item[(b2)] Let $Q_{\alpha+1}$ be a countable Janus forcing.
So $Q_{\alpha+1}$ forces $\dot X+\dot Z_\nabla=2^\omega$.
\item[(c)] We continue the iteration in a $\sigma$-centered way.
I.e., we use an almost FS iteration over $x$ of
ultralaver forcings and countable Janus forcings,
using trivial $Q_\zeta$ for all $\zeta\notin M^x$.
\item[(d)] So $P_\beta$ still
forces that $\dot X+\dot Z_\nabla=2^\omega$, and in particular that $\dot
r^x\in \dot X+\dot Z_\nabla$.
\end{enumerate}
We now pick (in $V^*$) a countable $N\prec H(\chi^*)$ containing
everything and ord-collapse $(N,\bar P)$ to $y\leq x$, by \eqref{claim:44}.
The HCON object $y$ is of course in $V$ (and even in $\mathbb{R}$), but we can say more: Since the iteration $\bar P$ starts with the $(\mathbb{R}\mathord\restriction\alpha^*)$-generic iteration $\bar\mathbf{P}^*$, the condition $y$ will be in the quotient forcing $\mathbb{R}^*$.\\
Set $\dot X^y\coloneqq \dot X\cap M^y$ (which is
the image of $\dot X$ under the collapse, since we view $\dot X$ as a set of
HCON-names).
By elementarity, $M^y$ thinks that (a)--(d) above holds for $\bar P^y$
and that $\dot X^y$ is forced to be uncountable.
Note that $\dot X^y\subseteq \dot X$ in the following sense:
Whenever $G^**H$ is $\mathbb{R}^**\mathbf{P}_{\om2}$-generic over $V^*$, and $y\in G^*$, then the evaluation of $\dot X^y $ in $M^y[H^y]$ is a subset of the evaluation of $\dot X$ in $V^*[G^**H]$.
\item Change: No change here.\\
For dBC: We get our desired contradiction as follows:\\
Let $G^*$ be $\mathbb{R}^*$-generic over $V^*$ and contain $y$.
Let $H_\beta$ be $\mathbf{P}_\beta$-generic over $V^*[G^*]$ and contain $p$.
So $M^y[H^y_\beta]$ thinks that
$ r=x+ z$ for some $x\in X^y\subseteq X$
and\footnote{Note
that we get the same Borel code, whether we evaluate $\dot Z_\nabla$ in $M^y[H^y_{\beta}]$ or in $V^*[G^**H_\beta]$. Accordingly,
the actual Borel set of reals coded by $Z_\nabla$ in the smaller universe
is a subset of the corresponding Borel set in the larger universe.}
$z \in Z_\nabla$, contradicting~(2).
\end{enumerate}
\section{A word on variants of the definitions}\label{sec:alternativedefs}
The following is not needed for understanding the paper, we just
briefly comment on alternative ways some notions could be defined.
\subsection{Regarding \qemph{alternating iterations}} \label{sec:7a}
We call the set of $\alpha\in\om2$ such that $Q_\alpha$ is (forced to be) nontrivial
the \qemph{true domain} of $\bar P$ (we use this notation in this
remark only). Obviously $\bar P$ is naturally isomorphic
to an iteration whose length is the order type of its true domain.
In Definitions~\ref{def:alternating} and~\ref{def:prep}, we could have
imposed
the following additional requirements. All these variants lead
to equivalent forcing notions.
\begin{enumerate}
\item $M^x$ is (an ord-collapse of) an
{\em elementary} submodel of $H(\chi^*)$.
\\
This is equivalent, as conditions coming from
elementary submodels are dense in our $\mathbb{R}$, by
Fact~\ref{fact:esmV}.
\\
While this definition looks much simpler and therefore
nicer (we could replace ord-transitive models
by the better understood elementary models), it would not make
things easier and just ``hides'' the point of the
construction: For example, we use models $M^x$ that
are (an ord-collapse of) an
elementary submodel of~$H^{V'}(\chi^*)$ for some forcing extension
$V'$ of~$V$.
\item Require that ($M^x$ thinks that) the true domain of $\bar P^x$
is $\om2$.
\\
This is equivalent for the same reason as (1) (and this
requirement is compatible with (1)).
\\
This definition would allow to drop the ``trivial'' option
from the definition. The whole proof would still work with
minor modifications --- in particular, because of the following
fact:
\footnote{We are grateful to Stefan
Geschke and Andreas Blass for pointing out
this fact. The only references we are aware
of are \cite[proof of Lemma 2]{MR1179593} and \cite{MO84129}.}
\proofclaim{eq:blass}{The finite support iteration of
$\sigma$-centered forcing notions of
length $<(2^{\aleph_0})^+$ is again
$\sigma$-centered.}
We chose our version for two reasons: first, it seems
more flexible, and second, we were initially not aware of
\eqref{eq:blass}.
\item Alternatively, require that ($M^x$ thinks that) the true
domain of $\bar P^x$ is countable.
\\
Again, equivalence can be seen as in~(1), again (3) is compatible with~(1)
but obviously not with~(2).
\\
This requirement would not make the definition easier,
so there is no reason to adopt it. It would have the slight
inconvenience
that instead of using ord-collapses as in Fact~\ref{fact:esmV},
we would have to put another model on top to make the iteration
countable. Also, it would have the (purely aesthetic) disadvantage that
the generic iteration itself does not satisfy this
requirement.
\item \label{item:nonproper}
Also, we could have dropped the requirement that the iteration
is proper. It is never directly used, and ``densely''
$\bar P$ is proper anyway. (E.g., in Lemma~\ref{lem:6.1}(4)(a),
we would just construct $\bar P$ up to $\alpha$ to
be proper or even ccc, so that $X$ remains uncountable.)
\end{enumerate}
\subsection{Regarding \qemph{almost CS iterations and separative iterands}}
\label{sec:7b}
Recall that in Definition~\ref{partial_CS} we
required that each iterand $Q_\alpha$ in a partial CS iteration
is separative. This implies the property (actually: the three equivalent
properties) from Fact~\ref{fact:suitable.equivalent}.
Let us call this property \qemph{suitability} for now.
Suitability is a property of the limit
$P_\varepsilon$ of $\bar P$. Suitability always holds for finite support
iterations and for countable support iterations. However, if we do not assume
that each $Q_\alpha$ is separative, then suitability may fail
for partial CS iterations.
We could drop the separativity assumption, and instead
add suitability as an additional natural requirement to the definition of partial CS limit.
The disadvantage of this approach is that we would have to check in all
constructions of partial CS iterations that suitability is
indeed satisfied
(which we found to be straightforward but rather cumbersome, in
particular in the case of the almost CS iteration).
In contrast, the disadvantage of assuming that $Q_\alpha$ is separative
is minimal and purely cosmetic: It is well known that every quasiorder $Q$
can be made into a separative one
which is forcing equivalent to the original~$Q$ (e.g., by just
redefining the order to be $\leq^*_Q$).
\subsection{Regarding \qemph{preservation of random and quick sequences}}
Recall Definition~\ref{def:locally.random} of local preservation of random reals
and Lemma~\ref{lem:4.28}.
In some respect the dense sets $D_n$ are unnecessary. For
ultralaver forcing $\mathbb{L}_{\bar D}$, the notion of a ``quick'' sequence
refers to the sets $D_n$ of conditions with stem of length at
least~$n$.
We could define a new partial order on $\mathbb{L}_{\bar D}$ as
follows:
\[ q\le' p \ \Leftrightarrow \ (q=p) \ \text{or} \ (q\le p \text{ and the
stem of $q$ is strictly longer than the stem of~$p$}).
\]
Then $(\mathbb{L}_{\bar D}, \le)$ and
$(\mathbb{L}_{\bar D}, \le')$ are forcing equivalent, and any
$\le'$-interpretation of a new real will automatically be quick.
Note however that $(\mathbb{L}_{\bar D}, \le')$ is now not separative any more.
Therefore we chose not to take this approach, since losing separativity
causes technical inconvenience, as described in \ref{sec:7b}.
\end{document} | math | 258,693 |
\begin{document}
\title{Measurement incompatibility and Channel Steering}
\author{Manik Banik}
\email{[email protected]}
\affiliation{Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India.}
\author{Subhadipa Das}
\email{[email protected] }
\author{A. S. Majumdar}
\email{[email protected]}
\affiliation{S. N. Bose National Centre for Basic Sciences
Block JD, Sector III, Salt Lake, Kolkata 700098, India}
\begin{abstract}
Incompatible measurements in quantum theory always lead to
Einstein-Podolsky-Rosen (EPR)-Schr\"{o}dinger steering. Channel steering which
is a generalized notion of EPR-Schr\"{o}dinger steering, has been introduced
recently. Here we establish a connection between lack of joint
measurability and channel steering.
\pacs{03.65.Ud, 03.67.Ac}
\end{abstract}
\maketitle
One of the important features of quantum theory is that not all measurements
are compatible, i.e., they cannot be carried out simultaneously.
Such a counter intuitive aspect makes quantum physics distinct from classical
physics. This property is intimately connected to central tenets in the theory,
such as Heisenberg's uncertainty principle \cite{Heisenberg'27}, and Bohr's
complementarity principle \cite{Bohr'28}. In the case of von Neuman
measurements (projective measurements), compatibility is uniquely
captured by the notion of commutativity. Non-commuting observables in quantum
mechanics do not admit unambiguous joint measurement \cite{Varadarajan'85}.
With the introduction of generalized measurements, i.e., positive
operator-valued measures (POVMs) \cite{Kraus'83,Nielsen'10}, it was shown
that observables which do not admit perfect joint measurement, may allow
sufficiently fuzzy joint
measurement \cite{Busch'85}. Since for general measurements
there is no unique notion of compatibility, here we focus on the
well-defined criterion of joint measurability \cite{Busch'96}.
The optimal degree of unsharpness that guarantees joint measurement for all
possible pairs of dichotomic observables of a theory may be considered as the
degree of complementarity of the theory, which quantitatively binds the amount
of optimal violation of the Bell-Clauser-Horne-Shimony-Holt (CHSH) inequality
for any theory which satisfies
the no-signaling principle \cite{Banik'13}. It is also known
that any set of two incompatible POVMs with binary outcomes may lead to a
violation of the Bell-CHSH inequality \cite{Wolf'09}. However, this may not
be extended to the general case of an arbitrary number of POVMs with
arbitrarily many outcomes, since pairwise joint measurability does not imply
full joint measurability in general \cite{Kraus'83}. On the other hand, it has
been shown recently that measurement incompatibility in quantum theory always
leads to
EPR-Schr\"{o}dinger steering \cite{Quintino'2014}. It has been further shown
by one of the authors of this article that the connection between measurement incompatibility
and steering holds in a class of tensor product theories rather than just Hilbert space quantum mechanics \cite{Banik'2015}.
Steering \cite{EPR'35} refers to the scenario where one party,
usually called Alice, wishes to convince the other party, called Bob, that she
can steer the state at Bob’s side by making measurements on her side.
Steering has attracted much attention in recent years with the formulation
of its information theoretic perspective \cite{Wiseman'07}, as well
as the subsequent
development \cite{walborn'11} and
applications \cite{schn'13} of steering inequalities. Experimental
demonstrations of steering have followed using different settings and loophole
free arrangements \cite{saunders'10}. Practical
applications of steering have been suggested in one-sided device-independent
quantum key distribution \cite{Branciard'12} and sub-channel
discrimination \cite{Piani'14}. A resource theory of steering has also been
proposed \cite{Gallego'14}.
For the present purpose it is important to note that a set of POVMs in
finite dimensions is not jointly measurable if and only if the set can be used
to show the steerability of some quantum state \cite{Quintino'2014}.
Recently, the notion of steerability of quantum channels has been introduced
by Piani \cite{Piani'14(1)}, generalizing EPR-Schr\"{o}dinger steerability.
Consider that there is a quantum transformation (a quantum channel) from Charlie
to Bob, which may applied/used by Bob. Such transformation is in general
noisy with information leaking to the environment (Alice). The relevant
question here is the following: is Alice coherently connected to the
input-output of the channel, or can she be effectively considered just a
``classical bystander", with at most access to classical information about
the transformation that affected the input of the channel ? Steerability of a
channel has been defined as the possibility for Alice to prove to
Bob that she is not a ``classical bystander", i.e., she is
coherently connected with
the input-output of the channel from Charlie to Bob. The way for Alice to
prove so is by informing Bob of the choice of measurements performed by her
and their outcomes.
In this work we show that Alice is required to perform incompatible
measurements in order to demonstrate channel steering.
We begin by first briefly discussing the mathematical framework of POVMs
required for studying the notion of steerability for channels as introduced
by Piani \cite{Piani'14(1)} as a generalization of the EPR-Schr\"{o}dinger
steering scenario. A POVM consists of a collection of operators $\{M_{a|x}\}_a$
which are positive, $M_{a|x}\ge 0~\forall~a$, and sum up to the identity,
$\sum_aM_{a|x}=\mathbf{1}$. Here $a$ denotes measurement outcome and $x$
denotes measurement choice. A POVM may be realized physically by first
letting the physical system interact with an auxiliary system and then
measuring an ordinary observable on the auxiliary system. Let
$\{M_{\vec{a}}\}$ be a set of measurements with outcome $\vec{a}=[a_{x=1},a_{x=2}, . . . ,a_{x=m}]$, where $a_x\in\{0,1,..,n\}$ is the outcome of the $x^{th}$
measurement.
A set of $m$
POVMs $\{M_{a|x}\}_a$ is called jointly measurable if
\begin{equation}
M_{\vec{a}}\ge 0,~~\sum_{\vec{a}}M_{\vec{a}}=\mathbf{1},~~\sum_{\vec{a}\backslash a_x}M_{\vec{a}}=M_{a|x}~\forall~x,
\label{joint-meas}
\end{equation}
where $\vec{a}\backslash a_x$ stands for the elements of $\vec{a}$ except for
$a_x$. All POVM elements $M_{a|x}$ are recovered as marginals of the
observable $M_{\vec{a}}$.
The EPR-Schr\"{o}dinger steering experiment can be completely characterized by
specifying an `assemblage' $\{\sigma_{a|x}\}_{a,x}$, the set of sub-normalized
states which Alice steers Bob into, given her choice of measurement $x$ and
outcome $a$. She can choose to perform one measurement from a set of $m$
choices, each of which has $n$ possible outcomes. The assemblage encodes
the conditional probability distribution of her outcomes given her inputs
$p(a|x) = \mbox{Tr}(\sigma_{a|x})$, as well as the conditional states prepared
for Bob given Alice's input and outcome $\hat{\sigma}_{a|x} = \sigma_{a|x}/p(a|x)$. All valid assemblages satisfy the consistency requirements, $\sum_a\sigma_{a|x}=\sum_a\sigma_{a|x'},~\forall~x\ne x'$ and $\mbox{Tr}(\sum_a\sigma_{a|x})=1$.
This encodes the fact that Alice cannot signal to Bob, and that without any
knowledge about Alice, Bob still holds a valid quantum state. We denote this
set of valid assemblages as $\Sigma^S$.
Assemblages which can be created via classical strategies (without using
entanglement) are called unsteerable and denoted as $\Sigma^{US}$. Unsteerable
assemblages can be expressed in the form
\begin{eqnarray}
\sigma_{a|x}=\sum_{\lambda}p(a|x,\lambda)\sigma_{\lambda},~~\forall~a,x
\end{eqnarray}
such that $\mbox{Tr}(\sum_{\lambda}\sigma_{\lambda}=1),~~\sigma_{\lambda}\ge 0~\forall\lambda$,
where $\lambda$ is a (classical) random variable held by Alice, $p(a|x,\lambda)$
are conditional probability distributions for Alice, and $\sigma_{\lambda}$ are
the states held by Bob. Collection of unsteerable assemblages form a convex set
\cite{Pusey'13}. Any assemblage that cannot be written in the
above form is called steerable. For such assemblages there is no classical
explanation as to how the different conditional states held by Bob could
be prepared by Alice.
EPR and Schr\"{o}dinger \cite{EPR'35} observed that by
performing measurements on her part of entangled quantum state shared with Bob,
Alice can remotely prepare steerable assemblages on Bob's side. Let us denote
the measurement assemblage on Alice's side as $\{M^A_{a|x}\}_{a,x}$, where
$M^A_{a|x}\ge 0~\forall~a,x$ and $\sum_aM^A_{a|x}=\mathbf{1}~\forall~x$.
This measurement assemblage whenever performed on Alice's part of a bipartite
quantum state $\rho^{AB}$ shared between Alice and Bob, gives rise the to
the sub-states assemblage $\{\sigma_{a|x}\}_{a,x}$ with $\sigma_{a|x}=\mbox{Tr}_A(M^A_{a|x}\otimes\mathbf{1}^B\rho^{AB})$ and $\sum_a\sigma_{a|x}=\mbox{Tr}_A(\rho^{AB})$ on Bob's side. Though Schr\"{o}dinger
pointed out steerability of bipartite
pure entangled states in the very early days of quantum theory, it took a long
time to establish that there exist mixed entangled states which exhibit this
property \cite{Wiseman'07}.
A quantum channel $\Lambda^{S\rightarrow S'}:\mathcal{D}(\mathcal{H}_S)\longrightarrow\mathcal{D}(\mathcal{H}_{S'})$ is a completely-positive trace-preserving
linear map \cite{Wilde'13}, where $S$ and $S'$,
respectively, are the input and output quantum systems of the channel,
and $\mathcal{H}_*$ denotes the Hilbert space associated with the system.
$\mathcal{D}(\mathcal{H}_*)$ denotes the set of density operators acting
on $\mathcal{H}_*$. We will denote a channel simply by $\Lambda$, whenever it
is not required to specify the input-output system.
The collection of completely-positive maps $\Lambda_a$ is called an instrument
$\mathcal{I}$, if $\sum_a\Lambda_a$ is a channel. In such a case, each
$\Lambda_a$ is a subchannel, i.e., a completely positive trace-non-increasing
linear map. A channel assemblage $\mathcal{CA}:= \{\mathcal{I}_x\}_x=\{\Lambda_{a|x}\}_{a,x}$ for a channel $\Lambda$ is a collection of instruments $\mathcal{I}_x$ for $\Lambda$, i.e., $\sum_a\Lambda_{a|x}=\Lambda$ for all $x$.
Consider a noisy quantum channel from $C$ to $B$, `leaking' information to
the environment. Suppose that Alice has access to some part $A$ of said
environment. The situation can be modeled by quantum broadcast channels with
one sender and two receivers \cite{Yard'11}.
This broadcast channel $\Lambda^{C\rightarrow AB}$ is a channel extension of the
given quantum channel
$\Lambda^{C\rightarrow B}$.
A channel extension $\Lambda^{C\rightarrow AB}$ of a channel
$\Lambda^{C\rightarrow B}$ is called an incoherent extension if there exists an
instrument $\{\Lambda^{C\rightarrow B}_{\lambda}\}_{\lambda}$ with $\sum_{\lambda}\Lambda^{C\rightarrow B}_{\lambda}=\Lambda^{C\rightarrow B}$, and normalized (unit trace)
quantum states $\{\sigma^A_{\lambda}\}$, such that
\begin{equation}
\Lambda^{C\rightarrow AB}=\sum_{\lambda}\Lambda^{C\rightarrow B}_{\lambda}\otimes\sigma^A_{\lambda}.
\end{equation}
A channel extension is called a coherent extension if it is not incoherent.
We now address the issue as to under
what circumstances is Alice coherently connected to the input-output of the
channel. In such a case the map from $C$ to $AB$ is a coherent extension of
the channel from $C$ to $B$. Steerability of a channel extension is defined
as the possibility for Alice to prove to Bob that she is not a classical
bystander, or in other words that the leakage of information from $C$ to $A$
cannot be described in terms of a classical channel.
As in the case of EPR-Schr\"{o}dinger steering, here Alice is untrusted in the
sense that we have no knowledge of either the state that Alice holds, or the
measurements she performs.
Note that one does not need to rely on the details/implementation of Alice's
measurements, i.e., the situation is device-independent on Alice's side. Thus,
the verification procedure does not require Bob to trust Alice's measurement
devices. Every choice of measurement by Alice corresponds to a different
decomposition into subchannels of the channel used by Bob. A channel
assemblage $\mathcal{CA}=\{\Lambda_{a|x}\}_{a,x}$ is unsteerable if
there exists an instrument $\{\Lambda_{\lambda}\}_{\lambda}$, and conditional
probability distributions
$p(a|x,\lambda)$, such that
\begin{equation}
\Lambda_{a|x}=\sum_{\lambda}p(a|x,\lambda)\Lambda_{\lambda},~~~\forall~a,x.
\end{equation}
An unsteerable channel assemblage is denoted as $\Lambda^{US}$. A channel
assemblage is steerable if it cannot be expressed in the above form.
In the following we show that Alice is able to produce a steerable
channel assemblage if and only if the measurements she performs are
incompatible.
{\bf Theorem:} The channel assemblage $\{\Lambda^{C\rightarrow B}_{a|x}\}_{a,x}$ for a channel $\Lambda=\Lambda^{C\rightarrow B}$, with $\Lambda^{C\rightarrow B}_{a|x}=\mbox{Tr}_A(M^A_{a|x}\Lambda^{C\rightarrow AB}[*])$, is unsteerable for any channel extension $\Lambda^{C\rightarrow AB}$ of $\Lambda^{C\rightarrow B}$ if and only if the set
of POVMs $\{M^A_{a|x}\}_x$ applied by Alice on $A$ is jointly measurable.
{\bf Proof}:
We first prove that joint measurability implies no channel steering.
Let, $\{M^A_{a|x}\}_{a,x}$ be jointly measurable, with the joint measurement
operator denoted as $M^A_{\vec{a}}$, i.e.,
\begin{equation*}
M^A_{\vec{a}}\ge 0,~~\sum_{\vec{a}}M^A_{\vec{a}}=\mathbf{1},~~\sum_{\vec{a}\backslash a_x}M^A_{\vec{a}}=M^A_{a|x},
\end{equation*}
where $\vec{a}=[a_{x=1},a_{x=2},...,a_{x-m}]$.
Our aim is to show that the channel assemblage $\{\Lambda^{C\rightarrow B}_{a|x}\}_{a,x}$ resulting from the measurement assemblage $\{M^A_{a|x}\}_{a,x}$ on Alice's
side for any channel extension (incoherent as well as coherent)
$\Lambda^{C\rightarrow AB}$ of $\Lambda^{C\rightarrow B}$ is unsteerable, or in other
words, there exists an instrument $\{\Lambda^{C\rightarrow B}_{\lambda}\}_{\lambda}$,
with $\sum_{\lambda}\Lambda^{C\rightarrow B}_{\lambda}=\Lambda^{C\rightarrow B}$, and a
conditional probability distribution $p(a|x,\lambda)$, such that
\begin{equation}
\Lambda^{C\rightarrow B}_{a|x}=\sum_{\lambda}p(a|x,\lambda)\Lambda^{C\rightarrow B}_{\lambda}, ~~~\forall~~a,x.
\end{equation}
Let, $\lambda=\vec{a}$,
$\Lambda^{C\rightarrow B}_{\lambda}=\Lambda^{C\rightarrow B}_{\vec{a}}=\mbox{Tr}_A(M^A_{\vec{a}}\Lambda^{C\rightarrow AB}[*])$ and
$p(a|x,\lambda)=p(a|x,\vec{a})=\delta_{a,a_x}$.
Clearly we have,
\begin{eqnarray}
\sum_{\lambda}p(a|x,\lambda)\Lambda^{C\rightarrow B}_{\lambda}&=& \sum_{\vec{a}}p(a|x,\vec{a})\Lambda^{C\rightarrow B}_{\vec{a}}\nonumber\\
&=& \sum_{\vec{a}}\delta_{a,a_x}\mbox{Tr}_A(M^A_{\vec{a}}\Lambda^{C\rightarrow AB}[*])\nonumber\\
&=& \mbox{Tr}_A(\sum_{\vec{a}}\delta_{a,a_x}M^A_{\vec{a}}\Lambda^{C\rightarrow AB}[*])\nonumber\\
&=& \mbox{Tr}_A(M^A_{a|x}\Lambda^{C\rightarrow AB}[*])\nonumber\\
&=& \Lambda^{C\rightarrow B}_{a|x}=(\Lambda^{C\rightarrow B})^{US}.
\end{eqnarray}
In Ref.\cite{Piani'14(1)} it was shown that every unsteerable channel
assemblage can be thought as arising from an incoherent channel extension.
We can therefore conclude that by performing compatible measurements Alice
cannot convince Bob that she is coherently connected with the input-output of
the noisy channel applied by Bob.
We now prove the converse of the above result that
if the channel assemblage $\{\Lambda^{C\rightarrow B}_{a|x}\}_{a,x}$ for a channel $\Lambda=\Lambda^{C\rightarrow B}$ with $\Lambda^{C\rightarrow B}_{a|x}=\mbox{Tr}_A(M^A_{a|x}\Lambda^{C\rightarrow AB}[*])$ is unsteerable for any channel extension, then
the measurement assemblage $\{M^A_{a|x}\}_{a,x}$ applied by Alice is jointly
measurable. In order to do so we use the Choi-Jamio\l{}kowski representation
\cite{Jamio'72} of channels. The Choi-Jamio\l{}kowski isomorphic
operator of the channel $\Lambda^{C\rightarrow AB}$ is given by
\begin{equation}
J_{C'AB}(\Lambda^{C\rightarrow AB}):=\Lambda^{C\rightarrow AB}[\psi^{CC'}_+],
\end{equation}
where $\psi^{CC'}_+$ is the density matrix corresponding to a fixed maximally
entangled state of systems $C$ and $C'$, with $C'$ a copy of $C$.
The measurement assemblage $\{M^A_{a|x}\}_{a,x}$ performed by Alice on her part
of the extended channel $\Lambda^{C\rightarrow AB}$ results in the channel
assemblage $\{\Lambda^{C\rightarrow B}_{a|x}\}_{a,x}$ for the channel $\Lambda^{C\rightarrow B}$, where $\Lambda^{C\rightarrow B}_{a|x}=\mbox{Tr}_A(M^A_{a|x}\Lambda^{C\rightarrow AB}[*])$ with the Choi-Jamio\l{}kowski operator $J_{C'B}(\Lambda^{C\rightarrow B}_{a|x})$. In \cite{Piani'14(1)}, it has also been proved that the channel
extension of a channel is steerable if and only if its Choi-Jamio\l{}kowski
operator is steerable.
Now, if the channel assemblage $\{\Lambda^{C\rightarrow B}_{a|x}\}_{a,x}$ is
unsteerable, there exists an instrument $\{\Lambda^{C\rightarrow B}_{\lambda}\}_{\lambda}$, with $\sum_{\lambda}\Lambda^{C\rightarrow B}_{\lambda}=\Lambda^{C\rightarrow B}$,
and conditional probability distribution $p(a|x,\lambda)$, such that
$\Lambda^{C\rightarrow B}_{a|x}=\sum_{\lambda}p(a|x,\lambda)\Lambda^{C\rightarrow B}_{\lambda}$ for all $a,x$. Clearly, the Choi-Jamio\l{}kowski operator assemblage
$\{J_{C'B}(\Lambda^{C\rightarrow B}_{a|x})\}_{a,x}$ of the Choi-Jamio\l{}kowski
operator $J_{C'B}(\Lambda^{C\rightarrow B})$ is also unsteerable, i.e.,
\begin{equation}
J_{C'B}(\Lambda^{C\rightarrow B}_{a|x})=\sum_{\lambda}p(a|x,\lambda)J_{C'B}(\Lambda^{C\rightarrow B}_{\lambda}),~~~\forall~a,x.
\end{equation}
It now follows from the result of Refs.\cite{Quintino'2014} that
one can construct joint measurements for the measurement assemblage $\{M^A_{a|x}\}_{a,x}$. $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\blacksquare$
To, summarize, in the present work we have studied the link between
lack of joint measurability and channel steering.
An important connection was established earlier between
EPR-Schr\"{o}dinger steering and the joint measurement of quantum observables.
It was shown in Refs.\cite{Quintino'2014} that incompatible
measurements are needed to be performed for demonstrating EPR-Schr\"{o}dinger
steering. A generalization of the notion of EPR-Schr\"{o}dinger steering
has been introduced recently through the concept of channel steering
\cite{Piani'14(1)}. Here one considers a noisy quantum transformation
or channel between two parties (say, Charlie and Bob), leaking some
information to the environment which is accessible to another party (say,
Alice). The task of channel steering is for Alice to convince Bob that she is
coherently connected to the input-output of the channel. In this work we
have shown that Alice needs to perform incompatible measurements to
succeed in her aim. By performing measurements that are
jointly measurable Alice succeeds to produce only unsteerable channel
assemblages of the noisy channel from Charlie to Bob.
Our result establishes that non-joint measurability and channel steering
imply each other. The connection between the two may have implications
\cite{pussey'15} for a resource theory of measurement incompatibility.
{\emph Acknowledgments:} A.S.M. acknowledges support from the project SR/S2/LOP-08/2013 of DST, India.
\end{document} | math | 19,612 |
\begin{document}
\begin{center}
{ \LARGE
On the theory of relaxation in nonlinear elasticity
with constraints on the determinant
\\[5mm]}
{\today}\\[5mm]
Sergio Conti$^{1}$ and Georg Dolzmann$^{2}$\\[2mm]
{\em $^1$ Institut f\"ur Angewandte Mathematik,
Universit\"at Bonn\\ 53115 Bonn, Germany }\\
{\em $^{2}$ Fakult\"at f\"ur Mathematik, Universit\"at Regensburg,\\
93040
Regensburg, Germany}
\\[3mm]
\begin{minipage}[c]{0.8\textwidth}\small
We consider vectorial variational problems in nonlinear elasticity of the
form $\Enog[u]=\int W(Du)dx$, where
$W$ is continuous on matrices with positive determinant
and diverges to infinity along sequences of matrices whose determinant is positive and
tends to zero.
We show that, under suitable growth assumptions, the
functional $\int W^\qc(Du)dx$
is an upper bound on the relaxation of
$\Enog$, and coincides with the relaxation if the quasiconvex envelope $W^\qc$ of
$W$ is polyconvex and has $p$-growth from below with $p\ge n$. This includes several
physically
relevant examples. We also show how a constraint of incompressibility can be
incorporated in our results.
\end{minipage}
\end{center}
\section{Introduction}
Starting with the work of Morrey \cite{Morrey1952}, the concept of
quasiconvexity has been
fundamental in the study of the relaxation of vectorial problems in the
calculus of variations,
see for example \cite{Dacorogna1989,MuellerLectureNotes,RoubicekBook1997}. In particular,
if $W:\R^{n\times
m}\to\R$ is continuous and has $p$-growth then the relaxation of the
functional $\Enog: W^{1,p}(\Omega;\R^m)\to\R$,
\begin{equation}\label{eqintW}
\Enog[u]=\int_\Omega W(Du) dx
\end{equation}
is given by
\begin{equation}\label{eqintWqc}
\Enogrel[u]=\int_\Omega W^\qc(Du) dx\,,
\end{equation}
where $W^\qc$ is the quasiconvex envelope of $W$, see
\cite{Morrey1952,AcerbiFusco84,Dacorogna1989}. Here $\Omega\subset\R^n$ is a bounded Lipschitz set,
$W^\qc$ is defined by~(\ref{eqdefwqcdetp}) below.
The computation of $W^\qc$ is in general difficult, but it was performed in a
number of special cases with high symmetry, see for example
\cite{DeSimoneDolzmannARMA2002,ContiTheil2005,Silhavy2007,ContiDolzmanntwowell}.
The key strategy is to construct specific test functions using
lamination and rank-one convexity to prove an upper bound, and then to show
that the resulting expression is polyconvex, which delivers the lower bound.
One of the main applications of the vectorial calculus of variations is
nonlinear elasticity with $m=n$. The physical constraint of non-interpenetration of
matter leads naturally to the requirement of injectivity of the
deformation $u$, a complex nonlocal condition which is often replaced by the
simpler condition that $\det Du>0$ almost everywhere. Correspondingly, one
assumes that the energy density $W$ diverges when the determinant of the
argument is positive and tends to zero. Such energy
densities are not continuous on $\R^{n\times n}$ and do not have $p$-growth
from above for any $p$, hence the general relaxation theorem is not
applicable.
Starting with the work of Ball
\cite{Ball1977} a large body of work developed with the aim of proving
existence of minimizers for variational problems with the constraint $\det
Du>0$, mainly building upon the concept of polyconvexity. In contrast,
to the best of our knowledge, there is no physically-relevant functional
incorporating the nonlinear constraint $\det Du>0$ for which a nontrivial
relaxation is known.
The significance of the constraint $\det Du>0$ depends dramatically on the
growth
exponent $p$ of the energy, and two main regimes emerge.
If $p\ge n$ the deformation $u$ is necessarily continuous. This follows from
the Sobolev embedding theorem for
$p>n$ and in the case $p=n$ from the work of
Vodop$'$janov and Gol$'$d{\v{s}}te{\u\i}n
\cite{VodopjanovGoldstein1976}, see also \cite{Sverak1988}. Further, if a
sequence $u_j$ converges weakly to some $u$ in $W^{1,n}$ and $\det Du_j>0$
almost everywhere, then necessarily $\det Du\ge 0$ almost everywhere.
For $p>n$ this follows directly from the properties of the determinant,
which in particular give $\det Du_j\weakto \det Du$ in $L^1$
\cite[Cor. 6.2.2]{Ball1977}, and still
holds for $p=n$, see \cite[Th. 4.1(ii)]{BallMurat1984}. A related treatment
with the distributional determinant instead of the pointwise determinant is
still possible if $p>n-1$ and a generalized invertibility condition is used
instead of $\det Du>0$ \cite{MuellerSpector1995},
see also \cite{MuellerSivaloganathanSpector1999,ContiDeLellis2003,FonsecaLeoniMaly2005,HenaoMoracorral2010} for subsequent developments.
The case $p<n$ is substantially different, since the deformations are
not continuous and can develop holes, as was first shown by Ball
\cite{Ball1982}.
Correspondingly, the constraint of having positive determinant
does not pass to the limit and the relaxed problem has a substantially
different structure, see for example
\cite{BallMurat1984,KoumatosRindlerWiedemann1,KoumatosRindlerWiedemann2}
for further developments. Relaxation in a related situation in
which the constraints are lost after rank-one convexification was
discussed in \cite{Benbelgacem2000}. We shall not discuss these cases
further here.
The proof of the classical relaxation theorem for continuous integrands is
based on a truncation procedure in
which one replaces a sequence $u_j\in W^{1,p}$ which converges weakly to an
affine function by a sequence with the same weak limit, the same energy, and
which is affine on the boundary. It is currently unknown if a similar
construction can be done if a constraint on the determinant has to be
preserved. In two spatial dimensions and under the
assumption that $u$ is bilipschitz
a solution was given in
\cite{BenesovaKruzik}
building upon an involved
construction of bilipschitz extensions by Daneri and Pratelli
\cite{DaneriPratelli1}.
The approximation of Hölder-continuous homeomorphisms was obtained in
\cite{BellidoMoracorral2011}. The situation with Sobolev functions is
substantially more complex, and was up to now only solved in the case
$p=n=2$, see \cite{IwaniecKovalevOnninen2011}.
A related problematic arises in the relaxation of problems with mixed
growth, see for example \cite{Kristensenlsc}.
In this paper we prove that the functional (\ref{eqintWqc}) gives an upper
bound on the relaxation of (\ref{eqintW}) for a class of energy densities
which are infinite on matrices $F$ with $\det F\le 0$ and have $p$-growth for
some $p\ge 1$ on the set $\{\det F>0\}$, see Theorem \ref{theoorientation} below.
If $W^\qc$
is polyconvex and $p\ge n$ then
(\ref{eqintWqc}) coincides with the relaxation of
(\ref{eqintW}), see Theorem \ref{theocorollaryorientat} below. The
growth assumption can be somewhat relaxed if a suitable integrability of the cofactor is assumed, see Remark \ref{remarkgener}. Since all known explicit quasiconvex
envelopes $W^\qc$ are
quasiconvex, our result fully characterizes the relaxation in all cases
where
$W^\qc$ has been computed and the pointwise determinant constraint survives the
relaxation.
We also show that our results can be generalized to problems where $\det Du=1$
almost everywhere, see Section \ref{subsecapplicincompr} below.
Notation: We denote
by $\R^{n\times n}_+=\{F\in \R^{n\times n}: \det F>0\}$ the set of orientation-preserving matrices, by $B(r,x_0)$ the open ball in $\R^n$ and set
$B_r=B(r,0)$. Finally $\fint_E f\, dx$ denotes the mean value of $f$ on $E$.
We define
the quasiconvex envelope $W^\qc:\R^{n\times n}\to[0,\infty]$ of a Borel-measurable function
$W:\R^{n\times n}\to[0,\infty]$ by
\begin{equation}\label{eqdefwqcdetp}
W^\qc(F)=\inf\{\fint_{B_1} W(D\varphi)\, dx: \varphi\in
W^{1,\infty}(B_1;\R^n), \varphi(x)=Fx \text{ for } x\in \partial B_1 \}\,.
\end{equation}
This is not necessarily the same as the largest finite-valued quasiconvex function below $W$, see Remark \ref{remarkwqc} below.
We say that a function $f:\R^{n\times n}\to [0,\infty]$ is polyconvex if
there is a lower semicontinuous and convex function $g:\R^{\tau(n)}\to[0,\infty]$ such
that $f(F)=g(M(F))$, where $M(F)$ denotes all minors of $F$
\cite{Ball1977,Dacorogna1989,MuellerLectureNotes}.
In particular, if $n=2$ then
$M(F)=(F,\det F)$, if $n=3$ then $M(F)=(F,\cof F, \det F)$.
The requirement of lower semicontinuity of $g$ is often not included
in the definition but instead enforced through appropriate growth conditions.
For finite-valued functions this makes no difference, but for
extended-valued we believe the present one to be the definition more naturally
related to lower semicontinuity, as the example
$g(\det F)=0$ if $\det F>0$, $g(\det F)=\infty$ otherwise, with the
sequence $u(x)=x/j$ shows, see also the discussion in \cite{Mielke2005}.
\section{Main results}
\subsection{Relaxation of orientation-preserving models}
\label{secapplicaorientpres}
The main result of the paper is a relaxation theorem
for coercive variational problems
in nonlinear elasticity incorporating a constraint on the determinant, see Theorem \ref{theocorollaryorientat} below.
Our key new contribution is a construction for the upper bound which preserves the positive-determinant
constraint and leads to the following statement.
\begin{theorem}\label{theoorientation}
Let $W\in C^0(\R^{n\times n}_+,[0,\infty))$ obey
\begin{equation}\label{eqgrowthwpd}
\frac1c |F|^p +\frac1c\theta(\det F)-c\le W(F)\le c|F|^p+c\theta(\det F)+c
\end{equation}
for some $p\ge 1$ and $c>0$, where
$\theta:(0,\infty)\to[0,\infty)$ is convex and satisfies
\begin{equation}\label{eqasstheta}
\theta(xy)\le c(1+\theta(x))(1+\theta(y)) \text{ for all
$x,y\in(0,\infty)$},
\end{equation}
and extend $W$ to $\R^{n\times n}$ by $W(F)=\infty$ if $\det F\le 0$.
Let $W^\qc$ be defined
as in (\ref{eqdefwqcdetp}), $\Omega\subset\R^n$ open, bounded and
Lipschitz. For any $u\in W^{1,p}(\Omega;\R^n)$ there is a sequence $u_j\in
W^{1,p}(\Omega;\R^n)$
which converges weakly to $u$ such that $u_j-u\in W^{1,p}_0$ for all $j$ and
\begin{equation*}
\limsup_{j\to\infty} \int_\Omega W(Du_j)dx\le \int_\Omega W^\qc(Du)dx\,.
\end{equation*}
\end{theorem}
\begin{proof}
If $\det Du>0$ almost everywhere the statement follows from Lemma \ref{lemmaconstr2} in Section \ref{secorientationpres} below.
From the definition one immediately obtains $W^\qc(F)=\infty$ if $\det F\le 0$, therefore in the
other case a constant sequence will do.
\end{proof}
If the coercivity exponent $p$ is at least $n$,
then the determinant is an $L^1$ function and weakly continuous in compact
subsets \cite{Mueller1990}, therefore the constraint on the
determinant passes to the limit.
Complementing Theorem~\ref{theoorientation}
with existing compactness and lower semicontinuity results,
based on the concept of polyconvexity \cite{Ball1977},
leads to a full relaxation and existence statement.
\begin{theorem}\label{theocorollaryorientat}
Let $W\in C^0(\R^{n\times n}_+,[0,\infty))$ obey
(\ref{eqgrowthwpd}--\ref{eqasstheta})
with $p\ge n$ and
\begin{equation*}
\lim_{t\to0}\theta(t)=\infty\,,
\end{equation*}
and extend $W$ by $W(F)=\infty$ to the set $\{\det F\le 0\}$.
Let $\Omega\subset\R^n$ be an open, bounded,
Lipschitz, connected set,
\begin{align*}
X=\{u\in W^{1,p}(\Omega;\R^n): \det Du>0 \text{ a.e. }\}\,,
\end{align*}
and $f\in C^0(\R^n)$ with $|f(t)|\le c (1+|t|^q)$ for some $q\in[0,p)$.
We define $W^\qc$ as in (\ref{eqdefwqcdetp})
and the functionals
$\Emitg, \Emitgrel:L^1(\Omega;\R^n)\to \R\cup\{\infty\}$ by
\begin{equation*}
\Emitg[u]=\int_\Omega \left(W(Du) + f(u)\right) dx \text{ and }
\Emitgrel[u]=\int_\Omega \left(W^\qc(Du) + f(u)\right) dx
\end{equation*}
for $u\in X$, and $\Emitg=\Emitgrel=\infty$ on $L^1\setminus X$.
Finally assume that $W^\qc = W^\pc$.
Then the following assertions hold:
\begin{enumerate}
\item
$\Emitgrel$ is the relaxation of $\Emitg$ with respect to strong $L^1$ convergence, in the sense that
\begin{equation*}
\Emitgrel[u]=\inf\{\liminf_{j\to\infty} \Emitg[u_j]: u_j \in L^1(\Omega;\R^n), u_j\to u \text{ in } L^1\}\,.
\end{equation*}
\item The same holds if,
for any given relatively open set
$\Gamma_D\subset\partial\Omega$ and $u_0\in X$,
the functionals $\Emitg$ and $\Emitgrel$ are set to be $\infty$ outside
\begin{equation*}
\widetilde X=X\cap \{u=u_0\text{ on }\Gamma_D\}\,.
\end{equation*}
\item The functional
$\Emitgrel$ has a minimizer in the space $\widetilde X$.
\end{enumerate}
\end{theorem}
\begin{proof}
By Theorem \ref{theoorientation} for any $u\in X$ there is $(u_j)_{j\in \N}\subset X$ with $u_j=u$ on $\partial\Omega$, $u_j\to u$ for $j\to\infty$ in $L^p$,
and $\limsup_{j\to\infty} \Emitg[u_j]\le \Emitgrel[u]$. This proves the upper bound in both cases.
Let now $(u_j)_{j\in \N}$ be a sequence in $X$ with $\Emitgrel[u_j]\le C<\infty$ for all
$j$. From~(\ref{eqgrowthwpd}) one immediately obtains $W^\qc(F)\ge |F|^p/c-c$.
The growth condition on $f$ ensures, since $q<p$, that $\int_\Omega W^\qc(Du)dx\le
C'<\infty$ for all $j$.
Taking a subsequence we can assume $u_j\weakto u$ in $W^{1,p}$ for some
$u\in W^{1,p}(\Omega;\R^n)$.
By the continuity of the trace, if $u_j=u_0$ on $\Gamma_D$ for all $j$ then $u=u_0$ on $\Gamma_D$.
In order to show that $u\in X$ it only remains to prove the condition on the determinant.
By \cite{Mueller1990} we have $\det Du_j\weakto \det Du$ in $L^1(K)$ for all $K\subset\subset\Omega$. Since $\theta$ diverges at 0 the weak limit is positive almost everywhere and $u\in X$.
To prove lower semicontinuity we
let $W^\qc(F)=g(M(F))$, with $g$ convex and lower semicontinuous, and
fix a compact set $K\subset\Omega$. As discussed above,
we have $\det Du^j\weakto\det Du$ in
$L^1(K)$ for all $K\subset\subset\Omega$. The other minors converge also weakly in $L^1$, since
$u_j\weakto u$ in $W^{1,p}$ with $p>n-1$. Therefore $M(Du^j)\weakto M(Du)$ in $L^1(K)$ and using
Jensen's inequality and the
convexity of $g$ we obtain
\begin{alignat*}1
\int_K W^\qc(Du)\, dx&=\int_K g(M(Du))\, dx\le\liminf_{j\to\infty} \int_K g(M(Du_j))\, dx
\\& \le \liminf_{j\to\infty} \int_\Omega W^\qc(Du_j)\, dx\,.
\end{alignat*}
The term $\int_\Omega f(u) dx$ is continuous.
Taking the supremum over all compact subsets $K\subset\Omega$ gives $\Emitgrel[u]\le \liminf \Emitgrel[u_j]\le \liminf \Emitg[u_j]$ and
concludes the proof.
\end{proof}
\begin{remark}\label{remarkgener}
\begin{enumerate}
\item The result of Theorem \ref{theoorientation} can be extended to functions $W$ which obey
\begin{equation}\label{eqwprod1}
W(FG)\le c (1+W(F))(1+W(G))
\end{equation}
and
\begin{equation}\label{eqwprod2}
\frac1c |F|-c\le W(F)\le c W^\qc(F)+c
\end{equation}
instead of (\ref{eqgrowthwpd}) and (\ref{eqasstheta}). We discuss in Appendix \ref{sectmultipl} the required modifications to the proof.
\item Using this generalization, one can extend Theorem \ref{theocorollaryorientat} to a situation in which $W$ obeys (\ref{eqwprod1}), (\ref{eqwprod2}) and the growth condition
\begin{equation*}
\frac1c |F|^p +\frac1c|\cof F|^q+\frac1c\theta(\det F)-c\le W(F)\,,
\end{equation*}
corresponding to the spaces $\mathcal{A}_{p,q}$ introduced by Ball \cite{Ball1977}, see also \cite{Sverak1988}.
\item A different picture arises if one instead uses a constraint on
the pointwise determinant, with the material becoming substantially softer,
see \cite{Ball1982,BallMurat1984} and \cite{KoumatosRindlerWiedemann1,KoumatosRindlerWiedemann2,KoumatosRindlerWiedemann3}.
\end{enumerate}
\end{remark}
\subsection{Relaxation of incompressible models}
\label{subsecapplicincompr}
We deal with integrands which are defined on the set of volume-preserving
matrices
$\Sigma=\{F\in \R^{n\times n}: \det F=1\}$
and which have $p$-growth. In this framework, we prove the following result.
\begin{theorem}\label{theovolume}
Let $W\in C^0(\Sigma,[0,\infty))$ obey
\begin{equation}\label{eqgrowthwincompr}
\frac1c |F|^p-c\le W(F)\le c|F|^p+c
\end{equation}
for some $p\ge 1$ and $c>0$, and set $W(F)=\infty$ if $\det F\ne 1$. Let $W^\qc$ be defined as in (\ref{eqdefwqcdetp}), $\Omega\subset\R^n$ open, bounded and
Lipschitz. For any $u\in W^{1,p}(\Omega;\R^n)$ there is a sequence $u_j\in
W^{1,p}(\Omega;\R^n)$
which converges weakly to $u$ such that $u_j-u\in W^{1,p}_0$ and
\begin{equation*}
\limsup_{j\to\infty} \int_\Omega W(Du_j)dx\le \int_\Omega W^\qc(Du_j)dx\,.
\end{equation*}
\end{theorem}
\begin{proof}
The statement follows from Lemma \ref{lemmarecoveryincompr} below.
\end{proof}
Also in this case, if coercivity is sufficient a full relaxation statement follows.
\begin{theorem}
Let $W\in C^0(\Sigma,[0,\infty))$ obey
(\ref{eqgrowthwincompr}) for some $p\ge n$, let $\Omega\subset\R^n$ be open bounded,
Lipschitz, connected,
\begin{align*}
X=\{u\in W^{1,p}(\Omega;\R^n): \det Du=1 \text{ a.e.}\}\,.
\end{align*}
We set $W(F)=\infty$ if $\det F\ne 1$, define $W^\qc$ as in (\ref{eqdefwqcdetp})
and, for $f\in C^0(\R^n)$ with $|f(t)|\le c (1+|t|^q)$ for some $q<p$,
\begin{equation*}
\Emitg[u]=\int_\Omega \left(W(Du) + f(u)\right) dx \text{ and }
\Emitgrel[u]=\int_\Omega \left(W^\qc(Du) + f(u)\right) dx
\end{equation*}
for $u\in X$, and $\Emitg=\Emitgrel=\infty$ on $L^1\setminus X$.
Finally assume that $W^\qc=W^\pc$.
Then the following assertions hold.
\begin{enumerate}
\item
$\Emitgrel$ is the relaxation of $\Emitg$ with respect to strong $L^1$ convergence, in the sense that
\begin{equation*}
\Emitgrel[u]=\inf\{\liminf_{j\to\infty} \Emitg[u_j]: u_j \in L^1(\Omega;\R^n), u_j\to u \text{ in } L^1\}\,.
\end{equation*}
\item The same holds if,
for any given relatively open set
$\Gamma_D\subset\partial\Omega$ and $u_0\in X$,
the functionals $\Emitg$ and $\Emitgrel$ are set to be $\infty$ outside
\begin{equation*}
\widetilde X=X\cap \{u=u_0\text{ on }\Gamma_D\}\,.
\end{equation*}
\item The functional
$\Emitgrel$ has a minimizer in the space $\widetilde X$.
\end{enumerate}
\end{theorem}
\begin{proof}
The proof is analogous to the proof of Theorem \ref{theocorollaryorientat}.
\end{proof}
\subsection{Examples}
We first consider the two-well problem in two dimensions, a classical model for
microstructure in martensite \cite{BallJames87,BallJames92,Bhatta,Ball2002}.
Precisely, we define
\begin{equation*}
W_{2W}(F)=
\mathrm{dist}^2(F, SO(2) U_1 \cup SO(2) U_2 ) + \theta(\det F)\,,
\end{equation*}
where
\begin{align*}
U_1=
\begin{pmatrix}
\lambda&0\\0&1/\lambda
\end{pmatrix}\,,\quad
U_2=
\begin{pmatrix}
1/\lambda&0\\0&\lambda
\end{pmatrix}\,,
\end{align*}
for some fixed $\lambda>1$ are the eigenstrains of the two martensitic phases
and $\theta\in C^0((0,\infty),[0,\infty))$ is a convex function which obeys
(\ref{eqasstheta}) and with $\lim_{t\to0} \theta(t)=\infty$, for example
$f(t)=(t-1/t)^2$, extended with $\theta=\infty$ on $(-\infty,0]$. One is then interested in the functional
$\Enog_{2W}: W^{1,2}(\Omega;\R^2)\to[0,\infty]$,
\begin{equation*}
\Enog_{2W}[u]=\int_\Omega W_{2W} (Du)\, dx\,.
\end{equation*}
In \cite{ContiDolzmanntwowell} it was shown that the quasiconvex envelope
of $W_{2W}$ is given by
\begin{equation*}
W_{2W}^\qc(F) =
h(|Fv|, |Fw|, \det(F)) + \theta(\det F)\,,
\end{equation*}
where $v=(e_1+e_1)/\sqrt2$,
$w=(e_1-e_1)/\sqrt2$ and
$h$ is defined by
\begin{equation*}
h(x,y,d)= \min_{\xi\in [x,\infty),\,\eta\in [y,\infty)} \left(
\xi^2+\eta^2+|U_1|^2 -2 \sqrt{A(\xi,\eta,d)} \right)\,,
\end{equation*}
with
\begin{alignat*}1
A(x,y,d)& =(x^2+y^2)\frac{|U_1|^2}2 +
(\lambda^2-\frac1{\lambda^2}) \sqrt{x^2y^2-d^2} + 2 d \,,
\end{alignat*}
and that $W_{2W}^\qc$ is polyconvex. Theorem \ref{theocorollaryorientat}
then shows that the relaxation of $\Enog_{2W}$ is given by
\begin{equation*}
\Enogrel_{2W}[u]=\int_\Omega W^\qc_{2W} (Du)\, dx\,.
\end{equation*}
A corresponding result holds for models with one potential well only, which can be recovered setting $\lambda=1$ in the previous expressions, the quasiconvex envelope is given for example in \cite{Silhavy1999}.
A related situation with the incompressibility constraint can be obtained from
the study of nematic elastomers \cite{DesimoneDolzmann00,Silhavy1999,DeSimoneDolzmannARMA2002,WarnerTerentjevBook,Silhavy2007}. They are composite materials in which a rubber (polymer) matrix is coupled to a nematic liquid crystal. The rubber has entropic elasticity and is usually modeled as incompressible; the ordering of the nematic liquid crystal leads to elongation in the direction of the nematic order parameter. After minimizing out the nematic director, the standard model \cite{WarnerTerentjevBook} can be cast in the form
\begin{equation*}
W_\nem(F)=
\begin{cases}
\sum_{i=1}^n \left(\frac{\lambda_i(F)}{\gamma_i}\right)^p
& \text{ if $\det F=1$}\\
\infty & \text{ if } \det F\ne 1
\end{cases}
\end{equation*}
where $\lambda_i(F)$ are the singular values of $F$, i.e., the eigenvalues of $(F^TF)^{1/2}$, and $\gamma_i\in (0,\infty)$ are material parameters with $\prod_{i=1}^n \gamma_i=1$.
In two dimensions one can assume $\gamma_2=1/\gamma_1>1$ and, taking the natural exponent $p=2$, one can show that \cite{DesimoneDolzmann00}
\begin{equation*}
W_\nem^\qc(F)=
\begin{cases}
\infty & \text{ if } \det F\ne 1\\
2 & \text{ if } \lambda_2(F)\le \gamma_2 \text{ and } \det F=1\\
W_\nem(F) & \text{ otherwise.}
\end{cases}
\end{equation*}
Further, the function $W_\nem^\qc$ is polyconvex \cite{DesimoneDolzmann00}.
Therefore Theorem \ref{theovolume} shows that the relaxation of $\Enog_\nem$ is given by $\Enogrel_\nem$.
In the physically relevant situation $n=3$ a similar expression is
analytically known and turns out to be polyconvex for all $p\ge 1$
\cite{DeSimoneDolzmannARMA2002}. Theorem \ref{theocorollaryorientat} then states that
$\Enogrel_\nem$ is the relaxation of $\Enog_\nem$ for all $p\ge 3$. This does not include,
however, the physically relevant exponent $p=2$. In this case, as noted in the
introduction, the variational model with the pointwise constraint on the determinant would
predict cavitation under tension.
Tensile experiments are however carried out in a regime
in which cavitation
does not occur, possibly due to metastability or to additional energy
contributions not modeled by $W_\nem$ \cite{WarnerTerentjevBook}. Indeed,
numerical simulations based on the quasiconvex envelope $W_\nem^\qc$ and
excluding cavitation via the choice of a the finite-element space of
continuous function lead to good agreement with experimental observations
\cite{ContiDesimoneDolzmann2002}.
\subsection{Strategy of the proof}
The ``classical'' construction of the upper bound for relaxation theorems is
based on density and interpolation. Given a function $u\in W^{1,p}$
one first approximates it strongly in $W^{1,p}$ by a piecewise affine
function and then uses a ``good'' test function $\varphi$ in each of the
pieces where $u$ is affine. In doing this it is important that the energy is
continuous along the approximating sequence. In our
case the energy is infinite on all deformations which are not orientation-preserving, therefore one would need to approximate orientation-preserving
$W^{1,p}$ maps with piecewise affine orientation-preserving maps,
a problem which, as we discussed
above, is very difficult and in general unsolved.
Here we show that the passage through piecewise affine maps is not needed.
The key idea is to locally use the {\em composition} of the limiting map $u$
with the map with oscillating gradient $\varphi$. Taking the composition preserves the
conditions on the determinant, and the map $u\circ\varphi$ belongs to $W^{1,p}$
since $u\in
W^{1,p}$ and $\varphi$ is Lipschitz. The key point is to prove that the unrelaxed energy
of the composition is close to the relaxed energy of $u$.
For simplicity we focus here on a small ball $B$ where $Du$ is close
to the
identity (after a change of variables this is true around any Lebesgue point
of $Du$). Let $\varphi\in W^{1,\infty}(B,\R^n)$ be a map from
(\ref{eqdefwqcdetp}), which in particular is the identity on the boundary of $B$. The key idea
is to define $z=u\circ \varphi$, so that $Dz=Du\circ\varphi D\varphi$. If $Du$ is close to the identity and $Dz$
is close to $D\varphi$ one then (heuristically) obtains
\begin{equation*}
\int_B W(Dz)\, dx\sim \int_B W(D\varphi)\, dx \sim |B|W^\qc(\Id)\, dx\sim \int_B
W^\qc(Du)\, dx\,.
\end{equation*}
This estimate can be made precise if $Du$ is uniformly close to the identity and $W$ is
uniformly continuous. Both properties are true only in appropriate subsets,
and hence the main part of this work is devoted to the treatment of the
exceptional sets, under the assumption that $\|Du-\Id\|_{L^p(B)}$ is small.
In the incompressible case, since $\varphi$ is a volume-preserving bilipschitz
map a change of variables shows that the smallness of $Du-\Id$ in $L^p$ immediately translates into the
smallness of $(Du-\Id)\circ \varphi$ in $L^p$ and hence of the contribution
of the set where $Dz-D\varphi$ is large. This renders the volume-preserving construction in Lemma \ref{lemmaconstr1} simpler than the corresponding one in the orientation-preserving case.
In the orientation-preserving case the situation is more complex, since a
factor $\det D\varphi$ arises from the change of variables formula. One of the problematic terms is
\begin{equation*}
\int_{\{|Du-\Id|\circ \varphi >\eps\}} |Du-\Id|^p (\varphi(x)) dx
= \int_{\{|Du-\Id| >\eps\}}\frac{ |Du-\Id|^p(y)}{\det D\varphi(y)} dy \,.
\end{equation*}
The fact that $\varphi$ is Lipschitz provides a bound on $\det D\varphi$ but
not on its inverse. However, by the very same change of variables formula we
know that $1/\det D\varphi\in L^1$, therefore the above expression has the
form of the integral of the product of two $L^1$ functions. This is
in general not defined, but can be controlled if one of the two functions is
first shifted by an ``appropriate amount'', as done for example when taking the
convolution of two $L^1$ functions. Precisely, this means that for every choice
of $f,g\in L^1(B_1)$ there are many $a_0\in B_{1/2}$ such that
\begin{equation*}
\int_{B_{1/2}} f(x) g(x-a_0) dx \le \|f\|_{L^1(B_1)} \|g\|_{L^1(B_1)}\,,
\end{equation*}
see Lemma \ref{lemmachoicex1} below for details.
In both cases the local construction is then extended to a global one by a covering
argument, see Lemma \ref{lemmarecoverydp} below.
\section{Construction of orientation-preserving maps}
\label{secorientationpres}
\subsection{Local construction}
Before presenting the construction we show how the translation is exploited.
We focus here on the derivation of the estimates, and address measurability and weak differentiability issues in Appendix \ref{secappchainrule}.
\begin{lemma}\label{lemmachoicex1}
Let $\psi\in W^{1,\infty}(B_r; \overline{B_r})$, $g\in L^1(B_{r})$, $f\in L^1(B(x_0, 2r))$ for some $x_0\in \R^n$, $r>0$. Then there
exists a measurable set $E\subset B(x_0,r)$ of positive $\calL^n$ measure
with the following property.
For $a_0\in E$
the function
\begin{equation*}
\widetilde f(x)=f(\psi(x-a_0) + a_0)g(x-a_0)
\end{equation*}
belongs to $L^1(B(a_0,r))$ with
\begin{equation*}
\|\widetilde f\|_{L^1(B(a_0,r))} \le \frac{1}{|B_r|} \|f\|_{L^1(B(x_0,2r))} \|g\|_{L^1(B_{r})} \,.
\end{equation*}
\end{lemma}
\begin{proof}
We can assume without loss of generality that $f,g\ge0$.
The function $(x,a_0)\mapsto \widetilde f(x)$ is $\calL^{2n}$-measurable by Lemma \ref{lemmacont}.
We define
$h:B(x_0,r)\to\R\cup\{\infty\}$ by
\begin{equation*}
h(a_0)=\int_{B(a_0,r)} \widetilde f(x)\, dx=
\int_{B(a_0,r)} f(\psi(x-a_0)+a_0)g(x-a_0)\,dx
\end{equation*}
and change variables to
\begin{alignat*}1
h(a_0)&=\int_{B_r} f(\psi(x')+a_0) g(x')dx'\,.
\end{alignat*}
We integrate over all $a_0\in B(x_0,r)$ and interchange the order of integration to obtain
\begin{alignat*}1
\int_{B(x_0,r)} h(a_0) da_0& =\int_{B_r}\left(\int_{B(x_0,r)}
f(\psi(x')+a_0) g(x') da_0 \right)dx'\\
&\le \|f\|_{L^1(B(x_0,2r))} \|g\|_{L^1(B_r)}\,.
\end{alignat*}
To conclude we observe that $h$ cannot be almost everywhere larger than its average.
\end{proof}
\begin{lemma}\label{lemmaconstr2}
Assume that the function $W\in
C^0(\R^{n\times n}_+,[0,\infty))$ satisfies
the growth condition (\ref{eqgrowthwpd}) with $p\geq 1$ and the structure condition
(\ref{eqasstheta}) and fix
$F\in \R^{n\times n}_+$ and $\eta>0$. Then there is $\delta>0$ such that for any
$B=B(x_0,r)$ and $u\in W^{1,p}(B,\R^n)$ with
\begin{equation}\label{eqlemmacostr2deltadef}
\fint_B \left(|Du-F|^p+|\theta(\det Du)-\theta(\det F)|\right)dx
\le \delta \text{ and } \det Du>0 \text{ a.e.}
\end{equation}
there are $a_0\in B(x_0,r/2)$
and $z\in W^{1,p}(B,\R^n)$ with
$\det Dz>0$ a.e.,
$z=u$ on $B(x_0,r)\setminus B(a_0,r/2)$ and
\begin{equation}\label{eqlemmacostrrisb1}
\int_{B(a_0,r/2)} W(Dz)\, dx\le \int_{B(a_0,r/2)}
(W^\qc(Du)+ \eta) \, dx \,.
\end{equation}
Additionally,
\begin{equation*}
\int_B |u-z|^pdx \le
c r^p \int_B (W^\qc(Du)+1)dx\,.
\end{equation*}
If $u$ is Lipschitz, then the same is true for $z$.
\end{lemma}
\begin{proof}
The $L^p$ bound follows from the bound on $W(Dz)$
using Poincar\'e and the growth condition, hence we only need to prove (\ref{eqlemmacostrrisb1}).
By the definition of $W^\qc(F)$ there is
$\varphi_\eta\in W^{1,\infty}(B_{r/2},\R^n)$ such that
$\varphi_\eta(x)=Fx$ on
$\partial B_{r/2}$ and
\begin{equation}\label{eqdefvarphieta}
\fint_{B_{r/2}} W(D\varphi_\eta)dx \le W^\qc(F) + \eta\,.
\end{equation}
By the growth condition (\ref{eqgrowthwpd}) we have
$\theta(\det D\varphi_\eta)\in L^1(B_{r/2})$, and with (\ref{eqasstheta})
also $\theta(\det (F^{-1}D\varphi_\eta))\in L^1(B_{r/2})$.
Since $\det D\varphi_\eta>0$ almost everywhere
there is $\gamma>0$ (depending on $F$ and $\eta$) such that
\begin{alignat}1\nonumber
&\int_{B_{r/2}\cap\{\det D\varphi_\eta<\gamma\}} (1+\theta(\det (F^{-1}D\varphi_\eta))) dx \\
\label{eqdefagmma}
&\hskip1cm \le \frac{|B_{r/2}| \eta}{(1+\|F^{-1}D\varphi_\eta\|_{L^\infty}^p)(1 + |F|^p + \theta(\det F))}\,.
\end{alignat}
The choice of the constant on the right-hand side will become clear after
(\ref{eqwdzomegad}).
The function $F^{-1}\varphi_\eta$ is Lipschitz continuous and therefore, by
\cite[Theorem 1]{Ball1981}, $F^{-1}\varphi_\eta(B_{r/2})\subset \overline{B_{r/2}}$.
For some
$a_0\in B(x_0,r/2)$ chosen below, we construct the function $z:B=B(x_0,r)\to\R^n$ by
\begin{equation*}
z(x)=
\begin{cases}
u(F^{-1}\varphi_\eta(x-a_0)+a_0) &\text{ if } x\in B'=B(a_0,r/2)\,, \\
u(x) &\text{ otherwise.}
\end{cases}
\end{equation*}
By Lemma \ref{lemmachainrule}
(with $\psi=F^{-1}\varphi_\eta$), there exists a null set $N$ such
for all choices of $a_0\not\in N$ the first expression belongs to $W^{1,1}$.
Further we can compute its weak derivative by the usual chain rule,
and the traces on $\partial B'$ of the two expressions coincide.
In particular, $z\in W^{1,1}(B';\R^n)$.
In order to obtain an estimate on the derivative we choose $a_0\in E\setminus N$ via
Lemma \ref{lemmachoicex1}, applied to the ball
$B$ with
$f=|Du-F|^p+|\theta(\det Du)-\theta(\det F)|$ and $g=1+\theta(\det(F^{-1}D\varphi_\eta))$. Then
\begin{equation}\label{eqlemmaghd}
\fint_{B'} (1+\theta(\det Dv))\, (|Du-F|^p+|\theta(\det Du)-\theta(\det F)|)\circ v \, dx \le c_{\eta}\delta \,,
\end{equation}
where $v(x)=F^{-1}\varphi_\eta(x-a_0)+a_0$ and
\begin{equation*}
c_{\eta}=2^n \fint_{B_{r/2}} (1+\theta(\det F^{-1}D\varphi_\eta))\, dx
\end{equation*}
(since $W$ and $F$ are fixed for the entire proof, we emphasize the dependence of the constants on $\eta$).
For the rest of the proof we only need to deal with the
fixed inner ball $B'$.
Let $R_\eta=\|Dv\|_{L^\infty}$, $M_\eta=\|D\varphi_\eta\|_{L^\infty}$.
Since $W$ is continuous in $\R^{n\times n}_+$ there
is $\varepsilon\in(0,1)$ such that
\begin{align}\begin{aligned}
|W(\xi)-W(\zeta)|\le \eta \text{ for all } &\xi,\zeta\in
\R^{n\times n}_+ \text{ with } |\zeta|\le M_\eta, \\
&\det\zeta\ge\gamma \text{ and }
|\xi-\zeta|\le \varepsilon R_\eta\,,
\end{aligned}
\label{eqcontwabr}
\end{align}
where $\gamma$ was defined in (\ref{eqdefagmma}) and depends only on $W$, $F$ and $\eta$.
Moreover, $W^\qc$ is continuous in $\R^{n\times n}_+$. This is proven for example in
\cite[Th. 2.4 and Prop. 2.3]{Fonseca1988} by showing that $W^\qc$ is
rank-one convex, and hence separately convex, in the open set $\R^{n\times n}_+$.
Hence we may assume additionally that
\begin{equation}\label{eqcontwAF}
|W^\qc(\xi)-W^\qc(F)|+|\theta(\det\xi)-\theta(\det F)|\le \eta \text{ for all $\xi$ with } |\xi-F|\le
\varepsilon \,.
\end{equation}
The parameter $\varepsilon$ depends on $\eta$,
but not on $u$ and $\delta$.
We compute, with $\widehat\varphi_\eta(x)=\varphi_\eta(x-a_0)$,
\begin{alignat}1\nonumber
\int_{B'} (W(Dz)-W^\qc(Du))dx
=& \int_{B'} (W(Dz)-W(D\widehat\varphi_\eta))dx\\
\nonumber
&+\int_{B'}(W(D\widehat\varphi_\eta)-W^\qc(F))dx \\
&+\int_{B'}(W^\qc(F)-W^\qc(Du))dx \label{eqwdzwqcduing}
\end{alignat}
and estimate the three terms separately.
The second term in (\ref{eqwdzwqcduing}) is bounded by $\eta|B'|$ by the
definition of $\varphi_\eta$, see (\ref{eqdefvarphieta}).
To treat the third one we use (\ref{eqcontwAF}) to obtain
\begin{equation*}
W^\qc(F)\le W^\qc(Du)+ \eta
\hskip1cm\text{ on the set where $|Du-F|\le \eps$} \,.
\end{equation*}
The complement is small, indeed, from
(\ref{eqlemmacostr2deltadef}) we obtain
\begin{alignat*}1
\int_{B'} (W^\qc(F)-W^\qc(Du))dx &\le \eta |B'| + W^\qc(F)
\calL^n(|Du-F|>\eps)\\
& \le \eta |B'| + W^\qc(F) \frac{1}{\eps^p} |B| \delta \,.
\end{alignat*}
To estimate the first term in (\ref{eqwdzwqcduing}) we distinguish three subsets:
$\omega=B'\cap \{|Du-F|\circ v \ge\varepsilon\}$,
$\omega_d=B'\cap\{\det D\widehat\varphi_\eta<\gamma\}\setminus\omega$
and the rest $B'\setminus\omega\setminus\omega_d$.
In $B'\setminus\omega\setminus\omega_d$ we have $|Dv|\le R_\eta$, $\det D\widehat\varphi_\eta\ge\gamma$, $|Du-F|\circ
v\le \eps$ and therefore, since
\begin{equation*}
Dz = Du \circ v Dv = (Du-F)\circ v Dv + D\widehat\varphi_\eta
\end{equation*}
we obtain
\begin{equation*}
|Dz-D\widehat\varphi_\eta|\le |Du-F|\circ v \, |Dv| \le \eps R_\eta\,.
\end{equation*}
By the continuity estimate (\ref{eqcontwabr}) we obtain
\begin{equation*}
|W(Dz)-W(D\widehat\varphi_\eta)| \le \eta
\end{equation*}
and therefore
\begin{equation*}
\int_{B'\setminus\omega_d\setminus\omega} (W(Dz)-W(D\widehat\varphi_\eta)) dx\le \eta|B'|\,.
\end{equation*}
In the two error sets we use the growth estimate (\ref{eqgrowthwpd}), which gives
\begin{alignat}1\label{eqwdzfehlerter1}
W(Dz)&\le c (1+|Du|^p\circ v \, |Dv|^p + \theta(\det Du\circ v\, \det Dv))\,.
\end{alignat}
With $|Dv|\le R_\eta$ and (\ref{eqasstheta}) we obtain
\begin{alignat}1\label{eqwdzfehlerter}
W(Dz)&\le c (1+R_\eta^p |Du|^p\circ v + (1+\theta(\det Du)\circ v) \,(1+\theta(\det Dv))\,,
\end{alignat}
where $c$ only depends on $W$.
At this point we treat the two error sets separately. For the estimate on
$\omega$ we observe that
$|Du-F| \ge\varepsilon$ implies
\begin{equation*}
|Du|+1\le |Du-F|+|F|+1 \le \left( \frac{|F|+1}{\eps}+1\right) |Du-F|
\end{equation*}
and
\begin{equation*}
\theta(\det Du)\le |\theta(\det Du)-\theta(\det F)|+ \frac{\theta(\det F)}{\eps^p}|Du-F|^p\,.
\end{equation*}
Therefore (\ref{eqwdzfehlerter}) gives
\begin{alignat*}1
\int_\omega W(Dz)&\le
c \int_{\omega} (1+\theta(\det Dv))(1+R_\eta^p|Du|^p+\theta(\det Du))\circ v
\, dx\\
\le
c_{\eta}& \int_{\omega} (1+\theta(\det Dv)) (|Du-F|^p+|\theta(\det Du)-\theta(\det F)|)\circ v\, dx\\
\le c_{\eta}& |B'| \delta\,.
\end{alignat*}
where in the last step we used (\ref{eqlemmaghd}).
The constant depends on $W$, $F$ and $\eta$ (via $\eps$), but not
on $\delta$ and $u$.
In $\omega_d$ instead we have $|Du-F|\circ v\le \eps$.
Then, recalling the continuity estimate (\ref{eqcontwAF}), we have $|Du|\circ v\le |F|+1$ and
$\theta(\det Du\circ v)\le \theta(\det F)+1$ and therefore
(\ref{eqwdzfehlerter}) reduces to
\begin{alignat}1\nonumber
W(Dz)&\le c (1+ R_\eta^p (1+|F|^p) + (1+\theta(\det F)) (1+\theta(\det Dv))\\
&\le c_*(1+\|F^{-1}D\varphi_\eta\|_\infty^p)
(1+|F|^p+\theta(\det F))(1+\theta(\det Dv))
\,,
\label{eqwdzomegad}
\end{alignat}
with a constant $c_*>0$ which depends only on $W$.
With (\ref{eqdefagmma}) we conclude
\begin{equation*}
\int_{\omega_d} W(Dz)dx\le c_* |B'| \eta\,.
\end{equation*}
Adding all terms we obtain
\begin{equation*}
\int_{B'} \left(W(Dz)-W^\qc(Du)\right)dx \le (
3\eta + 2^n\frac{W^{\qc}(F)}{\eps^p} \delta +
c_{\eta} \delta + c_*\eta) |B'|\,.
\end{equation*}
Since $c_*$ depends only on $W$, choosing $\delta$ sufficiently small the proof is concluded.
\end{proof}
\subsection{Upper bound}
\begin{lemma}[Recovery sequence]\label{lemmarecoverydp} Let $\Omega\subset\R^n$ open, Lipschitz, bounded,
let $W\in C^0(\R^{n\times n}_+,[0,\infty))$ obey
(\ref{eqgrowthwpd}--\ref{eqasstheta}). Then for any
$u\in
W^{1,p}(\Omega;\R^n)$ with $\det Du>0$ almost everywhere there is
a sequence $u_j\rightharpoonup u$ in $W^{1,p}$ such that $\det Du_j>0$
almost everywhere, $u_j=u$ on $\partial\Omega$ and
\begin{equation*}
\limsup_{j\to\infty}\int_\Omega W(Du_j)dx\le \int_\Omega W^\qc(Du)dx\,.
\end{equation*}
If additionally $u\in W^{1,\infty}$ then also $u_j\in
W^{1,\infty}$.
\end{lemma}
\begin{proof}
Fix $\eta>0$. It suffices to construct $w$ with $\|u-w\|_p\le \eta$,
$w=u$ on $\partial\Omega$,
and $\int_\Omega W(Dw)dx\le \int_\Omega W^\qc(Du)dx+\eta$.
If $\int_\Omega W^\qc(Du)dx=\infty$ the constant sequence will do, hence we
can assume that $W^\qc\circ Du\in L^1$.
By convexity of $\theta$ and of the $p$-norm the definition of $W^\qc(F)$ gives
\begin{equation*}
\frac1c |F|^p +\frac1c\theta(\det F)-c\le W^\qc(F)\text{ for all $F$},
\end{equation*}
therefore $|Du|^p$ and
$\theta(\det Du)$ are also integrable. We denote by $E$ the set of
Lebesgue points of $Du$ and $\theta(\det Du)$.
For every $x\in E$ we set $F(x)=Du(x)$ and choose $\delta(x)$ as in
Lemma \ref{lemmaconstr2}
for this $F$ and $\eta$ as above.
The construction is done by successive application of
Lemma \ref{lemmaconstr2}. We set $w_0=u$, $\Omega_0=\Omega$ and describe
how to pass from $(w_j,\Omega_j)$ to $(w_{j+1},\Omega_{j+1})$.
For all $x\in E\cap \Omega_j$ we choose
$r_j(x)\in(0,\eta)$ such that $B(x,r_j(x))\subset \Omega_j$ and
\begin{equation*}
\fint_{B(x,r)} \left(|Dw_j-F(x)|^p+|\theta(\det Dw_j)-\theta(\det F(x))|\right)
dx' \le \delta(x)
\end{equation*}
for all $r<r_j(x)$.
This gives a fine cover of $E\cap\Omega_j$. We extract a disjoint subcover
$B(x_k,r_k)_{k\in\N}$ and from this subcover finitely many balls
$B(x_k,r_k)_{k=0,\dots, M}$ which cover at least half the volume of
$\Omega_j$.
We set $w_{j+1}=w_j$ on $\Omega\setminus \cup_{k=0}^M B(x_k,r_k)$
and define $w_{j+1}$ as the result of Lemma \ref{lemmaconstr2} in each of
the
balls. Then $w_{j+1}\in W^{1,p}(\Omega;\R^n)$ and $w_{j+1}=w_j=u$
on $\partial\Omega$. Further,
the smaller balls $B(x_k',r_k/2)\subset B(x_k,r_k)$ obey
\begin{equation}\label{eqwdwj1}
\int_{B(x_k',r_k/2)} W(Dw_{j+1})dx\le
\int_{B(x_k',r_k/2)} (W^\qc(Du)+\eta) dx
\end{equation}
and
\begin{equation}\label{eqconvlp}
\int_{B(x_k',r_k/2)} |w_{j+1}-u|^pdx\le c \eta^p
\int_{B(x_k',r_k/2)} (1+W^\qc(Du))dx\,,
\end{equation}
with $w_{j+1}=w_j$ outside these balls.
Finally we set $\Omega_{j+1}=\Omega_j\setminus \cup_{k=0}^M
\overline B(x_k',r_k/2)$, so that $|\Omega_{j+1}|\le (1-2^{-n-1}) |\Omega_j|$,
and iterate. We remark that $w_{j+1}=u$ on the open set $\Omega_{j+1}$,
hence there is no need to redefine $E$, $F$ and $\delta$ at each step.
This concludes the construction of the sequence $w_j$.
It remains to show that $w_j$, for $j$ sufficiently large, has the desired
properties. Each of these functions coincides with $u$ outside a finite number
of disjoint balls, and has been modified exactly once in each of those balls.
By (\ref{eqconvlp}) we have
\begin{equation*}
\int_{\Omega} |w_j-u|^pdx \le c \eta^p \int_\Omega (1+W^\qc(Du))dx
\end{equation*}
hence $w_j$ is close to $u$ in $L^p$, independently of $j$.
Analogously from (\ref{eqwdwj1}) we deduce, for the union of the balls
$\Omega\setminus \Omega_j$,
\begin{equation*}
\int_{\Omega\setminus\Omega_{j}} W(Dw_{j})dx\le
\int_{\Omega\setminus\Omega_{j}} (W^\qc(Du)+\eta)dx
\end{equation*}
which implies
\begin{equation*}
\int_\Omega W(Dw_j)dx\le \int_{\Omega\setminus\Omega_j} (W^\qc(Du)+\eta)dx
+\int_{\Omega_j} W(Du)dx\,.
\end{equation*}
Since $|\Omega_j|\le (1-2^{-n-1})^j|\Omega|\to0$ and by the growth condition
$W(Du)\in L^1(\Omega)$, for
sufficiently large $j$ we have
\begin{equation*}
\int_\Omega W(Dw_j)dx\le \int_{\Omega} (W^\qc(Du)+2\eta)dx\,,
\end{equation*}
as required.
\end{proof}
\begin{lemma}[Quasiconvexity]\label{lemmaqcorientpres}
The function $W^\qc$ is quasiconvex.
\end{lemma}
\begin{remark}\label{remarkwqc}
$W^\qc$ is the largest (extended-valued) quasiconvex function
below $W$, hence in this sense its quasiconvex envelope. This function does not necessarily coincide with the supremum of all finite-valued quasiconvex functions below $W$; in particular, this is not true for the function discussed in \cite[Example 3.5]{BallMurat1984}.
\end{remark}
\begin{proof}
Fix $F$ with $\det F>0$, $\Omega=B_1$, $\psi\in W^{1,\infty}(B_1,\R^n)$
with $\psi(x)=Fx$ on $\partial B_1$. We need to show that
\begin{equation*}
W^\qc(F)\le \fint_{B_1} W^\qc(D\psi)dx\,.
\end{equation*}
By Lemma \ref{lemmarecoverydp}
there is a sequence $\varphi_j\in W^{1,\infty}$
with $\varphi_j(x)=\psi(x)=Fx$ on $\partial B_1$ and such that
\begin{equation*}
\limsup_{j\to\infty} \int_{B_1} W(D\varphi_j)dx\le \int_{B_1}
W^\qc(D\psi)dx\,.
\end{equation*}
Since every $\varphi_j$ is admissible in the definition of $W^\qc(F)$ we
obtain
\begin{equation*}
W^\qc(F)\le \fint_{B_1} W(D\varphi_j) dx
\end{equation*}
for all $j$, and in particular
\begin{equation*}
W^\qc(F)\le\fint_{B_1}
W^\qc(D\psi)dx
\end{equation*}
as desired.
\end{proof}
\section{Construction of volume-preserving maps}
\label{secvolumepres}
In this case the translation is not needed, and correspondingly the proof of Lemma \ref{lemmaconstr1} is simpler than the one of Lemma \ref{lemmaconstr2}; we give it in detail since it illustates in a compact way the key ideas of our constrruction. At the same time the continuity of $W^\qc$ is less clear than for orientation-preserving maps.
It essentially follows from the results of \cite{MuellerSverak,Conti2008}. Since it was not stated there we briefly show how it can be derived from the construction in \cite{Conti2008}.
\begin{lemma}
Given $W:\Sigma\to[0,\infty)$, extended by $\infty$ elsewhere, we define
$W^\qc$ by (\ref{eqdefwqcdetp}).
The function $W^\qc$ is rank-one convex and hence continuous on $\Sigma$.
\end{lemma}
\begin{proof}
We first observe that, by general scaling and covering arguments, the definition of $W^\qc$ does not depend on the domain, and in particular
\begin{equation}\label{eqwqcpolyhedron}
W^\qc(F)=\inf\{\fint_{\omega} W(D\varphi)dx: \varphi\in
W^{1,\infty}(\omega;\R^n), \varphi(x)=Fx \text{ for } x\in \partial \omega \}
\end{equation}
for any bounded open nonempty polyhedron $\omega\subset\R^n$.
To prove rank-one convexity we fix $A,B\in \Sigma$ with
$\rank(A-B)=1$ and $\lambda\in(0,1)$. We define $F=\lambda A+ (1-\lambda) B$.
By the construction in \cite[Th. 2.1]{Conti2008}
(with $n=m=r$, $P=Q=\Id$, $t=1$, $\eps=1$) there is a finite
set $K\subset\Sigma$ such that for any $\delta>0$ one can find
a polyhedron $\Omega$ and a piecewise affine function $u\in
W^{1,\infty}(\Omega;\R^n)$ such that $u(x)=Fx$ on $\partial \Omega$,
$Du \in K\subset\Sigma$ almost everywhere,
$|\{Du\not\in\{A,B\}\}|\le \delta|\Omega|$.
The latter, together with the boundary data, implies
\begin{equation*}
|\{Du=A\}|\le (\lambda+c\delta)|\Omega| \text{ and }
|\{Du=B\}|\le (1-\lambda+c\delta)|\Omega|\,,
\end{equation*}
with $c$ depending on $A$ and $B$.
Further, the set $\{Du=A\}$ is a finite union of simplexes $\omega^A_j$. For
each of
them there is, by (\ref{eqwqcpolyhedron}) with $F=A$, a Lipschitz function $v^A_j$
with $v^A_j=u$ on $\partial \omega^A_j$ and
\begin{equation*}
\int_{ \omega^A_j} W(Dv^A_j)dx \le |\omega^A_j|( W^\qc(A) + \delta)\,.
\end{equation*}
The same holds for the set $\{Du=B\}$. We set $w=v^A_j$ on each
$\omega^A_j$, $w=v^B_j$ on each
$\omega^B_j$, $w=u$ on the rest.
Since $w(x)=Fx$ on $\partial\Omega$ we have
\begin{alignat*}1
W^\qc(F)\le & \fint_\Omega W(Dw)dx \le \frac{|\{Du=A\}|}{|\Omega|}( W^\qc(A) + \delta)\\
& +\frac{|\{Du=B\}|}{|\Omega|}( W^\qc(B) + \delta)
+\frac{|\{Du\not\in \{A,B\}\}|}{|\Omega|} \max W(K)\\
&\le \lambda W^\qc(A)+(1-\lambda)W^\qc(B) + c\delta \max W(K)\,,
\end{alignat*}
with $c$ depending on $A$ and $B$.
Taking $\delta\to0$ (with fixed $K$) this implies the desired inequality
$W^\qc(F)\le \lambda W^\qc(A)+(1-\lambda) W^\qc(B)$.
Since $W^\qc$ is rank-one convex, it is separately
convex in suitable variables and hence continuous (for details see, e.g.,
\cite[Step 2 in the proof of Th. 3.1]{Conti2008}).
\end{proof}
\begin{lemma}\label{lemmaconstr1}
Let
$W\in C^0(\Sigma;[0,\infty))$ obey
(\ref{eqgrowthwincompr}) for some $p\ge 1$.
Then for any $F\in \R^{n\times n}$ and $\eta>0$
there is $\delta>0$ such that the following holds:
For any ball $B=B(x_0,r)$ and any function $u\in W^{1,p}(B,\R^n)$
with
\begin{equation*}
\fint_B |Du-F|^pdx\le \delta \text{ and } Du\in \Sigma \text{ a.e.}
\end{equation*}
one can find $z\in W^{1,p}(B,\R^n)$ with $u=z$ on $\partial B$,
\begin{equation*}
\fint_B W(Dz)dx\le \fint_B (W^\qc(Du)+ \eta)dx \text{ and } Dz
\in \Sigma \text{ a.e.}
\end{equation*}
Additionally,
\begin{equation*}
\fint_B |u-z|^p dx\le
c r^p \fint_B (1+W^\qc(Du))dx\,.
\end{equation*}
\end{lemma}
\begin{proof}
Let $\varphi_\eta\in W^{1,\infty}(B,\R^n)$ be such that $\varphi_\eta(x)=Fx$ on
$\partial B$ and
\begin{equation*}
\fint_B W(D\varphi_\eta)dx \le W^\qc(F) + \eta\,.
\end{equation*}
We define
\begin{equation*}
v=F^{-1}\varphi_\eta
\end{equation*}
and observe that, by \cite[Theorem 2]{Ball1981},
$v$ is a bilipschitz
map from $B$ onto itself. Therefore we can define
\begin{equation*}
z=u\circ v \in W^{1,p}(B,\R^n)
\end{equation*}
and compute its gradient
\begin{equation*}
Dz = Du \circ v Dv = (Du-F)\circ v Dv + D\varphi_\eta\,.
\end{equation*}
We set $R_\eta=\|Dv\|_\infty$, $M_\eta=\|D\varphi_\eta\|_\infty$ and choose $\varepsilon\in(0,1)$ such that
\begin{equation}\label{eqcontinc1}
|W(\xi)-W(\zeta)|\le \eta \text{ whenever } |\zeta|\le M_\eta\text{ and } |\xi-\zeta|\le
\varepsilon R_\eta
\end{equation}
and
\begin{equation}\label{eqcontinc2}
|W^\qc(\xi)-W^\qc(F)|\le \eta \text{ whenever } |\xi-F|\le \varepsilon \,.
\end{equation}
In order to estimate the integral
\begin{alignat}1\nonumber
\int_B (W(Dz)-W^\qc(Du))dx
=& \int_B (W(Dz)-W(D\varphi_\eta))dx\\ \nonumber
&+\int_B(W(D\varphi_\eta)-W^\qc(F))dx \\
&+\int_B(W^\qc(F)-W^\qc(Du))dx
\label{eqedzdfincomprs}
\end{alignat}
we consider the three terms separately.
The second integral in (\ref{eqedzdfincomprs}) is bounded by $\eta |B|$ by the definition of
$\varphi_\eta$.
In order to estimate the last integral in (\ref{eqedzdfincomprs}) we use
(\ref{eqcontinc2}) to obtain
\begin{equation*}
W^\qc(F)\le W^\qc(Du)+ \eta
\hskip1cm\text{ on the set where $|Du-F|\le \eps$} \,.
\end{equation*}
The complement is small and gives a small contribution. Precisely,
\begin{alignat*}1
\int_{B} (W^\qc(F)-W^\qc(Du))dx &\le \eta |B| + W^\qc(F)
\calL^n(|Du-F|>\eps)\\
& \le \eta |B| + W^\qc(F) \frac{1}{\eps^p} |B| \delta \,.
\end{alignat*}
In order to estimate the first integral in (\ref{eqedzdfincomprs}) we
distinguish the set
$\omega=\{|Du-F|\circ v >\varepsilon\}$ and the rest. On $B\setminus\omega$,
from the explicit expression for $Dz$ we obtain
\begin{equation*}
|Dz-D\varphi_\eta| \le \|Dv\|_\infty |Du-F|\circ v \le \eps R_\eta
\end{equation*}
and recalling (\ref{eqcontinc1}) we can estimate
\begin{equation*}
|W(Dz)-W(D\varphi_\eta)| \le \eta \text{ on } B\setminus \omega\,.
\end{equation*}
Since $|Du-F|\ge \eps$ implies
\begin{equation*}
|Du|+1\le |Du-F|+|F|+1 \le \left( \frac{|F|+1}{\eps}+1\right) |Du-F|\,,
\end{equation*}
the contribution of $\omega$ can be estimated by
\begin{alignat*}1
\int_\omega W(Dz)dx \le
c\int_{\omega} (R_\eta^p |Du|^p\circ v +1) dx
\le & c_{F,\eps} \int_{\omega} |Du-F|^p\circ v \, dx
\end{alignat*}
where the constant depends on $F$, $\eta$ and $\eps$. Finally,
$v$ is a bilipschitz
map from $B$ onto itself with $\det Dv=1$ almost everywhere
(see \cite[Theorem 2]{Ball1981}) and therefore
\begin{alignat*}1
\int_\omega W(Dz)dx\le
& c_{F,\eps} \int_{B} |Du-F|^p\circ v \, dx
= c_{F,\eps} \int_{B} |Du-F|^p dx\le c_{F,\eps} \delta |B|\,.
\end{alignat*}
Collecting terms we conclude
\begin{equation*}
\fint_B (W(Dz)-W^\qc(Du))dx\le 3\eta +
\frac{W^\qc(F)}{\eps^p}\delta
+ c_{F,\eps}\delta\,.
\end{equation*}
Since $\eps$ depends on $\eta$ and $F$ but not on $\delta$ and $u$, choosing $\delta$ sufficiently small the
proof is concluded. The $L^p$ estimate follows from the growth estimate and Poincar\'e's inequality.
\end{proof}
\begin{lemma}[Recovery sequence]\label{lemmarecoveryincompr}
Let $\Omega\subset\R^n$ open, Lipschitz, bounded, $u\in
W^{1,p}(\Omega;\R^n)$ with $Du\in \Sigma$ almost everywhere. Then there is
a sequence $u_j\rightharpoonup u$ in $W^{1,p}$ such that $\det Du_j\in \Sigma$
almost everywhere and
\begin{equation*}
\limsup_{j\to\infty}\int W(Du_j)dx\le \int W^\qc(Du)dx\,.
\end{equation*}
If additionally $u\in W^{1,\infty}$ then also $u_j\in
W^{1,\infty}$.
\end{lemma}
\begin{proof}
The proof is just like the one in Lemma \ref{lemmarecoverydp}, for brevity we do not repeat it.
\end{proof}
\begin{lemma}[Quasiconvexity]
The function $W^\qc$ is quasiconvex.
\end{lemma}
\begin{proof}
The proof is just like the one of Lemma \ref{lemmaqcorientpres}, for brevity we do not repeat it.
\end{proof}
\appendix
\section{Composition of Sobolev functions with Lipschitz functions}
\label{secappchainrule}
The composition of a Lipschitz with a Sobolev function and the composition of a Sobolev with a bilipschitz function are standard.
Although there is a substantial literature on the subject, see for example \cite{GoldsteinReshetnyak1990,LeoniMorini2007} and references therein, we have been unable to find the statement needed here on the composition of a Sobolev with a Lipschitz function, hence we give a short self-contained proof.
To see the difficulty with measurability one can consider the example $\psi(x_1,x_2)=(x_1,0)$,
$f(x_1,x_2)=h(x_1)\chi_{\{0\}}(x_2)$, with $h:\R\to\R$ not measurable. Then $f=0$ $\calL^2$-almost everywhere but $f\circ\psi$ is not measurable. To see the difficulty with integrability one can consider $f(x)=|x|^{-1/2}$ in the unit ball of $\R^2$, with $\psi(x)=|x|x$ around the origin. Then $f$ is in $W^{1,1}$ but $(f\circ \psi)(x)=|x|^{-1}$ is not.
\defg{g}
\begin{lemma}\label{lemmacont}
Let $\psi\in W^{1,\infty}(B_1;\overline B_1)$,
$f_k\in L^1(B_2;\R^m)$ with $\sum_k\|f_k\|_{L^1(B_2)}<\infty$.
Then the maps
$(x,a_0)\mapsto f_{k}(a_0+\psi(x-a_0))$ are $\calL^{2n}$ measurable and
for almost all $a_0\in B_1$
the functions
\begin{equation*}
z_{k}(x)=f_{k}(a_0+\psi(x-a_0))
\end{equation*}
are in $L^1(B(a_0,1))$ with $\sum_k \|z_{k}\|_{L^1(B(a_0,1))}<\infty$.
\end{lemma}
\begin{proof}
We define the continuous function $g:B_1\times B_1\to B_2$ by $g(x,y)=x+\psi(y)$ and show that for any $k$ the function $f_k\circ g$ is $\calL^{2n}$-measurable.
Let $A\subset\R^m$ be open. Then $f_k^{-1}(A)\subset B_2$ is $\calL^n$-measurable, therefore $f_k^{-1}(A)=E\setminus N$, with $E$ Borel and $N$ a null set.
Since $g$ is continuous, $g^{-1}(E)$ is Borel. It remains to show that $|N|=0$ implies $g^{-1}(N)=0$.
Let $F\subset B_2$ be Borel with $N\subset F$ and $|F|=0$.
Then $g^{-1}(F)$ is Borel and $\calL^{2n}$-measurable. For any $y\in\R^n$, the set $T_y=\{x\in \R^n: g(x,y)\in F\}=F-\psi(y)$ is a $\calL^n$-null set. By Fubini's theorem
\begin{equation*}
\calL^{2n}(g^{-1}(F))=\int_{\R^n} \calL^n(T_y) dy = 0\,.
\end{equation*}
Therefore $f_k\circ g$ is measurable.
A second application of Fubini's theorem shows that for almost all $a_0\in B_1$ each function $x\mapsto f_k(g(a_0,x))$ is measurable; clearly
the same holds for the translations
$z_k(x)=f_k(g(a_0,x-a_0))$. We conclude that for almost all $a_0\in B_1$ all the functions $z_k$ are $\calL^n$-measurable.
We define $A:B_1\to[0,\infty]$ by
\begin{alignat*}1
A(a_0)&=\sum_{k\in\N} \|z_k\|_{L^1(B(a_0,1))}
=\sum_{k\in\N} \int_{B(a_0,1)} |f_k|(a_0+\psi(x-a_0)) dx \\
&=\sum_{k\in\N} \int_{B_1} |f_k|(a_0+\psi(x')) dx'\,.
\end{alignat*}
The integrand is nonnegative and measurable, hence we can interchange
the order of summation and integration. Since the integrand is measurable as a function on $\R^{2n}$, the function $A$ is measurable.
Integrating and changing variables as usual,
\begin{equation*}
\int_{B_1} A(a_0) da_0 \le \sum_{k\in\N} \int_{B_1} \|f_k\|_{L^1(B_2)} dx'
\le |B_1| \sum_{k\in \N}\|f_k\|_{L^1(B_2)}<\infty \,.
\end{equation*}
Therefore $A(a_0)<\infty$ almost everywhere, which concludes the proof.
\end{proof}
\begin{lemma}[Chain rule]\label{lemmachainrule}
Let $\psi\in W^{1,\infty}(B_1;\overline B_1)$, $u\in W^{1,1}(B_2;\R^m)$.
Then for almost all $a_0\in B_1$
the function $w(x)=u(a_0+\psi(x-a_0))$ belongs to $W^{1,1}(B(a_0,1);\R^m)$ with
\begin{equation*}
Dw(x)=Du(a_0+\psi(x-a_0))D\psi(x-a_0)\,.
\end{equation*}
If $\psi(x)=x$ on $\partial B_1$ then $w=u$ (as traces) on $\partial B(a_0,1)$.
\end{lemma}
\begin{proof}
We choose a sequence $u_k\in C^\infty(\overline{B_2};\R^m)$ such that $\|u_k-u\|_{W^{1,1}(B_2)}\le 2^{-k}$ and
apply Lemma \ref{lemmacont} to the sequence
$f_k=(u_k-u,Du_k-Du)\in L^1(B_2;\R^{m}\times \R^{m\times n})$,
which obeys $\sum \|f_k\|_{L^1}\le 2$.
For any fixed $a_0$ not in the null set given by the lemma,
we obtain the corresponding sequence $z_k$ with the properties asserted in
Lemma \ref{lemmacont}. Additionally
we define $w_k$ by $w_k(x)=u_k(a_0+\psi(x-a_0))$ and $w$ as in the statement.
Each of the functions $z_k$ with values in $\R^m\times \R^{m\times n}$ is measurable,
therefore the first $m$ components which are given by $w_k-w$ are measurable.
The continuity of
$w_k$ implies the measurability of $w$. Furthermore,
\begin{alignat*}1
\|w_k-w\|_{L^1(B(a_0,1))} &= \int_{B(a_0,1)} |u_k-u|(a_0+\psi(x-a_0)) dx\\
& \le \int_{B(a_0,1)} |f_k|(a_0+\psi(x-a_0)) dx =\|z_k\|_{L^1(B(a_0,1))}\to0\,.
\end{alignat*}
We conclude $w\in L^1$ and $w_k\to w$ in $L^1$.
We now repeat the procedure for the gradient.
We denote by $F(x)=Du(a_0+\psi(x-a_0))D\psi(x-a_0)$ the expression given in the statement.
Since every $u_k$ is smooth by the usual chain rule we obtain
\begin{equation*}
Dw_k(x) = Du_k(a_0+\psi(x-a_0))D\psi(x-a_0)\,,
\end{equation*}
which is the product of a continuous and an $L^\infty$ function and therefore measurable. Further,
\begin{equation*}
(Dw_k-F)(x) = (Du_k-Du)(a_0+\psi(x-a_0))D\psi(x-a_0)\,.
\end{equation*}
The first factor is the second component of $z_k$ hence measurable by Lemma \ref{lemmacont}, the second belongs to $L^\infty$. Continuity of $Dw_k$ gives measurability of $F$. Further,
\begin{alignat*}1
\|Dw_k-F\|_{L^1(B(a_0,1))} &\le \|D\psi\|_\infty \int_{B(a_0,1)} |Du_k-Du|(a_0+\psi(x-a_0)) dx\\
& \le \|D\psi\|_\infty\int_{B(a_0,1)} |f_k|(a_0+\psi(x-a_0)) dx\to0 \,.
\end{alignat*}
Therefore $F\in L^1$ and $Dw_k\to F$ in $L^1$. Continuity of the distributional derivative implies $F=Dw$ distributionally and $w\in W^{1,1}$.
To obtain the condition on the trace it suffices to extend $\psi$ to be the identity outside $B_1$, $u$ to a function in $W^{1,1}(\R^n;\R^m)$ and work on a larger ball.
\end{proof}
\section{Construction for submultiplicative integrands}
\label{sectmultipl}
We show here how our construction of the recovery sequence can be extended to the more general situation discussed in Remark \ref{remarkgener}. We focus on the orientation-preserving case, the other one is simpler. For brevity we only show how the basic construction step is modified, the covering of Lemma \ref{lemmarecoverydp} is not significantly changed. Indeed, it suffices to use $p=1$ and takes Lebesgue points of $Du$ and $W(Du)$ instead of Lebesgue points of $Du$ and $\theta(\det Du)$; $W(Du)\in L^1$ by the growth condition (\ref{eqwprod2}).
\begin{lemma}\label{lemmaconstr2prod}
Assume that $W\in
C^0(\R^{n\times n}_+,[0,\infty))$ satisfies
\begin{equation}\label{eqwgrowthproduct1}
\frac1c|F|-c\le W(G)
\end{equation}
and
\begin{equation}\label{eqwgrowthproduct}
W(FG)\le c_W (1+W(F))(1+W(G))
\end{equation}
for all $F,G\in \R^{n\times n}_+$, with a fixed $c_W>0$.
Then for any
$F\in \R^{n\times n}_+$ and $\eta>0$ there is $\delta>0$ such that for any
$B=B(x_0,r)$ and $u\in W^{1,1}(B,\R^n)$ with
\begin{equation*}
\fint_B ( |Du-F|+|W(Du)-W(F)|)\, dx\le \delta \text{ and } \det Du>0 \text{ a.e.}
\end{equation*}
there are $a_0\in B(x_0,r/2)$
and $z\in W^{1,1}(B,\R^n)$ with
$\det Dz>0$ a.e.,
$z=u$ on $B(x_0,r)\setminus B(a_0,r/2)$ and
\begin{equation*}
\int_{B(a_0,r/2)} W(Dz)dx\le \int_{B(a_0,r/2)}
(W^\qc(Du)+ \eta) dx \,.
\end{equation*}
Additionally,
\begin{equation*}
\int_B |u-z|dx \le
c r \int_B (W^\qc(Du)+1)dx\,.
\end{equation*}
If $u$ is Lipschitz, then so is $z$.
\end{lemma}
\begin{proof}
This is very similar to the proof of Lemma \ref{lemmaconstr2}, we only discuss the differences.
After (\ref{eqdefvarphieta}), (\ref{eqwgrowthproduct}) implies $W(F^{-1} D\varphi_\eta)\in L^1$ and
equation (\ref{eqdefagmma}) is replaced by
\begin{equation}\label{eqdefagmmaprod}
\int_{B_{r/2}\cap\{\det D\varphi_\eta<\gamma\}} (1+W(F^{-1}D\varphi_\eta))\, dx \le \frac{|B_{r/2}|}{c_W(2+W(F))} \eta\,.
\end{equation}
In Lemma \ref{lemmachoicex1} we use
$f=|Du-F|+|W(Du)-W(F)|$ and $g=1+W(F^{-1}D\varphi_\eta)$,
(\ref{eqlemmaghd}) is replaced by
\begin{equation}\label{eqlemmaghdprod}
\fint_{B'} (1+W(F^{-1}D\widehat\varphi_\eta))\, (|Du-F|+|W(Du)-W(F)|)\circ v \, dx
\le c_{\eta}\delta \,,
\end{equation}
where $c_{\eta}= 2^n \fint_{B_{r/2}} (1+W(F^{-1}D\varphi_\eta))\,dx<\infty$.
In (\ref{eqcontwAF}) we use continuity of $W$ instead of $\theta$.
The remaining differences are in the treatment of the two error sets.
We replace (\ref{eqwdzfehlerter1}) by
\begin{equation}\label{eqwdzfehlerterprod}
W(Dz) \le c_W (1+W(Du)\circ v) (1+W(Dv))\,.
\end{equation}
We start from $\omega$.
From $|Du-F|\circ v\ge\eps$ we deduce
\begin{alignat*}1
1+W(Du)\circ v \le& 1+W(F)+|W(Du)-W(F)|\circ v\\
\le &c_{F,\eps} (|Du-F|+|W(Du)-W(F)|)\circ v
\end{alignat*}
where $c_{F,\eps}=1+(1+W(F))/\eps$.
Therefore the estimate (\ref{eqwdzfehlerterprod}) gives
\begin{alignat*}1
\int_\omega W(Dz)dx \le & c_Wc_{F,\eps}\int_{B'} (1+W(Dv)) \, (|Du-F|+|W(Du)-W(F)|)\circ v dx
\end{alignat*}
Recalling (\ref{eqlemmaghdprod}), which had been obtained by the choice of $a_0$, we get
\begin{alignat}1\nonumber
\int_\omega W(Dz)dx\le& c_W c_{F,\eps} c_{\eta}\delta\,.
\end{alignat}
The constant depends on $\eps$ and $F$ (and hence on $\eta$) but not
on $\delta$ and $u$.
In $\omega_d$ instead we have $|Du-F|\circ v\le \eps$.
The continuity estimate (\ref{eqcontwAF}) gives then $W(Du\circ v)\le W(F)+1$
and therefore
(\ref{eqwdzfehlerterprod}) reduces to
\begin{equation*}
W(Dz)\le c_W (2 + W(F))(1+W(Dv)) \,.
\end{equation*}
With (\ref{eqdefagmmaprod}) we conclude
\begin{equation*}
\int_{\omega_d} W(Dz)dx\le |B'| \eta \,.
\end{equation*}
The conclusion is the same.
\end{proof}
\end{document} | math | 61,013 |
\begin{document}
\title{A geometric decomposition of real diagonalizable matrices with complex eigenvalues.}
\begin{abstract}
For any real diagonalizable matrix with complex eigenvalues we provide a real, coordinate free decomposition with a clear geometric interpretation.
\end{abstract}
Imaginary numbers have been a source of puzzlement for centuries.
Square roots of negative numbers had crept into solution formulas for cubic equations in the XVI
century, and it took some years until Rafael Bombelli managed to show how those formulas lead to real solutions, providing along the way the correct multiplication rules of complex numbers~\cite{CrossleyEmergenceNumber,gonzalez2011journey}.
This remarkable feat of algebra had to wait for more than two hundred years for its geometric counterpart when, around the turn of the XIX century, multiplication and addition of complex numbers could be visualized in terms of vectors in the complex plane~\cite{gonzalez2011journey,nahin1998imaginary}.
This includes, of course, the interpretation of the imaginary unit $\sqrt{-1}$ as an operator which rotates the vector of a complex number by $\pi/2$ in the complex plane.
It would still take until the work of an influential figure like Gauss, who pondered about the use of the term `lateral' instead of `imaginary' or `impossible'~\cite{nahin1998imaginary}, for complex numbers to gain broad acceptance in mainstream Mathematics~\cite{gonzalez2011journey}.
Despite this acceptance, the question of the side ({\it latus}), if any, to which this `lateral' number relates is often shrugged off.
More precisely, the visualization of complex numbers in relation to real problems in which they appear is not always clear.
This is particularly surprising in solutions to problems in which one may expect to be able to think geometrically, like that of the eigenvalues of linear transformations of real vector spaces.
\subsection*{Matrix Diagonalization}
Consider a matrix $M \in \mathbb{R}^{n\times n}$ with a complete set of eigenvectors $\{b_i\}$ with eigenvalues $\lambda_i$ satisfying
\begin{equation} \label{eq:eigenEq}
M b_i = \lambda_i b_i,
\end{equation}
for $i=1,\ldots, n.$
The matrix $M$ is diagonalizable~\cite{horn_johnson_2012}, meaning that it can be decomposed as
\begin{equation} \label{eq:eigenMatrixDiaglze}
M = B \Lambda B^{-1},
\end{equation}
where $\Lambda$ is a diagonal matrix with the eigenvalues $\lambda_i$ and the columns of $B$ contain the corresponding eigenvectors $b_i.$
While the matrix $M$ is real, the eigenvalues and eigenvectors can generally come in complex conjugate pairs.
Of course, as described in ref. \cite{MurWin31PNAS} in a closely related context nearly a century ago, the decomposition can be written in purely real terms.
The real canonical form is obtained by replacing in $\Lambda$ the diagonal block of each pair of complex conjugate eigenvalues as
\begin{equation}
\left(\begin{array}{cc}
\lambda_i & 0 \\
0 & \lambda_i^*
\end{array} \right)
\longrightarrow
\left(\begin{array}{cc}
\sigma_i & \omega_i \\
-\omega_i & \sigma_i
\end{array} \right),
\end{equation}
where $\lambda_i=\sigma_i+\sqrt{-1}\,\omega_i,$ and the corresponding $b_i$ and $b_i^*$ in the columns of $B$ by their real and imaginary parts, respectively.
As it is commonly understood, the imaginary part of the eigenvalue $\omega_i$ has to do with a rotation in the plane determined by the real and imaginary parts of the complex eigenvector $b_i.$
A more detailed geometric understanding of the action of the real matrix $M$ on a real vector ${\bf x}$ is attained by considering left eigenvectors $d_i$ satisfying
\begin{equation} \label{eq:leftEigenEq}
d_i^{\dag} M = \lambda_i d_i^{\dag},
\end{equation}
where $d^\dag=d^{*T}$ is the transpose conjugate of $d.$
The left and right eigenvectors satisfy a biorthogonality property whereby $d_i^{\dag}b_j=0$ when $\lambda_i\neq\lambda_j.$
The eigenvalue decomposition of the real matrix $M$ can be written as
\begin{equation} \label{eq:eigenMatrixAdDecomp}
M=\sum_{i=1}^n \lambda_i \frac{b_i d_i^{\dag}}{d_i^{\dag} b_i},
\end{equation}
so that its action on a real vector ${\bf x}$ becomes
\begin{equation}
M{\bf x}= \sum_{i=1}^n \lambda_i b_i \frac{d_i^{\dag}{\bf x}}{d_i^{\dag} b_i},
\end{equation}
explicitly showing that the effect of the change of base matrix $B^{-1}$ in \eqref{eq:eigenMatrixDiaglze} is the scalar projection of $\bf x$ along the left eigenvectors normalized by the scalar product between the right and left eigenvectors $d_i^{\dag} b_i.$
For the case in which the eigenvalues are complex the geometric interpretation of these straightforward operations is not completely transparent.
\section*{Geometric $\sqrt{-1}.$}
A geometric understanding of possible origins of a $\sqrt{-1}$ in an $n-$dimensional real vector space is provided by Geometric Algebra.
Extensively developed and forcefully advocated by Hestenes (see for example \cite{HestenesReform} and references therein), Geometric Algebra has attracted efforts from a variety of scholars who have embarked in its development and promotion in different contexts such as Physics~\cite{gull1993imaginary,DoranLasenby2003}, Computer Science~\cite{Dorst2009geometric} and undergraduate education~\cite{macdonaldTextbook}.
While this is no place for an exposition of Geometric Algebra (see \cite{macdonald2017survey} for a brief survey), we give a brief description of some of its aspects which are relevant here.
Geometric Algebra (GA) is an associative algebra (a Clifford algebra) whose product is the geometric product, which in GA is implied by yuxtaposition but we will denote it here by $\odot.$
Its result between two vectors yields the (direct) sum of a scalar and a {\it 2-blade} or {\it bivector} as
\begin{equation} \label{eq:geomProduct}
{\bf u}\odot{\bf v} = {\bf u}\cdot{\bf v} + {\bf u} \wedge {\bf v},
\end{equation}
where $\cdot$ is the usual dot product and $\wedge$ represents the exterior or wedge product, commonly referred to in GA as the outer product (not to be confused with the tensor product).
For our purposes, it will suffice to define it in terms of standard matrix multiplication as
\begin{equation} \label{eq:wedgeDef}
{\bf u} \wedge {\bf v} \equiv {\bf uv}^T-{\bf vu}^T.
\end{equation}
A 2-blade or bivector can be geometrically interpreted as an oriented area, that is, a piece of the plane spanned by ${\bf u}$ and ${\bf v}$ with an associated number (with magnitude equal to the area of the parallelogram defined by ${\bf u}$ and ${\bf v}),$ but with no particular shape.
In addition, a 2-blade can also represent the plane $\operatorname{span}({\bf u},{\bf v});$ it is a (real) scalar factor of the outer product of any pair of vectors forming a base of that plane.
As exemplified after eq. \ref{eq:wedgeDef}, the exterior product $\wedge$ generates $k-$blades, geometrical objects representing $k-$dimensional subspaces $(k\leq n)$ spanned by its linearly independent factors.
Here we will not need to go beyond $k=2.$
As exemplified by eq. \ref{eq:geomProduct}, the geometric product $\odot$ generates different types of geometrical objects from its factors and adds them together; the example in eq. \ref{eq:geomProduct} includes the addition of a scalar projection and a $2-$blade.
While the addition of two quantitites of different nature may appear unusual, it is similar to the addition of real and imaginary parts of complex numbers.
In fact, the similiarities between GA and complex numbers go further.
If ${\bf e}_1$ and ${\bf e}_2$ are any pair of orthonormal vectors in $\mathbb{R}^n,$ then
\begin{align}
({\bf e}_1\odot{\bf e}_2)\odot({\bf e}_1\odot{\bf e}_2)&=({\bf e}_1\odot{\bf e}_2)\odot(-{\bf e}_2\odot{\bf e}_1) \nonumber\\
&=-{\bf e}_1\odot({\bf e}_2\odot{\bf e}_2)\odot{\bf e}_1\\
&=-({\bf e}_2\cdot{\bf e}_2){\bf e}_1\odot{\bf e}_1=-{\bf e}_1\cdot{\bf e}_1=-1, \nonumber
\end{align}
where we have used eqs. \ref{eq:geomProduct} and \ref{eq:wedgeDef}, and the associativity of the geometric product.
Therefore, under the geometric product, any 2-blade in GA is proportional to a $\sqrt{-1}.$
The fact that GA gives a geometric origin and interpretation to the imaginary unit is often highlighted by advocates of GA; the geometric products \ref{eq:geomProduct} of any two vectors in a given plane have been called `GA complex numbers' since they are isomorphic to the usual complex numbers~\cite{macdonald2017survey}.
This, among other possible types of $\sqrt{-1}$ in GA, is the kind of imaginary unit that we will consider.
These 2-blades, together with the usual scalars and vectors, are the only objects from GA that we will need;
there will be no further explicit use of the geometric product $\odot.$
For any vector ${\bf x},$ and any pair of
vectors ${\bf u}$ and ${\bf v},$
we have
\begin{equation} \label{eq:bladeAction}
{\bf x}\cdot[({\bf u}\wedge{\bf v}) {\bf x}]=0.
\end{equation}
We mention two aspects about the action of ${\bf u}\wedge{\bf v}$ on ${\bf x}$ that appears in the square bracket.
First, as seen from definition \ref{eq:wedgeDef}, it is a vector contained in the plane $\operatorname{span}({\bf u},{\bf v}),$ with no contribution from ${\bf x}_{\perp},$ the part of ${\bf x}$ which is orthogonal to that plane.
Second, ${\bf x}_{\parallel},$ the part of ${\bf x}$ in the plane $\operatorname{span}({\bf u},{\bf v}),$ is made orthogonal to itself by the multiplication by ${\bf u}\wedge{\bf v};$ therefore the blade rotates ${\bf x}_{\parallel}$ by $\pi/2.$
This is the role of the imaginary unit as an operator on complex vectors in the complex plane, but it happens here with real vectors on a real geometric plane.
Inspection shows that this rotation is in the direction that brings ${\bf v}$ towards ${\bf u}.$
Given that much emphasis to geometric interpretation of imaginary units is given in GA, one expects the subject of complex eigenvalues to have been addressed.
This has been done by extending the notion of eigenvector to that of {\it eigenblade}, which represent invariant subspaces associated to linear operators (whose domain has been extended to all elements of GA, \cite{hestenes1984clifford,DoranLasenby2003}).
The eigenvalues associated to such eigenblades are real, and the precise geometric connection to the usual complex eigenvalues and eigenvectors has, to the best of our knowledge, not been explicitly established.
This is possibly the case because GA does not directly build simple linear operators like ${\bf u}{\bf v}^T,$ which involve the symmetric part of the tensor product.
While we borrow from GA the exterior (or outer) product as defined in eq. \ref{eq:wedgeDef} and its interpretation, we proceed with standard matrix notation to make the result more broadly accessible.
\section*{Real Decomposition with Complex Eigenvalues}
Let us consider a pair of complex conjugate eigenvalues $(\lambda,\lambda^{*})$ with their corresponding right and left eigenvectors $(b,b^{*})$ and $(d,d^{*})$ satisfying eqs. \ref{eq:eigenEq} and \ref{eq:leftEigenEq}, respectively.
We write the right and left eigenvectors in terms of their real and imaginary parts as
\begin{subequations}
\begin{align}
b &= {\bf b} + \sqrt{-1}\, {\bf p},\\
d &= {\bf d} + \sqrt{-1}\, {\bf q}.
\end{align}
\end{subequations}
The orthogonality satisfied between $d^*$ and $b$, i.e., $d^{*\dag}b=0,$ translates into the conditions
\begin{subequations}\label{eq:biorthogonalities}
\begin{align}
{\bf d} \cdot {\bf b} &= {\bf q} \cdot {\bf p}, \\
{\bf q} \cdot {\bf b} &= -{\bf d}\cdot {\bf p}.
\end{align}
\end{subequations}
If $b$
and $d$
were associated to different pairs of complex conjugate eigenvalues, the biorthogonality property
implies that each side of these equations would be zero.
The four conditions that define the right (eq. \ref{eq:eigenEq}) and left (eq. \ref{eq:leftEigenEq}) eigenvectors associated to the same complex eigenvalue $\lambda=\sigma + \sqrt{-1}\,\omega$ are simultaneously satisfied by
\begin{subequations}\label{eq:realDecomposition}
\begin{align}
M_{\lambda}&\equiv \frac{1}{\nu}({\bf b} \wedge {\bf p})\left[\omega \{{\bf dd}^T+{\bf qq}^T\} - \sigma ({\bf d} \wedge {\bf q})\right],\label{eq:realDecLeftBlade}\\
&= \frac{1}{\nu}\left[\omega \{{\bf bb}^T+{\bf pp}^T\} - \sigma ({\bf b} \wedge {\bf p})\right]({\bf d} \wedge {\bf q}),\label{eq:realDecRightBlade}
\end{align}
\end{subequations}
where the normalization is given by
\begin{equation}
\nu\equiv({\bf d\cdot b})^2+ ({\bf q\cdot b})^2.
\end{equation}
The equality between the right hand side expressions in eqs. \ref{eq:realDecLeftBlade} and \ref{eq:realDecRightBlade} follows from conditions \ref{eq:biorthogonalities}.
We note that every bracket in \ref{eq:realDecomposition} is invariant to the complex phase of the eigenvectors, and that the same expression is obtained if written in terms of the complex conjugate eigenvalue $\lambda^*.$
Interchanging the positions of the vectors $({\bf b},{\bf p})$ of the right eigenspace with those of the vectors $({\bf d},{\bf q})$ of the left corresponds to a transposition $(\,\cdot\,)^T$ and the substitution $\omega\rightarrow-\omega.$
There is a clear geometric interpretation of eqs. \ref{eq:realDecomposition} for the action of $M_{\lambda}$ on real vectors.
To aid the interpretation, it is useful to keep in the back of one's mind the case of a normal matrix (i.e. when $MM^{T}=M^{T}M),$ for which one can identify the right and left eigenvectors.
As seen from eq. \ref{eq:realDecRightBlade}, a column vector multiplied on the right is first projected on the plane of the left eigenvector represented by $({\bf d} \wedge {\bf q})$ and rotated by $\pi/2$ in the direction going from ${\bf q}$ to ${\bf d}.$
The resulting vector is then brought to the plane of the right eigenvector in two different ways, as given by the two terms in the square bracket of eq. \ref{eq:realDecRightBlade}.
The real part $\sigma$ of the eigenvalue multiplies the vector as is rotated back towards its original orientation by $-({\bf b} \wedge {\bf p}).$
The imaginary part $\omega$ of the eigenvalue multiplies the term in the curly brackets which is similar to a projection on the plane $\operatorname{span}({\bf b},{\bf p}),$ thus keeping the vector close to its rotated orientation.
In the case of a normal matrix, ${\bf b}$ and ${\bf p}$ are orthogonal to each other and of the same norm (as seen from conditions \ref{eq:biorthogonalities} with $d=b),$ so the curly brackets in eq. \ref{eq:realDecomposition} are proportional to the identity of the plane; the orientation is then preserved exactly after the first rotation by $\pi/2.$
As seen in eq. \ref{eq:realDecLeftBlade}, essentially the same description applies to a multiplication by a row vector on the left.
As mentioned above, $M_\lambda$ given in eq. \ref{eq:realDecomposition} reproduces the action of the full matrix $M$ on (left and right) eigenvectors associated to a complex eigenvalue $\lambda$ (eqs. \ref{eq:eigenEq} and \ref{eq:leftEigenEq}).
The biorthogonality property guarantees that right (left) multiplication of $M_{\lambda}$ by other right (resp. left) eigenvectors of $M$ is zero (which may require orthogonalization when there are repeated eigenvalues).
If the basis of eigenvectors is complete, then eq. \ref{eq:realDecomposition} can be used to provide a real geometric decomposition of the full matrix $M$ by replacing the complex conjugate pairs in eq. \ref{eq:eigenMatrixAdDecomp}.
To be explicit, and using ${\bf a}_i$ and ${\bf c}_i$ to denote right and left eigenvectors associated to, say, $j$ real eigenvalues $\alpha_i,$ a diagonalizable matrix can be decomposed as
\begin{equation}
M=\sum_{i=1}^j \alpha_i \frac{{\bf a}_i {\bf c}_i^T}{{\bf c}_i\cdot {\bf a}_i}+ \sum_{i=1}^k M_{\lambda_i},
\end{equation}
where $k$ is the number of complex conjugate pairs so that $j+2k=n,$ and the $M_{\lambda_i}$ are given by eq. \ref{eq:realDecomposition}.
This provides a real, coordinate free representation of real diagonalizable linear operators.
\section*{Acknowledgments}
The author acknowledges support of Nordita and the Swedish Research Council Grant No. 2018-04290; Nordita is partially supported by Nordforsk.
\input{references.bbl}
\end{document} | math | 16,395 |
\begin{document}
\begin{abstract}
Let $p>5$ be a fixed prime and assume that $\alpha_1,\alpha_2,\alpha_3$ are coprime to $p$. We study the asymptotic behavior of small solutions of congruences of the form $\alpha_1x_1^2+\alpha_2x_2^2+\alpha_3x_3^2\equiv 0\bmod{q}$ with $q=p^n$, where $\max\{|x_1|,|x_2|,|x_3|\}\le N$ and $(x_1x_2x_3,p)=1$. (In fact, we consider a smoothed version of this problem.) If $\alpha_1,\alpha_2,\alpha_3$ are fixed and $n\rightarrow \infty$, we establish an asymptotic formula (and thereby the existence of such solutions) under the condition $N\gg q^{1/2+\varepsilon}$. If these coefficients are allowed to vary with $n$, we show that this formula holds if $N\gg q^{11/18+\varepsilon}$. The latter should be compared with a result by Heath-Brown who established the existence of non-zero solutions under the condition $N \gg q^{5/8+\varepsilon}$ for odd square-free moduli $q$.
\end{abstract}
\title{Asymptotic behavior of small solutions of quadratic congruences in three variables modulo prime powers}
\tableofcontents
\section{Introduction and main results}
Recently, the authors published a short article \cite{arx} titled ``Pythagorean triples modulo prime powers'' on the arXiv preprint server. In this article, we studied small solutions of quadratic congruences of the form
$$
x_1^2+x_2^2-x_3^2\equiv 0\bmod{q},
$$
where $q=p^n$ is a power of a fixed prime $p$ and $n\rightarrow \infty$.
In the present paper, we investigate, more generally,
small solutions of quadratic congruences of the form
\begin{equation} \label{pythpn}
\alpha_1x_1^2+\alpha_2x_2^2+\alpha_3x_3^2\equiv 0\bmod{q}
\end{equation}
with prime power moduli $q$.
First, we give a brief review of some history of this problem.
Let $Q(x_1,...,x_n)$ be a quadratic form with integer coefficients. The question of detecting small solutions of congruences of the form
$$
Q(x_1,...,x_k)\equiv 0 \bmod{q}
$$
has received a lot of attention (see, in particular, \cite{Hea1}, \cite{Hea2} and \cite{Hea3}). Here we focus on the case $k=3$. In this case, a result by Schinzel, Schlickewei and Schmidt \cite{SSS} for general moduli $q$ implies that there is a non-zero solution $(x_1,x_2,x_3)\in \mathbb{Z}^3$ such that $\max\{|x_1|,|x_2|,|x_3|\}=O(q^{2/3})$, where the $O$-constant is absolute. The exponent $2/3$ was improved to $5/8+\varepsilon$ by Heath-Brown \cite{Hea3} for forms with $(\det Q,q)=1$ and $q$ odd and square-free.
A result by Cochrane \cite{Coc} for general moduli $q$ implies that for any {\it fixed} form $Q(x)$, there is a non-zero solution with $\max\{|x_1|,|x_2|,|x_3|\}=O(q^{1/2})$, where the $O$-constant may depend on the form. Of particular interest is the case when $q=p^n$ is a prime power. This was considered by Hakimi in \cite{Hak}, with emphasis on quadratic forms with a large number $k$ of variables.
In the present paper, we study the {\it asymptotic behavior} of small solutions of diagonal quadratic congruences \eqref{pythpn}
with $\max\{|x_1|,|x_2|,|x_3|\}\le N$ and $(x_1x_2x_3,q)=1$ if $q=p^n$ is a power of a fixed odd prime $p$. The condition $(x_1x_2x_3,q)=1$ automatically excludes the trivial solution $(0,0,0)$. We also assume that $(\alpha_1\alpha_2\alpha_3,q)=1$. For convenience, we consider a smoothed version of this problem (i.e., the solutions are suitably weighted). We study this problem both for fixed and arbitrary coefficients $\alpha_i$. In the case of fixed coefficients $\alpha_i$ and $n\rightarrow\infty$, we obtain an asymptotic formula if $N\gg q^{1/2+\varepsilon}$, and in the case of coefficients which are allowed to vary with $n$, we obtain such a formula for $N\gg q^{11/18+\varepsilon}$. This should be compared with Heath-Brown's above-mentioned result from \cite{Hea3}. He focused on the ``orthogonal'' situation when $q$ is an odd square-free number and obtained the slightly weaker exponent $5/8=0.625$ in place of $11/18=0.6\overline{1}$, only addressing the {\it existence} of non-zero solutions. As pointed out by Heath-Brown in \cite{Hea3}, the existence of {\it non-zero} solutions $\ll q^{\theta}$ for {\it all} odd moduli follows if one has established it for all {\it square-free} odd moduli, using the following simple observation: If $q=q_0^2q_1$ with $q_1$ square-free and $Q(x_1,x_2,x_3)\equiv 0 \bmod{q_1}$, then $Q(q_0x_1,q_0x_2,q_0x_3)\equiv 0 \bmod{q}$. For powers $q=p^n$ of a fixed odd prime $p$, this argument even gives the existence of {\it non-zero} solutions $\ll_{\varepsilon} q^{1/2+\varepsilon}$ if $n$ is large enough (providing only a small fraction of all solutions). More precisely, we get a non-zero solution $\ll q^{1/2+1/(2n)}$. However, if we restrict ourselves to solutions satisfying $(x_1x_2x_3,q)=1$, then this argument does not work any longer since $q_0$ is itself a power of $p$ if $q=p^n$.
Before we state our results, we explain why the exponent $1/2$ is the limit of our method by looking at the case $\alpha_1=1,\alpha_2=1,\alpha_3=-1$ of Pythagorean triples modulo prime powers (see also the discussion in \cite{arx}).
Small solutions of the congruence
\begin{equation} \label{pythcongruence}
x_1^2+x_2^2-x_3^2\equiv 0 \bmod{p^n}
\end{equation}
arise immediately from Pythagorean triples in $\mathbb{Z}^3$, provided that $p>5$. (If $p=2,3,5$, we have $x_1^2+x_2^2-x_3^2\not\equiv 0\bmod{p}$ if $(x_1x_2x_3,p)=1$.) It is known that the number of Pythagorean triples $(x_1,x_2,x_3)\in \mathbb{Z}^3$ satisfying $x_1^2+x_2^2=x_3^2$ such that $|x_3|\le N$ is $\sim cN\log N$ with $c=\pi/4$. It should not be difficult to modify this into an asymptotic of the form $\sim c_pN\log N$ with $c_p$ depending on $p$ if one includes the restriction $(x_1x_2x_3,p)=1$. If $N\le \sqrt{q/2}$ with $q=p^n$, then any solution $(x_1,x_2,x_3)$ of the congruence \eqref{pythpn} is in fact a Pythagorean triple. Hence, in this case, one expects an asymptotic of the form $\sim c_pN\log N$ for the number of solutions satisfying $(x_1x_2x_3,p)=1$ and $\max\{|x_1|,|x_2|,|x_3|\}\le N$ of the said congruence. In contrast, for much larger $N$, the expected number of solutions should be $\sim C_pN^3/q$ for a suitable constant $C_p>0$. In particular, one may expect this to hold for $N\ge q^{1/2+\varepsilon}$. Hence, there should be a transition between two different asymptotic formulas around the point $N=q^{1/2}$. Indeed, we will work out an asymptotic of the said form $\sim C_p\cdot N^3/q$ for $N\ge q^{\nu}$ with $\nu>1/2$ for general congruences of the form in \eqref{pythpn}, where $C_p$ depends on $p$ and the coefficients $\alpha_i$.
To state our results, we define the following quantity
$$
C_p(\alpha_1,\alpha_2,\alpha_3):=\frac{(p-s_p(\alpha_1,\alpha_2,\alpha_3))(p-1)}{p^2},
$$
where
\begin{equation} \label{sdef}
s_p(\alpha_1,\alpha_2,\alpha_3):=2+\left(\frac{-\alpha_1\alpha_2}{p}\right)+
\left(\frac{-\alpha_1\alpha_3}{p}\right)+\left(\frac{-\alpha_2\alpha_3}{p}\right).
\end{equation}
We note that if $q=p$ is a prime, then the total number $(x_1,x_2,x_3)$ of solutions to the congruence \eqref{pythpn} satisfying $(x_1x_2x_3,p)=1$ turns out to be $(p-1)(p-s_p(\alpha_1,\alpha_2,\alpha_3))$. So a solution exists if $p>s_p(\alpha_1,\alpha_2,\alpha_3)$. Our first main result is as follows.
\begin{theorem} \label{mainresult}
Let $\varepsilon>0$ be fixed, $p>2$ be a fixed prime and $\alpha_1,\alpha_2,\alpha_3$ be fixed integers which are coprime to $p$. Let $\Phi:\mathbb{R}\rightarrow \mathbb{R}_{\ge 0}$ be a Schwartz class function. Set $q:=p^n$. Then as $n\rightarrow \infty$, we have the asymptotic formula
\begin{equation} \label{main}
\sum\limits_{\substack{(x_1,x_2,x_3)\in \mathbb{Z}^3\\ (x_1x_2x_3,p)=1\\ \alpha_1x_1^2+\alpha_2x_2^2+\alpha_3x_3^2 \equiv 0 \bmod{q}}} \Phi\left(\frac{x_1}{N}\right)
\Phi\left(\frac{x_2}{N}\right)\Phi\left(\frac{x_3}{N}\right)\sim
\hat{\Phi}(0)^3\cdot C_p(\alpha_1,\alpha_2,\alpha_3)\cdot \frac{N^3}{q},
\end{equation}
provided that $N\ge q^{1/2+\varepsilon}$ and $p>s_p(\alpha_1,\alpha_2,\alpha_3)$.
\end{theorem}
Secondly, we establish the following result.
\begin{theorem} \label{mainresult2}
Let the conditions in Theorem \ref{mainresult} be kept except that $\alpha_1,\alpha_2,\alpha_3$ are no longer fixed but allowed to vary with $n$. Also suppose that $p>s_p(\alpha_1,\alpha_2,\alpha_3)$. Then the asymptotic formula \eqref{main} holds if $N\ge q^{11/18+\varepsilon}$.
\end{theorem}
Using the rapid decay of the weight function $\Phi$, we obtain the following existence result as a corollary of Theorems \ref{mainresult} and \ref{mainresult2} above.
\begin{corollary} \label{mainresult3} Let $\varepsilon>0$ be fixed and $p>2$ be a fixed prime. Set $q:=p^n$. For $\alpha_1,\alpha_2,\alpha_3$ being integers such that the congruence $\alpha_1x_1^2+\alpha_2x_2^2+\alpha_3x_3^3\equiv 0\bmod q$ is solvable in integers with $p\not| x_1x_2x_3$, let $m(\alpha_1,\alpha_2,\alpha_3;q)$ be the smallest value of $\max\{|x_1|,|x_2|,|x_3|\}$ for such a solution. If no such solution exists, set
$m(\alpha_1,\alpha_2,\alpha_3;q)=0$. Then we have the following.
\\
(i) If $\alpha_1,\alpha_2,\alpha_3$ are fixed and satisfy $(\alpha_1\alpha_2\alpha_3,p)=1$, then, as $n\rightarrow\infty$,
\begin{equation} \label{parti}
m(\alpha_1,\alpha_2,\alpha_3;q)\ll q^{1/2+\varepsilon},
\end{equation}
where the implied constant depends only on $p$, $\alpha_1,\alpha_2,\alpha_3$ and $\varepsilon$.
\\
(ii) As $n\rightarrow\infty$,
\begin{equation} \label{partii}
\max\limits_{\substack{\alpha_1,\alpha_2,\alpha_3 \bmod{q}\\ (\alpha_1\alpha_2\alpha_3,p)=1}} m(\alpha_1,\alpha_2,\alpha_3;q)\ll q^{11/18+\varepsilon},
\end{equation}
where the implied constant depends only on $p$ and $\varepsilon$.
\end{corollary}
$ $\\
{\bf Comments:}
\\
(a) Legendre \cite{Leg} gave a criterion for the non-trivial representability of 0 by a diagonal ternary quadratic form. If, in particular, there exists a solution $(x_1,x_2,x_3)$ of the equation $Q(x_1,x_2,x_3)=0$ with $(x_1x_2x_3,p)=1$, then \eqref{parti} holds trivially. Thus, part (i) of Corollory \ref{mainresult} is of interest only if $Q(x_1,x_2,x_3)$ does not represent 0 in the above form. Part (ii) of Corollary \ref{mainresult} is of general interest.
\\
(b) With some extra effort, it is possible to sharpen the bound in part (i) of the above Corollary \ref{mainresult3} to
$$
m(\alpha_1,\alpha_2,\alpha_3;q)\le C(p,\varepsilon)\max\{|\alpha_1|,|\alpha_2|,|\alpha_3|\}q^{1/2+\varepsilon},
$$
where the function $C$ depends on the weight function $\Phi$. Similarly, the implied constant in part (ii) is of the form $D(p,\varepsilon)$, where the function $D$ depends on $\Phi$. \\
We begin with proving Theorem \ref{mainresult} in two parts. In the first part we deal with the case when one of $-\alpha_i\alpha_j$ with $i\not= j$ is a quadratic residue modulo $p$ (without loss of generality, we may take $i=2$ and $j=3$). In the second part we cover the complementary case when none of $-\alpha_i\alpha_j$ with $i\not=j$ is a quadratic residue modulo $p$.
Key ingredients in our method are a parametrization of $\mathbb{Q}_p$-rational points $(z_1,z_2)$ on the conic
$$
\alpha_1z_1^2+\alpha_2z_2^2=-\alpha_3,
$$
repeated use of Poisson summation and an explicit evaluation of complete exponential sums with rational functions to prime power moduli due to Cochrane \cite{CoZ}. This transforms the problem into a dual problem which in the case of fixed coefficients amounts to counting solutions of quadratic Diophantine {\it equations} (rather than {\it congruences}). Our method generalizes that in \cite{arx}.
To prove Theorem \ref{mainresult2}, we will observe that here our dual problem essentially amounts to bounding from above the number of solutions of {\it congruences} of the form
$$
\beta_1x_1^2+\beta_2x_2^2+\beta_3x_3^2\equiv 0 \bmod{q'}
$$
with $q'|q$ in boxes $\max\{|x_1|,|x_2|,|x_3|\}\le M$ with $M$ of size roughly $q'/N$,
where $\beta_1,\beta_2,\beta_3$ depend on $\alpha_1,\alpha_2,\alpha_3$. So these new boxes are much smaller than those in the original problem, but we don't need to establish an asymptotic formula here. To obtain an upper bound which allows us to beat the exponent $2/3$, we consider two cases depending on Diophantine properties of the fractions $\beta_1\overline{\beta_3}/q'$ and $\beta_2\overline{\beta_3}/q'$, where $\overline{\beta_3}$ is a multiplicative inverse of $\beta_3$ modulo $q'$. In the first case, we shall reduce the problem using the Cauchy-Schwarz inequality to counting small solutions of certain linear congruences. In the second case we turn the problem into counting solutions of quadratic Diophantine {\it equations}, similarly as in our proof of Theorem \ref{mainresult}.
Instead of results with smooth weights, it should also be possible to produce similarly strong results with sharp cutoff, but the technical details become then more complicated.\\ \\
{\bf Acknowledgements.} The authors would like to thank the Ramakrishna Mission Vivekananda Educational and Research Institute for providing excellent working conditions. The second-named author would like to thank CSIR, Govt. of India for financial support in the form of a Junior Research Fellowship under file number 09/934(0016)/2019-EMR-I. \\ \\
{\bf Data availability statement:} This manuscript has no associated data.\\ \\
{\bf Conflict of interest statement:} The authors have no conflicts of interest to declare. All co-authors have seen and agree with the contents of the manuscript and there is no financial interest to report. We certify that the submission is original work and is not under review at any other publication.
\section{Preliminaries}
We will use the notation
$$
e_q(z):=e\left(\frac{z}{q}\right)=e^{2\pi i z/q}
$$
for $q\in \mathbb{N}$ and denote by $G_q$ the quadratic Gauss sum
$$
G_q=\sum\limits_{x=1}^q e_q(x^2).
$$
We recall that if $q$ is odd, then
$$
G_q=\sum\limits_{y=1}^q \left(\frac{y}{q}\right)e_q(y),
$$
where $\left(\frac{y}{q}\right)$ is the Jacobi symbol. We further recall that in this case, $|G_q|=\sqrt{q}$. Throughout the sequel, we will write $\overline{\alpha}$ for a multiplicative inverse of $\alpha$ to the relevant modulus, which will always be apparent from the context. Implicitly, we often make use of the fact that $\overline{\alpha}$ is a quadratic (non-)residue modulo $p$ if and only if $\alpha$ is.
The following preliminaries will be needed in the course of this paper.
\begin{Proposition}[Parametrization of points on a conic] \label{para}
Let $K$ be a field and assume that $\alpha_1,\alpha_2,\alpha_3\in K^{\ast}$ and $a,b\in K$ such that
$$
\alpha_1a^2+\alpha_2b^2=-\alpha_3.
$$
Then all $K$-rational points on the conic
\begin{equation*}
\alpha_1z_1^2+\alpha_2z_2^2=-\alpha_3
\end{equation*}
are parametrized in the form
$$
(z_1,z_2)=\left(a-2\alpha_2\frac{at^2-bt}{\alpha_1+\alpha_2t^2},-b-2\alpha_1\frac{at-b}{\alpha_1+\alpha_2t^2}\right),
$$
where $t\in K$ with $t^2\not=-\alpha_1/\alpha_2$. The map
\begin{equation} \label{bijection}
m : \left\{t\in K : t^2\not=-\frac{\alpha_1}{\alpha_2}\right\} \longrightarrow \left\{(z_1,z_2)\in K^2 : \alpha_1z_1^2+\alpha_2z_2^2=-\alpha_3\right\}
\end{equation}
defined by
\begin{equation} \label{mt}
m(t):=\left(a-2\alpha_2\frac{at^2-bt}{\alpha_1+\alpha_2t^2},-b-2\alpha_1\frac{at-b}{\alpha_1+\alpha_2t^2}\right)
\end{equation}
is bijective.
\end{Proposition}
\begin{proof}
We use a standard method of parametrization. Given a $K$-rational point $Q=(a,b)$ on the conic and $t\in K$, by B\'ezout's theorem, the line $\mathcal{L}(t)$ through $Q$ given by the equation $z_2-b=t(z_1-a)$ intersects the conic in $Q$ and at most one more point $m(t)$. This point may be $Q$ itself, in which case $\mathcal{L}(t)$ is the tangent to the conic at $Q$. Conversely, if $P$ is a $K$-rational point on the conic, then there exists precisely one line through $P$ and $Q$ (which is the tangent in the case $P=Q$) with rational slope $t$. Hence, we have a bijection between the $K$-rational points on the conic and the set of $t\in K$ for which $m(t)$ exists. To find $m(t)$, we write
$$\alpha_1z_1^2+\alpha_2z_2^2=\alpha_1a^2+\alpha_2b^2,$$
which implies
$$\alpha_1(z_1-a)(z_1+a)=\alpha_2(z_2-b)(z_2+b).$$
Now we plug in $z_2-b=t(z_1-a)$ and get
$$\alpha_1(z_1+a)+\alpha_2t(2b+t(z_1-a))=0.$$
Therefore
\begin{equation*}
\begin{split}
z_1=&\frac{-a\alpha_1-\alpha_2t(2b-at)}{\alpha_1+\alpha_2t^2}\\=&-a+2\alpha_2t\frac{at-b}{\alpha_1+\alpha_2t^2}
\end{split}
\end{equation*}
and
\begin{equation}
\begin{split}
z_2&=b+(z_1-a)t\\
&=b+t\left(-2a+2\alpha_2t\frac{at-b}{\alpha_1+\alpha_2t^2}\right)\\&=b-2t\frac{a\alpha_1+b\alpha_2t}{\alpha_1+\alpha_2t^2}\\&=-b-2\alpha_1\frac{at-b}{\alpha_1+\alpha_2t^2},
\end{split}
\end{equation}
which exist if $t^2\not= -\alpha_1/\alpha_2$.
This gives the desired parametrization in \eqref{mt}, and the map in \eqref{bijection} is bijective.
\end{proof}
\begin{Proposition} \label{nofroot}
Let $p>2$ be a prime and $n\in \mathbb{N}$. Then the number of solutions $(x_1,x_2) \bmod p^n$ of the congruence
$$
\gamma_1x_1^2+\gamma_2x_2^2\equiv 1\bmod{p^n}
$$
is $p^n+p^{n-1}$ if $(\gamma_1\gamma_2,p)=1$ and $-\gamma_1\gamma_2$ is a quadratic non-residue modulo $p$.
\end{Proposition}
\begin{proof}
This is given in \cite[Corollary 35]{NoS}.
\end{proof}
\begin{Proposition} \label{quadequations} Let $A,B,C\in \mathbb{Z}\setminus \{0\}$ and $x\ge 1$. Then the number of solutions $(X,Y)\in \mathbb{Z}$ with $\max\{|X|,|Y|\}\le x$ of the equation
$$
AX^2+BY^2=C
$$
is bounded by $O(|ABCx|^{\varepsilon})$.
\end{Proposition}
\begin{proof} We first multiply the equation in question by $A$ to get
$$
(AX)^2+ABY^2=AC.
$$
Hence, it suffices to prove that there are at most $O(|ABCx|^{\varepsilon})$ solutions of the equation
$$
U^2+ABV^2=AC
$$
with $\max\{|U|,|V|\}\le |A|x$. We may write the above equation as
$$
(U+V\sqrt{-AB})(U-V\sqrt{-AB})=AC.
$$
Let $\mathcal{O}_K$ be the ring of integers of the number field $K:=\mathbb{Q}(\sqrt{-AB})$. The number of divisors of the ideal $\mathfrak{C}=(AC)$ is $\ll \mathcal{N}_{K:\mathbb{Q}}(\mathfrak{C})^{\varepsilon}\le |AC|^{2\varepsilon}$. Hence, if $\mathfrak{A}$ is a principal ideal divisor of $\mathfrak{C}$, then it suffices to show that $\mathfrak{A}$ has at most $O(|ABCx|^{\varepsilon})$ generators of the form $U+V\sqrt{-AB}$ with $\max{|U|,|V|}\le |A|x$. Hence, we must show that if $U_0+V_0\sqrt{-AB}$ is such a generator, then there are at most $O(|ABCx|^{\varepsilon})$ units in $\mathcal{O}_K$ such that $u(U_0+V_0\sqrt{-AB})=U_1+V_1\sqrt{-AB}$, where
$\max\{|U_1|,|V_1|\}\le |A|x$. This is trivial if $-AB$ is a square or $AB>0$ since then the number of units in $\mathcal{O}_K$ is bounded by 6. Assume now that $AB<0$ is not a square and $-AB=st^2$, where $s$ is square-free. Then $K=\mathbb{Q}(\sqrt{s})$ is a real-quadratic field, and there is a fundamental unit of the form
$$
\epsilon=a+b\sqrt{s}>1
$$
with $a,b\in \mathbb{Z}/2$, where $a,b\not=0$. Hence, $u=\pm \epsilon^k$ for some $k\in \mathbb{Z}$. Let $R:=|A|^{3/2}|B|^{1/2}|x|$. We observe that
$$
\frac{1}{R}\ll U_1+V_1\sqrt{-AB}\ll R
$$
since
$$
|U_1+V_1\sqrt{-AB}|\cdot |U_1-V_1\sqrt{-AB}|=|AC|\ge 1.
$$
Hence,
$$
\frac{1}{R}\ll \epsilon^k|U_0+V_0\sqrt{-AB}| \ll R,
$$
and the number of possible $k$'s is bounded by
$$
\ll \log_{\epsilon} R^2\ll \log R \ll \log|2ABx|,
$$
using the well-known fact that $\epsilon\ge (1+\sqrt{5})/2$ for every $s$.
Thus, we get $O(\log|2ABx|)$ possible units $u$, which completes the proof.
\end{proof}
\begin{Proposition}[Poisson summation formula] \label{Poisson} Let $\Phi : \mathbb{R}\rightarrow \mathbb{R}$ be a Schwartz class function, $\hat\Phi$ its Fourier transform. Then
$$
\sum\limits_{n\in \mathbb{Z}} \Phi(n)=\sum\limits_{n\in \mathbb{Z}} \hat\Phi(n).
$$
\end{Proposition}
\begin{proof}
See \cite[section 3]{notes}.
\end{proof}
\begin{Proposition}[Evaluation of exponential sums with rational functions] \label{Expsums}
Let $p>2$ be a prime, $n\ge 2$ be a natural number and $f=F_1/F_2$ be a rational function where $F_1,F_2\in \mathbb{Z}[x]$. For a polynomial $G$ over $\mathbb{Z}$, let $\mbox{ord}_p(G)$ be the largest power of $p$ dividing all of the coefficients of $G$, and for a rational function $g=G_1/G_2$ with $G_1$ and $G_2$ polynomials over $\mathbb{Z}$, let $\mbox{ord}_p(g) := \mbox{ord}_p(G_1)-\mbox{ord}_p(G_2)$. Set
$$
r:=\mbox{ord}_p(f'),
$$
and
$$
S_{\alpha}(f;p^n):=\sum\limits_{\substack{x=1\\ x\equiv \alpha \bmod{p}}}^{p^n} e_{p^n}(f(x)),
$$
where $\alpha\in \mathbb{Z}$.
Then we have the following if $r\le n-2$ and $(F_2(\alpha),p)=1$.
\\
(i) If $p^{-r}f'(\alpha)\not\equiv 0\bmod{p}$, then $S_{\alpha}(f,p^n) = 0$.
\\
(ii) If $\alpha$ is a root of $p^{-r}f'(x)\equiv 0\bmod{p}$ of multiplicity one, then
$$
S_{\alpha}(f;p^n) =\begin{cases} e\left(f(\alpha^{\ast})\right)p^{(n+r)/2} & \mbox{ if } n-r \mbox{ is even,}\\
e\left(f(\alpha^{\ast})\right)p^{(n+r)/2}\left(\frac{A(\alpha)}{p}\right)\cdot \frac{G_p}{\sqrt{p}} & \mbox{ if } n-r \mbox{ is odd,}
\end{cases}
$$
where $\alpha^{\ast}$ is the unique lifting of $\alpha$ to a solution of the congruence $p^{-r}f'(x) \equiv 0 \bmod p^{[(n-r+1)/2]}$ and
$$
A(\alpha):=2p^{-r}f''(\alpha^{\ast}).
$$
\end{Proposition}
\begin{proof} This is \cite[Theorem 3.1(iii)]{CoZ}.
\end{proof}
\section{Proof of Theorem \ref{mainresult}}
\subsection{Parametrization of solutions - Case I}
In the following, we consider the case when $-\alpha_2\alpha_3$ is a quadratic residue modulo $p$.
The congruence in question resembles the equation
\begin{equation} \label{conic}
\alpha_1 x_1^2+\alpha_2 x_2^2+\alpha_3x_3^2=0
\end{equation}
of a conic in homogeneous coordinates. Let us first see that every solution $(x_1,x_2,x_3)\in \mathbb{Z}^3$ with $(x_1x_2x_3,p)=1$ of the congruence
\begin{equation} \label{congru1}
\alpha_1 x_1^2+\alpha_2 x_2^2+\alpha_3x_3^2\equiv 0 \bmod{p^n}
\end{equation}
comes from a solution of \eqref{conic} in the $p$-adic integers. To this end, we need to use a Hensel-type argument. Let $(x_1,x_2,x_3)$ be such a solution. We would like to lift it to a solution $(\tilde{x}_1,\tilde{x}_2,\tilde{x}_3)$ of the congruence
\begin{equation} \label{congru2}
\alpha_1\tilde{x}_1^2+\alpha_2\tilde{x}_2^2+\alpha_3\tilde{x}_3^2\equiv 0\bmod{p^{n+1}}.
\end{equation}
So we consider $\tilde{x}_i=x_i+k_ip^n$ with $k_i=0,...,p-1$, $i=1,2,3$ and satisfying
$$
\alpha_1(x_1+k_1p^n)^2+\alpha_2(x_2+k_2p^n)^2+\alpha_3(x_3+k_3p^n)^2\equiv 0 \bmod{p^{n+1}}.
$$
Expanding the squares and using $2n\ge n+1$, this is equivalent to
$$
\alpha_1x_1^2+\alpha_2x_2^2+\alpha_3x_3^2+2\alpha_1k_1p^nx_1+2\alpha_2k_2p^nx_2+2\alpha_3k_3p^nx_3\equiv 0 \bmod{p^{n+1}},
$$
which in turn is equivalent to
$$
\frac{\alpha_1x_1^2+\alpha_2x_2^2+\alpha_3x_3^2}{p^n}+2\alpha_1x_1k_1+2\alpha_2x_2k_2+2\alpha_3x_3k_3\equiv 0\bmod{p}.
$$
This linear congruence in $k_1,k_2,k_ 3$ has exactly $p^2$ solutions. So in particular, a solution $(x_1,x_2,x_3)$ of \eqref{congru1} lifts to a solution $(\tilde{x}_1,\tilde{x}_2,\tilde{x}_3)$ of \eqref{congru2}. So indeed, every solution of \eqref{congru1} arises from a solution of \eqref{conic} in $\mathbb{Z}_p$.
Now we parametrize these solutions. Since $-\alpha_2\alpha_3$ is a quadratic residue modulo $p$, there exists $b\in \mathbb{Q}_p$ such that $b^2=-\alpha_3/\alpha_2$. Then $(0,-b)$ is a point on the conic
\begin{equation} \label{conics}
\alpha_1z_1^2+\alpha_2z_2^2=-\alpha_3,
\end{equation}
and using Proposition \ref{para} with $(0,-b)$ in place of $(a,b)$, the $\mathbb{Q}_p$-rational points $(z_1,z_2)$ on this conic are parametrized as
$$\left(z_1,z_2\right)=\left(-\frac{2tb\alpha_2}{\alpha_1+\alpha_2t^2},-\frac{b(\alpha_1-\alpha_2t^2)}{\alpha_1+\alpha_2 t^2}\right),
$$
where $t\in \mathbb{Q}_p$ with $t^2\not=-\alpha_1/\alpha_2$.
Now if $(x_1,x_2,x_3)$ is a solution of \eqref{conic} in the $p$-adic integers, where $x_3\not=0$, then
$$
\left(\frac{x_1}{x_3},\frac{x_2}{x_3}\right)
$$
is a point on the conic in \eqref{conics}. Hence, we have
$$
\left(\frac{x_1}{x_3},\frac{x_2}{x_3}\right)=\left(-\frac{2tb\alpha_2}{\alpha_1+\alpha_2t^2},-\frac{b(\alpha_1-\alpha_2t^2)}{\alpha_1+\alpha_2 t^2}\right).
$$
But we restricted ourselves to triples $(x_1,x_2,x_3)$ with $|x_i|_p=1$, $i=1,2,3$. Hence, we have
$$
1=\left|\frac{x_1}{x_3}\right|_p=\left|\frac{2tb\alpha_2}{\alpha_1+\alpha_2t^2}\right|_p
$$
and
$$
1=\left|\frac{x_2}{x_3}\right|_p=\left|\frac{b(\alpha_1-\alpha_2t^2)}{\alpha_1+\alpha_2 t^2}\right|_p.
$$
We claim that $|t|_p=1$. Indeed,
if $|t|_p<1$, then
$$
\left|\frac{2tb\alpha_2}{\alpha_1+\alpha_2t^2}\right|_p=|t|_p<1,
$$
a contradiction, and if $|t|_p>1$, then
$$
\left|\frac{2tb\alpha_2}{\alpha_1+\alpha_2t^2}\right|_p=|t|_p^{-1}<1.
$$
Hence $|t|_p=|\alpha_1+\alpha_2t^2|_p=|\alpha_1-\alpha_2t^2|_p=1$, and
$$
x_1=-b(\alpha_1-\alpha_2t^2)u, \quad x_2=-2b\alpha_2tu, \quad x_3=(\alpha_1+\alpha_2t^2)u,
$$
where $u$ is a unit in the ring of $p$-adic integers $\mathbb{Z}_p$, i.e. $|u|_p$=1.
By reducing modulo $p^n$, we deduce that the solutions of the congruence
$$
\alpha_1y_1^2+\alpha_2y_2^2+\alpha_3\equiv 0 \bmod{p^n}
$$
with $(y_1y_2,p)=1$ are parametrized in the form
\begin{equation} \label{para1}
y_1=-\frac{b(\alpha_1-\alpha_2t^2)}{\alpha_1+\alpha_2t^2}, \quad y_2=-\frac{2b\alpha_2t}{\alpha_1+\alpha_2t^2}, \quad t \bmod{p^n},\ (t(\alpha_1-\alpha_2t^2)(\alpha_1+\alpha_2t^2),p)=1.
\end{equation}
Here $1/(\alpha_1+\alpha_2t^2)$ stands for a multiplicative inverse of $\alpha_1+\alpha_2t^2 \bmod{p^n}$. Moreover, the pairs $(y_1,y_2)$ given as in \eqref{para1} are distinct modulo $p^n$ by the following argument: Suppose that
\begin{equation} \label{t1t21}
\frac{b(\alpha_1-\alpha_2t_1^2)}{\alpha_1+\alpha_2t_1^2}\equiv \frac{b(\alpha_1-\alpha_2t_2^2)}{\alpha_1+\alpha_2t_2^2} \bmod{p^n}
\end{equation}
and
\begin{equation} \label{t1t22}
\frac{2b\alpha_2t_1}{\alpha_1+\alpha_2t_1^2}\equiv \frac{2b\alpha_2t_2}{\alpha_1+\alpha_2t_2^2} \bmod{p^n}.
\end{equation}
Then from \eqref{t1t21} it follows upon multiplying both sides with the denominators that
$$
b(\alpha_1^2-\alpha_2^2t_1^2t_2^2+\alpha_1\alpha_2t_2^2-\alpha_1\alpha_2t_1^2)\equiv b(\alpha_1^2-\alpha_2^2t_1^2t_2^2-\alpha_1\alpha_2t_2^2+\alpha_1\alpha_2t_1^2)\bmod{p^{n}}
$$
and hence
$$
t_1^2\equiv t_2^2\bmod{p^n}.
$$
So if $t_1\not\equiv t_2\bmod{p^n}$, then $t_1\equiv -t_2\bmod{p^n}$. However, in this case, \eqref{t1t22} implies $t_1\equiv t_2\equiv 0\bmod{p^n}$ contradicting the assumption that $(t_i,p)=1$ for $i=1,2$.
\subsection{Double Poisson summation}\label{fdob} We start by writing
\begin{equation*}
\begin{split}
T= & \sum\limits_{\substack{(x_1,x_2,x_3)\in \mathbb{Z}^3\\ (x_1x_2x_3,p)=1\\ \alpha_1x_1^2+\alpha_2x_2^2+\alpha_3x_3^2\equiv 0 \bmod{p^n}}} \Phi\left(\frac{x_1}{N}\right)\Phi\left(\frac{x_2}{N}\right)\Phi\left(\frac{x_3}{N}\right)\\
=& \sum\limits_{(x_3,p)=1} \Phi\left(\frac{x_3}{N}\right)\sum\limits_{\substack{y_1,y_2\bmod{p^n}\\ (y_1y_2,p)=1 \\ \alpha_1y_1^2+\alpha_2y_2^2+\alpha_3\equiv 0\bmod{p^n}}} \sum\limits_{\substack{x_1\equiv x_3y_1\bmod{p^n}\\ x_2\equiv x_3y_2\bmod{p^n}}} \Phi\left(\frac{x_1}{N}\right)\Phi\left(\frac{x_2}{N}\right).
\end{split}
\end{equation*}
Now we apply Poisson summation, Proposition \ref{Poisson}, after a linear change of variables to the inner double sum over $x_1$ and $x_2$, obtaining
\begin{equation*}
T= \frac{N^2}{p^{2n}}\sum\limits_{(x_3,p)=1} \Phi\left(\frac{x_3}{N}\right)\sum\limits_{(k_1,k_2)\in \mathbb{Z}^2} \hat{\Phi}\left(\frac{k_1N}{p^n}\right)\hat{\Phi}\left(\frac{k_2N}{p^n}\right)\sum\limits_{\substack{y_1,y_2\bmod{p^n}\\ (y_1y_2,p)=1 \\ \alpha_1y_1^2+\alpha_2y_2^2+\alpha_3\equiv 0\bmod{p^n}}} e_{p^n}\left(k_2x_3y_1+k_1x_3y_2\right).
\end{equation*}
Using our parametrization \eqref{para1}, we deduce that
\begin{equation*}
T= \frac{N^2}{p^{2n}}\sum\limits_{(x_3,p)=1} \Phi\left(\frac{x_3}{N}\right)\sum\limits_{(k_1,k_2)\in \mathbb{Z}^2} \hat{\Phi}\left(\frac{k_1N}{p^n}\right)\hat{\Phi}\left(\frac{k_2N}{p^n}\right)\sum\limits_{\substack{t \bmod{p^n}\\
(t(\alpha_1-\alpha_2t^2)(\alpha_1+\alpha_2t^2),p)=1 }} e_{p^n}\left(x_3\cdot \frac{2k_1b\alpha_2t+k_2b(\alpha_1-\alpha_2t^2)}{\alpha_1+\alpha_2t^2}\right).
\end{equation*}
We decompose $T$ into
\begin{equation} \label{divide}
T=T_0+U,
\end{equation}
where $T_0$ is the main term contribution of $(k_1,k_2)=(0,0)$. It follows that
\begin{equation*}
T_0= \hat{\Phi}(0)^2\cdot \frac{N^2}{p^{2n}}\sum\limits_{(x_3,p)=1} \Phi\left(\frac{x_3}{N}\right) \cdot p^{n-1}(p-s_p(\alpha_1,\alpha_2,\alpha_3)),
\end{equation*}
where $s_p(\alpha_1,\alpha_2,\alpha_3))$ is defined as in \eqref{sdef}. To see this, we note that the congruence $(\alpha_1-\alpha_2t^2)(\alpha_1+\alpha_2t^2)\equiv 0 \bmod{p}$ has four solutions modulo $p$ if
$$
\left(\frac{-\alpha_1\alpha_2}{p}\right)=1 \quad \mbox{and} \quad
\left(\frac{\alpha_1\alpha_2}{p}\right)=1,
$$
two solution if these Legendre symbols have opposite signs and no solution if they are both equal to $-1$. Moreover, since we assumed that
$$
\left(\frac{-\alpha_2\alpha_3}{p}\right)=1,
$$
we have
$$
\left(\frac{\alpha_1\alpha_2}{p}\right)=\left(\frac{-\alpha_1\alpha_3}{p}\right).
$$
In each case, the congruence
$t(\alpha_1-\alpha_2t^2)(\alpha_1+\alpha_2t^2)\equiv 0 \bmod{p}$
has precisely $s_p(\alpha_1,\alpha_2,\alpha_3)$ solutions.
If $N\ge p^{n\varepsilon}$ for any fixed $\varepsilon>0$, then the term $T_0$ can be simplified into
\begin{equation} \label{T0I}
\begin{split}
T_0= & \hat{\Phi}(0)^2 \cdot \frac{p-s_p(\alpha_1,\alpha_2,\alpha_3)}{p}\cdot \frac{N^2}{p^n} \cdot \left(\sum\limits_{x} \Phi\left(\frac{x}{N}\right) -\sum\limits_{x} \Phi\left(\frac{x}{N/p}\right)\right)\\
= & \hat{\Phi}(0)^2 \cdot \frac{p-s_p(\alpha_1,\alpha_2,\alpha_3)}{p}\cdot \frac{N^2}{p^n}\cdot \left(N\cdot \frac{p-1}{p}\cdot \hat\Phi(0)
+\sum\limits_{y\in \mathbb{Z}\setminus\{0\}} \left(N\hat\Phi(Ny)-\frac{N}{p}\cdot \hat\Phi\left(\frac{Ny}{p}\right)\right)\right)\\
= & \hat{\Phi}(0)^3 \cdot \frac{(p-s_p(\alpha_1,\alpha_2,\alpha_3))(p-1)}{p^2}\cdot \frac{N^3}{p^{n}}\cdot\left(1+o(1)\right)=\hat{\Phi}(0)^3 \cdot C_p(\alpha_1,\alpha_2,\alpha_3)\cdot \frac{N^3}{p^{n}}\cdot\left(1+o(1)\right)
\end{split}
\end{equation}
as $n\rightarrow\infty$,
where we again use Poisson summation for the sums over $x$ above and the rapid decay of $\hat\Phi$.
\subsection{Evaluation of exponential sums} Now we look at the error contribution
\begin{equation} \label{errorcont}
U= \frac{N^2}{p^{2n}}\sum\limits_{(x_3,p)=1} \Phi\left(\frac{x_3}{N}\right)\sum\limits_{(k_1,k_2)\in \mathbb{Z}^2\setminus \{(0,0)\}} \hat{\Phi}\left(\frac{k_1N}{p^n}\right)\hat{\Phi}\left(\frac{k_2N}{p^n}\right) E\left(k_1,k_2,x_3;p^n\right)
\end{equation}
with
\begin{equation*}
E\left(k_1,k_2,x_3;p^n\right):=\sum\limits_{\substack{t \bmod{p^n}\\
(t(\alpha_1-\alpha_2t^2)(\alpha_1+\alpha_2t^2),p)=1 }} e_{p^n}\left(x_3\cdot \frac{2k_1b\alpha_2t+k_2b(\alpha_1-\alpha_2t^2)}{\alpha_1+\alpha_2t^2}\right).
\end{equation*}
Assume that
$$
(k_1,k_2,p^n)=p^r.
$$
The contribution of $r=n-1,n$ to the right-hand side of \eqref{errorcont} is $O_{\varepsilon}(1)$ if $N\ge p^{n\varepsilon}$ by the rapid decay of $\hat\Phi$ since $(k_1,k_2)=(0,0)$ is excluded from the summation. In the following, we assume that $r\le n-2$ so that Proposition \ref{Expsums} is applicable.
We split the inner sum over $t$ into
\begin{equation*} \label{split}
E\left(k_1,k_2,x_3;p^n\right)=\sum\limits_{\substack{\alpha=1\\ \alpha \not\equiv 0 \bmod{p}\\\alpha^2\not\equiv \pm \alpha_1\overline{\alpha_2}\bmod{p}}}^p S_{\alpha}\left(f_{k_1,k_2};p^n\right),
\end{equation*}
where
\begin{equation*} \label{Salphadef}
S_{\alpha}\left(f_{k_1,k_2};p^n\right)=\sum\limits_{\substack{t \bmod{p^n}\\ t\equiv \alpha\bmod{p}}} e_{p^n}\left(f_{k_1,k_2}(t)\right)
\end{equation*}
with
\begin{equation*} \label{fdef}
f_{k_1,k_2}(t):=x_3\cdot \frac{2k_1b\alpha_2t+k_2b(\alpha_1-\alpha_2t^2)}{\alpha_1+\alpha_2t^2}.
\end{equation*}
Here, for convenience, we have suppressed the dependency on $x_3$ in our notations of $S_{\alpha}\left(f_{k_1,k_2};p^n\right)$ and $f_{k_1,k_2}(t)$.
We calculate that
\begin{equation*}
\begin{split}
f_{k_1,k_2,x_3}'(t)
= & 2x_3b\alpha_2\cdot \frac{k_1(\alpha_1-\alpha_2t^2)-2k_2\alpha_1t}{(\alpha_1+\alpha_2t^2)^2}.
\end{split}
\end{equation*}
Set
$$
l_1:=\frac{k_1}{p^r}, \quad l_2:=\frac{k_2}{p^r}.
$$
Then using Proposition \ref{Expsums}, if $\alpha_2\alpha^2+\alpha_1\not\equiv 0\bmod{p}$, we have $S_{\alpha}\left(f_{k_1,k_2,x_3};p^n\right)=0$ unless
\begin{equation} \label{keycong}
2\alpha_1l_2\alpha\equiv l_1(\alpha_1-\alpha_2\alpha^2) \bmod{p}.
\end{equation}
If $\alpha^2\not\equiv 0,\alpha_1\overline{\alpha_2}\bmod{p}$, then it follows that $(l_1l_2,p)=1$ and
$$
(k_1,p^n)=p^r=(k_2,p^n).
$$
In summary, we have
\begin{equation*}
U= \frac{N^2}{p^{2n}}\sum\limits_{(x_3,p)=1} \Phi\left(\frac{x_3}{N}\right)\sum\limits_{r=0}^{n-2} \sum\limits_{\substack{\alpha=1\\ \alpha^2 \not\equiv 0,\pm \alpha_1\overline{\alpha_2}
\bmod{p}}}^p \sum\limits_{\substack{(l_1,l_2)\in \mathbb{Z}^2\\ (l_1l_2,p)=1\\ 2\alpha_1l_2\alpha\equiv l_1(\alpha_1-\alpha_2\alpha^2) \bmod{p}}} \hat{\Phi}\left(\frac{l_1N}{p^{n-r}}\right)\hat{\Phi}\left(\frac{l_2N}{p^{n-r}}\right) S_{\alpha}\left(f_{p^rl_1,p^rl_2};p^n\right)+O_{\varepsilon}(1)
\end{equation*}
if $N\ge p^{n\varepsilon}$.
Let $D:=\alpha_1\alpha_2l_1^2+\alpha_1^2l_2^2$. The congruence \eqref{keycong} has a double root $\alpha \bmod{p}$ iff $D\equiv 0 \bmod{p}$, and in this case we get $\alpha^2\equiv -\alpha_1\overline{\alpha_2}\bmod{p}$ which is excluded from the summation over $\alpha$. Hence, only the case $D\not\equiv 0\bmod{p}$ occurs in which we have no root if $D$ is a quadratic non-residue modulo $p$ and two roots of multiplicity one if $D$ is a quadratic residue modulo $p$. Therefore, we may assume from now on that $D\not\equiv 0 \bmod{p}$ and $D$ is a quadratic residue modulo $p$. Then using Proposition \ref{Expsums}, if $\alpha$ satisfies \eqref{keycong}, we obtain
$$
S_{\alpha}\left(f_{p^rl_1,p^rl_2};p^n\right)=
\begin{cases}
e_{p^n}\left(f_{p^rl_1,p^rl_2}(\alpha^{\ast})\right)\cdot p^{(n+r)/2} & \mbox{ if } n-r \mbox{ is even,}\\
e_{p^n}\left(f_{p^rl_1,p^rl_2}(\alpha^{\ast})\right)\cdot \left(\frac{A(\alpha)}{p}\right)\cdot \frac{G_p}{\sqrt{p}}\cdot p^{(n+r)/2} & \mbox{ if } n-r \mbox{ is odd,}
\end{cases}
$$
where $\alpha^{\ast}$ is the unique lifting of $\alpha$ to a root of the congruence
$$
2\alpha_1l_2\alpha^{\ast}\equiv l_1(\alpha_1-\alpha_2(\alpha^{\ast})^2) \bmod{p^{n-r}}
$$
and
$$
A(\alpha)=\frac{2 f_{p^rl_1,p^rl_2}''(\alpha)}{p^r}.
$$
We calculate that
$$
\alpha^{\ast}\equiv (-\alpha_1l_2\pm\sqrt{D})\overline{\alpha_2l_1} \bmod{p^{n-r}},
$$
where $\sqrt{D}$ denotes one of the two roots of the congruence
$$
x^2\equiv D\bmod{p^{n-r}}.
$$
A short calculation gives
$$
e_{p^n}\left(f_{p^rl_1,p^rl_2}(\alpha^{\ast})\right)=e_{p^{n-r}}\left(\pm bx_3\overline{\alpha_1} \sqrt{D}\right).
$$
Further, we calculate the second derivative of $f_{p^rl_1,p^rl_2}$ to be
$$
f_{p^rl_1,p^rl_2}''(t)=4\alpha_1\alpha_2bx_3\cdot \frac{-2\alpha_2l_1t-l_2(\alpha_1-\alpha_2t^2)}{(\alpha_1+\alpha_2t^2)^3}\cdot p^{r}.
$$
Another short calculation gives
$$
A(\alpha)=-\frac{2bx_3(\alpha_2l_1)^2}{\alpha_2\alpha^2\sqrt{D}}
$$
and hence
$$
\left(\frac{A(\alpha)}{p}\right)=\left(\frac{-2bx_3\alpha_2\sqrt{D}}{p}\right).
$$
As noted above, the cases $\alpha^2\not\equiv 0,\pm \alpha_1\overline{\alpha_2}\bmod{p}$ cannot occur if $\alpha$ is a multiple root of the congruence \eqref{keycong} with $(l_1l_2,p)=1$. So altogether, we obtain
\begin{equation} \label{Uaftereva}
\begin{split}
U= & \frac{N^2}{p^{3n/2}}\sum\limits_{r=0}^{n-2} p^{r/2} \sum\limits_{\substack{(l_1l_2,p)=1\\ D=\Box\bmod{p}}} \hat{\Phi}\left(\frac{l_1N}{p^{n-r}}\right)\hat{\Phi}\left(\frac{l_2N}{p^{n-r}}\right)\times \\ & \sum\limits_{(x_3,p)=1} \Phi\left(\frac{x_3}{N}\right)\cdot C_{n-r}(x_3,D)\cdot
\left(e_{p^{n-r}}\left(bx_3\overline{\alpha_1}\sqrt{D}\right)+ e_{p^{n-r}}\left(-bx_3\overline{\alpha_1}\sqrt{D}\right)\right)+O_{\varepsilon}(1),
\end{split}
\end{equation}
where $D=\Box\bmod{p}$ means that $D$ is a quadratic residue modulo $p$ and
$$
C_{n-r}(x_3,D):=\begin{cases} 1 & \mbox{ if } n-r \mbox{ is even,}\\
\left(\frac{-2bx_3\alpha_2\sqrt{D}}{p}\right)\cdot \frac{G_p}{\sqrt{p}} & \mbox { if } n-r
\mbox{ is odd.} \end{cases}
$$
If $(x_3D,p)=1$, then
$$
C_{n-r}(x_3,D)= \frac{G_{p^{n-r}}}{p^{(n-r)/2}}\cdot
\left(\frac{-2bx_3\alpha_2\sqrt{D}}{p^{n-r}}\right)
$$
in each of the two cases above. Therefore, $U$ can be more compactly written as
\begin{equation} \label{compact}
\begin{split}
U= & \frac{N^2}{p^{3n/2}}\sum\limits_{r=0}^{n-2} p^{r/2} \cdot \frac{G_{p^{n-r}}}{p^{(n-r)/2}} \cdot \sum\limits_{\substack{D=1\\ D\equiv \Box \bmod{p}}}^{\infty} F_{n-r}(D) \times\\
& \sum\limits_{x_3\in \mathbb{Z}} \Phi\left(\frac{x_3}{N}\right)\cdot \left(\frac{x_3}{p^{n-r}}\right) \cdot
\left(e_{p^{n-r}}\left(bx_3\overline{\alpha_1}\sqrt{D}\right)+ e_{p^{n-r}}\left(-bx_3\overline{\alpha_1}\sqrt{D}\right)\right)+O_{\varepsilon}(1),
\end{split}
\end{equation}
where
\begin{equation} \label{FD}
F_{n-r}(D):= \left(\frac{-2b\alpha_2\sqrt{D}}{p^{n-r}}\right)\cdot \sum\limits_{\substack{(l_1l_2,p)=1\\ \alpha_1\alpha_2l_1^2+\alpha_1^2l_2^2=D}} \hat{\Phi}\left(\frac{l_1N}{p^{n-r}}\right)\hat{\Phi}\left(\frac{l_2N}{p^{n-r}}\right).
\end{equation}
\subsection{Single Poisson summation and final count}
Now we split the sum over $x_3$ in \eqref{compact} into sub-sums over residue classes modulo $p$ and perform Poisson summation, getting
\begin{equation*}
\begin{split}
\sum\limits_{x_3\in \mathbb{N}} \Phi\left(\frac{x_3}{N}\right)\cdot \left(\frac{x_3}{p^{n-r}}\right) \cdot e_{p^{n-r}}\left(\pm bx_3\overline{\alpha_1}\sqrt{D}\right) =
& \sum\limits_{u=1}^{p} \left(\frac{u}{p^{n-r}}\right)
\sum\limits_{x_3\equiv u\bmod{p}} \Phi\left(\frac{x_3}{N}\right)\cdot
e_{p^{n-r}}\left(b x_3\overline{\alpha_1}\sqrt{D}\right)\\
= & \frac{N}{p} \sum\limits_{v\in \mathbb{Z}} \left(\sum\limits_{u=1}^{p} \left(\frac{u}{p^{n-r}}\right) \cdot e_{p}(uv)\right)
\cdot \hat\Phi\left(\frac{N}{p}\left(\frac{\pm b\overline{\alpha_1}\sqrt{D}}{p^{n-r-1}}-v\right)\right).
\end{split}
\end{equation*}
Using the rapid decay of $\hat\Phi$, the above is $O(N)$ if $||b\overline{\alpha_1}\sqrt{D}/p^{n-r-1}||\le p^{1+n\varepsilon}N^{-1}$ and negligible otherwise, provided $n$ is large enough. We may constraint $b\overline{\alpha_1}\sqrt{D}$ to the range $0\le \sqrt{D}\le p^{n-r}$ and then write $b\overline{\alpha_1}\sqrt{D}=wp^{n-r-1}+l_3$, where $w=0,...,p-1$, $(l_3,p)=1$ and $|l_3|\le L_r$ with
$$
L_r:=p^{n-r+n\varepsilon}N^{-1}.
$$
Moreover, the summations over $l_1$ and $l_2$ in \eqref{FD} can be cut off at $|l_1|,|l_2|\le L_r$ at the cost of a negligible error if $n$ is large enough. It follows that
\begin{equation*}
\begin{split}
U\ll & \frac{N^3}{p^{3n/2}}\sum\limits_{r=0}^{n-2} p^{r/2} \sum\limits_{w=0}^{p-1} \sum\limits_{\substack{(l_1,l_2,l_3)\in \mathbb{Z}^3\\ (l_1l_2l_3,p)=1\\ |l_1|,|l_2|,|l_3|\le L_r\\ b^2\overline{\alpha_1}^{2}(\alpha_1\alpha_2l_1^2+\alpha_1^{2}l_2^2)\equiv (wp^{n-r-1}+l_3)^2\bmod{p^{n-r}}}} 1 +O_{\varepsilon}(1).
\end{split}
\end{equation*}
Recalling that $b^2\equiv -\alpha_3\overline{\alpha_2} \bmod{p^{n-r}}$, the congruence above implies
\begin{equation} \label{acongru}
\alpha_2\alpha_3l_1^2+\alpha_1\alpha_3l_2^2+\alpha_1\alpha_2l_3^2\equiv 0\bmod{p^{n-r-1}},
\end{equation}
and hence
\begin{equation} \label{UboundI}
\begin{split}
U\ll & \frac{N^3}{p^{3n/2}}\sum\limits_{r=0}^{n-2} p^{r/2} \sum\limits_{\substack{(l_1,l_2,l_3)\in \mathbb{Z}^3\\ (l_1l_2l_3,p)=1\\ |l_1|,|l_2|,|l_3|\le L_r\\ \alpha_2\alpha_3l_1^2+\alpha_1\alpha_3l_2^2+\alpha_1\alpha_2l_3^2\equiv 0\bmod{p^{n-r-1}}}} 1 +O_{\varepsilon}(1).
\end{split}
\end{equation}
Let $H:=\max\{|\alpha_1|,|\alpha_2|,|\alpha_3|\}$.
Now if $3H^2L_r^2< p^{n-r-1}$, then the congruence above can be replaced by the equation
$$
\alpha_2\alpha_3l_1^2+\alpha_1\alpha_3l_2^2+\alpha_1\alpha_2l_3^2=0.
$$
Certainly, this is the case if $N\ge p^{n/2+2n\varepsilon}$ and $n$ is large enough. (At this place, we use our assumption that $\alpha_1,\alpha_2,\alpha_3$ are {\it fixed}!) Hence, in this case, we have
\begin{equation*}
\begin{split}
U\ll & \frac{N^3}{p^{3n/2}}\sum\limits_{r=0}^{n-2} p^{r/2} \sum\limits_{\substack{(l_1,l_2,l_3)\in \mathbb{Z}^3\\ (l_1l_2l_3,p)=1\\ |l_1|,|l_2|,|l_3|\le L_r\\ \alpha_2\alpha_3l_1^2+\alpha_1\alpha_3l_2^2+\alpha_1\alpha_2l_3^2=0}} 1 +O_{\varepsilon}(1).
\end{split}
\end{equation*}
Now we apply Proposition \ref{quadequations} with $A=\alpha_2\alpha_3$, $B=\alpha_1\alpha_3$, $C=-\alpha_1\alpha_2l_3^2$, $X=l_1$ and $Y=l_2$ to bound $U$ by
\begin{equation*}
\begin{split}
U\ll & \frac{N^3}{p^{3n/2}}\sum\limits_{r=0}^{n-2} p^{r/2} L_r^{1+\varepsilon} \ll \frac{N^2}{p^{n/2}}\cdot p^{9n\varepsilon}.
\end{split}
\end{equation*}
This needs to be compared to the main term which is of size
$$
T_0\asymp \frac{N^3}{p^n}.
$$
If $N\ge p^{(1/2+10\varepsilon)n}$, then $U=o(T_0)$, which completes the proof of Theorem \ref{mainresult} in this case.
\subsection{Parametrization of solutions - Cases II} \label{2ndpara}
Now we consider the case when none of $-\alpha_i\alpha_j$ is a quadratic residue modulo $p$. By Proposition \ref{para}, the $\mathbb{Q}_p$-rational points $(y_1,y_2)$ on the conic
$$
\alpha_1y_1^2+\alpha_2y_2^2=-\alpha_3,
$$
are parametrized as
\begin{equation*}
\begin{split}
y_1=y_1(t)&:=a-2\alpha_2\frac{at^2-bt}{\alpha_1+\alpha_2t^2},\\
y_2=y_2(t)&:=-b-2\alpha_1\frac{at-b}{\alpha_1+\alpha_2t^2},
\end{split}
\end{equation*}
where $\alpha_1a^2+\alpha_2b^2=-\alpha_3$ (in particular, $\alpha_1a^2+\alpha_2b^2\equiv -\alpha_3\bmod{p^n}$). This equation is soluble in $(a,b)$ as a consequence of ....
We define
$$
M_0:=\{(y_1(t),y_2(t)) : t=1,2,...,p^n\},
$$
$$
M_s:=\left\{\left(y_1\left(\frac{t}{p^s}\right),y_2\left(\frac{t}{p^s}\right)\right):t=1,...p^{n-s},t\not\equiv0~~\text{mod}~~ p\right\} \mbox{ for } s=1,2...,n
$$
and
$$
M:=\bigcup\limits_{s=0}^n M_s.
$$
Noting that $y_1(t/p^s)$ and $y_2(t/p^s)$ are $p$-adic integers, we will view $y_1(t/p^s)$ and $y_2(t/p^s)$ as elements of $\mathbb{Z}/p^n\mathbb{Z}$.
It can be seen that the pairs $(y_1(t/p^s),y_2(t/p^s)$ in the above sets $M_s$ ($s=0,...,n$) are distinct and $M_i\cap M_j=\emptyset$ for $i\neq j$ by the following argument.
Suppose that $$
\left(y_1\left(\frac{t_1}{p^s_1}\right),y_2\left(\frac{t_1}{p^s_1}\right)\right)= \left(y_1\left(\frac{t_2}{p^s_2}\right),y_2\left(\frac{t_2}{p^s_2}\right)\right),
$$
which implies
\begin{equation*}
\begin{split}
a-2\alpha_2\frac{at_1^2-bt_1p^{s_1}}{\alpha_1p^{2s_1}+\alpha_2t_1^2}=&a-2\alpha_2\frac{at_2^2-bt_2p^{s_2}}{\alpha_1p^{2s_2}+\alpha_2t_2^2},\\
-b-2\alpha_1p^{s_1}\frac{at_1-bp^{s_1}}{\alpha_1p^{2s_1}+\alpha_2t_1^2}=&-b-2\alpha_1p^{s_2}\frac{at_2-bp^{s_2}}{\alpha_1p^{2s_1}+\alpha_2t_2^2}.
\end{split}
\end{equation*}
Then a short calculation gives
\begin{equation} \label{ok} t_2p^{s_1}=t_1p^{s_2}\bmod{p^n}.
\end{equation}
So if $0\leq s_1=s_2\leq n$ then $p^{s_1}(t_2-t_1)\equiv 0\bmod{p^n}$, which implies $t_2-t_1\equiv0\bmod{p^{n-s_1}}$. Hence $t_2=t_1$ because $1\leq t_1,t_2\leq p^{n-s_1}$.
If $s_2>s_1$ then we have $t_2\equiv 0 \bmod {p^{s_2-s_1}}$, which contradicts the fact that $t_2\not\equiv 0\bmod{p}$.
Therefore we get that $|M_0|=p^n,|M_n|=1$, $|M_s|=p^{n-s}-p^{n-s-1}$ for $s=1,2...n-1$, and $$|M|=\left|\bigcup_{s=0}^nM_s\right|=p^n+(p^{n-1}-p^{n-2})+....+(p^2-p)+(p-1)+1=p^n+p^{n-1}.$$
Now Proposition \ref{nofroot} tells us that $M=\bigcup_{s=0}^nM_n$ is a complete set of solutions of $\alpha_1y_1^2+\alpha_2y_2^2\equiv-\alpha_3$ mod $p^n$.
We also observe that if $(x_1,x_2)$ is a solution of $\alpha_1x_1^2+\alpha_2x_2^2\equiv-\alpha_3$ mod $p^n$ then $(x_1x_2,p)=1$ because of the fact that $-\alpha_1\alpha_2$ is a quadratic non-residue.
\subsection{Double Poisson summation} As in \eqref{fdob} we write
\begin{equation*}
T= \frac{N^2}{p^{2n}}\sum\limits_{(x_3,p)=1} \Phi\left(\frac{x_3}{N}\right)\sum\limits_{(k_1,k_2)\in \mathbb{Z}^2} \hat{\Phi}\left(\frac{k_1N}{p^n}\right)\hat{\Phi}\left(\frac{k_2N}{p^n}\right)\sum\limits_{\substack{y_1,y_2\bmod{p^n}\\ (y_1y_2,p)=1 \\ \alpha_1y_1^2+\alpha_2y_2^2+\alpha_3\equiv 0\bmod{p^n}}} e_{p^n}\left(k_1x_3y_1+k_2x_3y_2\right).
\end{equation*}
Using the above parametrization, we deduce that
\begin{equation*}
T= \frac{N^2}{p^{2n}}\sum\limits_{(x_3,p)=1} \Phi\left(\frac{x_3}{N}\right)\sum\limits_{(k_1,k_2)\in \mathbb{Z}^2} \hat{\Phi}\left(\frac{k_1N}{p^n}\right)\hat{\Phi}\left(\frac{k_2N}{p^n}\right)\sum_{s=0}^n\sum\limits_{(y_1,y_2)\in M_s} e_{p^n}\left(k_1x_3y_1+k_2x_3y_2\right).
\end{equation*}
We decompose $T$ into
\begin{equation*} \label{2divide}
T=T_0+U,
\end{equation*}
where $T_0$ is the main term contribution of $(k_1,k_2)=(0,0)$. Hence,
\begin{equation*}
\begin{split}
T_0&= \hat{\Phi}(0)^2\cdot \frac{N^2}{p^{2n}}\sum\limits_{(x_3,p)=1} \Phi\left(\frac{x_3}{N}\right) \cdot\left|M\right|\\
&= \hat{\Phi}(0)^2\cdot \frac{N^2}{p^{2n}}\sum\limits_{(x_3,p)=1} \Phi\left(\frac{x_3}{N}\right) \cdot(p^n+p^{n-1})\\
&=\hat{\Phi}(0)^2\cdot \frac{N^2}{p^{2n}}\sum\limits_{(x_3,p)=1} \Phi\left(\frac{x_3}{N}\right) \cdot p^{n-1}(p-s_p(\alpha_1,\alpha_2,\alpha_3)),
\end{split}
\end{equation*}
where $s_p(\alpha_1,\alpha_2,\alpha_3))$ is defined as in \eqref{sdef}. Now if $N\geq p^{n\varepsilon}$ for any fixed $\varepsilon>0$, the term $T_0$ can be further simplified as in \eqref{fdob}, and we obtain
\begin{equation*} \label{T0II}
\begin{split}
T_0= \hat{\Phi}(0)^3 \cdot \frac{(p-s_p(\alpha_1,\alpha_2,\alpha_3))(p-1)}{p^2}\cdot \frac{N^3}{p^{n}}\cdot\left(1+o(1)\right)= \hat{\Phi}(0)^3\cdot C_p(\alpha_1,\alpha_2,\alpha_3)\cdot \frac{N^3}{p^n}\cdot \left(1+o(1)\right).
\end{split}
\end{equation*}
\subsection{Evaluation of exponential sums} Now we look at the error contribution
\begin{equation} \label{2errorcont}
U= \frac{N^2}{p^{2n}}\sum\limits_{(x_3,p)=1} \Phi\left(\frac{x_3}{N}\right)\sum\limits_{(k_1,k_2)\in \mathbb{Z}^2\setminus \{(0,0)\}} \hat{\Phi}\left(\frac{k_1N}{p^n}\right)\hat{\Phi}\left(\frac{k_2N}{p^n}\right) E\left(k_1,k_2,x_3;p^n\right)
\end{equation}
with
\begin{equation*}
\begin{split}
E\left(k_1,k_2,x_3;p^n\right):=\sum_{s=0}^n\sum\limits_{(y_1,y_2)\in M_s} e_{p^n}\left(x_3(k_1y_1+k_2y_2)\right).
\end{split}
\end{equation*}
Assume that
$$
(k_1,k_2,p^n)=p^r.
$$
The contribution of $r=n-1,n$ to the right-hand side of \eqref{2errorcont} is $O_{\varepsilon}(1)$ if $N\ge p^{n\varepsilon}$ by the rapid decay of $\hat\Phi$ since $(k_1,k_2)=(0,0)$ is excluded from the summation. In the following, we assume that $r\le n-2$ so that Proposition \ref{Expsums} is applicable.
Let $$f_{s,k_1,k_2}(t)=x_3\left(k_1y_1\left(\frac{t}{p^s}\right)+k_2y_2\left(\frac{t}{p^s}\right)\right),$$ where $y_1(t),y_2(t)$ is defined in \eqref{2ndpara}.
We have $f_{s,k_1,k_2}(t)\equiv f_{s,k_1,k_2}(t+wt^{n-s})\bmod{p^n}$ for $w=1,2...,p^{s}$
and deduce that
\begin{equation}\label{newsum}
\begin{split}
E\left(k_1,k_2,x_3;p^n\right):=\sum_{t=0}^{p^n}e_{p^n}\left(f_{0,k_1,k_2}(t)\right)+\sum_{s=1}^n\frac{1}{p^s}\sum\limits_{\substack{t=1\\t\not\equiv 0~~\text{mod}~~p}}^{p^n} e_{p^n}\left( f_{s,k_1,k_2}(t)\right).
\end{split}
\end{equation}
The derivative of the amplitude function turns out to be
\begin{equation*}
\begin{split}
f'_{s,k_1,k_2}(t)&=2x_3\frac{k_1\alpha_2(-b\alpha_2t^2p^{2s}-2a\alpha_1tp^{3s}+\alpha_1bp^{4s})+k_2\alpha_1(a\alpha_2t^2p^{2s}-2b\alpha_2tp^{3s}-a\alpha_1p^{4s})}{(\alpha_1p^{2s}+\alpha_2t^2)^2 }\\
&=2x_3\frac{\alpha_2(k_2\alpha_1a-k_1\alpha_2b)t^2p^{2s}-2\alpha_1\alpha_2(ak_1+bk_2)tp^{3s}-\alpha_1(k_2\alpha_1a-k_1\alpha_2b)p^{4s}}{(\alpha_1p^{2s}+\alpha_2t^2)^2}.
\end{split}
\end{equation*}
If $s>0$ then $t\not\equiv 0$ mod $p$, which implies $\alpha_1p^{2s}+\alpha_2t^2\not\equiv 0 \bmod{p}$. If $s=0$ then also $\alpha_1+\alpha_2t\not\equiv 0\bmod{p}$ because of the fact that $-\alpha_1\alpha_2$ is a quadratic non-residue modulo $p$.
It is easy to see that $\left(k_1,k_2,p^n\right)=\left((k_2\alpha_1a-k_1\alpha_2 b),(a k_1+b k_2),p^n\right)$ and therefore $\mbox{ord}_p(f_{s,k_1,k_2}')=p^{2s+r}$.
We split the second sum over $t$ on the right-hand side of \eqref{newsum} into
\begin{equation*}
\begin{split}
\sum\limits_{\substack{t=1\\t\not\equiv 0~~\text{mod}~~p}}^{p^n} e_{p^n}\left( f_{s,k_1,k_2}(t)\right)=\sum\limits_{\substack{\alpha=1\\ \alpha \not\equiv 0 \bmod{p}}}^p S_{\alpha}\left(f_{s,k_1,k_2};p^n\right).
\end{split}
\end{equation*}
We see that $s>0$ and $p^{-r-2s}f'_{s,k_1,k_2}(t)\equiv 0\bmod{p}$ imply $t\equiv0\bmod{p}$. Then using Proposition \ref{Expsums}, we have $S_{\alpha}=0$ if $\alpha\neq0$, which implies that
\begin{equation*}
\sum_{s=1}^n\frac{1}{p^s}\sum\limits_{\substack{t=1\\t\not\equiv 0~~\text{mod}~~p}}^{p^n} e_{p^n}\left( f_{s,k_1,k_2}(t)\right)=0.
\end{equation*}
It follows that
\begin{equation*} \label{2nderrorcont}
U= \frac{N^2}{p^{2n}}\sum\limits_{(x_3,p)=1} \Phi\left(\frac{x_3}{N}\right)\sum\limits_{(k_1,k_2)\in \mathbb{Z}^2\setminus \{(0,0)\}} \hat{\Phi}\left(\frac{k_1N}{p^n}\right)\hat{\Phi}\left(\frac{k_2N}{p^n}\right) \sum\limits_{\substack{t=1}}^{p^n} e_{p^n}\left( f_{0,k_1,k_2}(t)\right).
\end{equation*}
We split the inner-most sum over $t$ into
\begin{equation*} \label{2split}
\sum\limits_{\substack{t=1}}^{p^n} e_{p^n}\left( f_{0,k_1,k_2}(t)\right)=\sum\limits_{\substack{\alpha=1}}^p S_{\alpha}\left(f_{0,k_1,k_2};p^n\right),
\end{equation*}
where
\begin{equation*} \label{2Salphadef}
S_{\alpha}\left(f_{0,k_1,k_2};p^n\right)=\sum\limits_{\substack{t \bmod{p^n}\\ t\equiv \alpha\bmod{p}}} e_{p^n}\left(f_{0,k_1,k_2}(t)\right).
\end{equation*}
Set
$$
l_1:=\frac{k_1}{p^r}, \quad l_2:=\frac{k_2}{p^r}.
$$
If $(k_1,p^n)>(k_2,p^n)$ then
$$p^{-r}f'_{0,k_1,K_2}\equiv 0\bmod{p},$$
which implies $$a\alpha_2t^2-2b\alpha_2t-a\alpha_1\equiv 0 \bmod{p}.$$
The above congruence relation has no solution because the determinant of the corresponding polynomial equals $-4\alpha_2\alpha_3$ and is therefore a quadratic non-residue. So by Proposition \ref{Expsums},
\begin{equation*}
\sum\limits_{\substack{t=1}}^{p^n} e_{p^n}\left( f_{0,k_1,k_2}(t)\right)=0.
\end{equation*}
Similarly, this sum is zero if $(k_1,p^n)>(k_2,p^n)$.
Thus it follows that $(l_1l_2,p)=1$ and $(k_1,p^n)=(k_2,p^n)=p^r$.
Using Proposition \ref{Expsums}, we have $S_{\alpha}(f_{0,k_1,k_2})=0$ unless
\begin{equation}\label{2keycong}
l_2\alpha_1(a\alpha_2\alpha^2-2b\alpha_2\alpha-a\alpha_1)\equiv l_1\alpha_2(b\alpha_2\alpha^2+2a\alpha_1\alpha-\alpha_1b)\bmod{p}.
\end{equation}
Set $$C(t):=l_2\alpha_1(a\alpha_2t^2-2b\alpha_2t-a\alpha_1)- l_1\alpha_2(b\alpha_2t^2+2a\alpha_1t-\alpha_1b).$$
In summary, we have
\begin{equation*}
U= \frac{N^2}{p^{2n}}\sum\limits_{(x_3,p)=1} \Phi\left(\frac{x_3}{N}\right)\sum\limits_{r=0}^{n-2} \sum\limits_{\substack{\alpha=1}}^p \sum\limits_{\substack{(l_1,l_2)\in \mathbb{Z}^2\\ (l_1l_2,p)=1\\ C(\alpha)\equiv 0\bmod{p} }} \hat{\Phi}\left(\frac{l_1N}{p^{n-r}}\right)\hat{\Phi}\left(\frac{l_2N}{p^{n-r}}\right) S_{\alpha}\left(f_{0,p^rl_1,p^rl_2};p^n\right)+O_{\varepsilon}(1)
\end{equation*}
if $N\ge p^{n\varepsilon}$.
Let
$$
D:=-\alpha_3\overline{\alpha_2}(\alpha_1\alpha_2l_1^2+\alpha_1^2l_2^2) \bmod{p^n}.
$$
The congruence \eqref{2keycong} has a double root $\alpha \bmod{p}$ iff $D\equiv 0 \bmod{p}$, and in this case we get $\alpha^2\equiv -\alpha_1\overline{\alpha_2}\bmod{p}$ which contradicts the fact that $-\alpha_1\overline{\alpha_2}$ is a quadratic non-residue. Hence, only the case $D\not\equiv 0\bmod{p}$ occurs in which we have no root if $D$ is a quadratic non-residue modulo $p$ and two roots of multiplicity one if $D$ is a quadratic residue modulo $p$. Therefore, we may assume from now on that $D\not\equiv 0 \bmod{p}$ and $D$ is a quadratic residue modulo $p$. Then using Proposition \ref{Expsums}, if $\alpha$ satisfies \eqref{2keycong}, we obtain
$$
S_{\alpha}\left(f_{p^rl_1,p^rl_2},p^n\right)=
\begin{cases}
e_{p^n}\left(f_{0,p^rl_1,p^rl_2}(\alpha^{\ast})\right)\cdot p^{(n+r)/2} & \mbox{ if } n-r \mbox{ is even,}\\
e_{p^n}\left(f_{0,p^rl_1,p^rl_2}(\alpha^{\ast})\right)\cdot \left(\frac{A(\alpha)}{p}\right)\cdot \frac{G_p}{\sqrt{p}}\cdot p^{(n+r)/2} & \mbox{ if } n-r \mbox{ is odd,}
\end{cases}
$$
where $\alpha^{\ast}$ is the unique lifting of $\alpha$ to a root of the congruence
\begin{equation}
l_2\alpha_1(a\alpha_2(\alpha^{\ast})^2-2b\alpha_2\alpha^{\ast}-a\alpha_1)\equiv l_1\alpha_2(b\alpha_2(\alpha^{\ast})^2+2a\alpha_1\alpha^{\ast}-\alpha_1b)\bmod{p^{n-r}}
\end{equation}
and
$$
A(\alpha)=\frac{2 f_{0,p^rl_1,p^rl_2}''(\alpha)}{p^r}.
$$
We calculate that
$$
\alpha^{\ast}\equiv \frac{\alpha_1(al_1+bl_2)\pm\sqrt{D}}{l_2\alpha_1a-l_1\alpha_2b} \bmod{p^{n-r}},
$$
where $\sqrt{D}$ denotes one of the two roots of the congruence
$$
x^2\equiv D\bmod{p^{n-r}}.
$$
A short calculation gives
$$
e_{p^n}\left(f_{0,p^rl_1,p^rl_2}(\alpha^{\ast})\right)=e_{p^{n-r}}\left(\pm x_3\overline{\alpha_1}\cdot \sqrt{D}\right).
$$
Further, we calculate the second derivative of $f_{p^rl_1,p^rl_2}$ to be
$$
f_{0,p^rl_1,p^rl_2}''(t)=4\alpha_1\alpha_2x_3\cdot \frac{\alpha_2D_2t^2+2D_1t-\alpha_1D_2}{(\alpha_1+\alpha_2t^2)^3}\cdot p^{r},
$$
where $D_1=l_1\alpha_1a-l_1\alpha_2b$ and $D_2=l_1a+l_2b$.
Another short calculation gives
$$
A(\alpha)=\frac{2x_3D_1^2}{\alpha_2\alpha^2\sqrt{D}}
$$
and hence
$$
\left(\frac{A(\alpha)}{p}\right)=\left(\frac{2x_3\alpha_2\sqrt{D}}{p}\right).
$$
So altogether, we obtain
\begin{equation*} \label{2Uaftereva}
\begin{split}
U= & \frac{N^2}{p^{3n/2}}\sum\limits_{r=0}^{n-2} p^{r/2} \sum\limits_{\substack{(l_1l_2,p)=1\\ D=\Box\bmod{p}}} \hat{\Phi}\left(\frac{l_1N}{p^{n-r}}\right)\hat{\Phi}\left(\frac{l_2N}{p^{n-r}}\right)\times \\ & \sum\limits_{(x_3,p)=1} \Phi\left(\frac{x_3}{N}\right)\cdot C_{n-r}(x_3,D)\cdot
\left(e_{p^{n-r}}\left(x_3\overline{\alpha_1}\sqrt{D}\right)+ e_{p^{n-r}}\left(-x_3\overline{\alpha_1}\sqrt{D}\right)\right)+O_{\varepsilon}(1),
\end{split}
\end{equation*}
where $D=\Box\bmod{p}$ means that $D$ is a quadratic residue modulo $p$ and
$$
C_{n-r}(x_3,D):=\begin{cases} 1 & \mbox{ if } n-r \mbox{ is even,}\\
\left(\frac{2x_3\alpha_2\sqrt{D}}{p}\right)\cdot \frac{G_p}{\sqrt{p}} & \mbox { if } n-r
\mbox{ is odd.} \end{cases}
$$
If $(x_3D,p)=1$, then
$$
C_{n-r}(x_3,D)= \frac{G_{p^{n-r}}}{p^{(n-r)/2}}\cdot
\left(\frac{2x_3\alpha_2\sqrt{D}}{p^{n-r}}\right)
$$
in each of the two cases above. Therefore, $U$ can be more compactly written as
\begin{equation*} \label{2compact}
\begin{split}
U= & \frac{N^2}{p^{3n/2}}\sum\limits_{r=0}^{n-2} p^{r/2} \cdot \frac{G_{p^{n-r}}}{p^{(n-r)/2}} \cdot \sum\limits_{\substack{D=1\\ D\equiv \Box \bmod{p}}}^{\infty} F_{n-r}(D) \times\\
& \sum\limits_{x_3\in \mathbb{Z}} \Phi\left(\frac{x_3}{N}\right)\cdot \left(\frac{x_3}{p^{n-r}}\right) \cdot
\left(e_{p^{n-r}}\left(x_3\overline{\alpha_1}\sqrt{D}\right)+ e_{p^{n-r}}\left(-x_3\overline{\alpha_1}\sqrt{D}\right)\right)+O_{\varepsilon}(1),
\end{split}
\end{equation*}
where
\begin{equation*} \label{2FD}
F_{n-r}(D):= \left(\frac{2\alpha_2\sqrt{D}}{p^{n-r}}\right)\cdot \sum\limits_{\substack{(l_1l_2,p)=1\\ -\alpha_3\overline{\alpha_2}(\alpha_1\alpha_2l_1^2+\alpha_1^2l_2^2)=D }} \hat{\Phi}\left(\frac{l_1N}{p^{n-r}}\right)\hat{\Phi}\left(\frac{l_2N}{p^{n-r}}\right).
\end{equation*}
This should be compared to \eqref{compact}. Similar calculations as in subsection 3.4. now lead to precisely the same bound \eqref{UboundI}, and the rest of the proof is then the same as in Case I. This completes the proof of Theorem \ref{mainresult} in Case II.
\section{Proof of Theorem \ref{mainresult2}}
Let $p>5$ be a prime and $q=p^n$.
We denote the quantity in question as
$$
\Sigma_{\alpha_1,\alpha_2,\alpha_3}(\Phi,N,q)=\sum\limits_{\substack{(x_1,x_2,x_3)\in \mathbb{Z}^3\\ (x_1x_2x_3,p)=1\\ \alpha_1x_1^2+\alpha_2x_2^2+\alpha_3x_3^2 \equiv 0 \bmod{q}}} \Phi\left(\frac{x_1}{N}\right)
\Phi\left(\frac{x_2}{N}\right)\Phi\left(\frac{x_3}{N}\right).
$$
In particular, letting $\chi_{[-1,1]}$ be the characteristic function of the interval $[-1,1]$, we have
$$
\Sigma_{\alpha_1,\alpha_2,\alpha_3}(\chi_{[-1,1]},N,q)=\sum\limits_{\substack{|x_1|,|x_2|,|x_3|\le N\\ \alpha_1x_1^2+\alpha_2x_2^2+\alpha_3x_3^3\equiv 0 \bmod{q}\\ (x_1x_2x_3,q)=1}} 1,
$$
which we denote just by $\Sigma_{\alpha_1,\alpha_2,\alpha_3}(N,q)$ throughout the sequel.
If $\Phi$ is a Schwartz class function, then Theorem \ref{mainresult} yields, for {\it fixed} $\alpha_1,\alpha_2,\alpha_3$ and $n\rightarrow \infty$, an asymptotic formula for $\Sigma_{\alpha_1,\alpha_2,\alpha_3}(\Phi,N,q)$ if $N\gg q^{1/2+\varepsilon}$.
Now we allow $\alpha_1,\alpha_2,\alpha_3$ to be {\it arbitrary} (i.e., to vary with $n$). In this situation, we will see that using our approach, it is relatively easy to get an asymptotic formula if $N\gg q^{2/3+\varepsilon}$. We will describe how to reach the exponent $2/3$ and then refine our method to beat it.
Combining \eqref{divide}, \eqref{T0I}, \eqref{UboundI} in Case I and the corresponding equations and inequalities in Case II, we get an asymptotic formula of the form
\begin{equation} \label{essential}
\Sigma_{\alpha_1,\alpha_2,\alpha_3}(\Phi,N,q)=\Phi(0)^3\cdot C_p(\alpha_1,\alpha_2,\alpha_3)\cdot \frac{N^3}{q} \cdot (1+o(1))+O\left(\frac{N^3}{q^{3/2}}\sum\limits_{r=0}^{n-2} p^{r/2} \Sigma_{\beta_1,\beta_2,\beta_3}(L_r,q_r) +O_{\varepsilon}(1)\right),
\end{equation}
where
$$
L_r:=p^{-r}q^{1+\varepsilon}N^{-1}, \quad q_r:=p^{-r-1}q,\quad \beta_1:=\alpha_2\alpha_3, \quad \beta_2:=\alpha_1\alpha_3,\quad \beta_3:=\alpha_1\alpha_2.
$$
Obviously, $\beta_1,\beta_2,\beta_3$ are also coprime to the modulus.
Now our task becomes to bound from above the quantity
$\Sigma_{\beta_1,\beta_2,\beta_3}(M,q')$, where $M=L_r$ and $q'=q_r$. We need an upper bound for this quantity in the situation when $M\le (q')^{1/2-\varepsilon}$.
If $\alpha_1,\alpha_2,\alpha_3$ and hence $\beta_1,\beta_2,\beta_3$ are {\it fixed}, then by our method in the previous section, we easily get
\begin{equation} \label{firstbound}
\Sigma_{\beta_1,\beta_2,\beta_3}(M,q')\ll M^{1+\varepsilon}
\end{equation}
in this situation, where the implied $\ll$-constant depends on $\alpha_1,\alpha_2,\alpha_3$. Hence, we then obtain
$$
\Sigma_{\alpha_1,\alpha_2,\alpha_3}(\Phi,N,q)=\hat\Phi(0)^3\cdot C_p(\alpha_1,\alpha_2,\alpha_3)\cdot \frac{N^3}{q}+O\left(\frac{N^2}{q^{1/2-\varepsilon}}\right),
$$
which gives an asymptotic if $N\gg q^{1/2+\varepsilon}$. We may conjecture that the bound \eqref{firstbound} holds with an {\it absolute} $\ll$-constant, but the dependence on $\beta_1,\beta_2,\beta_2$ (and hence $\alpha_1,\alpha_2,\alpha_3$) seems to be difficult to remove. What we can establish instead relatively easily, with an absolute $O$-constant, is
\begin{equation} \label{secondbound}
\Sigma_{\beta_1,\beta_2,\beta_3}(M,q')\ll M^{3/2+\varepsilon},
\end{equation}
again provided that $M\le (q')^{1/2-\varepsilon}$.
This implies then
$$
\Sigma_{\alpha_1,\alpha_2,\alpha_3}(N,q)=\hat\Phi(0)^3\cdot C_p(\alpha_1,\alpha_2,\alpha_3)\cdot \frac{N^3}{q}+O\left(N^{3/2}q^{\varepsilon}\right),
$$
which yields an asymptotic for {\it arbitrary} $\alpha_1,\alpha_2,\alpha_3$ if $N\gg q^{2/3+\varepsilon}$.
There are several ways to establish \eqref{secondbound}. Below we describe an elementary method which has space for improvements. We then refine this method to beat the exponent $3/2$ in \eqref{secondbound}. For easy of notation, we write $q$ in place of $q'$ in the following, bearing in mind that it is not the original modulus $q$. We also assume $M\le q^{1/2-\varepsilon}$ henceforth. Set
$$
\mathcal{M}:=\left\{-\beta_3x_3^2 : |x_3|\le M,\ (x_3,q)=1\right\}.
$$
Then
$$
\Sigma_{\beta_1,\beta_2,\beta_3}(M,q)=2\sum\limits_{m\in \mathcal{M}} \sum\limits_{\substack{|x_1|,|x_2|\le M\\ \beta_1x_1^2+\beta_2x_2^2\equiv m \bmod{q}\\ (x_1x_2,q)=1}} 1.
$$
Now we apply the Cauchy-Schwarz inequality, getting
\begin{equation*}
\begin{split}
\Sigma_{\beta_1,\beta_2,\beta_3}(M,q)^2\le & |\mathcal{M}| \sum\limits_{m\in \mathcal{M}} \left| \sum\limits_{\substack{|x_1|,|x_2|\le M\\ \beta_1x_1^2+\beta_2x_2^2\equiv m \bmod{q}\\ (x_1x_2,q)=1}} 1\right|^2\\
= & M \sum\limits_{m\in \mathcal{M}} \sum\limits_{\substack{|x_1|,|x_2|,|y_1|,|y_2|\le M\\ \beta_1x_1^2+\beta_2x_2^2\equiv m \bmod{q}\\
\beta_1y_1^2+\beta_2y_2^2\equiv m \bmod{q}\\
(x_1x_2y_1y_2,q)=1}} 1\\
= & M(\mathcal{D}+\mathcal{E}),
\end{split}
\end{equation*}
where
$$
\mathcal{D}:=\sum\limits_{m\in \mathcal{M}} \sum\limits_{\substack{|x_1|,|x_2|\le M\\ \beta_1x_1^2+\beta_2x_2^2\equiv m \bmod{q}\\
|y_1|=|x_1| \ \mbox{\scriptsize and } |y_2|=|x_2| \\
(x_1x_2,q)=1}} 1 = 4\Sigma_{\beta_1,\beta_2,\beta_3}(M,q)
$$
and
$$
\mathcal{E}:=\sum\limits_{m\in \mathcal{M}} \sum\limits_{\substack{|x_1|,|x_2|,|y_1|,|y_2|\le M\\ \beta_1x_1^2+\beta_2x_2^2\equiv m \bmod{q}\\
\beta_1y_1^2+\beta_2y_2^2\equiv m \bmod{q}\\|y_1|\not=|x_1| \ \mbox{\scriptsize or } |y_2|\not=|x_2|\\
(x_1x_2y_1y_2,q)=1}} 1.
$$
We observe that
\begin{equation*}
\begin{split}
\mathcal{E}\le & \sum\limits_{\substack{|x_1|,|x_2|,|y_1|,|y_2|\le M\\ \beta_1x_1^2+\beta_2x_2^2\equiv \beta_1y_1^2+\beta_2y_2^2 \bmod{q}\\|y_1|\not=|x_1| \ \mbox{\scriptsize or } |y_2|\not=|x_2|}} 1\\
= & \sum\limits_{\substack{|x_1|,|x_2|,|y_1|,|y_2|\le M\\ \beta_1(x_1-y_1)(x_1+y_1)\equiv \beta_2(y_2-x_2)(y_2+x_2) \bmod{q}\\ (x_1-y_1)(x_1+y_1)\not=0 \ \mbox{\scriptsize or } (y_2-x_2)(y_2+x_2)\not=0}} 1.
\end{split}
\end{equation*}
We further observe that if one of the numbers
$$
A_1=(x_1-y_1)(x_1+y_1) \quad \mbox{and} \quad A_2=(y_2-x_2)(y_2+x_2)
$$
in the summation condition above equals 0, then the other one equals 0 as well. To see this, note that if $\beta_i A_i\equiv 0 \bmod{q}$, then $p^s|(x_i-y_i)$ and $p^t|(x_i+y_i)$ with $s+t\ge n$, which is not possible if $M\le q^{1/2-\varepsilon}$ unless $x_i-y_i=0$ or $x_i+y_i=0$.
It follows that
\begin{equation*}
\begin{split}
\mathcal{E}\ll \sum\limits_{\substack{|x_1|,|x_2|,|y_1|,|y_2|\le M\\ \beta_1(x_1-y_1)(x_1+y_1)\equiv \beta_2(y_2-x_2)(y_2+x_2) \bmod{q}\\ (x_1-y_1)(x_1+y_1)\not=0 \ \mbox{\scriptsize and } (y_2-x_2)(y_2+x_2)\not=0}} 1\\
\le \sum\limits_{\substack{0<|A_1|,|A_2|\le 2M^2\\ \beta_1A_1\equiv \beta_2A_2\bmod{q}}} \tau(|A_1|)\tau(|A_2|),
\end{split}
\end{equation*}
where $\tau(k)$ denotes the number of divisors of $k\in \mathbb{N}$. Since we know that $\tau(k)\ll k^{\varepsilon}$, we deduce that
$$
\mathcal{E}\ll M^{\varepsilon}F_{\beta_1,\beta_2}(2M^2,q),
$$
where
$$
F_{\beta_1,\beta_2}(X,q):=\sum\limits_{\substack{0<|A_1|,|A_2|\le X\\ \beta_1A_1\equiv \beta_2A_2\bmod{q}}} 1.
$$
We observe that if $M\le q^{1/2-\varepsilon}$ and $q$ is large enough, then any given $A_2$ in the summation condition above fixes $A_1$, if it exists at all. Hence, we trivially get
\begin{equation} \label{trivial}
F_{\beta_1,\beta_2}(2M^2,q)\le 4M^2.
\end{equation}
Collecting everything above, we arrive at
$$
\Sigma_{\beta_1,\beta_2,\beta_3}(M,q)^2\ll M\left(\Sigma_{\beta_1,\beta_2,\beta_3}(M,q)
+M^{2+\varepsilon}\right),
$$
implying the claimed bound
$$
\Sigma_{\beta_1,\beta_2,\beta_3}(M,q)\ll M^{3/2+\varepsilon}.
$$
The bound \eqref{trivial} is sharp: If $\beta_1=\beta_2$, then necessarily $A_1=A_2$ and we get exactly
$$
F_{\beta_1,\beta_2}(2M^2,q)= 4M^2.
$$
However, for {\it generic} $\beta_1$ and $\beta_2$, we should expect a much better bound. Indeed, if $\beta_1\overline{\beta_2}/q$ satisfies certain Diophantine properties, then we can get a saving over the trivial bound. In \cite[equation (63)]{BD}, we established that
$$
F_{\beta_1,\beta_2}(X,q)\ll \left(\frac{X^2}{q}+\frac{rX}{q}+\frac{X}{r}+1\right)(rXq)^{\varepsilon},
$$
provided that
$$
\frac{\beta_1\overline{\beta_2}}{q}=\frac{a}{r}+O(r^{-2})
$$
with $a\in \mathbb{Z}$, $r\in \mathbb{N}$ and $(a,r)=1$. Again combining everything above, we obtain
\begin{equation*}
\begin{split}
\Sigma_{\beta_1,\beta_2,\beta_3}(M,q)^2\ll & M\left(\Sigma_{\beta_1,\beta_2,\beta_3}(M,q)+M^{\varepsilon}F_{\beta_1,\beta_2}(2M^2,q)\right)\\
\ll & M\left(\Sigma_{\beta_1,\beta_2,\beta_3}(M,q)+\left(\frac{M^4}{q}+\frac{rM^2}{q}+\frac{M^2}{r}+1\right)(rMq)^{2\varepsilon}\right),
\end{split}
\end{equation*}
which implies the bound
\begin{equation} \label{method1}
\Sigma_{\beta_1,\beta_2,\beta_3}(M,q)\ll \left(\frac{M^{5/2}}{q^{1/2}}+\frac{r^{1/2}M^{3/2}}{q^{1/2}}+\frac{M^{3/2}}{r^{1/2}}+M\right)(rMq)^{\varepsilon}.
\end{equation}
So if $r$ is not too small or too large, we may get a saving over the trivial bound $\ll M^{3/2+\varepsilon}$.
The same arguments as above can be applied for $\beta_1\overline{\beta_3}/q$ or $\beta_2\overline{\beta_3}/q$ in place of $\beta_1\overline{\beta_2}/q$. Now for $i=1,2$, Dirichlet's approximation theorem tells us that if $Q\in \mathbb{N}$, then there exists $r_i\le Q$ and $a_i\in \mathbb{Z}$ with $(a_i,r_i)=1$ such that
\begin{equation}\label{appro}
\left|\frac{\beta_i\overline{\beta_3}}{q}-\frac{a_i}{r_i}\right|\le \frac{1}{r_iQ}\le \frac{1}{r_i^2}.
\end{equation}
Suppose $R\in \mathbb{N}$ is another parameter. Both, $R$ and $Q$ will be fixed later. If one of $r_1$ and $r_2$ exceeds $R$, then we deduce that
\begin{equation} \label{Method1}
\Sigma_{\beta_1,\beta_2,\beta_3}(M,q)\ll \left(\frac{M^{5/2}}{q^{1/2}}+\frac{Q^{1/2}M^{3/2}}{q^{1/2}}+\frac{M^{3/2}}{R^{1/2}}+M\right)(QMq)^{\varepsilon}.
\end{equation}
If both $r_1$ and $r_2$ are less than $R$, then we will use the following different method. We first write our congruence in question as
$$
x_3^2\equiv -\beta_1\overline{\beta_3}x_1^2-\beta_2\overline{\beta_3}x_2^2\bmod{q}.
$$
Multiplying by $r:=r_1r_2$ gives
\begin{equation} \label{newcong}
rx_3^2\equiv -\beta_1\overline{\beta_3}rx_1^2-\beta_2\overline{\beta_3}rx_2^2\bmod{q}.
\end{equation}
By \eqref{appro}, we have
$$
-\beta_1\overline{\beta_3}r=-a_1r_2q + O\left(\frac{qr_2}{Q}\right)
$$
and
$$
-\beta_2\overline{\beta_3}r=-a_2r_1q+O\left(\frac{qr_1}{Q}\right).
$$
Now reducing the right-hand side of \eqref{newcong} modulo $q$, we deduce that
\begin{equation} \label{newcong1}
rx_3^2\equiv \gamma_1x_1^2+\gamma_2x_2^2\bmod{q},
\end{equation}
where
$$
\gamma_1:=-\beta_1\overline{\beta_3}r+a_1r_2q
$$
and
$$
\gamma_2:=-\beta_2\overline{\beta_3}r+a_2r_1q,
$$
and $\gamma_1$, $\gamma_2$ satisfy the bound
$$
\gamma_1,\gamma_2\ll \frac{q(r_1+r_2)}{Q}\ll \frac{qR}{Q}.
$$
We also have $r\le R^2$.
Now we write \eqref{newcong1} as an equation in the form
\begin{equation} \label{equ}
rx_3^2+kq=\gamma_1x_1^2+\gamma_2x_2^2,
\end{equation}
where
$$
k=O\left(\frac{R^2M^2}{q}+\frac{RM^2}{Q}\right)
$$
since $|x_1|,|x_2|,|x_3|\le M$. If the left-hand side of \eqref{equ} is fixed, then by Proposition \ref{quadequations} there are $O((RM)^{\varepsilon})$ solutions $(x_2,x_3)$ with $|x_2|,|x_3|\le M$.
Since we have
$$
O\left(\frac{R^2M^3}{q}+\frac{RM^3}{Q}+M\right)
$$
choices for the left-hand side of \eqref{equ}, it follows that
\begin{equation} \label{Method2}
\Sigma_{\beta_1,\beta_2,\beta_3}(M,q)\ll \left(\frac{R^2M^3}{q}+\frac{RM^3}{Q}+M\right)(RM)^{\varepsilon}.
\end{equation}
Now using \eqref{Method1} and \eqref{Method2} in the relevant complementary cases, we obtain
\begin{equation*}
\Sigma_{\beta_1,\beta_2,\beta_3}(M,q)\ll \left(\frac{M^{5/2}}{q^{1/2}}+\frac{Q^{1/2}M^{3/2}}{q^{1/2}}+\frac{M^{3/2}}{R^{1/2}}+\frac{R^2M^3}{q}+\frac{RM^3}{Q}+M\right)(QRMq)^{\varepsilon}.
\end{equation*}
Choosing
$$
R:=\lceil q^{2/5}M^{-3/5} \rceil \quad \mbox{and} \quad Q:=\lceil q^{3/5}M^{3/5} \rceil
$$
and recalling $M\le q^{1/2-\varepsilon}$ gives
$$
\Sigma_{\beta_1,\beta_2,\beta_3}(M,q)\ll \left(\frac{M^{5/2}}{q^{1/2}}+\frac{M^{9/5}}{q^{1/5}}+M\right)q^{\varepsilon}.
$$
It follows that
$$
p^{r/2}\Sigma_{\beta_1,\beta_2,\beta_3}\left(L_r,q_r\right)\ll \left(\frac{q^2}{N^{5/2}}+\frac{q^{8/5}}{N^{9/5}}+\frac{q}{N}\right)q^{\varepsilon}
$$
for $r=0,...,n-2$.
Plugging this into \eqref{essential} gives
$$
\Sigma_{\alpha_1,\alpha_2,\alpha_3}(N,q)=\hat\Phi(0)^3\cdot C_p(\alpha_1,\alpha_2,\alpha_3)\cdot \frac{N^3}{q}+O\left(\left(N^{1/2}q^{1/2}+N^{6/5}q^{1/10}+\frac{N^2}{q^{1/2}}\right)q^{\varepsilon}\right).
$$
The $O$-term is smaller than the main term if
$$
N\ge q^{11/18+\varepsilon}.
$$
So if this inequality is satisfied, we get an asymptotic. This completes the proof of Theorem \ref{mainresult2}.
\end{document} | math | 68,194 |
\begin{document}
\author{Rafael von K\"anel and Benjamin Matschke}
\title{{\Large Solving $S$-unit, Mordell, Thue, Thue--Mahler and generalized Ramanujan--Nagell equations via Shimura--Taniyama conjecture}}
\maketitle
\begin{abstract} {\scriptsize In the first part we construct algorithms (over $\QQ$) which we apply to solve $S$-unit, Mordell, cubic Thue, cubic Thue--Mahler and generalized Ramanujan--Nagell equations. As a byproduct we obtain alternative practical approaches for various classical Diophantine problems, including the fundamental problem of finding all elliptic curves over $\QQ$ with good reduction outside a given finite set of rational primes. The first type of our algorithms uses modular symbols, and the second type combines explicit height bounds with efficient sieves. In particular we construct a refined sieve for $S$-unit equations which combines Diophantine approximation techniques of de Weger with new geometric ideas. To illustrate the utility of our algorithms we determined the solutions of large classes of equations, containing many examples of interest which are out of reach for the known methods. In addition we used the resulting data to motivate various conjectures and questions, including Baker's explicit $abc$-conjecture
and a new conjecture on $S$-integral points of any hyperbolic genus one curve over $\QQ$.}
{\scriptsize In the second part we establish new results for certain old Diophantine problems (e.g. the difference of squares and cubes) related to Mordell equations, and we prove explicit height bounds for cubic Thue, cubic Thue--Mahler and generalized Ramanujan--Nagell equations. As a byproduct, we obtain here an alternative proof of classical theorems of Baker, Coates and Vinogradov--Sprind{\v{z}}uk. In fact we get refined versions of their theorems, which improve the actual best results in many fundamental cases. We also conduct some effort to work out optimized height bounds for $S$-unit and Mordell equations which are used in our algorithms of the first part. Our results and algorithms all ultimately rely on the method of Faltings (Arakelov, Par\v{s}in, Szpiro) combined with the Shimura--Taniyama conjecture, and they all do not use lower bounds for linear forms in (elliptic) logarithms.}
{\scriptsize In the third part we solve the problem of constructing an efficient sieve for the $S$-integral points of bounded height on any elliptic curve $E$ over $\QQ$ with given Mordell--Weil basis of $E(\QQ)$. Here we combine a geometric interpretation of the known elliptic logarithm reduction (initiated by Zagier) with several conceptually new ideas. The resulting ``elliptic logarithm sieve" is crucial for some of our algorithms of the first part. Moreover, it considerably extends the class of elliptic Diophantine equations which can be solved in practice: To demonstrate this we solved many notoriously difficult equations by combining our sieve with known height bounds based on the theory of logarithmic forms.}
\end{abstract}
{\scriptsize\tableofcontents}
\mathbbm{1}halfspacing
\section{Introduction}
In this paper we combine the method of Faltings~\cite{faltings:finiteness} (Arakelov, Par\v{s}in, Szpiro) with the Shimura--Taniyama conjecture \cite{wiles:modular,taywil:modular,breuil:modular} in order to study various classical Diophantine problems, including $S$-unit equations, Mordell equations, cubic Thue equations, cubic Thue--Mahler equations and generalized Ramanujan--Nagell equations.
We now begin to discuss the Diophantine equations. Let $S$ be a finite set of rational prime numbers. Write $N_S=1$ if $S$ is empty and $N_S=\prod_{p\in S} p$ otherwise. We denote by $\mathcal O^\times$ the units of
$\mathcal O =\ZZ[1/N_S]$ and we consider the $S$-unit equation
\begin{equation}\label{eq:sunit}
x+y=1, \ \ \ (x,y)\in\mathcal O^\times\times\mathcal O^\times.
\end{equation}
Many important Diophantine conjectures can be reduced to the study of $S$-unit equations. For example, the $abc$-conjecture of Masser--Oesterl\'e is equivalent to a certain height bound for the solutions of \eqref{eq:sunit}. On using Diophantine approximations in the style of Thue and Siegel, Mahler~\cite{mahler:approx1} showed that~\eqref{eq:sunit} has only finitely many solutions.
Furthermore there already exists a practical method of de Weger~\cite{deweger:phdthesis} which solves $S$-unit equations by using the theory of logarithmic forms~\cite{bawu:logarithmicforms}, see Section~\ref{sec:suheightalgo}. For a detailed discussion of (general) $S$-unit equations we refer to the recent book of Evertse--Gy{\H{o}}ry~\cite{evgy:bookuniteq}. Next we take a nonzero $a\in \mathcal O$ and we consider the Mordell equation
\begin{equation}\label{eq:mordell}
y^2=x^3+a, \ \ \ (x,y)\in\mathcal O\times\mathcal O.
\end{equation}
This Diophantine equation is a priori more difficult than~\eqref{eq:sunit}. Further, if $\mathcal O=\ZZ$ then resolving~\eqref{eq:mordell} is equivalent to solving the classical problem, going back at least to Bachet (1621), of finding all perfect squares and perfect cubes with given difference. In the case $\mathcal O=\ZZ$, Mordell \cite{mordell:1922,mordell:1923} showed finiteness of~\eqref{eq:mordell} via Diophantine approximation, and Baker--Davenport~\cite{baker:contributions,bada:diophapp} and Masser, Zagier~\cite{masser:ellfunctions,zagier:largeintegralpoints} introduced practical approaches solving \eqref{eq:mordell} via the theory of logarithmic forms (see Section~\ref{sec:malgo}). Furthermore we shall see that a special class of Mordell equations~\eqref{eq:mordell} covers in particular any generalized Ramanujan--Nagell equation discussed in \eqref{eq:rana} below. Finally, we let $m\in\mathcal O$ be nonzero and we suppose that $f\in \mathcal O[x,y]$ is a homogeneous polynomial of degree three with nonzero discriminant. Consider the cubic Thue equation
\begin{equation}\label{eq:thue}
f(x,y)=m, \ \ \ (x,y)\in \mathcal O\times \mathcal O.
\end{equation}
Thue (1909) proved that (\ref{eq:thue}) has only finitely many solutions in the case $\mathcal O=\ZZ$. In general, equation \eqref{eq:thue} is essentially equivalent to the cubic Thue--Mahler equation recalled in~\eqref{eq:thue-mahler} below. Baker--Davenport~\cite{baker:contributions,bada:diophapp} and Tzanakis--de Weger~\cite{tzde:thue,tzde:thuemahler} obtained practical approaches solving in particular cubic Thue and Thue--Mahler equations via the theory of logarithmic forms~\cite{bawu:logarithmicforms}, see also the discussions in Section~\ref{sec:thuealgo}.
\subsection{Algorithms}\label{sec:ia}
We construct two types of algorithms which we use to solve $S$-unit equations \eqref{eq:sunit}, Mordell equations \eqref{eq:mordell}, cubic Thue equations~\eqref{eq:thue}, cubic Thue--Mahler equations~\eqref{eq:thue-mahler} and generalized Ramanujan--Nagell equations~\eqref{eq:rana}. Both types do not use the theory of logarithmic forms. Before we discuss our algorithms in more detail, we describe the general strategy.
\subsubsection{General strategy}\label{sec:igeneralstrategy} As in \cite{rvk:modular} we use the method of Faltings (Arakelov, Par\v{s}in, Szpiro) which in our situation is applied as follows: Let $Y(\mathcal O)$ be the set of solutions of any of the above equations.
Then there is an effective map $\phi$ (Par\v{s}in{} construction) from~$Y(\mathcal O)$
to the set $M(T)$ of isomorphism classes of elliptic curves over a controlled open $T\subset\textnormal{Spec}(
\ZZ)$, $$\phi: Y(\mathcal O)\to M(T).$$ Here effective means that one can compute $\phi^{-1}(E)$ for each $E$ in $M(T)$. To determine $Y(\mathcal O)$, it thus suffices to compute $M(T)$ (effective Shafarevich theorem). For this purpose we use two types of algorithms: The first type applies Cremona's algorithm~\cite{cremona:algorithms} involving modular symbols, and the second type combines our optimized height bounds (see Section~\ref{sec:ihb}) with efficient sieves. Both types of algorithms crucially rely on a geometric version of the Shimura--Taniyama conjecture~\cite[Thm A]{breuil:modular} using inter alia the Tate conjecture~\cite[Thm 4]{faltings:finiteness}, and on isogeny estimates based on the method of Mazur~\cite{mazur:qisogenies,kenku:ellisogenies} or Faltings~\cite{faltings:finiteness,raynaud:abelianisogenies}. In fact the strategy of combining modularity with Faltings' method gives effective finiteness results for considerably more general Diophantine problems, see \cite{rvk:modular,vkkr:intpointsshimura}. However in the present paper we focus on optimizing the strategy for the fundamental Diophantine equations appearing in the title, and in the future we plan to work out algorithms for other Diophantine problems of interest.
\subsubsection{Algorithms via modular symbols} We next discuss in more detail our first type of algorithms. They crucially rely on Cremona's algorithm~\cite{cremona:algorithms} using modular symbols in order to compute all elliptic curves over $\QQ$ of given conductor. This allows to determine $M(T)$, since the curves in $M(T)$ have bounded conductor. Then we compute $Y(\mathcal O)$ by enumerating $\phi^{-1}(E)$ for each $E$ in $\phi(Y(\mathcal O))$. Here we exploit that the maps $\phi$ are effective by classical constructions going back at least to Cayley, Mordell and Frey--Hellegouarch. To illustrate the utility of our first type of algorithms we computed several examples. For instance we solved the $S$-unit equation~\eqref{eq:sunit} for all sets $S$ with $N_S\leq 20000$, and we solved several Mordell equations~\eqref{eq:mordell}. In fact, as already pointed out in \cite[Sect 1.1.2]{rvk:modular}, our first type of algorithms can in principle solve any Diophantine equation inducing integral points on a moduli scheme of elliptic curves with an effective Par\v{s}in{} construction $\phi$, see \eqref{eq:mdisplay} and Section~\ref{sec:moduli}. This class of equations contains in particular all equations considered in this paper. Here we mention that the possibility of solving certain cubic Thue--Mahler equations via modular symbols was already discussed in Bennett--Dahmen~\cite[$\mathsection$14]{beda:kleinsuperell}, see also the recent works of Kim~\cite{kimd:modularthuemahler} and Bennett--Billerey~\cite{bebi:sumsofunits}. Our first type of algorithms is very fast for ``small" parameters, since in this case we can use Cremona's database listing all elliptic curves over $\QQ$ of conductor at most 350000 (as of August 2014). In particular these algorithms directly benefit from the ongoing extension of such databases. However the approach via modular symbols can usually not compete with the actual most efficient methods solving our equations of interest. Thus we worked out a second type of algorithms.
\subsubsection{Algorithms via height bounds}\label{sec:ialgoheight} We now give a more detailed description of our second type of algorithms. They rely on an effective Shafarevich theorem in the form of explicit bounds for the Faltings height $h_F$ on~$M(T)$. The space $M(T)$ can be very complicated and it is usually a difficult task to compute $M(T)$, see the discussions surrounding \eqref{eq:mdisplay}. Hence instead of first computing $M(T)$ and then $Y(\mathcal O)=\phi^{-1}(M(T))$, we often directly work with $Y(\mathcal O)$ by using that the height $\phi^*h_F$ is bounded on~$Y(\mathcal O)$. This has the advantage that we can exploit extra structures on $Y(\mathcal O)$ in order to construct efficient sieves for solutions of bounded height.
\paragraph{$S$-unit equation.} To solve $S$-unit equations~\eqref{eq:sunit} it is natural to consider the set $\Sigma(S)$ of solutions of~\eqref{eq:sunit} modulo symmetry. Here solutions $(x,y)$ and $(x',y')$ of~\eqref{eq:sunit} are called symmetric if $x'$ or $y'$ lies in $\{x,\frac{1}{x},\frac{1}{1-x}\}$. One can directly write down all solutions which are symmetric to a given solution and thus it suffices to determine $\Sigma(S)$ in order to solve \eqref{eq:sunit}. In fact the number of solutions of~\eqref{eq:sunit} is either zero or $6\mathcal As{\Sigma(S)}-3$ where $\mathcal As{\Sigma}$ denotes the cardinality of a set $\Sigma$. For any $n\in\ZZ_{\geq 1}$ we denote by $S(n)$ the set of the $n$ smallest rational primes. Our Algorithm~\ref{algo:suheight} allows to efficiently solve~\eqref{eq:sunit}, even for sets $S$ with relatively large~$\mathcal As{S}$. To demonstrate this we solved large classes of $S$-unit equations~\eqref{eq:sunit} by using Algorithm~\ref{algo:suheight}. In particular, we obtained the following result.
\noindent{\bf Theorem~A.}
\emph{Suppose that $n\in\{1,2,\ldots,16\}$. Then the cardinality $\#$ of the set $\Sigma(S(n))$ is given in the following table.}
\begin{table}[h]
\begin{center}
{\small
\begin{tabular}{lccccccccccccccc}
$n$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$\\
\cmidrule(r){2-9}
\# & $1$ & $4$ & $17$ & $63$ & $190$ & $545$ & $1433$ & $3649$\\
\\
$n$ & $9$ & $10$ & $11$ & $12$ & $13$ & $14$ & $15$ & $16$\\
\cmidrule(r){2-9}
\# & $8828$ & $20015$ & $44641$ & $95358$ & $199081$ & $412791$ & $839638$ & $1234567$
\end{tabular}
}
\end{center}
\end{table}
\\
We mention that among all sets $S$ of cardinality $n$ the set $S(n)$ is usually the most difficult case for solving \eqref{eq:sunit}.
In \cite[$\mathsection$10]{zagier:polylogs}, Zagier explained that the cardinality of $\Sigma(S)$ plays an important role in certain questions on polylogarithms. In particular he states a table attributed to Gross--Vojta, which for each $n\in\{1,\dotsc,8\}$ lists a lower bound for the cardinality of $\Sigma(S(n))$. Theorem~A proves that the entries of this table are not only lower bounds, but in fact the correct values. Further we point out that the cases $n\in\{1,\dotsc,6\}$ in Theorem~A are not new. They were previously known by the work of de Weger \cite[Thm 5.4]{deweger:lllred}. Our Algorithm~\ref{algo:suheight} substantially improves de Weger's method in~\cite{deweger:lllred} in the following sense: Instead of using inequalities based on the theory of logarithmic forms as done by de Weger, we apply our optimized height bounds (see Section~\ref{sec:ihb}). These optimized bounds are strong enough such that we can omit de Weger's reduction process. Then to enumerate all solutions of~\eqref{eq:sunit} of bounded height, we use de Weger's sieve which is efficient as long as $\mathcal As{S}$ is small (e.g. $\mathcal As{S}<6$). To deal efficiently with sets $S$ of larger cardinality, we were forced to introduce new ideas: In Section \ref{sec:dwsieve+} we take into account certain geometric considerations to construct a refined sieve, and in Section~\ref{sec:suenum} we develop a refined enumeration algorithm for solutions of~\eqref{eq:sunit} with very small height. Our new ideas are crucial to efficiently solve \eqref{eq:sunit} for sets $S$ with $\mathcal As{S}\geq 6$. Furthermore we prove that our refinements are substantial in the sense that they considerably improve the running time in theory and in practice, see \eqref{eq:rbound} and Section~\ref{sec:suapplications}.
In general we conducted some effort to optimize Algorithm~\ref{algo:suheight}. We refer to Section~\ref{sec:sucomplexity} where we explain and motivate our optimizations. Also we developed a method which (automatically) chooses parameters that are close to optimal in the generic case. This was necessary to obtain our database $\mathcal D_1$ listing the solutions of the $S$-unit equation \eqref{eq:sunit} for many distinct sets $S$, including all sets $S$ with $N_S\leq 10^7$ and all sets $S\subseteq S(16)$.
\noindent{\bf Theorem~B.}
\emph{For each finite set of rational primes $S$ considered in $\mathcal D_1$, the database $\mathcal D_1$ contains all solutions of the $S$-unit equation \eqref{eq:sunit}.}
\noindent Another useful feature of Algorithm~\ref{algo:suheight} is that it allows to prove properties of $abc$-triples with bounded radical. For example, on using our algorithm we verified Baker's explicit $abc$-conjecture \cite[Conj 4]{baker:abcexperiments} for all $abc$-triples with radical at most $10^7$ or with radical composed of primes in $S(16)$. Furthermore we used our database $\mathcal D_1$ to motivate several new questions.
In particular, in view of the construction of the refined sieve, we make the following conjecture describing a property of integral points of $\mathbb P_\ZZ^1-\{0,1,\infty\}$ which is rather unexpected from a general Diophantine geometry perspective.
\noindent{\bf Conjecture 1.}
\emph{There exists $c\in\ZZ$ with the following property: If $n\in\ZZ_{\geq 1}$ then any finite set of rational primes $S$ with $\mathcal As{S}\leq n$ satisfies $\mathcal As{\Sigma(S)}\leq \mathcal As{\Sigma(S(n))}+c$.}
\noindent Theorem B shows that Conjecture~1 holds with $c=0$ for all sets $S$ in $\mathcal D_1$ and this motivates to ask whether any set of rational primes $S$ with $\mathcal As{S}\leq n$ satisfies $\mathcal As{\Sigma(S)}\leq \mathcal As{\Sigma(S(n))}$? We remark that additional applications of Algorithm~\ref{algo:suheight} are given in Section~\ref{sec:suapplications}. To conclude the discussion we point out that the geometric main idea (described in Section~\ref{sec:dwsieve+}) underlying our refined sieve is applicable in many other situations where sieves of de Weger type are applied. Here one can mention for example the practical resolution of $S$-unit and Thue--Mahler equations over number fields. We leave this for the future.
\paragraph{Mordell equation.} To solve the Mordell equation~\eqref{eq:mordell} via height bounds, we constructed Algorithm~\ref{algo:mheight}. Generically, this algorithm allows to deal efficiently with huge parameters.
To illustrate this feature we used Algorithm~\ref{algo:mheight} to create our database $\mathcal D_2$ listing the solutions of \eqref{eq:mordell} for large classes of pairs $(a,S)$ with $a\in\ZZ-\{0\}$, including the classes $\{\mathcal As{a}\leq 10,S\subseteq S(10^5)\}$, $\{\mathcal As{a}\leq 100,S\subseteq S(10^3)\}$ and $\{\mathcal As{a}\leq 10^4,S\subseteq S(300)\}$.
\noindent{\bf Theorem~C.}
\emph{For each pair $(a,S)$ considered in $\mathcal D_2$, the database $\mathcal D_2$ contains all solutions of the Mordell equation \eqref{eq:mordell} defined by $(a,S)$.}
\noindent Here we point out that Gebel--Peth{\H{o}}--Zimmer~\cite{gepezi:mordell} already established the important special case $\{\mathcal As{a}\leq 10^4, S=\emptyset \}$ by using their algorithm~\cite{gepezi:ellintpoints} based on the elliptic logarithm approach introduced by Masser and Zagier. Algorithm~\ref{algo:mheight} substantially improves the latter approach for \eqref{eq:mordell} in the following sense: Instead of using inequalities based on the theory of logarithmic forms, we apply our optimized height bounds (see Section~\ref{sec:ihb}). Our bounds are considerably stronger in practice, which leads to significant running time improvements as illustrated in Section~\ref{sec:minitbounds}. Then to enumerate all solutions of~\eqref{eq:mordell} with bounded height, we use the elliptic logarithm sieve constructed in Section~\ref{sec:elllogsieve}. Here our construction combines a geometric interpretation of the known elliptic logarithm reduction with conceptually new ideas described in Section~\ref{sec:setup}. The elliptic logarithm sieve is very efficient and it considerably improves in all aspects (see Section~\ref{sec:comparisonwithelr}) the known methods enumerating solutions of \eqref{eq:mordell}. However, our sieve requires an explicit Mordell--Weil basis of the group $E_a(\QQ)$ associated to the elliptic curve $E_a$ defined by \eqref{eq:mordell}. While it is usually possible to determine such a basis in practice, there is so far no general effective method. In fact the dependence on a Mordell--Weil basis is a disadvantage of Algorithm~\ref{algo:mheight} compared to the classical approach of Baker--Davenport which can be applied in the important case $S=\emptyset$. Their approach is very efficient in solving \eqref{eq:mordell} for varying $a\in\ZZ-\{0\}$ with $\mathcal As{a}$ at most some given bound, see Bennett--Ghadermarzi~\cite{begh:mordell}. On the other hand, an advantage of Algorithm~\ref{algo:mheight} over the known algorithms is that it can efficiently solve \eqref{eq:mordell} for large sets $S$. This feature allows to study the Diophantine problem for hyperbolic curves described in the next paragraph. Other important features of Algorithm~\ref{algo:mheight} are the following: It can efficiently solve \eqref{eq:mordell} for parameters $a$ with huge height and its underlying correctness proofs are complete (even when $2\in S$ or when $S$ contains bad reduction primes of $E_a$). For example these two features are crucial to efficiently determine all elliptic curves over $\textnormal{Spec}(\ZZ)-S$ by solving certain equations \eqref{eq:mordell}, see below.
\paragraph{Points of hyperbolic curves.} Suppose that $T$ and $B$ are nonempty open subschemes of $\textnormal{Spec}(\ZZ)$, and assume that $T\subseteq B$. Let $Y\to B$ be an arbitrary hyperbolic curve of genus $g$, see for example \cite[p.81]{mochizuki:absanab} for the definition. We denote by $Y(T)$ the set of $T$-points of $Y$ and we now consider the following Diophantine problem.
\noindent{\bf Problem.}
\emph{Describe the set $Y(T)$ in terms of $T$, with $T\subseteq B$ varying.}
\noindent If $g\geq 2$ then a result of Faltings \cite{faltings:finiteness} implies that the cardinality of $Y(T)$ is uniformly bounded in terms of $T$, which in some sense solves the problem for $g\geq 2$. Over the last decades the case $g=0$ was successfully studied by many authors, including Bombieri, Erd{\"o}s, Evertse, Gy{\H{o}}ry, Moree, Silverman, Stewart, Tijdeman \cite{evertse:sunits,erstti:manysol,evgystti:sunitclasses,bomupo:cluster,evmostti:manysol} and more recently Harper, Konyagin, Lagarias, Soundararajan \cite{koso:manysunits,laso:smoothabcsol,harper:manysunits}. However the situation completely changes for $g=1$. In this case, the problem is essentially not investigated in the literature and is widely open.
On using Algorithm~\ref{algo:mheight} we study the problem for the families of hyperbolic genus one curves defined by Mordell equations \eqref{eq:mordell} and cubic Thue equations \eqref{eq:thue}. In particular, motivated by Theorem~C and the construction of the elliptic logarithm sieve, we propose the following conjecture.
\noindent{\bf Conjecture 2.}
\emph{There are constants $c_a$ and $c_r$, depending only on $a$ and $r$ respectively, such that any finite nonempty set of rational primes $S$ satisfies $\mathcal As{Y_a(\mathcal O)}\leq c_a \mathcal As{S}^{c_r}$.}
\noindent Here $Y_a(\mathcal O)$ denotes the set of solutions of the Mordell equation \eqref{eq:mordell} defined by $(a,S)$ and $r$ is the rank of the free part of the finitely generated abelian group $E_a(\QQ)$. The conjectured bound is polynomial in terms of $\mathcal As{S}$, while as far as we know all conjectures and results in the literature provide exponential bounds such as in Evertse--Silverman~\cite{evsi:uniformbounds}. We construct an infinite family of sets $S[b]$ which shows that the exponent $c_r$ has to be at least $\tfrac{r}{r+2}$. Furthermore Theorem~C strongly indicates that $c_r=\tfrac{r}{r+2}$ would be still far from optimal for many sets $S$ of interest, including the sets $S(n)$. On taking into account Theorem~C, we ask whether one can replace in Conjecture 2 the quantity $\mathcal As{S}$ by the logarithm of the largest prime in $S$? We motivate this question by constructing a probabilistic model. Together with a classical Diophantine approximation result of Siegel~(1929) and known estimates for the de Bruijn function, this model predicts a bound for $\mathcal As{Y_a(\mathcal O)}$ in terms of $S$ which would be optimal in view of the family $S[b]$.
\paragraph{Effective Shafarevich theorem.} Now we take $T=\textnormal{Spec}(\ZZ)-S$ and we identify $M(T)$ with the set $M(S)$ of $\QQ$-isomorphism classes of elliptic curves over $\QQ$ with good reduction outside $S$. In the 1960s Shafarevich showed that $M(S)$ is finite: He reduced the problem to Mordell equations \eqref{eq:mordell} and then he applied Diophantine approximations. Coates~\cite{coates:shafarevich} made Shafarevich's proof effective by using the theory of logarithmic forms. In fact there already exist several practical methods which allow to determine the space $M(S)$. We refer to Section~\ref{sec:shaf} for an overview. On combining Shafarevich's reduction with our Algorithm~\ref{algo:mheight} for Mordell equations~\eqref{eq:mordell}, we obtain Algorithm~\ref{algo:shaf} which allows to compute $M(S)$. To illustrate the practicality of our approach, we determined the space $M(S)$ for each set $S\in\mathcal S$. Here $\mathcal S$ is a family of sets which contains in particular the set $S(5)$ and all sets $S$ with $N_S\leq 10^3$. Motivated by our data, we conjecture that one can replace in Conjecture~1 the moduli scheme $\mathbb P^1_{\ZZ[1/2]}-\{0,1,\infty\}$ of Legendre elliptic curves by the moduli stack $\mathcal M_{1,1}$ of elliptic curves. In other words for any $n\in\ZZ_{\geq 1}$ our conjecture says that among all sets $S$ with $\mathcal As{S}\leq n$ the cardinality of $M(S)$ is maximal (up to an absolute constant) when $S=S(n)$. For many sets $S\in\mathcal S$ it seems that computing the space $M(S)$ is out of reach for the known methods, see the discussions in Section~\ref{sec:shaf}. In particular, our approach is significantly more efficient than the method of Cremona--Lingham~\cite{crli:shafarevich}. They use a different reduction to Mordell equations \eqref{eq:mordell} which involves $j$-invariants, and then they solve \eqref{eq:mordell} via the algorithm of Peth{\H{o}}--Zimmer--Gebel--Herrmann \cite{pezigehe:sintegralpoints} based on the theory of logarithmic forms.
The input of Algorithm~\ref{algo:shaf} requires a Mordell--Weil basis of $E_a(\QQ)$ for $2\cdot 6^{\mathcal As{S}}$ distinct integers $a$. Thus our approach is not practical for large $\mathcal As{S}$. Next, we mention that the problem of explicitly describing the space
\begin{equation}\label{eq:mdisplay}
M(T)=M(S)
\end{equation}
is of interest for many reasons. For instance, in \cite{rvk:modular} the moduli formalism was used to reduce many Diophantine problems to the study of $M(T)$. On combining this strategy with our database listing the set $M(T)=M(S)$ for $S\in\mathcal S$, we can directly solve any Diophantine problem inducing $T$-points on moduli schemes $Y$ of elliptic curves with effective Par\v{s}in{} construction $\phi:Y(T)\to M(T)$; see Section~\ref{sec:moduli} for details and explicit examples.
Here it suffices to know the image of $\phi$ in $M(T)$, which is often much smaller than the whole space $M(T)$. Taking this into account, we simplified and optimized the strategy for several classical Diophantine problems. In particular, we worked out the cases of cubic Thue equations \eqref{eq:thue}, cubic Thue--Mahler equations~\eqref{eq:thue-mahler} and generalized Ramanujan--Nagell equations~\eqref{eq:rana}. This led to the following algorithms and results.
\paragraph{Thue equation.} We constructed Algorithm~\ref{algo:theight} which allows to solve the cubic Thue equation~\eqref{eq:thue}. Our approach is efficient in the generic case and it can deal with large sets $S$.
To illustrate this we used Algorithm~\ref{algo:theight} in order to compile the database $\mathcal D_3$ containing the solutions of \eqref{eq:thue} for large classes of parameter triples $(f,S,m)$, where $m\in\ZZ$ is nonzero and $f\in\ZZ[x,y]$ is homogeneous of degree three with nonzero discriminant $\Delta$ (see Section~\ref{sec:thueproofs}).
In particular our database $\mathcal D_3$ covers all $(f,S,m)$ such that $m=1$ and such that $(\Delta,S)$ lies in $\{\mathcal As{\Delta}\leq 10^4,S\subseteq S(100)\}$, $\{\mathcal As{\Delta}\leq 100,S\subseteq S(10^3)\}$ or $\{\mathcal As{\Delta}\leq 20,S\subseteq S(10^5)\}$; see Section~\ref{sec:ttmapp} for more information and additional examples.
\noindent{\bf Theorem~D.}
\emph{For each triple $(f,S,m)$ considered in $\mathcal D_3$, the database $\mathcal D_3$ contains all solutions of the cubic Thue equation~\eqref{eq:thue} defined by $(f,S,m)$.}
\noindent
This gives in particular a new proof of several results in the literature (see Section~\ref{sec:thuealgo}) which determined the solutions of specific cubic Thue equations~\eqref{eq:thue}. Furthermore Theorem~D motivates new conjectures and questions on the number of solutions of \eqref{eq:thue}, see Section~\ref{sec:ttmapp}. We next describe the main ingredients of Algorithm~\ref{algo:theight}. As in \cite[Sect 7.4]{rvk:modular} we reduce the problem to Mordell equations: This reduction uses classical invariant theory which provides an explicit morphism $\varphi:X\to Y$ over $T=\textnormal{Spec}(\mathcal O)$, where $X$ and $Y$ are the closed subschemes of $\mathbb A^2_T$ given by the Thue equation \eqref{eq:thue} and by the Mordell equation \eqref{eq:mordell} with $a=432\Delta m^2$ respectively. Then we compute $Y(T)$ using our Algorithm~\ref{algo:mheight} for Mordell equations and we apply triangular decomposition in order to finally determine $X(T)=\varphi^{-1}(Y(T))$. Here we recall that Algorithm~\ref{algo:mheight} requires a Mordell--Weil basis of $E_a(\QQ)$. Although it turned out that it is usually possible to determine such a basis in practice, the dependence on a Mordell--Weil basis is a disadvantage of our approach compared to the known methods discussed in Section~\ref{sec:thuealgo}. On the other hand, an advantage of our approach is that it can solve \eqref{eq:thue} for huge sets $S$. Here it seems that already sets $S$ with $\mathcal As{S}\geq 10$ are out of reach for the known methods solving \eqref{eq:thue}.
\paragraph{Thue--Mahler equation.} Let $f\in\mathcal O[x,y]$ be a homogeneous polynomial of degree three with nonzero discriminant $\Delta$, and let $m\in\mathcal O$ be nonzero. We constructed Algorithm~\ref{algo:tmheight} which allows in particular to solve the classical cubic Thue--Mahler equation
\begin{equation}\label{eq:thue-mahler}
f(x,y)=mz,
\end{equation}
where $x,y,z\in \ZZ$ with $z\in \mathcal O^\times$ and $\gcd(x,y)=1$.
To demonstrate the practicality of our approach, we used Algorithm~\ref{algo:tmheight} in order to create the database $\mathcal D_4$ listing the solutions of \eqref{eq:thue-mahler} for many triples $(f,S,m)$ with $m=1$ and $f\in\ZZ[x,y]$ as above. In particular $\mathcal D_4$ covers all such triples with $(\Delta,S)$ in $\{\mathcal As{\Delta}\leq 3000,S\subseteq S(2)\}$, $\{\mathcal As{\Delta}\leq 10^3,S\subseteq S(3)\}$, $\{\mathcal As{\Delta}\leq 100,S\subseteq S(4)\}$ or $\{\mathcal As{\Delta}\leq 16,S\subseteq S(5)\}$; see Section~\ref{sec:ttmapp} for more information.
\noindent{\bf Theorem~E.}
\emph{For each triple $(f,S,m)$ considered in $\mathcal D_4$, the database $\mathcal D_4$ contains all solutions of the cubic Thue--Mahler equation~\eqref{eq:thue-mahler} defined by $(f,S,m)$.}
\noindent
We mention that $\mathcal D_4$ contains in addition the solutions of \eqref{eq:thue-mahler} for various other $(f,S,m)$ of interest, including cases with $S=S(6)$. In fact Theorem~E gives in particular a new proof of several results in the literature (see Section~\ref{sec:thuealgo}) which solved specific equations~\eqref{eq:thue-mahler}.
We next describe the main ingredients of Algorithm~\ref{algo:tmheight}. On using an elementary standard reduction, we reduce \eqref{eq:thue-mahler} to $3^{\mathcal As{S}}$ distinct cubic Thue equations~\eqref{eq:thue} and these equations are then solved via Algorithm~\ref{algo:theight}. Here the applications of Algorithm~\ref{algo:theight} require $3^{\mathcal As{S}}$ distinct Mordell--Weil bases. Hence our approach is not practical when $\mathcal As{S}$ is large. However for small $\mathcal As{S}$ it turned out that it is usually possible to determine the required Mordell--Weil bases and then our approach is indeed efficient as illustrated in Section~\ref{sec:ttmapp}.
\paragraph{Generalized Ramanujan--Nagell equations.} Let now $b$ and $c$ be arbitrary nonzero elements of $\mathcal O$. On using our approach for Mordell equations~\eqref{eq:mordell}, we obtained Algorithm~\ref{algo:ranaheight} which allows to solve the generalized Ramanujan--Nagell equation
\begin{equation}\label{eq:rana}
x^2+b=cy, \ \ \ \ \ (x,y)\in\mathcal O\times \mathcal O^\times.
\end{equation}
There is a vast literature devoted to the study of (special cases of) this Diophantine problem. See for example the results, discussions and references in Bugeaud--Shorey~\cite{bush:rana},
Bennett--Skinner~\cite[Sect 8]{besk:ternary} and Saradha--Srinivasan~\cite{sasr:rana}. To illustrate the practicality of our approach, we used Algorithm~\ref{algo:ranaheight} in order to create the database $\mathcal D_5$ listing the solutions of \eqref{eq:rana} for many triples $(b,c,S)$ with $c=1$ and $b\in\ZZ-\{0\}$. In particular our database $\mathcal D_5$ covers all such triples with $(b,S)$ contained in $\{\mathcal As{b}\leq 12,S\subseteq S(5)\}$, $\{\mathcal As{b}\leq 35,S\subseteq S(4)\}$, $\{\mathcal As{b}\leq 250,S\subseteq S(3)\}$ or $\{\mathcal As{b}\leq 10^3,S\subseteq S(2)\}$.
\noindent{\bf Theorem~F.}
\emph{For each triple $(b,c,S)$ considered in $\mathcal D_5$, the database $\mathcal D_5$ contains all solutions of the generalized Ramanujan--Nagell equation~\eqref{eq:rana} defined by $(b,c,S)$.}
\noindent This theorem gives in particular a new proof of many results in the literature (see Section~\ref{sec:ranaalgoheight}) which solved special cases of~\eqref{eq:rana}. If $b\in\ZZ$ is nonzero and $c=1$, then Peth{\H{o}}--de Weger~\cite{pede:binaryrec1} obtained a practical approach to find all solutions $(x,y)$ of \eqref{eq:rana} with $x,y\in\ZZ_{\geq 0}$. Their method involves binary recurrence sequences and the theory of logarithmic forms. Our approach is completely different: On using an elementary construction, we reduce \eqref{eq:rana} to certain Mordell equations~\eqref{eq:mordell} which we then solve via Algorithm~\ref{algo:mheight}. Here the involved Mordell curves usually have huge height. This is no problem for Algorithm~\ref{algo:mheight} and it turned out that the bottleneck of our approach is finding the $3^{\mathcal As{S}}$ distinct Mordell--Weil bases required for the applications of Algorithm~\ref{algo:mheight}. In light of this, we worked out a refinement of Algorithm~\ref{algo:ranaheight} in the following special case of \eqref{eq:rana}. For arbitrary nonzero $b,c,d$ in $\ZZ$ with $d\geq 2$, consider the classical Diophantine problem
\begin{equation}\label{eq:rana2}
x^2+b=cd^n, \ \ \ \ \ (x,n)\in\ZZ\times \ZZ.
\end{equation}
Now, the crucial advantage of our refinement (see Algorithm~\ref{algo:rana2}) is that it only requires three distinct Mordell--Weil bases in order to find all solutions of \eqref{eq:rana2}. On using Algorithm~\ref{algo:rana2}, we solved \eqref{eq:rana2} for all triples $(7,1,d)$ with $d\leq 888$; we note that here the case $d=2$ corresponds to the classical Ramanujan--Nagell equation. Furthermore, in Section~\ref{sec:ranaapp} we worked out additional applications of Algorithms~\ref{algo:ranaheight} and \ref{algo:rana2}. For example, we apply our approach to the problem of finding all coprime $S$-units $x,y\in\ZZ$ with $x+y$ a square or a cube. Here we solve several new cases of this problem. Also, we show that our approach is a useful tool to study conjectures of Terai on Pythagorean triples.
\subsection{Diophantine problems related to Mordell equations}\label{sec:im}
We next discuss certain old Diophantine problems which are related to Mordell equations~\eqref{eq:mordell}. After presenting new results for primitive solutions of \eqref{eq:mordell}, we state a corollary on the greatest prime divisor of the difference of coprime squares and cubes. We also give new height bounds for the solutions of cubic Thue equations~\eqref{eq:thue}, cubic Thue--Mahler equations~\eqref{eq:thue-mahler} and generalized Ramanujan--Nagell equations~\eqref{eq:rana}. As a byproduct, we obtain in this section alternative proofs of classical theorems of Baker~\cite{baker:contributions}, Coates~\cite{coates:thue1,coates:thue2,coates:shafarevich} and Vinogradov--Sprind{\v{z}}uk~\cite{visp:thuemahler}.
\subsubsection{Primitive solutions of Mordell equations}\label{sec:introprimsol}
Following Bombieri--Gubler \cite[12.5.2]{bogu:diophantinegeometry}, we say that $(x,y)\in\ZZ\times \ZZ$ is primitive if $\pm 1$ are the only $n\in\ZZ$ with $n^{6}$ dividing $\gcd(x^3,y^2)$. In particular $(x,y)\in\ZZ\times\ZZ$ is primitive if $x,y$ are coprime. To measure the number $a\in\mathcal O$ and the finite set $S$, we take $$a_S=1728N_S^2\prod p^{\min(2,\ord_p(a))}$$ with the product extended over all rational primes $p\notin S$. Let $h$ be the usual logarithmic Weil height \cite[p.16]{bogu:diophantinegeometry}, with $h(n)=\log\mathcal As{n}$ for $n\in\ZZ-\{0\}$. Building on the arguments of \cite[Cor 7.4]{rvk:modular}, we establish the following result (take $\mu=0$ in Theorem~\ref{thm:m}).
\noindent{\bf Theorem~G.}
\emph{Let $a\in \ZZ$ be nonzero. Assume that $y^2=x^3+a$ has a solution in $\ZZ\times\ZZ$ which is primitive. Then any $(x,y)\in\mathcal O\times\mathcal O$ with $y^2=x^3+a$ satisfies}
$$\max\bigl(h(x),\tfrac{2}{3}h(y)\bigl)\leq a_S\log a_S.$$
\noindent We now discuss several aspects of this result. A useful feature of Theorem~G is that it does not involve~$\mathcal As{a}$. To illustrate this we take $n\in \ZZ_{\geq 1}$, we let $\mathcal F_n$ be the infinite family of integers $a$ with radical $\textnormal{rad}(a)$ at most~$n$, and we put $a_{*}=a_\emptyset$. Then it holds $a_*\leq 1728\textnormal{rad}(a)^2$ and Theorem~\ref{thm:m} with $\mu=0$ directly implies the following corollary.
\noindent{\bf Corollary~H.}
\emph{For any integer $n\geq 1$, the set of primitive $(x,y)\in\ZZ\times\ZZ$ with $y^2-x^3\in \mathcal F_n$ is finite and can in principle be determined. Furthermore if $a\in \ZZ$ satisfies $\log\mathcal As{a}\geq a_*\log a_*$, then there are no primitive $(x,y)\in\ZZ\times\ZZ$ with $y^2-x^3=a$.}
\noindent It holds that $(3m^{3n})^2=(2m^{2n})^3+m^{6n}$ for all $m,n\in\ZZ$. Hence we see that one can not remove the assumption in Theorem~G. However, one can weaken the assumption by considering a certain class of (almost primitive) solutions of~\eqref{eq:mordell} which fits into Szpiro's small points philosophy \cite{szpiro:lefschetz}; see Theorem~\ref{thm:m} and the discussions given there.
We also deduce Corollaries~\ref{cor:coates1} and~\ref{cor:coates2} on the difference of perfect squares and perfect cubes. On taking for example $\varepsilon=\frac{1}{10}$ in Corollary~\ref{cor:coates2}, one obtains the following result.
\noindent{\bf Corollary~I.}
\emph{Suppose that $x,y\in\ZZ$ are coprime, and write $X=\max(\mathcal As{x},\mathcal As{y})$. Then the greatest rational prime divisor $p$ of $y^2-x^3$ exceeds
$(1-\frac{1}{10})\log\log X-20.$}
\noindent This improves the old theorem of Coates \cite[Thm 2]{coates:shafarevich} in which he established the lower bound $10^{-3}(\log\log X)^{1/4}$. Similarly our Corollary~\ref{cor:coates1} refines \cite[Thm 1]{coates:shafarevich}.
\noindent \paragraph{Comparison with literature.} We point out that Coates' method, which uses early estimates for logarithmic forms, is completely different to the method applied in this paper. In fact it is possible to prove a weaker version of our Theorem~\ref{thm:m} and to improve Coates' results \cite[Thm 1 and 2]{coates:shafarevich} by using more recent estimates for logarithmic forms. However, it turns out that without introducing new ideas the actual best lower bounds for linear forms in logarithms (see Baker--W\"ustholz~\cite{bawu:logarithmicforms} for an overview) do not give inequalities as strong as those provided by Theorem~\ref{thm:m} and Corollaries~\ref{cor:coates1} and~\ref{cor:coates2}. For example, let us consider our asymptotic version of Corollary~I established in Corollary~\ref{cor:coates2}. For any $\varepsilon>0$ this version gives that the prime $p$ in Corollary~I exceeds
\begin{equation}\label{eq:asymptoticprimelowerbound}
\alpha\log\log X+\beta, \ \ \ \ \ \alpha=1-\varepsilon,
\end{equation}
where $\beta$ denotes an effective constant depending only on $\varepsilon$. This improves the actual best factor $\alpha=\tfrac{1}{84}-\varepsilon$ contained in the general result of Bugeaud~\cite[Thm 2]{bugeaud:greatestprimefactor}, which was proven by using inter alia a direct and ingenious reduction to lower bounds for logarithmic forms. Here it seems possible that one can slightly improve the factor $\alpha=\tfrac{1}{84}-\varepsilon$ by updating Bugeaud's approach with the actual best lower bounds for logarithmic forms. However, the presence of the usual quantity $h(\alpha_1)\cdot\dotsc\cdot h(\alpha_n)$ in these lower bounds shows that
this approach will always produce a factor $\alpha$ which is smaller than $\tfrac{1}{36}-\varepsilon$. In view of this, our result \eqref{eq:asymptoticprimelowerbound} seems to be out of reach for the present state of the art in the theory of logarithmic forms. See also the related discussions given at the end of Section~\ref{sec:mordellcoates}.
\paragraph{Idea of proof.} To prove our results for primitive solutions of \eqref{eq:mordell}, we go into the proof of \cite[Cor 7.4]{rvk:modular} which combines the Shimura--Taniyama conjecture with the method of Faltings (Arakelov, Par\v{s}in, Szpiro) as outlined in Section~\ref{sec:igeneralstrategy}; see also Section~\ref{sec:ihb}. Then we exploit that primitive solutions of \eqref{eq:mordell} induce, via the Par\v{s}in{}
construction $\phi$, elliptic curves with useful extra properties. For instance, to obtain the factor $\alpha=1-\varepsilon$ in \eqref{eq:asymptoticprimelowerbound}, we use that the Par\v{s}in{} construction $\phi$ maps coprime solutions to elliptic curves which have semistable reduction over $\ZZ[1/6]$. The corollaries are then direct consequences of our results for primitive/coprime solutions and of the prime number theorem.
\subsubsection{Height bounds for cubic Thue and Thue--Mahler equations}\label{sec:ithue}
\noindent Baker \cite{baker:contributions} applied his theory of logarithmic forms in order to establish in particular an effective finiteness result for any cubic Thue equation~\eqref{eq:thue} in the case when $\mathcal O=\ZZ$.
In the general case \eqref{eq:thue} is essentially equivalent to the cubic Thue--Mahler equation~\eqref{eq:thue-mahler}.
Mahler~\cite{mahler:approx1} showed via Diophantine approximations that \eqref{eq:thue-mahler} has only finitely many solutions.
Furthermore Coates~\cite{coates:thue1,coates:thue2} and Vinogradov--Sprind{\v{z}}uk~\cite{visp:thuemahler} independently
proved effective finiteness via the theory of logarithmic forms.
We refer to Baker--W\"ustholz \cite{bawu:logarithmicforms} and Evertse--Gy{\H{o}}ry~\cite{evgy:bookuniteq,evgy:bookdiscreq} for an overview on generalizations and improvements of finiteness results for Thue and Thue--Mahler equations.
\paragraph{Height bounds.} A new effective finiteness proof for any cubic Thue equation \eqref{eq:thue} was obtained in \cite{rvk:modular}. On working out explicitly the arguments of \cite[Sect 7.4]{rvk:modular}, we get explicit height bounds for the solutions of cubic Thue and Thue--Mahler equations. To state our results we denote by $h(f-m)$ the maximum of the logarithmic Weil heights of the coefficients of the polynomial $f-m\in\mathcal O[x,y]$. We put $a=432\Delta m^2$ with $\Delta$ the discriminant of $f$. The next corollary may be viewed as a refinement of \cite[Thm 7.1]{rvk:modular} in the case of moduli schemes (see Section~\ref{sec:moduli}) corresponding to \eqref{eq:thue} or \eqref{eq:thue-mahler}.
\noindent{\bf Corollary~J.}
\emph{The following statements hold.
\begin{itemize}
\item[(i)] Define the number $n$ by putting $n=2$ if $f,m\in\ZZ[x,y]$ and $n=10$ otherwise. Then any solution $(x,y)$ of the cubic Thue equation \eqref{eq:thue} satisfies $$\max\bigl(h(x),h(y)\bigl)\leq a_S\log a_S+43nh(f-m).$$
\item[(ii)] If $(x,y,z)$ is a solution of the cubic Thue--Mahler equation \eqref{eq:thue-mahler} then $$\max\bigl(h(x),h(y),\tfrac{1}{3}h(z)\bigl)\leq 2a_S\log a_S+86nh(f-m).$$
\end{itemize}}
\noindent
In Corollary~\ref{cor:precthue} we shall establish a more precise version of Corollary~J which provides sharper but more complicated bounds. Furthermore we shall show in Corollary~\ref{cor:precthue} that statement (ii) holds more generally for any primitive solution $(x,y,z)$ of the general cubic Thue--Mahler equation~\eqref{eq:thue-mahler}; the definition of such solutions is given in Definition~\ref{def:primsoltm}.
\paragraph{Comparison with literature.} We now compare Corollary~J with corresponding results in the literature. On using the theory of logarithmic forms, Bugeaud--Gy{\H{o}}ry~\cite[Thm 3 and 4]{bugy:thuemahler}, Bugeaud~\cite[Thm 3]{bugeaud:thuemahler}, Gy{\H{o}}ry--Yu~\cite[Thm 3]{gyyu:sunits} and Juricevic~\cite[$\mathsection$4.2]{juricevic:mordell} obtained the actual best height bounds\footnote{We point out that these results hold for Diophantine equations which are considerably more general than \eqref{eq:thue} and \eqref{eq:thue-mahler}, and some of these results deal moreover with arbitrary number fields.} for the solutions of \eqref{eq:thue} and \eqref{eq:thue-mahler}. We do not state these rather complicated bounds, but we mention that each of them has certain advantages and disadvantages. To compare these results with Corollary~J, we may and do assume that $f\in\ZZ[x,y]$ and $m\in\ZZ$. Then it follows that $$a_S\leq 2^83^5\Delta_2\bigl(\textnormal{rad}(m)N_S\bigl)^2$$ where $\Delta_2=\min\bigl(\textnormal{rad}(\Delta)^2,\mathcal As{\Delta}\bigl)$, and standard height arguments lead to $\mathcal As{\Delta}\leq 3^5H^4$ for $H=\max_i \mathcal As{a_i}$ the maximum of the absolute values of the coefficients $a_i$ of $f$. Therefore Corollary~J gives estimates which are asymptotically of the form $H^4\log H$, improving the actual best bounds $(H\log H)^4$ in terms of $H$. In particular in the classical case, when $m$ is fixed (usually $m=1$) and $\mathcal O=\ZZ$, our Corollary~J improves the actual best results in all aspects. Furthermore Corollary~J improves the known estimates in terms of $S$ for infinitely many sets $S$, including all sets $S$ with $\mathcal As{S}\leq 3$. On the other hand, our results are worse in terms of $m$ and the bound \cite[(12)]{gyyu:sunits} is significantly better in terms of $S$ for infinitely many sets $S$ including all sets $S=S(n)$ with $n$ large. Finally we mention that our estimates (see also Corollary~\ref{cor:precthue}) involve small absolute constants and hence they considerably improve the actual best height bounds for all parameters which are not that large. This might be of interest for the practical resolution of \eqref{eq:thue} and \eqref{eq:thue-mahler}.
\paragraph{Idea of proof.} Following \cite[Sect 7.4]{rvk:modular}, we deduce our height bounds for cubic Thue equations \eqref{eq:thue} from a result for Mordell equations (Theorem~\ref{thm:m}) discussed above. This deduction uses classical invariant theory which provides an explicit morphism $\varphi:X\to Y$ over $T=\textnormal{Spec}(\mathcal O)$, where $X$ and $Y$ are the closed subschemes of $\mathbb A^2_T$ given by the Thue equation \eqref{eq:thue} and by the Mordell equation \eqref{eq:mordell} with $a=432\Delta m^2$ respectively. Then in Proposition~\ref{prop:heightineq} we control the Weil height of any $P\in X(T)$ in terms of the Weil height of $\varphi(P)\in Y(T)$. To prove Proposition~\ref{prop:heightineq} we apply inter alia an effective arithmetic Nullstellensatz over the hypersurface in $\mathbb A_T^3$ given by $f-mz^3$. In fact we use here the Nullstellensatz of D'Andrea--Krick--Sombra~\cite{dakrso:nullstell} which leads to small constants. Finally, we deduce our height bounds for cubic Thue--Mahler equations \eqref{eq:thue-mahler} by invoking an elementary standard construction which reduces \eqref{eq:thue-mahler} to Thue equations~\eqref{eq:thue}. Alternatively, one can obtain explicit height bounds for the solutions of \eqref{eq:thue} and \eqref{eq:thue-mahler} by directly applying \cite[Thm 7.1]{rvk:modular} with suitable moduli problems; see Sections~\ref{sec:ihb} and \ref{sec:moduli}.
\subsubsection{Height bounds for generalized Ramanujan--Nagell equations}\label{sec:irana}
An elementary construction reduces the generalized Ramanujan--Nagell equation \eqref{eq:rana}, and the more classical special case \eqref{eq:rana2}, to Mordell equations \eqref{eq:mordell}. In light of this, results of Mordell~(1922) and Mahler~(1933) give finiteness for \eqref{eq:rana2} and \eqref{eq:rana} respectively. Moreover effective finiteness follows from Baker~\cite{baker:mordellequation} in the case \eqref{eq:rana2} and from Coates~\cite{coates:shafarevich} in the case \eqref{eq:rana}. Our height bounds for Mordell equations lead to the following result which may be viewed as a refinement of \cite[Thm 7.1]{rvk:modular} in the case of moduli schemes (see Section~\ref{sec:moduli}) corresponding to generalized Ramanujan--Nagell equations \eqref{eq:rana}.
\noindent{\bf Corollary~K.}
\emph{If $(x,y)$ satisfies the generalized Ramanujan--Nagell equation \eqref{eq:rana}, then} $$\max\bigl(2h(x),h(y)\bigl)\leq 2a_S+h(a)+3h(c), \ \ \ a=bc^2.$$
\noindent In Corollary~\ref{cor:ranabounds} we shall give a more precise version of this height bound. Furthermore we shall deduce Corollary~\ref{cor:sumsofunits} which provides explicit height bounds for ``coprime" $u,v\in\mathcal O^\times$ with $u+v$ a square or cube in $\QQ$. To discuss an application, we take arbitrary coprime $m,n\in\ZZ$ and we consider the following simple condition in terms of $r=\textnormal{rad}(mn)$:
\begin{itemize}
\item[$(*)$] The natural logarithm of $\mathcal As{m}$ or $\mathcal As{n}$ exceeds $(90r)^2\log (9r)$.
\end{itemize}
\noindent Now, the height bounds in Corollary~\ref{cor:sumsofunits} imply our next result which shows that condition~$(*)$ is in fact sufficient to rule out that $m+n$ is a perfect square or cube.
\noindent{\bf Corollary~L.}
\emph{Suppose that $m$ and $n$ are arbitrary coprime rational integers. If condition~$(*)$ holds, then $m+n$ is not a perfect square or cube.}
\noindent One can obtain versions of our corollaries by using height bounds for the solutions of Mordell or Thue--Mahler equations which are based on the actual state of the art in the theory of logarithmic forms. For a comparison of the resulting bounds with our estimates, we refer to the analogous discussions given above and in Section~\ref{sec:simpleboundsm}. We further mention that the strong $abc$-conjecture of Masser--Oesterl\'e in Remark~\ref{rem:abc} directly implies versions of our corollaries which are asymptotically considerably better; notice that these implications are (as usual) not compatible with any exponential version of the $abc$-conjecture.
\subsubsection{Proof of the height bounds}\label{sec:ihb}
There is a long tradition of proving effective height bounds for Thue equations, Mordell equations, Thue--Mahler equations and $S$-unit equations. In fact during the last few decades, one conducted quite some effort to refine the initial effective bounds of Baker~\cite{baker:contributions,baker:mordellequation}, Coates~\cite{coates:thue1,coates:thue2,coates:shafarevich} and Gy{\H{o}}ry~\cite{gyory:sunitshelvetica}.
See for example \cite{bawu:logarithmicforms,gyyu:sunits} for an overview on these refinements which all\footnote{Except Bombieri's refinement (see Bombieri--Cohen~\cite{boco:effdioapp2}) of Thue's method using Diophantine approximations. This method is relatively new and it is essentially self-contained. So far it leads to height bounds which are (slightly) worse compared to those coming from the theory of logarithmic forms.} ultimately rely on the theory of logarithmic forms. We now discuss the strategy underlying the proofs of our height bounds for the Diophantine problems considered in the present paper. The symbol (ST) refers to the geometric version of the Shimura--Taniyama conjecture~\cite[Thm A]{breuil:modular} which relies inter alia on the Tate conjecture~\cite[Thm 4]{faltings:finiteness}.
\paragraph{$S$-unit equation.} Let $(x,y)$ be a solution of the $S$-unit equation \eqref{eq:sunit}. In the 1990s Frey \cite[p.544]{frey:ternary} (see also Murty~\cite{murty:strongmodular}) remarked that (ST) provides an alternative approach to bound $h(x)$, and in 2011 it was independently shown by Murty--Pasten and by the first mentioned author that Frey's ideas together with (ST) lead to effective bounds for $h(x)$; see \cite[Thm 1.1]{mupa:modular} and \cite[Cor 7.2]{rvk:modular}. Here \cite{mupa:modular} works with the coprime Hecke algebra which leads to $h(x)\ll N_S\log N_S$, while \cite{rvk:modular} uses the full Hecke algebra which in general leads to the (slightly) weaker bound $h(x)\ll N_S(\log N_S)(\log\log N_S)$. Since there are also many situations in which the full Hecke algebra provides the best bounds, we use in this paper the coprime and the full Hecke algebra approach. To work out the optimized height bounds for Algorithm~\ref{algo:suheight}, we follow closely the arguments of \cite[Thm 1.1]{mupa:modular} and \cite[Cor 7.2]{rvk:modular} and we conduct some effort to refine the involved estimates.
Asymptotically, the actual best result $h(x)\ll N_S^{1/3}(\log N_S)^3$ is due to Stewart--Yu \cite{styu:abc2}. However, for all sets $S$ with $N_S\leq 2^{100}$ and for many other sets $S$ of practical interest, our optimized height bounds are considerably stronger than those coming from the theory of logarithmic forms; see Section~\ref{sec:simpleboundssu}. We note that this alternative approach for $S$-unit equations \eqref{eq:sunit} is in fact (see \cite{rvk:modular}) a special case of the method of Faltings (Arakelov, Par\v{s}in, Szpiro) combined with (ST) as described in Section~\ref{sec:igeneralstrategy} above.
\paragraph{Other Diophantine problems.} Many classical Diophantine problems can be reduced to $S$-unit equations. However in most cases the known (unconditional) reductions involve number fields larger than $\QQ$. In particular one can not combine these reductions with (unconditional) results for \eqref{eq:sunit} in order to deal with the Diophantine problems considered in the present paper. Instead we use that all these problems induce integral points on moduli schemes of elliptic curves. Given this observation, we can apply the strategy of \cite[Thm 7.1]{rvk:modular} which provides explicit height bounds for integral points on moduli schemes of elliptic curves. This strategy consists of combining (ST) with Faltings' method in a way which is similar as described in Section~\ref{sec:igeneralstrategy} above. Here important ingredients of the proof are the height-conductor inequality \cite[Prop 6.1]{rvk:modular} (proven independently in \cite[Thm 7.1]{mupa:modular}, see Section~\ref{sec:heightcondstatement}) and the moduli formalism. Besides providing a useful geometrical interpretation of various classical Diophantine problems, the moduli formalism allows to find new explicit applications of the method. Indeed we discovered many results of the present paper by searching for moduli schemes with ``interesting" defining equations: Given a priori the information that the equation defines a moduli scheme with effective Par\v{s}in{} construction $\phi$, one can explicitly work out the strategy of \cite[Thm 7.1]{rvk:modular} to get effective finiteness results; see \cite[Sect 7]{rvk:modular} and Section~\ref{sec:moduli}. Furthermore, a posteriori one can often obtain here simpler (but less conceptual) proofs by removing the moduli formalism. In light of the practical purpose of the present article, we conducted some effort to simplify our proofs as much as possible in the case of Mordell, cubic Thue, cubic Thue--Mahler and generalized Ramanujan--Nagell equations.
For instance on working out the method for the moduli schemes defined by Mordell equations \eqref{eq:mordell}, one obtains the actual best height bounds \cite[Cor 7.4]{rvk:modular} for the solutions of~\eqref{eq:mordell}. To prove the optimized height bounds for Algorithm~\ref{algo:mheight}, we follow and simplify the arguments of \cite[Cor 7.4]{rvk:modular}. Here we try to optimally estimate the involved quantities. We note that a priori the arguments of \cite[Sect 1-7]{rvk:modular} prove all our explicit simplified height bounds, but of the form $X(\log X)(\log\log X)$. To remove here in addition the factor $\log\log X$, we go into the proof of \cite[Lem 5.1]{rvk:modular} and we now use an idea of Murty--Pasten~\cite{mupa:modular} involving the coprime Hecke algebra. Their idea was not known to the author of \cite{rvk:modular}. However, for each considered Diophantine problem, there are also many situations in which the full Hecke algebra approach of \cite[Lem 5.1]{rvk:modular} provides the best bounds. Hence we work out all our height bounds using the coprime and the full Hecke algebra approach.
\subsection{Organization of the paper}
\paragraph{Plan of the paper.} In Section~\ref{sec:cremonas+st} we briefly discuss some tools which are crucial for our results and algorithms. In particular, we recall a geometric version of the Shimura--Taniyama conjecture which relies inter alia on the Tate conjecture.
In Section~\ref{sec:sunitalgo} we present two algorithms for $S$-unit equations. The first approach via modular symbols is worked out in Section~\ref{sec:sucremalgo}. Then in Section~\ref{sec:suheightalgo} we conduct some effort to construct the second algorithm which uses height bounds. Here, after discussing in Section~\ref{sec:dwsieve} a slight variation of de Weger's method, we construct our refined sieve in Section~\ref{sec:dwsieve+} and we develop a refined enumeration in Section~\ref{sec:suenum}. In Section~\ref{sec:suapplications} we present various applications of our algorithm via height bounds. In particular, we discuss our database $\mathcal D_1$ containing the solutions of large classes of $S$-unit equations and we use our data to motivate several Diophantine conjectures related to $S$-unit equations.
Section~\ref{sec:malgo}
contains two algorithms which allow to solve the Mordell equation. The first approach via modular symbols is worked out in Section~\ref{sec:mcremalgo}. Then in Section~\ref{sec:mheightalgo} we construct the second algorithm via height bounds. Here, after discussing the main ingredients of our algorithm, including our initial height bounds in Section~\ref{sec:minitbounds} and the elliptic logarithm sieve, we put everything together in Section~\ref{sec:mordellalgostat}. We also present various applications of our algorithm via height bounds. In Section~\ref{sec:shaf} we apply this algorithm to study the problem of finding all elliptic curves over $\QQ$ with good reduction outside $S$, and in Section~\ref{sec:moduli} we solve certain classes of Diophantine equations by combining our algorithm with the moduli formalism. Then in Section~\ref{sec:malgoapplications} we discuss our database $\mathcal D_2$ containing the solutions of large classes of Mordell equations and we motivate new conjectures/questions. In Section~\ref{sec:malgocomparison} we compare our algorithms with other methods.
In Section~\ref{sec:thuealgo} we present algorithms for cubic Thue and Thue--Mahler equations. Our algorithms use a construction from classical invariant theory which allows to reduce the Diophantine problems to Mordell equations. After working out some useful properties of this construction in Section~\ref{sec:talgoconst}, we discuss our algorithms via modular symbols in Section~\ref{sec:talgocremona}. Then in Section~\ref{sec:talgoheight} we explain our algorithms via height bounds and we give various applications. In particular, we discuss in Section~\ref{sec:ttmapp} parts of our databases $\mathcal D_3$ and $\mathcal D_4$ containing the solutions of large classes of cubic Thue and Thue--Mahler equations respectively. In Section~\ref{sec:talgocompa} we compare our algorithms with the known methods.
Section~\ref{sec:ranaalgo} contains two algorithms for the generalized Ramanujan--Nagell equation. After presenting our approach via modular symbols in Section~\ref{sec:ranaalgocremona}, we explain the algorithm via height bounds in Section~\ref{sec:ranaalgoheight} and we discuss some applications including our database $\mathcal D_5$. Then we compare our algorithms with the known methods in Section~\ref{sec:ranaalgocomp}.
In Sections~\ref{sec:mordellcoates}, \ref{sec:thueproofs} and \ref{sec:ranaheight} we consider certain classical Diophantine problems related to Mordell equations. In particular, in Section~\ref{sec:mordellcoates} we study properties of (almost) primitive solutions of Mordell equations and we deduce explicit lower bounds for the largest prime divisor of the difference of coprime squares and cubes. In Sections~\ref{sec:thueproofs} and \ref{sec:ranaheight} we give new explicit height bounds for the solutions of cubic Thue and Thue--Mahler equations and of generalized Ramanujan--Nagell equations. Then in Section~\ref{sec:heightbounds} we prove our results for (almost) primitive solutions and we work out the optimized height bounds for the solutions of Mordell and $S$-unit equations which are used in our algorithms.
In Section \ref{sec:elllogsieve} we construct the elliptic logarithm sieve. It allows to efficiently find all integral points of bounded height on any elliptic curve $E$ over $\QQ$ with given Mordell--Weil basis of $E(\QQ)$. We refer to the introduction of Section~\ref{sec:elllogsieve} for an overview of the main ideas of our construction. The elliptic logarithm sieve is of independent interest and thus we made the presentation of Section~\ref{sec:elllogsieve} independent of the rest of this paper.
\paragraph{Notation.} We shall use throughout the following (standard) notations and conventions. By $\log$ we mean the principal value of the natural logarithm. We define the product taken over the empty set as~$1$. For any set $M$, we denote by $\lvert M\rvert$ the (possibly infinite) number of distinct elements of~$M$. Let $f_1$ and $f_2$ be real valued functions on $M$. We write $f_1=O(f_2)$ if there is a constant $c$ such that $f_1\leq cf_2$. Further $f_1=O_\varepsilon(f_2^\varepsilon)$ means that for any real number $\varepsilon>0$ there is a constant $c(\varepsilon)$ depending only on $\varepsilon$ such that $f_1\leq c(\varepsilon) f_2^\varepsilon$. For any $n\in\ZZ_{\geq 1}$, we say that $\mathcal E\subset \mathbb R^n$ is an ellipsoid centered at the origin if $\mathcal E=\{x\in\mathbb R^n\,;\, q(x)\leq c\}$ for some positive definite quadratic form $q:\RR^n\to \RR$ and some positive real number $c$.
We denote by $\mathcal As{z}$ the usual complex absolute value of~$z\in\mathbb C$. If $m,n\in\ZZ$ then the symbol $m\mid n$ (resp. $m\nmid n)$ means that $m$ divides $n$ (resp. $m$ does not divide $n$). Further $\gcd(a_1,\dotsc,a_n)$ denotes the greatest common divisor of $a_1,\dotsc,a_n\in\ZZ$.
The radical of $n\in\ZZ$ is given by $\textnormal{rad}(n)=\prod p$ with the product taken over all rational primes $p$ dividing~$n$. If $\alpha\in \QQ$ is nonzero and if $p$ is a rational prime, then we write $\ord_p(\alpha)\in\ZZ$ for the order of $p$ in~$\alpha$. We denote by $h(\alpha)$ the usual absolute logarithmic Weil height of $\alpha\in\QQ$, with $h(0)=0$ and $h(\alpha)=\log\max(\mathcal As{m},\mathcal As{n})$ if $\alpha=m/n$ for coprime $m,n\in\ZZ$. Finally for any real number $x\in\mathbb R$, we write $\floor{x}=\max(n\in\ZZ\,;\, n\leq x)$ and $\ceil{x}=\min(n\in\ZZ\,;\, n\geq x)$.
\paragraph{Computer, software and algorithms.} Unless mentioned otherwise, we used a standard personal working computer at the MPI Bonn for our computations. Our algorithms are all implemented in Sage and we shall use functions of the computer algebra systems Pari~\cite{pari:parisystem}, Sage~\cite{sage:sagesystem} and Magma~\cite{magma:magmasystem}. In what follows, we shall sometimes refer by (PSM) to these computer packages in order to simplify the notation.
For each of our algorithms, we conducted some effort to motivate our constructions (in theory and in practice), to explain our choice of parameters, to discuss important complexity aspects, to give detailed correctness proofs and to circumvent potential numerical issues. We shall also list the running times of our algorithms for many examples. The listed times are always upper bounds. In fact some of them were obtained by using older versions of our algorithms, and in many cases the running times would now be significantly better when using the most recent versions (as of February 2016) of our algorithms.
\paragraph{Acknowledgements.}
The research presented in this paper was initiated when we were members at the IAS Princeton (2011/12), it was continued at the IH\'ES (2012/13) and it was completed at the MPIM Bonn (2013-15). We are grateful to these institutions for providing excellent working conditions. The authors were supported by the NSF grant No. DMS-0635607 (2011/12) and by EPDI fellowships (2012-14). We would like to thank the MPI, in particular Gerd Faltings and Pieter Moree, for support in (2013-15).
Further, we would like to thank Richard Taylor for motivating and very useful initial discussions. We are grateful to Yuri Bilu, Enrico Bombieri, Sander Dahmen, Jan-Hendrik Evertse, K\'alm\'an Gy{\H{o}}ry, Pieter Moree, Hector Pasten and Don Zagier for encouraging discussions and/or for informing us about useful literature. Also, we learned a lot from Yuri Bilu, John Cremona, Stephen Donnelly, Nuno Freitas, Steffen M\"uller, Martin Raum and Samir Siksek. We would like to thank all of them for explaining various aspects of computational number theory and/or for answering questions.
\paragraph{Data and earlier versions.} The present version of the paper (Feb. 2016) extends in particular all results and algorithms presented in our earlier versions of the paper (April 2013 and Nov. 2014). However we removed certain applications and discussions of our algorithms via modular symbols, since they are meanwhile obsolete in view of more recent results. Our data is uploaded on: \url{https://www.math.u-bordeaux.fr/~bmatschke/data/}.
\section{Shimura--Taniyama conjecture}\label{sec:cremonas+st}
A crucial ingredient for all of our results and algorithms is a (geometric) version of the Shimura--Taniyama conjecture which relies inter alia on the Tate conjecture. In the case of our first type of algorithms, another important ingredient is an algorithm of Cremona using modular symbols. In this section we first introduce some notation and then we briefly discuss these ingredients in order to emphasize that they do not depend on results proven by (classical) transcendence or Diophantine approximation techniques.
Let $N\geq 1$ be an integer. Consider the classical congruence subgroup $\Gamma_0(N)\subset\textnormal{SL}_2(\ZZ)$, let $X_0(N)=X(\Gamma_0(N))_\QQ$ be the smooth, projective and geometrically connected model over $\QQ$ of the modular curve associated to $\Gamma_0(N)$ and denote by $S_2(\Gamma_0(N))$ the complex vector space of cuspforms of weight 2 with respect to~$\Gamma_0(N)$. See for example~\cite{dish:modular} for the definitions. We denote by $J_0(N)=\textnormal{Pic}^0(X_0(N))$ the Jacobian variety of $X_0(N)$. Let $\mathbb T_\ZZ$ be the subring of the endomorphism ring of $J_0(N)$, which is generated over $\ZZ$ by
the usual Hecke operators $T_n$ for all $n\in \ZZ_{\geq 1}$.
For any $f\in S_2(\Gamma_0(N))$, we denote by $a_n(f)$ the $n$-th Fourier coefficient of $f$ and we say that $f$ is rational if $a_n(f)\in \QQ$ for all $n\in \ZZ_{\geq 1}$. Further, we say that $f\in S_2(\Gamma_0(N))$ is a newform (of level $N$) if $a_1(f)=1$, if $f$ lies in the new part of $S_2(\Gamma_0(N))$ and if $f$ is an eigenform for all Hecke operators on $S_2(\Gamma_0(N))$. We now suppose that $f\in S_2(\Gamma_0(N))$ is a rational newform. Let $I_f$ be the kernel of the ring homomorphism $\mathbb T_\ZZ\to \ZZ[\{a_n(f)\}]$ which
is induced
by $T_n\mapsto a_n(f)$.
It turns out that the image
$I_fJ_0(N)$ of $J_0(N)$ under $I_f$ is connected, and the corresponding quotient
\begin{equation*}
E_f=J_0(N)/I_fJ_0(N)
\end{equation*}
is an elliptic curve over $\QQ$ since $f$ is rational. We next define the modular degree and congruence number of~$f$. On composing the usual embedding $X_0(N)\hookrightarrow J_0(N)$, which sends the cusp $\infty$ of $X_0(N)$ to the zero element of $J_0(N)$, with the natural projection $J_0(N)\to E_f$, we obtain a finite morphism $\varphi:X_0(N)\to E_f$ of curves over~$\QQ$. The modular degree $m_f$ of $f$ is defined as the degree of~$\varphi$:
\begin{equation}\label{def:mf}
m_f=\textnormal{deg}(\varphi).
\end{equation}
The congruence number $r_f$ of $f$ is defined as the largest integer such that there exists a cusp form $f_c\in S_2(\Gamma_0(N))$ with all Fourier coefficients in $\ZZ$ and
\begin{equation}\label{def:rf}
(f,f_c)=0 \textnormal{ and } a_n(f)\equiv a_n(f_c)\textnormal{ mod } (r_f) \textnormal{ for all } n\in \ZZ_{\geq 1}.
\end{equation}
Here $(\ ,\,)$ denotes the usual Petersson inner product. Let $\mathcal E(N)$ be the set of all elliptic curves $E$ over $\QQ$ which are of the form $E=E_f$ for some rational newform $f\in S_2(\Gamma_0(N))$. We say that an elliptic curve $E$ over $\QQ$ is modular if there exists a positive integer $N$ such that the curve $E$ is $\QQ$-isogenous to some elliptic curve in~$\mathcal E(N)$.
For any given $N\in \ZZ_{\geq 1}$, Cremona's algorithm~\cite{cremona:algorithms} computes in particular the coefficients of minimal Weierstrass equations of all modular elliptic curves over~$\QQ$. A short description of this algorithm may be as follows: One considers~$X_0(N)$, computes its first homology using $M$-symbols (after Manin~\cite{manin:symbols}), computes the action of sufficiently many Hecke operators on it, and determines the one-dimensional eigenspaces with rational eigenvalues. By induction on the divisors of~$N$, this yields the rational newforms in $S_2(\Gamma_0(N))$, and their period lattices allow then to compute the set~$\mathcal E(N)$.
Finally on using a theorem of Mazur~\cite{mazur:qisogenies}, one computes all elliptic curves over $\QQ$ which are $\QQ$-isogenous to some curve in $\mathcal E(N)$ and one determines their minimal Weierstrass equations.
Building on the key breakthroughs by Wiles~\cite{wiles:modular} and by Taylor--Wiles~\cite{taywil:modular}, the Shimura--Taniyama conjecture was finally established by Breuil--Conrad--Diamond--Taylor~\cite{breuil:modular}. This conjecture implies (its geometric version saying) that any elliptic curve over $\QQ$ of conductor $N$ is $\QQ$-isogenous to some curve in~$\mathcal E(N)$. This implication uses the Tate conjecture~\cite[Thm 4]{faltings:finiteness}. We point out that Faltings' proof of the Tate's conjecture does not use transcendence theory or classical Diophantine approximations.
\section{Algorithms for $S$-unit equations}\label{sec:sunitalgo}
Let $S$ be a finite set of rational primes, write $N_S=\prod_{p\in S} p$ and denote by $\mathcal O^\times$ the group of units of
$\mathcal O =\ZZ[1/N_S]$. In this section, we are interested to solve the $S$-unit equation
\begin{equation}
x+y=1, \ \ \ (x,y)\in\mathcal O^\times\times \mathcal O^\times. \tag{\ref{eq:sunit}}
\end{equation}
If $\mathcal As{S}\leq 1$ then \eqref{eq:sunit} has either no solutions or $(2,-1)$, $(-1,2)$ and $(\frac{1}{2},\frac{1}{2})$ are the only solutions. Hence we may and do assume that $\mathcal As{S}\geq 2$ in this section.
As mentioned in the introduction, there already exists a practical method of de Weger \cite{deweger:lllred} which solves~\eqref{eq:sunit}. De Weger \cite[Thm 5.4]{deweger:lllred} used his method to completely solve~\eqref{eq:sunit} in the case $S=\{2,3,5,7,11,13\}$.
Further, Wildanger~\cite{wildanger:unitalgo} and afterwards Smart~\cite{smart:smallsol} generalized the ideas of de Weger and they obtained a practical algorithm which solves~\eqref{eq:sunit} over arbitrary number fields; see also Hajdu~\cite{hajdu:optimalsys} and Evertse--Gy{\H{o}}ry \cite{evgy:bookuniteq}. There is also the recent work of Dan-Cohen--Wewers~\cite{dawe:explicitkim,dawe:sunitalgomotivic}, with the ultimate goal to construct an algorithm \cite{dancohen:sunitalgo3} solving~\eqref{eq:sunit} via ``explicit motivic Chabauty--Kim theory". This method is inspired by Kim's ($p$-adic \'etale) approach \cite{kim:siegel}, see also the discussion in \cite[2.5.2]{dancohen:sunitalgo3} which mentions an additional method of Brown. So far all practical approaches solving \eqref{eq:sunit} crucially rely on the theory of logarithmic forms.
In the following Sections~\ref{sec:sucremalgo} and~\ref{sec:suheightalgo}, we present and discuss two alternative algorithms which solve $S$-unit equations~\eqref{eq:sunit}. Both of our algorithms do not use the theory of logarithmic forms. The first algorithm relies on Cremona's algorithm,
and we refer to the beginning of Section~\ref{sec:suheightalgo} for a short description of the ingredients of the second algorithm.
Before we begin to describe our algorithms, we discuss useful properties of symmetric solutions and we give a lower bound for the complexity of any algorithm solving \eqref{eq:sunit}.
\begin{definition}\label{def:symmetric}Suppose that $(x,y)$ and $(x',y')$ are solutions of \eqref{eq:sunit}. Then we say that $(x,y)$ and $(x',y')$ are symmetric solutions if $x'$ or $y'$ lies in $\{x,\tfrac{1}{x},\tfrac{1}{1-x}\}$.
\end{definition}
It turns out that this defines an equivalence relation on the set of solutions of \eqref{eq:sunit}, and hence we can consider the set $\Sigma(S)$ of solutions of \eqref{eq:sunit} up to symmetry. Suppose that $(x,y)$ is a solution of \eqref{eq:sunit}. Then one can directly determine all its symmetric solutions. In fact there are exactly six solutions of \eqref{eq:sunit} which are symmetric to $(x,y)$ provided that $(x,y)$ is not equal to $(2,-1)$, $(-1,2)$ or $(\tfrac{1}{2},\tfrac{1}{2})$. In particular, we see that the number of solutions of \eqref{eq:sunit} is either zero or $6\cdot \mathcal As{\Sigma(S)}-3$. The following remark shows that any algorithm solving the $S$-unit equation~\eqref{eq:sunit} can not be too fast in general.
\begin{remark}[Lower bound for complexity]\label{rem:sulowercomplexity}
The result of Erd{\"o}s--Stewart--Tijdeman \cite[Thm 4]{erstti:manysol}, see also the more recent work of Harper, Konyagin, Lagarias, Soundararajan \cite{koso:manysunits,laso:smoothabcsol,harper:manysunits}, implies the existence of an effective absolute constant $s_0$ with the following property. For any $s\in\ZZ_{\geq s_0}$ there exists a set $S$ with $\mathcal As{S}=s$ such that the $S$-unit equation~\eqref{eq:sunit} has at least $\exp((s/\log s)^{1/2})$ solutions.
Furthermore, there are infinitely many sets $S$ such that the $S$-unit equation~\eqref{eq:sunit} has a solution $(x,y)$ with $H(x)\geq N_S$ for $H(x)$ the multiplicative Weil height \cite[p.15]{bogu:diophantinegeometry} of~$x$.
Therefore we conclude that any algorithm solving~\eqref{eq:sunit} has running time which is in general not better than linear in $\log N_S$ and which is in general not better than $\exp((\mathcal As{S}/\log \mathcal As{S})^{1/2})$.
\end{remark}
\subsection{Algorithm via modular symbols}\label{sec:sucremalgo}
To assure that our algorithm really computes all solutions of the $S$-unit equation~\eqref{eq:sunit}, we use the following observations. We suppose that $(x,y)$ satisfies~\eqref{eq:sunit}. Then there exist nonzero $a,b,c\in\ZZ,$ with $\gcd(a,b,c)=1$ and $\textnormal{rad}(abc)\mid N_S$, such that $x=\frac{a}{c}$, $y=\frac{b}{c}$ and $a+b=c$. In other words, resolving~\eqref{eq:sunit} is equivalent to resolving
\begin{equation}
\label{eq:abc}
a+b = c, \quad a,b,c\in\ZZ-\{0\}, \quad \gcd(a,b,c)=1,\quad \textnormal{rad}(abc)\mid N_S.
\end{equation}
Further we observe that to find all solutions of~\eqref{eq:abc} it suffices to consider solutions $(a,b,c)$ of~\eqref{eq:abc} with $a,b,c$ all positive. Indeed if $(a,b,c)$ satisfies~\eqref{eq:abc} then there exists a solution $(\alpha,\beta,\gamma)$ of~\eqref{eq:abc} such that the sets $\{\alpha,\beta,\gamma\}$ and $\{\mathcal As{a},\mathcal As{b},\mathcal As{c}\}$ coincide.
For any elliptic curve $E$ over~$\QQ$, we denote by $\Delta_E$ the minimal discriminant of $E$ and we write $N_E$ for the conductor of~$E$. A construction of Frey--Hellegouarch~\cite{frey:curves,hellegouarch:curves} associates to any solution $(a,b,c)$ of~\eqref{eq:abc} an elliptic curve\footnote{We warn the reader that here $E_{abc}$ is not necessarily the usual Frey--Hellegouarch curve with Weierstrass equation $y^2=x(x-a)(x+b)$. See the proof of Lemma~\ref{lem:psu2} in which $E_{abc}$ is denoted by $E$.}
$E_{abc}$ over~$\QQ$. On taking the quotient of $E_{abc}$ by the ``subgroup" generated by a suitable 2-torsion point of~$E_{abc}$, one obtains an elliptic curve $E$ over $\QQ$ with the following properties (see Lemma~\ref{lem:psu2}).
\begin{lemma}\label{lem:psu1}
Suppose that $(a,b,c)$ is a solution of~\eqref{eq:abc}. Then there exists an elliptic curve $E$ over $\QQ$ such that $N_E$ divides $2^4N_S$ and $\Delta_{E}=2^{8-12m}\mathcal As{ab}c^4$ with $m\in\{0,1,2,3\}$.
\end{lemma}
In fact this lemma may be viewed (see \cite[Prop 3.2]{rvk:modular}) as an explicit Par\v{s}in{} construction for integral points on the Legendre moduli scheme $\mathbb P^1_{\ZZ[1/2]}-\{0,1,\infty\}$ of elliptic curves, which is induced by forgetting the Legendre level structure.
\begin{Algorithm}[$S$-unit equation via modular symbols]\label{algo:sucremona} The input is a finite set of rational primes $S$, and the output is the set of solutions $(x,y)$ of the $S$-unit equation~\eqref{eq:sunit}.
The algorithm\textnormal{:} If $2\notin S$ then output the empty set, and if $2\in S$ then do the following.
\begin{itemize}
\item[(i)] Use Cremona's algorithm, described in Section~\ref{sec:cremonas+st}, to compute the set $\mathcal T_\Delta$ of minimal discriminants of modular elliptic curves over $\QQ$ of conductor dividing~$2^4N_S$.
\item[(ii)] Let $\mathcal T=\cup_{m}\mathcal T_m$ be the union of the sets $\mathcal T_m=\{2^{12m-8}\Delta_E\,;\, \Delta_E\in \mathcal T_\Delta\}\cap\ZZ$ where $m=0,1,2,3$. For each $d\in\mathcal T$,
factor $d$ as $d=\prod_{p\in S}p^{n_p}$ with $n_p\in \ZZ$. Then for each disjoint
partition of the set $S=S_{\alpha}\dot\cup S_{\beta}\dot\cup S_{\gamma}$, with \
$S_{\gamma}\subseteq\{p\in S\,;\, 4\mid n_p\}$, define $\alpha=\prod_{p\in
S_{\alpha}}p^{n_p}$, $\beta=\prod_{p\in S_{\beta}}p^{n_p}$, and $\gamma=\prod_{p\in
S_{\gamma}}p^{n_p/4}$.
Then output all $(x,y)$ of the form $(x,y)=(\frac{a}{c},\frac{b}{c})$ with $a,b,c\in\ZZ$ satisfying $a+b=c$ and $\{\mathcal As{a},\mathcal As{b},\mathcal As{c}\}=\{\alpha,\beta,\gamma\}$.
\end{itemize}
\end{Algorithm}
\paragraph{Correctness.}We now verify that this algorithm indeed finds all solutions of any $S$-unit equation~\eqref{eq:sunit}. Let $(x,y)$ be a solution of~\eqref{eq:sunit}. As explained above~\eqref{eq:abc}, there is a corresponding solution $(a,b,c)$ of~\eqref{eq:abc} with $(x,y)=(\frac{a}{c},\frac{b}{c})$.
We write $d=\mathcal As{ab}c^4$. Lemma~\ref{lem:psu1} gives an elliptic curve $E$ over $\QQ$ such that $N_E\mid 2^4N_S$ and $\Delta_E=2^{8-12m}d$ with $m\in\{0,1,2,3\}$. The Shimura--Taniyama conjecture assures that $E$ is modular. This proves that $\Delta_E$ is contained in the set $\mathcal T_\Delta$ computed in step (i) and it follows that $d\in \mathcal T$. Furthermore,
the number $d=\mathcal As{ab}c^4\in \mathcal T$ factors in step (ii) as
$d=\prod_{p\in S}p^{n_p}$ with $n_p\in \ZZ$. Thus the disjoint partition $S=S_{\alpha}\dot\cup S_{\beta}\dot\cup S_{\gamma}$,
with $S_{\alpha}=\{p\,;\, p\mid a\}$,
$S_{\beta}=\{p\,;\, p\mid b\}$ and
$S_{\gamma}=S-(S_{\alpha}\cup S_{\beta})$, produces in step (ii) our solution $(x,y)=(\frac{a}{c},\frac{b}{c})$ as desired. Here we used that our coprime $a,b,c\in\ZZ$ satisfy $a+b=c$ and $\{\mathcal As{a},\mathcal As{b},\mathcal As{c}\}=\{\alpha,\beta,\gamma\}$.
\paragraph{Complexity.} In step (i) the algorithm has to compute all (modular) elliptic curves over $\QQ$ of conductor $N$ for precisely\footnote{Notice that $6\cdot 2^{\mathcal As{S}-1}$ is the number of positive rational integers dividing $2^4N_S$, since $2\in S$.} $6\cdot 2^{\mathcal As{S}-1}$ positive integers~$N$. Here we can exploit the fact that Cremona's algorithm proceeds by induction over the divisors of~$N$, see Section~\ref{sec:cremonas+st}. However at the time of writing it is not clear to us what is the running time of Cremona's algorithm. We also mention that step (i) greatly benefits from the ongoing extension of Cremona's tables which in particular list all (modular) elliptic curves over $\QQ$ of given conductor~$N$. As of August 2014, these tables are complete for all $N\leq 350 000$.
The complexity of step (ii) crucially depends on the size of~${\mathcal T}$. It is an open (Diophantine) problem to find a simple formula for $\mathcal As{\mathcal T}$ in terms of~$S$. However one can give an upper bound for $\mathcal As{\mathcal T}$ in terms of~$S$. For example, the work of Ellenberg, Helfgott and Venkatesh~\cite{heve:integralpoints,elve:classgroup} implies that $\mathcal As{\mathcal T}\ll N_S^{0.1689}$.
In step (ii) the algorithm first needs to compute the prime factorization of each $d\in\mathcal T$ and then it needs to compute certain integers $a,b,c$ for most of the disjoint 3-partitions of the set~$S$. Inequality~\eqref{eq:szpiro} implies that any $d\in\mathcal T$ satisfies $\log d=O(N_S^2)$, and the number of disjoint 3-partitions of $S$ is~$3^{\mathcal As{
S}}$. Hence the above discussions show that the running time of step (ii) is at most polynomial in terms of~$N_S$. This running time estimate can be considerably improved by assuming various conjectures. Indeed for each $d\in\mathcal T$ the $abc$-conjecture ($abc$) recalled in Remark~\ref{rem:abc} would provide that $\log d=O(\log N_S)$, and it follows from Brumer--Silverman \cite[Thm 4]{brsi:number} that the Birch--Swinnerton-Dyer conjecture (BSD) together with the General Riemann Hypothesis (GRH) would give $\mathcal As{\mathcal T}=O_\varepsilon(N_S^\varepsilon)$. Hence the running time of step (ii) is $O_\varepsilon(N_S^\varepsilon)$ if the three conjectures (BSD), (GRH) and ($abc$) all hold.
\begin{remark}[Variation]\label{rem:sulegendrealternative}
We now discuss a variation of Algorithm~\ref{algo:sucremona} which in practice provides an improvement for large~$\mathcal As{S}$. The idea is that instead of using in (ii) the curve of Lemma~\ref{lem:psu1}, one can work with Legendre curves as in \cite[Prop 3.2 (i)]{rvk:modular}.
\begin{itemize}
\item[(i)'] Use Cremona's algorithm to compute the set $\mathcal T_W$ of minimal Weierstrass models over $\ZZ$ of modular elliptic curves over $\QQ$ with conductor dividing~$2^4N_S$.
\item[(ii)'] For each $W\in \mathcal T_W$ determine its set $\lambda(W)=\{\lambda,1-\lambda,\lambda^{-1},\dotsc\}$ of Legendre parameters by computing the three roots of $f+f_2^2/4$, where $Y^2+f_2(X)Y=f(X)$ defines~$W$.
If $\lambda\in \lambda(W)$ and $(\lambda,1-\lambda)$ satisfies~\eqref{eq:sunit} then output $(x,y)=(\lambda,1-\lambda)$.
\end{itemize} This algorithm outputs the set of solutions of~\eqref{eq:sunit}. Indeed for any solution $(x,y)$ of~\eqref{eq:sunit} the proof of Lemma~\ref{lem:psu2} gives $W\in\mathcal T_W$ with $x\in\lambda(W)$. The running times of (i) and (i)' are essentially equal, and it follows for example from \cite[Cor 6.3]{rvk:modular} that the running time of step (ii)' is polynomial in terms of~$N_S$.
Furthermore one can show that the running time of step (ii)' is in fact~$O_\varepsilon(N_S^\varepsilon)$, provided that all three conjectures (BSD), (GRH) and $(abc)$ hold. If $\mathcal As{S}$ is large, then step (ii)' is in practice considerably faster than (ii) since the latter iterates in addition over many disjoint 3-partitions of $S$. However for large $\mathcal As{S}$ the bottleneck of the algorithm is the computation of $\mathcal T_W$ or~$\mathcal T_{\Delta}$, and for small $\mathcal As{S}$ it turns out that (ii)' and (ii) are essentially equally fast. Hence we implemented the algorithm involving (i) and (ii), since the implementation of (i) is simpler compared to (i)'.
\end{remark}
\paragraph{Applications.} Let $\Sigma(S)$ be the set of solutions of~\eqref{eq:sunit} up to symmetry. For any $N\in\ZZ_{\geq 1}$, consider the set $\Sigma(N)=\cup_S \Sigma(S)$ with the union taken over all sets $S$ with $N_S\leq N$. On using an implementation in Sage of the above Algorithm~\ref{algo:sucremona}, we computed the sets $\Sigma(N)$ for all $N\leq 20000$.
This computation was very fast for $N\leq 20000$, since for such $N$ part (i) of our Algorithm~\ref{algo:sucremona} can use Cremona's database which contains in particular the required data for all elliptic curves of conductor dividing $2^4N< 350 000$. On the other hand, if the required data of the involved elliptic curves is not already known, then our Algorithm~\ref{algo:sucremona} is often not practical anymore. Here the problem is the application of Cremona's algorithm (using modular symbols) in step (i) which requires a huge amount of memory in order to deal with medium sized or large conductors.
\begin{remark}[$abc$-triples]\label{rem:abc}
The $abc$-conjecture $(abc)$ states that for any real number $\varepsilon>0$ there are only a finite number of solutions of~\eqref{eq:abc} with quality larger than $1+\varepsilon$, see for example~\cite{masser:abc}. Here the quality $q=q(a,b,c)$ of a solution $(a,b,c)$ of~\eqref{eq:abc} is defined as $$q=\frac{\log\max(\mathcal As{a},\mathcal As{b},\mathcal As{c})}{\log \textnormal{rad}(abc)}.$$
Among all known high quality $abc$-triples the top two, as of October 2014, can be obtained with our method: $(2,3^{10}109,23^5)$ with $q=1.6299$ due to Reyssat, and $(11^2,3^25^67^3,2^{21}23)$ with $q=1.6260$ due to de Weger; see \url{abcathome.com}.
\end{remark}
\subsection{Algorithm via height bounds}\label{sec:suheightalgo}
In this section we use the optimized height bounds in Proposition~\ref{prop:algobounds} to construct an algorithm which practically resolves the $S$-unit equation~\eqref{eq:sunit}.
The decomposition of our algorithm is inspired by de Weger's classical method whose main ingredients are as follows:
\begin{itemize}
\item[(1)]\label{itDeWegerStepBaker} De Weger uses explicit height bounds for the solutions of \eqref{eq:sunit}, based on the theory of logarithmic forms, to rule out the existence of solutions with very ``large" height. See Section~\ref{sec:heightbounds} for references and more details.
\item[(2)]\label{itDeWegerStepLLL} Then de Weger tries to further reduce the height bounds in (1) by using the LLL lattice reduction algorithm applied to certain approximation lattices, which are defined using $p$-adic logarithms. If this reduction can be applied, then he can rule out in addition the existence of solutions with ``large" and ``medium sized" height.
\item[(3)]\label{itDeWegerStepSieve} To find most of the solutions with ``small" height de Weger applies a sieve which we call de Weger's sieve, see Section~\ref{sec:dwsieve}. He repeats this step as many times as required to make sure that the remaining solutions have ``tiny'' height.
\item[(4)] \label{itDeWegerStepEnumerationByHand}
Finally de Weger checks (by hand) all potential solutions of ``tiny" height.
\end{itemize}
In our algorithm we replace the height bounds in (1) by the optimized height bounds which we shall work out in Section~\ref{sec:heightbounds}. These optimized bounds are strong enough such that we can now omit the reduction step (2). In Section~\ref{sec:dwsieve} we discuss a slight variation of de Weger's sieve described in (3). Then in Section~\ref{sec:dwsieve+} we develop a refined sieve which has considerably improved running time in practice, in particular for all sets $S$ with $\mathcal As{S}>6$. Moreover, in Section~\ref{sec:suenum} we construct an enumeration algorithm which is faster than the standard algorithm in (4). In fact our enumeration is fast enough such that it is now beneficial to go from (3) to (4) at an earlier stage, which leads to additional running time improvements. Finally we present and discuss our Algorithm~\ref{algo:suheight} solving the $S$-unit equation~\eqref{eq:sunit}, and we also give various applications of our algorithm.
Before we begin to carry out the above program, we introduce some notation which will be used throughout Section~\ref{sec:suheightalgo}. Let $n\in \ZZ_{\geq 1}$ and consider two vectors $l,u\in\mathbb R^n$. We write $l\leq u$ if and only if $l_i\leq u_i$ for all $i\in\{1,\dotsc,n\}$. Further by $l<u$ we mean that $l\leq u$ with $l\neq u$, and we use the symbol $l\not\leq u$ in order to say that $l\leq u$ does not hold. In other words, we use poset (partially ordered set) and not
coordinate-wise notation for $``<"$ and $``\not\leq"$.
The use of poset notation for $``<"$ and $``\not\leq"$ will simplify our exposition. Next, we suppose that $(x,y)$ is a solution of the $S$-unit equation~\eqref{eq:sunit}. It will be convenient for the description of the algorithm to use the following quantities associated to~$(x,y)$. For any rational prime $p$ we put $m_p(x,y)=\max (\mathcal As{\ord_p(x)},\mathcal As{\ord_p(y)})$, and then we define
$$m(x,y)= (m_p(x,y))_{p\in S} \ \ \textnormal{ and } \ \ M(x,y)= \max(m_p(x,y)\log p\,;\, p\in S).$$
Proposition~\ref{prop:algobounds} gives an upper bound $M_0$ for~$M(x,y)$. Hence it remains to find the solutions $(x,y)$ of~\eqref{eq:sunit} with $M(x,y)$ between zero and $M_0$, or between $M_{k+1}$ and $M_{k}$ for $k=0,\dotsc,k_0$ and some convenient sequence $0=M_{k_0}<\ldots< M_{1}<M_0$. For this purpose we shall use the following refinements of steps (3) and (4) of de Weger's method.
\subsubsection{De Weger's sieve}
\label{sec:dwsieve}
We now begin to describe the sieve which is used in step (3) of de Weger's method. In fact we shall describe a slight variation of this sieve, see Remark~\ref{rem:teske}.
For any given $M', M''\in\RR_{\geq 0}$, we want to enumerate all solutions $(x,y)$ of~\eqref{eq:sunit} with $M'<M(x,y)\leq M''$. For this purpose, it suffices by \eqref{def:ulbounds} to solve the following problem: For any two vectors $l,u\in \ZZ^S$ with $0\leq l\leq u$, find all solutions $(x,y)$ of~\eqref{eq:sunit} which satisfy
\begin{equation}
\label{prob:mxy}
m(x,y)\not\leq l \ \ \ \textnormal{ and } \ \ \ m(x,y)\leq u.
\end{equation}
If a solution $(x,y)$ of \eqref{eq:sunit} satisfies \eqref{prob:mxy}, then all its symmetric solutions satisfy \eqref{prob:mxy} as well.
The condition $m(x,y)\not\leq l$ means that there exists at least one ``large" exponent in the prime factorization of $x$ or~$y$. We now exploit this to reduce (\ref{prob:mxy}) to a Diophantine problem whose solutions can be quickly enumerated. Suppose that $(x,y)$ is a solution of~\eqref{eq:sunit} which satisfies (\ref{prob:mxy}). Then there exists $q\in S$ with $m_q(x,y)\geq 1+l_q$.
Further the discussion given above~\eqref{eq:abc} delivers a solution $(a,b,c)$ of~\eqref{eq:abc} with $x=\frac{a}{c}$ and $y=\frac{b}{c}$.
After possibly replacing $(x,y)$ with a symmetric solution, we may and do assume that $q^{l_q+1}$ divides~$c$. Then it holds that $a+b=0\mod q^{l_q+1}$ and thus $(a/b)^2 = 1$ in $G$ for $$G=(\ZZ/q^{l_q+1}\ZZ)^\times.$$
Here we squared plainly in order to get rid of the minus sign, and we used that $a,b$ are both coprime to $q$ which provides that $a,b$ are in $G$.
Next we write $(a/b)^2 =\prod_{p\in {S\backslash q}} p^{2\gamma_p}$ with $\gamma_p=\ord_p(a/b)$. Then we see that any solution $(x,y)$ of~\eqref{eq:sunit} satisfying (\ref{prob:mxy}) induces a solution $\gamma=(\gamma_p)\in\ZZ^{S\backslash q}$ of the following problem: If $l,u\in\ZZ^{S}$ are given with $0\leq l\leq u$, then for each $q\in S$ find all $\gamma\in \ZZ^{{S\backslash q}}$ such that $\mathcal As{\gamma_p}\leq u_p$ for all $p\in{S\backslash q}$ and such that
\begin{equation}
\label{eqDeWegerSieve}
\prod_{p\in {S\backslash q}} p^{2\gamma_p} = 1 \ \textnormal{ in } \ G.
\end{equation}
Furthermore if $\gamma$ is a solution of~\eqref{eqDeWegerSieve} then one can quickly reconstruct all solutions of~\eqref{eq:sunit} satisfying (\ref{prob:mxy}) which map to $\gamma$ via the above construction. Indeed one defines $a=\prod_{\gamma_p>0}p^{\gamma_p}$, $b_{+}=\prod_{\gamma_p<0}p^{-\gamma_p}$ and $b_-=-b_+$, and then one checks for each $b\in\{b_+,b_-\}$ whether $\textnormal{rad}(a+b)\divides N_S$ and whether $(x,y)=(\frac{a}{a+b},\frac{b}{a+b})$ satisfies (\ref{prob:mxy}).
In particular, we see that we can quickly enumerate all solutions of~\eqref{eq:sunit} satisfying (\ref{prob:mxy}) provided that we know all solutions of~\eqref{eqDeWegerSieve}. Finally it remains to enumerate all solutions of~\eqref{eqDeWegerSieve}. For this purpose we observe that the set of vectors $\gamma\in\ZZ^{{S\backslash q}}$ satisfying~\eqref{eqDeWegerSieve} is the intersection of a certain lattice $\Gamma\subseteq \ZZ^{S\backslash q}$ with the cube $\{\gamma \,;\, |\gamma_p|\leq u_p\}\subset \mathbb R^{S\backslash q}$. To determine this intersection we combine the following two algorithms:
\begin{enumerate}
\item[(T)]
The first algorithm is an application of Teske's ``Minimize'' \cite[Algo 5.1]{teske:minimize}. It takes as the input a set of generators $g_1,\ldots,g_d$ of a finite abelian group\footnote{Here it is not required to know the group $G_0$ explicitly. In fact it suffices to know a number divisible by $\mathcal As{G_0}$ and to be able to compute the following operations in $G_0$: Multiplying elements, inverting elements and testing whether an element is the neutral element.} $G_0$,
and it outputs a basis for the lattice $\Gamma\subseteq\ZZ^d$ formed by those $\gamma\in\ZZ^d$ with $\prod_{i=1}^dg_i^{\gamma_i}=1$.
\item[(FP)]
The second algorithm is a version of the Fincke--Pohst algorithm, see Remark~\ref{rem:fp}. For any $d\in \ZZ_{\geq 1}$ it takes as the input a basis of a lattice $\Gamma\subseteq\ZZ^d$ together with an ellipsoid $\mathcal E\subset\RR^d$ centered at the origin, and
it outputs the inter\-section~$\Gamma\cap \mathcal E$.
\end{enumerate}
More precisely, we apply (T) and (FP) as follows. Let $G_0$ be the subgroup of $G$ which is generated by the squares of the elements in $S\backslash q$.
An application of (T) with $G_0$ gives a basis for the lattice $\Gamma$ underlying \eqref{eqDeWegerSieve}, and then (FP) computes the intersection of $\Gamma$ with the smallest ellipsoid $\mathcal E\subset \mathbb R^{S\backslash q}$ that contains the cube $\{\gamma\,;\, |\gamma_p|\leq u_p\}\subset \mathbb R^{S\backslash q}$.
\begin{Algorithm}[De Weger's sieve]
\label{algDeWegersSieve}
The input consists of a finite set of rational primes $S$ together with two vectors $l,u\in\ZZ^S$ such that $0\leq l\leq u$, and the output is the set of solutions $(x,y)$ of the $S$-unit equation~\eqref{eq:sunit} which satisfy~\eqref{prob:mxy}.
The algorithm\textnormal{:} After possibly shrinking $S$, we may and do assume that all $u_p\geq 1$. If $2\notin S$ then output the empty set, and if $2\in S$ then do the following for each $q\in S$.
\begin{itemize}
\item[(i)]
Use the application \textnormal{(T)} of Teske's algorithm in order to compute a basis for the lattice $\Gamma\subseteq \ZZ^{S\backslash q}$ of all $\gamma\in \ZZ^{S\backslash q}$ with $\prod_{p\in S\backslash q}p^{2\gamma_p}=1$ in $(\ZZ/q^{l_q+1}\ZZ)^\times$.
\item[(ii)] Define the ellipsoid $\mathcal E=\{x\in\RR^{S\backslash q}\,;\, \sum_{p\in S\backslash q} |x_p/u_p|^2 \leq |S|-1\}$. Then compute the intersection $\Gamma\cap \mathcal E$ using the version \textnormal{(FP)} of the Fincke--Pohst algorithm.
\item[(iii)] For each $\gamma\in \Gamma\cap \mathcal E$ lying in the cube $\{\gamma\,;\,|\gamma_p| \leq u_p\}\subset\mathbb R^{S\backslash q}$, define the numbers $a=\prod_{\gamma_p>0}p^{\gamma_p}$, $b_+=\prod_{\gamma_p<0}p^{-\gamma_p}$ and $b_-=-b_+$. For any $b\in\{b_{+},b_-\}$ such that $c=a+b$ satisfies $\ord_p(c)\leq u_p$ for all $p\in S$, $q^{l_q+1}\mid c$ and $\prod_{p\in S} p^{\ord_p(c)}=\mathcal As{c}$, output the solution $(x,y)=(\tfrac{a}{c},\tfrac{b}{c})$ together with all its symmetric solutions.
\end{itemize}
\end{Algorithm}
\paragraph{Correctness.} We now verify that this algorithm indeed finds all solutions of \eqref{eq:sunit} satisfying \eqref{prob:mxy}. Suppose that $(x,y)$ is such a solution. Then our assumption $m(x,y)\not\leq l$ gives $q\in S$ together with $\gamma\in\ZZ^{S\backslash q}$ which is associated to $(x,y)$ via the construction given above \eqref{eqDeWegerSieve}. This $\gamma$ lies in the lattice $\Gamma$ appearing in (i). Furthermore, our assumption $m(x,y)\leq u$ implies that $\gamma$ is also contained in the ellipsoid $\mathcal E$ from step (ii). In particular $\gamma$ lies in the intersection $\Gamma\cap\mathcal E$ computed in step (ii), and $m(x,y)\leq u$ provides that $\mathcal As{\gamma_p}\leq u_p$ for all $p\in S\backslash q$. Therefore on using that $\gamma$ is associated to $(x,y)$ via the construction described above \eqref{eqDeWegerSieve}, we see that step (iii) produces our solution $(x,y)$ as desired.
\paragraph{Complexity.}
Step (i) uses (T). The algorithm (T) is reminiscent of the discrete logarithm problem in the multiplicative group $G$, and the bottleneck for this is the prime factorization of $\mathcal As{G}$. In our situation it holds $\mathcal As{G} = q^{l_q}(q-1)$ and therefore it suffices to factor $q-1$, which is in general much easier than factoring an arbitrary number of size $\mathcal As{G}$. In fact step (i) is in practice not the bottleneck of Algorithm \ref{algDeWegersSieve}, except in the case when $S$ consists of the prime 2 together with a few large primes. Usually step (ii) is the bottleneck of Algorithm \ref{algDeWegersSieve}, and thus we shall refine this step in Section~\ref{sec:dwsieve+} below.
\begin{remark}\label{rem:teske}
To find a basis for the lattice $\Gamma$, de Weger used $q$-adic logarithms instead of the application (T) of Teske's algorithm.
In fact (T) was already used in generalizations of de Weger's method to number fields, see Wildanger~\cite{wildanger:unitalgo} and Smart~\cite{smart:smallsol}.
\end{remark}
\paragraph{Application.} For later use we now mention how we will apply Algorithm \ref{algDeWegersSieve} in order to find all solutions $(x,y)$ of~\eqref{eq:sunit} satisfying $M'< M(x,y)\leq M''$, for some given $M',M''$ in $\mathbb R_{\geq 0}$ with $M'<M''$. Consider the two vectors $l=(l_p)$ and $u=(u_p)$ in $\ZZ^S$, with
\begin{equation}\label{def:ulbounds}
l_p=\floor{M'/\log p} \ \ \ \textnormal{ and } \ \ \ u_p=\floor{M''/\log p}.
\end{equation}
We observe that $0\leq l\leq u$. Then an application of Algorithm \ref{algDeWegersSieve} with $l,u$ finds in particular all solutions $(x,y)$ of~\eqref{eq:sunit} satisfying $M'< M(x,y)\leq M''$, as desired.
\begin{remark}
According to~\eqref{def:m0m1} below, splitting the possible candidates with respect to values of $M(x,y)$ is in particular reasonable in the case when $\log p$ is small compared to $\mathcal As{S}$ for all $p\in S$. In this case, the iteration over all $q\in S$ in Algorithm~\ref{algDeWegersSieve} will take about equally long for each $q\in S$.
If $S$ contains primes which are exponentially large in terms of $\mathcal As{S}$, then one should split the initial space $0\leq M(x,y)\leq M_0$ into more general pieces of the form $\{m(x,y)\not\leq u(k+1) \textnormal{ and } m(x,y)\leq u(k)\}$ for suitable $u(k)\in\ZZ^{S}$ with $k=0,1,\dotsc,k_0$. Here the vectors $u(k)\in\ZZ^S$ should satisfy $0=u(k_0)<\ldots<u(1)<u(0)$ and $u(0)_p=\floor{M_0/\log p}$ for $p\in S$. In practice it is reasonable to choose greedily the next $u(k+1)<u(k)$, such that the subsequent sieving step is as fast as possible.
\end{remark}
\subsubsection{Refined sieve}
\label{sec:dwsieve+}
In this section we continue our notation introduced above. We begin with a short description of the geometric main idea underlying our refinement. Recall that in de Weger's sieve one needs to determine the intersection of a certain lattice $\Gamma\subseteq \ZZ^{S\backslash q}$ with the cube $\{x\,;\, \mathcal As{x_p}\leq u_p\}\subset \mathbb R^{S\backslash q}$, and for this purpose Algorithm~\ref{algDeWegersSieve}~(ii) first determines $\Gamma\cap \mathcal E$ for $\mathcal E$ the smallest ellipsoid containing the cube. However, in terms of the rank $s-1$ where $s=\mathcal As{S}$, the volume of $\mathcal E$ is exponentially larger than the volume of the cube. In our refinement we essentially truncate from the cube some regions near the faces of codimension $2,\ldots,\floor{s/3}$. For $s\geq 6$ the resulting geometric object is contained in a notably smaller ellipsoid $\mathcal E'$, which allows to determine $\Gamma\cap\mathcal E'$ considerably faster than $\Gamma\cap \mathcal E$.
To explain in more detail our refined sieve, we need to introduce additional notation. Let $(x,y)$ be a solution of the $S$-unit equation~\eqref{eq:sunit}, and let $(a,b,c)$ be a solution of \eqref{eq:abc} with $(x,y)=(\tfrac{a}{c},\tfrac{b}{c})$. We take $j\in\{1,\dotsc,t\}$ for $t=\max(1,\floor{\mathcal As{S}/3})$. This choice of $t$ takes into account that any prime appearing in the prime factorization of $x$ or $y$ divides one of the three coprime integers $a,b,c$. For any $n\in \ZZ$ with $\textnormal{rad}(n)\divides N_S$, we denote by $\mu_j(n)$ the $j$-th largest\footnote{More precisely, $\mu_j(n)$ is the $j$-th largest element of the ordered multi-set of cardinality $\mathcal As{S}$ obtained by ordering the $\mathcal As{S}$ non-negative real numbers $\ord_p(n)\log p$, $p\in S$, with respect to their absolute values.}
of the real numbers $\ord_p(n)\log p$, $p\in S$. Then we define
\[
\mu_j(x,y)=\max\bigl(\mu_j(a),\mu_j(b),\mu_j(c)\bigl) \ \ \ \textnormal{ and } \ \ \ \mu(x,y) = \bigl(\mu_1(x,y),\ldots,\mu_t(x,y)\bigl).
\]
We observe that $\mu_1(x,y)=M(x,y)$. However we point out that if $j\in\{2,\dotsc,t\}$ then $\mu_j(x,y)$ is not necessarily the $j$-th largest of the numbers $m_p(x,y)\log p$, $p\in S$.
Now we consider the following problem: For any given vectors $\mu', \mu''\in\ZZ^t$ having monotonously decreasing entries such that $0\leq \mu'< \mu''$, find all solutions $(x,y)$ of \eqref{eq:sunit} that satisfy
\begin{equation}\label{eqBoundsOnVectorMxy}
\mu(x,y) \not\leq\mu' \ \ \ \textnormal{ and } \ \ \ \mu(x,y) \leq\mu''.
\end{equation}
If a solution $(x,y)$ of \eqref{eq:sunit} satisfies \eqref{eqBoundsOnVectorMxy}, then all its symmetric solutions satisfy \eqref{eqBoundsOnVectorMxy} as well. Further we note that the condition $\mu(x,y)\not\leq \mu'$ implies, for some $j\in\{1,\dotsc,t\}$, the existence of at least $j$ ``large" exponents in the prime factorization of $x$ or $y$. In the following algorithm we exploit this in order to work with lattices of rank $\mathcal As{S}-j$.
\begin{Algorithm}[Refined sieve]
\label{algRefinedDeWegerSieve}
The input is a finite set of rational primes $S$, together with two vectors $\mu', \mu''\in\ZZ^t$ having monotonously decreasing entries such that $0\leq \mu'< \mu''$; where $t=\max(1,\floor{\mathcal As{S}/3})$.
The output is the set of solutions $(x,y)$ of the $S$-unit equation~\eqref{eq:sunit} which satisfy \eqref{eqBoundsOnVectorMxy}.
The algorithm\textnormal{:} If $2\notin S$ then output the empty set, and if $2\in S$ then do the following for each non-empty subset $T\subseteq S$ of cardinality $\mathcal As{T}\leq t$.
\begin{enumerate}
\item[(i)]
Put $n=\prod_{q\in T} q^{\floor{\mu'_{\mathcal As{T}}/\log q}+1}$, and use the application \textnormal{(T)} of Teske's algorithm to compute a basis of the lattice $\Gamma_T$ of all $\gamma\in \ZZ^{S\backslash T}$ with
$
\prod_{p\in S\backslash T} p^{2\gamma_p} = 1$ in $(\ZZ/n\ZZ)^\times.
$
\item[(ii)]
Then use the version \textnormal{(FP)} of the Fincke--Pohst algorithm in order to determine the intersection of the lattice $\Gamma_T\subseteq \ZZ^{S\backslash T}$ with the ellipsoid $\mathcal E_T\subset \mathbb R^{S\backslash T}$ defined by
\begin{equation}
\label{eqBoundForAlphaForImprovedDeWeger}
\mathcal E_T=\{x\in\mathbb R^{S\backslash T}\,;\,\sum_{p\in S\backslash T} |x_p \log p|^2 \leq \sum_{i=1}^{|S\backslash T|} (\mu''_{\min(\ceil{i/2},t)})^2\}.
\end{equation}
\item[(iii)]
For each $\gamma\in\Gamma_T\cap\mathcal E_T$, define $a=\prod_{\gamma_p>0}p^{\gamma_p}$, $b_+=\prod_{\gamma_p<0}p^{-\gamma_p}$ and $b_-=-b_+$. For any $b\in\{b_{+},b_-\}$ such that $c=a+b$ satisfies $\prod_{p\in S}p^{\ord_p(c)}=\mathcal As{c}$ and such that $(x,y)=(\tfrac{a}{c},\tfrac{b}{c})$ satisfies \eqref{eqBoundsOnVectorMxy}, output $(x,y)$ together with all its symmetric solutions.
\end{enumerate}
\end{Algorithm}
\paragraph{Correctness.} To see that this algorithm works correctly, we suppose that $(x,y)$ is a solution of the $S$-unit equation \eqref{eq:sunit} that satisfies \eqref{eqBoundsOnVectorMxy}. Let $(a,b,c)$ be a solution of \eqref{eq:abc} with $(x,y)=(\tfrac{a}{c},\tfrac{b}{c})$. Our assumption $\mu(x,y)\not\leq \mu'$ then gives a subset $T\subseteq S$ of cardinality $j=\mathcal As{T}$ in $\{1,\dotsc,t\}$ together with $C\in\{a,b,c\}$ such that $\ord_q(C)\log q> \mu'_j$ for all $q\in T$.
Furthermore after replacing $(x,y)$ by a symmetric solution, we may and do assume that $C=c$. It holds that $\ord_q(c)\geq \lfloor \mu'_{\mathcal As{T}}/\log q\rfloor+1$ for all $q\in T$. Hence $n$ divides $c$, and this implies that $a,b$ are both invertible in $\ZZ/n\ZZ$ since $\gcd(a,b,c)=1$. Therefore the equation $a+b=c$ leads to $(a/b)^2=1$ in $(\ZZ/n\ZZ)^\times$. Then on writing $(a/b)^2=\prod_{p\in S\backslash T}p^{2\gamma_p}$ with $\gamma_p=\ord_p(a/b)$, we see that the vector $\gamma=(\gamma_p)\in\ZZ^{S\backslash T}$ lies in the lattice $\Gamma_T\subseteq\ZZ^{S\backslash T}$ of (i). Moreover, if $i\in\{1,\dotsc,\mathcal As{S\backslash T}\}$ then the $i$-th largest of the real numbers $\mathcal As{\gamma_p\log p},$ $p\in S\backslash T$, is at most $\max(\mu_\iota(a),\mu_\iota(b))$ for $\iota=\min(\ceil{i/2},t)$, and our assumption $\mu(x,y)\leq \mu''$ implies that $\max(\mu_\iota(a),\mu_\iota(b))\leq \mu''_{\iota}$. Hence we deduce that $\gamma$ lies in the ellipsoid $\mathcal E_T$ of (ii) and then we conclude that (iii) produces our solution $(x,y)$ as desired.
\paragraph{Application.} For any $M\in \ZZ_{\geq 1}$, we would like to find all solutions $(x,y)$ of the $S$-unit equation~\eqref{eq:sunit} with $M(x,y)\leq M$. For this purpose we can use for example Algorithm~\ref{algRefinedDeWegerSieve}, which we successively apply with $\mu'(n),\mu''(n)\in\ZZ^t$
for $n=M+1, M,\dotsc,1$; where
\begin{equation}\label{def:mubounds}
\mu'(n)=\floor{(n-1)\cdot (1,1/2,\dotsc,1/t)} \textnormal{ for } n\in\{1,\dotsc, M+1\},
\end{equation}
$$\mu''(M+1)=M\cdot (1,\dotsc,1), \ \ \ \mu''(n)=\mu'(n+1) \textnormal{ for } n\in\{1,\dotsc, M\}.$$
Here $\floor{v}=(\floor{v_i})$ for $v=(v_i)$ a vector with entries $v_i\in\mathbb R$. Suppose that $\mathcal As{S}\geq 6$, $T=\{q\}$ and $\mu''=\mu''(n)$ with $n\in\{1,\dotsc,M\}$. Then the ``radius" $R$ of the ellipsoid in \eqref{eqBoundForAlphaForImprovedDeWeger} satisfies
\begin{equation}\label{eq:rbound}
R=\sum_{i=1}^{\mathcal As{S}-1}(\mu''_{\iota(i)})^2\leq n^2\left(\tfrac{t+1}{t^2}+4\sum_{i=1}^{2t}\tfrac{1}{i^2}\right)
\end{equation}
for $\iota(i)=\min(\ceil{i/2},t)$. It follows that $R\leq 7n^2$, and this uniform bound together with $R<(\mathcal As{S}-1)n^2$ shows that $R$ is considerably smaller (in particular for large $\mathcal As{S}$) than the ``radius" $(\mathcal As{S}-1)n^2$ of the ellipsoid in Algorithm~\ref{algDeWegersSieve} with $u=(\floor{n/\log p})$. The following complexity discussion provides some motivation for our choice of $t$ and $\mu'(n),\mu''(n)$.
\paragraph{Complexity.} To discuss the improvements provided by our refined algorithm for sets $S$ with $\mathcal As{S}\geq 6$, we consider $M\in\ZZ_{\geq 2}$ and we apply Algorithm~\ref{algRefinedDeWegerSieve} with $\mu'(n),\mu''(n)$ for some $n\in\{1,\dotsc, M\}$. Recall that Algorithm~\ref{algRefinedDeWegerSieve} needs to compute in particular $\Gamma_{T}\cap\mathcal E_T$ for all $T\subseteq S$ with $\mathcal As{T}\in\{1,\dotsc,t\}$. Here the cases $T=\{q\}$ are essentially an application of Algorithm~\ref{algDeWegersSieve} as in \eqref{def:ulbounds}, with $M'=n-1$ and $M''=n$. However in light of the discussions surrounding~\eqref{eq:rbound}, the crucial difference is that the volume of the ellipsoid $\mathcal E_{\{q\}}$ of our refined Algorithm~\ref{algRefinedDeWegerSieve} is considerably smaller than the volume of the corresponding ellipsoid $\mathcal E$ of Algorithm~\ref{algDeWegersSieve}. In practice, this is the reason for the significantly improved running time of Algorithm~\ref{algRefinedDeWegerSieve} for $\mathcal As{S}\geq 6$. We note that our refinement needs to iterate in addition over certain sets $T$ with $\mathcal As{T}\geq 2$. However these additional iterations have little influence on the running time, since the running time for $\mathcal As{T}\geq 2$ is in practice much better than for $\mathcal As{T}=1$. Indeed if $\mathcal As{T}\geq 2$ then the lattices $\Gamma_T$ have smaller rank $\mathcal As{S}-\mathcal As{T}$, which crucially improves the running time of (FP) in Algorithm~\ref{algRefinedDeWegerSieve}~(ii).
\begin{remark}[Implementation of Fincke--Pohst]\label{rem:fp}
To avoid numerical issues in Algorithms \ref{algDeWegersSieve} and \ref{algRefinedDeWegerSieve}, we use our own implementation of the version of the Fincke--Pohst algorithm \cite{fipo:algo} described in (FP). Our implementation only uses integer arithmetic, and in particular we do not take square roots.
For this purpose the coordinates and the bound in~\eqref{eqBoundForAlphaForImprovedDeWeger} have been scaled and rounded to integers in such a way, that (FP) will return possibly slightly more candidates $\gamma$ than those fulfilling~\eqref{eqBoundForAlphaForImprovedDeWeger}.
Furthermore we use an LLL improvement of the original Fincke--Pohst algorithm, such as for example the one in Cohen~\cite{cohen:compant}. This is important since the original implementation~\cite{fipo:algo} becomes in many instances very slow, in fact already for $|S|\geq 10$ it is too slow for our purpose.
\end{remark}
\subsubsection{Refined enumeration}
\label{sec:suenum}
We continue the notation introduced above and we take $u\in\ZZ^S$ with $u\geq 0$. To find all solutions $(x,y)$ of the $S$-unit equation~\eqref{eq:sunit} satisfying $m(x,y)\leq u$,
we use the following refined enumeration Algorithm \ref{algEnumerationOfTinySolutions}. In practice, our refined enumeration algorithm is considerably faster than the standard algorithm which is described in the complexity discussion below. For any subset $T\subseteq S$, we define its weight $w(T)=\prod_{p\in T}(1+u_p)$.
\begin{Algorithm}[Refined enumeration]
\label{algEnumerationOfTinySolutions}
The input is a finite set of rational primes $S$, together with a vector $u\in\ZZ^{S}$ such that $u\geq 0$. The output consists of all solutions $(x,y)$ of the $S$-unit equation~\eqref{eq:sunit} that satisfy $m(x,y)\leq u$.
The algorithm: For each subset $S_a\subseteq S$ with $w(S_a)\geq w(S)^{1/3}$, do the following.
\begin{enumerate}
\item[(i)]
Split the set $S_a=S_{a_1}\dotcup S_{a_2}$ into disjoint subsets $S_{a_1},S_{a_2}$ such that $w(S_{a_1})\leq w(S)^{1/2}$ is fulfilled as tight as possible.
Construct the set $X$, implemented as a hash, of all $a_1\in \ZZ$ such that $\mathcal As{a_1}=\prod_{p\in S_{a_1}} p^{v_p(a_1)}$ with $0\leq v_p(a_1)\leq u_p$ for $v_p=\ord_p$.
\item[(ii)]
Construct the set $Y=\cup Y(S_b,S_c)$ with the union taken over all pairs $(S_b,S_c)$ such that $S_a\dotcup S_b\dotcup S_c=S$ and such that either $w(S_b)<w(S_c)$ or $w(S_b)=w(S_c)$ and $\min S_b< \min S_c$. Here $Y(S_b,S_c)$ is the set of $(b,c,a_2)\in\ZZ^3$ such that $b=\prod_{p\in S_b} p^{v_p(b)}$ with $1\leq v_p(b)\leq u_p$, $c=\prod_{p\in S_c} p^{v_p(c)}$ with $1\leq v_p(c)\leq u_p$, $a_2=\prod_{p\in S_{a_2}} p^{v_p(a_2)}$ with $0\leq v_p(a_2)\leq u_p$ and such that $a_2$ divides $b+c$ or $b-c$.
\item[(iii)]
For each $(b,c,a_2)\in Y$, check if $a_1:=(b+\varepsilon c)/a_2\in X$ for some $\varepsilon\in\{1,-1\}$.
If so then output all $(x,y)=(\tfrac{\alpha}{\gamma},\tfrac{\beta}{\gamma})$ with $\alpha+\beta=\gamma$ and $\{\mathcal As{\alpha},\mathcal As{\beta},\mathcal As{\gamma}\}=\{\mathcal As{a_1}a_2,b,c\}$.
\end{enumerate}
\end{Algorithm}
In the implementation of the above algorithm, we simultaneously carry out steps (ii) and (iii) as follows. We iterate over all $(S_b,S_c)$ and over all $(b,c)$: For each $(b,c)$ we determine the divisors $a_2\in \ZZ_{\geq 1}$ of $b\pm c$ which are only divisible by primes in $S_{a_2}$ and which have exponents bounded by $u$. In the same iteration step, we also check whether $(b\pm c)/a_2\in X$ and if this is the case then we output the corresponding solutions.
\paragraph{Correctness.} We now show that Algorithm~\ref{algEnumerationOfTinySolutions} indeed finds all solutions $(x,y)$ of~\eqref{eq:sunit} with $m(x,y)\leq u$. Suppose that $(x,y)$ is such a solution, and let $(a,b,c)$ be a solution of~\eqref{eq:abc} with $(x,y)=(\tfrac{a}{c},\tfrac{b}{c})$. After replacing $(x,y)$ with a symmetric solution, we may and do assume that $w(S_a)\geq \max(w(S_b),w(S_c))$ for $S_b=\{p\,;\, p\divides b\}$, $S_c=\{p\,;\, p\divides c\}$ and $S_a=S- (S_b\cup S_c)$. This implies that $w(S_a)\geq w(S)^{1/3}$, since $w(S)=w(S_a)w(S_b)w(S_c)$. Let $S_{a_1},S_{a_2},X,Y$ be the sets appearing in steps (i) and (ii), and define $a_i=\prod_{p\in S_{a_i}}p^{\ord_p(a)}$ for $i=1,2$. Our assumption $m(x,y)\leq u$ implies that $a_1\in X$ and $(\mathcal As{b},\mathcal As{c},a_2)\in Y$ since $a_2\mid a=-(b-c)$. Further we observe that $(\mathcal As{b}+\varepsilon \mathcal As{c})/a_2=\pm a/a_2=\pm a_1\in X$ for some $\varepsilonilon\in\{1,-1\}$, and by construction it holds that $a+b=c$ with $\{\mathcal As{a},\mathcal As{b},\mathcal As{c}\}=\{a_1a_2,\mathcal As{b},\mathcal As{c}\}$. Therefore we see that step (iii) produces our solution $(x,y)$ as desired.
\paragraph{Complexity.} To explain the improvements provided by Algorithm~\ref{algEnumerationOfTinySolutions}, we recall that the standard enumeration of all solutions $(x,y)$ of~\eqref{eq:sunit} with $m(x,y)\leq u$ is as follows: Consider all coprime pairs $(a,b)\in\ZZ\times \ZZ$ with $\textnormal{rad}(ab)\divides N_S$, and with $\max\bigl(\ord_p(a),\ord_p(b)\bigl)\leq u_p$ for all $p\in S$.
If $c=a+b$ satisfies $\textnormal{rad}(c)\divides N_S$ and $\ord_p(c)\leq u_p$ for all $p\in S$, then output the solution $(x,y)=(\tfrac{a}{c},\tfrac{b}{c})$. Now the improved running time of Algorithm \ref{algEnumerationOfTinySolutions} has the following basic reason.
For fixed $S_a$, $S_b$ and $S_c$, we iterate over all $a_1$ and over all $(b,c)$ in a subsequent way; see the remark given below Algorithm~\ref{algEnumerationOfTinySolutions}. That is the running time of these two iterations adds, and it does not multiply as in the standard enumeration which iterates over all coprime pairs $(a,b)\in\ZZ\times\ZZ$. Here we note that the splitting of $S_a$ into $S_a=S_{a_1}\dotcup S_{a_2}$ assures that the running time of (i) does not differ too much from the running time of (ii) together with (iii). Indeed $S_{a_1}$ is chosen such that $w(S_{a_1})$ is approximately $w(S)^{1/2}$ and our assumption $w(S_a)\geq w(S)^{1/3}$ provides that $w(S_b)w(S_c)\leq w(S)^{2/3}$. In particular, for bounded $\mathcal As{S}$ the complexity of our refined enumeration is asymptotically the $2/3$-th power of the complexity of the standard enumeration algorithm.
\begin{remark}Due to the hash our refinement needs more memory than the standard algorithm.
In practice this is not much compared to the running time,
and in case this becomes an issue we can avoid creating the hash as follows. Iterating over all $(b,c)$ as above, we try to factor $(b\pm c)/a_2$ using only primes in~$S_{a_1}$ and if this succeeds then we output the corresponding solutions provided the exponents are bounded by~$u$.
\end{remark}
\paragraph{Application.}We shall apply Algorithm~\ref{algEnumerationOfTinySolutions} in order to enumerate the solutions $(x,y)$ of \eqref{eq:sunit} with bounded $\mu(x,y)$. More precisely, let $\mu\in\ZZ^t$ with $\mu\geq 0$ and suppose that we want to enumerate all solutions $(x,y)$ of \eqref{eq:sunit} with $\mu(x,y)\leq \mu$. For this purpose it suffices to apply Algorithm~\ref{algEnumerationOfTinySolutions} with $u\in\ZZ^S$ given by $u_p =\floor{\mu_1/\log p}$ for $p\in S$. Indeed any solution $(x,y)$ with $\mu(x,y)\leq \mu$ satisfies $m_p(x,y)\log p\leq\mu_1(x,y)\leq \mu_1$ for all $p\in S$ and thus $m(x,y)\leq u$. This application could output too many solutions $(x,y)$, since not any $(x,y)$ with $m(x,y)\leq u$ satisfies $\mu(x,y)\leq \mu$. To avoid this we can directly check the latter condition at each step of the recursions building $X$ and $Y$, and we do this in our implementation. In general an improvement could come from a choice of the weight $w$ making the cardinalities of $X$ and $Y$ even more balanced; we leave this for the future.
\subsubsection{The algorithm} \label{sec:sualgoresults}
We continue the notation introduced above. On putting everything together, we obtain the following algorithm which solves the $S$-unit equation~\eqref{eq:sunit}.
\begin{Algorithm}[$S$-unit equation via height bounds]\label{algo:suheight}
The input is a finite set of rational primes~$S$, and the output is the set of solutions $(x,y)$ of the $S$-unit equation~\eqref{eq:sunit}.
\begin{enumerate}
\item[(i)] To rule out solutions with ``large" height, use Proposition~\ref{prop:algobounds} in order to compute $M_0\in\ZZ_{\geq 1}$ such that all solutions $(x,y)$ of \eqref{eq:sunit} satisfy $M(x,y)\leq M_0$.
\item[(ii)] To find the solutions of ``medium sized" and ``small" height, apply de Weger's sieve.
\begin{enumerate}
\item
Find a small $M_1\in\ZZ_{\geq 1}$ such that an application of Algorithm~\ref{algDeWegersSieve}~(ii) as in \eqref{def:ulbounds}, with $M'=M_1$ and $M''=M_0$, only returns $\gamma=0$.
Here first try $M_1=10$. If this does not work then replace $M_1$ by $\floor{1.3M_1}$, and so on.
\item Having found such an $M_1$, successively apply Algorithm~\ref{algDeWegersSieve} as in \eqref{def:ulbounds}, with $M'=M_{k+1}$, $M''=M_{k}$ and $M_{k+1}=\floor{M_k/1.3}$, for $k=1,2,\dotsc$ until either $M_k=0$ or Algorithm~\ref{algDeWegersSieve}~(ii) returns more than $10^3$ candidates.
\end{enumerate}
\item[(iii)]
To find the remaining solutions, combine the refined sieve with the refined enumeration. The following two steps run simultaneously, or alternatingly, until the parameters $n,n'$ appearing in these two steps satisfy $n=n'$.
\begin{enumerate}
\item
To enumerate the remaining solutions from above, let $k_0$ be the $k$ where the above step (ii) ended. Then successively apply Algorithm \ref{algRefinedDeWegerSieve} as in \eqref{def:mubounds}, with $M=M_{k_0}$ and $\mu'(n),\mu''(n)$, for $n=M+1,M,\dotsc$ until $n=n'$.
\item
To enumerate the remaining solutions from below, successively apply the refined enumeration Algorithm~\ref{algEnumerationOfTinySolutions} with $\mu'(n')$ for $n'=1,2,\ldots$ until $n'=n$.
\end{enumerate}
\end{enumerate}
\end{Algorithm}
\paragraph{Correctness.} To verify that this algorithm works correctly, we let $(x,y)$ be a solution of~\eqref{eq:sunit} and we let $M$ be as in step (iii). The construction of $M_0$ in step (i) shows that $M(x,y)\leq M_0$ and hence our solution $(x,y)$ is found in step (ii) if $M(x,y)>M$.
On the other hand, if $M(x,y)\leq M$ then step (iii) produces our solution $(x,y)$ as desired.
\subsubsection{Complexity}\label{sec:sucomplexity}
We conducted some effort to optimize the running time in practice. To explain our optimizations, we now discuss the above composition of Algorithm~\ref{algo:suheight} and we motivate our choice of the parameters appearing therein.
Furthermore, we also consider some additional practical aspects of our algorithm. We continue the notation introduced above.
\paragraph{Step (i).} In this step we need to find a number $M_0$ with the property that any solution $(x,y)$ of \eqref{eq:sunit} satisfies $M(x,y)\leq M_0$. For this purpose we use Proposition~\ref{prop:algobounds} which requires to compute the number $\alpha(N)$ appearing therein, where $N=2^4N_S$. In case the computation of $\alpha(N)$ takes too long, we can replace $\alpha(N)$ by the slightly larger number $\bar{\alpha}(N)$ defined below \eqref{def:barb} and this $\bar{\alpha}(N)$ can always be computed very fast. Hence we see that Proposition~\ref{prop:algobounds} allows in any case to quickly determine a relatively small number $M_0$ with the desired property. In practice this $M_0$ is small enough such that we can skip de Weger's first reduction process (2) described at the beginning of Section~\ref{sec:suheightalgo}, and thereby remove an uncertainty in de Weger's original method which crucially relies on (2). Indeed the reduction process (2) is (a priori) an uncertainty, since it is not proved that it always works. However, we should mention that in practice it is (essentially) always possible to successfully apply (2) by adapting the parameters to the specific situation at hand.
\paragraph{Step (ii).} Here we apply Algorithm~\ref{algDeWegersSieve} as in \eqref{def:ulbounds}. For this purpose we divided the space $0\leq M(x,y)\leq M_0$ into subspaces $M'< M(x,y)\leq M''$, with $M'=M_{k+1}$ and $M''=M_k$ for $k=0,\dotsc,k_0$ and for some convenient sequence $M_{k_0}<\ldots< M_{1}<M_0$.
To explain for which choices of $M',M''$ the application of Algorithm~\ref{algDeWegersSieve} is fast in practice, we use the notation of Section~\ref{sec:dwsieve}. The efficiency of the sieve depends on the number of points in $\Gamma\cap \mathcal E$. In the best case, the sieve \eqref{eqDeWegerSieve} decreases the space of candidates by a factor which is approximately $\mathcal As{G}/2= q^{l_q}(q-1)/2$.
However in the worst case, when the square of each element in $S\backslash q$ is $1$ modulo~$q^{l_q+1}$, we obtain $\Gamma=\ZZ^{S\backslash q}$ and the sieve \eqref{eqDeWegerSieve} does not decrease the number of candidates at all.
In practice we are almost always close to the best case.
Let $V$ be the euclidean volume of the ball in $\RR^{s-1}$ of square radius $s-1$ for $s=\mathcal As{S}$, and let $\textnormal{covol}(\Gamma)$ denote the covolume of $\Gamma$.
The ellipsoid $\mathcal E$ has volume $\vol(\mathcal E)=V\prod_{p\in S\setminus q}u_p$,
and in the generic case the cardinality of $\Gamma\cap \mathcal E$ can be approximated by $\vol(\mathcal E)/\textnormal{covol}(\Gamma)$.
Thus the sieve is efficient in practice if the ratio $\vol(\mathcal E)/(\mathcal As{G}/2)$ is ``small". For example this ratio is strictly smaller than $1$ when $M'$ and $M''$ satisfy
\begin{equation}\label{def:m0m1}
M'>(s-1)\log M''+\log V+2\log 2-\sum_{p\in S\setminus q}\log\log p.
\end{equation}
Stirling's approximation leads to a simpler expression for $V$ in terms of $s$. If we now choose $M_1=(s-1)(\log M_0+\tfrac{1}{2}\log(2\pi e))$, then \eqref{def:m0m1} suggests that the sieve \eqref{eqDeWegerSieve} is strong enough such that Algorithm~\ref{algDeWegersSieve} (ii) only returns the trivial candidate $\gamma=0$. A relatively small $M_1$ with this property is produced in step~(a) where we start with a very optimistic choice $M_1=10$.
Step (a) is fast in practice and it improves the running time of (ii) as follows: Algorithm~\ref{algDeWegersSieve} (iii) is trivial\footnote{The construction of $M_1$ provides that here Algorithm~\ref{algDeWegersSieve} (iii) only needs to output the trivial solutions $(2,-1)$, $(-1,2)$ and $(\frac{1}{2},\frac{1}{2})$ of \eqref{eq:sunit}, which all come from the trivial candidate $\gamma=0$.} for the large space $M_1<M(x,y)\leq M_0$, and for the space $M_2<M(x,y)\leq M_1$ the running time of Algorithm~\ref{algDeWegersSieve} (iii) is considerably faster than for $M_2<M(x,y)\leq M_0$ since $M_0$ is much larger than $M_1$. Indeed Algorithm~\ref{algDeWegersSieve}~(ii) applied with $M_0$ can produce much larger candidates $\gamma$, and for large candidates $\gamma$ the reconstruction process in Algorithm~\ref{algDeWegersSieve}~(iii) becomes slow. We next discuss step (b). According to \eqref{def:m0m1}, the choices $M_{k+1}=\floor{M_{k}/1.3}$ for $k\geq 1$ should give a strong sieve in the range where step (b) is applied and this turned out to be true in practice. We apply step (b) for $k=1,2,\dotsc$ until Algorithm~\ref{algDeWegersSieve} (ii) returns more than $10^3$ candidates. Here the condition more than $10^3$ candidates means that our refined sieve can find these candidates considerably faster, and thus we switch at this point to step (iii).
\paragraph{Step (iii).} In this step we are in a situation where we can fully exploit our refinements worked out in the previous sections. To explain this more precisely, we now mention two points which significantly slow down Algorithm~\ref{algDeWegersSieve} in the situation of step (iii) where many solutions exist. The first point is the application of (FP) in Algorithm~\ref{algDeWegersSieve}~(ii), which is the bottleneck of Algorithm~\ref{algDeWegersSieve}. The second point is that Algorithm~\ref{algDeWegersSieve}~(ii) repetitively enumerates the ``same" candidate $\gamma$ in many steps $k$ with $M_k$ small. This is due to the large fraction between the volume of the ellipsoid and the volume of the cube appearing in Algorithm~\ref{algDeWegersSieve}; indeed this fraction depends exponentially on the cardinality of $S$.
To improve these two points we worked out the following refinements:
\begin{enumerate}
\item Concerning the first and second point, we developed our refined Algorithm~\ref{algRefinedDeWegerSieve} which works with smaller ellipsoids. This leads to less repetitions of the candidates $\gamma$, and it considerably improves the running time of the step in which we apply (FP). See also the complexity discussions given in Section \ref{sec:dwsieve+}.
\item Regarding the second point, we constructed the refined enumeration Algorithm~\ref{algEnumerationOfTinySolutions}. This enumeration is fast enough such that we can now skip the final applications ($M_k=1,2,\dotsc$) of Algorithm~\ref{algDeWegersSieve} which are very slow, and thereby we can in particular circumvent for many candidates $\gamma$ that they get enumerated repetitively.
\end{enumerate}
In (iii) we carry out steps (a) and (b) alternatingly, depending on which step took less time so far. In some cases this leads to a significantly improved running time of (iii). Further we mention that our choice $\mu''(n)=\mu'(n+1)$ is motivated by \eqref{def:m0m1}. Indeed according to \eqref{def:m0m1}, one should work with step size $1=(n+1)-n$ in the situation of (iii) where usually $M(x,y)\leq (s-1)(\log s+\log\log s+\tfrac{1}{2}\log(2\pi e))$. See also the discussions surrounding~\eqref{eq:rbound} which provide additional motivation for our definition of $\mu'(n),\mu''(n)$.
\paragraph{Bottleneck.} Despite our refinements which considerably improve the running time in practice, step (iii) still remains in general the bottleneck of Algorithm~\ref{algo:suheight}. However, for certain special sets $S$ the location of the bottleneck can change. For example if $S$ consists of the prime 2 together with a few very large primes, then Algorithm~\ref{algo:suheight} often finds all solutions already in step (ii). In this case the bottleneck of Algorithm~\ref{algo:suheight} is located in step (ii) where we apply Algorithm~\ref{algDeWegersSieve}~(i). The main reason is that here the application of (T) becomes slow since $S$ contains very large primes, and the application of (FP) in Algorithm~\ref{algDeWegersSieve}~(ii) becomes fast since the cardinality of $S$ is small.
\begin{remark}[Parallelization]\label{rem:parallelization}
In our implementation of Algorithm~\ref{algo:suheight} we successfully parallelized essentially everything, except Algorithm~\ref{algDeWegersSieve}~(i) and Algorithm~\ref{algRefinedDeWegerSieve}~(i) which both involve the application of Teske's algorithm described in (T).
\end{remark}
\subsubsection{Applications}\label{sec:suapplications}
In this section we give some applications of Algorithm~\ref{algo:suheight}. In particular we discuss parts of our database $\mathcal D_1$ containing the solutions of the $S$-unit equation \eqref{eq:sunit} for many distinct sets $S$. We also use our database to motivate various Diophantine questions related to \eqref{eq:sunit}, including Baker's explicit $abc$-conjecture and a new conjecture.
We continue the notation introduced above. Recall that $\Sigma(S)$ denotes the set of solutions of the $S$-unit equation~\eqref{eq:sunit} up to symmetry, and for any $n\in\ZZ_{\geq 1}$ we recall that $S(n)$ denotes the set of the $n$ smallest rational primes. Let $N\in\ZZ_{\geq 1}$ and define $\Sigma(N)=\cup\Sigma(S)$ with the union taken over all finite sets of rational primes $S$ with $N_S\leq N$.
\paragraph{The sets $\Sigma(S(n))$.} We determined the sets $\Sigma(S(n))$ for all $n\leq 16$. The cardinality of these sets is given in the table of Theorem~A stated in the introduction. As already mentioned, our Algorithm~\ref{algo:suheight} substantially improves de Weger's original method in \cite{deweger:lllred} which de Weger used to compute the set $\Sigma(S(6))$ in \cite[Thm 5.4]{deweger:lllred}. To illustrate our improvements in practice, we now compare Algorithm~\ref{algo:suheight} with de Weger's original method. For this purpose we implemented in Sage de Weger's original method in a slightly improved form (dW), which uses in addition our optimized height bounds. If $S=S(6)$ then (dW) took 21 seconds, while it took 6 seconds by using Algorithm~\ref{algo:suheight}. For larger $|S|$ our running time improvement significantly increases:
For example if $S=S(10)$ then (dW) takes four days, whereas this decreases to only 25 minutes by using Algorithm~\ref{algo:suheight}.
Roughly speaking, for large $\mathcal As{S}$ our refinements should save an exponential factor with respect to $|S|$ in comparison to de Weger's original method. Further if $n>10$ then (dW) becomes too slow and thus we did not try to use (dW) in order to compute $\Sigma(S(n))$ for $n>10$. To deal with $S=S(n)$ for $10<n\leq 16$ we additionally parallelized (see Remark~\ref{rem:parallelization}) our Algorithm~\ref{algo:suheight}. Then it took 8 days for $n=15$ and 34 days for $n=16$, using 30 CPU's.
\begin{remark}[Automatically choosing parameters]\label{rem:fullyautomatic}
De Weger's original method does not specify in general how to choose the involved parameters. For example de Weger's first reduction process requires to make a choice, and to efficiently apply de Weger's sieve one has to choose suitable subspaces dividing the initial space.
To compute the set $\Sigma(S(6))$, de Weger has chosen by hand the required parameters. Although we can skip de Weger's first reduction process using our optimized height bounds, we still need to make many choices in Algorithm~\ref{algo:suheight}.
In particular, it is now favourable that the choices required for Algorithm~\ref{algo:suheight} are made by an automatism.
We implemented such an automatism, which takes into account \eqref{def:m0m1} and which properly adjusts the parameters during run time. In view of \eqref{def:m0m1}, our automatism chooses parameters for Algorithm~\ref{algo:suheight} such that the running time is considerably less than twice as long as for the optimal parameters.
However, we do not claim and can not prove that our automatism is optimal.\end{remark}
\paragraph{The sets $\Sigma(N)$.}
We computed the sets $\Sigma(N)$ for all $N\leq 10^7$ in approximately 13 days. For this computation it was crucial that Algorithm~\ref{algo:suheight} automatically chooses all required parameters, as mentioned in Remark~\ref{rem:fullyautomatic}. Indeed to compute the sets $\Sigma(N)$ for all $N\leq 10^7$, we had to apply Algorithm~\ref{algo:suheight} with so many distinct sets $S$ such that it would have been impossible to suitably choose by hand all the involved parameters.
\paragraph{Explicit $abc$-conjecture.} Baker \cite[Conj 4]{baker:abcexperiments} proposed the following fully explicit version of the $abc$-conjecture: If $a,b,c\in \ZZ_{\geq 1}$ are coprime with $a+b=c$, then it holds $c\leq\tfrac{6}{5}N(\log N)^{\omega}/\omega!$ for $\omega$ the number of rational primes dividing the radical $N=\textnormal{rad}(abc)$; here one should exclude the triple $(a,b,c)=(1,1,2)$. On using our database $\mathcal D_1$ containing in particular the sets $\Sigma(N)$ for all $N\leq 10^7$, we verified Baker's explicit $abc$-conjecture for all coprime $a,b,c\in\ZZ_{\geq 1}$ with $\textnormal{rad}(abc)\leq 10^7$. Furthermore, Algorithm~\ref{algo:suheight} can be used to verify other properties of all $abc$-triples with bounded radical. In particular all existing $abc$-triples with $\textnormal{rad}(abc)\leq 10^7$ can be directly taken from our database $\mathcal D_1$.
\paragraph{Elliptic curves with full $2$-torsion.}
For any given $N\in\ZZ_{\geq 1}$ we denote by $\mathcal M$ the set of $\QQ$-isomorphism classes of elliptic curves over $\QQ$ of conductor $N$, with all two torsion points defined over $\QQ$. Algorithm~\ref{algo:suheight} allows to efficiently compute the set $\mathcal M$. To see this, we let $S$ be the set of rational primes dividing $2N$. Any elliptic curve in $\mathcal M$ admits a Weierstrass equation $y^2=x(x-a)(x+b)$ such that $a,b\in\ZZ$ have the following properties: $d=\gcd(a,b)$ divides $N_S$ and $(\frac{a}{a+b},\frac{b}{a+b})$ satisfies the $S$-unit equation~\eqref{eq:sunit}. An application of Algorithm~\ref{algo:suheight} with $S$ determines all possible values of $a/d$ and $b/d$, and this then allows to directly compute the set $\mathcal M$. In particular, we see that Algorithm~\ref{algo:suheight} provides an alternative way to check the completeness of a small part of Cremona's database \cite{cremona:algorithms}. This application of Algorithm~\ref{algo:suheight} already turned out to be useful in practice.
\paragraph{Conjecture and question.} We next use our database $\mathcal D_1$ to motivate various questions concerning the solutions of $S$-unit equations \eqref{eq:sunit}. First we recall Conjecture~1 which is motivated by our data and by the construction of the refined sieve in Section~\ref{sec:dwsieve+}.
\noindent{\bf Conjecture 1.}
\emph{There exists $c\in\ZZ$ with the following property: If $n\in\ZZ_{\geq 1}$ then any finite set of rational primes $S$ with $\mathcal As{S}\leq n$ satisfies $\mathcal As{\Sigma(S)}\leq \mathcal As{\Sigma(S(n))}+c$.}
\noindent In other words, if $\mathcal T$ denotes the collection of schemes $T$ such that $T$ can be obtained by removing $n$ closed points of $\textnormal{Spec}(\ZZ)$, then Conjecture~1 means the following: Among all $T\in \mathcal T$, the maximal number (up to some constant) of $T$-points of $\mathbb P^1_\ZZ-\{0,1,\infty\}$ is attained at the scheme in $\mathcal T$ which corresponds to the $n$ closed points of smallest norm. This conjectured property of $\mathbb P^1_\ZZ-\{0,1,\infty\}$ is rather unexpected from a general Diophantine geometry perspective. We further ask whether one can remove the constant.
\noindent{\bf Question 1.1.}
\emph{Does \textnormal{Conjecture}~$1$ hold with $c=0$?}
\noindent Here the main motivation is given by our data. Indeed Question~1.1 has a positive answer for all sets $S$ in our database $\mathcal D_1$. In view of Theorem A listing the cardinality of $\Sigma(S(n))$ for all $n\in\{1,\dotsc,16\}$, a positive answer to Question 1.1 would give an optimal upper bound for the number of solutions of any $S$-unit equation~\eqref{eq:sunit} with $\mathcal As{S}$ in $\{1,\dotsc,16\}$.
\subsection{Comparison of algorithms}
To compare Algorithm~\ref{algo:suheight} with our Algorithm \ref{algo:sucremona} using modular symbols, we continue the notation introduced above. Recall that we already computed the sets $\Sigma(N)$ for all $N\leq 20000$ by using Algorithm~\ref{algo:sucremona}, see the examples in Section~\ref{sec:sucremalgo}. Here it turned out that the output of Algorithm~\ref{algo:sucremona} agrees with the output of Algorithm~\ref{algo:suheight}.
To determine all solutions of the $S$-unit equation \eqref{eq:sunit}, we recommend to use Algorithm~\ref{algo:sucremona} as long as one already knows the set of elliptic curves over $\QQ$ of conductor dividing $2^4N_S$. If this set is not already known, then it is usually much more efficient to use Algorithm~\ref{algo:suheight}. In fact, as demonstrated in the previous sections, our Algorithm~\ref{algo:suheight} is substantially more efficient in all aspects than the known methods which practically resolve \eqref{eq:sunit}.
\section{Algorithms for Mordell equations}\label{sec:malgo}
Let $S$ be a finite set of rational primes, write $N_S=\prod_{p\in S}p$ and put $\mathcal O=\ZZ[1/N_S]$.
We take a nonzero $a\in \mathcal O$. In this section, we would like to solve the Mordell equation
\begin{equation}
y^2=x^3+a, \ \ \ (x,y)\in\mathcal O\times\mathcal O. \tag{\ref{eq:mordell}}
\end{equation}
This Diophantine problem is a priori more difficult than solving the $S$-unit equation~\eqref{eq:sunit}. Indeed elementary transformations reduce~\eqref{eq:sunit} to~\eqref{eq:mordell}, while the known reductions of~\eqref{eq:mordell} to~\eqref{eq:sunit} require to solve~\eqref{eq:sunit} in number fields or they require a height bound for the solutions of~\eqref{eq:sunit} which is equivalent to the $abc$-conjecture in Remark~\ref{rem:abc}.
So far all known practical methods solving~\eqref{eq:mordell} crucially rely on the theory of logarithmic forms \cite{bawu:logarithmicforms}, see below for an overview. In the following sections we present two alternative algorithms which allow to practically resolve~\eqref{eq:mordell}. They both do not use the theory of logarithmic forms. In Section~\ref{sec:mcremalgo} we describe the first algorithm which relies on Cremona's algorithm using modular symbols. Then in Section~\ref{sec:mheightalgo} we construct the second algorithm via height bounds. Here we also give several applications and we discuss various questions motivated by our results, see Sections~\ref{sec:shaf}-\ref{sec:malgoapplications}. Finally in Section~\ref{sec:malgocomparison} we compare our algorithms with the actual best practical methods solving \eqref{eq:mordell}.
\paragraph{Known methods.} We now discuss algorithms and methods in the literature which allow to solve \eqref{eq:mordell}. First we consider the classical case $\mathcal O=\ZZ$. Ellison et al~\cite{ellison:mordell} used the approach of Baker--Davenport~\cite{baker:contributions,bada:diophapp} to solve~\eqref{eq:mordell} for some $a$. Recently the latter approach was refined by Bennett--Ghadermarzi~\cite{begh:mordell} who applied their algorithm to find all solutions of~\eqref{eq:mordell} in $\ZZ\times\ZZ$ for any nonzero $a\in\ZZ$ with $\mathcal As{a}\leq 10^7$; see also the work of Wildanger and J\"atzschmann discussed in Fieker--Ga\'al--Pohst~\cite[p.739]{figapo:mordellcharp}. Alternatively, Masser~\cite{masser:ellfunctions}, Lang \cite{lang:diophantineanalysis}, W\"ustholz~\cite{wustholz:recentprogress} and Zagier \cite{zagier:largeintegralpoints} initiated a practical approach to solve arbitrary elliptic Weierstrass equations $(W)$ over $\ZZ$ via elliptic logarithms. On applying this approach with David's explicit bounds~\cite{david:elllogmemoir}, Stroeker--Tzanakis~\cite{sttz:elllogaa} and Gebel--Peth{\H{o}}--Zimmer~\cite{gepezi:ellintpoints} obtained independently a practical algorithm solving $(W)$ over $\ZZ$. Gebel--Peth{\H{o}}--Zimmer~\cite{gepezi:mordell} used this algorithm to determine all solutions of \eqref{eq:mordell} in $\ZZ\times\ZZ$ for any nonzero $a\in \ZZ$ with $\mathcal As{a}\leq 10^4$ and for most $a\in\ZZ$ with $\mathcal As{a}\leq 10^5$. Let $r$ be the Mordell--Weil rank of the group $E(\QQ)$ associated to the elliptic curve $E$ over $\QQ$ defined by $(W)$. In the important special case $r=1$, there exists in addition a practical approach of Balakrishnan--Besser--M\"uller \cite{babemu:qchabcrelle,babemu:qchabalgo} which is in the spirit of Kim's non-abelian Chabauty program initiated in \cite{kim:siegel}.
We now discuss practical methods in the literature solving \eqref{eq:mordell} over any ring $\mathcal O$ as above. In practice and in theory, this task is considerably more difficult
than solving \eqref{eq:mordell} in $\mathcal O=\ZZ$. The method of Bilu~\cite{bilu:superellalgo} and Bilu--Hanrot~\cite{biha:algosuperell} for superelliptic Diophantine equations allows in particular to solve \eqref{eq:mordell}.
Further, classical constructions reduce \eqref{eq:mordell} to Thue--Mahler equations which in turn can be solved using the method of Tzanakis--de Weger \cite{tzde:thuemahler}.
Smart~\cite{smart:sintegralpoints} extended the above mentioned elliptic logarithm approach to solve $(W)$ over $\mathcal O$. His algorithm is conditional on explicit lower bounds for linear forms in $p$-adic elliptic logarithms. R\'emond--Urfels~\cite{reur:padicelllog} proved such bounds\footnote{Hirata-Kohno~\cite{hirata-kohno:p-adicelllogs} recently established the general case. Tzanakis~\cite[Chapt 11]{tzanakis:book} combined her bounds with the elliptic logarithm reduction to solve $(W)$ over $\mathcal O$, see also Hirata-Kohno--Kov\'acs~\cite{hiko:rank3}.} for $r\leq 2$, and these bounds were then applied by Gebel--Peth{\H{o}}--Zimmer~\cite{gepezi:bordeaux,gepezi:barcelona} to solve \eqref{eq:mordell} for some nonempty $S$. Furthermore, in a joint work with Herrmann~\cite{pezigehe:sintegralpoints}, they obtained a variation of the elliptic logarithm approach which works also for $r\geq 3$, see Section \ref{sec:mheightalgo}. We point out that the elliptic logarithm approach requires a basis of $E(\QQ)$. There are methods which often can compute such a basis in practice, in particular in our case of elliptic curves defined by \eqref{eq:mordell}. However, these methods are not (yet) effective in general as discussed in Section~\ref{sec:tor+mwbasis}. For a detailed description of the elliptic logarithm approach, we refer to the excellent book of Tzanakis~\cite{tzanakis:book} which is devoted to this method.
\subsection{Algorithm via modular symbols}\label{sec:mcremalgo}
We continue our notation. For any elliptic curve $E$ over $\QQ$, we denote by $c_4$ and $c_6$ the usual quantities associated to a minimal Weierstrass model of $E$ over $\ZZ$; see~\cite{tate:aoe}. Write $N_E$ and $\Delta_E$ for the conductor and minimal discriminant of $E$ respectively. We define
\begin{equation}\label{def:as}
a_S=1728N_S^2\prod p^{\min(\ord_p(a),2)}
\end{equation}
with the product taken over all rational primes $p$ not in~$S$. If $(x,y)$ satisfies the Mordell equation~\eqref{eq:mordell}, then one can consider the elliptic curve $E$ over $\QQ$ which admits the Weierstrass equation $t^2 = s^3 - 27xs - 54y$ with ``indeterminates" $s$ and~$t$. This construction leads to the following lemma which will be proven in course of the proof of Lemma~\ref{lem:pm2}.
\begin{lemma}\label{lem:pm1}
Suppose that $(x,y)$ is a solution of the Mordell equation~\eqref{eq:mordell}. Then there exists an elliptic curve $E$ over $\QQ$ such that $N_E\mid a_S$ and such that $c_4 =u^4x$ and $c_6= u^6y$ for some $u\in\QQ$ with $u^{12}=1728\Delta_E\mathcal As{a}^{-1}$.
\end{lemma}
In fact this lemma may be viewed as an explicit Par\v{s}in{} construction for integral points on the moduli scheme of elliptic curves defined by~\eqref{eq:mordell}, see \cite[Prop 3.4]{rvk:modular} for details.
\begin{Algorithm}[Mordell equations via modular symbols]\label{algo:mcremona} The input consists of a finite set of rational primes $S$ together with a nonzero number $a\in\mathcal O$, and the output is the set of solutions $(x,y)$ of the Mordell equation \eqref{eq:mordell}.
\begin{itemize}
\item[(i)] Define $a_S$ as in~\eqref{def:as}. Then use Cremona's algorithm using modular symbols, described in Section~\ref{sec:cremonas+st}, to compute the set $\mathcal T\subset \ZZ\times\ZZ$ of quantities $(c_4,c_6)$ which are associated to some modular elliptic curve over $\QQ$ of conductor dividing~$a_S$.
\item[(ii)] For each $(c_4,c_6)\in\mathcal T$, write $\mathcal As{(c_4^3-c_6^2)/a}=\frac{m}{n}$ with coprime $m,n\in\ZZ_{\geq 1}$. Compute the subset $\mathcal T_{0}\subseteq \mathcal T$ of $(c_4,c_6)\in\mathcal T$ with $m=u_1^{12}$ and $n=u_2^{12}$ for $u_1,u_2\in\ZZ$.
\item[(iii)] For any $(c_4,c_6)\in \mathcal T_{0}$ take $u=\frac{u_1}{u_2}$ in $\mathbb \QQ$ with $u^{12}=\mathcal As{(c_4^3-c_6^2)/a}$ and define $x=u^{-4}c_4$ and $y=u^{-6}c_6$. If $x$ and $y$ are both in $\mathcal O$, then output $(x,y)$.
\end{itemize}
\end{Algorithm}
\paragraph{Correctness.}We now verify that this algorithm indeed finds all solutions of any Mordell equation~\eqref{eq:mordell}. Suppose that $(x,y)$ satisfies~\eqref{eq:mordell}. Then Lemma~\ref{lem:pm1} gives an elliptic curve $E$ over $\QQ$ such that $N_E\mid a_S$ and such that $c_4 =u^4x$ and $c_6= u^6y$ for some $u\in\QQ$ with $u^{12}=1728\Delta_E\mathcal As{a}^{-1}$. The Shimura--Taniyama conjecture assures that $E$ is modular. This proves that $(c_4,c_6)$ is contained in the set $\mathcal T$ computed in step (i). Furthermore we obtain that $(c_4,c_6)\in\mathcal T_{0}$, since it holds $u^{12}=1728\Delta_E\mathcal As{a}^{-1}=\mathcal As{(c_4^3-c_6^2)/a}$ by the definition of the discriminant. Any $u'\in\QQ$ with $u'^{12}=\mathcal As{(c_4^3-c_6^2)/a}$ satisfies $u'^4=u^4$ and $u'^6=u^6$. Therefore we see that step (iii) produces our solution $(x,y)=(u^{-4}c_4,u^{-6}c_6)$ as desired.
\paragraph{Complexity.} The discussion of the complexity of step (i) is analogous to the discussion of the complexity of Algorithm~\ref{algo:sucremona}~(i) and thus we refer the reader to Section~\ref{sec:sucremalgo}. For each $(c_4,c_6)\in\mathcal T$, step (ii) needs to check if there is $u=\frac{u_1}{u_2}\in\QQ$ with $u^{12}=\mathcal As{(c_4^3-c_6^2)/a}$ and then step (iii) needs to check if $u_1\in\ZZ$ is only divisible by primes in~$S$. Therefore we see that the complexity discussions of Algorithm~\ref{algo:sucremona} together with the arguments in Remark~\ref{rem:sulegendrealternative} imply the following: The running time of step (ii) and (iii) is at most polynomial in terms of $H(a)N_S$, and is at most $O_\varepsilon((H(a)N_S)^\varepsilon)$ if all three conjectures (BSD), (GRH) and $(abc)$ hold. Here $H(a)=\exp(h(a))$ denotes the multiplicative Weil height of~$a$.
\paragraph{Applications.} We recall from \cite[12.5.2]{bogu:diophantinegeometry} that a solution $(x,y)\in\ZZ\times\ZZ$ of~\eqref{eq:mordell} is called primitive if $\pm 1$ are the only $m\in\ZZ$ with $m^6\mid \gcd(x^3,y^2)$. For each nonzero $a\in\ZZ$ one can quickly enumerate all $(x,y)\in\ZZ\times\ZZ$ with $y^2=x^3+a$ if one knows the primitive solutions of any Mordell equation $y^2=x^3+a'$ with $a'\in\ZZ$ satisfying $a=a'm^6$ for some $m\in\ZZ$. On using an implementation in Sage of Algorithm~\ref{algo:mcremona}, we computed the set of primitive solutions of the family of Mordell equations~\eqref{eq:mordell} with parameter $a\in\ZZ-\{0\}$ satisfying $r_2(a)\leq 200$ for $r_2(a)=\prod p^{\min(2,\ord_p(a))}$ with the product taken over all rational primes~$p$. Here the computation of the solutions was very fast, since for each $a\in\ZZ-\{0\}$ with $r_2(a)\leq 200$ part~(i) of Algorithm~\ref{algo:mcremona} can use Cremona's database which contains in particular the required data for all elliptic curves over $\QQ$ of conductor dividing $1728r_2(a)< 350 000$. On the other hand, if the required data of the involved elliptic curves is not already known, then our Algorithm~\ref{algo:mcremona} is usually not practical anymore. Here the problem is the application of Cremona's algorithm (using modular symbols) in step (i), which requires a huge amount of memory to deal with conductors that are not small.
\subsection{Algorithm via height bounds}\label{sec:mheightalgo}
In this section we use the optimized height bounds of Proposition~\ref{prop:algobounds} to construct Algorithm~\ref{algo:mheight} which allows to solve the Mordell equation~\eqref{eq:mordell}. We continue our notation and we work with the following setup: We may and do view $y^2=x^3+a$ as a Weierstrass equation of an elliptic curve $E_a$ over $\QQ$, since our $a\in\mathcal O$ is nonzero. A classical result of Mordell gives that the abelian group $E_a(\QQ)$ is finitely generated. Let $P_1,\dotsc,P_r$ be a basis of the free part of $E_a(\QQ)$, and let $E_a(\QQ)_{\textnormal{tor}}$ be the torsion group of $E_a(\QQ)$. We call $r$ the Mordell--Weil rank of $E_a(\QQ)$ and we say that $P_1,\dotsc,P_r$ is a Mordell--Weil basis of $E_a(\QQ)$. Let $(x,y)$ be a solution of \eqref{eq:mordell}. The corresponding point $P$ in $E_a(\QQ)$ takes the form $P=Q+\sum_{i=1}^r n_i P_i$ with $n_i\in\ZZ$ and $Q\in E_a(\QQ)_{\textnormal{tor}}$, and we define $N(x,y)=\max_{i}\mathcal As{n_i}$.
\paragraph{Decomposition.} Before we describe our Algorithm~\ref{algo:mheight} in detail, we discuss its decomposition which is inspired by the elliptic logarithm approach introduced by Masser--Zagier. A variation of the latter approach was used for example in the algorithm of Peth{\H{o}}--Zimmer--Gebel--Herrmann \cite{pezigehe:sintegralpoints} whose main ingredients can be described as follows:
\begin{itemize}
\item[(1)]\label{itDeWegerStepBaker} First they try to find a Mordell--Weil basis of $E_a(\QQ)$.
\item[(2)]\label{itDeWegerStepLLL} On using the explicit result of Hajdu--Herendi~\cite{hahe:elliptic} which is based on the theory of logarithmic forms~\cite{bawu:logarithmicforms}, they determine an initial upper bound $N_0$ such that any solution $(x,y)$ of~\eqref{eq:mordell} satisfies $N(x,y)\leq N_0$.
\item[(3)]\label{itDeWegerStepSieve} Following Smart~\cite{smart:sintegralpoints} they apply the elliptic logarithm reduction in order to reduce the initial upper bound $N_0$ to a bound $N_1$ which is usually much smaller.
\item[(4)] \label{itDeWegerStepEnumerationByHand}
Finally they enumerate all solutions $(x,y)$ of \eqref{eq:mordell} with $N(x,y)\leq N_1$. Here in the case $\mathcal O=\ZZ$ the ``inequality trick" usually improves the enumeration process.
\end{itemize}
Our Algorithm~\ref{algo:mheight} substantially improves in all aspects the elliptic logarithm approach for \eqref{eq:mordell}. More precisely, without introducing new ideas we use the known methods for (1) described in Section~\ref{sec:tor+mwbasis}. To obtain the initial upper bound for Algorithm~\ref{algo:mheight}, we apply in Proposition~\ref{prop:mwbound} the optimized height bounds worked out in Section~\ref{sec:heightbounds}. In practice our initial bound is considerably stronger than the initial bound $N_0$ in (2) based on the theory of logarithmic forms, and this leads to significant running time improvements of the reduction process as illustrated in Section~\ref{sec:minitbounds}. Then to enumerate the solutions of bounded height, we apply the elliptic logarithm sieve constructed in Section~\ref{sec:elllogsieve}. This sieve is substantially more efficient than the reduction process (3) together with the subsequent enumeration (4). We now discuss the ingredients of our approach in more detail.
\subsubsection{Torsion group and Mordell--Weil basis}\label{sec:tor+mwbasis}
On using the notation introduced above, we next briefly discuss methods which allow to determine the torsion group $E_a(\QQ)_{\textnormal{tor}}$ of $E_a(\QQ)$ and a Mordell--Weil basis of $E_a(\QQ)$.
\paragraph{Torsion.} Fueter~\cite{fueter:kubische} completely determined $E_a(\QQ)_{\textnormal{tor}}$. To state his result we write $a=e/d$ with coprime $e,d\in\ZZ$. If $d^5e=k\cdot l^6$ for $k,l\in\ZZ$ with $k$ sixth power free, then
\begin{equation}\label{eq:mtorsion}
E_a(\QQ)_{\textnormal{tor}}\cong\begin{cases}
\ZZ/6\ZZ & \textnormal{if } k=1,\\
\ZZ/3\ZZ & \textnormal{if } k\neq 1 \textnormal{ is a square, or } k=-432,\\
\ZZ/2\ZZ & \textnormal{if } k\neq 1 \textnormal{ is a cube,}\\
0 & \textnormal{otherwise.}
\end{cases}
\end{equation}
This completely determines all solutions of \eqref{eq:mordell} in $\QQ\times\QQ$ in the case when the Mordell--Weil rank $r$ of $E_a(\QQ)$ satisfies $r=0$. Therefore for the purpose of determining all solutions of the Mordell equation \eqref{eq:mordell} we always may assume that $r\geq 1$.
\paragraph{Mordell--Weil basis.} The problem of finding a Mordell--Weil basis of $E_a(\QQ)$ is usually more difficult. If the elliptic curve $E_a$ satisfies the part of the Birch--Swinnerton-Dyer conjecture (BSD) predicting that $r$ coincides with the analytic rank of $E_a$, then a result of Manin \cite{manin:cyclomodcurves} leads to a practical algorithm (see for example Gebel--Zimmer~\cite{gezi:mwalgo})
which computes such a basis. On applying this algorithm and an algorithm of Cremona given in~\cite{cremona:algorithms}, Gebel--Peth{\H{o}}--Zimmer~\cite{gepezi:mordell} computed a Mordell--Weil basis of $E_a(\QQ)$ for most $a\in\ZZ$ with $\mathcal As{a}\leq 10^5$. Parts of their database (Mordell$\pm$) are uploaded on their homepage \url{tnt.math.se.tmu.ac.jp/simath/MORDELL}.
On using (PSM) we checked their data for all nonzero $a\in \ZZ$ with $\mathcal As{a}\leq 10^4$. Here it turned out that for $a = 7823$ a basis was missing and that for $a = -7086$ and $a=-6789$ the given bases were not saturated. In all these three cases we determined a Mordell--Weil basis of $E_a(\QQ)$ using (PSM) and now the (updated) database contains a correct Mordell--Weil basis of $E_a(\QQ)$ for any nonzero $a\in \ZZ$ with $\mathcal As{a}\leq 10^4$. We note that in our special case given by the Mordell elliptic curve $E_a$, one can often exploit isogenies to find a Mordell--Weil basis of $E_a(\QQ)$. In fact, it turned out in practice that the known techniques implemented in (PSM) (Generators, two-descent, HeegnerPoint, etc.) usually allow to quickly determine such a basis.
However we point out that in the case of an arbitrary nonzero $a\in\ZZ$ there is so far no unconditional method which allows to determine a Mordell--Weil basis of $E_a(\QQ)$.
\subsubsection{Elliptic logarithm sieve}
Starting with Zagier \cite{zagier:largeintegralpoints} and de Weger~\cite{deweger:phdthesis}, many authors developed over the last decades the elliptic logarithm reduction process; see for example Stroeker--Tzanakis \cite{sttz:elllogaa}, Gebel--Peth{\H{o}}--Zimmer~\cite{gepezi:ellintpoints} and Smart~\cite{smart:sintegralpoints}. In practice this process allows to show that the solutions in $\mathcal O\times\mathcal O$ of an arbitrary elliptic Weierstrass equation have either relatively small or huge height. In Section~\ref{sec:elllogsieve} we constructed the elliptic logarithm sieve which considerably improves the elliptic logarithm reduction and the subsequent enumeration of solutions of small height.
In particular, for any given bound $N\in\ZZ$, the elliptic logarithm sieve solves the problem of efficiently enumerating all solutions $(x,y)$ of \eqref{eq:mordell} with $N(x,y)\leq N$. The sieve combines the core idea of the elliptic logarithm reduction with several conceptually new ideas. We refer to Section~\ref{sec:elllogsieve} for an overview of the new ideas introduced by the elliptic logarithm sieve and for a detailed discussion of the practical and theoretical improvements provided by these ideas.
\subsubsection{Initial bounds}\label{sec:minitbounds}
We continue the notation introduced above. In this section we give an initial upper bound for various heights attached to the solutions of the Mordell equation \eqref{eq:mordell}. We also compare our bound with the actual best results in the literature and we explain how our result improves the running time of the reduction process in the elliptic logarithm sieve.
We recall that $P_1,\dotsc,P_r$ denotes a Mordell--Weil basis of $E_a(\QQ)$, and for any solution $(x,y)$ of \eqref{eq:mordell} we write as above $N(x,y)=\max \mathcal As{n_i}$ for the ``infinity norm" of the corresponding point $P=Q+\sum n_i P_i$ in $E_a(\QQ)$. Let $\hat{h}$ be the canonical N\'eron-Tate height on $E_a(\QQ)$. Here we work with the natural normalization of $\hat{h}$ which divides by the degree of the involved rational function, see for example \cite[p.248]{silverman:aoes}.
\paragraph{Initial bounds.} Let $(x,y)$ be a solution of \eqref{eq:mordell}, with corresponding point $P\in E_a(\QQ)$. To deduce an upper bound for $N(x,y)$, we recall standard properties of $\hat{h}$. One can control $\hat{h}(P)$ in terms of the usual logarithmic Weil height $h$ as follows
\begin{equation}\label{eq:canonheightcomp}
\hat{h}(P)\leq \tfrac{1}{2}h(x)+\tfrac{m}{6}h(a)+1.58, \ \ \ \ m =\begin{cases}
1 & \textnormal{if } a\in\ZZ,\\
12 & \textnormal{otherwise.}
\end{cases}
\end{equation}
Indeed in the integral case $a\in\ZZ$ the displayed inequality directly follows for example from Silverman~\cite[Thm 1.1]{silverman:heightcomparison}.
To deal with any nonzero $a\in\mathcal O$, we write $a=c/d$ with coprime $c,d\in\ZZ$ and we consider $E_{b}$ for $b=d^6a\in\ZZ$. There is an isomorphism $\varphi:E_a\to E_{b}$ induced by $x\mapsto d^2x$. Then an application of the already verified integral case of \eqref{eq:canonheightcomp}, with $E_b$ and $b\in\ZZ$, gives an upper bound for $\hat{h}(P)=\hat{h}(\varphi(P))$ in terms of $h(d^2x)$ and $h(b)$ which proves \eqref{eq:canonheightcomp} as desired.
Further it is known that $\hat{h}$ defines a positive definite quadratic form on the real vector space $E(\QQ)\otimes_\ZZ\RR$. On using the basis $P_1,\dotsc,P_r$ we identify $E(\QQ)\otimes_\ZZ\RR$ with $\RR^r$ and then we denote by $\lambda$ the smallest eigenvalue of the matrix defining the binary form associated to the quadratic form $\hat{h}$ on $\RR^r$.
We now use the optimized height bound in Proposition~\ref{prop:algobounds}, or the $abc$-conjecture of Masser-Oesterl\'e~\cite{masser:abc} stated in Remark~\ref{rem:abc}, in order to obtain initial bounds.
\begin{proposition}\label{prop:mwbound}
Suppose that $(x,y)$ is a solution of \eqref{eq:mordell}, and denote by $P$ the corresponding point in $E_a(\QQ)$. Then the following statements hold.
\begin{itemize}
\item[(i)] Let $\alpha=\alpha(a_S)$ be the number from Proposition~\ref{prop:algobounds}, and recall that $m=1$ if $a\in\ZZ$ and $m=12$ otherwise. It holds $\lambda N(x,y)^2\leq\hat{h}(P) \leq M_0$ for some $M_0\in\ZZ$ with $$M_0\leq \tfrac{m+1}{6}h(a)+2\alpha+\log(\alpha+16.52)+52.12.$$
\item[(ii)] Suppose that $a\in\ZZ$, and assume that the $abc$-conjecture holds. Then for any real number $\varepsilonilon>0$ there exists a constant $c_\varepsilonilon$ depending only on $\varepsilonilon$ such that $$\hat{h}(P)\leq (1+\varepsilonilon)(\log N_S+\tfrac{7}{6}h(a))+c_\varepsilonilon.$$
\end{itemize}
\end{proposition}
In case the computation of the number $\alpha$ from Proposition~\ref{prop:algobounds} takes too long, one can replace $\alpha$ by the (slightly) larger number $\bar{\alpha}=\bar{\alpha}(a_S)$ which is defined below \eqref{def:barb}. The number $\bar{\alpha}$ has the advantage that it can be quickly computed in all cases. We further mention that the bound in Proposition~\ref{prop:mwbound}~(ii) is a direct consequence of a result in Bombieri--Gubler~\cite[Thm 12.5.12]{bogu:diophantinegeometry}; this bound is optimal in terms of $N_S$.
\begin{proof}[Proof of Proposition~\ref{prop:mwbound}]
We first prove (i). Linear algebra leads to $\lambda N(x,y)^2\leq\hat{h}(P)$, since $\hat{h}$ is a quadratic form on $E(\QQ)\otimes_\ZZ\RR$. Further, the estimate for $M_0$ in (i) follows by combining \eqref{eq:canonheightcomp} with the upper bound for $h(x)$ given in Proposition~\ref{prop:algobounds}.
To show assertion (ii) we assume that $a\in \ZZ$ and we write $x=x_1/d^2$ and $y=y_1/d^3$ with $x_1,y_1,d\in\ZZ$ satisfying $\gcd(d,x_1y_1)=1$
and $d>0$. Let $n$ be the largest element in $\ZZ$ with $n^6\mid \gcd(x_1^3,y_1^2)$. On dividing the equation $x_1^3-y_1^2=-ad^6$ by $n^6$, we obtain a new equation of the form $u^3-v^2=w$ with $u,v,w\in\ZZ$ and $\gcd(u^3,v^2)=1$. Now, on assuming the $abc$-conjecture, we see that \cite[Thm 12.5.12]{bogu:diophantinegeometry} gives estimates for $\mathcal As{u},\mathcal As{v}$. These estimates lead to an upper bound for $h(x)$ which together with \eqref{eq:canonheightcomp} proves (ii).
\end{proof}
\paragraph{Comparison with literature.} There are several explicit bounds for $M_0$ and $N(x,y)$ in the literature. They are all based on the theory of logarithmic forms. In fact this theory allows to effectively solve Diophantine equations which are considerably more general than Mordell equations \eqref{eq:mordell}, see for example \cite{bawu:logarithmicforms}. To compare Proposition~\ref{prop:mwbound}~(i) with the actual best bounds for \eqref{eq:mordell} in the literature, we use a simpler but weaker version of our bound. On replacing in the proof of Proposition~\ref{prop:mwbound}~(i) our optimized height bounds by the simplified height bounds in Proposition~\ref{prop:m}, we obtain
\begin{equation}\label{eq:simplmwbound}
\lambda N(x,y)^2\leq M_0\leq \tfrac{m+1}{6}h(a)+\tfrac{1}{2}a_S\log a_S.
\end{equation}
For the purpose of the following discussion, we recall that in the case $a\in\ZZ$ it holds that $a_S\leq 1728\mathcal As{a}N_S^2$ and $m=1$.
The actual best effective upper bound for $N(x,y)$ and $M_0$ was established by Peth{\H{o}}--Zimmer--Gebel--Herrmann~\cite[Thm]{pezigehe:sintegralpoints}.
Their result is based on the work of Hajdu--Herendi~\cite{hahe:elliptic} which in turn relies on the theory of logarithmic forms. To state the rather complicated bound for $N(x,y)$ provided by Peth{\H{o}} et al, we need to introduce some notation.
As in~\cite[Thm]{pezigehe:sintegralpoints} we define the constants
$$k_3=\frac{32}{3}\Delta_0^{\frac{1}{2}}(8+\frac{1}{2}\log \Delta_0)^4, \ \ \ k_4=10^4\cdot 256\cdot \Delta_0^{\frac{2}{3}}, \ \ \ \Delta_0=27\lvert a\rvert^2.$$ Further, we write $s=\lvert S\rvert$ and $q=\max S$ (with $q=1$ if $S=\emptyset$). Then we define
$$\kappa_1=\tfrac{7}{2}\cdot 10^{38s+87}(s+1)^{20s+35}q^{24}\max(1,\log q)^{4s+2} k_3(\log k_3)^2(k_3+20sk_3+\log (ek_4)).$$
We mention that the result in \cite[Thm]{pezigehe:sintegralpoints} is stated under the assumption that the given Weierstrass equation over $\ZZ$ is minimal at all primes in $S$. However, on looking at the proof one sees that this minimality assumption is not necessary for the portion of the theorem which provides an upper bound for $N(x,y)$. We conclude that \cite[Thm]{pezigehe:sintegralpoints} provides in general that any solution $(x,y)$ of (\ref{eq:mordell}) with $a\in\ZZ-\{0\}$ satisfies
\begin{equation}\label{eq:pzghmwbound}
\lambda N(x,y)^2\leq M_0\leq \kappa_1+\tfrac{1}{3}\log \mathcal As{4\cdot 6^3a}.
\end{equation}
In our simplified bound \eqref{eq:simplmwbound} the dependence on $a\in \ZZ$ is of the form $\lvert a\rvert \log \lvert a\rvert$, while in \eqref{eq:pzghmwbound} it is of the weaker form $\lvert a \rvert^2(\log \lvert a\rvert)^{10}$. Further we see that \eqref{eq:simplmwbound} considerably improves \eqref{eq:pzghmwbound} for essentially all sets $S$ of practical interest, in particular for all sets $S$ with $N_S\leq 2^{1200}$ or $s\leq 12$ and for all sets $S$ of the form $S=S(n)$ where $S(n)$ denotes the set of the first $n$ primes for some $n\in\ZZ_{\geq 1}$.
We now choose parameters $\mathcal A$ and $\mathcal S$ as follows: The set $\mathcal S$ is given by $\{\emptyset,S(1),S(10)\}$, and the set $\mathcal A$ consists of 24 distinct nonzero $a\in\ZZ$ such that for each $r\in\{1,\dotsc,12\}$ there are precisely two $a$ in $\mathcal A$ with $E_a(\QQ)$ of rank $r$; here we tried\footnote{For $r\leq 6$ we found the ``smallest" possible $a$. Further, we note that for our purpose of illustrating the running time improvements of the reduction process it suffices to work with $r$ independent points of $E_a(\QQ)$; for $r\geq 9$ we could not prove (unconditionally) that our $r$ independent points form a basis.} to choose these elements $a\in\ZZ$ with $\mathcal As{a}$ as small as possible. To illustrate that our bound leads to significant running time improvements, we computed for all parameter pairs $(a,S)\in\mathcal A\times\mathcal S$ the running times $\rho$ and $\rho^*$ of the elliptic logarithm reduction in Algorithm~\ref{algo:elllogsieve}~(ii) using Proposition~\ref{prop:mwbound}~(i) and \eqref{eq:pzghmwbound} respectively. In the case $S=\emptyset$, it turned out that we obtain a running time improvement by a factor $\rho^*/\rho$ which is approximately $2$ for small/medium $\mathcal As{a}$ and which is close to $4$ for large $\mathcal As{a}$. The running time improvements become more significant in the case $S=S(1)$. Here the factor $\rho^*/\rho$ is approximately $30$ for small/medium $\mathcal As{a}$ and it is approximately $60$ for large $\mathcal As{a}$.
Finally we achieve big running time improvements when $S=S(10)$. In this case the factor $\rho^*/\rho$ is approximately $300$ for small $\mathcal As{a}$, it lies between $500$ and $10^3$ for medium sized $\mathcal As{a}$ and it varies between $10^3$ and $10^4$ for large $\mathcal As{a}$. For example if $a=-2520963512$ ($r=8$) and $S=S(10)$ then $\rho$ is less than 23 seconds while $\rho^*$ exceeds 2 days.
In the classical case $\mathcal O=\ZZ$, there is also a fully explicit estimate $N_0\geq N(x,y)$ which was independently established by Stroeker--Tzanakis~\cite{sttz:elllogaa} and Gebel--Peth{\H{o}}--Zimmer~\cite{gepezi:ellintpoints}. This estimate is based on lower bounds for linear forms in elliptic logarithms (see Masser~\cite{masser:ellfunctions}, W\"ustholz~\cite{wustholz:recentprogress}, Hirata-Kohno~\cite{hirata-kohno:ellloginvent} and David~\cite{david:elllogmemoir}). We do not state here $N_0$ in its precise form, since $N_0$ is even more complicated than the bound in \eqref{eq:pzghmwbound}. To see that our result improves $N_0$ for essentially all $a\in\ZZ$ of practical interest, it suffices to consider the following simpler lower bound
\begin{equation}\label{eq:tzmwbound}
N_0\geq \lambda^{-1/2}10^{3(r+2)}4^{(r+1)^2}(r+2)^{(r^2+13r+23.3)/2} \prod_{i=1}^{r} \max\bigl(\hat{h}(P_i)^{1/2},\log (4\mathcal As{a})\bigl).
\end{equation}
This lower bound follows for example from \cite[p.386-387]{pezigehe:sintegralpoints}, see also the recent book of Tzanakis~\cite{tzanakis:book}. In \eqref{eq:tzmwbound} we may and do assume that $r\geq 1$ by Fueter's result \eqref{eq:mtorsion}. Then we see that our simplified bound \eqref{eq:simplmwbound} improves \eqref{eq:tzmwbound} for all nonzero $a\in\ZZ$ with $\mathcal As{a}\leq 10^{40}$. In the case of arbitrary $\mathcal O\neq \ZZ$ and $r=2$, one can deduce an explicit estimate $N_0\geq N(x,y)$ by using lower bounds of David \cite{david:elllogmemoir} and R\'emond--Urfels~\cite{reur:padicelllog}; see also the recent work of Hirata-Kohno--Kov\'acs~\cite{hirata-kohno:p-adicelllogs,hiko:rank3} removing the assumption $r=2$. Here the quantity $N_0$ is very complicated and its dependence on $S$ is quite involved. In any case $N_0$ is larger than the lower bound in \eqref{eq:tzmwbound} and thus our result is better than the estimate $N_0\geq N(x,y)$ for all pairs $(a,S)$ with $a_S\leq 10^{40}$. On the other hand, for large $a_S$ it is rather difficult (when not impossible in general) to compare Proposition~\ref{prop:mwbound}~(i) with the corresponding results based on lower bounds for linear forms in elliptic logarithms. The reason is that the involved quantities are quite different. However, the dependence of \eqref{eq:tzmwbound} on the rank $r$ means that our result leads to significant running time improvements in the notoriously difficult case when $r$ is not small. To illustrate this we computed for all parameter pairs $(a,S)\in\mathcal A\times \mathcal S$ the running times $\rho$ and $\rho'$ of the elliptic logarithm reduction in Algorithm~\ref{algo:elllogsieve}~(ii) using Proposition~\ref{prop:mwbound}~(i) and $N_0$ respectively. We note that instead of implementing the very complicated estimate $N_0$ in its precise form, we used here the simpler lower bound in \eqref{eq:tzmwbound}. In other words the running time $\rho'$ is slightly too good, which means that our running time improvements are slightly better than illustrated by the numbers appearing in the following discussion. In the case $S=\emptyset$, we obtain a running time improvement by a factor $\rho'/\rho$ which is approximately $2$ when $2\leq r\leq 4$ and which lies between $3$ and $10$ in the range $5\leq r\leq 12$. The running time improvements become more significant in the case $S=S(1)$. Here the factor $\rho'/\rho$ lies between $2$ and $20$ when $2\leq r\leq 4$, it varies between $50$ and $100$ in the range $5\leq r\leq 8$, and it lies between $500$ and $10^3$ for $9\leq r\leq 12$.
Finally we obtain big running time improvements when $S=S(10)$. In this case the factor $\rho'/\rho$ varies between $2$ and $10$ in the range $2\leq r\leq 4$, it lies between $30$ and $500$ for $r\leq 5\leq 8$, and it varies between $700$ and $3000$ in the range $r\leq 9\leq 12$. For example, if $S=S(10)$ then there is an $a\in\mathcal A$ with $r=12$ such that our running time $\rho$ is less than 3 minutes while $\rho'$ is approximately 5 days.
\subsubsection{The algorithm}\label{sec:mordellalgostat}
We continue the notation introduced above. On combining the ingredients of the previous sections, we obtain an algorithm which allows to solve the Mordell equation \eqref{eq:mordell}. Here we point out that our algorithm requires an explicitly given Mordell--Weil basis of $E_a(\QQ)$. While it is usually possible to determine such a basis in practice (see Section~\ref{sec:tor+mwbasis}), there is so far no unconditional method which in principle works for an arbitrary nonzero $a\in\ZZ$. In view of this we included a Mordell--Weil basis of $E_a(\QQ)$ in the input.
\begin{Algorithm}[Mordell equations via height bounds]\label{algo:mheight} The inputs are a finite set of rational primes $S$, a nonzero number $a\in\mathcal O$ and the coordinates of a Mordell--Weil basis of $E_a(\QQ)$. The output is the set of solutions $(x,y)$ of \eqref{eq:mordell}.
\begin{itemize}
\item[(i)] Use Proposition~\ref{prop:mwbound}~(i) to compute an initial bound $M_0$ such that for any solution $(x,y)$ of \eqref{eq:mordell} the corresponding point $P\in E_a(\QQ)$ satisfies $\hat{h}(P)\leq M_0$.
\item[(ii)] Apply the elliptic logarithm sieve in Algorithm~\ref{algo:elllogsieve} in order to find all solutions $(x,y)$ of \eqref{eq:mordell} with corresponding point $P\in E_a(\QQ)$ satisfying $\hat{h}(P)\leq M_0$.
\end{itemize}
\end{Algorithm}
\paragraph{Correctness.} If $P\in E_a(\QQ)$ corresponds to a solution $(x,y)$ of \eqref{eq:mordell}, then Proposition \ref{prop:mwbound}~(i) gives that $\hat{h}(P)\leq M_0$. Thus the application of the elliptic logarithm sieve in step (ii) produces all solutions of \eqref{eq:mordell} as desired, see Remark~\ref{rem:elllogsievegen} when $a\notin \ZZ$.
\paragraph{Complexity.} We now discuss various aspects which significantly influence the running time of Algorithm~\ref{algo:mheight}. In view of the remark given below Proposition~\ref{prop:mwbound}~(i), the computation of the initial upper bound $M_0$ in step (i) is always very fast. The running time of step (ii) crucially depends on the size of $M_0$, the height $h(a)$, the rank $r$ and the cardinality of $S$. For a complexity discussion of the elliptic logarithm sieve used in step (ii) we refer to Section~\ref{sec:elllogsieve}. Therein we explain in detail various complexity aspects and we also discuss in detail the influence of the parameters $M_0,r,h(a)$ and $\mathcal As{S}$ on the running time in practice (and in theory). See also Section~\ref{sec:minitbounds} where we illustrated the improvements provided by the sharpened initial bound $M_0$ obtained in Proposition~\ref{prop:mwbound}~(i).
\begin{remark}[Generalizations]
Algorithm~\ref{algo:mheight} allows to solve more general Diophantine equations associated to a Mordell curve, that is an elliptic curve with vanishing $j$-invariant. Assume that we are given the coefficients $a_1\dotsc,a_6\in\QQ$ of a Weierstrass equation
\begin{equation}\label{eq:mweieq}
y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6
\end{equation}
of an elliptic curve $E$ with $j$-invariant $j=0$, and suppose that we know a basis of the free part of $E(\QQ)$. Then Algorithm~\ref{algo:mheight} allows to find all solutions $(x,y)$ of \eqref{eq:mweieq} in $\mathcal O\times\mathcal O$. Indeed there is an explicit isomorphism which transforms any such solution of \eqref{eq:mweieq} into a solution of \eqref{eq:mordell} for some explicit $a\in\ZZ-\{0\}$, and hence an application of Algorithm~\ref{algo:mheight} with this $a$ produces the set of solutions of \eqref{eq:mordell} and then of \eqref{eq:mweieq}. In fact we implemented this slightly more general version of Algorithm~\ref{algo:mheight}. To conclude we mention that further generalizations are possible by using the arguments in \cite[Chapt 8]{tzanakis:book}.
\end{remark}
\subsubsection{Elliptic curves with good reduction outside a given set of primes}\label{sec:shaf}
We continue the notation introduced above. Let $M(S)$ be the set of $\QQ$-isomorphism classes of elliptic curves over $\QQ$ with good reduction outside a given finite set of rational primes $S$. In this section we apply Algorithm~\ref{algo:mheight} in order to compute the set $M(S)$.
\paragraph{Known methods.} There are already several practical methods in the literature which allow to determine $M(S)$. Agrawal--Coates--Hunt--van der Poorten~\cite{agcohuva:conductor11} computed the semi-stable locus of $M(\{11\})$ by using an approach via Thue--Mahler equations which ultimately relies on the theory of logarithmic forms. Their work builds on Coates' effective proof \cite{coates:shafarevich} of Shafarevich's theorem mentioned in the introduction.
Alternatively one can compute $M(S)$ by using the Shimura--Taniyama conjecture and modular symbols, see Cremona~\cite{cremona:algorithms}. There are also two more recent approaches which ultimately rely on the theory of logarithmic forms: The method of Cremona--Lingham~\cite{crli:shafarevich} discussed in the introduction, and the very recent approach of Koutsianas~\cite{koutsianas:shafarevich} via $S$-unit equations over number fields. Furthermore, very recently Bennett--Rechnitzer~\cite{bere:compell,bere:compelloneprime} substantially refined (in particular for $\mathcal As{S}=1$) the above mentioned classical Thue--Mahler approach: In the irreducible case they use the Thue--Mahler algorithm of Tzanakis--de Weger~\cite{tzde:thuemahler} and in the rational two torsion case they apply the algorithm of de Weger~\cite{deweger:phdthesis,deweger:sumsofunits} for sums of units being a square.
Finally, several authors used ingenious ad hoc methods to determine $M(S)$ for specific sets $S$. For an overview, see for example the discussions and references in \cite[Sect 1]{cremona:condhistory} and \cite[Sect 1]{bere:compelloneprime}.
\paragraph{The algorithm.} On combining Shafarevich's classical reduction to Mordell equations \eqref{eq:mordell} with Algorithm~\ref{algo:mheight}, we obtain an alternative approach to determine $M(S)$. Here we do not use modular symbols or lower bounds for linear forms in logarithms. To state our algorithm, we introduce some terminology. We may and do identify any $[E]$ in $M(S)$ with the pair $(c_4,c_6)$ associated by Tate \cite[p.180]{tate:aoe} to a minimal Weierstrass model of $E$ over $\ZZ$. Further, for arbitrary $s,t\in\QQ$ we say that an elliptic curve over $\QQ$ is given by $(s,t)$ if the equation $y^2=x^3-27sx-54t$ defines an affine model of the curve. If $s,t$ are in $\QQ$ with $s^3-t^2$ nonzero and if $E$ denotes an elliptic curve over $\QQ$ given by $(s,t)$, then Tate's algorithm~\cite{tate:algo} allows to compute the pair $(c_4,c_6)$ associated to a minimal Weierstrass model of $E$ over $\ZZ$ and it allows to check whether $[E]$ lies in $M(S)$.
\begin{Algorithm}\label{algo:shaf} The inputs are a finite set of rational primes $S$ and a Mordell--Weil basis of $E_a(\QQ)$ for all $a=1728w$ with $w\in\ZZ$ dividing $N_S^5$. The output is the set $M(S)$.
The algorithm: For each $a=1728w$ with $w\in\ZZ$ dividing $N_S^5$, do the following.
\begin{itemize}
\item[(i)] Apply Algorithm~\ref{algo:mheight} in order to determine the set $Y_a(\mathcal O)$ formed by the solutions of the Mordell equation \eqref{eq:mordell} defined by the parameter pair $(a,S)$.
\item[(ii)] For each $(x,y)\in Y_a(\mathcal O)$ and for any $d\in\ZZ_{\geq 1}$ dividing $N_S$, let $E$ be the elliptic curve over $\QQ$ given by $(d^2x,d^3y)$ and output the pair $(c_4,c_6)$ associated to a minimal Weierstrass model of $E$ over $\ZZ$ provided that $[E]$ lies in $M(S)$.
\end{itemize}
\end{Algorithm}
\paragraph{Correctness.} We take $[E]=(c_4,c_6)$ in $M(S)$. The minimimal discriminant $\Delta$ of $E$ lies in $\mathcal O^\times$. Hence there are integers $u,w,d\in\mathcal O^\times$, with $d\in\ZZ_{\geq 1}$ dividing $N_S$ and $w$ dividing $N_S^5$, such that $\Delta=-wd^6u^{12}$. We define $x=\tfrac{c_4}{u^4d^2}$ and $y=\tfrac{c_6}{u^6d^3}$. The formula $1728\Delta=c_4^3-c_6^2$ shows that $(x,y)$ lies in the set $Y_a(\mathcal O)$ computed in step (i) for $a=1728w$, and the elliptic curve $E$ is given by $(d^2x,d^3y)$. Hence we see that step (ii) produces our $[E]$ as desired.
\paragraph{Complexity.} The running time of Algorithm~\ref{algo:shaf} is essentially determined by step~(i). Therein we compute the sets $Y_a(\mathcal O)$ for all $a=1728w$ with $w\in\ZZ$ dividing $N_S^5$ and for this purpose we need to apply Algorithm~\ref{algo:mheight} with $2\cdot 6^{\mathcal As{S}}$ distinct parameters $a$. Hence the running time of Algorithm~\ref{algo:shaf} crucially depends on $\mathcal As{S}$ and on the complexity of Algorithm~\ref{algo:mheight} which we already discussed in the previous section. In step (ii), it might be possible that one can omit to check whether $[E]$ lies in $M(S)$. In any case this check is always very quick and it has no influence on the running time in practice.
\paragraph{Input obstruction and the family $\mathcal S$.} The input of Algorithm~\ref{algo:shaf} requires $2\cdot 6^{\mathcal As{S}}$ distinct Mordell--Weil bases. In fact, for large $\mathcal As{S}$, it usually happens that one can not determine unconditionally all bases and then our Algorithm~\ref{algo:shaf} can not be used to compute $M(S)$. However, for small $\mathcal As{S}$ it turned out in practice that one can often efficiently compute the required bases by using the known techniques in (PSM). For example, without introducing crucial new ideas, we computed the required bases for each set $S$ in $\mathcal S$. Here $\mathcal S$ is a family of sets which contains in particular the set $S(5)$ and all sets $S$ with $N_S\leq 10^3$. We observe that any elliptic curve over $\QQ$ with good reduction outside $S$ has conductor dividing $N_S^{\textnormal{cond}}=\prod_{p\in S} p^{f_p}$, where $(f_2,f_3)=(8,5)$ and $f_p=2$ if $p\geq 5$. It holds that $N_S^2$ divides $N_S^{\textnormal{\textnormal{cond}}}$, and thus $\mathcal S$ contains in particular all sets $S$ with $N_S^{\textnormal{\textnormal{cond}}}\leq 10^6$.
\paragraph{Applications.} On using Algorithm~\ref{algo:shaf}, we determined the sets $M(S)$ for all $S\in \mathcal S$. This took less than $2.5$ hours for $S=S(5)$, and on average it took approximately 30 seconds for sets $S\in\mathcal S$ with $\mathcal As{S}=2$, roughly 2.5 minutes for sets $S\in\mathcal S$ with $\mathcal As{S}=3$ and approximately 8 minutes for sets $S\in\mathcal S$ with $\mathcal As{S}=4$. Here we did not take into account the time required to determine the Mordell--Weil bases for the input. In fact if the bases for the input are not already known, then their computation is usually the bottleneck of our approach to determine $M(S)$ via Algorithm~\ref{algo:shaf}.
Let $T$ be a nonempty open subscheme of $\textnormal{Spec}(\ZZ)$. Inspired by our Conjecture~1 on $T$-points of $\mathbb P^1_\ZZ-\{0,1,\infty\}$, we propose the following analogous conjecture on $T$-points of the moduli stack $\mathcal M_{1,1}$ of elliptic curves.
\noindent{\bf Conjecture 1 for $\mathcal M_{1,1}$.}
\emph{Does there exist $c\in\ZZ$ with the following property: If $n\in\ZZ_{\geq 1}$ then any set of rational primes $S$ with $\mathcal As{S}\leq n$ satisfies $\mathcal As{M(S)}\leq \mathcal As{M(S(n))}+c?$}
\noindent Our database listing the sets $M(S)$ for all $S\in\mathcal S$ shows the following: For any $n\in\ZZ_{\geq 1}$ and for each $S\in\mathcal S$ with $\mathcal As{S}\leq n$, it holds that $\mathcal As{M(S)}$ is at most $\mathcal As{M(S(n))}$. In light of this we ask whether the above conjecture is true with the optimal constant $c=0$?
We point out that for certain sets $S\in\mathcal S$ one can compute the spaces $M(S)$ by using different methods. For example Cremona--Lingham~\cite{crli:shafarevich} determined $M(S)$ for $S=\{2,p\}$ with $p\leq 23$, and Koutsianas~\cite{koutsianas:shafarevich} moreover computed $M(S)$ for $S=\{2,3,23\}$ and $S=\{2,p\}$ with $p\leq 127$. Further Cremona's database \cite{cremona:algorithms} allows to directly determine the space $M(S)$ for all sets $S$ with $N_S^{\textnormal{cond}}\leq 380000$ (as of February 2016). We also mention that the case $\mathcal As{S}=1$ was studied by Edixhoven--Groot--Top in \cite{edgrto:primecond}. In particular they showed that $M(\{p\})$ is empty for many rational primes $p$, see \cite[Cor 1]{edgrto:primecond} which explicitly lists such primes $p$. Furthermore very recently Bennett--Rechnitzer~\cite{bere:compelloneprime} determined $M(\{p\})$ for all primes $p< 2\cdot 10^9$, and for all $p< 10^{12}$ conditional on an explicit version of Hall's conjecture with ``Hall ratio" $10^{14}$. Here to prove their unconditional results, they exploit that $\mathcal As{S}=1$ in order to reduce to Thue equations which can be solved much more efficiently than Thue--Mahler equations. Their ingenious reduction uses in particular a result of Mestre--Oesterl\'e~\cite[Thm 2]{meoe:weilcurvediscriminants} which in turn relies (inter alia) on the geometric version of the Shimura--Taniyama conjecture.
\paragraph{Comparison.} We now briefly discuss advantages and disadvantages of the different methods which allow to compute $M(S)$ in practice. Our Algorithm~\ref{algo:shaf} significantly improves the method (CL) of Cremona--Lingham~\cite{crli:shafarevich}. Indeed our Algorithm~\ref{algo:mheight} is considerably more efficient in solving \eqref{eq:mordell} than the algorithm \cite{pezigehe:sintegralpoints} used in (CL). To illustrate that our improvements are significant, we used (CL) to determine $M(S)$ for $S=S(3)$. This took more than 35 minutes\footnote{This is a lower bound for the time required for (CL) to solve the involved equations \eqref{eq:mordell}. Here we used the official Sage implementation of \cite{pezigehe:sintegralpoints} which works with an ``absolute" reduction process. This means that the running times of (CL) are in fact larger than the numbers we listed.}, while it took Algorithm~\ref{algo:shaf} less than 2 minutes. Furthermore there are several sets $S\in\mathcal S$ which seem to be out of reach for (CL). For example in the case $S=S(4)$ it took Algorithm~\ref{algo:shaf} less than 20 minutes to compute $M(S)$, while (CL) did not terminate within 2 months. In comparison with the other practical methods which allow to compute $M(S)$, the main disadvantage of our approach and of (CL) is that they both require $2\cdot 6^{\mathcal As{S}}$ distinct Mordell--Weil bases. The modular symbols method (see Cremona \cite{cremona:algorithms}) can efficiently compute the curves in $M(S)$ with small conductor, while the curves of large conductor cause memory problems. We note that even for relatively small sets $S$ the maximal conductor $N_S^{\textnormal{cond}}$ can be large. For example if $S$ contains $\{2,3,p\}$ for some $p\geq 13$ then it holds that $N_S^{\textnormal{cond}}\geq 10^8$ and thus the practical computation of $M(S)$ seems to be out of reach for the modular symbol method. On the other hand, the modular symbol method deals much more efficiently with the important related problem of compiling a database which lists all elliptic curves over $\QQ$ of given conductor $N\leq 380000$. The efficiency of the approach of Koutsianas~\cite{koutsianas:shafarevich} strongly depends on the size of $\mathcal As{S}$ and on the involved number fields (quadratic, cubic, or $S_3$-extension) in which one has to solve the unit equations. We point out that (CL) and the method of Koutsianas \cite{koutsianas:shafarevich} both allow to deal with more general number fields $K$, while our approach currently only works in the considerably simpler case $K=\QQ$. As already mentioned, Bennett--Rechnitzer~\cite{bere:compell,bere:compelloneprime} substantially refined the classical Thue--Mahler approach in order to compute $M(\{p\})$ for all primes $p<2\cdot 10^9$. This computation is unfavorable for our method, since finding unconditionally all the required Mordell--Weil bases would (when possible) take a long time with the known techniques. In general, \cite{bere:compelloneprime} crucially depends on the algorithms \cite{tzde:thuemahler,deweger:sumsofunits} for which we are not aware of a complexity analysis. In particular, if $\mathcal As{S}\neq 1$ then it is not clear to us how efficient is the Thue--Mahler approach of \cite{bere:compell,bere:compelloneprime}.
To compare some data, we computed the space $M(S)$ for all sets $S$ considered in the papers of Cremona--Lingham~\cite{crli:shafarevich} and Koutsianas~\cite{koutsianas:shafarevich} and for all sets $S$ which can be covered by Cremona's database \cite{cremona:algorithms} (as of February 2016). In all cases it turned out that our Algorithm~\ref{algo:shaf} produced exactly the same number of curves.
\subsubsection{Integral points on moduli schemes}\label{sec:moduli}
We continue our notation. Many Diophantine equations can be reduced via the moduli formalism to the study of $M(S)$. To explain this more precisely, we use the notation and terminology of \cite[Sect 3]{rvk:modular}. The set $M(S)$ identifies with the set $M(T)$ of isomorphism classes of elliptic curves over the open subscheme $T$ of $\textnormal{Spec}(\ZZ)$ given by $T=\textnormal{Spec}(\ZZ)-S$. Let $Y$ be a $T$-scheme and suppose that $Y=M_\mathcal P$ is a moduli scheme of elliptic curves. We further assume that the Par\v{s}in{} construction $\phi:Y(T)\to M(T)$, induced by forgetting the level structure $\mathcal P$, is effective in the sense that for each $E\in M(T)$ one can determine the set $\mathcal P(E)$. Then \cite[Thm 7.1]{rvk:modular} and the discussions in \cite[Sect 3]{rvk:modular} show that one can in principle determine $Y(T)$. Furthermore, if $S\in \mathcal S$ then one can indeed determine $Y(T)$ by applying our explicit results for $M(T)=M(S)$. This strategy allows to efficiently solve various classical Diophantine problems, including the following equations.
\begin{itemize}
\item[(i)] One can directly solve the $S$-unit equation \eqref{eq:sunit} for any set $S\in\mathcal S$. Here one works with the moduli problem $\mathcal P=[Legendre]$ as in the proof of \cite[Prop 3.2]{rvk:modular}.
\item[(ii)] We can directly solve any Mordell equation \eqref{eq:mordell} defined by $(a,S)$ such that $6a$ is invertible in $\ZZ[1/N_{S'}]$ for some $S'\in \mathcal S$ with $S\subseteq S'$. Here one works with the moduli problem $\mathcal P_b=[\Delta=b]$ as in the proof of \cite[Prop 3.4]{rvk:modular}, where $1728b=-a$.
\item[(iii)] One can directly solve any cubic Thue equation~\eqref{eq:thue} defined by $(f,S,m)$ such that $6\Delta m$ is invertible in $\ZZ[1/N_{S'}]$ for some $S'\in\mathcal S$ with $S\subseteq S'$, where $\Delta$ denotes the discriminant of $f$. Here one works with the moduli problem obtained by pulling back the problem $\mathcal P_b$, with $4b=-\Delta m^2$, along the morphism $\varphi$ given in \eqref{eq:thuemap}.
\item[(iv)] We can directly solve any cubic Thue--Mahler equation~\eqref{eq:thue-mahler} defined by $(f,S,m)$ such that $6\Delta m$ is invertible in $\ZZ[1/N_{S'}]$ for some $S'\in\mathcal S$ with $S\subseteq S'$. Here we work with the moduli problem $\mathcal P$ represented by the elliptic curve $E$ over the moduli scheme $Y=\textnormal{Spec}\bigl(\mathcal O[x,y,\tfrac{1}{d}]\bigl)$ for $d=6\Delta f^2$, where $E$ is given by the closed subscheme of $\mathbb P^2_Y$ defined by $v^2w=u^3+3\mathcal Huw^2+Jw^3$ with $\mathcal H$ and $J$ the covariants (Hessian and Jacobian) of the cubic form $f$ normalized as in \eqref{def:covarpol2} and \eqref{def:covarpol3}.
\item[(v)] We can directly solve any generalized Ramanujan--Nagell equation~\eqref{eq:rana} defined by $(b,c,S)$ such that $2bc$ is invertible in $\ZZ[1/N_{S'}]$ for some $S'\in\mathcal S$ with $S\subseteq S'$. Here we work with the moduli problem $\mathcal P$ represented by the elliptic curve $E$ over the moduli scheme $Y=\textnormal{Spec}\bigl(\ZZ[a_2,a_4,\tfrac{1}{\delta}]\bigl)$ for $\delta=16a_4^2(a_2^2-4a_4)$, where $E$ is given by the closed subscheme of $\mathbb P^2_Y$ defined by $v^2w=u(u^2+a_2uw+a_4w^2)$. Note that $\mathcal P$ is related to the classical moduli problem $[\Gamma_1(2)]$, see for example \cite{kama:moduli}.
\end{itemize}
In (iv) and (v), any elliptic curve $E$ over $T$ has either no or infinitely many level $\mathcal P$-structures and Tate's formulas \cite[p.181]{tate:aoe} allow to explicitly determine the set of level $\mathcal P$-structures $\mathcal P(E)$ of $E$. Here, in (iv) one can proceed similarly as in (R1) of Section~\ref{sec:talgoconst} and in (v) we exploit that one can compute the two torsion of the group $E(T)$.
In fact for each moduli problem used in (i)-(v), one can quickly compute the preimage of the involved Par\v{s}in{} construction $\phi:Y(T)\to M(T)$ and one can directly determine whether a given point in $Y(T)$ corresponds to a solution of the considered Diophantine problem.
Hence, if $M(T)$ is known for some $T$ then one can directly solve the equations in (i)-(v) defined by parameters satisfying the mentioned conditions with respect to $T$; for example the parameters need to be invertible in $\mathcal O_T(T)$. On the other hand, if $M(T)$ is not already known, then our algorithms via height bounds are more efficient than first computing $M(T)$ and afterwards the preimage of $\phi$. Here the main reason is that these algorithms only need to compute the image of $\phi$ inside $M(T)$ and this image is usually much smaller than the whole space $M(T)$. We conclude by mentioning that in (i)
we do not use modular symbols as in Algorithm~\ref{algo:sucremona} or de Weger's sieve as in Algorithm~\ref{algo:suheight}.
\subsubsection{Applications}\label{sec:malgoapplications}
In this section we present other applications of Algorithm~\ref{algo:mheight}. We first discuss parts of our database $\mathcal D_2$ containing the solutions of large classes of Mordell equations~\eqref{eq:mordell}. Then we use $\mathcal D_2$ to motivate a conjecture and two questions on the number of solutions of \eqref{eq:mordell}. Here we also construct a probabilistic model providing additional motivation.
We continue the notation introduced above. Let $Y_a(\mathcal O)$ be the set of solutions of \eqref{eq:mordell} and recall that $S(n)$ denotes the set of the first $n$ rational primes.
To determine a Mordell--Weil basis of $E_a(\QQ)$ which is required in the input of Algorithm~\ref{algo:mheight}, we used the methods discussed in Section~\ref{sec:tor+mwbasis}. Further we mention that among all sets $S$ of cardinality $n$ the set $S(n)$ is usually the most difficult case to determine $Y_a(\mathcal O)$. In particular the following running times of Algorithm~\ref{algo:mheight} would be considerably better if $S(n)$ is replaced by any set $S$ of $n$ large rational primes. The reason is that the elliptic logarithm sieve becomes considerably stronger for large primes. In fact one would already obtain significant running time improvements by removing from $S(n)$ the notoriously difficult prime $2$.
\paragraph{The case $\mathcal As{a}\leq 10^4$.}
We solved the Mordell equation \eqref{eq:mordell} for all pairs $(a,S)$ such that $S\subseteq S(300)$ and such that $a\in\ZZ$ is nonzero with $\mathcal As{a}\leq 10^4$. Here the important special case $S=\emptyset$ was already established by Gebel--Peth{\H{o}}--Zimmer~\cite{gepezi:mordell} using a different algorithm. Further we mention that for many $a\in\ZZ$ with $\mathcal As{a}\leq 10^4$ we determined $Y_a(\mathcal O)$ for sets $S$ which are considerably larger than $S(300)$. For example, in the ranges $\mathcal As{a}\leq 10$ and $\mathcal As{a}\leq 100$ we computed $Y_a(\mathcal O)$ for all $S\subseteq S(10^5)$ and all $S\subseteq S(10^3)$ respectively.
\paragraph{Huge $a$ and $S$.} In practice the most common (nontrivial) case is when the Mordell--Weil rank of $E_a(\QQ)$ is one, and in this case our algorithm allows to deal efficiently with huge parameters $a$ and $S$. To illustrate this feature, we have randomly chosen 100 distinct rank one curves $E_a$ with $\mathcal As{a}\geq 10^{10}$ and for each of these curves we then determined the sets $Y_a(\mathcal O)$ for all $S\subseteq S(10^5)$. On average it took Algorithm~\ref{algo:mheight} approximately 0.15 seconds, 6 seconds and 5 hours for $S=\emptyset$, $S=S(100)$ and $S=S(10^5)$ respectively.
\paragraph{Small rank.} The efficiency of Algorithm~\ref{algo:mheight} crucially depends on the Mordell--Weil rank $r$ of $E_a(\QQ)$. We recall that Fueter's result \eqref{eq:mtorsion} completely determines the set $Y_a(\mathcal O)$ when $r=0$. Thus we assume that $r\geq 1$ in the following discussion. In the generic case, the Mordell curve $E_a$ has small rank $r$ and then our algorithm is very fast.
(Rank $1$). As already mentioned, in this situation our algorithm can deal efficiently with huge sets $S$. In particular for each rank one curve $E_a$ with $\mathcal As{a}\leq 10^4$ we computed the set $Y_a(\mathcal O)$ for all $S\subseteq S(10^4)$. There are 9546 such rank one curves and on average it took Algorithm~\ref{algo:mheight} approximately 20 minutes to determine $Y_a(\mathcal O)$ for $S=S(10^4)$.
(Rank $2$ and $3$). These cases also appear quite often in practice. For example in the range $\mathcal As{a}\leq 10^4$ there are 3426 curves $E_a$ of rank two and 478 curves $E_a$ of rank three. For these curves, we computed the set $Y_a(\mathcal O)$ for all $S\subseteq S(300)$ and on average it took less than 5 hours and 7 hours in the case of a curve of rank two and three respectively.
\paragraph{Large rank.} The situation $r\geq 4$ is rather uncommon in practice. However the notoriously difficult case of large rank $r$ is of particular interest, since it is the most challenging for the known methods computing $Y_a(\mathcal O)$ inside the Mordell--Weil group $E_a(\QQ)$. We mention that in the present case $r\geq 4$ the following running times can be significantly improved by parallelizing the elliptic logarithm sieve which is used in Algorithm~\ref{algo:mheight}.
(Rank $4$, $5$ and $6$). We computed $Y_a(\mathcal O)$ for $18$ rank four curves with $S=S(300)$, for $12$ rank five curves with $S= S(100)$ and for $2$ rank six curves with $S=S(50)$. On average the corresponding running time was roughly $4$ days, $2$ days and $19$ hours in the case of a curve of rank four, five and six respectively. The running times considerably increased for larger $S$. For example on enlarging $S(100)$ to $S(150)$ and $S(50)$ to $S(75)$, the running time was on average $6$ days and $5$ days in the case of a curve of rank five and six respectively.
(Rank $7$ and $8$). We determined the set $Y_a(\mathcal O)$ for $2$ rank seven curves with $S=S(40)$ and for $4$ rank eight curves with $S=S(30)$. On average the corresponding running time was less than $3$ days and $5$ days in the case of a curve of rank seven and eight respectively. Here again, the running times significantly increased for larger sets $S$. For instance on enlarging $S(40)$ to $S(50)$ and $S(30)$ to $S(40)$, the running time was on average approximately $5$ days and $14$ days in the case of a curve of rank seven and eight respectively.
(Rank at least $9$). This situation is extremely rare. However there exist Mordell curves $E_a$ with $E_a(\QQ)$ of rank at least nine. Unfortunately we could not find such a curve for which we were able to determine a Mordell--Weil basis of $E_a(\QQ)$; here we usually could only prove that our candidate ``basis" generates a subgroup of $E_a(\QQ)$ which has full rank.
\paragraph{Conjecture and questions.}
We next use our database $\mathcal D_2$ to motivate various questions on the cardinality of the set $Y_a(\mathcal O)$ of solutions of \eqref{eq:mordell}. First we recall Conjecture~2 which is motivated by our data and by the construction of the elliptic logarithm sieve; see also the discussion at the end of this paragraph for additional motivation.
\noindent{\bf Conjecture~2.}
\emph{There are constants $c_a$ and $c_r$, depending only on $a$ and $r$ respectively, such that any nonempty finite set of rational primes $S$ satisfies} $$\mathcal As{Y_a(\mathcal O)}\leq c_a \mathcal As{S}^{c_r}.$$
\noindent
We now discuss the exponent $c_r$ in this conjecture. For any $b\in\ZZ_{\geq 1}$ we denote by $S[b]$ the smallest set of rational primes such that for any nonzero $P\in E_a(\QQ)$ with $\hat{h}(P)\leq b$ the corresponding solution $(x,y)$ of $y^2=x^3+a$ lies in $Y_a(S[b])$. The N\'eron--Tate height $\hat{h}$ defines a positive definite quadratic form on $E_a(\QQ)\otimes_\ZZ\RR\cong\RR^r$. Therefore we obtain that $b^{r/2}=O(\mathcal As{Y_a(S[b])})$ and we deduce that $\mathcal As{S[b]}=O(b^{r/2}\cdot b)$ since all nonzero $P\in E_a(\QQ)$ satisfy $\mathcal As{\tfrac{1}{2}h(x)-\hat{h}(P)}=O(1)$; here the $O$ constants depend only on $a$. It follows that the exponent $c_r$ has to be at least $\tfrac{r}{r+2}$ and this leads us to the following question.
\noindent{\bf Question 2.1.}
\emph{What is the optimal exponent $c_r$ in Conjecture~$2$?}
\noindent
In addition our database $\mathcal D_2$ strongly indicates that the exponent $c_r=\tfrac{r}{r+2}$ is still far from optimal for many families of sets $S$ of interest, including the family $S(n)$ with $n\in\ZZ_{\geq 1}$. More precisely, together with the bound \eqref{refquestbound}, our database $\mathcal D_2$ motivates the following question concerning the dependence on $q=\max S$.
\noindent{\bf Question 2.2.}
\emph{Are there constants $c_a$ and $c_r$, depending only on $a$ and $r$ respectively, such that any nonempty finite set of rational primes $S$ with $q=\max S$ satisfies} $$\mathcal As{Y_a(\mathcal O)}\leq c_a(\log q)^{c_r} \, \textnormal{?}$$
\noindent In the case $S=S(n)$ with $n\geq 2$, one can replace here $q$ by $n\log n$ without changing the question. However the above discussion of Conjecture~2 shows that Question~2.2 has in general a negative answer when $q$ is replaced by any power of $\max(2,\mathcal As{S})$. Further, on considering again the family $S[b]$, we see that the exponent $c_r$ of Question~2.2 has to be at least $r/2$. Now we ask whether Question~2.2 has a positive answer for the exponent
\begin{equation}\label{refquest}
c_r=r/2 \, \textnormal{?}
\end{equation}
To motivate this refined question, we may and do assume that $a\in\ZZ$. Recall that $S$ is nonempty with $q=\max S$. Mahler's result (1933) gives that $\mathcal As{Y_a(S')}$ is bounded for all sets of rational primes $S'$ with $\max S'\leq q$. Thus we may and do assume in addition that $q$ is large. Now we take a nonzero point $P\in E_a(\QQ)$ and we denote by $(x,y)$ the corresponding solution of $y^2=x^3+a$. We write $x=x_1/d^2$ and $y=y_1/d^3$ with $x_1,y_1,d\in\ZZ$ satisfying $\gcd(d,x_1y_1)=1$
and $d>0$. Further we define $\rho(0)=0$ and $\rho(P)=\tfrac{n(P)}{d(P)}$, where $d(P)=d$ and $n(P)$ is the number of positive integers $n\in \mathcal O^\times$ with $n\leq d(P)$. It holds that $d(P)\in\mathcal O^\times$ if and only if $(x,y)$ lies in $Y_a(\mathcal O)$. In light of this we would like to interpret $\rho(P)$ as the probability of the event that $P\in E_a(\QQ)$ corresponds to some $(x,y)\in Y_a(\mathcal O)$. More precisely, putting $\mu_P(\{1\})=\rho(P)$ defines a probability measure $\mu_P$ on the space $\Omega_P=(\{0,1\},\mathcal P)$ for $\mathcal P$ the power set of $\{0,1\}$. Consider the associated product probability space $\Omega=(\prod\Omega_P,\prod \mu_P)$ with the product taken over all nonzero points $P\in E_a(\QQ)$.
It follows that the random variable $\mathcal As{\tilde{Y}_a(S)}=\sum \omega_P$ on $\Omega$ has expected value
$$\mathbb E\bigl(\mathcal As{\tilde{Y}_a(S)}\bigl)=\sum_{P\in E_a(\QQ)} \rho(P)$$
where $\omega_P:\Omega\to \Omega_P$ denotes the coordinate function.
We next estimate this expected value. For each $n\in\ZZ_{\geq 1}$ we denote by $\Psi(n,q)$ the de Bruijn function, that is the number of $q$-smooth numbers which are at most $n$. We observe that $\rho(P)\leq \Psi(d(P),q)/d(P)$ and de Bruijn (1951) gives absolute constants $c_1,c_2\in\RR_{>0}$ such that $\tfrac{1}{n}\Psi(n,q)\leq c_1 n^{-c_2/\log q}$.
Further, for each $\varepsilon>0$ a classical Diophantine approximation result of Siegel (1929) implies that $\hat{h}(P)\leq (1+\varepsilon)d(P)+c_3$ with a constant $c_3$ depending only on $a$ and $\varepsilon$. We also recall that $E_a(\QQ)_{\textnormal{tor}}$ has bounded cardinality and that $\hat{h}$ defines a positive definite quadratic form on $E_a(\QQ)\otimes_\ZZ \RR\cong \RR^r$. Therefore, on combining the above observations, we see that elementary analysis gives a constant $c_a$ depending only on $a$ such that
\begin{equation}\label{refquestbound}
\mathbb E\bigl(\mathcal As{\tilde{Y}_a(S)}\bigl)\leq c_a(\log q)^{r/2}.
\end{equation}
This motivates Question~2.2 and its refinement in \eqref{refquest}. Moreover, the above arguments allow to describe explicitly the constant $c_a$ of \eqref{refquestbound} in terms of $r$, $a$, the regulator of $E_a(\QQ)$, the cardinality of $E_a(\QQ)_{\textnormal{tor}}$ and a constant given by an effectively computable integral involving the Dickman function. To control here the constant $c_3$ in terms of $a$, one can use Baker's explicit abc-conjecture stated in Section~\ref{sec:suapplications}. We point out that all constructions of this paragraph do not use that $E_a$ is a Mordell curve. In fact they can be directly applied to motivate the corresponding conjecture and questions for any hyperbolic genus one curve over $\textnormal{Spec}(\ZZ)-S$. We refer to Section~\ref{sec:elllogsieveapp} for details.
\subsection{Comparison of algorithms}\label{sec:malgocomparison} In this section we discuss advantages and disadvantages of Algorithms~\ref{algo:mcremona} and \ref{algo:mheight}. We also compare our approach to the actual best methods solving \eqref{eq:mordell}.
\paragraph{Advantages and disadvantages.} Our Algorithm~\ref{algo:mcremona} via modular symbols is very fast for all parameters $S$ and~$a$ which are small enough such that the image of the Par\v{s}in{} construction $\phi$ is contained (see Sections~\ref{sec:ia} and \ref{sec:mcremalgo}) in a database listing all elliptic curves over $\QQ$ of given conductor. Unfortunately this image is usually not contained in the actual largest known database (due to Cremona) and then the computation of the required elliptic curves via modular symbols is not efficient; here the main problem is the memory. Thus in most cases Algorithm~\ref{algo:mcremona} can presently not compete with other approaches.
In the generic case, our Algorithm~\ref{algo:mheight} considerably improves the actual best methods resolving \eqref{eq:mordell}. In particular it is significantly faster than the known algorithms using the elliptic logarithm approach. Indeed our optimized height bounds are sharper in practice and our elliptic logarithm sieve substantially improves in all aspects the known enumerations. Furthermore, an important feature of Algorithm~\ref{algo:mheight} is that it allows to efficiently solve \eqref{eq:mordell} for large sets $S$. This seems to be out of reach for approaches via logarithmic forms which usually reduce to Thue(--Mahler) equations or to $S$-unit equations over number fields. Here we point out that in the important special case $S=\emptyset$ and varying $a\in\ZZ-0$ with $\mathcal As{a}\leq A$ for some given $A\in \ZZ_{\geq 1}$, the classical Baker--Davenport approach via logarithmic forms is very efficient. As already mentioned, Bennett--Ghadermarzi~\cite{begh:mordell} refined this approach and computed the solutions of \eqref{eq:mordell} in $\ZZ\times\ZZ$ for all nonzero $a\in\ZZ$ with $\mathcal As{a}\leq 10^7$. This computation involves many distinct parameters $a$, which is unfavorable for our approach since finding unconditionally all the required Mordell--Weil bases would (when possible) take a long time with the known techniques. In particular this highlights the disadvantage of Algorithm~\ref{algo:mheight} which is its dependence on an explicitly given Mordell-Weil basis. On the other hand, for $a\in\ZZ-0$ fixed one can usually determine a basis in practice and then Algorithm~\ref{algo:mheight} is very fast even when $\mathcal As{a}$ is huge; see Section~\ref{sec:malgoapplications}.
\paragraph{Comparison of data.} Some parts of our database $\mathcal D_2$ containing the solutions of large classes of Mordell equations \eqref{eq:mordell} were already computed by other authors using different methods; see the work of Gebel--Peth{\H{o}}--Zimmer~\cite{gepezi:bordeaux,gepezi:barcelona,gepezi:mordell} and Bennett--Ghadermarzi~\cite{begh:mordell}. On comparing the data in the overlapping cases, it turned out that our Algorithm~\ref{algo:mheight} never produced less solutions. In particular, for all parameters in the class $\{\mathcal As{a}\leq 10^4, S=\emptyset\}$ one verifies that our data coincides with the corresponding results in the database obtained by Bennett--Ghadermarzi~\cite{begh:mordell}.
\section{Algorithms for Thue and Thue--Mahler equations}\label{sec:thuealgo}
In \cite[Sect 7.4]{rvk:modular} an effective finiteness proof (see Section~\ref{sec:thueproofs}) for arbitrary cubic Thue equations was obtained by using inter alia an explicit reduction to a specific Mordell equation. In the present section we combine the same strategy with our algorithms for Mordell equations in order to solve cubic Thue and Thue--Mahler equations.
We continue the notation introduced in the previous sections. In particular we denote by $S$ an arbitrary finite set of rational prime numbers and we write $\mathcal O=\ZZ[1/N_S]$ for $N_S=\prod_{p\in S} p$. Let $f\in\mathcal O[x,y]$ be a homogeneous polynomial of degree 3 with nonzero discriminant and let $m\in\mathcal O$ be nonzero. We recall the cubic Thue equation
\begin{equation}
f(x,y)=m, \ \ \ (x,y)\in\mathcal O\times\mathcal O. \tag{\ref{eq:thue}}
\end{equation}
In theory, the problem of solving cubic Thue equations \eqref{eq:thue} is equivalent to the problem of finding all primitive solutions of general cubic Thue--Mahler equations~\eqref{eq:thue-mahler}.
\begin{definition}\label{def:primsoltm}
We say that $(x,y,z)$ is a primitive solution of the general cubic Thue--Mahler equation~\eqref{eq:thue-mahler} if $x,y,z\in\ZZ$ satisfy the equation $f(x,y)=mz$ with $z\in\mathcal O^\times$ and if $\pm 1$ are the only $d\in\ZZ$ with the property that $d\mid\gcd(x,y)$ and $d^3\mid z$.
\end{definition}
If $(x,y,z)$ is a solution of the cubic Thue--Mahler equation \eqref{eq:thue-mahler} discussed in the introduction, then $(x,y,z)$ is in particular a primitive solution in the sense of Definition~\ref{def:primsoltm}. In fact one can directly write down all solutions of the Thue--Mahler equation \eqref{eq:thue-mahler} if one knows all primitive solutions of the general cubic Thue--Mahler equation~\eqref{eq:thue-mahler}.
\paragraph{Known methods.} Baker--Davenport~\cite{baker:contributions,bada:diophapp} obtained a practical approach (see e.g. Ellison et al \cite{ellison:mordell}) to solve the cubic Thue equation \eqref{eq:thue} in $\ZZ\times\ZZ$. See also the variation of Peth{\H{o}}--Schulenberg~\cite{pesc:thue} which uses in addition the $L^3$ algorithm. Moreover, Tzanakis--de Weger~\cite{tzde:thue,tzde:thuemahler} and Bilu--Hanrot~\cite{biha:thuehighdeg,biha:thuecomposite} constructed practical algorithms solving Thue and Thue--Mahler equations of arbitrary degree by applying the theory of logarithmic forms~\cite{bawu:logarithmicforms}. We further remark that the classical $p$-adic method of Skolem often allows to find all solutions of the cubic Thue equation \eqref{eq:thue} in $\ZZ\times\ZZ$. In fact several authors used this method to practically resolve specific Thue equations. See for instance Stroeker--Tzanakis~\cite{sttz:skolem} and the references therein. There is also a recent algorithm for \eqref{eq:thue-mahler} due to Kim~\cite{kimd:modularthuemahler}, which we shall discuss in Section~\ref{sec:talgocompa}.
\subsection{Preliminary constructions}\label{sec:talgoconst}
In this section we discuss various constructions which shall be used in our algorithms for Thue and Thue--Mahler equations. We continue the notation introduced above.
\paragraph{Invariant theory.} To reduce our given cubic Thue equation \eqref{eq:thue} to some specific Mordell equation~\eqref{eq:mordell}, we use classical invariant theory for cubic binary forms going back at least to Cayley. We write $\Delta$ for the discriminant of $f$ and we denote by $\mathcal H$ and $J$ the covariant polynomials of $f$ of degree two and three respectively; see Section~\ref{sec:ans+covariants} for the definitions and for our normalizations. Classical invariant theory gives that $u=-4\mathcal H$ and $v=4J$ satisfy the relation
$v^2=u^3+432\Delta f^2$ in $\mathcal O[x,y]$. This induces a morphism
\begin{equation}\label{eq:thuemap}
\varphi: X\to Y
\end{equation}
of $\mathcal O$-schemes, where $X$ and $Y$ are the closed subschemes of $\mathbb A^2_\mathcal O$ associated to the Thue equation \eqref{eq:thue} and to the Mordell equation \eqref{eq:mordell} with $a=432\Delta m^2$ respectively. The solution sets of \eqref{eq:thue} and \eqref{eq:mordell} identify with the sets of sections $X(\mathcal O)$ and $Y(\mathcal O)$ of the $\mathcal O$-schemes $X$ and $Y$ respectively. Further, the projective closure inside $\mathbb P^2_\QQ$ of the generic fiber of $Y$ coincides with the elliptic curve $E_a$ over $\QQ$ appearing in previous sections.
\paragraph{The preimage of $\varphi$.} The above morphism $\varphi:X\to Y$ is effective in the following sense:
For any given $Q\in Y(\bar{\QQ})$, one can determine all $P\in X(\bar{\QQ})$ with $\varphi(P)=Q$. Indeed this follows for example directly from the explicit height inequality in Proposition~\ref{prop:heightineq}.
Alternatively, for any given point $Q\in Y(\QQ)$ one can efficiently determine all $P\in X(\QQ)$ with $\varphi(P)=Q$ by using triangular decomposition. In particular if we are given all points in $Y(\mathcal O)$, then we can efficiently reconstruct the set $X(\mathcal O)$ as follows:
\begin{itemize}
\item[(R1)]
For any given $Q\in Y(\mathcal O)$ do the following: First determine the set $Z(\QQ)$ by applying a function in Sage (Singular) based on triangular decomposition, where $Z$ is the spectrum of $\QQ[x,y]/I$ for $I=\bigl(4\mathcal H+u,v-4J\bigl)$ with $(u,v)$ the solution of \eqref{eq:mordell} corresponding to $Q$. Then output the points of $Z(\QQ)$ which are in $X(\mathcal O)$.
\end{itemize}
Here one can apply triangular decomposition with the affine scheme $Z$, since it has dimension zero. Indeed it turns out (see Section~\ref{sec:thueproofs}) that $\varphi$ induces a finite morphism $\bar{X}\to E_a$ of degree 3, where $\bar{X}$ is the projective closure inside $\mathbb P^2_\QQ$ of the generic fiber of $X$.
\paragraph{Reduction to Thue equations.} To find all primitive solutions of the general cubic Thue--Mahler equation \eqref{eq:thue-mahler}, it suffices to solve certain cubic Thue equations \eqref{eq:thue}. We now consider an elementary standard reduction: For any $w\in \ZZ_{\geq 1}$ dividing $N_S^2$, we denote by $X_w$ the closed subscheme of $\mathbb A^2_\mathcal O$ given by $f=mw$. Suppose that $(x,y,z)$ is a primitive solution of the general cubic Thue--Mahler equation~\eqref{eq:thue-mahler}. On using that the integer $z$ lies in $\mathcal O^\times$, we may and do write $z=w\varepsilonilon^3$ with an integer $\varepsilonilon\in \mathcal O^\times$ and $w\in\ZZ_{\geq 1}$ dividing $N_S^2$. Then $u=x/\varepsilonilon$ and $v=y/\varepsilonilon$ are elements in $\mathcal O$ which satisfy the Thue equation $f(u,v)=mw$. In other words $(u,v)$ lies in $X_w(\mathcal O)$. This motivates to consider the following reconstruction:
\begin{itemize}
\item[(R2)] For each $w\in\ZZ_{\geq 1}$ dividing $N_S^2$ and for any point $(u,v)$ in $X_w(\mathcal O)$, define $x=lu$, $y=lv$ and $z=l^3w$ for $l\in\ZZ_{\geq 1}$ the least common multiple of the denominators of $u$ and $v$ and output the two primitive solutions $\pm(x,y,z)$.
\end{itemize}
Here one verifies that $\pm(x,y,z)$ are indeed primitive solutions by using that the integer $w\mid N_S^2$ is cube free.
Suppose now that we are given the sets $X_w(\mathcal O)$ for all $w\in \ZZ_{\geq 1}$ dividing $N_S^2$. Then an application of (R2) produces all primitive solutions of the general cubic Thue--Mahler equation~\eqref{eq:thue-mahler}. To prove this statement, we assume that $(x,y,z)$ is such a primitive solution. Then the construction described above (R2) gives $w\in\ZZ_{\geq 1}$ dividing $N_S^2$ and $(u,v)\in X_w(\mathcal O)$. If $x',y',z'$ are the integers in (R2) associated to $w$ and $(u,v)$, then there exists $\delta\in\mathcal O^\times$ such that $(x,y,z)=(\delta x',\delta y',\delta^3z')$. We deduce that $\delta=\pm 1$, since the triples are primitive. Hence (R2) produces all primitive solutions as desired.
\subsection{Algorithms via modular symbols}\label{sec:talgocremona}
We continue the above notation. Further we denote by $\mathcal I(S,f,m)$ the data consisting of a finite set of rational primes $S$, the coefficients of a homogeneous polynomial $f\in\mathcal O[x,y]$ of degree three with nonzero discriminant $\Delta$ and a nonzero number $m\in\mathcal O$.
\begin{Algorithm}[Thue equation via modular symbols]\label{algo:tcremona} The input is the data $\mathcal I(S,f,m)$ and the output is the set of solutions $(x,y)$ of the Thue equation \eqref{eq:thue}.
The algorithm: First use Algorithm~\ref{algo:mcremona} in order to compute the set $Y(\mathcal O)$ and then apply the reconstruction algorithm described in \textnormal{(R1)}.
\end{Algorithm}
\begin{Algorithm}[Thue--Mahler equation via modular symbols]\label{algo:tmcremona} The input consists of the data $\mathcal I(S,f,m)$ and the output is the set formed by the primitive solutions $(x,y,z)$ of the general cubic Thue--Mahler equation \eqref{eq:thue-mahler}.
The algorithm: First use Algorithm~\ref{algo:tcremona} in order to determine the sets $X_w(\mathcal O)$ for all $w\in\ZZ_{\geq 1}$ dividing $N_S^2$ and then apply the reconstruction described in \textnormal{(R2)}.
\end{Algorithm}
\paragraph{Correctness.} The discussions surrounding the reconstruction (R1) imply that Algorithm~\ref{algo:tcremona} finds all solutions of the cubic Thue equation \eqref{eq:thue} as desired. Furthermore, in view of the arguments given below the reconstruction (R2), we see that Algorithm~\ref{algo:tmcremona} indeed produces all primitive solutions of the general cubic Thue--Mahler equation \eqref{eq:thue-mahler}.
\paragraph{Complexity.} The set $Y(\mathcal O)$ appearing in Algorithm~\ref{algo:tcremona} contains very few elements in practice and then the reconstruction (R1) is always very efficient. In fact the bottleneck of Algorithm~\ref{algo:tcremona} is usually the application of Algorithm~\ref{algo:mcremona} whose complexity is discussed in Section~\ref{sec:mcremalgo}. We further mention that the running time of Algorithm~\ref{algo:tmcremona} is essentially determined by the computation of the sets $X_w(\mathcal O)$ for all $w\in\ZZ_{\geq 1}$ dividing $N_S^2$.
\paragraph{Applications.} To discuss practical applications, we define $a=432\Delta m^2$ and we let $a_S$ be as in \eqref{def:as}. In the case $a_S\leq 350 000$, Algorithm~\ref{algo:tcremona} efficiently solves the cubic Thue equation \eqref{eq:thue} and Algorithm~\ref{algo:tmcremona} quickly finds all primitive solutions of the general cubic Thue--Mahler equation \eqref{eq:thue-mahler}. Indeed in this case the applications of Algorithm~\ref{algo:mcremona} are very efficient, since the involved elliptic curves are given in Cremona's database (see Section~\ref{sec:mcremalgo}). On the other hand, if the required data of the involved elliptic curves is not already known, then our Algorithms~\ref{algo:tcremona} and \ref{algo:tmcremona} are often not practical anymore. Here the problem is Cremona's algorithm involving modular symbols, which is used in Algorithm~\ref{algo:mcremona} and which requires a huge amount of memory for large parameters.
\subsection{Algorithms via height bounds}\label{sec:talgoheight}
We continue the above notation. In view of the discussions at the beginning of Section~\ref{sec:mordellalgostat}, we included the required Mordell--Weil bases in the input of the following algorithms. We refer to Section~\ref{sec:tor+mwbasis} for methods computing such a basis in practice.
\begin{Algorithm}[Thue equation via height bounds]\label{algo:theight} The input is the data $\mathcal I(S,f,m)$ together with the coordinates of a Mordell--Weil basis of $E_a(\QQ)$ for $a=432\Delta m^2$. The output is the set of solutions $(x,y)$ of the cubic Thue equation \eqref{eq:thue}.
The algorithm: First use Algorithm~\ref{algo:mheight} in order to compute the set $Y(\mathcal O)$ and then apply the reconstruction algorithm described in \textnormal{(R1)}.
\end{Algorithm}
\begin{Algorithm}[Thue--Mahler equation via height bounds]\label{algo:tmheight} The input consists of the data $\mathcal I(S,f,m)$ together with the coordinates of a Mordell--Weil basis of $E_a(\QQ)$ for all parameters $a=432\Delta (m w)^2$ with $w\in\ZZ_{\geq 1}$ dividing $N_S^2$. The output is the set of primitive solutions $(x,y,z)$ of the general cubic Thue--Mahler equation \eqref{eq:thue-mahler}.
The algorithm: First use Algorithm~\ref{algo:theight} in order to determine the sets $X_w(\mathcal O)$ for all $w\in\ZZ_{\geq 1}$ dividing $N_S^2$ and then apply the reconstruction described in \textnormal{(R2)}.
\end{Algorithm}
\paragraph{Correctness.} On using the arguments appearing in the correctness proof of Algorithms~\ref{algo:tcremona} and \ref{algo:tmcremona}, we see that Algorithms~\ref{algo:theight} and \ref{algo:tmheight} work correctly.
\paragraph{Complexity.} We first discuss aspects influencing the running time of Algorithm~\ref{algo:theight} in practice. In this algorithm the reconstruction \textnormal{(R1)} is always very fast, while the running time of the computation of $Y(\mathcal O)$ is determined by the efficiency of the application of Algorithm~\ref{algo:mheight} with $a=432\Delta m^2$. Here the efficiency crucially depends on $\mathcal As{S}$ and on the size of the Mordell--Weil rank of $E_a(\QQ)$, see the complexity discussions in Section~\ref{sec:mordellalgostat}. The computation of $Y(\mathcal O)$ is usually the bottleneck of Algorithm~\ref{algo:theight}. We next discuss Algorithm~\ref{algo:tmheight}. The running time of this algorithm is essentially determined by the computation of the sets $X_w(\mathcal O)$ for all $w\in\ZZ_{\geq 1}$ dividing $N_S^2$. For this computation we need to apply Algorithm~\ref{algo:theight} with $3^{\mathcal As{S}}$ distinct inputs $\mathcal I(S,f,m')$, where $m'$ is of the form $m'=mw$ with $w\in\ZZ_{\geq 1}$ dividing $N_S^2$. In particular the running time of Algorithm~\ref{algo:tmheight} crucially depends on $\mathcal As{S}$ and on the aspects influencing the complexity of Algorithm~\ref{algo:theight} as discussed above.
\paragraph{Input obstruction.}
The inputs of the above algorithms require a Mordell--Weil basis of $E_a(\QQ)$ for certain parameters $a$. In the case of Algorithm~\ref{algo:theight}, one needs to determine such a basis for only one parameter $a$ and this is usually possible in practice (see Section~\ref{sec:ttmapp}) by using the known techniques implemented in (PSM). On the other hand, the input of Algorithm~\ref{algo:tmheight} requires a Mordell--Weil basis of $E_a(\QQ)$ for $3^{\mathcal As{S}}$ distinct parameters $a$. Here, for large $\mathcal As{S}$, it often happens in practice that one can not determine unconditionally all required bases in an efficient way and then our Algorithm~\ref{algo:tmheight} can not be applied to find all primitive solutions of the general cubic Thue--Mahler equation~\eqref{eq:thue-mahler}. However for small $\mathcal As{S}$ it turned out in practice that the known techniques are usually efficient enough to determine unconditionally the required bases, see Section~\ref{sec:ttmapp}.
\subsubsection{Cubic forms of given discriminant}\label{sec:redcubicforms}
There are infinitely many cubic Thue and Thue--Mahler equations of some given nonzero discriminant. However, to solve all these equations, it essentially suffices to consider the equations up to the equivalence relation induced by the action of $\textnormal{GL}_2(\ZZ)$. In this section we discuss certain aspects of this equivalence relation and we explain how to efficiently determine an explicit equation in each equivalence class. We continue our notation.
\paragraph{Equivalence classes.} We say that a polynomial in $\QQ[x,y]$ is a cubic form if it is homogeneous of degree three with nonzero discriminant. The group $G=\textnormal{GL}_2(\ZZ)$ acts on the set of cubic forms in the usual way. If $f,f'\in\mathcal O[x,y]$ are cubic forms with $f'=g\cdot f$ for some $g\in G$, then their discriminants coincide and there is an explicit isomorphism between $X(\mathcal O)$ and $X'(\mathcal O)$ induced by $g$;
here $X$ and $X'$ are the closed subschemes of $\mathbb A^2_\mathcal O$ given by $f-m$ and $f'-m$ respectively. To determine the set of solutions $X(\mathcal O)$ of the cubic Thue equation~\eqref{eq:thue}, it now suffices to know $g$ together with the set $X'(\mathcal O)$. Similarly if one is given $g\in G$ together with the set of primitive solutions of the general cubic Thue--Mahler equation~\eqref{eq:thue-mahler} defined by $(f',S,m)$ with $f'=g\cdot f$, then one can directly write down all primitive solutions of the general cubic Thue--Mahler equation~\eqref{eq:thue-mahler} defined by $(f,S,m)$.
\paragraph{Reduced cubic forms.} The reduction theory of binary forms over $\ZZ$ is well-developed. See for example the recent book of Evertse--Gy{\H{o}}ry~\cite[Sect 13.1]{evgy:bookdiscreq}. Let $f\in \ZZ[x,y]$ be a cubic form. We next discuss how to obtain a cubic form in $G\cdot f$ which is reduced in some sense. The notion of a reduced cubic form varies a lot in the literature and therefore we now explain in detail the notion which we shall use in this paper.
We first consider the case when $f$ is irreducible in $\QQ[x,y]$. In this case, Belabas showed in \cite[Cor 3.3 and Lem 4.3]{belabas:cubicfields} that the orbit $G\cdot f$ contains a unique cubic form $f'\in\ZZ[x,y]$ which is reduced in the sense of \cite[Def 3.2 and 4.1]{belabas:cubicfields}; this notion of a reduced form is inspired by the work of Hermite (1848/1859) if the roots of $f(x,1)$ are all real and of Mathews (1912) otherwise. Furthermore the arguments given in Belabas~\cite[Sect 3 and 4]{belabas:cubicfields} can be transformed into a simple algorithm which allows to efficiently determine the reduced form $f'$ together with $g\in G$ satisfying $f'=g\cdot f$.
Suppose now that $f$ is reducible in $\QQ[x,y]$. In this case we work with a notion of a reduced form which is very simple and which is convenient in the sense that one can trivially determine such a form in each equivalence class.
More precisely, the orbit $G\cdot f$ contains a cubic form $\sum a_i x^{3-i}y^i$ in $\ZZ[x,y]$ which is reduced in our following sense:
\begin{equation}\label{def:reducedcubicforms}
0=a_3\leq a_1\leq a_2.
\end{equation}
To prove this statement we first observe that we may assume that $f$ is primitive, that is the greatest common divisor of the coefficients $a_i$ of our cubic form $f=\sum a_i x^{3-i}y^i\in\ZZ[x,y]$ is one. Hence we assume that $f$ is primitive. We next show that one can obtain that $a_3=0$. Suppose that $a_3$ is nonzero. After possibly exchanging $x$ and $y$, we can assure that $a_0$ is nonzero. Then $f(x,1)$ is reducible in $\QQ[x]$, and thus it is reducible in $\ZZ[x]$ by Gauss' lemma and by our assumption that $f$ is primitive. Hence, on exploiting again that $f$ is primitive, we see that the extended Euclidean algorithm provides a transformation in $G$ which makes $a_3=0$ as desired. Furthermore, after possibly replacing $x$ by $-x$, we can assure that $a_2\geq 0$. It follows that $a_2\geq 1$, since the discriminant of $f$ is nonzero and since $a_3=0$. Then on using that $a_2\in\ZZ_{\geq 1}$, we find $\alpha\in\ZZ$ depending on $a_1,a_2$ such that $(x,y)\mapsto (x,y+\alpha x)$ leads to $-a_2<a_1\leq a_2$. Finally, after possibly replacing $y$ by $-y$ we can assure that $a_1\geq 0$. We conclude that the orbit $G\cdot f$ indeed contains a cubic form in $\ZZ[x,y]$ which is reduced in the sense of \eqref{def:reducedcubicforms}. Here the reduced form may not be unique in its $G$-orbit, which is no disadvantage for our purpose of solving equations.
\paragraph{Given discriminant.} For any given nonzero $\Delta\in\ZZ$, we now explain how we determine all reduced cubic forms in $\ZZ[x,y]$ of discriminant $\Delta$. First we apply the results \cite[Lem 3.5 and 4.4]{belabas:cubicfields} of Belabas in order to list all desired forms which are irreducible in $\QQ[x,y]$. Then to find the remaining cubic forms we proceed as follows: If $\sum a_ix^{3-i}y^i$ is a reduced cubic form in $\ZZ[x,y]$ which is reducible in $\QQ[x,y]$, then the property $a_3=0$ assures that $a_2^2\mid \Delta$. Hence we can directly write down all possible values for $a_2$, which together with $0\leq a_1\leq a_2$ allows to list all possible values for $a_1$. Finally we find all possible values for $a_0$ by using an explicit expression for $a_0$ in terms of $\Delta,a_1,a_2$. Here the explicit expression for $a_0$ can be obtained by inserting $a_3=0$ in the discriminant equation.
\subsubsection{Applications}\label{sec:ttmapp}
In this section we discuss applications of Algorithms~\ref{algo:theight} and \ref{algo:tmheight}. After explaining the database $\mathcal D_3$ containing the solutions of large classes of cubic Thue equations~\eqref{eq:thue}, we motivate new conjectures and questions on the number of solutions of \eqref{eq:thue}. Then we discuss the database $\mathcal D_4$ listing the primitive solutions of many general cubic Thue--Mahler equations~\eqref{eq:thue-mahler} and we consider generalized superelliptic equations studied by Darmon--Granville~\cite{dagr:superell} and Bennett--Dahmen~\cite{beda:kleinsuperell}. We continue the above notation.
\paragraph{Preliminaries.} In our databases $\mathcal D_3$ and $\mathcal D_4$ we use the set $\mathcal F_\Delta$ of reduced cubic forms in $\ZZ[x,y]$ of given nonzero discriminant $\Delta$. This is sufficient to cover the general case of an arbitrary cubic form $f\in\ZZ[x,y]$ of discriminant $\Delta$. Indeed the arguments of the previous section allow to quickly find a reduced cubic form $f'\in\ZZ[x,y]$ and $g\in \textnormal{GL}_2(\ZZ)$ with $f'=g\cdot f$, and then one can directly write down the solutions with respect to $f$ using the solutions in $\mathcal D_3$ and $\mathcal D_4$. On applying the techniques described in Section~\ref{sec:redcubicforms}, we computed in 3 seconds the sets $\mathcal F_\Delta$ for all nonzero $\Delta\in\ZZ$ with $\mathcal As{\Delta}\leq 10^4$. The database $\mathcal F$ containing the 17044 reduced forms is uploaded on our homepage: There are $2683$ distinct $\textnormal{GL}_2(\ZZ)$-orbits of cubic forms in $\ZZ[x,y]$ which are irreducible in $\QQ[x,y]$ and $\mathcal F$ lists the reduced form of each such orbit. In addition $\mathcal F$ contains the 14361 reduced cubic forms in $\ZZ[x,y]$ which are reducible in $\QQ[x,y]$. Further, we determined the Mordell--Weil bases required in the inputs of Algorithms~\ref{algo:theight} and \ref{algo:tmheight} by using the known techniques implemented in (PSM) without introducing new ideas; see also Section~\ref{sec:tor+mwbasis}.
\paragraph{Thue equation.} For any $n\in\ZZ_{\geq 1}$ we recall that $S(n)$ denotes the set of the first $n$ rational primes. Our database $\mathcal D_3$ contains in particular the solutions of the cubic Thue equation \eqref{eq:thue} for all parameter triples $(f,S,m)$ such that $f\in\mathcal F_\Delta$ with $1\leq \mathcal As{\Delta}\leq d$, $S\subseteq S(n)$ and $m\in\ZZ-0$ with $\mathcal As{m}\leq \mu$, where $(d,n,\mu)$ is as in the following discussion.
In the case $(d,n,\mu)=(10^4,300,1)$,
we could quickly compute almost all of the required Mordell--Weil bases: If the rank was not one then this took a few seconds (in rare cases a few minutes), and also for most rank one curves we could instantly determine a generator. However there were a few rank one curves of large regulator for which it took several hours to compute a generator by using methods in (PSM) (2, 4 and 8-descent, Heegner points). On average it then took Algorithm~\ref{algo:theight} approximately 5 seconds and 5 minutes in order to solve \eqref{eq:thue} for $S$ the empty set and $S=S(100)$ respectively.
In certain situations we can make $S$ huge. For example in the case $(d,n,\mu)=(100,10^3,1)$,
we could compute the required bases in less than 1 minute and on average it then took approximately 1.3 hours and 12 hours in order to solve \eqref{eq:thue} for $S=S(500)$ and $S=S(10^3)$ respectively.
Furthermore, in the case $(d,n,\mu)=(20,10^5,1)$,
we instantly found the required Mordell--Weil bases and on average we then solved \eqref{eq:thue} for $S=S(10^4)$ and $S=S(10^5)$ in less than 20 minutes and 5 hours respectively.
Finally for the classical form $f=x^3+y^3$ we solved \eqref{eq:thue} for all $(S,m)$ as follows\footnote{For varying $m\in \ZZ$ with $\mathcal As{m}$ bounded, it suffices to consider the case $m\geq 1$. Indeed the polynomial $f$ is homogeneous of odd degree and thus the equation $f(x,y)=m$ is equivalent to $f(-x,-y)=-m$.}. In the case $(S(10^5),m)$ with $m\in\ZZ_{\geq 1}$ satisfying $m\leq 15$, we computed the required bases in less than 1 second and on average it then took roughly 2 hours to solve \eqref{eq:thue}. Further in the case $(S(10^3),m)$ with $m\in\ZZ_{\geq 1}$ satisfying $m\leq 100$, we computed the required bases in less than 1 second and on average it then took approximately 6 hours to solve \eqref{eq:thue}.
\begin{remark}[Dependence on rank]
We created our database $\mathcal D_3$ with $\Delta$, $S$ and $m$ in a given range. These parameters directly influence the efficiency of the known methods solving \eqref{eq:thue}. However for our approach the crucial parameter is the involved Mordell--Weil rank $r$, which does not depend on $S$ and which is usually small even for huge $\Delta$, $m$. Hence the discussions in Section~\ref{sec:malgoapplications}, containing running times for any given $r$, might be more meaningful than the above running times. Finally we mention that in the generic situation where $r\leq 2$, our Algorithm~\ref{algo:theight} is very fast even for huge parameters $\Delta$, $S$, $m$.
\end{remark}
\paragraph{Conjectures and questions.} The morphism $\varphi:X\to Y$ in \eqref{eq:thuemap} induces a finite morphism $\bar{X}\to \bar{Y}$ of degree 3, where $\bar{X}$ and $\bar{Y}$ are the projective closures inside $\mathbb P^2_\QQ$ of the generic fibers of $X$ and $Y$ respectively. It follows that $\mathcal As{X(\mathcal O)}\leq 3\mathcal As{Y(\mathcal O)}$.
Hence on applying our conjectures and questions in Section~\ref{sec:malgoapplications} with $Y=Y_a$ for $a=432\Delta m^2$, we directly obtain the analogous conjectures and questions on upper bounds for the number of solutions of the cubic Thue equation~\eqref{eq:thue} in terms of $S$ and the Mordell--Weil rank $r$ of $\textnormal{Pic}^0(\bar{X})(\QQ)$. Our database $\mathcal D_3$ motivates these analogous conjectures and questions for cubic Thue equations~\eqref{eq:thue}. Furthermore, it might be possible to obtain more precise conjectures for \eqref{eq:thue} by analyzing in addition the fibers of $\varphi$. We leave this for the future.
\paragraph{Thue--Mahler equation.} We next discuss our database $\mathcal D_4$. In what follows, by solving \eqref{eq:thue-mahler} for $(f,S)$ we mean finding all primitive solutions of the general cubic Thue--Mahler equation~\eqref{eq:thue-mahler} defined by $f$, $S$ and $m=1$; note that any such primitive solution $(x,y,z)$ satisfies $\gcd(x,y)=1$ provided that $f\in\ZZ[x,y]$. We solved \eqref{eq:thue-mahler} for all $(f,S)$ such that $f\in\mathcal F_\Delta$ with $1\leq\mathcal As{\Delta}\leq d$ and $S\subseteq S(n)$, where $(d,n)$ is of the form $(3000,2)$, $(1000,3)$, $(100,4)$ or $(16,5)$.
Here again we could quickly determine almost all of the required Mordell--Weil bases. However for increasing $\mathcal As{S}$ and $\mathcal As{\Delta}$ there were more and more rank one curves of large regulator, and finding a generator for these rank one curves was often the bottleneck of our approach. Given the input, Algorithm~\ref{algo:tmheight} was fast in all cases.
To give the reader an idea of our running times, we now discuss some equations appearing in the literature. We solved the equation of Tzanakis--de Weger~\cite{tzde:thuemahlercomp}, and we determined all solutions of the equation of Agraval--Coates--Hunt--van der Poorten~\cite{agcohuva:conductor11}.
Here our total running times were 3 minutes and 3 seconds, which includes the 2.5 minutes and 1.5 seconds that were required to compute the involved bases. In addition we solved the equation of Tzanakis--de Weger~\cite{tzde:thuemahler} which they used to illustrate the practicality of their method. Here we determined the required bases in less than a day and then it took Algorithm~\ref{algo:tmheight} approximately 1 minute to solve the Thue--Mahler equation.
Further, we also solved \eqref{eq:thue-mahler} for all $(f,S(6))$ with $f\in\mathcal F_\Delta$ and $\Delta\in\{-9,-1,3,27\}$. Here the case $\Delta=27$ covers in particular $f=x^3+y^3$, and the case $\Delta=-1$ corresponds to the $S$-unit equation \eqref{eq:sunit} which means that our Theorem~B solves in particular \eqref{eq:thue-mahler} for all $(f,S)$ with $f\in\mathcal F_{-1}$ and with $S$ satisfying $S\subseteq S(16)$ or $N_S\leq 10^7$.
To conclude we mention that our Algorithm~\ref{algo:tmheight} can be used to study properties of certain generalized superelliptic equations. More precisely, let $f\in\ZZ[x,y]$ be a cubic form with nonzero discriminant $\Delta$ and take $l\in\ZZ$ with $l\geq 4$. Darmon--Granville~\cite{dagr:superell} deduced from the Mordell conjecture \cite{faltings:finiteness} that the generalized superelliptic equation
\begin{equation}\label{eq:superelliptic}
f(x,y)=z^l, \ \ \ (x,y,z)\in\ZZ^3
\end{equation}
with $\gcd(x,y)=1$ has at most finitely many solutions. Moreover on using inter alia modularity of certain Galois representations, level lowering, classical invariant theory and properties of elliptic curves with isomorphic mod-$n$ Galois representations, Bennett--Dahmen~\cite[Thm 1.1]{beda:kleinsuperell} proved: The equation $f(x,y)=z^l$ has only finitely many solutions $(x,y,z,l)\in\ZZ^4$ with $l\geq 4$ and $\gcd(x,y)=1$ if the following condition $(*)$ holds.
\begin{itemize}
\item[$(*)$] The polynomial $f$ is irreducible
and there are no solutions of the Thue--Mahler equation \eqref{eq:thue-mahler} defined by $f$, $S=\{p\,;\, p\mid 2\Delta\}$ and $m=1$.
\end{itemize}
Bennett--Dahmen explicitly constructed in \cite[Thm 1.2]{beda:kleinsuperell} an infinite family of polynomials satisfying condition $(*)$ and they explained in \cite[Sect 12]{beda:kleinsuperell} a heuristic indicating that ``almost all" cubic forms should satisfy $(*)$. Now, for any given cubic form $f\in\ZZ[x,y]$ with nonzero discriminant $\Delta$, our Algorithm~\ref{algo:tmheight} allows to verify in practice whether condition $(*)$ holds. In other words, one can check condition $(*)$ without using algorithms which ultimately rely on the theory of logarithmic forms. For example, we used Algorithm~\ref{algo:tmheight} to verify that $3x^3+2x^2y+5xy^2+3y^3$ satisfies condition $(*)$; note that according to \cite[p.174]{beda:kleinsuperell} this is the cubic form of minimal $\mathcal As{\Delta}$ which satisfies condition $(*)$.
\subsection{Comparison of algorithms}\label{sec:talgocompa}
In this section we compare our algorithms for cubic Thue equations~\eqref{eq:thue} and cubic Thue--Mahler equations~\eqref{eq:thue-mahler} with the actual best practical methods in the literature.
\paragraph{Advantages and disadvantages.} We begin by discussing Algorithms~\ref{algo:tcremona} and \ref{algo:tmcremona} for \eqref{eq:thue} and \eqref{eq:thue-mahler} using modular symbols (Cremona's algorithm). In the recent work \cite{kimd:modularthuemahler}, Kim independently constructed an algorithm for cubic Thue--Mahler equations \eqref{eq:thue-mahler} using modular symbols and the Shimura--Taniyama conjecture. Kim's method differs from our strategy in the sense that he is not using the route via Thue and Mordell equations, but directly associates to each solution of \eqref{eq:thue-mahler} a certain elliptic curve. It turns out that his method is more efficient in terms of $S$ and our strategy is more efficient in terms of $\Delta$. In fact both approaches are very fast for all parameters such that the involved elliptic curves are already known. However, usually these curves are not already known and computing these curves via modular symbols is currently not efficient for large parameters; here the main problem is the memory. Thus in most cases the algorithms via modular symbols can presently not compete with approaches solving \eqref{eq:thue} and \eqref{eq:thue-mahler} via height bounds.
We next compare our Algorithm~\ref{algo:theight} with the actual best methods in the literature solving cubic Thue equations~\eqref{eq:thue} using height bounds. Our algorithm requires a Mordell--Weil basis in its input. In practice this basis can usually be computed and then our approach is very efficient. In the important special case when $S$ is empty, the already mentioned method of Tzanakis--de Weger~\cite{tzde:thue} works very well in practice and it usually allows to efficiently solve \eqref{eq:thue}. Their method has the advantage of not requiring a Mordell--Weil basis in the input. On the other hand, an advantage of Algorithm~\ref{algo:theight} is that it efficiently deals with large sets $S$. For example it seems that already sets $S$ with $\mathcal As{S}\geq 10$ are out of reach for the known methods solving \eqref{eq:thue}, while in the generic case Algorithm~\ref{algo:theight} allows to efficiently solve \eqref{eq:thue} for essentially all sets $S$ with $\mathcal As{S}\leq 10^3$.
It remains to discuss our Algorithm~\ref{algo:tmheight} for cubic Thue--Mahler equations \eqref{eq:thue-mahler}. Its input requires $3^{\mathcal As{S}}$ distinct Mordell--Weil bases and thus our approach is not practical when $\mathcal As{S}$ is large. However for small sets $S$ it turned out in practice that one can usually determine the required bases and then our approach is efficient as illustrated in Section~\ref{sec:ttmapp}. If $f\in\ZZ[x,y]$ is irreducible, then the above mentioned method (TW) of Tzanakis--de Weger~\cite{tzde:thuemahler} solves in particular any cubic Thue--Mahler equation \eqref{eq:thue-mahler}. We are not aware of a complexity analysis of (TW) and thus we restrict ourselves to the following comments. There are several results in the literature which resolved specific equations \eqref{eq:thue-mahler} using (TW). As far as we know, these results all involve small sets $S$ with $\mathcal As{S}\leq 4$ and (TW) is quite practical for such small sets. On the other hand, sets $S$ of large cardinality are also problematic for (TW) since this method needs to enumerate points in lattices of rank at least $\mathcal As{S}$.
\paragraph{Comparison of data.} We are not aware of any database in the literature which contains the solutions of large classes of cubic Thue equations \eqref{eq:thue} or cubic Thue--Mahler equations \eqref{eq:thue-mahler}. To compare at least some data, we solved the equations of \cite{agcohuva:conductor11,tzde:thuemahlercomp,tzde:thuemahler} and in all cases it turned out that we found the same set of solutions.
\section{Algorithms for generalized Ramanujan--Nagell equations}\label{sec:ranaalgo}
In the present section we use our approaches for Mordell equations~\eqref{eq:mordell} in order to construct algorithms for the generalized Ramanujan--Nagell equation~\eqref{eq:rana}.
We continue the notation of the previous sections. In particular we denote by $S$ an arbitrary finite set of rational prime numbers and we let $\mathcal O^\times$ be the group of units of $\mathcal O=\ZZ[1/N_S]$ for $N_S=\prod_{p\in S} p$. Further we suppose that $b$ and $c$ are arbitrary nonzero elements of $\mathcal O$. Now we recall the generalized Ramanujan--Nagell equation
\begin{equation}
x^2+b=cy, \ \ \ \ \ (x,y)\in\mathcal O\times \mathcal O^\times. \tag{\ref{eq:rana}}
\end{equation}
We observe that this Diophantine problem is equivalent to the a priori more general Diophantine problem obtained by replacing in \eqref{eq:rana} the polynomial $x^2+b$ by any given polynomial $f\in\mathcal O[x]$ of degree two with nonzero discriminant.
\paragraph{Known methods.} As mentioned in the introduction, if $b\in\ZZ$ is nonzero and $c=1$ then Peth{\H{o}}--de Weger~\cite{pede:binaryrec1} already obtained a practical method to find all solutions $(x,y)$ of \eqref{eq:rana} with $x,y\in\ZZ_{\geq 0}$. They use inter alia the theory of logarithmic forms and binary recurrence sequences; see also de Weger~\cite{deweger:phdthesis,deweger:sumsofunits}. In addition, Kim~\cite[Sect 8]{kimd:modularthuemahler} and Bennett--Billerey~\cite[Sect 5]{bebi:sumsofunits} recently obtained other practical approaches for \eqref{eq:rana} which are briefly discussed in Sections~\ref{sec:talgocompa} and \ref{sec:ranaapp} respectively.
\subsection{Algorithm via modular symbols}\label{sec:ranaalgocremona}
We continue the notation introduced above. For any nonzero $a\in\mathcal O$, we denote by $Y_a(\mathcal O)$ the set of solutions of the Mordell equation \eqref{eq:mordell} defined by $(a,S)$. The following algorithm is a direct application of our Algorithm~\ref{algo:mcremona} for Mordell equations~\eqref{eq:mordell}.
\begin{Algorithm}[Ramanujan--Nagell equation via modular symbols]\label{algo:ranacremona} The input consists of a finite set of rational primes $S$ together with nonzero $b,c\in\mathcal O$. The output is the set of solutions $(x,y)$ of the generalized Ramanujan--Nagell equation \eqref{eq:rana}.
The algorithm: For each $\varepsilonilon\in\ZZ_{\geq 1}$ dividing $N_S^2$, use Algorithm~\ref{algo:mcremona} to determine $Y_a(\mathcal O)$ with $a=-b(\varepsilonilon c)^2$ and for any $(u,v)\in Y_a(\mathcal O)$ output $(\tfrac{v}{\varepsilonilon c},\tfrac{u^3}{\varepsilonilon^2 c^3})$ if it satisfies \eqref{eq:rana}.
\end{Algorithm}
\paragraph{Correctness.} To show that this algorithm works correctly, we take a solution $(x,y)$ of \eqref{eq:rana}. We write $y=\varepsilonilon y'^3$ with $y'\in\mathcal O^\times$ and $\varepsilonilon\in\ZZ_{\geq 1}$ dividing $N_S^2$, and we define $a=-b(c\varepsilonilon)^2$. Further we put $u=\varepsilonilon c y'$ and $v=\varepsilonilon c x$. It follows that $(u,v)$ lies in $Y_a(\mathcal O)$ and thus we see that the above Algorithm~\ref{algo:ranacremona} indeed finds all solutions of \eqref{eq:rana} as desired.
\paragraph{Complexity.} The running time of Algorithm~\ref{algo:ranacremona} is essentially determined by the applications of Algorithm~\ref{algo:mcremona} whose complexity is discussed in Section~\ref{sec:mcremalgo}.
\paragraph{Applications.} To discuss practical applications of Algorithm~\ref{algo:ranacremona}, we define $a=bc^2$ and we let $a_S$ be as in \eqref{def:as}. In the case $a_S\leq 350 000$, our Algorithm~\ref{algo:ranacremona} allows to efficiently determine all solutions of the generalized Ramanujan--Nagell equation \eqref{eq:rana}. Indeed in this case the applications of Algorithm~\ref{algo:mcremona} are very efficient, since the involved elliptic curves can be found in Cremona's database (see Section~\ref{sec:mcremalgo}). For example, one can quickly resolve the classical Ramanujan--Nagell equation: $x^2+7=2^n$ with $x,n\in\ZZ_{\geq 1}$.
Resolving this Diophantine equation is equivalent to the problem of finding all triangular Mersenne numbers, and any solution lies in the set $\{(1,3),(3,4),(5,5),(11,7),(181,15)\}$. The latter assertion was conjectured by Ramanujan (1913) and was proven by Nagell (1948). One obtains an alternative proof of Nagell's result by using Algorithm~\ref{algo:ranacremona}. To conclude the discussion we mention that our Algorithm~\ref{algo:ranacremona} is often not practical anymore if the elliptic curves induced by the solutions of \eqref{eq:rana} need to be computed via Cremona's algorithm involving modular symbols (see Algorithm~\ref{algo:mcremona}). Here the problem is that Cremona's algorithm requires a huge amount of memory for all parameters which are not small.
\subsection{Algorithm via height bounds}\label{sec:ranaalgoheight}
We continue the above notation. In the next algorithm we apply Algorithm~\ref{algo:mheight} several times. These applications require certain Mordell--Weil bases which we included in the input. See Section~\ref{sec:tor+mwbasis} for methods computing such bases in practice.
\begin{Algorithm}[Ramanujan--Nagell equation via height bounds]\label{algo:ranaheight} The inputs are nonzero $b,c\in\mathcal O$, a finite set of rational primes $S$ and the coordinates of a Mordell--Weil basis of $E_a(\QQ)$ for all parameters $a=-b(\varepsilonilon c)^2$ with $\varepsilonilon\in\ZZ_{\geq 1}$ dividing $N_S^2$. The output is the set of solutions $(x,y)$ of the generalized Ramanujan--Nagell equation \eqref{eq:rana}.
The algorithm: For each $\varepsilonilon\in\ZZ_{\geq 1}$ dividing $N_S^2$, use Algorithm~\ref{algo:mheight} to determine $Y_a(\mathcal O)$ with $a=-b(\varepsilonilon c)^2$ and for any $(u,v)\in Y_a(\mathcal O)$ output $(\tfrac{v}{\varepsilonilon c},\tfrac{u^3}{\varepsilonilon^2 c^3})$ if it satisfies \eqref{eq:rana}.
\end{Algorithm}
\paragraph{Correctness.} The arguments given in the correctness proof of Algorithm~\ref{algo:ranacremona} show that the above Algorithm~\ref{algo:ranaheight} indeed finds all solutions of \eqref{eq:rana} as desired.
\paragraph{Complexity.} To compute in Algorithm~\ref{algo:ranaheight} the sets $Y_a(\mathcal O)$, we need to apply Algorithm~\ref{algo:mheight} with $3^{\mathcal As{S}}$ distinct parameters $a$. In particular the running time of Algorithm~\ref{algo:ranaheight} crucially depends on $\mathcal As{S}$ and on the complexity of Algorithm~\ref{algo:mheight} discussed in Section~\ref{sec:mordellalgostat}. Here we mention that the involved Mordell--Weil ranks are usually small in practice and then our Algorithm~\ref{algo:ranaheight} is very fast even for parameters $b,c$ with huge height.
\paragraph{Refinement.} The input of Algorithm~\ref{algo:ranaheight} requires $3^{\mathcal As{S}}$ distinct Mordell--Weil bases. In practice it turned out that the computation of these bases is currently the bottleneck of our approach solving \eqref{eq:rana} via height bounds. Consider arbitrary nonzero $b,c,d$ in $\ZZ$ with $d\geq 2$. We now work out a refinement of Algorithm~\ref{algo:ranaheight} which only requires three distinct Mordell--Weil bases to find all solutions of the classical Diophantine problem
\begin{equation}
x^2+b=c d^n, \ \ \ \ \ (x,n)\in\ZZ\times \ZZ \tag{\ref{eq:rana2}}.
\end{equation}
This is a special case of \eqref{eq:rana} with $y=d^n$ and $S$ given by $S_d=\{p\,;\, p\mid d\}$. In fact many authors refer by ``generalized Ramanujan--Nagell equation" to (special cases of) the Diophantine problem \eqref{eq:rana2}. We obtain the following algorithm for \eqref{eq:rana2}.
\begin{Algorithm}[Refinement]\label{algo:rana2}
The input consists of nonzero $b,c,d$ in $\ZZ$ such that $d\geq 2$ together with the coordinates of a Mordell--Weil basis of $E_a(\QQ)$ for all $a=-b(\varepsilonilon c)^2$ with $\varepsilonilon\in\{1,d,d^2\}$. The output is the set of solutions $(x,n)$ of \eqref{eq:rana2}.
The algorithm: For each $\varepsilonilon\in\{1,d,d^2\}$, use Algorithm~\ref{algo:mheight} to find $Y_a(\mathcal O)$ with $(a,S)=(-b(\varepsilonilon c)^2,S_d)$ and for any $(u,v)\in Y_a(\mathcal O)$ output $\bigl(\tfrac{v}{\varepsilonilon c},\log_d(\tfrac{u^3}{\varepsilonilon^2 c^3})\bigl)$ if it satisfies \eqref{eq:rana2}.
\end{Algorithm}
Here for any $z\in\RR$ we define $\log_d(z)=(\log z)/\log d$ if $z>0$ and $\log_d(z)=-\infty$ otherwise. To prove that the above algorithm indeed finds all solutions of \eqref{eq:rana2}, we suppose that $(x,n)$ is such a solution. We write $d^n=\varepsilonilon d^{3m}$ with $\varepsilonilon\in\{1,d,d^2\}$ and $m\in\ZZ$. Further we define $v=\varepsilonilon c x$ and $u= \varepsilonilon cd^m$. Then we observe that $(u,v)$ lies in $Y_a(\mathcal O)$ for $(a,S)=(-b(\varepsilonilon c)^2,S_d)$ and thus we see that Algorithm~\ref{algo:rana2} finds all solutions of \eqref{eq:rana2} as desired.
\subsubsection{Applications}\label{sec:ranaapp}
In this section we give some applications of Algorithms~\ref{algo:ranaheight} and \ref{algo:rana2}. In particular we discuss the database $\mathcal D_5$ containing the solutions of many generalized Ramanujan--Nagell equations~\eqref{eq:rana} and of many equations which are of the more classical form \eqref{eq:rana2}. We also explain how to apply our approach in order to study $S$-units $m,n\in\ZZ$ with $m+n$ a square or cube, and we provide some motivation for Terai's conjectures on Pythagorean numbers.
\paragraph{Preliminaries.} We continue the above notation. Further for any $n\in\ZZ_{\geq 1}$ we denote by $S(n)$ the set of the first $n$ rational primes. In what follows in this section, the running time $(t_1,t_2)$ of our approach via (Al) is given by the time $t_1$ which was required to compute via (PSM) the Mordell--Weil bases for the input of (Al) and the time $t_2$ which was required to solve via (Al) the discussed equation. Here (Al) is either Algorithm~\ref{algo:ranaheight} or \ref{algo:rana2}.
\paragraph{Generalized Ramanujan--Nagell equation.} Our database $\mathcal D_5$ contains in particular the solutions of the generalized Ramanujan--Nagell equation \eqref{eq:rana} for all parameter triples $(b,c,S)$ such that $b\in\ZZ$ is nonzero with $\mathcal As{b}\leq B$, $c=1$ and $S\subseteq S(n)$, where $(B,n)$ is of the form $(6,5)$, $(35,4)$, $(250,3)$ or $(10^3,2)$. For our algorithms via height bounds, the running time to solve the equation~\eqref{eq:rana} defined by $(b,c,S)$ is essentially the same as the one to solve the Thue--Mahler equation~\eqref{eq:thue-mahler} defined by $(f,S,m)$ with $m=c$ and $f\in\mathcal O[x,y]$ of discriminant $\Delta=b$.
Hence we refer to the discussions in Section~\ref{sec:ttmapp} which contain in particular our running times for many distinct Thue--Mahler equations \eqref{eq:thue-mahler}.
We next discuss a problem inspired by the original Ramanujan--Nagell equation. Recall that the original equation is $x^2+7=y$ with $(x,y)\in\ZZ\times\ZZ$ and $y=2^m$ for some $m\in\ZZ$. Now we put $b=7$, $c=1$ and $S=S(n)$ with $n\in\ZZ_{\geq 1}$ and we consider the problem of finding all solutions $(x,y)$ of \eqref{eq:rana} with $x,y\in\ZZ$. Here the assumption $x,y\in\ZZ$ considerably simplifies the problem in practice. Indeed we can remove all primes $p\in S$ with $-7$ not a square modulo $p$, and we know that $\ord_7(y)$ is either zero or one. Hence to solve the problem for $n=8$, it suffices to find all solutions of \eqref{eq:rana} for $(b,c,S)=(7,c,\{2,11\})$ with $c=1$ and $c=7$. These solutions were computed by Peth{\H{o}}--de Weger~\cite[Thm 5.1]{pede:binaryrec1}. We obtained an alternative proof of their theorem by using Algorithm~\ref{algo:ranaheight}. Indeed it took our approach less than (5 sec,\,4 sec) and (3 sec,\,4 sec) to find all solutions in the case $c=1$ and $c=7$ respectively. In particular, we solved the case $n=8$ of the problem in less than 16 seconds. To settle in addition the open case $n=9$, we need to find all solutions of \eqref{eq:rana} defined by $(b,c,S)=(7,c,\{2,11,23\})$ with $c=1$ and $c=7$. Here it took our approach less than (15 sec,\,15 sec) and (12 sec,\,15 sec) respectively, which means that we solved the case $n=9$ of the problem in less than 1 minute. However in the cases $n\geq 10$ the running times $t_1$ become significantly larger, since for increasing $n$ we have to deal with more and more rank one curves of large regulator. For example to establish the case $n=11$ of the problem, it took our approach approximately (5 hours, 2 minutes) in total. Here we needed to solve \eqref{eq:rana} for $(b,c,S)=(7,c,\{2,11,23,29\})$ with $c=1$ and $c=7$.
\paragraph{Classical case.} Our database $\mathcal D_5$ contains in addition the solutions of the more classical equation \eqref{eq:rana2} for all $(b,c,d)$ of the form $(7,1,d)$ with $d\leq 888$. To give the reader an idea of the running times of our approach via Algorithm~\ref{algo:rana2}, we determined the solutions of various equations \eqref{eq:rana2} of interest which were already solved by different methods. For instance, we found in (1 sec,\,1 sec) and (3 sec,\,4 sec) all solutions of the equations appearing in the title of the papers of Leu--Li~\cite{leli:rana} and Stiller~\cite{stiller:rana} respectively. The Diophantine problem \eqref{eq:rana2} was intensively studied in the literature when $d$ is prime, see for example \cite[Sect 2]{sasr:rana} and \cite[Sect 8.2]{besk:ternary} for an overview. Here we solved in (8 sec,\,8 sec) all four exceptional equations \cite[(2.7)]{sasr:rana} which appear in the classification of Le initiated in \cite{le:firstrana}. Further, it took less than (30 sec,\,20 sec) in total to solve all nine exceptional equations appearing in the classification of Bugeaud--Shorey~\cite[Thm 2]{bush:rana}. In particular, this includes the two exceptional equations $x^2+19=55^n$ and $x^2+341=377^n$ of \cite[Thm 2]{bush:rana} which we solved in (3 sec,\,4 sec) and (19 sec,\,4 sec) respectively.
\paragraph{Sums of units being a square or cube.} We now consider the problem of finding all integers $m,n\in\mathcal O^\times$ with $\gcd(m,n)$ square-free and $m+n$ a perfect square. On using inter alia the theory of logarithmic forms and generalized recurrences, de Weger~\cite{deweger:phdthesis,deweger:sumsofunits} obtained a practical approach for this problem which he used to settle the case $S=S(4)$. Suppose that $l\in\{2,3\}$. In a recent work, Bennett--Billerey~\cite{bebi:sumsofunits} show in particular how to practically solve the following problem (in which $l=2$ is the original problem)
\begin{equation}\label{eq:powersumsofunits}
m+n=z^l, \ \ \ (m,n,z)\in\ZZ^3,
\end{equation}
where $m,n\in\mathcal O^\times$ have $l$-th power free $\gcd(m,n)$. They use a different method which combines the Shimura--Taniyama conjecture, modular symbols (Cremona's algorithm) and Frey--Hellegouarch curves. On using in addition congruence arguments and Cremona's database of elliptic curves of given conductor, they solved \eqref{eq:powersumsofunits} for $S=S(4)$ and $S=\{2,3,p\}$ with $p\leq 100$. Here for various sets $S=\{2,3,p\}$ they moreover applied the archimedean elliptic logarithm approach in the form of \cite{sttz:elllogaa,gepezi:ellintpoints}.
In the case $l=2$, we directly obtain an alternative approach for \eqref{eq:powersumsofunits} by applying Algorithm~\ref{algo:ranaheight} with $(-b,1,S)$ for all $b\in\ZZ_{\geq 1}$ dividing $N_S$. Indeed for any solution $(m,n,z)$ of \eqref{eq:powersumsofunits} we may and do write $\max(m,n)=bu^2$ with $b,u$ positive integers in $\mathcal O^\times$ such that $b\mid N_S$ and then we see that $x=z/u$ and $y=\min(m,n)/u^2$ satisfy the generalized Ramanujan--Nagell equation \eqref{eq:rana} defined by $(-b,1,S)$. Here $z$ and $u$ are coprime, since $\gcd(m,n)$ is square-free.
Similarly in the case $l=3$, we directly obtain an alternative approach for \eqref{eq:powersumsofunits} by combining Algorithm~\ref{algo:mheight} with an elementary reduction to Mordell equations \eqref{eq:mordell} with parameter $a\in\ZZ$ dividing $N_S^5$; see the proof of Corollary~\ref{cor:sumsofunits}~(ii).
To illustrate the practicality of our approach, we solved \eqref{eq:powersumsofunits} for all $S\subseteq S(5)$ and all $S$ with $N_S\leq 10^3$. Given the required bases, this took less than 1 day in total. Here we could use our database which already contained the required bases. For $l=2$ (resp. $l=3$) we need in general $6^{\mathcal As{S}}$ (resp. $2\cdot 6^{\mathcal As{S}}$) distinct Mordell--Weil bases to solve \eqref{eq:powersumsofunits}, which means that our approach is not practical when $\mathcal As{S}$ is large. On the other hand, for small $\mathcal As{S}$ it turned out that one can usually determine the required bases in practice and then our approach is efficient. We compared our data with the known results, obtained by de Weger~\cite{deweger:phdthesis,deweger:sumsofunits} ($S=S(4)$, $l=2$) and Bennett--Billerey~\cite{bebi:sumsofunits} ($S=S(4)$ and $S=\{2,3,p\}$ with $p\leq 100$). In all cases it turned out that we found the same solutions.
We briefly discuss advantages and disadvantages of the different approaches. De Weger's method for $l=2$ is quite involved and we are not aware of its strengths and weaknesses. The strategy of Bennett--Billerey via modular symbols\footnote{In fact on replacing in our approach the involved algorithms via height bounds by the corresponding Algorithms~\ref{algo:ranacremona} and \ref{algo:mcremona} via modular symbols, we would directly obtain an alternative approach for \eqref{eq:powersumsofunits} via modular symbols. However we did not include this, since the arguments of Bennett--Billerey (using a careful analysis of conductors of Frey--Hellegouarch curves) are more direct and more efficient.} has the usual advantages and disadvantages of an effective method involving modular symbols (Cremona's algorithm), see the analogous discussions in Section~\ref{sec:malgocomparison}. A weakness of our approach is its dependence on many Mordell--Weil bases, and a strength is its efficiency in the case when these bases can be determined. We also mention that Bennett--Billerey used additional tools (such as for example level lowering and the theory of logarithmic forms) to moreover prove explicit finiteness results for \eqref{eq:powersumsofunits} for all $l\geq 4$, see \cite[Sect 7]{bebi:sumsofunits}. Without introducing crucial new ideas, such results are out of reach for our approach.
\paragraph{Pythagorean numbers.} We next illustrate that Algorithm~\ref{algo:ranaheight} is a useful tool to study certain classical Diophantine problem on Pythagorean numbers which appear in the literature. To state the first Diophantine equation we take coprime $a,b,c\in\ZZ_{\geq 1}$ with $a$ even, and we assume that $a^2+b^2=c^2$. Inspired by the works of Je\'smanovic and Sierpi\'nski published in 1956, Terai~\cite{terai:conjecture} conjectured that $(a,2,2)$ is the unique solution of
\begin{equation}\label{eq:teraiconj}
x^2+b^m=c^n, \ \ \ (x,m,n)\in\ZZ^3,
\end{equation}
with $x,m,n$ all positive. Several authors settled special cases of \eqref{eq:teraiconj}, see for example \cite[p.21]{terai:conjaust} for an overview. We observe that equation \eqref{eq:teraiconj} is a special case of \eqref{eq:powersumsofunits} with $l=2$ and hence the above described approach via Algorithm~\ref{algo:ranaheight} allows to solve \eqref{eq:teraiconj} for any given Pythagorean triple $(a,b,c)$.
For example we verified Terai's conjecture for all triples $(a,b,c)$ with $c\leq 85$. Given the required Mordell--Weil bases, this took less than 1 minute in total. In fact to deal with \eqref{eq:teraiconj} we used a modified version of Algorithm~\ref{algo:ranaheight}, which exploits the special shape of \eqref{eq:teraiconj} in order to reduce the number of required Mordell--Weil bases to 18. More recently, Terai~\cite{terai:conjaust} studied the following variation of \eqref{eq:teraiconj}. Let $d\geq 2$ be a rational integer and consider the Diophantine equation
\begin{equation}\label{eq:teraiconj2}
x^2+(2d-1)^m=d^n, \ \ \ (x,m,n)\in\ZZ^3,
\end{equation}
with $x,m,n$ all positive. Terai conjectured in \cite[Conj 3.1]{terai:conjaust} that $(d-1,1,2)$ is the unique solution of \eqref{eq:teraiconj2}. He verified his conjecture for certain values of $d$, including all $d\leq 30$ except the two cases $d=12,24$ which were both settled independently by Deng~\cite{deng:teraiconj} and Bennett--Billerey~\cite[Prop 5.5]{bebi:sumsofunits}. As above, we observe that our approach via Algorithm~\ref{algo:ranaheight} allows to solve \eqref{eq:teraiconj2} for any given $d$. To illustrate the utility of this strategy, we also verified \cite[Conj 3.1]{terai:conjaust} for all $d\leq 30$. Here it was no problem to compute the required bases and then it took the modified version of Algorithm~\ref{algo:ranaheight} less than 1 minute to solve \eqref{eq:teraiconj2} for all $d\leq 30$. We can also prove new cases of Terai's conjecture concerning \eqref{eq:teraiconj2}. However the running times to compute the required bases explode for larger $d$. For example, in the range $30< d< 35$ it took several days to compute the bases and then we solved all equations \eqref{eq:teraiconj2} in roughly 30 seconds by using the modified version of Algorithm~\ref{algo:ranaheight}. It turned out that Terai's conjecture holds in the range $30< d< 35$.
\subsection{Comparison of algorithms}\label{sec:ranaalgocomp}
In this section we compare our algorithms for the generalized Ramanujan--Nagell equation~\eqref{eq:rana} and its more classical form \eqref{eq:rana} with the known practical methods.
\paragraph{Advantages and disadvantages.}Algorithms solving \eqref{eq:rana} via modular symbols (Cremona's algorithm) have the usual strengths and weaknesses, see the analogous discussion in Section~\ref{sec:talgocompa}. We further mention that our Algorithm~\ref{algo:ranacremona} uses a reduction to Mordell equations~\eqref{eq:mordell}, while Kim's approach \cite[Sect 8]{kimd:modularthuemahler} for \eqref{eq:rana} works with a reduction to cubic Thue--Mahler equations~\eqref{eq:thue-mahler}. In fact Kim studies the special case $x^2+7=y$ with $x,y\in\ZZ$ and $y\in\mathcal O^\times$. Similar as in the case of \eqref{eq:thue-mahler}, Kim's method is more efficient in terms of $S$ and our Algorithm~\ref{algo:ranacremona} is more efficient in terms of $b$.
We next discuss approaches via height bounds. The discussion of advantages and disadvantages of Algorithm~\ref{algo:ranaheight} (resp. of Algorithm~\ref{algo:rana2}) is analogous to the corresponding discussion of Algorithm~\ref{algo:tmheight} for cubic Thue--Mahler equations \eqref{eq:thue-mahler} (resp. of Algorithm~\ref{algo:theight} for cubic Thue equations \eqref{eq:thue}). Hence we refer to Section~\ref{sec:talgocompa}. The approach of Peth{\H{o}}--de Weger~\cite{pede:binaryrec1}, using inter alia the theory of logarithmic forms and binary recurrence sequences, is quite involved and we are not aware of its strengths and weaknesses.
\paragraph{Comparison of data.} In the cases where our results were already known (see above), we compared the data. In all cases it turned out that we obtained the same solutions.
\section{Mordell equations and almost primitive solutions}\label{sec:mordellcoates}
In this section we state and discuss Theorem~\ref{thm:m} which gives results for almost primitive solutions of Mordell equations. We also deduce Corollaries~\ref{cor:coates1} and~\ref{cor:coates2} on the difference of perfect squares and perfect cubes, and we discuss how these corollaries improve old theorems of Coates~\cite{coates:shafarevich} which are based on the theory of logarithmic forms.
Let $S$ be a finite set of rational primes. We write $\mathcal O=\ZZ[1/N_S]$ for $N_S=\prod_{p\in S} p$ and we denote by $h$ the logarithmic Weil height. Let $a\in \mathcal O$ be nonzero and define
\begin{equation}\label{def:asr2}
a_S=1728N_S^2r_2(a), \ \ \ \ r_2(a)=\prod p^{\min(2,\ord_p(a))}
\end{equation}
with the product taken over all rational primes $p$ not in $S$. It holds that $\log r_2(a)\leq h(a)$, and if $a\in\ZZ-\{0\}$ then we observe that $r_2(a)\leq \min(\mathcal As{a},\textnormal{rad}(a)^2)$. In view of the definition of Bombieri--Gubler \cite[12.5.2]{bogu:diophantinegeometry}, we say that $(x,y)\in\ZZ\times\ZZ$ is primitive if $\pm 1$ are the only $n\in\ZZ$ with $n^{6}$ dividing $\gcd(x^3,y^2)$. We recall the Mordell equation
\begin{equation}
y^2=x^3+a, \ \ \ (x,y)\in\mathcal O\times\mathcal O. \tag{\ref{eq:mordell}}
\end{equation}
Inspired by Szpiro's small points philosophy (see e.g. \cite[Sect 2]{szpiro:lefschetz}) discussed below, we consider a certain class of solutions of~\eqref{eq:mordell} which contains all primitive solutions of~\eqref{eq:mordell}. For want of a
better name we call\footnote{We note that~\eqref{eq:mordell} defines naturally a moduli scheme of elliptic curves. Then one observes that the notions minimal and almost minimal solutions may be more appropriate (from a geometric point of view) than primitive and almost primitive solutions respectively.} these solutions ``almost primitive".
\begin{definition}\label{def:ap}
Let $\mu:\ZZ_{\geq 1}\to \mathbb R_{\geq 0}$ be an arbitrary function, and let $(x,y)\in\mathcal O\times\mathcal O$. We define $u_{x,y}=u=u_1/u_2$ with $u_1\in\ZZ_{\geq 1}$ minimal such that $u_1x,u_1y$ are in $\ZZ$ and with $u_2\in\ZZ$ maximal such that $u_2^6$ divides $\gcd\bigl((u_1^2x)^3,(u_1^3y)^2\bigl)$. Then $(u^2x,u^3y)\in\ZZ\times\ZZ$ is primitive and we say that $(x,y)$ is almost primitive with respect to $\mu$ if $h(u)\leq \mu(a_S)$.
\end{definition}
We notice that $(x,y)\in\ZZ\times\ZZ$ is primitive if and only if it is almost primitive with respect to all functions $\mu:\ZZ_{\geq 1}\to\mathbb R_{\geq 0}$. Intuitively, one may view almost primitive elements of $\mathcal O\times\mathcal O$ as those elements with ``non-primitive part" bounded in terms of $a_S$.
Building on the arguments of \cite[Cor 7.4]{rvk:modular}, we obtain the following result which depends on the quantity $a_S$ but which does not involve the height $h(a)$ of~$a$.
\begin{theorem}\label{thm:m}
The following statements hold.
\begin{itemize}
\item[(i)] Let $\mu:\mathbb \ZZ_{\geq 1}\to\mathbb R_{\geq 0}$ be an arbitrary function. Assume that~\eqref{eq:mordell} has a solution which is almost primitive with respect to~$\mu$. Then any solution $(x,y)$ of~\eqref{eq:mordell} satisfies $$\max\bigl(h(x),\tfrac{2}{3}h(y)\bigl)\leq \tfrac{2}{3}a_S\log a_S+\tfrac{1}{4}a_S\log\log\log a_S+\tfrac{3}{5}a_S+2\mu(a_S).$$
Moreover, if $(x,y)$ is in addition almost primitive with respect to $\mu$ then
$$\max\bigl(h(x),\tfrac{2}{3}h(y)\bigl)\leq \tfrac{2}{9}a_S\log a_S+\tfrac{1}{12}a_S\log\log\log a_S+\tfrac{1}{4}a_S+2\mu(a_S).$$
\item[(ii)] Suppose that $a\in\ZZ$ with $\mathcal As{a}\to\infty$ and define $a_*=1728\prod_{p\mid a} p^{\min(2,\ord_p(a))}$. Then any primitive solution $(x,y)\in\ZZ\times\ZZ$ of the Mordell equation~\eqref{eq:mordell} satisfies
$$\max\bigl(h(x),\tfrac{2}{3}h(y)\bigl)\leq \tfrac{1}{6}a_*\log a_* + \frac{(\frac{2}{9}\log 2+o(1))}{\log\log a_*}a_*\log a_*.$$
\end{itemize}
\end{theorem}
We now make some remarks which complement our discussions of Theorem~\ref{thm:m} given in the introduction: Any solution $(x,y)$ of~\eqref{eq:mordell} is by definition almost primitive with respect to the constant function $\mu=h(u_{x,y})$ on $\ZZ_{\geq 1}$, and therefore we see that Theorem~\ref{thm:m}~(i) provides in particular an explicit upper bound in terms of $a_S$ and~$h(u_{x,y})$.
Theorem~\ref{thm:m} and its proof fit into Szpiro's small points philosophy \cite[Sect 2]{szpiro:lefschetz} for hyperbolic curves $X$ of genus at least two defined over number fields. This philosophy says, roughly speaking, that the rational points of $X$ have small\footnote{Here small means that the Arakelov height is effectively bounded from above only in terms of the bad reduction places of~$X$, the genus of $X$ and the given base field over which $X$ is defined.} Arakelov height defined with the minimal regular model of~$X$. In our case of the hyperbolic genus one curve~\eqref{eq:mordell}, the existence of an almost primitive solution assures that~\eqref{eq:mordell} is sufficiently minimal, and then the height bound in Theorem~\ref{thm:m}~(i) shows that all solutions of~\eqref{eq:mordell} are small\footnote{A solution of~\eqref{eq:mordell} is small if its Weil height, given on the subvariety of $\mathbb A^2$ defined by~\eqref{eq:mordell}, is effectively bounded from above only in terms of the bad reduction places of the affine curve~\eqref{eq:mordell}. The Weil height is more suitable in our case of the affine genus one curve~\eqref{eq:mordell}, since Szpiro's result \cite[Thm 2]{szpiro:grothendieck} implies that the Arakelov height is in fact constant on the rational points of any elliptic curve.}. Furthermore our proof of Theorem~\ref{thm:m} uses inter alia~\eqref{eq:szpiro} which is in fact a version\footnote{It follows for example from \cite[Thm 2]{szpiro:grothendieck} that~\eqref{eq:szpiro} is indeed a version of Szpiro's small points conjecture for elliptic curves over~$\QQ$.} of Szpiro's small points conjecture \cite[p.101]{szpiro:faltings} for elliptic curves over~$\QQ$.
\subsection{The difference of perfect squares and perfect cubes}
Let $a\in\ZZ-\{0\}$ and let $S$ be a finite set of rational primes. Following Coates~\cite{coates:shafarevich}, we denote by $f\in\ZZ$ the largest divisor of $a$ which is only divisible by primes in~$S$. Then $\mathcal As{a/f}$ is the largest divisor of $a$ which is coprime to $N_S=\prod_{p\in S}p$. We define
$$\alpha_S=\frac{a_S}{N_S}=1728N_Sr_2(a)$$
for $a_S$ and $r_2(a)$ the quantities given in (\ref{def:asr2}). Now we can state the following corollary.
\begin{corollary}\label{cor:coates1}
If $(x,y)\in\ZZ\times\ZZ$ with $y^2-x^3=a$ and $\gcd(x,y,N_S)=1$, then $$\log \max(\mathcal As{x},\mathcal As{y})\leq \tfrac{1}{2}\log \mathcal As{a/f}+2\alpha_S\log\alpha_S+\tfrac{3}{4}\alpha_S\log\log\log \alpha_S+6\alpha_S.$$
\end{corollary}
\begin{proof}
It follows that $(x,y)$ is almost primitive with respect to $\mu=\frac{1}{6}\log\mathcal As{a/f}$. Then Theorem~\ref{thm:m}~(i) proves the desired bound, but with $a_S$ in place of $\alpha_S$. To obtain the bound involving $\alpha_S$ claimed by Corollary~\ref{cor:coates1}, one uses a version of Theorem~\ref{thm:m}~(i) which takes into account that $\gcd(x,y,N_S)=1$. See Section~\ref{sec:mproofs} for details.
\end{proof}
We write $s=\mathcal As{S}$ and we define $P=\max(S\cup\{2\})$. On using a completely different method, based on the theory of logarithmic forms, Coates \cite[Thm 1]{coates:shafarevich} obtained that any $(x,y)\in\ZZ\times\ZZ$ with $y^2-x^3=a$ and $\gcd(x,y,N_S)=1$ satisfies
\begin{equation}\label{eq:coates}
\log\max(\mathcal As{x},\mathcal As{y})\leq 2^{10^7(s+1)^4}P^{10^9(s+1)^3}\mathcal As{a/f}^{10^6(s+1)^2}.
\end{equation}
We observe that $\alpha_S\leq 1728N_S\mathcal As{a/f}$ and it holds that $N_S\leq P^s$. Therefore we see that Corollary~\ref{cor:coates1} improves Coates' result \cite[Thm 1]{coates:shafarevich} stated in~\eqref{eq:coates}.
We also obtain the following corollary on the size of the greatest rational prime divisor of the difference of (coprime) perfect squares and perfect cubes.
\begin{corollary}\label{cor:coates2}
For any real number $\varepsilon>0$ there is an effective constant~$c(\varepsilon)$, depending only on $\varepsilon$, with the following property: Suppose that $x$ and $y$ are coprime rational integers, and write $X=\max(\mathcal As{x},\mathcal As{y})$. Then the greatest rational prime factor of $y^2-x^3$ exceeds $$ (1-\varepsilon)\log\log X+c(\varepsilon).$$
For example, if $\varepsilon=\frac{1}{10}$ then one can take here the constant $c(\varepsilon)=-20$.
\end{corollary}
\begin{proof}
Let $S$ be the set of rational primes dividing $a=y^2-x^3$ and write $q=\max(S)$. The explicit version of the prime number theorem given in \cite[Thm 4]{rosc:formulas} shows that $\log N_S\leq \sum_{p\leq q}\log p\leq q\bigl(1+\frac{1}{2\log q}\bigl)$. Thus Corollary~\ref{cor:coates1} implies Corollary~\ref{cor:coates2}.
\end{proof}
We conclude with several remarks. Coates obtained in \cite[Thm 2]{coates:shafarevich} the weaker lower bound $10^{-3}(\log\log X)^{1/4}$ by using his result \cite[Thm 1]{coates:shafarevich} displayed in~\eqref{eq:coates}.
The proofs of Corollaries~\ref{cor:coates1} and~\ref{cor:coates2} show in addition that one can weaken the assumptions $\gcd(x,y,N_S)=1$ and $\gcd(x,y)=1$ in Corollaries~\ref{cor:coates1} and~\ref{cor:coates2} by slightly changing the bounds.
Further on using the link $(abc)\Rightarrow\eqref{eq:mordell}$ in \cite[p.429]{bogu:diophantinegeometry}, between the $abc$-conjecture $(abc)$ and height bounds for the solutions of~\eqref{eq:mordell}, one can show that $(abc)$ gives grosso modo our inequalities with the logarithmic Weil height $h$ replaced by~$\exp(h)$.
Finally we point out that the known links $(abc)\Rightarrow\eqref{eq:mordell}$ are not compatible with exponential versions of~$(abc)$. In particular one can not combine the exponential version of $(abc)$ given in Stewart--Yu \cite[Thm 1]{styu:abc2} with the known links $(abc)\Rightarrow\eqref{eq:mordell}$ in order to improve our results in this Section~\ref{sec:mordellcoates} or Proposition~\ref{prop:m} below.
\section{Height bounds for Thue and Thue--Mahler equations}\label{sec:thueproofs}
To deduce Corollary~J from our results for Mordell equations, we work out explicitly the arguments of \cite[Sect 7.4]{rvk:modular}. In fact in the present section we shall establish a more precise version of Corollary~J. This version provides optimized and sharper height bounds for the solutions of any cubic Thue and Thue--Mahler equation, see Corollary~\ref{cor:precthue}. We continue the terminology and notation introduced in Section~\ref{sec:thuealgo}.
\subsection{Reduction to Mordell equations}
We begin by recalling that $X$ is given by the Thue equation $f-m=0$, where $m\in\mathcal O$ is nonzero and $f\in \mathcal O[x,y]$ is a homogeneous polynomial of degree 3 with nonzero discriminant $\Delta$. Further we recall that $Y$ is given by the Mordell equation $y^2-(x^3+a)=0$ with parameter $a=432\Delta m^2$. We shall work with the following morphism $$\varphi:X\to Y$$ of $\mathcal O$-schemes, which was constructed in \eqref{eq:thuemap} using classical invariant theory. For any polynomial $g$ with rational coefficients $a_\alpha$, we define $h(g)=\max_\alpha h(a_\alpha)$. On recalling that $X$ and $Y$ are affine, we see that the usual logarithmic Weil height $h$ (see \cite[p.16]{bogu:diophantinegeometry}) naturally defines a height function $h$ on $X(\bar{\QQ})$ and $Y(\bar{\QQ})$. We obtain the following result.
\begin{proposition}\label{prop:heightineq}
Assume that $f,m\in\ZZ[x,y]$. Then any $P\in X(\bar{\QQ})$ satisfies $$h(P)\leq \tfrac{1}{3}h(\varphi(P))+12\bigl(h(f-m)+6h(f)+186\bigl).$$
\end{proposition}
We point out that the first term $\tfrac{1}{3}h(\varphi(P))$ is optimal here. The coefficients in the second term come from a recent version of the arithmetic Nullstellensatz discussed in the next paragraph. These coefficients can be slightly improved in certain cases. For example, the proof of Proposition~\ref{prop:heightineq} shows in addition that one can replace the coefficient 12 by the smaller number 6 in the case when $P\in X(\ZZ)$. We also mention that one can directly remove the assumption $f,m\in\ZZ[x,y]$ in Proposition~\ref{prop:heightineq} by replacing in the bound the quantity $h(f-m)+6h(f)$ by the larger expression $7h(f-m)+28h(f)$.
\subsubsection{Arithmetic Nullstellensatz and covariants of cubic forms}\label{sec:ans+covariants}
We continue our notation. In this section we collect some results which shall be used in the proof of Proposition~\ref{prop:heightineq}. In particular we discuss effective versions of the arithmetic Nullstellensatz and we recall classical properties of covariants of cubic forms.
\paragraph{Arithmetic Nullstellensatz.} An important ingredient for our proof of Proposition~\ref{prop:heightineq} is an effective version of the Nullstellensatz. Masser--W\"ustholz~\cite[Thm IV]{mawu:ellfunc} worked out a fully explicit version of the strong arithmetic Nullstellensatz. Their result would be sufficient for our purpose in the sense that it would give a version of Proposition~\ref{prop:heightineq} in which the coefficient $12$ is replaced by a number exceeding $24^{15}$. To obtain the considerably smaller number $12$, we shall apply a recent result of D'Andrea--Krick--Sombra~\cite[Thm 2]{dakrso:nullstell} providing an explicit version of the strong arithmetic Nullstellensatz over an affine variety of pure dimension. More precisely we shall work over the affine hypersurface $V\subset \mathbb A^3_\QQ$ defined by the polynomial $g=f-mz^3$. On using our assumptions that $m$ and the discriminant of $f$ are both nonzero, we see that $g$ is geometrically irreducible
and therefore the affine variety $V_{\bar{\QQ}}$ is of pure dimension two.
Write $h(V)$ for the projective logarithmic Weil height $h$ (see \cite[p.15]{bogu:diophantinegeometry}) of the point in projective space determined by the coefficient vector of the polynomial $g$, and let $\hat{h}(V)$ be the canonical height (see \cite[p.589]{dakrso:nullstell}) of the projective closure $\bar{V}$ of $V$ inside $\mathbb P^3_\QQ$. It holds
\begin{equation}\label{eq:canheightest}
\hat{h}(V)\leq h(V)+3\log 5.
\end{equation}
To verify this inequality, we temporarily write $R=\ZZ[x_1,\dotsc,x_4]$ and we let $D$ be the effective Cartier/Weil divisor of $\mathbb P^3_\QQ$ given by the irreducible hypersurface $\bar{V}\subset \mathbb P^3_\QQ$.
On using the terminology of \cite[$\mathsection$1.1]{dakrso:nullstell}, we denote by $f_{D}\in R$ a primitive polynomial determined by the Cartier divisor $D$.
The homogeneous polynomial $g$ is irreducible in $R_\QQ=R\otimes_\ZZ\QQ$ and it holds that $\bar{V}\cong\textnormal{Proj}\bigl(R_\QQ/(g)\bigl)$. It follows that there exists $\varepsilonilon\in\QQ^\times$ such that $\varepsilonilon f_{D}$ coincides with $g$ in $R_\QQ$, which implies that $h(f_D)=h(V)$ since $f_D$ is primitive.
Let $m(f_D)$ be the Mahler measure of $f_D$ defined in \cite[$\mathsection$2.2]{dakrso:nullstell}. An application of \cite[Prop 2.39]{dakrso:nullstell} with $D$ gives that $\hat{h}(V)=m(f_D)$, and we deduce from \cite[Lem 2.30]{dakrso:nullstell} that $m(f_D)\leq h(f_{D})+3\log 5$. Hence the equality $h(f_D)=h(V)$ proves \eqref{eq:canheightest}.
\paragraph{Covariants of cubic forms.} We next recall some properties of cubic forms which shall be used in the proof of Proposition~\ref{prop:heightineq}. Write $f(x,y)=ax^3+bx^2y+cxy^2+dy^3$ with $a,b,c,d\in\QQ$ and denote by $\mathcal H$ the covariant of $f$ of degree two. The form $\mathcal H$ is the Hessian of $f$ in the sense of \cite[p.175]{salmon:book}; we shall work with the normalization
\begin{equation}\label{def:covarpol2}
\mathcal H(x,y)=Ax^2+Bxy+Cy^2, \ \ \ A=3ac-b^2, \ B=9ad-bc, \ C=3bd-c^2.
\end{equation}
To avoid a collision of notation, we write throughout this section $k=432\Delta m^2$ for the parameter $a$ appearing in the equation \eqref{eq:mordell} which defines $Y$. Here $\Delta$ denotes the discriminant of $f$ defined in \cite[p.175]{salmon:book}; note the sign convention used in \cite{salmon:book}.
Further we denote by $J$ the covariant of $f$ of degree three. It is a binary form
\begin{equation}\label{def:covarpol3}
J(x,y)=\sum a_i x^{3-i}y^{i}
\end{equation}
given by the Jacobian of the forms $f$ and $\mathcal H$ in the sense of \cite{salmon:book}; for our purpose it will be convenient to normalize the coefficients $a_i$ of the polynomial $J$ as follows:
\begin{align*}
a_0&= 27a^2d-9abc+2b^3, & a_1&=3(b^2c+9abd-6ac^2),\\
a_2&= 3(6b^2d-bc^2-9acd), & a_3&=9bcd-27ad^2-2c^3.
\end{align*}
To determine the set $Z(f,J)\subset\bar{\QQ}^2$ of common zeroes of $f$ and $J$, we suppose that $(x,y)\in Z(f,J)$. A formula going back at least to Cayley (see \cite[p.177]{salmon:book}) shows that the polynomials $u=-4\mathcal H$ and $v=4J$ satisfy the relation $v^2=u^3+432\Delta f^2$. This implies that $(x,y)$ is a zero of $\mathcal H$. In the case $a=0=d$, we then see that our assumption $\Delta\neq 0$ together with \eqref{def:covarpol2} implies that $(x,y)=0$ and therefore we obtain
\begin{equation}\label{eq:zfjcomp}
Z(f,J)=0.
\end{equation}
To prove that \eqref{eq:zfjcomp} holds in general, we now may and do assume that the coefficient $a$ of $f$ is nonzero. Then the polynomial $f(t,1)=a\prod (t-\gamma_j)$ in $\QQ[t]$ has degree 3. Furthermore the roots $\gamma_j$ of $f(t,1)$ are distinct since $\Delta\neq 0$. Thus $\mathcal H(\gamma_j,1)$ is nonzero by the formula for the Hessian given in \cite[p.175]{salmon:book}. It follows that $(x,y)=0$, since $f,\mathcal H$ are homogeneous and since $a\neq 0$. Hence the set $Z(f,J)$ is trivial. Alternatively, one can use here invariant theory providing that the resultant of $f$ and $H$ is a power of the invariant $\Delta$ up to a sign.
\subsubsection{Proof of Proposition~\ref{prop:heightineq}} We continue our notation. In this section we use the above results to prove Proposition~\ref{prop:heightineq}.
\begin{proof}[Proof of Proposition~\ref{prop:heightineq}]
To obtain the desired height inequality, we work in the projective space. Let $\bar{X}$ and $\bar{Y}$ be the projective closures in $\mathbb P^2_\QQ$ of the affine curves $X_\QQ$ and $Y_\QQ$ respectively. It follows from \eqref{eq:zfjcomp} that $\varphi:X\to Y$ induces a finite morphism $$\bar{\varphi}:\bar{X}\to\bar{Y}$$ of degree three, which is given by $\varphi_1=-4x_3\mathcal H(x_1,x_2)$, $\varphi_2=4J(x_1,x_2)$ and $\varphi_3=x_3^3$ in terms of coordinates $x_i$ on $\mathbb P^2_\QQ$. To simplify the exposition, we write $R=\QQ[x_1,x_2,x_3]$ and we shall identify (when convenient) a polynomial in $R$ with its image in $\mathcal O_V(V)$.
We next apply the Nullstellensatz to express $x_i$ in terms of $\varphi_j$. Let $Z\subset V(\bar{\QQ})$ be the set of common zeroes of the two functions $\varphi_1,\varphi_2\in\mathcal O_V(V)$. Suppose that $(x,y,z)\in Z$. Then it holds that $z=0$ or $\mathcal H(x,y)=0$ and thus the identity $432\Delta f^2=\varphi_2^2-(\varphi_1/x_3)^3$ together with $\Delta\neq 0$ implies $(x,y)\in Z(f,J)$. We conclude that $(x,y)=0$, since $Z(f,J)$ is trivial by \eqref{eq:zfjcomp}. It follows that the functions $x_i$ vanish on $Z$ for $i=1,2$. Furthermore, our assumptions $f,m\in\ZZ[x,y]$ together with \eqref{def:covarpol2} and \eqref{def:covarpol3} show that $\varphi_1,\varphi_2$ have coefficients in $\ZZ$. Therefore applications of the strong arithmetic Nullstellensatz over $V$, with $\varphi_1, \varphi_2$ and $x_i$, give $\alpha_i,e_i\in\ZZ_{\geq 1}$ and $\rho_{ij}\in R$ such that the two functions $\alpha_i x_i^{e_i}$ and $\sum \rho_{ij}\varphi_j$ coincide on $V(\bar{\QQ})$ for $i=1,2$. In other words the difference of these two functions vanishes on $V(\bar{\QQ})$,
and then $V=\textnormal{Spec}\bigl(R/(g)\bigl)$ implies that this difference is divisible in $R$ by the geometrically irreducible polynomial $g$. Thus there exist $\rho_{i0}\in R$ such that
\begin{equation}\label{eq:nssatz}
x_i^{e_i}=\sum (\rho_{ij}/\alpha_i)\varphi_j, \ \ \ i=1,2,
\end{equation}
where $\varphi_0=g$. On multiplying here both sides with $x_i^{e-e_i}$ for $e=\max e_i$, we may and do assume that $e_i=e$. Furthermore we may and do assume that all $\rho_{ij}$ are homogeneous of degree $e-3$, since the polynomials $\varphi_j$ are all homogeneous of degree 3.
We now control the right hand side of \eqref{eq:nssatz} in terms of $\varphi$ and $g$. Put $h(\rho/\alpha)=h(Q)$ for $Q$ the point in projective space whose coordinates are given by the coefficients of the four polynomials $\rho_{ij}/\alpha_i\in R$ with $i,j\in\{1,2\}$. Each polynomial $\rho_{ij}$ has at most $\tbinom{e-1}{2}$ nonzero coefficients, and the function $\varphi_0$ vanishes on the $\bar{\QQ}$-points of $\bar{X}=\textnormal{Proj}\bigl(R/(\varphi_0)\bigl)$. Therefore on combining \eqref{eq:nssatz} with the above observations, we see that standard height arguments lead to the following statement: Any $\bar{\QQ}$-point $P$ of $\bar{X}$ satisfies
\begin{equation}\label{eq:projheightineq}
3h(P)\leq h(\bar{\varphi}(P))+h(\rho/\alpha)+\log \bigl(2\tbinom{e-1}{2}\bigl).
\end{equation}
In particular any $P\in X(\bar{\QQ})$ satisfies \eqref{eq:projheightineq} with $\bar{\varphi}$ replaced by $\varphi$. Now we see that the explicit version \cite[Thm 2]{dakrso:nullstell} of the strong arithmetic Nullstellensatz gives the following: In \eqref{eq:nssatz} one can choose $\alpha_i$, $e_i$ and $\rho_{ij}$, with $e_i\leq 54$ and $h(\rho/\alpha)$ explicitly bounded in terms of $\hat{h}(V)$ and $h(\varphi_j)$, such that \eqref{eq:projheightineq}, \eqref{eq:canheightest}, \eqref{def:covarpol2} and \eqref{def:covarpol3} lead to the desired height inequality. This completes the proof of Proposition~\ref{prop:heightineq}.\end{proof}
The above proof gives moreover a version of Proposition~\ref{prop:heightineq} for projective closures inside $\mathbb P^2_\QQ$. Indeed it follows from \eqref{eq:projheightineq} that any $\bar{\QQ}$-point $P$ of $\bar{X}$ satisfies the height inequality in Proposition~\ref{prop:heightineq} with $\varphi$ replaced by the morphism $\bar{\varphi}:\bar{X}\to \bar{Y}.$
\subsection{Optimized height bounds}
We continue the notation introduced above. In this section we give Corollary~\ref{cor:precthue} which provides optimized height bounds for the solutions of cubic Thue and Thue--Mahler equations. To state our height bounds, we recall that $k=432\Delta m^2$ and we denote by $\Omega_{\textnormal{sim}}$ the simplified height height bound given in Proposition~\ref{prop:m} with $a=k$. It holds $$\Omega_{\textnormal{opt}}\leq\Omega_{\textnormal{sim}}=\tfrac{1}{3}h(k)+\tfrac{4}{9}k_S\log k_S+\tfrac{1}{6}k_S\log\log\log k_S+\tfrac{2}{5}k_S$$
for $k_S$ as in \eqref{def:as} and $\Omega_{\textnormal{opt}}$ the optimized height bound provided by Proposition~\ref{prop:algobounds} with $a=k$.
The next result is a sharper (but more complicated) version of Corollary~J.
\begin{corollary}\label{cor:precthue}Assume that $f,m\in\ZZ[x,y]$. Then the following statements hold.
\begin{itemize}
\item[(i)] Suppose that $(x,y)$ is a solution of the cubic Thue equation \eqref{eq:thue}, and put $n=1$ if $(x,y)\in\ZZ\times\ZZ$ and $n=2$ otherwise. Then $h(x)$ and $h(y)$ are at most $$\tfrac{n}{2}\Omega_{\textnormal{opt}}+6n\bigl(h(f-m)+6h(f)+186\bigl).$$
\item[(ii)]Suppose that $(x,y,z)$ is a primitive solution of the general cubic Thue--Mahler equation \eqref{eq:thue-mahler}. Then $h(x)$, $h(y)$ and $\tfrac{1}{3}h(z)$ are at most $$2\Omega_{\textnormal{opt}}+51\log N_S+24\bigl(h(f-m)+6h(f)+186\bigl).$$
\end{itemize}
\end{corollary}
One can directly remove the extra assumption $f,m\in\ZZ[x,y]$ by multiplying the equation $f(x,y)=m$ with a suitable integer, see \eqref{eq:precthue} for the resulting bounds.
\begin{proof}[Proof of Corollary~\ref{cor:precthue}]
To prove (i) we take $P\in X(\mathcal O)$. Then $(u,v)=\varphi(P)$ satisfies the Mordell equation \eqref{eq:mordell} with parameter $a=k$. The number $k=432\Delta m^2$ is nonzero, since $m\Delta\neq 0$ by assumption. Hence an application of Propositions~\ref{prop:m}~and~\ref{prop:algobounds} with $(u,v)$ gives an upper bound for $h(\varphi(P))$ which together with Proposition~\ref{prop:heightineq} implies (i). Here we used that one can replace in Proposition~\ref{prop:heightineq} the coefficient 12 by 6 if $P\in X(\ZZ)$.
To show (ii) we assume that $(x,y,z)$ is a primitive solution of the general Thue--Mahler equation \eqref{eq:thue-mahler}. We write $z=z_0\varepsilonilon^3$ with $z_0,\varepsilonilon\in\ZZ$ such that $\pm 1$ are the only $l\in\ZZ$ with $l^{3}$ dividing $z_0$. Then $u=x/\varepsilonilon$ and $v=y/\varepsilonilon$ satisfy the Thue equation $f(u,v)=m'$ with $m'=mz_0$. On exploiting our assumption that $(x,y,z)$ is primitive, one controls the absolute values of $x,y,z$ in terms of the Weil heights of $u,v$ and then (ii) follows from the height bound for $(u,v)$ obtained in (i). Here we used that $k'=432\Delta m'^2$ satisfies $k_S=k'_S$ and that $h(k')$ is at most $h(k)+4\log N_S$. This completes the proof of Corollary~\ref{cor:precthue}.
\end{proof}
To remove the extra assumption $f,m\in\ZZ[x,y]$ in Corollary~\ref{cor:precthue}, we define $f^*=lf$ and $m^*=lm$ for $l$ the least common multiple of the denominators of the coefficients of the polynomial $f-m\in\mathcal O[x,y]$. Any solution $(x,y)$ of the Thue equation \eqref{eq:thue} satisfies $f^*(x,y)=m^*$, and any primitive solution $(x,y,z)$ of the general Thue--Mahler equation~\eqref{eq:thue-mahler} satisfies $f^*(x,y)=m^*z$. Therefore an application of Corollary~\ref{cor:precthue} with $f^*,m^*\in\ZZ[x,y]$ implies the following: Statements (i) and (ii) of Corollary~\ref{cor:precthue} hold without the extra assumption $f,m\in\ZZ[x,y]$ if the bounds in (i) and (ii) are replaced by
\begin{equation}\label{eq:precthue}
B_1=\Omega_{\textnormal{opt}}+86\bigl(4h(f)+h(m)+26\bigl) \ \ \ \textnormal{ and } \ \ \ 2B_1+51\log N_S
\end{equation}
respectively. Here we used that $h(f^*),h(m^*)$ are at most $4h(f)+h(m)$ and we exploited that $k^*_S=k_S$, where $k^*=432\Delta^* (m^*)^2$ for $\Delta^*$ the discriminant of $f^*$. Note that $k^*_S=k_S$ follows from $k^*=l^6k$ and from our assumptions $f,m\in\mathcal O[x,y]$ which assure that $l\in \mathcal O^\times$. Finally we deduce Corollary~J by simplifying the bounds in Corollary~\ref{cor:precthue}.
\begin{proof}[Proof of Corollary~J]
In the case $f,m\in\ZZ[x,y]$, we see that Corollary~\ref{cor:precthue} together with $h(\Delta)\leq 4h(f)+5\log 3$ implies (i) and (ii). In general, on following the proof of \eqref{eq:precthue} we reduce to the case $f^*,m^*\in\ZZ[x,y]$ and then we apply the already established case of Corollary~J with $f^*,m^*\in\ZZ[x,y]$. This completes the proof of Corollary~J.
\end{proof}
\section{Height bounds for Ramanujan--Nagell equations}\label{sec:ranaheight}
In this section we give explicit height bounds for the solutions of the generalized Ramanujan--Nagell equation. We also study pairs of units whose sum is a square or cube.
As in the previous sections let $S$ be a finite set of rational primes, write $h$ for the usual logarithmic Weil height and denote by $\mathcal O^\times$ the units of $\mathcal O=\ZZ[1/N_S]$ for $N_S=\prod_{p\in S} p$. Further let $b,c\in\mathcal O$ be nonzero and recall the generalized Ramanujan--Nagell equation
\begin{equation}
x^2+b=cy, \ \ \ \ \ (x,y)\in\mathcal O\times \mathcal O^\times. \tag{\ref{eq:rana}}
\end{equation}
To state our height bounds for the solutions of \eqref{eq:rana}, we define $a=bc^2$ and we denote by $\Omega_{\textnormal{sim}}=\Omega_{\textnormal{sim}}(a,S)$ the simplified height height bound given in Proposition~\ref{prop:m}. It holds $$\Omega_{\textnormal{opt}}\leq\Omega_{\textnormal{sim}}=\tfrac{1}{3}h(a)+\tfrac{4}{9}a_S\log a_S+\tfrac{1}{6}a_S\log\log\log a_S+\tfrac{2}{5}a_S$$
for $a_S$ as in \eqref{def:as} and $\Omega_{\textnormal{opt}}=\Omega_{\textnormal{opt}}(a,S)$ the optimized height bound in Proposition~\ref{prop:algobounds}.
The next result is a direct consequence of our height bounds for Mordell equations.
\begin{corollary}\label{cor:ranabounds}
If $(x,y)$ satisfies \eqref{eq:rana} then $h(x^2),h(y)\leq 3\Omega_{\textnormal{opt}}+3h(c)+8\log N_S.$
\end{corollary}
\begin{proof}
We write $y=\varepsilonilon y'^3$ with $y'\in\mathcal O^\times$ and $\varepsilonilon\in\ZZ_{\geq 1}$ dividing $N_S^2$. Then $u=\varepsilonilon c y'$ and $v=\varepsilonilon c x$ satisfy the Mordell equation $v^2=u^3+a'$ with $a'=-b(\varepsilonilon c)^2$. It holds that $a'\neq 0$ and $a'_S=a_S$, since $bc\neq 0$ and $\varepsilonilon\in\mathcal O^\times$. Thus Proposition~\ref{prop:algobounds} implies Corollary~\ref{cor:ranabounds}.
\end{proof}
It holds that $\Omega_{\textnormal{opt}}\leq \Omega_{\textnormal{sim}}$ and we observe that $3\Omega_{\textnormal{sim}}+8\log N_S$ is at most $2a_S+h(a)$. Therefore we see that Corollary~\ref{cor:ranabounds} proves Corollary~K stated in the introduction.
\begin{remark}[Generalization]Suppose that $f\in\mathcal O[x]$ is a polynomial of degree two, with nonzero discriminant $\Delta$. Then we claim that Corollary~\ref{cor:ranabounds} gives an explicit
height bound for any solution $(x,y)\in\mathcal O\times\mathcal O^\times$ of the more general Diophantine equation $f(x)=cy.$
To prove this claim we suppose that $(x,y)\in\mathcal O\times\mathcal O^\times$ satisfies $f(x)=cy$. We write $f(x)=a_1x^2+a_2x+a_3$ with $a_i\in\mathcal O$ and we put $x'=2a_1x+a_2$. It follows that $(x',y)$ is a solution of \eqref{eq:rana} with parameters $b=-\Delta$ and $c=4a_1c$. Hence an application of Corollary~\ref{cor:ranabounds} with $(x',y)$ gives an explicit upper bound for $h(x)$ and $h(y)$ as claimed.
\end{remark}
We next consider the problem of finding all $(u,v)\in\mathcal O^\times\times\mathcal O^\times$ with $u+v$ a square or cube in $\QQ$. To obtain here finiteness statements, one has to work modulo the actions of $\mathcal O^\times$ arising naturally in this context. The above Corollary~\ref{cor:ranabounds} leads to the following fully explicit results which involve the quantity $\Omega=3\Omega_{\textnormal{opt}}(1,S)+9\log N_S$.
\begin{corollary}\label{cor:sumsofunits}
Assume that $u,v$ are in $\mathcal O^\times$. Then the following statements hold.
\begin{itemize}
\item[(i)] If $u+v$ is a square in $\QQ$, then there is $\varepsilonilon\in\mathcal O^\times$ such that $h(\varepsilonilon^2u),h(\varepsilonilon^2v)\leq \Omega$.
\item[(ii)] If $u+v$ is a cube in $\QQ$, then there is $\delta\in\mathcal O^\times$ such that $h(\delta^3u),h(\delta^3v)\leq \Omega$.
\end{itemize}
\end{corollary}
\begin{proof} To prove (i) we assume that $u+v$ is a square in $\QQ$. Then there is $\varepsilonilon\in\mathcal O^\times$ such that $\varepsilonilon^2u$ and $\varepsilonilon^2v$ are in $\ZZ$, with $\varepsilonilon^2u+\varepsilonilon^2v$ a perfect square and $\gcd(\varepsilonilon^2u,\varepsilonilon^2v)$ square-free. If $m,n$ are in $\ZZ\cap\mathcal O^\times$ with $m+n$ a perfect square and $\gcd(m,n)$ square-free, then we claim that
\begin{equation}\label{claimsquare}
h(n)\leq \Omega.
\end{equation}
To prove this inequality we take $l\in\ZZ$ with $l^2=m+n$ and we write $m=m'^2m_0$ with $m',m_0\in\ZZ$ such that $m_0\mid N_S$. Further we define $x=l/m'$ and $y=n/m'^2$. Then $(x,y)$ satisfies \eqref{eq:rana} with $b=-m_0$ and $c=1$. Thus an application of Corollary~\ref{cor:ranabounds} with $(x,y)$ implies \eqref{claimsquare} as claimed. Here we used that $n$ and $m'$ are coprime which follows from our assumption that $\gcd(m,n)$ is square-free. Now, an application of \eqref{claimsquare} with $m=\varepsilonilon^2u$ and $n=\varepsilonilon^2v$ shows that $\varepsilonilon$ has the desired properties. This proves assertion (i).
The following proof of (ii) uses the arguments of (i) with some modifications. We assume that $u+v$ is a cube in $\QQ$. Then there is $\delta\in\mathcal O^\times$ such that $\delta^3u$ and $\delta^3v$ are in $\ZZ$, with $\delta^3u+\delta^3v$ a perfect cube and $\gcd(\delta^3u,\delta^3v)$ cube-free. We now consider the following claim: If $m,n$ are in $\ZZ\cap\mathcal O^\times$ with $m+n$ a perfect cube and $\gcd(m,n)$ cube-free, then
\begin{equation}\label{claimcube}
h(n)\leq \Omega.
\end{equation}
To prove this claim we take $l\in\ZZ$ with $l^3=m+n$ and we write $m=m'^3m_0$ with $m',m_0\in\ZZ$ such that $m_0\mid N_S^2$. Further we define $x=l/m'$ and $y=n/m'^3$. Then $(x,y)$ lies in $\mathcal O\times\mathcal O^\times$ and satisfies $x^3-m_0=y$. On writing $y=w y'^2$ with $y'\in\mathcal O^\times$ and $w\in\ZZ$ dividing $N_S$, we see that $(xw,y'w^2)$ is a solution of the Mordell equation \eqref{eq:mordell} with parameter $a=-m_0w^3$. Here $a\in \ZZ$ is nonzero. Thus on using that $\gcd(m,n)$ is cube-free, we see that Proposition~\ref{prop:algobounds} implies our claim in \eqref{claimcube}. Finally we deduce (ii) by applying \eqref{claimcube} with $m=\delta^3u$ and $n=\delta^3v$. This completes the proof of Corollary~\ref{cor:sumsofunits}.
\end{proof}
We now suppose that $m,n\in\ZZ$ are coprime. Then $\gcd(m,n)$ is in particular square-free and cube-free, and $m,n$ are in $\mathcal O^\times$ for $S=\{p\,;\, p\mid mn\}$ with $N_S=\textnormal{rad}(mn)$. Therefore \eqref{claimsquare} and \eqref{claimcube} imply Corollary~L stated in the introduction.
\section{Height bounds for Mordell and $S$-unit equations}\label{sec:heightbounds}
In this section we prove the results of Section~\ref{sec:mordellcoates}, and we establish in Proposition~\ref{prop:algobounds} the height bounds for Mordell and $S$-unit equations which are used in the algorithms of Sections~\ref{sec:suheightalgo} and~\ref{sec:mheightalgo}. The plan of Section~\ref{sec:heightbounds} is as follows: In Section~\ref{sec:simpleversions} we give simplified versions of the height bounds. Then we prove the results for Mordell and $S$-unit equations in Sections~\ref{sec:mproofs} and~\ref{sec:suproofs} respectively. Finally, in Section~\ref{sec:hc}, we work out the height conductor inequalities for elliptic curves over $\QQ$ which are used in our proofs.
\subsection{Simplified versions}\label{sec:simpleversions}
The precise form of our height bounds in Proposition~\ref{prop:algobounds} is fairly complicated.
To give the reader an idea of the size of the used height bounds, we worked out simplified versions of our bounds for Mordell equations (see Proposition~\ref{prop:m}) and for $S$-unit equations (see Propositions~\ref{prop:su} and~\ref{prop:abc}). These simplified versions slightly improve several estimates in the literature and they will allow us (up to some extent) to compare our optimized height bounds with results based on the theory of logarithmic forms.
As in the previous sections we let $S$ be a finite set of rational primes, we write $\mathcal O=\ZZ[1/N_S]$ for $N_S=\prod_{p\in S} p$, and we denote by $h$ the logarithmic Weil height.
\subsubsection{Mordell equation}\label{sec:simpleboundsm}
Let $a\in\mathcal O$ be nonzero and let $a_S=1728N_S^2\prod_{p\notin S}p^{\min(2,\ord_p(a))}$ be as in (\ref{def:asr2}). On using and refining the arguments of \cite[Cor 7.4]{rvk:modular}, we obtain the following result.
\begin{proposition}\label{prop:m}
If $x,y\in\mathcal O$ satisfy $y^2=x^3+a$, then
$$
\max\bigl(h(x),\tfrac{2}{3}h(y)\bigl)\leq \tfrac{1}{3}h(a)+\tfrac{4}{9}a_S\log a_S+\tfrac{1}{6}a_S\log\log\log a_S+\tfrac{2}{5}a_S.
$$
\end{proposition}
To compare this result with the actual best height bounds for Mordell equations~\eqref{eq:mordell}, we suppose that $x,y\in\mathcal O$ satisfy $y^2=x^3+a$ and we observe that Proposition~\ref{prop:m} implies
\begin{equation}\label{eq:simplemordell}
\max\bigl(h(x),h(y)\bigl)\leq \tfrac{1}{2}h(a)+a_S\log a_S.
\end{equation}
We first consider the classical case $\mathcal O=\ZZ$. The discussions in \cite[Sect 7.3]{rvk:modular} show that~\eqref{eq:simplemordell} improves the results of Baker~\cite{baker:mordellequation}, Stark~\cite{stark:mordell} and Juricevic~\cite{juricevic:mordell} which are all based on the theory of logarithmic forms~\cite{bawu:logarithmicforms}.
In fact Proposition~\ref{eq:simplemordell} provides the actual best height bound for Mordell equations~\eqref{eq:mordell}, since it updates the inequality $\max\bigl(h(x),h(y)\bigl)\leq h(a)+4\cdot 36a_S\log (36a_S)^2$ in \cite[Cor 7.4]{rvk:modular}.
We now discuss the case $\mathcal O\supsetneq \ZZ$. On using the theory of logarithmic forms, Hajdu--Herendi \cite[Thm 2]{hahe:elliptic} obtained explicit height bounds for the solutions in $\mathcal O$ of arbitrary elliptic Diophantine equations. The discussion in \cite[Sect 7.3.3]{rvk:modular} shows that Proposition~\ref{prop:m} improves the case of Mordell equations~\eqref{eq:mordell} in \cite[Thm 2]{hahe:elliptic} for ``small'' sets $S$, in particular for
any set $S$ with $N_S\leq 2^{1200}$ or
$\mathcal As{S}\leq 12$.
If $N_S\gg\lvert a\rvert$ then there are sets $S$ for which Proposition~\ref{prop:m} is better than \cite[Thm 2]{hahe:elliptic}, and vice versa. To conclude our comparisons we mention that the theory of logarithmic forms allows to deal with more general Diophantine equations over arbitrary number fields; see~\cite{bawu:logarithmicforms}.
\subsubsection{$S$-unit equations}\label{sec:simpleboundssu}
On using and refining the arguments of \cite[Thm 1.1]{mupa:modular} and \cite[Cor 7.2]{rvk:modular}, which were discovered independently in 2011 by Murty--Pasten and by the first mentioned author, we obtain the following update of the explicit height bounds in \cite[Thm 1.1]{mupa:modular} and \cite[Cor 7.2]{rvk:modular}; see also Frey's remark in \cite[p.544]{frey:ternary} and Proposition~\ref{prop:abc}.
\begin{proposition}
\label{prop:su}
Any solution $(x,y)$ of the $S$-unit equation~\eqref{eq:sunit} satisfies $$\max\bigl(h(x),h(y)\bigl) \leq \tfrac{5}{2}N_S\log N_S+9N_S \ \textnormal{ and }$$
$$\max\bigl(h(x),h(y)\bigl) \leq \tfrac{12}{5}N_S\log N_S+\tfrac{9}{10}N_S\log\log\log (16N_S)+8.26N_S+28.
$$
\end{proposition}
To compare this result with the actual best height bounds for $S$-unit equations~\eqref{eq:sunit}, we suppose that $(x,y)$ satisfies~\eqref{eq:sunit} and we write $h=\max(h(x),h(y))$. Starting with Gy{\H{o}}ry~\cite{gyory:sunitshelvetica} several authors proved explicit bounds for $h$ by using the theory of logarithmic forms~\cite{bawu:logarithmicforms}; see the references in~\cite{gyyu:sunits}. In particular Gy{\H{o}}ry--Yu \cite[Thm 2]{gyyu:sunits} obtained that
$h\leq 2^{10s+22}s^4q\prod\log p$
with the product taken over all rational primes $p\in S-\{q\}$ for $q=\max S$ and $s=\mathcal As{S}$. Further Proposition~\ref{prop:su} updates the inequalities $h\leq 4.8N_S\log N_S+13N_S+25$ in Murty--Pasten \cite[Thm 1.1]{mupa:modular} and $h\leq 3\cdot 2^6N_S\log(2^7N_S)^2+65$ in \cite[Cor 7.2]{rvk:modular}.
Hence Proposition~\ref{prop:su} establishes the actual best height bound for all sets $S$ with ``small"~$N_S$, in particular for all sets $S$ with $N_S\leq 2^{90}.$ Further we see that there are sets $S$ with arbitrarily large $N_S$ for which Proposition~\ref{prop:su} is sharper than \cite[Thm 2]{gyyu:sunits}, and vice versa. If $N_S\to \infty$ then our bounds (see also~\eqref{eq:asymptoticsu}) are worse than $h\leq O(N_S^{1/3}(\log N_S)^3)$ in Stewart--Yu \cite[Thm 1]{styu:abc2}. To conclude our comparison, we mention that the theory of logarithmic forms and Bombieri's refinement of the Thue--Siegel method both give effective height bounds for the solutions of $S$-unit equations in arbitrary number fields; see for example~\cite{gyyu:sunits,boco:effdioapp2}.
\subsection{Notation}\label{sec:notations}
In the remaining of Section~\ref{sec:heightbounds} we shall use throughout the following notation. Let $S$ be a finite set of rational primes. We write $\mathcal O=\ZZ[1/N_S]$ for $N_S=\prod_{p\in S}p$ and we denote by $h$ the usual logarithmic Weil height. For any elliptic curve $E$ over~$\QQ$, we denote by $c_4$ and $c_6$ the usual quantities (see for example~\cite{tate:aoe}) associated to a minimal Weierstrass equation of $E$ over~$\ZZ$, and we write $N_E$ and $\Delta_E$ for the conductor and minimal discriminant of $E$ respectively. We denote by $h(E)$ the relative Faltings height of~$E$, defined for example in \cite[Sect 2]{rvk:modular} using Faltings' original normalization \cite[p.354]{faltings:finiteness} of the metric.
\subsection{Mordell equation}\label{sec:mproofs}
We use the notation introduced in Section~\ref{sec:notations} above. Let $a\in \mathcal O-\{0\}$ and recall that $a_S=1728N_S^2r_2(a)$ for $r_2(a)=\prod p^{\min({2,\ord_p(a)})}$ with the product taken over all rational primes $p$ not in~$S$. Further we recall the Mordell equation
\begin{equation}
y^2=x^3+a, \ \ \ (x,y)\in\mathcal O\times\mathcal O.\tag{\ref{eq:mordell}}
\end{equation}
The following lemma gives an explicit Par\v{s}in{} construction for the solutions of this equation.
\begin{lemma}\label{lem:pm2}
Suppose that $(x,y)$ satisfies the Mordell equation~\eqref{eq:mordell}. Then there exists an elliptic curve $E$ over $\QQ$ with the following properties. It holds that $c_4=u^4x$ and $c_6=u^6y$ for some $u\in \QQ$ with $u^{12}=1728\Delta_E\mathcal As{a}^{-1}$, the conductor $N_E$ divides $a_S$ and $$\max\bigl(h(x),\tfrac{2}{3}h(y)\bigl)\leq\tfrac{1}{3}h(a)+8h(E)+2\log\max(1,h(E))+36.$$
\end{lemma}
We conducted here some effort to refine the weaker result $N_E\mid 2^23^2a_S$ which follows directly from \cite[Prop 3.4]{rvk:modular}. In fact Lemma~\ref{lem:pm2} is based on \cite[Prop 3.4]{rvk:modular}, and we mention that the height inequality in \cite[Prop 3.4]{rvk:modular} relies inter alia on explicit versions of certain results of Faltings~\cite{faltings:finiteness,faltings:arithmeticsurfaces} and Silverman~\cite{silverman:arithgeo}.
\begin{proof}[Proof of Lemma~\ref{lem:pm2}]
Suppose that $(x,y)$ is a solution of~\eqref{eq:mordell}. Then an application of \cite[Prop 3.4]{rvk:modular} with $S=\textnormal{Spec}(\mathcal O)$ and $T=\textnormal{Spec}(\mathcal O[1/(6a)])$ gives an elliptic curve over~$T$, with generic fiber $E=E_\QQ$, such that $N_E\mid 2^{2}3^2a_S$ and such that $h(x)$ is bounded in terms of $h(E)$ and $h(a)$ as desired. Let $c_4$ and $c_6$ be the quantities associated to~$E$. The proof of \cite[Prop 3.4]{rvk:modular} shows in addition that
$c_4=u^4x$ and $c_6=u^6y$ for some $u\in \QQ$ with $u^{12}=1728\Delta_E\mathcal As{a}^{-1}$. Thus on combining \cite[Lem 3.3]{rvk:modular} with \cite[Lem 3.5]{rvk:modular}, we deduce an upper bound for $h(y)=\frac{1}{2}h(u^{-12}c_6^2)$ in terms of $h(E)$ and $h(a)$ as desired.
Furthermore, it was shown in the proof of \cite[Prop 3.4]{rvk:modular} that $E$ admits a Weierstrass model
$W$ over $\mathcal O$ defined by the Weierstrass equation $t^2 = s^3 - 27xs - 54y$
with ``indeterminates'' $s$ and~$t$. This Weierstrass equation has discriminant $\Delta'=-2^63^9a$.
It remains to show that $N_E\mid a_S$. For this purpose, we observe that $2^83^5=2^23^2\cdot 1728$ and we recall that $N_E\mid 2^23^2a_S$. Hence it suffices to consider the exponents $f_2$ and $f_3$ of $N_E$ at the primes 2 and 3 respectively. They always satisfy $f_2\leq 8$ and $f_3\leq 5$, see for example \cite[Thm 6.2]{brkr:conductor}. Thus we obtain that $f_2\leq 8= \ord_2(a_S)$ if $2\in S$ and $f_3\leq 5= \ord_3(a_S)$ if $3\in S$, and then we see that the desired result $N_E\mid a_S$ holds if $2\in S$ or $3\in S$. Hence we are left to treat the case where 2 and 3 are both not in $S$, or equivalently that $2,3\in \textnormal{Spec}(\mathcal O)$. In this case, after localizing at 2 and 3, we may and do view the model $W$ over $\mathcal O$ as an integral model of $E$ over the local rings at 2 and 3 respectively.
We first consider~$f_2$. It holds that $\ord_2(\Delta_E)\leq \ord_2(\Delta')$, since $W$ is a Weierstrass model of $E$ over the local ring at 2 and since $\Delta_E$ is minimal at~2. Thus the formula of Ogg--Saito \cite[Cor 2]{saito:conductor} implies that $f_2\leq \ord_2(\Delta')=6+\ord_2(a)$, and then the inequality $f_2\leq 8$ proves that $f_2\leq 6+\min(2,\ord_2(a))=\ord_2(a_S)$ as desired.
We now consider~$f_3$. If $W$ is not the minimal Weierstrass model of $E$ over the local ring at~$3$, then we get that $\ord_3(\Delta')\geq 12$ and hence $\ord_3(a)\geq 3$. This together with $f_3\leq 5$ shows the desired inequality $f_3\leq 3+\min(2,\ord_3(a))=\ord_3(a_S)$ when $W$ is not minimal at~$3$. Hence we may and do assume in addition that $W$ is minimal at~$3$. Then we claim that the ``reduction" of $E$ at $3$ is of Kodaira type IV$^*$, III$^*$, or II$^*$ and that $f_3 = \ord_3(\Delta')-6$, $f_3=\ord_3(\Delta')-7$, or $f_3=\ord_3(\Delta')-8$ respectively. It follows that $f_3\leq 3+\ord_3(a)$ and this together with the inequality $f_3\leq 5$ shows that $f_3\leq 3+\min(2,\ord_3(a))=\ord_3(a_S)$ as desired. It remains to verify our claim.
For this purpose we use Tate's algorithm~\cite{tate:algo} and we let $a_1,\ldots,a_6$ and $b_2,\ldots,b_8$ denote the usual quantities associated to the Weierstrass equation $t^2 = s^3 - 27xs - 54y$ which is minimal at 3 by assumption.
We observe that
$3\divides\Delta'$,
$3\divides a_1=0$,
$3\divides a_2=0$,
$3^3\divides a_3=0$,
$3^2\divides a_4=-27 x$,
$3^3\divides a_6=-54y$,
$3\divides b_2=0$,
$3^3\divides b_6=-216y$, and
$3^3\divides b_8=-729x^2$.
This brings us into case 6) of Tate's algorithm.
Here one considers the polynomial $P(T):=T^3+a_{2,1}T^2+a_{4,2}T+a_{6,3}=T^3-3xT-2y$. In~$\FF_3$, it reduces to $T^3-2y$ which is purely inseparable with $3$-fold root~$2y$. Therefore 8), 9) and 10) of Tate's algorithm prove our claim. This completes the proof of Lemma~\ref{lem:pm2}.\end{proof}
In what follows we need to control the function $\nu(n)=n\nu^*(n)$ on $\ZZ_{\geq 1}$, where $\nu^*$ is defined in Proposition~\ref{prop:explbounds}~(ii). If $m,n$ are in $\ZZ_{\geq 1}$ with $m$ dividing $n$, then it holds
\begin{equation}\label{eq:nuineq}
\nu(m)\leq \nu(n).
\end{equation}
Indeed this inequality directly follows by unwinding the definitions and by taking into account rational primes $p$ with $\ord_p(m)=1$ and $\ord_p(n)\geq 2$. We now combine the above Lemma~\ref{lem:pm2} with Proposition~\ref{prop:explbounds} in order to prove Proposition~\ref{prop:m}.
\begin{proof}[Proof of Proposition~\ref{prop:m}]
Suppose that $(x,y)$ is a solution of Mordell's equation~\eqref{eq:mordell}. Then Lemma~\ref{lem:pm2} provides an elliptic curve $E$ over $\QQ$ such that $N_E\mid a_S$ and such that $$\max\bigl(h(x),\tfrac{2}{3}h(y)\bigl)\leq\tfrac{1}{3}h(a)+8h(E)+2\log\max(1,h(E))+36.$$
It follows from~\eqref{eq:nuineq} that $\nu(N_E)\leq \frac{2}{3}a_S$.
Therefore on combining the displayed inequality with the explicit bound for $h(E)$ in terms of $\nu(N_E)$ given in Proposition~\ref{prop:explbounds}~(ii), we deduce an estimate for $\max\bigl(h(x),\frac{2}{3}h(y)\bigl)$ in terms of $a_S$ and $h(a)$ as stated in Proposition~\ref{prop:m}.
\end{proof}
We recall that a solution $(x,y)\in\ZZ\times\ZZ$ of~\eqref{eq:mordell} is primitive if $\pm 1$ are the only $n\in\ZZ$ with $n^{6}\mid \gcd(x^3,y^2)$. The following result refines Lemma~\ref{lem:pm2} for primitive solutions.
\begin{lemma}\label{lem:pm3}
Suppose that $(x,y)\in\ZZ\times\ZZ$ is a primitive solution of~\eqref{eq:mordell}. If Lemma~\ref{lem:pm2} associates $(x,y)$ to the elliptic curve $E$, then $E$ has in addition the following properties.
\begin{itemize}
\item[(i)] It holds that $\Delta_E=2^m3^n\lvert a\rvert$ with $m\in\{-6,6\}$ and $n\in\{-3,9\}$.
\item[(ii)] The curve $E$ is semi-stable at each rational prime $p\geq 5$ with $\gcd(x,y,p)=1$.
\item[(iii)] There is the refined height inequality $$\max\bigl(h(x),\tfrac{2}{3}h(y)\bigl)\leq 4h(E)+2\log\max(1,h(E))+28.$$
\end{itemize}
\end{lemma}
\begin{proof}
Let $(x,y)\in\ZZ\times\ZZ$ be a primitive solution of~\eqref{eq:mordell}. Suppose that Lemma~\ref{lem:pm2} associates $(x,y)$ to the elliptic curve $E$ over $\QQ$. To prove (i) we recall from the proof of Lemma~\ref{lem:pm2} that $t^2 = s^3 - 27xs - 54y$ is a Weierstrass equation of $E$
with ``indeterminates'' $s$ and $t$ and with discriminant $\Delta'=-2^63^9a$. This Weierstrass equation has associated quantities $c_4'=6^4x$ and $c_6'=6^6y$. On using that $x,y$ are both in $\ZZ$ and that $\Delta_E$ is the discriminant of a minimal Weierstrass model of $E$ over $\ZZ$, we obtain a nonzero $u\in\ZZ$ with
\begin{equation}\label{eq:mc4c6}
u^4c_4=6^4x, \ \ \ u^6c_6=6^6y, \ \ \ u^{12}\Delta_E=2^63^9\mathcal As{a}.
\end{equation}
It follows that $u^{12}$ divides $6^{12}\gcd(x^3,y^2)$ and hence we deduce that $\mathcal As {u}\in\{1,2,3,6\}$ since $(x,y)$ is primitive by assumption. Therefore the equality $u^{12}\Delta_E=2^63^9\mathcal As{a}$ implies that $2^m3^n\lvert a\rvert=\Delta_E$ with $m\in\{-6,6\}$ and $n\in\{-3,9\}$. This proves assertion~(i).
To show the semi-stable properties of $E$ claimed in~(ii), we take a rational prime $p$. If $p\nmid \Delta_E$ then $E$ has good (and thus semi-stable) reduction at $p$. We now assume that $p\mid \Delta_E$, that $p\geq 5$ and that $\gcd(x,y,p)=1$.
Then~\eqref{eq:mc4c6} together with $\mathcal As{u}\in \{1,2,3,6\}$ implies that $\gcd(c_4,c_6,p)=1$. Hence we obtain that $p\nmid c_4$ since $p\geq 5$ divides $12^3\Delta_E=\mathcal As{c_4^3-c_6^2}$ by assumption. Therefore the semi-stable criterion in \cite[p.196]{silverman:aoes} shows that $E$ has semi-stable reduction at each $p\geq 5$ with $\gcd(x,y,p)=1$. This proves assertion~(ii).
It remains to show the refined height inequality in~(iii). The identities in~\eqref{eq:mc4c6} together with \cite[Lem 3.5]{rvk:modular} and $\mathcal As{u}\leq 6$ lead to an explicit upper bound for $\max\bigl(h(x),\frac{2}{3}h(y)\bigl)$ in terms of $h(E)$ as claimed in~(iii). This completes the proof of Lemma~\ref{lem:pm3}.
\end{proof}
The above proof shows in addition that Lemma~\ref{lem:pm3} holds more generally for all solutions $(x,y)\in\ZZ\times\ZZ$ of~\eqref{eq:mordell} such that $\pm 1$ are the only $n\in\ZZ$ with $n^{12}\mid \gcd(x^3,y^2)$. We now use Lemma~\ref{lem:pm3} and Proposition~\ref{prop:m} to prove Theorem~\ref{thm:m}.
\begin{proof}[Proof of Theorem~\ref{thm:m}]
We begin to prove the first part of assertion~(i). Let $\mu:\ZZ_{\geq 1}\to\mathbb R_{\geq 0}$ be an arbitrary function, and suppose that $(x,y)$ is a solution of~\eqref{eq:mordell} which is almost primitive with respect to~$\mu$. Let $u=u_{x,y}$ be as in Definition~\ref{def:ap}, and define $x'=u^2x$ and $y'=u^3y$. Then it follows that $(x',y')\in\ZZ\times\ZZ$ is a primitive solution of the Mordell equation $y'^2=x'^3+a'$ for $a'=u^6a\in\ZZ$. Therefore Lemmas~\ref{lem:pm2} and~\ref{lem:pm3} give an elliptic curve $E$ over $\QQ$ together with integers $m\in \{-6,6\}$ and $n\in\{-3,9\}$ such that
\begin{equation}\label{eq:deltaea}
\Delta_E=2^{m}3^n\mathcal As{a'} \ \ \ \textnormal{ and } \ \ \ N_E\mid a'_{S}.
\end{equation}
The construction of the number $u=u_1/u_2$ in Definition~\ref{def:ap} shows that any rational prime $p$ with $\ord_p(a')>\ord_p(a)$ satisfies $p\mid u_1$ and thus $p\in S$ since $x,y\in \mathcal O$. It follows that $\ord_p(a')\leq\ord_p(a)$ for all rational primes $p$ not in $S$ and therefore we find that $a'_S$ divides $a_S.$
Further it holds that $h(a)\leq 6h(u)+\log\mathcal As{a'}$ and $h(u)\leq \mu(a_S)$ since $(x,y)$ is almost primitive with respect to~$\mu$. Then on combining~\eqref{eq:deltaea} and $N_E\mid a'_S\mid a_{S}$ with the estimate for $\log \Delta_E$ in terms of $N_E$ given in~\eqref{eq:szpiro}, we derive an explicit upper bound for $h(a)$ in terms of $a_S$ and $\mu(a_S)$ which together with Proposition~\ref{prop:m} implies the first part of~(i).
We now show the second part of~(i). Since $(x,y)$ is almost primitive with respect to $\mu$ we obtain that $h(x)\leq 2\mu(a_S)+h(x')$ and $\frac{2}{3}h(y)\leq 2\mu(a_S)+\frac{2}{3}h(y')$. Hence an application of Lemma~\ref{lem:pm3}~(iii) with the primitive $(x',y')\in\ZZ\times\ZZ$ leads to
\begin{equation}\label{eq:primitivesolestimate}
\max\bigl(h(x),\tfrac{2}{3}h(y)\bigl)\leq 2\mu(a_S)+4h(E)+2\log\max(1,h(E))+28.
\end{equation}
Then on combining~\eqref{eq:deltaea} and $a'_S\mid a_{S}$ with the explicit estimate for $h(E)$ in terms of $N_E$ given in Proposition~\ref{prop:explbounds}, we deduce an inequality as claimed in the second part of~(i).
To prove~(ii) we may and do assume that $\mathcal As{a}\to \infty$ and that $S$ is empty. Let $(x,y)\in\ZZ\times\ZZ$ be a primitive solution of~\eqref{eq:mordell}, and take $(x',y')=(x,y)$ and $a'=a$ in the proof of~(i). Then~\eqref{eq:deltaea} and~\eqref{eq:szpiro} show that $\log\mathcal As{a}\leq O( N_E^2)$ and thus $\mathcal As{a}\to \infty$ forces $N_E\to\infty$. Hence on using~\eqref{eq:primitivesolestimate} with $\mu=0$ and on applying the asymptotic bound for $h(E)$ in terms of $N_E$ given in Proposition~\ref{prop:explbounds}, we see that~\eqref{eq:deltaea} together with $a_S=a_*$ leads to the asymptotic estimate claimed in~(ii). This completes the proof of Theorem~\ref{thm:m}.
\end{proof}
Next we give a proof of the inequality claimed in Corollary~\ref{cor:coates1} by using a version of Theorem~\ref{thm:m}~(i) which takes into account Lemma~\ref{lem:pm3}~(ii).
\begin{proof}[Proof of Corollary~\ref{cor:coates1}]
Recall that $a\in\ZZ$ is nonzero, $S$ is an arbitrary finite set of rational primes and $f\in\ZZ$ is the largest divisor of $a$ which is only divisible by primes in~$S$. We take $(x,y)\in\ZZ\times\ZZ$ with $y^2-x^3=a$ and $\gcd(x,y,N_S)=1$ as in the statement.
We first show that $(x,y)$ is almost primitive with respect to $\mu=\frac{1}{6}\log\mathcal As{a/f}$. Let $m=u_2\in\ZZ$ be maximal such that $m^{6}\mid\gcd(x^3,y^2)$, and define $x'=m^{-2}x$ and $y'=m^{-3}y$. Then $(x',y')\in\ZZ\times\ZZ$ is a primitive solution of the Mordell equation $y'^2=x'^3+a'$ for $a'=m^{-6}a\in\ZZ$. Further, $\gcd(x,y,N_S)=1$ together with $\textnormal{rad}(f)\mid N_S$ implies that $\gcd(x,y,f)=1$, and hence $\gcd(m,f)=1$ since $m$ divides $x$ and~$y$. Thus on using that $m^6\mid a$, we see that $m^{6}$ divides~$a/f$. This shows that the number $u_{x,y}=1/m$ in Definition~\ref{def:ap} satisfies $h(u_{x,y})\leq \mu$ as desired.
We now apply (a version of) Theorem~\ref{thm:m}~(i). Suppose that Lemma~\ref{lem:pm2} associates the primitive $(x',y')\in\ZZ\times\ZZ$ to the elliptic curve $E$ over $\QQ$. Lemma~\ref{lem:pm3}~(ii) gives that $E$ is semi-stable at each $p\in S$ with $p\geq 5$, since $\gcd(x',y',N_S)=1$. Then (the proof of) Lemma~\ref{lem:pm2} shows that $N_E$ divides $6\cdot 1728N_Sr_2(a')$, and hence we obtain that $N_E\mid 6\alpha_S$ since $a'\mid a$.
Thus, on replacing $a_S$ by $6\alpha_S=6a_S/N_S$ in the proof of Theorem~\ref{thm:m}~(i) and on taking $\mu=\frac{1}{6}\log\mathcal As{a/f}$, we deduce Corollary~\ref{cor:coates1}.
\end{proof}
\subsection{$S$-unit equations}\label{sec:suproofs}
We use the notation introduced in Section~\ref{sec:notations} above. The discussions in Section~\ref{sec:sucremalgo} show that bounding the Weil height of the solutions of the $S$-unit equation~\eqref{eq:sunit} is equivalent to estimating the absolute value of the integers satisfying the Diophantine equation
\begin{equation}
a+b = c, \quad a,b,c\in\ZZ-\{0\}, \quad \gcd(a,b,c)=1,\quad \textnormal{rad}(abc)\mid N_S. \tag{\ref{eq:abc}}
\end{equation}
We first prove Lemma~\ref{lem:psu2} which is used in the proof of Proposition~\ref{prop:su} given below.
\begin{lemma}\label{lem:psu2}
Suppose that $(a,b,c)$ is a solution of~\eqref{eq:abc}. Then there exist $\QQ$-isogenous elliptic curves $E$ and $E'$ over $\QQ$ such that $N_E$ divides $2^4N_S$, $\Delta_E=2^n(abc)^2$ with $n\in\{4,-8\}$, and $\Delta_{E'}=2^{8-12m}\mathcal As{ab}c^4$ with $m\in\{0,1,2,3\}$.
\end{lemma}
The proof of Lemma~\ref{lem:psu2} uses inter alia a formula of Diamond--Kramer~\cite{dikr:modularity} and classical results which are given for example in~\cite{silverman:aoes}. In fact the proof consists of explicit computations with Frey--Hellegouarch elliptic curves. A more conceptual proof of (parts of) Lemma~\ref{lem:psu2} can be given by using the point of view in \cite[Sect 3]{rvk:modular}.
\begin{proof}[Proof of Lemma~\ref{lem:psu2}]
We now use in particular several (known) reductions. To make sure that these reductions are compatible with each other we prefer to give all details.
We suppose that $(a,b,c)$ is a solution of~\eqref{eq:abc}. Then exactly one of the numbers $a, b, c$ is even, since they are coprime and satisfy $a+b=c$. We denote this number by $B'$. Now we define $A'=a$ if $B'=b$, $A'=b$ if $B'=a$, and $A'=-a$ if $B'=c$. Further we define $(A,B,C)=(A',B',A'+B')$ if $A'=-1\pmod{4}$, and $(A,B,C)=(-A',-B',-A'-B')$ otherwise. Let $E$ be the elliptic curve over $\QQ$, defined by the Weierstrass equation
$$y^2=x(x-A)(x+B)$$
with discriminant $\Delta=2^4(ABC)^2$.
We observe that $(A,B,C)$ has the following properties: $A,B,C$ are coprime, $A=-1\pmod{4}$, $B$ is even, $A+B=C$ and $(ABC)^2=(abc)^2$. Thus \cite[Lem 1 and Lem 2]{dikr:modularity} give that the conductor $N_E$ of $E$ divides $2^4\textnormal{rad}(abc)$. Our assumption, that $(a,b,c)$ satisfies~\eqref{eq:abc}, provides that $\textnormal{rad}(abc)\mid N_S$ and hence we obtain that $N_E$ divides~$2^4 N_S$.
It follows from \cite[Lem 2]{dikr:modularity} and \cite[p.257]{silverman:aoes} that the minimal discriminant $\Delta_E$ of $E$ satisfies $\Delta_E=\Delta$ if $\ord_2(abc)\leq 3$, and $\Delta_E=2^{-8}(abc)^2$ if $\ord_2(abc)\geq 4$. We conclude that $\Delta_E=2^n(abc)^2$ with $n\in\{4,-8\}$, and since $N_E$ divides $2^4N_S$ we see that the elliptic curve $E$ has all the desired properties.
To construct the elliptic curve $E'$, we notice that $A,B,C$ are in $\{\pm a,\pm b,\pm c\}$. We define $(\alpha,\beta,\gamma,x')=(A,B,C,x)$ if $C=\pm c$, $(\alpha,\beta,\gamma,x')=(C,-B,A,x+B)$ if $A=\pm c$, and $(\alpha,\beta,\gamma,x')=(-A,C,B,x-A)$ if $B=\pm c$. It follows that $\gamma=\pm c$, that $\alpha+\beta=\gamma$, that $(\alpha,\beta)=1$ and that $E$ admits the Weierstrass equation $y^2=x'(x'-\alpha)(x'+\beta).$
Then we obtain that $E$ is $\QQ$-isogenous to the elliptic curve $E'$ over $\QQ$, which is defined by the Weierstrass equation (see for example \cite[p.70]{silverman:aoes}) $$w^2=z^3-2(\beta-\alpha)z^2 +\gamma^2z$$
with discriminant $\Delta'=-2^8\alpha\beta\gamma^4$. This Weierstrass equation has associated quantities $c_4=16(\alpha^2-14\alpha\beta+\beta^2)$ and $c_6=64(\alpha^3+33\alpha^2\beta-33\alpha\beta^2-\beta^3)$. Further we observe that $\Delta'=-2^8abc^4$. To determine the minimal discriminant $\Delta_{E'}$ of $E'$ we use a strategy inspired by \cite[p.258]{silverman:aoes}. An application of the Euclidean algorithm gives the identities
\begin{align*}
4(395\alpha^2-430\alpha\beta-13\beta^2 )c_4 + (181\alpha - 13\beta)c_6 &= 2^{12} 3^2\alpha^4,\\
4(-13\alpha^2 - 430\alpha\beta + 395\beta^2)c_4 + (13\alpha - 181\beta)c_6 &= 2^{12} 3^2\beta^4.
\end{align*}
On using that $\Delta',c_4,c_6\in\ZZ$ and that $\Delta_{E'}$ is the absolute value of the discriminant of a minimal Weierstrass model of $E'$ over $\ZZ$, we obtain $u\in\ZZ$ such that $u^{12}\Delta_{E'}=\pm\Delta'$, $u^4\mid c_4$ and $u^6\mid c_6$.
Thus the displayed identities together with $(\alpha,\beta)=1$ imply that $u^4\mid 2^{12} 3^2$ and
hence $\pm u\in\{1,2,4,8\}$. We deduce that $\Delta_{E'}=2^{8-12m}\mathcal As{ab}c^4$ with $m\in\{0,1,2,3\}$, and therefore $E'$ has the desired properties. This completes the proof of Lemma~\ref{lem:psu2}.
\end{proof}
We remark that the application of \cite[Lem 2]{dikr:modularity} in the above proof shows in addition the following: Suppose that $(a,b,c)$ is a solution of~\eqref{eq:abc}. If Lemma~\ref{lem:psu2} associates $(a,b,c)$ to the elliptic curve $E$ over $\QQ$, then the conductor $N_E$ of $E$ satisfies
\begin{equation}\label{eq:refinedcondbound}
\ord_2(N_E)=\ee+1, \ \ \ (\ee,\lambda)=\begin{cases}
(4,12) & \textnormal{if }\ord_2(abc) = 1,\\
(2,3) & \textnormal{if }\ord_2(abc) = 2, 3,\\
(-1,\frac{1}{2}) & \textnormal{if }\ord_2(abc) = 4,\\
(0,1) & \textnormal{if }\ord_2(abc) \geq 5.\\
\end{cases}
\end{equation}
We note that it always holds that $\ord_2(abc)\geq 1$, since $a+b=c$ and $(a,b,c)=1$. The following result is in fact a (slightly) more precise version of Proposition~\ref{prop:su}.
\begin{proposition}\label{prop:abc}
Suppose that $(a,b,c)$ is a solution of~\eqref{eq:abc}, and let $(\ee,\lambda)$ be the numbers in~\eqref{eq:refinedcondbound} associated to $(a,b,c)$. Then it holds $$\log\max\left(\mathcal As{a},\mathcal As{b},\mathcal As{c}\right)\leq \tfrac{\lambda}{5}N_S\log(2^{\ee}N_S)+\tfrac{3\lambda}{40} N_S\log\log\log(2^{\ee}N_S)+\tfrac{2\lambda}{15}N_S+28.$$
\end{proposition}
\begin{proof}
We begin by noting that the elliptic curves appearing in Lemma~\ref{lem:psu2} have the same conductor since they are $\QQ$-isogenous. Thus an application of Lemma~\ref{lem:psu2} with the solution $(a,b,c)$ of~\eqref{eq:abc} gives an elliptic curve $E'$ over~$\QQ$, with conductor $N_{E'}$ and minimal discriminant~$\Delta_{E'}$, such that $\mathcal As{ab}c^4\leq 2^{28}\Delta_{E'}$ and such that $N_{E'}\mid 2^{\ee}N_S$ for $\ee$ as in~\eqref{eq:refinedcondbound}. The equality $a+b=c$ proves that $\mathcal As{c}-1\leq \mathcal As{ab}$
and then we deduce $$(\mathcal As{c}-1)^5\leq (\mathcal As{c}-1)c^4\leq \mathcal As{ab}c^4\leq 2^{28}\Delta_{E'}.$$
Further,
$N_{E'}\mid 2^{\ee}N_S$ with $\ord_2(N_{E'})=\ee+1$ implies that $\nu(N_{E'})\leq\lambda N_S$
for $\lambda$ as in~\eqref{eq:refinedcondbound} and for $\nu$ the function on $\ZZ_{\geq 1}$ defined in Proposition~\ref{prop:explbounds}~(ii). Then we see that~\eqref{eq:szpiro} leads to Proposition~\ref{prop:abc} provided that $H=\mathcal As{c}$ for $H=\max(\mathcal As{a},\mathcal As{b},\mathcal As{c})$. To deal with the remaining cases $H=\mathcal As{a}$ and $H=\mathcal As{b}$, we notice that $(b,-c,-a)$ and $(a,-c,-b)$ are solutions of~\eqref{eq:abc} as well. Thus applications of the above arguments with $(b,-c,-a)$ and $(a,-c,-b)$ prove Proposition~\ref{prop:abc} in the cases $H=\mathcal As{a}$ and $H=\mathcal As{b}$ respectively.
\end{proof}
Let $(a,b,c)$ be a solution of~\eqref{eq:abc}. Lemma~\ref{lem:psu2} associates to $(a,b,c)$ an elliptic $E$ over $\QQ$ with conductor $N_E$ that satisfies $N_E\to \infty $ if $\textnormal{rad}(abc)\to \infty$. Therefore the arguments of Proposition~\ref{prop:abc} together with the asymptotic bound obtained below~\eqref{eq:szpiro} show that any solution $(a,b,c)$ of~\eqref{eq:abc} with $\textnormal{rad}(abc)=r$ satisfies
\begin{equation}\label{eq:asymptoticsu}
\log \max(|a|,|b|,|c|)\leq \tfrac{9}{5}r\log r + O\Big(\frac{r\log r}{\log\log r}\Big) \ \ \textnormal{ for } r\to \infty.
\end{equation}
This (slightly) improves the bound $4r\log r+O(r\log\log r)$ obtained by Murty--Pasten in \cite[Thm 1.1]{mupa:modular}. However, our estimate displayed in~\eqref{eq:asymptoticsu} is still worse than the actual best asymptotic bound $O(r^{1/3}(\log r)^3)$ of Stewart--Yu \cite[Thm 1]{styu:abc2}.
\begin{proof}[Proof of Proposition~\ref{prop:su}] We suppose that $(x,y)$ satisfies the $S$-unit equation~\eqref{eq:sunit}. Then there exists a solution $(a,b,c)$ of~\eqref{eq:abc} with $(x,y)=(\frac{a}{c},\frac{b}{c})$. The number $\max(h(x),h(y))$ equals $\log\max(\mathcal As{a},\mathcal As{b},\mathcal As{c})$ and thus Proposition~\ref{prop:abc} implies Proposition~\ref{prop:su}.
\end{proof}
\subsection{Optimized height bounds and height conductor inequalities}\label{sec:hc}
We use the notation of Sections~\ref{sec:cremonas+st} and~\ref{sec:notations}. Let $N\geq 1$ be an integer. We now define constants $\alpha$, $\beta$ and $\beta^*$ depending on $N$, which will appear in the optimized height bounds.
\subsubsection{The constants $\alpha$, $\beta$, $\beta^*$ and optimized height bounds}\label{sec:optimizedbounds} To define $\beta$ and $\beta^*$ we let $m$ be the number of newforms of level dividing $N$, and we let $g=g(N)$ be the genus of $X_0(N)$. We write $l=\lfloor\frac{N}{6}\prod (p+1)\rfloor$ and $l^*=\lfloor\frac{N}{6}\prod (1+1/p)\rfloor$ with both products taken over all rational primes $p$ dividing $N$, where $\lfloor r\rfloor=\max(n\in\ZZ; \, n\leq r)$ for any $r\in\mathbb R$. Let $\tau(n)$ be the number of divisors of any $n\in\ZZ_{\geq 1}$. We define
\begin{equation}\label{def:bb*}
\beta=\tfrac{1}{2}m\log m+\max_J \sum_{j\in J}\log(\tau(j)j^{1/2}) \ \textnormal{ and } \ \beta^*=\tfrac{1}{2}g\log g+\max_J \sum_{j\in J}\log(\tau(j)j^{1/2})
\end{equation}
with the first maximum taken over all subsets $J\subseteq\{1,\dotsc,l\}$ of cardinality $m$ and with the second maximum taken over all subsets $J\subseteq\{1,\dotsc,l^*\}$ of cardinality~$g$. On comparing $\beta$ with $\beta^*$ we notice that $\beta$ involves the smaller number $m\leq g$ at the expense of depending on the larger parameter $l\geq l^*$. It turns out that $\beta\leq O(N\log N)$, while such an upper bound can not hold for $\beta^*$ since there is a constant $r\in\mathbb R$ such that infinitely many $n\in\ZZ_{\geq 1}$ satisfy $n\log\log n\leq rg(n)$.
On the other hand, it holds that $\beta^*< \beta$ for infinitely many $N$ and thus we shall work in our algorithms with the quantity $$\alpha=\min(\beta,\beta^*).$$
We define $\kappa=4\pi+\log(163/\pi)$ and we mention that in the statement of the following Proposition~\ref{prop:algobounds} we use the notation of Propositions~\ref{prop:m} and~\ref{prop:abc}.
\begin{proposition}\label{prop:algobounds}
Proposition~\ref{prop:abc} holds with the bound $\frac{6}{5}\alpha(2^{\ee}N_S)+28,$ and Proposition~\ref{prop:m} holds with the bound $\frac{1}{3}h(a)+4\alpha(a_S)+2\log(\alpha(a_S)+\kappa)+35+4\kappa$.
\end{proposition}
\begin{proof}
We observe that $\alpha(m)\leq \alpha(n)$ for all $m,n$ in $\ZZ_{\geq 1}$ with $m$ dividing $n$. Therefore Proposition~\ref{prop:algobounds} follows directly by using in the above proofs of Propositions~\ref{prop:m} and~\ref{prop:abc} the optimized bound given in Proposition~\ref{prop:explbounds}~(i).
\end{proof}
We remark that for any given $N$, one can practically compute $\alpha$ by using the formulas for $m$ and $g$ in \cite[Thm 4]{martin:dimension} and \cite[p.107]{dish:modular} respectively. However, if $N$ becomes large then the precise computation of $\alpha$ becomes slow and in this case we shall use
\begin{equation}\label{def:barb}
\bar{\beta}=\tfrac{1}{2}m\log m+\tfrac{5}{8}m(18+\log l) \ \textnormal{ and } \ \bar{\beta^*}=\tfrac{1}{2}g\log (gl^*)+\tfrac{1}{2}l^*\log(4+4\log l^*).
\end{equation}
Remark~\ref{rem:tau} gives that $\tau(j)\leq 45197j^{1/8}$ for all $j\in\ZZ_{\geq 1}$ and hence $\beta\leq\bar{\beta}$.
To prove that $\beta^*\leq \bar{\beta^*}$ we may and do assume that $g\geq 1$. Let $J\subseteq\{1,\dotsc,l^*\}$ be a subset of cardinality $g$ and observe that $g\leq n=\lfloor l^*/2\rfloor$.
Then the elementary inequalities $l^*\leq 4n$,
$\prod_{j\in J} \tau(j)\leq \bigl(\frac{1}{n}\sum_{j=1}^{l^*}\tau(j)\bigl)^n$
and $\frac{1}{l^*}\sum_{j=1}^{l^*}\tau(j)\leq 1+\log l^*$ show that $\sum_{j\in J}\log\tau(j)\leq n\log(4+4\log l^*)$. This implies that $\beta^*\leq\bar{\beta^*}$ as desired.
It follows that $\alpha\leq \bar{\alpha}$ for
$$\bar{\alpha}=\min(\bar{\beta},\bar{\beta^*}).$$
We take here the minimum
since there are infinitely many $N$ for which $\bar{\beta^*}<\bar{\beta}$ and vice versa. Finally we point out that the computation of $\bar{\alpha}$ is very fast, even for large~$N$.
It remains to work out the explicit height conductor inequalities for elliptic curves over $\QQ$ which are used in our proofs and this will be done in the next section.
\subsubsection{Height and conductor of elliptic curves over $\QQ$}\label{sec:heightcondstatement}
The geometric version of the Shimura--Taniyama conjecture gives that any elliptic curve $E$ over $\QQ$ of conductor $N$ is $\QQ$-isogenous to $E_f$ for some rational newform $f\in S_2(\Gamma_0(N))$, see Section~\ref{sec:cremonas+st}. We say that $f$ is the newform associated to $E$. Let $m_f$ be the modular degree of $f$ and let $r_f$ be the congruence number of $f$, defined in (\ref{def:mf}) and (\ref{def:rf}) respectively.
On using and refining the arguments\footnote{These arguments use an approach of Frey which involves the modular degree $m_f$ and the geometric version of the Shimura--Taniyama conjecture, see for example Frey~\cite[p.544]{frey:ternary}.} of \cite[Thm 7.1]{mupa:modular} and \cite[Prop 6.1]{rvk:modular}, which were discovered independently in 2011 by Murty--Pasten and by the first mentioned author, we obtain the following update of several results (see below) in~\cite{mupa:modular,rvk:modular}.
\begin{proposition}\label{prop:explbounds}
Let $\beta$ and $\beta^*$ be as in \textnormal{(\ref{def:bb*})}. Suppose that $E$ is an elliptic curve over $\QQ$ of conductor~$N$, with associated newform~$f$. Then the following statements hold.
\begin{itemize}
\item[(i)] Define $\kappa=4\pi+\log(163/\pi)$. There are inequalities
$$
2h(E)-\kappa\leq \log m_f \leq \log r_f\leq \alpha=\min(\beta,\beta^*).$$
\item[(ii)] Let $\nu^*$ be the multiplicative function on $\ZZ_{\geq 1}$ defined by $\nu^*(p)=1$ for $p$ a rational prime and $\nu^*(p^k)=1-1/p^{2}$ for $k\in\ZZ_{\geq 2}$, and put $\nu=N\nu^*(N)$. It holds $$\beta\leq \tfrac{1}{6}\nu\log N + \tfrac{1}{16}\nu\log\log\log N + \tfrac{1}{9}\nu, \ \ \ \nu\leq N.$$
\item[(iii)] If $N\to \infty$ then $\beta\leq \frac{1}{8}\nu\log N + \frac{(\frac{1}{6}\log 2+o(1))}{\log\log N}\nu\log N.$
\end{itemize}
\end{proposition}
It is known (see e.g.~\cite{frey:linksulm,silverman:arithgeo}) that $h(E)$ is related to~$m_f$, and a result attributed to Ribet provides that~$m_f\mid r_f$. We now compare Proposition~\ref{prop:explbounds} with the literature. It improves $h(E)\leq (2N)^{40^2}$ in\footnote{The result \cite[Thm 3.6]{rvk:thesis} bounds a ``naive" height of $E$. However, Silverman's arguments in \cite[Sect 2]{silverman:arithgeo} lead to an explicit upper bound for $h(E)$ in terms of the ``naive" height of $E$ used in~\cite{rvk:thesis}.} \cite[Thm 3.6]{rvk:thesis} and $h(E)\leq (25N)^{162}$ in \cite[Thm 3.1]{rvk:height} which were established by different methods: \cite{rvk:thesis} uses the effective reduction theory of Evertse--Gy{\H{o}}ry and~\cite{rvk:height} combines Legendre level structure with the theory of logarithmic forms. Furthermore, Proposition~\ref{prop:explbounds} updates $2h(E)\leq \frac{1}{5}N\log N+22$ in Murty--Pasten \cite[Thm 7.1]{mupa:modular} and $2h(E)\leq \frac{1}{2}N(\log N)^2+18$ in \cite[Prop 6.1]{rvk:modular}. It also updates the asymptotic bound $2h(E)\leq \frac{1}{6}N\log N+O(N\log\log N)$ in Murty--Pasten \cite[Thm 7.1]{mupa:modular} and the bounds for $m_f$ and $r_f$ in \cite[Thm 4.3]{mupa:modular} and \cite[Lem 5.1 (i)]{rvk:modular}.
We proved Proposition~\ref{prop:explbounds} in our unpublished 2014 preprint ``Solving S-unit and Mordell equations via Shimura--Taniyama conjecture". Hector Pasten told us that in Taipei (May 20, 2016) he will announce the following results which he obtained independently.
\begin{remark}[Recent results of Hector Pasten] Let $S$ be a finite set of rational primes. For all elliptic curves $E$ over $\QQ$ that are semistable outside S, we have $$h(E) \ll_S \phi(N)\log N.$$ Furthermore, if we assume GRH then we have $h(E) \ll_S \phi(N)\log \log N.$ Here $\phi$ denotes the Euler totient function, and the constants $\ll_S$ are effective, small and explicit. Further, it holds that $\log m_f\leq (o(1)+\tfrac{1}{24})N\log N$ as $N\to \infty$, and if $E$ is semistable then $\log m_f\leq (o(1)+\tfrac{1}{24})\phi(N)\log N $ as $N\to\infty$; both estimates are effective and can be made explicit with good constants. We thank Hector Pasten for sending us the statements of his results.
\end{remark}
\subsubsection{Proof of Proposition~\ref{prop:explbounds}}
As already mentioned, the proof of Proposition~\ref{prop:explbounds} uses the ideas of \cite[Thm 7.1]{mupa:modular} and \cite[Prop 6.1]{rvk:modular}; see also Frey \cite[p.544]{frey:ternary}. For example the arguments of \cite[Lem 5.1, Prop 6.1]{rvk:modular} directly lead to $2h(E)-\kappa\leq \log m_f \leq \log r_f\leq \beta^*.$
Then to replace here $\beta^*$ with $\beta$ we go through the proof of \cite[Lem 5.1]{rvk:modular} by using the ``coprime" matrix constructed in Murty--Pasten~\cite{mupa:modular}. To show (ii) we combine a formula of Martin~\cite{martin:dimension} with classical analytic number theory. The latter is used to explicitly bound the quantities $\prod_{p\mid n}(1+1/p)$ and $\tau(n)$ in terms of $n\in\ZZ_{\geq 1}$. Finally we deduce (iii) by replacing in the proof of (ii) our explicit estimate for $\tau(n)$ by Wigert's asymptotic bound.
\begin{proof}[Proof of Proposition~\ref{prop:explbounds}]
We first prove (i). To show that $\log r_f\leq \beta$ we denote by $\mathcal B=\{f_1,\ldots,f_g\}$ the Atkin--Lehner basis \cite[Thm 5]{atle:hecke} for $S_2(\Gamma_0(N))$, indexed in such a way such that $f_1=f$ and such that $f_1,\ldots,f_m$ are the newforms of level dividing $N$.
We write $I=\{1,\dotsc,m\}$ and $J'=\{j\in \{1,\dotsc,l\}; (j,N)=1\}$. On using Atkin--Lehner theory, Murty--Pasten proved in \cite[Prop 3.2]{mupa:modular} that the matrix $F'=(F_{ij})$ has full rank $m$ for $F_{ij}=a_j(f_i)$ with $i\in I$ and $j\in J'$.
Hence there is a subset $J\subseteq J'$ of cardinality $m$ such that the matrix $F=(F_{ij})$, with $i\in I$ and $j\in J$, is invertible. The Ramanujan--Petersson bounds for Fourier coefficients imply that $\lvert F_{ij}\rvert\leq\tau(j)j^{1/2}$ for all $i\in I$ and $j\in J$. Thus Hadamard's determinant inequality shows that $\log \det(F)\leq \beta$ and then the claim $r_f^2\mid \det(F)^2\in \ZZ$ proves that $\log r_f\leq \beta$ as desired. To verify the claim $r_f^2\mid \det(F)^2\in \ZZ$ we take $f_c\in S_2(\Gamma_0(N))$ as in (\ref{def:rf}). Then there exists $y=(y_j)\in\ZZ^J$ such that
\begin{equation}
\label{eq:coeffOfFandFc}
a_j(f_c)=a_j(f)+y_jr_f \ \textnormal{ for all } j\in J.
\end{equation}
We write $f_c=\sum_{i=1}^g k_if_i$ with $(k_i)\in\CC^g$.
It holds that $k_1=0$, since $\mathcal B$ is an orthogonal basis and since $(f,f_c)=0$ by (\ref{def:rf}).
Therefore on comparing Fourier coefficients we deduce
\begin{equation}
\label{eq:FcInTermsOfFis}
a_j(f_c)=\sum_{i=2}^{g} k_ia_j(f_i)=\sum_{i=2}^m k_ia_j(f_i) \ \textnormal{ for all } j\in J.
\end{equation}
Here we used that $a_j(f_i)=0$ for all $i\geq m+1$ and $j\in J$. To see that $a_j(f_i)=0$ for all $i\geq m+1$ and $j\in J$, one recalls that each $j\in J\subseteq J'$ is coprime to $N$ and any $f_i\in \mathcal B$ with $i\geq m+1$ is of the following form: $f_i(\tau)=f^*(n\tau)$
with $f^*\in S_2(\Gamma_0(M))$ a newform, $M$ a proper divisor of~$N$, and $n\geq 2$ a divisor of~$N/M$.
Now on using exactly the same arguments as in the proof of \cite[Lem 5.1 (ii)]{rvk:modular}, we see that the formulas~\eqref{eq:coeffOfFandFc} and~\eqref{eq:FcInTermsOfFis} imply the claim $r_f^2\mid \det(F)^2$. It follows that $\log r_f\leq \min(\beta,\beta^*)$ since \cite[Lem 5.1 (ii)]{rvk:modular} gives the upper bound $\log r_f\leq \beta^*$. Further \cite[Thm 2.1]{agrist:congruence} implies that $\log m_f\leq \log r_f$. Finally, the desired lower bound for $\log m_f$ follows for example from the explicit inequality $h(E)\leq \frac{1}{2}\log m_f+2\pi+\frac{1}{2}\log(163/\pi)$ which was obtained in course of the proof of \cite[Prop 6.1]{rvk:modular}. This completes the proof of assertion~(i).
We now prove statement~(ii).
Martin obtained in \cite[Thm 4]{martin:dimension} an explicit formula for $m$ in terms of $N$ and~$\nu$. This formula implies the estimate
\begin{equation}
\label{eq:martinbound}
m\leq \frac{\nu}{12}-\frac{1}{2}+\frac{1}{3}+\frac{1}{4}=\frac{\nu+1}{12}.
\end{equation}
We next work out an explicit upper bound for $l=\lfloor\frac{N}{6}\prod_{p\mid N}p(1+1/p)\rfloor$ with the product taken over all rational primes $p$ dividing~$N$.
Let $\gamma=0.577\dotsc$ denote the Euler--Mascheroni constant. In a first step we show that any $n\in\ZZ_{\geq 3}$ satisfies
\begin{equation}
\label{eq:keyexplbound}
\prod_{p\divides n} (1+1/p) \leq \frac{6 e^{\gamma}}{\pi^2}\left(\log\log n + \frac{2}{\log\log n}\right)
\end{equation}
with the product taken over all rational primes $p\mid n$.
To prove~\eqref{eq:keyexplbound} we may and do assume that $n$ is of the form $n=\prod_{p\leq x}p=e^{\vartheta(x)}$ with the product taken over all rational primes $p$ at most $x=x(n)\in\ZZ_{\geq 2}$. Indeed this follows by observing that the function $\log\log n+2/\log\log n$ is monotonously increasing for $n\geq 62$, by considering special cases such as for example the case when $n$ is a prime power and by checking (for example with Sage) all $n\leq 62$.
On writing $1+1/p = (1-1/p^2)/(1-1/p)$, we obtain
$$\prod_{p\leq x}(1+1/p)=\left(\prod_{p}(1-\frac{1}{p^2})\cdot \bigl(\prod_{p>x}(1-\frac{1}{p^2})\bigl)^{-1}\right)/\prod_{p\leq x}(1-1/p)$$
with the products taken over all rational primes $p$ satisfying the specified conditions. The effective version of Merten's theorem in \cite[Thm 7]{rosc:formulas} provides
\[
\prod_{p\leq x}(1-1/p) > \frac{e^{-\gamma}}{\log x}\Big(1-\dfrac{1}{2(\log x)^2}\Big) \ \textnormal{ if } x\geq 285.
\]
Euler's product formula gives that $\prod_p (1-1/p^2)=\zeta(2)^{-1}=6/\pi^2$ with the product taken over all rational primes $p$. Further, we deduce the inequalities
\[
\log\prod_{p>x}\Big(1-\frac{1}{p^2}\Big) \geq -\sum_{p>x}\frac{1}{p^2}\Big(1+\frac{1}{2p^2}\Big)\geq -\Big(1+\frac{1}{2x^2}\Big)\int_{x}^{\infty}\frac{1}{t^2}\,dt = -\frac{1}{x}\Big(1+\frac{1}{2x^2}\Big).
\]
On combining the results collected above with the effective prime number theorem of the following form (see \cite[Thm 4]{rosc:formulas})
\[
x-\frac{x}{2\log x} < \vartheta(x)=\log n < x+\frac{x}{2\log x} \ \textnormal{ if } x\geq 563,
\]
we see that the claimed inequality~\eqref{eq:keyexplbound} holds for all $n=e^{\vartheta(x)}$ with $x>10^4$.
Finally, one checks (for example with Sage) that~\eqref{eq:keyexplbound} holds in addition in the remaining cases $n=e^{\vartheta(x)}$ for $2\leq x\leq 10^4$. The conductor $N$ is at least $11$ and hence~\eqref{eq:keyexplbound} gives
\begin{equation}
\label{eq:lbound}
l\leq \frac{e^\gamma}{\pi^2}N^2\bigl(\log\log N + \frac{2}{\log\log N}\bigl).
\end{equation}
To estimate $\tau(n)$ we consider the real valued function $u(n)=\tau(n)/n^{1/4}$ on $\ZZ_{\geq 1}$. This function is multiplicative and satisfies $u(n)= \prod_{p}(n_p+1)p^{-n_p/4}$ with the product taken over all rational primes $p$, where $n_p=\ord_p(n)$ denotes the order of $p$ in $n\in\ZZ_{\geq 1}$. To find the maximum of $u$ we look at each factor separately.
It holds that $(n_p+1)p^{-n_p/4}=1$ when $n_p=0$, and if $n_p\geq 1$ then we observe that $(n_p+1)p^{-n_p/4}< 1$ provided that $p\geq 17$ or $n_p\geq 17$.
Thus after checking (for example with Sage) the remaining cases we find that $\sup(u)=8.44\dotsc$; in fact this supremum is attained at $n=2^5\cdot 3^3\cdot 5^2\cdot 7\cdot 11\cdot 13$.
Therefore we conclude that any $n\in \ZZ_{\geq 1}$ satisfies the inequality
\begin{equation}
\label{eq:tau}
\tau(n)\leq 8.5\, n^{1/4}.
\end{equation}
To put everything together we recall that in $\beta=\frac{m}{2}\log m+\max_J \sum_{j\in J}\log(\tau(j)j^{1/2})$ the maximum is taken over all subsets $J\subseteq \{1,\dotsc,l\}$ of cardinality~$m$. Hence~\eqref{eq:tau} gives
\begin{equation*}
\beta\leq \beta'=m\left(\tfrac{1}{2}\log m+\tfrac{3}{4}\log l+\log 8.5\right).
\end{equation*}
Further Euler's product formula shows that $\nu\geq 6N/\pi^2$ and then~\eqref{eq:martinbound} together with~\eqref{eq:lbound} leads to $\beta'\leq \frac{\nu}{6}\log N+\frac{\nu}{16}\log\log\log N+\frac{\nu}{9}$ for all $N\geq 23$.
Moreover, one checks (for example with Sage) that this upper bound holds in addition for all $N$ with $11\leq N< 23$. Therefore the displayed inequality $\beta\leq \beta'$ proves (ii).
To show~(iii) we observe that $N\to \infty$ implies $l\to \infty$ and $m\to \infty$. Hence we may and do assume that $l\to \infty$ and $m\to \infty$. If $n\in \ZZ_{\geq 1}$ with $n\to \infty$, then Wigert's bound gives $$\log \tau(n)\leq(\log 2+o(1))\frac{\log n}{\log\log n}.
$$
This implies that $\sum_{j\in J}\log(\tau(j))\leq m(\log 2+o(1))\frac{\log l}{\log\log l}$ for any subset $J\subseteq \{1,\dotsc,l\}$ of cardinality~$m$.
Therefore on recalling the definition of $\beta$ we obtain
\begin{equation*}
\beta\leq m\left(\tfrac{1}{2}\log m+\tfrac{1}{2}\log l+(\log 2+o(1))\frac{\log l}{\log\log l}\right).
\end{equation*}
Then we see that the above estimates for $m$ and $l$, given in~\eqref{eq:martinbound} and~\eqref{eq:lbound} respectively, lead to statement~(iii). This completes the proof of Proposition~\ref{prop:explbounds}.
\end{proof}
\begin{remark}\label{rem:tau}
To prove (ii) we used the explicit bound $\tau(n)\leq 8.5n^{1/4}$ obtained in~\eqref{eq:tau}, since the constant $8.5$ is reasonably small. On enlarging the constant $8.5$, one could replace the exponent $1/4$ by any other positive real number. However, for small exponents the corresponding (effective) constants become quite large. For example, if we take the exponent $1/8$ then the constant $8.5$ needs to be replaced by $45196.8$.\end{remark}
We mention that Proposition~\ref{prop:explbounds} allows to update several bounds in the literature. For example, Proposition~\ref{prop:explbounds} together with \cite[Lem 3.3]{rvk:modular} directly implies that any elliptic curve $E$ over $\QQ$ of conductor $N$ and minimal discriminant $\Delta_E$ satisfies
\begin{equation}\label{eq:szpiro}
\log\Delta_E\leq \nu\log N+\tfrac{3}{8}\nu\log\log\log N+\tfrac{2}{3}\nu+115.1
\end{equation}
and if $N\to\infty$ then $\log \Delta_E\leq \frac{3}{4}\nu\log N+\frac{\log 2+o(1)}{\log\log N}\nu\log N$. Here $\nu$ is as in Proposition~\ref{prop:explbounds}~(ii). These inequalities update the discriminant conductor inequalities in Murty--Pasten \cite[Thm 7.1]{mupa:modular} and in \cite[Cor 6.2]{rvk:modular}; see also \cite[Thm 3.3]{rvk:szpiro} for more general (but weaker) discriminant bounds based on the theory of logarithmic forms.
\section{The elliptic logarithm sieve}\label{sec:elllogsieve}
\subsection{Introduction}\label{sec:setup}
In this section we solve the problem of constructing an efficient sieve for the $S$-integral points of bounded height on any elliptic curve $E$ over $\QQ$ with given Mordell--Weil basis of $E(\QQ)$. Our construction combines a geometric interpretation of the known elliptic logarithm reduction (initiated by Zagier~\cite{zagier:largeintegralpoints}) with several conceptually new ideas. The resulting ``elliptic logarithm sieve" considerably extends the class of elliptic Diophantine equations which can be solved in practice. To illustrate this we solved many notoriously difficult equations by applying our sieve.
We also used the resulting data and our sieve to motivate new conjectures and questions on the number of $S$-integral points of $E$.
The precise construction of the elliptic logarithm sieve is rather technical. We now begin to describe the underlying ideas using a geometric point of view: After fixing the setup and briefly discussing the known elliptic logarithm approach, we explain the main ingredients of our sieve and we describe the improvements provided by our new ideas.
\paragraph{Setup.}Throughout this section we shall work with the following setup. Let $E$ be an elliptic curve over $\QQ$. We suppose that we are given an arbitrary Weierstrass equation of $E$, with coefficients $a_1,\dotsc,a_6$ in $\ZZ$, of the following form
\begin{equation}\label{eq:weieq}
y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.
\end{equation}
For any field $K$ containing $\QQ$, we often identify a nonzero point in $E(K)$ with the corresponding solution of \eqref{eq:weieq} and vice versa. We further assume that we are given a basis $P_1,\dotsc,P_r$ of the free part of the finitely generated abelian group $E(\QQ)$, see Section~\ref{sec:compmwbasis}.
Let $S$ be a finite set of rational primes and let $\Sigma(S)$ be the set of $(x,y)$ in $\mathcal O\times\mathcal O$ satisfying \eqref{eq:weieq}, where $\mathcal O=\ZZ[1/N_S]$ and $N_S=\prod_{p\in S} p$.
Finally, we suppose that we are given an initial bound $M_0$, that is $M_0\in\ZZ_{\geq 1}$ such that any $P\in \Sigma(S)$ satisfies $\hat{h}(P)\leq M_0$ for $\hat{h}$ the canonical N\'eron--Tate height of $E$. In fact everything works equally well with an initial bound for the usual Weil height or for the infinity norm $\|\cdot\|_\infty$, see Remark~\ref{rem:elllogsievegen}~(ii).
\subsubsection{Elliptic logarithm approach}\label{sec:elllintroelr}
Starting with Masser~\cite{masser:ellfunctions}, Lang~\cite{lang:diophantineanalysis} and W\"ustholz~\cite{wustholz:recentprogress}, many authors developed a practical approach to determine $\Sigma(S)$ using elliptic logarithms. Here a fundamental ingredient is a technique introduced by Zagier~\cite{zagier:largeintegralpoints} which we call the elliptic logarithm reduction.
In practice this technique allows to considerably reduce the initial bound $N_0$ coming from transcendence theory; $\|P\|_\infty\leq N_0$ for all $P\in \Sigma(S)$. More precisely on combining Zagier's arguments with de Weger's approach via $L^3$~\cite{lelelo:lll},
Stroeker--Tzanakis~\cite{sttz:elllogaa} and Gebel--Peth{\H{o}}--Zimmer~\cite{gepezi:ellintpoints,gepezi:mordell} showed independently the following when $\mathcal O=\ZZ$: The elliptic logarithm reduction produces a relatively small number $N_1<N_0$ such that any point $P\in \Sigma(S)$ with $N_1< \|P\|_\infty \leq N_0$ has to be exceptional (Definition~\ref{def:exceptpoint}). Smart~\cite{smart:sintegralpoints} extended
the method to general $\mathcal O$, see also \cite{pezigehe:sintegralpoints} and the recent book of Tzanakis~\cite{tzanakis:book} devoted to the elliptic logarithm approach.
If $N_1^r$ is small enough then $\Sigma(S)$ can be enumerated by checking all remaining candidates $P$ with $\|P\|_\infty\leq N_1$ and by finding the exceptional points. However there are many situations of interest in which $N_1^r$ is usually too large to determine $\Sigma(S)$ via the known methods. In view of this, resolving the following problem would be of fundamental importance.
\noindent{\bf Problem.}
\emph{Construct an efficient sieve for the points $\Sigma(S)\subseteq E(\QQ)$ of bounded height.}
\noindent Here the known sieves are often useless in practice. For example, working in the finite groups obtained by ``reducing the curve $E$ mod $p$" for suitable primes $p$ is usually not efficient (see Smart \cite[p.398]{smart:sintegralpoints}). In fact, since an efficient sieve for $\Sigma(S)$ inside $E(\QQ)$ was not available, various authors conducted some effort to develop other techniques to enumerate $\Sigma(S)$ in certain cases when $N_1^r$ is not small enough; see Section~\ref{sec:comparisonwithelr}.
\subsubsection{The elliptic logarithm sieve}
Building on the core idea of the elliptic logarithm reduction, we construct the elliptic logarithm sieve which resolves in particular the above problem. Here we introduce several conceptually new ideas. They all rely on a geometric point of view and to explain our ideas we now give a geometric interpretation of the known elliptic logarithm reduction: For each $v$ in $S^*=S\cup \{\infty\}$ one uses elliptic logarithms to construct a lattice $\Gamma_v\subset\ZZ^d$ of rank $d$ such that any non-exceptional $P\in \Sigma(S)$ with $\|P\|_\infty>N_1$ is essentially determined by a nonzero point in some $\Gamma_v$. Then
one tries to show via $L^3$ that a certain cube $Q_v\subset\RR^{d}$ satisfies $\Gamma_v\cap Q_v=0$ proving that all $P\in \Sigma(S)$ with $\|P\|_\infty>N_1$ are exceptional. Here $d$ equals $r$ or $r+1$, and for any $v\in S$ the cube $Q_v\subset\RR^d$ is given by $\{\|P\|_\infty\leq N_0\}$ inside $E(\QQ)\otimes_\ZZ\RR\cong \RR^d$. Further, increasing $N_1$ enlarges the co-volume of each $\Gamma_v$ and hence $\Gamma_v\cap Q_v$ is usually trivial for sufficiently large $N_1$. In fact the cube $Q_v$ always contains a certain ellipsoid $\mathcal E_v\subset \RR^d$ arising from $\hat{h}$. Now, our new ideas can be described as follows:
\paragraph{Global sieves.} We use $\Gamma_v\cap \mathcal E_v$ to construct various global sieves for $\Sigma(S)$ inside $E(\QQ)$. Here, instead of computing a lower bound for the length of the shortest nonzero vector in $\Gamma_v$, we actually determine the points in $\Gamma_v\cap \mathcal E_v$ using Fincke--Pohst~\cite{lelelo:lll,fipo:algo} and we check if these points come from $\Sigma(S)$. This has the following advantages:
\begin{itemize}
\item[(i)] We can further reduce $N_1$ in the crucial situation where the usual reduction is not working anymore (e.g. the shortest nonzero vector of $\Gamma_v$ actually lies in $Q_v$).
\item[(ii)] On ``covering" the set $\Sigma(S)$ by the local sieves $\Gamma_v\cap\mathcal E_v$, $v\in S^*$, we obtain a global sieve for $\Sigma(S)$ inside $E(\QQ)$ which is more efficient than the standard enumeration.
\end{itemize}
\paragraph{Refined coverings.} On using the geometric point of view, we construct in Proposition~\ref{prop:refinedcov} refined ``coverings" of $\Sigma(S)$ in order to improve the global-local passage in (ii). This leads to a refined sieve which enhances our sieve in (ii) and which allows to reduce $N_1$ even further. Here the construction is inspired by our refined sieve for $S$-unit equations in Section \ref{sec:dwsieve+}. However, in the present case of elliptic curves, the technicalities arising from $v$-adic elliptic logarithms at $v=2$ and $v=\infty$ are more involved.
\paragraph{Height-logarithm sieve.} We construct a sieve for $\Sigma(S)$ inside $E(\QQ)$ by exploiting that for any non-exceptional point $P\in \Sigma(S)$ the height $\hat{h}(P)$ is essentially determined by the local $v$-adic elliptic logarithms with $v\in S^*$. The height-logarithm sieve is a crucial ingredient of our global sieves discussed above. For many involved points $P$, it allows to avoid the slow process of testing whether the coordinates of $P$ are in fact $S$-integers.
\paragraph{Ellipsoids.} Instead of using the infinity norm $\|\cdot\|_\infty$ as done by all other authors, we work directly with the canonical height $\hat{h}$. Our approach using $\hat{h}$ is more efficient than the known improvements of the elliptic logarithm reduction (see Section~\ref{sec:comparisonwithelr}), since the cubes $Q_v$ arising from $\|\cdot\|_\infty$ always contain our ellipsoids $\mathcal E_v$ determined by $\hat{h}$. In fact working here with ellipsoids is optimal from a geometric point of view and it is crucial for the construction of our sieves. To circumvent issues with the real valued function $\hat{h}$, we constructed a suitable rational approximation of the quadratic form $\hat{h}$ on $E(\QQ)\otimes_\ZZ\RR$.
\paragraph{Exceptional points.} We conducted some effort to avoid as much as possible working with the coordinates of the points. For example to deal with exceptional points, we prove the crucial Proposition~\ref{prop:refinedcov} which allows here to work entirely in the finitely generated group $E(\QQ)$. This considerably improves the ``extra search" for exceptional points. In fact in most cases Proposition~\ref{prop:refinedcov} completely removes the ``extra search".
\paragraph{Generic situation.} The case $r\leq 1$ is of particular importance, since it represents the most common situation in practice; see also Katz--Sarnak~\cite{kasa:random} and Bhargava--Shankar~\cite{bhsh:avranklessthan1}. Furthermore, one can efficiently verify in practice whether $r\leq 1$ by using for example the work of Kolyvagin~\cite{kolyvagin:bsd} and Gross--Zagier--Zhang~\cite{zhang:gengz}.
Also one can directly determine $\Sigma(S)$ when $r=0$. In view of this we tried to further improve our sieves for $r=1$. On exploiting that $\Gamma_v$ has rank $r=1$ for $v\in S$, we optimized the reduction process at $v\in S$ and we enhanced our height-logarithm sieve for huge sets $S$.
\subsubsection{Discussion}
We shall motivate (using geometry) the ideas and constructions underlying the elliptic logarithm sieve. Further, for each of our algorithms, we conducted some effort to discuss important complexity aspects, to motivate our choice of parameters, to circumvent potential numerical issues, and to give detailed correctness proofs. In particular we prove in detail that our constructions involving $v$-adic elliptic logarithms have the required properties at the problematic places $v$ of $\QQ$, that is $v=\infty$, $v=2$ and bad reduction $v$.
\paragraph{Improvements.} The elliptic logarithm sieve improves in all aspects the known elliptic logarithm reduction and its subsequent enumeration. In Sections~\ref{sec:globalsieve} and \ref{sec:comparisonwithelr}, we shall demonstrate (in theory and in practice) that our improvements are substantial. In fact we obtain running time improvements by a factor which is exponential in terms of the rank $r$, and which is exponential in terms of $\mathcal As{S}$ when $\max\mathcal As{a_i}$ is large. Furthermore in the case of a generic Weierstrass equation \eqref{eq:weieq} we can efficiently determine all $S$-integral solutions for huge sets $S$. For example sets $S$ with $\mathcal As{S}=10^5$ are usually no problem here. Also, if $\mathcal As{S}$ is very small then our sieve allows to deal efficiently with large ranks $r$ such as $r=14,\dotsc,19$. In particular, in the case when $\mathcal O=\ZZ$, the sieve is practical even for huge ranks such as $r=28$. The elliptic logarithm sieve considerably extends the class of elliptic Diophantine equations which can be solved in practice. We shall demonstrate this by solving several notoriously difficult Diophantine problems which appear to be completely out of reach for the known methods, see Section~\ref{sec:elllogsieveapp} for explicit examples.
\subsubsection{Applications}\label{sec:elllintroapp}
We solved large classes of elliptic Diophantine equations by applying our sieve.
In particular, we efficiently solved several Diophantine problems in which the involved rank $r$ is large. Further, for each globally minimal Weierstrass equation \eqref{eq:weieq} of any elliptic curve over $\QQ$ of conductor at most $100$ (resp. $1000$), we determined its set of $S$-integral solutions with $S$ given by the first $10^4$ (resp. $20$) primes. See Section~\ref{sec:elllogsieveapp} for more information.
\paragraph{Conjecture and questions.} We used our data to motivate various questions on points of hyperbolic curves $Y=(X,D)$ of genus one. More precisely, let $B$ be a nonempty open subscheme of $\textnormal{Spec}(\ZZ)$ and let $X\to B$ be a smooth, proper and geometrically connected genus one curve. Let $Y\hookrightarrow X$ be an open immersion onto the complement $X-D$ of a nontrivial relative Cartier divisor $D\subset X$ which is finite \'etale over $B$. We now state the following conjecture involving the rank $r$ of the group formed by the $\QQ$-points of $\textnormal{Pic}^0(X_\QQ)$.
\noindent{\bf Conjecture.}
\emph{There are constants $c_Y$ and $c_r$, depending only on $Y$ and $r$ respectively, such that any nonempty finite set of rational primes $S$ with $T=\textnormal{Spec}(\ZZ)-S$ satisfies} $$\mathcal As{Y(T)}\leq c_Y \mathcal As{S}^{c_r}.$$
\noindent Our initial motivation for making this conjecture is explained in Section~\ref{sec:ialgoheight}. Further, the above conjecture generalizes our conjecture for Mordell equations which we discussed and motivated in Section~\ref{sec:malgoapplications}. In fact our discussion and motivation given there, including the construction of our probabilistic model, can be applied in exactly the same way in the case when $Y$ is a Weierstrass curve. Here $Y$ is a Weierstrass curve if the Cartier divisor $D$ is given by the image of a section of $X\to B$. We shall also discuss and motivate various questions related to the above conjecture. For example, we ask whether the above conjecture holds with $\mathcal As{S}$ replaced by the logarithm of the largest prime in $S$?
\subsubsection{Organization of the section}
\paragraph{Plan.} In Section~\ref{sec:heightselllog} we discuss a suitable rational approximation of the N\'eron--Tate height on $E(\QQ)$. The subsequent Sections~\ref{sec:archisieve} and \ref{sec:nonarchisieve} contain our construction of the local sieves at the archimedean place and the non-archimedean places. In Sections~\ref{sec:heightlogsieve} and \ref{sec:refinedenumell} we explain the height-logarithm sieve and the refined enumeration. Then we construct the refined sieve in Section~\ref{sec:refinedsieveell}. In Sections~\ref{sec:globalsieve} and \ref{sec:elllogsievealgo}, we put everything together to obtain the elliptic logarithm sieve. Here we also compare our sieve with the known approach. Then, after recalling in Section~\ref{sec:input} results and methods which allow to compute the required input data, we discuss applications of our sieve in Section~\ref{sec:elllogsieveapp}. Finally, we explain in Section~\ref{sec:compuaspects} computational aspects of our constructions.
\paragraph{Notation.} Throughout this section we shall use the following conventions. By $\log$ we mean the principal value of the natural logarithm. Unless mentioned otherwise, $\mathcal As{z}$ denotes the usual complex absolute value of $z\in\CC$ and the product taken over the empty set is~$1$. Further $\textnormal{lcm}(a_1,\dotsc,a_n)$ denotes the least common multiple of $a_1,\dotsc,a_n\in\ZZ$. For any real number $x\in\mathbb R$, we write $\floor{x}=\max(n\in\ZZ\,;\, n\leq x)$ and $\ceil{x}=\min(n\in\ZZ\,;\, n\geq x)$. We denote by $h(\alpha)$ the usual absolute logarithmic Weil height of $\alpha\in\QQ$, with $h(0)=0$ and $h(\alpha)=\log\max(\mathcal As{m},\mathcal As{n})$ if $\alpha=m/n$ for coprime $m,n\in\ZZ$. If $\alpha\in \QQ$ is nonzero and if $p$ is a rational prime, then we write $\ord_p(\alpha)\in\ZZ$ for the order of $p$ in~$\alpha$. For any set $M$, we denote by $\lvert M\rvert$ the (possibly infinite) number of distinct elements of~$M$. Finally, for any $n\in\ZZ_{\geq 1}$, we say that $\mathcal E\subset \mathbb R^n$ is an ellipsoid if $\mathcal E=\{x\in\mathbb R^n\,;\, q(x)\leq c\}$ for some positive definite quadratic form $q:\RR^n\to \RR$ and some positive real number $c$.
\paragraph{Computer, software and algorithms.} We implemented all our algorithms in Sage. Here a significant part of our program code is devoted to assure that the numerical aspects of the algorithms are all correct, see Section~\ref{sec:compuaspects} for certain important numerical aspects. We point out that we shall use various functions of Sage~\cite{sage:sagesystem} which in fact are direct applications of the corresponding functions of Pari~\cite{pari:parisystem}. Further, to compute the Mordell--Weil bases required for the input of the elliptic logarithm sieve, we used the techniques implemented in the computer packages Pari, Sage and Magma~\cite{magma:magmasystem}. For all our computations, we used a standard personal working computer at the MPIM Bonn. We shall list the running times of our algorithms for many examples. In fact the listed times are always upper bounds and some of them were obtained by using older versions of our algorithms. Here in many cases the running times would now be significantly better when using the most recent versions (as of February 2016) of our algorithms.
\paragraph{Acknowledgements.} Our construction of the elliptic logarithm sieve crucially builds on ideas and techniques of the authors who developed, generalized and/or refined the elliptic logarithm reduction over the last 30 years (see \cite{tzanakis:book} for an overview).
\subsection{Heights}\label{sec:heightselllog}
In this section we first recall useful results for heights of rational points on our given elliptic curve $E$ over $\QQ$ with Weierstrass equation \eqref{eq:weieq}. Then we construct a suitable rational approximation of the canonical N\'eron--Tate height on $E(\QQ)$, and we fix some terminology.
\paragraph{Canonical height.}Let $\hat{h}$ be the canonical N\'eron--Tate height on $E(\QQ)$. Here we use the natural normalization which divides by the degree of the involved rational function, see for example \cite[p.248]{silverman:aoes}. For any nonzero $P\in E(\QQ)$, it is known that the logarithmic Weil height $h(x)$ of the corresponding solution $(x,y)$ of \eqref{eq:weieq} can be explicitly compared with $\hat{h}(P)$. For instance Silverman~\cite[Thm 1.1]{silverman:heightcomparison} used an approach of Lang to obtain an explicit constant $\mu(E)$, depending only on the coefficients $a_i$ of \eqref{eq:weieq}, such that
\begin{equation}\label{eq:nthcompa}
-\tfrac{1}{24}h(j_E)-\mu\leq \hat{h}(P)-\tfrac{1}{2}h(x)\leq \mu
\end{equation}
for $\mu=\mu(E)+1.07$ and $j_E$ the $j$-invariant of $E$. Further it is known that $\hat{h}$ defines a positive semi-definite quadratic form on the geometric points of $E$, and for any point $P\in E(\QQ)$ it holds $\hat{h}(P)=0$ if and only if $P$ lies in the torsion subgroup $E(\QQ)_{\textnormal{tor}}$ of $E(\QQ)$. Therefore on identifying the real vector space $E(\QQ)\otimes_\ZZ \RR$ with $\RR^r$ via our given basis $P_1,\dotsc,P_r$ of the free part of $E(\QQ)$, we obtain that $\hat{h}$ extends to a positive definite quadratic form on $\RR^r$.
Let $\lambda$ be the smallest eigenvalue of the matrix $(\hat{h}_{ij})$ in $\RR^{r\times r}$ defining the bilinear form associated to $\hat{h}$. Linear algebra gives that any point $P$ in $E(\QQ)$ satisfies
\begin{equation}\label{eq:nthlowerbound}
\lambda \|P\|^2_\infty\leq \hat{h}(P)\leq r\lambda'\|P\|^2_\infty
\end{equation}
for $\lambda'$ the largest eigenvalue of $(\hat{h}_{ij})$. Here the infinity norm is defined by $\|P\|_\infty=\max \mathcal As{n_i}$, where $P=Q+\sum n_i P_i$ with $n_i\in\ZZ$ and $Q\in E(\QQ)_{\textnormal{tor}}$.
We point out that in practice it is always possible to quickly determine the points in $E(\QQ)_{\textnormal{tor}}$. For what follows we therefore always may assume that the rank $r\geq 1$.
To avoid numerical problems with the real valued function $\hat{h}$, we shall work with a rational approximation of $\hat{h}$.
\paragraph{Rational approximation.}
We next explain our construction of a suitable rational approximation of $\hat{h}$. Let $k\in\ZZ_{\geq 1}$ and define the norm $\|\hat{h}_{ij}\|$ of $(\hat{h}_{ij})$ by $\|\hat{h}_{ij}\|=\max \mathcal As{\hat{h}_{ij}}$. On using continued fractions we obtain $f\in\QQ$ which approximates the real number $2^{k}/\|\hat{h}_{ij}\|$ up to any required precision, see Section~\ref{sec:compuaspects}. We identify the rational vector space $E(\QQ)\otimes_\ZZ \QQ$ with $\QQ^r$ via the basis $P_1,\dotsc,P_r$ and we consider the quadratic form
\begin{equation}
\hat{h}_k:E(\QQ)\otimes_\ZZ\QQ\to \QQ
\end{equation}
associated to $\tfrac{1}{f}([f\hat{h}]-r\cdot \textnormal{id})\in \QQ^{r\times r}$. Here $[f\hat{h}]$ denotes the symmetric matrix in $\ZZ^{r\times r}$ with $ij$-th entry given by $[f\hat{h}_{ij}]$ for $[\cdot]$ the rounding ``function" defined in Section~\ref{sec:compuaspects}. The following lemma compares $\hat{h}$ with the natural extension of $\hat{h}_k$ to $E(\QQ)\otimes_\ZZ\RR=\RR^r$.
\begin{lemma}\label{lem:hk}
If $\|\cdot\|_2$ denotes the euclidean norm on $\RR^r$, then any $x\in \RR^r$ satisfies $$\hat{h}(x)-\tfrac{2r}{f}\|x\|_2^2\leq \hat{h}_k(x)\leq \hat{h}(x).$$
\end{lemma}
\begin{proof}
To simplify notation we write $V=\RR^r$ and we denote by $q$ the quadratic form on $V$ which is associated to $(\delta_{ij})=(\hat{h}_{ij})-\tfrac{1}{f}[f\hat{h}]$. We take $x\in V$ and we deduce
$$\hat{h}_k(x)=\hat{h}(x)-q(x)-\tfrac{r}{f}\|x\|_2^2.$$
It holds that $f\mathcal As{\delta_{ij}}\leq 1$ and the Cauchy--Schwarz inequality implies that $\sum_{ij}\mathcal As{x_ix_j}\leq r\|x\|_2^2$. Therefore we obtain $\mathcal As{q(x)}\leq \tfrac{r}{f}\|x\|_2^2$ and then we see that the displayed formula leads to the claimed inequality. This completes the proof of the lemma.
\end{proof}
If $k$ is sufficiently large then the above lemma implies that the quadratic form $\hat{h}_k$ is positive definite and is close to $\hat{h}$.
For what follows we fix an element $k\in\ZZ$ such that $\hat{h}_k$ is positive definite and is close to $\hat{h}$, see also the discussions in Section~\ref{sec:compuaspects}.
\paragraph{Terminology.} To introduce some terminology, we take $\sigma\in \RR_{>0}$ and we consider a place $v$ of $\QQ$. The $v$-adic elliptic logarithm is of local nature, while $\hat{h}$ and $\hat{h}_k$ are global height functions. In our global sieve we shall need to measure the ``weight" of the $v$-adic norm of the $v$-adic elliptic logarithm inside $\hat{h}_k$. For this purpose, we shall work with the set
\begin{equation}\label{def:sigmavsigma}
\Sigma(v,\sigma)
\end{equation}
formed by the nonzero points $P\in E(\QQ)$ whose corresponding solution $(x,y)$ of \eqref{eq:weieq} satisfies $\tfrac{1}{2}\log \mathcal As{x}_v\geq \tfrac{1}{\sigma}(\hat{h}_k(P)-\mu).$
Here we write $\mathcal As{x}_v=p^{-\ord_p(x)}$ if $v$ is a finite place given by the rational prime $p$, and if $v=\infty$ then $\mathcal As{x}_v$ is defined by $\mathcal As{x}_v=\mathcal As{x}$ for $\mathcal As{\cdot}$ the usual complex absolute value.
We note that one can define the set $\Sigma(v,\sigma)$ more intrinsically using (Arakelov) intersection theory. However, it is not clear to us if this provides a significant advantage in practice and thus we work with \eqref{def:sigmavsigma} using \eqref{eq:nthcompa}. For any rational integer $n\geq r$, we say that $P\in E(\QQ)$ is determined modulo torsion by $\gamma\in \ZZ^n$ if there exists $Q\in E(\QQ)_{\textnormal{tor}}$ such that $P=Q+\sum \gamma_i P_i$. Further we denote by $\Gamma_E=\ZZ^r$ the lattice inside $\RR^r$ given by the image of $E(\QQ)$ inside $E(\QQ)\otimes_\ZZ\RR=\RR^r$ using the identification via $P_i$.
\subsection{Archimedean sieve}\label{sec:archisieve}
Building on ideas of Zagier~\cite{zagier:largeintegralpoints}, de Weger~\cite{deweger:phdthesis}, Stroeker--Tzanakis~\cite{sttz:elllogaa} and Gebel--Peth{\H{o}}--Zimmer~\cite{gepezi:ellintpoints}, we construct in this section our archimedean sieve. We shall use this sieve to improve inter alia the known reduction process at infinity of the elliptic logarithm method, see the discussions in Sections~\ref{sec:globalsieve} and \ref{sec:comparisonwithelr}.
Throughout this Section~\ref{sec:archisieve} we use the setup of Section~\ref{sec:setup} and we continue the notation introduced above. Further throughout this section we write $\mathcal As{\cdot}=\mathcal As{\cdot}_\infty$.
\paragraph{Real elliptic logarithm.}We shall work with the following normalization of the elliptic logarithm on the identity component $E^0(\RR)$ of the real Lie group $E(\RR)$.
First we recall that the uniformization theorem for complex elliptic curves gives a lattice $\Lambda=\omega_1\ZZ+\omega_2\ZZ$ inside $\CC$ with $\omega_1\in\RR_{>0}$ and an isomorphism $\CC/\Lambda\xrightarrow{\sim} E(\CC)$ whose inverse we denote by $$\log: E(\CC)\xrightarrow{\sim} \CC/\Lambda.$$
To describe more explicitly the restriction to $E^0(\RR)$ of the displayed morphism, we write $x=x'-\tfrac{1}{12}b_2$ with $b_2=a_1^2+4a_2$ and $y=\tfrac{1}{2}(y'-a_1x-a_3)$ and we transform \eqref{eq:weieq} into the Weierstrass equation $y'^2=4x'^3-g_2x'-g_3$ whose complex solutions we identify with the nonzero points of $E(\CC)$.
We may and do assume
that $g_i=g_i(\Lambda)$ is associated to $\Lambda$ as in \cite[p.169]{silverman:aoes} and then the isomorphism $\CC/\Lambda\xrightarrow{\sim} E(\CC)$ is given outside zero by $z\mapsto (\wp(z),\wp'(z))$ for $\wp=\wp(\Lambda)$ the Weierstrass $\wp$-function and $\wp'$ its derivative. Hence
we deduce for example from \cite[p.174]{silverman:aoes} that the restriction of $\log: E(\CC)\xrightarrow{\sim} \CC/\Lambda$ to $E^0(\RR)$ is of the form $E^0(\RR)\xrightarrow{\sim} \RR/(\omega_1\ZZ)$,
which in turn induces a bijective map
\begin{equation}\label{def:reallog}
\log: E^0(\RR)\to \{z\in\RR\,;\, 0\leq z<\omega_1\}.
\end{equation}
Explicitly if $P\in E^0(\RR)$ corresponds to a real solution $(x,y)$ of \eqref{eq:weieq} then it holds that $\log(P)=\tfrac{y'}{\mathcal As{y'}} \int^{x'}_{\infty}\tfrac{dz}{f(z)^{1/2}}$ mod $(\omega_1\ZZ)$ for $f(z)=4z^3-g_2z-g_3$.
Here one can compute the real number $\log(P)$ up to any required precision, see for example Zagier~\cite[p.430]{zagier:largeintegralpoints}.
Further, we denote by $e_t$ the exponent of the finite group $E(\QQ)_{\textnormal{tor}}$ and we define
\begin{equation}\label{def:marchi}
m=\textnormal{lcm}(e_t,
\iota )
\end{equation}
for $\iota $ the index of $E^0(\RR)$ inside $E(\RR)$.
It holds that $\iota \in\{1,2\}$,
since $E(\RR)$ is either connected or isomorphic to $E^0(\RR)\times (\ZZ/2\ZZ)$.
We recall that the points $P_1,\dotsc,P_r$ form a basis of the free part of $E(\QQ)$. Now any $P=Q+\sum n_i P_i$ in $E(\QQ)$, with $Q\in E(\QQ)_{\textnormal{tor}}$ and $n_i\in\ZZ$, satisfies $mP=\sum n_i (mP_i)\in E^0(\RR)$.
Next we take $\kappa\in\ZZ_{\geq 1}$ and we define
\begin{equation}\label{def:x0}
x_0(\kappa)=(\kappa+1)(\mathcal As{b_2}/12+\max\mathcal As{\xi_i})
\end{equation}
for $\{\xi_i\}$ the set of roots of $f(z)=4z^3-g_2z-g_3$.
If $P\in E^0(\RR)$ corresponds to a solution $(x,y)$ of \eqref{eq:weieq}, then the next lemma allows to control $\log(P)$ in terms of $\mathcal As{x}$.
\begin{lemma}\label{lem:archiest}
The following statements hold.
\begin{itemize}
\item[(i)] Suppose that $P\in E(\RR)$ corresponds to a real solution $(x,y)$ of \eqref{eq:weieq} with $\mathcal As{x}\geq x_0(\kappa)$. Then $P$ lies in $E^{0}(\RR)$ and there is $\varepsilonilon\in\{0,-1\}$ such that any $n\in\ZZ$ satisfies
\begin{equation*}
\mathcal As{n\log(P)+n\varepsilonilon\omega_1}\leq \mathcal As{n}\bigl(1+\tfrac{1}{\kappa}\bigl)^2\mathcal As{x}^{-1/2}.
\end{equation*}
\item[(ii)]If $P=Q+\sum n_i P_i$ lies in $E^0(\RR)$ with $Q\in E(\QQ)_{\textnormal{tor}}$ and $n_i\in\ZZ$, then there exists $l\in\ZZ$ with $\mathcal As{l}\leq m+ \sum \mathcal As{n_i}$ such that $m\log(P)=\sum n_i\log(mP_i)+l\omega_1$.
\end{itemize}
\end{lemma}
\begin{proof}
We first prove assertion (i). Our assumption implies that $x'=x+\tfrac{1}{12}b_2$ is positive and that $x'$ strictly exceeds the largest real root of $f(z)=4z^3-g_2z-g_3$.
Hence we conclude that $P\in E^0(\RR)$.
To verify the second statement of (i) we observe that any $z\in\RR$ with $z\geq (\kappa+1)\max\mathcal As{\xi_i}$ satisfies $f(z)\geq 4\bigl(\tfrac{\kappa}{\kappa+1}\bigl)^3z^3$. It follows that $\mathcal As{\int^{x'}_\infty \tfrac{dz}{f(z)^{1/2}}}^2$ is at most $\bigl(\tfrac{\kappa+1}{\kappa}\bigl)^{3}\mathcal As{x'}^{-1}$, since our assumption gives $\mathcal As{x'}\geq (\kappa+1)\max\mathcal As{\xi_i}$. Furthermore our assumption
provides that $\mathcal As{x'}\geq \tfrac{\kappa}{\kappa+1}\mathcal As{x}$, and then we see that there exists $\varepsilonilon\in\{0,-1\}$ such that the claimed inequality holds for $n=1$ and thus for all $n\in\ZZ$.
It remains to show (ii). The points $mP_i$ are all in $E^0(\RR)$ since $\iota $ divides $m$, and the point $P$ is in $E^0(\RR)$ by assumption. Thus, on exploiting that the real elliptic logarithm is induced by a group isomorphism $E^0(\RR)\xrightarrow{\sim} \RR/(\omega_1\ZZ)$, we find $l',l''\in\ZZ$ with $m\log(P)=\log(mP)+l'\omega_1$ and $\log(mP)=\sum n_i\log(mP_i)+l''\omega_1$.
Then on using that $\log(P)$, $\log(mP)$ and $\log(mP_i)$ are in the interval $[0,\omega_1[$, we deduce that $\mathcal As{l'}\leq m$ and $\mathcal As{l''}\leq \sum \mathcal As{n_i}$.
Hence the integer $l=l'+l''$ has the desired property. This completes the proof of the lemma.
\end{proof}
\paragraph{Construction of $\Gamma$ and $\mathcal E$.}
Let $\sigma>0$ be a real number, let $\mu$ be as in \eqref{eq:nthcompa} and write $\Sigma$ for the set $\Sigma(\infty,\sigma)$ defined in \eqref{def:sigmavsigma}. Suppose that $M',M\in\ZZ$ with $\mu\leq M'<M$ and let $\kappa\in\ZZ_{\geq 1}$. We would like to construct a lattice $\Gamma\subset \ZZ^{r+1}$ and an ellipsoid $\mathcal E\subset \RR^{r+1}$ such that any $P\in \Sigma$ with $M'<\hat{h}_k(P)\leq M$ is determined modulo torsion by a point in $\Gamma\cap\mathcal E$. The following construction depends on a suitable choice of a parameter $c\in \ZZ_{\geq 1}$, which we shall explain below \eqref{archivolumecomp}. We write $\alpha_i=\log(mP_i)$ for $i\in\{1,\dotsc,r\}$ and we denote by $$\Gamma\subset\ZZ^{r+1}$$ the lattice formed by the elements $\gamma\in\ZZ^{r+1}$ such that $\gamma_{r+1}=l[c\omega_1]+\sum \gamma_i [c\alpha_i]$ for some $l\in\ZZ$.
Next we choose a positive number $\delta\in\QQ$ as explained in the discussion surrounding \eqref{def:delta12} and we denote by $q$ the positive definite quadratic form on $\RR^{r+1}$ which is given by $q(z)=\hat{h}_k(z_1,\dotsc,z_r)+(M/\delta^2)z_{r+1}^2$ for any $z\in \RR^{r+1}$. Now we define the ellipsoid
$$\mathcal E=\{z\in\mathbb R^{r+1}\,;\, q(z)\leq 2M\}.$$
Let $x_0=x_0(\kappa)$ be as in \eqref{def:x0} and let $\Sigma(x_0)$ be the set of points $P$ in $\Sigma$ with $\mathcal As{x}>x_0$, where $(x,y)$ is the solution of \eqref{eq:weieq} corresponding to $P$. We obtain the following lemma.
\begin{lemma}\label{lem:archicov}
Suppose that $P\in \Sigma(x_0)$ satisfies $M'<\hat{h}_k(P)\leq M$. Then the point $P$ is determined modulo torsion by some lattice point $\gamma$ in $\Gamma\cap \mathcal E$.
\end{lemma}
\begin{proof}
Let $(x,y)$ be the solution of \eqref{eq:weieq} corresponding to $P$, and write $P=Q+\sum n_i P_i$ with $Q\in E(\QQ)_{\textnormal{tor}}$ and $n_i\in\ZZ$.
It holds that $\mathcal As{x}\geq x_0$ since $P\in \Sigma(x_0)$ and therefore Lemma~\ref{lem:archiest}~(i) shows that $P\in E^0(\RR)$. Hence we see that Lemma~\ref{lem:archiest}~(ii) gives $l_0\in\ZZ$ with $\mathcal As{l_0}\leq m+\sum\mathcal As{n_i}$ such that $m\log(P)=\sum n_i\alpha_i+l_0\omega_1$. On inserting this into the inequality in Lemma~\ref{lem:archiest}~(i) with $n=m$, we obtain $l\in\ZZ$ with $\mathcal As{l}\leq 2m+\sum\mathcal As{n_i}$
such that
\begin{equation}\label{eq:archifundineq}
\mathcal As{\sum n_i\alpha_i+l\omega_1}\leq m\bigl(1+\tfrac{1}{\kappa}\bigl)^2e^{-\tfrac{1}{\sigma}(\hat{h}_k(P)-\mu)}.
\end{equation}
Here we used our assumption $P\in \Sigma(x_0)$, which provides that $\tfrac{1}{2}\log\mathcal As{x}\geq\tfrac{1}{\sigma}(\hat{h}_k(P)-\mu)$.
Next we define $d=l[c\omega_1]+\sum n_i [c\alpha_i]$ and we observe that $\gamma=((n_i),d)\in \ZZ^{r+1}$ lies in our lattice $\Gamma$. To show that $\gamma$ lies in addition in the ellipsoid $\mathcal E$, we use the mean inequality and linear algebra in order to deduce that $\lambda_k(\sum \mathcal As{n_i})^2\leq r\hat{h}_k(P)$ for $\lambda_k\in \QQ$ the smallest eigenvalue of the positive definite quadratic form $\hat{h}_k$ on $E(\QQ)\otimes_\ZZ\QQ$. Then we see that \eqref{eq:archifundineq} together with our assumption $M'<\hat{h}_k(P)\leq M$ implies that $\mathcal As{d}\leq \delta$. Here $\delta\in \QQ$ is chosen such that $\delta$ has ``small" height in the sense of Section~\ref{sec:compuaspects} and such that $\delta\geq \delta_1+\delta_2$ for
\begin{equation}\label{def:delta12}
\delta_1=2m+2\bigl(r\tfrac{M}{\lambda_k}\bigl)^{1/2} \ \ \ \textnormal{ and } \ \ \ \delta_2=cm\bigl(1+\tfrac{1}{\kappa}\bigl)^2e^{-\tfrac{1}{\sigma}(M'-\mu)}.
\end{equation}
On using again that $\hat{h}_k(P)\leq M$ we obtain $q(\gamma)\leq M+(M/\delta^2) d^2$. This together with $\mathcal As{d}\leq \delta$ implies that $\gamma\in\mathcal E$ and thus $P$ is determined modulo torsion by $\gamma\in \Gamma\cap \mathcal E$.
\end{proof}
This lemma provides a sieve for the points $P$ in $\Sigma(x_0)$ with $M'<\hat{h}_k(P)\leq M$. In the following paragraph we discuss the strength of the sieve depending on various parameters.
\paragraph{Strength of the sieve.} To make the sieve as efficient as possible, we would like choose the parameter $c$ such that the intersection $\Gamma\cap \mathcal E$ does not contain many points. In the generic case the cardinality of $\Gamma\cap \mathcal E$ can be approximated (for large $M$) by the euclidean volume of the ellipsoid $\mathcal E_\psi=\{z\in\RR^{r+1}\,;\, q_\psi(z)\leq 2M\}$ inside $\RR^{r+1}$. Here $q_\psi$ denotes the positive definite quadratic form obtained by pulling back $q$ along the linear transformation $\psi$ of $\RR^{r+1}$ which satisfies $\psi (\ZZ^{r+1})=\Gamma$ and which is explicitly given by
\[ \begin{pmatrix}
1 & & & 0 \\
& \ddots & & \vdots \\
& & 1 & 0 \\
[c\alpha_1] & \cdots & [c\alpha_r] & [c\omega_{1}]
\end{pmatrix}.\]
To compute the euclidean volume $\vol(\mathcal E_\psi)$ of $\mathcal E_\psi$, we let $R_{E}=2^r\det(\hat{h}_{ij})$ be the regulator of $E(\QQ)$ normalized as in \cite[p.253]{silverman:aoes} and we denote by $V_{r+1}$ the euclidean volume of the unit ball in $\RR^{r+1}$. Then the volume $\vol(\mathcal E_\psi)$ is approximately
\begin{equation}\label{archivolumecomp}
u\cdot M^{r/2}\tfrac{(\delta_1+\delta_2)}{c}, \ \ \ u=\tfrac{2^{r+1/2}V_{r+1}}{\omega_1R_{E}^{1/2}}.
\end{equation}
We note that $\delta_2/c$ does not depend on $c$. Hence in view of \eqref{archivolumecomp} we choose $c$ such that $u\cdot M^{r/2}\tfrac{\delta_1}{c}$ is smaller than $u\cdot M^{r/2}\tfrac{\delta_2}{c}$. For example $c$ should always dominate $M^{(r+1)/2}$ if $M$ is large. We next discuss the dependence of the sieve on $M'$ and $M$. For some large $M$, we choose $c$ as indicated above and we assume for a moment that $M'$ dominates
$
\frac{\sigma r}{2}\log M.
$
Then it follows that $M^{r/2}\tfrac{\delta_2}{c}$ is close to zero and hence \eqref{archivolumecomp} implies that the volume of $\mathcal E_\psi$ is small. In the generic case this assures that $\Gamma\cap \mathcal E$ has very little points or is even trivial. In particular, we see that the archimedean sieve is very efficient for such $M'$ and $M$. On the other hand, for small $M'$ our sieve is not that efficient in view of \eqref{archivolumecomp}.
\begin{remark}\label{rem:optellinf}
One can replace $\mathcal E$ by the more balanced ellipsoid $\mathcal E^*\subset \RR^{r+1}$ of the form $\mathcal E^*=\{z\in \RR^{r+1}\,;\, q^*(z)\leq M\}$ for $q^*(z)=\tfrac{r}{r+1}\hat{h}_k(z_1,\dotsc,z_r)+\tfrac{1}{r+1}(M/\delta^2) z_{r+1}^2$. Indeed this follows by observing that $\gamma$ appearing in the proof of Lemma~\ref{lem:archicov} satisfies $q^*(\gamma)\leq M$.
\end{remark}
\paragraph{Archimedean sieve.} In the following sieve, we use the version (FP) of the Fincke--Pohst algorithm described in Section~\ref{sec:compuaspects} in order to determine all points in $\Gamma\cap \mathcal E$.
\begin{Algorithm}[Archimedean sieve]\label{algo:archisieve} The inputs are $\kappa\in\ZZ_{\geq 1}$ and $M',M\in\ZZ$ with $\mu\leq M'< M$. The output is the set of points $P\in\Sigma(x_0)$ with $M'<\hat{h}_k(P)\leq M$.
\begin{itemize}
\item[(i)] First choose the parameter $c\in \ZZ_{\geq 1}$ as explained in the discussion surrounding \eqref{archivolumecomp}. Then compute the lattice $\Gamma\subset \ZZ^{r+1}$, by determining the period $\omega_1$ and the real elliptic logarithms $\alpha_i=\log(mP_i)$ up to the required precision for all $i\in\{1,\dotsc,r\}$.
\item[(ii)] Determine $\Gamma\cap \mathcal E$ by using the version of the Fincke--Pohst algorithm in \textnormal{(FP)}.
\item[(iii)] For each lattice point $\gamma$ in $\Gamma\cap \mathcal E$ and for each torsion point $Q$ in $E(\QQ)_{\textnormal{tor}}$, output the point $P=Q+\sum \gamma_i P_i$ if $M'<\hat{h}_k(P)\leq M$ and if $P$ is in $\Sigma(x_0)$.
\end{itemize}
\end{Algorithm}
\paragraph{Correctness.} Suppose that $P\in\Sigma(x_0)$ satisfies $M'<\hat{h}_k(P)\leq M$. Lemma~\ref{lem:archicov} gives that $P$ is determined modulo torsion by some $\gamma\in \Gamma\cap\mathcal E$. In other words, there is $Q\in E(\QQ)_{\textnormal{tor}}$ such that $P=Q+\sum \gamma_i P_i$ and hence step (iii) produces our point $P$ as desired.
\paragraph{Complexity.} We now discuss aspects of Algorithm~\ref{algo:archisieve} which significantly influence the running time. In step (i) the running time of the computation of the lattice $\Gamma=\psi(\ZZ^{r+1})$ crucially depends on the size of $c$. For example if $c$ is approximately $M^{(r+1)/2}$ then we need to know the real logarithms $\omega_1$ and $\log(mP_i)$ up to a number of decimal digits which is approximately $\tfrac{r+1}{2}\log_{10} M$, where $\log_b z=(\log z)/\log b$ for $z,b\in\RR_{>0}$. We shall apply the algorithm with huge parameters $M$. Therefore we need to compute $(r+1)$ real elliptic logarithms up to a very high precision and this can take a long time. Step (ii) is essentially always fast in practice. The reason is that the involved Mordell--Weil rank $r$ of $E(\QQ)$ is usually not that large and hence the application of (FP) with the lattice $\Gamma$ of rank $r+1$ is fast. Finally step (iii) needs to compute in particular the coordinates of certain points in $E(\QQ)$ and this can take some time if $\hat{h}(P)$ and $r$ are not small.
\paragraph{Comparison.} There are important differences between our approach and the known approach. In particular we work with the N\'eron--Tate height $\hat{h}$, while all other authors use the inequality $\lambda\|\cdot\|_\infty^2\leq \hat{h}(\cdot)$ to work with the norm $\|\cdot\|_\infty$. Also we actually determine the intersection $\Gamma\cap \mathcal E$, while the known approach computes a lower bound for the length of the shortest nonzero vector in $\Gamma$ in order to rule out non-trivial points in $\Gamma\cap\mathcal E$. Other, more technical, differences are the following: The parameter $\kappa$ allows us (up to a certain extent) to adapt the strength of the sieve to the given situation, and the construction of our ellipsoid $\mathcal E^*$ involving the weights $\tfrac{r}{r+1}$ and $\tfrac{1}{r+1}$ is more balanced in particular for large $r$. In Sections~\ref{sec:globalsieve} and \ref{sec:comparisonwithelr}, we shall further compare the two approaches and we shall explain in detail the improvements provided by our new ideas.
\subsection{Non-archimedean sieve}\label{sec:nonarchisieve}
Building on ideas of Smart~\cite{smart:sintegralpoints}, Peth{\H{o}}--Zimmer--Gebel--Herrmann~\cite{pezigehe:sintegralpoints} and Tzanakis~\cite{tzanakis:book}, we construct in this section the non-archimedean sieve. We shall use this sieve to improve inter alia the known reduction process of the elliptic logarithm method at non-archimedean primes, see the discussions in Sections~\ref{sec:globalsieve} and \ref{sec:comparisonwithelr}.
Throughout this Section~\ref{sec:nonarchisieve} we work with the setup of Section~\ref{sec:setup} and we continue the notation introduced above. Further we fix $p$ in $S$ and we assume that the Weierstrass model \eqref{eq:weieq} of our given elliptic curve $E$ is minimal at $p$. To simplify the notation of this section, we write $v(\cdot)=\ord_p(\cdot)$ and $\mathcal As{\cdot}=\mathcal As{\cdot}_p$ with $\mathcal As{x}_p=p^{-v(x)}$ for $x\in \QQ_p$.
\paragraph{The $p$-adic elliptic logarithm.}
Let $E_1(\QQ_p)$ be the subgroup of $E(\QQ_p)$ formed by the points $P$ in $E(\QQ_p)$ with $\pi(P)=0$ for $\pi:E(\QQ_p)\to E(\mathbb F_p)$ the reduction map. Here $E(\mathbb F_p)$ denotes the set of $\mathbb F_p$-points of the special fiber of the projective closure\footnote{Here we mean $\textnormal{Proj}\bigl(\ZZ_p[x,y,z]/(f)\bigl)$ for $f=y^2z+a_1xyz+a_3yz^2-(x^3+a_2x^2z+a_4xz^2+a_6z^3)$.} of $\eqref{eq:weieq}$ inside $\mathbb P^2_{\ZZ_p}$. Let $\hat{E}$ be the formal group over $\ZZ_p$ associated to $E_{\QQ_p}$. There is an isomorphism $E_1(\QQ_p)\xrightarrow{\sim} \hat{E}(p\ZZ_p)$ of abelian groups, which is given away from zero by $(x,y)\mapsto -\tfrac{x}{y}$.
Composing this isomorphism with the formal logarithm of $\hat{E}$ induces a morphism
\begin{equation}\label{def:plog}
\log:E_1(\QQ_p)\to \mathbb G_a(\QQ_p)
\end{equation}
of abelian groups.
We call the displayed morphism the $p$-adic elliptic logarithm. Explicitly if $P\in E_1(\QQ_p)$ is nonzero and corresponds to the solution $(x,y)$ of \eqref{eq:weieq}, then it holds that $\log(P)=z+\sum_{n\geq 2} \tfrac{b_{n}}{n} z^{n}$ with $z=-\tfrac{x}{y}$ and $b_n\in\ZZ_p$.
A priori the $p$-adic elliptic logarithm is only defined on the subgroup $E_1(\QQ_p)$ of $E(\QQ_p)$. One can somehow circumvent this problem by multiplying the points in $E(\QQ_p)$ with a suitable integer. To construct such an integer, let $E_{\textnormal{ns}}(\mathbb F_p)$ be the group formed by the nonsingular points in $E(\mathbb F_p)$ and consider the subgroup $E_0(\QQ_p)=\pi^{-1}(E_{\textnormal{ns}}(\mathbb F_p))$ of $E(\QQ_p)$.
We denote by $\iota$ the index of $E_0(\QQ_p)$ in $E(\QQ_p)$, and we write $e_t$ and $e_{ns}$ for the exponents of the finite groups $E(\QQ)_{\textnormal{tor}}$ and $E_{\textnormal{ns}}(\mathbb F_p)$ respectively. The short exact sequence $0\to E_1(\QQ_p)\to E_0(\QQ_p)\overset{\pi}{\to} E_{\textnormal{ns}}(\mathbb F_p)\to 0$ of abelian groups shows that $(\iota e_{ns})P\in E_1(\QQ_p)$ for all $P\in E(\QQ_p)$. We now define
\begin{equation}\label{def:mnonarchi}
m=\textnormal{lcm}\bigl(e_t,\iota e_{ns}\bigl).
\end{equation}
Recall that $P_1,\dotsc,P_r$ denotes our given basis of the free part of $E(\QQ)$. Any $P=Q+\sum n_i P_i$ in $E(\QQ)$, with $Q\in E(\QQ)_{\textnormal{tor}}$ and $n_i\in \ZZ$, satisfies $mP=\sum n_i (mP_i)\in E_1(\QQ_p)$.
The case distinction in the following lemma takes into account that in general the formal logarithm of $\hat{E}$ is not necessarily an isomorphism of formal groups over the given base.
\begin{lemma}\label{lem:nonarchiest}
Let $P\in E(\QQ_p)$ be nonzero, and suppose that $(x,y)$ is the solution of \eqref{eq:weieq} corresponding to $P$. Then the following two statements hold.
\begin{itemize}
\item[(i)] Assume that $p\geq 3$. If $P\notin E_1(\QQ_p)$ with $mP\neq 0$ then $\mathcal As{\log(mP)}^2<\mathcal As{x}^{-1}$, and if $P\in E_1(\QQ_p)$ then $\mathcal As{\log(nP)}^2=\mathcal As{n}^2\mathcal As{x}^{-1}$ for all $n\in\ZZ$.
\item[(ii)] If $p=2$ and $v(x)<-2$, then any $n\in\ZZ$ satisfies $\mathcal As{\log(nP)}^2=\mathcal As{n}^2\mathcal As{x}^{-1}$.
\end{itemize}
\end{lemma}
\begin{proof}
Our proof given below relies on the classical result that the formal logarithm is compatible with the valuation $v$ in the following sense:
For any $l\in\ZZ$ with $l>v(p)/(p-1)$, the restriction of the formal logarithm of $\hat{E}$ induces an isomorphism
\begin{equation}\label{eq:formalgpiso}
\hat{E}((p\ZZ_p)^l)\cong (p\ZZ_p)^l
\end{equation}
of abelian groups.
Further, we shall use below that if $P\in E_1(\QQ_p)$ then it holds that $3v(x)=2v(y)$, thus $v(x)$ is even and the number $z=-x/y$ satisfies $2v(z)=-v(x)$.
If $mP\neq 0$ then we denote by $(x_m,y_m)$ the solution of \eqref{eq:weieq} corresponding to $mP$.
To prove (i) we may and do assume that $p\geq 3$.
Then $p$ satisfies $1>v(p)/(p-1)$ and hence the isomorphism in \eqref{eq:formalgpiso} exists for all $l\geq 1$. This implies that $2v(\log(mP))=-v(x_m)$ since $mP\in E_1(\QQ_p)$ is nonzero by assumption.
If $P$ is not in $E_1(\QQ_p)$ then $v(x)\geq 0$, and $mP\in E_1(\QQ_p)$ thus shows that $v(x)\geq 0> v(x_m)$. This together with $2v(\log(mP))=-v(x_m)$ proves the claimed inequality if $P$ is not in $E_1(\QQ_p)$. Suppose now that $P\in E_1(\QQ_p)$. Then we obtain that $n\log(P)=\log(nP)$ for all $n\in \ZZ$ since the formal logarithm is a morphism of abelian groups, and the isomorphisms in \eqref{eq:formalgpiso} provide that $2v(\log(P))=-v(x)$. On combining these two equalities, we deduce the second statement of (i).
To show (ii) we may and do assume that $p=2$ and $v(x)<-2$. The latter assumption implies that $P\in E_1(\QQ_2)$ and $v(x)\leq -4$. We deduce that $v(z)\geq 2$ and hence $z$ lies in $(2\ZZ_2)^2$. Further, the isomorphism in~\eqref{eq:formalgpiso} exists for all $l\geq 2$ since $2>v(2)/(2-1)$. Thus we obtain that $2v(\log(P))=-v(x)$ and then the equality $n\log(P)=\log(nP)$, which holds for all $n\in \ZZ$ since $P\in E_1(\QQ_2)$, implies (ii). This completes the proof of the lemma.
\end{proof}
We remark that the assumptions $P\in E_1(\QQ_p)$ and $v(x)<-2$, in (i) and (ii) respectively, assure in particular that the point $P$ has infinite order in $E(\QQ_p)$.
\paragraph{Construction of $\Gamma$ and $\mathcal E$.}
Let $\sigma> 0$ be a real number and write $\Sigma$ for the set $\Sigma(v,\sigma)$ defined in \eqref{def:sigmavsigma}. Suppose that we are given $M',M\in\ZZ$ with $\mu\leq M'< M$ for $\mu$ as in \eqref{eq:nthcompa}. We would like to find a lattice $\Gamma\subset \ZZ^r$ such that any $P\in\Sigma$ with $M'<\hat{h}_k(P)\leq M$ is determined modulo torsion by some point in $\Gamma\cap\mathcal E$. Here $\mathcal E\subset \RR^{r}$ is the ellipsoid
$$
\mathcal E=\{z\in\RR^{r}\,;\, \hat{h}_k(z)\leq M\}.
$$
We identify $\alpha\in \ZZ_p$ with the corresponding element $(\alpha^{(1)},\alpha^{(2)},\dotsc)$ of the inverse limit $\lim \ZZ/(p^n\ZZ)$ and we set $\alpha_i=\log(mP_i)$ for each $i\in\{1,\dotsc,r\}$. If $p\geq 3$ then Lemma~\ref{lem:nonarchiest} implies that $\alpha_i\in \ZZ_p$. To deal with the general case, we choose $i^*\in\{1,\dotsc,r\}$ with $v(\alpha_{i^*})= \min v(\alpha_i)$ and then $\beta_i=\alpha_i/p^{v(\alpha_{i^*})}$ lies in $\ZZ_p$.
Now we denote by $$\Gamma\subset\ZZ^r$$ the lattice formed by the elements $\gamma\in\ZZ^r$ with $\sum \gamma_i\beta_i^{(c)}=0$ in $\ZZ/(p^c\ZZ)$, where $c\in \ZZ_{\geq 0}$ will be chosen in~\eqref{def:cnonarchi}.
Further we define the set $\Sigma^*$ by setting $\Sigma^*=\Sigma$ if $p\geq 3$ and $\Sigma^*=\Sigma(4)$ if $p=2$. Here $\Sigma(4)$ denotes the set of points $P$ in $\Sigma$ with $\mathcal As{x}>4$, where $(x,y)$ is the solution of \eqref{eq:weieq} corresponding to $P$. We obtain the following lemma.
\begin{lemma}\label{lem:nonarchicov}
Suppose that $P\in \Sigma^*$ satisfies $M'<\hat{h}_k(P)\leq M$. Then the point $P$ is determined modulo torsion by some lattice point in $\Gamma\cap\mathcal E$.
\end{lemma}
\begin{proof}
Let $(x,y)$ be the solution of \eqref{eq:weieq} which corresponds to $P$, and write $P=Q+\sum n_i P_i$ with $Q\in E(\QQ)_{\textnormal{tor}}$ and $n_i\in\ZZ$. On using that $P$ is in $\Sigma$ and that $\mu\leq M'<\hat{h}_k(P)$, we deduce that $\log\mathcal As{x}>0$ and thus our point $P$ lies in fact in $E_1(\QQ_p)$. Further, if $p=2$ then our additional assumption $P\in \Sigma(4)$ provides that $v(x)<-2$. Hence on recalling that $P\in \Sigma$, we see that Lemma~\ref{lem:nonarchiest} leads to the inequality
$$
\mathcal As{\sum n_i\alpha_i}\leq \mathcal As{m}e^{-\tfrac{1}{\sigma}(\hat{h}_k(P)-\mu)}.
$$
Here we used that $v(\log(mP))=v(\sum n_i \alpha_i)$, which in turn follows from $mP=\sum n_i(mP_i)$ and $mP_i\in E_1(\QQ_p)$. The displayed inequality together with $M'<\hat{h}_k(P)$ shows that $v(\sum n_i\beta_i)\geq c$, where $c$ is the smallest element of $\ZZ_{\geq 0}$ which exceeds
\begin{equation}\label{def:cnonarchi}
v(m)-v(\alpha_{i^*})+\tfrac{1}{\sigma \log p}(M'-\mu).
\end{equation}
It follows that $\sum n_i \beta_i^{(c)}=0$ in $\ZZ/(p^c\ZZ)$ and therefore $\gamma=(n_i)$ lies in $\Gamma$. Furthermore, our assumption $\hat{h}_k(P)\leq M$ assures that $\gamma$ lies in $\mathcal E$ and hence $P$ is determined modulo torsion by the lattice point $\gamma\in \Gamma\cap \mathcal E$. This completes the proof of the lemma.
\end{proof}
The above lemma provides a sieve for the points $P$ in $\Sigma^*$ with $M'<\hat{h}_k(P)\leq M$. The discussion of the strength of this sieve, depending on the parameters $M'$, $M$ and $p^c$, is analogous to the discussion of the strength of the archimedean sieve in Section~\ref{sec:archisieve}. However there are some minor differences. For example, in the non-archimedean case we can work entirely in dimension $r$ and the parameter $p^c$ is uniquely determined by \eqref{def:cnonarchi}; note that $p^c$ plays here the role of the parameter $c$ in the archimedean sieve.
\paragraph{Non-archimedean sieve.} The following sieve uses the version (FP) of the Fincke--Pohst algorithm described in Section~\ref{sec:compuaspects} in order to determine all points in $\Gamma\cap \mathcal E$.
\begin{Algorithm}[Non-archimedean sieve]\label{algo:nonarchsieve}
The inputs are $M',M\in\ZZ$ with $\mu\leq M'< M$. The output is the set of points $P$ in $\Sigma^*$ with $M'<\hat{h}_k(P)\leq M$.
\begin{itemize}
\item[(i)] To find the number $m$, determine $e_{ns}$, $e_t$ and $\iota $.
\item[(ii)]Determine the lattice $\Gamma\subset \ZZ^{r}$ by computing the $p$-adic elliptic logarithms $\log(mP_i)$ up to the required precision for all $i\in\{1,\dotsc,r\}$.
\item[(iii)] Use \textnormal{(FP)} to find all points in $\Gamma\cap\mathcal E$.
\item[(iv)] For each $\gamma$ in $\Gamma\cap \mathcal E$ and for each torsion point $Q\in E(\QQ)_{\textnormal{tor}}$, output the point $P=Q+\sum \gamma_i P_i$ if $M'<\hat{h}_k(P)\leq M$ and if $P\in \Sigma^*$.
\end{itemize}
\end{Algorithm}
\paragraph{Correctness.} We take a point $P\in \Sigma^*$ which satisfies $M'<\hat{h}_k(P)\leq M$ and we write $P=Q+\sum n_i P_i$ with $n_i\in\ZZ$ and $Q\in E(\QQ)_{\textnormal{tor}}$. Lemma~\ref{lem:nonarchicov} gives that $\gamma=(n_i)$ lies in $\Gamma\cap\mathcal E$ and hence we see that step (iv) produces our point $P$ as desired.
\paragraph{Complexity.} We now discuss the influence of each step on the running time in practice.
In step (i) standard results and algorithms allow to quickly compute the numbers $e_{ns},e_t$ and $\iota$. In fact the computation of (a suitable) $m$ is very fast in practice, even if $p$ is relatively large. Step (ii) needs to compute $r$ distinct $p$-adic elliptic logarithms up to a number of $p$-adic digits which is approximately $c+v(\alpha_{i^*})$. The efficiency of this computation crucially depends on the size of $c+v(\alpha_{i^*})$, which in turn depends in particular on the lower bound $M'$. If the number $M'$ is huge, then this step (ii) can become very slow in practice. Finally we mention that a complexity analysis of steps (iii) and (iv) is contained in the complexity discussions of the analogous steps of the archimedean sieve in Algorithm~\ref{algo:archisieve}.
\paragraph{Comparison.} Similarly as in the archimedean case, there are important differences between our approach and the known method. For instance, we work with the ellipsoid $\mathcal E$ arising from the N\'eron--Tate height $\hat{h}$ and we actually determine all points in the intersection $\Gamma\cap \mathcal E$. We refer to Sections~\ref{sec:globalsieve} and \ref{sec:comparisonwithelr} for a comparison of the two approaches and for a detailed discussion of the improvements provided by our approach.
\subsection{Height-logarithm sieve}\label{sec:heightlogsieve}
We work with the setup of Section~\ref{sec:setup}. The goal of this section is to construct a sieve which allows to efficiently determine the set of $S$-integral points in any given finite subset of $E(\QQ)$. The sieve exploits that the global N\'eron--Tate height is essentially determined by the various local elliptic logarithms and thus we call it the height-logarithm sieve. Throughout this section we use the notation introduced above and we assume that the Weierstrass model \eqref{eq:weieq} of our given elliptic curve $E$ is minimal at all $p\in S$.
\paragraph{Main idea.} To describe the main idea of the sieve, we take $P\in E(\QQ)$. For any finite place $v$ of $\QQ$, we define $\log_v(P)=\tfrac{1}{m_v}\log(m_vP)$ with $\log(\cdot)$ and $m_v=m$ as in Section~\ref{sec:nonarchisieve}. There are real valued functions $f$ and $f_\infty$ on $E(\QQ)$, with $f$ bounded and $f_\infty$ determined by the real elliptic logarithm, such that any non-exceptional\footnote{Here we exclude the exceptional points (Definition~\ref{def:exceptpoint}) in order to avoid the usual technical problems arising when working with the $v$-adic elliptic logarithm at $v=\infty$ and $v=2$.} point $P\in E(\QQ)$ satisfies
\begin{equation}\label{eq:heightlogformula}
\hat{h}(P)=f(P)+f_\infty(P)-\log\prod \mathcal As{\log_v(P)}_v
\end{equation}
with the product taken over certain finite places $v$ of $\QQ$.
Here if $P$ is an $S$-integral point then the product ranges only over $v$ in $S$, providing a strong condition for points in $E(\QQ)$ to be $S$-integral. Furthermore, one can check this condition requiring only to know the form of $P$ in $E(\QQ)_{\textnormal{tor}}\oplus \ZZ^r$. For most points $P$, this allows to circumvent the slow process of checking whether the coordinates of $P$ are $S$-integral, that is whether $P\in\Sigma(S)$.
\paragraph{Construction.} To transform the above idea into an efficient sieve for the set of $S$-integral points $\Sigma(S)$ inside $E(\QQ)$, we shall work with a slightly weaker version of \eqref{eq:heightlogformula} which is suitable for our purpose. More precisely, we shall work with an inequality of the form $\hat{h}_k(P)\leq L(P)$ involving an efficiently computable quantity $L(P)$ which is essentially determined by the right hand side of \eqref{eq:heightlogformula}. We begin to explain how to determine $L(P)$ for any $P\in E(\QQ)$. Suppose that $P=Q+\sum n_i P_i$ with $Q\in E(\QQ)_{\textnormal{tor}}$ and $n_i\in\ZZ$. If $v$ is a finite place of $\QQ$ and if $\alpha_{i,v}=m_v\log_v(P_i)$, then we define $$l_v(P)=\log\max\bigl(\mathcal As{\tfrac{1}{2}}_v,\mathcal As{\tfrac{1}{m_v}\sum n_i \alpha_{i,v}}_v^{-1}\bigl).$$
To give a similar definition at $v=\infty$, we take $\kappa\in\ZZ_{\geq 1}$ and we let $x_0=x_0(\kappa)$ be as in \eqref{def:x0}. Let $\omega_1$ be the period associated to \eqref{eq:weieq}, see Section~\ref{sec:archisieve}. If $v=\infty$ then we write $\alpha_{i,v}=\log(m_vP_i)$ with $\log(\cdot)$ and $m_v=m$ as in Section~\ref{sec:archisieve} and for any $l\in \ZZ$ we define
$$l_v(P,l)=\log\max\bigl( x_0^{1/2},(1+\tfrac{1}{\kappa})^2\mathcal As{\tfrac{1}{m_v}\bigl(l\omega_1+\sum n_i\alpha_{i,v}\bigl)}_v^{-1}\bigl).$$
Here we say that $l\in \ZZ$ is admissible for $P$ if $\mathcal As{l}\leq 2m_\infty+\sum \mathcal As{n_i}$. Let $\mu$ be as in \eqref{eq:nthcompa}. For any finite place $v$ of $\QQ$, we denote by $G_v$ the subgroup of $E(\QQ)$ formed by the points whose images in $E(\QQ_v)$ lie in fact in $E_1(\QQ_v)$. We shall use the following lemma.
\begin{lemma}\label{lem:height-log}
If $P\in \Sigma(S)$ then there is an admissible $l\in\ZZ$ such that $$\hat{h}_k(P)\leq \mu+l_\infty(P,l)+\sum_{v\in S\,;\, P\in G_v} l_v(P).$$
\end{lemma}
\begin{proof}
The statement follows by combining \eqref{eq:nthcompa} with Lemmas~\ref{lem:hk}, \ref{lem:archiest} and \ref{lem:nonarchiest}.
\end{proof}
In the next paragraph we shall explain how to control the quantities $l_\infty(P,l)$ and $\sum l_v(P)$ in order to obtain a suitable upper bound $L(P)$ for the right hand side of the inequality in Lemma~\ref{lem:height-log}. The resulting height-logarithm inequality $\hat{h}_k(P)\leq L(P)$ is the main ingredient of the following algorithm in which we identify $E(\QQ)$ with $E(\QQ)_{\textnormal{tor}}\oplus \Gamma_E$, where $\Gamma_E=\ZZ^r$ is the image of $E(\QQ)$ inside $E(\QQ)\otimes_\ZZ\RR\cong \RR^r$ as in Section~\ref{sec:heightselllog}.
\begin{Algorithm}[Height-logarithm sieve]\label{algo:heightlogsieve} The inputs are $\kappa\in\ZZ_{\geq 1}$ and a finite subset $\Sigma$ of $E(\QQ)_{\textnormal{tor}}\oplus \Gamma_E$. The output is the set $\Sigma\cap\Sigma(S)$ of $S$-integral points inside $\Sigma$.
Determine the set $S_E$ formed by the places $v\in S$ where the elliptic curve $E$ has bad reduction. Then for each nonzero point $P\in \Sigma$ do the following:
\begin{itemize}
\item[(i)] Use the arguments of \textnormal{(1)} below to determine an upper bound $l_\infty(P)\geq \max l_\infty(P,l)$ with the maximum taken over all $l\in \ZZ$ which are admissible for $P$.
\item[(ii)] Compute the set $S_P=\{v\in S\,;\, P\in G_v\}\cup S_E$ as described in \textnormal{(2)} below.
\item[(iii)] For each $v\in S_P$ determine $l_v(P)$ by using the arguments of \textnormal{(3)} below, and then set $L(P)=\mu+l_\infty(P)+\sum_{v\in S_P}l_v(P)$. Output $P$ if $\hat{h}_k(P)\leq L(P)$ and if $P\in \Sigma(S)$.
\end{itemize}
\end{Algorithm}
\paragraph{Correctness.} Suppose that $P$ lies in $\Sigma\cap\Sigma(S)$. For each $v\in S_E$ we obtain that $l_v(P)\geq 0$ and thus $\sum_{v\in S_P}l_v(P)$ exceeds the sum $\sum l_v(P)$ taken over all $v\in S$ with $P\in G_v$. Hence Lemma~\ref{lem:height-log} implies that $\hat{h}_k(P)\leq L(P)$ and thus (iii) produces our point $P$ as desired.
\paragraph{Computing $L(P)$.} We consider a nonzero point $P=(Q,(n_i))$ in $E(\QQ)_{\textnormal{tor}}\oplus \Gamma_E$; note that $P=Q+\sum n_i P_i$ in $E(\QQ)$. To compute the quantity $L(P)$ we proceed as follows:
\begin{itemize}
\item[(1)] To control $l_\infty(P,l)$ for any admissible $l\in \ZZ$, we compute the real elliptic logarithms $\omega_1$ and $\alpha_i=\alpha_{i,v}$ up to a certain precision with respect to $\mathcal As{\cdot}=\mathcal As{\cdot}_v$ for $v=\infty$. If the linear form $\Lambda=l\omega_1+\sum n_i\alpha_i$ is nonzero then the required precision can be obtained as follows: After choosing a sufficiently large $n\in\ZZ$, one determines approximations $\alpha_{i}'$ and $\omega_1'$ of $\alpha_{i}$ and $\omega_1$ respectively such that $\Lambda'=l\omega_1'+\sum n_i\alpha_i'$ satisfies $\mathcal As{\Lambda'}>\varepsilonilon=10^{-n}$ and such that the absolute differences $\mathcal As{\alpha_i-\alpha_i'}$ and $\mathcal As{\omega_1-\omega_1'}$ are at most $10^{-c}$ for some fixed integer $c\geq n+\log_{10} (2m_\infty+2\sum \mathcal As{n_i})$.
Then the proof of Lemma~\ref{lem:archicov} gives that $-\log \mathcal As{\Lambda}\leq -\log(\mathcal As{\Lambda'}-\varepsilonilon)$ and hence we can practically compute an upper bound $l_\infty(P)$ in $\RR\cup\{\infty\}$ for $\max l_\infty(P,l)$ with the maximum taken over all admissible $l\in \ZZ$. Here if $\Lambda$ is close to zero or if $c$ is too large, then we just put $l_\infty(P)=\infty$ to assure that the computation of $l_\infty(P)$ is always efficient.
\item[(2)] We would like to quickly compute the set $S_P=\{v\in S\,;\, P\in G_v\}\cup S_E$. Here one can directly determine $S_E$, since the Weierstrass model \eqref{eq:weieq} is minimal at all $p\in S$. It remains to deal with the places $p\in S_P-S_E$.
The elliptic curve $E$ has good reduction at $p$ and the canonical reduction map $E(\QQ)\hookrightarrow E(\QQ_p)\to E(\mathbb F_p)$ is a morphism of abelian groups. We determine the images $\bar{Q}$ and $\bar{P_i}$ in $E(\mathbb F_p)$ of all $Q\in E(\QQ)_{\textnormal{tor}}$ and all $P_i$. It follows that our point $P=Q+\sum n_i P_i$ lies in $G_v$ if and only if the point $\bar{Q}+\sum n_i\bar{P_i}$ is zero in $E(\mathbb F_p)$. Therefore we see that we can quickly determine the set $S_P$ provided we already know all $\bar{Q}$, all $\bar{P_i}$ and the group structure of $E(\mathbb F_p)$.
\item[(3)] We take $v\in S$ and we now explain how to efficiently determine $l_v(P)$. As already mentioned in Section~\ref{sec:nonarchisieve} one can always quickly compute the number $m_v$ in practice. We write $\alpha_i=m_v\log_v(P_i)$ and we define $\alpha=\sum n_i \alpha_i$.
To compute $v(\alpha)$ we need to know the $v$-adic elliptic logarithms $\alpha_i$ with a certain precision. In practice it usually suffices here to know $\alpha_i$ with a small $v$-adic precision. Indeed after computing $v(\alpha_{i^*})=\min v(\alpha_i)$, we consider $\beta=\sum n_i\beta_i$ for $\beta_i\in \ZZ_p$ of the form $\beta_i=\alpha_i/p^{v(\alpha_{i^*})}$. The integer $v(\beta)$ is almost always small in practice. Hence one can usually compute $v(\beta)$ and $v(\alpha)$ by knowing only the first coefficients of the $v$-adic power series of $\alpha_i$.
\end{itemize}
\paragraph{Huge and tiny parameters}
To assure that Algorithm~\ref{algo:heightlogsieve} is still fast for huge parameters, one can slightly weaken the sieve as follows: If one of the steps (except the final check whether $P\in \Sigma(S)$) should take too long for a point $P\in \Sigma$, then abort these steps and directly check whether $P\in \Sigma(S)$. To deal with the case of huge sets $S$ in which step (ii) becomes slow (see Remark~\ref{rem:rank1heightlog}), we can always replace $S_P$ by the usually much larger set $S$. The resulting sieve is still strong for points $P$ with $\hat{h}(P)\gg\log N_S$. However, replacing $S_P$ by $S$ considerably weakens the sieve for points of small height. We now discuss the case when the rank $r$ is small and the height of the involved point $P$ is tiny. Here one can quickly compute the coordinates of $P$ and the Weil height of these coordinates is not that large.
Hence in this case it is often faster to skip steps (i) and (ii) and to directly determine in (iii) whether the coordinates of $P$ are $S$-integers. In our implementation of the height-logarithm sieve we take into account the above observations to avoid that Algorithm~\ref{algo:heightlogsieve} is unnecessarily slow for huge or tiny parameters.
\paragraph{Complexity.} We now discuss aspects of Algorithm~\ref{algo:heightlogsieve} which considerably influence the running time in practice. In steps (i) and (iii) we need to compute various elliptic logarithms up to a certain precision depending on the height $\hat{h}(P)$ of the points $P\in \Sigma$. In practice we will apply the height-logarithm sieve only in situations in which the heights $\hat{h}(P)$ are not huge and in such situations steps (i) and (iii) are always very fast.
In step (ii) the running time of the computation of the set $S_P$ crucially depends on the number of primes in $S$. In practice it turned out that this step is fast when $\mathcal As{S}$ is small. However, if $\mathcal As{S}$ becomes huge then step (ii) can take a long time as explained in the following remark.
\begin{remark}[Rank $r=1$]\label{rem:rank1heightlog}
For huge sets $S$ the computation of $S_P$ takes a long time, since one has to compute with many large groups $E(\mathbb F_p)$. In the case $r=1$ the following observation considerably improves this process. Let $v\in S$ such that $E$ has good reduction at $v$ and write $p=v$. Let $e_v$ be the order of $P_1$ in the finite group $E(\mathbb F_p)$. Consider a point $P\in E(\QQ)$ with $P=Q+n_1P_1$ for $n_1\in\ZZ$ and $Q\in E(\QQ)_{\textnormal{tor}}$, and let $e_Q$ be the order of $Q$ in $ E(\QQ)_{\textnormal{tor}}$. If $P\in G_v$ then the points $-Q$ and $n_1P_1$ coincide in $E(\mathbb F_p)$ and hence $e_v$ divides $n_1e_Q$. In other words, if $e_v$ does not divide $n_1e_Q$ then $v$ is not in $S_P$ and therefore we obtain a sufficient criterion to decide whether $v\in S$ satisfies $v\notin S_P$.
\end{remark}
\remark[Inequality trick]\label{rem:inequtrick} In the case when $S$ is empty, one can use the known inequality trick \cite[p.147]{sttz:elllogoverview} which tests whether a given nonzero point $P\in E(\QQ)$ satisfies the inequality $\lambda\|P\|_\infty^2\leq \mu+ l_\infty(P)$. This inequality is weaker than $\hat{h}(P)\leq \mu+l_\infty(P)$ used in our height-logarithm sieve when $S$ is empty, since $\lambda\|P\|_\infty^2\leq \hat{h}(P)$. Hence our height-logarithm sieve is more efficient than the inequality trick, in particular in the case of large rank $r\geq 2$ where the function $\lambda\|\cdot\|_\infty^2$ is usually much smaller than $\hat{h}(\cdot)$. See also the examples in the next section. In the case when $S$ is nonempty, one could obtain in principle an inequality trick by testing whether $P$ satisfies the inequality $\lambda\|P\|_\infty^2\leq \mu+\sigma l_v(P)$ for some $v\in S^*=S\cup\{\infty\}$ and $\sigma=\mathcal As{S^*}$. However the resulting sieve is not that efficient (and often useless if $\sigma$ is large), since an arbitrary point $P$ usually satisfies at least one of these $\sigma$ different inequalities which are all considerably weakened by the factor $\sigma$.
\subsection{Refined enumeration}\label{sec:refinedenumell}
We work with the setup of Section~\ref{sec:setup}. The goal of this section is to construct a refined enumeration for the set of $S$-integral points $\Sigma(S)\subset E(\QQ)$ of bounded height which improves the standard enumeration. Throughout this section we assume that the Weierstrass model \eqref{eq:weieq} of $E$ is minimal at all $p\in S$ and we continue the notation introduced above.
Recall from Section~\ref{sec:heightselllog} that $\Gamma_E=\ZZ^r$ denotes the image of $E(\QQ)$ inside $E(\QQ)\otimes_\ZZ\RR\cong \RR^r$. For any given upper bound $b\in \RR_{\geq 1}$, consider the ellipsoid $\mathcal E_b=\{z\in\RR^r\,;\, \hat{h}_k(z)\leq b\}$ contained in $\RR^r$. We observe that the following algorithm works correctly.
\begin{Algorithm}[Refined enumeration]\label{algo:refenu} The input consists of $\kappa\in\ZZ_{\geq 1}$ together with an upper bound $b\in \RR_{\geq 1}$. The output is the set of points $P\in \Sigma(S)$ with $\hat{h}_k(P)\leq b.$
\begin{itemize}
\item[(i)] Use \textnormal{(FP)} to determine all points in the intersection $\Gamma_E\cap\mathcal E_b$.
\item[(ii)] For each $\gamma\in\Gamma_E\cap \mathcal E_{b}$ and for any $Q\in E(\QQ)_{\textnormal{tor}}$, output the point $P=Q+\sum \gamma_i P_i$ if $P$ lies in the set obtained by applying Algorithm~\ref{algo:heightlogsieve} with $\kappa=\kappa$ and $\Sigma=\{P\}$.
\end{itemize}
\end{Algorithm}
\paragraph{Complexity.} We now discuss various aspects which influence the running time of Algorithm~\ref{algo:refenu} in practice. As usual, the application of \textnormal{(FP)} in step (i) crucially depends on the rank $r$.
The running time of step (ii) depends on the cardinality of $\Gamma_E\cap \mathcal E_b$, which in turn depends on $r$, $b$ and the regulator of $E(\QQ)$. Here the application of the height-logarithm sieve efficiently throws away most points in $\Gamma_E\cap \mathcal E_b$, in particular essentially all points in $\Gamma_E\cap \mathcal E_b$ of large height. This considerably improves the running time.
\paragraph{Comparison.} We next compare our refined enumeration with the standard enumeration of the points $P\in\Sigma(S)$ with $\|P\|_\infty^2\leq b'$, where $b'=b/\lambda_k$ depends on the smallest eigenvalue $\lambda_k$ of $\hat{h}_k$.
Recall that the standard enumeration proceeds as follows: For any $\gamma\in\Gamma_E$ with $\max \mathcal As{\gamma_i}^2\leq b'$ and for each $Q\in E(\QQ)_{\textnormal{tor}}$, output the point $P=Q+\sum \gamma_i P_i$ if the coordinates of $P$ are $S$-integers. In the case when $S$ is empty, one can use here in addition the known inequality trick explained in Remark~\ref{rem:inequtrick}. In general we observe that our refined sieve working with the ellipsoid $\mathcal E_b$ is more efficient, in particular for large rank $r$. Indeed the cube $\{\|\cdot\|_\infty^2\leq b'\}\subset\RR^r$ always contains the ellipsoid $\mathcal E_b$ and then on comparing volumes we see that our refined sieve involves much fewer points. Furthermore the application of the height-logarithm sieve in the refined enumeration gives significant running time improvements. See Section~\ref{sec:comparisonwithelr} for examples and tables which illustrate in particular the running time improvements provided by our refined enumeration.
\subsection{Refined sieve}\label{sec:refinedsieveell}
In this section we work out a refinement of the global sieve obtained by patching together the archimedean sieve of Section~\ref{sec:archisieve} with the various non-archimedean sieves of Section~\ref{sec:nonarchisieve}. Throughout this section we work with the setup of Section~\ref{sec:setup} and we continue the notation introduced above. Furthermore we assume that the Weierstrass model \eqref{eq:weieq} of our given elliptic curve $E$ is minimal at all primes $p\in S$.
\paragraph{Main idea.} The main ingredient of the refined sieve is Proposition~\ref{prop:refinedcov}. Therein we construct a refined covering of certain subsets of the set of $S$-integral points $\Sigma(S)$, which allows to improve the global-local passage required to apply the local sieves obtained in Sections \ref{sec:archisieve} and \ref{sec:nonarchisieve}. The construction of the covering is inspired by the refined sieve for $S$-unit equations developed in Section~\ref{sec:dwsieve+}. However, in the present case of elliptic curves, everything is more complicated. For example, one has to distinguish archimedean and non-archimedean places and one has to take care of certain exceptional points (Definition~\ref{def:exceptpoint}) arising from technical issues of the $v$-adic elliptic logarithm at the places $v=\infty$ and $v=2$. To deal efficiently with the exceptional points, we conducted some effort to work entirely in the abelian group $E(\QQ)$. This allows here to avoid working with coordinate functions, which in turn is crucial to solve equations \eqref{eq:weieq} with huge parameters.
\paragraph{Construction of the covering.} For any given $M,M'$ in $\ZZ$ with $0\leq M'<M$, we would like to find the set of points $P\in \Sigma(S)$ which satisfy $M'<\hat{h}_k(P)\leq M$. For this purpose we ``cover" this set as follows.
Let $\kappa,n,\tau$ in $\ZZ_{\geq 1}$ with $\tau\leq n\leq s^*$ for $s^*=\mathcal As{S}+1$
and choose an admissible partition $\{S_j\}$ of $S^*=S\cup \{\infty\}$ into disjoint nonempty parts $S^*= S_1\dotcup\ldots\dotcup S_g$. Here admissible partition means that $|S_j|\leq n$ for all $j\in\{1,\dotsc,g\}$ and that $g\leq\ceil{\tfrac{s^*}{n}}+2$. For a motivation of working with admissible partitions of $S^*$, we refer to the efficiency discussion given below. Further we choose ``weights" $w_1,\dotsc,w_\tau$ in $\QQ$ with $w_1=1$ and $w_1\geq\dotsc\geq w_\tau>0$ and for any $t\in \{1,\dotsc,\tau\}$ we put
\begin{equation}\label{def:weightswt}
\sigma_t=\tfrac{w}{w_t}, \ \ \
w=\sum_{j=1}^g w(j), \ \ \ w(j)=
\begin{cases}
(|S_j|-\tau)w_\tau+\sum_{t\leq\tau} w_t & \textnormal{if } \tau\leq \mathcal As{S_j},\\
\sum_{t\leq \mathcal As{S_j}} w_t & \textnormal{if } \tau>\mathcal As{S_j}.
\end{cases}
\end{equation}
Next we take $j\in\{1,\dotsc,g\}$ and we consider a nonempty subset $T$ of $S_j$ with cardinality $\mathcal As{T}$ at most $\tau$. Write $t=\mathcal As{T}$ and suppose that $v\in T$. If $v\in S$ then we denote by $\Gamma_v\subseteq\ZZ^r$ the lattice constructed in Section~\ref{sec:nonarchisieve} with $\sigma=\sigma_t$, and if $v=\infty$ then $\Gamma_v\subseteq\ZZ^{r+1}$ denotes the lattice from Section~\ref{sec:archisieve} with $\sigma=\sigma_t$ and $\kappa=\kappa$. In the case $\mu>M'$, where $\mu$ is as in \eqref{eq:nthcompa}, we set here $\Gamma_v=\ZZ^r$ if $v\in S$ and $\Gamma_v=\ZZ^{r+1}$ if $v=\infty$. Now we define
$$\Gamma_T=\bigcap_{v\in T} \Gamma_v.$$
Here if $T$ contains $\infty$ then for any $v\in S\cap T$ we identify $\Gamma_v\subseteq \ZZ^r$ with the lattice inside $\ZZ^{r+1}$ given by $\phi(\Gamma_v)\oplus e\ZZ$, where $\phi$ denotes the canonical product embedding of $\ZZ^r$ into $\ZZ^r\times\ZZ=\ZZ^{r+1}$ and $e=(0,1)\in\ZZ^{r}\times\ZZ$.
Next we consider the ellipsoid $$\mathcal E_T$$
which is defined as follows: If $T\subseteq S$ then $\mathcal E_T\subset \RR^r$ is the ellipsoid appearing in Section~\ref{sec:nonarchisieve}, and if $T$ contains $\infty$ then $\mathcal E_T\subset \RR^{r+1}$ is the ellipsoid constructed in Section~\ref{sec:archisieve} with respect to the parameters $\sigma=\sigma_t$ and $\kappa=\kappa$. In the case $\mu>M'$ we define here $\mathcal E_T=\RR^r$ if $T\subseteq S$ and $\mathcal E_T=\RR^{r+1}$ if $T$ contains $\infty$. We next define the exceptional points.
\begin{definition}[Exceptional point]\label{def:exceptpoint}
Consider a point $P\in\Sigma(S)$ and denote by $(x,y)$ the corresponding solution of \eqref{eq:weieq}. We say that $P$ is an exceptional point if $\mathcal As{x}_2\leq 4$ or if $\mathcal As{x}_\infty\leq x_0$, where $x_0$ denotes the number $x_0(\kappa)$ defined in \eqref{def:x0}.
\end{definition}
To completely ``cover" our set of interest, we need to take into account the exceptional points.
For this purpose we let $b$ be the positive real number defined in \eqref{def:ellrad}, which depends inter alia on the parameters $\kappa,\tau,\{S_j\}$ and $w_t$, and we work with the ellipsoid $$\mathcal E_{b}=\{z\in\RR^r\,;\, \hat{h}_k(z)\leq b\}.$$
Recall from Section~\ref{sec:heightselllog} that $\Gamma_E=\ZZ^r$ denotes the image of $E(\QQ)$ inside $E(\QQ)\otimes_\ZZ\RR\cong \RR^r$. The following result shows that our set of interest can be ``covered" by the set $\Gamma_E\cap \mathcal E_{b}$ together with the sets $\Gamma_T\cap \mathcal E_T$ associated to some $T$ as above.
\begin{proposition}\label{prop:refinedcov}
Suppose that $P$ lies in $\Sigma(S)$ and assume that $M'<\hat{h}_k(P)\leq M$. Then at least one of the following statements holds.
\begin{itemize}
\item[(i)] The point $P$ is determined modulo torsion by some $\gamma\in\Gamma_E\cap\mathcal E_b$.
\item[(ii)] There is a nonempty set $T$ with $\mathcal As{T}\leq \tau$ such that $T\subseteq S_j$ for some $S_j$ in $\{S_j\}$ and such that $P$ is determined modulo torsion by an element $\gamma\in \Gamma_T\cap \mathcal E_T$.
\end{itemize}
\end{proposition}
\begin{proof}
If $M'<\mu$ then (ii) holds for example with $T=\{\infty\}$. Thus we may and do assume that $M'\geq \mu$. Let $(x,y)$ be the solution of \eqref{eq:weieq} corresponding to $P$. We claim that there exists a nonempty set $T$ with $\mathcal As{T}\leq \tau$ such that $T\subseteq S_j$ for some $S_j$ in $\{S_j\}$ and such that
\begin{equation}\label{coveringclaim}
P\in \bigcap_{v\in T}\Sigma(v,\sigma_t).
\end{equation}
Here $\Sigma(v,\sigma_t)$ is defined in \eqref{def:sigmavsigma} with $t=\mathcal As{T}$.
To prove this claim by contradiction, we assume that \eqref{coveringclaim} does not hold. Then for each $j$ and for any nonempty subset $T\subseteq S_j$ with $t=\mathcal As{T}\leq \tau$, there exists $v\in T$ such that $P\notin \Sigma(v,\sigma_t)$. In particular, for any $j$ and for each $t\in\ZZ_{\geq 1}$ with $t\leq \min(\tau,\mathcal As{S_j})$, it follows that the $t$-th largest of the real numbers $\tfrac{1}{2}\log\mathcal As{x}_v$, $v\in S_j,$ is strictly smaller than $\tfrac{1}{\sigma_t}(\hat{h}_k(P)-\mu)$.
We deduce that $\tfrac{1}{2}\sum\max(0,\log\mathcal As{x}_v)< \tfrac{w(j)}{w}(\hat{h}_k(P)-\mu)$ with the sum taken over all $v\in S_j$. Here we used that $\mu\leq M'<\hat{h}_k(P)$ and that the weights $w_t$ satisfy $w_1\geq \dotsc \geq w_\tau>0$. Then our assumption $P\in \Sigma(S)$ together with $S^*=\cup S_j$ implies that $\tfrac{1}{2}h(x)< \hat{h}_k(P)-\mu$. But this contradicts the inequality $\hat{h}_k(P)-\mu\leq \tfrac{1}{2}h(x)$ which follows by combining Lemma~\ref{lem:hk} with \eqref{eq:nthcompa}. Therefore we conclude that our claim \eqref{coveringclaim} holds as desired.
Let $\mathcal T$ be the nonempty set of all sets $T$ satisfying \eqref{coveringclaim}; put $t=\min\{\mathcal As{T}\,;\, T\in\mathcal T\}$ and define $\mathcal T_{\min}=\{T\in\mathcal T\,;\, \mathcal As{T}=t\}$. Further, on slightly abusing terminology, we write $\Sigma(4)$ for the subset of $\Sigma(2,\sigma_t)$ defined in Section~\ref{sec:nonarchisieve} and we denote by $\Sigma(x_0)$ the subset of $\Sigma(\infty,\sigma_t)$ from Section~\ref{sec:archisieve}. First we consider the case $t=1$. Suppose that we can choose $T\in\mathcal T_{\min}$ with $T\subseteq S-2$. Then we obtain that $T=\{p\}$ with $p\geq 3$. Thus on recalling that the Weierstrass model \eqref{eq:weieq} is minimal at all $p\in S$, we see that the inequalities $\mu\leq M'$ and $M'<\hat{h}_k(P)\leq M$ together with \eqref{coveringclaim} show that $P$ satisfies the assumptions of Lemma~\ref{lem:nonarchicov}. Hence Lemma~\ref{lem:nonarchicov} implies (ii). Suppose now that there is no $T\in\mathcal T_{\min}$ with $T\subseteq S-2$. Then $\{\infty\}$ or $\{2\}$ lies in $\mathcal T_{\min}$ and any $v\in S-2$ satisfies
\begin{equation}\label{eq:extrasearch}
\tfrac{1}{2}\log |x|_v< \tfrac{1}{\sigma_1}(\hat{h}_k(P)-\mu).
\end{equation}
To complete the proof for $t=1$, it remains to establish (i) or (ii) in the following cases (a), (b) and (c). Before we go into these cases we define the number $b$ appearing in $\mathcal E_b$: If $t^*=\min(\tau,\max_{v=2,\infty}\mathcal As{S_{j(v)}})$ with $S_{j(v)}$ denoting the set $S_j$ which contains $v$, then
\begin{equation}\label{def:ellrad}
b=\mu+\tfrac{1}{2w_{t^*}}(1+s_{t^*})\log \max(x_0,4).
\end{equation}
Here $x_0=x_0(\kappa)$ is defined in \eqref{def:x0} and $s_{t^*}$ is the number of $p\in S$ with $p^{2w_{t^*}}\leq \max(x_0,4)$. In what follows we shall use that $P\in \Sigma(S)$, that for each finite place $v$ of $\QQ$ it holds $v(x)\leq -2$ if $P\in E_1(\QQ_v)$, that $1=w_1\geq\dotsc\geq w_\tau>0$, and that $\hat{h}_k\leq \hat{h}$ by Lemma~\ref{lem:hk}.
\begin{itemize}
\item[(a)]Case $\{2\}\in\mathcal T_{\min}$ and $\{\infty\}\in\mathcal T_{\min}$. If $\mathcal As{x}_\infty>x_0$ or $\mathcal As{x}_2> 2^2$, then \eqref{coveringclaim} implies that $P\in\Sigma(x_0)$ or $P\in\Sigma(4)$ and thus Lemma~\ref{lem:archicov} or Lemma \ref{lem:nonarchicov} shows (ii) for $T=\{\infty\}$ or $T=\{2\}$ respectively. On the other hand, if $\mathcal As{x}_\infty\leq x_0$ and $\mathcal As{x}_2\leq 2^2$ then \eqref{eq:extrasearch} and \eqref{coveringclaim} lead to
an upper bound for $h(x)$ which together with \eqref{eq:nthcompa} proves (i).
\item[(b)] Case $\{2\}\in \mathcal T_{\min}$ and $\{\infty\}\notin\mathcal T_{\min}$. Here inequality \eqref{eq:extrasearch} holds in addition for $v=\infty$, since $\{\infty\}$ is not in $\mathcal T_{\min}$. Therefore, if $\mathcal As{x}_2\leq 2^2$ then we see as above that \eqref{eq:extrasearch}, \eqref{coveringclaim} and \eqref{eq:nthcompa} imply statement (i). If $\mathcal As{x}_2> 2^ 2$ then \eqref{coveringclaim} gives that $P\in \Sigma(4)$ and hence Lemma~\ref{lem:nonarchicov} shows statement (ii) with $T=\{2\}$.
\item[(c)]Case $\{2\}\notin \mathcal T_{\min}$ and $\{\infty\}\in\mathcal T_{\min}$. Now \eqref{eq:extrasearch} holds in addition for $v=2$, since $\{2\}$ is not in $\mathcal T_{\min}$. Thus as above we deduce (i) if $\mathcal As{x}_\infty\leq x_0$. If $\mathcal As{x}_\infty> x_0$ then \eqref{coveringclaim} gives $P\in \Sigma(x_0)$ and hence Lemma~\ref{lem:archicov} proves (ii) with $T=\{\infty\}$.
\end{itemize}
We now establish the case $t\geq 2$. If we can choose $T\in\mathcal T_{\min}$ with $T\subseteq S-2$, then \eqref{coveringclaim} together with Lemma~\ref{lem:nonarchicov} implies (ii). Suppose now that there is no $T\in\mathcal T_{\min}$ with $T\subseteq S-2$.
Then each $T$ in $\mathcal T_{\min}$ contains $\infty$ or $2$. Furthermore any $v\in S^*$ satisfies \eqref{eq:extrasearch}, since $t\geq 2$. To complete the proof it thus suffices to consider the following cases:
\begin{itemize}
\item[(d)] Case when each $T\in\mathcal T_{\min}$ contains $2$ and $\infty$. If $\mathcal As{x}_\infty > x_0$ and $\mathcal As{x}_2> 2^2$, then \eqref{coveringclaim} gives that $P\in \Sigma(x_0)\cap \Sigma(4)$. Thus on recalling the construction of $\Gamma_T$ and $\mathcal E_T$, we see that Lemmas~\ref{lem:archicov} and \ref{lem:nonarchicov} together with \eqref{coveringclaim} show that (ii) holds for any $T\in \mathcal T_{\min}$.
On the other hand, if $\mathcal As{x}_\infty\leq x_0$ or $\mathcal As{x}_2\leq 2^2$ then \eqref{eq:extrasearch}, \eqref{coveringclaim} and \eqref{eq:nthcompa} prove (i).
\item[(e)] Case when there is $T\in \mathcal T_{\min}$ with $2\in T$ and $\infty\notin T$. If $\mathcal As{x}_2> 2^2$ then \eqref{coveringclaim} gives that $P\in \Sigma(4)$ and thus we deduce (ii) by using $\infty\notin T$, \eqref{coveringclaim} and Lemma~\ref{lem:nonarchicov}. On the other hand, if $\mathcal As{x}_2\leq 2^2$ then \eqref{eq:extrasearch}, \eqref{coveringclaim} and \eqref{eq:nthcompa} imply (i).
\item[(f)] Case when there exists $T\in \mathcal T_{\min}$ with $2\notin T$ and $\infty\in T$. If $\mathcal As{x}_\infty\leq x_0$ then \eqref{eq:extrasearch}, \eqref{coveringclaim} and \eqref{eq:nthcompa} imply (i). Finally, if $\mathcal As{x}_\infty> x_0$ then \eqref{coveringclaim} gives $P\in \Sigma(x_0)$. Therefore on using $2\notin T$ and \eqref{coveringclaim}, we see that Lemmas~\ref{lem:archicov} and \ref{lem:nonarchicov} prove (ii).
\end{itemize}
Hence we conclude that in all cases (i) or (ii) holds. This completes the proof.
\end{proof}
The arguments used to prove \eqref{coveringclaim} show in addition that one can further refine the covering in Proposition~\ref{prop:refinedcov} by working with the ellipsoids $\mathcal E_T^*$ discussed in Remark~\ref{rem:refellips}.
\paragraph{Refined Sieve.} A collection of sieve parameters $\mathcal P$ consists of the following data: Parameters $\kappa,\tau,n\in\ZZ_{\geq 1}$ with $\tau\leq n\leq s^*$, an admissible partition $\{S_j\}$ of $S^*=S_1\dotcup\dotsc \dotcup S_g$ with respect to $n$, and weights $w_1,\dotsc,w_\tau$ as in \eqref{def:weightswt}. We denote by $b(\mathcal P)$ the number associated to $\mathcal P$ as in \eqref{def:ellrad} and we obtain the following algorithm.
\begin{Algorithm}[Refined sieve]\label{algo:refinedsieve} The input is a collection of sieve parameters $\mathcal P$ together with bounds $M',M\in\ZZ_{\geq 0}$ satisfying $M'< M$. Put $M'_b=\max(b(\mathcal P),M')$. The output is the set of points $P\in \Sigma(S)$ with $M'_b<\hat{h}_k(P)\leq M$.
For any $j\in \{1,\dotsc,g\}$ and for each $T\subseteq S_j$ with $1\leq \mathcal As{T}\leq \tau$, do the following:
\begin{itemize}
\item[(i)] Determine a basis of $\Gamma_T$ and then compute $\Gamma_T\cap \mathcal E_T$ by using the version of the Fincke--Pohst algorithm in \textnormal{(FP)}.
\item[(ii)] For each $\gamma\in\Gamma_T\cap \mathcal E_T$ and for any $Q\in E(\QQ)_{\textnormal{tor}}$, output the point $P=Q+\sum \gamma_i P_i$ if $P$ satisfies $M'_b<\hat{h}_k(P)\leq M$ and if $P$ lies in the set produced by an application of Algorithm~\ref{algo:heightlogsieve} with $\Sigma=\{P\}$ and $\kappa=1$.
\end{itemize}
\end{Algorithm}
\paragraph{Correctness.} Assume that $P\in \Sigma(S)$ satisfies $M'_b<\hat{h}_k(P)\leq M$. Then it holds that $\hat{h}_k(P)>b(\mathcal P)$ and hence there is no lattice point $\gamma$ in $\Gamma_E\cap \mathcal E_{b(\mathcal P)}$ such that $P$ is determined modulo torsion by $\gamma$. Furthermore, our assumption provides that $M'<\hat{h}_k(P)\leq M$. Therefore Proposition~\ref{prop:refinedcov} shows that step (ii) produces our point $P$ as desired.
\paragraph{Efficiency.} We now discuss the efficiency of the refined sieve and we motivate several concepts appearing therein. First we observe that the case $n=1$ in the refined sieve corresponds to the non-refined sieve obtained by patching together the local sieves at $v\in S^*$ with $\sigma=s^*$. Suppose now that $n\geq\tau\geq 2$ in the refined sieve. Then the iteration over the sets $T$ ranges in particular over all sets $T=\{v\}$ with $v\in S^*$. However, compared with the non-refined sieve $n=1$, there is the following fundamental difference: If $T=\{v\}$ then the discussions in Sections~\ref{sec:archisieve} and \ref{sec:nonarchisieve} together with $\sigma_1\leq s^*$ show that the refined sieve involving $\Gamma_T\cap \mathcal E_T$ with $\sigma_1$ is usually much stronger than the non-refined sieve $n=1$ involving the local sieve at $v$ with $\sigma=s^*$. Furthermore, if $\mathcal As{T}\geq 2$ then the intersection $\Gamma_T=\cap \Gamma_v$ is usually considerably smaller than each part $\Gamma_v$. These observations suggest that the improvements coming from $\mathcal As{T}=1$ are significant enough to absorb the additional iterations over sets $T$ with $\mathcal As{T}\geq 2$. In practice this turned out to be correct in many fundamental situations (see Section~\ref{sec:globalsieve}), showing that the refined sieve provides significant running time improvements. To deal with huge sets $S$, we introduced admissible coverings of $S^*$ which allow to control the number of additional iterations over sets $T$ with $\mathcal As{T}\geq 2$. Indeed the conditions $\mathcal As{S_j}\leq n$ and $\mathcal As{T}\leq \tau$ assure that the number of additional iterations are controlled in terms of $n,\tau$. Further we point out that a canonical choice for the weights $w_t$ would be $w_t=\tfrac{1}{t}$. However in practice it turned out that for $t\geq 2$ it would be better to choose $w_t$ slightly larger than $\tfrac{1}{t}$. In fact this is the reason for working with the more general weights $w_t$ defined above \eqref{def:weightswt}. Finally we mention that the discussion of the influence of the parameters $M'$, $M$ and $\sigma_t$ on the strength of the sieve $\Gamma_T\cap\mathcal E_T$ with $\mathcal As{T}=t$ is similar to the corresponding discussions in Sections~\ref{sec:archisieve} and \ref{sec:nonarchisieve}.
\paragraph{Complexity.} We do not try to analyse the complexity of the refined sieve in general, since it depends on too many parameters. However, in Sections~\ref{sec:globalsieve} and \ref{sec:comparisonwithelr} we shall discuss aspects influencing the complexity of the refined sieve and we shall illustrate the running improvements provided by the refined sieve in various fundamental cases.
\begin{remark}[Refined ellipsoids]\label{rem:refellips}
For any set $T$ as above with $\mathcal As{T}\geq 2$, the arguments used in the proof of \eqref{coveringclaim} show in addition the following: Instead of using in Algorithm~\ref{algo:refinedsieve} the ellipsoids $\mathcal E_T$, one can work with the ellipsoids $\mathcal E_T^*$ obtained by replacing in the definition of $\mathcal E_T$ the bound $M$ by the possibly much smaller number $$M_T=\min\left(M,w_T(M'-\mu)+\mu\right), \ \ \ w_T=\tfrac{1}{w}\sum_{j=1}^g w^*(j).$$ Here $w^*(j)$ is obtained by replacing in the definition of $w(j)$ the number $\tau$ by $\mathcal As{T}-1$. Note that $\mathcal E_T^*\subseteq\mathcal E_T$ and $w_T$ does not depend on $M$. Now if $M>M_T$ then $\mathcal E_T^*$ is strictly contained in $\mathcal E_T$ and hence using $\mathcal E_T^*$ improves Algorithm~\ref{algo:refinedsieve}~(i). In principle further refinements are possible by taking into account the part $S_{j}$ of $S^*$ which contains $T\subseteq S_{j}$.
\end{remark}
\subsection{Global sieve}\label{sec:globalsieve}
We continue the setup, notation and assumptions of the previous section. After choosing suitable collections of sieve parameters, we combine in this section the refined enumeration with the refined sieve: For any given upper bound $M_1\in\ZZ_{\geq 1}$, we obtain a global sieve which allows to efficiently determine all points $P\in \Sigma(S)$ with $\hat{h}_k(P)\leq M_1$.
\paragraph{Sieve parameters.} We shall apply our refined sieve with the following collections of sieve parameters. Choose $\kappa\in\ZZ_{\geq 1}$ such that $\mathcal As{b-10}$ is as small as possible, where $b$ is defined in \eqref{def:ellrad} with $\tau=1$ and $\kappa=\kappa$. Then let $\mathcal P(1)$ be the collection of sieve parameters determined by $\kappa=\kappa$ and $n=1$. For any $i\in \{2,3,4\}$ we define $\mathcal P(i)$ as follows: We take $\kappa=\kappa$, $\tau=i$ and $n=10$, and we choose weights $w_1,\dotsc,w_\tau$ as in \eqref{def:weightswt} such that each $w_t$ is slightly larger than $\tfrac{1}{t}$ for $t\geq 2$. Further we use here an admissible covering $\{S_j\}$ of $S^*=S\cup\{\infty\}$ with $S_1=\{\infty\}$ and with the following properties: If $2\notin S$ then $\mathcal As{S_j}=n$ for each $j\in\{2,g-1\}$, and if $2\in S$ then $S_2=\{2\}$ and $\mathcal As{S_j}=n$ for any $j\in\{3,g-1\}$. We shall motivate our choice of sieve parameters in the discussions below.
\begin{remark}\label{rem:equalrad}
For each $i\in\{1,\dotsc,4\}$ the number $b_i$, associated to $\mathcal P(i)$ in \eqref{def:ellrad}, satisfies $b=b_i$. Indeed on using that $\tau=1$ in $\mathcal P(1)$ and that $\max_{v=2,\infty}\mathcal As{S_{j(v)}}=1$ in $\mathcal P(i)$ with $i\geq 2$, we see in all four cases that $t^*=1$ and hence we obtain that $b=b_i$ as desired.
\end{remark}
\paragraph{Global sieve.} For any given $M_1\in\ZZ_{\geq 1}$, we would like to efficiently determine the set of points $P\in \Sigma(S)$ with $\hat{h}_k(P)\leq M_1$. For this purpose we enumerate these points from below and from above, using the refined enumeration in Algorithm~\ref{algo:refenu} and the refined sieve in Algorithm~\ref{algo:refinedsieve} respectively. More precisely we proceed as follows:
\begin{itemize}
\item[(a)] Let $b$ be as in the above paragraph, and define parameters $\nu'=b$, $\nu=M_1$ and $f(\nu)=\min(\nu-1,\lfloor 0.99\nu\rfloor)$. Then apply the following sieves $(i=1,\dotsc,4)$:
\begin{itemize}
\item[0.] Apply Algorithm~\ref{algo:refenu} with $b=\nu'$ and $\kappa=1$, and let $\rho_0$ be the running time divided by the euclidean volume $\vol(\mathcal E_{\nu'})$. Put $\nu'= 4^{1/r}\nu'$.
\item[i.] If $\mathcal As{S^*}\geq i$ and $\nu>\nu'/4^{1/r}$ then apply Algorithm~\ref{algo:refinedsieve} with the parameters $\mathcal P=\mathcal P(i)$, $M'=f(\nu)$ and $M=\nu$, and let $\rho_i$ be the running time divided by the euclidean volume $\vol(\mathcal E_{M}\setminus\mathcal E_{M'})$. Put $\nu=f(\nu)$.
\end{itemize}
\item[(b)] As long as $\nu>\nu'/4^{1/r}$ continue with the most efficient sieve in (a), that is the sieve $i^*\in \{0,\dotsc,4\}$ for which the ``efficiency measure" $1/\rho_{i^*}$ is maximal.
\end{itemize}
This algorithm outputs the set of points $P\in \Sigma(S)$ with $\hat{h}_k(P)\leq M_1$. Indeed Remark~\ref{rem:equalrad} gives that $b=b_i$ for each $i\in\{1,\dotsc,4\}$, and therefore we see that the whole space $\{P\in \Sigma(S)\,;\, \hat{h}_k(P)\leq M_1\}$ is covered by an application of steps (a) and (b).
\paragraph{Decomposition.} We now motivate the decomposition of the above algorithm and we explain our choice of the parameters appearing therein. First we discuss the step size functions. In the enumeration from below, we double the volume of the ellipsoid in each step by working with the step size function given by multiplication with $4^{1/r}$. This assures that the repeated enumerations of candidates with tiny height are not that significant for the running time, see also the remark at the end of this paragraph. In the enumeration from above, we work with the step size function $f(\nu)=\min(\nu-1,\lfloor 0.99\nu\rfloor)$. If $\nu$ is large then the height lower bound $M'=f(\nu)$ is still large and thus the refined sieve is strong in view of the complexity discussions in Sections~\ref{sec:archisieve} and \ref{sec:nonarchisieve}. Hence, to accelerate the enumeration from above, we work with the relatively big step size $\nu-f(\nu)\geq 10^{-2}\nu$ for large $\nu$; here the factor $0.99$ turned out to be suitable in practice where usually $M_1\leq 10^9$. On the other hand, if $\nu\leq 100$ is small then we work with the tiny step size $\nu-f(\nu)=1$ to assure that $M'=\nu-1$ is relatively large making the sieves stronger.
We next motivate our choice of the sieve parameters $\mathcal P(i)$. If the parameter $\kappa$ becomes larger then the refined sieve becomes stronger at $v=\infty$. On the other hand, we can not choose $\kappa$ arbitrarily large since the ``square-radius" $b$ of the ellipsoid $\mathcal E_b$ satisfies $b\geq \log(\kappa)$. Now, choosing $\kappa$ such that $\mathcal As{b-10}$ is as small as possible, assures that the refined enumeration via $\Gamma_E\cap \mathcal E_b$ is efficient in practice where usually $r\leq 12$. To explain our choices for $n$ and $\tau$, we recall that the efficiency of the refined sieve depends inter alia on $\sigma_1=\sigma_1(n,\tau)$ and the number of additional iterations over subsets $T$ with $\mathcal As{T}\leq \tau$. Here it is not clear to us what are the optimal choices for $\tau$ and $n$. In practice it turned out that for $\tau\geq 5$ or $n\geq 11$ there are usually too many additional iterations and thus we only work with $\tau\leq 4$ and $n=10$. The reason for using admissible partitions $\{S_j\}$ of $S^*$ with $S_1=\{\infty\}$ and $S_2=\{2\}$ if $2\in S$, is to assure that $b=b_i$ (see Remark~\ref{rem:equalrad}) which means that the ellipsoids $\mathcal E_{b_i}$ are not larger than the minimal involved ellipsoid $\mathcal E_{b}$. In fact controlling the ellipsoids $\mathcal E_{b_i}$ is crucial for dealing efficiently with huge parameters.
Finally we mention that in the enumeration from below, the application of (FP) repetitively enumerates candidates. Here it is not clear to us how to avoid these repeated enumerations of candidates, since (FP) is not faster for circular discs than for the whole ellipsoid. In any case these repeated enumerations of candidates have a small influence on the running time in practice, since (FP) is very fast in our situations of interest where usually the rank $r$ is small. Furthermore, to assure that the height-logarithm sieve is applied at most once for each candidate, we order the candidates with respect to their height $\hat{h}_k$ in the implementation of the enumeration from below.
\paragraph{Main features.} We now discuss the main features of the global sieve. The first steps of the enumeration from above may be viewed as a reduction of $M_1$. Indeed these steps are usually very fast in practice and they often allow to considerably reduce $M_1$.
Further we point out that each step of the global sieve is more efficient than the standard enumeration. In fact in general it is not clear to us in which situation which sieve of (a) is the most efficient. To overcome this problem, we work with the quantities $1/\rho_i$ in order to ``measure" the efficiency of the sieves in the given situation. In practice this allows our algorithm to choose a suitable sieve in each step. This is very important for the running time, since the efficiency of the involved sieves strongly depends on the given situation. We also mention that Section~\ref{sec:comparisonwithelr} contains explicit examples which illustrate (up to some extent) the improvements in practice provided by our global sieve.
\subsection{Elliptic logarithm sieve}\label{sec:elllogsievealgo}
We work with the setup of Section~\ref{sec:setup} and we continue the notation introduced above. In the first part of this section we construct the elliptic logarithm sieve by putting together the sieves obtained in the previous sections. In the second part we compare the elliptic logarithm sieve with the known approach and we explain in detail our improvements.
We recall that in the setup of Section~\ref{sec:setup} we are given the following information: The coefficients $a_i\in\ZZ$ of a Weierstrass equation \eqref{eq:weieq} of an elliptic curve $E$ over $\QQ$, a finite set of rational primes $S$, a basis $P_1,\dotsc,P_r$ of the free part of $E(\QQ)$ and a number $M_0\in\ZZ$ such that any $P\in\Sigma(S)$ satisfies $\hat{h}(P)\leq M_0$. Given this information, the following algorithm completely determines the set $\Sigma(S)$ formed by the $S$-integral solutions of \eqref{eq:weieq}.
\begin{Algorithm}[Elliptic logarithm sieve]\label{algo:elllogsieve} The inputs are the coefficients $a_i$ of \eqref{eq:weieq}, the set $S$, the basis $P_1,\dotsc,P_r$ and the initial bound $M_0$. The output is the set $\Sigma(S)$.
\begin{itemize}
\item[(i)] Compute the following additional input data.
\begin{itemize}
\item[(a)] Determine the equation of an affine Weierstrass model $W$ of $E$ over $\ZZ$, which is minimal at all primes in $S$, together with an isomorphism $\varphi$ over $\mathcal O=\ZZ[1/N_S]$ from $W$ to the affine model defined by \eqref{eq:weieq}.
\item[(b)] Compute a suitable rational approximation $\hat{h}_k$ of $\hat{h}$.
\item[(c)] Determine the torsion subgroup $E(\QQ)_{\textnormal{tor}}$ of $E(\QQ)$, and compute the numbers $b$ and $\kappa$ appearing in the collections of sieve parameters from Section~\ref{sec:globalsieve}.
\end{itemize}
\item[(ii)] To reduce locally the initial bound $M_0$, apply the archimedean sieve and the non-archimedean sieve. More precisely, work with the set $\Sigma'(S)$ formed by the $\mathcal O$-points of $W$ and for each place $v$ in $S^*=S\cup \{\infty\}$ do the following:
\begin{itemize}
\item[(a)] Find an integer $M_1(v)\geq b$ with the following property: If $M'=M_1(v)$, $M=M_0$, $\kappa=\kappa$ and $\sigma=\mathcal As{S}+1$, then Algorithm~\ref{algo:archisieve}~(ii) outputs only $0$ when $v=\infty$ or $0$ is the output of Algorithm~\ref{algo:nonarchsieve}~(iii) when $v\in S$. Here first try $M_1(v)=b$. If this does not work then try a slightly larger number, and so on.
\item[(b)] Having found such an $M_1(v)$, try to reduce it further by repeating (a) with different parameters $M'$ and $M$. Let $M_1(v)$ be the final reduced bound at $v$.
\end{itemize}
\item[(iii)] Determine the set $\Sigma'(S)$ by applying the global sieve from Section~\ref{sec:globalsieve} with $M_1=\max_{v\in S^*}M_1(v)$. Then output the set $\varphi(\Sigma'(S))$.
\end{itemize}
\end{Algorithm}
\paragraph{Correctness.} To prove that this algorithm works correctly, we recall that $\varphi$ is an isomorphism of affine Weierstrass models of $E$ over $\mathcal O$. This shows that $\varphi(\Sigma'(S))=\Sigma(S)$ and hence it remains to verify that $\Sigma'(S)$ is completely determined. Lemma~\ref{lem:hk} gives that $\hat{h}_k\leq\hat{h}$, and any $P\in \Sigma'(S)$ satisfies $\hat{h}(P)\leq M_0$ since $\hat{h}$ is invariant under isomorphisms. Therefore on using that $b\leq M_1$, we see that the arguments of Proposition~\ref{prop:refinedcov}, with $n=1$ and $\kappa=\kappa$, prove that $\hat{h}_k(P)\leq M_1$ for all $P\in \Sigma'(S)$.
It follows that the application of the global sieve with $M_1$ produces the set $\Sigma'(S)$ in step (iii) as desired.
\paragraph{Complexity.} We now discuss various aspects which influence the running time in practice. The computation of the additional input data in step (i) is always very fast. More precisely, in (a) we use Tate's algorithm to transform \eqref{eq:weieq} into a globally minimal Weierstrass model of $E$ over $\ZZ$ which then can be used to directly determine a pair $(W,\varphi)$ with the desired properties. To construct the quadratic form $\hat{h}_k$ in (b) we proceed as described in Section~\ref{sec:heightselllog}; see also Section~\ref{sec:compuaspects} for computational aspects. Finally in (c) the numbers $b$ and $\kappa$ can be directly determined and the computation of $E(\QQ)_{\textnormal{tor}}$ is always efficient.
The running time of step (ii) crucially depends on the initial upper bound $M_0$, see the complexity discussions in Sections~\ref{sec:archisieve} and \ref{sec:nonarchisieve} for details. Here step (a) can take a long time for huge $M_0$, while the repetitions in step (b) are then quite fast since at this point we have already computed the involved elliptic logarithms. Step (ii) usually allows to avoid the process of testing whether candidates of huge height have $S$-integral coordinates. This process is so slow that it is beneficial in (b) to make as many repetitions as required to obtain a reduced global bound $M_1$ which is as small as possible.
The running time of the application of the global sieve in step (iii) crucially depends on the cardinality of $S$ and the rank $r$. For example, as already explained in previous sections, the rank $r$ has a huge influence on the running time of the refined enumeration and on the refined sieve. We also recall that the cardinality of $S$ significantly influences the efficiency of the height-logarithm sieve, which in turn is used in the refined sieve and the refined enumeration. See also the discussions in Sections~\ref{sec:globalsieve} and \ref{sec:comparisonwithelr}.
\paragraph{Bottleneck.} In practice the bottleneck of the elliptic logarithm sieve crucially depends on the situation, in particular on the Mordell--Weil rank $r$, the cardinality of $S$ and the size of the initial bound $M_0$. We first suppose that $M_0$ is huge, say $M_0$ is the initial bound coming from the theory of logarithmic forms (see Section~\ref{sec:input}). If $S$ is empty or when $r\leq 1$, then either the elliptic logarithm sieve is fast or the bottleneck is step (iii). Assume now that $r\geq 2$ and that $S$ is nonempty. If in addition $r\leq 4$ then the bottleneck is usually part (a) of step (ii), in particular when $\mathcal As{S}$ is large. On the other hand, if in addition $r\geq 5$ then the bottleneck is either step (iii) or part (a) of step (ii). If $M_0$ is not that large then the bottleneck is usually step (iii). For example, in the case when \eqref{eq:weieq} is a Mordell equation, the initial bounds of Proposition~\ref{prop:mwbound} are strong enough such that either the elliptic logarithm sieve is fast or the bottleneck is step (iii).
\begin{remark}[Generalizations]\label{rem:elllogsievegen}
(i) Algorithm~\ref{algo:elllogsieve} allows to solve Diophantine equations which are a priori more general than \eqref{eq:weieq}. For example, our algorithm can be applied to find all $S$-integral solutions with bounded height of any Weierstrass equation \eqref{eq:weieq} of $E$ with coefficients $a_i$ in $\QQ$. Indeed on multiplying equation \eqref{eq:weieq} with the sixth power $u^6$ of the least common multiple $u$ of the denominators of the $a_i$, one obtains a Weierstrass equation \eqref{eq:weieq}$^*$ with coefficients $a_i^*=u^ia_i$ in $\ZZ$ and then one checks for each $(x,y)\in \Sigma^*(S)$ whether $u^{-2}x$ and $u^{-3}y$ are in $\mathcal O$. Here $\Sigma^*(S)$ denotes the set of solutions of \eqref{eq:weieq}$^*$ in $\mathcal O\times\mathcal O$ obtained by applying Algorithm~\ref{algo:elllogsieve} with the coefficients $a_i^*$, with the same initial bound $M_0$ and with the transformed coordinates of the same basis $P_i$.
(ii) The above Algorithm~\ref{algo:elllogsieve} works equally well with any initial bound for the usual Weil height $h$ or for the infinity norm $\|\cdot\|_\infty$. Indeed the explicit inequalities \eqref{eq:nthcompa} and \eqref{eq:nthlowerbound} translate any initial bound for $h$ or $\|\cdot\|_\infty$ into an initial bound for $\hat{h}$.
(iii) We mention that various authors (including Stroeker, Tzanakis and de Weger) generalized and modified the known elliptic logarithm approach in order to efficiently solve more general Diophantine equations defining genus one curves. For an overview we refer to the discussions in Stroeker--Tzanakis~\cite{sttz:genus1} and Tzanakis~\cite{tzanakis:book}.
\end{remark}
\subsubsection{Comparison with the known approach}\label{sec:comparisonwithelr}
To discuss the improvements provided by the elliptic logarithm sieve, we now compare our sieve with the known approach via elliptic logarithms. We recall that the main steps of the known method are as follows (see for example \cite[Sect 4]{pezigehe:sintegralpoints} or \cite{tzanakis:book}):
\begin{itemize}
\item[(1)] As explained in Section~\ref{sec:elllintroelr}, one tries to obtain a reduced bound $N_1$ which is as small as possible such that any non-exceptional point $P\in \Sigma(S)$ satisfies $\|P\|_\infty\leq N_1$.
\item[(2)] One goes through all points $P\in E(\QQ)$ with $\|P\|_\infty\leq N_1$ and one tests whether $P$ lies in fact in $\Sigma(S)$. In the case $S^*=\{\infty\}$, one can apply in addition the known inequality trick (see Remark~\ref{rem:inequtrick}) before one tests whether $P$ lies in $\Sigma(S)$.
\item[(3)] One makes a so-called extra search to find all exceptional points.
\end{itemize}
\paragraph{Reduction.} Steps (i)+(ii) of our elliptic logarithm sieve may be viewed as an analogue of (1). Here the running times of (1) and (i)+(ii) are essentially equal, since the aspects in which the two approaches differ are irrelevant for the running time. Indeed in both approaches the running time is essentially determined by the computations of the involved elliptic logarithms and these computations are the same in both approaches. On the other hand, in view of the subsequent enumeration, an important difference is that (1) uses the inequality $\lambda\|P\|_\infty^2\leq \hat{h}(P)$ in order to work with $\|\cdot\|_\infty$, while our reduction in (ii) directly works with $\hat{h}$. Geometrically, this means that (1) uses a cube which always contains the ellipsoids $\mathcal E$ used in (ii). In fact there exist non-trivial improvements of (1), see Stroeker--Tzanakis~\cite{sttz:elllogoverview} which optimizes the Mordell--Weil basis and see Hajdu--Kov\'acs \cite{hako:elllog} which in the case $S=\emptyset$ intersects cubes containing $\mathcal E$. Our reduction in (ii) is always as good as these improvements of (1), since we work directly with the ellipsoid $\mathcal E$ which is optimal from a geometric point of view. We define $N_{\textnormal{opt}}=\lfloor(M_1/\lambda)^{1/2}\rfloor$ with $\lambda$ coming from a Mordell--Weil basis which is optimized in the sense of \cite{sttz:elllogoverview}. For instance, the usual elliptic logarithm reduction can not reduce anymore (\cite[p.400]{pezigehe:sintegralpoints}) the bound $N_1=17$ in the example of \cite{pezigehe:sintegralpoints} involving an optimized Mordell--Weil basis ($r=4$). On the other hand, our reduction in (ii) gives $M_1\leq 61$ which implies that $N_{\textnormal{opt}}=12$. Furthermore, in many important situations, the reduction in (ii) is considerably stronger than the known improvements of (1). For example if $r\geq 2$ becomes large then the volume of $\mathcal E$ becomes much smaller than the volume of any cube containing $\mathcal E$. Hence (ii) is significantly more efficient when $r\geq 2$ is large. In particular, in the generic case we obtain here a running time improvement by a factor which is exponential in terms of $r$. Furthermore, in the most common nontrivial case (where $r=1$), we obtain huge running time improvements for large $\mathcal As{S}$ by using the following trick: The idea is that we do not need to know the involved $v$-adic elliptic logarithms $(\beta_{1,v}^{(c)}, v\in S)$ appearing in Section~\ref{sec:nonarchisieve}. Indeed on exploiting that the involved lattice has rank $r=1$, one observes that it suffices here to know the orders $(v(\alpha_{1,v}),v\in S)$. These orders can always be efficiently computed in practice.
\paragraph{Enumeration.} Step (iii) of our sieve plays the role of (2)+(3). In the following comparison, we denote by $t$ and $t^*$ the running times of (iii) and (2)+(3) respectively.
Comparing $t$ with $t^*$ is suitable to illustrate our improvements. Indeed it takes into account that the running times of (i)+(ii) and (1) are essentially equal and it makes the comparison independent of the initial bound. We denote by $t_2$ the running time of our refined enumeration in Algorithm~\ref{algo:refenu}~(i) applied with our reduced initial bound $b=M_1$ from (ii). In practice the running time of (2) always exceeds $t_2$, which means that $t^*> t_2$. In fact, in many cases of interest (e.g. when $r$, $\mathcal As{S}$ or $\max \mathcal As{a_i}$ is not small), the time $t^*$ is often considerably larger than $t_2$. To explain the improvements provided by our sieve, we mention four situations in which the known enumeration (2)+(3) usually becomes very slow:
\begin{itemize}
\item[(S1)] Suppose that $r$ is not small and $S$ is nonempty. Then the enumeration in (2) becomes very slow or even hopeless, since one has to consider $(2N_1+1)^r$ points of $E(\QQ)$.
\item[(S2)] Assume that $r$ is large and $S$ is empty. Then the enumeration in (2) becomes often hopeless, since one has to go through $(2N_1+1)^r$ points of $E(\QQ)$.
\item[(S3)] Suppose that the height $\max \mathcal As{a_i}$ of \eqref{eq:weieq} is large. Then the extra search in (3) has to test many pairs $(x,y)$ of $S$-integers whether they satisfy \eqref{eq:weieq}. Thus (3) becomes very slow when $\max\mathcal As{a_i}$ is large, in particular if $S$ is nonempty.
\item[(S4)] Suppose that $\mathcal As{S}$ and $N_1$ are both large. Then (2) becomes very slow, since one has to compute many rational numbers $x(P)$ of huge height and one has to test whether they are $S$-integral. Here $x(P)$ is the $x$-coordinate of some $P\in E(\QQ)$.
\end{itemize}
To deal with situation (S1) we developed the global sieve which combines our refined sieve and our refined enumeration. We recall from the discussions in previous sections that the first steps of the global sieve are essentially a further reduction of $M_1$, while the subsequent steps of the global sieve are also more efficient than the corresponding enumerations in (2) and (3). To illustrate our improvements in practice, we consider the Mordell curve $y^2=x^3+1358556$ of rank $r=6$ and three additional examples which shall be further discussed in Section~\ref{sec:largerankexamples} below. These three additional examples are given by Kretschmer's curve in \cite[Ex.3.3]{siksek:infdescent} with $r=8$, Mestre's curve in \cite[Ex.3.2]{siksek:infdescent} with (conditional) $r=12$ and the curve of Fermigier with $r=14$. In the following table, the entries of the first and second row are rounded up and down respectively.
\begin{table}[h]
\begin{center}
{\small
\begin{tabular}{lccccccccccccccc}
$(r,S)$ & $(6,S(10))$ & $(6,S(20))$ & $(8,S(10))$ & $(12,S(5))$ & $(14,S(2))$\\
\cmidrule(r){2-6}
$t$ & 3.5m & 24m & 3.4h & 12h & 15h\\
$t_2$ & 115m & 1116m & ? & ? & ?
\\
$N_{\textnormal{opt}}$ & 15 & 21 & 36 & 36 & 16
\end{tabular}
}
\end{center}
\end{table}
\noindent We point out that here the running times of (2) would be significantly larger than the listed times $t_2$, since $N_1\geq N_{\textnormal{opt}}$ and since $(2N_{\textnormal{opt}}+1)^r$ is huge in each case. In particular the enumeration (2) would be very slow in the above cases involving $r=6$, while the cases involving $r=8,12,14$ seem to be completely out of reach for (2).
Furthermore our global sieve leads in addition to a significant improvement in situation (S2) where $r$ is large and $S$ is empty. To illustrate our improvements in practice, we consider again Mestre's curve of rank $r=12$, Fermigier's curve of rank $r=14$ and two curves of Elkies of rank $r=17,19$ respectively. These four curves shall be discussed in more detail in Section~\ref{sec:largerankexamples} below. In the case when $r=12,14,17,19$, it turned out that $t$ is less than 2 minutes, 13 minutes, 26 hours, 73 hours respectively and that $N_{\textnormal{opt}}= 14,10,22,15$ respectively. It seems that all these cases are out of reach for the enumeration (2).
To deal with situation (S3) where $\max \mathcal As{a_i}$ is large, we constructed the refined covering in Proposition~\ref{prop:refinedcov}. This result allows us to work entirely in the finitely generated abelian group $E(\QQ)$ in order to find the exceptional points. We illustrate our improvements in practice by considering a rather randomly chosen Mordell equation \eqref{eq:weieq} of rank $r=1$ with $a_6=-17817895$. This choice is suitable, since the assumption $r=1$ is satisfied in the nontrivial generic case and since our sieve does not exploit special properties of Mordell equations. Furthermore, if the rank $r$ is large then $\max \mathcal As{a_i}$ is large and the above table already contains times $t$ for $r\geq 6$.
To obtain the lower bounds for $t^*$ listed in the following table, we used the running times of the extra search (3); this search was implemented in Sage by Cremona, Mardaus and Nagel following the presentation in \cite[p.393]{pezigehe:sintegralpoints}.
\begin{table}[h]
\begin{center}
{\small
\begin{tabular}{lccccccccccccccc}
$S$ & $S(1)$ & $S(2)$ & $S(3)$ & $S(4)$ & $S(5)$ & $S(10)$ & $S(100)$\\
\cmidrule(r){2-8}
$t $ & 1s & 1.1s & 1.1s & 1.1s & 1.2s & 1.6s & 7s \\
$t^*$ & 1.3s & 34s & 15m & 12.5h & ? & ? & ?
\end{tabular}
}
\end{center}
\end{table}
\noindent Here the entries of the first and second row are upper and lower bounds for $t$ and $t^*$ respectively. We note that $t^*$ explodes when $\max\mathcal As{a_i}$ becomes larger. For example, let us consider the Mordell equation \eqref{eq:weieq} of rank $r=1$ with $a_6=- 4211349581402184375$. Here our running times $t$ essentially coincide with the times displayed in the above table, while the extra search (3) did not terminate within 48 hours in the simplest case $S=S(1)$. We mention that our improvements in the case of large $\max\mathcal As{a_i}$ are crucial for the computation (see Section~\ref{sec:shaf}) of elliptic curves over $\QQ$ with good reduction outside $S$. Indeed these computations usually require to find all $S'$-integral solutions of many distinct Mordell equations $y^2=x^3+a$, with $a\in\ZZ$ having huge $\mathcal As{a}\geq 10^{15}$ and $S'=S\cup \{2,3\}$.
We developed various global constructions to efficiently deal with situation (S4) where $\mathcal As{S}$ and $N_1$ are large. In particular, for many involved points $P\in E(\QQ)$ our height-logarithm sieve allows to avoid the slow process of testing whether the coordinates of $P$ are $S$-integral. To demonstrate our improvements in practice, we considered the curves 37a1, 389a1, 5077a1 in Cremona's database with rank $r=1,2,3$ respectively. They have minimal conductor among all elliptic curves over $\QQ$ of rank $r=1,2,3$ respectively.
\begin{table}[h]
\begin{center}
{\small
\begin{tabular}{lccccccccccccccc}
$(r,S)$ & $(1,S(10^5))$ & $(1,S(2\cdot 10^5))$ & $(2,S(10^4))$ & $(3,S(500))$ \\
\cmidrule(r){2-5}
$t$ & 3.4h & 9.4h & 18d & 7.6h \\
$t_2$ & 150h? & 900h? & 58d & 85h \\
$N_{\textnormal{opt}}$ & 11263 & 16015 & 1467 & 246
\end{tabular}
}
\end{center}
\end{table}
\noindent Here the entries of the first row are rounded up. Further, it is reasonable to expect that the first two entries in the $t_2$ row are larger than 150 hours and 900 hours respectively. Indeed, it took 192 seconds (resp. 777 seconds) to determine whether the coordinates of the point $11000P$ (resp. $16015P$) are $S(10^5)$-integral (resp. $S(2\cdot 10^5)$-integral), where $P\ZZ=E(\QQ)$ and $E$ denotes the rank one curve 37a1 used in the above table. We mention that the ability of Algorithm~\ref{algo:elllogsieve} to solve \eqref{eq:weieq} for large sets $S$ was crucial for obtaining data motivating our conjectures in Section~\ref{sec:malgoapplications} and in Section~\ref{sec:elllogsieveapp} below.
\subsection{Input data}\label{sec:input}
We continue our notation. The elliptic logarithm sieve requires an initial height bound for the points in $\Sigma(S)$ and a Mordell--Weil basis of $E(\QQ)$. In this section we recall some results and techniques which allow to compute the required input data in practice.
\subsubsection{Mordell--Weil basis}\label{sec:compmwbasis}
The problem of finding a Mordell--Weil basis of $E(\QQ)$ is difficult in theory and in practice. In fact in the case of an arbitrary elliptic curve $E$ over $\QQ$ there is so far no unconditional method which allows in principle to determine a Mordell--Weil basis of $E(\QQ)$. However, thanks to the work of many authors, it is usually possible to compute such a basis in practice. In fact it turned out in practice that the methods implemented in Pari, Sage and Magma are remarkably efficient in computing such a basis, even in the case when the height $\max\mathcal As{a_i}$ of \eqref{eq:weieq} is large. Furthermore, Cremona's database contains a Mordell--Weil basis of $E(\QQ)$ for each elliptic curve $E$ over $\QQ$ with conductor at most $350 000$. Unless mentioned otherwise, we shall use below these bases of Cremona's database.
\subsubsection{Initial height bounds}
Starting with the works of Baker~\cite{baker:contributions,baker:mordellequation,baker:elliptic}, there is a long tradition of establishing explicit height bounds for the points in $\Sigma(S)$ using lower bounds for linear forms in logarithms. See for example Baker--W\"ustholz~\cite{bawu:logarithmicforms} for an overview and a discussion of the state of the art. Furthermore Masser~\cite{masser:ellfunctions} and W\"ustholz~\cite{wustholz:recentprogress} initiated an approach which provides explicit height bounds for the points in $\Sigma(S)$ using lower bounds for linear forms in elliptic logarithms. Here the actual best lower bounds can be found in the works of David~\cite{david:elllogmemoir} and Hirata-Kohno~\cite{hirata-kohno:p-adicelllogs}.
To obtain our results discussed in the next section, we compute two initial height bounds for the points in $\Sigma(S)$ and then we take the minimum of these two bounds. More precisely, the first initial height bound is a direct consequence of the results of Hajdu--Herendi~\cite{hahe:elliptic} combined with a height comparison in the style of \eqref{eq:nthcompa}, see Peth{\H{o}}--Zimmer--Gebel--Herrmann~\cite[Thm]{pezigehe:sintegralpoints}. Here the results of Hajdu--Herendi ultimately rely on lower bounds for linear forms in complex and $p$-adic logarithms. The second initial height bound depends on the above mentioned lower bounds for linear forms in elliptic logarithms due to David in the archimedean case and due to\footnote{In fact explicit lower bounds for linear forms in two nonarchimedean elliptic logarithms were established in the work of R\'emond--Urfels~\cite{reur:padicelllog}.} Hirata-Kohno in the nonarchimedean case. See for example the proof of Tzanakis~\cite[Thm 11.2.6]{tzanakis:book}; here one has to take into account that in some cases the normalizations in \cite{tzanakis:book} do not coincide with our corresponding normalizations used in Sections~\ref{sec:archisieve} and \ref{sec:nonarchisieve}. In fact, unless mentioned otherwise, we obtained all applications in Section~\ref{sec:elllogsieveapp} by using the first height bound.
\subsection{Applications}\label{sec:elllogsieveapp}
In this section we discuss additional applications of the elliptic logarithm sieve. To obtain the input data required for the applications of our sieve, we used unless mentioned otherwise the results described in the previous section. We continue the notation introduced above and for any $n\in\ZZ_{\geq 1}$ we denote by $S(n)$ the set of the first $n$ rational primes.
\subsubsection{Elliptic curves database}
For any elliptic curve $E$ over $\QQ$ of conductor at most 1000, Cremona's database contains in particular a minimal Weierstrass equation \eqref{eq:weieq} of $E$. We used Algorithm~\ref{algo:elllogsieve} to compute the $S$-integral solutions of each of these equations with $S=S(20)$. Moreover, for any of these minimal equations defining an elliptic curve over $\QQ$ of conductor at most 100, we used Algorithm~\ref{algo:elllogsieve} to determine its set of $S$-integral solutions with $S=S(10^4)$.
\subsubsection{Elliptic curves of large rank}\label{sec:largerankexamples}
In addition, we used the elliptic logarithm sieve (Algorithm~\ref{algo:elllogsieve}) in order to determine the set of $S$-integral solutions of various Weierstrass equations \eqref{eq:weieq} for which the involved Mordell--Weil rank $r$ of $E(\QQ)$ is relatively large. We now discuss some examples.
\paragraph{Mordell curves.} Recall that \eqref{eq:weieq} is called a Mordell equation if the coefficients $a_1,\dotsc,a_5$ are all zero. In Section~\ref{sec:malgoapplications} we used the elliptic logarithm sieve to find all $S$-integral solutions of two (resp. four) Mordell equations with $S=S(50)$ and $r=7$ (resp. $S=S(40)$ and $r=8$). Instead of using initial bounds coming from the theory of logarithmic forms, we applied here our optimized height bound in Proposition~\ref{prop:mwbound} which is based on the method of Faltings (Arakelov, Par\v{s}in, Szpiro) \cite{faltings:finiteness} combined with the Shimura--Taniyama conjecture~\cite{wiles:modular,taywil:modular,breuil:modular}. Further, we determined here the required Mordell--Weil bases by using techniques implemented in Pari, Sage and Magma.
\paragraph{Rank eight.} Let $E_{\textnormal{Kr}}$ be the elliptic curve over $\QQ$ considered in \cite[Example 5.3]{siksek:infdescent}, with $E_{\textnormal{Kr}}(\QQ)$ of rank $r=8$ by Kretschmer \cite{kretschmer:largerank}. Siksek~\cite{siksek:infdescent} combined his refined descent techniques with Cremona's ``mwrank" to find a Mordell--Weil basis of $E_{\textnormal{Kr}}(\QQ)$. On using Siksek's basis as an input for our sieve, we determined all $S$-integral solutions of the minimal Weierstrass equation \eqref{eq:weieq} of $E_{\textnormal{Kr}}$ with $S=S(10)$. This took our sieve less than 35 seconds, 17 hours and 75 hours for $S=\emptyset$, $S=S(8)$ and $S=S(10)$ respectively. Here we notice that our running times for $E_{\textnormal{Kr}}$ are significantly worse than for the four Mordell curves of rank $r=8$ discussed in Section~\ref{sec:malgoapplications}. The reason is that in the case of $E_{\textnormal{Kr}}$ we need to use initial height bounds based on the theory of logarithm forms. These initial bounds are substantially weaker (see Section~\ref{sec:minitbounds}) than our optimized height bound in Proposition~\ref{prop:mwbound} which is currently only available for Mordell curves.
\paragraph{Rank twelve.} Mestre~\cite{mestre:rank12} constructed an elliptic curve $E_{\textnormal{Me}}$ over $\QQ$ of analytic rank 12, together with 12 independent points in $E_{\textnormal{Me}}(\QQ)$ of infinite order. Here again Siksek~\cite[Example 5.2]{siksek:infdescent} applied his refined descent techniques to find a Mordell--Weil basis of $E_{\textnormal{Me}}(\QQ)$. On using Siksek's basis as an input for our sieve, we determined the set of $S$-integral solutions of a minimal Weierstrass equation \eqref{eq:weieq} of $E_{\textnormal{Me}}$ with $S=S(7)$. This took our sieve less than 2 minutes, 4 days and 16 days for $S=\emptyset$, $S=S(6)$ and $S=S(7)$ respectively. We point out that the completeness of our solution sets are here conditional on Siksek's assumption that $E_{\textnormal{Me}}(\QQ)$ has rank $r=12$ which he used in his construction of a basis of $E_{\textnormal{Me}}(\QQ)$. For example, this assumption is satisfied if $r$ is at most the analytic rank of $E_{\textnormal{Me}}$ as predicted by the rank part of the Birch--Swinnerton-Dyer conjecture.
\paragraph{Rank at most 28.} At the time of writing, we are not aware of an elliptic curve over $\QQ$ of rank $r\geq 13$ for which an explicit Mordell--Weil basis can be computed explicitly. If $r\leq 28$ and $S$ is empty then the elliptic logarithm sieve would allow to determine $\Sigma(S)$ for such large rank elliptic curves $E$ over $\QQ$, provided that one knows an explicit Mordell--Weil basis of $E(\QQ)$. To demonstrate this feature, we work with independent points $Q_1,\dotsc,Q_r$ generating a rank $r$ subgroup $\Lambda$ of the free part of $E(\QQ)$. Dujella lists in particular such points (see \url{web.math.pmf.unizg.hr/~duje}) for three elliptic curves over $\QQ$ of rank $r=14,17,19$ respectively. Here the curve of rank $r=14$ was constructed by Fermigier, while the other two curves were found by Elkies. We denote by $\Sigma_\Lambda(S)$ the intersection of $\Sigma(S)$ with $\Lambda\oplus E(\QQ)_{\textnormal{tor}}$. On using the basis $Q_1,\dotsc,Q_r$ of $\Lambda$ in the input of Algorithm~\ref{algo:elllogsieve}, we determined the set $\Sigma_\Lambda(S)$. If $S$ is empty then this took less than 13 minutes, 26 hours, 73 hours for $r=14,17,19$ respectively, and it took less than 18 hours when $r=14$ and $S=S(2)$. Now, given a Mordell--Weil basis, one can expect similar running times of Algorithm~\ref{algo:elllogsieve} when computing the full set $\Sigma(S)\supseteq \Sigma_\Lambda(S)$. Indeed the index of $\Lambda$ in the free part of $E(\QQ)$ is not that large for these three curves.
Finally, we consider Elkies' elliptic curve $E_{\textnormal{El}}$ over $\QQ$ of rank $r\geq 28$. In the case when $S$ is empty, we applied Algorithm~\ref{algo:heightlogsieve} with the 28 independent points $Q_i$ constructed by Elkies and we computed the
intersection of $\Sigma(S)$ with $(\oplus_{i}Q_i\ZZ)\oplus E_{\textnormal{El}}(\QQ)_{\textnormal{tor}}$ in less than 75 days.
\subsubsection{Conjectures and questions}
In this section we generalize our conjectures and questions for Mordell curves (see Section~\ref{sec:malgoapplications}) to hyperbolic genus one curves. We then provide some motivation by using our data and by generalizing our constructions for Mordell curves. See also Section~\ref{sec:ia} which contains the initial motivation for our conjectures and questions.
We continue our notation. As in Section~\ref{sec:elllintroapp}, we let $Y=(X,D)$ be a hyperbolic genus one curve over some open subscheme $B$ of $\textnormal{Spec}(\ZZ)$ and we denote by $r$ the rank of the group formed by the $\QQ$-points of $\textnormal{Pic}^0(X_\QQ)$. Now we recall our conjecture.
\noindent{\bf Conjecture.}
\emph{There are constants $c_{Y}$ and $c_r$, depending only on $Y$ and $r$ respectively, such that any nonempty finite set of rational primes $S$ with $T=\textnormal{Spec}(\ZZ)-S$ satisfies} $$\mathcal As{Y(T)}\leq c_{Y} \mathcal As{S}^{c_r}.$$
\noindent If $D$ is given by a section of $X\to B$, then $Y$ identifies with a closed subscheme of $\mathbb A^2_B$ defined by a Weierstrass equation~\eqref{eq:weieq}.
Hence the family $S[b]$ constructed in Section~\ref{sec:malgoapplications} shows that the exponent $c_r$ has to be at least $\tfrac{r}{r+2}$ when $Y$ is a Weierstrass curve.
\noindent{\bf Question 1.}
\emph{What is the optimal exponent $c_r$ in the above conjecture?}
\noindent Our data strongly indicates that the exponent $c_r=\tfrac{r}{r+2}$ is still far from optimal for many families of sets $S$ of interest, including the family $S(n)$ with $n\in\ZZ_{\geq 1}$. Further, our data motivates in addition the following question on the dependence on $q=\max S$.
\noindent{\bf Question 2.}
\emph{Are there constants $c_{Y}$ and $c_r$, depending only on $Y$ and $r$ respectively, such that any nonempty finite set of rational primes $S$ with $T=\textnormal{Spec}(\ZZ)-S$ satisfies} $$\mathcal As{Y(T)}\leq c_{Y}(\log \max S)^{c_r} \, \textnormal{?}$$
\noindent In the case when $S=S(n)$ with $n\in\ZZ_{\geq 2}$, one can replace here $q$ by $n\log n$ without changing the content of the question. However Question~2 has in general a negative answer when $q$ is replaced by any power of $\max(2,\mathcal As{S})$. On using again the arguments of Section~\ref{sec:malgoapplications}, we see that the exponent $c_r$ of Question~2 has to be at least $r/2$ if $Y$ is a Weierstrass curve. In light of this we ask whether Question~2 has a positive answer with the exponent
\begin{equation*}
c_r=r/2 \, \textnormal{?}
\end{equation*}
Our probabilistic model constructed in the discussion surrounding \eqref{refquestbound} predicts that this question has a positive answer when $Y$ is a Weierstrass curve. To conclude we mention that additional motivation for our conjecture is given by the theory of logarithmic forms \cite{bawu:logarithmicforms}, which was applied in \cite{rvk:ed} to obtain new bounds for the number of integral points on arbitrary hyperbolic genus one curves over any number field. These bounds establish in particular our Conjecture for certain sets $S$ of interest, including the sets $S(n)$.
\subsection{Computational aspects}\label{sec:compuaspects}
In this final section we discuss various computational aspects of our algorithms. In particular we explain the numerical details of constructions used in previous sections.
\paragraph{Interval arithmetic.} Most real numbers $x$ cannot be explicitly presented by a computer. Hence we apply standard interval arithmetic which uses lower and upper bounds for $x$. This allows us to control numerical errors. In particular, one can detect when the error explodes and in this case one can restart the computation with higher precision.
\paragraph{Rounding real numbers.} We consider a real number $x\in\RR$. In this paper the symbol $[x]$ denotes an element of $\ZZ$ with $\mathcal As{x-[x]}<1$. There might be two distinct choices for $[x]\in \ZZ$ with the desired property. However, for our purpose any choice will be sufficient. In the implementation, real numbers $x$ are only stored up to some chosen precision. Suppose that $x$ is stored with absolute precision $m\in\ZZ_{\geq 1}$, that is the computer stores $x$ as $x'\in\QQ$ with $\mathcal As{x-x'} \leq \tfrac{1}{2^{m}}$. Then we can compute $[x]$ as a closest integer to $x'$; there may be two choices, in which case we pick one. This $[x]\in\ZZ$ moreover satisfies $\mathcal As{x-[x]}\leq \tfrac{1}{2}+\tfrac{1}{2^m}$.
\paragraph{Construction and eigenvalues of $\hat{h}_k$.} To discuss the numerical details required for the construction of $\hat{h}_k$, we continue the notation and terminology of Section~\ref{sec:heightselllog}. We recall that we need to find a suitable $k\in\ZZ_{\geq 1}$ for which $\hat{h}_k$ is close enough to $\hat{h}$. To find such a $k$, we start with some $k$ such as for example $k=10$. We first determine a reasonably good rational approximation $f\in\QQ$ of $2^k/\|\hat{h}_{ij}\|$. After computing a lower bound for the smallest eigenvalue $\lambda_k$ of $\hat{h}_k$, we can check whether $\lambda_k>0$ and $r<\tfrac{1}{100}f\lambda_k$. If both conditions are satisfied, then $k$ is suitable. Otherwise we increase $k$ and we repeat the above procedure as long as required to find a $k$ with the desired properties. Lemma~\ref{lem:hk} implies that this procedure terminates; in fact it terminates always very quickly in practice. While the condition $r<\tfrac{1}{100}f\lambda_k$ is in principle not required for the correctness of our algorithms, it assures via Lemma~\ref{lem:hk} that $\hat{h}_k$ is relatively close to $\hat{h}$, which is reasonable in practice.
To explain how we compute a reasonably good lower bound for $\lambda_k$, we recall that the quadratic form $\hat{h}_k$ is given by $A/f$ with $A=([f\hat{h}]-r\cdot \textnormal{id})\in\ZZ^{r\times r}$. First, we apply Newton's method to compute a root of the characteristic polynomial of $A$. Here we start at minus the Cauchy bound for the largest absolute value of its roots.
At each step of Newton's method we stay below all eigenvalues. Indeed
the characteristic polynomial is a convex or concave function on the interval $(-\infty,f\lambda_k)$, since all $r$ eigenvalues of $A$ are real as $A$ is symmetric.
In order to remove numerical issues, we use interval arithmetic. After each step of Newton's method, we moreover replace the point by its lower bound. Once this value is smaller than or equal to the one in the previous step, we terminate. An upper bound for the largest eigenvalue of $A$ can be obtained in an analogous way.
\paragraph{Choice of $\delta$.} To explain our choice of the parameter $\delta$ in \eqref{def:delta12}, we continue the notation and terminology of Section~\ref{sec:archisieve}. We choose $a,b\in\ZZ$ such that $\tfrac{a}{b}$ is a good lower approximation of $M/(\delta_1+\delta_2)^2$, such that $\tfrac{a}{b}=M/\delta^2$ for some positive $\delta\in \QQ$ and such that $\mathcal As{a},\mathcal As{b}$ are not too large. These are two simultaneous objective functions that we try to optimize: 1. Making the inequality $\tfrac{a}{b} \leq M/(\delta_1+\delta_2)^2$ as sharp as possible assures that the Fincke--Pohst algorithm does not return too many additional candidates. 2. The smaller $|a|$ and $|b|$, the smaller are the entries of the quadratic form on which we run the Fincke--Pohst algorithm. Further, this construction of $\delta$ (that is of the integers $a,b$) allows us to apply our implementation of the Fincke--Pohst lattice point enumeration which works with integral quadratic forms. Indeed on multiplying by the (controlled) denominator $bf$ of the quadratic form $q$ used in the definition of $\mathcal E$, one obtains an integral quadratic form.
\paragraph{Lattice point enumeration.}
To enumerate lattice points in an ellipsoid, we apply the version (FP) of Fincke--Pohst~\cite{fipo:algo} which in turn uses $L^3$ \cite{lelelo:lll}. More precisely, for any $d\in \ZZ_{\geq 1}$ the algorithm (FP) takes as the input a basis of a lattice $\Gamma\subseteq\ZZ^d$ together with an ellipsoid $\mathcal E\subset\RR^d$ centered at the origin, and
it outputs the inter\-section~$\Gamma\cap \mathcal E$. In fact we use here our own implementation of (FP) which is described in Remark~\ref{rem:fp}.
\paragraph{Elliptic logarithms.}
Information on the computation of the elliptic logarithms can be found in Zagier~\cite{zagier:largeintegralpoints} for real elliptic logarithms and in Peth{\H{o}} et al~\cite{pezigehe:sintegralpoints} for $p$-adic elliptic logarithms, see also Tzanakis~\cite{tzanakis:book}.
To compute the elliptic logarithms we use Sage which in turn is based on Pari. Our normalizations of the real and $p$-adic elliptic logarithm in Sections~\ref{sec:archisieve} and \ref{sec:nonarchisieve} respectively coincide with the normalizations of Pari.
{\scriptsize
}
\noindent Rafael von K\"anel, MPIM Bonn, Vivatsgasse 7, 53111 Bonn, Germany
\noindent Current affiliation: Princeton University, Mathematics, Fine Hall, NJ 08544-1000, USA
\noindent E-mail address: {\sf [email protected]}
\noindent Benjamin Matschke, MPIM Bonn, Vivatsgasse 7, 53111 Bonn, Germany
\noindent Current affiliation: Institut de Math\'ematiques de Bordeaux, Universit\'e de Bordeaux,
351, cours de la Lib\'eration,
33405 Talence, France
\noindent E-mail address: {\sf [email protected]}
\end{document} | math | 486,245 |
\begin{document}
\title{\bf\normalsize\MakeUppercase{Classification of Cohen--Macaulay} $t$--\MakeUppercase{spread lexsegment ideals via simplicial complexes}}
\author{Marilena Crupi, Antonino Ficarra}
\newcommand{\Addresses}{{
\footnotesize
\textsc{Department of Mathematics and Computer Sciences, Physics and Earth Sciences, University of Messina, Viale Ferdinando Stagno d'Alcontres 31, 98166 Messina, Italy}
\begin{center}
\textit{E-mail addresses}: \texttt{[email protected]}; \texttt{[email protected]}
\end{center}
}}
\date{}
\maketitle
\Addresses
\begin{abstract}
We study the minimal primary decomposition of completely $t$--spread lexsegment ideals via simplicial complexes.
We determine some algebraic invariants of such a class of $t$--spread ideals. Hence, we classify all $t$--spread lexsegment ideals which are Cohen--Macaulay.
\blfootnote{
\hspace{-0,3cm} \emph{Keywords:} Betti numbers, $t$--spread ideals, primary decomposition, simplicial complexes, Cohen--Macaulay rings.\\
\emph{2020 Mathematics Subject Classification:} 05E40, 13B25, 13D02, 16W50.
}
\end{abstract}
\section*{Introduction}
In combinatorial commutative algebra, one often associates to a combinatorial object $X$ a suitable monomial ideal $I_X$ in a polynomial ring $S=K[x_1,\dots,x_n]$ in finitely many variables with coefficients in a field $K$. Distinguished combinatorial properties of the object $X$ are reflected by algebraic properties of the ideal $I_X$, and conversely. One says that $X$ is Cohen--Macaulay if the ring $S/I_X$ is Cohen--Macaulay. A typical task is to classify all Cohen--Macaulay objects $X$ of a given class. For instance, in combinatorics, to a simplicial complex $\Delta$ on the vertex set $[n]$ one associates the Stanley--Reisner ideal $I_\Delta$, \emph{i.e.}, the ideal generated by all squarefree monomials $x_{i_1}\cdots x_{i_r}$, $r\le n$, such that $\{i_1, \ldots, i_r\}\notin \Delta$.
A simplicial complex $\Delta$ is said Cohen--Macaulay if the ideal $I_\Delta$ is Cohen--Macaulay, \emph{i.e.}, $S/I_\Delta$ is a Cohen--Macaulay ring.
To classify all Cohen--Macaulay simplicial complexes is an hopeless task. However, many results can be found, for instance, in \cite{JT}.
The aim of this article is to classify all Cohen--Macaulay $t$--spread lexsegment ideals for $t\ge1$.
Let $S=K[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables with coefficients over a fixed field $K$.
Let $t\ge 0$ be an integer. A monomial $u=x_{i_1}x_{i_2}\cdots x_{i_d}$ is \textit{$t$--spread} if $i_{j+1}-i_j\ge t$, for all $j=1,\dots,d-1$.
If $t\ge1$, a $t$--spread monomial is a squarefree monomial.
A monomial ideal is $t$--spread if it is generated by $t$--spread monomials. Such a topic introduced in \cite{EHQ} has immediately captured the attention of many algebraists (see, for instance, \cite{LA, AFC1, AFC2, ACF3, AEL, CAC, RD, FC1}).
Let $t\ge1$. A $t$--spread ideal $I$ of $S$ is said a \emph{$t$--spread lexsegment ideal} if for all $t$--spread monomials $u\in I$ and all $t$--spread monomials $v\in S$ with $\deg(u)=\deg(v)$ and such that $v\ge_{\slex}u$, then $v\in I$ \cite{CAC}, where $\ge_{\slex}$ is the squarefree lexicographic order \cite{AHH2, JT}. Therefore, for $t=1$, such a definition coincides with the \emph{classical} definition of squarefree lexsegment ideal \cite{AHH2}.
Let $t\ge 1$ and let $M_{n,d,t}$ be the set of all $t$--spread monomials of $S$ of degree $d$. Assume that $M_{n,d,t}$ is endowed with the squarefree lexicographic order $\ge_{\slex}$. Let $u,v\in M_{n,d,t}$. If $u\ge_{\slex}v$, the set $\mathcal{L}_t(u,v)=\{w\in M_{n,d,t}:u\ge_{\slex}w\ge_{\slex}v\}$ is called an \emph{arbitrary $t$--spread lexsegment}.
If $u$ is the maximum monomial of $M_{n,d,t}$, or $v$ is the minimum monomial of $M_{n,d,t}$, we say that $\mathcal{L}_t(u,v)$ is an \emph{initial} or a \emph{final $t$--spread lexsegment}, respectively.
The ideal generated by an arbitrary $t$--spread lexsegment is called \emph{arbitrary $t$--spread lexsegment ideal}. It is clear that the notion of initial $t$--spread lexsegment ideal coincides with the notion of $t$--spread lexsegment ideal in \cite{CAC}.
An arbitrary $t$--spread lexsegment ideal $I = (\mathcal{L}_t(u,v))$ is said \emph{completely $t$--spread lexsegment} if $I=(\mathcal{L}_t(\max(M_{n,d,t}),v))\cap(\mathcal{L}_t(u,\min(M_{n,d,t})))$, \cite{FC1}.
One can easily observe that every initial and every final $t$--spread lexsegment ideal is a completely $t$--spread lexsegment, but not all arbitrary $t$--spread lexsegment ideals are completely $t$--spread lexsegment ideals. In \cite{FC1}, the authors characterized all completely $t$--spread lexsegment ideals and classified all completely $t$--spread lexsegment ideals with a linear resolution. The case $t=0$ has been faced by De Negri and Herzog in \cite{DH}.
In this article we determine the minimal primary decomposition of completely $t$--spread lexsegment ideals and some algebraic invariants of such a class of ideals by tools from the simplicial complex theory.
Hence, we classify all $t$--spread lexsegment ideals which are Cohen--Macaulay. The case $t=0$ has been considered in \cite{EOS2010} and the case $t=1$ has been analyzed in \cite{BST}.
The article is organized as follows. Section \ref{sec1} contains preliminary notions and results.
In Section \ref{sec2}, we explicitly compute the minimal primary decomposition for
initial and final $t$--spread lexsegment ideals (Theorems \ref{primdecompinitialtspreadlex}, \ref{primdecompfinaltspreadlex}). As a consequence we are able to compute the standard primary decomposition for the class of completely $t$--spread lexsegments (Theorem \ref{primdecompgeneraltspreadlex}). Moreover, we obtain formulas for
the Krull dimension, the projective dimension and the depth of $S/I$ when $I$ is an initial or a final $t$--spread lexsegment ideal.
In Section \ref{sec3}, we classify all $t$--spread lexsegment ideals which are Cohen--Macaulay by the results stated in Section \ref{sec2}. The notion of Betti splitting will be a crucial tool. Finally, Section \ref{sec4} contains our conclusions and perspectives.
All the examples are constructed by means of \emph{Macaulay2} packages one of which developed in \cite{LA}. Furthermore, some functions described in \cite{LA} have allowed us to test classes of Cohen--Macaulay $t$--spread lexsegment ideals whose determination has been fundamental for the development of Section \ref{sec3}.
\section{Preliminaries}\label{sec1}
Let $S=K[x_1,\dots,x_n]$ be the polynomial ring in $n$ indeterminates with coefficients in a field $K$. $S$ is a ${\NZQ N}$--graded $K$--algebra with $\deg(x_i)=1$ for all $i$.
If $I$ is a monomial ideal of $S$, we denote by $G(I)$ the unique minimal set of monomial generators of $I$. Moreover, we set $G(I)_j=\{u\in G(I):\deg(u)=j\}$.
Given a non empty subset $A\subseteq[n]$, we set ${\bf x}_A=\mathfrak{p}rod_{i\in A}x_i$.
Moreover, we set ${\bf x}_{\emptyset}=1$. Every monomial $w\in S$ can be written as $w=x_{i_1}x_{i_2}\cdots x_{i_d}$, with sorted indexes $1\le i_1\le i_2\le\dots\le i_d\le n$. We define the \textit{support} of $w$ as follows:
$$
\supp(w)=\{j: x_j \,\mbox{divides $w$} \}= \{i_1,i_2,\dots,i_d\}.
$$
With the notation above, $\min(w)=\min\big\{i:i\in\supp(u)\big\}=i_1$, $\max(w)=\max\big\{i:i\in\supp(u)\big\}= i_d$. Moreover, we set $\max(1)=\min(1)=0$.
If $i_1\ne i_2\ne\dots\ne i_d$ we say that $u$ is \textit{squarefree}. In such a case, $u$ can be written as $u={\bf x}_A$ with $A=\supp(u)$.\\
We quote next definition from \cite{EHQ}.
\begin{Def}
\rm Given $n\ge1,t\ge0$ and $u=x_{i_1}x_{i_2}\cdots x_{i_d}\in S$,
with $1\le i_1\le i_2\le\ldots\le i_d\le n$, we say that $u$ is \textit{$t$--spread} if $i_{j+1}-i_j\ge t$, for all $j=1,\dots,d-1$. We say that a monomial ideal $I$ of $S$ is \textit{$t$--spread} if it is generated by $t$--spread monomials.
\end{Def}
We note that any monomial ideal of $S$ is a $0$--spread monomial ideal, and any squarefree monomial ideal of $S$ is a $1$--spread monomial ideal.\\
Let $n,d,t\ge 1$. We denote by $M_{n,d,t}$ the set of all $t$--spread monomials of $S$ of degree $d$. Throughout this paper we tacitly assume that $n\ge 1+(d-1)t$, otherwise $M_{n,d,t}=\emptyset$. From \cite{EHQ}, we have that $|M_{n,d,t}|=\binom{n-(t-1)(d-1)}{d}$. The monomial ideal generated by $M_{n,d,t}$ is denoted by $I_{n,d,t}$ and is called the \textit{$t$--spread Veronese ideal of degree $d$} of the polynomial ring $S=K[x_1,\dots,x_n]$ \cite{EHQ}.\\
From now on we consider $t\ge 1$ and assume that $M_{n,d,t}$ is endowed with the \emph{squarefree lexicographic order}, $\ge_{\slex}$, \cite{JT}. Recall that given $u=x_{i_1}x_{i_2}\cdots x_{i_d}, v=x_{j_1}x_{j_2}\cdots x_{j_d} \in M_{n,d,t}$, with $1\le i_1<i_2<\dots< i_d\le n$, $1\le j_1<j_2<\dots<j_d\le n$, then $u>_{\slex}v$ if
$$i_1=j_1,\ \dots,\ i_{s-1}=j_{s-1}\ \ \text{and}\ \ i_s<j_s$$
for some $1\le s\le d$.
Let $U$ be a not empty subset of $M_{n,d,t}$. We denote by $\max(U)$ ($\min(U)$, respectively) the maximum (minimum, respectively) monomial $w\in U$, with respect to $\ge_{\slex}$. One has
\begin{align*}
\max(M_{n,d,t})& = x_1x_{1+t}x_{1+2t}\cdots x_{1+(d-1)t}, \\
\min(M_{n,d,t})& = x_{n-(d-1)t}x_{n-(d-2)t}\cdots x_{n-t}x_n.
\end{align*}
In \cite{FC1}, the following definitions have been introduced.
\begin{Def}\rm
Let $u,v\in M_{n,d,t}$, $u\ge_{\slex}v$.
The set
\[\mathcal{L}_t(u,v) = \big\{w\in M_{n,d,t}:u\ge_{\slex}w\ge_{\slex}v\big\}\]
is called an \emph{arbitrary $t$--spread lexsegment set}, or simply a \emph{$t$--spread lexsegment set}. The set
\[\mathcal{L}_t^{i}(v) = \big\{w\in M_{n,d,t}:w\ge_{\slex}v\big\}=\mathcal{L}_t(\max(M_{n,d,t}),v)\]
is called an \emph{initial $t$--spread lexsegment} and the set
\[\mathcal{L}_t^{f}(u)=\big\{w\in M_{n,d,t}:w\le_{\slex}u\big\}=\mathcal{L}_t(u,\min(M_{n,d,t}))
\]
is called a \emph{final $t$--spread lexsegment}.
\end{Def}
One can observe that, if $u,v\in M_{n,d,t}$, with $u\ge_{\slex}v$, then $\mathcal{L}_t(u,v)=\mathcal{L}_t^i(v)\cap\mathcal{L}_t^f(u)$.
\begin{Def}\label{def:comp}\rm
A $t$--spread monomial ideal $I$ of $S$ is called an \textit{arbitrary $t$--spread lexsegment ideal}, or simply a \textit{$t$--spread lexsegment ideal} if it is generated by an (arbitrary) $t$--spread lexsegment.\\ A $t$--spread lexsegment ideal $I=(\mathcal{L}_t(u,v))$ is a \textit{completely $t$--spread lexsegment ideal} if
$$
I=J\cap T,
$$
where $J=(\mathcal{L}_t^i(v))$ and $T=(\mathcal{L}_t^f(u))$.
\end{Def}
In \cite{FC1}, all completely $t$--spread lexsegment ideals have been characterized.
Now, we recall some important notions that will be useful in the sequel.\\
A $t$--spread monomial ideal $I$ is a \textit{$t$--spread strongly stable ideal} if for all $t$--spread monomial $u\in I$, all $j\in \supp(u)$ and all $1\le i< j$ such that $x_i(u/x_j)$ is $t$--spread, it follows that $x_i(u/x_j)\in I$.
Every initial $t$--spread lexsegment ideal $I$ of $S$ is a $t$--spread strongly stable ideal. Hence, in order to compute the graded Betti numbers of $I$ one may use the Ene, Herzog, Qureshi's formula (\cite[Corollary 1.12]{EHQ}):
\begin{equation}
\label{eq1}
\beta_{i,i+j}(I)\ =\ \sum_{u\in G(I)_j}\binom{\max(u)-t(j-1)-1}{i}.
\end{equation}
A $t$--spread final lexsegment ideal $I$ of $S$ is also a $t$--spread strongly stable ideal, but with the order of the variables reversed, \emph{i.e.}, $x_n> x_{n-1} > \cdots >x_1$. Hence, in order to compute their graded Betti numbers one may use the following \emph{modified} Ene, Herzog, Qureshi's formula:
\begin{equation}\label{Bettinumbersreversetspread}
\beta_{i,i+j}(I)\ =\ \sum_{u\in G(I)_j}\binom{n-\min(u)-t(j-1)}{i}.
\end{equation}
Let $>_{\lex}$ be the usual \textit{lexicographic order} on $S$, with $x_1>x_2>\cdots>x_n$, \cite{JT}. The next theorem collects some results from \cite{FC1} that will be pivotal for the aim of this article.
\begin{Thm}\textup{\cite{FC1}}.\label{thm:compltspreadlex}
Given $n,d,t\ge1$, let $u=x_{i_1}x_{i_2}\cdots x_{i_d}$ and $v=x_{j_1}x_{j_2}\cdots x_{j_d}$ be $t$--spread monomials of degree $d$ of $S$ such that $u\ge_{\slex}v$.
\begin{enumerate}\item[\em(1)] Let $I=(\mathcal{L}_t(u,v))$. The following conditions are equivalent:
\begin{enumerate}
\item[\em(a)] $I$ is a completely $t$--spread lexsegment ideal, \emph{i.e.}, $I=J\cap T$;
\item[\em(b)] for every $w\in M_{n,d,t}$ with $w<_{\slex}v$ there exists an integer $s>i_1$ such that $x_s$ divides $w$ and $x_{i_1}(w/x_{s})\le_{\lex}u$.
\end{enumerate}
\item[\em(2)] Let $I=(\mathcal{L}_t(u,v))$ be a completely $t$--spread lexsegment ideal with $\min(v)>\min(u)=1$. $I$ has a linear resolution if and only if one of the following conditions hold:
\begin{enumerate}
\item[\em(i)] $i_2=1+t$;
\item[\em(ii)] $i_2>1+t$ and for the largest $w\in M_{n,d,t}$, $w<_{\lex}v$, we have $x_1(w/x_{\max(w)})\le_{\lex}x_1x_{i_2-t}x_{i_3-t}\cdots x_{i_d-t}$.
\end{enumerate}
\item[\em(3)] Let $I=(\mathcal{L}_t(u,v))$ be a completely $t$--spread lexsegment ideal and suppose $I=(\mathcal{L}_t(u,v))$ has a linear resolution. Then, for all $i\ge0$,
$$
\beta_{i}(I)=\sum_{w\in\mathcal{L}_t^f(u)}\binom{n-\min(w)-(d-1)t}{i}-\sum_{\substack{w\in\mathcal{L}_t^f(v)\\ w\ne v}}\binom{\max(w)-(d-1)t-1}{i}.
$$
\end{enumerate}
\end{Thm}
We close the section recalling some notions about simplicial complexes.
Given $n\in\mathbb{N}$, we set $[n]=\{1,2,\dots,n\}$. We recall that
a \textit{simplicial complex} on the \textit{vertex set} $[n]$ is a family of subsets of $[n]$ such that
\begin{enumerate}
\item[-] $\{i\}\in\Delta$ for all $i\in[n]$, and
\item[-] if $F\subseteq[n]$, $G\subseteq F$, we have $G\in\Delta$.
\end{enumerate}
The dimension of $\Delta$ is the number $d=\max\{|F|-1:F\in\Delta\}$. Any $F\in\Delta$ is called a \textit{face} and $|F|-1$ is the \textit{dimension} of $F$. A \textit{facet} of $\Delta$ is a maximal face with respect to the inclusion. The set of facets of $\Delta$ is denoted by $\mathcal{F}(\Delta)$.
It is well known that for any squarefree ideal ($1$--spread ideal) $I$ of $S$ there exists a unique simplicial complex $\Delta$ on $[n]$ such that $I=I_\Delta$, where
$$
I_\Delta=({\bf x}_F:F\subseteq[n],F\notin\Delta)
$$
is the Stanley--Reisner ideal of $\Delta$ \cite{JT}. In the sequel when we say that a simplicial complex $\Delta$ is associated to a squarefree ideal $I$ of $S$, we mean that $I_{\Delta}= I$.
A simplicial complex $\Delta$ on $[n]$ is called \textit{pure} if its facets have the same dimension. We say that $\Delta$ is \textit{Cohen--Macaulay} if $K[\Delta]=S/I_{\Delta}=K[x_1,\dots,x_n]/I_\Delta$ is a Cohen--Macaulay ring. If $\Delta$ is Cohen--Macaulay then $\Delta$ is pure (see, for instance, \cite{JT}).
It is well known that any monomial ideal (not necessarily squarefree) has a unique minimal primary decomposition. We refer to it as the \textit{standard primary decomposition}. Moreover, for a squarefree monomial ideal, its standard primary decomposition is $I=\bigcap_{\mathfrak{p}\in\textup{Min}(I)}\mathfrak{p}$, \cite[Corollary 1.3.6]{JT}, where $\text{Min}(I)$ is the set of minimal primes of $I$. Every minimal prime is a monomial prime ideal, \emph{i.e.}, $\mathfrak{p}=(x_{i_1},x_{i_2},\dots,x_{i_s})$, for some $A=\{i_1<i_2<\dots<i_s\}\subseteq[n]$. To denote such an ideal we also use the notation $\mathfrak{p}_A$.
\begin{Prop}\label{prop:AlexDuality1}
\textup{\cite[Lemma 1.5.4]{JT}} The standard primary decomposition of $I_\Delta$ is
$$
I_{\Delta}=\bigcap_{F\in\mathcal{F}(\Delta)}\mathfrak{p}_{[n]\setminus F}.
$$
\end{Prop}
Given two positive integers $j\ge k$, we set $[j,k]=\{\ell\in\mathbb{N}:j\le\ell\le k\}$ and we associate to a $t$--spread monomial two special sets of integers.
Let $t\ge1$ and let $w=x_{\ell_1}x_{\ell_2}\cdots x_{\ell_d}$ be a $t$--spread monomial of $S$ of degree $d$.
If $\max(w)\le n+1-t$, we define the \textit{$t$--spread support} of $w$ as follows:
\begin{equation}
\label{eq:suppt}
\supp_t(w)=\bigcup_{r=1}^d[\ell_r,\ell_r+(t-1)];
\end{equation}
whereas, if $\min(w)\ge t$, we define the \textit{$t$--spread cosupport} of $w$ as follows:
\begin{equation}
\label{eq:cosuppt}\cosupp_t(w)=\bigcup_{r=1}^{d}[\ell_r-(t-1),\ell_r].
\end{equation}
Moreover, we set $\supp_t(1)=\cosupp_t(1)=\emptyset$.
\begin{Rem}\em Let $w=x_2x_5x_{10}x_{13}\in S=K[x_1, \ldots, x_{13}]$ be a 3--spread monomial. One can note that $\max(w)=13 >13+1-3=11$. Furthermore, $\supp_3(w/x_{\max(w)})=\{2,3,4,5,6,7,10,11,12\}$ and $\cosupp_3(w/x_{\min(w)})=\{3,4,5,8,9,10,11,12,13\}$.
Observe that for all $w\in M_{n,d,t}$, we can always compute $\supp_t(w/x_{\max(w)})$ and $\cosupp_t(w/x_{\min(w)})$.
\end{Rem}
In \cite[Theorem 2.3]{EHQ}, Ene, Herzog, Qureshi computed the standard primary decomposition of the $t$--spread Veronese ideal of degree $d$, $I_{n,d,t}$,
\begin{equation}\label{eq:decVeronese}
I_{n,d,t}=\bigcap_{D\in\mathcal{D}}\mathfrak{p}_{[n]\setminus D},
\end{equation}
where
$$
\mathcal{D}=\Big\{\bigcup_{r=1}^{d-1}[\ell_r,\ell_r+(t-1)]\subseteq[n]\ :\ \ell_{r+1}-\ell_{r}\ge t,\ r=1,\dots,d-2\Big\}.
$$
Using our notation, each set $D\in\mathcal{D}$ may be written as $\supp_t(x_{\ell_1}\cdots x_{\ell_{d-1}})$, with
$x_{\ell_1}\cdots x_{\ell_{d-1}}\in M_{n+1-t,d-1,t}$. Hence, the primary decomposition (\ref{eq:decVeronese}) becomes
\begin{equation}\label{eq:primdecompIndt}
I_{n,d,t}=\bigcap_{w\in M_{n+1-t,d-1,t}}\mathfrak{p}_{[n]\setminus\supp_t(w)}.
\end{equation}
In particular, $I_{n,d,t}$ is an \textit{height--unmixed} ideal, \emph{i.e.}, all its minimal primes have the same height, $n-(d-1)t$ \cite[Theorem 2.3]{EHQ}.
Furthermore, $I_{n,d,t}$ is a completely $t$--spread lexsegment ideal \cite{FC1}.\\
\section{The primary decomposition of\\ completely $t$--spread lexsegment ideals}\label{sec2}
In this section we study the primary decomposition of a completely $t$--spread lexsegment ideal of $S=K[x_1, \ldots, x_n]$. The case $t=1$ has been analyzed in \cite{OO}. The sets defined in (\ref{eq:suppt}) and (\ref{eq:cosuppt}) will be pivotal for our aim.\\
We start the section with some comments and remarks.
Let $n,d,t\ge 1$, $u=x_{i_1}x_{i_2}\cdots x_{i_d}$ and
$v=x_{j_1}x_{j_2}\cdots x_{j_d}$ monomials of $M_{n,d,t}$, with $u\ge_{\slex}v$.
Set $L=\mathcal{L}_t(u,v)$, $I=(L)$. We assume that $I$ is a completely $t$--spread lexsegment ideal, \emph{i.e.}, $I\ =\ J\cap T$,
with $J=(\mathcal{L}_t^i(v))$ and $T=(\mathcal{L}_t^f(u))$.
\begin{enumerate}
\item[-] If $u=v$, $I=(L)=(u)$ is a principal ideal, and its standard primary decomposition is $I=(x_{i_1})\cap\dots\cap(x_{i_d})$. Therefore, we may assume $u>_{\slex}v$.
\item[-] If $\deg(u)=\deg(v)=d=1$, then $I=(L)=(x_{i_1},x_{i_1+1},\dots,x_{{j_1}-1},x_{j_1})=\mathfrak{p}_{[i_1,j_1]}$ is the standard primary decomposition of $I$. Thus, we may assume $d\ge2$.
\item[-] If $I$ is initial, \emph{i.e.}, $I=J=(\mathcal{L}_t^i(v))$, we may assume $\min(v)=j_1\ge 2$, otherwise
$$
J=(x_1)\cap(\mathcal{L}_t(x_{1+t}\cdots x_{1+(d-1)t},v/x_{1})),
$$
and to determine the standard primary decomposition of $J$, it suffices to determine that of $(\mathcal{L}_t(x_{1+t}\cdots x_{1+(d-1)t},v/x_{j_1}))$ which is an initial $t$--spread lexsegment ideal in the polynomial ring $K[x_{1+t},\dots,x_n]$ with fewer indeterminates than $S$.
\item[-] If $I$ is final, \emph{i.e.}, $I=T=(\mathcal{L}_t^f(u))$, we can assume $\min(u)=i_1=1$. Indeed, if $i_1>1$, none of the variables $x_1,\dots,x_{i_1-1}$ divides any minimal monomial generator of $T$. So computing the standard primary decomposition of $T$ is equivalent to computing that of $T\cap K[x_{i_1},\dots,x_n]$ in the polynomial ring $K[x_{i_1},\dots,x_n]$ with fewer indeterminates than $S$.
\end{enumerate}
Hence, from now on, we assume that $I=(\mathcal{L}_t(u,v))=J\cap T$, with $u>_{\slex}v$, $\min(u)=1$, $\min(v) \ge 2$ and $\deg(u)=\deg(v)=d\ge 2$.\\
To compute the standard primary decomposition of $I=J\cap T$, we can proceed as follows.
Firstly, we determine the decomposition of $J$, and if $I=J$ we are done. Secondly, we determine that of $T$, and if $I=T$ we are done. Finally, knowing the primary decompositions of $J$ and $T$ we take into account the intersection $J\cap T=I$, and deleting the non minimal primes, we obtain the standard primary decomposition of $I$ in the general case.
\subsection{The initial $t$--spread lexsegment case}
In this subsection we analyze the case of the standard primary decomposition of an initial $t$--spread lexsegment ideal of $S=K[x_1,\dots,x_n]$.
\begin{Thm}\label{primdecompinitialtspreadlex}
Let $v=x_{j_1}x_{j_2}\dots x_{j_d}$ be a $t$--spread monomial of degree $d\ge2$ of $S$ with $j_1\ge 2$
and let $J=(\mathcal{L}_t^i(v))$ be an initial $t$--spread lexsegment ideal of $S$. Then, the standard primary decomposition of $J$ is
$$
J\ =\ \bigcap_{p=1}^d\mathfrak{p}_{F_p}\cap\bigcap_{F\in\mathcal{F}}\mathfrak{p}_{[n]\setminus F},
$$
where
\begin{align*}
F_p\ &=[j_p]\setminus\supp_t(x_{j_1}x_{j_2}\cdots x_{j_{p-1}}),\ p=1,\dots,d,\\
\mathcal{F}\ &=\ \big\{\supp_t(w):w\in M_{n+1-t,d-1,t},\ w >_{\slex} v/x_{\max(v)}\big\}.
\end{align*}
\end{Thm}
\begin{proof}
Let $\Delta$ be the simplicial complex on the vertex set $[n]$ associated to $J$.
From Proposition \ref{prop:AlexDuality1}, we have to prove that the facets of $\Delta$ are exactly the sets $[n]\setminus F_{p}$, $p=1,\dots,d$, together with the sets of $\mathcal{F}$, \emph{i.e.},
\begin{equation}\label{facetprimdecomplexinit}
\mathcal{F}(\Delta)=\big\{[n]\setminus F_{p}:p=1,\dots,d\big\}\cup\mathcal{F}.
\end{equation}
Observe that if $G\in\Delta$, $G\ne\emptyset$, then $G$ is a facet of $\Delta$ if and only if $G\cup\{i\}\notin\Delta$, for all $i\in[n]\setminus G$, \emph{i.e.}, if and only if ${\bf x}_{G\cup\{i\}}\in J$, for all $i\in[n]\setminus G$.
We show that each $F\in\mathcal{F}$ is a facet of $\Delta$. Let $F\in\mathcal{F}$, then
\begin{align*}
F\ &=\ \supp_t(x_{\ell_1}\cdots x_{\ell_{d-1}})\ =\ \bigcup_{r=1}^{d-1}[\ell_r,\ell_r+(t-1)]\\
&=\ \big\{\ell_1,\ell_1+1,\dots,\ell_1+(t-1),\ \dots,\ \ell_{d-1},\ell_{d-1}+1,\dots,\ell_{d-1}+(t-1) \big\},
\end{align*}
with $x_{\ell_1}\cdots x_{\ell_{d-1}}\in M_{n+1-t,d-1,t}$ and $x_{\ell_1}\cdots x_{\ell_{d-1}}>_{\slex}v/x_{\max(v)}=x_{j_1}\cdots x_{j_{d-1}}$.
Clearly, ${\bf x}_F\notin J$, since
${\bf x}_F$ is not a multiple of any $t$--spread monomial of degree $d$ of $S$. To prove that $F$ is a facet, it suffices to show that ${\bf x}_{F\cup\{i\}}\in J$, for all $i\in[n]\setminus F$.
Let $i\in[n]\setminus F$. We have:
\begin{enumerate}
\item[-] if $i<\ell_1$, then ${\bf x}_{F\cup\{i\}}$ is divided by the $t$--spread monomial $x_ix_{\ell_1+(t-1)}\cdots x_{\ell_{d-1}+(t-1)}$;
\item[-] if $\ell_j+(t-1)<i<\ell_{j+1}$, for some $1\le j\le d-2$, then ${\bf x}_{F\cup\{i\}}=$ ${\bf x}_Fx_i$ is divided by the $t$--spread monomial $\big(\mathfrak{p}rod\limits_{r=1}^jx_{\ell_r}\big)x_i\big(\mathfrak{p}rod\limits_{r=j+1}^{d-1}x_{\ell_r+(t-1)}\big)$;
\item[-] if $\ell_{d-1}+(t-1)<i$, then ${\bf x}_{F\cup\{i\}}$ is is divided by the monomial $x_{\ell_1}\cdots x_{\ell_{d-1}}x_i$.
\end{enumerate}
More in detail, in every case, ${\bf x}_{F\cup\{i\}}$ is a multiple of a $t$--spread monomial of degree $d$ strictly greater than $v$, with respect to $>_{\slex}$. Therefore ${\bf x}_{F\cup\{i\}}\in J$ and $F$ is a facet.
Now, we determine all the facets of $\Delta$. Let $G\in\Delta$ and define the following integers:
\begin{enumerate}
\item[] $\ell_1=\min(G)$, and
\item[] $\ell_i=\min\big\{r\in G:r\ge\ell_{i-1}+t\big\}$, for $i\ge 2$.
\end{enumerate}
Let $\ell_k$ be the last element of the sequence $\ell_1 < \ell_2 < \cdots$.\\
If $\ell_i\le j_i$ for all $i$, then $k\le d-1$, otherwise $x_{\ell_1}x_{\ell_2}\cdots x_{\ell_d}\in J$. Against the fact that $G\in\Delta$. Then,
\begin{align*}
G\ &\subseteq\ \bigcup_{i=1}^{k}[\ell_i,\ell_i+(t-1)]\cup\bigcup_{i=k+1}^{d-1}[j_i,j_i+(t-1)]\\
&=\ \supp_t(x_{\ell_1}\cdots x_{\ell_k}x_{j_{k+1}}\cdots x_{j_{d-1}})=F \in\mathcal{F}.
\end{align*}
Assume there exists an integer $1\le p\le d$ such that $\ell_p>j_p$ and let $p$ be minimal for such a property. By the meaning of $p$, for $q<p$, $\ell_q\le j_q$. We need to distinguish two cases.\\
\textsc{Case 1.} There exists an integer $q<p$ such that $\ell_q<j_q$.\\
In such a case, $x_{\ell_1}x_{\ell_2}\cdots x_{\ell_{p-1}}>_{\slex}x_{j_1}x_{j_2}\cdots x_{j_{p-1}}$. Moreover, $k\le d-1$, otherwise $x_{\ell_1}x_{\ell_2}\cdots x_{\ell_{p-1}}x_{\ell_p}\cdots x_{\ell_d}>_{\slex}v$ and so ${\bf x}_G\in J$. A contradiction.
Hence,
$$
G\subseteq\bigcup_{i=1}^{k}[\ell_i,\ell_i+(t-1)]
$$
and we can quickly find a facet $F\in\mathcal{F}$ which contains $\bigcup_{i=1}^{k}[\ell_i,\ell_i+(t-1)]$ and consequently $G$.\\
\textsc{Case 2.} For all integers $q<p$, $\ell_q=j_q$.\\
In such a case, since $\ell_p>j_p$, we have $x_{j_1}\cdots x_{j_p}>_{\slex}x_{\ell_1}\cdots x_{\ell_p}$, and
$$
G\subseteq\bigcup_{i=1}^{p-1}[j_i,j_i+(t-1)]\cup[j_p+1,n]=[n]\setminus F_p
$$
and $[n]\setminus F_p$ is clearly a facet.
Finally, all the facets of $\Delta$ are those described in (\ref{facetprimdecomplexinit}).
\end{proof}
\begin{Rem}\label{Rem:FacetsIniTSpread}
\rm We note that for an initial $t$--spread lexsegment ideal $J=I_\Delta$, all the facets $F\in\mathcal{F}(\Delta)$ have cardinality $|F|\ge(d-1)t$. Indeed, from Theorem \ref{primdecompinitialtspreadlex}, $\mathcal{F}(\Delta)=\big\{[n]\setminus F_{p}:p=1,\dots,d\big\}\cup\mathcal{F}$. If $F\in\mathcal{F}$, then $|F|=(d-1)t$. Otherwise, if $F=[n]\setminus F_p$ for some $p$, then $|F|=|[n]\setminus F_p|=n-|F_p|=n-(j_p-(p-1)t)$. We observe that $j_p\le n-(d-p)t$. Thus
$$
|F|=n+(p-1)t-j_p\ge n+(p-1)t-n+(d-p)t=(d-1)t,
$$
as desired.
\end{Rem}
We illustrate the previous result with an example.
\begin{Expl}\label{ex:primdecompJ}
\rm Let $v=x_2x_5x_7\in S=\mathbb{Q}[x_1,\dots,x_7]$ be a $2$--spread monomial and consider the initial $2$--spread ideal $J=(\mathcal{L}_2^i(v))$ of $S$:
$$
J=(x_{1}x_{3}x_{5},x_{1}x_{3}x_{6},x_{1}x_{3}x_{7},x_{1}x_{4}x_{6},x_{1}x_{4}x_{7},x_{1}x_{5}x_{7},
x_{2}x_{4}x_{6},x_{2}x_{4}x_{7},x_{2}x_{5}x_{7}).
$$
We have $n=7,d=3,t=2$. The monomials $w\in M_{n+1-t,d-1,t}=M_{6,2,2}$ such that $w>_{\slex}v/x_{\max(v)}=x_2x_5$ are the following
$$
x_1x_3,\ \ x_1x_4,\ \ x_1x_5,\ \ x_1x_6,\ \ x_2x_4.
$$
Therefore,
\begin{align*}
\mathcal{F}\ &=\ \big\{\supp_2(x_1x_3),\supp_2(x_1x_4),\supp_2(x_1x_5),\supp_2(x_1x_6),\supp_2(x_2x_4)\big\}\\
&=\ \big\{\{1,2,3,4\},\{1,2,4,5\},\{1,2,5,6\},\{1,2,6,7\},\{2,3,4,5\}\big\}.
\end{align*}
Moreover, $v=x_{j_1}x_{j_2}x_{j_3}=x_2x_5x_7$, hence
\begin{align*}
F_1&=[j_1]\setminus\supp_2(1)=[2]\setminus\emptyset=\{1,2\},\\
F_2&=[j_2]\setminus\supp_2(x_{j_1})=[5]\setminus\{2,3\}=\{1,4,5\},\\
F_3&=[j_3]\setminus\supp_2(x_{j_1}x_{j_2})=[7]\setminus\{2,3,5,6\}=\{1,4,7\}.
\end{align*}
Finally, the standard decomposition of $J$ is
\begin{align*}
J&=(x_{1},x_{2})\cap (x_{1},x_{4},x_{5})\cap(x_{1},x_{4},x_{7})\cap(x_{1},x_{6},x_{7})\cap(x_{3},x_{4},x_{5})\\
&\mathfrak{p}hantom{=..}\cap
(x_{3},x_{4},x_{7})\cap(x_{3},x_{6},x_{7})\cap(x_{5},x_{6},x_{7}).
\end{align*}
\end{Expl}
Here are some corollaries of Theorem \ref{primdecompinitialtspreadlex}.
\begin{Cor}\label{cor:JinitialInvariants}
In the hypotheses of Theorem \ref{primdecompinitialtspreadlex}, $\mathfrak{p}d(S/J)=n-(d-1)t$, $\depth(S/J)=(d-1)t$ and $\dim(S/J)=n-j_1$.
\end{Cor}
\begin{proof}
Since $J$ is a $t$--spread strongly stable ideal, by (\ref{eq1}), $\mathfrak{p}d(S/J)=\mathfrak{p}d(J)+1=n-(d-1)t$. So, by the Auslander--Buchsbaum formula, $\depth(S/J)=\depth(S)-\mathfrak{p}d(S/J)=n-(n-(d-1)t)=(d-1)t$.
Let $\Delta$ be the simplicial complex associated to $J$. By Theorem \ref{primdecompinitialtspreadlex}, the facets of $\Delta$ are those described in (\ref{facetprimdecomplexinit}), and so, since $j_1\le n-(d-1)t$, and $j_1+(i-1)t\le j_i$, for $i=2,\dots,d$, we have
\begin{align*}
\textup{height}(J)\ &=\ \min\big\{ n-(d-1)t, |F_\ell|\ :\ \ell=1,\dots,d\big\}\\
&=\ \min\big\{ n-(d-1)t,j_1,j_2-t,\dots,j_{d}-(d-1)t\big\}\ =\ j_1.
\end{align*}
Therefore, $\dim(K[\Delta])=\dim(S/J)=n-j_1$.
\end{proof}
\begin{Cor}\label{cor:JCohenMac}
An initial $t$--spread lexsegment ideal generated in degree $d\ge2$ is Cohen--Macaulay if and only if it is the $t$--spread Veronese ideal of degree $d$.
\end{Cor}
\begin{proof}
Let $J=(\mathcal{L}_t^i(v))$, $v=x_{j_1}x_{j_2}\cdots x_{j_d}$ and let $\Delta$ be the simplicial complex associated to $J$. Assume $J$ is Cohen--Macaulay, then $\Delta$ is pure. By Theorem \ref{primdecompinitialtspreadlex}, the facets of $\Delta$ are those described in (\ref{facetprimdecomplexinit}), and so $\Delta$ is pure if and only if
$$
(d-1)t=n-j_1=n-(j_2-t)=\cdots=n-(j_d-(d-1)t).
$$
Hence $v=x_{n-(d-1)t}x_{n-(d-2)t}\cdots x_{n-t}x_n$ and $J=I_{n,d,t}$. Conversely, $I_{n,d,t}$ is Cohen--Macaulay. Indeed, $\dim(S/J)=\textup{depth}(S/J)=(d-1)t$ (Corollary \ref{cor:JinitialInvariants}).
\end{proof}
\subsection{The final $t$--spread lexsegment case}
In this subsection we determine the standard primary decomposition of a final $t$--spread lexsegment ideal of $S=K[x_1,\dots,x_n]$.
The next result will be crucial.
\begin{Prop}\label{propcardfacetslexfin}
Let $u$ be a $t$--spread monomial of degree $d$ of $S$ such that $\min(u)=1$ and let $T=(\mathcal{L}_t^f(u))$ be a final $t$--spread lexsegment of $S$.
Let $\Delta$ be the simplicial complex on $[n]$ associated to $T$.
Then $|F|\in\{(d-1)t,1+(d-1)t\}$, for all $F\in\mathcal{F}(\Delta)$.
\end{Prop}
\begin{proof}
Since we may see $T$ as an initial $t$--spread lexsegment ideal with the order on the variables reversed $x_n>x_{n-1}>\cdots>x_1$, then each facet of $\Delta$ has cardinality $\ge(d-1)t$ (Remark \ref{Rem:FacetsIniTSpread}). To prove that $|F|\le 1+(d-1)t$ for all $F\in\mathcal{F}(\Delta)$, it is enough to show that each subset $G\subseteq[n]$ with cardinality $|G|=2+(d-1)t$ does not belong to $\Delta$, \emph{i.e.}, ${\bf x}_G\in T$.
Let $G\subseteq[n]$ with $|G|=2+(d-1)t$. Define
\begin{enumerate}
\item[] $\ell_1=\min\{g\in G:g>1\}$, and
\item[] $\ell_j=\min\big\{k\in G:k\ge \ell_{j-1}+t\big\}$, for $j\ge 2$.
\end{enumerate}
Since $|G|=2+(d-1)t$, the set $\{g\in G:g>1\}$ has at least $1+(d-1)t$ elements. Hence, the sequence of integers $\ell_1<\ell_2<\cdots$ has at least $d$ terms. Therefore, $H=\{\ell_1,\ell_2,\dots,\ell_d\}\subseteq G$ and $u>_{\slex}{\bf x}_H$. Indeed, $\min(u)=1$ and $\min({\bf x}_H)=\ell_1>1$ and so
${\bf x}_H\in T$. It follows that ${\bf x}_G\in T$, as desired.
\end{proof}
In the next theorem, we assume that $T\ne I_{n,d,t}$, as the case of the Veronese ideal has been covered in Theorem \ref{primdecompinitialtspreadlex}.
\begin{Thm}\label{primdecompfinaltspreadlex}
Let $u=x_{i_1}x_{i_2}\dots x_{i_d}$ be a $t$--spread monomial of degree $d\ge2$ of $S$ with $i_1=1$ and let $T=(\mathcal{L}_t^f(u))$ be a final $t$--spread lexsegment ideal of $S$, with $T\ne I_{n,d,t}$. Then, the standard primary decomposition of $T$ is
$$
T\ =\ \bigcap_{G\in\mathcal{G}}\mathfrak{p}_{[n]\setminus G}\cap\bigcap_{H\in\mathcal{H}}\mathfrak{p}_{[n]\setminus H},
$$
where
\begin{align*}
\mathcal{G}\ &=\ \big\{\cosupp_t(w)\cup\{1\}:x_1w\in M_{n,d,t},\ w>_{\slex} u/x_1\big\},\\
\mathcal{H}\ &=\ \big\{\cosupp_t(w):x_1w\in M_{n,d,t},\ w\le_{\slex}u/x_1\big\}.
\end{align*}
\end{Thm}
\begin{proof}
Let $\Delta$ be a simplicial complex on the vertex set $[n]$ associated to $T$. By Proposition \ref{prop:AlexDuality1}, we have to prove that
\begin{equation}\label{facetprimdecomplexfint}
\mathcal{F}(\Delta)=\mathcal{G}\cup\mathcal{H}.
\end{equation}
By Proposition \ref{propcardfacetslexfin},
the facets of $\Delta$ have cardinality $(d-1)t$ or $1+(d-1)t$.
Let $G\in\mathcal{F}(\Delta)$ such that $|G|=1+(d-1)t$. We prove that $G\in\mathcal{G}$. First of all, $\min(G)=1$. Indeed, if $\min(G)\ge 2$, then setting $s_1=\min(G)$, and
$$
s_j=\min\big\{s\in G:s\ge s_{j-1}+t\big\},\,\, \mbox{for $j\ge2$},
$$
the sequence $s_1<s_2<\cdots$ has at least $d$ elements, otherwise $|G|<1+(d-1)t$. Hence, $\{s_1,s_2,\dots,s_d\}=U\subseteq G$ and ${\bf x}_U\in T$, as $\min({\bf x}_U)=\min(G)\ge 2>1=\min(u)$. It follows that $G\notin\Delta$. A contradiction. Therefore, $\min(G)=1$.
Consider the following integers:
\begin{enumerate}
\item[] $\ell_d =\max(G)$, and
\item[] $\ell_j =\max\big\{\ell\in G:\ell\le\ell_{j+1}-t\big\}$, for $j<d$.
\end{enumerate}
The sequence $\ell_d>\ell_{d-1}>\dots>\ell_k$ has at least $d$ terms, otherwise $|G|<1+(d-1)t$. Moreover it has at most $d$ terms, otherwise $\ell_1>1$, $x_{\ell_1}x_{\ell_2}\cdots x_{\ell_d}\in T$ and then $G\notin\Delta$. A contradiction. Hence, $k=1$.
Finally,
\begin{align*}
G\subseteq F\ &=\ \big\{\ell_d,\ell_d-1,\dots,\ell_d-(t-1),\ \ldots,\ \ell_2,\ell_2-1,\dots,\ell_2-(t-1),\ \ell_1 \big\}\\
&=\ \ \bigcup_{r=2}^{d}[\ell_r-(t-1),\ell_r]\cup\{\ell_1\}\ =\ \cosupp_t(x_{\ell_2}\cdots x_{\ell_d})\cup\{\ell_1\}.
\end{align*}
Moreover $1+(d-1)t=|G|\le|F|=1+(d-1)t$ and $\ell_1=1$, and so $G=F\in\mathcal{G}$. Clearly, $G$ is a facet. Indeed ${\bf x}_G\notin T$.\\
Now, let us determine the facets of $\Delta$ with cardinality $(d-1)t$. Let $H\in\mathcal{F}(\Delta)$ with $|H|=(d-1)t$. We prove that $H\in\mathcal{H}$. Consider the following integers
\begin{enumerate}
\item[] $\ell_d =\max(H)$, and
\item[] $\ell_j=\max\big\{\ell\in H:\ell\le\ell_{j+1}-t\big\}$, for $j<d$.
\end{enumerate}
The sequence $\ell_d>\ell_{d-1}>\dots>\ell_k$ has at least $d-1$ terms, otherwise $|H|<(d-1)t$, against our assumption. In fact, we have $k=2$. Indeed, suppose $k=1$.
If $\ell_1=1$, then $H\subseteq G\in\mathcal{G}$. A contradiction since $H$ is a facet. On the other hand, if $\ell_1>1$, then $x_{\ell_1}x_{\ell_2}\cdots x_{\ell_d}\in T$, and $H\notin\Delta$. A contradiction.
Finally, setting $U=\{\ell_2,\dots,\ell_d\}\subseteq H$, if $x_1{\bf x}_U>_{\slex}u$, then $H$ is contained in a facet $G\in\mathcal{G}$. A contradiction since $H$ is a facet. If $x_1{\bf x}_U\le_{\slex}u$, then $H\in\mathcal{H}$ and $H$ is clearly a facet. The proof is complete.
\end{proof}
\begin{Expl}\label{ex:primdecompT}
\rm Let $u= x_1x_4x_6$ be a $2$--spread monomial of degree $3$ of $S=\mathbb{Q}[x_1,\dots,x_7]$. Consider the ideal $T=(\mathcal{L}_2^f(u))$ of $S$:
$$
T=(x_{1}x_{4}x_{6},x_{1}x_{4}x_{7},x_{1}x_{5}x_{7},x_{2}x_{4}x_{6},x_{2}x_{4}x_{7},x_{2}x_{5}x_{7},
x_{3}x_{5}x_{7}).
$$
The monomials $w\in M_{7,2,2}$ such that $x_1w\in M_{7,3,2}$ and $w>_{\slex} u/x_1 = x_4x_6$ are
$$
x_3x_5,\ \ x_3x_6,\ \ x_3x_7;
$$
whereas, the monomials $w\in M_{7,2,2}$ such that $x_1w\in M_{7,3,2}$ and $w\le_{\slex}u/x_1 = x_4x_6$ are
$$
x_4x_6,\ \ x_4x_7,\ \ x_5x_7.
$$
Therefore,
\begin{align*}
\mathcal{G}\ &=\ \big\{\cosupp_2(x_3x_5)\cup\{1\}, \cosupp_2(x_3x_6)\cup\{1\}, \cosupp_2(x_3x_7)\cup\{1\}\big\}\\
&=\ \big\{\{1,2,3,4,5\},\{1,2,3,5,6\},\{1,2,3,6,7\} \big\},
\\
\mathcal{H}\ &=\ \big\{\cosupp_2(x_4x_6), \cosupp_2(x_4x_7), \cosupp_2(x_5x_7)\big\}\\
&=\ \big\{\{3,4,5,6\},\{3,4,6,7\},\{4,5,6,7\} \big\}.
\end{align*}
Thus, the standard primary decomposition of $T$ is
$$
T=(x_{4},x_{5})\cap(x_{4},x_{7})\cap(x_{6},x_{7})\cap(x_{1},x_{2},x_{3})\cap(x_{1},x_{2},x_{5})\cap(x_{1},x_{2},x_{7}).
$$
\end{Expl}
Here are some corollaries of Theorem \ref{primdecompfinaltspreadlex}.
\begin{Cor}\label{cor:TinitialInvariants}
Let $T=(\mathcal{L}_t^f(u))$ be a final $t$--spread lexsegment ideal of $S$. Then $\mathfrak{p}d(S/T)=n-(d-1)t$, $\depth(S/T)=(d-1)t$ and
$$
\dim(S/T)=\begin{cases}
(d-1)t&\textup{if}\ u=x_1x_{1+t}\cdots x_{1+(d-1)t},\\
1+(d-1)t&\textup{otherwise}.
\end{cases}
$$
\end{Cor}
\begin{proof}
The ideal $T$ is $t$--spread strongly stable but with the order on the variables reversed, $x_n>x_{n-1}>\dots>x_1$. Hence, by (\ref{Bettinumbersreversetspread}), as $\min(u)=1$ and $u\in T$, $\mathfrak{p}d(S/T)=\mathfrak{p}d(T)+1=n-1-(d-1)t+1=n-(d-1)t$. By the Auslander--Buchsbaum formula, $\depth(S/T)=(d-1)t$.
Suppose $T\ne I_{n,d,t}$. Let $\Delta$ the simplicial complex associated to $T$. By Theorem \ref{primdecompfinaltspreadlex}, the facets of $\Delta$ are those in (\ref{facetprimdecomplexfint}). Now, $\mathcal{H}$ is always non empty, as $u\in\mathcal{L}_t^f(u)$. Moreover, $\mathcal{G}$ is non empty too, as $u\ne\min(M_{n,d,t})$, otherwise $T=I_{n,d,t}$. Thus
\begin{align*}
\textup{height}(T)\ &=\ \min\big\{\textup{height}(\mathfrak{p}_{[n]\setminus G}),\textup{height}(\mathfrak{p}_{[n]\setminus H}):G\in\mathcal{G},H\in\mathcal{H}\big\}\\
&=\ \min\big\{n-(d-1)t,n-1-(d-1)t\big\}\ =\ n-1-(d-1)t.
\end{align*}
Thus, $\dim(K[\Delta])=1+(d-1)t$.\\
Otherwise, if $T=I_{n,d,t}$, \emph{i.e.}, $u=\max(M_{n,d,t})=x_1x_{1+t}\cdots x_{1+(d-1)t}$, we have $\dim(K[\Delta])$ $=(d-1)t$ (Corollary \ref{cor:JinitialInvariants}).
\end{proof}
\begin{Cor}\label{cor:TCohenMac}
A final $t$--spread lexsegment ideal of $S$ generated in degree $d\ge2$ is Cohen--Macaulay if and only if it is the $t$--spread Veronese ideal of degree $d$.
\end{Cor}
\begin{proof}
Let $T=(\mathcal{L}_t^f(u))$ and let $\Delta$ be the simplicial complex associated to $T$. Assume $T$ is Cohen--Macaulay, then $\Delta$ is pure. By contradiction, suppose $T\ne I_{n,d,t}$. By Theorem \ref{primdecompfinaltspreadlex}, the facets of $\Delta$ are those described in (\ref{facetprimdecomplexfint}). The families $\mathcal{G}$ and $\mathcal{H}$ are both non empty, the first as $u\in\mathcal{L}_t^f(u)$, the second as $u\ne\max(M_{n,d,t})$. But, any $G\in\mathcal{G}$ has cardinality $|G|=1+(d-1)t$ and any $H\in\mathcal{H}$ has cardinality $|H|=(d-1)t$. So $\Delta$ should not be pure. A contradiction. Thus $T=I_{n,d,t}$. The converse easily follows.
\end{proof}
\begin{Rem}\label{rem:unmixed}
\em A simplicial complex $\Delta$ is pure if and only if the Stanley--Reisner ideal $I_\Delta$ is unmixed, in the sense that all associated prime ideals have the same height (see, for instance, \cite{JT}).
One can observe that the proofs of Corollary \ref{cor:JCohenMac} and Corollary \ref{cor:TCohenMac} have pointed out that the classification for initial Cohen--Macaulay $t$--spread lexsegment ideal and for final Cohen--Macaulay $t$--spread lexsegment ideal is equivalent to the classification for initial unmixed $t$--spread lexsegment ideal and for final unmixed $t$--spread lexsegment ideal.
\end{Rem}
\subsection{The case of a completely $t$--spread lexsegment ideal}
The aim of this subsection is to determine the minimal primary decomposition of a completely $t$--spread lexsegment ideal $I=(\mathcal{L}_t(u,v))$ of $S=K[x_1,\dots,x_n]$, $u, v \in M_{n,d,t}$, $u>_{\slex} v$ with $\min(u)=1$ and $\min(v)\ge 2$.
\begin{Thm}\label{primdecompgeneraltspreadlex}
Let $u=x_{i_1}x_{i_2}\dots x_{i_d}, v=x_{j_1}x_{j_2}\cdots x_{j_d}$ be $t$--spread monomials of degree $d\ge2$ of $S$ with $i_1=1$ and $j_1\ge 2$. Let $I=(\mathcal{L}_t(u,v))$ be a completely $t$--spread lexsegment ideal of $S$, $I\ne I_{n,d,t}$. Then, the standard primary decomposition of $I$ is
$$
I\ =\ \bigcap_{G\in\mathcal{G}}\mathfrak{p}_{[n]\setminus G}\cap\bigcap_{p\in\mathcal{I}}\mathfrak{p}_{F_p}\cap\bigcap_{F\in\widetilde{\mathcal{F}}}\mathfrak{p}_{[n]\setminus F},
$$
where $F_p$ $(p=1,\dots,d)$ and $\mathcal{G}$
are the families described in Theorems \ref{primdecompinitialtspreadlex} and \ref{primdecompfinaltspreadlex},
\begin{enumerate}
\item[-] $\mathcal{I}=[d]\setminus\big\{p\in[d]:\big|[n]\setminus F_{p}\big|=(d-1)t \,\,\mbox{and $v/x_{j_p} >_{\slex} u/x_1$}\big\}$,
\item[-] $\widetilde{\mathcal{F}}$ is the family consisting of the sets $F=\supp_t(w)\in\mathcal{F}$ with $w=x_{\ell_1}\cdots x_{\ell_{d-1}}$ such that
\begin{enumerate}
\item[-] $1\notin F$ and $x_{\ell_1+(t-1)}\cdots x_{\ell_{d-1}+(t-1)}\le_{\slex}u/x_1$, or
\item[-] $1\in F$ and $F\cup\{j\}\notin\mathcal{G}$, for all $j\notin F$,
\end{enumerate}
\end{enumerate}
where $\mathcal{F}$ is the family described in Theorem \ref{primdecompinitialtspreadlex}.
\end{Thm}
\begin{proof}
By hypothesis, $I=J\cap T$ with $J=(\mathcal{L}^i_t(v))$ and $T=(\mathcal{L}_t^f(u))$. By Theorems \ref{primdecompinitialtspreadlex} and \ref{primdecompfinaltspreadlex} we have
\begin{equation}\label{eq:notminimalprimdecom}
I=T\cap J=\bigcap_{G\in\mathcal{G}}\mathfrak{p}_{[n]\setminus G}\cap\bigcap_{H\in\mathcal{H}}\mathfrak{p}_{[n]\setminus H}\cap\bigcap_{p=1}^d\mathfrak{p}_{F_p}\cap\bigcap_{F\in\mathcal{F}}\mathfrak{p}_{[n]\setminus F}.
\end{equation}
To find the standard primary decomposition of $I$, it suffices to delete from this presentation those primes that are not minimal. Equivalently, if $\Delta$ is the simplicial complex on vertex set $[n]$ associated to $I$, we must determine the facets of $\Delta$. By (\ref{eq:notminimalprimdecom}) we have that
\begin{equation}\label{eq:4families}
\mathcal{F}(\Delta)\subseteq\mathcal{G}\cup\mathcal{H}\cup\big\{[n]\setminus F_p:p=1,\dots,d\big\}\cup\mathcal{F}.
\end{equation}
where $\mathcal{G}$, $\mathcal{H}$, $F_p$ ($p=1,\dots,d$), $\mathcal{F}$
are the families described in Theorems \ref{primdecompinitialtspreadlex} and \ref{primdecompfinaltspreadlex}.\\\\
\textsc{Claim.} $\mathcal{F}(\Delta)=\mathcal{G}\cup\big\{[n]\setminus F_{\ell}:\ell\in\mathcal{I}\big\}\cup\widetilde{\mathcal{F}}$.\\
\textsc{Proof of the Claim}.
We analyze the four families in (\ref{eq:4families}), and in so doing we determine the facets of $\Delta$.\\
\begin{enumerate}
\item[-] Let $G\in\mathcal{G}$. Since $I\subseteq T$ and ${\bf x}_G\notin T$, it follows that ${\bf x}_G\notin I$, \emph{i.e.}, $G\in\Delta$. Let us prove that $G$ is a facet. Indeed, $G\not\subseteq F, H$, for all $F\in\mathcal{F}$ and all $H\in\mathcal{H}$. Indeed, $|F|=|H|=(d-1)t<1+(d-1)t=|G|$. Moreover, $G\not\subseteq [n]\setminus F_{p}$, as $1\in G$ and $1\in F_{p}$. In fact, $\min(v)=j_1\ge2$.\\
\item[-] Let $H\in\mathcal{H}$. Arguing as before, $H\in\Delta$. We show that we can ``eliminate" the family $\mathcal{H}$. In fact, we can write $H=\cosupp_t(x_{\ell_2}\cdots x_{\ell_d})$, with $x_1x_{\ell_2}\cdots x_{\ell_d}\in M_{n,d,t}$ and $x_{\ell_2}\cdots x_{\ell_d}\le_{\slex}u/x_1$. Let
$$
\widetilde{H}=\{\ell_2-(t-1),\ell_3-(t-1),\ldots,\ell_{d}-(t-1)\}\subseteq H.
$$
Setting $h_i = \ell_{i+1}-(t-1)$, for $i=1, \ldots, d-1$, then $\widetilde H=\{h_1<h_2<\dots<h_{d-1}\}$. We distinguish three cases.
\begin{enumerate}
\item[-] \textsc{Case 1.} Let ${\bf x}_{\widetilde{H}}>_{\slex}v/x_{\max(v)}$. Since $\supp_t({\bf x}_{\widetilde{H}}) = \cosupp_t(x_{\ell_2}\cdots x_{\ell_d})=H$, then $H\in\mathcal{F}$ (we will determine when $H\in\mathcal{F}$ is a facet of $\Delta$ later).
\item[-] \textsc{Case 2.} Let ${\bf x}_{\widetilde{H}}<_{\slex}v/x_{\max(v)}$. Then there exists an integer $s\in[d-1]$ such that $j_1=h_1,\ldots,j_{s-1}=h_{s-1}$ and $j_s<h_s$. Hence
\begin{align*}
H\ &\subseteq\ \big\{j_1,\dots,j_1+(t-1),\ldots,j_{s-1},\dots,j_{s-1}+(t-1),\ j_s+1,\dots,n\big\}\\
&=\ [n]\setminus F_s=F.
\end{align*}
Moreover, ${\bf x}_F\notin I$, as $I\subseteq J$ and ${\bf x}_F\notin J$. So $H$ is included in $F\in\Delta$.
\item[-] \textsc{Case 3.} Let ${\bf x}_{\widetilde{H}}=v/x_{\max(v)}$. Then we have $H\subseteq[n]\setminus F_d=F$, and also $F\in\Delta$. Thus $H$ is included in the face $F\in\mathcal{F}$.
\end{enumerate}
\item[-] Let $F=[n]\setminus F_{p}$, some some $1\le p\le d$. Since $I\subseteq J$ and ${\bf x}_F\notin J$, then ${\bf x}_F\notin I$ and so $F\in\Delta$. If $|F|\ge1+(d-1)t$, then $F$ is a facet as $F$ is not contained in any other set of the families $\mathcal{F},\mathcal{H}$. Moreover, $F\ne G$ for all $G\in\mathcal{G}$, as $1\in G\setminus F$.\\
Suppose $|F|=(d-1)t$. Then $F$ is not a facet if and only if $F\subsetneq G$, for some $G\in\mathcal{G}$. Therefore, if and only if $F=G\setminus\{1\}$, for some $G\in\mathcal{G}$. On the other hand, $G=\big\{\ell_d,\ell_d-1,\dots,\ell_d-(t-1),\ \ldots,\ \ell_2,\ell_2-1,\dots,\ell_2-(t-1),\ 1 \big\}$ is characterized by the conditions $x_1x_{\ell_2}\cdots x_{\ell_d}\in M_{n,d,t}$ and $x_1x_{\ell_2}\cdots x_{\ell_d}>_{\slex}u$. We observe that $\supp(v/x_{j_p})\subseteq F=[n]\setminus F_{p}$. Hence, if $F\subseteq G$, then
$$
v/x_{j_p}\ge_{\slex}x_{\ell_2}\cdots x_{\ell_d}>_{\slex}u/x_1.
$$
Thus, if $F\subseteq G$, then $v/x_{j_p}>_{\slex}u/x_1$. On the other hand, if $v/x_{j_p}\le_{\slex}u/x_1$, then $x_{\ell_2}\cdots x_{\ell_d}\le_{\slex}v/x_{j_p}\le_{\slex}u/x_1$ and $G$ is not a facet.
Finally, we have verified that $F=[n]\setminus F_{p}$ is a facet if and only if $p\in\mathcal{I}$.\\
\item[-] Let $F\in\mathcal{F}$. Clearly ${\bf x}_F\notin I$ and so $F\in\Delta$. We have that $$F=\big\{\ell_1,\ell_1+1,\ \ldots,\ \ell_1+(t-1),\dots,\ell_{d-1},\ell_{d-1}+1,\dots,\ell_{d-1}+(t-1)\big\},$$
with $x_{\ell_1}\cdots x_{\ell_{d-1}}\in M_{n+1-t,d-1,t}$ and $x_{\ell_1}\cdots x_{\ell_{d-1}}>_{\slex}v/x_{\max(v)}$. If $F$ is not a facet, then $F\subsetneq G$, for some $G\in\mathcal{G}$.
\begin{enumerate}
\item[-] \textsc{Case 1.} Let $1\notin F$. Then we have $F=G\setminus\{1\}$. $G$ is characterized by the condition $x_{\ell_1+(t-1)}\cdots x_{\ell_{d-1}+(t-1)}>_{\slex}u/x_1$. Hence, in such a case, $F$ is a facet if and only if $x_{\ell_1+(t-1)}\cdots x_{\ell_{d-1}+(t-1)}\le_{\slex}u/x_1$.\\
\item[-] \textsc{Case 2.} Let $1\in F$. Set $\ell=\ell_j+t$, if $j=\min\big\{j\in [d-2]:\ell_j+(t-1)<\ell_{j+1}\big\}$ does exist, otherwise set $\ell=\ell_{d-1}+t$. Then $F\subseteq F\cup\{\ell\}$.\\
\end{enumerate}
Finally $F$ is a facet if and only if $F\cup\{\ell\}\notin\mathcal{G}$.
Thus, if and only if
\begin{align*}
x_{\ell_1+t}\cdots x_{\ell_{j-1}+t}x_{\ell_j+t}\cdots x_{\ell_{d-1}+(t-1)}\le_{\slex}u/x_1,&\ \ \text{if}\ \ell=\ell_j+t,\\
x_{\ell_1+t}\cdots x_{\ell_{j-1}+t}x_{\ell_j+t}\cdots x_{\ell_{d-1}+t}\le_{\slex}u/x_1,&\ \ \text{if}\ \ell=\ell_{d-1}+t.
\end{align*}
\end{enumerate}
The claim follows.
\end{proof}
\begin{Expl}
\rm Let $u=x_1x_4x_6$ and $v=x_2x_5x_7$ be $2$--spread monomials of degree $3$ of $S=\mathbb{Q}[x_1,\dots,x_7]$. Consider the $t$--spread lexsegment ideal
$$
I=(\mathcal{L}_2(u,v))= (x_1x_4x_6,x_1x_4x_7,x_1x_5x_7,x_2x_4x_6,x_2x_4x_7,x_2x_5x_7)
$$
of $S$. Firstly, we verify that $I$ is a completely $2$--spread lexsegment ideal, \emph{i.e.}, $I=J\cap T$, where $J$ and $T$ are the $2$--spread lexsegment ideals in Examples \ref{ex:primdecompJ} and \ref{ex:primdecompT}, respectively. By \cite[Theorem 3.7. (b)]{FC1}, $I$ is a completely $2$--spread lexsegment ideal if and only if for all $w\in M_{7,3,2}$ with $w<_{\slex}v=x_2x_5x_7$, there exists $s\in\supp(w)$ such that $x_{\min(u)}w/x_s=x_1w/x_s\le_{\slex}u=x_1x_4x_6$.\\
We have $w=x_3x_5x_7$ and we may choose $s=\min(w)$. Thus, $I$ is a completely $2$--spread lexsegment ideal.\\
Setting, $u=x_{i_1}x_{i_2}x_{i_3} = x_1x_{i_2}x_{i_3}$ and $v=x_{j_1}x_{j_2}x_{j_3} = x_2x_{j_2}x_{j_3}$, we have
$$
\mathcal{I}=[3]\setminus\big\{p\in[3]:\big|[7]\setminus F_{p}\big|=2\cdot2\ \text{and}\ u/x_1=x_4x_6<_{\slex}v/x_{j_p}=x_2x_5x_7/x_{j_{\ell}}\big\}.
$$
The only sets $F_{p}$ with $\big|[7]\setminus F_{p}\big|=2\cdot2=4$ are $F_2=\{1,4,5\}$ and $F_3=\{1,4,7\}$ (Example \ref{ex:primdecompJ}). Moreover, $v/x_{j_2}=x_2x_7>_{\slex}x_4x_6$ and $v/x_{j_3}=x_2x_5>_{\slex}x_4x_6$. Thus, $\mathcal{I}=[3]\setminus\{2,3\}=\{1\}$.
From Examples \ref{ex:primdecompJ} and \ref{ex:primdecompT}, we have:
$$\mathcal{G}=\big\{\{1,2,3,4,5\},\{1,2,3,5,6\},\{1,2,3,6,7\} \big\}$$ and
$$
\mathcal{F}=\big\{\{1,2,3,4\},\{1,2,4,5\},\{1,2,5,6\},\{1,2,6,7\},\{2,3,4,5\}\big\}.
$$
Let $F\in\mathcal{F}$ such that $1\notin F$. Then $F=\{2,3,4,5\}=\supp_2(x_2x_4)$. We have that
$x_3x_5>_{\slex}u/x_1=x_4x_6$, so $F\notin\widetilde{\mathcal{F}}$.\\
Let $F\in\mathcal{F}$ such that $1\in F$. Then $F\in\widetilde{\mathcal{F}}$ if and only if $F\not\subseteq G$, for all $G\in\mathcal{G}$. Since such sets do not exist, $\widetilde{\mathcal{F}}=\emptyset$.
Finally, the standard primary decomposition of $I$ is
$$
I=(x_1,x_2)\cap(x_4,x_5)\cap(x_4,x_7)\cap(x_6,x_7).
$$
\end{Expl}
\section{Cohen--Macaulay $t$--spread lexsegment ideals} \label{sec3}
In Section \ref{sec2}, we have seen that if $I$ is an initial or a final $t$--spread lexsegment ideal of $S=K[x_1,\dots,x_n]$, $t\ge1$, then $I$ is Cohen--Macaulay if and only if $I=I_{n,d,t}$ is the Veronese $t$--spread ideal of degree $d$ of $S$ (Corollaries \ref{cor:JCohenMac}, \ref{cor:TCohenMac}). In this section, we consider the more general problem of classifying all Cohen--Macaulay $t$--spread lexsegment ideals $I=(\mathcal{L}_t(u,v))$ of $S$, with $t\ge1$. The case $t=1$ has been examined also in \cite{BST} by using Serre's condition $(S_2)$.
As in Section \ref{sec2}, we may assume that $I=(\mathcal{L}_t(u,v))$ with $u>_{\slex}v$, $\min(u)=1$, $\min(v)\ge 2$ and $\deg(u)=\deg(v)=d\ge 2$. In order to achieve our purpose, we will distinguish two cases: $\min(v)=2$ and $\min(v)>2$.
First, we note that we can always suppose $n>2+(d-1)t$. Indeed, for $n=1+(d-1)t$, there is only one $t$--spread lexsegment ideal, namely $I=(x_1x_{1+t}\cdots x_{1+(d-1)t})=I_{1+(d-1)t,d,t}$ which is always a Cohen--Macaulay ideal; whereas, for $n=2+(d-1)t$, $I=(\mathcal{L}_t(u,v))$ with $v=x_2x_{2+t}\cdots x_{2+(d-1)t}$, necessarily. In fact, in such a case
there does exist only one $t$--spread monomial whose minimum is equal to $2$. Hence, $I$ is final and therefore Cohen--Macaulay if and only if $I=I_{2+(d-1)t,d,t}$ (Corollary \ref{cor:TCohenMac}).
In what follows we denote the squarefree lex order $\ge_{\slex}$ simply by $\ge$.
\subsection{The height two case}
In this subsection we study the Cohen--Macaulayness of a $t$--spread lexsegment ideal $I=(\mathcal{L}_t(u,v))$ of $S$ with $\min(v)=2$. We will distinguish two cases: $3+(d-1)t\le n\le 3+(2d-3)t$ and $n\ge 4+(2d-3)t$.
We start with the case $3+(d-1)t\le n\le 3+(2d-3)t$. The notion of \textit{Betti splitting} \cite{EK} (see also \cite{FHT2009} and the reference therein) will be crucial.
Let $I$, $P$, $Q$ be monomial ideals of $S=K[x_1,\dots,x_n]$ such that $G(I)$ is the disjoint union of $G(P)$ and $G(Q)$. We say that $I=P+Q$ is a \textit{Betti splitting} if
$$
\beta_{i,j}(I)=\beta_{i,j}(P)+\beta_{i,j}(Q)+\beta_{i-1,j}(P\cap Q), \ \ \ \textup{for all}\ i,j.
$$
If $I=P+Q$ is a Betti splitting \cite[Corollary 2.2]{FHT2009}, then
\begin{align}
\label{eq:pd(I)j1=2BettiSplit}\mathfrak{p}d(I)\ &=\ \max\{\mathfrak{p}d(P),\mathfrak{p}d(Q),\mathfrak{p}d(P\cap Q)+1\}.
\end{align}
\begin{Lem}\label{Lem:BettiSplitxi}
\textup{\cite[Corollary 2.7]{FHT2009}} Let $I$ be a monomial ideal of $S$. Let $Q$ be the ideal generated
by all elements of $G(I)$ divisible by $x_i$ and let $P$ be the ideal generated by all other elements of $G(I)$. If the ideal $Q$ has a linear resolution, then $I=P+Q$ is a Betti splitting.
\end{Lem}
For $I$ a monomial ideal of $S$, we set $\gcd(I)=\gcd(u:u\in G(I))$.
\begin{Thm}\label{Thm:gcd(I)PcapQ}
Let $u=x_{i_1}x_{i_2}\dots x_{i_d}$, $v=x_{j_1}x_{j_2}\cdots x_{j_d}$ be $t$--spread monomials of degree $d\ge 2$ of $S$ with $i_1=1$, $j_1=2$ and let $I$ be the $t$--spread lexsegment ideal $I=(\mathcal{L}_t(u,v))$ of $S$. Assume $3+(d-1)t\le n\le 3+(2d-3)t$ and set
$$
P=(\mathcal{L}_t(u,x_1x_{n-(d-2)t}\cdots x_{n-t}x_n)),\ \ \ \ Q=(\mathcal{L}_t(x_2x_{2+t}\cdots x_{2+(d-1)t},v)).
$$
Then $I$ is Cohen--Macaulay if and only if $\gcd(I)=1$ and $P\cap Q$ is a principal ideal.
\end{Thm}
\begin{proof}
Let $I$ be Cohen--Macaulay and let $\Delta$ be the simplicial complex on the vertex set $[n]$ associated to $I$. Then $\Delta$ is pure. Since each monomial in $G(I)$ has minimum equal to 1 or 2, the set $[n]\setminus\{1,2\}$ is a facet of $\Delta$. Thus, $\Delta$ must be pure of dimension $\dim(\Delta)=n-3$.
Now, we prove that $v\in \{v_1, v_2\}$, where
\begin{equation}\label{moncase2.2}
\begin{aligned}
v_1&=\textstyle\mathfrak{p}hantom{\big(}\mathfrak{p}rod_{s=0}^{d-1}x_{2+st},\\
v_2&=\textstyle\big(\mathfrak{p}rod_{s=0}^{d-2}x_{2+st}\big)x_{3+(d-1)t}.
\end{aligned}
\end{equation}
First, note that $v_1>v_2$ are the greatest monomials of $M_{n,d,t}$ with minimum 2. So $v_1\in I$. Let $E=\big\{\max(w):w\in\mathcal{L}_t(v_1,v)\big\}$ and set $k=\max(E)$. Then $k\ge 2+(d-1)t$. Indeed $v_1\in I$. If $v=v_1$, there is nothing to prove. Assume $v>v_1$, then $|E|\ge2$. We show that $v=v_2$. The monomial prime ideal $\mathfrak{p}_{\{1\}\cup E}$ contains $I$, and $\textup{height}(\mathfrak{p}_{\{1\}\cup E})=1+|E|\ge 3$. Since $I$ is Cohen--Macaulay, thus height--unmixed of height two, there must be a minimal prime $\mathfrak{p}\in\Min(I)$ of height $2$ that contains properly $\mathfrak{p}_{\{1\}\cup E}$. For all $j\in E$, $\mathfrak{p}_{\{1\}\cup(E\setminus\{j\})}$ does not contain $I$. Indeed, the monomial $\big(\mathfrak{p}rod_{s=0}^{d-2}x_{2+st}\big)x_j\in\mathcal{L}_t(v_1,v)$ is not in $\mathfrak{p}_{\{1\}\cup(E\setminus\{j\})}$. Hence, $x_1$ can be omitted and $\mathfrak{p}_{E}$ must contain $I$. One can quickly observe that no variable $x_j$, $j\in E$, can be omitted again. Hence, $\mathfrak{p}_{E}$ is a minimal prime with $\textup{height}(\mathfrak{p}_{E})=|E|=2$, \emph{i.e.}, $k=2+(d-1)t$. So $v\in\{v_1,v_2\}$ as desired.\\
For $\ell=0,\dots,d-1$, let us define the following monomials:
\begin{equation}\label{moncase2.3}
\begin{aligned}
u_\ell\ &=\ x_1\big(\textstyle\mathfrak{p}rod_{s=\ell}^{d-2}x_{n-st-1}\big)\big(\mathfrak{p}rod_{s=0}^{\ell-1}x_{n-st}\big)\\
&=\ x_1x_{n-(d-2)t-1}\cdots x_{n-\ell t-1}x_{n-(\ell-1)t}\cdots x_{n-t}x_n.
\end{aligned}
\end{equation}
One can observe that $\mathcal{L}_t(u_0,u_{d-1})=\{u_0>u_1>\dots>u_{d-1}\}$ and furthermore $u_{d-1}$ is the smallest monomial of $M_{n,d,t}$ whose minimum is equal to $1$. Thus $u>u_{d-1}>v$ and $u_{d-1}\in I$.
We prove that $u=u_\ell$, for some $\ell\in\{0,\dots,d-1\}$.\\
Indeed, writing $u=x_1x_{i_2}\cdots x_{i_d}$, we note that $i_2\le n-(d-2)t$. The condition $u\le u_0$ is equivalent to $i_2\ge n-(d-2)t-1$. As before, we consider a suitable monomial prime ideal that contains $I$, namely $\mathfrak{p}_{[i_2,n-(d-2)t]\cup\{2\}}$. None of $x_j$ with $j\in[i_2,n-(d-2)t]$ can be omitted from $\mathfrak{p}_{[i_2,n-(d-2)t]\cup\{2\}}$. Thus, $\mathfrak{p}_{[i_2,n-(d-2)t]}$ must be a minimal prime of height two. Hence, $i_2=n-(d-2)t-1$, as desired.
By what shown so far, if $I$ is Cohen--Macaulay, then $I=(\mathcal{L}_t(u_\ell,v_k))$, for some $\ell\in\{0,\dots,d-1\}$ and $k\in\{1,2\}$.
On the other hand, $I$ is Cohen--Macaulay if and only if $\dim(S/I)=\textup{depth}(S/I)$. We note that $\dim(S/I)=n-2$. Indeed, we have shown previously that $\textup{height}(I)=2$.
By the Auslander--Buchsbaum formula, $\textup{depth}(S/I)=n-\mathfrak{p}d(S/I)=n-1-\mathfrak{p}d(I)$. Hence, $I$ is Cohen--Macaulay if and only if $\textup{height}(I)=2$ and $\mathfrak{p}d(I)=1$.
Hence, in order to establish the theorem we need to prove the next facts.\\
\textsc{Claim 1.} $\height(I)=2$ if and only if $\gcd(I)=\gcd(w:w\in G(I))=1$.\\
\textsc{Proof of Claim 1.} Indeed, $\mathfrak{p}_{[2]}=(x_1,x_2)$ is a minimal prime of $I$ having height two, so $\height(I)\le\height(\mathfrak{p}_{[2]})=2$. Moreover, $\height(I)=1$ if and only if there is a minimal prime $\mathfrak{q}$ of $I$ whose height is $1$, \emph{i.e.}, there exists a variable $x_s$ dividing all generators of $I$. The claimed statement follows.\\
\textsc{Claim 2.} $\mathfrak{p}d(I)=1$ if and only if the ideal $(\mathcal{L}_t(u,u_{d-1}))\cap(\mathcal{L}_t(v_1,v))=P\cap Q$ is a principal ideal.
\\
\textsc{Proof of Claim 2.} We can observe that
$$
P=(w\in G(I):\min(w)=1),\ \ \ Q=(w\in G(I):\min(w)=2).
$$
Clearly $I=P+Q$. We claim that $I=P+Q$ is a Betti splitting of $I$. Indeed using Lemma \ref{Lem:BettiSplitxi}, it sufficies to note that $Q=(w\in G(I):x_2\ \textup{divides}\ w)$ and that $Q$ has a linear resolution. For all monomials $w\in G(I)$ with $\min(w)=1$, $\min(w/x_1)\ge n-(d-2)t-1>2$, as by hypothesis $n\ge 3+(d-1)t>3+(d-2)t$. Moreover $Q$ is an equigenerated initial $t$--spread lexsegment ideal in $K[x_2,\dots,x_n]$, so it has a linear resolution. Hence, by (\ref{eq:pd(I)j1=2BettiSplit}),
$$
\mathfrak{p}d(I)\ =\ \max\{\mathfrak{p}d(P),\mathfrak{p}d(Q),\mathfrak{p}d(P\cap Q)+1\}.
$$
We have $\mathfrak{p}d(P)\le1$. Indeed, $P$ is isomorphic to the final $t$--spread lexsegment ideal $(\mathcal{L}_t(u/x_1,x_{n-(d-2)t}\cdots x_{n-t}x_n))$ generated in degree $d-1$, and formula (\ref{Bettinumbersreversetspread}) gives
$$
\mathfrak{p}d(I)=\begin{cases}
n-(n-(d-2)t-1)-(d-2)t=1&\textup{if}\ u>u_{d-1},\\
n-(n-(d-2)t)-(d-2)t=0&\textup{if}\ u=u_{d-1}.
\end{cases}
$$
Analogously, as $Q$ is an initial $t$--spread lexsegment ideal in $K[x_2,\dots,x_n]$, $\mathfrak{p}d(Q)=0$ if $v=v_1$, or $\mathfrak{p}d(Q)=1$ if $v=v_2$. Thus, (\ref{eq:pd(I)j1=2BettiSplit}) implies that $\mathfrak{p}d(I)=1$ if and only if $\mathfrak{p}d(P\cap Q)=0$, \emph{i.e.}, if and only if $P\cap Q$ is a principal ideal.
\end{proof}
The next result points out that for $n\gg0$, precisely $n\ge 4+(2d-3)t$, the Cohen--Macaulay $t$--spread lexsegment ideals with $j_1=2$ are just complete intersection ideals.
\begin{Thm} \label{thm:unmixed}
Let $u=x_{i_1}x_{i_2}\dots x_{i_d}$ and $v=x_{j_1}x_{j_2}\cdots x_{j_d}$ be $t$--spread monomials of degree $d\ge 2$ of $S$ with $i_1=1$, $j_1=2$, and let $I$ be the $t$--spread lexsegment ideal $I=(\mathcal{L}_t(u,v))$ of $S$. Assume $n\ge 4+(2d-3)t$, then the following conditions are equivalent:
\begin{enumerate}
\item[\textup{(a)}] $I$ is Cohen--Macaulay;
\item[\textup{(b)}] $u=x_{1}x_{n-(d-2)t}\cdots x_{n-t}x_n$, $v=x_{2}x_{2+t}\cdots x_{2+(d-1)t}$.
\end{enumerate}
\end{Thm}
\begin{proof}
Let $\Delta$ be the simplicial complex on vertex set $[n]$ associated to $I$.
\\
(a)$\mathcal{L}ongrightarrow$(b). If $I=I_\Delta$ is Cohen--Macaulay, then $\Delta$ is pure. The set $[n]\setminus\{1,2\}$ is a facet of $\Delta$. Indeed, any monomial $w\in I$ has $1\in\supp(w)$ or $2\in\supp(w)$. Therefore, $\Delta$ pure implies $\dim(\Delta)=n-3$.
Consider the set $G=[n]\setminus\{2,i_2,i_2+1,\dots,n-(d-2)t\}$. Observe that $G\in\Delta$, indeed $\mathfrak{p}_{[n]\setminus G}=(x_2,x_{i_2},x_{i_2+1},\dots,x_{n-(d-2)t})$ is a monomial prime ideal that contains $I$. Now, the condition $\Delta$ is pure implies that there exists a minimal prime $\mathfrak{p}\in\Min(I)$ of height two such that $\mathfrak{p}\subseteq\mathfrak{p}_{[n]\setminus G}$. The $t$--spread monomial $w=x_2x_{2+t}\cdots x_{2+(d-1)t}\in I$ as $u>w\ge v$. We have $x_2\in\mathfrak{p}$. In fact, if $x_2\notin\mathfrak{p}$, then $\mathfrak{p}\subseteq\mathfrak{p}_{[i_2,n-(d-2)t]}$. We must have $\mathfrak{p}=\mathfrak{p}_{[i_2,n-(d-2)t]}$. Indeed, if we omit some $x_j$, $j\in[i_2,n-(d-2)t]$, then we can find a monomial $z\in G(I)$, with $\min(z)=1$ and $\min(z/x_1)=j$, and this monomial does not belong to $\mathfrak{p}$, a contradiction. $\Delta$ pure implies $\textup{height}(\mathfrak{p})=2$, thus $i_2=n-(d-2)t-1$. But $[n-(d-2)t-1,n-(d-2)t]\cap\supp(w)=\emptyset$. Indeed, $n\ge 4+(2d-3)t$, so
\begin{equation}\label{eq:calc4+(2d-3)t}
\begin{aligned}
n-(d-2)t-2\ &\ge\ 4+(2d-3)t-(d-2)t-2=2+(d-1)t\\
&=\ \max(w).
\end{aligned}
\end{equation}
Thus $\max(w)<n-(d-2)t-1$. This implies that $w\in I\setminus\mathfrak{p}$, a contradiction. Hence, $x_2\in\mathfrak{p}$. Therefore, $\textup{height}(\mathfrak{p})=2$ and $\mathfrak{p}=(x_2,x_{i_2})$. The monomial $z=x_{1}x_{n-(d-2)t}\cdots x_{n-t}x_n$ belongs to $I$. We have $z\in\mathfrak{p}$ if and only if $x_{i_2}$ divides $z$. Thus, as $1+t\le i_2\le n-(d-2)t$, we have $i_2=n-(d-2)t$ and $u=x_{1}x_{n-(d-2)t}\cdots x_{n-t}x_n$.\\
Let $r=\max\{\max(w):w\in G(I)\,\, \mbox{and $\min(w)=2$}\}$. Observe that $r\ge 2+(d-1)t$, as $x_2x_{2+t}\cdots x_{2+(d-1)t}\in I$. The ideal $\mathfrak{q}=(x_1,x_{2+(d-1)t},x_{3+(d-1)t},\dots,x_{r})$ is a minimal prime of $I$. The condition $\Delta$ pure implies $\textup{height}(\mathfrak{q})=2$ and $r=2+(d-1)t$. Thus, $\max(v)=2+(d-1)t$ and $v=x_{2}x_{2+t}\cdots x_{2+(d-1)t}$.
\\
(b)$\mathcal{L}ongrightarrow$(a). Let $u=x_1x_{n-(d-2)t}\cdots x_{n-t}x_{n}$ and $v=x_2x_{2+t}\cdots x_{2+(d-1)t}$. By (\ref{eq:calc4+(2d-3)t}), we have $\supp(u)\cap\supp(v)=\emptyset$. Moreover $I=(u,v)$. Thus, $I$ is a complete intersection and so it is Cohen--Macaulay.
\end{proof}
\begin{Rem}
\em One can observe that under the hypotheses of Theorem \ref{thm:unmixed}, the classification for Cohen--Macaulay $t$--spread lexsegment ideal is equivalent to the classification for unmixed $t$--spread lexsegment ideal (see also, Remark \ref{rem:unmixed}).
\end{Rem}
\subsection{The general case}
In this subsection, we classify all Cohen--Macaulay $t$--spread lexsegment ideals $I=(\mathcal{L}_t(u,v))$ of $S= K[x_1, \ldots, x_n]$ with $\min(v)>2$.
\subsubsection{The underlying idea behind the classification}
Let $I=(\mathcal{L}_t(u,v))$ be a Cohen--Macaulay $t$--spread lexsegment ideal of $S$ with $u=x_{i_1}x_{i_2}\dots x_{i_d}$, $v=x_{j_1}x_{j_2}\cdots x_{j_d}\in M_{n, d, t}$ such that
$\min(u)=i_1=1$, $\min(v)=j_2>2$, $d\ge2$. If $\Delta$ is the simplicial complex on the vertex set $[n]$ associated to $I$, then $\Delta$ is pure. This is a key property in order to get the classification.
First, note that since $\min(u)=1$, then
$$u\in\mathcal{L}_t(x_1x_{1+t}\cdots x_{1+(d-1)t},x_1x_{n-(d-2)t}x_{n-(d-3)t}\cdots x_{n-t}x_{n}).$$
Assume $u=x_1x_{1+t}\cdots x_{1+(d-1)t} = \max(M_{n, d, t})$, then $I=(\mathcal{L}_t(u,v))=(\mathcal{L}_t^i(v))$ is an initial $t$--spread lexsegment ideal. Hence, $\Delta$ is Cohen--Macaulay if and only if $I=I_{n,d,t}$ (Corollary \ref{cor:JCohenMac}). \\
Now, let $u<x_1x_{1+t}\cdots x_{1+(d-1)t}$. Then, $F=[1+(d-1)t]\in\Delta$. Moreover, $[1+(d-1)t]\cup\{k\}\notin\Delta$, for all $k\notin[1+(d-1)t]$. In fact, $j_1>2$ implies that $w=x_2x_{2+t}\cdots x_{2+(d-2)t}x_k\in I=I_\Delta$, thus $G=\supp(w)\notin\Delta$ and $G\subseteq F\cup\{k\}$. Therefore, $F$ is a facet of $\Delta$ and by the purity of $\Delta$ we have that
\begin{equation}\label{eq:dim(D)tLex(d-1)t}
\dim(\Delta)=|F|-1=(d-1)t.
\end{equation}
We show that $v\le x_{n-1-(d-1)t}x_{n-1-(d-2)t}\cdots x_{n-1-t}x_{n-1}$. Suppose on the contrary that $v> x_{n-1-(d-1)t}x_{n-1-(d-2)t}\cdots x_{n-1-t}x_{n-1}$, then $A=[n-1-(d-1)t,n]$ is a face of $\Delta$ and $\dim(\Delta)\ge\dim A=|A|-1=(d-1)t+1>(d-1)t=\dim(\Delta)$, a contradiction. Thus, $v\le x_{n-1-(d-1)t}x_{n-1-(d-2)t}\cdots x_{n-1-t}x_{n-1}$. Hence,
$$v \in \{v_0>v_1>\dots>v_{d-1}>v_d=\min(M_{n,d,t})\},$$
where
\begin{equation}\label{eq:v_ellmons}
v_\ell=\Big(\mathfrak{p}rod_{s=\ell}^{d-1}x_{n-st-1}\Big)\Big(\mathfrak{p}rod_{s=0}^{\ell-1}x_{n-st}\Big), \mathfrak{q}uad \ell\in\{0,\dots,d\}.
\end{equation}
We can observe that $v\ne v_d$. Otherwise, $I=(\mathcal{L}_t(u,v))=(\mathcal{L}_t^f(u))$ is a final $t$--spread lexsegment ideal and so $I$ is Cohen--Macaulay if and only if $I$ is the $t$--spread Veronese ideal $I_{n,d,t}$ (Corollary \ref{cor:TCohenMac}). Thus $u=\max(M_{n,d,t})$, against the assumption that $u<\max(M_{n,d,t})$.\\
Finally, for $v$ there do exist three admissible choices:
\begin{enumerate}
\item[-] $v=v_{d-1}$,
\item[-] $t=1$ and $v=v_\ell$, for $\ell\in\{0,\dots,d-2\}$,
\item[-] $t>1$ and $v=v_\ell$, for $\ell\in\{0,\dots,d-2\}$.
\end{enumerate}
In particular, if $v=v_\ell=\big(\textstyle\mathfrak{p}rod_{s=\ell}^{d-1}x_{n-st-1}\big)\big(\mathfrak{p}rod_{s=0}^{\ell-1}x_{n-st}\big)$ ($\ell\in\{0,\dots,d-2\}$), then
\begin{equation}\label{eq:importset2}
H=[n-(d-1)t-1,n]\setminus\{n-\ell t-1\}\in \Delta.
\end{equation}
Moreover, $|H|=1+(d-1)t$ and $\Delta$ pure implies that $H\cup\{1\}\notin\Delta$, \emph{i.e.}, ${\bf x}_{H\cup\{1\}}\in I=I_\Delta$.
\subsubsection{The classification}
Now we are in position to prove the main result in the article.
\begin{Thm}\label{Teor:ItLexCM>2}
Let $u=x_{i_1}x_{i_2}\dots x_{i_d}$, $v=x_{j_1}x_{j_2}\cdots x_{j_d}$ be $t$--spread monomials of degree $d\ge 2$ of $S$
with $i_1=1$, $j_1>2$ and let $I$ be the $t$--spread lexsegment ideal $I=(\mathcal{L}_t(u,v))$ of $S$. Assume $I\ne I_{n,d,t}$. Then $I$ is Cohen--Macaulay if and only if one of the following conditions holds true
\begin{enumerate}
\item[\textup{(a)}] $u=x_1x_{n-(d-2)t}\cdots x_{n-t}x_n$ and $v=x_{n-(d-1)t-1}\big(\textstyle\mathfrak{p}rod_{s=0}^{d-2}x_{n-st}\big)$;
\item[\textup{(b)}] $t=1$ and for some integer $\ell\in\{0,\dots,d-2\}$,
\begin{align*}
u&=x_1\big(\textstyle\mathfrak{p}rod_{s=\ell+1}^{d-2}x_{n-s-1}\big)\big(\mathfrak{p}rod_{s=0}^{\ell}x_{n-s}\big),\ \ \ \ &v=\big(\textstyle\mathfrak{p}rod_{s=\ell}^{d-1}x_{n-s-1}\big)\big(\mathfrak{p}rod_{s=0}^{\ell-1}x_{n-s}\big);
\end{align*}
\item[\textup{(c)}] $t>1$ and for some integer $\ell\in\{0,\dots,d-2\}$,
\begin{align*}
u&\in\mathcal{L}_t\big(x_1\big(\textstyle\mathfrak{p}rod_{s=\ell}^{d-2}x_{n-st-1}\big)\big(\mathfrak{p}rod_{s=0}^{\ell-1}x_{n-st}\big),x_1\big(\mathfrak{p}rod_{s=0}^{d-2}x_{n-st}\big)\big),\\
v&=\big(\textstyle\mathfrak{p}rod_{s=\ell}^{d-1}x_{n-st-1}\big)\big(\mathfrak{p}rod_{s=0}^{\ell-1}x_{n-st}\big).
\end{align*}
\end{enumerate}
\end{Thm}
\begin{proof} (a) Let $v=v_{d-1}=x_{n-(d-1)t-1}\big(\mathfrak{p}rod_{s=0}^{d-2}x_{n-st}\big)$.
The only monomial $w\in M_{n,d,t}$ with $w< v$ is $w=\min(M_{n,d,t})=\big(\textstyle\mathfrak{p}rod_{s=0}^{d-1}x_{n-st}\big)$. Since $x_1(w/x_{n-(d-1)t})\le u$, then $I$ is a completely $t$--spread lexsegment ideal (Theorem \ref{thm:compltspreadlex}(2)). Moreover, $I$ has a linear resolution (Theorem \ref{thm:compltspreadlex}(2)). Indeed, setting $u=x_{1}x_{i_2}\cdots x_{i_d}$, we have $i_2>1+t$ and
$$
x_1x_{i_2-t}x_{i_3-t}\cdots x_{i_d-t}\ge x_1x_{n-(d-1)t}x_{n-(d-2)t}\cdots x_{n-t}=x_1(w/x_{\max(w)}).
$$
Note that the Cohen--Macaulayness of $\Delta$ implies $\dim(S/I)=\depth(S/I)$. From (\ref{eq:dim(D)tLex(d-1)t}), $\dim(S/I)=n-\height(I)=1+(d-1)t = \depth(S/I)$.
On the other hand, by the Auslander--Buchsbaum formula, $\depth(S/I)=n-\mathfrak{p}d(S/I)=n-1-\mathfrak{p}d(I)$. Thus, $S/I$ is Cohen--Macaulay if and only if $\mathfrak{p}d(I)=n-(d-1)t-2$.
Setting $a=n-1-(d-1)t$ and $b=a+d$, we can note that $\mathfrak{p}d(I)\le n-(d-1)t-1$ if and only if (Theorem \ref{thm:compltspreadlex}(3))
\begin{align*}
\beta_{a}(I)= \beta_{a,b}(I)&=\sum_{w\in\mathcal{L}_t^f(u)}\binom{n-\min(w)-(d-1)t}{n-1-(d-1)t}-\sum_{\substack{w\in\mathcal{L}_t^f(v)\\ w\ne v}}\binom{\max(w)-(d-1)t-1}{n-1-(d-1)t}\\
&=\big|\big\{ w\in\mathcal{L}_t^f(u):\min(w)=1\big\}\big|-\big|\big\{w\in\mathcal{L}_t^f(v)\setminus\{v\}:\max(w)=n\big\}\big|\\
&=\big|\big\{ w\in\mathcal{L}_t^f(u):\min(w)=1\big\}\big|-1 = 0,
\end{align*}
\emph{i.e.}, $u=x_1\big(\mathfrak{p}rod_{s=0}^{d-2}x_{n-st}\big)$. Condition (a) follows.
Conversely, let $u=x_1x_{n-(d-2)t}\cdots x_{n-t}x_n$ and $v=x_{n-(d-1)t-1}\big(\textstyle\mathfrak{p}rod_{s=0}^{d-2}x_{n-st}\big)$. We prove that $\dim(S/I) =\depth(S/I)$.
First, using the same arguments as before, we can verify that $I$ is a completely $t$--spread lexsegment ideal which has a linear resolution.
Setting $a = n-(d-1)t-1$ and $c=a-1=n-(d-1)t-2$, we have $m=\big|\{w\in\mathcal{L}_t^f(u):\min(w)=2\}\big|>1$ and
\begin{align*}
\beta_{c}(I)&=\sum_{w\in\mathcal{L}_t^f(u)}\binom{n-\min(w)-(d-1)t}{n-2-(d-1)t}-\sum_{\substack{w\in\mathcal{L}_t^f(v)\\ w\ne v}}\binom{\max(w)-(d-1)t-1}{n-2-(d-1)t}
\\
&=\binom{a}{a-1}+m\binom{a-1}{a-1}-\binom{a}{a-1}=m>0.
\end{align*}
by Theorem \ref{thm:compltspreadlex}(3). Thus, $\mathfrak{p}d(I)=c=n-(d-1)t-2$. Hence, from the Auslander--Buchsbaum formula, $\depth(S/I)= 1+(d-1)t$.
Let us prove that $\dim(S/I) = 1+(d-1)t$. It is sufficient to verify that $\dim(\Delta)=(d-1)t$.
Using the notation of Theorem \ref{primdecompgeneraltspreadlex}, since $I$ is a completely $t$--spread lexsegment ideal, we have $\mathcal{F}(\Delta)=\mathcal{G}\cup\{[n]\setminus F_p:p\in\mathcal{I}\}\cup\widetilde{\mathcal{F}}$.
If $F\in\mathcal{G}$, then $|F|=1+(d-1)t$.
If $F\in\widetilde{\mathcal{F}}$, then $|F|=(d-1)t<1+(d-1)t$. Finally, we check that $\mathcal{I}=\{1\}$ and $|[n]\setminus F_1|=1+(d-1)t$. Indeed, for $p=1$, $[n]\setminus F_1=[j_1+1,n]=[n-(d-1)t,n]$ has cardinality $1+(d-1)t$, so $1\in\mathcal{I}$. Let $2\le p\le d$, then
\begin{align*}
[n]\setminus F_p&=\bigcup_{i=1}^{p-1}[j_i,j_i+(t-1)]\cup[j_p+1,n]\\
&=\bigcup_{i=1}^{p-1}[j_i,j_i+(t-1)]\cup[n-(d-p)t+1,n]
\end{align*}
has cardinality $(p-1)t+n+1-(n-(d-p)t+1)=(d-1)t$ and moreover,
$$
\min(v/x_{j_p})=\min(v)=n-(d-2)t-1<n-(d-2)t-1\le\min(u/x_1).
$$
Thus $v/x_{j_p}>u/x_1$ and $p\notin\mathcal{I}$ and so $\mathcal{I}=\{1\}$ and $|[n]\setminus F_1|=1+(d-1)t$. Hence, $\dim(\Delta)=(d-1)t$ and consequently $\dim(S/I)=n-\height(I)=1+(d-1)t = \depth(S/I)$ and $I$ is Cohen--Macaulay.\\\\
(b) Let $t=1$. Let us consider the set $H$ defined in (\ref{eq:importset2}). We note that the smallest $1$--spread (squarefree) monomial of degree $d$ that we can ``extract" from $H\cup\{1\}=\{1,n-d,n-d+1,\dots,n-\ell-2,n-\ell,\dots,n-1,n\}$ is
\begin{align*}
z&=x_1x_{n-d+1}\cdots x_{n-\ell-2}\cdot x_{n-\ell}\cdots x_{n-1}x_n\\
&=x_1\big(\textstyle\mathfrak{p}rod_{s=\ell+1}^{d-2}x_{n-s-1}\big)\big(\mathfrak{p}rod_{s=0}^{\ell}x_{n-s}\big).
\end{align*}
Thus, we must have $z\in I$, and so $u\ge z$, \emph{i.e.},
$$
u\in\mathcal{L}_t\big(x_1\big(\textstyle\mathfrak{p}rod_{s=\ell}^{d-2}x_{n-s-2}\big)\big(\mathfrak{p}rod_{s=0}^{\ell-1}x_{n-s-1}\big),x_1\big(\mathfrak{p}rod_{s=\ell}^{d-2}x_{n-s-1}\big)\big(\mathfrak{p}rod_{s=0}^{\ell+1}x_{n-s}\big)\big).
$$
Let $v=v_\ell$, for some $\ell\in \{0, \ldots, d-2\}$. The monomials $w\in M_{n,d,1}$ with $w< v$ are $v_{\ell+1}>\dots>v_d$. Since, for $r=\ell+1,\dots,d$, $$x_1(v_r/x_{n-(d-1)})\le x_1\big(\textstyle\mathfrak{p}rod_{s=\ell+1}^{d-2}x_{n-s-1}\big)\big(\mathfrak{p}rod_{s=0}^{\ell}x_{n-s}\big)\le u,$$ $I$ is a completely $1$--spread lexsegment ideal. Moreover, $I$ has a linear resolution. Indeed, setting $u=x_{1}x_{i_2}\cdots x_{i_d}$, we have $i_2>1+t=2$ and
$$
x_1x_{i_2-1}x_{i_3-1}\cdots x_{i_d-1}\ge x_1\big(\textstyle\mathfrak{p}rod_{s=\ell+1}^{d-2}x_{n-s-2}\big)\big(\mathfrak{p}rod_{s=0}^{\ell}x_{n-s-1}\big)> x_1(v_{\ell+1}/x_{\max(w)}).
$$
From (\ref{eq:dim(D)tLex(d-1)t}), $\dim(S/I) = 1+(d-1)t = d$ and, since we are assuming $I$ Cohen--Macaulay, then $\depth(I) =d$.
On the other hand from the Auslander--Buchsbaum formula, $S/I$ is Cohen--Macaulay if and only if $\mathfrak{p}d(I)=n-d-1$.
Setting $a=n-d$ and $b=a+d$,
$\mathfrak{p}d(I)\le n-d$,
$S/I$ is Cohen--Macaulay if and only if
\begin{align*}
\beta_{a}(I)=\beta_{a,b}(I)&=\big|\big\{ w\in\mathcal{L}_t^f(u):\min(u)=1\big\}\big|-\big|\big\{w\in\mathcal{L}_t^f(v)\setminus\{v\}:\max(w)=n\big\}\big|\\
&=\big|\big\{ w\in\mathcal{L}_t^f(u):\min(w)=1\big\}\big|-\big|\big\{v_{\ell+1}>\dots>v_d\big\}\big|\\
&=\big|\big\{ w\in\mathcal{L}_t^f(u):\min(w)=1\big\}\big|-(d-\ell)=0,
\end{align*}
\emph{i.e.}, $u=x_1\big(\textstyle\mathfrak{p}rod_{s=\ell+1}^{d-2}x_{n-s-1}\big)\big(\mathfrak{p}rod_{s=0}^{\ell}x_{n-s}\big)$. The assertion (b) follows.
Conversely, let $u=x_1\big(\textstyle\mathfrak{p}rod_{s=\ell+1}^{d-2}x_{n-s-1}\big)\big(\mathfrak{p}rod_{s=0}^{\ell}x_{n-s}\big)$ and $v=v_\ell$, for some $\ell$ $\in\{0,\dots,d-2\}$.
We compute $\dim(S/I)$ and $\depth(S/I)$.
Using the notation of Theorem \ref{primdecompgeneraltspreadlex}, $\mathcal{F}(\Delta)=\mathcal{G}\cup\{[n]\setminus F_p:p\in\mathcal{I}\}\cup\widetilde{\mathcal{F}}$. We show that $\dim(\Delta)= d$.
If $F\in\mathcal{G}$, then $|F|=d$.
Arguing as in the proof of condition (b), one can observe that $\mathcal{I}=[d-\ell]$ and for $p\in\mathcal{I}$, $|[n]\setminus F_p|=d$.
Furthermore, for any $F\in{\mathcal{F}}$, $|F|=d-1<d$. Hence $\dim(\Delta)=d$
and so $\dim(S/I)=d$.\\
Setting $a=n-d$
and $c=a-1=n-d-1$,
we have $\big|\{w\in \mathcal{L}_t^f(u):\min(w)=1\}\big|=d-\ell$, $m=\big|\{w\in\mathcal{L}_t^f(u):\min(w)=2\}\big|>1$ and
\begin{align*}
\beta_{c}(I)&=\sum_{w\in\mathcal{L}_t^f(u)}\binom{n-\min(w)- d+1}{n-d-1}-\sum_{\substack{w\in\mathcal{L}_t^f(v)\\ w\ne v}}\binom{\max(w)-d}{n-d-1}
\\
&=(d-\ell)\binom{a}{a-1}+m\binom{a-1}{a-1}-(d-\ell)\binom{a}{a-1}=m>0.
\end{align*}
Thus, $\mathfrak{p}d(I)=n-d-1$
and from the Auslander--Buchsbaum formula $\depth(I)=d$ and $I$ is Cohen--Macaulay.
\end{proof}
For the proof of the last case (c), we need some more preparations.
First, recall that if $0\rightarrow L\rightarrow M\rightarrow N\rightarrow0$ is a short exact sequence of finitely generated graded $S$--modules, then $\mathfrak{p}d(M)\le\max\{\mathfrak{p}d(L),\mathfrak{p}d(N)\}$. Moreover, given a monomial ideal $I$ of $S$ and a monomial $u$ of $S$, there exists the short exact sequence
$$
0\rightarrow S/(I:u)\rightarrow S/I\rightarrow S/(I,u)\rightarrow0.
$$
Therefore $\mathfrak{p}d(S/I)\le\max\{\mathfrak{p}d(S/(I:u)),\mathfrak{p}d(S/(I,u))\}$.
\begin{Lem}\label{lem:pdJ<I}
Let $I,J$ equigenerated monomial ideals of $S$ having initial degree $d$. Suppose $J\subseteq I$, then
$$
\beta_{i,i+d}(J)\le\beta_{i,i+d}(I),
$$
for all $i\ge0$. Moreover, if $J$ has a linear resolution, $\mathfrak{p}d(I)\ge\mathfrak{p}d(J)$.
\end{Lem}
\begin{proof} The first assertion follows from \cite[Lemma 10.3.5]{JT}. For the second assertion, assume $J$ has a linear resolution, and let $\mathfrak{p}d(J)=s$. Then $\beta_s(J)=\beta_{s,s+d}(J)\le\beta_{s,s+d}(I)$. Finally, $\mathfrak{p}d(I)\ge s=\mathfrak{p}d(J)$.
\end{proof}
Now we are in position to prove the last statement of Theorem \ref{Teor:ItLexCM>2}.\\\\
(c) Let $t\ge2$. From (\ref{eq:importset2}), the smallest $t$--spread monomial we can ``extract" from $H\cup\{1\}$ is
$$
z=x_1x_{n-(d-2)t}x_{n-(d-3)t}\cdots x_{n-t}x_n.
$$
Since $z$ is the smallest $t$--spread monomial of $S$ with minimum equal to $1$, then $z\in I$, and there are no further conditions to impose on the monomial $u$. Thus
\begin{equation}\label{eq:Setcase(d)}
u\in\mathcal{L}_t\big(x_1\big(\textstyle\mathfrak{p}rod_{s=\ell}^{d-2}x_{n-st-2}\big)\big(\mathfrak{p}rod_{s=0}^{\ell-1}x_{n-st-1}\big),x_1\big(\mathfrak{p}rod_{s=0}^{d-2}x_{n-st}\big)\big).
\end{equation}
Setting $$
u_p=x_1\Big(\mathfrak{p}rod_{s=p}^{d-2}x_{n-st-1}\Big)\Big(\mathfrak{p}rod_{s=0}^{p-1}x_{n-st}\Big), \,\,\, p=\ell,\dots,d-1,
$$
we have that $$\mathcal{L}_t\big(x_1\big(\textstyle\mathfrak{p}rod_{s=\ell}^{d-2}x_{n-st-1}\big)\big(\mathfrak{p}rod_{s=0}^{\ell-1}x_{n-st}\big),x_1\big(\mathfrak{p}rod_{s=0}^{d-2}x_{n-st}\big)\big)=\big\{u_\ell>u_{\ell+1}>\dots>u_{d-1}\big\}.$$
We prove that if
\begin{equation}\label{eq:noCM}
u >u_\ell=x_1\big(\textstyle\mathfrak{p}rod_{s=\ell}^{d-2}x_{n-st-1}\big)\big(\mathfrak{p}rod_{s=0}^{\ell-1}x_{n-st}\big),
\end{equation}
then $I$ is not Cohen--Macaulay.
By hypothesis $v=v_\ell$ ($\ell =0, \ldots, d-2$) and from (\ref{eq:noCM})
$$
u\ge x_1\big(\textstyle\mathfrak{p}rod_{s=\ell-1}^{d-2}x_{n-st-1}\big)\big(\mathfrak{p}rod_{s=0}^{\ell-2}x_{n-st}\big),
$$
where $x_1\big(\textstyle\mathfrak{p}rod_{s=\ell-1}^{d-2}x_{n-st-1}\big)\big(\mathfrak{p}rod_{s=0}^{\ell-2}x_{n-st}\big)$ is the greatest monomial with minimum equal to $1$ preceding $u_\ell$, with respect to the squarefree lex order. The monomials $w\in M_{n,d,t}$ smaller than $v$ are $v_{\ell+1}>\dots>v_d$ and for each $v_q$, $q=\ell+1, \ldots, d$, we have
$$
u\ge x_1\big(\textstyle\mathfrak{p}rod_{s=\ell-1}^{d-2}x_{n-st-1}\big)\big(\mathfrak{p}rod_{s=0}^{\ell-2}x_{n-st}\big)=x_1(v_{\ell+1}/x_{\min(v_{\ell+1})})>x_1(v_q/x_{\min(v_q)}).
$$
Hence, setting $u=x_1x_{i_2}\cdots x_{i_d}$, we have $i_2>1+t$ and
\begin{align*}
x_1x_{i_2-t}\cdots x_{i_d-t}&\ge x_1x_{n-(d-1)t-1}\cdots x_{n-\ell t-1}\cdot x_{n-(\ell-1)t}\cdots x_{n-t}\\
&=x_1(v_{\ell}/x_{\max(v_\ell)})\ge x_1(v_{\ell+1}/x_{\max(v_{\ell+1})}).
\end{align*}
Therefore, $I$ is a completely $t$--spread lexsegment ideal with a linear resolution. Setting $a=n-1-(d-1)t$ and $b=a+d$, we have
\begin{align*}
\beta_{a}(I)=\beta_{a,b}(I)&=\big|\big\{ w\in\mathcal{L}_t^f(u):\min(u)=1\big\}\big|-\big|\big\{w\in\mathcal{L}_t^f(v)\setminus\{v\}:\max(w)=n\big\}\big|\\
&\ge\big|\big\{ u>u_{\ell}>\dots>u_{d-1}\}\big|-(d-\ell)= d+1-\ell-(d-\ell)= 1.
\end{align*}
It follows that $\mathfrak{p}d(I)=n-1-(d-1)t$ and consequently $I$ is not Cohen--Macaulay. Condition (c) follows.
Conversely, assume (c) holds. We prove that $\dim(S/I)=1+(d-1)t$. This is equivalent to verify that $\dim(\Delta)=(d-1)t$. For this aim it sufficies to show the following facts:
\begin{enumerate}
\item[(i)] there exists $F\in\mathcal{F}(\Delta)$ with cardinality $|F|=1+(d-1)t$;
\item[(ii)] for any $A\subseteq[n]$ with $|A|=2+(d-1)t$, then $A\notin\Delta$.
\end{enumerate}
\noindent
(i) Let us consider $F=[n-(d-1)t,n]$. By the structure of $v$, it follows that $F\in\Delta$ and furthermore $|F|=1+(d-1)t$, as desired.
\\
(ii) Let $A\subseteq[n]$ with $|A|=2+(d-1)t$. We have $1\le\min(A)\le n-(d-1)t-1$.
If $\min(A)=n-(d-1)t-1$, then $A=[n-(d-1)t-1,n]\notin\Delta$, as
$$
x_{n-(d-1)t-1}x_{n-(d-2)t-1}\cdots x_{n-t-1}x_{n-1}\in I.
$$
If $1<\min(A)<n-(d-1)t-1$, setting $\ell_1=\min(A)$ and $\ell_r=\min\{\ell\in A:\ell\ge\ell_{r-1}+t\}$, for $r\ge 2$, the sequence $L:\ell_1>\ell_2>\cdots$ has at least $d$ terms, and ${\bf x}_L\in I$. Thus, $A\notin\Delta$.
If $\min(A)=1$, we consider the sequence $L$ above defined. If ${\bf x}_L\le u$, then $A\notin\Delta$. Otherwise, set $B=A\setminus\{1\}$.
If $\min(B)=n-(d-1)t$, then $A=\{1\}\cup B=\{1\}\cup[n-(d-1)t,n]\notin\Delta$, as $x_1x_{n-(d-2)t}\cdots x_{n-t}x_{n}\in I$. If $\min(B)<n-(d-1)t-1$, setting $\nu_1=\min(B)$, and $\nu_r=\min\{\nu\in B:\nu\ge\nu_{r-1}+t\}$, for $r\ge 2$, the sequence $N:\nu_1>\nu_2>\cdots$ has at least $d$ terms, and ${\bf x}_N\in I$, implying that $A\notin\Delta$.
Let $\min(B)=n-(d-1)t-1$. Then $A=\{1\}\cup B=\{1\}\cup[n-(d-1)t-1,n]\setminus\{m\}$, for some $m>n-(d-1)t-1$.
We can write $m=n-pt-q$, with $0\le p\le d-3$ and $0\le q\le t-1$. If $q\ge2$, then $\supp(v)\subseteq B\subseteq A$, and $A\notin\Delta$. If $q=1$, then $\{1,n-(d-2)t,\dots,n-t,n\}\subseteq A$, and $x_1x_{n-(d-2)t}\cdots x_{n-t}x_n\in I$, thus $A\notin\Delta$. If $q=0$, we consider $p$. If $p<\ell-1$, the structure of $v$ implies that $\supp(v)\subseteq A$, and once again $A\notin\Delta$. If $p\ge\ell-1$, then $C=\{n-kt-1:k=0,\dots,d-1\}\subseteq A$, and ${\bf x}_C>v$, so ${\bf x}_C\in I$, implying $A\notin\Delta$. Finally, in all possible cases, $A\notin\Delta$, and (ii) follows.
Finally, $\dim(S/I)=1+(d-1)t$.
Now, set $I_{u_p}=(\mathcal{L}_t(u_p,v))$, $p=\ell,\dots,d-1$.
We prove that $I_{u_{p}}$ is Cohen--Macaulay,
\emph{i.e.}, $\mathfrak{p}d(I_{u_p})=n-2-(d-1)t$.
First, assume $p=\ell$. We have
$$
u=u_\ell=x_1(v_\ell/x_{\min(v_\ell)})>x_1(v_q/x_{\min(v_q)}),\,\,\, q=\ell+1,\dots,d
$$
and setting
$u=x_1x_{i_2}\cdots x_{i_d}$, it follows that
$$
x_1x_{i_2-t}\cdots x_{i_d-t}=x_1(v_{\ell}/x_{\max(v_\ell)}).
$$
Thus, $I_{u_\ell}$ is a completely $t$--spread lexsegment ideal with a linear resolution. Set $a=n-1-(d-1)t$, $b=a+d$, we have
\begin{align*}
\beta_{a}(I_{u_\ell})=\beta_{a,b}(I_{u_\ell})&=\big|\big\{ w\in\mathcal{L}_t^f(u):\min(u)=1\big\}\big|-\big|\big\{w\in\mathcal{L}_t^f(v)\setminus\{v\}:\max(w)=n\big\}\big|\\
&=\big|\big\{u_\ell>u_{\ell+1}>\dots >u_{d-1}\}\big|-|\{v_{\ell+1}>\dots>v_d\}|=0,
\end{align*}
and, as $m=\big|\{w\in\mathcal{L}_t^f(u):\min(w)=2\}\big|>1$, then
\begin{align*}
\beta_{a-1}(I_{u_\ell})&=\sum_{w\in\mathcal{L}_t^f(u)}\binom{n-\min(w)-(d-1)t}{n-2-(d-1)t}-\sum_{\substack{w\in\mathcal{L}_t^f(v)\\ w\ne v}}\binom{\max(w)-(d-1)t-1}{n-2-(d-1)t}
\\
&=(d-\ell)\binom{a}{a-1}+m\binom{a-1}{a-1}-(d-\ell)\binom{a}{a-1}=m>0.
\end{align*}
Thus, $\mathfrak{p}d(I_{u_\ell})=a-1=n-(d-1)t-2$, and $I_{u_\ell}$ is Cohen--Macaulay, as desired.
Now let $p=\ell+1$. Firstly, observe that $J=(\mathcal{L}_t(x_2x_{2+t}\cdots x_{2+(d-1)t},v))\subseteq I_{u_{\ell+1}}$. Moreover, $J$ is an initial $t$--spread lexsegment ideal in $K[x_2,\dots,x_{n}]$ with $\mathfrak{p}d(J)=n-2-(d-1)t$. Thus $J$ has a linear resolution, and Lemma \ref{lem:pdJ<I} implies $\mathfrak{p}d(I_{u_{\ell+1}})\ge n-2-(d-1)t$. For the other inequality, consider the short exact sequence:
$$
0\rightarrow S/(I_{u_{\ell+1}}:u_{\ell})\rightarrow S/I_{u_{\ell+1}}\rightarrow S/(I_{u_{\ell+1}},u_\ell)\rightarrow0.
$$
Observe that $S/(I_{u_{\ell+1}},u_\ell)=S/I_{u_{\ell}}$, and $\mathfrak{p}d(S/I_{u_{\ell}})=\mathfrak{p}d(I_{u_{\ell}})+1=n-1-(d-1)t$.
Let us verify that $I_{u_{\ell+1}}:u_\ell=\mathfrak{p}_{[2,n-(d-1)t-1]\cup\{n-\ell t\}}$ is a complete intersection and $\mathfrak{p}d(S/(I_{u_{\ell+1}}:u_\ell))=n-1-(d-1)t$.
By \cite[Proposition 1.2.2]{JT}, a set of generators for $I_{u_{\ell+1}}:u_\ell$ is given by $$\big\{w/\gcd(w,u_\ell):w\in G(I_{u_{\ell+1}})\big\}.$$
If $w=u_p$, $p=\ell+1,\dots,d-1$, we have $u_{p}/\gcd(u_{p},u_\ell)=x_{n-(p-1)t}\cdots x_{n-\ell t}$ and for $p=\ell+1$, $u_{\ell+1}/\gcd(u_{\ell+1},u_\ell)=x_{n-\ell t}$, thus $x_{n-\ell t}$ divides all these generators.
If $w\in\mathcal{L}_t(x_2x_{2+t}\cdots x_{2+(d-1)t},v)$, then $\min(w)\in[2,n-(d-1)t-1]$. Note that $[2,n-(d-1)t-1]\cap\supp(u_\ell)=\emptyset$, thus $x_{\min(w)}$ divides $w/\gcd(w,u_\ell)$ and for $z=x_{\min(w)}(u_\ell/x_{1})\in G(I_{u_{\ell+1}})$, we have that $z/\gcd(z,u_\ell)=x_{\min(w)}\in I_{u_{\ell+1}}:u_\ell$.
Finally, we have verified that $I_{u_{\ell+1}}:u_\ell=\mathfrak{p}_{[2,n-(d-1)t-1]\cup\{n-\ell t\}}$ and consequently $\mathfrak{p}d(S/(I_{u_{\ell+1}}:u_\ell))=n-1-(d-1)t$.
Hence,
$$
\mathfrak{p}d(S/I_{u_{\ell+1}})\le\max\big\{ \mathfrak{p}d(S/I_{u_{\ell}}), \mathfrak{p}d(S/(I_{u_{\ell+1}}:u_\ell))\big\}=n-1-(d-1)t,
$$
so $\mathfrak{p}d(I_{u_{\ell+1}})\le n-2-(d-1)t$. Thus $\mathfrak{p}d(I_{u_{\ell+1}})=n-2-(d-1)t$ and $I_{u_{\ell+1}}$ is Cohen--Macaulay.
For $p\in \{\ell+2,\dots,d\}$ the same arguments work and the proof of Theorem \ref{Teor:ItLexCM>2} is complete.
\section{Conclusions and Perspectives}\label{sec4}
In this article, we have investigated the Cohen--Macaulayness of $t$--spread lexsegment ideals using the theory of simplicial complexes. The completely $t$--spread lexsegment ideals have played an essential role. Non--completely $t$--spread lexsegment ideals are much less understood. Here are some possible questions to investigate.
\begin{Que}
\rm Determine the standard primary decomposition of non--completely $t$--spread lexsegment ideals.
\end{Que}
\begin{Que}
\rm Classify the pure simplicial complexes associated to $t$--spread lexsegment ideals.
\end{Que}
In \cite{OO}, Olteanu classified all \textit{sequentially Cohen--Macaulay} squarefree completely lexsegment ideals. To the best of our knowledge, a classification for squarefree non--completely lexsegment ideals is unknown. More generally, we ask the following
\begin{Que}
\rm Classify all sequentially Cohen--Macaulay $t$--spread lexsegment ideals.
\end{Que}
Finally, due to many examples that we performed and our results on completely $t$--spread lexsegment ideals, we are led to conjecture the following:
\begin{Conj}\label{Conj:dimpdtspreadLex}
\rm Let $I$ be a $t$--spread lexsegment ideal generated in degree $d\ge2$. Then $\dim(S/I)\ge(d-1)t$.
\end{Conj}
Note that Conjecture \ref{Conj:dimpdtspreadLex} is trivially true when $t=1$. Indeed, if $\Delta$ is the simplicial complex associated to $I$, then each set $A\subseteq[n]$, with $|A|=d-1$, is in $\Delta$. Thus $\dim(S/I)\ge d-1$. Conjecture \ref{Conj:dimpdtspreadLex} is also true for completely $t$--spread lexsegment ideals by virtue of Theorems \ref{primdecompinitialtspreadlex}, \ref{primdecompfinaltspreadlex}, \ref{primdecompgeneraltspreadlex}.\\\\
\emph{Acknowledgement}. We thank the the referee for his/her helpful suggestions that allowed us to improve the quality of the paper.
\end{document} | math | 84,137 |
\begin{equation}gin{document}
\title{Measuring the purity of a qubit state:
entanglement estimation with fully separable measurements}
\author{E.~Bagan}
\affiliation{Grup de F{\'\i}sica Te{\`o}rica \& IFAE, Facultat de Ci{\`e}ncies,
Edifici Cn, Universitat Aut{\`o}noma de Barcelona, 08193 Bellaterra
(Barcelona) Spain}
\author{M.~A.~Ballester}
\affiliation{Department of Mathematics, University of Utrecht, Box
80010, 3508 TA Utrecht, The Netherlands}
\author{R.~Mu{\~n}oz-Tapia}
\affiliation{Grup de F{\'\i}sica Te{\`o}rica \& IFAE, Facultat de Ci{\`e}ncies,
Edifici Cn, Universitat Aut{\`o}noma de Barcelona, 08193 Bellaterra
(Barcelona) Spain}
\author{O.~Romero-Isart}
\affiliation{Grup de F{\'\i}sica Te{\`o}rica \& IFAE, Facultat de Ci{\`e}ncies,
Edifici Cn, Universitat Aut{\`o}noma de Barcelona, 08193 Bellaterra
(Barcelona) Spain}
\date{\today}
\begin{equation}gin{abstract}
Given a finite number $N$ of copies of a qubit state we compute
the maximum fidelity that can be attained using joint-measurement
protocols for estimating its purity. We prove that in the
asymptotic $N\to\infty$ limit, separable-measurement protocols can
be as efficient as the optimal joint-measurement one if classical
communication is used. This in turn shows that the optimal
estimation of the entanglement of a two-qubit state can also be
achieved asymptotically with fully separable measurements. Thus, quantum memories
provide no advantage in this situation.
The relationship between our global Bayesian approach and the quantum
Cram\'er-Rao bound is also discussed.
\end{abstract}
\!\mbox{conv}dot\!acs{03.67.Hk, 03.65.Ta}
\mbox{\scriptsize max}ketitle
The ultimate goal of quantum state estimation is to determine the
value of the parameters that fully characterize a given unknown
quantum state. However, in practical applications,
a partial characterization
is often all one needs. Thus, e.g., knowing the purity of a qubit
state or the degree of entanglement of a bipartite state may be
sufficient to determine whether it can perform some particular
task~\mbox{conv}ite{white} ---See Ref.~\mbox{conv}ite{gisin} for recent experimental progress on estimating the degree of polarization (the purity) of light beams. This paper concerns this type of situation.
To be more specific, assume we are given $N$ identical copies of
an unknown qubit mixed state $\rho({\bf v}ec r)$, so that the state of
the total system is $\rho^N({\bf v}ec r)\equiv[\rho({\bf v}ec r)]^{\otimes
N}$. The set of all such density matrices $\{\rho({\bf v}ec r)\}$ can
be mapped into the Bloch sphere ${\mbox{conv}al B}=\{{\bf v}ec r :\ r\equiv|{\bf v}ec
r|\le1\}$ through the relation $\rho({\bf v}ec r)=(\openone+{\bf v}ec
r\mbox{conv}dot{\bf v}ec\sigma)/2$, where
${\bf v}ec\sigma=(\sigma_x,\sigma_y,\sigma_z)$ is a vector made out of
the three standard Pauli matrices. Our aim is to estimate the purity, $r$, as
accurately as possible by performing suitable measurements on the
$N$ copies, i.e., on $\rho^N({\bf v}ec r)$. This problem can also be
viewed as the parameter estimation of a depolarizing
channel~\mbox{conv}ite{depolarizing} when it is fed with $N$ identical states.
The estimation protocols are broadly divided into two classes
depending on the type of measurements they use: joint and
separable. The former treats the system of $N$ qubits as a whole,
allowing for the most general measurements, and leads to the most
accurate estimates or, equivalently, to the largest fidelity
(properly defined below). The latter, treats each copy separately
but classical communication can be used in the measurement
process. This class is particularly important because it is
feasible with nowadays technology and it offers an economy of
resources. In this paper we show that for a sufficiently large
$N$, separable measurement protocols for purity estimation can
attain the optimal joint-measurement fidelity bound. The power of
separable measurement protocols in achieving optimal performance
has also been demonstrated in other
contexts~\mbox{conv}ite{us-local,others, discrim}.
It has been shown~\mbox{conv}ite{vidal} that given $N$ copies of a
bipartite qubit pure state, $|\Psi\rangle_{AB}$, the optimal
protocol for measuring its entanglement consists in estimating the
purity of $\rho({\bf v}ec r)\equiv{\rm
tr}_B(|\Psi\rangle_{AB}\langle\Psi|)$, where ${\rm tr}_B$ is the
partial trace over the Hilbert space of party $B$
(see~\mbox{conv}ite{susana,horodecki} for related work on bipartite mixed
states). We thus show that for {\em large $N$} this entanglement
can be optimally estimated by performing just {\em separable}
measurements on {\em one} party (party $A$ in this discussion) of
{\em each} of the $N$ copies of~$|\Psi\rangle_{AB}$.
Though many of our results here concern finite $N$, special
attention is paid to the asymptotic regime, when $N$ is large.
There are several reasons for this. First, in this limit, formulas
greatly simplify and usually reveal important features of the
estimation protocol. Second, the asymptotic theory of quantum
statistical inference, which has become in recent years a very
active field in mathematical statistics~\mbox{conv}ite{masahito-book},
deals with problems such as the one at hand. Our results give
support to some quantum statistical methods for which only
heuristic proofs exist; e.g., the applicability of the integrated
quantum Cram\'er-Rao bound in the Bayesian approach (which is
formulated below)~\mbox{conv}ite{us-prep}.
In the first part of this paper we obtain the optimal joint
estimation protocols and the corresponding fidelity bounds. In
addition to the general case of states in $\mbox{conv}al B$, which was
partially addressed in~\mbox{conv}ite{vidal}, we also discuss the situation
when the unknown state is constrained to lie on the equatorial
plane $\mbox{conv}al E$ of the Bloch sphere $\mbox{conv}al B$. In the second part,
we discuss separable measurement protocols, we prove that they
saturate the joint-measurement bound asymptotically and we state our
conclusions.
Mathematically, the problem of estimating the purity of $\rho({\bf v}ec
r)$ can be formulated within the Bayesian framework as follows (see~\mbox{conv}ite{keyl} for an alternative
approach).
Let ${\mbox{conv}al R}_{\mbox{conv}al O}=\{R_\mbox{conv}hi\}$ be the set of estimates of $r$,
each of them based on a particular outcome $\mbox{conv}hi$ of some
generalized measurement, $\mbox{conv}al O$, over $\rho^N({\bf v}ec r)$. In full
generality, we assume that such measurement is characterized by a
Positive Operator Valued Measure (POVM), namely, by a set of
positive operators ${\mbox{conv}al O}=\{O_\mbox{conv}hi\}$ that satisfy $\sum_\mbox{conv}hi
O_\mbox{conv}hi=\openone$ ($\mbox{conv}hi$ can be a continuous variable, in which
case the sum becomes an integral over $\mbox{conv}hi$).
A separable measurement is a particularly interesting instance of a POVM for
which each $O_\mbox{conv}hi$ is a tensor product of $N$ individual operators
(usually projectors) each one of them acting on $\rho({\bf v}ec r)$.
Next, a figure of merit, $f(r,R_\mbox{conv}hi)$, is introduced as a
quantitative way of expressing the quality of the purity
estimation. Throughout this paper we use
\begin{equation}gin{eqnarray}
f(r,R_\mbox{conv}hi)&\equiv&2\mbox{\scriptsize max}x_{{\bf v}ec m} \left[{\rm tr}\sqrt{\rho^{1/2}({\bf v}ec r)\rho(R_\mbox{conv}hi {\bf v}ec m)
\rho^{1/2}({\bf v}ec r)}\right]^2-1\nonumber\\
&=&rR_\mbox{conv}hi+\sqrt{1-r^2}\sqrt{1-R_\mbox{conv}hi^2}={\bf r}\mbox{conv}dot{\bf R}_\mbox{conv}hi,
\label{fidelity}
\end{eqnarray}
where $|{\bf v}ec m|=1$, i.e., $[1+f(r,R_\mbox{conv}hi)]/2$ is the standard
fidelity~\mbox{conv}ite{fuchs} (see also \mbox{conv}ite{fid}) between $\rho({\bf v}ec r)$
and $\rho(R_\mbox{conv}hi {\bf v}ec n)$, where we have defined ${\bf v}ec n={\bf v}ec
r/r$.
Throughout this paper we refer to $f(r,R_\mbox{conv}hi)$ also as
fidelity for short. Its values are in the range $[0,1]$, where unity corresponds to perfect determination. It is interesting to note that in Uhlmann's geometric representation of the set of density matrices as the hemisphere $(1/2){\mbox{\scriptsize max}thbb S}^3\subset{\mbox{\scriptsize max}thbb R}^4$, the function $D(r,R_\mbox{conv}hi)=(1/2)\arccos f(r,R_\mbox{conv}hi)$ is the geodesic (Bures) distance~\mbox{conv}ite{som}
between two sets (two parallel 2-dimensional spheres) characterized by the purities~$r$ and~$R_\mbox{conv}hi$ respectively.
In the same spirit as
in~\mbox{conv}ite{us-prep,alberto}, we have written $f(r,R_\mbox{conv}hi)$ as a
scalar product of the two unit vectors ${\bf a}=(\sqrt{1-a^2},a)$;
$a=r, \,R_\mbox{conv}hi$. The optimal protocol is obtained by maximizing
\begin{equation}gin{equation}
F({\mbox{conv}al O},{\mbox{conv}al R}_{\mbox{conv}al O})=\sum_\mbox{conv}hi\int d\rho f(r,R_\mbox{conv}hi) {\rm
tr}[\rho^N({\bf v}ec r) O_\mbox{conv}hi], \label{averaged fidelity}
\end{equation}
where $d\rho$ is the prior probability distribution of $\rho({\bf v}ec
r)$, and we identify the trace as the probability of obtaining the
outcome $\mbox{conv}hi$ given that the state we measure upon is
$\rho^N({\bf v}ec r)$. Thus, $F$ is the average fidelity. The
maximization is over the estimator (guessed purity) ${\mbox{conv}al
R}_{\mbox{conv}al O}$ and the POVM ${\mbox{conv}al O}$. Using Schwarz inequality the
optimal estimator is easily seen to be
\begin{equation}gin{equation}
R_\mbox{conv}hi^{\rm opt}={V_\mbox{conv}hi\over\sqrt{{\bf V}_\mbox{conv}hi\mbox{conv}dot{\bf
V}_\mbox{conv}hi}}; \quad {\bf V}_\mbox{conv}hi=\int d\rho\; {\bf r} \, {\rm
tr}[\rho^N({\bf v}ec r) O_\mbox{conv}hi], \label{optimal guess}
\end{equation}
and
\begin{equation}gin{equation}
F({\mbox{conv}al O})\equiv \mbox{\scriptsize max}x_{\{{\mbox{conv}al R}_{\mbox{conv}al O}\}}F({\mbox{conv}al O},{\mbox{conv}al
R}_{\mbox{conv}al O})=\sum_\mbox{conv}hi \sqrt{{\bf V}_\mbox{conv}hi\mbox{conv}dot{\bf V}_\mbox{conv}hi} \ .
\label{optimal fidelity}
\end{equation}
We are still left with the task of computing $F^{\rm
max}=\mbox{\scriptsize max}x_{{\mbox{conv}al O}} F({\mbox{conv}al O})$.
In this formulation, we need to provide a prior probability
distribution (prior for short) $d\rho$, which encodes our initial
knowledge about $\rho({\bf v}ec r)$. Here we assume to be completely
ignorant of both ${\bf v}ec n$ and $r$. Our lack of knowledge about the
former is properly represented with the choice $d\rho \!\mbox{conv}dot\!ropto
d\Omega$ (solid angle element), which states that {\em \`a priori}
${\bf v}ec n$ is isotropically distributed on ${\mbox{conv}al B}$. Therefore, we
write
\begin{equation}gin{equation}
d\rho={d\Omega\over4\!\mbox{conv}dot\!i} w(r)dr;\quad \int_0^1dr\, w(r)=1.
\label{measure}
\end{equation}
While there is wide agreement on this respect, the $r$-dependence
of the prior is controversial and so far we will not stick to any
particular choice.
Nevertheless, it is worth keeping in mind that the hard sphere prior $w(r)=3 r^2$ shows
up in the context of entanglement estimation~\mbox{conv}ite{zycz},
whereas the Bures
prior $w(r)=(4/\!\mbox{conv}dot\!i) r^2 (1-r^2)^{-1/2}$ is most
natural in connection with distinguishability of density matrices~\mbox{conv}ite{fuchs,fid,prior}.
We are now in a position to compute $F^{\mbox{\scriptsize max}x}$. We first assume no
constraint on $\mbox{conv}al O$, thus allowing for the most general
measurement setup. The density matrix $\rho^N({\bf v}ec r)$ can be
written in a block-diagonal form, where each block,
$\rho_{Nj\alpha}({\bf v}ec r)$, transforms with a corresponding
spin~$\bf j$ irreducible representation of $SU(2)$ and $\alpha$
($\alpha=1,2,\dots, n_j$) labels the different $n_j$ occurrences
of the same block~\mbox{conv}ite{cirac, us-prep}. This implies that each
element, $O_\mbox{conv}hi$, of the optimal POVM can be likewise chosen to
have the same block-diagonal structure.
Given a POVM $\tilde{\mbox{conv}al O}$ of this type, we consider the two-stage measurement protocol ${\mbox{conv}al O}$ consisting of
({\em i}\hspace{.1em})~a `preliminary' measurement of
the projection of the state $\rho^N({\bf v}ec r)$ onto the $SU(2)$
irreducible subspaces, followed by ({\em ii\hspace{.1em}})~the measurement defined by~$\tilde{\mbox{conv}al O}$.
The outcomes of $\mbox{conv}al O$ are thus labeled by three indexes $\mbox{conv}hi=(j,\alpha,{\bf x}i)$, and the corresponding operators are defined by $O_{j\alpha{\bf x}i}=\openone_{j\alpha}{\tilde O}_{\bf x}i\openone_{j\alpha}$.
Since the
projector on each irreducible subspace, $\openone_{j\alpha}\equiv\sum_m |jm;\alpha\rangle
\langle jm;\alpha|$,
commutes with $\rho^N({\bf v}ec r)$, the probabilities ${\rm
tr}[\rho^N({\bf v}ec r)\, \tilde O_{\bf x}i]$ are the marginals of ${\rm
tr}[\rho^N({\bf v}ec r)\, O_{j\alpha{\bf x}i }]$ and the fidelity cannot decrease by using $\mbox{conv}al O$ instead of the original~$\tilde{\mbox{conv}al O}$.
In our quest for optimality, we thus stick to these two-stage measurements.
We next recall that $\rho({\bf v}ec r)=U\rho(r {\bf v}ec z)U^\dagger$ for a
suitable $SU(2)$ transformation $U$, where ${\bf v}ec z$ is the unit
vector along the $z$ axis, and that $d\Omega$ can be replaced by
the Haar measure of $SU(2)$. Using Schur's lemma the integral
in~(\ref{optimal guess}) gives
\begin{equation}gin{equation}
{\bf V}_{j\alpha{\bf x}i}={{\rm tr}(
O_{j\alpha{\bf x}i})\over2j+1}\int dr\,w(r)\,{\bf r}\,{\rm tr}
[\rho_{Nj\alpha}(r{\bf v}ec z)] . \label{Vjchi}
\end{equation}
Hence, the estimate $R^{\rm opt}_{\mbox{conv}hi}=R^{\rm opt}_{j\alpha{\bf x}i}$ turns out to be
independent of the outcomes~${\bf x}i$ (of~$\tilde{\mbox{conv}al O}$), and we can write $R^{\rm
opt}_{j\alpha}$ instead. This, in turn, renders the maximization
in~(\ref{optimal fidelity}) trivial, since, using the relation $\sum_{\bf x}i
O_{j\alpha{\bf x}i}=\openone_{j\alpha}$, we see that the right
hand side of~(\ref{optimal fidelity}) becomes also independent of
$\tilde {\mbox{conv}al O}$, and we can drop the
subscript ${\bf x}i$ from now on.
The bottom line is that, assuming an isotropic prior, the optimal
purity estimation is entirely based on the outcomes of $\mbox{conv}al I$
(no additional information about the purity can be extracted from the
state) and we might as well choose not to perform any further
measurement ($\{\tilde O_{\bf x}i\}\to\openone$). With this choice, the
prefactor in~(\ref{Vjchi}) becomes unity. Since the $n_j$ spin
$\bf j$ blocks $\rho_{Nj\alpha}$ all give an identical
contribution
\begin{equation}gin{equation}
{\rm tr}[\rho_{Nj\alpha}(r{\bf v}ec z)]=\sum_{m=-j}^j p_r^{{N\over2}-m}
q_r^{{N\over2}+m}, \label{trace}
\end{equation}
where $p_r=(1-r)/2$, $q_r=1-p_r$, the left hand side of~(\ref{Vjchi})
can be simply called ${\bf V}_{j}$,
The maximal fidelity is
thus given by
\begin{equation}gin{equation}
F^{\rm
max}=\!\mbox{conv}dot\!matrix{N\mbox{conv}r{N\over2}-j}{2j+1\over{N\over2}+j+1}\sum_j\sqrt{{\bf
V}_j\mbox{conv}dot{\bf V}_j} \ , \label{Fmax}
\end{equation}
where the coefficient in front of the sum
is~$n_j$~\mbox{conv}ite{cirac,us-prep}. This, along with~(\ref{trace})
and~(\ref{Vjchi}), provides an explicit expression of~$F^{\rm
max}$. For large $N$, this can be computed~to~be~\mbox{conv}ite{details}
\begin{equation}gin{equation}
F^{\rm max}= 1-{1\over2N}+ o(N^{-1}) . \label{Fasymp}
\end{equation}
One can also check that at leading order in $1/N$ the optimal
guess is $R^{\rm opt}_j=2j/N$, as one would intuitively expect.
These asymptotic results hold for any prior $w(r)$.
\begin{equation}gin{figure}
\!\mbox{conv}dot\!sfrag{N}{$N$}
\!\mbox{conv}dot\!sfrag{P}[bc]{$N(1-F^{\rm max})$}
\includegraphics[width=8cm]{figura-1}
\mbox{conv}aption {A log-linear plot of $N(1-F^{\rm max})$ in terms of the number $N$ of copies for
the optimal joint measurement and for the Bures (solid line)
and hard sphere (dashed line) priors.}\label{fig}
\end{figure}
In Fig.~\ref{fig}, we plot $N(1-F^{\rm max})$ as a function of $N$
in the range $10$--$5000$ for states in $\mbox{conv}al B$ and for the Bures (solid line)
and the
hard sphere (dashed
line) priors. The two lines are seen to approach the
asymptotic value $1/2$ [which can be read off from
Eq.~(\ref{Fasymp}) ] for large~$N$ at a similar rate.
It is also interesting to analyze the case where ${\bf v}ec r$ is known
to lie on the equatorial plane $\mbox{conv}al E$. With this information,
the prior probability distribution becomes
$d\rho=(d\!\mbox{conv}dot\!hi/2\!\mbox{conv}dot\!i)w(r)dr$, where $\!\mbox{conv}dot\!hi$ is the polar angle of the
spherical coordinates. Though it is still possible to use the
block-diagonal decomposition discussed above, the individual
blocks are now reducible under the unitary symmetry
transformations on~$\mbox{conv}al E$, i.e., under a $U(1)$ subgroup of
$SU(2)$.
In full analogy to the general case,
the optimal POVM is given by the set
of one-dimensional projectors over the $U(1)$-invariant subspaces, $\{\openone_{j\alpha
m}\equiv|jm;\alpha\rangle\langle jm;\alpha|\}$,
and, as above, the equivalent representations,
labelled by $\alpha$, contribute a multiplicative factor~$n_j$.
The analogous of~(\ref{trace}) is now
\begin{equation}gin{equation}
[\rho_{Nj\alpha}(r{\bf v}ec x)]_{mm}=\!\!\sum_{m'=-j}^j \left[{\rm
d}_{mm'}^{(j)}(\mbox{${\!\mbox{conv}dot\!i\over2}$})\right]^2 p_r^{{N\over2}-m'}
q_r^{{N\over2}+m'}, \label{trace2D}
\end{equation}
where ${\rm
d}_{mm'}^{(j)}(\mbox{${\begin{equation}ta}$})$ are the standard Wigner d-matrices~\mbox{conv}ite{edmonds}.
From~(\ref{trace2D}) we can compute ${\bf V}_{jm}$ and $F^{\rm max}$, as
in~(\ref{Fmax}), where in this case the sum extends over $j$ and
$m$.
The resulting
expression can be evaluated for small $N$ but it is not very
enlightening. The corresponding plots for the
analogous of Bures and hard sphere priors are indistinguishable
from those in~Fig.~\ref{fig}. Far more interesting is the large $N$ regime. It
turns out that $F^{\rm max}$ is also given by~(\ref{Fasymp}) and
the optimal guess becomes $m$ independent, $R^{\rm
opt}_{jm}=2j/N+\dots$. Therefore, we see that the information
about~${\bf v}ec n$ becomes irrelevant in the asymptotic limit.
A word regarding quantum statistical inference is in order here.
It is often argued that the quantum Cram\'er-Rao
bound~\mbox{conv}ite{holevo} can be integrated to provide an attainable
asymptotic lower bound for some averaged figures of merit, such as
the fidelity~(\ref{fidelity}). Ours is a so-called one parameter
problem for which the quantum Cram\'er-Rao bound takes the simple
form ${\rm Var}\, R\ge H^{-1}({\bf v}ec r)/N$, where ${\rm Var}\,
R\equiv\langle (R_\mbox{conv}hi-\langle R_\mbox{conv}hi\rangle)^2\rangle$ is the
variance of the estimator $R_\mbox{conv}hi$, the average is over the
outcomes $\mbox{conv}hi$ of a measurement, $H({\bf v}ec r)$ is the quantum
information matrix~\mbox{conv}ite{holevo}, and $R_\mbox{conv}hi$ is assumed to be
unbiased: $ \langle R_\mbox{conv}hi\rangle=r $. In our case $H({\bf v}ec
r)=(1-r^2)^{-1}$, and the bound is attainable. This provides in
turn an attainable asymptotic upper bound for the
fidelity~(\ref{fidelity}), since $\langle
f(r,R_\mbox{conv}hi)\rangle\approx
1-\raisebox{.12em}{\mbox{\tiny$1\over2$}}H({\bf v}ec r)\,{\rm Var}\,
R+\dots$. Assuming one can integrate this relations over the
whole of~$\mbox{conv}al B$ (including the region $r\approx1$, where $H({\bf v}ec
r)$ is singular), with a weight function given by the
prior~(\ref{measure}), we obtain Eq.~(\ref{Fasymp}).
Unfortunately, there are only heuristic arguments supporting this
assumption, but so far no rigorous proof exists in the
literature~\mbox{conv}ite{van-trees}.
We now abandon the joint protocols to dwell on separable
measurement strategies for the rest of the paper. Here we focus
on the asymptotic regime, but some brief comments concerning small
$N$ can be found in the conclusions.
In previous work~\mbox{conv}ite{alberto}, some of the authors showed that
the maximum fidelity one can achieve in estimating both $r$ and
${\bf v}ec n$ (full estimation of a qubit mixed state) assuming the
Bures prior and using tomography
behaves as
\begin{equation}gin{equation}
F^{\rm max}_{\rm full}=1-{{\bf x}i\over N^{3/4}}+o(N^{-3/4}) ,
\label{Ffull}
\end{equation}
where ${\bf x}i$ is a positive constant. The same behavior one should
expect for our fidelity $F^{\rm max}$, since the effect of the
purity estimation is dominant in~(\ref{Ffull}). This strange power
law, somehow unexpected on statistical grounds, is caused by the
behavior of $w(r)$ in a small region \mbox{$r\approx 1$}. Indeed,
it is not difficult to convince oneself that if $w(r) \!\mbox{conv}dot\!ropto
(1-r^2)^{-\lambda}\approx 2(1-r)^{-\lambda}$ for $r\approx 1$, one
should expect $1-F^{\rm max}\!\mbox{conv}dot\!ropto N^{\lambda/2-1}+\dots$, for
$0<\lambda<1$ (for $\lambda=0$, hard sphere prior, one should
expect logarithmic corrections). This differs drastically
from~(\ref{Fasymp}) which, as stated above, holds for
{\rm any} such values of $\lambda$. Would classical communication
be enough to restore the right power law $N^{-1}$ for $1-F^{\rm
max}$ and, moreover, saturate the bound of the optimal joint protocol?
On quantum statistical grounds, one should expect a positive
answer to this question since the quantum Cram\'er-Rao bound is
attained by a separable protocol consisting in performing the (von
Neumann) measurements ${\mbox{conv}al M}=\{(\openone\!\mbox{conv}dot\!m{\bf v}ec
n\mbox{conv}dot\sigma)/2\}$ on each copy. Note, however, that $\mbox{conv}al M$
depends on ${\bf v}ec n={\bf v}ec r/r$, which is, of course, unknown {\em
\`a priori}. This protocol can only make sense if we are ready to
spend a fraction of the $N$ copies of $\rho({\bf v}ec r)$ to obtain an
estimate of~${\bf v}ec n$, use this classical information to design
$\mbox{conv}al M$ and, finally, perform this adapted measurement on the
remaining copies. This protocol was successfully applied to pure
states by Gill and Massar in~\mbox{conv}ite{gill-massar}. We extend it to
purity estimation below.
Let us consider a family of priors of the form
\begin{equation}gin{equation}
w(r)={4\over\sqrt\!\mbox{conv}dot\!i}{\Gamma(5/2-\lambda)\over\Gamma(1-\lambda)}{r^2
(1-r^2)^{-\lambda}} , \label{gen prior}
\end{equation}
which includes both the Bures ($\lambda=1/2$) and the hard sphere ($\lambda=0$) metrics. Despite
of this particular $r$ dependence, the final results apply to any
prior whose behavior near $r=1$ is given by~(\ref{gen prior}).
We now proceed {\em \`a la} Gill-Massar~\mbox{conv}ite{gill-massar} and
consider the following one-step adaptive protocol: we take a
fraction $N^\alpha\equiv N_0$ ($0<\alpha<1$) of the $N$ copies
of $\rho({\bf v}ec r)$ and we use them to estimate ${\bf v}ec n$. Tomography
along the three orthogonal axis $x$, $y$ and $z$, together with a
very elementary estimation based on the relative frequencies of
the outcomes~\mbox{conv}ite{us-local}, enables us to estimate ${\bf v}ec n$
with an accuracy given by
\begin{equation}gin{equation}
{\langle\Theta^2_r\rangle\over
2}\approx1-\langle\mbox{conv}os\Theta_r\rangle={3\over N_0}\left({1\over
r^2}-{1\over5}\right)+o(N_0^{-1}), \label{Theta}
\end{equation}
where $\Theta_r$ is the angle between ${\bf v}ec n$ and its estimate.
Here and below $\langle\mbox{conv}dots\rangle$ is not only the average over
the outcomes of this tomography measurements, but also contains an
integration over the prior angular distribution $d\Omega/(4\!\mbox{conv}dot\!i)$
for fixed $r$. We see from~(\ref{Theta}) that the pure state limit is
\mbox{$\langle\Theta_{r\to1}^2\rangle\approx24/(5N_0)+\dots$}, and
one can compute the fidelity, as defined in~\mbox{conv}ite{us-local}, to
check that it agrees with the result therein. This concludes the
first step of the protocol.
In a second step, we measure the projection of ${\bf v}ec\sigma$ along
the estimated ${\bf v}ec n$ obtained in the previous step. We perform
this von Neumann measurement on each of the remaining $N-N_0\equiv
N_1$ copies of the state $\rho({\bf v}ec r)$. We estimate the purity to
be $R=2N_+/N_1-1$, where $N_\!\mbox{conv}dot\!m/N_1$ is the relative frequency of
$\!\mbox{conv}dot\!m1$ outcomes, and we drop the $N_+$ dependence of $R$ to
simplify the notation.
Obviously, as a random variable and for large $N_1$, $R$~is normally
distributed as $R\sim{\rm N}(r c_r,\sqrt{1-r^2
c^2_r}/\sqrt{N_1})$, where $c_r=\mbox{conv}os\Theta_r$. Hence, for large
$N_0$ and $N_1$ it makes sense to expand $f(r,R)$,
Eq.~(\ref{fidelity}), around $R= r c_r$, and thereafter, because
of~(\ref{Theta}), expand the resulting expression around~$c_r=1$.
We obtain
\begin{equation}gin{equation}
F(r)= 1-{1\over 2
N_1}+{r^2\over1-r^2}\left({\langle\Theta^2_r\rangle\over4N_1}-{\langle\Theta^4_r\rangle\over8}\right)+\dots
, \label{<f>}
\end{equation}
where $F(r)$ is the average fidelity for fixed $r$, i.e., $\int
dr\,w(r) F(r)=F$. In view of~(\ref{Theta}),
$\langle\Theta_r^4\rangle\sim N_0^{-2}=N^{-2\alpha}$. Hence, the
two terms in parenthesis in~(\ref{<f>}) can only be dropped if
$\alpha>1/2$. Provided $w(r)$ vanishes as in~(\ref{gen prior})
with $\lambda<0$, we can integrate $r$ in~(\ref{<f>}) over the
unit interval to obtain
\begin{equation}gin{equation}
F=1-{1\over2N(1-N^{\alpha-1})}+o(N^{-1}) , \label{F in I}
\end{equation}
and we conclude that this protocol attains asymptotically the
joint-measurement bound~(\ref{Fasymp}).
However, most of the physically interesting priors~\mbox{conv}ite{prior,zycz}, $w(r)$, not
only do not vanish as $r\to1$, but often diverge like~(\ref{gen
prior}) with $0<\lambda<1$. In this case (\ref{<f>}) cannot be
integrated, as the last term does not lead to a convergent
integral. This signals that the series expansion around $c_r=1$
leading to~(\ref{<f>}) is not legitimated in the whole of~$\mbox{conv}al
B$.
To fix the problem, we split $\mbox{conv}al B$ in two regions. A sphere
of radius $1-\epsilon$, $\epsilon>0$, which we call ${\mbox{conv}al B}^{\rm
I}$, and a spherical sheet of thickness $\epsilon$: ${\mbox{conv}al
B}^{\rm II}=\{{\bf v}ec r: 1-\epsilon<r\le 1\}$. The fidelity can thus
be written as the sum of the corresponding two contributions:
$F=F^{\rm I}+F^{\rm II}$. While $F^{\rm I}$ can be obtained by
simply integrating~(\ref{<f>}) over ${\mbox{conv}al B}^{\rm I}$, where this
expansion is valid, some care must be taken in the region~${\mbox{conv}al B}^{\rm II}$. There, we proceed
as follows.
We compute the fidelity as if all the states in ${\mbox{conv}al B}^{\rm II}$ had the lowest possible purity ($r=1-\epsilon$)
when the first-step tomography was performed. This leads to a lower bound for $F^{\rm II}$,
because the lower the purity of a state the less accurately ${\bf v}ec n$ can be determined [see Eq.~(\ref{Theta})], and hence, the worse its purity can be estimated in the
second step.
The trick, which amounts to replacing $c_r$ by $c_{1-\epsilon}$, enables us to perform the $r$-integration prior
to $\langle\mbox{conv}dots\rangle$. We simply expand $f(r,R)$,
Eq.~(\ref{fidelity}), around $R= r c_{1-\epsilon}$ to obtain
\begin{equation}gin{eqnarray}
F(r)&\gtrsim& \Bigg\langle\sqrt{(1-r^2)(1-r^2c^2_{1-\epsilon})}\nonumber\\
&-&{1\over2N_1}\sqrt{{1-r^2\over 1-r^2 c_{1-\epsilon}^2}} +\dots\Bigg\rangle
,
\end{eqnarray}
where the dots stand for additional terms that are irrelevant to
the problem we are addressing here. Integrating this expression
and expanding around $c_{1-\epsilon}=1$ we obtain
\begin{equation}gin{eqnarray}
&&\kern-4em\int_{1-\epsilon}^1\kern-1.3em dr\,w(r) F(r)\gtrsim 1-{1\over2N_1}-
k_\lambda\left\langle(1-c_{1-\epsilon})^{2-\lambda}\right\rangle\nonumber\\
&&\kern3.4em-\left(1-{1\over2N_1}\right)\int_0^{1-\epsilon}\kern-1.3em
dr\,w(r)+\dots,
\end{eqnarray}
where $
k_\lambda={2^{2-\lambda}\Gamma({5\over2}-\lambda)\Gamma({3\over2}-\lambda)\Gamma(\lambda-2)/[\!\mbox{conv}dot\!i\Gamma(1-\lambda)]}
$. Putting together the different pieces of the calculation we
have
\begin{equation}gin{equation}
F\gtrsim 1-{1\over2N_1}-2^{\lambda-2}k_\lambda
\langle\Theta_{1-\epsilon}^2\rangle^{2-\lambda} +\dots,
\label{fidelity ok}
\end{equation}
$0<\lambda<1$, where now we can safely take the limit
$\epsilon\to0$. We see that by choosing
\begin{equation}gin{equation}
{\rm
\mbox{\scriptsize max}x}\left\{{1\over2},{1\over2-\lambda}\right\}<\alpha< 1
\label{alpha}
\end{equation}
we ensure that the joint-measurement
bound~(\ref{Fasymp}) is attained. It is worth
emphasizing that the last term in~(\ref{fidelity ok}), which is
completely missing in~(\ref{F in I}), is actually the dominant
contribution if $\alpha<1/(2-\lambda)$. For $\lambda=0$ we have
\begin{equation}gin{equation}
F^{\rm hard}\gtrsim1-{1\over2N_1}-{3 \langle\Theta^2_1\rangle
\log\langle\Theta^2_1\rangle\over8N_1}+\dots ,
\end{equation}
and we again conclude that the protocol presented here attains the
joint-measurement bound.
Two comments about the choice of $\alpha$ are in order. First, numerical simulations show that the optimal value of $\alpha$ is very close to the lower bound in~(\ref{alpha}). Second, we see that the lower bound in~(\ref{alpha})
increases with increasing~$\lambda$. This can be understood by recalling that for large $N$, the estimated purity $R$
is normally distributed with a variance of ${\rm Var}\,R=(1-r^2 c_r^2)/N_1$. For $\lambda\ll1$, the prior is a rather flat function of $r$ and, on average, ${\rm Var}\,R=a/N_1$, where $a$ is a constant. Increasing
the accuracy by which ${\bf v}ec n$ is determined does not improve significantly the estimation of~$r$. Hence, using a small fraction of the number of copies at the first stage of the protocol should be enough.
This suggest that $\alpha$ must be relatively small. In contrast, for $\lambda\approx1$ the prior peaks at $r=1$ and ${\rm Var}\,R=\Theta^2_r/N_1$. Hence, it pays to
spend a large fraction of $N$ to estimate ${\bf v}ec n$ with high accuracy (as this drastically reduces ${\rm Var}\,R$), for which we need that $\alpha\approx 1$.
At this point one may wonder if the conclusions above depend upon
our particular choice of figure of merit. To get a grasp on this,
it is worth using again the standard pointwise approach to quantum
statistics. There, one is interested in the mean square error
${\rm MSE}\,R=\langle(R-r)^2\rangle$ for fixed $r$, where now the
average $\langle\mbox{conv}dots\rangle$ is over the outcomes of {\em all}
measurements for a fixed ${\bf v}ec r$. One can write ${\rm
MSE}\,R={\rm Var}\, R + (\langle R\rangle-r)^2 $, where the second
term is the \emph{bias}. Using the same one-step adaptive protocol
described above, we get that the mean square error after step two
is
\begin{equation}gin{equation}
{\rm
MSE}\,R=\frac{H^{-1}(r)}{N_1}+{r^2\over4}\langle\Theta_r^4\rangle+
\dots .
\end{equation}
As above, the last term can be dropped if $\alpha>1/2$,
and
\begin{equation}gin{equation}
{\rm MSE}\,R=\frac{H(r)^{-1}}{N} +o[N^{-1}],
\end{equation}
saturating the quantum Cram\'er-Rao bound. This protocol is,
therefore, also asymptotically optimal in the present context.
Though the argumentation above is somehow heuristic, it can be
made fully rigorous~\mbox{conv}ite{ballester}.
In summary, we have addressed the problem of optimally estimating
the purity of a qubit state of which $N$ identical copies are
available. The optimal estimation of the entanglement of a
bipartite qubit state can be reduced to this problem. Though the
absolute bounds for the average fidelity involve joint
measurements, these bounds can be obtained asymptotically with
separable measurements. This requires classical communication
among the sequential von Neumann measurements performed on each of
the $N$ individual copies of the state. This result, which has
been speculated on quantum statistical grounds, is here proved for
the first time by a direct calculation. This leads to a very
surprising result: in the asymptotic limit of many copies,
bipartite entanglement, a genuinely non-local property, can be
optimally estimated by performing fully separable measurements.
This meaning that measurements can be performed not only on copies
of {\em one} of the two entangled parties, but on {\em each} of
these copies {\em separately}. This avoids the necessity of quantum memories.
For finite (but otherwise arbitrary) $N$, finding the optimal
separable measurement protocol is an open problem. Interestingly
enough, a `greedy' protocol designed to be optimal at each
measurement step~\mbox{conv}ite{us-local, others} leads to an unacceptably
poor estimation. Notice that in the one-step adaptive protocol
described above, part of the copies were spent (`wasted' from a
`greedy' point of view) in estimating ${\bf v}ec n$.
We have seen that this strategy pays in the long run.
However, the `greedy' strategy optimizes measurements in the short run,
which translates into measuring ${\bf v}ec\sigma$ along the same arbitrarily fixed axis on each copy of~$\rho({\bf v}ec r)$.
This yields a low value for the fidelity, which does not
even converge to unity in the strict limit $N\to\infty$.
This counterintuitive behavior of the `greedy' protocol also appears in
other contexts as, e.g., economics, biology or social sciences
(see~\mbox{conv}ite{parrondo} for a nice example).
We acknowledge useful conversations with Antonio Ac\'{\i}n, Richard Gill and Juanma
Parrondo. This work is supported by the Spanish Ministry of
Science and Technology project BFM2002-02588, CIRIT project
SGR-00185, Netherlands Organization for Scientific Research NWO,
the European Community projects QUPRODIS contract no.
IST-2001-38877 and RESQ contract no IST-2001-37559.
\newcommand{\PRL}[3]{Phys.~Rev. Lett.~\textbf{#1}, #2~(#3)}
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\newcommand{\JOB}[3]{J.~Opt. B~\textbf{#1}, #2~(#3)}
\newcommand{\JMP}[3]{J.~Math.~Phys.~\textbf{#1}, #2~(#3)}
\newcommand{\JMO}[3]{J.~Mod.~Opt.~\textbf{#1}, #2~(#3)}
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\end{thebibliography}
\end{document} | math | 37,399 |
\begin{document}
\textitle{An asymptotic approach in Mahler's method}
\author{Michael Coons}
\address{School of Math.~and Phys.~Sciences\\
University of Newcastle\\
Callaghan\\
Australia}
\email{[email protected]}
\thanks{The research of M.~Coons was supported by ARC grant DE140100223.}
\keywords{Algebraic independence, Mahler functions, radial asymptotics}
\subjclass[2010]{Primary 11J85; Secondary 11J91, 30B30}
\date{\today}
\begin{abstract} We provide a general result for the algebraic independence of Mahler functions by a new method based on asymptotic analysis. As a consequence of our method, these results hold not only over $\mathbb{C}(z)$, but also over $\mathbb{C}(z)(\mathcal{M})$, where $\mathcal{M}$ is the set of all meromorphic functions. Several examples and corollaries are given, with special attention to nonnegative regular functions.
\end{abstract}
\title{An asymptotic approach in Mahler's method}
\section{Introduction}
Mahler's method is a method in number theory wherein one answers questions surrounding the transcendence and algebraic independence of both functions $F(z)\in\mathbb{C}[[z]]$, which satisfy the functional equation \begin{equation}\label{MFE}a_0(z)F(z)+a_1(z)F(z^k)+\cdots+a_d(z)F(z^{k^d})=0\end{equation} for some integers $k\geqslant 2$ and $d\geqslant 1$ and polynomials $a_0(z),\ldots,a_d(z)\in\mathbb{C}[z]$, and their special values $F(\alpha)$, typically at algebraic numbers $\alpha$. Functions $F(z)$ satisfying a functional equation of the type in \eqref{MFE} are called {\em $k$-Mahler} (or simply {\em Mahler}, when $k$ is understood); the minimal $d$ for which $F(z)$ satisfies \eqref{MFE} is called the {\em degree} of $F(z)$. Such functions can be considered in a vector setting as well, wherein one considers a vector of functions ${\bf F}(z)=[F_1(z),\ldots,F_d(z)]^T\in\mathbb{C}[[z]]^d$ for some integer $d\geqslant 1$ for which there is a matrix of rational functions ${\bf A}(z)\in\mathbb{C}(z)^{d\times d}$ and an integer $k$ such that \begin{equation}\label{vecmahl} {\bf F}(z)={\bf A}(z){\bf F}(z^k).\end{equation}
Questions and results concerning the transcendence of Mahler functions and their special values were studied in depth by Mahler in the late 1920s and early 1930s \cite{M1929, M1930a, M1930b}, though the study of special Mahler functions dates back to at least the beginning of the XXth century with the publication of Whittaker and Watson's classic text, ``A Course of Modern Analysis'' \cite[Section~5$\cdot$501]{WW1902}. Therein the Mahler function $\sum_{n\geqslant 0}z^{2^n}$ is presented as an example of a function having the unit circle as a natural boundary.
Mahler's early results focused on degree-$1$ Mahler functions, his most famous result in this area being the transcendence of the Thue-Morse number $T(1/2)$, which is a special value of the function $T(z)$ satisfying $T(z)-(1-z)T(z^2)=0$. According to Waldschmidt \cite{W2009}, after Mahler's initial results his method was forgotten; the resurgence waited nearly forty years, following the publication of Mahler's paper ``Remarks on a paper of W.~Schwarz'' \cite{M1969} in 1969. Mahler's method was then extended by Kubota, Loxton, Ke.~Nishioka, Ku.~Nishioka, and van der Poorten among others; see \cite{K1977b, K1977, L1984, LvdP1976, LvdP1977, LvdP1977b, LvdP1978, LvdP1982, LvdP1988, KeN1984, KeN1985, KuN1982, KuN1990, KuN1996, KuN1996book}, though this list is certainly not exhaustive. Much of the continuing interest is connected with the fact that if the sequence $\{f(n)\}_{n\geqslant 0}$ is output by a deterministic finite automaton, then its generating function $F(z)=\sum_{n\geqslant 0}f(n)z^n$ is a Mahler function.
Arguably, the most celebrated result in this area is due to Ku.~Nishioka \cite{KuN1990}, who proved that if $F_1(z),\ldots,F_d(z)$ are components of a vector of Mahler functions with algebraic coefficients satisfying \eqref{vecmahl}, then for all but finitely many algebraic numbers $\alpha$ in the common disc of convergence of $F_1(z),\ldots,F_d(z)$, we have $$\trdeg_{\mathbb{Q}}\mathbb{Q}(F_1(\alpha),\ldots,F_d(\alpha))=\trdeg_{\mathbb{C}(z)}\mathbb{C}(z)(F_1(z),\ldots,F_d(z)).$$ Ku.~Nishioka's result fully reveals the heart of Mahler's method, {\em one can obtain an algebraic independence result for the special values of Mahler functions by producing the result at the function level.} Of course, to gain full use of this theorem, one must produce a function-level result.
While there are several results concerning specific functions of degrees $1$ and $2$ (see in particular the recent work of Bundschuh and V\"a\"an\"anen \cite{B2012, B2013, BV2014, BV2015c, BV2015d, BV2015a, BV2015b}), there is a lack of general results for the algebraic independence of Mahler functions. For degree-$1$ Mahler functions, general results have been given by Kubota \cite{K1977} and Ke.~Nishioka \cite{N1984}, though the criteria they provide can be quite hard to check, making their results difficult to apply.
In this paper, we provide a general algebraic independence result for Mahler functions of arbitrary degree. Our result is based on properties of the eigenvalues of Mahler functions, a concept we recently introduced with Bell \cite{BCpre} in order to produce a quick transcendence test for Mahler functions. To formalise this notion here, suppose that $F(z)$ satisfies \eqref{MFE}, set $a_i:=a_i(1)$, and form the characteristic polynomial of $F(z)$, $$p_F(\lambda):=a_0\lambda^d+a_1\lambda^{d-1}+\cdots+a_{d-1}\lambda+a_d.$$ In the above-mentioned work with Bell, we showed that if $p_F(\lambda)$ has $d$ distinct roots, then there exists an eigenvalue $\lambda_F$ with $p_F(\lambda_F)=0$, which is naturally associated to $F(z)$. We use the term `eigenvalue' to denote the root of a characteristic polynomial.
Our first result is the following.
\begin{theorem}\label{main} Let $k\geqslant 2$ be an integer, $F_1(z),\ldots,F_d(z)\in\mathbb{C}[[z]]$ be $k$-Mahler functions convergent in the unit disc for which the eigenvalues $\lambda_{F_1},\ldots,\lambda_{F_d}$ exist, and let $\mathcal{M}$ denote the set of meromorphic functions. If $k,\lambda_{F_1},\ldots,\lambda_{F_d}$ are multiplicatively independent, then $$\trdeg_{\mathbb{C}(z)(\mathcal{M})}\mathbb{C}(z)(\mathcal{M})(F_1(z),\ldots,F_d(z))=d.$$ In particular, the functions $F_1(z),\ldots,F_d(z)$ are algebraically independent over $\mathbb{C}(z)$.
\end{theorem}
As the title of this paper suggests, our results are obtained by an asymptotic argument. Indeed, the ability to include meromorphic functions in Theorem~\ref{main} is a by-product of our method being analytic and not heavily dependent on algebra. Though our result adds general meromorphic functions in the context of algebraic independence, the comparison of Mahler functions with meromorphic functions is not new. B\'ezivin \cite{B1994} showed that a Mahler function that satisfies a homogeneous linear differential equation with polynomial coefficients is necessarily rational. Taking this further, in his thesis (and unpublished otherwise), Rand\'e \cite{R1992} proved that a Mahler function is either rational or has a natural boundary; see our paper with Bell and Rowland \cite{BCR2013} for a more recent proof of this result.
While Theorem \ref{main} is quite general, if we focus on a certain subclass of Mahler functions, the nonnegative $k$-regular functions, we can remove the existence assumption on the eigenvalues $\lambda_{F_i}$ for $i=1,\ldots,d$.
An integer-valued sequence $\{f(n)\}_{n\geqslant 0}$ is called {\em $k$-regular} provided there exist a positive integer $d$, a finite set of matrices $\{{\bf A}_0,\ldots,{\bf A}_{k-1}\}\subseteq \mathbb{Z}^{d\times d}$, and vectors ${\bf v},{\bf w}\in \mathbb{Z}^d$ such that $$f(n)={\bf w}^T {\bf A}_{i_0}\cdots{\bf A}_{i_s} {\bf v},$$ where $(n)_k={i_s}\cdots {i_0}$ is the base-$k$ expansion of $n$. The notion\footnote{Our definition is not the definition of Allouche and Shallit, though a result of theirs \cite[Lemma~4.1]{AS1992} gives the equivalence.} of $k$-regularity is due to Allouche and Shallit \cite{AS1992}, and is a direct generalisation of automaticity; in fact, a $k$-regular sequence that takes finitely many values can be output by a deterministic finite automaton. We call the generating function $F(z)=\sum_{n\geqslant 0}f(n)z^n$ of a $k$-regular sequence $\{f(n)\}_{n\geqslant 0}$, a {\em $k$-regular function} (or just {\em regular}, when the $k$ is understood). Establishing the relationship to Mahler functions, Becker \cite{pgB1994} proved that a $k$-regular function is also a $k$-Mahler function.
In order to prove an algebraic independence result for regular functions, we prove the following result on the asymptotics of $k$-regular sequences.
\begin{theorem}\label{fNlogN} Let $k\geqslant 2$ be a integer and $\{f(n)\}_{n\geqslant 0}$ be a nonnegative integer-valued $k$-regular sequence, which is not eventually zero. Then there is a real number $\alpha_f\geqslant 1$ and a nonnegative integer $m_f$ such that as $N\to\infty$, $$\alpha_f^{-1}(1+o(1))\leqslant \frac{\sum_{n\leqslant N} f(n)}{N^{\log_k\alpha_f} \log^{m_f}N}\leqslant \alpha_f(1+o(1)).$$
\end{theorem}
We stress that Theorem \ref{fNlogN} provides the existence of a constant $\alpha_f$, which essentially takes the place of the Mahler eigenvalue for regular functions. We use these asymptotics to give the following result for $k$-regular functions.
\begin{theorem}\label{mainreg} Let $k\geqslant 2$ be an integer, $F_1(z),\ldots,F_d(z)\in\mathbb{Z}_{\geqslant 0}[[z]]$ be $k$-regular functions with $F_i(z):=\sum_{n\geqslant 0}f_i(n)z^n$ for $i=0,\ldots,d$, and let $\mathcal{M}$ denote the set of meromorphic functions. If the numbers $k,\alpha_{f_1},\ldots,\alpha_{f_d}$ are multiplicatively independent, where the $\alpha_{f}$ are provided by Theorem \ref{fNlogN}, then $$\trdeg_{\mathbb{C}(z)(\mathcal{M})}\mathbb{C}(z)(\mathcal{M})(F_1(z),\ldots,F_d(z))=d.$$ In particular, the functions $F_1(z),\ldots,F_d(z)$ are algebraically independent over $\mathbb{C}(z)$.
\end{theorem}
Theorems \ref{main} and \ref{mainreg} have some interesting corollaries; we list three here. The first concerns the derivatives of regular functions.
\begin{corollary} Let $k\geqslant 2$ be an integer and $F(z)$ be a $k$-regular function with $k$ and $\alpha_f$ multiplicatively independent. If $n_1$ and $n_2$ are any two distinct nonnegative integers, then $$\trdeg_{\mathbb{C}(z)}\mathbb{C}(z)(e^z,F^{(n_1)}(z),F^{(n_2)}(z))=3,$$ where $F^{(n)}(z)$ denotes the $n$th derivative of $F(z)$.
\end{corollary}
The next corollary demonstrates that Theorems \ref{main} and \ref{mainreg} can be used to give results for infinite sets of functions.
\begin{corollary}\label{DN} Let $p$ be an odd prime, let $\Phi_p(z)$ be the $p$th cyclotomic polynomial, let $k\geqslant 2$ be an integer, and set $F_p(z):=\prod_{n\geqslant 0}\Phi_p(z^{k^n}).$ Then the functions $$F_3(z),F_5(z),F_7(z),\ldots,F_p(z),\ldots,$$ with indices odd primes $p$ coprime to $k$, are algebraically independent over $\mathbb{C}(z)$.
\end{corollary}
\noindent The functions considered in Corollary \ref{DN} were recently studied by Duke and Nguyen~\cite{DN2015}.
Our last corollary in this Introduction concerns the algebraic independence of Mahler functions of different degrees.
\begin{corollary}\label{FS} Let $S(z)$ be Stern's function satisfying $zS(z)-(1+z+z^2)S(z^2)=0$, and $F(z)$ be the function of Dilcher and Stolarsky \cite{DS2009}, which has $0,1$-coefficients and satisfies $F(z)-(1+z+z^2)F(z^4)+z^4F(z^{16})=0.$ Then $$\trdeg_{\mathbb{C}(z)}\mathbb{C}(z)(S(z), F(z))=2.$$
\end{corollary}
Corollary \ref{FS} holds also for any pair of derivatives of $S(z)$ and $F(z)$. We note that the algebraic independence over $\mathbb{C}(z)$ of the Dilcher-Stolarsky function $F(z)$ and its derivative $F'(z)$ follows from our recent joint work with Brent and Zudilin \cite{BCZ2015}.
The remainder of this paper is organised as follows. In Section \ref{SecMahl}, we prove Theorem \ref{main}. Section \ref{SecReg} contains the proofs of Theorems \ref{fNlogN} and \ref{mainreg}. In the final section, we present an extended `illustrative' example as well as a few corollaries and questions.
\section{Algebraic independence of Mahler functions}\label{SecMahl}
In this section, we prove Theorem \ref{main}. Our proof relies heavily on the use of the radial asymptotics of Mahler functions as $z$ approaches various roots of unity. In joint work with Bell, we recently provided the initial case of the more general result to follows here (see Theorem \ref{xi}). We record the special case here as a proposition.
\begin{proposition}[Bell and Coons \cite{BCpre}]\label{initial} Let $F(z)$ be a $k$-Mahler function satisfying \eqref{MFE} whose characteristic polynomial $p_F(\lambda)$ has $d$ distinct roots. Then there is an eigenvalue $\lambda_F$ with $p_F(\lambda_F)=0$, such that as $z\to 1^-$ \begin{equation}\label{Fzto1}F(z)=\frac{C_F(z)}{(1-z)^{\log_k \lambda_F}} (1+o(1)),\end{equation} where $\log_k$ denotes the principal value of the base-$k$ logarithm and $C_F(z)$ is a real-analytic nonzero oscillatory term, which on the interval $(0,1)$ is bounded away from $0$ and $\infty$, and satisfies $C_F(z)=C_F(z^k)$.
\end{proposition}
For the purposes of transcendence, Proposition \ref{initial} is enough; this was the purpose of our joint work with Bell \cite{BCpre}. To gain algebraic independence results, we additionally require the asymptotics as $z$ approaches a general root of unity of degree $k^n$ for any $n\geqslant 0$. Concerning these asymptotics, we give the following result.
\begin{theorem}\label{xi} Let $F(z)$ be a $k$-Mahler function satisfying \eqref{MFE} whose characteristic polynomial $p_F(\lambda)$ has $d$ distinct roots and let $\xi$ be a root of unity of degree $k^n$ for some $n\geqslant 0.$ Then as $z\to 1^-$, there is an integer $m_\xi$ and a nonzero number $\Lambda_F(\xi)$ such that $$F(\xi z)=\frac{\Lambda_F(\xi)C_F(z)}{(1-z)^{\log_k \lambda_F-m_\xi}}(1+o(1)),$$ where $C_F(z)$ is the function of Theorem \ref{initial}.
\end{theorem}
\begin{proof} If $\xi_1$ is a root of unity of degree $k$, then using the functional equation \eqref{MFE} and Proposition \ref{initial}, \begin{align} \nonumber F(\xi_1 z)&=\frac{-1}{a_0(\xi_1 z )}\sum_{j=1}^d a_j(\xi_1 z)F(z^{k^j})\\
\label{ratxiz}&=\left(-\sum_{j=1}^d \frac{a_j(\xi_1 z)}{a_0(\xi_1z)}\lambda_F^{-j}\right)\frac{C_F(z)}{(1-z)^{\log_k \lambda_F}}(1+o(1))\\
\nonumber &=\frac{\Lambda_F(\xi_1)C_F(z)}{(1-z)^{\log_k \lambda_F-m_1}}(1+o(1)),
\end{align} where we have used the fact that as $z\to 1^-$, the rational function in \eqref{ratxiz} can be written $$-\sum_{j=1}^d \frac{a_j(\xi_1 z)}{a_0(\xi_1z)}\lambda_F^{-j}=\Lambda_F(\xi_1)(1-z)^{m_1}(1+o(1)),$$ for some nonzero complex number $\Lambda_F(\xi_1)$ and some integer $m_1$ that depends on $\xi_1$ and the polynomials $a_i(z)$, for $i=0,\ldots,d$. The fact that $\sum_{j=1}^d \frac{a_j(\xi_1 z)}{a_0(\xi_1z)}\lambda_F^{-j}\neq 0$ follows from the assumption that the characteristic polynomial $p_F(\lambda)$ has $d$ distinct roots.
Note that we can continue this process iteratively for any root of unity $\xi$ of degree $k^n$, as $z\to \xi$ radially.
The result is now a direct consequence of the above argument with the additional realisation that for roots of unity $\xi$ of degree $k^n$ with $n$ large enough, $a_i(\xi)\neq 0$ for $i=0,\ldots,d$.
\end{proof}
For a special case of Theorem \ref{xi}, see our recent work with Brent and Zudilin \cite[Theorem 3 and Lemma 5]{BCZ2015}, in which we used radial asymptotics to extend the work of Bundschuh and V\"a\"an\"anen \cite{BV2014}.
With these asymptotic results established, we may now prove Theorem \ref{main}.
\begin{proof}[Proof of Theorem \ref{main}] Towards a contradiction, assume the theorem is false, so that we have an algebraic relation \begin{equation}\label{ar}\sum_{{\bf m}=(m_1,\ldots,m_d)\in{\bf M}}p_{\bf m}(z,G_1(z),\ldots,G_s(z))F_1(z)^{m_1}\cdots F_d(z)^{m_d}=0,\end{equation} where the set ${\bf M}\subseteq \mathbb{Z}^{d}_{\geqslant 0}$ is finite and none of the polynomials $p_{\bf m}(z,G_1(z),\ldots,G_s(z))$ in $\mathbb{C}[z][\mathcal{M}]$ is identically zero. Moreover, without loss of generality, we may suppose that the polynomial $\sum_{{\bf m}}p_{\bf m}(z,w_1\ldots,w_s)y_1^{m_1}\cdots y_d^{m_d}$ in $d+s+1$ variables is irreducible.
Pick a $z_0\in(0,1)$ and note that as $z\to 1^-$ along the sequence $\{z_0^{k^m}\}_{m\geqslant 0}$, for $\xi$ any root of unity of degree $k^n$, with $n$ large enough, in the notation of Theorem \ref{xi}, we have \begin{equation}\label{algterm}F_1(\xi z)^{m_1}\cdots F_d(\xi z)^{m_d}= \frac{C_{{\bf m}} \left(\prod_{i=1}^d \Lambda_{F_i}(\xi)^{m_i}\right)}{(1-z)^{m_1\log_k\lambda_{F_1}+\cdots+m_d\log_k\lambda_{F_d}+|{\bf m}|\cdot m}}(1+o(1)),\end{equation} where $|{\bf m}|=m_1+\cdots+m_d$, $m$ is an integer, and $C_{\bf m}\neq 0$ depends on the choice of $z_0$, but is independent of $\xi$ and $z$.
Let ${\bf M}_{\max}\subseteq{\bf M}$ be the (nonempty) set of indices ${\bf m}=(m_1,\ldots,m_d)$ such that the quantity $$\delta:= m_1\log_k\lambda_{F_1}+\cdots+m_d\log_k\lambda_{F_d}+|{\bf m}|\cdot m$$ is maximal.
We claim that the set ${\bf M}_{\max}$ contains only one element. To see this, suppose that ${\bf m},{\bf m}'\in{\bf M}_{\max}.$ Then $$m_1\log_k\lambda_{F_1}+\cdots+m_d\log_k\lambda_{F_d}+|{\bf m}|\cdot m=m_1'\log_k\lambda_{F_1}+\cdots+m_d'\log_k\lambda_{F_d}+|{\bf m}'|\cdot m=\delta,$$ and so $$(m_1-m_1')\log_k\lambda_{F_1}+\cdots+(m_d-m_d')\log_k\lambda_{F_d}+(|{\bf m}|-|{\bf m}'|)m=0.$$ Since the numbers $k,\lambda_{F_1},\ldots,\lambda_{F_d}$ are multiplicatively independent, the numbers $\log_k\lambda_{F_1},\ldots,\log_k\lambda_{F_d},m$ are linearly independent. Thus $m_i=m_i'$ for each $i\in\{1,\ldots,d\}$, and we have ${\bf m}={\bf m}'.$
Using the uniqueness of the term of index ${\bf m}_{\max}$ with maximal asymptotics, we multiply the algebraic relation \eqref{ar} by $(1-z)^\delta$ and send $z\to 1^-$ along the sequence $\{z_0^{k^m}\}_{m\geqslant 0}$. Then for a root of unity $\xi$ of degree $k^n$, for $n$ large enough and for which $G_1(\xi),\ldots,G_s(\xi)$ each exist, we gain the equality $$p_{{\bf m}_{\max}}(\xi,G_1(\xi)\ldots,G_s(\xi))\cdot C_{{\bf m}_{\max}}\cdot \left(\prod_{i=1}^d \Lambda_{F_i}(\xi)^{m_i}\right)=0.$$ This implies that $$p_{{\bf m}_{\max}}(\xi,G_1(\xi)\ldots,G_s(\xi))=0,$$ for each choice of such $\xi$. Since there are infinitely many such $\xi$ that are dense on the unit circle and $p_{{\bf m}_{\max}}(z,G_1(z),\ldots,G_s(z))$ is a meromorphic function, it must be that $p_{{\bf m}_{\max}}(z,G_1(z),\ldots,G_s(z))=0$ identically, contradicting our original assumption.
\end{proof}
\section{Algebraic independence of regular functions}\label{SecReg}
In this section, we prove Theorems \ref{fNlogN} and \ref{mainreg}. As stated in the Introduction, focusing on the subclass of nonnegative regular functions allows us a bit more freedom in the results. While we will still use the fact that regular functions $F(z)$ are Mahler functions, we no longer require the existence of the eigenvalue $\lambda_F$. For nonnegative regular functions, the role of the eigenvalue will be played by a different constant $\alpha_f$, some properties of which are discussed in what follows.
To establish Theorem \ref{fNlogN}, we require a few preliminary results, the first of which separates out a special linear recurrent subsequence of a regular sequence.
\begin{lemma}\label{sigma} If $f$ is a $k$-regular sequence, then $\{f(k^\ell)\}_{\ell\geqslant 0}$ is linearly recurrent.
\end{lemma}
\begin{proof} Recalling the definition of regular sequences in the Introduction, let $\{{\bf A}_0,\ldots,$ ${\bf A}_{k-1}\}$, ${\bf v}$, and ${\bf w}$ be such that $f(m)={\bf w}^T {\bf A}_{i_0}\cdots{\bf A}_{i_s} {\bf v},$ where $(m)_k={i_s}\cdots {i_0}$ is the base-$k$ expansion of $m$. Then we have $$f(k^\ell)={\bf w}^T {\bf A}_{0}^{\ell}{\bf A}_{1} {\bf v},$$ which proves the lemma.
\end{proof}
Though unneeded for our purposes, it is worth noting that one may strengthen the above lemma to show that for any choice of $n$ and $r$, the sequence $\{f(k^\ell n+r)\}_{\ell\geqslant 0}$ is linearly recurrent.
We require the following classical result of Allouche and Shallit \cite[Theorem 3.1]{AS1992}.
\begin{proposition}[Allouche and Shallit \cite{AS1992}]\label{AS} Let $k\geqslant 2$ be an integer. Then the set of $k$-regular sequences is closed under (Cauchy) convolution. In particular, if $\{f(n)\}_{n\geqslant 0}$ is $k$-regular, then so is the sequence $\{g(n)\}_{n\geqslant 0}$, where $$g(n)=\sum_{j\leqslant n}f(j).$$
\end{proposition}
By applying Lemma \ref{sigma} and Proposition \ref{AS}, we now prove Theorem \ref{fNlogN}.
\begin{proof}[Proof of Theorem \ref{fNlogN}] Combining Lemma \ref{sigma} and Proposition \ref{AS}, we have that $\sigma_f(r):=\sum_{n\leqslant k^r} f(n)$ is linearly recurrent. Further, since $\{f(n)\}_{n\geqslant 0}$ is nonnegative, the sequence $\{\sigma_f(r)\}_{r\geqslant 0}$ is increasing. Thus using the eigenvalue representation of the linear recurrence $\sigma_f(r)$, as $r\to\infty$, we have $$\sigma_f(r)=c_1r^{m_f}\alpha_f^r(1+o(1)),$$ for some integer $m_f\geqslant 0$ and $\alpha_f\geqslant 1$. The lower bound on $\alpha_f$ follows as $\sigma_f(r)$ is increasing and integer-valued.
The result of the theorem now follows quite quickly. To see this, let $N$ be large enough. Then $N\in(k^{r},k^{r+1}]$ for $r=\lfloor\log_k N\rfloor$. So for any $\varepsilon>0$, $$(c_1-\varepsilon)r^{m_f}\alpha_f^r<\sigma_f(r)\leqslant\sum_{n\leqslant N} f(n)\leqslant\sigma_f(r+1)<(c_1+\varepsilon)(r+1)^{m_f}\alpha_f^{r+1}.$$ Using the trivial upper and lower bounds $\log_k N-1\leqslant r<\log_k N,$ we then have \begin{multline*}(c_1-\varepsilon)\alpha_f^{-1}\left(1-\frac{1}{\log_k N}\right)^{m_f} N^{\log_k \alpha_f}\log_k^{m_f} N\\ \leqslant (c_1-\varepsilon)r^{m_f}\alpha_f^r<\sum_{n\leqslant N} f(n)<(c_1+\varepsilon)(r+1)^{m_f}\alpha_f^{r+1}\\ <(c_1+\varepsilon)\alpha_f\left(1+\frac{1}{\log_k N}\right)^{m_f}N^{\log_k \alpha_f}\log_k^{m_f}N,\end{multline*} which finishes the proof of the lemma.
\end{proof}
While the statement of Theorem \ref{fNlogN} is very precise, we use it in a less technical way; we need only the fact that $$\sum_{n\leqslant N} f(n)\asymp N^{\log_k\alpha_f} \log^{m_f}N.$$
In order to prove Theorem \ref{mainreg}, we determine an asymptotic result that mimics Theorem \ref{xi} for nonnegative regular functions, making sure to avoid the need of a Mahler eigenvalue. As in the proof of Theorem \ref{fNlogN} above, nonnegativity remains an important assumption.
\begin{proposition} Let $k\geqslant 2$ be an integer and $\{f(n)\}_{n\geqslant 0}$ be a nonnegative integer-valued $k$-regular sequence, which is not eventually zero. Let $\alpha_f\geqslant 1$ be as given by Theorem \ref{fNlogN}. If $F(z)=\sum_{n\geqslant 0}f(n)z^n$, then for any $\varepsilon>0$, as $z\to 1^-$, $$\frac{1}{(1-z)^{\log_k\alpha_f+\varepsilon}}\leqslant F(z)\leqslant \frac{1}{(1-z)^{\log_k\alpha_f-\varepsilon}}.$$
\end{proposition}
\begin{proof} Let $k\geqslant 2$ be a integer and $\{f(n)\}_{n\geqslant 0}$ be a nonnegative integer-valued $k$-regular sequence, which is not eventually zero. Set $$G(z):=\frac{F(z)}{1-z}.$$ Let $\alpha_f\geqslant 1$ and $m_f\geqslant 0$ be as given in Theorem \ref{fNlogN}. For $z\in(0,1)$ define the function $$C(z):=\frac{G(z)}{1+\sum_{n\geqslant 1} n^{\log_k\alpha_f}(\log_k n)^{m_f} z^n}.$$ By Theorem \ref{fNlogN} and the fact that the series in the denominator is nonzero and differentiable on $(0,1)$, we have that on $(0,1)$ the function $C(z)$ is nonzero, differentiable, and bounded above and below by positive constants.
Set $$D(z):=1+\sum_{n\geqslant 1} n^{\log_k\alpha_f}(\log_k n)^{m_f} z^n.$$ We continue by finding asymptotic bounds on the function $D(z)$. To this end, note that for any positive real number $r$, we have $$\frac{1}{(1-z)^r}
=\sum_{n\geqslant 0}\frac{\Gamma(r+n)}{\Gamma(r)n!}z^n,$$ where $\Gamma(z)$ is the Euler $\Gamma$-function. By Sterling's formula, we have that $$
\frac{\Gamma(r+n)}{\Gamma(r)n!}\sim \frac{n^{r-1}}{\Gamma(r)}.$$ It then follows from a classical result of C\'esaro (see P\'olya and Szeg\H{o} \cite[Problem 85 of Part I]{PS1}, that for any given $\varepsilon>0$, as $z\to 1^-$, \begin{equation}\label{Dmajmin}\frac{1}{(1-z)^{\log_k\alpha_f+1+\varepsilon}}\leqslant D(z)\leqslant \frac{1}{(1-z)^{\log_k\alpha_f+1-\varepsilon}}.\end{equation}
Recall $C(z)\asymp 1$, so we have $G(z)\asymp D(z)$, and thus \eqref{Dmajmin} holds with $D(z)$ replaced by $G(z)$. The result now follows since $F(z)=(1-z)G(z)$.
\end{proof}
In order to simplify our further exposition, we make the following definition.
\begin{definition} We call a function $H(z)$ an {\em $\varepsilon$-function}, if there is an $a>0$ such that $H(z)$ is defined on the interval $(1-a,1)$, and as $z\to 1^-$, either $H(z)$ is bounded away from zero and infinity, or the function satisfies $H(z)=o((1-z)^{\varepsilon})$ or $H(z)=o((1-z)^{-\varepsilon})$ for any $\varepsilon>0$.
\end{definition}
\begin{corollary}\label{FL} Let $k\geqslant 2$ be an integer and $\{f(n)\}_{n\geqslant 0}$ be a nonnegative integer-valued $k$-regular sequence, which is not eventually zero. Let $\alpha_f\geqslant 1$ be as given by Theorem \ref{fNlogN}. If $F(z)=\sum_{n\geqslant 0}f(n)z^n$, then there is an $\varepsilon$-function $L(z)$ such that $$F(z)=\frac{L(z)}{(1-z)^{\log_k\alpha_f}}(1+o(1)),$$ as $z\to 1^-$.
\end{corollary}
We now extend Corollary \ref{FL} to include all radial limits as $z$ approaches a root of unity of degree $k^n$ for $n$ large enough. In this way, the following result is the analogue of Theorem \ref{xi} for nonnegative regular functions.
\begin{theorem}\label{regxizto1} Let $k\geqslant 2$ be an integer and $\{f(n)\}_{n\geqslant 0}$ be a nonnegative integer-valued $k$-regular sequence, which is not eventually zero. Let $\alpha_f\geqslant 1$ be as given by Theorem~\ref{fNlogN} and set $F(z)=\sum_{n\geqslant 0}f(n)z^n$. If $\xi$ is a root of unity of degree $k^n$, with $n$ large enough, then there is an integer $d_\xi$ and an $\varepsilon$-function $L_\xi(z)$ such that $$F(\xi z)=\frac{L_\xi(z)}{(1-z)^{\log_k\alpha_f+d_\xi}}(1+o(1)),$$ as $z\to 1^-$.
\end{theorem}
\begin{proof} By Corollary \ref{FL}, there is a real number $\alpha_f\geqslant 1$ and an $\varepsilon$-function $L_0(z)$ such that as $z\to 1^-$, we have $$F(z)=\frac{L_0(z)}{(1-z)^{\log_k\alpha_f}}(1+o(1)).$$
Recall that any $k$-regular function is a $k$-Mahler function; using this fact, let us suppose that $F(z)$ satisfies \eqref{MFE}.
Let $\xi_1$ be a $k$-th root of unity. Then as $z\to 1^-$, we have $$ F(\xi_1 z)=-\sum_{i=1}^d \frac{a_i(\xi_1 z)}{a_0(\xi_1 z)}F(x^{k^i})=\frac{-\sum_{i=1}^d\frac{a_i(\xi_1 z)}{a_0(\xi_1 z)}\alpha_f^{-i}L_0(x^{k^i})}{(1-z)^{\log_k\alpha_f}}(1+o(1)).$$ Note that since $L_0(z)$ is an $\varepsilon$-function, so is $L_0(z^{k^i})$. Thus as $z\to 1^-$, we have $$-\sum_{i=1}^d\frac{a_i(\xi_1 z)}{a_0(\xi_1 z)}\alpha_f^{-i}L_0(x^{k^i})=(1-z)^{d_1}L_1(z)(1+o(1)),$$ for some $\varepsilon$-function $L_1(z)$ and some integer $d_1\in\mathbb{Z}$, which depends on the polynomials $a_i(z)$ ($i=0,\ldots,d$), $\alpha_f$, and $\xi_1$. Thus $$F(\xi_1 z)=\frac{L_1(z)}{(1-z)^{\log_k \alpha_f-d_1}}(1+o(1)),$$ as $z\to 1^-$.
Continuing in this way, if $\xi$ is a root of unity of degree $k^n$, with $n$ large enough so that $a_i(\xi)\neq 0$ for each $i=0,\ldots,d$, then there is an $\varepsilon$-function $L_\xi(z)$ and an integer $d_\xi$, such that $$F(\xi z)=\frac{L_\xi(z)}{(1-z)^{\log_k\alpha_f+d_\xi}}(1+o(1))$$ as $z\to 1^-$, which is the desired result.
\end{proof}
With our asymptotic results in place, we can now prove Theorem \ref{mainreg}.
\begin{proof}[Proof of Theorem \ref{mainreg}] This proof follows very close our proof of Theorem \ref{main}, but with the $\varepsilon$-functions $L(z)$ given by Theorem \ref{regxizto1} in place of the functions $C(z)$ from Theorem \ref{xi}.
To start, as in our proof of Theorem \ref{main}, and towards a contradiction, assume the theorem is false, so that we have an algebraic relation \begin{equation}\label{arreg}\sum_{{\bf m}=(m_1,\ldots,m_d)\in{\bf M}}p_{\bf m}(z,G_1(z),\ldots,G_s(z))F_1(z)^{m_1}\cdots F_d(z)^{m_d}=0,\end{equation} where the set ${\bf M}\subseteq \mathbb{Z}^{d}_{\geqslant 0}$ is finite and none of the polynomials $p_{\bf m}(z,G_1(z),\ldots,G_s(z))$ in $\mathbb{C}[z][\mathcal{M}]$ is identically zero. Again, without loss of generality, we may suppose that the polynomial $\sum_{{\bf m}}p_{\bf m}(z,w_1\ldots,w_s)y_1^{m_1}\cdots y_d^{m_d}$ in $d+s+1$ variables is irreducible.
As $z\to 1^-$, for $\xi$ any root of unity of degree $k^n$, with $n$ large enough, in the notation of Theorem \ref{regxizto1}, we have \begin{equation}\label{algterm}F_1(\xi z)^{m_1}\cdots F_d(\xi z)^{m_d}=\frac{ L_{\xi, {\bf m}}(z)}{(1-z)^{m_1\log_k\lambda_{F_1}+\cdots+m_d\log_k\lambda_{F_d}+|{\bf m}|\cdot m}}(1+o(1)),\end{equation} where $|{\bf m}|=m_1+\cdots+m_d$, $m$ is an integer, and $$L_{\xi,{\bf m}}(z):=\prod_{i=1}^d L_{F_i,\xi}(z)^{m_i}$$ is an $\varepsilon$-function. Note that $L_{\xi,{\bf m}}(z)$ is an $\varepsilon$-function since a product of $\varepsilon$-functions is again an $\varepsilon$-function.
Let ${\bf M}_{\max}\subseteq{\bf M}$ be the (nonempty) set of indices ${\bf m}=(m_1,\ldots,m_d)$ such that the quantity $$\delta:= m_1\log_k\lambda_{F_1}+\cdots+m_d\log_k\lambda_{F_d}+|{\bf m}|\cdot m$$ is maximal. Note here that the asymptotic properties of the function $L_{\xi,{\bf m}}(z)$ do not effect the maximal asymptotics in a way that changes the set ${\bf M}_{\max}$.
Again, we claim that the set ${\bf M}_{\max}$ contains only one element. To see this, suppose that ${\bf m},{\bf m}'\in{\bf M}_{\max}.$ Then $$m_1\log_k\lambda_{F_1}+\cdots+m_d\log_k\lambda_{F_d}+|{\bf m}|\cdot m=m_1'\log_k\lambda_{F_1}+\cdots+m_d'\log_k\lambda_{F_d}+|{\bf m}'|\cdot m=\delta,$$ and so $$(m_1-m_1')\log_k\lambda_{F_1}+\cdots+(m_d-m_d')\log_k\lambda_{F_d}+(|{\bf m}|-|{\bf m}'|)m=0.$$ Since the numbers $k,\lambda_{F_1},\ldots,\lambda_{F_d}$ are multiplicatively independent, the numbers $\log_k\lambda_{F_1},\ldots,\log_k\lambda_{F_d},m$ are linearly independent. Thus $m_i=m_i'$ for each $i\in\{1,\ldots,d\}$ and ${\bf m}={\bf m}'.$
Using the uniqueness of the term of index ${\bf m}_{\max}$ with maximal asymptotics, we multiply the algebraic relation \eqref{ar} by $(1-z)^\delta/L_{\xi,{\bf m}_{\max}}(z)$ and send $z\to 1^-$. Then for a root of unity $\xi$ of degree $k^n$, for $n$ large enough and for which $G_1(\xi),\ldots,G_s(\xi)$ each exist, we gain the equality $$p_{{\bf m}_{\max}}(\xi,G_1(\xi),\ldots,G_s(\xi))=0.$$ Since there are infinitely many such $\xi$ which are dense on the unit circle, and $p_{{\bf m}_{\max}}(z,G_1(z),\ldots,G_s(z))$ is a meromorphic function, it must be that $$p_{{\bf m}_{\max}}(z,G_1(z),\ldots,G_s(z))=0$$ identically, contradicting our original assumption.
\end{proof}
\section{Concluding remarks}\label{SecCon}
In this paper, we presented general algebraic independence results for Mahler functions and nonnegative regular functions. We did this by making use of the arithmetic properties of the eigenvalues of Mahler functions, and by the arithmetic properties of certain constants, which we denoted $\alpha_f$, associated to regular sequences $\{f(n)\}_{n\geqslant 0}$.
Before ending our paper, we provide an extended example that illustrates some of the objects that we have considered as well as a few corollaries and questions that may be of interest.
\subsection{An extended example: Stern's sequence} Let $\{s(n)\}_{n\geqslant 0}$ be {\em Stern's diatomic sequence}. Stern's sequence is determined by the relations $s(0)=0$, $s(1)=1$, and for $n\geqslant 0$, by $$s(2n)=s(n), \quad\mbox{and}\quad s(2n+1)=s(n)+s(n+1).$$ This sequence is $2$-regular and is determined by the vectors and matrices $${\bf w}={\bf v}=[1\ 0]^T\quad\mbox{and}\quad ({\bf A}_0,{\bf A}_1)=\left(\left[\begin{array}{rr} 1&1\\ 0&1\end{array}\right],\left[\begin{array}{rr} 1&0\\ 1&1\end{array}\right]\right).$$ Further, the generating function $S(z)=\sum_{n\geqslant 0}s(n)z^n$ is $2$-Mahler and satisfies the functional $$zS(z)-(1+z+z^2)S(z^2)=0.$$ The characteristic polynomial for $S(z)$ is linear; it is $p_S(\lambda)=\lambda-3.$ Thus $\lambda_S=3$. Of course, since $S(z)$ is regular, the constant $\alpha_s$ also exists, and in this case $\alpha_s=\lambda_S=3.$ In fact, this equality holds for all regular functions; that is, if $F(z)=\sum_{n\geqslant 0}f(n)z^n$ is regular and $\lambda_F$ exists, then $\lambda_F=\alpha_f$.
To illustrate the effect of these constants, we use two figures. In Figure \ref{Stern15}, we have plotted the values of the Stern sequence in the interval $[2^{15}, 2^{16}]$. As the power of two is increased this picture will fill out with more values, but already at these values it is quite stable.
\begin{figure}
\caption{Stern's diatomic sequence in the interval $[2^{15}
\label{Stern15}
\end{figure}
\noindent Notice that the Stern sequence, while definitely exhibiting structure is quite erratic as well. But if we consider the weighted partial sums $N^{-\log_2 3}\sum_{n\leqslant N}s(n)$ for $N$ between two large powers of two, a whole other structure arises; see Figure \ref{Sternperiod}.
\begin{figure}
\caption{The weighted partial sums $N^{-\log_2 3}
\label{Sternperiod}
\end{figure}
\noindent Figure \ref{Sternperiod} illustrates Theorem \ref{fNlogN}. In the case of the Stern sequence, the upper and lower (asymptotic) bounds given are $\alpha_s=3$ and $\alpha_s^{-1}=1/3,$ though it is quite clear from Figure \ref{Sternperiod} that these bounds are not optimal.
Many results have been proven regarding the transcendence and algebraic independence of the generating function of the Stern sequence, so that providing interesting corollaries of Theorems \ref{main} and \ref{mainreg} using this function is a bit of a challenge. We proved \cite{C2010} the transcendence of $S(z)$, and Bundschuh \cite{B2012} extended that result by showing that the derivatives $$S(z),S'(z),S^{(2)}(z),\ldots,S^{(n)}(z),\ldots$$ are algebraically independent over $\mathbb{C}(z)$. But if we throw a wrench in the works, things get more interesting. We can pair the Stern function with other functions and give results which previous methods could not attack. Corollary \ref{FS} stated in the Introduction is a good example of this. We give another example here.
The {\em Baum-Sweet sequence} is given by the recurrences $b_0=1,$ $b_{4n}=b_{2n+1}=b_n$, and $b_{4n+2}=0.$ The sequence $\{b_n\}_{n\geqslant 0}$ is $2$-automatic (and so also $2$-regular) and as a consequence of the above relations, its generating function $B(z)=\sum_{n\geqslant 0}b_n z^n$ satisfies the $2$-Mahler equation $$B(z)-zB(z^2)-B(z^4)=0.$$ The function $B(z)$ has the characteristic polynomial $p_B(\lambda)=\lambda^2-\lambda-1$ and $\lambda_B=(1+\sqrt{5})/2$. We thus have the following corollary to Theorem \ref{main}.
\begin{corollary} For $S(z)$ the Stern's function and $B(z)$ the Baum-Sweet function, $$\trdeg_{\mathbb{C}(z)}\mathbb{C}(z)(S(z),B(z))=2.$$
\end{corollary}
\subsection{Hypertranscendence of Mahler functions} As mentioned in the Introduction, B\'ezivin \cite{B1994} proved that a Mahler function that is $D$-finite (that is, satisfies a homogeneous linear differential equation with polynomial coefficients) is rational. Of course, the general question of hypertranscendence of irrational Mahler functions remains open. Recall that a function is called {\em hypertranscendental} if it does not satisfy an algebraic differential equation with polynomial coefficients. We reiterate the following question.
\begin{question} Is it true that an irrational Mahler function is hypertranscendental?
\end{question}
Partial progress has been made in the case of Mahler functions of degree $1$ (see Bundschuh \cite{B2012, B2013}), though there is not a single known example of a hypertranscendental Mahler function of degree $2$ or greater.
Towards this question, we can offer the following modest result, which is a corollary to Theorems \ref{main} and \ref{mainreg}.
\begin{corollary}\label{deg3} Let $F(z)=\sum_{n\geqslant 0}f(n)z^n$ be either a Mahler function for which $\lambda_F$ exists and $k$ and $\lambda_F$ are multiplicatively independent, or a nonnegative regular function with $k$ and $\alpha_f$ multiplicatively independent. If $p(z,z_0,\ldots,z_n)\in\mathbb{C}[z,z_0,\ldots,z_n]$ is any polynomial with $p(z,F(z),F'(z),\ldots,F^{(n)}(z))=0$, then for any $c\in\mathbb{C}$ $$\deg p(c,z_0,\ldots,z_n)\geqslant 3.$$
\end{corollary}
\noindent Note that Corollary \ref{deg3} also holds with the addition of any number of meromorphic functions. In this way, it seems natural to consider as well the hypertranscendence of Mahler functions over polynomials in meromorphic functions.
\subsection{A question about $e$ and $\pi$ and Mahler numbers} We end this paper with one more corollary (more a novelty of our method) as well as a question, which we hope will stimulate further research using possibly a combination of algebraic and asymptotic techniques for algebraic independence.
\begin{corollary}\label{Szeta} If $F(z)$ is a $k$-Mahler function for which $\lambda_F$ exists and $k$ and $\lambda_F$ are multiplicatively independent, then $$\trdeg_{\mathbb{C}(z)}\mathbb{C}(z)(F(z),G(z),\zeta(z))=3,$$ where $G(z)$ is any $D$-finite function and $\zeta(z)$ is Riemann's zeta function.
\end{corollary}
\noindent The proof of this corollary, follows from the fact that $\trdeg_{\mathbb{C}(z)(G(z),\zeta(z))}(F(z))=1$, and $\trdeg_{\mathbb{C}(z)}(G(z),\zeta(z))=2$, which is implied by the work of Ostrowski \cite{O1920} and methods that can be found in Kaplansky's short book \cite{K1976}.
Corollary \ref{Szeta} can be viewed as a sort of classification, which is saying that Mahler functions, functions that satisfy linear differential equations, and the zeta function are very different sorts of functions. It would be extremely interesting interesting if one could prove an algebraic independence result about the special values of such functions. In particular, one would like to answer the following question.
\begin{question}\label{ez} Is it true for a $k$-Mahler function $F(z)$ that $\trdeg_{\mathbb{Q}}\mathbb{Q}(F(\alpha),e)=2$ and/or $\trdeg_{\mathbb{Q}}\mathbb{Q}(F(\alpha),\pi)=2$ for any reasonable choice of algebraic $\alpha$?
\end{question}
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\end{document} | math | 38,943 |
\begin{document}
\title[Two Dimensional Diamond-alpha Inequalities on Time
Scales]{H\"{o}lder's and Hardy's Two Dimensional Diamond-alpha Inequalities on Time Scales\footnote{Accepted for publication (October 21, 2009) in the journal \emph{Annals of the University
of Craiova, Mathematics and Computer Science Series}
(\url{http://inf.ucv.ro/~ami}).}}
\author[M. R. Sidi Ammi, D. F. M. Torres]{Moulay Rchid Sidi
Ammi$^1$, Delfim F. M. Torres$^2$}
\address{$^{1}$ Department of Mathematics,
Fac. des Sci. et Tech. (FST-Errachidia),
University My Ismail, BP: 509 Boutalamine,
Errachidia, 52000, Morocco.}
\email{[email protected]}
\address{$^{2}$ African Institute for Mathematical Sciences,
6-8 Melrose Road, Muizenberg 7945, Cape Town, South Africa;
\newline
On leave of absence from:\newline
Department of Mathematics,
University of Aveiro, 3810-193 Aveiro, Portugal.}
\email{[email protected], [email protected]}
\subjclass[2000]{Primary 26D15; Secondary 39A13.}
\keywords{Time scales, diamond-alpha integrals,
dynamic inequalities, two-dimensional H\"{o}lder's inequalities, Hardy's inequalities.}
\date{}
\begin{abstract}
We prove a two dimensional H\"{o}lder and reverse-H\"{o}lder
inequality on time scales via the diamond-alpha integral. Other
integral inequalities are established as well, which have as
corollaries some recent proved Hardy-type inequalities on time
scales.
\end{abstract}
\maketitle
\section{Introduction}
The theory and applications of dynamic derivatives on time scales
is receiving an increase of interest and attention. This relative
new area was created in order to unify and generalize discrete and
continuous analysis. It was introduced by Stefan Hilger
\cite{h2,h3}, then used as a tool in several computational and
numerical applications \cite{abra,b1,b2}. One important and very
active subject being developed within the theory of time scales is
the study of inequalities
\cite{abp,ozkan,Stef,srd,sidel,adnan,wong}.
The primary purpose of
this paper is to prove more general two dimensional
reverse-H\"{o}lder's and H\"{o}lder's inequalities on time scales,
using the recent theory of combined dynamic derivatives and the
more general notion of diamond-$\alpha$ integral
\cite{Rogers,Sheng,sfhd}. As particular cases, we get Hardy's
inequalities \cite{krm,adnan}.
H\"{o}lder's inequalities and their extensions have received
considerable attention in the theory
of differential and difference equations,
as well as other areas of mathematics
\cite{ozkan,srd,adnan,wong}. Recently, authors in
\cite{adnan} proved a time scale version of H\"{o}lder's
inequality in the two dimensional case, by using the
$\Delta$-integral. Here we extend this result to more general
diamond-$\alpha$ integral inequalities. The results in
\cite{adnan} are obtained choosing $\alpha = 1$; different
inequalities on time scales follow by choosing $0 \le \alpha < 1$
(\textrm{e.g.}, for $\alpha = 0$ one gets new
$\nabla$-integral inequalities).
\section{Preliminaries}
\label{sec:Prel}
A time scale $\mathbb{T}$ is an arbitrary nonempty closed subset
of the real numbers. Let $\mathbb{T}$ be a time scale with the
topology that it inherits from the real numbers. For $t \in
\mathbb{T}$, we define the forward jump operator $\sigma:
\mathbb{T} \rightarrow \mathbb{T}$ by $\sigma(t)= \inf \left \{s
\in \mathbb{T}: s >t \right\}$, and the backward jump operator
$\rho: \mathbb{T} \rightarrow \mathbb{T}$ by $\rho(t)= \sup \left
\{s \in \mathbb{T}: s < t \right \}$.
If $\sigma(t) > t$ we say that $t$ is right-scattered, while if
$\rho(t) < t$ we say that $t$ is left-scattered. Points that are
simultaneously right-scattered and left-scattered are said to be
isolated. If $\sigma(t)=t$, then $t$ is called right-dense; if
$\rho(t)=t$, then $t$ is called left-dense. Points that are
right-dense and left-dense at the same time are called dense. The
mappings $\mu, \nu: \mathbb{T} \rightarrow [0, +\infty)$ defined
by $\mu(t):=\sigma(t)-t$ and $\nu(t):=t-\rho(t)$ are called,
respectively, the forward and backward graininess function.
Given a time scale $\mathbb{T}$, we introduce the sets
$\mathbb{T}^{\kappa}$, $\mathbb{T}_{\kappa}$, and
$\mathbb{T}^{\kappa}_{\kappa}$ as follows. If $\mathbb{T}$ has a
left-scattered maximum $t_{1}$, then $\mathbb{T}^{\kappa}=
\mathbb{T}-\{t_{1} \}$, otherwise $\mathbb{T}^{\kappa}=
\mathbb{T}$. If $\mathbb{T}$ has a right-scattered minimum
$t_{2}$, then $\mathbb{T}_{\kappa}= \mathbb{T}-\{t_{2} \}$,
otherwise $\mathbb{T}_{\kappa}= \mathbb{T}$. Finally,
$\mathbb{T}_{\kappa}^{\kappa}= \mathbb{T}^{\kappa} \bigcap
\mathbb{T}_{\kappa}$.
Let $f: \mathbb{T}\rightarrow \mathbb{R}$ be a real valued
function on a time scale $\mathbb{T}$. Then, for $t \in
\mathbb{T}^{\kappa}$, we define $f^{\Delta}(t)$ to be the number,
if one exists, such that for all $\epsilon >0$, there is a
neighborhood $U$ of $t$ such that for all $s \in U$,
$$
\left|f(\sigma(t))- f(s)-f^{\Delta}(t)(\sigma(t)-s)\right| \leq
\epsilon |\sigma(t)-s|.
$$
We say that $f$ is delta differentiable on $\mathbb{T}^{\kappa}$
provided $f^{\Delta}(t)$ exists for all $t \in
\mathbb{T}^{\kappa}$. Similarly, for $t \in \mathbb{T}_{\kappa}$
we define $f^{\nabla}(t)$ to be the number, if one exists, such
that for all $\epsilon >0$, there is a neighborhood $V$ of $t$
such that for all $s \in V$
$$
\left|f(\rho(t))- f(s)-f^{\nabla}(t)(\rho(t)-s)\right| \leq
\epsilon |\rho(t)-s|.
$$
We say that $f$ is nabla differentiable on $\mathbb{T}_{\kappa}$,
provided that $f^{\nabla}(t)$ exists for all $t \in
\mathbb{T}_{\kappa}$.
For $f:\mathbb{T}\rightarrow \mathbb{R}$ we define the function
$f^{\sigma}: \mathbb{T}\rightarrow \mathbb{R}$ by
$f^{\sigma}(t)=f(\sigma(t))$ for all $t \in \mathbb{T}$, that is,
$f^{\sigma}= f\circ \sigma$. Similarly, we define the function
$f^{\rho}: \mathbb{T}\rightarrow \mathbb{R}$ by
$f^{\rho}(t)=f(\rho(t))$ for all $t \in \mathbb{T}$, that is,
$f^{\rho}= f\circ \rho$.
A function $f: \mathbb{T} \rightarrow \mathbb{R} $ is called
rd-continuous, provided it is continuous at all right-dense points
in $\mathbb{T}$ and its left-sided limits finite at all left-dense
points in $\mathbb{T}$. A function $f: \mathbb{T} \rightarrow
\mathbb{R} $ is called ld-continuous, provided it is continuous at
all left-dense points in $\mathbb{T}$ and its right-sided limits
finite at all right-dense points in $\mathbb{T}$.
A function $F: \mathbb{T} \rightarrow \mathbb{R} $ is called a
delta antiderivative of $f: \mathbb{T} \rightarrow \mathbb{R}$,
provided $F^{\Delta}(t)=f(t)$ holds for all $t \in
\mathbb{T}^{\kappa}$. Then the delta integral of $f$ is defined by
$$\int^b_a f(t)\Delta t=F(b)-F(a) \, .$$
A function $G: \mathbb{T} \rightarrow \mathbb{R} $ is called a
nabla antiderivative of $g: \mathbb{T} \rightarrow \mathbb{R}$,
provided $G^{\nabla}(t)=g(t)$ holds for all $t \in
\mathbb{T}_{\kappa}$. Then the nabla integral of $g$ is defined by
$\int^b_a g(t)\nabla t=G(b)-G(a)$. For more on the delta and nabla
calculus on time scales, we refer the reader to \cite{abra,b1,b2}.
We review now the recent diamond-$\alpha$ derivative and
integral \cite{Rogers,Sheng,sfhd}.
Let $\mathbb{T}$ be a time scale and $f$ differentiable on
$\mathbb{T}$ in the $\Delta$ and $\nabla$ senses. For $t \in
\mathbb{T}$, we define the diamond-$\alpha$ dynamic derivative
$f^{\diamondsuit_{\alpha}}(t)$ by
$$
f^{\diamondsuit_{\alpha}}(t)= \alpha
f^{\Delta}(t)+(1-\alpha)f^{\nabla}(t), \quad 0 \leq \alpha \leq 1.
$$
Thus, $f$ is diamond-$\alpha$ differentiable if and only if $f$ is
$\Delta$ and $\nabla$ differentiable. The diamond-$\alpha$
derivative reduces to the standard $\Delta$ derivative for
$\alpha =1$, or the standard $\nabla$ derivative for $\alpha =0$.
On the other hand, it represents a ``weighted derivative'' for
$\alpha \in (0,1)$. Diamond-$\alpha$ derivatives have shown in
computational experiments to provide efficient and balanced
approximation formulas, leading to the design of more reliable
numerical methods \cite{Sheng,sfhd}.
Let $f, g: \mathbb{T} \rightarrow \mathbb{R}$ be diamond-$\alpha$
differentiable at $t \in \mathbb{T}$. Then,
\begin{itemize}
\item[(i)] $f+g: \mathbb{T} \rightarrow \mathbb{R}$ is
diamond-$\alpha$ differentiable at $t \in \mathbb{T}$ with
$$ (f+g)^{\diamondsuit^{\alpha}}(t)=
(f)^{\diamondsuit^{\alpha}}(t)+(g)^{\diamondsuit^{\alpha}}(t).
$$
\item[(ii)] For any constant $c$, $cf: \mathbb{T} \rightarrow
\mathbb{R}$
is diamond-$\alpha$ differentiable at $t \in \mathbb{T}$ with
$$
(cf)^{\diamondsuit^{\alpha}}(t)= c(f)^{\diamondsuit^{\alpha}}(t).
$$
\item[(ii)] $fg: \mathbb{T} \rightarrow \mathbb{R}$ is
diamond-$\alpha$ differentiable at $t \in \mathbb{T}$ with
$$
(fg)^{\diamondsuit^{\alpha}}(t)=
(f)^{\diamondsuit^{\alpha}}(t)g(t)+ \alpha
f^{\sigma}(t)(g)^{\Delta}(t) +(1-\alpha)
f^{\rho}(t)(g)^{\nabla}(t).
$$
\end{itemize}
Let $a, t \in \mathbb{T}$, and $h: \mathbb{T} \rightarrow
\mathbb{R}$. Then, the diamond-$\alpha$ integral from $a$ to $t$
of $h$ is defined by
$$
\int_{a}^{t}h(\tau) \diamondsuit_{\alpha} \tau = \alpha
\int_{a}^{t}h(\tau) \Delta \tau +(1- \alpha) \int_{a}^{t}h(\tau)
\nabla \tau, \quad 0 \leq \alpha \leq 1 \, ,
$$
provided that there exist delta and nabla integrals of $h$ on
$\mathbb{T}$. It is clear that the diamond-$\alpha$ integral of
$h$ exists when $h$ is a continuous function. Let $a$, $b$, $t \in
\mathbb{T}$, $c \in \mathbb{R}$, and $f$ and $g$ be continuous
functions on $[a,b] \cap \mathbb{T}$. Then (\textrm{cf.}
\cite[Theorem~3.7]{sfhd} and \cite[Lemma~2.2]{srd}), the following
properties hold:
\begin{itemize}
\item[(a)] $\int_{a}^{t}\left( f(\tau)+g(\tau) \right)
\diamondsuit_{\alpha} \tau = \int_{a}^{t} f(\tau)
\diamondsuit_{\alpha} \tau + \int_{a}^{t} g(\tau)
\diamondsuit_{\alpha} \tau$;
\item[(b)] $\int_{a}^{t} c f(\tau) \diamondsuit_{\alpha} \tau = c
\int_{a}^{t} f(\tau) \diamondsuit_{\alpha} \tau$;
\item[(c)] $\int_{a}^{t} f(\tau) \diamondsuit_{\alpha} \tau =
\int_{a}^{b} f(\tau) \diamondsuit_{\alpha} \tau + \int_{b}^{t}
f(\tau) \diamondsuit_{\alpha} \tau$.
\item[(d)] If $f(t)\geq 0$ for all $t\in[a,b]_{\mathbb{T}}$,
then $\int_a^b f(t)\Diamond_\alpha t\geq 0$.
\item[(e)] If $f(t)\leq g(t)$ for all $t\in[a,b]_{\mathbb{T}}$,
then $\int_a^b f(t)\Diamond_\alpha t\leq\int_a^b g(t)\Diamond_\alpha t$.
\item[(f)] If $f(t)\geq 0$ for all $t\in[a,b]_{\mathbb{T}}$, then
$f(t)=0$ if and
only if $\int_a^b f(t)\Diamond_\alpha t=0$.
\end{itemize}
\section{Main Results}
\label{sec:MR}
We prove new diamond-$\alpha$ inequalities. As
particular cases we get $\Delta$-inequalities on time scales for
$\alpha = 1$, and $\nabla$-inequalities on time scales when
$\alpha = 0$. In the sequel we use $[a, b]$ to denote
$[a, b] \cap \mathbb{T}$. We also suppose that all integrals converge.
\begin{theorem}[reverse diamond-$\alpha$ H\"{o}lder's inequality]
\label{thm1} Let $\mathbb{T}$ be a time scale, $a$, $b \in
\mathbb{T}$ with $a < b$, and $f$ and $g$ be two positive
functions satisfying $0< m \leq \frac{f^{p}}{g^{q}} \leq M <
+\infty $ on the set $[a, b]$. If $\frac{1}{p}+ \frac{1}{q}=1$
with $p> 1$, then
\begin{equation}
\label{eq:rdaHi} \left(\int_{a}^{b} f^{p}(t) \diamondsuit_{\alpha}
t \right)^{\frac{1}{p}} \left ( \int_{a}^{b} g^{q}(t)
\diamondsuit_{\alpha} t \right)^{\frac{1}{q}} \leq
\left(\frac{M}{m}\right)^{\frac{1}{pq}} \int_{a}^{b} f(t) g(t)
\diamondsuit_{\alpha} t.
\end{equation}
\end{theorem}
\begin{proof}
We have $\frac{f^{p}}{g^{q}} \leq M$. Then, $ f^{\frac{p}{q}} \leq M^{\frac{1}{q}} g$. Multiplying by $f>0$, it follows that
$$
f^{p}= f^{1+\frac{p}{q}}\leq M^{\frac{1}{q}}fg.
$$
Using properties $(e)$ and $(b)$, we can write that
\begin{equation} \label{eq1}
\left ( \int_{a}^{b} f^{p}(t) \diamondsuit_{\alpha} t
\right)^{\frac{1}{p}} \leq M^{\frac{1}{pq}} \left ( \int_{a}^{b}
f(t) g(t) \diamondsuit_{\alpha} t \right)^{\frac{1}{p}}.
\end{equation}
In the same manner, we have $m^{\frac{1}{p}} g^{\frac{q}{p}} \leq
f$. Then,
$$
\int_{a}^{b} m^{\frac{1}{p}} g^{q}(t) \diamondsuit_{\alpha} t =
m^{\frac{1}{p}} \int_{a}^{b} g^{1+\frac{q}{p}}(t)
\diamondsuit_{\alpha} t \leq \int_{a}^{b} f(t) g(t)
\diamondsuit_{\alpha} t.
$$
We obtain that
\begin{equation} \label{eq2}
m^{\frac{1}{pq}} \left(\int_{a}^{b} g^{q}(t) \diamondsuit_{\alpha}
t \right)^{\frac{1}{q}} \leq \left ( \int_{a}^{b} f(t) g(t)
\diamondsuit_{\alpha} t \right)^{\frac{1}{q}}.
\end{equation}
Gathering \eqref{eq1} and \eqref{eq2}, the intended
inequality \eqref{eq:rdaHi} is proved.
\end{proof}
\begin{remark}
For the particular case $\mathbb{T}=\mathbb{R}$,
Theorem~\ref{thm1} gives \cite[Theorem~2.1]{krm}. For $\alpha =
1$, Theorem~\ref{thm1} coincides with \cite[Lemma~1]{adnan}.
\end{remark}
We now define the diamond-$\alpha$ integral for a function of two variables.
The double integral is defined as an iterated integral.
Let $\mathbb{T}$ be a time scale with $a, b \in \mathbb{T}$,
$a < b$, and $f$ be a real-valued function on $\mathbb{T} \times \mathbb{T}$.
Because we need notation for partial derivatives with respect to time scale
variables $x$ and $y$ we denote the time scale partial derivative of $f(x,y)$
with respect to $x$ by $f^{\diamondsuit_{\alpha}^1}(x,y)$ and let
$f^{\diamondsuit_{\alpha}^2}(x,y)$ denote the time scale partial derivative
with respect to $y$. Definition of these partial derivatives are now given.
Fix an arbitrary $y \in \mathbb{T}$. Then the diamond-$\alpha$ derivative
of function
\begin{gather*}
\mathbb{T} \rightarrow \mathbb{R} \\
x \mapsto f(x,y)
\end{gather*}
is denoted by $f^{\diamondsuit_{\alpha}^1}$.
Let now $x \in \mathbb{T}$. The diamond-$\alpha$ derivative
of function
\begin{gather*}
\mathbb{T} \rightarrow \mathbb{R} \\
y \mapsto f(x,y)
\end{gather*}
is denoted by $f^{\diamondsuit_{\alpha}^2}$.
If function $f$ has a $\diamondsuit_{\alpha}^1$ antiderivative $A$, \textrm{i.e.},
$A^{\diamondsuit_{\alpha}^1} = f$, and $A$ has a $\diamondsuit_{\alpha}^2$
antiderivative $B$, \textrm{i.e.}, $B^{\diamondsuit_{\alpha}^2} = A$, then
\begin{equation*}
\begin{split}
\int_{a}^{b} \int_{a}^{b} f(x, y) \diamondsuit_{\alpha} x \diamondsuit_{\alpha} y
&:= \int_{a}^{b} \left( A(b,y) - A(a,y)\right) \diamondsuit_{\alpha} y \\
&= B(b,b) - B(b,a) - B(a,b) + B(a,a) \, .
\end{split}
\end{equation*}
Note that $\left(B^{\diamondsuit_{\alpha}^2}\right)^{\diamondsuit_{\alpha}^1} = f$.
\begin{theorem}[two dimensional diamond-$\alpha$ H\"{o}lder's inequality]
\label{thm:2} Let $\mathbb{T}$ be a time scale, $a$, $b \in
\mathbb{T}$ with $a < b$, $f, g, h : [a, b] \times [a, b]
\rightarrow \mathbb{R}$ be $\diamondsuit_{\alpha}$-integrable
functions, and $\frac{1}{p}+ \frac{1}{q}=1$ with $ p> 1$. Then,
\begin{multline}\label{eq3}
\int_{a}^{b} \int_{a}^{b}|h(x, y)f(x, y)g(x, y)|
\diamondsuit_{\alpha} x \diamondsuit_{\alpha} y \\
\leq \left ( \int_{a}^{b} \int_{a}^{b}|h(x, y)|f(x, y)|^{p}
\diamondsuit_{\alpha} x \diamondsuit_{\alpha} y
\right)^{\frac{1}{p}} \left ( \int_{a}^{b} \int_{a}^{b}|h(x,
y)|g(x, y)|^{q} \diamondsuit_{\alpha} x \diamondsuit_{\alpha} y
\right)^{\frac{1}{q}}.
\end{multline}
\end{theorem}
\begin{proof}
Inequality \eqref{eq3} is trivially true in the case when $f$ or
$g$ or $h$ is identically zero. Suppose that
$$
\left (\int_{a}^{b} \int_{a}^{b}|h(x, y)||f(x,
y)|^{\frac{1}{p}}\diamondsuit_{\alpha} x \diamondsuit_{\alpha} y
\right ) \left ( \int_{a}^{b} \int_{a}^{b}|h(x, y)||g(x,
y)|^{\frac{1}{q}} \diamondsuit_{\alpha} x \diamondsuit_{\alpha} y
\right)\neq 0 \, ,
$$
and let
$$
A(x, y)= \frac{|h^{\frac{1}{p}}(x, y)||f(x, y)|}{\int_{a}^{b}
\int_{a}^{b}|h(x, y)||f(x, y)|^{\frac{1}{p}} \diamondsuit_{\alpha}
x \diamondsuit_{\alpha} y} \, ,
$$
and
$$
B(x, y)= \frac{|h^{\frac{1}{q}}(x, y)||g(x, y)|}{\int_{a}^{b}
\int_{a}^{b}|h(x, y)||g(x, y)|^{\frac{1}{q}} \diamondsuit_{\alpha}
x \diamondsuit_{\alpha} y} \, .
$$
From the well-known Young's inequality $\xi \lambda \leq
\frac{\xi^{p}}{p}+ \frac{\lambda^{q}}{q}$, valid for nonnegative
real numbers $\xi$ and $\lambda$, we have that
\begin{equation*}
\begin{split}
\int_{a}^{b} & \int_{a}^{b} A(x, y) B(x, y) \diamondsuit_{\alpha}
x
\diamondsuit_{\alpha} y \\
& \leq \int_{a}^{b} \int_{a}^{b} \left [ \frac{A^{p}(x, y)}{p}+
\frac{B^{q}(x, y)}{q}\right ]
\diamondsuit_{\alpha} x \diamondsuit_{\alpha} y\\
& \leq \frac{1}{p}\int_{a}^{b} \int_{a}^{b}
\frac{|h||f|^{p}\diamondsuit_{\alpha} x \diamondsuit_{\alpha}
y}{\int_{a}^{b} \int_{a}^{b}|h||f|^{p}} + \frac{1}{q}\int_{a}^{b}
\int_{a}^{b} \frac{|h||g|^{q}\diamondsuit_{\alpha} x
\diamondsuit_{\alpha} y}{\int_{a}^{b} \int_{a}^{b}|h||g|^{q}}\\
& \leq \frac{1}{p}+ \frac{1}{q}=1 \, ,
\end{split}
\end{equation*}
and the desired result follows.
\end{proof}
\begin{remark}
For the particular case $\alpha=1$, Theorem~\ref{thm:2} coincides
with \cite[Theorem~4]{adnan}.
\end{remark}
\begin{theorem}[two dimensional diamond-$\alpha$ Cauchy-Schwartz's inequality]
Let $\mathbb{T}$ be a time scale, $a$, $b \in \mathbb{T}$ with $a
< b$. For $\diamondsuit_{\alpha}$-integrable functions $f, g, h:
[a, b] \times [a, b] \rightarrow \mathbb{R}$, we have:
\begin{multline}\label{eq4}
\int_{a}^{b} \int_{a}^{b}|h(x, y)f(x, y)g(x, y)|
\diamondsuit_{\alpha} x \diamondsuit_{\alpha} y \\
\leq \sqrt{ \left ( \int_{a}^{b} \int_{a}^{b}|h(x, y)||f(x,
y)|^{2} \diamondsuit_{\alpha} x \diamondsuit_{\alpha} y \right)
\left (
\int_{a}^{b} \int_{a}^{b}|h(x, y)||g(x, y)|^{2}
\diamondsuit_{\alpha} x \diamondsuit_{\alpha} y \right)}.
\end{multline}
\end{theorem}
\begin{proof}
The Cauchy-Schwartz inequality \eqref{eq4} is the particular case
$p=q=2$ of \eqref{eq3}.
\end{proof}
We now obtain some general results for estimating the diamond-alpha double integral
$\int_{a}^{b} \int_{a}^{b} K(x, y) f(x)g(y) \diamondsuit_{\alpha}
x \diamondsuit_{\alpha} y $.
\begin{theorem}[diamond-$\alpha$ Hardy-type inequalities]
\label{thm:da:Hineq} Let $\mathbb{T}$ be a time scale, $a$, $b \in
\mathbb{T}$ with $a < b$, and $K(x, y)$, $f(x)$, $g(y)$,
$\varphi(x)$, and $\psi(y)$ be nonnegative functions. Let $$F(x)=\int_{a}^{b} K(x, y)
\psi^{-p}(y)\diamondsuit_{\alpha}y$$ and $$G(y)=\int_{a}^{b} K(x,y) \varphi^{-q}(x) \diamondsuit_{\alpha} x \, ,$$
where
$\frac{1}{p}+\frac{1}{q}=1$, $p> 1$. Then, the two inequalities
\begin{multline}\label{eq5}
\int_{a}^{b} \int_{a}^{b} K(x, y) f(x)g(y) \diamondsuit_{\alpha} x
\diamondsuit_{\alpha} y \\
\leq \left (\int_{a}^{b} \varphi^{p}(x) F(x) f^{p}(x)
\diamondsuit_{\alpha} x \right)^{\frac{1}{p}} \left (
\int_{a}^{b} \psi^{q}(y) G(y) g^{q}(y) \diamondsuit_{\alpha} y
\right)^{\frac{1}{p}}
\end{multline}
and
\begin{equation}\label{eq6}
\int_{a}^{b} G^{1-p}(y) \psi^{-p}(y) \left ( \int_{a}^{b} K(x, y)
f(x) \diamondsuit_{\alpha} x \right)^{p} \diamondsuit_{\alpha} y
\leq \int_{a}^{b} \varphi^{p}(x) F(x)
f^{p}(x)\diamondsuit_{\alpha} x
\end{equation}
hold and are equivalent.
\end{theorem}
Equation \eqref{eq6} is the
diamond-$\alpha$ Hardy's inequality.
\begin{proof}
First, we prove that \eqref{eq5} hold. Write
\begin{equation*}
\int_{a}^{b} \int_{a}^{b} K(x, y) f(x)g(y) \diamondsuit_{\alpha} x
\diamondsuit_{\alpha} y = \int_{a}^{b} \int_{a}^{b} K(x, y)
f(x)\frac{\varphi(x)}{\psi(y)}g(y) \frac{\psi(y)}{\varphi(x)}
\diamondsuit_{\alpha} x \diamondsuit_{\alpha} y.
\end{equation*}
Applying H\"{o}lder's inequality on time scale, we have
\begin{multline*}
\int_{a}^{b} \int_{a}^{b} K(x, y) f(x)g(y) \diamondsuit_{\alpha} x
\diamondsuit_{\alpha} y \\
\leq \left (\int_{a}^{b} \varphi^{p}(x) F(x) f^{p}(x)
\diamondsuit_{\alpha} x \right)^{\frac{1}{p}} \left (
\int_{a}^{b} \psi^{q}(y) G(y) g^{q}(y) \diamondsuit_{\alpha} y
\right)^{\frac{1}{p}}.
\end{multline*}
Now we show that \eqref{eq5} is equivalent to \eqref{eq6}.
Suppose that inequality \eqref{eq5} is verified. Set
$$
g(y)= G^{1-p}(y) \psi^{-p}(y) \left ( \int_{a}^{b} K(x, y) f(x)
\diamondsuit_{\alpha} x \right)^{p-1} \, .
$$
Using \eqref{eq5} and the fact that $\frac{1}{p}+\frac{1}{q}=1$,
we obtain:
\begin{equation*}
\begin{split}
&\int_{a}^{b} G^{1-p}(y) \psi^{-p}(y) \left ( \int_{a}^{b} K(x, y)
f(x) \diamondsuit_{\alpha} x \right)^{p} \diamondsuit_{\alpha} y
\\
&= \int_{a}^{b} \int_{a}^{b} K(x, y) f(x) g(y)
\diamondsuit_{\alpha} x \diamondsuit_{\alpha} y \\
& \leq \left( \int_{a}^{b} \varphi^{p}(x) F(x) f^{p}(x)
\diamondsuit_{\alpha} x
\right)^{\frac{1}{p}} \left( \int_{a}^{b} \psi^{q}(y) G(y) g^{q}(y) \diamondsuit_{\alpha}
y \right)^{\frac{1}{q}} \\
& = \left( \int_{a}^{b} \varphi^{p}(x) F(x) f^{p}(x)
\diamondsuit_{\alpha} x
\right)^{\frac{1}{p}} \\
& \qquad \qquad \cdot \left ( \int_{a}^{b} G^{1-p}(y) \psi^{-p}(y) \left( \int_{a}^{b} K(x, y) f(x)
\diamondsuit_{\alpha} x\right)^{p} \diamondsuit_{\alpha} y \right
)^{\frac{1}{q}}.
\end{split}
\end{equation*}
Inequality \eqref{eq6} is obtained by dividing both sides of the
previous inequality by
$$
\left ( \int_{a}^{b} G^{1-p}(y) \psi^{-p}(y) \left( \int_{a}^{b}
K(x, y) f(x)
\diamondsuit_{\alpha} x\right)^{p} \diamondsuit_{\alpha} y \right
)^{\frac{1}{q}}.
$$
Reciprocally, suppose that \eqref{eq6} is valid. From
H\"{o}lder's inequality we can write that
\begin{equation*}
\begin{split}
\int_{a}^{b} & \int_{a}^{b} K(x, y) f(x) g(y)
\diamondsuit_{\alpha} x \diamondsuit_{\alpha} y \\
&= \int_{a}^{b} \left ( \psi^{-1}(y) G^{\frac{-1}{q}}(y)
\int_{a}^{b} K(x, y) f(x) \diamondsuit_{\alpha} x \right )
\psi(y) G^{\frac{1}{q}}(y) g(y) \diamondsuit_{\alpha} y \\
& \leq \left( \int_{a}^{b} G^{1-p}(y) \psi^{-p}(y) \left (
\int_{a}^{b} K(x, y) f(x) \diamondsuit_{\alpha} x \right
)^{p}\diamondsuit_{\alpha} y \right )^{\frac{1}{p}} \\
& \qquad \qquad \cdot \left (
\int_{a}^{b} \psi^{q}(y) G(y) g^{q}(y) \diamondsuit_{\alpha} y
\right)^{\frac{1}{q}} \, .
\end{split}
\end{equation*}
Using \eqref{eq6}, we get that
\begin{multline*}
\int_{a}^{b} \int_{a}^{b} K(x, y) f(x) g(y) \diamondsuit_{\alpha}
x \diamondsuit_{\alpha} y \\
\leq \left ( \int_{a}^{b}
\varphi^{p}(x) F(x) f^{p}(x)\diamondsuit_{\alpha} x \right
)^{\frac{1}{q}} \left ( \int_{a}^{b} \psi^{q}(y) G(y) g^{q}(y)
\diamondsuit_{\alpha} y \right)^{\frac{1}{q}} \, ,
\end{multline*}
which completes the proof.
\end{proof}
\begin{remark}
Choose $\mathbb{T}= \mathbb{R}$. In this particular case the inequalities
\eqref{eq5} and \eqref{eq6} give the Hardy type inequalities
proved in \cite{krm}. If
\begin{equation}
\label{eq:rem:krm}
\left ( f(x)\frac{\varphi(x)}{\psi(y)}\right )^{p} = K\left (
g(y)\frac{\psi(y)}{\varphi(x)}\right )^{q},
\end{equation}
then \eqref{eq5} takes the form of equality.
In this case there exist arbitrary constants $A$ and $B$,
not both zero, such that
$$
f^{p}(x)= A \varphi^{-(p+q)}(x) \mbox{ and } g^{q}(y)= B
\psi^{-(p+q)}(y).
$$
This is possible only if
$$
\int_{a}^{b} F(x) \varphi^{-q}(x) \diamondsuit_{\alpha} x < \infty
\mbox{ and } \int_{a}^{b} G(y) \psi^{-p}(y) \diamondsuit_{\alpha}
y < \infty \, .
$$
If \eqref{eq:rem:krm} does not hold,
inequalities in Theorem~\ref{thm:da:Hineq} are strict.
\end{remark}
As corollaries of Theorem~\ref{thm:da:Hineq}
we have the following results.
\begin{corollary}
\label{thm37} Let $\mathbb{T}$ be a time scale, $a$, $b \in
\mathbb{T}$ with $a < b$, $h(y)$, $f(x)$, $g(y)$, $\varphi(x)$,
and $\psi(y)$ be nonnegative functions, and
$\frac{1}{p}+\frac{1}{q}=1$ with $p>1$. Setting $H(y)= h(y)
\psi^{-p}(y)$, then the two inequalities
\begin{multline*}
\int_{a}^{b} \int_{a}^{y} h(y) f(x)g(y) \diamondsuit_{\alpha} x
\diamondsuit_{\alpha} y \\
\leq \left( \int_{a}^{b} \varphi^{p}(x) f^{p}(x) \left(
\int_{x}^{b} H(y) \diamondsuit_{\alpha} y \right )
\diamondsuit_{\alpha} x \right )^{\frac{1}{p}} \\
\qquad \qquad \left( \int_{a}^{b}
\psi^{q}(y) g^{q}(y) h(y) \left( \int_{a}^{y} \varphi^{-q}(x)
\diamondsuit_{\alpha} x \right ) \diamondsuit_{\alpha} y \right
)^{\frac{1}{q}}
\end{multline*}
and
\begin{multline*}
\int_{a}^{b} H(y) \left ( \int_{a}^{y} \varphi^{-q}
\diamondsuit_{\alpha}x \right)^{1-p} \left ( \int_{a}^{y} f(x)
\diamondsuit_{\alpha}x\right)^{p}\diamondsuit_{\alpha}y \\
\leq \left( \int_{a}^{b} \varphi^{p}(x) f^{p}(x)
\left(
\int_{x}^{b} H(y) \diamondsuit_{\alpha} y \right )
\diamondsuit_{\alpha} x \right )^{\frac{1}{p}}
\end{multline*}
hold and are equivalent.
\end{corollary}
\begin{proof}
Use Theorem~\ref{thm:da:Hineq} with $
K(x, y)=
\left\{
\begin{array}{rll}
h(y), & \mbox{ if } & x \leq y \\
0, & \mbox{ if } & x > y \, .
\end{array}
\right. $
\end{proof}
\begin{corollary}
\label{thm38} Let $\mathbb{T}$ be a time scale, $a$, $b \in
\mathbb{T}$ with $a < b$, $h(y)$, $f(x)$, $g(y)$, $\varphi(x)$,
and $\psi(y)$ be nonnegative, and
$\frac{1}{p}+\frac{1}{q}=1$ with $p> 1$. Then, the two
inequalities
\begin{multline*}
\int_{a}^{b} \int_{y}^{b} h(y) f(x)g(y) \diamondsuit_{\alpha} x
\diamondsuit_{\alpha} y \\
\leq \left( \int_{a}^{b} \varphi^{p}(x) f^{p}(x) \left(
\int_{a}^{x} H(y) \diamondsuit_{\alpha} y \right )
\diamondsuit_{\alpha} x \right )^{\frac{1}{p}} \\
\left( \int_{a}^{b}
\psi^{q}(y) g^{q}(y) h(y) \left( \int_{y}^{b} \varphi^{-q}(x)
\diamondsuit_{\alpha} x \right ) \diamondsuit_{\alpha} y \right
)^{\frac{1}{q}},
\end{multline*}
and
\begin{multline*}
\int_{a}^{b} H(y) \left ( \int_{y}^{b} \varphi^{-q}
\diamondsuit_{\alpha}x \right)^{1-p} \left ( \int_{y}^{b} f(x)
\diamondsuit_{\alpha}x\right)^{p}\diamondsuit_{\alpha}y \\
\leq \left( \int_{a}^{b} \varphi^{p}(x) f^{p}(x) \left(
\int_{a}^{x} H(y) \diamondsuit_{\alpha} y \right )
\diamondsuit_{\alpha} x \right )^{\frac{1}{p}}
\end{multline*}
hold and are equivalent.
\end{corollary}
\begin{proof}
Use Theorem~\ref{thm:da:Hineq} with $
K(x, y)=
\left\{
\begin{array}{rll}
0, & \mbox{ if } & x \leq y \\
h(y), & \mbox{ if } & x > y \, .
\end{array}
\right. $
\end{proof}
\begin{remark}
When $\alpha = 1$, Corollaries~\ref{thm37} and \ref{thm38}
coincide, respectively, with Theorems~7 and 8 in \cite{adnan}.
In the particular case $\mathbb{T}=\mathbb{R}$, they give
Theorem~3 and Theorem~4 of \cite{krm}.
\end{remark}
It is interesting to consider the case when functions $F(x)$
and $G(y)$ of Theorem~\ref{thm:da:Hineq} are bounded. We then
obtain the following:
\begin{theorem}
Let $\frac{1}{p}+ \frac{1}{q}=1$ with $p>1, K(x, y), f(x), g(y),
\varphi(x), \psi(x)$ be nonnegative functions and $F(x) =
\int_{a}^{b} \frac{K(x, y)}{\psi^{p}(y)} \diamondsuit_{\alpha} y
\leq F_{1}(x) \, , G(y) = \int_{a}^{b} \frac{K(x,
y)}{\varphi^{q}(x)} \diamondsuit_{\alpha} x \leq G_{1}(y) $. Then, the inequalities
\begin{multline}\label{eq7}
\int_{a}^{b} \int_{a}^{b} K(x, y) f(x)g(y) \diamondsuit_{\alpha} x
\diamondsuit_{\alpha} y \\
\leq \left (\int_{a}^{b} \varphi^{p}(x) F_{1}(x) f^{p}(x)
\diamondsuit_{\alpha} x \right)^{\frac{1}{p}} \left (
\int_{a}^{b} \psi^{q}(y) G_{1}(y) g^{q}(y) \diamondsuit_{\alpha} y \right)^{\frac{1}{p}}
\end{multline}
and
\begin{equation}\label{eq8}
\int_{a}^{b} G_{1}^{1-p}(y) \psi^{-p}(y) \left ( \int_{a}^{b} K(x,
y) f(x) \diamondsuit_{\alpha} x \right)^{p} \diamondsuit_{\alpha}
y \leq \int_{a}^{b} \varphi^{p}(x) F_{1}(x)
f^{p}(x)\diamondsuit_{\alpha} x
\end{equation}
hold and are equivalent.
\end{theorem}
The following result extends
the one found in \cite{sulaiman}.
\begin{theorem}
\label{lst:thm}
Let $F, G, L(f, g), M(f)$, and $N(g)$ be positive functions,
$p> 1$, $\frac{1}{p}+ \frac{1}{q}=1$, such that
$$
0 < \int_{a}^{b} M^{p}(f(t)) F^{p}(t) \diamondsuit_{\alpha} t <
\infty \, , \quad
0 < \int_{c}^{d} N^{q}(g(t)) G^{q}(t)
\diamondsuit_{\alpha} t < \infty \, .
$$
Then, the inequalities
\begin{multline}\label{eq9}
\int_{a}^{b} \int_{c}^{d} \frac{F(x)G(y)}{L(f(x), g(y))}
\diamondsuit_{\alpha}x \diamondsuit_{\alpha} y \\
\leq C \left (\int_{a}^{b} M^{p}(f(t))
F^{p}(t)\diamondsuit_{\alpha} t \right )^{\frac{1}{p}} \left
(\int_{c}^{d} N^{q}(g(t)) G^{q}(t)\diamondsuit_{\alpha} t \right
)^{\frac{1}{q}}
\end{multline}
and
\begin{equation}\label{eq10}
\int_{c}^{d} N^{-p}(g(y)) \left ( \int_{a}^{b} \frac{F(x)}{L(f(x),
g(y))} \diamondsuit_{\alpha} x \right )^{p} \diamondsuit_{\alpha}
y \leq C^{p}\int_{a}^{b} M^{p}(f(t)) F^{p}(t)
\diamondsuit_{\alpha} t \, ,
\end{equation}
where $C$ is a constant, are equivalent.
\end{theorem}
\begin{proof}
Suppose that the inequality \eqref{eq10} is valid. Then,
\begin{equation*}
\begin{split}
& \int_{a}^{b} \int_{c}^{d} \frac{F(x)G(y)}{L(f(x), g(y))}
\diamondsuit_{\alpha}x \diamondsuit_{\alpha} y \\
& = \int_{c}^{d} N(g(y)) G(y) \left ( N^{-1}(g(y)) \int_{a}^{b}
\frac{F(x)}{L(f(x), g(y))} \diamondsuit_{\alpha}x
\right)\diamondsuit_{\alpha}y \\
&\leq \left ( \int_{c}^{d} N^{q}(g(y)) G^{q}(y)
\diamondsuit_{\alpha} y \right )^{\frac{1}{q}} \left (
\int_{c}^{d} N^{-p}(g(y)) \left ( \int_{a}^{b} \frac{F(x)}{L(f(x),
g(y))} \diamondsuit_{\alpha}x
\right )^{p}\diamondsuit_{\alpha}y \right)^{\frac{1}{p}}\\
& \leq C^{p}\left ( \int_{a}^{b} M^{p}(f(t)) F^{p}(t)
\diamondsuit_{\alpha} t \right )^{\frac{1}{p}} \left (
\int_{c}^{d} N^{q}(g(t)) G^{q}(t) \diamondsuit_{\alpha} t \right
)^{\frac{1}{q}}.
\end{split}
\end{equation*}
We just proved inequality \eqref{eq9}.
Let us now suppose that the inequality \eqref{eq9} is valid.
By setting $G(y)= N^{-p}(g(y)) \left (\int_{a}^{b}
\frac{F(x)}{L(f(x), g(y))} \diamondsuit_{\alpha}x \right
)^{\frac{p}{q}} \diamondsuit_{\alpha} y $ and applying \eqref{eq9}, we obtain that
\begin{equation*}
\begin{split}
& \int_{c}^{d} N^{-p}(g(y)) \left ( \int_{a}^{b}
\frac{F(x)}{L(f(x), g(y))} \diamondsuit_{\alpha} x \right )^{p}
\diamondsuit_{\alpha} y \\
& = \int_{c}^{d} \left ( \int_{a}^{b} \frac{F(x)}{L(f(x), g(y))}
\diamondsuit_{\alpha}x \right ) N^{-p}(g(y)) \left (\int_{a}^{b}
\frac{F(x)}{L(f(x), g(y))} \diamondsuit_{\alpha}x \right
)^{\frac{p}{q}} \diamondsuit_{\alpha} y \\
& \leq C \left ( \int_{a}^{b} M^{p}(f(x)) F^{p}(x)
\diamondsuit_{\alpha} x \right )^{\frac{1}{p}}\\
& \qquad \times \left ( \int_{c}^{d} N^{q}(g(y)) N^{-pq}(g(y)) \left
(\int_{a}^{b} \frac{F(x)}{L(f(x), g(y))} \diamondsuit_{\alpha}x
\right
)^{p}\diamondsuit_{\alpha}y \right )^{\frac{1}{q}}\\
& = C \left ( \int_{a}^{b} M^{p}(f(x)) F^{p}(x)
\diamondsuit_{\alpha} x \right )^{\frac{1}{p}} \\
& \qquad \qquad \left (
\int_{c}^{d} N^{-p}(g(y)) \left ( \int_{a}^{b}
\frac{F(x)}{L(f(x), g(y))} \diamondsuit_{\alpha}x \right)^{p}
\diamondsuit_{\alpha}y \right )^{\frac{1}{q}}.
\end{split}
\end{equation*}
It follows \eqref{eq10}:
\begin{equation*}
\int_{c}^{d} N^{-p}(g(y)) \left ( \int_{a}^{b} \frac{F(x)}{L(f(x),
g(y))} \diamondsuit_{\alpha} x \right )^{p} \diamondsuit_{\alpha}
y \leq C^{p} \int_{a}^{b} M^{p}(f(t)) F^{p}(t)
\diamondsuit_{\alpha} t.
\end{equation*}
\end{proof}
\section{Conclusion}
The study of integral inequalities on time scales via the
diamond-$\alpha$ integral, which is defined as a linear
combination of the delta and nabla integrals, plays an important
role in the development of the theory of time scales
\cite{ozkan,Stef,srd,sidel}. In this paper we generalize some
delta-integral inequalities on time scales to diamond-$\alpha$
integrals. As special cases, one obtains previous H\"{o}lder's
and Hardy's inequalities.
\\
\\
{\bf Acknowledgements:}
Work partially supported by the {\it Centre for Research on
Optimization and Control} (CEOC) from the {\it Portuguese
Foundation for Science and Technology} (FCT), cofinanced by the
European Community Fund FEDER/POCI 2010.
\end{document} | math | 31,156 |
\begin{document}
\title{Robust Weak-Measurement Protocol for Bohmian Velocities}
\author{F. L. Traversa}
\email{[email protected]}
\affiliation{Departament d'Enginyeria Electr\`onica, Universitat Aut\`onoma de Barcelona, 08193-Bellaterra (Barcelona), Spain}
\affiliation{Department of Physics, University of California, San Diego, La Jolla, California 92093, USA}
\author{G. Albareda}
\affiliation{Departament d'Enginyeria Electr\`onica, Universitat Aut\`onoma de Barcelona, 08193-Bellaterra (Barcelona), Spain}
\author{M. Di Ventra}
\affiliation{Department of Physics, University of California, San Diego, La Jolla, California 92093, USA}
\author{X. Oriols}
\email{[email protected]}
\affiliation{Departament d'Enginyeria Electr\`onica, Universitat Aut\`onoma de Barcelona, 08193-Bellaterra (Barcelona), Spain}
\begin{abstract}
We present a protocol for measuring Bohmian - or the mathematically equivalent hydrodynamic - velocities based on an ensemble of two position measurements, defined from a Positive Operator Valued Measure, separated by a finite time interval. The protocol is very accurate and robust as long as the first measurement uncertainty divided by the finite time interval between measurements is much larger than the Bohmian velocity, and the system evolves under flat potential between measurements. The difference between the Bohmian velocity of the unperturbed state and the measured one is predicted to be much smaller than $1 \%$ in a large range of parameters. Counter-intuitively, the measured velocity is that at the final time and not a time-averaged value between measurements.
\end{abstract}
\maketitle
\section{Introduction}
The velocity of a classical object, requiring two position measurements, is trivially implemented in many apparati which control our daily activity. On the contrary, in the quantum world, such measurements are much more complicated. The first position measurement implies a perturbation on the quantum system so that the knowledge of the velocity without perturbation is hardly accessible. One can minimize the back-action of the measurement on the system using weak measurements. Such measurements were initially developed by Aharonov, Albert and Vaidman (AAV) \cite{AAV} more than two decades ago and they are receiving increasing attention \cite{weak5,Wiseman, weak2,weak4,Bohmphoton,weak3,Hiley,Vaidman,Garretson} nowadays. As a relevant example, the spatial distribution of \emph{velocities} of relativistic photons in a double slit scenario has been measured, and the associated quantum trajectories reconstructed \cite{Bohmphoton}. However, we may ask the question: \emph{Does the ensemble velocity obtained from weak measurements have a clear physical meaning?} A partial answer was provided recently by Wiseman \cite{Wiseman}. Using the weak AAV value \cite{AAV}, he showed that the ensemble velocity constructed from an arbitrarily pre-selected state and a post-selected position eigenstate, with an infinitesimal temporal separation between position measurements, exactly corresponds to the Bohmian velocity \cite{Bohm} of the unperturbed state. Note that Wiseman's answer is only valid for non-relativistic scenarios (thus, strictly speaking, excluding \cite{Bohmphoton}).
We emphasize that two weak position measurements on an individual state do not provide the Bohmian velocity because of the unavoidable back-action \cite{Golstein}. However, for an idealized scenario, Wiseman showed that when the individual measurements are repeated over an ensemble of identical states, the final ensemble velocity is identical to the Bohmian velocity of the unperturbed state \cite{Wiseman}. These ensemble velocities can be interpreted either as the orthodox hydrodynamic velocity \cite{Madelung,Di_Ventra} or as a genuine measurement of the Bohmian velocity \cite{Golstein}. Following the recent literature \cite{Bohmphoton,Wiseman,Golstein}, we will refer to these ensemble velocities as Bohmian velocities, however the adjectives \emph{Bohmian} and \emph{hydrodynamic} are fully interchangeable in this work.
The practical conditions for measuring Bohmian velocities in a laboratory are different from the idealized theoretical scenario studied by Wiseman \cite{Wiseman} (implying discrepancies between the measured velocity and the expected one). First, {\it weak} measurements in a laboratory can be outside the linear-response regime assumed in the AAV development \cite{nori}. Second, position measurements have a small but finite uncertainty, meaning that the post-selected state is not an exact position eigenstate. Third, the time-separation between measurements must be finite. In this paper we bring the original Wiseman's conclusions about the measurement of Bohmian velocities into practical laboratory conditions, free from previous idealized assumptions. We will use the Positive Operator Valued Measure (POVM) framework \cite{nori} (instead of the AAV value) allowing positions uncertainties in both measurements and we will consider a finite time interval between position measurements.
\section{Ensemble velocity}
\subsection {Definition of ensemble velocity}
>From a large set of measured positions, $x_w$ at time $t_w$ and $x_s$ at $t_s=t_w+\tau$, we construct the experimental velocity as:
\begin{equation}
\label{vexp}
v_{e}(x_s,t_s)=\frac { E[(x_s - x_w)|x_s]} {\tau},
\end{equation}
being $E[(x_s - x_w)|x_s]$ the ensemble average of the distance $x_s-x_w$, conditioned to the fact that $x_s$ is effectively measured.
Since $E[x_s|x_s]=x_s$, the theoretical computation of the velocity $v_e$ does only require evaluating $\text{E}[x_w | x_s]$ using standard probability calculus,
\begin{equation}
\label{cond}
\text{E}[x_w | x_s] = \frac {\int dx_w x_w P(x_w\cap x_s)} {P( x_s)},
\end{equation}
with $P(x_w \cap x_s)$ the joint probability of the sequential measurements of $x_w$ and $x_s$, and $P(x_s)$ of $x_s$. After properly modeling the system perturbation due to the measurement, both probabilities can be computed.
\subsection{Two consecutive POVMs separated by a finite time interval}
The POVM appears as a natural modeling of a measuring process \cite{weak} when the laboratory is divided into the quantum system and the rest (including the measuring apparatus).
Thus, the perturbation of the state due to the measurement of the first position $x_w$ can be defined through POVMs. In this treatment we chose the Gaussian measurement Krauss operators
\begin{eqnarray}
\label{W}
\hat{W}a=C_w \int dx e^{-\frac {(x_w-x)^2} {2\sigma_w^2}} \ket x \bra x,
\end{eqnarray}
where $\sigma_w$ is the experimental uncertainty. The measured position $x_w$ belongs to the set $\mathfrak{M}$ of all possible measurement outputs of the apparatus. For simplicity, we assume $\mathfrak{M}\equiv\mathbb{R}$ in a 1D system, being the extension to the 3D spatial domain straightforward. Then, the normalization coefficient $C_w=(\sqrt\hat{p}i\sigma_w)^{-1/2}$ is fixed by the condition $\int dx_w \hat{W}ad \hat{W}a=I$. Due to the unavoidable uncertainty on any position measurement, we consider an equivalent operator for the second position measurement of $x_s$:
\begin{eqnarray}
\label{S}
\hat{S}_s=C_s \int dx e^{-\frac {(x_s-x)^2} {2\sigma_s^2}} \ket x \bra x.
\end{eqnarray}
We remark here that the choice of Gaussian measurement operators is not the only possible one that leads to our results. In fact, it can be proven that any POVM that weakly perturbs the wave function only in a neighborhood of $x_w$ ($x_s$) of radius $\sigma_w$ ($\sigma_s$), and \emph{cancels} the wave function in any other position leads to equivalent results. Thus the choice of Gaussian POVM is purely formal. It allows a simple analytical treatment.
Now, using the definitions in \hat{p}ref{W} and \hat{p}ref{S}, we can compute $P(x_w \cap x_s)$ and $P(x_s)$ from the Born rule, as:
\begin{align}
\label{prob1}
P(x_w \cap x_s)=&\bra{\Psi}\hat{W}adU_\taud\hat{S}_sd\hat{S}_sU_\tau\hat{W}a\ket{\Psi} \\
\label{prob2}
P(x_s)=&\int dx_w P(x_w \cap x_s).
\end{align}
being $\ket {\Psi(t_w)}\equiv \ket {\Psi}$ the initial state. Strictly speaking, contrarily to the AAV expression \cite{AAV}, we are using a weak measurement without post-selection. The final state of the system (determined by the time-evolution of the initial state $\ket {\Psi}$ and the measurement processes) has no relevant effect when computing \hat{p}ref{prob1} and \hat{p}ref{prob2}.
\subsection{Calculation of the ensemble velocity}
\label{sec_velocity}
Let us now analyze $P(x_s)$ in detail by substituting \eref{W} and \hat{p}ref{S} into \eref{prob2}. Then, we have
\begin{multline}
\label{Pxs}
P(x_s)=C_w^2 \iiint dx_wdx\rq{}dx\rq{}\rq e^{-\frac{(
x_w-x\rq)^2}{2\sigma_w^2}} e^{-\frac{(x_w-x\rq{}\rq)^2}{2\sigma_w^2}}\times\\
\times \braket {\Psi} {x\rq} \bra{x\rq}U_\taud\hat{S}_sd\hat{S}_sU_\tau\ket{x\rq{}\rq} \braket {x\rq{}\rq} {\Psi}.
\end{multline}
Integrating over $x_w$ and using \eref{S}, we can rewrite \eref{Pxs} as:
\begin{multline}
\label{Pxs1}
P(x_s)= C_s^2\iint dx\rq dx\rq{}\rq\braket {\Psi} {x\rq} e^{-\frac{(x'-x'')^2}{4\sigma_w^2}} \braket {x\rq{}\rq} {\Psi} \times \\
\times\left(\int dxe^{-\frac{(x_s-x) ^2}{\sigma_s^2}}\bra{x\rq}U_\taud\ket{x}\bra{x}U_\tau\ket{x\rq{}\rq}\right).
\end{multline}
For a particle of mass $m$ that evolves under a flat potential during $\tau$, we can evaluate $\bra x U_\tau \ket{x\rq}$ using \cite{shankar}
\begin{equation}
\label{evo}
\bra x U_\tau \ket{x\rq}=\left( i\hat{p}i(2\hbar \tau/m)\right)^{-1/2} e^{\frac{i(x-x\rq) ^{2}}{(2\hbar \tau/m)}}.
\end{equation}
Substituting \eref{evo} into \hat{p}ref{Pxs1} and solving the integral between parenthesis, we have
\begin{multline}
\label{Pxs2}
P(x_s)= \iint dx\rq dx\rq{}\rq e^{-\frac{(x'-x'')^2}{4\sigma_w^2}} e^{-\left(\frac{\sigma_s m}{2\hbar \tau}\right)^2(x\rq-x\rq{}\rq) ^2} \times \\
\braket {\Psi} {x\rq} \bra{x\rq}U_\taud\ket{x_s}\bra{x_s}U_\tau\ket{x\rq{}\rq} \braket {x\rq{}\rq} {\Psi}.
\end{multline}
One easily realizes that the probability in \hat{p}ref{Pxs2} can be computed as $P(x_s)=\bra\PsiU_\taud\hat{S}_sd\hat{S}_sU_\taud\ket\Psi$ when the following limit is satisfied,
\begin{equation}
\label{lim}
\frac {\sigma_w} {\tau} \gg \frac{\hbar}{m\sigma_{s}}.
\end{equation}
Let us emphasize that this condition, includes Wiseman\rq{}s result \cite{Wiseman} as a particular case: $\sigma_w \rightarrow \infty$, $\sigma_s \rightarrow 0$ and $\tau \rightarrow 0$. Our development will justify the effective measurement of the Bohmian velocity (up to a negligible error) for a broad range of $\sigma_w$, $\sigma_s$ and $\tau$.
Identical steps can be done for the evaluation of ${\int dx_w x_w P(x_w\cap x_s)}$ in \eref{cond}. The only difference resides on the integration on $x_w$, which in this case gives $(x\rq{}+x\rq{}\rq{})/2\exp[-(x'-x'')^2/4\sigma_w^2]$.
Using $\int dx \hspace{0.02in}x \ket {x}\bra {x} =\hat{x}$, under the limit \hat{p}ref{lim}, we obtain ${\int dx_w x_w P(x_w\cap x_s)}={\operatorname{Re}(\bra\PsiU_\taud\hat{S}_sd\hat{S}_s U_\tau\hat{x}\ket\Psi )}$ . Finally, we can rewrite \eref{cond} as:
\begin{equation}
\label{AAV1}
\text{E}[x_w|x_s]=\textstyle\frac{\operatorname{Re}(\bra\PsiU_\taud\hat{S}_sd\hat{S}_sU_\tau\hat{x}\ket\Psi )}{\bra\PsiU_\taud\hat{S}_sd\hat{S}_sU_\tau\ket\Psi} .
\end{equation}
Next, we define the following (averaged) position $\bar x_s=\bra\PsiU_\taud\hat{S}_sd\hat{S}_s\hat{x}U_\tau\ket\Psi/\bra\PsiU_\taud\hat{S}_sd\hat{S}_sU_\tau\ket\Psi$, so that using \eref{AAV1} and the commutator $[U_\tau,\hat{x}]$, we get:
\begin{equation}
\label{AAV2}
\bar x_s-\text{E}[x_w|x_s]=\textstyle\frac {\operatorname{Re}(\bra\PsiU_\taud\hat{S}_sd\hat{S}_s [U_\tau,\hat{x}]\ket\Psi )}{\bra\PsiU_\taud\hat{S}_sd\hat{S}_sU_\tau\ket\Psi},
\end{equation}
without any reference to $\hat{W}a$.
To further develop \eref{AAV2}, we evaluate the commutator $[U_\tau,\hat{x}]$ using the Maclaurin series for $U_\tau$:
\begin{equation}
\label{maclaurin}
[U_\tau,\hat{x}] =\textstyle\sum\limits_{n=1}^{\infty}\frac{(-i) ^n\tau^n}{n!\hbar^n}[\hat{H}^n ,\hat{x}],
\end{equation}
where $\hat{H}=\hat{p}^2/2m+V$ is the system Hamiltonian with $V$ a flat potential at the spatial region where the wave function is different from zero during the time between measurements. No restriction on $V$ for other regions and times. Given two operators $\hat A$ and $\hat B$, it can be proven that $[\hat A^n,\hat B]=\textstyle\sum_{j=1}^{n}\hat A^{j-1}[\hat A,\hat B] \hat A^{n-j}$. Then, being $[\hat{H},\hat{x}]=-i\hbar/m\hat{p}$ and $[\hat{H},\hat{p}]=0$, the commutator $[\hat{H}^n,\hat{x}]$ gives:
\begin{equation}
\label{Hanx}
[\hat{H}^n,\hat{x}] =-\textstyle\frac{i\hbar n}{m}\hat{p}\hat{H}^{n-1},
\end{equation}
and substituting \eref{Hanx} into \eref{maclaurin} we obtain:
\begin{equation}
\label{commutator}
[U_\tau,\hat{x}]=-{\frac{\tau}{m}}\hat{p}U_\tau,
\end{equation}
without considering the limit $\tau\rightarrow0$.
Using \eref{commutator} and the definition \hat{p}ref{S}, a straightforward calculation for the numerator of \eref{AAV2} gives:
\begin{eqnarray}
\operatorname{Re}(\bra\PsiU_\taud\hat{S}_sd\hat{S}_s[U_\tau,\hat{x}]\ket\Psi ) \equiv \tau \bar J(x_s,t_s) =\nonumber\\
\tau C_s^2 \int dx J(x,t_s)\exp[-(x_s-x)^2/\sigma_s^2],
\label{nolabel}
\end{eqnarray}
where $J(x,t_s)$ is the standard quantum current probability density \cite{libro}. Similarly, we define ${\bra\PsiU_\taud\hat{S}_sd\hat{S}_sU_\tau\ket\Psi} = C_s^2 \int dx |\Psi(x,t_s)|^2\exp[-(x_s-x)^2/\sigma_s^2]\equiv |\bar \Psi(x_s,t_s)|^2$ for the denominator. Finally, the velocity, defined as \eref{AAV2} divided by $\tau$, gives:
\begin{equation}
\label{velobohm_s}
\bar v(x_s,t_s)=\frac {\bar x_s - E[x_w|x_s]} {\tau}=\frac {\bar J(x_s,t_s)} { |\bar{\Psi}(x_s,t_s)|^2}.
\end{equation}
This expression is just the Gaussian-spatially-averaged current density $\bar J(x_s,t_s)$ inside a tube of diameter $\sigma_s$ divided by the corresponding Gaussian-spatially-averaged probability $|\bar \Psi(x_s,t_s)|^2$.
Whether or not the Gaussian-spatially-averaged value \hat{p}ref{velobohm_s} is identical to the Bohmian velocity depends on the measuring apparatus resolution, i.e. $\sigma_s$, and the de Broglie wavelength $\lambda$ associated to $\ket \Psi$. Under the limit
\begin{equation}
\label{lim2}
\sigma_s< \lambda,
\end{equation}
one can assume $\Psi(x,\tau) \approx \Psi(x_s,t_s)$ for $x\in [x_s-\sigma_s,x_s+\sigma_s]$, so that $\bar \Psi(x_s,t_s)\approx \Psi(x_s,t_s)$. Identically, $\bar J(x_s,t_s) \approx J(x_s,t_s)$ and $\bar x_s=x_s$. Then, \eref{velobohm_s} directly recovers the Bohmian velocity $\bar v(x_s,t_s) \approx v$ with:
\begin{equation}
\label{velobohm}
v \equiv v(x_s,t_s) =\frac {J(x_s,t_s)}{|\Psi(x_s,t_s)|^2}.
\end{equation}
Let us mention that the consideration $\sigma_s \approx \lambda$ and the momentum $p=h/\lambda$ implies $\hbar/(m\sigma_s) \approx v$ in the limit \hat{p}ref{lim}.
>From the definition of velocity in \hat{p}ref{vexp}, one could reasonably expect to get a value associated to the velocity \emph{averaged} during the time interval $\tau$ and associated to a \emph{perturbed} wave function. However, under the conditions \hat{p}ref{lim} and \hat{p}ref{lim2}, the result \hat{p}ref{velobohm} is clearly identified as the \emph{instantaneous} (bohmian) velocity associated with an \emph{unperturbed} wave function at the final time $t_s$. The mathematical reasons leading to \hat{p}ref{velobohm} are fully detailed in the previous calculations. Here, we try to provide some physical insights. It is well known that a measurement process induces a perturbation on the wave function, breaking the symmetry in its time evolution. In our case, because of the imposed conditions \hat{p}ref{lim} and \hat{p}ref{lim2}, the roles of the first and second measurements are very different. The condition \hat{p}ref{lim} implies that the first measurement perturbs very weakly the wave function in a neighborhood $I_w$ of radius $\sigma_w$ around $x_w$, while the second limit \hat{p}ref{lim2} implies a very strong perturbation of the wave function during the second measurement process. As a result, when constructing \hat{p}ref{vexp}, only the position eigenstates belonging to $I_w$ (where the wave function remains mainly \emph{unperturbed} by the first measurement) are used. In fact, the ensemble average \hat{p}ref{AAV1} has no memory of the first measurement process (i.e., of the first POVM). Moreover, the condition of flat potential between the two measurements that leads to Eq. \hat{p}ref{commutator} implies explicit independence of $\tau$ because it provides \emph{free} evolution of the \emph{unperturbed} wave function. In this regard, the first measurement does not actually break the symmetry. The obvious consequence (supported by our calculation) is that the velocity in \hat{p}ref{vexp} is independent of the time $\tau$ between the two measurements. Finally, since the symmetry is broken essentially by the second measurement, the velocity that we obtain is the one associated with an \emph{unperturbed} wave function at the last time $t_s$.
Another way of explaining our results is by noticing that the identity \hat{p}ref{commutator} can be used for a finite $\tau$ because we assume that the potential is flat at the spatial region where the wave function is different from zero. For a classical system evolving under a flat potential from $t_w$ till $t_s=t_w+\tau$, the instantaneous velocity at $t_s$ is exactly equal to the averaged velocity during $\tau$. The classical velocity remains constant during this time interval because the classical acceleration is zero. In the quantum counterpart, from Ehrenfest theorem, we know that the ensemble momentum with a flat potential is constant during $t_w< t\le t_s$. Using the limit \hat{p}ref{lim}, the ensemble momentum can be defined as $\bra{\Psi(t)}\hat{p} \ket{\Psi(t)}=\int\bra{\Psi(t_w)}\hat{W}ad U_{t-t_w}^\dagger\hat{p} U_{t-t_w}\hat{W}a\ket{\Psi(t_w)}dx_w$ which corresponds to \hat{p}ref{nolabel} without performing the second measurement. This again justifies why the resulting velocity evaluated with our protocol is independent of $\tau$ and exactly equal to the (Bohmian) velocity measured at $t_s$.
\subsection{Calculation of the ensemble velocity variance}
Let us now compute the velocity variance. Since $x_s$ and $\tau$ are fixed in \eref{vexp}, $var(v_{e})=var(x_w)/\tau^2$. Thus, $var(x_w)=E[x_w^2|x_s]-(E[x_w|x_s])^2$ where $E[x_w|x_s]$ defined in \eref{cond} is obtained from \eref{velobohm}. The evaluation of ${\int dx_w x_w^2 P(x_w\cap x_s)}$ follows identical steps as in the computation of $P(x_s)$, where again the only difference resides in the integral in $x_w$ that now gives $( \sigma_w^2/2+(x\rq{}+x\rq{}\rq{})^2/4) \exp[-(x\rq{}-x\rq{}\rq{})^2/4 \sigma_w^2]$.
Using again $\int dx \hspace{0.02in}x \ket {x}\bra {x} =\hat{x}$ and $\int dx \hspace{0.02in}x^2 \ket {x}\bra {x} =\hat{x}^2$, the final result, under the limit \hat{p}ref{lim}, is:
\begin{multline}
\label{cond4}
\text{E}[x_w^2 | x_s] =\textstyle\frac 1 2 \sigma_w^2+\frac 1 2 \frac{\operatorname{Re}(\bra\PsiU_\taud\hat{S}_sd\hat{S}_sU_\tau\hat{x}^2\ket\Psi )}{\bra\PsiU_\taud\hat{S}_sd\hat{S}_sU_\tau\ket\Psi}\\
+ \textstyle\frac 1 2 \frac{\operatorname{Re}(\bra\Psi\hat{x}U_\taud\hat{S}_sd\hat{S}_sU_\tau\hat{x}\ket\Psi )}{\bra\PsiU_\taud\hat{S}_sd\hat{S}_sU_\tau\ket\Psi},
\end{multline}
which, as detailed in \sref{appendixA} in the Appendix, finally gives
\begin{equation}
\label{var}
var(v)= \frac {\sigma_w^2} {2\tau^2}+\textstyle\frac 2 m Q_B(x_s)+O\left(\frac{\hbar}{m\tau}\right),
\end{equation}
where $Q_B(x_s)$ is the (local) Bohmian quantum potential \cite{Bohm, libro}. Under the limits \hat{p}ref{lim} and \hat{p}ref{lim2}, the term ${\sigma_w^2} /(2{\tau^2})$ in Eq.~(\ref{var}) will be orders of magnitude greater than the other two. For an experimentalist, this means that the presence of the quantum potential on the spatial fluctuations of \eref{var} will be hardly accessible, and that $var(v)$ provides basically the value $\sigma_w$ of the apparatus. Using the well know result from the probability calculus $\varepsilon(N)={\sqrt{var(v)}}/{\sqrt{N}}\approx {\sigma_w}/({\tau\sqrt{2N}})$, such variance can be used to evaluate the number $N$ of measurements needed to obtain \hat{p}ref{velobohm} with a given error $\varepsilon(N)$.
\subsection{Error analysis}
In order to test how robust (i.e. how independent of $\sigma_w$, $\sigma_s$ and $\tau$) is the possibility of measuring the Bohmian velocity in a laboratory, we compute the (local) error $\varepsilon_w(x_s) \equiv \abs{v_{e}(x_s)-\bar {v}(x_s)}$. The details of the calculation are reported in \sref{appendixB} in the Appendix:
\begin{equation}
\label{errorW}
\varepsilon_w(x_s)=\frac{\tau\hbar^2}{4m^2\sigma_w^2}\abs{
\dfrac{2( 1-\tau\hat{p}artial_x v)\hat{p}artial_x \rho-\tau\rho\hat{p}artial_x^2 v} {\rho+\frac{\tau^{2}\hbar^{2}
}{4m^{2}\sigma_w^{2}}\hat{p}artial_x^{2}\rho}},
\end{equation}
where $\rho=\abs{\hat{p}si(x_s,t_s)}^2$. We further define the measuring apparatus error $\varepsilon_{s}(x_s) \equiv \abs{v(x_s)-\bar {v}(x_s)}$ deriving from the requirement \hat{p}ref{lim2}. The calculation reported in \sref{appendixC} in the Appendix gives:
\begin{equation}
\label{errorS}
\varepsilon_{s}(x_s) =\sigma_s^2\abs{ \frac{\frac{2}{\tau}\hat{p}artial_x\rho+( 2\hat{p}artial_x\rho-\rho \hat{p}artial_x ) \hat{p}artial_x v}{4\rho+\sigma_s^2\hat{p}artial_x^2\rho}}.
\end{equation}
It is worth noticing that, by construction, the total error $\varepsilon(x_s) \equiv \abs{v(x_s)-v_e(x_s)}$ accomplishes $\varepsilon(x_s) \leq \varepsilon_s(x_s)+\varepsilon_w(x_s)$.
\section{Ensemble current density}
We observe that the same set of measured values $x_w$ and $x_s$ can be used to define an experimental current density:
\begin{equation}
\label{encurrentg}
J_e(x_s,t_s) = \frac {P(x_s)x_s-\int dx_wx_wP(x_w\cap x_s)} {\tau}.
\end{equation}
The get experimental value $J_e(x_s,t_s)$, we do only need to change how the measured data $x_w$ and $x_s$ is treated. The fact that expression \hat{p}ref{encurrentg} provides the expected theoretical definition of the current density (within a negligible error) can be straightforwardly computed following previous developments of ${P(x_s)}$ and ${\int dx_w x_w P(x_w\cap x_s)}$ in \sref{sec_velocity}. Identically, all the previous calculations for the variance of the current density and their errors can be then repeated for the current in a similar way.
\begin{figure}
\caption{\label{vel}
\label{vel}
\end{figure}
\section{Numerical results and discussion}
As a numerical test of our prediction, we consider an electron passing through a double slit. For simplicity, the time evolution of two 1D initial Gaussian wave-packets with zero central momenta and central positions separated a distance of 100 nm are explicitly simulated. This roughly corresponds to the evolution of the quantum state after crossing the double-slit at $t=0$s. From \fref{vel}(a) the agreement between the exact Bohmian velocity $v$ in \hat{p}ref{velobohm} and $v_e$ [numerically evaluated from \hat{p}ref{vexp}, \hat{p}ref{cond}, \hat{p}ref{prob1} and \hat{p}ref{prob2} without any limit or approximation] is excellent and it is highlighted by the inset \ref{vel}(a\rq{}) where the total error \hat{p}ref{errorW} plus \hat{p}ref{errorS} is reported.
\begin{figure}
\caption{\label{tau_sigma_w}
\label{tau_sigma_w}
\end{figure}
\begin{figure}
\caption{\label{tau_sigma_s}
\label{tau_sigma_s}
\end{figure}
In \fref{tau_sigma_w}, we plot the normalized value of the error $\varepsilon_w(x_s)$ integrated over $x_s$ as $\varepsilon_w=(\int dx_s \varepsilon_w(x_s)^2/\int dx_s v(x_s)^2)^{1/2}$. The main conclusion extracted from \fref{tau_sigma_w} is that a large set of parameters (large $\sigma_w/\tau$ values) allows a very accurate measurement of the Bohmian velocity, justifying the robustness of our proposal.
At this point, we emphasize some relevant issues. First, we have shown theoretically and numerically that the Bohmian velocity of an unperturbed state under general laboratory conditions can be obtained from two POVM measurements separated by a finite $\tau$. Unlike the results derived from the AAV formulation \cite{AAV}, the limits \hat{p}ref{lim} and \hat{p}ref{lim2} provide a simple quantitative explanation of the experimental conditions for an accurate and robust measurement of the Bohmian velocity.
On the other hand, the error $\varepsilon_s(x_s)$ in \hat{p}ref{errorS} has a term that diverges as $\sigma_s^2/\tau$, meaning that a $\tau$ close to zero will produce an inaccurate measurement of the velocity for finite $\sigma_s$. This regime is reported in the right inset of \fref{tau_sigma_s}. Roughly speaking, for $\tau \rightarrow 0$, the wave packet moves a distance $v \tau$. When $v \tau < \sigma_s$ the measured position $x_s$ has no relation to the velocity. We emphasize again that Wiseman's result \cite{Wiseman} does not suffer from this inaccuracy because he considers, both, $\sigma_s \rightarrow 0$ and $\tau \rightarrow 0$.
A closer look at the expressions \hat{p}ref{errorW} and \hat{p}ref{errorS} shows that the error diverges when $\rho$ has oscillations with minima tending to zero. This can be clearly seen in \fref{vel}(a) and (a\rq{}) where the highest peak of the velocity corresponds to a minimum of $\rho$ very close to zero. This situation is reversed when we evaluate the current $J$ [see \fref{vel}(b) and (b\rq{})]. In fact, in these critical points, $J\rightarrow 0$ and even the corresponding errors become very small. In \fref{tau_sigma_s} it is evident the shift of the $<1\%$ region due to this error reduction.
Perhaps, the most surprising feature of our protocol is that a local (in time and position) Bohmian velocity can be measured with a large temporal separation between measurements, while one would expect a time-averaged value as discussed at the end of section \ref{sec_velocity}.
This is highly counter-intuitive because we are in a scenario where the time-evolving interferences implies large acceleration of the Bohmian particle in order to rapidly avoid the nodes of the wave function.
Finally, another relevant result is that the accuracy of the Bohmian velocity is obtained at the prize of increasing the dispersion on $x_w$ (as seen in \eref{var} for large $\sigma_w$). Therefore, the fact that we can obtain the Bohmian velocity is not because the system remains unperturbed after one position measurement, rather because of the ability of the ensemble average done in the $x_w$ integrals on \eref{prob1} and \eref{prob2} to compensate for the different perturbations. The fact that a very large perturbation of the state is fully compatible with a negligible error can be easily seen in our numerical data. The {\it measured} state is roughly equal to the product of the unperturbed wave function (whose support is $L\approx 2000 $nm at time $t_w=11$ps in \fref{vel}) by a Gaussian function centered at the measured position with a dispersion equal to $\sigma_w$ (for example, $\sigma_w \approx 150 $nm for $\tau=1$ps in \fref{tau_sigma_w}). Even for $\sigma_w << L$ (i.e. a large perturbation), the velocity error is negligible in \fref{tau_sigma_w}.
\section{Conclusions}
The work presented here explains a protocol for measuring Bohmian velocities. It is based on using an ensemble of two position measurements separated by a finite time interval. The perturbation of each position measurements on the state is modeled by a POVM. The difference between the Bohmian velocity of the unperturbed state and the ensemble Bohmian velocity of the two-times measured state is predicted to be much smaller than $1 \%$ in a large range of parameters. The work clarifies the laboratory conditions necessary for measuring Bohmian velocities, while relaxing the experimental setup by allowing reasonable position uncertainties and a finite time interval between measurements. Following the same ideas presented in this work (with two POVM for position measurements) an equivalent analysis for the case of combined POVM momentum plus POVM position measurements can be carried out for particles with mass. This case, experimentally tested also for relativistic photons \cite{Bohmphoton}, could be of major interest for several experiments. In this sense, a clear and feasible proposal has been recently presented for the demonstration of the nonlocal character of Bohmian mechanics by measuring the ensemble velocities of path-entangled particles \cite{nou}. Finally, as mentioned in the introduction, the present work is fully developed within orthodox quantum mechanics. However, we emphasize that this works opens relevant and unexplored possibilities for understanding quantum phenomena through the quantitative comparison between simulated and measured Bohmian (or hydrodynamic) trajectories \cite{libro,Oriols,turbulence}, instead of using the wave function and its related parameters.
\section{Derivation of the variance}
\label{appendixA}
In order to evaluate the variance $var(v)=var(x_{w}^{2})$ defined as
\[
var(x_{w}^{2})=\frac{\int dx_{w}x_{w}^{2}P(x_{w}\cap x_{s})}{P(x_{s}
)}-(E[x_{w}|x_{s}])^{2},
\]
where $P(x_{w}\cap x_{s})$ and $P(x_{s})$ are given respectively by Eq. \hat{p}ref{prob1} and \hat{p}ref{prob2}, we calculate
\begin{multline}
\int dx_{w}x_{w}^{2}P(x_{w}\cap x_{s})=\dfrac{\sigma_{w}^{2}}{2}\langle
\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}|\Psi\rangle+\\
+C_{s}^{2}\iiint dxdx^{\hat{p}rime}dx^{\hat{p}rime\hat{p}rime}\left( \frac{x^{\hat{p}rime
}+x^{\hat{p}rime\hat{p}rime}}{2}\right) ^{2}\times \\
\times e^{-\frac{\left( x^{\hat{p}rime}-x^{\hat{p}rime
\hat{p}rime}\right) ^{2}}{4\sigma_{w}^{2}}}e^{-\frac{\left( x_{s}-x\right) ^{2}
}{\sigma_{s}^{2}}}|x^{\hat{p}rime}\rangle\langle x^{\hat{p}rime}|U^{\dag}|x\rangle
\langle x|U|x^{\hat{p}rime\hat{p}rime}\rangle\langle x^{\hat{p}rime\hat{p}rime}|,
\end{multline}
where the integral over $x_{w}$ has been already evaluated. From Eq. \hat{p}ref{evo} and
accounting for the limit \hat{p}ref{lim} we have
\begin{multline}
\int dx_{w}x_{w}^{2}P(x_{w}\cap x_{s})=\dfrac{\sigma_{w}^{2}}{2}\langle
\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}|\Psi\rangle+\\
+\tfrac{1}{2}\operatorname{Re}(\langle\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag
}\hat{S}_{s}U_{\tau}\hat{x}^{2}|\Psi\rangle)+\tfrac{1}{2}\langle\Psi|\hat
{x}U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}\hat{x}|\Psi
\rangle.\label{x2}
\end{multline}
Under the limit \hat{p}ref{lim} we have shown in the text that $P(x_{s})=\langle
\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}|\Psi\rangle$. Moreover using Eq. (16) we have
\begin{multline}
\langle\Psi|\hat{x}U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}\hat
{x}|\Psi\rangle=\operatorname{Re}(\langle\Psi|U_{\tau}^{\dag}\hat{S}_{s}
^{\dag}\hat{S}_{s}U_{\tau}\hat{x}^{2}|\Psi\rangle+\\
+\dfrac{\tau}{m}\langle
\Psi|U_{\tau}^{\dag}[\hat{S}_{s}^{\dag}\hat{S}_{s},\hat{p}]U_{\tau}\hat
{x}|\Psi\rangle),
\end{multline}
that substituted in Eq. \hat{p}ref{cond4} gives
\begin{multline}
var(x_{w}^{2})=\dfrac{\sigma_{w}^{2}}{2}+\frac{\operatorname{Re}(\langle
\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}\hat{x}^{2}
|\Psi\rangle)}{\langle\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}_{s}
U_{\tau}|\Psi\rangle}+\\
+\frac{\tau}{2m}\frac{\operatorname{Re}(\langle
\Psi|U_{\tau}^{\dag}[\hat{S}_{s}^{\dag}\hat{S}_{s},\hat{p}]U_{\tau}\hat
{x}|\Psi\rangle)}{\langle\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}
_{s}U_{\tau}|\Psi\rangle}-(E[x_{w}|x_{s}])^{2}.\label{var1}
\end{multline}
The difference between the second and the fourth terms on the r.h.s. of Eq.
(\ref{var1}) can be rewritten using again Eq. \hat{p}ref{AAV1} and \hat{p}ref{commutator} as
\begin{multline}
\frac{\operatorname{Re}(\langle\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat
{S}_{s}U_{\tau}\hat{x}^{2}|\Psi\rangle)}{\langle\Psi|U_{\tau}^{\dag}\hat
{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}|\Psi\rangle}-(E[x_{w}|x_{s}])^{2}
=\tfrac{\tau^{2}}{m^{2}}\times\\
\times \left( \dfrac{\operatorname{Re}\langle\Psi|U_{\tau
}^{\dag}\hat{S}_{s}^{\dag}\hat{S}_{s}\hat{p}^{2}U_{\tau}|\Psi\rangle}
{\langle\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}|\Psi\rangle
}-\left( \dfrac{\operatorname{Re}\langle\Psi|U_{\tau}^{\dag}\hat{S}_{s}
^{\dag}\hat{S}_{s}\hat{p}U_{\tau}|\Psi\rangle}{\langle\Psi|U_{\tau}^{\dag}
\hat{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}|\Psi\rangle}\right) ^{2}\right)
.\label{var2}
\end{multline}
Using in \hat{p}ref{var2} the relations $\langle x|\hat{p}U_{\tau}|\Psi\rangle=-i\hbar\hat{p}artial
_{x}\Psi(x,\tau)$ and $\langle x|\hat{p}^{2}U_{\tau}|\Psi\rangle=-\hbar
^{2}\hat{p}artial_{x}^{2}\Psi(x,\tau)$ and the limit \hat{p}ref{lim2}, we can rewrite (\ref{var2}) as:
\begin{multline}
var(x_{w}^{2})=\dfrac{\sigma_{w}^{2}}{2}+2\dfrac{\tau^{2}}{m}Q_{B}(x_{s}
,\tau)+\\
+\frac{\tau}{2m}\frac{\operatorname{Re}(\langle\Psi|U_{\tau}^{\dag}
[\hat{S}_{s}^{\dag}\hat{S}_{s},\hat{p}]U_{\tau}\hat{x}|\Psi\rangle)}
{\langle\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}|\Psi\rangle
}.\label{var3}
\end{multline}
We further evaluate the commutator $[\hat{S}_{s}^{\dag}\hat{S}_{s},\hat{p}]$
as
\begin{multline}
\lbrack\hat{S}_{s}^{\dag}\hat{S}_{s},\hat{p}]|\Psi\rangle=-i\hbar C_{s}
^{2}\int dx\left( e^{-\frac{\left( x_{s}-x\right) }{\sigma_{s}^{2}}^{2}
}\left( \hat{p}artial_{x}\Psi(x)\right) |x\rangle-\right. \\
-\left.\left[ \hat{p}artial_{x}\left(
e^{-\frac{\left( x_{s}-x\right) }{\sigma_{s}^{2}}^{2}}\Psi(x)\right)
\right] |x\rangle\right) =-i\hbar\hat{p}artial_{x_{s}}\left( \hat{S}_{s}^{\dag
}\hat{S}_{s}\right) |\Psi\rangle,\label{commutator_appendix}
\end{multline}
and using Eq. (\ref{commutator_appendix}) in the last term of Eq. (\ref{var3}) we have
\begin{multline}
var(x_{w}^{2})=\dfrac{\sigma_{w}^{2}}{2}+2\dfrac{\tau^{2}}{m}Q_{B}(x_{s}
,\tau)+\\
+\frac{\tau\hbar}{2m}\frac{\hat{p}artial_{x_{s}}\operatorname{Im}(\langle
\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}\hat{x}|\Psi\rangle
)}{\langle\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}
|\Psi\rangle}.\label{var4}
\end{multline}
>From the limits \hat{p}ref{lim} and \hat{p}ref{lim2} we have
\begin{equation}
\frac{\tau\hbar}{m}\ll\sigma_{w}\sigma_{s}\ll\sigma_{w}^{2},
\end{equation}
and we can conclude that both the last two terms of the r.h.s. of Eq. (\ref{var4})
are much smaller than $\sigma_{w}^{2}$.
\section{Derivation of the error $\varepsilon_{s}(x_{s})$}
\label{appendixB}
The definition of $\varepsilon_{s}(x_{s})$ is:
\begin{multline}
\varepsilon_{s}(x_{s})=|v(x_{s})-\bar{v}(x_{s})|=\tau^{-1}\left\vert
\dfrac{\operatorname{Re}\langle\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat
{S}_{s}U_{\tau}\hat{x}|\Psi\rangle}{\langle\Psi|U_{\tau}^{\dag}\hat{S}
_{s}^{\dag}\hat{S}_{s}U_{\tau}|\Psi\rangle}-\right.\\
\left.-\dfrac{\operatorname{Re}
\langle\Psi|U_{\tau}^{\dag}|x_{s}\rangle\langle x_{s}|U_{\tau}\hat{x}
|\Psi\rangle}{\langle\Psi|U_{\tau}^{\dag}|x_{s}\rangle\langle x_{s}|U_{\tau
}|\Psi\rangle}\right\vert .\label{epss}
\end{multline}
We can easily take the limit of (\ref{epss}) for $\sigma_{s}$ small using a Taylor
series,
\begin{multline}
\langle\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}|\Psi\rangle=\\%
={\textstyle\sum_{n=0}^{2}}
\frac{\hat{p}artial_{x}^{n}\rho}{n!}C_{s}^{2}\int e^{-\frac{\left( x_{s}-x\right)
}{\sigma_{s}^{2}}^{2}}\left( x-x_{s}\right) ^{n}dx=\\
=\rho+\frac{\sigma_{s}
^{2}}{4}\hat{p}artial_{x}^{2}\rho\label{1}
\end{multline}
and in the same way using Eq. \hat{p}ref{commutator}
\begin{multline}
\operatorname{Re}\langle\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}
_{s}U_{\tau}\hat x|\Psi\rangle=\\
=\operatorname{Re}\langle\Psi|U_{\tau}^{\dag}\hat
{S}_{s}^{\dag}\hat{S}_{s}\hat x U_{\tau}|\Psi\rangle-\frac{\tau}{m}\operatorname{Re}
\langle\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}_{s}\hat p U_{\tau}|\Psi
\rangle=\label{2}\\
x_{s}\rho+\frac{\sigma_{s}^{2}}{2}\hat{p}artial_{x}\rho+x_{s}\frac{\sigma_{s}}
{4}\hat{p}artial_{x}^{2}\rho-\tau J-\tau\frac{\sigma_{s}^{2}}{4}\hat{p}artial_{x}^{2}J.
\end{multline}
Being $\operatorname{Re}\langle\Psi|U_{\tau}^{\dag}|x_{s}\rangle\langle
x_{s}|U_{\tau}X|\Psi\rangle=x_{s}\rho-\tau J$, and substituting Eq. (\ref{1}) and
(\ref{2}) into Eq. (\ref{epss}), we finally have
\begin{multline}
\varepsilon_{s}(x_{s})=\\=\tau^{-1}\left\vert \dfrac{4x_{s}\rho+2\sigma_{s}
^{2}\hat{p}artial_{x}\rho+x_{s}\sigma_{s}\hat{p}artial_{x}^{2}\rho-4\tau J-\tau
\sigma_{s}^{2}\hat{p}artial_{x}^{2}J}{4\rho+\sigma_{s}^{2}\hat{p}artial_{x}^{2}\rho
}-\right.\\ \left.-\dfrac{x_{s}\rho-\tau J}{\rho}\right\vert =\\
=\tau^{-1}\left\vert \dfrac{2\sigma_{s}^{2}\hat{p}artial_{x}\rho+\tau v\sigma
_{s}^{2}\hat{p}artial_{x}^{2}\rho-\tau\sigma_{s}^{2}\hat{p}artial_{x}^{2}J}{4\rho
+\sigma_{s}^{2}\hat{p}artial_{x}^{2}\rho}\right\vert =\\=\sigma_{s}^{2}\left\vert
\dfrac{\frac{2}{\tau}\hat{p}artial_{x}\rho+\left( 2\hat{p}artial_{x}\rho-\rho
\hat{p}artial_{x}\right) \hat{p}artial_{x}v}{4\rho+\sigma_{s}^{2}\hat{p}artial_{x}^{2}\rho
}\right\vert
\end{multline}
\section{Derivation of the error $\varepsilon_{w}(x_{s})$}
\label{appendixC}
The definition of $\varepsilon_{w}(x_{s})$ is:
\begin{multline}
\varepsilon_{w}(x_{s})=\tau^{-1}\left\vert \dfrac{\int dx_{w}x_{w}\langle
\Psi|\hat{W}_{w}^{\dag}U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}_{s}U_{\tau
}\hat{W}_{w}\Psi\rangle}{\int dx_{w}\langle\Psi|\hat{W}_{w}U_{\tau}^{\dag}
\hat{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}\hat{W}_{w}|\Psi\rangle}-\right.\\-\left.\dfrac
{\operatorname{Re}\langle\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}
_{s}U_{\tau}\hat{x}|\Psi\rangle}{\langle\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag
}\hat{S}_{s}U_{\tau}|\Psi\rangle}\right\vert .\label{espw}
\end{multline}
Under the limit (11) and after the integration over $x_{w}$ we can expand $\exp\left[ -\left(
x^{\hat{p}rime\hat{p}rime}-x^{\hat{p}rime}\right) ^{2}/4\sigma_{w}^{2}\right]$ in Taylor series
in the numerator and denominator of (\ref{espw}) to get
\begin{multline}
\label{ap01}
\int dx_{w}\langle\Psi|\hat{W}_{w}^{\dag}U_{\tau}^{\dag}\hat{S}_{s}^{\dag}
\hat{S}_{s}U_{\tau}\hat{W}_{w}\Psi\rangle=
\langle\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}|\Psi
\rangle-\\-\frac{1}{2\sigma_{w}^{2}}\left( \operatorname{Re}\langle\Psi|U_{\tau
}^{\dag}\hat{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}\hat{x}^{2}|\Psi\rangle
-\langle\Psi|\hat{x}U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}
\hat{x}|\Psi\rangle\right)
\end{multline}
and
\begin{multline}
\int dx_{w}x_{w}\langle\Psi|\hat{W}_{w}^{\dag}U_{\tau}^{\dag}\hat{S}_{s}
^{\dag}\hat{S}_{s}U_{\tau}\hat{W}_{w}\Psi\rangle=\label{ap02}\\
=\operatorname{Re}\langle\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}
_{s}U_{\tau}\hat{x}|\Psi\rangle-\frac{1}{4\sigma_{w}^{2}}\times\\ \times\left(
\operatorname{Re}\langle\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}
_{s}U_{\tau}\hat{x}^{3}|\Psi\rangle-\operatorname{Re}\langle\Psi|\hat
{x}U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}\hat{x}^{2}|\Psi
\rangle\right) .
\end{multline}
Moreover using twice Eq. \hat{p}ref{commutator} we have
\begin{multline}
\langle\Psi|\hat{x}U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}\hat
{x}|\Psi\rangle=\operatorname{Re}\left( \langle\Psi|U_{\tau}^{\dag}\hat
{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}\hat{x}^{2}|\Psi\rangle+\right. \\ \left. +\frac{\tau}{m}
\langle\Psi|U_{\tau}^{\dag}[\hat{S}_{s}^{\dag}\hat{S}_{s},\hat{p}]U_{\tau}
\hat{x}|\Psi\rangle\right) \label{ap1}\\
\end{multline}
and
\begin{multline}
\operatorname{Re}\langle\Psi|\hat{x}U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat
{S}_{s}U_{\tau}\hat{x}^{2}|\Psi\rangle=\operatorname{Re}\left( \langle
\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}\hat{x}^{3}
|\Psi\rangle+\right. \\ \left. +\frac{\tau}{m}\langle\Psi|U_{\tau}^{\dag}[\hat{S}_{s}^{\dag}
\hat{S}_{s},\hat{p}]U_{\tau}\hat{x}^{2}|\Psi\rangle\right) .\label{ap2}
\end{multline}
Putting Eq. (\ref{commutator_appendix}) into Eqs. (\ref{ap1}) and (\ref{ap2}) and
substituting them into Eqs. (\ref{ap01}) and (\ref{ap02}) we have
\begin{multline}
\int dx_{w}\langle\Psi|\hat{W}_{w}^{\dag}U_{\tau}^{\dag}\hat{S}_{s}^{\dag}
\hat{S}_{s}U_{\tau}\hat{W}_{w}\Psi\rangle=\\=\langle\Psi|U_{\tau}^{\dag}\hat
{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}|\Psi\rangle+\frac{\tau\hbar}{2m\sigma
_{w}^{2}}\hat{p}artial_{x_{s}}\operatorname{Im}\langle\Psi|U_{\tau}^{\dag}\hat
{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}\hat{x}|\Psi\rangle
\end{multline}
and
\begin{multline}
\int dx_{w}x_{w}\langle\Psi|\hat{W}_{w}^{\dag}U_{\tau}^{\dag}\hat{S}_{s}
^{\dag}\hat{S}_{s}U_{\tau}\hat{W}_{w}\Psi\rangle=\\=\operatorname{Re}\langle
\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}X|\Psi\rangle
+\\+\frac{\tau\hbar}{4m\sigma_{w}^{2}}\hat{p}artial_{x_{s}}\operatorname{Im}
\langle\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}\hat{x}
^{2}|\Psi\rangle.
\end{multline}
Using again Eqs. \hat{p}ref{commutator} and (\ref{commutator_appendix}) we realize that
\begin{align}
\operatorname{Im}\langle\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}
_{s}U_{\tau}\hat{x}|\Psi\rangle & =\frac{\hbar\tau}{2m}\hat{p}artial_{x_{s}}
\langle\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}|\Psi\rangle\\
\operatorname{Im}\langle\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}
_{s}U_{\tau}\hat{x}^{2}|\Psi\rangle & =\frac{\hbar\tau}{m}\hat{p}artial_{x_{s}
}\operatorname{Re}\langle\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}
_{s}U_{\tau}\hat{x}|\Psi\rangle
\end{align}
so finally we can write
\begin{multline}
\int dx_{w}\langle\Psi|\hat{W}_{w}^{\dag}U_{\tau}^{\dag}\hat{S}_{s}^{\dag}
\hat{S}_{s}U_{\tau}\hat{W}_{w}\Psi\rangle =\\=\left( 1+\frac{\tau^{2}\hbar
^{2}}{4m^{2}\sigma_{w}^{2}}\hat{p}artial_{x_{s}}^{2}\right) \langle\Psi|U_{\tau
}^{\dag}\hat{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}|\Psi\rangle\label{ap21}
\end{multline}
and
\begin{multline}
\int dx_{w}x_{w}\langle\Psi|\hat{W}_{w}^{\dag}U_{\tau}^{\dag}\hat{S}_{s}
^{\dag}\hat{S}_{s}U_{\tau}\hat{W}_{w}\Psi\rangle =\\=\left( 1+\frac{\tau
^{2}\hbar^{2}}{4m^{2}\sigma_{w}^{2}}\hat{p}artial_{x_{s}}^{2}\right)
\operatorname{Re}\langle\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}
_{s}U_{\tau}\hat{x}|\Psi\rangle\label{ap22}
\end{multline}
Evaluating the derivatives in (\ref{ap21}) and (\ref{ap22}), we have
\begin{multline}
\hat{p}artial_{x_{s}}^{2}\langle\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}
_{s}U_{\tau}|\Psi\rangle=C_{s}^{2}\hat{p}artial_{x_{s}}^{2}\int e^{-\frac{\left(
x_{s}-x\right) ^{2}}{\sigma_{s}^{2}}}\rho(x)dx=\\=-C_{s}^{2}\frac{4}{\sigma
_{s}^{4}}\int e^{-\frac{\left( x_{s}-x\right) ^{2}}{\sigma_{s}^{2}}}\left(
-(x_{s}-x)^{2}+\frac{\sigma_{s}^{2}}{2}\right) \rho(x)dx\label{ap31}
\end{multline}
and
\begin{multline}
\hat{p}artial_{x_{s}}^{2}\operatorname{Re}\langle\Psi|U_{\tau}^{\dag}S^{\dag
}SU_{\tau}X|\Psi\rangle=\\=-C_{s}^{2}\frac{4}{\sigma_{s}^{4}}\int e^{-\frac
{\left( x_{s}-x\right) ^{2}}{\sigma_{s}^{2}}}\left( -(x_{s}-x)^{2}
+\frac{\sigma_{s}^{2}}{2}\right) \times\\ \times\left( x\rho(x)-\tau J(x)\right)
dx\label{ap32}
\end{multline}
which, both can be rewritten in a compact way as
\begin{multline}
-C_{s}^{2}\frac{4}{\sigma_{s}^{2}}\int e^{-\frac{\left( x_{s}-x\right) ^{2}
}{\sigma_{s}^{2}}}\left( -(x_{s}-x)^{2}+\frac{\sigma_{s}^{4}}{2}\right)
\alpha(x)dx \approx \\ \approx \hat{p}artial_{x_{s}}^{2}\alpha(x_{s})\label{der}
\end{multline}
where we keep only the first three terms in the Taylor expansion.
Using Eq. (\ref{der}) in Eqs. (\ref{ap21}) and (\ref{ap22}) and plugging them into expression (\ref{espw}) we have
\begin{multline}
\varepsilon(\sigma_{w})=\\=\tau^{-1}\frac{\tau^{2}\hbar^{2}}{4m^{2}\sigma_{w}
^{2}}\left\vert \dfrac{\hat{p}artial_{x}^{2}\left( x\rho-\tau J\right)
-\tfrac{\operatorname{Re}\langle\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat
{S}_{s}U_{\tau}\hat{x}|\Psi\rangle}{\langle\Psi|U_{\tau}^{\dag}\hat{S}
_{s}^{\dag}\hat{S}_{s}U_{\tau}|\Psi\rangle}\hat{p}artial_{x}^{2}\rho}{\langle
\Psi|U_{\tau}^{\dag}\hat{S}_{s}^{\dag}\hat{S}_{s}U_{\tau}|\Psi\rangle
+\frac{\tau^{2}\hbar^{2}}{4m^{2}\sigma_{w}^{2}}\hat{p}artial_{x}^{2}\rho
}\right\vert
\end{multline}
which can be finally rewritten using equations (\ref{1}) and (\ref{2}) as
\begin{multline}
\varepsilon(\sigma_{w})=\\=\frac{\tau\hbar^{2}}{4m^{2}\sigma_{w}^{2}}\left\vert
\dfrac{2\hat{p}artial_{x}\rho-\tau\hat{p}artial_{x}^{2}J-\tfrac{2\sigma_{s}^{2}
\hat{p}artial_{x}^{2}\rho-4\tau J-\tau\sigma_{s}^{2}\hat{p}artial_{x}^{2}J}{4\rho
+\sigma_{s}^{2}\hat{p}artial_{x}^{2}\rho}\hat{p}artial_{x}^{2}\rho}{\rho+\frac
{\sigma_{s}^{2}}{4}\hat{p}artial_{x}^{2}\rho+\frac{\tau^{2}\hbar^{2}}{4m^{2}
\sigma_{w}^{2}}\hat{p}artial_{x}^{2}\rho}\right\vert .
\end{multline}
In the limit of small $\sigma_{s}$ we finally get
\begin{multline}
\varepsilon(\sigma_{w})=\\=\frac{\tau\hbar^{2}}{4m^{2}\sigma_{w}^{2}}\left\vert
\dfrac{2\hat{p}artial_{x}\rho-\tau\hat{p}artial_{x}^{2}J+\tau v\hat{p}artial_{x}^{2}\rho
}{\rho+\frac{\tau^{2}\hbar^{2}}{4m^{2}\sigma_{w}^{2}}\hat{p}artial_{x}^{2}\rho
}\right\vert =\\=\frac{\tau\hbar^{2}}{4m^{2}\sigma_{w}^{2}}\left\vert
\dfrac{2\left( 1-\tau\hat{p}artial_{x}v\right) \hat{p}artial_{x}\rho-\tau\rho
\hat{p}artial_{x}^{2}v}{\rho+\frac{\tau^{2}\hbar^{2}}{4m^{2}\sigma_{w}^{2}}
\hat{p}artial_{x}^{2}\rho}\right\vert .
\end{multline}
\end{document} | math | 45,288 |
\begin{document}
\begin{abstract}
Let $\lambda$ be a cardinal with $\lambda=\lambda^{\aleph_0}$ and $p$ be either $0$ or a prime number. We show that there are fields $K_0$ and $K_1$ of cardinality $\lambda$
and characteristic $p$ such that the automorphism group of $K_0$ is a free group of cardinality $2^\lambda$ and the automorphism group of $K_1$ is a free abelian group of
cardinality $2^\lambda$.
This partially answers a question from \cite{MR1736959} and complements results from \cite{MR1934424}, \cite{MR2773054} and \cite{MR1720580}.
The methods developed in the proof of the above statement also allow us to show that the above cardinal arithmetic assumption is consistently not necessary for the existence
of such fields and that
the existence of a cardinal $\lambda$ of uncountable cofinality with the property that there is no field of cardinality $\lambda$ whose automorphism group
is a free group of cardinality greater than $\lambda$ implies the existence of large cardinals in certain inner models of set theory.
\end{abstract}
\title{Free groups and automorphism groups of infinite fields}
\section{Introduction}\label{section:Introduction}
The work of this paper is motivated by questions of the following type: \emph{given an abstract group $G$ and an infinite cardinal $\lambda$, is $G$ isomorphic to the
automorphism group of a field of cardinality $\lambda$?} We start by presenting some known results related to this kind of problem.
If $K$ is an infinite field of cardinality $\lambda$, then the group $\Aut{K}$ consisting of all automorphisms of $K$ can be embedded into the group $\Sym{\kappa}$
of all permutations of $\lambda$ and therefore has cardinality at most $2^\lambda$.
It is well known (see \cite{MR660867}, \cite{Sh913} and Section \ref{section:MI} of this paper) that, given an infinite cardinal $\lambda$,
a first-order language $\mathcal{L}$ of cardinality at most $\lambda$ and an $\mathcal{L}$-model $\mathcal{M}$ of cardinality at most $\lambda$,
there is a field $K$ of arbitrary characteristic and cardinality $\lambda$ whose automorphism group is isomorphic to $\Aut{\mathcal{M}}$.
Given an infinite group $G$ of cardinality $\lambda$, it is easy to construct a first-order language $\mathcal{L}$ of cardinality $\lambda$ and an $\mathcal{L}$-model $\mathcal{M}$
of cardinality $\lambda$ such that the groups $G$ and $\Aut{\mathcal{M}}$ are isomorphic.
In particular, every infinite group is isomorphic to the automorphism group of a field of the same cardinality.
In contrast, for every infinite cardinal $\lambda$ there are groups of cardinality $\lambda^+$ that are not isomorphic to automorphism groups of fields of cardinality $\lambda$.
For example, De Bruijn showed in {\cite[Theorem 5.1]{MR0098127}} that the group $\Fin{\lambda^+}$ consisting of all
finite permutations of $\lambda^+$ cannot be embedded into the group $\Sym{\lambda}$.
In this paper, we focus on free groups and the following instances of the above problem.
\begin{question}\label{question:Motiv}
Is there a field $K$ whose automorphism group is a free group of cardinality greater than the cardinality of $K$?
More specifically, given an infinite cardinal $\lambda$, is there a field of cardinality $\lambda$ whose automorphism group is a free group of cardinality greater than $\lambda$?
\end{question}
The above question was first asked by David Evans for the case $\lambda=\aleph_0$.
The results of \cite{MR1736959} motivate its generalizations to uncountable cardinalities.
The following results due to the second author show that the second part of Question \ref{question:Motiv}
has a negative answer for $\lambda=\aleph_0$ and singular strong limit cardinals of countable cofinality.
\begin{theorem}[{\cite[Theorem 1]{MR1934424}}]\label{theorem:ST1}
Let $\mathcal{L}$ be a countable first-order language and $\mathcal{M}$ be a countable $\mathcal{L}$-model. Then $\Aut{\mathcal{M}}$ is not an uncountable free group.
\end{theorem}
\begin{theorem}[{\cite[Remark 5.2]{MR1934424}}]\label{theorem:ST2}
Let $\seq{\lambda_n}{n<\omega}$ be a sequence of infinite cardinals with $2^{\lambda_n}<2^{\lambda{n+1}}$ for all $n<\omega$, $\lambda=\sum_{n<\omega}\lambda_n$ and
$\mu=\sum_{n<\omega}2^{\lambda_n}$. If $\mathcal{L}$ is a first-order language of cardinality $\lambda$ and $\mathcal{M}$ is an $\mathcal{L}$-model of cardinality $\lambda$ such that
$\Aut{\mathcal{M}}$ has cardinality greater than $\mu$, then $\Aut{\mathcal{M}}$ is not a free group.
\end{theorem}
In contrast, Just, Thomas and the second author showed in {\cite[Theorem 1.14]{MR1736959}} that, given a regular uncountable cardinal $\lambda$ with
$\lambda=\lambda^{{<}\lambda}$ and $\nu>\lambda$, there is a cofinality preserving forcing extension of the ground model that adds no new sequences of ordinals
of length less than $\lambda$ and contains a field of cardinality $\lambda$ whose automorphism group is a free group of cardinality $\nu$.
In particular, it is consistent with the axioms of ${\rm{ZFC}}$ that the above question has a positive answer.
The following main result of this paper shows that the axioms of ${\rm{ZFC}}$ already imply a positive answer to the above question
for large class of cardinals of uncountable cofinality.
\begin{theorem}\label{theorem:Main1}
Let $\lambda$ be a cardinal with $\lambda=\lambda^{\aleph_0}$ and $p$ be either $0$ or a prime number. Then there is a field $K$ of characteristic $p$ and cardinality $\lambda$
whose automorphism group is a free group of cardinality $2^\lambda$.
\end{theorem}
Since the axioms of ${\rm{ZFC}}$ prove the existence of a cardinal $\lambda$ with $\lambda=\lambda^{\aleph_0}$, this results answers the first part of Question \ref{question:Motiv} positively.
Moreover, a combination of the above results allows us to completely answer the second part of the question under certain cardinal arithmetic assumptions.
The following corollary is an example of such an application.
\begin{corollary}\label{corollary:CHSCHAnswer}
Assume that the \emph{Continuum Hypothesis} and the \emph{Singular Cardinal Hypothesis} hold. Then the following statements are equivalent for every infinite cardinal $\lambda$.
\begin{enumerate}
\item There is a field of cardinality $\lambda$ whose automorphism group is a free group of cardinality greater than $\lambda$.
\item There is a cardinal $\kappa\leq\lambda$ with $2^\kappa>\lambda$ and $\cof{\kappa}>\omega$.
\end{enumerate}
\end{corollary}
We outline the proof of Theorem \ref{theorem:Main1}: In Section \ref{section:MI}, we will show that it suffices to construct an inverse system groups satisfying certain
cardinality assumptions whose inverse limit is a free group of large cardinality. We will construct such systems of groups assuming the existence of certain inverse systems of
sets in Section \ref{section:freegroups}. Finally, we will use the assumption $\lambda=\lambda^{\aleph_0}$ to construct suitable inverse systems of sets
in Section \ref{section:system}.
The methods developed in the proof of the above result also allow us to produce uncountable fields whose automorphism group is a free \emph{abelian} group of large cardinality.
\begin{theorem}\label{theorem:Main2}
Let $\lambda$ be a cardinal with $\lambda=\lambda^{\aleph_0}$ and $p$ be either $0$ or a prime number. Then there is a field $K$ of characteristic $p$ and cardinality
$\lambda$ whose automorphism group is a free abelian group of cardinality $2^\lambda$.
\end{theorem}
Again, this drastically contrasts the countable setting as the following result due to S\l awomir Solecki shows.
\begin{theorem}[{\cite[Remark 1.6]{MR1720580}}]
Let $\mathcal{L}$ be a countable first-order language and $\mathcal{M}$ be an $\mathcal{L}$-model. Then $\Aut{\mathcal{M}}$ is not an uncountable free abelian group.
\end{theorem}
In another direction, the methods developed in the proofs of the above results also allow us to show that the cardinal arithmetic assumption $\lambda=\lambda^{\aleph_0}$
is consistently not necessary for the existence of a field of cardinality $\lambda$ whose automorphism group is a free group of cardinality greater than $\lambda$.
This is an implication of the following result. Given a cardinal $\lambda$, we use $\Add{\omega}{\lambda}$ to denote the forcing that adds $\lambda$-many Cohen reals to the ground model.
\begin{theorem}\label{theorem:OuterModelFields}
Let $\lambda$ be a cardinal with $\lambda=\lambda^{\aleph_0}$ and $p$ be either $0$ or a prime number.
If $G$ is $\Add{\omega}{\kappa}$-generic over the ground model ${\rm{V}}$ for some cardinal $\kappa$, then there is a field $K$ of characteristic $p$ and cardinality $\lambda$
contained in ${\rm{V}}[G]$ whose automorphism group is a free group of cardinality greater than or equal to $(2^\lambda)^{\rm{V}}$ in ${\rm{V}}[G]$.
\end{theorem}
The above results raise the question whether the existence of a cardinal $\lambda$ of uncountable cofinality with the property that there is no field of cardinality $\lambda$
whose automorphism group is a free group of cardinality greater than $\lambda$ is even consistent with the axioms of ${\rm{ZFC}}$.
Another byproduct of our constructions is the observation that the existence of such a cardinal has consistency strength strictly greater than that of ${\rm{ZFC}}$.
This observation is a consequence of the next result.
Remember that a partial order ${\mathbb{T}}=\langle T,\leq_{\mathbb{T}}\rangle$ is a \emph{tree} if
${\mathbb{T}}$ has a unique minimal element and the set $prec_{\mathbb{T}}(t)=\Set{s\in T}{s\leq_{\mathbb{T}} t,~s\neq t}$ is a well-ordered by $\leq_{\mathbb{T}}$ for every $t\in T$.
Given such a tree ${\mathbb{T}}$ and $t\in T$, we define $\rank{t}{{\mathbb{T}}}$ to be the order-type of $\langle prec_{\mathbb{T}}(t),\leq_{\mathbb{T}}\rangle$. We call the ordinal
$\height{{\mathbb{T}}}={\rm{lub}}\Set{\rank{t}{{\mathbb{T}}}}{t\in T}$ the \emph{height} of ${\mathbb{T}}$. Finally, a subset $B$ of $T$ is a \emph{cofinal branch through ${\mathbb{T}}$}
if $B$ is $\leq_{\mathbb{T}}$-downwards closed and $B$ is well-ordered by $\leq_{\mathbb{T}}$ with order-type $\height{{\mathbb{T}}}$.
\begin{theorem}\label{theorem:InnerModelFields}
Let $\lambda$ be a cardinal of uncountable cofinality.
If there is a tree of cardinality and height $\lambda$ with more than $\lambda$-many cofinal branches, then there is a field of cardinality $\lambda$
whose automorphism group is a free group of cardinality bigger than $\lambda$.
\end{theorem}
By considering the tree $\langle({}^{{<}\lambda}2)^M,\subseteq\rangle$ for some inner model $M$, this result directly implies the following corollary.
\begin{corollary}
Let $\lambda$ be a cardinal of uncountable cofinality and $M$ be an inner model ${\rm{ZFC}}$ with $(\lambda^+)^M=\lambda^+$.
If $\lambda=(\lambda^{{<}\lambda})^M$, then there is a field of cardinality $\lambda$
whose automorphism group is a free group of cardinality bigger than $\lambda$. \qed
\end{corollary}
This statement directly allows us to derive large cardinal strength from the non-existence of certain fields.
\begin{corollary}
Let $\lambda$ be a regular uncountable cardinal such that there is no field of cardinality $\lambda$ whose automorphism group is a free group of cardinality greater than
$\lambda$. Then $\lambda^+$ is an inaccessible cardinal in ${\rm{L}}[x]$ for every $x\subseteq\lambda$.
\end{corollary}
\begin{proof}
Assume, towards a contradiction, that $\lambda^+$ is not an inaccessible cardinal in ${\rm{L}}[x]$ for some $x\subseteq\lambda$. Then there is a $y\subseteq\lambda$ with
$\lambda^+=(\lambda^+)^{{\rm{L}}[y]}$ and $\langle({}^{{<}\kappa}2)^{{\rm{L}}[y]},\subseteq\rangle$ is a tree of cardinality and height $\lambda$, because our assumptions imply
$(\lambda^{{<}\lambda})^{{\rm{L}}[y]}=\lambda$. But the set of cofinal branches through this tree has cardinality at least $(2^\lambda)^{{\rm{L}}[y]}=(\lambda^+)^{{\rm{L}}[y]}=\lambda^+$,
a contradiction.
\end{proof}
Note that Mitchell used an inaccessible cardinal to constructed a model of ${\rm{ZFC}}$ in which every tree of cardinality and height $\omega_1$ has at most $\aleph_1$-many cofinal
branches (see {\cite[Section 8]{MR823775}} and \cite{MR0313057}). This statement is also a consequence of the \emph{Proper Forcing Axiom} (see {\cite[Theorem 7.10]{MR776640}}).
In the case of singular cardinals of uncountable cofinality, it is possible to use \emph{core model theory}
(see, for example, \cite{MR2768699}) to obtain inner models containing much larger large cardinals from the above assumption.
\begin{corollary}
Let $\lambda$ be a singular cardinal of uncountable cofinality such that there is no field of cardinality $\lambda$ whose automorphism group is a free group of cardinality
greater than $\lambda$. Then there is an inner model with a Woodin cardinal.
\end{corollary}
\begin{proof}
Assume, towards a contradiction, that there is no inner model with a Woodin cardinal. Then we can construct the \emph{core model ${\rm{K}}$ below one Woodin cardinal}.
It satisfies the \emph{Generalized Continuum Hypothesis} and has the \emph{covering property}. In particular, we have
$\lambda^+=(\lambda^+)^{{\rm{K}}}$. But this means that $\langle({}^{{<}\lambda}2)^{{\rm{K}}},\subseteq\rangle$ is a tree of cardinality and height $\lambda$ and the set of cofinal
branches through this tree has cardinality at least $(2^\lambda)^{{\rm{K}}}=(\lambda^+)^{{\rm{K}}}=\lambda^+$, a contradiction.
\end{proof}
The results of {\cite[Section 2]{MR1812172}} show that the non-existence of such trees at a singular cardinal of uncountable cofinality is equivalent to a
\rm{PCF}-theoretic statement that is not known to be consistent. Related questions can also be found in {\cite[Chapter II, Section 6]{MR1318912}}.
\section{Representing inverse limits as automorphism groups}\label{section:MI}
In this section, we start from an inverse system of groups ${\mathbb{I}}$ to construct a first-order language $\mathcal{L}$ and an $\mathcal{L}$-model $\mathcal{M}$ such that $\Aut{M}$
is an inverse limit of ${\mathbb{I}}$ and the cardinalities of $\mathcal{L}$ and $\mathcal{M}$ only depend on the cardinalities of the groups in ${\mathbb{I}}$ and the cardinality of the
underlying directed set.
We start by recalling some standard definitions and presenting the relevant examples.
We call a pair ${\mathbb{D}}=\langle D,\leq_{\mathbb{D}}\rangle$ a \emph{directed set} if $\leq_{\mathbb{D}}$ is a reflexive, transitive binary relation on the set $D$
with the property that for all $p,q\in D$ there is a $r\in D$ with $p,q\leq_{\mathbb{D}} r$.
Given a directed set ${\mathbb{D}}=\langle D,\leq_{\mathbb{D}}\rangle$, we call a pair
\begin{equation*}
{\mathbb{I}}=\langle\seq{A_p}{p\in D},\seq{f_{p,q}}{p,q\in D,~p\leq_{\mathbb{D}} q}\rangle
\end{equation*}
an \emph{inverse system of sets over ${\mathbb{D}}$} if the following statements hold for all $p,q,r \in D$ with $p\leq_{\mathbb{D}} q\leq_{\mathbb{D}} r$.
\begin{enumerate}
\item $A_p$ is a non-empty set and $\map{f_{p,q}}{A_q}{A_p}$ is a function.
\item $f_{p,p}={\rm{id}}_{A_p}$ and $f_{p,q}\circ f_{q,r}=f_{p,r}$.
\end{enumerate}
Given such an inverse system ${\mathbb{I}}$, we call the set
\begin{equation*}
A_{\mathbb{I}} = \BigSet{(a_p)_{p\in D}}{\textit{$f_{p,q}(a_q)=a_p$ for all $p,q\in D$ with $p\leq_{\mathbb{D}} q$}}
\end{equation*}
the \emph{inverse limit of ${\mathbb{I}}$}.
\begin{example}\label{example:LambdaCountableInverseLimit}
Let $\lambda$ be an infinite cardinal and let $[\lambda]^{\aleph_0}$ denote the set of all countable subsets of $\lambda$.
Given $u,v\in[\lambda]^{\aleph_0}$ with $u\subseteq v$,
set $A_u={}^u 2$ and define $\map{f_{u,v}}{A_v}{A_u}$ by $f_{u,v}(s)=s\restriction u$ for all $s\in{}^v 2$.
Let
\begin{equation*}
{\mathbb{I}}_\lambda ~ = ~ \langle\seq{A_u}{u\in[\lambda]^{\aleph_0}},\seq{f_{u,v}}{u,v\in[\lambda]^{\aleph_0},~u\subseteq v}\rangle
\end{equation*}
denote the resulting inverse system of sets over the directed set $\langle [\lambda]^{\aleph_0},\subseteq\rangle$.
Then it is easy to see that
\begin{equation*}
\Map{b}{{}^\lambda 2}{A_{{\mathbb{I}}_\lambda}}{x}{(x\restriction u)_{u\in[\lambda]^{\aleph_0}}}
\end{equation*}
is a well-defined bijection between the sets ${}^\lambda 2$ and $A_{{\mathbb{I}}_\lambda}$.
\end{example}
\begin{example}\label{example:TreesInverseSystems}
Let ${\mathbb{T}}=\langle T,\leq_{\mathbb{T}}\rangle$ be a tree. Given $\alpha<\height{{\mathbb{T}}}$, we let ${\mathbb{T}}(\alpha)$ denote the set of all $t\in T$ with $\rank{t}{{\mathbb{T}}}=\alpha$.
If $t\in T$ and $\alpha\leq\rank{t}{{\mathbb{T}}}$, then we let $t\restriction\alpha$ denote the unique element $s\in \{t\}\cup prec_{\mathbb{T}}(t)$ with $\rank{s}{{\mathbb{T}}}=\alpha$.
Given $\alpha\leq\beta<\height{{\mathbb{T}}}$, set $A_\alpha={\mathbb{T}}(\alpha)$ and
\begin{equation*}
\Map{f_{\alpha,\beta}}{A_\beta}{A_\alpha}{t}{t\restriction\alpha}.
\end{equation*}
We let
\begin{equation*}
{\mathbb{I}}_{\mathbb{T}} ~ = ~ \langle\seq{A_\alpha}{\alpha<\height{{\mathbb{T}}}},\seq{f_{\alpha,\beta}}{\alpha\leq\beta<\height{{\mathbb{T}}}}\rangle
\end{equation*}
denote the resulting inverse system of sets over the directed set $\langle\height{{\mathbb{T}}},\leq\rangle$.
It is easy to see that the induced map
\begin{equation*}
\Map{b}{A_{{\mathbb{I}}_{\mathbb{T}}}}{[{\mathbb{T}}]}{(a_\alpha)_{\alpha<\lambda}}{\Set{a_\alpha}{\alpha<\lambda}}
\end{equation*}
is a bijection between the inverse limit $A_{{\mathbb{I}}_{\mathbb{T}}}$ and the set $[{\mathbb{T}}]$ consisting of all cofinal branches through ${\mathbb{T}}$.
\end{example}
We now consider inverse limits in the category of groups.
Given a directed set ${\mathbb{D}}=\langle D,\leq_{\mathbb{D}}\rangle$, a pair
\begin{equation*}
{\mathbb{I}}=\langle\seq{G_p}{p\in D},\seq{h_{p,q}}{p,q\in D,~p\leq_{\mathbb{D}} q}\rangle
\end{equation*}
is an \emph{inverse system of groups over ${\mathbb{D}}$} if the following statements hold.
\begin{enumerate}
\item $G_p$ is a group with underlying set $X_p$ for all $p\in D$.
\item If $p,q\in D$ with $p\leq_{\mathbb{D}} q$, then $\map{h_{p,q}}{G_q}{G_p}$ is a homomorphism of groups.
\item The pair
\begin{equation*}
\langle\seq{X_p}{p\in D},\seq{h_{p,q}}{p,q\in D,~p\leq_{\mathbb{D}} q}\rangle
\end{equation*}
is an inverse system of sets.
\end{enumerate}
Given such a system ${\mathbb{I}}$, we call the subgroup
\begin{equation*}
G_{{\mathbb{I}}}=\BigSet{(g_p)_{p\in D}\in\prod_{p\in D}G_p}{\textit{$h_{p,q}(g_q)=g_p$ for all $p,q\in D$ with $p\leq_{\mathbb{D}} q$}}
\end{equation*}
of the direct product of the $G_p$'s the \emph{inverse limit of ${\mathbb{I}}$}.
\begin{example}\label{example:LimitOfFreeGroups}
Let ${\mathbb{D}}=\langle D,\leq_{\mathbb{D}}\rangle$ be a directed set and
\begin{equation*}
{\mathbb{I}}=\langle\seq{A_p}{p\in D},\seq{f_{p,q}}{p,q\in D,~p\leq_{\mathbb{D}} q}\rangle
\end{equation*}
be an inverse system of sets over ${\mathbb{D}}$. For each $p\in D$, let $G_p$ be the free group with generators $\Set{x_{p,a}}{a\in A_p}$.
Given $p,q\in D$ with $p\leq_{\mathbb{D}} q$, we let $\map{h_{p,q}}{G_q}{G_p}$ denote the unique homomorphism of groups with
$h_{p,q}(x_{q,a})=x_{p,f_{p,q}(a)}$ for all $a\in A_q$. Let
\begin{equation*}
{\mathbb{I}}_{gr} ~ = ~ \langle\seq{G_p}{p\in D},\seq{h_{p,q}}{p,q\in D,~p\leq_{\mathbb{D}} q}\rangle
\end{equation*}
denote the resulting inverse system of groups over ${\mathbb{D}}$.
\end{example}
It is an obvious question whether the corresponding inverse limit $G_{{\mathbb{I}}_{gr}}$ is itself a free group. In Section \ref{section:freegroups}
we will present conditions that imply this statement. These implications will allow us to prove Theorem \ref{theorem:Main1}.
\begin{example}
Pick ${\mathbb{D}}$ and ${\mathbb{I}}$ as in Example \ref{example:LimitOfFreeGroups}.
For each $p\in D$, let $H_p$ be the free abelian group with basis $\Set{x_{p,a}}{a\in A_p}$.
Define $\map{h_{p,q}}{H_q}{H_p}$ for all $p,q\in D$ with $p\leq_{\mathbb{D}} q$ as above and let
\begin{equation*}
{\mathbb{I}}_{ab} ~ = ~ \langle\seq{H_p}{p\in D},\seq{h_{p,q}}{p,q\in D,~p\leq_{\mathbb{D}} q}\rangle
\end{equation*}
denote the resulting inverse system of groups over ${\mathbb{D}}$. Since every $H_p$ is an abelian group, the group $G_{{\mathbb{I}}_{ab}}$ is also abelian.
\end{example}
This section focuses on the proof of the following result.
\begin{theorem}\label{theorem:ModelIAut}
Let ${\mathbb{I}}=\langle\seq{G_q}{q\in D},\seq{h_{q,r}}{q,r\in D,~q\leq_{\mathbb{D}} r}\rangle$
be an inverse system of groups over a directed set ${\mathbb{D}}=\langle D,\leq_{\mathbb{D}}\rangle$ and $p$ be either $0$ or a prime number. Then there is a field $K$ of characteristic $p$ with the following properties.
\begin{enumerate}
\item The groups $\Aut{K}$ and $G_{{\mathbb{I}}}$ are isomorphic.
\item $\betrag{K} \leq \max\{\aleph_0,\sum_{q\in D}\betrag{G_q}\}$.
\end{enumerate}
\end{theorem}
In the remainder of this section, we fix ${\mathbb{D}}$ and ${\mathbb{I}}$ as in the statement of the theorem.
We define $\lambda$ to be the cardinal $\max\{\aleph_0,\sum_{q\in D}\betrag{G_q}\}$.
The following results show that it suffices to find a first-order language $\mathcal{L}_{\mathbb{I}}$ of cardinality $\lambda$ and an $\mathcal{L}_{\mathbb{I}}$-model $\mathcal{M}_{\mathbb{I}}$ of cardinality $\lambda$
such that the groups $\Aut{\mathcal{M}_{\mathbb{I}}}$ and $G_{{\mathbb{I}}}$ are isomorphic.
\begin{proposition}
Let $\lambda$ be an infinite cardinal and $\mathcal{L}$ be a first order language of cardinality at most $\lambda$.
If $\mathcal{M}$ is an $\mathcal{L}$-model of cardinality at most $\lambda$, then there is a connected graph $\Gamma=\langle X,E\rangle$ such that $\betrag{X}=\lambda$
and the groups $\Aut{\mathcal{M}}$ and $\Aut{\Gamma}$ are isomorphic.
\end{proposition}
\begin{proof}
The above statement can be derived from the results of {\cite[Section 5.5]{MR1221741}} and {\cite[Section 3]{Sh913}}.
\end{proof}
\begin{theorem}[\cite{MR660867} and {\cite[Main Theorem B]{Sh913}}]
Let $\Gamma=\langle X,E\rangle$ be a connected graph and $p$ be either $0$ or a prime number. Then there is a field $K$ of characteristic $p$ with the following properties.
\begin{enumerate}
\item The groups $\Aut{\Gamma}$ and $\Aut{K}$ are isomorphic.
\item $\betrag{K}\leq\max\{\aleph_0,\betrag{X}\}$.
\end{enumerate}
\end{theorem}
We are now ready to construct $\mathcal{L}_{\mathbb{I}}$ and $\mathcal{M}_{\mathbb{I}}$ with the properties stated above.
Define $\mathcal{L}_{\mathbb{I}}$ to be first-order language with the following symbols.
\begin{itemize}
\item Constant symbols $\dot{c}_{g,q}$ for all $q\in D$ and $g\in G_q$.
\item Unary relation symbols $\dot{P}_q$ for all $q\in D$.
\item Binary relation symbols $\dot{H}_{q,r}$ for all $q,r\in D$ with $q\leq_{\mathbb{D}} r$.
\item Ternary relations symbols $\dot{F}_q$ for all $q\in D$.
\end{itemize}
Let $\mathcal{M}_{\mathbb{I}}$ denote the unique $\mathcal{L}_{\mathbb{I}}$-model with the following properties.
\begin{itemize}
\item The domain of $\mathcal{M}_{\mathbb{I}}$ is the set
\begin{equation*}
M_{\mathbb{I}}=\Set{\langle g,q,i\rangle}{q\in D,~g\in G_q,~i<2}.
\end{equation*}
\item $\dot{c}_{g,q}^{\mathcal{M}_{\mathbb{I}}}=\langle g,q,1\rangle$ for all $q\in D$ and $g\in G_q$.
\item $\dot{P}_q^{\mathcal{M}_{\mathbb{I}}}=\Set{\langle g,q,0\rangle}{g\in G_q}$ for all $q\in D$.
\item $\dot{H}_{q,r}^{\mathcal{M}_{\mathbb{I}}}=\Set{\langle\langle g,r,0\rangle,\langle h_{q,r}(g),q,0\rangle\rangle}{g\in G_r}$ for all $q,r\in D$ with $q\leq_{\mathbb{D}} r$.
\item $\dot{F}_q^{\mathcal{M}_{\mathbb{I}}}=\Set{\langle\langle g,q,0\rangle,\langle h,q,1\rangle,\langle g\cdot h,q,0\rangle\rangle}{g,h\in G_q}$ for all $q\in D$.
\end{itemize}
\begin{proposition}\label{proposition:SigmaRestrictionQ}
If $\sigma\in\Aut{\mathcal{M}_{\mathbb{I}}}$, $q\in D$ and $g\in G_q$, then $\sigma(\langle g,q,1\rangle)=\langle g,q,1\rangle$ and $\map{\sigma\restriction\dot{P}_q^{\mathcal{M}_{\mathbb{I}}}}{\dot{P}_q^{\mathcal{M}_{\mathbb{I}}}}{\dot{P}_q^{\mathcal{M}_{\mathbb{I}}}}$. \qed
\end{proposition}
\begin{proposition}\label{proposition:proposition:SigmaRestrictionP}
If $\sigma\in\Aut{\mathcal{M}_{\mathbb{I}}}$ and $q\in D$, then there is a unique $c_{\sigma,q}\in G_q$ with $\sigma(\langle g,q,0\rangle)=\langle c_{\sigma,q}\cdot g,q,0\rangle$
for all $g\in G_q$.
\end{proposition}
\begin{proof}
By Proposition \ref{proposition:SigmaRestrictionQ}, there is a unique $c_{\sigma,q}\in G_q$ such that $\sigma(\langle {1{\rm\hspace{-0.5ex}l}} _{G_q},q,0\rangle)=\langle c_{\sigma,q},q,0\rangle$.
Given $g\in G_q$, we have $\dot{F}_q^{\mathcal{M}_{\mathbb{I}}}(\langle c_{\sigma,q},q,0\rangle,\langle g,q,1\rangle,\sigma(\langle g,q,0\rangle)\rangle)$
and this implies $\sigma(\langle g,q,0\rangle)=\langle c_{\sigma,q}\cdot g,q,0\rangle$.
\end{proof}
\begin{proposition}
If $\sigma\in\Aut{\mathcal{M}_{\mathbb{I}}}$ and $q,r\in D$ with $q\leq_{\mathbb{D}} r$, then $h_{q,r}(c_{\sigma,r})=c_{\sigma,q}$.
In particular, the sequence $(c_{\sigma,q})_{q\in D}$ is an element of $G_{\mathbb{I}}$ for every $\sigma\in\Aut{\mathcal{M}_{\mathbb{I}}}$.
\end{proposition}
\begin{proof}
The definition of $\mathcal{M}_{\mathbb{I}}$ yields $\dot{H}_{q,r}^{\mathcal{M}_{\mathbb{I}}}(\langle {1{\rm\hspace{-0.5ex}l}} _{G_r},r,0\rangle,\langle {1{\rm\hspace{-0.5ex}l}} _{G_q},q,0\rangle)$. We can conclude
$\dot{H}_{q,r}^{\mathcal{M}_{\mathbb{I}}}(\langle c_{\sigma,r},r,0\rangle,\langle c_{\sigma,q},q,0\rangle)$ and $h_{q,r}(c_{\sigma,r})=c_{\sigma,q}$.
\end{proof}
\begin{lemma}\label{lemma:IsomorphismModelIInverseLimit}
The map
\begin{equation*}
\Map{\Phi}{\Aut{\mathcal{M}_{\mathbb{I}}}}{G_{\mathbb{I}}}{\sigma}{(c_{\sigma_q})_{p\in D}}
\end{equation*}
is an isomorphism of groups.
\end{lemma}
\begin{proof}
Given $\sigma_0,\sigma_1\in\Aut{\mathcal{M}_{\mathbb{I}}}$ and $q\in D$, we have
\begin{equation*}
(\sigma_1\circ\sigma_0)(\langle {1{\rm\hspace{-0.5ex}l}} _{G_q},q,0\rangle) = \sigma_1 (\langle c_{\sigma_0,q},q,0\rangle) = \langle c_{\sigma_1,q}\cdot c_{\sigma_0,q},q,0\rangle
\end{equation*}
and therefore $c_{\sigma_1\circ\sigma_0,q}=c_{\sigma_1,q}\cdot c_{\sigma_0,q}$. This shows that $\Phi$ is a homomorphism.
Given $\vec{g}=(g_q)_{q\in D}\in G_{\mathbb{I}}$, we define $\map{\sigma_{\vec{g}}}{M_{\mathbb{I}}}{M_{\mathbb{I}}}$ by the following clauses.
\begin{enumerate}
\item $\sigma_{\vec{g}}(\langle g,q,0\rangle)=\langle g_q\cdot g,q,0\rangle$ for all $q\in D$ and $g\in G_q$.
\item $\sigma_{\vec{g}}(\langle g,q,1\rangle)=\langle g,q,1\rangle$ for all $q\in D$ and $g\in G_q$.
\end{enumerate}
Then $\sigma_{\vec{g}}\in\Aut{\mathcal{M}_{\mathbb{I}}}$ and $c_{\sigma_{\vec{g}},q}=g_q$ for all $q\in D$. This shows that $\Phi$ is surjective.
Since Propositions \ref{proposition:SigmaRestrictionQ} and \ref{proposition:proposition:SigmaRestrictionP}
imply that $\Phi$ is also injective, this concludes the proof of the lemma.
\end{proof}
\begin{proof}[Proof of Theorem \ref{theorem:ModelIAut}]
By our assumptions, both $\mathcal{L}_{\mathbb{I}}$ and $\mathcal{M}_{\mathbb{I}}$ have cardinality at most $\lambda$.
By the results mentioned above, there is a field $K$ of characteristic $p$ and cardinality $\lambda$ such that the groups
$\Aut{\mathcal{M}_{\mathbb{I}}}$ and $\Aut{K}$ are isomorphic. By Lemma \ref{lemma:IsomorphismModelIInverseLimit}, this completes the proof of the theorem.
\end{proof}
\section{Representing free groups as inverse limits}\label{section:freegroups}
This sections shows how free groups can be represented as inverse limits of systems of groups assuming the existence of certain \emph{suitable} inverse systems of sets.
In the following, we prove statements from assumptions much weaker than the ones present in Theorem \ref{theorem:Main1} to motivate possible strengthenings of this result.
Let ${\mathbb{D}}=\langle D,\leq_{\mathbb{D}}\rangle$ be a directed set. We define an infinite game $\mathcal{G}({\mathbb{D}})$ of perfect information between Player I and Player II:
in the $i$-th round of this game Player I chooses an element $p_{2i}$ from $D$ and then Player II chooses an element $p_{2i+1}$ from $D$.
Player I wins a run $(p_i)_{i<\omega}$ of $\mathcal{G}({\mathbb{D}})$ if and only if either there is an $i<\omega$ with $p_{2i}\not\leq_{\mathbb{D}} p_{2i+1}$ or $p_{2i+1}\leq_{\mathbb{D}} p_{2i+2}$
holds for all $i<\omega$ and there is a $p\in D$ with $p_i\leq_{\mathbb{D}} p$ for all $i<\omega$.\footnote{A similar game can be used to characterize the $\sigma$-distributivity of Boolean algebras. See {\cite{MR739910}}.}
A \emph{winning strategy} for Player II is a function $\map{s}{{}^{{<}\omega}D}{D}$ with the property that Player II wins every run $(p_i)_{i<\omega}$ that is played
according to $s$, in the sense that $s(\langle p_0,\dots,p_{2i}\rangle)=p_{2i+1}$ holds for all $i<\omega$.
\begin{theorem}\label{theorem:FreeGroupFromSystem}
Let ${\mathbb{D}}=\langle D,\leq_{\mathbb{D}}\rangle$ be a directed set with the property that Player II has no winning strategy in $\mathcal{G}({\mathbb{D}})$.
If ${\mathbb{I}}$ is an inverse system of sets over ${\mathbb{D}}$ with $A_{\mathbb{I}}\neq\emptyset$,
then the inverse limit $G_{{\mathbb{I}}_{gr}}$ is a free group of cardinality $\max\{\aleph_0,\betrag{A_{\mathbb{I}}}\}$.
\end{theorem}
\begin{proof}
Let ${\mathbb{I}}=\langle\seq{A_p}{p\in D},\seq{f_{p,q}}{p,q\in D,~p\leq_{\mathbb{D}} q}\rangle$.
If $\vec{g}=(g_p)_{p\in D}\in G_{{\mathbb{I}}_{gr}}$ and $p\in D$, then we let
\begin{itemize}
\item $n(\vec{g},p)<\omega$,
\item $k(\vec{g},p,1), ~ \dots ~ ,k(\vec{g},p,n(\vec{g},p))\in{\mathbb{Z}}\setminus\{0\}$,
\item $a(\vec{g},p,1), ~ \dots ~ ,a(\vec{g},p,n(\vec{g},p))\in A_p$
\end{itemize}
denote the uniquely determined objects with the property that the word
\begin{equation*}
w_{\vec{g},p} ~ = ~ x_{p,a(\vec{g},p,1)}^{k(\vec{g},p,1)} ~ \dots ~ x_{p,a(\vec{g},p,n(\vec{g},p))}^{k(\vec{g},p,n(\vec{g},p))}
\end{equation*}
is the unique reduced word representing $g_p$, i.e. $w_{\vec{g},p}$ represents $g_p$ and
\begin{equation*}
a(\vec{g},p,i)\neq a(\vec{g},p,i+1)
\end{equation*}
for all $1\leq i<n(\vec{g},p)$
(see {\cite[2.1.2]{MR1357169}}).
\begin{claim}\label{claim:One}
If $\vec{g}=(g_p)_{p\in D}\in G_{{\mathbb{I}}_{gr}}$ and $p,q\in D$ with $p\leq_{\mathbb{D}} q$, then $n(\vec{g},p)\leq n(\vec{g},q)$.
\end{claim}
\begin{proof}[Proof of the Claim]
Since $h_{p,q}(g_q)=g_p$, we know that the word
\begin{equation}\label{equation:AlterWord}
w ~ = ~ x_{p,f_{p,q}(a(\vec{g},q,1))}^{k(\vec{g},q,1)} ~ \dots ~ x_{p,f_{p,q}(a(\vec{g},q,n(\vec{g},q)))}^{k(\vec{g},q,n(\vec{g},q))}
\end{equation}
also represents $g_p$. Hence $w_{\vec{g},p}$ can be obtained from $w$ by a finite number of reductions. This implies $n(\vec{g},p)\leq n(\vec{g},q)$.
\end{proof}
\begin{claim}
If $\vec{g}\in G_{{\mathbb{I}}_{gr}}$, then there are $p_{\vec{g}}\in D$ and $n_{\vec{g}}<\omega$ such that $n_{\vec{g}}=n(\vec{g},p)$ for all $p\in D$ with $p_{\vec{g}}\leq_{\mathbb{D}} p$.
\end{claim}
\begin{proof}[Proof of the Claim]
Let $\vec{g}=(g_p)_{p\in D}$ and assume, toward a contradiction, that for every $p\in D$ there is a $q\in D$ with $p\leq_{\mathbb{D}} q$ and $n(\vec{g},p)<n(\vec{g},q)$. Then there is a function $\map{s}{{}^{{<}\omega}D}{D}$ such that $p_{2i}\leq_{\mathbb{D}} s(\langle p_0,\dots,p_{2i}\rangle)$ and $n(\vec{g},p_{2i})<n(\vec{g},s(\langle p_0,\dots,p_{2i}\rangle))$ for all $i<\omega$ and $p_0,\dots,p_{2i}\in D$. By our assumption, $s$ is not a winning strategy for Player II and there is a run $(p_i)_{i<\omega}$ of $\mathcal{G}({\mathbb{D}})$ played according to $s$ that is won by Player I. This gives us a $p\in D$ with $p_i\leq_{\mathbb{D}} p$ for all $i<\omega$. By Claim \ref{claim:One}, we have $n(\vec{g},p)>i$ for all $i<\omega$, a contradiction.
\end{proof}
\begin{claim}\label{claim:Three}
If $\vec{g}=(g_p)_{p\in D}\in G_{{\mathbb{I}}_{gr}}$ and $p,q\in D$ with $p_{\vec{g}}\leq_{\mathbb{D}} p\leq_{\mathbb{D}} q$, then $a(\vec{g},p,i)=f_{p,q}(a(\vec{g},q,i))$ and
$k(\vec{g},p,i)=k(\vec{g},q,i)$ for all $1\leq i<n_{\vec{g}}$.
\end{claim}
\begin{proof}[Proof of the Claim]
Let $w$ be the word defined in (\ref{equation:AlterWord}). Then $w$ is reduced, because otherwise there would be a reduced word
$x^{k_0}_{p,a_1}\dots x^{k_{l-1}}_{p,a_l}$ with $l<n(\vec{g},q)=n(\vec{g},p)$ representing $g_p$
and this would contradict the choice of $w_{\vec{g},p}$. We can conclude $w=w_{\vec{g},p}$ and, again by the uniqueness of $w_{\vec{g},p}$, this yields the statements of the claim.
\end{proof}
Given $\vec{a}=(a_p)_{p\in D}\in A_{\mathbb{I}}$, we define
\begin{equation*}
\vec{g}_{\vec{a}} ~ = ~ (x_{p,a_p})_{p\in D} ~ \in ~ \prod_{p\in D}G_p.
\end{equation*}
It is easy to see that $\vec{g}_{\vec{a}}$ is an element of $G_{{\mathbb{I}}_{gr}}$.
\begin{claim}
The group $G_{{\mathbb{I}}_{gr}}$ is generated by the set $\Set{\vec{g}_{\vec{a}}}{\vec{a}\in A_{\mathbb{I}}}$.
\end{claim}
\begin{proof}[Proof of the Claim]
Let $\vec{g}=(g_p)_{p\in D}\in G_{{\mathbb{I}}_{gr}}$. Set $n=n_{\vec{g}}$ and $k_i=k(\vec{g},p_{\vec{g}},i)$ for all $1\leq i\leq n$.
For each $p\in D$, we fix an element $\bar{p}$ of $D$ with $p,p_{\vec{g}}\leq_{\mathbb{D}} \bar{p}$. Given $p\in D$ and $1\leq i\leq n$,
define $a_{\vec{g},p,i}=f_{p,\bar{p}}(a(\vec{g},\bar{p},i)) \in A_p$.
Let $p,q\in D$ with $p\leq_{\mathbb{D}} q$. Fix an $r\in D$ with $\bar{p},\bar{q}\leq_{\mathbb{D}} r$.
By Claim \ref{claim:Three}, we have $a(\vec{g},\bar{p},i)=f_{\bar{p},r}(a(\vec{g},r,i))$ and $a(\vec{g},\bar{q},i)=f_{\bar{q},r}(a(\vec{g},r,i))$. This implies
\begin{equation*}
f_{p,q}(a_{\vec{g},q,i}) ~ = ~ f_{p,q}(f_{q,\bar{q}}(a(\vec{g},\bar{q},i))) ~ = ~ f_{p,r}(a(\vec{g},r,i)) ~ = ~ f_{p,\bar{p}}(a(\vec{g},\bar{p},i)) ~ = ~ a_{\vec{g},p,i}
\end{equation*}
and we can conclude
\begin{equation*}
\vec{a}_{\vec{g},i} ~ = ~ (a_{\vec{g},p,i})_{p\in D} ~ \in ~ A_{\mathbb{I}}.
\end{equation*}
By the above computations, we know that
\begin{equation*}
g_p ~ = ~ h_{p,\bar{p}}(g_{\bar{p}}) ~ = ~ h_{p,\bar{p}}\big(x^{k_1}_{\bar{p},a(\vec{g},\bar{p},1)}\cdot ~ \dots ~ \cdot x^{k_n}_{\bar{p},a(\vec{g},\bar{p},n)}\big) ~ = ~ x^{k_1}_{p,a_{\vec{g},p,1}} \cdot ~ \dots ~ \cdot x^{k_n}_{p,a_{\vec{g},p,n}}
\end{equation*}
holds for all $p\in D$ and this shows
\begin{equation*}
\vec{g} ~ = ~ \vec{g}^{~ k_1}_{\vec{a}_{\vec{g},1}} ~ \cdot ~ \dots ~ \cdot ~ \vec{g}^{~ k_n}_{\vec{a}_{\vec{g},n}}.
\end{equation*}
\end{proof}
\begin{claim}
The group $G_{{\mathbb{I}}_{gr}}$ is freely generated by the set $\Set{\vec{g}_{\vec{a}}}{\vec{a}\in A_{\mathbb{I}}}$.
\end{claim}
\begin{proof}[Proof of the Claim]
Assume, toward a contradiction, that we can find $1\leq n<\omega$, $\vec{a}_1,\dots,\vec{a}_n\in A_{\mathbb{I}}$ and $k_1,\dots,k_n\in{\mathbb{Z}}\setminus\{0\}$ with
\begin{equation*}
\vec{g}_{\vec{a}_1}^{~k_1} ~ \cdot ~ \dots ~ \cdot ~ \vec{g}_{\vec{a}_n}^{~k_n} ~ = ~ {1{\rm\hspace{-0.5ex}l}} _{G_{{\mathbb{I}}_{gr}}}
\end{equation*}
and $\vec{a}_i\neq \vec{a}_{i+1}$ for all $1\leq i<n$. Let $\vec{a}_i=(a_{p,i})_{p\in D}$.
Then there are $p_1,\dots,p_{n-1}\in D$ with $a_{p_i,i}\neq a_{p_i,i+1}$ for all $1\leq i<n$ and we can find a $p\in D$ with $p_1,\dots,p_{n-1}\leq_{\mathbb{D}} p$.
This means $a_{p,i}\neq a_{p,i+1}$, because otherwise
\begin{equation*}
a_{p_i,i} ~ = ~ h_{p_i,p}(a_{p,i}) ~ = ~ h_{p_i,p}(a_{p,i+1}) ~ = ~ a_{p_i,i+1}.
\end{equation*}
By our assumption, the word $w = x^{k_1}_{p,a_1}\cdot ~ \dots ~ \cdot x^{k_n}_{p,a_n}$ is equivalent to the trivial word.
But this yields a contradiction, because $w$ is reduced and not trivial.
\end{proof}
This completes the proof of the theorem.
\end{proof}
A small modification of the above proof yields the corresponding result for free abelian group.
As usual, we let $[A]^{{<}\aleph_0}$ denote the set of all finite subsets of a given set $A$.
\setcounter{name}{0}
\begin{theorem}\label{theorem:AbelianFreeGroupFromSystem}
Let ${\mathbb{D}}=\langle D,\leq_{\mathbb{D}}\rangle$ be a directed set with the property that Player II has no winning strategy in $\mathcal{G}({\mathbb{D}})$.
If ${\mathbb{I}}$ is an inverse system of sets over ${\mathbb{D}}$ with $A_{\mathbb{I}}\neq\emptyset$,
then the inverse limit $G_{{\mathbb{I}}_{ab}}$ is a free abelian group of cardinality $\max\{\aleph_0,\betrag{A_{\mathbb{I}}}\}$.
\end{theorem}
\begin{proof}
Let ${\mathbb{I}}=\langle\seq{A_p}{p\in D},\seq{f_{p,q}}{p,q\in D,~p\leq_{\mathbb{D}} q}\rangle$.
If $\vec{g}=(g_p)_{p\in D}\in G_{{\mathbb{I}}_{ab}}$, then we let
\begin{itemize}
\item $I_{\vec{g},p}\in[A_p]^{{<}\aleph_0}$,
\item $\seq{k(\vec{g},p,a)\in{\mathbb{Z}}\setminus\{0\}}{a\in I_{\vec{g},p}}$
\end{itemize}
denote the uniquely determined objects such that
\begin{equation*}
g_p ~ = ~ \sum_{a\in I_{\vec{g},p}}~k(\vec{g},p,a)\cdot x_{p,a}
\end{equation*}
holds for all $p\in D$. The following claims can be derived in the same way as the corresponding claims in the proof of Theorem \ref{theorem:FreeGroupFromSystem}.
\begin{claim}
If $\vec{g}=(g_p)_{p\in D}\in G_{{\mathbb{I}}_{ab}}$ and $p,q\in D$ with $p\leq_{\mathbb{D}} q$, then $\betrag{I_{\vec{g},p}}\leq\betrag{I_{\vec{g},q}}$. \qed
\end{claim}
\begin{claim}
If $\vec{g}\in G_{{\mathbb{I}}_{ab}}$, then there are $p_{\vec{g}}\in D$ and $n_{\vec{g}}<\omega$, such that $n_{\vec{g}}=\betrag{I_{\vec{g},p}}$
for all $p\in D$ with $p_{\vec{g}}\leq_{\mathbb{D}} p$. \qed
\end{claim}
\begin{claim}
If $\vec{g}=(g_p)_{p\in D}\in G_{{\mathbb{I}}_{ab}}$ and $p,q\in D$ with $p_{\vec{g}}\leq_{\mathbb{D}} p\leq_{\mathbb{D}} q$, then $f_{p,q}\restriction I_{\vec{g},q}$ is a bijection of
$I_{\vec{g},q}$ and $I_{\vec{g},p}$ and $k(\vec{g},q,a)=k(\vec{g},p,f_{p,q}(a))$ for all $a\in I_{\vec{g},q}$. \qed
\end{claim}
Given $\vec{a}=(a_p)_{p\in D}\in A_{\mathbb{I}}$, we define
\begin{equation*}
\vec{g}_{\vec{a}} ~ = ~ (x_{p,a_p})_{p\in D} ~ \in ~ \prod_{p\in D}G_p.
\end{equation*}
It is easy to see that $\vec{g}_{\vec{a}}$ is an element of the inverse limit $G_{{\mathbb{I}}_{ab}}$.
As in the proof of Theorem \ref{theorem:FreeGroupFromSystem}, we can use the above claims to show that $G_{{\mathbb{I}}_{ab}}$ is
a free abelian group with basis $\Set{\vec{g}_{\vec{a}}}{\vec{a}\in A_{\mathbb{I}}}$.
\end{proof}
\section{Good inverse systems of sets}\label{section:system}
In this section, we complete the proofs of the results listed in Section \ref{section:Introduction} by construction \emph{suitable} inverse systems of sets from the assumptions appearing in the statements of those results. The next definition precises the notion of \emph{suitable inverse system}.
\begin{definition}\label{definition:GoodInverseSystem}
Let $\lambda$ and $\nu$ be infinite cardinals. We say that an inverse system ${\mathbb{I}}=\langle\seq{A_p}{p\in D},\seq{f_{p,q}}{p,q\in D,~p\leq_{\mathbb{D}} q}\rangle$ of sets
over a directed set ${\mathbb{D}}=\langle D,\leq_{\mathbb{D}}\rangle$ is $(\lambda,\nu)$-good if the following statements hold.
\begin{enumerate}
\item Player II has no winning strategy in $\mathcal{G}({\mathbb{D}})$.
\item $\betrag{D}\leq\lambda$ and $\betrag{A_p}\leq \lambda$ for all $p\in D$.
\item $\betrag{A_{{\mathbb{I}}}}=\nu$.
\end{enumerate}
\end{definition}
The following proposition summarizes the results of the previous sections.
\begin{proposition}\label{proposition:FieldsFromGoodSystems}
If ${\mathbb{I}}$ is a $(\lambda,\nu)$-good inverse system of sets and $p$ is either $0$ or a prime number,
then there are fields $K_0$ and $K_1$ of characteristic $p$ and cardinality $\lambda$
with the property that $\Aut{K_0}$ is a free group of cardinality $\nu$ and $\Aut{K_1}$ is a free abelian group of cardinality $\nu$.
\end{proposition}
\begin{proof}
Our assumptions imply that $A_{\mathbb{I}}\neq\emptyset$ and all groups appearing in the corresponding inverse systems of groups ${\mathbb{I}}_{gr}$ and ${\mathbb{I}}_{ab}$
have cardinality at most $\lambda$.
We can now apply Theorem \ref{theorem:ModelIAut} to find fields $K_0$ and $K_1$ of characteristic $p$ and cardinality $\lambda$
such that the group $\Aut{K_0}$ is isomorphic to the inverse limit $G_{{\mathbb{I}}_{gr}}$ and the group $\Aut{K_1}$ is isomorphic to the inverse limit $G_{{\mathbb{I}}_{ab}}$.
By our assumptions, Theorem \ref{theorem:FreeGroupFromSystem} implies that $G_{{\mathbb{I}}_{gr}}$ is a free group of cardinality $\nu$ and
Theorem \ref{theorem:AbelianFreeGroupFromSystem} implies that $G_{{\mathbb{I}}_{ab}}$ is a free abelian group of cardinality $\nu$.
\end{proof}
In order to prove Theorem \ref{theorem:Main1}, we now construct $(\lambda,2^\lambda)$-good inverse system from the assumption $\lambda=\lambda^{\aleph_0}$.
\begin{proposition}\label{proposition:ClosedDirectedSet}
Let ${\mathbb{D}}=\langle D,\leq_{\mathbb{D}}\rangle$ be a directed set with the property that for every $P\in[D]^{\aleph_0}$ there is a $q\in D$ with $p\leq_{\mathbb{D}} q$ for all $p\in P$.
Then Player I has a winning strategy in $\mathcal{G}({\mathbb{D}})$ and hence Player II has no winning strategy in $\mathcal{G}({\mathbb{D}})$. \qed
\end{proposition}
\begin{lemma}\label{lemma:GoodInverseSystemFromAssumption}
If $\lambda$ is a cardinal with $\lambda=\lambda^{\aleph_0}$, then ${\mathbb{I}}_\lambda$ is a $(\lambda,2^\lambda)$-good inverse system of sets.
\end{lemma}
\begin{proof}
By Proposition \ref{proposition:ClosedDirectedSet}, Player II has no winning strategy in $\langle[\lambda]^{\aleph_0},\subseteq\rangle$.
It is shown in Example \ref{example:LambdaCountableInverseLimit} that the direct limit $A_{{\mathbb{I}}_\lambda}$ has cardinality $2^\lambda$.
The other cardinality requirements follow directly from the assumption $\lambda=\lambda^{\aleph_0}$.
\end{proof}
The statements of Theorem \ref{theorem:Main1} and Theorem \ref{theorem:Main2} now follow directly from the combination of Proposition \ref{proposition:FieldsFromGoodSystems}
and Lemma \ref{lemma:GoodInverseSystemFromAssumption}. Next, we show that Corollary \ref{corollary:CHSCHAnswer} is a direct consequence of Theorem \ref{theorem:Main1} and the
results presented in the first two sections.
\begin{proof}[Proof of Corollary \ref{corollary:CHSCHAnswer}]
If $\lambda$ is an infinite cardinal with $\cof{\lambda}>\omega$, then our assumptions and {\cite[Theorem 5.22]{MR1940513}} imply $\lambda=\lambda^{\aleph_0}$ and
we can apply Theorem \ref{theorem:Main1} to find a field of cardinality $\lambda$ with the desired properties.
Next, if $\lambda=\aleph_0$ or $\lambda$ is a singular strong limit cardinal of countable cofinality, then Theorem \ref{theorem:ST1}
and Theorem \ref{theorem:ST2} imply that the automorphism group of a field of cardinality $\lambda$ either has cardinality at most $\lambda$ or is not a free group.
Finally, let $\lambda$ be a singular cardinal and $\kappa<\lambda$ be an uncountable regular cardinal with $2^\kappa>\lambda$. By the above arguments, we can apply
Theorem \ref{theorem:Main1} to find a field $K$ of cardinality $\kappa$ whose automorphism group is a free group of cardinality $2^\kappa$. Then we can construct
a first-order language $\mathcal{L}$ of cardinality $\lambda$ and an $\mathcal{L}$-model $\mathcal{M}$ whose automorphism group is isomorphic to $\Aut{K}$. By the results presented
Section \ref{section:MI}, this allows us to produce a field of cardinality $\lambda$ with the desired properties.
\end{proof}
The following proposition will allow us to prove Theorem \ref{theorem:OuterModelFields}.
\begin{proposition}\label{proposition:CoverInnerNiceSystem}
Let $M$ be an inner model of ${\rm{ZFC}}$ with the property that every countable set of ordinals in ${\rm{V}}$ is contained in a set that is an element of $M$ and countable in $M$.
If $\lambda$ is a cardinal with $\lambda=(\lambda^{\aleph_0})^M$, then there is a $(\lambda,\nu)$-good inverse system for some cardinal $\nu$
with $(2^\lambda)^M\leq\nu\leq 2^\lambda$.
\end{proposition}
\begin{proof}
Set $D=([\lambda]^{\aleph_0})^M$ and ${\mathbb{D}}=\langle D,\subseteq\rangle$. Given a sequence $\seq{u_n\in D}{n<\omega}$, our assumption implies that there is a
$u\in D$ with $\bigcup_{n<\omega}u_n\subseteq u$ and Proposition \ref{proposition:ClosedDirectedSet} shows that Player II has no winning strategy in $\mathcal{G}({\mathbb{D}})$.
Let ${\mathbb{I}}={\mathbb{I}}_\lambda^M$ and $\nu$ be the cardinality of $A_{\mathbb{I}}$. Then every element of $({}^\lambda 2)^M$ gives rise to a distinct element of $A_{\mathbb{I}}$ and
$\nu$ is an infinite cardinal greater than or equal to $(2^\lambda)^M$. We can conclude that ${\mathbb{I}}$ is $(\lambda,\nu)$-good.
\end{proof}
Since partial orders of the form $\Add{\omega}{\kappa}$ satisfy the countable chain condition
and therefore every countable set of ordinals in an $\Add{\omega}{\kappa}$-generic extension of the ground model is covered by a set countable set of ordinals
from the ground model, we can directly derive the statement of Theorem \ref{theorem:OuterModelFields} from Proposition \ref{proposition:FieldsFromGoodSystems}
and Proposition \ref{proposition:CoverInnerNiceSystem}.
\begin{lemma}\label{lemma:GoodSystemFromTree}
Let $\lambda$ be a cardinal of uncountable cofinality and ${\mathbb{T}}$ be a tree of cardinality and height $\lambda$ with the property that the set $[{\mathbb{T}}]$ of cofinal branches
through ${\mathbb{T}}$ has infinite cardinality $\nu$. Then ${\mathbb{I}}_{\mathbb{T}}$ is a $(\lambda,\nu)$-good inverse system of sets.
\end{lemma}
\begin{proof}
By Proposition \ref{proposition:ClosedDirectedSet}, the assumption $\cof{\lambda}>\omega$ implies that Player II has no winning strategy in $\mathcal{G}(\langle\lambda,\leq\rangle)$.
The computations in Example \ref{example:TreesInverseSystems} show that the inverse limit $A_{{\mathbb{I}}_{\mathbb{T}}}$ also has cardinality $\nu$.
Since the cardinality requirements of Definition \ref{definition:GoodInverseSystem} are obviously satisfied, this completes the proof of the lemma.
\end{proof}
The statement of Theorem \ref{theorem:InnerModelFields} now follows directly from Proposition \ref{proposition:FieldsFromGoodSystems} and Lemma \ref{lemma:GoodSystemFromTree}.
\section{Some questions}
We close this paper with questions raised by the above results.
\begin{question}
Is it consistent with the axioms of ${\rm{ZFC}}$ that there is a cardinal $\lambda$ of uncountable cofinality with the property that no free group of cardinality $2^\lambda$ is isomorphic to the automorphism group of a field of cardinality $\lambda$?
\end{question}
\begin{question}
Is it consistent with the axioms of ${\rm{ZFC}}$ that there is a cardinal $\lambda$ of uncountable cofinality with the property that no free group of cardinality greater than
$\lambda$ is isomorphic to the automorphism group of a field of cardinality $\lambda$?
\end{question}
\begin{question}
Is it consistent with the axioms of ${\rm{ZFC}}$ that there is a singular cardinal $\lambda$ of uncountable cofinality with the property that there is no tree of cardinality and
height $\lambda$ with more than $\lambda$ many cofinal branches?
\end{question}
\end{document} | math | 47,494 |
\begin{document}
\title{On-flow and strong solutions\
to Killing-type equations}
\begin{abstract}
If we impose infinitesimal invariance up to a boundary term of the action functional for Lagrangian ordinary differential equations, we are led to Killing-type equations, which are related to first integrals through Noether theorem. We review the ``on-flow'' and ``strong'' interpretations of the Killing-type equation, and for each we detail the complete explicit structure of the solution set in terms of the associated first integral. We give examples that reappraise the usefulness of the ``on-flow'' solutions. Finally, we describe an equivalent alternative approach to variational invariance.
\end{abstract}
Keywords: Noether variational theorem; Killing-type equations; Laplace-Runge-Lenz vector.
AMS subject classification: 34C14; 70H33.
This work was done under the auspices of the INDAM (Istituto Nazionale di Alta Matematica). The authors are grateful to Prof.~Giuseppe Gaeta for helpful discussion.
\section{Introduction}\label{introduction}
Suppose we are given a smooth Lagrangian function $L(t,q,\dot q)$, with $t\in\mathbb{R}$, $q,\dot q\in\mathbb{R}^n$. The variational principle for Lagrangian dynamics posits that
\begin{equation}\label{variationalPrinciple}
\delta\int_{t_1}^{t_2}L\bigl(t,q(t),\dot q(t)\bigr)dt=0,
\end{equation}
which is equivalent to the \emph{Euler-Lagrange equation}
\begin{equation}\label{Lagrange}
\frac{d}{dt}\partial_{\dot q}L\bigl(t,q(t),\dot q(t)\bigr)
-\partial_{q}L\bigl(t,q(t),\dot q(t)\bigr)=0.
\end{equation}
We will assume that the Euler-Lagrange equation can be put into normal form
\begin{equation}\label{normalformLagrange}
\ddot q=\Lambda(t,q,\dot q),
\end{equation}
Following closely the notation of Sarlet and Cantrijn~\cite{SarletCantrijn} (except that $\dot q$ is an independent variable from the outset), we consider an \emph{infinitesimal transformation} in the $(t,q)$ space given by
\begin{equation}\label{infinitesimalTransformation}
\bar t=t+\varepsilon \tau(t,q,\dot q),\qquad
\bar q=q+\varepsilon\xi(t,q,\dot q).
\end{equation}
This transformation is said to leave the action integral (infinitesimally) \emph{invariant up to boundary terms} (using the nomenclature recommended by P.G.L.~Leach), if a function $f(t,q,\dot q)$ exists, such that for the given smooth curve $t\mapsto q(t)$ we have
\begin{multline}\label{infinitesimalInvariance}
\int_{\bar t_1}^{\bar t_2}L\Bigl(\bar t,\bar q(\bar t),
\frac{d\bar q}{d\bar t}(\bar t)\Bigr)d\bar t=
\int_{t_1}^{t_2}L\bigl(t,q(t),\dot q(t)\bigr)dt+{}\\
+\varepsilon\int_{t_1}^{t_2}
\frac{df}{dt}\bigl(t,q(t),\dot q(t)\bigr)dt
+O(\varepsilon^2),
\end{multline}
which is equivalent to the following \emph{Killing-type equation} for ODEs:
\begin{equation}\label{Killing-type}
\tau\partial_t L
+\partial_q L\cdot \xi
+\partial_{\dot q}L\cdot
\bigl(\dot \xi-\dot q\dot\tau\bigr)
+L\dot\tau=
\dot f.
\end{equation}
This is the same formula as Sarlet and Cantrijn~\cite{SarletCantrijn}, formula~(9) p.~471. We will write $\partial_{t}, \partial_{q}, \partial_{\dot q}$ for the partial derivative and gradients, $\dot x$ for the total time derivative of the function~$x$, and $x\cdot y$ will denote the ordinary scalar product of $x,y\in\mathbb{R}^n$.
Noether's theorem states that equation~\eqref{Killing-type} is a sufficient condition so that the function
\begin{equation}\label{firstintegral}
N=f-L\tau-\partial_{\dot q}L\cdot\bigl(\xi-\dot q\tau\bigr)
\end{equation}
(\cite[p.~471, formula~(11)]{SarletCantrijn}) be a \emph{constant of motion} for the solutions to the Lagrange equation~\eqref{Lagrange}.
The function~$L(t,q,\dot q)$ will be given, and we solve Killing-type equation~\eqref{Killing-type} for the triple $(\tau,\xi,f)$. The equation in the terse form~\eqref{Killing-type} is open to at least three interpretations that we know of, differing on what the independent variables are and on how to treat the $\ddot q$ terms.
The most restrictive approach is when we seek $\tau,\xi, f$ as functions of $(t,q)$ only: ($\tau(t,q),\xi(t,q),f(t,q)$). Since in this case $\ddot q$ does not appear, the independent variables are $t,q,\dot q$ only, and we want the equation to hold identically. We will be concerned with this approach mainly in Section~\ref{strongIndependentOfDotQ}.
If we instead allow full dependence of $\tau,\xi,f$ on~$\dot q$, the expanded-out form of the total time derivatives $\dot\xi,\dot \tau,\dot f$ will contain~$\ddot q$. We distinguish two alternatives:
\begin{itemize}
\item \emph{Strong form}: we treat $\ddot q$ as just another independent variable and require the equation to hold for all $t,q,\dot q,\ddot q$. This way the Hamiltonian action functional will be infinitesimally invariant along all smooth trajectories~$q(t)$. The strong form was introduced by Djukic~\cite{Djukic}, and studied also by Kobussen~\cite{Kobussen}. Since the equation depends linearly on~$\ddot q$, it is equivalent to a system of $n+1$ equations that do not contain~$\ddot q$, but we will not exploit this fact in the sequel.
\item \emph{On-flow form}: we replace every occurrence of $\ddot q$ with $\Lambda(t,q,\dot q)$ from the normalized Lagrange equation~\eqref{normalformLagrange}, and require the resulting equation to hold for all $t,q,\dot q$. This way the Hamiltonian action functional will be infinitesimally invariant as in~\eqref{infinitesimalInvariance} only along the Lagrangian motions~$q(t)$, a condition which is however enough for the expression~\eqref{firstintegral} to be a first integral.
\end{itemize}
In Sections~\ref{KillingSection} and~\ref{strongIndependentOfDotQ} of this work we give a full description of the structure of the solution sets of the Killing-like equation, in the above senses, assuming that we are given both $L$ and the first integral~$N$. This problem is also known as ``reverse Noether theorem''. The main results of Sections~\ref{KillingSection} and~\ref{strongIndependentOfDotQ}, specially for the strong form, are basically contained already in Sarlet and Cantrijn's 1981 paper~\cite{SarletCantrijn}, where they are deduced as a corollaries of the theory of~$d\theta$-symmetries. Here we make a direct derivation, which we hope will be helpful to some readers.
In Section~\ref{examplesSection} we illustrate the general theory with examples. In Subsection~\ref{FreeParticle} we re-examine the free particle's notorious ``non-noetherian'' symmetries that were ``noetherized'' by P.G.L.~Leach~\cite{Leach2} by substituting a new Lagrangian for the usual one. We will see what we can say about those symmetries from the point of view of on-flow and strong solutions without switching Lagrangian.
In Subsection~\ref{superintegrableSubSection} we search for classes of superintegrable systems among the ones of the form $\ddot x=-G(x)$, $\ddot y=-G'(x)y$, by making a suitable ansatz on the solution triple $(\tau,\xi,f)$ in the on-flow form. For the sake of clarity we will stop short of pursuing the method beyond known territory. The on-flow form of Killing-like equation was dismissed by most authors, the reason being that it has too many solution. Our point is that a large solution set can work to our advantage when we do not know exactly the form of the system and of the first integral, because an ansatz has more chances of catching a solution when the solutions are plenty.
Subsection~\ref{LRLSubSection} is devoted to the Laplace-Runge-Lenz vector conservation of the classical Kepler problem. We review some known formulas for solutions to Killing-like equation in either on-flow or strong interpretation, and propose some new ones in dimension~3, that feel simpler to us.
In Section~\ref{differentTimeChangeSection} we return to an alternative way of performing time change in the action integral, that we proposed in a recent paper~\cite{GorniZampieri}. We show how this approach leads to a different, but equivalent, Killing-like equation and to some formulas that match closely with formulas written in a 1972 paper~\cite{Candotti} by Candotti, Palmieri and Vitale.
\section{Structure of the solution sets}\label{KillingSection}
Equation~\eqref{Killing-type} is referred to as ``Killing-type'' because of the important particular case when the Lagrangian function is a quadratic form in the~$\dot q$ variable: $L=\frac12\dot q\cdot A(q)\dot q$, with $A(q)$ symmetric $n\times n$ non-singular matrix. The Lagrange equation~\eqref{normalformLagrange} reduces to the equation of geodesics. Equation~\eqref{Killing-type} with $\tau\equiv 0$, $f\equiv 0$, and $\xi(q)$ as a function of $q$~only, becomes $\partial_q L\cdot \xi(q)+ \partial_{\dot q}L\cdot\xi'(q)\dot q=0$, which is quadratic homogeneous in~$\dot q$: if we equate to zero the coefficients, we get the well-known Killing equations of Differential Geometry. The first integral~\eqref{firstintegral} simplifies to $-\partial_{\dot q} L\cdot \xi(q)=-A(q)\dot q\cdot \xi(q)=-\dot q \cdot A(q) \xi(q)$.
Back to the general Killing-type equation~\eqref{Killing-type}, we will tacitly assume that $L\in C^3$ and that the following usual regularity condition is satisfied:
\begin{equation}\label{Legendre}
\det g\ne0\qquad
\text{where }g:=\partial^2_{\dot q,\dot q}L(t,q,\dot q).
\end{equation}
The notation $\partial^2_{\dot q,\dot q}L$ means the Hessian matrix of the second derivatives of~$L$ with respect to~$\dot q$. This will ensure that the Lagrange equation can indeed be put into normal form~\eqref{normalformLagrange}, and that there is existence and uniqueness of the solutions to the Cauchy problems.
The following result was basically found by Lutzky~\cite[formula~(19)]{Lutzky}, who uses it to argue that it is not useful to allow $f$ to depend on~$\dot q$.
\begin{Teorema}[General solution of the on-flow equation]\label{familyOfSolutionsWithLagrange}
Let $L(t,q,\dot q)$ be a Lagrangian function. Suppose that the Lagrange equation has the $C^1$ first integral $N(t,q,\dot q)$. Then a triple $(\tau,\xi,f)$ is a solution of the on-flow version of the Killing-type equation with~$N$ as associated first integral if and only if
\begin{equation}\label{familyFormula}
f=\tau L+N+\partial_{\dot q}L\cdot(\xi-\tau\dot q).
\end{equation}
\end{Teorema}
\begin{proof}
Equation~\eqref{familyFormula} is simply a rearrangement of formula~\eqref{firstintegral}. If the triple $(\tau,\xi,f)$ is a solution associated with~$N$ then it must satisfy~\eqref{familyFormula}.
Conversely, suppose that the triple satisfies~\eqref{familyFormula} and let us check that it is an on-flow solution. Taking the time derivative of~$f=\tau L+N+ \partial_{\dot q} L\cdot(\xi-\tau\dot q)$ along a solution of Lagrange equation and replacing into Killing-type equation~\eqref{Killing-type}
\begin{multline}
\tau\partial_t L
+\partial_q L\cdot \xi
+\partial_{\dot q}L\cdot
\bigl(\dot \xi-\dot q\dot \tau\bigr)
+L\dot \tau=\\
=\dot f=\dot \tau L+\tau \dot L+0+
\Bigl(\frac{d}{dt}\partial_{\dot q} L\Bigr)\cdot(\xi-\tau \dot q)+
\partial_{\dot q} L\cdot(\dot\xi-\dot \tau \dot q-\tau \ddot q)
\end{multline}
Canceling out the common terms and using Lagrange equation we get
\begin{equation}
\tau\partial_t L
+\partial_q L\cdot \xi
=\tau\dot L+
\Bigl(\frac{d}{dt}\partial_{\dot q} L\Bigr)\cdot(\xi-\tau \dot q)+
\partial_{\dot q} L\cdot(-\tau \ddot q).
\end{equation}
Using Lagrange equation~\eqref{Lagrange} this becomes
\begin{equation}
\tau\partial_t L
+\partial_q L\cdot \xi
=\tau \dot L+
\partial_{q} L\cdot(\xi-\tau \dot q)-
\tau \partial_{\dot q} L\cdot\ddot q.
\end{equation}
Canceling out and rearranging we get
\begin{equation}
\tau \bigl(\partial_t L+\partial_{q} L\cdot q+
\partial_{\dot q} L\cdot\ddot q\bigr)
=\tau\dot L,
\end{equation}
which is simply the chain rule.
\end{proof}
From formula~\eqref{familyFormula} we can express $\tau $ as a function of $f,\xi$, at least when $L-\partial_{\dot q}L\cdot\dot q\ne0$. Of particular interest are the solutions with $f=0$:
\begin{Corollario}[Simplest on-flow solution with $f=0$]\label{existenceOnFlow}
Let $L$ be a Lagrangian function. Suppose that the Lagrange equation has the $C^1$ first integral $N$. Then the on-flow version of the Killing-type equation~\eqref{Killing-type} is satisfied by the triple
\begin{equation}\label{solutionTriple}
\tau =-\frac{N}{L},\qquad
\xi=-\frac{N}{L}\dot q,\qquad
f\equiv 0.
\end{equation}
The corresponding first integral~\eqref{firstintegral} is precisely~$N$.
\end{Corollario}
Around points where $L=0$ we can take $\tau =-N/(L+c)$, $\xi=- N/ (L+c)$, for a constant $c\ne0$.
\begin{Corollario}[More general on-flow solution with $f=0$]\label{existenceOnFlow2}
Let $L(t,q,\dot q)$ be a Lagrangian function. Suppose that the Lagrange equation has the $C^1$ first integral $N(t,q,\dot q)$. Let $R(t,q,\dot q)$ be an arbitrary smooth function with values in~$\mathbb{R}^n$. Then the on-flow version of the Killing-type equation~\eqref{Killing-type} is satisfied by the triple
\begin{equation}\label{solutionTriple2}
\tau (t,q,\dot q)=
-\frac{N+\partial_{\dot q}L\cdot R}{L},\qquad
\xi(t,q,\dot q)=R-\dot q
\frac{N+\partial_{\dot q}L\cdot R}{L},\qquad
f\equiv 0.
\end{equation}
The corresponding first integral~\eqref{firstintegral} is precisely~$N$.
\end{Corollario}
The solutions of the strong Killing equation were given indirectly by Sarlet and Cantrijn~\cite[Th.~6.1]{SarletCantrijn} as a consequence of a result on $d\theta$-symmetries. Here we give a direct formulation and proof.
\begin{Teorema}[General solution of the strong equation]\label{familyOfSolutionsWithoutLagrange}
Let $L(t,q,\dot q)$ be a Lagrangian function. Suppose that the Lagrange equation has the $C^2$ first integral $N(t,q,\dot q)$. Then a triple $(\tau ,\xi,f)$ is a solution of the strong version of the Killing-type equation with~$N$ as associated first integral if and only if
\begin{gather}\label{strongSolutionCondition}
\xi= \tau \dot q-g^{-1}\partial_{\dot q}N,\\
f=\tau L+N-\partial_{\dot q}L\cdot g^{-1}\partial_{\dot q}N,
\label{strongSolutionGauge}
\end{gather}
where $g=\partial^2_{\dot q,\dot q}L$ is the Hessian matrix as in~\eqref{Legendre}.
\end{Teorema}
\begin{proof}
Let us establish some formulas first. The function~$\Lambda$ appearing in the normal form of Lagrange equation~\eqref{normalformLagrange} can be made explicit this way:
\begin{equation}
\Lambda\equiv g^{-1}\bigl(\partial_{q}L-\partial^2_{\dot q,t}L-
\partial^2_{\dot q,q}L\;\dot q\bigr).
\end{equation}
For any smooth $q(t)$, not necessarily a Lagrangian motion, the following relations holds:
\begin{gather}
\partial_qL-\frac{d}{dt}\partial_{\dot q}L=g(\Lambda-\ddot q),
\label{differenceOfLagrangeSides}\\
\dot L=\partial_tL+\partial_qL\cdot\dot q
+\partial_{\dot q}L\cdot\ddot q\label{lDot}
\end{gather}
Since $N$ is a first integral, again for any smooth $q(t)$ we have
\begin{equation}\label{nDot}\begin{split}
\dot N={}&\partial_tN+\partial_qN\cdot\dot q+
\partial_{\dot q}N\cdot\ddot q=\\
={}&\underbrace{\partial_tN+\partial_qN\cdot\dot q+
\partial_{\dot q}N\cdot \Lambda}_{=0}+
\partial_{\dot q}N\cdot(\ddot q-\Lambda)=\\
={}&\partial_{\dot q}N\cdot(\ddot q-\Lambda).
\end{split}
\end{equation}
A solution of the strong form with $N$ as associated first integral must in particular satisfy relation~\eqref{familyFormula}. Suppose that the triple $(\tau ,\xi,f)$ satisfies~\eqref{familyFormula} and let us impose that it solves the strong version of Killing equation. The total time derivative of~$f$ along a generic smooth $q(t)$ is:
\begin{align}
\dot f={}&
\frac{d}{dt}\bigl(\tau L+N+\partial_{\dot q}L\cdot(\xi-\tau \dot q)
\bigr)=\\
={}&\dot \tau L+\tau \dot L+\dot N+
\Bigl(\frac{d}{dt}\partial_{\dot q}L\Bigr)\cdot(\xi-\tau \dot q)+
\partial_{\dot q}L\cdot
(\dot\xi-\dot \tau \dot q-\tau \ddot q).
\end{align}
Equating this with the left-hand side of the strong Killing-like equation we get
\begin{multline}
\tau\partial_t L
+\partial_q L\cdot \xi
+\partial_{\dot q}L\cdot
\bigl(\dot \xi-\dot q\dot\tau \bigr)
+L\tau =\\
=\dot \tau L+\tau \dot L+\dot N+
\Bigl(\frac{d}{dt}\partial_{\dot q}L\Bigr)\cdot(\xi-\tau \dot q)+
\partial_{\dot q}L\cdot
(\dot\xi-\dot \tau \dot q-\tau \ddot q)
\end{multline}
which simplifies immediately to
\begin{equation}
\tau\partial_t L
+\partial_q L\cdot \xi
=\tau \dot L+\dot N+
\Bigl(\frac{d}{dt}\partial_{\dot q}L\Bigr)\cdot(\xi-\tau \dot q)-
\tau \partial_{\dot q}L\cdot
\ddot q.
\end{equation}
Using~\eqref{lDot} to replace~$\dot L$ it becomes
\begin{equation}
\tau\partial_t L
+\partial_q L\cdot \xi
=(\partial_tL+\partial_qL\cdot\dot q)\tau +\dot N+
\Bigl(\frac{d}{dt}\partial_{\dot q}L\Bigr)\cdot(\xi-\tau \dot q),
\end{equation}
which further simplifies to
\begin{equation}
\partial_q L\cdot \xi
=\tau \partial_qL\cdot\dot q+\dot N+
\Bigl(\frac{d}{dt}\partial_{\dot q}L\Bigr)\cdot(\xi-\tau \dot q),
\end{equation}
which can be rearranged to
\begin{equation}\label{productNdot}
\Bigl(\partial_q L-\frac{d}{dt}\partial_{\dot q}L\Bigr)
\cdot(\xi-\tau \dot q)=
\dot N.
\end{equation}
Using~\eqref{differenceOfLagrangeSides} and~\eqref{nDot}, formula~\eqref{productNdot} becomes
\begin{equation}
\bigl(g(g-\ddot q)\bigr)\cdot
(\xi-\tau \dot q)=\partial_{\dot q}N(\ddot q-\Lambda)=
-\partial_{\dot q}N\cdot(\Lambda-\ddot q).
\end{equation}
Since the hessian matrix~$g$ is symmetric, this becomes
\begin{equation}
\bigl(g(\xi-\tau \dot q)\bigr)\cdot
(\Lambda-\ddot q)=-\partial_{\dot q}N\cdot(\Lambda-\ddot q).
\end{equation}
Finally, since $\ddot q$ is arbitrary, we conclude that
\begin{equation}
g(\xi-\tau \dot q)=-\partial_{\dot q}N,
\end{equation}
which is equivalent to~\eqref{strongSolutionCondition}. Equation~\eqref{strongSolutionGauge} is simply a consequence of~\eqref{strongSolutionCondition} and~\eqref{familyFormula}.
\end{proof}
A direct proof of the ``if'' part of Theorem~\ref{familyOfSolutionsWithoutLagrange} can be found in a paper by Boccaletti and Pucacco~\cite[Sec.~2]{BoccalettiPucacco}.
If we know a solution to the Killing-type equation, either on-flow or strong, we can easily generate infinitely many others, parameterized by an arbitrary function:
\begin{Corollario}[Multiplicity for both on-flow and strong equation]\label{multiplicityBoth}
Let $L(t,q,\dot q)$ be a Lagrangian function. Suppose that the triple $\bigl(\tau (t,q,\dot q), \xi(t,q,\dot q),\allowbreak f(t,q,\dot q))$ satisfies the Killing-type equation~\eqref{Killing-type} in either the strong or the on-flow version. Take an arbitrary smooth function $h(t,q,\dot q)$. Then also the following triple
\begin{equation}\label{equivalenttriples}
\tilde \tau =\tau +\frac{h-f}{L},\qquad
\tilde\xi=\xi+\dot q\,\frac{h-f}{L},\qquad
\tilde f=h
\end{equation}
satisfies the Killing-type equation of the same form. The corresponding first integral \eqref{firstintegral} is the same.
\end{Corollario}
\begin{proof}
If we assume that any of the equations~\eqref{familyFormula}, \eqref{strongSolutionGauge} and~\eqref{strongSolutionCondition} holds for the triple $(\tau ,\xi,f)$, a simple replacement shows that the equation holds also for $(\tilde \tau ,\tilde\xi,\tilde f)$.
\end{proof}
Within the family of solutions given by Theorem~\ref{multiplicityBoth} there is always one with trivial (i.e., zero) time change and another one with trivial boundary term. This simple fact was already established in a more general setting (including, for example, nonlocal constants of motion) and different notations by the authors~\cite[Theorem~10]{GorniZampieri}.
\begin{Corollario}[Trivializing either time-change or gauge]\label{trivializationCorollary}
Let $L(t,q,\dot q)$ be a Lagrangian function. Suppose that the triple $\bigl(\tau (t,q,\dot q), \xi(t,q,\dot q),\allowbreak f(t,q,\dot q))$ satisfies the Killing-type equation~\eqref{Killing-type} in either the strong or the on-flow form. Then also the following two triples are solutions:
\begin{equation}\label{trivializationFormulas}
(0,\;\xi-\dot q\tau ,\;f-L\tau ),\qquad
\Bigl(\tau -\frac{f}{L},\;\xi-\dot q\,\frac{f}{L},\;0\Bigr).
\end{equation}
The corresponding first integrals are the same.
\end{Corollario}
\begin{proof}
Simply take either $h=f-L\tau $ or $h=f$ in Corollary~\ref{multiplicityBoth}.
\end{proof}
\section{Strong solutions independent of $\dot q$}
\label{strongIndependentOfDotQ}
Given $\xi$ and~$\tau $ that do not depend on~$\dot q$ there is a simple necessary condition for them to be part of a solution triple $(\tau ,\xi,f)$ of the strong form of Killing-like equation, regardless of~$N$.
\begin{Proposizione}\label{independenceOfVelocity}
Suppose that $(\tau ,\xi,f)$ solves the strong form of Killing equation, and that $\xi(t,q)$ and~$\tau (t,q)$ do not depend on~$\dot q$. Then $f$ does not depend on~$\dot q$ either. Moreover, the left-hand side of the Killing-like equation
\begin{equation}\label{leftHandSide}
\tau\partial_t L
+\partial_q L\cdot \xi
+\partial_{\dot q}L\cdot
\bigl(\dot \xi-\dot q\dot\tau \bigr)
+L\dot \tau
\end{equation}
after replacing with the given $L(t,q,\dot q),\xi(t,q),\tau (t,q)$, depends linearly on~$\dot q$.
\end{Proposizione}
\begin{proof}
Starting from formula~\eqref{familyFormula}, which holds in the strong case too,
\begin{equation}
f=\tau L+N+\partial_{\dot q}L\cdot(\xi-\tau \dot q)
\end{equation}
and taking the gradient with respect to~$\dot q$ we get
\begin{equation}\begin{split}
\partial_{\dot q}f={}&\partial_{\dot q}N+
\partial_{\dot q}\bigl(
\tau L+\partial_{\dot q}L\cdot (\xi -\tau \dot q)\bigr)=\\
={}&
-g(\xi -\tau \dot q)+
\partial_{\dot q}\bigl(
\tau L+\partial_{\dot q}L\cdot (\xi -\tau \dot q)\bigr),
\end{split}
\end{equation}
where we have used the replacement $\partial_{\dot q}N=-g(\xi -\tau \dot q)$, which is a rearrangement of~\eqref{strongSolutionCondition}.
Using now the assumption that $\xi,\tau $ do not depend on~$\dot q$ we can carry on the calculation
\begin{equation}\begin{split}
\partial_{\dot q}f={}&
-g(\xi -\tau \dot q)+
\tau \partial_{\dot q}L+
g (\xi -\tau \dot q)+
\partial_{\dot q}L(-\tau )
\equiv0.
\end{split}
\end{equation}
We deduce that $f$ does not depend on~$\dot q$ either. Hence $\dot f$ is linear in~$\dot q$. We conclude that expression~\eqref{leftHandSide}, which is identically equal to~$\dot f$, must be linear in~$\dot q$ too.
\end{proof}
Only some first integrals $N$ can be deduced from a triple $(\tau (t,q),\xi(t,q),\allowbreak f(t,q))$ independent of~$\dot q$.
\begin{Proposizione}\label{conditionOnTXiForIndependenceOfVelocity}
A first integral $N(t,q,\dot q)$ can be deduced from a triple $(\tau ,\xi, f)$ that does not depend on~$\dot q$ if and only if $g^{-1} \partial_{ \dot q}N=a(t,q)+b(t,q)\dot q$, where $a(t,q)$ is vector-valued and $b(t,q)$ is scalar-valued.
\end{Proposizione}
\begin{proof}
If $N$ can be deduced from $\tau (t,q),\xi(t,q),f(t,q)$, then from Theorem~\ref{familyOfSolutionsWithoutLagrange}, formula~\eqref{strongSolutionCondition}, $g^{-1}\partial_{\dot q}N= -\xi(t,q)+\tau (t,q)\dot q$. Conversely, if $g^{-1} \partial_{ \dot q}N=a(t,q)+b(t,q)\dot q$, we can choose $\xi=-a$, $\tau =b$ and $f$ given by equation~\eqref{strongSolutionGauge}, so that the triple $(\tau ,\xi, f)$ is a solution of Killing-like equation in the strong sense (Theorem~\ref{familyOfSolutionsWithoutLagrange}). Finally, $f$~does not depend on~$\dot q$ because of Proposition~\ref{independenceOfVelocity}.
\end{proof}
\section{Examples}\label{examplesSection}
\subsection{The free particle}
\label{FreeParticle}
In one of his papers~\cite{Leach2}, Leach argues that Lie point symmetries that are usually called ``nonnoetherian'' are indeed fully Noetherian, provided that we switch from the obvious Lagrangian to some other Lagrangian which retains the same equations of motion. The point is illustrated with the example of the free particle in one dimension: the equation of motion is $\ddot q=0$, whose Lie symmetries $\xi\partial_q +\tau \partial_t$ are an 8-dimensional space (Table~\ref{simmetrie}). If we examine these symmetries in the Noetherian sense together with the ``natural'' Lagrangian~$L=\dot q^2/2$, we see that only five of them can be completed to a Noetherian triple $(\tau ,\xi,f)$. Let us see what we can say about the three remaining ``nonnoetherian'' symmetries from the point of view of the strong and on-flow solutions, without resorting to a different Lagrangian.
\begin{table}
\begin{equation*}
\begin{array}{lccc}
\text{Lie symmetry}&\text{Lie 1st integral}&
f&
\text{Noether 1st int.}\\\hline
\mathstrut\Gamma_1=\partial_q & \dot q & 0 & -\dot q\\
\Gamma_2=t\partial_q & t\dot q-q & q & q-t\dot q\\
\Gamma_3=\partial_t & \dot q & 0 & \dot q^2/2\\
\Gamma_4=2t\partial_t+q\partial_q & (t\dot q-q)\dot q & 0 &
(t\dot q-q)\dot q\\
\Gamma_5=t^2\partial_t+tq\partial_q & t\dot q-q & q^2/2 &
(q-t\dot q)^2/2\\[3pt]
\Gamma_6=q\partial_q & (t\dot q-q)/\dot q\\
\Gamma_7=q\partial_t & (t\dot q-q)/\dot q\\
\Gamma_8=qt\partial_t+q^2\partial_q & (t\dot q-q)/\dot q
\end{array}
\end{equation*}
\begin{center}
\caption{Lie and Noether symmetries of the free particle}
\label{simmetrie}
\end{center}
\end{table}
For the Lie symmetries $\Gamma_6,\Gamma_7,\Gamma_8$ the Killing-like equation becomes respectively
\begin{equation}
\dot q^2=\dot f_6,\qquad
-\frac{\dot q^3}{2}=\dot f_7,\qquad
\frac{3q\dot q^2-t\dot q^3}{2}=\dot f_8.
\end{equation}
These equations have no solution in~$f$ in the strong sense, because the left-hand sides are not linear in~$\dot q$ (Proposition~\ref{conditionOnTXiForIndependenceOfVelocity}).
As for the on-flow version of the Killing-like equation, given any arbitrary couple $(\tau ,\xi)$ and a first integral~$N$, formula~\eqref{familyFormula} of Theorem~\ref{familyOfSolutionsWithLagrange} immediately gives a boundary term~$f$ that completes to a solution triple. Specifically:
\begin{gather}
\text{for }\Gamma_6\qquad
f_6=q\dot q+N
\\
\text{for }\Gamma_7\qquad
f_7=-\frac{q\dot q^2}{2}+N
\\
\text{for }\Gamma_8\qquad
f_8=\frac{1}{2}(2q-t\dot q)q\dot q+N.
\end{gather}
Of course, these boundary terms depend on~$\dot q$. With solutions in the on-flow solutions we can recover all first integral of the system, i.e., all function of the form $k(\dot q,q-t\dot q)$. Using Proposition~\ref{conditionOnTXiForIndependenceOfVelocity}, since $g^{-1}=1$, we can say that with solutions $(\tau (t,q),\xi(t,q),f(t,q))$ in the strong sense we cannot obtain first integrals that are not quadratic in~$\dot q$, for example $(q-t\dot q)^3$.
\subsection{Superintegrable systems related to isochrony}
\label{superintegrableSubSection}
The authors are familiar with with the following system of two scalar differential equations
\begin{equation}\label{isoch}
\ddot x=-G(x),\quad
\ddot y=-G'(x)y,
\end{equation}
which are the Lagrange equations of the Lagrangian
\begin{equation}\label{LagrangianoIsoch}
L(t,q,\dot q)=
\dot x\,\dot y-G(x)y,
\qquad\text{where }q=\binom{x}{y},\
\dot q=\binom{\dot x}{\dot y}
\end{equation}
This system is rich with first integrals. One is $N_1=\dot x\dot y+G(x)y$. Since $g^{-1}\partial_{\dot q}N_1\allowbreak=\dot q$, following Proposition~\ref{conditionOnTXiForIndependenceOfVelocity} we can deduce $N_1$ from the following triple independent of~$\dot x,\dot y$:
\begin{equation}
\tau =1,\quad
\xi=0,\quad
f=0,
\end{equation}
which is a solution in the strong sense.
Another first integral is $N_2=\dot x^2/2+\int G(x)dx$. Since $g^{-1}\partial_{\dot q}N_2=(\begin{smallmatrix}0&0\\ -1&0 \end{smallmatrix})\dot q$ and because of Proposition~\ref{conditionOnTXiForIndependenceOfVelocity}, to deduce $N_2$ we must accept dependence on~$\dot q$. According to Theorem~\ref{familyOfSolutionsWithoutLagrange}, all solutions in the strong sense are given by an arbitrary~$\tau $ and
\begin{gather}
\xi=\tau \dot q-g^{-1}\partial_{\dot q}N_2=
\binom{\tau \dot x}{\tau \dot y-\dot x},\\
f=\tau L+N_2-\partial_{\dot q}g^{-1}\partial_{\dot q}N_2=
\tau \bigl(\dot x\dot y-G(x)y\bigr)-\frac{1}{2}\dot x^2+\int G(x)dx
\end{gather}
The special feature of the system~\eqref{isoch} is that for some classes of function~$G$ the system has a third, independent, first integral. One way to detect some of these superintegrable system is by making a plausible ansatz on the triple $(\tau ,\xi,f)$ and solving the Killing-like equation for $G$ as an additional unknown function. We think it is preferable to use the on-flow version of the equation, simply because it has so many more solution, heightening the chances that the ansatz may catch one.
Our starting ansatz is
\begin{equation}\label{ansatzH}
\tau \equiv0,\qquad
\xi=\binom{h(x,\dot x)}{0}.
\end{equation}
In keeping with the on-flow version, the first and second total time derivatives of~$h$ will take the Lagrange equations into account:
\begin{equation}
\dot h=\dot x\partial_x h+\ddot x\partial_{\dot x}h=
\dot x\partial_x h-G(x)\partial_{\dot x}h,\qquad
\ddot h=\dot x\partial_x \dot h-G(x)\partial_{\dot x}\dot h
\end{equation}
The Killing-like equation becomes
\begin{equation}
-yG'(x)h+\dot y\dot h=\dot f.
\end{equation}
With the further ansatz that
\begin{equation}\label{ansatzF}
f=y\dot h
\end{equation}
the equation is
\begin{equation}
-yG'(x)h+\dot y\dot h=\dot y\dot h+y\ddot h
\end{equation}
which simplifies to
\begin{equation}\label{killingWithH}
-G'(x)h=\ddot h.
\end{equation}
We can make a third ansatz by setting $h$ to be a polynomial in~$\dot x$ of the form $h=\alpha(x)+\beta(x)\dot x^2$. Replacing into~\eqref{killingWithH} we obtain a polynomial of degree~4 in~$\dot x$ equated to~0:
\begin{multline}\label{polyInXdot}
\beta ''(x)\dot x^4 +
\bigl(\alpha''(x)-\beta(x)G'(x)-5 G(x)\beta'(x)\bigr)\dot x^2+{}\\
+2 G(x)^2\beta(x)-G(x)\alpha '(x)
+\alpha (x) G'(x)=0.
\end{multline}
The coefficient of~$\dot x^4$ is $\beta''(x)$, which must be~0. Let us simply take $\beta(x)\equiv x$. Equating the coefficient of~$\dot x^0$ in~\eqref{polyInXdot} to~0 we get
\begin{equation}
\alpha(x) G'(x)-G(x)\alpha '(x)+2 x G(x)^2=0,
\end{equation}
which can be solved for~$\alpha$ as $\alpha(x)=(c+x^2)G(x)$. Replacing into the coefficient of~$\dot x^2$ we get the second order linear equation in~$G$
\begin{equation}\label{equationInG}
(c+x^2) G''(x)+3 x G'(x)-3 G(x)=0,
\end{equation}
whose linear space of real solutions around $x=0$ is generated by $G(x)=x$ and by
\begin{equation}
\frac{1}{x^3}\quad\text{if }c=0,\qquad
\frac{c+2x^2}{\sqrt{c+x^2}}\quad\text{if }c>0,
\qquad
\frac{-c-2x^2}{\sqrt{-c-x^2}}\quad\text{if }c<0,
\end{equation}
If we take any $G$ in this space, and set $h=(c+x^2)G(x)+x\dot x^2$, the triple given by equations~\eqref{ansatzH} and~\eqref{ansatzF} is an on-flow solution to Killing-like equation, and the associated first integral is
\begin{equation}\begin{split}
N_3={}&f-L\tau -\partial_{\dot q}L\cdot\bigl(\xi-\tau \dot q\bigr)=\\
={}&y\dot h-\partial_{\dot q}L\cdot\xi=\\
={}&(c+x^2)G'(x)\dot xy-(c+x^2) G(x)\dot y-x
\dot x^2 \dot y+\dot x^3 y.
\end{split}
\end{equation}
It can be verified that the three first integrals $N_1,N_2,N_3$ are functionally independent. The triple $(\tau ,\xi,f)$ that we have found is not a solution in the strong sense, since $\xi-(\tau \dot q-g^{-1}\partial_{\dot q}N_3)=(0, (c+x^2) G'(x)y+(3 \dot{x} y-2 x\dot{y})\dot{x} )$ does not vanish identically.
Now that we know the expression of the first integral~$N_3$ we can construct the solution triples $(\tau ,\Xi,f)$ of the Killing-like equation in the strong sense:
\begin{gather}
\tau =T,\qquad
\Xi=\binom{(c+x^2) G(x)+
(T +x \dot{x})\dot{x}}{
-y(c+x^2) G'(x)-3 \dot{x}^2 y+2 x \dot{x} \dot{y}
+T \dot{y}},\\
f=(T +2 x \dot{x})\dot{x} \dot{y}
-2 \dot{x}^3 y-G(x)T y.
\end{gather}
Summing up, we have used the on-flow version of the Killing-like equation to detect a class of superintegrable systems:
\begin{Proposizione}
The Lagrangian system given by equations~\eqref{isoch} and~\eqref{LagrangianoIsoch} is superintegrable whenever the function~$G$ satisfies equation~\eqref{equationInG}.
\end{Proposizione}
This class of systems with the parameter $c>0$ overlaps with the one that was found by the second author~\cite[Sec.~5]{Zampieri} using a totally different line of reasoning. The on-flow method that we have illustrated here was pushed further (albeit with a different language) by the two authors~\cite[Sec.~13]{GorniZampieri}, using with the more general ansatz $h=\alpha(x)+\beta(x)\dot x^2+\gamma(x)\dot x^4$. We expect that a larger class of superintegrable systems can readily be found by increasing the degree of~$h$ with respect to~$\dot x$.
\subsection{The Laplace-Runge-Lenz vector for Kepler's problem}\label{LRLSubSection}
Consider the Lagrangian function and Lagrange equation of Kepler's problem in dimension~3
\begin{gather}\label{LKepler}
L(t,\vec r, \vec v)=
\frac{1}{2} \lVert\vec v\rVert^2
+\frac{\mu}{\lVert \vec r\rVert},\quad
\vec r\in\mathbb{R}^3\setminus\{0\},
\\
\label{LagrangeKepler}
\ddot{\vec r}=-\frac{\mu}{\lVert \vec r\rVert^3}\,\vec r\,.
\end{gather}
Here we depart from the $q,\dot q$ notation and use $\vec r,\vec v$ instead, as done in common introductory mechanics textbooks. The vector product~``$\times$'' for 3-dimensional vectors will allow more compact formulas than what we get in the otherwise equivalent 2-dimensional treatment we gave in an earlier paper~\cite{GorniZampieri}.
Besides energy and angular momentum, the Kepler system has the LRL vector first integral
\begin{equation}
\vec A:= \vec v\times
(\vec r\times \vec v)-
\frac{\mu}{\lVert \vec r\rVert}\,\vec r.
\end{equation}
Fix an arbitrary vector $\vec u\in\mathbb{R}^3$ and consider the scalar first integral
\begin{equation}
N:=-\vec u\cdot \vec A.
\end{equation}
If we check the condition of Proposition~\ref{conditionOnTXiForIndependenceOfVelocity} we see that $N$ cannot be obtained from a triple $(\tau,\xi,f)$ which is independent of~$\vec v$. Let us see what we can do with either on-flow or strong solutions involving~$\vec v$.
Theorem~\ref{familyOfSolutionsWithLagrange} gives us so many on-flow solutions that we may be choosy and aim for subjectively simple, elegant formulas. One that is simple enough is
\begin{equation}
\tau _0=\frac{\vec u\cdot \vec v\times
(\vec r\times \vec v)}{L},\qquad
{\vec \xi}_0=\frac{\vec u\cdot \vec v\times(\vec r\times \vec v)}{L}
\vec v,\qquad
f_0=\frac{\mu}{\lVert\vec r\rVert}\vec r\cdot\vec u.
\end{equation}
Levy-Leblond~\cite{Leblond} proposed the following one, without explanation as to how he came up with the formula:
\begin{gather}
\tau _L=0,\qquad
{\vec\xi}_L=-\frac{1}{2}\partial_{\vec v}N=
(\vec r\cdot \vec u)\vec v
-\frac{1}{2}(\vec v\cdot \vec u)\vec r
-\frac{1}{2}(\vec v\cdot \vec r)\vec u,\\
f_L=\tau_L L-N+\partial_{\dot q}L\cdot(\xi_L-\tau_L\dot q)=
\frac{\mu}{\lVert\vec r\lVert}\vec r\cdot\vec u=f_0,
\end{gather}
To find different solutions with trivial first order time variation $\tau \equiv 0$, we write the Killing-type equation~\eqref{Killing-type} within the current setting:
\begin{equation}\label{KillingWithZeroT}
\partial_{\vec r} L\cdot \vec\xi
+\partial_{\vec v}L\cdot
\dot {\vec\xi}=
\dot f,
\end{equation}
and we impose that the first order space variation $\vec \xi$ be such that
\begin{equation}\label{imposition}
\partial_{\vec v}L\cdot \vec \xi=\vec u\cdot \vec v\times
(\vec r\times \vec v).
\end{equation}
Our favourite way to satisfy this condition is
\begin{equation}
{\vec \xi}_Z=
(\vec r\times \vec v)\times\vec u=
(\vec r\cdot \vec u)\vec v-(\vec v\cdot \vec u)\vec r.
\end{equation}
This choice is not the only possible: $\vec\xi_L$ satisfies the same condition:
\begin{equation}
\partial_{\vec v}L\cdot\vec \xi_{L}
=\vec v\cdot\vec \xi_{L}=
\vec u\cdot \vec v\times
(\vec r\times \vec v)
=\lVert\vec v\rVert^2\vec u\cdot\vec r
-(\vec v\cdot \vec r)(\vec v\cdot \vec u).
\end{equation}
Using Lagrange equation \eqref{LagrangeKepler} we have
\begin{equation}
{\dot {\vec\xi}}_Z=
\bigl(\partial_{\vec r}{\vec \xi}_Z\bigr) \vec v
+\partial_{\vec v}{\vec \xi}_Z\,
\Bigl(-\frac{\mu}{\lVert \vec r\rVert^3}\, \vec r\Bigr)
=\vec 0.
\end{equation}
Using equation~\eqref{imposition}, the left-hand side of equation~\eqref{KillingWithZeroT} becomes
\begin{align*}
\partial_{\vec r} L\cdot {\vec \xi}_Z
+\partial_{\vec {}v}L\cdot
{\dot {\vec \xi}}_Z={}&
\partial_{\vec r} L\cdot {\vec \xi}_Z
+\partial_{\vec v}L\cdot \vec 0=
-\frac{\mu}{\lVert \vec r\rVert^3}\vec r\cdot
(\vec r\times \vec v)\times\vec u=\\
={}&-\frac{\mu}{\lVert \vec r\rVert^3}\vec r\times
(\vec r\times \vec v)\cdot\vec u=\\
={}&-
\frac{\mu}{\lVert \vec r\rVert^3}\,\vec u\cdot
\bigl(\vec r(\vec r\cdot \vec v)-
\vec v\Vert \vec r\rVert^2\bigr)=
\vec v\cdot\partial_{\vec r}
\Bigl(\frac{\mu}{\lVert \vec r\rVert}\vec r\cdot\vec u\Bigr)=\\
={}&\frac{d}{dt}
\Bigl(\frac{\mu}{\lVert \vec r\rVert}\vec r\cdot\vec u\Bigr)=
\dot f_0.
\end{align*}
We can complete the solution triple as follows:
\begin{equation}\label{LRLgauge}
\tau _Z\equiv0,\qquad
{\vec \xi}_Z=
(\vec r\times \vec v)\times\vec u,\qquad
f_Z=f_0=\frac{\mu}{\lVert \vec r\rVert}\,\vec r\cdot\vec u.
\end{equation}
The resulting first integral of formula~\eqref{firstintegral} is what we expected:
\begin{align*}
f_Z-\partial_{\dot \vec v}L\cdot{\vec \xi}_Z={}&
\frac{\mu}{\lVert \vec r\rVert}\vec r\cdot\vec u
-\vec v\cdot (\vec r\times \vec v)\times\vec u=\\
={}&-\Bigl(\vec v\times (\vec r\times \vec v)
-\frac{\mu}{\lVert \vec r\rVert}\vec r\Bigr)\cdot\vec u=
N.
\end{align*}
The triple~\eqref{LRLgauge} belongs to the family of Theorem~\ref{familyOfSolutionsWithLagrange}, as can be checked by direct computation.
Corollary~\ref{multiplicityBoth}, formula~\eqref{equivalenttriples}, applied to the triple~\eqref{LRLgauge}, gives a whole family of solution triples, depending on an arbitrary function~$h(t,\vec r,\vec v)$:
\begin{equation}\label{familyForLRL}
\tau =\frac{1}{L}\Bigl(h-\frac{\mu}{\lVert \vec r\rVert}
\vec u\cdot\vec r\Bigr),\quad
\vec \Xi= (\vec r\times \vec v)\times\vec u
+\frac{1}{L}\Bigl(h-\frac{\mu}{\lVert \vec r\rVert}
\vec u\cdot\vec r\Bigr)\vec v,\quad
f=h.
\end{equation}
As in Corollary~\ref{trivializationCorollary}, the choice $h\equiv0$ will trivialize the boundary term.
Using a computer algebra system the reader can directly check all these solutions to the Killing-type equation, independently of the theorems in Section~\ref{KillingSection}. For example here is some simple code written for Wolfram \emph{Mathematica} that implements the solution triple~\eqref{familyForLRL} and then checks that the on-flow Killing-type equation is satisfied and that the first integral is the LRL vector:
\begin{verbatim}
(*Defining the variables*)
r = {r1, r2, r3};
v = {v1, v2, v3};
L = v.v/2 + mu/Sqrt[r.r];
A = Cross[v, Cross[r, v]] - mu*r/Sqrt[r.r];
u = {u1, u2, u3};
arbitrary = h[t, r1, r2, r3, v1, v2, v3];
f0 = mu*(u.r)/Sqrt[r.r];
T = (arbitrary - f0)/L;
Xi = Cross[Cross[r, v], u] + T*v;
f = arbitrary;
(*the time dot derivatives are on-flow*)
rDotDot = -mu*(u.r)*r/(r.r)^(3/2);
Tdot = D[T, t] + D[T, {r}].v + D[T, {v}].rDotDot;
XiDot = D[Xi, t] + D[Xi, {r}].v + D[Xi, {v}].rDotDot;
fDot = D[f, t] + D[f, {r}].v + D[f, {v}].rDotDot;
(*checking on-flow Killing-type equation*)
Simplify[
D[L, t]*T + D[L, {r}].Xi + D[L, {v}].(XiDot - v*Tdot)
+ L*Tdot == fDot]
(*checking LRL vector as first integral*)
Simplify[
f - L*T - D[L, {v}].(Xi - T*v) == -A.u]
\end{verbatim}
\noindent
Upon evaluation, the code gives \verb!True! and \verb!True! in an instant.
Solutions of the Killing-type equation in the strong sense are fewer, and there is less freedom to simplify formulas. The explicit triples given by Sarlet and Cantrijn~\cite[Sec.~6]{SarletCantrijn}, and by the authors~\cite[Sec.~12]{GorniZampieri} are for dimension~2. Boccaletti an Pucacco~\cite[Sec.~2.2]{Boccaletti} deduce their solution, also in dimension~2, by assuming $\tau\equiv0$ and $\xi$ to be a bilinear function of~$q,\dot q$ and then working out the coefficients. Here we contribute a triple written for dimension~3, where the vector cross product again leads to elegant formulas for the solution $(\tau ,\Xi,f)$, and where we incorporate the multiplicity Corollary~\ref{multiplicityBoth}:
\begin{gather}
\vec b:=-\vec u \bigl(\vec r\cdot \vec v\bigr)
-\vec r\bigl(\vec v\cdot \vec u\bigr)
+\vec v\bigl(\vec u\cdot \vec r\bigr),\\
\tau =\frac{1}{L}\biggl(h-\vec u\cdot
\Bigl(\vec v\times \bigl(\vec r\times \vec v\bigr)
+\frac{\mu}{\lVert \vec r\rVert}\, \vec r\Bigr)\biggr),\\
\vec \Xi=\frac{1}{L}
\Bigl(h\,\vec v+\frac12
\vec v\times\bigl(\vec b\times\vec v\bigr)
+\frac{\mu}{\lVert \vec r\rVert}\,\vec b\Bigr),\qquad
f=h,
\end{gather}
where $\vec u\in\mathbb{R}^3$ is an arbitrary parameter vector, as in the previous section. The first integral associated to the triple through Noether's theorem~\eqref{firstintegral} is the same as before:
\begin{equation}
L\tau +\partial_{\vec v}L\cdot
\bigl(\vec \Xi-\tau \vec v\bigr)=
-\vec u\cdot\Bigl(\vec v\times
\bigl(\vec r\times \vec v\bigr)-
\frac{\mu}{\lVert \vec r\rVert}\,\vec r\Bigr)=-\vec u\cdot\vec A.
\end{equation}
Again we provide below some \emph{Mathematica} code that implements the solution triple and checks that it solves the Killing-type equation in the strong sense, and that it gives the Laplace-Runge-Lenz first integral.
\begin{verbatim}
(*Defining the variables*)
r = {r1, r2, r3};
v = {v1, v2, v3};
L = v.v/2 + mu/Sqrt[r.r];
A = Cross[v, Cross[r, v]] - mu*r/Sqrt[r.r];
u = {u1, u2, u3};
arbitrary = h[t, r1, r2, r3, v1, v2, v3];
T = (arbitrary - u.(Cross[v, Cross[r, v]] + mu*r/Sqrt[r.r]))/L;
b = -u*(r.v) - r*(u.v) + v*(r.u);
Xi = (arbitrary*v + Cross[v, Cross[b, v]]/2 + mu*b/Sqrt[r.r])/L;
f = arbitrary;
(*the time dot derivatives are generic, not on-flow*)
rDotDot = {a1, a2, a3};
Tdot = D[T, t] + D[T, {r}].v + D[T, {v}].rDotDot;
XiDot = D[Xi, t] + D[Xi, {r}].v + D[Xi, {v}].rDotDot;
fDot = D[f, t] + D[f, {r}].v + D[f, {v}].rDotDot;
(*checking Killing-type equation*)
Simplify[
D[L, t]*T + D[L, {r}].Xi + D[L, {v}].(XiDot - v*Tdot)
+ L*Tdot == fDot]
(*checking LRL vector as first integral*)
Simplify[
f - L*T - D[L, {v}].(Xi - T*v) == -A.u]\end{verbatim}
\noindent
The evaluation gives \verb!True!, as expected.
\section{Killing-like equation for a different time\\ change}
\label{differentTimeChangeSection}
Infinitesimal invariance up to boundary terms usually refers to the dependence of the quantity
\begin{equation}\label{actionWithInverseTimeChange}
\int_{\bar t_1}^{\bar t_2}L\Bigl(\bar t,\bar q(\bar t),
\frac{d\bar q}{d\bar t}(\bar t)\Bigr)d\bar t
\end{equation}
with respect to $\varepsilon$, as in equation~\eqref{infinitesimalInvariance} of Section~\ref{introduction}, where again
\begin{equation}\label{timeAndSpaceChange}
\bar t_\varepsilon(t)=t+\varepsilon
\tau\bigl(t,q(t),\dot q(t)\bigr),\qquad
\bar q_\varepsilon(t)=q+\varepsilon
\xi\bigl(t,q(t),\dot q(t)\bigr).
\end{equation}
In an earlier paper on Noether's theorem~\cite{GorniZampieri} we proposed, among other things, a generalization of infinitesimal invariance that leads to nonlocal constants of motion, and also, more to the point here, that infinitesimal invariance with time change can be based on the following expression
\begin{equation}\label{actionWithNewTimeChange}
\int_{\bar t_\varepsilon(t_1)}^{
\bar t_\varepsilon(t_2)}L\Bigl(t,\bar q_\varepsilon(t),
\frac{d\bar q_\varepsilon}{dt}(t)\Bigr)dt.
\end{equation}
instead of~\eqref{actionWithInverseTimeChange} \cite[Sec.~4]{GorniZampieri}. What is different is that the time derivative of $\bar q_\varepsilon$ and the integration in~\eqref{actionWithNewTimeChange} is made by respect to the original time~$t$, whilst in Section~\ref{introduction} the derivative was made with respect to the transformed time~$\bar t$.
We will say that the transformation~\eqref{timeAndSpaceChange} leaves the action integral \emph{alternatively}-invariant up to boundary terms if a function $f(t,q,\dot q)$ exists, such that for all $t_1,t_2$ we have
\begin{multline}\label{infinitesimalInvarianceAlternative}
\int_{\bar t_\varepsilon(t_1)}^{
\bar t_\varepsilon(t_2)}L\Bigl(t,\bar q_\varepsilon(t),
\frac{d\bar q_\varepsilon}{dt}(t)\Bigr)dt+{}\\
+\varepsilon\int_{t_1}^{t_2}
\frac{df}{dt}\bigl(t,q(t),\dot q(t)\bigr)dt
+O(\varepsilon^2)
\quad\text{as }\varepsilon\to0.
\end{multline}
To translate this condition into a differential equation, we take the integral
\begin{equation}
\int_{\bar t_\varepsilon(t_1)}^{
\bar t_\varepsilon(t_2)}L\Bigl(t,\bar q_\varepsilon(t),
\frac{d\bar q_\varepsilon}{dt}(t)\Bigr)dt
\end{equation}
and replace the $t$ variable with $\bar t_\varepsilon(t)$. The integral becomes with fixed extrema $t_1,t_2$:
\begin{equation}
\int_{t_1}^{t_2}L\bigl(\bar t_\varepsilon(t),
\bar q_\varepsilon(\bar t_\varepsilon(t)),
\dot{\bar q}(\bar t_\varepsilon(t))\bigr)
\dot{\bar t}_\varepsilon(t)\,dt.
\end{equation}
The first-order expansion of this expression as $\varepsilon\to0$ is
\begin{multline}
\int_{t_1}^{t_2}L\bigl(t,q(t),\dot q(t)\bigr)dt+{}\\
+\varepsilon\int_{t_1}^{t_2}
\bigl(\tau\partial_t L
+\partial_q L\cdot (\xi+\tau\dot q)
+\partial_{\dot q}L\cdot(\dot\xi+\tau\ddot q)
+L\dot\tau\bigr)dt
+O(\varepsilon^2).
\end{multline}
Hence the alternative invariance~\eqref{infinitesimalInvarianceAlternative} is equivalent to the following \emph{alternative} Killing-type equation for ODEs:
\begin{equation}\label{Killing-typeAlternative}
\tau\partial_t L
+\partial_q L\cdot (\xi+\tau\dot q)
+\partial_{\dot q}L\cdot(\dot\xi+\tau\ddot q)
+L\dot\tau=
\dot f.
\end{equation}
The commonly made assumption that $\tau,\xi,f$ only depend on~$(t,q)$ eliminated~$\ddot q$ for the standard Killing-type equation~\eqref{Killing-type}, collapsing the on-flow and the strong interpretation. The alternative equation~\eqref{Killing-typeAlternative} does not lend itself to this simplification, so that we are forced to take position on how to understand~$\ddot q$: either as an independent $n$-dimensional variable (strong form), or as a shorthand for the $\Lambda(t,q,\dot q)$ of the Lagrange equation~\eqref{normalformLagrange} (on-flow form).
If a triple $(\tau,\xi,f)$ solves the alternative equation~\eqref{Killing-typeAlternative} in either sense, then the following function is a first integral for the Lagrangian system:
\begin{equation}\label{firstintegralAlternative}
N=f-L\tau-\partial_{\dot q}L\cdot\xi.
\end{equation}
The expression is different from the corresponding formula~\eqref{firstintegral} for the standard Killing-like equation. Compare however our formula~\eqref{firstintegralAlternative} with formula~(I.11) from Candotti, Palmieri and Vitale~\cite{Candotti}.
There is a simple correspondence between the solutions to the standard and the alternative Killing-type equations, either in the strong or in the on-flow interpretation: if $(\tau,\xi,f)$ solves the alternative version~\eqref{Killing-typeAlternative} then $(\tau,\xi+\tau\dot q,f)$ solves the standard~\eqref{Killing-type}. Conversely, if $(\tau,\xi,f)$ solves the standard~\eqref{Killing-type} then $(\tau,\xi-\tau\dot q,f)$ solves the alternative~\eqref{Killing-typeAlternative}. This is basically Theorem~8 of the previous paper~\cite{GorniZampieri}, except that the ``alternative'' tag is used in the opposite sense.
Given a first integral~$N$, the associated general solution for the strong interpretation of the standard Killing-type equation are given by Theorem~\ref{familyOfSolutionsWithoutLagrange}. The equivalent statement for the solutions to the alternative Killing-type equation is simply obtained by replacing equations~(\ref{strongSolutionCondition}--\ref{strongSolutionGauge}) with
\begin{gather}\label{strongSolutionConditionAlternative}
\xi= -g^{-1}\partial_{\dot q}N,\\
f=\tau L+N-\partial_{\dot q}L\cdot g^{-1}\partial_{\dot q}N.
\label{strongSolutionGaugeAlternative}
\end{gather}
If we choose $\tau$ so as to get trivial $f\equiv0$, we obtain the alternative strong solution
\begin{equation}
\label{strongSolutionConditionAlternativeWithTrivialGauge}
\tau=-\frac{1}{L}\Bigl(N-
\partial_{\dot q}L\cdot g^{-1}\partial_{\dot q}N\Bigr),\qquad
\xi= -g^{-1}\partial_{\dot q}N\qquad
f=0.
\end{equation}
Compare with equation~(I.17) from Candotti, Palmieri and Vitale~\cite{Candotti}.
\end{document} | math | 48,653 |
\begin{document}
\title{
Characterizations of exchangeable partitions and random discrete distributions by deletion properties \thanks{
This research is supported in parts by N.S.F.\
Award 0806118. }}
\author{
Alexander Gnedin
\thanks{
University of Utrecht; email [email protected]}
\and
Chris Haulk
\thanks{
University of California at Berkeley; email [email protected]}
\and
Jim Pitman
\thanks{
University of California at Berkeley; email [email protected]}
}
\date{\today}
\maketitle
\begin{abstract}
\Mathbb Noindent
We prove a long-standing conjecture which characterises the Ewens-Pitman
two-parameter family of exchangeable random partitions, plus a short list of limit
and exceptional cases, by the following property: for each $n = 2,3, \ldots$, if one of $n$ individuals is chosen uniformly at random,
independently of the random partition $\pi_n$ of these individuals into various types, and all individuals of the same type as the
chosen individual are deleted, then for each $r > 0$, given that $r$ individuals remain, these individuals are partitioned
according to $\pi_r'$ for some sequence of random partitions $(\pi_r')$
which does not depend on $n$.
An analogous result characterizes the associated Poisson-Dirichlet family of random discrete distributions
by an independence property
related to random deletion of a frequency chosen by a size-biased pick. We also survey the regenerative properties of members of the two-parameter family, and settle a question regarding the explicit arrangement of intervals with lengths given by the terms of
the Poisson-Dirichlet random sequence into the interval partition induced by the range of a homogeneous
neutral-to-the right process.
\end{equation}d{abstract}
\tableofcontents
\section{Introduction}
Kingman \cite{MR509954}
introduced the concept of a {\em partition structure}, that is a
family of probability distributions for random partitions $\pi_n$ of a positive integer $n$, with a sampling
consistency property as $n$ varies.
Kingman's work was motivated by applications in population
genetics, where the partition of $n$ may be the allelic partition generated by randomly sampling a set of $n$ individuals from
a population of size $N \gg n$, considered in a large $N$ limit which implies
sampling consistency.
Subsequent authors have established the importance of Kingman's theory of partition structures,
and representations of these structures in terms of exchangeable random partitions and random discrete
distributions \cite{jp.isbp}, in a number of other settings, which include the theory of species sampling
\cite{jp96bl},
random trees and associated random processes of fragmentation and coalescence
\cite{jp97cmc,hpw07,BertoinFragCoag,BertoinCoagFrag},
Bayesian statistics and machine learning
\cite{teh06,09memoizer}.
Kingman
\cite{MR0526801}
showed that the Ewens sampling formula from population genetics defines a particular partition
structure $(\pi_n)$, which he characterized by the following property, together
with the regularity condition that $\mathbb P(\pi_n = \lambda) >0 $ for every
partition $\lambda$ of $n$:
\begin{quote}
for each $n = 2,3, \ldots$, if an individual is chosen uniformly at random
independently
of a random partitioning of these individuals into various types according to $\pi_n$, and all individuals of the same type as the
chosen individual are deleted, then conditonally given that the number of remaining individuals is $r >0$,
these individuals are partitioned according to a copy of $\pi_r$.
\end{equation}d{quote}
We establish here a conjecture of Pitman \cite{jp.epe} that if this property is weakened by replacing $\pi_r$ by $\pi_r'$
for some sequence of
random partitions $(\pi_r')$, and a suitable regularity condition is imposed,
then
$(\pi_n)$ belongs to the two-parameter family of partition structures
introduced in \cite{jp.epe}.
Theorem \ref{2pchar1} below provides a more careful statement.
We also present a corollary of this result, to characterize
the two-parameter family of Poisson-Dirichlet distributions by an independence property of a single size-biased pick,
thus improving upon \cite{jp.isbp}.
\par
Kingman's characterization of the Ewens family of partition structures by
deletion of a type has been extended in another direction by allowing
other deletion algorithms but continuing to require that the distribution
of the partition structure be preserved.
The resulting theory of {\em regenerative partition structures} \cite{rps04},
is connected to the theory of regenerative sets, including Kingman's regenerative phenomenon \cite{KingmanReg}, on a multiplicative scale.
In the last section of the paper we review such deletion properties of the two-parameter family of partition structures, and offer a new proof of a result of
Pitman and Winkel \cite{PitmanWinkel} regarding the
explicit arrangement of intervals with lengths given by the terms of the Poisson-Dirichlet random sequence into the interval partition induced by a
multiplicatively regenerative set.
\section{Partition Structures}
This section briefly reviews Kingman's theory of partition structures, which provides the general context of this article.
To establish some terminology and notation for use throughout the paper, recall
that a {\em composition } $\lambda$ of a positive integer $n$
is a sequence of positive integers
$\lambda = (\lambda_1, \ldots, \lambda_k)$, with $\sum_{i = 1}^{k} \lambda_i = n$.
Both $k = k_\lambda$ and $n = n_\lambda$ may be regarded as functions of $\lambda$. Each term $\lambda_i$ is called a
{\em part} of $\lambda$.
A {\em partition $\lambda$ of $n$} is a multiset of positive integers whose sum is $n$, commonly identified with the composition of $n$ obtained by putting its positive
integer parts in decreasing order, or with the infinite
sequence of non-negative integers obtained by appending an infinite string of zeros to this composition of $n$. So
$$\lambda = (\lambda_1, \lambda_2, \ldots) \mbox{~ with~ } \lambda_1 \ge \lambda_2 \ge \cdots \ge 0 $$
represents a partition of $n = n_\lambda$ into $k = k_\lambda$ parts, where
$$
n_\lambda := \sum_i \lambda_i \mbox{ and } k_\lambda:= \max \{ i : \lambda_i > 0 \}.
$$
Informally,
a partition $\lambda$ describes an unordered collection of $n_\lambda$ balls of
$k_\lambda$ different colors, with $\lambda_i$ balls of the $i$th most frequent color.
A {\em random partition of $n$} is a random variable $\pi_n$ with values in the finite set of all partitions $\lambda$ of $n$.
Kingman \cite{MR509954}
defined a {\em partition structure} to be a sequence of random partitions $(\pi_n)_{n \in \mathbb N}$
which is {\em sampling consistent} in the following sense:
\begin{quote}
if a ball is picked uniformly at random and deleted from $n$ balls randomly colored according to $\pi_n$, then
the random coloring of the remaining $n-1$ balls is distributed according to $\pi_{n-1}$.
\end{equation}d{quote}
As shown by Kingman
\cite{MR671034},
the theory of partition structures and associated partition-valued processes is best developed in terms of random partitions of the
set of positive integers. Our treatment here follows \cite{jp.epe}.
If we regard a random partition $\pi_n$ of a positive integer $n$ as a random coloring of $n$ unordered balls, an associated
random partition $\mathbb Pi_n$ of the set $[n]:= \{1, \ldots, n \}$ may be obtained by placement of the colored balls in a row.
We will assume for the rest of this introduction that this placement is made by a random permutation which given $\pi_n$ is uniformly distributed
over all $n!$ possible orderings of $n$ distinct balls.
Formally, a partition of $[n]$ is a collection of disjoint non-empty blocks
$\{B_1, \ldots, B_k\}$ with $\cup_{i=1}^k B_i = n$
for some $1 \le k \le n$, where each $B_i \subseteq [n]$ represents the set
of places occupied by balls of some particular color.
We adopt the convention that the blocks $B_i$ are listed {\em in order of appearance}, meaning that $B_i$ is the set of places in the row occupied by balls
of the $i$th color to appear.
So $1 \in B_1$, and if $k \ge 2$ the least element of $B_2$ is the
least element of $[n]\setminus B_1$, if $k \ge 3$ the least element of $B_3$ is the least element of $[n]\setminus(B_1 \cup B_2)$, and so on.
This enumeration of blocks identifies each partition of $[n]$ with an {\it ordered partition} $(B_1,\ldots,B_k)$,
subject to these constraints.
The sizes of parts $(|B_1|,\ldots, |B_k|)$ of this partition form a composition of $n$.
The notation $\mathbb Pi_n = (B_1, \ldots, B_k)$ is
used to signify that $\mathbb Pi_n = \{B_1, \ldots, B_k\}$ for some particular sequence of blocks
$(B_1, \ldots, B_k)$ listed in order of appearance.
If $\mathbb Pi_n$ is derived from $\pi_n$ by uniform random placement of balls in a row, then $\mathbb Pi_n$ is {\em exchangeable},
meaning that its distribution is invariant under every deterministic rearrangement of places by a permutation of $[n]$.
Put another way,
for each partition $(B_1, \ldots, B_k)$ of $[n]$, with blocks in order of appearance,
\begin{equation}
\label{EPF}
\mathbb P( \mathbb Pi_n = (B_1, \ldots, B_k) ) = p ( |B_1|, \ldots, |B_k|)
\end{equation}d{equation}
for a function $p = p(\lambda)$ of compositions $\lambda$ of $n$ which is a symmetric function of its $k$ arguments for each $1 \le k \le n$.
Then $p$ is called the {\em exchangeable partition probability function} (EPPF) associated with $\mathbb Pi_n$, or with $\pi_n$, the partition of $n$
defined by the unordered sizes of blocks of $\mathbb Pi_n$.
As observed by Kingman
\cite{MR671034},
$(\pi_n)$ is sampling consistent if and only if the sequence of partitions $(\mathbb Pi_n)$ can be
constructed to be consistent in the sense that for $m<n$ the restriction of $\mathbb Pi_n$ to $[m]$ is $\mathbb Pi_m$. This amounts to a simple recursion
formula satisfied by $p$, recalled later as \re{add-rule}.
The sequence $\mathbb Pi = (\mathbb Pi_n)$ can then be interpreted as a random partition of the set $\mathbb N$ of all positive integers, whose restriction to $[n]$ is $\mathbb Pi_n$ for every $n$.
Such $\mathbb Pi$ consists of a sequence of blocks ${\cal B}_1, {\cal B}_2, \ldots$, which may be identified as random disjoint subsets of $\mathbb N$, with
$\cup_{i = 1}^\infty {\cal B}_i = \mathbb N$, where the nonempty blocks
are
arranged by increase of their minimal elements, and
if the number of nonempty blocks is some $K < \infty$, then by convention ${\cal B}_i = \varnothing$ for $ i > K$.
Similarly, $\mathbb Pi_n$ consists of a sequence of blocks ${\cal B}_{ni}:={\cal B}_{i}\cap [n]$,
where
$\cup_i{\cal B}_{ni}=[n]$, and
the nonempty blocks are consistently arranged
by increase of their minimal elements,
for all $n$.
These considerations are summarized by the following proposition:
\begin{proposition}
\label{kinp}{\rm (Kingman
\cite{MR671034}
)}
The most general partition structure,
defined by a sampling consistent collection of distributions for
partitions $\pi_n$ of integers $n$, is associated with a unique
probability distribution of an exchangeable partition of positive integers $\mathbb Pi = (\mathbb Pi_n)$, as determined by an EPPF $p$ according to {\rm \re{EPF}}.
\end{equation}d{proposition}
We now recall a form of {\em Kingman's paintbox construction} of such an exchangeable random partition $\mathbb Pi$ of
positive integers.
Regard the unit interval $[0,1)$ as a continuous spectrum of distinct colors, and suppose given
a sequence of random variables $(P^{\downarrow}_1,P^{\downarrow}_2,\ldots)$ called
{\em ranked frequencies},
subject to the constraints
\begin{equation}
\label{part1}
1\geq P^{\downarrow}_1\geq P^{\downarrow}_2\geq \ldots\geq0,~~~P_*:=1-\sum_{j=1}^\infty P^{\downarrow}_j\geq 0.
\end{equation}
The color spectrum is partitioned into a sequence of intervals $[l_i, r_i)$ of lengths $P^{\downarrow}_i$, and
in case $P_* >0$ a further interval
$[1-P_*, 1)$ of length $P_*$.
Each point $u$ of $[0,1)$ is assigned the color $c(u) = l_i$ if $u \in [l_i, r_i)$ for some $i = 1,2, \ldots$,
and $c(u) = u$ if $u \in [1- P_*, 1)$. This coloring of points of $[0,1)$, called {\em Kingman's paintbox}
associated with $(P^{\downarrow}_1,P^{\downarrow}_2,\ldots)$,
is sampled by an infinite sequence of independent uniform$[0,1]$ variables $U_i$,
to assign a color $c(U_i)$ to the $i$th ball in a row of balls indexed by $i = 1,2, \ldots$.
The associated {\em color partition} of $\mathbb N$ is generated
by the random equivalence relation $\sim$ defined by $i \sim j $ if and only if $c(U_i) = c(U_j)$,
meaning that either $U_i$ and $U_j$ fall in the same compartment of the paintbox, or that $i = j$ and $U_i$ falls in $[1-P_*,1)$.
\begin{theorem}\label{Kpaintbox}
{\em (Kingman's paintbox representation of exchangeable partitions
\cite{MR509954})}
Each exchangeable partition $\mathbb Pi$ of $\mathbb N$ generates a sequence of
ranked frequencies $(P^{\downarrow}_1,P^{\downarrow}_2,\ldots)$ such that the conditional distribution of $\mathbb Pi$ given these frequencies is that of the color partition of $\mathbb N$
derived from
$(P^{\downarrow}_1,P^{\downarrow}_2,\ldots)$ by Kingman's paintbox construction.
The exchangeable partition probability function $p$ associated with $\mathbb Pi$ determines the distribution of $(P^{\downarrow}_1,P^{\downarrow}_2,\ldots)$, and vice versa.
\end{equation}d{theorem}
The distributions of ranked frequencies $(P^{\downarrow}_1,P^{\downarrow}_2,\ldots, )$ associated with naturally arising partition structures
$(\pi_n)$ are quite difficult to deal with analytically. See for instance \cite{py95pd2}. Still, $(P^{\downarrow}_1,P^{\downarrow}_2,\ldots)$ can be constructed as the
decreasing rearrangement of the frequencies $P_i$ of blocks ${\cal B}_i$ of $\mathbb Pi$
defined as the almost sure limits
\begin{equation}
\label{freqs}
P_i=\lim_{n\to\infty} n^{-1} |{\cal B}_{ni}|
\end{equation}d{equation}
where $i = 1,2, \ldots$ indexes the blocks in order of appearance,
while
$$
P_* = 1 - \sum_{i = 1}^\infty P_i = 1 - \sum_{i = 1}^\infty P^{\downarrow}_i
$$
is the asymptotic frequency of the union of singleton blocks
$${\cal B}_*:= \cup_{\{i : |{\cal B}_i| = 1\}} {\cal B}_i,$$
so that \re{freqs} holds also for $i = *$.
The frequencies are called {\em proper} if $P_* = 0$ a.s.;
then almost surely every nonempty block ${\cal B}_i$ of $\mathbb Pi$ has a
strictly positive frequency, hence $|{\cal B}_i| = \infty$, while every block ${\cal B}_i$ with $0 < |{\cal B}_i| < \infty$ is a singleton block.
The ranked frequencies $P^{\downarrow}_1, P^{\downarrow}_2, \ldots $ appear
in the sequence $(P_j)$ in the order in which intervals of these lengths
are discovered by a process of uniform random sampling, as in Kingman's paintbox construction. If $P_* >0$ then in addition to the strictly positive
terms of $P^{\downarrow}_1, P^{\downarrow}_2, \ldots$ the sequence $(P_i)$ also contains infinitely many zeros which correspond to singletons in $\mathbb Pi$. The conditional distribution of $(P_j$) given $(P^{\downarrow}_j)$ can also be described in terms of iteration of a single size-biased pick, defined as follows. For a sequence of non-negative random variables $(X_i)$ with $\sum_i X_{i}\leq 1$
and a random index $J\in \{1,2,\ldots,\infty\}$,
call $X_J$ a {\em size-biased pick} from $(X_i)$ if
$X_J$ has value $X_j$ if $J= j < \infty$ and $X_J = 0$ if $J = \infty$,
with
\begin{equation}
\label{sbpick}
\mathbb P(J = j\, |\, (X_i, i\in \mathbb Nat)) = X_j
~~~( 0 < j < \infty )
\end{equation}d{equation}
(see \cite{GnedinSBP}
for this and another definition of size-biased pick in the case of improper frequencies).
The {\em sequence derived from $(X_i)$ by deletion
of $X_J$ and renormalization} refers to the sequence $(Y_i)$ obtained from $(X_i)$ by
first deleting the $J$th term $X_J$, then closing up the gap if $J\Mathbb Neq\infty$, and finally normalizing each term by $1 - X_J$.
Here by convention, $(Y_i)$ = $(X_i)$ if $X_J= 0$ and $(Y_i)$ is the the zero sequence if $X_J=1$.
Then $P_1$ is a size-biased pick from $(P^{\downarrow}_j)$, $P_2$ is a size-biased pick from the sequence derived from $(P^{\downarrow}_j)$ by deletion of $P_1$ and renormalization, and so on. For this reason, $(P_i)$ is said to be a \emph{size-biased permutation} of $(P^{\downarrow}_i)$.
\paragraph{The two-parameter family}
It was shown in \cite{jp.epe} that for each pair of real parameters $(\alpha,\theta)$ with
\begin{equation}\label{2Prange}
0\leq\alpha <1,~\theta>-\alpha
\end{equation}
the formula
\begin{equation}\label{EPPF2Par}
p_{\alpha,\theta} (\lambda) := {\prod_{i=1}^{k-1}(\theta+i\alpha)\over(\theta+1)_{n -1}}\prod_{j=1}^{k}(1-\alpha)_{\lambda_j - 1}
\end{equation}
where $k = k_\lambda$, $n = n_\lambda$, and
$$(x)_n := x (x+1) \ldots (x+n-1) = \frac{\Gamma(x +n)}{\Gamma(x)}$$
is a rising factorial, defines the EPPF of an exchangeable random partition of positive integers whose block frequencies $(P_i)$ in order
of appearance admit the {\em stick-breaking representation}
\begin{equation}\label{St-Br}
P_i=W_{i}\prod_{j=1}^{i-1} (1-W_j)
\end{equation}d{equation}
for random variables $W_j$ such that
\begin{equation}
\label{wind2}
W_1, W_2, \ldots \mbox{ are mutually independent}
\end{equation}d{equation}
with
\begin{equation}
\label{W-beta}
W_k\ed \beta_{1-\alpha,\theta+k\alpha}
\end{equation}
where $\ed$ indicates equality in distribution, and $\beta_{a,b}$ for $a,b >0$ denotes a random variable
with the beta$(a,b)$ density
\begin{equation}
\label{betadens}
\mathbb P(\beta_{a,b}\in {\rm d}u ) = {\Gamma(a+b) \over \Gamma(a) \Gamma(b) } u ^{a-1}(1-u)^{b-1}{\rm d}u ~~~~~~(0 < u < 1)
\end{equation}d{equation}
which is also characterized by the moments
\begin{equation}
\label{betamoms}
{\mathbb E} [\beta_{a,b}^i (1 - \beta_{a,b})^j ] = \frac{ (a)_i (b)_j }{ (a + b)_{i+j} } ~~~~~~~(i,j = 0,1, 2, \ldots ).
\end{equation}d{equation}
Formula \re{EPPF2Par} also defines an EPPF for $(\alpha,\theta)$ in the range
\begin{equation}\label{alphaneg}
\alpha<0,~\theta=-M \alpha ~{\rm for~some~}M \in \mathbb Nat,
\end{equation}
in which case the stick-breaking representation \re{St-Br} with factors as in \re{W-beta} makes sense for
$1 \le k \le M $, with the last factor $W_M = 1$.
The frequencies $(P_1, \ldots, P_M)$ in this case are a size-biased random permutation of $(Q_1, \ldots, Q_M)$ with
the symmetric Dirichlet distribution with $M$ parameters equal to $\Mathbb Nu:= - \alpha >0$. It is well known that the $Q_i$ can be
constructed as $Q_i = \gamma_{\Mathbb Nu}^{(i)}/\Sigma, 1 \le i \le M$,
where $\Sigma = \sum_{i=1}^M \gamma_\Mathbb Nu^{(i)}$ and the $\gamma_{\Mathbb Nu}^{(i)}$ are independent and identically distributed copies of a gamma variable $\gamma_\Mathbb Nu$ with density
\begin{equation}
\label{gammadens}
\mathbb P( \gamma_\Mathbb Nu \in {\rm d}x ) = \Gamma(\Mathbb Nu)^{-1} x ^{\Mathbb Nu-1} e^{-x} {\rm d}x ~~~~~~( x > 0 ).
\end{equation}d{equation}
As shown by Kingman \cite{MR0526801},
the $(0,\theta)$ EPPF \re{EPPF2Par} for $\alpha = 0, \theta >0$ arises in the limit of random sampling from such symmetric Dirichlet frequencies as $\Mathbb Nu = -\alpha \downarrow 0$ and $M \uparrow \infty$ with
$\Mathbb Nu M = \theta$ held fixed. In this case, the distribution of the partition $\pi_n$ is that determined by the Ewens sampling formula with parameter $\theta$, the residual fractions
$W_i$ in the stick-breaking representation are identically distributed like $\beta_{1,\theta}$, and the ranked frequencies $P^{\downarrow}_i$ can be obtained by normalization of the jumps of a
gamma process with stationary independent increments $(\gamma_\Mathbb Nu, 0 \le \Mathbb Nu \le \theta)$. Perman, Pitman and Yor \cite{ppy92} gave extensions of this description to the case
$0 < \alpha < 1$ when the distribution of ranked frequencies can be derived from the jumps of a stable subordinator of index $\alpha$.
See also \cite{py95pd2,pitman02pk,csp} for further discussion and applications to the description of ranked lengths of excursion intervals of Brownian motion and Bessel processes.
In the limit case when $\Mathbb Nu = - \alpha \to \infty$ and $\theta = M \Mathbb Nu \to \infty$, for a fixed positive integer $M$, the EPPF \re{EPPF2Par}
converges to
\begin{equation}\label{Coupon}
p_{M}
(\lambda):= \frac{ M (M -1) \cdots (M -k+1) }{M ^n}\,,
\end{equation}
corresponding to sampling from $M$ equal frequencies
$$
P_1 = P_2 = \cdots = P_M = 1/M
$$
as in the classical coupon collector's problem with some fixed number $M$ of equally frequent types of coupon.
We refer to the collection of partition structures defined by \re{EPPF2Par} for the parameter
ranges (\ref{2Prange}) and (\ref{alphaneg}),
as well as the limit cases \re{Coupon}, as the
{\em extended two-parameter family}.
\par The partition $\mathbf{0}$ of $\mathbb{N}$ into singletons and the partition $\mathbf{1}$ of $\mathbb{N}$ into a single block
both belong to the closure of the two-parameter family.
As noticed by Kerov \cite{Kerov}, a mixture of these two trivial partitions with mixing proportions $t$ and $1-t$
also belongs to the closure, as is seen
from (\ref{EPPF2Par}) by letting $\alpha \to 1$ and $\theta \to -1$ in such a way that
$(1- \alpha)/(\theta+1)\to t\,$ and $(\theta+\alpha)/(\theta +1)\to 1-t$.
\paragraph{Characterizations by deletion properties}
The main focus of this paper is the following result, which was first conjectured by Pitman \cite{jp.epe}.
For convenience in presenting this result, we impose the following mild {\em regularity condition}
on the EPPF $p$ associated with a partition structure $(\pi_n)$:
\begin{equation}
\label{p221}
p(2,2,1) > 0 \mbox{~ and~ } \lim_{n \rightarrow \infty}p(n) = 0.
\end{equation}
Equivalently, in terms of the frequencies $P_i$ in order of appearance,
\begin{equation}
\label{p221x}
\mathbb P(0 < P_1 < P_1 + P_2 < 1 ) > 0 \mbox{ and } \mathbb P(P_1 = 1) = 0,
\end{equation}
or again, in terms of the ranked frequencies $P^{\downarrow}_i$,
\begin{equation}
\label{p221r}
\mathbb P(0 < P^{\downarrow}_2 , ~P^{\downarrow}_1 + P^{\downarrow}_2 < 1 ) > 0 \mbox{ and } \mathbb P(P^{\downarrow}_1 = 1) = 0.
\end{equation}
Note that this regularity condition does not rule out the case of improper frequencies.
See Section \ref{secfails} for discussion of how the following results can be modified to accomodate partition
structures not satisfying the regularity condition.
\begin{theorem}\label{2pchar1}
Among all partition structures $(\pi_n)$
with
EPPF $p$ subject to {\rm \re{p221}},
the extended two-parameter family is characterized by the following property:
\begin{quote}
if one of $n$ balls is chosen uniformly at random, independently of a random coloring of these balls according to $\pi_n$, then
given the number of other balls of the same color as the chosen ball is $m-1$, for some $1 \le m < n$, the coloring of
the remaining $n-m$ balls is distributed according to $\pi_{n-m}'$ for some sequence of partitions
$(\pi_1',\pi'_2,\ldots)$ which does not depend on $n$.
\end{equation}d{quote}
Moreover, if $(\pi_n)$ has the $(\alpha,\theta)$ EPPF {\rm\re{EPPF2Par}},
then $(\pi_n')$ has the $(\alpha,\theta +\alpha )$ EPPF {\rm\re{EPPF2Par}},
whereas if $(\pi_n)$ has the EPPF {\rm \re{Coupon}} for some $M$,
then the EPPF of $(\pi_n')$ has the same form except with $M$ decremented by $1$.
\end{equation}d{theorem}
\Mathbb Noindent
Note that it is not assumed as part of the property that $(\pi_n')$ is a
partition structure. Rather, this is implied by the conclusion.
Our formulation of Theorem \ref{2pchar1} was inspired by Kingman
\cite{MR0526801}
who assumed also that $\pi_n' \stackrel{d}{=} \pi_n$ for all $n$. The conclusion then holds with $\alpha = 0$, in which case the distribution of
$\pi_n$ is that determined by the Ewens sampling formula from population genetics.
In Section \ref{sect:ExchPart} we offer a proof of Theorem \ref{2pchar1} by purely combinatorial methods.
Some preliminary results which we develop in Section \ref{sect:PEP} allow
Theorem \ref{2pchar1} to be reformulated in terms of frequencies as in the following Corollary:
\begin{corollary}\label{2pcharcor}
Let the asymptotic frequencies $(P_i)$ of an exchangeable random partition of positive integers $\mathbb Pi$ be represented in the
stick-breaking form {\rm \re{St-Br}} for some sequence of random variables $W_1<1, W_2, \ldots$.
The condition
\begin{equation}
\label{wind1}
W_1 \mbox{ is independent of } (W_2, W_3, \ldots )
\end{equation}d{equation}
obtains if and only if
\begin{equation}
\label{windm}
\mbox{ the $W_i$ are mutually independent. }
\end{equation}d{equation}
If in addition to $(\ref{wind1})$ the regularity condition {\rm \re{p221}} holds, then
$\mathbb Pi$ is governed by the extended two-parameter family, either with $W_i \ed \beta_{1-\alpha,\theta+i\alpha}$, or with $W_i = 1/(M - i +1)$ for $1 \le i \le M$, as in the limit case {\rm \re{Coupon}}, for some
$M = 3,4, \ldots$.
\end{equation}d{corollary}
\Mathbb Noindent
The characterization of the two-parameter family using \re{windm} rather than the weaker condition \re{wind1}
was provided by Pitman \cite{jp.isbp}.
As we show in Section \ref{sect:ExchPart}, it is possible to derive \re{windm} directly from \re{wind1}, without passing via Theorem \ref{2pchar1}.
The law of frequencies $(P_i)$ defined by the stick-breaking scheme
\re{St-Br} for independent factors $W_i$ with
$W_i \ed \beta_{1-\alpha,\theta+i\alpha}$ is
known as the the two-parameter Griffiths-Engen-McCloskey distribution,
denoted ${\rm GEM}(\alpha,\theta)$.
The property of the independence of residual proportions $W_i$, also known as
{\it complete neutrality}, has also been studied extensively in connection
with finite-dimensional Dirichlet distributions \cite{BobWes}.
The above results can also be expressed in terms of ranked frequencies. Recall that the distribution of ranked frequencies $(P^{\downarrow}_k)$ of an $(\alpha,\theta)$-partition is known as the
{\em two-parameter Poisson-Dirichlet distribution ${\rm PD}(\alpha,\theta)$}.
According the the previous discussion, a random sequence $(P^{\downarrow}_k)$ with
${\rm PD}(\alpha,\theta)$ distribution is obtained by ranking
a sequence $(P_i)$ with ${\rm GEM}(\alpha,\theta)$ distribution.
The ${\rm PD}(\alpha,\theta)$ distribution was systematically studied in \cite{py95pd2},
and has found numerous further applications to random trees and associated processes of fragmentation and coagulation
\cite{jp97cmc,hpw07,BertoinCoagFrag}.
\begin{corollary}\label{corextra}
Let $(P^{\downarrow}_k)$ be a decreasing sequence of ranked frequencies subject to the regularity condition {\rm(\ref{part1})} and {\rm(\ref{p221r})}.
For $P^{\downarrow}_J$, a size-biased pick from $(P^{\downarrow}_k)$, let $(Q^{\downarrow}_k)$ be derived from
$(P^{\downarrow}_k)$ by deletion of $P^{\downarrow}_J$ and renormalization. The random variable $P^{\downarrow}_J$ is independent of the sequence $(Q^{\downarrow}_k)$ if and only if either the distribution of
$(P^{\downarrow}_k)$ is ${\rm PD}(\alpha,\theta)$ for some $(\alpha,\theta)$, or $P^{\downarrow}_k = 1/M $ for all $1 \le k \le M $, for some $M
\ge 3$. In the former case, the distribution of
$(Q^{\downarrow}_k)$ is ${\rm PD}(\alpha,\theta + \alpha)$,
whereas in the latter case, the deletion and renormalization simply decrements $M$ by one.
\end{equation}d{corollary}
The `if' part of this Corollary is Proposition 34 of Pitman-Yor \cite{py95pd2}, while the `only if' part follows
easily from Corollary \ref{2pcharcor}, using Kingman's paintbox representation.
\section{Partially Exchangeable Partitions}\label{sect:PEP}
We start by recalling from \cite{jp.epe} some basic properties of {\em partially exchangable partitions of positive
integers}, which are consistent sequences $\mathbb Pi = (\mathbb Pi_n)$, where $\mathbb Pi_n$ is a partition of $[n]$ whose probability
distribution is of the form \re{EPF} for some function $p = p(\lambda)$ of compositions $\lambda$ of positive
integers. The consistency of $\mathbb Pi_n$ as $n$ varies amounts to the {\em addition rule}
\begin{equation}\label{add-rule}
p(\lambda)=\sum_{j =1}^{k + 1 } p(\lambda^{(j)}),
\end{equation}
where $k = k_\lambda$ is the number of parts of $\lambda$, and $\lambda^{(j)}$ is the composition of $n_\lambda +1$ derived from $\lambda$ by incrementing $\lambda_j$ to $\lambda_j+1$,
and leaving all other components of $\lambda$ fixed. In particular,
for $j = k_\lambda +1$ this means appending a $1$ to $\lambda$.
There is also the normalization condition $p(1)=1$. To illustrate \re{add-rule} for $\lambda = (3,1,2)$:
$$
p(3,1,2) = p(4,1,2) + p(3,2,2) + p(3,1,3) + p(3,1,2,1).
$$
The following proposition recalls the analog of Kingman's representation
for partially exchangeable partitions:
\begin{proposition}\label{pep} {\rm (Corollary 7 from \cite{jp.epe})}
Every partially exchangeable partition of positive integers
$\mathbb Pi$ is such that for each $k \ge 1$, the $k$th block ${\cal B}_k$ has an almost sure limit frequency $P_k$.
The partition probability function $p$ can then be presented as
\begin{equation}\label{Eqn:EPPFMomentFormula}
p(\lambda)
=
{\mathbb E}\left[ \prod_{i=1}^k
P_i^{\lambda_i-1}\prod_{j=1}^{k-1} R_j \right] ,
\end{equation}d{equation}
where $k =k_\lambda$ and $R_j := (1 - P_1 - \cdots -P_j)$.
Alternatively, in terms of
the residual fractions $W_k$ in the stick-breaking representation {\rm \re{St-Br}}:
\begin{equation}\label{Eqn:EPPFMomentFormula2}
p(\lambda) = {\mathbb E}\left[\prod_{i=1}^k W_i^{\lambda_{i}-1}\overline{W}_i^{\Lambda_{i+1}}\right],
\end{equation}d{equation}
where $\overline{W}_i:= 1- W_i$, $\Lambda_j:=\sum_{i\geq j}\lambda_i$.
This formula sets up a correspondence between the probability distribution of $\mathbb Pi$,
encoded by the partition probability function $p$,
and an arbitrary joint distribution of a sequence of random variables
$(W_1, W_2, \ldots)$ with $0 \le W_i \le 1$ for all $i$.
\end{equation}d{proposition}
In terms of randomly coloring a row of $n_\lambda$ balls, the product whose expectation appears in \re{Eqn:EPPFMomentFormula2} is the conditional probability given $W_1, W_2, \ldots$ of the event that
the first $\lambda_1$ balls are colored one color, the next $\lambda_2$ balls another color, and so on. So \re{Eqn:EPPFMomentFormula2} reflects the fact that conditionally given
$W_1, W_2, \ldots$ the process of random coloring of integers occurs according to the following {\rm residual allocation scheme}
\cite[Construction 16]{jp.epe}:
\begin{quote}
Ball $1$ is painted a first color,
and so is each subsequent ball according to a sequence of independent trials with probability $W_1$ of painting with color 1.
The set of balls so painted defines the first block ${\cal B}_1$ of $\mathbb Pi$.
Conditionally given ${\cal B}_1$, the first unpainted ball is painted a second color,
and so is each subsequent unpainted ball according to a sequence of independent trials with probability $W_2$ of painting with color 2.
The balls colored 2 define ${\cal B}_2$, and so on.
Given an arbitrary sequence of random variables $(W_k)$ with $0 \le W_k \le 1$, this coloring scheme shows how to construct a partially exchangeable partition of $\mathbb N$ whose asymptotic block
frequencies are given by the stick-breaking scheme \re{St-Br}.
\end{equation}d{quote}
Note that the residual allocation scheme terminates at the first $k$, if any, such that $W_k = 1$, by painting all remaining balls color $k$.
The values of $W_i$ for $i$ larger than such a $k$ have no effect on the construction of $\mathbb Pi$, so cannot be recovered from its almost sure limit frequencies.
To ensure that a unique joint distribution of $(W_1, W_2, \ldots)$ is associated with each $p$,
the convention may be adopted that the sequence $(W_i)$ terminates at the first $k$ if any such that $W_k = 1$.
This convention will be adopted in the following discussion.
For $W_i$ which are independent, formula (\ref{Eqn:EPPFMomentFormula2}) factorizes as
\begin{equation}\label{Eqn:EPPFMomentFormula22}
p(\lambda) = \prod_{i=1}^k
\mathbb E( W_i^{\lambda_{i}-1}\overline{W}_i^{\Lambda_{i+1}} ) .
\end{equation}d{equation}
In particular, for independent $W_i$ with the beta distributions (\ref{W-beta}),
this formula is readily evaluated using \re{betamoms}
to obtain \re{EPPF2Par}. Inspection of \re{EPPF2Par} shows that this function of compositions $\lambda$ is a symmetric function of its parts. Hence the
associated random partition $\mathbb Pi$ is exchangeable.
There is an alternate sequential construction of the two-parameter family of partitions which has become known as the ``Chinese Restaurant Process''
(see \cite{csp}, {Chapter 3}).
Instead of coloring rows of balls,
imagine customers entering a restaurant with an unlimited number of tables. Initially customer $1$ sits at table $1$. At stage $n$, if there are $k$ occupied tables, the $i$th of them occupied by $\lambda_i$
customers for $1 \le i \le k$, customer $n+ 1$ sits at one of the
previously occupied tables with probability $(\lambda_i - \alpha)/(n + \theta)$, and occupies a new table $k+1$ with probability
$(\theta + k\alpha)/(n + \theta)$.
It is then readily checked that for each partition of $[n]$ into blocks $B_i$ with $|B_i| = \lambda_i$, after $n$ customers labeled by $[n]$ have entered the restaurant, the probability
that those customers
labeled by $B_i$ sat at table $i$ for each $1 \le i \le k_\lambda$ is given by the product formula \re{EPPF2Par}.
Moreover, the stick-breaking description of the limit frequencies $P_i$ is readily derived
from the P{\'o}lya urn-scheme description of exchangeable trials which given a beta$(a,b)$-distributed variable $S$,
are independent with success probability $S$.
Continuing the consideration of a partially exchangeable partition $\mathbb Pi$ of positive integers, we record the following Lemma.
\begin{lemma}\label{p.prime}
Let $\mathbb Pi$ be a partially exchangeable random partition of $\mathbb N$ with partition probability function $p$, and with blocks ${\cal B}_1, {\cal B}_2, \ldots$
and residual frequencies $W_1, W_2, \ldots$
such that $W_1 < 1$ almost surely.
Let $\mathbb Pi '$ denote the partition of $\mathbb N$ derived from $\mathbb Pi$ by deletion of the block ${\cal B}_1$ containing $1$ and re-labeling of $\mathbb N - {\cal B}_1$ by the increasing bijection with $\mathbb N$.
Then the following hold:
\begin{itemize}
\item[\rm(i)] The partition $\mathbb Pi'$ is partially exchangeable, with partition probability function
\begin{equation}
\label{piprime}
p'(\lambda_2,\ldots,\lambda_k)=\sum_{\lambda_1=1}^\infty {\lambda_1+\ldots+\lambda_k-2\choose \lambda_1-1}p(\lambda_1,\lambda_2,\ldots,\lambda_k)
\end{equation}
and residual frequencies $W_2, W_3, \ldots$.
\item[\rm(ii)] If $\mathbb Pi$ is exchangeable, then so is $\mathbb Pi'$.
\item[\rm(iii)] For $1 \le m \le n$
\begin{equation}
\label{qnm}
q(n:m) := \mathbb P( {\cal B}_1\cap [n]=[m]) = {\mathbb E}(W_1^{m-1}\overline{W}_1^{n-m}),
\end{equation}
and there is the addition rule
\begin{equation}\label{add-rule-q}
q(n:m)=q(n+1:m+1)+q(n+1:m).
\end{equation}
\item[\rm(iv)] Let $T_n := \inf \{m : |[n+m]\setminus{\cal B}_1| = n\}$ which is the number of balls of the first
color preceding the $n$th ball not of the first color. Then
\begin{equation}
\label{negbin}
\mathbb P (T_n = m) = {m+n-2\choose m-1} q(n+m:m),
\end{equation}
and consequently
\begin{equation}\label{sumone}
\sum_{m=1}^\infty {m+n-2\choose m-1} q(n+m:m) =1.
\end{equation}d{equation}
\end{equation}d{itemize}
\end{equation}d{lemma}
\proof
Formula \re{qnm} is read from the general construction of ${\cal B}_1$ given $W_1$ by assigning each $i \ge 2$
to ${\cal B}_1$ independently with the same probability $W_1$.
The formulas \re{piprime} and \re{negbin} are then seen to be marginalizations of the following
expression for the joint distribution of $T_n$ and $\mathbb Pi_n'$, the restriction of $\mathbb Pi'$ to $[n]$:
\begin{equation}
\label{tnprime}
\mathbb P( T_n = m, \mathbb Pi_n' = (C_1, \ldots, C_{k-1} ) )
= {m+n-2\choose m-1} q(n+m:m) p(m, |C_1|, \ldots, |C_{k-1}|)
\end{equation}
for every partition $(C_1, \ldots, C_{k-1})$ of $[n]$. To check \re{tnprime}, observe that the
event in question occurs if and only if $\mathbb Pi_{n+m} = (B_1, \ldots, B_{k})$ for some blocks $B_i$ with
$|B_1| = m$ and $|B_i| = |C_{i-1}|$ for $2 \le i \le k$. Once $B_1$ is chosen, each $B_i$ for
$2 \le i \le k$ is the image of $C_{i-1}$ via the increasing bijection from $[n]$ to $[n+m] \setminus B_1$.
For prescribed $C_{i-1}, 2 \le i \le k$, the choice of $B_1 \subset [n+m]$ is arbitrary subject to the constraint that
$1 \in B_1$ and $n+m \Mathbb Notin B_1$. The number of choices is the binomial coefficient in \re{tnprime}, so the
conclusion is evident.
\end{equation}dpf
The connection between Theorem \ref{2pchar1} and Corollary \ref{2pcharcor} is established by the following Lemma:
\begin{lemma}
\label{lemma4equiv}
Let $\mathbb Pi$ be a partially exchangeable partition of $\mathbb N$ with residual frequencies $W_i$ such that
$\mathbb P (W_1 < 1 ) = 1$, with the convention that the sequence terminates at the first $k$ (if any) such that $W_k = 1$,
so the joint distribution of $(W_i)$ is determined uniquely by the partition probability function $p$ of $\mathbb Pi$,
and vice versa, according to formula
{\rm \re{Eqn:EPPFMomentFormula2}}.
For ${\cal B}_1$ the first block of $\mathbb Pi$ with frequency $W_1$,
let $\mathbb Pi'$ be derived from $ \mathbb Pi$ by deleting block ${\cal B}_1$
and relabeling the remaining elements
as in {\rm Lemma \ref{p.prime}}.
The following four conditions on $\mathbb Pi$ are equivalent:
\begin{itemize}
\item[\rm(i)] $W_1$ is independent of $(W_2, W_3, \ldots)$.
\item[\rm(ii)] The partition probability function $p$ of $\mathbb Pi$ admits
a factorization of the following form, for all compositions $\lambda$ of positive integers with $k \ge 2$ parts:
\begin{equation}
\label{factor}
p(\lambda) = q( n_\lambda: \lambda_1) p'(\lambda_2, \ldots, \lambda_k)
\end{equation}
for some non-negative functions $q(n:m)$ and $p'(\lambda_2, \ldots, \lambda_k)$.
\item[\rm(iii)]
For each $1 \le m < n$, the conditional distribution of $\mathbb Pi_{n-m}'$ given $|{\cal B}_1 \cap [n]| = m$ depends only on $n-m$.
\item[\rm(iv)]
The random set ${\cal B}_1$ is independent of the random partition $\mathbb Pi'$ of $\mathbb N$.
\end{equation}d{itemize}
Finally, if these conditions hold, then {\rm (ii)} holds in particular for $q(n:m)$ as in {\rm \re{qnm}}
and $p'(\lambda_2, \ldots, \lambda_k)$
the partition probability function of $\mathbb Pi'$.
\end{equation}d{lemma}
\proof
That (i) implies (ii) is immediate by combination of the moment formula
\re{Eqn:EPPFMomentFormula2}, \re{piprime} and
\re{qnm}.
Conversely, if (ii) holds for some $q(n:m)$ and $p'(\lambda_2, \ldots, \lambda_k)$, Lemma \ref{p.prime}
implies easily that (ii) holds for $q$ and $p'$ as in that
Lemma. So (ii) gives a formula of the form
\begin{equation}
\label{fg}
{\mathbb E} [ f(W_1) g( W_2, W_3, \ldots ) ] = {\mathbb E} [ f(W_1) ] {\mathbb E} [ g(W_2, W_3, \ldots )],
\end{equation}
where $g$ ranges over a collection of bounded measurable functions whose expectations determine the law of
$W_2, W_3, \ldots$, and for the $g$ associated with $\lambda_2, \ldots, \lambda_k$, the function $f(w)$
ranges over the polynomials $w^{m-1} (1-w)^n$ where $m = \lambda_1 \in \mathbb N$
and $n = n_\lambda -\lambda_1 = \sum_{j=2}^k \lambda_j$. But linear combinations of these polynomials can be used to
uniformly approximate any bounded continuous function of $w$ on $[0,1]$ which vanishes in a neighbourhood of $1$.
It follows that \re{fg} holds for all such $f$, for each $g$, hence the full independence condition (i).
Lastly, the equivalence of (ii), (iii) and (iv) is easily verified.
\end{equation}dpf
\section{Exchangeable Partitions}\label{sect:ExchPart}
For a block $B$ of a random partition $\mathbb Pi_n$ of $[n]$ with $|B| = m$, let $\mathbb Pi_n \setminus B$ denote the partition of $[n-m]$ obtained by first deleting the
block $B$ of $\mathbb Pi_n$, then mapping the restriction of $\mathbb Pi_n$ to $[n] \setminus B$ to a partition of $[n-m]$ via the increasing bijection
between $[n] \setminus B$ and $[n-m]$. In terms of a coloring of $n$ balls in a row, this means deleting all $m$ balls
of some color, then closing up the gaps between remaining balls, to obtain a coloring of $n-m$ balls in a row.
Theorem \ref{2pchar1} can be formulated a little more sharply as follows:
\begin{theorem}\label{Kchar1}
Among all exchangeable partitions $(\mathbb Pi_n)$ of positive integers with EPPF $p$ subject to {\rm\re{p221}},
the extended two-parameter family is characterized by the following property:
\begin{quote}
if ${\cal B}_{n1}$ denotes the random block of $\mathbb Pi_n$
containing $1$, then for each $1 \le m < n$,
conditionally given ${\cal B}_{n1}$ with $|{\cal B}_{n1}| = m$,
the partition $\mathbb Pi_n \setminus {\cal B}_{n1}$ has the same distribution as $\mathbb Pi_{n-m}'$
for some sequence of partitions $\mathbb Pi'_1,\mathbb Pi'_2,\ldots$ which does not depend on $n$.
\end{equation}d{quote}
Moreover, if $(\mathbb Pi_n)$ is an $(\alpha,\theta)$ partition, then we can take for $(\mathbb Pi_n')$
the exchangeable $(\alpha,\theta +\alpha )$ partition of $\mathbb Nat$.
\end{equation}d{theorem}
For an arbitrary partition $\mathbb Pi_n$ of $[n]$ with blocks listed in the order of appearance,
define $J_n$ as the index of the block containing an element chosen from $[n]$ uniformly
at random, independently of $\mathbb Pi_n$.
We call the block ${\cal B}_{nJ_n}$ a {\it size-biased pick} from the sequence of blocks.
Note that this definition agrees with (\ref{sbpick}) in the sense that the number
$|{\cal B}_{nJ_n}|/n$ is a size-biased pick from the numerical sequence $(|{\cal B}_{nj}|/n, ~j=1,2,\dots)$,
because
given a sequence of blocks of partition $\mathbb Pi_n$
the value $J_n=j$ is taken with probability $|{\mathcal B}_{nj}|/n$.
Assuming $\mathbb Pi_n$ exchangeable, the size of the block $|{\mathcal B}_{n1}|$
has the same distribution as $|{\cal B}_{nJ_n}|$ conditionally given the ranked sequence of block-sizes,
and the reduced partitions $\mathbb Pi_n\setminus{\cal B}_{n1}$ and $\mathbb Pi_n\setminus{\cal B}_{nJ_n}$
also have the same distributions.
The equivalence of Theorem \ref{2pchar1} and Theorem \ref{Kchar1} is evident from these considerations.
We turn to the proof of Theorem \ref{Kchar1}.
The condition considered in Theorem \ref{Kchar1} is just that considered in Lemma \ref{lemma4equiv}(iii),
so we can work with the equivalent factorization condition \re{factor}.
We now invoke the symmetry of the EPPF for an exchangeable $\mathbb Pi$.
Suppose that an EPPF $p$ admits the factorization \re{factor}, and re-write
the identity \re{factor} in the form
$$p(m,\lambda)={q(|\lambda|+m:m)\over q(|\lambda|+1:1)} p(1,\lambda).$$
For this expression we must have non-zero denominator, but this is assured by $\mathbb P(0 < W_1 < 1) >0$, which
is implied by the regularity condition (\ref{p221}).
Instead of part $m$ in $p(m,\lambda)$, we have now $1$ in $p(1,\lambda)$.
But $p$ is symmetric, hence
we can iterate, eventually reducing each part to $1$.
\par
Let $\lambda=(\lambda_1,\ldots,\lambda_k)$ be a generic composition, and denote $\Lambda_j=\lambda_j+\cdots+\lambda_k$ the tail
sums, thus $\Lambda_1=|\lambda|$. Iteration yields
\begin{equation}\label{iterate}
p(\lambda)={q(\Lambda_1:\lambda_1)\over q(1+\Lambda_2:1)}{q(1+\Lambda_2:\lambda_2)\over q(2+\Lambda_3:1)}\cdots
{q(k-2+\Lambda_{k-1}:\lambda_{k-1})\over q(k-1+\Lambda_k:1)}
{q(k-1+\Lambda_{k}:\lambda_{k})\over q(k:1)}\,p(1^k),
\end{equation}
where $p(1^k)$ is the probability of the singleton partition of $[k]$.
This leads to the following lemma, which is a simplification of \cite[Lemma 12]{jp.isbp}:
\begin{lemma} \label{beta}
Suppose that an EPPF $p$ satisfies the factorization condition {\rm (\ref{factor})} and
the regularity condition {\rm (\ref{p221})}.
Then
\begin{itemize}
\item[{\rm (i)}]
either
$$q(n:m)={(a)_{m-1}(b)_{n-m}\over (a+b)_{n-1}}$$
for some $a,b>0$, corresponding to $W_1$ with beta$(a,b)$ distribution,
\item[{\rm (ii)}] or
$$q(n:m)= c^{m-1}(1-c)^{n-m}$$
for some $0<c<1$, corresponding to $W_1 = c$, in which case necessarily $c = 1/M$ for some $M \ge 3$.
\end{equation}d{itemize}
\end{equation}d{lemma}
\proof
By symmetry and the assumption that $p(2,2,1) >0$, it is easily seen from Kingman's paintbox representation that
for each $m = 1,2, \ldots$ there is some composition $\mu$ of $m$ such that
$$
p(3,2,\mu)=p(2,3,\mu)>0,
$$
where for instance $(3,2,\mu)$ means the composition of $5 + m$ obtained by concatenation of $(3,2)$ and $\mu$.
Indeed, it is clear that one can take either $\mu = 1^m$ or $\mu$ to be a single part of size $m$, according to
whether the probability of at least three non-zero frequencies is zero or greater than zero.
Applying (\ref{iterate}) for suitable $k \ge 3$ with $p(1^k) >0$, and cancelling some common factors of
the form $q(n',m')$, which are all strictly positive because $p(2,2,1) >0$ implies $\mathbb P(0 < W_1 < 1) >0$, we
see that for every $m = 1,2, \ldots$
\begin{equation}\label{rec-start}
{q(m+5:3)q(m+3:2)\over q(m+3:1)}={q(m+5:2)q(m+4:3)\over q(m+4:1)}.
\end{equation}d{equation}
We have by the addition rule (\ref{add-rule-q})
$$q(m+1:2)=q(m:1)-q(m+1:1),~~~q(m+2:3)=q(m:1)-2q(m+1:1)+q(m+2:1),$$
and introducing variables
$x_m=q(m:1)$, $n=m+2$
$$ {(x_{n+1}-2x_{n+2}+x_{n+3})(x_n-x_{n+1})\over x_{n+1}}={(x_{n+2}-x_{n+3})(x_n-2x_{n+1}+x_{n+2})\over x_{n+2}}.$$
The recursion is homogeneous, to pass to inhomogeneous variables
divide both sides of the equality by $x_n$, then
set $y_n:=x_{n+1}/x_n$ and rewrite as
$$(1-2y_{n+1}+y_{n+2}y_{n+1})(1-y_n)=(1-y_{n+2})(1-2y_n +y_ny_{n+1}),$$
which simplifies as
$$-2y_{n+1}+y_{n+1}y_{n+2}+y_ny_{n+1}=-y_n-y_{n+2}+2y_n y_{n+2}.$$
Finally,
use substitution
$$y_n=1-{1\over z_n}$$
to arrive at
$${z_n-2 z_{n+1}+z_{n+2}\over z_nz_{n+1}z_{n+2}}=0.$$
From this, $z_n$ is a linear function of $n$, which must be nondecreasing to agree with $0<y_n<1$.
\par If $z_n$ is not constant, then going back to $x_n$'s we obtain
$$q(n:1)= c_0 {(b)_{n-1}\over (a+b)_{n-1}}, ~~~n\geq 3,$$
for some $a,b,c_0$, where the factor $c_0$ appears since the relation (\ref{rec-start}) is homogeneous.
It is seen from the moments representation
$$q(n:1)=\int_{[0,1]} (1-x)^{n-1}{\mathbb P}({P}_1\in {\rm d}x),~~~~n\geq 3,$$
that when $a,b$ are fixed, the factor $c_0$ is determined from the normalization by choosing a value of ${\mathbb P}({P}_1=1)$.
The condition $p(n)\to 0$ means that ${\mathbb P}({P}_1=1)=0$, in which case
$c_0=1$ and the distribution of ${P}_1$ is beta$(a,b)$ with some positive $a,b$.
\par If $(z_n, ~n\geq 3)$ is a constant sequence, then
$q(n:1)$ is a geometric progression, and a similar argument shows that the case (ii) prevails.
That $c = 1/M$ for some $M \ge 3$ is quite obvious: the only way that a size-biased choice of a frequency can be constant
is if there are $M$ equal frequencies for some $M \ge 1$. The regularity assumption
\re{p221} rules out the cases $M =1,2$.
\end{equation}dpf
\Mathbb Noindent
\paragraph{Proof of Theorem \ref{Kchar1}}
In the case (i) of Lemma \ref{beta}, substituting in (\ref{iterate}) yields
$$\frac{p(\lambda)}{p(1^k)} ={(a)_{\lambda_1-1}(b)_{\Lambda_2}\over(a+b)_{\Lambda_1-1}}{(a+b)_{\Lambda_2}\over(b)_{\Lambda_2}}
{(a)_{\lambda_2-1}(b)_{\Lambda_3+1}\over(a+b)_{\Lambda_2}}{(a+b)_{\Lambda_3+1}\over(b)_{\Lambda_3+1}}\cdots
{(a)_{\lambda_k-1}(b)_{k-1}\over (a+b)_{\Lambda_k+k-2}}{(a+b)_{k-1}\over (b)_{k-1}},
$$
provided $p(1^k) >0$. After cancellation this becomes
$$\frac{p(\lambda)}{p(1^k)} ={(a+b)_{k-1} \over (a+b)_{n-1}}\prod_{j=1}^k (a)_{\lambda_j-1},$$
where $n=\Lambda_1=\lambda_1+\ldots+\lambda_k=|\lambda|$.
Specializing,
$${p(2,1^{k-1})\over p(1^k)}={a\over a+b+k-1}$$
and using the addition rule \re{add-rule}
$$p(1^k)=k p(2,1^{k-1})+p(1^{k+1}),$$
we obtain the recursion
$${p(1^{k+1})\over p(1^k)}={a+b+k(1-a)-1\over a+b+k-1},~~~p(1)=1.$$
Now (\ref{EPPF2Par}) follows readily by re-parametrisation $\theta=a+b-1,\,\alpha=1-a$.
\par The case (ii) of Lemma \ref{beta} is even simpler, as it is immediate that $W_1 = 1/M$ implies that the
partition is generated as if by coupon collecting with $M$ equally frequent coupons.
\end{equation}dpf
\paragraph{Proof of Corollary \ref{2pcharcor}}
As observed earlier, Corollary \ref{2pcharcor} characterizing the extended two-parameter family by the condition that
\begin{equation}
\label{w123}
W_1 \mbox{ and } (W_2, W_3, \ldots) \mbox{ are independent}
\end{equation}
can be read from Theorem \ref{Kchar1} and Lemma \ref{lemma4equiv}.
We find it interesting nonetheless to provide another proof of Corollary \ref{2pcharcor} based on analysis of the limit frequencies rather than the EPPF.
This was in fact the first argument we found, without which we might not have persisted with the algebraic approach of the previous section.
Suppose then that $W_1,W_2, W_3, \ldots$ is the sequence of residual fractions associated with an EPPF $p$, and that \re{w123} holds.
The symmetry condition $p(r+1,s+1) = p(s+1,r+1)$
and the moment formula \re{Eqn:EPPFMomentFormula2} give
\begin{equation}
\label{momeq}
{\mathbb E}(W_1^{r} \overline{W}_1^{s+1}){\mathbb E}( W_2^{s}) = {\mathbb E}(W_1^{s} \overline{W}_1^{r+1}){\mathbb E}( W_2^{r})
\end{equation}
for non-negative integers $r$ and $s$. Setting $r=0$, this expresses moments of $W_2$ in terms of the moments of $W_1$.
So the distribution of $W_1$ determines that of $W_2$.
Assume now the regularity condition {\rm (\ref{p221})}. According to Lemma \ref{beta} we are reduced either to the case
with $M$ equal frequencies with sum $1$, or to the case where $W_1$ has a beta distribution, and hence so does $W_2$, by
consideration of \re{momeq}. There is nothing more to discuss in the first case, so we assume for the rest of this section
that
\begin{equation}
\label{beta12}
\mbox{ each of $W_1$ and $W_2$ has a non-degenerate beta distribution, with possibly different parameters.}
\end{equation}
Recall that
$$
P_1 = W_1 \mbox{ and } P_2 = (1-W_1) W_2.
$$
As observed in \cite{jp.isbp},
$$
\mbox{the conditional distribution of $(P_3, P_4, \ldots)$ given $P_1$ and $P_2$ depends symmetrically on $P_1$ and $P_2$.}
$$
This can be seen from Kingman's paintbox representation, which implies that conditionally given $P^{\downarrow}_1, P^{\downarrow}_2, \ldots$, as well as $P_1$ and $P_2$, the sequence $(P_3, P_4, \ldots)$
is derived by a process of random sampling from the frequencies $(P^{\downarrow}_i)$ with the terms $P_1$ and $P_2$ deleted. No matter what $(P^{\downarrow}_i)$ this process depends
symmetrically on $P_1$ and $P_2$, so the same is true without the extra conditioning on $(P^{\downarrow}_i)$.
Since $P_1+P_2$ is a symmetric function of $P_1$ and $P_2$, and $(W_3,W_4, \ldots$) is a measurable function of $P_1 + P_2$ and $(P_3, P_4, \ldots)$,
$$
\mbox{ the conditional distribution of $W_3,W_4, \ldots$ given $(P_1, P_2)$ depends symmetrically on $P_1$ and $P_2$. }
$$
The condition that $W_1$ is independent of $(W_2,W_3, W_4, \ldots)$ implies easily that
\begin{quote}
$W_1$ is conditionally independent of $(W_3,W_4, \ldots)$ given $W_2$.
\end{equation}d{quote}
Otherwise put:
\begin{quote}
$P_1$ is conditionally independent of $(W_3,W_4, \ldots)$ given $P_2/(1-P_1)$,
\end{equation}d{quote}
hence by the symmetry discussed above
\begin{quote}
$P_2$ is conditionally independent of $(W_3, W_4, \ldots)$ given $P_1/(1-P_2)$.
\end{equation}d{quote}
Let $X:= P_2/(1-P_1)$, $Y:= P_1/(1-P_2)$ and $Z:= (W_3,W_4, \ldots)$. Then we have both
\begin{equation}
\label{XZY}
\mbox{ $X$ is conditionally independent of $Z$ given $Y$,}
\end{equation}
and
\begin{equation}
\label{YZX}
\mbox{ $Y$ is conditionally independent of $Z$ given $X$,}
\end{equation}
from which it follows under suitable regularity conditions (see Lemma \ref{JPLemma} below) that
\begin{equation}
\label{XYZ}
\mbox{ $(X,Y)$ is independent of $Z$, }
\end{equation}
meaning in the present context that
\begin{equation}
\label{WWW}
\mbox{ $W_1$, $W_2$ and $(W_3, W_4, \ldots)$ are independent. }
\end{equation}
Lauritzen \cite[Proposition 3.1]{Lauritzen96} shows that \re{XZY} and \re{YZX} imply \re{XYZ}
under the assumption that $(X,Y,Z)$ has a positive and continuous joint density relative to a product measure.
From (\ref{beta12}) and strict positivity of the beta densities on $(0,1)$,
we see that $(X,Y)$ has a strictly positive and continuous density relative to Lebesgue measure on $(0,1)^2$.
We are not in a position to assume that $(X,Y,Z)$ has a density relative to a product measure. However, the passage from
\re{XZY} and \re{YZX} to \re{XYZ} is justified by Lemma \ref{JPLemma} below without need for a trivariate density.
So we deduce that \re{WWW} holds.
By Lemma \ref{p.prime}, $(W_2, W_3, \ldots)$ is the sequence of residual fractions of an exchangeable partition $\mathbb Pi'$, and $W_2$ has a beta density. So either $W_3 = 1$ and we are in the case
\re{alphaneg} with $M = 3$, or $W_3$ has a beta density, and the previous argument applies to show that
\begin{quote}
$W_1$, $W_2$, $W_3$ and $( W_4, W_5, \ldots)$ are independent.
\end{equation}d{quote}
Continue by induction to conclude the independence of $W_1, W_2, \ldots W_k$ for all $k$ such that $p(1^k) >0$.
\end{equation}dpf
\begin{lemma}\label{JPLemma}
Let $X,Y$ and $Z$ denote random variables with values in arbitrary measurable spaces, all defined on a common probability space,
such that {\rm \re{XZY}} and {\rm \re{YZX}} hold.
If the joint distribution of the pair $(X,Y)$ has a strictly positive probability density
relative to some product probability measure, then {\rm \re{XYZ}} holds.
\end{equation}d{lemma}
\proof
Let $p(X,Y)$ be a version of $\mathbb P(Z\in B \mid X,Y)$ for $B$ a measurable set in
the range of $Z$.
By standard measure theory (e.g. Kallenberg \cite[6.8]{KallenbergFMP}) the first conditional independence assumption gives
$\mathbb P(Z\in B\mid X,Y) = \mathbb P(Z \in B \mid X)$ a.s. so that
\begin{quote}
$p(X,Y)=g(X)$ a.s. for some measurable function $g$.
\end{equation}d{quote}
Similarly from the second conditional independence assumption,
\begin{quote}
$p(X,Y)=h(Y)$ a.s. for some measurable function $h$,
\end{equation}d{quote}
and we wish to conclude that
\begin{quote}
$p(X,Y) = c $ a.s. for some constant $c$.
\end{equation}d{quote}
To complete the argument it suffices to draw this conclusion from the above two assumptions
about a jointly measurable function $p$, with $(X, Y)$ the identity map on
the product
space of pairs $\mathcal{X} \times \mathcal{Y}$, and the two
almost sure equalities holding with respect to some probability measure $P$ on this space,
with $P$ having a strictly positive density relative to a product probability measure $\mu \otimes \Mathbb Nu$.
Fix $u \in (0,1)$, from the previous assumptions it follows that
\begin{equation}\label{JPLemma1}
\{ p(X,Y) > u\} = \{X \in A_u\} = \{Y \in C_u\} ~~~~~ {\rm a.s.}
\end{equation}d{equation}
for some measurable sets $A_u$, $C_u$, whence
\begin{equation}\label{JPLemma2}
\{ p(X,Y) > u\} = \{X \in A_u\} \cap \{Y \in C_u\} ~~~~~{\rm a.s.},
\end{equation}d{equation}
where the almost sure equalities hold both with respect to the joint distribution $P$ of $(X,Y)$, and with
respect to a product probability measure $\mu \otimes \Mathbb Nu$ governing $(X,Y)$. But under $\mu \otimes \Mathbb Nu$ the random variables $X$ and $Y$
are independent. So if $q:= ( \mu \otimes \Mathbb Nu)( p(X,Y) > u)$, then (\ref{JPLemma1}) and
(\ref{JPLemma2}) imply that $q=q^2$, so $q =0$ or $q=1$.
Thus $p(X,Y)$ is constant a.s. with respect to $\mu \otimes \Mathbb Nu$, hence also constant with respect to $P$.
\end{equation}dpf
\section{The deletion property without the regularity condition}
\label{secfails}
Observe that the property required in Theorem \ref{2pchar1} is void if $\pi_n$ happens to be the one-block partition $(n)$.
This readily implies that mixing with
the trivial one-block partition $\bf 1$ does not destroy the property.
Therefore the $\bf 1$-component may be excluded from the consideration, meaning that it is enough to focus on the case
\begin{equation}
\label{nicecase}
\mbox{ $P_1<1$ a.s., or equivalently $P^{\downarrow}_1<1$ a.s., or equivalently $\lim_{n\to\infty} p(n)=0$. }
\end{equation}
Suppose then that this condition holds,
but that the first condition in (\ref{p221}) does not hold, so that $p(2,2,1)=0$.
Then
$$\mathbb P(P^{\downarrow}_2=1-P^{\downarrow}_1>0) +\mathbb P(P^{\downarrow}_1<1,~P^{\downarrow}_2=0)=1.$$
If both terms have positive probability then $\mathbb P(W_2=1\,|\,W_1=0)=0$ but
$\mathbb P(W_2=1\,|\,W_1>0)>0$, so the independence of $W_1$ and $W_2$ fails.
Thus the independence forces either
$\mathbb P(P^{\downarrow}_2=1-P^{\downarrow}_1>0) =1$ or $\mathbb P(P^{\downarrow}_1<1,~P^{\downarrow}_2=0)=1$.
The two cases are readily treated:
\begin{itemize}
\item[(i)]
If $\mathbb P(P^{\downarrow}_2=1-P^{\downarrow}_1>0)=1$ then $W_2=1$ a.s. and the independence trivially holds.
This is the case when $\mathbb Pi$ has two blocks almost surely.
\item[(ii)]
If $\mathbb P(P^{\downarrow}_1<1,~P^{\downarrow}_2=0)=1$ and $\mathbb P(P^{\downarrow}_1>0)>0$ then
$\mathbb P(W_2>0\,|\,W_1>0)=0$ but $\mathbb P(W_2>0\,|\,W_1=0)>0$, hence $W_1$ and $W_2$ are not independent.
Therefore $\mathbb P(P^{\downarrow}_1<1,~P^{\downarrow}_2=0)=1$ and the independence imply $P^{\downarrow}_1=0$ a.s., meaning that $\mathbb Pi={\bf 0}$.
\end{equation}d{itemize}
\par We conclude that the most general exchangeable partition $\mathbb Pi$ which has the property
in Theorem \ref{Kchar1} is a two-component mixture, in which the first component
is either a partition from the extended two-parameter family, or a two-block partition as in (i) above, or $\bf 0$,
and the second component
is the trivial partition $\bf 1$.
\section{Regeneration and $\tau$-deletion }
In this section we partly survey and partly extend
the results from
\cite{RegenComp,rps04} concerning characterizations of $(\alpha,\theta)$ partitions
by regeneration properties.
As in Kingman's study of the regenerative processes \cite{KingmanReg},
subordinators (increasing L{\'e}vy processes) appear naturally in our framework of
{\it multiplicative} regenerative phenomena.
Following \cite{rps04},
we call a partition structure $(\pi_n)$ {\em regenerative} if
\begin{quote}
for each $n$ it is possible to delete a randomly chosen part of $\pi_n$ in such a way that for each $0 < m < n$, given the deleted part is of size $m$, the remaining parts form a partition of $n-m$ with the same distribution as $\pi_{n-m}$.
\end{equation}d{quote}
In terms of an exchangeable partition $\mathbb Pi = (\mathbb Pi_n)$ of $\mathbb{N}$, the
associated partition structure $(\pi_n)$ is regenerative if and only if
\begin{quote}
for each
$n$ it is possible to select a random block ${\cal B}_{nJ_n}$ of $\mathbb Pi_n$ in such a way that for each $0 < m < n$, conditionally given that $|{\cal B}_{nJ_n}| = m$ the partition $\mathbb Pi_n \setminus {\cal B}_{nJ_n}$ of $[n-m]$ is distributed according to the unconditional distribution of $\mathbb Pi_{n-m}$:
\end{equation}d{quote}
\begin{equation}
\label{regendef}
\mbox{ ($\mathbb Pi_n\setminus B_{nJ_n}$ given $|B_{nJ_n}|=m)$ } \ed \mathbb Pi_{n-m}
\end{equation}
where $\mathbb Pi_n \setminus {\cal B}_{nJ_n}$ is defined as in the discussion preceding Theorem \ref{Kchar1}. Moreover, there is no loss of generality in supposing
further that the conditional distribution of $J_n$ given $\mathbb Pi_n$ is of the form
\begin{equation}
\label{jnjb}
\mathbb P(J_n = j \,|\, \mathbb Pi_n = \{B_1, \ldots, B_k\} ) = d(|B_1|,\ldots, |B_k|;j )
\end{equation}
for some {\em symmetric deletion kernel} $d$, meaning a non-negative
function of a composition $\lambda$ of $n$ and $1 \le j \le k_\lambda$
such that
\begin{equation}
\label{dnu}
d(\lambda_1,\lambda_2, \ldots, \lambda_k;j )
= d(\lambda_{\sigma(1)},\lambda_{\sigma(2)}, \ldots, \lambda_{\sigma(k)};1 )
\end{equation}
for every permutation $\sigma$ of $[k]$ with $\sigma(1) = j$.
To determine a symmetric deletion kernel, is suffices to specify
$d(\lambda;1 )$,
which is the conditional probability, given blocks of sizes $\lambda_1,\lambda_2, \ldots, \lambda_k$,
of picking the first of these blocks. This is a non-negative symmetric function of
$(\lambda_2, \ldots, \lambda_k)$, subject to the further constraint that its extension
to arguments $j \Mathbb Ne 1$ via \re{dnu} satisfies
$$
\sum_{j = 1}^{k_\lambda} d(\lambda; j ) = 1
$$
for every composition $\lambda$ of $n$.
The regeneration condition can now be reformulated
in terms of the EPPF $p$ of $\mathbb Pi$ in a manner similar to \re{factor}:
\begin{lemma}
An exchangeable random partition $\mathbb Pi$ with EPPF $p$ is regenerative
if and only if there exists a symmetric deletion kernel $d$ such that
\begin{equation}
\label{dnu1}
p( \lambda )
d( \lambda ; 1)
= q(n,\lambda_1)
{n \choose \lambda_1}^{-1} p(\lambda_2, \ldots, \lambda_k)
\end{equation}
for every composition $\lambda$ of $n$ into at least two parts and some non-negative function $q$.
Then
\begin{equation}
\label{decmat}
q(n,m) = \mathbb P(|{\cal B}_{n,J_n}| = m ) ~~~~(m \in [n])
\end{equation}
for $J_n$ as in {\rm \re{jnjb}}.
\end{equation}d{lemma}
\proof
Formula \re{dnu1} offers two different ways of computing the probability
of the event that $\mathbb Pi_n = \{B_1, \ldots, B_k \}$ and $J_n = 1$ for
an arbitrary partition $\{B_1, \ldots, B_k \}$ of $[n]$ with $|B_i| = \lambda_i$
for $i \in [k]$: on the left side, by definition of the symmetric deletion
kernel, and on the right side by conditioning on the event
${\cal B}_{n,J_n} = B_1$ and appealing to the regeneration property and
exchangeability.
\end{equation}dpf
Consider now the question of whether an $(\alpha,\theta)$ partition with EPPF
$p = p_{\alpha,\theta}$ as in \re{EPPF2Par} is regenerative with respect to some
deletion kernel. By the previous lemma and cancellation of common factors, the question is whether there exists
a symmetric deletion kernel $d(\lambda;j)$ such that the value of
\begin{equation}
\label{f1}
q(n,\lambda_1)= d(\lambda; 1) {n \choose \lambda_1} \frac{
(1 - \alpha)_{\lambda_1 - 1}
( \theta + (k-1) \alpha )
}{ ( \theta + n - \lambda_1 )_{\lambda_1} }
\end{equation}
is the same for all compositions $\lambda$ of $n$ with $k$ parts and a prescribed
value of $\lambda_1$. But it is easily checked that the formula
\begin{equation}
\label{f2}
d_{\alpha,\theta}(\lambda ; j) =
\frac{ \theta \lambda_j + \alpha (n - \lambda_j) }{ n (\theta + \alpha (k-1) ) }
\end{equation}
provides just such a symmetric deletion kernel. Note that the kernel depends
on $(\alpha,\theta)$ only through the ratio $\tau:= \alpha/(\alpha + \theta)$,
and that the kernel is non-negative for all compositions $\lambda$ only if both $\alpha$ and $\theta$
are non-negative.
To provide a more general context for this and later discussions, let $( x_1, \ldots, x_k)$ be a fixed sequence of
positive numbers with sum $s = \sum_{j=1}^k x_j$. For a fixed parameter $\tau\in [0,1]$, define a random variable $T$ with values in $[k]$ by
\begin{equation}\label{T-dist}
\mathbb P\left(T=j \mid (x_1, \ldots, x_k)\right)=
{(1-\tau)x_j + \tau (s-x_j)
\over s(1-\tau+\tau (k-1) )},
\end{equation}d{equation}
The random variable $x_{T}$ is called a {\it $\tau$-biased pick} from $x_1, \ldots, x_k$.
The law of $x_{T}$ does not depend on the order of the sequence $(x_1, \ldots, x_k)$, and there is also a scaling invariance: $s^{-1}x_T$ is a $\tau$-biased pick from $(s^{-1}x_1, \ldots, s^{-1}x_k)$.
Note that a $0$-biased pick is a size-biased pick from $(x_1, \ldots, x_k)$, choosing any particular element with probability proportional to its size. A $1/2$-biased pick is
a uniform random choice from the list, as (\ref{T-dist}) then equals $1/k$ for all $j$. And
a $1$-biased pick may be called a co-size biased pick, as it
chooses $j$ with probability proportional to its co-size $s-x_j$.
\par These definitions are now applied to the sequence of block sizes $x_j$ of the restriction to $[n]$ of an an exchangeable partition
$\mathbb Pi$ of $\mathbb{N}$.
We denote by $T_n$ a random variable whose conditional distribution
given $\mathbb Pi_n$ with $k$ blocks and $|{\cal B}_{nj}| = x_j$ for $j \in [k]$ is
defined by (\ref{T-dist}), and denote by ${\cal B}_{nT_n}$ the $\tau$-biased pick from
the sequence of blocks of $\mathbb Pi_n$. We call $\mathbb Pi$ {\it $\tau$-regenerative} if
$\mathbb Pi_n$ is regenerative with respect to deletion of the $\tau$-biased pick ${\cal B}_{nT_n}$.
\begin{theorem}{\rm \cite{ RegenComp, rps04}} \label{T-char}
For each $\tau \in [0,1]$,
apart from the constant partitions $\mathbf{0}$ and $\mathbf{1}$,
the only exchangeable partitions of $\mathbb{N}$
that are $\tau$-regenerative are the members of the two parameter family with parameters in the range
$$\{(\alpha,\theta)\in [0,1]\times [0,\infty]: ~ \alpha/(\alpha+\theta)=\tau\}.$$
Explicitly, the distribution of the $\tau$-biased pick for such $(\alpha,\theta)$ partitions of $[n]$ is
\begin{equation}
\label{2paramdec}
\mathbb P(|{\cal B}_{nT_n}|=m)={n\choose m}{(1-\alpha)_{m-1}\over (\theta+n-m)_{m}}
{(n-m)\alpha+ m\theta\over n},~~~~~~m\in[n].
\end{equation}
\end{equation}d{theorem}
\proof
The preceding discussion around \re{f1} and \re{f2} shows that
members of the two parameter family with parameters in the indicated
range are $\tau$-regenerative, and gives the formula \re{2paramdec} for the decrement matrix.
See \cite{rps04} for the proof that these are the only non-degenerate
exchangeable partitions of $\mathbb{N}$ that are $\tau$-regenerative.
\end{equation}dpf
In particular, each $(\alpha,\alpha)$ partition is $1/2$-regenerative,
meaning regenerative with respect to deletion of a
block chosen uniformly at random.
The constant partitions $\mathbf{0}$ and $\mathbf{1}$
are obviously $\tau$ regenerative for every $\tau \in [0,1]$. This is consistent with the characterization above because the $(1,\theta)$ partition is the $\mathbf{0}$ partition for every $\theta \geq 0$, and because the partition $\mathbf{1}$ can be reached as a limit of $(\alpha, \theta)$ partitions as $\alpha, \theta \downarrow 0$
with $\alpha(\alpha+\theta)^{-1}$ held fixed.
\paragraph{Multiplicative regeneration}
By Corollary \ref{2pcharcor}, if $(P_i)$ is the sequence of limit frequencies for a $(0,\theta)$ partition for some $\theta>0$ and if the first limit frequency $P_1$ is deleted and the other frequencies renormalized to sum to 1, then the resulting sequence $(Q_j)$ is independent of $P_1$ and has the same
distribution as $(P_j)$. Because $P_1$ is a size-biased pick from the sequence $(P_i)$, this regenerative property of the frequencies $(P_i)$ can be seen as an analogue of the $0$-regeneration property of the $(0, \theta)$ partitions.
If $(P_i)$ is instead the sequence of limit frequencies of an $(\alpha, \theta)$ partition $\mathbb Pi$ for parameters satisfying $0<\alpha<1, ~\alpha/(\alpha+\theta)=\tau$,
a question arises: does the regenerative property of $\mathbb Pi_n$ with respect to a $\tau$-biased pick have an analogue in terms of a $\tau$-biased pick from the frequencies $(P_i)$?
This cannot be answered straightforwardly as in the $\tau=0$ case,
because when $\tau>0$ the formula
(\ref{T-dist}) defines a proper probability distribution
only for series $(x_j)$ with some finite number $k$
of positive terms.
For instance, in the case $\tau=1/2$ there is no such analogue of (\ref{T-dist})
as `uniform random choice' from infinitely many terms.
\par Still, Ewens' case provides a clue if we turn to a {\it bulk deletion}.
Let $P_J$ be a size-biased pick from the frequencies $(P_j)$, as
defined by (\ref{sbpick}), and let $(Q_j)$ be a sequence obtained from $(P_j)$ by deleting all $P_1,\ldots,P_J$ and renormalizing.
Then $(Q_j)$ is independent of $P_1,\dots,P_J$, and $(Q_j)\ed(P_j)$. The latter assertion follows from the i.i.d. property
of the residual fractions and by noting that
(\ref{sbpick}) is identical with
$$\mathbb P(J=j\,|\, (W_i, i\in \mathbb Nat))= W_j\prod_{i=1}^{j-1}(1-W_i).$$
A similar bulk deletion property holds for partitions in the Ewens' family, in the form:
$$
\mbox{ ($\mathbb Pi_n\setminus ({\cal B}_{n1}\cup\dots\cup{\cal B}_{nJ_n})$ given $|{\cal B}_1\cup\dots\cup{\cal B}_{nJ_n}|=m)$ } \ed \mathbb Pi_{n-m}
$$
for all $1 \leq m \leq n$, where $B_{nJ_n}$ is a size-biased pick from the
blocks.
To make the ansatz of bulk deletion work for $\tau\Mathbb Neq 0$ it is necessary to arrange the frequencies in a
more complex manner.
To start with, we modify the paintbox construction.
Let ${\cal U}\subset [0,1]$ be a random open set canonically represented as the union of its disjoint open
component intervals.
We suppose that the Lebesgue measure of $\cal U$, equal to the sum of lengths of the components,
is $1$ almost surely. We associate with $\cal U$ an exchangeable partition
$\mathbb Pi$
exactly as in Kingman's representation
in Theorem \ref{Kpaintbox}.
For each component interval $G \subset \cal U$ there is an index
$i_G:=
\min\{n: U_n \in G \}$ that is the minimal index
of a sequence $(U_i)$ of iid uniform[0,1] points hitting the interval,
and for all $j$, $P_{j}$ is the length of the $j$th component interval when the intervals
are listed in order of increasing minimal indices. So $(P_{j})$ is a size-biased permutation of
the lengths of interval components of $\cal U$.
Let $\triangleleft$ be the linear order on $\mathbb Nat$ induced by the
{\it interval order} of the components of $\cal U$, so $j\triangleleft k$ iff
the interval of length $P_j$, which is the home interval of the $j$th block ${\cal B}_j$ to appear in the process of uniform random
sampling of intervals, lies to the left of the interval of length $P_k$ associated with block ${\cal B}_k$.
A convergence argument shows that
$\cal U$ is uniquely determined by $(P_j)$ and $\triangleleft$.
In loose terms, $\cal U$ is an arrangement of a sequence of tiles of sizes $P_j$ in the order on indices $j$
prescribed by $\triangleleft$, and this arrangement is constructable by sequentially
placing the tile $j$ in the position prescribed by the order $\triangleleft$ restricted to $[j]$.
For $x\in [0,1)$ let $(a_x,b_x)\subset{\cal U}$ be the component interval
containing $x$. Define $\mathcal{V}_x$ as the open set
obtained by deleting
the bulk of component intervals to the left of $b_x$, then linearly rescaling the remaining set
${\mathcal U}\cap [b_x,1]$ to $[0,1]$.
We say that $\mathcal U$ is {\it multiplicatively regenerative} if for each $x\in [0,1)$,
${\mathcal{V}_x}$ is independent of $\mathcal{U} \cap [0,b_x]$ and $\mathcal{V}_x \ed {\cal U}$.
\par An ordered version of the paintbox correspondence yields:
\begin{theorem}\label{PartPa}{\rm \cite{ RegenComp, rps04}} An exchangeable partition $\mathbb Pi$ is regenerative if and only if
it has a paintbox representation in terms of some multiplicatively regenerative set $\cal U$.
The deletion operation is then defined by classifying $n$ independent uniform points
from $[0,1]$ according to the intervals of ${\cal U}$ into which they fall, and deleting
the block of points in the leftmost occupied interval.
\end{equation}d{theorem}
A property of the frequencies $(P_j)$ of an exchangeable regenerative partition $\mathbb Pi$ of $\mathbb Nat$
now emerges: there exists a strict total order $\triangleleft$ on $\mathbb Nat$, which is a random order, which has some joint distribution with
$(P_j)$ such that arranging the intervals of sizes $(P_j)$ in order $\triangleleft$ yields a multiplicatively regenerative set ${\cal U}$.
Equivalently, there exists a multiplicatively regenerative set ${\cal U}$ that induces a partition
with frequencies $(P_j)$ and an associated order $\triangleleft$. This set ${\cal U}$ is then necessarily unique
in distribution as a random element
of the space of open subsets
of $[0,1]$ equipped with the Hausdorff metric \cite{RegenComp} on the complementary closed subsets.
A subtle point here is that the joint distribution of $(P_j)$ and $\triangleleft$ is not unique,
and neither is the joint distribution of $(P_j)$ and ${\cal U}$, unless further conditions are imposed.
For instance, one way to generate $\triangleleft$ is to suppose that the $(P_j)$ are generated by a process of uniform random
from ${\cal U}$. But for a $(0,\theta)$ partition, we know that another way is to construct ${\cal U}$ from
$(P_j)$ by simply placing the intervals in deterministic order $P_1, P_2, \ldots$ from left to right.
In the construction by uniform random sampling from $\cal U$ the interval of length $P_1$ discovered by the first sample point
need not be the leftmost, and need not lie to the left of the second discovered interval $P_2$.
In \cite{RegenComp} we showed that the multiplicative regeneration of $\mathcal{U}$ follows from
an apparently weaker property:
if $(a_U, b_U)$ is the component interval of $\cal U$ containing an uniform[0,1]
sample $U$ independent of $\mathcal{U}$, and if $\mathcal{V}$ is defined as
the open set obtained by deleting the component intervals to the left of $b_U$
and linearly rescaling the remaining set $\mathcal{U}\cap [b_U,1]$ to [0,1], then given $b_U<1$,
$\mathcal{V}$ is independent of $b_U$
(hence, as we proved, independent of $\mathcal{U}\cap [0,b_U]$ too!)
and has distribution equal
to the unconditional distribution of $\mathcal{U}$.
This independence is the desired analogue for more general regenerative partitions of the
bulk-deletion property of Ewens' partitions.
\par The fundamental representation of multiplicatively regenerative sets
involves a random process $F_t$ known in statistics as a neutral-to-the right distribution function.
\begin{theorem}\label{m-reg-rep} {\rm \cite{RegenComp}}
A random open set $\cal U$ of Lebesgue measure $1$ is multiplicatively regenerative
if and only if there exists a drift-free subordinator $S=(S_t,t\geq 0)$ with $S_0=0$ such that $\cal U$
is the complement to the closed range of the process $F_t=1-\mathbb Ep(-S_t),~t\geq 0$.
The L{\'e}vy measure of $S$ is determined uniquely up to a positive factor.
\end{equation}d{theorem}
According to Theorems \ref{PartPa} and \ref{m-reg-rep}, regenerative partition structures with
proper frequencies are parameterised by a measure $\tilde{\Mathbb Nu}({\rm d}u)$ on $(0,1]$ with
finite first moment, which is the image
via the transformation from $s$ to $1 - \mathbb Ep(-s)$
of the L{\'e}vy measure $\Mathbb Nu({\rm d}s)$ on $(0,\infty]$
associated with the subordinator $S$.
The Laplace exponent $\mathbb Phi$ of the subordinator,
defined
by the L\'evy-Khintchine formula
$$
\mathbb E [ \mathbb Ep ( - a S_t ) ] = \mathbb Ep [ - t \mathbb Phi(a) ], ~~~~~~~~a \ge 0
$$
determines the L\'evy measure $\Mathbb Nu({\rm d}s)$ on $(0,\infty]$
and its image $\tilde{\Mathbb Nu}({\rm d}u)$ on $(0,1]$ via the formulae
$$
\mathbb Phi(a)=
\int_{(0,\infty]} (1 - e^{-a x}) \Mathbb Nu(dx)
= \int_{]0,1]}(1-(1-x)^a)\tilde{\Mathbb Nu}({\rm d}x).
$$
As shown in \cite{RegenComp}, the decrement matrix $q$ of the regenerative partition structure,
as in \re{decmat}, is then
$$
q(n,m)={\mathbb Phi(n,m)\over \mathbb Phi(n)}\,,\qquad 1\leq m\leq n\,,~n=1,2,\ldots
$$
where
$$
\mathbb Phi(n,m)= {n\choose m}\int_{]0,1]} x^m(1-x)^{n-m}\tilde{\Mathbb Nu}({\rm d}x)\,.
$$
Uniqueness of the parameterisation is achieved by a normalisation condition, such as $\mathbb Phi(1)=1$.
In \cite{RegenComp} the subordinator $S^{\alpha,\theta}$
which produces $\cal U$
as in Theorem \ref{m-reg-rep} for the $(\alpha, \theta)$ partition was identified by the following formula
for the right tail of
its L{\'e}vy measure:
\begin{equation}\label{LM}
\Mathbb Nu^{}(x,\infty]= (1-e^{-x })^{-\alpha} e^{-x\theta}, ~~~x>0.
\end{equation}d{equation}
The subordinator $S^{(0,\theta)}$
is a compound Poisson process whose jumps are exponentially distributed
with rate $\theta$.
For $\theta=0$ the L{\'e}vy measure has a unit mass at $\infty$, so
the subordinator $S^{(\alpha,0)}$ is killed at unit rate.
The $S^{(\alpha,\alpha)}$
subordinator belongs to the class of Lamperti-stable processes recently studied in \cite{Lstable}.
For positive parameters the subordinator $S^{(\alpha,\theta)}$ can be constructed from
the $(0,\theta)$ and $(\alpha,0)$ cases, as follows.
First split ${\mathbb R}_+$ by the range of $S^{(0,\theta)}$, that is at points $E_1<E_2<\dots$
of a Poisson process with rate $\theta$. Then run an independent copy of $S^{(\alpha,0)}$
up to the moment the process crosses $E_1$ at some random time,
say $t_1$. The level-overshooting value is neglected and the process is stopped. At the same time
$t_1$ a new independent copy of $S^{(\alpha,0)}$
is started at value $E_1$ and run until crossing $E_2$ at some random time $t_2$, and so on.
In terms of $F_t=1-\mathbb Ep(-S_t)$, the range of the process in the $(0,\theta)$ case is a
stick-breaking set $\{1-\prod_{i=1}^{j-1} (1-V_i), i=0,1,\ldots\}$
with i.i.d. beta$(1,\theta)$ factors $V_i$.
In the case $(\alpha, 0)$
the range of $(F_t)$ is
the intersection of $[0,1]$ with the $\alpha$-stable set
(the range of $\alpha$-stable subordinator).
In other cases $\cal U$ is constructable as a cross-breed of the cases $(\theta,0)$ and
$(0,\alpha)$:
first $[0,1]$ is partitioned in subintervals by the beta$(1,\theta)$ stick-breaking,
then each subinterval $(a,b)$ of this partition is further split by independent copy of the multiplicatively regenerative
$(\alpha,0)$ set, shifted to start at $a$ and truncated at $b$.
\paragraph{Constructing the order}
Following \cite{rps04, PitmanWinkel}, we shall describe
an arrangement
which allows us to pass from $(\alpha, \theta)$ frequencies $(P_j)$ to the multiplicatively regenerative set
associated with the subordinator $S^{(\alpha,\alpha)}$.
The connection between size-biased permutation with $\tau$-deletion (Lemma \ref{Leem}) is new.
A linear order $\triangleleft$ on $\mathbb Nat$ is conveniently described by a sequence of the initial ranks
$(\rho_j)\in [1]\times[2]\times\cdots$, with $\rho_j=i$ if and only if $j$ is ranked $i$th smallest
in the order $\triangleleft$ among the integers $1,\dots,j$.
For instance, the initial ranks $1,2,1,3\dots$ appear when $3\triangleleft 1\triangleleft 4\triangleleft 2$.
For $\xi\in[0,\infty]$ define a random order $\triangleleft_\xi$ on $\mathbb Nat$ by
assuming that the initial ranks $\rho_k, k\in \mathbb Nat,$ are independent, with distribution
$$\mathbb P(\rho_k=j)=
{1\over k+\xi-1}{\tt 1}(0 < j<k)
+
{\xi\over k+\xi-1}{\tt 1}(j=k) ~~~~~~~~~~~,k>1.
$$
The edge cases $\xi=0,\infty$ are defined by continuity. The order $\triangleleft_1$
is a `uniformly random order', in the sense that restricting to $[n]$ we have all $n!$
orders equally likely, for every $n$. The order $\triangleleft_\infty$ coincides
with the standard order $<$ almost surely.
For every permutation $i_1,\ldots,i_n$ of $[n]$, we have
$$\mathbb P(i_1\triangleleft_\xi\dots\triangleleft_\xi i_n)={\xi^r\over \xi(\xi+1)\dots(\xi+n-1)}$$
where $r$ is the number of upper records in the permutation.
See \cite{Coherent} for this and more general permutations with tilted record statistics.
\begin{theorem}\label{newthm}
{\rm \cite[Corollary 7]{PitmanWinkel} }
For $0\leq \alpha<1, \theta\geq0$ the arrangement
of $GEM(\alpha,\theta)$ frequencies $(P_j)$ represented as open intervals in an independent random order $\triangleleft_{\theta/\alpha}$ is a
multiplicatively regenerative open set ${\cal U}\subset[0,1]$,
where $\cal U$ is representable
as the complement of the closed range of
the process $F_t=1-\mathbb Ep(-S_t), t\geq 0,$ for $S$ the subordinator with L{\'e}vy measure {\rm (\ref{LM})}.
\end{equation}d{theorem}
This result was presented without proof as \cite[Corollary 7]{PitmanWinkel},
in a context where the regenerative ordering of frequencies was motivated by
an application to a tree growth process.
Here we offer a proof which exposes the combinatorial structure of the
composition of size-biased permutation and a $\triangleleft_{\theta/\alpha}$ ordering of frequencies.
For a sequence of positive reals $(x_1,\dots,x_k)$, define the {\it $\tau$-biased permutation}
of this sequence, denoted ${\rm perm}_\tau(x_1,\ldots,x_k)$,
by iterating a single $\tau$-biased pick, as follows. A number $x_{T}$ is chosen from $x_1,\dots,x_k$
without replacement,
with $T$ distributed on $[k]$ according to (\ref{T-dist}), and $x_T$ is placed in position $1$.
Then the next number is chosen from $k-1$ remaining numbers
using again the rule of $\tau$-biased pick,
and placed in position 2, etc.
The instance ${\rm perm}_0$ is the size-biased permutation, which
is defined more widely for
finite or infinite summable sequences $(x_1,x_2,\ldots)$, and shuffles them
in the same way as it shuffles $(s^{-1}x_1, s^{-1}x_2,\dots)$ where $s=\sum_{j} x_j$.
Denote by $\triangleleft_\xi(x_1,\dots,x_k)$ the arrangement of $x_1,\ldots,x_k$ in succession according to the
$\triangleleft_\xi$-order on $[k]$.
\begin{lemma}\label{Leem} For $\xi=(1 - \tau)/\tau$ there is the compositional formula
\begin{equation}\label{coin}
{\rm perm}_\tau(x_1,\ldots,x_k)\ed {\triangleleft}_\xi({\rm perm}_0(x_1,\ldots,x_k)),
\end{equation}
where on the right-hand side
$\triangleleft_\xi$
and
${\rm perm}_0$
are independent.
\end{equation}d{lemma}
\proof
On each side of this identity, the distribution of the random permutation remains the same if the
sequence $x_1, \ldots, x_k$ is permuted. So it suffices to check that each scheme
returns the identity permutation with the same probability. If on the right hand side we set
$$
{\rm perm}_0(x_1,\ldots,x_k) = ( x_{\sigma(1)}, \ldots, x_{\sigma(k)})
$$
then the right hand scheme generates the identity permutation with probability
\begin{equation}
\label{ExiR}
\frac{ \mathbb E \xi^{R} } {\xi ( \xi +1 ) \cdots (\xi + k - 1 )}
\end{equation}
where $R$ is the number of upper records in the sequence of ranks which generated $\sigma^{-1}$, which equals the number of upper records in $\sigma$. Now
$R = \sum_{j = 1}^k X_j$ where $X_j$ is the indicator of the event $A_j$ that $j$ is an upper record level for $\sigma$, meaning that
there is some $1 \le i \le n$ such that
$$
\mbox{ $\sigma(i') < j $ for all $i' < i$ and $\sigma(i) = j$.}
$$
Equivalently, $A_j$ is the event that
$$
\mbox{ $\sigma^{-1}(j) < \sigma^{-1}(\ell)$ for each $j < \ell \le k$. }
$$
Or again, assuming for simplicity
that the $x_i$ are all distinct, which involves no loss of generality, because the probability in question depends continuously on $(x_1, \ldots, x_k)$,
$A_j$ is the event that $x_j$ precedes $x_\ell$ in the permutation $( x_{\sigma(1)}, \ldots, x_{\sigma(k)})$
for each $j < \ell \le k$.
Now it is easily shown that $( x_{\sigma(1)}, \ldots, x_{\sigma(k)})$ with $x_1$ deleted is a size-biased permutation of $(x_2, \ldots, x_k)$, and
that the same is true conditionally given $A_1$. It follows by induction that the events $A_j$ are mutually independent, with
$$
\mathbb P(A_j) = x_j/(x_j + \cdots + x_k) \mbox{ for } 1 \le j \le k.
$$
This allows the probability in \re{ExiR} to be evaluated as
$$
\prod_{j = 1}^k \frac{(\xi x_j + x_{j+1} + \cdots + x_k ) } { ( x_j + x_{j+1} + \cdots + x_k)(\xi + j -1 ) }
$$
This is evidently the probability that ${\rm perm}_\tau(x_1,\ldots,x_k)$ generates the identity permutation,
and the conclusion follows.
\end{equation}dpf
\par The $\tau$-biased arrangement cannot be defined for infinite positive summable sequence
$(x_1,x_2,\dots)$, since the `$k=\infty$' instance of (\ref{T-dist}) is not a proper distribution for $\tau\Mathbb Neq 0$.
But the right-hand side of (\ref{coin}) is well-defined
as arrangement of $x_1,x_2,\dots$ in some total order,
hence
the composition ${\triangleleft}_\xi\circ{\rm perm}_0$ is the natural extension of
the $\tau$-biased arrangement to infinite series.
\vskip0.3cm
\Mathbb Noindent
{\it Proof of Theorem \ref{newthm}.} We represent a finite or infinite positive sequence $(x_j)$ whose sum is $1$
as an open subset of $[0,1]$ composed of contiguous intervals of sizes $x_j$. The space of open subsets of $[0,1]$
is endowed with the Hausdorff distance on the complementary compact sets. This topology is weaker than the product topology
on positive series summable to $1$.
The limits below are understood as $n\to\infty$.
\par We know by a version of
Kingman's correspondence \cite{jp.epe} that $(|{\cal B}_{nj}|/n, j\geq 1)\to (P_j)$ a.s. in the product topology.
This readily implies $\triangleleft_\xi (|{\cal B}_{nj}|/n, j\geq 1)\to \triangleleft_\xi(P_j)$ a.s.
in the Hausdorff topology, by looking at the $M$ first terms for $M$ such that these terms sum to at least $1-\epsilon$
with probability at least $1 -\epsilon$, then sending $\epsilon \to 0$ and $M\to\infty$.
In \cite{RegenComp} we showed that ${\rm perm}_\tau(|{\cal B}_{nj}|, j\geq 1)\to {\cal U}$ a.s. in the Hausdorff topology.
(Here the definition of the ${\rm perm}_\tau$ is coupled with $(|{\cal B}_{nj}|, j\geq 1)$ by putting these blocks in the order
determined by uniform sampling from ${\cal U}$).
The missing link is provided by Lemma \ref{Leem}, from which we obtain
$${\rm perm}_\tau(|{\cal B}_{nj}|, j\geq 1)\ed \triangleleft_\xi(|{\cal B}_{nj}|,j\geq 1),$$
with the $\tau$-biased permutation ${\rm perm}_\tau$ applied to the {\it finite} sequence of positive
block-sizes $(|{\cal B}_{nj}|, j\geq 1)$.
Putting things together we conclude that
$\triangleleft_\xi(P_j, j\ge 1)\ed{\cal U}$.
\end{equation}dpf
In three special cases, already identified in the previous work \cite{rps04},
the arrangement of PD$(\alpha,\theta)$ (or GEM$(\alpha,\theta)$) frequencies
in a multiplicatively regenerative set has a simpler description:
in the $(0,\theta)$ case the frequencies are placed in the size-biased order;
in the $(\alpha,\alpha)$ case the frequencies are `uniformly randomly shuffled';
and in the $(\alpha,0)$ case a size-biased pick is placed
contiguously to 1, while the other frequencies are `uniformly randomly shuffled'.
The latter is an infinite analogue of the co-size biased arrangement ${\rm perm}_1$.
\par We refer to \cite{hpw07, PitmanWinkel} for further recent developments related to
ordered $(\alpha,\theta)$ partitions and their regenerative properties.
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\title[Fractional Schr\"odinger-Poisson-Slater system in one dimension]
{Fractional Schr\"odinger-Poisson-Slater system in one dimension}
\author[A. R. Giammetta]{Anna Rita Giammetta}
\address{Department of Mathematics, University of Pisa,
Italy}
\email{[email protected]}
\subjclass{}
\keywords{Fractional Poisson equation, $1D$ Schr\"odinger-Poisson-Slater, Hartree equations}
\date{\today}
\betagin{abstract}
In this paper we study local and global well-posedness of the following Cauchy problem:
\betagin{subnumcases}
{\,}
i\partialartial_t\Psi+\frac{1}{2}\Deltalta_{x}\Psi = A_0\Psi +\alphapha |\Psi|^{\gammamma-1}\Psi \,\,\,\,\,\,\,\,\,\,(t,x)\in{\mathbb R}\times{\mathbb R} \notag\\
(-\Deltalta_{x})^{\sigmagma/2}A_0= |\Psi|^2 \notag\\
\Psi(0,\mathcal Dot)=f, \notag
\end{subnumcases}
with $\sigmagma\in(0,1)$, $\alphapha=\partialm 1$, $1<\gammamma\leq 5$, in the spaces $L^2({\mathbb R})$ and $H^1({\mathbb R})$. \\
\end{abstract}
\maketitle
\section{Introduction}
The nonlinear Schr\"odinger equation
\betagin{equation*}
i\partialartial_t\Psi+\frac{1}{2}\Deltalta_{x}\Psi = \alphapha |\Psi|^{\gammamma-1}\Psi,
\end{equation*}
with $\alphapha=\partialm 1$,
is one of the universal model to describe the evolution of a wave packet in a weakly nonlinear and dispersive media. In particular, the case $\gammamma=3$ occurs to model different physical phenomena: the propagation of waves in optical fibers for $n=1$, the focusing of laser beams for $n=2$, the Bose-Einstain condensation phenomenon for $n=3$, see C^{\infty}te{sulemsulem}, C^{\infty}te{karlsson} and references therein.\\
In the construction of a mathematical model, many physical laws are simplified, so, it is essential to deal with well posed problems: existence of the solution indicates that the model is coherent, uniqueness and stability are related to the problem of approximate the solution with numerical algorithms. The math problem of well-posedness of NLS has been studied for a long time and we can find its history and its current state of the art at the web page C^{\infty}te{tao} \href{http://www.math.ucla.edu/~tao/Dispersive/}{"Local and global well-posedness for non-linear dispersive and wave equations"} manteined by Colliander, Keel, Staffilani, Takaoka and Tao. In addition, we mention two fondamental monographs specialized in the nonlinear Schr\"odinger equation: Cazenave C^{\infty}te{cazenave} and Sulem Sulem C^{\infty}te{sulemsulem}.
Schr\"odinger-Poisson-Slater system is a nonlinear Schr\"odinger mixed–system which combines
the nonlinear and nonlocal Coulomb interaction, $A_0$, with a local potential nonlinearity
known as the ”Slater exchange term”:
\betagin{subnumcases}
{\,(SPS)}
\partialartial_t \partialsi+\frac{1}{2}\Deltalta \partialsi =A_0 \partialsi -C |\partialsi|^{\gammamma-1}\partialsi\notag\\
-\Deltalta A_0 = |\partialsi|^2,\notag
\end{subnumcases}
with $C\geq 0$. Such a model appears in studying of quantum transport in semiconductor devices as a correction to the
Schr\"odinger-Poisson (SP) system ( $C = 0$). For $3D$ well-posedness results of (SPS) in $L^2$ and $H^1$ we mention C^{\infty}te{bokanowskilopezsoler}. For asymptotic behaviour of $3D$ (SPS) solutions, we mention C^{\infty}te{sanchez}. One can see a broad literature also about problems concerning the existence and stability
of standing waves for systems like SPS: C^{\infty}te{fortunato}, C^{\infty}te{Ruiz}, C^{\infty}te{nonlinRuiz} , C^{\infty}te{visciglia}, C^{\infty}te {georgiev}, C^{\infty}te{coclitegeorgiev} and references therein.\\
Although there are many papers concerning (SPS) and similar systems in 3D, for the case of one space dimension the literature is narrower. The first 1D global results were established for the Maxwell-Schr\"odinger system, which is a generalization of (SP) system that includes the magnetic field. The first result is due to
Nakamitsu and Tsutsumi C^{\infty}te{nakamitsutsutsumi} and uses the Lorentz gauge and high regularity of initial data. Those assumptions imply also the following boundary condition on the electric potential:
\betagin{equation*}
A_0(t,x)\rightarrow 0 \,\,\,\,(|x| \rightarrow \infty).
\end{equation*}
Later, Tsutsumi, in C^{\infty}te{tsutsumi95}, proved that this condition can be relaxed to
\betagin{equation}\lambdabel{ultraviolet}
A_0(t,x)\rightarrow c_0 |x| \,\,\, (|x| \rightarrow \infty),
\end{equation}
where,
\betagin{equation*}\lambdabel{ultraviolet1}
c_0=\frac{1}{2}\int_{{\mathbb R}}|\Psi(0,x)|^2\,dx,
\end{equation*}
and the initial datum is in $H^1({\mathbb R})\mathcal Ap L^2({\mathbb R},|x|\,dx)$.\\
Recently, in 1D context, the global well-posedness of the Cauchy problem for the \emph{Hartree equations}
\betagin{subnumcases}{ \, }
i\partialartial_t\Psi+\frac{1}{2}\Deltalta_{x}\Psi = \lambdambda A_0\Psi \,\,\,& $\lambdambda\in{\mathbb R}$ \notag\\
(-\Deltalta_{x})^{\sigmagma/2}A_0= |\Psi|^2 \,\,\,& $\sigmagma\in (0,1)$ \notag
\end{subnumcases}
with initial data in $H^s({\mathbb R}),s\geq 0$, was studied in C^{\infty}te{2007hartreewp} and in C^{\infty}te{ozawa} (with exchange-correlation correction).\\
In this work we study, in one dimensional space, a model like (SPS) with the fractional Poisson equation of Hartree model. Therefore, our attempt is to study a fractional Schr\"odinger-Poisson-Slater (FSPS) system
\betagin{subnumcases}
{ \, }
i\partialartial_t\Psi+\frac{1}{2}\Deltalta_{x}\Psi = A_0\Psi +\alphapha |\Psi|^{\gammamma-1}\Psi \lambdabel{1}\\
(-\Deltalta_{x})^{\sigmagma/2}A_0= |\Psi|^2 \lambdabel{2},
\end{subnumcases}
with $\alphapha=\partialm 1$ and where $\sigmagma\in(0,1)$ and $\gammamma\in(1,5]$ are chosen such that no boundary condition of type \eqref{ultraviolet} are required.\\
By physical viewpoint, fractional powers of the Laplacian are important in many situations in which one has to consider long-range interaction and anomalous phenomena, see C^{\infty}te{valdinoci} and references therein. On the other hand, by a mathematical viewpoint, the fractional Poisson equation brings some significant difficulties in the analysis of the well-posedness and allows us to obtain a well-posedness result in one dimension when long range interactions are taken into account.\\
Our goal is to establish existence and uniqueness results about the Cauchy problem (FSPS) with initial data in $L^2({\mathbb R})$ and $H^1{({\mathbb R})}$. We give a sketch of the plan of the work.\\
At first, following the work of Kato C^{\infty}te{kato}, we rewrite the Cauchy problem (FSPS) as the integral equation
\betagin{align*}\lambdabel{integralform1}
\Psi(t)= S(t)f-i\int_{0}^{t}S(t-s)& A_0(\Psi(s))\Psi(s)\,ds\\
&-i\alphapha \int_{0}^{t}S(t-s)|\Psi|^{\gammamma-1}(s)\Psi(s)\,ds,
\end{align*}
where $S(t)$ denotes the Schr\"odinger group $ {\rm e} ^{i\frac{\Deltalta}{2}t}$ and the electric potential $A_0$ solves the fractional Poisson equation \eqref{2}.\\
We deal with local solvability of the initial value problem in $L^2({\mathbb R})$ with standard \emph{contraction argument} obtained by linear techniques (\emph{Strichartz estimates}). The problem is finding at least one admissible pair, $(q,r)$, for which the classical contraction argument works at the same time for the nonlocal and for the local nonlinearity. Indeed, the nonlocal term required to introduce some convolution estimates. We have the following main result:
\betagin{thm}\lambdabel{11}
Let $f\in L^2({\mathbb R})$ and $\gammamma\neq 5$. \\
Then, there exists an interval $I_\gammamma\subseteq (0,1)$, such that, for all $\sigmagma\in I_\gammamma$, one can find a time $T=T(\|f\|_{L^2})>0$ and a unique wave function $\Psi$,
$$\PsiC^{1}lon[0,T]\times {\mathbb R}\to \C,$$
solution of the Cauchy problem \eqref{1}-\eqref{2}. \\
In addition, we have
\betagin{equation*}\lambdabel{}
\Psi \in C([0,T],L^2)\mathcal Ap L^q([0,T],L^r),
\end{equation*}
for any $(q,r)$ admissible pair.
\end{thm}
We note that, if $\gammamma=3$, the Theorem \ref{11} holds for $\sigmagma\in(0,\frac{1}{2}]$.\\
The problem of extending the local solution to all times can be solved thanks to \emph{conservation laws} of $L^2$-norm (charge or mass conservation):
\betagin{thm}\lambdabel{21}
Let $f\in L^2({\mathbb R})$, $\sigmagma\in I_\gammamma$ and $\gammamma\neq 5$. Then, the Cauchy problem \eqref{1}-\eqref{2} has a unique global solution $\Psi \in C({\mathbb R},L^2({\mathbb R}))\mathcal Ap L^q({\mathbb R},L^r({\mathbb R}))$, for any $(q,r)$ admissible pair.
\end{thm}
The critical case, $\gammamma=5$ is more delicate, but the problem lies only in the local nonlinearity. So, we have local well-posedness for large data and global well-posedness for small data:
\betagin{thm}\lambdabel{3}
Let $f\in L^2({\mathbb R})$, $\sigmagma\in I_5$ and $\gammamma=5$. There exists a maximal interval $(-T_{min},T_{max})$, $T_{min}=T_{min}(f)$ and $T_{max}=T_{max}(f)$, such that the Cauchy problem \eqref{1}-\eqref{2} has a unique solution
$$\Psi \in C([-T_{min},T_{max}],L^2({\mathbb R}))\mathcal Ap L^q([-T_{min},T_{max}],L^r({\mathbb R})),$$
for any $(q,r)$ admissible pair.
\end{thm}
\betagin{thm}\lambdabel{4}Let $f\in L^2({\mathbb R})$, $\sigmagma\in I_5$ and $\gammamma=5$. There exists a small $\deltalta$ such that, if $\|f\|_{L^2}\leq \deltalta$, then the Cauchy problem \eqref{1}-\eqref{2} has a unique global solution $\Psi \in C({\mathbb R},L^2({\mathbb R}))\mathcal Ap L^q({\mathbb R},L^r({\mathbb R}))$, for any $(q,r)$ admissible pair.
\end{thm}
Next we would like to perform the same previous result with initial data in $H^1({\mathbb R})$. The $H^1({\mathbb R})$ theory distinguishes the defocusing case ($\alphapha=1$) and the focusing case ($\alphapha=-1$). In particular the second case is more delicate. We have the following results:
\betagin{thm}\lambdabel{5}
Let $f\in H^1({\mathbb R})$, $\sigmagma\in I_\gammamma$, $\gammamma\neq 5$ and $\alphapha=\partialm 1$. \\
Then, there exists a time $T=T(\|f\|_{H^1})>0$, such that one can find a unique wave function $\Psi$,
$$
\PsiC^{1}lon[0,T]\times {\mathbb R}\to \C,
$$
solution of the Cauchy problem \eqref{1}-\eqref{2} and
\betagin{equation*}\lambdabel{}
\Psi \in C([0,T],H^1({\mathbb R}))\mathcal Ap L^q([0,T],W^{1,r}({\mathbb R})),
\end{equation*}
for any $(q,r)$ admissible pair.
\end{thm}
\betagin{thm}\lambdabel{6}
Let $f\in H^1({\mathbb R})$, $\sigmagma\in I_\gammamma$, $\gammamma\neq 5$ and $\alphapha=-1$. \\
Then, the (FSPS) system has a unique global solution $\Psi \in C({\mathbb R},H^1({\mathbb R}))\mathcal Ap L^q({\mathbb R},W^{1,r}({\mathbb R}))$, for any $(q,r)$ admissible pair.\\
Otherwise, if $\gammamma=5$, there exists $\deltalta>0$ such that, if $\|f\|_{L^2}<\deltalta$ then the Cauchy problem \eqref{1}-\eqref{2} has a unique global solution $\Psi \in C({\mathbb R},H^1({\mathbb R}))\mathcal Ap L^q({\mathbb R},W^{1,r}({\mathbb R}))$, for any $(q,r)$ admissible pair.
\end{thm}
Our
plan
in
this
paper
is
as
follows. In Section $2$ we introduce some notations and basic fact about LS and NLS: decay estimates, Strichartz estimates, NLS well-posedness results in $L^2$ and $H^1$. The Section $3$ is devoted to well posed problem of (FSPS) system: at first we treat the $L^2$ theory (proof of the Theorems \eqref{11}, \eqref{21}, \eqref{3}, \eqref{4}) and then the $H^1$ theory (proof of the Theorems \eqref{5}, \eqref{6}). In the Section $4$, we establish some decay estimates for the solution of (FSPS): we control the $L^4 L^\infty$-norm of the solution with initial data in $L^2({\mathbb R})$ and cubic nonlinearity. Lastly, we get a control estimate for the speed of the oscillation of the solution with initial data in $H^1({\mathbb R})$.\\
\textbf{Acknoledgment.} It is a pleasure to acknowledge the interesting conversations about the one dimension Maxwell-Schr\"odinger system C^{\infty}te{tsutsumi95} with T. Ozawa.\\
The author has been supported by Comenius project "Dynamat" 2010, Universit\`{a} di Pisa and FIRB "Dinamiche Dispersive: Analisi di Fourier e Metodi Variazionali" 2012.
\section{Preliminaries}
We first introduce some notations.\\
Let $\varphi\in \mathscr S({\mathbb R}^n)$, a Schwartz function. We define the Fourier transform of $\varphi$ and its inverse as follows:
\betagin{align*}
\mathscr F [\varphi](\xi)&= \hat \varphi (\xi)= \frac{1}{(2\partiali)^{n/2}}\int_{{\mathbb R}^n}{\rm e}^{-ix\mathcal Dot\xi}\varphi(x)\,dx,\\
\mathscr F^{-1} [\varphi](x)&= \mathcal Heck \varphi(x)= \frac{1}{(2\partiali)^{n/2}}\int_{{\mathbb R}^n}{\rm e}^{ix\mathcal Dot\xi}\varphi(\xi)\,d\xi,
\end{align*}
and then we can extend this operator on tempered distribution $S'({\mathbb R}^n)$.
The fractional Laplacian $(-\Deltalta)^{\sigmagma/2}$ is a pseudo-differential operator defined as:
\betagin{equation}\lambdabel{fraclapl}
(-\Deltalta)^{\sigmagma/2}A(x)= \mathscr F^{-1}[|\xi|^{\sigmagma}\hat{A}](x),
\end{equation}
with $A\in S'({\mathbb R}^n) $
and $0<\sigmagma<n$. One can see Stein C^{\infty}te{stein} for a detailed theory on Riesz potentials.
In this work we will consider the Lebesgue spaces $L^p({\mathbb R})$, the Sobolev spaces $H^s({\mathbb R})$, and some Bochner spaces, $L^q([0,T],L^r({\mathbb R}))$, $L^q([0,T],W^{1,r}({\mathbb R}))$ and $C([0,T],L^2({\mathbb R}))$. \\
For the Borel-mesaurable functions $g(t,x)C^{1}lon [0,T]\times{\mathbb R}\to\C$, $f(x)C^{1}lon {\mathbb R}\to\C$, we define the norms of the spaces listed above:
\betagin{gather*}
\|f\|_{L^p} =\left(\int_{\mathbb R} |f|^p\,dx \right)^{1/p}, 1\leq p<\infty;\\
\|f\|_{L^\infty} =\supess_{\mathbb R} |f| ;\\
\|f\|_{H^s} = \|\mathscr F^{-1}[ \lambdangle \xi \rangle^s\hat f]\|_{L^2}= \|\lambdangle \xi \rangle^s\hat f\|_{L^2}, \,\,\,\,s\in{\mathbb R};\\
\|g\|_{L^q([0,T],L^r({\mathbb R}))} =\left(\int_{0}^{T}\|g(t)\|_{L^r}^{q} \,dt\right)^{1/q};\,\, 1\leq q,r<\infty\\
\|g\|_{L^q([0,T],W^{1,r}({\mathbb R}))} =\left(\int_{0}^{T}\|g\|_{W^{1,r}}^{q} \,dt\right)^{1/q};\,\, 1\leq q,r<\infty\\
\|g\|_{C([0,T],L^r({\mathbb R}))} = \sup_{[0,T]} \|g(t,\mathcal Dot)\|_{L^r}, \,\, 1\leq r\leq\infty.
\end{gather*}
\subsection{Linear estimates of the free Schr\"odinger equation}
We introduce some basic fact about linear Schr\"odinger equation
\betagin{subnumcases}
{(LS)}
i\partialartial_t\Psi+\frac{1}{2}\Deltalta\Psi =0 \notag\\
\Psi(0,\mathcal Dot)=f, \notag
\end{subnumcases}
If $f\in S({\mathbb R}^n)$, the Cauchy problem (LS) has a unique solution, $$\Psi(t)=S(t)f,$$ where $$S(t)={\rm e}^{i\frac{\Deltalta}{2}t}C^{1}lon S({\mathbb R}^n)\to S({\mathbb R}^n),$$ is defined by Fourier transform:
\betagin{equation*}
S(t)f= \mathscr F^{-1}({\rm e}^{-i\frac{|\xi|^2}{2}t}\hat f).
\end{equation*}
By duality we can extend $S(t)$ to $S'({\mathbb R}^n)$.\\
In addition, by the proprieties of Fourier transform in $S'({\mathbb R})$, we can rewrite the solution $\Psi$ as following
\betagin{equation*}
\Psi(t)= S(t)f= \mathscr F^{-1}({\rm e}^{-i\frac{|\xi|^2}{2}t})* f= \frac{1}{(2\partiali i t)^{n/2}}{\rm e}^{i\frac{|\mathcal Dot|^2}{2t}}*f.
\end{equation*}
We summarize the \emph{time-dispersion} estimates of linear Schr\"odinger in the following Lemma.
\betagin{lem}
Let $2 \leq p \leq \infty$, $\frac{1}{p}+\frac{1}{p'}= 1$. There exists $C>0$, such that, for all $f\in L^1 \mathcal Ap L^2$,
\betagin{align*}
\|S(t)f\|_{L^p} & \leq C \frac{1}{t^{n/2-n/p}}\|f\|_{L^{p'}}.
\end{align*}
\end{lem}
\betagin{remark}
The Schr\"odinger group, $S(t)$, generates a dispersive effect on initial data, i.e, the initial pulse spreads out after a while because of plane waves with large wave
number travel faster than those with a smaller one.
\end{remark}
Now we summarize the decay estimates for the linear nonhomogeneous Schr\"odinger equation
\betagin{subnumcases}
{\,}
i\partialartial_t\Psi+\frac{1}{2}\Deltalta\Psi =F(t,x) \,\,\,\,\,\,(t,x)\in {\mathbb R}\times {\mathbb R}\notag\\
\Psi(0,\mathcal Dot)=f, \notag
\end{subnumcases}
in the following Lemma:
\betagin{lem}
Let $2\leq p\leq \infty $, $f\in L^{p'}$ and $F\in {L^\infty [(0,T),L^{p'}]}$ for $T>0$. Then there exists a constant $C=C(p)>0$ such that
\betagin{equation*}
\|\Psi(t)\|_{L^p}\leq t^{-1/2+1/p}\|f\|_{L^{p'}}+C \int_{0}^{t}(t-s)^{-1/2+1/p}\|F(s)\|_{L^{p'}}\,ds,
\end{equation*}
for $t\in (0,T)$.
\end{lem}
These dispersive estimates are remarkable but is not quite handy for solving the nonlinear problems. In a perturbative regime we need to space-time estimates. We begin by introducing the notion of admissible pair.
\betagin{definition}
We say that a pair $(q,r)$, is admissible if
\betagin{equation*}\lambdabel{admiss}
\frac{2}{q}=\frac{n}{2}-\frac{n}{r},
\end{equation*}
and
\betagin{align*}
2\leq r \leq \frac{2n}{n-2} \,\,\,&\text{ if } n\geq 3, \\
2\leq r <\infty \,\,\,&\text{ if } n=2 ,\\
2\leq r \leq \infty \,\,\,&\text{ if } n=1.
\end{align*}
\end{definition}
\betagin{remark}
Scaling argument for Strichartz estimates say us that these restrictions on the pair $(q,r)$ are necessary. The pairs $(2,\frac{2n}{n-2})$, $n\geq 3$, are called \emph{endpoint}.
\end{remark}
\betagin{thm}[Strichartz's estimates]
Let $(q,r)$, $(\tilde q,\tilde r)$ be two Schr\"odinger admissible pairs. Then, the following estimates hold:
\betagin{gather}
\| S(t)f\|_{L^q({\mathbb R} ,L^r({\mathbb R}^n))}\leq C \|f\|_{L^2({\mathbb R}^n)},\lambdabel{1str}\\
\| \int_{{\mathbb R}}S^{*}(t)F(t)\,dt\|_{L^2({\mathbb R}^n)}\leq C \|F\|_{L^{\tilde q'}({\mathbb R} ,L^{\tilde r'}({\mathbb R}^n))},\lambdabel{2str}\\
\|\int_{0}^{t}S(t-s)F(s)\,ds \|_{L^q_t({\mathbb R} ,L^r_x({\mathbb R}))}\leq C \| F \|_{L^{\tilde q'}({\mathbb R} ,L^{\tilde r'}({\mathbb R}^n))}\lambdabel{3str}.
\end{gather}
With $S^{*}(t)={\rm e}^{-i\frac{\Deltalta}{2}t}$ we denote the adjoint of $S(t)= {\rm e}^{i\frac{\Deltalta}{2}t}$.
\end{thm}
For a complete proof of the Theorem one can see C^{\infty}te{keeltao}.
\betagin{remark}\lambdabel{crucial}
The pairs $(q,r)$, $(\tilde q,\tilde r)$ are not related to each other in
the Strichartz's estimates. This turns out to be a crucial fact for the nonlinear applications.
\end{remark}
\subsection{A class of semilinear Schr\"odinger equations}
One of the most important class of nonlinear Schr\"odinger equations are the following:
\betagin{equation}\lambdabel{nonlschr}
i\partialartial_t\Psi+\frac{1}{2}\Deltalta\Psi =\partialm |\Psi|^{\gammamma-1}\Psi,
\end{equation}
with $\gammamma>1$.\\
As we can see in \eqref{nonlschr}, the evolution is a competition between the linear part and the nonlinear one. So we can expect that the evolution has \emph{linearly dominated behavior} or \emph{nonlinearly dominated behavior} or \emph{intermediate behavior}. Nonlinear physics phenomena are characterized by a variety
of complex phenomena; e.g. shock-waves, solitons and instabilities, hence, in a predominantly nonlinear regime we can expect a tricky scenario.\\
So, we are interested in classifying the nonlinearity. Two basic features are crucial: the conservation laws and the natural scale-invariance of the equation.\\
Thanks to the structure of the equation \eqref{nonlschr}, in $H^1({\mathbb R})$, the following conservation laws hold:
\betagin{itemize}
\item Mass conservation:
\betagin{equation*}
\|\Psi(t)\|_{L^2}= \|\Psi(0)\|_{L^2},
\end{equation*}
\item Energy conservation:
\betagin{equation*}
E(\Psi(t))= \frac{1}{4}\|\nabla \Psi (t)\|_{L^2}^2\partialm\frac{1}{\gammamma+1}\|\Psi(t)\|_{L^{\gammamma+1}}^{\gammamma+1}=E(\Psi(0)),
\end{equation*}
\item Momentum conservation:
\betagin{equation*}
\rm{Im} \left( \int \nabla \Psi (t,x)\bar{\Psi}(t,x)\,dx\right)=\rm{Im} \left( \int \nabla \Psi (0,x)\bar{\Psi}(0,x)\,dx\right).
\end{equation*}
\end{itemize}
Using the scale-invariance for \eqref{nonlschr}
\betagin{equation}\lambdabel{scalingsolution}
\Psi_{\lambdambda}(t,x)=\lambdambda^{2/(1-\gammamma)}\Psi(\frac{t}{\lambdambda^2},\frac{x}{\lambdambda}),
\end{equation}
for $\lambdambda>0$, we can classify the conservation laws as \emph{subcritical}, \emph{critical} (scale-invariant), or \emph{supercritical}. \\
In particular, in one dimension, using $L^2$-conservation (similarly for $H^s$ conservation), we have
\betagin{equation}\lambdabel{rescaling}
\|\Psi_{\lambdambda}(t,\mathcal Dot)\|_{L^2}=\lambdambda^\frac{5-\gammamma}{2(1-\gammamma)}\|\Psi(t,\mathcal Dot)\|_{L^2}.
\end{equation}
We can give the following definition:
\betagin{definition}
Let $\gammamma >1$, we say that
\betagin{itemize}
\item $\gammamma$ is $L^2$-subcritical if $1<\gammamma< 5$,
\item $\gammamma$ is $L^2$-critical if $\gammamma= 5$,
\item $\gammamma$ is $L^2$-supercritical if $\gammamma>5$.
\end{itemize}
\end{definition}
The rescaling relation \eqref{rescaling}, maight be interpreted as following: in subcritical case, the norm of the initial data can be made small while the interval of time is made longer; in supercritical case, the norm grows as the time interval gets longer; finally, in the critical case, the norm is invariant while the interval of time is made longer or shorter: this looks like a limit situation for well-posedness results.
Another most important distinction is whether the equation is \emph{focusing} ($\alphapha=-1$) or \emph{defocusing} ($\alphapha=1$). We can not make an exact distinction, but, broadly, in a defocusing case, the nonlinearity has the same sign as the linear component, thus, the dispersive effects of the linear equation are amplified. On the contrary, in the focusing case the dispersive effects can be attenuated, halted (stationary or travelling waves can occur) or even reversed (blow up of solution in finite time can occur).
Except to $1$-dim cubic NLS, the equations are not completly integrable. We are interestested in the fundamental question of \emph{well-posedness} that is often closely intertwined with the quantitative estimates (\emph{a priori estimates}).
For some literature on local existence results in the subcritical case, one can see C^{\infty}te{ginibrevelo79}, C^{\infty}te{kato}, C^{\infty}te{tsutsumi87} and C^{\infty}te{cazenave}. For local existence in the critical case, one can see C^{\infty}te{cazenave90}, C^{\infty}te{cazenave}. Finally, for the global critical case one can see C^{\infty}te{ginibrevelo}, C^{\infty}te{cazenave}. \\
Now we state some of the results that will come in handy later.
\betagin{thm}[$L^2$
well-posedness] (Cazenave C^{\infty}te{cazenave}, Section 4.6)
Let $f\in L^2({\mathbb R})$. The following statements hold:
\betagin{itemize}
\item Let $1<\gammamma<5$ and $\alphapha=\partialm 1$. Then there exists a unique global solution $\Psi\in C({\mathbb R},L^2)\mathcal Ap L^q ({\mathbb R},L^r({\mathbb R}))$;
\item Let $\gammamma=5$ and $\alphapha=\partialm 1$. Then there exists $\deltalta>0$, quite small, such that, if $\|f\|_{L^2}\leq \deltalta$ we have a unique global solution $\Psi\in C({\mathbb R},L^2)\mathcal Ap L^q ({\mathbb R},L^r({\mathbb R}))$.
\end{itemize}
\end{thm}
\betagin{thm}[$H^1$ well-posedness](Cazenave C^{\infty}te{cazenave}, Section 4.4)
Let $f\in H^1({\mathbb R})$. The following statements hold:
\betagin{itemize}
\item Let $1<\gammamma <5$ and $\alphapha=\partialm 1$. Then there exists a unique global solution $\Psi\in \C({\mathbb R},H^2)\mathcal Ap L^q ({\mathbb R},W^{1,r}({\mathbb R}))$;
\item Let $\gammamma=5$ and $\alphapha=-1$. Then there exists $\deltalta>0$, quite small, such that, if $\|f\|_{L^2}\leq \deltalta$ we have a unique global solution $\Psi\in C({\mathbb R},H^1)\mathcal Ap L^q({\mathbb R}, W^{1,r}({\mathbb R}))$.
\end{itemize}
\end{thm}
In $H^1$-theory, about global existence results for small data when $\gammamma=5$, looking for the sharp mass $\deltalta$ which allows to obtain global well-posedness is interlinked with the problem of the best constant in the Gagliardo-Nirenberg interpolation estimates.
\section {Well-posedness of the Cauchy problem}
We now turn to the system (FSPS). \\
We say that an initial-value problem for a partial differential equation is well-posed in $H^s$ if:
\betagin{itemize}
\item there exists a time interval, $[0,T]$, in which the problem in fact has an $H^s$ solution,
\item the solution is unique,
\item the solution depends continuosly on the initial data.
\end{itemize}
We seek for an handy formulation of the Cauchy problem $(FSPS)$ to begin with.
\betagin{rem}\lambdabel{propsps} (Stein C^{\infty}te{stein}, Section 5.1)
Let $0<\sigmagma <1$. Then
\betagin{equation}\lambdabel{trasformatafraz}
\mathscr F (|x|^{-\sigmagma})(\xi)=C(\sigmagma) |\xi|^{-(1-\sigmagma)},
\end{equation}
where
$$
C(\sigmagma)= \sqrt{\partiali}2^{1-\sigmagma}\frac{\Gammamma(\frac{1-\sigmagma}{2})}{\Gammamma(\frac{\sigmagma}{2})}.
$$
The equation \eqref{trasformatafraz} is understood in the sense of the tempered distributions.
\end{rem}
\betagin{lem}[Hardy-Littlewood-Sobolev Inequality]\lambdabel{hls}
Let $fC^{1}lon {\mathbb R}^n\to\C$ a mesaurable function, $1<r<p<\infty$ and $f\in L^r({\mathbb R}^n)$. Let $0<\betata <n$. Then there exists $C=C_{p,\betata,n}$ a positive constant such that
\betagin{equation}\lambdabel{hlsi}
\left\|\frac{1}{|y|^{\betata}}*f\right\|_{L^p}\leq C \|f\|_{L^r},
\end{equation}
where,
$\frac{1}{p}=\frac{\betata}{n}+\frac{1}{r}-1$.
\end{lem}
For a proof of this Lemma see Stein C^{\infty}te{stein}.
\betagin{prop}
Let $0<\sigmagma<1$, $T>0$ and let $(q,r)$ be an admissible pair with $2<r<\frac{2}{\sigmagma}$. Suppose that $\PsiC^{1}lon[0,T]\times{\mathbb R}\to\C$ a known function and $\Psi\in L^q([0,T],L^r)$.\\
Then, there exists a unique electric potential $A_0$,
$$A_0C^{1}lon[0,T]\times {\mathbb R}\to {\mathbb R},$$
\betagin{equation}\lambdabel{convolform}
A_0(t,x)=C(\sigmagma)[|\mathcal Dot|^{-(1-\sigmagma)}*|\Psi|^2](t,x),
\end{equation}
\betagin{equation}
A_0 \in L^{q/2}([0,T],L^{r/(2-r\sigmagma)}({\mathbb R})),
\end{equation}
solution of the fractional Poisson equation
\betagin{equation}\lambdabel{fractionalpoisson}
(-\Deltalta)^{-\sigmagma/2}A_0=|\Psi|^2.
\end{equation}
\end{prop}
\betagin{proof}
By the \eqref{fraclapl} and by \eqref{fractionalpoisson} we have that
\betagin{equation}
\hat A_0(\xi)=|\xi|^{-\sigmagma} (|\Psi|^2 \hat)(\xi).
\end{equation}
Thanks to the Lemma \ref{propsps} and passing under Fourier antitransform, we have that
\betagin{equation*}
A_0(t,x)=C(\sigmagma)\left[ |\mathcal Dot|^{-(1-\sigmagma)}*|\Psi|^2(t,\mathcal Dot)\right] \,(x).
\end{equation*}
Hence the equality \eqref{convolform} has been proved.\\
By the hypothesis on $\Psi$ and by the H\"older inequality we have that $|\Psi|^2\in L^{q/2}([0,T],L^{r/2}_x({\mathbb R}))$. So, the Lemma \ref{hlsi} tells us that $A_0 \in L^{q/2}([0,T],L^{\frac{r}{2-r\sigmagma}}_x({\mathbb R}))$. \\
The unicity of the electric potential $A_0$ is guaranteed by the unicity of the wave function $\Psi$ and by injectivity of Fourier transform.
\end{proof}
Now we bring us back to study the following Cauchy problem:
\betagin{subnumcases}
{\,}
i\partialartial_t\Psi+\frac{1}{2}\Deltalta\Psi =C(\sigmagma)[|\mathcal Dot|^{-(1-\sigmagma)}*|\Psi|^2]\Psi +\alphapha |\Psi|^{\gammamma-1}\Psi \, \lambdabel{sps1}\\
\Psi(0,\mathcal Dot)=f. \lambdabel{sps3}
\end{subnumcases}
First of all we specify which kind of solutions we are searching for. We give the following definition:
\betagin{definition}
Let $X_0$ be a Banach space, $f\in X_0$ and $T>0$. We consider the map
\betagin{equation}\lambdabel{mappa}
\mathscr H[\Psi](t)=S(t)f-iC(\sigmagma)\int_{0}^{t}S(t-s)[|\mathcal Dot|^{-(1-\sigmagma)}*|\Psi|^2]\Psi(s)\,ds-i\alphapha\int_{0}^{t}S(t-s) |\Psi|^{\gammamma-1}\Psi(s) \,ds,
\end{equation}
with $t\in[0,T]$.\\
We say that $\Psi\in C([0,T],X_0)$ is a local solution of \eqref{sps1}-\eqref{sps3} if $\Psi$ is a fixed point of the map $\mathscr H$, i.e. $\Psi=\mathscr H (\Psi)$.
\end{definition}
\subsection{Well-posed problems with initial data in $L^2$}
We start with $L^2$-theory. By means of the contraction theory, we can prove, in subcritical case, a local existence result (Theorem \ref{lex}). Actually, in this case , thanks to the mass conservation, we can extend the local result to a global one (Corollary 4.9). On the other hand, in the $L^2$-critical case we can prove a local result with large data (Theorem \ref{lecld}) and a global result with small data (Theorem \ref{gecsd}). \\
Notice that the $L^2$-theory does not see the difference between the defocusing case ($\alphapha=1$) and the focusing case ($\alphapha=-1$).
\betagin{thm}[local existence $L^2$-subcritical]\lambdabel{lex}
Let $1<\gammamma<5$, $\alphapha=\partialm1$ and $f\in L^2({\mathbb R})$. \\
Then, there exists an interval $I_\gammamma \subseteq (0,1)$, such that, if $\sigmagma\in I_\gammamma$, one can find a time $T=T(\|f\|_{L^2})>0$ and a unique wave function $\Psi$,
$$\PsiC^{1}lon[0,T]\times {\mathbb R} \to \C ,$$
solution of the problem \eqref{sps1}-\eqref{sps3}. \\
In addition, we have
\betagin{equation}\lambdabel{regolonda}
\Psi \in C([0,T],L^2({\mathbb R}))\mathcal Ap L^q([0,T],L^r({\mathbb R})),
\end{equation}
for any $(q,r)$ admissible pair.
\end{thm}
\betagin{proof}
As we mentioned before, we are going to proof the theorem by means of a contraction argument. Hence, we have to introduce a suitable Banach space, $X_{0}$, and then we have to prove that $\mathscr HC^{1}lon X_0\to X_0$, defined in \eqref{mappa}, is a contraction.\\
Unlike classical NLS, we have also a nonlocal term
\betagin{equation*}\lambdabel{nonlocalterm}
C(\sigmagma)[|\mathcal Dot|^{-(1-\sigmagma)}*|\Psi|^2]\Psi,
\end{equation*}
that will bring necessary modification.\\
Let $f\in L^2$ be the initial data. \\
Let $T=T(\|f\|_{L^2})>0$ and $M=M(\|f\|_{L^2})$ be two positive constant which will be defined later and, let $(q,r)$ be an admissible pair
\betagin{equation}
\frac{1}{q}=\frac{1}{4}-\frac{1}{2r}.
\end{equation} \\
We will denote the spaces $ L^\infty([0,T],L^2({\mathbb R}))$ and $L^q([0,T],L^r({\mathbb R}))$ as $L^\infty L^2$ and $L^q L^r$ respectively to simplify the notation. \\
Let $X_0$ be the Banach space defined as follows:
\betagin{equation*}
X_0=\left\lbrace \Psi \in L^\infty L^2\mathcal Ap L^q L^r | \,\Psi(0)=f, \,\|\Psi\|_{X_0}= \|\Psi\|_{L^\infty L^2}+ \|\Psi\|_{L^q L^r}\leq M\right\rbrace.
\end{equation*}
We will prove that $\mathscr H$ is a contraction on $X_0$.\\
\textbf{Step 1.} (Looking for a working admissible pair)\\
To apply classical estimates that work also on nonlocal term, we have to make some considerations.\\
In order that the Strichartz estimates might give back the desired norm, we have to choose the admissible pair, $(q,r)$, such that we can be able to construct the pairs $(\tilde{q}',\tilde{r}')$ and $(\tilde{q}'_1,\tilde{r}'_1)$ as follows.
At first we consider the nonlocal term. By the Hardy-Littlewood-Sobolev hypothesis and by the condition of Schr\"odinger admissibility on $(\tilde{q}',\tilde{r}')$ we have:
\betagin{subnumcases}
{\,}
0<\sigmagma<1, \,\,\,\notag\\
2<r<\frac{2}{\sigmagma}\notag\\
\frac{3}{1+\sigmagma} \leq r \leq \frac{6}{1+2\sigmagma} \notag\\
\frac{1}{\tilde{r}'}=\frac{3}{r}-\sigmagma\notag\\
\frac{1}{\tilde{q}'}= \frac{1}{x}+\frac{3}{q}= \frac{1+\sigmagma}{2}+\frac{3}{q}\notag\\
\frac{1}{\tilde{q}'}=\frac{5}{4}-\frac{1}{2\tilde{r}'}.\notag
\end{subnumcases}
On the other hand, the local nonlinearity brings the following conditions:
\betagin{subnumcases}
{\,}
\gammamma\leq r \leq 2\gammamma, \notag\\
\frac{1}{\tilde{r}'_1}=\frac{\gammamma}{r}\notag\\
\frac{1}{\tilde{q}'_1}= \frac{1}{x}+\frac{\gammamma}{q}= \frac{5-\gammamma}{2}+\frac{3}{q}\notag\\
\frac{1}{\tilde{q}'_1}=\frac{5}{4}-\frac{1}{2\tilde{r}'_1}\notag
\end{subnumcases}
Hence, we would of course want the interval
\betagin{equation*}
I_{\sigmagma,\gammamma}= (2,\frac{2}{\sigmagma})\mathcal Ap[\frac{3}{1+\sigmagma}, \frac{6}{1+2\sigmagma}]\mathcal Ap [\gammamma,2\gammamma]
\end{equation*}
to be not empty for all $\sigmagma\in(0,1)$ and for all $\gammamma\in(0,5)$. It is not possible. Indeed, if $\gammamma=3$, we have that the set $I_{\sigmagma,\gammamma}\neq \emptyset$ iff $\sigmagma\in (0,1/2]$.\\
We can represent the relations between $\gammamma$ and $\sigmagma$ such that $I_{\sigmagma,\gammamma}$ is not empty. In particular, in the picture below, the coloured region rapresents the set of the pairs $(\sigmagma,\gammamma)$ for wich our proof works.
\betagin{figure}[h!hhh]
\includegraphics[width=0.8\textwidth]{sigmagammalastversion.jpg}
\end{figure}
Hence, if we fix $\gammamma\in (1,5)$, there exists an interval $I_\gammamma$, such that, if $\sigmagma\in I_\gammamma$, then we can construct the admissible pairs $(\tilde{q}',\tilde{r}')$ and $(\tilde{q}'_1,\tilde{r}'_1)$ as specified above.
Without loss of generality, in the following, we will consider $\sigmagma$ small enough, in particular, $\sigmagma\in(0,1/10]$ and $3/2\leq\gammamma<5$.
\textbf{Step 2.}\,($\mathscr H$ is a contraction on $X_0$)\\
Proof that $X_0$ is mapped into itself by $\mathscr H$:
\betagin{align}
\|\mathscr H \Psi\|_{X_0}&\leq \| S(t)f \|_{X_0} +C\|\int_{0}^{t}S(t-s) [|\mathcal Dot|^{-(1-\sigmagma)}*|\Psi|^2](s,\mathcal Dot)\Psi(s)\,ds\|_{X_0}\lambdabel{pallainpalla}\\
\,\,\,\,\,\,&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, +\|\int_{0}^{t}S(t-s) |\Psi|^{\gammamma-1}\Psi(s)\,ds \|_{X_0}\notag\\
&\leq C \|f\|_{L^2} +C\left\|[|\mathcal Dot|^{-(1-\sigmagma)}*|\Psi|^2] \Psi \right\|_{L^{\tilde q'}L^{\tilde r'}}+C\||\Psi|^{\gammamma-1}\Psi\|_{L^{\tilde q'_1}L^{\tilde r'_1}} \notag\\
&\leq C \|f\|_{L^2}+C\left\| \||\mathcal Dot|^{-(1-\sigmagma)}*|\Psi|^2 \|_{L^{r/(2-r\sigmagma)}}\| \Psi\|_{L^r} \right\|_{L^{\tilde q'}}+C\left\| \| \Psi\|_{L^r}^\gammamma \right\|_{L^{\tilde q'_1}}\notag\\
&\leq C \|f\|_{L^2}+C \left\| \| \Psi\|^3_{L^r} \right\|_{L^{\tilde q'}}+C\left\| \| \Psi\|_{L^r}^\gammamma \right\|_{L^{\tilde q'_1}}\notag\\
&\leq C \|f\|_{L^2}+C T^{\frac{1}{2}+\frac{\sigmagma}{2}}\left\| \Psi\right\|^3_{L^{q}L^r}+CT^{\frac{5-\gammamma}{2}}\left\| \Psi\right\|^\gammamma_{L^{q}L^r},\notag
\end{align}
where we have used the Strichartz estimates, the H\"older inequality and the Lemma \ref{hlsi}. Note that $C$ depends on the constants involved in \eqref{1str}-\eqref{3str} and \eqref{hlsi}.\\
We put $M=3C\|f\|_{L^2}$. If $T=T(\|f\|_{L^2})$ is quite small, then we get
\betagin{equation}\lambdabel{finita}
\| \mathscr H \Psi\|_{X_0}\leq 3C\|f\|_{L^2}=M.
\end{equation}
Now we want to prove that $\mathscr H$ is a contraction.\\
We have that the following estimates hold:
\betagin{align*}
\left| (|\mathcal Dot|^{-(1-\sigmagma)}*|\Psi_1|^2) \Psi_1-(|\mathcal Dot|^{-(1-\sigmagma)}*|\Psi_2|^2) \Psi_2 \right|\leq& (|\mathcal Dot|^{-(1-\sigmagma)}*|\Psi_1|^2)\left|\Psi_1-\Psi_2 \right| \\ &+(|\mathcal Dot|^{-(1-\sigmagma)}*\left[ (|\Psi_1|-|\Psi_2|)(|\Psi_1|+|\Psi_2|)\right])|\Psi_2|
\end{align*}
and
\betagin{equation*}
\left|\Psi_{1}|\Psi_{1}|^{\betata-1} -\Psi_{2}|\Psi_{2}|^{\betata-1}\right| \leq C |\Psi_1-\Psi_2|(|\Psi_1|^{\betata-1}+|\Psi_2|^{\betata-1}),
\end{equation*}
for $\betata>1$. As in \eqref{pallainpalla}, we can prove that $\mathscr H$ is a contraction. \\
Indeed, let $\Psi_1,\Psi_2\in X_0$, we have that
\betagin{align*}
\| \mathscr H (\Psi_1)-\mathscr H (\Psi_2)\|_{X_0}&\leq C T ^{1/2+\sigmagma/2}\|\Psi_1-\Psi_2\|_{X_0}(\|\Psi_1\|^2_{X_0}+\|\Psi_2\|^2_{X_0})\\
&\,\,\,\,\,+CT^{\frac{5-\gammamma}{2}}\|\Psi_1-\Psi_2\|_{X_0}(\|\Psi_1\|^{\gammamma-1}_{X_0}+\|\Psi_2\|^{\gammamma-1}_{X_0})\\
& \leq 2CT^{1/2+\sigmagma/2}(3C\|f\|_{L^2})^2\|\Psi_1-\Psi_2\|_{X_0}\\
& \,\,\,\,+ 2CT^{\frac{5-\gammamma}{2}}(3C\|f\|_{L^2})^{\gammamma-1}\|\Psi_1-\Psi_2\|_{X_0}.
\end{align*}
Choosing $T=T(\|f\|_{L^2})$ small enough, we get
\betagin{equation}\lambdabel{getcontraction}
\| \mathscr H (\Psi_1)-\mathscr H (\Psi_2)\|_{X_0}\leq \frac{1}{2}\|\Psi_1-\Psi_2\|_{X_0}.
\end{equation}
The Banach fixed point theorem guarantees the existence and uniqueness of $\Psi\in X_{0}$, such that $\mathscr H (\Psi)=\Psi$.\\
Hence, there exists a unique wave function $\Psi$, solution of Cauchy problem \eqref{sps1}-\eqref{sps3} and its continuity in time is immediate a posteriori by the \eqref{mappa}. Actually, we have had an additional regularity information: $\Psi\in L^\gammamma([0,T],L^\rho({\mathbb R}))$ for any $(\gammamma,\rho)$ admissible pair. It follows by Strichartz estimates \eqref{1str}-\eqref{3str} and by the \eqref{finita}:
\betagin{equation*}
\| \Psi\|_{L^\gammamma([0,T],L^\rho({\mathbb R}))}\leq 3C\|f\|_{L^2}.
\end{equation*}
\textbf{Step 3.} (Continuos dependence by initial data)\\
Now we deduce the continuous dependence from initial data to complete the local well-posedness of the Cauchy problem \eqref{sps1}-\eqref{sps3}.\\
Let $f,g\in L^2({\mathbb R})$. Let $\Psi(f)$ and $\Psi(g)$ be the solutions of the Cauchy problem \eqref{sps1} with initial data $f$ and $g$ respectively. Writing the solutions in the integral form \eqref{mappa}, a computation like in \eqref{getcontraction} tells us
\betagin{equation}
\| \Psi(f)-\Psi(g)\|_{X_0}\leq C\|f-g\|_{L^2}.
\end{equation}
\end{proof}
\betagin{rem}
Note that if $\sigmagma=\frac{2(3-\gammamma)}{\gammamma-1}$ the problem \eqref{sps1} is scale invariant. That is, if $\Psi $ solves \eqref{sps1}, then $\Psi_{\lambdambda}$, defined as in \eqref{scalingsolution}, is still a solution. So, in subcritical case, we may expect well-posedness beyond yellow region, at least on the path $\sigmagma=\frac{2(3-\gammamma)}{\gammamma-1}$ (green path in the picture above), with $\sigmagma\in(0,1)$. This may suggest looking for other skills to proof local well posed results.
\end{rem}
Once local existence is established, some natural issues are the following. What is the existence time of the solution? Can we extend the local solution to global one? Can blow-up phenomena occur? We will try to answer them.\\
At first, we state a conservation law.
\betagin{lem}[Conservation mass]\lambdabel{conservationmass}
Let $\Psi \in C([0,T],L^2({\mathbb R})) $ be a local solution of the Cauchy problem \eqref{sps1}-\eqref{sps3}. Then
\betagin{equation}\lambdabel{massa}
\|\Psi (t,\mathcal Dot)\|_{L^2}= \|f\|_{L^2},
\end{equation}
for all times $t\in [0,T]$.
\end{lem}
\betagin{proof}
We assume $\Psi \in C([0,T],H^1({\mathbb R})) $ and we multiply by $\bar{\Psi}$ the equation \eqref{sps1}. The \eqref{massa} follows by integration by parts. Density arguments give us the general statement (for a detailed proof see C^{\infty}te{cazenave} Section 4.6).
\end{proof}
Our goal is to try to extend the solution to all times. At first we define the \emph{maximal solution} using the uniqueness for small time.
\betagin{definition}
Let $f\in L^2$. Let
\betagin{align*}
T_{max}&= \sup \{T>0, \text{such that \eqref{sps1} has a solution in } [0,T]\},\\
T_{min}&= \sup \{T>0, \text{such that \eqref{sps1} has a solution in } [-T,0]\}.
\end{align*}
The uniqueness for small time allows us to define the \emph{maximal solution} $$\Psi \in C([-T_{min},T_{max}],L^2).$$
\end{definition}
\betagin{prop}[Blow-up alternative]
Let $T_{max}<\infty$ (respectively , if $T_{min}<\infty$ ), then, under the hypothesis of the Theorem \ref{lex} we have
\betagin{equation}\lambdabel{blowup}
L^{\infty}m_{t \nearrow T_{max}}\|\Psi(t,\mathcal Dot)\|_{L^2({\mathbb R})}=\infty \,\,\,(L^{\infty}m_{t \searrow T_{min}}\|\Psi(t,\mathcal Dot)\|_{L^2({\mathbb R})}=\infty).
\end{equation}
\end{prop}
\betagin{proof}
Let $T_{max}<\infty$. Assume that there exist $M<\infty$ and a sequence $t_n \nearrow T_{max}$ such that $\|\Psi (t_n)\|_{L^2({\mathbb R})}\leq M$. \\
We consider $ k \in \mathbf N $, such that $t_k+T(M)>T_{max}$, where $[0,T(M)]$ denotes the maximal existence interval of a solution with initial data of $L^2$-norm equals to $M$. \\
By Theorem \ref{lex} and starting from $f=\Psi(t_k)$, we can extend $\Psi$ up to $t_k+T(M)$, which contradicts maximality.
\end{proof}
\betagin{cor}\lambdabel{gser}
Let $f\in L^2$, $1<\gammamma<5$ and $\sigmagma\in I_\gammamma$. The Cauchy problem \eqref{sps1}-\eqref{sps3} has a unique global solution $\Psi \in C({\mathbb R},L^2({\mathbb R}))\mathcal Ap L^q({\mathbb R},L^r({\mathbb R}))$, for any $(q,r)$ admissible pair.
\end{cor}
\betagin{proof}
By the mass conservation and by the blow-up alternative we have that the local solution is actually global.
\end{proof}
\betagin{thm}[local existence $L^2$-critical with large data]\lambdabel{lecld}
Let $f\in L^2$, $\gammamma=5$ and $\sigmagma\in(0,1/10]$. There exists a maximal interval $(-T_{min},T_{max})$, $T_{min}=T_{min}(f)$ and $T_{max}=T_{max}(f)$, such that the Cauchy problem \eqref{sps1}-\eqref{sps3} has a unique solution ,$\Psi$, such that
$$\Psi \in C([-T_{min},T_{max}],L^2({\mathbb R}))\mathcal Ap L^q([-T_{min},T_{max}],L^r({\mathbb R})),$$
for any $(q,r)$ admissible pair.
\end{thm}
\betagin{proof}
We proceed in a similar way to how we did in the subcritical case and we use the same notations of the Theorem \ref{lex}. The difficulty, in the critical case, lies in the local nonlinear term $|\Psi|^{4}\Psi$. Indeed, let $T>0$, by Strichartz estimates we have that
\betagin{equation*}
\| \mathscr H (\Psi)\|_{L^{q}([0,T],L^r)}\leq \|S(t)f\|_{L^{q}([0,T],L^r)}+C T^{\frac{1+\sigmagma}{2}}\left\| \Psi\right\|^{3}_{L^{q}([0,T],L^r)} +C\left\| \Psi\right\|^{5}_{L^{q}([0,T],L^r)}.
\end{equation*}
By the Strichartz estimate \eqref{1str} and by absolute continuity of the Lebesgue integral, if $T$ is suitably small, we have that $\|S(t)f\|_{L^{q}([0,T],L^r)}<\deltalta$, for some small $\deltalta$ depending on $f$ and on the constant in the Strichartz estimates. Hence,
\betagin{equation}\lambdabel{palla}
\| \mathscr H (\Psi)\|_{L^{q}([0,T],L^r)}\leq \deltalta+\deltalta +C\left\| \Psi\right\|_{L^{q}([0,T],L^r)}^{5}.
\end{equation}
We choose $M=3\deltalta$, for small $\deltalta$, i.e. for time interval sufficiently small. So, we have a unique fixed point $\Psi\in L^{q}([0,T],L^r) $, which locally solves the Cauchy problem. \\
In order to conclude the proof we will prove that $\Psi$ is actually also $L^{\infty}([0,T],L^2)$.\\
For the \eqref{palla}, we have that $\|\Psi\|_{L^{q}([0,T],L^r)}<\infty$. By Strichartz estimates we have
\betagin{equation*}
\|\Psi\|_{L^\infty([0,T],L^2)}\leq \|f\|_{L^2}+C T^{\frac{1+\sigmagma}{2}}\left\| \Psi\right\|^{3}_{L^{q}([0,T],L^r)}+C\left\| \Psi\right\|_{L^{q}([0,T],L^r)}^{5}.
\end{equation*}
\end{proof}
\betagin{thm}[global existence $L^2$-critical small data]\lambdabel{gecsd}
Let $f\in L^2({\mathbb R})$, $\gammamma=5$ and $\sigmagma\in(0,1/10]$. There exists a small $\deltalta>0$ such that, if $\|f\|_{L^2}\leq \deltalta$, then the Cauchy problem \eqref{sps1}-\eqref{sps3} has a unique global solution $\Psi \in C({\mathbb R},L^2({\mathbb R}))\mathcal Ap L^q({\mathbb R},L^r({\mathbb R}))$, for any $(q,r)$ admissible pair.
\end{thm}
\betagin{proof}
By the condition $\gammamma=5$ follows that
\betagin{equation*}
\|\mathscr H \Psi \|_{X_0}\leq \deltalta +C T^{\frac{1+\sigmagma}{2}}\left\| \Psi\right\|^{3}_{X_0}+\|\Psi\|_{X_0}^5.
\end{equation*}
So, if we choose $T=T(\deltalta)$ and $M= 3C \deltalta$, for $\deltalta$ sufficiently small we have that the ball with radius $M$ in $X_0$ is mapped into itself by $\mathscr H$. Similarly we prove that $\mathscr H $ is a contraction. As in subcritical case we deduce first the local well-posedness and then the global result.
\end{proof}
\subsection{Well-posed problems with initial data in $H^1$}
Here we want to perform the same previous results about well-posedness in the space $H^1({\mathbb R})$. In this case, with regard to global well-posed problem, the defocusing and focusing case are situations more different. In the defocusing case, thanks to contraction arguments and energy conservation, we have the same results of $L^2$-theory. Therefore, we focus our attenction on focusing case that, already in subcritical case, is quite complicated.\\
At first we construct the local solution in $C([0,T],H^1({\mathbb R}))$, in subcritical focusing and defocusing case, with a fixed point argument.
\betagin{thm}[local existence $H^1$-subcritical]\lambdabel{lth1}
Let $f\in H^1({\mathbb R})$, $1<\gammamma<5$ and $\alphapha=\partialm1$.\\
Then, there exists an interval $I_\gammamma\subseteq (0,1)$, such that, for all $\sigmagma\in I_\gammamma$ one can find a time $T=T(\|f\|_{H^1})>0$ and a unique wave function $\Psi$,
$$\PsiC^{1}lon[0,T]\times {\mathbb R}\to \C,$$
solution of the problem \eqref{sps1}-\eqref{sps3}. \\
In addition, we have
\betagin{equation}\lambdabel{regolonda}
\Psi \in C([0,T],H^1({\mathbb R}))\mathcal Ap L^q([0,T],W^{1,r}({\mathbb R})),
\end{equation}
for any $(q,r)$ admissible pair.
\end{thm}
\betagin{proof}
The construction of the local $H^1$-solution is entirely similar to the construction of the local solution in $L^2$-theory. We give only a sketch of the proof.
We introduce the Banach space $X_0$ defined as following:
\betagin{equation*}
X_0=\left\lbrace \Psi \in L^\infty H^1\mathcal Ap L^q W^{1,r}, \Psi(0)=f, \|\Psi\|_{X_0}= \|\Psi\|_{L^\infty H^1}+ \|\Psi\|_{L^q W^{1,r}}\leq M\right\rbrace.
\end{equation*}
Let $\Psi_1,\Psi_2\in X_0$. We have that the inequalities
\betagin{equation*}
\left|\nabla[\Psi_{1}|\Psi_{1}|^{\gammamma-1}] \right| \leq 2C |\Psi_1|^{\gammamma-1}|\nabla\Psi_1|,
\end{equation*}
and
\betagin{equation*}
\left|\nabla[ (|\mathcal Dot|^{-(1-\sigmagma)}*|\Psi_{1}|^2)\Psi_{1}] \right| \leq 2 [|\mathcal Dot|^{-(1-\sigmagma)}*(\Psi_1\nabla\Psi_1)]|\Psi_1|+(|\mathcal Dot|^{-(1-\sigmagma)}*|\Psi_1|^2)|\nabla\Psi_1|
\end{equation*}
come true almost everywhere.\\
Moreover, we have similar estimates for the difference of the gradients.
Thanks to these inequalities we construct the solution as a fixed point of the contraction map $\mathscr H$ in $X_0$ with the same arguments of the Theorem \ref{lex}.
\end{proof}
\betagin{definition}[Energy]
Let $\Psi \in C([0,T],H^1)$, with $T>0$. We define the \textit{energy} of the system \eqref{sps1}-\eqref{sps3} as follows:
\betagin{equation}\lambdabel{energy}
E(t)= \frac{1}{4}\|\nabla \Psi(t)\|_{L^2}^2+\frac{1}{4} \int_{{\mathbb R}}A_0|\Psi|^2(t,x)\,dx +\frac{\alphapha}{\gammamma+1}\int_{{\mathbb R}}|\Psi|^{\gammamma+1}(t,x)\,dx,
\end{equation}
for any $t\in [0,T]$. \\
Sobolev embedding ensure that the energy is well-defined.
\betagin{lem}[Conservation energy]
Let $\Psi \in C([0,T],H^1({\mathbb R})) $ be a local solution of the Cauchy problem \eqref{sps1}-\eqref{sps3}. Then
\betagin{equation}\lambdabel{energycons}
E(t)=E(0),
\end{equation}
for all times $t\in [0,T]$.
\end{lem}
\betagin{proof}
We assume $\Psi \in C^{1}([0,T],H^2)$. Multiplying by $\partialartial_t\bar{\Psi}$ the equation \eqref{sps1}, similarly to Lemma \ref{conservationmass}, we deduce the conservation energy. Thanks to continuous dependence on initial data guaranteed by the local well-posedness (Theorem \ref{lth1}), density arguments prove that the quantity \eqref{energy} is a constant during the evolution of the system.
\end{proof}
\end{definition}
\betagin{prop}[Blow-up alternative]\lambdabel{bualternative}
Let $T_{max}<\infty$ (respectively , if $T_{min}<\infty$ ), then, under the hypothesis of the Theorem \ref{lth1} we have
\betagin{equation}\lambdabel{blowup}
L^{\infty}m_{t \nearrow T_{max}}\|\Psi(t,\mathcal Dot)\|_{H^1({\mathbb R})}=\infty \,\,\,(L^{\infty}m_{t \searrow T_{min}}\|\Psi(t,\mathcal Dot)\|_{H^1({\mathbb R})}=\infty).
\end{equation}
\end{prop}
\betagin{rem}
In defocusing case, $\alphapha=1$, we have that the energy, $E(t)=E(0)$, is a positive constant. \\
So we have that
\betagin{equation}\lambdabel{h1finita}
\|\nabla \Psi\|_{L^2}^2 \leq E(0).
\end{equation}
Hence, thanks to \eqref{h1finita}, Lemma \ref{conservationmass} and by Proposition \ref{bualternative} we can extend the $H^1$-local solution to global one.
\end{rem}
\betagin{cor}[$H^1$ global existence - defocusing case]\lambdabel{gser}
Let $f\in H^1$, $1<\gammamma<5$, $\sigmagma\in I_\gammamma$ and $\alphapha=+1$. The Cauchy problem \eqref{sps1}-\eqref{sps3} has a unique global solution $\Psi \in C({\mathbb R},H^1({\mathbb R}))\mathcal Ap L^q({\mathbb R},W^{1,r}({\mathbb R}))$, for any $(q,r)$ admissible pair.
\end{cor}
Now we give more attenction to focusing case.
We have the following result.
\betagin{thm}\lambdabel{h1focus}
Let $f\in H^1({\mathbb R})$ and $\alphapha=-1$. \\
If $1<\gammamma <5$ and $\sigmagma\in I_\gammamma $, then, the Cauchy problem \eqref{sps1}-\eqref{sps3} has a unique global solution $\Psi \in C({\mathbb R},H^1({\mathbb R}))\mathcal Ap L^q({\mathbb R},W^{1,r}({\mathbb R}))$, for any $(q,r)$ admissible pair.\\
Otherwise, if $\gammamma=5$ and $\sigmagma\in I_5$, then, there exists $\deltalta>0$ such that, if $\|f\|_{L^2}<\deltalta$ then the Cauchy problem \eqref{sps1}-\eqref{sps3} has a unique global solution $\Psi \in C({\mathbb R},H^1({\mathbb R}))\mathcal Ap L^q({\mathbb R},W^{1,r}({\mathbb R}))$, for any $(q,r)$ admissible pair.
\end{thm}
\betagin{proof}
The local solution is found by means of a point fix argument in the proof of the Theorem \ref{lth1}. To conclude that the solution actually is global, since $L^2$-norm is conserved, it is sufficient to prove that the norm $\|\nabla \Psi(t,\mathcal Dot)\|_{L^2}$ does not blow up.\\
By the Gagliardo-Nirenberg inequality, there exists $C_{GN}>0$ (the sharp constant) such that
\betagin{equation}\lambdabel{gagliardon}
\|f\|_{\betata+1}^{\betata+1}\leq C_{GN} \|\nabla f\|_{L^2}^{(\betata-1)/2}\|f\|_{L^2}^{(\betata+3)/2},
\end{equation}
for $1\leq \betata <\infty$.\\
So, choosing $\betata=\gammamma$, we obtain that
\betagin{align*}
E(0)=E(t)\geq& \frac{1}{4}\|\nabla \Psi(t)\|_{L^2}^2+\frac{1}{4}\int_{{\mathbb R}}A_0(t)|\Psi(t)|^2\,dx-\frac{C_{GN} }{\gammamma+1} \|\nabla \Psi(t)\|_{L^2}^{(\gammamma-1)/2}\|\Psi(t)\|_{L^2}^{(\gammamma+3)/2}\\
\geq& \frac{1}{4}\|\nabla \Psi(t)\|_{L^2}^2\left( 1-4\frac{C_{GN}}{\gammamma+1}\|\nabla \Psi(t)\|_{L^2}^{(\gammamma-5)/2}\|\Psi(t)\|_{L^2}^{(\gammamma+3)/2} \right).
\end{align*}
Since the mass is costant, $\|\Psi(t)\|_{L^2}=\|f\|_{L^2}$, if $1<\gammamma <5$ ($\sigmagma\in I_\gammamma$), we have that the $H^1$-norm cannot blow up. Indeed, if $\|\nabla \Psi\|_{L^2}$ was large we would control it with the energy:
\betagin{equation}\lambdabel{wpH1}
\|\nabla \Psi(t)\|_{L^2}\leq C E(0),
\end{equation}
for some $C>0$. \\
So, the proof in subcritical case is complete.
In the critical case, $\gammamma= 5$ ($\sigmagma\in I_5$), we have that
\betagin{align*}
E(0)\geq& \frac{1}{4}\|\nabla \Psi(t)\|_{L^2}^2\left( 1-4\frac{C_{GN} }{\gammamma+1}\|\Psi(t)\|_{L^2}^{4} \right)\\
=& \frac{1}{4}\|\nabla \Psi(t)\|_{L^2}^2\left( 1-\frac{\|f\|_{L^2}^{4} }{\left( \sqrt[4]{\frac{2}{3}}\sqrt{\frac{\partiali}{2}}\right)^4}\right) .
\end{align*}
As a consequence, if we choose initial data with $L^2$-norm suitably small, $\|f\|_{L^2}<\deltalta$, with $\deltalta = \sqrt[4]{\frac{2}{3}}\sqrt{\frac{\partiali}{2}} $, we have the \eqref{wpH1}, and the proof is complete.
\end{proof}
\betagin{rem}
The sharp constant for the Gagliardo-Nirenberg inequality in one dimensional setting was derived by Nagy in 1941; Weinstein in 1983 solved the problem for higher dimensions.
\end{rem}
\betagin{rem}
The physical meaning of the Theorem \ref{h1focus} is that for waves propagating in a weakly focusing medium ($1<\gammamma<5$), the potential term in the energy, $ E_{pot}=\frac{-1}{\gammamma+1}\|\Psi\|^{\gammamma+1}_{L^{\gammamma+1}}$, is dominated by interaction term, $E_{int}=\frac{1}{4} \int_{{\mathbb R}}A_0|\Psi|^2(t,x)\,dx $, and by kinetic term, $ E_{kin}=\frac{1}{4}\|\nabla \Psi(t)\|_{L^2}^2$, in according to Gagliardo-Nirenberg inequality. \\
When $\gammamma=5$, the potential energy and the kinetic one, seems to balance out, so, global results, at least in the case of focusing nonlinear Schr\"odinger, are not guaranteed.
\end{rem}
\section{$L^4-L^\infty$ estimates of the solution of cubic (FSPS)}
In this section we will use $L^p-L^q$ Gronwall's inequalities to establish a decay estimate for the solution of the (FSPS) system. Compare with Cazenave, we have the following result.
\betagin{lem}\lambdabel{2gron}
Let $1\leq q<p\leq \infty$, $1\leq \rho <\infty$ with $\frac{1}{\rho}=\frac{1}{q}-\frac{1}{p}$, $C_1>0$ and $0<T\leq \infty$. We consider $a\in L^{\rho}(0,T)$ and $v$ a function that satisfies the following inequality:
\betagin{equation}\lambdabel{ipotesigro}
\|v\|_{L^p(0,t)}\leq C_1+\|av\|_{L^q(0,t)},
\end{equation}
for all $t\in (0,T)$. \\
Then
\betagin{equation}\lambdabel{gammagronwall}
\|v\|_{L^p(0,t)}\leq 2C_1 \Gammamma(2+2^{\rho}\|a\|^{\rho}_{L^{\rho}(0,t)}),
\end{equation}
for all $t\in (0,T]$.
\end{lem}
\betagin{proof}
Suppose $\|a\|_ {L^{\rho}(0,T)}\geq1/2$. We can partition the interval $(0,T)$ into $n$ parts, with $n\geq 2$, such that $(\tau_{k})_{\{0\leq k\leq n\}}$ is an increasing sequence of time, $\tau_{0}=0$, $\tau_{n}=T$ and
\betagin{equation}
\|a\|_{L^{\rho}(\tau_{k-1},\tau_{k})}=\frac{1}{2},\,\, \text{$1\leq k\leq n-1$}; \, \text{ } \, \|a\|_{L^{\rho}(\tau_{n-1},\tau_{n})}\leq \frac{1}{2}.
\end{equation}
So, we have that
\betagin{align*}
\int_{0}^{T}|a|^{\rho}\,ds=& \int_{0}^{\tau_1}|a|^{\rho}\,ds+\dots +\int_{\tau_{n-1}}^{T}|a|^{\rho}\,ds\\
\leq& \frac{1}{2^{\rho}}+\dots + \frac{1}{2^{\rho}}= \frac{n}{2^{\rho}}.
\end{align*}
We put $n= [2^\rho \|a\|_{L^\rho(0,T)}^\rho]+1$.\\
Set $a_0=0$ and $a_{k}= \|v\|_{L^p(0,\tau_k)}$.
By the \eqref{ipotesigro} and by the H\"older inequality, we have
that
\betagin{align}
a_{k+1}=& \|v\|_{L^p(0,\tau_{k+1})}\leq C_1+ \|av\|_{L^q(0,\tau_{k})}+\|av\|_{L^q(\tau_{k},\tau_{k+1})}\notag\\
\leq& C_1 + \|a\|_{L^\rho(0,\tau_{k})}\|v\|_{L^p(0,\tau_{k})}+\|a\|_{L^\rho(\tau_{k},\tau_{k+1})}\|v\|_{L^p(\tau_{k},\tau_{k+1})}\notag\\
\leq& C_1+\frac{k}{2} a_{k} + \frac{1}{2} a_{k+1}\notag
\end{align}
So we have that
\betagin{equation}
a_{k+1}\leq 2C_1+ k a_{k},
\end{equation}
hence
\betagin{align*}
a_{k+1}\leq& 2C_1\left( 1+k+ k(k-1)+ k(k-1)(k-2)+ \dots + k(k-1)(k-2)\dots 2\mathcal Dot 1 \right)\\
\leq& 2C_1 (k+1)k!\leq 2C_1( k+1)!,
\end{align*}
Let $t\in[\tau_{k},\tau_{k+1}]$. Then
\betagin{equation}
\|a\|_{L^\rho(0,t)}\geq \|a\|_{L^\rho(0,\tau_{k})}= \frac{k^{1/\rho}}{2},
\end{equation}
and so
\betagin{equation}
k\leq 2^\rho \|a\|^{\rho}_{L^\rho(0,t)}.
\end{equation}
A simple working gets the thesis. Actually, we have
\betagin{equation}
\|v\|_{L^\rho(0,t)}\leq a_{k+1}\leq 2C_1( k+1)!= 2C_1\Gammamma(k+2) \leq 2C_1 \Gammamma (2+2^\rho\|a\|^\rho_{L^\rho(0,t)}).
\end{equation}
Else if $\|a\|_ {L^{\rho}(0,T)}<1/2$, trivially, we have that
\betagin{equation*}
\|v\|_{L^{p}(0,T)}\leq 2C_1.
\end{equation*}
\end{proof}
In different context the next Lemma is useful.
\betagin{lem}\lambdabel{1gron}
Let $C_2>0$, $ 1 \leq q < p \leq \infty$ and $v(t) \in C([0,+\infty))$, $a(t) \in L^\infty_{loc}((0,+\infty))$ are positive functions that satisfy the following inequalities
\betagin{equation}\lambdabel{eq.a2.1}
\|v\|_{L^p(0,1)}^q \leq C_2,
\end{equation}
\betagin{equation}\lambdabel{eq.a2.2}
\|v\|_{L^p(0,t)}^q \leq C_2 + \int_1^t a(\tau) v(\tau)^q \,d\tau \,\,\,\text{\,\, $t>1$,}
\end{equation}
then
\betagin{equation}\lambdabel{eq.a2.5}
\|v\|_{L^p(0,t)} \leq \left(\frac{p}{p-q} \right)^{1/p}C_2^{1/q} \exp\left( \frac{1}{p} \left(\frac{p}{q} \right)^{p/(p-q)}\|a\|_{L^{\frac{p}{p-q}}(1,t)}^{\frac{p}{p-q}} \right),
\end{equation}
with the obvious modifications for $p=\infty$.
\end{lem}
\betagin{proof}
By the \eqref{eq.a2.2} we have that
\betagin{equation}\lambdabel{eq.a2.2a}
t > 1 \Longrightarrow \|v\|_{L^p(0,t)}^q \leq C_2 + \int_1^t a(\tau) v^q(\tau) d\tau .
\end{equation}
Set
$$ \varphi(t) = C_2+\int_1^t a(\tau) v^q(\tau) d\tau.$$
Then we have the relation
$$ v^q (t) = \frac{\varphi^\partialrime(t)}{a(t)}, $$
so we have
\betagin{equation}\lambdabel{eq.a2.2b}
\left\|\frac{\varphi^\partialrime}{a}\right\|_{L^{p/q}(1,t)} \leq \varphi(t).
\end{equation}
To simplify the notation, we set $\alphapha=p/q$ and $\alphapha^\partialrime= p/(p-q)$.\\
We can use the inequality \eqref{eq.a2.2b} to derive the estimates
\betagin{alignat}{3} \nonumber
\varphi^\alphapha(t) =& \varphi^\alphapha(1) + \alphapha\int_1^t \varphi^\partialrime(\tau) \varphi^{\alphapha-1}(\tau) d \tau \\
\nonumber
=& C_2^\alphapha +\alphapha\int_1^t \frac{\varphi^\partialrime(\tau)}{a(\tau)} \varphi^{\alphapha-1}(\tau)a(\tau) d \tau \\
\nonumber
\leq& C_2^\alphapha + \alphapha \left\|\frac{\varphi^\partialrime}{a} \right\|_{L^{\alphapha}(1,t)} \left\|a\varphi^{\alphapha-1} \right\|_{L^{\alphapha'}(1,t)}\\
\nonumber
\leq& C_2^\alphapha+\frac{\varphi^\alphapha(t) }{\alphapha} + \frac{\alphapha^{\alphapha'}\left\|a\varphi^{\alphapha-1} \right\|_{L^{\alphapha'}(1,t)}^{\alphapha' }}{\alphapha'}.
\end{alignat}
We can rewrite the inequality above as
\betagin{equation*}
\frac{\varphi^\alphapha(t)}{\alphapha'}\leq C_2^\alphapha+ \frac{\alphapha^{\alphapha'}}{\alphapha'}\int_1^t a^{\alphapha'}(\tau) \varphi^\alphapha (\tau) \,d\tau,
\end{equation*}
so, we are in position to apply classical Gronwall's inequality and derive that
\betagin{equation*}
\varphi^\alphapha(t)\leq \alphapha' C_2^{\alphapha} \exp\left(\alphapha^{\alphapha'} \int_1^t a^{\alphapha'}(\tau) \,d\tau \right).
\end{equation*}
Rise to $\frac{1}{\alphapha}$ and by the \eqref{eq.a2.2a} we obtain the \eqref{eq.a2.5}.
\end{proof}
Actually, if we have $a\in\L^\infty(0,T)$, for large time $T$, the inequality \eqref{eq.a2.5} is better than \eqref{gammagronwall}.\\
Now we consider the particular case $\gammamma=3$ and $\sigmagma \in (0,1/2]$.
\betagin{prop}[$L^4(0,T)L^\infty$ no blow-up result]
Let $T>0$ and let $\Psi\in C([0,T];L^2)$ be the solution of \eqref{sps1}-\eqref{sps3} with initial data $f\in L^2$. Then we have that
\betagin{equation}
\|\Psi\|_{L^4(0,t)L^\infty}<\infty
\end{equation}
for all $t\in (0,T)$.
\end{prop}
\betagin{proof}
Let $T>0$ and $t\in(0,T)$.
By the Strichartz estimates combined with H\"older inequality and Hardy-Littlewood-Sobolev inequality we get:
\betagin{align*}
\|\Psi\|_{L^4(0,t)L^\infty } &\leq C\|f\|_{L^2}+ \|A_0\Psi+\alphapha |\Psi|^2\Psi\|_{L^{4/3}(0,t)L^1}\\
&\leq C\|f\|_{L^2}+\|A_0\Psi\|_{L^{4/3}(0,t)L^1}+\| |\Psi|^2\Psi\|_{L^{4/3}(0,t)L^1}\\
&\leq C\|f\|_{L^2}+C\|f\|_{L^2}^{2(1+\sigmagma)}\|\lambdangle s \rangle^{\frac{3}{2}\frac{\sigmagma}{1-2\sigmagma}} \Psi \|^{1-2\sigmagma}_{L^{4/3}(0,t)L^\infty }+C\|f\|_{L^2}^2 \|\Psi\|_{L^{4/3}(0,t)L^\infty}.
\end{align*}
Putting $C_{f}=\max \left\lbrace C\|f\|_{L^2},C\|f\|_{L^2} ^{2(1+\sigmagma)},C\|f\|_{L^2}^2 \right\rbrace $, we have that:
\betagin{equation*}
\|\Psi\|_{L^4(0,t)L^\infty } \leq C_f+C_f \|\lambdangle s \rangle^{\frac{3}{2}\frac{\sigmagma}{1-2\sigmagma}} \Psi \|_{L^{4/3}(0,t)L^\infty }.
\end{equation*}
By the $L^p-L^q$ Gronwall's inequality \eqref{gammagronwall} (similarly with \eqref{eq.a2.5}), we get the thesis:
\betagin{equation}
\|\Psi\|_{L^4(0,t)L^\infty_{x} }\leq 2C_f \Gammamma \left( 2+ (2C_f)^{\frac{2}{3}\frac{1-2\sigmagma}{\sigmagma}}t^2 \right) .
\end{equation}
\end{proof}
We conclude with a remark on the \emph{speed of the oscillation} of the solution of (FSPS).
\betagin{lem}
Let $T>0$, $ \rho, SC^{1}lon (0,T)\times {\mathbb R} \to{\mathbb R}$ and let $\Psi=\rho(t,x){\rm e}^{iS(t,x)}$ be the solution of \eqref{sps1}-\eqref{sps3}, with initial data $f\in H^1$ and $\alphapha=-1$. Define the functions $hC^{1}lon {\mathbb R}\times {\mathbb R} \to {\mathbb R}$ and $\thetaC^{1}lon {\mathbb R} \to {\mathbb R}$ in according to C^{\infty}te{cazenavebl}:
\betagin{gather}
h(t)=\partialartial_tS\\
\theta(t)=\int_{{\mathbb R}}|\Psi(t,x)|^2h(t,x)\,dx,
\end{gather}
with $t\in (0,T)$. Then, the following statements on the speed of the oscillation, $\theta(t)$, hold:
\betagin{itemize}
\item[1)] If $1<\gammamma<5$ and $\sigmagma\in I_\gammamma$ then $\theta(t)$ cannot blow up for all $t\in(0,T)$;
\item[2)] elseif $\gammamma=5$ and $\sigmagma\in I_5$, then there exists $\deltalta >0$ such that, if $\|f\|_2<\deltalta$, $\theta(t)$ cannot blow up for all $t\in(0,T)$.
\end{itemize}
\end{lem}
\betagin{proof}
Let $1<\gammamma<5$ and $\sigmagma\in I_\gammamma$. Suppose $\|f\|_{L^2}=1$.
A simple computation shows that
\betagin{equation}\lambdabel{veloscill1}
h(t)= \frac{\Im (\bar{\Psi}\partialartial_t\Psi)}{|\Psi|^{2}}.
\end{equation}
Multlipying by $\bar{\Psi}$ the equation \eqref{sps1}, integrating by parts and by the \eqref{veloscill1} we have that
\betagin{equation*}
\theta(t)=\int_{{\mathbb R}} h(t)|\Psi|^2\,dx= -2E(0)-\frac{1}{2}\int A_0 |\Psi|^2\,dx -\alphapha\frac{\gammamma}{\gammamma+1}\|\Psi\|_{L^{\gammamma+1}}^{\gammamma+1}.
\end{equation*}
Thanks to Gagliardo Nirenberg inequality \eqref{gagliardon} and by finiteness of the energy we have that $\|\nabla\Psi\|_{L^2}$ cannot blow up.
Hence, $\|\Psi\|_{\gammamma+1}$ cannot blow up. So, we can conclude that $\int A_0\Psi\,dx$ does not blow up. \\
In particular, the following inequality holds
\betagin{align*}\lambdabel{veloscilla}
|\theta(t)|\leq & 2|E(0)|+(\frac{1}{2}\|f\|_2+\frac{C_{GN}^2}{2}\|\nabla \Psi\|_{L^2}^{1-\frac{1}{p'}}\|f\|_2^{1+\frac{1}{p'}})+\frac{\gammamma C_{GN}}{\gammamma+1}\|\nabla\Psi\|_{L^2}^{\frac{\gammamma-1}{2}} \\
= & 2|E(0)|+(\frac{1}{2}+C\|\nabla \Psi\|_{L^2}^{1-\frac{1}{p'}})+C\|\nabla\Psi\|_{L^2}^{\frac{\gammamma-1}{2}},
\end{align*}
where $1\leq p<\frac{1}{1-\sigmagma}$, and $\frac{1}{p}+\frac{1}{p'}=1$.
Hence, the speed of the oscillations (on average) cannot blow up if the initial energy is finite.\\
Note that in defocusing case $\alphapha=1$, we get easily
\betagin{equation*}
|\theta(t)|\leq C|E(0)|,
\end{equation*}
with $C$ a positive constant.\\
If $\gammamma=5$ and $\sigmagma\in I_5$, we need the smallness of the initial mass $\|f\|_2$ (see the proof of the Theorem \ref{h1focus}) to get the same conclusion.\\
\betagin{rem}
The blow-up as a consequence of rotational proprieties of the solution concerning the nonlinear Schr\"odinger problem is studied in C^{\infty}te{cazenavewebl} and C^{\infty}te{cazenavebl}.
\end{rem}
\end{proof}
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\end{document} | math | 74,184 |
\begin{document}
\title{The method of hypergraph containers}
\author{J\'ozsef Balogh}
\address{Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA}
\email{[email protected]}
\author{Robert Morris}
\address{IMPA, Estrada Dona Castorina 110, Jardim Bot\^anico, Rio de Janeiro, 22460-320, Brazil}
\email{[email protected]}
\author{Wojciech Samotij}
\address{School of Mathematical Sciences, Tel Aviv University, Tel Aviv 6997801, Israel}
\email{[email protected]}
\thanks{JB is partially supported by NSF Grant DMS-1500121 and by the Langan Scholar Fund (UIUC); RM is partially supported by CNPq (Proc.~303275/2013-8), by FAPERJ (Proc.~201.598/2014), and by ERC Starting Grant 680275 MALIG; WS is partially supported by the Israel Science Foundation grant 1147/14.}
\begin{abstract}
In this survey we describe a recently-developed technique for bounding the number (and controlling the typical structure) of finite objects with forbidden substructures. This technique exploits a subtle clustering phenomenon exhibited by the independent sets of uniform hypergraphs whose edges are sufficiently evenly distributed; more precisely, it provides a relatively small family of `containers' for the independent sets, each of which contains few edges. We attempt to convey to the reader a general high-level overview of the method, focusing on a small number of illustrative applications in areas such as extremal graph theory, Ramsey theory, additive combinatorics, and discrete geometry, and avoiding technical details as much as possible.
\end{abstract}
\maketitle
\section{Introduction}
Numerous well-studied problems in combinatorics concern families of discrete objects which avoid certain forbidden configurations, such as the family of $H$-free graphs\footnote{A graph is $H$-free if it does not contain a subgraph isomorphic to $H$.} or the family of sets of integers containing no $k$-term arithmetic progression. The most classical questions about these families relate to the size and structure of the extremal examples; for example, Tur\'an~\cite{T41} determined the unique $K_r$-free graph on $n$ vertices with the most edges and Szemer\'edi~\cite{Sz75} proved that every set of integers of positive upper density contains arbitrarily long arithmetic progressions. In recent decades, partly motivated by applications to areas such as Ramsey theory and statistical physics, there has been increasing interest in problems relating to the typical structure of a (e.g., uniformly chosen) member of one of these families and to extremal questions in (sparse) random graphs and random sets of integers. Significant early developments in this direction include the seminal results obtained by Erd\H{o}s, Kleitman, and Rothschild~\cite{EKR}, who proved that almost all triangle-free graphs are bipartite, by Kleitman and Winston~\cite{KW82}, who proved that there are $2^{\Theta(n^{3/2})}$ $C_4$-free graphs on $n$ vertices, and by Frankl and R\"odl~\cite{FR}, who proved that if $p \gg 1/\sqrt{n}$, then with high probability every 2-colouring of the edges of $G(n,p)$ contains a monochromatic triangle.
An important recent development in this area was the discovery that, perhaps surprisingly, it is beneficial to consider such problems in the more abstract (and significantly more general) setting of independent sets in hypergraphs. This approach was taken with stunning success by Conlon and Gowers~\cite{CG}, Friedgut, R\"odl, and Schacht~\cite{FRS}, and Schacht~\cite{Sch} in their breakthrough papers on extremal and Ramsey-type results in sparse random sets. To give just one example of the many important conjectures resolved by their work, let us consider the random variable
\[
\mathrm{ex}\big( G(n,p), H \big) = \max\big\{ e(G) \,\colon H \not\subset G \subset G(n,p) \big\},
\]
which was first studied (in the case $H = K_3$) by Frankl and R\"odl~\cite{FR}. The following theorem was conjectured by Haxell, Kohayakawa, and \L uczak~\cite{HKL-odd, HKL-even} and proved (independently) by Conlon and Gowers~\cite{CG} and by Schacht~\cite{Sch}.
\begin{thm}
\label{thm:Turan:Gnp}
Let $H$ be a graph with at least two edges and suppose that $p \gg n^{-1/m_2(H)}$, where $m_2(H)$ is the so-called $2$-density\footnote{To be precise, $m_2(H) = \max\big\{ \frac{e(F) - 1}{v(F) - 2} \,\colon F \subset H, \, v(F) \geqslantslant 3 \big\}$.} of $H$. Then
\begin{equation*}\label{eq:thm:Turan:Gnp}
\mathrm{ex}\big( G(n,p), H \big) = \bigg( 1 - \frac{1}{\chi(H) - 1} + o(1) \bigg) p {n \choose 2}
\end{equation*}
asymptotically almost surely (a.a.s.), that is, with probability tending to $1$ as $n \to \infty$.
\end{thm}
It is not hard to show that $\mathrm{ex}\big( G(n,p), H \big) = \big( 1 + o(1) \big) p {n \choose 2}$ a.a.s.\ if $n^{-2} \ll p \ll n^{-1/m_2(H)}$ and so the assumption on $p$ in Theorem~\ref{thm:Turan:Gnp} is optimal. We remark that in the case when $H$ is a clique even more precise results are known, due to work of DeMarco and Kahn~\cite{DK2, DK1}, who proved that if $p \gg n^{-1/m_2(H)} (\log n)^{2/(r+1)(r - 2)}$, then with high probability the largest $K_{r+1}$-free subgraph of $G(n,p)$ is $r$-partite, which is again essentially best possible. We refer the reader to an excellent recent survey of R\"odl and Schacht~\cite{RS} for more details on extremal results in sparse random sets.
In this survey we will describe an alternative approach to the problem of understanding the family of independent sets in a hypergraph, whose development was inspired by the work in~\cite{CG,FRS,Sch} and also strongly influenced by that of Kleitman and Winston~\cite{KW82} and Sapozhenko~\cite{Sap01,Sap03,Sap05}. This technique, which was developed independently by the authors of this survey~\cite{BMS} and by Saxton and Thomason~\cite{ST}, has turned out to be surprisingly powerful and flexible. It allows one to prove enumerative, structural, and extremal results (such as Theorem~\ref{thm:Turan:Gnp}) in a wide variety of settings. It is known as the \emph{method of hypergraph containers}.
To understand the essence of the container method, it is perhaps useful to consider as an illustrative example the family $\mathcal{F}_n(K_3)$ of triangle-free graphs on (a given set of) $n$ vertices. Note that the number of such graphs is at least $2^{\lfloor n^2/4 \rfloor}$, since every bipartite graph is triangle-free.\footnote{In particular, every subgraph of the complete bipartite graph with $n$ vertices and $\lfloor n^2/4 \rfloor$ edges is triangle-free.} However, it turns out that there exists a vastly smaller family $\mathcal{G}_n$ of graphs on $n$ vertices, of size $n^{O(n^{3/2})}$, that forms a set of \emph{containers} for $\mathcal{F}_n(K_3)$, which means that for every $H \in \mathcal{F}_n(K_3)$, there exists a $G \in \mathcal{G}_n$ such that $H \subset G$. A remarkable property of this family of containers is that each graph $G \in \mathcal{G}_n$ is `almost triangle-free' in the sense that it contains `few' triangles. It is not difficult to use this family of containers, together with a suitable `supersaturation' theorem, to prove Theorem~\ref{thm:Turan:Gnp} in the case $H = K_3$ or to show, using a suitable `stability' theorem, that almost all triangle-free graphs are `almost bipartite'. We will discuss these two properties of the family of triangle-free graphs in much more detail in Section~\ref{basic:sec}.
In order to generalize this container theorem for triangle-free graphs, it is useful to first restate it in the language of hypergraphs. To do so, consider the 3-uniform hypergraph $\mathcal{H}$ with vertex set $V(\mathcal{H}) = E(K_n)$ and edge set
\[
E(\mathcal{H}) = \big\{ \{e_1,e_2,e_3\} \subset E(K_n) \,\colon \textup{$e_1, e_2, e_3$ form a triangle} \big\}.
\]
We shall refer to $\mathcal{H}$ as the `hypergraph that encodes triangles' and emphasize that (somewhat confusingly) the vertices of this hypergraph are the edges of the complete graph $K_n$. Note that $\mathcal{F}_n(K_3)$ is precisely the family $\mathcal{I}(\mathcal{H})$ of independent sets of $\mathcal{H}$, so we may rephrase our container theorem for triangle-free graphs as follows:
\begin{align*}
& \text{``There exists a relatively small family $\mathcal{C}$ of subsets of $V(\mathcal{H})$, each containing only few}\\
& \text{edges of $\mathcal{H}$, such that every independent set $I \in \mathcal{I}(\mathcal{H})$ is contained in some member of $\mathcal{C}$."}
\end{align*}
There is nothing special about the fine structure of the hypergraph encoding triangles that makes the above statement true. On the contrary, the method of containers allows one to prove that a similar phenomenon holds for a large class of $k$-uniform hypergraphs, for each $k \in \mathbb{N}$. In the case $k=3$, a sufficient condition is the following assumption on the distribution of the edges of a~$3$-uniform hypergraph $\mathcal{H}$ with average degree $d$: each vertex of $\mathcal{H}$ has degree at most $O(d)$ and each pair of vertices lies in at most $O(\sqrt{d})$ edges of $\mathcal{H}$. For the hypergraph that encodes triangles, both conditions are easily satisfied, since each edge of $K_n$ is contained in exactly $n-2$ triangles and each pair of edges is contained in at most one triangle. The conclusion of the container lemma (see Sections~\ref{basic:sec} and~\ref{keylemma:sec}) is that each independent set $I$ in a $3$-uniform hypergraph $\mathcal{H}$ satisfying these conditions has a \emph{fingerprint} $S \subset I$ of size $O\big( v(\mathcal{H}) / \sqrt{d} \big)$ that is associated with a set $X(S)$ of size $\Omega\big( v(\mathcal{H}) \big)$ which is disjoint from $I$. The crucial point is that the set $X(S)$ depends only on $S$ (and not on $I$) and therefore the number of sets $X(S)$ is bounded from above by the number of subsets of the vertex set $V(\mathcal{H})$ of size $O\big( v(\mathcal{H}) / \sqrt{d} \big)$. In particular, each independent set of $\mathcal{H}$ is contained in one of at most $v(\mathcal{H})^{O(v(\mathcal{H})/\sqrt{d})}$ sets of size at most $(1 - \delta)v(\mathcal{H})$, for some constant $\delta > 0$. By iterating this process, that is, by applying the container lemma repeatedly to the subhypergraphs induced by the containers obtained in earlier applications, one can easily prove the container theorem for triangle-free graphs stated (informally) above.
Although the hypergraph container lemma (see Section~\ref{keylemma:sec}) was discovered only recently (see~\cite{BMS,ST}), several theorems of the same flavour (though often in very specific settings) appeared much earlier in the literature. The earliest container-type argument of which we are aware appeared (implicitly) over 35 years ago in the work of Kleitman and Winston on bounding the number of lattices~\cite{KW80} and of $C_4$-free graphs~\cite{KW82}, which already contained some of the key ideas needed for the proof in the general setting; see~\cite{Sa15} for details. Nevertheless, it was not until almost 20 years later that Sapozhenko~\cite{Sap01,Sap03,Sap05} made a systematic study of containers for independent sets in graphs (and coined the name \emph{containers}). Around the same time, Green and Ruzsa~\cite{GrRu04} obtained (using Fourier analysis) a container theorem for sum-free subsets of $\mathbb{Z}/p\mathbb{Z}$.
More recently, Balogh and Samotij~\cite{BSmm,BSst} generalized the method of~\cite{KW82} to count $K_{s,t}$-free graphs, using what could be considered to be the first container theorem for hypergraphs of uniformity larger than two. Finally, Alon, Balogh, Morris, and Samotij~\cite{ABMS1,ABMS2} proved a general container theorem for 3-uniform hypergraphs and used it to prove a sparse analogue of the Cameron--Erd\H{o}s conjecture. Around the same time, Saxton and Thomason~\cite{SaTh12} developed a simpler version of the method and applied it to the problem of bounding the list chromatic number of hypergraphs. In particular, the articles~\cite{ABMS1} and~\cite{SaTh12} can be seen as direct predecessors of~\cite{BMS} and~\cite{ST}.
The rest of this survey is organised as follows. In Section~\ref{basic:sec}, we warm up by stating a container lemma for 3-uniform hypergraphs, giving three simple applications to problems involving triangle-free graphs and a more advanced application to a problem in discrete geometry that was discovered recently by Balogh and Solymosi~\cite{BS}. Next, in Section~\ref{keylemma:sec}, we state the main container lemma and provide some additional motivation and discussion of the statement and in Section~\ref{counting:Hfree:sec} we describe an application to counting $H$-free graphs. Finally, in Sections~\ref{sec:many-colours}--\ref{more:applications:sec}, we state and discuss a number of additional applications, including to multi-coloured structures (e.g., metric spaces), asymmetric structures (e.g., sparse members of a hereditary property), hypergraphs of unbounded uniformity (e.g., induced Ramsey numbers, $\varepsilon$-nets), number-theoretic structures (e.g., Sidon sets, sum-free sets, sets containing no $k$-term arithmetic progression), sharp thresholds in Ramsey theory, and probabilistic embedding in sparse graphs.
\section{Basic applications of the method}\label{basic:sec}
In this section we will provide the reader with a gentle introduction to the container method, focusing again on the family of triangle-free graphs. In particular, we will state a version of the container lemma for 3-uniform hypergraphs and explain (without giving full details) how to deduce from it bounds on the largest size of a triangle-free subgraph of the random graph $G(n,p)$, statements about the typical structure of a (sparse) triangle-free graph, and how to prove that every $r$-colouring of the edges of $G(n,p)$ contains a monochromatic triangle. To give a simple demonstration of the flexibility of the method, we will also describe a slightly more complicated application to a problem in discrete geometry.
In order to state the container lemma, we need a little notation. Given a hypergraph $\mathcal{H}$, let us write $\Delta_\ell(\mathcal{H})$ for the maximum degree of a set of $\ell$ vertices of $\mathcal{H}$, that is,
\[
\Delta_\ell(\mathcal{H}) = \max\big\{ d_\mathcal{H}(A) \,\colon A \subset V(\mathcal{H}), \, |A| = \ell \big\},
\]
where $d_\mathcal{H}(A) = \big| \big\{ B \in E(\mathcal{H}) \,\colon A \subset B \big\} \big|$, and $\mathcal{I}(\mathcal{H})$ for the collection of independent sets of $\mathcal{H}$.
\begin{HCL3}
For every $c > 0$, there exists $\delta > 0$ such that the following holds. Let $\mathcal{H}$ be a $3$-uniform hypergraph with average degree $d \geqslantslant \delta^{-1}$ and suppose that
\[
\Delta_1(\mathcal{H}) \leqslantslant c \cdot d \qquad \text{and} \qquad \Delta_2(\mathcal{H}) \leqslantslant c \cdot \sqrt{d}.
\]
Then there exists a collection $\mathcal{C}$ of subsets of $V(\mathcal{H})$ with
\[
|\mathcal{C}| \leqslantslant \binom{v(\mathcal{H})}{v(\mathcal{H})/\sqrt{d}}
\]
such that
\begin{enumerate}
\item[$(a)$] for every $I \in \mathcal{I}(\mathcal{H})$, there exists $C \in \mathcal{C}$ such that $I \subset C$,
\item[$(b)$] $|C| \leqslantslant (1 - \delta) v(\mathcal{H})$ for every $C \in \mathcal{C}$.
\end{enumerate}
\end{HCL3}
In order to help us understand the statement of this lemma, let us apply it to the hypergraph $\mathcal{H}$ that encodes triangles in $K_n$, defined in the Introduction. Recall that this hypergraph satisfies
\[
v(\mathcal{H}) = {n \choose 2}, \qquad \Delta_2(\mathcal{H}) = 1, \qquad \text{and} \qquad d_\mathcal{H}(v) = n - 2
\]
for every $v \in V(\mathcal{H})$. We may therefore apply the container lemma to $\mathcal{H}$, with $c = 1$, to obtain a collection $\mathcal{C}$ of $n^{O(n^{3/2})}$ subsets of $E(K_n)$ (that is, graphs on $n$ vertices) with the following properties:
\begin{enumerate}
\item[$(a)$] Every triangle-free graph is a subgraph of some $C \in \mathcal{C}$.
\item[$(b)$] Each $C \in \mathcal{C}$ has at most $(1 - \delta) e(K_n)$ edges.
\end{enumerate}
Now, if there exists a container $C \in \mathcal{C}$ with at least $\varepsilon n^3$ triangles, then take each such $C$ and apply the container lemma to the subhypergraph $\mathcal{H}[C]$ of $\mathcal{H}$ induced by $C$, i.e., the hypergraph that encodes triangles in the graph $C$. Note that the average degree of $\mathcal{H}[C]$ is at least $6\varepsilon n$, since each triangle in $C$ corresponds to an edge of $\mathcal{H}[C]$ and $v(\mathcal{H}[C]) = |C| \leqslantslant e(K_n)$. Since (trivially) $\Delta_\ell(\mathcal{H}[C]) \leqslantslant \Delta_\ell(\mathcal{H})$, it follows that we can apply the lemma with $c = 1/\varepsilon$ and replace $C$ by the collection of containers for $\mathcal{I}(\mathcal{H}[C])$ given by the lemma.
Let us iterate this process until we obtain a collection $\mathcal{C}$ of containers, each of which has fewer than $\varepsilon n^3$ triangles. How large is the final family $\mathcal{C}$ that we obtain? Note that we apply the lemma only to hypergraphs with at most $\binom{n}{2}$ vertices and average degree at least $6\varepsilon n$ and therefore produce at most $n^{O(n^{3/2})}$ new containers in each application, where the implicit constant depends only on~$\varepsilon$. Moreover, each application of the lemma shrinks a container by a factor of $1 - \delta$, so after a bounded (depending on $\varepsilon$) number of iterations every container will have fewer than $\varepsilon n^3$ triangles (since $\Delta_1(\mathcal{H}) < n$, then every graph with at most $\varepsilon n^2$ edges contains fewer than $\varepsilon n^3$ triangles).
The above argument yields the following container theorem for triangle-free graphs.
\begin{thm}\label{thm:CT:triangles}
For each $\varepsilon > 0$, there exists $C > 0$ such that the following holds. For each $n \in \mathbb{N}$, there exists a collection $\mathcal{G}$ of graphs on $n$ vertices, with
\begin{equation}\label{eq:CT:triangles:size}
|\mathcal{G}| \leqslantslant n^{C n^{3/2}},
\end{equation}
such that
\begin{itemize}
\item[$(a)$] each $G \in \mathcal{G}$ contains fewer than $\varepsilon n^3$ triangles;
\item[$(b)$] each triangle-free graph on $n$ vertices is contained in some $G \in \mathcal{G}$.
\end{itemize}
\end{thm}
In order to motivate the statement of Theorem~\ref{thm:CT:triangles}, we will next present three simple applications: bounding the largest size of a triangle-free subgraph of the random graph $G(n,p)$, determining the typical structure of a (sparse) triangle-free graph, and proving that $G(n,p)$ cannot be partitioned into a bounded number of triangle-free graphs.
\subsection{Mantel's theorem in random graphs} \label{Mantel:sec}
The oldest result in extremal graph theory, which states that every graph on $n$ vertices with more than $n^2/4$ edges contains a triangle, was proved by Mantel~\cite{Ma07} in 1907. The corresponding problem in the random graph $G(n,p)$ was first studied by Frankl and R\"odl~\cite{FR}, who proved the following theorem (cf.~Theorem~\ref{thm:Turan:Gnp}).
\begin{thm}\label{thm:Mantel:Gnp}
For every $\alpha > 0$, there exists $C > 0$ such that the following holds. If $p \geqslantslant C / \sqrt{n}$, then a.a.s.\ every subgraph $G \subset G(n,p)$ with
\[
e(G) \geqslantslant \bigg( \frac{1}{2} + \alpha \bigg) p \binom{n}{2}
\]
contains a triangle.
\end{thm}
As a simple first application of Theorem~\ref{thm:CT:triangles}, let us use it to prove Theorem~\ref{thm:Mantel:Gnp} under the marginally stronger assumption that $p \gg \log n / \sqrt{n}$. The proof exploits the following crucial property of $n$-vertex graphs with $o(n^3)$ triangles: each such graph has at most $\big(\frac{1}{2}+o(1)\big) \binom{n}{2}$ edges. This statement is made rigorous in the following supersaturation lemma for triangles, which can be proved by simply applying Mantel's theorem to each induced subgraph of $G$ with $O(1)$ vertices.
\begin{lemma}[Supersaturation for triangles]\label{lem:supersat:triangles}
For every $\delta > 0$, there exists $\varepsilon > 0$ such that the following holds. If $G$ is a graph on $n$ vertices with
\[
e(G) \geqslantslant \bigg( \frac{1}{4} + \delta \bigg) n^2,
\]
then $G$ has at least $\varepsilon n^3$ triangles.
\end{lemma}
Applying Lemma~\ref{lem:supersat:triangles} with $\delta = \alpha/2$ and Theorem~\ref{thm:CT:triangles} with $\varepsilon = \varepsilon(\delta)$ given by the lemma, we obtain a family of containers $\mathcal{G}$ such that each $G \in \mathcal{G}$ has fewer than $\varepsilon n^3$ triangles and thus
\[
e(G) \leqslantslant \bigg( \frac{1 + \alpha}{2} \bigg) \binom{n}{2}
\]
for every $G \in \mathcal{G}$. Since every triangle-free graph is a subgraph of some container, if $G(n,p)$ contains a triangle-free graph with $m$ edges, then in particular $e\big( G \cap G(n,p) \big) \geqslantslant m$ for some $G \in \mathcal{G}$. Noting that $e\big( G \cap G(n,p) \big) \sim \textup{Bin}\big( e(G), p \big)$, standard estimates on the tail of the binomial distribution yield
\[
\mathcal{P}r\bigg( e\big( G \cap G(n,p) \big) \geqslantslant \bigg( \frac{1}{2} + \alpha \bigg) p \binom{n}{2} \bigg) \leqslantslant e^{- \beta pn^2},
\]
for some constant $\beta = \beta(\alpha) > 0$. Therefore, taking a union bound over all containers $G \in \mathcal{G}$ and using the bound~\eqref{eq:CT:triangles:size}, we have (using the notation of Theorem~\ref{thm:Turan:Gnp})
\begin{equation}
\label{eq:Mantel:Gnp}
\mathcal{P}r\bigg( \mathrm{ex}\big( G(n,p), K_3 \big) \geqslantslant \bigg( \frac{1}{2} + \alpha \bigg) p {n \choose 2} \bigg) \leqslantslant n^{O(n^{3/2})} \cdot e^{- \beta pn^2} \to 0
\end{equation}
as $n \to \infty$, provided that $p \gg \log n / \sqrt{n}$. This gives the conclusion of Theorem~\ref{thm:Mantel:Gnp} under a slightly stronger assumption on $p$. In Section~\ref{keylemma:sec}, we show how to remove the extra factor of $\log n$.
We remark here that Theorem~\ref{thm:Mantel:Gnp}, as well as numerous results of this type that now exist in the literature, cannot be proved using standard first moment estimates. Indeed, since there are at least $\binom{\lfloor n^2/4 \rfloor}{m}$ triangle-free graphs with $n$ vertices and $m$ edges, then letting $X_m$ denote the number of such graphs that are contained in $G(n,p)$, we have
\[
\mathbb{E}[X_m] \geqslantslant p^m \binom{\lfloor n^2/4 \rfloor}{m} = \leqslantslantft(\frac{(e/2+o(1))p\binom{n}{2}}{m}\right)^m \gg 1
\]
if $m \leqslantslant \big( e/2+o(1) \big) p\binom{n}{2} = o(n^2)$. This means that a first moment estimate would yield an upper bound on $\mathrm{ex}\big(G(n,p), K_3\big)$ that is worse than the trivial upper bound of $\big( 1+o(1) \big) p\binom{n}{2}$.
\subsection{The typical structure of a sparse triangle-free graph}
A seminal theorem of Erd\H{o}s, Kleitman, and Rothschild~\cite{EKR} states that almost all triangle-free graphs are bipartite. Our second application of Theorem~\ref{thm:CT:triangles} is the following approximate version of this theorem for sparse graphs, first proved by {\L}uczak~\cite{Lu00}. Let us say that a graph $G$ is \emph{$t$-close to bipartite} if there exists a bipartite subgraph $G' \subset G$ with $e(G') \geqslantslant e(G) - t$.
\begin{thm}\label{thm:structure:trianglefree}
For every $\alpha > 0$, there exists $C > 0$ such that the following holds. If $m \geqslantslant C n^{3/2}$, then almost all triangle-free graphs with $n$ vertices and $m$ edges are $\alpha m$-close to bipartite.
\end{thm}
We will again (cf.~the previous subsection) prove this theorem under the marginally stronger assumption that $m \gg n^{3/2}\log n$. To do so, we will need a finer characterisation of graphs with $o(n^3)$ triangles that takes into account whether or not a graph is close to bipartite. Proving such a result is less straightforward than Lemma~\ref{lem:supersat:triangles}; for example, one natural proof combines the triangle removal lemma of Ruzsa and Szemer\'edi~\cite{RuSz} with the classical stability theorem of Erd\H{o}s and Simonovits~\cite{Er67, Si68}. However, an extremely simple, beautiful, and elementary proof was given recently by F\"uredi~\cite{Fur} (see also~\cite{BBCLMS}).
\begin{lemma}[Robust stability for triangles]\label{lem:superstability:triangles}
For every $\delta > 0$, there exists $\varepsilon > 0$ such that the following holds. If $G$ is a graph on $n$ vertices with
\[
e(G) \geqslantslant \bigg( \frac{1}{2} - \varepsilon \bigg) {n \choose 2},
\]
then either $G$ is $\delta n^2$-close to bipartite or $G$ contains at least $\varepsilon n^3$ triangles.
\end{lemma}
Applying Lemma~\ref{lem:superstability:triangles} with $\delta = \delta(\alpha) > 0$ sufficiently small and Theorem~\ref{thm:CT:triangles} with $\varepsilon = \varepsilon(\delta)$ given by the lemma, we obtain a family of containers $\mathcal{G}$ such that every $G \in \mathcal{G}$ is either $\delta n^2$-close to bipartite or
\begin{equation}\label{eq:container:with:few:edges}
e(G) \leqslantslant \bigg( \frac{1}{2} - \varepsilon \bigg) {n \choose 2}.
\end{equation}
Let us count those triangle-free graphs $H$ with $n$ vertices and $m$ edges that are not $\alpha m$-close to bipartite; note that each such graph is a subgraph of some container $G \in \mathcal{G}$.
Suppose first that $G$ satisfies~\eqref{eq:container:with:few:edges}; in this case we simply use the trivial bound
\[
{e(G) \choose m} \leqslantslant \binom{\leqslantslantft(\frac{1}{2}-\varepsilon\right)\binom{n}{2}}{m} \leqslantslant (1 - \varepsilon)^m {n^2/4 \choose m}
\]
for the number of choices for $H \subset G$. On the other hand, if $G$ is $\delta n^2$-close to bipartite, then there is some bipartite $G' \subset G$ with $e(G') \geqslantslant e(G) - \delta n^2$. Since $e(H \cap G') \leqslantslant (1-\alpha)m$ by our assumption on $H$, we bound the number of choices for $H$ by
\[
{e(G)-e(G') \choose \alpha m} {e(G) \choose (1 - \alpha) m} \leqslantslant \binom{\delta n^2}{\alpha m} \binom{\binom{n}{2}}{(1-\alpha)m} \leqslantslant 2^{-m} \binom{n^2/4}{m},
\]
provided that $\delta = \delta(\alpha)$ is sufficiently small. Summing over all choices of $G \in \mathcal{G}$ and using~\eqref{eq:CT:triangles:size}, it follows that if $m \gg n^{3/2} \log n$, then there are at most
\[
n^{O(n^{3/2})} \cdot (1 - \varepsilon)^m \binom{n^2/4}{m} \ll \binom{\lfloor n^2/4 \rfloor}{m}
\]
triangle-free graphs $H$ with $n$ vertices and $m$ edges that are not $\alpha m$-close to bipartite. However, there are clearly at least ${\lfloor n^2/4 \rfloor \choose m}$ triangle-free graphs $H$ with $n$ vertices and $m$ edges, since every bipartite graph is triangle-free, so the conclusion of Theorem~\ref{thm:structure:trianglefree} holds when $m \gg n^{3/2} \log n$. We again postpone a discussion of how to remove the unwanted factor of $\log n$ to Section~\ref{keylemma:sec}.
\subsection{Ramsey properties of sparse random graphs}
A folklore fact that is presented in each introduction to Ramsey theory states that every $2$-colouring of the edges of $K_6$ contains a monochromatic triangle. With the aim of constructing a small $K_4$-free graph that has the same property, Frankl and R\"odl~\cite{FR} proved that if $p \gg 1/\sqrt{n}$, then a.a.s.\ every $2$-colouring of the edges of $G(n,p)$ contains a monochromatic triangle. Ramsey properties of random graphs were later thorougly investigated by R\"odl and Ruci\'nski~\cite{RoRu93, RoRu94, RoRu95}. The following theorem is the main result of~\cite{RoRu94}.
\begin{thm}
\label{thm:triangle-Ramsey}
For every $r \in \mathbb{N}$, there exists $C > 0$ such that the following holds. If $p \gg C/\sqrt{n}$, then a.a.s.\ every $r$-colouring of the edges of $G(n,p)$ contains a monochromatic triangle.
\end{thm}
We will present a simple proof of this theorem that was discovered recently by Nenadov and Steger~\cite{NS}. For the sake of simplicity, we will again use the marginally stronger assumption that $p \gg \log n / \sqrt{n}$. The proof exploits the following property of $n$-vertex graphs with $o(n^3)$ triangles: the union of any bounded number of such graphs cannot cover a $\big( 1 - o(1) \big)$-proportion of the edges of $K_n$. This property is a straightforward corollary of the following lemma, which can be proved by applying Ramsey's theorem to the colourings induced by all subsets of $V(K_n)$ of size $O(1)$.
\begin{lemma}\label{lemma:Ramsey-supersat}
For every $r \in \mathbb{N}$, there exist $n_0$ and $\varepsilon > 0$ such that for all $n \geqslantslant n_0$, every $(r+1)$-colouring of the edges of $K_n$ contains at least $(r+1) \varepsilon n^3$ monochromatic triangles.
\end{lemma}
Applying Theorem~\ref{thm:CT:triangles} with $\varepsilon = \varepsilon(r)$ given by the lemma, we obtain a family of containers $\mathcal{G}$ such that every $G \in \mathcal{G}$ has fewer than $\varepsilon n^3$ triangles. If $G(n,p)$ does not have the desired Ramsey property, then there are triangle-free graphs $H_1, \dotsc, H_r$ such that $H_1 \cup \dotsc \cup H_r = G(n,p)$. It follows that $G(n,p) \subset G_1 \cup \dotsc \cup G_r$, where each $G_i \in \mathcal{G}$ is a container for $H_i$. Since each $G_i$ has fewer than $\varepsilon n^3$ triangles, then Lemma~\ref{lemma:Ramsey-supersat} implies that $K_n \setminus (G_1 \cup \dotsc \cup G_r)$ contains at least $\varepsilon n^3$ triangles.\footnote{To see this, consider an $(r+1)$-colouring of the edges of $K_n$ that assigns to each edge $e \in G_1 \cup \dotsc \cup G_r$ some colour~$i$ such that $e \in G_i$ and assigns colour $r+1$ to all edges of $K_n \setminus (G_1 \cup \dotsc \cup G_r)$.} Since each edge of $K_n$ belongs to fewer than $n$ triangles, we must have $e\big(K_n \setminus (G_1 \cup \dotsc \cup G_r)\big) \geqslantslant \varepsilon n^2$. Consequently, for each fixed $G_1, \dotsc, G_r \in \mathcal{G}$,
\[
\mathcal{P}r\big( G(n,p) \subset G_1 \cup \dotsc \cup G_r \big) = (1-p)^{e(K_n \setminus (G_1 \cup \cdots \cup G_r))} \leqslantslant (1-p)^{\varepsilon n^2} \leqslantslant e^{- \varepsilon pn^2}.
\]
Taking a union bound over all $r$-tuples of containers, we conclude that
\[
\mathcal{P}r\big( \text{$G(n,p)$ admits a `bad' $r$-colouring} \big) \leqslantslant n^{O(n^{3/2})} \cdot e^{- \varepsilon pn^2} \to 0
\]
as $n \to \infty$, provided that $p \gg \log n / \sqrt{n}$. As before, the unwanted factor of $\log n$ can be removed with a somewhat more careful analysis that we shall discuss in Section~\ref{keylemma:sec}.
\subsection{An application in discrete geometry}\label{geometry:sec}
In order to give some idea of the flexibility of the container method, we will next present a somewhat more elaborate application of the container lemma for 3-uniform hypergraphs, which was discovered recently by Balogh and Solymosi~\cite{BS}, to the following question posed by Erd\H{o}s~\cite{Er88}. Given $n$ points in the Euclidean plane $\mathbb{R}^2$, with at most three on any line, how large a subset are we guaranteed to find in general position (i.e., with at most two on any line)? F\"uredi~\cite{Fur91} proved that one can always find such a subset of size $\Omega\big(\sqrt{n\log n}\big)$ and gave a construction (which relied on the density Hales--Jewett theorem of Furstenberg and Katznelson~\cite{FuKa}) in which the largest such set has size $o(n)$. Using the method of hypergraph containers, Balogh and Solymosi~\cite{BS} obtained the following stronger upper bound.
\begin{thm}\label{thm:geometry}
There exists a set $S \subset \mathbb{R}^2$ of size $n$, containing no four points on a line, such that every subset of $S$ of size $n^{5/6+o(1)}$ contains three points on a line.
\end{thm}
The key idea in~\cite{BS} is to first construct a set $P$ of points that contains `few' collinear quadruples, but such that every `large' subset of $P$ contains `many' collinear triples. Then a random subset $R$ of $P$ of a carefully chosen density will typically contain only $o(|R|)$ collinear quadruples, since the density is not too large and there are few collinear quadruples. On the other hand, every subset of $R$ with more than $|R|^{5/6+o(1)}$ elements will still contain a collinear triple; this follows from the hypergraph container lemma, as large sets contain many collinear triples and the density is not too small. Removing one element from each collinear quadruple in $R$ gives the desired set~$A$.
Formally, we first define the following 3-uniform hypergraph $\mathcal{H}$. We let $V(\mathcal{H}) = [m]^3$ (so the vertices are lattice points in $\mathbb{R}^3$) and let $E(\mathcal{H})$ be the collection of triples of points that lie on a common line. Thus, a subset of $V(\mathcal{H})$ is in general position if and only if it is an independent set of $\mathcal{H}$. The following lemma was proved in~\cite{BS}.
\begin{lemma}[Supersaturation for collinear triples]\label{lem:supersat:triples:on:lines}
For every $0 < \gamma < 1/2$ and every $S \subset [m]^3$ of size at least $m^{3-\gamma}$, there exist at least $m^{6 - 4\gamma - o(1)}$ collinear triples of points in $S$.
\end{lemma}
We now repeatedly apply the hypergraph container lemma for 3-uniform hypergraphs to subhypergraphs of $\mathcal{H}$. Suppose that $s \geqslantslant m^{8/3+o(1)}$ and let $S \subset [m]^3$ be an arbitrary $s$-element set. Lemma~\ref{lem:supersat:triples:on:lines} gives
\[
e\big( \mathcal{H}[S] \big) \geqslantslant s^4/m^{6+o(1)} \qquad \text{and} \qquad \Delta_2\big( \mathcal{H}[S] \big) \leqslantslant \Delta_2(\mathcal{H}) \leqslantslant m.
\]
Moreover, it is not difficult to deduce that there exists a subhypergraph $\mathcal{H}' \subset \mathcal{H}[S]$ with
\[
v(\mathcal{H}') = |S| = s, \quad e(\mathcal{H}') = s^4 / m^{6+o(1)}, \quad \text{and} \quad \Delta_1(\mathcal{H}') = O\big(e(\mathcal{H}') / v(\mathcal{H}')\big).
\]
We may therefore apply the container lemma for 3-uniform hypergraphs to $\mathcal{H}'$ to obtain a collection $\mathcal{C}$ of at most $\mathrm{ex}p\big( m^{3+o(1)}/\sqrt{s} \big)$ subsets of $S$ with the following properties:
\begin{enumerate}
\item[$(a)$] Every set of points of $S$ in general position is contained in some $C \in \mathcal{C}$,
\item[$(b)$] Each $C \in \mathcal{C}$ has size at most $(1 - \delta) |S|$.
\end{enumerate}
Starting with $S = [m]^3$ and iterating this process for $O(\log m)$ steps, we obtain the following container theorem for sets of points in general position.
\begin{thm}\label{thm:CT:points:gen:position}
For each $m \in \mathbb{N}$, there exists a collection $\mathcal{C}$ of subsets of $[m]^3$ with
\begin{equation}\label{eq:CT:points:gen:position:size}
|\mathcal{C}| \leqslantslant \mathrm{ex}p\big( m^{5/3 + o(1)} \big)
\end{equation}
such that
\begin{itemize}
\item[$(a)$] $|C| \leqslantslant m^{8/3+o(1)}$ for each $C \in \mathcal{C}$;
\item[$(b)$] each set of points of $[m]^3$ in general position is contained in some $C \in \mathcal{C}$.
\end{itemize}
\end{thm}
Now, let $p = m^{-1+o(1)}$ and consider a $p$-random subset $R \subset [m]^3$, that is, each element of $[m]^3$ is included in $R$ independently at random with probability $p$. Since $[m]^3$ contains $m^{6+o(1)}$ sets of four collinear points\footnote{This is because there are $O(m^6/t^4)$ lines in $\mathbb{R}^3$ that contain more than $t$ points of $[m]^3$.}, it follows that, with high probability, $|R| = pm^{3+o(1)} = m^{2+o(1)}$ and $R$ contains $p^4 m^{6+o(1)} = o(|R|)$ collinear 4-tuples. Moreover, since $|C| \leqslantslant m^{8/3+o(1)}$ for each $C \in \mathcal{C}$, it follows from~\eqref{eq:CT:points:gen:position:size} and standard estimates on the tail of the binomial distribution that with high probability we have $|R \cap C| \leqslantslant m^{5/3 + o(1)}$ for every $C \in \mathcal{C}$. In particular, removing one element from each collinear 4-tuple in $R$ yields a set $A \subset [m]^3$ of size $m^{2+o(1)}$ with no collinear 4-tuple and containing no set of points in general position of size larger than $m^{5/3 + o(1)}$. Finally, project the points of~$A$ to the plane in such a way that collinear triples remain collinear, and no new collinear triple is created. In this way, we obtain a set of $n = m^{2+o(1)}$ points in the plane, no four of them on a line, such that no set of size greater than $n^{5/6+o(1)} = m^{5/3+o(1)}$ is in general position, as required.
\section{The key container lemma}\label{keylemma:sec}
In this section, we state a container lemma for hypergraphs of arbitrary uniformity. The version of the lemma stated below, which comes from~\cite{MSS}, differs from the statement originally proved by the authors of this survey~\cite[Proposition~3.1]{BMS} only in that the dependencies between the various constants have been made more explicit here; a careful analysis of the proof of~\cite[Proposition~3.1]{BMS} will yield this slightly sharper statement.\footnote{A complete proof of the version of the container lemma stated here can be found in~\cite{MSS}.} Let us recall that for a hypergraph $\mathcal{H}$ and an integer $\ell$, we write $\Delta_\ell(\mathcal{H})$ for the maximum degree of a set of $\ell$ vertices of $\mathcal{H}$, that is,
\[
\Delta_\ell(\mathcal{H}) = \max\big\{ d_\mathcal{H}(A) \,\colon A \subset V(\mathcal{H}), \, |A| = \ell \big\},
\]
where $d_\mathcal{H}(A) = \big| \big\{ B \in E(\mathcal{H}) \,\colon A \subset B \big\} \big|$, and $\mathcal{I}(\mathcal{H})$ for the collection of independent sets of $\mathcal{H}$. The lemma states, roughly speaking, that each independent set $I$ in a uniform hypergraph $\mathcal{H}$ can be assigned a \emph{fingerprint} $S \subset I$ in such a way that all sets with the same fingerprint are contained in a single set $C = f(S)$, called a \emph{container}, whose size is bounded away from $v(\mathcal{H})$. More importantly, the sizes of these fingerprints (and hence also the number of containers) can be bounded from above (in an optimal way!) by basic parameters of $\mathcal{H}$.
\begin{HCL}
Let $k \in \mathbb{N}$ and set $\delta = 2^{-k(k+1)}$. Let $\mathcal{H}$ be a $k$-uniform hypergraph and suppose that
\begin{equation}\label{eq:containers:condition}
\Delta_\ell(\mathcal{H}) \leqslantslant \bigg( \frac{b}{v(\mathcal{H})} \bigg)^{\ell-1} \, \frac{e(\mathcal{H})}{r}
\end{equation}
for some $b,r \in \mathbb{N}$ and every $\ell \in \{1, \dotsc, k\}$. Then there exists a collection $\mathcal{C}$ of subsets of $V(\mathcal{H})$ and a function $f \mathrm{col}on \mathcal{P}\big( V(\mathcal{H}) \big) \to \mathcal{C}$ such that:
\begin{enumerate}
\item[$(a)$] for every $I \in \mathcal{I}(\mathcal{H})$, there exists $S \subset I$ with $|S| \leqslantslant (k-1)b$ and $I \subset f(S)$;
\item[$(b)$] $|C| \leqslantslant v(\mathcal{H}) - \delta r$ for every $C \in \mathcal{C}$.
\end{enumerate}
\end{HCL}
The original statement of the container lemma~\cite[Proposition~3.1]{BMS} had $r = v(\mathcal{H}) / c$ for some constant $c$, since this choice of parameters is required in most standard applications. In particular, the simple container lemma for $3$-uniform hypergraphs presented in Section~\ref{basic:sec} is easily derived from the above statement by letting $b = v(\mathcal{H}) / (2\sqrt{d})$ and $r = v(\mathcal{H}) / (6c)$, where $d = 3e(\mathcal{H})/v(\mathcal{H})$ is the average degree of $\mathcal{H}$. There are, however, arguments that benefit from setting $r = o(v(\mathcal{H}))$; we present one of them in Section~\ref{sec:many-colours}.
Even though the property $|C| \leqslantslant v(\mathcal{H}) - \delta r$ that is guaranteed for all containers $C \in \mathcal{C}$ seems rather weak at first sight, it can be easily strengthened with repeated applications of the lemma. In particular, if for some hypergraph $\mathcal{H}$, condition~\eqref{eq:containers:condition} holds (for all $\ell$) with some $b = o(v(\mathcal{H}))$ and $r = \Omega(v(\mathcal{H}))$, then recursively applying the lemma to subhypergraphs of $\mathcal{H}$ induced by all the containers $C$ for which $e(\mathcal{H}[C]) \geqslantslant \varepsilon e(\mathcal{H})$ eventually produces a collection $\mathcal{C}$ of containers indexed by sets of size $O(b)$ such that $e(\mathcal{H}[C]) < \varepsilon e(\mathcal{H})$ for every $C \in \mathcal{C}$. This is precisely how (in Section~\ref{basic:sec}) we derived Theorem~\ref{thm:CT:triangles} from the container lemma for $3$-uniform hypergraphs. For a formal argument showing how such a family of `tight' containers may be constructed, we refer the reader to~\cite{BMS}.
One may thus informally say that the hypergraph container lemma provides a covering of the family of all independent sets of a uniform hypergraph with `few' sets that are `almost independent'. In many natural settings, these almost independent sets closely resemble truly independent sets. In some cases, this is a straightforward consequence of corresponding removal lemmas. A more fundamental reason is that many sequences of hypergraphs $\mathcal{H}_n$ of interest possess the following self-similarity property: For all (or many) pairs $m$ and $n$ with $m < n$, the hypergraph $\mathcal{H}_n$ admits a very uniform covering by copies of $\mathcal{H}_m$. For example, this is the case when $\mathcal{H}_n$ is the hypergraph encoding triangles in $K_n$, simply because every $m$-element set of vertices of $K_n$ induces $K_m$. Such self-similarity enables one to use elementary averaging arguments to characterise almost independent sets; for example, the standard proof of Lemma~\ref{lem:supersat:triangles} uses such an argument.
The fact that the fingerprint $S$ of each independent set $I \in \mathcal{I}(\mathcal{H})$ is a subset of $I$ is not merely a by-product of the proof of the hypergraph container lemma. On the contrary, it is an important property of the family of containers that can be often exploited to make union bound arguments tighter. This is because each $I \in \mathcal{I}(\mathcal{H})$ is sandwiched between $S$ and $f(S)$ and consequently when enumerating independent sets one may use a union bound over all fingerprints $S$ and enumerate only over the sets $I \setminus S$ (which are contained in $f(S)$). In particular, such finer arguments can be used to remove the superfluous logarithmic factor from the assumptions of the proofs outlined in Section~\ref{basic:sec}.
For example, in the proof of Theorem~\ref{thm:Mantel:Gnp} presented in Section~\ref{Mantel:sec}, the fingerprints of triangle-free subgraphs of $K_n$ form a family $\mathcal{S}$ of $n$-vertex graphs, each with at most $C_\varepsilon n^{3/2}$ edges. Setting $m = \big( \frac{1}{2} + \alpha \big) p \binom{n}{2}$, this allows us to replace~\eqref{eq:Mantel:Gnp} with the following estimate:
\begin{equation}\label{eq:fingerprint}
\mathcal{P}r\Big( \mathrm{ex}\big( G(n,p), K_3 \big) \geqslantslant m \Big) \, \leqslantslant \, \sum_{S \in \mathcal{S}} \mathcal{P}r\Big( S \subset G(n,p) \textup{ and } e\big( \big(f(S)\setminus S\big) \cap G(n,p) \big) \geqslantslant m - |S| \Big).
\end{equation}
Since the two events in the right-hand side of~\eqref{eq:fingerprint} concern the intersections of $G(n,p)$ with two disjoint sets of edges of $K_n$, they are independent. If $p \gg n^{-1/2}$, then $|S| \ll p \binom{n}{2}$ and consequently, recalling that $e(f(S)) \leqslantslant \big(\frac{1+\alpha}{2}\big) \binom{n}{2}$, we may bound the right-hand side of~\eqref{eq:fingerprint} from above by
\[
\sum_{S \in \mathcal{S}} p^{|S|} e^{-\beta pn^2} \leqslantslant \sum_{s \leqslantslant C_\varepsilon n^{3/2}} \binom{\binom{n}{2}}{s} \cdot p^s e^{-\beta pn^2} \leqslantslant \sum_{s \leqslantslant C_\varepsilon n^{3/2}} \leqslantslantft(\frac{e\binom{n}{2}p}{s}\right)^s e^{-\beta pn^2} \leqslantslant e^{-\beta pn^2/2}
\]
for some $\beta = \beta(\alpha) > 0$.
Finally, what is the intuition behind condition~\eqref{eq:containers:condition}? A natural way to define $f(S)$ for a given (independent) set $S$ is to let $f(S) = V(\mathcal{H}) \setminus X(S)$, where $X(S)$ comprises all vertices $v$ such that $A \subset S \cup \{v\}$ for some $A \in E(\mathcal{H})$. Indeed, every independent set $I$ that contains $S$ must be disjoint from $X(S)$. (In reality, the definition of $X(S)$ is -- and has to be -- more complicated than this, and some vertices are placed in $X(S)$ simply because they do not belong to $S$.) Suppose, for the sake of argument, that $S$ is a random set of $b$ vertices of $\mathcal{H}$. Letting $\tau = b/v(\mathcal{H})$, we have
\begin{equation}
\label{eq:fingerprint-intuition}
\mathbb{E}\big[|X(S)|\big] \leqslantslant \sum_{A \in E(\mathcal{H})} \mathcal{P}r\big(|A \cap S| = k-1\big) \leqslantslant k \cdot \tau^{k-1} \cdot e(\mathcal{H}).
\end{equation}
Since we want $X(S)$ to have at least $\delta r$ elements for every fingerprint $S$, it seems reasonable to require that
\[
\Delta_k(\mathcal{H}) = 1 \leqslantslant \frac{k}{\delta} \cdot \tau^{k-1} \cdot \frac{e(\mathcal{H})}{r},
\]
which is, up to a constant factor, condition~\eqref{eq:containers:condition} with $\ell = k$. For some hypergraphs $\mathcal{H}$ however, the first inequality in~\eqref{eq:fingerprint-intuition} can be very wasteful, since some $v \in X(S)$ may have many $A \in E(\mathcal{H})$ such that $A \subset S \cup \{v\}$. This can happen if for some $\ell \in \{1, \dotsc, k-1\}$, there is an $\ell$-uniform hypergraph $\mathcal{G}$ such that each edge of $\mathcal{H}$ contains an edge of $\mathcal{G}$; note that $e(\mathcal{G})$ can be as small as $e(\mathcal{H}) / \Delta_\ell(\mathcal{H})$. Our assumption implies that $\mathcal{I}(\mathcal{G}) \subset \mathcal{I}(\mathcal{H})$ and thus, letting $Y(S)$ be the set of all vertices $w$ such that $B \subset S \cup \{w\}$ for some $B \in E(\mathcal{G})$, we have $X(S) \subset Y(S)$. In particular, we want $Y(S)$ to have at least $\delta r$ elements for every fingerprint $S$ of an independent set $I \in \mathcal{I}(\mathcal{G})$. Repeating~\eqref{eq:fingerprint-intuition} with $X$ replaced by $Y$, $\mathcal{H}$ replaced by $\mathcal{G}$, and $k$ replaced by $\ell$, we arrive at the inequality
\[
\delta r \leqslantslant \ell \cdot \tau^{\ell-1} \cdot e(\mathcal{G}) = \ell \cdot \tau^{\ell-1} \cdot \frac{e(\mathcal{H})}{\Delta_\ell(\mathcal{H})},
\]
which is, up to a constant factor, condition~\eqref{eq:containers:condition}.
One may further develop the above argument to show that condition~\eqref{eq:containers:condition} is asymptotically optimal, at least when $r = \Omega(v(\mathcal{H}))$. Roughly speaking, one can construct $k$-uniform hypergraphs that have $\binom{(1-o(1)) v(\mathcal{H})}{m}$ independent $m$-sets for every $m = o(b)$, where $b$ is minimal so that condition~\eqref{eq:containers:condition} holds, whereas the existence of containers of size at most $(1-\delta) v(\mathcal{H})$ indexed by fingerprints of size $o(b)$ would imply that the number of such sets is at most $\binom{(1-\varepsilon) v(\mathcal{H})}{m}$ for some constant $\varepsilon > 0$.
\section{Counting $H$-free graphs}\label{counting:Hfree:sec}
How many graphs are there on $n$ vertices that do not contain a copy of $H$? An obvious lower bound is $2^{\mathrm{ex}(n,H)}$, since each subgraph of an $H$-free graph is also $H$-free. For non-bipartite graphs, this is not far from the truth. Writing $\mathcal{F}_n(H)$ for the family of $H$-free graphs on $n$ vertices, if $\chi(H) \geqslantslant 3$, then
\begin{equation}\label{eq:EFR}
|\mathcal{F}_n(H)| = 2^{(1 + o(1))\mathrm{ex}(n,H)}
\end{equation}
as $n \to \infty$, as was first shown by Erd\H{o}s, Kleitman, and Rothschild~\cite{EKR} (when $H$ is a complete graph) and then by Erd\H{o}s, Frankl, and R\"odl~\cite{EFR}. For bipartite graphs, on the other hand, the problem is much more difficult. In particular, the following conjecture (first stated in print in~\cite{KW82}), which played a major role in the development of the container method, remains open.
\begin{conj}\label{conj:counting}
For every bipartite graph $H$ that contains a cycle, there exists $C > 0$ such that
\[
|\mathcal{F}_n(H)| \leqslantslant 2^{C\mathrm{ex}(n,H)}
\]
for every $n \in \mathbb{N}$.
\end{conj}
The first significant progress on Conjecture~\ref{conj:counting} was made by Kleitman and Winston~\cite{KW82}. Their proof of the case $H = C_4$ of the conjecture introduced (implicitly) the container method for graphs. Nevertheless, it took almost thirty years\footnote{An unpublished manuscript of Kleitman and Wilson from 1996 proves that $|\mathcal{F}_n(C_6)| = 2^{O(\mathrm{ex}(n,C_6))}$.} until their theorem was generalized to the case $H = K_{s,t}$, by Balogh and Samotij~\cite{BSmm,BSst}, and then (a few years later) to the case $H = C_{2k}$, by Morris and Saxton~\cite{MS}. More precisely, it was proved in~\cite{BSst,MS} that
\[
|\mathcal{F}_n(K_{s,t})| = 2^{O(n^{2-1/s})} \qquad \textup{and} \qquad |\mathcal{F}_n(C_{2k})| = 2^{O(n^{1+1/k})}
\]
for every $2 \leqslantslant s \leqslantslant t$ and every $k \geqslantslant 2$, which implies Conjecture~\ref{conj:counting} when $t > (s-1)!$ and $k \in \{2,3,5\}$, since in these cases it is known that $\mathrm{ex}(n,K_{s,t}) = \Theta(n^{2-1/s})$ and $\mathrm{ex}(n,C_{2k}) = \Theta(n^{1+1/k})$.
Very recently, Ferber, McKinley, and Samotij~\cite{FMS}, inspired by a similar result of Balogh, Liu, and Sharifzadeh~\cite{BLS} on sets of integers with no $k$-term arithmetic progression, found a very simple proof of the following much more general theorem.
\begin{thm}\label{thm:FMS}
Suppose that $H$ contains a cycle. If $\mathrm{ex}(n,H) = O(n^\alpha)$ for some constant $\alpha$, then
\[
|\mathcal{F}_n(H)| = 2^{O(n^\alpha)}.
\]
\end{thm}
Note that Theorem~\ref{thm:FMS} resolves Conjecture~\ref{conj:counting} for every $H$ such that $\mathrm{ex}(n,H) = \Theta(n^\alpha)$ for some constant $\alpha$. Moreover, it was shown in~\cite{FMS} that the weaker assumption that $\mathrm{ex}(n,H) \gg n^{2 - 1/m_2(H) + \varepsilon}$ for some $\varepsilon > 0$ already implies that the assertion of Conjecture~\ref{conj:counting} holds for infinitely many $n$; we refer the interested reader to~\cite{FMS} for details. Let us also note here that, while it is natural to suspect that in fact the stronger bound~\eqref{eq:EFR} holds for all graphs $H$ that contain a cycle, this is false for $H = C_6$, as was shown by Morris and Saxton~\cite{MS}. However, it may still hold for $H = C_4$ and it would be very interesting to determine whether or not this is indeed the case.
The proof of Theorem~\ref{thm:FMS} for general $H$ is somewhat technical, so let us instead sketch the proof in the case $H = C_4$. In this case, the proof combines the hypergraph container lemma stated in the previous section with the following supersaturation lemma.
\begin{lemma}\label{lem:C4:super}
There exist constants $\beta > 0$ and $k_0 \in \mathbb{N}$ such that the following holds for every $k \geqslantslant k_0$ and every $n \in \mathbb{N}$. Given a graph~$G$ with~$n$ vertices and $k \cdot \mathrm{ex}(n,C_4)$ edges, there exists a collection $\mathcal{H}$ of at least $\beta k^5 \cdot \mathrm{ex}(n,C_4)$ copies of $C_4$ in $G$ that satisfies:
\begin{itemize}
\item[$(a)$] Each edge belongs to at most $k^4$ members of $\mathcal{H}$.
\item[$(b)$] Each pair of edges is contained in at most $k^2$ members of $\mathcal{H}$.
\end{itemize}
\end{lemma}
The proof of Lemma~\ref{lem:C4:super} employs several simple but important ideas that can be used in a variety of other settings, so let us sketch the details. The first key idea, which was first used in~\cite{MS}, is to build the required family $\mathcal{H}$ one $C_4$ at a time. Let us say that a collection~$\mathcal{H}$ of copies of $C_4$ is \emph{legal} if it satisfies conditions $(a)$ and $(b)$ and suppose that we have already found a legal collection~$\mathcal{H}_m$ of $m$ copies of $C_4$ in $G$. Note that we are done if $m \geqslantslant \beta k^5 \cdot \mathrm{ex}(n,C_4)$, so let us assume that the reverse inequality holds and construct a legal collection~$\mathcal{H}_{m+1} \supset \mathcal{H}_m$ of $m+1$ copies of $C_4$ in $G$.
We claim that there exists a collection $\mathcal{A}_m$ of $\beta k^5 \cdot \mathrm{ex}(n,C_4)$ copies of $C_4$ in $G$, any of which can be added to $\mathcal{H}_m$ without violating conditions $(a)$ and $(b)$, that is, such that $\mathcal{H}_m \cup \{ C \}$ is legal for any $C \in \mathcal{A}_m$. (Let us call these \emph{good} copies of $C_4$.) Since $m < \beta k^5 \cdot \mathrm{ex}(n,C_4)$, then at least one element of $\mathcal{A}_m$ is not already in $\mathcal{H}_m$, so this will be sufficient to prove the lemma.
To find $\mathcal{A}_m$, observe first that (by simple double-counting) at most $4\beta k \cdot \mathrm{ex}(n,C_4)$ edges of $G$ lie in exactly $k^4$ members of $\mathcal{H}_m$ and similarly at most $6\beta k^3 \cdot \mathrm{ex}(n,C_4)$ pairs of edges of $G$ lie in exactly $k^2$ members of $\mathcal{H}_m$. Now, consider a random subset $A \subset V(G)$ of size $pn$, where $p = D/k^2$ for some large constant $D$. Typically $G[A]$ contains about $p^2 k \cdot \mathrm{ex}(n,C_4)$ edges. After removing from $G[A]$ all \emph{saturated} edges (i.e., those belonging to $k^4$ members of $\mathcal{H}_m$) and one edge from each \emph{saturated} pair (i.e., pair of edges that is contained in $k^2$ members of $\mathcal{H}_m$), we expect to end up with at least
\begin{equation*}
p^2 k \cdot \mathrm{ex}(n,C_4) - 4\beta p^2 k \cdot \mathrm{ex}(n,C_4) - 6\beta p^3 k^3 \cdot \mathrm{ex}(n,C_4) \geqslantslant \frac{p^2 k \cdot \mathrm{ex}(n,C_4)}{2} \geqslantslant 2 \cdot \mathrm{ex}( pn, C_4)
\end{equation*}
edges, where the first inequality follows since $p = D / k^2$ and $\beta$ is sufficiently small, and the second holds because $\mathrm{ex}(n,C_4) = \Theta(n^{3/2})$ and $D$ is sufficiently large. Finally, observe that any graph on $pn$ vertices with at least $2 \cdot \mathrm{ex}( pn, C_4)$ edges contains at least
$$\mathrm{ex}\big( pn, C_4 \big) \, = \, \Omega\Big( p^{3/2} \cdot \mathrm{ex}(n,C_4) \Big)$$
copies of~$C_4$. But each copy of $C_4$ in $G$ was included in the random subgraph $G[A]$ with probability at most $p^4$ and hence (with a little care) one can show that there must exist at least $\Omega\big( p^{-5/2} \cdot \mathrm{ex}(n, C_4) \big)$ copies of $C_4$ in $G$ that avoid all saturated edges and pairs of edges. Since $p^{-5/2} = k^5/D^{5/2}$ and $\beta$ is sufficiently small, we have found $\beta k^5 \cdot \mathrm{ex}(n, C_4)$ good copies of $C_4$ in $G$, as required.
We now show how one may combine Lemma~\ref{lem:C4:super} and the hypergraph container lemma to construct families of containers for $C_4$-free graphs. Let $\beta$ and $k_0$ be the constants from the statement of Lemma~\ref{lem:C4:super} and assume that $G$ is an $n$-vertex graph with at least $k \cdot \mathrm{ex}(n, C_4)$ and at most $2k \cdot \mathrm{ex}(n, C_4)$ edges, where $k \geqslantslant k_0$. Denote by $\mathcal{H}_G$ the 4-uniform hypergraph with vertex set $E(G)$, whose edges are the copies of $C_4$ in $G$ given by Lemma~\ref{lem:C4:super}. Since
\[
v(\mathcal{H}_G) = e(G), \quad e(\mathcal{H}_G) \geqslantslant \beta k^5 \cdot \mathrm{ex}(n,C_4), \quad \Delta_1(\mathcal{H}_G) \leqslantslant k^4, \quad \Delta_2(\mathcal{H}_G) \leqslantslant k^2,
\]
and $\Delta_3(\mathcal{H}_G) = \Delta_4(\mathcal{H}_G) = 1$, the hypergraph $\mathcal{H}_G$ satisfies the assumptions of the container lemma with $r = \beta k \cdot \mathrm{ex}(n, C_4)$ and $b = 2k^{-1/3} \cdot \mathrm{ex}(n, C_4)$. Consequently, there exist an absolute constant~$\delta$ and a collection $\mathcal{C}$ of subgraphs of $G$ with the following properties:
\begin{enumerate}
\item[$(a)$] every $C_4$-free subgraph of $G$ is contained in some $C \in \mathcal{C}$,
\item[$(b)$] each $C \in \mathcal{C}$ has at most $(1 - \delta) e(G)$ edges,
\end{enumerate}
and moreover
\[
|\mathcal{C}| \leqslantslant \sum_{s=0}^{3b} \binom{e(G)}{s} \leqslantslant \leqslantslantft(\frac{e(G)}{b}\right)^{3b} \leqslantslant k^{4b} \leqslantslant \mathrm{ex}p\Big( 8 k^{-1/3} \log k \cdot \mathrm{ex}(n,C_4) \Big).
\]
Note that we have just replaced a single container for the family of $C_4$-free subgraphs of $G$ (namely $G$ itself) with a small collection of containers for this family, each of which is somewhat smaller than~$G$. Since every $C_4$-free graph with $n$ vertices is contained in $K_n$, by repeatedly applying this `breaking down' process, we obtain the following container theorem for $C_4$-free graphs.
\begin{thm}\label{thm:C4containers:weak}
There exist constants $k_0 > 0$ and $C > 0$ such that the following holds for all $n \in \mathbb{N}$ and $k \geqslantslant k_0$. There exists a collection~$\mathcal{G}(n,k)$ of at most
\[
\mathrm{ex}p\bigg( \frac{C \log k}{k^{1/3}} \cdot \mathrm{ex}(n,C_4) \bigg)
\]
graphs on $n$ vertices such that
\[
e(G) \leqslantslant k \cdot \mathrm{ex}(n,C_4)
\]
for every $G \in \mathcal{G}(n,k)$ and every $C_4$-free graph on $n$ vertices is a subgraph of some~$G \in \mathcal{G}(n,k)$.
\end{thm}
To obtain the claimed upper bound on $|\mathcal{G}(n,k)|$, note that if $k \cdot \mathrm{ex}(n,C_4) \geqslantslant \binom{n}{2}$ then we may take $\mathcal{G}(n,k) = \{K_n\}$, and otherwise the argument presented above yields
\[
|\mathcal{G}(n,k)| \leqslantslant \big|\mathcal{G}\big(n, k/(1-\delta)\big)\big| \cdot \mathrm{ex}p\Big(8k^{-1/3}\log k \cdot \mathrm{ex}(n, C_4)\Big).
\]
In particular, applying Theorem~\ref{thm:C4containers:weak} with $k = k_0$, we obtain a collection of $2^{O(\mathrm{ex}(n,C_4))}$ containers for $C_4$-free graphs on $n$ vertices, each with $O\big( \mathrm{ex}(n,C_4) \big)$ edges. This immediately implies that Conjecture~\ref{conj:counting} holds for $H = C_4$. The proof for a general graph $H$ (under the assumption that $\mathrm{ex}(n,H) = \Theta(n^\alpha)$ for some $\alpha \in (1,2)$) is similar, though the details are rather technical.
\subsection{Tur\'an's problem in random graphs}
Given that the problem of estimating $|\mathcal{F}_n(H)|$ for bipartite graphs $H$ is notoriously difficult, it should not come as a surprise that determining the typical value of the Tur\'an number $\mathrm{ex}\big(G(n,p), C_4\big)$ for bipartite $H$ also poses considerable challenges. Compared to the non-bipartite case, which was essentially solved by Conlon--Gowers~\cite{CG} and Schacht~\cite{Sch}, see Theorem~\ref{thm:Turan:Gnp}, the typical behaviour of $\mathrm{ex}\big(G(n,p), H \big)$ for bipartite graphs $H$ is much more subtle.
For simplicity, let us restrict our attention to the case $H = C_4$. Recall from Theorem~\ref{thm:Turan:Gnp} that the typical value of $\mathrm{ex}\big(G(n,p), C_4\big)$ changes from $\big( 1 + o(1) \big) p\binom{n}{2}$ to $o(pn^2)$ when $p = \Theta(n^{-2/3})$, as was first proved by Haxell, Kohayakawa, and {\L}uczak~\cite{HKL-even}. However, already several years earlier F\"uredi~\cite{Fur91} used the method of Kleitman and Winston~\cite{KW82} to prove\footnote{To be precise, F\"uredi proved that, if $m \geqslantslant 2n^{4/3} (\log n)^2$, then there are at most $(4n^3 / m^2)^m$ $C_4$-free graphs with $n$ vertices and $m$ edges, which implies the upper bounds in Theorem~\ref{thm:randomturan}. For the lower bounds, see~\cite{KKS,MS}.} the following much finer estimates of this extremal number for $p$ somewhat above the threshold.
\begin{thm}\label{thm:randomturan}
Asymptotically almost surely,
\[
\mathrm{ex} \big( G(n,p), C_4 \big) =
\begin{cases}
\big( 1 + o(1) \big) p {n \choose 2} & \textup{if $n^{-1} \ll p \ll n^{-2/3}$}, \\
n^{4/3} (\log n)^{O(1)} & \textup{if $n^{-2/3} \leqslantslant p \leqslantslant n^{-1/3} (\log n)^4$}, \\
\Theta\big( \sqrt{p} \cdot n^{3/2} \big) & \textup{if $p \geqslantslant n^{-1/3} (\log n)^4$}.
\end{cases}
\]
\end{thm}
We would like to draw the reader's attention to the (somewhat surprising) fact that in the middle range $n^{-2/3+o(1)} \leqslantslant p \leqslantslant n^{-1/3+o(1)}$, the typical value of $\mathrm{ex} \big( G(n,p), C_4 \big)$ stays essentially constant. A similar phenomenon has been observed in random Tur\'an problems for other forbidden bipartite graphs (even cycles~\cite{KKS, MS} and complete bipartite graphs~\cite{MS}) as well as Tur\'an-type problems in additive combinatorics~\cite{DeKoLeRoSa-Bh, DeKoLeRoSa-B3}. It would be very interesting to determine whether or not a similar `long flat segment' appears in the graph of $p \mapsto \mathrm{ex}\big( G(n,p), H \big)$ for every bipartite graph $H$. We remark that the lower bound in the middle range is given (very roughly speaking) by taking a random subgraph of $G(n,p)$ with density $n^{-2/3+o(1)}$ and then finding\footnote{One easy way to do this is simply to remove one edge from each copy of $C_4$. A more efficient method, used by Kohayakawa, Kreuter, and Steger~\cite{KKS} to improve the lower bound by a polylogarithmic factor, utilizes a version of the general result of~\cite{AKPSS} on independent sets in hypergraphs obtained in~\cite{DuLeRo95}; see also~\cite{FMS}.} a large $C_4$-free subgraph of this random graph; the lower bound in the top range is given by intersecting $G(n,p)$ with a suitable blow-up of an extremal $C_4$-free graph and destroying any $C_4$s that occur; see~\cite{KKS, MS} for details.
Even though Theorem~\ref{thm:C4containers:weak} immediately implies that $\mathrm{ex}\big( G(n,p), C_4\big) = o(pn^2)$ if $p \gg n^{-2/3} \log n$, it is not strong enough to prove Theorem~\ref{thm:randomturan}. A stronger container theorem for $C_{2\ell}$-free graphs (based on a supersaturation lemma that is sharper than Lemma~\ref{lem:C4:super}) was obtained in~\cite{MS}. In the case $\ell = 2$, the statement is as follows.
\begin{thm}\label{thm:C4containers:turan}
There exist constants $k_0 > 0$ and $C > 0$ such that the following holds for all $n \in \mathbb{N}$ and $k_0 \leqslantslant k \leqslantslant n^{1/6} / \log n$. There exists a collection~$\mathcal{G}(n,k)$ of at most
\[
\mathrm{ex}p\bigg( \frac{C \log k}{k} \cdot \mathrm{ex}(n,C_4) \bigg)
\]
graphs on $n$ vertices such that
\[
e(G) \leqslantslant k \cdot \mathrm{ex}(n,C_4)
\]
for every $G \in \mathcal{G}(n,k)$ and every $C_4$-free graph on $n$ vertices is a subgraph of some~$G \in \mathcal{G}(n,k)$.
\end{thm}
Choosing $k$ to be a suitable function of $p$, it is straightforward to use Theorem~\ref{thm:C4containers:turan} to prove a slightly weaker version of Theorem~\ref{thm:randomturan}, with an extra factor of $\log n$ in the upper bound on $\mathrm{ex} \big( G(n,p), C_4 \big)$. As usual, this logarithmic factor can be removed via a more careful application of the container method, using the fact that the fingerprint of an independent set is contained in it, cf.\ the discussion in Section~\ref{keylemma:sec}; see~\cite{MS} for the details. However, we are not able to determine the correct power of $\log n$ in $\mathrm{ex} \big( G(n,p), C_4 \big)$ in the middle range $n^{-2/3 + o(1)} \ll p \ll n^{-1/3 + o(1)}$ and we consider this to be an important open problem. It would also be very interesting to prove similarly sharp container theorems for other bipartite graphs $H$.
\section{Containers for multicoloured structures}
\label{sec:many-colours}
All of the problems that we have discussed so far, and many others, are naturally expressed as questions about independent sets in various hypergraphs. There are, however, questions of a very similar flavour that are not easily described in this way but are still amenable to the container method. As an example, consider the problem of enumerating large graphs with no \emph{induced} copy of a given graph $H$. We shall say that a graph $G$ is \emph{induced-$H$-free} if no subset of vertices of $G$ induces a subgraph isomorphic to $H$. As it turns out, it is beneficial to think of an $n$-vertex graph $G$ as the characteristic function of its edge set. A function $g \mathrm{col}on E(K_n) \to \{0, 1\}$ is the characteristic function of an induced-$H$-free graph if and only if for every set $W$ of $v(H)$ vertices of $K_n$, the restriction of $g$ to the set of pairs of vertices of $W$ is not the characteristic function of the edge set of $H$. In particular, viewing $g$ as the set of pairs $\big\{(e, g(e)) \,\colon e \in E(K_n)\big\}$, we see that if $g$ represents an induced-$H$-free graph, then it is an independent set in the $\binom{v(H)}{2}$-uniform hypergraph $\mathcal{H}$ with vertex set $E(K_n) \times \{0, 1\}$ whose edges are the characteristic functions of all copies of $H$ in $K_n$; formally, for every injection $\varphi \mathrm{col}on V(H) \to V(K_n)$, the set
\[
\Big\{\big(\varphi(u)\varphi(v), 1\big) \,\colon uv \in E(H)\Big\} \cup \Big\{\big(\varphi(u)\varphi(v), 0\big) \,\colon uv \notin E(H)\Big\}
\]
is an edge of $\mathcal{H}$. Even though the converse statement is not true and not every independent set of $\mathcal{H}$ corresponds to an induced-$H$-free graph, since we are usually interested in bounding the number of such graphs from above, the above representation can be useful. In particular, Saxton and Thomason~\cite{ST} applied the container method to the hypergraph $\mathcal{H}$ described above to reprove the following analogue of~\eqref{eq:EFR}, which was originally obtained by Alekseev~\cite{Al92} and by Bollob\'as and Thomason~\cite{BoTh95, BoTh97}. Letting $\mathcal{F}_n^{\textup{ind}}(H)$ denote the family of all induced-$H$-free graphs with vertex set $\{1, \dotsc, n\}$, we have
\[
|\mathcal{F}_n^{\textup{ind}}(H)| = 2^{(1-1/\mathrm{col}(H)) \binom{n}{2} + o(n^2)},
\]
where $\mathrm{col}(H)$ is the so-called \emph{colouring number}\footnote{The \emph{colouring number} of a graph $H$ is the largest integer $r$ such that for some pair $(r_1, r_2)$ satisfying $r_1 + r_2 = r$, the vertex set of $H$ cannot be partitioned into $r_1$ cliques and $r_2$ independent sets.} of $H$.
This idea of embedding non-monotone properties (such as the family of induced-$H$-free graphs) into the family of independent sets of an auxiliary hypergraph has been used in several other works. In particular, K\"uhn, Osthus, Townsend, and Zhao~\cite{KuOsToZh17} used it to describe the typical structure of oriented graphs without a transitive tournament of a given order. The recent independent works of Falgas-Ravry, O'Connell, Str\"omberg, and Uzzell~\cite{FROCStUz} and of Terry~\cite{Te} have developed a general framework for studying various enumeration problems in the setting of multicoloured graphs~\cite{FROCStUz} and, more generally, in the very abstract setting of finite (model theoretic) structures~\cite{Te}. In order to illustrate some of the ideas involved in applications of this kind, we will discuss the problem of counting finite metric spaces with bounded integral distances.
\subsection{Counting metric spaces}
Let $\mathcal{M}_n^M$ denote the family of metric spaces on a given set of $n$ points with distances in the set $\{1,\ldots,M\}$. Thus $\mathcal{M}_n^M$ may be viewed as the set of all functions $d \mathrm{col}on E(K_n) \to \{1, \dotsc, M\}$ that satisfy the triangle inequality $d(uv) \leqslantslant d(uw) + d(wv)$ for all $u, v, w$. Since $x \leqslantslant y + z$ for all $x, y, z \in \{\lceil M/2 \rceil, \dotsc, M\}$, we have
\begin{equation}
\label{eq:MnM-lower}
\big|\mathcal{M}_n^M\big| \geqslantslant \leqslantslantft|\leqslantslantft\{\leqslantslantft\lceil \frac{M}{2} \right\rceil, \dotsc, M\right\}\right|^{\binom{n}{2}} = \leqslantslantft\lceil \frac{M+1}{2} \right\rceil^{\binom{n}{2}}.
\end{equation}
Inspired by a continuous version of the model suggested Benjamini (and first studied in~\cite{KoMePeSa}), Mubayi and Terry~\cite{MuTe} proved that for every fixed even $M$, the converse of~\eqref{eq:MnM-lower} holds asymptotically, that is, $|\mathcal{M}_n^M| \leqslantslant \big( 1 + o(1) \big) \big\lceil \frac{M+1}{2} \big\rceil^{\binom{n}{2}}$ as $n \to \infty$. The problem becomes much more difficult, however, when one allows $M$ to grow with $n$. For example, if $M \gg \sqrt{n}$ then the lower bound
\[
\big|\mathcal{M}_n^M\big| \geqslantslant \leqslantslantft[\leqslantslantft(\frac{1}{2} + \frac{c}{\sqrt{n}} \right) M \right]^{\binom{n}{2}}
\]
for some absolute constant $c > 0$, proved in~\cite{KoMePeSa}, is stronger than~\eqref{eq:MnM-lower}. Balogh and Wagner~\cite{BaWa16} proved strong upper bounds on $|\mathcal{M}_n^M|$ under the assumption that $M \ll n^{1/3} / (\log n)^{4/3+o(1)}$. The following almost optimal estimate was subsequently obtained by Kozma, Meyerovitch, Peled, and Samotij~\cite{KoMePeSa}.
\begin{thm}\label{thm:KMPS}
There exists a constant $C$ such that
\begin{equation}\label{eq:thm:KMPS}
\big|\mathcal{M}_n^M\big| \leqslantslant \leqslantslantft[\leqslantslantft(\frac{1}{2} + \frac{2}{M} + \frac{C}{\sqrt{n}} \right) M \right]^{\binom{n}{2}}
\end{equation}
for all $n$ and $M$.
\end{thm}
Here, we present an argument due to Morris and Samotij that derives a mildly weaker estimate using the hypergraph container lemma. Let $\mathcal{H}$ be the 3-uniform hypergraph with vertex set $E(K_n) \times \{1, \dotsc, M\}$ whose edges are all triples $\big\{(e_1, d_1), (e_2, d_2), (e_3, d_3)\big\}$ such that $e_1, e_2, e_3$ form a triangle in $K_n$ but $d_{\sigma(1)} + d_{\sigma(2)} < d_{\sigma(3)}$ for some permutation $\sigma$ of $\{1, 2, 3\}$. The crucial observation, already made in~\cite{BaWa16}, is that every metric space $d \mathrm{col}on E(K_n) \to \{1, \dotsc, M\}$, viewed as the set of pairs $\big\{(e, d(e)) \,\colon e \in E(K_n)\big\}$, is an independent set of $\mathcal{H}$. This enables the use of the hypergraph container method for bounding $\big|\mathcal{M}_n^M\big|$ from above. Define the volume of a set $A \subset E(K_n) \times \{1, \dotsc, M\}$, denoted by $\mathrm{vol}(A)$, by
\[
\mathrm{vol}(A) = \prod_{e \in E(K_n)} \Big|\Big\{ d \in \big\{1, \dotsc, M \big\} \,\colon (e, d) \in A \Big\}\Big|
\]
and observe that $A$ contains at most $\mathrm{vol}(A)$ elements of $\mathcal{M}_n^M$. The following supersaturation lemma was proved by Morris and Samotij.
\begin{lemma}\label{lemma:metric-space-supsat}
Let $n \geqslantslant 3$ and $M \geqslantslant 1$ be integers and suppose that $A \subset E(K_n) \times \{1, \dotsc, M\}$ satisfies
\[
\mathrm{vol}(A) = \bigg[ \bigg( \frac{1}{2} + \varepsilon \bigg) M \bigg]^{\binom{n}{2}}
\]
for some $\varepsilon \geqslantslant 10/M$. Then there exist $m \leqslantslant M$ and a set $A' \subset A$ with $|A'| \leqslantslant mn^2$, such that the hypergraph $\mathcal{H}' = \mathcal{H}[A']$ satisfies
\[
e(\mathcal{H}') \geqslantslant \frac{\varepsilon m^2 M}{50 \log M} \binom{n}{3}, \qquad \Delta_1(\mathcal{H}') \leqslantslant 4m^2n, \qquad \text{and} \qquad \Delta_2(\mathcal{H}') \leqslantslant 2m.
\]
\end{lemma}
It is not hard to verify that the hypergraph $\mathcal{H}'$ given by Lemma~\ref{lemma:metric-space-supsat} satisfies the assumptions of the hypergraph container lemma stated in Section~\ref{keylemma:sec} with $r = \varepsilon n^2 M / (2^{11} \log M)$ and $b = O(n^{3/2})$. Consequently, there exist an absolute constant $\delta$ and a collection $\mathcal{C}$ of subsets of $A'$, with
$$|\mathcal{C}| \leqslantslant \mathrm{ex}p\Big( O\big( n^{3/2}\log(nM) \big) \Big),$$
such that, setting $A_C = C \cup (A \setminus A') = A \setminus (A' \setminus C)$ for each $C \in \mathcal{C}$, the following properties hold:
\begin{enumerate}
\item[$(a)$] every metric space in $A$, viewed as a subset of $E(K_n) \times \{1, \dotsc, M\}$, is contained in $A_C$ for some $C \in \mathcal{C}$, and
\item[$(b)$] $|C| \leqslantslant |A'| - \delta r$ for every $C \in \mathcal{C}$.
\end{enumerate}
Observe that
\begin{align*}
\mathrm{vol}(A_C) & \, \leqslantslant \, \leqslantslantft(\frac{M-1}{M}\right)^{|A' \setminus C|} \mathrm{vol}(A) \, \leqslantslant \, e^{-\delta r / M} \, \mathrm{vol}(A) \\
& \, \leqslantslant \, e^{-\delta\varepsilon n^2/(2^{11} \log M)} \, \mathrm{vol}(A) \, \leqslantslant \, \leqslantslantft[\leqslantslantft(\frac{1}{2}+\leqslantslantft(1 - \frac{\delta}{2^{12}\log M}\right)\varepsilon\right)M\right]^{\binom{n}{2}}.
\end{align*}
Since every metric space in $\mathcal{M}_n^M$ is contained in $E(K_n) \times \{1, \dotsc, M\}$, by recursively applying this `breaking down' process to depth $O(\log M)^2$, we obtain a family of
$$\mathrm{ex}p\Big( O\big( n^{3/2} (\log M)^2 \log (nM) \big) \Big)$$
subsets of $E(K_n) \times \{1, \dotsc, M\}$, each of volume at most $\big(M/2+10\big)^{\binom{n}{2}}$, that cover all of $\mathcal{M}_n^M$. This implies that
\[
\big|\mathcal{M}_n^M\big| \leqslantslant \leqslantslantft[\leqslantslantft(\frac{1}{2} + \frac{10}{M} + \frac{C(\log M)^2\log(nM)}{\sqrt{n}} \right) M \right]^{\binom{n}{2}},
\]
which, as promised, is only slightly weaker than~\eqref{eq:thm:KMPS}.
\section{An asymmetric container lemma}\label{sec:asymm-cont-lemma}
The approach to studying the family of induced-$H$-free graphs described in the previous section has one (rather subtle) drawback: it embeds $\mathcal{F}_n^{\textup{ind}}(H)$ into the family of independent sets of a $\binom{v(H)}{2}$-uniform hypergraph with $\Theta(n^2)$ vertices. As a result, the hypergraph container lemma produces fingerprints of the same size as for the family of graphs without a clique on $v(H)$ vertices. This precludes the study of various threshold phenomena in the context of sparse induced-$H$-free graphs with the use of the hypergraph container lemma presented in Section~\ref{keylemma:sec}; this is in sharp contrast with the non-induced case, where the container method proved very useful.
In order to alleviate this shortcoming, Morris, Samotij, and Saxton~\cite{MSS} proved a version of the hypergraph container lemma for $2$-coloured structures that takes into account the possible asymmetries between the two colours. We shall not give the precise statement of this new container lemma here (since it is rather technical), but we would like to emphasize the following key fact: it enables one to construct families of containers for induced-$H$-free graphs with fingerprints of size $\Theta(n^{2-1/m_2(H)})$, as in the non-induced case.
To demonstrate the power of the asymmetric container lemma, the following application was given in~\cite{MSS}. Let us say that a graph $G$ is \emph{$\varepsilon$-close to a split graph} if there exists a partition $V(G) = A \cup B$ such that $e(G[A]) \geqslantslant (1-\varepsilon)\binom{|A|}{2}$ and $e(G[B]) \leqslantslant \varepsilon e(G)$.
\begin{thm}\label{thm:MSS}
For every $\varepsilon > 0$, there exists a $\delta > 0$ such that the following holds. Let $G$ be a~uniformly chosen induced-$C_4$-free graph with vertex set $\{1, \dotsc, n\}$ and $m$ edges.
\begin{enumerate}
\item[$(a)$] If $n \ll m \ll \delta n^{4/3} (\log n)^{1/3}$, then a.a.s.\ $G$ is not $1/4$-close to a split graph.
\item[$(b)$] If $n^{4/3} (\log n)^4 \leqslantslant m \leqslantslant \delta n^2$, then a.a.s.\ $G$ is $\varepsilon$-close to a split graph.
\end{enumerate}
\end{thm}
Theorem~\ref{thm:MSS} has the following interesting consequence: it allows one to determine the number of edges in (and, sometimes, also the typical structure of) the binomial random graph $G(n,p)$ conditioned on the event that it does not contain an induced copy of $C_4$. Let us denote by $G _{n,p}^{\mathrm{ind}}(C_4)$ the random graph chosen according to this conditional distribution.
\begin{cor}\label{cor:GnpC4free}
The following bounds hold asymptotically almost surely as $n \to \infty$:
\[
e \big( G _{n,p}^{\mathrm{ind}}(C_4) \big) \, = \,
\begin{cases}
\, \big( 1 + o(1) \big) p {n \choose 2} & \textup{if $n^{-1} \ll p \ll n^{-2/3}$}, \\
\, n^{4/3} (\log n)^{O(1)} & \textup{if $n^{-2/3} \leqslantslant p \leqslantslant n^{-1/3} (\log n)^4$}, \\
\, \Theta\big( p^2 n^2 / \log(1/p) \big) & \textup{if $p \geqslantslant n^{-1/3} (\log n)^4$}.
\end{cases}
\]
\end{cor}
We would like to emphasize the (surprising) similarity between the statements of Theorem~\ref{thm:randomturan} and Corollary~\ref{cor:GnpC4free}. In particular, the graph of $p \mapsto e \big( G _{n,p}^{\mathrm{ind}}(C_4) \big)$ contains exactly the same `long flat segment' as the graph of $p \mapsto \mathrm{ex} \big( G(n,p), C_4 \big)$, even though the shape of the two graphs above this range is quite different. We do not yet fully understand this phenomenon and it would be interesting to investigate whether or not the function $p \mapsto e \big( G _{n,p}^{\mathrm{ind}}(H) \big)$ exhibits similar behaviour for other bipartite graphs $H$.
\section{Hypergraphs of unbounded uniformity}\label{unbounded:sec}
Since the hypergraph container lemma provides explicit dependencies between the various parameters in its statement, it is possible to apply the container method even when the uniformity of the hypergraph considered is a growing function of the number of its vertices. Perhaps the first result of this flavour was obtained by Mousset, Nenadov, and Steger~\cite{MoNeSt14}, who proved an upper bound of $2^{\mathrm{ex}(n, K_r) + o(n^2/r)}$ on the number of $n$-vertex $K_r$-free graphs for all $r \leqslantslant (\log_2n)^{1/4}/2$. Subsequently, Balogh, Bushaw, Collares, Liu, Morris, and Sharifzadeh~\cite{BBCLMS} strengthened this result by establishing the following precise description of the typical structure of large $K_r$-free graphs.
\begin{thm}
If $r \leqslantslant (\log_2 n)^{1/4}$, then almost all $K_r$-free graphs with $n$ vertices are $(r-1)$-partite.
\end{thm}
Around the same time, the container method applied to hypergraphs with unbounded uniformity was used to analyse Ramsey properties of random graphs and hypergraphs, leading to improved upper bounds on several well-studied functions. In particular, R\"odl, Ruci\'nski, and Schacht~\cite{RoRuSc17} gave the following upper bound on the so-called Folkman numbers.
\begin{thm}
For all integers $k \geqslantslant 3$ and $r \geqslantslant 2$, there exists a $K_{k+1}$-free graph with
$$\mathrm{ex}p\big( Ck^4\log k + k^3r\log r \big)$$
vertices, such that every $r$-colouring of its edges contains a monochromatic copy of $K_k$.
\end{thm}
The previously best known bound was doubly exponential in $k$, even in the case $r = 2$. Not long afterwards, Conlon, Dellamonica, La Fleur, R\"odl, and Schacht~\cite{CoDeLFRoSc17} used a similar method to prove the following strong upper bounds on the induced Ramsey numbers of hypergraphs. Define the tower functions $t_k(x)$ by $t_1(x) = x$ and $t_{i+1}(x) = 2^{t_i(x)}$ for each $i \geqslantslant 1$.
\begin{thm}
For each $k \geqslantslant 3$ and $r \geqslantslant 2$, there exists $c$ such that the following holds. For every $k$-uniform hypergraph $F$ on $m$ vertices, there exists a $k$-uniform hypergraph $G$ on~$t_k(cm)$ vertices, such that every $r$-colouring of $E(G)$ contains a monochromatic induced copy of~$F$.
\end{thm}
Finally, let us mention a recent result of Balogh and Solymosi~\cite{BS}, whose proof is similar to that of Theorem~\ref{thm:geometry}, which we outlined in Section~\ref{geometry:sec}. Given a family $\mathcal{F}$ of subsets of an $n$-element set $\Omega$, an \emph{$\varepsilon$-net} of $\mathcal{F}$ is a set $A \subset \Omega$ that intersects every member of $\mathcal{F}$ with at least $\varepsilon n$ elements. The concept of an $\varepsilon$-net plays an important role in computer science, for example in computational geometry and approximation theory. In a seminal paper, Haussler and Welzl~\cite{HaWe87} proved that every set system with VC-dimension\footnote{The VC-dimension (VC stands for Vapnik--Chervonenkis) of a family $\mathcal{F}$ of subsets of $\Omega$ is the largest size of a~set $X \subset \Omega$ such that the set $\{A \cap X \,\colon A \in \mathcal{F}\}$ has $2^{|X|}$ elements.} $d$ has an $\varepsilon$-net of size $O\big( (d/\varepsilon) \log(d/\varepsilon) \big)$. It was believed for more than twenty years that for `geometric' families, the $\log(d/\varepsilon)$ factor can be removed; however, this was disproved by Alon~\cite{A12}, who constructed, for each $C > 0$, a set of points in the plane such that the smallest $\varepsilon$-net for the family of lines (whose VC-dimension is $2$) has size at least~$C/\varepsilon$.
By applying the container method to the hypergraph of collinear $k$-tuples in the $k$-dimensional $2^{k^2} \times \cdots \times 2^{k^2}$ integer grid, Balogh and Solymosi~\cite{BS} gave the following stronger lower bound.
\begin{thm}\label{thm:epsnets}
For each $\varepsilon > 0$, there exists a set $S \subset \mathbb{R}^2$ such that the following holds. If $T \subset S$ intersects every line that contains at least $\varepsilon |S|$ elements of $S$, then
\[
|T| \geqslantslant \frac{1}{\varepsilon} \leqslantslantft( \log \frac{1}{\varepsilon} \right)^{1/3 + o(1)}.
\]
\end{thm}
It was conjectured by Alon~\cite{A12} that there are sets of points in the plane whose smallest $\varepsilon$-nets (for the family of lines) contain $\Omega\big(1/\varepsilon \log(1/\varepsilon)\big)$ points.
\section{Some further applications}\label{more:applications:sec}
There are numerous applications of the method of containers that we do not have space to discuss in detail. Still, we would like to finish this survey by briefly mentioning just a few of them.
\subsection{List colouring}
A hypergraph $\mathcal{H}$ is said to be \emph{$k$-choosable} if for every assignment of a list $L_v$ of $k$ colours to each vertex $v$ of $\mathcal{H}$, it is possible to choose for each $v$ a colour from the list $L_v$ in such a way that no edge of $\mathcal{H}$ has all its vertices of the same colour. The smallest $k$ for which $\mathcal{H}$ is $k$-choosable is usually called the \emph{list chromatic number} of $\mathcal{H}$ and denoted by $\chi_\ell(\mathcal{H})$. Alon~\cite{Al93,Al00} showed that for graphs, the list chromatic number grows with the minimum degree, in stark contrast with the usual chromatic number; more precisely, $\chi_\ell(G) \geqslantslant \big( 1/2 + o(1) \big) \log_2 \delta(G)$ for every graph $G$. The following generalisation of this result, which also improves the constant $1/2$, was proved by Saxton and Thomason~\cite{ST}, see also~\cite{SaTh12,ST16}.
\begin{thm}
Let $\mathcal{H}$ be a $k$-uniform hypergraph with average degree $d$ and $\Delta_2(\mathcal{H}) = 1$. Then, as $d \to \infty$,
\[
\chi_\ell(\mathcal{H}) \geqslantslant \leqslantslantft(\frac{1}{(k-1)^2} + o(1)\right) \log_k d.
\]
Moreover, if $\mathcal{H}$ is $d$-regular, then
\[
\chi_\ell(\mathcal{H}) \geqslantslant \leqslantslantft(\frac{1}{k-1} + o(1)\right) \log_kd.
\]
\end{thm}
We remark that proving lower bounds for the list chromatic number of simple hypergraphs was one of the original motivations driving the development of the method of hypergraph containers.
\subsection{Additive combinatorics}
The method of hypergraph containers has been applied to a number of different number-theoretic objects, including sum-free sets~\cite{ABMS1,ABMS2,BLST2,BLST1}, Sidon sets~\cite{ST16}, sets containing no $k$-term arithmetic progression~\cite{BLS,BMS}, and general systems of linear equations~\cite{ST16}. (See also~\cite{Gr04,GrRu04,Sap03} for early applications of the container method to sum-free sets and~\cite{DeKoLeRoSa-B3, DeKoLeRoSa-Bh-16, DeKoLeRoSa-Bh, KoLeRoSa} for applications of graph containers to $B_h$-sets.) Here we will mention just three of these results.
Let us begin by recalling that a {\it Sidon set} is a set of integers containing no non-trivial solutions of the equation $x+y=z+w$. Results of Chowla, Erd\H os, Singer, and Tur\'an from the 1940s imply that the maximum size of a Sidon set in $\{1,\ldots,n\}$ is $\big( 1 + o(1) \big) \sqrt n$ and it was conjectured by Cameron and Erd{\H{o}}s~\cite{CamErdos} that the number of such sets is $2^{(1+o(1))\sqrt n}$. This conjecture was disproved by Saxton and Thomason~\cite{ST16}, who gave a construction of $2^{(1+\varepsilon) \sqrt n}$ Sidon sets (for some $\varepsilon > 0$), and also used the hypergraph container method to reprove the following theorem, which was originally obtained in~\cite{KoLeRoSa} using the graph container method.
\begin{thm}\label{sidon}
There are $2^{O(\sqrt n)}$ Sidon sets in $\{1,\ldots,n\}$.
\end{thm}
Dellamonica, Kohayakawa, Lee, R\"odl, and Samotij~\cite{DeKoLeRoSa-Bh} later generalized these results to $B_h$-sets, that is, set of integers containing no non-trivial solutions of the equation $x_1+ \ldots + x_h = y_1 + \ldots + y_h$.
The second result we would like to state was proved by Balogh, Liu, and Sharifzadeh~\cite{BLS}, and inspired the proof presented in Section~\ref{counting:Hfree:sec}. Let $r_k(n)$ be the largest size of a subset of $\{1, \dotsc, n\}$ containing no $k$-term arithmetic progressions.
\begin{thm}\label{thm:BLS}
For each integer $k \geqslantslant 3$, there exist a constant $C$ and infinitely many $n \in \mathbb{N}$ such that there are at most $2^{C r_k(n)}$ subsets of $\{1, \dotsc, n\}$ containing no $k$-term arithmetic progression.
\end{thm}
We recall (see, e.g.,~\cite{GowersErdos}) that obtaining good bounds on $r_k(n)$ is a well-studied and notoriously difficult problem. The proof of Theorem~\ref{thm:BLS} avoids these difficulties by exploiting merely the `self-similarity' property of the hypergraph encoding arithmetic progressions in $\{1,\ldots,n\}$, cf.~the discussion in Section~\ref{keylemma:sec} and the proof of Lemma~\ref{lem:C4:super}.
The final result we would like to mention was one of the first applications of (and original motivations for the development of) the method of hypergraph containers. Recall that the Cameron--Erd\H{o}s conjecture, proved by Green~\cite{Gr04} and, independently, by Sapozhenko~\cite{Sap03}, states that there are only $O(2^{n/2})$ sum-free subsets of $\{1,\ldots,n\}$. The following sparse analogue of the Cameron--Erd\H{o}s conjecture was proved by Alon, Balogh, Morris, and Samotij~\cite{ABMS2} using an early version of the hypergraph container lemma for 3-uniform hypergraphs.
\begin{thm}\label{CEthm}
There exists a constant $C$ such that, for every $n \in \mathbb{N}$ and every $1 \leqslantslant m \leqslantslant \lceil n/2 \rceil$, the set $\{1,\ldots,n\}$ contains at most $2^{Cn/m} {\lceil n/2 \rceil \choose m}$ sum-free sets of size $m$.
\end{thm}
We remark that if $m \geqslantslant \sqrt{n}$, then Theorem~\ref{CEthm} is sharp up to the value of $C$, since in this case there is a constant $c > 0$ such that there are at least $2^{c n/m} {n/2 \choose m}$ sum-free $m$-subsets of $\{1,\ldots,n\}$. For smaller values of $m$ the answer is different, but the problem in that range is much easier and can be solved using standard techniques. Let us also mention that in the case $m \gg \sqrt{n \log n}$, the structure of a typical sum-free $m$-subset of $\{1,\ldots,n\}$ was also determined quite precisely in~\cite{ABMS2}.
Finally, we would like to note that, although the statements of Theorems~\ref{sidon},~\ref{thm:BLS} and~\ref{CEthm} are somewhat similar, the difficulties encountered during their proofs are completely different.
\subsection{Sharp thresholds for Ramsey properties}
Given an integer $k \geqslantslant 3$, let us say that a set $A \subset \mathbb{Z}_n$ has the \emph{van der Waerden property} for $k$ if every 2-colouring of the elements of $A$ contains a monochromatic $k$-term arithmetic progression; denote this by $A \to (k\text{-AP})$. R\"odl and Ruci\'nski~\cite{RoRu95} determined the threshold for the van der Waerden property in random subsets of $\mathbb{Z}_n$ for every~$k \in \mathbb{N}$. Combining the sharp threshold technology of Friedgut~\cite{F99} with the method of hypergraph containers, Friedgut, H\'an, Person, and Schacht~\cite{FrHaPeSc16} proved that this threshold is sharp. Let us write $\mathbb{Z}_{n,p}$ to denote a $p$-random subset of $\mathbb{Z}_n$ (i.e., each element is included independently with probability $p$).
\begin{thm}
For every $k \geqslantslant 3$, there exist constants $c_1 > c_0 > 0$ and a function $p_c \mathrm{col}on \mathbb{N} \to [0,1]$ satisfying $c_0 n^{-1/(k-1)} < p_c(n) < c_1 n^{-1/(k-1)}$ for every $n \in \mathbb{N}$, such that, for every $\varepsilon > 0$,
\[
\mathcal{P}r\Big( \mathbb{Z}_{n,p} \to \big( \text{$k$-AP} \big) \Big) \, \to \,
\begin{cases}
\, 0 & \text{if } \, p \leqslantslant (1-\varepsilon) \, p_c(n),\\[+0.05ex]
\, 1 & \text{if } \, p \geqslantslant (1+\varepsilon) \, p_c(n),
\end{cases}
\]
as $n \to \infty$.
\end{thm}
The existence of a sharp threshold in the context of Ramsey's theorem for the triangle was obtained several years earlier, by Friedgut, R\"odl, Ruci\'nski, and Tetali~\cite{FRRT}. Very recently, using similar methods to those in~\cite{FrHaPeSc16}, Schacht and Schulenburg~\cite{SchSch} gave a simpler proof of this theorem and also generalised it to a large family of graphs, including all odd cycles.
\subsection{Maximal triangle-free graphs and sum-free sets}
In contrast to the large body of work devoted to counting and describing the typical structure of $H$-free graphs, relatively little is known about $H$-free graphs that are \emph{maximal} (with respect to the subgraph relation). The following construction shows that there are at least $2^{n^2/8}$ maximal triangle-free graphs with vertex set $\{1, \dotsc, n\}$. Fix a partition $X \cup Y = \{1, \dotsc, n\}$ with $|X|$ even. Define $G$ by letting $G[X]$ be a perfect matching, leaving $G[Y]$ empty, and adding to $E(G)$ exactly one of $xy$ or $x'y$ for every edge $xx' \in E(G[X])$ and every $y \in Y$. It is easy to verify that all such graphs are triangle-free and that almost all of them are maximal.
Using the container theorem for triangle-free graphs (Theorem~\ref{thm:CT:triangles}), Balogh and Pet\v{r}\'\i\v{c}kov\'a~\cite{BaPe14} proved that the construction above is close to optimal by showing that there are at most $2^{n^2/8 + o(n^2)}$ maximal triangle-free graphs on $\{1, \dotsc, n\}$. Following this breakthrough, Balogh, Liu, Pet\v{r}\'\i\v{c}kov\'a, and Sharifzadeh~\cite{BaLiPeSh15} proved the following much stronger theorem, which states that in fact almost all maximal triangle-free graphs can be constructed in this way.
\begin{thm}
For almost every maximal triangle-free graph $G$ on $\{1, \dotsc, n\}$, there is a vertex partition $X \cup Y$ such that $G[X]$ is a perfect matching and $Y$ is an independent set.
\end{thm}
A similar result for sum-free sets was obtained by Balogh, Liu, Sharifzadeh, and Treglown~\cite{BLST2,BLST1}, who determined the number of maximal sum-free subsets of $\{1, \dotsc, n\}$ asymptotically. However, the problem of estimating the number of maximal $H$-free graphs for a general graph $H$ is still wide open. In particular, generalizing the results of~\cite{BaLiPeSh15,BaPe14} to the family of maximal $K_k$-free graphs seems to be a very interesting and difficult open problem.
\subsection{Containers for rooted hypergraphs}
A family $\mathcal{F}$ of finite sets is \emph{union-free} if $A \cup B \ne C$ for every three distinct sets $A, B, C \in \mathcal{F}$. Kleitman~\cite{K76} proved that every union-free family in $\{1,\dotsc,n\}$ contains at most $\big( 1 + o(1) \big) \binom{n}{n/2}$ sets; this is best possible as the family of all $\lfloor n/2 \rfloor$-element subsets of $\{1, \dotsc, n\}$ is union-free. Balogh and Wagner~\cite{BaWaUnion} proved the following natural counting counterpart of Kleitman's theorem, confirming a conjecture of Burosch, Demetrovics, Katona, Kleitman, and Sapozhenko~\cite{BDKKS}.
\begin{thm}
There are $2^{( 1 + o(1)) \binom{n}{n/2}}$ union-free families in $\{1,\ldots,n\}$.
\end{thm}
It is natural to attempt to prove this theorem by applying the container method to the 3-uniform hypergraph $\mathcal{H}$ that encodes triples $\{A,B,C\}$ with $A \cup B = C$. However, there is a problem: for any pair $(B,C)$, there exist $2^{|B|}$ sets $A$ such that $A \cup B = C$ and this means that $\Delta_2(\mathcal{H})$ is too large for a naive application of the hypergraph container lemma. In order to overcome this difficulty, Balogh and Wagner developed in~\cite{BaWaUnion} a new container theorem for `rooted' hypergraphs (each edge has a designated root vertex) that exploits the asymmetry of the identity $A \cup B = C$. In particular, note that while the degree of a pair $(B, C)$ can be large, the pair $\{A,B\}$ uniquely determines $C$; it turns out that this is sufficient to prove a suitable container theorem. We refer the reader to~\cite{BaWaUnion} for the details.
\subsection{Probabilistic embedding in sparse graphs}
The celebrated regularity lemma of Szemer\'edi~\cite{Sz78} states that, roughly speaking, the vertex set of every graph can be divided into a bounded number of parts in such a way that most of the bipartite subgraphs induced by pairs of parts are pseudorandom; such a partition is called a~\emph{regular partition}. The strength of the regularity lemma stems from the so-called counting and embedding lemmas, which tell us approximately how many copies of a particular subgraph a graph $G$ contains in terms of basic parameters of the regular partition of $G$. While the original statement of the regularity lemma applied only to dense graphs (i.e., $n$-vertex graphs with $\Omega(n^2)$ edges), the works of Kohayakawa~\cite{Ko97}, R\"odl~(unpublished), and Scott~\cite{Sc11} provide extensions of the lemma that are applicable to sparse graphs. However, these extensions come with a major caveat: the counting and embedding lemmas do not extend to sparse graphs; this unfortunate fact was observed by \L uczak. Nevertheless, it seemed likely that such atypical graphs that fail the counting or embedding lemmas are so rare that they typically do not appear in random graphs. This belief was formalised in a conjecture of Kohayakawa, \L uczak, and R\"odl~\cite{KLR2}, which can be seen as a `probabilistic' version of the embedding lemma.
The proof of this conjecture, discovered by the authors of this survey~\cite{BMS} and by Saxton and Thomason~\cite{ST}, was one of the original applications of the hypergraph container lemma. Let us mention here that a closely related result was proved around the same time by Conlon, Gowers, Samotij, and Schacht~\cite{CoGoSaSc14}. A~strengthening of the K\L R conjecture, a `probabilistic' version of the counting lemma, proposed by Gerke, Marciniszyn, and Steger~\cite{GeMaSt07}, remains open.
\end{document} | math | 94,727 |
\begin{document}
\title {Secondary Brown-Kervaire Quadratic forms and $\pi$-manifolds}
\author{Fuquan Fang}
\address{ Nankai Institute of Mathematics, Nankai University,
Tianjin 300071, P.R.C }
\email{ [email protected]}
\author{Jianzhong Pan }
\address{Institute of Math.,Academia Sinica ,Beijing 100080 ,China
and \newline Department of Mathematics Education , Korea University , Seoul , Korea }
\email{[email protected]}
\thanks{The first author was supported in part by NSFC 1974002, Qiu-Shi Foundation and CNPq
and the second author is partially supported by the
NSFC project 19701032 and ZD9603 of Chinese Academy of Science and
Brain Pool program of KOSEF }
\subjclass{}
\keywords{}
\date{Dec. 20,1999}
\begin{abstract}
In this paper we assert that for each $\Phi$-oriented
$2n$-manifold (c.f : Definition 1.1) $M$ where $n\ge 4$ and
$n\ne 3(mod 4)$, there is a well-defined quadratic function
$\phi_M: H^{n-1}(M, {\mathbb Z}_4)\to {\mathbb Q}/{\mathbb Z}$, we call the secondary Brown-Kervaire quadratic forms, so that
\begin{itemize}
\item{ $\phi _{M}(x+y)=\phi _{M}(x)+\phi _{M}(y)+j(x\cup Sq^2y)[M]$},
\item{ the Witt class of $\phi _M$ is a homotopy invariant, if
the Wu class $ v_{n+2-2^i}(\nu _M)=0$ for all $i$.}
\end{itemize}
where $j: {\mathbb Z}_2 \to {\mathbb Q}/{\mathbb Z}$ is the inclusion homomorphism and
$\nu _M$ the stable normal bundle of $M$.
Among the applications we obtain a complete classification of
$(n-2)$-connected $2n$-dimensional $\pi$-manifolds up to
homeomorphism and homotopy equivalence, where $n\geq 4$ and
$n+2\neq 2^i$ for any $i$. In particular, we prove that the
homotopy type of such manifolds determine their homeomorphism
type.
\end{abstract}
\maketitle
\section{Introduction}\label{S:intro}
Let $M$ be a $2n$-dimenisonal framed manifold (i.e. a $\pi$-manifold with a framing) where $n=1(mod2)$.
The Kervaire invariant of $M$ is the Arf invariant of a ${\mathbb Z}_2$-valued Kervaire quadratic form of $M$
$$q_M: H^{n}(M, {\mathbb Z}_2) \to {\mathbb Z}_2$$
satisfying
$$q_M(x+y)=q_M(x)+q_M(y)+(x\cup y)[M]_2 \hspace{2cm} (1.1) $$
It was invented by Kervaire to find the first example of non-smoothable PL-manifold. Kervaire invariants
and its various generalizations, e.g. the Brown-Kervaire invariants[4], play very important roles in
geometric topology. Formally, $q_M$ is a ``quadratic form'' subject to the symmetric bilinear form
$$
\begin{array}{cc}
H^n(M,{\mathbb Z}_2)\times H^n(M,{\mathbb Z}_2)\to {\mathbb Z}_2\\
(x,y)\to x\cup y[M]_2
\end{array}
$$
For a Spin manifold of even dimension, there is another symmetric bilinear form $\mu _M$ studied by
Landweber and Stong [16]:
$$
\begin{array}{cc}
\mu _M: H^{n-1}(M,{\mathbb Z}_2)\times H^{n-1}(M,{\mathbb Z}_2)\to {\mathbb Z}_2\\
(x,y)\to Sq^2(x)\cup y[M]_2
\end{array}
$$
A natural algebraic question to ask is whether there is an intrinsic ``quadratic form'' of $M$ subject to
$\mu _M$.
To answer this turns out to be the main novelty of this paper. For a large family of Spin manifolds
including all $\pi$-manifolds, the so called
$\Phi$-oriented manifolds, we will define a ${\mathbb Q}/{\mathbb Z}$-form subject to $\mu _M$, which resembles to the
Brown-Kervaire quadratic forms in the formulation. It has the most similar properties of the Brown-Kervaire
quadratic forms, e.g., the isomorphism class of the form is a homotopy invariant if the manifold has
vanishing Wu classes. A bit surprising to us, this invariant applies to give a classification of
$(n-2)$-connected $2n$-dimensional $\pi$-manifolds up to homotopy equivalence and homeomorphism ($n\ge 4$).
To state our main results, let us start with some notations.
Let $\{ Y_k\}_{k\in {\mathbb N}} $ be a connected spectrum with $U\in H^{0}(Y)\cong {\mathbb Z}$ a generator so that
$i^{*}U\in H^0(S^0)$ a generator, where $i: S^0 \to Y$ is the inclusion map of the spectrum.
\begin{defn}
(i) $\{ Y_k\}_{k\in {\mathbb N}} $ is called $\Phi$-{\it
orientable} if $Sq^2U=0$, $\chi (Sq^{n+2})(U)=0$ and $0\in \Phi
(U)$, where $\Phi$ is a secondary cohomology operator associated
with the Adem relation (see Section \ref{S:2} for the definition):
$$
\begin{array}{ll}
\chi(Sq^n)Sq^3+\chi(Sq^{n+2})Sq^1+Sq^1\chi(Sq^{n+2})=0 & n=2(mod
4) \\ \chi(Sq^n)Sq^3+Sq^1\chi(Sq^{n+2})=0 &n=0(mod 4) \\
\chi(Sq^{n+1})Sq^2+Sq^1\chi(Sq^{n+2})=0 & n=1(mod 4)
\end{array}
$$ where $\chi: \mathcal{A}_2\to \mathcal{A}_2$ is the anti-automorphism of the
Steenrod algebra $\mathcal{A}_2$ \cite{adams}.
\end{defn}
A spherical fibration $\xi$ (a manifold) is called $\Phi
$-{\it orientable} if its Thom spectrum $T\xi $ (stable normal bundle
$\nu _M$) is. We define the {\it universal} $\Phi $-orientable $\Omega$-spectrum
$\widetilde W(n)$ by setting $\widetilde {W}_k{(n)}$ to be the total space of the following
Postnikov tower:
\newline
$$
\begin{array}{ccccc}
& & \widetilde{W}_{k}(n) & & \\ & & \downarrow {\Pi _2} & &
\\ & & W_{k}(n) & \stackrel {k_2}{\longrightarrow} & K_{k+n+2}
\\ & & \downarrow {\Pi _1} & & \\ & & K({\mathbb Z},k) &
\stackrel {Sq^2\times \chi (Sq^{n+2}) } {\longrightarrow} &
K_{k+2}\times K_{k+n+2}
\end{array}
$$
where $K_i=K({\mathbb Z}_2, i)$, $K({\mathbb Z}, i)$ are the Eilenberg-Maclane spaces, $k_2\in \Phi ({\Pi _1 } ^*l_{k})$
and $l_k$ is the basic class.
Note that a spectrum $Y$ is $\Phi$-orientable if and only if $U\in H^0(Y)$ can be lifted to a map
$w: Y\to \widetilde W(n)$. We call such a lifting a $\Phi$-{\it orientation} of $Y$. A $\Phi$-orientation of
a manifold is understood as a $\Phi$-orientation of its Thom spectrum.
\begin{rem}\label{T:1.2}
The sphere spectrum $S^0$ is $\Phi $-orientable. Thus
stably parallelizable manifolds are $\Phi$-orientable.
\end{rem}
Our main results are:
\begin{thm}\label{T:0.6}
{\it Let $M$ be a $\Phi$-oriented manifold of dimension $2n$, where $n\ne 3(\mbox{mod } 4)$. Then there is
a function $\phi_{M}: H^{n-1}(M, {\mathbb Z}_4)\to {\mathbb Q}/{\mathbb Z}$ such that, for all $x, y\in H^{n-1}(M,{\mathbb Z}_4)$,
$$\phi _{M}(x+y)=\phi _{M}(x)+\phi _{M}(y)+j(x\cup Sq^{2}y)[M],$$ where
$j: {\mathbb Z}_2 \to {\mathbb Q}/{\mathbb Z} $ is the inclusion.}
\end{thm}
\begin{rem} In general, $\phi _M$ depends on the $\Phi$-orientation, just like the Kervaire quadratic
form depends on the framing of the manifold. We will prove that $\phi _{M}(x)$ depends only
on the $\Phi $-oriented bordism class $[M,x]$.
\end{rem}
\begin{rem} If $n=3(\mbox{mod }4)$, the analogous definition gives only a linear function.
\end{rem}
Let $BSpin _G$ be the classifying space for spherical Spin fibrations. By Brown[4], a {\it Wu orientation }
of a Spin spherical fibration $\xi \searrow M$ is a lifting of the classifying map $\xi : M\to BSpin_G$ to
$BSpin_G\langle v_{n+2}\rangle $. A {\it Wu orientation } of $\nu _M$, the
stable normal bundle of $M$, is understood as a Wu orientation of
$M$, where $BSpin_G\langle v_{n+2}\rangle \to BSpin_G$ is a
principal fibration with $v_{n+2}\in H^{n+2}(BSpin_G, {\mathbb Z}_2)$ as the
$k$-invariant.
We call quadratic forms $\phi _{M_i}: H^{n-1}(M_i, {\mathbb Z}_4)\to
{{\mathbb Q}/{\mathbb Z}}$, $i=1,2$ {\it Witt equiavlent} if there exists an
isomorphism $\tau : H^{n-1}(M_1, {\mathbb Z}_4)\to H^{n-1}(M_2, {\mathbb Z}_4)$ so
that $\phi _{M_2}(\tau x)=\phi _{M_1}(x) $ for all $x\in
H^{n-1}(M_1, {\mathbb Z}_4)$.
\begin{thm}\label{T:0.11}
{\it Let $M_1$ and $M_2$ be $\Phi$-oriented $2n$-manifolds.
Suppose that the Wu classes $v_{n+2-2^j}(\nu _{M_i})=0$ for all $2^j\le n+2$. If $f: M_1\to
M_2$ is a homotopy equivalence preserving the spin structure
(resp. Wu orientation) if $n=0, 1(mod 4)$ (resp. $n=2(mod 4)$).
Then $$\phi _{M_1}(f^*x)=\phi _{M_2}(x)$$ for all $x\in
H^{n-1}(M_2, {\mathbb Z}_4)$.}
\end{thm}
Since the Wu class $v_0=1$, the assumption in the above theorem implies that $n+2\ne 2^i$ for any integer
$i$.
For framed manifolds, the Brown-Kervaire secondary quadratic forms have the following property:
\begin{prop}\label{T:0.9}
If $M$ is a framed manifold of dimension $2n$, where $n\ne 3(mod 4)$. Then $\phi _M$
factors through ${\mathbb Z}_4\subset {{\mathbb Q}/{\mathbb Z}}$ $(\text{resp. } {\mathbb Z}_2 \subset {{\mathbb Q}/{\mathbb Z}})$,
provided $n=2(mod4)$ $(\text{resp. }n=0, 1(mod4))$.
\end{prop}
To state the next results, we need some preliminaries.
Let $H$ be a finitely generated abelian group, and
$$\begin{array}{rll}\mu : Hom(H, {\mathbb Z}_2 ) \otimes Hom(H, {\mathbb Z}_2)
\to {\mathbb Z}_2
\end{array}$$
be a symmetric bilinear form. We say that $\mu $ is of \underline
{\it diagonal zero} if $\mu (x,x)=0$ for each $x \in Hom(H, {\mathbb Z}_2)$.
A function $\phi : Hom(H, {\mathbb Z}_4 ) \to {\mathbb Q}/{\mathbb Z}$ is called \underline
{\it quadratic} with respect to $\mu $ if $$
\begin{array}{rll}
\phi (x+y) = \phi (x) + \phi (y) + j( \mu (x, y))
\end{array}$$
where $j : {\mathbb Z}_2 \to {\mathbb Q}/{\mathbb Z} $ is the inclusion. This gives a triple
$(H, \mu , \phi )$. We say triples $(H_1, \mu _1, \phi _1)$, and
$(H_2, \mu _2, \phi _2)$ are {\it isometric} if there exists an
isomorphism $\tau: H_1\to H_2$ such that $\mu _1(x, y)=\mu _2(\tau
x, \tau y)$ and $\phi _1(x)=\phi _2(\tau x)$ for all $x, y$. We
denote by $[H, \mu , \phi ]$ the isometry class of a triple.
\begin{rem}
Since the natural map $ Hom(H_{n-1}(M), {\mathbb Z}_2 ) \to H^{n-1}(M,
{\mathbb Z}_2)$ is not an isomorphism in general, the notions of isometry
associated with $\mu$ and $\mu_M$ as above are different. They do
agree however for $(n-2)$-connected manifolds which we will assume
in the later application. We will use both of them when necessary.
\end{rem}
Let ${i}$ denote the maximal exponent of the $2$-torsion subgroup of
$H_{n-1}(M)$ and let $Sq_{i}^1 \in H^n(K( {\mathbb Z}_{2^{i}}, n-1),
{\mathbb Z}_2)\cong {\mathbb Z}_2$ be the unique generator. Considering $Sq_{i}^1 $ as
a cohomology operation we get a function $$\begin{array}{rll}
q_{M}(Sq_{i}^1): H^{n-1}(M, {\mathbb Z}_{2^i}) \to {\mathbb Z}_2.
\end{array}$$
This gives a homomorphism since $Sq^1_ix\cup Sq^1_iy=Sq_i^1(x\cup
Sq_i^1y)=0$ for $x, y\in H^{n-1}(M, {\mathbb Z}_2)$. We denote by
$[H_{n-1}(M), \mu _M, q_M(Sq^1_i)]$ for the isometry class of the
triple. By \cite{browd2}, the Kervaire invariant of a smooth
framed manifold of dimension $2n$, where $n\ne 2^i-1$, is zero.
For $i\leq 5$, there are smooth manifolds of dimension $2^{i+1}-2$
of Kervaire invariant $1$. It is still an open problem whether
there is such a manifold for $i\geq 6$.
The Kervaire invariant does not depend on the framings
of the underlying $2n$-manifold if $n\ne 1, 3, 7$ and the manifold
is highly connected, e.g. $(n-2)$-connected. Moreover, by
\cite{brown} the Kervaire form is a homotopy invariant if $n\ne 1, 3, 7$ and $(n-2)$-connected.
Let $M$ be a $(n-2)$-connected $2n$-dimensional $\pi$-manifold.
Observe that if $n\ge 3$, there exists a $(n-2)$-connected
$\pi$-manifold, $N$, so that $M=N\# X$ and $H_{n}(N, {\mathbb Q})=0$, where
$X$ is a $(n-1)$-connected $2n$-manifold. Since the classification
of $(n-1)$-connected $2n$-manifolds is well understood
\cite{wall2}, for convenience in the following theorem we
assume that $H_{n}(M, {\mathbb Q})=0$. For such a manifold, consider the
correspondence $$\pi: M \longmapsto [H_{n-1}(M), \mu _{M}, \phi
_M] (\mbox{ resp.}\text{ } [H_{n-1}(M), \mu _{M}, \phi _M,
q_M(Sq^1_{2^i})])$$ if $n=0(mod 2)$ (resp. $n=1(mod 2)$).
In the following theorem let $\alpha(n+2)$ be the number of $1's$
in the binary expansion of $n+2$.
\begin{thm}\label{T:0.13}
{\it Suppose $n\geq 4$ and $\alpha (n+2)\geq 2$. Then $\pi$ gives
a 1-1 correspodence between the homeomorphism types (resp.
homotopy types) of $(n-2)$-connected $2n$-dimensional
$\pi$-manifolds $M$ so that $H_n(M, {\mathbb Q})=0$ with the following
algebraic data
\noindent (a) $\wp _n=\{ [H, \mu , \phi ]: \mbox{ diag }\mu =0
\mbox{ and } \phi \mbox{ factors through } j: {\mathbb Z}_4\to {\mathbb Q}/{\mathbb Z}\}$
\mbox{ if $n=2(mod 4)$},\newline (b) $\wp _n=\{ [H, \mu , \phi ]:
\phi \mbox{ factors through }j: {\mathbb Z}_2\to {\mathbb Q}/{\mathbb Z}\}$ if $n=0(mod
4)$,\newline (c) $\wp _n=\{ [H, \mu , \phi , \omega ]: \omega \in
Hom(tor (H)\otimes {\mathbb Z}_{2^i}, {\mathbb Z}_2), \phi $ factors through $j:
{\mathbb Z}_2\to {\mathbb Q}/{\mathbb Z} \}$ if $n=1(mod 4)$,\newline (d) $\wp _n=\{ [H, \mu
, \omega ]: \omega \in Hom(tor H\otimes
{\mathbb Z}_{2^i}, {\mathbb Z}_2)\}$ if $n=3(mod 4)$.
\noindent where $i$ is the highest exponent of the $2$-cyclic
subgroup of $H$ and if $n=1(mod 2)$, the pairing $\mu (x, x)=0$
(resp. $\delta \omega (x)$) if $x$ can be lifted to a ${\mathbb Z}_4$ class
with order $4$ (resp. $x$ is of order $2$), $\delta \in \{ 0, 1\}$
is ambiguous. }
\end{thm}
\begin{rem}
The classification of $(n-2)$-connected $2n$-manifolds with torsion
free homology groups has been given by Ishimoto[9][10]. But his method does not work
if the homology group has torsion.
\end{rem}
The organization of this paper is as follows.
In $\S$\ref{S:1} we define the secondary Brown-Kervaire form and state its basic properties.
In $\S$\ref{S:2}, we set up the necessary foundations on the stable homotopy theory of the
Eilenberg-Maclane spaces.
In $\S$\ref{S:3}, we are addressed to show Theorems 1.3 and 1.6.
In $\S$5, we prove Theorem \ref{T:0.13}.
\section{A ${\mathbb Q}/{\mathbb Z}$-quadratic form of $\Phi$-oriented manifolds}\label{S:1}
Let us begin with some conventions. All homology/cohomology
groups will be with integral coefficients unless otherwise stated.
All spaces will have base points. Let\newline (i) $[X, Y]$ denote
the set of homotopy classes of pointed maps from $X$ to $Y$.
\newline (ii) $\{ X, Y\} =lim[S^kX, S^kY]$.\newline (iii)
$\pi_*^s(X)$ be its 2-localization to simplify the notation.
Let $\kappa : K({\mathbb Z}_4, n-1) \times K({\mathbb Z}_4, n-1) \to K({\mathbb Z}_4, n-1)$ be
the multiplication of $K({\mathbb Z}_4, n-1)$ and let $H(\kappa )$ be the
Hopf construction of $\kappa$.
\begin{prop}\label{T:0.2}
{\it The homomorphism $$H(\kappa )_*: \pi ^s_{2n}(K({\mathbb Z}_4, n-1)\land
K({\mathbb Z}_4, n-1))\to \pi_{2n}^s(K({\mathbb Z}_4, n-1)) $$ is injective if
$n\neq 3(mod 4)$, and zero if $n=3(mod4)$.}
\end{prop}
\begin{rem}
If ${\mathbb Z}_4$ is replaced by ${\mathbb Z}_2$, then $H(\kappa )_*$
is trivial.
\end{rem}
By Theorem \ref{T:stable} and the proof of it, we obtain
$$\begin{array}{lcc} \pi ^s_{2n}(K({\mathbb Z}_4, n-1)\land K({\mathbb Z}_4,
n-1))\cong {\mathbb Z}_2\mbox{ if $n\geq 4$,}\\ \pi ^s_{2n}(K({\mathbb Z}_4,
n-1))\cong {\mathbb Z}_4 \mbox{ if $n=2(mod 4)$.}
\end{array}
$$
Let $\lambda _0$ be a generator of Im $H(\kappa )_*$ if $n\neq
2(mod4)$, and a specified generator of $\pi ^s_{2n}(K({\mathbb Z}_4,
n-1))\cong {\mathbb Z}_4$ otherwise. For a given spectrum $Y$, let
$$H_{*}(K({\mathbb Z}_{4}, n-1); Y)=\mbox{lim }\pi _{*+k}
(K({\mathbb Z} _4, n-1)\wedge Y_k).$$
\begin{thm}\label{T:0.3}
{\it Suppose that $\{Y_k\}_{k\in {\mathbb N}}$ is a $\Phi$-orientable
spectrum. Then there exists a homomorphism $$h: H_{2n}(K({\mathbb Z}_4,
n-1); Y) \to {\mathbb Q}/{\mathbb Z} $$ such that $h(\lambda)=\frac {1}{4}$
$(\text{resp. } \frac{1}{2})$ if $n=2(mod 4)$ $(\text{resp. }n=0,
1(mod 4))$, where $\lambda =i_*(\lambda_0)$ and $i_*: H_{2n}(K({\mathbb Z}_{4},
n-1); S^0)\to H_{2n}(K({\mathbb Z}_{4}, n-1); Y) $ is induced by the inclusion.}
\end{thm}
\begin{defn} A Poincar\'e triple $(M, \xi , \alpha )$ of
dimension $2n$ consists of \newline (i) A CW complex $M$ with
finitely generated homology.\newline (ii) A fibration $\xi $ over
$M$ with fiber homotopy equivalent to $S^{k-1}$, $k$ large.
\newline (iii) $\alpha \in \pi _{2n+k}(T\xi)$ such that an $(2n+k)$
Spanier-Whitehead S-duality is given by
$$S^{2n+k} \stackrel {\alpha }{\longrightarrow} T\xi
\stackrel {\Delta }{\longrightarrow}T\xi\land M^+$$
where $T\xi$ is the Thom complex of $\xi$ and $\Delta $ is the diagonal map.
\end{defn}
Let $A_{\alpha}: \{M_{+}, K({\mathbb Z}_4, n-1)\} \to \{S^{2n+k}, T\xi\wedge
K({\mathbb Z}_4, n-1)\}$ be the $S$-duality map.
\begin{defn}
Let $(M, \xi, \alpha)$ be a Poincar\'e triple and
$w$ is a $\Phi$-orientation of the Thom spectrum $T\xi $. For a homomorphism $h$ in Theorem 2.3,
let $$\phi_{w,h} : H^{n-1}(M, {\mathbb Z}_4)\to {\mathbb Q}/{\mathbb Z}$$ be defined by setting
$$\phi
_{w,h}(x)=h([(w\wedge id) A_{\alpha}(x)]).
$$
\end{defn}
\begin{thm}
{\it Let $\phi _{w,h}$ be defined as above. Then for all $x, y\in H^{n-1}(M,{\mathbb Z}_4)$, \newline
(i) If $n\ne 3(\mbox{mod } 4)$, the function is quadratic, i.e.
$$\phi _{w,h}(x+y)=\phi _{w,h}(x)+\phi _{w,h}(y)+j(x\cup Sq^{2}y)[M]$$
where $j: {\mathbb Z}_2 \to {\mathbb Q}/{\mathbb Z} $ is the inclusion;\newline
(ii) If $n=3(mod 4)$, $\phi _{w,h}$ is linear, i.e.
$$\phi _{w,h}(x+y)=\phi _{w,h}(x)+\phi _{w,h}(y).$$}
\end{thm}
Now we want to study how the function $\phi_{w,h}$ depends on the
choice of the orientation of the Thom spectrum $T\xi$.
Let $w_i$, $i=1, 2$ are orientations of the Thom spectrum $T\xi$.
Let $$d_{1}(w_1, w_2)\in H^1(T\xi)\oplus H^{n+1}(T\xi )$$ denote
the difference of the composition maps $\Pi _2$$w_1$ and
$\Pi_2$$w_2$, where $\Pi _2$ is as in the definition of the
universal $\Omega$-spectrum $\widetilde W(n)$. Clearly, $w_1$ and
$w_2$ are homotopy if and only if $d_1(w_1, w_2)=0$ and a
secondary obstruction vanishes. The following theorem shows that
the secondary obstruction does not affect our quadratic function
$\phi _{w,h}$.
\begin{thm} Let $\phi _{w_i,h}$ be the quadratic forms associated with $(w_i, h)$, $i=1, 2$.
If $d_1(w_1, w_2)=0$, then $\phi _{w_1,h} (x) = \phi _{w_2,h}(x) $ for all
$x\in H^{n-1}(M, {\mathbb Z}_4)$.
\end{thm}
In general, the quadratic form $\phi _{w,h}$ does depend on the choice of $w$ and $h$. In order to obtain a
well-defined invariant of the $\Phi$-oriented manifold, we now choose certain type of $\Phi$-orientations
of the Thom spectrum $T\xi$ in an universal way and then define the Brown-Kervaire secondary quadratic forms
to be the quadratic functions associated to those $\Phi$-orientations.
Let $\gamma \searrow BSpin_G $ be the universal Spin spherical fibration and
$U\in H^0(MSpin_G, {\mathbb Z}_2)$ the universal Thom class. Note that
$$\begin{array}{lccc} \chi
(Sq^{n+2})U =\chi (Sq^{n+1})Sq^1U =0\mbox{ if $n$ is odd}\\ \chi
(Sq^{n+2})U =\chi (Sq^n)Sq^2U =0\mbox{ if $n=0(mod 4)$},
\end{array}
$$
Thus $U$ lifts to a map $f: MSpin_G\to W (n)$. By the Thom isomorphism, $f^*k_2$ gives an element of
$\bar {k_2} \in H^{n+2}(BSpin_G, {\mathbb Z}_2)$. Let $\pi :
BSpin_G\langle \bar{k_2}\rangle \to BSpin_G $ be the principal fibration with $k$-invariant
$\bar{k_2}$.
If $n=2(mod 4)$, we get a similar principal fibration $\pi :
BSpin_G\langle \bar{k_2}\rangle \to BSpin_G \langle v_{n+2}\rangle$, where $BSpin_G\langle
v_{n+2}\rangle \to BSpin_G $ is the fibration with fibre $K_{n+1}$
and $k$-invariant $v_{n+2}$.
It is easy to see that the fibration $\pi ^*\gamma $ is $\Phi$-orientable. Clearly the classifying map
of every $\Phi$-orientable stable spherical fibration lifts
to $BSpin_G\langle \bar{k_2}\rangle $.
\begin{defn}
The fibration $\pi ^*\gamma $ is called the {\it universal $\Phi$-orientable spherical Spin
fibration}. Its Thom spectrum, $MSpin_G\langle \bar{k_2}\rangle$, is called the {\it universal
$\Phi$-orientable Thom spectrum}.
\end{defn}
For a closed $\Phi$-orientable manifold $M^{2n}$, there is a Poincar\'e triple $(M, \nu _M, \alpha )$
where $\nu _M$ is the stable normal bundle and $\alpha \in \pi _{2n+k}(T\nu _M )$ is the normal invariant
of $M$ (obtained by the Thom-Pontryagin construction.)
\begin{defn}
Fix a connected spectral map ${\bf u}: MSpin_G\langle
\bar{k_2}\rangle \to \widetilde {W}(n)$ and a homomorphism $h$ in
Theorem 2.3. For a $\Phi$-orientable manifold $M$, let $$\phi
_M=\phi _{w,h}$$ where $w={\bf u}\circ T(v)$ and $T(v)$ is the
Thom map of a classifying bundle map of the stable bundle $\nu
_M$.
\end{defn}
Now we prove Theorem 1.6 assuming Theorem 2.7.
\begin{proof}[Proof of Theorem \ref{T:0.11}]
Let $\xi _i=\nu _{M_i}$ be the stable normal bundle of $M_i$ and $\alpha _i\in \pi _{2n+k}(T\xi _i)$ be the
normal invariant, $i=1,2$. By the definition, $\phi _{M_i}=\phi _{w_i, h}$ where $w_i={\bf u}\circ T(v_i)$
and $T(v_i): T(\xi _i)\to MSpin _G\langle \bar k_2\rangle$ the Thom map.
Let $\tilde f: f^*\xi _2\to \xi _2$ be a bundle map over the homotopy equivalence
$f$. Let $\alpha _3=T(\tilde f)^{-1}_*\alpha _2$, where $T(\tilde f)$ is the Thom map of $\tilde f$.
The Poincar\'e triple $(M_1, f^*\xi _2, \alpha _3)$ together with the $\Phi$-orientation
$w_2\circ T(\tilde f)$ gives a quadratic form $\phi _3$, where $w_2={\bf u}\circ T(v_2)$ is a
$\Phi$-orientation of $M_2$. By 2.5 we get that
$$\phi _3(f^*x)=\phi _{M_2}(x)$$
for all $x\in H^{n-1}(M_2, {\mathbb Z}_4)$.
To prove the desired result, it suffices to prove $\phi _3=\phi _{M_1}$.
Note that $f^*\xi _2$ and $\xi _1$ are stably equivalent as
spherical fibration since $f$ is a homotopy equivalence. Thus we can regard $f^*\xi _2$ and $\xi _1$ as the
the same and so get two orientations for $\xi _1$, $({\bf u}\circ T(v_1), h)$ and $({\bf u}\circ T(v_2)
\circ T(\tilde f), h)$. Since $f$ preserves the Spin structures/Wu orientations, $\pi \circ
v_2\circ f\simeq \pi \circ v_1$, where $\pi: BSpin_G \langle \bar
{k_2}\rangle \to BSpin_G$/$BSpin_G \langle v_{n+2} \rangle$ is the
principal fibration as above. This clearly implies that there exists a fibre automorphism
$g\in Aut (\xi _1)$ over the identity such that
$$T(\pi \circ v_2\circ \tilde f) \simeq
T(\pi \circ v_1)\circ T(g).$$
Notice that $g$ gives a unique element $g _0\in [M_1, G_k]$, where $G_k$ is the space of self homotopy
equivalences of $S^k$. By a formula in Brown \cite{brown}, the $(n+1)$-dimensional component
of $d_1({\bf u}\circ T(v_1)\circ T(g),{\bf u}\circ T(v_1))$ is
$\sum v_{n+2-2^i}\cup g_0^*u_{2^i-1}$, where $u_{2^i-1}$ is the transgression of $w_{2^i}\in
H^{2i}(BG_k, Z_2)$. By assumption, it must vanish since the Wu classes vanish.
On the other hand, the $1$-dimensional component of $d_1({\bf u}\circ T(v_1)\circ T(g),{\bf u}\circ T(v_1))$
is determined by the Spin structures and so it vanishes since $f$ preserves the Spin structures.
By Theorem 2.7 it follows that
$$\phi _{M_1}=\phi _4,$$
the quadratic form associated with the Poincar\'e triple $(M_1, \xi _1, \alpha _1)$ and the
$\Phi$-orientation $w_2\circ T(\tilde f)$.
Note that in the definitions of $\phi _3$ and $\phi _4$
the only different ingradients are the normal invariants, after identifying $\xi _1$ with
$f^*\xi _2$. By Theorem 2.7 once again $\phi _3=\phi _4$. This implies the desired result.
\end{proof}
Now we prove Proposition 1.7.
\begin{proof}[Proof of Proposition 1.7]
Since $M$ is a framed manifold, the stable normal bundle is trivial, i.e. the classifying map
of $\nu _M$ factors through a point. Choose a $\Phi$-orientation $w={\bf u}\circ T(v): \nu _M$ with
$v$ the bundle map of $\nu _M$ to the trivial $k$-bundle on a point, then $\phi _M(x)$ factors through
the stable homotopy group $\pi _{2n}^s(K({\mathbb Z}_4, n-1))$. By Theorem 3.1 $\pi _{2n}^s(K({\mathbb Z}_4, n-1))\cong {\mathbb Z}_4$
if $n=2(mod 4)$ and the order of elements in $\pi _{2n}^s(K({\mathbb Z}_4, n-1)$ is at most $2$ if
$n=0, 1(mod 4)$. On the other hand, by Theorem 1.6 the definition of $\phi _M$ does not depend on the choice of the $\Phi$-orientations since $M$ is a framed manifold. This completes the proof.
\end{proof}
\section{Some preliminaries on stable homotopy theory}\label{S:2}
In this section we calculate the stable homotopy groups $\pi
^s_{2n}(K(\pi , n-1))$ (see Theorem \ref{T:stable}). We will also
introduce some $2$-stage Postnikov tower which will give the
secondary cohomology operation $\Phi$ used in Section
$\S$\ref{S:intro}.
\begin{thm}\label{T:stable}
{\it The $2n$-th stable homotopy group of $K(\pi , {n-1})$
for $n\geq 4$ is as follows:
{\small \begin{tabular}{|r|c|c|c|c|c|}\hline
$n\geq 4$ & $0(mod 4)$ & $1(mod4)$ & $2(mod4)$ & $3(mod4)$
\\ \hline
$\pi_{2n}^s(K(\pi ,n-1))$ & $ ({\mathbb Z}_2)^{2(t+k)+s+p} $ & $
({\mathbb Z}_2)^{t+2k+s+p} $ & $ ({\mathbb Z}_4)^{t+k}\oplus ({\mathbb Z}_2)^{s+p} $& $
({\mathbb Z}_2)^{k+s+p} $ \\ \hline
\end{tabular} }
\newline where $p={{t+k+s}\choose{2}}$ and $\pi =G_0\times {\mathbb Z}^t\times
{\mathbb Z}_{2^{i_1}} \times \cdots \times {\mathbb Z}_{2^{i_k}} \times {\mathbb Z}_2^s$,
$i_j\geq 2$ if $1\leq j\leq k$ and $G_0\otimes {\mathbb Z}_2=0$.}
When $\pi={\mathbb Z}$, Theorem \ref{T:stable} follows from
\cite{mahowald1}.
\end{thm}
\begin{proof}
It is easy to know that(since we are computing the 2-localization)
\[
\pi ^s_{2n}(K(\pi , n-1)) =\pi ^s_{2n}(K(\pi/G_0 , n-1))
\]
Assume $G_0=0$ from now on. If $\pi=\pi_1 \bigoplus \pi_2$ with
$\pi_1$ nontrivial and $\pi_2$ a nontrivial cyclic group, then
\[
K(\pi , n-1)=K(\pi_1 , n-1) \times K(\pi_2 , n-1)
\]
and we have by a result in \cite{barcus} that
\[
\pi ^s_{2n}(K(\pi , n-1))=\bigoplus_{i=1,2}\pi ^s_{2n}(K(\pi_i ,
n-1)) \bigoplus \pi ^s_{2n}(K(\pi_1 , n-1)\wedge K(\pi_2 , n-1))
\]
\[
=\bigoplus_{i=1,2}\pi ^s_{2n}(K(\pi_i , n-1)) \bigoplus
H_{n+1}(K(\pi_1 , n-1),\pi_2)
\]
An easy calculation shows that $ H_{n+1}(K(G_1 , n-1),G_2)=Z_2$ if
$G_1,G_2$ are nontrivial cyclic groups and thus $ H_{n+1}(K(\pi_1
, n-1),\pi_2)=Z_2^{t+k+s-1}$.
On the other hand we know groups $\pi ^s_{2n}(K({\mathbb Z} , n-1))$ and
$\pi ^s_{2n}(K({\mathbb Z}_2 , n-1))$ by results in
\cite{milg},\cite{mahowald1}. To complete the proof it remains to
calculate $\pi ^s_{2n}(K({\mathbb Z}_{2^i} , n-1))$ for $2\leq i < \infty$
which will be given in the following results.
\end{proof}
Recall that for each locally finite connected CW complex $X$ we
can define a space $$\Gamma _qX= S^{q-1}\propto_{T}X\land X=
S^{q-1}\times (X\land X) /\{ (x,y,z)\sim (-x,z,y); (x, *)\sim *
\}$$ for every $q\in Z_+$. By \cite{milg} Theorem 1.11, for a
$(n-2)$-connected space $X$, $\Gamma _qX$ is $(2n-3)$-connected.
Moreover, if $X=K(\pi , n-1)$, we have a fibration $$G_q\to
\Sigma ^qK(\pi , n-1) \to K(\pi , q+n-1)$$ where $G_q\simeq \Sigma
^q\Gamma _q(K(\pi , n-1))$ through dimension $(3n+q-3)$. Thus
$\pi _i^s(K(\pi , n-1))\cong \pi _i^s(\Gamma _q(K(\pi , n-1))$
for $n< i<3n-3$.
When $q=1$, $\Gamma _qX= X\land X$ .The corresponding sequence is
:
\[
\Sigma F_{n-1}(\pi) \overset{H(\kappa)}{\to} \Sigma K(\pi , n-1)
\to K(\pi , n)
\]
where $F_{n-1}(\pi)=K(\pi , n-1) \land K(\pi , n-1)$. After
$q-1$ time suspensions we get a fibration sequence at least in
dimensions less than $3n+q-4$
\[
\Sigma^q F_{n-1}(\pi) \overset{\Sigma^{q-1}H(\kappa)}{\to}
\Sigma^q K(\pi , n-1) \to \Sigma^{q-1}K(\pi , n)
\]
Let $q$ be large enough so that we are always in the stable range
and let $r=q+2n$ ,then we have an exact sequence
\[ \label{T:sequence}
\cdots \to \pi ^s_{2n+2}(K(\pi , n))\overset{\partial}{\to}\pi
_{r}(\Sigma^qF_{n-1}(\pi))\to
\]
\[
\to \pi ^s_{2n}(K(\pi ,n-1))\to
\pi ^s_{2n+1}(K(\pi , n))\overset{\partial}{\to} \cdots
\]
Since we know that $\pi _{r}(\Sigma^qF_{n-1}(\pi))={\mathbb Z}_2$ for
$\pi={\mathbb Z}$ and ${\mathbb Z}_{2^i}$ , we can determine inductively $ \pi
^s_{2n}(K(\pi ,n-1))$ up to extension if we know the map
$\partial$.
\begin{lem}\label{T:2.2}
For $\pi={\mathbb Z}$ or ${\mathbb Z}_{2^i}$, a homotopy class $[g]\in \pi
^s_{2n+2}(K(\pi ,n))$ has $\partial g \neq 0$ iff
$g^*(\Sigma^{q-1}(\iota \cup Sq^2\iota))\neq 0$ where $g:S^{r+1}
\to \Sigma^{q-1}K(\pi,n)$,$\iota \in H^n(K(\pi,n),{\mathbb Z}_2)$ is the
generator and $g^*:H^*(\Sigma^{q-1}K(\pi,n),{\mathbb Z}_2) \to
H^*(S^{r+1},{\mathbb Z}_2)$
\end{lem}
The proof is similar to that of Lemma 1.3 in \cite{mahowald1}. The
key points are the followings:
\begin{itemize}
\item{$g^*$ can be nonzero only on element $\Sigma^{q-1}(\iota \cup Sq^2\iota)$}
\item{the Hurewicz homomorphism $H:{\mathbb Z}_2 \cong\pi
_r(\Sigma^qF_{n-1}(\pi)) \to H_r(\Sigma^qF_{n-1}(\pi)) $ is
nonzero}
\end{itemize}
The first statement is clear while the second is an easy
consequence of the Whitehead exact sequence(c.f. \cite{white},
page 555)
With the lemma above we can now prove the Proposition\ref{T:0.2}.
\begin{proof}[Proof of Proposition\ref{T:0.2}]
It suffices to prove that $\partial$ is trivial if $n=
0,1,2(mod4)$ and nontrivial if $n= 3(mod4)$.
For $n= 0,1,2(mod4)$ there is no $g:S^{r+1} \to
\Sigma^{q-1}K({\mathbb Z},n)$ such that $g^*(\Sigma^{q-1}(\iota \cup
Sq^2\iota))\neq 0$ since $\Sigma^{q-1}(\iota \cup Sq^2\iota)$ is
detected by the secondary cohomology operation $\varphi_n$ in
\cite{mahowald2}. Thus there is no $g:S^{r+1} \to
\Sigma^{q-1}K(Z_{2^i},n)$ such that $g^*(\Sigma^{q-1}(\iota \cup
Sq^2\iota))\neq 0$ by the naturality of secondary cohomology
operation and the fact that $\rho_{2^i}:K({\mathbb Z},n)\to K({\mathbb Z}_{2^i},n)$
corresponding to mod $2^i$ reduction induces a homomorphism
sending $\Sigma^{q-1}(\iota \cup Sq^2\iota)$ to the corresponding
element. It follows that $\partial = 0$
When $n= 3(mod4)$,there is a map $g:S^{r+1} \to
\Sigma^{q-1}K({\mathbb Z},n)$ such that $g^*(\Sigma^{q-1}(\iota \cup
Sq^2\iota))\neq 0$ since otherwise $\pi_{2n}^s(K({\mathbb Z},n-1))\neq 0$.
By the above fact on map $\rho_{2^i}$ it is easy to see that there
is a map $h:S^{r+1} \to \Sigma^{q-1}K({\mathbb Z}_{2^i},n)$ such that
$h^*(\Sigma^{q-1}(\iota \cup Sq^2\iota))\neq 0$.It follows from
the lemma above that $\partial h\neq 0$.
\end{proof}
With the help of Proposition\ref{T:0.2} and the known results
about $\pi _{2n+j}^s(K(\pi ,n))$ for $j=0,1$, we can now determine
the group $\pi _{2n}^s(K({\mathbb Z}_{2^i} ,n-1))$.
Assume $i \geq 2$ in the following unless otherwise stated.
\begin{prop}
If $n= 0 (mod2)$,then $${\rho_{2^i}}_*: \pi _{2n}^s(K({\mathbb Z} ,n-1))
\to \pi _{2n}^s(K({\mathbb Z}_{2^i} ,n-1))$$ is an isomorphism.
\end{prop}
Before the proof of the Proposition, let's give two remarks which
are clear from the proof of the Proposition.
\begin{rem}\label{T:remark}
If $i=1$, ${\rho_{2^i}}_*$ is onto.
\end{rem}
\begin{rem}\label{T:remark1}
If $n=0 (mod4)$, then the spherical cohomology class in \newline $
\pi _{2n}^s(K({\mathbb Z} ,n-1))$ does not belongs to the image of the
natural map: \newline $\pi_{r}(\Sigma^qF_{n-1}({\mathbb Z})) \to \pi
_{2n}^s(K({\mathbb Z} ,n-1))$.
\end{rem}
\begin{proof}
Note that we have a commutative diagram
\[
\begin{CD}
\pi_{r}(\Sigma^qF_{n-1}({\mathbb Z})) @>>> \pi _{2n}^s(K(Z ,n-1)) @>>> \pi
_{2n+1}^s(K({\mathbb Z} ,n))
\\ @V{\rho_{2^i}}_*VV @V{\rho_{2^i}}_*VV @V{\rho_{2^i}}_*VV \\
\pi_{r}(\Sigma^qF_{n-1}({\mathbb Z}_{2^i})) @>>> \pi_{2n}^s(K({\mathbb Z}_{2^i} ,n-1))
@>>> \pi_{2n+1}^s(K({\mathbb Z}_{2^i} ,n))
\end{CD}
\]
In the above diagram,the two left horizontal maps are injective by
Lemma\ref{T:2.2}, the left vertical map is obviously an
isomorphism while the fact that the right vertical one is also an
isomorphism follows by comparing the Whitehead exact sequences of
$\Gamma_{q-1}(K({\mathbb Z},n))$ and $\Gamma_{q-1}(K({\mathbb Z}_{2^i},n))$.On the
other hand ,the fact that the right horizontal map on the bottom
line is onto follows from the long exact sequence and the known
results about $\pi _{2n+j}^s(K({\mathbb Z}_{2^i} ,n))$ for $j=0,1$.
\end{proof}
\begin{prop}\label{T:reduction}
For $n= 1(mod2)$, $\pi _{2n}^s(K({\mathbb Z}_{2^i} ,n-1))=\pi _{2n}^s(K({\mathbb Z}
,n-1))\bigoplus {\mathbb Z}_2$.
\end{prop}
\begin{proof}
The relevant commutative diagram in this case is
\[
\begin{CD}
\pi_{r}(\Sigma^qF_{n-1}({\mathbb Z})) @>>> \pi _{2n}^s(K({\mathbb Z} ,n-1))@>>> \pi
_{2n+1}^s(K({\mathbb Z} ,n))=0 \\ @V{\rho_{2^i}}_*VV @V{\rho_{2^i}}_*VV
@V{\rho_{2^i}}_*VV \\ \pi_{r}(\Sigma^qF_{n-1}({\mathbb Z}_{2^i})) @>>>
\pi_{2n}^s(K({\mathbb Z}_{2^i} ,n-1)) @>>> \pi_{2n+1}^s(K({\mathbb Z}_{2^i} ,n))
@>\partial>>
\end{CD}
\]
By the same argument as in the last Proposition , we know the map
$\partial$ is onto. If $n = 3(mod4)$,the two left horizontal maps
are trivial by Lemma\ref{T:2.2}, thus $\pi_{2n}^s(K({\mathbb Z}_{2^i}
,n-1))\cong \text{coker}\partial\cong {\mathbb Z}_2 $.
If $n = 1(mod4)$, what we can get is an exact sequence
\[
0 \rightarrow \pi_{2n}^s(K({\mathbb Z} ,n-1))
\rightarrow\pi_{2n}^s(K({\mathbb Z}_{2^i} ,n-1)) \rightarrow {\mathbb Z}_2 \rightarrow
0
\]
To complete the proof, it suffices to prove the last map in the
above sequence has a section.
To do this we need another diagram
\[
\begin{CD}
\pi _{2n}^s(K_{n-1})@>>> \pi _{2n+1}^s(K_n) @>\partial>>
\pi_{r-1}(\Sigma^qF_{n-1}({\mathbb Z}_2)) \\ @Vj_*VV @Vj_*VV @Vj_*VV
\\ \pi_{2n}^s(K({\mathbb Z}_{2^i} ,n-1)) @>>> \pi_{2n+1}^s(K({\mathbb Z}_{2^i}
,n)) @>\partial>> \pi_{r-1}(\Sigma^qF_{n-1}({\mathbb Z}_{2^i}))
\end{CD}
\]
where $j:{\mathbb Z}_2 \to {\mathbb Z}_{2^i}$ is the natural inclusion.
The same argument as above combined with the proof of Theorem 10.9
in\cite{milg} shows that the two $\partial$'s are onto and $j_*$
induces an isomorphism between kernels of two $\partial$'s.
Finally we get the following diagram which gives the desired
section.
\[
\begin{CD}
\pi _{2n}^s(K_{n-1})@>\cong>> {\mathbb Z}_2 \\ @Vj_*VV @V{\cong}VV
\\ \pi_{2n}^s(K({\mathbb Z}_{2^i} ,n-1)) @>>> {\mathbb Z}_2
\end{CD}
\]
\end{proof}
\begin{lem}\label{T:3.7}
{\it If $n$ is odd and $Sq^1_i\in H^n( K({\mathbb Z}_{2^i}, n-1),
{\mathbb Z}_2)$ is a generator. Then $$(Sq^1_i )_*: \pi
_{2n}^s(K({\mathbb Z}_{2^i}, n-1))\to \pi _{2n}^s (K({\mathbb Z}_2, n))\cong {\mathbb Z}_2$$ is
an epimorphism.}
\end{lem}
\begin{proof} It suffices to prove that the following map
$Sq^1:K({\mathbb Z}_2,n-1) \to K({\mathbb Z}_2,n)$ induces an isomorphism on $2n$-th
stable homotopy group. By the calculation in Milgram's
book\cite{milg}, the first group is generated by the class
corresponding to $Sq^1(t)\bigotimes Sq^1(t)$ and the second by $s
\bigotimes s$ where $s,t$ are the fundamental classes of the
corresponding groups. Now what we want follows from the fact that
$Sq^1$ induces a homomorphism mapping $s \bigotimes s$ to
$Sq^1(t)\bigotimes Sq^1(t)$.
\end{proof}
\begin{prop}\label{T:post}
Let $\widetilde{E}_{n+q}$ ($q$ large) be the following $2$-stage
Postnikov tower. Then there is a map $f:\Sigma^qK({\mathbb Z}_4,n-1)\to
\widetilde{E}_{n+q}$ such that the composite $(\Sigma^q F_{n-1}({\mathbb Z})
\to)$ $\Sigma^q K({\mathbb Z},n-1)\overset{\Sigma^q \rho_4}{\to} \Sigma^q
K({\mathbb Z}_4,n-1)\to \widetilde{E}_{n+q}$ if $n=1,2(mod4)$$(\text{or, }
n=0(mod4))$
induces an isomorphism on $\pi_r$ where $r=q+2n$ as
above.
(1). $n=2(mod4)$ $$
\begin{array}{ccccc}
K_{r} & \stackrel{i_2}{\longrightarrow } &\widetilde{E}_{n+q} & &
\\ & & \downarrow {\Pi _2} & & \\ K_{r-2}\times
K_{r}& \stackrel{i_1}{\longrightarrow } & E_{n+q} & \stackrel
{\omega _2 }{\longrightarrow} & K_{r+1} \\ & & \downarrow { \Pi
_1} & & \\ \Sigma ^{q}K({\mathbb Z}_4, n-1)& \stackrel {\Sigma ^ql_{n-1} }
{\longrightarrow}& K({\mathbb Z}_4,q+n-1) & \stackrel {Sq^n\times Sq^{n+2} }
{\longrightarrow} & K_{r-1}\times K_{r+1}
\end{array}
$$ where $i_1^*(\omega _2)=Sq^2Sq^1l_{r-2} +Sq^1l_{r}$.
(2). $n=0(mod4)$
$$
\begin{array}{ccccc}
K_{r} \stackrel{i_2}{\longrightarrow } & \widetilde{E}_{n+q} & &
\\ & \downarrow { \Pi _2} & & \\ K_{r-2}
\stackrel{i_1}{\longrightarrow } & E_{n+q} & \stackrel {\omega _2
}{\longrightarrow} & K_{r+1} \\ & \downarrow { \Pi _1} & &
\\ \Sigma ^qK({\mathbb Z}_4, n-1)\stackrel {\Sigma
^ql_{n-1}}{\longrightarrow} & K({\mathbb Z}_4,q+n-1) & \stackrel {Sq^n }
{\longrightarrow} & K_{r-1}
\end{array}
$$ where $i_1^*(\omega _2)=Sq^2Sq^1l_{r-2}$.
(3). $n=1(mod4)$
$$
\begin{array}{ccccc}
K_{r} \stackrel{i_2}{\longrightarrow }& \widetilde{E}_{n+q} & &
\\ & \downarrow {\Pi _2} & & \\ K_{r}
\stackrel{i_1}{\longrightarrow }& E_{n+q} & \stackrel {\omega _2
}{\longrightarrow}& K_{r+1} \\ & \downarrow { \Pi _1} & &
\\ \Sigma ^qK({\mathbb Z}_4, n-1) \stackrel {\Sigma
^ql_{n-1}}{\longrightarrow}& K({\mathbb Z}_4,q+n-1)& \stackrel { Sq^{n+1}}
{\longrightarrow} & K_{r}
\end{array} $$
where $i_1^*(\omega _2)=Sq^2l_{r-1} $ .
\end{prop}
\begin{proof}
Denote the tower in the Proposition by
$\widetilde{E}_{n+q}({\mathbb Z}_4)$. Denote by $\widetilde{E}_{n+q}({\mathbb Z})$ a
similar tower in which $K({\mathbb Z}_4,n+q-1)$ is replaced by $K({\mathbb Z},n+q-1)$.
By Remark \ref{T:remark1}, it is easy to see that there is a map
from $\Sigma^q F_{n-1}({\mathbb Z})$ to $\widetilde{E}_{n+q}({\mathbb Z})$ which
induces an isomorphism on $\pi_r$ when $n=0(mod4)$. On the other
hand it is not difficult to see that there is a map from the tower
$\widetilde{E}_{n+q}({\mathbb Z})$ to the tower $\widetilde{E}_{n+q}({\mathbb Z}_4)$
which induces an isomorphism on $\pi_r$. It remains to prove that
the natural map $\Sigma^q\iota_{n-1}:\Sigma^qK({\mathbb Z}_4,n-1)\to
K({\mathbb Z}_4,n+q-1)$ can be lifted to $\widetilde{E}_{n+q}({\mathbb Z}_4)$ and the
lifting is compatible to that of the map
$\Sigma^q\iota_{n-1}:\Sigma^qK({\mathbb Z},n-1)\to K({\mathbb Z},n+q-1)$ to
$\widetilde{E}_{n+q}({\mathbb Z})$.
We will give a proof only for $n= 2(mod4)$ , the other cases are
similar. Consider the fiber inclusion map $h: \Sigma ^q\Gamma
_{q}\to \Sigma ^qK(Z_4, n-1)$, we have the following
Peterson-Stein formula
$$Sq^2Sq^1Sq^n_{h} (\Sigma
^ql_{n-1})+Sq^1Sq^{n+2}_{h} (\Sigma ^ql_{n-1})=h^*\Psi(\Sigma
^ql_{n-1})\in H^{r+1}(\Sigma ^q \Gamma _{q}, {\mathbb Z}_2)/Q$$ where $Q=
Sq^2Sq^1(Im h^*)+Sq^1(Im h^*)=Sq^1(Im h^*)$.
By Theorem 4.6 \cite{milg} and a familiar diagram chase argument
as in the proof of Proposition 2 in Chap.16 \cite{mosh}(see also
\cite{thomas}) , we have $\Sigma ^q(\theta \otimes \theta )\in
Sq^n_h(\Sigma ^ql_{n-1})$ and $\Sigma ^q{e^2\cup(\theta \otimes
\theta )}\in Sq^{n+2}_h(\Sigma ^ql_{n-1})$. It follows easily that
$h^*\Psi(\Sigma ^ql_{n-1})=0 \in H^{r+1}(\Sigma ^q \Gamma _{q},
{\mathbb Z}_2)/Q$. It is not difficult to see from this and a simple
computation that $\Psi(\Sigma ^ql_{n-1})=0$ and a lifting can be
chosen such that $\omega_2$ lies in its kernel.
To complete the proof, note that , as mentioned before , there is
a commutative diagram up to homotopy
\[
\begin{CD}
\widetilde{E}_{n+q}({\mathbb Z}) @>\rho_4>> \widetilde{E}_{n+q}({\mathbb Z}_4) \\
@VVV @VVV \\
E_{n+q}({\mathbb Z}) @>\rho_4>> E_{n+q}({\mathbb Z}_4) \\
@VVV @VVV \\
K({\mathbb Z},n+q-1) @>\rho_4>> K({\mathbb Z}_4,n+q-1)\\
@A\Sigma ^ql_{n-1}AA @A\Sigma ^ql_{n-1}AA \\
\Sigma^qK({\mathbb Z},n-1) @>\rho_4>> \Sigma^qK({\mathbb Z}_4,n-1)
\end{CD}
\]
The lifting from $\Sigma^qK({\mathbb Z},n-1)$ of $\Sigma^ql_{n-1}$ and the
lifting from $\Sigma^qK({\mathbb Z}_4,n-1)$ of $\Sigma^ql_{n-1}$ can be made
compatible by a modification of the lifting from
$\Sigma^qK({\mathbb Z}_4,n-1)$ of $\Sigma^ql_{n-1}$. The same way the
liftings to $\widetilde{E}_{n+q}$ can also be made compatible.
Thus we have the following commutative diagram up to homotopy
which completes the proof.
\[
\begin{CD}
\widetilde{E}_{n+q}({\mathbb Z}) @>\rho_4>> \widetilde{E}_{n+q}({\mathbb Z}_4) \\
@AAA @AAA \\
\Sigma^qK({\mathbb Z},n-1) @>\rho_4>> \Sigma^qK({\mathbb Z}_4,n-1)
\end{CD}
\]
\end{proof}
\begin{rem}
The $2^{nd}$ k-invariant $\omega _2$ in the Postnikov tower above
gives an unique secondary cohomology operator $\Psi$ (with
$Z_4$-coefficients) associated with the Adem relation
$$\begin{array}{ll} Sq^{2}Sq^{1}Sq^{n}+Sq^{1}Sq^{n+2}=0 &n=2(mod
4)\\ Sq^{2}Sq^{1}Sq^{n}=0 &n=0(mod 4)\\ Sq^{2}Sq^{n+1}=0 &n=1(mod
4)\\
\end{array}$$
Note that $E_{n+q}$ is the universal example of the operator
$\Psi$.
By Peterson-Stein\cite{stein1}, there are operators $\Phi $ which are S-dual to
$\Psi $(which is uniquely determined by $\Psi$) so it is a
secondary operator associated with the Adem relations: $$
\begin{array}{ll}
\chi(Sq^n)Sq^3+\chi(Sq^{n+2})Sq^1+Sq^1\chi(Sq^{n+2})=0 &
n=2(\bmod 4) \\ \chi(Sq^n)Sq^3+Sq^1\chi(Sq^{n+2})=0 & n=0(\bmod
4)
\\ \chi(Sq^{n+1})Sq^2+Sq^1\chi(Sq^{n+2})=0 & n=1(\bmod 4)
\end{array}
$$ as we stated in $\S \ref{S:intro}$.
\end{rem}
\section{Proofs of Theorems 2.3, 2.6 and 2.7}\label{S:3}
\begin{proof}[Proof of Theorem \ref{T:0.3}] First note that
it suffices to show this for the universal
spectrum $\widetilde {W}(n)$ since the map $i: S^0\to \widetilde
{W}(n)$ factors through $i: S^0\to Y$. Notice that $H_i(\widetilde
{W}_k(n)/S^k)=0$ for $i\leq k+2$. Thus in the following proof, we
may assume that $Y_k/S^k$ satisfies the same for $k$ large.
Assuming $k$ large, without loss of generality we can assume that
$Y_k$ is a finite complex. Write $Y^{\ast}_k$ for the $m$
$S$-dual of $Y_k$ and $g: Y^{\ast}_k \to S^{m-k}$ for the $S$-dual
of the inclusion $i: S^k \to Y_k$. Note that $g^*(\varsigma
_{S^{m-k}})\neq 0$, where $ \varsigma _{S^{m-k}} $ is the
cohomology fundamental class of the sphere. By the $S$-duality we
get a commutative diagram
\[
\begin{array}{ccc}
\{ S^{2n+k}, S^{k} \wedge K( {\mathbb Z}_4, n-1) \} &
\stackrel{i_{\ast}}{\longrightarrow} & \{ S^{2n+k}, Y_{k} \wedge
K( {\mathbb Z}_4, n-1) \} \\ \downarrow {\cong}& & \downarrow {\cong} \\
\{ S^{2n+m}, S^{m} \wedge K( {\mathbb Z}_4, n-1) \} & \stackrel{g^{\ast}}{
\longrightarrow} & \{ S^{2n+k}\wedge Y^{\ast}_{k} , S^{m}\wedge
K({\mathbb Z}_4, n-1) \} \\
\downarrow & & \downarrow {q_2}_* \\ \lbrack S^{2n+m},
\widetilde{E}_{n+m} \rbrack & \stackrel{g^{\ast}}
{\longrightarrow} & \lbrack S^{2n+k}\wedge Y^{\ast}_{k},
\widetilde{E}_{n+m}\rbrack
\end{array}
\]
where $\widetilde {E}_{m+n}$ is the tower in
Proposition\ref{T:post} and $q_2: S^m\land K({\mathbb Z}_4, n-1)\to
\widetilde {E}_{n+m}$ is a lifting of
$\Sigma ^ml_{n-1}$. From the
diagram above and Proposition \ref{T:post},it suffices to show
that the homomorphism $g^*$ at the bottom line is injective.From
now on we will restrict to the case $n\equiv 2(mod4)$. The other
cases are similar. Let $i_{0}: F \to \widetilde{E}_{n+m}$ be the
fibre of the composite $\Pi_1\circ \Pi_2$. Note that $F$ can be
viewed as a fibration over $K_{2n+m-2}$ with fibre $K( {\mathbb Z}_4, 2n+m)$
and $k$-invariant $j_{\ast}(Sq^{2}Sq^{1})(l)$; where $$j_{\ast} :
H^{m+2n+1}(-, {\mathbb Z}_2) \to H^{m+2n+1}(-, {\mathbb Z}_4)$$ is the homomorphism
induced by the inclusion $ {\mathbb Z}_2 \subset {\mathbb Z}_4$ and $l$ is the basic
class of $K_{m+2n-2}$.
Consider
the following commutative diagrams
\[
\begin{array}{ccccc}
& & [S^{2n+m}, F] & \stackrel{{\cong }_{\ast}}{\longrightarrow} &
[S^{2n+m}, \widetilde{E}_{n+m}] \\
& & \downarrow {J:= g^*} & & \downarrow
{g^*} \\
\lbrack S^{2n+k} \wedge Y_{k}^{\ast}, K( {\mathbb Z}_4, n+m-2)\rbrack &
\stackrel{{i_{1}}_ {\ast}}{\longrightarrow} & \lbrack S^{2n+k}
\wedge Y_{k}^{\ast}, F \rbrack &
\stackrel{{i_{0}}_{\ast}}{\longrightarrow} & \lbrack S^{2n+k}
\wedge Y_{k}^ {\ast}, \widetilde E_{n+m}\rbrack
\end{array}
\]
and
\[
\begin{array}{ccccc}
& [S^{2n+m}, K( {\mathbb Z}_4, 2n+m)] & \stackrel{{\cong}}{\longrightarrow}
& [S^{2n+m}, F] & \\ & \downarrow
{g^*} & & \downarrow {J} &\\
\lbrack S^{2n+k}\land Y_k^*, K_{2n+m-3}\rbrack &
\stackrel{{j_*(Sq^2Sq^1)} }{\longrightarrow} \lbrack S^{2n+k}
\wedge Y_{k}^{\ast}, K( {\mathbb Z}_4, 2n+m)\rbrack & \stackrel{{\cong}
}{\longrightarrow} & \lbrack S^{2n+k} \wedge Y_{k}^{\ast},
F\rbrack
\end{array}
\]
where $i_{1} : K({\mathbb Z}_4, n+m-2) \to F$ is the homotopy fibre of
$i_{0}$. $j_*(Sq^2Sq^1)$ in the second diagram above is zero
since $Sq^3U_k=0$ and thus by duality $\chi
(Sq^3)H^{m-k-3}(Y_k^*)=Sq^2Sq^1 H^{m-k-3}(Y_k^*)=0$. Thus the
second diagram implies that $J$ is a monomorphism. To complete the
proof, it suffices to show $Ker(i_{0})_{\ast} = Im(i_1)_{\ast}
=0$ in the first diagram above.
Let $q=m-n-k-1$, if $x\in H^{q-1}(Y^{\ast}_k, {\mathbb Z}_4)$,then
$Sq^n(x)\in H^{n+q-1}(Y_k^*, {\mathbb Z}_2) \cong
(H^{k+2}(Y_k, {\mathbb Z}_2))^*=0$. On the other hand, by duality $\chi
(Sq^{n+2})U_k=0$ implies that $Sq^{n+2}H^{q-1}(Y^{\ast}_k,
{\mathbb Z}_2)=0$. Thus $$x \in \text{Ker}{Sq^n}\cap \text{Ker}{Sq^{n+2}}$$
Since ${ Y_k}$ is $\Phi$-orientable, i.e, $0\in\Phi (U_k)$. By
\cite{stein1} that $0\in \Psi (x)$. Thus $x$ can be lifted to
$\widetilde{E}_{q-1} $ and so $(i_1)_{\ast}(x)=0$. This completes
the proof.
\end{proof}
For simplicity,denote by $F_{n-1}({\mathbb Z}_4)$ the space $K( {\mathbb Z}_4,
n-1)\wedge K( {\mathbb Z}_4, n-1)$ as before in the following proof.
\begin{proof}[Proof of Theorem 2.6] For $x\in H^{n-1}(M, {\mathbb Z} _4)$, let $f(x)=
(w\wedge id)A_\alpha (x)\in H_{2n}(K({\mathbb Z} _4, n-1); \widetilde {W}(n))$. For $k$ large,
$f(x+y)$ is the following composition of maps
$S^1\land S^{2n+k} \stackrel{id\land \Delta
\alpha}{\longrightarrow} S^1\land T\xi\wedge M_{+}
\stackrel{id\land {w\land (x\times y)}}{\longrightarrow}\newline
\rightarrow S^1\land\widetilde {W}(n)_k \wedge (K( {\mathbb Z}_4, n-1)\times K( {\mathbb Z}_4, n-1))=
\newline = \widetilde {W}(n)_{k}\wedge S^1\land (K( {\mathbb Z}_4,
n-1)\times K( {\mathbb Z}_4, n-1)) \stackrel{id\land \kappa
}{\longrightarrow} \widetilde {W}(n)_{k}\wedge S^1\land K( {\mathbb Z}_4,
n-1), $
\noindent where $\kappa^*(l)=l\otimes 1+1\otimes l $ for the
basic class $l\in H^{n-1}(K({\mathbb Z}_4, n-1), {\mathbb Z}_4)$.
Identifying
$\widetilde {W}(n)_{k}\wedge S^1\land (K( {\mathbb Z}_4, n-1)\times K( {\mathbb Z}_4,
n-1))$ with $$\{\widetilde {W}(n)_{k}\wedge S^1\land K( {\mathbb Z}_4,
n-1)\} \lor \{ \widetilde {W}(n)_{k}\wedge S^1\land K( {\mathbb Z}_4,
n-1)\} \lor $$ $$\lor \{ \widetilde {W}(n)_{k} \wedge S^1\wedge
F_{n-1}({\mathbb Z}_4)\}.$$ It is readily to see that
$f(x+y)=f(x)+f(y)+g$, here $g$ is the composition
{\small
$S^{2n+k+1} \stackrel{id \wedge \Delta \alpha}{\longrightarrow}
S^1\wedge T\xi\wedge M_{+} \stackrel{id\wedge w\land
\Delta} {\longrightarrow} S^1\land \widetilde {W}(n)_{k} \wedge
M_{+} \wedge M_{+} \stackrel{id\land x\land y}{\longrightarrow}
\widetilde {W}(n)_k \wedge S^1\land K( {\mathbb Z}_4, n-1) \wedge K( {\mathbb Z}_4,
n-1) \stackrel{id\land H(\kappa )}{\longrightarrow} \widetilde
{W}(n)_{k}\wedge S^1\land K( {\mathbb Z}_4, n-1), $ }
\newline where $H(\kappa )$ is the Hopf constuction of $\kappa$.
Now the cofibration $$ S^{k+1}\wedge F_{n-1}({\mathbb Z}_4)\overset{\Sigma i\wedge id}{\to} S^1\wedge \widetilde
{W}(n)_{k}\wedge F_{n-1}({\mathbb Z}_4) \to S^1\wedge(\widetilde
{W}(n)_{k}/S^{k})\wedge F_{n-1}({\mathbb Z}_4) $$ is also a fibration at
least in the stable range. It follows immediately that $$(\Sigma
i\wedge id)_{\ast}: \pi_{2n+k+1}(S^{k+1}\wedge F_{n-1}({\mathbb Z}_4))\to
\pi _{2n+k+1}(S^1\wedge \widetilde {W}(n)_{k}\wedge F_{n-1}({\mathbb Z}_4))
$$ is surjective. On the other hand, it is easy to know that the
generator $\beta \in \pi ^s_{2n}(F_{n-1}({\mathbb Z}_4)) \cong {\mathbb Z}_2 $
satisfies $\beta ^*(l\otimes Sq^2l)\neq 0$. Thus, for the
inclusion map $i$, the composition $(\Sigma i\land id) \circ \beta
\in \pi_{2n+k+1}(S^1\wedge \widetilde {W}(n)_{k}\wedge
F_{n-1}({\mathbb Z}_4))$ induces a nontrivial homomorphism on the
$(2n+k)$-th homology and thus ($\Sigma i\wedge id)_{\ast} $ is an
isomorphism. Moreover, the generator $g_0 \in \pi
^s_{2n}(\widetilde {W}(n)_k \land F_{n-1}({\mathbb Z}_4)) $ satisfies that
$g_{0} ^*(U_{k}\wedge Sq^2l_{n-1} \wedge l_{n-1})\ne 0$. Thus the
composition $(id\wedge x\wedge y)( w\wedge \Delta
)(\Delta \alpha )$ is null homotopy if and only if $ \langle x\cup
Sq^{2}y, [M]_2\rangle = 0 $. By Proposition \ref{T:0.2}, the proof
now follows by the commutative diagram
\newline
\[
\begin{array}{ccc}
S^{k}\wedge \Sigma F_{n-1}({\mathbb Z}_4) & \stackrel{i\wedge
id}{\longrightarrow} & \widetilde {W}(n)_{k}\wedge \Sigma
F_{n-1}({\mathbb Z}_4)\\ \downarrow {id\wedge H(\kappa )} & & \downarrow {id
\wedge H(\kappa )}\\ S^{k} \wedge \Sigma K( {\mathbb Z}_4, n-1) &
\stackrel{i\wedge id} {\longrightarrow} & \widetilde {W}(n)_{k}
\wedge \Sigma K( {\mathbb Z}_4, n-1).
\end{array}
\]
\end{proof}
\begin{proof}[Proof of Theorem 2.7] Let $\mu : K_{n+k+1}\times \widetilde W_k(n)\to \widetilde
W_k(n)$ denote the fiber multiplication. Since
$d_1(w_1, w_2)=0$, $w_2$ is the composition $$T\xi \stackrel
{\Delta }{\longrightarrow} T\xi \times T\xi \stackrel {w_1\times
vU_k}{\longrightarrow} \widetilde W_k(n)\times K_{n+k+1}\stackrel
{\mu}{\longrightarrow} \widetilde W_k(n),$$ where $vU_k\in
H^{k+n+1}(T\xi , Z_2)$ is the second difference of $w_1$ and $w_2$, i.e, the secondary obstruction
to deform $w_1$ to $w_2$. Consider the commutative diagram:
\begin{displaymath}
\begin{array}{cccc}
S^{2n+k} & \stackrel {\alpha '}{\longrightarrow} & (T\xi \land
M_+)\lor (T\xi \land M_+) & \stackrel {a}{\longrightarrow}
\widetilde {W}_k(n)\land K(Z_4, n-1)\\
\parallel & & {\bigcap} & \parallel \\
S^{2n+k} & \stackrel {\Delta \alpha }{\longrightarrow} & (T\xi
\times T\xi ) \land M_+ & \stackrel {b}{\longrightarrow}
\widetilde {W}_k(n)\land K(Z_4, n-1)
\end{array}
\end{displaymath}
where $\alpha '$ is a lifting of $\Delta \alpha$, $b=\mu (w_1\times
vU_k)\land x$, $a=(w_1\land x )\lor c$, and $c=i(vU_k)\land x$, $i:
K_{n+k+1}\to \widetilde W_k(n)$ the inclusion of the fibre. Write
$\alpha '= \alpha _1+\alpha _2$, here $\alpha _1$
and $\alpha _2$ are the factors of the wedge. Note that
$\phi _ 2(x)=h(b\circ \Delta \alpha )=h(a\alpha _1)+h(a\alpha _2)=
\phi _ 1(x)+h(a \alpha _2)$. \newline
We are going to show $h(a\alpha _2)=0$.
As $a\alpha _2$ factors through the map $i\land id: K_{n+k+1}
\land K({\mathbb Z}_4, n-1) \to \widetilde W_k(n)\land K({\mathbb Z}_4, n-1)$, it
suffices to prove that $$(i\land id)_*: \pi _{2n+k}(K_{n+k+1}\land
K({\mathbb Z}_4, n-1)) \to \pi _{2n+k}( \widetilde W_k(n)\land K({\mathbb Z}_4,
n-1))$$ is zero. Note the homomorphism $$(Sq^1\land id)_* : \pi
_{2n+k}(K_{n+k} \land K({\mathbb Z}_4, n-1))\to \pi _{2n+k}(K_{n+k+1} \land
K({\mathbb Z}_4, n-1)) \cong {\mathbb Z}_2 $$ is an isomorphism as it induces an
ismomorphism on the $(2n+k)$-th homology groups. The composition
$K_{n+k}\stackrel {Sq^1} {\longrightarrow} K_{n+k+1}\stackrel {i
}{\longrightarrow} \widetilde W_k(n)$ is null homotopy. Thus
$(i\land id)_*=0$. This completes the proof.
\end{proof}
\section{Proof of Theorem 1.9}
\vskip 4mm
In this section we prove Theorem 1.9. We first study the properties of the invariants $\mu _M$ and
$q_M(Sq^1_i)$ defined in $\S$1.
\begin{lem}\label{T:1.6}
{\it Let $M$ be a framed manifold of dimension $2n$ with $n$ odd. Let
$q_M: H^n(M, {\mathbb Z}_2)\to {\mathbb Z}_2$ be the Kervaire quadratic form. For
$x\in H^{n-1}(M, {\mathbb Z}_{2^i})$,
\newline
(i) $n=3(mod 4)$, $[M, x]$ is reduced bordant to zero iff
$q_M(Sq^1_i)x=0.$\newline (ii) $n=1(mod 4)$, $[M, x]$ is reduced
bordant to $[M', x']$ where $x'\in H^ {n-1}(M')$ iff
$q_M(Sq^1_i x)=0$.}
\end{lem}
\begin{proof}
Identify the reduced framed bordism group $\widetilde{\Omega }_{2n}^{fr}(-)$ with the stable homotopy
group $\pi _{2n}^{s}(-)$. Recall that $ \pi _{2n}^s (K({\mathbb Z}_2,n)) ={\mathbb Z}_2$. By [4] it is easy to see that
the homomorphism
$$(Sq_i^1)_*: \pi _{2n}^{s}(K({\mathbb Z} _{2^i},n-1)\to \pi _{2n}^s (K({\mathbb Z}_2,n))$$
is identified with the following geometrically defined homomorphism
$$
\begin{array}{lcc}
\widetilde {\Omega }_{2n}^{fr}(K({\mathbb Z} _{2^i},n-1)) & \to & {\mathbb Z}_2 \\
\ [M , x] & \to & q_{M}(Sq^1_i)x
\end{array}
$$ By Theorem 3.1 and Lemma 3.7 (i) follows since $(Sq^1_i)_*$ is
an isomorphism. To prove (ii), note that there is an exact
sequence by Proposition\ref{T:reduction} and Lemma\ref{T:3.7}$$\pi
_{2n}^s(K({\mathbb Z},n-1))\to \pi _{2n}^s(K({\mathbb Z}
_{2^i},n-1))\stackrel{(Sq^1_i)_*} \longrightarrow \pi
_{2n}^s(K_n).$$ This completes the proof.
\end{proof}
Now we want to study which bilinear forms $\mu$ can be realized by $(n-2)$-connected $2n$-dimensional
$\pi$-manifolds. Note that a sphere bundle over $S^{n+1}$ with fiber $S^{n-1}$ is a $\pi$-manifold
if the characteristic map of the bundle, $\theta \in \pi _n(SO(n))$, belongs to the kernel of the
stablization homomorphism $S_*: \pi _n(SO(n))\to \pi _n(SO)$.
Recall that the homotopy groups of $\pi _n(SO(n))$ are as follows
(c.f: [11]):
\vskip 3mm
\small{
\centerline {$\pi _n(SO(n))$, $n\geq 3, \ne 6$ }
\vskip 1mm
{\footnotesize \begin{tabular}{|r|c|c|c|c|c|c|c|c|} \hline
$n\geq 3, \ne 6$ & $8s $ & $8s+1$ & $8s+2$ & $8s+3$ &$8s+4$
& $8s+5$ & $ 8s+6$ & $8s+7$
\\ \hline
$\pi _n(SO(n))$ & ${\mathbb Z}_2\oplus {\mathbb Z}_2\oplus {\mathbb Z}_2 $
& $ {\mathbb Z}_2\oplus {\mathbb Z}_2 $ &
${\mathbb Z}_4 $&
${\mathbb Z} $ & ${\mathbb Z}_2\oplus {\mathbb Z}_2$& ${\mathbb Z}_2$ & ${\mathbb Z}_4$ & $ {\mathbb Z}$ \\ \hline
\end{tabular} }}
\noindent and $\pi _6(SO(6))=0$.
Let $\pi : SO(n)\to S^{n-1}$ be the canoincal $SO(n-1)$-fiberation. For a $S^{n-1}$-bundle
over $S^{n+1}$ with characteristic map $\theta \in \pi _{n}(SO(n))$, say $M_\theta$, it is easy to see
that $Sq^2: H^{n-1}(M_\theta ,{\mathbb Z}_2)\to H^{n+1}(M_\theta , {\mathbb Z}_2 )$ is an isomorphism if and only if
$\pi _*(\theta )\in \pi _n(S^{n-1})={\mathbb Z}_2$ is nonzero. By duality this implies that $z\cup Sq^2z=0$ for all
$z\in H^{n-1}(M_\theta , {\mathbb Z}_2)$ if and only if $\pi _*(\theta )=0$. The latter is equivalent to the fact of
that the bundle has a section.
\begin{lem}\label{T:0.12}
{\it Let $M$ be a $\pi$-manifold of dimension $2n$.
Then\newline (i) $\mu _M(x, x) =0$, $\forall x\in H^{n-1}(M, {\mathbb Z}_2)$
if $n=2(mod 4)$. \newline (ii) $\mu _M(x, x)=0$, $\forall x\in Im
(\rho _2:T\subset H^{n-1}(M, {\mathbb Z}_4)\to
H^{n-1}(M, {\mathbb Z}_2))$, \newline if $n$ is odd where $T$ is the set of
elements of order $4$.\newline
(iii) If $n=0(mod 4)$, then there is a $S^{n-1}$-bundle over
$S^{n+1}$, $M$, so that $\mu _{M}(x,x)\neq 0$,
where $x\in H^{n-1}(M, {\mathbb Z}_2)$ is a generator.}
\end{lem}
\begin{proof} For each $x\in H^{n-1}(M, {\mathbb Z}_2)$, consider the reduced
bordism class $[M, x]\in \widetilde {\Omega
}_{2n}^{fr}(K_{n-1})\cong {\mathbb Z}_2$. It is easy to see that $x\cup Sq^2x[M]$ is
a bordism invariant. One verifies the following map defines a homomorphism
$$
\begin{array}{lcc}
\widetilde {\Omega }_{2n}^{fr}(K_{n-1})& \to &{\mathbb Z}_2 \\
\ [M , x] & \to &
x\cup Sq^2x[M]
\end{array}
$$
By Remark \ref{T:remark} the reduction homomorphism
$$\widetilde {\Omega }_{2n}^{fr}(K({\mathbb Z}, {n-1}))\to
\widetilde {\Omega }_{2n}^{fr}(K_{n-1})$$
is surjective if $n$ is even.
If $n=2(mod 4)$, let $\theta\in \pi _{n}(SO(n))$ be a generator. By the tables (I)(II) of \cite{kerv1}
it follows that $\theta $ lies in the image of the inclusion map $\pi _n(SO(n-1))\to \pi _{n}(SO(n))$. By
the remark above this implies that the sphere bundle $M_\theta $ has a section. Therefore $z\cup Sq^2z=0$
for all $z\in H^{n-1}(M_\theta , {\mathbb Z}_2)$. On the other hand, one can verify that $[M_\theta , z]\in
\widetilde {\Omega}_{2n}^{fr}(K_{n-1})$ is a generator if $z\in H^{n-1}(M_\theta , {\mathbb Z}_2)$ is nonzero.
This proves (i).
If $n=0(mod 4)$, by \cite{kerv1} there is an element $\beta \in kerS_* : \pi
_{n}(SO(n))\to \pi _n(SO)$ so that $\pi _*(\beta )$ is nonzero. This proves (iii).
If $n$ is odd, by Lemma \ref{T:1.6} the homomorphism
$$
\begin{array}{ccc}
q(Sq^1): \widetilde {\Omega }_{2n}^{fr}(K_{n-1}) \to {\mathbb Z}_2\\ \ [M,
x] \to q_M(Sq^1x)
\end{array}
$$ is an isomorphism. Thus there is a $\delta \in {\mathbb Z}_2$ so that
$\delta q_M(Sq^1x)=x\cup Sq^2x[M]$ for all $[M, x]$. In
particular, if $x$ can be lifted to the ${\mathbb Z}_4$-coefficient class
with order $4$, $Sq^1x=0$ and so $x\cup Sq^2x=0$. This completes
the proof.
\end{proof}
Now we are ready to prove Theorem 1.9.
\vskip 2mm
\begin{proof}[Proof of Theorem \ref{T:0.13}]
By Theorem 1.6 the data of invariants are homotopy invariants of the manifolds.
Thus the homotopy and homeomorphism classification of such manifolds are the same.
There is an
isomorphism
$$\widetilde{\Omega} _{2n}^{fr}(K(H, n-1))\cong \pi _{2n}^s(K(H, n-1)).$$
Therefore from Theorem 3.1 there is a reduced framed bordism class $[M,f]\in \Omega _{2n}^{fr}(K(H, n-1))$
corresponding to the given algebraic data $[H,\mu , \phi ]$ (resp. $[H, \mu , \phi ,\omega ]$ ) if
$n$ is even (resp. odd). This together with Lemmas 5.1 and 5.2 implies this is an 1-1 correspondence.
Add some $S^{n-1}\times S^{n+1}$ to $M$ if necessary so that $f_*: H_{n-1}(M)\to H$ is surjective. By surgery
on $M$ we may assume further that $f_*: H_{n-1}(M)\to H$ is an isomorphism and $H_{n}(M,{\mathbb Q})=0$. Therefore
the data can be realized by a $(n-2)$-connected $2n$-dimensional $\pi$-manifold,
$M$, so that $H_n(M,{\mathbb Q})=0$ and $\pi (M)=[H, \mu ,\phi ]$ (resp.
$[H, \mu ,\phi ,\omega ]$.
Now it suffices to prove that the map $\pi$ is injective.
Suppose that $M_i$, $i=1,2$, are two framed smooth manifolds with
the same data (for TOP manifold, the similar argument works
identically). Note that the Kervaire invariants of $M_i$ must
vanish since $H_n(M_i,{\mathbb Q})=0$. By the assumption there are maps
$f_i: M_i\to K(H, n-1)$, so that $(M_1, f_1)$ and $(M_2, f_2)$ are
reduced framed bordant, where $f_i$ induces an isomorphism on the
$(n-1)$-th homology groups. Since both $M_i$ framed cobordant to
some homtotopy spheres, there is a framed homotopy sphere,
$\Sigma$, so that $(M_1, f_1)$ and $(M_2\# \Sigma , f_2)$ are
framed bordant. By Freedman \cite{freed} or Kreck \cite{kreck} it
follows that $M_1$ and $M_2\# \Sigma $ are diffeomorphic since
$H_n(M_i, {\mathbb Q})=0$. Therefore $M_1$ and $M_2$ are almost
diffeomorphic. The same argument as above applies to show that
$M_1$ and $M_2$ are homeomorphic to each other. This completes the
proof.
\end{proof}
\vskip 4mm
{\bf Acknowledgements:} This paper is a revised version of
\cite{ff}. This work began during first author's studys and visits
at Jilin University, Nankai Institute of Mathematics,
Universit\"at Mainz, Universit\"at Bielefeld, the
Max-Planck-Institut f\"ur Mathematik and I.H.E.S. He would like to
express his sincere thanks to all of those Institutions and
to Yifeng Sun and Xueguang Zhou for their encouragements and supports,
to Matthias Kreck for teaching him his surgery theory \cite{kreck}. The second author joins the project
at the later part of
the work mainly for clarifying the argument. Part of the work was
done during his visit to Korea University. He would like to thank
Prof.Woo Mooha and Department of Mathematics Education for the
hospitality.
\vskip 5mm
\end{document} | math | 58,364 |
\begin{document}
\title{Secure quantum channels with correlated twin laser beams}
\author{Constantin V. Usenko\dag\ and Vladyslav C. Usenko\ddag
\footnote[3]{To whom correspondence should be addressed ([email protected])} }
\address{\dag\ National Shevchenko University of Kyiv, Department of Theoretical Physics, Kyiv, Ukraine}
\address{\ddag\ Institute of Physics of National Academy of Science, Kyiv, Ukraine}
\begin{abstract}
This work is the development and analysis of the recently proposed quantum cryptographic protocol,
based on the use of the two-mode coherently correlated states. The protocol is supplied with the
cryptographic control procedures. The quantum noise influence on the channel error properties is examined.
State detection features are proposed.
\end{abstract}
\pacs{03.67.Dd, 03.67.Hk, 42.50.Ar,42.50.Dv}
\section{Introduction.}
The goals of quantum cryptography and secure quantum communications
\cite{qc1, qc2, qc3} can be achieved using various protocols, which were
developed and realized \cite{entprot, fourexp} in the past years on the basis
of the quantum entanglement \cite{ent1, entprot}
of weak beams and the single \cite{single1, single2} or few photon states \cite{four},
mostly by means of adjusting and detecting their polarization angles \cite{polar}.
Another method, based on the usage of the two-mode coherently correlated (TMCC) beams
was proposed recently \cite{tmcc}. In this case the secure cryptographic key is generated
by the laser shot noise and duplicated through the quantum channel. Unlike the single or few photon schemes,
which require large numbers of transmission reiterations to obtain the statistically significant
results, the TMCC beam can be intensive enough to make each single measurement
statistically significant and thus to use single impulse for each
piece of information, and remain cryptographically steady.
In this work we analyse the error properties of the secure quantum channels, based on the
TMCC-beams and propose some additions to the TMCC-based cryptographic protocol.
The two-mode coherently correlated state is the way we refer to the
generalized coherent state in the meaning by Perelomov \cite{per}. Such state can
be described by its presentation through series by Fock states:
\begin{equation}
\label{eq:tmcc}
\left| \lambda \right\rangle =\frac{1}{\sqrt{I_0\left(2\left|\lambda\right|\right)}}\sum_{n=0}^\infty {\frac{\lambda ^n}{n!}\left| {nn}
\right\rangle }
\end{equation}
Here we use the designation $\left| {nn} \right\rangle = \left| n
\right\rangle _1 \otimes \left| n \right\rangle _2 $, where $\left| n
\right\rangle _1 $ and $\left| n \right\rangle _2 $stand for the states of
the $1^{st}$ and $2^{nd}$ modes accordingly, represented by their photon
numbers. The states (\ref{eq:tmcc}) are not the eigenstates for each of the operators
separately, but are the eigenstates for the product of annihilation
operators:
\begin{equation}
a_1 a_2 \left| \lambda \right\rangle = \lambda \left| \lambda
\right\rangle .
\end{equation}
Such states can also be obtained from the zero state:
\begin{equation}
\label{eq:tmccground}
\left| \lambda \right\rangle = \frac{1}{\sqrt{I_0\left(2\left|\lambda\right|\right)}}I_0 (\lambda a_1^ + a_2^ + )\left| 0
\right\rangle
\end{equation}
In this work we assume that two laser beams, which are
propagating independently from each other, correspond to the two modes of
the TMCC state. States of beams are mutually correlated (surely, the TMCC
state can also be represented in another way, for example, as a beam
consisting of two correlated polarizations).
\section{Beam measurement}
Let's examine any of the two TMCC beams separately.
The intensity of the beam's radiation, registered by an observer is proportional to the
mean of the $N = a^ + a$ operator, which is the number of the photons
in the corresponding mode.
The mean observable values, which characterize the results of the
measurements of the beam are:
\begin{equation}
\label{eq:meann}
\left\langle {N } \right\rangle = \left\langle \lambda \right|a^
+ a \left| \lambda \right\rangle , \left\langle {N^2 } \right\rangle =
\left\langle \lambda \right|a^ + a a^ + a \left|
\lambda \right\rangle
\end{equation}
These characteristics are squared in field, and thus their mean values don't turn to
zero (this fact is not specific for the TMCC-states, because the usual non-correlated
states and processes, like the heat propagation, show the same properties).
Assuming the state expression (\ref{eq:tmcc}) we obtain
\begin{equation}
\label{eq:meann2}
\langle {N } \rangle = \frac{1}
{I_0 ( 2| \lambda | )}\sum_{n=0}^\infty n \frac{| \lambda |^{2n}}{n!^2} , \langle {N^2 } \rangle = \frac{1}
{I_0 ( 2| \lambda | )}\sum_{n=0}^\infty n^2 \frac{| \lambda |^{2n}}{n!^2}
\end{equation}
The mean number of registered photons is
\begin{equation}
\langle N \rangle = \sum_{n=0}^\infty nP_n (\lambda)
\end{equation}
The probability of registering n photons depends on the intensity of a beam:
\begin{equation}
\label{eq:nphotprob}
P_n(\lambda) = \frac{1}{I_0 (2|\lambda|)}\frac{|\lambda|^{2n}}{n!^2}
\end{equation}
An important feature of this distribution is the quick (proportional to $n!^2$) decreasing dependence of the registration probability
on the photon number.
This circumstance makes the experimental identification of the
TMCC-states quite convenient. The distribution of the probability of different photon
numbers registration along with the analogous distribution for a usual coherent beam
are given at the \fref{plot_pn}. One can see that there are significant differences
for the TMCC and the Poisson beam distributions - the TMCC-beam distribution is relatively
sharp and narrow.
\begin{figure}
\caption{ Probability of different photon numbers registration distribution for the TMCC-beam (circles, solid line) and the analogous distribution for a usual coherent beam (squares, dotted line)}
\label{plot_pn}
\end{figure}
Taking into account (\ref{eq:nphotprob}) the expressions for the mean and mean
square values of the registered photon numbers (\ref{eq:meann2}) turn to:
\begin{equation}
\label{eq:meann3}
\langle N \rangle = \frac{|\lambda|^2 I_1(2|\lambda|)}{I_0 (2|\lambda|)} ,
\left\langle {N^2 } \right\rangle = \left| \lambda \right|^2
\end{equation}
The measurements have the statistical uncertainty, caused by quantum
fluctuations. This uncertainty can be characterized
by the corresponding dispersion:
\begin{equation}
\sigma ^2 = \langle {N^2} \rangle - \langle
{N} \rangle ^2
\end{equation}
Taking into account (\ref{eq:meann3}) we get the following
expression:
\begin{equation}
\label{eq:sigma}
\sigma ^2 = \left| \lambda \right|^2\left( {1 - \left( {\frac{I_1
\left( {2\left| \lambda \right|} \right)}{I_0 \left( {2\left| \lambda
\right|} \right)}} \right)^2} \right)
\end{equation}
The dependencies of the measurement results uncertainty on the mean photon number for
the TMCC-beam and a usual correlated beam are given at \fref{plot_dispers}.
The difference between these dependencies can also be used for the TMCC-states identification.
\begin{figure}
\caption{The dependencies of the measurement results uncertainty on the mean photon number for the TMCC-beam (solid line) and a usual coherent beam (dotted line)}
\label{plot_dispers}
\end{figure}
\section{Communication via quantum channel}
Let we have to establish a secure quantum channel between two parties (\fref{scheme1}). Alice has the laser on her side, which produces two beams in the TMCC
state. The optical channel is organized in such a way, that Alice receives one
of the modes, the first, for example, i.e. $\varphi _A \equiv \varphi _1
$,$\varphi _A (x_A ,t_0 ) = 1$ , and Bob receives another one, i.e. $\varphi _B
\equiv \varphi _2 $ ,$\varphi _B (x_B ,t_0 ) = 1$ at any moment of
measurement $t_0 $, where $x_A $and $x_B $are Alice's and Bob's locations
respectively. Accordingly, Alice cannot measure the Bob's beam and vice
versa:$\varphi _B (x_A ,t_0 ) = 0$, $\varphi _A (x_B ,t_0 ) = 0$. At that
the vector-potential of the TMCC-beam is:
\begin{equation}
A = \varphi _A^\ast (x,t)a_A^ + + \varphi _A (x,t)a_A + \varphi _B^\ast
(x,t)a_B^ + + \varphi _B (x,t)a_B
\end{equation}
Unlike the usual non-correlated coherent states, which
show their quasiclassical properties in the fact, that the mean value of a vector-potential of
a corresponding beam is not equal to 0, the mean value of a vector-potential of a TMCC-beam
and any other characteristic, which is linear in field, turns to be equal to 0, because during
the averaging by the
1$^{st}$ mode, for example, the $a_1$ converts $\left| {n,n} \right\rangle $ to $\left|
{n - 1,n} \right\rangle $, which is orthogonal to all the present state
terms, so $\left\langle {\lambda _i } \right|a_i \left| {\lambda _i }
\right\rangle = 0$. So the quasiclassical properties in their usual meaning are absent in
the case of a TMCC-beam. But they become apparent in the non-zero value of the spatial correlation function,
which characterizes the interdependence of the results of measurements
taken by Alice and Bob:
\begin{equation}
g_{AB} = < N_A N_B > - < N_A > < N_B >
\end{equation}
\begin{figure}
\caption{Quantum channel between two parties with a TMCC source}
\label{scheme1}
\end{figure}
It's useful to describe the channel quality by the relative correlation,
which is
\begin{equation}
\rho _{AB} = \frac{ < N_A N_B > - < N_A > < N_B > }{\sigma _A \sigma _B }
\end{equation}
The main feature of the TMCC state is that the value $\rho _{AB} $ is
exactly equal to 1, while in the case of non-correlated beams we would get
$\rho _{AB} = 0$. This means that the measurements of the photon numbers,
obtained by Alice and Bob, each with her/his own detector, not only show the same
mean values, but
even have the same deflection from the mean values.
The laser beam is the semi-classical radiation with well defined phase, but
due to the uncertainty principle for the number of photons and the
phase of the radiation, there is a large enough uncertainty in the photon
numbers, this can be seen from the dispersion expression (\ref{eq:sigma}). Thus one can
observe the noise, which is similar to the shot noise in an electron tube.
In the TMCC radiation the characteristics of such noise for each of the
modes are amazingly well correlated to each other. This fact
enables the use of such radiation for generation of a random code, which
will be equally good received by two mutually remote detectors.
\subsection{The protocol}
The following scheme can be used for the TMCC-based protocol. The laser is
set up to produce the constant mean number of photons during the session and
both parties know this number. At some moment Alice and Bob start the
measurements. They detect the number of photons at unit time by measuring
the integrated intensity of the corresponding incoming beam. If the number
of photons for the specific unit time is larger than the known expected mean
(which is due to the shot noise), the next bit of the generated code is
considered to have the value ``1''. If the measured number is less than
the expected mean, the next bit is considered to be equal to ``0'':
\begin{equation}
\label{protocol}
B ={\left\{
\begin{array}{l}
\{n \leq [<N>]\} \rightarrow 0 \\ \{n > [<N>]\} \rightarrow 1
\end{array}\right.}
\end{equation}
Upon the receipt of a sufficient number of bits (the code), both Bob and Alice
divide them in half, each obtaining two bit sequences (half-codes). Bob encodes one
sequence with another, using the "eXclusive OR" logical operation (XOR, $B_i\bigoplus B_j$),
and sends this encoded half-code to Alice using any public channel.
Alice uses any of her half-codes to
decode the code she has got from Bob using the same XOR operation. She compares
the result of this operation with another of her half-codes. If
all the bits coincide, this means that Alice and Bob both have the same
code, which can be used as a cryptographic key for encoding their communication.
Otherwise they have to repeat the key generation and transfer procedure and check
the channel for the possible eavesdropping if the procedure fails again.
The stability of the protocol against the basic beam splitting attacks was
examined in \cite{tmcc} and it was shown that any successful
attempt destroys the channel and cancels the key distribution session.
Besides the basic listening-in, Eve may carry out a more advanced
state cloning eavesdropping attack by detecting the overall photon
number in the Bob's mode for each upcoming bit and then re-emiting the
same number again, which requires Eve to have the same laser source on her site.
Though, due to the laser shot noise Eve can't be sure if she is
producing exactly the same number of photons, she can set up her
laser just to produce some mean photon number. In case she will
adjust her laser to produce the mean photon number, which is equal to
the current photon number measured in the Bob's mode for a next
bit, she may probably be successfull by repeating some bits. But in
this case she will change the Bob's measurement results ditribution.
This can be checked by expanding a protocol with a post-measurement
analysis of the measurement results, which can be done by both of
the trusted parties. It should consist of the comparison of an
obtained photon numbers distribution by the frequencies of their
detection with the expected one for the known constant mean photon number
value which is actually a task of comparing two numerical arrays
The difference between expected and obtained distributions will
reveal a state cloning attempt.
\subsection{Quantum channel error analysis}
Let the parties of the secret key transmission procedure are using the protocol
described above, thus they estimate the value of the next bit by comparing the
actual registered photon number to the average. The probability of detecting "0"
bit value then is
\begin{equation}
P_{(0)} (\lambda) = \sum_{n=0}^{[\langle N \rangle]}P_n(\lambda)
\end{equation}
The noise is present in the channel and it may increase the number of the registered
photons. We suppose that the noise is thermal and assume that
it may, with some probability, cause an appearance of one and no more than one
additional photon in any of the modes during the time of a bit detection.
We will denote the probability of a noise photon detection as $\epsilon$
and refer to it as the noise factor.
We suppose that the channel is qualitative enough to transfer the impulse at the
required distance without losing any single photon, thus errors are possible only
due to the appearance of the noise photons.
An error, when Alice registers "0" bit value and Bob registers "1" may occur upon
the joint realization of two events. The first is that Alice detects the maximum possible
number of these, corresponding to the "0" bit value, which is, according to the proposed protocol,
equal to the integer part of $\langle n \rangle$. The second is that in addition to this number Bob
detects the appearance of a noise photon. The opposite situation, when Alice gets the
"1" bit value and Bob registers "0" is possible when the noise photon was detected
by Alice, the probability of such error is the same.
The probability of the realization of a state, which consists of the maximum possible
for the "0" bit value photons and, at the same time, is detected as "0", is the
relation between the corresponding probabilities:
\begin{equation}
P_{max(0)} = \frac{P_{[\langle N \rangle]}}{P_{(0)}} = \frac{P_{[\langle N \rangle]}}{\sum_{n=0}^{[\langle N \rangle]}P_n(\lambda)}
\end{equation}
We will refer to this probability as to the error factor.
So the probability of an error during the bit registration is equal to the product
of the noise and error factors:
\begin{equation}
P_{err(0)}(\lambda)=\epsilon P_{max(0)}
\end{equation}
One can easily see that upon the intensity increase the error factor becomes less and
so the channel tends to a self-correction if the beam gets more intensive.
\section{Conclusions}
Correlated coherent states of the two-mode laser beam (TMCC states) show
interesting properties, which can be used, in particular, for the tasks of
the quantum communication and cryptography.
The TMCC-beams can be identified due to the special form of the registration
probabilities distribution for different photon numbers in the corresponding beam
and the dependence of the dispersion on the mean photon numbers value.
On the one hand, each of the
modes looks like a flow of the independent photons rather then a coherent
beam, since mean values of the operators, which are linear in field, are
equal to 0 for each mode separately.
On the other hand, the strong
correlation between the results of measurements for each of the modes takes
place. This correlation shows itself in the fact that in each of the modes numbers
of photons are the same and even the shot noise shows itself equally in the both
modes. This enables the use of the TMCC state as the generator and carrier
of random keys in a quantum channel which is stable against the eavesdropping \cite{tmcc}.
Thus, the TMCC-laser generates and transmits exactly the 2 copies of a
random key. Unlike the single or two-photon schemes, which require large numbers of
transmission reiterations to obtain the statistically significant
results, the TMCC beam can be intensive enough to make each single measurement statistically
significant and thus to use single impulse for each
piece of information, and remain cryptographically steady. This allows to essentially increase
the effective data transfer rate and distance. Analysis of the noise influence on
the channel properties shows that the channel tends to a self-correction upon the beam intensity increase.
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\end{document} | math | 20,047 |
{\underline b}egin{document}
\title{Iterated Functions Systems, Blenders and Parablenders}
{\underline a}uthor{Pierre Berger, \; Sylvain Crovisier, Enrique Pujals\thanks{This work is partially supported by the project BRNUH of Sorbonne Paris Cit\'e University and the French-Brazilian network.}}
\date{May 30, 2016}
\maketitle
{\underline b}egin{abstract}
We recast the notion of parablender introduced in~\cite{BE15} as a parametric IFS. This is done using the concept of open covering property and looking to parametric IFS as systems acting on jets.
\end{abstract}
Given a contractive IFS on $\mathbb R^n$, it is well known that for any point $x$
in the limit set and any admissible backward itinerary $\underline x$, it is possible to
consider its continuation for any IFS nearby.
In particular, if we consider parametric perturbations, then the
continuation of the point (provided an itinerary) is given by a smooth
curve $a\mapsto \underline x(a)$. In this note we study the $r$-jets of such a continuation,
(i.e. the $r$ first derivatives at the zero parameter).
As it is explained in section~\ref{s.paraIFS} (see also proposition~\ref{p.paraIFS})
the $r$-jet can be viewed as a point in the limit set of a new multidimensional
contractive IFS acting on $\mathbb R^{n.(r+1)}$ .
For certain type of parametric IFS as the one introduced in~\cite{BE15} (as a
matter of fact in a more general setting), a further property holds:
\emph{ the $r$-jet
of any $C^r-$curve (with appropriate bounds in the derivative),
coincide with the $r$-jets of the continuation of some points of the IFS}.
We show that this property can be recast
as saying that the limit set of the IFS acting on the jets has
interior. As we want that this property remains valid for nearby
parametric families, we consider below IFS such that their actions on
$r$-jets exhibits a \emph{covering property} (see definition~\ref{d.covering} in section~\ref{s.IFS}).
In section~\ref{section3} we give an example of such a contractive IFS. The
example may seem extremely restrictive but as it is shown in theorem A
their action on $r$-jets has the covering property.
In sections~\ref{s.blender} and~\ref{s.parablender},
we recall that the covering property for IFS is related to the notion of blender
for hyperbolic sets. Our purpose is to explain how the present approach for IFS can be use
to revisit the notion of parablender that was introduced in~\cite{BE15}.
{\mathfrak e}ction{Iterated Functions Systems}\label{s.IFS}
{\underline b}egin{defi}
A (contracting) \emph{Iterated Functions System (IFS)} is the data of a finite family $(f_{\mathfrak b})_{{\mathfrak b}\in {\mathfrak B}}$ of contracting maps on $\mathbb R^n$.
The IFS is of class $C^r$, $r\ge 1$, if each $f_{\mathfrak b}$ is of class $C^r$.
\end{defi}
The topology on the set of IFS of class $C^r$ (with $\operatorname{Card}\, {\mathfrak B} $ elements) is given by the product strong topology $\prod_{\mathfrak B} C^r(\mathbb R^n,\mathbb R^n)$.
The limit set of an IFS is:
\[\Lambda:= \{x\in \mathbb R^n: \exists ({\mathfrak b}_i)_i\in {\mathfrak B}^{\mathbb Z^-}, \; x= \lim_{k\to +\infty} f_{{\mathfrak b}_{-1}}\circ \cdots \circ f_{{\mathfrak b}_{-k}} (0)\}.\]
The limit set $\Lambda$ is compact. One is usually interested in its geometry. Natural questions are:
{\underline b}egin{question}
Under which condition the limit set $\Lambda$ has non-empty interior?
Under which condition the limit set has $C^r$-robustly non empty interior ?
\end{question}
Let us recall that a system satisfies a property \emph{$C^r$-robustly} if the property holds also for any $C^r$-perturbations of the system.
Both questions are still open, although there are already partial answers to them. Let us state a classical sufficient property:
{\underline b}egin{defi}\label{d.covering} The IFS $(f_{\mathfrak b})_{{\mathfrak b}\in {\mathfrak B}}$ satisfies the \emph{covering property} if there exists a non-empty open set $U$ of $\mathbb R^n$ such that:
\[\operatorname{Closure}(U)\subset {\underline b}igcup_{{\mathfrak b}\in {\mathfrak B}} f_{\mathfrak b}(U).\]
\end{defi}
{\underline b}egin{exam} For $\lambda \in (1/2,1)$, the IFS spanned by the two following one-dimensional maps
\[f_{1}\colon x\mapsto \lambda x+1,\]
\[f_{-1} \colon x\mapsto \lambda x-1,\]
satisfies the covering property since
$[-2,2]\subset \phi_{1}((-2,2)) \cup \phi_{-1}((-2,2))$.
\end{exam}
One easily proves the following:
{\underline b}egin{prop}
If the IFS $ (f_{\mathfrak b})_{{\mathfrak b}\in {\mathfrak B}}$ satisfies the covering property with the open set $U$, then the limit set of the IFS contains $C^1$-robustly $U$.
\end{prop}
Hence the covering property is a sufficient condition for an IFS to have $C^r$-robustly non-empty interior, for every $r\ge 1$.
{\underline b}egin{question}\label{quesIFS}
Is the covering property a necessary condition to have
$C^r$-robustly non-empty interior?
\end{question}
The answer to this question is not known even when $n=1$. It is not clear to us that the answer would be independent of $r$. Indeed there are phenomena which occur for $r>1$ and not for $r=1$, such as the stable intersection of regular cantor set \cite{GY01, Ne79, Gu11}.
Nevertheless the case $n=1$ and $Card\; {\mathfrak B}=2$ is simple.
Indeed consider the IFS generated by two contracting maps $f_1,f_2$ of $\mathbb R$. Let $I$ be the convex hull of $\Lambda$: the endpoints of $I$ are the fixed points of $f_1$ and $f_2$.
If the interiors of $f_1(I)$ and $f_2(I)$ are disjoint, then a small perturbation makes $f_1(I)$ and $f_2(I)$ disjoint. Then it is easy to see that the IFS has empty interior. Otherwise the interiors of $f_1(I)$ and $f_2(I)$ are not disjoint. They cannot coincide since the interior of $I$ is non empty and $f_1,f_2$ are contracting.
By removing the $\epsilon$-neighborhood of the endpoints of $I$,
one gets an interval $I_\epsilon$ which is covered by the images by $f_1$ and $f_2$ of its interior.
Consequently, if $\Lambda$ has robustly non empty interior it must satisfy the covering property.
{\mathfrak e}ction{IFS with parameters}\label{s.paraIFS}
We denote $\mathbb I^k= [-1,1]^k$.
One considers $C^r$ parametrized families, i.e. elements in the
the Banach space $C^r(\mathbb I^k\times \mathbb R^n,\mathbb R^n)$, that we denote
$(f_a)_{a\in \mathbb I^k}$.
For each parameter $a_0\in \mathbb I^k$, one introduces the jet space
$J^r_{a_0}(\mathbb I^k,\mathbb R^n)$, whose elements are the Taylor series
$(x_a, \partial_ax_a, \dots, \partial_a^rx_a)_{|a=a_0}$ at $a=a_0$ of $C^r$ functions $a\mapsto x_a$
in $C^r(\mathbb I^k,\mathbb R^n)$. Each $C^r$ family of maps $(f_a)_{a\in \mathbb I^k}\in C^r(\mathbb I^k\times \mathbb R^n,\mathbb R^n)$ acts on the space of jets as a map $\widehat f$ defined by:
\[
\widehat f\; \colon \;
(x_a, \partial_ax_a, \dots, \partial_a^rx_a)_{|a=a_0}\mapsto
(f_a(x_a),\partial_a(f_a(x_a)),\dots,\partial^r_a (f_a(x_a)))_{|a=a_0}.\]
Our goal is to study parametrized IFS:
{\underline b}egin{defi}
An \emph{Iterated Functions System (IFS) with parameter} is the data of a finite families $(f_{{\mathfrak b}, a})_{{\mathfrak b}\in {\mathfrak B}}$ of contracting maps on $\mathbb R^n$ depending on a parameter $a\in \mathbb I^k$.
The IFS with parameter is of class $C^r$, $r\ge 1$, if each $(f_{{\mathfrak b}, a})_{a\in \mathbb I^k}$ is in red $C^r(\mathbb I^k\times \mathbb R^n,\mathbb R^n)$.
\end{defi}
For every $a\in \mathbb I^k$, we consider the limit set $\Lambda_a$ associated to the system $(f_{{\mathfrak b},a})_{{\mathfrak b}\in {\mathfrak B}}$ equal to
the set of points $X_a(\underline {\mathfrak b}):= \lim_{k\to +\infty} f_{{\mathfrak b}_{-1}, a}\circ \cdots \circ f_{{\mathfrak b}_{-k}, a} (0)$, among all $\underline {\mathfrak b}= ({\mathfrak b}_i)_i\in {\mathfrak B}^{\mathbb Z^-}$. These points admit a continuation
when $a$ varies: each function
$a\mapsto X_a(\underline {\mathfrak b})$ is of class $C^r$.
One can consider its jet:
$$J^r_{a_0}X(\underline {\mathfrak b}):=(X_a(\underline {\mathfrak b}),\partial_a X_a(\underline {\mathfrak b}),\dots,
\partial^r_a X_a(\underline {\mathfrak b}))_{|a=a_0}.$$
Let us observe that the image of $X_a(\underline {\mathfrak b})$ by $f_{{\mathfrak b}, a}$ is the continuation of $X_a({\mathfrak b}\underline {\mathfrak b})$ (where ${\mathfrak b}\underline {\mathfrak b}$ means a new sequence such that the first element now is $b$) i.e.: $f_{{\mathfrak b}, a}(X_a(\underline {\mathfrak b}))=X_a({\mathfrak b}\underline {\mathfrak b}).$ In particular, its partial derivatives respect to the parameter at $a=a_0$ is nothing else that the image of $\widehat f_{{\mathfrak b}}$ on the jets of $X_a(\underline {\mathfrak b})$; therefore, the jets of the continuations are the limit set of $(\widehat f_{{\mathfrak b}})_{{\mathfrak b}\in {\mathfrak B}}$. More precisely:
{\underline b}egin{prop}\label{p.paraIFS}
The set $J^r_{a_0}\Lambda$ is the limit set of the IFS $(\widehat f_{{\mathfrak b}})_{{\mathfrak b}\in {\mathfrak B}}$
acting on the $r$-jet space $J^r_{a_0}\mathbb R^n$ which is generated by the finite collection of maps
$\{(f_{{\mathfrak b}, a})_{a\in \mathbb I^k},\,{\mathfrak b}\in {\mathfrak B}\}$ .
\end{prop}
It is natural to wonder if the following set has (robustly) non-empty interior:
\[J^r_{a_0}\Lambda := \left\{ J^r_{a_0}X(\underline {\mathfrak b})\;:\; \underline {\mathfrak b} \in {\mathfrak B}^{\mathbb Z^-}\right\}\; .\]
We notice that if $J^r_0\Lambda$ has non empty interior, then there exists a non-empty open subset $U\subset C^r(\mathbb I^k, \mathbb R^n)$ such that for every $(x_a)_{a\in \mathbb I^k}\in U$ there is $\underline {\mathfrak b}\in {\mathfrak B}^{\mathbb Z^-}$ satisfying:
\[ x_a= X_a(\underline {\mathfrak b})+o(\|a\|^r)\; .\]
{\mathfrak e}ction{The covering property for an affine IFS acting on the jet space} \label{section3}
In this note we study a simple IFS with parameter. We set $k=n=1$
and choose $r\geq 1$.
Let:
\[f_{+1, a}\colon x\mapsto \lambda_a x+1,\]
\[f_{-1, a} \colon x\mapsto \lambda_a x-1,\]
where $(\lambda_a)_{a\in\mathbb I}\in C^r(\mathbb I, (-1,1))$
satisfies $(\partial_a \lambda_a)_{|a=0} {\underline n}ot= 0$.
{\underline b}egin{rema}
Any system in an open and dense set of
parametrized IFS generated by a pair of contracting affine maps with the same contraction,
can be conjugated to a system which coincides with $(f_{+1,a}, f_{-1,a})$ for $a$ close to $0$.
\end{rema}
As $k=1$, after coordinate change on the parameter space, we can also assume that $\lambda_a= \lambda+a$ for $a$ in a neighborhood of $0$. Note that the maps induced on $r$-jet space $J^r_0(\mathbb I,\mathbb R)$ are now:
\[\widehat f_{+1}\colon (x_a, \partial_ax_a, \dots, \partial_a^r x_a)_{|a=0}
\mapsto (\lambda x_a +1, \lambda \partial_a x_a + x_a ,
\dots,
\lambda \partial_a^r x_a + r \partial_a^{r-1} x_a)_{|a=0}.\]
\[\widehat f_{-1}\colon (x_a, \partial_ax_a, \dots, \partial_a^r x_a)_{|a=0}
\mapsto (\lambda x_a-1, \lambda \partial_a x_a + x_a ,
\dots,
\lambda \partial_a^r x_a + r \partial_a^{r-1} x_a)_{|a=0}.\]
In \cite{HS2014,HS2015}, its is proved that the IFS generated by $(\widehat f_{+ 1}, \widehat f_{-1})$ has non empty interior. Let us adapt their proof to obtain the following stronger result.
{\underline b}egin{theo}\label{paraIFS}
For any $r\geq1$, if $\lambda\in (0,1)$ is close enough to $1$,
then the IFS generated by $(\widehat f_{+ 1}, \widehat f_{-1})$
acting on the $r$-jet space $J^r_0(\mathbb I,\mathbb R)$
satisfies the open covering property.
\end{theo}
{\underline b}egin{cor}
Any IFS with parameter generated by two families of maps $C^r$-close to $(f_{+1,a})_a$ and
$(f_{-1,a})_a$ induces an IFS on the $r$-jet space $J^r_0(\mathbb I,\mathbb R)$
whose limit set has non-empty interior.
\end{cor}
{\underline b}egin{proof}[Proof of theorem~\ref{paraIFS}]
Let us remark that the IFS generated by $(\widehat f_{+ 1}, \widehat f_{-1})$ is conjugated (via affine coordinates change) to the IFS on $\mathbb R^{N}$, $N=r+1$,
generated by the maps
$$F_{+1}\colon X\mapsto J X+ T,\quad F_{-1}\colon X\mapsto J X- T,$$
where $T= (0,\dots, 0,1)$ and
$$J={\underline b}egin{pmatrix}
\lambda & {N-1} & & & 0\\
& \lambda & \dots & & \\
& & \dots & \dots &\\
& & & \lambda & 1\\
0 & & & & \lambda
\end{pmatrix}.$$
One introduces a polynomial $P(x)=b_nx^n+b_{n-1}x^{n-1}+\dots+b_0$ with large degree $n$ which satisfies:
{\underline b}egin{enumerate}
\item[i.] $b_0{\underline n}eq 0$, $b_n=1$,
\item[ii.] $\sum_{j=0}^{n-1}|b_j|<2,$
\item[iii.] $P^{(i)}(1/\lambda)=0$ for $0\leq i\leq N-1$ (where $P^{(i)}(x)$ denotes the $i^\text{th}$
deviated polynomial of $P$),
\item[iv.] $P$ induces a projection $\pi\colon \mathbb R^n\to \mathbb R^N$ with rank $N$, defined by
$$\pi(u_{-n},\dots,u_{-1})={\underline b}igg(\sum_{k=0}^{n-1}u_{k-n}.B_k^{(N-i)}(\lambda){\underline b}igg)_{1\leq i\leq N}\quad \text{with} \quad
B_k(x)=\sum_{j=0}^kb_jx^{k-j}.$$
\end{enumerate}
{\underline b}egin{prop}
For any $N\geq 1$, if $\lambda\in (0,1)$ is close enough to $1$,
there exists a polynomial $P$ satisfying conditions (i)--(iv).
\end{prop}
{\underline b}egin{proof}
From \cite[Theorem 3.4]{HS2014}, there exists a monic polynomial
$Q(x)=x^n+a_{n-1}x^{n-1}+\dots+a_0$ such that $\sum_{i=0}^{n-1}|a_i|<2$
and $(x-1)^N|Q(x)$. Dividing by some $x^k$, one can assumes that $a_0{\underline n}eq 0$.
One then sets $P(x)=\lambda^{-n}Q(\lambda\cdot x)$. Provided $\lambda$ is close enough to $1$,
it satisfies the conditions (i)--(iii). In order to check the last item, it is enough to check that
the following matrix has rank $N$:
$${\underline b}egin{pmatrix}
B_0 & B_1 & \cdots & B_{n-1}\\
B^{(1)}_0 & B^{(1)}_1 & \cdots & B^{(1)}_{n-1}\\
\vdots & \vdots & \ddots & \vdots &\\
B^{(N-1)}_0 & B^{(N-1)}_1 & \cdots & B^{(N-1)}_{n-1}
\end{pmatrix}.$$
This can be easily deduced from the fact that $B_0,B_1^{(1)},\dots,B_{N-1}^{(N-1)}$
are constant and non-zero polynomials and that $B_k^{(i)}=0$ when $k<i$.
\end{proof}
Let $S_{-1},S_1$ be the linear automorphisms of $\mathbb R^n$ defined for $\delta\in \{+1,-1\}$ by:
$$S_{\delta}\colon (u_{-n+1},\dots,u_{0})\mapsto (u_{-n},\dots,u_{-1})\quad \text{with} \quad
u_{-n}=\frac1{b_0} ( \delta-\sum_{j=1}^{n} b_{j} u_{j-n}).$$
{\underline b}egin{prop}\label{p.sc}
$\pi$ is a semi-conjugacy:
$F_\delta\circ \pi=\pi\circ S_{\delta}$.
\end{prop}
Before proving the proposition, one checks easily the following relations.
{\underline b}egin{lemm}\label{l1} If $1\leq k\leq n$ and $i\geq 1$,
$$B_k(x)=x\cdot B_{k-1}(x)+b_k \quad \text{and} \quad
B_k^{(i)}(x)=x\cdot B_{k-1}^{(i)}(x)+i\cdot B_{k-1}^{(i-1)}(x).$$
\end{lemm}
Since $B_n(x)=x^n\cdot P(1/x)$ and $P^{(i)}(1/\lambda)=0$ for $0\leq i\leq N-1$ one gets
{\underline b}egin{lemm}\label{l2} If $0\leq i\leq N-1$,
$$B^{(i)}_n(\lambda)=0.$$
\end{lemm}
{\underline b}egin{proof}[Proof of the Proposition]
One has to check $F_\delta\circ \pi \circ S_{\delta}^{-1}=\pi$.
One fixes $(u_{-n},\dots,u_{-1})$.
It is sent by $S_{\delta}^{-1}$ to $(u_{-n+1},\dots,u_0)$ with
$u_0=\delta-\sum_{j=0}^{n-1}b_ju_{j-n}.$
Then by $\pi$ to
${\underline b}igg(\sum_{k=0}^{n-1}u_{k+1-n}\cdot B_k^{(N-i)}(\lambda){\underline b}igg)_{1\leq i\leq N}$.
Applying $F_{\delta}$, one gets a vector $(v_1,\dots,v_N)$ whose $i^\text{th}$ coordinate coincides with
$$v_i=\lambda \sum_{k=0}^{n-1}u_{k+1-n}\cdot B_k^{(N-i)}(\lambda)+(N-i)\sum_{k=0}^{n-1}u_{k+1-n}\cdot B_k^{(N-i-1)}(\lambda)\quad
\text{if} \quad i{\underline n}eq N,$$
$$v_N=\lambda \sum_{k=0}^{n-1}u_{k+1-n}\cdot B_k(\lambda)+\delta\quad
\text{otherwise}.$$
For the $N-1$ first coordinates, from lemma~\ref{l1} one gets
$$v_i=\lambda\cdot \sum_{k=1}^{n}u_{k-n}\cdot B_{k-1}^{(N-i)}(\lambda)+(N-i)\sum_{k=1}^{n}u_{k-n}\cdot B_{k-1}^{(N-i-1)}(\lambda)=\sum_{k=1}^{n}u_{k-n}\cdot B_{k}^{(N-i)}(\lambda)$$
$$= \sum_{k=0}^{n-1}u_{k-n}\cdot B_{k}^{(N-i)}(\lambda) \quad\quad\quad \quad\quad\quad \text{since $B_0^{(N-i)}(\lambda)=B_n^{(N-i)}(\lambda)=0$}.$$
For the last coordinate, one gets similarly from lemmas~\ref{l1} and~\ref{l2}
$$v_N=\lambda\cdot \sum_{k=1}^{n}u_{k-n}\cdot B_{k-1}(\lambda)+\delta=\lambda\cdot \sum_{k=1}^{n}u_{k-n}\cdot B_{k-1}(\lambda)+\sum_{k=0}^n u_{k-n}\cdot b_k=
\sum_{k=1}^{n}u_{k-n}\cdot B_{k}(\lambda)+u_{-n}\cdot b_0$$
$$= \sum_{k=0}^{n-1}u_{k-n}\cdot B_{k}(\lambda) \quad\quad\quad\quad\quad\quad\quad\quad \text{since $B_0(\lambda)=b_0$ and $B_n(\lambda)=0$}.$$
This gives $(v_1,\dots,v_N)=\pi(u_{-n},\dots,u_{-1})$ as required.
\end{proof}
{\underline b}igskip
Since $\sum_{j=0}^{n-1}|b_j|<2$, one can choose
$\eta>1$ such that ${ \eta^n} \sum_{j=0}^{n-1}|b_j| <\eta+1$ and let $A$ be the image by $\pi$
of
$$\Delta:=(-{\eta^n},{ \eta^n})\times (-{ \eta^{n-1}},{\eta^{n-1}})\times\dots\times(-\eta,\eta).$$
{\underline b}egin{prop} The subset $A$ is open and satisfies: $\operatorname{Closure}(A)\subset F_{-1}(A)\cup F_{1}(A)$.
\end{prop}
{\underline b}egin{proof}
The linear map $\pi$ is open since it has rank $N$.
Since $\pi$ sends compact sets to compact sets, it is enough to prove
$$\pi(\operatorname{Closure}(\Delta))\subset F_{+1}\circ \pi (\Delta)\cup
F_{-1}\circ \pi (\Delta).$$
By proposition~\ref{p.sc}, one has to check the following inclusion:
$$\operatorname{Closure}(\Delta)\subset S_{+1}(\Delta)\cup
S_{-1}(\Delta).$$
Consider any point $(u_{-n},\dots,u_{-1})$ in $\operatorname{Closure}(\Delta)$.
By our choice of $\eta$ and since $|u_j|\le{ \eta^n}$ for each $-n\leq j\leq -1$,
there exists $u_0\in (-\eta,\eta)$ and $\delta\in \{-1,1\}$ satisfying the relation
$u_0=\delta-\sum_{j=0}^{n-1}b_ju_{j-n}$.
{ Since $|u_i|\leq \eta^i$ we get $|u_{i-1}|<\eta^{i}$.}
One deduces that $(u_{-n+1},\dots,u_{-1},u_0)$ belongs to
$\Delta$. Since $b_n=1$,
one has $\sum_{j=0}^{n}b_ju_{j-n}=\delta$, so
$S_{\delta}(u_{-n+1},\dots,u_{-1},u_0)=(u_{-n},\dots,u_{-1})$.
This proves the required inclusion.
\end{proof}
The covering property is thus satisfied and the Theorem is proved.
\end{proof}
{\mathfrak e}ction{Blenders for endomorphisms}\label{s.blender}
Our motivations for studying the action of IFS on jet spaces come from hyperbolic differentiable dynamics, and more specifically from the study of blenders and para-blenders that we explain in these two last sections.
If $f\colon M\to M$ is a $C^1$-map on a manifold $M$,
a compact subset $K\subset M$ is \emph{hyperbolic} if:
{\underline b}egin{itemize}
\item[--] $f$ is a local diffeomorphism on a neighborhood of $K$,
\item[--] $K$ is invariant (i.e. $f(K)= K$),
\item[--] there exists an invariant sub-bundle $E^s\subset TM_{|K}$ and $N\geq1$ so that $\forall x\in K$:
\end{itemize}
\[ D_xf(E_x^s)\subset E^s_{f(x)},\quad \| D_xf^N|E^s_x\| <1,\quad
{\| p_{E^s_{\underline b}ot}\circ (D_xf^N)^{-1}|E^s_{{\underline b}ot,x}\|<1}, \]
where $E^s_{{\underline b}ot,x}$ is the orthogonal complement of $E^s_x$ and $p_{E^s_{\underline b}ot}$ the orthogonal projection onto it.
Note that the map $f$ is in general not invertible.
Hence one can define an unstable space at any $x\in K$,
but it is in general not unique: it depends on the choice of a preorbit of $x$.
We recall that the inverse limit $\overleftarrow K$ is the set of preorbits:
\[\overleftarrow K:= \{(x_i)_{i\le 0}\in K^{\mathbb Z^-} :\; f(x_i)=x_{i+1},\; \forall i<0\}.\]
The map $f$ induces a map $\overleftarrow f\colon (x_i)\mapsto (f(x_{i}))$ on $\overleftarrow K$.
For every preorbit $\underline x =(x_i)_{i\le 0}\in \overleftarrow K$ and for every $\epsilon>0$ small enough, the following set is a submanifold of dimension $\operatorname{Codim}(E^s)$:
\[W^u(\underline x, \epsilon)=\{x'\in M:\exists \underline x'\in \overleftarrow K \text{ s.t. }
x'_0= x',\; \forall i\;d(x'_i ,x_i)<\epsilon \text{ and } \lim_{i\to -\infty} d(x'_i ,x_i)=0\},\]
and is called \emph{local unstable manifold}
(also denoted by $W^u(\underline x, \epsilon,f)$ when one specifies the map $f$).
We recall that $K$ is \emph{inverse-limit stable}: for every $C^1$-perturbation $f'$ of $f$, there exists a unique map $\pi_{f'}\colon \overleftarrow K\to M$ which is $C^0$ close to the zero coordinate projection $\pi \colon (x_i)_i \in \overleftarrow K \mapsto x_0\in M$ so that the following diagram commutes:
\[f'\circ \pi_{f'}=\pi_{f'}\circ \overleftarrow f\; .\]
Moreover $\pi_{f'}(\overleftarrow K)$ is hyperbolic for $f'$ and is called the \emph{hyperbolic continuation of $K$ for $f'$}.
In particular any $\underline x$ has a continuation, that is the sequence $\underline x'=(x'_i)$
in $\pi_{f'}(\overleftarrow K)$ such that
$$x'_i=\pi_{f'}((x_{i+k})_{k\leq i}).$$
The local unstable manifold of $\underline x'$ will be denoted
$W^u(\underline x, \epsilon, f')$. When $\epsilon$ is implicit,
the local unstable manifolds are also denoted by $W^u_{loc}(\underline x)$ and $W^u_{loc}(\underline x, f')$.
The notion of blender was first introduced in the invertible setting by \cite{BD96} to construct robustly transitive diffeomorphisms, and then \cite{BD99, DNP} to construct locally generic diffeomorphism with infinitely many sinks. The work \cite{BE15} deals with blenders for endomorphisms.
{\underline b}egin{defi} A \emph{$C^r$-blender} for a $C^r$-endomorphism is a hyperbolic set $K$ such that the union of its local unstable manifolds has $C^r$-robustly a non-empty interior:
there exists a non-empty open set $U\subset M$ which is contained in the
union of the local unstable manifolds of the hyperbolic continuation of $K$ for any
endomorphism $f'$ $C^r$-close to $f$.
\end{defi}
The classical definition for diffeomorphisms is more general:
fixing an integer $d$ smaller than the stable dimension of $K$,
it asserts that there exists an open collection $U$ of embeddings of the $d$-dimensional disc
in $M$ such that any $D\in U$ intersects the union of the local unstable manifolds of the hyperbolic continuation of $K$ for any diffeomorphism $f'$ $C^r$-close to $f$.
{\underline b}egin{exam}\label{e.blender} For $\lambda \in (1/2,1)$, we consider a local diffeomorphism $f$ of $\mathbb R^2$ whose restriction to
$([-2,-1]\cup[1,2])\times [-1/(1-\lambda),1/(1-\lambda)]$ is:
\[(x,y)\mapsto (4 |x|-6, \lambda y+\operatorname{sgn}(x)),\]
where $\operatorname{sgn}(x)$ is equal to $\pm 1$ following the sign of $x$.
The set of points $(x,y)$ whose iterates are all contained in
$([-2,-1]\cup[1,2])\times [-1/(1-\lambda),1/(1-\lambda)]$ is a hyperbolic set $K$
which is a $C^1$-blender.
{\underline b}egin{proof}
Note that $K$ is locally maximal: any orbit $(x_n,y_n)_{n\in \mathbb Z}$ contained in a
small neighborhood of $([-2,-1]\cup[1,2])\times [-1/(1-\lambda),1/(1-\lambda)]$ belongs to $K$. For diffeomorphisms $C^1$-close,
such an orbit is contained in the hyperbolic continuation of $K$.
For $\eta>0$ small, let
$\Delta_\eta:= ([-2-\eta,-1+\eta]\cup [1-\eta, 2+\eta])\times [-2,2]$.
For every $C^1$-perturbation $f'$ of $f$, it holds:
\[f'(\operatorname{Interior}(\Delta_\eta))\supset [-2-\eta,2+\eta]\times [-2,2]\; .\]
Hence every point $(x,y)\in [-2,2]\times [-2,2]$ admits an $f'$-preorbit $(\underline x,\underline y)=(x_n,y_n)_{n< 0}$
in $\Delta_\eta$. It shadows a unique preorbit $\underline z\in \overleftarrow K$.
Consequently we have $(x,y)\in W^u_{loc}(\underline z, f')$
and this shows that $K$ is a $C^1$-blender.
\end{proof}
\end{exam}
{\underline b}igskip
More generally, given a finite set ${\mathfrak B}$, one can construct
disjoint intervals $\sqcup_{{\mathfrak b}\in {\mathfrak B}} I_{\mathfrak b}\subset [-1,1]$ and an expanding map $q\colon \sqcup_{{\mathfrak b}\in {\mathfrak B}} I_{\mathfrak b}\to [-1,1]$ so that $q(I_{\mathfrak b})$ is equal to $[-1,1]$.
Then given an IFS $(f_{\mathfrak b})_{{\mathfrak b}\in {\mathfrak B}}$ by contracting diffeomorphisms of $\mathbb R^n$, we can define
a map:
\[f\colon (x,y)\in \sqcup_{{\mathfrak b}\in {\mathfrak B}} I_{\mathfrak b}\times \mathbb R^n\mapsto (q(x), f_{\mathfrak b}(y)),\quad \text{if } x\in I_{\mathfrak b}\; .\]
whose maximal invariant set $K$ is hyperbolic. The second coordinate projection of $K$ is the limit set $\Lambda$ of the IFS. Also if the IFS satisfies the covering property, then $K$ is a blender (see also~\cite{BKR14}).
Despite its fundamental aspect, our specific interest for question \ref{quesIFS} is to know whereas a covering-like property is equivalent to the above definition of blender.
{\mathfrak e}ction{Parablenders}\label{s.parablender}
In this section we deal with $C^r$-families of endomorphisms of a compact manifold $M$, with $k\geq 1$ parameters, that is
elements in { $C^r(\mathbb I^k\times M,M)$}, denoted as $(f_a)_{a\in \mathbb I^k}$.
Let $f$ be a local diffeomorphism on $M$ with a hyperbolic set $K$.
Given { a $C^r$-family} $(f_a)_{a\in \mathbb I^k}$ $C^0$-close to the constant family $(f)_{a\in \mathbb I^k}$, for each $\underline x\in \overleftarrow K$ we can consider the family of local unstable manifolds { $(W^u_{loc}(\underline x,f_a))_{a\in \mathbb I^k}$}. It is easy to see that this family is $C^0$-close to {
the constant family} $(W^u_{loc}(\underline x, f))_{a\in \mathbb I^k}$. Actually it is much more:
{ {\underline b}egin{prop}\label{prop16}
For every $\underline x\in \overleftarrow K$, the set $\cup_{a\in \mathbb I^k } \{a\}\times W^u_{loc}(\underline x,f_a)$ is a $C^r$-submanifold of $\mathbb I^k\times M$ which depends continuously on $\underline x$. In other words, the family of submanifolds $(W^u_{loc}(\underline x,f_a))_{a\in \mathbb I^k}$ is of class $C^r$ and depends continuously on $\underline x\in \overleftarrow K$ for the $C^r$-topology.
\end{prop}
{\underline b}egin{proof}
The submanifolds $\{(a, \pi_{f_a}(\underline x)): \; a \in \mathbb I^k\}$ among $\underline x\in \overleftarrow K$ form the leaves of lamination immersed in $\mathbb I^k\times M$.
The dynamics $(a,x)\mapsto(a, f_a(x))$ leaves invariant this lamination, and is $r$-normally hyperbolic at it.
By Proposition 9.1 of \cite{BEsbm}, the local unstable set of each of these leaves is a $C^r$-submanifold which depends continuously on $\underline x\in \overleftarrow K$.
\end{proof}}
We are now able to state:
{\underline b}egin{defi} A \emph{$C^r$-parablender} at $a_0\in \mathbb I^k$ for a family of endomorphisms
{ $(f_a)_a\in C^r(\mathbb I^k\times M,M)$} is a hyperbolic set $K$ for $f_{a_0}$ such that:
{\underline b}egin{itemize}
\item[--] for every $\gamma$ in a non-empty open subset $U$ of $C^r(\mathbb I^k,M)$,
\item[--] and every $(f'_a)_a$ in a neighborhood $V$ of $(f_a)_a$ in { $C^r(\mathbb I^k\times M,M)$},
\end{itemize}
there exist $\underline x\in \overleftarrow K$ and $\zeta\in C^r(\mathbb I^k,M)$ satisfying:
{\underline b}egin{itemize}
\item[--] $\zeta(a)$ belongs to $W^u_{loc}(\pi_{f_a}(\underline x),f_a)$ for every $a\in \mathbb I^k$,
\item[--] the $r$-first derivatives of $\gamma$ and $\zeta$ are equal at $a_0$:
\[\zeta(a_0)= \gamma(a_0),\quad D\zeta(a_0)= D\gamma(a_0),\quad \dots, \quad D^r\zeta(a_0)= D^r\gamma(a_0)\; .\]
\end{itemize}
\end{defi}
In particular $K$ is a $C^r$-blender.
The concept of parablender was introduced in \cite{BE15} to prove that the diffeomorphisms with finitely many attractors are \emph{not typical} in the sense of Kolmogorov, a result in the opposite direction to a conjecture of Pugh-Shub \cite{PS95} and to the main conjecture of Palis \cite{Pa00}.
The parablenders defined therein are based on IFS of $\mathbb R$ generated by
$2^{\dim\, \{P\in \mathbb R[X_1,\dots ,X_{k}]:\; \deg(P)\le r\}}$ elements.
We give here a new example in the case $k=1$ and $n=2$, based on the above IFS theory.
Note that the number of elements is reduced to $2$ and is now independent from the smoothness $r$.
{\underline b}egin{theo}\label{thm.parblender}
For any surface $M$ and any $r\geq 1$,
there exists a family $(F_a)_a\in { C^r(\mathbb I\times M,M)}$
which admits a $C^r$-parablender induced by an IFS with $2$ elements.
\end{theo}
{\underline b}egin{proof}
The construction is realized inside a disc and can be extended to any surface $M$.
It uses the previous examples:
for $\lambda\in (1/2, 1)$, we consider a family $(F_a)_{a\in \mathbb I}\in { C^r(\mathbb I\times \mathbb R^2,\mathbb R^2)}$ whose restrictions to
$([-2,-1]\cup[1,2])\times [-1/(1-\lambda),1/(1-\lambda)]$ is:
\[F_a\colon (x,y)\mapsto (4 |x|-6, (\lambda+a) y+\operatorname{sgn}(x)).\]
The set of orbits $(x_n,y_n)_{n\in \mathbb Z}$ of $F_0$
that are contained in $([-2,-1]\cup[1,2])\times [-1/(1-\lambda),1/(1-\lambda)]$
project through the map $(x_n,y_n)_n\mapsto (x_0,y_0)$ on a hyperbolic set $K$.
Denoting $f_{\pm1,a}\colon y\mapsto (\lambda+a) y\pm 1$ the maps introduced in section~\ref{section3},
and $g_{\pm1}\colon x\mapsto \pm 4x-6$, we get:
\[F_a\colon (x,y)\mapsto (g_{\operatorname{sgn}(x)}(x), f_{\operatorname{sgn}(x),a}(y).\]
The families $(f_{\pm1,a})_{a\in \mathbb I}$ induce an IFS on the jet space $J_0^r(\mathbb I,\mathbb R)$
generated by two maps $(\widehat f_{\pm 1})$.
Theorem \ref{paraIFS} states that (for $\lambda<1$ close enough to $1$)
there exists a non-empty open set $A\subset J_0^r(\mathbb I,\mathbb R)$ such that $\widehat f_{+1}( A)\cup \widehat f_{-1}( A)$ contains the closure of $A$.
Let $\delta>0$ be the Lebesgue number of this covering: every point in $\operatorname{Closure}(A)$ is the center of a { closed } $\delta $- ball contained in $\widehat f_{+1}( A)$ or in $\widehat f_{-1}( A)$.
Let $A_+$ and $A_-$ be the subsets of $A$ formed by points whose $\delta$-neighborhoods are contained in respectively $\widehat f_{+1}( A)$ and $\widehat f_{-1}( A)$.
Note that $A_+$ and $A_-$ are open sets and:
\[A = A_+\cup A_-\; \text{and}\;
\operatorname{Closure}( \widehat f_{+1}^{-1}(A_+)\cup \widehat f_{-1}^{-1}(A_-))\subset A \;.\]
On the other hand, $g_{+1}$ and $g_{-1}$ act on the $C^r$-jet space $J_0^r(\mathbb I,\mathbb R){\underline a}pprox \mathbb R^{r+1}$ as maps
$$\widehat g_{\pm1}\colon (\partial^i_a x_a)\mapsto (g_{\pm1}(x_a), \pm 4 \partial_a x_a, \dots, \pm 4 \partial^r_a x_a).$$
Let $B$ be the open subset of $J_0^r(\mathbb I,\mathbb R)$ equal to:
\[B:= (-2-\eta,2+\eta)\times (-\eta,\eta)^r\; ,\]
for $\eta>0$ small enough so that each of the inverse maps $\widehat g_{\pm1}^{-1}$ sends the closure of $B$ into $B$.
We notice that the action $\widehat F$ of $(F_a)_a$ on the $r$-jets $J^r_0(\mathbb I,\mathbb R^2)$ has two inverse branches: $\widehat F^{-1}_{-1}:=(\widehat g_{-1}^{-1},\widehat f_{-1}^{-1})$ and $\widehat F^{-1}_{+1}:=(\widehat g_{+1}^{-1},\widehat f_{+1}^{-1})$ satisfying with the open subsets $W_{\pm 1}:=B\times A_{\pm 1}$ and $W:=B\times A=W_{+1}\cup W_{-1}$ of $J^r_0(\mathbb I,\mathbb R^2)$ the following:
{\underline b}egin{equation}\tag{$\star$}\operatorname{Closure} (\widehat F^{-1}_{+1}(W_{+1})\cup \widehat F^{-1}_{-1}(W_{-1}))\subset W=W_{+1}\cup W_{-1}\; .
\end{equation}
For an open set $V$ of $C^r$-perturbations $(F'_a)_a$ of $(F_a)_a$, the inverse of branches $\widehat F'^{-1}_{+1}$ and $\widehat F'^{-1}_{-1}$ of the induced action on the $r$-jets still satisfy ($\star$).
Let $U$ be the non-empty open set of curves $a\mapsto \gamma(a)\in C^r(\mathbb I, \mathbb R^2)$ so that the $r$-jet
$\widehat \gamma:=(\gamma,\partial_a\gamma,\dots,\partial^r_a\gamma)_{|a=0}$ of $\gamma$ at $a=0$ lies in $W$. By the latter inclusion $(\star)$, we can define inductively a sequence
$\underline \delta := (\delta_i)_{i\le 0}\in \{-1,+1\}^{\mathbb Z^-}$
and
$(\widehat \gamma_i)_{i\le 0}\in \prod_{i\le 0} W_{\delta_i}$ so that $\widehat \gamma_0 =\widehat \gamma$, and for $i\le 0$, $\widehat \gamma_{i} = \widehat F'_{\delta_{i}}(\widehat \gamma_{i-1})$. Note that given a $\widehat \gamma$, the sequences $\underline \delta$ and $(\widehat \gamma_i)_{i\le 0}$ are in general not uniquely defined.
We remark that $\widehat \gamma_i$ is the $r$-jet at $a=0$ of the curve $a\mapsto \gamma_i(a)$
defined by
\[\gamma_i(a) := (F'_{a}|Y_{\delta_i})^{-1}
\circ \cdots \circ
(F'_{a}|Y_{\delta_0})^{-1}(\gamma(a)),\]
\[\text{where } Y_{+1}:=[1-\eta, 2+\eta]\times \left[\frac{-1-\eta}{1-\lambda},\frac{1+\eta}{1-\lambda}\right]
\text{ and }
Y_{-1}:=[-2-\eta, -1+\eta]\times \left[\frac{-1-\eta}{1-\lambda},\frac{1+\eta}{1-\lambda}\right] .\]
The sequence $\underline \delta$ defines a local unstable manifold of $K$
\[W^u_{loc}(\underline \delta; F_0):= {\underline b}igcap_{j\ge 0} F^{j+1}_0(Y_{\delta_j}).\]
It admits a continuation $W^u_{loc}(\underline \delta; F'_a)$ for any family $(F'_a)_a$ close to $(F_a)_a$
and any parameter $a$ close to $0$. This unstable manifold also contains
the projection of a point $\underline x\in \overleftarrow K$ so that
$W^u_{loc}(\underline \delta; F'_a)={ W^u_{loc}(\underline x, F'_a)}$.
Let $\zeta(a)$ be the vertical projection of $\gamma(a)$ into $W^u(\underline \delta; f_a)$ for every $a\in \mathbb I$. As $(W^u(\underline \delta; f_a))_a$ is of class $C^r$ by proposition \ref{prop16}, the function $a\mapsto \zeta(a)$ is of class $C^r$.
We now consider the vertical segment $C(a):= [\zeta(a),\gamma(a)]$. Up to shrinking slightly $V$, we can assume that the stable cone field $\mathcal{C}:= \{(u,v)\colon |u|\le \eta |v|\}$ is backward invariant by each $F'_a$ and $(\lambda+\eta)$-contracted by $DF'_a$, for every $a$ close to $0$. Thus the curve
$$C_i(a) := (F'_a| Y_{\delta_{i}})^{-1}\circ \cdots \circ (F'_a | Y_{\delta_0})^{-1}(C(a))$$
has its tangent space in $\mathcal{C}$ and connects $\gamma^i(a)$ to the local unstable manifold { $W^u_{loc}\underline x, F'_a)$}.
By proposition \ref{prop16}, the $r$ first derivatives of { $(W^u_{loc}(\underline x, F'_a))_a$}
are uniformly bounded.
By assumption $(\gamma_i(a))_i$ has its $r$-first derivatives for $a$ small enough contained in $W$,
hence uniformly bounded.
Thus there exists $A>0$ independent of $i$ and so that for any
$a$ small enough (depending on $i$), the length of $C_i(a)$ is at most $ A\sum_{j=0}^r |a|^j+|a|^{r}\rho_i(a)$, with $\rho_i$ a continuous function equal to $0$ at $a=0$.
We recall that $DF'_a|\mathcal{C} $ is $(\lambda+\eta)$-contracting. As $C_i(a)$ has its tangent space in $\mathcal{C}$ it comes that the length of $C(a)$ is at most $(\lambda+\eta)^{|i|} [A\sum_{j=0}^r |a|^j+|a|^r\rho_i(a)]$ for every $i\le 0$ and $a$ small enough in function of $i$.
This proves that the length of $C(a)$ and its $r$ first derivative w.r.t. $a$
at $a=0$ are smaller than $(\lambda+\eta)^{|i|}$, for every $i\le 0$.
Hence they vanish all and so the $r$-first derivatives of $\zeta$ and $\gamma$ are equal at $a=0$. \end{proof}
{
{\mathfrak e}ction{Nearly affine blenders and parablenders}
The previous constructions may be realized in the following more general setting.
Let us fix $r\geq 1$ and choose $\lambda<1$ close to $1$.
{\underline b}egin{defi}
For $\varepsilon>0$ small, we say that
a local diffeomorphism defined on a neighborhood of the rectangle
$R:=[-2,2]\times [-1/(1-\lambda), 1/(1-\lambda)]$
is a $\varepsilon$-\emph{nearly affine blender with contraction $\lambda$} if
there exists two inverse branches $g_+,g_-$ for $f^{-1}$:
{\underline b}egin{itemize}
\item[--] $g_+$ is defined on a neighborhood of
$Y_+:=[-2,2]\times [(1-2\lambda)/(1-\lambda),1/(1-\lambda)]$ and is $\varepsilon$-close to the map
$(x,y)\mapsto (0,(y-1)/\lambda)$ for the $C^1$-topology;
\item[--] $g_-$ is well defined on a neighborhood of
$Y_-:=[-2,2]\times [-1/(1-\lambda),(2\lambda-1)/(1-\lambda)]$ and is $\varepsilon$-close to the map
$(x,y)\mapsto (0,(y+1)/\lambda)$ for the $C^1$-topology;
\item[--]
$g_+([-2,2]\times \{(1-2\lambda)/(1-\lambda),1/(1-\lambda)\})$ and
$g_-([-2,2]\times \{-1/(1-\lambda),(2\lambda-1)/(1-\lambda)\})$ are disjoint from $R$.
\end{itemize}
\end{defi}
The last item implies that there exist two maps $\psi_-<\psi_+:=[-2,2]\to [-1/(1-\lambda), 1/(1-\lambda)]$
whose graphs are contained in $Y_-$ and $Y_+$ respectively and contracted by the respective
branches of $f^{-1}$, so that the strip of $[-2,2]\to [-1/(1-\lambda), 1/(1-\lambda)]$ bounded by these two graphs
is contained in its image. Arguing as in the example~\ref{e.blender}, one shows that the maximal invariant
set in $R$ is a blender.
{\underline b}egin{figure}[ht]
{\underline b}egin{center}
\includegraphics[scale=0.4]{blender.pdf}
{\underline b}egin{picture}(0,0)
\put(-325,65){$Y_+$}
\put(-325,30){$Y_-$}
\put(-145,30){$g_-$}
\put(-145,80){$g_+$}
\end{picture}
\end{center}\end{figure}
{\underline b}egin{defi}
For $\varepsilon>0$ small, we say that
a $C^r$-family $(f_a)_{a\in \mathbb I}$ is a $\varepsilon$-\emph{nearly affine parablender with contraction $\lambda$
and $a=0$} if:
{\underline b}egin{itemize}
\item[--] $f_0$ is a $\varepsilon$-\emph{nearly affine blender with contraction $\lambda$};
\item[--] for some ${\underline a}lpha>0$, the family $(f^{-1}_a)_{|a|\leq {\underline a}lpha}$ on a neighborhood of $Y_+$
is $\varepsilon$-close in $C^r$-topology to
$$(a,x,y)\mapsto (0,(y-1)/(\lambda+a));$$
\item[--] for some ${\underline a}lpha>0$, the family $(f^{-1}_a)_{|a|\leq {\underline a}lpha}$ on a neighborhood of $Y_-$
is $\varepsilon$-close in $C^r$-topology to
$$(a,x,y)\mapsto (0,(y+1)/(\lambda+a)).$$
\end{itemize}
\end{defi}
The same proof as for theorem~\ref{thm.parblender}
shows that the maximal invariant set for $f_0$ in $R$ is a $C^r$-parablender for $(f_a)_{a\in \mathbb I}$
at $a=0$ provided $\lambda$ has been chose close enough to $1$ in function of $r$ and provided
$\varepsilon>0$ has been chosen small enough in function of $r$ and $\lambda$.
}
{\underline b}ibliographystyle{alpha}
\def$'$} \def\cprime{$'$} \def\cprime{$'${$'$} \def$'$} \def\cprime{$'$} \def\cprime{$'${$'$} \def$'$} \def\cprime{$'$} \def\cprime{$'${$'$}
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{\underline b}igskip
{\underline n}oindent \emph{Pierre Berger}, {\small LAGA, CNRS - UMR 7539, Universit\'e Paris 13, 93430 Villetaneuse, France.}
\vskip 3pt
{\underline n}oindent \emph{Sylvain Crovisier}, {\small LMO, CNRS - UMR 8628, Universit\'e Paris-Sud 11, 91405 Orsay, France.}
\vskip 3pt
{\underline n}oindent
\emph{Enrique Pujals}, {\small IMPA,
Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, Brazil.}
\end{document} | math | 40,750 |
\begin{document}
\title{From nominal sets binding to functions and $\leftarrowmbda$-abstraction: connecting the logic of permutation models with the logic of functions}
\author{Gilles Dowek}
\address{\href{http://www-roc.inria.fr/who/Gilles.Dowek/}{www-roc.inria.fr/who/Gilles.Dowek}}
\author{Murdoch J. Gabbay}
\address{\href{http://www.gabbay.org.uk}{gabbay.org.uk}}
\begin{abstract}
Permissive-Nominal Logic (PNL) extends first-order predicate logic with term-formers that can bind names in their arguments.
It takes a semantics in (permissive-)nominal sets.
In PNL, the $\forall$-quantifier or $\leftarrowmbda$-binder are just term-formers satisfying axioms, and their denotation is functions on nominal atoms-abstraction.
Then we have higher-order logic (HOL) and its models in ordinary (i.e. Zermelo-Fraenkel) sets; the denotation of $\forall$ or $\leftarrowmbda$ is functions on full or partial function spaces.
This raises the following question: how are these two models of binding connected?
What translation is possible between PNL and HOL, and between nominal sets and functions?
We exhibit a translation of PNL into HOL, and from models of PNL to certain models of HOL.
It is natural, but also partial: we translate a restricted subsystem of full PNL to HOL.
The extra part which does not translate is the symmetry properties of nominal sets with respect to permutations.
To use a little nominal jargon: we can translate names and binding, but not their nominal equivariance properties.
This seems reasonable since HOL---and ordinary sets---are not equivariant.
Thus viewed through this translation, PNL and HOL and their models do different things, but they enjoy non-trivial and rich subsystems which are isomorphic.
\end{abstract}
\begin{keyword}
Permissive-nominal logic, higher-order logic, nominal sets, nominal renaming sets, mathematical foundations of programming.
\\
\emph{MSC-class:} 03B70 (primary), 68Q55 (secondary)
\\
\emph{ACM-class:} F.3.0; F.3.2
\end{keyword}
\maketitle
\tableofcontents
\section{Introduction}
Permissive-Nominal Logic (PNL) extends first-order predicate logic with term-formers that can bind names in their arguments.
For instance, arithmetic, set theory, and functions axiomatise naturally in PNL; their binders are modelled as ordinary PNL term-formers and their axioms look very much like the axioms normally written in informal practice.
PNL is sound and complete for a first-order style semantics in (permissive-nominal) sets \cite{gabbay:pernl-jv,gabbay:nomtnl}.
This captures the essence of nominal techniques, whose initial motivation has been to handle names and binding in a first-order framework.
Higher-order logic (HOL) also has binding \cite{miller:logho,farmer:sevvst}.
This has been used to encode other binders, e.g. the Church encoding of quantifiers as constants of higher type such as $\forall:(\iota{\to} o){\to} o$ \cite{andrews:intmlt,church:forstt}; higher-order abstract syntax (HOAS) encoding term-formers of an encoded syntax with binders as constants of higher type such as $\forall:(\iota{\to} \rho){\to}\rho$ or $\forall:(\nu{\to}\rho){\to}\rho$ (strong vs. weak HOAS)\footnote{A word of clarification here: we take $o$ to be a type of truth-values, $\iota$ to be a type of terms, and $\rho$ to be a type of predicates. $\forall$-the-quantifier generates truth-values, whence the type headed by $o$, namely $\forall:(\iota{\to} o){\to} o$. $\forall$-the-syntax-building-constant in HOAS generats \emph{terms}, whence the types headed by $\rho$, namely $\forall:(\iota{\to} \rho){\to}\rho$ or $\forall:(\nu{\to}\rho){\to}\rho$. Do not confuse a HOL constant for a HOAS-style binder (a way to give meaning to building syntax with binding) with a HOL constant for the corresponding quantifier (a way to give meaning to what that that syntax is intended to denote; namely, actual quantification).}
\cite{despeyroux94higherorder,pfenning:hoas}; and higher-order rewrite systems \cite{mayr:horwc}.
This paper is not about how PNL and HOL can be used as meta-mathematical reasoning frameworks, or about what models look like expressed as nominal sets or as functions.
The deeper point is that we have before us two foundations for mathematics.
The question we address is then as follows:
\emph{There is a `nominal' model of names and binding which can be applied in various ways, and also a functional model which can also be applied in various ways.
These are captured by two logics---PNL and HOL---and by their nominal and functional denotations respectively.
We observe that these are clearly different, yet their applications just as clearly overlap.
So, what positive and mathematically precise statements we now make about their relationship?
}
Since PNL is first-order and has a sound and complete semantics (so expressivity and models are fairly `small'), whereas HOL is higher-order (so expressivity and models are fairly `large'), the natural direction for a translation is from nominal sets and PNL, to functions and HOL.\footnote{In other words, we want a \emph{shallow embedding} of PNL into HOL. A \emph{deep embedding} e.g. of HOL in PNL is an answer to a different question; for more on this direction, see \cite{gabbay:unialt}.}
This raises the question of how PNL translates to HOL, and how PNL models translate to functional models.
In this paper we translate a subsystem of PNL into HOL and prove it sound and complete using arguments on nominal sets and and nominal renaming sets models \cite{gabbay:nomrs}.
The proof of completeness involves giving a functional semantics to nominal terms, and a nominal semantics to $\leftarrowmbda$-terms in the spirit of Henkin models \cite{andrews:intmlt,benzmuller:higose}.
This involves a construction on nominal sets models corresponding to a free extension to \emph{nominal renaming sets}, as previously considered by the second author with Hofmann \cite{gabbay:nomrs}.
The partiality of the translation seems to be inherent and reflects natural differences in structure between nominal and `ordinary' sets.
That is, it is not the case that nominal techniques are `just' a concise presentation of HOL with a weakened $\beta$-equivalence (e.g. higher-order patterns \cite{miller:logpll}).
There is that, but there is also more.
Thus, the nominal and functional models of names and binding are distinct, but they have non-trivial and rich subsystems which are isomorphic in a sense made precise in this paper.
\subsection{Some background on PNL}
We study PNL for its own sake in this paper, but the interested reader can find example nominal theories in the literature.
PNL is designed as a first-order logic for denotations with binding.
The reader can find sound and complete nominal algebra theories for substitution, $\beta$-equivalence, and first-order logic \cite{gabbay:capasn,gabbay:nomalc,gabbay:oneaah} (nominal algebra can be viewed as the equality fragment of PNL).
Not all PNL theories are expressed in the equality fragment.
For instance, in the paper which introduced PNL \cite{gabbay:pernl} we included theories of first-order logic and arithmetic which put universal quantification to the left of an implication.
This cannot be done in nominal algebra because it is a purely equational logic.
To give some idea of what this family of logics looks like in practice, assume a name-sort $\nu$ and a base sort $\iota$ and term-formers $\tf{lam}:([\nu]\iota)\iota$,\ \ $\tf{app}:(\iota,\iota)\iota$,\ and $\tf{var}:(\nu)\iota$.
(Full definitions are in the body of the paper.)
We sugar $\tf{lam}([a]r)$ to $\leftarrowm{a}r$ and $\tf{app}(r',r)$ to $r'r$ and $\tf{var}(a)$ to $a$.
Atoms in PNL are a form of data and populate their own sort $\nu$; so $\tf{var}$ serves to map them into the sort $\iota$, where they represent object-level variables.
Here is $\eta$-equivalence, written out as it would be informally:
$$
\leftarrowm{x}(tx)=t\text{\ \ \ if $x$ is not free in $t$}
$$
Here is a PNL axiom for $\eta$-equivalence, written out formally:
$$
\Forall{Z}(\leftarrowm{a}(Za)=Z)\quad (a\not\in\pmss(Z))
$$
(See \cite{gabbay:nomalc} for a detailed study of this axiom in a nominal context.)
$a$ is an \emph{atom} and corresponds to the \emph{object-level variable} $x$; $a$ is not a PNL variable but it \emph{represents} a variable of the object level system being axiomatised.
$Z$ is an \emph{unknown} and correspond to the \emph{meta-level variable} $t$; $Z$ is a variable in PNL and may be instantiated.
The reader can see how similar the two axioms look.
Their status is different in the following sense: whereas $t$ is typically taken to range over terms, $Z$ ranges over elements of nominal sets (via a valuation; see Definition~\ref{defn.valuation}).
This is possible because nominal sets have a notion of \emph{supporting set of atoms} which mirrors the free variables of a term.
The condition $a\not\in\pmss(Z)$ is a \emph{typing condition} in PNL.
The types, or \emph{permission sets} as we call them, restrict the support of denotations associated to $Z$ by a valuation.
They correspond to freshness side-conditions in nominal terms from \cite{gabbay:nomu-jv} and to informal freshness conditions of the form `$x$ not free in $t$' in informal practice.
To see this intuition made formal see a translation from nominal terms to permissive-nominal terms in \cite{gabbay:perntu-jv}.
There is no requirement to axiomatise $\alpha$-equivalence because this is done automatically by the PNL system.
Sugar $(\leftarrowm{a}r)r'$ to $r[a\sm r']$.
Then axioms for $\beta$-equivalence are:
$$
\begin{array}{l@{\ }l@{\ =\ }l@{\ }l}
\Forall{Y}&a[a\sm Y]&Y
\\
\Forall{Z,X}&Z[a\sm X]&Z &(a\not\in\pmss(Z))
\\
\Forall{X',X,Y}&(X'X)[a\sm Y]&(X'[a\sm Y])(X[a\sm Y])
\\
\Forall{X,Z}&(\leftarrowm{a}X)[b\sm Z]&\leftarrowm{a}(X[b\sm Z]) &(a\not\in\pmss(Z))
\\
\Forall{X}&X[a\sm a]&X
\end{array}
$$
Thus, the design philosophy of PNL is that axioms should look like what we would write informally anyway, where variables map to atoms, meta-variables to unknowns, binding to atoms-abstraction, and capture-avoidance conditions to choice of permission sets.
Note that in the axioms above, $a$ and $b$ cannot be equal because they are distinct atoms, and atoms are data, not variables ($a$ is $a$, and $b$ is $b$, and they are distinct).
More on this and on the use of permutations in the body of the paper.\footnote{The axioms above also have typing constraints, because unknowns are typed with their permission set. These typing constraints turn out not to be so restrictive, for quite subtle reasons. The interested reader can find a discussion in \cite[Subsection~2.7]{gabbay:pernl-jv}. For the purposes of the discussion here, it is not important.}
Equality reasoning is not necessary to $\alpha$-rename atoms in PNL; we can quotient by $\alpha$-equivalence so that we can rename $\Forall{a}\tf P(a)$ to $\Forall{b}\tf P(b)$ without proving a logical equivalence.
This is unlike other `nominal' reasoning systems, such as Fraenkel-Mostowski set theory as used by the author with Pitts to introduce nominal techniques in \cite{gabbay:newaas-jv},
nominal rewriting by Fern\'andez and the second author \cite{gabbay:nomr-jv}, nominal algebra by the second author with Mathijssen \cite{gabbay:noma-nwpt,gabbay:forcie,gabbay:nomuae},
$\alpha$Prolog by Cheney and Urban \cite{cheney:nomlp}, and other systems in the same spirit.
\subsection{Map of the paper}
This paper has a lot of technical ground to cover.
This is unavoidable, because we need to deal with two logics (restricted PNL and HOL) and two semantics (nominal sets, and the hand-crafted Henkin models in nominal renaming sets used in the completeness proof), as well as two translations (from logic to logic, and from models to models).
For the reader's convenience, we provide an overview of the main technical points with brief justifications for their design:
\begin{itemize*}
\item
Section~\ref{sect.pnl} introduces permissive-nominal logic.
This comes from previous work into `nominal' axiomatisations of systems with binding \cite{gabbay:pernl,gabbay:pernl-jv}.\footnote{Note that PNL is not only about nominal abstract syntax as considered in e.g. \cite{gabbay:newaas-jv,gabbay:fountl}. Nominal abstract syntax is a denotation for syntax with binding. PNL and its models are a (more general) syntax and semantics for denotations with binding in general, which are not all necessarily datatypes of abstract syntax.}
In fact, we need to introduce two logics: full PNL and also a \emph{restricted} version which has a weaker non-equivariant axiom rule.
We write the entailment relations $\cent$ and $\nopicent$ respectively.
It is the restricted version that we will eventually translate to HOL.
\item
Section~\ref{sect.hol} introduces higher-order logic as a theory over the syntax of the simply-typed $\leftarrowmbda$-calculus.
We write the entailment relation $\holcent$.
\item
Section~\ref{sect.translation.sound} defines the translation from restricted PNL to HOL, and proves it sound using arguments on syntax.
In order to do the translation, we need to introduce a \emph{capture typing} ${D\cent r:A}$ which is a measure of how many functional abstractions are required to translate a given nominal term without losing information; that is, of the functional complexity of a nominal term.
\item
Our goal is then to prove completeness of the translation.
We do this by transforming models of PNL into models of HOL.
So Section~\ref{sect.semantics} introduces two categories: \theory{PmsPrm} of permissive-nominal sets and \theory{PmsRen} of permissive-nominal renaming sets.
We also give a \emph{free} construction, transforming a permissive-nominal set into a permissive-nominal renaming set.
\item
In Section~\ref{sect.permissive-nominal.sets} we interpret full and restricted PNL in \theory{PmsPrm}.
In Section~\ref{sect.interp.hol} we interpret HOL in \theory{PmsRen}.
\item
Finally, in Section~\ref{sect.pnl.hol.complete} we use the free construction of Section~\ref{sect.semantics} to map a model of PNL in \theory{PmsPrm} to a model in \theory{PmsRen}, and because the free construction does not `make anything equal' this is sufficient to prove completeness.
\item
As one further mathematical note, the results in the literature concern full PNL and not restricted PNL.
So in Appendix~\ref{sect.completeness} we sketch proofs of soundness, cut-elimination, and completeness of restricted PNL with respect to non-equivariant models in \theory{PmsPrm}.
These are modest, if not entirely direct, modifications of the existing definitions and proofs for full PNL and equivariant models in \theory{PmsPrm}.
\end{itemize*}
Quite a number of new ideas are required to make this all work.
The highlights are: permissive-nominal renaming sets and their application to give non-standard `nominal' Henkin models for higher-order logic; restricted PNL and its semantics; the free construction; and the technical arguments as discussed in Section~\ref{sect.pnl.hol.complete}.
\ \\
Given that the proofs and constructions in this paper are non-trivial and involve an effort to extend existing machinery, we should pause to ask again why doing this is justified, even necessary.
Nominal techniques were designed originally to reason on syntax-with-binding (see the original journal paper \cite{gabbay:newaas-jv} or a recent survey paper \cite{gabbay:fountl}).
But since then this remit has expanded to reasoning about denotations with binding more generally (an overview of which is in \cite{gabbay:nomtnl}).
In doing this, we have created a whole new syntax and semantics for meta-mathematics.
We will not argue for or against either the nominal foundation or the higher-order foundation for mathematics.\footnote{There has been more than enough of that already, and anyway, because truth is free, proving theorems is never a zero sum game.}
Our question is: given that these two foundations exist, how do they relate?
In fact, questions have been asked about how nominal names and binding are related to functions, ever since nominal techniques were conceived in the second author's thesis.
Since then, the development of PNL \cite{gabbay:pernl-jv} and nominal renaming sets \cite{gabbay:nomrs} has given us two powerful new tools with which to address these questions: a proof-theory for a logic in which nominal reasoning so far can be formalised, and a visibly nominal semantics which is not based on permutations but on possibly non-bijective renamings on atoms, so that atoms-abstraction can be considered as a function in that semantics.
In this paper, we leverage this to give a precise, concrete, and mathematically detailed account of how these two worlds really stand in relation to one another---and how they differ.
In conclusion we speculate that there is some potential (not explored in this paper) that our translations might be used to piggyback nominal techniques on the substantial implementational efforts that have gone into developing HOL over the past seventy years.
\section{Permissive-Nominal Logic}
\leftarrowbel{sect.pnl}
Permissive-nominal logic is a first-order logic for nominal terms quotiented by $\alpha$-equivalence.
Doing this is not entirely trivial; the interested reader can find more on this elsewhere \cite{gabbay:nomu-jv,gabbay:pernl,gabbay:pernl-jv,gabbay:nomtnl}.
\subsection{Syntax}
\begin{defn}
\leftarrowbel{defn.sort.sig}
A \deffont{sort-signature} is a pair $(\mathcal A,\mathcal B)$ of \deffont{name} and \deffont{base sorts}.
$\nu$ will range over name sorts; $\basesort$ will range over base sorts.
A \deffont{sort language} is then defined by
\begin{frameqn}
\alpha ::= \nu \mid (\alpha,\dots,\alpha) \mid [\nu]\alpha \mid \basesort
.
\end{frameqn}
\end{defn}
\begin{rmrk}
Examples of base sorts are: `$\leftarrowmbda$-terms',\ `formulae',\ `$\pi$-calculus processes',\ and `program environments', `functions', `truth-values', `behaviours',\ and `valuations'.
Examples of name sorts are `variable symbols',\ `channel names',\ or `memory locations'.
$[\nu]\alpha$ is an \emph{abstraction sort}.
This does a similar job to function-types in higher-order logic but note that $\nu$ must always be a name-sort.
The behaviour of a term of sort $[\nu]\alpha$ corresponds to `bind a name of sort $\nu$ in a term of sort $\alpha$'.
Such a term does not denote a function, though later on in our completeness proof we will deliberately undermine that intuition to obtain our completeness result.
\end{rmrk}
\begin{defn}
\leftarrowbel{defn.term.signature}
A \deffont{term-signature} over a sort-signature $(\mathcal A,\mathcal B)$ is a tuple $(\mathcal F,\mathcal P,\f{ar},\mathcal X)$ where:
\begin{itemize*}
\item
$\mathcal F$ and $\mathcal P$ are disjoint sets of \deffont{term-}
and \deffont{proposition-formers}.
$\tf f$ will range over term-formers.
$\tf P$ will range over proposition-formers.
\item
$\f{ar}$ assigns to each ${\tf f\in\mathcal F}$ a
\deffont{term-former arity} $(\alpha)\tau$
and to each $\tf P\in\mathcal P$ a \deffont{proposition-former arity}
$\alpha$, where $\alpha$ and $\tau$ are in the sort-language
determined by $(\mathcal A,\mathcal B)$.
We will write $((\alpha_1,\ldots,\alpha_n))\tau$ just as $(\alpha_1,\ldots,\alpha_n)\tau$.
\item
$\mathcal X$ is a set of \deffont{unknowns} $X$, each of which has a sort $\sort(X)$ and a permission set $\pmss(X)$, such that for each sort $\alpha$ and permission set $S$ the set $\{X\in\mathcal X\mid \sort(X)=\alpha,\ \pmss(X)=S\}$ is countably infinite.
$X,Y,Z$ will range over distinct unknowns.
\end{itemize*}
\leftarrowbel{defn.signature}
A \deffont{signature} $\mathcal S$ is then a tuple $(\mathcal A,\mathcal B,\mathcal F,\mathcal P,\f{ar},\mathcal X)$.
\end{defn}
We write $\tf f:(\alpha)\tau$ for $\f{ar}(\tf f)=(\alpha)\tau$ and similarly we write $\tf P:\alpha$ for $\f{ar}(\tf P)=\alpha$.
\begin{xmpl}
\leftarrowbel{xmpl.lam.sig}
The signature for the $\leftarrowmbda$-calculus from the Introduction has a name-sort for $\leftarrowmbda$-calculus object-level variables, a base sort for $\leftarrowmbda$-terms, and appropriate term-formers:
\begin{itemize*}
\item
$\tf{var}:(\nu)\iota$ to form $\leftarrowmbda$-calculus variables in $\iota$ out of names in $\nu$,
\item
$\tf{app}$ for application, and
\item
$\tf{lam}$ taking an abstraction in $[\nu]\iota$ and forming from it a $\leftarrowmbda$-abstraction term in $\iota$.
\end{itemize*}
\end{xmpl}
\begin{defn}
\leftarrowbel{defn.atoms}
For each $\nu$ fix a disjoint countably infinite set of \deffont{atoms} $\atoms_\nu$, and an arbitrary bijection $f_\nu$ between $\atoms_\nu$ and the integers $\mathbb Z=\{0,\text{-}1,1,\text{-}2,2,\ldots\}$.
Write
$$
\atomsdown_\nu=\{f_\nu(i)\mid i<0\}
\qquad
\atomsup_\nu=\{f_\nu(i)\mid i\geq 0\}.
$$
Finally, write
$$
\atomsdown=\bigcup\atomsdown_\nu
\qquad
\atomsup=\bigcup\atomsup_\nu
\qquad
\mathbb A=\bigcup \mathbb A_\nu
$$
$a,b,c,\ldots$ will range over \emph{distinct} atoms (we call this the \deffont{permutative} convention).
A \deffont{permission set} has the form $(\atomsdown \cup A)\setminus B$ where $A\subseteq\atomsup$ and $B\subseteq\atomsdown$ are finite (and a permission set may be finitely represented by the pair $(A,B)$).
$S$, $T$, and $U$ will range over permissions sets.
\end{defn}
The use of $\atomsdown$ and $\atomsup$ ensures that permission sets are infinite and also co-infinite (their complement is also infinite).
\begin{frametxt}
\begin{defn}
\leftarrowbel{defn.permutation}
A \deffont{permutation} is a bijection $\pi$ on $\mathbb A$ such that $a\in\mathbb A_\nu\liff \pi(a)\in\mathbb A_\nu$ and $\f{nontriv}(\pi)=\{a\mid \pi(a)\neq a\}$ is finite.
Write $\mathbb P$ for the set of permutations.
Given $a,b\in\mathbb A_\nu$ let a \deffont{swapping} $(a\ b)$ be the bijection on atoms that maps $a$ to $b$, $b$ to $a$, and all other $c$ to themselves.
\end{defn}
\end{frametxt}
\begin{nttn}
\leftarrowbel{nttn.permutations}
We use the following notation:
\begin{itemize*}
\item
Write $\pi\circ\pi'$ for \deffont{functional composition}, so $(\pi\circ\pi')(a)=\pi(\pi'(a))$).
\item
Write $\id$ for the \deffont{identity permutation}, so $\id(a)=a$ always.
\item
Write $\pi^\mone$ for \deffont{inverse}, so $\pi\circ\pi^\mone=\id$.
\end{itemize*}
\end{nttn}
\begin{defn}
For each signature $\mathcal S$, define \deffont{terms} and \deffont{propositions} over $\mathcal S$ by:
\begin{frameqn}
\begin{array}{c@{\qquad}c@{\qquad}c}
\begin{prooftree}
(a\in\mathbb A_\nu)
\justifies
a:\nu
\end{prooftree}
&
\begin{prooftree}
\rightarrowwr_1:\alpha_1 \ \ldots\ \rightarrowwr_n:\alpha_n
\justifies
(\rightarrowwr_1,\ldots,\rightarrowwr_n):(\alpha_1,\ldots,\alpha_n)
\end{prooftree}
&
\begin{prooftree}
\rightarrowwr:\alpha\quad (\f{ar}(\tf f)=(\alpha)\tau)
\justifies
\tf f(\rightarrowwr):\tau
\end{prooftree}
\\[4ex]
\begin{prooftree}
\rightarrowwr:\alpha\quad (a\in\mathbb A_\nu)
\justifies
[a]\rightarrowwr:[\nu]\alpha
\end{prooftree}
&
\begin{prooftree}
(\sort(X)=\alpha)
\justifies
\pi\act X:\alpha
\end{prooftree}
\\[4ex]
\begin{prooftree}
\phantom{h}
\justifies
\bot\text{ prop.}
\end{prooftree}
&
\begin{prooftree}
\rightarrowwphi\text{ prop.}\ \ \rightarrowwpsi\text{ prop.}
\justifies
\rightarrowwphi\limp\rightarrowwpsi\text{ prop.}
\end{prooftree}
&
\begin{prooftree}
\rightarrowwr:\alpha\ \ (\f{ar}(\tf P)=\alpha)
\justifies
\tf P(\rightarrowwr)\text{ prop.}
\end{prooftree}
\\[4ex]
\begin{prooftree}
\rightarrowwphi\text{ prop.}
\justifies
\Forall{X}\rightarrowwphi\text{ prop.}
\end{prooftree}
\end{array}
\end{frameqn}
\end{defn}
\begin{xmpl}
Continuing Example~\ref{xmpl.lam.sig}, we have the following terms and propositions:
\begin{itemize*}
\item
$\tf{var}(a):\iota$ where $a\in\mathbb A_\nu$.
\item
$[a]X:[\nu]\iota$ where $a\in\mathbb A_\nu$ and $\sort(X)=\iota$, and $\tf{lam}([a]X):\iota$.
\item
$\Forall{X}\tf P(\tf{lam}([a]X),X)$ is a proposition if $\tf P$ is a proposition-former and $\tf P:(\iota,\iota)$.
\end{itemize*}
\end{xmpl}
\subsection{Permutation, substitution, and so on}
These definitions are all needed for the rest of the paper, starting with $\alpha$-equivalence in Subsection~\ref{subsect.aeq}.
We need them at both levels; both for atoms and for unknowns.
\begin{defn}
\leftarrowbel{defn.permutation.action}
Define a (level 1) \deffont{permutation action} on syntax by:
$$
\begin{array}{r@{\ }l@{\qquad}r@{\ }l}
\pi\act a=& \pi(a)
&
\pi\act (\rightarrowwr_1,\ldots,\rightarrowwr_n) =& (\pi\act \rightarrowwr_1,\ldots,\pi\act \rightarrowwr_n)
\\
\pi\act [a]\rightarrowwr =& [\pi(a)]\pi\act \rightarrowwr
&
\pi\act(\pi'\act X) =& (\pi{\circ}\pi')\act X
\\
\pi\act \tf f(\rightarrowwr) =& \tf f(\pi\act \rightarrowwr)
\\
\pi\act\bot =& \bot
&
\pi\act (\rightarrowwphi\limp\rightarrowwpsi)=& (\pi\act \rightarrowwphi)\limp(\pi\act \rightarrowwpsi)
\\
\pi\act \tf P(\rightarrowwr)=& \tf P(\pi\act \rightarrowwr)
&
\pi\act (\Forall{X}\rightarrowwphi) =& \Forall{X}\pi\act\rightarrowwphi
\end{array}
$$
\end{defn}
\begin{defn}
\leftarrowbel{defn.permutation.action.2}
Let $\Pi$ range over sort- and permission-set-preserving bijections on unknowns
(so $\sort(\Pi(X)){=}\sort(X)$ and $\pmss(\Pi(X)){=}\pmss(X)$)
such that $\{X\mid \Pi(X)\neq X\}$ is finite.
Write $\Pi\circ\Pi'$ for functional composition,\ $\Id$ for the identity permutation, and $\Pi^\mone$ for inverse, much as in Notation~\ref{nttn.permutations}.
Define a (level 2) \deffont{permutation action} by:
{
$$
\begin{array}{r@{\ }l@{\qquad}r@{\ }l}
\Pi\act a=& a
&
\Pi\act (\rightarrowwr_1,\ldots,\rightarrowwr_n) =& (\Pi\act \rightarrowwr_1,\ldots,\Pi\act \rightarrowwr_n)
\\
\Pi\act [a]\rightarrowwr =& [a]\Pi\act \rightarrowwr
&
\Pi\act(\pi\act X) =& \pi\act(\Pi(X))
\\
\Pi\act \tf f(\rightarrowwr) =& \tf f(\Pi\act \rightarrowwr)
\\
\Pi\act\bot =& \bot
&
\Pi\act (\rightarrowwphi\limp\rightarrowwpsi)=& (\Pi\act \rightarrowwphi)\limp(\Pi\act \rightarrowwpsi)
\\
\Pi\act \tf P(\rightarrowwr)=& \tf P(\Pi\act \rightarrowwr)
&
\Pi\act (\Forall{X}\rightarrowwphi) =& \Forall{\Pi(X)}\Pi\act\rightarrowwphi
\end{array}
$$
}
\end{defn}
\begin{defn}
\leftarrowbel{defn.pointwise}
Suppose $A$ is a set of atoms and $\pi$ is a level 1 permutation.
Suppose $U$ is a set of unknowns and $\Pi$ is a level 2 permutation.
Define $\pi\act A$ and $\Pi\act U$ by
$$
\pi\act A = \{\pi(a)\mid a\in A\} \qquad\text{and}\qquad
\Pi\act U = \{\Pi(X)\mid X\in U\}.
$$
This is the standard \deffont{pointwise} permutation action on sets.
\end{defn}
\begin{defn}
\leftarrowbel{defn.fa}
Define \deffont{free atoms} $\fa(\rightarrowwr)$ and $\fa(\rightarrowwphi)$ by:
$$
\begin{array}{r@{\ }l@{\quad}r@{\ }l@{\quad}r@{\ }l}
\fa(\pi\act X)=& \pi\act\pmss(X)
&
\fa([a]\rightarrowwr)=& \fa(\rightarrowwr)\setminus\{a\}
&
\fa(a)=& \{a\}
\\
\fa(\tf f(\rightarrowwr)) =& \fa(\rightarrowwr)
&
\fa((\rightarrowwr_1,\ldots,\rightarrowwr_n)) =&
\bigcup\fa(\rightarrowwr_i)
&&
\\[1.5ex]
\fa(\bot) =& \varnothing
&
\fa(\rightarrowwphi\limp\rightarrowwpsi)=& \fa(\rightarrowwphi)\cup \fa(\rightarrowwpsi)
\\
\fa(\tf P(\rightarrowwr)) =& \fa(\rightarrowwr)
&
\fa(\Forall{X}\rightarrowwphi)=& \fa(\rightarrowwphi)
\end{array}
$$
Define \deffont{free unknowns} $\f{fV}(r)$ and $\f{fV}(\rightarrowwphi)$ by:
$$
\begin{array}{r@{\ }l@{\quad}r@{\ }l@{\quad}r@{\ }l}
\f{fV}(a)=& \varnothing
&
\f{fV}(\pi\act X)=& \{X\}
&
\f{fV}(\tf f(\rightarrowwr)) =& \f{fV}(\rightarrowwr)
\\
\f{fV}([a]\rightarrowwr)=& \f{fV}(\rightarrowwr)
&
\f{fV}((\rightarrowwr_1,\ldots,\rightarrowwr_n)) =&
\bigcup\f{fV}(\rightarrowwr_i)
\\[1.5ex]
\f{fV}(\bot) =& \varnothing
&
\f{fV}(\rightarrowwphi\limp\rightarrowwpsi)=& \f{fV}(\rightarrowwphi)\cup \f{fV}(\rightarrowwpsi)
\\
\f{fV}(\tf P(\rightarrowwr)) =& \f{fV}(\rightarrowwr)
&
\f{fV}(\Forall{X}\rightarrowwphi)=& \f{fV}(\rightarrowwphi)\setminus\{X\}
\end{array}
$$
\end{defn}
\begin{lemm}
\leftarrowbel{lemm.fa.pi.r}
$\fa(\pi\act \rightarrowwr)=\pi\act \fa(\rightarrowwr)$ and $\fa(\pi\act\rightarrowwphi)=\pi\act\fa(\rightarrowwphi)$.
Also,
$\f{fV}(\Pi\act \rightarrowwr)=\Pi\act \f{fV}(\rightarrowwr)$ and $\f{fV}(\Pi\act\rightarrowwphi)=\Pi\act\f{fV}(\rightarrowwphi)$.
\end{lemm}
\begin{proof}
By routine inductions on $\rightarrowwr$.
\end{proof}
\subsection{$\alpha$-equivalence}
\leftarrowbel{subsect.aeq}
The use of permissive-nominal terms allows us to `just quotient' syntax by $\alpha$-equivalence.
We can do this for both level 1 variable symbols (atoms) and level 2 variable symbols (unknowns).
\begin{defn}
Call a relation $\somerel$ on terms and on propositions a \deffont{congruence} when it is closed under the following rules:\footnote{We do not assume a congruence is an equivalence relation. This is because in a more general context we are interested in rewriting relations, which satisfy the rules below but are not equivalence relations.}
$$
\begin{array}{c@{\qquad}c}
\begin{prooftree}
\rightarrowwr_i\somerel \rightarrowws_i\quad 1\leq i\leq n
\justifies
(\rightarrowwr_1,\ldots,\rightarrowwr_n)\somerel (\rightarrowws_1,\ldots,\rightarrowws_n)
\end{prooftree}
&
\begin{prooftree}
\rightarrowwr\somerel \rightarrowws\ \ (\tf f:(\alpha)\tau,\ \rightarrowwr,\rightarrowws:\alpha)
\justifies
\tf f(\rightarrowwr)\somerel\tf f(\rightarrowws)
\end{prooftree}
\\[3ex]
\begin{prooftree}
\rightarrowwr\somerel \rightarrowws
\justifies
[a]\rightarrowwr\somerel [a]\rightarrowws
\end{prooftree}
&
\begin{prooftree}
\rightarrowwphi\somerel\rightarrowwphi'\quad \rightarrowwpsi\somerel\rightarrowwpsi'
\justifies
\rightarrowwphi\limp\rightarrowwpsi\somerel \rightarrowwphi'\limp\rightarrowwpsi'
\end{prooftree}
\\[3ex]
\begin{prooftree}
\rightarrowwr\somerel \rightarrowws\quad (\tf P:\alpha,\ \rightarrowwr,\rightarrowws:\alpha)
\justifies
\tf P(\rightarrowwr)\somerel \tf P(\rightarrowws)
\end{prooftree}
&
\begin{prooftree}
\rightarrowwphi\somerel \rightarrowwphi'
\justifies
\Forall{X}\rightarrowwphi\somerel \Forall{X}\rightarrowwphi'
\end{prooftree}
\end{array}
$$
\end{defn}
\begin{defn}
\leftarrowbel{defn.aeq}
Write $(a\ b)$ for the \deffont{(level 1) swapping} permutation which maps $a$ to $b$ and $b$ to $a$ and all other $c$ to themselves.
Similarly, provided $\sort(X)=\sort(Y)$ and $\pmss(X)=\pmss(Y)$, write $(X\ Y)$ for the \deffont{(level 2) swapping}.
Define \deffont{$\alpha$-equivalence} $\aeq$ on terms and propositions to be the least equivalence relation that is a congruence and is such that:
\begin{frameqn}
\begin{array}{c@{\qquad}c}
\begin{prooftree}
(a,b\not\in\fa(\rightarrowwr))
\justifies
(b\ a)\act \rightarrowwr \aeq r
\end{prooftree}
&
\begin{prooftree}
(X,Y\not\in\f{fV}(\rightarrowwphi))
\justifies
(Y\ X)\act\rightarrowwphi\aeq \rightarrowwphi
\end{prooftree}
\end{array}
\end{frameqn}
\end{defn}
\begin{xmpl}
We $\alpha$-convert $X$ and $a$ in $\Forall{X}\tf P([a]X)$.
Let $\sort(Y)=\sort(X)$ and $\pmss(Y)=\pmss(X)$.
Suppose $b\not\in\pmss(X)$.
Using $(a\ b)$ and $(X\ Y)$ we deduce:
$$
\begin{array}{r@{\quad}c@{\quad}l}
\Forall{X}\tf P([a]X) &\stackrel{(a\ b)}{\aeq}& \Forall{X}\tf P([b](b\ a)\act X)
\\
&\stackrel{(X\ Y)}{\aeq}& \Forall{Y}\tf P([b](b\ a)\act Y) .
\end{array}
$$
It is routine to convert this sketch into a full derivation-tree.
\end{xmpl}
\begin{frametxt}
\begin{defn}
\leftarrowbel{defn.terms.and.propositions}
For each signature $\mathcal S$, we take terms and propositions quotiented by $\alpha$-equivalence.
\end{defn}
\end{frametxt}
\subsection{Substitution}
\begin{frametxt}
\begin{defn}
A (level 2) \deffont{substitution} $\theta$ is a function from unknowns to terms such that:
\begin{itemize*}
\item
For all $X$, $\theta(X):\sort(X)$ and $\fa(\theta(X))\subseteq \pmss(X)$.
\item
$\theta(X)= \id\act X$ for all but finitely many $X$.
\end{itemize*}
$\theta$ will range over substitutions.
\end{defn}
\end{frametxt}
\begin{defn}
Define $\f{nontriv}(\theta)$ by:
$$
\f{nontriv}(\theta)=
\{X\mid \theta(X){\not=} \id\act X \text{ or } X{\in}\f{fV}(\theta(Y))\text{ for some }Y\}
$$
\end{defn}
$\f{nontriv}(\theta)$ is unknowns that can be produced or consumed by $\theta$, other than in the trivial manner that $\theta(X)=\id\act X$.
\begin{defn}
\leftarrowbel{defn.subst.action}
Define a \deffont{substitution action} by:
\begin{frameqn}
\begin{array}{r@{\ }l@{\qquad}r@{\ }l}
a\theta=& a
&
(r_1,\ldots,r_n)\theta=& (r_1\theta,\ldots,r_n\theta)
\\
([a]r)\theta=& [a](r\theta)
&
(\pi\act X)\theta=& \pi\act \theta(X)
\\
\tf f(r)\theta=& \tf f(r\theta)
\\
\bot\theta=& \bot
&
(\phi\limp\psi)\theta=& (\phi\theta)\limp\psi\theta
\\
(\tf P(r))\theta=& \tf P(r\theta)
&
(\Forall{X}\phi)\theta =& \Forall{X}(\phi\theta) \quad (X\not\in\f{nontriv}(\theta))
\end{array}
\end{frameqn}
\end{defn}
\begin{rmrk}
Level 2 substitution $r\theta$ is capturing for level 1 abstraction $[a]\text{-}$.
For example if $\theta(X)=a$ then $([a]X)\theta= [a]a$.
This is the behaviour displayed by the informal meta-level when we write ``take $t$ to be $x$ in $\leftarrowm{x}t$''.
\end{rmrk}
\subsection{Sequents and derivability}
\begin{defn}
\leftarrowbel{defn.seq}
$\Phi$ and $\Psi$ will range over sets of propositions.
We may write $\phi,\Phi$ and $\Phi,\phi$ as shorthand for $\{\phi\}\cup\Phi$ (where we do not insist that $\phi\not\in\Phi$, that is, the union need not be disjoint).
\begin{itemize*}
\item
A \deffont{sequent} of restricted PNL is a pair $\Phi\nopicent\Psi$.
\item
A \deffont{sequent} of full PNL is a pair $\Phi\cent\Psi$.
\end{itemize*}
Write
$\f{fV}(\Phi,\Psi)=\bigcup\{\f{fV}(\phi)\mid \phi\in\Phi\}\cup\bigcup\{\f{fV}(\psi)\mid\psi\in\Psi\}$.
\end{defn}
\begin{frametxt}
\begin{defn}[Derivable sequents]
Define the \deffont{derivable sequents} of full PNL and restricted PNL by the rules in Figures~\ref{Seq} and~\ref{rSeq} respectively.
\end{defn}
\end{frametxt}
\noindent The sole difference between Figures~\ref{Seq} and~\ref{rSeq} is in the axiom rule, and is highlighted with a light blue rectangle.
\begin{figure*}
\caption{Sequent derivation rules of full Permissive-Nominal Logic}
\end{figure*}
\begin{figure*}
\caption{Sequent derivation rules of restricted Permissive-Nominal Logic}
\end{figure*}
\begin{nttn}
We may write $\Phi\nopicent\Psi$ as shorthand for `$\Phi\nopicent\Psi$ is a derivable sequent'.
We may write $\Phi\not\nopicent\Psi$ as shorthand for `$\Phi\nopicent\Psi$ is not a derivable sequent'.
Similarly for $\Phi\cent\Psi$ and $\Phi\not\cent\Psi$.
\end{nttn}
Figure~\ref{Seq} is the logic of \cite{gabbay:pernl-jv,gabbay:nomtnl}.
Figure~\ref{rSeq} is the logic we translate to HOL in this paper.
The only difference is the `$\pi$' in the axiom rule: full PNL has it (see \rulefont{Ax}), and restricted PNL does not (see \rulefont{Ax^\nopi}).
Restricted PNL is a subset of full PNL, in the sense that (obviously) $\Phi\nopicent\Psi$ implies $\Phi\cent\Psi$ (this suggests that the models of restricted PNL should be a superset of those of full PNL, which will indeed turn out to be the case; see Appendix~\ref{sect.completeness}).
Why the difference?
Because the translation to HOL identifies atoms with functional arguments.
Atoms are symmetric up to permutation in full PNL; this is built into \rulefont{Ax} in Figure~\ref{Seq}.
Functional arguments are typically not symmetric.
We might try to translate full PNL to HOL by translating $n!$ permutation instances of each $r$ or $\phi$, where $n$ is some notion of the number of atoms in $r$ or $\phi$ (cf. \emph{capture typings} in Definition~\ref{defn.capture.typing}); but that would be `cheating' in the sense that most of the syntax would then be generated by a meta-level `macro' which does $n!$ amount of work.
The issue here is not whether PNL can be encoded in HOL; the issue is whether it can be cleanly translated into HOL.
These are related but distinct questions.
To quickly see the difference in derivational power between full and restricted PNL, assume a name sort $\nu$, a proposition-former $\tf P:\nu$, and two atoms $a,b:\nu$.
Then the difference in the entailment relations of PNL and restricted PNL can be summed up as follows:
\begin{itemize*}
\item
$\tf P(a)\cent \tf P(a)$\ and\ $\tf P(a)\nopicent \tf P(a)$.
\item
$\tf P(a)\cent \tf P(b)$\ but\ \sout{$\tf P(a)\nopicent \tf P(b)$}.
\end{itemize*}
In Appendix~\ref{sect.completeness} we see that this difference corresponds in models to proposition-formers being interpreted by equivariant functions (for full PNL) or not necessarily equivariant functions (for restricted PNL).
It has to be this way:
Definition~\ref{defn.translation} translates PNL terms and predicates to HOL terms and predicates.
In Lemma~\ref{lemm.it.has.to.be} we illustrate why only restricted PNL can be translated to HOL by our translation: the derivability of full PNL is too strong for HOL derivability and the translation would not be sound.
Note that this does not prove that other translations to HOL do not exist, but (as the discussion of $n!$ above suggests) we speculate that they would be significantly less natural.
\section{HOL syntax and derivability}
\leftarrowbel{sect.hol}
Higher-order logic (HOL) syntax and derivability should be familiar \cite{miller:logho,farmer:sevvst,andrews:intmlt,church:forstt}.
We give the basics.
\subsection{Syntax}
We present HOL as a derivation system over simply-typed $\leftarrowmbda$-terms with constants and types for logical reasoning (like a type of truth-values and constant symbols like $\limp$ and $\forall$).
This is all standard.
\begin{defn}
\leftarrowbel{defn.hol.sort.sig}
A \deffont{HOL signature} is a set $\mathcal D$ of \deffont{base types}, which includes a distinguished base type of \deffont{truth-values} $o\in\mathcal D$.
$\basetype$ will range over base types.
A \deffont{type-language} is defined by
\begin{frameqn}
\beta ::= \basetype \mid
(\beta,\ldots,\beta) \mid \beta\to\beta
.
\end{frameqn}
\end{defn}
It is not necessary to include products $(\beta_1,\ldots,\beta_n)$, but for the purposes of translating PNL into HOL doing this is convenient.
\begin{defn}
\leftarrowbel{defn.hol.term.signature}
A \deffont{term-signature} over a HOL signature $\mathcal D$ is a tuple $(\mathcal G,\type)$ where:
\begin{itemize*}
\item
$\mathcal G$ is a set of \deffont{constants}, which must contain elements $\bot$, $\limp$, and $\forall_\beta$ for every type $\beta$.
\item $\type$ assigns to each ${\tf g\in\mathcal G}$ a
type $\beta$
in the type-language
determined by $\mathcal D$, such that $\type(\bot)=o$, $\type(\limp)=o\to o\to o$, and $\type(\forall_\beta)=(\beta\to o)\to o$.\footnote{The authors deprecate calling this `higher-order abstract syntax' (HOAS), as sometimes happens. We should reserve that term for inductive types with binding constructed using constants of higher type like $(\Lambda\to\Lambda)\to\Lambda$ (strong HOAS) or $(\nu\to\Lambda)\to\Lambda$ (weak HOAS) \cite{despeyroux94higherorder,pfenning:hoas}.
A term $\forall_\beta:(\beta\to o)\to o$ (plus axioms) expresses the \emph{meaning} of $\forall$ \cite[Section~2]{church:forstt} and would still have meaning if our syntax was, e.g. combinators. In contrast, the \emph{syntax} of combinators could be represented without any need for higher-order syntax, since it does not have binders \cite[Section~2]{hindley:lamcci}.}
\end{itemize*}
A \deffont{signature} $\mathcal T$ is then a tuple $(\mathcal D,\mathcal G,\type)$.
\end{defn}
We write $\tf g:\beta$ for $\type(\tf g)=\beta$.
\begin{defn}
\leftarrowbel{defn.hol.terms.sorts}
For each signature $\mathcal T=(\mathcal D,\mathcal G,\type)$ and each type $\beta$ over $\mathcal D$ fix a countably infinite set of \deffont{variables} of that type.
$X,Y,Z$ will range over distinct HOL variables.\footnote{This means that if the reader sees `$X$' this could refer either to a HOL variable or---recalling Definition~\ref{defn.term.signature}---to a PNL unknown.
We will make sure that it is always clear from context which is meant.}
Write $\type(X)$ for the type of $X$.
\end{defn}
\begin{defn}
For each signature $\mathcal T$ define \deffont{HOL terms} over $\mathcal T$ by
$$
t::= X \mid \leftarrowm{X}t \mid tt \mid (t,\ldots,t) \mid \tf g
$$
and a \deffont{typing} relation by:
\begin{frameqn}
\begin{array}{c@{\qquad}c@{\qquad}c@{\qquad}c}
\begin{prooftree}
\rightarrowwt:\beta\ \ (\type(X){=}\beta')
\justifies
\leftarrowm{X}\rightarrowwt:\beta'{\to}\beta
\end{prooftree}
&
\begin{prooftree}
\rightarrowwt':\beta'\quad \rightarrowwt:\beta'{\to}\beta
\justifies
\rightarrowwt'\rightarrowwt:\beta
\end{prooftree}
&
\begin{prooftree}
\rightarrowwt_1:\beta_1 \ \ldots\ \rightarrowwt_n:\beta_n
\justifies
(\rightarrowwt_1,\ldots,\rightarrowwt_n):(\beta_1,\ldots,\beta_n)
\end{prooftree}
&
\begin{prooftree}
(\type(\tf g){=}\mu)
\justifies
\tf g:\mu
\end{prooftree}
\end{array}
\end{frameqn}
\end{defn}
We now define $\alpha$-equivalence.
We would not normally be so detailed about this, but when we map PNL terms and propositions to HOL later, it will be useful to have been precise here:
\begin{defn}
\leftarrowbel{defn.hol.perm}
A \deffont{permutation} of HOL variables is a bijection $\varpi$ such that $\f{nontriv}(\varpi)=\{X\mid \varpi(X)\neq X\}$ is finite.
Give HOL terms a permutation action $\varpi\act t$ defined by:
\maketab{tab4}{R{6.5em}@{}C{1em}@{}L{6em}@{\quad}R{4em}@{}C{1em}@{}L{5em}@{\quad}R{3em}@{}C{1em}@{}L{7em}}
\begin{tab4}
\varpi\act X&=&\varpi(X)
&
\varpi\act \leftarrowm{X}t&=&\leftarrowm{\varpi(X)}\varpi\act t
&
\varpi\act (t't)&=&(\varpi\act t')(\varpi\act t)
\\
\varpi\act(t_1,\dots,t_n)&=&(\varpi\act t_1,\dots,\varpi\act t_n)
&
\varpi\act \tf g&=&\tf g
\end{tab4}
Free variables are defined by:
\begin{tab4}
\f{fv}(X)&=&\{X\}
&
\f{fv}(\leftarrowm{X}t)&=&\f{fv}(t)\setminus\{X\}
&
\f{fv}(t't)&=&\f{fv}(t')\cup\f{fv}(t)
\\
\f{fv}((t_1,\dots,t_n))&=&\bigcup_i\f{fv}(t_i)
&
\f{fv}(\tf g)&=&\varnothing
\end{tab4}
Call a relation $\somerel$ on HOL terms a \deffont{congruence} when it is closed under the following rules:
$$
\begin{prooftree}
t\somerel u
\justifies
\leftarrowm{X}t\somerel \leftarrowm{X}u
\end{prooftree}
\qquad
\begin{prooftree}
t'\somerel u'\quad t\somerel u
\justifies
t't\somerel u'u
\end{prooftree}
\qquad
\begin{prooftree}
t_i\somerel u_i \quad (1\leq i\leq n)
\justifies
(t_1,\dots,t_n)\somerel (u_1,\dots,u_n)
\end{prooftree}
$$
Define $\alpha$-equivalence to be the least congruence that is an equivalence relation and is such that:
$$
\begin{prooftree}
(X,Y\not\in\f{fv}(t))
\justifies
(Y\ X)\act t\aeq t
\end{prooftree}
$$
We quotient terms by $\alpha$-equivalence and define \deffont{capture-avoiding substitution} $\rightarrowwt[X\ssm u]$ as usual.
\end{defn}
\begin{defn}
We write $\rightarrowwt{\,:\,}\beta$ for \emph{$\rightarrowwt$ is a term and has type $\beta$}.
We call $\rightarrowwt$ \deffont{typable} when $\rightarrowwt:\beta$ for some type $\beta$.
We call a term a \deffont{HOL proposition} when it has type $o$.
$\xi$ and $\chi$ will range over HOL propositions.
\end{defn}
\begin{defn}
\leftarrowbel{defn.hol.seq}
$\Xi$ and $\Chi$ will range over sets of HOL propositions.
We may write $\xi,\Xi$ and $\Xi,\xi$ as shorthand for $\{\xi\}\cup\Xi$.
Write
$\f{fV}(\Xi,\Chi)=\bigcup\{\f{fV}(\xi)\mid \xi\in\Xi\}\cup\bigcup\{\f{fV}(\chi)\mid\chi\in\Chi\}$.
A \deffont{sequent} is a pair $\Xi\holcent\Chi$.
\end{defn}
\begin{frametxt}
\begin{defn}[Derivable sequents]
The \deffont{derivable sequents} are defined in Figure~\ref{hol.Seq}.
\end{defn}
\end{frametxt}
\begin{figure*}
\caption{Sequent derivation rules of Higher-Order Logic}
\end{figure*}
\section{The translation from nominal to functional syntax, and its soundness}
\leftarrowbel{sect.translation.sound}
\subsection{Translation from PNL to higher-order logic}
\leftarrowbel{subsect.pnl.to.hol}
In this subsection we show how to translate a PNL signature $\mathcal S$ and propositions and terms in that signature, to a higher-order logic (HOL) signature and propositions and terms in that signature.
We start by translating a PNL signature $\mathcal S$ to a HOL signature $\mathcal T_{\mathcal S}$.
First, we set up some notation:
\begin{nttn}
\leftarrowbel{nttn.finite.lists}
Let $D$ range over finite lists of distinct atoms.
\begin{itemize*}
\item
Write $a\in D$ when $a$ occurs in $D$.
\item
Write $D'\subseteq D$ when every element in $D'$ occurs in $D$ (disregarding order).
Similarly if $S$ is a set of atoms write $D\subseteq S$ when every element in $D$ occurs in $S$.
\item
If $S$ is a set of atoms write $D\cap S$ for the list obtained by removing from $D$ just those atoms not in $S$.
Also write $D_X$ as shorthand for $D\cap\pmss(X)$.
\item
Write $\pi\act D$ for the list obtained by applying $\pi$ pointwise to the elements of $D$ in order.
\item
Write $D,a$ for the list obtained by appending $a$; when we write this we include an assumption that $a\not\in D$.
\item
Write $\leftarrowm{D}t$ for $\leftarrowm{d_1}\dots\leftarrowm{d_n}t$ where $D=[d_1,\dots,d_n]$.
\end{itemize*}
\end{nttn}
\begin{defn}
\leftarrowbel{defn.TS}
\leftarrowbel{defn.hol.translation}
From a PNL signature $\mathcal S$ determine a HOL signature $\mathcal T_{\mathcal S}$ by the following specification:
\begin{itemize*}
\item
For every atoms-sort $\nu$ in $\mathcal S$ assume a HOL base type $\mu_\nu$.
\item
For every base sort $\tau$ assume a HOL type $\mu_\tau$.
\end{itemize*}
Translate sorts in $\mathcal S$ to types in $\mathcal T_{\mathcal S}$ as follows:
\begin{frameqn}
\begin{array}{r@{\ }l@{\qquad}r@{\ }l@{\qquad}r@{\ }l}
\hol{}{\nu}=&\mu_\nu
&
\hol{}{\tau}=&\mu_\tau
&
\hol{}{(\alpha_1,\ldots,\alpha_n)}=&(\hol{}{\alpha_1},\cdots,\hol{}{\alpha_n})
\\
\hol{}{[\nu]\alpha}=&\nu\to\hol{}{\alpha}
\end{array}
\end{frameqn}
\begin{itemize*}
\item
For every term-former $\tf f:(\alpha)\tau$ assume a HOL constant $\tf g_{\smtf f}:\hol{}{\alpha}\to\tau$.
\item
For every proposition-former $\tf P:\alpha$ assume a HOL constant $\tf g_{\smtf P}:\hol{}{\alpha}\to o$.
\item
For every atom $a:\nu$ assume a HOL variable $a:\nu$.
It is convenient to assume this correspondence is a literal identity; i.e. that $\mathbb A_\nu$ is actually a subset of the set of HOL variables of type $\nu$, and that there are countably infinitely many HOL variables of type $\nu$ that are not atoms.
In particular, this means that every permutation $\pi$ in the sense of Definition~\ref{defn.permutation} is also a permutation $\varpi$ in the sense of Definition~\ref{defn.hol.perm}.
\item
For every unknown $X:\alpha$ and list $D$ assume a distinct HOL variable $X_D$ that is not an atom\footnote{So $X$ is one of the countably infinitely many HOL variables that are not atoms.} of type $\nu_{\GammaX}\to\hol{}\alpha$ where $\nu_{\GammaX}$ is the sorts of the atoms in $\GammaX$, in order.
\end{itemize*}
\end{defn}
\begin{frametxt}
\begin{defn}
\leftarrowbel{defn.translation}
Given a list $D$ translate PNL terms and propositions in $\mathcal S$ to HOL terms and propositions in $\mathcal T_{\mathcal S}$ (Definition~\ref{defn.TS}) by the rules in Figure~\ref{fig.hol.translation}.
\end{defn}
(The notation $\pi\act\GammaX$ is defined in Notation~\ref{nttn.finite.lists}.)
\end{frametxt}
\begin{figure}
\caption{Translation from PNL to HOL}
\end{figure}
\begin{xmpl}
\leftarrowbel{xmpl.why.capturable}
Suppose $\GammaX$ (Notation~\ref{nttn.finite.lists})
is the list $[a]$ and write $X$ for $X_D$.
Assume a proposition-former $\tf{equal}$ of appropriate arity.
Then:
$$
\begin{gathered}
\hol{D}{\id\act X}=Xa
\quad
\hol{D}{(b\ a)\act X}=Xb
\quad
\hol{D}{[a]\id\act X}=\leftarrowm{a}(Xa)
\quad
\hol{D}{[b](b\ a)\act X}=\leftarrowm{b}(Xb)
\\
\hol{D}{\Forall{X}\tf{equal}([a]X,[b](b\ a)\act X)}=\forall\,\leftarrowm{X}(\tf{equal}(\leftarrowm{a}(Xa))(\leftarrowm{b}(Xb)))
\end{gathered}
$$
Assuming appropriate axioms for $\tf{equal}$, we would expect this to be true.
Now assume $\GammaY$ is the list $[a,b]$ and write $Y$ for $Y_{\GammaY}$.
Then:
$$
\begin{gathered}
\hol{D}{\id\act Y}=Yab
\quad
\hol{D}{(b\ a)\act Y}=Yba
\quad
\hol{D}{[a]\id\act Y}=\leftarrowm{a}(Yab)
\quad
\hol{D}{[b](b\ a)\act Y}=\leftarrowm{b}(Yba)
\\
\hol{D}{\Forall{Y}\tf{equal}([a]Y,[b](b\ a)\act Y)}=\forall\,\leftarrowm{Y}(\tf{equal}(\leftarrowm{a}(Yab))(\leftarrowm{b}(Yba)))
\end{gathered}
$$
We would expect this to be false.
What has changed with respect to the previous case, is that $b$ is fresh for $X$ but not for $Y$.
\end{xmpl}
\begin{lemm}
\leftarrowbel{lemm.hol.gamma.fa}
\begin{itemize*}
\item
Suppose $a$ is an atom.
Then if $a\in\f{fv}(\hol{D}{r})$ then $a\in\fa(r)$.
\item
$\hol{D}{\pi\act r}=\pi\act\hol{D}{r}$ (for $\pi$ on the right-hand side considered as a permutation of HOL variables).
\end{itemize*}
As a corollary, the translation $\hol{D}{r}$ is well-defined.
That is, if $r$ and $s$ are $\alpha$-equivalent then $\hol{D}{r}=\hol{D}{s}$.
\end{lemm}
\begin{proof}
By routine inductions on $r$.
The proof that $\fa(\pi\act X)\subseteq\f{fv}(\hol{D}{\pi\act X})$ uses the assumption that $\GammaX\subseteq\pmss(X)$.
The corollary follows; for more details see \cite[Section~8]{gabbay:perntu-jv}.
\end{proof}
\subsection{Capture typing}
In order to translate to HOL, some atoms are `important' and others are not.
This is expressed by a \emph{capture typing}, an idea going back to \cite{gabbay:perntu,gabbay:perntu-jv}.
\begin{defn}
\leftarrowbel{defn.capture.typing}
Define \deffont{capture typings} $D\cent r:A$ and $D\cent\phi:A$ inductively by the rules in Figure~\ref{fig.capture.typings}.
Here $D$ ranges over finite lists of distinct atoms as described in Notation~\ref{nttn.finite.lists}, and $A$ ranges over finite sets of atoms.
If $A=\varnothing$ then we may omit the `${:}A$' and write just $D\cent r$ and $D\cent\phi$.
Write $D\cent\Psi$ when $D\cent\psi$ for every $\psi\in\Psi$.
\end{defn}
\begin{figure}
\caption{Capture typing}
\end{figure}
\begin{rmrk}
\leftarrowbel{rmrk.capture.closed}
The interesting case in Figure~\ref{fig.capture.typings} is the rule for $\pi\act X$.
This ensures that $D$ is large enough to record all the important atoms in $\pi$ or abstracted further up in the term---that is, those permitted in $X$---so that we do not lose information when we form $\hol{D}{\pi\act X}=X\pi\act\GammaX$.
This is made formal in Proposition~\ref{prop.capturable.minimal}, which is Theorems~8.12 and~8.14 of \cite{gabbay:perntu-jv}:
\end{rmrk}
\begin{prop}
\leftarrowbel{prop.capturable.minimal}
\begin{itemize*}
\item
If $D\cent r$ and $D\cent s$ then $\hol{D}{r}=\hol{D}{s}$ implies $r=s$ (note that $=$ denotes $\alpha$-equality, because we quotiented terms by this relation), and similarly for $\phi$ and $\psi$.
\item
If $D\not\cent r$ then there exists $s$ such that $\hol{D}{r}=\hol{D}{s}$ yet $r\neq s$, and similarly for $\phi$.
\end{itemize*}
\end{prop}
Definition~\ref{defn.translation} maps PNL terms and predicates to typable HOL terms:
\begin{prop}
\leftarrowbel{prop.typable.hol.gamma.r}
If $r:\alpha$ then for any $D$,\ $\hol{D}{r}:\hol{}{\alpha}$,
\ and $\hol{D}{\phi}:o$.
\end{prop}
\begin{proof}
By inductions on $r$ and $\phi$.
\begin{itemize*}
\item
\emph{The case $a\in\mathbb A_\nu$.}\quad
$a:\nu$ by definition.
\item
\emph{The case $[a]r$ where $a\in\mathbb A_\nu$.}\quad
By inductive hypothesis $\hol{D}{r}:\beta$ for some type $\beta$.
It follows that $\hol{D}{[a]r}=\leftarrowm{a}\hol{D}{r}:\nu\to\beta$.
\item
\emph{The case $\pi\act X$.}\quad
Suppose $D\cent \pi\act X$.
It is routine to check that $X_\Gamma\pi\act\GammaX:\hol{}{\sort(X)}$.
\qedhere\end{itemize*}
\end{proof}
\subsection{Re-indexing capture contexts}
When we prove soundness of the translation (Theorem~\ref{thrm.soundness}) there will be a problem, because we are interested in proving soundness of translating a sequent $\Phi\nopicent\Psi$ but because we work by induction on derivations $\Pi$ we may have to translate all sequents in $\Pi$, some of which might have `extra' capturable atoms.
We need to translate using a large $\Gamma'$ and then re-index to $\Gamma$:
\begin{defn}
\leftarrowbel{defn.thetaC}
Define a substitution $\inter{\Gamma'\sm\Gamma}$ by:
$$
\begin{array}{l@{\ =\ }l@{\qquad}l}
\inter{\Gamma'\sm\Gamma}(X_{\Gamma'}) & \leftarrowm{\GammapX}(X_{\Gamma}\GammaX)
\\
\inter{\Gamma'\sm\Gamma}(Y)&Y &\text{all other}\ Y
\end{array}
$$
\end{defn}
\maketab{tab3}{@{\hspace{-2em}}R{10em}@{\ }L{10em}@{\quad}L{17em}}
\maketab{tab7}{@{\hspace{-2em}}R{10em}@{\ }L{12em}L{14em}}
\begin{thrm}
\leftarrowbel{thrm.sub.composition}
If $D'\cent r:A$ then $\hol{D}{r}\abeq \hol{D'}{r}\inter{D'\sm D}$.
Similarly, if $D'\cent \phi:A$ then $\hol{D}{\phi}\abeq\hol{D'}{\phi}\inter{D'\sm D}$.
\end{thrm}
\begin{proof}
By inductions on $r$ and $\phi$.
We consider a selection of cases:
\begin{itemize*}
\item
The case $\pi \act X$.\quad
We reason as follows:
\begin{tab3}
\hol{D'}{\pi\act X}\inter{D'\sm D}=&(X_{D'}\pi\act\GammapX)\inter{D'\sm D}
& \text{Definition~\ref{defn.translation}}
\\
=&(\leftarrowm{\GammapX}(X_{D}\GammaX))\pi\act\GammapX
&\text{Definition~\ref{defn.thetaC}}
\\
=&X_{D}\pi\act\GammaX &\nontriv(\pi)\cap\pmss(X)\subseteq\GammapX
\end{tab3}
\item
The case $[a]r$.\quad
We reason as follows:
\begin{tab3}
\hol{D'}{[a]r}\inter{D'\sm D}
=&(\leftarrowm{a}\hol{D'}{r})\inter{D'\sm D} &\text{Definition~\ref{defn.translation}}
\\
=&\leftarrowm{a}(\hol{D'}{r}\inter{D'\sm D}) &\text{taking $a\not\in D,D'$}
\\
=&\leftarrowm{a}(\hol{D}{r})
&\text{ind. hyp.}
\\
=&\hol{D}{\leftarrowm{a}r}
&\text{Definition~\ref{defn.translation}}
\end{tab3}
\item
The case $\Forall{X}\phi$.\quad
We reason as follows:
\begin{tab3}
\hol{D'}{\Forall{X}\phi}\inter{D'\sm D}
=&(\forall \leftarrowm{X}\hol{D'}{\phi})\inter{D'\sm D}
&\text{Definition~\ref{defn.translation}}
\\
=&\forall \leftarrowm{X}(\hol{D'}{\phi}\inter{D'\sm D})
&\text{fact}
\\
=&\forall \leftarrowm{X}\hol{D}{\phi}
&\text{ind. hyp.}
\\
=&\hol{D}{\Forall{X}\phi}
&\text{Definition~\ref{defn.translation}}
\end{tab3}
\end{itemize*}
\end{proof}
\subsection{Soundness of the translation}
Recall that HOL terms have a permutation action $\pi\act t$ given by considering $\pi$ as a permutation on HOL variables and using Definition~\ref{defn.hol.perm}.
Then:
\begin{lemm}
\leftarrowbel{lemm.hol.pi}
If $\f{nontriv}(\pi)\cap\f{fv}(t)\subseteq D$ then $(\leftarrowm{D}t)\pi\act D\abeq\pi\act t$ (see Notation~\ref{nttn.finite.lists}).
\end{lemm}
\begin{proof}
A fact of $\alpha\beta$-conversion \cite[Lemma~9.2]{gabbay:perntu-jv}.
\end{proof}
\begin{defn}
Write $r':X$ when $r':\sort(X)$ and $\fa(r')\subseteq\pmss(X)$.
\end{defn}
\begin{lemm}
\leftarrowbel{lemm.hol.sub}
Suppose $D\cent r$ and $D\cent\phi$.
Suppose $r':X$.
Then:
\begin{itemize*}
\item
$\hol{D}{r[X\ssm r']} \abeq \hol{D}{r}[X\ssm \leftarrowm{\GammaX}\hol{D}{r'}]$.
\item
$\hol{D}{\phi[X\ssm r']} \abeq \hol{D}{\phi}[X\ssm \leftarrowm{\GammaX}\hol{D}{r'}]$.
\end{itemize*}
\end{lemm}
\begin{proof}
By routine inductions on $r$ and $\phi$.
We sketch two cases:
\begin{itemize}
\item
\emph{The case $(\pi\act X)[X\ssm r']$.}\quad
We must prove that
$$
\hol{D}{\pi\act r'}\abeq \bigl(\leftarrowm{\GammaX}\hol{D}{r'}\bigr)\pi\act\GammaX.
$$
This follows by Lemmas~\ref{lemm.hol.gamma.fa} and~\ref{lemm.hol.pi}.
\item
\emph{The case $\tf P(r)[X\ssm r']$.}\quad
We must prove that
$$
\hol{D}{\tf P(r[X\ssm r'])}\abeq \tf g_{\smtf P}(\hol{D}{r})[X\ssm\leftarrowm{\GammaX}\hol{D}{r'}].
$$
This follows directly from the first part.
\qedhere\end{itemize}
\end{proof}
\begin{prop}
\leftarrowbel{prop.hol.forall.sound}
Suppose $D\cent\phi$ and $D\cent r':X$.
Then $\hol{D}{\Forall{X}\phi}\holcent \hol{D}{\phi[X\ssm r']}$.
\end{prop}
\begin{proof}
Using Lemma~\ref{lemm.hol.sub} and \rulefont{h\forall L} from Figure~\ref{hol.Seq}.
\end{proof}
\begin{frametxt}
\begin{thrm}
\leftarrowbel{thrm.soundness}
The interpretation is sound: if $\Phi\nopicent\Psi$ and $D\cent\Phi$ and $D\cent\Psi$ then $\hol{D}{\Phi}\holcent\hol{D}{\Psi}$.
\end{thrm}
\end{frametxt}
\begin{proof}
Choose $D'$ such that $D'\cent\Psi'$ and $D'\cent\Phi'$ for every sequent $\Psi'\cent\Phi'$ appearing in $\Pi$---it is not hard to verify that some such $D'$ must exist.
It is routine to verify by induction on $\Pi$ that $\hol{D'}{\Phi'}\holcent\hol{D'}{\Psi'}$ is derivable; the case of \rulefont{\forall R} uses Proposition~\ref{prop.hol.forall.sound}.
So in particular $\hol{D'}{\Phi}\holcent\hol{D}{\Psi'}$.
It follows, applying the substitution $\inter{D'\sm D}$ to both sides and using Theorem~\ref{thrm.sub.composition}, that $\hol{D}{\Phi}\holcent\hol{D}{\Psi}$.
\end{proof}
\begin{lemm}
\leftarrowbel{lemm.it.has.to.be}
The interpretation for full PNL (Figure~\ref{Seq}, with the stronger axiom rule) would not be sound.
That is, there exist $\Phi$ and $\Psi$ and $D$ such that $D\cent\Phi$, $D\cent\Psi$, and $\Phi\cent\Psi$, but $\hol{D}{\Phi}\not\holcent\hol{D}{\Psi}$.
\end{lemm}
\begin{proof}
Consider a name sort $\nu$ and a unary predicate $\tf P:\nu$.
Then $\tf P(a)\cent\tf P(b)$ in full PNL, but it is not the case that $\tf g_{\tf P}a \cent\tf g_{\tf P}b$ in HOL.
\end{proof}
\section{Semantics}
\leftarrowbel{sect.semantics}
For the reader's convenience we will clarify one aspect of the coming notation now: if the reader sees $\ns X$ this is a set with a permutation action; if the reader sees $\rs X$ this is a set with a renaming action.
There is no particular connection between $\ns X$ and $\rs X$.
A typical renaming is $[a\ssm b]$ (instead of a typical permutation $(a\ b)$).
Formal definitions are in Definition~\ref{defn.permutation} and~\ref{defn.renaming}.
The reader may not be surprised by the use of sets with a permutation action---nominal techniques are based on these \cite{gabbay:newaas-jv}.
But why the renaming action?
We need renamings to make a function out of an atoms-abstraction, mirroring the clause $\hol{D}{[a]r}=\leftarrowm{a}\hol{D}{r}$ in Definition~\ref{defn.translation}.
In PNL models, an abstraction $[a]r$ is modelled as Gabbay-Pitts atoms-abstraction $[a]x$, a sets-based construction from \cite{gabbay:newaas-jv} (Definition~\ref{defn.abstraction.sets}, in this paper).
This is constructed like a pair, from $a$ and $x$, but destructed like a \emph{partial function} the graph of which is evident in Definition~\ref{defn.abstraction.sets}.
It is defined for fresh $b$ but not for $b\in\supp(x)\setminus\{a\}$.
When we translate $[a]r$ to HOL we interpret $[a]r$ as a function using $\leftarrowmbda$-abstraction.
This suggests of our models that we translate a \emph{partial} function $[a]x$ to a total function.
But then we have to give meaning to $[a]x$ applied to $b$ where $b$ is not fresh.
This is where renaming sets are used.
We can then conclude by noting that every model of PNL can be transformed into a model of HOL, and in a compositional manner (Lemma~\ref{lemm.commuting.square}).
Completeness quickly follows.
\subsection{Categories of finitely-supported permutation and renaming sets}
\subsubsection{Permutation and renaming sets}
\begin{defn}
\leftarrowbel{defn.renaming}
Suppose $\rho$ is a map from $\mathbb A$ to $\mathbb A$.
Define $\dom(\rho)$ and $\img(\rho)$ by
$$
\dom(\rho)=\{a\mid \rho(a)\neq a\}
\quad\text{and}\quad
\img(\rho)=\{\rho(a)\mid a\in\dom(\rho)\} .
$$
Echoing Definition~\ref{defn.permutation}, a \deffont{renaming} is a map $\rho$ from $\mathbb A$ to $\mathbb A$ such that $a\in\mathbb A_\nu\liff \rho(a)\in\mathbb A_\nu$ and
$\f{nontriv}(\rho)=\dom(\rho)\cup\img(\rho)$ is finite.
Write $\mathbb R$ for the set of renamings.
For $a,b\in\mathbb A_\nu$ let an \deffont{atomic renaming} $[a\ssm b]$ map $a$ to $b$, $b$ to $b$, and other $c$ to themselves.
$\rho$ will range over renamings.
\end{defn}
\begin{frametxt}
\begin{defn}
\leftarrowbel{defn.perm.set}
\begin{itemize*}
\item
A \deffont{permutation set} is a pair $\ns X=(|\ns X|,\act)$ of an \deffont{underlying set} $|\ns X|$ and a \deffont{permutation action} $(\mathbb P\times|\ns X|)\to |\ns X|$ which is a group action; write it infix.
(So $\id\act x=x$ and $\pi\act(\pi'\act x)=(\pi\circ\pi')\act x$.)
\item
A \deffont{renaming set} is a pair $\rs X=(|\rs X|,\act)$ of an \deffont{underlying set} $|\rs X|$ and a \deffont{renaming action} $(\mathbb R\times|\rs X|)\to |\rs X|$ which is a monoid action; write it infix.
(So $\id\act x=x$ and $\rho\bigact(\rho'\bigact x)=(\rho\circ\rho')\bigact x$.)
\end{itemize*}
\end{defn}
\end{frametxt}
\begin{defn}
\leftarrowbel{defn.finsupp}
\begin{itemize*}
\item
Suppose $\ns X$ is a permutation set.
Say that $A\subseteq \mathbb A$ \deffont{supports} $x\in|\ns X|$ when for all $\pi,\pi'\in\mathbb P$, if $\Forall{a\in A}\pi(a)=\pi'(a)$ then $\pi\act x=\pi'\act x$.
\item
Suppose $\rs X$ is a renaming set.
Say that $A\subseteq \mathbb A$ \deffont{supports} $x\in|\rs X|$ when for all $\rho,\rho'\in\mathbb P$, if $\Forall{a\in A}\rho(a)=\rho'(a)$ then $\rho\bigact x=\rho'\bigact x$.
\end{itemize*}
\end{defn}
\begin{lemm}
If $x\in |\ns X|/|\rs X|$ has a supporting permission set (Definition~\ref{defn.atoms}) then it has a unique least supporting set which is equal to the intersection of all permission sets supporting $x$.
We call this the \deffont{support} of $x$ when it exists, and write it $\supp(x)$.
\end{lemm}
\begin{frametxt}
\begin{defn}
\begin{itemize*}
\item
Call $x\in |\ns X|/|\rs X|$ \deffont{supported} when $\supp(x)$ exists.
\item
Call $\ns X$/$\rs X$ \deffont{supported} when every element $x\in|\ns X|/|\rs X|$ is supported.
\end{itemize*}
\end{defn}
\end{frametxt}
\begin{lemm}
\leftarrowbel{lemm.supp.subsets}
\begin{itemize*}
\item
If $x\in|\ns X|$ then $\supp(\pi\act x)=\pi\act\supp(x)$.
\item
If $x\in|\rs X|$ then $\supp(\rho\bigact x)\subseteq\rho\bigact\supp(x)$.
As a corollary, if $\rho$ is injective on $\supp(x)$ then $\supp(\rho\bigact x)=\rho\bigact\supp(x)$.
\end{itemize*}
\end{lemm}
\begin{proof}
By routine calculations using the group/monoid action.
\end{proof}
\begin{xmpl}
The reverse subset inclusion in Lemma~\ref{lemm.supp.subsets} would not work.
For instance, consider $\mathbb A\times\mathbb A\cup\{\ast\}$ with the `exploding' renaming action such that:
\begin{itemize*}
\item
$\rho(\ast)=\ast$.
\item
$\rho\bigact(a,a)=(\rho(a),\rho(a))$.
\item
$\rho\bigact(a,b)=(\rho(a),\rho(b))$ if $\rho(a)\neq\rho(b)$.\footnote{Recall from Definition~\ref{defn.atoms} that by convention $a$ and $b$ are distinct.}
\item
$\rho\bigact(a,b)=\ast$ if $\rho(a)=\rho(b)$.
\end{itemize*}
Then $\supp([a\ssm b]\bigact (a,b))=\varnothing\subsetneq \{a\}=[a\ssm b]\bigact\supp((a,b))$.
\end{xmpl}
\subsubsection{Equivariant elements and maps}
\begin{defn}
\leftarrowbel{defn.equivariant.element}
Call an element $x$ in $|\ns X|/|\rs X|$ \deffont{equivariant} when $\supp(x)=\varnothing$.
\end{defn}
$x$ is equivariant when $\pi\act x=x$ for all $\pi$, or $\rho\bigact x=x$ for all $\rho$, respectively.
\begin{defn}
\leftarrowbel{defn.equivariant}
\begin{itemize*}
\item
Call a function $F\in |\ns X|\to|\ns Y|$ \deffont{equivariant} when
$$
\Forall{\pi{\in}\mathbb P}\Forall{x{\in}|\ns X|}F(\pi\act x)=\pi\act F(x).
$$
\item
Call a function $G\in |\rs X|\to|\rs Y|$ \deffont{equivariant} when
$$
\Forall{\rho{\in}\mathbb R}\Forall{x{\in}|\rs X|}G(\rho\bigact x)=\rho\bigact G(x).
$$
\end{itemize*}
$F$ and $G$ will range over equivariant functions between pairs of permutation and renaming sets respectively.
\end{defn}
\begin{lemm}
\leftarrowbel{lemm.equivar.reduces.supp}
\begin{enumerate*}
\item
Suppose $F\in |\ns X|\to|\ns Y|$ is equivariant.
Then $\supp(F(x))\subseteq \supp(x)$ for every $x\in|\ns X|$.
\item
Suppose $G\in |\rs X|\to|\rs Y|$ is equivariant.
Then $\supp(G(x))\subseteq \supp(x)$ for every $x\in|\rs X|$.
\end{enumerate*}
\end{lemm}
\begin{proof}
We consider only the second part.
Suppose $S$ supports $x$ so that for all $\rho$ and $\rho'$, if $\Forall{a\in S}\rho(a)=\rho'(a)$ then $\rho\bigact x=\rho'\bigact x$.
The result follows if we note that $\rho\bigact G(x)=G(\rho\bigact x)$ and $\rho'\bigact G(x)=G(\rho'\bigact x)$.
\end{proof}
\begin{frametxt}
\begin{defn}
\leftarrowbel{defn.fps}
\begin{itemize*}
\item
Write \theory{PmsPrm} for the category with objects supported permutation sets and arrows equivariant functions between them.
Henceforth, $\ns X$ and $\ns Y$ will range over objects in \theory{PmsPrm}.
\item
Write \theory{PmsRen} for the category with objects supported renaming sets and arrows equivariant functions between them.
Henceforth, $\rs X$ and $\rs Y$ will range over objects in \theory{PmsPrm}.
\end{itemize*}
\end{defn}
\end{frametxt}
\subsection{The exponential in \theory{PmsRen}}
\leftarrowbel{subsect.exp}
\theory{PmsPrm} and \theory{PmsRen} are both cartesian closed, but we only discuss exponentials for \theory{PmsRen} in this paper.
The reader can find the constructions for \theory{PmsPrm} e.g. in \cite[Section~9]{gabbay:fountl}.
\theory{PmsPrm} is used to give denotation to PNL only, while \theory{PrmRen} is used to give a denotation to PNL and also to HOL.
For this reason, the exponentials of \theory{PmsRen} are of specific and immediate importance to us, but not those of \theory{PmsPrm}.
\subsubsection{Functions}
Recall the definitions of $\dom$ and $\img$ from Definition~\ref{defn.renaming}.
\begin{frametxt}
\begin{defn}
\leftarrowbel{defn.exp.ren}
\begin{itemize*}
\item
Suppose $\ns X,\ns Y\in\theory{PmsPrm}$.
Suppose $f\in |\ns X|\to|\ns Y|$ ($f$ is not necessarily equivariant).
Call $f$ \deffont{supported} when there exists a permission set $S_f\subseteq\mathbb A$ such that for every $x\in |\ns X|$ and permutation $\pi\in\mathbb P$, if $\nontriv(\pi)\cap S_f=\varnothing$ then
$$
\pi\bigact(f(x)) = f(\pi\bigact x) .
$$
\item
Suppose $\rs X,\rs Y\in\theory{PmsRen}$.
Suppose $f\in |\rs X|\to|\rs Y|$ ($f$ is not necessarily equivariant).
Call $f$ \deffont{supported} when there exists a permission set $S_f\subseteq\mathbb A$ such that for every $x\in |\rs X|$ and renaming $\rho\in\mathbb R$, if $\dom(\rho)\cap S_f=\varnothing$ then
$$
\rho\bigact(f(x)) = f(\rho\bigact x) .
$$
\end{itemize*}
\end{defn}
\end{frametxt}
\begin{rmrk}
Definition~\ref{defn.exp.ren} uses a word `supported' for $f$, suggestive of Definition~\ref{defn.finsupp}, even though $f$ has no permutation/renaming action.
It \emph{will} have a permutation/renaming action (Remark~\ref{rmrk.conj.action} and Definition~\ref{defn.exp.ren.action}), and then the terminologies will coincide (see Lemma~\ref{lemm.OK}).
\end{rmrk}
\begin{rmrk}
\leftarrowbel{rmrk.conj.action}
It is a fact that \theory{PmsPrm} is cartesian closed and functions have the \emph{conjugation action}
$$
\leftarrowbel{Conjugation action} (\pi\act f)(x)=\pi\act(f(\pi^\mone\act x)).
$$
and $f$ is supported in the sense of Definition~\ref{defn.exp.ren} if and only if it is supported as an element of $|\ns X|\to|\ns Y|$ with the conjungation action.
For more on this see \cite{gabbay:fountl,gabbay:newaas-jv}.
Renamings $\rho$ are not invertible, so we must work a little harder to define a renaming action.
This is Definition~\ref{defn.exp.ren.action}.
However, the end result is similar to the conjugation action, in a sense made formal in Lemma~\ref{lemm.renaming.distribute} which is similar to an immediate corollary of the conjugation action that $\pi\act f(x) =(\pi\act f)(\pi\act x)$.
\end{rmrk}
\begin{lemm}
\leftarrowbel{lemm.supp.supported.f.bound}
If $f$ is supported then $\supp(f(x))\subseteq S_f\cup\supp(x)$ for every $x\in|\rs X|$.
\end{lemm}
\begin{proof}
By contradiction.
Suppose there exists $a\in \supp(f(x))\setminus(S_f\cup\supp(x))$.
Choose $b$ fresh (so $b\not\in\supp(f(x))\cup S_f\cup\supp(x)$).
Then $(b\ a)\bigact (f(x))=f((b\ a)\bigact x)$ since $a,b\not\in S_f$
and $f((b\ a)\bigact x)=f(x)$ since $b,a\not\in\supp(x)$.
It follows by Lemma~\ref{lemm.supp.subsets} that $(b\ a)\bigact\supp(f(x))=\supp(f(x))$, which is impossible.
\end{proof}
\begin{defn}
\leftarrowbel{defn.freshening.pair}
Suppose $S\subseteq\mathbb A$ is a permission set and $A\subseteq\mathbb A$ is finite.
Call $\rho_1$ and $\rho_2$ a \deffont{freshening pair} of renamings for $A$ with respect to $S$ when:
\begin{itemize*}
\item
$\dom(\rho_1)=A$ and $\dom(\rho_2)=\img(\rho_1)$.
\item
$(\rho_2\circ\rho_1)(a)=a$ for all $a\in A$.
\item
$\dom(\rho_2)\cap (S\cup A)=\varnothing$.
\end{itemize*}
\end{defn}
In words, $\rho_1$ maps the atoms in $A$ to be outside $S$ (and $A$), and $\rho_2$ is an `inverse' to $\rho_1$ that puts them back.
\subsubsection{Renaming action}
\begin{defn}
\leftarrowbel{defn.exp.ren.action}
(We continue the notation of Definition~\ref{defn.exp.ren}.)
If $f$ is supported then define $\rho\bigact f$ by
\begin{frameqn}
(\rho\bigact f)(x) = (\rho_2\circ\rho)\bigact f(\rho_1\bigact x)
\end{frameqn}
for some/any freshening pair of renamings $\rho_1$ and $\rho_2$ for $\nontriv(\rho)$ (which is finite), with respect to $\supp(x)\cup S_f$.
\end{defn}
\begin{lemm}
Definition~\ref{defn.exp.ren.action} is well-defined.
That is, it does not matter which freshening pair of renamings we choose.
\end{lemm}
\begin{proof}
Consider two freshening pairs of renamings $\rho_1,\rho_2$ and $\rho_1',\rho_2'$.
Let $\rho_1''$ map $\img(\rho_1)$ to $\img(\rho_1')$ and $\rho_2''$ map $\dom(\rho_2')=\img(\rho_1')$ to $\dom(\rho_2)=\img(\rho_1)$ in such a way that
\begin{itemize*}
\item
$\rho_1'(a)=(\rho_1''\circ\rho_1)(a)$ for all $a\in\dom(\rho_1')$,
\item
$\rho_2'(a)=(\rho_2\circ\rho_2'')(a)$ for all $a\in\dom(\rho_2')$, and
\item
$\nontriv(\rho_1'')=\img(\rho_1)\cup\img(\rho_1')$ and $\nontriv(\rho_2'')=\dom(\rho_2')\cup\dom(\rho_2)$.
\end{itemize*}
We reason as follows:
\begin{tab7}
(\rho_2'\circ\rho)\bigact f((\rho_1'\circ\rho)\bigact x)=&
(\rho_2\circ\rho_2''\circ\rho)\bigact f((\rho_1''\circ\rho_1\circ\rho)\bigact x)
&\text{Lems.~\ref{lemm.supp.supported.f.bound} \& \ref{lemm.supp.subsets}, Def.~\ref{defn.finsupp}}
\\
=&(\rho_2\circ\rho_2''\circ\rho\circ\rho_1'')\bigact f((\rho_1\circ\rho)\bigact x)
&\dom(\rho_1'')\cap S_f=\varnothing
\\
=&(\rho_2\circ\rho_2''\circ\rho_1''\circ\rho)\bigact f((\rho_1\circ\rho)\bigact x)
&\nontriv(\rho_1'')\cap\nontriv(\rho)=\varnothing
\\
=&(\rho_2\circ\rho)\bigact f((\rho_1\circ\rho)\bigact x)
&\text{Lems.~\ref{lemm.supp.supported.f.bound} \& \ref{lemm.supp.subsets}, Def.~\ref{defn.finsupp}}
\end{tab7}
\end{proof}
\begin{lemm}
\leftarrowbel{lemm.renaming.distribute}
Suppose $x\in|\rs X|$ and $\rho$ is a renaming.
Suppose $f\in|\rs X|\to|\rs Y|$ is supported.
Then $\rho\bigact(f(x))=(\rho\bigact f)(\rho\bigact x)$.
\end{lemm}
\begin{proof}
Let $\rho_1$ and $\rho_2$ be a freshening pair of renamings of $\nontriv(\rho)$ with respect to $S_f\cup\supp(x)$.
Let $\rho'$ be a renaming with $\nontriv(\rho')=\img(\rho_1)$ such that $\rho_1\circ\rho=\rho'\circ\rho_1$; this exists since $\rho_1$ is injective on $\nontriv(\rho)$ and `freshens' this set to some fresh set of atoms.
We reason as follows:
\begin{tab7}
(\rho\bigact f)(\rho\bigact x)=&(\rho_2\circ\rho)\bigact f((\rho_1\circ\rho)\bigact x)
&\text{Definition~\ref{defn.exp.ren.action}}
\\
=&(\rho_2\circ\rho)\bigact f((\rho'\circ\rho_1)\bigact x)
&\text{Definition~\ref{defn.finsupp}}
\\
=&(\rho_2\circ\rho\circ\rho')\bigact f(\rho_1\bigact x)
&\nontriv(\rho')\cap S_f=\varnothing
\\
=&(\rho\circ\rho_2)\bigact f(\rho_1\bigact x)
&\text{Lem.~\ref{lemm.supp.supported.f.bound}, Def.~\ref{defn.finsupp}}
\\
=&\rho\bigact f((\rho_2\circ\rho_1)\bigact x)
&\dom(\rho_2)\cap S_f=\varnothing
\\
=&\rho\bigact f(x)
&\text{Definition~\ref{defn.finsupp}}
\end{tab7}
\end{proof}
\subsubsection{Definition of the exponential}
\begin{frametxt}
\begin{defn}
\leftarrowbel{defn.frs.exp}
Write $\rs X\Rightarrow\rs Y$ for the renaming set with underlying set those $f\in|\rs X|\to|\rs Y|$ that are supported in the sense of Definition~\ref{defn.exp.ren}, and renaming action as defined in Definition~\ref{defn.exp.ren.action}.
\end{defn}
\end{frametxt}
\begin{lemm}
\leftarrowbel{lemm.OK}
If $f$ is supported in the sense of Definition~\ref{defn.exp.ren} then it is supported by $S_f$ in the sense of Definition~\ref{defn.finsupp}.
Thus, $\rs X\Rightarrow\rs Y$ is indeed a permissive-nominal renaming set.
\end{lemm}
\begin{proof}
It suffices to show that if $a\not\in S_f$ then $([a\ssm b]\bigact f)(x)=f(x)$.
This follows by routine calculations.
\end{proof}
\begin{lemm}
\theory{PmsRen} (Definition~\ref{defn.fps}) is cartesian closed:
\begin{itemize*}
\item
The exponential is $\rs X\Rightarrow\rs Y$ from Definition~\ref{defn.frs.exp}.
\item
Products are given pointwise as in Definition~\ref{defn.times}.
\item
The terminal object $\rs 1$ is the singleton set $\{0\}$ with the trivial action $\rho\bigact 0=0$.
\end{itemize*}
\end{lemm}
\begin{proof}
The bijection between $(\rs X\times\rs Y)\to\rs Z$ and $\rs X\to (\rs X\Rightarrow\rs Y)$ is given by currying and uncurrying as usual.
Thus $G:(\rs X\times\rs Y)\to\rs Z$ maps to $x\mapsto \leftarrowm{y}G(x,y)$.
It is not hard to verify that if $\dom(\rho)\cap \supp(x)=\varnothing$ then
$$
(\rho\bigact \leftarrowm{y}F(x,y))(y) = \rho\bigact F(x,y) = F(x,\rho\bigact y) = (\leftarrowm{y}F(x,y))(\rho\bigact y) .
$$
Thus $\leftarrowm{y}G(x,y)$ is supported by $\supp(x)$ and is in $\rs Y\Rightarrow\rs Z$.
\end{proof}
We take a moment to build a particular exponential which will be useful later.
\begin{defn}
\leftarrowbel{defn.lambda.a}
Suppose $x\in|\rs X|$ and $a\in\mathbb A_\nu$.
Write $\leftarrowm{a}x\in|\mathbb A_\nu|\to|\rs X|$ for the function mapping $a$ to $x$ and $b$ to $[a\ssm b]\bigact x$.
\end{defn}
\begin{lemm}
$\leftarrowm{a}x\in |\mathbb A_\nu\Rightarrow\rs X|$.
\end{lemm}
\begin{proof}
It suffices to show that $\leftarrowm{a}x$ is supported by $\supp(x)$ (in fact, it is also supported by $\supp(x){\setminus}\{a\}$).
Suppose $\dom(\rho)\cap\supp(x)=\varnothing$ and $z\in\mathbb A_\nu$ ($z$ is not necessarily distinct from $a$).
Write $\rho\text{-}a$ for the renaming such that $(\rho\text{-}a)(b)=\rho(b)$ and $(\rho\text{-}a)(a)=a$.
We sketch the relevant reasoning:
$$
\rho\bigact((\leftarrowm{a}x)z) = (\rho\circ [a\ssm z])\bigact x
=([a\ssm\rho(z)]\circ(\rho\text{-}a))\bigact x=[a\ssm\rho(z)]\bigact x=(\leftarrowm{a}x)(\rho\bigact z)
$$
\end{proof}
\subsection{Atoms, products, atoms-abstraction, and functions out of atoms}
\leftarrowbel{subsect.atoms.ren.example}
\subsubsection{Atoms}
\begin{defn}
\leftarrowbel{defn.bool}
Write $\mathbb B$ for the nominal set and the permutation/renaming set with underlying set $\{0,1\}$ and the \deffont{trivial} permutation/renaming action such that $\pi\act x=x$/$\rho\bigact x=x$ always.
We will be lax and write $x\in\mathbb B$ for $x\in|\mathbb B|$.
Write $\mathbb A_\nu$ for the permutation set and the renaming set with underlying set $\mathbb A_\nu$ and the natural permutation/renaming action such that $\pi\act x=\pi(x)$/$\rho\bigact x=\rho(x)$ always.
We will be lax and write $x\in\mathbb A_\nu$ for $x\in|\mathbb A_\nu|$.
\end{defn}
\subsubsection{Atoms-abstraction in permutation and renaming sets}
\begin{defn}
\leftarrowbel{defn.abstraction.sets}
Suppose $\ns X$ is a supported permutation set.
Suppose $x\in |\ns X|$ and $a\in\mathbb A_\nu$.
Define \deffont{atoms-abstraction} $[a]x$ and $[\mathbb A_\nu]\ns X$ by:
\begin{frameqn}
\begin{array}{r@{\ }l}
[a]x =& \{(a,x)\}\cup \{(b,(b\ a)\act x)\mid b\in\mathbb A_\nu{\setminus} \f{supp}(x)\}
\\
|[\mathbb A_\nu]\ns X| =& \{[a]x\mid a\in\mathbb A_\nu,\ x\in|\ns X|\}
\\
\pi\act [a]x =& [\pi(a)]\pi\act x
\end{array}
\end{frameqn}
\end{defn}
\begin{lemm}
\leftarrowbel{lemm.supp.abstraction}
Suppose $\ns X$ is a supported permutation set.
\begin{enumerate*}
\item
$[\mathbb A_\nu]\ns X$ is a supported permutation set.
\item
$[a]x{=}[a]x'$ if and only if $x{=}x'$, for $a{\in}\mathbb A_\nu$ and $x{\in} |\ns X|$.
\item
$[a]x{=}[a']x'$ if and only if $a'{\not\in}\f{supp}(x)$ and $(a'\, a)\act x{=}x'$, for $a,a'{\in}\mathbb A_\nu$ and $x,x'{\in}|\ns X|$.
\end{enumerate*}
\end{lemm}
We do not need Definition~\ref{defn.abstraction.sets'} for the completeness proof but we include it for the interested reader to compare and constrast with Definition~\ref{defn.abstraction.sets}.
\begin{defn}
\leftarrowbel{defn.abstraction.sets'}
Suppose $\rs X$ is a supported renaming set.
Suppose $x\in |\rs X|$ and $a\in\mathbb A_\nu$.
Define \deffont{atoms-abstraction} $[a]x$ and $[\mathbb A_\nu]\rs X$ by:
\begin{frameqn}
\begin{array}{r@{\ }l}
[a]x =& \{(a,x)\}\cup \{(b,[a\ssm b]\bigact x)\mid b\in\mathbb A_\nu{\setminus} \supp(x)\}
\\
|[\mathbb A_\nu]\rs X| =& \{[a]x\mid a\in\mathbb A_\nu,\ x\in|\rs X|\}
\\
\rho\bigact [a]x =& [a]\rho\bigact x\quad (a\not\in\f{nontriv}(\rho))
\end{array}
\end{frameqn}
\end{defn}
\begin{rmrk}
\leftarrowbel{rmrk.total-partial}
Definitions~\ref{defn.abstraction.sets} and~\ref{defn.abstraction.sets'} look similar; both define graphs of partial functions defined on $\supp(x)\setminus\{a\}$.
However, the critical difference is that in renaming sets, this partial function can be extended to a total function in $\mathbb A_\nu\to\rs X$.
That is, $[a]x\in[\mathbb A_\nu]\rs X$ determines a total function which we could write $\leftarrowm{a}x$, mapping $a$ to $x$ and any other $b$ to $[a\ssm b]\bigact x$.
We return to this in Lemma~\ref{lemm.non-iso} where we show that the natural map from $[\mathbb A_\nu]\rs X$ to $\mathbb A_\nu\Rightarrow\rs X$ is not surjective; so Definition~\ref{defn.abstraction.sets'} identifies a `small' and `well-behaved' subset of the function space.
\end{rmrk}
A cognate of Lemma~\ref{lemm.supp.abstraction} also holds for $[\mathbb A_\nu]\rs X$:
\begin{lemm}
\leftarrowbel{lemm.supp.abstraction'}
Suppose $\rs X$ is a finitely-supported permutation set.
\begin{enumerate*}
\item
$[\mathbb A_\nu]\rs X$ is a permissive-nominal set.
\item
$[a]x{=}[a]x'$ if and only if $x{=}x'$, for $a{\in}\mathbb A_\nu$ and $x{\in} |\rs X|$.
\item
$[a]x{=}[a']x'$ if and only if $a'{\not\in}\f{supp}(x)$ and $(a'\, a)\bigact x{=}x'$ (or equivalently $[a\ssm a']\bigact x{=}x'$), for $a,a'{\in}\mathbb A_\nu$ and $x,x'{\in}|\ns X|$.
\end{enumerate*}
\end{lemm}
\subsubsection{Product}
\begin{defn}
\leftarrowbel{defn.times}
If $\ns X_i$ and $\rs X_i$ are supported permutation sets for $1\leq i\leq n$ then define $\ns X_1\times\ldots\times \ns X_n$ and $\rs X_1\times\ldots\times\rs X_n$ by:
$$
\begin{array}{r@{\ }l}
|\ns X_1\times\ldots\times\ns X_n|=&|\ns X_1|\times\ldots\times|\ns X_n|
\\
\pi\act (x_1,\ldots,x_n)=&(\pi\act x_1,\ldots,\pi\act x_n)
\end{array}
\quad\quad
\begin{array}{r@{\ }l}
|\rs X_1\times\ldots\times\rs X_n|=&|\rs X_1|\times\ldots\times|\rs X_n|
\\
\rho\bigact (x_1,\ldots,x_n)=&(\rho\bigact x_1,\ldots,\rho\bigact x_n)
\end{array}
$$
\end{defn}
\begin{lemm}
\leftarrowbel{lemm.properties.of.support}
\begin{itemize*}
\item
$\f{supp}(a)=\{a\}$.
\item
$\f{supp}([a]x)=\f{supp}(x)\setminus\{a\}$.
\item
$\f{supp}((x_1,\ldots,x_n))=\bigcup\{\f{supp}(x_i)\mid 1\leq i\leq n\}$.
\end{itemize*}
\end{lemm}
\begin{proof}
By routine arguments like those in \cite{gabbay:newaas-jv} or \cite[Corollary~2.30 \& Theorem~3.11]{gabbay:fountl}.
\end{proof}
\subsection{The free extension of a permutation set to a renaming set}
\leftarrowbel{subsect.free.ext}
\begin{nttn}
If $\sim$ is an equivalence relation, $[\text{-}]_\sim$ will denote the equivalence class of $\text{-}$ in $\sim$.
\end{nttn}
\begin{defn}
\leftarrowbel{defn.free.ren}
We define a functor $\Ren{\text{-}}$ from \theory{PmsPrm} to \theory{PmsRen} as follows:
\begin{itemize*}
\item
\emph{Action of $\Ren{\text{-}}$ on objects.}
$\ns X$ maps to $\Ren{\ns X}=((\mathbb R_{\text{fin}}\times|\ns X|)/{\sim},\bigact)$ where $\rho\bigact [(\rho',x)]_\sim = [(\rho\circ\rho',x)]_\sim$ and $\sim$ is the least equivalence relation such that:
\begin{frametxt}
\begin{enumerate*}
\item
If $\rho(a)=\rho'(a)$ for every $a\in\supp(x)$ then $(\rho,x)\sim (\rho',x)$.
\item
$(\rho\circ\pi,x)\sim(\rho,\pi\act x)$.
\end{enumerate*}
\end{frametxt}
For convenience we will write $[(\rho,x)]_\sim$ as $\rho\bigact x$.
\item
\emph{Action of $\Ren{\text{-}}$ on arrows.}
An arrow $F:\ns X\longrightarrow\ns Y$ maps to $\Ren F:\Ren{\ns X}\longrightarrow\Ren{\ns Y}$ given by:
$$
\Ren F(\rho\bigact x)=\rho\bigact F(x)
$$
\end{itemize*}
\end{defn}
\begin{lemm}
$\Ren F$ is well-defined; that is, that if $(\rho,x)\sim(\rho',x')$ then $\Ren F((\rho,x))\sim \Ren F((\rho',x'))$.
\end{lemm}
\begin{proof}
Induction on the derivation that $(\rho,x){\sim}(\rho',x')$.
We consider the two base cases:
\begin{itemize*}
\item
\emph{The case $\rho(a)=\rho'(a)$ for every $a\in\supp(x)$.}\quad
By part~2 of Lemma~\ref{lemm.equivar.reduces.supp} also $\rho(a)=\rho'(a)$ for every $a\in\supp(F(x))$.
\item
\emph{The case $(\rho\circ\pi,x)\sim (\rho,\pi\act x)$.}\quad
Then also $(\rho\circ\pi,F(x))\sim (\rho,\pi\act F(x))$ and by equivariance $\pi\act F(x)=F(\pi\act x)$.
\qedhere\end{itemize*}
\end{proof}
\begin{rmrk}
\leftarrowbel{rmrk.alpha}
Rules~2 and~1 of Definition~\ref{defn.free.ren} can be viewed as $\alpha$-conversion and garbage-collection respectively.
Thus in $\rho\bigact x\in\Ren{\ns X}$ we may without loss of generality (using rule~2) assume that $\dom(\rho)\cap S=\varnothing$ for any permission set $S$, and we may also assume (using rule~1) that $\dom(\rho)\subseteq\supp(x)$.
\end{rmrk}
\begin{lemm}
\leftarrowbel{lemm.BA.ren.isos}
\begin{enumerate*}
\item
$\Ren{\mathbb B}$ (for $\mathbb B$ considered a set with the trivial permutation action) is isomorphic to $\mathbb B$ (for $\mathbb B$ considered a set with a trivial renaming action).
\item
$\Ren{\mathbb A_\nu}$ (for $\mathbb A_\nu$ with its natural permutation action) is isomorphic to $\mathbb A_\nu$ (for $\mathbb A_\nu$ with its natural renaming action).
\end{enumerate*}
\end{lemm}
\begin{proof}
We consider only the second part.
This follows if we note that according to the rules for $\sim$ in Definition~\ref{defn.free.ren},\
$$
(\rho,a)\stackrel{\text{\it rule~1}}{\sim} ((\rho(a)\ a),a)\stackrel{\text{\it rule~2}}{\sim} (\id,\rho(a)).
$$
\end{proof}
Where we are dealing with more than zero or one atoms at a time, isomorphisms like those in Lemma~\ref{lemm.BA.ren.isos} may fail:
\begin{lemm}
$\Ren{\mathbb A_\nu{\times}\mathbb A_\nu}$ is not isomorphic to $\Ren{\mathbb A_\nu}{\times}\Ren{\mathbb A_\nu}$ (which is isomorphic to $\mathbb A_\nu{\times}\mathbb A_\nu$).
\end{lemm}
\begin{proof}
Consider the element $[a\ssm b]\bigact(a,b)$.
\end{proof}
\section{Interpretation of permissive-nominal logic}
\leftarrowbel{sect.permissive-nominal.sets}
\subsection{Interpretation of signatures}
\begin{defn}
\leftarrowbel{defn.interpretation}
Suppose $(\mathcal A,\mathcal B)$ is a sort-signature (Definition~\ref{defn.sort.sig}).
A \deffont{PNL interpretation} $\mathcal I$ for $(\mathcal A,\mathcal B)$ consists of an assignment of a nonempty supported permutation set $\basesort^\iden$ to each $\basesort\in\mathcal B$.
We extend an interpretation $\mathcal I$ to sorts by:
\begin{frameqn}
\begin{array}{r@{\ }l@{\qquad}r@{\ }l}
\model{\basesort}=&\basesort^\iden
&
\model{(\alpha_1,\ldots,\alpha_n)}=&\model{\alpha_1}\times\ldots\times\model{\alpha_n}
\\
\model{\nu}=&\mathbb A_\nu
&
\model{[\nu]\alpha}=&[\mathbb A_\nu]\model{\alpha}
\end{array}
\end{frameqn}
\end{defn}
\begin{defn}
\leftarrowbel{defn.interpret.I}
Suppose $\mathcal S=(\mathcal A,\mathcal B,\mathcal F,\mathcal P,\f{ar},\mathcal X)$ is a signature (Definition~\ref{defn.signature}).
A \deffont{(non-equivariant) PNL interpretation} $\mathcal I$ for $\mathcal S$ consists of the following data:
\begin{itemize*}
\item
An interpretation for the sort-signature $(\mathcal A,\mathcal B)$ (Definition~\ref{defn.interpretation}).
\item
For every $\tf f\in\mathcal F$ with $\f{ar}(\tf f)=(\alpha')\alpha$ an equivariant function $\tf f^\iden$ from $\model{\alpha'}$ to $\model{\alpha}$ (Definition~\ref{defn.equivariant}).
\item
For every $\tf P\in\mathcal P$ with $\f{ar}(\tf P)=\alpha$ a supported function $\tf P^\iden$ from $\model{\alpha}$ to $\{0,1\}$.
\end{itemize*}
If every $\tf P^\iden$ is equivariant, then call $\mathcal I$ a \deffont{fully equivariant} interpretation.\footnote{A non-equivariant PNL interpretation still interprets term-formers equivariantly. Only the predicates might not be equivariant.
We do this in order to completely model \rulefont{Ax^\nopi} from Figure~\ref{rSeq}, so that $\tf P(r)\not\liff\tf P(\pi\act r)$; see Theorem~\ref{thrm.reduced.pnl.completeness}.
Of course it is possible to imagine a notion of non-equivariant interpretation where term-formers are interpreted as non-equivariant functions. This would correspond to something else: namely, to losing the property that $\pi\act\tf f(r)=\tf f(\pi\act r)$.}
\end{defn}
\subsection{Interpretation of terms}
\leftarrowbel{subsect.interpret.pnl.terms}
\begin{defn}
\leftarrowbel{defn.valuation}
Suppose $\mathcal I$ is an interpretation for $\mathcal S$.
A \deffont{valuation} $\varsigma$ to $\mathcal I$ is a map on unknowns such that for each unknown $X$,\
\begin{itemize*}
\item
$\varsigma(X)\in\model{\sort(X)}$,\ and\
\item
$\f{supp}(\varsigma(X))\subseteq \pmss(X)$.
\end{itemize*}
$\varsigma$ will range over valuations.
\end{defn}
\begin{defn}
\leftarrowbel{defn.interpret.terms}
Suppose $\mathcal I$ is an interpretation of a signature $\mathcal S$.
Suppose $\varsigma$ is a valuation to $\mathcal I$.
Define an \deffont{interpretation}
$\denot{\mathcal I}{\varsigma}{r}$ in $\mathcal S$ by:
\begin{frameqn}
\begin{array}{r@{\ }l@{\qquad}r@{\ }l}
\denot{\mathcal I}{\varsigma}{a} =& a
&
\denot{\mathcal I}{\varsigma}{[a]r} =& [a]\denot{\mathcal I}{\varsigma}{r}
\\
\denot{\mathcal I}{\varsigma}{\tf f(r)} =&
\tf f^\iden(\denot{\mathcal I}{\varsigma}{r})
&
\denot{\mathcal I}{\varsigma}{\pi\act X} =& \pi\act\varsigma(X)
\\
\denot{\mathcal I}{\varsigma}{(r_1,\ldots,r_n)} =&
(\denot{\mathcal I}{\varsigma}{r_1},\ldots,\denot{\mathcal I}{\varsigma}{r_n})
\end{array}
\end{frameqn}
\end{defn}
\begin{lemm}
\leftarrowbel{lemm.sort.r}
If $r:\alpha$ then $\denot{\mathcal I}{\varsigma}{r}\in\model{\alpha}$.
\end{lemm}
\begin{proof}
By a routine induction on $r$.
\end{proof}
\begin{lemm}
\leftarrowbel{lemm.pi.r.model}
$\pi\act\denot{\mathcal I}{\varsigma}{r} = \denot{\mathcal I}{\varsigma}{\pi\act r}$.
\end{lemm}
\begin{proof}
By a routine induction on $r$.
We consider one case:
\begin{itemize}
\item
\emph{The case $\pi'\act X$.}\quad
By Definition~\ref{defn.interpret.terms}
$\denot{\mathcal I}{\varsigma}{\pi'\act X} = \pi'\act \varsigma(X)$.
Therefore $\pi\act
\denot{\mathcal I}{\varsigma}{\pi'\act X} = \pi\act(\pi'\act \varsigma(X))$.
It is a fact of the group action (Definition~\ref{defn.perm.set}) that $\pi\act(\pi'\act\varsigma(X))=(\pi\circ\pi')\act\varsigma(X)$, and of the permutation action (Definition~\ref{defn.permutation.action}) that $\pi\act(\pi'\act X)= (\pi\circ\pi')\act X$.
The result follows.
\qedhere
\end{itemize}
\end{proof}
\begin{lemm}
\leftarrowbel{lemm.supp.r}
$\f{supp}(\denot{\mathcal I}{\varsigma}{r})\subseteq\fa(r)$.
\end{lemm}
\begin{proof}
By a routine induction on $r$.
We consider one case in detail:
\begin{itemize}
\item
\emph{The case $\pi\act X$.}\quad
$\fa(\pi\act X)=\pi\act\pmss(X)$ by Definition~\ref{defn.fa}.
By assumption in Definition~\ref{defn.valuation} $\f{supp}(\varsigma(X))\subseteq\pmss(X)$.
\end{itemize}
The cases of $a$, $[a]r$, and $[a]r$ use parts~1, 2, and 3 of Lemma~\ref{lemm.properties.of.support}.
The case of $\tf f$ uses part~1 of Lemma~\ref{lemm.equivar.reduces.supp}.
\end{proof}
\subsection{Interpretation of propositions}
\leftarrowbel{subsect.interpret.prop}
\begin{defn}
\leftarrowbel{defn.varsigma.sub}
Suppose $\varsigma$ is a valuation to an interpretation $\mathcal I$.
Suppose $X$ is an unknown and $x\in \model{\sort(X)}$ is such that $\f{supp}(x)\subseteq \pmss(X)$.
Define $\varsigma[X\ssm x]$ by
$$
(\varsigma[X\ssm x])(Y)=\varsigma(Y)
\quad\text{and}\quad
(\varsigma[X\ssm x])(X)=x .
$$
\end{defn}
It is easy to verify that $\varsigma[X\ssm x]$ is also a valuation to $\mathcal I$.
\begin{defn}
\leftarrowbel{defn.truth}
Suppose $\mathcal I$ is an interpretation.
Define an \deffont{interpretation of propositions} by:
\begin{frameqn}
\begin{array}{r@{\ }l}
\denot{\mathcal I}{\varsigma}{\tf P(r)} =&
\tf P^\iden(\denot{\mathcal I}{\varsigma}{r})
\\
\denot{\mathcal I}{\varsigma}{\bot}=& 0
\\
\denot{\mathcal I}{\varsigma}{\phi\limp\psi}=&
\f{max}\{1{-}\denot{\mathcal I}{\varsigma}{\phi},\denot{\mathcal I}{\varsigma}{\psi}\}
\\
\denot{\mathcal I}{\varsigma}{\Forall{X}\phi}=&\f{min}\{\denot{\mathcal I}{\varsigma[X{\ssm} x]}{\phi}\mid x{\in} \model{\sort(X)},\, \f{supp}(x){\subseteq} \pmss(X)
\}
\end{array}
\end{frameqn}
\end{defn}
We may identify $\denot{\mathcal I}{}{\phi}$ with a set of valuations $\{\varsigma\mid\denot{\mathcal I}{\varsigma}{\phi}=1\}$.
We discuss soundness and completeness in Appendix~\ref{sect.completeness}.
\begin{lemm}
\leftarrowbel{lemm.denotsub}
\begin{itemize*}
\item
$\denot{\mathcal I}{\varsigma[X\ssm \denot{\mathcal I}{\varsigma}{r'}]}{r}
=\denot{\mathcal I}{\varsigma}{r[X\ssm r']}$.
\item
$\denot{\mathcal I}{\varsigma[X\ssm \denot{\mathcal I}{\varsigma}{r'}]}{\phi} =
\denot{\mathcal I}{\varsigma}{\phi[X\ssm r']}$.
\end{itemize*}
\end{lemm}
\begin{proof}
By routine inductions on the definitions of
$\denot{\mathcal I}{\varsigma}{r}$ and
$\denot{\mathcal I}{\varsigma}{\phi}$ in
Definitions~\ref{defn.interpret.terms} and~\ref{defn.truth}.
We consider two cases:
\begin{itemize*}
\item
The case of $\denot{\mathcal I}{\varsigma[X\ssm r']}{\pi\act X}$.\quad
We reason as follows:
\begin{tab3}
\denot{\mathcal I}{\varsigma[X\ssm \denot{\mathcal I}{\varsigma}{r'}]}{\pi\act X}
=& \pi\act \denot{\mathcal I}{\varsigma}{r'}
&\text{Definition~\ref{defn.interpret.terms}}
\\
=& \denot{\mathcal I}{\varsigma}{\pi\act r'}
&\text{Lemma~\ref{lemm.pi.r.model}}
\\
=& \denot{\mathcal I}{\varsigma}{(\pi\act X)[X\ssm r']}
&\text{Definition~\ref{defn.subst.action}} .
\end{tab3}
\item
The case of $\denot{\mathcal I}{\varsigma[X\ssm r']}{\tf P(r)}$.
\quad
We reason as follows:
\begin{tab3}
\denot{\mathcal I}{\varsigma[X\ssm \denot{\mathcal I}{\varsigma}{r'}]}{\tf P(r)}
=&
{\tf P}^\iden(\denot{\mathcal I}{\varsigma[X\ssm \denot{\mathcal I}{\varsigma}{r'}]}{r})
&\text{Definition~\ref{defn.truth}}
\\
=&
{\tf P}^\iden(\denot{\mathcal I}{\varsigma}{r[X\ssm r']})
&\text{Part~1 of this result}
\\
=&
\denot{\mathcal I}{\varsigma}{\tf P(r)[X\ssm r']}
&\text{Definition~\ref{defn.truth}} .
\end{tab3}
\end{itemize*}
\end{proof}
\begin{lemm}
\leftarrowbel{lemm.fV.denot}
If $\varsigma(X)=\varsigma'(X)$ for all $X\in\f{fV}(r)$ then $\denot{\mathcal I}{\varsigma}{r}=\denot{\mathcal I}{\varsigma'}{r}$, and similarly for $\phi$.
\end{lemm}
\begin{proof}
By a routine induction on $r$ and $\phi$.
\end{proof}
\section{Interpretation of HOL}
\leftarrowbel{sect.interp.hol}
For this section fix some PNL interpretation $\mathcal I$ of a PNL signature $\mathcal S$.
Recall from Definition~\ref{defn.TS} the definition of the corresponding HOL signature $\mathcal T_{\mathcal S}$.
We have our interpretation of PNL and we have from Definition~\ref{defn.translation} a translation of PNL syntax to HOL syntax.
We also have a functor from nominal sets to renaming sets (Definition~\ref{defn.free.ren}).
It remains to interpret HOL in renaming sets consistent with these interpretations and translations.
This is Definitions~\ref{defn.hol.interpretation} and~\ref{defn.hol.interpret.terms}, and the key technical result Lemma~\ref{lemm.commuting.square}.
Completeness follows quickly as a corollary (Theorem~\ref{thrm.PNL.HOL.complete}).
Note that in the interpretation (Definition~\ref{defn.hol.interpretation}) the type $\mu_\nu\to\beta$ is not necessarily interpreted as the set of all functions; it may be interpreted as a small subset of this function space.
This is an old idea: since Henkin, models of HOL have been constructed to cut down on the full function-space (e.g. to create a complete semantics \cite[Section~55]{andrews:intmlt}; see also \cite{benzmuller:higose} for a survey of non-standard semantics for HOL).
What we need to prove completeness of the syntactic translation $\hol{D}{\text{-}}$ is the existence of \emph{some} interpretation of HOL with certain properties.
This should not be mistaken as a commitment of nominal techniques to using this model of HOL always (unless we want to).
\subsection{Interpretation of types}
Recall the definition of a valuation $\varsigma$ (Definition~\ref{defn.valuation}) to an intepretation $\mathcal I$ for the PNL signature $\mathcal S$.
Recall the definition of $\varsigma[X\ssm x]$ (Definition~\ref{defn.varsigma.sub}), and the interpretations of terms $\denot{\mathcal I}{\varsigma}{r}$ (Definition~\ref{defn.interpret.terms}) and propositions $\denot{\mathcal I}{\varsigma}{\phi}$ (Definition~\ref{defn.truth}).
We give similar definitions for HOL and renaming sets, culminating with Theorem~\ref{thrm.hol.soundness} (soundness).
\begin{defn}
\leftarrowbel{defn.hol.interpretation}
We provide an interpretation $\mathcal H$ of $\mathcal T_{\mathcal S}$ by:
\begin{frameqn}
\begin{array}{r@{\ }l@{\qquad}l}
\holmodel{\hol{}{\alpha}}=&\Ren{\model{\alpha}}
\\
\holmodel{o}=&\mathbb B
\\
\holmodel{(\beta_1,\ldots,\beta_n)}=&\holmodel{\beta_1}\times\ldots\times\holmodel{\beta_n}
& (\beta_i\text{ not of the form }\hol{}{\alpha}\text{ for at least one }i)
\\
\holmodel{\beta'\to\beta}=&\holmodel{\beta'}\Rightarrow\holmodel{\beta}
& (\beta'\text{ or }\beta\text{ not of the form }\hol{}{\alpha})
\end{array}
\end{frameqn}
\end{defn}
Recall $\rs X\Rightarrow\rs Y$ from Definition~\ref{defn.frs.exp} and $\rs X\times\rs Y$ from Definition~\ref{defn.times}.
\begin{rmrk}
\leftarrowbel{rmrk.case-split}
Not all function types are interpreted equally by Definition~\ref{defn.hol.interpretation}.
If a type is the image of a PNL sort then we handle it using the first clause by wrapping it up in $\Ren{\text{-}}$.
Otherwise the interpretation is as standard: pairs to product; function types to the (supported) function set.
This case-split makes Lemma~\ref{lemm.commuting.square} work, which is central to Corollary~\ref{corr.notsubseteq} and to Completeness (Theorem~\ref{thrm.PNL.HOL.complete}).
Why Lemma~\ref{lemm.commuting.square} \emph{could not} work if we did not do this, is indicated in Lemma~\ref{lemm.non-iso}.
Briefly,
$\mathbb A_\nu\Rightarrow\text{-}$ contains `exotic elements' making it bigger than $[\mathbb A_\nu]\text{-}$, which readers familiar with higher-order abstract syntax would expect \cite[\emph{exotic terms}]{despeyroux94higherorder}.
Perhaps less familiar from Lemma~\ref{lemm.commuting.square} is that
$\Ren{\text{-}}$ does not commute with atoms-abstraction or even with cartesian product.
That is, even e.g. $\mathbb A_\nu\times\mathbb A_\nu$ in \theory{PmsRen} has an `exotic element'.
\end{rmrk}
\begin{lemm}
\leftarrowbel{lemm.non-iso}
\begin{enumerate*}
\item
The natural map from $\Ren{\mathbb A_\nu}$ to $\mathbb A_\nu$ mapping $\rho\bigact a$ to $\rho(a)$, is a bijection (cf. Lemma~\ref{lemm.BA.ren.isos}).
\item
The natural map from $\Ren{\ns X\times\ns Y}$ to $\Ren{\ns X}\times\Ren{\ns Y}$ mapping $\rho\bigact(x,y)$ to $(\rho\bigact x,\rho\bigact y)$ is neither surjective nor injective.
\item
The natural map from $\Ren{[\mathbb A_\nu]\ns X}$ to $[\mathbb A_\nu]\Ren{\ns X}$ mapping $\rho\bigact [a]x$ where $a\not\in\f{nontriv}(\rho)$ to $[a]\rho\bigact x$, is not surjective.
\item
The natural map from $[\mathbb A_\nu]\rs Y$ to $\mathbb A_\nu\Rightarrow\rs Y$ mapping $[a]x$ to $\leftarrowm{a}x$ (Definition~\ref{defn.lambda.a}),
is not surjective.
\end{enumerate*}
\end{lemm}
\begin{proof}
\begin{enumerate*}
\item
By rule~2 of Definition~\ref{defn.free.ren}.
\item
Take $\ns X=\ns Y=\mathbb A_\nu$.
The natural map from $\Ren{\ns X\times\ns Y}$ to $\Ren{\ns X}\times\Ren{\ns Y}$ takes $\id\bigact (a,b)$ to $(\id\bigact a,\id\bigact b)$.
By equivariance it must map $[a\ssm b]\bigact(a,b)$ to $(\id\bigact b,\id\bigact b)$.
But then it is not injective, since $[a\ssm b]\bigact(a,b)\neq \id\bigact(b,b)$ in $\Ren{\ns X\times\ns Y}$.
Now take $\ns X=\ns Y=\mathbb A_\nu\times\mathbb A_\nu$.
It is not hard to see that $([a\ssm b]\bigact(a,b),\id\bigact(b,b))$ is not in the image of the natural map, so the map is also not surjective.
\item
Take $\ns X=\mathbb A_\nu\times\mathbb A_\nu$ and consider $[a][a\ssm b]\bigact(a,b)\in[\mathbb A_\nu]\Ren{\ns X}$.
\item
Take $\rs X=\rs Y=\mathbb A_\nu$ for $\mathbb A_\nu$ considered a renaming set as in Definition~\ref{defn.bool}.
Consider the function $[a\ssm b]\in\mathbb A_\nu\Rightarrow\mathbb A_\nu$, mapping $a$ to $b$, $b$ to $b$, and all other $c$ to $c$.
\qedhere\end{enumerate*}
\end{proof}
\subsection{Interpretation of terms}
\begin{defn}
\leftarrowbel{defn.hol.valuation}
A \deffont{(HOL) valuation} $\varrho$ to $\mathcal H$ is a map on variables $X:\beta$
such that
$\varrho(X)\in\holmodel{\beta}$.
$\varrho$ will range over valuations.
\end{defn}
\begin{defn}
\leftarrowbel{defn.varrho.update}
Suppose $\varrho$ is a valuation.
Suppose $X$ is a variable and $x\in \holmodel{\type(X)}$.
Define a function $\varrho[X\ssm x]$ by:
\begin{frameqn}
(\varrho[X\ssm x])(b)=\varrho(b)
\qquad
(\varrho[X\ssm x])(Y)=\varrho(Y)
\quad\text{and}\quad
(\varrho[X\ssm x])(X)=x
\end{frameqn}
\end{defn}
It is easy to verify that $\varrho[X\ssm x]$ is also a valuation to $\mathcal H$.
\begin{defn}
\leftarrowbel{defn.hol.interpret.terms}
Extend $\mathcal H$ to terms as follows:
\begin{itemize*}
\item
$\holmodel{a}(\varrho) = \varrho(a)$.
\item
$\holmodel{X}(\varrho) = \varrho(X)$.
\item
$\holmodel{\tf g_{\smtf f}} = \Ren{\tf f^\iden}$ and $\holmodel{\tf g_{\smtf P}} = \Ren{\tf P^\iden}$ (Definition~\ref{defn.free.ren}).
\item
$\holmodel{\bot}(\varrho)=0$.
\item
$\holmodel{\limp}(\varrho)=\leftarrowm{x\in\mathbb B,y\in\mathbb B}\f{max}\{1{-}x,y\}$.
\item
$\holmodel{\forall_{\beta})}(\varrho)=
\leftarrowm{x\in\holmodel{\beta\Rightarrow\mathbb B}}\f{min}\{xy\mid y\in\holmodel{\beta}\}$.
\item
$\holmodel{\leftarrowm{a}t}(\varrho)=\rho\bigact [a]x$ where $\holmodel{t}(\varrho[a\ssm a])=\rho\bigact x$ provided that $t:\hol{}{\alpha}$ for some PNL sort $\alpha$ and $a\in\mathbb A_\nu$ for some name sort $\nu$ and ($\alpha$-converting if necessary) $a\not\in\bigcup_{X\in\f{fv}(t)\setminus\{a\}}\supp(\varrho(X))$.
\item
$\holmodel{\leftarrowm{X}t}(\varrho) = \leftarrowm{x}\holmodel{t}(\varrho[X\ssm x])$ provided that $\leftarrowm{X}t:\beta'\to\beta$ where $\beta'\to\beta$ is not equal to $\hol{}{[\mathbb A_\nu]\alpha}$ for any $\nu$ or $\alpha$.
\item
$\holmodel{tu}(\varrho) = ([a\ssm b]\circ\rho)\bigact x$ provided that $t:\hol{}{\alpha}$ for some PNL sort $\alpha$, where $\holmodel{u}(\varrho)=\id\bigact b$ (by construction some such $b$ always exists) and $\holmodel{t}(\varrho)=\rho\bigact [a]x$, and (renaming if necessary) $a\not\in\f{nontriv}(\rho)\cup\{b\}$.
\item
$\holmodel{tu}(\varrho) = \holmodel{t}(\varrho)\holmodel{u}(\varrho)$ provided that $t:\beta$ for $\beta$ not equal to $\hol{}{\alpha}$ for any PNL sort $\alpha$.
\item
$\holmodel{(t_1,\ldots,t_n)}(\varrho) = (\bigcup \rho_i)\bigact (x_1,\ldots,x_n)$ provided that $t_i:\hol{}{\alpha_i}$ for $1\leq i\leq n$, where $\holmodel{t_i}=\rho_i\bigact x_i$, and we choose represenatives such that $\dom(\rho_i)\cap\dom(\rho_j)=\varnothing$ for all $1\leq i\neq j\leq n$.
\item
$\holmodel{(t_1,\ldots,t_n)}(\varrho) = (\holmodel{t_1}(\varrho),\ldots,\holmodel{t_n}(\varrho))$ provided that there exists some $i$ and $\beta$ such that $t_i:\beta$ and $\beta$ is not equal to $\hol{}{\alpha}$ for any PNL sort $\alpha$.
\end{itemize*}
\end{defn}
\begin{rmrk}
\leftarrowbel{rmrk.outline}
Definition~\ref{defn.hol.interpret.terms} propagates to terms the case-split noted in Remark~\ref{rmrk.case-split}.
We treat terms differently depending on whether they populate the translation of a PNL sort, or not.
We must do this because of how we interpreted types in Definition~\ref{defn.hol.interpretation}.
Just to locate where we are, here is an schematic of the overall structure of the proof of completeness:
$$
\xymatrix@=6em{
\text{PNL syntax} \ar[r]^{\hol{D}{\text{-}}}\ar[d]_{\denot{\mathcal I}{\varsigma}{\text{-}}} & \text{HOL syntax} \ar[d]^{\holmodel{\text{-}}(D(\varsigma))}\ar@{-->}[dl]^{\text{\em not possible}}
\\
\theory{PmsPrm} \ar[r]_{\Ren{\text{-}}} & \theory{PmsRen}
}
$$
We translated PNL to HOL using $\hol{D}{\text{-}}$ in Definition~\ref{defn.translation}.
Ideally, to prove completeness we would give HOL a denotation directly in \theory{PmsRen}.
Unfortunately this is not possible (the dashed arrow) because $[a]r$ translates to $\leftarrowm{a}\hol{D}{r}$ and has nominal denotation as an atoms-abstraction $[a]\denot{\mathcal I}{\varsigma}{r}$; atoms-abstraction (Definition~\ref{defn.abstraction.sets}) is the graph of a partial function, whereas $\leftarrowm{a}\hol{D}{r}$ `wants' to take denotation as a total function.
So we use a commuting square as illustrated, and in \theory{PmsRen} atoms-abstraction can be viewed as a total function, as noted in Remark~\ref{rmrk.total-partial}.
Definition~\ref{defn.hol.interpret.terms} uses this, and fills in the right-hand arrow.
Note that by forming this diagram we give a new semantics to PNL in \theory{PmsRen}, and thus in particular give a semantics to nominal atoms-abstraction in which it becomes interpreted as a total function.
The top arrow is Definition~\ref{defn.translation}; the left-hand arrow is Definition~\ref{defn.interpret.terms}; and the bottom arrow is Definition~\ref{defn.free.ren}.
Lemma~\ref{lemm.abs.conc.pi} proves commutativity of the square.
\end{rmrk}
\begin{lemm}
\leftarrowbel{lemm.hol.denotren}
Suppose $a\in\mathbb A_\nu$ and $b\in\mathbb A_\nu$.
Suppose $a\not\in\supp(\varrho(X))$ for every $X\in\f{fv}(r)\setminus\{a\}$ (including $b$).
Then $\holmodel{t}(\varrho[a\ssm \id\bigact b]) =[a\ssm b]\bigact(\holmodel{t}(\varrho))$.
\end{lemm}
\begin{proof}
By a routine induction on $t$.
We mention two cases:
\begin{itemize*}
\item
The case $t$ is $a$.\quad
Using the fact that $\id\bigact b=[a\ssm b]\bigact a$ in $\mathbb A_\nu$ with the action described in Definition~\ref{defn.bool}.
\item
The case $t$ is $X$ for some HOL variable that is not an atom.\quad
By assumption $a\not\in\supp(\varrho(X))$ and so by Definition~\ref{defn.finsupp},\ $\varrho(X)=[a\ssm b]\bigact\varrho(X)$.
The result follows.
\qedhere
\end{itemize*}
\end{proof}
\begin{rmrk}
Lemma~\ref{lemm.hol.denotren} may fail if $a\in\supp(\varrho(X))$.
For instance, if $\varrho(X)=a$ where $a\in\mathbb A_\nu$ and $\type(X)=\mu_\nu$ and $X$ is not itself an atom, then $\holmodel{X}(\varrho[a\ssm \id\bigact b])=\id\bigact a$ yet $[a\ssm b]\bigact(\holmodel{X}(\varrho))=[a\ssm b]\bigact(\id\bigact a)=\id\bigact b$.
\end{rmrk}
We need to check that the denotation of terms populates the denotation of their types, and that $\beta$-equivalent terms receive equal denotations.
\begin{lemm}
\leftarrowbel{lemm.type.t}
If $t:\beta$ then $\holmodel{t}(\varrho)\in\holmodel{\beta}$.
\end{lemm}
\begin{thrm}
$\holmodel{(\leftarrowm{X}t)u}(\varrho) = \holmodel{t}(\varrho[X\ssm\holmodel{u}(\varrho)])$.
\end{thrm}
\begin{proof}
There are two cases, depending on whether $\leftarrowm{X}t:\hol{}{[\mathbb A_\nu]\alpha}$ for some PNL sort, or not.
\begin{itemize*}
\item
\emph{The case $t:\hol{}{\alpha}$.}\quad
By Definition~\ref{defn.hol.interpret.terms} $\holmodel{u}(\varrho)=\id\bigact b$ and $\holmodel{\leftarrowm{X}t}(\varrho) =\rho\bigact [a]x$, for some $b$, $a$, and $x$.
$\alpha$-converting if necessary assume $X$ is equal to $a$ which we choose fresh (so $a\not\in\f{nontriv}(\rho)\cup\{b\}$ and $a\not\in\supp(\varrho(Y))$ for every $Y\in\f{fv}(t)\setminus\{a\}$).
Then also by definition $\holmodel{(\leftarrowm{a}t)u}(\varrho) = ([a\ssm b]\circ\rho)\bigact x$.
Thus it suffices to check that $([a\ssm b]\circ\rho)\bigact x=\holmodel{t}(\varrho[a\ssm b])$.
This follows using Lemma~\ref{lemm.hol.denotren}.
\item
\emph{The case $t:\beta$ where $\beta$ is not equal to $\hol{}{\alpha}$ for any PNL sort $\alpha$.}\quad
This is as standard.
\qedhere
\end{itemize*}
\end{proof}
\subsection{Soundness}
\begin{lemm}
\leftarrowbel{lemm.fV.hol.denot}
If $\varrho(X)=\varrho'(X)$ for all $X\in\f{fV}(t)$ then $\holmodel{t}(\varrho)=\holmodel{t}(\varrho')$.
\end{lemm}
\begin{proof}
By a routine induction on terms.
\end{proof}
\begin{lemm}
\leftarrowbel{lemm.hol.denotsub}
$\holmodel{\rightarrowwt}(\varrho[X\ssm \holmodel{\rightarrowwu}(\varrho)]) =\holmodel{\rightarrowwt[X\ssm \rightarrowwu]}(\varrho)$.
\end{lemm}
\begin{proof}
By a routine induction on $\rightarrowwt$.
We mention two cases (bearing in mind that in HOL, a variable $X:\nu$ may be an atom in $\mathbb A_\nu$):
\begin{itemize*}
\item
\emph{The case $\rightarrowwt$ equals $X$ equals $a\in\mathbb A_\nu$ for some atom $a$.}\quad
By Definition~\ref{defn.hol.interpret.terms},\ $\holmodel{a}(\varrho[a\ssm\holmodel{\rightarrowwu}(\varrho)])= \holmodel{\rightarrowwu}(\varrho)$.
\item
\emph{The case $\rightarrowwt$ equals $\leftarrowm{Y}\rightarrowwt'$.}\quad
We assume ${Y\not\in\f{fv}(\rightarrowwu)}$, so $(\leftarrowm{Y}\rightarrowwt')[X\ssm \rightarrowwu]=\leftarrowm{Y}(\rightarrowwt'[X\ssm \rightarrowwu])$,\ and use the inductive hypothesis.
\qedhere
\end{itemize*}
\end{proof}
\begin{defn}[Validity]
\leftarrowbel{defn.hol.ment}
Call the proposition $\xi$ \deffont{valid} in ${\mathcal H}$ when
$\holmodel{\xi}(\varrho) = 1$ for all $\varrho$.
Call the sequent $\xi_1, ..., \xi_n \holcent \chi_1, ..., \chi_p$ \deffont{valid}
in ${\mathcal H}$ when
$(\xi_1 \wedge ... \wedge \xi_n) \Rightarrow
(\chi_1 \vee ... \vee \chi_p)$ is valid.
If this is true for all ${\mathcal H}$ then write $\xi_1,\dots,\xi_n\holment\chi_1,\dots,\chi_p$.
\end{defn}
\begin{thrm}[Soundness]
\leftarrowbel{thrm.hol.soundness}
If $\Xi\holcent\Chi$ is derivable then $\Xi\holment\Chi$.
\end{thrm}
\begin{proof}
Fix some interpretation $\mathcal H$.
We work by induction on derivations (Figure~\ref{rSeq}).
We sketch the two non-trivial cases:
\begin{itemize*}
\item
\emph{The case of \rulefont{h\forall L}.}\quad
We check that $u:\type(X)$ implies $\holmodel{\Forall{X}\xi}(\varrho)\leq \holmodel{\xi[X\ssm u]}(\varrho)$.
We reason as follows:
$$
\begin{array}{r@{\ }l@{\quad}l}
\holmodel{\Forall{X}\xi}(\varrho)=&\f{min}\{\holmodel{\leftarrowm{X}\xi}(\varrho)y \mid y\in\holmodel{\type(X)}\}
&\text{Definition~\ref{defn.hol.interpret.terms}}
\\
=&\f{min}\{\holmodel{\xi}(\varrho[X\ssm y]) \mid y\in\holmodel{\type(X)}\}
&\text{Definition~\ref{defn.hol.interpret.terms}}
\\
\leq&\holmodel{\xi}(\varrho[X\ssm\holmodel{u}(\varrho)])
&\text{Fact}
\\
=&\holmodel{\xi[X\ssm u]}(\varrho)
&\text{Lemma~\ref{lemm.hol.denotsub}}
\end{array}
$$
In the second use of Definition~\ref{defn.hol.interpret.terms} above, note that $[\mathbb A_\nu]o$ is never of the form $\hol{}{[\mathbb A_\nu]\alpha}$ for any $\alpha$.
\item
\emph{The case of \rulefont{h\forall R}.}\quad
We use Lemma~\ref{lemm.fV.hol.denot} and routine calculations on truth-values.
\qedhere
\end{itemize*}
\end{proof}
\section{Completeness of the translation of PNL to HOL}
\leftarrowbel{sect.pnl.hol.complete}
We are now ready to prove completeness (Theorem~\ref{thrm.PNL.HOL.complete}) of the translation from Definition~\ref{defn.translation}.
The proof is subtle; notably Lemma~\ref{lemm.rho.varrho} and the case of $\Forall{X}\phi$ in Lemma~\ref{lemm.commuting.square} are non-trivial.
Some mathematical action also takes place in Lemma~\ref{lemm.abs.conc.pi} and the case of $\pi\act X$ in Lemma~\ref{lemm.commuting.square}.
\subsection{Renamings and HOL propositions}
We need a few technical observations about how renamings interact with the denotations of HOL propositions:
\begin{lemm}
\leftarrowbel{lemm.equivar.to.triv}
Suppose $G:\rs X\longrightarrow\mathbb B$.
Then for every $\rho$, $G(x)=1$ implies $G(\rho\bigact x)=1$.
\end{lemm}
\begin{proof}
From equivariance and the fact that $\rho\bigact 1=1$ in $\mathbb B$.
\end{proof}
\begin{corr}
\leftarrowbel{corr.unren.prop}
Suppose $F:\ns X\longrightarrow\mathbb B$.
Then $\Ren{F}(\rho\bigact x)=F(x)$.
\end{corr}
\begin{nttn}
Write $\rho\bigact\varrho$ for the valuation mapping $X$ to $\rho\bigact\varrho(X)$.
\end{nttn}
\begin{lemm}
\leftarrowbel{lemm.rho.varrho}
Suppose $\xi$ is a HOL proposition.
Then
\begin{itemize*}
\item
$\holmodel{\xi}(\rho\bigact\varrho)=\holmodel{\xi}(\varrho)$ for every $\rho$ and $\varrho$, and
\item
as a corollary, if $X:\beta$ and $x\in\holmodel{\beta}$
then $\holmodel{\xi}(\varrho[X\ssm x])=\holmodel{\xi}(\varrho[X\ssm\rho\bigact x])$.
\end{itemize*}
\end{lemm}
\begin{proof}
We work by induction on $\xi$.
For each $\xi$ the corollary follows from the first part using a freshening pair of renamings (see Definition~\ref{defn.freshening.pair}).
For the first part, the case of $\tf g_{\smtf P}$ is by Corollary~\ref{corr.unren.prop}.
The case of $\forall$ follows using the second part and some routine calculations.
The cases of $\bot$ and $\limp$ are immediate.
\end{proof}
\begin{rmrk}
Lemma~\ref{lemm.rho.varrho} expresses that $\holmodel{\xi}$ does not examine atoms for inequality across its arguments (if it did then Lemma~\ref{lemm.rho.varrho} could not hold, because $\rho$ can identify atoms---make them become equal---in the denotations of variables in $\xi$).
The corollary is even more powerful: we can even apply renamings to the denotations of individual free variables, and still not affect validity.
We use this in the case of $\Forall{X}\phi$ in Lemma~\ref{lemm.commuting.square} to `jettison' unwanted $\rho$ in the denotation of the quantified variable.
\end{rmrk}
\subsection{The completeness proof}
\begin{nttn}
\leftarrowbel{nttn.D}
Suppose $D=[d_1,\ldots,d_n]$ is a finite list of distinct atoms in $\mathbb A_{\nu_1}$, \ldots, $\mathbb A_{\nu_n}$ respectively.
Suppose $r:\alpha$ is a PNL term.
Then:
\begin{itemize*}
\item
Write $[D]r$ for the PNL term $[d_1]\ldots[d_n]r$.
\item
Write $[\mathbb A_D]\alpha$ for the PNL sort $[\mathbb A_{\nu_1}]\ldots[\mathbb A_{\nu_n}]\alpha$.
\end{itemize*}
\end{nttn}
\begin{defn}
\leftarrowbel{defn.epsilond}
Given a finite list of distinct atoms $D$, map a PNL valuation $\varsigma$ to a HOL valuation $D(\varsigma)$ defined by
\begin{frameqn}
D(\varsigma)\quad\text{maps}\quad
\begin{array}[t]{l@{\quad\text{to}\quad}l}
X:\alpha & \id\bigact [\GammaX]\varsigma(X)\in\holmodel{\hol{}{[\mathbb A_{\GammaX}]\alpha}}\quad\text{and}
\\
a:\nu & a\in\mathbb A_\nu
\end{array}
\end{frameqn}
\end{defn}
\begin{lemm}
\leftarrowbel{lemm.always.id}
Suppose $D\cent r$.
Then $\holmodel{\hol{D}{r}}(D(\varsigma))=\id\bigact x$ for some $x\in\holmodel{\hol{}{\sort(r)}}$.\footnote{The point here is that $\holmodel{\hol{D}{r}}(D(\varsigma))$ is \emph{not} equal to $\rho\bigact x$ for any $\rho$ that is non-injective on $\supp(x)$.}
\end{lemm}
\begin{proof}
By a routine induction on Definition~\ref{defn.hol.interpret.terms} using Definition~\ref{defn.epsilond} for the case that $r$ is a variable $X$.
\end{proof}
Compare Lemma~\ref{lemm.abs.conc.pi} with Lemma~\ref{lemm.hol.pi}:
\begin{lemm}
\leftarrowbel{lemm.abs.conc.pi}
If $\f{nontriv}(\pi)\cap\supp(x)\subseteq D'$ then $(\id\bigact[D']x)\pi\act D' =\id\bigact \pi\act x$.
\end{lemm}
\begin{proof}
From Definition~\ref{defn.hol.interpret.terms} and rule~2 of Definition~\ref{defn.free.ren}.
\end{proof}
Lemma~\ref{lemm.commuting.square} proves that the schematic diagram of Remark~\ref{rmrk.outline} does indeed commute:
\begin{lemm}
\leftarrowbel{lemm.commuting.square}
Suppose $r:\alpha$ and $\phi:\alpha$.
Then:
\begin{itemize*}
\item
If $D\cent r$ then $\holmodel{\hol{D}{r}}(D(\varsigma))=\id\bigact\model{r}(\varsigma)$.
\item
If $D\cent\phi$ then $\holmodel{\hol{D}{\phi}}(D(\varsigma))=\model{\phi}(\varsigma)$.
\end{itemize*}
\end{lemm}
\begin{proof}
By inductions on $r$ and $\phi$.
\begin{itemize*}
\item
\emph{The case $\pi\act X$.}\quad
We reason as follows, where $\alpha=\sort(X)$ and $S=\pmss(X)$:
\begin{tab3}
\holmodel{\hol{D}{\pi\act X}}(D(\varsigma))
=& \holmodel{X\pi\act\GammaX}(D(\varsigma))
&\text{Definition~\ref{defn.translation}}
\\
=& D(\varsigma)(X)\pi\act\GammaX
&\text{Definition~\ref{defn.hol.interpret.terms}}
\\
=& (\id\bigact[\GammaX]\varsigma(X))\pi\act\GammaX
&\text{Definition~\ref{defn.epsilond}}
\\
=& \id\bigact\pi\act\varsigma(X)
&\text{Lemma~\ref{lemm.abs.conc.pi}},\ \supp(\varsigma(X)){\subseteq} S
\\
=& \id\bigact \model{\pi\act X}(\varsigma)
&\text{Definition~\ref{defn.interpret.terms}}
\end{tab3}
Note of the penultimate step that by assumption $D\cent r$, so by Definition~\ref{defn.capture.typing} $\f{nontriv}(\pi)\cap S\subseteq \GammaX=D\cap S$.
\item
\emph{The case $[a]r$.}\quad
We reason as follows:
\begin{tab3}
\holmodel{\hol{D}{[a]r}}(D(\varsigma))
=&
\holmodel{\leftarrowm{a}\hol{D}{r}}(D(\varsigma))
&
\text{Definition~\ref{defn.translation}}
\\
=&\rho\bigact [a]x
&
\text{Definition~\ref{defn.hol.interpret.terms}},\ a\text{ fresh},
\\
&&
\quad\rho\bigact x = \holmodel{\hol{D}{r}}(D(\varsigma)[a\ssm a])
\\
=&\id\bigact [a]x
&\text{Wlog }\rho=\id\text{ by Lemma~\ref{lemm.always.id}}
\\
=&\id\bigact [a]\model{r}(\varsigma)
&\text{ind. hyp.}
\\
=&\id\bigact \model{[a]r}(\varsigma)
&\text{Definition~\ref{defn.interpret.terms}}
\end{tab3}
\item
\emph{The case $\tf P(r)$.}\quad
We reason as follows:
\begin{tab3}
\holmodel{\hol{D}{\tf P(r)}}(D(\varsigma))
=& \holmodel{\tf g_{\smtf P}(\hol{D}{r})}(D(\varsigma))
&\text{Definition~\ref{defn.translation}}
\\
=& \tf g_{\smtf P}^\hiden(\holmodel{\hol{D}{r}}(D(\varsigma)))
&\text{Definition~\ref{defn.truth}}
\\
=& \tf g_{\smtf P}^\hiden(\id\bigact\model{r}(\varsigma))
&\text{part~1}
\\
=& \Ren{\tf P^\iden}(\id\bigact\model{r}(\varsigma))
&\text{Definition~\ref{defn.hol.interpret.terms}}
\\
=& \tf P^\iden(\model{r}(\varsigma))
&\text{Corollary~\ref{corr.unren.prop}}
\\
=& \model{\tf P(r)}(\varsigma)
&\text{Definition~\ref{defn.truth}}
\end{tab3}
\item
\emph{The case $\Forall{X}\phi$.}\quad
Write $\alpha=\sort(X)$ and $S=\pmss(X)$.
From Definition~\ref{defn.hol.interpret.terms}
$$
\holmodel{\hol{D}{\Forall{X}\phi}}(D(\varsigma))
=\f{min}\{\holmodel{\hol{D}{\phi}}(D(\varsigma)[X\ssm x])\mid x\in\holmodel{\hol{}{[\mathbb A_{\GammaX}]\alpha}}\}
$$
By construction in Definition~\ref{defn.hol.interpretation} every $x\in\holmodel{\hol{}{[\mathbb A_{\GammaX}]\alpha}}$ has the form $\rho\bigact x'$ for $x'\in [\GammaX]\model{\alpha}$.
By Lemma~\ref{lemm.rho.varrho} we have
\begin{multline*}
\f{min}\{\holmodel{\hol{D}{\phi}}(D(\varsigma)[X\ssm x])\mid x\in\holmodel{\hol{}{[\mathbb A_{\GammaX}]\alpha}}\}
\\
=
\f{min}\{\holmodel{\hol{D}{\phi}}(D(\varsigma)[X\ssm \id\bigact x'])\mid x'\in\model{[\mathbb A_{\GammaX}]\alpha}\}
\end{multline*}
Using Lemma~\ref{lemm.rho.varrho} again we assume without loss of generality that $\supp([\GammaX]x')\subseteq\pmss(X)\setminus\GammaX$, and so:
\begin{multline*}
\f{min}\{\holmodel{\hol{D}{\phi}}(D(\varsigma)[X\ssm \id\bigact [\GammaX]x'])\mid x'\in\model{[\mathbb A_{\GammaX}]\alpha}\}
\\
=\f{min}\{\holmodel{\hol{D}{\phi}}(D(\varsigma)[X\ssm \id\bigact x''])\mid x''\in\model{\alpha},\ \supp(x''){\subseteq}\pmss(X)\}
\end{multline*}
Now we unfold definitions and use the inductive hypothesis that $D(\varsigma)[X\ssm \id\bigact [\GammaX]x'']=D(\varsigma[X\ssm x''])$, and we obtain:
$$
\hspace{-2em}\begin{array}{r@{}l}
\f{min}\{\holmodel{\hol{D}{\phi}}(D(\varsigma)[X\ssm \id&\bigact [\GammaX]x''])\mid x''\in\model{\alpha},\ \supp(x''){\subseteq}\pmss(X)\}
\\
&=\f{min}\{\holmodel{\hol{D}{\phi}}(D(\varsigma[X\ssm x'']))\mid x''\in\model{\alpha},\ \supp(x''){\subseteq}\pmss(X)\}
\\
&=\f{min}\{\model{\phi}(\varsigma[X\ssm x''])\mid x''\in\model{\alpha},\ \supp(x''){\subseteq}\pmss(X)\}
\\
&=\model{\Forall{X}\phi}(\varsigma)
\end{array}
$$
\end{itemize*}
\end{proof}
\begin{corr}
\leftarrowbel{corr.notsubseteq}
Suppose $\Phi=\{\phi_1,\dots,\phi_n\}$ and $\Psi=\{\psi_1,\dots,\psi_p\}$ and $D\cent\Phi$, and $D\cent\Psi$ (Definition~\ref{defn.capture.typing}).
Suppose $\mathcal I$ is a PNL interpretation and suppose $\phi_1,\dots,\phi_n\nopicent\psi_1,\dots,\psi_p$ is not valid in $\mathcal I$.
Then $\mathcal H$ from Definition~\ref{defn.hol.interpretation} is a HOL interpretation and
$\hol{D}{\phi_1},\dots,\hol{D}{\phi_n}\holcent\hol{D}{\psi_1},\dots,\hol{D}{\psi_p}$ is not valid in $\mathcal H$.
\end{corr}
\begin{proof}
Suppose $\varsigma$ is such that $\model{\phi_1\leftarrownd\dots\leftarrownd\phi_n}(\varsigma)=1$ and $\model{\psi_1\lor\dots\lor\psi_p}(\varsigma)=0$.
We use Lemma~\ref{lemm.commuting.square} for $D(\varsigma)$ (Definition~\ref{defn.epsilond}).
\end{proof}
\begin{thrm}[Completeness]
\leftarrowbel{thrm.PNL.HOL.complete}
Suppose $D\cent\Phi$ and $D\cent\Psi$.
If $\Phi\not\nopicent\Psi$ then $\hol{D}{\Phi}\not\holcent\hol{D}{\Psi}$.
\end{thrm}
\begin{proof}
We use the contrapositive of completeness of restricted PNL (Theorem~\ref{thrm.reduced.pnl.completeness}), then Corollary~\ref{corr.notsubseteq}, then the contrapositive of HOL soundness (Theorem~\ref{thrm.hol.soundness}).
\end{proof}
\section{Conclusions}
We have translated a logic with its own proof-theory, syntax, and sound and complete semantics.
Any formal theory specified in the PNL fragment of this paper can be systematically, soundly, and completely translated to HOL.
For the reader interested in nominal techniques, the main contribution of this paper is that in
proving completeness of the translation, we have given another
semantics of permissive nominal logic, besides the `obvious' one
in nominal sets. In this new semantics, a term of
the form $[a]t$ is interpreted as a function, like $\leftarrowm{a}t$ would be in higher-order logic.
This shows at the semantic level an implicit similarity between PNL and HOL (we discuss presheaves in the next Subsection).
For the reader interested in higher-order logic, this paper is of interest because its image is readily identified with the \emph{higher-order patterns} developed by Miller \cite{miller:logpll} (so that, intuitively, restricted PNL could be thought of as a compact first-order logic and nominal semantics for higher-order patterns).
In this semantics the sort $[\mathbb A]\alpha$ is not interpreted as the set of all
functions from atoms to the interpretation of $\alpha$,
but as a small subset of this function space.
This is an old idea: since Henkin, models of HOL have been constructed to cut down on the full function-space (e.g. to create a complete semantics \cite[Section~55]{andrews:intmlt}).
Moreover in weak HOAS to avoid so-called \emph{exotic terms}, function
existence axioms must be weakened in HOL: for instance, the
description axiom that entails the existence of a function for all
functional relations has to be dropped (an alternative is to introduce an explicit modality \cite{despeyroux:prirh-jv}).
We now have a new view of these `smaller' function-spaces as being the image of nominal atoms-abstractions via the semantic operations considered in this paper.
\subsection{Permissive nominal logic in perspective}
Permissive-nominal logic is the endpoint---so far---of an evolution as follows:
\begin{itemize*}
\item
Fraenkel-Mostowski set theory and a first-order axiomatisation by Pitts introduced and described the underlying nominal sets models in first-order logic \cite{gabbay:newaas-jv,pitts:nomlfo-jv}.
\item
Nominal terms introduced a dedicated syntax with two-levels of variable and freshness side-conditions \cite{gabbay:nomu-jv}.
\item
Nominal algebra and $\alpha$Prolog inserted nominal terms syntax into formal reasoning systems \cite{gabbay:nomuae,cheney:alppl}.
\item
Permissive-nominal terms introduced permission sets \cite{gabbay:perntu-jv}.
\item
PNL introduced a proof-theory and universal quantifier for nominal terms unknowns \cite{gabbay:pernl,gabbay:pernl-jv}.
\end{itemize*}
Meanwhile in the semantics
\begin{itemize*}
\item
Nominal renaming sets extended nominal sets from a permutation action to a renaming action \cite{gabbay:nomrs}.
\item
A permissive version of nominal algebra (an equality fragment of PNL) was given semantics in \theory{PmsPrm} and theories were translated from HOL \cite{gabbay:unialt}, but this was done purely syntactically without using nominal renaming sets and without considering universal quantification.
\end{itemize*}
The categories \theory{PmsPrm} and \theory{PmsRen} from Definition~\ref{defn.fps} are identical to the categories of nominal sets and nominal renaming sets from \cite{gabbay:newaas-jv} and \cite{gabbay:nomrs}, except that here we insist on supporting \emph{permission} sets instead of supporting \emph{finite} sets.
The reader familiar with presheaf techniques will see in \theory{PmsRen} the category $\mathsf{Sets}^{\mathbb F}$ (presheaves over the category of finite sets and functions between them).
\theory{PmsRen} corresponds to presheaves (not quite over $\mathbb F$, as discussed in the previous paragraph) that preserve pullbacks of pairs of monos \cite{gabbay:nomrs} and because of this it admits an arguably preferable sets-based presentation.
(In the same sense, \theory{PmsPrm} corresponds to $\mathsf{Sets}^{\mathbb I}$.)
If for the sake of argument we set aside the issues of finiteness and preserving pullbacks of monos, then this paper can be summed up as follows: PNL, and thus nominal terms, can be given a semantics in something that looks like $\mathsf{Sets}^{\mathbb F}$.
This semantics is functional in that atoms-abstractions in $\mathsf{Sets}^{\mathbb F}$ can be naturally identified with total functions, though not all of them, which is good.
HOL can also be given a semantics in something that looks like $\mathsf{Sets}^{\mathbb F}$, and in such a way that it overlaps with the semantics of PNL, as described in Definition~\ref{defn.hol.interpret.terms} and~\ref{lemm.commuting.square}.
We describe and exploit that overlap, in this paper.
\theory{PmsRen} from Definition~\ref{defn.fps} is related to the category of (finitely-supported) nominal renaming sets from \cite{gabbay:nomrs}.
Here, the difference that $x\in|\rs X|$ need not have finite support is significant because it is impossible with a finite renaming to rename $\supp(x)$ to be entirely disjoint for some other permission set $S$.
The definitions and proofs in Subsection~\ref{subsect.exp} are delicately revised with respect to those in \cite[Section~3]{gabbay:nomrs}.
Thus this paper contributes to the use of non-finitely-supported objects in nominal techniques, building on \cite{gabbay:nomrs} and also on Cheney's and the second author's considerations of infinitely supported permutation sets \cite{cheney:comhtn,gabbay:genmn}.
A similar construction as in Subsection~\ref{subsect.free.ext} has been considered, also in the context of names, though tersely, in Fiore and Turi's paper on the semantics of name and value passing \cite{fiore:semnvp}.
The reader can compare for example the final two paragraphs of Subsection~1.3 in \cite{fiore:semnvp} with Definition~\ref{defn.free.ren} from Subsection~\ref{subsect.free.ext}.
Fiore and Turi want substitutions to model bisimulation in the presence of name-generation and message-passing; we want renamings to model function application on names.
The underlying technical demands overlap and are similar.
Fiore and Turi's framework includes the possibility of arbitrary substitutions for atoms (not just what we call renamings: substitution of atoms for atoms).
This was apparent in \cite{fiore:semnvp} and is developed greatly in subsequent work by Fiore and Hur \cite{fiore:secoel}.
We hypothesise that from the point of view of PNL, their logic and semantics correspond to PNL enriched with substitution actions like those in \cite{gabbay:pernl,gabbay:capasn}, but this remains to be checked.\footnote{Conversely, Fiore and Hur would view PNL as a restriction of their logic \emph{without} substitution. The two points of view are consistent with each other, of course, and it is interesting that different authors are converging on similar systems.
It might be worth mentioning that \emph{deduction modulo} by the first author with Hardin and Kirchner was designed to mediate between these kinds of design decisions while retaining proof-theory \cite{dowek:dedm}.}
Levy and Villaret translated nominal unification problems to higher-order unification problems \cite{levy:nomufh}.
A similar but more detailed analysis, translating solutions and introducing the same notion of capturable atoms as used in the capture typings in this paper, appears in the paper which introduced permissive nominal terms \cite{gabbay:perntu-jv}.
See also a journal version of Levy and Villaret's paper \cite{levy:nomufh-jv}, which expanded on their previous work by eliminating freshness contexts (in a similar spirit to PNL, we feel, though the details are different).
This paper can be viewed as a very considerable extension, refinement, and generalisation of these works: this paper is their grandchild, so to speak, via two other papers \cite{gabbay:pernl,gabbay:unialt}.
The extension of nominal sets to nominal renaming sets is free.
This is touched on in Lemma~\ref{lemm.non-iso} when we note that $[a\ssm b]\bigact(a,b)$ and $\id\bigact(b,b)$ are distinct elements in $\Ren{\mathbb A_\nu\times\mathbb A_\nu}$ in \theory{PmsRen}; this happens because the free construction `suspends the non-injectivity' of $[a\ssm b]$ on $(a,b)$.
This is as things should be, in order to obtain completeness.
The second author has considered a more radical non-free construction \cite{gabbay:stusun}, which has the effect of extending atoms-abstraction to a total function and in which $[a\ssm b]\act x$ really does identify $a$ with $b$ in $x$ in a suitable sense.
As we have emphasised, we translate a fragment of PNL to HOL.
In \cite{gabbay:pernl} we considered full PNL with \emph{equivariance}, which corresponds to strengthening the axiom rule \rulefont{Ax^{\nopi}} in Figure~\ref{rSeq} from
$\begin{prooftree}
\justifies
\Phi,\,\phi\nopicent \phi,\,\Psi
\end{prooftree}
$
to
$
\begin{prooftree}
\justifies
\Phi,\,\phi\cent \pi\act\phi,\,\Psi
\end{prooftree}
$
as illustrated in Figure~\ref{Seq}.
This internalises the equivariance assumed in Definition~\ref{defn.interpret.I} and allows us to derive e.g. $\tf P(a) \cent \tf P(b)$.
In the journal version \cite{gabbay:pernl-jv} of \cite{gabbay:pernl} we strengthen PNL further by allowing a \emph{shift}-permutation.
This is a non-finitely-supported bijection on $\mathbb A$ similar to a \emph{de Bruijn shift function} $\uparrow$ \cite[Subsection~2.2]{abadi:exps}.
Its effect in this paper is to make all permission sets isomorphic up to bijection (e.g. $\atomsdown\cup\{a\}=\pi\act\atomsdown$ for some $\pi$, where $a\not\in\atomsdown$) and this deals with a subtle restriction in the power of universal quantification discussed for instance in \cite[Example~2.29]{gabbay:pernl}.
Briefly, \emph{shift} lets us derive $\Forall{X}\tf P(X)\cent \tf P(Z)$ where $\pmss(X)=\atomsdown$ and $\pmss(Z)=\atomsdown\cup\{a\}$ where $a\not\in\atomsdown$, which was not possible in the PNL from \cite{gabbay:pernl}.
Neither equivariance nor \emph{shift} are translated to HOL in this paper; more on this in the next subsection.
\subsection{Future work}
We have translated Permissive-Nominal Logic
to Higher-Order Logic. The translation is not surjective: all variables are at most
second-order; all constants are at most third-order; higher types
are not used; and in fact all terms in the image of the translation are \emph{$\leftarrowmbda$-patterns} \cite{miller:logpll}.
In addition, the translation is not total: we have dropped equivariance.
This is with good reason.
We have not been able to simulate equivariance in HOL---not without `cheating' by simply adding it (and causing a blowup in the size of propositions).
We have not proved this impossible, but we hypothesise that it cannot be done.
We further hypothesise (based on preliminary calculations not included in this paper) that HOL augmented with the $\nabla$-quantifier from \cite{Miller:protgj} would allow us to express equivariance.
It is not currently clear how to extend HOL with a \emph{shift}-like
permutation as discussed in \cite{gabbay:pernl-jv,gabbay:nomtnl}.
This seems reasonable since $\f{shift}$ would correspond to an infinite renaming.
Some natural theories in PNL might correspond to other fragments of HOL.
Notably, it is not known what relation exists between HOL and PNL with the theory of atoms-substitution from \cite{gabbay:capasn-jv,gabbay:pernl-jv}.
\hyphenation{Mathe-ma-ti-sche}
\appendix
\section{Soundness and completeness of restricted PNL with respect to non-equivariant models}
\leftarrowbel{sect.completeness}
\subsection{Validity and soundness}
\begin{defn}[Validity]
\leftarrowbel{defn.pnl.ment}
Suppose $\mathcal I$ is a non-equivariant interpretation of a signature $\mathcal S$ (Definition~\ref{defn.interpret.I}).
Call the proposition $\phi$ \deffont{valid} in ${\mathcal I}$ when
$\denot{\mathcal I}{\varsigma}{\phi} = 1$ for all $\varsigma$.
Call the sequent
$\phi_1, ..., \phi_n \cent \psi_1, ..., \psi_p$
\deffont{valid}
in ${\mathcal I}$ when
$(\phi_1 \wedge ... \wedge \phi_n) \Rightarrow (\psi_1 \vee ... \vee \psi_p)$ is valid.
If this is true for all non-equivariant ${\mathcal I}$ then write $\phi_1,\dots,\phi_n\nopiment\psi_1,\dots,\psi_p$.
If this is true for all equivariant ${\mathcal I}$ then write $\phi_1,\dots,\phi_n\ment\psi_1,\dots,\psi_p$.
\end{defn}
\begin{thrm}[Soundness]
\leftarrowbel{thrm.pnl.soundness}
\begin{enumerate*}
\item
If $\Phi\nopicent\Psi$ is derivable then $\Phi\nopiment\Psi$.
\item
If $\Phi\cent\Psi$ is derivable then $\Phi\ment\Psi$.
\end{enumerate*}
\end{thrm}
\begin{proof}
Fix some interpretation $\mathcal I$.
We work by induction on derivations.
The case of \rulefont{\forall L} uses Lemma~\ref{lemm.denotsub}.
The case of \rulefont{\forall R} uses Lemma~\ref{lemm.fV.denot}.
Other rules are routine by unpacking definitions.
If the interpretation $\mathcal I$ is fully equivariant then it can further be proved that $\denot{\mathcal I}{\varsigma}{\phi}=\denot{\mathcal I}{\varsigma}{\pi\act\phi}$ always, so that \rulefont{Ax} is valid.
If $\mathcal I$ is not fully equivariant, then just \rulefont{Ax^{\nopi}} is valid.
\end{proof}
\begin{thrm}
\leftarrowbel{thrm.rPNL.cut}
\rulefont{Cut} is admissible in both full and restricted PNL.
\end{thrm}
\begin{proof}
The proof for full PNL is in \cite[Section~7]{gabbay:pernl-jv} or \cite[Subsection~11.2]{gabbay:nomtnl}; the derivation rules are almost exactly those of first-order logic, and so is the proof of cut-elimination.
The argument for restricted PNL is identical; we note that none of the cut-eliminating transformations add $\pi$ to axiom rules unless they are already there, so the same reductions on derivations work also for the restricted system.
\end{proof}
\subsection{Completeness}
In \cite{gabbay:pernl-jv,gabbay:nomtnl} we prove completeness of full PNL with respect to equivariant models, by means of a Herbrand construction (a model built out of syntax).
We can leverage this result to concisely prove completeness of restricted PNL with respect to non-equivariant models, without having to repeat the model constructions.
For this subsection, fix the following data:
\begin{itemize*}
\item
A signature $\mathcal S=(\mathcal A,\mathcal B,\mathcal F,\mathcal P,\f{ar},\mathcal X)$.
\item
A formula $\phi$ such that $\not\nopicent\phi$.
\end{itemize*}
\begin{defn}
\leftarrowbel{defn.S.pi}
Define a new signature $\mathcal S^\pi$ as follows:
\begin{itemize*}
\item
$\mathcal A^\pi=\mathcal A$ and $\mathcal B^\pi=\mathcal B\cup\{\tau^\pi\}$ (so we have the same atom sorts and the same base sorts, plus one extra base sort $\tau^\pi$).
\item
$\mathcal F^\pi=\mathcal F$ and $\mathcal P^\pi=\mathcal P$ (so we have the same term- and proposition-formers).
\item
If $\tf f\in\mathcal F$ then $\f{ar}^\pi(\tf f)=\f{ar}(\tf f)$ (the term-formers are identical).
\item
If $\tf P\in\mathcal P$ and $\f{ar}(\tf P)=\alpha$ then $\f{ar}^\pi(\tf P)=(\tau^\pi,\alpha)$ (so proposition-formers take one extra argument of sort $\tau^\pi$).
\item
$\mathcal X^\pi=\mathcal X\cup\{Z_{i,S}^\pi\mid i\in\mathbb N,\ S\text{ a permission set}\}$ where $\sort(Z_{i,S}^\pi)=\tau^\pi$ (so we add unknowns of sort $\tau^\pi$).
\end{itemize*}
Now fix some particular unknown $Z^\pi$ with $\sort(Z^\pi)=\tau^\pi$ and such that $\fa(\phi)\subseteq\pmss(Z^\pi)$.
\end{defn}
\begin{defn}
Define a translation $\text{-}^\pi$ from PNL propositions in the signature $\mathcal S$ to PNL propositions in the signature $\mathcal S^\pi$ by mapping $\tf P(r)$ to $\tf P(Z^\pi,r)$ and extending this in the natural way to all predicates.
\end{defn}
Our proof depends on the following technical lemma about restricted PNL:
\begin{lemm}
\leftarrowbel{lemm.fa.restrict}
If $\Phi\cent\Psi$ is derivable in full PNL then there exists a derivation $\Pi$ such that every sequent $\Phi'\cent\Psi'$ in $\Pi$ satisfies $\fa(\Phi')\cup\fa(\Psi')\subseteq\fa(\Phi)\cup\fa(\Psi)$.
\end{lemm}
\begin{proof}
By cut-elimination of restricted PNL (Theorem~\ref{thrm.rPNL.cut}) if a derivation of $\Phi\cent\Psi$ exists then a cut-free derivation exists.
We now examine the derivation rules in Figure~\ref{rSeq} and the definition of free atoms in Definition~\ref{defn.fa} and note that the rules \rulefont{{\limp}L}, \rulefont{{\limp}R}, \rulefont{\forall L}, and \rulefont{\forall R} do not increase the free atoms moving from below the line to above the line.\footnote{\rulefont{\forall R} and \rulefont{\forall L} can increase the free \emph{unknowns}---but not the free atoms.}
\end{proof}
\begin{lemm}
\leftarrowbel{lemm.pi.r.fa}
$\pi\act r=\pi'\act r$ if and only if $\pi(a)=\pi'(a)$ for every $a\in\fa(r)$, and similarly for $\phi$.
\end{lemm}
See \cite[Lemma~3.2.9]{gabbay:nomtnl} or \cite[Lemma~4.15]{gabbay:perntu-jv}.
\begin{prop}
\leftarrowbel{prop.completeness.lemma}
If $\Phi^\pi\cent\Psi^\pi$ in PNL and $\fa(\Phi)\cup\fa(\Psi)\subseteq \pmss(Z^\pi)$ then $\Phi\nopicent\Psi$.
\end{prop}
\begin{proof}
Using cut-elimination of full PNL (Theorem~\ref{thrm.rPNL.cut}) assume a cut-free PNL derivation $\Pi$ of $\Phi^\pi\cent\Psi^\pi$.
Because of Lemma~\ref{lemm.fa.restrict}, the condition on free atoms holds of every sequent in $\Pi$.
Because of the form of the derivation rules in Figure~\ref{Seq}, $\Pi$ cannot instantiate $Z^\pi$.
So we can go through the entire syntax of $\Pi$ and delete $Z^\pi$ to obtain a structure that is a candidate for being a derivation in restricted PNL of $\Phi\nopicent\Psi$.
The only non-trivial thing to check is that valid instances of \rulefont{Ax} are transformed to valid instances of \rulefont{Ax^\pi}.
Suppose we deduce $\Phi^\pi,\psi^\pi\cent\pi'\act\psi^\pi,\Psi^\pi$ using \rulefont{Ax}.
By assumption $\pi'\act\psi^\pi={\psi'}^\pi$ for some $\psi'$.
It follows that $\pi'\act Z^\pi=\id\act Z^\pi$ (recall from Subsection~\ref{subsect.aeq} that we quotient by $\alpha$-equivalence) and so by Lemma~\ref{lemm.pi.r.fa} that $\pi'(a)=a$ for all $a\in\pmss(Z^\pi)$.
By assumption $\fa(\Phi)\cup\fa(\Psi)\cup\fa(\psi)\cup\fa(\psi')\subseteq \pmss(Z^\pi)$ and so by Lemma~\ref{lemm.pi.r.fa} $\psi=\psi'$, and we are done.
\end{proof}
\begin{thrm}
\leftarrowbel{thrm.reduced.pnl.completeness}
If $\Phi\nopiment\Psi$ then $\Phi\nopicent\Psi$.
\end{thrm}
\begin{proof}
We prove the contrapositive, that if $\Phi\not\nopicent\Psi$ then $\Phi\not\nopiment\Psi$.
Suppose $\Phi\not\nopicent\Psi$.
Using the constructions above we augment to a signature $\mathcal S^\pi$ (Definition~\ref{defn.S.pi}) with some $Z^\pi$ with $\fa(\Phi)\cup\fa(\Psi)\subseteq\pmss(Z^\pi)$.
Thus by Proposition~\ref{prop.completeness.lemma} $\Phi^\pi\not\cent\Psi^\pi$.
By completeness of full PNL with respect to equivariant models (\cite[Theorem~3.45]{gabbay:pernl-jv}, \cite[Theorem~9.4.15]{gabbay:nomtnl}) we have that $\Phi^\pi\not\ment\Psi^\pi$.
So there exists an equivariant model $\mathcal I$ and valuation $\varsigma$ to $\mathcal I$ such that $\denot{\mathcal I}{\varsigma}{\Phi}=1$ and $\denot{\mathcal I}{\varsigma}{\Psi}=0$.
It is now routine to convert $\mathcal I$ into a non-equivariant model of the original signature $\mathcal S$ by taking $\tf P^\hden(x)=\tf P^\iden(\varsigma(Z^\pi),x)$.
\end{proof}
\end{document} | math | 136,470 |
\begin{document}
\title{Intersection of continua and rectifiable curves}
\author{Rich\'ard Balka}
\address{Alfr\'ed R\'enyi Institute of Mathematics,
PO Box 127, 1364 Budapest, Hungary}
\email{[email protected]}
\thanks{We gratefully acknowledge the support of the
Hungarian Scientific Research Fund grant no.~72655.}
\author{Viktor Harangi}
\address{Alfr\'ed R\'enyi Institute of Mathematics,
PO Box 127, 1364 Budapest, Hungary}
\email{[email protected]}
\date{}
\begin{abstract}
We prove that for any non-degenerate continuum $K \subseteq \mathbb{R}^d$
there exists a rectifiable curve such that
its intersection with $K$ has Hausdorff dimension $1$.
This answers a question of B.~Kirchheim.
\end{abstract}
\keywords{Continuum, rectifiable curve, Hausdorff dimension}
\subjclass[2010]{28A78}
\maketitle
\section{Introduction}
A topological space $K$ is called a \emph{continuum}
if it is compact and connected.
The following question was asked by B.~Kirchheim \cite{BK}.
\begin{question} \label{q}
Does there exist a non-degenerate curve (or more generally, a continuum)
$K\subseteq \mathbb{R}^{d}$ such that every rectifiable curve intersects $K$
in a set of Hausdorff dimension less than $1$?
\end{question}
The motivation behind this question was that
in \cite[Examples (b), p. 208.]{G}
Gromov implicitly suggested that such curves exist.
In this paper we answer Question \ref{q} in the negative.
\begin{xx}
For any non-degenerate continuum $K\subseteq \mathbb{R}^{d}$ there exists
a rectifiable curve such that its intersection with $K$
has Hausdorff dimension $1$.
\end{xx}
\begin{remark}
Finding a $1$-dimensional intersection is the best we can hope for,
since any purely unrectifiable curve $K$ in the plane
(e.g., the Koch snowflake curve) has the property that
the intersection of $K$ and a rectifiable curve has zero $\mathcal{H}^{1}$ measure.
\end{remark}
\section{Preliminaries} \label{s:prelim}
The diameter and the boundary of a set $A$ are denoted by
$\diam A$ and $\partial A$, respectively.
For $A\subseteq \mathbb{R}^{d}$ and $s \ge 0$ the
\emph{$s$-dimensional Hausdorff measure} is defined as
\begin{align*}
\mathcal{H}^{s}(A)&=\lim_{\delta\to 0+}\mathcal{H}^{s}_{\delta}(A)
\mbox{, where}\\
\mathcal{H}^{s}_{\delta}(A)&=\inf \left\{ \sum_{i=1}^\infty (\diam
A_{i})^{s}: A \subseteq \bigcup_{i=1}^{\infty} A_{i},~
\forall i \diam A_i \le \delta \right\}.
\end{align*}
Then the \emph{Hausdorff dimension} of $A$ is
\[
\dim_{H} A = \sup\{s \ge 0: \mathcal{H}^{s}(A)>0\}.
\]
Let $A\subseteq \mathbb{R}^{d}$ be non-empty and bounded, and let $\delta>0$. Set
$$N(A,\delta)=\min\left \{k: A\subseteq \bigcup_{i=1}^{k} A_i,~
\forall i \ \diam A_i\leq \delta \right\}.$$
The \emph{upper Minkowski dimension} of $A$ is defined as
$$\overline{\dim}_{M}(A)=\limsup_{\delta \to 0+}
\frac{\log N(A,\delta)}{-\log \delta}.$$
If $A\subseteq \mathbb{R}^{d}$ is non-empty and bounded, then it follows
easily from the above definitions that
$$\dim_{H}A\leq \overline{\dim}_{M}(A).$$
For more information on these concepts see \cite{F} or \cite{Ma}.
A continuous map $f\colon [a,b]\to \mathbb{R}^{d}$ is called a \emph{curve}.
Its \emph{length} is defined as
$$\length (f)=\sup \left\{\sum_{i=1}^{n} |f(x_{i})-f(x_{i-1})|:
n\in \mathbb{N}^{+},~ a=x_0<\dots<x_n=b \right\}.$$
If $\length(f)<\infty$, then $f$ is said to be \emph{rectifiable}.
We say that $f$ is \emph{naturally parametrized}
if for all $x,y\in [a,b]$, $x\leq y$ we have
$$\length \left(f|_{[x,y]}\right)=|x-y|.$$
We simply write $\Gamma=f([a,b])$ instead of $f$
if the parametrization is obvious or not important for us.
For every non-degenerate rectifiable curve
$\Gamma$ we have $0<\mathcal{H}^{1} (\Gamma)<\infty$, so $\dim_{H} \Gamma=1$.
If $|f(x)-f(y)|\leq |x-y|$ for all $x,y\in [a,b]$,
then $f$ is called \emph{1-Lipschitz}.
Every naturally parametrized curve is clearly 1-Lipschitz.
\section{The proof} \label{s:proof}
First we need some lemmas. The following lemma is probably known,
but we could not find a reference, so we outline its proof.
\begin{lemma} \label{l:cover}
If a non-empty bounded set $A \subseteq \mathbb{R}^d$ has
upper Minkowski dimension less than $1$,
then a rectifiable curve covers $A$.
\end{lemma}
\begin{proof} We can assume that $A$ is compact and $A \subseteq [0,1]^d$,
since we can take its closure and transform it into the unit
cube with a similarity, this does not change
the upper Minkowski dimension of the set
and the fact whether it can be covered by a rectifiable curve.
For every $n\in \mathbb{N}$ we divide $[0,1]^d$ into non-overlapping
cubes with edge length $2^{-n}$ in the natural way,
and we denote the cubes that intersect $A$ by
$$ Q_{n, 1}, Q_{n, 2}, \ldots, Q_{n, r_n} ,$$
where $r_n$ is the number of such cubes.
As every set with diameter at most $2^{-n}$ can intersect
at most $3^d$ of the above cubes, we obtain $r_n\leq 3^d N(A,2^{-n})$.
Let us fix $s$ such that
$\overline{\dim}_{M}(A)<s<1$. By the definition of upper Minkowski
dimension there exists a constant $c_1\in \mathbb{R}$ such that for all $n\in
\mathbb{N}$
\begin{equation} \label{rn} r_n \leq c_1 \cdot 2^{sn}.
\end{equation}
Let $n\in \mathbb{N}$ and $i\in \{1,\dots,r_n\}$ be arbitrarily fixed.
Let $P_{n,i}$ be the vertex of $Q_{n,i}$ that is the closest to the origin.
If $Q_{n+1, j_1}, \ldots, Q_{n+1, j_m}$ are the next level
cubes contained by $Q_{n,i}$, then consider the broken line
$$\Gamma_{n,i}=P_{n,i} P_{n+1, j_1} P_{n+1, j_2} \ldots P_{n+1,j_m} P_{n,i} .$$
Thus
\begin{equation} \label{leng} \length(\Gamma_{n,i})\leq
(m+1) \diam Q_{n,i} \leq 2m \sqrt{d} 2^{-n}.
\end{equation}
Let $l_n$ be the sum of these lengths for all $i\in \{1,\dots,r_n\}$.
Then \eqref{leng} and \eqref{rn} imply
\begin{equation} \label{ln}
l_n\leq 2 r_{n+1} \sqrt{d} 2^{-n}
\leq 2c_1 \cdot 2^{s(n+1)} \sqrt{d} 2^{-n} = c_2 2^{(s-1)n},
\end{equation}
where $c_2=c_1 \sqrt{d} 2^{s+1}$. We set
$$L_n=\sum_{k=0}^{n}l_k \quad \textrm{and} \quad L=\sum_{k=0}^{\infty} l_k.$$
Since $s<1$, \eqref{ln} implies $L < \infty$.
Now we define the rectifiable curve covering $A$.
First we take the broken line $\Gamma_0=\Gamma_{0,1}$
with its natural parametrization $g_{0}\colon [0,L_0]\to \Gamma_0$.
Assume that the curves $g_{k}\colon [0,l_k]\to \Gamma_k$ are
already defined for all $k<n$. At every point $P_{n,i}$,
$i\in \{1,\dots, r_n\}$, we insert the broken line
$\Gamma_{n,i}$ in $\Gamma_{n-1}$, so we obtain
a naturally parametrized curve $g_{n}\colon [0,L_{n}]\to \Gamma_{n}$.
For every $n\in \mathbb{N}$ let us define $f_{n}\colon [0,L]\to \Gamma_n$ such that
$$f_{n}(x)= \begin{cases} g_{n}(x) & \textrm{ if } x\in [0,L_n], \\
g_{n}(L_n) & \textrm{ if } x\in [L_n,L].
\end{cases} $$
Now we prove that the sequence $\langle f_n \rangle$ uniformly converges.
Let us fix $n\in \mathbb{N}$ and $x\in [0,L]$ arbitrarily.
As $\sum_{n=0}^{\infty} l_n<\infty$, it is enough to prove that
$|f_{n+1}(x)-f_{n}(x)|\leq l_{n+1}$. By construction
there exists $y\in [0,L]$ such that
$f_{n}(x)=f_{n+1}(y)$ and $|x-y| \leq l_{n+1}$.
Since $g_{n+1}$ is naturally parametrized, we obtain that
$$|f_{n+1}(x)-f_{n}(x)|= |f_{n+1}(x)-f_{n+1}(y)|\leq |x-y|\leq l_{n+1}.$$
Therefore $\langle f_n \rangle$ uniformly converges to
some $f:[0,L] \to \mathbb{R}^d$. As a uniform limit of 1-Lipschitz functions
$f$ is also 1-Lipschitz, thus rectifiable.
It remains to prove that $A\subseteq f([0,L])$. Let $\vec{z} \in A$.
We need to show that there is $x\in [0,L]$ such that $f(x)=\vec{z}$.
For every $n\in \mathbb{N}$ there exists $i_n\in \{1,\dots, r_n\}$
such that $\vec{z} \in Q_{n,i_n}$. Let $x_n\in [0,L]$ such that
$f_{n}(x_n)=P_{n,i_n}$ for all $n\in \mathbb{N}$. By choosing a subsequence
we may assume that $x_{n}$ converges to some $x \in [0,L]$.
Therefore
$$f(x)=\lim_{n\to \infty} f_{n} (x_{n})=
\lim_{n\to \infty} P_{n,i_{n}}=\vec{z}.$$
The proof is complete.
\end{proof}
The next lemma is \cite[Lemma 6.1.25]{E}.
\begin{lemma} \label{l:comp}
If $A$ is a closed subspace of a continuum $X$
such that $\emptyset \neq A\neq X$,
then for every connected component $C$ of $A$
we have $C\cap \partial A\neq \emptyset$.
\end{lemma}
We also need the following technical lemma.
\begin{lemma} \label{l:cube}
Suppose that $K\subseteq \mathbb{R}^{d}$ is a continuum
contained by a unit cube $Q$ and
$K$ has a point on each of two opposite sides of $Q$.
Then for any positive integer $N$
we can find $N$ pairwise non-overlapping cubes $Q_1, \ldots, Q_N$
with edge length $\frac 1N$ such that for each $i\in \{1,\dots, N\}$
there exists a continuum $K_i \subseteq K \cap Q_i$ with the property that
$K_i$ has a point on each of two opposite sides of $Q_i$.
\end{lemma}
\begin{proof}
Let $N\in \mathbb{N}^{+}$ be fixed. Set $S_0=\{0\}\times [0,1]^{d-1}$
and for all $i\in \{1,\dots, N\}$ consider
$$S_i=\{i/N\}\times [0,1]^{d-1} \quad \textrm{and} \quad
T_i=\left[(i-1)/N,i/N\right]\times [0,1]^{d-1}.$$
We may assume that $Q=[0,1]^{d}$ and
that the two opposite sides intersecting $K$
are $S_0$ and $S_N$. Let $\vec{x}\in K \cap S_0$
and $\vec{y} \in K \cap S_N$.
Now we prove that for each $i\in \{1,\dots ,N\}$
there is a continuum $C_i\subseteq K\cap T_i$ such that
$C_i\cap S_{i-1} \neq \emptyset$ and $C_i\cap S_i\neq \emptyset$.
Let $C_1$ be the component of $K\cap T_1$ containing $\vec{x}$.
Applying Lemma \ref{l:comp} for $X=K$, $A=K \cap T_1$, and $C=C_1$
yields that $C_1 \cap S_1 \neq \emptyset$. Let $C_2'$ be the component of
$K\cap \left( T_2\cup \dots \cup T_N\right)$ containing $\vec{y}$.
Similarly as above, we obtain $C_2'\cap S_1\neq \emptyset$.
If we continue this process, we get the required continua $C_2,\dots, C_N$.
Finally, for each $i\in \{1,\dots,N\}$ we construct
a cube $Q_i \subseteq T_i$ with edge length $\frac 1N$ and
a continuum $K_i\subseteq Q_i$ such that
$K_i$ has a point on each of two opposite sides of $Q_i$.
Clearly, the cubes $Q_i$ will be pairwise non-overlapping,
and it is enough to construct $Q_1$ and $K_1$
(one can get $Q_i, K_i$ similarly).
Let us consider the standard basis of $\mathbb{R}^d$:
$\vec{e}_1=(1,0,\dots,0),\dots,\vec{e}_d=(0,0,\dots,1)$.
Set $A_1=C_1$, $V_1=\{0\}\times \mathbb{R}^{d-1}$, $W_1=\{1/N\}\times \mathbb{R}^{d-1}$,
$Z_1=[0,1/N]\times \mathbb{R}^{d-1}$, and $m(1)=1$.
Then the definitions yield that $A_1$ has a point
on both $V_{m(1)}$ and $W_{m(1)}$.
Let $j\in \{2,\dots, d\}$ and assume that
$A_{k}$, $V_{k}$, $W_{k}$, $Z_{k}$, and $m(k)$
are already defined for all $k<j$ such that
$A_k$ has a point on both $V_{m(k)}$ and $W_{m(k)}$.
Let $\vec{x}_j\in A_{j-1}$ be a point which has minimal $j$th coordinate,
and let $V_j$ be the affine hyperplane that is orthogonal to
$\vec{e}_j$ and contains $\vec{x}_j$.
Set $W_j=V_j+\frac {1}{N} \vec{e}_j$,
and let $Z_j$ be the closed strip between $V_j$ and $W_j$.
If $A_{j-1}\subseteq Z_j$ then let $A_j=A_{j-1}$ and $m(j)=m(j-1)$.
If $A_{j-1} \nsubseteq Z_j$ then let $A_j$ be
the component of $\vec{x}_j$ in $A_{j-1}\cap Z_j$ and $m(j)=j$,
in this case Lemma \ref{l:comp} yields $A_{j}\cap W_j\neq \emptyset$.
Thus $A_j$ has a point on both $V_{m(j)}$ and $W_{m(j)}$.
Let $Q_1=\bigcap _{j=1}^{d} Z_j$ and $K_1=A_d$.
Then $Q_1\subseteq S_1$ is a cube with edge length $\frac 1N$
and $K_1\subseteq Q_1$ is a continuum.
As $K_1$ has a point on both $V_{m(d)}$ and $W_{m(d)}$,
we obtain that $K_1$ has a point on each of two opposite sides of $Q_1$.
The proof is complete.
\end{proof}
Now we are ready to prove Theorem \ref{thm}.
\begin{theorem} \label{thm}
For any non-degenerate continuum $K\subseteq \mathbb{R}^{d}$ there exists
a rectifiable curve such that its intersection with $K$
has Hausdorff dimension $1$.
\end{theorem}
\begin{proof} By considering a similar copy of $K$
we may assume that $K$ is contained by a unit cube $Q$
and $K$ has a point on each of two opposite sides of $Q$.
Let $\varepsilon>0$ be arbitrary. First we prove the weaker result
that there exists $A\subseteq K$ such that
$ 1-\varepsilon \leq \dim_{H}A= \overline{\dim}_{M}(A)<1$.
By Lemma \ref{l:cover} $A$ is covered by a rectifiable curve.
Let us fix an integer $N\geq 2$ for which
$s:=\frac {\log (N-1)}{\log N} \geq 1-\varepsilon$.
We construct $A\subseteq K$ such that $\dim_{H}A=\overline{\dim}_{M}(A)=s$.
Set $\mathcal{I}_{n}=\{1,\dots, N-1\}^{n}$ for every $n\in \mathbb{N}^{+}$.
Iterating Lemma \ref{l:cube} implies that for all $n\in \mathbb{N}^{+}$ and
$(i_1,\dots, i_n)\in \mathcal{I}_{n}$
there are cubes $Q_{i_1 \dots i_{n}}$ in $Q$
with edge length $\frac{1}{N^{n}}$ such that
$Q_{i_1 \dots i_{n}}\subseteq Q_{i_1 \dots i_{n-1}}$,
and there are continua $K_{i_1 \dots i_{n}} \subseteq K$ such that
$K_{i_1 \dots i_{n}}\subseteq Q_{i_1 \dots i_{n}}\cap K_{i_1\dots i_{n-1}}$
and $K_{i_1 \dots i_{n}}$ has a point on each of
two opposite sides of $Q_{i_1 \dots i_{n}}$. Set
\begin{equation*}
A_n=\bigcup_{i_1=1}^{N-1} \! \cdots \!
\bigcup_{i_n=1}^{N-1} K_{i_1 \dots i_{n}},
\end{equation*}
and let
\begin{equation*}
A=\bigcap_{n=1}^{\infty} A_n.
\end{equation*}
Clearly, $A\subseteq K$ is compact.
On the one hand, as $A\subseteq A_n$ and $A_n$ is covered by $(N-1)^{n}$
many cubes of edge length $\frac{1}{N^{n}}$, we obtain that
$N(A_n, \sqrt{d}/N^{n})\leq (N-1)^{n}$ for all $n\in \mathbb{N}^{+}$.
Therefore $\overline{\dim}_{M}(A)\leq \frac {\log (N-1)}{\log N}=s$.
On the other hand, we prove that $\mathcal{H}^{s}(A)>0$.
Assume that $A\subseteq \bigcup_{j=1}^{\infty} U_j$,
it is enough to prove that
$ \sum_{j=1}^{\infty} (\diam U_j)^{s}\geq \frac{1}{2^d(N-1)}$.
Clearly, we may assume that $U_j$ is a non-empty open set
with $\diam U_j < 1$ for each $j$,
and the compactness of $A$ implies that
there is a finite subcover $A\subseteq \bigcup_{j=1}^{k} U_j$.
Let us fix $n_0\in \mathbb{N}^{+}$ such that
$\frac{1}{N^{n_0}}<\min_{1\leq j\leq k} \diam U_j$.
For $j\in \{1,\dots,k\}$ let
$$ t_j = \# \left\{(i_1,\dots,i_{n_0})\in \mathcal{I}_{n_0}:
U_j\cap K_{i_1\dots i_{n_0}}\neq \emptyset\right\} .$$
Since $A\subseteq \bigcup_{j=1}^{k}U_j$, we have
\begin{equation} \label{eq:tj} \sum_{j=1}^{k} t_{j}\geq
(N-1)^{n_0}.
\end{equation}
Now we show that for all $j\in \{1,\dots,k\}$
\begin{equation} \label{eq:Uj} (\diam U_j)^{s} \geq
\frac{t_j}{2^{d}(N-1)^{n_0+1}}.
\end{equation}
Let us fix $j\in \{1,\dots, k\}$. There exists $0\leq m<n_0$ such that
$\frac{1}{N^{m+1}}\leq \diam U_j < \frac{1}{N^{m}}$.
Clearly, the number of cubes $Q_{i_1 \dots i_m}$ at level $m$ that
intersect $U_j$ is at most $2^d$. Therefore $t_j\leq 2^{d} (N-1)^{n_0-m}$.
On the other hand, $\diam U_j\geq \frac{1}{N^{m+1}}$
implies $(\diam U_j)^{s}\geq \frac{1}{(N-1)^{m+1}}$,
and \eqref{eq:Uj} follows. By \eqref{eq:tj} and \eqref{eq:Uj} we obtain
$$ \sum_{j=1}^{k} (\diam U_j)^{s} \geq
\sum_{j=1}^{k} \frac{t_j}{2^{d}(N-1)^{n_0+1}}\geq \frac{1}{2^d(N-1)}.$$
Hence $\mathcal{H}^{s} (A)>0$. Therefore $\dim_{H} A\geq s$,
so $s\leq \dim_{H}A \leq \overline{\dim}_{M}(A)\leq s$.
Thus $1-\varepsilon \leq \dim_{H} A=\overline{\dim}_{M}(A)<1$.
Now we are in a position to prove that there exists
a rectifiable curve $\Gamma$ with $\dim_{H}(\Gamma \cap K) = 1$.
Pick an arbitrary point $\vec{x} \in K$
and let $K_n$ be the intersection of $K$ and
the closed ball of radius $1/2^n$ centered at $\vec{x}$.
Let $C_n$ denote the component of $K_n$ containing $\vec{x}$.
Since $C_n$ is a non-degenerate continuum by Lemma \ref{l:comp}, we know that
there exists $A_n\subseteq C_n$ such that
$1-\frac{1}{n}\leq \dim_{H} A_n=\overline{\dim}_{M}(A)<1$.
Therefore Lemma \ref{l:cover} implies that
there exist rectifiable curves $\Gamma_n$ covering $A_n$.
We may assume that the endpoints of $\Gamma_n$ are in $A_n$.
We can also assume that the length of $\Gamma_n$ is at most $1/2^n$.
(Otherwise we split up $\Gamma_n$ into finitely many parts,
each having length at most $1/2^n$; then one of these parts
intersects $A_n$ in a set of Hausdorff dimension at least $1-\frac{1}{n}$.)
Let us concatenate the curves $\Gamma_n$ with line segments.
Then the full length of the line segments is
at most $2\sum_{n=1}^{\infty} \frac{1}{2^n}=2$,
the full length of the curves $\Gamma_n$ is at most
$\sum_{n=1}^{\infty} \frac{1}{2^n}=1$,
so we get a rectifiable curve $\Gamma$ that
covers $\bigcup_{n=1}^{\infty} A_n$.
As $\dim_{H}\left(\bigcup_{n=1}^{\infty} A_n\right)=1$,
the intersection $\Gamma \cap K$ has Hausdorff dimension $1$.
The proof is complete.
\end{proof}
\end{document} | math | 16,421 |
\begin{document}
\begin{center}
{{\large\bf Uniform spanning forests associated with biased random walks\\ on Euclidean lattices}\footnote{The project is supported partially by CNNSF (No.~11671216).}}
{\mathrm{e}}nd{center}
\vskip 2mm
\begin{center}
Z.
Shi, V.
Sidoravicius, H.
Song, L. Wang, K. Xiang
{\mathrm{e}}nd{center}
\vskip 2mm
\begin{abstract}
The uniform spanning forest measure (${\mathsf{USF}}$) on a locally finite, infinite connected graph $G$ with {conductance} $c$ is defined as a
weak limit of uniform spanning tree measure on finite subgraphs. Depending
on the underlying graph and conductances, the corresponding ${\mathsf{USF}}$ is not necessarily concentrated on the set of spanning trees.
Pemantle~\cite{PR1991} showed that on ${\mathbb{Z}}^d$, equipped with the {\it unit }conductance $ c=1$, ${\mathsf{USF}}$ is concentrated on spanning trees if and only if $d \leq 4$. In this work we study the ${\mathsf{USF}}$ associated with conductances induced by $\lambda$--biased random walk on ${\mathbb{Z}}^d$, $d \geq 2$, $0 < \lambda < 1$, {\it i.e.} conductances are set to be $c(e) = \lambda^{-|e|}$, where $|e|$ is the graph distance of the edge $e$ from the origin. Our main result states that in this case ${\mathsf{USF}}$ consists
of finitely many trees if and only if $d = 2$ or $3$. More precisely, we prove that the uniform spanning forest has $2^d$ trees if $d = 2$ or $3$,
and infinitely many trees if $d \geq 4$. Our method relies on the analysis of the spectral radius and the speed of the $\lambda$--biased random walk on ${\mathbb{Z}}^d$.
\noindent{\it AMS 2010 subject classifications}. Primary 60J10, 60G50, 05C81; secondary 60C05, 05C63, 05C80.
\noindent{\it Key words and phrases}. Biased random walk, spectral radius, speed, free uniform spanning forest, wired uniform spanning forest.
{\mathrm{e}}nd{abstract}
\section{Introduction and main results}
Let $G:= (V(G), \, E(G))$ be a locally finite, connected infinite graph and fix a vertex $o$ in $G$ as root.
For $x \in V(G)$, let $|x|$ be the graph distance of $x$ from $o$. We define, for $n \ge 0$,
$$
B_G(n)
:=
\{x\in V(G):\ |x| \le n\},
\qquad
\partial B_G(n)
:=
\{x\in V(G):\ |x| =n\}.
$$
\noindent Let $\lambda>0$. The $\lambda$-biased random walk, or ${\mathbb{R}}W_\lambda$, is a random walk on $(G,\, o)$ with transition probabilities: for $y$ adjacent to $x$,
\begin{eqnarray}
\label{(1.1)}
p_\lambda(x,y)=
\begin{cases}
\frac{1}{d_o} &{\rm if}\ x=o,\\
\frac{\lambda}{d_x+\left(\lambda-1\right)d_x^-} &{\rm if} \ x\neq o \text{ and } |y| = |x|-1, \\
\frac{1}{d_x+\left(\lambda-1\right)d_x^-} & \text{otherwise.}
{\mathrm{e}}nd{cases}
{\mathrm{e}}nd{eqnarray}
\noindent Here $d_x$ is the degree of vertex $x$, and $d_x^-$ (resp. $d_x^0$) is the number of edges connecting $x$ to $\partial B_G(|x|-1)$ (resp. $\partial B_G(|x|)$). Note that $d_x^{-}\ge 1$ if $x\not=o$, and $d_o^-=d_o^0=0$. When $\lambda = 1$, ${\mathbb{R}}W_\lambda$ is the usual simple random walk on $G$. For general properties of biased random walks on graphs we refer to \cite{LR-PY2016} and \cite{SSSWX2017a+}.
In this work we study the uniform spanning forest on the network associated with ${\mathbb{R}}W_\lambda$. It relies on the analysis of the spectral radius and the speed of the walk. More specifically, we focus on the spectral radius and the speed of $\lambda$--biased random walk on the $d$-dimensional lattice ${\mathbb{Z}}^d$, and always assume $0 < \lambda < 1$, unless it is stated otherwise.
From {\mathrm{e}}qref{(1.1)} one can see that ${\mathbb{R}}W_\lambda$ on ${\mathbb{Z}}^d$ is closely related to the drifted random walk on ${\mathbb{Z}}^d$, whose distribution is given by convolutions of step distribution
\begin{equation}
\label{e:mu}
\mu({\mathbf{e}}_1) = \cdots = \mu({\mathbf{e}}_d) =
\frac{1}{d(1+\lambda)}, \quad \mu(-{\mathbf{e}}_1) = \cdots =
\mu(-{\mathbf{e}}_d) = \frac{\lambda}{d(1+\lambda)},
{\mathrm{e}}nd{equation}
\noindent where $\{{\mathbf{e}}_1, \ldots, {\mathbf{e}}_d\}$ is the standard basis of ${\mathbb{Z}}^d$. Before exiting from one of the $2^d$ open orthants, the $\lambda$--biased random walk and drifted random walk have the same distributions. This fact is crucial for the analysis of spectral radius, speed and intersection properties of $\lambda$--biased random walks on ${\mathbb{Z}}^d$.
However, $\lambda$--biased random walks exhibit quite different behavior from drifted random walk when they hit some axial hyperplane or the boundary of the orthant.
Let $(X_n)$ be the ${\mathbb{R}}W_\lambda$ on ${\mathbb{Z}}^d$ and $p^{(n)}_\lambda (x,\, y) = {\mathbb{P}}_x (X_n = y)$ be the $n$-step transition probability of $X_n$,
where $\mathbb{P}_x$ is the law of ${\mathbb{R}}W_\lambda$ starting at $x$.
The spectral radius $\rho(\lambda)$ of ${\mathbb{R}}W_\lambda$ is defined to be the reciprocal of the convergence radius for the Green function
$$
\mathbb{G}_\lambda(x,y | z)
:=
\sum_{n = 0}^\infty p^{(n)}_\lambda(x,y) z^n.
$$
\noindent Clearly, $\rho(\lambda)$ does not depend on the choices of $x$ and $y$, and can be expressed as
$$
\rho(\lambda)
:=
\limsup_{n \to \infty} [p^{(n)}_\lambda (o,\, o)]^{1/n}.
$$
\noindent Define the speed $\mathcal{S}(\lambda)$ of ${\mathbb{R}}W_{\lambda}$ by
$$
\mathcal{S}(\lambda)
:=
\lim_{n \to \infty} \frac{|X_n|}{n},
$$
\noindent provided the limit exists almost surely.
There are many deep and important questions related to how the spectral radius and the speed depend on the bias parameter $\lambda$.
Lyons, Pemantle and Peres \cite{LR-PR-PY1996b} asked whether the speed of ${\mathbb{R}}W_\lambda$ on the supercritical Galton--Watson tree without leaves is strictly decreasing. This has been confirmed for $\lambda$ lying in some regions (cf.\ \cite{BG-FA-SV2014,AE2014,AE2013,SH-WL-XK2015}), but still remains open for general values of $\lambda$. For the supercritical Galton--Watson tree with leaves, the speed is expected (\cite[Section~3]{BG-FA2014}) to be unimodal in $\lambda$ (due to presence of traps). On lamplighter graph ${\mathbb{Z}} \ltimes \sum_{x \in {\mathbb{Z}}} {\mathbb{Z}}_2$, the speed of ${\mathbb{R}}W_\lambda$ is positive if and only if $1 < \lambda < (1+\sqrt{5})/2$; see \cite{LR-PR-PY1996a}.
We are ready to state our first main result. Its proof is given in Section~{\mathrm{Re}}f{s:spectral}.
\begin{thm}\label{T:main1}
Let $\lambda \in (0, \, 1)$ and let $(X_n)$ be ${\mathbb{R}}W_\lambda$ on ${\mathbb{Z}}^d$.
\begin{enumerate}[(i)]
\item
The spectral radius is $\rho(\lambda) = \frac{2\sqrt{\lambda}}{1+\lambda}<1$.
\item The speed exists, and equals ${\mathcal{S}}(\lambda)=\frac{1-\lambda}{1+\lambda}$.
{\mathrm{e}}nd{enumerate}
{\mathrm{e}}nd{thm}
It is straightforward from the expressions above that the spectral radius is strictly increasing in $\lambda$
and speed is strictly decreasing in $\lambda$.
We now turn to the main topic of the paper, namely, the study of the uniform spanning forest of the network associated with the ${\mathbb{R}}W_\lambda$, by applying Theorem~{\mathrm{Re}}f{T:main1}. Viewing $G=(V(G), \, E(G))$ as an infinite network with appropriate conductances on its edges, the uniform spanning forest measures are defined as weak limit of uniform spanning tree measures of finite subgraphs of $G$. The limit can be taken with either free or wired boundary conditions, yielding the free uniform spanning forest measure (denoted by ${\mathfrak{F}}SF$) and the wired uniform spanning forest measure (${\mathsf{WSF}}$), respectively. In general, ${\mathfrak{F}}SF$ stochastically dominates ${\mathsf{WSF}}$ on any infinite network. If they coincide, we call them the uniform spanning forests (${\mathsf{USF}}$) for simplicity. For more details, see Section {\mathrm{Re}}f{s:usf} (or \cite{LR-PY2016}).
Both ${\mathfrak{F}}SF$ and ${\mathsf{WSF}}$ on an infinite network are concentrated on the set of spanning forests with the property that every tree (i.e., the connected component) in the forest is infinite.
When $\lambda=1$, the remarkable result of Pemantle~\cite{PR1991} (see also \cite{LR-PY2016}) states that ${\mathsf{USF}}$ on ${\mathbb{Z}}^d$ has a single tree for $d\le 4$ and has infinitely many trees for $d \ge 5$. By \cite[Theorem~9.4]{BI-LR-PY2001}, this type of phase transition depending on the dimension has a deep connection with the well-known intersection property of independent simple random walks on ${\mathbb{Z}}^d$, namely, two independent simple random walks on ${\mathbb{Z}}^d$ intersect infinitely often if $d \le 4$ and finitely many times if $d \ge 5$; see for example Lawler \cite{LG1980,LG1991}. We show that there is a phase transition for the number of trees in the USF on the network associated with ${\mathbb{R}}W_\lambda$ on ${\mathbb{Z}}^d$ with $0<\lambda < 1$, while the critical dimension is reduced from $4$ to $3$.
\begin{thm}
\label{T:USF}
Let $\lambda \in (0, \, 1)$. Almost surely, the number of trees in the uniform spanning forest associated with ${\mathbb{R}}W_{\lambda}$ on ${\mathbb{Z}}^d$ is $2^d$ if $d = 2$ or $3$,
and is infinite if $d \ge 4$.
{\mathrm{e}}nd{thm}
Theorem~{\mathrm{Re}}f{T:USF} is (restated and proved) in Section~{\mathrm{Re}}f{s:usf}.
As we mentioned before, an important step in the proof is to
determine the number of intersections of two independent random walks. We state the result below and its proof is given in Section~{\mathrm{Re}}f{s:intersection}.
\begin{thm}
\label{T:intersection}
Assume $\lambda\in (0,\, 1)$. Let $(Z_n)_{n=0}^{\infty}$ and $(W_n)_{n=0}^{\infty}$ be independent drifted random walks on ${\mathbb{Z}}^d$ with the same step distribution $\mu$ given by {\mathrm{e}}qref{e:mu}, starting at $z_0$ and $w_0$ respectively. Then almost surely,
$$
| \{Z_m;\ m\ge 0\} \cap \{W_n;\ n\ge 0\} |
\ \text{is finite for}\ d\ge 4\ \text{and infinite for}\ d\le 3.
$$
{\mathrm{e}}nd{thm}
The rest of the paper is organised as follows. In Section~{\mathrm{Re}}f{s:spectral}, we prove sharp estimates for the $n$-step transition probability and the strong law of large numbers of ${\mathbb{R}}W_\lambda$ on ${\mathbb{Z}}^d$. The statements for the spectral radius and the speed in Theorem~{\mathrm{Re}}f{T:main1} are direct consequences. The number of intersections of two independent drifted (or biased) random
walks is studied in Section~{\mathrm{Re}}f{s:intersection}. In Section~{\mathrm{Re}}f{s:usf} we consider the uniform spanning forests associated with ${\mathbb{R}}W_\lambda$, and prove Theorem~{\mathrm{Re}}f{T:USF}.
\section{Spectral radius and speed}
\label{s:spectral}
In this section, we prove Theorem {\mathrm{Re}}f{T:main1} for ${\mathbb{R}}W_\lambda$ on $\mathbb{Z}^d$. In fact, we obtain sharp estimates for the $n$-step transition probability (Theorem~{\mathrm{Re}}f{T:hkZd}), and establish a strong law of large numbers (Theorem~{\mathrm{Re}}f{T:speed}). Theorem~{\mathrm{Re}}f{T:main1} is a straightforward consequence of Theorems~{\mathrm{Re}}f{T:hkZd} and
{\mathrm{Re}}f{T:speed}.
For positive functions $f$ and $g$ on ${\mathbb{N}}$, we write $f\asymp g$ if there is a constant $c>0$ such that $c^{-1} g(n) \le f(n) \le c g(n)$ for all $n \in {\mathbb{N}}$, and write $f \sim g$ if $\lim_{n\to\infty} \frac{f(n)}{g(n)} = 1$.
\begin{thm}
\label{T:hkZd}
Let $\lambda \in (0, \, 1)$, and let $(X_n)$ be ${\mathbb{R}}W_\lambda$ on ${\mathbb{Z}}^d$. Then
\begin{equation}
\label{e:hk}
p^{(2n)}_\lambda(o,\, o)
\asymp
\Big(\frac{2\sqrt{\lambda}}{1+\lambda}\Big)^{\! 2n} \frac{1}{n^{3d/2}}.
{\mathrm{e}}nd{equation}
\noindent In particular, the spectral radius equals $\rho(\lambda) = \frac{2\sqrt{\lambda}}{1+\lambda}<1$, and is strictly increasing in $\lambda$.
{\mathrm{e}}nd{thm}
The proof of Theorem {\mathrm{Re}}f{T:hkZd} relies on the following lemma, which is motivated by \cite[Exercise 1.7]{PG2015}. For $0 \le k < n$, let $B_{n,k}$ be the set of paths $(x_0, \ldots, x_{2n})$ taking values in ${\mathbb{Z}}$ with $x_0=0=x_{2n}$ and $\# \{ i: \ 1\le i\le 2n, \, x_i = 0\} = k+1$. Here and throughout, by a path we mean $|x_i-x_{i-1}|=1$ for all $i$ (in other words, it is a possible trace of a simple random walk), and $2n$ is called the length of the path.
\begin{lem}
\label{Lem5.1}
There is a positive constant $c$ such that $|B_{n,k}|\le \frac{c \, k^{5/2}4^n}{n^{3/2}}$ for $n\in\mathbb{N}$ and $k\in [0,\, n]$.
{\mathrm{e}}nd{lem}
\begin{proof}
The lemma holds if $k=0$ ($|B_{n,k}|=0$ in this case), or if $k \ge \frac{n}{2}$ (using the trivial inequality $|B_{n,k}| \le 2^{2n}$).
Assume now $0<k<\frac{n}{2}$. For ${\mathrm{e}}ll\ge 1$, let $C_{\mathrm{e}}ll= \frac{1}{{\mathrm{e}}ll+1} {2{\mathrm{e}}ll \choose {\mathrm{e}}ll}$ be the ${\mathrm{e}}ll$-th Catalan number. The number of paths $(x_0, \ldots, x_{2{\mathrm{e}}ll})$ on $\mathbb{Z}$ with length $2{\mathrm{e}}ll$ such that $x_0=x_{2{\mathrm{e}}ll}=0\in {\mathbb{Z}}$ and that $x_i \not= 0$ for $1\le i\le 2{\mathrm{e}}ll-1$ is $2C_{{\mathrm{e}}ll-1}$; such paths are the so-called excursions. By splitting the paths in $B_{n,k}$ into excursions, we see that
$$
|B_{n,k}|
\le
\sum_{\substack{n_1 + \cdots + n_k = n \\ n_i \ge 1,\, 1\le i\le k}}
(2C_{n_1-1})(2C_{n_2-1})\cdots (2C_{n_k-1})
=
2^k \sum_{\substack{n_1 + \cdots + n_k = n \\ n_i \ge 1,\, 1\le i\le k}}
C_{n_1-1} C_{n_2-1} \cdots C_{n_k-1} \, .
$$
\noindent
Since $n_i \geq \frac{n}{k}$ for some $i$, we have that
\begin{align*}
|B_{n,k}|
&\le k \, 2^k \sum_{\substack{n_1 + \cdots + n_k = n \\ n_i \ge 1,\, 2\le i\le k, \; n_1 \ge n/k}}
C_{n_1-1} C_{n_2-1} \cdots C_{n_k-1}
\\
&= k \, 2^k \sum_{\substack{n_2 + \cdots + n_k \le n-(n/k)\\ n_i \ge 1,\, 2\le i\le k}}
C_{n-n_2-\cdots-n_k-1} C_{n_2-1} \cdots C_{n_k-1} \, .
{\mathrm{e}}nd{align*}
\noindent Recall that (\cite{DR-BR1986}) $C_{\mathrm{e}}ll < \frac{4^{\mathrm{e}}ll}{({\mathrm{e}}ll+1)\, (\pi {\mathrm{e}}ll)^{1/2}}$ for all ${\mathrm{e}}ll$. So $C_{n-n_2-\cdots-n_k-1} < \frac{4^{n-n_2-\cdots-n_k-1}}{(n-n_2-\cdots-n_k)\, (\pi (n-n_2-\cdots-n_k-1))^{1/2}}$, which is bounded by $\frac{4^{n-n_2-\cdots-n_k-1}}{\frac{n}{k}\, (\pi (\frac{n}{k}-1))^{1/2}}$ if $n_2 + \cdots + n_k \le n - \frac{n}{k}$. Accordingly,
$$
|B_{n,k}|
\le
\frac{k \, 2^k \, 4^{n-1}}{\frac{n}{k}\, (\pi (\frac{n}{k}-1))^{1/2}}
\sum_{\substack{n_2 + \cdots + n_k \le n-(n/k) \\ n_i \ge 1,\, 2\le i\le k}} \frac{C_{n_2-1}}{4^{n_2}} \cdots \frac{C_{n_k-1}}{4^{n_k}}
\le
\frac{k \, 2^k \, 4^{n-1}}{\frac{n}{k}\, (\pi (\frac{n}{k}-1))^{1/2}} \Big( \sum_{m=1}^\infty \frac{C_{m-1}}{4^m} \Big)^{k-1} .
$$
\noindent Recall that the generating function of $C_{\mathrm{e}}ll$ is
$$
\sum_{{\mathrm{e}}ll=0}^\infty C_{\mathrm{e}}ll x^{\mathrm{e}}ll
=
\frac{1 - 2\sqrt{1-x}}{2 x},
\qquad
x \in \Big[ - \frac{1}{4}, \, \frac{1}{4} \Big],
$$
\noindent from which it follows that $\sum_{{\mathrm{e}}ll=0}^\infty \frac{C_{\mathrm{e}}ll}{4^{{\mathrm{e}}ll+1}} = \frac12$. Hence
$$
|B_{n,k}|
\le
\frac{k \, 2^k \, 4^{n-1}}{\frac{n}{k}\, (\pi (\frac{n}{k}-1))^{1/2}} \, \frac{1}{2^{k-1}}
=
\frac{2k \, 4^{n-1}}{\frac{n}{k}\, (\pi (\frac{n}{k}-1))^{1/2}} .
$$
\noindent Since $0<k<\frac{n}{2}$, we have $(\pi (\frac{n}{k}-1))^{1/2} = (\frac{\pi n}{k})^{1/2} \, (1-\frac{k}{n})^{1/2} \ge (\frac{\pi n}{2k})^{1/2}$, so that $|B_{n,k}| \le \frac{2^{3/2}k^{5/2} \, 4^{n-1}}{\pi^{1/2}n^{3/2}}$ as desired.
{\mathrm{e}}nd{proof}
\begin{proof}[Proof of Theorem~{\mathrm{Re}}f{T:hkZd}]
\textbf{Step 1.} We first show the lower bound for $p^{(2n)}_\lambda(o,\, o)$.
Let $n>d$. We get a lower bound for $p^{(2n)}_\lambda(o,\, o)$ by considering only the paths starting at $o$ that reach $(1, \ldots, 1)$ at step $d$ (which happens with probability greater than or equal to $(\frac{1}{d(1+\lambda)})^d$), then stay in the first open orthant $\{ x = (x_1, \ldots, x_d) \in {\mathbb{Z}}^d: \, x_i > 0, \; 1 \le i \le d\}$ for the next $2(n-d)$ steps and end up at $(1, \ldots, 1)$ again (of which we are going to estimate the probability), and finally return to $o$ at step $2n$ (which happens with probability greater than or equal to $(\frac{\lambda}{d(1+\lambda)})^d$). To compute the probability that, starting at $(1, \ldots, 1)$, the walk stays in the first open orthant for $2(n-d)$ steps and ends up at $(1, \ldots, 1)$, we observe, by decomposing the paths into excursions as in the proof of Lemma {\mathrm{Re}}f{Lem5.1}, that the total number of possible such paths is at least
$$
\sum_{\substack{n_1 + \cdots + n_d = n - d \\ n_i \ge 0,\; 1\le i\le d}} {2n \choose 2 n_1,\cdots, 2n_d} C_{n_1} \cdots C_{n_d} ,
$$
\noindent where $C_{\mathrm{e}}ll$ denotes as before the Catalan number, and ${2n \choose 2 n_1,\cdots, 2n_d} := \frac{(2n)!}{(2n_1)! \cdots (2n_d)!}$ is the multinomial coefficient. By definition, the transition probability that ${\mathbb{R}}W_{\lambda}$, along such paths, steps forward (resp.\ backward) along each coordinate in the first open
orthant is $\frac{1}{d(1+\lambda)}$ (resp.\ $\frac{\lambda}{d(1+\lambda)}$), with the number of both forward and backward steps being $n-d$. Consequently,
\begin{align*}
p^{(2n)}_\lambda(o,\, o)
&\ge \left(\frac{1}{d(1+\lambda)}\right)^d \left(\frac{\lambda}{d(1+\lambda)}\right)^d\\
&\ \ \ \
\sum_{\substack{n_1 + \cdots + n_d = n - d \\ n_i \ge 0,\; 1\le i\le d}} {2n \choose 2 n_1,\cdots, 2n_d} C_{n_1} \cdots C_{n_d} \left(\frac{1}{d(1+\lambda)}\right)^{n-d} \left(\frac{\lambda}{d(1+\lambda)}\right)^{n-d}\\
&= \frac{\lambda^n}{[d(1+\lambda)]^{2n}} \sum_{\substack{n_1 + \cdots + n_d = n - d \\ n_i \ge 0,\; 1\le i\le d}} {2n \choose 2 n_1,\cdots, 2n_d} C_{n_1} \cdots C_{n_d} .
{\mathrm{e}}nd{align*}
\noindent Since $C_{\mathrm{e}}ll = \frac{1}{{\mathrm{e}}ll+1} {2{\mathrm{e}}ll \choose {\mathrm{e}}ll}$, and $(n_1+1) \cdots (n_d +1) \le (n+1)^d$, we get
$$
p^{(2n)}_\lambda(o,\, o)
\ge
\frac{\lambda^n}{[d(1+\lambda)]^{2n}} \frac{1}{(n+1)^d} \sum_{\substack{n_1 + \cdots + n_d = n - d \\ n_i \ge 0,\; 1\le i\le d}} {2n \choose 2 n_1,\cdots, 2n_d} {2n_1 \choose n_1} \cdots {2n_d \choose n_d}
$$
\noindent Note that $\sum_{\substack{n_1 + \cdots + n_d = n - d \\ n_i \ge 0,\; 1\le i\le d}} {2n \choose 2 n_1,\cdots, 2n_d} {2n_1 \choose n_1} \cdots {2n_d \choose n_d}$ equals $(2d)^{2n-2d} \, q^{(2n - 2 d)}(o, \, o)$, where $q^{(2k)}(o,\, o)$ is the $(2k)$-step transition probability, from $o$ to $o$, of the (unbiased) simple random walk on ${\mathbb{Z}}^d$. Since $k^{d/2} q^{(2k)}(o,\, o)$ converges, as $k\to \infty$, to a strictly positive limit (\cite[Theorem~1.2.1]{LG1991}), it follows that for some constant $c_1>0$ (depending on $d$ and on $\lambda$) and all sufficiently large $n$,
$$
p^{(2n)}_\lambda(o,\, o)
\ge
c_1 \, \Big( \frac{2\lambda^{1/2}}{1+\lambda} \Big)^{\! 2n} \frac{1}{(n+1)^d\, n^{d/2} } ,
$$
\noindent yielding the desired lower bound for $p^{(2n)}_\lambda(o,\, o)$.
\textbf{Step 2.} It remains to prove the upper bound for $p^{(2n)}_\lambda(o,o)$.
Let ${\mathcal{P}}_{2n}$ be the set of paths from $o$ to $o$ on ${\mathbb{Z}}^d$ with length $2n$. For $\gamma := o \, \omega_1 \, \omega_2 \cdots \omega_{2n-1} \, o \in {\mathcal{P}}_{2n}$, let
$$
{\mathbb{P}}(\gamma, \, \lambda)
:=
p_{\lambda}(o,\, \omega_1) \, p_{\lambda}(\omega_1, \, \omega_2) \ldots p_{\lambda}(\omega_{2n-1}, \, o) ,
$$
\noindent which stands for the transition probability of ${\mathbb{R}}W_{\lambda}$ along $\gamma$. Define, for $1\le i\le d$, the projection $\phi_i$: ${\mathbb{Z}}^d \to {\mathbb{Z}}$ by $\phi_i(y) := y_i$ for $y:= (y_1,\ldots,y_d)\in {\mathbb{Z}}^d$. Let $\gamma_i$ be the path on $\mathbb{Z}$ obtained from $\phi_i(\gamma) := \phi_i(o)\, \phi_i(\omega_1)\cdots \phi_i(\omega_{2n-1})\, \phi_i(o)$ by deleting all null moves. Let $n(\gamma)$ and $n(\gamma_i)$ be respectively the numbers of hits (but excluding the initial hit) to the axial hyperplanes of $\gamma$ and $\gamma_i$; hence $n(\gamma)$ and $n(\gamma_i)$ are odds numbers, with $n(\gamma) \ge 2$ (due to the initial and ending positions), and
$$
n(\gamma) \ge n(\gamma_1) + \cdots + n(\gamma_d).
$$
Consider the first $2n$ steps of ${\mathbb{R}}W_{\lambda}$ along the path $\gamma$. Each time the walk is inside some open orthant, the transition probability for the next step is either $\frac{1}{d (1+\lambda)}$ or $\frac{\lambda}{d(1+\lambda)}$, whereas each time it hits an axial hyperplane (which happens $n(\gamma)$ times by definition), the transition probability is of the form $\frac{1}{d + k + (d-k)\lambda}$ (with $1\le k\le d$) or $\frac{\lambda}{d+k + (d-k)\lambda}$ (with $1\le k\le d-1$). Note that $d+k + (d-k)\lambda \ge d ( 1 +\lambda) + 1 - \lambda$. The total number of probability terms of the forms $\frac{1}{d + k + (d-k)\lambda}$ or $\frac{1}{d (1+\lambda)}$ is exactly $n$, so is the total number of probability terms of the forms $\frac{\lambda}{d + k + (d-k)\lambda}$ or $\frac{\lambda}{d (1+\lambda)}$. Therefore, writing ${\mathrm{e}}ta := \frac{d(1 + \lambda)}{d (1 + \lambda) + 1 - \lambda} \in (0, \, 1)$, we get, for $\gamma \in {\mathcal{P}}_{2n}$,
\begin{equation}
\label{e:Pgammalambda}
{\mathbb{P}}(\gamma, \, \lambda)
\le
{\mathrm{e}}ta^{n(\gamma)} \left( \frac{1}{d (1+\lambda)} \right)^{\! n} \left( \frac{\lambda}{d (1+\lambda)} \right)^{\! n}
\le
{\mathrm{e}}ta^{n(\gamma_1) + \cdots + n(\gamma_d)}
\Big( \frac{\sqrt{\lambda}}{d(1 + \lambda)} \Big)^{\! 2n}.
{\mathrm{e}}nd{equation}
Let ${\mathcal{P}}_{2n}^0 \subset {\mathcal{P}}_{2n}$ be the set of paths $\gamma$ that is contained in the hyperplane $\{ (x_1, \ldots, x_d)\in {\mathbb{Z}}^d: \ x_i = 0\}$ for some $1 \le i \le d$. By definition, $n(\gamma) = 2n$ for $\gamma \in {\mathcal{P}}_{2n}^0$. Since $\# {\mathcal{P}}_{2n}^0 \le \# {\mathcal{P}}_{2n} \le (2d)^{2n}$, we have
\begin{equation}
\label{e:P2n0}
\sum_{\gamma \in {\mathcal{P}}_{2n}^0} {\mathbb{P}}(\gamma, \, \lambda)
\le
(2d)^{2n} {\mathrm{e}}ta^{2n} \left( \frac{1}{d (1+\lambda)} \right)^{\! n} \left( \frac{\lambda}{d (1+\lambda)} \right)^{\! n}
=
\Big( \frac{2 \sqrt{\lambda}}{1+\lambda} \Big)^{\! 2n} {\mathrm{e}}ta^{2n}.
{\mathrm{e}}nd{equation}
We now consider the case $\gamma \in {\mathcal{P}}_{2n}\setminus{\mathcal{P}}_{2n}^0$. By Lemma {\mathrm{Re}}f{Lem5.1},
\begin{align*}
\sum_{\gamma \in {\mathcal{P}}_{2n}\setminus{\mathcal{P}}_{2n}^0}
{\mathrm{e}}ta^{n(\gamma_1) + \cdots + n(\gamma_d)}
&\le c \sum_{\substack{n_1 + \cdots + n_d = n \\ n_i \ge 1, \; 1\le i\le d}} {2n \choose 2n_1, \ldots, 2n_d}
\prod_{j=1}^d \sum_{k_j = 1}^{n_j} {\mathrm{e}}ta^{k_j}
\frac{k_j^{5/2} 4^{n_j}}{n_j^{3/2}}
\\
&\le c \left(\sum_{k=1}^{\infty} {\mathrm{e}}ta^k k^{5/2}\right)^{\! d}
4^n \sum_{\substack{n_1 + \cdots + n_d = n \\ n_i \ge 1, \; 1\le i\le d}} {2n \choose 2n_1, \ldots, 2n_d}
\prod_{j=1}^d n_j^{-3/2}\, .
{\mathrm{e}}nd{align*}
\noindent In view of {\mathrm{e}}qref{e:Pgammalambda}, we obtain, with $c_2 := c (\sum_{k=1}^{\infty} {\mathrm{e}}ta^k k^{5/2})^d <\infty$,
\begin{equation}
\sum_{\gamma \in {\mathcal{P}}_{2n}\setminus{\mathcal{P}}_{2n}^0}
{\mathbb{P}}(\gamma, \, \lambda)
\le
c_2 \, \Big( \frac{2\sqrt{\lambda}}{d(1 + \lambda)} \Big)^{\! 2n}
\sum_{\substack{n_1 + \cdots + n_d = n \\ n_i \ge 1, \; 1\le i\le d}} {2n \choose 2n_1, \ldots, 2n_d}
\prod_{j=1}^d n_j^{-3/2}\, .
\label{e:Pgammalambda2}
{\mathrm{e}}nd{equation}
\noindent To study the expression on the right-hand side, we consider (unbiased) simple random walk on ${\mathbb{Z}}^d$, and let $S_i$ be the number of steps among the first $2n$ steps that are taken in the $i$-th coordinate. For $n_1+\cdots+n_d=n$ with $n_i\in\mathbb{Z}_{+}$ (for all $i$),
$$
{\mathbb{P}} (S_i = 2 n_i, \ 1\le i\le d)
=
d^{-2n} {2n \choose 2n_1, \ldots, 2n_d}.
$$
\noindent By \cite[Lemma 1.4]{PG2015}, there exist constants $c_3>0$ and $c_4>0$, depending only on $d$, such that
$$
\sum_{\substack{n_1+\cdots + n_d=n \\ {\mathrm{e}}xists n_i \not\in [\frac{n}{d},\, \frac{3n}{d}]}} {\mathbb{P}} (S_i = 2 n_i, \ 1\le i\le d)
\le
c_3{\mathrm{e}}xp(-c_4n).
$$
\noindent Hence
\begin{align*}
&d^{-2n} \sum_{\substack{n_1 + \cdots + n_d = n \\ n_i \ge 1, \; 1\le i\le d}} {2n \choose 2n_1, \ldots, 2n_d}
\prod_{j=1}^d n_j^{-3/2}
\\
&\le c_3{\mathrm{e}}xp(-c_4n)
+
\sum_{\substack{n_1+\cdots + n_d=n \\ n_i \in [\frac{n}{d},\, \frac{3n}{d}], \; 1\le i\le d}} {\mathbb{P}} (S_i = 2 n_i, \ 1\le i\le d)
\prod_{j=1}^d n_j^{-3/2} \, .
{\mathrm{e}}nd{align*}
\noindent Consider the sum on the right-hand side. Since $n_i \in [\frac{n}{d},\, \frac{3n}{d}]$ for all $1\le i\le d$, we argue that $\prod_{j=1}^d n_j^{-3/2} \le (\frac{d}{n})^{3d/2}$, so the sum is bounded by $(\frac{d}{n})^{3d/2} \sum_{\substack{n_1+\cdots + n_d=n \\ n_i \in [\frac{n}{d},\, \frac{3n}{d}], \; 1\le i\le d}} {\mathbb{P}} (S_i = 2 n_i, \ 1\le i\le d) \le (\frac{d}{n})^{3d/2}$. Consequently,
$$
d^{-2n} \sum_{\substack{n_1 + \cdots + n_d = n \\ n_i \ge 1, \; 1\le i\le d}} {2n \choose 2n_1, \ldots, 2n_d}
\prod_{j=1}^d n_j^{-3/2}
\le
c_3{\mathrm{e}}xp(-c_4n)
+
(\frac{d}{n})^{3d/2}
\le
c_5\, n^{-3d/2} \, ,
$$
\noindent for some constant $c_5>0$ depending on $d$. Going back to {\mathrm{e}}qref{e:Pgammalambda2}, we obtain
$$
\sum_{\gamma \in {\mathcal{P}}_{2n}\setminus{\mathcal{P}}_{2n}^0}
{\mathbb{P}}(\gamma, \, \lambda)
\le
c_2 c_5 \Big( \frac{2\sqrt{\lambda}}{1 + \lambda} \Big)^{\! 2n} \, n^{-3d/2}.
$$
\noindent In view of {\mathrm{e}}qref{e:P2n0}, and since ${\mathrm{e}}ta <1$ and $p^{(2n)}_{\lambda}(o, \, o) = \sum_{\gamma \in {\mathcal{P}}_{2n}} {\mathbb{P}}(\gamma, \, \lambda)$, this yields the desired upper bound for $p^{(2n)}_{\lambda}(o, \, o)$.
{\mathrm{e}}nd{proof}
Let ${\mathcal{X}} := \{ (x_1, \ldots, x_d) \in {\mathbb{Z}}^d: \, x_i =0 \hbox{ \rm for some } i\}$.
\begin{lem}
\label{P:axialplane}
Almost surely, ${\mathbb{R}}W_{\lambda}$ with $\lambda\in (0, \, 1)$ visits ${\mathcal{X}}$ only finitely many times.
{\mathrm{e}}nd{lem}
\begin{proof} In dimension $d=2$, the lemma is a consequence of \cite[Proposition 2.1]{KI-MV1998}, whose proof relies on properties of Riemann surfaces, and does not seem to be easily extended to higher dimensions.
Let $\lambda\in (0,\, 1)$. Let $(X_n)_{n=0}^{\infty} := ( (X_n^1, \ldots, X_n^d ))_{n=0}^{\infty}$ be ${\mathbb{R}}W_{\lambda}$ on ${\mathbb{Z}}^d$. Write $Y_n := (|X_n^1|, \ldots, |X_n^d|)$ for $n\in\mathbb{Z}_{+}$. Then $(Y_n)_{n=0}^{\infty}$ is a Markov chain on the first orthant ${\mathbb{Z}}^d_+$. Define
$$
\sigma_1
:=
\inf \{ n > 0: \ Y_n \in {\mathbb{Z}}^d_+ \setminus {\mathcal{X}} \},
\qquad
\tau_1
:=
\inf \{ n > \sigma_1 : \ Y_n \in {\mathcal{X}} \},
$$
\noindent and recursively for $i\ge 2$,
$$
\sigma_i
:=
\inf \{n > \tau_{i-1} :\ Y_n \in {\mathbb{Z}}^d_+ \setminus {\mathcal{X}} \},
\qquad
\tau_i
:=
\inf \{n> \sigma_i:\ Y_n \in {\mathcal{X}} \} ,
$$
\noindent with the convention that $\inf {\mathrm{e}}mptyset := \infty$. Let
$({\mathcal{F}}_n)_{n=0}^{\infty}$ be the filtration generated by $(Y_n)_{n=0}^{\infty}$, i.e., ${\mathcal{F}}_n := \sigma(Y_1, \ldots, Y_n)$. We claim that
\begin{enumerate}[{\bf (i)}]
\item For any $i>1$, conditioned on $\{\tau_{i-1}<\infty\}$ and ${\mathcal{F}}_{\tau_{i-1}}$, $\sigma_i < \infty$ a.s.
\item There exists a constant $0<q<1$ such that for any $i\ge 1$, ${\mathbb{P}} (\tau_i<\infty \, |\, \sigma_i<\infty, \; {\mathcal{F}}_{\sigma_i} ) \le q$.
{\mathrm{e}}nd{enumerate}
Indeed, conditionally on $\{\tau_{i-1} < \infty\}$ and ${\mathcal{F}}_{\tau_{i-1}}$, $(Y_{\tau_{i-1} + n})_{n=0}^{\infty}$ is a Markov chain starting at $Y_{\tau_{i-1}}$ with the same transition probability as that of $(Y_n)_{n=0}^{\infty}$. At each step, the transition probability from a state in ${\mathcal{X}} \setminus \{o\}$ to another state in ${\mathcal{X}}$ is $\frac{d-k + (d-k) \lambda}{d+k + (d-k) \lambda}$ for some $1 \le k \le d-1$, which is at most $\frac{(d-1) (1+\lambda)}{(d-1) (1+\lambda) + 2} < 1$. Since the number of visits to $o$ in the first $2n$ steps is at most $n$, we have
\begin{align*}
&{\mathbb{P}} (\sigma_i-\tau_{i-1}>2n\, | \, \tau_{i-1}<\infty, \; {\mathcal{F}}_{\tau_{i-1}})
\\
&={\mathbb{P}} (Y_{\tau_{i-1}+k}\in {\mathcal{X}} \ \text{for}\ 1 \le k \le 2n \, | \,\tau_{i-1}<\infty, \; {\mathcal{F}}_{\tau_{i - 1}} )
\le \left( \frac{(d-1) (1+\lambda)}{(d-1) (1+\lambda) + 2}\right)^n.
{\mathrm{e}}nd{align*}
\noindent We get (i) by sending $n$ to $\infty$.
Let $(Z_n)_{n=0}^{\infty}$ be a drifted random walk on ${\mathbb{Z}}^d$, starting inside the first open orthant, with the step distribution $\mu$ given by $\mu({\mathbf{e}}_1) = \cdots = \mu({\mathbf{e}}_d) = \frac{1}{d(1+\lambda)}$ and $\mu(-{\mathbf{e}}_1) = \cdots =\mu(-{\mathbf{e}}_d) = \frac{\lambda}{d(1+\lambda)}$ (where $\{{\mathbf{e}}_1, \ldots, {\mathbf{e}}_d\}$ is the standard basis in ${\mathbb{Z}}^d$). Let $\tau := \inf \{n\geq 0:\ Z_n\in{\mathcal{X}} \}$. Since the walk has a constant drift whose components are all strictly positive, ${\mathbb{P}}(\tau<\infty) \le q <1$ where $q$ depends on $d$ and $\lambda$.
Conditioned on $\sigma_i<\infty$ and ${\mathcal{F}}_{\sigma_i}$, $(Y_{\sigma_i+n}, \, 0\le n<\tau_i-\sigma_i)$ has the same distribution as $(Z_n, 0\le n<\tau)$. Now (ii) follows readily.
By (i) and (ii), for $i\ge 2$, ${\mathbb{P}}(\tau_i < \infty) \le q \, {\mathbb{P}}(\tau_{i-1} < \infty)$, hence ${\mathbb{P}}(\tau_i < \infty) \le q^i$. The Borel--Cantelli lemma implies that a.s.\ there are only finitely many $i$'s such that $\tau_i < \infty$. Let $m$ be the largest one. The total number of visits to ${\mathcal{X}}$ of $(Y_n)_{n=0}^{\infty}$ is $\sigma_1 + (\sigma_2 - \tau_1) + \cdots + (\sigma_m - \tau_{m-1})$, which is a.s.\ finite.
{\mathrm{e}}nd{proof}
\begin{thm}\label{T:speed}
Let $\lambda \in (0, \, 1)$ and let $(X_n)$ be ${\mathbb{R}}W_\lambda$ on ${\mathbb{Z}}^d$. Then
$$
\lim_{n\to\infty} \frac{1}{n} \left(\left|X_n^1\right|,
\ldots, \left|X_n^d\right|\right) = \frac{1-\lambda}{1+\lambda}
\left(\frac1d, \, \ldots, \, \frac1d \right)
\qquad
\hbox{\rm a.s.}
$$
\noindent In particular, the speed ${\mathcal{S}}(\lambda)=\frac{1-\lambda}{1+\lambda}$ of ${\mathbb{R}}W_{\lambda}$ is positive and strictly decreasing in $\lambda \in (0,\, 1)$.
{\mathrm{e}}nd{thm}
\begin{proof}
For simplicity, we only prove the theorem for $d=2$. Define functions $f_1$ and $f_2$ on ${\mathbb{Z}}^2$ by
$$
f_1(x)
:=
\begin{cases}
0, & x_1 = 0, \\
\frac{1-\lambda}{3+\lambda}, & x_1 \neq 0, \ x_2 = 0, \\
\frac{1-\lambda}{2(1+\lambda)}, & \text{otherwise},
{\mathrm{e}}nd{cases}
\qquad
f_2(x)
:=
\begin{cases}
\frac{1-\lambda}{3+\lambda}, & x_1 = 0, \ x_2 \neq 0, \\
0, & x_2 = 0, \\
\frac{1-\lambda}{2(1+\lambda)}, & \text{otherwise}.
{\mathrm{e}}nd{cases}
$$
\noindent It is easily seen that $( |X_n^1| - |X_{n-1}^1| - f_1(X_{n-1}), \, |X_n^2| - |X_{n-1}^2| - f_2(X_{n-1}) )_{n=1}^{\infty}$ is a martingale-difference sequence. By the strong law of large numbers (cf.\ \cite[Theorem 13.1]{LR-PY2016}),
$$
\lim_{n \to \infty} \frac1n \left( |X_n^1| - \sum_{k=0}^{n-1} f_1(X_k) \right)
=
\lim_{n \to \infty} \frac1n \left( |X_n^2| - \sum_{k=0}^{n-1} f_2(X_k) \right)
=
0
\qquad
\hbox{\rm a.s.}
$$
\noindent Since $|X_n| = |X_n^1| + |X_n^2|$, the theorem follows from Lemma {\mathrm{Re}}f{P:axialplane} and the definitions of $f_1$ and $f_2$.
{\mathrm{e}}nd{proof}
\section{Intersections of two independent random walks}
\label{s:intersection}
In this section, we consider the number of intersections of two
independent drifted or biased random walks on ${\mathbb{Z}}^d$. As we mentioned in the introduction, these results are crucial in the forthcoming computation in Section~{\mathrm{Re}}f{s:usf} of the number of trees in the uniform spanning forests of ${\mathbb{Z}}^d$.
\subsection{Intersections of drifted random walks: Proof of Theorem~{\mathrm{Re}}f{T:intersection}}
Let $(Z_n)_{n=0}^{\infty}$ and $(W_n)_{n=0}^{\infty}$ be two independent drifted random walks on ${\mathbb{Z}}^d$ with the same step distribution $\mu$ given by {\mathrm{e}}qref{e:mu}, starting at $z_0$ and $w_0$ respectively.
Without loss of generality, let us assume $z_0=w_0=0$. The expectation of the intersection number for $(Z_m)_{m=0}^{\infty}$ and $(W_n)_{n=0}^{\infty}$ is
\begin{equation}
\label{e:expectation}
\sum_{m=0}^{\infty} \sum_{n = 0}^{\infty}
{\mathbb{P}} ( Z_m = W_n )
=
\sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \sum_{x \in {\mathbb{Z}}^d}
p^{(m)}(o,\, x) \, p^{(n)}(o,\, x),
{\mathrm{e}}nd{equation}
\noindent where $p^{(n)}(x,\, y)$ is the $n$-step transition probability for $(Z_m)_{m=0}^{\infty}$ from $x$ to $y$. By
\cite[Theorem 10.24]{LR-PY2016}, to prove Theorem {\mathrm{Re}}f{T:intersection}, it suffices to prove that the sum on the right-hand side of {\mathrm{e}}qref{e:expectation} is finite if $d \ge 4$, and is infinite if $d \le 3$.
Let $\mathbf{m}$ and $\Sigma = (\Sigma_{ij})$ be respectively the mean and the covariance matrix of $\mu$. Then $\mathbf{m} = \frac{1-\lambda}{d(1+\lambda)}(1, \ldots, 1)$ and $\Sigma_{ij} = \frac{1}{d} \delta_{ij} - \frac{(1-\lambda)^2}{d^2 (1+\lambda)^2}$ for $1\le i, \, j \le n$. By the local limit theorem (\cite[Theorem~2]{SC1966}),\begin{equation}
\label{e:llt}
p^{(n)}(o,\, x)
=
\frac{1}{(2 \pi n)^{d/2} (\det \Sigma)^{1/2}} {\mathrm{e}}xp \left( - \frac{ (x - n \mathbf{m}) \cdot \Sigma^{-1} (x - n \mathbf{m})}{2n} \right) + o(n^{-d/2}),
{\mathrm{e}}nd{equation}
\noindent where $n^{d/2} o(n^{-d/2}) \to 0$ as $n \to \infty$ uniformly in $x \in {\mathbb{Z}}^d$. Since the largest eigenvalue of $\Sigma$ is $\frac1d$, we have
$$
(x - n \mathbf{m}) \cdot \Sigma^{-1} (x - n \mathbf{m})
\ge
d | x - n \mathbf{m}|
$$
\noindent for $x\in {\mathbb{Z}}^d$. The local limit theorem {\mathrm{e}}qref{e:llt} immediately implies the following result.
\begin{lem}
\label{L:hkdriftupper}
{\bf (i)} There exists a constant $c>0$ such that
\begin{equation}
\label{e:hkup}
\sup_{x\in {\mathbb{Z}}^d} p^{(n)}(0, \, x) \le
c n^{-d/2},
\qquad
\forall n\ge 1.
{\mathrm{e}}nd{equation}
\label{L:hklower}
{\bf (ii)} For $\sigma>0$, define
$$
R_{n,\sigma}
:=
\{ x\in {\mathbb{Z}}^d :\ | x_i - \frac{1-\lambda}{d(1+\lambda)} n | \le \sigma n^{1/2},\ 1\le i\le d \}.
$$
\noindent Then there exists a constant $c>0$, depending on $\sigma$, $\lambda$ and $d$ such that for any $n\in\mathbb{N}$ with $R_{n,\sigma} \not= \varnothing$,
\begin{equation}
\label{e:hklower}
p^{(n)}(0,\, x)
\ge
c n^{-d/2}
\qquad
\text{for $x \in R_{n,\sigma}$ with $n + |x|$ being even.}
{\mathrm{e}}nd{equation}
{\mathrm{e}}nd{lem}
We need another preliminary result.
\begin{lem}
\label{L:concatenation}
Let $\varepsilon > 0$. For any $n\in\mathbb{N}$, define
$$
Q_n(\varepsilon)
:=
\{ x = (x_1,\cdots, x_d) \in {\mathbb{Z}}^d :\ | x_i - \frac{1-\lambda}{d(1+\lambda)} n | < n^{(1+\varepsilon)/2},\ 1\le i\le d \}.
$$
\noindent Then there exists a constant $c>0$, depending on $\lambda\in (0, \, 1)$ and $d$, such that
\begin{equation}
\sum_{x \in {\mathbb{Z}}^d \setminus Q_n(\varepsilon)} p^{(n)}(0,\, x)
\le
2 d {\mathrm{e}}xp (-c n^{\varepsilon} ),
\qquad
\forall n\in\mathbb{N}.
\label{e:concentration}
{\mathrm{e}}nd{equation}
{\mathrm{e}}nd{lem}
\begin{proof}
By the Azuma--Hoeffding inequality, there exists a constant
$c_0>0$, depending only on $\lambda$ and $d$, such that
$$
{\mathbb{P}} ( \max_{1\le i\le d} |Z_n^i - \frac{1-\lambda}{d(1+\lambda)} n | \ge t )
\le
2 d {\mathrm{e}}xp ( - \frac{c_0 t^2}{n} ),
\qquad
t > 0 ,
$$
\noindent where $Z_n^i$ is the $i$-th coordinate component of $Z_n\in{\mathbb{Z}}^d$. The lemma follows by taking $t = n^{(1+\varepsilon)/2}$.
{\mathrm{e}}nd{proof}
\begin{comment}
\begin{lem}
\label{L:hklower}
For $\sigma>0$, define
$$
R_{n,\sigma}
:=
\{ x\in {\mathbb{Z}}^d :\ | x_i - \frac{1-\lambda}{d(1+\lambda)} n | \le \sigma n^{1/2},\ 1\le i\le d \}.
$$
\noindent Then there exists a constant $c>0$, depending on $\sigma$, $\lambda$ and $d$ such that for any $n\in\mathbb{N}$ with $R_{n,\sigma} \not= \varnothing$,
\begin{equation}
\label{e:hklower}
p_n(0,\, x)
\ge
c n^{-d/2}
\qquad
\text{for $x \in R_{n,\sigma}$ with $n + |x|$ being even.}
{\mathrm{e}}nd{equation}
{\mathrm{e}}nd{lem}
\begin{proof}
{\tt (xxxx Lemma {\mathrm{Re}}f{L:hklower} should also be a consequence of Stone's local limit theorem,\break lattice version.)}
We use the same notations as in Lemma {\mathrm{Re}}f{L:hkdriftupper}. If $\left| n_i - \frac{n}{d} \right| \leq
n^{1/2}$ and $\left| x_i - \frac{1-\lambda}{d(1+\lambda)}n\right| \leq
\sigma n^{1/2},$ then for some positive constant $c_1$ depending on $\sigma,\lambda$ and $d,$
\begin{align*}
\left| x_i - \frac{1-\lambda}{1+\lambda} n_i \right| &\leq \left|
x_i - \frac{1-\lambda}{d(1+\lambda)} n \right| +
\frac{1-\lambda}{1+\lambda} \left| n_i - \frac{n}{d} \right| \\
&\leq \sigma n^{1/2}+\frac{1-\lambda}{1+\lambda}n^{1/2}\leq c_1n_i^{1/2}.
{\mathrm{e}}nd{align*}
By the Stirling's formula, for some positive constant $c_2$ depending on $\sigma,\lambda$ and $d,$
$$\psi (n_i,x_i)\geq c_2\frac{\sqrt{n_i}}{\sqrt{\{(n_i+x_i)\vee 1\}\{(n_i-x_i)\vee 1\}}}{\mathrm{e}}xp\left(\phi_{x_i}(n_i)\right)\ \mbox{when}\ n_i\geq \vert x_i\vert.$$
Since for any $y\geq\vert x_i\vert$ with $\left\vert y-\frac{n}{d}\right\vert\leq n^{1/2},$
$\phi_{x_i}''(y)\geq -c_3\vert x_i\vert^{-1}$ for some positive constant $c_3$ depending on $\sigma,\lambda$ and $d;$
we get that $\psi(n_i,x_i)\geq c_4n_i^{-1/2}$ when $n_i\geq 1$ for some positive constant $c_4$ depending on $\sigma,\lambda$ and $d.$
Note that when $n\geq 4d^2$ and $\left\vert t-\frac{n}{d}\right\vert\leq\sqrt{n},$ there exists $\xi\in \left(\frac{n}{d}-\sqrt{n},\frac{n}{d}+\sqrt{n}\right)$ such that
\begin{align*}
t\log t&=\frac{n}{d}\log\frac{n}{d}+\left(1+\log\frac{n}{d}\right)\left(t-\frac{n}{d}\right)+\frac{1}{2}\xi^{-1}\left(t-\frac{n}{d}\right)^2\\
&\leq \frac{n}{d}\log\frac{n}{d}+\left(1+\log\frac{n}{d}\right)\left(t-\frac{n}{d}\right)+\frac{1}{2}\frac{n}{n/d-\sqrt{n}}\\
&\leq \frac{n}{d}\log\frac{n}{d}+\left(1+\log\frac{n}{d}\right)\left(t-\frac{n}{d}\right)+d.
{\mathrm{e}}nd{align*}
Therefore, when $n\geq 4d^2$ and $\left\vert n_i-\frac{n}{d}\right\vert\leq\sqrt{n},\ 1\leq i\leq d,$
\begin{align*}
n_1^{n_1}\cdots n_d^{n_d}&={\mathrm{e}}xp\left(n_1\log n_1+\cdots+n_d\log n_d\right)\\
&\leq{\mathrm{e}}xp\left(\frac{n}{d}\log\frac{n}{d}+\left(1+\log\frac{n}{d}\right)\sum\limits_{j=1}^d\left(n_j-\frac{n}{d}\right)+d^2\right)\\
&={\mathrm{e}}xp\{d^2\}d^{-n}n^n.
{\mathrm{e}}nd{align*}
Clearly $R_{n,\sigma}\not={\mathrm{e}}mptyset$ for large enough $n.$ Thus there are positive constants $c_5,c_6$ and $c_7$ depending on $\sigma,\lambda$ and $d$ satisfying that when $n$ is sufficiently large, for any $x\in R_{n,\sigma}$ with $n+\vert x\vert$ being even,
\begin{align*}
p_n(0,x)=& d^{-n} \sum_{(n_1,\cdots,n_d)\in\mathcal{I}_n(x)} {n \choose n_1,\cdots,n_d} \psi(n_1,x_1) \cdots
\psi(n_d,x_d) \\
\geq & d^{-n} \sum_{\substack{(n_1,\cdots,n_d)\in\mathcal{I}_n(x) \\ \left| n_i -
n/d \right| \leq n^{1/2},1\leq i\leq d}} {n \choose n_1,\cdots, n_d}\psi(n_1,x_1) \cdots
\psi(n_d,x_d)\\
\geq & c_4^d d^{-n} \sum_{\substack{(n_1,\cdots,n_d)\in\mathcal{I}_n(x) \\ \left| n_i -
n/d \right| \leq n^{1/2},1\leq i\leq d}} {n \choose n_1,\cdots, n_d}
n_1^{-1/2}\cdots n_d^{-1/2}\\
\geq & c_5n^{-d/2}d^{-n}\sum_{\substack{(n_1,\cdots,n_d)\in\mathcal{I}_n(x) \\ \left| n_i -
n/d \right| \leq n^{1/2},1\leq i\leq d}} {n \choose n_1,\cdots, n_d}\\
\geq & c_6 n^{-d/2} d^{-n} \sum_{\substack{(n_1,\cdots,n_d)\in\mathcal{I}_n(x) \\ \left| n_i -
n/d \right| \leq n^{1/2},1\leq i\leq d}}\frac{n^{n + 1/2}}{n_1^{n_1 + 1/2}
\cdots n_d^{n_d + 1/2}} \ \ \ \ (\mbox{by the Stirling's formula})\\
\geq & c_6n^{-d/2}{\mathrm{e}}xp\left\{-d^2\right\}\sum_{\substack{(n_1,\cdots, n_d)\in\mathcal{I}_n(x) \\ \left| n_i -
n/d \right| \leq n^{1/2},1\leq i\leq d}}\frac{n^{1/2}}{n_1^{1/2}\cdots n_d^{1/2}}\\
\geq & c_6n^{-d/2}{\mathrm{e}}xp\left\{-d^2\right\}\sum_{\substack{(n_1,\cdots, n_d)\in\mathcal{I}_n(x) \\ \left| n_i -
n/d \right| \leq n^{1/2},1\leq i\leq d}}\frac{n^{1/2}}{(n/d+\sqrt{n})^{d/2}}\\
\geq & c_6n^{-d/2}{\mathrm{e}}xp\left\{-d^2\right\}(1/d+1)^{-d/2}\sum_{\substack{(n_1,\cdots, n_d)\in\mathcal{I}_n(x) \\ \left| n_i -
n/d \right| \leq n^{1/2},1\leq i\leq d}}n^{-\frac{d-1}{2}}\\
\geq & c_7 n^{-d/2}.
{\mathrm{e}}nd{align*}
This implies the lemma holds true.
{\mathrm{e}}nd{proof}
{\mathrm{e}}nd{comment}
Now we are ready to prove Theorem {\mathrm{Re}}f{T:intersection}.
\begin{proof}[Proof of Theorem {\mathrm{Re}}f{T:intersection}]
\textbf{Case 1: $d \ge 4$.}
Fix a small enough $\varepsilon\in (0,\, 1)$. Let $n_{\varepsilon} := \max \{ n - \frac{2d (1+\lambda)}{1-\lambda} n^{(1+\varepsilon)/2}, \, 0\}$. Note that if $1\le m < n_{\varepsilon}$, then
$$
\frac{1 - \lambda}{d(1+\lambda)} n - n^{(1+\varepsilon)/2}
>
\frac{1-\lambda}{d(1+\lambda)}m + m^{(1+\varepsilon)/2}.
$$
\noindent This implies $Q_m(\varepsilon)\cap Q_n(\varepsilon) =\varnothing$. In particular,
$$
\sum_{n\in\mathbb{N}} \sum_{1\le m < n_{\varepsilon}}
\sum_{x \in {\mathbb{Z}}^d} p^{(m)}(0,\, x) p^{(n)}(0,\, x)
\le
\sum_{n\in\mathbb{N}} \sum_{1\le m < n_{\varepsilon}}
\left( \sum_{x \in {\mathbb{Z}}^d \setminus Q_n(\varepsilon)} + \sum_{x \in {\mathbb{Z}}^d \setminus Q_m(\varepsilon)} \right)
p^{(m)}(0,\, x) p^{(n)}(0,\, x) .
$$
\noindent By Lemmas {\mathrm{Re}}f{L:hkdriftupper} and {\mathrm{Re}}f{L:concatenation}, $\sum_{x \in {\mathbb{Z}}^d \setminus Q_n(\varepsilon)} p^{(m)}(0,\, x) p^{(n)}(0,\, x)$ and $\sum_{x \in {\mathbb{Z}}^d \setminus Q_m(\varepsilon)} p^{(m)}(0,\, x) p^{(n)}(0,\, x)$ are bounded by $2d {\mathrm{e}}xp (-c_2 n^{\varepsilon} ) c_1m^{-d/2}$ and $2d {\mathrm{e}}xp (-c_2 m^{\varepsilon} ) c_1 n^{-d/2}$, respectively. Hence
\begin{align*}
&\sum_{n\in\mathbb{N}} \sum_{1\le m < n_{\varepsilon}}
\sum_{x \in {\mathbb{Z}}^d} p^{(m)}(0,\, x) p^{(n)}(0,\, x)
\\
&\le \sum_{n\in\mathbb{N}} \sum_{1\le m < n_{\varepsilon}}
\Big( 2d {\mathrm{e}}xp (-c_2 n^{\varepsilon} ) c_1m^{-d/2} + 2d {\mathrm{e}}xp (-c_2 m^{\varepsilon} ) c_1 n^{-d/2} \Big)
< \infty.
{\mathrm{e}}nd{align*}
\noindent On the other hand, by Lemma {\mathrm{Re}}f{L:hkdriftupper},
\begin{align*}
\sum_{n\in\mathbb{N}} \sum_{n_{\varepsilon}\le m\le n}
\sum_{x \in {\mathbb{Z}}^d} p^{(m)}(0,\, x) p^{(n)}(0,\, x)
&\le \sum_{n\in\mathbb{N}} \sum_{n_{\varepsilon}\le m\le n}
\sum_{x \in {\mathbb{Z}}^d} p^{(m)}(0,\, x) c_1 n^{-d/2}
\\
&= \sum_{n\in\mathbb{N}} \sum_{n_{\varepsilon}\le m\le n}
c_1 n^{-d/2}
\le
\sum_{n\in\mathbb{N}} c_3 n^{- (d-1-\varepsilon)/2}
<\infty.
{\mathrm{e}}nd{align*}
\noindent Moreover, by transience of $(Z_n)_{n=0}^{\infty}$,
$$
\sum_{n\in\mathbb{N}} \sum_{x \in {\mathbb{Z}}^d} p^{(1)}(0,\, x) p^{(n)}(0,\, x)
\le
\sum_{n\in\mathbb{N}} \sum_{x \in {\mathbb{Z}}^d} p^{(1)}(0,\, x) \, c_1 n^{-d/2}
=
\sum_{n\in\mathbb{N}} c_1 n^{-d/2}
<\infty.
$$
\noindent Assembling these pieces yields $\sum_{n\in\mathbb{N}} \sum_{m\in\mathbb{N}} \sum_{x \in {\mathbb{Z}}^d} p^{(m)}(0,\, x) p^{(n)}(0,\, x) <\infty$. A fortiori, we obtain $\sum_{m=0}^{\infty} \sum_{n = 0}^{\infty} {\bf 1}_{\{ Z_m = W_n\} } <\infty$ a.s., as desired.
\textbf{Case 2: $d \le 3.$}
By {\mathrm{e}}qref{e:hklower} in Lemma {\mathrm{Re}}f{L:hkdriftupper}, there exist constants $c_4>0$ and $c_5>0$ such that
\begin{align*}
\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}
\sum_{x \in {\mathbb{Z}}^d}
p^{(n)}(0,\, x) p^{(m)}(0,\, x)
&\ge \sum_{n=2}^{\infty} \sum_{n - n^{1/2}\le m\le n}
\sum_{x \in R_{n,1}}
p^{(n)}(0,\, x) p^{(m)}(0,\, x)
\\
&\ge \sum_{n=2}^{\infty} \sum_{n - n^{1/2}\le m\le n}
\sum_{x \in R_{n,1}}
c_4\, n^{-d/2} m^{-d/2}
\\
&\ge c_5 \sum_{n=2}^{\infty} \sum_{n - n^{1/2}\le m\le n}
m^{-d/2}
\ge
c_5\sum_{n=2}^{\infty} n^{-\frac{d-1}{2}},
{\mathrm{e}}nd{align*}
\noindent which is infinity. By \cite[Theorem 10.24]{LR-PY2016}, $\sum_{m=0}^{\infty} \sum_{n = 0}^{\infty} {\bf 1}_{\{ Z_m = W_n\} } =\infty$ a.s.
{\mathrm{e}}nd{proof}
\subsection{Intersections of biased random walks}
For $x = (x_1, \ldots, x_d) \in {\mathbb{Z}}^d$, define $\phi(x) := (|x_1|, \ldots, |x_d|) \in {\mathbb{Z}}_{+}^d$. We start by studying the number of intersections of the reflecting random walks $(\phi(X_n))_{n=0}^{\infty}$ and $(\phi(Y_n))_{n=0}^{\infty}$,
where $(X_n)_{n=0}^{\infty}$ and $(Y_n)_{n=0}^{\infty}$ are independent ${\mathbb{R}}W_\lambda$'s on ${\mathbb{Z}}^d$. By Theorem~{\mathrm{Re}}f{T:intersection}, with positive probability, the number of intersections is infinite if $d \le 3$, and is finite if $d \ge 4$. The Liouville property below will ensure that the probability
is indeed one.
\begin{lem}
\label{L:poisson}
Let $(X_n)_{n=0}^{\infty}$ be ${\mathbb{R}}W_{\lambda}$ on ${\mathbb{Z}}^d$ with $\lambda\in (0,\, 1)$. The Poisson boundary for $(\phi(X_n))_{n=0}^{\infty}$ is trivial, i.e., all bounded harmonic functions are constants.
{\mathrm{e}}nd{lem}
\begin{proof}
When $d = 2$, the lemma is a special case of the main result in \cite{KI-1999}. Our proof is essentially a reproduction of the argument of \cite{KI-1999}, formulated for all $d$.
Following \cite{BD1955}, a subset $C \subset {\mathbb{Z}}^d_+$ is said to be {\mathrm{e}}mph{almost closed} with respect to $(\phi(X_n))_{n=0}^{\infty}$ if
$$
{\mathbb{P}}(\phi(X_n) \in C \text{ for all sufficiently large } n) = 1.
$$
\noindent A set $C$ is called {\mathrm{e}}mph{atomic} if $C$ does not contain two disjoint almost closed subsets. By \cite{BD1955}, there exists a collection $\{C_1, \, C_2, \cdots \}$ of disjoint almost closed sets such that
\begin{eqnarray*}
&&{\bf (i)}\ \mbox{every $C_i$ except at most one is atomic},\\
&&{\bf (ii)}\ \mbox{the non-atomic $C_i$, if present, contains no atomic subsets,}\\
&&{\bf (iii)}\ \sum_i {\mathbb{P}} ( \lim_{n\to \infty} \{ \phi(X_n) \in C_i \} ) =1.
{\mathrm{e}}nd{eqnarray*}
\noindent Furthermore, $(\phi(X_n))_{n=0}^{\infty}$ has the Liouville property if and only if it is simple and atomic in the sense that the decomposition consists of a single atomic set $C_1$.
Let
$$
{\mathcal{X}} := \{ x = (x_1, \ldots, x_d) \in {\mathbb{Z}}^d_+ : \
x_i = 0 \text{ for some } 1 \le i \le d \},
$$
\noindent be the boundary of ${\mathbb{Z}}^d_+$. Starting at $x \in {\mathbb{Z}}^d_+ \setminus {\mathcal{X}}$, $(\phi(X_n))_{n=0}^{\infty}$ has the same
distribution as the drifted random walk $(Z_n)_{n=0}^{\infty}$ driven by $\mu$ specified in Theorem {\mathrm{Re}}f{T:intersection}, before hitting the boundary ${\mathcal{X}}$.
Define
$$
\tau^X
:=
\inf \{n :\ \phi(X_n) \in {\mathcal{X}} \}
\ \mbox{and}\
\tau^Z = \inf \{n:\ Z_n \in {\mathcal{X}}\}.
$$
\noindent Let $J \subset {\mathbb{Z}}^d_+ \setminus {\mathcal{X}}$ be an almost closed set with respect to $(\phi(X_n))_{n=0}^{\infty}$, i.e.,
$$
{\mathbb{P}}(\phi(X_n) \in J \text{ for all sufficiently large } n ) = 1.
$$
\noindent Since $( \phi(X_n), \, n < \tau^X)$ is distributed as $(Z_n, \, n < \tau^Z$, and ${\mathbb{P}}(\tau^X = \infty)> 0$, we have
\begin{align}
&{\mathbb{P}} (Z_n \in J \text{ for all sufficiently large } n )
\nonumber
\\
&\ge {\mathbb{P}} (\phi(X_n)\in J \text{ for all sufficiently large }n,\ \tau^X = \infty ) > 0.
\label{e:positive}
{\mathrm{e}}nd{align}
\noindent By \cite[Theorem 3]{BD1955}, $(Z_n)_{n=0}^{\infty}$ has the Liouville property, thus
$$
{\mathbb{P}}(Z_n \in J \text{ for all sufficiently large } n) \in\{0, \, 1\}.
$$
\noindent Combining this with {\mathrm{e}}qref{e:positive}, we see that $J$ is
also almost closed with respect to $(Z_n)_{n=0}^{\infty}$.
Thus, the decomposition for $(\phi(X_n))_{n=0}^{\infty}$ is automatically the unique decomposition for $(Z_n)_{n=0}^{\infty}$. Since $(Z_n)_{n=0}^{\infty}$ is simple and atomic, so is $(\phi(X_n))_{n=0}^{\infty}$, which is equivalent to the aforementioned
Liouville property.
{\mathrm{e}}nd{proof}
\begin{lem}
\label{L:intersect2}
Let $(X_n)_{n=0}^{\infty}$ and $(Y_n)_{n=0}^{\infty}$ be independent ${\mathbb{R}}W_\lambda$'s on ${\mathbb{Z}}^d$ with $\lambda\in (0, \, 1)$. Then almost
surely the number of intersections of $(\phi(X_n))_{n=0}^{\infty}$
and $(\phi(Y_n))_{n=0}^{\infty}$ is infinite if $d\le 3$ and is finite if $d\ge 4$.
{\mathrm{e}}nd{lem}
\begin{proof}
By Lemma {\mathrm{Re}}f{L:poisson} and the proof of \cite[Theorem
1.1]{BI-CN-GA2012}, the probability that $(\phi(X_n))_{n=0}^{\infty}$ and $(\phi(Y_n))_{n=0}^{\infty}$ intersect infinitely often is either $0$ or $1$.
Before hitting any axial hyperplanes, $(\phi(X_n))_{n=0}^{\infty}$ and $(\phi(Y_n))_{n=0}^{\infty}$ has the same joint distribution as that of $(Z_n)_{n=0}^{\infty}$, $(W_n)_{n=0}^{\infty}$, where
$(Z_n)_{n=0}^{\infty}$ and $(W_n)_{n=0}^{\infty}$ are independent drifted random walks on ${\mathbb{Z}}^d$ with step distribution $\mu$ described in Theorem {\mathrm{Re}}f{T:intersection}, and $Z_0=\phi(X_0)$, $W_0=\phi(Y_0)$.
Let $T$ be the first time either $(Z_n)_{n=0}^{\infty}$ or $(W_n)_{n=0}^{\infty}$ hits an hyperplane. By Theorem {\mathrm{Re}}f{T:intersection}, on the event $\{T =\infty\}$, which has positive
probability, the number of intersections between $(Z_n)_{n=0}^{\infty}$ and $(W_n)_{n=0}^{\infty}$ is infinite if $d\le 3$, and is finite if $d\ge 4$. In view of the aforementioned $0$--$1$ law above prove this lemma.
{\mathrm{e}}nd{proof}
\section{Uniform spanning forests associated with ${\mathbb{R}}W_\lambda$}
\label{s:usf}
Let $G=(V(G), E(G))$ be a locally finite, connected infinite graph, rooted at $o$. To each edge $e=(x,\, y) \in E(G)$, we assign a weight or conductance $c(e) = c(x,y) = c(y,\, x)$. The weighted graph $(G,\, c)$ is called an electrical network. Consider a Markov chain on $G$ with transition probability $p(x,\, y) = \frac{c(x,\, y)}{\sum_{z \sim x} c(x,\, z)}$, where $z \sim x$ means that $z$ and $x$ are adjacent vertices in $G$. The chain is referred to as a random walk on $G$ with conductance $c$.
Biased random walk ${\mathbb{R}}W_\lambda$ on $G$ is a random walk on $G$ with conductance defined by $c(e) = c_\lambda(e) := \lambda^{-|e|}$.
For any finite network $(G,\, c)$, we consider associated spanning trees, i.e., subgraphs that are trees and that include every vertex. We define the uniform spanning tree measure ${\mathsf{UST}}_G$ to be the probability measure on spanning trees of $G$ such that the measure of each tree is proportional to the product of conductances of the edges in the tree.
An exhaustion of an infinite graph $G$ is a sequence $\{ V_n \}_{n \geq 1}$ of finite, connected subsets of $V(G)$ such that $V_n \subset V_{n+1}$ for all $n\ge 1$ and $\cup_n V_n = V(G)$. Given such an exhaustion, we define the network $G_n$ to be the subgraph of $G$ induced by $V_n$ together with the conductances inherited from $G$. The free uniform spanning forest measure ${\mathfrak{F}}SF$ is defined to be the
weak limit of the sequence $\{ {\mathsf{UST}}_{G_n} \}_{n \ge 1}$ in the sense that
$$
{\mathfrak{F}}SF ( S \subset {\mathfrak{F}} )
=
\lim_{n\to \infty} {\mathsf{UST}}_{G_n} (S \subset T) ,
$$
\noindent for each finite set $S \subset E(G)$. For each $n$, we can also construct a network $G_n^*$ from $G$ by gluing (= wiring) every vertex of $G \setminus G_n$ into a single vertex, denoted by $\partial_n$, and deleting all the self-loops that are created. The set of edges of $G_n^*$ is identified with the set of edges of $G$ having at least one endpoint in $V_n$. The wired uniform spanning forest measure ${\mathsf{WSF}}_G$ is defined to be the weak limit of the sequence $\{{\mathsf{USF}}_{G_n^*}\}_{n\ge 1}$ so that
$$
{\mathsf{WSF}}_G (S \subset {\mathfrak{F}})
=
\lim_{n\to\infty} {\mathsf{UST}}_{G_n^*} (S \subset T),
$$
\noindent for each finite set $S \subset E(G)$. [For the existence of both ${\mathfrak{F}}SF$ and ${\mathsf{WSF}}$, see \cite[Chapter 10]{LR-PY2016}.] Both measures ${\mathfrak{F}}SF$ and ${\mathsf{WSF}}$ are easily seen to be concentrated on the set of uniform spanning forests of $G$ with the property that every
connected component is infinite. It is also easy to see that ${\mathfrak{F}}SF$ stochastically dominates ${\mathsf{WSF}}$ for any infinite network $G$.
The number of trees in the wired uniform spanning forest is a.s.\ a constant; see {\mathrm{e}}qref{e:treenum} below. In \cite{BI-LR-PY2001}, it is asked whether ${\mathfrak{F}}SF$ and ${\mathsf{WSF}}$ are mutually singular (also formulated in \cite[Question 10.59]{LR-PY2016}) and whether the number of trees in the free uniform spanning forest is a.s.\ constant (also formulated in \cite[Question 10.28]{LR-PY2016}) if ${\mathfrak{F}}SF \not= {\mathsf{WSF}}$.\footnote{For a group acting on a network so that every vertex has an infinite orbit, it is known (\cite[Corollary 10.19]{LR-PY2016}) that the action is mixing and ergodic for both ${\mathfrak{F}}SF$ and ${\mathsf{WSF}}$ (so the number of trees in the uniform spanning forest is a.s.\ a constant), and if ${\mathfrak{F}}SF$ and ${\mathsf{WSF}}$ are distinct, they are mutually singular. It is unknown whether this remains true without the assumption that each vertex has an infinite orbit.}
To answer these questions, the first step is to know whether ${\mathfrak{F}}SF$ and ${\mathsf{WSF}}$ are identical.
When the electric network is not transitive and ${\mathfrak{F}}SF \not= {\mathsf{WSF}}$, it seems interesting to study whether ${\mathfrak{F}}SF$ and ${\mathsf{WSF}}$ are singular. A simple situation is when $G$ is a tree, in which case the free uniform spanning forest has one tree (which is the singleton $\{ G\}$), whereas the number of trees in the wired uniform spanning forest can be higher if the constant $K$ defined in {\mathrm{e}}qref{e:treenum} below is at least 2.
Let $\lambda > 0$. Let $c_\lambda(\cdot)$ be the conductances associated with ${\mathbb{R}}W_\lambda$ on graph $G$, and $r_\lambda(\cdot) :=\frac{1}{c_\lambda(\cdot)}$ being the corresponding resistance. Write ${\mathfrak{F}}SF_\lambda$ and ${\mathsf{WSF}}_\lambda$ for the free and wired uniform spanning forest measures. [When they are identical, we use the notation ${\mathsf{USF}}_\lambda$ instead.]
We give a criterion to determine whether ${\mathfrak{F}}SF_\lambda = {\mathsf{WSF}}_\lambda$, compute the number of trees in ${\mathsf{USF}}_\lambda$ on ${\mathbb{Z}}^d$, and consider the singularity problem when ${\mathfrak{F}}SF_\lambda \not= {\mathsf{WSF}}_\lambda$.
\subsection{${\mathsf{USF}}_\lambda$ on ${\mathbb{Z}}^d$}
On any graph $G$, if $\lambda > \lambda_c(G)$, then ${\mathbb{R}}W_\lambda$ is recurrent, so ${\mathfrak{F}}SF_\lambda = {\mathsf{WSF}}_\lambda$. The following theorem deals with the case $0< \lambda < \lambda_c(G)$. Recall (\cite[Section 6.5]{LR-PY2016}) that a graph is said to have one end if the deletion of any finite set of vertices leaves exactly one infinite component.
\begin{thm}
\label{thm6.1}
Let $G$ be a graph with one end such that
\begin{equation}
\label{(6.1)}
\lim_{n\to\infty} \Big( \sum_{x\in\partial B_G(n)} (d_x^{+}+d_x^0)\Big)^{1/n}
=1.
{\mathrm{e}}nd{equation}
\noindent Then for $0<\lambda<\lambda_c(G)=1$
we have ${\mathfrak{F}}SF_\lambda = {\mathsf{WSF}}_\lambda$. In particular, for any $d\ge 2$ and any Cayley graph of additive group $\mathbb{Z}^d$, ${\mathfrak{F}}SF_\lambda = {\mathsf{WSF}}_\lambda$ for $\lambda\in (0,\, 1)$.
{\mathrm{e}}nd{thm}
The proof of Theorem {\mathrm{Re}}f{thm6.1} shows that for any graph $G$ with one end and such that
$$
\mathrm{gr}_{*}(G)
:=
\limsup_{n\to\infty} \Big( \sum_{x\in\partial B_G(n)} (d_x^{+}+d_x^0) \Big)^{1/n}
\in
[1, \, \infty),
$$
\noindent
we have ${\mathfrak{F}}SF_\lambda = {\mathsf{WSF}}_\lambda$ for any $0<\lambda<\frac{1}{\mathrm{gr}_{*}(G)^2}$.
\begin{proof}[{\bf Proof of Theorem {\mathrm{Re}}f{thm6.1}}]
For any function $f: \, V \to {\mathbb{R}}$, let ${\rm d}f$ be the antisymmetric function on oriented edges defined by
$$
{\rm d} f(e)
:=
f(e^-) - f(e^+),
$$
\noindent where $e^-$ and $e^+$ are respectively the tail and head of
$e$. Define the space of Dirichlet functions as
$$
{\mathbf D}_\lambda
:=
\Big\{ f: \ ({\rm d}f,{\rm d}f)_{c_\lambda} := \sum_{e\in \mathbf{E}} |{\rm d}f(e) |^2c_\lambda(e)<\infty \Big\} ,
$$
\noindent where $\mathbf{E}$ is the set of all oriented edges of $G$.
By
\cite[Theorem 7.3]{BI-LR-PY2001},
\begin{align*}
{\mathfrak{F}}SF_\lambda = {\mathsf{WSF}}_\lambda\
&\; {\mathbb{L}}ongleftrightarrow \;
\mbox{all harmonic functions in $\mathbf{D}_\lambda$ are constant}.
{\mathrm{e}}nd{align*}
Clearly $\lambda_c(G)=1$. Let $\lambda\in (0,\, 1)$. Let $f$ be a harmonic function in $\mathbf{D}_\lambda$. We need to prove that $f$ is a constant.
By the maximum principle, for every $n\ge 1$, there are $v_1(n)$, $v_2(n)\in \partial B_G(n)$ such that $f$ takes its maximum
at $v_1(n)$ and minimum at $v_2(n)$ over all vertices in $B_G(n)$. By the assumption,
$$
({\rm d}f, \, {\rm d}f)_{c_\lambda}
=
\sum_{e\in\mathbf{E}} |{\rm d}f(e)|^2 \, \lambda^{-|e|}
<
\infty,
$$
\noindent where $|e|$ is the distance for $e$ from $o$. Hence for some constant $C>0$, $\sup_{e\in\mathbf{E}} |{\rm d}f(e)|^2 \lambda^{-|e|} \le C$, i.e.,
$$
|{\rm d}f(e)|
\le
C^{1/2}\lambda^{|e|/2},
\qquad
\forall e\in\mathbf{E}.
$$
\noindent Combined with ({\mathrm{Re}}f{(6.1)}), we see that
\begin{equation}
\label{(6.2)}
\sum_{e\in\mathbf{E}} |{\rm d}f(e)|
\le
C^{1/2} \sum_{n=0}^\infty \lambda^{n/2}\sum_{e\in\mathbf{E}, \; |e|=n} 1
<
\infty.
{\mathrm{e}}nd{equation}
Let $n\ge 1$. Since $G$ has one end, $G\setminus B_G(n)$ is a connected graph, so there is a finite path $u_0^nu_1^n\cdots u_{k_n}^n$ in $G\setminus B_G(n)$ such that $u_0^n=v_1(n+1)$, $u_{k_n}^n=v_2(n+1)$. As such,
$$
0
\le
f(v_1(n+1))-f(v_2(n+1))
=
\sum_{j=1}^{k_n} [ f(u_{j-1}^n)-f(u_j^n) ]
\le
\sum_{e\in\mathbf{E}, \, |e| \ge n+1} |{\rm d}f(e)|.
$$
\noindent By ({\mathrm{Re}}f{(6.2)}),
$$
\lim_{n\to\infty} \{f(v_1(n+1))-f(v_2(n+1))\}=0,
$$
\noindent which implies that $f$ is constant.
{\mathrm{e}}nd{proof}
Let us consider the uniform spanning forests associated with
${\mathbb{R}}W_{\lambda}$ on ${\mathbb{Z}}^d$. Theorem {\mathrm{Re}}f{thm6.1} says that
${\mathfrak{F}}SF_{\lambda} = {\mathsf{WSF}}_{\lambda}$ on ${\mathbb{Z}}^d$ ($d\ge 2$) for $\lambda \in (0, \, 1)$. For $\lambda = 1$, the two measures are also known to be identical (
\cite{PR1991}). In these cases, we denote both of them by ${\mathsf{USF}}_{\lambda}$.
When $\lambda=1$, the uniform spanning forest on ${\mathbb{Z}}^d$ has one tree a.s.\ for $d \le 4$ and has infinitely many trees a.s.\ for $d \ge 5$; see \cite{PR1991} or \cite[Theorem 10.30]{LR-PY2016}. When $0<\lambda<1$, Theorem {\mathrm{Re}}f{thm6.2} below reveals the existence of a novel phase transition, with the critical dimension reduced to $3$.
\begin{thm}\label{thm6.2}
Let $0 < \lambda < 1$.
{\bf (i)} Almost surely, the number of trees in the uniform spanning forest associated with ${\mathbb{R}}W_{\lambda}$ on ${\mathbb{Z}}^d$ is $2^d$ if $d = 2$ or $3$,
and is infinite if $d \ge 4$. Moreover, when $d\ge 2$, ${\mathsf{USF}}_\lambda$-a.s.\ every tree has one end.
{\bf (ii)} On $\mathbb{Z}^1$, ${\mathfrak{F}}SF_\lambda \not= {\mathsf{WSF}}_\lambda$: the free uniform spanning forest is the singleton of the tree $\mathbb{Z}^1$, whereas the wired uniform spanning forest has two trees and satisfies
\begin{equation}
{\mathsf{WSF}}_{\lambda} [\mathfrak{F} = \{T_{i-1}^{-},\, T_i^{+}\} ]
=
\frac{1}{2}\, (1-\lambda)\lambda^{|i|\wedge |i-1|},
\qquad i\in\mathbb{Z}.
\label{(6.13)}
{\mathrm{e}}nd{equation}
\noindent Here, $\mathfrak{F}$ has the distribution ${\mathsf{WSF}}_\lambda$, $T^{-}_{i-1}$ and $T^+_i$ are subtrees of $\mathbb{Z}^1$ with vertex sets $\{i-1, \, i-2,\, \ldots\}$ and $\{i,\, i+1, \, \ldots\}$, respectively.
{\mathrm{e}}nd{thm}
\begin{proof}
(i) The proof relies on the following general result (\cite[Theorem~9.4]{BI-LR-PY2001}):
Let $G$ be a connected network, and let $\alpha(w_1, \ldots, w_k)$
denote the probability that $k$ independent RW's on the network started at $w_1$, $\ldots$, $w_k$ have no pairwise intersections. Then the number of trees in the wired uniform spanning forest is a.s.\
\begin{equation}
\label{e:treenum}
K=
\sup\{ k:\ {\mathrm{e}}xists w_1, \ldots, w_k,\ \alpha(w_1,\cdots,w_k)>0 \}.
{\mathrm{e}}nd{equation}
We first study the number of trees in the uniform spanning forest.
The case $d \ge 4$ is easy: According to Theorem {\mathrm{Re}}f{T:intersection}, two independent ${\mathbb{R}}W_\lambda$'s on ${\mathbb{Z}}^d$ intersect finitely often a.s., so by
{\mathrm{e}}qref{e:treenum}, the number of trees in the uniform spanning forest associated with ${\mathbb{R}}W_{\lambda}$ on ${\mathbb{Z}}^d$ is a.s.\ infinite.
Consider now the case $d = 2$ or $3$. Let $(X_n^{(j)})_{n=0}^{\infty}$, $1 \le j \le 2^d$, be independent ${\mathbb{R}}W_\lambda$'s on ${\mathbb{Z}}^d$ starting at $o$. Note that the lower limit
$$
\liminf_{n \to \infty} \alpha (X^{(1)}_n, \ldots, X^{(2^d)}_n)
$$
\noindent is a.s.\ greater than or equal to the probability that $(X_n^{(j)})_{n=0}^{\infty}$, $1 \le j \le 2^d$, eventually direct into different orthants. The latter probability is strictly positive according to Theorem {\mathrm{Re}}f{T:hkZd}(ii). Consequently, there exist $\varepsilon_0 >0$ and $n_0 \in \mathbb{N}$ such that
$$
{\mathbb{P}} \{ \alpha (X^{(1)}_{n_0}, \ldots, X^{(2^d)}_{n_0}) > \varepsilon_0 \} >0.
$$
\noindent A fortiori, there are $v_1$, $\ldots$, $v_{2^d}$ such that $\alpha(v_1, \ldots, v_{2^d}) > \varepsilon_0$. By {\mathrm{e}}qref{e:treenum}, there are at least $2^d$ trees in the uniform spanning forest associated with ${\mathbb{R}}W_{\lambda}$ on ${\mathbb{Z}}^d$.
To prove that the number of trees is at most $2^d$, let us consider $2^d + 1$ independent ${\mathbb{R}}W_\lambda$'s on ${\mathbb{Z}}^d$ starting at any initial points. Since there are $2^d$ orthants in ${\mathbb{Z}}^d$, Lemma {\mathrm{Re}}f{P:axialplane} implies that a.s.\ there are at least two of them eventually directing into a common orthant. By Lemma {\mathrm{Re}}f{L:intersect2}, these two ${\mathbb{R}}W_\lambda$'s intersect i.o. with probability $1$. Therefore,
\[\sup\{k:\ {\mathrm{e}}xists w_1,\cdots,w_k,\ \alpha(w_1,\cdots,w_k)>0\} \leq
2^d. \]
Therefore the number of trees in the uniform spanning forest is exactly $2^d$ by
{\mathrm{e}}qref{e:treenum}.
Now fix $d\geq 2$ and $\lambda\in (0, \, 1)$. Write
\begin{align*}
|F|_{c_\lambda}
&= \sum\limits_{e\in F}c_\lambda(e),\ F\subset E\left(\mathbb{Z}^d\right),
\\
|K|_\pi
&= \sum\limits_{x\in K}
\pi(x), \ K\subset\mathbb{Z}^d,
\\
\psi (\mathbb{Z}^d,\, t)
&= \inf\left\{ |\partial_E K|_{c_\lambda}:\ t\leq\vert K\vert_\pi<\infty\right\},\ t>0;
{\mathrm{e}}nd{align*}
\noindent where $\partial_EK=\left\{\{x,y\}\in E\left(\mathbb{Z}^d\right):\ x\in K,y\notin K\right\}$, and $\pi(x) := \left(d_x^++d_x^-\lambda\right)\lambda^{-\vert x\vert}$, $x\in \mathbb{Z}^d$, is an invariant measure of the walk.
Recall from Theorem~{\mathrm{Re}}f{T:main1} that $\rho_\lambda=\frac{2\sqrt{\lambda}}{1+\lambda}<1$. By \cite[Theorem 6.7]{LR-PY2016},
$$\inf\left\{\frac{\vert\partial _EK\vert_{c_\lambda}}{\vert K\vert_\pi};\ {\mathrm{e}}mptyset\not= K\subseteq\mathbb{Z}^d\ \mbox{is finite}\right\}\geq
1-\rho_\lambda>0.$$
Thus for any $t>0,$ $\psi\left(\mathbb{Z}^d,t\right)\geq (1-\rho_\lambda)t.$ Since
$$\inf\limits_{x\in\mathbb{Z}^d}\left(d_x^++d_x^-\lambda\right)\lambda^{-\vert x\vert}>2d\lambda>0,$$
by \cite[Theorem 10.43]{LR-PY2016}, ${\mathsf{USF}}_\lambda$-a.s. every tree has only one end.
(ii) It remains to prove ({\mathrm{Re}}f{(6.13)}). For any $n\in\mathbb{N},$ let $G_n=[-n,n]\cap\mathbb{Z}^1$ be the induced subgraph of tree $\mathbb{Z}^1,$ and $G_n^*$ the graph obtained from $G_n$ by identifying all vertices of $\mathbb{Z}^1\setminus G_n$ to a single vertex $z_n$ and deleting all the self-loops.
Note $G_n^*$ is a simple cycle of length $2(n+1),$ and $z_n$ is adjacent to $n$ and $-n.$ Endow $G_n^*$ with the following edge conductance function $c_\lambda(\cdot):$
\begin{align*}
c_\lambda(\{i-1,i\})
&= \lambda^{-\left(\vert i\vert\wedge\vert i-1\vert\right)}, \qquad i\in [- (n-1), \, n] \cap {\mathbb{Z}}^1,
\\
c_\lambda(\{z_n,n\})
&= c_\lambda(\{z_n,-n\})=\lambda^{-n}.
{\mathrm{e}}nd{align*}
Clearly all spanning trees of $G_n^*$ are of the form $G_n^*\setminus\{e\}$ for some edge $e$ of $G_n^*.$ Let
$$\Xi\left(G_n^*\setminus\{e\}\right)=\prod\limits_{f\in E(G_n^*)\setminus\{e\}}c_\lambda(f)=\frac{1}{c_\lambda(e)}\prod\limits_{f\in E(G_n^*)}c_\lambda(f).$$
By the definition of ${\mathsf{WSF}}$, for any $i\in\mathbb{Z}^1,$
\begin{align*}
{\mathsf{WSF}}_\lambda\left[\mathfrak{F}= \left\{T_{i-1}^{-},T_i^{+}\right\}\right]
&= \lim\limits_{n\rightarrow\infty}\frac{\Xi\left(G_n^*\setminus\{i-1,i\}\right)}
{\sum\limits_{e\in E(G_n^*)}\Xi\left(G_n^*\setminus\{e\}\right)}
=\lim\limits_{n\rightarrow\infty}\frac{c_\lambda\left(\{i-1,i\}\right)^{-1}}
{\sum\limits_{e\in E(G_n^*)}c_\lambda\left(\{e\}\right)^{-1}}
\\
&= \lim\limits_{n\rightarrow\infty}\frac{\lambda^{\vert i\vert\wedge\vert i-1\vert}}
{2\sum\limits_{k=0}^n\lambda^k}=\frac{1}{2}(1-\lambda)\lambda^{\vert i\vert\wedge\vert i-1\vert} ,
{\mathrm{e}}nd{align*}
\noindent as desired.
{\mathrm{e}}nd{proof}
\subsection{Discussions of the singularity problem}
Recall that both ${\mathfrak{F}}SF$ and ${\mathsf{WSF}}$ are determinantal point processes (DPPs) on the set of all edges of a graph (\cite{LR2003}). For the singularity problem,
the following general version is false: Given an infinite countable set $E$ and any two closed subspaces $H_1$ and $H_2$ of ${\mathrm{e}}ll^2(E)$ with $H_1\subsetneq H_2,$ the distributions of DPPs corresponding to $H_1$ and $H_2$ are mutually singular. See \cite{LR2003} p.~203.
\begin{thm}\label{thm6.3}
Let $G$ be any graph whose simple cycles are of uniformly bounded lengths. For any network on $G$ with positive conductances,
the corresponding ${\mathfrak{F}}SF$ has only one tree.
{\mathrm{e}}nd{thm}
\noindent {\it Proof.}
Let $G=(V,E)$ be any graph whose simple cycles are of uniformly bounded lengths. Given any positive conductance function $c(\cdot)$ on $E$.
Consider the exhaustion $G_n=B_G(n)$, $n\in\mathbb{N}$, of $G$. Let ${\mathrm{e}}ll\in\mathbb{N}$ be the maximal length of all simple cycles of $G.$
Given any two distinct vertices $x,y\in V.$ Choose $n_0\in\mathbb{N}$ such that $x,y\in G_n,\ \forall n\geq n_0;$ and let $d_G(x,y)$
be the graph distance between $x$ and $y$ in $G.$ Then for any $n\geq n_0$ and any spanning tree $T_n$ of $G_n,$ the distance $d_{T_n}(x,y)$ between
$x$ and $y$ in $T_n$ is at most ${\mathrm{e}}ll d_G(x,y).$
{Indeed}, suppose conversely
$$d_{T_n}(x,y)\geq {\mathrm{e}}ll d_G(x,y)+1,$$
and let $\gamma_1=x_0x_1\cdots x_{n_1}$ (resp. $\gamma_2=y_0y_1\cdots y_{d_G(x,y)}$) be the geodesic from $x$ to $y$ in $T_n$ (resp. $G$).
Here
$$x_0=y_0=x,\ x_{n_1}=y_{d_G(x,y)}=y,\ n_1=d_{T_n}(x,y).$$
Assume successive intersection points of $\gamma_1$ and $\gamma_2$ are
$$x_{i_0}=y_{j_0},\ x_{i_1}=y_{j_1},\ \cdots,\ x_{i_k}=y_{j_k},$$
where $i_0=0<i_1<\cdots<i_k=n_1,\ j_0=0<j_1<\cdots<j_k=d_G(x,y)$ and $1\leq k\leq d_G(x,y).$ Note for each $0\leq r\leq k-1,$
the segments of $\gamma_1$ and $\gamma_2$
between $x_{i_r}=y_{j_r}$ and $x_{i_{r+1}}=y_{j_{r+1}}$ forms a simple cycle $C_r$ in $G;$ and the total length of all these $C_r$s
is $n_1+d_G(x,y).$ Hence there is a simple cycle $C_r$ in $G$ whose length is at least
$$\frac{n_1+d_G(x,y)}{k}\geq\frac{{\mathrm{e}}ll d_G(x,y)+1+d_G(x,y)}{d_G(x,y)}>{\mathrm{e}}ll+1;$$
which is a contradiction to the definition of ${\mathrm{e}}ll.$
Hence for any $n\geq n_0,$ $\mu_n^F$-a.s. $d_{T_n}(x,y)\leq {\mathrm{e}}ll d_G(x,y),$ where $T_n$ has the law $\mu_n^F$. Taking limit $n\rightarrow\infty$,
we have that
$${\mathfrak{F}}SF\text{-a.s.}\ T,\ d_T(x,y)\leq {\mathrm{e}}ll d_G(x,y)<\infty,$$
where $T$ obeys the law ${\mathfrak{F}}SF$, $d_T(x,y)$ is the graph distance between $x$ and $y$ in $T.$ This means that any two distinct vertices $x$ and
$y$ in $G$ is connected in $T$ for ${\mathfrak{F}}SF$-a.s.\ $T$. Therefore, $T$ is ${\mathfrak{F}}SF$-almost surely a tree, namely ${\mathfrak{F}}SF$ has only one tree.
\rule{4pt}{7pt}\\
\begin{remark}
Let $(G,\, c)$ be a network such that there exists a number $n_0$ with the property that at least two disjoint components of $G \setminus B_G(n_0)$ are transient (i.e., the associated random walks are transient). By {\mathrm{e}}qref{e:treenum}, the ${\mathsf{WSF}}$ a.s. has at least two trees. If all the simple cycles have uniformly bounded lengths, then we have from Theorem~{\mathrm{Re}}f{thm6.3} that the ${\mathfrak{F}}SF$ and ${\mathsf{WSF}}$ are mutually singular.
{\mathrm{e}}nd{remark}
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{\mathrm{e}}nd{thebibliography}
\flushleft{Zhan Shi\\
LPMA, Universit\'{e} Paris VI\\
4 place Jussieu, F-75252 Paris Cedex 05\\
France\\
E-mail: \texttt{[email protected]}}\\
\flushleft{Vladas Sidoravicius\\
NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai\\
\& Courant Institute of Mathematical Sciences\\
New York, NY 10012, USA\\
E-mail: \texttt{[email protected]}}\\
\flushleft{He Song\\
Department of Mathematical Science, Taizhou University\\
Taizhou 225300, P. R. China\\
Email: \texttt{[email protected]}}
\flushleft{Longmin Wang and Kainan Xiang\\
School of Mathematical Sciences, LPMC, Nankai University\\
Tianjin 300071, P. R. China\\
E-mails: \texttt{[email protected]} (Wang)\\
\hskip 1.4cm \texttt{[email protected]} (Xiang)}
{\mathrm{e}}nd{document} | math | 73,460 |
\begin{document}
\author[1]{Alex J. Chin}
\author[2]{Gary Gordon}
\author[3]{Kellie J. MacPhee}
\author[2]{Charles Vincent}
\affil[1]{North Carolina State University, Raleigh, NC}
\affil[2]{Lafayette College, Easton, PA}
\affil[3]{Dartmouth College, Hanover, NH}
\title{Random subtrees of complete graphs}
\maketitle
\begin{abstract} We study the asymptotic behavior of four statistics associated with subtrees of complete graphs: the uniform probability $p_n$ that a random subtree is a spanning tree of $K_n$, the weighted probability $q_n$ (where the probability a subtree is chosen is proportional to the number of edges in the subtree) that a random subtree spans and the two expectations associated with these two probabilities. We find $p_n$ and $q_n$ both approach $e^{-e^{-1}}\approx .692$, while both expectations approach the size of a spanning tree, i.e., a random subtree of $K_n$ has approximately $n-1$ edges.
\end{abstract}
\section{Introduction} We are interested in the following two questions:
\begin{center}
\begin{itemize}
\item [Q1.] What is the asymptotic probability that a random subtree of $K_n$ is a spanning tree?
\item [Q2.] How many edges (asymptotically) does a random subtree of $K_n$ have?
\end{itemize}
\end{center}
In answering both questions, we consider two different probability measures: a uniform random probability $p_n$, where each subtree has an equal probability of being selected, and a weighted probability $q_n$, where the probability a subtree is selected is proportional to its size (measured by the number of edges in the subtree).
As expected, weighting subtrees by their size increases the chances of selecting a spanning tree, i.e., $p_n<q_n$. Table~\ref{globalprobdata} gives data for these values when $ n \leq 100$.
\begin{table}[htdp]
\begin{center}
\begin{tabular}{c|ll}
$n$ & $p_n$ & $q_n$ \\ \hline
10 & 0.617473 & 0.652736 \\
20 & 0.657876 & 0.672725 \\
30 & 0.669904 & 0.679294 \\
40 & 0.675689 & 0.682552 \\
50 & 0.67909 & 0.684497 \\
60 & 0.681329 & 0.685789 \\
70 & 0.682915 & 0.686711 \\
80 & 0.684097 & 0.687401 \\
90 & 0.685012 & 0.687936 \\
100 & 0.685741 & 0.688365 \\
\end{tabular}
\caption{Probabilities of selecting a spanning tree using uniform and weighted probabilities.}
\label{globalprobdata}
\end{center}
\end{table}
The (somewhat) surprising result is that $p_n$ and $q_n$ approach the same limit as $n \to \infty$. This is our first main result, completely answering Q1.
\begin{thm}\label{T:main1}
\begin{enumerate}
\item Let $p_n$ be the probability of choosing a spanning tree among all subtrees of $K_n$ with uniform probability, i.e., the probability any subtree is selected is the same. Then $$\lim_{n \to \infty} p_n=e^{-e^{-1}}=0.692201\dots$$
\item Let $q_n$ be the probability of choosing a spanning tree among all subtrees of $K_n$ with weighted probability, i.e., the probability any subtree is selected is proportional to its number of edges. Then $$\lim_{n \to \infty} q_n=e^{-e^{-1}}=0.692201\dots$$
\end{enumerate}
\end{thm}
For the second question Q2, the expected number of edges of a random subtree of $K_n$ is $\sum pr(T) |E(T)|$, where $pr(T)$ is the probability a tree $T$ is selected, $E(T)$ is the edge set of $T$, and the sum is over all subtrees $T$ of $K_n$. Since we have two distinct probability functions $p_n$ and $q_n$, we obtain two distinct expected values.
The relation between these two expected values is equivalent to a famous example from elementary probability:
\begin{quote}
All universities report ``average class size.'' However, this average depends on whether you first choose a class at random, or first select a student at random, and then ask that student to randomly select one of their classes.
\end{quote}
Our uniform expectation is exactly analogous to the first situation, and our weighted expectation is equivalent to the student weighted average. In this context, edges play the role of students and the subtrees are the classes. It is a straightforward exercise to show the student weighting always produces a larger expectation. This was first noticed by Feld and Grofman in 1977 in \cite{fg}. For our purposes, this result will show the weighted expectation is always greater than the uniform expectation.
Since both of these expected values are obviously bounded above by $n-1$, the size of a spanning tree, we use a variation of {\it subtree density} first defined in \cite{jam1}. We divide our expected values by $n-1$, the number of edges in a spanning tree, to convert our expectations to densities. Letting $a_k$ equal the number of $k$-edge subtrees in $K_n$, this gives us two formulas for subtree density, one using uniform probability and one using weighted probability:
$$\mbox{Uniform density: } \mu_p(n)=\frac{\sum_{k=1}^{n-1}ka_k}{(n-1)\sum_{k=1}^{n-1}a_k} \hskip.5in \mbox{Weighted density: } \mu_q(n)=\frac{\sum_{k=1}^{n-1}k^2a_k}{(n-1)\sum_{k=1}^{n-1}ka_k}$$
Table~\ref{globaldensities} gives data for these two densities when $n \leq 100$.
\begin{table}[htdp]
\begin{center}
\begin{tabular}{c|ll}
$n$ & $\mu_p(n)$ & $\mu_q(n)$ \\ \hline
10 & 0.945976 & 0.952436 \\
20 & 0.977928 & 0.97912 \\
30 & 0.986177 & 0.986661 \\
40 & 0.989945 & 0.990205 \\
50 & 0.9921 & 0.992263 \\
60 & 0.993496 & 0.993607 \\
70 & 0.994472 & 0.994553 \\
80 & 0.995194 & 0.995255 \\
90 & 0.995749 & 0.995797 \\
100 & 0.996189 & 0.996228 \\
\end{tabular}
\caption{Subtree densities using uniform and weighted probabilities.}
\label{globaldensities}
\end{center}
\end{table}
Evidently, both of these densities approach 1. This is our second main result, and our answer to Q2.
\begin{thm}\label{T:main2}
\begin{enumerate}
\item $\displaystyle{\lim_{n\to\infty}\mu_p(n) = 1,}$
\item $\displaystyle{\lim_{n\to\infty} \mu_q(n) = 1.}$
\end{enumerate}
\end{thm}
The fact that the probabilities and the densities do not depend on which probability measure we use is an indication of the dominance of the number of spanning trees in $K_n$ compared to the the number of non-spanning trees. Theorems~\ref{T:main1} and \ref{T:main2} are proven in Section~\ref{S:global}. The proofs follow from a rather detailed analysis of the growth rate of individual terms in the sums that are used to compute all of the statistics. But we emphasize that these proof techniques are completely elementary.
Subtree densities have been studied before, but apparently only when the graph is itself a tree. Jamison introduced this concept in \cite{jam1} and studied its properties in \cite{jam2}. A more recent paper of Vince and Wang \cite{vw} characterizes extremal families of trees with the largest and smallest subtree densities, answering one of Jamison's questions. A recent survey of results connecting subtrees of trees with other invariants, including the Weiner index, appears in \cite{sw}.
There are several interesting directions for future research in this area. We indicate some possible projects in Section~\ref{S:future}.
\section{Global probabilities}\label{S:global}
Our goal in this section is to provide proofs of Theorems~\ref{T:main1} and \ref{T:main2}. As usual, $K_n$ represents the complete graph on $n$ vertices. We fix notation for the subtree enumeration we will need.
\begin{notation} Assume $n$ is fixed. We define $a_k, b_k, A$ and $B$ as follows.
\begin{itemize}
\item Let $a_k$ denote the number of $k$-edge subtrees in $K_n$. (We ignore subtrees of size 0, although setting $a_0=n$ will not change the asymptotic behavior of any of our statistics.)
\item Let $\displaystyle{A=\sum_{k=1}^{n-1}a_k}$ be the total number of subtrees of all sizes in $K_n$.
\item Let $b_k=ka_k$ denote the number of edges used by all of the $k$-edge subtrees in $K_n$.
\item Let $\displaystyle{B=\sum_{k=1}^{n-1}b_k}$ be the sum of the sizes (number of edges) of all the subtrees of $K_n$.
\end{itemize}
\end{notation}
It is immediate from Cayley's formula that $\displaystyle{ a_k= \binom{n}{k+1} (k+1)^{k-1}.}$ We can view $B$ as the sum of all the entries in a 0--1 edge--tree incidence matrix.
The four statistics we study here, $p_n, q_n, \mu_p(n)$ and $\mu_q(n)$, can be computed using $A, B, a_k$ and $b_k$. We omit the straightforward proof of the next result.
\begin{lem}\label{L:global} Let $p_n, q_n, \mu_p(n)$ and $\mu_q(n)$ be as given above. Then
\begin{enumerate}
\item $\displaystyle{p_n=\frac{a_{n-1}}{A}}$,
\item $\displaystyle{q_n=\frac{b_{n-1}}{B}}$,
\item $\displaystyle{\mu_p(n)=\frac{B}{(n-1)A}}$,
\item $\displaystyle{\mu_q(n)=\frac{\sum_{k=1}^{n-1}kb_k}{(n-1)B}=\frac{\sum_{k=1}^{n-1}k^2a_k}{(n-1)B}}$.
\end{enumerate}
\end{lem}
\begin{ex} We compute each of these statistics for the graph $K_4$. In this case, there are 6 subtrees of size one (the 6 edges of $K_4$), 12 subtrees of size two and 16 spanning trees (of size three). Then we find
$\displaystyle{p_4=\frac{16}{34}=.471\dots}$,
$\displaystyle{q_4=\frac{48}{78}=.615\dots}$,
$\displaystyle{\mu_p(4)=\frac{78}{102}=.768\dots}$,
and $\displaystyle{\mu_q(4)=\frac{198}{234}=.846\dots}$.
\end{ex}
The remainder of this section is devoted to proofs of Theorems~\ref{T:main1} and \ref{T:main2}. We first prove part (1) of Theorem~\ref{T:main2}, then use the bounds from Lemmas~\ref{lemmatop} and \ref{lemmabottom} to help prove both parts of Theorem~\ref{T:main1}. Lastly, we prove part (2) of Theorem~\ref{T:main2}.
Thus, our immediate goal is to prove that the uniform density $\displaystyle{\mu_p(n)=\frac{B}{(n-1)A}}$ approaches 1 as $n \to \infty$. Our approach is as follows: We bound the numerator $B$ from below and the term $A$ in the denominator from above so that $\mu_p(n)$ is bounded below by a function that approaches 1 as $n \to \infty$. Lemma~\ref{lemmatop} establishes the lower bound for $B$ and Lemma~\ref{lemmabottom} establishes the upper bound for $A$, from which the result follows.
\begin{lem}
\label{lemmatop} Let $\displaystyle{B=\sum_{k=1}^{n-1}k\binom{n}{k+1} (k+1)^{k-1}}$, as above. Then
\begin{equation}
B > (n-1)n^{n-2} \left( \frac{n-3}{n-1}\right) e^{e^{-1}}.
\label{topresult}
\end{equation}
\end{lem}
\begin{proof}
Recall $b_{n-1} = (n-1)n^{n-2}$ counts the total number of edges used in all the spanning trees of $K_n$. We examine the ratio $b_{i}/b_{i-1}$ in order to establish a lower bound for each $b_i$ in terms of $b_{n-1}$.
\begin{equation}
\frac{b_i}{b_{i-1}} = \frac{\binom{n}{i+1} (i+1)^{i-1} i}{\binom{n}{i} i^{i-2} (i-1)} = \frac{n-i}{i+1}\cdot \frac{i}{i-1}\cdot \frac{(i+1)^{i-1}}{i^{i-2}} = (n-i)\cdot\frac{i}{i-1} \cdot\left(\frac{i+1}{i}\right)^{i-2}
\label{topratio}
\end{equation}
We use the fact that $e$ is the least upper bound for the sequence $\left\{\left(\frac{i+1}{i}\right)^{i-2}\right\}$ to rewrite \eqref{topratio} as the inequality
$$
b_{i-1} \geq \frac{i-1}{i(n-i)e} \cdot b_i,
$$
and this is valid for $i = 2, 3,\dots,n-1$. In general, for $i<n$, an inductive argument on $n-i$ establishes the following:
\begin{equation}
b_{i} \geq \frac{i}{(n-i-1)!(n-1)e^{n-i-1}} \cdot b_{n-1}.
\label{topinequality}
\end{equation}
Then equation \eqref{topinequality} gives
\begin{equation}
B = \sum_{i=1}^{n-1} b_i \geq \frac{b_{n-1}}{n-1} \left((n-1) + \frac{n-2}{e} + \frac{n-3}{2e^2} + \frac{n-4}{6e^3} + \dots + \frac{1}{(n-2)!e^{n-2}}\right).
\end{equation}
We bound this sum below using standard techniques from calculus. Let
$$h(k) = 1 + \frac{1}{e} + \frac{1}{2e^2} + \frac{1}{6e^3} + \dots + \frac{1}{k!e^{k}}.$$
Then
\begin{equation}
B \geq \frac{b_{n-1}}{n-1}(h(n-2) + h(n-3) + \dots + h(0)).
\label{deriv}
\end{equation}
Now $\displaystyle{e^{x} = \sum_{i=0}^{k}\frac{x^i}{i!} + R_k(x)}$, where $\displaystyle{R_k(x) = \frac{e^y}{(k+1)!} x^{k+1}}$
for some $y \in (0,x)$. We are interested in this expression when $x=e^{-1}$. Then $R_k(x)$ is maximized when $x=y= e^{-1}$. So, using $e^{(e^{-1})} \approx 1.44 \ldots<2$, we have
$$
R_k\left(e^{-1}\right) \leq \frac{e^{(e^{-1})}}{{(k+1)!} }(e^{-1})^{k+1} \leq \frac{2}{(k+1)!e^{k+1}}.
$$
Now, using this upper bound on the error in the Maclaurin polynomial for $e^x$ at $x = e^{-1}$ gives
$$
e^{e^{-1}} = h(k) + R_k\left(e^{-1}\right) \leq h(k) + \frac{2}{(k+1)!e^{k+1}},
$$
so
$$
h(k) \geq e^{e^{-1}} - \frac{2}{(k+1)!e^{k+1}}.
$$
Substituting into \eqref{deriv},
\begin{align*}
B &\geq \frac{b_{n-1}}{n-1}\sum_{k=0}^{n-2} h(k)
\geq \frac{b_{n-1}}{n-1} \left((n-1)e^{e^{-1}} - 2\sum_{k=0}^{n-2}\frac{1}{(k+1)!e^{k+1}}\right) \\
&> \frac{b_{n-1}}{n-1} \left((n-1)e^{e^{-1}} - 2\sum_{i=0}^\infty\frac{1}{i!e^i}\right)
= \frac{b_{n-1}}{n-1} \left((n-1)e^{e^{-1}} - 2e^{e^{-1}}\right) = b_{n-1} \left( \frac{n-3}{n-1}\right)e^{e^{-1}}.
\end{align*}
\end{proof}
We now give an upper bound for $A$, the total number of subtrees of $K_n$.
\begin{lem}
\label{lemmabottom}
Let $\displaystyle{A=\sum_{k=1}^{n-1}a_k}$ be the total number of subtrees of all sizes in $K_n$, as above. Then, for every $\varepsilon >0$, there is a positive integer $r(\varepsilon) \in \mathbb{N}$ so that, for all $n>r(\varepsilon)$,
\begin{equation}
A <n^{n-2} \left(e^{(e-\varepsilon)^{-1}} + \frac{e}{r(\varepsilon)!}\right)
\label{bottomresult}
\end{equation}
\end{lem}
\begin{proof}
Recall $a_{n-1}=n^{n-2}$ is the number of spanning trees in $K_n$. As in Lemma~\ref{lemmatop}, we examine ratios of consecutive terms, but this time we need to establish an upper bound for the $a_i$ in terms of $a_{n-1}$. Now
\begin{equation}
\frac{a_i}{a_{i-1}} = \frac{\binom{n}{i+1} (i+1)^{i-1}}{\binom{n}{i} i^{i-2}} = \frac{n-i}{i+1}\cdot \frac{(i+1)^{i-1}}{i^{i-2}} = (n-i) \left(\frac{i+1}{i}\right)^{i-2}
\label{bottomratio}
\end{equation}
Let $\varepsilon > 0$. Since $\lim_{i\to\infty} \left(\frac{i+1}{i}\right)^{i-2} = e$, there exists a $k(\varepsilon)$ such that
$$
a_{i-1} \leq \frac{a_i}{(n-i)(e-\varepsilon)}
$$
for every $k(\varepsilon) < i \leq n-1$.
As in the proof of Lemma~\ref{lemmatop}, an inductive argument can be used to show
$$
a_i \leq \frac{a_{n-1}}{(n-i-1)!(e-\varepsilon)^{n-i-1}}
$$
for all $i$ such that $k(\varepsilon) < i \leq n-1$.
On the other hand, if $i\leq k(\varepsilon)$, then
$$
a_{i-1}= \frac{a_i}{(n-i)\left(\frac{i+1}{i}\right)^{i-2}} \leq \frac{a_i}{n-i} \leq \frac{a_{n-1}}{(n-i)!}
$$
where we have bounded $\left(\frac{i+1}{i}\right)^{i-2}$ below by 1 and the final inequality follows by a similar inductive argument.
Therefore,
\begin{equation*}
A = \sum_{i=1}^{n-1}a_i \leq a_{n-1} (f(n,k(\varepsilon)) + g(n,k(\varepsilon)))
\end{equation*}
where
\begin{equation*}
f(n,k(\varepsilon)) = 1 + \frac{1}{e-\varepsilon} + \frac{1}{2!(e-\varepsilon)^2} + \dots + \frac{1}{(n-k(\varepsilon))!(e-\varepsilon)^{n-k(\varepsilon)}},
\end{equation*}
corresponding to those terms where $i>k(\varepsilon)$, and
\begin{equation*}
g(n,k(\varepsilon)) = \frac{1}{(n-k(\varepsilon)+1)!} + \frac{1}{(n-k(\varepsilon)+2)!} + \dots + \frac{1}{(n-2)!} + \frac{1}{(n-1)!}
\end{equation*}
corresponds to the terms where $i\leq k(\varepsilon)$.
Using the Maclaurin expansion for $e^x$ evaluated at $x = (e-\varepsilon)^{-1}$ gives an upper bound for $f(n,k(\varepsilon))$:
\begin{equation*}
f(n,k(\varepsilon)) = \sum_{i=0}^{n-k(\varepsilon)} \frac{1}{i!(e-\varepsilon)^i} < \sum_{i=0}^{\infty} \frac{1}{i!(e-\varepsilon)^i} = e^{(e-\varepsilon)^{-1}}.
\end{equation*}
For $g(n,k(\varepsilon))$, we have
\begin{equation*}
g(n,k(\varepsilon)) = \sum_{i = n-k(\varepsilon)+2}^{n} \frac{1}{(i-1)!} < \sum_{i=n-k(\varepsilon)+2}^{\infty} \frac{1}{(i-1)!} <\frac{e}{r(\varepsilon)!},
\end{equation*}
where $r(\varepsilon) = n-k(\varepsilon) + 1$.
Therefore,
\begin{equation*}
A \leq a_{n-1}(f(n,k(\varepsilon)) + g(n,k(\varepsilon))) < a_{n-1}\left(e^{(e-\varepsilon)^{-1}} + \frac{e}{r(\varepsilon)!}\right).
\end{equation*}
\end{proof}
We can now prove part (1) of Theorem~\ref{T:main2}.
\begin{proof} [Proof: Theorem~\ref{T:main2} (1)]
Recall $b_{n-1} = (n-1)n^{n-2}$ and $a_{n-1} = n^{n-2}$, so
\begin{equation*}
\frac{1}{n-1} \cdot \frac{b_{n-1}}{a_{n-1}} = 1.
\end{equation*}
Therefore, \eqref{topresult} and \eqref{bottomresult} imply
\begin{equation*}
\mu_p(n) = \frac{1}{n-1} \cdot \frac{B}{A} > \frac{1}{n-1} \cdot \frac{b_{n-1}}{a_{n-1}} \cdot \frac{\left( \frac{n-3}{n-1} \right) e^{e^{-1}} }{e^{(e-\varepsilon)^{-1}} + \frac{e}{r(\varepsilon)!}} =\left( \frac{n-3}{n-1}\right) \cdot \frac{e^{e^{-1}} }{e^{(e-\varepsilon)^{-1}} + \frac{e}{r(\varepsilon)!}}.
\end{equation*}
Then
$\displaystyle{\lim_{n\to\infty} \mu_p(n) = 1}$ as $n \to \infty$ since $\varepsilon$ can be chosen arbitrarily small and $r(\varepsilon)$ can be made arbitrarily large.
\end{proof}
We now prove Theorem~\ref{T:main1}.
\begin{proof}[Proof: Theorem~\ref{T:main1}]
\begin{enumerate}
\item Recall $p_n=\frac{a_{n-1}}{A}$, where $a_{n-1}$ is the number of spanning trees in $K_n$ and $A$ is the total number of subtrees of $K_n$. Then the argument in Lemma~\ref{lemmabottom} can be modified to prove
$$A \geq a_{n-1}\sum_{i=0}^{n-1}\frac{1}{i!e^i}.$$
This follows by bounding $\displaystyle{ \left(\frac{i+1}{i}\right)^{i-2}}$ below by $e$ in equation~\eqref{bottomratio} -- this is the same bound we needed in our proof of Lemma~\ref{lemmatop}. Then, bounding $\displaystyle{ \frac{1}{p_n}=\frac{A}{a_{n-1}} }$, we have
$$e^{e^{-1}}-\varepsilon' \leq \sum_{1=0}^{n-1}\frac{1}{i!e^i} \leq \frac{A} {a_{n-1}}\leq \left(e^{(e-\varepsilon)^{-1}} + \frac{e}{r(\varepsilon)!}\right),$$ where $\varepsilon$ and $\varepsilon'$ can be made as small as we like, and $r(\varepsilon)$ can be made arbitrarily large.
Hence $$p_n=\frac{a_{n-1}}{A} \to \frac{1}{e^{e^{-1}}} \approx 0.6922\dots$$ as $n \to \infty$.
\item Note that $$q_n=\frac{(n-1)a_{n-1}}{B}=\frac{(n-1)A}{B}\cdot \frac{a_{n-1}}{A} = \frac{p_n}{\mu_p(n)}.$$
By Theorem~\ref{T:main2}(1), $\mu_p(n)\to 1$ and, by part (1) of this theorem, $p_n \to e^{-e^{-1}}$. The result now follows immediately.
\end{enumerate}
\end{proof}
An interesting consequence of our proof of part (2) of Theorem~\ref{T:main1} is that $p=q\mu_p(n)$ for any graph $G$, so $p<q$. Thus, if $G=C_n$ is a cycle, then the (uniform) density is approximately $\frac12$, so we immediately get the weighted probability that a random subtree spans is approximately twice the probability for the uniform case (although both probabilities approach 0 as $n \to \infty$).
We state this observation as a corollary.
\begin{cor}\label{C:pnqn} Let $G$ be any connected graph, let $p(G)$ and $q(G)$ be the uniform and weighted probabilities (resp.) that a random subtree is spanning, and let $\mu(G)$ be the uniform subtree density. Then $p(G)=q(G)\mu(G).$
\end{cor}
If $\mathcal{G}$ is an infinite family of graphs, then we can interpret Cor.~\ref{C:pnqn} asymptotically. In this case, the two probabilities $p$ and $q$ coincide (and are non-zero) in the limit if and only if the density approaches 1.
We conclude this section with a very short proof of part (2) of Theorem~\ref{T:main2}.
\begin{proof}[Proof: Theorem~\ref{T:main2} (2)] We have $\mu_p(n)<\mu_q(n)<1$ for all $n$ by a standard argument in probability (see \cite{fg}). Since $\mu_p(n) \to 1$ as $n \to \infty$, we are done.
\end{proof}
\section{Conjectures, extensions and open problems}\label{S:future}
We believe the study of subtrees in arbitrary graphs is a fertile area for interesting research questions. We outline several ideas that should be worthy of future study. Many of these topics are addressed in \cite{cgmv}.
\begin{enumerate}
\item {\bf Local statistics.} Compute ``local'' versions of the four statistics given here. Fix an edge $e$ in $K_n$. Then it is straightforward to compute $p_n', q_n', \mu_{p'}(n)$ and $\mu_{q'}(n)$, where each of these is described below.
\begin{itemize}
\item $p_n'$ is the (uniform) probability that a random subtree {\it containing the edge $e$} is a spanning tree. This is given by
$$p_n'=\frac{n^{n-3}}{\sum_{k=0}^{n-2}{n-2 \choose k}(k+2)^{k-1}}.$$
The number of spanning trees containing a given edge is $2n^{n-3}$ -- this is easy to show using the tree-edge incidence matrix. Incidence counts can also show the total number of subtrees containing the edge $e$ is given by $\displaystyle{\sum_{k=0}^{n-2}2{n-2 \choose k}(k+2)^{k-1}}$. This can also be derived by using the {\it hyperbinomial transform} of the sequence of 1's.
\item $q_n'$ is the (weighted) probability that a random subtree {\it containing the edge $e$} is a spanning tree. This time, we get
$$q_n'=\frac{(n-1)n^{n-3}}{\sum_{k=0}^{n-2}{n-2 \choose k}(k+2)^{k-1}}.$$
\item $ \mu_{p'}(n)$ is the local (uniform) density, so
$$\mu_{p'}(n)=\frac{\sum_{k=0}^{n-2}(k+1){n-2 \choose k}(k+2)^{k-1}}{(n-1)\sum_{k=0}^{n-2}{n-2 \choose k}(k+2)^{k-1}}.$$
\item $ \mu_{q'}(n)$ is the local (weighted) density, which gives
$$\mu_{q'}(n)=\frac{\sum_{k=0}^{n-2}(k+1)^2{n-2 \choose k}(k+2)^{k-1}}{(n-1)\sum_{k=0}^{n-2}(k+1){n-2 \choose k}(k+2)^{k-1}}.$$
\end{itemize}
It is not difficult to prove the same limits that hold for the global versions of these statistics also hold for the local versions: $p'_n$ and $q'_n$ both approach $e^{-e^{-1}}$ and $ \mu_{p'}(n)$ and $\mu_{q'}(n)$ both approach 1 as $n \to \infty$. (All of these can be proven by arguments analogous to the global versions.)
For $p'_n$, however, the connection to the global statistics is even stronger: $p_n'=q_n$ for all $n$, i.e., the global weighted probability exactly matches the local uniform probability of selecting a spanning tree. (This can be proven by a direct calculation, but it is immediate from the ``average class size'' formulation of the weighted probability.)
Other statistics that might be of interest here include larger local versions of these four: suppose we consider only subtrees that contain a given pair of adjacent edges, or a given subtree with 3 edges. Will the limiting probabilities match the ones given here?
\item {\bf Non-spanning subtrees.} Explore the probabilities that a random subtree of $K_n$ has exactly $k$ edges for specified $k<n-1$. The analysis given here can be used to show the uniform and weighted probabilities of choosing a subtree with $n-2$ edges (one less than a spanning tree) approaches $\displaystyle{\left(e^{-1-e^{-1}}\right) =0.254646\dots.}$ This follows by observing the ratio $a_{n-1}/a_{n-2} \to e$ as $n \to \infty$. (It is interesting that both sequences are decreasing here, in contrast to the sequences $p_n$ and $q_n$.)
\item {\bf Other graphs.} Explore these statistics for other classes of graphs. For instance, when $G=K_{n,n}$ is a complete bipartite graph, we get limiting values for the probability (uniform or weighted) that a random subtree spans is
$$e^{-2e^{-1}} = 0.478965\dots,$$
the square of the limiting value we obtained for the complete graph. Both the uniform and weighted densities approach 1 in this case (forced by Cor.~\ref{C:pnqn}).
On the other hand, consider the theta graph $\theta_{n,n}$, formed adding an edge ``in the middle'' of a $2n-2$ cycle (see Fig.~\ref{F:theta}). Then the uniform density approaches $\frac23$ as $n \to \infty$, and both the uniform and weighted probabilities that a random subtree spans tend to 0 as $n$ tends to $\infty$ \cite{cgmv}.
\begin{figure}
\caption{$\theta_{4,4}
\label{F:theta}
\end{figure}
We conjecture that, for any given graph, the number of edges determines whether or not these limiting densities are 0.
\begin{conj}
If $|E|=O(n^2)$, then the probability (either uniform or weighted) of selecting a spanning tree is non-zero (in the limit).
\end{conj}
\begin{conj}
If $|E|=O(n)$, then the probability (either uniform or weighted) of selecting a spanning tree is zero (in the limit).
\end{conj}
\item {\bf Optimal graphs.} Determine the ``best'' graph on $n$ vertices and $m$ edges. There are many possible interpretations for ``best'' here: for instance, we might investigate which graph maximizes $p(G)$ and which maximizes the density? Must the graph that maximizes one of these statistics maximize all of them? This is closely related to the work in \cite{vw}, where extremal classes of trees are determined for uniform density.
\item {\bf Subtree polynomial.} We can define a polynomial to keep track of the number of subtrees of size $k$. If $G$ is any graph with $n$ vertices, let $a_k$ be the number of subtrees of size $k$. Then define a {\it subtree polynomial} by
$$s_G(x) = \sum_{k=0}^{n-1} a_k x^k.$$
We can compute the subtree densities directly from this polynomial and its derivatives:
$$\mu_p(G) = \frac{s_G'(1)}{(n-1)s_G(1)} \hskip.2in \mbox{and} \hskip.2in \mu_q(G) = \frac{s_G'(1)+s''_G(1)}{(n-1)s'_G(1)}$$
It would be worthwhile to study the roots and various properties of this polynomial. In particular, we conjecture that the coefficients of the polynomial are unimodal.
\begin{conj}
The coefficients of $s_G(x)$ are unimodal.
\end{conj}
As an example of an interesting infinite 2-parameter family of graphs, the coefficients of the $\theta$-graph $s_{\theta_{a,b}}(x)$ are unimodal (see Fig.~\ref{F:thetacoef}).
\begin{figure}
\caption{Coefficients of $\theta_{24,48}
\label{F:thetacoef}
\end{figure}
The mode of the sequence of coefficients gives another measure of the subtree density. For the $\theta$-graph $\theta_{n,n}$, we can show the mode is approximately $\sqrt{2}n$ (in \cite{cgmv}).
Stating the unmodality conjecture in terms of a polynomial has the advantage of potentially using the very well developed theory of polynomials. Surveys that address unmodality of coefficients in polynomials include Stanley's classic paper \cite{st} and a recent paper of Pemantle \cite{pe}. In particular, if all the roots of $s_G(x)$ are negative reals, then the coefficients are unimodal. (Such polynomials are called {\it stable}.) Although $s_G(x)$ is not stable in general, perhaps it is for some large class of graphs.
It is not difficult to find different graphs (in fact, different trees) with the same subtree polynomial.
\begin{conj}
For any $n$, construct $n$-pairwise non-isomorphic graphs that all share the same subtree polynomial.
\end{conj}
This is expected by pigeonhole considerations -- there should be far more graphs on $n$ vertices than potential polynomials.
\item {\bf Density monotonicity.} Does adding an edge always increase the density? This is certainly false for disconnected graphs -- simply add an edge to a small component; this will lower the overall density. But we conjecture this cannot happen if $G$ is connected:
\begin{conj}
Suppose $G$ is a connected graph, and $G+e$ is obtained from $G$ by adding an edge between two distinct vertices of $G$. Then $\mu(G)<\mu(G+e)$.
\end{conj}
One consequence of this conjecture would be that, starting with a tree, we can add edges one at a time to create a complete graph, increasing the density at each stage. This could be a useful tool in studying optimal families.
\item {\bf Matroid generalizations.} Instead of using subtrees of $G$, we could use {\it subforests}. This has the advantage of being well behaved under deletion and contraction. In particular, the total number of subforests of a graph $G$ is an evaluation of its Tutte polynomial: $T_G(2,1)$. The number of spanning trees is also an evaluation of the Tutte polynomial: $T_G(1,1)$ (In fact, this is how Tutte defined his {\it dichromatic} polynomial originally.)
All of the statistics studied here would then have direct analogues to the subtree problem. This entire approach would then generalize to matroids. In this context, subforests correspond to independent sets in the matroid, and spanning trees correspond to bases. It would be of interest to study basis probabilities and densities for the class of binary matroids, for example.
\end{enumerate}
\end{document} | math | 27,835 |
\begin{document}
\selectlanguage{french}
\textbf{
{}} \textit{To appear in Journal of Mathematical Sciences}
\selectlanguage{english}
\title{Application of a Bernstein type inequality to rational interpolation
in the Dirichlet space}
\author{Rachid Zarouf}
\begin{abstract}
We prove a Bernstein-type inequality involving the Bergman and the
Hardy norms, for rational functions in the unit disc $\mathbb{D}$
having at most $n$ poles all outside of $\frac{1}{r}\mathbb{D}$,
$0<r<1$. The asymptotic sharpness of this inequality is shown as
$n\rightarrow\infty$ and $r\rightarrow1^{-}.$ We apply our Bernstein-type
inequality to an effective Nevanlinna-Pick interpolation problem in
the standard Dirichlet space, constrained by the $H^{2}$- norm.
\end{abstract}
\maketitle
\section*{Introduction}
\subsection*{a. Statement of the problems}
\begin{flushleft}
Let $\mathbb{D}=\{z\in\mathbb{C}:\,\vert z\vert<1\}$ be the unit
disc of the complex plane and let ${\rm Hol}\left(\mathbb{D}\right)$
be the space of holomorphic functions on $\mathbb{D}.$ Let also $X$
and $Y$ be two Banach spaces of holomorphic functions on the unit
disc $\mathbb{D},$ $X,\, Y\subset{\rm Hol}\left(\mathbb{D}\right).$
Here and later on, $H^{\infty}$ stands for the space (algebra) of
bounded holomorphic functions in the unit disc $\mathbb{D}$ endowed
with the norm $\left\Vert f\right\Vert _{\infty}=\sup_{z\in\mathbb{D}}\left|f(z)\right|.$
We suppose that $n\geq1$ is an integer, $r\in[0,\,1)$ and we consider
the two following problems.
\par\end{flushleft}
\textbf{Problem 1.} Let $\mathcal{P}_{n}$ be the complex space of
analytic polynomials of degree less or equal than $n$, and \[
\mathcal{R}_{n,\, r}=\left\{ \frac{p}{q}\,:\; q\in\mathcal{P}_{n},\; d^{\circ}p<d^{\circ}q,\; q(\zeta)=0\Longrightarrow\left|\zeta\right|\geq\frac{1}{r}\right\} ,\]
(where $d^{\circ}p$ means the degree of any $p\in\mathcal{P}_{n}$)
be the set of all rational functions in $\mathbb{D}$ of degree less
or equal than $n\ge1$, having at most $n$ poles all outside of $\frac{1}{r}\mathbb{D}.$
Notice that for $r=0$, we get $\mathcal{R}_{n,\,0}=\mathcal{P}_{n-1}$.
Our first problem is to search for the {}``best possible'' constant
$\mathcal{C}_{n,\, r}(X,\, Y)$ such that \[
\left\Vert f'\right\Vert _{X}\leq\mathcal{C}_{n,\, r}(X,\, Y)\left\Vert f\right\Vert _{Y}\]
for all $f\in\mathcal{R}_{n,\, r}.$
\textbf{Problem 2.} Let $\sigma=\left\{ \lambda_{1},...,\lambda_{n}\right\} $
be a finite subset of $\mathbb{D}$. What is the best possible interpolation
by functions of the space $Y$ for the traces $f_{\vert\sigma}$ of
functions of the space $X$, in the worst case? The case $X\subset Y$
is of no interest, and so one can suppose that either $Y\subset X$
or $X$ and $Y$ are incomparable. More precisely, our second problem
is to compute or estimate the following interpolation constant\[
I\left(\sigma,\, X,\, Y\right)=\sup_{f\in X,\,\parallel f\parallel_{X}\leq1}\mbox{inf}\left\{ \left\Vert g\right\Vert _{Y}:\, g_{\vert\sigma}=f_{\vert\sigma}\right\} .\]
We also define \[
\mathcal{I}_{n,\, r}(X,\, Y)=\mbox{sup}\left\{ I(\sigma,\, X,\, Y)\,:\,{\rm card}\,\sigma\leq n\,,\,\left|\lambda\right|\leq r,\,\forall\lambda\in\sigma\right\} .\]
\subsection*{b. Motivations}
$\,$
\textbf{Problem 1. }Bernstein-type inequalities for rational functions
are applied
\textbf{1.1.} in matrix analysis and in operator theory (see {}``Kreiss
Matrix Theorem'' {[}LeTr, Sp{]} or {[}Z1, Z4{]} for resolvent estimates
of power bounded matrices),
\textbf{1.2.} to {}``inverse theorems of rational approximation''
using the \textit{classical Bernstein decomposition} (see {[}Da, Pel,
Pek{]}),
\textbf{1.3.} to effective $H^{\infty}$ interpolation problems (see
{[}Z3{]} and our Theorem B below in Subsection d), and more generally
to our Problem 1.
\textbf{Problem 2.} We can give three main motivations for Problem
2.
\textbf{2.1.} It is explained in {[}Z3{]} (the case $Y=H^{\infty})$
why the classical interpolation problems, those of Nevanlinna-Pick
(1908) and Carathéodory-Schur (1916) (see {[}N2{]} p.231 for these
two problems), on the one hand and Carleson's free interpolation problem
(1958) (see {[}N1{]} p.158) on the other hand, are of the nature of
our interpolation problem.
\textbf{2.2.} It is also explained in {[}Z3{]} why this constrained
interpolation is motivated by some applications in matrix analysis
and in operator theory.
\textbf{2.3.} It has already been proved in {[}Z3{]} that for $X=H^{2}$
(see Subsection c. for the definition of $H^{2}$) and $Y=H^{\infty},$
\def${21}$}\begin{equation{${1}$}\begin{equation}
\frac{1}{4\sqrt{2}}\frac{\sqrt{n}}{\sqrt{1-r}}\leq\mathcal{I}_{n,\, r}\left(H^{2},\, H^{\infty}\right)\leq\sqrt{2}\frac{\sqrt{n}}{\sqrt{1-r}}.\label{eq:-2-1}\end{equation}
The above estimate (1) answers a question of L. Baratchart (private
communication), which is part of a more complicated question arising
in an applied situation in {[}BL1{]} and {[}BL2{]}: given a set $\sigma\subset\mathbb{D}$,
how to estimate $I\left(\sigma,\, H^{2},\, H^{\infty}\right)$ in
terms of $n=\mbox{card}(\sigma)$ and $\mbox{max}{}_{\lambda\in\sigma}\left|\lambda\right|=r$
only?
\subsection*{c. The spaces $X$ and $Y$ considered here}
\begin{flushleft}
Now let us define some Banach spaces $X$ and $Y$ of holomorphic
functions in $\mathbb{D}$ which we will consider throughout this
paper. From now on, if $f\in{\rm Hol}(\mathbb{D})$ and $k\in\mathbb{N}$,
$\hat{f}(k)$ stands for the $k^{th}$ Taylor coefficient of $f.$
\par\end{flushleft}
\textbf{1.} The standard Hardy space $H^{2}=H^{2}(\mathbb{D}),$ \[
H^{2}=\left\{ f\in{\rm Hol}\left(\mathbb{D}\right):\:\left\Vert f\right\Vert _{H^{2}}^{2}=\sup_{0\leq r<1}\int_{\mathbb{T}}\left|f(rz)\right|^{2}{\rm d}m(z)<\infty\right\} ,\]
where $m$ stands for the normalized Lebesgue measure on $\mathbb{T}=\{z\in\mathbb{C}:\,\vert z\vert=1\}.$
An equivalent description of the space $H^{2}$ is \[
H^{2}=\left\{ f=\sum_{k\geq0}\hat{f}(k)z^{k}:\,\,\left\Vert f\right\Vert _{H^{2}}=\left(\sum_{k\geq0}\left|\hat{f}(k)\right|^{2}\right)^{\frac{1}{2}}<\infty\right\} .\]
\textbf{2}. The standard Bergman space $L_{a}^{2}=L_{a}^{2}\left(\mathbb{D}\right),$
\[
L_{a}^{2}=\left\{ f\in{\rm Hol}\left(\mathbb{D}\right):\:\left\Vert f\right\Vert _{L_{a}^{2}}^{2}=\frac{1}{\pi}\int_{\mathbb{D}}\left|f(z)\right|^{2}{\rm d}A(z)<\infty\right\} ,\]
where $A$ is the standard area measure, also defined by \[
L_{a}^{2}=\left\{ f=\sum_{k\geq0}\hat{f}(k)z^{k}:\,\left\Vert f\right\Vert _{L_{a}^{2}}\,=\left(\sum_{k\geq0}\left|\hat{f}(k)\right|^{2}\frac{1}{k+1}\right)^{\frac{1}{2}}<\infty\right\} .\]
\textbf{3.} The analytic Besov space $B_{2,\,2}^{\frac{1}{2}}$ (also
known as the standard Dirichlet space) defined by \[
B_{2,\,2}^{\frac{1}{2}}=\left\{ f=\sum_{k\geq0}\hat{f}(k)z^{k}:\,\left\Vert f\right\Vert _{B_{2,\,2}^{\frac{1}{2}}}=\left(\sum_{k\geq0}(k+1)\left|\hat{f}(k)\right|^{2}\right)^{\frac{1}{2}}<\infty\right\} .\]
Then if $f\in B_{2,\,2}^{\frac{1}{2}},$ we have the following equality
\def${21}$}\begin{equation{${2}$}\begin{equation}
\left\Vert f\right\Vert _{B_{2,\,2}^{\frac{1}{2}}}^{2}=\left\Vert f'\right\Vert _{L_{a}^{2}}^{2}+\left\Vert f\right\Vert _{H^{2}}^{2},\label{eq:-2}\end{equation}
which establishes a link between the spaces $B_{2,\,2}^{\frac{1}{2}}$
and $L_{a}^{2}$.
\subsection*{d. The results}
Here and later on, the letter $c$ denotes a positive constant that
may change from one step to the next. For two positive functions $a$
and $b$, we say that $a$ is dominated by $b$, denoted by $a=O(b),$
if there is a constant $c>0$ such that $a\leq cb;$ and we say that
$a$ and $b$ are comparable, denoted by $a\asymp b$, if both $a=O(b)$
and $b=O(a)$ hold.
\textbf{Problem 1.} Our first result (Theorem A, below) is a partial
case ($p=q=2$, $s=\frac{1}{2}$) of the following K. Dyakonov's result
{[}Dy{]}: if $p\in[1,\,\infty),$ $s\in(0,\,+\infty),\; q\in[1,\,+\infty]$,
then there exists a constant $c_{p,\, s}>0$ such that
\def${21}$}\begin{equation{${3}$}\begin{equation}
\mathcal{C}_{n,\, r}\left(B_{p,\, p}^{s-1},\, H^{q}\right)\leq c_{p,\, s}\sup\left\Vert B'\right\Vert _{H^{\gamma}}^{s},\label{eq:-5}\end{equation}
where $\gamma$ is such that $\frac{s}{\gamma}+\frac{1}{q}=\frac{1}{p},$
and the supremum is taken over all finite Blaschke products $B$ of
order $n$ with $n$ zeros outside of $\frac{1}{r}\mathbb{D}.$ Here
$B_{p,\, p}^{s}$ stands for the Hardy-Besov space which consists
of analytic functions $f$ on $\mathbb{D}$ satisfying\[
\left\Vert f\right\Vert _{B_{p,\, p}^{s}}=\sum_{k=0}^{n-1}\left|f^{(k)}(0)\right|+\int_{\mathbb{D}}\left(1-\left|w\right|\right)^{(n-s)p-1}\left|f^{(n)}(w)\right|^{p}{\rm d}A(w)<\infty.\]
For the (tiny) partial case considered here, our proof is different
and the constant $c_{2,\,\frac{1}{2}}$ is asymptotically sharp as
$r$ tends to $1^{-}$ and $n$ tends to $+\infty$.
\begin{flushleft}
\textbf{Theorem A.} \textit{Let $n\geq1$ and $r\in[0,\,1).$ We have}
\par\end{flushleft}
\textit{(i)}
\def${21}$}\begin{equation{${4}$}
\textit{\begin{equation}
\widetilde{a}(n,\, r)\sqrt{\frac{n}{1-r}}\leq\mathcal{C}_{n,\, r}\left(L_{a}^{2},\, H^{2}\right)\leq\widetilde{A}(n,\, r)\sqrt{\frac{n}{1-r}},\label{eq:-4}\end{equation}
where \[
\widetilde{a}(n,\, r)\geq\left(1-\frac{1-r}{n}\right)^{\frac{1}{2}}\; and\;\widetilde{A}(n,\, r)\leq\left(1+r+\frac{1}{\sqrt{n}}\right)^{\frac{1}{2}}.\]
}
\textit{(ii) Moreover, the sequence \[
\left(\frac{\mathcal{C}_{n,\, r}\left(L_{a}^{2},\, H^{2}\right)}{\sqrt{n}}\right)_{n\geq1}\]
is convergent and there exists a limit}
\textit{\def${21}$}\begin{equation{${5}$}\begin{equation}
\lim_{n\rightarrow\infty}\frac{\mathcal{C}_{n,\, r}\left(L_{a}^{2},\, H^{2}\right)}{\sqrt{n}}=\sqrt{\frac{1+r}{1-r}}.\label{eq:-4}\end{equation}
for all $r\in[0,\,1)$. }
$\,$
Notice that it has already been proved in {[}Z2{]} that there exists
a limit
\def${21}$}\begin{equation{${6}$}\begin{equation}
\lim_{n\rightarrow\infty}\frac{\mathcal{C}_{n,\, r}\left(H^{2},\, H^{2}\right)}{n}=\frac{1+r}{1-r},\label{eq:-6}\end{equation}
for every $r,\;0\leq r<1$.
\textbf{Problem 2.} Looking at motivation 2.3, we replace the algebra
$H^{\infty}$ by the Dirichlet space $B_{2,\,2}^{\frac{1}{2}}.$ We
show that the {}``gap'' between $X=H^{2}$ and $Y=H^{\infty}$ (see
(1)) is asymptotically the same as the one which exists between $X=H^{2}$
and $Y=B_{2,\,2}^{\frac{1}{2}}.$ In other words,
\def${21}$}\begin{equation{${7}$}\textit{\begin{equation}
\mathcal{I}_{n,\, r}\left(H^{2},\, B_{2,\,2}^{\frac{1}{2}}\right)\asymp\mathcal{I}_{n,\, r}\left(H^{2},\, H^{\infty}\right)\asymp\sqrt{\frac{n}{1-r}}.\label{eq:-1}\end{equation}
}More precisely, we prove the following Theorem B, in which the right-hand
side inequality of (10) is a consequence of the right-hand side inequality
of (4) in the above Theorem A.
\textbf{Theorem B. }\textit{Let $n\geq1$, and $r\in[0,\,1).$ Then,}
\def${21}$}\begin{equation{${8}$}\textit{\begin{equation}
\mathcal{I}_{n,\, r}\left(H^{2},\, B_{2,\,2}^{\frac{1}{2}}\right)\leq\left[\left(\mathcal{C}_{n,\, r}\left(L_{a}^{2},\, H^{2}\right)\right)^{2}+1\right]^{\frac{1}{2}}.\label{eq:-3}\end{equation}
}
\begin{flushleft}
\textit{Let $\lambda\in\mathbb{D}$ and the corresponding one-point
interpolation set $\sigma_{n,\,\lambda}=\underbrace{\{\lambda,\lambda,...,\lambda\}}_{n}.$
We have,}
\par\end{flushleft}
\def${21}$}\begin{equation{${9}$}\textit{\begin{equation}
I\left(\sigma_{n,\,\lambda},H^{2},\, B_{2,\,2}^{\frac{1}{2}}\right)\geq\sqrt{\frac{n}{1-\left|\lambda\right|}}\left[\frac{(1+\left|\lambda\right|)^{2}-\frac{2}{n}-\frac{2\left|\lambda\right|}{n}}{2(1+\left|\lambda\right|)}\right]^{\frac{1}{2}}.\label{eq:-3}\end{equation}
}
\begin{flushleft}
\textit{In particular,}
\par\end{flushleft}
\def${21}$}\begin{equation{${10}$}\textit{\begin{equation}
\left[\frac{1+r}{2}\left(1-\frac{1}{n}\right)\right]^{\frac{1}{2}}\sqrt{\frac{n}{1-r}}\leq\mathcal{I}_{n,\, r}\left(H^{2},\, B_{2,\,2}^{\frac{1}{2}}\right)\leq\left(1+r+\frac{1}{\sqrt{n}}+\frac{1-r}{n}\right)^{\frac{1}{2}}\sqrt{\frac{n}{1-r}},\label{eq:-3}\end{equation}
}
\def${21}$}\begin{equation{${11}$}\begin{equation}
\sqrt{\frac{\frac{1+r}{2}}{1-r}}\leq\liminf_{n\rightarrow\infty}\frac{\mathcal{I}_{n,\, r}\left(H^{2},\, B_{2,\,2}^{\frac{1}{2}}\right)}{\sqrt{n}}\leq\limsup_{n\rightarrow\infty}\frac{\mathcal{I}_{n,\, r}\left(H^{2},\, B_{2,\,2}^{\frac{1}{2}}\right)}{\sqrt{n}}\leq\sqrt{\frac{1+r}{1-r}},\label{eq:-3}\end{equation}
\textit{and }
\def${21}$}\begin{equation{${12}$}\begin{equation}
\frac{\sqrt{2}}{2}\leq\liminf_{r\rightarrow1^{-}}\liminf_{n\rightarrow\infty}\sqrt{\frac{1-r}{n}}\mathcal{I}_{n,\, r}\left(H^{2},\, B_{2,\,2}^{\frac{1}{2}}\right)\leq\limsup_{r\rightarrow1^{-}}\limsup_{n\rightarrow\infty}\sqrt{\frac{1-r}{n}}\mathcal{I}_{n,\, r}\left(H^{2},\, B_{2,\,2}^{\frac{1}{2}}\right)\leq\sqrt{2}.\label{eq:-3}\end{equation}
In the next Section, we first give some definitions introducing the
main tools used in the proofs of Theorem A and Theorem B. After that,
we prove these theorems.
\section*{Proofs of Theorems A and B}
From now on, if $\sigma=\left\{ \lambda_{1},\,...,\,\lambda_{n}\right\} \subset\mathbb{D}$
is a finite subset of the unit disc, then \[
B_{\sigma}={\displaystyle \prod_{j=1}^{n}}b_{\lambda_{j}}\]
is the corresponding finite Blaschke product where $b_{\lambda}=\frac{\lambda-z}{1-\overline{\lambda}z},$
$\lambda\in\mathbb{D}$. In Definitions 1, 2, 3 and in Remark 4 below,
$\sigma=\left\{ \lambda_{1},\,...,\,\lambda_{n}\right\} $ is a sequence
in the unit disc $\mathbb{D}$ and $B_{\sigma}$ is the corresponding
Blaschke product.
\begin{flushleft}
\textbf{Definition 1.}\textit{ Malmquist family. }For $k\in[1,\, n]$,
we set $f_{k}=\frac{1}{1-\overline{\lambda_{k}}z},$ and define the
family $\left(e_{k}\right)_{1\leq k\leq n}$, (which is known as Malmquist
basis, see {[}N1, p.117{]}), by
\par\end{flushleft}
\def${21}$}\begin{equation{${13}$}\begin{equation}
e_{1}=\frac{f_{1}}{\left\Vert f_{1}\right\Vert _{2}}\,\,\,\mbox{and}\,\,\, e_{k}=\left({\displaystyle \prod_{j=1}^{k-1}}b_{\lambda_{j}}\right)\frac{f_{k}}{\left\Vert f_{k}\right\Vert _{2}}\,,\label{eq:}\end{equation}
for $k\in[2,\, n]$; we have $\left\Vert f_{k}\right\Vert _{2}=\left(1-\vert\lambda_{k}\vert^{2}\right)^{-1/2}.$
\begin{flushleft}
\textbf{Definition 2.}\textit{ The model space $K_{B_{\sigma}}$.
}We define $K_{B_{\sigma}}$ to be the $n$-dimensional space:
\par\end{flushleft}
\def${21}$}\begin{equation{${14}$}\begin{equation}
K_{B_{\sigma}}=\left(B_{\sigma}H^{2}\right)^{\perp}=H^{2}\ominus B_{\sigma}H^{2}.\label{eq:}\end{equation}
\begin{flushleft}
\textbf{Definition 3.}\textit{ The orthogonal projection ~$P_{B_{\sigma}}$on
$K_{B_{\sigma}}.$ }We define $P_{B_{\sigma}}$ to be the orthogonal
projection of $H^{2}$ on its $n$-dimensional subspace $K_{B_{\sigma}}.$
\par\end{flushleft}
\begin{flushleft}
\textbf{Remark 4.} The Malmquist family $\left(e_{k}\right)_{1\leq k\leq n}$
corresponding to $\sigma$ is an orthonormal basis of $K_{B_{\sigma}}.$
In particular,
\par\end{flushleft}
\def${21}$}\begin{equation{${15}$}\begin{equation}
P_{B_{\sigma}}=\sum_{k=1}^{n}\left(\cdot,\, e_{k}\right)_{H^{2}}e_{k},\label{eq:}\end{equation}
\begin{flushleft}
where $\left(\cdot,\,\cdot\right)_{H^{2}}$ means the scalar product
on $H^{2}$.
\par\end{flushleft}
\begin{flushleft}
\textbf{Proof of Theorem A.}
\par\end{flushleft}
\begin{flushleft}
\textit{Proof of (i).} 1) We fist prove the the right-hand side inequality
of (4). Using both Cauchy-Schwarz inequality and the fact that $\widehat{f'}(k)=(k+1)\widehat{f}(k+1)$
for all $k\geq0,$ we get \[
\left\Vert f'\right\Vert _{L_{a}^{2}}^{2}=\sum_{k\geq0}\frac{\left|\widehat{f'}(k)\right|^{2}}{k+1}=\sum_{k\geq0}\frac{(k+1)^{2}\left|\widehat{f}(k+1)\right|^{2}}{k+1}=\]
\[
=\sum_{k\geq1}k\left|\widehat{f}(k)\right|^{2}\leq\left(\sum_{k\geq1}k^{2}\left|\widehat{f}(k)\right|^{2}\right)^{\frac{1}{2}}\left(\sum_{k\geq1}\left|\widehat{f}(k)\right|^{2}\right)^{\frac{1}{2}}=\]
\[
=\left\Vert f'\right\Vert _{H^{2}}\left\Vert f\right\Vert _{H^{2}}\leq\mathcal{C}_{n,\, r}\left(H^{2},\, H^{2}\right)\left\Vert f\right\Vert _{H^{2}}^{2},\]
and hence, \[
\left\Vert f'\right\Vert _{L_{a}^{2}}\leq\sqrt{\mathcal{C}_{n,\, r}\left(H^{2},\, H^{2}\right)}\left\Vert f\right\Vert _{H^{2}},\]
which means \[
\mathcal{C}_{n,\, r}\left(L_{a}^{2},\, H^{2}\right)\leq\sqrt{\mathcal{C}_{n,\, r}\left(H^{2},\, H^{2}\right)}.\]
Then it remains to use {[}Z2, p.2{]}: \[
\mathcal{C}_{n,\, r}\left(H^{2},\, H^{2}\right)\leq\left(1+r+\frac{1}{\sqrt{n}}\right)\frac{n}{1-r},\]
for all $n\geq1$ and $r\in[0,\,1)$.
\par\end{flushleft}
\begin{flushleft}
2) The proof of the left-hand side inequality of (4) repeates the
one of {[}Z2, (i){]} (for the left-hand side inequality) excepted
that this time, we replace the Hardy norm $\left\Vert \cdot\right\Vert _{H^{2}}$
by the Bergman one $\left\Vert \cdot\right\Vert _{L_{a}^{2}}$. Indeed,
we use the same test function $e_{n}=\frac{\left(1-r^{2}\right)^{\frac{1}{2}}}{1-rz}b_{r}^{n-1}$
(the $n^{th}$ vector of the Malmquist family associated with the
one-point set $\sigma_{n,\, r}=\underbrace{\{r,\, r,...,\, r\}}_{n}$
see Definition 1) and prove by the same changing of variable $\circ b_{r}$
(in the integral on the unit disc $\mathbb{D}$ which defines the
$L_{a}^{2}-$norm) that \[
\left\Vert e_{n}'\right\Vert _{L_{a}^{2}}^{2}=\frac{n}{1-r}\left(1-\frac{1-r}{n}\right),\]
which gives \[
\mathcal{C}_{n,\, r}\left(L_{a}^{2},\, H^{2}\right)\geq\sqrt{\frac{n}{1-r}}\left(1-\frac{1-r}{n}\right)^{\frac{1}{2}}.\]
Here are the details of the proof. We have $e_{n}\in K_{b_{r}^{n}}$
and $\left\Vert e_{n}\right\Vert _{H^{2}}=1,$ (see {[}N1{]}, Malmquist-Walsh
Lemma, p.116). Moreover,\[
e_{n}'=\frac{r\left(1-r^{2}\right)^{\frac{1}{2}}}{\left(1-rz\right)^{2}}b_{r}^{n-1}+(n-1)\frac{\left(1-r^{2}\right)^{\frac{1}{2}}}{1-rz}b_{r}'b_{r}^{n-2}=\]
\[
=-\frac{r}{\left(1-r^{2}\right)^{\frac{1}{2}}}b_{r}'b_{r}^{n-1}+(n-1)\frac{\left(1-r^{2}\right)^{\frac{1}{2}}}{1-rz}b_{r}'b_{r}^{n-2},\]
since $b_{r}'=\frac{r^{2}-1}{\left(1-rz\right)^{2}}$. Then,\[
e_{n}'=b_{r}'\left[-\frac{r}{\left(1-r^{2}\right)^{\frac{1}{2}}}b_{r}^{n-1}+(n-1)\frac{\left(1-r^{2}\right)^{\frac{1}{2}}}{1-rz}b_{r}^{n-2}\right],\]
and \[
\left\Vert e_{n}'\right\Vert _{L_{a}^{2}}^{2}=\frac{1}{2\pi}\int_{\mathbb{D}}\left|b_{r}'(w)\right|^{2}\left|-\frac{r}{\left(1-r^{2}\right)^{\frac{1}{2}}}\left(b_{r}(w)\right)^{n-1}+(n-1)\frac{\left(1-r^{2}\right)^{\frac{1}{2}}}{1-rw}\left(b_{r}(w)\right)^{n-2}\right|^{2}\mbox{d}m(w)=\]
\[
=\frac{1}{2\pi}\int_{\mathbb{D}}\left|b_{r}'(w)\right|^{2}\left|\left(b_{r}(w)\right)^{n-2}\right|^{2}\left|-\frac{r}{\left(1-r^{2}\right)^{\frac{1}{2}}}b_{r}(w)+(n-1)\frac{\left(1-r^{2}\right)^{\frac{1}{2}}}{1-rw}\right|\mbox{d}m(w),\]
which gives, using the variables $u=b_{r}(w)$,\[
\left\Vert e_{n}'\right\Vert _{L_{a}^{2}}^{2}=\frac{1}{2\pi}\int_{\mathbb{D}}\left|u^{n-2}\right|^{2}\left|-\frac{r}{\left(1-r^{2}\right)^{\frac{1}{2}}}u+(n-1)\frac{\left(1-r^{2}\right)^{\frac{1}{2}}}{1-rb_{r}(u)}\right|^{2}\mbox{d}m(u).\]
But $1-rb_{r}=\frac{1-rz-r(r-z)}{1-rz}=\frac{1-r^{2}}{1-rz}$ and
$b_{r}'\circ b_{r}=\frac{r^{2}-1}{\left(1-rb_{r}\right)^{2}}=-\frac{\left(1-rz\right)^{2}}{1-r^{2}}$.
This implies \[
\left\Vert e_{n}'\right\Vert _{L_{a}^{2}}^{2}=\frac{1}{2\pi}\int_{\mathbb{D}}\left|u^{n-2}\right|^{2}\left|-\frac{r}{\left(1-r^{2}\right)^{\frac{1}{2}}}u+(n-1)\frac{\left(1-r^{2}\right)^{\frac{1}{2}}}{1-r^{2}}(1-ru)\right|^{2}\mbox{d}m(u)=\]
\[
=\frac{1}{\left(1-r^{2}\right)}\frac{1}{2\pi}\int_{\mathbb{D}}\left|u^{n-2}\right|^{2}\left|\left(-ru+(n-1)(1-ru)\right)\right|^{2}\mbox{d}m(u),\]
which gives\[
\left\Vert e_{n}'\right\Vert _{L_{a}^{2}}=\frac{1}{\left(1-r^{2}\right)^{\frac{1}{2}}}\left\Vert \varphi_{n}\right\Vert _{2},\]
where $\varphi_{n}=z^{n-2}\left(-rz+(n-1)(1-rz)\right).$ Expanding,
we get \[
\varphi_{n}=z^{n-2}\left(-rz+n-1+rz-nrz\right)=\]
\[
=z^{n-2}\left(-nrz+n-1\right)=(n-1)z^{n-2}-nrz^{n-1},\]
and \[
\left\Vert e_{n}'\right\Vert _{L_{a}^{2}}^{2}=\frac{1}{\left(1-r^{2}\right)}\left(\frac{(n-1)^{2}}{n-1}+\frac{n^{2}}{n}r^{2}\right)=\frac{1}{\left(1-r^{2}\right)}\left(n(1+r)-1\right)\]
\[
=\frac{n}{\left(1-r\right)(1+r)}\left((1+r)-\frac{1}{n}\right)=\frac{n}{1-r}\left(1-\frac{1-r}{n}\right)\,,\]
which gives \[
\mathcal{C}_{n,\, r}\left(L_{a}^{2},\, H^{2}\right)\,\geq\sqrt{\frac{n}{1-r}}\left(1-\frac{1-r}{n}\right)^{\frac{1}{2}}.\]
\par\end{flushleft}
\begin{flushleft}
\par\end{flushleft}
\begin{flushleft}
\textit{Proof of (ii).} This is again the same proof as {[}Z2, (ii){]}
(the three steps). More precisely in Step 2, we use the same test
function \[
f=\sum_{k=0}^{s+2}(-1)^{k}e_{n-k},\]
(where $s=\left(s_{n}\right)$ is defined in {[}Z2, p.8{]}), and the
same changing of variable $\circ b_{r}$ in the integral on $\mathbb{D}$.
Here are the details of the proof.
\par\end{flushleft}
\begin{flushleft}
\textbf{Step 1. }We first prove the right-hand-side inequality:\[
\limsup_{n\rightarrow\infty}\frac{1}{\sqrt{n}}\mathcal{C}_{n,\, r}\left(L_{a}^{2},\, H^{2}\right)\,\leq\sqrt{\frac{1+r}{1-r},}\]
which becomes obvious since \[
\frac{1}{\sqrt{n}}\mathcal{C}_{n,\, r}\left(L_{a}^{2},\, H^{2}\right)\,\leq\frac{1}{\sqrt{n}}\sqrt{\mathcal{C}_{n,\, r}\left(H^{2},\, H^{2}\right)}\,,.\]
and \[
\frac{1}{\sqrt{n}}\sqrt{\mathcal{C}_{n,\, r}\left(H^{2},\, H^{2}\right)}\,\rightarrow\sqrt{\frac{1+r}{1-r}},\]
as $n$ tends to infinity, see {[}Z1{]} p. 2.
\par\end{flushleft}
\begin{flushleft}
\textbf{Step 2.} We now prove the left-hand-side inequality:\[
\liminf_{n\rightarrow\infty}\frac{1}{\sqrt{n}}\mathcal{C}_{n,\, r}\left(L_{a}^{2},\, H^{2}\right)\,\geq\sqrt{\frac{1+r}{1-r}.}\]
More precisely, we show that\[
\liminf_{n\rightarrow\infty}\frac{1}{\sqrt{n}}\left\Vert D\right\Vert _{\left(K_{b_{r}^{n}},\,\left\Vert \cdot\right\Vert _{L_{a}^{2}}\right)\rightarrow H^{2}}\geq\sqrt{\frac{1+r}{1-r}}.\]
Let $f\in K_{b_{r}^{n}}.$ Then, \[
f'=\left(f,\, e_{1}\right)_{H^{2}}\frac{r}{\left(1-rz\right)}e_{1}+\sum_{k=2}^{n}(k-1)\left(f,\, e_{k}\right)_{H^{2}}\frac{b_{r}'}{b_{r}}e_{k}+r\sum_{k=2}^{n}\left(f,\, e_{k}\right)_{H^{2}}\frac{1}{\left(1-rz\right)}e_{k}=\]
\[
=r\sum_{k=1}^{n}\left(f,\, e_{k}\right)_{H^{2}}\frac{1}{\left(1-rz\right)}e_{k}+\frac{1-r^{2}}{(1-rz)(z-r)}\sum_{k=2}^{n}(k-1)\left(f,\, e_{k}\right)_{H^{2}}e_{k}=\]
\[
=\frac{r\left(1-r^{2}\right)^{\frac{1}{2}}}{\left(1-rz\right)^{2}}\sum_{k=1}^{n}\left(f,\, e_{k}\right)_{H^{2}}b_{r}^{k-1}+\frac{\left(1-r^{2}\right)^{\frac{3}{2}}}{(1-rz)^{2}(z-r)}\sum_{k=2}^{n}(k-1)\left(f,\, e_{k}\right)_{H^{2}}b_{r}^{k-1}=\]
\[
=-b_{r}'\left[\frac{r}{\left(1-r^{2}\right)^{\frac{1}{2}}}\sum_{k=1}^{n}\left(f,\, e_{k}\right)_{H^{2}}b_{r}^{k-1}+\frac{\left(1-r^{2}\right)^{\frac{1}{2}}}{z-r}\sum_{k=2}^{n}(k-1)\left(f,\, e_{k}\right)_{H^{2}}b_{r}^{k-1}\right].\]
Now using the change of variables $v=b_{r}(u),$ we get\[
\left\Vert f'\right\Vert _{L_{a}^{2}}^{2}=\int_{\mathbb{D}}\left|b_{r}'(u)\right|^{2}\left|\frac{r}{\left(1-r^{2}\right)^{\frac{1}{2}}}\sum_{k=1}^{n}\left(f,\, e_{k}\right)_{H^{2}}b_{r}^{k-1}+\frac{\left(1-r^{2}\right)^{\frac{1}{2}}}{u-r}\sum_{k=2}^{n}(k-1)\left(f,\, e_{k}\right)_{H^{2}}b_{r}^{k-1}\right|^{2}\mbox{d}u=\]
\[
=\int_{\mathbb{D}}\left|\frac{r}{\left(1-r^{2}\right)^{\frac{1}{2}}}\sum_{k=1}^{n}\left(f,\, e_{k}\right)_{H^{2}}v^{k-1}+\frac{\left(1-r^{2}\right)^{\frac{1}{2}}}{b_{r}(v)-r}\sum_{k=2}^{n}(k-1)\left(f,\, e_{k}\right)_{H^{2}}v^{k-1}\right|^{2}\mbox{d}v.\]
Now, $b_{r}-r=\frac{r-z-r(1-rz)}{1-rz}=\frac{z(r^{2}-1)}{1-rz},$
which gives\[
\left\Vert f'\right\Vert _{L_{a}^{2}}^{2}=\int_{\mathbb{D}}\left|\frac{r}{\left(1-r^{2}\right)^{\frac{1}{2}}}\sum_{k=1}^{n}\left(f,\, e_{k}\right)_{H^{2}}v^{k-1}+\frac{\left(1-r^{2}\right)^{\frac{1}{2}}}{v(r^{2}-1)}(1-rv)\sum_{k=2}^{n}(k-1)\left(f,\, e_{k}\right)_{H^{2}}v^{k-1}\right|^{2}\mbox{d}v=\]
\[
=\frac{1}{1-r^{2}}\int_{\mathbb{D}}\left|r\sum_{k=1}^{n}\left(f,\, e_{k}\right)_{H^{2}}v^{k-1}-(1-rv)\sum_{k=2}^{n}(k-1)\left(f,\, e_{k}\right)_{H^{2}}v^{k-2}\right|^{2}\mbox{d}v=\]
\[
=\frac{1}{1-r^{2}}\int_{\mathbb{D}}\left|r\sum_{k=0}^{n-1}\left(f,\, e_{k+1}\right)_{H^{2}}v^{k}-(1-rv)\sum_{k=0}^{n-2}(k+1)\left(f,\, e_{k+2}\right)_{H^{2}}v^{k}\right|^{2}\mbox{d}v.\]
Thus,
\par\end{flushleft}
\def${21}$}\begin{equation{${16}$}\begin{equation}
\frac{1}{^{\left\Vert f\right\Vert _{H^{2}}\sqrt{n(1+r)}}}\left[\left\Vert (1-rv)\sum_{k=0}^{n-2}(k+1)\left(f,\, e_{k+2}\right)_{H^{2}}v^{k}\right\Vert _{L_{a}^{2}}+\left\Vert r\sum_{k=0}^{n-1}\left(f,\, e_{k+1}\right)_{H^{2}}v^{k}\right\Vert _{L_{a}^{2}}\right]\geq\label{eq:-7-1}\end{equation}
\begin{flushleft}
\[
\geq\sqrt{\frac{1-r}{n}}\frac{\left\Vert f'\right\Vert _{L_{a}^{2}}}{\left\Vert f\right\Vert _{H^{2}}}\geq\]
\[
\geq\frac{1}{^{\left\Vert f\right\Vert _{H^{2}}\sqrt{n(1+r)}}}\left[\left\Vert (1-rv)\sum_{k=0}^{n-2}(k+1)\left(f,\, e_{k+2}\right)_{H^{2}}v^{k}\right\Vert _{L_{a}^{2}}-\left\Vert r\sum_{k=0}^{n-1}\left(f,\, e_{k+1}\right)_{H^{2}}v^{k}\right\Vert _{L_{a}^{2}}\right].\]
Now, \[
(1-rv)\sum_{k=0}^{n-2}(k+1)\left(f,\, e_{k+2}\right)_{H^{2}}v^{k}=\]
\[
=\sum_{k=0}^{n-2}(k+1)\left(f,\, e_{k+2}\right)_{H^{2}}v^{k}-r\sum_{k=0}^{n-2}(k+1)\left(f,\, e_{k+2}\right)_{H^{2}}v^{k+1}=\]
\[
=\sum_{k=0}^{n-2}(k+1)\left(f,\, e_{k+2}\right)_{H^{2}}v^{k}-r\sum_{k=1}^{n-1}k\left(f,\, e_{k+1}\right)_{H^{2}}v^{k}=\]
\[
=\left(f,\, e_{2}\right)_{H^{2}}+2\left(f,\, e_{3}\right)_{H^{2}}v+\sum_{k=2}^{n-2}\left[(k+1)\left(f,\, e_{k+2}\right)_{H^{2}}-rk\left(f,\, e_{k+1}\right)_{H^{2}}\right]v^{k}+\]
\[
-r\left[\left(f,\, e_{2}\right)_{H^{2}}v+(n-1)\left(f,\, e_{n}\right)_{H^{2}}v^{n-1}\right]=\]
\[
=\left(f,\, e_{2}\right)_{H^{2}}+\left[\left(f,\, e_{3}\right)_{H^{2}}-r\left(f,\, e_{2}\right)_{H^{2}}\right]v+\sum_{k=2}^{n-2}\left[(k+1)\left(f,\, e_{k+2}\right)_{H^{2}}-rk\left(f,\, e_{k+1}\right)_{H^{2}}\right]v^{k}+\]
\[
-r(n-1)\left(f,\, e_{n}\right)_{H^{2}}v^{n-1},\]
which gives
\par\end{flushleft}
\def${21}$}\begin{equation{${17}$}\begin{equation}
\left\Vert (1-rv)\sum_{k=0}^{n-2}(k+1)\left(f,\, e_{k+2}\right)_{H^{2}}v^{k}\right\Vert _{L_{a}^{2}}^{2}=\label{eq:-7-1-1}\end{equation}
\begin{flushleft}
\[
=\left|\left(f,\, e_{2}\right)_{H^{2}}\right|^{2}+\frac{1}{2}\left|\left(f,\, e_{3}\right)_{H^{2}}-r\left(f,\, e_{2}\right)_{H^{2}}\right|^{2}+\]
\[
+\frac{1}{n}r^{4}(n-1)^{2}\left|\left(f,\, e_{n}\right)_{H^{2}}\right|^{2}+\sum_{k=2}^{n-2}\left|\left(f,\, e_{k+2}\right)_{H^{2}}-\frac{rk}{k+1}\left(f,\, e_{k+1}\right)_{H^{2}}\right|^{2}.\]
On the other hand,
\par\end{flushleft}
\def${21}$}\begin{equation{${18}$}\begin{equation}
\left\Vert r\sum_{k=0}^{n-1}\left(f,\, e_{k+1}\right)_{H^{2}}v^{k}\right\Vert _{L_{a}^{2}}\leq r\left(\sum_{k=0}^{n-1}\frac{1}{k+1}\left|\left(f,\, e_{k+1}\right)_{H^{2}}\right|^{2}\right)^{1/2}\leq r\left\Vert f\right\Vert _{H^{2}},\label{eq:-7}\end{equation}
\begin{flushleft}
Now, let $s=\left(s_{n}\right)$ be a sequence of even integers such
that\[
\mbox{lim}{}_{n\rightarrow\infty}s_{n}=\infty\:\mbox{and}\; s_{n}=o(n)\;\mbox{as}\; n\rightarrow\infty.\]
Then we consider the following function $f$ in $K_{b_{r}^{n}}$:
\[
f=\sum_{k=0}^{s+2}(-1)^{k}e_{n-k}.\]
Applying (17) with such an $f$, we get \[
\left\Vert (1-rv)\sum_{k=0}^{n-2}(k+1)\left(f,\, e_{k+2}\right)_{H^{2}}v^{k}\right\Vert _{L_{a}^{2}}^{2}=\]
\[
=r^{4}\frac{(n-1)^{2}}{n}+\]
\[
+\sum_{l=2}^{n-2}(n-l+1)\left|\left(f,\, e_{n-l+2}\right)_{H^{2}}-\frac{r(n-l)}{n-l+1}\left(f,\, e_{n-l+1}\right)_{H^{2}}\right|^{2},\]
setting the change of index $l=n-k$ in the last sum. This finally
gives\[
\left\Vert (1-rv)\sum_{k=0}^{n-2}(k+1)\left(f,\, e_{k+2}\right)_{H^{2}}v^{k}\right\Vert _{L_{a}^{2}}^{2}=\]
\[
=r^{4}\frac{(n-1)^{2}}{n}+\sum_{l=2}^{s+1}(n-l+1)\left|1+\frac{r(n-l)}{n-l+1}\right|^{2}=\]
\[
=r^{4}\frac{(n-1)^{2}}{n}+\sum_{l=2}^{s+1}(n-l+1)\left[1+r\left(1-\frac{1}{n-l+1}\right)\right]^{2},\]
and\[
\left\Vert (1-rv)\sum_{k=0}^{n-2}(k+1)\left(f,\, e_{k+2}\right)_{H^{2}}v^{k}\right\Vert _{L_{a}^{2}}^{2}\geq\]
\[
\geq r^{4}\frac{(n-1)^{2}}{n}+(s+1-2+1)(n-(s+1)+1)\left[1+r\left(1-\frac{1}{n-(s+1)+1}\right)\right]^{2}=\]
\[
=r^{4}\frac{(n-1)^{2}}{n}+s(n-s)\left[1+r\left(1-\frac{1}{n-s}\right)\right]^{2}.\]
In particular,
\par\end{flushleft}
\[
\left\Vert (1-rv)\sum_{k=0}^{n-2}(k+1)\left(f,\, e_{k+2}\right)_{H^{2}}v^{k}\right\Vert _{L_{a}^{2}}^{2}\geq s(n-s)\left[1+r\left(1-\frac{1}{n-s}\right)\right]^{2}.\]
Now, since $\left\Vert f\right\Vert _{H^{2}}^{2}=s_{n}+3,$ we get\[
\liminf_{n\rightarrow\infty}\frac{1}{n\left\Vert f\right\Vert _{H^{2}}^{2}}\left\Vert (1-rv)\sum_{k=0}^{n-2}(k+1)\left(f,\, e_{k+2}\right)_{H^{2}}v^{k}\right\Vert _{2}^{2}\geq\]
\[
\geq\liminf_{n\rightarrow\infty}\frac{1}{n\left\Vert f\right\Vert _{H^{2}}^{2}}\left\Vert f\right\Vert _{H^{2}}^{2}\left(n-\left\Vert f\right\Vert _{H^{2}}^{2}\right)\left[1+r\left(1-\frac{1}{n-s}\right)\right]^{2}=\]
\[
=\lim_{n\rightarrow\infty}\left(1-\frac{s_{n}}{n}\right)\left[1+r\left(1-\frac{1}{n-s}\right)\right]^{2}=(1+r)^{2}.\]
On the other hand, applying (18) with this $f,$ we obtain\[
\lim_{n\rightarrow\infty}\frac{1}{\sqrt{n}\left\Vert f\right\Vert _{H^{2}}}\left\Vert r\sum_{k=0}^{n-1}\left(f,\, e_{k+1}\right)_{H^{2}}v^{k}\right\Vert _{L_{a}^{2}}=0.\]
Thus, we can conclude passing after to the limit as $n$ tends to
$+\infty$ in (16), that\[
\liminf_{n\rightarrow\infty}\sqrt{\frac{1-r}{n}}\frac{\left\Vert f'\right\Vert _{L_{a}^{2}}}{\left\Vert f\right\Vert _{H^{2}}}=\frac{1}{\sqrt{1+r}}\liminf_{n\rightarrow\infty}\frac{1}{^{\left\Vert f\right\Vert _{H^{2}}\sqrt{n}}}\left\Vert (1-rv)\sum_{k=0}^{n-2}(k+1)\left(f,\, e_{k+2}\right)_{H^{2}}v^{k}\right\Vert _{L_{a}^{2}}\geq\]
\[
\geq\frac{1+r}{\sqrt{1+r}}=\sqrt{1+r},\]
and\[
\liminf_{n\rightarrow\infty}\sqrt{\frac{1-r}{n}}\left\Vert D\right\Vert _{K_{b_{r}^{n}}\rightarrow H^{2}}\geq\liminf_{n\rightarrow\infty}\sqrt{\frac{1-r}{n}}\frac{\left\Vert f'\right\Vert _{L_{a}^{2}}}{\left\Vert f\right\Vert _{H^{2}}}\geq\sqrt{1+r}.\]
\textbf{Step 3. Conclusion.} Using both \textbf{Step 1 }and\textbf{
Step 2}, we get\textbf{ }\[
\limsup_{n\rightarrow\infty}\sqrt{\frac{1-r}{n}}\mathcal{C}_{n,\, r}\left(L_{a}^{2},\, H^{2}\right)=\liminf_{n\rightarrow\infty}\sqrt{\frac{1-r}{n}}\mathcal{C}_{n,\, r}\left(L_{a}^{2},\, H^{2}\right)\,=1+r,\]
which means that the sequence $\left(\frac{1}{\sqrt{n}}\mathcal{C}_{n,\, r}\left(L_{a}^{2},\, H^{2}\right)\right)_{n\geq1}$
is convergent and \[
\lim_{n\rightarrow\infty}\frac{1}{\sqrt{n}}\mathcal{C}_{n,\, r}\left(L_{a}^{2},\, H^{2}\right)=\sqrt{\frac{1+r}{1-r}}.\]
\begin{flushright}
$\square$
\par\end{flushright}
\begin{flushleft}
\textbf{Proof of Theorem B.}
\par\end{flushleft}
\textit{Proofs of inequality (8) and of the right-hand side inequality
of (10).} Let $\sigma$ be a sequence in $\mathbb{D},$ and $B=B_{\sigma}$
the finite Blaschke product corresponding to $\sigma$. If $f\in H^{2},$
we use the same function $g$ as in {[}Z3{]} which satisfies $g_{|\sigma}=f_{|\sigma}.$
More precisely, let $g=P_{B}f\in K_{B}$ (see Definitions 2, 3 and
Remark 4 above for the definitions of $K_{B}$ and $P_{B}$). Then
$g-f\in BH^{2}$ and using the definition of $\mathcal{C}_{n,\, r}\left(L_{a}^{2},\, H^{2}\right),$
\[
\left\Vert g'\right\Vert _{L_{a}^{2}}^{2}\leq\left(\mathcal{C}_{n,\, r}\left(L_{a}^{2},\, H^{2}\right)\right)^{2}\left\Vert g\right\Vert _{H^{2}}^{2}.\]
Now applying the identity (2) to $g$ we get \[
\left\Vert g\right\Vert _{B_{2,\,2}^{\frac{1}{2}}}^{2}\leq\left[\left(\mathcal{C}_{n,\, r}\left(L_{a}^{2},\, H^{2}\right)\right)^{2}+1\right]\left\Vert g\right\Vert _{H^{2}}^{2}.\]
Using the fact that $\left\Vert g\right\Vert _{H^{2}}=\left\Vert P_{B}f\right\Vert _{H^{2}}\leq\left\Vert f\right\Vert _{H^{2}},$
we finally get \[
\left\Vert g\right\Vert _{B_{2,\,2}^{\frac{1}{2}}}\leq\left[\left(\mathcal{C}_{n,\, r}\left(L_{a}^{2},\, H^{2}\right)\right)^{2}+1\right]^{\frac{1}{2}}\left\Vert f\right\Vert _{H^{2}},\]
and as a result,\[
I\left(\sigma,\, H^{2},\, B_{2,\,2}^{\frac{1}{2}}\right)\leq\left[\left(\mathcal{C}_{n,\, r}\left(L_{a}^{2},\, H^{2}\right)\right)^{2}+1\right]^{\frac{1}{2}}.\]
It remains to apply the right-hand side inequality of (4) in Theorem
A to prove the right-hand side one of (10).
\textit{Proof of inequality (9).} 1) We use the same test function\[
f=\sum_{k=0}^{n-1}(1-\vert\lambda\vert^{2})^{\frac{1}{2}}b_{\lambda}^{k}\left(1-\overline{\lambda}z\right)^{-1},\]
as the one used in the proof of {[}Z3, Theorem B{]} (the lower bound,
page 11 of {[}Z3{]}). $f$ being the sum of $n$ elements of $H^{2}$
which are an orthonormal family known as Malmquist's basis (associated
with $\sigma_{n,\,\lambda}=\underbrace{\{\lambda,\lambda,...,\lambda\}}_{n}$,
see Remark 4 above or {[}N1, p.117{]}) , we have $\Vert f\Vert_{H^{2}}^{2}=n$.
2) Since the spaces $H^{2}$ and $B_{2,\,2}^{\frac{1}{2}}$ are rotation
invariant, we have $I\left(\sigma_{n,\,\lambda},H^{2},\, B_{2,\,2}^{\frac{1}{2}}\right)=I\left(\sigma_{n,\,\mu},H^{2},\, B_{2,\,2}^{\frac{1}{2}}\right)$
for every $\lambda,\,\mu$ with $\vert\lambda\vert=\vert\mu\vert=r$.
Let $\lambda=-r$. To get a lower estimate for $\Vert f\Vert_{B_{2,\,2}^{\frac{1}{2}}/b_{\lambda}^{n}B_{2,\,2}^{\frac{1}{2}}}$
consider $g$ such that $f-g\in b_{\lambda}^{n}{\rm Hol}(\mathbb{D})$,
i.e. such that $f\circ b_{\lambda}-g\circ b_{\lambda}\in z^{n}{\rm Hol}(\mathbb{D})$.
3) First, we notice that\[
\left\Vert g\circ b_{\lambda}\right\Vert _{B_{2,\,2}^{\frac{1}{2}}}^{2}=\left\Vert (g\circ b_{\lambda})^{'}\right\Vert _{L_{a}^{2}}^{2}+\left\Vert g\circ b_{\lambda}\right\Vert _{H^{2}}^{2}=\left\Vert b_{\lambda}.(g'\circ b_{\lambda})\right\Vert _{L_{a}^{2}}^{2}+\left\Vert g\circ b_{\lambda}\right\Vert _{H^{2}}^{2}=\]
\[
=\int_{\mathbb{D}}\left|b_{\lambda}(u)\right|^{2}\left|g'(b_{\lambda}(u))\right|^{2}du+\left\Vert g\circ b_{\lambda}\right\Vert _{H^{2}}^{2}=\int_{\mathbb{D}}\left|g'(w)\right|^{2}dw+\left\Vert g\circ b_{\lambda}\right\Vert _{H^{2}}^{2},\]
using the changing of variable $w=b_{\lambda}(u)$. We get\[
\left\Vert g\circ b_{\lambda}\right\Vert _{B_{2,\,2}^{\frac{1}{2}}}^{2}=\left\Vert g'\right\Vert _{L_{a}^{2}}^{2}+\left\Vert g\circ b_{\lambda}\right\Vert _{H^{2}}^{2}=\left\Vert g\right\Vert _{B_{2,\,2}^{\frac{1}{2}}}^{2}+\left\Vert g\circ b_{\lambda}\right\Vert _{H^{2}}^{2}-\left\Vert g\right\Vert _{H^{2}}^{2}\,,\]
and\[
\left\Vert g\right\Vert _{B_{2,\,2}^{\frac{1}{2}}}^{2}=\left\Vert g\right\Vert _{H^{2}}^{2}+\left\Vert g\circ b_{\lambda}\right\Vert _{B_{2,\,2}^{\frac{1}{2}}}^{2}-\left\Vert g\circ b_{\lambda}\right\Vert _{H^{2}}^{2}=\]
\[
\geq\left\Vert g\circ b_{\lambda}\right\Vert _{B_{2,\,2}^{\frac{1}{2}}}^{2}-\left\Vert g\circ b_{\lambda}\right\Vert _{H^{2}}^{2}.\]
Now, we notice that
\[
f\circ b_{\lambda}=\sum_{k=0}^{n-1}z^{k}\frac{(1-\vert\lambda\vert^{2})^{\frac{1}{2}}}{1-\overline{\lambda}b_{\lambda}(z)}=\left(1-\vert\lambda\vert^{2}\right)^{-\frac{1}{2}}\left(1+(1-\overline{\lambda})\sum_{k=1}^{n-1}z^{k}-\overline{\lambda}z^{n}\right)=\]
\[
=(1-r^{2})^{-\frac{1}{2}}\left(1+(1+r)\sum_{k=1}^{n-1}z^{k}+rz^{n}\right)\,.\]
4) Next,\[
\left\Vert g\circ b_{\lambda}\right\Vert _{B_{2,\,2}^{\frac{1}{2}}}^{2}-\left\Vert g\circ b_{\lambda}\right\Vert _{H^{2}}^{2}=\sum_{k\geq1}k\left|\widehat{g\circ b_{\lambda}}(k)\right|^{2}\geq\]
\[
\geq\sum_{k=1}^{n-1}k\left|\widehat{g\circ b_{\lambda}}(k)\right|^{2}=\sum_{k=1}^{n-1}k\left|\widehat{f\circ b_{\lambda}}(k)\right|^{2},\]
since $\widehat{g\circ b_{\lambda}}(k)=\widehat{f\circ b_{\lambda}}(k)\,,\;\forall\, k\in[0,\, n-1].$
This gives\[
\left\Vert g\circ b_{\lambda}\right\Vert _{B_{2,\,2}^{\frac{1}{2}}}^{2}-\left\Vert g\circ b_{\lambda}\right\Vert _{H^{2}}^{2}\geq\frac{1}{1-r^{2}}\left((1+r)^{2}\sum_{k=1}^{n-1}k\right)=\]
\[
=\frac{(1+r)^{2}}{1-r^{2}}\frac{n(n-1)}{2}=\frac{1+r}{1-r}\frac{n(n-1)}{2}=\frac{1+r}{1-r}\frac{(n-1)}{2}\left\Vert f\right\Vert _{H^{2}}^{2},\]
for all $n\geq2$ since $\left\Vert f\right\Vert _{H^{2}}^{2}=n.$
Finally, \[
\left\Vert g\right\Vert _{B_{2,\,2}^{\frac{1}{2}}}^{2}\geq\frac{n}{1-r}\frac{1+r}{2}\left(1-\frac{1}{n}\right)\left\Vert f\right\Vert _{H^{2}}^{2}.\]
In particular, \[
\mathcal{I}_{n,\, r}\left(H^{2},\, B_{2,\,2}^{\frac{1}{2}}\right)\geq\sqrt{\frac{n}{1-r}}\left[\frac{1+r}{2}\left(1-\frac{1}{n}\right)\right]^{\frac{1}{2}}.\]
\begin{flushright}
$\square$
\par\end{flushright}
\subsection*{Some comments}
$\,$
\textbf{a. Extension of Theorem A to spaces $B_{2,\,2}^{s},\, s\geq0$}.
Using the techniques developped in the proof of our Theorem A (combined
with complex interpolation (between Banach spaces) and a reasoning
by induction), it is possible both to precise the sharp numerical
constant $c_{2,\, s}$ in K. Dyakonov's result (3) (mentioned above
in paragraph d. of the Introduction) and to prove the asymptotic sharpness
(at least for $s\in\mathbb{N}\cup\frac{1}{2}\mathbb{N})$ of the right-hand
side inequality of (3). In the same spirit, we would obtain that there
exists a limit: \def${21}$}\begin{equation{${19}$}\begin{equation}
\lim_{n\rightarrow\infty}\frac{\mathcal{C}_{n,\, r}\left(B_{2,\,2}^{s-1},\, H^{2}\right)}{n^{s}}=\left(\frac{1+r}{1-r}\right)^{s}.\label{eq:-5-1}\end{equation}
Our Theorem A corresponds to the case $s=\frac{1}{2}.$
\textbf{b. Extension of Theorem B to spaces $B_{2,\,2}^{s},\, s\geq0$.}
The proof of the upper bound in our Theorem B can be extended so as
to give an upper (asymptotic) estimate of the interpolation constant
$\mathcal{I}_{n,\, r}\left(H^{2},\, B_{2,\,2}^{s}\right),\: s\geq0.$
More precisely, applying K. Dyakonov's result (3) (mentioned above
in paragraph d. of the Introduction) we get\def${21}$}\begin{equation{${20}$}\begin{equation}
\mathcal{I}_{n,\, r}\left(H^{2},\, B_{2,\,2}^{s}\right)\leq\tilde{c}_{s}\left(\frac{n}{1-r}\right)^{s},\;{\rm with}\;\tilde{c_{s}}\asymp c_{2,\, s},\label{eq:-5-1-1}\end{equation}
where $c_{2,\, s}$ is defined in (3) and precised in (19). Looking
at the above comment 1, $\tilde{c}_{s}\asymp(1+r)^{s}$ for sufficiently
large values of $n$. Our Theorem B corresponds again to the case
$s=\frac{1}{2}.$ In this Theorem B, we prove the sharpness of the
right-hand side inequality in (20) for $s=\frac{1}{2}.$ However,
for the general case $s\geq0,$ the asymptotic sharpness of $\left(\frac{n}{1-r}\right)^{s}$
as $r\rightarrow1^{-}$ and $n\rightarrow\infty$ is less obvious.
Indeed, the key of the proof (for the sharpness) is based on the property
that the Dirichlet norm (the one of $B_{2,\,2}^{1/2}$) is {}``nearly''
invariant composing by an elementary Blaschke factor $b_{\lambda},$
as this is the case for the $H^{\infty}$ norm. A conjecture given
by N. K. Nikolski is the following: \def${21}$}\begin{equation{${21}$}\begin{equation}
\mathcal{I}_{n,\, r}\left(H^{2},\, B_{2,\,2}^{s}\right)\asymp\left\{ \begin{array}{c}
\frac{n^{s}}{\sqrt{1-r}}\;\mbox{if}\; s\geq\frac{1}{2}\\
\left(\frac{n}{1-r}\right)^{s}\;\mbox{if}\;0\leq s\leq\frac{1}{2}\end{array},\right.\label{eq:-5-1-1-1}\end{equation}
and is due to the position of the spaces $B_{2,\,2}^{s},\, s\geq0$
with respect to the algebra $H^{\infty}.$
\noun{CMI-LATP, UMR 6632, Université de Provence, 39, rue F.-Joliot-Curie,
13453 Marseille cedex 13, France}
\textit{E-mail address} : [email protected]
\end{document} | math | 39,105 |
\begin{document}
\title{Producing and Detecting Correlated Atoms}
\classification{03.75.Dg, 03.75.Ss}
\keywords {atom interferometry, atom correlations, Hanbury Brown Twiss effect }
\author{C. I. Westbrook}{
address={Laboratoire Charles Fabry de l'Institut d'Optique, F-91403 Orsay Cedex, France}}
\author{M. Schellekens}{
address={Laboratoire Charles Fabry de l'Institut d'Optique, F-91403 Orsay Cedex, France}}
\author{A. Perrin}{
address={Laboratoire Charles Fabry de l'Institut d'Optique, F-91403 Orsay Cedex, France}}
\author{V. Krachmalnicoff}{
address={Laboratoire Charles Fabry de l'Institut d'Optique, F-91403 Orsay Cedex, France}}
\author{J. Viana Gomes}{
address={Laboratoire Charles Fabry de l'Institut d'Optique, F-91403 Orsay Cedex, France}
,altaddress={Departamento de Fisica, Universidade do Minho, Campus
de Gualtar, 4710-057 Braga, Portugal}}
\author{J.-B. Trebbia}{
address={Laboratoire Charles Fabry de l'Institut d'Optique, F-91403 Orsay Cedex, France}}
\author{J. Est\`eve}{
address={Laboratoire Charles Fabry de l'Institut d'Optique, F-91403 Orsay Cedex, France}}
\author{H. Chang}{
address={Laboratoire Charles Fabry de l'Institut d'Optique, F-91403 Orsay Cedex, France}}
\author{I. Bouchoule}{
address={Laboratoire Charles Fabry de l'Institut d'Optique, F-91403 Orsay Cedex, France}}
\author{D. Boiron}{
address={Laboratoire Charles Fabry de l'Institut d'Optique, F-91403 Orsay Cedex, France}}
\author{A. Aspect}{
address={Laboratoire Charles Fabry de l'Institut d'Optique, F-91403 Orsay Cedex, France}}
\author{T. Jeltes}{
address={Laser Center Vrije Universiteit, 1081 HV Amsterdam, the Netherlands}}
\author{J. McNamara}{
address={Laser Center Vrije Universiteit, 1081 HV Amsterdam, the Netherlands}}
\author{W. Hogervorst}{
address={Laser Center Vrije Universiteit, 1081 HV Amsterdam, the Netherlands}}
\author{W. Vassen}{
address={Laser Center Vrije Universiteit, 1081 HV Amsterdam, the Netherlands}}
\begin{abstract}
We discuss experiments to produce and detect atom correlations in a degenerate or nearly degenerate gas of neutral atoms. First we treat the atomic analog of the celebrated Hanbury Brown Twiss experiment, in which atom correlations result simply from interference effects without any atom interactions. We have performed this experiment for both bosons and fermions. Next we show how atom interactions produce correlated atoms using the atomic analog of spontaneous four-wave mixing. Finally, we briefly mention experiments on a one dimensional gas on an atom chip in which correlation effects due to both interference and interactions have been observed.
\end{abstract}
\maketitle
\section{Introduction}
Recent years have seen a blossoming in the use of experimental techniques sensitive to atom correlation in the study of ultra-cold atomic gases. After a pioneering experiment in 1996 \cite{Yasuda:96}, in which the atomic Hanbury Brown Twiss (HBT) experiment was first observed, there were many analyses and proposed extensions to other situations \cite{naraschewski:99, cahill:99, Grondalski:99,Altman:04}. In the past two years several experimental realizations have been reported \cite{Foelling:05,Oettl:05,Greiner:05,Schellekens:05,Esteve:06}. At the same time, the theoretical community has shown much interest in correlated pairs of atoms, either from collisions or from the breakup of molecules\cite{ Duan:00, Moore:00, Zin:06,Norrie:06,Kheruntsyan:05,Deuar:06}. Here again, the year 2005 saw the report of an experimental realization. In this paper we will discuss some new experiments concerning atom correlations. We refer the reader to the contribution of I. Bloch in this volume for additional ones. All this activity promises to provide much new information about the behavior of cold quantum gases, but we also emphasize that in the field of nuclear and particle physics, Hanbury Brown Twiss correlations are a well established experimental technique and we recomend Ref.~\cite{Boal:90} for a review.
\section{The atomic Hanbury Brown Twiss experiment}
\subsection{Intuitive picture}
At this conference, the contributions of R. Glauber and that of E. Demler discuss the theoretical interpretation of the correlation experiments. Here we will give a less general but intuitive point of view due to Fano \cite{Fano:61}. The HBT effect necessarily involves the detection of two particles at different space-time points. Thus one is led to consider, two source points, A and B, which emit particles detected at two detection points,
$C$ and $D$. One must consider the quantum mechanical amplitude
for the process ($A\rightarrow C$ and $B \rightarrow D$) as well
as that for ($A \rightarrow D$ and $B \rightarrow C$). If the
two processes are indistinguishable, the amplitudes interfere. For
bosons, the interference is constructive resulting in a joint
detection probability which is enhanced compared to that of two
statistically independent detection events, while for fermions the
joint probability is lowered. For a detector separation larger than the aperture which
would permit the resolution of the structure of the source, the average over different points in the
source washes out the interference and one recovers
the situation for uncorrelated particles.
In the case of a chaotic source of light the correlations are also easily understood in terms of speckle, without reference to the concept of photons. But, as in optics with light, we are capable of producing sources for which a classical analysis is not adequate. The experiment discussed below using fermions is an example. The idea of speckle can still be useful however, because it reminds us that these experiments can be analyzed from the point of view of noise or fluctuation phenomena.
\subsection{Experiment with bosons}
The detection of HBT correlations has presented a significant experimental challenge, because the correlation is only maximal if the two detectors occupy the same $\hbar^3$ volume in phase space. In other words, the correlation length, ${\hbar t}\over{m s}$ where $s$ is the size of the source and $t$ is the time of flight to the detector\cite{Gomes:06}, is generally quite small. In addition, the signal to noise ratio in the experiment is proportional to the phase space density of the source. Thus, the detailed study of the HBT effect was greatly facilitated by the advent of degenerate quantum gases. Figure~\ref{detector} shows our detector. A micro-channel plate (MCP) amplifies the electron ejected upon impact by a metastable helium atom (20 eV internal energy). The charge pulse is recorded with ns time resolution and in the horizontal plane a delay line anode and timing electronics provides position resolution\footnote{Available from Roentdek \url{http://www.roendtek.com}. Our time to digital converter is manufactured by ISITech \url{http://www.opticsvalley.org/data/isitech.pdf}}. Although the detector actually measures arrival times, we conventionally convert the arrival time into a vertical position by multiplying by the velocity at arrival at the detector. This velocity has a spread of less than 1\% over the atomic sample.
In the spring of 2005, we successfully used this detector to observe the HBT effect using an evaporatively cooled sample of $^4$He\cite{Schellekens:05,Westbrook:05}. We refer the reader to those papers for more information. The shape, width and height of the correlation signal can be quantitavely understood by a simple ideal gas treatment of the atoms.
\subsection{Experiment with fermions}
Only weeks before the ICAP2006 conference, the metastable helium groups in Orsay and in Amsterdam began a collaboration to observe the analagous effect with the fermionic isotope $^3$He. Instead of a bump, we expect a dip. The Amsterdam group had already demonstrated the production of a degenerate gas of $^3$He \cite{McNamara:06}, and the setup was sufficiently similar to the one in Orsay that it was possible to install the Orsay detector in the Amsterdam apparatus with some minor modifications to the vacuum system. It took only two weeks of (albeit intense) work to see an unambiguous anti-bunching signal. One of the first of these is shown in Fig.~\ref{detector}. Unlike the published boson data, the data shown are not normalized. Thus one sees a broad structure in the peak corresponding to the Gaussian shape of the cloud as it arrives at the detector (or more precisely its auto-convolution). The anti-bunching signal is the small dip for pair separations below 1 ms. At this writing, quantitative analysis of these data is still in progress, but roughly speaking, the antibunching signal corresponds closely to our expectations. Under identical conditions, the width and amplitude of the signal for bosons and fermions are of similar magnitude.
\begin{figure}
\caption{(A). Drawing of the cloverleaf trap and the position sensitive detector. The detector consists of two 8 cm diameter micro channel plates. The charge is collected by two delay-line anodes to give both the arrival time (or equivalently the vertical position) and the position of the particles in the x-y plane. The vertical resolution is determined by the time resolution (1 ns) while the horizontal resolution is about 500~$\mu$m. (B). Pair distribution histogram for $^3$He falling on the detector. A time separation of 1 ms corresponds to about 3.5 mm. The broad overall shape of the distribution (HWHM about 6 ms) is due to the approximately Gaussian temporal shape of the cloud. The dip for times below 1 ms is the antibunching or Fermibury effect.}
\label{detector}
\end{figure}
\section{Observation of correlated atom pairs}
Interactions between atoms also produce correlated atoms. The correlations are particularly simple for elastic collisions, and obviously the study of collision products has occupied much of atomic, nuclear and particle physics for the last century. The collisions we study here however are a little different because the source is a Bose-Einstein condensate (BEC), and therefore approximates a "single mode" source. Thus, analogies with four-wave mixing are very apt. Indeed stimulated four wave mixing using BEC's has already been observed\cite{Deng:99,Vogels:02}. Collisions between BEC's have also been studied by two other groups\cite{Buggle:04,Thomas:04}. In these experiments collision velocities were sufficiently high that not only s-waves but also d-waves were involved. The experiments we describe here are limited to the s-wave regime.
The point of departure for the experiment is the Orsay apparatus used in the HBT experiment described above. To this apparatus we have added two more laser beams to drive stimulated Raman transitions between the trapped ($m=1$) and field insensitive ($m=0$) states of the $2^3S_1$ energy level. The quantization axis is defined by the magnetic field which is along the x-axis. The two lasers are detuned by 400 MHz from the $2^3S_1$ - $2^3P_0$ transition and have a relative detuning of about 600 kHz in order to be in Raman resonance for atoms in the bias field of the magnetic trap. The laser beams propagate at small angles to the $z$ and $x$ axes of Fig. 1 and thus a momentum transfer of $\hbar k(\mathbf e_x+{\mathbf e_z})$, where $\mathbf e_x$ is a unit vector along the x-axis, accompanies the Raman transition. The beam along the $x$-axis is retro-reflected resulting in a second possible Raman transition with a momentum transfer of $\hbar k(-\mathbf e_x+{\mathbf e_z})$.
\begin{figure}
\caption{Images of the collision of two condensates. Each frame represents successive slices of the atomic cloud as it passes the plane of the detector. The two colliding condensates and the s-wave collision sphere are clearly visible.}
\label{circle}
\end{figure}
Most of the atoms are transfered to the $m=0$ state with one of the two possible momentum components. These atoms collide with a relative velocity of twice the recoil velocity and, in the absence of gravity, would be scattered into a spherical shell whose radius is ${\hbar k \over m}t$ where $t$ is the time after the collision. In the presence of gravity, the kinematics are the same except that the spherical shell accelerates downward. Atoms which are not transferred to the $m=0$ state remain in the trap. Figure~\ref{circle} shows some data taken with the delay line detector under these conditions. The figure shows successive slices of the three-dimensional reconstruction of the atoms' positions. The sphere into which the atoms are scattered, as well as the two colliding clouds are clearly visible. Less than 10\% of the atoms are scattered from the colliding clouds.
\begin{figure}
\caption{Atom correlation signals: (A), for opposite momenta, (B), for colinear momenta. The left image shows the signal in the x-y plane. The next two in each row show the signals along one axis and integrated along the other. The second line (B) is simply a manifestation of the Hanbury Brown Twiss effect. In both cases the correlation function is anisotropic because of the initial anisotropy of the source. The normalization is such that unity corresponds to no correlation.}
\label{pi-corr}
\end{figure}
The correlations on the sphere are shown in Fig.~\ref{pi-corr}. We see a correlation signal both for pairs of opposite momenta in the center of mass frame, $(p,-p)$ as well as for pairs with the same momentum $(p,p)$. The latter effect is simply another manifestation of the HBT effect\cite{Zin:06,Deuar:06} and may prove useful because it allows us to characterize the size of the source. The size of signal for opposite momenta can in principle be many times the background level. Its small size here appears to be due to a rather poor quantum efficiency of the detector (on the order of 5\% averaged over the detector). The width of the anti-colinear correlation peak is clearly larger than the colinear peak. We believe that its width is partly due to the mean field energy acquired by the atoms during their expansion\cite{Zin:06,Norrie:05,Kheruntsyan:05,Deuar:06}. A quantitative study of the width and the detector quantum efficiency is in progress.
\section{Correlations on an atom chip}
As has been shown in other recent experiments, single atom counting is not necessary to observe atom correlation effects. Absorptive imaging, when performed in sufficiently low noise conditions can also be sensitive to atom number or atom density fluctuations\cite{Altman:04,Foelling:05, Greiner:05}. Inspired by these experiments, we have also examined number fluctuations in a nearly one dimensional gas on an atom chip. A difficulty in absorption imaging is the fact that the necessary integration over one direction can average out the desired signal. The one dimensional geometry is particularly favorable in this respect because this integration can be avoided. Atom chips are also advantageous for this sort of study because they permit the use of a compact, and mechanically stable apparatus. In our experiment we had little difficulty in taking images at the photon shot-noise limit. Multiple averages (several hundreds) of these images permitted an accurate subtraction of the photon shot-noise to reveal the atom number fluctuations\cite{Esteve:06}.
In spite of the fact that the resolution of our imaging system was significantly larger than the correlation length of the sample, these measurements had two new features. First, the fluctuations were observed without releasing the atoms from the trap. They were thus sensitive to correlations in position rather than momentum. Second, and more importantly, we were able to identify a regime of high density in which interactions between the atoms suppressed the fluctuations which would be expected for a non-interacting gas. A careful analysis of the density profile of such a gas also reveals that the profile cannot be explained by a Hartree-Fock calculation that neglects correlations between the particles \cite{Trebbia:06}. Improved calculations, including correlation effects promise to give a better account of the observations \cite{Blakie:05, Proukakis:06}.\\
We hope that the experiments discussed above have given a taste of the rich possibilities in correlation measurements. The main conclusion with which we would like to leave the reader is that, in the field of degenerate quantum gases, treating correlations between particles remains an important challenge. But our increasingly sophisticated experimental techniques are beginning provide a window on these phenomena, and we can expect great progress in the near future.
\begin{theacknowledgments}
The Laboratoire Charles Fabry is a member of the CNRS Federation LUMAT (FR2764), and of the Institut Francilien pour la Recherche en Atomes Froids (IFRAF). We acknowledge the technical help provided by the DPTI Technology Platform of the CNRS and the Universit\'e de Paris-sud. We acknowledge the support of the EU under grants MRTN-CT-2003-505032 (Atom Chips) and IP-CT-015714 (SCALA) and of the ANR under contract 05-NANO-008-01.The Amsterdam/Orsay collaboration is supported by the European Community Integrated Infrastructure Initiative action (RII3-CT-2003-506350) and by the Netherlands Foundation for Fundamental Research of Matter (FOM).
\end{theacknowledgments}
\end{document} | math | 17,361 |
\betagin{document}
\author{Yuxiang Ji, Yanir A. Rubinstein, Kewei Zhang}
\title{Eguchi--Hanson metrics arising from\ K\"ahler-Einstein edge metrics}
\centerline{\it Dedicated to Scott Wolpert on the occassion of his retirement }
\betagin{abstract}
Calabi--Hirzebruch manifolds are higher-dimensional generalizations of both the football and Hirzebruch surfaces.
We construct a family of K\"{a}hlerE edge metrics singular along two disjoint divisors on the Calabi--Hirzebruch manifolds and study their Gromov--Hausdorff limits when either cone angle tends to its extreme value. As a very special case, we show that the
celebrated Eguchi--Hanson metric arises in this way naturally as a Gromov--Hausdorff limit. We also completely describe all other (possibly rescaled) Gromov--Hausdorff limits which exhibit a wide range of behaviors, resolving in this setting a conjecture of Cheltsov--Rubinstein. This gives a new interpretation of
both the Eguchi--Hanson space
and Calabi's Ricci flat spaces
as limits of compact singular Einstein spaces.
\end{abstract}
\tableofcontents
\section{Motivation}
The main motivation for this
work is a program of Cheltsov--Rubinstein
concerning the small angle deformation of K\"{a}hlerEE (KEE) metrics,
which in particular makes the following prediction \cite[Conjecture 1.11]{CR15}:
\betagin{conj}
\label{GeneralConj}
Suppose that $(X,D)$ is strongly asymptotically log
Fano manifold with $D$ smooth and irreducible.
Suppose that
$
\kappa:=\inf\{\NN\ni k\le\dim X \,:\, (K_X+D)^k=0\}\le\dim X,
$
and that there exist KEE metrics $\omega_\beta, \beta\in(0,\epsilon)$ on $(X,D)$
for some $\epsilon>0$.
Then, $(X,D,\omega_\beta)$
converges in an appropriate sense as $\beta$ tends to zero to a
generalized KE metric $\omega_\infty$
that is Calabi--Yau along its generic $(\dim X+1-\kappa)$-dimensional
fibers.
\end{conj}
In this article
we actually treat a slightly more general situation where $D$ is allowed
to have two disjoint smooth components $D=D_1+D_2$, but the setting is
essentially identical to that of Conjecture \ref{GeneralConj} since
the angle $2\pi\betata_1$ along $D_1$ of the KEE metric $\omega_{\beta_1,\beta_2}$
actually determines the angle $2\pi\betata_2$ along $D_2$
and, importantly, vice versa.
A second motivation for this article is given by a prediction posed by two of the present authors in a previous work
concerning the {\it large} angle limit of a family of
K\"{a}hlerEE metrics constructed on the second Hirzebruch surface ${\mathbb F}_2$. On that surface let $D_1:=Z_{-2}$ denote the $-2$-curve and $D_2:=Z_2$ the smooth
infinity section, a $2$-curve
satisfying $Z_{-2}\cap Z_2=\emptyset$.
According to \cite[Theorem 1.2]{RZ21} there exist for each $\betata_1\in(0,1)$
a unique K\"{a}hlerEE metric $\omega_{\beta_1,\beta_2}$ with angle $2\pi\betata_1$ along $Z_{-2}$ and angle
$2\pi\betata_2=2\pi(2\beta_1-3+\sqrt{9+12\betata_1-12\betata_1^2})/4$ along $Z_2$
and cohomologous to $\frac{1+\beta_2}{1-\beta_1}[Z_2]-[Z_{-2}]$.
The following prediction was made \cite[Remark 5.1]{RZ21}:
\betagin{conj}
\label{RZConj}
As $\betata_1$ tends to $1$, an appropriate limit of
$({\mathbb F}_2,\omega_{\beta_1,\beta_2})$ converges to the Eguchi--Hanson metric.
\end{conj}
The virtue of Conjecture \ref{RZConj} is that it proposes
a remarkable new interpretation of the Eguchi--Hanson space
from mathematical physics
as an angle deformation limit of compact singular Einstein spaces with edge singularities.
The goal of the present article is to treat both conjectures
and a bit more.
We solve Conjecture \ref{GeneralConj} in the
setting of Calabi--Hirzebruch manifolds as well as solve
Conjecture \ref{RZConj}. In fact we solve a
generalized version of Conjecture \ref{RZConj}
that interprets Calabi's Ricci flat spaces
as limits of compact KEE spaces.
Moreover,
due to the existence of
{\it two} disjoint divisors
there are also interesting limits
to study that are not part of
the above conjectures:
these limits
are studied in Theorems
\ref{thm: main} and
\ref{thm: main relation} below.
\section{Results}
Let $M$ be a compact K\"{a}hler manifold and $D=D_1+\cdots D_r$ a simple normal crossing divisor in $M$. A K\"{a}hler metric $\omega$ is said to have edge singularity along $D$ if $\omega$ is smooth on $M\setminus D$ and asymptotically equivalent to the model edge metric along $D$ \cite[Definition 3.1]{Y14}. The study of K\"{a}hlerE edge metrics dates back to Tian \cite{Tian96} where he considered applications of such metrics to algebraic geometry. Cheltsov--Rubinstein \cite{CR15} initiated the program of studying small angle limits of K\"{a}hlerE edge metrics. In previous works, two of us treated the Riemann surface footballs case and Hirzebruch surfaces case by first constructing K\"{a}hlerE edge metrics on the manifolds using Calabi ansatz
and then studying their limiting behaviors when the cone angles tend to $0$
\cite{RZ21, RZ20}.
In this paper, we consider a more general setting, Calabi--Hirzebruch manifolds.
To construct these KEE metrics we use
the standard Calabi ansatz
in Section \ref{sec: general}, generalizing \cite{RZ21}.
The angle at either divisor
$D_1:=Z_{n,k}$ or $D_2:=Z_{n,-k}$
then determines the angle
on the other divisor which leads to two families of KEE
metrics $\eta_{\betata_1}$ and $\xi_{\betata_2}$
on $\mathbb{F}_{n, k}$.
\betagin{rmk}
Very recently,
Biquard--Guenancia completely
solved the folklore case $\kappa=1$
of Conjecture \ref{GeneralConj}
\cite{BG21}
(that case of the conjecture
seems to have been
already conjectured
by Tian and Mazzeo in the 90's
and then explicitly stated
around 2009 by Donaldson
\cite{Mazzeo,JMR16,DonConic}),
and interestingly
the Calabi ansatz makes an appearance in their proof as well.
It would be interesting to explore
whether some of the rather
elementary ideas here
can be combined with some of
their deep estimates to
attack the general case
of Conjecture \ref{GeneralConj}.
\end{rmk}
A common feature for footballs,
Hirzebruch surfaces, and Calabi--Hirzebruch manifolds is
that there are two smooth disjoint divisors and hence two angle parameters $\betata_1,\betata_2$.
In Section \ref{sec: model}, we review the construction of Calabi--Hirzebruch manifolds \cite{Calabi82, H51}, denoted by $\mathbb{F}_{n, k}$ for $\NN\ni n\geq 2$ and $k\in\mathbb{N}$, and define two (families of) model metrics. The first
are Ricci-flat edge metrics on the total space of the (non-compact) line bundle $-kH_{\mathbb{P}^{n-1}}$,
\betagin{equation*}
\omega_{\mathrm{eh}, n, k},
\end{equation*}
and an edge singularity of the angle $2\pi n/k$ along $Z_{n, k}$.
The second are
compact K\"{a}hlerEE spaces with positive Ricci curvature $(n+1)/k$ and an edge singularity of angle $2\pi/k$,
\betagin{equation*}
\omega_{\mathrm{orb}, n, k},
\end{equation*}
on the weighted projective space $\mathbb{P}^n(1,\dots, 1,k)$.
These two model spaces turn out to be the
different Gromov--Hausdorff limits
of large-angle
limits of the KEE metrics we construct
on the Calabi--Hirzebruch
manifolds.
\subsection{Large angle limits}
The following result describes precisely
the different large-angle limits that arise from these KEE
metrics by either using different pointed limits or else
parametrizing the angles in different ways.
Note that $\betata_1$ ranges in $(0,n/k)$ and
$\betata_2$ ranges in $(0,1/k)$.
\betagin{thm}\label{thm: main}
Fix a base point $p$ on the zero section of $\mathbb{F}_{n, k}$ and $q$ on the infinity section. The pointed metric space $(\mathbb{F}_{n, k}, \eta_{\betata_1}, p)$ converges in the pointed Gromov--Hausdorff sense to $(-kH_{\mathbb{P}^{n-1}}, \omega_{\mathrm{eh}, n, k}, p)$ as $\betata_1$ tends to $n/k$. On the other hand, $(\mathbb{F}_{n, k}, \xi_{\betata_2}, q)$ converges in the pointed Gromov--Hausdorff sense to $(\mathbb{P}^n(1,\dots, 1, k), \omega_{\mathrm{orb}, n, k}, q)$ as $\betata_2\to 1/k$.
\end{thm}
In particular, when $n=k=2$, Theorem \ref{thm: main}
resolves
Conjecture \ref{RZConj}.
Theorem \ref{thm: main}
amounts to saying that the famous Eguchi--Hanson metric from mathematical physics is the Gromov--Hausdorff limit of {\it compact} K\"{a}hlerE edge metrics that we construct on the second Hirzebruch surface (see Remark \ref{rmk: eh as limit} for more details or see Appendix \ref{app: eh another discussion} for another proof).
More generally, when $n=k$, Theorem \ref{thm: main} recovers a family of Ricci-flat metrics on the total space of canonical bundle of $\mathbb{P}^{n-1}$ that was constructed by Calabi \cite{Calabi79},
once again as a limit of
{\it compact} Einstein spaces.
\betagin{figure}[!htbp]
\centering
\includegraphics[width=9.3cm]{ccc.png}
\caption{The upper part shows the K\"ahler edge structure on the Calabi--Hirzebruch manifold $\mathbb{F}_{n, k}$. When $n=2$, $F_{2,k}$ is the $k$-th
Hirzebruch surface and $Z_{2,\pm k}$ is a $\mp k$-curve. The lower part shows the different limits described in Theorem \ref{thm: main}. Note that on the left, we get a non-compact limit, with $q$ (together with all of $Z_{n,-k}$) pushed-out to infinity. On the right we get a compact limit, with $p$ limiting (together with all of $Z_{n,k}$) to an isolated orbifold point.}
\label{fig:my_label}
\end{figure}
\betagin{rmk}
It would be
interesting to find physical
interpretations of Theorem
\ref{thm: main} in the case
$n=k=2$ of the Eguchi--Hanson
metric.
\end{rmk}
The elementary proof of Theorem \ref{thm: main} is divided into two parts. In section \ref{sec: asym of kee}, we study asymptotic behaviors of K\"{a}hlerE edge metrics $\eta_{\betata_1}$ and $\xi_{\betata_2}$ when $\betata_1$, or respectively $\betata_2$, is close to $n/k$ or $1/k$. In section \ref{sec: gh kee}, we prove the convergence results by studying a family of ODEs that arises from the construction of $\eta_{\betata_1}$ and $\xi_{\betata_2}$.
\betagin{rmk}\label{rmk_twocases}
The two families of K\"{a}hlerE edge metrics $\eta_{\betata_1}$ and $\xi_{\betata_2}$ are
related via a simple, but important, rescaling. In Theorem \ref{thm: main}, we see
that different limits airse for those two family of metrics. The normalization factor can be obtained by studying the asymptotic behavior of $\xi_{\betata_2}$: see Proposition \ref{prop: length finite nkb2 case} for details.
\end{rmk}
\betagin{rmk}
As noted before Theorem \ref{thm: main}, $\betata_1$ ranges in $(0,n/k)$ and
$\betata_2$ ranges in $(0,1/k)$.
In proving Theorem \ref{thm: main}
for the family $\eta_{\betata_1}$
one obtains the asymptotic
dependence of $\betata_2$ on $\betata_1$ in terms of a parameter
that controls the length of the
fibers in the Hirzebruch fibration---see \eqref{Tbeta1beta2Eq}. This
shows that as $\betata_1$ tends
to its maximal value $n/k$,
$\betata_2$ tends to its maximal
value as well, $1/k$, and
the divisor $Z_{n,k}$ gets
pushed-off to infinity.
It thus comes a bit as a surprise
that when we consider the family
$\xi_{\betata_2}$ parametrized in terms of $\betata_2$, and we let
$\betata_2$ tend towards $1/k$,
while the parameter $\betata_1$ still tends towards its maximal
value $n/k$ we get a completely
different limiting behavior:
instead of a non-compact limit we get a compact limit via
a metric degeneration along $Z_{n, k}$.
Thus, studying the two families
is an essential feature of the setting and leads to two completely different Gromov--Hausdorff limits as in Theorem \ref{thm: main}.
It would be interesting to
generalize this phenomenon to
other settings.
\end{rmk}
\subsection{Small angle limits}
Next, we resolve Conjecture \ref{GeneralConj}
in the setting of $\mathbb{F}_{n, k}$.
Denote by $\widetilde{\eta_{\betata_1}}$ and
$\widetilde{\xi_{\betata_2}}$
fiber-wise rescalings of $\eta_{\betata_1}, \xi_{\betata_2},$ respectively
(see \eqref{eq: fibrescal} and \S\ref{Sec7}).
\betagin{thm}\label{thm: main relation}
Both $(\mathbb{F}_{n, k}, \eta_{\betata_1})$ and
$(\mathbb{F}_{n, k}, \xi_{\betata_2})$ converge in the Gromov--Hausdorff sense to
$(\mathbb{P}^{n-1}, k\omega_{\operatorname{FS}})$ as $\betata_1$ or $\betata_2$
tends to $0$. Moreover,
as $\betata_1\searrow 0$, $(\mathbb{F}_{n, k}, \widetilde{\eta_{\betata_1}}, p)$ converges in the pointed Gromov--Hausdorff sense to $(\mathbb{P}^{n-1}\times \mathbb{C}^*, \frac{k}{n}(n\pi_1^*\omega_{\operatorname{FS}}+\pi_2^*\omega_{\operatorname{Cyl}}), p)$
and similarly for $(\mathbb{F}_{n, k}, \widetilde{\xi_{\betata_2}}, q)$,
where $p,q\in \mathbb{F}_{n, k}$ are as in Theorem \ref{thm: main}.
\end{thm}
A compendium of the limit theorems in \cite{RZ20, RZ21} and the
present work is shown in Table 1.
{
\betagin{table}[htbp]
\textspace{-2em}
\resizebox{1.1\textwidth}{!}{
\betagin{tabular}{|c|c|c|c|c|c|c|}
\textline
$n$ & $k$ & $\betata_1$ & $\betata_2$ & Sequence & Limit & Reference \\
\textline\textline
$1$ & $k$ & $\betata_1\searrow 0$ & $\betata_2(\betata_1)\equiv\betata_1$ & football metrics &$(\mathbb{C}^*, \omega_{\mathrm{Cyl}})$ & folklore, \cite[Thm 1.3]{RZ20}\\
\textline
$1$ & $k$ & $\betata_1\nearrow 1$ & $\betata_2(\betata_1)\equiv\betata_1$ & football metrics & $\left(\mathbb{P}^1, \omega_{\operatorname{FS}}\right)$ & folklore, \cite[Thm 1.2]{RZ20}\\
\textline
$2$ & $k$ & $\betata_1\searrow 0$ & $\betata_2(\betata_1)=\betata_1+O(\betata_1^2)$ & KEE metrics & $\left(\mathbb{P}^1, k\omega_{\operatorname{FS}}\right)$ & \cite[Thm 1.2]{RZ21}\\
\textline
$2$ & $k$ & $\betata_1\searrow 0$ & $\betata_2(\betata_1)=\betata_1+O(\betata_1^2)$ & rescaled KEE metrics & $\left(\mathbb{P}^1\times \mathbb{C}^*, k\pi_1^*\omega_{\operatorname{FS}}+k\pi_2^*\omega_{\operatorname{Cyl}}\right)$ & \cite[Thm 1.2]{RZ21}\\
\textline
$2$ & $1$ & $\betata_1\to 1$ & $\betata_2(\betata_1)\to \sqrt{3}-1$ & KEE metrics & KEE metric on $(\mathbb{F}_1, Z_{-1})$ & \cite[Cor 1.5]{RZ21}\\
\textline
$2$ & $1$ & $\betata_1\nearrow 2$ & $\betata_2(\betata_1)\nearrow 1$ & KEE metrics & Ricci-flat metric on $(-H_{\mathbb{P}^1}, Z_1)$ & Thm \ref{thm: limit n2k1 case}\\
\textline
$2$ & $1$ & the metric degenerates on $Z_1$ & $\betata_2\nearrow 1$ & KEE metrics & $(\mathbb{P}^2, \omega_{\operatorname{FS}})$& Thm \ref{thm: appthm2}.\\
\textline
$2$ & $1$ & $\betata_1(\betata_2)\nearrow 2$ & $\betata_2\nearrow 1$ & rescaled KEE metrics & Ricci-flat metric on $(-H_{\mathbb{P}^1}, Z_1)$ & Thm \ref{thm: rescale simcase}\\
\textline
$2$ & $2$ & $\betata_1\nearrow 1$ & $\betata_2(\betata_1)\nearrow 2$ & KEE metrics & Eguchi--Hanson metric ($\epsilon=1$) & Thm \ref{thm: metric limit nk case}\\
\textline
$2$ & $k$ & the metric degenerates on $Z_{2, k}$ & $\betata_2(\betata_1)\nearrow 1/k$ & KEE metrics & $(\mathbb{P}^2(1,1,k), \omega_{\mathrm{orb},2,k})$ & Thm \ref{thm: conv mod2}\\
\textline
$n$ & $1$ & the metric degenerates on $Z_{n, 1}$ & $\betata_2(\betata_1)\nearrow 1$ & KEE metrics & $(\mathbb{P}^n, \omega_{\mathrm{FS}})$ & Thm \ref{thm: conv mod2}\\
\textline
$n$ & $k$ & $\betata_1\nearrow n/k$ & $\betata_2(\betata_1)\nearrow 1/k$ & KEE metrics & $(-kH_{\mathbb{P}^{n-1}}, \omega_{\mathrm{eh}, n, k})$ & Thm \ref{thm: metric limit nk case}\\
\textline
$n$ & $k$ & $\betata_1\searrow 0$ & $\betata_2(\betata_1)=\betata_1+O(\betata_1^2)$ & rescaled KEE metrics & $(\mathbb{P}^{n-1}\times \mathbb{C}^*, \frac{k}{n}(n\pi_1^*\omega_{\operatorname{FS}}+\pi_2^*\omega_{\operatorname{Cyl}}))$ & Thm \ref{thm: ResB1Case}\\
\textline
$n$ & $k$ & the metric degenerates on $Z_{n,k}$ & $\betata_2\nearrow 1/k$ & KEE metrics & $(\mathbb{P}^{n}(1,\dots,1,k), \omega_{\mathrm{orb}, n, k})$ & Thm \ref{thm: conv mod2}\\
\textline
$n$ & $k$ & $\betata_1(\betata_2)\nearrow n/k$ & $\betata_2\nearrow 1/k$ & rescaled KEE metrics & $(-kH_{\mathbb{P}^{n-1}}, \omega_{\mathrm{eh}, n, k})$ & Cor \ref{thm: ResB2}\\
\textline
$n$ & $k$ & $\betata_1(\betata_2)=\betata_2+O(\betata_2^2)$ & $\betata_2\searrow 0$ & rescaled KEE metrics & $(\mathbb{P}^{n-1}\times \mathbb{C}^*, \frac{k}{n}(n\pi_1^*\omega_{\operatorname{FS}}+\pi_2^*\omega_{\operatorname{Cyl}}))$ & Thm \ref{thm: ResB2Zero}\\
\textline
\end{tabular}
}
\caption{Limits of K\"{a}hlerEE metrics on $\mathbb{F}_{n, k}$, the $k$-th Calabi--Hirzebruch manifold of dimension $n$.}
\label{summary}
\end{table}
}
\subsection{Organization}
In Section \ref{sec: model}, we first review the construction of Calabi--Hirzebruch manifolds and then define two families of model metrics respectively on the total space of line bundles $-kH_{\mathbb{P}^{n-1}}$ and the Calabi--Hirzebruch manifolds. In Section \ref{sec: general}, we construct K\"{a}hlerE edge metrics on the Calabi--Hirzebruch manifolds following Calabi ansatz. In Section \ref{sec: asym of kee}, we study the asymptotic behaviors of K\"{a}hlerE edge metrics by reducing it to the study of some ODEs. In Section \ref{sec: gh kee}, we consider the Gromov--Hausdorff limits of the K\"{a}hlerE edge metrics on the Calabi--Hirzebruch manifolds and find out the model metrics in the limit. Theorem \ref{thm: main} is then proved in two parts: Theorem \ref{thm: metric limit nk case} and Theorem \ref{thm: conv mod2}.
The first statement of Theorem \ref{thm: main relation} about the non-rescaled limits
is proved in Theorems \ref{thm: b1 0 case} and \ref{thm: b2 0 case}.
The second statement of Theorem \ref{thm: main relation} concerning
rescaled limits and pointed limits is contained in Theorems \ref{thm: ResB1Case}
and \ref{thm: ResB2Zero}. We also discuss the relation between Theorem \ref{thm: metric limit nk case} and Theorem \ref{thm: conv mod2} as mentioned in Remark \ref{rmk_twocases} in Corollary \ref{thm: ResB2}.
Appendices \ref{app: review EH} and \ref{app: eh another discussion} provide a brief review on basic properties of Eguchi--Hanson metrics and explain how to understand Eguchi--Hanson metrics as Gromov--Hausdorff limits of K\"{a}hlerE edge metrics.
In Appendix \ref{app: more eg}, we provide more examples of K\"{a}hlerE edge metrics and their Gromov--Hausdorff limit metrics.
\paragraph{Acknowledgments.}
Research supported by NSF grant 1906370, NSFC grant 12101052, the Fundamental Research Funds 2021NTST10 for the Central Universities, and a Brin Graduate
Fellowship at the University of
Maryland.
This article is dedicated to Scott Wolpert that aside from
his seminal mathematical contributions to complex geometry
has been instrumental in forging the University of Maryland
as a leader in that field, and has positively impacted the
careers of all three authors
as either graduate students
or faculty in the department
(and, in particular, in all likelihood this article
would not have appeared
had it not been for his constant support).
\section{Model metrics}\label{sec: model}
We first review the construction of Calabi--Hirzebruch manifolds \cite{Calabi82, H51}.
\betagin{defi}\label{defi: CH manifolds}
The Calabi--Hirzebruch manifold, denoted by $\mathbb{F}_{n, k}$, where $n, k\in\mathbb{N}$, is defined as follows:
\betagin{equation*}
\mathbb{F}_{n, k}:=\mathbb{P}(-kH_{\mathbb{P}^{n-1}}\oplus\mathbb{C}_{\mathbb{P}^{n-1}}),\quad n, k\in\mathbb{N},
\end{equation*}
where $H_{\mathbb{P}^{n-1}}$ is the hyperplane bundle over $\mathbb{P}^{n-1}$ and $\mathbb{C}_{\mathbb{P}^{n-1}}$ is the trivial one.
An alternative way to define $\mathbb{F}_{n, k}$ is by adding an infinity section to the blow up of $\mathbb{C}^n/\mathbb{Z}_k$ at the origin \cite[Lemma 2.1]{RZ21}.
\end{defi}
In this section, we introduce two model metrics $\omega_{\mathrm{eh}, n, k}$ and $\omega_{\mathrm{orb}, n, k}$, that are defined on the total space of line bundles $-kH_{\mathbb{P}^{n-1}}$ and the Calabi--Hirzebruch manifolds $\mathbb{F}_{n, k}$ respectively. They will serve as the candidates of limit metrics in the latter sections.
Denote by $[Z_1:\cdots:Z_n]$ the homogeneous coordinates on $\mathbb{P}^{n-1}$. Working on the chart $\{Z_i\neq 0\}$, we shall use the nonhomogeneous coordinates $z_j:=Z_j/Z_i$ for all $j\neq i$. Denote by $w$ the coordinate along each fiber on $-kH_{\mathbb{P}^{n-1}}$ or $\mathbb{F}_{n, k}$. Then $w\in\mathbb{C}$ for $-kH_{\mathbb{P}^{n-1}}$ and $w\in \mathbb{C}\cup \{\infty\}$ for $\mathbb{F}_{n, k}$. In particular, we have two divisors on $\mathbb{F}_{n, k}$: the zero section
\betagin{equation*}
Z_{n, k}:=\{w=0\}
\end{equation*}
and the infinity section
\betagin{equation*}
Z_{n, -k}:=\{w=\infty\}.
\end{equation*}
Consider the Hermitian norm on $-kH_{\mathbb{P}^{n-1}}$ (and also $\mathbb{F}_{n, k}$)
\betagin{equation*}
||(z_1,\dots, \textat{z}_i,\dots, z_n, w)||:=|w|^2\left(1+\sum_{j\neq i}^{n}|z_j|^2\right)^k,
\end{equation*}
on the chart $\{Z_i\neq 0\}$. Let $s$ be the logarithm of this fiberwise norm, i.e.,
\betagin{equation}\label{eq: def s}
s:=\log|w|^2+k\log\left(1+\sum_{j\neq i}|z_j|^2 \right).
\end{equation}
Next, we define model metrics that depend only on $s$ on $-kH_{\mathbb{P}^{n-1}}$ and $\mathbb{F}_{n, k}$.
\subsection{Non-compact case: model metrics on $-kH_{\mathbb{P}^{n-1}}$}
The first model metric, $\omega_{\mathrm{eh}, n, k}$, is a Ricci-flat edge metric with edge singularity of angle $2n\pi/k$ along the zero section $Z_{n, k}$ of $-kH_{\mathbb{P}^{n-1}}$. In particular, when $n=k$, this metric is smooth. Indeed, when $n=k$, $\omega_{\mathrm{eh}, n, k}$ coincides with the Ricci-flat metric on the canonical bundle of $\mathbb{P}^{n-1}$ that was constructed by Calabi \cite{Calabi79}.
\betagin{defi}
Let $s$ be defined in \eqref{eq: def s}. To define $\omega_{\mathrm{eh}, n, k}$, introduce another coordinate $\lambda\in(1, +\infty)$ such that
\betagin{equation}\label{eq: lbd for c1}
\lambda=(e^{\frac{n}{k}s}+1)^{\frac{1}{n}}.
\end{equation}
Thus $\{\lambda=1\}$ corresponds to the zero section $Z_{n, k}=\{w=0\}$ on $-kH_{\mathbb{P}^{n-1}}$. We define $\omega_{\mathrm{eh}, n, k}$ by giving its potential function as follows:
\betagin{equation}\label{eq: poten c1}
f(\lambda):=k\lambda +k\int\frac{1}{\lambda^n-1}\;\textrm{d}\lambda,
\end{equation}
where the last term denotes an indefinite integral, and
\betagin{equation*}
\omega_{\mathrm{eh}, n, k}:=\sqrt{-1}\partial\bar{\partial}f(s).
\end{equation*}
$\omega_{\mathrm{eh}, n, k}$ is a Ricci-flat metric on $-kH_{\mathbb{P}^{n-1}}$ with edge singularity of angle $2n\pi/k$ along the zero section (See Theorem \ref{thm: metric limit nk case} for details). It can be seen as a generalization of Eguchi--Hanson metrics to higher dimensional manifolds (See Remark \ref{rmk: eh spec case} for details). More precisely, in local coordinates,
\betagin{align*}
\omega_{\mathrm{eh}, n, k}=k (e^{\frac{n}{k}s}+1)^{\frac{1}{n}} \pi_1^*\omega_{\operatorname{FS}}+\frac{e^{\frac{n}{k}s}}{k(e^{\frac{n}{k}s}+1)^{\frac{n-1}{n}}}&\left(\frac{\sqrt{-1}\textrm{d}w\wedge\textrm{d}\bar{w}}{|w|^2}+\sqrt{-1}\alphapha\wedge\bar{\alphapha}\right.+\\
&\left.\sqrt{-1}\alphapha\wedge\frac{\textrm{d}\bar{w}}{\bar{w}}+\sqrt{-1}\frac{\textrm{d}w}{w}\wedge\bar{\alphapha}\right),
\end{align*}
where
\betagin{equation*}
\alphapha=k\frac{\sum_{i\neq j}\bar{z}_i\textrm{d}z_i}{1+\sum_{i\neq j}|z_i|^2},\quad \textup{on the chart}\;\{z_j\neq 0\},
\end{equation*}
and
\betagin{equation*}
\pi_1([Z_1:\cdots:Z_n], w)=[Z_1:\cdots:Z_n]
\end{equation*}
is the projection map from the total space $-kH_{\mathbb{P}^{n-1}}$ to the base space $\mathbb{P}^{n-1}$.
\end{defi}
\betagin{rmk}\label{rmk: eh spec case}
Fixing $n=k=2$ in \eqref{eq: lbd for c1} and \eqref{eq: poten c1}, we obtain that on $-2H_{\mathbb{P}^1}$ the model metric $\omega_{\mathrm{eh}, n, k}$ has the expression
\betagin{equation}\label{eq: mod sim eh}
\sqrt{-1}\partial\bar{\partial}(2\lambda+\log(\lambda-1)-\log(\lambda+1)),\quad \lambda>1.
\end{equation}
Set
\betagin{equation}\label{eq: def r for eh}
r:=e^{\frac{1}{4}s},\quad r>0.
\end{equation}
Plugging \eqref{eq: lbd for c1} and \eqref{eq: def r for eh} in \eqref{eq: mod sim eh}, one finds $\omega_{\mathrm{eh}, n, k}$ has the following expression on $-2H_{\mathbb{P}^1}$:
\betagin{equation}\label{eq: der eh}
\sqrt{-1}\partial\bar{\partial}[\sqrt{r^4+1}+\log r^2-\log(\sqrt{r^4+1}+1)].
\end{equation}
\eqref{eq: der eh} coincides with \eqref{eq: eh metric potential form} in the Appendix where we put $\epsilon=1$. In other words, when $n=k=2$, $\omega_{\mathrm{eh}, n, k}$ recovers the famous Eguchi--Hanson metric \cite{EH79} that is constructed on the total space of $-2H_{\mathbb{P}^1}$.
It is recalled in Proposition \ref{prop: eh ricflat} that the Eguchi--Hanson metric is Ricci-flat. $\omega_{\mathrm{eh}, n, k}$, which is also Ricci-flat but with edge singularities along $Z_{n, k}$ when $n\neq k$, can be seen as a generalization of Eguchi--Hanson metrics to the total space of $-kH_{\mathbb{P}^{n-1}}$ for arbitrary $n, k\in\mathbb{N}_{>0}$.
\end{rmk}
\subsection{Compact case: model metrics on $\mathbb{P}^{n}(1,\dots, 1,k)$}
In this section, we introduce another model metric $\omega_{\mathrm{orb}, n, k}$ that is defined on the weighted projective space
$\mathbb{P}^{n}(1,\dots,1,k)$. We first recall the construction of the weighted projective space and its orbifold structure.
\betagin{defi}
For $k\in\mathbb{N}$, consider the group action of $\mathbb{C}^*$ on $\mathbb{C}^{n+1}\setminus\{0\}$ given by
\betagin{equation*}
\lambda \cdot (z_0,\dots,z_{n-1},z_n) = (\lambda z_0,\dots, \lambda z_{n-1},\lambda^k z_n),\quad \lambda\in\mathbb{C}^*,\; (z_0,\dots, z_{n-1}, z_n) \in \mathbb{C}^{n+1}\setminus \{0\}.
\end{equation*}
Then the weighted projective space $\mathbb{P}^{n}(1,\dots,1,k)$ is defined as the quotient of this group action:
\betagin{equation*}
\mathbb{P}^{n}(1,\dots,1,k) := (\mathbb{C}^{n+1}\setminus\{0\})/\mathbb{C}^*.
\end{equation*}
We use homogeneous coordinates on $\mathbb{P}^n(1,\dots, 1,k)$. A point $[x_0:\cdots:x_{n-1}:x_n]\in\mathbb{P}^n(1,\dots,1,k)$ corresponds to an equivalence class in $\mathbb{C}^{n+1}\setminus \{0\}$. More precisely,
\betagin{equation}\label{eq_PonWps}
[x_0:\cdots:x_{n-1}:x_n] := \{\lambda x_0,\dots, \lambda x_{n-1}, \lambda^k x_{n}: \lambda\in\mathbb{C}^*\}.
\end{equation}
\end{defi}
Consider the $\mathbb{Z}_k$ action on $\mathbb{C}^n$ such that the $\mathbb{Z}_k$ orbit of a point $(z_1,\dots, z_n)\in\mathbb{C}^n$ is
\betagin{equation*}
\{e^{\frac{2\pi\sqrt{-1}\ell}{k}}z_1,\dots, e^{\frac{2\pi\sqrt{-1}\ell}{k}}z_n: \ell = 0,\dots, k-1\}.
\end{equation*}
Then near the point $[0:0\cdots:0:1]\in\mathbb{P}^n(1,\dots, 1,k)$ the local structure is $\mathbb{C}^n/\mathbb{Z}_k$. The next lemma shows the relation between $\mathbb{P}^n(1,\dots,1,k)$ and $\mathbb{F}_{n, k}$.
\betagin{lem}\label{lem_WpsAndCh}
The total space of the line bundle $kH_{\mathbb{P}^{n-1}}$ can be embedded in the weighted projective space $\mathbb{P}^{n}(1,\dots,1,k)$ and the complement is a single point $p=[0:\cdots:0:1]\in\mathbb{P}^n(1,\dots,1,k)$. Moreover, $\mathbb{F}_{n, k}$ is the blow up of $\mathbb{P}^{n}(1,\dots,1,k)$ at $p$.
\end{lem}
\betagin{proof}
Since for any vector bundle $A$ and line bundle $L$ we have $\mathbb{P}(A\otimes L)=\mathbb{P}(A)$, it follows
that $\mathbb{F}_{n, -k}$ is biholomorphic to $\mathbb{F}_{n, k}$ by taking $L=2kH_{\mathbb{P}^{n-1}}$ in Definition \ref{defi: CH manifolds} with the biholomorphism exchanging the
zero and the infinity sections $Z_{n, k}$ and $Z_{n, -k}$. Thus, we identify $\mathbb{F}_{n, k}$ as
\betagin{equation}\label{eq_ExCh}
\mathbb{F}_{n, k} = \mathbb{P}(kH_{\mathbb{P}^{n-1}}\oplus \mathbb{C}_{\mathbb{P}^{n-1}}).
\end{equation}
We first show the total space of $kH_{\mathbb{P}^{n-1}}$ can be naturally embedded in $\mathbb{P}^n(1,\dots,1,k)$.
Consider $\mathbb{P}^{n}(1,\dots,1,k)$ with homogeneous coordinate $[x_0:\cdots, x_n]$ and embed $\mathbb{P}^{n-1}$ in $\mathbb{P}^n$ as $\{x_n=0\}$. Recall the transition function at a point $[x_0:\cdots:x_{n-1}]\in\mathbb{P}^{n-1}$ of the line bundle $kH_{\mathbb{P}^{n-1}}$ is given by
\betagin{equation}\label{eq_TranFunc}
g_{ij}([x_0:\cdots:x_{n-1}]) = \left(\frac{x_j}{x_i}\right)^k.
\end{equation}
Then the fiber of $kH_{\mathbb{P}^{n-1}}$ at an arbitrary point $[x_0:\cdots,x_{n-1}:0]\in\mathbb{P}^{n}(1,\dots,1,k)$ can be identified as the set of all $[x_0:\cdots:x_{n-1}:\lambda]$ for $\lambda\in\mathbb{C}$, where $[x_0:\cdots:x_{n-1}:\lambda]$ is defined in \eqref{eq_PonWps}. By definition of the weighted projective space and \eqref{eq_TranFunc}, this is well-defined. Thus, $kH_{\mathbb{P}^{n-1}}$ can be naturally embedded in $\mathbb{P}^n(1,\dots, 1,k)$ and the complement is the point $[0:\cdots:0:1]$. We have mentioned that the local structure of $\mathbb{P}^n(1,\dots, 1, k)$ near this point is $\mathbb{C}^n/\mathbb{Z}_k$.
Next, we blow up $\mathbb{P}^n(1,\dots,1,k)$ at $[0:\cdots:0:1]$. The exceptional divisor can be identified as adding an infinity section that is biholomorphic to $\mathbb{P}^{n-1}$ to the total space $kH_{\mathbb{P}^{n-1}}$. Combining this observation with \eqref{eq_ExCh}, one realizes the manifold upstairs is the Calabi--Hirzebruch manifold $\mathbb{F}_{n, k}$. We henceforth denote by $\pi$ the blow down map.
\end{proof}
Now we are ready to define $\omega_{\mathrm{orb},n,k}$. To do so, we first define another family of model metrics $\tilde{\omega}_{\mathrm{orb}, n, k}$ on $\mathbb{F}_{n, k}$. $\tilde{\omega}_{\mathrm{orb}, n, k}$ is a K\"{a}hlerE edge metric with Ricci curvature $(n+1)/k$ and an edge singularity of angle $2\pi/k$ along $Z_{n, -k}$ that degenerates on $Z_{n, k}$. In particular, this metric is smooth along $Z_{n, -k}$ when $k=1$ and collapses along $Z_{n, k}$. Indeed, when $k=1$ this metric coincides with the Fubini--Study metric on $\mathbb{P}^n$. We will then define $\omega_{\mathrm{orb},n,k}$ as the pull-back of the metric $\tilde{\omega}_{\mathrm{orb},n,k}$ under the blow up map.
\betagin{defi}\label{def_ModOnFnk}
Let $s$ be defined in \eqref{eq: def s}. To define $\tilde{\omega}_{\mathrm{orb}, n, k}$, introduce another coordinate $\nu\in(0, 1)$ such that
\betagin{equation*}
\nu=1-(e^{\frac{s}{k}}+1)^{-1}.
\end{equation*}
Note that $\{\nu=0\}$ and $\{\nu=1\}$ correspond to the zero section $Z_{n, k}$ and the infinity section $Z_{n, -k}$ respectively. We define $\tilde{\omega}_{\mathrm{orb}, n, k}$ by giving its potential function as follows:
\betagin{equation*}
g(\nu):=-k\log(1-\nu), \quad \nu\in(0, 1),
\end{equation*}
and
\betagin{equation*}
\tilde{\omega}_{\mathrm{orb}, n, k}:=\sqrt{-1}\partial\bar{\partial}g(s).
\end{equation*}
Note that $\tilde{\omega}_{\mathrm{orb}, n, k}$ is a degenerate K\"{a}hlerE edge metric on $\mathbb{F}_{n, k}$ with Ricci curvature $(n+1)/k$ and an edge singularity of the angle $2\pi/k$ along $Z_{n, -k}$, while $\tilde{\omega}_{\mathrm{orb}, n, k}$ collapses along $Z_{n, k}$ (See Theorem \ref{thm: conv mod2} for details). More precisely, in local coordinates,
\betagin{align*}
\tilde{\omega}_{\mathrm{orb}, n, k}=\frac{ke^{\frac{s}{k}}}{e^{\frac{s}{k}}+1}\pi_1^*\omega_{\operatorname{FS}}+\frac{e^{\frac{s}{k}}}{k(e^{\frac{s}{k}}+1)}&\left(\frac{\sqrt{-1}\textrm{d}w\wedge\textrm{d}\bar{w}}{|w|^2}+\sqrt{-1}\alphapha\wedge\bar{\alphapha}+\right.\\
&\left.\sqrt{-1}\alphapha\wedge\frac{\textrm{d}\bar{w}}{\bar{w}}+\sqrt{-1}\frac{\textrm{d}w}{w}\wedge\bar{\alphapha}\right),
\end{align*}
where
\betagin{equation*}
\alphapha=k\frac{\sum_{i\neq j}\bar{z}_i\textrm{d}z_i}{1+\sum_{i\neq j}|z_i|^2},\quad \textup{on the chart}\;\{z_j\neq 0\},
\end{equation*}
and
\betagin{equation*}
\pi_1([Z_1:\cdots:Z_n], w)=[Z_1:\cdots:Z_n]
\end{equation*}
is the projection map from $\mathbb{F}_{n, k}$ to the zero section $Z_{n, k}$ identified as $\mathbb{P}^{n-1}$.
\end{defi}
\betagin{defi}\label{def_ModOnWps}
For $k>1$, the model metric $\omega_{\mathrm{orb},n,k}$ on $\mathbb{P}^n(1,\dots,1,k)$ is defined away from $[0:\cdots:0:1]$ as the pull-back of the metric $\tilde{\omega}_{\mathrm{orb},n,k}$ on $\mathbb{F}_{n, k}\setminus\{Z_{n, k}\}$ under the blow up map. For $k=1$, we define $\omega_{\mathrm{orb},n,1}$ as the Fubini--Study metric on $\mathbb{P}^n(1,\dots,1,1)=\mathbb{P}^n$.
\end{defi}
\section{Constructions of K\"{a}hler--Einstein edge metrics }\label{sec: general}
In this section, we aim to construct K\"{a}hlerE edge metrics defined on the Calabi--Hirzebruch manifold $\mathbb{F}_{n, k}$ with edge singularities along the zero section $Z_{n, k}$ and the infinity section $Z_{n, -k}$ using Calabi ansatz. By K\"{a}hlerE edge metrics we mean K\"{a}hler edge currents that satisfy the K\"{a}hlerE equation on smooth locus. The reader may refer to \cite{Y14} for detailed exposition of K\"{a}hler edge metrics.
\subsection{Constructions of $\eta_{\betata_1}$ on $\mathbb{F}_{n, k}$}
Recall, the coordinate $s$ is defined in \eqref{eq: def s}, where $w$ and $z_j$, $j\neq i$ are coordinates we use when working on the chart $\{Z_i\neq 0\}\subset \mathbb{F}_{n, k}$. We seek a K\"{a}hler--Einstein edge metric
\betagin{equation*}
\eta:=\sqrt{-1}\partial\bar{\partial}f(s)
\end{equation*}
for some smooth function $f$ on $\mathbb{F}_{n, k}$. Define
\betagin{align}
\betagin{aligned}\label{eq: defi of tau and varphi nk case}
\tau(s):&=f'(s),\quad s\in(-\infty, +\infty)\\
\varphi(s):&=f''(s)=\tau'(s),\quad s\in(-\infty, +\infty).
\end{aligned}
\end{align}
Let $\pi_1$ and $\pi_2$ be projections from $\mathbb{F}_{n, k}$ to the zero section $Z_{n, k}$ and each fiber respectively, i.e.,
\betagin{align*}
\pi_1([Z_1,\cdots:Z_n], w)&=[Z_1,\cdots:Z_n],\\ \pi_2([Z_1,\cdots:Z_n], w)&=w.
\end{align*}
Denote by $\omega_{\operatorname{FS}}$ the Fubini--Study metric on $\mathbb{P}^{n-1}$ and define
\betagin{align*}
\omega_{\operatorname{Cyl}}&:=\frac{\sqrt{-1}\textrm{d}w\wedge\textrm{d}\bar{w}}{|w|^2},\\
\alphapha&:=\frac{k\sum_{j\neq i}\bar{z}_j\textrm{d}z_j}{1+\sum_{j\neq i}|z_j|^2},\quad \text{on the chart}\; \{Z_i\neq 0\}.
\end{align*}
Then direct calculation yields
\betagin{equation}\label{eq: eta in nk case}
\betagin{aligned}
\eta=k\tau\pi_1^*\omega_{\operatorname{FS}}+\varphi&\left(\pi_2^*\omega_{\operatorname{Cyl}}+\sqrt{-1}\alphapha\wedge\bar{\alphapha}+\right.\\
&\left.\sqrt{-1}\alphapha\wedge\frac{\textrm{d}\bar{w}}{\bar{w}}+\sqrt{-1}\frac{\textrm{d}w}{w}\wedge\bar{\alphapha}\right).
\end{aligned}
\end{equation}
Positive definiteness of $\eta$ then implies $f'\geq 0$ and $f''\geq 0$ for $s\in(-\infty, +\infty)$.
\betagin{prop}\label{prop: inf sup cond}
Assume $Z_{n, k}$ is non-collapsed, then $\inf_{s\in\mathbb{R}} \tau(s)>0$. Moreover, $\sup_{s\in\mathbb{R}} \tau(s)<+\infty$.
\end{prop}
\betagin{proof}
Assume by contradiction that $\inf_{s\in\mathbb{R}} \tau(s)=0$. Then by \eqref{eq: eta in nk case}, $\eta$ is identically zero when restricted to $Z_{n, k}$. However, we assume $Z_{n, k}$ is non-collapsed. Thus we must have $\inf_{s\in\mathbb{R}} \tau(s)>0$. To see $\sup_{s\in\mathbb{R}}\tau(s)<+\infty$, we follow the arguments in \cite[Lemma 3.2]{RZ21}. Indeed, restricting $\eta$ to the fiber $\{Z_j = 0, \forall j\neq i\}$, in \eqref{eq: eta in nk case} we get
\betagin{equation}
\betagin{aligned}\label{eq: restriction}
\eta &= \varphi \cdot \pi_2^* \omega_{\operatorname{Cyl}}\\
&= \varphi \frac{\sqrt{-1}\textrm{d}w\wedge\textrm{d}\bar{w}}{|w|^2},
\end{aligned}
\end{equation}
and also in this case $s = \log|w|^2$. Then we use the coordinate $w=e^{s/2+\sqrt{-1}\theta}$ on this fiber, and by \eqref{eq: restriction} $\eta$ restricted on the fiber gives a metric
\betagin{equation}\label{eq: metric res}
g_\eta = \frac{1}{2\varphi(\tau)} \textrm{d}\tau^2+2\varphi(\tau)\textrm{d}\theta^2.
\end{equation}
Thus, the volume form on the fiber is $\textrm{d}\tau\wedge \textrm{d}\theta$, and the volume of the fiber is $2\pi (\sup_{s\in\mathbb{R}}\tau(s)-\inf_{s\in\mathbb{R}}\tau(s))$. Since for any K\"{a}hler edge metric the volume of a complex submanifold is finite, we conclude $\sup_{s\in\mathbb{R}}\tau(s)<\infty$.
\end{proof}
By Proposition \ref{prop: inf sup cond}, we may rescale $f$ by a positive constant such that $\inf \tau=1$ and $\sup \tau=T<\infty$. Thus, we henceforth assume that $\tau$ ranges from $[1, T]$.
\betagin{prop}\label{prop: bd cond b1 case}
If $\eta$ is a K\"{a}hler edge metric with conic singularities along $Z_{n, k}$ and $Z_{n, -k}$, then
\betagin{align}
\betagin{aligned}\label{eq: bd condition nk case}
\varphi(1)&=0,\quad \frac{\textrm{d}\varphi}{\textrm{d}\tau}(1)=\betata_1,\\
\varphi(T)&=0,\quad \frac{\textrm{d}\varphi}{\textrm{d}\tau}(T)=-\betata_2,
\end{aligned}
\end{align}
if we denote by $2\pi\betata_1$ and $2\pi\betata_2$ respectively the angles of $\eta$ along $Z_{n, k}$ and $Z_{n, -k}$.
\end{prop}
\betagin{proof}
Recall we assume $\tau$ ranges from $[1, T]$. In particular, this implies that
\betagin{equation*}
\lim_{s\to \pm \infty}\frac{\textrm{d}\tau}{\textrm{d}s}=0,
\end{equation*}
which combines with \eqref{eq: defi of tau and varphi nk case} show that
\betagin{equation*}
\varphi(1)=\varphi(T)=0.
\end{equation*}
Next, we follow the arguments in the proof of \cite[Proposition 3.3]{RZ21}. Indeed, it follows from \cite[Theorem 1, proposition 4.4]{JMR16} that the potential function $f$ of $\eta$ has complete asymptotic expansions near both $w=0$ and $w=\infty$. Near $w=0$, the leading term in the expansion is $|w|^{2\betata_1}$. More precisely, by \eqref{eq: def s},
\betagin{equation*}
\betagin{aligned}
\varphi &\sim C_1 + C_2|w|^{2\betata_1}+(C_3\sin\theta+C_4\cos\theta)|w|^2+O(|w|^{2+\epsilon})\\
&= C_1 + C_2 e^{\betata_1 s}+(C_3\sin\theta+C_4\cos\theta)e^s+O(e^{(1+\epsilon)s}).
\end{aligned}
\end{equation*}
We first find $C_1=0$ by the fact that $\varphi(1)=\varphi(T)=0$ in \eqref{eq: bd condition nk case}. Moreover, the expansion can be differentiated term-by-term as $|w|\to 0$ or $s\to -\infty$. As $\varphi'(\tau)=\frac{\partial \varphi}{\partial s}\frac{\textrm{d}s}{\textrm{d}\tau}=\frac{\partial \varphi}{\partial s}/\varphi$, we obtain
\betagin{equation*}
\varphi'(1) = \betata_1.
\end{equation*}
The same arguments imply that
\betagin{equation*}
\varphi'(T) = -\betata_2,
\end{equation*}
where the minus sign comes from the fact that the leading term in this expansion is $|w|^{-2\betata_2}=e^{-\betata_2 s}$.
\end{proof}
\betagin{defi}\label{defi: KEE}
A K\"{a}hler--Einstein edge metric $\omega$ is a K\"{a}hler edge metric that has constant Ricci curvature away from the singular locus.
\end{defi}
By Definition \ref{defi: KEE}, $\eta$ is a K\"{a}hlerE edge metric if and only if it satisfies the following K\"{a}hlerE edge equation:
\betagin{equation}\label{eq: eta kee}
\operatorname{Ric}\eta=\lambda\eta+(1-\betata_1)[Z_{n, k}]+(1-\betata_2)[Z_{n, -k}],
\end{equation}
where by $\lambda$ we denote the Ricci curvature.
The next proposition shows that \eqref{eq: eta kee} is equivalent to an ODE satisfied by $\tau$ and $\varphi$ with boundary conditions given in \eqref{eq: bd condition nk case}.
\betagin{prop}\label{prop: red to ode}
The K\"{a}hler\; edge metric $\eta$ given in \eqref{eq: eta in nk case} satisfies the K\"{a}hlerE edge equation \eqref{eq: eta kee} if and only if
\betagin{align}
&\lambda=\frac{n}{k}-\betata_1,\quad \betata_1\in\left(0, \frac{n}{k}\right), \label{eq: ric curv gene}\\
&\varphi(\tau)=\frac{1}{k}\frac{\tau^n-1}{\tau^{n-1}}+\frac{1}{n+1}(\betata_1-\frac{n}{k})\frac{\tau^{n+1}-1}{\tau^{n-1}}, \quad \tau\geq 1.\label{eq: varphi general expression nk case}
\end{align}
Moreover, $\betata_2$ and $T$ are determined by $\betata_1$ such that $T>1$ and $\betata_2 > 0$.
\end{prop}
\betagin{proof}
By \eqref{eq: eta in nk case}, we calculate $\operatorname{Ric}\eta$:
\betagin{align}
\betagin{aligned}\label{eq: ric eta nk case}
\operatorname{Ric}\eta&=-\sqrt{-1}\partial\bar{\partial}\log \eta^n\\
&=-\sqrt{-1}\partial\bar{\partial} \log \left(k^{n-1}\tau^{n-1}\varphi(\pi_1^* \omega_{\operatorname{FS}})^{n-1}\wedge\pi_2^*\omega_{\operatorname{Cyl}} \right)\\
&=(1-\betata_1)[Z_{n, k}]+(1-\betata_2)[Z_{n, -k}]+\left(n-k(n-1)\frac{\varphi}{\tau}-k\frac{\textrm{d}\varphi}{\textrm{d}\tau}\right)\pi_1^*\omega_{\operatorname{FS}}\\
&-\varphi \frac{\textrm{d}}{\textrm{d}\tau}\left((n-1)\frac{\varphi}{\tau}+\frac{\textrm{d}\varphi}{\textrm{d}\tau} \right)\left(\pi_2^*\omega_{\operatorname{Cyl}}+\sqrt{-1}\alphapha\wedge\bar{\alphapha}+\right.\\
&\left.\sqrt{-1}\alphapha\wedge\frac{\textrm{d}\bar{w}}{\bar{w}}+\sqrt{-1}\frac{\textrm{d}w}{w}\wedge\bar{\alphapha} \right).
\end{aligned}
\end{align}
Plugging \eqref{eq: eta in nk case} and \eqref{eq: ric eta nk case} in \eqref{eq: eta kee}, \eqref{eq: eta kee} is equivalent to
\betagin{align}
n-k(n-1)\frac{\varphi}{\tau}-k\frac{\textrm{d}\varphi}{\textrm{d}\tau}&=\lambda k \tau,\label{eq: l1 ode}\\
-\varphi\frac{\textrm{d}}{\textrm{d}\tau}((n-1)\frac{\varphi}{\tau}+\frac{\textrm{d}\varphi}{\textrm{d}\tau})&=\lambda \varphi.\label{eq: l2 ode}
\end{align}
Now we first observe that \eqref{eq: l1 ode} and \eqref{eq: l2 ode} are equivalent since taking derivative of \eqref{eq: l1 ode} with respect to $\tau$ gives \eqref{eq: l2 ode}. Thus we conclude \eqref{eq: eta kee} is equivalent to the ODE \eqref{eq: l1 ode} together with boundary conditions \eqref{eq: bd condition nk case}. Solving this ODE with boundary conditions gives \eqref{eq: ric curv gene} and \eqref{eq: varphi general expression nk case}. We need the assumption $\betata_1\in(0, n/k)$ to ensure the existence of $T$ and $\betata_2$ that satisfy \eqref{eq: bd condition nk case}. Indeed, $\betata_2$ and $T$ are determined by $\betata_1$ due to \eqref{eq: bd condition nk case} and \eqref{eq: varphi general expression nk case}. More precisely, $T$ should be a root of the polynomial that appears in the right hand side of $\eqref{eq: varphi general expression nk case}$ and $-\betata_2$ should be determined by the derivative $\textrm{d}\varphi/\textrm{d}\tau$ at $T$. Writing \eqref{eq: varphi general expression nk case} as $\varphi(\tau)=P(\tau)/\tau^{n-1}$, we factor $P(\tau)$ as
\betagin{equation}\label{eq: polyforT}
\betagin{aligned}
P(\tau) &= (\tau-1)\left[\frac{1}{k}(\tau^{n-1}+\cdots+1)+\frac{1}{n+1}\left(\betata_1-\frac{n}{k}\right)(\tau^n+\cdots+1)
\right]\\
&= \frac{(\tau-1)}{n+1}\left[
\left(\betata_1-\frac{n}{k}\right)\tau^n+\left(\frac{1}{k}+\betata_1\right)(\tau^{n-1}+\cdots +1)
\right].
\end{aligned}
\end{equation}
Under the assumption $\betata_1\in(0, n/k)$, one finds $P(\tau)>0$ for $1<\tau\ll 2$ and $P(\tau)<0$ for $\tau\to \infty$. Thus, $P(\tau)$ has at least one real root that is greater than $1$. Denote by $T$
the first root of $P$ after $1$. In particular, $\textrm{d}\varphi/\textrm{d}\tau(T)<0$. By \eqref{eq: bd condition nk case}, this implies that $\betata_2>0$ as claimed.
\end{proof}
By \eqref{eq: defi of tau and varphi nk case}, there holds
\betagin{equation}\label{eq: der sty}
\frac{\textrm{d}s}{\textrm{d}\tau}=\frac{1}{\varphi(\tau)}.
\end{equation}
Combining \eqref{eq: eta in nk case}, \eqref{eq: der sty} and Proposition \ref{prop: red to ode} altogether, we realize that given $\betata_1\in(0, n/k)$ we can construct a K\"{a}hlerE edge metric $\eta$ on $\mathbb{F}_{n, k}$ using coordinates $\tau$ and $\varphi$. This K\"{a}hlerE edge metric has Ricci curvature $\lambda=n/k-\betata_1$, and has edge singularities of angle $2\pi\betata_1$ along $Z_{n, k}$ and angle $2\pi\betata_2$ along $Z_{n, -k}$. Note that $\betata_2$ is determined by $\betata_1$. In other words, we can construct a family of K\"{a}hlerE edge metrics on $\mathbb{F}_{n, k}$ parametrized by $\betata_1\in (0, n/k)$. In Section \ref{sec: asym of kee}, we study the asymptotic behavior of this family of metrics when $\betata_1$ approaches the two extremes: $n/k$ or $0$.
Recall, in the construction of $\eta$ we rescale the metric such that $\tau$ ranges from $[1, T]$. Note that $T$ is determined by $\betata_1$. By Proposition \ref{prop: inf sup cond}, we may also rescale the metric such that $\tau$ ranges from $[t, 1]$ for some $0<t<1$. In such a way, we construct another family of K\"{a}hlerE edge metrics on $\mathbb{F}_{n, k}$ parametrized by $\betata_2$ in the remainder of this section. Such metrics can be seen as obtained after renormalizing metrics in the family that is parametrized by $\betata_1$. However, when studying their asymptotic behaviors they give rise to different limit metric.
\subsection{Constructions of $\xi_{\betata_2}$ on $\mathbb{F}_{n, k}$}
Now consider a change of coordinate $u:=1/w$. Then \eqref{eq: def s} can be written as
\betagin{equation*}
s=-\log|u|^2+k\log\left(1+\sum_{j\neq i}^n |z_j|^2 \right).
\end{equation*}
We still denote by $f(s)$ a smooth function on $\mathbb{F}_{n, k}$ and seek K\"{a}hlerE edge metrics that have the form $\sqrt{-1}\partial\bar{\partial} f(s)$. Recall $\tau(s)$ and $\varphi(s)$ are defined in \eqref{eq: defi of tau and varphi nk case}. Let
\betagin{align}
\xi:&=\sqrt{-1}\partial\bar{\partial}f(s)\notag\\
&\betagin{aligned}
&=k\tau\pi_1^*\omega_{\operatorname{FS}}+\varphi\left(\pi_2^*\omega_{\operatorname{Cyl}}+\sqrt{-1}\alphapha\wedge\bar{\alphapha}-\right.\\
&\left.\sqrt{-1}\alphapha\wedge\frac{\textrm{d}\bar{u}}{\bar{u}}-\sqrt{-1}\frac{\textrm{d}u}{u}\wedge\bar{\alphapha} \right),\label{eq: xi metric}
\end{aligned}
\end{align}
where $\pi_1$, $\pi_2$ and $\alphapha$ are the same as those in \eqref{eq: eta in nk case}. Recall by Proposition \ref{prop: inf sup cond},
\betagin{equation*}
0<\inf_{s\in\mathbb{R}}\tau(s)<\sup_{s\in\mathbb{R}}\tau(s)<+\infty.
\end{equation*}
We rescale the potential function $f$ such that for some $t>0$,
\betagin{equation}\label{eq: new iod tau}
\sup_{s\in\mathbb{R}} \tau(s)=1,\quad \inf_{s\in\mathbb{R}} \tau(s)=t.
\end{equation}
Assume $\xi$ is a K\"{a}hler\;edge metric on $\mathbb{F}_{n, k}$. Then, under the renormalization given by \eqref{eq: new iod tau}, Proposition \ref{prop: bd cond b1 case} and Proposition \ref{prop: red to ode}, which respectively describe the boundary conditions for $\varphi$ and the ODE satisfied by $\varphi$, translate to the following two Propositions.
\betagin{prop}\label{prop: bd cond b2 case}
Assume $\xi$ has edge singularities of angle $2\pi\betata_1$ and $2\pi\betata_2$ respectively along $Z_{n, k}$ and $Z_{n, -k}$. Recall $\varphi$ in \eqref{eq: xi metric}. Then,
\betagin{align}
\betagin{aligned}\label{eq: bd cond nk b2 case}
\varphi(t)&=0,\quad \frac{\textrm{d}\varphi}{\textrm{d}\tau}(t)=\betata_1,\\
\varphi(1)&=0,\quad \frac{\textrm{d}\varphi}{\textrm{d}\tau}(1)=-\betata_2.
\end{aligned}
\end{align}
\end{prop}
\betagin{prop}\label{prop: red to ode b2 case}
Under the same assumptions in Proposition \ref{prop: bd cond b2 case}, the K\"{a}hler\;edge metric $\xi$ satisfies the K\"{a}hlerE edge equation if and only if
\betagin{align}
&\mu=\frac{n}{k}+\betata_2,\quad \betata_2\in\left(0, \frac{1}{k}\right),\label{eq: beta2range}\\
&\varphi(\tau)=\frac{1}{k}\frac{\tau^n -1}{\tau^{n-1}}-\frac{1}{n+1}\left(\frac{n}{k}+\betata_2\right)\frac{\tau^{n+1}-1}{\tau^{n-1}},\quad \tau\in(t, 1),\notag
\end{align}
where by $\mu$ we denote the Ricci curvature of $\xi$. Moreover, $\betata_1$ and $t$ are determined by $\betata_2$.
\end{prop}
By Propositions \ref{prop: bd cond b2 case} and \ref{prop: red to ode b2 case}, we realize that given $\betata_2$ we can construct a family of K\"{a}hlerE edge metrics on $\mathbb{F}_{n, k}$ parametrized by $\betata_2$ such that $\betata_1$ and $t$ are determined by $\betata_2$. This family of metrics can be obtained by renormalizing the family of metrics parametrized by $\betata_1$.
\section{Angle asymptotics
}\label{sec: asym of kee}
In this section, we study the asymptotic behaviors of K\"{a}hlerE edge metrics $\eta$ in \eqref{eq: eta in nk case} (respectively, $\xi$ in \eqref{eq: xi metric}) as $\betata_1$ (respectively, $\betata_2$) approaches either of its extremes.
We first study the limit behavior of $\eta$ when $\betata_1\nearrow n/k$. It is enough to study the limiting behavior of $\varphi$ and $\tau$ thanks to Proposition \ref{prop: red to ode}.
\betagin{prop}\label{prop: T infty nk case}
When $\betata_1$ is close to $n/k$, we have $T>1$ and $T\to \infty$ as $\betata_1\nearrow n/k$.
\end{prop}
\betagin{proof}
Recall by \eqref{eq: bd condition nk case}, $1$ and $T$ are both roots of $\varphi(\tau)$. By Proposition \ref{prop: red to ode}, $T>1$. To prove $T\to\infty$ as $\betata_1\nearrow n/k$, we write \eqref{eq: varphi general expression nk case} as \betagin{equation}\label{eq: varphi roots expre}
\varphi(\tau)=\frac{1}{\tau^{n-1}}\cdot\frac{1}{n+1}(\betata_1-\frac{n}{k})(\tau-1)(\tau-\alphapha_1)\cdots(\tau-\alphapha_{n-1})(\tau-T),
\end{equation}
where $\alphapha_1, \dots, \alphapha_{n-1}\in\mathbb{C}$. Comparing \eqref{eq: varphi general expression nk case} to \eqref{eq: varphi roots expre}, we have
\betagin{equation*}
(\tau-\alphapha_1)\cdots(\tau-\alphapha_{n-1})(\tau-T)=\tau^n+\frac{1+k\betata_1}{k\betata_1-n}(\tau^{n-1}+\cdots+1).
\end{equation*}
By Vieta's formulas,
\betagin{align}
(-1)^n\alphapha_1\cdots\alphapha_{n-1}\cdot T&=\frac{1+k\betata_1}{k\betata_1-n},\label{eq: product}\\
\alphapha_1+\cdots+\alphapha_{n-1}+T&=\frac{1+k\betata_1}{n-k\betata_1}.\label{eq: sum}
\end{align}
By \eqref{eq: varphi general expression nk case}, $\varphi(\tau)$ has $n+1$ (possibly complex) roots. Since
\betagin{equation*}
\varphi(\tau)\to \frac{1}{k}\frac{\tau^n-1}{\tau^{n-1}},\quad\;\textrm{as}\;\betata_1\nearrow n/k,
\end{equation*}
we conclude that the $n$ roots of $\varphi$ except $T$ converge to the $n$th root of unity as $\betata_1\nearrow n/k$. In particular, $|\alphapha_1|,\dots, |\alphapha_{n-1}|$ converge to $1$ as $\betata_1\nearrow n/k$. However, by \eqref{eq: product} we see
\betagin{equation*}
|\alphapha_1|\cdots|\alphapha_{n-1}||T|\to +\infty,\quad\textrm{as}\;\betata_1\nearrow n/k.
\end{equation*}
Thus we must have $T\to +\infty$ as $\betata_1\nearrow n/k$.
\end{proof}
\betagin{prop}\label{prop: length to infty nk case}
As $\betata_1\nearrow n/k$, the length of the path on each fiber between the intersection point of the fiber with $Z_{n, k}$ and that of the fiber with $Z_{n, -k}$ tends to infinity. In other words, $Z_{n, -k}$ gets pushed--off to infinity as $\betata_1\nearrow n/k$ if we fix a base point on $Z_{n, k}$.
\end{prop}
\betagin{proof}
Restricted to each fiber, by \eqref{eq: metric res}, $\eta=\varphi \sqrt{-1}\textrm{d}w\wedge\textrm{d}\bar{w}/|w|^2$ gives a metric
\betagin{equation*}
g=\frac{1}{2\varphi(\tau)}\textrm{d}\tau^2+2\varphi(\tau)\textrm{d}\theta^2.
\end{equation*}
Up to some constant, the distance between $\{\tau=1\}$ and $\{\tau=T\}$ is given by
\betagin{equation}\label{eq: length int nk case}
\betagin{aligned}
&\int_1^T \frac{1}{\sqrt{\varphi(\tau)}}\;\textrm{d}\tau=\\
&\int_1^T \frac{\tau^{\frac{n-1}{2}}}{\sqrt{\frac{n/k-\betata_1}{n+1}}\cdot\sqrt{(\tau-1)(\tau-\alphapha_1)\cdots(\tau-\alphapha_{n-1})(T-\tau)}}\;\textrm{d}\tau.
\end{aligned}
\end{equation}
Recall in the proof of Proposition \ref{prop: T infty nk case}, we have shown
\betagin{equation}\label{eq: T asm behav}
T\sim \frac{1+k\betata_1}{n-k\betata_1}\sim\frac{1+n}{k}\cdot\frac{1}{n/k-\betata_1},\quad \textrm{as}\;\betata_1\nearrow n/k.
\end{equation}
For any fixed $\epsilon>0$, consider
\betagin{equation*}
\int_{T-\epsilon}^T \frac{\tau^{\frac{n-1}{2}}}{\sqrt{\frac{n/k-\betata_1}{n+1}}\cdot\sqrt{(\tau-1)(\tau-\alphapha_1)\cdots(\tau-\alphapha_{n-1})(T-\tau)}}\;\textrm{d}\tau.
\end{equation*}
We can find $\betata_1$ close to $n/k$ such that
\betagin{equation*}
T-\epsilon>\frac{1}{2}T.
\end{equation*}
Then in $[T-\epsilon, T]$, we have
\betagin{align}
\betagin{aligned}\label{eq: items in length int}
&\tau^{\frac{n-1}{2}}>(T-\epsilon)^{\frac{n-1}{2}}>\left(\frac{1}{2}T\right)^{\frac{n-1}{2}},\\
&\sqrt{(\tau-1)(\tau-\alphapha_1)\cdots(
\tau-\alphapha_{n-1})}<\sqrt{2(\tau^n-1)}<\sqrt{2T^n}.
\end{aligned}
\end{align}
By \eqref{eq: T asm behav} and \eqref{eq: items in length int}, there exists a constant $C>0$ that is independent of $\epsilon$ such that
\betagin{align*}
&\int_{T-\epsilon}^T \frac{\tau^{\frac{n-1}{2}}}{\sqrt{\frac{n/k-\betata_1}{n+1}}\cdot\sqrt{(\tau-1)(\tau-\alphapha_1)\cdots(\tau-\alphapha_{n-1})(T-\tau)}}\;\textrm{d}\tau\\
&>C\cdot \frac{T^{\frac{n-1}{2}}}{\sqrt{n/k-\betata_1}\cdot T^{\frac{n}{2}}} \int_{T-\epsilon}^T \frac{1}{\sqrt{T-\tau}}\;\textrm{d}\tau\\
&>C\epsilon.
\end{align*}
Since $\epsilon$ is arbitrarily chosen, the integral in \eqref{eq: length int nk case} diverges as $\betata_1\nearrow n/k$. Thus, we have shown $Z_{n, -k}$ gets pushed-off to infinity as $\betata_1\nearrow n/k$ if we choose a base point on $Z_{n, k}$.
\end{proof}
\betagin{prop}\label{prop: b2 limit nk case}
$\displaystyle\lim_{\betata_1\nearrow \frac{n}{k}} \betata_2(\betata_1)=\frac{1}{k}$.
\end{prop}
\betagin{proof}
We calculate
\betagin{equation*}
\frac{\textrm{d}\varphi}{\textrm{d}\tau}=\frac{1}{k}\frac{\tau^{n-2}}{(\tau^{n-1})^2}(n+\tau^n-1)+\frac{1}{n+1}(\betata_1-\frac{n}{k})\frac{\tau^{n-2}}{(\tau^{n-1})^2}(2\tau^{n+1}+n-1).
\end{equation*}
By $\varphi_\tau(T)=-\betata_2$ we have
\betagin{equation}\label{eq: phi T general}
\frac{1}{k T^n}(n+T^n-1)+\frac{1}{n+1}(\betata_1-\frac{n}{k})\frac{1}{T^n}(2T^{n+1}+n-1)=-\betata_2.
\end{equation}
Recall $\varphi(T)=0$, then we have
\betagin{equation}\label{eq: phi T zero general}
\frac{T^n-1}{k}+\frac{1}{n+1}(\betata_1-\frac{n}{k})(T^{n+1}-1)=0.
\end{equation}
Combining \eqref{eq: phi T general} and \eqref{eq: phi T zero general},
\betagin{equation}
\label{Tbeta1beta2Eq}
\Big(\frac{1}{k}-\betata_2\Big)T^n=\betata_1+\frac{1}{k}.
\end{equation}
Since $T\to \infty$ as $\betata_1\nearrow n/k$, there holds $\betata_2\to1/k$ as $\betata_1\nearrow n/k$.
\end{proof}
In Section \ref{sec: gh kee}, we use asymptotic behaviors discussed above to study the limit metric of $\eta$ as $\betata_1$ approaches $n/k$ or $0$. In the remainder of this section, we focus on the family of metrics $\xi$ that is parametrized by $\betata_2$. Inspired by Proposition \ref{prop: b2 limit nk case}, we study the asymptotic behaviors of $\xi$ as $\betata_2\nearrow 1/k$.
\betagin{prop}
\label{PropBeta2Beta1LimitAsymp}
When $\betata_2 \nearrow \frac{1}{k}$, as an analogue of \eqref{Tbeta1beta2Eq}, there holds
\betagin{equation*}
\left(\frac{1}{k}+\betata_1\right)t^n=\frac{1}{k}-\betata_2.
\end{equation*}
\end{prop}
\betagin{proof}
Let $\varphi(\tau)$ and $t$ be as in \eqref{eq: bd cond nk b2 case} and \eqref{eq: beta2range}. We calculate
\betagin{equation*}
\frac{\textrm{d} \varphi}{\textrm{d}\tau} = \frac{1}{k}\frac{\tau^{n-2}}{(\tau^{n-1})^2}(\tau^n+n-1)-\frac{1}{n+1}\left(\frac{n}{k}+\betata_2\right)\frac{\tau^{n-2}}{(\tau^{n-1})^2}(2\tau^{n+1}+n-1).
\end{equation*}
By the fact that $\textrm{d}\varphi/\textrm{d}\tau(t)=\betata_1$,
\betagin{equation*}
\betata_1 = \frac{1}{k t^n}(n+t^n-1)-\frac{1}{n+1}\left(\frac{n}{k}+\betata_2\right)\frac{1}{t^n}(2t^{n+1}+n-1).
\end{equation*}
Combining this with the fact that $\varphi(t)=0$, there holds
\betagin{equation*}
\betagin{aligned}
\betata_1 t^n &= \frac{1}{k}(n+t^n-1)-\frac{2(t^n-1)}{k}-\frac{n}{k}-\betata_2\\
& = -\frac{1}{k}t^n+\frac{1}{k}-\betata_2,
\end{aligned}
\end{equation*}
which implies
\betagin{equation*}
\left(\betata_1+\frac{1}{k}\right)t^n=\frac{1}{k}-\betata_2
\end{equation*}
as claimed.
\end{proof}
\betagin{prop}\label{prop: length finite nkb2 case}
As $\displaystyle \betata_2\nearrow \frac{1}{k}$, $\displaystyle t = O\left(\left(\frac{k}{n+1}\right)^{\frac{1}{n}}\left(\frac{1}{k}-\betata_2\right)^{\frac{1}{n}}\right)$. In particular, $t$ tends to $0$ when $\betata_2\nearrow 1/k$. Moreover, the length of the path on each fiber between the intersection point of the fiber with $Z_{n, k}$ and that of the fiber with $Z_{n, -k}$ converges to a finite number as $\betata_2\nearrow 1/k$.
\end{prop}
\betagin{proof}
Recall $t$ is a root of
\betagin{equation*}
\varphi(\tau)=\frac{1}{k}\frac{\tau^n -1}{\tau^{n-1}}-\frac{1}{n+1}\left(\frac{n}{k}+\betata_2\right)\frac{\tau^{n+1}-1}{\tau^{n-1}}=0.
\end{equation*}
Direct calculations yield
\betagin{equation*}
\frac{\varphi(\tau)}{\tau-1}=\frac{1}{\tau^{n-1}}\left(-\frac{1}{n+1}\left(\frac{n}{k}+\betata_2\right)\tau^n+\frac{1/k-\betata_2}{n+1}(\tau^{n-1}+\cdots+\tau+1)\right).
\end{equation*}
Then it is easy to see that every root of $\varphi(\tau)$ except for $\tau=1$, denoted by $\alphapha$, satisfies that $|\alphapha| = O((k/(n+1))^{1/n}(1/k-\betata_2)^{1/n})$ when $\betata_2\nearrow 1/k$. In particular, as $\betata_2\nearrow 1/k$, all roots of $\varphi(\tau)=0$ except for $\tau=1$ converge to $0$. We conclude that $t\to 0$.
To see that the length between $Z_{n, k}$ and $Z_{n, -k}$ converges to a finite number in the limit, one follows the arguments in the proof of Proposition \ref{prop: length to infty nk case} and Proposition \ref{prop: length finite k1 case}.
\end{proof}
\section{Large angle limits}\label{sec: gh kee}
In this section, we first study the Gromov--Hausdorff limit of the family of metrics $\eta$ when the parameter $\betata_1$ approaches $n/k$.
\betagin{thm}\label{thm: metric limit nk case}
Fix an arbitrary base point $p$ on $Z_{n, k}$. As $\betata_1\nearrow n/k$, the K\"{a}hler--Einstein edge metric $(\mathbb{F}_{n, k}, \eta_{\betata_1}, p)$ converges in the pointed Gromov--Hausdorff sense to a Ricci-flat K\"{a}hler edge metric $(-kH_{\mathbb{P}^{n-1}}, \eta_\infty, p)$ described in \eqref{eq: limit eta nk case}. This limit metric coincides with the model metric $\omega_{\mathrm{eh}, n, k}$ defined in Section \ref{sec: model}.
\end{thm}
\betagin{proof}
We use notation $\eta_{\betata_1}, \tau(\betata_1)$ and $\varphi(\betata_1)$ to emphasize the dependence of metrics and coordinates on $\betata_1$. Combining \eqref{eq: defi of tau and varphi nk case} and \eqref{eq: varphi general expression nk case}, we have
\betagin{equation}\label{eq: relation s and tau}
\frac{\textrm{d}s}{\textrm{d}\tau(\betata_1)}=\frac{1}{\varphi(\betata_1)}=\frac{\tau(\betata_1)^{n-1}}{\frac{1}{k}(\tau(\betata_1)^n-1)+\frac{\betata_1-n/k}{n+1}(\tau(\betata_1)^{n+1}-1)}.
\end{equation}
\betagin{claim}\label{clm: tpinfty}
The pointwise limits of functions
\betagin{align*}
\tau_\infty(s):&=\lim_{\betata_1\nearrow n/k}\tau(\betata_1, s),\quad s\in(-\infty, +\infty),\\
\varphi_\infty(s):&=\lim_{\betata_1\nearrow n/k}\varphi({\betata_1}, s),\quad s\in(-\infty, +\infty)
\end{align*}
exist. Moreover, $\tau_\infty(s)$ and $\varphi_\infty(s)$ are smooth in $s$.
\end{claim}
\betagin{proof}[Proof of the Claim]
By the continuous dependence of ODEs in \eqref{eq: relation s and tau} on the parameter $\betata_1$, the pointwise limit function $\tau_\infty$ exists. Since $\tau(\betata_1, s)$ is smooth in $s$ for each $\betata_1$, $\tau_\infty(s)$ is smooth in $s$. By \eqref{eq: varphi general expression nk case} and the existence of $\tau_\infty(s)$, $\varphi_\infty(s)$ also exists. Moreover, $\varphi_\infty(s)$ is smooth in $s$ due to the smoothness of $\tau_\infty(s)$.
\end{proof}
Thanks to Claim \ref{clm: tpinfty}, $\tau_\infty$ and $\varphi_\infty$ satisfy
\betagin{align}
\betagin{aligned}\label{eq: tau infty and varphi infty}
\frac{\textrm{d}s}{\textrm{d}\tau_{\infty
}}&=\frac{\tau^{n-1}_\infty}{\frac{1}{k}(\tau_\infty^n-1)},\\
\varphi_\infty(\tau_\infty)&=\frac{1}{k}\frac{\tau^n_\infty-1}{\tau^{n-1}_\infty}.
\end{aligned}
\end{align}
Solving the first equation in \eqref{eq: tau infty and varphi infty}, we get
\betagin{equation}\label{eq: taui ex formula}
\tau_\infty(s) = (1+e^{(s-C)\frac{n}{k}})^{1/n}, \quad \text{for some constant}\; C.
\end{equation}
By considering a change of coordinate $w' = C' w$ for some appropriate $C'$ in \eqref{eq: def s}, we may choose $C$ in \eqref{eq: taui ex formula} to be $0$. Plugging \eqref{eq: taui ex formula} into the second equation in \eqref{eq: tau infty and varphi infty}, one finds
\betagin{equation}\label{eq: varphiinf ex for}
\varphi_\infty(s)=\frac{1}{k}\cdot \frac{e^{s\cdot\frac{n}{k}}}{(e^{s\cdot\frac{n}{k}}+1)^{\frac{n-1}{n}}}.
\end{equation}
Plugging \eqref{eq: taui ex formula} and \eqref{eq: varphiinf ex for} into
\eqref{eq: eta in nk case}, we obtain the convergence of K\"{a}hler--Einstein edge metric $\eta_{\betata_1}$ on any compact subsets of $\mathbb{F}_{n, k}$ in every $C^k$-norm to the following metric:
\betagin{align}
\betagin{aligned}\label{eq: limit eta nk case}
\eta_\infty:&=\lim_{\betata_1\nearrow n/k}\eta_{\betata_1}\\
&=
k (1+e^{s\cdot\frac{n}{k}})^{1/n} \pi_1^*\omega_{\operatorname{FS}} +
\frac{1}{k}\cdot \frac{e^{s\cdot\frac{n}{k}}}{(e^{s\cdot\frac{n}{k}}+1)^{\frac{n-1}{n}}} \vphantom{\frac{\textrm{d} w}{w}}\left(\pi_2^*\omega_{\operatorname{Cyl}}+\sqrt{-1}\alphapha\wedge\bar{\alphapha}\right.\\
&\left.+\sqrt{-1}\alphapha\wedge\frac{\textrm{d}\bar{w}}{\bar{w}}+\sqrt{-1}\frac{\textrm{d}w}{w}\wedge\bar{\alphapha}
\right).
\end{aligned}
\end{align}
Recall by \eqref{eq: ric curv gene}, the Ricci curvature of $\eta_{\betata_1}$ is given by $\lambda_{\betata_1}=n/k-\betata_1$, which converges to $0$ as $\betata_1 \nearrow n/k$. Thus, $\eta_\infty$ is a Ricci-flat K\"{a}hler edge metric on $-kH_{\mathbb{P}^{n-1}}$. $\eta_\infty$ has edge singularity of angle $2n\pi/k$ along $Z_{n, k}\subset -kH_{\mathbb{P}^{n-1}}$. Indeed, $\eta_\infty$ coincides with the model metric $\omega_{\mathrm{eh}, n, k}$ defined in Section \ref{sec: model}. To obtain the convergence in the pointed Gromov--Hausdorff sense, we first recall by Proposition \ref{prop: length to infty nk case} the distance between $Z_{n, -k}$ and $Z_{n, k}$ tends to infinity as $\betata_1 \nearrow n/k$. Once we choose a base point on $Z_{n, k}$. Since $\eta_{\betata_1}$ converges to $\eta_\infty$ on any compact geodesic balls centered at the base point, we conclude that $\eta_{\betata_1}$ converges in the pointed Gromov--Hausdorff sense to $\eta_\infty$ on $-kH_{\mathbb{P}^{n-1}}$.
\end{proof}
\betagin{rmk}\label{rmk: eh as limit}
If we let $n=k=2$ in Theorem \ref{thm: metric limit nk case}, then by Remark \ref{rmk: eh spec case} we obtain in the limit the Eguchi--Hanson metric with parameter $\epsilon$ set as $1$ (see \eqref{eq: eh metric potential form}). In other words, the Eguchi--Hanson metric arises as the pointed Gromov--Hausdorff limit of K\"{a}hlerE edge metrics $\eta_{\betata_1}$ when $\betata_1\nearrow 1$. This interesting observation has been conjectured in our previous work \cite[Remark 5.1]{RZ21} and provided some of the
motivation for the present article.
\end{rmk}
Next, we fix a base point on the infinity section $Z_{n, -k}$ to study the Gromov--Hausdorff limit of the family of K\"{a}hlerE edge metrics $\xi$ on $\mathbb{F}_{n, k}$. In the limit, we obtain an orbifold K\"{a}hlerE edge metric instead of the Ricci-flat edge metric obtained in Theorem \ref{thm: metric limit nk case}.
From now on, we use $\xi_{\betata_2}, \tau(\betata_2)$ and $\varphi(\betata_2)$ to emphasize the dependence of metrics and coordinates on $\betata_2$. We consider the case $\betata_2\nearrow 1/k$.
\betagin{thm}\label{thm: conv mod2}
Fix an arbitrary base point $p$ on the infinity section $Z_{n, -k}$. As $\betata_2\nearrow 1/k$, the K\"{a}hlerE edge metric $(\mathbb{F}_{n, k}, \xi_{\betata_2}, p)$ on $\mathbb{F}_{n, k}$ converges in the pointed Gromov--Hausdorff sense to an orbifold K\"{a}hlerE edge metric $(\mathbb{P}^n(1,\dots,1,k), \xi_\infty, p)$ on the weighted projective space $\mathbb{P}^n(1,\dots,1,k)$ with an edge singularity of angle $2\pi/k$ along $Z_{n, -k}$. This limit metric coincides with the model metric $\omega_{\mathrm{orb}, n, k}$ defined in Definition \ref{def_ModOnWps}.
\end{thm}
\betagin{proof}
By similar notation and calculations as in the proof of Theorem \ref{thm: metric limit nk case}, we have
\betagin{equation*}
\frac{\textrm{d}s}{\textrm{d}\tau(\betata_2)} = \frac{1}{\varphi(\betata_2)} = \frac{\tau(\betata_2)^{n-1}}{\frac{1}{k}(\tau(\betata_2)^n-1)-\frac{n/k+\betata_2}{n+1}(\tau(\betata_2)^{n+1}-1)}.
\end{equation*}
As $\betata_2\nearrow 1/k$, we have
\betagin{equation}\label{eq_OdeTs}
\betagin{aligned}
\frac{\textrm{d}s}{\textrm{d}\tau_\infty} &= \frac{k}{\tau_\infty-\tau_\infty^2},\\
\varphi_\infty(\tau_\infty) &= \frac{1}{k}(\tau_\infty-\tau_\infty^2).
\end{aligned}
\end{equation}
Solving \eqref{eq_OdeTs} and considering a change of coordinate $u'=C' u$ for some appropriate $C$, we have
\betagin{align*}
\tau_\infty(s)&=1-\frac{1}{e^{\frac{s}{k}}+1},\quad s\in(-\infty, +\infty),\\
\varphi_\infty(s)&=\frac{1}{k}\frac{e^{s/k}}{(e^{s/k}+1)^2},\quad s\in(-\infty, +\infty).
\end{align*}
Thus, the limit metric on $\mathbb{F}_{n, k}$ is as follows:
\betagin{align*}
\tilde{\xi}_\infty&=k\frac{e^{s/k}}{e^{s/k}+1}\pi_1^*\omega_{\operatorname{FS}}+\frac{1}{k}\frac{e^{s/k}}{(e^{s/k}+1)^2}\left(\pi_2^* \omega_{\operatorname{Cyl}}+\sqrt{-1}\alphapha\wedge\bar{\alphapha}\right.\\
&\left.+\sqrt{-1}\alphapha\wedge\frac{\textrm{d}\bar{w}}{\bar{w}}+\sqrt{-1}\frac{\textrm{d}w}{w}\wedge\bar{\alphapha}\right).
\end{align*}
Recall the Ricci curvature $\mu_{\betata_2}$ is given in \eqref{eq: beta2range} by $n/k+\betata_2$ and converges to $(n+1)/k$ in the limit. Thus $\tilde{\xi}_{\infty}$ has Ricci curvature $(n+1)/k$. Moreover, $\tilde{\xi}_{\infty}$ has an edge singularity of angle $2\pi/k$ along $Z_{n, -k}$. It degenerates on $Z_{n, k}$ since $\tau\equiv 0$ on $Z_{n, k}$. Indeed, $\tilde{\xi}_\infty$ coincides with the model metrics defined in Definition \ref{def_ModOnFnk}. Then by Definition \ref{def_ModOnWps}, we denote by $\xi_\infty$ the model metric on $\mathbb{P}^n(1,\dots, 1,k)$ that is the pull-back of $\tilde{\xi}_\infty$ under the blow up map.
We have shown that $\tilde{\xi}_\infty$ is the limit of $\xi_{\betata_2}$ as tensors in the pointwise smooth sense. Next, fix an arbitrary base point on $Z_{n, -k}$. By Proposition \ref{prop: length finite nkb2 case} and the local smooth convergence result, we conclude that $(\mathbb{P}^{n}(1,\dots,1,k), \xi_\infty)$ is the limit of $(\mathbb{F}_{n, k}, \xi_{\betata_2})$ in the pointed Gromov--Hausdorff sense when $\betata_2\nearrow 1/k$. Moreover, the limit metric coincides with the model metric $\omega_{\mathrm{orb}, n, k}$ defined in Section \ref{sec: model}.
\end{proof}
As we pointed out in Section \ref{sec: asym of kee}, the family of metrics $\xi_{\betata_2}$ can be obtained by renormalizing the family of metrics $\eta_{\betata_1}$. Comparing Theorem \ref{thm: metric limit nk case} to Theorem \ref{thm: conv mod2}, we obtain different limit metrics for those two family of metrics. However, we show that after a proper normalization of $\xi_{\betata_2}$, we obtain the same limit metric for both $\xi_{\betata_2}$ and $\eta_{\betata_1}$. The normalization factor is actually given by Proposition \ref{prop: length finite nkb2 case}.
\betagin{cor}\label{thm: ResB2}
Rescale the K\"{a}hlerE edge metric $\xi_{\betata_2}$ by $((n+1)/k)^{1/n}/(1/k-\betata_2)^{1/n}$, then the normalized metric converges in the pointed Gromov--Hausdorff sense to a Ricci-flat metric on $-kH_{\mathbb{P}^{n-1}}$ when $\betata_2\nearrow 1/k$, where the base point is chosen from $Z_{n, k}$. See the proof for a more precise explanation. Moreover, this Ricci-flat metric coincides with the one obtained in Theorem \ref{thm: metric limit nk case}, i.e., the model metric $\omega_{\mathrm{eh}, n, k}$.
\end{cor}
\betagin{proof}
Consider a change of coordinate
\betagin{equation*}
y({\betata_2}):=\left(\frac{n+1}{k}\right)^{\frac{1}{n}}\cdot \frac{\tau({\betata_2})}{(\frac{1}{k}-\betata_2)^{\frac{1}{n}}}.
\end{equation*}
By Proposition \ref{prop: length finite nkb2 case}, the interval of definition of $y({\betata_2})$ converges to $[1, +\infty]$ as $\betata_2\nearrow 1/k$. The rescaled metric reads
\betagin{align*}
&\left(\frac{n+1}{k}\right)^{\frac{1}{n}}\cdot\frac{\xi_{\betata_2}}{(\frac{1}{k}-\betata_2)^{\frac{1}{n}}}\\
&=ky\pi_1^*\omega_{\operatorname{FS}}+\left(\frac{n+1}{k}\right)^{\frac{1}{n}}\cdot\frac{\varphi({\betata_2})}{(\frac{1}{k}-\betata_2)^{\frac{1}{n}}}\left(\pi_2^*\omega_{\operatorname{Cyl}}+\sqrt{-1}\alphapha\wedge\bar{\alphapha}-\right.\\
&\left.\sqrt{-1}\alphapha\wedge\frac{\textrm{d}\bar{u}}{\bar{u}}-\sqrt{-1}\frac{\textrm{d}{u}}{{u}}\wedge\bar{\alphapha} \right).
\end{align*}
Recall
\betagin{equation}\label{eq: rescaled phi nk case}
\betagin{aligned}
&\left(\frac{n+1}{k}\right)^{\frac{1}{n}}\cdot\frac{\varphi({\betata_2})}{(\frac{1}{k}-\betata_2)^{\frac{1}{n}}}=\\
&\left(\frac{n+1}{k}\right)^{\frac{1}{n}}\cdot\frac{\frac{1}{k}(\tau({\betata_2})^n-1)-\frac{1}{n+1}(\frac{n}{k}+\betata_2)(\tau({\betata_2})^{n+1}-1)}{(\tau({\betata_2})^{n-1})(\frac{1}{k}-\betata_2)^{\frac{1}{n}}}.
\end{aligned}
\end{equation}
Denote by $y$ the coordinate in the limit. Letting $\betata_2\nearrow 1/k$, the right hand side of \eqref{eq: rescaled phi nk case} converges to
\betagin{equation*}
\frac{y^n-1}{ky^{n-1}},\quad y\in[1, +\infty].
\end{equation*}
Solving
\betagin{equation*}
\frac{\textrm{d}s}{\textrm{d}\tau_{\betata_2}}=\frac{1}{\varphi_{\betata_2}(\tau_{\betata_2})}\\
\Rightarrow \frac{\textrm{d}s}{\textrm{d}y_{\betata_2}}\frac{1}{(\frac{1}{k}-\betata_2)^{\frac{1}{n}}}\cdot \left(\frac{n+1}{k}\right)^{\frac{1}{n}}=\frac{1}{\varphi_{\betata_2}(y_{\betata_2})},
\end{equation*}
we obtain in the limit
\betagin{equation}\label{eq: y hacase}
s=\frac{k}{n}\log\left(y^n-1\right),\quad y\in(1, +\infty).
\end{equation}
Thus the limit metric is given by
\betagin{equation}\label{eq: lim metric nkb2 renor case}
\betagin{aligned}
\tilde{\xi}_\infty=ky\pi_1^*\omega_{\operatorname{FS}}+\frac{y^n-1}{ky^{n-1}}&\left(\pi_2^*\omega_{\operatorname{Cyl}}+\sqrt{-1}\alphapha\wedge\bar{\alphapha}-\right.\\
&\left.\sqrt{-1}\alphapha\wedge\frac{\textrm{d}\bar{u}}{\bar{u}}-\sqrt{-1}\frac{\textrm{d}{u}}{{u}}\wedge\bar{\alphapha} \right),
\end{aligned}
\end{equation}
where $y$ and $s$ satisfy \eqref{eq: y hacase}. This limit metric is Ricci-flat. And it coincides with the limit metric in Theorem \ref{thm: metric limit nk case}, i.e., the model metric $\omega_{\mathrm{eh}, n, k}$ defined in Section \ref{sec: model}. Fix an arbitrary base point on $Z_{n, k}$. By Proposition \ref{prop: length finite nkb2 case}, the distance between $Z_{n, k}$ and $Z_{n, -k}$ tends to $+\infty$ in the limit under the renormalized metric. Thus, $Z_{n, -k}$ gets pushed-off to infinity in the limit. We obtain the pointed Gromov--Hausdorff convergence of $(\mathbb{F}_{n, k}, ((n+1)/k)^{1/n}\xi_{\betata_2}/(\frac{1}{k}-\betata_2)^{1/n})$
to $-kH_{\mathbb{P}^{n-1}}$ with the metric obtained in \eqref{eq: lim metric nkb2 renor case}.
\end{proof}
\section{Small angle limits
and fiberwise rescaling}
\label{Sec7}
In this section we first consider the $\betata_1\searrow 0$ case for $\eta_{\betata_1}$.
\betagin{thm}\label{thm: b1 0 case}
As $\betata_1$ tends to $0$, $(\mathbb{F}_{n, k}, \eta_{\betata_1})$ converges in the Gromov--Hausdorff sense to $(\mathbb{P}^{n-1}, k\omega_{\operatorname{FS}})$.
\end{thm}
\betagin{proof}
As $\betata_1 \searrow 0$, by \eqref{eq: varphi general expression nk case} we have
\betagin{align*}
\varphi_0:=\lim_{\betata_1\searrow 0}\varphi_{\betata_1}&=\frac{1}{\tau^{n-1}}\left(\frac{1}{k}(\tau^n-1)-\frac{n}{k(n+1)}(\tau^{n+1}-1)\right)\\
&=\frac{1}{k(n+1)}\cdot\frac{1}{\tau^{n-1}}(-n\tau^{n+1}+(n+1)\tau^n-1).
\end{align*}
Then we observe $\varphi_0$ does not have any root greater than $1$. Indeed, notice that
\betagin{align*}
-n\tau^{n+1}+(n+1)\tau^n-1&=(\tau-1)(1+\cdots+\tau^{n-1}-n\tau^n)\\
&=(\tau-1)(1-\tau^n+\tau-\tau^n+\cdots+\tau^{n-1}-\tau^n).
\end{align*}
Thus $\varphi_0$ is always positive when $\tau>1$. However, combining this fact with \eqref{eq: varphi roots expre}, we conclude that
\betagin{equation}
\label{TbetatozeroEq}
\lim_{\betata_1\searrow 0}T=1.
\end{equation}
Since $\varphi(1)=\varphi(T)=0$, we have $\varphi({\betata_1})\to 0$ as $\betata_1 \searrow 0$. Since $\tau$ ranges from $1$ to $T$, by \eqref{eq: eta in nk case} we conclude that as $\betata_1\searrow 0$, $\eta_{\betata_1}$ converges to $k\pi_1^* \omega_{\operatorname{FS}}$. Thus we have shown $(\mathbb{F}_{n, k}, \eta_{\betata_1})$ converges in the Gromov--Hausdorff sense to $(\mathbb{P}^{n-1}, k\omega_{\operatorname{FS}})$ when $\betata_1\searrow 0$.
\end{proof}
Roughly speaking, Theorem \ref{thm: b1 0 case} says that as $\betata_1\searrow 0$, the fibers collapse to the zero section. This motivates us to rescale the metric $\eta_{\betata_1}$ along the fiber so that we can obtain a non-collapsed metric in the limit. We first need the following lemma.
\betagin{lem}\label{lem: b1Tasy}
For $\betata_1>0$ and close to zero, $T=T(\betata_1)=1 + O(\betata_1).
$
\end{lem}
\betagin{proof}
By Proposition \ref{prop: red to ode}, $T(\betata_1)$ is determined by $\betata_1$, and $T(\betata_1)$ is the first root of the polynomial in \eqref{eq: polyforT}. We rewrite the polynomial in \eqref{eq: polyforT} as \betagin{equation*}
P(\tau) = \frac{k\betata_1-n}{k(n+1)}(\tau-1)\left(\tau^n+\frac{k\betata_1+1}{k\betata_1-n}(\tau^{n-1}+\cdots+1)\right).
\end{equation*}
Letting $y=\tau-1$,
\betagin{equation}\label{eq: fapT}
\betagin{aligned}
P(y) &= \frac{k\betata_1-n}{k(n+1)} y \left((y+1)^n + \frac{k\betata_1+1}{k\betata_1-n}\big((y+1)^{n-1}+\cdots+(y+1)+1\big)\right)\\
&= \frac{k\betata_1-n}{k(n+1)} y \bigg(y^n+\ldots+
y\Big(n+\frac{k\betata_1+1}{k\betata_1-n}(1+\ldots+n-1)\Big)+ \frac{k(n+1)\betata_1}{k\betata_1-n}
\bigg)\\
&=
\frac{k\betata_1-n}{k(n+1)} y \left(y Q(y) + \frac{k(n+1)\betata_1}{k\betata_1-n}
\right),
\end{aligned}
\end{equation}
where $Q(y)$ is a polynomial of
degree $n-1$ whose coefficients
depend on $\betata_1$ and
whose constant
term is
\betagin{equation}
\label{QzeroEq}
Q(0)=n
+
\frac{k\betata_1+1}{k\betata_1-n}
\big(1+\ldots+n-1)
=
n\frac{(n-1)(k\betata_1+1)}{2(k\betata_1-n)}
=
\frac{n+1}2
\frac{nk\betata_1-n}{k\betata_1-n}
.
\end{equation}
By Proposition \ref{prop: red to ode}, $T-1$ is a root of the term in the parenthesis of the second equation in \eqref{eq: fapT}, i.e.,
$$
0=(T-1)Q(T-1) + \frac{k(n+1)\betata_1}{k\betata_1-n}.
$$
In particular, it
follows that $Q(T-1)\neq0$
for small
enough $\betata_1$. Thus,
dividing
we obtain
$$
T-1=
\frac{k(n+1)\betata_1}{(-k\betata_1+n)Q(T-1)}.
$$
By \eqref{TbetatozeroEq}
$\lim_{\betata_1\searrow0}
T=1$, and so
$\lim_{\betata_1\searrow0}
Q(T-1)=(n+1)/2$
by
\eqref{QzeroEq}.
Altogether,
\betagin{equation}
\label{Tminus1AsymEq}
T-1=\frac{2k}n\betata_1+o(\betata_1),
\end{equation}
as claimed.
\end{proof}
\betagin{rmk}
The last display generalizes
\cite[(5.1)]{RZ21} from
the surface case $n=2$ to any dimension.
Note that in op. cit. $n$ corresponds
to our $k$
\end{rmk}
It follows that in the small
angle limit, both angles approach
zero at the same rate:
\betagin{lem}
\label{lem: beta2beta1small}
For $\betata_1>0$ and close to zero, $\betata_2=\betata_2(\betata_1)=\betata_1 + O(\betata_1^2).
$
\end{lem}
\betagin{proof}
Combining \eqref{TbetatozeroEq}
and \eqref{Tbeta1beta2Eq}
it follows that
$\lim_{\betata_1\searrow0}
\betata_2=0$.
Using this, and plugging \eqref{Tminus1AsymEq},
in \eqref{Tbeta1beta2Eq}
we find that
$$
\betata_1+\frac1k=\frac1k+2\betata_1-\betata_2+o(\betata_1),
$$
so $\betata_2=\betata_1+o(\betata_1)$,
and so bootstrapping we obtain
$\betata_2=\betata_1+O(\betata_1^2)$,
as claimed.\end{proof}
Lemma \ref{lem: beta2beta1small}
motivates treating the angles
$2\pi\betata_1$ and $2\pi\betata_2$ on
the same footing in the small angle regime, so that it reasonable to
hope that under some appropriate
rescaling the fibers converge to
cylinders, as in \cite{RZ20,RZ21}.
This is precisely what we prove next.
We change variable from $\tau\in(1, T)$ in \eqref{eq: eta in nk case} and \eqref{eq: varphi general expression nk case} to
\betagin{equation}\label{eq: newcoord}
x:= \frac{\tau-1-\frac{k\betata_1}{n}}{\frac{k\betata_1^2}{n}},
\end{equation}
with $x\in\left(-\frac{1}{\betata_1}, \frac{1}{\betata_1}+O(1)\right)$ by \eqref{Tminus1AsymEq}. Note that $x=0$ roughly corresponds to the middle section between $Z_{n, k}$ and $Z_{n, -k}$. By \eqref{eq: varphi general expression nk case} and \eqref{eq: newcoord},
\betagin{equation}\label{eq: varphi in x}
\varphi(x) = \frac{k}{2n}\betata_1^2 + \frac{k}{n}\betata_1^3 x + o(\betata_1^2), \quad x\in\left(-\frac{1}{\betata_1}, \frac{1}{\betata_1}+O(1)\right).
\end{equation}
Let $p\in \mathbb{F}_{n, k}$ be a fixed base point chosen from the section $\{x=0\}$, which will serve as the base point we use later for pointed Gromov--Hausdorff convergence.
To find a fiberwise-rescaled limit, we next rescale the metric $\eta_{\betata_1}$ in \eqref{eq: eta in nk case} along each fiber, i.e., we define
\betagin{equation}\label{eq: fibrescal}
\widetilde{\eta_{\betata_1}}:=k\tau \pi_1^*\omega_{\operatorname{FS}}+\frac{1}{\betata_1^2}\varphi\pi_2^*\omega_{\operatorname{Cyl}} + \varphi(\sqrt{-1}\alphapha\wedge\bar{\alphapha}+\sqrt{-1}\alphapha\wedge \overline{\textrm{d}w/w}+\sqrt{-1}{\textrm{d}w/w}\wedge\bar{\alphapha}).
\end{equation}
This fiberwise rescaled metric may no longer be KEE.
\betagin{thm}\label{thm: ResB1Case}
As $\betata_1\searrow 0$, $(\mathbb{F}_{n, k}, \widetilde{\eta_{\betata_1}}, p)$ converges in the pointed Gromov--Hausdorff sense to $(\mathbb{P}^{n-1}\times \mathbb{C}^*, \frac{k}{n}(n\pi_1^*\omega_{\operatorname{FS}}+\pi_2^*\omega_{\operatorname{Cyl}}), p)$.
\end{thm}
\betagin{proof}
We first show on compact subsets, the following pointwise convergence holds:
\betagin{claim}
The restriction of $\widetilde{\eta_{\betata_1}}$ to a fiber converges to a cylindrical metric pointwise on compact subsets. More precisely,
\betagin{equation*}
\lim_{\betata_1\searrow 0} \frac{1}{\betata_1^2}\varphi \pi_2^* \omega_{\operatorname{Cyl}} = \frac{k}{n} \pi_2^* \omega_{\operatorname{Cyl}}.
\end{equation*}
\end{claim}
\betagin{proof}[Proof of the Claim]
As shown in \eqref{eq: metric res}, the restriction of ${\eta_{\betata_1}}$ is given by
\betagin{equation*}
\frac{1}{2\varphi(\tau)}\textrm{d}\tau^2 + 2\varphi(\tau) \textrm{d}\theta^2,
\end{equation*}
thus the restriction of $\widetilde{\eta_{\betata_1}}$ to a fiber, using the new coordinates \eqref{eq: newcoord} is given by
\betagin{equation}\label{eq: ResVarphiFib}
\frac{k^2\betata_1^2}{2n^2\varphi(x)}\textrm{d}x^2 + \frac{2\varphi(x)}{\betata_1^2}\textrm{d}\theta^2.
\end{equation}
As $\betata_1 \searrow 0$, by \eqref{eq: varphi in x}, \eqref{eq: ResVarphiFib} converges pointwise on compact subsets to
\betagin{equation*}
\frac{k}{n}\textrm{d}x^2 + \frac{k}{n}\textrm{d}\theta^2 = \frac{k}{n}\omega_{\operatorname{Cyl}},
\end{equation*}
as claimed.
\end{proof}
By the collapsing arguments in the proof of Theorem \ref{thm: b1 0 case} and the claim above, we have shown $\widetilde{\eta_{\betata_1}}$ converges pointwise to $\frac{k}{n}(n\pi_1^* \omega_{\operatorname{FS}}+\pi_2^* \omega_{\operatorname{Cyl}})$ on compact subsets as $\betata_1\searrow 0$. It remains to prove the pointed Gromov--Hausdorff convergence. Indeed, by arguments in the proof of Proposition \ref{prop: length to infty nk case}, the distance between $Z_{n, k}$ and $\{x=0\}$ and the distance between $Z_{n, -k}$ and $\{x=0\}$ tend to infinity under the metric $\widetilde{\eta_{\betata_1}}$. Thus in the limit $\betata_1\searrow 0$, we get the product differential structure on $\mathbb{P}^{n-1}\times \mathbb{C}^*$ as claimed. Choosing the point $p$ as the base point, the pointwise convergence result implies that
\betagin{equation*}
\lim_{\betata_1\searrow 0} \widetilde{\eta_{\betata_1}} = \frac{k}{n}(n\pi_1^* \omega_{\operatorname{FS}}+\pi_2^* \omega_{\operatorname{Cyl}})
\end{equation*}
in the pointed Gromov--Hausdorff sense.
\end{proof}
The $\betata_2\searrow 0$ case for $\xi_{\betata_2}$ is similar to Theorem \ref{thm: b1 0 case} and Theorem \ref{thm: ResB1Case}.
The asymptotic behaviors of $t(\betata_2)$ and $\betata_1(\betata_2)$ when $\betata_2 \searrow 0$ are similar to those described in Lemma \ref{lem: b1Tasy} and Lemma \ref{lem: beta2beta1small}. We collected them as follows.
\betagin{lem}\label{lem: b2tasy}
For $\betata_2>0$ and close to zero,
\betagin{equation*}
t = t(\betata_1) = 1-\frac{2k}{n}\betata_2+o(\betata_2).
\end{equation*}
\end{lem}
\betagin{lem}\label{lem: b1b2s}
For $\betata_2>0$ and close to zero,
\betagin{equation*}
\betata_1 = \betata_1(\betata_2)=\betata_2+O(\betata_2^2).
\end{equation*}
\end{lem}
Now we state the non-rescaling limit of $\xi_{\betata_2}$ as $\betata_2\searrow 0$.
\betagin{thm}\label{thm: b2 0 case}
As $\betata_2$ tends to $0$, $(\mathbb{F}_{n, k}, \xi_{\betata_2})$ converges in the Gromov--Hausdorff sense to $(\mathbb{P}^{n-1}, k\omega_{\operatorname{FS}})$.
\end{thm}
\betagin{proof}
By Proposition \ref{prop: bd cond b2 case} and Proposition \ref{prop: red to ode b2 case}, $t$ tends to $1$ as $\betata_2\searrow 0$. The remaining proof is similar to that of Theorem \ref{thm: b1 0 case}.
\end{proof}
To obtain a non-collapsed metric in the limit, we consider rescaling $\xi_{\betata_2}$ in the way of \eqref{eq: fibrescal} and denote the rescaled metric by $\widetilde{\xi_{\betata_2}}$:
\betagin{equation*}
\widetilde{\xi_{\betata_2}}:=k\tau \pi_1^*\omega_{\operatorname{FS}}+\frac{1}{\betata_2^2}\varphi\pi_2^*\omega_{\operatorname{Cyl}} + \varphi(\sqrt{-1}\alphapha\wedge\bar{\alphapha}-\sqrt{-1}\alphapha\wedge \overline{\textrm{d}u/u}-\sqrt{-1}{\textrm{d}u/u}\wedge\bar{\alphapha}).
\end{equation*}
Moreover, we consider a change of variable as \eqref{eq: newcoord}:
\betagin{equation*}
u := \frac{\tau-1+\frac{k}{n}\betata_2}{\frac{k}{n}\betata_2^2}.
\end{equation*}
As before, we choose a fixed base point from the section $\{u=0\}$.
\betagin{thm}\label{thm: ResB2Zero}
As $\betata_2\searrow 0$, $(\mathbb{F}_{n, k}, \widetilde{\xi_{\betata_2}}, q)$ converges in the pointed Gromov--Hausdorff sense to $(\mathbb{P}^{n-1}\times \mathbb{C}^*, \frac{k}{n}(n\pi_1^*\omega_{\operatorname{FS}}-\pi_2^*\omega_{\operatorname{Cyl}}), q)$.
\end{thm}
\betagin{proof}
By Lemma \ref{lem: b2tasy} and Lemma \ref{lem: b1b2s}, we have similar asymptotic behaviors as Lemma \ref{lem: b1Tasy} and Lemma \ref{lem: beta2beta1small} in the $\betata_2$ case. Then we can apply similar arguments as in the proof of Theorem \ref{thm: ResB1Case}.
\end{proof}
\appendix
\section{A brief review on Eguchi--Hanson metrics}\label{app: review EH}
In this section, we give a brief review of the construction of Eguchi--Hanson metrics \cite{EH79}. They are Ricci-flat K\"{a}hler metrics defined on the total space of the line bundle $-2H_{\mathbb{P}^1}$.
For $(x_1+\sqrt{-1}y_1, x_2+\sqrt{-1}y_2)\in\mathbb{C}^2$, the Hopf coordinates are defined as:
\betagin{align*}
x_1+\sqrt{-1}y_1&=\xi\cos\frac{\theta}{2} e^{\frac{\sqrt{-1}}{2}(\psi+\phi)},\\
x_2+\sqrt{-1}y_2&=\xi\sin\frac{\theta}{2} e^{\frac{\sqrt{-1}}{2}(\psi-\phi)},
\end{align*}
where $\xi\geq 0$, $\theta\in[0, \pi]$, $\psi\in[0, 4\pi]$ and $\phi\in[0, 2\pi]$. Define one-forms on $\mathbb{C}^2$ by
\betagin{align}
\betagin{aligned}\label{eq: defi of sigma}
\sigma_1:&=\frac{1}{\xi^2}(x_1\textrm{d}y_2-y_2\textrm{d}x_1+y_1\textrm{d}x_2-x_2\textrm{d}y_1)=\frac{1}{2}(\sin\psi\textrm{d}\theta-\sin\theta\cos\psi\textrm{d}\phi),\\
\sigma_2:&=\frac{1}{\xi^2}(y_1\textrm{d}y_2-y_2\textrm{d}y_1+x_2\textrm{d}x_1-x_1\textrm{d}x_2)=\frac{1}{2}(-\cos\psi\textrm{d}\theta-\sin\theta\sin\psi\textrm{d}\phi),\\
\sigma_3:&=\frac{1}{\xi^2}(x_2\textrm{d}y_2-y_1\textrm{d}x_2+x_1\textrm{d}y_1-y_1\textrm{d}x_1)=\frac{1}{2}(\textrm{d}\psi+\cos\theta\textrm{d}\phi).
\end{aligned}
\end{align}
Direct calculations yield:
\betagin{align}
\betagin{aligned}\label{eq: sigma cyclic}
\textrm{d}\sigma_1=2\sigma_2\wedge\sigma_3,\\
\textrm{d}\sigma_2=2\sigma_3\wedge\sigma_1,\\
\textrm{d}\sigma_3=2\sigma_1\wedge\sigma_2.
\end{aligned}
\end{align}
The standard Euclidean metric on $\mathbb{C}^2$ can be written as
\betagin{equation*}
\textrm{d}x_1^2+\textrm{d}y_1^2+\textrm{d}x_2^2+\textrm{d}y_2^2=\textrm{d}\xi^2+\xi^2(\sigma_1^2+\sigma_2^2+\sigma_3^2).
\end{equation*}
The Eguchi--Hanson metric with parameter $\epsilon>0$ is defined by
\betagin{equation}\label{eq: EH epsilon}
g_{\operatorname{EH}, \epsilon}:=\left(1-\frac{\epsilon^4}{\xi^4}\right)^{-1}\textrm{d}\xi^2+\xi^2\left(\sigma_1^2+\sigma_2^2+\left(1-\frac{\epsilon^4}{\xi^4}\right)\sigma_3^2\right),\quad \xi\geq \epsilon.
\end{equation}
\betagin{prop}\label{prop: eh ricflat}
Eguchi--Hanson metrics defined in \eqref{eq: EH epsilon} are Ricci-flat.
\end{prop}
\betagin{proof}
We provide a proof by directly calculating connection forms and curvature forms of the metric. See Remark \ref{rmk: ricci flat using kahler form} for another proof using K\"{a}hler forms of Eguchi--Hanson metrics. Consider a change of variable $\zeta=\zeta(\xi)$ such that $\textrm{d}\zeta=(1-(\epsilon/\xi)^4)^{-1/2}d\xi$. Then we can write \eqref{eq: EH epsilon} in the form
\betagin{equation*}
g_{\operatorname{EH}, \epsilon}=\textrm{d}\zeta^2+f^2(\zeta)(\sigma_1^2+\sigma_2^2+g^2(\zeta)\sigma_3^2),
\end{equation*}
where $f=\xi$ and $g=(1-(\epsilon/\xi)^4)^{1/2}$. Consider an orthonormal basis
\betagin{equation*}
(\omega^0, \omega^1, \omega^2, \omega^3)=(\textrm{d}\zeta, fg\sigma_3, f\sigma_1, f\sigma_2).
\end{equation*}
Then $g_{\operatorname{EH}, \epsilon}=\sum_{i=0}^3(\omega^i)^2$ and $\{\omega^i\}_{i=0}^3$ satisfy the following equations:
\betagin{align*}
&\textrm{d}\omega^i=\omega^j\wedge\omega_{j}^i,\quad \text{for}\;i=0, 1,2,3,\\
&\omega_i^j+\omega_j^i=0,\quad \text{for}\;i, j=0, 1,2,3,
\end{align*}
where $\{\omega_i^j\}_{i, j=0}^3$ are connection forms with respect to $\{\omega^i\}_{i=0}^3$. Next let us determine connection forms. For $i=1$, we have
\betagin{equation*}
\textrm{d}\omega^1=\frac{f'g+fg'}{fg}\omega^0\wedge\omega^1+\frac{2g}{f}\omega^2\wedge\omega^3,
\end{equation*}
where $f'$ and $g'$ denote the derivative with respect to $\zeta$. Without loss of generality, we let
\betagin{align}
\betagin{aligned}\label{eq:omega sub 1}
\omega_0^1&=\frac{f'g+fg'}{fg}\omega^1,\\
\omega_2^1&=\frac{g}{f}\omega^3,\\
\omega_3^1&=-\frac{g}{f}\omega^2.
\end{aligned}
\end{align}
It remains to find $\omega_0^2$, $\omega_0^3$ and $\omega_2^3$ due to the skew-symmetry of connection forms. By similar calculations we have
\betagin{align}
\betagin{aligned}\label{eq:d of omega others}
\textrm{d}\omega^2=\frac{f'}{f}\omega^0\wedge\omega^2+\frac{2}{fg}\omega^3\wedge\omega^1,\\
\textrm{d}\omega^3=\frac{f'}{f}\omega^0\wedge\omega^3+\frac{2}{fg}\omega^1\wedge\omega^2.
\end{aligned}
\end{align}
Thus, combining $\eqref{eq:omega sub 1}$ and \eqref{eq:d of omega others} we obtain
\betagin{align}
\betagin{aligned}\label{eq: omega others}
\omega_0^2&=\frac{f'}{f}\omega^2,\\
\omega_0^3&=\frac{f'}{f}\omega^3,\\
\omega_2^3&=\frac{g^2-2}{fg}.
\end{aligned}
\end{align}
Notice that
\betagin{align}
\betagin{aligned}\label{eq: f and g}
f'&=\frac{\textrm{d}\xi}{\textrm{d}\zeta}=(1-(\epsilon/\xi)^4)^{1/2}=g,\\
g'&=2\epsilon^4f^{-5}.
\end{aligned}
\end{align}
Combining \eqref{eq:omega sub 1}, \eqref{eq: omega others} and \eqref{eq: f and g} we see
\betagin{align}
\betagin{aligned}\label{eq: connection form selfdual}
\omega_0^1&=-\omega_2^3,\\
\omega_2^1&=\omega_0^3,\\
\omega_3^1&=-\omega_0^2.
\end{aligned}
\end{align}
By \eqref{eq: connection form selfdual} and the fact that $R_i^j=\textrm{d}\omega_i^j-\omega_i^k\wedge\omega_k^j$, for $i, j=0, 1,2,3$, we obtain that the curvature forms also satisfy
\betagin{align*}
R_0^1&=-R_2^3,\\
R_2^1&=R_0^3,\\
R_3^1&=-R_0^2.
\end{align*}
Finally, the Ricci-flatness comes from the formula $\operatorname{Ric}_{ij}=\sum_{k=0}^3R_{ikj}^{k}$ and the first Bianchi identity.
\end{proof}
Introduce
\betagin{equation*}
r^4=\xi^4-\epsilon^4, \quad \xi\geq\epsilon,
\end{equation*}
then \eqref{eq: EH epsilon} can be written as
\betagin{equation}\label{eq: eh metric in r}
g_{\operatorname{EH}, \epsilon}=\frac{r^2}{(\epsilon^4+r^4)^{\frac{1}{2}}}(\textrm{d}r^2+r^2\sigma_3^2)+(\epsilon^4+r^4)^{\frac{1}{2}}(\sigma_1^2+\sigma_2^2),\quad r\geq 0.
\end{equation}
From \eqref{eq: eh metric in r} we are able to convert Eguchi--Hanson metrics into complex form by letting $(z_1, z_2)\in\mathbb{C}^2$ satisfy
\betagin{align}
\betagin{aligned}\label{complex coordinates for EH}
z_1&=r\cos\frac{\theta}{2}e^{\frac{\sqrt{-1}}{2}(\psi+\phi)},\\
z_2&=r\sin\frac{\theta}{2}e^{\frac{\sqrt{-1}}{2}(\psi-\phi)}.
\end{aligned}
\end{align}
Denote by $\omega_{\operatorname{EH}, \epsilon}$ the K\"{a}hler form corresponding to $g_{\operatorname{EH}, \epsilon}$. Then by \eqref{eq: eh metric in r} we have in complex coordinates,
\betagin{align}
\omega_{\operatorname{EH}, \epsilon}
&=\sqrt{-1}\partial\bar{\partial}[\sqrt{r^4+\epsilon^4}+\log r^2-\log(\epsilon^2+\sqrt{r^4+\epsilon^4})]\label{eq: eh metric potential form}\\
&=\frac{\sqrt{-1}r^2}{\sqrt{r^4+\epsilon^4}}(\textrm{d}z_1\wedge\textrm{d}\bar{z}_1+\textrm{d}z_2\wedge\textrm{d}\bar{z}_2)+\frac{\epsilon^4}{\sqrt{r^4+\epsilon^4}}\sqrt{-1}\partial\bar{\partial}\log(r^2).\label{eq: eh metric kahler form}
\end{align}
\betagin{rmk}\label{rmk: ricci flat using kahler form}
By calculating the Ricci form of $\omega_{\operatorname{EH}, \epsilon}$ as in \eqref{eq: eh metric kahler form}, we can also derive the Ricci-flatness of Eguchi--Hanson metrics. Indeed, from \eqref{eq: eh metric kahler form} we calculate
\betagin{align*}
\operatorname{Ric}\omega_{\operatorname{EH}, \epsilon}&=\operatorname{Ric}\sqrt{-1}\Bigg(
\left[\frac{r^2}{\sqrt{r^4+\epsilon^4}}+\frac{\epsilon^4|z_2|^2}{\sqrt{r^4+\epsilon^4}r^4}\right]\textrm{d}z_1\wedge\textrm{d}\bar{z}_1\\
&-\frac{\epsilon^4z_2\bar{z}_1}{\sqrt{r^4+\epsilon^4}r^4}\textrm{d}z_1\wedge\textrm{d}\bar{z}_2-\frac{\epsilon^4z_1\bar{z}_2}{\sqrt{r^4+\epsilon^4}r^4}\textrm{d}z_2\wedge\textrm{d}\bar{z}_1\\
&+\left[\frac{r^2}{\sqrt{r^4+\epsilon^4}}+\frac{\epsilon^4|z_1|^2}{\sqrt{r^4+\epsilon^4}r^4}\right]\textrm{d}z_2\wedge\textrm{d}\bar{z}_2 \Bigg)\\
&=-\sqrt{-1}\partial\bar{\partial}\log\left(
\left[\frac{r^2}{\sqrt{r^4+\epsilon^4}}+\frac{\epsilon^4|z_2|^2}{\sqrt{r^4+\epsilon^4}r^4}\right]^2\right.\\
&-\left.\frac{\epsilon^4|z_1|^2|z_2|^2}{(r^4+\epsilon^4)r^8}
\right)\\
&=-\sqrt{-1}\partial\bar{\partial}\log1\\
&=0.
\end{align*}
\end{rmk}
From \eqref{eq: eh metric in r} one finds that $g_{\operatorname{EH}, \epsilon}$ is defined on $\mathbb{C}^2$ with possible singularity at $r=0$. Since $g_{\operatorname{EH},\epsilon}$ is invariant under the antipodal reflection, we have an induced metric on $(\mathbb{C}^2\setminus\{0\})/\mathbb{Z}_2$ that admits no singularity. Consider the blow-up of $(\mathbb{C}^2\setminus\{0\})/\mathbb{Z}_2$ at the origin, which is biholomorphic to the total space of the line bundle $-2H_{\mathbb{P}^1}$, then a calculation in \eqref{eq: sigma1 plus 2} below shows that $\sigma_1^2+\sigma_2^2$ is the pull-back of the Fubini--Study metric from the exceptional divisor. Hence $g_{\operatorname{EH}, \epsilon}$ extends to a metric on the total space $-2H_{\mathbb{P}^1}$ by letting $r=0$ when restricting $g_{\operatorname{EH}, \epsilon}$ to the exceptional divisor.
\section{Eguchi--Hanson metrics as Gromov--Hausdorff limits of K\"{a}hler--Einstein edge metrics}\label{app: eh another discussion}
In this section we give a direct proof of a special case of Theorem \ref{thm: metric limit nk case} when $n=k=2$. We already know we will obtain Eguchi--Hanson metrics in the limit.
Recall in \eqref{eq: eta in nk case} we denote by $\eta$ a K\"{a}hler--Einstein edge metric on Calabi--Hirzebruch manifolds $\mathbb{F}_{n, k}$ that has the following form:
\betagin{equation}\label{eq: eta origin expression}
\eta=k\tau \pi_1^*\omega_{\operatorname{FS}}+\varphi\left(\pi_2^*\omega_{\operatorname{Cyl}}+\sqrt{-1}\alphapha\wedge\bar{\alphapha}+\sqrt{-1}\alphapha\wedge\frac{\textrm{d}\bar{w}}{\bar{w}}+\sqrt{-1}\frac{\textrm{d}w}{w}\wedge\bar{\alphapha}\right),
\end{equation}
where $\pi_1$, $\pi_2$ and $\alphapha$ are defined below \eqref{eq: eta in nk case}.
From now on, we assume $k=2$ and $n=2$, i.e., consider the second Hirzebruch surface $\mathbb{F}_2$. To build a connection between $g_{\operatorname{EH}, \epsilon}$ and the K\"{a}hler edge metric $\eta$ on $\mathbb{F}_2$, we first write $\eta$ in terms of one forms introduced in \eqref{eq: defi of sigma}.
Consider a change of coordinate $w=v^2$. The reason to do this is that $w$ is the coordinate along each fiber of the line bundle $-2H_{\mathbb{P}^1}$. Recall \eqref{complex coordinates for EH}, then we have the following correspondence:
\betagin{align}
\betagin{aligned}\label{eq: coordinates uz}
z_1=vz&=r\cos\frac{\theta}{2}e^{\frac{\sqrt{-1}}{2}(\psi+\phi)},\\
z_2=v&=r\sin\frac{\theta}{2}e^{\frac{\sqrt{-1}}{2}(\psi-\phi)}.
\end{aligned}
\end{align}
In particular,
\betagin{equation}
\betagin{aligned}\label{eq: relation r u z}
r^2&=|v|^2(1+|z|^2)\\
&=|w|(1+|z|^2).
\end{aligned}
\end{equation}
By the definition of $\alphapha$ in \eqref{eq: eta in nk case}, $(1, 1)$-forms that appear in $\eta$ are
\betagin{equation*}
\frac{\textrm{d}z\wedge \textrm{d}\bar{z}}{(1+|z|^2)^2},\; \alphapha\wedge\bar{\alphapha},\; \frac{4\textrm{d}v\wedge\textrm{d}\bar{v}}{|v|^2},\;\alphapha\wedge\frac{2\textrm{d}\bar{v}}{\bar{v}},\;\frac{2\textrm{d}v}{v}\wedge\bar{\alphapha}.
\end{equation*}
Let us first calculate $\sigma_1^2+\sigma_2^2$ in terms of the coordinate $z$ and $v$. By \eqref{eq: defi of sigma} and \eqref{eq: coordinates uz} we have
\betagin{align}
\betagin{aligned}\label{eq: sigma1 plus 2}
\sigma_1^2+\sigma_2^2&=\frac{1}{4}\textrm{d}\theta^2+\frac{1}{4}\sin^2\theta\textrm{d}\phi^2\\
&=\frac{1}{4}\left(-\frac{|z|\textrm{d}z}{(1+|z|^2)z}-\frac{|z|\textrm{d}\bar{z}}{(1+|z|^2)\bar{z}}\right)^2+\\
&\frac{1}{4}\left(\frac{2|z|}{1+|z|^2}\right)^2\cdot\left(\frac{1}{2\sqrt{-1}}\left(\frac{\textrm{d}z}{z}-\frac{\textrm{d}\bar{z}}{\bar{z}}\right)\right)^2\\
&=\operatorname{Re}\frac{\textrm{d}z\otimes\textrm{d}\bar{z}}{(1+|z|^2)^2}.
\end{aligned}
\end{align}
For $\textrm{d}r^2$ and $\sigma_3$, By \eqref{eq: defi of sigma} and \eqref{eq: coordinates uz} we have
\betagin{align}
\betagin{aligned}\label{eq:r and sigma3}
4r^2\textrm{d}r^2&=\frac{r^4}{4}\left(\alphapha+\bar{\alphapha}+\frac{2\textrm{d}v}{v}+\frac{2\textrm{d}\bar{v}}{\bar{v}}\right)^2,\\
\sigma_3^2&=-\frac{1}{16}\left(\alphapha-\bar{\alphapha}+\frac{2\textrm{d}v}{v}-\frac{2d\bar{v}}{\bar{v}}\right)^2.\\
\textrm{d}r^2+r^2\sigma_3^2&=\frac{r^2}{4}\operatorname{Re}\left(\frac{4\textrm{d}v\otimes\textrm{d}\bar{v}}{|v|^2}+\alphapha\otimes\bar{\alphapha}+\alphapha\otimes\frac{2\textrm{d}\bar{v}}{\bar{v}}+\frac{2\textrm{d}v}{v}\otimes\bar{\alphapha}\right).
\end{aligned}
\end{align}
Denote by $g_\eta$ the corresponding Riemannian metric on $-2H_{\mathbb{P}^1}$ with respect to $\eta$. Then combining \eqref{eq: eta origin expression}, \eqref{eq: sigma1 plus 2} and \eqref{eq:r and sigma3} we obtain
\betagin{equation}\label{eq: eta in limit}
g_\eta=2\tau(\sigma_1^2+\sigma_2^2)+\varphi\cdot \frac{4}{r^2}(\textrm{d}r^2+r^2\sigma_3^2).
\end{equation}
For $\tau$ and $\varphi$ in \eqref{eq: eta in limit}, results in Section \ref{sec: general} apply after we fix $n=k=2$ there. A key feature when $n=2$ is the right hand side in \eqref{eq: varphi general expression nk case} is a cubic polynomial, which is easy to handle. In other words, we will be able to derive more precise dependence of $T$ and $\betata_2$ on $\betata_1$ comparing to results in Proposition \ref{prop: T infty nk case} and Proposition \ref{prop: b2 limit nk case}. Indeed, this was done in the work of \cite{RZ21}. In this and the next section, we fix $n=2$ and make use of several results obtained in \cite{RZ21}.
Since $n=k=2$, we find $\betata_1\in(0, 1)$. We will study the asymptotic behaviors of K\"{a}hlerE edge metrics $\eta$ when $\betata_1\to 1$. Recall \cite[(4.20)]{RZ21}
\betagin{equation*}
\betata_2=\frac{1}{4}(2\betata_1+\sqrt{3(3-2\betata_1)(1+2\betata_1)}-3).
\end{equation*}
So we have $\betata_2\to \frac{1}{2}$ as $\betata_1\to 1$. Recall, $\tau$ ranges from $[1, T]$ and $T$ is given by \cite[(5.1)]{RZ21}
\betagin{equation*}
T=1+3\frac{\sqrt{1+ \frac{4}{3}\betata_1 -\frac{4}{3}\betata_1^2}+2\betata_1-1}{4-4\betata_1}.
\end{equation*}
Thus, $T\to +\infty$ as $\betata_1\to 1$. Moreover, recall in \eqref{eq: varphi general expression nk case} $\varphi(\tau)$ is given by
\betagin{align}
\varphi(\tau)&=\frac{1}{2}\frac{\tau^2-1}{\tau}+\frac{1}{3}(\betata_1-1)\frac{\tau^3-1}{\tau} \label{eq:varphi and tau}\\
&=\frac{1}{3}(\betata_1-1)(\tau-1)(\tau-\alphapha_1)(\tau-T)/\tau, \quad\textrm{for}\;\tau\in[1, T],\notag
\end{align}
where $\alphapha_1$ is given in \cite[(5.2)]{RZ21} by $\displaystyle \alphapha_1=1+3\frac{-\sqrt{1+\frac{4}{3}\betata_1-\frac{4}{3}\betata_1^2}+2\betata_1-1}{4-4\betata_1}$ and $\alphapha_1$ tends to $-1$ as $\betata_1\to 1$.
Below we first show as $\betata_1\to 1$, the divisor $Z_{-2}$ gets pushed-off to infinity. This result is a special case of Proposition \ref{prop: length to infty nk case}, and for the reader's convenience we include a proof here.
\betagin{prop}\label{prop: length to infty n2 case}
The length of the path on each fiber between the intersection point of the fiber with $Z_2$ and that of the fiber with $Z_{-2}$ tends to infinity as $\betata_1\to 1$.
\end{prop}
\betagin{proof}
Restricted to the fiber $\{z=0\}$, by \eqref{eq: eta origin expression} $\eta=\varphi\sqrt{-1}\textrm{d}w\wedge\textrm{d}\bar{w}/{|w|^2}$ gives a metric
\betagin{equation*}
g=\frac{1}{2\varphi(\tau)}\textrm{d}\tau^2+2\varphi(\tau)\textrm{d}\theta^2.
\end{equation*}
Up to some constant, the distance between $\{\tau=1\}$ and $\{\tau=T\}$ is given by
\betagin{align}
\int_{1}^{T} \frac{1}{\sqrt{\varphi(\tau)}}\;\textrm{d}\tau&=\int_1^T \frac{\sqrt{\tau}\;\textrm{d}\tau}{\sqrt{\frac{1}{3}(\betata_1-1)(\tau-1)(\tau-\alphapha_1)(\tau-T)}}\notag\\
&=\frac{1}{\sqrt{\frac{1}{3}(1-\betata_1)}}\int_1^T\frac{\sqrt{\tau}\;\textrm{d}\tau}{\sqrt{(\tau-1)(\tau-\alphapha_1)(T-\tau)}}\notag\\
&\overset{\xi=\tau-1}{=\joinrel=}\frac{1}{\sqrt{\frac{1}{3}(1-\betata_1)}}\int_0^{T-1}\frac{\sqrt{\xi+1}\;\textrm{d}\xi}{\sqrt{\xi(\xi+1-\alphapha_1)(T-1-\xi)}}\notag\\
&=:\int_{0}^{T-1} I\;\textrm{d}\xi.\label{eq: estimate I}
\end{align}
Near $\xi=0$, terms $\sqrt{\xi+1}$ and $\sqrt{\xi+1-\alphapha_1}$ are uniformly bounded as $\betata_1\to 1$. \eqref{eq: estimate I} satisfies
\betagin{equation*}
\int_0^\epsilon I\;\textrm{d}\xi\leq C\cdot\frac{1}{\sqrt{1-\betata_1}}\cdot\frac{1}{\sqrt{\frac{1}{1-\betata_1}}}\cdot \sqrt{\xi}|_{0}^\epsilon,
\end{equation*}
for some uniform constant $C>0$ and any small $\epsilon>0$. Thus the integration in \eqref{eq: estimate I} does not blow up near $\xi=0$. Near $\xi=T-1$, in \eqref{eq: estimate I} for any fixed $\epsilon>0$, we can find $\betata_1$ close to $1$ such that for $\xi\in(T-1-\epsilon, T-1)$, we have
\betagin{align*}
&\sqrt{\xi+1}\geq \sqrt{T-\epsilon}\geq \frac{1}{2}T,\\
&\sqrt{\xi(\xi+1-\alphapha_1)}\leq \sqrt{(T-1)(T-\alphapha_1)}\leq \sqrt{2}T.
\end{align*}
Then we have
\betagin{align}
\int_{T-1-\epsilon}^{T-1}I\;\textrm{d}\xi&\geq C\cdot\frac{1}{\sqrt{1-\betata_1}}\cdot \frac{1-\betata_1}{\sqrt{1-\betata_1}}\cdot \int_{T-1-\epsilon}^{T-1}\frac{\textrm{d}\xi}{\sqrt{T-1-\xi}}\notag\\
&=C\cdot\frac{1}{\sqrt{1-\betata_1}}\cdot \frac{1-\betata_1}{\sqrt{1-\betata_1}}\cdot \sqrt{\xi}|_{0}^{\epsilon}\label{eq: near T-1}
\end{align}
for a uniform constant $C>0$ and arbitrary $\epsilon>0$. Since we can choose arbitrary large $\epsilon$ in \eqref{eq: near T-1}, the integration in \eqref{eq: estimate I} does not converge as $T\to \infty$. Combining the discussions above we see that $\int_0^{T-1}I\;\textrm{d}\xi$ tends to $\infty$ as $\betata_1\to 1$, i.e., $Z_{-2}$ gets pushed-off to infinity.
\end{proof}
From now on, we use $\betata_1$ as a subscript to emphasize the dependence on angles.
By \eqref{eq: eta origin expression} and \eqref{eq:varphi and tau}, on any compact subsets of $\mathbb{F}_2$ we have the following convergence in the $C^k$-norm for every $k$:
\betagin{align}
\eta_\infty:&=\lim_{\betata_1\to1}\eta_{\betata_1}\notag\\
&=2\tau_{\betata_1} \frac{\sqrt{-1}\textrm{d}z\wedge\textrm{d}\bar{z}}{(1+|z|^2)^2}+\frac{1}{2}\left(\tau_{\betata_1}-\frac{1}{\tau_{\betata_1}}\right)\left(\frac{\sqrt{-1}\textrm{d}w\wedge\textrm{d}\bar{w}}{|w|^2}+\sqrt{-1}\alphapha\wedge \bar{\alphapha}\right.\notag\\
&\left.+\sqrt{-1}\alphapha\wedge \frac{\textrm{d}\bar{w}}{\bar{w}}+\sqrt{-1}\frac{\textrm{d}w}{w}\wedge \bar{\alphapha}\right).\label{eq:convergence of eta}
\end{align}
Recall the Ricci curvature tensor of $\eta_{\betata_1}$ is given by
\betagin{equation}\label{eq: ric tensor eta}
\betagin{aligned}
\operatorname{Ric}\eta_{\betata_1}&=(1-\betata_1)[C_1]+(1-\betata_2)[C_2]+2\frac{\sqrt{-1}\textrm{d}z\wedge\textrm{d}\bar{z}}{(1+|z|^2)^2}-\sqrt{-1}\partial\bar{\partial}\log\tau_{\betata_1}\\
&-\sqrt{-1}\partial\bar{\partial}\log\varphi_{\betata_1}.
\end{aligned}
\end{equation}
The Ricci curvature $\lambda_{\betata_1}$ of $\eta_{\betata_1}$ is given by $\displaystyle \lambda_{\betata_1}=1-\betata_1$. Notice that $\lambda_{\betata_1}\to0$ as $\betata_1\to1$. Hence, by \eqref{eq: ric tensor eta} and facts that $\varphi\to(\tau^2-1)/2\tau$ in the limit and $\tau$ is a function of $r^4=|w|^2(1+|z|^2)^2$ (recall \eqref{eq: relation r u z}) we have
\betagin{align}
&2\frac{\sqrt{-1}\textrm{d}z\wedge\textrm{d}\bar{z}}{(1+|z|^2)^2}-\sqrt{-1}\partial\bar{\partial}\log\tau_{\infty}-\sqrt{-1}\partial\bar{\partial}\log\varphi_{\infty}=0,\notag\\
\Rightarrow &\tau_\infty \varphi_\infty=C|w|^2(1+|z|^2)^2,\notag\\
\Rightarrow&\tau_\infty=C^{-\frac{1}{2}}(C+r^4)^{\frac{1}{2}},\quad\textrm{for some constant}\;C>0.\label{eq: limiting tau}
\end{align}
Replacing $\tau$ and $\varphi$ in \eqref{eq: eta in limit} using \eqref{eq: limiting tau}, we have
\betagin{equation}\label{eq: eta infty}
\eta_\infty=2C^{-\frac{1}{2}}\left((C+r^4)^{\frac{1}{2}}(\sigma_1^2+\sigma_2^2)+\frac{r^2}{(C+r^4)^{\frac{1}{2}}}(\textrm{d}r^2+r^2\sigma_3^2)\right).
\end{equation}
Comparing \eqref{eq: eta infty} to \eqref{eq: eh metric in r} we see $\eta$ converges on compact subsets to an Eguchi--Hanson metric as $\betata_1\to 1$. Summarizing discussions above, we have shown the following result. It is a special case of Theorem \ref{thm: metric limit nk case} and provides a new way of understanding Eguchi--Hanson metrics.
\betagin{thm}\label{thm: limit n2 case}
Fix an arbitrary base point $p$ on the zero section. The K\"{a}hler--Einstein edge metric $(\mathbb{F}_2, \eta_{\betata_1}, p)$ on $\mathbb{F}_2$ converges in the pointed Gromov--Hausdorff sense
to the following Eguchi--Hanson metric $(-2H_{\mathbb{P}^1}, \eta_\infty), p$ on $-2H_{\mathbb{P}^1}$ as $\betata_1\to 1$:
\betagin{equation*}
\eta_\infty=2C^{-\frac{1}{2}}\left((C+r^4)^{\frac{1}{2}}(\sigma_1^2+\sigma_2^2)+\frac{r^2}{(C+r^4)^{\frac{1}{2}}}(\textrm{d}r^2+r^2\sigma_3^2)\right),
\end{equation*}
where $C>0$ is a constant.
\end{thm}
\betagin{proof}
The convergence on any compact subset of $-2H_{\mathbb{P}^1}$ of such $\eta$ to an Eguchi--Hanson metric in $C^k$-norm for every $k$ can be seen from \eqref{eq: eta infty}. Combining this fact, the fact that $Z_{-2}$ gets pushed-off to infinity as $\betata_1\to1$ and choosing an arbitrary base point from the exceptional divisor $Z_2$, we obtain the convergence in the pointed Gromov--Hausdorff sense by considering convergence of $\eta$ to an Eguchi--Hanson metric on compact geodesic balls centered at the base point.
\end{proof}
\section{Examples of limit of K\"{a}hlerE edge metrics}\label{app: more eg}
In this section, we fix $k=1$ and $n=2$. Then we follow Section \ref{sec: asym of kee} and Section \ref{sec: gh kee} to give more concrete examples as limit of K\"{a}hlerE edge metrics. In the previous work, we treated the case $\betata_1\searrow 0$ and $\betata_1 \to 1$ \cite{RZ21}. In this section, we consider the several cases: $\betata_1\nearrow 2$, $\betata_2 \nearrow 1$ with no rescaling and $\betata_2\nearrow 1$ with rescaling. The asymptotic behaviors in such cases are summarized in Table \ref{summary}.
Under the assumption $k=1$ and $n=2$, $\betata_1$ ranges from $(0, 2)$. $\tau$ ranges from $[1, T]$. We will study the limiting behaviors of K\"{a}hlerE edge metrics when $\betata_1\to 2$.
By \eqref{eq: varphi general expression nk case} $\varphi(\tau)$ satisfies
\betagin{align*}
\varphi(\tau)&=\frac{\tau^2-1}{\tau}+\frac{1}{3}(\betata_1-2)(\tau^3-1)\\
&=\frac{1}{3}(\betata_1-2)(\tau-1)(\tau-\alphapha_1)(
\tau-T)/\tau,
\end{align*}
where $T$ and $\alphapha_1$ satisfy \cite[(5.1), (5.2)]{RZ21}
\betagin{align}
\betagin{aligned}\label{eq: T a1 simcase}
T&=1+3\frac{\sqrt{1+\frac{2}{3}\betata_1-\frac{1}{3}\betata_1^2}+\betata_1-1}{4-2\betata_1},\\
\alphapha_1&=1+3\frac{-\sqrt{1+\frac{2}{3}\betata_1-\frac{1}{3}\betata_1^2}+\betata_1-1}{4-2\betata_1}.
\end{aligned}
\end{align}
Obviously $T\to +\infty$ and $\alphapha_1\to-1$ as $\betata_1\to2$. $\betata_2$ is given by \cite[(4.20)]{RZ21}
\betagin{equation*}
\betata_2=\frac{\betata_1-3+3\sqrt{1+\frac{2}{3}\betata_1-\frac{1}{3}\betata_1^2}}{2}.
\end{equation*}
Thus $\betata_2\to 1$ as $\betata_1\to 2$. More precisely, we have the following asymptotic behavior of $\betata_2(\betata_1)$:
\betagin{lem}
For $\betata_1 < 2$ and close to $2$, $\betata_2(\betata_1)=\frac{1}{2}\betata_1+o(1)$.
\end{lem}
The length of the path on each fiber between the intersection point of the fiber with $Z_1$ and that of the fiber with $Z_{-1}$ is given by the integration of $\varphi(\tau)$ from $1$ to $T$. A similar calculation as in the proof of Proposition \ref{prop: length to infty n2 case} shows the following result.
\betagin{prop}\label{prop: length infty n1 case}
The length of the path on each fiber between the intersection point of the fiber with $Z_1$ and that of the fiber with $Z_{-1}$ tends to infinity as $\betata_1\to 2$.
\end{prop}
Recall we assume K\"{a}hler--Einstein edge metrics on $\mathbb{F}_1$ have the form as in \eqref{eq: eta origin expression}. The Ricci curvature form of such K\"{a}hlerE edge metrics, denoted by $\eta$, is given by \eqref{eq: ric tensor eta}. From now on, we denote by $\eta_{\betata_1}$, $\tau_{\betata_1}$ and $\varphi_{\betata_1}$ to emphasize the dependence of metrics and coordinates on $\betata_1$. Let us consider on any compact subsets of $\mathbb{F}_1$,
\betagin{align*}
\lim_{\betata_1\to 2}\eta_{\betata_1}&=\lim_{\betata_1\to 2}\tau\frac{\sqrt{-1}\textrm{d}z\wedge\textrm{d}\bar{z}}{(1+|z|^2)^2}+\left(\frac{\tau^2-1}{\tau}+\frac{1}{3}(\betata_1-2)(\tau^3-1)\right)\left(\frac{\sqrt{-1}\textrm{d}w\wedge\textrm{d}\bar{w}}{|w|^2}\right.\\
&\left.+\sqrt{-1}\alphapha\wedge\bar{\alphapha}+\sqrt{-1}\alphapha\wedge\frac{\textrm{d}\bar{w}}{\bar{w}}+\sqrt{-1}\frac{\textrm{d}w}{w}\wedge\bar{\alphapha}\right)\\
&=\tau\frac{\sqrt{-1}\textrm{d}z\wedge\textrm{d}\bar{z}}{(1+|z|^2)^2}+\frac{\tau^2-1}{\tau}\left(\frac{\sqrt{-1}\textrm{d}w\wedge\textrm{d}\bar{w}}{|w|^2}+\sqrt{-1}\alphapha\wedge\bar{\alphapha}+\sqrt{-1}\alphapha\wedge\frac{\textrm{d}\bar{w}}{\bar{w}}\right.\\
&\left.+\sqrt{-1}\frac{\textrm{d}w}{w}\wedge\bar{\alphapha}\right).\\
&=:\eta_\infty
\end{align*}
The Ricci curvature $\lambda_{\betata_1}$ of $\eta_{\betata_1}$ is given by $\lambda_{\betata_1}=2-\betata_1$. As $\betata_1\to2$, the Ricci curvature $\lambda_{\betata_1}$ tends to $0$. Thus, the limit metric $\eta_\infty$ has Ricci curvature $0$. Hence by \eqref{eq: ric tensor eta}, $\tau_\infty$ and $\varphi_\infty$ satisfy
\betagin{align}
&2\frac{\sqrt{-1}\textrm{d}z\wedge\textrm{d}\bar{z}}{(1+|z|^2)^2}-\sqrt{-1}\partial\bar{\partial}\log\tau_{\infty}-\sqrt{-1}\partial\bar{\partial}\log\varphi_{\infty}=0,\notag\\
\Rightarrow &\tau_\infty \varphi_\infty=C|w|^4(1+|z|^2)^2,\notag\\
\Rightarrow&\tau_\infty=(1+C|w|^4(1+|z|^2)^2)^{\frac{1}{2}},\quad\textrm{for some constant}\;C>0.\label{eq: tau infty n1 case}
\end{align}
Thus we obtain the following theorem, which is a special case of Theorem \ref{thm: metric limit nk case}.
\betagin{thm}\label{thm: limit n2k1 case}
Fix an arbitrary base point $p$ on $Z_1\subset \mathbb{F}_1$. As $\betata_1\to2$, the K\"{a}hler--Einstein edge metric $(\mathbb{F}_1, \eta_{\betata_1}, p)$ on $\mathbb{F}_1$ converges in the pointed Gromov--Hausdorff sense to a Ricci-flat metric $(-H_{\mathbb{P}^1} , \eta_\infty, p)$ on $-H_{\mathbb{P}^1}$ with conic singularity of angle $4\pi$ along $Z_1$.
\end{thm}
\betagin{proof}
By \eqref{eq: tau infty n1 case}, the limit metric $\eta_\infty$ has the form
\betagin{align*}
\eta_\infty=&(1+C|w|^4(1+|z|^2)^2)^{\frac{1}{2}}\frac{\sqrt{-1}\textrm{d}z\wedge\textrm{d}\bar{z}}{(1+|z|^2)^2}\\
&+\frac{C|w|^4(1+|z|^2)^2}{(1+C|w|^4(1+|z|^2)^2)^{\frac{1}{2}}}\left(\frac{\sqrt{-1}\textrm{d}w\wedge\textrm{d}\bar{w}}{|w|^2}+\sqrt{-1}\alphapha\wedge\bar{\alphapha}+\sqrt{-1}\alphapha\wedge\frac{\textrm{d}\bar{w}}{\bar{w}}\right.\\
&\left.+\sqrt{-1}\frac{\textrm{d}w}{w}\wedge\bar{\alphapha}\right),
\end{align*}
for some constant $C>0$. Thus $\eta_\infty$ has edge singularity of angle $4\pi$ along $Z_1$. For a fixed base point on $Z_1$, Proposition \ref{prop: length infty n1 case} shows that $Z_{-1}$ gets pushed-off to infinity in the limit. The remaining proof is the same as that of Theorem \ref{thm: limit n2 case}.
\end{proof}
\subsection{Calculations in terms of $\betata_2$}
In this section, we choose a base point from the infinity section and then study the limiti behavior of the K\"{a}hlerE edge metrics. In the language of Section \ref{sec: asym of kee}, we will consider the K\"{a}hlerE edge metrics that are parametrized by $\betata_2$.
As in Section \ref{sec: asym of kee}, we consider $u:=1/w$ as the fiber coordinate. Then $\{u=0\}$ is the infinity section and $\{u=\infty\}$ is the zero section. We still define $s$ as follows:
\betagin{equation*}
s=\log(1+|z|^2)-\log|u|^2.
\end{equation*}
Then as before $\{s=-\infty\}$ still corresponds to the zero section while $\{s=+\infty\}$ corresponds to the infinity section.
Denote now by $\xi$ the K\"{a}hlerE edge metric that we seek on $\mathbb{F}_1$. Assume $\xi=\sqrt{-1}\partial\bar{\partial} f(s)$ for some smooth function $f(s)$. As \eqref{eq: xi metric} we calculate
\betagin{equation*}
\eta=\sqrt{-1}\partial\bar{\partial}f(s)=\tau\pi_1^*\omega_{\operatorname{FS}}+\varphi\left(\frac{\sqrt{-1}\textrm{d}u\wedge\textrm{d}\bar{u}}{|u|^2}+\alphapha\wedge\bar{\alphapha}-\alphapha\wedge\frac{\textrm{d}\bar{u}}{\bar{u}}-\frac{\textrm{d}u}{u}\wedge\bar{\alphapha}\right),
\end{equation*}
where $\alphapha:=\bar{z}\textrm{d}z/1+|z|^2$, $\tau=f'(s)$ and $\varphi=f''(s)$ as before. As in Section \ref{sec: asym of kee}, after a renormalization of the metric we may assume
\betagin{equation*}
\sup f'(s)=1.
\end{equation*}
We also assume $\inf f'(s)=t$, for some $t>0$. In other words, $\tau$ ranges from $[t, 1]$.
Now we calculate the Ricci curvature form $\eta$ using coordinates $u$ and $z$:
\betagin{align*}
\operatorname{Ric}\eta&=-\sqrt{-1}\partial\bar{\partial}\log\eta^2\\
&=-\sqrt{-1}\partial\bar{\partial}\log \frac{\tau\varphi}{|u|^2(1+|z|^2)^2}\\
&=(1-\betata_1)[Z_1]+(1-\betata_2)[Z_{-1}]+2\pi^*_1 \omega_{\operatorname{FS}}-\sqrt{-1}\partial\bar{\partial}\log\tau\\
&-\sqrt{-1}\partial\bar{\partial}\log\varphi.
\end{align*}
Denote the Ricci curvature by $\mu$. The K\"{a}hlerE edge equation
\betagin{equation*}
\operatorname{Ric}\eta=\mu \eta+(1-\betata_1)[Z_1]+(1-\betata_2)[Z_{-1}]
\end{equation*}
is equivalent to
\betagin{equation}\label{eq: ode in b2 case}
2-\varphi_{\tau}-\varphi/\tau=\mu\tau,\quad\tau\in[t, 1].
\end{equation}
Note that \eqref{eq: ode in b2 case} gives us the same ODE derived in \cite[(4.12)]{RZ21}.
Now let us determine boundary conditions satisfied by $\varphi(\tau)$. The same arguments in \cite[Proposition 3.3]{RZ21} give us that
\betagin{equation}\label{eq: bd cond n1k1b2}
\varphi(t)=\varphi(1)=0,\quad \varphi'(t)=\betata_1,\quad \varphi'(1)=-\betata_2.
\end{equation}
Plugging boundary conditions in \eqref{eq: ode in b2 case} implies that
\betagin{equation*}
\mu=2+\betata_2.
\end{equation*}
The solution to \eqref{eq: ode in b2 case} is
\betagin{equation}\label{eq: sol n1k1b2}
\varphi(\tau)=\frac{\tau^2-1}{\tau}+\frac{2+\betata_2}{3}\frac{1-\tau^3}{\tau}.
\end{equation}
Combining \eqref{eq: bd cond n1k1b2} and \eqref{eq: sol n1k1b2}, we obtain the dependence of $t$ and $\betata_1$ on $\betata_2$:
\betagin{align}
\betagin{aligned}\label{eq: tb1 simp case}
t&=\frac{1-\betata_2+\sqrt{(\betata_2-1)(-3\betata_2-9)}}{2(2+\betata_2)},\\
\betata_1&=\frac{3}{2}+\frac{1}{2}\betata_2-\frac{1}{2}\sqrt{(1-\betata_2)(3\betata_2+9)}.
\end{aligned}
\end{align}
Next, inspired by the result in Theorem \ref{thm: limit n2k1 case}, we study the limiting behavior of K\"{a}hlerE edge metrics when $\betata_2$ tends to $1$.
\betagin{prop}\label{prop: length finite k1 case}
The length of the path on each fiber between the intersection point of the fiber with $Z_1$ and that of the fiber with $Z_{-1}$ converges to a finite number as $\betata_2\to 1$.
\end{prop}
\betagin{proof}
As shown in Proposition \ref{prop: length to infty n2 case}, when restricted to the fiber $\{z=0\}$, the distance between $\{\tau=t\}$ and $\{\tau=1\}$ is given by
\betagin{equation}\label{eq: integrad n2k1 b2 case}
\int_t^1 \frac{1}{\sqrt{\varphi(\tau)}}\;\textrm{d}\tau=\int_t^1 \frac{\sqrt{\tau}}{\sqrt{(2+\betata_2)/3}\sqrt{(1-\tau)(\tau-\alphapha_1)(\tau-t)}}\;\textrm{d}\tau.
\end{equation}
It remains to show the integral in \eqref{eq: integrad n2k1 b2 case} converges uniformly as $\betata_2\to 1$. Near $\tau=1$, $\sqrt{\tau}/(\tau-\alphapha_1)(\tau-t)$ in \eqref{eq: integrad n2k1 b2 case} is uniformly bounded as $\betata_2\to 1$. Thus for $\epsilon>0$,
\betagin{align*}
&\int_{1-\epsilon}^1 \frac{\sqrt{\tau}}{\sqrt{(2+\betata_2)/3}\sqrt{(1-\tau)(\tau-\alphapha_1)(\tau-t)}}\;\textrm{d}\tau\\
&\leq C\int_{1-\epsilon}^1 \frac{1}{\sqrt{1-\tau}}\;\textrm{d}\tau<\infty
\end{align*}
for some uniform constant $C>0$. In other words, the integral \eqref{eq: integrad n2k1 b2 case} does not blow up near $\tau=1$. It remains to study \eqref{eq: integrad n2k1 b2 case} near $\tau=t$. We consider a change of coordinate $\xi:=\tau-t$. Then for $\epsilon>0$,
\betagin{equation*}
\int_t^{t+\epsilon} \frac{\sqrt{\tau}}{\sqrt{(2+\betata_2)/3}\sqrt{(1-\tau)(\tau-\alphapha_1)(\tau-t)}}\;\textrm{d}\tau\leq C\int_0^\epsilon \frac{\sqrt{\xi+t}}{\sqrt{\xi(\xi+t-\alphapha_1)}},
\end{equation*}
for some uniform constant $C>0$. Since
\betagin{equation*}
\lim_{\xi\to 0}\frac{\sqrt{\xi+t}}{\xi+t-\alphapha_1}=\frac{\sqrt{t}}{\sqrt{t-\alphapha_1}}\leq C,
\end{equation*}
for some uniform constant $C>0$ when $\betata_2\to 1$, we conclude that the integral \eqref{eq: integrad n2k1 b2 case} converges as $\int_0^\epsilon 1/\sqrt{\xi}$ near $\tau=t$. Hence we have finished the proof.
\end{proof}
From now on, we denote by $\xi_{\betata_2}$, $\tau_{\betata_2}$ and $\varphi_{\betata_2}$ to indicate the dependence of metrics and coordinates on $\betata_2$.
\betagin{thm}\label{thm: appthm2}
Fix an arbitrary base point $p$ on $Z_{-1}\subset \mathbb{F}_1$. As $\betata_2\to 1$, the K\"{a}hlerE edge metric $(\mathbb{F}_1, \xi_{{\betata}_2}, p)$ on $\mathbb{F}_1$ converge in the pointed Gromov--Hausdorff sense to the Fubini--Study metric $(\mathbb{P}^2, \omega_{\mathrm{FS}},p)$.
We will show that $\xi_{\betata_2}$ converges pointwise smoothly to a degenerate metric tensor that is the pull-back of the Fubini--Study metric under the blow-up map on $\mathbb{F}_1$.
\end{thm}
\betagin{proof}
Denote by $\xi_\infty$, $\tau_\infty$ and $\varphi_\infty$ the metric and coordinates in the limit when $\betata_2\to 1$. To find a relation between $\tau_\infty$, $\varphi_\infty$ and $s$, consider the ODE satisfied by $s$ and $\tau_{\betata_2}$:
\betagin{equation}\label{eq: ODE n1k1b2}
\frac{\textrm{d}s}{\textrm{d}\tau_{\betata_2}}=\frac{1}{\varphi_{\betata_2}}=\frac{\tau_{\betata_2}}{(\frac{2+\betata_2}{3})(1-\tau_{\betata_2}^3)+\tau_{\betata_2}^2-1}.
\end{equation}
Letting $\betata_2\to1$ in \eqref{eq: ODE n1k1b2}, we obtain (up to a constant that can be chosen to be $0$)
\betagin{equation}\label{eq: ot r1}
s=\log\frac{\tau_{\infty}}{1-\tau_\infty},\quad\tau_\infty\in(0, 1),
\end{equation}
where $\tau_\infty$ ranges from $(0, 1)$ since $t\to 0$ as $\betata_2\to 1$. Obviously there holds
\betagin{equation}\label{eq: ot r2}
\varphi_\infty=\tau_\infty(1-\tau_\infty).
\end{equation}
Recall $s=\log(1+|z|^2)-\log|u|^2$. Combining \eqref{eq: ot r1} and \eqref{eq: ot r2}, the limit metric $\xi_\infty$ has the form:
\betagin{equation}\label{eq_XiInfN2K1}
\betagin{aligned}
\xi_\infty&=\tau_\infty \pi_1^*\omega_{\operatorname{FS}}+\varphi_\infty \left(\pi_2^*\omega_{\operatorname{Cyl}}+\sqrt{-1}\alphapha\wedge\bar{\alphapha}-\sqrt{-1}\alphapha\wedge\frac{\textrm{d}\bar{u}}{\bar{u}}-\sqrt{-1}\frac{\textrm{d}u}{u}\wedge\bar{\alphapha}\right)\\
&=\frac{1+|z|^2}{|u|^2+1+|z|^2}\pi_1^*\omega_{\operatorname{FS}}+\frac{1+|z|^2}{(|u|^2+1+|z|^2)^2}\sqrt{-1}\textrm{d}u\wedge\textrm{d}\bar{u}\\
&+\frac{|u|^2(1+|z|^2)}{(|u|^2+1+|z|^2)^2}\left(\sqrt{-1}\alphapha\wedge\bar{\alphapha}-\sqrt{-1}\alphapha\wedge\frac{\textrm{d}\bar{u}}{\bar{u}}-\sqrt{-1}\frac{\textrm{d}u}{u}\wedge\bar{\alphapha}\right).
\end{aligned}
\end{equation}
Next, we derive the explicit formula for $\xi_\infty$. Recall,
\betagin{equation}\label{eq_AlpDefN2K1}
\alphapha = \frac{\bar{z}\textrm{d}z}{1+|z|^2}.
\end{equation}
Plugging \eqref{eq_AlpDefN2K1} in \eqref{eq_XiInfN2K1} and by calculations:
\betagin{equation}\label{eq_XiInfForm}
\betagin{aligned}
\xi_\infty &=\frac{1+|z|^2}{|u|^2+1+|z|^2} \cdot \frac{\sqrt{-1}\textrm{d}z\wedge \textrm{d}\bar{z}}{(1+|z|^2)^2} +\frac{1+|z|^2}{(|u|^2+1+|z|^2)^2}\sqrt{-1}\textrm{d}u\wedge\textrm{d}\bar{u}\\
&+\frac{|u|^2(1+|z|^2)}{(|u|^2+1+|z|^2)^2}\left(\sqrt{-1} \cdot \frac{|z|^2\textrm{d}z\wedge \textrm{d}\bar{z}}{(1+|z|^2)^2}
-\sqrt{-1}\frac{\bar{z}\textrm{d}z}{1+|z|^2}\wedge\frac{\textrm{d}\bar{u}}{\bar{u}}-\sqrt{-1}\frac{\textrm{d}u}{u}\wedge\frac{{z}\textrm{d}\bar{z}}{1+|z|^2}\right)\\
& = \sqrt{-1}\left(\frac{1+|u|^2}{(1+|u|^2+|z|^2)^2} \textrm{d}z\wedge \textrm{d}\bar{z} - \frac{\bar{z}u}{(1+|u|^2+|z|^2)^2}\textrm{d}z\wedge \textrm{d}\bar{u} - \frac{\bar{u}z}{(1+|u|^2+|z|^2)^2}\textrm{d}u\wedge \textrm{d}\bar{z}\right. \\
& + \left.\frac{1+|z|^2}{(1+|u|^2+|z|^2)^2}\textrm{d}u\wedge \textrm{d}\bar{u}
\right)\\
&=\sqrt{-1}\partial\bar{\partial} \log (1+|u|^2+|z|^2).
\end{aligned}
\end{equation}
This limit metric does not have singularity along $Z_-1$ since $\betata_2\to 1$ in the limit. Moreover, $\xi_\infty$ has Ricci curvature $3$ since $\mu_{\betata_2}$ tends to $3$ when $\betata_2\to 1$. Moreover, the metric $\xi_\infty$ degenerates along $Z_1 = \{u=\infty\}$ as $\xi_\infty\to 0$ as $|u|\to +\infty$.
We next show that $\xi_\infty$ is the pull-back of the Fubini--Study metric $\omega_{\operatorname{FS}}$ on $\mathbb{P}^2$ under the blow-down map $\pi: \mathbb{F}_1\to \mathbb{P}^2$. Figure \ref{fig_BlowUp} shows the blow up of $\mathbb{P}^2(1, 1, k)$ at $p$ giving rise to $\mathbb{F}_1$. To see this, we regard $\mathbb{F}_1$, which is the blow-up of $\mathbb{P}^2$ at one point $p$ (WLOG assuming $p=[1:0:0]$), as the variety embedded in $\mathbb{P}^2\times \mathbb{P}^1$:
\betagin{equation*}
\mathbb{F}_1 = \{([x_0:x_1:x_2], [y_0:y_1])\in \mathbb{P}^2\times \mathbb{P}^1: x_1 y_1 = x_2 y_0\}.
\end{equation*}
Then the blow-down map $\pi$ is given by:
\betagin{equation*}
\betagin{aligned}
\pi: \mathbb{F}_1 &\to \mathbb{P}^2\\
([x_0:x_1:x_2], [y_0:y_1])&\mapsto [x_0:x_1:x_2].
\end{aligned}
\end{equation*}
Restricted on the chart $\{x_1\neq 0\}$, there hold:
\betagin{equation*}
u = \frac{x_0}{x_1}, \quad z = \frac{x_2}{x_1}
\end{equation*}
and
\betagin{equation}\label{eq_PiForm}
\pi(u, z) = (u, z)\in\mathbb{P}^2.
\end{equation}
Recall the Fubini--Study metric on $\mathbb{P}^2$ (restriced to the chart $\{x_1\neq 0\}$) has the formula
\betagin{equation*}
\omega_{\operatorname{FS}} = \sqrt{-1}\log(1+|u|^2+|z|^2).
\end{equation*}
Thus by \eqref{eq_XiInfForm} and \eqref{eq_PiForm} we have shown,
\betagin{equation*}
\xi_\infty = \pi^* \omega_{\operatorname{FS}},
\end{equation*}
confirming that $\xi_\infty$ degenerates along the exceptional curve $Z_1$.
We have shown $\xi_{\betata_2}$ converges on any compact subset of $\mathbb{F}_1$ to $\xi_\infty$. Now fix an arbitrary base point on $Z_{-1}\subset\mathbb{F}_1$. By Proposition \ref{prop: length finite k1 case}, the length between $Z_1$ and $Z_{-1}$ remains to be finite when $\betata_2\to 1$. Thus $(\mathbb{F}_1, \xi_{\betata_2}, p)$ converges in the pointed Gromov--Hausdorff sense to $(\mathbb{P}^2, \omega_{\mathrm{FS}}, p)$ when $\betata_2\to 1$.
\end{proof}
\betagin{figure}[!htbp]
\centering
\includegraphics[width=10cm]{new_blow.png}
\caption{Blow up of $\mathbb{P}^2(1,1,k)$ at $p$ (with $k>1$).}
\label{fig_BlowUp}
\end{figure}
Next, we consider the limiting behavior of properly renormalized K\"{a}hlerE edge metrics on $\mathbb{F}_1$ when $\betata_2 \to 1$.
\betagin{lem}\label{lem: two t relations simcase}
Rescaling the metric $\xi$ by a factor $2(2+\betata_2)/(1-\betata_2+\sqrt{(1-\betata_2)(3\betata_2+9)})$, the interval of definition of $\tau$ will change from $[t, 1]$ to $[1, T]$ as in \eqref{eq: T a1 simcase}.
\end{lem}
\betagin{proof}
When we calculate in terms of $\betata_1$, we have
\betagin{align}
\betata_2&=\frac{\betata_1-3+3\sqrt{1+\frac{2}{3}\betata_1-\frac{1}{3}\betata_1^2}}{2},\label{eq: b2 compare}\\
T&=1+3\frac{\sqrt{1+\frac{2}{3}\betata_1-\frac{1}{3}\betata_1^2}+\betata_1-1}{4-2\betata_1}.\label{eq: big t compare}
\end{align}
When we calculate in terms of $\betata_2$, we have
\betagin{align}
\betata_1&=\frac{3}{2}+\frac{1}{2}\betata_2-\frac{1}{2}\sqrt{(\betata_2-1)(-3\betata_2-9)},\label{eq: b1 compare}\\
t&=\frac{1-\betata_2+\sqrt{(\betata_2-1)(-3\betata_2-9)}}{2(2+\betata_2)}.\label{eq: lil t compare}
\end{align}
Direct calculation shows \eqref{eq: b2 compare} and \eqref{eq: b1 compare} are equivalent. Combining \eqref{eq: b2 compare} and \eqref{eq: big t compare}, we have
\betagin{equation*}
T=\frac{2(2+\betata_2)}{4-2\betata_1}.
\end{equation*}
Combining \eqref{eq: b1 compare} and \eqref{eq: lil t compare}, we have
\betagin{equation*}
t=\frac{4-2\betata_1}{2(2+\betata_2)}.
\end{equation*}
Thus, $T=1/t$ and we can rescale the metric $\eta$ by the factor $1/t$ to change the domain of $\tau$ from $[t, 1]$ to $[1, T]$.
\end{proof}
Inspired by Lemma \ref{lem: two t relations simcase}, we normalize $\xi_{\betata_2}$ by the factor $\sqrt{3}/\sqrt{1-\betata_2}$ and study its limiting behavior when $\betata_2$ tends to $1$.
\betagin{thm}\label{thm: rescale simcase}
Rescaling the metric $\xi_{\betata_2}$ by $\sqrt{3}/\sqrt{1-\betata_2}$, then as $\betata_2\to 1$, the renormalized metric converges in the pointed Gromov--Hausdorff sense to a Ricci-flat metric on $-H_{\mathbb{P}^1}$, where the base point is chosen from $Z_1$. See the proof for an explicit explanation. This metric coincides with the one obtained in Theorem \ref{thm: limit n2k1 case}.
\end{thm}
\betagin{proof}
Consider the change of coordinate $y_{\betata_2}:=\sqrt{3}\tau_{\betata_2}/\sqrt{1-\betata_2}$ in the following ODE:
\betagin{equation*}
\frac{\textrm{d}s}{\textrm{d}\tau_{\betata_2}}=\frac{1}{\varphi_{\betata_2}(\tau_{\betata_2})},
\end{equation*}
In the following equations, we omit the subscript $\betata_2$:
\betagin{align*}
\frac{\textrm{d}s}{\textrm{d}y}\frac{\textrm{d}y}{\textrm{d}\tau}&=\frac{1}{\varphi(y)}\\
\Rightarrow \frac{\textrm{d}s}{\textrm{d}y}&=\frac{\sqrt{3}}{3}\cdot\frac{(1-\betata_2)y}{(\sqrt{1-\betata_2}y)^2-1+\frac{2+\betata_2}{3}(1-(\sqrt{1-\betata_2}y)^3)}.
\end{align*}
By an abuse of notation, still denote by $y$ the coordinate in the limit. As $\betata_2\to 1$, there holds
\betagin{equation}\label{eq: sy mid case}
\frac{\textrm{d}s}{\textrm{d}y}=\frac{y}{y^2-1},\quad y\in\left(1, +\infty\right),
\end{equation}
where the range of $y$ comes from \eqref{eq: lil t compare}. Recall the renormalized metric $\sqrt{3}\xi_{\betata_2}/\sqrt{1-\betata_2}$ reads
\betagin{equation}\label{eq: metric mid case}
\betagin{aligned}
\frac{\sqrt{3}\xi_{\betata_2}}{\sqrt{1-\betata_2}}&=\frac{\sqrt{3}\tau_{\betata_2}}{\sqrt{1-\betata_2}}\pi_1^*\omega_{\operatorname{FS}}+\frac{\sqrt{3}\varphi_{\betata_2}}{\sqrt{1-\betata_2}}\left(\pi_2^*\omega_{\operatorname{Cyl}}+\sqrt{-1}\alphapha\wedge\bar{\alphapha}-\sqrt{-1}\alphapha\wedge\frac{\textrm{d}\bar{u}}{\bar{u}}\right.\\
&\left.-\sqrt{-1}\frac{\textrm{d}{u}}{{u}}\wedge\alphapha\right).
\end{aligned}
\end{equation}
Combining \eqref{eq: sy mid case} and \eqref{eq: metric mid case} we obtain the limit metric
\betagin{equation*}
\betagin{aligned}
\tilde{\xi}_\infty&=\sqrt{e^{2s}+1}\pi_1^*\omega_{\operatorname{FS}}+\frac{e^{2s}}{\sqrt{e^{2s}+1}}\left(\pi_2^*\omega_{\operatorname{Cyl}}+\sqrt{-1}\alphapha\wedge\bar{\alphapha}-\sqrt{-1}\alphapha\wedge\frac{\textrm{d}\bar{u}}{\bar{u}}\right.\\
&\left.-\sqrt{-1}\frac{\textrm{d}{u}}{{u}}\wedge\alphapha\right).
\end{aligned}
\end{equation*}
Note that this limit metric is Ricci-flat. It coincides with the metric obatined in Theorem \ref{thm: limit n2k1 case} if we choose $C=1$ there. We have shown the renormalized K\"{a}hlerE edge metrics converge to $\tilde{\xi}_\infty$ in smooth local sense. Next, fix a base point from the zero section $Z_1$. Then by Proposition \ref{prop: length finite k1 case}, we conclude that the infinity section $Z_{-1}$ gets pushed-off to infinity in the limit. Combining this fact with the local smooth convergence, we conclude the pointed Gromov--Hausdorff limit of $(\mathbb{F}_1, \sqrt{3}\eta_{\xi_2}/\sqrt{1-\betata_2})$ is $(-H_{\mathbb{P}^1}, \tilde{\xi}_\infty)$.
\end{proof}
{\sc University of Maryland}
{\tt [email protected], [email protected]}
{\sc Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, P. R. China.}
{\tt [email protected]}
\end{document} | math | 126,965 |
\begin{document}
\title[Robust unfoldings]{~
Robust degenerate unfoldings of cycles and tangencies}
\author[Barrientos]{Pablo G. Barrientos}
\address{\centerline{Instituto de Matem\'atica e Estat\'istica, UFF}
\centerline{Rua M\'ario Santos Braga s/n - Campus Valonguinhos, Niter\'oi, Brazil}}
\email{[email protected]}
\author[Raibekas]{Artem Raibekas~
}
\address{\centerline{Instituto de Matem\'atica e Estat\'istica, UFF}
\centerline{Rua M\'ario Santos Braga s/n - Campus Valonguinhos, Niter\'oi, Brazil}}
\email{[email protected]}
\subjclass[2010]{Primary 58F15, 58F17; Secondary: 53C35.}
\keywords{Homoclinic tangencies, heterodimensional cycle,
blenders, parablenders}
\begin{abstract}
We construct open sets of {degenerate} unfoldings
of heterodimensional cycles of any co-index $c>0$ and homoclinic
tangencies of arbitrary codimension $c>0$.
These {type of} sets are known to be the support
of unexpected phenomena in families of diffeomorphisms, such as
the Kolmogorov typical co-existence of infinitely many attractors.
{As a prerequisite}, we also construct robust homoclinic
tangencies of large codimension
{which cannot be inside a strong partially
hyperbolic set}.
\end{abstract}
\setcounter{tocdepth}{1}
\maketitle
\section{Introduction}
Robust homoclinic tangencies and robust heterodimensional cycles
are, in general, pre\-requisites for obtaining abundant
complicated dynamical systems~\cite{New70,GS72,New74,BD08,BD12}.
Both configurations imply the existence of a non-transversal
intersection between the stable and unstable manifolds of points
in the same or in different transitive hyperbolic sets. A priori,
the non-transverse intersection could be destroyed by a small
perturbation. But since it is robust, this means that a new
non-transverse intersection is created between the manifolds of
the continuation of the hyperbolic sets. The unfolding of these
bifurcations yields a great number of changes in the dynamics. For
instance, infinitely many saddle periodic points and sinks appear
in the unfolding of homoclinic tangencies. Hence, the persistence
of these bifurcations allowed to get a generic coexistence of
infinitely many periodic attractors~\cite{New79,GTS93,PV94,GST08}.
The construction of robust tangencies in lower dimension is based
on the creation of thick horseshoes involving distortion estimates
which are typically $C^2$. However, in higher dimensions it was
possible to construct robust homoclinic tangencies in the
$C^1$-topology using blenders~\cite{Ao08,BD12}. Blenders are
hyperbolic sets having a thicker invariant manifold than initially
expected. They were discovered by Bonatti and Diaz~\cite{BD96} and
now are essential objects in the study of non-hyperbolic dynamics.
On the other hand, all of the above mentioned constructions are of
codimension one. That is, the dimension of the coincidence of the
tangent spaces at the tangent point. Recently in~\cite{BR17}, the
authors gave the first examples of $C^2$-robust tangencies of
large codimension. The novelty in the construction was the use of
the blender for the dynamics induced in the tangent bundle.
A different approach, when compared to the generic results
mentioned above, is to look for bifurcations of homoclinic
tangencies in \textit{parametric families} of diffeomorphisms. For
decades it was thought that the coexistence of infinitely many
hyperbolic attractors was meager in families of dynamical
systems~\cite{PS95}. However, recently and far from intuition,
Berger showed in~\cite{Ber16} that actually these phenomena form a
residual set.
Behind this result was the construction of open sets of families
of endomorphisms with robust homoclinic tangencies and with an
extra property: the family unfolds {degenerately} a tangency.
This means the unfolding is slow in the sense of the zeroing of
the first terms in a certain Taylor polynomial describing the
local separation between the manifolds. Although a {degenerate}
unfolding of a tangency could be destroyed by a small perturbation
of the family, this perturbation has another tangency which
unfolds {also degenerately}. The mechanism involved in these
constructions of Berger~\cite{Ber16} is also the blender, but now
constructed for the dynamics induced in the space of jets (the
space of velocities). See also \cite{BCP17,Ber17}.
The objective of the present work is to unite the construction
of~\cite{BR17} and \cite{Ber16} to obtain robust {degenerate}
unfoldings of homoclinic tangencies of large codimension for
families of diffeomorphisms. Only this will not be done by merely
combining the two previous results. Here we present a new method
of construction of robust tangencies of large codimension,
different from the one in~\cite{BR17}. This is a generalization to
higher codimension of the construction in~\cite{BD12} using
folding manifolds.
It is expected that the unfolding of these robust degenerated
tangencies gives new dynamical consequences. For example, the
existence of residual sets with infinitely many attracting
invariant tori of large dimension.
\subsection{Dimension and codimension of an intersection}
{Let us begin with some definitions on the intersections of
submanifolds.} Let $\mathcal{L}$ and $\mathcal{S}$ be submanifolds
of $\mathcal{M}$. {We say that $\mathcal{L}$ and $\mathcal{S}$
have an intersection of \emph{dimension} $d\geq 0$ at $x\in
\mathcal{L}\cap \mathcal{S}$, if
$$
d=d_x(\mathcal{L},\mathcal{S})\eqdef \dim T_x\mathcal{L}\cap
T_x\mathcal{S}.
$$
Notice that $d$ is the maximum number of common linearly
independent tangent directions in $T_x\mathcal{M}$. However, this
number is not enough to measure how far the intersection is from
being transverse. In order to quantify this
we say that
$\mathcal{L}$ and $\mathcal{S}$ have an intersection of
\emph{codimension} $c\geq 0$ if
\begin{equation*}
\label{codimension}
c=c_x(\mathcal{L},\mathcal{S})\eqdef \dim \mathcal{M}- \dim
\left(T_x\mathcal{L}+T_x\mathcal{S}\right).
\end{equation*}
An intersection of codimension zero is called \emph{transverse}
and is said to be \emph{tangencial} otherwise. Observe that the definition of
a tangency ($c>0$) includes the case $d=0$ in which
$T_x\mathcal{L}+T_x\mathcal{S}=T_x\mathcal{L}\oplus T_x
\mathcal{S}$. In the literature this case is called as a
\emph{quasi-transverse} intersection. On the other hand, the sum
of the dimension and the codimension of an intersection between
submanifolds is in general not equal to the dimension of
$\mathcal{M}$. Moreover, when the codimension of $\mathcal{L}$
coincides with the dimension of $\mathcal{S}$, then if $x\in
\mathcal{L}\cap \mathcal{S}$
$$
c_x(\mathcal{L},\mathcal{S})= \dim \mathcal{M}- (\dim T_x\mathcal{L} +
\dim T_x \mathcal{S} - \dim T_x\mathcal{L} \cap T_x
\mathcal{S}) = d_x(\mathcal{L},\mathcal{S}).
$$
}
\begin{figure}
\caption{
Intersection of two submanifolds in a three dimensional space.
Figure~\ref{fig1a}
\label{fig1a}
\label{fig1b}
\label{fig11}
\end{figure}
\subsection{Heterodimensional cycles and homoclinic tangencies}
A $C^r$-diffeomorphism $f$ of a manifold $\mathcal{M}$ has a \emph{homoclinic tangency of codimension $c>0$}
if there is a pair of points $P$ and $Q$, in the same transitive
hyperbolic set, so that the unstable invariant manifold of $P$ and
the stable invariant manifold of $Q$ have {an intersection of
codimension~$c$} at a point $Y$. That~is,
$$
Y \in W^u(P)\cap W^s(Q) \quad \text{and} \quad c= c_Y( W^u(P),W^s(Q)).
$$
{Observe that actually, as the codimension of
$W^u(P)$ coincides with the dimension of $W^s(Q)$ we have that also in
this case
$$
c=d_Y(W^u(P),W^s(Q)).
$$}
Similarly, $f$ has a \emph{heterodimensional cycle of co-index
$c>0$} if there exist two transitive hyperbolic sets $\Lambda$ and
$\Gamma$ such that their invariant manifolds meet cyclically and
$|\mathrm{ind}^s(\Lambda)-\mathrm{ind}^s(\Gamma)|=c$. Here
$\mathrm{ind}^s(\cdot)$ denotes the dimension of the stable bundle
of the respective set.
{By means of an arbitrarily small perturbation if
necessary, the stable and unstable manifolds have a
transverse intersection of dimension $c$ and a tangency which is
a quasi-transverse intersection of codimension $c$. Indeed,
we can assume that
$\mathrm{ind}^s(\Lambda)-\mathrm{ind}^s(\Gamma)=c$
and for $P\in
\Lambda$, $Q\in \Gamma$ suppose that $Y$ belongs to
$W^s(P)\cap W^u(Q)$. Then $\dim T_YW^s(P)+\dim T_YW^u(Q)=\dim
\mathcal{M}+c$ and thus, in general,
$$
d_Y(W^s(P),W^u(Q))=c \quad \text{and} \quad c_Y(W^s(P),W^u(Q))=0.
$$
On the other hand, if $Y\in W^u(P)\cap W^s(Q)$ then $\dim
T_YW^u(P)+\dim T_YW^s(Q)=\dim \mathcal{M}-c$ and hence,
$$
d_Y(W^u(P),W^s(Q))=0 \quad \text{and} \quad
c_Y(W^u(P),W^s(Q))=c.
$$
}
\subsection{Robust tangencies of large codimension}
{By Kupka-Smale's theorem, $C^r$-generically, the stable and
unstable manifolds of a pair of saddle hyperbolic periodic points
meet transversally. Hence, tangencies associated with saddles
occurs in the complement of a residual set of diffeomorphisms and
thus are non-generic dynamical configurations. However, the
situation is different if instead of the periodic saddles we
consider non-trivial hyperbolic sets.
It is well-known the existence of open sets of diffeomorphisms
displaying non-transverse intersections between the stable and
unstable manifolds of points in the continuations of these
hyperbolic sets.
That is, the so-called $C^r$-open sets of diffeomorphisms with
robust tangencies or robust heterodimensional cycles.
See~\cite{New70,GTS93,PV94,BD12} for robust homoclinic tangencies
of codimension one and \cite{BD08} and reference therein for
robust heterodimensional cycles.}
Robust homoclinic tangencies of large codimension in the
$C^2$-topology were recently discovered in~\cite{BR17} inside
\emph{strong partially hyperbolic sets}. That is, invariant sets
with a dominated splitting of the form $E^s\oplus E^c \oplus E^u$,
where $E^s$ and $E^u$ are the non-trivial contracting and
expanding bundles respectively. Here, we will construct new
examples
of a different nature from the robust tangencies of large
codimension showed in~\cite{BR17}. This is because they cannot be
embedded inside a strong partially hyperbolic set. At the point of
tangency,
the splitting is of the form $E^s\oplus E^c$, where $E^c$ cannot
be divided into neither contracting nor expanding subbundles. In
this case, we say that the tangency is inside a \emph{weak
partially hyperbolic set}.
\begin{mainthm}
\label{thmD} Every manifold of dimension $m> c^2+c$ admits a
diffeomorphism having a $C^{2}$-robust homoclinic tangency of
codimension $c>0$ inside a weak partially hyperbolic set.
\end{mainthm}
Notice that the above theorem gives as a particular case the
well-known results~\cite{GTS93,PV94,Ro95} about $C^2$-robust
homoclinic tangencies of codimension one in higher dimensions.
Here, we provide a different proof inspired by the construction of
$C^1$-robust homoclinic tangencies of Bonatti and D\'iaz
in~\cite{BD12}. The concepts of \emph{folding manifolds} and
\emph{blenders} constructed in the \emph{tangent bundle} allows us
to extend their result to large codimension.
\subsection{{Degenerate} unfoldings}
A {tangency} at a point $Y$ between the unstable manifold
$W^u(P)$ and the stable manifold $W^s(Q)$ of a
$C^r$-diffeomorphism $f$ can be unfolded by considering
$C^d$-families $(f_a)_a$ of $C^r$-diffeomorphisms
pa\-ra\-me\-te\-ri\-zed by $a\in \mathbb{I}^k$ with $f_0=f$ and
$\mathbb{I}=[-1,1]$. {We will suppose $0< d\le r<\infty$ and
$k\geq 1$.}
Many articles usually impose a generic condition on the velocity
of the unfolding. {They assume} that the distance between the
manifolds has positive derivative with respect to the parameter:
$$
\frac{d\delta}{da}(0)\not=0 \quad \text{where} \ \
\delta(a)=\min \{d(x,y): x\in W^u(P_a)\cap U, \ \ y\in W^s(Q_a)\cap U\}.
$$
Here $P_a$, $Q_a$ are the continuations of the hyperbolic saddles
$P_0=P$ and $Q_0=Q$ for $f_a$ respectively and $U$ is a small
neighborhood of $Y$. However, in this work we are interested in
studying {unfoldings} where this generic assumption fails.
{Let us consider first the case that the family $f=(f_a)_a$
unfolds a heterodimensional cycle of co-index $c>0$ at $a=0$. That
is,
the above points $P_0$ and $Q_0$ now belong, respectively, to
transitive hyperbolic sets $\Lambda_0$ and $\Gamma_0$ of $f_0$
with co-index $c$ and whose stable and unstable manifolds
intersect cyclically. Moreover, say that $Y\in W^u(P_0)\cap
W^s(Q_0)$ is the tangency of the cycle (i.e, the intersection that
in general could be assumed quasi-transverse and of codimension
$c$). The unfolding of this heterodimensional cycle
is said to be \emph{$C^d$-{degenerate}} at $a=0$
if there exist
$$
p_a \in W^u_{loc}(P_a) \ \ \text{and} \ \ q_a\in W^s(Q_a) \ \
\text{so that} \ \ d(p_a,q_a)=o(\|a\|^{d})\ \ \text{at $a=0$}
$$
where
$p_0=q_0=Y$ and $p_a$, $q_a$ vary $C^d$-continuously with respect
to the parameter $a\in \mathbb{I}^k$.}
{Now we consider that the family $f=(f_a)_a$
unfolds a homoclinic tangency of codimension $c>0$ at $a=0$. That
is, we assume
the points $P_0$ and $Q_0$ belong to the same hyperbolic set
$\Lambda_0$ of $f_0$ and that the homoclininc tangency $Y \in
W^u(P_0)\cap W^s(Q_0)$ has {codimension} $c>0$. The unfolding of
this homoclinic tangency of codimension $c>0$ is said to be
\emph{$C^d$-{degenerate}} at $a=0$ if there are points $p_a \in
W^u_{loc}(P_a)$, $q_a \in W^s(Q_a)$ and $c$-dimensional subspaces
$E_a$ and $F_a$ of $T_{p_a}W^u(P_a)$ and $T_{q_a}W^s(Q_a)$
respectively such that
$$
d(p_a,q_a)=o(\|a\|^{d}) \quad \text{and} \quad
d(E_a,F_a)={o(\|a\|^{d})} \quad \text{at $a=0$}.
$$
Here,} $p_0=q_0=Y$ and $(p_a,E_a)$, $(q_a,F_a)$ vary
$C^d$-continuously with respect to the parameter $a\in
\mathbb{I}^k$. Observe that in this case it is necessary to assume
that $d<r$ because the above definition involves the dynamics of
the family $(f_a)_a$ in the tangent bundle (in fact, in certain
Grassmannian bundles). In~\cite{Ber16} $C^d$-degenerate unfoldings
of homoclinic tangencies were called for short
\emph{$C^d$-paratangencies}. {The key consequence of having a
$C^d$-paratangency at $a=0$ is that one can perturb the family and
obtain a new family which now has a \emph{persistent homoclinic
tangency} in the sense of~\cite{Ber17}. That is, a tangency point
$Y_a$ between the stable and unstable manifold which varies
$C^d$-continuously with respect to $a$ in an open set of
parameters $J\subset \mathbb{I}^k$ containing $a=0$.
}
\subsection{Open sets of families with {degenerate} unfoldings}
{In order to be more precise, we now introduce the following
definitions. The exact notion of a $C^d$-family of
$C^r$-diffeomorphisms and the $C^{d,r}$-topology considered is
going to be defined in \S\ref{sec:topology}. }
A $k$-parameter $C^{d}$-family $f=(f_a)_a$ of
$C^r$-diffeomorphisms $f_a$ {displays} a
\begin{enumerate}[leftmargin=0.9cm,itemsep=0.2cm]
\item[-] \emph{$C^{d,r}$-robust $C^d$-{degenerate} unfolding of
a homoclinic tangency of codimension $c$} at $a=0$ if there are a
transitive hyperbolic set $\Lambda_0$ of $f_0$ and a
$C^{d,r}$-neighborhood $\mathscr{U}$ of $f$, such that any
$g=(g_a)_a \in \mathscr{U}$ {displays} a $C^d$-{degenerate}
unfolding of a homoclinic tangency of codimension $c>0$ at $a=0$
associated with the continuations of $\Lambda_{0}$ for $g_0$.
\item[-] \emph{$C^{d,r}$-robust $C^d$-{degenerate} unfolding of a
heterodimensional cycle of co-index $c$} at $a=0$ if there are
transitive hyperbolic sets $\Lambda_0$ and $\Gamma_0$ of $f_0$
with co-index $c$ and a $C^{d,r}$-neighborhood $\mathscr{U}$ of
$f$, such that any $g=(g_a)_a \in \mathscr{U}$ {displays} a
$C^d$-{degenerate} unfolding of a heterodimensional cycle of
co-index $c>0$ at $a=0$ associated with the continuations of
$\Lambda_{0}$ and $\Gamma_0$ for $g_0$.
\end{enumerate}
For simplicity, we have chosen $a=0$ as the critical parameter of
the unfolding. However, {degenerate} unfoldings can also be
introduced at any other parameter $a=a_0$ with $a_0\in
\mathbb{I}^k$. We say that $f=(f_a)_a$ {displays} a
$C^{d,r}$-robust $C^d$-{degenerate} unfolding of a
heterodimensional cycle (or a tangency) at \emph{any parameter}
when any $g=(g_a)_a \in \mathscr{U}$ has a heterodimensional cycle
(or a {homoclinic} tangency) at $a=a_0$ which unfolds
$C^d$-{degenerate} for all $a_0\in \mathbb{I}^k$. Robust
{degenerate} unfoldings at any parameter are involved in
unexpected phenomena as the typical coexistence of infinitely many
sinks~\cite{Ber16,Ber17}, infinitely many non-hyperbolic strange
attractors~\cite{R17} and fast growth of periodic
points~\cite{Ber19} among others.
\begin{mainthm}
\label{thmC} Any manifold of dimension $m> c^2+c$ admits a
$k$-parameter $C^d$-family of $C^r$-diffeomor\-phisms with
{$0<d<r-1$}, which {displays} a $C^{d,r}$-robust
$C^d$-{degenerate} unfolding of a homoclinic tangency of
codimension $c>0$ at any parameter.
\end{mainthm}
We will also show the existence of robust {degenerate}
unfoldings of heterodimensional cycles of any co-index at any
parameter:
\begin{mainthm}
\label{thmA} Any manifold of dimension $m> 1+c$ admits a
$k$-parameter $C^d$-family of $C^r$-diffeomor\-phisms with {$0<d<
r$} which {displays} a $C^{d,r}$-robust $C^d$-{degenerate}
unfolding of a heterodimensional cycle of co-index $c>0$ at any
parameter.
\end{mainthm}
{The differences in the regularity and dimension that appear in
the above theorems come from the nature of the unfolding of the
{tangency}, as we now explain. With respect to the
regularity assumption in Theorem~\ref{thmA}, the unfolding of a
heterodimensional cycle only deals with the distance in the
ambient manifold. There is a loss of a derivative ($d<r$) in the
moment that we pass to study the kinematic of the movement by
lifting the family to the space of velocities (jet space). This is
because we will need that the induced dynamics in the jet space is
a $C^1$-diffeomorphism. On the other hand, the unfolding of
tangencies in Theorem~\ref{thmC} requires first to lift the
dynamics to the space where the bifurcation is produced, that is
to the Grassmannian bundle.
After that one needs to perform a similar analysis in the space of
velocities to study the corresponding $C^d$-{degenerate}
unfolding. This provides a loss of two degrees of regularity
($d<r-1$) as is claimed in Theorem~\ref{thmC}. Moreover, we will
need to use Theorem~\ref{thmD} to obtain Theorem~\ref{thmC}, thus
explaining the dimension of the manifold, $m> c^2+c$, as is
related to the codimension of the tangency.}
\subsection{Topology of families of diffeomorphisms}
\label{sec:topology} Set $\mathbb{I}=[-1,1]$. Given $0<d \leq
r\leq \infty$, $k\geq 1$ and a manifold $\mathcal{M}$, we denote
by $C^{d,r}(\mathbb{I}^k,\mathcal{M})$ the space of $C^d$-families
$f=(f_a)_a$ of $C^r$-diffeomorphisms $f_a$ of $\mathcal{M}$
parameterized by $a\in\mathbb{I}^k$ such that
$$
\partial^i_a \partial^j_x f_a(x) \ \ \text{exists continuously
for all $0\leq i\leq d$, \ \ $0\leq i+j\leq r$ \ \ and \ \
$(a,x)\in \mathbb{I}^k\times \mathcal{M}.$}
$$
{We endow this space with the topology given by the
$C^{d,r}$-norm
$$
\|f\|_{{C}^{d,r}}=\max\{\sup \|\partial^i_a\partial_x^j f_a (x):
\, 0\leq i \leq d, \ 0\leq i + j \leq r\} \quad \text{where \
$f=(f_a)_a \in {C}^{d,r}(\mathbb{I}^k,\mathcal{M})$.}
$$
}
In what follows we restrict our attention to $C^{d}$-families
$f=(f_a)_a$ of $C^r$-diffeomorphisms $f_a$ of a manifold
$\mathcal{M}$ of dimension $m\geq 3$.
\subsection{Structure of the paper}
Section \S\ref{sec:blender} contains the definition of a
\emph{blender}, one of the main tools in this paper.
In section~\S\ref{sec:tangencias} we prove Theorem~\ref{thmD}.
After that, we describe formally the notion of {degenerate}
unfoldings in section~\S\ref{sec:unfoding}. In
\S\ref{sec:parablender} we recall and develop the notion of
\emph{parablenders}, the second main tool of the paper. Finally in
sections \S\ref{sec:cycles} and \S\ref{sec:final} we prove
{Theorems}~\ref{thmA} and~\ref{thmC} respectively.
\section{Blenders}
\label{sec:blender} We attribute the following definition to
Bonatti and D\'iaz (see~\cite{BBD16}). Blenders were initially
defined having {central dimension $c=1$}
(see~\cite{BD96,BDV05,BD12}) and blenders with large
{central dimension} were first studied
in~\cite{NP12,BKR14,BR17}. {After that, they also appeared
in~\cite{BR18,ACW17} and in holomorphic dynamics
in~\cite{biebler2016persistent,dujardin2017non,taflin2017blenders}.}
\begin{defi} \label{def:blender} Let $f$ be a $C^r$-diffeomorphism of a manifold $\mathcal{M}$.
A {non-empty} compact set $\Gamma \subset \mathcal{M}$ is a
\emph{$cs$-blender} of {central dimension $c\geq 1$} if
\begin{enumerate}
\item $\Gamma$ is a transitive, {maximal invariant hyperbolic set
in the closure of a neighborhood $\mathcal U$ having a partially
hyperbolic splitting}
$$
\text{$T_\Gamma \mathcal{M}=
E^{ss} \oplus E^c \oplus E^{u}$}
$$
where $E^s=E^{ss}\oplus E^c$ is the stable bundle,
${d_{ss}}=\dim E^{ss}\geq 1$ and $c=\dim E^{c}\geq 1$,
\item there exists a {non-empty} open set $\mathscr{D}$ of $C^1$-embeddings of
$d_{ss}$-dimensional discs into $\mathcal{M}$, and
\item there exists a $C^1$-neighborhood $\mathscr{U}$ of $f$,
\end{enumerate}
such that
$$
W^u_{loc}(\Gamma_g) \cap \mathcal{D} \not = \emptyset \quad \text{for all
$\mathcal{D}\in \mathscr{D}$ and $g\in \mathscr{U}$}
$$
where $\Gamma_g$ is the continuation of $\Gamma$ for $g$ and
{$W^u_{loc}(\Gamma_g)=\{x\in \mathcal{U}: g^{-n}(x)\in \mathcal{U}
\ \text{for all $n\geq 0$}\}$}. The set $\mathscr{D}$ is called
{a} \emph{superposition region} of the blender. Finally, a
\emph{$cu$-blender of {central dimension} $c$} is $cs$-blender of
{central dimension} $c$ for $f^{-1}$.
\end{defi}
The hyperbolicity of $\Gamma$ implies that given a point $x\in
W^u_{loc}(\Gamma) \cap \mathcal{D}$, there is a point $z\in
\Gamma$ such that $x \in W^u_{loc}(z)\cap \mathcal{D}$. Observe
that the local unstable manifold of $z$ is a $C^1$-embedded disc
of dimension {$d_u=\dim E^u$} and $\mathcal{D}$ is a
{$d_{ss}$}-dimensional disc. These two discs are
in {\emph{relative general position}} if it holds that
$$
T_x W^u_{loc}(z) + T_x \mathcal{D} = T_x W^u_{loc}(z)\oplus
T_x\mathcal{D}.
$$
In this case,
we have an {intersection} of codimension
$$
c_x(W^u_{loc}(z),\mathcal{D})=\dim \mathcal{M} - \dim(T_x W^u_{loc}(z)+ T_x\mathcal{D}) =
\dim \mathcal{M} - {(d_u+ d_{ss})}=c \geq 1.
$$
{Thus, $W^u_{loc}(z)$ and $\mathcal{D}$
have a tangency of codimension at least $c$, which is in general,
a quasi-transverse intersection of codimension exactly $c$.}
\subsection{Covering criterium} \label{sec:covering-criterium}
In \cite{BKR14,BR17} blenders of large {central dimension} were
constructed by using the covering criterium. Namely, we {consider}
$C^1$-diffeomorphisms
which are locally defined as a skew-product as explained below.
First, we consider a $C^1$-diffeomorphism $F$ of a manifold
$\mathsf{N}$ having a horseshoe $\mathsf{\Lambda}$ {contained in a
local chart} which is the maximal $F$-invariant set in the
closure of some bounded open set $\mathsf{R}$ of $\mathsf{N}$. The
horseshoe has stable index (dimension of the stable bundle) equal
to $d_{ss}=\mathrm{ind}^s(\mathsf{\Lambda})>0$ and satisfies that
\begin{enumerate}
\item $F|_{\Lambda}$ is conjugate to a shift of $\kappa$-symbols
and
\item there exists $0<\nu<1$ such that
\begin{equation} \label{eq:nu}
m(DF(x)) \leq \nu < 1<\nu^{-1} \leq \|DF(x)\| \qquad \text{for all $x\in
\mathsf{\Lambda}$.}
\end{equation}
\end{enumerate}
Here $m(T)=\|T^{-1}\|^{-1}$ denotes the co-norm of a linear
operator $T$. Let $\{\mathsf{R}_1,\dots,\mathsf{R}_\kappa\}$ be an
open covering of $\Lambda$, whose intersection with $\Lambda$ is a
Markov partition. There is no loss of generality in assuming that
$\mathsf{R}=\mathsf{R}_1\cup\dots\cup \mathsf{R}_\kappa$
{with} $\mathsf{R}_\ell=(-2,2)^{d_{ss}}\times
I_\ell$, where $I_\ell$ is a product of {$d_u$} open intervals in
$[-2,2]$ with $\dim \mathsf{N}= d_{ss}+ {d_{u}}$ for
$\ell=1,\dots,\kappa$. {Moreover, from the hyperbolicity of
$\Lambda$, we can assume that there is a $DF^{-1}$-invariant
cone-field on $\mathsf{R}$:
\begin{enumerate}[start=3]
\item there exist $\alpha>0$ such that
$$DF^{-1}(x)\mathcal{C}^{ss}_\alpha \subset
\mathcal{C}^{ss}_{\nu^2\alpha} \quad \text{for all
$x\in \mathsf{R}$.}
$$
\end{enumerate}
Here, for a given $\theta>0$, we denote
\begin{equation} \label{strong-stable-cone}
\mathcal{C}^{ss}_\theta\eqdef
\{(u,v)\in
\mathbb{R}^{d_{ss}}\oplus
\mathbb{R}^{d_u}: \, \|v\|<\theta\|u\|
\} \cup \left\{0\right\}.
\end{equation}
We will call $\mathcal{C}^{ss}_\alpha$ as a \emph{stable cone-field
on $\mathsf{R}$ of $F$} and refer to the parameter
$\alpha$ as the \emph{width} of the cone. }
Now take $C^1$-diffeomorphisms $\phi_1,\dots,\phi_\kappa$ of
another manifold $M$ of dimension $c>0$, which are local
$(\lambda,\beta)$-contractions in a bounded open set $D \subset
M$, with $0<\lambda<\beta<1$:
$$
\phi_\ell(\overline{D})\subset D \quad \text{and} \quad
\lambda<m(D\phi_\ell(y))<\|D\phi_\ell(y)\|<\beta<1 \quad \text{for
all $y\in \overline{D}$ \ and \ $\ell=1,\dots,\kappa$.}
$$
Finally, we consider a $C^1$-diffeomorphism $\Phi$ of
$\mathcal{M}=\mathsf{N}\times M$ locally defined as a skew-product
$$
\Phi=F\ltimes (\phi_1,\dots,\phi_\kappa) \quad \text{on} \ \
\mathcal{U}=(\mathsf{R}_1 \times D) \cup \dots \cup (\mathsf{R}_\kappa \times D)
$$
so that
$$
\Phi(x,y)=(F(x),\phi(x,y)) \quad \text{with \ \ $\phi(x,y)=\phi_\ell(y)$ \ \ if \ \ $(x,y)\in \mathsf{R}_\ell\times
D$.}
$$
\begin{notation}
In the rest of the paper, we will use the notation
$$
\Psi=G\ltimes(\psi_1,\dots,\psi_\kappa) \quad \text{on} \ \
\mathcal{V}=\mathcal{V}_1\cup \dots \cup \mathcal{V}_\kappa
$$
to define the skew-product map $\Psi(x,y)=(G(x),\psi_\ell(y))$
with $(x,y)\in \mathcal{V}_\ell$ for $\ell=1,\dots,\kappa$, {where
$\mathcal{V}_1, \dots, \mathcal{V}_\kappa$ are pairwise disjoint
sets.}
\end{notation}
The following theorem {from~\cite[Thm.~C]{BKR14} and
\cite[Thm.~3.8]{BR17}} shows that under the assumption of
domination and the covering criterium, the map $\Phi$ has a
$cs$-blender of {central dimension $c\geq 1$}.
\begin{thm}
\label{thmBKR} Let $\Phi$ be a $C^1$-diffeomorphism of a manifold
$\mathcal{M}$ locally defined as a skew-product $\Phi=F\ltimes
(\phi_1,\dots,\phi_\kappa)$ on $\mathcal{U}$ as above. Assume that
\begin{enumerate}[itemsep=0.05cm]
\item \label{item:domination} the hyperbolic base $F|_{\mathsf{\Lambda}}$ dominates the
fiber dynamics $\phi_\ell$, i.e, it holds that $\nu<\lambda$,
\item \label{item:cover} there exists an open
set $B \subset D$ such that $
\overline{B} \subset \phi_1(B) \cup \dots \cup \phi_\kappa(B).
$
\end{enumerate}
Then the maximal invariant set $\Gamma$ of $\Phi$ in
$\overline{\mathcal{U}}$ is a $cs$-blender of
{central dimension} $c$. {The superposition
region of the blender is the family of $( {\alpha,}
\nu,\delta)$-horizontal, $d_{ss}$-dimensional $C^1$-discs into
$\mathcal{B}=\mathsf{R}\times B$, where $0<\delta<\lambda L/2$
and $L>0$ is the Lebesgue number of the cover of $B$
in~\eqref{item:cover}. }
\end{thm}
The open set $\mathcal{B}=\mathsf{R}\times B$ of $\mathcal{M}$ is
called {a \emph{superposition domain}. Also, in the above theorem
appears the notion of a family of $( {\alpha,}
\nu,\delta)$-horizontal discs in $\mathcal{B}$ that we define as
follows.}
{
A proper $C^r$-embedded $d_{ss}$-dimensional
disc $\mathcal{D}$ into $\mathcal{B}=\mathsf{R}\times B$ (or a
\emph{$d_{ss}$-dimensional $C^r$-disc in $\mathcal{B}$} for short)
will be an injective $C^r$-immersion $\mathcal{D}: [-2,2]^{d_{ss}}
\to \overline{\mathcal{B}}$ of the form,
$$
\mathcal{D}(\xi)=(\xi,g(\xi),h(\xi))\in [-2,2]^{d_{ss}}\times I_\ell \times
B \quad \text{for} \ \ \xi \in [-2,2]^{d_{ss}} \ \text{and some} \ \ell\in \{1,\dots,\kappa\}.
$$
As usual, we will identify the embedding $\mathcal{D}$ with its
image $\mathcal{D}([-2,2]^{d_{ss}})$.}
\begin{defi} \label{def:almost-horizontal-disc}
We say that a $d_{ss}$-dimensional $C^1$-disc $\mathcal{D}$ in
$\mathcal{B}=\mathsf{R}\times B$ is \emph{{$( {\alpha,}
\nu,\delta)$}-horizontal} if
\begin{enumerate}[itemsep=0.1cm]
\item \label{disc1} {$\|Dg\|_\infty\leq \alpha$,}
\item \label{disc2} there is a point $y \in B$ such that
$d(y,h(\xi))<\delta$ for all $\xi\in [-2,2]^{d_{ss}}$,
\item \label{disc3} ${C\cdot\nu} < \delta$ where $C\geq 0$ is a Lipschitz constant
of $h$, i.e.,
$$d(h(\xi),h(\xi'))\leq C\, d(\xi,\xi') \quad \text{for all
$\xi,\xi'\in [-2,2]^{d_{ss}}$.}
$$
\end{enumerate}
\end{defi}
Since $\mathcal{D}$ is a $C^1$-disc notice that $C$ is any
positive constant satisfying $\|Dh\|_{\infty} \leq C$. If {$C=0$}
we say that $\mathcal{D}$ is \emph{horizontal}.
{Notice that by condition~\eqref{disc1}, the disc
$\mathcal{D}$ is tangent to the stable cone-field defined on
$\mathcal{B}$. Moreover, from condition~\eqref{disc2},
$\mathcal{D}$ is $C^0$-close to an horizontal disc. On the other
hand, although $\mathcal{D}$ may not be $C^1$-close to a
horizontal disc, condition~\eqref{disc3} asks that we still have a
good control of the distortion. Finally,} for a fixed
{$\alpha$,} $\nu$ and $\delta$ under
{the conditions in Theorem~\ref{thmBKR}}, the set
of $( {\alpha,}\nu,\delta)$-horizontal discs in $\mathcal{B}$ is
said to be, for short, the family of \emph{almost-horizontal
discs}.
In the next sections we {construct} diffeomorphisms having robust
{tangencies} in any manifold of dimension $m\geq
3$.
Our constructions will use the following particular class of
blenders {obtained} from the covering criterium.
\subsection{Affine blender}
\label{sec:affine-blender} {We will introduce a class of
$C^r$-diffeomorphims $f$ of $\mathbb{R}^m = \mathbb{R}^n \times
\mathbb{R}^c$ with $r\geq 1$, $n=ss+u\geq 2$ and $c\geq 1$. To do
this, consider first} a $C^r$-diffeomorphism $F$ of $\mathbb{R}^n$
having a horseshoe $\mathsf{\Lambda}$ in the the open cube
$\mathsf{V}=(-2,2)^n$. The horseshoe has stable index
$ss=\mathrm{ind}^s(\mathsf{\Lambda})>0$ and {$F|_{\Lambda}$ is
conjugate to a full shift of a large number $\kappa$ of symbols to
be specified later.} We notice that this number will depend only
on the dimension $c$. For simplicity, assume that
$$
\mathsf{R}_\ell= (-2,2)^{ss} \times I_\ell, \quad
\ell=1,\dots,\kappa
$$ is a Markov partition of $\mathsf{\Lambda}$
where $I_\ell$ is an open disc in $[-2,2]^u$ and $F$ is affine on
each rectangle $\mathsf{R}_\ell$. {More precisely}, there are
$0<\nu<1$ and linear maps $S_\ell: \mathbb{R}^{ss}\to
\mathbb{R}^{ss}$ and $U_\ell: \mathbb{R}^{u} \to \mathbb{R}^u$
such that
$$
DF=
\begin{pmatrix}
S_\ell & 0 \\
0 & U_\ell
\end{pmatrix}
\quad \text{on \ \ $\mathsf{R}_\ell$ \ \ where \ \ $\|S_\ell \|,\
\|U_\ell^{-1}\| < \nu$ \quad for $\ell=1,\dots,\kappa$.}
$$
{Notice that,
$$DF^{-1}(x)\mathcal{C}^{ss}_{\alpha}
\subset \mathcal{C}^{ss}_{\nu^2\alpha} \quad \text{for all $x\in
\mathsf{R}=\mathsf{R}_1\cup\dots\cup \mathsf{R}_\kappa$ \ \ and \
\ $\alpha>0$,}
$$
where the cones $\mathcal{C}^{ss}_{\alpha}$ and
$\mathcal{C}^{ss}_{\nu^2\alpha}$ are defined as
in~\eqref{strong-stable-cone}.
In particular, the cone-field $\mathcal{C}^{ss}_\alpha$ is
$DF^{-1}$-invariant.}
Take affine $(\lambda,\beta)$-contractions
$\phi_1,\dots,\phi_\kappa$ on $D=(-2,2)^c$ with
$\nu<\lambda<\beta<1$. That is, $C^r$-diffeo\-morphisms
$\phi_\ell$ of $\mathbb{R}^c$ such that
$\phi_\ell(\overline{D})\subset D$ and there are linear maps
$T_\ell:\mathbb{R}^c\to \mathbb{R}^c$~so~that
$$
\text{$D\phi_\ell(y)=T_\ell$ for all $y\in\overline{D}$ and
$\lambda<m(T_\ell)\leq \|T_\ell\|<\beta$ for
$\ell=1,\dots,\kappa$.}
$$
Moreover, { we ask that there is an open set $B \subset D$}
containing the origin such that
\begin{equation}
\label{eq:cover}
\overline{B}\subset \phi_1(B) \cup \dots \cup \phi_\kappa(B).
\end{equation}
\begin{exap} \label{exa-afin-blender}
Take $\phi_{\pm}(t)=\lambda t \pm (1-\lambda)$ for $t\in [-2,2]$
with $1/2<\lambda<1$ and consider
$$
\phi_\ell = \phi_{\ell_1}\times \dots\times \phi_{\ell_c} \quad \text{on \ $D=(-2,2)^c$ } \quad \text{for any \ $\ell=(\ell_1,\dots,\ell_c)\in \{-,+\}^c$.}
$$
Observe that here $\kappa=2^c$. It is not difficult to see that
$B=(-1,1)^c$ satisfies~\eqref{eq:cover}.
\end{exap}
Finally we consider a $C^r$-diffeomorphism $\Phi$ of
$\mathbb{R}^m$ locally defined as the skew-product
$$
\Phi=F\ltimes (\phi_1,\dots,\phi_\kappa) \quad \text{on} \ \
\mathcal{U}=(\mathsf{R}_1 \times D) \cup \dots \cup (\mathsf{R}_\kappa \times
D).
$$
According to Theorem~\ref{thmBKR}, the maximal invariant set
$\Gamma$ in $\overline{\mathcal{U}}$ is a $cs$-blender of
{central dimension}~$c>0$. {
Moreover, the superposition region is the family of almost-horizontal $d_{ss}$-dimensional
$C^1$-discs in $\mathcal{B}=\mathsf{R}\times B$, where
$\mathsf{R}=\mathsf{R}_1\cup\dots\cup \mathsf{R}_\kappa$.}
\section{Robust homoclinic tangencies}
\label{sec:tangencias} In this section
we prove Theorem~\ref{thmD}. We provide the existence of
$C^r$-diffeomorphisms with { $r\geq 2$ having $C^2$-robust}
homoclinic tangencies {of large codimension} by constructing
these objets in local coordinates. Thus, we {may consider}
$\mathbb{R}^m = \mathbb{R}^n \times \mathbb{R}^c$ with $n\geq 2$
and $c \geq 1$. Throughout this section, we ask that $n=ss+u$ and
$c=u^2$ but we keep the notation $u$, $c$ in order to distinguish
coordinates. {We divide the proof into several parts and for the
convenience of the reader will explain next the ideas involved.}
{We study the homoclinic tangencies of
$f$ by analyzing the induced map $f^G$ on Grassmannian manifolds,
and we would like for this induced map to have a blender.
The main idea is to obtain robust tangencies
for $f$ by means of a robust intersection between the local unstable
manifolds of a blender (for the induced dynamics)
and a
particular disc in the superposition region.
Hence,
in~\S\ref{sec:affine-blenders-grasmannian} we will construct a
class of $C^r$-diffeomorphisms of $\mathbb{R}^m$ which induces a
$cs$-blender $\Gamma^G$ on the Grassmannian manifold. This class
of diffeomorphisms are the locally defined skew-product maps
having an affine blender $\Gamma$ introduced
in~\S\ref{sec:affine-blender} with some additional restrictions.
Afterwards, we introduce in~\S\ref{sec:folding-manifold} the
notion of a folding manifold $\mathcal{S}$ in $\mathbb{R}^m$,
having the
main property
of inducing a
disc $\mathcal{S}^G$ in the superposition region of $\Gamma^G$.
Finally, in \S\ref{sec:robust-tangency-folding-manifold} we show
how the robust intersection between the local unstable manifolds of
$\Gamma^G$ and the induced disc $\mathcal{S}^G$ provides a robust
tangency between the unstable manifolds of $\Gamma$ and the
folding manifold $\mathcal{S}$. One can see the folding manifold as a
piece of a leaf of the stable manifold of $\Gamma$ and then the proof of
Theorem~\ref{thmD} can be concluded in~\S\ref{sec:prove-ThmD}.}
\subsection{Grassmannian manifold}
\label{sec:grasmanian} Let $f$ be a $C^r$-diffeomorphism of
$\mathbb{R}^m$. We will consider an induced map by $f$ on the
Grassmannian manifold $G_u(\mathbb{R}^m)=\mathbb{R}^m\times
G(u,m)$ given by
$$
{f}^{{G}} : G_u(\mathbb{R}^m) \to G_u(\mathbb{R}^m), \qquad
{f}^{{G}}(x,E)=(f(x),Df(x)E)
$$
where $G(u,m)$ is the set of $u$-planes in $\mathbb{R}^m$. Notice
that $f^{{G}}$ is a $C^{r-1}$-diffeomorphism of
${G}_u(\mathbb{R}^m)$.
\subsection{Blender induced on the Grassmannian manifold} \label{sec:affine-blenders-grasmannian}
Fix $r\geq 2$. We will start by considering a
$C^{r}$-diffeomorphism $\Phi$ of $\mathbb{R}^m$ locally defined as
a skew-product $\Phi=F\ltimes (\phi_1,\dots,\phi_\kappa)$ and
having an affine $cs$-blender $\Gamma$, as
in~\S\ref{sec:affine-blender}. Notice that for each
$\ell=1,\dots,\kappa$, the differential map $D\Phi(x,y)$ is the
same linear map $D\Phi_\ell$ for all $(x,y)\in
\mathsf{R}_\ell\times D$. Moreover, $E^u=\{0^{ss}\}\times
\mathbb{R}^u \times \{0^c\}$ is an attracting fixed point of the
action of these maps on $G(u,m)$ with eigenvalues less than
$\beta\nu<1$. Let $\mathcal{C}^{{G}}$ be an open neighborhood of
$E^u$ in $G(u,m)$ so that $D\Phi_\ell \cdot \mathcal{C}^{{G}}
\subset \mathrm{int}(\mathcal{C}^{{G}})$ {for all
$\ell=1,\dots,\kappa$}. The Grassmannian induced map $\Phi^{{G}}$
restricted to $\mathsf{R}_\ell\times D\times \mathcal{C}^{{G}}$ is
given by
$$
\Phi^{{G}}(x,y,E)=(F(x),\phi_i(y),D\Phi_\ell \cdot E) \qquad
\text{for all \ $\ell=1,\dots,\kappa$.}
$$ By a change of coordinates we can write $\Phi^{{G}}$ restricted to
$\mathcal{U}^{{G}}=\mathcal{U}\times \mathcal{C}^{{G}}$ as the
skew-product
$$
\Phi^{{G}}=F^{{G}} \ltimes (\phi_1,\dots,\phi_\kappa)
\quad
\text{on \ \
${\mathcal{U}}^{{G}}=(\mathsf{R}^{{G}}_1 \times D)\cup\dots\cup(\mathsf{R}^{{G}}_\kappa\times
D$)}
$$
where
$$
F^{{G}}=F \ltimes (D\Phi_1,\dots,D\Phi_\kappa) \quad \text{on \ \
$\mathsf{R}^{{G}}=\mathsf{R}^{{G}}_1\cup\dots\cup\mathsf{R}^{{G}}_\kappa=(\mathsf{R}_1\times\mathcal{C}^{{G}})
\cup \dots \cup (\mathsf{R}_\kappa\times\mathcal{C}^{{G}})$.}
$$
Moreover, $F^{{G}}$ has a horseshoe
${\mathsf{\Lambda}}^{{G}}=\mathsf{\Lambda} \times \{E^u\}$ with
stable index
$$
d^{{G}}_{ss}\eqdef\mathrm{ind}^s({\Lambda}^{{G}})=ss+\dim
G(u,m)=ss+u(m-u).
$$
{Observe that shrinking $\mathcal{C}^G$ if necessary, the contraction of $D\Phi_\ell$ on $\mathcal{C}^G$ dominates the contraction
of $F$ (that is $\beta\nu<\nu$). Then, the width $\alpha>0$ of the
stable cone-field on $\mathsf{R}\times \mathcal{C}^G$ of $F^G$ is
the same width $\alpha$ that we have for the stable cone-field on
$\mathsf{R}$ of $F$.}
Since $\beta\nu < \nu<\lambda$ then
$F^{{G}}|_{{\mathsf{\Lambda}}^{{G}}}$ dominates the fiber dynamics
given by $\phi_1,\dots,\phi_\kappa$. By Theorem~\ref{thmBKR}, we
have that $\Gamma^{{G}}=\Gamma \times \{E^u\}$ is a $cs$-blender
of {central dimension} $c>0$ of $\Phi^{{G}}$.
{Moreover, the family of
{$( {\alpha,}\nu,\delta)$}-horizontal, $d^{{G}}_{ss}$-dimensional
$C^1$-discs in ${\mathcal{B}}^{{G}}=\mathcal{B} \times
\mathcal{C}^{{G}}$ is a superposition region of the blender
$\Gamma^{{G}}$. Here, as in~\S\ref{sec:affine-blender},
$0<\delta<\lambda L /2$ whereas $L$ is the Lebesgue number of the
cover~\eqref{eq:cover}.}
\subsection{Folding manifold with respect to the affine blender}
\label{sec:folding-manifold} Next we introduce the notion of a
folding manifold.
To do this,
we will consider a submanifold $\mathcal{S}$ of $\mathbb{R}^m$ of
dimension $ss+c$.
In what follows we {identify canonically the tangent space
$T_z\mathcal{S}$ of $\mathcal{S}$ at $x$ with a} subspace of
$\mathbb{R}^m$.
\begin{figure}
\caption{
Folding manifold with respect to $\mathcal{B}
\label{fig1}
\end{figure}
\begin{defi} \label{def:dobra}
We say that $\mathcal{S}$ is a {$( {\alpha,}
\nu,\delta)$}-\emph{folding $C^r$-manifold} with respect to
$\mathcal{B}^{{G}}=\mathcal{B}\times \mathcal{C}^{{G}}$
if {there is $\epsilon>0$ such that}
\begin{enumerate}[leftmargin=0.45cm,itemsep=0.2cm]
\item {$\mathcal{S}$ is parameterized as a
$(ss+c)$-dimensional $C^r$-embedding
$\mathcal{S}: [-2,2]^{ss}\times [-\epsilon,\epsilon]^c \to
\overline{\mathcal{B}}$, of the form}
$$ {\mathcal{S}(x,t)=(x,(t_1,\dots,t_u),h(x,t))\in \mathbb{R}^{ss}\times \mathbb{R}^{u}\times\mathbb{R}^c}$$
{with $x\in [-2,2]^{ss}$ and $t=(t_1,\dots,t_u,\dots,t_c)\in
\mathbb{R}^c$;}
\item there is $y\in B$ such that
$d(h(x,t),y)<\delta$ for all $(x,t)\in [-2,2]^{ss}\times
[-\epsilon,\epsilon]^c$;
\item \label{foldingCu}{for all $x\in [-2,2]^{ss}$ and
$E\in {\overline{\mathcal{C}^G}}$ there is a unique $t\in
[-\epsilon,\epsilon]^c$ such that $E$ is a subspace of $T_z
\mathcal{S}$ with $z= \mathcal{S}(x,t)$. Moreover, $t=t(x,E)$
varies $C^{r-1}$-continuously} with $(x,E)$ and
$$
(\|Dh\|_{\infty} \cdot \max\{1,\|Dt\|_{\infty}\})\cdot \nu <\delta
\quad {\text{and} \quad \|Dt\|_\infty \leq \alpha}.
$$
\end{enumerate}
\end{defi}
{Let us explain geometrically the above notion of a folding
manifold. First define the unstable cone for some small $\theta>0$
as
\begin{align*}
\mathcal{C}^{u}_\theta&=\{(u,v,w)\in
\mathbb{R}^m=\mathbb{R}^{ss}\oplus \mathbb{R}^u \oplus
\mathbb{R}^c: \ \ \|u+w\| < \theta \|v \|\}\cup\{0\}.
\end{align*}
Each vector subspace $E$ of dimension $u$ contained in the cone
$\mathcal{C}^{u}_\theta$ can be identified with an element of
$\mathcal{C}^{G}$ and vice-versa. Then condition~\eqref{foldingCu}
implies that for every $x\in [-2,2]^{ss}$,
\begin{equation}
\label{eq:cover-Cu}
\overline{\mathcal{C}^{u}_\theta} \subset \bigcup_{t\in [-\epsilon,\epsilon]^c}
T_{\gamma(t)}\mathcal{S} \quad \text{where $\gamma(t)=\mathcal{S}(x,t)$.}
\end{equation}
In fact, the uniqueness in condition~\eqref{foldingCu} implies the
injectivity of the map $t\mapsto T_{\gamma(t)}\mathcal{S}$ and
thus, the parameters $\epsilon$ and $\theta$ can be interpreted as
the size of the neighborhood $\mathcal{C}^G$ of
$E^u=\{0^{ss}\}\times \mathbb{R}^u \times \{0^c\}$ in $G(u,m)$. }
{Next we will show an example of a $(
{\alpha,}\nu,\delta)$-folding manifold with respect to
$\mathcal{B}^G=\mathcal{B}\times \mathcal{C}^G$. Recall that $B$
is an open set of $\mathbb{R}^c$ containing the origin.
{Up to a conjugacy with a translation}, we can
assume that $\mathcal{B}=\mathsf{R}\times B$ contains
$(-2,2)^{ss}\times \{0^u\} \times \{0^c\}$. }
\begin{exap} \label{exap:dobra} {
Consider the $(ss+c)$-dimensional embedding given by
$$
\mathcal{S}:[-2,2]^{ss}\times [-\epsilon,\epsilon]^c \to
\mathbb{R}^m, \ \ \mathcal{S}(x,t)=(x,(t_1,\dots,t_u),
( {H}(t), {t_{u+1},\dots,t_c}))
\in \mathbb{R}^{ss}\times\mathbb{R}^u\times \mathbb{R}^c
$$
where $x\in \mathbb{R}^{ss}$, $t=(t_1,\dots,t_u,t_{u+1},\dots,t_c)
\in [-\epsilon,\epsilon]^c$ and $ {H}(t)=( {H}_1(t),\dots,
{H_u}(t))$ with
\begin{align*}
{H}_i(t)= \sum_{j=0}^{u-1} t_{j+1} t_{ju+i} \quad \text{for $i=1,\dots,u$.}
\end{align*}
For a fixed $\delta>0$, we will prove that $\mathcal{S}$ is a $(
{\alpha,}\nu,\delta)$-folding {$C^\infty$}-manifold
{for any $\alpha>0$ large enough} and
$\epsilon,\nu>0$ small enough. To do this, we will show that
$\mathcal{S}$ satisfies all the conditions of
Definition~\ref{def:dobra}.}
{It is straightforward that $\mathcal{S}$ is a
$(ss+c)$-dimensional {$C^\infty$}-embedding. Since $\mathcal{B}$
contains $(-2,2)^{ss}\times \{0^u\} \times \{0^c\}$, then
$\mathcal{S}([-2,2]^{ss}\times [-\epsilon,\epsilon]^c)\subset
\mathcal{B}$ for any $\epsilon>0$ small enough, concluding the
first condition in Definition~\ref{def:dobra}. Observe now that
the central coordinate of $\mathcal{S}$, i.e., the map
$h(t)=\mathscr{P} \circ \mathcal{S}(x,t)$ does not depend on $x$.
Moreover, if $\epsilon>0$ is small enough then
$d(h(t),0^c)<\delta$ for all $(x,t)\in [-2,2]^{ss}\times
[-\epsilon,\epsilon]^c$, as is required by the second condition in
Definition~\ref{def:dobra}.} {To conclude that $\mathcal{S}$ is a
folding manifold, it only remains to prove the last condition in
Definition~\ref{def:dobra}, which is somewhat longer and will be
done in the next paragraphs.
By a direct computation, $T_z\mathcal{S}$ at $z=\mathcal{S}(x,t)$
is given by
$$
T_z\mathcal{S}(x',t')=(x',(t'_1,\dots,t'_u),(g(t',t), {t'_{u+1},\dots,t'_{c}}))\in
\mathbb{R}^{ss}\times \mathbb{R}^u\times \mathbb{R}^c
$$
where $x'\in \mathbb{R}^{ss}$,
$t'=(t'_1,\dots,t'_u,t'_{u+1},\dots,t'_c)\in \mathbb{R}^c$ and
$g(t',t)=(g_1(t',t),\dots,g_{ {u}}(t',t))$ with
\begin{align*}
g^{}_i(t',t)= \sum_{j=0}^{u-1} t'_{j+1} t^{}_{ju+i} + t^{}_{j+1}
t'_{ju+i} \quad \text{for $i=1,\dots,u$.}
\end{align*}
We want to prove that for any $x\in [-2,2]$ and $E\in\overline{
\mathcal{C}^G}$, there is a unique $t\in [-\epsilon,\epsilon]^c$
such that $E$ is a subspace of $T_z\mathcal{S}$ for
$z=\mathcal{S}(x,t)$. Observe that if $E=\langle v_k:
k=1,\dots,u\rangle$ is generated by linearly independent vectors
$v_k$, then $E$ is a subspace of $T_z\mathcal{S}$ if and only if
$v_k\in T_z\mathcal{S}$ for all $k=1,\dots,u$. Denoting
$v_k=(a_k,b_k,c_k)\in \mathbb{R}^{ss}\times \mathbb{R}^u\times
\mathbb{R}^c$, the above condition is equivalent to the existence
of $t\in [-\epsilon,\epsilon]^c$ such that for every
$k=1,\dots,u$, {there are $x_k'\in\mathbb{R}^{ss}$, and
$t'_k\in\mathbb{R}^c$ satisfying:}
$$
x'_k=a^{}_k \ \ \ \ t'_k=
(b^{}_{k1},\dots,b^{}_{ku},c^{}_{k\,u+1},\dots,c^{}_{kc}) \ \ \ \text{and}
\ \ c^{}_{ki}= g^{}_i(t'_k,t) \ \ \ \text{for $i=1,\dots,u$}.
$$
Notice that $g^{}_i(t'_k,t)$ can be written as a scalar product of
$t$ by a vector {$\vec{a}_{ki}\in \mathbb{R}^c$} that depends
on~$t'_k$. Thus, {having into account that $c=u^2$,} we can write
the relation $c_{ki}=g_i(t'_k,t)$ for $k=1,\dots,u$ and
$i=1,\dots,u$ as a matrix product
$$
At=\vec{c} \ \ \text{where} \ \vec{c}=(c_{11},\dots,c_{1
{u}},\dots,c_{u1},\dots,c_{u {u}})^T
$$
and $A
{=[\vec{a}_{11};\dots;\vec{a}_{1u};\dots;\vec{a}_{u1};\dots;\vec{a}_{uu}]}$
is a $c$-by-$c$ matrix that depends on $t'_k$ for $k=1,\dots,u$.
In fact, since $t'_k$ form part of the coordinates of the vector
$v_k$, then $A=A(E)$ depends on the vector space $E$. Similarly
this holds for $\vec{c}=\vec{c}(E)$. Hence, to find the required
$t\in [-\epsilon,\epsilon]^c$ we only need to show that the linear
system $A(E)\cdot t = \vec{c}(E)$ is uniquely solved.}
{ To do this, we will analyze the determinant of $A$ at
$E^u=\{0^{ss}\}\times\mathbb{R}^u\times\{0^c\}$ in which the open
set $\mathcal{C}^G$ is centered. Observe that $E^u$ is generated
by the vectors $e^u_k=(0^{ss},e_k,0^c)$ for $k=1,\dots,u$, where
$e_k$ is the $k$-th canonical vector in $\mathbb{R}^u$. Thus,
$t'_k=(e_k,0^{c-u})$ and then
$$
g^{}_i(t'_k,t)=t_{(k-1)u+i}+\delta_{ki} \, t_1 \ \ \text{for} \ \
k=1,\dots,u \ \ \text{and} \ \ i=1,\dots,u,
$$ where $\delta_{ki}$ is the Kronecker delta. In view
of this, $A(E^u)=\mathrm{Id}+\mathrm{L}$ where $\mathrm{Id}$ is
the identity matrix and $\mathrm{L}$ is a matrix whose first
column is given by $(e_1,e_2,\dots,e_u)^T$ and the rest of the
elements are zero. Hence $A(E^u)$ is a triangular matrix with
$\det A(E^u) =2 \not =0$. Thus, we get that $A(E)\cdot t =
\vec{c}(E)$ is uniquely solved for any $E$ close enough to $E^u$.}
{ This shows the first part of the last condition in
Definition~\ref{def:dobra} but still we need to prove that $C \nu
<\delta$ where $C=\|Dh\|_\infty \cdot \max\{1,\|Dt\|_\infty\}$
{and $\|Dt\|_\infty\leq \alpha$}. Since both
$h=h(t)$ and $t=t(E)$ are functions of class $C^{ {\infty}}$, then
$C<\infty$ over $[-2,2]^{ss}\times \mathcal{C}^{G}$ and
$\|Dt\|_\infty<\infty$. Thus, this condition trivially holds by
taking $\nu>0$ small enough {and $\alpha$ large enough.}}
\end{exap}
\begin{rem}
{ In Proposition~\ref{prop1:appendix} in Appendix~\ref{appendix}
we show that actually $\mathcal{S}$ in the above example is a $(
{\alpha,}\nu,\delta)$-folding $ {C^\infty}$-manifold for any $\nu<
\delta$ {and $\alpha>1$}.}
\end{rem}
\begin{rem} \label{rem:rob-dobra}
{Fixing a small enough $\epsilon>0$, for which $\mathcal{S}$ is a
$( {\alpha,}\nu,\delta)$-folding manifold, the above example is
$C^2$-robust in the following sense. Let $\tilde{\mathcal{S}}:
[-2,2]^{ss}\times [-\epsilon,\epsilon]^c \to
\overline{\mathcal{B}}$
be $(ss+c)$-dimensional $C^2$-embedding, which is $C^2$-
sufficiently close to $\mathcal{S}$. Then
$\tilde{\mathcal{S}}$ is also a $( {\alpha,}\nu,\delta)$-folding manifold.
Let us comment on why this is true. What has to be shown is
basically condition~\eqref{foldingCu} of
Definition~\ref{def:dobra}. The function $\tilde{\mathcal{S}}$ can
be written in the form
$$
(x,t)\mapsto (x,(t_1,\dots,t_u),(H(x,t),t_{u+1},\dots,t_c)), \
\text{where} \ H(x,t)=H(t)+\kappa(x,t).
$$
Here $H(t)$ comes from the embedding $\mathcal{S}$, and
$\kappa(x,t)$ is a small $C^2$-perturbation. Following the
notation of the previous example, the non-linear equations that
now have to be solved take the form
$$
x'_k=a_k, \ \ t'_k=(b_{k1},\dots,b_{ku},c_{k\, u+1},\dots,c_{kc}),
\ \text{and} \ c_{ki}=g_i(t'_k,t)+\varkappa_{i} \quad \text{for
$i,k=1,\dots,u$}
$$
where $\varkappa_{i}$ are functions which depend on $(x,t,x',t')$
with small $C^1$-derivative.
The solution of these equations for $t$ given $x$ and $E$, is then
guaranteed by an application of the Implicit Function Theorem. }
\end{rem}
Let $\mathcal{S}$ be a $( {\alpha,}\nu,\delta)$-folding
$C^r$-manifold with respect to
$\mathcal{B}^{{G}}=\mathcal{B}\times\mathcal{C}^{{G}}$.
Consider
$$
\mathcal{S}^{{G}} = \{ (z,E): z\in \mathcal{S}, \ E \in G(u,m)
\ \
\text{and} \ \ {E {\subset} T_z\mathcal{S}} \} \subset
G_u(\mathbb{R}^m).
$$
One can see $\mathcal{S}^{{G}}$ as a fiber bundle over
$\mathcal{S}$ with fibers
$$
(\mathcal{S}^{{G}})_z=\{E\in G(u,m): E {\subset} T_z\mathcal{S} \}.
$$
Notice that $(\mathcal{S}^{{G}})_z$ is a compact manifold of
dimension $ \dim G(u,ss+c)= u(ss+c-u)$. Then, the dimension of
$\mathcal{S}^{{G}}$ is $ss+c + u(ss+c-u) = {ss+}
u(m-u)+c-u^2$. In fact, since $c=u^2$ we have that this dimension
coincides with $d^{{G}}_{ss}=ss+u(m-u)$.
\begin{lem} \label{lem-induced-folding-manifold}
The set $\mathcal{H}^{{G}}=\mathcal{S}^{{G}} \cap
\overline{\mathcal{B}^{{G}}}$ is a {($
{\alpha,}\nu,\delta)$}-horizontal $d^{{G}}_{ss}$-dimensional
$C^{{r-1}}$-disc in $\mathcal{B}^{{G}}$.
\end{lem}
\begin{proof} First notice that $\mathcal{H}^{{G}}$ is
a $d^{{G}}_{ss}$-dimensional $C^{{r-1}}$-disc in
$\mathcal{B}^{{G}}$. This follows from Definition~\ref{def:dobra},
since given any $x\in [-2,2]^{ss}$ and any $E\in
\overline{\mathcal{C}^{{G}}}$ we have a unique $t=t(x,E)\in
[{-\epsilon,\epsilon}]^c$ which varies $C^r$-continuously with $x$
and $E$ such that
$E\subset T_{z}\mathcal{S}$ with $z=\mathcal{S}(x,t)$. {Thus we
can parametrize $\mathcal{H}^G$ as}
$$
\mathcal{H}^{{G}}: [-2,2]^{ss}\times \overline{\mathcal{C}^{{G}}} \longrightarrow
G_u(\mathbb{R}^m), \qquad \mathcal{H}^{{G}}(x,E)=(\mathcal{S}(x,t), E)
\in \overline{\mathcal{B}}\times \overline{\mathcal{C}^{{G}}}=
\overline{\mathcal{B}^{{G}}}
$$
is a $C^{{r-1}}$-disc in $\mathcal{B}^{{G}}$. On the other hand,
the {unstable and} central {coordinates} of this disc
{are} given by
\begin{align*}
{g^{{G}}(x,E)}& {=\mathscr{P}_u \circ
\mathcal{H}^{{{G}}}(x,E)=g(t(x,E)), \ \ \text{and}} \\
h^{{G}}(x,E)&=\mathscr{P}_{ {c}} \circ
\mathcal{H}^{{{G}}}(x,E)=h(x,t(x,E)) \qquad \text{for} \ \
(x,E)\in [-2,2]^{ss}\times\overline{\mathcal{C}^{{G}}},
\end{align*}
where
$$
{(t_1,\dots,t_u)=g(t)=\mathscr{P}_u\circ
\mathcal{S}(x,t) \quad \text{and}} \quad h(x,t)=\mathscr{P}_{
{c}}\circ \mathcal{S}(x,t)
$$
{are} the {unstable and} central
{coordinates} of the folding manifold
$\mathcal{S}$. Here $\mathscr{P}_u$ and $\mathscr{P}_c$ denote
the {canonical} projections on {$\mathbb{R}^u$ and} $\mathbb{R}^c$
{respectively}. Hence, again, by the definition of
folding manifold we have $y\in B$ such that
$d(h^{{G}}(x,E),y)=d(h(x,t(x,E)),y)<\delta$,
$$
\|Dh^{{G}}\|_{\infty} \leq \|Dh\|_{\infty}
\max \{1, \|Dt\|_{\infty}\}=C \quad \text{with \ \ $C\nu < \delta$.}
$$
and {$$
\|Dg^G\|_\infty \leq \|Dg\|_\infty \|Dt\|_\infty \leq
\|Dt\|_\infty \leq \alpha.
$$}
This proves that $\mathcal{H}^{{G}}$ is
$( {\alpha,}\nu,\delta)$-horizontal disc concludes the proof.
\end{proof}
\begin{rem}
The {$( {\alpha,}\nu,\delta)$}-horizontal $C^{{r-1}}$-disc
$\mathcal{H}^{{G}}$ obtained from the $C^r$-folding manifold in
Example~\ref{exap:dobra} is $C^0$-close to a horizontal disc but
$C^1$-far from it.
\end{rem}
\subsection{Robust tangencies with a folding manifold}
\label{sec:robust-tangency-folding-manifold} {Recall that the
$C^r$-diffeomorphism $\Phi$ of $\mathbb{R}^m$ we are considering
in this section was introduced
in~\S\ref{sec:affine-blenders-grasmannian}. This map has a
$cs$-blender $\Gamma$ of {central dimension} $c>0$, where a
superposition region contains the family of $(
{\alpha,}\nu,\delta)$-horizonal discs in $\mathcal{B}$ with
$\delta<\lambda L/2$.} Now, we will prove the following key
result:
\begin{prop}
\label{prop-tangency-folding} There is a {$C^2$-neighborhood}
$\mathscr{U}$ of $\Phi$ such that {for any $(
{\alpha,}\nu,\delta)$-folding $C^r$-manifold $\mathcal{S}$ with
respect to $\mathcal{B}^G=\mathcal{B}\times \mathcal{C}^G$ it
holds that}
{for any} $g\in \mathscr{U}$
there are points $z\in\Gamma_g$ and $x\in W^u_{loc}(z) \cap
\mathcal{S}$ such that
\begin{equation}
\label{eq:tangencia}
\dim T_x W^u_{loc}(z) \cap T_x \mathcal{S} = u
\quad \text{or equivalently, \ \ $T_xW^u_{loc}(z) \subset T_x
\mathcal{S}$}.
\end{equation}
{In particular, since the codimension of
$W^u_{loc}(z)$ coincides with the dimension of $\mathcal{S}$,
these two manifolds intersect at $x$ in a tangency of codimension
$u$.}
\end{prop}
\begin{proof} We recall that $\Gamma^{{G}}=\Gamma\times\{E^u\}$
is a $cs$-blender of {central dimension} $c>0$ for the induced
$C^{1}$-diffeomorphism $\Phi^{{G}}$, whose superposition region
contains the set $\mathscr{D}$ of $(
{\alpha,}\nu,\delta)$-horizontal $d^{{G}}_{ss}$-dimensional
$C^1$-discs in $\mathcal{B}^{{G}}=\mathcal{B} \times
\mathcal{C}^{{G}}$.
Hence, {by definition of a blender}, there is a $C^1$-neighborhood
$\mathscr{U}^{{G}}$ of $\Phi^{{G}}$
where for {each map $\Psi \in \mathscr{U}^G$} we have an
{intersection} between {{each} disc
in $\mathscr{D}$ and the local} unstable manifold of
the continuation of $\Gamma^{{G}}$ for $\Psi$. We take a
{$C^2$-neighborhood} $\mathscr{U}$ of $\Phi$ so that for every
$g\in\mathscr{U}$ its induced {$C^1$-diffeomorphism} $g^{{G}}$ on
$G_u(\mathbb{R}^m)$ belongs to $\mathscr{U}^{{G}}$. Hence, {the
continuation $\Gamma^{G}_g$ of $\Gamma^G$ for $g^{{G}}$ is a
$cs$-blender}. Moreover, $W^u_{loc}(\Gamma_g^{{G}})$ is {
{laminated by} plaques of dimension $u$ which project one-to-one
onto $W^u_{loc}(\Gamma_g)$}. In particular,
{shrinking $\mathscr{U}$ if necessary},
\begin{equation}\label{eq:implication}
\text{if $(x,E)\in W^u_{loc}(\Gamma_g^{{G}})$ then there is $z\in
\Gamma_g$ such that $x\in W^u_{loc}(z)$ and $E=T_xW^u_{loc}(z)\in
\mathcal{C}^{{G}}$.}
\end{equation}
On the hand, {if $\mathcal{S}$ is a $(
{\alpha,}\nu,\delta)$-folding manifold with respect to
$\mathcal{B}^G$}, then by
Lemma~\ref{lem-induced-folding-manifold}, the manifold
$\mathcal{S}^{{G}}$ contains a ${(
{\alpha,}\nu,\delta)}$-horizontal $d^{{G}}_{ss}$-dimensional
$C^1$-disc $\mathcal{H}^{{G}}$ in $\mathcal{B}^{{G}}$. Hence
$\mathcal{H}^{{G}} \in \mathscr{D}$. Thus,
$W^u_{loc}(\Gamma_g^{{G}})\cap \mathcal{H}^{{G}} \not =
\emptyset$. Consequently, there is $(x,E)$ belonging to
$\mathcal{H}^{{G}}\subset \mathcal{S}^{{G}}$ and
$W^u_{loc}(\Gamma_g^{{G}})$. In particular,
from~\eqref{eq:implication}, we get that $x\in \mathcal{S}\cap
W^u_{loc}(\Gamma_g)$ and $T_xW^u_{loc}(z)=E\subset T_x
\mathcal{S}$ for some $z\in \Gamma_g$. This completes the proof.
\end{proof}
\subsection{Proof of Theorem~\ref{thmD}}
\label{sec:prove-ThmD} Finally we prove Theorem~\ref{thmD} by
assuming that the global stable manifold of a periodic point $P$
in the affine $cs$-blender $\Gamma$ contains {the folding manifold
with respect to
$\mathcal{B}^{{G}}=\mathcal{B}\times\mathcal{C}^{{G}}$ given in
Example~\ref{exap:dobra}. As was explained in
Remark~\ref{rem:rob-dobra} this folding manifold is $C^2$-robust.}
Thus, the stable manifold $W^s(P_g)$ of the continuation $P_g$ of
$P$ contains a folding manifold with respect to
$\mathcal{B}^{{G}}$ for all small enough $C^2$-perturbations $g$
of $\Phi$. Then, Proposition~\ref{prop-tangency-folding} implies
that {there is $z\in \Gamma_g$ such that $W^u_{loc}(z)$ and
$W^s(P_g)$ have a tangency of codimension $u>0$}. Thus, we get
that $\Phi$ has a $C^2$-robust homoclininc tangency of codimension
$u$. Moreover, using ~\eqref{eq:tangencia} we can conclude that
the tangency {cannot be inside a strong partially hyperbolic set.
To see this, notice that $T_x\mathcal{S}=E^{ss}\oplus F$ where
$E^{ss}=\mathbb{R}^{ss}\times\{0^{u}\}\times\{0^c\}$,
$T_xW^u_{loc}(z)\subset F$ and $T_xW^u_{loc}(z)\in \mathcal{C}^G$.
Oberve that $E^{ss}$ is in the strong stable cone-field on
$\mathcal{B}$ of $\Phi$, while $\mathcal{C}^G$ can be identified
with the unstable cone on $\mathcal{B}$ of $\Phi$. Since both
cone-fields are disjoint, we obtain that $T_xW^u_{loc}(z)$ cannot
be the strong stable direction. This proves that the tangency must
be inside a weak partially hyperbolic set.} Finally, recall that
$c=u^2$ and then $m=ss+c+u>
u^2+u$, completing the proof.
\section{ {Degenerate} unfoldings of {tangencies}}
\label{sec:unfoding}
{ Recall that by a tangency we understand the
opposite of a transverse intersection.}
We will introduce the notion of {degenerate}
unfoldings
of a {tangency}
between two submanifolds $\mathcal{L}_0$ {and} $\mathcal{S}_0$.
Let $\mathcal{L}=(\mathcal{L}_a)_a$ and
$\mathcal{S}=(\mathcal{S}_a)_a$ be $k$-parameter families of
submanifolds $\mathcal{L}_a$ and $\mathcal{S}_a$ of $\mathcal{M}$
diffeomorphic to $\mathcal{L}_0$ and $\mathcal{S}_0$ respectively
by families of diffeomorphisms $C^{d,r}$-close to the identity
{with $0<d\leq r$.}
\begin{defi}
We say that $\mathcal{L}$ {and} $\mathcal{S}$ has a
{tangency}
at $a=0$
which unfolds \emph{$C^d$-{degenerate}} if there exist $x=(x_a)_a,
y=(y_a)_a \in C^d(\mathbb{I}^k,\mathcal{M})$ such that
$$
x_a \in \mathcal{L}_a \ \ \text{and} \ \ y_a \in \mathcal{S}_a \ \
\text{so that}
\ \
d(x_a,y_a)=o(\|a\|^{ {d}}) \ \ \text{at $a=0$}.
$$
\end{defi}
A useful formalism to define a $C^d$-{degenerate} unfolding of a
{tangency} is to consider the spa\-ce of jets
$J^d_0(\mathbb{I}^k,\mathcal{M})$ whose elements are the
coefficients of the truncated~Taylor~series~at~$a=0$,
$$
J_0^d(z)=(z_a,\partial^1_a z_a,\partial^2_a
z_a,\dots,\partial^d_az_a)_{|_{\,a=0}}
\ \ \text{with} \ \ z=(z_a)_a \in
C^d(\mathbb{I}^k,\mathcal{M}).
$$
For a more precise definition see~\S\ref{sec:jets}. Then
$$
d(x_a,y_a)=o(\|a\|^{{ {d}}}) \ \ \text{at $a=0$} \quad \text{if and only
if} \quad J^d_0(x)=J_0^d(y).
$$
The set $J_0^d(\mathbb{I}^k,\mathcal{M})$ can be endowed with a
smooth manifold structure sometimes called the \emph{manifold of
$(d,k)$-velocities over $\mathcal{M}$}.
Next, we will be interested in unfoldings which control not only
the separation of points on the manifold, but also the separation
of the tangent spaces. {Hence, to control this separation, we
assume $d<r$ and introduce the following definition.}
Let $G_{ {\ell}}(\mathcal{M})$ be the $ {\ell}$-th Grassmannian
bundle of $\mathcal{M}$. That is, the fiber bundle over
$\mathcal{M}$ whose fibers are the $ {\ell}$-th Grassmannian
manifold of the tangent space $T_p\mathcal{M}$, i.e.,
$$
G_{ {\ell}}(\mathcal{M}) = \bigsqcup_{p\in \mathcal{M}} G_ { {\ell}}(\mathcal{M})_p =
\bigcup_{p\in \mathcal{M}} \{p\}\times G({ {\ell}},T_p\mathcal{M})
$$
where $G({ {\ell}},T_p\mathcal{M})$ is the set of all ${
{\ell}}$-dimensional linear subspaces of $T_pM$.
\begin{defi} \label{def:tangencia}
We say that $\mathcal{L}$ {and} $\mathcal{S}$ has a tangency of
{dimension $\ell>0$} at $a=0$ which unfolds
$C^d$-{degenerate} if there exist $x=(x_a)_a, y=(y_a)_a \in
C^d(\mathbb{I}^k,G_{ {\ell}}(\mathcal{M}))$ such that
$$
x_a \in G_{ {\ell}}(\mathcal{L}_a) \ \ \text{and} \ \ y_a \in G_{ {\ell}}(\mathcal{S}_a) \ \ \text{so that}
\ \
d(x_a,y_a)=o(\|a\|^{ {d}}) \ \ \text{at $a=0$.}
$$
Using the formalism of jets, the unfolding is $C^d$-{degenerate}
if and only if $
J^d_0(x)=J_0^d(y)$.
\end{defi}
\begin{rem}
In the terminology of~\cite{Ber16}, $C^d$-{degenerate} unfoldings
of a tangency {of dimension one (between curves in dimension
two)} are called \emph{$C^d$-paratangencies.}
\end{rem}
\section{Parablenders}
\label{sec:parablender} The concept of parablender was initially
introduced by Berger~\cite{Ber16} for endomorphisms (see
also~\cite{BCP17,Ber17,Ber19}). The following generalizes both,
the blender (Definition~\ref{def:blender}) and the definition of
parablender for diffeomorphisms given {also by Berger}
in~\cite[Example~1.21,Def.~1.23]{Ber19}.
\begin{defi} \label{def:parablender}
Let $\Gamma_0$ be a $cs$-blender {of central dimension} $c\geq 1$
{and strong stable dimension
$d_{ss}=\mathrm{Ind}^s(\Gamma_0)-c$} of a $C^r$-diffeomorphism
$f_0$ of $\mathcal{M}$. Consider a $k$-parameter $C^d$-family
$f=(f_a)_a$ of $C^r$-diffeomorphisms of $\mathcal{M}$ unfolding
$f_0$ at $a=0$. A~family $\Gamma=(\Gamma_a)_a$ of compact sets
$\Gamma_a$ of $\mathcal{M}$ is said to be \emph{
{$C^d$-}$cs$-parablender {at $a=0$} of {central dimension} $c$}
for $f$ if
\begin{enumerate}[itemsep=0.1cm]
\item $\Gamma_a$ is the continuation for $f_a$
of the $cs$-blender $\Gamma_0$ for all $a\in\mathbb{I}^k$,
\item there exists an open set $\mathscr{D}$ of $k$-parameter
$C^d$-families $\mathcal{D}=(\mathcal{D}_a)_a$,
{where each $\mathcal{D}_a$ is a $C^r$-embedded
$d_{ss}$-dimensional disc into $\mathcal{M}$,}
\item there exists a $C^{d,r}$-neighborhood $\mathscr{U}$ of
$\Phi$,
\end{enumerate}
such that for every $g=(g_a)_a \in \mathscr{U}$ and
$\mathcal{D}=(\mathcal{D}_a)_a \in \mathscr{D}^{ss}$ it holds that
$$
\text{$W^u_{loc}(\Gamma_g)$ {and} $\mathcal{D}$ has a
{tangency} at $a=0$ which unfolds
$C^d$-{degenerate}.}
$$
That is, there are $x=(x_a)_a$, $y=(y_a)_a$, $z=(z_a)_a\in
C^d(\mathbb{I}^k,\mathcal{M})$ with
$$
z_a \in \Gamma_{a,g}, \ \ x_a \in W^u_{loc}(z_a) \ \ \text{and} \
\ y_a\in \mathcal{D}_a \ \ \text{such that} \ \
d(x_a,y_a)=o(\|a\|^{ {d}}) \ \ \text{at $a=0$}
$$
where $\Gamma_{a,g}$ is the continuation for $g_a$ of the
$cs$-blender $\Gamma_a$ for all $a\in \mathbb{I}^k$.
A \emph{$C^d$-$cu$-parablender} {at $a=0$} of
{central dimension $c$} is
{$C^d$-}$cs$-parablender for
$f^{-1}=(f^{-1}_a)^{}_a$.
\end{defi}
\begin{rem}
For $k=0$ and $d=r=1$, i.e., when there are no parameters and the
class is $C^1$, the above definition of a parablender coincides
with the definition of a blender. {As was mentioned after
Definition~\ref{def:blender}, the tangency between
$W^u_{loc}(z_0)$ and $\mathcal{D}_0$ has codimension at least $c$.
In general, it is a quasi-transverse intersection of codimension
exactly $c$.}
\end{rem}
\begin{rem}
For simplicity, to introduce parablenders, we have chosen the
parameter $a=0$. However, we can also define a parablender at any
other parameter $a=a_0$ with $a_0\in \mathbb{I}^k$. Moreover, we
will say that a $k$-parameter family $f=(f_a)_a$ has a parablender
$\Gamma=(\Gamma_a)_a$ at \mbox{\emph{any parameter}} when $\Gamma$
is a parablender for $f$ at $a=a_0$ for all $a_0\in \mathbb{I}^k$
with $\mathscr{D}$ and $\mathscr{U}$ independent of the value
$a_0$.
\end{rem}
Parablenders are a mechanism to provide $C^{d,r}$-open sets of
families of diffeomorphisms which are $C^d$-{degenerate}
unfoldings of {tangencies (of dimension zero in general)}. The
following theorem proves the existence of such open sets.
\begin{thm} \label{thm-parablender}
Any manifold of dimension $m> c+1$ admits a $k$\,-\,parameter
$C^d$-family of $C^r$-diffeomorphisms with { $0<d < r$} having a
{$C^d$-}parablender of {central
dimension $c\geq 1$} \mbox{at any parameter.}
\end{thm}
We split the proof of this theorem into several parts.
{The basic idea to obtain a parablender is by
constructing a blender for the induced dynamics $\widehat{f}$ in
the space of jets using a parametric family $f=(f_a)_a$ of
diffeomorphisms. To do this, first in~\S\ref{sec:jets}, we
introduce the jet space and the induced dynamics. After that, we
consider in~\S\ref{sec:affine-blenders-family} a class of
parametric families of diffeomorphisms with a family
$\Gamma=(\Gamma_a)_a$, where each $\Gamma_a$ is an affine
$cs$-blender as constructed in~\S\ref{sec:affine-blender}. In
order to see that this family of blenders is, indeed, a
parablender we show in~\S\ref{sec:blender-jets} that the induced
dynamics on the jet space has a $cs$-blender $\widehat{\Gamma}$.
To conclude that $\Gamma$ is a parablender we also need to provide
an open set $\mathscr{D}$ of $k$-parametric families of discs.
This is done in~\S\ref{sec:discos-afin-jet-blender} where
additionally we show that each family of discs
$\mathcal{D}=(\mathcal{D}_a)_a$ in $\mathscr{D}$ induces a disc
$\widehat{D}$ of jets in a superposition region of the blender
$\widehat{\Gamma}$. Finally, we show
in~\S\ref{sec:parablender-from-blender} that the robust
intersection between each of these discs $\widehat{\mathcal{D}}$
of jets and the local unstable manifolds of $\widehat{\Gamma}$
implies a degenerate unfolding of a tangency between the family
$\mathcal{D}$ and the local unstable manifold of $\Gamma$ in the
sense of Definition~\ref{def:parablender}.}
First of all, notice that we will provide the existence of
parablenders by constructing these objets in local coordinates.
Thus, again we will work in an open set of
$\mathbb{R}^m=\mathbb{R}^n\times \mathbb{R}^c$ with $n\geq 2$ and
$c\geq 1$. We also ask that $n=ss+u$.
\subsection{Jet space}
\label{sec:jets} Let $f=(f_a)_a$ be a $k$-parameter $C^{d}$-family
of $C^r$-diffeomorphisms of $\mathbb{R}^m$ with $0<d \leq r$. To
analyze
the unfolding of $f_a$ for $a\in\mathbb{I}^k$,
we will consider on $J^d_0(\mathbb{I}^k,\mathbb{R}^m)$ the map
$\widehat{f}$ \ induced
by the family $f=(f_a)_a$ and given by
$$
\widehat{f}(J_0^d(z))=J_0^d(f\circ z)=(f_a(z_a),\partial^1_a
f_a(z_a), \dots, \partial^d_a f_a(z_a))_{|_{\,a=0}}
\ \ \text{with} \ \ z=(z_a)_a \in
C^d(\mathbb{I}^k,\mathbb{R}^m).
$$
Here $J^d_0(\mathbb{I}^k,\mathbb{R}^m)$ denotes the $d$-th order
jet space at $a=0$, i.e., the set of equivalence classes
$J^d_0(z)$ where $z=(z_a)_a\in C^d(\mathbb{I}^k,\mathbb{R}^m)$.
The equivalent relation is defined by declaring that
$J^d_0(u)=J^d_0(v)$ if the functions $u$ and $v$ have all of their
partial derivatives equal at $a=0$ up to $d$-th order. A useful
choice of a representative for $J_0^d(z)$ is the $d$-th order
Taylor approximation of $z$ at $a=0$. This polynomial is
completely determined by the derivatives of $z$ at $a=0$, a finite
list of numbers. Therefore, it make sense to identify
$J_0^d(\mathbb{I}^k,\mathbb{R}^m)$ with $\mathbb{R}^m \times
\mathcal{J}^d(k,m)$ where
$$
\mathcal{J}^d(k,m) =
\prod_{i=1}^{d}\mathscr{L}^i_{sym}(\mathbb{R}^k,\mathbb{R}^m)
$$
and $\mathscr{L}^i_{sym}(\mathbb{R}^k,\mathbb{R}^m)$ denotes the
space of symmetric $i$-linear maps from $\mathbb{R}^k$ to
$\mathbb{R}^m$. Hence, clearly $J_0^d(\mathbb{I}^k,\mathbb{R}^m)$
is a Euclidian vector space of dimension
$$
\dim J_0^d(\mathbb{I}^k,\mathbb{R}^m)= m \cdot \binom{d+k}{d} = \frac{m \cdot (d+k)!}{d! \cdot
k!}.
$$
\begin{rem}
\label{rem:parablender-regularidad} Notice that the map
$\widehat{f}$ is of class $C^{r-d}$.
\end{rem}
\begin{notation} In order to simplify notation write
$$
J(\mathbb{R}^m) \eqdef J_0^d(\mathbb{I}^k,\mathbb{R}^m) \qquad
\text{and} \qquad J(z)\eqdef J^d_0(z)=(z_a,\partial^1_a z_a,\dots,\partial^d_a z_a)_{|_{\,a=0}}.
$$
Sometimes, by
considering $z_a=(x_a,y_a) \in \mathbb{R}^n\times \mathbb{R}^c$,
we will split the manifold of $(d,k)$-velocities over
$\mathbb{R}^m$ (i.e., the space of $d$-jets from $\mathbb{I}^k$ to
$\mathbb{R}^m$ at $a=0$) in the form of $
J(\mathbb{R}^m)=J(\mathbb{R}^n)\times J(\mathbb{R}^c)
$~and
$$
J(z)=(J(x),J(y)) \quad \text{where $x=(x_a)_a \in
C^d(\mathbb{I}^k,\mathbb{R}^n)$ and $y=(y_a)_a \in
C^d(\mathbb{I}^k,\mathbb{R}^c)$.}
$$
Moreover, denote by $J({\Lambda})$ the subset of $J(\mathbb{R}^*)$
of $d$-jets $J(z)$ at $a=0$ of families of points $z=(z_a)_a\in
C^{d}(\mathbb{I}^k,\mathbb{R}^*)$ such that $z_0 \in \Lambda$,
where $\Lambda\subset \mathbb{R}^*$ and $*\in \{m,n,c\}$. Also,
denote by $
\text{$\mathscr{P}_*:J(\mathbb{R}^m)\to \mathbb{R}^*$
the canonical projection onto $\mathbb{R}^*$ with $*\in
\{ss,u,n,c,m\}$.}$
\end{notation}
\subsection{A family of affine blenders} \label{sec:affine-blenders-family}
We will take a $C^r$-diffeomorphism $\Phi_0$ of $\mathbb{R}^m$
locally defined as the skew-product given
in~\S\ref{sec:affine-blender}.
In particular, we have an affine $cs$-blender $\Gamma_0$ for
$\Phi_0$ in the cube $[-2,2]^{m}$ {having the family of
almost-horizonal $C^1$-discs in $\mathcal{B}=\mathsf{R}\times B$
as a superposition region}. Here $B$ is an open neighborhood of
$0$ in $D=(-2,2)^c$ satisfying
the covering property~\eqref{eq:cover} and
$\mathsf{R}=\mathsf{R}_1\cup\dots\cup\mathsf{R}_\kappa$. Now, we
will take a particular family $\Phi=(\Phi_a)_a$, unfolding
$\Phi_0$ at $a=0$. Namely, we consider $C^r$-diffeomorphisms
$\Phi_a$ locally defined in a similar way by means of
skew-products of the form
$$
\Phi_a=F\ltimes(\phi_{1,a},\dots,\phi_{\kappa,a})
\quad \text{on} \ \ \mathcal{U}=(\mathsf{R}_1\times
D)\cup\dots\cup (\mathsf{R}_\kappa\times D)
$$
where $\phi_{\ell,a}$ are $k$-parameter $C^d$-families of affine
$(\lambda,\beta)$-contractions on $D$ for $\nu<\lambda<\beta<1$.
That is, $\phi_\ell=(\phi_{\ell,a})_a$ is a $k$-parameter
$C^d$-family of $C^r$-diffeomorphisms $\phi_{\ell,a}$ of
$\mathbb{R}^c$ such that $\phi_{\ell,a}(\overline{D})\subset D$
and there are linear maps $T_{\ell,a}:\mathbb{R}^c\to
\mathbb{R}^c$ so that
$$
\text{$D\phi_{\ell,a}(y)=T_{\ell,a}$ for all $y\in\overline{D}$ \
\ and \ \ $\lambda<m(T_{\ell,a})\leq \|T_{\ell,a}\|<\beta$ for
$\ell=1,\dots,\kappa$ and $a\in \mathbb{I}^k$.}
$$
Moreover, we ask that { a bounded open neighborhood $\widehat{B}$
of the $d$-jet $0$ in $J(\mathbb{R}^c)$} such that
\begin{equation}
\label{eq:cover-jets}
\overline{\widehat{B}} \subset \widehat{\phi}_1(\widehat{B})\cup \dots\cup
\widehat{\phi}_\kappa(\widehat{B})
\end{equation}
where $\widehat{\phi}_\ell$ is the induced map on $J(D)$ by the
family $\phi_\ell=(\phi_{\ell,a})^{}_a$, i.e.,
\begin{equation*}
\widehat{\phi}_\ell(J(y))=J(\phi_\ell\circ y)=
(\phi_{\ell,a}(y_a), {\partial^1_a \phi_{\ell,a}(y_a),\dots,\partial^d_a \phi_{\ell,a}(y_a)})_{|_{\,a=0}}
\end{equation*}
with $y=(y_a)_a \in C^d(\mathbb{I}^k,\mathbb{R}^c)$ such that $y_0
\in D$.
Without restriction of generality we can assume that
$B\times\{0\}\subset \widehat{B}$ where $B$ is the open set given
in~\eqref{eq:cover}.
On the other hand, let $\Gamma_a$ be the affine $cs$-blender
continuation of $\Gamma_0$ for $\Phi_a$. To conclude the proof we
need to prove that $\Gamma=(\Gamma_a)_a$ is a $cs$-parablender of
$\Phi=(\Phi_a)_a$ at $a=0$.
\begin{rem} \label{rem:any-parameter}
The family $\Phi=(\Phi_a)_a$ can be seen as an unfolding of
$\Phi_{a_0}$ for any $a_0 \in \mathbb{I}^k$. Since $\phi_{\ell,a}$
varies $C^d$-continuously with $a\in \mathbb{I}^k$, a similar
covering property as in~\eqref{eq:cover-jets} holds for the maps
$\widehat{\phi}_\ell=J_{a_0}^d[\phi_\ell]$. These are induced by
the families of fiber maps $\phi_{\ell}=(\phi_{\ell,a})_a$ on the
$d$-jet space $J_{a_0}^d(\mathbb{I}^k,\mathbb{R}^c)$ at $a=a_0$
for all sufficiently small parameter $a_0$. That is,
$$
\widehat{\phi}_\ell(J(y))=
(\phi_{\ell,a}(y_a), {\partial^1_a \phi_{\ell,a}(y_a),\dots,\partial^d_a \phi_{\ell,a}(y_a)})_{|_{\,a=a_0}}
$$
with $y=(y_a)_a \in C^d(\mathbb{I}^k,\mathbb{R}^c)$ such that
$y_{a_0} \in D$. In what follows, we will show that $\Gamma$ is a
{$C^d$-}$cs$-parablender of $\Phi$ at $a=0$.
However, the choice of $a=0$ is only for convenience to fix an
unfolding parameter (and thus a jet space). The same argument
works to prove that $\Gamma$ is a
{$C^d$-}$cs$-parablender of $\Phi$ at $a=a_0$ for
any $a_0$ close enough to~$0$. In fact, by continuity with respect
to the parameter, we can take an uniform open set $\mathscr{D}$ of
families of discs and an uniform neighborhood $\mathscr{U}$ of the
family $\Phi$ for all $a_0$ close to~$0$. Therefore
$\Gamma=(\Gamma_a)_a$ will be, up to scaling the parametrization,
a {$C^d$-}$cs$-parablender of the $k$-parametric family
$\Phi=(\Phi_a)_a$ at any value of the parameter $a\in
\mathbb{I}^k$.
\end{rem}
\begin{exap}
\label{exa}
Let $\phi(t)=\lambda t$ for $t\in [-2,2]$ with $1/2<\lambda<1$.
Set
$$
\Upsilon=\{\iota=(\iota_1,\dots,\iota_k)\in
\{0,1,\dots,d\}^k \ \ \text{with} \ \
|\iota|=\iota_1+\dots+\iota_k \leq d\} \ \ \text{and} \ \
\Delta=(1-\lambda)\cdot\{-1,+1\}^\Upsilon.
$$
Each $\delta \in \Delta$ is seen as a function which maps $\iota
\in \Upsilon$ to $\delta(\iota)\in \{-(1-\lambda),+(1-\lambda)\}$.
Take
\begin{align*}
\phi_{\delta,a}(t)= \phi(t)+ P_{\delta}(a) \quad
\text{for \ $\delta \in \Delta$, \ $a\in \mathbb{I}^k$ \ and \ $t \in [-2,2]$ }
\end{align*}
where
$$
P_{\delta}(a) =\sum_{\iota \in \Upsilon} \delta(\iota) \
a^\iota \quad \text{with \ \ $a^\iota = a^{\iota_1}_1\cdots a^{\iota_k}_k$ \ and \ $\iota=(\iota_1,\dots,\iota_k)$.}
$$
Finally, consider
\begin{align}
\label{eq:multi-index}
\phi_{\ell,a} = \phi_{\ell_1,a}\times \dots\times \phi_{\ell_c,a}
\quad
\text{on \ $D$ } \ \
\text{for any \ $\ell=(\ell_1,\dots,\ell_c)\in \Delta^c$.}
\end{align}
Here, $\kappa=\varrho^c$ where $\varrho$ is the cardinal of
$\Delta$. When there are no parameters, i.e., for $k=0$, we
recover the Example~\ref{exa-afin-blender}. Moreover,
$D\phi_{\ell,a}(y)$ is the diagonal matrix $\lambda I$ where $I$
is the identity matrix and thus it does not depend on $a$ for all
$y\in \overline{D}$. Hence, we can rewrite~\eqref{eq:multi-index}
as
\begin{equation}
\label{eq:mult0}
\phi_{\ell,a}(y)= \lambda y + P_\ell(a) \quad \text{for \ $y\in D$, \
$a\in \mathbb{I}^k$ \ and \ $\ell=(\ell_1,\dots,\ell_c) \in \Delta^c$}
\end{equation}
where using multilinear algebra
$$
P_\ell(a)= \partial^0\ell + \partial^1 \ell \cdot a +
\frac{1}{2!} \, \partial^2\ell \cdot a^2 +\dots +
\frac{1}{d!} \, \partial^d\ell \cdot a^d
$$
with
$\partial^i\ell \in
\mathscr{L}^i_\mathrm{sym}(\mathbb{R}^k,\mathbb{R}^c)$ determined
by
$$
\partial^i\ell \cdot a^i \eqdef \partial^i\ell(a,\dots,a)=
\sum_{|\iota|=i} \ell(\iota) \, a^\iota \quad \text{ {for all $i=0,1,\dots,d$.}}
$$
Now, we can easily compute the induced map $\widehat{\phi}_{\ell}$
on $J(\mathbb{R}^c)$.
To do this, {consider $y=(y_a)_a \in
C^d(\mathbb{I}^k,\mathbb{R}^c)$ such that $y_0 \in D$. Denoting by
$$
J(y)=(y^{}_a,\partial^1_ay^{}_a,\dots,\partial^d_ay^{}_a)_{|_{\,a=0}}\eqdef
(\partial^0y,\partial^1y,\dots,\partial^dy),$$}
from~\eqref{eq:mult0} we get that
$$
\partial^i_a\phi_{\ell,a}(y^{}_a)_{|_{\,a=0}} = {\lambda\cdot (\partial^i_a y_a)_{|_{\,a=0}}} +
\partial^i_a P_\ell(a)_{|_{\,a=0}}=\lambda \partial^i y + \partial^i\ell \quad \text{ {for all $i=0,1,\dots,d$.}}
$$
Thus,
$$
\widehat{\phi}_\ell(\partial^0y, {\dots,\partial^d y})=
\lambda \cdot(\partial^0y, {\dots,\partial^d y})
+(\partial^0\ell, {\dots,\partial^d\ell}).
$$
Consequently, $\widehat{\phi}_\ell$ is the composition of a
contracting hyperbolic linear map on $J(\mathbb{R}^c)$ with a
translation by the jet
$(\partial^0\ell,\partial^1\ell,\dots,\partial^i\ell)$. Since
$\ell$ runs over $\Delta^c$, we find that the open neighborhood
$\widehat{B}=(-1,1)^{\widehat{d}_c}$ of $0$ in $J(\mathbb{R}^c)$
satisfies~\eqref{eq:cover-jets}, where $\widehat{d}_c=\dim
J(\mathbb{R}^c)$.
\end{exap}
\subsection{Parablenders in the $C^{d,r}$-topology for $0<d<r$}
According to Remark~\ref{rem:parablender-regularidad}, in order to
construct blenders for the induced map $\widehat{\Phi}$ by the
$C^{d,r}$-family $\Phi=(\Phi_a)_a$ we {will} restrict our
analysis to $0<d< r$ to obtain that $\widehat{\Phi}$ is at least
$C^1$.
\subsubsection{Blender induced in the jet space}
\label{sec:blender-jets} {Consider $z=(z_a)_a \in
C^d(\mathbb{I}^k,\mathbb{R}^{m})$ and write $z_a=(x_a,y_a)$ with
$x=(x_a)_a\in C^d(\mathbb{I}^k,\mathbb{R}^n)$ and $y=(y_a)_a\in
C^d(\mathbb{I}^k,\mathbb{R}^c)$.} For each $i=1,\dots,d$, since
$F$ is an affine map which does not depend on $a$, the partial
derivative~is
$$
\partial^i_a \Phi_a(z_a)_{|_{\,
a=0}}=(DF(x_a)\, \partial^i_a x_a, \
\partial^i_a\phi_{\ell,a}(y_a))_{|_{\,a=0}} \ \ \text{where
$z_0=(x_0,y_0)\in \mathsf{R}_\ell\times D$ for
$\ell=1,\dots,\kappa$.}
$$
Hence, the map $\widehat\Phi$ on $J(\mathbb{R}^m)$ induced by the
family $\Phi=(\Phi_a)_a$ restricted to
$J(\mathcal{U})=J(\mathsf{R})\times J(D)$ is given by the
skew-product
$$
\widehat{\Phi}=\widehat F \ltimes
(\widehat{\phi}_1,\dots,\widehat{\phi}_\kappa) \quad \text{on} \ \
J(\mathcal{U})=J(\mathsf{R}_1) \times J(D)\cup \dots \cup
J(\mathsf{R}_\kappa) \times J(D),
$$ where $\widehat F $ acts on
$J({\mathsf{R}})$ given by
$$
\widehat{F}(J(x))=(F(x_a), \ {DF(x_a)\,
\partial^1_a
x^{}_a,\dots, DF(x_a)\,\partial^d_a
x^{}_a)_{|_{\,a=0}}} \ \ \text{where
$x_0 \in \mathsf{R}$}
$$
and $\widehat{\phi}_\ell$ is the induced map on $J(D)$ by the
family $\phi_{\ell}=(\phi_{\ell,a})_a$ for $\ell=1,\dots,\kappa$.
Moreover, $\widehat F$ has a horseshoe
$\widehat{\mathsf{\Lambda}}=\mathsf{\Lambda} \times \{0\}$
with, except for multiplicity, the same eigenvalues of $F$ and~stable~index
$$
\widehat{d}_{ss}\eqdef\mathrm{ind}^s(\widehat{\Lambda})= \dim
J(\mathbb{R}^{ss})= ss \cdot \binom{d+k}{d}.
$$
{Also the width of the stable cone-field on
$J(\mathsf{R})$ of $\hat{F}$ is the same width $\alpha$ that the
stable cone-field of $F$ has with respect to $\mathsf{R}$.}
Similarly, $\widehat{\phi}_\ell$ on $J(D)$ has also, except
multiplicity, the same eigenvalues of $\phi_{\ell,0}$ on $D$ for
all $\ell=1,\dots, \kappa$. Thus,
$\widehat{F}|_{\widehat{\Lambda}}$ dominates the fiber dynamics
$\widehat{\phi}_1,\dots,\widehat{\phi}_\kappa$ and also by
assumption the covering property~\eqref{eq:cover-jets} holds.
Hence, according to Theorem~\ref{thmBKR}, we have a $cs$-blender
$\widehat{\Gamma}$ of
{central dimension} $\widehat{d}_{c}$
for $\widehat{\Phi}$, where
$$\widehat{d}_{c}\eqdef \dim J(\mathbb{R}^c)=c\cdot\binom{d+k}{d}.$$
{Additionally, the family of
$( {\alpha},\nu,\delta)$-horizonal discs in
$\widehat{\mathcal{B}}= \widehat{\mathsf{R}} \times \widehat{B}$
is a superposition region of $\widehat{\Gamma}$, where now $\delta
< \lambda \hat{L}/2$ and $0<\hat{L}\leq L$ is the Lebesgue number
of the cover~\eqref{eq:cover-jets}}. Here $\widehat{\mathsf{R}}$
is a bounded open neighborhood on $J(\mathsf{R})$ of
$\widehat{\Lambda}=\Lambda\times\{0\}$. Moreover, by construction,
$\mathscr{P}_m(\widehat{\Gamma})=\Gamma_0$ where $\mathscr{P}_m:
J(\mathbb{R}^m) \to \mathbb{R}^m$ is the canonical projection.
\subsubsection{An open set of families of discs for the
family of affine blenders} \label{sec:discos-afin-jet-blender}
Recall that $\widehat{\mathsf{R}}$ was taken as a bounded
neighborhood on $J(\mathsf{R})$ of
$\widehat{\Lambda}=\Lambda\times\{0\}$. Since
$J(\mathbb{R}^n)=J(\mathbb{R}^{ss})\times J(\mathbb{R}^u)$, there
is no loss of generality in assuming that
\begin{equation} \label{eq:hatR}
\widehat{\mathsf{R}}=\widehat{\mathsf{R}}_{ss} \times
\widehat{\mathsf{R}}_{u} \quad \text{with \
$\widehat{\mathsf{R}}_{ss} \subset J(\mathbb{R}^{ss})$ \ \ and \ \
$\widehat{\mathsf{R}}_{u} \subset J(\mathbb{R}^{u})$.}
\end{equation}
In fact, we can assume that
$\widehat{\mathsf{R}}_{ss}=(-2,2)^{ss} \times B_{ {\rho}}(0)$,
where {$B_\rho(0)$ denotes the subset of $\mathcal{J}^d(k,ss)$ so
that the symmetric $i$-linear maps have all norm less than
$\rho$.}
Notice that the closure of $\widehat{\mathsf{R}}_{ss}$
{can be identified with $[-2,2]^{d_{ss}}\times
[-\rho,\rho]^{\widehat{d}_{ss}-d_{ss}}$}. Hence, without loss of
generality, this set can be used to parameterize the
$(\textcolor{blue}{\alpha,}\nu,\delta)$-horizontal
$\widehat{d}_{ss}$-dimensional discs in
$\widehat{\mathcal{B}}=\widehat{\mathsf{R}}\times\widehat{B}$.
Consider a $( {\alpha,}\nu,\delta)$-horizontal $ss$-dimensional
$C^r$-disc $\mathcal{H}_0$ in $\mathcal{B}=\mathsf{R}\times B$.
Take the $k$-parametric constant family associated with
$\mathcal{H}_0$ given by
$$
\mathcal{H}=(\mathcal{H}_a)_a \quad \text{where \ \
$\mathcal{H}_a=\mathcal{H}_0$~for~all~$a\in \mathbb{I}^k$}.
$$
\begin{lem} \label{lem:disc-jets}
The set $\widehat{\mathcal{H}}$ in $J(\mathbb{R}^m)$ parameterized
by
$$
\widehat{\mathcal{H}}(J(\xi))
=J(\mathcal{H}\circ \xi)
\quad \text{for $\xi=(\xi_a)_a\in C^d(\mathbb{I}^k,\mathbb{R}^{ss})$ with
$J(\xi) \in \overline{\widehat{\mathsf{R}}_{ss}}$. }
$$
is a $( {\alpha,}\nu,\delta)$-horizontal
$\widehat{d}_{ss}$-dimensional $C^1$-disc in
$\widehat{\mathcal{B}}=\widehat{\mathsf{R}}\times \widehat{B}$ for
any {$\alpha>0$ large enough} and $\nu>0$ small enough.
\end{lem}
\begin{proof}
According to Definition~\ref{def:almost-horizontal-disc}, we need
to show that $\widehat{h}=\mathscr{P}\circ \widehat{\mathcal{H}}$
is $\delta$-close in the $C^0$-topology to a constant function on
$\widehat{B}$ where $\mathscr{P}$ the canonical projection on the
central coordinate, i.e., onto $J(\mathbb{R}^c)$. Moreover, we
also need to show that $\widehat{h}$ is $C^1$-dominated by a
constant $C>0$ so that $C\nu < \delta$ {and
$\|D\widehat{g}\|_\infty \leq \alpha$ where $\widehat{g}$ is the
$J(\mathbb{R}^u)$-coordinate of $\widehat{\mathcal{H}}$}. Since
$\widehat{h}$ {and $\widehat{g}$ are} of class $C^{r-d}$ with
$r>d$, then $C=\|D\widehat{h}\|_{\infty} <\infty$
{and $D=\|D\widehat{g}\|_\infty<\infty$} over the
closure of $\widehat{\mathsf{R}}_{ss}$. Thus taking $\nu>0$ small
enough {and $\alpha>0$ large enough}, we can guarantee that $C\nu
<\delta$ {and $D\leq \alpha$}. So, we only need to prove that
there is a point $\widehat{y}\in \widehat{B}$ such that
$$
d(\widehat{h}(J(\xi) {)},\widehat{y})<\delta \quad \text{for all
$J(\xi) \in \overline{\widehat{\mathsf{R}}_{ss}}$.}
$$
Since $\mathcal{H}=(\mathcal{H}_a)_a$ is a constant family of
discs,
then~$\widehat{h}=\widehat{h}_0$~where~$h_0=\mathscr{P}\circ
\mathcal{H}_0$~and
$$
\widehat{h}_0(J(\xi))=
(h_0(\xi_a), {\partial_a^1h_0(\xi_a),\dots,\partial_a^d
h_0(\xi_a)})_{|_{\,a=0}}
\quad \text{for $\xi=(\xi_a)_a\in C^d(\mathbb{I}^k,\mathbb{R}^{ss})$ with
$J(\xi) \in \overline{\widehat{\mathsf{R}}_{ss}}$. }
$$
Moreover, as $\mathcal{H}_0$ is a $(
{\alpha,}\nu,\delta)$-horizontal disc in $\mathcal{B}$, there is
$y\in B$ such that $d(h_0(x),y)<\delta$ for all $x \in
[-2,2]^{ss}$. Set $\widehat{y}=(y,0)\in \widehat{B}$. For any
$x\in [-2,2]^{ss}$, we consider $\xi=(\xi_a)_a$ given by $\xi_a=x$
for all $a\in \mathbb{I}^k$. Then $J(\xi)=(x,0)$ and
{since $h_0(\xi_a)=h_0(x)$ for all $a\in
\mathbb{I}^k$ we have $\widehat{h}_0(J(\xi))=(h_0(x),0)$.}
Therefore $d(\widehat{h}(J(\xi)),\widehat{y})<\delta$ for all
$J(\xi) \in [-2,2]^{ss}\times \{0\}$. By continuity, and since $
{\rho}>0$ can be taken arbitrarily small, it follows that
$d(\widehat{h}(J(\xi)),\widehat{y})<\delta$ for all $J(\xi)$ in
the closure of $\widehat{\mathsf{R}}_{ss}=(-2,2)^{ss}\times B_{
{\rho}}(0)$. This completes the proof of the lemma.
\end{proof}
\begin{rem} If $\mathcal{H}_0$ is a horizontal $ss$-dimensional $C^r$-disc in
$\mathcal{B}$ then $\widehat{\mathcal{H}}$ is also a horizontal
$\widehat{d}_{ss}$-dimensional $C^1$-disc in
$\widehat{\mathcal{B}}$. Thus, in this case, we do not need a
strong contraction for the dynamics on the base. It is only
required the domination assumption $\nu <\lambda$.
\end{rem}
Since being an almost-horizontal $C^1$-disc is an open property,
any small enough $C^{d,r}$-perturbation
$\mathcal{D}=(\mathcal{D}_a)_a$ of $\mathcal{H}=(\mathcal{H}_a)_a$
still provides an almost-horizontal $\widehat{d}_{ss}$-dimensional
$C^1$-disc $\widehat{\mathcal{D}}$ in $\widehat{\mathcal{B}}$
close to $\widehat{\mathcal{H}}$ given by
$$
\widehat{\mathcal{D}}(J(\xi))=J(\mathcal{D}\circ \xi)
\quad
\text{for $\xi=(\xi_a)_a \in C^d(\mathbb{I}^k,\mathbb{R}^{ss})$ with
$J(\xi)\in \overline{\widehat{\mathsf{R}}^{ss}}$.}
$$
In fact, taking $\xi_a \in [-2,2]^{ss}$ and
$z_a=\mathcal{D}_a(\xi_a)$ for all $a\in \mathbb{I}^k$, it is not
difficult to see that the image of this embedding is given by
$$
\widehat{\mathcal{D}}=\{ \, J(z)\in J(\mathbb{R}^m):
z=(z_a)\in C^d(\mathbb{I}^k,\mathbb{R}^m)
\ \ \text{with} \ \ z_a\in \mathcal{D}_a \ \
\text{for all $a\in \mathbb{I}^k$} \}\cap \overline{\widehat{\mathcal{B}}}.
$$
In this way, we take $\mathscr{D}=\mathscr{D}({\mathcal{H}})$, a
small enough $C^{d,r}$-neighborhood of $\mathcal{H}$.
\begin{rem} The superposition region $\mathscr{D}$ of an affine $cs$-parablender
contains the open set of
\emph{almost-constant $k$-parameter $C^d$-families of
almost-horizontal $ss$-dimensional $C^r$-discs in $\mathcal{B}$.}
\end{rem}
\subsubsection{Parablenders from blenders in the jet space} \label{sec:parablender-from-blender} We will
get that $\Gamma=(\Gamma_a)_a$ is a $cs$-parablender of
$\Phi=(\Phi_a)_a$ as a consequence of the following general
result.
\begin{prop} \label{prop:parablender}
Let $\Gamma_0$ be a $cs$-blender of {central dimension} $c$
{and strong stable dimension
$d_{ss}=\mathrm{ind}^s(\Gamma_0)-c$} of a $C^r$-diffeomorphism
$f_0$ of a manifold~$\mathcal{M}$. Consider a $k$-parameter
$C^d$-family $f=(f_a)_a$ unfolding $f_0$ at $a=0$ such that the
induced map $\widehat{f}$ on the manifold of $(d,k)$-velocities
$J(\mathcal{M})=J^d_0(\mathbb{I}^k,\mathcal{M})$ over
$\mathcal{M}$, {which is} given by
$$
\widehat{f}(J(z))=J(f\circ z)=(f_a(z_a), {\partial^1_a f_a(z_a),\dots,\partial^d_a f_a(z_a)})_{|_{\,a=0}}
\ \ \text{with} \ \ z=(z_a)_a \in
C^d(\mathbb{I}^k,\mathcal{M}),
$$
has a $cs$-blender $\widehat{\Gamma}$ satisfying the following
assumptions:
\begin{enumerate}[itemsep=0.1cm]
\item $\widehat{\Gamma}$ projects on $\mathcal{M}$ onto $\Gamma_0$;
\item there is a $k$-parameter $C^d$-family $\mathcal{H}=(\mathcal{H}_a)_a$ of
$d_{ss}$-dimensional $C^r$-embedded discs $\mathcal{H}_a$ into
$\mathcal{M}$ so that $\widehat{\mathcal{H}}_0 \in
\widehat{\mathscr{D}}$, where
$\widehat{\mathcal{H}}_0$ is
contained in
$$
\widehat{\mathcal{H}}=\{ \, J(z)\in J(\mathcal{M}):
z=(z_a)\in C^d(\mathbb{I}^k,\mathcal{M})
\ \ \text{with} \ \ z_a\in \mathcal{H}_a \ \
\text{for all $a\in \mathbb{I}^k$} \}
$$
and $\widehat{\mathscr{D}}$ is {a} superposition region of the
blender $\widehat{\Gamma}$.
\end{enumerate}
Then $\Gamma=(\Gamma_a)_a$ is a
{$C^d$-}$cs$-parablender {at $a=0$}
of {central dimension $c$} for $f$, where $\Gamma_a$ is the
continuation of $\Gamma_0$ for~$f_a$.
\end{prop}
\begin{proof}
First of all, we will provide the open set of embedded discs. To
do this, similarly as in \S\ref{sec:discos-afin-jet-blender}, we
take a small $C^{d,r}$-neighborhood
$\mathscr{D}=\mathscr{D}(\mathcal{H})$ of the family
$\mathcal{H}$, so that any family $\mathcal{D}$ in $\mathscr{D}$
still gives a disc $\widehat{\mathcal{D}}_0 \in
\widehat{\mathscr{D}}$ contained in $\widehat{\mathcal{D}}$.
Next, we will construct the open set of families of
diffeomorphisms. Consider the neighborhood $\widehat{\mathscr{U}}$
of the induced map $\widehat{f}$ coming from the definition of the
blender. Take the $C^{d,r}$-neighborhood $\mathscr{U}$ of the
family $f=(f_a)_a$, so that for every $g=(g_a)_a \in \mathscr{U}$
its induced map $\widehat{g}$ on $J(\mathcal{M})$ belongs to
$\widehat{\mathscr{U}}$.
Now, we will prove the existence of a {degenerate} unfolding
{at $a=0$ of a tangency} between any family of
$ss$-dimensional discs $\mathcal{D}=(\mathcal{D}_a)_a \in
\mathscr{D}$ and the unstable manifold of
$\Gamma_g=(\Gamma_{a,g})_a$ for any $g=(g_a)_a \in \mathscr{U}$,
where $\Gamma_{a,g}$ is the continuation of $\Gamma_a$ for $g_a$.
Since $\widehat{\mathcal{D}}$ contains a disc
$\widehat{\mathcal{D}}_0$ in the superposition region
$\widehat{\mathscr{D}}$ of the $cs$-blender $\widehat{\Gamma}$ of
$\widehat{f}$, then
$$W^u_{loc}(\widehat{\Gamma}_g)\cap
\widehat{\mathcal{D}}_0\not =\emptyset$$ where
$\widehat{\Gamma}_g$ is the continuation of $\widehat{\Gamma}$ for
the induced map $\widehat{g}$. It is clear that
$\mathscr{P}_m(\widehat{\Gamma}_{g})=\Gamma_{0,g}$ and that
$\widehat{\Gamma}_g$ is a hyperbolic set of $\widehat{g}$.
If $J(z)\in \widehat{\Gamma}_g$, where $z=(z_a)_a \in
C^d(\mathbb{I}^k,\mathcal{M})$, then $z_0 \in \Gamma_{0,g}$ and
the point $z_a$ must be the continuation in $\Gamma_{a,g}$ of
$z_0$ for $g_a$. Similarly,
$\mathscr{P}_m(W^u_{loc}(J(z)))=W^u_{loc}(z_0)$ and if
$\widehat{x}\in W^u_{loc}(J(z))$ then
$$
\text{$x=(x_a)_a\in C^d(\mathbb{I}^k,\mathcal{M})$ \ so that \
$x_a\in W^u_{loc}(z_a)$ for all $a\in \mathbb{I}^k$ and
$J(x)=\widehat{x}$.}
$$
In summary, we can find a point $\widehat{q} \in
W^{u}_{loc}(\widehat{\Gamma}_g) \cap \widehat{\mathcal{D}}_0$.
Since $\widehat{q}$ belongs to the local unstable manifold of
$\widehat{\Gamma}_{g}$ there are functions $x=(x_a)_a,
z=(z_a)_a\in C^d(\mathbb{I}^k,\mathcal{M})$ such that
$$
\text{$x_a\in W^u_{loc}(z_a)$ with $z_a\in\Gamma_{a,g}$ for all
$a\in \mathbb{I}^k$ and $\widehat{q}=J(x)$.}
$$
On the other hand, since $\widehat{q}\in \widehat{\mathcal{D}}$,
$$\text{there is $y=(y_a)_a \in
C^d(\mathbb{I}^k,\mathcal{M})$ such that $y_a\in \mathcal{D}_a$
for all $a\in \mathbb{I}^k$ and $\widehat{q}=J(y)$.}$$ Thus
$J(x)=J(y)$. This concludes that $\mathcal{W}=(W^u_{loc}(z_a))_a$
and $\mathcal{D}=(\mathcal{D}_a)_a$ {has a tangency} at $a=0$
which unfolds $C^d$-{degenerately}. Therefore
$\Gamma=(\Gamma_a)_a$ is a {$C^d$-}$cs$-parablender
{at $a=0$} of {central dimension $c
\geq 1$} for $f=(f_a)_a$ and we complete the proof of the
proposition.
\end{proof}
\subsubsection{Proof of Theorem~\ref{thm-parablender}} Take the
family of $cs$-blenders $\Gamma=(\Gamma_a)_a$ of
{central dimension} $c>0$ of the particular family
of locally defined affine skew-products $\Phi=(\Phi_a)_a$
constructed in~\S\ref{sec:affine-blenders-family}. From
\S\ref{sec:blender-jets}, we get a $cs$-blender $\widehat{\Gamma}$
for the induced map $\widehat{\Phi}$ on $J(\mathbb{R}^m)$ which
projects in $\mathbb{R}^m$ onto $\Gamma_0$. In
\S\ref{sec:discos-afin-jet-blender} it was obtained that any
$k$-parameter constant family of horizontal discs induced a
$\widehat{d}_{ss}$-dimensional $C^1$-disc into the superposition
domain $\widehat{\mathcal{B}}$ of $\widehat{\Gamma}$. Thus, this
disc belongs to the superposition region of the induced blender.
Hence, according to Proposition~\ref{prop:parablender}, $\Gamma$
is a {$C^d$}-$cs$-parablender of {central dimension} $c>0$ at
$a=0$. Finally, by Remark~\ref{rem:any-parameter} and
reparameterizing if necessary, $\Phi=(\Phi_a)_a$ is $k$-parametric
$C^d$-family of $C^r$-diffeomorphisms having a
{$C^d$-}$cs$-parablender $\Gamma=(\Gamma_a)_a$ of
{central dimension $c\geq 1$} at any parameter.
This
completes the proof.
\section{Robust {degenerate} unfolding of heterodimensional cycles}
\label{sec:cycles} Now we will prove Theorem~\ref{thmA}. We will
consider a $C^d$-family $f=(f_a)_a$ of $C^r$-diffeomor\-phisms of
a manifold $\mathcal{M}$ parameterized by $a\in\mathbb{I}^k$ with
a {$C^d$-}$cs$-parablender $\Gamma=(\Gamma_a)_a$ of codimension
$c\geq 1$ at any parameter. For simplicity, we will assume that
$\Gamma$ is the family of affine blenders constructed to prove
Theorem~\ref{thm-parablender}. We will assume that $f_0$ has a
heterodimensional cycle of co-index $c\geq 1$ associated with
$\Gamma_0$ and another hyperbolic periodic point $P_0$. We suppose
that $W^s(P_0)$ contains a $ss$-dimensional horizontal disc
$\mathcal{H}_0$ in the superposition domain $\mathcal{B}$ of the
$cs$-blender $\Gamma_0$ of $f_0$. Moreover, as the construction is
local, we ask that $W^s(P_a)$ contains the same disc
$\mathcal{H}_0$ for all $a\in\mathbb{I}^k$ where $P_a$ denotes the
continuation of $P_0$ for $f_a$. Hence the constant family of
discs $\mathcal{H}=(\mathcal{H}_a)$ where
$\mathcal{H}_a=\mathcal{H}_0$ for all $a\in \mathbb{I}^k$ belongs
to the open set $\mathscr{D}$ of families of embedded discs
associated with the {$C^d$-}$cs$-parablender $\Gamma$. Thus, for
every $C^{d,r}$-close enough family $g=(g_a)_a$ of $f=(f_a)_a$ the
family of stable manifolds $W^s(P_g)=(W^s(P_{a,g}))_a$ of the
continuation $P_{a,g}$ of $P_a$ contains a family of discs
$\mathcal{D}=(D_a)_a \in \mathscr{D}$. Therefore,
{$W^u_{loc}(\Gamma_g)$ and $W^s(P_g)$} has a
{tangency} at $a=0$ which unfolds
$C^d$-{degenerately}. In fact, since $\Gamma$ is a
{$C^d$-}$cs$-parablender at any parameter, the same
argument also works for any parameter $a=a_0$. This concludes the
proof of the theorem.
\section{Robust {degenerate} unfoldings of homoclinic tangencies}
\label{sec:final} In this section we prove Theorem~\ref{thmC}. We
begin by mentioning a few words about the strategy of the proof.
Recall that to prove Theorem~\ref{thmA} we first show that any
manifold $\mathcal{M}$ of dimension at least $3$ admits a family
$f=(f_a)_a$ of diffeomorphisms $f_a$ of $\mathcal{M}$ having a
parablender at any parameter (see Theorem~\ref{thm-parablender}).
Now, to prove Theorem~\ref{thmC}, we will proceed similarly by
showing first the following result:
\begin{thm}
\label{thm-paratangencies} Any manifold $\mathcal{M}$ of dimension
$m> c+u$ admits a $k$-parameter $C^d$-family $f=(f_a)_a$ of
\mbox{$C^r$-diffeomorphisms} with { $0<d<r-1$} such that the
$k$-parameter induced $C^d$-family $f^{{G}}=(f^{{G}}_a)^{}_a$ of
$C^{r-1}$-diffeomorphisms on the $u$-th Grassmannian bundle of
$\mathcal{M}$,
$$
f^{{G}}_a : G_u(\mathcal{M})\longrightarrow G_u(\mathcal{M}), \qquad
f^{{G}}_a(x,E)=(f_a(x),Df_a(x)E)
$$
has a $C^d$-parablender $\Gamma^{{G}}=(\Gamma^{{G}}_a)^{}_a$ of
{central dimension $c\geq 1$} at any parameter.
\end{thm}
As in the proof of Theorem~\ref{thm-parablender}, we will obtain a
parablender for the $k$-parameter family
$f^{{G}}=(f_a^{{G}})^{}_a$ for the induced dynamics by
constructing a blender with respect to the induced dynamics
$\widehat{f}^{{G}}$ on the manifold of $(c,k)$-velocities over
$G_u(\mathcal{M})$, i.e., on the jet space
$J^d_0(\mathbb{I}^k,G_u(\mathcal{M}))$.
\begin{rem}
\label{rem:paratangencias-regularidad} Notice that the map
$\widehat{f}^{{G}}$ is of class $C^{r-1-d}$.
\end{rem}
In what follows, we fix { $0<d<r-1$}. As in the previous section,
we will provide the proof of Theorem~\ref{thm-paratangencies}
using the local coordinates in $\mathcal{M}$. Thus, as usual, we
will work in $\mathbb{R}^m=\mathbb{R}^{ss}\times\mathbb{R}^u\times
\mathbb{R}^c$ with $u,c\geq 1$ and $n=ss+u$.
We also recall some notation
from~\S\ref{sec:affine-blenders-grasmannian}:
\begin{gather*}
\mathsf{R}=\mathsf{R}_1\cup\dots\cup \mathsf{R}_\kappa \ \ \
\text{and} \ \ \ \mathsf{R}^{{G}} = \mathsf{R}\times \mathcal{C}^{{G}} =
\mathsf{R}^{{G}}_1\cup \dots \cup \mathsf{R}^{{G}}_\kappa \ \ \text{with} \ \
\mathsf{R}_i^{{G}}=\mathsf{R}_i\times \mathcal{C}^{{G}} \\
\mathcal{U}=\mathsf{R}\times D = (\mathsf{R}_1\times D)\cup\dots\cup
(\mathsf{R}_\kappa\times D) \ \ \ \text{and} \ \ \
\mathcal{U}^{{G}}=\mathcal{U}\times \mathcal{C}^{{G}}.
\end{gather*}
Sometimes, when no confusion arises, we write $\mathcal{U}^{{G}}$
by a change of coordinates as
$$
\mathcal{U}^{{G}}=\mathsf{R}^{{G}}\times D = (\mathsf{R}^{{G}}_1\times D)\cup\dots\cup
(\mathsf{R}^{{G}}_\kappa\times D).
$$
\subsection{A parablender on the manifold of velocities over the Grassmanian manifold}
\label{sec:parablender-grasmmannian}
We will start considering the $k$-parameter $C^{d}$-family of
locally defined affine $C^{r}$ skew-product maps
$$
\Phi_a=F\ltimes(\phi_{1,a},\dots,\phi_{\kappa,a}) \quad \text{on
\ \ $\mathcal{U}=(\mathsf{R}_1\times D)\cup\dots\cup
(\mathsf{R}_\kappa\times D)$}
$$
introduced in~\S\ref{sec:affine-blenders-family}. For simplicity,
we assume that there are $0<\nu<\lambda<\beta<1$ and diagonal
linear maps $S: \mathbb{R}^{ss}\to \mathbb{R}^{ss}$, $U:
\mathbb{R}^{u} \to \mathbb{R}^u$ and
$T:\mathbb{R}^c\to\mathbb{R}^c$ such that
$$
DF(x)=
\begin{pmatrix}
S & 0 \\
0 & U
\end{pmatrix}
\quad \text{for all \ \ $ x\in \mathsf{R}$ \ \ where \ \ $\|S
\|,\ \|U^{-1}\| < \nu$}
$$
and
$$
\text{$D\phi_{i,a}(y)=T$ \ \ for all \ $y\in \mathbb{R}^c$, \
$a\in\mathbb{I}^k$ \ and \ $i=1,\dots,\kappa$ \ \ where \ \
$\lambda <m(T)\leq \|T\|<\beta$.}
$$
Under theses assumptions, we get that $D\Phi_a(x,y)$ is the same
linear map $D\Phi$ for all $(x,y)\in \mathsf{R}\times D$ which has
$E^u_a=E^u$ with $E^u=\{0^{ss}\}\times \mathbb{R}^u \times
\{0^c\}$ as a fixed point and $\mathcal{C}^{{G}}$ as a
neighborhood of attraction for all $a\in \mathbb{I}^k$. Thus,
following \S\ref{sec:affine-blenders-grasmannian}, the induced
$C^{r}$-diffeomorphism $\Phi^{{G}}_a$ of $\Phi_a$ on the
Grassmannian manifold $G_u(\mathbb{R}^m)$ is given by
$$
\Phi_a^{{G}} = F^{{G}} \ltimes (\phi_{1,a},\dots,\phi_{\kappa,a})
\quad
\text{on \ \
$\mathcal{U}^{{G}}=(\mathsf{R}^{{G}}_1 \times D)\cup\dots\cup(\mathsf{R}^{{G}}_\kappa\times
D$)}
$$
where $F^{{G}}=F \times D\Phi$ on
$\mathsf{R}^{{G}}=\mathsf{R}\times \mathcal{C}^{{G}}$. Moreover,
for each $a\in \mathbb{I}^k$ we have a $cs$-blender
$\Gamma^{{G}}_a=\Gamma_a \times \{E^u\}$ of
{central dimension $c\geq 1$} where $\Gamma_a$ is
the $cs$-blender of $\Phi_a$. Now, we will show that the family
$\Gamma^{{G}}=(\Gamma^{{G}}_a)^{}_a$ is a $cs$-parablender of
{central dimension} $c$ at $a=0$ for
${\Phi}^{{G}}=({\Phi}^{{G}}_a)^{}_a$.
To prove this, we {will} work with the induced $C^1$-map on
$J(G_u(\mathbb{R}^m))\eqdef J^d_0(\mathbb{I}^k,G_u(\mathbb{R}^m))$
by the family $\Phi^{{G}}=({\Phi}^{{G}}_a)^{}_a$ given by
$$
\widehat{\Phi}^{{G}}: J(G_u(\mathbb{R}^m)) \to
J(G_u(\mathbb{R}^m)), \quad
\widehat{\Phi}^{{G}}(J(z))=J(\Phi^{{G}}\circ
z)
\ \ \text{where $z=(z_a)_a \in
C^d(\mathbb{I}^k,G_u(\mathbb{R}^m))$.}
$$
According to Proposition~\ref{prop:parablender} to prove that
$\Gamma^{{G}}=(\Gamma^{{G}}_a)^{}_a$ is a
{$C^d$-}$cs$-parablender for
$\Phi^{{G}}=(\Phi^{{G}}_a)^{}_a$ at $a=0$ we need to show the
following. First, we must prove that $\widehat{\Phi}^{{G}}$ has a
$cs$-blender $\widehat{\Gamma}^{{G}}$ which projects onto
$\Gamma^{{G}}_{0}$ and after provide a particular $C^d$-family
$\mathcal{H}^{{G}}=(\mathcal{H}^{{G}}_a)_a$ of $C^{r-1}$-discs
which induce a disc $\widehat{\mathcal{H}}^{{G}}$ in the open set
of $C^1$-discs. {This will be done in the two next sections. }
\subsubsection{Blender}
\label{sec:blender-grasma-jets} Using local coordinates
(c.f.~\cite{M80,KK00}) in the manifold of $(d,k)$-velocities over
$G_u(\mathbb{R}^m)=\mathbb{R}^m\times G(u,m)$ we can identify
$J(\mathcal{U}^{{G}}) = J(\mathsf{R}^{{G}}) \times J(D)$. Thus,
it is not difficult to see that $\widehat{\Phi}^{{G}}$ restricted
to $J(\mathcal{U}^{{G}})$ can be written as a skew-product map
$$
\widehat{\Phi}^{{G}}=\widehat{F}^{{G}} \ltimes
(\widehat{\phi}_1,\dots,\widehat{\phi}_\kappa) \quad
\text{on \ \
$J(\mathcal{U}^{{G}})=J(\mathsf{R}^{{G}}_1) \times
J(D) \cup \dots \cup J(\mathsf{R}^{{G}}_\kappa)
\times
J(D)$}
$$
where
$\widehat{F}^{{G}}$ is the induced map on $J(\mathsf{R}^{{G}})$ by
the map $F^{{G}}$ and $\widehat{\phi}_\ell$ are the induced maps
on $J(D)$ by the family $\phi_\ell=(\phi_{\ell,a})_a$ for
$\ell=1,\dots,\kappa$. Then, according to~\eqref{eq:cover-jets}
and Theorem~\ref{thmBKR} we only need to
prove that the base dynamics
of $\widehat{\Phi}^{{G}}$ has a horseshoe which dominates the fiber dynamics.
To do this, first we identify $J(\mathsf{R}^{{G}})=
J(\mathsf{R}) \times J(\mathcal{C}^{{G}})$. In this way,
we write the base dynamics of $\widehat{\Phi}^{{G}}$ as a
direct product map
$$
\widehat{F}^{{G}}=\widehat{F} \times \widehat{D\Phi} \quad
\text{on \ \ $ J(\mathsf{R}) \times J(\mathcal{C}^{{G}})$}
$$
where $\widehat F $ acts on
$J(\mathsf{R})$ by means of
$$
\widehat{F}(J(x))=
(F(x_a), {\partial^1_a F(x_a),\dots,\partial^d_a F(x_a)})_{|_{\,
a=0}}
\ \ \text{with $x=(x_a)_a \in C^d(\mathbb{I}^k,\mathbb{R}^n)$ such
that $x_0 \in \mathsf{R}$}
$$
and $\widehat{D\Phi}$ acts on $J(\mathcal{C}^{{G}})$ defined as
$$
\widehat{D\Phi}(J(E))=
(D\Phi \cdot E_a, \ {\partial_a^1(D\Phi \cdot
E_a),\dots,\partial_a^d(D\Phi \cdot
E_a)})_{|_{\, a=0}}
$$
with $E=(E_a)_a \in C^d(\mathbb{I}^k,G(u,m))$ and $E_0 \in
\mathcal{C}^{{G}}$.
As in \S\ref{sec:blender-jets}, using that $F$ is an affine map
and is independent of $a$, we have that
$$
\widehat{F}(J(x))=(F(x_a), {
DF(x_a)\,\partial^1_ax_a,\dots,DF(x_a)\,\partial^d_ax_a})_{|_{\,a=0}}
$$
with $x=(x_a)_a \in C^d(\mathbb{I}^k,\mathbb{R}^n)$ such that
$x_0\in \mathsf{R}$. From here we get that $\widehat{F}$ has a
horseshoe $\widehat{\Lambda}=\Lambda \times \{0\}$ as an invariant
set. Moreover, the eigenvalues of the linear part of $\widehat{F}$
are the same as of $DF$ at $\mathsf{R}$ and thus, as in
\S\ref{sec:blender-jets}, they dominate the fiber dynamics.
On the other hand, it is not difficult to see that
$\widehat{D\Phi}$ has the fixed point $J(E^u)$ where
$E^u=(E^u_a)^{}_a \in C^d(\mathbb{I}^k,G(u,m))$ is given by
$E^u_a=\{0^{ss}\}\times \mathbb{R}^u\times\{0^c\}$ for all $a\in
\mathbb{I}^k$. Hence
$$
\widehat{\Lambda}^{{G}}=\widehat{\Lambda} \times \{J(E^u)\}
\subset J(\mathsf{R}) \times
J(\mathcal{C}^{{G}})
$$
is a horseshoe for $\widehat{F}^{{G}}$ {with~stable~index
$$
\widehat{d}^G_{ss}\eqdef\mathrm{ind}^s(\widehat{\Lambda}^G)= \dim
J(\mathbb{R}^{ss})+\dim J(G(u,m)).
$$
Also, analogously with~\S\ref{sec:blender-jets}
and~\S\ref{sec:affine-blenders-grasmannian}, the width of the
stable cone-field of $\widehat{F}^G$ on $J(\mathsf{R}^G)$
coincides with the width $\alpha$ of the stable cone-field of $F$ on
$\mathsf{R}$.}
Now we need to prove that $\widehat{F}^{{G}}$ restricted to
$\widehat{\Lambda}^{{G}}$ dominates $\widehat{\phi}_1, \dots,
\widehat{\phi}_\kappa$. Since, $\widehat{F}$ dominates the fiber
dynamics, it suffices to show that $\widehat{D\Phi}$ at $J(E^u)$
also dominates $\widehat{\phi}_1, \dots, \widehat{\phi}_\kappa$.
In local coordinates around $E^u$ we can write
\begin{equation} \label{eq:DPhi-E}
D\Phi \cdot E \equiv P \, e + O(e^2), \qquad E\in \mathcal{C}^{{G}}, \ \
e \in \mathbb{R}^{d_u} \ \ \text{and} \ \ E\equiv e \ \
\text{with} \ \ E^u \equiv 0
\end{equation}
being $P$ a diagonal $(d_u\times d_u)$-matrix whose eigenvalues
are dominated by $\beta \nu< 1$ and $d_u=\dim G(u,m)=u(m-u)$.
Similarly, we can identify $J(E) \equiv (e, {\partial^1
e,\dots,\partial^d e}) \in \mathbb{R}^{d_u}\times
\mathcal{J}^d(c,d_u)$ and take as a representative of $J(E)$ the
function $E=(E_a)_a$ given by
\begin{equation} \label{eq:Ea}
E_a \equiv e+
\partial e \cdot a + \frac{1}{2!} \ \partial^2 e \cdot a^2 + \dots +
\frac{1}{d!} \ \partial^d e \cdot a^d.
\end{equation}
Substituting~\eqref{eq:Ea} into~\eqref{eq:DPhi-E}, in local
coordinates we have that
\begin{equation}
\label{eq2}
\partial_a^i(D\Phi \cdot
E_a)_{|_{\, a=0}} \equiv \ P \, \partial^i e + O(2) \quad
{\text{for $i=1,\dots,d$}}
\end{equation}
where $O(2)$ is a function that envolves the products of
$\partial^s e \cdot \partial^te$ with $s+t=2$. In local
coordinates $J(E^u)\equiv(e,\partial^1 e,\dots,\partial^de)=0$ is
a fixed point of $\widehat{D\Phi}$. Moreover, from~\eqref{eq2} the
linear part at this point is given by a triangular matrix whose
diagonal elements are the eigenvalues of $P$. Since these
eigenvalues are dominated by $\beta\nu <\nu < \lambda$, then
$\widehat{D\Phi}$ dominates the fiber dynamics. This concludes the
proof of the existence of a $cs$-blender $\widehat{\Gamma}^{{G}}$
of $\widehat{\Phi}^{{G}}$ projecting on
$\Gamma^{{G}}_0=\Gamma_0\times \{E^u\}$.
{Moreover, the family of
$(\alpha,\nu,\delta)$-horizontal $C^1$-discs in
$\widehat{\mathcal{B}}^{{G}}=\widehat{\mathsf{R}}^{{G}}\times
\widehat{B}$ is a superposition region of $\widehat{\Gamma}^G$,
where $0<\delta<\hat{L}\lambda/2$}.
\subsubsection{Discs on the manifold of velocities induced by
folding manifolds} \label{sec:disc-folding-manifold-jets} Let
$\widehat{\mathcal{B}}^{{G}}=\widehat{\mathsf{R}}^{{G}}\times
\widehat{B}$ be the superposition domain of the blender
$\widehat{\Gamma}^{{G}}$ where $\widehat{\mathsf{R}}^{{G}}$ is a
neighborhood on $J(\mathsf{R}^{{G}})=J(\mathsf{R})\times
J(\mathcal{C}^{{G}})$ of $\widehat{\Lambda}^{{G}}$. Similar as
in~\S\ref{sec:discos-afin-jet-blender}, since $\mathsf{R}\subset
\mathbb{R}^n=\mathbb{R}^{ss}\times \mathbb{R}^u$ and
$\mathcal{C}^{{G}} \subset G(u,m)$ we can take
$$
\widehat{\mathsf{R}}^{{G}}=\widehat{\mathsf{R}}_{ss} \times
\widehat{\mathsf{R}}_{u} \times \widehat{\mathsf{R}}_{G} \quad
\text{with \quad $\widehat{\mathsf{R}}_{ss} \subset
J(\mathbb{R}^{ss})$, \ \ $\widehat{\mathsf{R}}_{u} \subset
J(\mathbb{R}^u)$ \ \ and \ \ $\widehat{\mathsf{R}}_{G} \subset
J(G(u,m))$.}
$$
In fact, we have that ${\widehat{\mathsf{R}}_{G}}$ can be taken as
an arbitrarily small neighborhood in $J(G(u,m))$ of
$J(E^u)\equiv(e, {\partial^1e,\dots,\partial^de})=0$ and
$\widehat{\mathsf{R}}^{ss}=(-2,2)^{ss} \times B_{ {\rho}}(0)$.
Again here denotes {$B_\rho(0)$ denotes the subset of
$\mathcal{J}^d(k,ss)$ so that the symmetric $i$-linear maps have
all norm less than $\rho$.}
Now, fix a $( {\alpha,}\nu,\delta)$-folding $C^r$-manifold
$\mathcal{S}_0$ with respect to
$\mathcal{B}^{{G}}=\mathcal{B}\times \mathcal{C}^{{G}}$, where
$\mathcal{B}=\mathsf{R}\times B$ is the superposition domain of
the blender $\Gamma_0$ of $\Phi_0$. Consider the $k$-parametric
constant family of $( {\alpha,}\nu,\delta)$-folding
$C^r$-manifolds associated with $\mathcal{S}_0$ given by
$$
\mathcal{S}=(\mathcal{S}_a)_a \quad \text{where \ \
$\mathcal{S}_a=\mathcal{S}_0$ for all $a\in \mathbb{I}^k$}.
$$
According to Lemma~\ref{lem-induced-folding-manifold}, the set
$$
\mathcal{H}^{{G}}_0=\mathcal{S}^{{G}}_0 \cap
\overline{\mathcal{B}^{{G}}} \quad \text{with} \quad
\mathcal{S}^{{G}}_0=\{ (z,E): z\in \mathcal{S}_0, \ E \in G(u,m)
\ \
\text{and} \ \dim E \cap T_z\mathcal{S}_0=u \}
$$
is an almost-horizonal $d^{{G}}_{ss}$-dimensional $C^{r-1}$-disc
in $\mathcal{B}^{{G}}$ where $d^{{G}}_{ss}=ss+u(m-u)$. Hence, the
constant family of folding $C^r$-manifolds
$\mathcal{S}=(\mathcal{S}_a)_a$ induces a constant family
$\mathcal{H}^{{G}}=(\mathcal{H}^{{G}}_a)^{}_a$ of $C^r$-discs in
$\mathcal{B}^{{G}}$ given by
$\mathcal{H}^{{G}}_a=\mathcal{H}^{{G}}_0$ for all $a\in
\mathbb{I}^k$. By means of a similar argument as in
Lemma~\ref{lem:disc-jets} we obtain the following:
\begin{lem}
The set $\widehat{\mathcal{H}}^{{G}}$ in $J(G_u(\mathbb{R}^m))$
parameterized by
$$
\widehat{\mathcal{H}}^{{G}}(J(\xi))=J(\mathcal{H}^{{G}}_a\circ
\xi)=(\mathcal{H}^{{G}}_a(\xi_a), {
\partial_a^1\mathcal{H}^{{G}}_a(\xi_a),\dots,\partial_a^d\mathcal{H}^{{G}}_a(\xi_a)})_{|_{\,a=0}}
$$
for $\xi=(\xi_a)_a\in C^d(\mathbb{R}^k,\mathbb{R}^{ss}\times
G(u,m))$ with $J(\xi)$ belongs to the closure of $
\widehat{\mathsf{R}}_{ss}\times \widehat{\mathsf{R}}_{G}$, is a
$( {\alpha,}\nu,\delta)$-horizontal
$\widehat{d}_{ss}^{{G}}$-dimensional $C^1$-disc in
$\widehat{\mathcal{B}}^{{G}}=\widehat{\mathsf{R}}^{{G}}\times
\widehat{B}$ for any {$\alpha>0$ large enough} and $\nu>0$ small
enough where
$$\widehat{d}^{{G}}_{ss} \eqdef
\mathrm{ind}^{s}(\widehat{\Lambda}^{{G}}) = \dim
J(\mathbb{R}^{ss}) + \dim J(G(u,m)).$$
\end{lem}
\begin{proof}
Since $d<r-1$, it is straight forward that
$\widehat{\mathcal{H}}^{{G}}$ is a
$\widehat{d}_{ss}^{{G}}$-dimensional $C^1$-disc in
$\widehat{\mathcal{B}}^{{G}}$. Let $\widehat{h}^{{G}}$
{and $\widehat{g}\ ^G$ be, respectively, the
$J(\mathbb{R}^c)$-coordinate and $J(\mathbb{R}^u)$-coordinate of
$\widehat{\mathcal{H}}^{{G}}$ which correspond with the central
and unstable coordinates of the disc.} In order to prove that
$\widehat{\mathcal{H}}^{{G}}$ is a $(
{\alpha,}\nu,\delta)$-horizontal disc, notice that
$$
C=\|D\widehat{h}^{{G}}\|_{\infty} <\infty \quad
{\text{and} \quad D=\|D\widehat{g}\ ^G\|_\infty
<\infty} \quad \text{over the closure of
$\widehat{\mathsf{R}}_{ss}\times \widehat{\mathsf{R}}_G$.}
$$
Hence, by taking {$\alpha>0$ large enough and} $\nu>0$ small
enough we can always guarantee that $C\nu <\delta$
{and $D\leq \alpha$}. Thus, we only need to show
that there is a point $\widehat{y}\in \widehat{B}$ such that
$$
d(\widehat{h}^{{G}}(J(\xi),\widehat{y})<\delta \quad \text{for all
$J(\xi^{{G}})=(J(\xi),J(E)) \in
\overline{\widehat{\mathsf{R}}_{ss}}\times
\overline{\widehat{\mathsf{R}}_G}$}.
$$
Since $\mathcal{H}^{{G}}=(\mathcal{H}^{{G}}_a)^{}_a$ is a constant
family of discs
then~$\widehat{h}^{{G}}=\widehat{h}^{{G}}_0$~where~$h^{{G}}_0=\mathscr{P}\circ
\mathcal{H}^{{G}}_0$~and
$$
\widehat{h}^{{G}}_0(J(\xi^{{G}}))=J(h^{{G}}_0\circ
\xi^{{G}})
\quad \text{for \ \ $\xi^{{G}}=(\xi^{{G}}_a)_a\in C^d(\mathbb{R}^k,\mathbb{R}^{ss}\times
G(u,m))$}
$$
with $J(\xi^{{G}})$ belonging to the closure of
$\widehat{\mathsf{R}}_{ss}\times \widehat{\mathsf{R}}_G$. The same
computation as in Lemma~\ref{lem:disc-jets} proves that
$$
\widehat{h}^{{G}}_0(J(\xi^{{G}}))=(h^{{G}}_0(\xi^{{G}}_0),0) \in
J(\mathbb{R}^c) \quad \text{for all
$J(\xi^{{G}})
\in
([-2,2]^{ss}\times \{E^u\}) \times \{0\}$.}
$$
Hence $d(\widehat{h}^{{G}}(J(\xi^{{G}})),\widehat{y})<\delta$ for
all $J(\xi^{{G}}) \in ([-2,2]^{ss}\times \{E^u\})\times \{0\}$,
where $\widehat{y}=(y,0)\in \widehat{B}$ and $y\in B$ comes from
the definition of the folding $C^r$-manifold. By continuity and
since the neighborhood $\widehat{\mathsf{R}}_G$ of
$J(E^u)=(E^u,0)$ can be taken arbitrarily small, it follows
$$
d(\widehat{h}(J(\xi^{{G}})),\widehat{y})<\delta \quad \text{for
all \ \ $J(\xi^{{G}})=(J(\xi),J(E)) \in
\overline{\widehat{\mathsf{R}}_{ss}} \times
\overline{\widehat{\mathsf{R}}_G}$.}
$$
This completes the proof of the lemma.
\end{proof}
\begin{rem}
Let $\mathcal{S}_0$ be the $( {\alpha,}\nu,\delta)$-folding
$C^r$-manifold with respect to
$\mathcal{B}^{{G}}=\mathcal{B}\times \mathcal{C}^{{G}}$ introduced
in Example~\ref{exap:dobra}. Proposition~\ref{prop2:appendix} in
Appendix~\ref{appendix} proves that for any $0<\nu<\delta$ the
constant family $\mathcal{S}=(\mathcal{S}_a)_a$ of $(
{\alpha,}\nu,\delta)$-folding $C^r$-manifolds
induces a $( {\alpha,}\nu,\delta)$-horizontal
$C^1$-disc $\widehat{\mathcal{H}}^{{G}}$ in
$\widehat{\mathcal{B}}^{{G}}$ from the constant family of
$C^{r-1}$-discs $\mathcal{H}^{{G}}=(\mathcal{H}^{{G}}_a)^{}_a$
given by
$\mathcal{H}^{{G}}_a=\mathcal{H}_0^{{G}}=\mathcal{S}^{{G}}_0\cap\overline{\mathcal{B}^{{G}}}$.
\end{rem}
The previous lemma implies that $\widehat{\mathcal{H}}^{{G}}$
belongs to {a} superposition region of the
blender~$\widehat{\Gamma}^{{G}}$.
This completes the proof of the particular $C^d$-family
$\mathcal{H}^{{G}}=(\mathcal{H}^{{G}}_a)_a$ of $C^{r-1}$-discs.
\subsubsection{Proof of Theorem~\ref{thm-paratangencies}}
The proof will follow from Proposition~\ref{prop:parablender} and
using a similar construction as in the proof of
Theorem~\ref{thmD}. Indeed, we take a $k$-parameter family of
$cs$-blenders $\Gamma=(\Gamma_a)_a$ of {central dimension $c\geq
1$} for a $C^d$-family $\Phi=(\Phi_a)_a$ of
$C^{r}$-diffeomorphisms of $\mathcal{M}$ locally defined as affine
skew-product maps given at the beginning
of~\S\ref{sec:parablender-grasmmannian}. As we showed
in~\S\ref{sec:blender-grasma-jets}, these maps provide a family
$\Gamma^{{G}}=(\Gamma^{{G}}_a)^{}_a$ of $cs$-blenders of
{central dimension $c\geq 1$} for the induced
dynamics $\Phi^{{G}}_a$ on $G_u(\mathcal{M})$ and as well as a
$cs$-blender $\widehat{\Gamma}^{{G}}$ for the map
$\widehat{\Phi}^{{G}}$ on $J(G_u(\mathcal{M}))$. Similarly, as in
the proof of Theorem~\ref{thmD}, we take a $C^d$-family
{$\mathcal{S}=(\mathcal{S}_a)_a$ of $C^2$-robust
folding $C^r$-manifolds $\mathcal{S}_a$ with respect to the
superposition domain $\mathcal{B}^{{G}}$ of $\Phi^{{G}}$ in the
sense of Remark~\ref{rem:rob-dobra}.} Moreover, we assume that the
$C^d$-family $\mathcal{S}$ induces a $C^1$-disc in the
superposition region of $\widehat{\Gamma}^{{G}}$. This was done
in~\S\ref{sec:disc-folding-manifold-jets} by taking a constant
family of folding manifolds. Then, according to
Proposition~\ref{prop:parablender}
we have that
$\Gamma^{{G}}=(\Gamma^{{G}}_a)^{}_a$ is a
{$C^d$}-parablender {at $a=0$} of
{central dimension $c\ge 1$} of
$\Phi^{{G}}=(\Phi^{{G}}_a)^{}_a$ at $a=0$. As in
Remark~\ref{rem:any-parameter}, we can extend the result for any
parameter $a_0$ close to $a=0$. This completes the proof of
Theorem~\ref{thm-paratangencies}.
\subsection{Proof of Theorem~\ref{thmC}} {The following} result
is, {basically}, a consequence of
Theorem~\ref{thm-paratangencies}.
\begin{thm} \label{thm-BR19}
For any $0<d<r-1$ and $k\geq 1$, there exists a $C^d$-family
$f=(f_a)_a$ of locally defined $C^r$-diffeomorphisms of
$\mathcal{M}$ having a family of $cs$-blenders
$\Gamma=(\Gamma_a)_a$ with unstable dimension $u\geq 1$
and a family of folding manifolds $\mathcal{S}=(\mathcal{S}_a)_a$
satisfying the following:
For any $a_0\in \mathbb{I}^k$, any family $g=(g_a)_a$ close enough
to $f$ in the $C^{d,r}$-topology and any $C^{d,r}$-perturbation
$\mathcal{L}=(\mathcal{L}_a)_a$ of $\mathcal{S}$ there exists
$z=(z_a)_a\in C^d(\mathbb{I}^k,\mathcal{M})$ such that
{
\begin{enumerate}
\item $z_a\in\Gamma_{g,a}$, where $\Gamma_{g,a}$ denotes the
continuation for $g_a$ of the blender $\Gamma_a$,
\item the family of local unstable manifolds
$\mathcal{W}=(W^u_{loc}(z_a;g_a))_a$ and $\mathcal{L}$ have a
tangency
of dimension $u$ at $a=a_0$ which unfolds $C^d$-degenerately.
\end{enumerate}
}
\end{thm}
\begin{rem} \label{rem-BR19} The central dimension of the blenders $\Gamma_a$ is $c=u^2$ and the folding manifold $\mathcal{S}_a$ has dimension $ss+c$
where $ss\geq 1$ is the strong stable dimension of the blenders
$\Gamma_a$. Thus the dimension of the manifold $\mathcal{M}$ is
$m>u+u^2$. Theorem~\ref{thmC} follows immediately from the above
result assuming that each folding manifold $\mathcal{S}_a$ is part
of the stable manifold of a point of $\Gamma_a$.
\end{rem}
\begin{proof}[Proof of Theorem~\ref{thm-BR19}]
Let $\Gamma=(\Gamma_a)_a$ be the family of $cs$-blenders of the
$C^d$-family $f=(f_a)_a$ of $C^r$-diffeomorphisms of $\mathcal{M}$
given in Theorem~\ref{thm-paratangencies}. Consider the
{$C^d$-}$cs$-parablender
$\Gamma^{{G}}=(\Gamma^{{G}}_a)^{}_a$ {at any parameter} of
{central dimension $c\geq 1$} for the induced
dynamics $f^{{G}}=(f^{{G}}_a)^{}_a$ in ${G}_u(\mathcal{M})$. From
the proof of Theorem~\ref{thm-paratangencies}, we have
a $C^d$-family of $C^2$-robust folding
$C^r$-manifolds $\mathcal{S}=(\mathcal{S}_a)_a$ with respect to
the superposition domain $\mathcal{B}^{{G}}$ of $f^{{G}}$ such
$\mathcal{S}$ induces a family of $C^r$-discs in
$\mathcal{B}^{{G}}$ contained in
$\mathcal{S}^{{G}}=(\mathcal{S}^{{G}}_a)^{}_a$ where
$$
\mathcal{S}^{{G}}_a=\{ (z,E): z\in \mathcal{S}_a, \ E \in G(u,m)
\ \
\text{and} \ E \subset T_z\mathcal{S}_a \}.
$$
According to the proof of Proposition~\ref{prop:parablender}, the
open set of $k$-parameter $C^d$-families of $C^r$-discs contains a
$C^{d,r}$-neighborhood $\mathscr{D}(\mathcal{S}^{{G}})$ of
$\mathcal{S}^{{G}}=(\mathcal{S}^{{G}}_a)^{}_a$. Denote by
$\mathscr{D}(\mathcal{S})$ a $C^{d,r}$-neighborhood of
$\mathcal{S}=(\mathcal{S}_a)_a$, so that if
$\mathcal{L}=(\mathcal{L}_a)_a \in \mathscr{D}(\mathcal{S})$ then
the induced family of discs
$\mathcal{L}^{{G}}=(\mathcal{L}^{{G}}_a)^{}_a \in
\mathscr{D}(\mathcal{S}^{{G}})$. Similarly, let $\mathscr{U}(f)$
be a $C^{d,r}$-neighborhood of $f=(f_a)_a$ so that if $g=(g_a)_a
\in \mathscr{U}(f)$ then the induced family of maps
$g^{{G}}=(g^{{G}}_a)^{}_a \in \mathscr{U}(f^{{G}})$. Here,
$\mathscr{U}(f^{{G}})$ comes from the definition of a parablender
as the neighborhood of the $k$-parameter family
$f^{{G}}=(f^{{G}}_a)^{}_a$. From Definition~\ref{def:parablender}
{applied at $a=a_0$}, we have
$x=(x_a)_a,y=(y_a)_a,z=(z_a)_a\in
C^d(\mathbb{I}^k,G_u(\mathcal{M}))$ such that
$$
z_a \in \Gamma^{{G}}_{a,g} \ \ x_a \in W^u_{loc}(z_a) \ \
\text{and} \ \ y_a\in \mathcal{L}^{{G}}_a \ \ \text{so that} \ \
d(x_a,y_a)=o(\|a-a_0\|^{d}) \ \ \text{at $a=a_0$}
$$
where $\Gamma^{{G}}_{a,g}$ is the continuation for $g^{{G}}_a$ of
the $cs$-blender $\Gamma^{{G}}_a$ for all $a\in \mathbb{I}^k$. Set
$\mathcal{W}_a=W^u_{loc}(\tilde{z}_a)$ where
$z_a=(\tilde{z}_a,W_a) \in \Gamma_a \times G(u,m)$
and denote
$\mathcal{W}=(\mathcal{W}_a)_a$. Hence,
$$
\text{$x_a=(\tilde{x}_a,E_a) \in W^u_{loc}(z_a)$ implying that
$E_a = T_{\tilde{x}_a}\mathcal{W}_a$ and thus $x_a \in
G_u(\mathcal{W}_a)$.}
$$
Similarly,
$$
\text{$y_a=(\tilde{y}_a,F_a) \in \mathcal{L}_a$ implies that $F_a
\subset T_{\tilde{y}_a}\mathcal{L}_a$ and thus $y_a \in
G_u(\mathcal{L}_a)$.}
$$
Therefore, by Definition~\ref{def:tangencia}, $\mathcal{L}$
{and} $\mathcal{W}$ has a tangency of
{dimension} $u>0$ at $a=a_0$ which unfolds
$C^d$-{degenerately}.
This completes the proof.
\end{proof}
\appendix
\addtocontents{toc}{\protect\setcounter{tocdepth}{-1}}
\renewcommand{A}{A}
\renewcommand{A.\arabic{equation}}{A.\arabic{equation}}
\setcounter{equation}{0} \setcounter{thm}{0}
\section{Estimates for the folding manifold of Example~\ref{exap:dobra}}
\label{appendix} We consider the $(ss+c)$-dimensional
$C^\infty$-manifold of Example~\ref{exap:dobra} given by
$$
\mathcal{S}:[-2,2]^{ss}\times [-\epsilon,\epsilon]^c \to
\mathbb{R}^m, \ \ \mathcal{S}(x,t)=(x,(t_1,\dots,t_u),h(t))
\in \mathbb{R}^{ss}\times\mathbb{R}^u\times \mathbb{R}^c
$$
where $t=(t_1,\dots,t_u,\dots,t_c) \in [-\epsilon,\epsilon]^c$ and
$h(t)=(h_1(t),\dots,h_c(t))$ with
\begin{align*}
h_i(t)= \sum_{j=0}^{u-1} t_{j+1} t_{ju+i} \quad \text{for $i=1,\dots,u$ \quad and} \quad h_i(t)=t_i \quad \text{for $i=u+1,\dots,c$.}
\end{align*}
Let $t=t(E)$ be the $C^{r-1}$-function on $\mathcal{C}^{{G}}$
computes in Example~\ref{exap:dobra}. We have that:
\begin{prop} \label{prop1:appendix}
For every $\varepsilon>0$ suffices small there is a neighborhood
$\mathcal{C}^{{G}}$ of $E^u$ such that
$$
C=\|Dh\|_\infty \cdot
\max\{1,\|Dt\|_\infty\} \leq 1+\varepsilon \quad \text{over
$\mathcal{C}^{{G}}$.}
$$
Thus, $\mathcal{S}$ is a
$(\textcolor{blue}{\alpha,}\nu,\delta)$-folding manifold for any
$\nu>0$ such that $(1+\varepsilon)\nu<\delta$.
\end{prop}
\begin{proof}
First of all, notice that for all $t\in [-\epsilon,\epsilon]^c$,
it holds that
$$
\|Dh(t)\|_{\infty} \eqdef \max_{i=1,\dots, c} |
Dh_i(t)| =\max \bigg\{ \,1, \ \max_{i=1,\dots,u}
\big|\sum_{j=0}^{u-1} t_{j+1}+t_{ju+i}
\big|\,\bigg\}.
$$
Hence, taking $\epsilon>0$ small enough we have that
$\|Dh\|_\infty =1$. On the other hand, by Cramer's rule we get
that the solution of the linear system $At= \vec{c}$ \ is given by
$$
t_i = \frac{\det A_i^*}{\det A} \qquad \text{for
$i=1,\dots,c$}
$$
where $A^*_{i}$ is the matrix formed by replacing the $i$-th
column of $A$ by the column~vector~$\vec{c}$. In order to compute
$Dt$ we write the variable of $t$ by $E=\langle
v_{1},\dots,v_u\rangle$ with $v_k=(a_k,b_k,c_k)\in
\mathbb{R}^{ss}\times\mathbb{R}^{u}\times\mathbb{R}^c$ and then
$$
Dt=(\partial_{v_1}t,\dots,\partial_{v_u}t) \quad \text{with} \ \ \
\partial_{v_k}
t=(\partial_{a_k}t,\partial_{b_k}t,\partial_{c_k}t) \quad
\text{for $k=1,\dots,u$}.
$$
Moreover, since the matrix $A$ does not depend on the variables
$a_k$ we get that $\partial_{a_k}t=0$. In the sequel we will use
the symbol $D_{k}$ to denote any partial derivative of the
form~$\partial_{b_{k\iota}}$~or~$\partial_{c_{k\iota}}$. For each
$i=1,\dots,c$, using Jacobi's formula it follows that
$$
D_kt_i =\frac{D_k(\det A^*_i)-\mathrm{tr}(A^{-1} \cdot D_kA ) \cdot \det A^*_i}{\det
A}.
$$
In particular, since $\vec{c}(E^u)=0$ then $\det A^*_i(E^u)=0$,
and $\det A(E^u)=2$, we obtain that
$$
D_kt_i(E^u)=\frac{1}{2} \, D\det A^*_i|_{E^u} = \frac{1}{2} \,
\mathrm{tr}(\mathrm{Adj}(A^*_i) \cdot D_kA^*_i)|_{E^u}
$$
where $\mathrm{Adj}(A^*_i)$ is the adjugate matrix of $A^*_i$.
Notice that
$$
\mathrm{Adj}(A^*_i(E^u))=(C_{\ell j})^T \quad \text{with \ \
$C_{\ell j} =0$ \ if \ $j\not=i$ \ \ and \ \ $C_{\ell i}=(-1)^{\ell+i}\cdot \sigma_i$ \ \ for $\ell=1,\dots,c$}
$$
where $\sigma_i=2$ if $i\not=1$ and $\sigma_i=1$ otherwise
($i=1$). From this follows that
$$
\mathrm{tr}(\mathrm{Adj}(A^*_i) \cdot D_kA^*_i)|_{\,(E^u)}=
(C_{1i},\dots,C_{ci})\cdot D_k\vec{c}(E^u).
$$
If $D_k$ is either, $\partial_{b_{k\iota}}$ for $\iota=1,\dots,u$
or $\partial_{c_{k\iota}}$ for $\iota=u+1,\dots,c$ then
$D_k\vec{c}=0$ and thus $D_kt_i(E^u)=0$. Otherwise,
$$
(C_{1i},\dots,C_{ci})\cdot D_k\vec{c}(E^u) = C_{(k-1)u+\iota \, i}
\quad \text{and hence} \quad D_kt_i(E^u)=\frac{1}{2} \,
C_{(k-1)u+\iota \, i}.
$$
Therefore
$$
\|Dt(E^u)\|_{\infty}=\frac{1}{2} \ \max_{i=1,\dots,c} \ \max_{k=1,\dots,u} \ \big|\sum_{\iota=1}^{u}
C_{(k-1)u+\iota \, i} \big|= \frac{1}{2} \
\max_{i=1,\dots,c} \ \max_{k=1,\dots,u} \
\big|\sum_{\iota=1}^u (-1)^{i+(k-1)u+\iota} \sigma_i \big|
\leq 1.
$$
Since $Dt$ varies continuously with respect to $E$ we have that
$\|Dt(E)\|_\infty$ is close to $\|Dt(E^u)\|_\infty$ for any $E$
close enough to $E^u$. Thus, shrinking $\mathcal{C}^{{G}}$ if
necessary, this implies that $\|Dt\|_\infty \leq 1+\varepsilon$
over $ \mathcal{C}^{{G}}_\alpha$. Hence, $C=\|Dh\|_\infty \cdot
\max\{1,\|Dt\|_\infty\} \leq 1+\varepsilon$ for a fixed but
arbitrarily small $\varepsilon>0$.
\end{proof}
This $(\textcolor{blue}{\alpha,}\nu,\delta)$-folding
$C^r$-manifold $\mathcal{S}$ induces a $C^{r-1}$-disc
$$
\mathcal{H}^{{G}}: [-2,2]^{ss}\times \mathcal{C}^{{G}}_\alpha \longrightarrow
G_u(\mathbb{R}^m), \qquad \mathcal{H}^{{G}}(x,E)=(\mathcal{S}(x,t), E)
\in \overline{\mathcal{B}}\times \mathcal{C}^{{G}}=
\overline{\mathcal{B}^{{G}}}
$$
Set $h^{{G}}=\mathscr{P}\circ \mathcal{H}^{{G}}$. Here
$\mathscr{P}$ denotes the standard projection onto $\mathbb{R}^c$.
We consider
$$
\widehat{h}^{{G}}(J(\xi^{{G}}))=J(h^{{G}}\circ
\xi^{{G}})
\quad \text{for \ \ $\xi^{{G}}=(\xi^{{G}}_a)_a\in C^d(\mathbb{R}^k,\mathbb{R}^{ss}\times
G(u,m))$}
$$
with
$$
J(\xi^{{G}})=((\xi_0,E_0),\partial^1\xi^{{G}},\dots,\partial^d\xi^{{G}})
\in \big([-2,2]^{ss}\times \mathcal{C}^{{G}}\big) \times
\overline{B}_\rho(0)$$ where $\overline{B}_\rho(0)$ denotes a
closed ball of radius $\rho>0$ at $0$ velocity of the jets over
$\mathbb{R}^{ss}\times G(u,m)$.
\begin{prop} \label{prop2:appendix}
For every $\varepsilon>0$ suffices small there are $\rho>0$ and a
neighborhood $\mathcal{C}^{{G}}$ of $E^u$ so that
$$
\|D\widehat{h}^{{G}}\|_\infty \leq 1+\varepsilon \quad \text{over \ \
$ \big([-2,2]^{ss}\times \mathcal{C}^{{G}}\big) \times
\overline{B}_\rho(0)$.}
$$
Thus, $\widehat{\mathcal{H}}^{{G}}$ is a
$(\textcolor{blue}{\alpha,}\nu,\delta)$-horizontal disc for any
$\nu>0$ such that $(1+\varepsilon)\nu<\delta$.
\end{prop}
\begin{proof}
Observe that $h^{{G}}(E)=h(t(E))$. In this way,
\begin{equation}
\label{eqA1}
\widehat{h}^{{G}}(J(E))=J(h^{{G}}\circ
E)
=(h(t(E_a)),\partial^1_a h(t(E_a)),\dots,\partial^d_a h(t(E_a)))_{|_{\,a=0}}
\end{equation}
for $E=(E_a)_a \in C^d(\mathbb{I}^k,G(u,m))$ with $E_0\in
\mathcal{C}^{{G}}_\alpha$. Denoting $t_a=t(E_a)$ for all $a\in
\mathbb{I}^k$, we can rewrite~\eqref{eqA1} as
$$
\widehat{h}^{{G}}(J(t))=(h(t_a),\partial^1_ah(t_a),\dots,\partial^d_ah(t_a))_{|_{\,a=0}}
\quad \text{where $t=(t_a)_a \in C^d(\mathbb{I}^k,\mathbb{R}^c)$}
$$
with $t_0$ small enough in norm. Therefore,
\begin{equation}
\label{eqA2} D\widehat{h}^{{G}}= \frac{d\widehat{h}^{{G}}}{dJ(t)}
\cdot \frac{dJ(t)}{dJ(E)}= \frac{d\widehat{h}^{{G}}}{dJ(t)} \cdot
\frac{d \widehat t}{dJ(E)}
\end{equation}
where $\widehat{t}(J(E))=J(t\circ E)$. We want to compute
$\|D\widehat{h}^{{G}}(J(E^u))\|_{\infty}$ where $E^u=(E^u_a)^{}_a$
with $E^u_a=E^u_0=\{0^{ss}\}\times \mathbb{R}^u\times \{0^c\}$ for
all $a\in \mathbb{I}^k$. Hence $J(E^u)=(E^u_0,0)$. By a
straightforward calculation using Fa\`a di Bruno's formula, we
have
\begin{equation}
\label{eqA3} \big\| \frac{d\widehat{t}}{dJ(E)}(J(E^u))\big \|_{\infty} =
\|Dt(E^u_0)\|_\infty \leq 1.
\end{equation}
By means of a similar computation we can show that
\begin{equation}
\label{eqA4} \big\| \frac{d\widehat{h}^{G}}{dJ(t)}(J(t^{u}))\big \|_{\infty} =
\|Dh(0)\|_\infty = 1
\end{equation}
where $t^{u}=(t^{u}_a)^{}_a$ with $t^{u}_a=t(E^u_a)=t(E^u_0)=0$
for all $a\in \mathbb{I}^k$
and hence $J(t^{u})=0$.
Finally putting together \eqref{eqA2}-\eqref{eqA4} we get that
$$
\|D\widehat{h}^{{G}}(J(E^u))\|_\infty \leq \big\|
\frac{d\widehat{h}^{{G}}}{dJ(t)}(J(t^{{G}})) \big\| \cdot \big\|
\frac{dJ(t)}{dJ(E)} (J(E^u)) \big\| \leq 1.
$$
By continuity with respect to $J(E)$, shrinking
$\mathcal{C}^{{G}}$ if necessary and taking $\rho>0$ small enough,
we have $\|D\widehat{h}^{{G}}\|_{\infty} \leq 1+\varepsilon$ over
$\big([-2,2]^{ss}\times \mathcal{C}^{{G}}_\alpha\big) \times
\overline{B}_\rho(0)$ for a fixed but arbitrarily small
$\varepsilon>0$. This completes the proof.
\end{proof}
\end{document} | math | 125,308 |
\begin{document}
\title{Atomic noise spectra in nonlinear
magneto-optical rotation in a rubidium vapor}
\author{Hebin Li,$^{1}$ Vladimir A. Sautenkov,$^{1,2}$ Tigran S. Varzhapetyan,$^{1,3}$ Yuri V. Rostovtsev,$^1$ and Marlan O. Scully$^{1,4}$}
\address{$^1$Institute for Quantum Studies and Department of Physics,
Texas A\&M University, College Station, Texas 77843, USA}
\address{$^2$Lebedev Institute of Physics, Moscow 119991, Russia}
\address{$^3$Institute for Physical Research of NAS of Armenia,
Ashtarak-2 378410, Armenia}
\address{$^4$Princeton Institute for the Science and Technology of
Materials and Department of Mechanical \& Aerospace Engineering,
Princeton University, Princeton, New Jersey 08544, USA}
\begin{abstract}
We have studied the noise spectra in a nonlinear magneto-optical
rotation experiment in a rubidium vapor. We observed the reduction
of noise in the intensity difference of two orthogonally polarized
components of the laser beam. The dependence of the noise level on
both the frequency and the longitudinal magnetic field has been
studied. We found that the optimal condition for the noise reduction
is to work around zero longitudinal magnetic field, where the
intensity correlation between the two orthogonally polarized
components is maximum. Our results can be used to reduce or
eliminate the atomic excess noise, therefore improving the
sensitivity of nonlinear magneto-optical rotation magnetometers and
other atom-optical based applications.
\end{abstract}
\maketitle
\section{INTRODUCTION}
Improving the sensitivity of magnetometers is important both for
practical applications and for fundamental research. Magnetometers
based on atom-optical techniques, such as the optical pumping
magnetometers \cite{Alexandrov1996} and the nonlinear
magneto-optical rotation (NMOR) magnetometers
\cite{Budker2000,Sautenkov2000,Kominis2003}, have achieved
sensitivities of the order of 10$^{-15}$ T Hz$^{-1/2}$. Quantum
noise starts to play a crucial role in obtaining higher sensitivity,
which approaches the atom shot-noise-limited sensitivity
\cite{Budker2002}. One of the contributions to the quantum noise is
the increase of laser beam intensity fluctuation due to the laser
interacting with an atomic vapor (atomic excess noise). Possible
processes that are responsible for the generation of the atomic
excess noise include the conversion of laser phase noise to
intensity noise
\cite{Yabuzaki1991,Mcintyre1993,Camparo1998,Camparo1999} and the
four-wave mixing process \cite{Agarwal1995}. Although the atomic
excess noise could be a useful spectroscopic tool
\cite{Yabuzaki1991,Walser1994}, it is usually not desirable in
atom-optical based applications, such as atom-optical magnetometers,
atomic frequency references \cite{Kitching2001}, and the generation
of squeezed light
\cite{Hetet2007,Matsko2002,Ries2003,Hsu2006,Mikhailov2008}.
To reduce or eliminate the influence of atomic excess noise, one can
take advantage of the intensity correlation properties of the
optical fields passing through an atomic vapor. As is shown in
\cite{Sautenkov2005}, an electromagnetically induced transparency
(EIT) experiment was performed by coupling two beams from one laser
with an excited state and Zeeman sublevels of the ground state in a
rubidium vapor. The authors observed the intensity correlation and
anticorrelation between two circularly polarized laser beams. More
generally, a similar effect was also observed with two beams from
two independent lasers \cite{Cruz2007}. In the case of correlation
or anticorrelation, the intensity noise in each of two laser fields
is fluctuating with a phase difference of 0 or 180$^\circ$. A simple
summation or subtraction of these two signals can suppress the
noise. Experiments have shown the ability of reducing the noise to
the shot-noise level by taking the difference of two laser beams in
an EIT configuration \cite{Sautenkov2007}. In NMOR experiments, the
recent observation of the intensity correlation \cite{Tigran2008},
along with the power spectra study of the noise at 2.5 MHz
\cite{Martinelli2004}, indicate the possibility of using the
intensity correlation to reduce or eliminate the atomic excess
noise.
In this paper, we report the experimental study of the noise spectra
in a nonlinear magneto-optical rotation experiment in a rubidium
vapor. We show that the atomic excess noise in NMOR can be
essentially reduced to the shot-noise level because of the intensity
correlation of two orthogonally polarized components. The dependence
of the noise in the difference signal of these two components on
both frequency and magnetic field, has been studied. Our results
show that the optimal working condition for reducing the atomic
excess noise is to work near zero longitudinal magnetic field.
\section{EXPERIMENTAL SETUP}
\begin{figure}
\caption{\label{1}
\label{1}
\end{figure}
The experimental schematic is illustrated in Fig. 1. The laser
source is an external cavity diode laser (ECDL) described in
\cite{Vassiliev2006}. The laser is tuned to the rubidium D$_1$ line
(795 nm), specifically at the transition 5S$_{1/2}$ (F=2)
$\leftrightarrow$ 5P$_{1/2}$ (F=1) of $^{87}$Rb, referenced to the
Doppler-free saturation resonance in a rubidium cell at room
temperature. The frequency drift is less than 30 MHz per hour after
a sufficient warm-up time. The linewidth of the laser emission is
less than 1 MHz. The laser beam has a diameter of 1 mm, and it is
linearly polarized. After passing through an optical isolator, the
beam proceeds through a polarizing beam splitter (PBS) and possesses
a polarization parallel to the optical table.
The beam goes into a glass cell filled with a rubidium vapor that
contains the natural isotope abundance of rubidium atoms. The cell
has the length of 7.5 cm, and it is heated to reach an atomic
density of 10$^{12}$ cm$^{-3}$. A two-layer magnetic shield isolates
the cell from environmental magnetic fields in the lab, while a
solenoid inside the magnetic shield provides an adjustable
longitudinal magnetic field. The linearly polarized beam is a
combination of the left- and right-circularly polarized components.
The two circular components are coupled to the energy levels of
$^{87}$Rb as shown in the energy diagram in Fig. 1.
The output beam from the rubidium cell is analyzed by a half-wave
plate ($\lambda$/2) and a PBS. The half-wave plate is set to rotate
the polarization by 45$^\circ$, such that without the rubidium cell,
the PBS equally splits the intensity of the beam. If the rubidium
cell is placed in the system, a rotation angle of the beam
polarization will be introduced that depends on the magnitude of the
longitudinal magnetic field \cite{Budker2002}. With a nonzero
magnetic field (B$\neq$0), the two beams coming out from the PBS do
not have equal intensities. Recording the intensities of two beams
as $I_1$ and $I_2$, the polarization rotation due to rubidium atoms
can be calculated using the following equation
\begin{equation}
\phi=\arcsin(\frac{I_1-I_2}{I_1+I_2})\ .
\end{equation}
To study the power spectra of the atomic excess noise, a balanced
photo detector (BPD) with a sensitivity of 2$\times$10$^4$ V/W and a
bandwidth from DC to 100 MHz is used to register the intensities of
two laser beams. The optical path lengths of the beams going into
two channels of the BPD are chosen to be the same so that no
additional time delay between the two channels is introduced. The
signal is analyzed by an RF spectrum analyzer. In the case of a zero
magnetic field (B=0), for example, each channel of the BPD records
an intensity
\begin{equation}
I_i=I_0+I(t)+\delta I_i(t)\ ,\ (i=1,2)\ ,
\end{equation}
where $I_0$ is the average intensity, $I(t)$ is the low frequency
intensity fluctuations, and $\delta I_i(t)$ is the atomic excess
noise. Then, the difference signal $\Delta I$ from the BPD is given
by $\Delta I(t) =\delta I_1(t)-\delta I_2(t)$. The spectrum analyzer
gives the Fourier transform of the time dependence of the signal.
\section{EXPERIMENTAL RESULTS AND DISCUSSION}
\begin{figure}
\caption{\label{2}
\label{2}
\end{figure}
We begin the presentation of the results by showing the power
spectra of noise with no magnetic field (B=0). Figure 2 shows the
noise spectra for different input laser intensities. The left and
right figures display the spectra for input laser powers of 0.24 mW
and 0.49 mW, respectively. Traces (a2) and (b2) are recorded with
two laser beams sent to the BPD, and they show the noise spectra of
the difference signal. The noise is larger in the low frequency
region. The noise level approaches the shot-noise level
\cite{Sautenkov2007} at higher frequencies. For comparison, traces
(a1) and (b1) are recorded with only one laser beam sent to the BPD.
They represent the noise spectra of the laser beam passing through
the rubidium vapor. Before entering the cell, the laser beam has
small intensity noise but large phase noise. The phase noise is
converted into the intensity noise due to the laser interacting with
the atoms. This process causes a substantial increase of intensity
fluctuations in the laser beam coming out of the cell
\cite{Yabuzaki1991}. Our results show that these intensity
fluctuations can be suppressed by subtracting the intensity of one
laser beam from the other. Comparing the noise spectra of one laser
beam and of the difference signal, (a1) and (a2) for instance, the
noise level of the difference signal is dramatically reduced. When
the input laser power is doubled (0.49 mW), the corresponding
spectra presented as (b1) and (b2) show the same behavior, although
the shot-noise level increases approximately two times because of
the higher laser power. These results can be understood as a
consequence of the intensity correlation between the two output
laser beams from the PBS. As is shown in \cite{Tigran2008}, the
intensities of two beams in an NMOR experiment are highly correlated
(correlation function $G^{(2)}(0)\approx 0.9$) at zero magnetic
field (B=0). The fluctuations $\delta I_1(t)$ and $\delta I_2(t)$
are varying simultaneously, and thus $\Delta I(t)$ will be small.
\begin{figure}
\caption{\label{1}
\label{1}
\end{figure}
The intensity correlation and the substantial reduction of noise in
NMOR experiments is not trivial in terms of various behaviors at
different magnitudes of the magnetic field B. To show this, besides
the preceding results with a zero magnetic field, we have also
studied the noise spectra in NMOR at magnetic fields of various
magnitudes. Prior to showing these results, a typical measurement of
the polarization rotation in our experiment (laser power P=0.24 mW)
is presented in Fig. 3, to remind us of the rotation dependence on
the magnetic field.
\begin{figure}
\caption{\label{1}
\label{1}
\end{figure}
We record the noise spectra of the difference signal with two output
beams sent to the BPD for several magnetic fields. The results are
presented as three dimensional plots in Fig. 4 (laser power P=0.24
mW). The magnetic field varies from 0 to 184 mG. The spectra
corresponding to magnetic fields ranging from 0 to 26.3 mG are shown
in plot (a), and the ones corresponding to magnetic field ranging
from 26.3 mG to 184 mG are shown in plot (b). The spectra are sorted
by the magnetic field ascending along the arrows shown in the
figure. The magnetic field in plot (a) steps by about 2.6 mG, while
it steps by about 26 mG in plot (b). Note that the noise spectra
corresponding to negative magnetic fields, which are not plotted
here, have the symmetric behaviors. From these results, we see that
the reduction of high frequency noise is nearly the same for
different magnetic fields, but the low frequency noise is not
appreciably reduced. This shows that the low frequency noise is not
correlated at all magnitudes of the magnetic field, but the high
frequency noise is better correlated. A detailed study of the noise
spectra dependence on the magnetic field for each individual NMOR
system can provide a guideline for choosing optimal working
parameters to reduce or eliminate the atomic excess noise.
\begin{figure}
\caption{\label{1}
\label{1}
\end{figure}
To better demonstrate how the noise at a certain frequency depends
on the magnitude of the magnetic field, we cut the three dimensional
plots in Fig. 4 at specific frequencies along the plane of the
magnetic field axis and the intensity axis. The cross sections
picked up are shown in Fig. 5(a), in which the symbols square,
hollow square, triangle, hollow triangle, dot and circle represent
the level of noise at frequencies of 2 MHz, 5 MHz, 10 MHz, 15 MHz,
20 MHz and 30 MHz, respectively. At the zero magnetic field, the
noise is suppressed to the shot-noise level over the entire
frequency region. For the frequencies lower than 20 MHz, the noise
level increases quickly within about 50 mG, and comes back close to
the shot-noise level within about 130 mG. For the frequencies higher
than 20 MHz, the noise level essentially remains close to the
shot-noise level. To show the details of the rising slope in the
dashed square in Fig. 5(a), a magnification of this region is shown
in Fig. 5(b). For low frequency noise, the noise level remains
nearly minimum value not only at the zero magnetic field but also in
a small region around zero magnetic field. In our data, the noise
level of 2 MHz and 5 MHz noise is almost flat for magnetic fields
ranging from 0 to 2.5 mG.
These results show that, for the best reduction of atomic excess
noise in atom-optical applications, one should work near a zero
longitudinal magnetic field. However, a rigorously exact zero
magnetic field is not necessary, because the same reduction of noise
can be obtained in a region around zero magnetic field. This
property makes the implementation relatively easier and more
reliable. As an example, to obtain the best reduction of the atomic
excess noise in NMOR magnetometers, one might use an external
calibrated magnetic field to compensate \cite{Budker2002}, so that
longitudinal magnetic field is close to zero.
\section{CONCLUSION}
We have experimentally studied the noise spectra in a nonlinear
magneto-optical rotation experiment in rubidium vapor. We have shown
that a detailed study of noise reduction, due to the intensity
correlation between two orthogonally polarized components of the
laser beam, can suggest the optimal working conditions for reducing
atomic excess noise. The noise in the difference signal of two
orthogonal components at different frequencies has been studies as a
function of magnetic field. The study of the noise dependence on
both the noise frequency and the magnetic field shows that the
maximum reduction of noise can be obtained around zero longitudinal
magnetic field.
Our results can be used to reduce or eliminate atomic excess noise,
and thus improve the sensitivity of NOMR magnetometers. The study
also indicates the potential importance of the intensity
correlations in other atom-optical applications such as atomic
frequency references and the generation of squeezed light.
\end{document} | math | 15,099 |
\begin{document}
\title[A formula for the number of partitions of $n$]
{A formula for the number of partitions of $n$ in terms of the partial Bell polynomials}
\begin{abstract}
We derive a formula for $p(n)$ (the number of partitions of $n$) in terms of the partial Bell polynomials using Fa\`{a} di Bruno's formula and Euler's pentagonal number theorem.
\end{abstract}
\maketitle
\section{Main result}
Recall the classical partition function, denoted by $p(n)$, gives the number of ways of writing the integer $n$ as a sum of positive integers, where the order of summands is not considered significant. For example, $p(4)=5$, since there are $5$ ways to represent 4 as
sum of positive integers, namely, $4= 3+1=2+2=2+1+1=1+1+1+1$.\par
We also recall another classical statistic, the {\it $(n,k)$th partial Bell polynomial} in the variables $x_{1},x_{2},\dotsc,x_{n-k+1}$, denoted by $B_{n,k}\equiv\textup{B}_{n,k}(x_1,x_2,\dotsc,x_{n-k+1})$ (\cite[p. 134]{Comtet}, \cite[Ch. 12]{Andrews}), defined by
\begin{equation*}
\textup{B}_{n,k}(x_1,x_2,\dotsc,x_{n-k+1})=\sum_{\substack{1\le i\le n,\ell_i\in\mathbb{N}\\ \sum_{i=1}^ni\ell_i=n\\ \sum_{i=1}^n\ell_i=k}}\frac{n!}{\prod_{i=1}^{n-k+1}\ell_i!} \prod_{i=1}^{n-k+1}\Bigl(\frac{x_i}{i!}\Bigr)^{\ell_i}.
\end{equation*}
Cvijovi\'{c} \cite{Bell} gives the following formula for calculating these polynomials
\begin{align}
\label{explicit}
B_{n, k + 1} = & \frac{1}{(k+1)!} \underbrace{\sum_{\alpha_1\,= k}^{n-1} \, \sum_{\alpha_2\,= k-1}^{\alpha_1-1} \cdots \sum_{\alpha_k\, = 1}^{\alpha_{k-1}-1} }_{k }
\overbrace{\binom{n}{\alpha_1} \binom{\alpha_1}{\alpha_2} \cdots \binom{\alpha_{k-1}}{\alpha_k}}^{k}\nonumber \times \cdots
\\
&\times x_{n-\alpha_1} x_{\alpha_1 -\alpha_2} \cdots x_{\alpha_{k-1}-\alpha_k} x_{\alpha_k} \qquad(n\geq k+1, k\,=1, 2, \ldots)
\end{align}
We prove the following here.
\begin{theorem}
We have
\begin{equation}
\label{explicit2}
p(n)=\frac{1}{n!}\sum_{k=0}^{n}(-1)^{k}\, k!\, B_{n,k}(\lambda_{1},\lambda_{2},\cdots,\lambda_{n-k+1})
\end{equation}
where
\begin{equation}
\normalfont
\label{lamb}
\lambda_{m}=
\begin{cases}
(-1)^{\frac{1+\sqrt{1+24m}}{6}}\, m! & \text{if $\frac{1+\sqrt{1+24m}}{6}\in \mathbb{Z}$,}\\
(-1)^{\frac{1-\sqrt{1+24m}}{6}}\, m! & \text{if $\frac{1-\sqrt{1+24m}}{6}\in \mathbb{Z}$,}\\
0 & \text{otherwise.}
\end{cases}
\end{equation}
\end{theorem}
\begin{proof}
We begin by the following generating function \cite[Equation 22.13]{Fine}
\begin{equation}
\sum_{n\geq 0}p(n)q^{n}=\prod_{j=1}^{\infty}\frac{1}{1-q^{j}}.
\end{equation}
We recall the Euler's pentagonal number theorem \cite[Equation 7.8]{Fine}
\begin{align}
\label{pent}
E(q)&:=\prod_{j=1}^{\infty}(1-q^{j})=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{\frac{3n^{2}+n}{2}}\\
\nonumber &=1-q-q^2+q^5+q^7-q^{12}-q^{15}+q^{22}+q^{26}-\cdots.
\end{align}
Let $f(q)=1/q$. Using Fa\`{a} di Bruno's formula (\cite[p. 137]{Comtet}, \cite[Ch. 12]{Andrews}) we have
\begin{equation}
\label{faa}
{d^n \over dq^n} f(E(q)) = \sum_{k=0}^n f^{(k)}(E(q))\cdot B_{n,k}\left(E'(q),E''(q),\dots,E^{(n-k+1)}(q)\right).
\end{equation}
Since $f^{(k)}(q)=\frac{(-1)^{k}\,k!}{q^{k+1}}$ and $E(0)=1$, letting $q\rightarrow 0$ in the above equation gives
$$
p(n)\, n! = \sum_{k=0}^n (-1)^{k}\, k!\, B_{n,k}\left(E'(0),E''(0),\dots,E^{(n-k+1)}(0)\right).
$$
Then Euler's pentagonal number theorem \eqref{pent} gives us
$$
E^{(m)}(0)=\lambda_{m}
$$
where $\lambda_{m}$ is as defined in \eqref{lamb}.
\end{proof}
Combining equations \eqref{explicit} and \eqref{explicit2} we can conclude that
$$
p(n)=-\theta_{n}+\sum_{k=1}^{n-1}(-1)^{k-1} \underbrace{\sum_{\alpha_1\,= k}^{n-1} \, \sum_{\alpha_2\,= k-1}^{\alpha_1-1} \cdots \sum_{\alpha_k\, = 1}^{\alpha_{k-1}-1}}_{k}\theta_{n-\alpha_1} \theta_{\alpha_1 -\alpha_2} \cdots \theta_{\alpha_{k-1}-\alpha_k} \theta_{\alpha_k}
$$
where
\begin{equation*}
\theta_{m}=
\begin{cases}
(-1)^{\frac{1+\sqrt{1+24m}}{6}} & \text{if $\frac{1+\sqrt{1+24m}}{6}\in \mathbb{Z},$}\\
(-1)^{\frac{1-\sqrt{1+24m}}{6}} & \text{if $\frac{1-\sqrt{1+24m}}{6}\in \mathbb{Z},$}\\
0 & \text{otherwise.}
\end{cases}
\end{equation*}
\begin{corollary}
Let $E(q)^{r}:=\prod_{j=1}^{\infty}(1-q^{j})^{r}=\sum_{n=0}^{\infty}p_{r}(n)q^{n}$ with $p_{r}(0)=1$ (see \cite{Atkin}). Then
$$
p(n)=\sum_{r=0}^{n}(-1)^{r}\,\binom{n+1}{r+1}\,p_{r}(n),
$$
where by the virtue of the Fa\`{a} di Bruno's formula \eqref{faa} with $f(q)=q^{l}$ we have
\begin{align*}
p_{l}(n)&=\frac{1}{n!}\sum_{k=0}^{l}\binom{l}{k}\,k!\, B_{n,k}(\lambda_{1},\ldots ,\lambda_{n-k+1}).
\end{align*}
\end{corollary}
\begin{proof}
We start with the generating function for the partial Bell polynomials \cite[Equation (3a') on p. 133]{Comtet} as follows
\begin{align*}
{\displaystyle \sum _{n=k}^{\infty }B_{n,k}(\lambda_{1},\ldots ,\lambda_{n-k+1}){\frac {q^{n}}{n!}}}
&= {\frac {1}{k!}}\left(\sum _{j=1}^{\infty }\lambda_{j}{\frac {q^{j}}{j!}}\right)^{k} \\
&=\frac{1}{k!}(E(q)-1)^{k}\\
&=\frac{1}{k!}\sum_{r=0}^{k}(-1)^{k-r}\binom{k}{r}E(q)^{r}\\
&=\frac{1}{k!}\sum_{r=0}^{k}(-1)^{k-r}\binom{k}{r}\sum_{n=0}^{\infty}p_{r}(n)q^{n}
\end{align*}
to conclude that
$$
B_{n,k}(\lambda_{1},\ldots ,\lambda_{n-k+1})=\frac{n!}{k!}\sum_{r=0}^{k}(-1)^{k-r}\binom{k}{r}p_{r}(n).
$$
The above equation, together with the formula \eqref{explicit2}, gives us
\begin{align*}
p(n)&=\sum_{k=0}^{n}\sum_{r=0}^{k}(-1)^{r}\, \binom{k}{r}\,p_{r}(n)\\
&=\sum_{r=0}^{n}(-1)^{r}p_{r}(n)\sum_{k=r}^{n}\binom{k}{r}\\
&=\sum_{r=0}^{n}(-1)^{r}\,\binom{n+1}{r+1}\,p_{r}(n).
\end{align*}
\end{proof}
\begin{remark}
Note that similar ideas are used in \cite{Ono-etc} with relation to partition zeta functions.
\end{remark}
\end{document} | math | 5,631 |
\begin{document}
\ca
\centerline{\Large The New Quantum Logic}
\xa
\cb
\title{The New Quantum Logic}
\author{Robert B. Griffiths
\theta anks{Electronic mail: [email protected]}\\
Department of Physics,
Carnegie-Mellon University,\\
Pittsburgh, PA 15213, USA}
\date{Version of 27 June 2014}
\maketitle
\ca
\centerline{Robert B. Griffiths}
\centerline{Physics Department}
\centerline{Carnegie-Mellon University}
\cb
\ca
\centerline{Version of 11 November 2013}
\cb
\xb
\xa
\begin{abstract}
It is shown how all the major conceptual difficulties of standard (textbook)
quantum mechanics, including the two measurement problems and the (supposed)
nonlocality that conflicts with special relativity, are resolved in the
consistent or decoherent histories interpretation of quantum mechanics by
using a modified form of quantum logic to discuss quantum properties
(subspaces of the quantum Hilbert space), and treating quantum time
development as a stochastic process. The histories approach in turn gives
rise to some conceptual difficulties, in particular the correct choice of a
framework (probabilistic sample space) or family of histories, and these are
discussed. The central issue is that the principle of unicity, the idea that
there is a unique single true description of the world, is incompatible with
our current understanding of quantum mechanics.
\end{abstract}
\xb
\tableofcontents
\xa
\xb
\section{Introduction}
\label{sct1}
\xa
\xb
\outl{Students find standard (textbook) QM difficult; textbooks lack some
basic principles}
\xa
The conceptual difficulties of standard quantum mechanics, defined as what one
finds in standard textbooks, are encountered by students in their very first
course in the subject. Part of the problem is unfamiliar mathematics, but even
when the mathematics has been (more or less) mastered a serious problem remains
in relating the mathematical formalism to some physical understanding of
quantum systems. And it is no wonder that students are having difficulty if
even textbook writers do not really understand the subject, and are sometimes
bold enough to admit it \cite{Lloe12}. Are there some basic
principles which are missing from the textbooks, ideas which were they included
therein would clear up quantum paradoxes?
\xb
\outl{Principles missing from textbooks are in widely ignored CH approach}
\xa
\xb
\outl{CH resolves all major Qm conceptual difficulties including measurement
problem}
\xa
\xb
\outl{CH has been criticized. Mermin: CH vs SR difficulties $\leftrightarrow $ elephant vs
gnat}
\xa
The thesis of this paper is that there are such basic principles, which have
been around in some form or another for nearly thirty years, and they deserve
to be more widely known. Although ignored in much of the current literature,
the (consistent or decoherent) histories approach appears capable of resolving
\emptyset h{every} major conceptual difficulty of quantum mechanics, not least the
infamous measurement problem. To be sure, the histories approach has not been
completely ignored; a small but distinguished group of critics---who may
possibly outnumber the advocates---have not hesitated to point out what they
consider serious flaws; see the references given in \cite{Grff13}. One of the
more generous of these critics, N.\ David Mermin, expressed what has probably
troubled many others when he made the following comparison with special
relativity, see p.~281 of \cite{Schl11} or p.~16 of \cite{Mrmn13}:
\begin{quote}
[But] I am disconcerted by the reluctance of some consistent historians to
acknowledge the utterly radical nature of what they are proposing. The
relativity of time was a pretty big pill to swallow, but the relativity of
reality itself is to the relativity of time as an elephant is to a
gnat.
\end{quote}
\xb
\outl{CH is radical. Feynman thot QM much more difficult than SR.}
\xa
\xb
\outl{Revised reasoning more radical than SR. More like earth beginning to
move}
\xa
What the consistent historians are proposing is indeed radical, which should
surprise no one familiar with Feynman's famous remark that ``no one understands
quantum mechanics'' (p.~129 of \cite{Fynm65}). It occurs in a context where
Feynman makes it abundantly clear that he considers quantum theory much more
difficult than special relativity, though he does not quantify this by means of
a zoological analogy. The historians' proposal to revise some of the rules of
reasoning which before the arrival of quantum mechanics were thought to apply
universally, both in scientific reasoning and in everyday human affairs, is
obviously more difficult to accept than the move from pre-relativistic to
relativistic physics. It is much more like the transition our intellectual
ancestors made when they abandoned the notion that the earth is motionless at
the center of the universe in favor of the radical proposal that it moves
around the sun as well as spinning around its axis. The question physicists
should be asking is not whether the ideas in the consistent histories approach
are radical, but rather whether they are internally consistent, and whether
they genuinely resolve the serious conceptual issues which have beset quantum
theory ever since its development in the mid 1920s.
\xb
\outl{Birkhoff and von Neumann Qm logic not very successful. Physicists not
smart enough?}
\xa
However radical it may seem, the idea that quantum theory requires a new mode
of reasoning is itself not at all new. In 1936, just four years after the
appearance of von Neumann's famous book, Birkhoff and von Neumann \cite{BrvN36}
published their proposal for a \emptyset h{quantum logic} as a replacement in the
quantum domain for ordinary propositional logic. Through the years there has
been a continuing, albeit modest, research effort attempting to develop quantum
logic in hopes that it would lead to a solution of the quantum conceptual
difficulties. Despite some early enthusiasm, e.g. \cite{Ptnm75}, this program
has not made a great deal of progress; for some discussion of the current
situation see \cite{Mdln05,Bccg09}. It may be that we physicists are simply
not smart enough to reason in this fashion, and the quantum mysteries will have
to rest until the day when superintelligent robots (with access to quantum
computers?) can make sense of the quantum world. But will they be able (or even
want to) explain it to us?
\xb
\outl{New logic less radical, more successful than Birkhoff and von
Neumann }
\xa
\xb
\outl{Outline of paper. QM difficulties in Tbl. 1, Sec.~\ref{sct2}; CH in
Sec.~\ref{sct3} resolves these; conceptual problems of CH in Tbl. 2,
Sec.~\ref{sct4};
conclusion in Sec.~\ref{sct5}.}
\xa
What is here called the \emptyset h{new} quantum logic has the same motivation and
shares important ideas with the proposal of Birkhoff and von Neumann. It is in
some respects a less radical break with conventional reasoning than the older
quantum logic, and has turned out to be much more useful in terms of allowing
human beings, including college seniors and beginning graduate students, to
understand the quantum world in a consistent and coherent way. The present
paper is devoted to explaining how this approach resolves the major conceptual
problems of standard quantum mechanics listed in Table~\ref{tbl1} and discussed
in some detail in Sec.~\ref{sct2}. Following that, Sec.~\ref{sct3} summarizes
the histories approach and how it addresses these difficulties. Next,
Table~\ref{tbl2} lists and Sec.~\ref{sct4} discusses various conceptual
problems raised by the histories approach itself. A brief conclusion follows
in Sec.~\ref{sct5}.
\xb
\section{Quantum Conceptual Difficulties}
\label{sct2}
\xa
\xb
\outl{Table with Conceptual Difficulties of QM}
\xa
Table~\ref{tbl1} is a list of major conceptual difficulties of standard quantum
mechanics; that is, the treatment currently found in most textbooks. These are
topics which have given rise to a lengthy and continuing discussion in the
quantum foundations literature. While no such list can claim to be complete,
the author believes that most of the significant interpretational problems fall
in one or another of these categories.
\xb
\begin{table}[h]
\caption{Major Conceptual Difficulties of Quantum Mechanics}
\label{tbl1}
\begin{center}
\begin{tabular}{l l l}
\hline
\\
1. & \multicolumn{2}{l}{ Meaning of the wave function}\\
& a. & Ontological\\
& b. & Time development\\
& c. & Epistemological\\[1ex]\hline
\\
2. & \multicolumn{2}{l}{ Measurements}\\
& a. & Outcomes (pointer states)\\
& b. & What was measured?\\
& c. & Wave function collapse\\[1ex] \hline
\\
3. & \multicolumn{2}{l}{Inteference}\\
& a. & Particle vs.\ wave\\
& b. & Delayed choice\\[1ex]\hline
\\
4. & \multicolumn{2}{l}{Locality}\\
& a. & Bell inequalities\\
& b. & GHZ and Hardy\\[1ex] \hline
\end{tabular}
\end{center}
\end{table}
\xa
\xb
\subsection{Meaning of a wave function}
\label{sbct2.1}
\xa
\xb
\outl{Ontological: Wave function like a point in phase space, a quantum beable}
\xa
Students are taught that a wave function or wave packet or ket in the quantum
Hilbert space is analogous to a point in a classical phase space; e.g., for
a single particle it contains information about both the position and the
momentum. Wave packets have a unitary time development governed by the
Hamiltonian through Schr\"odinger's equation,
and under suitable circumstances the wave packet can be seen to ``move''
somewhat like a a classical particle---one thinks of the well-known Ehrenfest
relations. Thus it is rather natural to conclude that the wave function
represents whatever it is in the quantum world that is ``really there,'' a
\emptyset h{beable} in Bell's terminology \cite{Bll04}. Let us call this the
\emptyset h{ontological} perspective.
\xb
\outl{Epistemic: Wave fn $\rightarrow $ probabilities, which can suddenly change. It
provides information, but information about what? How is discontinuous change
related to Schr Eq?}
\xa
There are other circumstances in which a wave function seems to play a
different role. It can be used to calculate probabilities of the outcomes of a
measurement. Students learn that the process of measurement makes a wave
function collapse. This can, it seems, take place instantly, which for a wave
function with significant extension in space might violate special
relativity. It is certainly contrary to the unitary continuous time development
induced by Schr\"odinger's equation. Probabilities can be instantly updated
according to new knowledge, and relativity theory need not be violated in such
updating. So if a wave function is just a means of calculating the probability
of something, there is no reason why it should not suddenly change. Such is the
\emptyset h{epistemic} understanding of wave functions: rather than actually
representing the physical state of affairs they only provide \emptyset h{information}
about a quantum system. But what is this information \emptyset h{about}? Presumably
quantum theory is able in principle, even if in practice the calculations may
be very difficult, to tell us something about what is going on in systems which
have been probed experimentally leading to the conclusion that classical
mechanics is not an adequate representation of atomic systems. Is there
something really \emptyset h{there}, the way experimentalists seem to think, and if
so how is \emptyset h{it} related to the wave function? And how is discontinuous
time development related to Schr\"odinger's equation?
\xb
\outl{Reconciling ontological \& epistemic a serious problem not helped by
replacing $\ket{\psi}$ with $\rho$}
\xa
Reconciling the ontological and epistemic points of view is a serious
conceptual problem, and it does not disappear when one replaces a wave function
with a density operator, somewhat analogous to a classical probability
distribution. A classical distribution provides a probability of something
definite, which either occurs or does not occur. But what is the referent of a
quantum probability distribution?
\xb
\subsection{Measurements}
\label{sbct2.2}
\xa
\xb
\outl{Measurement problem: provide fully Qm description of entire process}
\xa
Measurements play a central role in textbook expositions of quantum theory.
The students are suspicious, and rightly so. After all, the measuring
apparatus is itself constructed from a large collection of atoms whose behavior
is governed by quantum laws, and therefore it should be possible, at least in
principle, to describe the entire measuring process, both the system being
measured and the measuring apparatus, in fully quantum mechanical
terms. Providing an adequate and fully \emptyset h{quantum} description constitutes
the infamous \emptyset h{measurement problem} of quantum foundations.
\xb
\outl{Two measurement problems. \#1. Schr cat state for pointer. Decoherence
does not help}
\xa
There are actually two distinct measurement problems. The \emptyset h{first}, the one
most often discussed in the foundations literature, has to do with the fact
that a measuring process amplifies microscopic differences in such a way as to
make them macroscopic. In the dated but picturesque terminology of this field,
these difference are ultimately revealed through different positions occupied
by a macroscopic \emptyset h{pointer} that indicates the measurement outcome. When
unitary time evolution is applied to both the microscopic system being measured
and the apparatus the result will often be a quantum wave function which is a
superposition of different macroscopic pointer positions. (See
Sec.~\ref{sbct3.5} below for a particular measurement model.) How is such a
quantum state, nowadays often called a Schr\"odinger cat, to be understood? Is
the pointer oscillating back and forth between different positions unable,
so-to-speak, to make up its mind? Superficial invocations of decoherence do
not really resolve the problem \cite{Adlr03}.
\xb
\outl{Measurement problem \#2. Infer prior property of measured system}
\xa
If one can somehow get the pointer to stop wiggling and settle down in a
definite position, the \emptyset h{second} measurement problem remains: how is
this position related to the microscopic state of affairs that the apparatus
was designed to measure? Experimenters typically claim that the outcomes of
their experiments tell them something about a prior state of affairs. E.g., a
gamma ray was detected coming form a decaying nucleus, or a neutrino from a
distant supernova was detected by the apparatus. This seems directly contrary
to the claim found in some textbooks that measurements tell one nothing about
what was there before the measurement took place.
\xb
\outl{Proper analysis of measurements should explain wave fn collapse}
\xa
The third item under the measurement heading in Table~\ref{tbl1}, wave function
collapse, has already been mentioned in Sec.~\ref{sbct2.1}. Obviously, an
adequate and fully quantum mechanical description of the measurement process
should provide some insight into why collapse can be a useful epistemic
perspective, or else replace it with something else which will accomplish
the same purpose.
\xb
\subsection{Interference}
\label{sbct2.3}
\xa
\xb
\outl{Double slit, MZ interference. Delayed choice, indirect measurement}
\xa
Double-slit interference leads to a well-known paradox in which one must
understand a quantum particle as a wave that is sufficiently delocalized that
it can in some sense pass through both of the slits in order to produce an
interference pattern. However, if detectors are placed immediately behind the
slits only one, not both, will be triggered, indicating that the particle
passed through only one slit. And despite its wavelike character the particle
can arrive at a quite specific location in the interference region. Feynman's
superb discussion in Ch.~1 of \cite{FyLS65} can be recommended to any reader
who has not yet encountered it. A very similar paradox occurs in a
Mach-Zehnder interferometer where a photon must in some sense be moving through
both arms in order to produce the expected interference at the second beam
splitter, whereas a measurement inside the interferometer will detect the
photon in just one arm, not both. The paradox is even more striking in
Wheeler's delayed choice version \cite{Whlr78}, where the final beam splitter
in the Mach-Zehnder is either left in place or else suddenly removed at a time
when the photon has already passed through the first beam splitter. In yet
another version \cite{ElVd93} the fact that one arm of the interferometer is
blocked can seemingly be detected by a photon which passes through the other
arm a long distance away.
\xb
\subsection{Locality}
\label{sbct2.4}
\xa
\xb
\outl{Bell inequalities, also Hardy \& GHZ, violate QM, which is supported by
experiments. So something must be wrong with derivations of Bell
inequalities, etc.}
\xa
Bell inequalities \cite{Bll64b,Bll90c} apply to a situation where two quantum
particles, typically two photons, are prepared in an entangled state and
various measurements are used to determine the statistical correlations of some
of their properties. By making certain assumptions about the presence of
physical properties in the particles before measurement, and assuming locality,
which is that influences only travel at a finite speed from one point to
another, Bell derived certain inequalities these correlations should satisfy.
The inequalities are violated by the predictions of quantum mechanics, and
numerous experiments of increasing precision all agree with quantum theory and
disagree with Bell's inequalities. This has convinced most physicists that one
or the other of Bell's assumptions must be wrong. Hardy's paradox
\cite{Hrdy92}, which also applies to correlations of properties of two
separated particles, is to some degree more straightforward than Bell's work as
it is easier to see that quantum predictions for the correlations, again in
accord with experiments, contradict what one might naively expect if
measurements reveal prior properties. The paradox of Greenberger, Horne, and
Zeilinger \cite{GrHZ89,GHSZ90} is similar to Hardy's and preceded it in time,
but refers to three particles rather than two. Once again, the predictions of
quantum mechanics are supported by experiment, suggesting that something must
be wrong with the reasoning that leads to these paradoxes.
\xb
\outl{Frequent claim: preceding implies there exist nonlocal Qm influences }
\xa
\xb
\outl{These influences cannot transmit info, so experimentally undetectable}
\xa
The claim has often been made that the only reasonable conclusion to be drawn
from the experimental violation of Bell inequalities and these other paradoxes
is that the quantum world must be nonlocal, and allow for instantaneous
interactions or influences between spatially separated systems, even in
situations where this conflicts with special relativity. However, even those
who believe in the existence of such influences agree that they cannot be used
to transmit information, which conveniently makes them experimentally
unobservable.
\xb
\section{The New Quantum Logic}
\label{sct3}
\xa
\xb
\outl{Problems resolved using frameworks + stochastic time development }
\xa
\xb
\outl{Brief comments on contents of following subsections}
\xa
In resolving the conceptual difficulties listed in Table~\ref{tbl1} the
histories approach uses two main tools. The first is the new quantum logic,
employed in Sec.~\ref{sbct3.1} to discuss physical properties of a quantum
system at a single instant of time, and extended to probabilities in
Sec.~\ref{sbct3.2}. The second is stochastic time development, the
subject of Sec.~\ref{sbct3.3}. Together these provide a way to
understand the dynamics of macroscopic systems using quasiclassical frameworks,
Sec.~\ref{sbct3.4}, and thus resolve both measurement problems, as
discussed in Sec.~\ref{sbct3.5}. Interference and the locality of quantum
mechanics are taken up in Secs.~\ref{sbct3.6} and \ref{sbct3.7}. Some brief
remarks on approximations in Sec.~\ref{sbct3.8} complete the discussion of the
new logic and how it resolves quantum paradoxes.
\xb
\subsection{Properties}
\label{sbct3.1}
\xa
\xb
\outl{Old and new Qm logic: property $\leftrightarrow $ subspace ${\mathcal P}$, projector $P$}
\xa
\xb
\outl{Single $\ket{\psi}$ $\rightarrow $ one-d subspace projector $[\psi]=\dya{\psi}$}
\xa
\xb
\outl{Subspaces of dimension greater than 1 also represent Qm properties}
\xa
The new quantum logic shares with its older counterpart a very fundamental
idea, consistent with but seldom sufficiently emphasized in quantum textbooks.
A quantum \emptyset h{physical property}, something which can be true or false---such
as ``the energy is between $2$ and $3$ J''---is represented in quantum
mechanics by a (closed) \emptyset h{subspace} ${\mathcal P}$ of the quantum Hilbert space or,
equivalently, by the \emptyset h{projector} $P$ (orthogonal projection operator) onto
this subspace. Subspaces (or their projectors) represent the quantum ontology,
they are the mathematical counterparts of Bell's ``beables.'' Note that a wave
packet or any nonzero ket $\ket{\psi}$ in the Hilbert space corresponds to, or
generates, a one-dimensional subspace consisting of all its multiples: kets of
the form $c \ket{\psi}$, where $c$ is any complex number. When $\ket{\psi}$ is
normalized we denote the projector onto this subspace by $[\psi] =
\dya{\psi}$. Used in this way a ket or wave function has an ontological
meaning. However, kets can also play an epistemological role, as discussed
below in Secs.~\ref{sbct3.2} and \ref{sbct3.3}. It is important to note that
subspaces of dimension greater than one also represent quantum properties.
\xb
\outl{Cl phase space $\Gamma $, property ${\mathcal P}$, indicator function $P(\gamma )$
$\leftrightarrow $ Qm projector }
\xa
A classical phase space $\Gamma $ with a point in the phase space $\gamma $
representing the actual physical state, provides a useful analogy for the
quantum Hilbert space. A collection of points ${\mathcal P}$ in $\Gamma $ represents a
classical property, and this property is true for a given system if the point
$\gamma $ representing its physical state is in the set ${\mathcal P}$. There is a
one-to-one correspondence between the subset ${\mathcal P}$ and the corresponding
\emptyset h{indicator function} $P(\gamma )$, equal to 1 for $\gamma \in {\mathcal P}$ and 0
otherwise. This classical indicator is thus analogous to a quantum projector,
whose eigenvalues are 1 and 0.
\xb
\outl{Cl physical variable = real valued fn on phase space}
\xa
\xb
\outl{Qm analog: observable $A=\sum_j a_j P_j$; $\{P_j\}$ form PDI }
\xa
\xb
\outl{(Property $A=a_j$) $\leftrightarrow $ $P_j$. (Measurement of $A$) $\leftrightarrow $ determine
$P_j$}
\xa
In addition, classical mechanics employs various \emptyset h{physical variables}
represented by real-valued functions on the phase space; e.g., energy,
momentum, angular momentum. In quantum mechanics a physical variable, referred
to as an \emptyset h{observable}, is represented by a Hermitian operator which can be
written in the form
\begin{equation}
A = \sum a_j P_j,\quad P_j = P_j^\dagger = P_j^2,\quad \sum_j P_j = I,
\label{eqn1}
\end{equation}
where each eigenvalue $a_j$ of $A$ occurs but once in the sum: $j\neq k$
implies
$a_j\neq a_k$. Here the $\{P_j\}$ are a collection of projectors that form a
\emptyset h{projective decomposition of the identity operator} $I$, sometimes
called a projector-valued measure or PVM. The property that the physical
variable $A$ takes on or possesses the value $a_j$, thus $A=a_j$, corresponds
to the projector $P_j$ or, equivalently, the subspace ${\mathcal P}_j$ onto which $P_j$
projects. While measurement will be discussed in more detail below in
Sec.~\ref{sbct3.5}, it is worth remarking that in quantum mechanics the
measurement of an observable is the same thing as measuring the corresponding
decomposition of the identity, determining which property ${\mathcal P}_j$, equivalently
$P_j$, is, in fact true.
\xb
\outl{Negation, Cl, Qm. For Qm, use ${\mathcal P}^\perp$ or $I-P$}
\xa
The negation of a classical property ${\mathcal P}$ corresponds to the complementary
subset ${\mathcal P}^c$ in the phase space: the set of points in $\Gamma $ which are not in
${\mathcal P}$. Its indicator, $P^c$ or $\lnot P$, is $I-P$, where $I$ is the identity:
$I(\gamma )=1$ for every $\gamma \in\Gamma $. Von Neumann proposed that in quantum theory
the negation $\lnot P$ should be represented not by the set-theoretical
complement of the corresponding Hilbert subspace, but instead by its
\emptyset h{orthogonal complement}, the collection ${\mathcal P}^\perp$ of all kets which are
orthogonal to every ket in ${\mathcal P}$. This is a subspace with projector $I-P$,
where $I$ is the identity operator on the Hilbert space. This identification
is consistent with textbook quantum mechanics, though often it is not properly
discussed.
\xb
\outl{Expls: 1. Spin 1/2. Negation of $[z^+]$ is $[z^-]$, consistent with SG
expt. 2. Harmonic oscillator}
\xa
Let us consider two examples. For the two-dimensional Hilbert space
representing a spin-half particle the one-dimensional subspaces corresponding
to the orthogonal kets $\ket{z^+}$ and $\ket{z^-}$ (denoted by $\ket{0}$ and
$\ket{1}$ in quantum information theory), ``spin up'' and ``spin down'', are
associated with the projectors
\begin{equation}
[z^+]=\dya{z^+},\quad [z^-]=\dya{z^-}.
\label{eqn2}
\end{equation}
As they sum to $I$ they are negations of each other, and together form a
projective decomposition of the identity. If the spin is not ``down'' it is
``up'', consistent with what Stern and Gerlach observed in their famous
experiment. If $S_z=+1/2$ (in units of $\hbar$) is true, then $S_z=-1/2$ is
false, and vice versa. In the case of a quantum harmonic oscillator with energy
eigenstates $\ket{n}$, eigenvalues $(n+\hf)\hbar\omega $, and projectors
$[n]=\dya{n}$, ``the energy is less than $2\hbar\omega $'' is represented by the
projector $P= [0]+[1]$, and its negation, ``the energy is greater than
$2\hbar\omega $'', by the projector $I-P = [3]+[4]+\cdots$. We shall return to
these examples later.
\xb
\outl{ $P\land Q$: old vs new Qm logic. New logic requires $PQ=QP$ to define
$P\land Q$}
\xa
With reference to quantum properties and their negations the old and the new
quantum logic are identical. The difference begins to emerge when one
considers the conjunction $P\land Q$, ``$P$ {\sim\!\!all AND}\ $Q$'', of two
properties. Birkhoff and von Neumann defined it using the set-theoretical
intersection ${\mathcal P}\cap{\mathcal Q}$ of the two subspaces, itself a (closed)
subspace. This has a precise analog in the classical phase space, where the
conjunction of two properties is represented by the intersection of the two
subsets ${\mathcal P}$ and ${\mathcal Q}$. However, if one follows the analogy of indicator
functions and projectors there is an important difference. The indicator for a
classical conjunction $P\land Q$ is the product $PQ$ of the indicators. But in
the quantum case the product $PQ$ of two projectors is a projector \emptyset h{if and
only if} $PQ=QP$, i.e., the two projectors \emptyset h{commute}, in which case the
projectors, or the corresponding quantum properties, are said to be
\emptyset h{compatible}. For compatible properties the product $PQ$ of the
projectors projects on the subspace ${\mathcal P}\cap{\mathcal Q}$. However, when $P$ and $Q$ do
not commute, that is, the quantum properties or their projectors are
\emptyset h{incompatible}, neither $PQ$ nor $QP$ is a projector, and there is no
simple relationship between either $PQ$ or $QP$ and the projector onto the
subspace ${\mathcal P}\cap{\mathcal Q}$. The new quantum logic differs from the old in that it
\emptyset h{does not define} $P\land Q$ when the projectors do not commute; the
expression ``$P\land Q$'' in such a case is \emptyset h{meaningless}: quantum
mechanics does not assign it a meaning.
\xb
\outl{Disjunction $P\lor Q$: old vs new Qm logic }
\xa
The old quantum logic defines the disjunction ``$P$ {\sim\!\!all OR}\ $Q$'' (or both),
$P\lor Q$, as the \emptyset h{span} of ${\mathcal P}\cup{\mathcal Q}$, the union of the two collections
of kets. (Note that ${\mathcal P}\cup{\mathcal Q}$ is in general not a subspace.) The indicator
for the classical property ${\mathcal P}\cup{\mathcal Q}$ is $P+Q-PQ$, and same expression works
for the quantum projector \emptyset h{when $PQ=QP$}, but not otherwise. Again, the
new quantum logic only defines the disjunction $P\lor Q$ when $P$ and $Q$
commute Otherwise the disjunction is undefined, thus meaningless.
\xb
\outl{Syntactical rule prohibits combining incompatible properties; compare
$P\land\lor Q$ in std logic}
\xa
\xb
\outl{Negation of false statement is true, negation of meaningless statement is
meaningless}
\xa
The prohibition of conjunctions and disjunctions when $P$ and $Q$ do not
commute is an application of the \emptyset h{single framework rule}, which plays a
central role in the new quantum logic, and about which more will be said in
Sec.~\ref{sbct3.2}. This prohibition is a \emptyset h{syntactical} rule governing
the combination of meaningful expressions (propositions or properties) to form
other meaningful expressions. Thus, for example, in ordinary logic the
combination $P\land\lor\, Q$ has no meaning because it has not been constructed
according to the rules for meaningful expressions. In a similar way the new
logic forbids the combinations $P\land Q$ and $P\lor Q$ when $PQ\neq QP$; they
are meaningless. Note the difference between a statement which is meaningful
but \emptyset h{false} and one which is \emptyset h{meaningless}. The negation of a false
statement is true, whereas the negation of a meaningless statement is equally
meaningless.
\xb
\outl{Spin half illustration: $[x^+]$ and $[z^+]$ don't commute.}
\xa
\xb
\outl{Old vs new logic for $[z^+]\land[x^+]$, $[z^-]\lor[x^-]$ }
\xa
\xb
\outl{Birkhoff and von Neumann had to remove distributive laws of logic}
\xa
The spin half example introduced above in \eqref{eqn2} provides a useful
illustration. If we define
\begin{equation}
\ket{x^+} = (1/\sqrt{2})(\ket{z^+} + \ket{z^-},\quad
\ket{x^-} = (1/\sqrt{2})(\ket{z^+} - \ket{z^-},
\label{eqn3}
\end{equation}
with $[x^+]$ and $[x^-]$ the projectors for the properties $S_x=\pm 1/2$, it is
easily checked that these do not commute with $[z^+]$ and $[z^-]$. In the old
quantum logic the statement ``$S_z=+1/2$ {\sim\!\!all AND}\ $S_x=+1/2$'' corresponds to the
zero operator, which is always false, and consistent with this its negation
``$S_z=-1/2$ {\sim\!\!all OR}\ $S_x=-1/2$'' is always true. This last does not seem to make
much physical sense, and will indeed lead to a contradiction if one follows the
rules of ordinary reasoning---see Sec.~4.6 of \cite{Grff02c}. Consequently, as Birkhoff
and von Neumann \cite{BrvN36} pointed out, it is necessary to modify the rules
of ordinary reasoning, by removing the distributive laws, in order to construct
a quantum logic free of contractions.
By contrast, in the new quantum logic, since ``$S_z=+1/2$ {\sim\!\!all AND}\ $S_x=+1/2$'' is
meaningless, its negation is equally meaningless, and thus no contradiction
arises. The approach in textbook quantum theory is to say that $S_x$ and $S_z$
cannot be simultaneously measured. This is quite true, and one wishes that the
textbooks would go on and state the reason for this: even the most skilled
experimental physicists cannot measure that which does not exist!
\xb
\outl{New logic a subset of the old logic}
\xa
\xb
\outl{Advantage of new logic: ordinary rules of reasoning, probability}
\xa
The new quantum logic is in a sense a ``subset'' or restricted part of
the old quantum logic, as the former accepts only a special collection of the
formulas which are valid (constructed according to syntactical rules) for the
latter. What is gained by adding this restriction is the ability to use the
ordinary rules of reasoning---and, as discussed below, the ordinary rules of
probability---to understand the quantum world without encountering
contradictions and paradoxes.
\xb
\subsection{Probabilities and the single framework rule}
\label{sbct3.2}
\xa
\xb
\outl{CH time development stochastic, not limited to measurements}
\xa
\xb
\outl{Sample space ${\mathcal S}$, event algebra ${\mathcal E}$, probability measure ${\mathcal M}$;
finite or countable ${\mathcal S}$ will do.}
\xa
In the histories interpretation the time development of a quantum system is a
stochastic process. Always, not just when measurements are being made.
Furthermore, the probabilities in question obey the standard rules of
probabilistic reasoning found in textbooks on probability theory (and which
ought to be found in quantum textbooks). Three things are needed: a sample
space ${\mathcal S}$ of mutually exclusive possibilities, one and only one of which is
true, a Boolean event algebra ${\mathcal E}$, and a probability measure ${\mathcal M}$
that assigns probabilities to the elements of ${\mathcal E}$. For present
purposes we do not need sophisticated concepts. A finite, or at most countable,
sample space will do very well, and ${\mathcal E}$ can consist of all the subsets of
${\mathcal S}$, including ${\mathcal S}$ and the empty set.
\xb
\outl{Qm ${\mathcal S}$ always a PDI. ${\mathcal E}$ = sums of items in ${\mathcal S}$, framework =
${\mathcal S}$ or ${\mathcal E}$. All its projectors commute}
\xa
\xb
\outl{Elements of ${\mathcal S}$ are \emptyset h{elementary} events}
\xa
A quantum sample space is always a projective decomposition of the identity,
and any such projection can serve as a sample space. The event algebra consists
of all projectors which are sums of some of the projectors in the sample space,
including the zero projector 0 and the identity $I$. Since the elements of a
projective decomposition of the identity commute with each other, so do all the
projectors in ${\mathcal E}$. The term \emptyset h{framework} will be used either for ${\mathcal S}$ or
${\mathcal E}$; given the close relationship between the two this ambiguity should not
matter. When a distinction is important the elements of ${\mathcal S}$ will be called
\emptyset h{elementary} events or projectors.
\xb
\outl{(In)compatible frameworks defined using (non)commuting projectors}
\xa
\xb
\outl{Common refinement of compatible frameworks; incompatible frameworks
do not have one}
\xa
Two frameworks with sample spaces $\{P_j\}$ and $\{Q_k\}$ are said to be
\emptyset h{compatible} if every $P_j$ commutes with every $Q_k$; otherwise they are
\emptyset h{incompatible}. One arrives at exactly the same definition using
projectors belonging to the two event algebras: either they all commute
(compatible) or some do not (incompatible). In the compatible case there is
always a smallest \emptyset h{common refinement} of the two frameworks with a sample
space consisting of all nonzero products of the form $P_jQ_k$, with duplicates
eliminated. The event algebra of the refinement includes all the
projectors in the two original event algebras. Two incompatible frameworks do
not have a common refinement, and thus there is no way to combine the event
algebras.
\xb
\outl{Probability of event using sum of appropriate collection of $p_j$}
\xa
\xb
\outl{$p_j$ arbitrary except $p_j\geq 0,\,\sum p_j=1$. Qm time development
gives certain conditional probs (Born rule). Probabilities can apply to
a single system.}
\xa
To assign probabilities we start with a collection of nonnegative numbers
$\{p_j\}$, one for each projector in ${\mathcal S}$, which sum to 1. Probabilities of
events in ${\mathcal E}$ are calculated in the obvious way; e.g., $\Pr(P_2 + P_3 + P_5)
= p_2 + p_3 + p_5$. Where do the $p_j$ come from? For ordinary (classical)
probabilistic models they are simply parameters chosen by guesswork, or to
agree with experiment: there are no hard and fast rules. The same is true in
quantum theory \emptyset h{except} that for the time development of a closed system
the Born rule and its extensions, Sec.~\ref{sbct3.3}, provide certain
conditional probabilities which combined with appropriate assumptions (e.g., an
initial state) yield a probability distribution. There is no reason quantum
probabilities should not be applied to single systems as is done for the
weather or to estimate the probability that the earth will collide with an
asteroid of a given size during the next millennium.
\xb \outl{SFR defined. It rules out $P$ {\sim\!\!all AND}\ $Q$ if $PQ\neq QP$ (projectors
in framework must commute)} \xa
The \emptyset h{single framework rule} is a central principle of the new quantum
logic. What it says, in brief, is that a probabilistic calculation or a logical
argument must be carried out using a single framework, a single event algebra
generated by a single sample space, a single projective decomposition of the
identity. Carrying out half of the reasoning or calculation using one framework
and then transferring the result to a different framework for additional
reasoning or calculations is prohibited. Since all projectors in a framework
commute with each other, this immediately rules out combinations of
noncommuting projectors using {\sim\!\!all AND}\ or {\sim\!\!all OR}\, as discussed in
Sec.~\ref{sbct3.1}.
\xb \outl{Harmonic oscillator example. $P=[0]+[1]$. Cannot use framework ${\mathcal S}_1
= \{P, I-P\}$ for $P\Rightarrow $ energy is $1/2$ or $3/2\hbar\omega $. Instead use
${\mathcal S}_2=\{[0],[1],I-P\}$ } \xa
A useful illustration is provided by the harmonic oscillator,
Sec.~\ref{sbct3.1}, where $P=[0]+[1]$ is the property that the energy is not
greater than $2\hbar\omega $. It might seem obvious that if $P$ is true then the
energy is either $\hbar\omega /2$ or $3\hbar\omega /2$, the values when $n=0$ or 1.
However, this is \emptyset h{not} a consequence of $P$ being true if we employ the
smallest framework ${\mathcal F}_1$ which contains $P$, with sample space ${\mathcal S}_1 = \{P,
I-P\}$. The corresponding event algebra ${\mathcal E}_1=\{0,P,I-P,I\}$ does not contain
either $[0]$ or $[1]$, so there is no way to discuss them. However, they are
included in the alternative framework ${\mathcal F}_2$ with sample space
${\mathcal S}_2=\{[0],[1],I-P\}$, a refinement of ${\mathcal F}_1$. The event algebra ${\mathcal E}_2$
includes both $[0]$ and $[1]$ as well as their sum $P$, and within this
framework one can use ordinary logic to infer that if $P$ is true then either
$[0]$ or $[1]$ is true, the energy is either $\hbar\omega /2$ or $3\hbar\omega /2$.
\xb
\outl{Difference ${\mathcal F}_1$ vs ${\mathcal F}_2$ is important; consider
${\mathcal S}_3= \{[+],[-],I-P\}$}
\xa
\xb
\outl{Example of incorrect reasoning blocked by the SFR}
\xa
That the distinction between ${\mathcal F}_2$ and ${\mathcal F}_1$ is not just a matter of
nitpicking can be seen by introducing a third framework ${\mathcal F}_3$ whose sample
space is ${\mathcal S}_3 = \{[+],[-],I-P\}$, where $[+]$ and $[-]$ are projectors onto
the states $\ket{+}=(\ket{0}+\ket{1})/\sqrt{2}$ and $\ket{-}=(\ket{0}-\ket{1})/\sqrt{2}$.
Note that $[+]+[-]=P$, so ${\mathcal F}_3$ is another refinement of ${\mathcal F}_1$. If we use
${\mathcal F}_3$ the truth of $P$ implies that the oscillator has either the property
$[+]$ or the property $[-]$, neither of which has a well-defined energy. The
frameworks ${\mathcal F}_2$ and ${\mathcal F}_3$ are mutually incompatible because $[+]$ and
$[-]$ do not commute with $[0]$ and $[1]$---the relationship is formally the
same, if one restricts attention to the subspace $P$, as that between the $S_z$
and $S_x$ eigenstates of a spin-half particle. The single framework rule
allows the use of either ${\mathcal F}_2$ or ${\mathcal F}_3$, but insists that they not be
combined. Here is an argument that violates that rule: ``Let us suppose the
the oscillator is in the state $[+]$. From this we infer (framework ${\mathcal F}_3$)
that it possesses the property $P$. But it is obvious (framework ${\mathcal F}_2$) that
a system with the property $P$ has an energy of either $\hbar\omega /2$ or
$3\hbar\omega /2$. Consequently the state $[+]$ has one of these two energies.''
It is this sort of reasoning, which in the quantum domain can lead to
paradoxes, that is blocked by the single framework rule.
\xb
\outl{Utility of ${\mathcal F}_3$: oscillator in superposition of $\ket{0}$, $\ket{1}$}
\xa
But why would one ever want to use a framework such as ${\mathcal F}_3$? It is nowadays
possible to prepare a harmonic oscillator, either a mechanical oscillator or
the electromagnetic field inside a cavity, in a superposition of the ground and
first excited state, and ${\mathcal F}_3$ might be useful in describing such a
situation. Thus in quantum mechanics it is quite possible to say that ``the
energy is less than $2\hbar\omega $'' \emptyset h{without} implying that the energy is
equal to either of the two possible energies that are less than this value, and
this is precisely the significance of the projector $P$.
\xb
\outl{SFR means: Liberty, Equality, Incompatibility, Utility}
\xa
\xb
\outl{Physicist's choice of framework does not affect reality}
\xa
Four principles provide a compact summary of what the single framework rule
does and does not mean. First, the physicist has perfect Liberty to construct
different, perhaps incompatible, frameworks when analyzing and describing a
quantum system. No law of nature singles out a particular quantum
framework as the ``correct'' one; from a fundamental point of view there is
perfect Equality among different possibilities. However, the principle of
Incompatibility prohibits \emptyset h{combining} incompatible frameworks into a
single description, or in employing them for a single logical argument leading
from premisses to conclusions. The last principle is Utility: not every
framework is useful for understanding a particular physical situation or
addressing certain scientific questions. In addition it is important to avoid
thinking that the physicist's choice of framework somehow influences reality.
Instead, quantum reality allows a variety of alternative descriptions, useful
for different purposes, which when they are incompatible cannot be
combined.
\xb
\subsection{Time development}
\label{sbct3.3}
\xa
\xb
\outl{Unitary vs stochastic time evolution}
\xa
\xb
\outl{Von Neumann, Everett, CH:stochastic evolution, textbooks}
\xa
Von Neumann's quantum mechanics had two distinct sorts of time evolution:
unitary evolution, based on Schr\"odinger's equation, and a separate stochastic
evolution associated with measurements, Sec.~V.1 of \cite{vNmn32b}. Few have
found this satisfactory, but devising something better has proven difficult. In
the Everett or many worlds interpretation \cite{Evrt57,DWGr73} there is only
unitary time development: a single unitarily evolving wave function of the
universe, or \emptyset h{uniwave} in the terminology of \cite{Grff13}. Proponents of
this approach then have to explain the probabilistic behavior of quantum
systems observed in the laboratory, a not altogether easy task. The histories
interpretation takes the opposite approach: all quantum time development is
\emptyset h{stochastic}, and the deterministic Schr\"odinger equation is used to
calculate probabilities. This is also what is done in textbooks, where physics
is extracted from the formalism using absolute squares of transition
amplitudes, though the whole matter is obscured through frequent (and
unnecessary) references to measurements.
\xb \outl{History $Y$: sequence of
projectors (Qm properties) at successive times} \xa
\xb
\outl{$Y$ a projector on history Hilbert space $\breve{\mathcal H}$}
\xa
\xb
\outl{Physical interpretation of $Y$: $F_0$ at $t_0$, $F_1$ at $t_1$\dots}
\xa
Stochastic time evolution requires a sample space with events at successive
times, and in the histories approach each event is a quantum property. Thus
for a sequence of times $t_0<t_1<\cdots t_f$ the sequence of properties
\begin{equation}
Y = F_0\odot F_1\odot \cdots F_f,
\label{eqn4}
\end{equation}
where each $F_j$ is a projector, is a \emptyset h{history} to which under appropriate
conditions one can assign a probability. The $\odot $ in \eqref{eqn4} indicates a
tensor product. (While it would be perfectly correct to use the standard symbol
symbol $\otimes $, it is often helpful when considering the time development of a
quantum system possessing subsystems to employ a distinct symbol that separates
situations at different times.) The operator $Y$ is a projector on a subspace
of the \emptyset h{history Hilbert space}
\begin{equation}
\breve{\mathcal H} = {\mathcal H}\odot {\mathcal H}\odot \cdots {\mathcal H}
\label{eqn5}
\end{equation}
formed from the tensor product of copies of the Hilbert space ${\mathcal H}$ that
describes the system at a single time. Its physical interpretation is that
that the (quantum) event $F_0$ occurred or, equivalently, the property $F_0$
was true at the time $t_0$, $F_1$ at the time $t_1$, and so forth.
\xb
\outl{Sample space ${\mathcal S}$ of elementary histories: projectors sum to $\breve I$}
\xa
\xb
\outl{Both ${\mathcal S}$ \& ${\mathcal E}$ are called ``family of histories''}
\xa
\xb
\outl{Prob of history in ${\mathcal E}$: sum of probs of ${\mathcal S}$ elements it contains}
\xa
\xb
\outl{One and only one elementary history actually occurs in any exptl run}
\xa
The sample space ${\mathcal S}$ consists of a collection of orthogonal projectors of the
kind shown in \eqref{eqn4}, the \emptyset h{elementary histories}, that sum to the
history identity,
\begin{equation}
\breve I = I\odot I\odot \cdots I,
\label{eqn6}
\end{equation}
and thus constitute a projective decomposition of $\breve I$. The
corresponding event algebra ${\mathcal E}$ consists of projectors which are sums of some
of the projectors that make up ${\mathcal S}$, and the probability of any history in
${\mathcal E}$ is the sum of the probabilities of the elementary histories of which it
is composed. The term ``family of histories'' is often employed in place
``framework'', and depending on the context can refer to either ${\mathcal S}$ or ${\mathcal E}$.
As in any probabilistic model, one and only one of the elementary histories,
which are mutually exclusive, occurs in any given situation or ``experimental
run.''
\xb
\outl{Family with fixed initial state $[\psi_0]$}
\xa
A simple but fairly useful family employs a set of elementary histories
\begin{equation}
Y^\alpha = [\psi_0]\odot P_1^{\alpha _1}\odot P_2^{\alpha _2}\odot \cdots P_f^{\alpha _f},
\label{eqn7}
\end{equation}
where $[\psi_0]=\dya{\psi_0}$ is a fixed initial state at $t_0$, and at the
later time $t_m$ the projector $P_m^{\alpha _m}$, where $\alpha _m$ is a label not an
exponent, belongs to a fixed decomposition of the (single time) identity,
\begin{equation}
\sum_{\alpha _m} P_m^{\alpha _m} = I,
\label{eqn8}
\end{equation}
and $\alpha = (\alpha _1,\alpha _2,\dots \alpha _f)$ is the label for the history $Y^\alpha $. If
one includes along with the $Y^\alpha $ in \eqref{eqn7} a special history $Y^0 =
(I-[\psi_0])\odot I\odot I \odot \cdots I$, the result is a family for which the
projectors add to $\breve I$, as required for a sample space.
\xb
\outl{Assigning probabilities for 2-time histories: Born rule using $T(t,t')$}
\xa
The rules for assigning probabilities to elementary histories of a closed
quantum system whose unitary time development is generated by a Hermitian
Hamiltonian are best explained using examples. The simplest situation is that
in which $f=1$, so only two times $t_0$ and $t_1$ are involved. Let $T(t,t')$
be the unitary time development operator for the time interval from $t'$ to
$t$; for a time-independent Hamiltonian $H$ it is
$T(t_1,t_0)=\exp[-i(t_1-t_0)H/\hbar]$. At time $t_1$ assume that the projective
decomposition of the identity
corresponds to an orthonormal basis $\{\ket{\phi_1^k}\}$, and let $P_1^k =
[\phi_1^k]$. The Born rule assigns a conditional probability
\begin{equation}
p_k = \Pr(P_1^k\,|\,B [\psi_0]) = |\mapsto ed{\phi_1^k}{T(t_1,t_0)}{\psi_0}|^2
\label{eqn9}
\end{equation}
to $P_1^k$ at $t_1$ given the initial state $[\psi_0]$ at $t_0$.
If we assign a probability of 1 to $[\psi_0]$ and 0 to $I-[\psi_0]$, which is
to say we assume the system starts at $t_0$ in the initial state $[\psi_0]$,
then $p_k$ is the probability assigned to the elementary history
$Y^k=[\psi_0]\odot [\phi^k]$.
\xb
\outl{Textbooks: use uniwave $\ket{\psi(t)}$ to calculate Born probabilities}
\xa
Textbooks use the same rule, but tend to word it differently. Solving
Schr\"odinger's equation with initial state $\ket{\psi_0}$ at $t_0$ yields
\begin{equation}
\ket{\psi(t)} = T(t,t_0)\ket{\psi_0},
\label{eqn10}
\end{equation}
at time $t$. Setting $t=t_1$ allows us to write $p_k$ in \eqref{eqn9} as
\begin{equation}
p_k = |\inpd{\phi_1^k}{\psi(t_1)}|^2.
\label{eqn11}
\end{equation}
That is, students are taught to first calculate the uniwave \eqref{eqn10}, and
then use \eqref{eqn11} to find probabilities.
(The other difference is that the textbooks generally refer to $p_k$ as the
probability of a (macroscopic) measurement outcome, the pointer position, rather
than as the probability of the (microscopic) property the measurement apparatus
was designed to measure. The connection between the two will be discussed in
Sec.~\ref{sbct3.5} below.)
\xb
\outl{Two reasons why $\ket{\psi(t)}$ is not a physical property}
\xa
While this is an excellent calculational technique and
gives the right answers, it has unfortunately given rise to the idea
that the uniwave constitutes the fundamental quantum ontology, it represents
quantum reality. There are two ways to see that this is mistaken. First,
if one regards $[\psi(t_1)]$ as a quantum property, then it will not commute
with any $[\phi_1^k]$ for which $0<p_k<1$. Thus except in special cases the
single framework rule prevents
$[\psi(t_1)]$ from being added to the set of
properties $\{[\phi_1^k]\}$ under consideration. Second, note that
$p_k$ can be calculated using the formula
\begin{equation}
p_k = |\inpd{\psi_0}{\hat\phi^k(t_0)}|^2;\quad
\ket{\hat\phi^k(t)} = T(t,t_1)\ket{\phi_1^k}.
\label{eqn12}
\end{equation}
That is, start with $\ket{\phi_1^k}$ at time $t_1$ and integrate
Schr\"odinger's equation backwards in time to obtain $\ket{\hat\phi^k(t)}$ at
$t=t_0$. In this procedure (which is not particularly efficient, since to
obtain probabilities for several different $k$ requires integrating
Schr\"odinger's equation a comparable number of times) the uniwave never
appears, which shows that it was merely a convenient calculational tool.
\xb
\outl{$\ket{\psi(t)}$ is a pre-probability; its use is epistemic}
\xa
Referring to $\ket{\psi(t)}$ as a ``pre-probability'' (something used to
calculate a probability), as in Sec.~9.4 of \cite{Grff02c}, serves to emphasize
that, at least in the situation under consideration, it is \emptyset h{not} to be
regarded as a quantum property, a genuine ``beable.'' (Similarly,
$\ket{\hat\phi^k(t)}$ in \eqref{eqn12} is a pre-probability.)
Consequently, the appropriate use of ``the wave function'' obtained by solving
Schr\"odinger's equation is, at least in general, epistemic: it is employed to
compute probabilities. The claim, as found for example in
\cite{PsBR12,ClRn12,Hrdy13,PtPM13}, that the wave function cannot be used in
this way seems to be based on the use of classical hidden variables, which are
inconsistent with Hilbert-space quantum mechanics \cite{Grff13d}.
\xb
\outl{Chain kets, consistency for histories of 3 or more times}
\xa
\xb
\outl{Reduces to previous formula for 2 times ($f=1$)}
\xa
When histories involve three or more times an extension of the simple Born rule
is needed in order to generate a consistent set of probabilities for quantum
histories. The essential features appear already in the case of three times,
$f=2$, but it will be convenient to consider the general case of a family of
the type \eqref{eqn7}, For each elementary history $Y^\alpha $ define the
\emptyset h{chain ket}
\begin{equation}
\ket{\alpha } = \ket{(\alpha _1,\alpha _2,\ldots \alpha _f)}
=P_f^{\alpha _f}T(t_f,t_{f-1})P_{f-1}^{\alpha _{f-1}}T(t_{f-1},t_{f-2})\cdots
P_1^{\alpha _1} T(t_1,t_0)\ket{\psi_0}.
\label{eqn13}
\end{equation}
Provided these chain kets are mutually orthogonal, which is to say
\begin{equation}
\inpd{\alpha }{\alpha '} = 0 \text{ whenever } \alpha \neq \alpha '
\label{eqn14}
\end{equation}
where $\alpha = \alpha '$ if and only if $\alpha _j=\alpha '_j$ for every $j$, then the
history $Y^\alpha $ is assigned the probability
\begin{equation}
\Pr(\alpha \,|\,B [\psi_0]) = \inp{\alpha }
\label{eqn15}
\end{equation}
conditioned on the initial state $[\psi_0]$. The \emptyset h{consistency conditions}
\eqref{eqn14} are automatically satisfied for histories only involving two
times, the case $f=1$, and then \eqref{eqn15} gives the same Born probability as
\eqref{eqn9}. However, for $f=2$ or more the restriction \eqref{eqn14}, which
depends both on the projectors making up the family and the unitary dynamics,
is needed and is not trivial.
\xb
\outl{Extended SFR. Families can be combined only if commensurate: consistency
still satisfied}
\xa
As noted previously, the single framework rule prohibits combining two sample
spaces when the projectors do not commute. This applies also to combining two
families of histories: the history projectors must commute with each other for
the combination to be possible, and if this is not so we say the history
families are \emptyset h{incompatible}. However, even if all the projectors in the
two families commute, it may be the case that the common refinement fails to
satisfy the consistency conditions, so there is no way to assign probabilities
to this family using the extended Born rule for a closed quantum system. In
that case we say the two families are \emptyset h{incommensurate}. It is then a
natural extension of the single framework rule to prohibit incommensurate as
well as incompatible families of histories. Or, to put it another way, the
usual rules of probabilistic quantum dynamics can only be applied to a single
consistent family, and results from two incommensurate, as well as from two
incompatible, families cannot be combined. For an example of incommensurate
families see the discussion of families ${\mathcal A}$ and ${\mathcal B}$ for the Aharonov and
Vaidman three-box paradox given in Sec.~22.5 of \cite{Grff02c}.
\xb
\subsection{Quasiclassical frameworks}
\label{sbct3.4}
\xa
\xb
\outl{GMH proposal: Coarse-grained projectors, almost deterministic dynamics}
\xa
\xb
\outl{Exception: In chaotic Cl regime quasiclassical Qm histories will not be
deterministic}
\xa
It has been argued by Omn\`es \cite{Omns99,Omns99b}, and Gell-Mann and Hartle
\cite{GMHr93,GMHr07,Hrtl11} (also see Ch.~26 of \cite{Grff02c}) that classical
mechanics for macroscopic systems emerges as a good approximation to a more
exact but unwieldy quantum description. The idea is to use a
\emptyset h{quasiclassical} quantum framework employing coarse-grained projectors
that project onto Hilbert subspaces of enormous, albeit finite, dimension,
suitably chosen so as to be counterparts of classical properties such as those
used in macroscopic hydrodynamics. The stochastic quantum dynamics associated
with a family of histories constructed using these coarse-grained
quasiclassical projectors gives rise, in suitable circumstances, to individual
histories which occur with high probability and are quantum counterparts of the
trajectories in phase space predicted by classical Hamiltonian mechanics. There
are exceptions. For example, in a system whose classical dynamics is chaotic
with sensitive dependence upon initial conditions one does not expect the
quantum histories to be close to deterministic.
\xb
\outl{Qcl framework not unique, but this not a great concern}
\xa
\xb
\outl{One qcl framework suffices for CP; SFR not needed, ordinary logic OK}
\xa
A quasiclassical family can hardly be unique given the enormous size of the
corresponding Hilbert subspaces, but this is of no great concern provided
classical mechanics is reproduced to a good approximation, in the sense just
discussed, by any of them. Therefore all discussions which involve nothing but
classical physics can, from the quantum perspective, be carried out using a
single quasiclassical framework. As long as reasoning and descriptions are
restricted to this one framework there is no need for the single framework
rule, which explains why a central principle of quantum mechanics is absent
from classical physics. And why ordinary propositional logic is
adequate for the macroscopic world of everyday affairs.
\xb
\subsection{Measurements}
\label{sbct3.5}
\xa
\xb
\outl{Measurement model of vN}
\xb
\outl{Particle, apparatus states; unitary
evolution $\ket{\Psi_0}\rightarrow \ket{\Psi_1}\rightarrow \ket{\Psi_2}$}
\xa
A simple measurement model based on the one proposed by von Neumann in Ch.~V of
\cite{vNmn32b} will illustrate how the histories approach addresses the
measurement problems listed in Table~\ref{tbl1}. Suppose properties of a system
with Hilbert space ${\mathcal H}_s$, hereafter referred to as a ``particle'', are to be
measured by an apparatus, Hilbert space ${\mathcal H}_m$, with ${\mathcal H}_s\otimes {\mathcal H}_m$ the
Hilbert space of the combined closed system. Let
\begin{equation}
\ket{s_0} = \sum_j c_j \ket{s^j},
\label{eqn16}
\end{equation}
be the initial state of the particle, where $\{\ket{s^j}\}$ is an orthonormal
basis of ${\mathcal H}_s$, and $\ket{m_0}$ the ``ready'' state of the apparatus at the
initial time $t_0$. During the interval from $t_0$ to $t_1$ the particle and
apparatus do not interact, so we set $T(t_1,t_0) = I = I_s\otimes I_m$
corresponding to a trivial dynamics. For the time interval from $t_1$ to $t_2$
during which they interact we assume that
\begin{equation}
T(t_2,t_1) \bigl( \ket{s^j}\otimes \ket{m_0}\bigr) = \ket{s^j}\otimes \ket{m^j},
\label{eqn17}
\end{equation}
where the $\ket{m^j}$, associated with different pointer positions, are
normalized and mutually orthogonal. Thus under unitary time evolution the
initial state
\begin{equation}
\ket{\Psi_0} = \ket{s_0}\otimes \ket{m_0}
\label{eqn18}
\end{equation}
develops into
\begin{equation}
\ket{\Psi_1} = T(t_1,t_0)\ket{\Psi_0} = \ket{\Psi_0},\quad
\ket{\Psi_2} = T(t_2,t_1)\ket{\Psi_1} = \sum_j c_j\ket{s^j}\otimes \ket{m^j}
\label{eqn19}
\end{equation}
at the times $t_1$ and $t_2$.
\xb
\outl{Unitary family ${\mathcal F}_u$ cannot be used to discuss measurement
outcomes}
\xa
\xb
\outl{First measurement problem insoluble in interpretations using uniwave}
\xa
Now consider various history families of the form \eqref{eqn7}, with the
initial state $[\Psi_0]=\dya{\Psi_0}$ at time $t_0$ given by \eqref{eqn18}.
One possibility is unitary time development:
\begin{equation}
{\mathcal F}_u:\;\;[\Psi_0]\;\odot \; \{[\Psi_1],I-[\Psi_1]\}\;\odot \;
\{[\Psi_2], I-[\Psi_2]\}
\label{eqn20}
\end{equation}
for times $t_0<t_1<t_2$, with different histories in the sample space
constructed by choosing one of the projectors inside the curly brackets at each
of the later times. Since the events $I-[\Psi_1]$ and $I-[\Psi_2]$ occur with
zero probability they can be ignored, and the single history $[\Psi_0]\odot
[\Psi_1]\odot [\Psi_2]$ occurs with probability 1. While ${\mathcal F}_u$ is perfectly
acceptable as a family of quantum histories, it cannot be used to discuss
possible outcomes of the measurement because it does not include the projectors
$\{[m^j]\}$ for the pointer positions at time $t_2$, nor can it be refined to
include them, because $[\Psi_2]$ will not commute with some of the $[m^j]$,
assuming at least two of the $c_j$ in \eqref{eqn16} are nonzero. Thus the first
measurement problem cannot be solved if all time development is unitary. This
is a basic difficulty facing all quantum interpretations that make the uniwave
fundamental to their ontology.
\xb
\outl{Family ${\mathcal F}_1$ with pointer basis at $t_2$ resolves first measurement
problem; $\ket{\Psi_2}$ can serve as pre-probability}
\xa
The histories approach can solve the first measurement problem by
replacing ${\mathcal F}_u$ with the family
\begin{equation}
{\mathcal F}_1:\;\;[\Psi_0]\;\odot \;[\Psi_1] \;\odot \; \{[m^j]\}.
\label{eqn21}
\end{equation}
That is, the different histories agree at $t_0$ and $t_1$, but correspond to
different pointer positions at $t_2$. The alternative $I-[\Psi_1]$, which
occurs with zero probability, has been omitted at $t_1$, and at time $t_2$ we
employ the usual physicist's convention that $[m^j]=\dya{m^j}$ means
$I_s\otimes [m^j]$ on the full Hilbert space ${\mathcal H}_s\otimes {\mathcal H}_m$. An additional
projector $R'=I -\sum_j [m^j]$ should be included at the final time in
\eqref{eqn12} so that the total sum is the $I$, but, again, it has probability
zero. While $[\Psi_2]$ cannot be one of the properties at time $t_2$ in family
${\mathcal F}_1$, see the discussion of ${\mathcal F}_u$ above, it can be used as a
pre-probability (see the discussion following \eqref{eqn12}) to calculate
probabilities of the different pointer positions at time $t_2$:
\begin{equation}
\Pr([m^j]_2) = {\rm Tr}(\;[\Psi_2]\;[m^j]\;) = \mapsto e{\Psi_2}{\,[m^j]\,}.
\label{eqn22}
\end{equation}
(Here and below we omit from $\Pr()$ the condition $[\Psi_0]$ at $t_0$, as it
applies in all cases.)
\xb
\outl{Family ${\mathcal F}_2$ solves 2d measurement problem: infer prior state from
measurement outcome}
\xa
\xb
\outl{Textbook `` probability of measurement outcome'' correct but
confusing}
\xa
In order to relate the measurement outcome, the pointer position, to a prior
property of the measured particle and thus solve the second measurement problem
yet another family is needed:
\begin{equation}
{\mathcal F}_2:\;\;[\Psi_0]\;\odot \; \{ [s^j]\}\;\odot \; \{[m^k]\}.
\label{eqn23}
\end{equation}
The decomposition $\{ [s^j]\}$ at $t_1$ refers to
properties of the particle, $[s^j]$ means $[s^j]\otimes I_m$,
without reference to the apparatus. It is straightforward to show that
${\mathcal F}_2$ is consistent, leading to a joint probability
distribution
\begin{equation}
\Pr(\,[s^j]_1,[m^k]_2) = |c_j|^2 \delta _{jk},
\label{eqn24}
\end{equation}
where the subscripts on $[s^j]$ and $[m^k]$ indicate the time. The marginals
are:
\begin{equation}
\Pr([s^j]_1) = \Pr([m^j]_2) = |c_j|^2.
\label{eqn25}
\end{equation}
Thus if $|c_k|^2 > 0$ the conditional probability
\begin{equation}
\Pr([s^j]_1 \,|\,B [m^k]_2) = \delta _{jk}
\label{eqn26}
\end{equation}
implies that from the (macroscopic) measurement outcome or pointer position
$[m^k]$ at time $t_2$ we can infer, using standard statistical inference, that
the particle had the (microscopic) property $[s^k]$ at the earlier time $t_1$.
This solves the second measurement problem. And because the probability of
the $[s^j]$ at $t_1$ is the same as $[m^j]$ at $t_2$, textbooks in which
students are taught to calculate $|c_j|^2$ for the particle alone and then
ascribe the resulting probability to the outcome of a measurement, are not
wrong. They would be less confusing if they provided a proper quantum
analysis of the measurement process, such as given here.
\xb
\outl{CH allows retrodiction so sometimes confused with HV approaches}
\xa
Because it solves the second measurement problem, a measurement actually
\emptyset h{measures} something, the histories approach is sometimes confused with
the sorts of \emptyset h{hidden variables} approach studied by Bell and his
followers. However, the basic hypothesis underlying most hidden variables
schemes is that \emptyset h{every} property which could possibly be measured already
``exists'' in some sense in the particle before measurement. This leads to
various difficulties such as the Bell-Kochen-Specker paradox, for which see the
discussion in Sec.~22.1 of \cite{Grff02c}. Suppose, for example, that a
measurement is to be carried out on a spin half particle. Since this might be a
measurement of $S_x$ or of $S_y$ or of $S_z$, it then seems natural to suppose
that all three values are somehow ``present'' in the particle before it is
measured. But this is contrary to Hilbert space quantum mechanics, since the
different projectors do not commute; see the discussion in Sec.~\ref{sbct3.1}.
The histories approach avoids the Bell-Kochen-Specker paradox by applying the
single framework rule \cite{Grff00b}. This point will come up again in the
discussion of locality in Sec.~\ref{sbct3.7} below. For the same reason the
histories approach rejects the notion that quantum mechanics is contextual; see
the detailed discussion in \cite{Grff13b}.
\xb
\outl{${\mathcal F}_u$,${\mathcal F}_1$, ${\mathcal F}_2$ all legitimate, but have different utility}
\xa
It is to be noted that all three history families or frameworks, ${\mathcal F}_u$,
${\mathcal F}_1$, and ${\mathcal F}_2$ employed above satisfy the consistency conditions and thus
provide legitimate quantum descriptions. There is no reason a priori to prefer
one to another; at a fundamental level there is Equality. However, Utility
plays a significant role. If measurement outcomes (pointer positions) at $t_2$
are under discussion, ${\mathcal F}_u$ is unsatisfactory, as its event algebra does not
contain them. Both ${\mathcal F}_1$, and ${\mathcal F}_2$ allow discussion of measurement
outcomes, but if one is interested in how the outcomes are related to the
properties the device was designed to measure, the $[s^j]$ at time $t_1$,
${\mathcal F}_2$ makes this possible, whereas ${\mathcal F}_1$ does not.
\xb \outl{${\mathcal F}_1$, ${\mathcal F}_2$ both useful if Alice prepares $S_x$ \& Bob measures
$S_z$} \xa
This does not mean that ${\mathcal F}_2$ is always the ``right'' framework. Consider a
situation in which Alice prepares a spin half particle with $S_x=+1/2$ and
sends it through a field-free region to an apparatus that Bob has set up to
measure $S_z$, and which yields the value $S_z=-1/2$. At the intermediate time
the value of $S_x$ may be of interest to Alice---did the preparation device
work as intended?---in which case the ${\mathcal F}_1$ family is appropriate. However,
if Bob is concerned with how his apparatus has functioned, ${\mathcal F}_2$ is more
relevant. Both frameworks are valid tools for quantum analysis, and they could
both be used by the same person, e.g., someone who sets up both the preparation
and the measurement apparatus. The restriction which the new logic imposes is
that they \emptyset h{cannot be combined} into a single description
\xb
\outl{Family ${\mathcal F}_3$: later particle state correlated with pointer position
$\leftrightarrow $ wave function collapse}
\xa
\xb
\outl{Collapse not an independent principle. Family ${\mathcal F}_3$ is a
\emptyset h{preparation} }
\xa
It is worth mentioning yet another family
\begin{equation}
{\mathcal F}_3:\;\;[\Psi_0]\;\odot \;[\Psi_1] \;\odot \; \{[s^k]\otimes [m^j]\},
\label{eqn27}
\end{equation}
which is similar ${\mathcal F}_1$ except that at time $t_2$ we have added the final
particle states to the description. It is easy to show, using $\ket{\Psi_2}$
from \eqref{eqn19} as a pre-probability, that the probability of any history
with $k\neq j$ is 0, and hence if $c_j$ in \eqref{eqn16} is nonzero,
\begin{equation}
\Pr([s^k]_2\,|\,B [m^j]) = \delta _{jk}
\label{eqn28}
\end{equation}
That is to say, if at time $t_2$ the pointer is in the position $[m^j]$ the
particle is in the state $[s^j]$. This is von Neumann's (and the textbooks')
``wave function collapse''. But now it is simply an ordinary probabilistic
inference using a conditional probability, so there is nothing at all
mysterious about it, and it definitely does not have to be added to quantum
theory as an independent principle. Other cases of wave function collapse
(when it is being properly used) can also be replaced by conditional
probabilities, thus eliminating another of the conceptual difficulties in
Table~\ref{tbl1}. Note that ${\mathcal F}_3$ is a family useful for analyzing a
\emptyset h{preparation} procedure for producing a particle in a well-defined initial
state.
\xb
\outl{More realistic measurement models discussed in Ch.~17 of CQT}
\xa
Our discussion has employed various simplifications not present in real
measurements. In particular, pointer positions will always be associated with
macroscopic properties, thus projectors onto enormous subspaces and not the
pure states $\ket{m^k}$ assumed above. Similarly, the initial state of a
macroscopic apparatus should be described using a macroscopic projector, or an
appropriate density operator. The discussion given above is extended in Ch.~17
of \cite{Grff02c} to include these more realistic features, along with
irreversible (in the thermodynamic sense) behavior of the apparatus. None of
the conclusions discussed above is undermined by this extension.
\xb
\subsection{Interference}
\label{sbct3.6}
\xa
\xb
\outl{Double slit, Mach-Zehnder paradoxes; latter discussed in detail in CQT
Ch.~13 }
\xa
\xb
\outl{End results of CQT analysis applied to double slit}
\xa
The double slit and the similar Mach-Zehnder paradoxes were introduced in
Sec.~\ref{sbct2.3}. The histories approach in the case of the Mach-Zehnder
interferometer is discussed in considerable detail in Ch.~13 of \cite{Grff02c},
and the same principles apply to the double slit. Here are the end results of
that analysis. A family of histories referring to the particle needs to
include both its existence in a coherent state $\ket{\psi_0}$ at a time $t_0$
before it encounters the slit system, and then its presence in a reasonably
compact region in the interference zone at a later time $t_2$. One can then
argue that a family of histories in which the particle passes through one or
the other of the two slits at the intermediate time $t_1$ fails to satisfy the
consistency conditions, and thus cannot be discussed in appropriate
(probabilistic) quantum terms, in agreement with Feynman's conclusion based on
(excellent) physical intuition. There is, on the other hand, an alternative
family in which the particle does pass through a definite slit and nothing is
said about what happens later. Still another possibility is that the particle
passes through a definite slit and the quantum detectors in the later
interference region are left in a macroscopic quantum superposition
(Schr\"odinger cat) state. Various possibilities are discussed in Ch.~13 of
\cite{Grff02c} using simplified models which are very useful for gaining a
better intuitive understanding of quantum interference.
\xb
\outl{Consistency conditions interpreted as ``absence of interference''
does not prevent CH discussion of situations where Qm interference occurs.}
\xa
The consistency condition \eqref{eqn14}, the requirement that the inner product
of chain kets for different elementary histories be zero, can be understood as
requiring an ``absence of interference'' when calculating probabilities. This
should not be misinterpreted to mean that the histories approach cannot be
applied to physical situations, such as the double slit, where quantum
interference is central to the phenomena under consideration. Instead, for any
given physical situation, whether or not there is some form of interference,
the histories approach and in particular the consistency condition singles out
physically sensible and consistent ways of discussing what is going.
\xb
\subsection{Locality}
\label{sbct3.7}
\xa
\xb
\outl{Analogy of colored slips of paper sent by Charlie to Alice, Bob}
\xa
The following analogy shows how the histories approach counters the widespread
claim that quantum mechanics is nonlocal because it violates Bell inequalities.
Charlie in Chicago places a red slip of paper in an opaque envelope, a green
slip in another, and shuffles the two before mailing one to Alice in Atlanta
and the other to Bob in Boston. Knowing the protocol followed by Charlie, if
Alice opens her envelope and sees a red slip of paper she can immediately
conclude that Bob's envelope contains (or contained) a green slip. This
inference has nothing to do with whether Bob opens the envelope earlier or
later than Alice, or simply throws it away unopened. Furthermore, Alice's
``measurement'' by opening and looking in the envelope has absolutely no
influence on the color, or any other property, of the slip in Bob's
envelope.
\xb
\outl{Spin singlet, Alice measures $S_{az}$, can infer $S_{bz}$}
\xa
How are things different if Charlie prepares two spin half particles $a$ and
$b$ in a singlet state, and pushes a button that sends $a$ towards Alice's
apparatus set up to measure $S_{az}$ for this particle, and $b$ towards Bob's
apparatus set up to measure $S_{bz}$? If Alice's measurement outcome,
indicated by a suitable pointer, corresponds to $S_{az}=+1/2$ she is entitled,
as a competent experimentalist who knows how her equipment functions, to infer
that particle $a$ had this property before the measurement. By using a suitable
framework that includes both $S_{az}$ and $S_{bz}$ values at this earlier time,
and knowing the protocol followed by Charlie, she can infer the value
$-1/2$ for $S_{bz}$.
\xb \outl{Framework: $\ket{\Psi_0}$ at $t_0$ includes singlet state; at $t_1$
use $S_z$ basis for $a$, $b$ particles} \xa
\xb
\outl{Joint distribution of $S_{az}$,
$S_{bz}$ $\rightarrow $ $\Pr(S_{bz}\,|\,B S_{az})$}
\xa
\xb
\outl{Bob can calculate probabilities and conditionals, but does not know
$S_{az}$ }
\xa
\xb
\outl{Later measurement by Bob will confirm Alice's inference}
\xa
To be more specific, use the ${\mathcal F}_2$ type of framework discussed in
Sec.~\ref{sbct3.5}, with $\ket{\Psi_0}$ the initial state of Alice's apparatus
tensored with the spin singlet state of the two particles, and at the
intermediate time $t_1$ use projectors onto the orthonormal basis (with a
notation similar to \eqref{eqn2})
\begin{equation}
\ket{z_a^+,z_b^+},\;\ket{z_a^+,z_b^-},\;\ket{z_a^-,z_b^+},\;
\ket{z_a^-,z_b^-}
\label{eqn29}
\end{equation}
of particle spin states. Their joint probability distribution at time $t_1$
given the singlet state at $t_0$ (and assuming no magnetic fields are present)
is
\begin{align}
\Pr(S_{az}=+\hf,S_{bz}=+\hf) = 0&\quad \Pr(S_{az}=+\hf,S_{bz}=-\hf) = \hf,
\notag\\
\Pr(S_{az}=-\hf,S_{bz}=+\hf) = \hf&,\quad \Pr(S_{az}=-\hf,S_{bz}=-\hf) =0,
\label{eqn30}
\end{align}
which gives conditional probabilities
\begin{equation}
\Pr(S_{bz}=-\hf\,|\,B S_{az}=+\hf) = 1 = \Pr(S_{bz}=+\hf\,|\,B S_{az}=-\hf).
\label{eqn31}
\end{equation}
Since Alice knows on the basis of her measurement outcome that $S_{az}$ had the
value $+1/2$ at $t_1$, she can infer with certainty using the first equality in
\eqref{eqn31} that $S_{bz}$ had the value $-1/2$. Note that Bob, who also
knows the protocol, can write down exactly the same probability formulas
\eqref{eqn30} and \eqref{eqn31}. The only difference is that Alice because she
knows the outcome of her measurement can use \eqref{eqn31} to infer the value
of $S_{bz}$. Bob of course will get this result if he carries out a later
measurement.
\xb
\outl{Wave function collapse not needed. Time of Alice's vs Bob's measurement
unimportant}
\xa
Wave function collapse never appears in the foregoing argument. It could be
used as a calculational tool, as in ${\mathcal F}_3$ in Sec.~\ref{sbct3.5}, to compute a
conditional probability, but computational tools are not to be confused with
physical processes, and wave functions serving as pre-probabilities should be
carefully distinguished from quantum properties. Note also that the time at
which Alice carries out a measurement, relative to when Bob does or does not
carry out a measurement, is of no importance. (See \cite{Grff11b} for a more
extended discussion of this point.) There are no mysterious influences of
Alice's measurement on Bob's particle, and thus no hint of any violation of the
principles of special relativity.
\xb \outl{What if Alice measures $S_{ax}$? Similar, but cannot combine $S_x$
and $S_z$ framework} \xa
And what if if Alice measures some other component of spin angular momentum,
say $S_{ax}$ rather than $S_{az}$? In that case she can use an appropriate
framework to infer the value of $S_{ax}$ before the measurement was made, and
from it deduce the value of $S_{bx}$ for particle $b$. But that framework
cannot be combined with the one which allows her to infer the earlier value of
$S_{az}$ when that is measured, since the $x$ and $z$ components of angular
momentum are incompatible quantum variables. Alice, a competent
experimentalist, cannot measure both $S_{ax}$ and $S_{az}$ on the same particle
for there is no combined property to be measured.
\xb
\outl{Alice could have measured $S_x$ instead of $S_z$. Qm counterfactuals must
use single framework}
\xa
But even if Alice measures $S_{az}$ on this occasion, surely she could have
instead measured $S_{ax}$ on the very same particle, and in that case the
measurement would surely have revealed either $S_{ax}=+1/2$ or $-1/2$. And
therefore before the measurement the particle must have had both a definite
value of $S_{ax}$ as well as $S_{az}$. What is wrong with this argument? The
``if\dots would'' construction betrays the presence of \emptyset h{counterfactual}
reasoning: something actually happened, $S_{az}$ was measured, but one imagines
a different world in which $S_{ax}$ was measured instead. As discussed in
Sec.~19.4 of \cite{Grff02c}, it is important to subject counterfactual
reasoning about quantum systems to the single framework rule. There is no
difficulty imagining a counterfactual world with distinct macroscopic
measurement setting, for these represent mutually exclusive alternatives
represented in quantum theory by commuting projectors (their product is 0).
But there is no room in the Hilbert space for simultaneous values of $S_{az}$
and $S_{ax}$, as noted earlier in Sec.~\ref{sbct3.1}.
\xb
\outl{Bell inequalities use HVs not satisfying Qm Hilbert space rules}
\xa
Here, indeed, is the point where derivations of Bell inequalities are
inconsistent with Hilbert space quantum mechanics: the inequalities are
obtained using ``hidden variables'' not subject to the rules appropriate to a
quantum Hilbert space. The fundamental issue has nothing to do with locality,
for it already arises when considering measurements by just one party, in our
case Alice, of two incompatible physical variables belonging to
incompatible frameworks. For further discussion see \cite{Grff11b,Grff11}.
\xb
\subsection{Approximations}
\label{sbct3.8}
\xa
\xb
\outl{Is it reasonable to use approximate compatibility or consistency?}
\xa
The condition for compatibility of two frameworks for a quantum system at a
single time was stated in Sec.~\ref{sbct3.1} in terms of commutation of the
projectors from both collections. Would not approximate commutation suffice?
The consistency condition for a family of histories at three or more times is
rather stringent; would it suffice if it were approximately satisfied?
\xb \outl{Physicists prefer simple precise theories, but often must use
approximations} \xa
\xb
\outl{Adequacy of approximation a matter of judgment}
\xa
\xb \outl{New logic is similar: exact rules given above. Now a few words re
approximations } \xa
As physicists we prefer to have theories stated in precise terms, especially if
the mathematical expressions are simple and ``clean'', even if in practice it
is almost always necessary to make approximations to what we believe are exact
laws in order to have a theory which can be related to the real world of
experience and experiments. Whether a particular approximation is adequate is
an element of judgment, and it is often difficult if not impossible to provide
precise error bounds. Quantum theory interpreted using the new logic is no
different from other physical theories in this respect, and the material given
above has deliberately been expressed in terms of exact rules. Nonetheless,
since the new logic is intended to assist in providing a \emptyset h{physical}
interpretation of quantum mechanics, let us add a couple of comments that may
be helpful.
\xb
\outl{Plausibility of physical equivalence of nearby Hilbert space rays}
\xa
First, it is plausible that two rays $[\psi]$ and $[\phi]$ in the Hilbert space
that are ``near'' each other in the sense that $|\inpd{\psi}{\phi}|$ is close
to 1 should have a very similar physical interpretation. For example, the spin
of a spin-half particle has a positive component in a direction which is close
to the $z$ axis but not exactly aligned with it. Then we can, at least for
certain purposes, think of it has having the property $S_z=+1/2$. For example,
if the spin is measured in the $S_z$ basis the result will be $S_z=+1/2$
most of the time, with a probability of $1-|\inpd{\psi}{\phi}|^2$ of obtaining
$S_z=-1/2$. Nor will the difference between $[\psi]$ and $[\phi]$ increase in
time under unitary time evolution. It is considerations of this sort that
suggest that the condition that exact orthogonality of the projectors making up
a quantum sample space can be relaxed in some cases without seriously
distorting the physical interpretation.
\xb
\outl{Plausibility of almost consistent family}
\xa
A similar intuition applies to the consistency condition \eqref{eqn14} for a
family of histories of the form \eqref{eqn7}. If consistency is satisfied with
small errors, then it can be argued that small alterations of the projectors in
the sample space, with but small shifts in their physical interpretation, will
yield a family that exactly satisfies the consistency conditions. In this case
the matter is not as obvious as for a simple collection of almost orthogonal
projectors, but at least it is plausible, see \cite{DwKn96}.
\xb
\section{Conceptual Difficulties of the New Logic}
\label{sct4}
\xa
\xb
\outl{CH difficulties known to RBG are listed in Table~\ref{tbl2} \& discussed
below}
\xa
The histories approach gives rise to various conceptual difficulties, and this
section discusses those listed in Table~\ref{tbl2}. While it may not be
complete, no important difficulty known to the author has been omitted from
this list, which helps organize the material that follows. As the new logic is
a scheme of reasoning, the issues it raises are conveniently divided into two
categories. First, is it internally consistent, free from contradictions?
This is addressed in Sec.~\ref{sbct4.1}. Second, assuming consistency, does it
provide a good way to think about quantum mechanics? From the perspective of
the physicist the second question is just as important as, and perhaps even more
important than, the first: a scheme which is logically sound but does not help
us understand the world will not resolve the problems plaguing quantum
foundations. Sections~\ref{sbct4.2} to \ref{sbct4.5} address issues of the
second type, those numbered 2 to 5 in the table.
\xb
\begin{table}[h]
\caption{Conceptual Difficulties of the New Logic}
\label{tbl2}
\begin{center}
\begin{tabular}{l l l}
\hline
\\
1. & \multicolumn{2}{l}{ Internal consistency}\\[1ex]\hline
\\
2. & \multicolumn{2}{l}{Stochastic time development}\\
& a. & Determinism abandoned\\
& b. & The uniwave\\[1ex]\hline
\\
3. & \multicolumn{2}{l}{ Framework selection}\\
& a. & Numerous frameworks\\
& b. & Incompatible frameworks\\
& c. & Selection based on utility\\
& d. & Single framework rule\\
& e. & Choice influences reality?\\[1ex]\hline
\\
4. & \multicolumn{2}{l}{ Particular histories}\\
& a. & Which history occurs?\\
& b.& Retrodiction from different measurements\\[1ex]\hline
\\
5. & \multicolumn{2}{l}{ Truth and reality}\\
& a. & Framework dependence of truth\\
& b. & Unicity \\[1ex] \hline
\end{tabular}
\end{center}
\end{table}
\xa
\subsection{Internal consistency}
\label{sbct4.1}
\xb \outl{Rules for new logic: Choose framework. Projectors commute so ordinary
reasoning works} \xa
Let us start by summarizing the new logic's rules for probabilistic reasoning
about the quantum world. First, choose a quantum framework. By definition this
consists of a quantum sample space, a collection of projectors on the
appropriate quantum Hilbert space that sum to the identity, together with the
associated event algebra composed of all projectors made up of sums of sample
space projectors. Within this framework the usual laws of \emptyset h{classical}
probability theory and ordinary propositional logic apply without any change,
as discussed in Secs.~\ref{sbct3.1} and \ref{sbct3.2}, because the projectors
all commute with each other. And one can use the same intuition---e.g., the
sample space is a collection of mutually exclusive possibilities, one and only
of which is correct---employed in other uses of ordinary (Kolmogorov)
probability theory. (The nontrivial issue of \emptyset h{how} to go about choosing a
quantum framework is taken up in Sec.~\ref{sbct4.3} below.)
\xb
\outl{Single framework rule $\Rightarrow $ consistency, by usual (Cl) arguments}
\xa
\xb
\outl{Incorrect claims that CH inconsistent ignored single framework
rule}
\xa
The single framework rule, which is a central principle of the new logic,
prohibits combining frameworks: any sort of probabilistic argument from
premises to conclusions, including propositional logic as a special case when
probabilities are 0 or 1, must employ \emptyset h{just one} framework. From this it
follows that arguments that prove the consistency of ordinary probabilistic or
propositional reasoning also demonstrate the consistency of quantum reasoning
based on the new logic. In particular, any contradiction that might arise
within a fixed quantum framework will also have a counterpart in standard
(classical) reasoning. Claims made in the literature that the histories
approach is inconsistent, in the sense of leading to contradictions
\cite{Knt97,BsGh00}, are flawed in that the authors have not taken the single
framework rule seriously; see \cite{GrHr98,Grff00b}.
\xb \outl{Probabilistic arguments (Cl \& Qm) begin with a framework, sometimes
chosen implicitly, that includes all events/histories of interest. } \xa
\xb
\outl{Some set of probabilities assumed.}
\xa
\xb
\outl{Initial data $\rightarrow $ final conclusions:``initial'' = beginning, ``final''
= end of argument, not necessarily earliest and latest time}
\xa
\xb
\outl{If probs are all 0 or 1 $\leftrightarrow $ ordinary logical argument}
\xa
It may help to supplement the preceding remarks with some comments about how
probabilistic reasoning is actually carried out. Classical
applications of probability theory also begin with a framework, which is to say
a sample space and an event algebra, though this is sometimes done
implicitly---e.g., random variables are introduced without bothering to say
which space they are functions on, since the knowledgeable reader ought to know
how to construct it. In quantum mechanics one needs to be a bit more careful,
since many---perhaps most---quantum paradoxes are constructed by combining
incompatible frameworks. The framework chosen must, of course, include all the
events or histories one is interested in. Next some set of probabilities are
assumed: the only strict rule for assigning them is that they must be additive
and sum to 1. Then the typical argument proceeds from some \emptyset h{initial data},
assumed to be correct or perhaps assigned some initial probabilities, to
\emptyset h{final conclusions}, also expressed using probabilities. If all the
probabilities are 0 or 1, one has an ordinary logical argument. The
``initial'' in ``initial data'' refers to something assumed at the beginning of
the argument, not necessarily properties of a physical system at the earliest
time of interest, though these are often included in the initial
data. Similarly, ``final'' refers to the end of the argument, not necessarily
the latest time.
\xb \outl{Example: Family ${\mathcal F}_2$ in Sec.~\ref{sbct3.5}: data =
(initial state + measurement outcome) $\Rightarrow $ particle state
at intermediate time} \xa
For a specific quantum example see the discussion of measurements in
Sec.~\ref{sbct3.5} using the family ${\mathcal F}_2$. The state $[\Psi_0]$ at the
earliest time $t_0$ together with the measurement outcome at time $t_2$
constitute the initial data needed to infer a property of the particle at an
intermediate time $t_1$, which is the final conclusion. (In this instance the
inference requires the use of probabilities obtained by applying the extended
Born rule to a closed system in a situation involving three times, so an
acceptable quantum framework must be a family of histories satisfying the
consistency conditions.)
\xb
\outl{Different frameworks with same data \& conclusions yield same
probabilities for conclusions}
\xa
\xb
\outl{Heuristic: use coarsest possible framework}
\xa
Often there is more than one framework which will contain the events of
specific interest along with other events, and since two such frameworks could
be incompatible, one might be concerned that they would lead to different
results, i.e., different outcome probabilities. However, a basic consistency
rule, discussed in more detail in Ch.~16 of \cite{Grff02c}, shows that the
probabilities linking a particular set of conclusions to a specific collection
of initial data are always the same for any framework that includes both. A
useful heuristic in constructing arguments of this sort is to employ the
coarsest possible framework, i.e., the smallest number of projectors in the
quantum sample space, that can accommodate all data and conclusions, since
adding refinements is usually more work, and there is the danger that in
constructing a complicated argument one may overlook something, such as a
consistency condition, and arrive at incorrect conclusions.
\xb
\subsection{Stochastic time development}
\label{sbct4.2}
\xa
\xb
\outl{Qm stochastic time evolution not a severe difficulty}
\xa
\xb
\outl{Can imagine a Cl nondeterministic world; Cl chaos effectively
indeterministic }
\xa
\xb
\outl{In Qcl regime can have QM $\Rightarrow $ determinism FAPP}
\xa
\xb
\outl{Even Einstein might have accepted Qm indeterminism}
\xa
A stochastic (probabilistic) time development in place of determinism should
not represent a serious conceptual problem. One can without difficulty imagine
a classical world in which the fundamental dynamical law has some stochastic
element, and in the regime of classical chaos even deterministic equations can
lead to behavior which is for all practical purposes indeterministic.
Furthermore, the study of quasiclassical frameworks, Sec.~\ref{sbct3.4}, shows
how fundamentally indeterministic quantum laws can, under suitable
circumstances, give rise to what is for all practical purposes deterministic
behavior for a macroscopic system. Even Einstein might have been willing to
abandon determinism in order to achieve a theory which contains no mysterious
nonlocal influences, no longer has measurement as a fundamental principle, and
allows one to say the moon is there even when it is not being observed.
\xb
\outl{Sociological barrier: students taught to reverence Schr Eqn}
\xa
\xb
\outl{Fix this by teaching uniwave tool for finding probs, how measurements
work }
\xa
\xb
\outl{Abandoning the uniwave $\leftrightarrow $ recognizing proper role of Schr Eqn for
finding probabilities}
\xa
There is, however, another barrier, as much sociological as scientific.
Students in their first quantum course are taught to reverence the
deterministic Schr\"odinger equation as the central principle of quantum
dynamics, whereas probabilities are treated as somewhat of an embarrassment, a
necessary evil when measurements interfere with the ``correct'' time
dependence, which however will resume again once the nasty measurement is
over. Were students at the beginning of the course taught that the unitarily
developing wave function, the uniwave, is simply a tool for calculating
probabilities, not a representation of reality, and during the course supplied
with a schematic but fully quantum description of measurements, this particular
difficulty would likely disappear.
\xb
\subsection{Framework selection}
\label{sbct4.3}
\xa
\xb
\outl{Theoretical physics uses approximate models constructed using certain
rules}
\xa
\xb
\outl{Framework choice occurs in CP; e.g., sample space for probabilistic
model}
\xa
\xb \outl{Will explore Qm situation by comparing with Cl relative to Liberty,
Equality\dots} \xa
Quantum mechanics is similar to other branches of theoretical physics in that
in order to apply it to a particular system one must construct a conceptual
model following certain rules. The choice of which model to use
depends on what one wants to discuss, but a variety of other considerations can
enter that choice. All physical models are approximate in one way or another,
and in this sense have varying degrees of ``reality'' associated with them.
Simplifications are often introduced so as to allow a easier mathematical
analysis, or in the hopes of gaining physical insight into the problem under
discussion. Consequently the task of choosing a framework in which to carry
out a discussion is not absent from classical physics, though it is generally
simpler than in the quantum case. In particular, whenever probabilities are
used, there must be either an explicit or implicit choice of a sample space and
an event algebra. Difficulties arise in the quantum case both because there
are a large number of possibilities, and also because one cannot combine
incompatible alternatives into a single description. A useful way of exploring
the quantum difficulties is to consider some classical systems, and ask which
of the principles for quantum frameworks---Liberty, Equality,
Incompatibility, Utility---introduced in Sec.~\ref{sbct3.2} have classical
analogs.
\xb
\outl{Cl analogy: Coffee cup seen from above and below}
\xa
Let us begin with an everyday classical example. It is possible to view a
coffee cup from below as well as from above, and the two perspectives give
different types of information about, or describe different aspects of, a
single object. One can choose either, and neither perspective is more
fundamental than the other, so Liberty and Equality are represented in this
mundane example. The Utility of each depends on what one is interested in
learning: Is there coffee in the cup? Is there a crack on the bottom surface?
Both perspectives are \emptyset h{compatible}: they can (in principle) be
harmoniously combined into a single, more detailed, more refined description,
containing all the information present in the views from above and below. And
this compatibility is consistent with quantum theory: the relevant projectors,
should one be so foolish as to attempt a quantum mechanical description of the
coffee cup, will form a commuting set that is part of a quasiclassical
framework.
\xb
\outl{Compare with Sec. 3.5 example: $S_x$ prepared, $S_z$ later measured}
\xa
Contrast this with the example in Sec.~\ref{sbct3.5} where Alice prepares a
spin-half particle in a state $S_x=+1/2$ and Bob later measures it and finds
$S_z=-1/2$. To describe the particle at an intermediate time between
preparation and measurement one can use either a consistent family of histories
which contains the value of $S_x$ at this time, or one which contains the
value of $S_z$. Either is perfectly acceptable from the perspective of quantum
theory; there is no fundamental law that says that one should be used rather
than the other. However, they are \emptyset h{incompatible} with each other and
cannot be combined. Each family has its uses;
e.g., in addressing the question of whether the preparation was successful, or
whether the measurement apparatus performed what it was designed to do.
\xb
\outl{Compare golf ball prepared with Sx positive; Sz measured}
\xa
It is Incompatibility that most clearly marks the border between classical and
quantum physics, as can be seen by comparing the preceding example with a
situation where the spin-half particle is replaced by a golf ball which Alice
prepares with a positive $x$ component of spin angular momentum, and Bob later
measures the $z$ component and finds it is negative. Again two valid
descriptions at the intermediate time, but now they can be combined. Since the
typical angular momentum of a spinning golf ball is on the order of $10^{30}$
in units of $\hbar$ there is no problem constructing a quasiclassical framework
in which both $x$ and $z$ components are represented approximately with a
precision much more than adequate for all practical purposes.
\xb
\outl{Lorentz frames not a good analogy for Qm frameworks}
\xa
Another partial analogy is provided by Lorentz transformations in classical
special relativity. The physicist is at Liberty to choose different Lorentz
frames, with none being more ``fundamental'' than another, and Utility may
determine the choice; e.g., in scattering problems there is an advantage to
using the center of mass. However, the situation is unlike quantum mechanics
in that every Lorentz frame contains the same information as any other, since
there is a well-defined means of transforming positions and momenta between
different frames. Distinct quantum frameworks which are mutually incompatible
obviously do \emptyset h{not} contain the same information. (As an aside we note
that there is no particular problem in constructing a relativistic version of
the histories approach; see, e.g., \cite{Grff02b}.)
\xb
\outl{Coarse grainings in Cl Stat Mech a better analogy}
\xa
Perhaps a closer analogy is provided by classical statistical mechanics where
it is sometimes useful, for purposes of discussing irreversibility or the
origin of hydrodynamic laws, to introduce a coarse graining of the classical
phase space into nonoverlapping cells, with the coarse-grained description
providing not the actual phase point of the system, but instead the label of
the cell in which it is located. Here the choice of coarse graining is clearly
one made by the physicist on the basis of its utility for the calculation he
has in mind, and of course no coarse graining is more ``fundamental'' than any
other. In addition, two coarse grainings do not in general contain the same
information. However, given any two coarse grainings there is always a common
refinement using the intersection of the cells, so with respect to
Incompatibility the analogy with (incompatible) quantum frameworks breaks down.
\xb \outl{Cl analogies imply: frameworks must be chosen, are not mutually
exclusive, choice of framework does not influence reality} \xa
\xb
\outl{FORTRAN vs C provides partial analog of incompatibility}
\xa
To summarize the situation, classical physics provides analogies of many of the
features which need to be taken into account when thinking about quantum
frameworks. Frameworks are not automatic: they must be chosen. There are
multiple possibilities, and alternative frameworks are not mutually exclusive
in the sense that if one is right the others must be wrong. The physicist's
choice of framework has no influence on the reality being described, and is
generally motivated by the desire to understand or describe a particular aspect
of the system of interest. What classical physics does not provide is a good
analogy for incompatible frameworks and the single framework rule that
prohibits combining them. Computer languages provide a partial analogy: woe be
to the programmer who mixes FORTRAN with C. But since a particular algorithm
can be expressed using either, this analogy, while it may be helpful, is not
exact.
\xb
\subsection{Particular histories}
\label{sbct4.4}
\xa
\xb
\outl{Objection to CH: Many families, many histories, which is the TRUE one?}
\xa
A common objection to the histories approach is that there many consistent
families of histories, and even if in each family only one elementary history
can occur, this still leaves a large number of possibilities. How does one
know which of these histories \emptyset h{actually} occurred? What is the one
\emptyset h{true} history? What rule selects the \emptyset h{correct} family that contains
it?
\xb
\outl{This issue, viewed formally, is one of framework selection}
\xa
From a formal perspective the issue raised here is a particular instance of the
framework selection problem discussed above. Quantum mechanics allows many
different frameworks any one of which can be chosen by the physicist for
constructing a description, as long as they are not combined in a way that
violates the single framework rule. That prohibition includes, in the case of
consistent families of histories, the combining of incommensurate
families. Within a consistent family the elementary histories (those belonging
to the sample space) are mutually exclusive, so one and only one of them occurs
or is correct, even though quantum theory can in general only provide
probabilities for different possibilities.
\xb
\outl{Histories written by historians are not defective because they are
different }
\xa
\xb
\outl{Should not physicist have similar liberty in choosing material?}
\xa
A formal statement is often insufficient to resolve some intuitive difficulty,
and here is where a classical analogy may be helpful. History books written by
professional historians are often quite different, but this by itself does not
mean they are defective. The historian chooses material that
provides a coherent narrative for the time period of interest while still
remaining consistent with the facts insofar as they are known. No one would
expect a history of the United States to cover the same territory as a history
of Great Britain. The historian has Liberty to choose material that
best serves his purpose, and there seems no reason to deny the quantum
physicist a similar freedom.
\xb
\outl{Histories should be consistent when discussing the same event; Qm
histories satisfy this }
\xa
To be sure we expect different histories of the world to be consistent to the
extent that they deal with the same events, and it seems reasonable to expect
quantum descriptions to satisfy similar conditions of consistency. And indeed
they do. Given the same input data, without which one cannot assign
probabilities, two consistent families of histories that include this data
will always assign the same probabilities to other events that occur in both
families; this is a consequence of the internal consistency of the histories
approach discussed earlier in Sec.~\ref{sbct4.1}.
\xb \outl{Spin-half: prepare $S_x$, measure $S_z$; incompatible histories at
intermediate time} \xa
\xb
\outl{A \& V 3-box paradox discussed in CQT. SFR removes apparent contradiction}
\xa
The example discussed earlier in which Alice prepares $S_x=+1/2$ and Bob
measures $S_z=-1/2$ may help to illustrate this point. There are two
incompatible consistent families, one containing $S_x$ and the other $S_z$ at
the intermediate time $t_1$, and in each family one draws some conclusion about
the spin angular momentum at $t_1$. The conclusions are indeed different,
but they are not contradictory, for events involving $S_x$ cannot be combined
with those involving $S_z$. A more striking example is provided by the
three-box paradox of Aharonov and Vaidman \cite{AhVd91}, discussed in detail in
Sec.~22.5 of \cite{Grff02c}, where again the single framework rule removes the
apparent contradiction resulting from applying classical reasoning in a
situation where it violates quantum principles.
\xb
\outl{Insisting that ``there just HAS to be a SINGLE true history''. See
following section}
\xa
To be sure, some may want to insist that ``there just \emptyset h{has} to be a
\emptyset h{single} history a single true story.'' This is as much a philosophical
position as a scientific objection, which does not mean it can simply be
dismissed. The discussion in the following section is an attempt to get to the
bottom of what is here at issue.
\xb
\subsection{Truth and reality}
\label{sbct4.5}
\xa
\xb
\outl{Examples discussed earlier point to central conceptual difficulty}
\xa
\xb
\outl{Consistency of CH probabilities ensured by single framework rule}
\xa
\xb
\outl{Incompatible frameworks $\Rightarrow $ probs relative to framework; no single prob
distribution}
\xa
The preceding examples and discussions help identify what is perhaps the
central conceptual difficulty of the new logic. In quantum mechanics
interpreted in this way any description of nature must be formulated using a
framework of commuting Hilbert space projectors which can be assigned
probabilities in a consistent fashion. Consistency is ensured by the single
framework rule, which prohibits combining incompatible frameworks or
incommensurate families of histories. Consequently, probability distributions
are relative to frameworks; there is not a single probability distribution that
can be used for every framework or every family of histories.
\xb
\outl{True/false $\leftrightarrow $ probability 1,0; implies True/false relative to
framework}
\xa
\xb
\outl{No single universally true state of affairs: a major block to accepting
CH}
\xa
In a probabilistic model the limiting cases of probability 1 and 0 correspond
to statements which in propositional logic are true and false,
respectively. This same interpretation is employed in histories quantum
mechanics. But then ``true'' and ``false'' \emptyset h{must be understood
relative to a framework}. There is no single universally true state of affairs
in histories quantum mechanics. This has undoubtedly been a major stumbling
block standing in the way of its more general acceptance by the physics
community, despite the fact that it provides a consistent resolution of all the
usual quantum paradoxes, something which cannot be said of any other
interpretation of quantum mechanics currently available. And it seems to be
at the heart of Mermin's objection, see Sec.~\ref{sct1}.
\xb
\outl{Unicity a deeply-rooted faith called into question by QM}
\xa
Indeed, there is a deep-rooted faith or intuition, shared by scientists as well
as the ordinary man on the street, that at any point in time there is a
particular state of affairs which exists, which is ``true'', and to which every
true description of the world must conform. No one claims to know what this
exact truth is, and indeed it must, if it exists, be beyond human knowledge.
Let us call this belief the principle of \emptyset h{unicity}. Calling it into
question seems heretical, contrary to both common sense and sound science.
Nonetheless, in the quantum world it does not seem to be valid.
\xb
\outl{Classical phase space provides mathematical picture of unicity}
\xa
To see how and why it breaks down, recall that a classical space can be divided
into finer and finer regions until one arrives at a single point representing
the precise state of a physical system. All subsets of the phase space that
contain this point represent properties that are simultaneously true; their
intersection is the point itself, which is the ultimate truth. Hence classical
mechanics provides a convenient mathematical picture for visualizing unicity,
and the enormous success of classical mechanics lends support to its validity.
\xb
\outl{Hilbert space: finest property $\leftrightarrow $ 1d subspace $\leftrightarrow $ point in Cl
phase space}
\xa
\xb \outl{But then: Incompatible Qm properties completely different from CP} \xa
\xb
\outl{Unicity runs into logical problems seen by Birkhoff, vN}
\xa
\xb
\outl{CH abandons unicity; other Qm interpretations ignore the logical problem}
\xa
In the quantum Hilbert space the description of properties is ``quantized'':
subspaces have integer dimensions, and the smallest subspace representing a
property which could possibly be true has dimension 1. So this ought to be the
quantum analog of a single point in the classical phase space. But then one
finds, as discussed in detail in Sec.~\ref{sbct3.1}, that at this level the
structure of Hilbert space is significantly different from that of classical
phase space. In particular one has properties that are incompatible---their
projectors do not commute---in a manner which is completely foreign to
classical physics. The obvious extrapolation of classical unicity, the notion
of a single true property, runs into the logical difficulties understood by
Birkhoff and von Neumann. Abandoning unicity, as in the new logic, may not be
the only way to solve the problem, but simply ignoring it, which is what one
finds in much modern work on quantum foundations, is not likely to result in
progress.
\xb
\outl{Qcl frameworks explain intuition behind unicity in everyday life \& why
it fails in QM}
\xa
To put the matter in a slightly different way, if we assume that the world
is governed by classical principles we eventually run into disagreement with
experiment. However, by assuming that it is governed by quantum principles,
with quasiclassical frameworks explaining the success of classical physics at
the macroscopic level, we can begin to understand the deep intuition that lies
behind the notion of unicity, based on everyday human experience in the
classical world. But at the same time we can understand why and in what
circumstances this intuition breaks down.
\xb
\outl{Cl truth and reality as self evident as the fact that the earth is at
rest}
\xa
\xb
\outl{History of science: self-evident things replaced by alternatives}
\xa
To those who claim that classical notions of truth and reality are necessary
truths, which are self evident, the appropriate response is to say that they
are just as self evident as the fact, accepted by our intellectual ancestors,
that the earth is at rest at the center of the universe. The history of
science is marked by a set of important revolutions in thought in which things
thought to be intuitively obvious and self-evident have been replaced by
alternative explanations in better agreement with empirical observation. Why
should quantum theory be different?
\xb
\outl{Abandoning unicity does not imply abandoning logical thought}
\xa
\xb
\outl{Need new intuition to go along with the new logic}
\xa
\xb
\outl{Abandoning unicity $\not\Rightarrow $ giving up notion of real world ``out
there''}
\xa
Abandoning unicity is not equivalent to abandoning logical thought, and it is
worth stressing that the histories approach to quantum interpretation, the new
quantum logic, is entirely consistent provided one pays attention to the
rules, discussed with various examples in Sec.~\ref{sct3} and summarized in
Sec.~\ref{sbct4.1} for constructing quantum descriptions. Nor does abandoning
unicity mean that one has to give up on physical intuition about what is
``going on'' in the quantum world. True, classical thinking is no longer
satisfactory in the quantum domain, and the physicist has to develop an
appropriate quantum intuition in its place. This takes effort, but it is not
impossible.
Nor does abandoning unicity require giving up the notion of a real world ``out
there'', one whose existence is independent of our thoughts, wishes, and
beliefs. What the development of quantum mechanics and its consistent
interpretation using the new logic indicates is that the certain features of
this reality differ from what was thought to be the case before quantum theory
was developed and successfully applied to understanding phenomena in the
microscopic world.
\xb
\section{Conclusion}
\label{sct5}
\xa
\xb
\outl{Claim: new logic has successful resolved Qm conceptual
difficulties}
\xa
\xb
\outl{It is radical break with many long accepted ideas}
\xa
\xb
\outl{It is consistent with logical thot and an independent reality}
\xa
The fundamental thesis of this paper is that the conceptual difficulties of
quantum foundations listed in Table~\ref{tbl1} and discussed in
Sec.~\ref{sct2}, can be, and in fact have been, successfully resolved using the
new quantum logic embodied in the histories approach, as summarized in
Sec.~\ref{sct3}. The new logic, while not as radical as the older quantum
logic, still represents an important break with ideas which have long seemed
central to human thought in general and to the natural sciences in particular.
The most significant changes, and the conceptual difficulties that they in turn
give rise to, are indicated in Table~\ref{tbl2} and discussed in
Sec.~\ref{sct4}. However novel it may seem, it is worth remembering that the
new logic is consistent both with logical thought and with an independent
reality that does not require human thought (or measurements) to bring it into
existence.
\xb
\outl{CH differs from other proposals in (i) role played by noncommutation;
(ii) stochastic time development; (iii) resolves ALL the paradoxes}
\xa
\xb
\outl{Supposed superluminal influences are simply fudge factors}
\xa
There might be yet better ways of resolving quantum conceptual difficulties
than those provided by the histories approach. What distinguishes it from
alternative proposals at the present time is the combination of (i) the central
role played by the noncommutation of quantum operators, in particular
projectors, in the conceptual foundations of the subject; (ii) its insistence
that \emptyset h{all} quantum time development is stochastic, not just when
measurements take place; (iii) its success in resolving not just one or two,
but \emptyset h{all} of the standard quantum paradoxes. In particular, the supposed
superluminal influences that have infested quantum foundations for the last 50
years and make some other interpretations of quantum mechanics difficult to
reconcile with relativity theory are absent from the histories approach; such
influences are nothing but fudge factors needed to compensate for a lack of
understanding of what quantum measurements measure, and the failure to use a
fully consistent set of quantum principles when discussing entangled states.
\xb
\outl{Accept vs reject scientific ideas a judgment call}
\xa
\xb
\outl{Cannot prove that earth moves; assuming it does makes things simpler}
\xa
\xb
\outl{Critical scrutiny of new logic is welcome. That it's radical $\not\Rightarrow $
it's wrong}
\xa
In the end the acceptance or rejection of a set of ideas by individual
scientists and by the scientific community is a matter of scientific judgment;
there are no overwhelming arguments that establish the proof of any scientific
theory. That scientists today believe that the earth moves, around its axis
and around the sun, rather than lying fixed in place at the center of the
universe, is a consequence not of rigorous logical proofs of the sort that
appeal to some philosophers, but instead the fact that this way of looking at
things clears up a number of conceptual difficulties in a way much simpler and
seemingly more satisfactory than can be done by assuming the earth is fixed.
The author believes that the same is true of the new quantum logic, and
welcomes critical scrutiny by those who are willing to examine it in
detail before publishing their conclusions. That the histories approach to
interpreting quantum theory is radical must be acknowledged. This does not
mean it is wrong.
\section*{Acknowledgments}
The work described here is based on research supported by the National Science
Foundation through Grant PHY-1068331.
\end{document}
\xb
\end{document} | math | 116,227 |
\begin{document}
\title{Mixing operators on spaces with weak topology}
\begin{abstract} We prove that a continuous linear operator $T$ on a
topological vector space $X$ with weak topology is mixing if and
only if the dual operator $T'$ has no finite dimensional invariant
subspaces. This result implies the characterization of hypercyclic
operators on the space $\omega$ due to Herzog and Lemmert and
implies the result of Bayart and Matheron, who proved that for any
hypercyclic operator $T$ on $\omega$, $T\oplus T$ is also
hypercyclic.
\end{abstract}
\small \noindent{\bf MSC:} \ \ 47A16, 37A25
\noindent{\bf Keywords:} \ \ Hypercyclic operators; transitive
operators; mixing operators; weak topology \normalsize
\section{Introduction \label{s1}}\rm
All topological vector spaces in this article {\it are assumed to be
Hausdorff} and are over the field ${\mathbb K}$, being either the field ${\mathbb C}$
of complex numbers or the field ${\mathbb R}$ of real numbers. As usual, ${\mathbb Z}$
is the set of integers and ${\mathbb N}$ is the set of positive integers. If
$X$ and $Y$ are vector spaces over the same field ${\bf k}$, symbol
$L(X,Y)$ stands for the space of ${\bf k}$-linear maps from $X$ to $Y$.
If $X$ and $Y$ are topological vector spaces, then ${\mathcal L}(X,Y)$ is the
space of continuous linear operators from $X$ to $Y$. We write
$L(X)$ instead of $L(X,X)$, ${\mathcal L}(X)$ instead of ${\mathcal L}(X,X)$ and $X'$
instead of ${\mathcal L}(X,{\mathbb K})$. For each $T\in {\mathcal L}(X)$, the dual operator
$T':X'\to X'$ is defined as usual: $(T'f)(x)=f(Tx)$ for $f\in X'$
and $x\in X$. We say that the topology $\tau$ of a topological
vector space $X$ {\it is weak} if $\tau$ is exactly the weakest
topology making each $f\in Y$ continuous for some linear space $Y$
of linear functionals on $X$ separating points of $X$. We use symbol
$\omega$ to denote the product of countably many copies of ${\mathbb K}$. It
is easy to see that $\omega$ is a separable complete metrizable
topological vector space, whose topology is weak.
Let $X$ be a topological vector space and $T\in{\mathcal L}(X)$. A vector
$x\in X$ is called a {\it hypercyclic vector} for $T$ if
$\{T^nx:n\in{\mathbb N}\}$ is dense in $X$ and $T$ is called {\it
hypercyclic} if it has a hypercyclic vector. Recall also that $T$ is
called {\it hereditarily hypercyclic} if for each infinite subset
$A$ of ${\mathbb N}$, there is $x\in X$ such that $\{T^nx:n\in A\}$ is dense
in $X$. Next, $T$ is called {\it transitive} if for any non-empty
open subsets $U$ and $V$ of $X$, there is $n\in{\mathbb N}$ for which
$T^n(U)\cap V\neq\varnothing$ and $T$ is called {\it mixing} if for
any non-empty open subsets $U$ and $V$ of $X$, there is $n\in{\mathbb N}$
such that $T^m(U)\cap V\neq\varnothing$ for each $n\geqslant m$. It is
well-known and easy to see that any hypercyclic operator (on any
topological vector space) is transitive and any hereditarily
hypercyclic operator is mixing. If $X$ is complete separable and
metrizable, then the converse implications hold: any transitive
operator is hypercyclic and any mixing operator is hereditarily
hypercyclic. For the proof of these facts as well as for any
additional information on the above classes of operators we refer to
the book \cite{bama-book} and references therein. Herzog and Lemmert
\cite{gele} characterized hypercyclic operators on $\omega$.
\begin{thmhl}Let ${\mathbb K}={\mathbb C}$ and $T\in {\mathcal L}(\omega)$. Then $T$ is
hypercyclic if and only if the point spectrum $\sigma_p(T')$ of $T'$
is empty.
\end{thmhl}
Another result concerning hypercyclic operators on $\omega$ is due
to Bayart and Matheron \cite{bama}.
\begin{thmbm}For any hypercyclic operator $T\in {\mathcal L}(\omega)$, $T\oplus
T$ is also hypercyclic.
\end{thmbm}
We refer to \cite{beco,peter1} for results on the structure of the
set of hypercyclic vectors of operators on $\omega$ and to
\cite{chan,shk1} for results on hypercyclicity of operators on
Banach spaces endowed with its weak topology. We characterize
transitive and mixing operators on spaces with weak topology.
\begin{theorem}\label{omegA} Let $X$ be a topological vector space,
whose topology is weak and $T\in {\mathcal L}(X)$. Then the following
conditions are equivalent$:$
\begin{itemize}\itemsep=-2pt
\item[\rm(\ref{omegA}.1)]$T'$ has no non-trivial finite dimensional invariant subspaces$;$
\item[\rm(\ref{omegA}.2)]$T$ is transitive$;$
\item[\rm(\ref{omegA}.3)]$T$ is mixing$;$
\item[\rm(\ref{omegA}.4)]for any non-empty open subsets $U$ and $V$
of $X$, there is $k\in{\mathbb N}$ such that $p(T)(U)\cap V\neq\varnothing$
for any polynomial $p$ of degree $\geqslant k$.
\end{itemize}
\end{theorem}
Since $\omega$ is complete, separable, metrizable and carries weak
topology, we obtain the following corollary.
\begin{corollary}\label{omeg} Let $T\in {\mathcal L}(\omega)$. Then the following conditions
are equivalent
\begin{itemize}\itemsep=-2pt
\item[\rm(\ref{omeg}.1)]$T'$ has no non-trivial finite dimensional invariant subspaces$;$
\item[\rm(\ref{omeg}.2)]$T$ is hypercyclic$;$
\item[\rm(\ref{omeg}.3)]$T$ is hereditarily hypercyclic$;$
\item[\rm(\ref{omeg}.4)]for any sequence $\{p_k\}_{k\in{\mathbb N}}$ of
polynomials with $\hbox{\tt deg}\, p_k\to\infty$, there is $x\in \omega$ such
that $\{p_k(T)x:k\in{\mathbb N}\}$ is dense in $\omega$.
\end{itemize}
\end{corollary}
Note that in the case ${\mathbb K}={\mathbb C}$, $T'$ has no non-trivial finite
dimensional invariant subspaces if and only if
$\sigma_p(T')=\varnothing$. Moreover, the direct sum of two mixing
operators is always mixing. Thus Theorem~HL and Theorem~BM follow
from Theorem~\ref{omegA}.
\begin{remark}\label{Rem2} Chan and Sanders \cite{chan} observed
that on $(\ell_2)_\sigma$, being $\ell_2$ with the weak topology,
there is a transitive non-hypercyclic operator. Theorem~\ref{omegA}
provides a huge supply of such operators. For instance, the backward
shift $T$ on $\ell_2$ is mixing on $(\ell_2)_\sigma$ since $T'$ has
no non-trivial finite dimensional invariant subspaces and $T$ is
non-hypercyclic since each its orbit is bounded.
\end{remark}
Since each weak topology is determined by the corresponding space of
linear functionals, it comes as no surprise that Theorem~\ref{omegA}
is algebraic in nature. Indeed, we derive it from the following
characterization of linear maps without finite dimensional invariant
subspaces. The idea of the proof is close to that Herzog and Lemmert
\cite{gele}, although by reasoning on a more abstract level, we were
able to get a result, which is simultaneously stronger and more
general.
We start by introducing some notation. Let ${\bf k}$ be a field. Symbol
${\mathcal P}$ stands for the algebra ${\bf k}[t]$ of polynomials in one variable
over ${\bf k}$, while ${\mathcal R}$ is the field ${\bf k}(t)$ of rational functions
in one variable over ${\bf k}$. Consider the ${\bf k}$-linear map
\begin{equation*}
M:{\mathcal R}\to{\mathcal R},\qquad Mf(z)=zf(z).
\end{equation*}
If $A$ is a set and $X$ is a vector space, then $X^{(A)}$ stands for
the algebraic direct sum of copies of $X$ labeled by $A$:
$$
X^{(A)}=\bigoplus_{\alpha\in A}{\mathcal R}=\bigl\{x\in X^A:\{\alpha\in
A:x_\alpha\neq 0\}\ \ \text{is finite}\bigr\}.
$$
Symbol $M^{(A)}$ stands for the linear operator on ${\mathcal R}^{(A)}$,
being the direct sum of copies of $M$ labeled by $A$. That is,
$$
M^{(A)}\in L({\mathcal R}^{(A)}),\quad (M^{(A)}f)_\alpha=Mf_\alpha\ \ \
\text{for each}\ \ \alpha\in A.
$$
It is easy to see that each $M^{(A)}$ has no non-trivial finite
dimensional invariant subspaces. Obviously, the same holds true for
each restriction of $M^{(A)}$ to an invariant subspace.
\begin{theorem}\label{inva} Let $X$ be a vector space
over a field ${\bf k}$ and $T\in L(X)$. Then $T$ has no non-trivial
finite dimensional invariant subspaces if and only if $T$ is similar
to a restriction of some $M^{(A)}$ to an invariant subspace.
\end{theorem}
The above theorem is interesting on its own right. It also allows us
to prove the following lemma, which is the key ingredient in the
proof of Theorem~\ref{omegA}.
\begin{lemma} \label{empty1}
Let $X$ be a non-trivial vector space over a field ${\bf k}$ and $T:X\to
X$ be a linear map with no non-trivial finite dimensional invariant
subspaces. Then for any finite dimensional subspace $L$ of $X$,
there is $m=m(L)\in {\mathbb N}$ such that $p(T)(L)\cap L=\{0\}$ for each
$p\in{\mathcal P}$ with $\hbox{\tt deg}\, p\geqslant m$.
\end{lemma}
\section{Linear maps without finite dimensional invariant subspaces}
Throughout this section ${\bf k}$ is a field, $X$ is a non-trivial
linear space over ${\bf k}$ and $T:X\to X$ is a ${\bf k}$-linear map. We
also denote ${\mathcal P}^*={\mathcal P}\setminus\{0\}$.
\begin{lemma} \label{inj} Let $T$ be a linear operator on a linear space
$X$. Then $T$ has no non-trivial finite dimensional invariant
subspaces if and only if $p(T)$ is injective for any non-zero
polynomial $p$.
\end{lemma}
\begin{proof} If $p$ is a non-zero polynomial and $p(T)$ is
non-injective, then there is non-zero $x\in X$ such that $p(T)x=0$.
Let $k=\hbox{\tt deg}\, p$. It is straightforward to verify that
$E=\hbox{\tt span}\,\{x,Tx,\dots,T^{k-1}x\}$ is a non-trivial finite
dimensional invariant subspace for $T$. Assume now that $T$ has a
non-trivial finite dimensional invariant subspace $L$ and $p$ is the
characteristic polynomial of the restriction of $T$ to $L$. By the
Hamilton--Cayley theorem, $p(T)$ vanishes on $L$. Hence $p(T)$ is
non-injective.
\end{proof}
\begin{definition}\label{Tind}
For a linear operator $T$ on a vector space $X$ we say that vectors
$x_1,\dots,x_n$ in $X$ are $T$-{\it independent} if for any
polynomials $p_1,\dots,p_n$, the equality
$p_1(T)x_1+{\dots}+p_n(T)x_n=0$ implies $p_j=0$ for $1\leqslant j\leqslant n$.
Otherwise, we say that $x_1,\dots,x_n$ are $T$-{\it dependent}. A
set $A\subset X$ is called $T$-{\it independent} if any pairwise
different vectors $x_1,\dots,x_n\in A$ are $T$-{\it independent}.
\end{definition}
For a subset $A$ of a vector space $X$ and $T\in L(X)$, we denote
\begin{equation}\label{efat}
E(A,T)=\hbox{\tt span}\,\biggl(\bigcup_{n=0}^\infty T^n(A)\biggr)\ \
\text{and}\ \ F(A,T)=\bigcup_{p\in{\mathcal P}^*} p(T)^{-1}(E(A,T)).
\end{equation}
Clearly, $E(A,T)$ is the smallest subspace of $X$, containing $A$
and invariant with respect to $T$ and $F(A,T)$ consists of all $x\in
X$ for which
\begin{equation}\label{xat}
q(T)x=\sum_{a\in A} p_a(T)a
\end{equation}
for some $q\in{\mathcal P}^*$ and $p=\{p_a\}_{a\in A}\in {\mathcal P}^{(A)}$. Since
$p_a\neq 0$ for finitely many $a\in A$ only, the sum in the above
display is finite.
\begin{lemma}\label{tin} Let $T\in L(X)$ be a linear operator with
no non-trivial finite dimensional invariant subspaces and $A\subset
X$ be a $T$-independent set. Then $F(A,T)$ is a linear subspace of
$X$ invariant for $T$ and for every $x\in F(A,T)$, the rational
functions $f_{x,a}=\frac{p_a}{q}$ with $p\in {\mathcal P}^{(A)}$ and
$q\in{\mathcal P}^*$ satisfying $(\ref{xat})$ are uniquely determined by $x$
and $a\in A$. Moreover, the map
\begin{equation}\label{J}
J:F(A,T)\to {\mathcal R}^{(A)},\quad J_x=\{f_{x,a}\}_{a\in A}
\end{equation}
is linear, injective and satisfies $JTx=M^{(A)}Jx$ for any $x\in
F(A,T)$. In particular, the restriction $T_A=T\bigr|_{F(A,T)}\in
L(F(A,T))$ is similar to the restriction of $M^{(A)}$ to the
invariant subspace $J(F(A,T))$.
\end{lemma}
\begin{proof} First, we show that the rational functions
$f_{x,a}=\frac{p_a}q$ for $a\in A$ are uniquely determined by $x\in
F(A,T)$. Assume that $q_1,q_2\in {\mathcal P}^*$ and $\{p_{1,a}\}_{a\in
A},\{p_{2,a}\}_{a\in A}\in {\mathcal P}^{(A)}$ are such that
$$
q_1(T)x=\sum_{a\in A} p_{1,a}(T)a\quad\text{and}\quad
q_2(T)x=\sum_{a\in A} p_{2,a}(T)a.
$$
Applying $q_2(T)$ to the first equality and $q_1(T)$ to the second,
we get
$$
(q_1q_2)(T)x=\sum_{a\in A} (q_2p_{1,a})(T)a=\sum_{a\in A}
(q_1p_{2,a})(T)a.
$$
Since $A$ is $T$-independent, $q_2p_{1,a}=q_1p_{2,a}$ for each $a\in
A$. That is, $\frac{p_{1,a}}{q_1}=\frac{p_{2,a}}{q_2}$. Thus the
rational functions $f_{x,a}=\frac{p_a}q$ for $a\in A$ are uniquely
determined by $x$. It is also clear that the set $\{a\in
A:f_{x,a}\neq 0\}$ is finite for each $x\in X$. Thus the formula
(\ref{J}) defines a map $J:F(A,T)\to {\mathcal R}^{(A)}$.
Our next step is to show that $F(A,T)$ is a linear subspace of $X$
and that the map $J$ is linear. Let $x,y\in F(A,T)$ and $t,s\in{\bf k}$.
Pick $q_1,q_2\in {\mathcal P}^*$ and $\{p_{1,a}\}_{a\in A},\{p_{2,a}\}_{a\in
A}\in {\mathcal P}^{(A)}$ such that
$$
q_1(T)x=\sum_{a\in A} p_{1,a}(T)a\quad\text{and}\quad
q_2(T)y=\sum_{a\in A} p_{2,a}(T)a.
$$
Hence
$$
(q_1q_2)(T)(tx+sy)=\sum\limits_{a\in B}
(tp_{1,a}q_2+sp_{2,a}q_1)(T)a.
$$
It follows that $tx+sy\in F(A,T)$ and therefore $F(A,T)$ is a linear
subspace of $X$. Moreover, by definition of the rational functions
$f_{x,a}$, we have $f_{x,a}=\frac{p_{1,a}}{q_1}$,
$f_{y,a}=\frac{p_{2,a}}{q_2}$ and
$$
f_{tx+sy,a}=\frac{tp_{1,a}q_2+sp_{2,a}q_1}{q_1q_2}=tf_{x,a}+sf_{y,a}\quad
\text{for any $a\in A$,}
$$
which proves linearity of $J$. Since $Jx=0$ if and only if $q(T)x=0$
for some $q\in{\mathcal P}^*$, Lemma~\ref{inj} implies that $\hbox{\tt ker}\, J=\{0\}$.
That is, $J$ is injective.
Now let us show that $F(A,T)$ is invariant for $T$ and that
$JTx=M^{(A)}Jx$ for any $x\in F(A,T)$. Let $x\in F(A,T)$ and $q\in
{\mathcal P}^*$ and $\{p_{a}\}_{a\in A}\in {\mathcal P}^{(A)}$ be such that
$$
q(T)x=\sum_{a\in A} p_a(T)a.\ \ \ \text{Then}\ \ \
q(T)(Tx)=\sum\limits_{a\in B}p_{1,a}(T)a,\ \ \text{where
$p_{1,a}(z)=zp_a(z)$.}
$$
Hence $Tx\in F(A,T)$ and therefore $F(A,T)$ is invariant for $T$.
Moreover, $f_{Tx,a}=\frac{p_{1,a}}q=Mf_{x,a}$ for any $x\in F(A,T)$
and $a\in A$. That is, $JTx=M^{(A)}Jx$ for any $x\in F(A,T)$. Since
$J$ is injective it is a linear isomorphism of $F(A,T)$ and
$Y=J(F(A,T))$. Then the equality $JTx=M^{(A)}Jx$ for $x\in F(A,T)$
implies that $Y$ is invariant for $M^{(A)}$ and that $T_A$ is
similar to $M^{(A)}\bigr|_{Y}$ with the linear map $J$ providing the
similarity.
\end{proof}
\subsection{Proof of Theorem~\ref{inva}}
Let $T\in L(X)$ be without non-trivial finite dimensional invariant
subspaces. A standard application of the Zorn lemma allows us to
take a maximal by inclusion $T$-independent subset $A$ of $X$. From
the definition of the spaces $F(B,T)$ it follows that if $B\subset
X$ is $T$-independent and $x\in X\setminus F(B,T)$, then
$B\cup\{x\}$ is also $T$ independent. Thus maximality of $A$ implies
that $X=F(A,T)$. By Lemma~\ref{tin}, $T=T_A$ is similar to a
restriction of $M^{(A)}$ to an invariant subspace.
\subsection{Proof of Lemma~\ref{empty1}}
Let $A\subset L$ be a linear basis of $L$. Since $L$ is finite
dimensional, $A$ is finite. Pick a maximal by inclusion
$T$-independent subset $B$ of $A$ (since $A$ is finite, we do not
need the Zorn lemma to do that). Now let $F=F(B,T)$ be the subspace
of $X$ defined in (\ref{efat}). Since $B$ is a maximal
$T$-independent subset of a basis of $L$, $L\subseteq F$. By
Lemma~\ref{tin}, $F$ is invariant for $T$. Thus we can without loss
of generality assume that $X=F$. Then by Lemma~\ref{tin}, we can
assume that $T$ is a restriction of $M^{(B)}$ to an invariant
subspace. Since extending $T$ beyond $X$ is not going to change the
spaces $L\cap p(T)(L)$, we can assume that $T=M^{(B)}$. Since $B$ is
finite, without loss of generality, $X={\mathcal R}^n$ and
$T=M\oplus{\dots}\oplus M$, where $n\in{\mathbb N}$.
Consider the degree function $\hbox{\tt deg}\,:{\mathcal R}\to{\mathbb Z}\cup\{-\infty\}$. We set
$\hbox{\tt deg}\,(0)=-\infty$ and let $\hbox{\tt deg}\,(p/q)=\hbox{\tt deg}\, p-\hbox{\tt deg}\, q$, where $p$ and
$q$ are non-zero polynomials and the degrees in the right hand side
are the conventional degrees of polynomials. Clearly this function
is well-defined and is a grading on ${\mathcal R}$. That is,
\begin{itemize}\itemsep=-2pt
\item[(g1)]$\hbox{\tt deg}\,(f_1f_2)=\hbox{\tt deg}\,(f_1)\!+\!\hbox{\tt deg}\,(f_2)$ and $\hbox{\tt deg}\,(f_1\!+\!f_2)\leqslant \max\{\hbox{\tt deg}\, f_1,\hbox{\tt deg}\,
f_2\}$ for any $f_1,f_2\in{\mathcal R}$;
\item[(g2)]if $f_1,f_2\in{\mathcal R}$ and $\hbox{\tt deg}\, f_1\neq \hbox{\tt deg}\, f_2$, then
$\hbox{\tt deg}\,(f_1+f_2)=\max\{\hbox{\tt deg}\, f_1,\hbox{\tt deg}\, f_2\}$.
\end{itemize}
By (g1), $\hbox{\tt deg}\,(Mf)=1+\hbox{\tt deg}\, f$ for each $f\in{\mathcal R}$. For $f\in X={\mathcal R}^n$,
we write
$$
\delta(f)=\max_{1\leqslant j\leqslant n}\hbox{\tt deg}\, f_j.
$$
Clearly $\delta(0)=-\infty$ and $\delta(f)\in{\mathbb Z}$ for each $f\in
X\setminus \{0\}$. Let also
$$
{\mathbb D}elta^+=\sup_{f\in L} \delta(f)\ \ \text{and}\ \
{\mathbb D}elta^-=\inf_{f\in L\setminus\{0\}} \delta(f).
$$
Then ${\mathbb D}elta^+$ and ${\mathbb D}elta^-$ are finite. Indeed, assume that
either ${\mathbb D}elta^+=+\infty$ or ${\mathbb D}elta^-=-\infty$. Then there exists a
sequence $\{u_l\}_{l\in{\mathbb N}}$ in $L\setminus\{0\}$ such that
$\{\delta(u_l)\}_{l\in{\mathbb N}}$ is strictly monotonic. For each $l$ we
can pick $j(l)\in\{1,\dots,n\}$ such that $\delta(u_l)=\hbox{\tt deg}\,
(u_l)_{j(l)}$. Then there is $\nu\in \{1,\dots,n\}$ for which the
set $B_\nu=\{l\in{\mathbb N}:j(l)=\nu\}$ is infinite. It follows that the
degrees of $(u_l)_\nu$ for $l\in B_\nu$ are pairwise different.
Property (g2) of the degree function implies that the rational
functions $(u_l)_\nu$ for $l\in B_\nu$ are linearly independent.
Hence the infinite set $\{u_l:l\in B_\nu\}$ is linearly independent
in $X$, which is impossible since all $u_l$ belong to the finite
dimensional space $L$. Thus ${\mathbb D}elta^+$ and ${\mathbb D}elta^-$ are finite.
Now let $p\in{\mathcal P}^*$ and $d=\hbox{\tt deg}\, p$. By (g1) and the equality
$(Tf)_j=Mf_j$, $\delta(p(T)f)=\delta(f)+d$ for each $f\in X$.
Therefore, $\inf\bigl\{\delta(f):f\in
p(T)(L)\setminus\{0\}\bigr\}={\mathbb D}elta^-+d$. In particular, if
$d>{\mathbb D}elta^+-{\mathbb D}elta^-$, then
$$
\inf_{f\in
p(T)(L)\setminus\{0\}}\delta(f)={\mathbb D}elta^-+d>{\mathbb D}elta^+=\sup_{f\in L}
\delta(f).
$$
Thus $\delta(u)>\delta(v)$ for any non-zero $u\in P(T)(L)$ and $v\in
L$, which implies that $p(T)(L)\cap L=\{0\}$ whenever $\hbox{\tt deg}\,
p>{\mathbb D}elta^+-{\mathbb D}elta^-$. Thus the number $m={\mathbb D}elta^+-{\mathbb D}elta^-+1$
satisfies the desired condition. The proof of Lemma~\ref{empty1} is
complete.
\section{Proof of Theorem~\ref{omegA}}
The implications
$(\ref{omegA}.4)\Longrightarrow(\ref{omegA}.3)\Longrightarrow(\ref{omegA}.2)$
are trivial. Assume that $T$ is transitive and $T'$ has a
non-trivial finite dimensional invariant subspace. Then $T$ has a
non-trivial closed invariant subspace of finite codimension. Passing
to the quotient by this subspace, we obtain a transitive operator on
a finite dimensional topological vector space. Since there is only
one Hausdorff vector space topology on a finite dimensional space,
we arrive to a transitive operator on a finite dimensional Banach
space. Since transitivity and hypercyclicity for operators on
separable Banach spaces are equivalent, we obtain a hypercyclic
operator on a finite dimensional Banach space. On the other hand, it
is well known that such operators do not exist, see, for instance,
\cite{ww}. Thus (\ref{omegA}.2) implies (\ref{omegA}.1). It remains
to show that (\ref{omegA}.1) implies (\ref{omegA}.4).
Assume that (\ref{omegA}.1) is satisfied and (\ref{omegA}.4) fails.
Then there exist non-empty open subsets $U$ and $V$ of $X$ and a
sequence $\{p_l\}_{l\in{\mathbb N}}$ of polynomials such that $\hbox{\tt deg}\,
p_l\to\infty$ and $p_l(T)(U)\cap V=\varnothing$ for each $l\in{\mathbb N}$.
Since $X$ carries weak topology, there exist two finite linearly
independent sets $\{f_1,\dots,f_n\}$ and $\{g_1,\dots,g_m\}$ in $X'$
and two vectors $(a_1,\dots,a_n)\in{\mathbb K}^n$ and
$(b_1,\dots,b_m)\in{\mathbb K}^m$ such that $U_0\subseteq U$ and
$V_0\subseteq V$, where
\begin{equation*}
U_0=\{u\in X:f_k(u)=a_k\ \ \text{for}\ \ 1\leqslant k\leqslant n\}\ \
\text{and}\ \ V_0=\{u\in X:g_j(u)=b_j\ \ \text{for}\ \ 1\leqslant j\leqslant
m\}.
\end{equation*}
Let $L=\hbox{\tt span}\,\{f_1,\dots,f_n,g_1,\dots,g_m\}$. Since $T'$ has no
non-trivial finite dimensional invariant subspaces, by
Lemma~\ref{empty1}, $p_l(T')(L)\cap L=\{0\}$ for any sufficiently
large $l$. For such an $l$, the equality $p_l(T')(L)\cap L=\{0\}$
together with the injectivity of $p_l(T')$, provided by
Lemma~\ref{inj}, and the definition of $L$ imply that the vectors
$p_l(T')g_1,\dots,p_l(T')g_m,f_1,\dots,f_n$ are linearly
independent. Hence there exists $u\in X$ such that
$$
\text{$p_l(T')g_j(u)=b_j$ \ for \ $1\leqslant j\leqslant m$\ \ and\ \
$f_k(u)=a_k$ \ for $1\leqslant k\leqslant n$.}
$$
Since $p_l(T')g_j(u)=g_j(p_l(T)u)$, the last display implies that
$u\in U_0\subseteq U$ and $p_l(T)u\in V_0\subseteq V$. Hence
$p_l(T)(U)\cap V$ contains $p_l(T)u$ and therefore is non-empty.
This contradiction completes the proof of Theorem~\ref{omegA}.
\begin{remark}\label{last} The only place in the proof of
Theorem~\ref{omegA}, where we used the nature of the underlying
field, is the reference to the absence of transitive operators on
non-trivial finite dimensional spaces. Thus Theorem~\ref{omegA}
extends to topological vector spaces with weak topology over any
topological field ${\bf k}$ provided there are no transitive operators
on non-trivial finite dimensional topological vector spaces over
${\bf k}$.
\end{remark}
The author would like to thank the referee for helpful comments.
\small\rm
\vskip1truecm
\scshape
\noindent Stanislav Shkarin
\noindent Queens's University Belfast
\noindent Department of Pure Mathematics
\noindent University road, Belfast, BT7 1NN, UK
\noindent E-mail address: \qquad {\tt [email protected]}
\end{document} | math | 21,878 |
\begin{example}in{equation}gin{document}
\begin{example}in{equation}gin{abstract}
We describe recent results with A. Pogan developing dynamical systems tools for a class of
degenerate evolution equations arising in kinetic theory, including
the steady Boltzmann and BGK equations.
These yield information on structure of large- and small-amplitude kinetic shocks,
the first steps in a larger program toward time-evolutionary stability and asymptotic behavior.
\varepsilonnd{abstract}
\title{Invariant manifolds for a class of degenerate evolution equations and structure of kinetic shock layers
}
\wti{\alpha}bleofcontents
\section{Introduction}
In these notes, we describe recent results \cite{PZ1,PZ2} with Alin Pogan developing a set of dynamical
systems tools suitable for the study of existence and structure of shock and boundary layer solutions
arising in Boltzmann's equation and related kinetic models. These represent the first steps in a larger program to develop
dynamical systems methods like those used in the study of finite-dimensional viscous and relaxation shocks in
\cite{GZ,MasZ2,Z2,Z3,Z4,Z5,ZH,ZS}, suitable for treatment of one- and multi-dimensional stability of large-amplitude kinetic shock and boundary layers.
\subsection{Equations and assumptions}
Our goal is the study of shock or boundary layer solutions
\begin{example}in{equation}\lambdaabel{wave}
\mathbf{u}(x,t)= \check {\mathbf{u}}(x), \qquad \lambdaim_{x\tauo\pm\infty} \check {\mathbf{u}}(x)=\mathbf{u}^\pm,
\varepsilonnd{equation}
of kinetic-type relaxation systems
\begin{example}in{equation}gin{equation}\lambdaabel{Relax}
A^0 \mathbf{u}_t + A\mathbf{u}_x = Q(\mathbf{u})
\varepsilonnd{equation}
on a Hilbert space $\bH$, where $A^0$ and $A$ are constant bounded
linear operators, and $Q$, the {\it collision operator}, is a bounded bilinear map.
This leads us to the study of the associated {\it steady equation}
\begin{example}in{equation}\lambdaabel{steady}
A\mathbf{u}'=Q(\mathbf{u}).
\varepsilonnd{equation}
Following \cite{MZ,PZ1,PZ2}, we make the following structural assumptions.
{\varepsilonnsuremath{{\scriptscriptstyle -}}}allskip
\notagindent{\bf Hypothesis (H1)} (i) The linear operator $A$ is
bounded, self-adjoint, and one-to-one
on the Hilbert space $\bH$, but {\it not boundedly invertible}. (ii) There exists $\bV$ a proper, closed subspace of $\bH$ with $\dim\bV^\perp<\infty$ and $B:\bH\tauildemes\bH\tauo\bV$ is a bilinear, symmetric, continuous map such that $Q(\mathbf{u})=B(\mathbf{u},\mathbf{u})$.
{\varepsilonnsuremath{{\scriptscriptstyle -}}}allskip
\notagindent{\bf Hypothesis (H2)} There exist an equilibrium $\overline{\mathbf{u}}\in\ker Q$ satisfying
\begin{example}in{equation}gin{enumerate}
\item[(i)] $Q'(\overline{\mathbf{u}})$ is self-adjoint and $\ker Q'(\overline{\mathbf{u}})=\bV^\perp$;
\item[(ii)] There exists $\delta>0$ such that $Q'(\overline{\mathbf{u}})_{|\bV}\lambdaeq -\delta I_{\bV}$;
\varepsilonnd{enumerate}
The class of system so described includes in particular our
main example, of {\it Boltzmann's equation} with hard-sphere potential, written in appropriate coordinates
\cite{MZ}; see Section \ref{s:reduction}.
As regards \varepsilonqref{steady}, the main novelty is that $A$ by (H1)(i) has an {\it essential singularity}, i.e., essential
spectrum at the origin,
hence \varepsilonqref{steady} is a {\it degenerate evolution equation} to which invariant manifold results
of standard dynamical systems theory do not immediately apply.
Our purpose here is precisely the construction of invariant manifolds for the class of degenerate equations
\varepsilonqref{steady} satisfying (H1)-(H2),
and the application of these tools toward existence and structure of kinetic shock and boundary layers.
\br\lambdaabel{kawrmk}
We do not assume as in \cite{MZ} the ``genuine coupling'' or ``Kawashima'' condition that
no eigenvector of $A$ lie in the kernel of $Q'(\overline{\mathbf{u}})$.
The assumption $A$ one-to-one implies (trivially) the weaker condition, sufficient for our analysis,
that no zero eigenvector of $A$ lie in the kernel of $Q'(\overline{\mathbf{u}})$.
\varepsilonr
\subsection{Chapman-Enskog expansion and canonical form}\lambdaabel{s:ce}
Our starting point is the formal {\it Chapman-Enskog} expansion designed to approximate near-equilibrium flow \cite{L}.
Near $\overline{\mathbf{u}}$, (H1)-(H2) yields by the Implicit Function Theorem
existence of a (Fr\'echet) $C^\infty$ manifold of equlibria
\begin{example}in{equation} \lambdaabel{eq}
\cE= \ker Q, \qquad \dim \cE=\dim \bV^\perp=:r,
\varepsilonnd{equation}
tangent to $\bV^\perp$ at $\overline{\mathbf{u}}$, expressible in coordinates $\mathbf{w}:=\mathbf{u}-\overline{\mathbf{u}}$ as a $C^\infty$
graph
\begin{example}in{equation}\lambdaabel{v*}
v_*:\bV^\perp \tauo\bV.
\varepsilonnd{equation}
Denote $u=P_{\bV^\perp}\mathbf{u}$, $v=P_{\bV}\mathbf{u}$, where $P_{\bV^\perp}$ and $P_{\bV}$ are the orthogonal
projections onto $\bV^\perp$ and $\bV$ associated with the decomposition $\bH=\bV^\perp \oplus \bV$.
The second-order Chapman-Enskog approximation, or ``hydrodynamic limit,'' of \varepsilonqref{Relax} is then
$h_*(u)_t + f_*(u)_x=D_*u_{xx}$, with associated steady equation
\begin{example}in{equation}\wti{\alpha}g{${\rm CE}$}\lambdaabel{ce2}
f_*(u)_x=D_*u_{xx},
\varepsilonnd{equation}
where $ h_*(u):= P_{\bV^\perp} A^0 (u^T, v_*(u)^T)^T$ and
\begin{example}in{equation}\lambdaabel{D}
f_*(u):= P_{\bV^\perp} A (u^T, v_*(u)^T)^T,
\quad D_*:=A_{12} E^{-1} A_{12}^T,
\varepsilonnd{equation}
with $A_{12}:=P_{\bV^\perp}A P_{\bV}$ and $E:= Q'(\overline{\mathbf{u}})_{|\bV}$. See \cite{L,MZ,PZ2} for further details.
From (H1)(ii), $P_{\bV^\perp} (A \mathbf{u})'= P_{\bV^\perp}Q\varepsilonquiv 0$. Integrating, we find that
\varepsilonqref{steady} admits a {\it conservation law}
\begin{example}in{equation}\lambdaabel{cons}
P_{\bV^\perp} A \mathbf{u}\varepsilonquiv q=\tauext{\rm constant}.
\varepsilonnd{equation}
By the definition of $f_*$, $v_*$, equilibria $\mathbf{u}_\pm= (u^T, v_*(u)^T)^T_\pm$ satisfy the
{\it Rankine-Hugoniot} condition
\begin{example}in{equation}\wti{\alpha}g{RH}\lambdaabel{rh}
f_*(u_+)=f_*(u_-)=q
\varepsilonnd{equation}
associated with viscous shock profiles of the Chapman-Enskog system \varepsilonqref{ce2}, giving a rigorous connection
at the inviscid level between shock or boundary layer profiles of the two systems \varepsilonqref{Relax} and \varepsilonqref{ce2}.
A further connection, between the types of the equilibria $\overline{\mathbf{u}}=(\bar u^T, v_*(\bar u)^T)^T$
and $\bar u$ with respect to their associated flows, is given by the following
key observation proved in Section \ref{s:reduction}.
\begin{example}in{equation}gin{lemma}\lambdaabel{l:canon} System \varepsilonqref{steady} may, by an invertible change of coordinates, be put
in canonical form
\ba\lambdaabel{canon}
w_c'&=Jw_c + \tauildelde Q_c(w_c,w_h)\\
\,\mbox{\bf G}amma_0 w_h' &= -w_h + \tauildelde Q_h(w_c,w_h),
\varepsilona
$w_c$ and $w_h$ parametrizing center and hyperbolic (i.e., stable/unstable) subspaces, $\dim w_c=m+r$,
$m=\dim\ker f_*'(\bar u)$, $r=\dim \bV^\perp$,
where $J=\bp 0 & I_m & 0 \\ 0&0 & 0\\ 0&0&0 \varepsilonp$ is a nilpotent block-Jordan form,
$\,\mbox{\bf G}amma_0$ is a constant, bounded symmetric operator, and $\tauildelde Q_j(w_c,w_h)=O(|w_c,w_h|^2)$.
In case $m=0$, $J, \tauildelde Q_c\varepsilonquiv 0$.
\varepsilonnd{lemma}
One may compute that the perturbation equations for \varepsilonqref{ce2} about $\bar u$ have the same canonical
form (noting $f_*'(\bar u)= P_{\bV^\perp} AP_{\bV^\perp}$, $D_*$ symmetric) with
$\,\mbox{\bf G}amma_0$ finite-dimensional, invertible \cite{MaP,Pe}.
\subsection{Dichotomies vs. direct $L^p$ estimate} \lambdaabel{s:dichotomy}
Lemma \ref{l:canon} effectively reduces the study of near-equilibrium flow of \varepsilonqref{steady}
to understanding the hyperbolic operator $(\,\mbox{\bf G}amma_0{\partial}rtial_x + {\rm Id })$,
specifically, obtaining bounds on solutions of the degenerate inhomogeneous linear evolution system
\begin{example}in{equation}\lambdaabel{inhom}
(\,\mbox{\bf G}amma_0{\partial}rtial_x + {\rm Id }) w_c=g,
\varepsilonnd{equation}
where $\,\mbox{\bf G}amma_0$ is bounded, symmetric, and one-to-one, but (by (H1)) {\it not boundedly invertible:}
formally,
\begin{example}in{equation}\lambdaabel{formal}
({\partial}rtial_x + \,\mbox{\bf G}amma_0^{-1}) w_c= \tauildelde g,
\varepsilonnd{equation}
where $\,\mbox{\bf G}amma_0^{-1}$ is an unbounded self-adjoint operator and $\tauildelde g:= \,\mbox{\bf G}amma_0^{-1} g$.
As $\,\mbox{\bf G}amma_0$ is indefinite, \varepsilonqref{formal} is {\it ill-posed} with respect to the Cauchy problem,
featuring unbounded growth in both directions.
Ill-posed equations, and the derivation of associated resolvent bounds, have been treated in a variety
of contexts via {\it generalized exponential dichotomies}:
for example, modulated waves on cylindrical domains
\cite{PSS,SS1,SS2}, Morse theory \cite{AM1,AM2,RobSal}, PDE Hamiltonian systems
\cite{BjornSand}, and the functional-differential equations of mixed type \cite{Mallet-Paret}.
It is not difficult to see, either by spectral decomposition of $\,\mbox{\bf G}amma_0$, or by Galerkin approximation,
that $({\partial}rtial_x + \,\mbox{\bf G}amma_0^{-1})$ generates a {\it stable bi-semigroup} \cite{BGK,LP2},
the infinite-dimensional analog of an exponential dichotomy,
that is, there exist bounded projections on whose range the homogeneous flow
is exponentially decaying in forward/backward direction, in this case with rate $|\,\mbox{\bf G}amma_0|_{\bH}^{-1}$,
where $|\cdot|_{\bH}$ denotes operator norm; see \cite{PZ1} for details.
This, however, yields only $ \|u\|\lambdaeq C\|\tauildelde g\|=\|\,\mbox{\bf G}amma_0^{-1}g\|, $
the intervention of the unbounded operator $\,\mbox{\bf G}amma_0^{-1}$ making these bounds useless for our analysis.
Thus, the present problem differs from the above-mentioned ones in that
{\it exponential dichotomies are inadequate to bound the resolvent}
$(\,\mbox{\bf G}amma_0 {\partial}rtial_x+{\rm Id })^{-1}$.
Indeed, we have the following striking result obtained by direct estimate in Section \ref{s:linear},
showing that our situation is one of {\it maximal regularity}.
In this sense, our analysis is related in flavor to construction of center manifolds for quasilinear systems;
see \cite{HI,Mi}, and references therein.
\begin{example}in{equation}gin{lemma}\lambdaabel{l:linbd}
Assuming (H1)-(H2), $|(\,\mbox{\bf G}amma_0 {\partial}rtial_x + {\rm Id })^{-1}|_{L^p(\R)}<\infty$ for $1<p<\infty$,
but \varepsilonmph{not} for $p=1,\infty$.
\varepsilonnd{lemma}
An important consequence is that usual weighted $L^\infty$ constructions of invariant manifolds are unavailable.
We work instead in {\it weighted $H^1$ spaces},
with accompanying new technical issues.
\subsection{Results}
We are now ready to state our main results.
Assuming (H1)-(H2), from \varepsilonqref{canon} and symmetry of $\,\mbox{\bf G}amma_0$ we readily obtain a decomposition
$\bH=\bH_{\mathrm{c}}\oplus \bH_{\mathrm{c}}\oplus \bH_{\mathrm{u}}$ of $\bH$ into stable, center, and unstable subpaces invariant
under the homogeneous linearized flow of \varepsilonqref{steady} about the equilibrium $\overline{\mathbf{u}}$.
Let $H^1_\varepsilonnd{theorem}a(\R,\bH)$ denote the space of functions bounded in the exponentially weighted $H^1$ norm
\begin{example}in{equation}\lambdaabel{norm}
\|f\|_{H^1_\varepsilonnd{theorem}a(\R,\bH)}:= \|e^{\varepsilonnd{theorem}a \lambdaangle \cdot \tauext{\rm{ran}}gle}f(\cdot)\|_{L^2(\R,\bH)}
+\|e^{\varepsilonnd{theorem}a \lambdaangle \cdot \tauext{\rm{ran}}gle}f'(\cdot)\|_{L^2(\R,\bH)},
\varepsilonnd{equation}
where $\lambdaangle x\tauext{\rm{ran}}gle:=(1+|x|^2)^{1/2}$ and $\varepsilonnd{theorem}a\in \R$ may be positive or negative according to our needs.
Following \cite{LP2}, we define solutions of \varepsilonqref{steady} using Lemma \ref{l:linbd} as
$H^1_{loc}$ solutions of the fixed-point equation $w_h=(\,\mbox{\bf G}amma_0{\partial}rtial_x+{\rm Id })^{-1}g_c(w)$ and the finite-dimensional ODE
$({\partial}rtial_x-J)w_c=g_c(w)$ in $w_c$; see \cite{PZ1,PZ2}.
\subsubsection{$H^1$ stable manifold and exponential decay of large-amplitude shock and boundary layers}\lambdaabel{s:invariant}
Our first observation is that for singular $\,\mbox{\bf G}amma_0$ the $H^1$ stable subspace of \varepsilonqref{canon}, defined as the trace at $x=0$ of
solutions $w_h$ bounded in $H^1(\R^+,\bH)$, {\it is a dense proper subspace of $\bH_{\mathrm{s}}$,}
related to the domain of the generator $\,\mbox{\bf G}amma_0^{-1}$ of the bi-semigroup associated with
homogeneous linearized flow.
\begin{example}in{equation}gin{lemma}\lambdaabel{l:dom}
Assuming (H1)-(H2),
the $H^1$ stable subspace of the linearized equations of \varepsilonqref{steady} about $\overline{\mathbf{u}}$
(equivalently, the linearization of \varepsilonqref{canon} about $0$)
is $\tauext{\rm{dom}}(|\,\mbox{\bf G}amma_0|^{-1/2})\cap \bH_{\mathrm{s}} \subset \bH_{\mathrm{s}}$.
\varepsilonnd{lemma}
\begin{example}in{equation}gin{proof}
The $H^1$ stable subspace consists of $f\in \bH_{\mathrm{s}}$ such that
$ \int_0^\infty \lambdaangle {\partial}rtial_x e^{\,\mbox{\bf G}amma_0^{-1} x}f, {\partial}rtial_x e^{\,\mbox{\bf G}amma_0^{-1} x}f\tauext{\rm{ran}}gle dx <\infty, $
or, equivalently,
$ -(1/2)\int_0^\infty {\partial}rtial_x \lambdaangle e^{\,\mbox{\bf G}amma_0^{-1} x} |\,\mbox{\bf G}amma_0|^{-1/2}f,
e^{\,\mbox{\bf G}amma_0^{-1} x}|\,\mbox{\bf G}amma_0|^{-1/2}f\tauext{\rm{ran}}gle dx <\infty. $
Integrating, and observing that the boundary term at infinity vanishes, gives condition
$\lambdaangle |\,\mbox{\bf G}amma_0|^{-1/2}f , |\,\mbox{\bf G}amma_0|^{-1/2}f \tauext{\rm{ran}}gle <\infty. $
Alternatively, this may be deduced by spectral decomposition of $\,\mbox{\bf G}amma_0$ and direct computation \cite{PZ1}.
\varepsilonnd{proof}
We have accordingly the following modification of the usual stable manifold theorem.
\begin{example}in{equation}gin{theorem}\lambdaabel{t1.3}
Assuming (H1)-(H2), for any $0<\alpha<\tauildelde \nu <\nu:=|\,\mbox{\bf G}amma_0|_{\bH}^{-1}$,
there exists a local stable manifold $\cM_{\mathrm{s}}$ near $\overline{\mathbf{u}}$,
expressible in coordinates $w=\mathbf{u}-\overline{\mathbf{u}}$ as a $C^1$ embedding tangent to $\bH_s$
of $\tauext{\rm{dom}}(\,\mbox{\bf G}amma_0^{-1/2})\cap \bH_{\mathrm{s}}$ with (graph) norm induced by $\,\mbox{\bf G}amma_0^{-1/2}$ into $\bH$,
locally invariant under the flow of \varepsilonqref{steady}, containing the orbits of all solutions $w$ with
$ H^1_{\alpha}({\mathbb R}_+,\bH)$ norm sufficiently small,
with solutions $w$ initiating in $\cM_{\mathrm{s}}$ at $x=0$ lying in $ H^1_{\tauildelde \nu}({\mathbb R}_+,\bH)$.
In case $\det f_*'(u_+)\neq 0$, $\alpha$ may be taken to be zero.
\varepsilonnd{theorem}
We obtain as a consequence exponential decay of noncharacteristic shock or boundary layers.
\begin{example}in{equation}gin{corollary}\lambdaabel{c5.9}
Assuming (H1)-(H2), let $\overline{\mathbf{u}}$ be a noncharacteristic equilibrium
in the sense of \varepsilonqref{ce2}, $\det f_*'(\bar u)\neq 0$, and $\tauildelde \nu <\nu= 1/|\,\mbox{\bf G}amma_0|_{\bH}$.
Then, for any solution $\check{\mathbf{u}}$ of \varepsilonqref{steady} converging to $\overline{\mathbf{u}}$ as $x\tauo +\infty$ in the sense that
$\check {\mathbf{u}}- \overline{\mathbf{u}}$ is eventually bounded in $H^1([x,\infty),\bH)$, we have \varepsilonmph{exponential decay}:
\begin{example}in{equation}\lambdaabel{expdecay}
|\check u-\overline{\mathbf{u}}|_{\bH}(x)\lambdaesssim e^{-\tauildelde \nu x}
\quad \hbox{\rm as $x\tauo +\infty$.}
\varepsilonnd{equation}
\varepsilonnd{corollary}
\subsubsection{Center manifold and structure of small-amplitude shock layers}
We have, similarly, the following modification of the usual center manifold theorem (cf. \cite{B,HI,Z1,Zode}).
\begin{example}in{equation}gin{theorem}\lambdaabel{t1.1}
Let $\overline{\mathbf{u}}$ be an equilibrium satisfying (H1)-(H2). Then, for any integer $k\gammaeq2$
there exists local to $\overline{\mathbf{u}}$ a $C^k$ center manifold $\cM_{\mathrm{c}}$, tangent at $\overline{\mathbf{u}}$ to $\bH_{\mathrm{c}}$,
expressible in coordinates $\mathbf{w}:=\mathbf{u}-\overline{\mathbf{u}}$ as a $C^k$ graph $\cJ_{\mathrm{c}}:\bH_{\mathrm{c}}\tauo\bH_{\mathrm{s}}\oplus\bH_{\mathrm{u}}$,
that is locally invariant under the flow of \varepsilonqref{steady} and contains all solutions
that remain sufficiently close to $\overline{\mathbf{u}}$ in forward and backward $x$.
Moreover, $\cM_{\mathrm{c}}$ has the \varepsilonmph{$H^1$ exponential approximation property}: for any $0<\tauildelde \nu< \nu=1/|\,\mbox{\bf G}amma_0|_{\bH}$,
a solution ${\mathbf{u}}$ of \varepsilonqref{steady}
with $\|{\mathbf{u}}-\overline{\mathbf{u}}\|_{H^1_{-\alpha}\cap L^\infty([M,\infty),\bH)}$ and $\alpha>0$ sufficiently small approaches a solution
$\mathbf{z}$ with orbit lying in $\cM_{\mathrm{c}}$ as $x\tauo +\infty$ at exponential rate $\|{\mathbf{u}}-\mathbf{z}\|_{\bH}\lambdaesssim e^{-\tauildelde \nu x}$,
with also $\|{\mathbf{u}}-\mathbf{z}\|_{H^1_{\tauildelde \nu}([M,\infty),\bH)}<\infty$.
\varepsilonnd{theorem}
Here, the only difference from the standard center manifold theorem \cite{B}
is the weakened, $H^1$, version of the exponential approximation property.
For applications involving normal form reduction, they are essentially equivalent; in particular, the
formal Taylor expansion for center graph $w_h=\Xi(w_c)$ may be computed to arbitrary order
in coordinates \varepsilonqref{canon} by successively matching terms of increasing order in
the defining relation $\,\mbox{\bf G}amma_0 \Xi(w_c)' =-\Xi(w_h)+\tauildelde Q_h$, or equivalently
$ \Xi(w_c)= -\,\mbox{\bf G}amma_0 \Xi'(w_c)(Jw_c + \tauildelde Q_c)+ \tauildelde Q_h $,
exactly as in the usual (nonsingular $A$, $\,\mbox{\bf G}amma_0$) case \cite{Carr,HI}.
\br\lambdaabel{cmdecay}
In the noncharacteristic case, the center manifold, by dimensional count and the fact that it must contain all local equilibria,
is uniquely determined as the manifold of equilibria $\cE$.
In this case, the exponential approximation property improves slightly the result of Corollary \ref{c5.9},
yielding that solutions $\check {\mathbf{u}}$ of \varepsilonqref{steady} lying sufficiently close to $\overline{\mathbf{u}}$ in $L^\infty(\R^+,\bH)$
and sufficiently slowly {exponentially growing} in $H^1$, converges to an equilibrium
at exponential rate $e^{-\tauildelde \nu x}$, $0<\tauildelde \nu<1/|\,\mbox{\bf G}amma_0|_{\bH}$.
\varepsilonr
Denote the characteristics of Chapman-Enskog system \varepsilonqref{ce2}, or eigenvalues of $f_*'(u)$,
by
$$
\lambdaambda_1(u)\lambdaeq \dots \lambdaeq \lambdaambda_r(u).
$$
The {\it noncharacteristic case} $f_*'(\bar u)\neq 0$ is the case that no characteristic velocity $\lambdaambda_j(\bar u)$
vanishes, in which case,
by the Inverse Function Theorem, the Rankine-Hugoniot equations \varepsilonqref{rh} admit a single nearby solution for each value of $q$,
hence no local shock connections occur.
To study small-amplitude shock profiles, we focus therefore on the {\it characteristic case} $f_*'(\bar u)=0$, specifically
on the generic case that $\lambdaambda_j(\bar u)=0$ for a single characteristic velocity $\lambdaambda_p$,
with associated unit eigenvector $\overline{\mathbf{r}}$, that is {\it genuinely nonlinear} in the sense of Lax \cite{La,Sm}:
\begin{example}in{equation}\wti{\alpha}g{GNL}\lambdaabel{gnl}
\Lambda:=\overline{\mathbf{r}}\cdot f_*''(\bar u)(\overline{\mathbf{r}},\overline{\mathbf{r}})\neq 0.
\varepsilonnd{equation}
In this case, it is well known \cite{La,Sm,MaP} that there exists a family of small-amplitude shock profiles
$\check{\bar u}$ of \varepsilonqref{ce2} connecting
endstates $\bar u_\pm\tauo \bar u$, with $(\bar u_+-\bar u_-)$ lying in approximate direction $\overline{\mathbf{r}}$,
with $\lambdaambda:=\lambdaambda_p(\check{\bar u})$ satisfying an approximate Burgers equation
\begin{example}in{equation}\lambdaabel{burgers}
\delta \lambdaambda'=-\varepsilon^2 + \lambdaambda^2/2 + O(|\varepsilon, \lambdaambda|^3),
\varepsilonnd{equation}
$\Lambda$ as in \varepsilonqref{gnl}, $\varepsilon>0$ parametrizing amplitude,
provided there holds the {\it stable viscosity criterion}
$\delta:=\overline{\mathbf{r}} \cdot D_* \overline{\mathbf{r}}>0,$
as may be readily seen to hold for $D_*$ using \varepsilonqref{D} and (H1) (cf. Rmk. \ref{kawrmk}).
Our final result gives a corresponding characterization of small-amplitude kinetic shocks of \varepsilonqref{steady}
bifurcating from a simple genuinely nonlinear eigenvalue of $f_*'(\bar u)$. The complementary case of bifurcation from a
multiple, linearly degenerate eigenvalue of $f_*'(\bar u)$ \cite{La,Sm} is treated also in \cite[Thm. 1.5]{PZ2} (not stated here);
in that case, no nontrivial shock or boundary layer connections exist.
\begin{example}in{equation}gin{corollary}\lambdaabel{c1.3}
Let $\overline{\mathbf{u}}$ be an equilibrium satisfying (H1)-(H2)
in the characteristic case \varepsilonqref{gnl}, $\lambdaambda_p(\bar u)=0$ a simple eigenvalue, and $k$ an integer $\gammaeq 2$.
Then, local to $\overline{\mathbf{u}}$, $\bar u$, each pair of points $u_\pm$ satisfying the Rankine-Hugoniot condition \varepsilonqref{rh}
has a corresponding viscous shock solution $u_{CE}$ of \varepsilonqref{ce2} and
relaxation shock solution $\mathbf{u}_{REL}=(u_{REL},v_{REL})$ of \varepsilonqref{steady}, satisfying for all $j\lambdaeq k-2$:
\begin{example}in{equation}gin{equation}\lambdaabel{finalbds}
\begin{example}in{equation}gin{aligned}
\bibitemg|{\partial}rtial_x^j ( u_{REL}- u_{CE})\bibitemg| &\lambdae C \varepsilon^{j+2}e^{-\mu \varepsilon|x|},\\
\bibitemg|{\partial}rtial_x^j \bibitemg(v_{REL}-v_*(u_{CE})\bibitemg)\bibitemg| &\lambdae C \varepsilon^{j+2}e^{-\mu \varepsilon|x|},\\
|{\partial}rtial_x^j (u_{REL}-u_\pm)|&\lambdae C \varepsilon^{j+1}e^{-\mu \varepsilon|x|}, \quad x\gammatrless 0,\\
\varepsilonnd{aligned}
\varepsilonnd{equation}
$\mu>0$, $C>0$, $\varepsilon:=|u_+-u_-|$,
unique up to translation,
with $\lambdaambda_p(u_{REL})$ and $\lambdaambda_p(u_{CE})$ both satisfying approximate Burgers equations \varepsilonqref{burgers}:
in particular, both monotone decreasing in $x$.
\varepsilonnd{corollary}
\subsection{Discussion and open problems}\lambdaabel{s:discussion}
Corollary \ref{c1.3} recovers under slightly weakened assumptions, the result of \cite[Prop. 5.4]{MZ},
which, applied to Boltzmann's equation, in turn recovers and sharpens
the fundamental result \cite{CN} of existence of small-amplitude Boltzmann shocks with standard, square-root
Maxwellia-weighted $L^2$ norm in velocity \cite{G}.
With further effort, one may show \cite[Prop. 1.8]{PZ2} (not stated here) that the center manifold of Theorem \ref{t1.1},
hence also the small-amplitude shock profiles obtained of Corollary \ref{c1.3},
are contained in a stronger space of near Maxwellian-weighted $L^2$ norm in velocity,
recovering the strongest current existence result for Boltzmann shocks \cite[Thm. 1.1]{MZ},
plus the additional dynamical information of \varepsilonqref{burgers} and monotonicity of $\lambdaambda_p(u_{REL}(x))$-
neither available by the Sobolev-based fixed point iteration arguments of \cite{CN,MZ}.
To our knowledge, {Theorems \ref{t1.3} and \ref{t1.1} are the first results on existence of invariant manifolds} for any
system of form \ref{Relax}, (H1)-(H2) in either Hilbert or Banach space setting,
{in particular for the steady Boltzmann equation with hard sphere potential}.
Liu and Yu \cite{LiuYu} have studied existence of invariant manifolds for Boltzmann's equation
in a weighted $L^\infty$ (in both velocity and $x$) Banach space setting, using
rather different methods of time-regularization and detailed pointwise bounds,
pointing out that monotonicity of $\lambdaambda_p(\bar u)$ follows from center manifold reduction
and describing physical applications of center manifold theory
to condensation and subsonic/supersonic transition in Milne's problem.
However, their claimed linearized bounds, based on exponential dichotomies,
hence also their arguments for existence of invariant manifolds, were incorrect \cite{Z6}; see Remark \ref{LYrmk}.
Our results among other things repair this gap, validating their larger program/physical conclusions.
A longer term program is to develop further dynamical systems tools for kinetic systems \varepsilonqref{Relax} with
structure (H1)-(H2), sufficient to treat {\it time-evolutionary stability} of shock and boundary layers by the
methods used for viscous/relaxation shocks in \cite{GZ,MasZ2,Z2,Z3,Z4,Z5,ZH,ZS}.
Besides unification/simplification, this approach has the advantage of applying in principle to multi-dimensional and/or
large-amplitude waves, each of these long-standing open problems in the area.
These techniques have the further advantages of separating the issues of existence, spectral stability,
and linearized/nonlinear stability, with the first two often treated by a combination of analytical and numerical methods,
up to and including (see, e.g., \cite{Ba,BZ}) interval arithmetic-based rigorous numerical proof.
The development of numerical and or analytical methods for the treatment of existence of large-amplitude
kinetic shocks we regard as a further, very interesting open problem.
Indeed, the {\it structure problem} discussed by Truesdell, Ruggeri, Boillat, and others, of existence and description of large-amplitude Boltzmann shocks, is perhaps {\it the} fundamental open problems in the theory, and
one of the main motivations for their study.
As discussed, e.g., in \cite{BR}, Navier-Stokes theory
well-describes behavior of shocks of Mach number $M\lambdaessapprox 2$, but inaccurately predicts shock width/structure
at large Mach numbers; by contrast, Boltzmann's equation (numerically and via various formal approximations) appears to
match experiment in the large-$M$ regime.
\section{Reductions and main example}\lambdaabel{s:reduction}
We begin by carrying out various reductions, first from Boltzmann's equation to the abstract form \varepsilonqref{steady}, (H1)-(H2),
then the abstract equation to the canonical form \varepsilonqref{canon}.
\subsection{Boltzmann's equation}\lambdaabel{s:boltz}
(Following \cite{MZ})
Our main interest is Boltzmann's equation with hard-sphere potential (or Grad hard cutoff potential as in \cite{CN}):
\begin{example}in{equation}\lambdaabel{Boltz}
f_t + \xi_1 {\partial}rtial_x f= \cQ(f,f),
\varepsilonnd{equation}
where $f(x,t,\xi)\in \R$ is the distribution of velocities $\xi\in \R^3$ at $x$, $t\in \R$, and
\begin{example}in{equation}gin{equation}
\lambdaabel{colop}
\cQ (g, h) := \int \bibitemg( g( \xi') h (\xi'_*) - g(\xi) h(\xi_*) \bibitemg) C(\Omega, \xi - \xi_*) d \Omega d\xi_*
\varepsilonnd{equation}
is the collision operator, with collision kernel $C (\Omega, \xi) = \bibitemg| \Omega \cdot \xi \bibitemg|$ for hard-sphere case.
The space of \varepsilonmph{collision invariants} $\lambdaangle \psi \tauext{\rm{ran}}gle $, $ \int_{\R^3}\psi(\xi)\cQ(g, g)( \xi)d\xi \varepsilonquiv 0 $,
of \varepsilonqref{Boltz} is spanned by
\begin{example}in{equation}gin{equation}
\lambdaabel{defR}
Rf: = \int \Psi (\xi ) f( \xi ) d\xi \in \R^5 , \qquad
\Psi (\xi ) = (1 ,\xi_1 , \xi_2 , \xi_3 , \varepsilonnsuremath{\mathrm{e}}z | \xi|^2 )^T.
\varepsilonnd{equation}
(Here, we are assuming that distributions $f(x,t,\cdot)$ are confined to a space $\bH$
to be specified later such that the integral converges.)
The associated {macroscopic (fluid-dynamical) variables} are
\begin{example}in{equation}gin{equation}\lambdaabel{fvar}
{\mathrm{u}} := Rf=: (\rho , \rho v_1 , \rho v_2 , \rho v_3 , \rho E )^T,
\varepsilonnd{equation}
where $\rho$ denotes density, $v=(v_1,v_2,v_3)$ velocity,
$E= e + \varepsilonnsuremath{\mathrm{e}}z | v|^2 $ total energy density, and $e$ internal energy density.
The set of \varepsilonmph{equilibria} ($\ker \cQ$) consists of the \varepsilonmph{Maxwellian distributions}:
\begin{example}in{equation}gin{equation}\lambdaabel{max}
M_{\mathrm{u}}(\xi) = \fracrac{\rho}{\sqrt{(4\pi e/3)^3}}
e^{- \fracrac{| \xi - v |^2}{4 e/3} } .
\varepsilonnd{equation}
\subsubsection{Symmetry, boundeness, and spectral gap}\lambdaabel{s:sym}
{Boltzmann's $H$-theorem} \cite{G,Gl,Ce} (equivalent to existence of a thermodynamical entropy in the sense of
\cite{CLL}) asserts the variational principle
$$
\int \lambdaog f \cQ(f,f) d\xi \lambdae 0 ,
$$
with equality on the set of Maxwellians $\underline M$.
Taylor expanding about a local maximum $\underline M$, we obtain symmetry and nonnegativity of the Hessian
$ \int \underline M^{-1} ( {\partial}rtial \cQ|_{\underline M} h) h d\xi\lambdae 0. $
giving symmetry and nonnegativity of ${\partial}rtial \cQ|_{\underline M} $ on the space $\bH$ defined by
the square root Maxwellian-weighted norm
\begin{example}in{equation}\lambdaabel{mnorm}
\|f\|_{\bH}:=\|f \underline M^{-1/2}\|_{L^2(\R^3)}.
\varepsilonnd{equation}
Making the coordinate change
\begin{example}in{equation}\lambdaabel{change}
\mathbf{u}= \lambdaangle \xi \tauext{\rm{ran}}gle^{1/2}f, \quad Q(\mathbf{u}): = \lambdaangle \xi^{-1/2} \tauext{\rm{ran}}gle^{-1}\cQ(\lambdaangle \xi\tauext{\rm{ran}}gle^{-1/2}\mathbf{u}),
\qquad \lambdaangle \xi\tauext{\rm{ran}}gle:=\sqrt{1+|\xi|^2},
\varepsilonnd{equation}
and definining multiplication operators $A^0=\lambdaangle \xi\tauext{\rm{ran}}gle^{-1}$ and $A=\xi_1/\lambdaangle \xi\tauext{\rm{ran}}gle$,
we find that \varepsilonqref{Boltz} may be put in form \varepsilonqref{Relax}, for $\mathbf{u}\in \bH$, with $A^0$, $A$ evidently
symmetric and bounded, $A^0>0$, and $Q'(\overline{\mathbf{u}})$ symmetric nonpositive
at any equilibrium $\overline{\mathbf{u}}= \lambdaangle \xi\tauext{\rm{ran}}gle^{1/2}\underline M$.
By \cite[Cor. 2.4]{MZ}, $Q$ is bounded as a bilinear map on $\bH$.
Moreover, by \cite[Prop. 3.5]{MZ}, $Q'(\overline{\mathbf{u}})$ is negative definite with respect to $\bH$ on its range, this
last being a straightforward consequence of Carleman's theorem \cite{C} that
${\partial}rtial \cQ|_{\underline M} $ acting on $\bH$ may be decomposed as the sum of a multiplication operator
$\nu(\xi)\sim \lambdaangle -\xi\tauext{\rm{ran}}gle$ and a compact operator $K$, whence $Q'(\overline{\mathbf{u}})$ is the sum of a multiplication
operator $\tauildelde \nu(\xi)\sim -1$ and the compact operator
$\tauildelde K=\lambdaangle \xi\tauext{\rm{ran}}gle^{-1/2} K\lambdaangle \xi\tauext{\rm{ran}}gle^{-1/2}$, Weyl's Theorem
thereby implying existence of a spectral gap.
Collecting information, we find that we have reduced to a system of form \varepsilonqref{Relax} satisfying (H1)-(H2),
with $\bV:=\lambdaangle \xi\tauext{\rm{ran}}gle^{1/2}({\rm Range } R)^\perp$, $R$ as in \varepsilonqref{defR}, $\dim \bV^\perp=5$, and
$\overline{\mathbf{u}}=\lambdaangle \xi\tauext{\rm{ran}}gle^{1/2}\underline M$ for any Maxwellian $\underline M$.
Note that $A$ has no kernel on $\bH$, but essential spectra
$\xi_1/\lambdaangle \xi\tauext{\rm{ran}}gle \tauo 0$ as $\xi_1\tauo 0$: an {\it essential singularity}.
A consequence is that {\it small velocities $\xi_1\tauo 0$ constitute the main difficulties
in our analysis,} large-velocities issues having been subsumed in the reduction \cite{MZ} to form \varepsilonqref{Relax}.
\subsubsection{Hydrodynamic limit}\lambdaabel{s:hydro}
The formal Chapman-Enskog expansion \varepsilonqref{ce2}, or hydrodynamic limit, being independent of coordinate representation, is the
same in our variables $\mathbf{u}$, $Q$ as in the standard Boltzmann variables $f$, $\cQ$.
As computed, e.g., in \cite{Ce,LiuYu}, this appears in fluid variables \varepsilonqref{fvar} as the
{\it compressible Navier-Stokes} equations with temperature-dependent viscosity and heat conduction:
\begin{example}in{equation}\wti{\alpha}g{cNS}\lambdaabel{cns}
\begin{example}in{equation}gin{aligned}
\rho_t + (\rho v_1)_x&=0,\\
(\rho v_1)_t + (\rho v_1^2 + p)_x&= ((4/3)\mu v_{1,x})_x ,\\
(\rho _2)_t + (\rho v_1v_2)_x&= (\mu v_{2,x})_x ,\\
(\rho _3)_t + (\rho v_1v_3)_x&= (\mu v_{3,x})_x ,\\
(\rho E)_t + (\rho v_1 \rho E + v_1 p)_x&= ( \kappa T_x + (4/3)\mu v_1 v_{1,x} )_{x},\\
\varepsilonnd{aligned}
\varepsilonnd{equation}
where $T$ denotes temperature, with monatomic equation of state $p=\,\mbox{\bf G}amma \rho e$, $T=c_v^{-1} e$, with
\begin{example}in{equation}\lambdaabel{mukappa}
\,\mbox{\bf G}amma= 2/3,\quad c_v=3/4, \quad
\mu=\mu(T)= (5/16)\sqrt{T/\pi} ,\quad \kappa=\kappa(T)=(75/16)\sqrt{T/\pi}.
\varepsilonnd{equation}
As computed in, e.g., \cite{Sm}, the hyperbolic (i.e., lefthand side) part of \varepsilonqref{cns} has characteristics
\begin{example}in{equation}\lambdaabel{bchar}
\lambdaambda_1= v_1-c, \quad \lambdaambda_2=\lambdaambda_3=\lambdaambda_4= v_1, \quad \lambdaambda_5= v_1+c,
\varepsilonnd{equation}
where $c:=\sqrt{\,\mbox{\bf G}amma(1+\,\mbox{\bf G}amma) e}>0$ denotes sound speed, with ``acoustic modes'' $v_1\pm c$ simple
and satisfying \varepsilonqref{gnl}, and ``entropic/vorticity modes'' $v_1$ multiplicity three and linearly degenerate in
the sense of Lax \cite{La,Sm} (not addressed here; see \cite{PZ2} for discussion of the linearly degenerate case).
\subsection{Macro-micro decomposition}\lambdaabel{s:macro}
Next, starting with form \varepsilonqref{steady}, (H1)-(H2), coordinatize as in Section \ref{s:ce}
$\mathbf{u}$ as $(u,v)$, $u=P_{\bV^\perp}\mathbf{u}$, $v=P_{\bV}\mathbf{u}$, where $P_{\bV^\perp}$ and $P_{\bV}$ are the orthogonal
projections associated with orthogonal decomposition $\bH=\bV^\perp \oplus \bV$, to obtain the block decomposition
\begin{example}in{equation}\lambdaabel{mm}
\bp A_{11}& A_{12}\\A_{21}& A_{22} \varepsilonp \bp u\\v\varepsilonp'= \bp 0 & 0\\ 0 & E\varepsilonp \bp u\\v\varepsilonp +\bp 0\\frac\varepsilonp,
\varepsilonnd{equation}
into ``macro'' and ``micro'' variables $u$ and $v$ similarly as in \cite{LiuYu,MZ},
with forcing term $f=B(\mathbf{u},\mathbf{u})$, where $B$ is a bounded bilinear map and $E<0$ is symmetric negative definite on $\bH$.
The following further reduction greatly simplifies computations later on; {hereafter we take $E=-{\rm Id }$}.
\begin{example}in{equation}gin{obs}\lambdaabel{idobs}
By the change of variables $v\tauo (-E)^{1/2}$ combined with left-multiplication of the $v$-equation by $(-E)^{-1}$,
we may take without loss of generality $E={\rm Id }$.
\varepsilonnd{obs}
\subsection{Reduction to canonical form}\lambdaabel{s:canred}
Since $A$ and $P_{|\bV^\perp}$ are self-adjoint on $\bH$,
$A_{11}=P_{\bV^\perp}A_{|\bV^\perp}$ is self-adjoint on $\bV^\perp$, hence
$\bV^\perp=\ker A_{11}\oplus{\rm im } A_{11}$.
Denote by $P_{\ker A_{11}}$ and $P_{{\rm im } A_{11}}$ the associated orthogonal projections onto $\ker A_{11}$ and ${\rm im } A_{11}$,
and $\widetilde{A}_{12}:\bV\tauo{\rm im } A_{11}$ and $T_{12}:\bV\tauo\ker A_{11}$ the operators defined by $\widetilde{A}_{12}=P_{{\rm im } A_{11}}A_{12}$ and $T_{12}=P_{\ker A_{11}}A_{12}$.
From the assumption that $A$ is one-to-one, we readily obtain the following; see \cite[Lemma 2.1]{PZ2} for details.
\begin{example}in{equation}gin{lemma}\lambdaabel{r2.1} Assuming (H1)-(H2),
(i) $\ker T_{12}^*=\{0\}$, ${\rm im } T_{12}=\ker A_{11}$, $\ker T_{12}\ne\{0\}$, and
(ii) The linear operator $\widetilde{A}_{11}=(A_{11})_{|{\rm im } A_{11}}$ is self-adjoint and invertible on ${\rm im } A_{11}$.
\varepsilonnd{lemma}
Introduce now orthogonal subspaces $\bV_1={\rm im } T_{12}^*$ and $\widetilde{\bV}=\ker T_{12}$ decomposing $\bV$,
with associated projectors $P_{\bV_1}$ and $P_{\widetilde{\bV}}$.
Denoting
\begin{example}in{equation}\lambdaabel{coordn}
u_1=P_{\ker A_{11}}u, \quad \widetilde{u}=P_{{\rm im } A_{11}}u, \quad v_1=P_{\bV_1}v, \quad \hbox{\rm and $\widetilde{v}=P_{\widetilde{\bV}}v$,}
\varepsilonnd{equation}
and applying $P_{\ker A_{11}}$ and $P_{{\rm im } A_{11}}$ to the first equation of \varepsilonqref{mm} we obtain
\begin{example}in{equation}gin{equation}\lambdaabel{u1-tildeu}
T_{12}v'= 0,\quad\widetilde{A}_{11}\widetilde{u}'+\widetilde{A}_{12}\widetilde{v}'= 0.
\varepsilonnd{equation}
Moreover, by $(A_{21})_{|\ker A_{11}}=T_{12}^*$, $(A_{21})_{|{\rm im } A_{11}}=\widetilde{A}_{12}^*$ the second equation of \varepsilonqref{mm} is equivalent to
\begin{example}in{equation}gin{equation}\lambdaabel{inter-eq2}
T_{12}^*u_1'+\widetilde{A}_{12}^*\widetilde{u}'+A_{22}v'=Ev+f.
\varepsilonnd{equation}
Since $v_1\in\bV_1={\rm im } T_{12}^*$, from \varepsilonqref{u1-tildeu} we conclude $v_1'=0$. In addition, since $\widetilde{A}_{11}$ is invertible on ${\rm im } A_{11}$ by Lemma~\ref{r2.1}(ii), we have $\widetilde{u}'=-\widetilde{A}_{11}^{-1}\widetilde{A}_{12}\widetilde{v}'$. Summarizing, \varepsilonqref{u1-tildeu} is equivalent to
\begin{example}in{equation}gin{equation}\lambdaabel{u1-tildeu2}
v_1'=0,\quad (\widetilde{u} +\widetilde{A}_{11}^{-1}\widetilde{A}_{12}\widetilde{v})'=0.
\varepsilonnd{equation}
Next, taking without loss of generality $E={\rm Id }$, we obtain from \varepsilonqref{inter-eq2} evidently
\begin{example}in{equation}gin{equation}\lambdaabel{u1-1}
T_{12}^*u_1'+P_{\bV_1}(A_{22}-\widetilde{A}_{12}^*\widetilde{A}_{11}^{-1}\widetilde{A}_{12})\widetilde{v}'=-v_1 +P_{\bV_1}f
\varepsilonnd{equation}
and
\begin{example}in{equation}gin{equation}\lambdaabel{prelim-tilde-v}
P_{\widetilde{\bV}}(A_{22}-\widetilde{A}_{12}^*\widetilde{A}_{11}^{-1}\widetilde{A}_{12})\widetilde{v}'=-\widetilde{v}+P_{\widetilde{\bV}}Ev_1+P_{\widetilde{\bV}_1}f.
\varepsilonnd{equation}
From Lemma~\ref{r2.1}(i), $(T_{12}^*)^{-1}$ is well-defined and bounded, hence we obtain from \varepsilonqref{u1-1}
\begin{example}in{equation}gin{equation}\lambdaabel{u1-2}
(u_1- \,\mbox{\bf G}amma_1\widetilde{v})'= -(T_{12}^*)^{-1}v_1 + (T_{12}^*)^{-1}P_{\bV_1}f,
\quad
\,\mbox{\bf G}amma_0\widetilde{v}'=\widetilde{v}+P_{\widetilde{\bV}}f,
\varepsilonnd{equation}
where $\,\mbox{\bf G}amma_1=(T_{12}^*)^{-1}(\widetilde{A}_{12}^*\widetilde{A}_{11}^{-1}\widetilde{A}_{12}-A_{22})\in\mathcal{B}(\bV,\ker A_{11})$ and
\begin{example}in{equation}\lambdaabel{Gamma0}
\,\mbox{\bf G}amma_0=P_{\widetilde{\bV}}(A_{22}-\widetilde{A}_{12}^*\widetilde{A}_{11}^{-1}\widetilde{A}_{12})_{|\widetilde{\bV}}\in\mathcal{B}(\widetilde{\bV})
\; \hbox{\rm is symmetric.}
\varepsilonnd{equation}
Summarizing, we have that \varepsilonqref{mm} is equivalent to the system
\begin{example}in{equation}\lambdaabel{sys1}
(u_1- \,\mbox{\bf G}amma_1\widetilde{v})'= (T_{12}^*)^{-1}v_1 + (T_{12}^*)^{-1}P_{\bV_1}f, \quad
(\widetilde{u} + \widetilde{A}_{11}^{-1}\widetilde{A}_{12}\widetilde{v})'=0, \quad
v_1'=0, \quad
\,\mbox{\bf G}amma_0\widetilde{v}'=\widetilde{v}+P_{\widetilde{\bV}}f.
\varepsilonnd{equation}
By the invertible change of coordinates
\ba\lambdaabel{coord}
w_c=\Big( ( u_1 - \,\mbox{\bf G}amma_1\tauildelde v)^T,
(-(T_{12}^*)^{-1}v_1)^T,
(\tauildelde u + \tauildelde A_{11}^{-1}\tauildelde A_{12}\tauildelde v)^T\Big)^T ,
\quad
w_h=\tauildelde v,
\varepsilona
we reduce \varepsilonqref{mm} finally to the canonical form of Lemma \ref{l:canon}:
\ba\lambdaabel{canon2}
w_c'=Jw_c + g_c, \quad
\,\mbox{\bf G}amma_0 w_h' = w_h + g_h,
\varepsilona
where $J=\bp 0 & I_m & 0 \\ 0&0 & 0\\ 0&0&0 \varepsilonp$ and
$g_c=\tauildelde Q_c(w,w)$ and $g_h=\tauildelde Q_h(w,w)$ are bounded bilinear maps.
\begin{example}in{equation}gin{obs}\lambdaabel{fibers}
We record for later that the tangent subspace $(u,v)=(\zeta, 0)$ to equilibrium manifold $\cE=\{(u,v)\in\bV^\perp\oplus\bV: Q(u,v)=0\}$
is given in coordinates \varepsilonqref{coord} by
$w_c=(\zeta_1, 0, \tauildelde \zeta)$, $w_h=0$,
as can also be seen by computing the subspace of equilibria of \varepsilonqref{canon} with $g=(g_c,g_h)=0$.
\varepsilonnd{obs}
\section{Linear resolvent estimates}\lambdaabel{s:linear}
The starting point for construction of invariant manifolds is the study of the solution operator for the decoupled
linear inhomogeneous equations \varepsilonqref{canon2} with arbitrary forcing terms $g_c$, $g_h$.
The ``center,'' $w_c$ equation is of standard finite-dimensional type, so may be treated by usual methods.
Evidently, then, the key issue is treatment of the degenerate ``hyperbolic,'' $w_h$ equation.
\subsection{Symmetric degenerate evolution equations} Consider a degenerate inhomogeneous evolution equation
$(\,\mbox{\bf G}amma_0{\partial}rtial_x + {\rm Id }) w_c=g,$
with $\,\mbox{\bf G}amma_0$ (recalling \varepsilonqref{Gamma0} and (H1)-(H2)) symmetric and one-to-one but not boundedly invertible,
with the goal to obtain bounds on the resolvent operator
\begin{example}in{equation}\lambdaabel{res}
\mathcal{R}:=(\,\mbox{\bf G}amma_0 {\partial}rtial_x -{\rm Id })^{-1}.
\varepsilonnd{equation}
As discussed in the introduction, the inhomogeneous flow $u'+\,\mbox{\bf G}amma_0^{-1}u=0$ possesses generalized exponential dichotomies,
but the resulting bounds on $({\partial}rtial_x-\,\mbox{\bf G}amma_0^{-1})^{-1}$
are insufficient to bound the inhomogeneous solution operator
$(\,\mbox{\bf G}amma_0 {\partial}rtial_x-{\rm Id })^{-1}= ({\partial}rtial_x -\,\mbox{\bf G}amma_0^{-1})^{-1}\,\mbox{\bf G}amma_0^{-1}$.
That is, \varepsilonqref{inhom} represents an interesting new class of symmetric degenerate evolution equations
{for which construction of dichotomies is inadequate to bound the resolvent \varepsilonqref{res}.}
{A key observation} of \cite{PZ1} is that $L^2$ bounds may be obtained {\it directly}, using symmetry.
In \cite{PZ1}, we use for technical reasons a frequency domain/Fourier transform formulation
following \cite{LP2,LP3}; however, this can be seen at formal level through
an a priori energy estimate
\begin{example}in{equation}\lambdaabel{key}
|\lambdaangle u, g\tauext{\rm{ran}}gle|= |\lambdaangle u, \,\mbox{\bf G}amma_0 u'\tauext{\rm{ran}}gle -\lambdaangle u, u\tauext{\rm{ran}}gle |=
|\lambdaangle u, u\tauext{\rm{ran}}gle| = \|u\|^2 \Rightarrow \|u\|\lambdaeq C\|g\|,
\varepsilonnd{equation}
reminiscent of Friedrichs estimates for symmetric hyperbolic PDE,
where $\|\cdot\|$ and $\lambdaangle \cdot, \cdot \tauext{\rm{ran}}gle$ denote $L^2$ norm and inner product;
indeed, one could view \varepsilonqref{inhom} as a ``symmetric hyperbolic'' analog for ODE.
As in the PDE setting, the crucial property of symmetry of $\,\mbox{\bf G}amma_0$
is guaranteed by existence of a convex entropy for \varepsilonqref{Relax} \cite{CLL}, e.g., the Boltzmann H-Theorem
as discussed in Section \ref{s:sym}.
\subsection{Details/counterexamples}
Viewing the constant-coefficient operator $\mathcal{R}$,
\varepsilonqref{res}, as a Fourier multiplier with symbol $\hat{\mathcal{R}}(\omega)=(i\omega \,\mbox{\bf G}amma_0 -{\rm Id })^{-1}$,
and computing the uniform estimates
\begin{example}in{equation}\lambdaabel{Rbds}
|\hat{\mathcal{R}}(\omega)|\lambdaeq C,\quad
|\hat{\mathcal{R}}'(\omega)|=|-\hat{\mathcal{R}} i\,\mbox{\bf G}amma_0 \hat{\mathcal{R}}|
\lambdaeq C_2(1+|\omega|)^{-1},
\varepsilonnd{equation}
we find by the Mikhlin-Hormander multiplier theorem that $\mathcal{R}$ is bounded on $L^p$, $1<p<\infty$.
Further detail may be obtained by spectral decomposition of $\,\mbox{\bf G}amma_0$, converting
$(\,\mbox{\bf G}amma_0{\partial}rtial_x + {\rm Id }) w_c=g$ into a family of scalar equations
$(\alpha_\lambdaambda {\partial}rtial_x-1)u_\lambdaambda=g_\lambdaambda$, with
$u_\lambdaambda$ the coordinate associated with spectrum $ \alpha_\lambdaambda$ and
$\|u\|_{\bH}^2=\int |u_\lambdaambda|^2d\mu_\lambdaambda$.
The associated (scalar) resolvent operators $\mathcal{R}_\lambdaambda=(\alpha_\lambdaambda {\partial}rtial_x+1)^{-1}$
have explicit kernels
\begin{example}in{equation}\lambdaabel{sR}
R_\lambdaambda (\tauheta)= \alpha_\lambdaambda^{-1}e^{(\tauheta)/\alpha_\lambdaambda^{-1}},
\, \tauheta \alpha_\lambdaambda<0;
\qquad
(\mathcal{R}_\lambdaambda h) (x)= \int_{\R} R_\lambdaambda (x-y)h(y)dy,
\varepsilonnd{equation}
that are evidently integrable with respect to $x$, so bounded coordinate-wise on
any $L^p(\R_+)$.
However, explicit example \cite[Eg. 4.7, p. 23]{PZ1} shows that the full operator $\mathcal{R}$ is not
bounded on $L^\infty(\R,\bH)$ (resp. $L^1(\R,\bH)$); that is, {\it it is not an $L^\infty$ (resp. $L^1$)
multiplier.}
This has the important consequence that our dynamical theory must be carried out in $H^1$ (bounding
$L^\infty$) rather than the usual $C^0(\R)$
setting costing a surprising amount of technical difficulty.
The above shows also that the full resolvent kernel $R(\tauheta)$ determined by \varepsilonqref{sR}, considered
as an operator-valued function from $\bH\tauo \bH$, {\it is not integrable}, since otherwise $\mathcal{R}$ by standard
convolution bounds would be a bounded multiplier on all $L^p$.
Likewise, the computation
\begin{example}in{equation}\lambdaabel{supcalc}
|R(\tauheta)|_{\bH}= \sup_{\alpha_\lambdaambda(\tauheta<0}
|\alpha_\lambdaambda^{-1}e^{-\tauheta/\alpha_\lambdaambda^{-1}}| \sim C/|\tauheta|
\, \hbox{\rm as $\tauheta\tauo 0$}
\varepsilonnd{equation}
shows that $|R(\tauheta)|_{\bH}$ {\it is not bounded}.
This indicates the delicacy of, and cancellation involved in, the bounds on $\mathcal{R}$
obtained above through energy estimate \varepsilonqref{key}/resolvent bounds \varepsilonqref{Rbds}.
\br\lambdaabel{emprmk}
We emphasize that $L^p$ multiplier theory/spectral decomposition is used here only to construct counterexamples,
our construction of invariant manifolds relying on Parseval's identity.
\varepsilonr
\subsection{The Banach space setting}
(Following \cite{Z6})
Weighted $L^\infty$ spaces $L^\infty_{r,\xi}$ in velocity
$\xi$, defined by norms $\|f\|_{L^\infty_{r,\xi}}:=\sup_{\xi\in \R^3}(1+|\xi|)^r M(\xi)^{-1/2}|f(\xi)|$, $r\gammaeq 0$,
where $M(\xi)=e^{-c_0|\xi-v|^2}$ is the Maxwellian corresponding to equilibrium $\overline{\mathbf{u}}$,
have been used in the study of Boltzmann's equation in, e.g., \cite{LiuYu,LY3}.
Though resolvent bounds appear more difficult to obtain in this context, we can establish
that {\it $|R(\tauheta)|_{L^\infty_{r,\xi}}$ is not bounded,} similarly as in the Hilbert case.
Recall \cite{G} that the linearized collision operator $L$ appearing in the
linearized inhomogeneous steady Boltzmann equation $\xi_1 f'- L f= \tauildelde g$
may be decomposed as $ \tauildelde L=\tauildelde{\nu}(\xi) + \tauildelde {K}$,
where $\tauildelde \nu(\xi)$ is a multiplication operator with $\tauildelde \nu(\xi)\sim \lambdaangle\xi\tauext{\rm{ran}}gle$
and $\tauildelde{K}$ has kernel
$|\tauildelde k(\xi,\xi_*)|\lambdaeq C|\xi-\xi_*|^{-1}e^{-c|\xi-\xi_*|^2}$,
$(\tauildelde {K}h)(\xi)=\int_{\R^3} \tauildelde k(\xi,\xi_*)h(\xi_*)d\xi_*.$
By $\||\xi|^{-1}e^{-c|\xi|^2}\|_{L^1}<\infty$ and standard convolution bounds,
$\tauildelde{K}$ is bounded on $L^\infty(\xi)$, hence, by
$\lambdaangle \xi\tauext{\rm{ran}}gle/\lambdaangle \xi-\xi_* \tauext{\rm{ran}}gle\lambdaeq C\lambdaangle \xi_*\tauext{\rm{ran}}gle$, on $L^\infty_{r,\xi}$.
In our coordinates \varepsilonqref{change}, $A f'- Q'f=g$,
$A= \fracrac{\xi_1}{\lambdaangle\xi\tauext{\rm{ran}}gle}$, $Q'=-\nu(\xi)+{K}$, where $\nu(\xi)\sim 1$ and
${ K}=
\lambdaangle\xi\tauext{\rm{ran}}gle^{-1/2}\hat { K}\lambdaangle \xi\tauext{\rm{ran}}gle^{-1/2}$
is bounded from
$L^\infty_{r,\xi}\tauo L^\infty_{r,\xi}$.
The reduced equation $\,\mbox{\bf G}amma_0 u'-Eu=g$
of \varepsilonqref{inhom} corresponds to the restriction of $g$ to a finite-codimension subspace $\Sigma$ of ``hyperbolic modes,''
where $E:=Q'|_\Sigma<0$ \cite{PZ1}.
That
\begin{example}in{equation}\lambdaabel{Rinf}
|R(\cdot)|_{L^\infty_{r,\xi}}=\infty
\varepsilonnd{equation}
thus follows (by contradiction, using
standard convolution bounds) from the following slightly stronger statement.
\begin{example}in{equation}gin{lemma}[adapted from \cite{Z6}]\lambdaabel{claim1}
{The solution of $Au'-Q'(\overline{\mathbf{u}})u=g$,
with data $g$ valued in a finite-codimension subspace $\Sigma$ of $L^\infty_{r,\xi}$
does not satisfy a uniform bound} $|u|_{L^\infty(x,L^\infty_{r,\xi})}\lambdaeq C |g|_{L^1(x,L^\infty_{r,\xi})}.$
\varepsilonnd{lemma}
\begin{example}in{equation}gin{proof}
Defining $\mathcal{S}=\bibitemg( (\xi_1/\lambdaangle \xi\tauext{\rm{ran}}gle ){\partial}rtial_x - \nu(\xi) \bibitemg)^{-1}$, we have
the explicit solution formula
\begin{example}in{equation}\lambdaabel{Sform}
(\mathcal{S}g)(x,\xi)= \int_{\R}S_\xi(x-y)g(y,\xi) dy;\qquad
S_\xi(\tauheta)=(\xi_1/\lambdaangle \xi\tauext{\rm{ran}}gle)^{-1}e^{-\nu(\xi)/(\xi_1/\lambdaangle \xi\tauext{\rm{ran}}gle)^{-1}},
\varepsilonnd{equation}
where scalar kernels $S_\xi(\cdot)$ are integrable, hence $\mathcal{S}$ is bounded on
$L^\infty(x, L^\infty_{r,\xi})= L^\infty_r(\xi, (L^\infty(x)$.
Writing $Au'-Q'(\overline{\mathbf{u}})u=g$ as
$\bibitemg( (\xi_1/\lambdaangle \xi\tauext{\rm{ran}}gle ){\partial}rtial_x - \nu(\xi) \bibitemg)u= {K}u + g$, applying
$\mathcal{S}$, and rearranging, we obtain
$ \mathcal{S}g= u- \mathcal{S}{K}u$, hence $| \mathcal{S}g |_{L^\infty(x,L^\infty_{r,\xi})}\lambdaeq C| u |_{L^\infty(x,L^\infty_{r,\xi})}$
by boundedness of $|K|_{L^\infty_{r,\xi}}$, $|\mathcal{S}|_{L^\infty(x,L^\infty_{r,\xi})}$.
Thus, $|u|_{L^\infty(x,L^\infty_{r,\xi})}\lambdaeq C |g|_{L^1(x,L^\infty_{r,\xi})}$ would imply
$|\mathcal{S}g|_{L^\infty(x,L^\infty_{r,\xi})}\lambdaeq C |g|_{L^1(x,L^\infty_{r,\xi})}$,
or, taking $g\tauo \delta(x) h(\xi)$,
$|S(\tauheta)|_{L^\infty_{r,\xi}}\lambdaeq C$ for the full kernel $S$ of $\mathcal{S}$.
But, direct calculation as in \varepsilonqref{supcalc} shows
$|S(\tauheta)|_{L^\infty_{r,\xi}}\sim |\tauheta|^{-1}$ as $\tauheta\tauo 0$, a contradiction.
\varepsilonnd{proof}
\br\lambdaabel{LYrmk}
In our notation, the bound asserted in \cite{LiuYu} is
$ |R(\tauheta )|_{L^\infty_{5/2,\xi}}\lambdaeq Ce^{-\begin{example}in{equation}ta |x|}, $
in contradiction with \varepsilonqref{Rinf}.
We conjecture that $|R(\tauheta)|_{L^\infty_{r,\xi}} \gammatrsim |\tauheta|^{-1}$ as $\tauheta\tauo 0$ similarly as for its principal part $ S(\tauheta)$,
and similarly as in the Hilbert space setting \varepsilonqref{supcalc}, so that
$|R(\cdot )|_{L^\infty_{r,\xi}} \notagt \in L^p(\R)$ for any $1\lambdaeq p\lambdaeq \infty$.
\varepsilonr
\section{$H^1$ stable manifold theorem}\lambdaabel{s:stable}
We now outline the argument for construction of the stable manifold; for details, see \cite{PZ1}.
\begin{example}in{equation}gin{proof}[Proof of Theorem \ref{t1.3}]
For clarity, we first treat the {\it noncharacteristic case} $m=0$, $\dim w_c=\dim \cE=r$, for which \varepsilonqref{canon} becomes
$ w_c\varepsilonquiv \tauext{\rm constant}$, $(\,\mbox{\bf G}amma_0 {\partial}rtial_x + {\rm Id }) w_h= \tauildelde Q_c(w)$,
and the equation for the stable manifold reduces to $w_c\varepsilonquiv 0$ and
$(\,\mbox{\bf G}amma_0 {\partial}rtial_x + {\rm Id }) w_h= B(w_h,w_h)$, $B(w_h,w_h):=\tauildelde Q_c((0,w_h)$ a bounded bilinear map.
Inverting, we deduce (see \cite{PZ1}) the fixed-point formulation
\begin{example}in{equation}\lambdaabel{fixedstable}
u(\wti{\alpha}u)= T_S(\wti{\alpha}u) \Pi_S u_0 + ( \,\mbox{\bf G}amma_0 {\partial}rtial_x-{\rm Id })^{-1} B(u,u)(\wti{\alpha}u),
\varepsilonnd{equation}
where $\Pi_S$, $T_S$ denote projection and semigroup associated with the stable
subspace of homogeneous flow $\,\mbox{\bf G}amma_0 u'=-u$, so that
$T_S(\wti{\alpha}u) \Pi_S u_0$ is a homogeneous solution with data $\Pi_Su_0$ lying in the
stable subspace at $\wti{\alpha}u=0$, and $\Pi_Su(0)=\Pi_Su_0$.
This can be recognized as a concise, frequency-domain version of the usual variation
of constants formula for finite-dimensional ODE.
However, significant new difficulties arise from the fact that, due to the properties
of $(\,\mbox{\bf G}amma_0 {\partial}rtial_x+{\rm Id })^{-1}$ described in Section \ref{s:linear}, we must carry out the analysis in
weighted $H^1$ rather than standard $L^\infty$ spaces.
For example, for the unbounded formal generator $-\,\mbox{\bf G}amma_0^{-1}$, the $H^1$-stable subspace
is strictly contained in the $L^2$-stable one, so that we must seek a graph not over the
entire stable subspace but only the $H^1$ part, conveniently
conveniently characterized as $\tauext{\rm{dom}}(\Pi_S (-\,\mbox{\bf G}amma_0)^{-1/2})$.
Moreover, differentiating the equation gives
$u'(\wti{\alpha}u)= T_S'(\wti{\alpha}u) \Pi_S (u_0 -B(u(0),u(0)))
( \,\mbox{\bf G}amma_0 {\partial}rtial_x-{\rm Id })^{-1} B(u,u)'(\wti{\alpha}u)$
(noting $u'(0)= -\,\mbox{\bf G}amma_0^{-1} \bibitemg( u(0)-B(u(0),u(0)) \bibitemg)$) by the equation)
so that the ``homogeneous term'' involving $T_S'$ lies in $L^2$
when $v_0:=u_0 -B(u(0),u(0))$, not $u_0$, lies in the $H^1$-stable subspace.
Our solution is to introduce the {\it modified fixed-point equation}
\begin{example}in{equation}\lambdaabel{modfixedstable}
u(\wti{\alpha}u)= T_S(\wti{\alpha}u) \Pi_S\bibitemg( v_0-B(u(0),u(0)) \bibitemg) + ( \,\mbox{\bf G}amma_0 {\partial}rtial_x-{\rm Id })^{-1} B(u,u)(\wti{\alpha}u)
\varepsilonnd{equation}
parametrized by elements $v_0$ in the $H^1$-stable
subspace, for which the derivative equation is the harmless
$u'(\wti{\alpha}u)= T_S'(\wti{\alpha}u) \Pi_S v_0+ ( \,\mbox{\bf G}amma_0 {\partial}rtial_x-{\rm Id })^{-1} B(u,u)'(\wti{\alpha}u)$. Observing that
the trace $u\tauo u(0)$ is bounded on $H^1$ by 1D Sobolev embedding, as is $(\,\mbox{\bf G}amma_0{\partial}rtial_x + {\rm Id })^{-1}$
by $L^2$-boundedness plus commutation of constant coefficient operators with derivatives, we find that
\varepsilonqref{modfixedstable} is contractive, yielding existence/uniqueness in $H^1$ (and
exponentially weighted $H^1$) norm, and thereby existence of an (exponentially decaying)
stable manifold expressed as a graph over the $H^1$ stable subspace, Fr\'echet-differentiable
from $\tauext{\rm{dom}}(-(\Pi_S \,\mbox{\bf G}amma_0)^{-1/2})$ with norm induced by
$(-\Pi_S \,\mbox{\bf G}amma_0)^{-1/2}$ to the full space $\bH$ with its original norm.
A novel aspect is that {the graph lies above the $H^1$-stable subspace not only in unstable
directions, but also in stable directions lying in the stable but not $H^1$-stable subspace.}
In the noncharacteristic case, there is a nontrivial center equation $w_c'=Jw_c + B_c(w,w)$, coupled
to the hyperbolic equation $\,\mbox{\bf G}amma_0 w_h'=-w_h + B_h(w,w)$.
This may be treated, setting $w=(z,u)$, by the larger fixed-point equation
appending to \varepsilonqref{modfixedstable} a standard finite-dimensional $z$ equation:
\ba\lambdaabel{fixsys}
z(\wti{\alpha}u)&= -\int_\wti{\alpha}u^{+\infty} e^{J(\wti{\alpha}u-\tauheta)}B_c(w,w)(\tauheta)d\tauheta,\\
u(\wti{\alpha}u)&= T_S(\wti{\alpha}u) \Pi_S\bibitemg( v_0-B_h(u(0),u(0)) \bibitemg) + ( \,\mbox{\bf G}amma_0 {\partial}rtial_x-{\rm Id })^{-1} B_h(w,w)(\wti{\alpha}u).
\varepsilona
\varepsilonnd{proof}
\begin{example}in{equation}gin{proof}[Proof of Corollary \ref{c5.9}]
Because the stable manifold contains the forward orbits of all solutions with $H^1(\R_+,\bH))$ norm sufficiently small,
it contains the orbit on $\R_+$ of $\check{\mathbf{u}}_M:=\check{\mathbf{u}}(\cdot + M)$ for $M$ sufficiently large,
whence $\check{\mathbf{u}}_M\in H^1_{\tauildelde \nu}(\R_+,\bH)$ by Theorem \ref{t1.3}.
It follows that $e^{\tauildelde \nu |\cdot|}\mathbf{u}_M \in H^1(\R_+,\bH)$, hence, by Sobolev embedding,
$|e^{\tauildelde \nu |x|}\mathbf{u}(x)|\lambdaeq C$, or
$|\mathbf{u}(x)|\lambdaeq C|e^{\tauildelde \nu |x|}$, for $x\gammaeq M$.
\varepsilonnd{proof}
\br\lambdaabel{stablermk}
The key technical points in the above construction are the use of $H^1$ rather than sup norms
to bound the resolvent, and the ``integration by parts'' parametrization by $v_0$ in \varepsilonqref{modfixedstable}.
\varepsilonr
\section{Existence of a center manifold}\lambdaabel{s:cm}
Next, we outline the argument for existence of a an $H^1$ center manifold; for details, see \cite{PZ2}.
The translation from standard $C^0$ to $H^1$ framework again introduces interesting new difficulties:
surprisingly, different from those encountered in the stable manifold case.
\begin{example}in{equation}gin{proof}[Proof of Theorem \ref{t1.1}]
Following the standard approach to construction of center manifolds \cite{VI,B,Z1},
we first replace $\tauildelde Q$ by a truncated nonlinearity
$N_\varepsilon(w):=\rho(w/\varepsilon)\tauildelde Q(w)$,
where $\rho$ is a smooth cutoff function equal to $1$ for $|w|\lambdaeq 1$ and $0$ for $|w|\gammaeq 2$.
The truncated nonlinearity satisfies bounds
\begin{example}in{equation}\lambdaabel{Nbds}
|N_\varepsilon| \lambdaeq {\mathrm{c}}\varepsilon^2, \quad
|N_\varepsilon'|\lambdaeq {\mathrm{c}}\varepsilon, \quad
|N_\varepsilon''| \lambdaeq {\mathrm{c}} \qquad
(|\cdot|=|\cdot|_\bH),
\varepsilonnd{equation}
and agrees with the original one locally to $\overline{\mathbf{u}}$.
Translating the usual sup-norm approach to the $H^1$ setting, we
seek solutions to the modified (truncated) equation in a negatively weighted space
$H^1_{-\alpha}$, for $\alpha>0$ sufficiently small.
Similarly as in \varepsilonqref{fixedstable}, this yields the fixed point formulation
\begin{example}in{equation}\lambdaabel{cfixed}
w(\wti{\alpha}u)= T_{\mathrm{c}}(\wti{\alpha}u) \Pi_{\mathrm{c}} w_0 +\int_0^\wti{\alpha}u T_{\mathrm{c}}(\wti{\alpha}u -\tauheta) \Pi_{\mathrm{c}} N_\varepsilon(w(\tauheta))d\tauheta
+ ( \,\mbox{\bf G}amma_0 {\partial}rtial_x+ {\rm Id })^{-1} \Pi_{\mathrm{h}} N_\varepsilon(w)(\wti{\alpha}u)
\varepsilonnd{equation}
for solution $w=(w_{\mathrm{c}}^T, w_{\mathrm{h}}^T)^T$, where $\Pi_{\mathrm{c}}$ denotes projection onto the $w_c$ component,
$T_{\mathrm{c}}(\cdot)=e^{J (\cdot)}$ the associated (nondegenerate) flow,
and $\Pi_{\mathrm{h}}$ denotes projection onto the $w_{\mathrm{h}}$ component.
The difficulty in this case is not with the ``homogeneous'' term $T_{\mathrm{c}}(\wti{\alpha}u) \Pi_{\mathrm{c}} w_0$ as in the stable
manifold case (since derivatives on $\Sigma_{\mathrm{c}}$ are bounded) nor
$\int_0^\wti{\alpha}u T_{\mathrm{c}}(\wti{\alpha}u -\tauheta) \Pi_{\mathrm{c}} N_\varepsilon(w(y))dy$,
but the formerly harmless $ ( A {\partial}rtial_x-{\rm Id })^{-1} \Pi_H N_\varepsilon(w,w)(\wti{\alpha}u)$, specifically,
the ``substitution operator'' $\mathcal{N}_\varepsilon: w\tauo N_\varepsilon(w)$. Bounds \varepsilonqref{Nbds}
yield readily that \varepsilonqref{cfixed} is contractive in $L^2_{-\alpha}$ and bounded in $H^1_{-\alpha}$,
$\|f\|_{H^s_{-\alpha}}:=\|e^{-\alpha \lambdaangle \cdot \tauext{\rm{ran}}gle} f(\cdot)\|_{H^s}$,
giving existence and uniqueness of a $C^{0+1/2}$ center manifold $\Pi_{\mathrm{c}} w_0\tauo w(0)$
via the trace map $w\tauo w(0)$ and the 1-d Sobolev estimate
$|f(0)|\lambdaeq \|f\|_{L^2_{-\alpha}}^{1/2}\|{\partial}rtial_x f\|_{L^2_{-\alpha}}$.
However, higher (even Lipshitz) regularity seems to require contraction in $\|\cdot\|_{H^1_{-\alpha}}$,
the difficulty lying in term
$$
\|{\partial}rtial_x(N_\varepsilon(v_1)-N_\varepsilon(v_2))\|_{L^2_{-\alpha}}
\sim \| \max_j (|N_\varepsilon''(v_j)||{\partial}rtial_x v_j|)|v_2-v_1| \|_{L^2_{-\alpha}}
\sim \sum_j \| |{\partial}rtial_x v_j||v_2-v_1| \|_{L^2_{-\alpha}} ,
$$
for which the obvious Sobolev embedding estimate gives
$ \|v_1-v_2\|_{H^1_{-\alpha}} \sum_j(\int_\R |{\partial}rtial_x v_j|^2)^{1/2}=+\infty$.
{A key observation} is that, for $0<\alpha_1\lambdal \alpha \lambdal \alpha_2\lambdal 1$,
\varepsilonqref{cfixed} is contractive in the {\it mixed norm}
\begin{example}in{equation}\lambdaabel{mixed}
\|f\|:=\|f\|_{L^2_{-\alpha}}+ \|{\partial}rtial_x f\|_{L^2_{-\alpha_2}}
\varepsilonnd{equation}
and bounded in $H^1_{-\alpha_1}$ for $\|w\|_{H^1_{-\alpha_1}}\lambdal 1$. For, the Sobolev bound
$$
e^{-2\alpha_2 \lambdaangle x \tauext{\rm{ran}}gle}|f(x)|^2 \lambdaeq \|f\|_{L^2_{-\alpha_2}(x,\infty)} \|{\partial}rtial_x f\|_{L^2_{-\alpha_2}(x,\infty)}
\lambdaeq e^{-(\alpha_2-\alpha)\lambdaangle x \tauext{\rm{ran}}gle} \|f\|_{L^2_{-\alpha}(x,\infty)} \|{\partial}rtial_x f\|_{L^2_{-\alpha_2}(x,\infty)}
$$
gives
\ba\lambdaabel{obs}
&\|{\partial}rtial_x(N_\varepsilon(v_1)-N_\varepsilon(v_2))\|_{L^2_{-\alpha_2}}^2 \lambdaesssim
\int_{\mathbb R} e^{-2\alpha_2\lambdaangle x\tauext{\rm{ran}}gle}|v_1(x)-v_2(x)|^2|{\partial}_xv_1(x)|^2 dx\\
&\qquad \lambdaesssim \Big(\int_{{\mathbb R}} e^{-(\alpha_2-\alpha)\lambdaangle x\tauext{\rm{ran}}gle} |{\partial}_xv_1(x)|^2 dx\Big)\|v_1-v_2\|^2
\lambdaesssim \|{\partial}_xv_1(x)\|^2_{H^1_{-\alpha_1}} \|v_1-v_2\|^2.
\varepsilona
With this observation, working in norm $\|\cdot\|$,
we obtain essentially immediately
existence and uniqueness of a global center
manifold for the truncated equation/ local center manifold
for the exact equation that is {\it Lipschitz continuous,}
as a graph over the center subspace $\Sigma_{\mathrm{c}}$.
$C^r$ (Fr\'echet) regularity, $r\gammaeq 1$ may then be obtained similarly
as in the finite-dimensional case \cite{VI,B,Z1,HI},
by a bootstrap argument,
using a nested sequence of mixed-weight norms together with
a general result on smooth dependence with
respect to parameters of a fixed point mapping $y=T(x,y)$ that is
Fr\'echet differentiable in $y$ from a stronger to a weaker Banach space,
with differential $T_y$ extending to a bounded, contractive map on the
weaker space \cite[Lemma 2.5, p. 53]{Z1} (\cite[Lemma 3,p. 132]{Z2}).
See \cite[Appendix A]{PZ2}, for further details.
The $H^1$ exponential approximation property (not discussed in \cite{PZ2}) follows by
transcription to the $H^1$ setting of the finite-dimensional argument given in \cite[Step 7, p. 9]{B}.
\varepsilonnd{proof}
\br\lambdaabel{obsrmk}
The estimate \varepsilonqref{obs}, and introduction of norm \varepsilonqref{mixed}, we view as the crucial technical points in our construction of center manifolds, and the
main novelty in this part of the analysis.
\varepsilonr
\section{Structure of small-amplitude kinetic shocks}\lambdaabel{s:prof}
Given existence of a center manifold, on may in principle obtain
an arbitrarily accurate description of near-equilibrium dynamics via formal Taylor expansion/reduction to normal form.
We give here a particularly simple normal form argument
describing bifurcation of stationary shock profiles from a simple genuinely nonlinear characteristic equilibrium,
adapting more general center manifold arguments of \cite{MaP,MasZ2} in the finite-dimensional case.
Similarly as in \cite{MaP,MasZ2}, the main idea is to use the fact that equilibria are predicted by the Rankine-Hugoniot shock
conditions \varepsilonqref{rh} to deduce normal form information from the structure of the Chapman-Enskog approximation \varepsilonqref{ce2}.
\begin{example}in{equation}gin{lemma}\lambdaabel{t1.2}
Let $\overline{\mathbf{u}}\in\ker Q$ be an equilibrium satisfying (H1)-(H2).
In the simple genuinely nonlinear characteristic case \varepsilonqref{gnl}, $m=1$, the center manifolds of \varepsilonqref{steady} and
\varepsilonqref{ce2} both consist of the union of one-dimensional fibers parametrized by $q\in \R^r$ as in \varepsilonqref{rh} and
coordinatized by $u_1$ as in \varepsilonqref{coordn}, satisfying an approximate Burgers flow:
without loss of generality
\begin{example}in{equation}\lambdaabel{burgers2}
\tauildelde q=0,\qquad
u_1'=\delta^{-1}\bibitemg(-q_1 + \Lambda u_1^2/2\bibitemg) + O(|u_1|^3 + |q_1||u_1|+ |q_1|^2),
\varepsilonnd{equation}
where $\delta:=\overline{\mathbf{r}}^T D_* \overline{\mathbf{r}}>0$ with $\overline{\mathbf{r}}$, $D_*$ as in \varepsilonqref{gnl}, \varepsilonqref{ce2}.
In particular, under the normalization $\tauildelde q=0$, there exist local heteroclinic (Lax shock) connections
for $q_1 \Lambda<0$ between endstates $u_1^\pm \approx \sqrt{-2 q_1/\Lambda}$.
\varepsilonnd{lemma}
\begin{example}in{equation}gin{proof}
First, note that $T_{12}v_1$ in the original coordinates of \varepsilonqref{sys1}
is exactly the first component $q_1$ of $q$ in \varepsilonqref{rh}, or $v_1=T_{12}^{-1} q_1$.
By Observation \ref{fibers} and the Implicit Function Theorem, we may take without loss of generality
$\tauildelde q=0$ by a shift along equilibrium manifold $\cE$ of the background equilibrium $\overline{\mathbf{u}}$.
By \varepsilonqref{canon}, therefore, the flow on the $(r+1)$-dimensional center manifold has an $r$-dimensional constant of motion
\begin{example}in{equation}\lambdaabel{zetaeq}
\bibitemg(w_{c,2},w_{c,3}\bibitemg)\varepsilonquiv (\zeta, \gammaamma)=\bibitemg(-(T_{12}^*)^{-1}v_1, \tauildelde q\bibitemg)= \bibitemg(( -T_{12}^*)^{-1}T_{12}^{-1}q_1,0 \bibitemg),
\varepsilonnd{equation}
$w$ as in \varepsilonqref{coord}, with flow along one-dimensional fibers coordinatized by
$ w_{c,1}= u_1 - \,\mbox{\bf G}amma_1\tauildelde v= u_1-\,\mbox{\bf G}amma_1 w_{\mathrm{h}} $
given by the $w_{{\mathrm{c}},1}$ equation of \varepsilonqref{canon}:
\begin{example}in{equation}\lambdaabel{fibeq}
w_{c,1}'= \zeta+ \phi(w_{c,1},\zeta),
\quad
\phi(w_{c,1},\zeta):=g_{c,1}\bibitemg((w_{c,1},\zeta, 0), \Xi(w_{c,1},\zeta, 0)\bibitemg)=\mathcal{O}(\|w_{c,1}\|^2,\|\zeta\|^2).
\varepsilonnd{equation}
The factor $ (T_{12}^*)^{-1}T_{12}^{-1}>0$ in term $\zeta= -(T_{12}^*)^{-1}T_{12}^{-1}q_1$
is easily recognized as $\delta^{-1}$, where
$\delta:= T_{12}T_{12}^{*}>0$, or, using $\overline{\mathbf{r}}=e_1$,
$\delta= \overline{\mathbf{r}} \cdot D_* \overline{\mathbf{r}}$ with $D_*$ as in \varepsilonqref{ce2}.
Using the fact that $w_{\mathrm{h}}=\cJ(w_{\mathrm{c}})=O(|w_{\mathrm{c}}|^2)$ along the center manifold to trade $w_{c,1}$ for
$u_1$ by an invertible coordinate change preserving the order of error terms, we may thus rewrite \varepsilonqref{fibeq} as
\begin{example}in{equation}\lambdaabel{tempfibeqT}
u_1'= \delta^{-1} (-q_1 + \delta \chi u_{1}^2) +O(|u_{1}|^3 +|u_{1}||q_1|+ |q_1|^2),
\varepsilonnd{equation}
where $\chi$, hence the product $\delta \chi$, is yet to be determined.
On the other hand, performing Lyapunov-Schmidt reduction for the equilibrium problem \varepsilonqref{rh}, we obtain the normal form
$$
0= (-q_1 + \fracrac12 \Lambda u_{1}^2) +O(|u_{1}|^3 +|u_{1}||q_1|+ |q_1|^2),
$$
where $\Lambda$ is as in \varepsilonqref{gnl}.
Using the fact that equilibria for \varepsilonqref{steady} and \varepsilonqref{rh} agree, we find that $\delta \chi$ must be
equal to $\fracrac12 \Lambda$, yielding a final normal form consisting of the approximate Burgers flow \varepsilonqref{burgers}.
A similar computation yields the same normal form for fibers of the center manifold of the formal viscous problem \varepsilonqref{ce2};
see also the more detailed computations of \cite{MaP} yielding the same result.
For $q_1 \Lambda>0$, the scalar equation \varepsilonqref{burgers} evidently possesses equilibria $\sim \mp\sqrt{2 q_1/\Lambda}$,
connected (since the equation is scalar) by a heteroclinic profile.
Since $\tauext{\rm sgn} u_1'=-\tauext{\rm sgn} \Lambda$ for $u_1$ between the equilibria, so that
$\bibitemg(\lambdaambda(u)\bibitemg)'\sim \Lambda u_1'$ has sign of $-\Lambda^2<0$,
the connection is in the direction of decreasing characteristic $\lambdaambda(u)$, corresponding to a Lax-type solution
of \varepsilonqref{rh} (cf. \cite{MaP,MasZ2}).
\varepsilonnd{proof}
\br\lambdaabel{lamrmk}
Using $\lambdaambda(u_1)\sim \Lambda u_1$, we may rewrite \varepsilonqref{burgers2} as \varepsilonqref{burgers} as in the introdiuction,
eliminating the $\tauildelde q$-dependent term $\Lambda$.
However, the ``effective viscosity'' $\delta$ remains dependent on $\tauildelde q$.
\varepsilonr
Having determined the normal form \varepsilonqref{burgers}, we establish closeness
of profiles of \varepsilonqref{steady} and \varepsilonqref{ce2} by comparing
their $u_1$ coordinates, separately, to an exact Burgers shock, then showing that differences in remaining, slaved,
coordinates, since vanishing at both endstates, are negligibly small.
\begin{example}in{equation}gin{lemma}[\cite{Li,PlZ}]\lambdaabel{lburgers}
Let $\varepsilonnd{theorem}a\in \R^1$ be a heteroclinic connection of an approximate Burgers equation
\begin{example}in{equation}\lambdaabel{approx}
\delta \varepsilonnd{theorem}a'= \fracrac12 \Lambda(-\varepsilon^2 + \varepsilonnd{theorem}a^2) + S(\varepsilon,\varepsilonnd{theorem}a),
\quad
S=O(|\varepsilonnd{theorem}a|^3+|\varepsilon|^3)\in C^{k+1}(\R^2), \quad k\gammaeq 0,
\varepsilonnd{equation}
and $\bar \varepsilonnd{theorem}a:= -\varepsilon\wti{\alpha}nh(\Lambda \varepsilon x/2\delta)$ a connection of the exact Burgers equation
$\delta\bar \varepsilonnd{theorem}a'= \fracrac12 \Lambda(-\varepsilon^2 + \bar \varepsilonnd{theorem}a^2)$.
Then,
\ba\lambdaabel{etacomp}
|\varepsilonnd{theorem}a_{\pm}-\bar{\varepsilonnd{theorem}a}_{\pm}| &\lambdaeq C\varepsilonilon^2,\\
|{\partial}rtial_x^k \bibitemg(\bar \varepsilonnd{theorem}a - \bar \varepsilonnd{theorem}a_\pm)(x)| &\sim \varepsilon^{k+1}e^{-\delta \varepsilon|x|},
\quad x\gammatrless 0, \quad \delta>0,\\
\bibitemg|{\partial}rtial_x^k \bibitemg((\varepsilonnd{theorem}a- \varepsilonnd{theorem}a_\pm)- (\bar \varepsilonnd{theorem}a - \bar \varepsilonnd{theorem}a_\pm)\bibitemg)(x)\bibitemg| &\lambdae C \varepsilon^{k+2}e^{-\delta \varepsilon|x|},
\quad x\gammatrless 0,
\varepsilona
uniformly in $\varepsilon>0$, where $\varepsilonnd{theorem}a_\pm:=\varepsilonnd{theorem}a(\pm \infty)$, $\bar \varepsilonnd{theorem}a_\pm:=\bar \varepsilonnd{theorem}a(\pm \infty)=\mp\varepsilon$ denote
endstates of the connections.
\varepsilonnd{lemma}
\begin{example}in{equation}gin{proof}(From \cite{PZ2}, following \cite{Li})
Rescaling ${\varepsilonnd{theorem}a} \tauo {\varepsilonnd{theorem}a}/\varepsilonilon, {x} \tauo \Lambda \varepsilonilon \tauildelde{x}/\begin{example}in{equation}ta$,
we obtain the blowup equations
$$
\varepsilonnd{theorem}a'=\fracrac{1}{2}(\varepsilonnd{theorem}a^2-1)+\varepsilonilon \tauildelde{S}(\varepsilonnd{theorem}a,\varepsilonilon)
\quad \tauildelde{S} \in C^{k+1}(\mathbb{R}^2)
$$
and $\bar{\varepsilonnd{theorem}a}'=\fracrac{1}{2}(\bar{\varepsilonnd{theorem}a}^2-1)$, for which estimates \varepsilonqref{approx} translate to
\ba\lambdaabel{scaleetacomp}
|\varepsilonnd{theorem}a_{\pm}-\bar{\varepsilonnd{theorem}a}_{\pm}| &\lambdaeq C\varepsilonilon,\\
|{\partial}rtial_x^k (\bar \varepsilonnd{theorem}a - \bar \varepsilonnd{theorem}a_\pm)(x)| &\sim C\varepsilon^{k}e^{-\tauheta |x|},
\quad x\gammatrless 0, \quad \tauheta>0,\\
|{\partial}rtial_x^k \bibitemg((\varepsilonnd{theorem}a- \varepsilonnd{theorem}a_\pm)- (\bar \varepsilonnd{theorem}a - \bar \varepsilonnd{theorem}a_\pm)\bibitemg)(x)| &\lambdae C \varepsilon^{k+1}e^{-\tauheta |x|},
\quad x\gammatrless 0.
\varepsilona
The estimates \varepsilonqref{scaleetacomp} follow readily from the implicit function theorem and stable manifold theorems together with
smooth dependence on parameters of solutions of ODE, giving the result.
\varepsilonnd{proof}
Setting $q_1=\Lambda \varepsilon^2/2$, and either $\varepsilonnd{theorem}a=u_{REL,1}$ or $\varepsilonnd{theorem}a=u_{CE,1}$, we obtain approximate Burgers
equation \varepsilonqref{approx}, and thereby estimates \varepsilonqref{etacomp} relating $\varepsilonnd{theorem}a=u_{REL,1}$, $u_{CE,1}$ to an exact
Burgers shock $\bar \varepsilonnd{theorem}a$.
\begin{example}in{equation}gin{corollary}[\cite{PZ2}]\lambdaabel{ctri}
Let $\overline{\mathbf{u}}\in\ker Q$ be an equilibrium satisfying (H1)-(H2),
in the noncharacteristic case \varepsilonqref{gnl}, and $k$ and integer $\gammaeq 2$.
Then, local to $\overline{\mathbf{u}}$ ($\bar u$), each pair of points $u_\pm$ corresponding to a standing Lax-type shock of \varepsilonqref{rh}
has a corresponding viscous shock solution $u_{CE}$ of \varepsilonqref{ce2} and
relaxation shock solution $\mathbf{u}_{REL}=(u_{REL},v_{REL})$ of \varepsilonqref{steady}, satisfying for all $j\lambdaeq k-2$:
\ba\lambdaabel{u1comp}
|{\partial}rtial_x^j ( u_{REL,1} - u_{REL,1}^\pm )(x)| &\sim C \varepsilon^{j}e^{-\tauheta |x|},
\quad x\gammatrless 0, \quad \tauheta>0,\\
|{\partial}rtial_x^j (u_{REL,1}-u_{CE,1})(x)| &\lambdae C \varepsilon^{j+1}e^{-\tauheta |x|},
\quad x\gammatrless 0.
\varepsilona
\varepsilonnd{corollary}
\begin{example}in{equation}gin{proof}
Immediate, by \varepsilonqref{scaleetacomp}, Lemma \ref{lburgers} and the triangle inequality, together with the observation that,
as equilibria of \varepsilonqref{ce2} and \varepsilonqref{steady}, hence solutions of \varepsilonqref{rh},
endstates $u_{REL,1}^\pm=u_{CE,1}^\pm$ agree.
\varepsilonnd{proof}
\begin{example}in{equation}gin{proof} [Proof of Corollary \ref{c1.3}](\cite{PZ2})
Noting that the ${\rm im } A_{11}$ and $\bV$ components of $\mathbf{u}_{REL}$ are the $C^1$ functions $\Psi(u_{REL,1})$, $\Phi(u_{REL,1})$ of $u_{REL,1}$ along
the fiber \varepsilonqref{burgers}, we obtain \varepsilonqref{finalbds}(iii) immediately from \varepsilonqref{u1comp}(i).
Denote by $\Psi_{CE}$ the map describing the dependence of ${\rm im } A_{11}$ component of $u_{CE}$ on $u_{CE,1}$
on the corresponding fiber of \varepsilonqref{ce2}.
Noting that $\Psi-\Psi_{CE}$ and $\Phi- v_*$ both vanish at the endstates $u_{REL,1}^\pm$, we have by smoothness
of $\Psi$, $\Psi_{CE}$, $\Phi$, $v_*$ that
$$
|\Psi-\Psi_{CE}|, \, |\Psi-v_*|=\mathcal{O}(|u_{REL,1}- u_{REL,1}^+|,|u_{REL,1}- u_{REL,1}^-|),
$$
giving \varepsilonqref{finalbds}(i)-(ii) by \varepsilonqref{u1comp}(i)-(ii).
\varepsilonnd{proof}
\br\lambdaabel{lastrmk}
Applied to Boltzmann's equation, Corollary \ref{c1.3} yields existence/convergence to hydrodynamic shock
profiles in the square root Maxwellian-weighted norm \varepsilonqref{mnorm}.
Using a bootstrap argument analogous to that of \cite[Prop. 3.1]{MZ},
one can show \cite[Prop. 1.8]{PZ2} that the center manifold of Theorem \ref{t1.1}
lies in the stronger spaces determined by near-Maxwellian weighted norms
$\|f\|_{\bH^s}:=\|f \underline M^{-s}\|_{L^2(\R^3)}$, $1/2\lambdaeq s<1$, yielding further information on
localization of velocity in small-amplitude shock profiles.
This, and the streamlined proof of existence above, are the main novelties in our
treatment by center manifold techniques
of existence and structure of kinetic shocks.
\varepsilonr
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\varepsilonnd{thebibliography}
\varepsilonnd{document} | math | 84,009 |
\begin{document}
\begin{center}
{\bf The Gessel Correspondence and the Partial $\gamma$-Positivity of
the Eulerian Polynomials
on Multiset Stirling Permutations}
\vskip 6mm
William Y.C. Chen$^1$, Amy M. Fu$^2$
and Sherry H.F. Yan$^3$
\vskip 3mm
$^{1}$Center for Applied Mathematics\\
Tianjin University\\
Tianjin 300072, P.R. China
\vskip 3mm
$^{2}$School of Mathematics\\
Shanghai University of Finance and Economics\\
Shanghai 200433, P.R. China
\vskip 3mm
$^3$Department of Mathematics\\
Zhejiang Normal University\\
Jinhua, Zhejiang 321004, P.R. China
Emails: { [email protected], [email protected], [email protected] }
\end{center}
\begin{center}
{\bf Abstract}
\end{center}
Pondering upon the grammatical labeling
of 0-1-2 increasing plane trees, we come to the
realization that
the grammatical labels play a role
as records of chopped off leaves of
the original increasing binary trees. While such an
understanding is purely psychological, it does
give rise to an efficient apparatus to tackle
the partial $\gamma$-positivity
of the Eulearian polynomials on
multiset Stirling permutations, as long as
we bear in mind the combinatorial meanings of
the labels $x$ and $y$ in the Gessel representation of
a $k$-Stirling permutation by means of an
increasing $(k+1)$-ary tree. More precisely, we introduce a Foata-Strehl
action on the Gessel trees resulting in an interpretation
of the partial $\gamma$-coefficients of the aforementioned
Eulerian polynomials, different from
the ones found by Lin-Ma-Zhang and
Yan-Huang-Yang. In particular, our strategy
can be adapted to deal with
the partial $\gamma$-coefficients of the
second order Eulerian polynomials, which in turn can be
readily converted to
the combinatorial formulation due to Ma-Ma-Yeh in connection
with certain statistics of Stirling permutations.
\noindent
{\bf Keywords:} Eulerian polynomials,
Stirling permutations on a multiset, $\gamma$-positivity,
increasing plane trees
\noindent
{\bf AMS MSC:} 05A15, 05A19
\section{Introduction}
This work is concerned with the partial
$\gamma$-coefficients of the
Eulerian polynomials on multiset
Stirling permutations,
which are also called the Stirling polynomials.
For $n\geq 1$, let ${S}_n$ denote the set of
permutations of $[n]=\{1, 2, \ldots, n\}$.
For a permutation $\sigma=\sigma_1\sigma_2\cdots \sigma_n$
in $S_n$, we adopt the convention that
a zero is patched both at the beginning and
at the end of $\sigma$, that is,
$\sigma_0=\sigma_{n+1}=0$.
An ascent of $\sigma$ is
defined to be an index $i$ $(1\leq i \leq n)$
such that $\sigma_{i-1}<\sigma_i$,
whereas a descent is
defined to be an index $i$ $(1\leq i \leq n)$
such that $\sigma_i > \sigma_{i+1}$.
The numbers of ascents and descents of
$\sigma$ are denoted by ${\rm asc}(\sigma)$
and ${\rm des}(\sigma)$, respectively.
The
bivariate Eulerian polynomials $A_n(x, y)$ are defined by
\begin{equation}\label{A_n}
A_n(x,y)=\sum_{\sigma\in {S}_n} x^{{\rm asc}(\sigma)}y^{{\rm des}(\sigma)}.
\end{equation}
Setting $y=1$, $A_n(x,y)$ takes the form
of the usual Eulerian polynomials, or
the descent polynomials of $S_n$.
One of the most remarkable facts about the Eulerian polynomials is the
$\gamma$-positivity discovered by
Foata and Sch\"uzenberger \cite{FS-70},
which has been extensively studied ever since,
see, for example, \cite{A-2018, Branden-2008, Chow-2008, Elizalde-2021}.
We shall choose to work with the
bivariate version $A_n(x,y)$.
The following expression of
$A_n(x, y)$ is called $\gamma$-expansion:
\begin{equation}\label{gammaA}
A_n(x,y)= \sum_{k=1}^{\lfloor (n+1)/2\rfloor} \gamma_{n,k} (xy)^k (x+y)^{n+1-2k}.
\end{equation}
The coefficients $\gamma_{n,k}$ are
called the $\gamma$-coefficients.
Foata and Sch\"uzenberger
discovered a combinatorial interpretation of the $\gamma$-coefficients
implying the positivity. More precisely, it has been shown that
\begin{equation}
\gamma_{n,k}=|\{\sigma\in {S}_n\mid {\rm des}(\sigma)=k,
{\rm ddes}(\sigma)=0\}|,
\end{equation}
where ${\rm ddes}(\sigma)$ means the number of double descents of $\sigma$,
that is, the number of indices $i$
such that $\sigma_{i-1}> \sigma_i >\sigma_{i+1}$.
As a notable extension of the Eulerian polynomials,
Gessel and Stanley
\cite{Gessel-Stanley-1978} introduced the
notion of Stirling permutations whose descent
polynomials have been called the second order
Eulerial polynomials.
For $n\geq 1$, let $[n]_2$ denote
the multiset $\{1^2, 2^2, \ldots, n^2\}$,
where $i^2$ signifies two occurrences of $i$. A permutation $\sigma$
on $[n]_2$ is said to be a Stirling
permutation if for any $i$,
the elements between the two occurrences
of $i$ in $\sigma$, if any, are greater than $i$. For $n\geq 1$, the set of
Stirling permutations of $[n]_2$ is usually denoted by $Q_n$. As before,
we assume that a Stirling permutation
is patched a zero both at the beginning
and at the end. The statistics ${\rm asc}$ and ${\rm des}$ can be analogously defined for Stirling permutations.
The number of Stirling permutations in $Q_n$ with $k+1$ descents,
often denoted by $C(n,k)$, is called the second order Eulerian number.
The generating function $C_n(x)$ of $C(n,k)$ has been referred to as
the second order Eulerian polynomial.
For Stirling permutations,
one more statistic naturally comes on the scene, that is, the number of plateaux. It appears that the notion
of a plateau was first introduced by
Dumont \cite{Dumont-1980} under the name of
a repitition. Let $\sigma=\sigma_1\sigma_2\cdots \sigma_{2n} \in Q_n$. An index $i$
$(1\leq i \leq 2n)$ is called a plateau
if $\sigma_{i}=\sigma_{i+1}$. The
number of plateaux of $\sigma$ is
denoted by ${\rm plat}(\sigma)$.
B\'ona \cite{Bona-2008} proved that the three
statistics ${\rm asc}$, ${\rm plat}$ and ${\rm des}$
are equidistributed over $Q_n$.
Janson \cite{Janson-2008} constructed an urn model to prove the symmetry of the joint distribution of the three statistics.
In fact, Dumont \cite{Dumont-1980}
defined the trivariate second order Eulerian polynomials
\begin{equation}\label{C-n}
C_{n}(x,y,z)=\sum_{\sigma\in {Q}_{n}}
x^{{\rm asc}(\sigma)}y^{{\rm des}(\sigma)}z^{{\rm plat}(\sigma)},
\end{equation}
which can be regarded as an extension
to the second order Eulerian polynomials
of Gessel-Stanley and the bivariate
Eulerian polynomials. Apparently, when
a Stirling permutation $\sigma \in Q_n$
has $n$ plateaux, it can be considered
as a permutation on $[n]$ with each element $i$ replaced by $ii$.
It was noticed by Dumont that
$C_n(x,y,z)$
are symmetric in $x,y,z$.
The question of $\gamma$-positivity
for $C_n(x,y,z)$ has been studied by
Ma-Ma-Yeh \cite{Ma-Ma-Yeh-2019}. Write
\begin{equation}\label{c-n-gamma-1}
C_n(x,y,z) = \sum_{i=1}^n z^i \sum_{j=0}^{\lfloor (2n+1-i)/2\rfloor}
\gamma_{n, i, j} (xy)^j (x+y)^{2n+1-i-2j},
\end{equation}
which is called the partial $\gamma$-expansion
of $C_n(x,y,z)$. The coefficients $\gamma_{n,i,j}$
are called the partial $\gamma$-coefficients.
Making use of a context-free grammar argument,
Ma-Ma-Yeh showed that $C_n(x,y,z)$ are partial $\gamma$-positive
in the sense that the coefficients $\gamma_{n,i,j}$ are nonnegative.
Moreover, they obtained a combinatorial interpretation
of $\gamma_{n,i,j}$ resorting to certain statistics on
Stirling permutations.
The structure of a Stirling permutation can be
further extended to a multiset. Throughout this paper,
we assume that $n\geq 1$. Unless specified otherwise, we always assume that
\begin{equation}\label{m}
M = \{ 1^{k_1}, 2^{k_2}, \ldots, n^{k_n} \},
\end{equation}
where $k_i\geq 1$ for all $i$ and $i^{k_i}$
stands for $k_i$ occurrences of $i$.
Moreover, we always designate $K$ to denote $k_1+k_2+\cdots + k_n$.
A permutation $\sigma=\sigma_1\sigma_2\ldots \sigma_K$ of $M$
is said to be a { Stirling } permutation if $\sigma_i=\sigma_j$ with $i<j$, then $\sigma_k \geq \sigma_i$ for any $i<k<j$.
The set of Stirling permutations of $M$
will be denoted by $Q_M$.
For $M=[n]_k=\{ 1^k, 2^k, \ldots, n^k\}$,
a Stirling permutation on $M$ is called
a $k$-Stirling permutation.
The statistics ${\rm asc} $, ${\rm des} $ and ${\rm plat}$ for Stirling permutations
in $Q_n$ can be literally carried over to
$Q_M$. Then
the Eulerian
polynomials
on Stirling permutations of a multiset $M$ can be defined by
\begin{equation}\label{Q_n}
C_{M}(x,y,z)=\sum_{\sigma\in {Q}_{M}}x^{{\rm asc}(\sigma)}y^{{\rm des}(\sigma)}z^{{\rm plat}(\sigma)},
\end{equation}
see also \cite{Lin-Ma-Zhang-2021}. In particular, the descent polynomial
over $Q_M$ is often denoted by
\begin{equation}
Q_M(x)= \sum_{\sigma\in {Q}_{M}}x^{{\rm des}(\sigma)},
\end{equation}
see also \cite{Yan-Zhu-2022}. As shown by Frenti \cite{Brenti-1989},
$Q_M(x)$ has only real roots for any multiset $M$.
While $C_M(x,y,z)$ are no longer symmetric
in general, they are symmetric in $x$ and $y$. This means that
$C_{M}(x,y,z)$ can be expressed as
\begin{equation}\label{gammaQ}
C_{M}(x,y,z)=\sum_{i=0}^{K-n}z^i \sum_{j=1}^{\lfloor (K+1-i)/2 \rfloor} \gamma_{M, i,j}(xy)^j(x+y)^{K+1-i-2j}.
\end{equation}
The above relation (\ref{gammaQ}) is
called the { partial $\gamma$-expansion} of $C_{M}(x,y,z)$. The coefficients $\gamma_{M,i,j}$ are called the { partial $\gamma$-coefficients} of $C_{M}(x,y,z)$. The nonnegativity of the coefficients $\gamma_{M, i,j}$ is referred to as the {partial $\gamma$-positivity}.
The partial $\gamma$-positivity
for several multivariate ploynomials associated various classes of permutations
has recently been studied in
Ma-Ma-Yeh \cite{Ma-Ma-Yeh-2019},
Lin-Ma-Zhang \cite{Lin-Ma-Zhang-2021} and Yan-Huang-Yang \cite{Yan-Huang-Yang-2021}.
The objective of this paper
is to present a combinatorial treatment of
the partial $\gamma$-coefficients of
$C_M(x,y,z)$.
In particular, our strategy
can be adapted to deal with
the partial $\gamma$-coefficients of the
second order Eulerian polynomials, which in turn can be
readily converted to
the combinatorial formulation obtained
by Ma-Ma-Yeh \cite{Ma-Ma-Yeh-2019}.
Our combinatorial interpretation
of $\gamma_{M,i,j}$ is built on the Gessel trees
which are increasing plane trees in which the internal vertices are
represented by distinct numbers,
whereas in
the combinatorial framework of Yan, Huang and Yang
\cite{Yan-Huang-Yang-2021}, two vertices
are allowed to be represented by the
same number.
The underlying combinatorial structure
that is concerned with in this work is that of
a Gessel tree for a Stirling permutation of a multiset.
Our main result (Theorem \ref{mainthT}) is a combinatorial interpretation
in light of canonical Gessel trees. A closely related approach is
to utilize context-free grammars, which leads to the notion of pruned Gessel
trees. {A careful study of the Gessel correspondence
reveals the properties needed to
turn Theorem \ref{mainthT}
into an equivalent
statement on multiset
Stirling permutations (Theorem \ref{mainthQ}).}
Specializing to the set $Q_n$ of
Stirling permutations, we are led to the combinatorial interpretation
of the partial $\gamma$-coefficients (Theorem \ref{thma})
due to Ma-Ma-Yeh \cite{Ma-Ma-Yeh-2019}.
The rest of this paper is organized as follows. In Section 2,
we give an overview of the Gessel correspondence
between increasing trees and multiset Stirling permutations. We also address a refined property of the Gessel correspondence. Section 3 is devoted to
a classification of Gessel trees based
on the structure of canonical Gessel trees. An operation in the spirit of the
Foata-Strehl action \cite{Foata-Strehl-1974}
is introduced to serve the purpose. As a main result of this work, we present a combinatorial
interpretation of the partial $\gamma$-coefficients of $C_M(x,y,z)$. In Section 4, we present a context-free grammar approach. In fact, such a consideration has spurred the notion of
pruned Gessel trees and has provided a
motivation for the Foata-Strehl action in Section 3. The aim of Section 5 is to present a combinatorial explanation
of the
partial $\gamma$-coefficients in terms
of multiset Stirling permutations. In Section 6, we explain how to get to the
result of Ma-Ma-Yeh. It turns out that the
symmetry property of $C_n(x,y,z)$ is
required to fulfill the task.
\section{The Gessel correspondence}
The Gessel correspondence \cite{Gessel-2020} is an
extension of the classical
bijection between permutations and
increasing binary trees to Stirling permutations on a multiset, see also \cite{Chow-2008,
JKP-2011, Elizalde-2021}.
It is the aim of this paper to
utilize this correspondence to
study the $\gamma$-positivity and
the partial $\gamma$-positivity of the Eulerian polynomials on
multiset Stirling permutations.
Gessel \cite{Gessel-2020} established a bijection
$\phi$ between $k$-Stirling permutations and $(k+1)$-ary
increasing trees in which only the
internal vertices are labeled and represented by
solid dots, whereas the external
vertices (leaves) are not labeled and represented by circles.
For a Stirling permutation $\sigma$ on $M$. If $\sigma=\emptyset$, let $\phi(\sigma)$ be the tree with only one (unlabeled) vertex. If $\sigma\neq \emptyset$, let $i$ be the smallest element of $\sigma$. Then $\sigma$ can be uniquely decomposed as $w_0 \,i \, w_1 \, i \, \cdots \, w_{k-1} \,
i \, w_k$, where $w_j$ is either empty or a Stirling permutation for all $0 \leq j\leq k$.
Set $i$ to be the root of $\phi(T)$,
and set $\phi(T_j)$ to be the $j$-th subtree (counting from left to right) of $i$. This procedure yields a recursive construction
of $\phi(\sigma)$.
For example,
let $\sigma= 33552217714664 \in {Q}_{7}$, the corresponding ternary
increasing tree is illustrated in Figure \ref{f1}.
\begin{figure}
\caption{A ternary
increasing tree.}
\label{f1}
\end{figure}
Recall that for the classical representation of
permutations by increasing
binary trees, every vertex (regardless
of an internal vertex or a leaf) is labeled,
whereas in the Gessel representation, only the
internal vertices are labeled, and external
vertices (leaves) are not labeled, where a leaf
is drawn as a circle.
In the Gessel representation, a leaf
is called an {$x$-leaf} if it is the first
child, and is called a {$y$-leaf} if it is
the last child; otherwise, it is called a
$z$-leaf. Such leaves are crucial
for the study of the $\gamma$-positivity. As will be seen,
such leaves are the natural ingredients of a
Foata-Strehl action.
Furthermore, Janson-Kuba-Panholzer \cite{JKP-2011} introduced the notion of a
$j$-plateau of a $k$-Stirling permutaiton.
In this event, an index $i$ is said to be a
{$j$-plateau} of $\sigma$
if $\sigma_i=\sigma_{i+1}=r$ and $\sigma_{i}$
is the $j$-th occurrence of $r$ in $\sigma$.
If a leaf $v$ of $T$ is the $j$-th child for some $2\leq j\leq k$, then $v$ is called a { $z_j$-leaf}. Janson-Kuba-Panholzer \cite{JKP-2011} showed that a $z_j$-leaf in
a $(k+1)$-ary increasing tree
corresponds to a $j$-plateau of $\sigma$.
By a Gessel tree on $M$ we mean a
plane tree with internal vertices $1, 2, \ldots, n$
along with unlabeled external vertices such that the internal
vertex $i$ has exactly $k_i+1$
children and the internal vertices form an increasing tree.
For instance, Figure \ref{f2} exhibits a Gessel
tree on $M =\{1^2, 2,3^2,4^2,5^2, 6^3, 7\}$.
\begin{figure}
\caption{A Gessel tree on $M =\{1^2, 2,3^2,4^2,5^2, 6^3, 7\}
\label{f2}
\end{figure}
Denote by ${\rm xleaf}(T)$, ${\rm yleaf}(T)$ and ${\rm zleaf}(T)$ the numbers of $x$-leaves, $y$-leaves and $z$-leaves, respectively.
The following property can be easily deduced from the
recursive construction of the Gessel correspondence.
\begin{proposition}\label{lem1}
Let $n$ and $M$ be given as before.
The Gessel map $\phi$ establishes a one-to-one
correspondence between Stirling permutations of $M$ and Gessel
trees on $M$. Moreover, let
$\sigma\in {Q}_{M}$ and $T=\phi(\sigma)$. Then
\begin{equation}\label{sigma-T}
({\rm asc}(\sigma), {\rm des}(\sigma), {\rm plat}(\sigma))=({\rm xleaf}(T), {\rm yleaf}(T), {\rm zleaf}(T)).
\end{equation}
\end{proposition}
Instead of reproducing a proof of the above property,
we discuss a refined description of the Gessel correspondence that
will be needed later in this paper. For this purpose, we shall
introduce the notion of the Gessel decomposition of a
Stirling permutation on $M$.
Let $\sigma$ be a Stirling permutation on $M$.
For any $1\leq i \leq n$, the $i$-segment of $\sigma$, denoted by
$S_i(\sigma)$, is defined to be
the unique sequence $\sigma_r \sigma_{r+1} \cdots \sigma_s$
containing the element $i$, where
$1\leq r \leq s \leq K$, such that
$\sigma_{r-1} < \sigma_{r}$ and $\sigma_{s} > \sigma_{s+1}$
with the convention $\sigma_0=\sigma_{K+1}=0$.
It is clear from the definition of
a Stirling permutation of
$M$ that the $i$-segment of $\sigma$ is well-defined. In fact, one sees that
$S_i(\sigma)$ contains all the
occurrences of $i$ in $\sigma$.
For example, for the Stirling permutation $\sigma=55 33 2 11 4 666 7 4$ of $M =\{1^2, 2,3^2,4^2,5^2, 6^3, 7\}$ corresponding to the Gessel tree in Figure 2, we have
\begin{eqnarray*}
S_1(\sigma)&= & 55 33 2 11 4 666 7 4, \\[1pt]
S_2(\sigma) & = & 55332, \\[1pt]
S_3(\sigma) & = & 5533, \\[1pt]
S_4 (\sigma)& = & 4 666 7 4 , \\[1pt]
S_5(\sigma) & = & 55, \\[1pt]
S_6 (\sigma)&= & 6667 , \\[1pt]
S_7(\sigma) & = & 7.
\end{eqnarray*}
The idea of the Gessel correspondence can be perceived
as a decomposition of an $i$-segment of a Stirling permutation. Let $\sigma$ be
a Stirling permutation on a multiset $M$. The Gessel
decomposition of the $i$-segament $S_i(\sigma)$ is
defined to be a decomposition
\begin{equation}\label{GD} S_i(\sigma)=w_0\,i\, w_1 \, i \, w_2 \, i \, \cdots \, i \, w_{k_i},
\end{equation}
where for each $0\leq t \leq k_i$, $w_t$ is either empty or
a $j$-segment for some $j$.
We now come to the following refined property
of the Gessel correspondence.
\begin{proposition}
\label{RGC}
Let $n$ and $M$ be given as before.
Let $\sigma$ be a Stirling permutation on $M$ and let
$T$ be the corresponding Gessel tree. Assume that
$\sigma_p$ is the first occurrence of $i$ in $\sigma$ and
$\sigma_q$ is the last occurrence of $i$ in $\sigma$.
Then the vertex $i$ has an $x$-leaf in $T$ if and only if
$p$ is an ascent and $i$ has a
$y$-leaf if and only if $q$
is a descent of $\sigma$.
\end{proposition}
\noindent{\em Proof.}
First, assume
that $i$ has an $x$-leaf in $T$.
By the Gessel correspondence,
$w_0$ in the Gessel decomposition of $\sigma$ as
given in (\ref{GD}) is empty.
Since an $i$-segment of $\sigma$ is surrounded by two elements smaller than $i$, $\sigma_p$ is
directly preceded by a smaller
element. That means that $p$ is
an ascent of $\sigma$. Conversely, if $p$ is an ascent of $\sigma$, then $w_0$ must be
empty, and so $i$ has an $x$-leaf in $T$. The same reasoning applies to a descent
of $\sigma$ involving the last
occurrence of $i$ and a $y$-leaf of $i$ in $T$, and hence the proof is complete.
\rule{4pt}{7pt}
\section{A Foata-Strehl action on
Gessel trees}
To give a combinatorial interpretation of the $\gamma$-coefficients,
we shall define an action on a Gessel tree, which plays the same role
as the original Foata-Strehl group
action for the evaluation of the $\gamma$-coefficients. Such an action
is often called a modified Foata-Strehl action.
Let $T$ be a Gessel tree
on $M$.
We say that an $x$-leaf in $T$ is balanced if its parent has a $y$-leaf; otherwise, we say that the $x$-leaf is unbalanced. Similarly, a $y$-leaf is said to be
balanced if its parent has an $x$-leaf; otherwise, it is said to be unbalanced.
For $1\leq i \leq n$, the Foata-Strehl
action $\psi_i$ is defined as follows. It does nothing to $T$ unless the internal vertex $i$ possesses an unbalanced $y$-leaf. In case
the vertex $i$ has unbalanced $y$-leaf,
then
$\psi_i(T)$ is defined to be the Gessel tree obtained from $T$ by interchanging the first child
(along with its the subtree) and the last child of vertex $i$, and keeping the order of the other children unchanged. As a result, the unbalanced $y$-leaf becomes an unbalanced $x$-leaf. Figure \ref{f3} depicts the action of $\psi_2$ on the Gessel tree in
Figure \ref{f2}.
\begin{figure}
\caption{ Action of $\psi_2$
on the Gessel tree in Figure \ref{f2}
\label{f3}
\end{figure}
\begin{figure}
\caption{A canonical Gessel tree.}
\label{f4}
\end{figure}
Let $G_M$ be the set of Gessel
trees on $M$. By Proposition \ref{lem1},
the Eulerian polynomials $C_M(x,y,z)$
as defined in (\ref{C-n}) can be expressed
in terms of the Gessel trees, namely,
\begin{equation}
C_M(x,y,z)= \sum_{T\in G_M }x^{ {{\rm xleaf}(T)}}y^{{\rm yleaf}(T)}z^{{\rm zleaf}(T)}.
\end{equation}
The above connection is our starting
point to arrive at a combinatorial
interpretation of the partial $\gamma$-coefficients of $C_M(x,y,z)$
in the context of Gessel trees.
We need a special class of Gessel trees,
which we call canonical Gessel trees.
To be more specific, we say that
a Gessel tree is canonical if it does not contain any internal vertex having an unbalanced $y$-leaf.
\begin{theorem}\label{mainthT}
Let $n$, $M$ and $K$ be given as before. Then
\begin{equation}
\label{gammaT}
C_{M}(x,y,z)=\sum_{i=0}^{K-n}z^i \sum_{j=1}^{\lfloor (K+1-i)/2 \rfloor} \gamma_{M, i,j}(xy)^j(x+y)^{K+1-i-2j},
\end{equation}
where
$\gamma_{M, i,j}$ is the number of
canonical Gessel trees on $M$ with $i$ $z$-leaves, $j$ $y$-leaves.
\end{theorem}
\noindent{\em Proof}.
Let $G_{M, i}$ denote the set of Gessel
trees on $M$ with $i$ $z$-leaves.
Then Theorem \ref{mainthT} is equivalent to
\begin{equation}\label{eqT}
\sum_{T\in G_{M,i}}x^{{\rm xleaf}(T)}y^{{\rm yleaf}(T)}
z^{{\rm zleaf}(T)}
= \sum_{T\in H_{M,i}}(xy)^{{\rm yleaf}(T)}(x+y)^{K+1-i-2{\rm yleaf}(T)} z^i,
\end{equation}
where $H_{M,i}$ denotes the
set of canonical trees $T\in G_{M, i}$ without any unbalanced $y$-leaves.
For each $T\in H_{M,i}$, we define ${\rm Orbit}(T)$ to be the
set of Gessel trees $S$ that can be transformed to $T$
via the Foata-Strehl action.
Clearly, the number of $z$-leaves is invariant under any
Foata-Strehl action.
For any internal vertex $j$ of $T$ with an unbalanced $x$-leaf, there is a Gessel tree $S$
such that $\psi_j(S)=T$. Let ${\rm uxleaf}(T)$ and ${\rm bxleaf}(T)$ denote the
numbers of unbalanced $x$-leaves and
balanced $x$-leaves of $T$, respectively.
Since there are no unbalanced $y$-leaves in $T$,
\[ {\rm bxleaf}(T) = {\rm yleaf}(T).\]
Given that the total number of leaves in $T$ equals $K+1$,
we find that \[
2\, {\rm yleaf}(T)+ {\rm uxleaf}(T) ={\rm xleaf}(T) +{\rm yleaf}(T) = K+1-i,
\]
that is,
\begin{equation}
{\rm uxleaf}(T) = K+1-i-2 {\rm yleaf}(T).
\end{equation}
Therefore,
$$
\sum_{S\in {\rm Orbit}(T)}x^{{\rm xleaf(T)}}y^{{\rm yleaf(T)}}
z^{{\rm zleaf}(T)} = (xy)^{{\rm yleaf}(T)}(x+y)^{K+1-i-2{\rm yleaf}(T)} z^i.
$$
Summing over all
canonical Gessel trees in
$H_{M,i}$ yields (\ref{eqT}), and hence the proof is complete.
\rule{4pt}{7pt}
\section{A context-free grammar approach}
In this section, we present
a context-free grammar approach to the partial
$\gamma$-positivity of
$C_{M}(x,y,z)$. Observe that the construction of
Gessel trees implies a recursive
formula for the computation of the Eulerian polynomials $C_M(x,y,z)$. For the case
of
Stirling permutations, the trivariate second order Eulerian
polynomials $C_n(x,y,z)$ are defined by
(\ref{C-n}). Dumont
\cite{Dumont-1980} deduced the
recursion
\begin{equation}
C_n(x,y,z) = xyz
\left(
\frac{\partial}{\partial x}
+
\frac{\partial}{\partial y} +
\frac{\partial}{\partial z}
\right) C_{n-1}(x,y,z),
\end{equation}
where $n\geq 1$ and $C_0(x,y,z)=x$,
see also Haglund-Visontai \cite{Haglund-Visontai-2012}.
\begin{theorem}
Let $n$ and $M$ be given as before. Set $M'=\{1^{k_1}, 2^{k_2}, \ldots, (n-1)^{k_{n-1}}\}$. Then
\begin{equation} \label{C-M-r}
C_M(x,y,z) = xyz^{k_n-1}\left(
\frac{\partial}{\partial x} +
\frac{\partial}{\partial y} +
\frac{\partial}{\partial z}\right) C_{M'}(x,y,z)
\end{equation}
with $C_\emptyset (x,y,z)=x$.
\end{theorem}
In the language of context-free grammars, for $k\geq 1$, define the grammar
\begin{equation}
G_k = \{ x \rightarrow xy z^{k-1}, \quad y \rightarrow xy
z^{k-1}, \quad z \rightarrow xy
z^{k-1} \}.
\end{equation}
Let $D_k$ denote the formal
derivative with respect to the
grammar $G_k$. Then the above
relation (\ref{C-M-r}) can be rewritten
as
\begin{equation}
C_{M}(x,y,z) = D_{k_n}D_{k_{n-1}} \cdots
D_{k_1}(x),
\end{equation}
where $D_{k_n}D_{k_{n-1}} \cdots
D_{k_1}$ is meant to apply $D_{k_1}$ first,
followed by the
applications of $D_{k_2}$ and so on.
The grammatical expression
is informative to establish a connection to the
partial $\gamma$-positivity of $C_M(x,y,z)$. Thanks to the
idea of the change of variables due to Ma-Ma-Yeh
\cite{Ma-Ma-Yeh-2019}, we set
\[ u=xy, \quad v=x+y \]
to get
\begin{equation}
D_k(u) = D_k(xy) = D_k(x)y + x D_k(y) = xy(x+y) z^{k-1} = uv z^{k-1},
\end{equation}
\begin{equation}
D_k(v) =D_k(x+y)=2xyz^{k-1}= 2u z^{k-1},
\end{equation}
and
\begin{equation}
D_k(z) = xyz^{k-1}=u z^{k-1}.
\end{equation}
Now, for the variables $u,v, z$, the grammar $G_k$ can be recast as
\begin{equation}
\label{gk}
G_k=\{ u \rightarrow uv z^{k-1},
\quad v \rightarrow 2u z^{k-1},
\quad z\rightarrow u z^{k-1}\}.
\end{equation}
The following theorem can be
viewed as an equivalent form of
Theorem \ref{mainthT} reconstructed
on a variation of Gessel trees, called
pruned Gessel trees,
along with a grammatical labeling
by using the variables $u$ and $v$.
It is worth mentioning that the grammatical labeling can be
thought as a guideline to generate the $\gamma$-coefficients. Indeed,
it drops a hint in search for
a Foata-Strehl action.
Intuitively, a pruned Gessel tree
is obtained from a Gessel tree by chopping off the
$x$-leaves and $y$-leaves. We have to say
that this viewpoint alone is of no substantial help.
Here comes the idea of characterizing pruned Gessel trees. Assume that
$T$ is a Gessel tree on $M$. Then
internal vertices of the
pruned Gessel tree obtained from $T$
are of the following four types.
\noindent Type 1: $i$ has $k_i+1$ children, and neither
the first nor the last child is a leaf.
This means that $i$ has neither an $x$-leaf
nor a $y$-leaf in $T$.
\noindent Type 2: $i$ has $k_i$ children, and the first child
of $i$ is not a leaf. This means that $i$
has a $y$-leaf, but no $x$-leaf, in $T$.
In this case, the vertex $i$ is associated with a label $y$.
\noindent Type 3: $i$ has $k_i$ children, and the last child
of $i$ is not a leaf. This means that $i$
has an $x$-leaf, but no $y$-leaf, in $T$.
In this case, the vertex is associated with a label
$x$.
\noindent Type 4: $i$ has $k_i-1$ children. This means
that $i$ has both an $x$-leaf and a $y$-leaf
in $T$. In this case, the vertex $i$ is
associated a label $xy$.
Conversely, the four types of vertices
are sufficient for the generation of
pruned Gessel trees.
Figure \ref{f5} displays the
pruned tree of the canonical Gessel tree in Figure \ref{f4}
along with the $(x,y)$-labeling and the
$(u,v)$-labeling.
When restricted to pruned canonical Gessel trees,
Type 2 vertices are not allowed to show up.
This property enables us to
substitute the label $xy$ with $u$, and the label $x$
with $v$.
\begin{figure}
\caption{A pruned canonical Gessel tree.}
\label{f5}
\end{figure}
The weight of
a pruned canonical
Gessel tree $T$, denoted by $w(T)$,
is defined to be the product of the
$(u,v)$-labels. As usual, the
empty product is meant to be
1. For example, the weight of the
pruned Gessel tree in Figure \ref{f5}
equals $u^3v^3z^5$.
The $\gamma$-polynomial $\gamma_M(u,v,z)$, called the
$\gamma$-polynomial of
$M$, is defined to be the
generating function of the
partial $\gamma$-coefficients
in (\ref{gammaT}). To be more specific,
let
\begin{equation} \gamma_M(u,v,z) = \sum_{i=0}^{K-n}z^i \sum_{j=1}^{\lfloor
(K+1-i)/2 \rfloor}
\gamma_{M, i,j}u^jv^{K+1-i-2j}.
\end{equation}
\begin{theorem}
Let $n$ and $M$ be given as before. Then
\begin{equation}
\gamma_M(u,v,z) = \sum_{T\in P_M} w(T),
\end{equation}
where $P_M$ denotes the set of
pruned canonical Gessel trees on $M$.
\end{theorem}
For
the case of Stirling permutations,
the above grammar $G_k$ reduces to the grammar
given by Ma-Ma-Yeh \cite{Ma-Ma-Yeh-2019}, namely,
\begin{equation} \label{G-MMY}
G= \{ u \rightarrow uv z,
\quad v \rightarrow 2u z,
\quad z\rightarrow u z\},
\end{equation}
where we have used $z$ in place of $w$ in \cite{Ma-Ma-Yeh-2019}.
The above grammatical labeling of
pruned canonical Gessel trees for Stirling
permutations serve as an underlying combinatorial
structure for the grammar in (\ref{G-MMY}).
The following theorem asserts that
pruned canonical Gessel trees
on a multiset $M$ with
the $(u,v)$-labeling
can be generated in the same way as successively applying the
formal derivatives $D_{k_1}$,
$D_{k_2}$, $\ldots$, $D_{k_n}$ to $x$.
\begin{theorem}
Let $n$ and $M$ be given as before.
Then \begin{equation}\gamma_M(u,v,z)
= D_{k_n}D_{k_{n-1}} \cdots
D_{k_1}(x).
\end{equation}
\end{theorem}
The proof is in the same
lines as the argument in \cite{Chen-Fu-2022} for
the grammatical labeling of 0-1-2
plane trees. Instead of presenting a proof in full detail,
it suffices to focus on the
action corresponding to the
rule $v\rightarrow 2uz^{k-1}$
of the grammar $G_k$ as in
(\ref{gk}).
Assume that $T$ is a pruned
canonical Gessel tree on
$$
M'=\{1^{k_1}, 2^{k_2}, \ldots, (n-1)^{k_{n-1}}\}.
$$
As before, we have
$M=\{1^{k_1}, 2^{k_2},
\ldots, n^{k_{n}}\}$. Put $k=k_n$. Suppose that $T$
has a vertex $i$ with the label $v$. This means that
$i$ has $k$ children with the
last child not being a leaf.
There are two ways to generate
a pruned canonical Gessel tree
$S$ from $T$ by appending the
vertex $n$ to $T$ as a child of $i$.
\noindent Case 1. Make $n$ the first child of $i$.
Then $i$ will no longer has a label, and $n$ will
be assigned the label $u$.
Meanwhile, $n$ will have $k-1$ $z$-leaves.
We see that this operation corresponds to
the rule $v\rightarrow uz^{k-1}$.
\noindent Case 2. Swap the
first child and the last
child along with their subtrees in the pruned
canonical Gessel tree $S$
obtained in Case 1. Then
we get a pruned
canonical Gessel tree on $M$. Summing up, these two
cases correspond to the
rule $v\rightarrow 2uz^{k-1}$.
For example, for the pruned
canonical Gessel tree $T$ on
$M'=\{1^2, 2, 3^2, 4^2, 5^2, 6\}$
in Figure \ref{f6},
there are two ways to append the
vertex $7$ to $T$ as a child of $4$
to produce a pruned canonical
Gessel tree on $M=M' \cup \{ 7^3\}$.
\begin{figure}
\caption{The two possibilities for the rule
$v\rightarrow 2uz^{k-1}
\label{f6}
\end{figure}
It must be understood, however, that
the notion of pruned
canonical Gessel trees is
somewhat cosmetic, in the sense that whereas it
does not change anything in nature, it may have an effect on the impression.
In fact, we might
as well keep the original
$(x,y)$-leaves, and transport
the labels of the $(x,y)$-leaves to their parents. Nevertheless, the pruned version bears the advantage of
taking a simpler form
especially for permutations where the structure
of 0-1-2 increasing plane trees comes into play.
To conclude this section, we claim that
the grammar $G_k$ captures all the possibilities of constructing pruned canonical
Gessel trees on $M$ from the ones on $M'$. It is only a matter of exercise to verify
that this is indeed the case.
\section{The partial $\gamma$-coefficients}
In this section, we demonstrate that
the combinatorial interpretation
of the partial $\gamma$-coefficients
of the Eulerian polynomials of Stirling permutations
on a multiset falls into the scheme of
canonical Gessel trees.
First, we need to translate the
defining property of an unbalanced $y$-leaf
into the language of Stirling permutations. This goal can be
achieved with the aid of Proposition
\ref{RGC} on the implications of
the $x$-leaves and $y$-leaves to
Stirling permutations.
Let $\sigma$ be a Stirling permutation of $M$.
Observe that a descent $i$ arises only when
$\sigma_i$ is the last occurrence. Similarly, an ascent
$j$ arises only when $\sigma_i$ is the first occurrence.
Assume that an index $i$ is a descent of $\sigma$, and assume that
$\sigma_p$ is the first occurrence of $\sigma_i$, where
$p\leq i$.
As an extension of the notion of a double descent of a
permutation of $[n]$, we say that $i$ is a double fall
of $\sigma$ if $i$ is a descent and $p-1$ is a descent
as well. Keep in mind the convention that
$\sigma_0=\sigma_{K+1}=0$.
Denote by ${\rm dfall}(\sigma)$ the number of double falls of $\sigma$.
For example, for the Stirling permutation $\sigma=2533114664$, the descents $2,9,10$ are not double falls, whereas the descent $4$ is a double fall.
\begin{proposition}
\label{p-5-1}
Let $n$ and $M$ be given as before.
Assume that $\sigma$ is a
Stirling permutation on $M$ and $T$ is
the corresponding Gessel tree of $\sigma$. Then
an index $i$ is a double fall
of $\sigma$ if and only if
the vertex $\sigma_i$ in $T$ has an unbalanced
$y$-leaf.
\end{proposition}
\noindent{\em Proof.}
Let $j=\sigma_i$, and let $S_j(\sigma)$
be the $j$-segment with the first occurrence
of $j$ at position $p$ and the last occurrence
of $j$ at position $q$. Moreover,
let
\begin{equation}\label{sj}
S_j(\sigma) = w_0 \, j \, w_1 \, j\,
\cdots \, w_{k_j -1} \, j \, w_{k_j}
\end{equation}
be the Gessel decomposition of $S_j(\sigma)$.
Assume that $j$ has an
unbalanced $y$-leaf. We proceed to show that $i$ is a
double fall of $\sigma$. The $y$-leaf indicates that
the last factor $w_{k_j}$ is empty, from which it
follows that $i$ is a descent, since $S_j(\sigma)$ is
surrounded by smaller elements at both ends. Suppose the first occurrence
of $j$ appears at position $p$. It remains to confirm that $p-1$
is also a descent. The $y$-leaf is
unbalanced means that the segment $w_0$
in (\ref{sj}) is nonempty. By the definition
of the Gessel decomposition, any element
in $w_0$ is greater than $j$, so that
$p-1$ must be a descent. This completes
the proof.
\rule{4pt}{7pt}
Given the above characterization
of unbalanced $y$-leaves in terms
of Stirling permutations, we
obtain the following interpretation
of the partial $\gamma$-coefficients.
\begin{theorem}\label{mainthQ}
Let $n$, $M$ and $K$ be given as before.
For $0\leq i\leq K-n $ and $1\leq j\leq {\lfloor(K+1-i)/2 \rfloor}$, we have
\begin{equation}\label{M-i-j}
\gamma_{M, i,j}=|\{\sigma\in {Q}_{M}\mid {\rm plat}(\sigma)=i,\, {\rm des}(\sigma)=j,\, {\rm dfall}(\sigma)=0\}|.
\end{equation}
\end{theorem}
Notice that a double fall of a Stirling permutation of
a multiset boils down to a double descent of a permutation of $[n]$.
Therefore, when specialized to $M=[n]$, Theorem \ref{mainthQ}
reduces to the $\gamma$-expansion (\ref{gammaA})
of the bivariate Eulerian polynomials $A_n(x, y)$
due to Foata and Sch\"uzenberger \cite{FS-70}.
\section{A theorem of Ma-Ma-Yeh}
In this section, we show
that our combinatorial interpretation of the partial
$\gamma$-coefficients given the preceding section reduces to a theorem of Ma-Ma-Yeh
\cite{Ma-Ma-Yeh-2019} subject to a restatement prompted by a symmetry consideration.
Recall that the partial
$\gamma$-coefficients $\gamma_{n,i,j}$ for $C_n(x,y,z)$ are defined by
\begin{equation}\label{c-n-gamma}
C_n(x,y,z) = \sum_{i=1}^n z^i \sum_{j=0}^{\lfloor (2n+1-i)/2\rfloor}
\gamma_{n, i, j} (xy)^j (x+y)^{2n+1-i-2j}.
\end{equation}
Let $\sigma\in {Q}_{n}$. An index $1 \leq i \leq 2n$ is called
an {ascent-plateau} if $\sigma_{i-1}< \sigma_{i}=\sigma_{i+1}$ and a
{ descent-plateau} if $\sigma_{i-1}> \sigma_{i}=\sigma_{i+1}$.
Let ${\rm aplat}(\sigma)$ and ${\rm dplat}(\sigma)$
denote the numbers of ascent-plateaux and descent-plateaux, respectively.
The following finding is due to Ma-Ma-Yeh \cite{Ma-Ma-Yeh-2019}.
\begin{theorem} \label{thma}
For $n\geq 1$, $0\leq i\leq n$ and
$1\leq j\leq \lfloor (2n+1-i)/2\rfloor$, we have
\begin{equation} \label{gnij}
\gamma_{n,i,j} = | \{ \sigma \in {Q}_n \, | \,
{\rm des} (\sigma) =i, \;
{\rm aplat} (\sigma) = j, \;
{\rm dplat} (\sigma)=0\} |.
\end{equation}
\end{theorem}
For the case of Stirling permutations, that is, $M=[n]_2$, we
write $G_n$ for $G_M$.
By Theorem \ref{mainthT},
$\gamma_{n, i, j}$ equals
the number of trees $T\in G_n$
with $i$ $z$-leaves and $j$ $y$-leaves
but with no unbalanced $y$-leaves.
In the meantime, Theorem \ref{mainthQ} provides
a formulation resorting to the notion
of a double fall of a Stirling permutation.
In this situation, a double fall can be described as follows.
Let $n\geq 1$ and $\sigma\in Q_n$.
Assume that $i$ is a descent of $\sigma$. Let $\sigma_i=j$.
Then $\sigma_i$ must be the second occurrence of $j$ in
$\sigma$. Assume that $\sigma_p$ is the first
occurrence of $j$ in $\sigma$. Now, the descent
$i$ is called a double fall if $p-1$ is a descent of
$\sigma$ as well.
While there is no doubt that Theorem \ref{mainthQ} offers a
legitimate combinatorial statement, one could not
help wondering about the
connection to the
Ma-Ma-Yeh story. It turns out that the answer
lies in the symmetry of the polynomials $C_n(x,y,z)$.
As far as
Stirling permutations are concerned, Theorem
\ref{mainthQ} can be reassembled with regards to
a slight twist of canonical Gessel trees.
In fact, one only needs to interchange the roles of the $x$-leaves and the $y$-leaves.
Equivalently, we define a canonical Gessel tree on $[n]_2$,
or a canonical ternary increasing tree,
by imposing the constraint that there are no internal vertices having
a $z$-leaf, but no $x$-leaf.
\begin{theorem}
For $n\geq 1$, the partial $\gamma$-coefficient
$\gamma_{n,i,j}$ for Stirling permutations as defined
by (\ref{c-n-gamma}) equals the number of canonical
ternary increasing trees on $[n]_2$ with $i$ $y$-leaves and $j$ $x$-leaves.
\end{theorem}
Notice that the above explanation of $\gamma_{n,i,j}$
can be directly justified in the same manner as the proof of Theorem
\ref{mainthT}.
The following proposition gives a characterization
of Stirling permutations corresponding to
canonical ternary increaing trees.
\begin{proposition} Let $n\geq 1$ and $\sigma \in Q_n$.
Let $T$ be the corresponding increasing ternary increasing tree
of $\sigma$.
Then $\sigma$ has a descent-plateau if and only if
$T$ contains an internal vertex having a $z$-leaf, but no $x$-leaf.
That is to say, $\sigma$ has no descent-plateaux
if and only if $T$ is canonical.
\end{proposition}
For example, the vertex $2$ in the ternary increasing tree
in Figure 1 has a $z$-leaf, but no $x$-leaf. The corresponding
Stirling permutation $\sigma= 33552217714664 $ has a descent-plateau
$522$. On the other hand, Figure 7 furnishes a canonical
ternary increasing tree. The corresponding Stirling
permutation $\sigma= 22 33 55 1 77 1 4 66 4 $ contains no descent-plateaux.
\begin{figure}
\caption{A canonical ternary
increasing tree.
}
\label{f7}
\end{figure}
The proof of the above proposition is analogous to that of
Proposition \ref{p-5-1}. One more thing, we must add that
the statistic ${\rm aplat}(\sigma)$ reflects the number
of internal vertices in the corresponding
ternary increasing tree that have an $x$-leaf and a $z$-leaf
simultaneously. Thus we have reconfirmed the assertion of Ma-Ma-Yeh
in the context of canonical ternary increasing trees
in connection with Stirling permutations.
\vskip 5mm \noindent{\large\bf Acknowledgments.}
We wish to thank the referees for helpful suggestions.
This work was supported
by the National
Science Foundation of China.
\end{document} | math | 40,144 |
\betaegin{document}
\title{Isometric immersions in products of many space forms: an introductory study}
\betaegin{abstract}
This article begins the theory of submanifolds in products of 2 or more space forms. The tensors $\mathbf{R}$, $\mathbf{S}$ and $\mathbf{T}$ defined by Lira, Tojeiro and Vitório at \cite{LTV} and the Bonnet theorem proved by them are generalized for the product of many space forms. Besides, some examples given by Mendonça and Tojeiro at \cite{MT} and the reduction of codimension theorem proved by them are also generalized.
\varepsilonnd{abstract}
\nablaoindent{\betaf Key words:} \mathrm{t}extit{product of space forms, isometric immersions, reduction of codimension, Bonnet theorem}.
\mathrm{t}ableofcontents
\section{Introduction}
A \mathrm{t}extbf{space form} is a (semi-)riemannian manifold whose sectional curvature is constant. To us, a space form is also connected and simply connected. Thus, up to isometries, we just have three kinds of space forms:
\betaegin{enumerate}
\item the euclidean space $\mathbb{R}^n$, which is the space form whose sectional curvature is $0$;
\item the sphere $\mathbb{S}_k^n$, which is the space form whose sectional curvature $k$ is positive;
\item and the hyperbolic space $\mathbb{H}_k^n$, which os the space form whose sectional curvature $k$ is negative.
\varepsilonnd{enumerate}
We will use the following models:
\betaegin{align*}
\mathbb{S}_k^n = \set{x \in \mathbb{R}^{n+1}}{\|x\|^2 = \varphirac{1}{k}}, && \mathrm{t}ext{if} \ k > 0;\\
\mathbb{S}_k^n = \set{x \in \mathbb{L}^{n+1}}{\|x\|^2 = \varphirac{1}{k}}, && \mathrm{t}ext{if} \ k < 0;\\
\mathbb{H}_k^n = \set{x \in \mathbb{L}^{n+1}}{\|x\|^2 = \varphirac{1}{k} \ \mathrm{t}ext{and} \ x_0 > 0}, && \mathrm{t}ext{if} \ k < 0.
\varepsilonnd{align*}
Here, $\mathbb{L}^{n+1}$ is the vector space $\mathbb{R}^{n+1}$ with a the following inner product: $\interno{x}{y} := -x_0y_0 + \sum\lambdaimits_{i=1}^n x_iy_i$. To make things easier, we will denote the space form with dimension $n$ and sectional curvature $k$ by $\o{}$
The authors of \cite{LTV} defined the tensors $\mathbf{R}$, $\mathbf{S}$ and $\mathbf{T}$ associated to an isometric immersion $f \colon M^m \mathrm{t}o \o{i}\mathrm{t}imes\o{2}$. They proved some results involving this tensors and they found the fundamental equations for isometric immersions in products of two space forms. Using these fundamental equations, they proved a Bonnet theorem for isometric immersions in products of two space forms.
The authors of \cite{MT} constructed some examples of isometric immersions in $\o{1}\mathrm{t}imes\o{2}$, they proved some reduction of codimension results, they classified all parallel immersions in $\o{1}\mathrm{t}imes\o{2}$ and they reduced the classification of umbilical immersions $f \colon M^m \mathrm{t}o \o{1}\mathrm{t}imes\o{2}$ (with $m \gammaeq 3$ and $k_1+k_2 \nablae 0$) to the classification of isometric immersions in $\mathbb{S}^n\mathrm{t}imes\mathbb{R}$ and $\mathbb{H}^n\mathrm{t}imes\mathbb{R}$. The umbilical immersions in $\mathbb{S}^n\mathrm{t}imes\mathbb{R}$ were classified in \cite{MT2}.
The present article is a beginning of studying isometric immersions in product of many space forms, 2 or more. So here the tensors $\mathbf{R}$, $\mathbf{S}$ and $\mathbf{T}$ must be defined for isometric immersions $f \colon M^m \mathrm{t}o \o{1} \x \cdots \x \o{\ell}$. Section 2 takes care of these generalized definitions and of the equations involving the new tensors.
Section 3 generalizes some examples given at \cite{MT} and Section 4 generalizes the reduction of codimension theorem of \cite{MT}.
Finally, the last section of this article generalizes the Bonnet theorem of \cite{LTV} for isometric immersion in $\o{1} \x \cdots \x \o{\ell}$.
\section{Isometric immersions in $\o{1} \x \cdots \x \o{\ell}$}
\subsection{The inclusion function $\imath \colon \o{1} \x \cdots \x \o{\ell} \hookrightarrow \mathbb{R}N$}
Let $k_1$, $\cdots$, $k_\varepsilonll$ be real numbers and let $\upsilon, \mathrm{t}au \colon \mathbb{R} \mathrm{t}o \{0,1\}$ be the functions given by
\[\upsilon(x) :=
\betaegin{cases}
1, \ \mathrm{t}ext{if} \ x \nablae 0;\\
0, \ \mathrm{t}ext{if} \ x = 0;
\varepsilonnd{cases} \quad \mathrm{t}ext{and} \quad
\mathrm{t}au(x) := \betaegin{cases}
1, \ \mathrm{t}ext{if} \ x < 0;\\
0, \ \mathrm{t}ext{if} \ x \gammaeq 0.
\varepsilonnd{cases}\]
Let $\mathbb{R}^{N_i} = \mathbb{R}_{\mathrm{t}au(k_i)}^{n_i+\upsilon(k_i)}$, for each $i \in \{1, \cdots, \varepsilonll\}$, and let $\mathbb{R}N = \mathbb{R}^{N_1} \mathrm{t}imes \cdots \mathrm{t}imes \mathbb{R}^{N_\varepsilonll}$. Here, $\mathbb{R}^n_0 = \mathbb{R}^n$ and $\mathbb{R}^n_1 = \mathbb{L}^n$. So there exists a canonical inclusion $\imath \colon \o{1} \x \cdots \x \o{\ell} \hookrightarrow \mathbb{R}N$ given by $\imath(x_1, \cdots, x_\varepsilonll) = (x_1, \cdots, x_\varepsilonll)$, where $x_i \in \o{i}\subset \mathbb{R}^{N_i}$.
Lets denote $\hat{\mathbb{O}} = \o{1} \x \cdots \x \o{\ell}$. Then the codimension of $\imath \colon \hat{\mathbb{O}} \mathrm{t}o \mathbb{R}N$ is the number of elements of the set $J := \set{i \in \{1, \cdots, \varepsilonll\}}{k_i \nablae 0}$.
\betaegin{lem}\lambdaabel{tangentei}
If $J := \set{i \in \{1, \cdots, \varepsilonll\}}{k_i \nablae 0}$ and $x = (x_1, \cdots, x_\varepsilonll) \in \hat{\mathbb{O}}$, with $x_i \in \o{i}$, then
\[T_x \hat{\mathbb{O}} = \set{X \in \mathbb{R}N}{X_i \rhoerp x_i, \varphiorall i \in J},\]
where $X = (X_1, \cdots, X_\varepsilonll)$, with $X_i \in \mathbb{R}^{N_i}$.
\varepsilonnd{lem}
\betaegin{proof}
Let $x \in \hat{\mathbb{O}}$ and $X \in T_x \hat{\mathbb{O}}$. Thus there is a differentiable curve $\beta \colon I \mathrm{t}o \mathbb{O}$ such that $\beta(0) = x$ and $\beta'(0) = X$.
But $\beta(t) = \betaig( \beta_1(t), \cdots, \beta_\varepsilonll(t) \betaig)$ and $\|\beta_i(t)\|^2 = \varphirac{1}{k_i}$, for all $i \in J$. Then $\interno{\beta_i(t)}{\beta_i'(t)} = 0$, for all $i \in J$. It follows that $\interno{x_i}{X_i} = \interno{\beta_i(0)}{\beta_i'(0)} = 0$, for each $i \in J$. Thus $T_x \hat{\mathbb{O}} \subset \set{X \in \mathbb{R}N}{X_i \rhoerp x_i, \varphiorall i \in \{1, \cdots, \varepsilonll\}}$.
Lets consider
\[V_i = \betaegin{cases}
\{x_i\}^\rhoerp, & \mathrm{t}ext{if} \ i \in J; \\
\mathbb{R}^{N_i}, & \mathrm{t}ext{if} \ i \nablaot\in J;
\varepsilonnd{cases}\]
where $\{x_i\}^\rhoerp$ is the orthogonal complement of $\spa\{x_i\}$ in $\mathbb{R}^{N_i}$. Thus
\[\set{X \in \mathbb{R}N}{X_i \rhoerp x_i, \varphiorall i \in J} = V_1 \mathrm{t}imes \cdots \mathrm{t}imes V_\varepsilonll.\]
Since $T_x \hat{\mathbb{O}}$ and $\set{X \in \mathbb{R}N}{X_i \rhoerp x_i, \varphiorall i \in J}$ have the same dimension, they are the same subspace of $\mathbb{R}N$.
\varepsilonnd{proof}
\betaegin{lem} \lambdaabel{lem: pi_j}
Let $\mathbb{R}N = \mathbb{R}^{N_1} \mathrm{t}imes \cdots \mathrm{t}imes \mathbb{R}^{N_\varepsilonll}$ and $\rhoi_i \colon \mathbb{R}N \mathrm{t}o \mathbb{R}N$ be the orthogonal projections given by $\rhoi_i(x_1, \cdots, x_\varepsilonll) = (0, \cdots, 0, x_i, 0, \cdots, 0)$. If $k_i \nablae 0$, then the field $\lambdaeft.\rhoi_i\right|_{\hat{\mathbb{O}}} = \rhoi_i \circ \imath$ is normal to $\hat{\mathbb{O}}$.
\varepsilonnd{lem}
\betaegin{proof}
Let $X \in T_x \hat{\mathbb{O}}$ a tangent vector, thus $X_i \rhoerp x_i$ (by Lemma \ref{tangentei}). It follows that $\interno{(\rhoi_i \circ \imath)(x)}{X} = \interno{x_i}{X_i} = 0$. Therefore $(\rhoi_i \circ \imath)$ is normal to $\hat{\mathbb{O}}$.
\varepsilonnd{proof}
Since the codimension of $\hat{\mathbb{O}}$ in $\mathbb{R}N$ is equal to the number of elements of the set $J := \set{i \in \{1, \cdots, \varepsilonll\}}{k_i \nablae 0}$, then
\betaegin{equation}\lambdaabel{Tiperp}
T_x^\rhoerp \hat{\mathbb{O}} = \spa\set{\rhoi_i(x)}{i \in J} = \spa\{-k_1 \rhoi_1(x), \cdots, -k_\varepsilonll \rhoi_\varepsilonll(x)\}.
\varepsilonnd{equation}
Now, let $\nablabar$ and $\nablatil$ be the Levi-Civita connections in $\hat{\mathbb{O}}$ and $\mathbb{R}N$ respectively, and let $\alpha_{\imath}$, $\betaar{\mathcal{R}}$ and $A_\varepsilonta^\imath$ be (respectively) the second fundamental form of $\imath$, the curvature tensor of $\hat{\mathbb{O}}$ and the shape operator in the normal direction $\varepsilonta$ given by $\interno{A_\varepsilonta^\imath X}{Y} = \interno{\ai{X}{Y}}{\varepsilonta}$.
\betaegin{lem}\lambdaabel{lem: ai}
For all $X, Y \in \Gamma \lambdaeft(T \hat{\mathbb{O}} \right)$, it holds that
\betaegin{equation}\lambdaabel{eq: ai}
\ai{X}{Y} = -\sum_{i=1}^\varepsilonll \interno{\rhoi_i X}{Y}k_i(\rhoi_i \circ \imath),
\varepsilonnd{equation}
Besides, if $k_i \nablae 0$, then $A^{\imath}_{\rhoi_i \circ \imath} = -\rhoi_i$.
\varepsilonnd{lem}
\betaegin{proof}
Let $x \in \hat{\mathbb{O}}$ and $X \in T_x \hat{\mathbb{O}}$. If $k_i \nablae 0$, then $X_i \rhoerp x_i$ and $\rhoi_i X = (0, \cdots, X_i, \cdots, 0) \in T_x \hat{\mathbb{O}}$.
On the other side,
\[\nablatil_X (\rhoi_i \circ \imath)(x) = \mathrm{d} {\rhoi_i}_{\imath(x)}\lambdaeft( \mathrm{d} \imath_x X \right) = \rhoi_i \imath_*X = \imath_* \rhoi_i X \ \mathbb{R}ightarrow \ A^{\imath}_{\rhoi_i \circ \imath} = -\rhoi_i.\]
Let $J = \set{i \in \{1, \cdots, \varepsilonll\}}{k_i \nablae 0}$. Then
\betaegin{align*}
& \ai{X}{Y} = \sum_{i\in J} \interno{\nablatil_X \imath_* Y}{\rhoi_i\circ \imath}\varphirac{\rhoi_i\circ \imath}{\|\rhoi_i\circ \imath\|^2} = \sum_{i\in J} \interno{\ai{X}{Y}}{\rhoi_i\circ \imath}k_i(\rhoi_i\circ \imath) = \\
& = -\sum_{i \in J} \interno{\rhoi_i X}{Y} k_i(\rhoi_i\circ \imath) = -\sum_{i=1}^\varepsilonll \interno{\rhoi_i X}{Y} k_i(\rhoi_i\circ \imath). \qedhere
\varepsilonnd{align*}
\varepsilonnd{proof}
\betaegin{lem} \lambdaabel{lem: curv1}
$\mathrm{d}isplaystyle \betaar{\mathcal R}(X,Y)Z = \sum\lambdaimits_{i=1}^\varepsilonll k_i \lambdaeft(\rhoi_i X \wedge \rhoi_i Y \right) Z$, where $(A\wedge B) C := \interno{B}{C}A - \interno{A}{C}B$.
\varepsilonnd{lem}
\betaegin{proof}
Using Gauss equation,
\betaegin{align*}
& \betaar{\mathcal R}(X,Y)Z = A_{\ai{Y}{Z}}^{\imath} X - A_{\ai{X}{Z}}^{\imath} Y = \\
& \stackrel{\varepsilonqref{eq: ai}}{=} -\sum_{i=1}^\varepsilonll \interno{\rhoi_i Y}{Z} A^{\imath}_{k_i(\rhoi_i \circ \imath)} X + \sum_{i=1}^\varepsilonll \interno{\rhoi_i X}{Z} A^{\imath}_{k_i(\rhoi_i \circ \imath)} Y = \\
& = \sum_{i=1}^\varepsilonll \interno{\rhoi_i Y}{Z} k_i \rhoi_i X - \sum_{i=1}^\varepsilonll \interno{\rhoi_i X}{Z} k_i \rhoi_i Y = \\
&= \sum_{i=1}^\varepsilonll k_i\lambdaeft[\interno{\rhoi_i Y}{Z} \rhoi_i X - \interno{\rhoi_i X}{Z} \rhoi_i Y \right]. \qedhere
\varepsilonnd{align*}
\varepsilonnd{proof}
\betaegin{lem}\lambdaabel{pi_i parallel}
The tensor $\rhoi_i \colon T \hat{\mathbb{O}} \mathrm{t}o T \hat{\mathbb{O}}$ is parallel, that is, $(\nabla_X \rhoi_i) Y = 0, \ \varphiorall X, Y \in \Gamma\lambdaeft(T \hat{\mathbb{O}} \right)$.
\varepsilonnd{lem}
\betaegin{proof}
Here we use $\rhoi_i$ to denote the orthogonal projection $\rhoi_i \colon \mathbb{R}N \mathrm{t}o \mathbb{R}N$ and also the restrictions $\lambdaeft.\rhoi_i\right|_{\hat{\mathbb{O}}} \colon \hat{\mathbb{O}} \mathrm{t}o \hat{\mathbb{O}}$ and $\lambdaeft.\rhoi_i\right|_{T_x \hat{\mathbb{O}}} \colon T_x \hat{\mathbb{O}} \mathrm{t}o T_x \hat{\mathbb{O}}$.
Let $X,Y \in \Gamma\lambdaeft(T \hat{\mathbb{O}} \right)$, thus
\betaegin{align*}
& \imath_* \rhoi_i\nablabar_X Y = \rhoi_i \lambdaeft[\imath_* \nablabar_X Y \right] = \rhoi_i \lambdaeft[ \nablatil_X \imath_*Y \right] - \rhoi_i \lambdaeft[\ai{X}{Y} \right] = \\
& = \rhoi_i \lambdaeft[ \nablatil_X \imath_*Y \right] - \rhoi_i\lambdaeft[ -\sum_{j=1}^\varepsilonll \interno{\rhoi_j X}{Y}k_j(\rhoi_j \circ \imath) \right] = \\
&= \rhoi_i \lambdaeft[ \nablatil_X \imath_*Y \right] + \interno{\rhoi_i X}{Y}k_i(\rhoi_i \circ \imath).
\varepsilonnd{align*}
But $Y \colon \hat{\mathbb{O}} \mathrm{t}o \mathbb{R}N$ and $\imath_* Y = Y$. Thus
\betaegin{align*}
& \rhoi_i \lambdaeft[ \nablatil_X \imath_*Y \right] = \rhoi_i\lambdaeft[\mathrm{d} (\imath_* Y) X \right] = \rhoi_i\lambdaeft[\mathrm{d} Y \cdot X \right] = \mathrm{d} \lambdaeft(\rhoi_i \circ Y \right)\cdot X = \nablatil_X \rhoi_i Y = \\
& = \nablatil_X \imath_* \rhoi_i Y = \imath_* \nablabar_X \rhoi_i Y + \ai{X}{\rhoi_i Y} \stackrel{\varepsilonqref{eq: ai}}{=} \imath_* \nablabar_X \rhoi_i Y - \interno{\rhoi_i X}{\rhoi_i Y}k_i (\rhoi_i \circ \imath) = \\
& = \imath_* \nablabar_X \rhoi_i Y - \interno{\rhoi_i X}{Y}k_i (\rhoi_i \circ \imath).
\varepsilonnd{align*}
Therefore $\imath_* \rhoi_i\nablabar_X Y = \imath_* \nablabar_X \rhoi_i Y$, and hence $\lambdaeft( \nabla_X \rhoi_i \right) Y = 0$.
\varepsilonnd{proof}
\subsection{The $\mathbf{R}_i$, $\mathbf{S}_i$ and $\mathbf{T}_i$ tensors}
Let $M^m$ be a riemannian manifold and $f\colon M^m \mathrm{t}o \hat{\mathbb{O}}$ be an isometric immersion. We will call $\mathcal R$ and $\mathcal{R}^\rhoerp$ the curvature tensors on the tangent bundle ($TM$) and on the normal bundle ($T^\rhoerp M$) of $f$, respectively. Let also $\alpha := \alpha_f \colon (TM \mathrm{t}imes TM) \mathrm{t}o T_f^\rhoerp M$ be the second fundamental form of $f$ and $A_\varepsilonta := A^f_\varepsilonta \colon TM \mathrm{t}o TM$ be the shape operator in the normal direction $\varepsilonta$, given by $\interno{A_\varepsilonta X}{Y} = \interno{\al{X}{Y}}{\varepsilonta}$, for all $X, Y \in TM$.
\betaegin{df}
Let \index{$\mathbf{L}_i$} $\mathbf{L}_i \colon TM \mathrm{t}o T \hat{\mathbb{O}}$, $\mathbf{K}_i\index{$\mathbf{K}_i$} \colon T^\rhoerp M \mathrm{t}o T \hat{\mathbb{O}}$, \index{$\mathbf{R}_i$} $\mathbf{R}_i \colon TM \mathrm{t}o TM$, \index{$\mathbf{S}_i$} $\mathbf{S}_i \colon TM \mathrm{t}o T^\rhoerp M$ and \index{$\mathbf{T}_i$} $\mathbf{T}_i \colon T^\rhoerp M \mathrm{t}o T^\rhoerp M$ be given by
\betaegin{center}
\betaegin{tabular}{ll}
$\mathbf{L}_i := \mathbf{L}_i^f := \rhoi_i\circ {f}_*$; & $\mathbf{K}_i := \mathbf{K}_i^{f} := \lambdaeft.\rhoi_i\right|_{T^\rhoerp M}$; \\
$\mathbf{R}_i := \mathbf{R}_i^f := \mathbf{L}_i^\mathrm{t}\mathbf{L}_i$ ; & $\mathbf{S}_i := \mathbf{S}_i^f := \mathbf{K}_i^\mathrm{t}\mathbf{L}_i$; \\
$\mathbf{T}_i := \mathbf{T}_i^f := \mathbf{K}_i^\mathrm{t} \mathbf{K}_i$.
\varepsilonnd{tabular}
\varepsilonnd{center}
\varepsilonnd{df}
Lets remark that $\mathbf{R}_i$ and $\mathbf{T}_i$ are self-adjoint tensors. Besides, given $X \in T_xM$, $\mathrm{t}imesi \in T_x^\rhoerp M$ and $Z \in T_{f(x)} \hat{\mathbb{O}}$, then
\[\betaegin{cases}
\interno{\mathbf{L}_i^\mathrm{t} Z}{X} = \interno{Z}{\mathbf{L}_i X} = \interno{Z}{\rhoi_if_*X} = \interno{\rhoi_i Z}{f_*X} = \interno{\lambdaeft(\rhoi_i Z \right)^T}{f_* X}, \\
\interno{\mathbf{K}_i^\mathrm{t} Z}{\mathrm{t}imesi} = \interno{Z}{\mathbf{K}_i \mathrm{t}imesi} = \interno{Z}{\rhoi_i\mathrm{t}imesi} = \interno{\rhoi_i Z}{\mathrm{t}imesi} = \interno{\lambdaeft(\rhoi_i Z \right)^\rhoerp}{\mathrm{t}imesi},
\varepsilonnd{cases}\]
where $\lambdaeft( \rhoi_i Z \right)^T$ and $\lambdaeft( \rhoi_i Z \right)^\rhoerp$ are the orthogonal projections of $\rhoi_i Z$ on $TM$ and $T^\rhoerp M$, respectively. Thus $f_* \mathbf{L}_i^\mathrm{t} Z = \lambdaeft(\rhoi_i Z \right)^T$ and $\mathbf{K}_i^\mathrm{t} Z = \lambdaeft(\rhoi_i Z \right)^\rhoerp$. Consequently
\betaegin{align*}
& f_* \mathbf{R}_i X = f_* \mathbf{L}_i^\mathrm{t}\mathbf{L}_i X = \lambdaeft( \rhoi_i \mathbf{L}_i X \right)^T = (\mathbf{L}_i X)^T, \\
& \mathbf{S}_i X = \mathbf{K}_i^\mathrm{t}\mathbf{L}_i X = \lambdaeft( \rhoi_i \mathbf{L}_i X \right)^\rhoerp = (\mathbf{L}_i X)^\rhoerp, \\
& f_*\mathbf{S}_i^\mathrm{t} \mathrm{t}imesi = f_* \mathbf{L}_i^\mathrm{t}\mathbf{K}_i \mathrm{t}imesi = \lambdaeft( \rhoi_i \mathbf{K}_i \mathrm{t}imesi \right)^T = (\mathbf{K}_i \mathrm{t}imesi)^T, \\
& \mathbf{T}_i \mathrm{t}imesi = \mathbf{K}_i^\mathrm{t}\mathbf{K}_i \mathrm{t}imesi = \lambdaeft( \rhoi_i \mathbf{K}_i \mathrm{t}imesi \right)^\rhoerp = (\mathbf{K}_i \mathrm{t}imesi)^\rhoerp.
\varepsilonnd{align*}
Therefore
\betaegin{equation}\lambdaabel{eq: K L}
\rhoi_if_* X = \mathbf{L}_i X = f_*\mathbf{R}_i X + \mathbf{S}_i X \quad \mathrm{t}ext{and} \quad \rhoi_i \mathrm{t}imesi = \mathbf{K}_i \mathrm{t}imesi = f_*\mathbf{S}_i^\mathrm{t} \mathrm{t}imesi + \mathbf{T}_i \mathrm{t}imesi.
\varepsilonnd{equation}
\betaegin{obs} If $X \in T_x M$ and $\mathrm{t}imesi \in T_x^\rhoerp M$, then
\betaegin{align*}
& f_* \mathbf{R}_i X + \mathbf{S}_i X = \rhoi_i f_* X = f_* X - \sum_{\substack{j=1\\ j\nablae i}}^\varepsilonll \rhoi_j f_* X = f_* X - \sum_{\substack{j=1\\ j\nablae i}}^\varepsilonll \lambdaeft( f_* \mathbf{R}_j X + \mathbf{S}_j X \right). \\
& f_* \mathbf{S}_i^\mathrm{t} \mathrm{t}imesi + \mathbf{T}_i \mathrm{t}imesi = \rhoi_i \mathrm{t}imesi = \mathrm{t}imesi - \sum_{\substack{j=1\\ j\nablae i}}^\varepsilonll \rhoi_j \mathrm{t}imesi = \mathrm{t}imesi - \sum_{\substack{j=1\\ j\nablae i}}^\varepsilonll \lambdaeft( f_* \mathbf{S}_j^\mathrm{t} \mathrm{t}imesi + \mathbf{T}_j \mathrm{t}imesi \right).
\varepsilonnd{align*}
Thus
\betaegin{align*}
\mathbf{R}_i &= \id - \sum_{\substack{j=1\\ j\nablae i}}^\varepsilonll \mathbf{R}_j, & \mathbf{S}_i &= - \sum_{\substack{j=1\\ j\nablae i}}^\varepsilonll \mathbf{S}_j, & \mathbf{T}_i &= \id - \sum_{\substack{j=1\\ j\nablae i}}^\varepsilonll \mathbf{T}_j,
\varepsilonnd{align*}
that is,
\betaegin{align}\lambdaabel{somas}
\sum_{i=1}^\varepsilonll \mathbf{R}_i &= \id|_{T_x M}, & \sum_{i=1}^\varepsilonll \mathbf{S}_i &= 0, & \sum_{i=1}^\varepsilonll \mathbf{T}_i &= \id|_{T_x^\rhoerp M}.
\varepsilonnd{align}
\varepsilonnd{obs}
\betaegin{lem}\lambdaabel{lem: R S T}
The following equations hold:
\betaegin{align}
& \mathbf{S}_i^\mathrm{t}\mathbf{S}_i = \mathbf{R}_i(\id - \mathbf{R}_i), && \mathbf{T}_i\mathbf{S}_i = \mathbf{S}_i( \id - \mathbf{R}_i), && \mathbf{S}_i\mathbf{S}_i^\mathrm{t} = \mathbf{T}_i(\id - \mathbf{T}_i), \lambdaabel{eq: RST}\\
& \mathbf{S}_i^\mathrm{t}\mathbf{S}_j \stackrel{i \nablae j}{=} -\mathbf{R}_i\mathbf{R}_j, && \mathbf{T}_i \mathbf{S}_j \stackrel{i \nablae j}{=} -\mathbf{S}_i\mathbf{R}_j, && \mathbf{S}_i \mathbf{S}_j^\mathrm{t} \stackrel{i\nablae j}{=} -\mathbf{T}_i\mathbf{T}_j. \lambdaabel{eq: RST2}
\varepsilonnd{align}
\varepsilonnd{lem}
\betaegin{proof}
Let $X, Y \in TM$ and $\mathrm{t}imesi \in T^\rhoerp M$. Hence
\betaegin{multline*}
f_* \mathbf{R}_i X + \mathbf{S}_i X = \rhoi_if_* X = {\rhoi_i}^2 f_* X = \rhoi_i (f_*\mathbf{R}_i X + \mathbf{S}_i X) = \\
= f_* \mathbf{R}_i^2 X + \mathbf{S}_i\mathbf{R}_i X + f_*\mathbf{S}_i^\mathrm{t}\mathbf{S}_i X + \mathbf{T}_i\mathbf{S}_i X.
\varepsilonnd{multline*}
Consequently, $\mathbf{R}_i X = \mathbf{R}_i^2 X + \mathbf{S}_i^\mathrm{t}\mathbf{S}_i X$ and $\mathbf{S}_i X = \mathbf{S}_i\mathbf{R}_i X + \mathbf{T}_i\mathbf{S}_i X$. Therefore $\mathbf{S}_i^\mathrm{t}\mathbf{S}_i = \mathbf{R}_i(\id - \mathbf{R}_i)$ and $\mathbf{T}_i\mathbf{S}_i = \mathbf{S}_i(\id - \mathbf{R}_i)$.
Analogously,
\betaegin{multline*}
f_* \mathbf{S}_i^\mathrm{t} \mathrm{t}imesi + \mathbf{T}_i \mathrm{t}imesi = \rhoi_i \mathrm{t}imesi = {\rhoi_i}^2 \mathrm{t}imesi = \rhoi_i \lambdaeft(f_* \mathbf{S}_i^\mathrm{t}\mathrm{t}imesi + \mathbf{T}_i\mathrm{t}imesi \right) = \\
= f_* \mathbf{R}_i\mathbf{S}_i^\mathrm{t} \mathrm{t}imesi + \mathbf{S}_i\mathbf{S}_i^\mathrm{t} \mathrm{t}imesi + f_*\mathbf{S}_i^\mathrm{t}\mathbf{T}_i \mathrm{t}imesi + \mathbf{T}_i^2 \mathrm{t}imesi.
\varepsilonnd{multline*}
Therefore, $\mathbf{T}_i \mathrm{t}imesi = \mathbf{S}_i\mathbf{S}_i^\mathrm{t} \mathrm{t}imesi + \mathbf{T}_i^2 \mathrm{t}imesi$ and $\mathbf{S}_i\mathbf{S}_i^\mathrm{t} = \mathbf{T}_i(\id - \mathbf{T}_i)$.
On the other side,
\betaegin{multline*}
0 = \rhoi_i \rhoi_j f_* X = \rhoi_i \lambdaeft( f_* \mathbf{R}_j X + \mathbf{S}_j X \right) = \\
= f_* \mathbf{R}_i\mathbf{R}_j X + \mathbf{S}_i\mathbf{R}_j X + f_* \mathbf{S}_i^\mathrm{t}\mathbf{S}_j X + \mathbf{T}_i\mathbf{S}_j X.
\varepsilonnd{multline*}
Thus $\mathbf{S}_i^\mathrm{t}\mathbf{S}_j = -\mathbf{R}_i\mathbf{R}_j$ and $\mathbf{T}_i\mathbf{S}_j = -\mathbf{S}_i\mathbf{R}_j$.
Analogously,
\[0 = \rhoi_i \rhoi_j \mathrm{t}imesi = \rhoi_i \lambdaeft( f_* \mathbf{S}_j^\mathrm{t} \mathrm{t}imesi + \mathbf{T}_j \mathrm{t}imesi \right) = f_* \mathbf{R}_i\mathbf{S}_j^\mathrm{t} \mathrm{t}imesi + \mathbf{S}_i\mathbf{S}_j^\mathrm{t} \mathrm{t}imesi + f_* \mathbf{S}_i^\mathrm{t}\mathbf{T}_j \mathrm{t}imesi + \mathbf{T}_i\mathbf{T}_j \mathrm{t}imesi.\]
Hence $\mathbf{S}_i^\mathrm{t}\mathbf{T}_j = -\mathbf{R}_i\mathbf{S}_j^\mathrm{t}$ and $\mathbf{T}_i\mathbf{T}_j = -\mathbf{S}_i\mathbf{S}_j^\mathrm{t}$.
\varepsilonnd{proof}
\betaegin{obs}\lambdaabel{observação: autovalor R}
If $\lambda$ is an eigenvalue of $\mathbf{R}_i$ and $X$ is an eigenvector associated to $\lambda$, then
\[ \lambda (1 - \lambda) \|X\|^2 = \interno{\mathbf{R}_i (\id - \mathbf{R}_i) X}{X}\stackrel{\varepsilonqref{eq: RST}}{=} \interno{\mathbf{S}_i^\mathrm{t}\mathbf{S}_i X}{X} = \| \mathbf{S}_i X\|^2 \gammaeq 0.\]
Therefore $\lambda \in [0,1]$. Analogously, using the third equation of \varepsilonqref{eq: RST}, the eigenvectors of $\mathbf{T}_i$ are also in $[0,1]$.
\varepsilonnd{obs}
\betaegin{lem}
The following equations hold:
\betaegin{align}
& (\nabla_X \mathbf{R}_i)Y = A_{\mathbf{S}_i Y}X + \mathbf{S}_i^\mathrm{t} \al{X}{Y}, \lambdaabel{eq: derivada R} \\
& \lambdaeft. \betaegin{aligned}
& (\nabla_X \mathbf{S}_i)Y = \mathbf{T}_i \al{X}{Y} - \al{X}{\mathbf{R}_i Y}, \\
& \betaig(\nabla_X \mathbf{S}_i^\mathrm{t} \betaig) \mathrm{t}imesi = A_{\mathbf{T}_i \mathrm{t}imesi} X - \mathbf{R}_i A_\mathrm{t}imesi X,
\varepsilonnd{aligned}\right\} \lambdaabel{eq: derivada S} \\
& (\nabla_X \mathbf{T}_i)\mathrm{t}imesi = -\mathbf{S}_i A_\mathrm{t}imesi X - \al{X}{\mathbf{S}_i^\mathrm{t}\mathrm{t}imesi}. \lambdaabel{eq: derivada T}
\varepsilonnd{align}
\varepsilonnd{lem}
\betaegin{proof}
Let $X,Y \in \Gamma(TM)$ and $\mathrm{t}imesi \in \Gamma\lambdaeft( T^\rhoerp M \right)$. Because of Lemma \ref{pi_i parallel}, we have that
\betaegin{align*}
&0 = \lambdaeft( \nabla_{f_*X} \rhoi_i \right) f_* Y = \nablabar_{f_* X} \rhoi_if_* Y - \rhoi_i \nablabar_{f_* X} f_* Y = \\
& = \nablabar_X (f_* \mathbf{R}_i Y + \mathbf{S}_i Y) - \rhoi_i \lambdaeft[ f_* \nabla_X Y + \al{X}{Y} \right] = \\
& = \nablabar_X f_* \mathbf{R}_i Y + \nablabar_X \mathbf{S}_i Y - f_*\mathbf{R}_i \nabla_X Y - \mathbf{S}_i \nabla_X Y - \\
& \quad - f_*\mathbf{S}_i^\mathrm{t}\al{X}{Y} - \mathbf{T}_i \al{X}{Y} = \\
& = f_* \nabla_X \mathbf{R}_i Y + \al{X}{\mathbf{R}_i Y} - f_*A_{\mathbf{S}_i Y} X + \nablaperp_X \mathbf{S}_i Y - \\
& \quad - f_*\lambdaeft[\mathbf{R}_i \nabla_X Y + \mathbf{S}_i^\mathrm{t}\al{X}{Y} \right] - \mathbf{S}_i \nabla_X Y - \mathbf{T}_i \al{X}{Y} = \\
& = f_* \lambdaeft[ \lambdaeft(\nabla_X \mathbf{R}_i \right) Y - A_{\mathbf{S}_i Y} X - \mathbf{S}_i^\mathrm{t} \al{X}{Y} \right] + \al{X}{\mathbf{R}_i Y} + \\
& \quad + \lambdaeft( \nabla_X \mathbf{S}_i \right) Y - \mathbf{T}_i \al{X}{Y}.
\varepsilonnd{align*}
Therefore
\[\lambdaeft(\nabla_X \mathbf{R}_i \right) Y = A_{\mathbf{S}_i Y} X + \mathbf{S}_i^\mathrm{t} \al{X}{Y} \ \ \mathrm{t}ext{and} \ \ \lambdaeft( \nabla_X \mathbf{S}_i \right) Y = \mathbf{T}_i \al{X}{Y} - \al{X}{\mathbf{R}_i Y}.\]
Analogously,
\betaegin{align*}
& 0 = \lambdaeft( \nabla_{f_*X} \rhoi_i \right) \mathrm{t}imesi = \\
&= \nablabar_{f_* X} \rhoi_i\mathrm{t}imesi - \rhoi_i \nablabar_{f_* X} \mathrm{t}imesi = \nablabar_{f_* X} \lambdaeft( f_* \mathbf{S}_i^\mathrm{t}\mathrm{t}imesi + \mathbf{T}_i \mathrm{t}imesi \right) - \rhoi_i \lambdaeft( - f_* A_\mathrm{t}imesi X + \nablaperp_X \mathrm{t}imesi \right) = \\
& = \nablabar_X f_* \mathbf{S}_i^\mathrm{t}\mathrm{t}imesi + \nablabar_X \mathbf{T}_i\mathrm{t}imesi + f_*\mathbf{R}_i A_{\mathrm{t}imesi} X + \mathbf{S}_i A_{\mathrm{t}imesi} X - f_* \mathbf{S}_i^\mathrm{t}\nablaperp_X \mathrm{t}imesi - \mathbf{T}_i \nablaperp_X \mathrm{t}imesi = \\
& = f_*\nabla_X \mathbf{S}_i^\mathrm{t}\mathrm{t}imesi + \al{X}{\mathbf{S}_i^\mathrm{t} \mathrm{t}imesi} - f_*A_{\mathbf{T}_i\mathrm{t}imesi} X + \nablaperp_X \mathbf{T}_i\mathrm{t}imesi + f_*\lambdaeft( \mathbf{R}_i A_\mathrm{t}imesi X - \mathbf{S}_i^\mathrm{t}\nablaperp_X \mathrm{t}imesi \right) + \\
& \quad + \mathbf{S}_i A_{\mathrm{t}imesi} X - \mathbf{T}_i \nablaperp_X \mathrm{t}imesi = \\
& = f_* \lambdaeft[ \lambdaeft( \nabla_X \mathbf{S}_i^\mathrm{t} \right)\mathrm{t}imesi - A_{\mathbf{T}_i \mathrm{t}imesi} X + \mathbf{R}_i A_\mathrm{t}imesi X \right] + \al{X}{\mathbf{S}_i^\mathrm{t} \mathrm{t}imesi} + \lambdaeft(\nabla_X \mathbf{T}_i\right) \mathrm{t}imesi + \mathbf{S}_i A_\mathrm{t}imesi X.
\varepsilonnd{align*}
Therefore
\[\lambdaeft(\nabla_X \mathbf{S}_i^\mathrm{t}\right)\mathrm{t}imesi = A_{\mathbf{T}_i \mathrm{t}imesi} X - \mathbf{R}_i A_\mathrm{t}imesi X \quad \mathrm{t}ext{and} \quad \lambdaeft(\nabla_X \mathbf{T}_i\right) \mathrm{t}imesi = - \mathbf{S}_i A_\mathrm{t}imesi X - \al{X}{\mathbf{S}_i^\mathrm{t} \mathrm{t}imesi}. \qedhere\]
\varepsilonnd{proof}
\betaegin{obs}
Making the inner product of both sides of the first equation in \varepsilonqref{eq: derivada S} by $\mathrm{t}imesi$, and comparing to the inner product of both sides of the second equation in \varepsilonqref{eq: derivada S} by $Y$, we can conclude that both equations in \varepsilonqref{eq: derivada S} are equivalent. We decided to write down both in order to use the most convenient form wen we need it.
\varepsilonnd{obs}
\subsubsection{Gaus equation}
\betaegin{align*}
& f_* \mathcal{R}(X,Y)Z = \lambdaeft( \betaar{\mathcal R}(f_*X,f_*Y)f_*Z \right)^T + f_*A_{\al{Y}{Z}}X - f_*A_{\al{X}{Z}}Y \\
& \stackrel{\mathrm{t}ext{Lemma \ref{lem: curv1}}}{=} \lambdaeft( \sum_{i=1}^\varepsilonll k_i \lambdaeft[ \interno{\rhoi_if_*Y}{f_*Z}\rhoi_if_*X - \interno{\rhoi_if_*X}{f_*Z}\rhoi_if_*Y \right] \right)^T + \\
& \quad + f_*A_{\al{Y}{Z}}X - f_*A_{\al{X}{Z}}Y = \\
& = \sum_{i=1}^\varepsilonll k_i \lambdaeft[ \interno{\mathbf{R}_i Y}{Z}f_*\mathbf{R}_i X - \interno{\mathbf{R}_i X}{Z}f_*\mathbf{R}_i Y \right] + f_*A_{\al{Y}{Z}}X - f_*A_{\al{X}{Z}}Y.
\varepsilonnd{align*}
Therefore
\betaegin{equation}\lambdaabel{eq: Gauss}
\mathcal{R}(X,Y)Z = \sum_{i=1}^\varepsilonll k_i \lambdaeft( \mathbf{R}_i X \wedge \mathbf{R}_i Y\right)Z + A_{\al{Y}{Z}}X - A_{\al{X}{Z}}Y,
\varepsilonnd{equation}
where $(A \wedge B) C := \interno{B}{C} A - \interno{A}{C} B$.
\subsubsection{Codazzi equation}
\betaegin{align*}
& \lambdaeft(\nablaabla^\rhoerp_X \alpha \right)(Y,Z) - \lambdaeft(\nablaabla^\rhoerp_Y \alpha \right)(X,Z) = \lambdaeft(\betaar{\mathcal R}(f_*X,f_*Y)f_*Z \right)^\rhoerp = \\
& \stackrel{\mathrm{t}ext{Lemma \ref{lem: curv1}}}{=} \lambdaeft( \sum_{i=1}^\varepsilonll k_i \lambdaeft[ \interno{\rhoi_if_*Y}{f_*Z}\rhoi_if_*X - \interno{\rhoi_if_*X}{f_*Z}\rhoi_if_*Y \right] \right)^\rhoerp = \\
& = \sum_{i=1}^\varepsilonll k_i \lambdaeft[ \interno{\mathbf{R}_i Y}{Z}\mathbf{S}_i X - \interno{\mathbf{R}_i X}{Z}\mathbf{S}_i Y \right].
\varepsilonnd{align*}
\betaegin{align*}
& f_* (\nabla_Y A)(X,\mathrm{t}imesi) - f_*(\nabla_X A)(Y, \mathrm{t}imesi) = \lambdaeft(\betaar{\mathcal R}(f_*X,f_*Y)\mathrm{t}imesi \right)^\mathrm{T} = \\
& \stackrel{\mathrm{t}ext{Lemma \ref{lem: curv1}}}{=} \lambdaeft( \sum_{i=1}^\varepsilonll k_i \lambdaeft[ \interno{\rhoi_if_*Y}{\mathrm{t}imesi}\rhoi_if_*X - \interno{\rhoi_if_*X}{\mathrm{t}imesi}\rhoi_if_*Y \right] \right)^\mathrm{T} = \\
& = \sum_{i=1}^\varepsilonll k_i \lambdaeft[ \interno{\mathbf{S}_i Y}{\mathrm{t}imesi}f_*\mathbf{R}_i X - \interno{\mathbf{S}_i X}{\mathrm{t}imesi}f_*\mathbf{R}_i Y \right].
\varepsilonnd{align*}
Therefore,
\betaegin{equation}\lambdaabel{eq: Codazzi}
\lambdaeft. \betaegin{aligned}
\lambdaeft(\nablaabla^\rhoerp_X \alpha \right)(Y,Z) - \lambdaeft(\nablaabla^\rhoerp_Y \alpha \right)(X,Z) = \sum_{i=1}^\varepsilonll k_i \lambdaeft[ \interno{\mathbf{R}_i Y}{Z}\mathbf{S}_i X - \interno{\mathbf{R}_i X}{Z}\mathbf{S}_i Y \right] \\
(\nabla_Y A)(X,\mathrm{t}imesi) - (\nabla_X A)(Y, \mathrm{t}imesi) = \sum_{i=1}^\varepsilonll k_i \lambdaeft[ \interno{\mathbf{S}_i Y}{\mathrm{t}imesi}\mathbf{R}_i X - \interno{\mathbf{S}_i X}{\mathrm{t}imesi}\mathbf{R}_i Y \right]
\varepsilonnd{aligned} \right\}
\varepsilonnd{equation}
\subsubsection{Ricci equation}
\betaegin{align*}
& \mathcal{R}^\rhoerp(X,Y)\mathrm{t}imesi = \lambdaeft(\betaar{\mathcal R}(f_*X,f_*Y)\mathrm{t}imesi \right)^\rhoerp + \al{X}{A_\mathrm{t}imesi Y} - \al{A_\mathrm{t}imesi X}{Y} = \\
& \stackrel{\mathrm{t}ext{Lemma \ref{lem: curv1}}}{=} \lambdaeft( \sum_{i=1}^\varepsilonll k_i \lambdaeft[ \interno{\rhoi_if_*Y}{\mathrm{t}imesi}\rhoi_if_*X - \interno{\rhoi_if_*X}{\mathrm{t}imesi}\rhoi_if_*Y \right] \right)^\rhoerp + \\
& \quad + \al{X}{A_\mathrm{t}imesi Y} - \al{A_\mathrm{t}imesi X}{Y} = \\
& = \sum_{i=1}^\varepsilonll k_i \lambdaeft[ \interno{\mathbf{S}_i Y}{\mathrm{t}imesi}\mathbf{S}_i X - \interno{\mathbf{S}_i X}{\mathrm{t}imesi}\mathbf{S}_i Y \right] + \al{X}{A_\mathrm{t}imesi Y} - \al{A_\mathrm{t}imesi X}{Y}.
\varepsilonnd{align*}
Therefore,
\betaegin{equation}\lambdaabel{eq: Ricci}
\lambdaeft. \betaegin{aligned}
\mathcal{R}^\rhoerp(X,Y)\mathrm{t}imesi = \al{X}{A_\mathrm{t}imesi Y} - \al{A_\mathrm{t}imesi X}{Y} + \sum_{i=1}^\varepsilonll k_i \lambdaeft( \mathbf{S}_i X \wedge \mathbf{S}_i Y \right) \mathrm{t}imesi \\
\interno{\mathcal{R}^\rhoerp(X,Y)\mathrm{t}imesi}{\zeta} = \interno{\lambdaeft[A_\mathrm{t}imesi, A_\zeta \right]X}{Y} + \interno{\sum_{i=1}^\varepsilonll k_i \lambdaeft( \mathbf{S}_i X \wedge \mathbf{S}_i Y \right) \mathrm{t}imesi}{\zeta}
\varepsilonnd{aligned} \right\}
\varepsilonnd{equation}
\subsection{The immersion $F = \imath \circ f$}
Let $F := \imath \circ f \colon M \mathrm{t}o \mathbb{R}N$ and, for each $i \in \{1, \cdots, \varepsilonll\}$, let $\nablau_i = -k_i(\rhoi_i\circ F)$. Let also $\nablabar$, $\nablatil$ and $\nablabarperp$ be the connexions on $\hat{\mathbb{O}}$, $\mathbb{R}N$ and $T_F^\rhoerp M$, respectively.
\betaegin{lem}
$T_F^\rhoerp M = T_f^\rhoerp M \obot \spa\{\nablau_1, \cdots, \nablau_\varepsilonll\}$. Besides,
\betaegin{align}
& \nablabarperp_X \nablau_i = -k_i\imath_* \mathbf{S}_i X, && A^F_{\nablau_i} = k_i \mathbf{R}_i, \lambdaabel{eq: nui} \\
& A_{\imath_*\mathrm{t}imesi}^F = A_\mathrm{t}imesi^f, && \nablabarperp_X \imath_* \mathrm{t}imesi = \imath_*\nablaperp_X \mathrm{t}imesi + \sum_{i=1}^\varepsilonll \interno{\mathbf{S}_i X}{\mathrm{t}imesi}\nablau_i, \lambdaabel{nbarperpxi}
\varepsilonnd{align}
for any $X \in \Gammaamma(TM)$ and every $\mathrm{t}imesi \in \Gammaamma\lambdaeft( T^\rhoerp M\right)$.
\varepsilonnd{lem}
\betaegin{proof}
By Lemma \ref{lem: pi_j}, the vector field $\nablau_i := -k_i(\rhoi_i \circ F)$ is normal to $F$. Hence
\betaegin{align*}
& T_{F(p)}^\rhoerp M = T_{f(p)}^\rhoerp M \obot T_{\imath(F(p))}^\rhoerp \hat{\mathbb{O}} \stackrel{\varepsilonqref{Tiperp}}{=} \\
& = T_{f(p)}^\rhoerp M \obot \spa\lambdaeft\{-k_1(\rhoi_1\circ F)(p), \cdots, -k_\varepsilonll(\rhoi_\varepsilonll\circ F)(p)\right\} = \\
& = T_{f(p)}^\rhoerp M \obot \spa\lambdaeft\{\nablau_1(p), \cdots, \nablau_\varepsilonll(p)\right\}
\varepsilonnd{align*}
Let $X \in \Gammaamma(TM)$, thus
\[\nablatil_X\nablau_i = -k_i \rhoi_i F_* X = -k_i\imath_*\rhoi_i f_* X = -k_i\imath_* \lambdaeft[ f_*\mathbf{R}_i X + \mathbf{S}_i X\right].\]
Therefore, $\nablabarperp_X \nablau_i = -k_i\imath_* \mathbf{S}_i X$ and $A^F_{\nablau_i} = k_i \mathbf{R}_i$.
Last, if $\mathrm{t}imesi \in \Gammaamma\lambdaeft(T^\rhoerp M \right)$, then
\betaegin{align*}
&\nablatil_X \imath_*\mathrm{t}imesi (p) = \imath_* \nablabar_X \mathrm{t}imesi(p) + \ai{f_*X}{\mathrm{t}imesi}(F(p)) = \\
& \stackrel{\varepsilonqref{eq: ai}}{=} \imath_*\lambdaeft(-f_*A^f_\mathrm{t}imesi X + \nablaperp_X \mathrm{t}imesi(p) \right) - \sum_{i=1}^\varepsilonll \interno{\rhoi_if_* X}{\mathrm{t}imesi}k_i \rhoi_i(F(p)) = \\
& = -F_*A^f_\mathrm{t}imesi X + \imath_* \nablaperp_X \mathrm{t}imesi(p) + \sum_{i=1}^\varepsilonll \interno{\mathbf{S}_i X}{\mathrm{t}imesi}\nablau_i(p).
\varepsilonnd{align*}
Therefore $A_{\imath_*\mathrm{t}imesi}^F = A_\mathrm{t}imesi^f$ and $\nablabarperp_X \imath_* \mathrm{t}imesi = \imath_*\nablaperp_X \mathrm{t}imesi + \sum\lambdaimits_{i=1}^\varepsilonll \interno{\mathbf{S}_i X}{\mathrm{t}imesi}\nablau_i$.
\varepsilonnd{proof}
\betaegin{lem}\lambdaabel{lem: aF}
$\aF{X}{Y} = \imath_* \af{X}{Y} + \sum\lambdaimits_{i=1}^\varepsilonll \interno{\mathbf{R}_i X}{Y} \nablau_i$, for all $X,Y \in TM$.
\varepsilonnd{lem}
\betaegin{proof}
If $X, Y \in TM$, then
\betaegin{align*}
& \aF{X}{Y} = \imath_*\af{X}{Y} + \ai{f_*X}{f_*Y} = \\
& \stackrel{\mathrm{t}ext{Lemma \ref{lem: ai}}}{=} \imath_* \af{X}{Y} - \sum_{i=1}^\varepsilonll \interno{\rhoi_i f_* X}{f_* Y} k_i(\rhoi_i \circ F) = \\
& = \imath_* \af{X}{Y} + \sum_{i=1}^\varepsilonll \interno{\mathbf{R}_i X}{Y} \nablau_i. \qedhere
\varepsilonnd{align*}
\varepsilonnd{proof}
\section{Examples of isometric immersions in $\o{1} \x \cdots \x \o{\ell}$}
\subsection{Products of isometric immersions}\lambdaabel{sec: R S T nulos}
Let $i \in \{1, \cdots, \varepsilonll\}$ and $z \in \rhorod\lambdaimits_{\substack{j=1\\ j \nablae i}}^\varepsilonll \o{j}$ be a fixed point, $\betaar f \colon M \mathrm{t}o \o{i}$ be an isometric immersion and $\imath_i^z \colon \o{i} \mathrm{t}o \hat{\mathbb{O}}$ be the totally geodesic embedding given by $\imath_i^z(x) := (z,x)$. Simple examples of isometric immersions in $\hat{\mathbb{O}}$ can be made with compositions like $\imath_i^z \circ \betaar f$:
\[\betaegin{matrix}
M & \stackrel{\betaar f}{\lambdaongrightarrow} & \o{i} & \stackrel{\imath_i^z}{\lambdaongrightarrow} & \lambdaeft( \rhorod\lambdaimits_{\substack{j=1\\ j \nablae i}}^\varepsilonll \o{j}\right) \mathrm{t}imes \o{i} \\
x & \lambdaongmapsto & \betaar f(x) & \lambdaongmapsto & \lambdaeft(z, \betaar f(x) \right)
\varepsilonnd{matrix}\]
\betaegin{lem}\lambdaabel{lem: R=0}
Let $f \colon M^m \mathrm{t}o \hat{\mathbb{O}}$ be an isometric immersion. Then
\betaegin{enum}
\item $f(M) \subset \{z\} \mathrm{t}imes \o{i}$ for some $z \in \rhorod\lambdaimits_{\substack{j=1\\ j \nablae i}}^\varepsilonll \o{j}$ if, and only if, $\mathbf{R}_i = \id$.
\item $f(M) \subset \lambdaeft( \rhorod\lambdaimits_{\substack{j=1\\ j \nablae i}}^\varepsilonll \o{j} \right)\mathrm{t}imes \{z\}$, for some $z\in \o{i}$ if, and only if, $\mathbf{R}_i = 0$.
\varepsilonnd{enum}
\varepsilonnd{lem}
\betaegin{proof}
\nablaoindent We know that $f(M) \subset \{z\}\mathrm{t}imes\o{i}$ if, and only if, $\rhoi_if_* X = f_* X$, for any $X \in TM$. But $\rhoi_i f_* = \mathbf{L}_i$ and $\mathbf{R}_i = \mathbf{L}_i^\mathrm{t} \mathbf{L}_i$, thus
\betaegin{multline*}
\rhoi_i f_* X = f_* X, \ \varphiorall X \in TM \mathbb{L}eftrightarrow \\
\mathbb{L}eftrightarrow \interno{\mathbf{R}_i X}{Y} = \interno{\mathbf{L}_i X}{\mathbf{L}_i Y} = \interno{X}{Y}, \ \varphiorall X,Y \in TM \mathbb{L}eftrightarrow \mathbf{R}_i = \id.
\varepsilonnd{multline*}
On the other side, $f(M) \subset \lambdaeft( \rhorod\lambdaimits_{\substack{j=1\\ j \nablae i}}^\varepsilonll \o{j}\right) \mathrm{t}imes \{z\}$ if, and only if, $\rhoi_i f_* X =0$, for all $X \in TM$. But
\betaegin{multline*}
\rhoi_i f_* X = 0, \ \varphiorall X \in TM \mathbb{L}eftrightarrow \\
\mathbb{L}eftrightarrow \interno{\mathbf{R}_i X}{Y} = \interno{\mathbf{L}_i X}{\mathbf{L}_i Y} = 0, \ \varphiorall X, Y \in TM \mathbb{L}eftrightarrow \mathbf{R}_i = 0. \qedhere
\varepsilonnd{multline*}
\varepsilonnd{proof}
Let $I = \{i_1, \cdots, i_l\} \subsetneq \{1, \cdots, \varepsilonll\}$, with $i_1 < i_2 < \cdots < i_l$. Now, lets consider the totally geodesic embedding
\[\jmath \colon \rhorod_{\substack{i=1\\i \nablaotin I}}^\varepsilonll \o{i} \mathrm{t}o \lambdaeft( \rhorod\lambdaimits_{\substack{i=1\\i \nablaotin I}}^\varepsilonll \o{i} \right) \mathrm{t}imes \lambdaeft( \rhorod_{i \in I} \o{i} \right) = \hat{\mathbb{O}},\]
given by $\jmath(x) = (x,z)$, where each $z \in \rhorod\lambdaimits_{j=1}^l \o{i_j}$ is a fixed point.
As a consequence of Lemma \ref{lem: R=0}, we have the following Corollary.
\betaegin{cor}
Let $I = \{i_1, \cdots, i_l\} \subsetneq \{1, \cdots, \varepsilonll\}$, with $i_1 < i_2 < \cdots < i_l$, and let $f \colon M \mathrm{t}o \hat{\mathbb{O}}$ be an isometric immersion. Then the following sentences are equivalent:
\betaegin{enum}
\item $f = \jmath \circ \betaar f$, where $\betaar f \colon M \mathrm{t}o \rhorod\lambdaimits_{\substack{i=1\\i \nablaotin I}}^\varepsilonll \o{i}$
is an isometric immersion and $\jmath$ is the totally geodesic embedding given above.
\item $\mathbf{R}_{i_1}= \cdots = \mathbf{R}_{i_l} = 0$.
\varepsilonnd{enum}
\varepsilonnd{cor}
We can build other examples of isometric immersions $f \colon M_1\mathrm{t}imes M_2 \mathrm{t}o \hat{\mathbb{O}}$, by $f(x,y) := (f_1(x), f_2(y))$, where
\[\betaegin{matrix}
f_1 : & M_1 & \mathrm{t}o & \rhorod\lambdaimits_{\substack{j=1\\ j \nablae i}}^\varepsilonll \o{j}\\
& x & \mapsto & f_1(x)
\varepsilonnd{matrix}\quad \mathrm{t}ext{and} \quad
\betaegin{matrix}
f_2 : & M_2 & \mathrm{t}o & \o{i} \\
& y & \mapsto & f_2(y)
\varepsilonnd{matrix}\]
are isometric immersions. In order to study these examples, we need some new results.
\betaegin{lem}\lambdaabel{lem: ker S}
$\ker \mathbf{S}_i = \ker \mathbf{R}_i \obot \ker (\id - \mathbf{R}_i)$ and $\mathbf{R}_i(\ker \mathbf{S}_i) = \ker(\id - \mathbf{R}_i)$.
\varepsilonnd{lem}
\betaegin{proof}
Since $\ker \mathbf{S}_i = \ker \mathbf{S}_i^\mathrm{t}\mathbf{S}_i$ and $\mathbf{S}_i^\mathrm{t}\mathbf{S}_i \stackrel{\varepsilonqref{eq: RST}}{=} \mathbf{R}_i(\id - \mathbf{R}_i)$, then
\[\ker \mathbf{S}_i = \set{ X \in T_xM}{\mathbf{R}_i X = \mathbf{R}_i^2 X}.\]
Hence $\ker \mathbf{R}_i \subset \ker \mathbf{S}_i$ and $\ker(\id - \mathbf{R}_i) \subset \ker \mathbf{S}_i$.
\betaegin{afi}{$\mathbf{R}_i(\ker \mathbf{S}_i) \subset \ker \mathbf{S}_i$.}
If $X \in \ker \mathbf{S}_i$, then $\mathbf{R}_i X = \mathbf{R}_i^2 X$. So $\mathbf{R}_i(\mathbf{R}_i X) = \mathbf{R}_i \lambdaeft( \mathbf{R}_i^2 X\right) = \mathbf{R}_i^2 (\mathbf{R}_i X)$, that is, $\mathbf{R}_i X \in \ker \mathbf{S}_i$.
\varepsilonnd{afi}
So $\mathbf{R}_i|_{\ker \mathbf{S}_i} = \mathbf{R}_i^2|_{\ker \mathbf{S}_i}$ and we know that $\mathbf{R}_i$ is self-adjoint, then $\mathbf{R}_i|_{\ker \mathbf{S}_i}$ is an orthogonal projection. Therefore $\ker\mathbf{S}_i = \ker \mathbf{R}_i|_{\ker \mathbf{S}_i} \obot \mathbf{R}_i(\ker \mathbf{S}_i)$. Now we have to show that $\mathbf{R}_i(\ker \mathbf{S}_i) = \ker (\id - \mathbf{R}_i)$.
Indeed, if $Y \in \mathbf{R}_i(\ker \mathbf{S}_i)$, then $Y = \mathbf{R}_i X$, for some $X \in \ker \mathbf{S}_i$. Hence $Y = \mathbf{R}_i X = \mathbf{R}_i^2 X = \mathbf{R}_i Y$, that is, $Y \in \ker (\id - \mathbf{R}_i)$.
On the other side, if $Y \in \ker(\id - \mathbf{R}_i)$, then $Y = \mathbf{R}_i Y$. Thus $\mathbf{R}_i Y = \mathbf{R}_i^2 Y$. Therefore $Y \in \ker \mathbf{S}_i$ and $Y = \mathbf{R}_i Y$, so $Y \in \mathbf{R}_i( \ker \mathbf{S}_i)$.
\varepsilonnd{proof}
\betaegin{lem}\lambdaabel{lem: S=0}
Let $f\colon M^m \mathrm{t}o \hat{\mathbb{O}}$ be an isometric immersion with $M$ connected. If $\mathbf{S}_i = 0$, then $\ker \mathbf{R}_i$ and $\ker (\id - \mathbf{R}_i)$ have constant dimension on $M$.
\varepsilonnd{lem}
\betaegin{proof}
If $\mathbf{S}_i = 0$ then, by Lemma \ref{lem: ker S}, $TM = \ker \mathbf{S}_i = \ker \mathbf{R}_i \obot \ker (\id - \mathbf{R}_i) = \ker \mathbf{R}_i \obot \mathbf{R}_i(TM)$.
Now, for each $j \in \{0, \cdots, m\}$, let $A_j := \set{x \in M}{\mathrm{d}im \lambdaeft(\ker \lambdaeft.\mathbf{R}_i\right|_{T_x M} \right) = j}$.
\betaegin{afi}{Each $A_j$ is an open set.}
If $A_j = \varnothing$, then $A_j$ is open. So lets suppose that $\mathrm{d}im \lambdaeft( \ker \lambdaeft.\mathbf{R}_i\right|_{T_p M} \right) = j$, for some $p \in M$.
If $j = 0$, then there are vector fields $X_1, \cdots, X_m$, defined in a neighborhood $U$ of $p$, such that $\mathbf{R}_i X_1, \cdots, \mathbf{R}_i X_m$ are LI in $U$. Hence $\mathbf{R}_i\lambdaeft(T_x M \right) = T_x M$, for every $x \in U$, that is, $\mathrm{d}im \lambdaeft(\ker \lambdaeft.\mathbf{R}_i\right|_{T_x M} \right) = 0$, for every $x \in U$.
If $j = m$, then $\mathrm{d}im \lambdaeft[ \ker\lambdaeft.(\id - \mathbf{R}_i)\right|_{T_p M} \right] = 0$. Hence there are $X_1, \cdots, X_m$, defined in a neighborhood $U$ of $p$, such that $(\id - \mathbf{R}_i)X_1, \cdots, (\id - \mathbf{R}_i)X_m$ are LI in $U$. Thus $\mathrm{d}im \lambdaeft[ \ker \lambdaeft.(\id - \mathbf{R}_i)\right|_{T_x M} \right] = 0$, for every $x \in U$, that is, $m = \mathrm{d}im \lambdaeft( \ker\lambdaeft.\mathbf{R}_i\right|_{T_x M} \right)$, for all $x \in U$.
Lets suppose now that $0 < j < m$. In this case there are vector fields $X_1, \cdots, X_m$, defined in a neighborhood $U$ of $p$, such that $(\id - \mathbf{R}_i)X_1$, $\cdots$, $(\id - \mathbf{R}_i)X_j$ are LI in $U$ and $\mathbf{R}_i X_{j+1}$, $\cdots$, $\mathbf{R}_i X_m$ are also LI in $U$. Thus, $\mathrm{d}im \lambdaeft(\ker \mathbf{R}_i|_{T_x M} \right) \gammaeq j$ and $\mathrm{d}im \lambdaeft[ \ker (\id - \mathbf{R}_i)|_{T_x M} \right] \gammaeq m-j$, for any $x \in U$.
But $m = \mathrm{d}im (\ker\mathbf{R}_i) + \mathrm{d}im [\ker (\id - \mathbf{R}_i)]$, then $\mathrm{d}im \lambdaeft( \lambdaeft.\ker \mathbf{R}_i\right|_{T_x M} \right) = j$, for all $x \in U$. Therefore $A_j$ is open, for any $j \in \{0, \cdots, m\}$.
\varepsilonnd{afi}
We know that $M = \betaigcup\lambdaimits_{j=0}^m A_j$ and $A_j \cap A_o = \varnothing$, if $j \nablae o$. Since $M$ is connected, $M = A_j$, for some $j \in \{0, \cdots, m\}$, that is, $\mathrm{d}im (\ker \mathbf{R}_i)$ is constant in $M$.
\varepsilonnd{proof}
\betaegin{prop}\lambdaabel{prop: S=0}
Let $f \colon M^m \mathrm{t}o \hat{\mathbb{O}}$ be an isometric immersion with, $M$ connected. Thus the following claims are equivalent:
\betaegin{enum}
\item $M$ is locally (isometric to) a product manifold $M_1^{m_1}\mathrm{t}imes M_2^{m_2}$, with $0 < m_1 < m$, and $f$ is locally a product immersion
\[\betaegin{matrix}
f : & M_1 \mathrm{t}imes M_2 & \lambdaongrightarrow & \lambdaeft(\rhorod\lambdaimits_{\substack{j=1 \\ j \nablae i}}^\varepsilonll \o{j} \right) \mathrm{t}imes \o{i} \\
& (x,y) & \lambdaongmapsto & \betaig(f_1(x), f_2(y)\betaig).
\varepsilonnd{matrix}\]
\item $\mathbf{S}_i = 0$ and $\mathrm{d}im (\ker \mathbf{R}_i) = m_1$, with $0 < m_1 < m$.
\varepsilonnd{enum}
Besides that, if $M$ is complete and simply connected and the second claim is true, then $M$ is globally isometric to a product manifold $M_1^{m_1} \mathrm{t}imes M_2^{m_2}$ and $f$ is globally a product immersion like in \mathrm{t}extsl{(I)}.
\varepsilonnd{prop}
\betaegin{proof}
\mathrm{t}extbf{\mathrm{t}extsl{(I)} $\mathbb{R}ightarrow$ \mathrm{t}extsl{(II)}:} Lets suppose that $M^m$ like in \mathrm{t}extsl{(I)}. Hence, $T_{(x,y)} M = {\imath_1^y}_* T_x M_1 \obot {\imath_2^x}_* T_y M_2$, where $\imath_1^y \colon M_1 \mathrm{t}o M$ and $\imath_2^x \colon M_2 \mathrm{t}o M$ are given by $\imath_1^y(z) = (z,y)$ and $\imath_2^x (z) = (x,z)$.
If $X \in {\imath_1^y}_* T_x M_1$, then $X = {\imath_1^y}_* \mathrm{t}ilde X = \lambdaeft(\mathrm{t}ilde X, 0 \right) \in T_{(x,y)} \lambdaeft(\rhorod\lambdaimits_{\substack{j=1 \\ j \nablae i}}^\varepsilonll \o{j} \right) \mathrm{t}imes \o{i}$ and
$\rhoi_i f_* X = \rhoi_if_* \lambdaeft( \mathrm{t}ilde X, 0 \right) = \rhoi_i\lambdaeft({f_1}_* \mathrm{t}ilde X, {f_2}_* 0 \right) = 0$, thus $\mathbf{R}_i X = 0$.
Analogously, if $Y \in {\imath_2^x}_* T_y M_2$ then $(\id - \mathbf{R}_i)Y = 0$. Thus
\[T_{(x,y)} M = {\imath_1^y}_* T_x M_1 \obot {\imath_2^x}_* T_y M_2 = \ker \mathbf{R}_i|_{T_{(x,y)} M} \obot \ker (\id - \mathbf{R}_i)|_{T_{(x,y)} M}.\]
Therefore $\mathrm{d}im (\ker \mathbf{R}_i) = m_1$ and from Lemma \ref{lem: S=0} we conclude that $\mathbf{S}_i = 0$.
\nablaoindent \mathrm{t}extbf{\mathrm{t}extsl{(II)} $\mathbb{R}ightarrow$ \mathrm{t}extsl{(I)}:} Lets suppose now that $\mathbf{S}_i = 0$ and $\mathrm{d}im (\ker \mathbf{R}_i) = m_1$, with $0 < m_1 < m$. Hence, by Lemma \ref{lem: ker S}, we know that $TM = \ker \mathbf{R}_i \obot \ker(\id - \mathbf{R}_i)$, and it follows from Lemma \ref{lem: S=0} that $\ker \mathbf{R}_i$ and $\ker (\id-\mathbf{R}_i)$ are distributions on $M$.
\betaegin{afi}{$\ker \mathbf{R}_i$ and $\ker (\id - \mathbf{R}_i)$ are parallel distributions.}
Let $Y \in \Gamma(\ker \mathbf{R}_i)$ and $X \in \Gamma(TM)$. Thus $\mathbf{R}_i \nabla_X Y \stackrel{\varepsilonqref{eq: derivada R}}{=} \nabla_X \mathbf{R}_i Y = 0$. Therefore $\ker \mathbf{R}_i$ and $(\ker \mathbf{R}_i)^\rhoerp = \ker (\id - \mathbf{R}_i)$ are parallel distributions.
\varepsilonnd{afi}
Now, for each $x \in M$, let $L_1^{m_1}(x)$ and $L_2^{m_2}(x)$ be integral submanifolds of $\ker \mathbf{R}_i$ and $(\ker \mathbf{R}_i)^\rhoerp$, respectively, at the point $x$. The De Rham's Theorem (see \cite{dR} and \cite{RS}) assure us that for each $x \in M$, there is a neighborhood $U$ of $x$ and there are open sets $M_1^{m_1}\subset L_1^{m_1}(x)$ and $M_2^{m_2} \subset L_2^{m_2}(x)$ and an isometry $\rhosi \colon M_1\mathrm{t}imes M_2 \mathrm{t}o U$ such that $x \in U \cap M_1 \cap M_2$ and, for each $(x_1,x_2) \in M_1\mathrm{t}imes M_2$, $\rhosi \lambdaeft( M_1 \mathrm{t}imes\{x_2\} \right)$ is a leaf of $\ker \mathbf{R}_i$ and $\rhosi \lambdaeft(\{x_1\}\mathrm{t}imes M_2 \right)$ is a leaf of $(\ker \mathbf{R}_i)^\rhoerp$.
Thus we can identify $U$ with $M_1^{m_1} \mathrm{t}imes M_2^{m_2}$, $\ker \mathbf{R}_i$ with $T M_1$ and $(\ker \mathbf{R}_i)^\rhoerp$ with $T M_2$ and we can consider the applications $f \colon M_1^{m_1} \mathrm{t}imes M_2^{m_2} \mathrm{t}o \hat{\mathbb{O}}$ and $F = \imath \circ f \colon M_1^{m_1}\mathrm{t}imes M_2^{m_2} \mathrm{t}o \mathbb{R}N$, where $\imath \colon \hat{\mathbb{O}} \hookrightarrow \mathbb{R}N$ is the canonical inclusion.
\betaegin{afi}{If $X \in \ker \mathbf{R}_i$ and $Y \in (\ker \mathbf{R}_i)^\rhoerp$, then $\aF{X}{Y} = 0$.}
If $X \in \ker \mathbf{R}_i$ and $Y \in \mathbf{R}_i(TM)$, then $Y = \mathbf{R}_i Z$, for some $Z \in TM$. Hence $\af{X}{Y} = \af{X}{\mathbf{R}_i Z} \stackrel{\varepsilonqref{eq: derivada S}}{=} \mathbf{T}_i \af{X}{Z} \stackrel{\varepsilonqref{eq: derivada S}}{=} \af{\mathbf{R}_i X}{Z} = 0$. Thus
\betaegin{multline*}
\aF{X}{Y} = \imath_* \af{X}{Y} + \sum_{j=1}^\varepsilonll \interno{\mathbf{R}_j X}{Y}\nablau_j = \sum_{\substack{j=1\\ j\nablae i}}^\varepsilonll \interno{\mathbf{R}_j X}{\mathbf{R}_i Z}\nablau_j = \\
= \sum_{\substack{j=1\\ j \nablae i}}^\varepsilonll\interno{X}{\mathbf{R}_j \mathbf{R}_i Z}\nablau_j \stackrel{\varepsilonqref{eq: RST2}}{=} 0.
\varepsilonnd{multline*}
Therefore the claim holds.
\varepsilonnd{afi}
Now we can apply Moore's Lemma (see \cite{Detc}) and conclude that there is an orthogonal decomposition $\mathbb{R}N = V_0 \obot V_1 \obot V_2$ and a vector $v_0 \in V_0$ such that $F \colon M_1 \mathrm{t}imes M_2 \mathrm{t}o \mathbb{R}N$ is given by $F(x,y) = (v_0, F_1(x), F_2(y))$. Besides,
\betaegin{align*}
& V_1 = \spa\set{F_*(p) X}{p \in M_1 \mathrm{t}imes M_2 \ \mathrm{t}ext{e} \ X \in \ker \mathbf{R}_i|_{T_p M}}, \\
& V_2 = \spa\set{F_*(p) Y}{p \in M_1 \mathrm{t}imes M_2 \ \mathrm{t}ext{e} \ Y \in \mathbf{R}_i\lambdaeft(T_p M \right)}
\varepsilonnd{align*}
and $F_i(M_i) \subset V_i$.
On the other side, $\rhoi_if_*(\ker \mathbf{R}_i) = \{0\}$ and $\lambdaeft.\rhoi_i\right|_{f_*\ker (\id - \mathbf{R}_i)} = \id$, hence
\betaegin{align*}
& \spa\set{F_*(p) X}{p \in M_1 \mathrm{t}imes M_2 \ \mathrm{t}ext{e} \ X \in \ker \mathbf{R}_i|_{T_p M}} \subset \rhorod_{\substack{j=1\\j \nablae i}}^\varepsilonll \mathbb{R}^{N_j},\\
& \spa\set{F_*(p) Y}{p \in M_1 \mathrm{t}imes M_2 \ \mathrm{t}ext{e} \ Y \in \mathbf{R}_i\lambdaeft(T_p M \right)} \subset \mathbb{R}^{N_i},
\varepsilonnd{align*}
therefore $F_1(M_1)\subset \rhorod\lambdaimits_{\substack{j=1\\j \nablae i}}^\varepsilonll \mathbb{R}^{N_j}$ and $F_2(M_2) \subset \mathbb{R}^{N_i}$.
Lets define $\mathrm{t}ilde f_1 \colon M_1 \mathrm{t}o \rhorod\lambdaimits_{\substack{j=1\\j \nablae i}}^\varepsilonll \mathbb{R}^{N_j}$ and $\mathrm{t}ilde f_2 \colon M_2 \mathrm{t}o \mathbb{R}^{N_i}$ by $\mathrm{t}ilde f_1(x) := \Pi(v_0) + F_1(x)$ and $\mathrm{t}ilde f_2(y) = \rhoi_i(v_0) + F_2(y)$, where $\Pi \colon \mathbb{R}N \mathrm{t}o \rhorod\lambdaimits_{\substack{j=1\\j \nablae i}}^\varepsilonll \mathbb{R}^{N_j}$ is the orthogonal projection. Hence
\[F(x,y) = \betaig(\mathrm{t}ilde f_1(x), \mathrm{t}ilde f_2(y)\betaig) \in \lambdaeft( \rhorod_{\substack{j=1\\ j \nablae i}}^\varepsilonll \o{j}\right) \mathrm{t}imes \o{i}.\]
Now, if $f_1 \colon M_1 \mathrm{t}o \rhorod\lambdaimits_{\substack{j=1\\ j \nablae i}}^\varepsilonll \o{j}$ and $f_2 \colon M_2 \mathrm{t}o \o{i}$ are given by $f_1(x) = \mathrm{t}ilde f_1(x)$ and $f_2(y) = \mathrm{t}ilde f_2(y)$, then $f = f_1 \mathrm{t}imes f_2$.
If $M$ is complete and simply connected, the De Rham's Lemma assure us that $M$ is (globally) isometric to $L_1 \mathrm{t}imes L_2$, where $L_1$ and $L_2$ are the leafs of $\ker \mathbf{R}_i$ and $(\ker \mathbf{R}_i)^\rhoerp$ (respectively) at the same point. In this case, considering $f \colon L_1 \mathrm{t}imes L_2 \mathrm{t}o \hat{\mathbb{O}}$, the calculations made above show us that $f$ is globally a product immersion.
\varepsilonnd{proof}
\betaegin{cor}
Let $f \colon M^m \mathrm{t}o \hat{\mathbb{O}}$ be an isometric immersion. The following claims are equivalent:
\betaegin{enum}
\item $M$ is locally (isometric to) a product manifold $M_1^{m_1}\mathrm{t}imes \cdots \mathrm{t}imes M_\varepsilonll^{m_\varepsilonll}$, with $0 < m_i < m$, and $f$ is locally a product immersion $f|_{M_1\mathrm{t}imes \cdots \mathrm{t}imes M_\varepsilonll} = f_1 \mathrm{t}imes \cdots \mathrm{t}imes f_\varepsilonll$, where each $f_i \colon M_i \mathrm{t}o \o{i}$ is an isometric immersion.
\item For each $i \in \{1, \cdots, \varepsilonll\}$, $\mathbf{S}_i = 0$ and $\mathrm{d}im (\ker \mathbf{R}_i) = m_i$, with $0 < m_i < m$.
\varepsilonnd{enum}
Besides, if $M$ is complete and simply connected and the second claim holds, then $M$ is globally an isometric product $M_1^{m_1} \mathrm{t}imes \cdots \mathrm{t}imes M_\varepsilonll^{m_\varepsilonll}$ and $f$ is globally a product immersion.
\varepsilonnd{cor}
\subsection{Other products of isometric immersions}
Let $I = \{i_1, \cdots, i_{\varepsilonll_1}\} \subsetneq \{1, \cdots, \varepsilonll\}$, with $i_1 < i_2 < \cdots < i_{\varepsilonll_1}$. Other products of isometric immersions are the following kind of immersions
\[\betaegin{matrix}
f_1 \mathrm{t}imes f_2 : & M_1 \mathrm{t}imes M_2 & \lambdaongrightarrow & \lambdaeft(\rhorod\lambdaimits_{i\in I} \o{i} \right) \mathrm{t}imes \lambdaeft(\rhorod\lambdaimits_{\substack{i=1 \\ i \nablaotin I}}^\varepsilonll \o{i} \right) \\
& (x,y) & \lambdaongmapsto & \betaig(f_1(x), f_2(y)\betaig),
\varepsilonnd{matrix}\]
where $f_1 \colon M_1 \mathrm{t}o \rhorod\lambdaimits_{i\in I} \o{i}$ and $f_2 \colon M_2 \mathrm{t}o \rhorod\lambdaimits_{\substack{i=1 \\ i \nablaotin I}}^\varepsilonll \o{i}$ are isometric immersions.
Let $\Pi_1$ and $\Pi_2$ be the orthogonal projections given by
\betaegin{align*}
\betaegin{matrix}
\Pi_1 : & \mathbb{R}N & \lambdaongrightarrow & \lambdaeft(\rhorod\lambdaimits_{i \in I} \mathbb{R}^{N_i} \right) \mathrm{t}imes \lambdaeft(\rhorod\lambdaimits_{\substack{i=1 \\ i \nablaotin I}}^\varepsilonll \mathbb{R}^{N_i}\right) \\
& (x,y) & \lambdaongmapsto & (x,0)
\varepsilonnd{matrix} \\
\betaegin{matrix}
\Pi_2 : & \mathbb{R}N & \lambdaongrightarrow & \lambdaeft(\rhorod\lambdaimits_{i \in I} \mathbb{R}^{N_i} \right) \mathrm{t}imes \lambdaeft(\rhorod\lambdaimits_{\substack{i=1 \\ i \nablaotin I}}^\varepsilonll \mathbb{R}^{N_i}\right) \\
& (x,y) & \lambdaongmapsto & (0,y);
\varepsilonnd{matrix}
\varepsilonnd{align*}
Hence, $\Pi_1 = \sum\lambdaimits_{i \in I} \rhoi_i$ and $\Pi_2 = \sum\lambdaimits_{\substack{i=1 \\ i \nablaotin I}}^\varepsilonll \rhoi_i$.
If $X \in T_x M$ and $\mathrm{t}imesi \in T_x^\rhoerp M$, then
\[\betaegin{aligned}
& \Pi_1 f_* X = \sum_{i \in I} \rhoi_i f_* X = \sum_{i \in I} \lambdaeft[ f_* \mathbf{R}_i X + \mathbf{S}_i X \right]= f_* \sum_{i \in I} \mathbf{R}_i X + \sum_{i \in I} \mathbf{S}_i X.\\
& \Pi_1 \mathrm{t}imesi = \sum_{i \in I} \rhoi_i \mathrm{t}imesi = \sum_{i \in I} \lambdaeft[ f_* \mathbf{S}^\mathrm{t}_i X + \mathbf{T}_i X \right]= f_* \sum_{i \in I} \mathbf{S}^\mathrm{t}_i X + \sum_{i \in I} \mathbf{T}_i X.
\varepsilonnd{aligned}\]
Analogously $\Pi_2 f_* X = f_* \sum\lambdaimits_{\substack{i=1 \\ i \nablaotin I }}^\varepsilonll \mathbf{R}_i X + \sum\lambdaimits_{\substack{i=1 \\ i \nablaotin I }}^\varepsilonll \mathbf{S}_i X$, and $\Pi_2 \mathrm{t}imesi = f_* \sum\lambdaimits_{\substack{i=1 \\ i \nablaotin I }}^\varepsilonll \mathbf{S}^\mathrm{t}_i X + \sum\lambdaimits_{\substack{i=1 \\ i \nablaotin I }}^\varepsilonll \mathbf{T}_i X$. Consequently, the following equations hold
\betaegin{align*}
\lambdaeft(\Pi_1 f_* X \right)^T &= f_* \sum_{i \in I} \mathbf{R}_i X, & \lambdaeft( \Pi_1 f_* X\right)^\rhoerp &= \sum_{i \in I} \mathbf{S}_i X,\\
\lambdaeft(\Pi_2 f_* X \right)^T &= f_* \sum_{\substack{i=1 \\ i \nablaotin I }}^\varepsilonll \mathbf{R}_i X, & \lambdaeft( \Pi_2 f_* X\right)^\rhoerp &= \sum_{\substack{i=1 \\ i \nablaotin I }}^\varepsilonll \mathbf{S}_i X, \\
\lambdaeft(\Pi_1 \mathrm{t}imesi \right)^T &= \sum_{i \in I} \mathbf{S}^\mathrm{t}_i X, & \lambdaeft(\Pi_1 \mathrm{t}imesi \right)^\rhoerp &= \sum_{i=1}^{\varepsilonll_1} \mathbf{T}_i \mathrm{t}imesi, \\
\lambdaeft(\Pi_2 \mathrm{t}imesi \right)^T &= \sum_{\substack{i=1 \\ i \nablaotin I }}^\varepsilonll \mathbf{S}^\mathrm{t}_i X, & \lambdaeft(\Pi_2 \mathrm{t}imesi \right)^\rhoerp &= \sum_{\substack{i=1 \\ i \nablaotin I }}^\varepsilonll \mathbf{T}_i \mathrm{t}imesi.
\varepsilonnd{align*}
So, for each $i\in \{1, 2\}$, lets consider the tensors $\mathrm{t}ilde{\mathbf{L}}_i \colon TM \mathrm{t}o T\hat{\mathbb{O}}$ and $\mathrm{t}ilde{\mathbf{K}}_i \colon T^\rhoerp M \mathrm{t}o T\hat{\mathbb{O}}$ given by $\mathrm{t}ilde{\mathbf{L}}_i X = \Pi_i f_*X$ and $\mathrm{t}ilde{\mathbf{K}}_i \mathrm{t}imesi = \Pi_i \mathrm{t}imesi$. Let also $\mathrm{t}ilde{\mathbf{R}}_i := \mathrm{t}ilde{\mathbf{L}}^\mathrm{t}_i \mathrm{t}ilde{\mathbf{L}}_i$, $\mathrm{t}ilde{\mathbf{S}}_i := \mathrm{t}ilde{\mathbf{K}}_i^\mathrm{t} \mathrm{t}ilde{\mathbf{L}}_i$ and $\mathrm{t}ilde{\mathbf{T}}_i := \mathrm{t}ilde{\mathbf{K}}_i^\mathrm{t}\mathrm{t}ilde{\mathbf{K}}_i$. So, from calculations analogous to those made for the tensors $\mathbf{R}_i$, $\mathbf{S}_i$ and $\mathbf{T}_i$, it follows that:
\betaegin{align*}
&\mathrm{t}ilde{\mathbf{L}}_i X = f_*\mathrm{t}ilde{\mathbf{R}}_i X + \mathrm{t}ilde{\mathbf{S}}_i X, && \mathrm{t}ilde{\mathbf{K}}_i \mathrm{t}imesi = f_*\mathrm{t}ilde{\mathbf{S}}_i^\mathrm{t} \mathrm{t}imesi + \mathrm{t}ilde{\mathbf{T}}_i \mathrm{t}imesi, \\
& \mathrm{t}ilde{\mathbf{R}}_1 + \mathrm{t}ilde{\mathbf{R}}_2 = \id|_{TM}, && \mathrm{t}ilde{\mathbf{S}}_1 = -\mathrm{t}ilde{\mathbf{S}}_2, && \mathrm{t}ilde{\mathbf{T}}_1 + \mathrm{t}ilde{\mathbf{T}}_2 = \id|_{T^\rhoerp M}.
\varepsilonnd{align*}
Hence
\betaegin{align*}
& \mathrm{t}ilde{\mathbf{R}}_1 = \sum_{i \in I} \mathbf{R}_i, && \mathrm{t}ilde{\mathbf{S}}_1 = \sum_{i \in I} \mathbf{S}_i, && \mathrm{t}ilde{\mathbf{T}}_1 = \sum_{i \in I} \mathbf{T}_i,\\
& \mathrm{t}ilde{\mathbf{R}}_2 = \sum_{\substack{i=1 \\ i \nablaotin I }}^\varepsilonll \mathbf{R}_i, && \mathrm{t}ilde{\mathbf{S}}_2 = \sum_{\substack{i=1 \\ i \nablaotin I }}^\varepsilonll \mathbf{S}_i, && \mathrm{t}ilde{\mathbf{T}}_2 = \sum_{\substack{i=1 \\ i \nablaotin I }}^\varepsilonll \mathbf{T}_i.
\varepsilonnd{align*}
Also from calculations analogous to those made for $\mathbf{R}_i$, $\mathbf{S}_i$ and $\mathbf{T}_i$, we have the following equations:
\betaegin{align}
& \mathrm{t}ilde{\mathbf{S}}_i^\mathrm{t}\mathrm{t}ilde{\mathbf{S}}_i = \mathrm{t}ilde{\mathbf{R}}_i(\id - \mathrm{t}ilde{\mathbf{R}}_i), && \mathrm{t}ilde{\mathbf{T}}_i\mathrm{t}ilde{\mathbf{S}}_i = \mathrm{t}ilde{\mathbf{S}}_i( \id - \mathrm{t}ilde{\mathbf{R}}_i), && \mathrm{t}ilde{\mathbf{S}}_i\mathrm{t}ilde{\mathbf{S}}_i^\mathrm{t} = \mathrm{t}ilde{\mathbf{T}}_i(\id - \mathrm{t}ilde{\mathbf{T}}_i), \lambdaabel{equação: TR TS TT} \\
& \mathrm{t}ilde{\mathbf{S}}_i^\mathrm{t}\mathrm{t}ilde{\mathbf{S}}_j \stackrel{i \nablae j}{=} -\mathrm{t}ilde{\mathbf{R}}_i\mathrm{t}ilde{\mathbf{R}}_j, && \mathrm{t}ilde{\mathbf{T}}_i \mathrm{t}ilde{\mathbf{S}}_j \stackrel{i \nablae j}{=} -\mathrm{t}ilde{\mathbf{S}}_i\mathrm{t}ilde{\mathbf{R}}_j, && \mathrm{t}ilde{\mathbf{S}}_i \mathrm{t}ilde{\mathbf{S}}_j^\mathrm{t} \stackrel{i\nablae j}{=} -\mathrm{t}ilde{\mathbf{T}}_i\mathrm{t}ilde{\mathbf{T}}_j.
\varepsilonnd{align}
\betaegin{lem}\lambdaabel{lem: TR=0}
Let $f \colon M^m \mathrm{t}o \hat{\mathbb{O}}$ be an isometric immersion. Then
\betaegin{enum}
\item $f(M) \subset \{z\} \mathrm{t}imes \lambdaeft(\rhorod\lambdaimits_{\substack{i=1 \\ i \nablaotin I}}^\varepsilonll \o{i} \right)$ for some $z \in \rhorod\lambdaimits_{i \in I} \o{i}$ if, and only if, $\sum\lambdaimits_{i \in I} \mathbf{R}_i = 0$.
\item $f(M) \subset \lambdaeft(\rhorod\lambdaimits_{i \in I} \o{i} \right) \mathrm{t}imes \{z\}$, for some $z \in \rhorod\lambdaimits_{\substack{i=1 \\ i \nablaotin I}}^\varepsilonll \o{i}$ if, and only if, $\sum\lambdaimits_{i \in I} \mathbf{R}_i = \id$.
\varepsilonnd{enum}
\varepsilonnd{lem}
\betaegin{proof}
The proof is analogous to the proof of Lemma \ref{lem: R=0}.
\varepsilonnd{proof}
\betaegin{lem}\lambdaabel{lem: ker TS}
For each $i \in \{1,2\}$,
\[\ker \mathrm{t}ilde{\mathbf{S}}_i = \ker \mathrm{t}ilde{\mathbf{R}}_i \obot \ker \lambdaeft(\id - \mathrm{t}ilde{\mathbf{R}}_i\right) \quad \mathrm{t}ext{and} \quad \mathrm{t}ilde{\mathbf{R}}_i\lambdaeft(\ker \mathrm{t}ilde{\mathbf{S}}_i\right) = \ker\lambdaeft(\id - \mathrm{t}ilde{\mathbf{R}}_i\right).\]
\varepsilonnd{lem}
\betaegin{proof}
The proof is analogous to the proof of Lemma \ref{lem: ker S}
\varepsilonnd{proof}
\betaegin{lem}\lambdaabel{lem: TS=0}
Let $f\colon M^m \mathrm{t}o \hat{\mathbb{O}}$ be a isometric immersion. If $M$ is connected and $\sum\lambdaimits_{i \in I}\mathbf{S}_i = 0$, then $\ker \sum\lambdaimits_{i \in I} \mathbf{R}_i$ and $\ker \sum\lambdaimits_{\substack{i=1 \\ i \nablaotin I}}^\varepsilonll \mathbf{R}_i$ have constant dimension in $M$.
\varepsilonnd{lem}
\betaegin{proof}
If $\sum\lambdaimits_{i \in I}\mathbf{S}_i = 0$ , then $\mathrm{t}ilde{\mathbf{S}}_1 = \sum\lambdaimits_{i \in I} \mathbf{S}_i = 0$, and, by Lemma \ref{lem: ker TS}, $TM = \ker \mathrm{t}ilde{\mathbf{R}}_1 \obot \ker \lambdaeft(\id - \mathrm{t}ilde{\mathbf{R}}_1\right)$.
For each $j \in \{0, \cdots, m\}$, let $A_j := \set{x \in M}{\mathrm{d}im \lambdaeft(\ker \lambdaeft.\mathrm{t}ilde{\mathbf{R}}_1\right|_{T_x M} \right) = j}$.
\betaegin{afi}{$A_j$ is open.}
Analogous to the proof of Claim 1 of Lemma \ref{lem: S=0}.
\varepsilonnd{afi}
But $M = \betaigcup\lambdaimits_{j=0}^m A_j$ and $A_j \cap A_o = \varnothing$, if $j \nablae o$. Then, since $M$ is connected, $M = A_j$, for some $j \in \{0, \cdots, m\}$, that is, $\ker \mathrm{t}ilde{\mathbf{R}}_1$ have constant dimension in $M$. Therefore $\ker \sum\lambdaimits_{i \in I} \mathbf{R}_i$ and $\ker \sum\lambdaimits_{\substack{i=1 \\ i \nablaotin I}}^\varepsilonll \mathbf{R}_i$ have constant dimension in $M$
\varepsilonnd{proof}
\betaegin{prop}
Let $f \colon M^m \mathrm{t}o \hat{\mathbb{O}}$ be an isometric immersion, with $M$ connected. Thus the following claims are equivalent
\betaegin{enum}
\item $M$ is locally (isometric to) a product manifold $M_1^{m_1}\mathrm{t}imes M_2^{m_2}$, with $0 < m_2 < m$, and $f$ is locally a product immersion $f|_{M_1\mathrm{t}imes M_2} = f_1 \mathrm{t}imes f_2$, given by $f(x,y) = (f_1(x), f_2(y))$, where
\[f_1 \colon M_1 \mathrm{t}o \rhorod_{i \in I} \o{i} \quad \mathrm{t}ext{and} \quad f_2 \colon M_2 \mathrm{t}o \rhorod_{\substack{i=1 \\ i \nablaotin I}}^\varepsilonll \o{i}\]
are isometric immersions.
\item $\sum\lambdaimits_{i \in I} \mathbf{S}_i = 0$, $\mathrm{d}im \lambdaeft[\ker \lambdaeft(\sum\lambdaimits_{i \in I} \mathbf{R}_i \right)\right] = m_2$ and $0 < m_2 < m$.
\varepsilonnd{enum}
Besides, if \mathrm{t}extsl{(II)} holds and $M$ is complete and simply connected, then $M$ is globally isometric to product manifold $M_1^{m_1} \mathrm{t}imes M_2^{m_2}$ and $f$ is globally an product immersion like in \mathrm{t}extsl{(I)}.
\varepsilonnd{prop}
\betaegin{obs}\lambdaabel{obsevação: soma Si}
$\sum\lambdaimits_{i \in I} \mathbf{S}_i = 0 \stackrel{\varepsilonqref{somas}}{\mathbb{L}eftrightarrow} \sum\lambdaimits_{\substack{i=1 \\ i \nablaotin I}}^\varepsilonll \mathbf{S}_i =0$.
\varepsilonnd{obs}
\betaegin{proof}
\mathrm{t}extbf{\mathrm{t}extsl{(I)} $\mathbb{R}ightarrow$ \mathrm{t}extsl{(II)}:} Lets suppose that $M$ is locally a product manifold $M_1^{m_1}\mathrm{t}imes M_2^{m_2}$ and that
\[\betaegin{matrix}
f|_{M_1\mathrm{t}imes M_2} = f_1 \mathrm{t}imes f_2 : & M_1 \mathrm{t}imes M_2 & \lambdaongrightarrow & \lambdaeft(\rhorod\lambdaimits_{i \in I} \o{i} \right) \mathrm{t}imes \lambdaeft(\rhorod\lambdaimits_{\substack{i=1 \\ i \nablaotin I}}^\varepsilonll \o{i} \right) \\
& (x,y) & \lambdaongmapsto & \betaig(f_1(x), f_2(y)\betaig).
\varepsilonnd{matrix}\]
Let $\imath_1^y \colon M_1 \mathrm{t}o M$ and $\imath_2^x \colon M_2 \mathrm{t}o M$ be given by $\imath_1^y(z) = (z,y)$ and $\imath_2^x (z) = (x,z)$. Hence, $T_{(x,y)} M = {\imath_1^y}_* T_x M_1 \obot {\imath_2^x}_* T_y M_2$.
If $X \in {\imath_1^y}_* T_x M_1$, then $X = {\imath_1^y}_* \mathrm{t}ilde X = \lambdaeft(\mathrm{t}ilde X, 0 \right) \in T_{(x,y)} \lambdaeft(\rhorod\lambdaimits_{i \in I} \o{j} \right) \mathrm{t}imes \lambdaeft(\rhorod\lambdaimits_{\substack{i=1 \\ i \nablaotin I}}^\varepsilonll \o{i} \right)$ and
$\Pi_2 f_*X = \Pi_2 f_* \lambdaeft( \mathrm{t}ilde X, 0 \right) = \Pi_2\lambdaeft({f_1}_* \mathrm{t}ilde X, {f_2}_* 0 \right) = 0$, thus $\mathrm{t}ilde{\mathbf{R}}_2 X = 0$, $\mathrm{t}ilde{\mathbf{S}}_2 X = 0$ and $\mathrm{t}ilde{\mathbf{S}}_1X= 0$.
Analogously, if $Y \in {\imath_2^x}_* T_y M_2$ then $\mathrm{t}ilde{\mathbf{R}}_1 Y = 0$, $\mathrm{t}ilde{\mathbf{S}}_1 Y = 0$ and $\mathrm{t}ilde{\mathbf{S}}_2 Y = 0$. Thus $\sum\lambdaimits_{i \in I} \mathbf{S}_i = 0$, ${\imath_1^y}_* T_x M_1 \subset \ker \mathrm{t}ilde{\mathbf{R}}_2$ and ${\imath_2^x}_* T_y M_2 \subset \ker \mathrm{t}ilde{\mathbf{R}}_1$.
So $TM = \ker \mathrm{t}ilde{\mathbf{S}}_1 = \ker \mathrm{t}ilde{\mathbf{R}}_1 \obot \ker\lambdaeft( \id - \mathrm{t}ilde{\mathbf{R}}_1 \right) = \ker \mathrm{t}ilde{\mathbf{R}}_1 \obot \ker \mathrm{t}ilde{\mathbf{R}}_2$. Therefore ${\imath_1^y}_* T_x M_1 = \ker\mathrm{t}ilde{\mathbf{R}}_2$, ${\imath_2^x}_* T_x M_2 = \ker\mathrm{t}ilde{\mathbf{R}}_1$ and $\mathrm{d}im \ker\lambdaeft( \sum\lambdaimits_{i \in I} \mathbf{R}_i \right) = m_2$.
\nablaoindent \mathrm{t}extbf{\mathrm{t}extsl{(II)} $\mathbb{R}ightarrow$ \mathrm{t}extsl{(I)}:} By Lemma \ref{lem: ker TS}, $TM = \lambdaeft( \ker\mathrm{t}ilde{\mathbf{R}}_2 \right) \obot \lambdaeft( \ker \mathrm{t}ilde{\mathbf{R}}_1\right)$ and, by Lemma \ref{lem: TS=0}, $\ker \mathrm{t}ilde{\mathbf{R}}_2 = \ker \sum\lambdaimits_{\substack{i=1 \\ i \nablaotin I }}^{\varepsilonll_1} \mathbf{R}_i$ and $\ker \mathrm{t}ilde{\mathbf{R}}_1= \ker \sum\lambdaimits_{i \in I} \mathbf{R}_i$ are distributions on $M$.
\betaegin{afi}{$\ker \mathrm{t}ilde{\mathbf{R}}_2$ and $\ker \mathrm{t}ilde{\mathbf{R}}_1$ are parallel distributions.}
Let $Y \in \Gamma(\ker \mathrm{t}ilde{\mathbf{R}}_1)$ and $X \in \Gamma(TM)$. Hence,
\betaegin{align*}
& \nabla_X \mathrm{t}ilde{\mathbf{R}}_1 Y = \nabla_X \sum_{i \in I} \mathbf{R}_i Y \stackrel{\varepsilonqref{eq: derivada R}}{=} \sum_{i \in I} \lambdaeft( \mathbf{R}_i \nabla_X Y + A_{\mathbf{S}_i Y} X + \mathbf{S}^\mathrm{t}_i \al{X}{Y} \right) = \\
&= \lambdaeft(\sum_{i \in I} \mathbf{R}_i \right)\nabla_X Y + A_{\sum\lambdaimits_{i \in I} \mathbf{S}_i Y} X + \lambdaeft(\sum_{i \in I} \mathbf{S}_i \right)^\mathrm{t} \al{X}{Y} =\\
& = \mathrm{t}ilde{\mathbf{R}}_1 \nabla_X Y + A_{\mathrm{t}ilde{\mathbf{S}}_1 Y} X +\mathrm{t}ilde{\mathbf{S}}_1^\mathrm{t} \al{X}{Y} = \mathrm{t}ilde{\mathbf{R}}_1 \nabla_X Y.
\varepsilonnd{align*}
Therefore $\ker \mathrm{t}ilde{\mathbf{R}}_1$ is parallel and the same holds for $\lambdaeft(\ker \mathrm{t}ilde{\mathbf{R}}_1\right)^\rhoerp = \ker \mathrm{t}ilde{\mathbf{R}}_2$.
\varepsilonnd{afi}
For each $x \in M$, let $L_1^{m_1}(x)$ and $L_2^{m_2}(x)$ be integral submanifolds of $\ker\mathrm{t}ilde{\mathbf{R}}_2$ and $\ker\mathrm{t}ilde{\mathbf{R}}_1$, respectively, at $x$. Thus, by De Rham's Lemma (see \cite{dR} and \cite{RS}), for each $x \in M$, there is a neighborhood $U$ of $x$, and open sets $M_1^{m_1}$ and $M_2^{m_2}$ of $L_1^{m_1}(x)$ and $L_2^{m_2}(x)$ (respectively) and there is an isometry $\rhosi \colon M_1\mathrm{t}imes M_2 \mathrm{t}o U$ such that $x \in U \cap M_1 \cap M_2$ and, for each $(x_1,x_2) \in M_1\mathrm{t}imes M_2$, $\rhosi \lambdaeft( M_1 \mathrm{t}imes\{x_2\} \right)$ is a leaf of $\ker\mathrm{t}ilde{\mathbf{R}}_2$ and $\rhosi \lambdaeft(\{x_1\}\mathrm{t}imes M_2 \right)$ is a leaf of $\ker\mathrm{t}ilde{\mathbf{R}}_1$.
So we can identify $U$ with $M_1^{m_1} \mathrm{t}imes M_2^{m_2}$, $\ker\mathrm{t}ilde{\mathbf{R}}_2$ with $T M_1$ and $\ker\mathrm{t}ilde{\mathbf{R}}_1$ with $T M_2$ and we can consider the applications $f \colon M_1^{m_1} \mathrm{t}imes M_2^{m_2} \mathrm{t}o \hat{\mathbb{O}}$ and $F = \imath \circ f \colon M_1^{m_1}\mathrm{t}imes M_2^{m_2} \mathrm{t}o \mathbb{R}N$, where $\imath \colon \hat{\mathbb{O}} \hookrightarrow \mathbb{R}N$ is the canonical inclusion.
\betaegin{afi}{If $X \in \ker\mathrm{t}ilde{\mathbf{R}}_2$ and $Y \in \ker\mathrm{t}ilde{\mathbf{R}}_1$, then $\aF{X}{Y} = 0$.}
Let $X \in \ker\lambdaeft.\mathrm{t}ilde{\mathbf{R}}_2\right|_{T_xM}$ and $Y \in \ker\lambdaeft.\mathrm{t}ilde{\mathbf{R}}_1\right|_{T_xM}$. We know that $\mathrm{t}ilde{\mathbf{R}}_1$ is a orthogonal projection in $T_x M$, because it is self-adjoint and $\mathrm{t}ilde{\mathbf{R}}_1^2 = \mathrm{t}ilde{\mathbf{R}}_1$, thus $\ker\mathrm{t}ilde{\mathbf{R}}_2 = \ker\lambdaeft( \id - \mathrm{t}ilde{\mathbf{R}}_1 \right) = \mathrm{t}ilde{\mathbf{R}}_1(TM)$. Hence, $X = \mathrm{t}ilde{\mathbf{R}}_1 Z$, for some $Z \in TM$, and
\betaegin{multline*}
\al{X}{Y} = \al{\mathrm{t}ilde{\mathbf{R}}_1 Z}{Y} = \al{\sum_{i \in I} \mathbf{R}_i Z}{Y} \stackrel{\varepsilonqref{eq: derivada S}}{=} \sum_{i \in I}\mathbf{T}_i \al{Z}{Y} = \\
\stackrel{\varepsilonqref{eq: derivada S}}{=} \sum_{i \in I}\al{Z}{\mathbf{R}_i Y} = \al{Z}{\mathrm{t}ilde{\mathbf{R}}_1 Y} = 0.
\varepsilonnd{multline*}
On the other side, $\mathrm{t}ilde{\mathbf{S}}_1 = 0 = \mathrm{t}ilde{\mathbf{S}}_2$, thus $\Pi_1 f_* X = f_* \mathrm{t}ilde{\mathbf{R}}_1 X + \mathrm{t}ilde{\mathbf{S}}_1 X = f_* X$ and $\Pi_2 f_* Y = f_* \mathrm{t}ilde{\mathbf{R}}_2 Y + \mathrm{t}ilde{\mathbf{S}}_2 Y = f_* Y$,
that is, $f_* X \in \rhorod\lambdaimits_{i \in I} \mathbb{R}^{N_i}$ and $f_* Y \in \rhorod\lambdaimits_{\substack{i=1 \\ i \nablaotin I}}^\varepsilonll \mathbb{R}^{N_i}$. It follows that $\rhoi_i f_* X = 0$, $\varphiorall i \nablaotin I$, and $\rhoi_i f_* Y = 0$, $\varphiorall i \in I$. Thus
\betaegin{multline*}
\aF{X}{Y} = \imath_* \af{X}{Y} + \sum_{i=1}^\varepsilonll \interno{\rhoi_i f_* X}{f_*Y}\nablau_i = \\
= \sum_{i \in I} \interno{f_* X}{\rhoi_i f_*Y}\nablau_i + \sum_{\substack{i=1 \\ i \nablaotin I} }^\varepsilonll \interno{\rhoi_i f_*X}{f_*Y}\nablau_i = 0.
\varepsilonnd{multline*}
Therefore the claim holds.
\varepsilonnd{afi}
Now, by Moore's Lemma (see \cite{Detc}), there is a orthogonal decomposition $\mathbb{R}N = V_0 \obot V_1 \obot V_2$ and a vector $v_0 \in V_0$ such that $F \colon M_1 \mathrm{t}imes M_2 \mathrm{t}o \mathbb{R}N$ is given by $F(x,y) = (v_0, F_1(x), F_2(y))$. Besides
\betaegin{align*}
& V_1 = \spa\set{F_*(p) X}{p \in M_1 \mathrm{t}imes M_2 \ \mathrm{t}ext{and} \ X \in \lambdaeft.\ker\mathrm{t}ilde{\mathbf{R}}_2\right|_{T_p M}}, \\
& V_2 = \spa\set{F_*(p) Y}{p \in M_1 \mathrm{t}imes M_2 \ \mathrm{t}ext{and} \ Y \in \lambdaeft.\ker\mathrm{t}ilde{\mathbf{R}}_1\right|_{T_p M}},
\varepsilonnd{align*}
and $F_i(M_i) \subset V_i$.
But, $\Pi_if_*(\ker \mathrm{t}ilde{\mathbf{R}}_i) = \{0\}$ and $\lambdaeft.\Pi_i\right|_{f_*\ker (\id - \mathrm{t}ilde{\mathbf{R}}_i)} = \id$, thus
\betaegin{align*}
& \spa\set{F_*(p) X}{p \in M_1 \mathrm{t}imes M_2 \ \mathrm{t}ext{and} \ X \in \ker \lambdaeft.\mathrm{t}ilde{\mathbf{R}}_2\right|_{T_p M}} \subset \rhorod_{i \in I} \mathbb{R}^{N_i} \quad \mathrm{t}ext{and}\\
& \spa\set{F_*(p) Y}{p \in M_1 \mathrm{t}imes M_2 \ \mathrm{t}ext{and} \ Y \in \ker \lambdaeft.\mathrm{t}ilde{\mathbf{R}}_1\right|_{T_p M}} \subset \rhorod_{\substack{i=1 \\ i \nablaotin I}}^\varepsilonll \mathbb{R}^{N_i}.
\varepsilonnd{align*}
Therefore $F_1(M_1)\subset\rhorod\lambdaimits_{i \in I} \mathbb{R}^{N_i}$ and $F_2(M_2) \subset \rhorod\lambdaimits_{\substack{i=1 \\ i \nablaotin I }}^\varepsilonll \mathbb{R}^{N_i}$. Hence we can define $f_1 \colon M_1 \mathrm{t}o \rhorod\lambdaimits_{i \in I} \mathbb{R}^{N_i}$ and $f_2 \colon M_2 \mathrm{t}o \rhorod\lambdaimits_{\substack{i=1 \\ i \nablaotin I}}^\varepsilonll \mathbb{R}^{N_i}$ by $f_1(x) := \Pi_1(v_0) + F_1(x)$ and $f_2(y) = \Pi_2(v_0) + F_2(y)$. Consequently
\[f(x,y) = \betaig(f_1(x), f_2(y)\betaig) \in \lambdaeft( \rhorod_{i \in I} \o{i}\right) \mathrm{t}imes \lambdaeft( \rhorod_{\substack{i=1 \\ i \nablaotin I }}^\varepsilonll\o{i}\right).\]
If $M$ is complete and simply connected, De Rham's Lemma assure us that $M$ is (globally) isometric to $L_1 \mathrm{t}imes L_2$, where $L_1$ and $L_2$ are leafs of $\ker \mathrm{t}ilde{\mathbf{R}}_2$ and $(\ker \mathbf{T}_2)^\rhoerp$ (respectively) at the same point. In this case, considering $f \colon L_1 \mathrm{t}imes L_2 \mathrm{t}o \hat{\mathbb{O}}$, the calculations made above show that $f$ is globally a product immersion.
\varepsilonnd{proof}
\betaegin{cor}
Let $f \colon M \mathrm{t}o \hat{\mathbb{O}}$ be an isometric immersion and let $\varepsilonll_1$, $\cdots$, $\varepsilonll_n$ be positive natural numbers such that $\sum\lambdaimits_{j=1}^n \varepsilonll_j = \varepsilonll$. Let also $I_1 := \{i_1^1, \cdots, i_{\varepsilonll_1}^1\}$, $\cdots$, $I_n := \{i_1^n, \cdots, i_{\varepsilonll_n}^n\}$ be disjoint sets such that $\betaigcup\lambdaimits_{j=1}^n I_j = \{1, \cdots, \varepsilonll\}$. Then the following claims are equivalent:
\betaegin{enum}
\item $M$ is locally (isometric to) a product manifold $M_1^{m_1}\mathrm{t}imes \cdots \mathrm{t}imes M_n^{m_n}$, with $0 < m_j < m$, and $f$ is locally a product immersion $f|_{M_1\mathrm{t}imes \cdots \mathrm{t}imes M_n} = f_1 \mathrm{t}imes \cdots \mathrm{t}imes f_n$, given by $f(x_1,\cdots, x_n) = (f_1(x_1), \cdots, f_n(x_n))$, where $f_j \colon M_j \mathrm{t}o \rhorod\lambdaimits_{i \in I_j} \o{i}$ is a isometric immersion, for each $j \in \{1, \cdots, n\}$.
\item For each $j \in \{1, \cdots, n\}$, $\sum\lambdaimits_{i \in I_j} \mathbf{S}_i = 0$ and $0< \mathrm{d}im \ker \lambdaeft(\sum\lambdaimits_{i \in I_j} \mathbf{R}_i \right) < m$.
\varepsilonnd{enum}
Besides, if \mathrm{t}extsl{(II)} holds and $M$ is complete and simply connected, then $M$ is globally isometric to product manifold $M_1^{m_1} \mathrm{t}imes \cdots \mathrm{t}imes M_n^{m_n}$ and $f$ is globally an product immersion like in \mathrm{t}extsl{(I)}.
\varepsilonnd{cor}
\subsection{Weighted sums}
Let $k_1, \cdots, k_\varepsilonll \in \mathbb{R}$ and $a_1, \cdots, a_\varepsilonll\in \mathbb{R}^*$ be such that $\sum\lambdaimits_{i=1}^\varepsilonll a_i^2 = 1$. For each $i$, let $\mathrm{t}ilde{k}_i := a_i^2 k_i$ and $f_i \colon M^m \mathrm{t}o \mathbb{O}_{\mathrm{t}ilde{k}_i}^{n_i}$ be a isometric immersion. We can define $f \colon M^m \mathrm{t}o \hat{\mathbb{O}}$ by $f(x) := \lambdaeft(a_1 f_1(x), \cdots, a_\varepsilonll f_\varepsilonll(x)\right)$. Thus $f$ is a isometric immersion, because
\[\interno{f_* X}{f_* Y} = \sum_{i=1}^\varepsilonll a_i^2 \interno{{f_i}_*X}{{f_i}_*Y} = \interno{X}{Y}.\]
This immersion $f$ is called \mathrm{t}extbf{weighted sum} of the immersions $f_1$, $\cdots$, $f_\varepsilonll$ and the numbers $a_1$, $ \cdots$, $a_\varepsilonll$ are called the \mathrm{t}extbf{weights} of $f$.
Now, lets consider $F = \imath \circ f$. We know that the fields $\nablau_i = -k_i(\rhoi_i \circ F)$ are normal to $F$ and that $\mathbb{R}N = F_*T M \obot \imath_*T^\rhoerp M \obot \spa\{\nablau_1, \cdots, \nablau_\varepsilonll\}$, where $T_x^\rhoerp M \subset T_{f(x)}\hat{\mathbb{O}}$ is the normal space of $f$ at $x$. But since $\sum\lambdaimits_{i=1}^\varepsilonll a_i^2 = 1$ and $F_*T_xM = \set{\lambdaeft(a_1 {f_1}_* X, \cdots, a_\varepsilonll{f_\varepsilonll}_*X \right)}{X \in T_x M}$, then $\lambdaeft( a_1{f_1}_* X, \cdots, \varphirac{a_i^2-1}{a_i}{f_i}_* X, \cdots, a_\varepsilonll {f_\varepsilonll}_* X\right) \in \imath_* T_x^\rhoerp M \rhoerp F_* T_xM \obot \spa\{\nablau_1, \cdots, \nablau_1\varepsilonll\}$.
Now we will study the tensors $\alpha$, $\mathbf{R}_i$, $\mathbf{S}_i$ and $\mathbf{T}_i$ of $f$.
\betaegin{lem}\lambdaabel{lem:RS weighted}
If $f$ is a weighted sum and $a_1, \cdots, a_\varepsilonll$ are its weights, then $\mathbf{R}_i = a_i^2 \id$ and $\mathbf{S}_i X = -a_i^2\lambdaeft(a_1{f_1}_*X, \cdots, \varphirac{a_i^2-1}{a_i}{f_i}_*X, \cdots, a_\varepsilonll{f_\varepsilonll}_*X\right)$.
\varepsilonnd{lem}
\betaegin{proof}
\betaegin{align*}
& \rhoi_if_*X = \lambdaeft(0, \cdots, a_i{f_i}_*X, \cdots, 0\right) = \lambdaeft(0, \cdots, \lambdaeft(\sum_{j=1}^\varepsilonll a_j^2 \right)a_i{f_i}_*X, \cdots, 0 \right) = \\
& = a_i^2 \lambdaeft( 0, \cdots, a_i {f_i}_*X, \cdots, 0 \right) + \lambdaeft(0, \cdots, a_i \lambdaeft(\sum_{\substack{j=1\\ j\nablae i}}^\varepsilonll a_j^2 \right){f_i}_*X, \cdots 0\right) = \\
& = a_i^2 \lambdaeft( a_1{f_1}_* X, \cdots, a_i {f_i}_*X, \cdots, a_\varepsilonll {f_\varepsilonll}_* X \right) - \\
& \quad - a_i^2 \lambdaeft( a_1{f_1}_* X, \cdots, 0, \cdots, a_\varepsilonll {f_\varepsilonll}_* X \right) + a_i\lambdaeft(0, \cdots, \lambdaeft(1-a_i^2\right){f_i}_*X, \cdots, 0 \right) = \\
& = a_i^2 f_* X - a_i^2\lambdaeft(a_1{f_1}_*X, \cdots, \varphirac{a_i^2-1}{a_i}{f_i}_*X, \cdots, a_\varepsilonll{f_\varepsilonll}_*X\right). \qedhere
\varepsilonnd{align*}
\varepsilonnd{proof}
\betaegin{obs}
From Lemma \ref{lem:RS weighted}
\[\mathbf{S}_i\lambdaeft(T_x M \right) = \set{a_i^2\lambdaeft(a_1{f_1}_*X, \cdots, \varphirac{a_i^2-1}{a_i}{f_i}_*X, \cdots, a_\varepsilonll{f_\varepsilonll}_*X\right)}{X \in T_x M},\]
and $\mathrm{d}im \mathbf{S}_i\lambdaeft(T_x M \right) = m$. Besides, $\mathbf{T}_i \mathbf{S}_i = \mathbf{S}_i (\id - \mathbf{R}_i) = \lambdaeft(1-a_i^2 \right) \mathbf{S}_i = \sum\lambdaimits_{\substack{j=1\\ j\nablae i}}^\varepsilonll a_j^2 \mathbf{S}_i$.
\varepsilonnd{obs}
\betaegin{lem}\lambdaabel{lem: segunda forma}
If $f$ is a weighted sum then
\[\af{X}{Y} = \betaig(a_1\alpha_{f_1}(X,Y), \cdots, a_\varepsilonll\alpha_{f_\varepsilonll}(X,Y) \betaig).\]
\varepsilonnd{lem}
\betaegin{proof}
For each $i$, let $\mathrm{t}ilde{\imath}_i \colon \mathbb{O}_{\mathrm{t}ilde{k}_i}^{n_i} \mathrm{t}o \mathbb{R}^{N_i}$ be the canonical inclusion and let $\mathcal{J}_i \colon \mathbb{R}^{N_i} \mathrm{t}o \mathbb{R}N$ be totally geodesic immersion given by $\mathcal{J}_i(x) := (0, \cdots, x, \cdots, 0)$, with $x$ in the $i$th position. Thus, each $\mathrm{t}ilde{\imath}_i$ is the restriction of the identity $I_i \colon \mathbb{R}^{N_i} \mathrm{t}o \mathbb{R}^{N_i}$ and $T_x \mathbb{O}_{\mathrm{t}ilde{k}_i}^{n_i} = T_{a_i x} \o{i}$. So, if $X \in T_x \mathbb{O}_{\mathrm{t}ilde{k}_i}^{n_i}$, then $\lambdaeft.\mathrm{t}ilde{\imath}_i\right._* X = {I_i}_* X = {\imath_i}_* X$, where $\imath_i \colon \o{i} \mathrm{t}o \mathbb{R}^{N_i}$ is the canonical inclusion.
Besides, $\imath = (\imath_1 \mathrm{t}imes \cdots \mathrm{t}imes \imath_\varepsilonll)$ and
\betaegin{multline*}
F_* X = \imath_* \betaig(a_1{f_1}_* X, \cdots, a_\varepsilonll {f_\varepsilonll}_*X \betaig) = \betaig({\imath_1}_* a_1 {f_1}_* X, \cdots, {\imath_\varepsilonll}_* a_\varepsilonll {f_2}_*X \betaig) = \\
= \betaig( a_1 \lambdaeft. \mathrm{t}ilde{\imath}_1 \right._* {f_1}_* X, \cdots, a_\varepsilonll \lambdaeft.\mathrm{t}ilde{\imath}_2\right._* {f_2}_* X \betaig).
\varepsilonnd{multline*}
Let $\nablatil^i$ be the Levi-Civita connexion in $\mathbb{R}^{N_i}$. Hence
\betaegin{align*}
& \nablatil_X F_* Y = \nablatil_X \betaig( a_1 \lambdaeft. \mathrm{t}ilde{\imath}_1 \right._* {f_1}_* Y, \cdots, a_\varepsilonll \lambdaeft.\mathrm{t}ilde{\imath}_2\right._* {f_\varepsilonll}_* Y \betaig) = \\
&= \nablatil_X \lambdaeft[ a_1 {\mathcal{J}_1}_* \lambdaeft.\mathrm{t}ilde{\imath}_1\right._* {f_1}_* Y + \cdots + a_\varepsilonll {\mathcal{J}_\varepsilonll}_* \lambdaeft.\mathrm{t}ilde{\imath}_2\right._* {f_\varepsilonll}_* Y \right] = \\
& = a_1 {\mathcal{J}_1}_* \nablatil^1_X \lambdaeft( \mathrm{t}ilde{\imath}_1 \circ f_1\right)_* Y + \cdots + a_\varepsilonll {\mathcal{J}_\varepsilonll}_* \nablatil^\varepsilonll_X \lambdaeft( \mathrm{t}ilde{\imath}_2 \circ f_\varepsilonll\right)_* Y = \\
& = a_1 {\mathcal{J}_1}_* \lambdaeft.\mathrm{t}ilde{\imath}_1\right._* \lambdaeft[ {f_1}_* \nabla_X Y + \alpha_{f_1}(X,Y) \right] + a_1 {\mathcal{J}_1}_* \alpha_{\mathrm{t}ilde{\imath}_1}\lambdaeft({f_1}_*X, {f_1}_* Y\right) + \cdots + \\
& \quad + a_\varepsilonll {\mathcal{J}_\varepsilonll}_* \lambdaeft.\mathrm{t}ilde{\imath}_\varepsilonll\right._* \lambdaeft[ {f_\varepsilonll}_* \nabla_X Y + \alpha_{f_\varepsilonll}(X,Y) \right] + a_\varepsilonll {\mathcal{J}_\varepsilonll}_* \alpha_{\mathrm{t}ilde{\imath}_\varepsilonll}\lambdaeft({f_\varepsilonll}_*X, {f_\varepsilonll}_* Y\right) = \\
& = \betaig( a_1 \lambdaeft.\mathrm{t}ilde{\imath}_1\right._* \lambdaeft[{f_1}_* \nabla_X Y + \alpha_{f_1}(X, Y) \right], 0, \cdots, 0 \betaig) - a_1 \interno{X}{Y} \lambdaeft( \mathrm{t}ilde{k}_1 f_1, 0, \cdots, 0 \right) + \\
& \quad + \cdots + \\
& \quad + \betaig( 0, \cdots, 0, a_\varepsilonll \lambdaeft.\mathrm{t}ilde{\imath}_\varepsilonll\right._* \lambdaeft[{f_\varepsilonll}_* \nabla_X Y + \alpha_{f_\varepsilonll}(X,Y) \right]\betaig) - a_\varepsilonll \interno{X}{Y} \lambdaeft( 0, \cdots, 0, \mathrm{t}ilde{k}_\varepsilonll f_\varepsilonll \right)= \\
& = \betaig( {\imath_1}_* a_1 {f_1}_* \nabla_X Y , \cdots, {\imath_\varepsilonll}_* a_\varepsilonll {f_\varepsilonll}_* \nabla_X Y \betaig) + \\
& \quad + \betaig( {\imath_1}_* a_1 \alpha_{f_1}(X,Y), \cdots, {\imath_\varepsilonll}_* a_\varepsilonll \alpha_{f_\varepsilonll}(X,Y) \betaig) - \interno{X}{Y} \lambdaeft( a_1 \mathrm{t}ilde{k}_1 f_1, \cdots, a_\varepsilonll \mathrm{t}ilde{k}_\varepsilonll f_\varepsilonll \right).
\varepsilonnd{align*}
But, $a_i \mathrm{t}ilde{k_i} = a_i \cdot a_i^2 k_i$ and $F = (a_1f_1, \cdots, a_\varepsilonll f_\varepsilonll)$, thus
\[\lambdaeft( a_1 \mathrm{t}ilde{k}_1 f_1, \cdots, a_\varepsilonll \mathrm{t}ilde{k}_\varepsilonll f_\varepsilonll \right) = \lambdaeft(a_1^2 \cdot a_1 k_1 f_1, \cdots, a_\varepsilonll^2 \cdot a_\varepsilonll k_\varepsilonll f_\varepsilonll\right) = \sum_{i=1}^\varepsilonll a_i^2 k_i(\rhoi_i \circ F).\]
Therefore
\betaegin{multline*}
\nablatil_X F_* Y = F_* \nabla_X Y + \imath_* \betaig( a_1 \alpha_{f_1}(X,Y), \cdots, a_\varepsilonll \alpha_{f_\varepsilonll}(X,Y) \betaig) + \interno{X}{Y} \sum_{i=1}^\varepsilonll a_i^2 \nablau_i.
\varepsilonnd{multline*}
On the oder side, $\nablatil_X F_* Y = F_* \nabla_X Y + \imath_* \af{X}{Y} + \sum\lambdaimits_{i=1}^\varepsilonll \interno{\mathbf{R}_i X}{Y} \nablau_i$. Since $\mathbf{R}_i X = a_i^2 X$, it follows that $\af{X}{Y} = \betaig(a_1\alpha_{f_1}(X,Y), \cdots, a_\varepsilonll\alpha_{f_\varepsilonll}(X,Y) \betaig)$.
\varepsilonnd{proof}
\betaegin{cor}\lambdaabel{cor: f_i umbílica}
Let $f = (a_1f_1, \cdots, a_\varepsilonll f_\varepsilonll)$ be a weighted sum of isometric immersions. The following claims are equivalent:
\betaegin{enum}
\item $f$ is umbilical.
\item $F = \imath \circ f$ is umbilical.
\item Each $f_i$ é umbilical.
\varepsilonnd{enum}
\varepsilonnd{cor}
\betaegin{proof}
It follows directly from Lemma \ref{lem:RS weighted}.
\varepsilonnd{proof}
The follow proposition characterizes weighted sums.
\betaegin{prop}\lambdaabel{prop: weighted}
Let $f \colon M^m \mathrm{t}o \hat{\mathbb{O}}$ be an isometric immersion, then $f$ is a weighted sum of isometric immersions with positive if, and only if, there are $a_1, \cdots, a_\varepsilonll \in (0,1)$ such that $\mathbf{R}_i = a_i^2 \id$ and $\sum\lambdaimits_{i=1}^\varepsilonll a_i^2 = 1$.
\varepsilonnd{prop}
\betaegin{proof}
If $f \colon M \mathrm{t}o \hat{\mathbb{O}}$ is a wighted sum with weights $a_1, \cdots, a_\varepsilonll \in \mathbb{R}_+^*$, then $\sum\lambdaimits_{i=1}^\varepsilonll a_i^2 = 1$, $a_i \in (0,1)$ and we already know that $\mathbf{R}_i = a_i^2 \id$, for each $i \in \{1, \cdots, \varepsilonll\}$.
Lets suppose that there are $a_1, \cdots, a_\varepsilonll \in (0,1)$ such that $\mathbf{R}_i = a_i^2 \id$ and $\sum\lambdaimits_{i=1}^\varepsilonll a_i^2 =1$. Thus $a_i^2 \interno{X}{Y} = \interno{\mathbf{R}_i X}{Y} = \interno{\mathbf{L}_i^\mathrm{t} \mathbf{L}_i X}{Y} = \interno{\rhoi_i f_* X}{\rhoi_i f_* Y}$, as a consequence each $\rhoi_i\circ f$ is a similarity with ratio $a_i$ and $f_i := a_i^{-1}(\rhoi_i \circ f)$ is an isometric immersion in $\mathbb{R}^{N_i}$.
Let $\mathrm{t}ilde{k}_i := a_i^2 k_i$. If $k_i = 0$ then $\mathrm{t}ilde{k}_i = 0$, $\mathbb{O}_{\mathrm{t}ilde{k}_i}^{n_i} = \mathbb{E}^{N_i}$ and $f_i(x) \in \mathbb{O}_{\mathrm{t}ilde{k}_i}^{n_i}$. If $k_i \nablae 0$, then $\|f_i(x)\|^2 = \varphirac{1}{a_i^2 k_i}$, thus $f_i(x) \in \mathbb{O}_{\mathrm{t}ilde{k}_i}^{n_i} \subset \mathbb{R}^{N_i}$.
But $f = (a_1 f_1, \cdots, a_\varepsilonll f_\varepsilonll)$, therefore $f$ is a weighted sum.
\varepsilonnd{proof}
\subsection{A particular weighted sum}\lambdaabel{seção: exemplo}
Let $k_1, \cdots, k_\varepsilonll$ be real numbers such that $k_ik_j > 0$, for every $i, j \in \{1, \cdots, \varepsilonll\}$. Lets denote $\varepsilon := \varphirac{k_1}{|k_1|}$, $M_i := \rhorod\lambdaimits_{ \substack{j=1\\ j\nablae i} }^\varepsilonll k_j$, $\lambda := \sum\lambdaimits_{i=1}^\varepsilonll M_i$, $a_i := \lambdaeft(\varphirac{M_i}{\lambda}\right)^{\varphirac{1}{2}}$ and $\mathrm{t}ilde{k}_i := a_i^2 k_i$.
Thus, $\varepsilon = \varphirac{k_i}{|k_i|}$, $\sum\lambdaimits_{i=1}^\varepsilonll a_i^2 = 1$ and $\mathrm{t}ilde{k}_i = a_i^2 k_i = \varphirac{\rhorod\lambdaimits_{\substack{j=1\\ j\nablae i}}^\varepsilonll k_j}{\lambda}k_i = \varphirac{\rhorod\lambdaimits_{j=1}^\varepsilonll k_j}{\lambda} = \mathrm{t}ilde k_1$.
Lets denote $k := \mathrm{t}ilde{k}_1$ and let $T_1, \cdots, T_\varepsilonll \in \mathrm{O}_{\mathrm{t}au(k)}(n+1)$ be such that $T_i \lambdaeft(\o{i}\right) = \o{i}$. We know that the restrictions $\lambdaeft.T_i\right|_{\o{}} \colon \o{} \mathrm{t}o \o{}$ are isometries, hence the function
\[\betaegin{matrix}
g : & \o{} & \lambdaongrightarrow & \o{1} \x \cdots \x \o{\ell} \\
& x & \lambdaongmapsto & \betaig(a_1 T_1(x), \cdots, a_\varepsilonll T_\varepsilonll(x) \betaig),
\varepsilonnd{matrix}\]
is a weighted sum of $T_1$, $\cdots$, $T_\varepsilonll$ with weights $a_1$, $\cdots$, $a_\varepsilonll$.
By Lemma \ref{lem: segunda forma}, $g$ is a totally geodesic immersion. Besides, the following equations hold
\betaegin{align*}
& \mathbf{R}_i = a_i^2 \id, \quad \mathbf{S}_i X = -a_i^2\lambdaeft(a_1T_1 X, \cdots, \varphirac{a_i^2-1}{a_i}T_i X, \cdots, a_\varepsilonll T_\varepsilonll X\right),\\
& k_i \mathbf{R}_i = k_i a_i^2 \id = k \id, \quad \mathbf{T}_i\mathbf{S}_i = \lambdaeft(1-a_i^2 \right) \mathbf{S}_i \quad \mathrm{t}ext{and} \\
& \mathbf{S}_i\lambdaeft(T_x \o{} \right) = \set{a_i^2\lambdaeft(a_1T_1 X, \cdots, \varphirac{a_i^2-1}{a_i}T_i X, \cdots, a_\varepsilonll T_\varepsilonll X\right)}{X \in T_x \o{}}.
\varepsilonnd{align*}
The weighted sum $g$ plays an important hole at the theory of isometric immersions in $\hat{\mathbb{O}}$. This hole is a kind of reduction of codimension theorem (Theorem \ref{teo: Phi=0}). But first we need some results.
\betaegin{lem}\lambdaabel{lem: V}
Let $k_1, \cdots, k_\varepsilonll \in \mathbb{R}^*$ be such that $k_ik_j > 0$, for all $i,j \in \{1, \cdots, \varepsilonll\}$, and lets denote $M_i := \rhorod\lambdaimits_{\substack{j=1\\ j\nablae i}}^\varepsilonll k_j$, $\lambda := \sum\lambdaimits_{i=1}^\varepsilonll M_i$, $a_i := \lambdaeft(\varphirac{M_i}{\lambda}\right)^{\varphirac{1}{2}}$ and $\quad k := \mathrm{t}ilde{k}_i := a_i^2 k_i$. Let also $V^{n+1} \subset \rhorod\lambdaimits_{i=1}^\varepsilonll \mathbb{R}_{\mathrm{t}au(k)}^{n+1}$ be a vector subspace isomorphic to $\mathbb{R}^{n+1}_{\mathrm{t}au(k)}$. With the assumptions above, the following claims are equivalent:
\betaegin{enum}
\item $V \cap \mathbb{S}_k^{n\varepsilonll + \varepsilonll-1} \subset \rhorod\lambdaimits_{i=1}^\varepsilonll \mathbb{S}_{k_i}^n \subset \rhorod\lambdaimits_{i=1}^\varepsilonll \mathbb{R}_{\mathrm{t}au(k)}^{n+1}$.
\item $\lambdaeft.\rhoi_i\right|_V$ is a similarity with ratio $a_i$.
\item There are linear isometries $T_1, \cdots, T_\varepsilonll \in \mathrm{O}_{\mathrm{t}au(k)}(n+1)$ such that
\[V = \set{\lambdaeft(a_1 T_1(x), \cdots, a_\varepsilonll T_\varepsilonll(x)\right)}{x \in \mathbb{R}_{\mathrm{t}au(k)}^{n+1}}.\]
\varepsilonnd{enum}
\varepsilonnd{lem}
\betaegin{proof}
Since $k_ik_j > 0$, then $k_1, \cdots, k_\varepsilonll$ are all positive or all negative. Thus $k$ and $k_i$ have the same sign.
\nablaoindent \underline{\mathrm{t}extsl{(I)} $\mathbb{R}ightarrow$ \mathrm{t}extsl{(II)}:} We have two cases: $k >0$ or $k < 0$.
\nablaoindent
\mathrm{t}extbf{First case: $k > 0$.}
In this case $\mathbb{R}_{\varepsilonll\mathrm{t}au(k)}^{\varepsilonll n+\varepsilonll} = \mathbb{R}^{\varepsilonll n+\varepsilonll}$ is an euclidean space. Thus,
\betaegin{align*}
& V \cap \mathbb{S}_k^{n\varepsilonll + \varepsilonll-1} \subset \rhorod_{i=1}^\varepsilonll \mathbb{S}_{k_i}^n \mathbb{L}eftrightarrow \varphirac{x}{\hksqrt{k} \|x\|} \in \rhorod_{i=1}^\varepsilonll \mathbb{S}_{k_i}^n, \ \varphiorall x \in V\setminus\{0\} \mathbb{L}eftrightarrow \\
& \mathbb{L}eftrightarrow \varphirac{\|\rhoi_i x\|^2}{k \|x\|^2} = \varphirac{1}{k_i}, \ \varphiorall x \in V\setminus\{0\} \mathbb{L}eftrightarrow \ \|\rhoi_i x\|^2 = \varphirac{k \|x\|^2}{k_i} = a_i^2 \|x\|^2, \ \varphiorall x \in V.
\varepsilonnd{align*}
Therefore \mathrm{t}extsl{(I) $\mathbb{R}ightarrow$ (II)}.
\nablaoindent \mathrm{t}extbf{Second case: $k < 0$.}
In this case $\mathbb{R}_{\varepsilonll \mathrm{t}au(k)}^{\varepsilonll n+\varepsilonll} = \mathbb{R}_\varepsilonll^{\varepsilonll n+\varepsilonll}$ and
$V \cap \mathbb{S}_k^{n\varepsilonll + \varepsilonll -1} = \set{\varphirac{x}{\hksqrt{|k|} \|x\|}}{x \in V \ \mathrm{t}ext{and} \ \|x\|^2 < 0}$.
Hence,
\betaegin{align*}
& V \cap \mathbb{S}_k^{n\varepsilonll + \varepsilonll -1} \subset \rhorod_{i=1}^\varepsilonll \mathbb{S}_{k_i}^n \mathbb{L}eftrightarrow \varphirac{x}{\hksqrt{|k|} \|x\|} \in \rhorod_{i=1}^\varepsilonll \mathbb{S}_{k_i}^n, \ \varphiorall x \in V \ \mathrm{t}ext{with} \ \|x\|^2 < 0 \mathbb{L}eftrightarrow \\
& \mathbb{L}eftrightarrow \varphirac{\|\rhoi_i x\|^2}{k \|x\|^2} = \varphirac{1}{k_i}, \ \varphiorall x \in V \ \mathrm{t}ext{with} \ \|x\|^2 < 0 \mathbb{L}eftrightarrow \\
& \mathbb{L}eftrightarrow \|\rhoi_i x\|^2 = \varphirac{k \|x\|^2}{k_i} = a_i^2\|x\|^2, \ \varphiorall x \in V \ \mathrm{t}ext{with} \ \|x\|^2 < 0.
\varepsilonnd{align*}
Lets suppose that $V \cap \mathbb{S}_k^{n\varepsilonll + \varepsilonll -1} \subset \rhorod\lambdaimits_{i=1}^\varepsilonll \mathbb{S}_{k_i}^n$. We will show that each $\lambdaeft.\rhoi_i\right|_V$ is a similarity of ratio $a_i$.
Let $\{e_0, \cdots, e_n\}$ be a orthonormal base of $V$ with $e_0$ timelike. Then $\|\rhoi_ie_0\|^2 = -a_i^2$. Let $\alpha, \betaeta \in \mathbb{R}$ be such that $-\alpha^2 + \betaeta^2 < 0$, and let $j \in \{1, \cdots, n\}$. Thus $\alpha e_0 + \betaeta e_j$ is timelike and, by the equivalences above, $\lambdaeft\|\rhoi_i\lambdaeft(\alpha e_0 + \betaeta e_j \right) \right\|^2 = a_i^2 \lambdaeft(-\alpha^2 + \betaeta^2 \right)$.
On the other side,
\betaegin{align*}
& \lambdaeft\|\rhoi_i\lambdaeft(\alpha e_0 + \betaeta e_j \right) \right\|^2 = \alpha^2 \|\rhoi_ie_0\|^2 + 2\alpha\betaeta \interno{\rhoi_ie_0}{\rhoi_ie_j} + \betaeta^2 \|\rhoi_ie_j\|^2 = \\
& =-a_i^2\alpha^2 + 2\alpha\betaeta \interno{\rhoi_ie_0}{\rhoi_ie_j} + \betaeta^2 \|\rhoi_ie_j\|^2.
\varepsilonnd{align*}
Hence
\betaegin{align*}
& -a_i^2\alpha^2 + 2\alpha\betaeta \interno{\rhoi_ie_0}{\rhoi_ie_j} + \betaeta^2 \|\rhoi_ie_j\|^2 = a_i^2 \lambdaeft(-\alpha^2 + \betaeta^2 \right) \ \mathbb{R}ightarrow \\
& \mathbb{R}ightarrow \ 2\alpha\betaeta \interno{\rhoi_ie_0}{\rhoi_ie_j} + \betaeta^2 \|\rhoi_ie_j\|^2 = a_i^2 \betaeta^2 \ \mathbb{R}ightarrow \\
& \mathbb{R}ightarrow \interno{\rhoi_ie_0}{\rhoi_ie_j} = \varphirac{\betaeta}{2\alpha}\lambdaeft(a_i^2 - \|\rhoi_ie_j\|^2 \right), \ \mathrm{t}ext{if} \ |\alpha|>|\betaeta|.
\varepsilonnd{align*}
Thus, if we first take $\alpha=2$ and $\betaeta =1$, and then we take $\alpha=2$ and $\betaeta = -1$, we will conclude that
\[\interno{\rhoi_ie_0}{\rhoi_ie_j} = \varphirac{1}{4}\lambdaeft(a_i^2 - \|\rhoi_ie_j\|^2 \right) \quad \mathrm{t}ext{and} \quad \interno{\rhoi_ie_0}{\rhoi_ie_j} = \varphirac{-1}{4}\lambdaeft(a_i^2 - \|\rhoi_ie_j\|^2 \right).\]
Therefore $\interno{\rhoi_ie_0}{\rhoi_i e_j} = 0$ and $\|\rhoi_ie_j\|^2 = a_i^2$.
Now, let $\alpha$, $\betaeta$, $\gamma \in \mathbb{R}$ such that $\alpha^2 > \betaeta^2 + \gamma^2$, and let $j_1, j_2 \in \{1, \cdots, n\}$ with $j_1\nablae j_2$. Thus $\alpha e_0 + \betaeta e_{j_1} + \gammaamma e_{j_2}$ is timelike and
\betaegin{multline*}
a_i^2\lambdaeft(-\alpha^2 + \betaeta^2 + \gamma^2 \right) = \lambdaeft\|\rhoi_i\lambdaeft(\alpha e_0 + \betaeta e_{j_1} + \gamma e_{j_2}\right) \right\|^2 =\\
= a_i^2\lambdaeft(-\alpha^2 + \betaeta^2 + \gamma^2 \right) + 2 \betaeta \gamma \interno{\rhoi_ie_i}{\rhoi_ie_j}.
\varepsilonnd{multline*}
Therefore, $\interno{\rhoi_ie_{j_1}}{\rhoi_ie_{j_2}} = 0$ and we conclude that $\lambdaeft.\rhoi_i\right|_V$ is a similarity with ratio $a_i$. {\scriptsize\mathrm{t}extbullet}
\nablaoindent \underline{\mathrm{t}extsl{(II)} $\mathbb{R}ightarrow$ \mathrm{t}extsl{(III)}:} Lets suppose that each $\lambdaeft.\rhoi_i\right|_V$ be a similarity of ratio $a_i$, thus each $a_i^{-1}\cdot \lambdaeft.\rhoi_i\right|_V$ is a linear isometry on its image. So, given an orthonormal base $\{v_0, \cdots, v_n\}$ of $V$, let $G \colon \mathbb{R}_{\mathrm{t}au(k)}^{n+1} \mathrm{t}o V$ be the linear isometry such that $G(e_i) = v_i$, where $\{e_0, \cdots, e_n\}$ is the canonical base of $\mathbb{R}_{\mathrm{t}au(k)}^{n+1}$ (with $e_0$ timelike, if $k < 0$). Now let $T_i$ be given by $T_i := a_i^{-1} \cdot \lambdaeft(P_i \circ G \right)$, where $P_i$ is the projection of $\mathbb{R}_{\varepsilonll \mathrm{t}au(k)}^{\varepsilonll n+\varepsilonll} = \rhorod\lambdaimits_{i=1}^\varepsilonll \mathbb{R}_{\mathrm{t}au(k)}^{n+1}$ on its $i$th factor. Hence each $T_i$ is linear and
\[V = \set{G(x)}{x \in \mathbb{R}_{\mathrm{t}au(k)}^{n+1}} = \set{\betaig(a_1 T_1(x), \cdots, a_\varepsilonll T_\varepsilonll(x) \betaig)}{x \in \mathbb{R}_{\mathrm{t}au(k)}^{n+1}}.\]
On the other side, $\interno{T_i(x)}{T_i(y)} = \varphirac{1}{a_i^2}\interno{\rhoi_i(G(x))}{\rhoi_i(G(y))} = \interno{x}{y}$, therefore each $T_i$ is an isometry. {\scriptsize\mathrm{t}extbullet}
\nablaoindent \underline{\mathrm{t}extsl{(III)} $\mathbb{R}ightarrow$ \mathrm{t}extsl{(I)}:}
Lets suppose that there are $T_1, \cdots, T_\varepsilonll \in \mathrm{O}_{\mathrm{t}au(k)}(n+1)$ such that
\[V = \set{ \betaig(a_1 T_1(x), \cdots, a_\varepsilonll T_\varepsilonll(x) \betaig)}{x \in \mathbb{R}_{\mathrm{t}au(k)}^{n+1}}.\]
If $y \in V \cap \mathbb{S}(0,\varepsilon\hksqrt{|k|^{-1}})$, then $y = \betaig(a_1 T_1(x), \cdots, a_\varepsilonll T_\varepsilonll(x) \betaig)$, for some $x \in \mathbb{R}_{\mathrm{t}au(k)}^{n+1}$ and $\|y\|^2 = \varphirac{1}{k}$. Since $\sum\lambdaimits_{i=1}^\varepsilonll a_i^2 = 1$, then $\|x\|^2 = \|T_i(x)\|^2 = \|y\|^2 = \varphirac{1}{k}$. Thus $\|\rhoi_i(y)\|^2 = a_i^2 \|T_i(x)\|^2 = \varphirac{a_i^2}{k} = \varphirac{1}{k_i}$. Therefore $y \in \mathbb{S}_{k_1}^n \mathrm{t}imes \cdots \mathrm{t}imes \mathbb{S}_{k_\varepsilonll}^n$.
\varepsilonnd{proof}
\betaegin{cor}\lambdaabel{cor: V}
Let $k_1, \cdots, k_\varepsilonll \in \mathbb{R}_-^*$ and let $a_i$ and $k$ be like in Lemma \ref{lem: V}. Let also $V^{n+1} \subset \rhorod\lambdaimits_{i=1}^\varepsilonll \mathbb{L}^{n+1}$ be a vector subspace isomorphic to $\mathbb{L}^{n+1}$ and $C$ one of the two connected components of $V \cap \mathbb{S}_k^{n\varepsilonll + \varepsilonll -1}$. With the assumptions above, if $C \subset \rhorod\lambdaimits_{i=1}^\varepsilonll \mathbb{O}_{k_i}^n$, then there are linear isometries $T_1, \cdots, T_\varepsilonll \in \mathrm{O}_{\mathrm{t}au(k)}(n+1)$ such that $V = \set{\lambdaeft(a_1 T_1(x), \cdots, a_\varepsilonll T_\varepsilonll(x)\right)}{x \in \mathbb{R}_{\mathrm{t}au(k)}^{n+1}}$ and $T_i\lambdaeft(\mathbb{O}_{k_i}^n \right) = \mathbb{O}_{k_i}^n$.
\varepsilonnd{cor}
\betaegin{proof}
If $C \subset \rhorod\lambdaimits_{i=1}^\varepsilonll \mathbb{O}_{k_i}^n$, then
\[V \cap \mathbb{S}_k^{n\varepsilonll + \varepsilonll -1} = C \cup (-C) \subset \lambdaeft( \rhorod_{i=1}^\varepsilonll \mathbb{O}_{k_i}^n \right) \cup \lambdaeft(-\rhorod_{i=1}^\varepsilonll \mathbb{O}_{k_i}^n \right) = \rhorod_{i=1}^\varepsilonll \mathbb{S}_{k_i}^\varepsilonll.\]
Thus, by Lemma \ref{lem: V}, $V = \set{\lambdaeft(a_1 T_1(x), \cdots, a_\varepsilonll T_\varepsilonll(x)\right)}{x \in \mathbb{R}_{\mathrm{t}au(k)}^{n+1}}$, for some linear isometries $T_1, \cdots, T_\varepsilonll \in \mathrm{O}_{\mathrm{t}au(k)}(n+1)$.
So, if $y \in C$, there is an $x \in \mathbb{R}_{\mathrm{t}au(k)}^{n+1}$ such that $ y = \lambdaeft(a_1 T_1(x), \cdots, a_\varepsilonll T_\varepsilonll(x)\right)$. But $P_i(y) = a_i T_i(x) \in \mathbb{O}_{k_i}^n$, for all $i \in \{1, \cdots, \varepsilonll\}$, where $P_i$ is the projection on the $i$th coordinate. Hence $T_i(x) \in \mathbb{O}_k^n$, for all $i \in \{1, \cdots, n \}$, and $x \in \rhom \mathbb{O}_k^n$.
If $x \in \mathbb{O}_k^n$, then $T_i\lambdaeft(\mathbb{O}_k^n \right) = \mathbb{O}_k^n$ and $T_i \lambdaeft( \mathbb{O}_{k_i}^n \right) = \mathbb{O}_{k_i}^n$, for all $i$, otherwise, taking $\mathrm{t}ilde T_i = -T_i$, we conclude that $V = \set{\lambdaeft(a_1 \mathrm{t}ilde T_1(x), \cdots, a_\varepsilonll \mathrm{t}ilde T_\varepsilonll(x)\right)}{x \in \mathbb{R}_{\mathrm{t}au(k)}^{n+1}}$ and $\mathrm{t}ilde T_i \lambdaeft(\mathbb{O}_{k_i}^n\right) = \mathbb{O}_{k_i}^n$.
\varepsilonnd{proof}
\betaegin{df}
Let $f \colon M \mathrm{t}o N$ be an isometric immersion. The \mathrm{t}extbf{first normal space} of $f$ at $x$, is the space $\mathcal{N}_1(x) := \spa\set{\al{X}{Y}}{X, Y \in T_x M}$.
\varepsilonnd{df}
\betaegin{lem}\lambdaabel{lem: sinal}
Let $a_1, \cdots a_n \in \mathbb{R}^*$ and $A_i := \rhorod\lambdaimits_{\substack{j=1\\ j\nablae i}}^n a_j$. If $A_iA_j > 0$, for all $i,j \in \{1, \cdots, n\}$, then $a_ia_j >0$, for all $i,j \in \{1, \cdots, n\}$.
\varepsilonnd{lem}
\betaegin{proof}
Lets suppose, by absurd, that $a_1, \cdots, a_n$ do not have all the same sign. Hence, reordering if necessary, we can suppose that $a_1, \cdots, a_m$ are negative and that $a_{m+1}, \cdots, a_n$ are positive. Thus
\betaegin{align*}
A_1 &= \rhorod_{j=2}^n a_j = \rhorod_{j=2}^m a_j \cdot \rhorod_{j=m+1}^n a_j = (-1)^{m-1}\rhorod_{j=2}^n|a_j|, \\
A_n &= \rhorod_{j=1}^{n-1} a_j = \rhorod_{j=1}^m a_j \cdot \rhorod_{j=m+1}^{n-1} a_j = (-1)^m\rhorod_{j=1}^{n-1}|a_j|.
\varepsilonnd{align*}
Therefore $A_1A_n < 0$, which is a contradiction.
\varepsilonnd{proof}
\betaegin{prop}\lambdaabel{prop: Phi=0}
Let $f \colon M \mathrm{t}o \hat{\mathbb{O}}$ be an isometric immersion with $k_1$, $\cdots$, $k_\varepsilonll \in \mathbb{R}^*$. Thus, if $k_i \mathbf{R}_i = k_j \mathbf{R}_j$, for every $i, j$, then the following claims hold:
\betaegin{enum}
\item $\lambda := \sum\lambdaimits_{i=1}^\varepsilonll M_i \nablae 0$, where $M_i := \rhorod\lambdaimits_{\substack{j=1\\ j \nablae i}}^\varepsilonll k_j$.
\item $\mathbf{R}_i = \lambda_i \id$, where $\lambda_i := \varphirac{M_i}{\lambda}$.
\item $k_ik_j > 0$.
\item $\mathbf{L}_i$ is a similarity of ratio $a_i := \hksqrt{\lambda_i}$.
\item $\|\rhoi_i F\|^2 = a_i^2 \|F\|^2$.
\item $\mathcal{N}_1 \rhoerp \mathbf{S}(TM)$.
\item If $\mathcal{N}_1$ is parallel in the normal connexion of $f$, then each $\lambdaeft.\rhoi_i\right|_{\mathcal{N}_1}$ is a similarity of ratio $a_i$.
\varepsilonnd{enum}
\varepsilonnd{prop}
\betaegin{prova} From $k_i \mathbf{R}_i = k_j \mathbf{R}_j$, we know that $\mathbf{R}_j = \varphirac{k_i}{k_j} \mathbf{R}_i$.
\nablaoindent \mathrm{t}extbf{\mathrm{t}extsl{(I)} and \mathrm{t}extsl{(II)}:} Using the first equation in \varepsilonqref{somas},
\betaegin{multline*}
\id = \sum_{j=1}^n \mathbf{R}_j = \mathbf{R}_i + \sum_{\substack{j=1,\\ j\nablae i}}^\varepsilonll \mathbf{R}_j = \lambdaeft[1 + \sum_{\substack{j=1\\ j\nablae i}}^\varepsilonll \varphirac{k_i}{k_j}\right]\mathbf{R}_i = \lambdaeft[1 + k_i \varphirac{\sum\lambdaimits_{\substack{j=1\\j \nablae i}}^\varepsilonll \rhorod\lambdaimits_{\substack{l=1\\ l \nablaotin\{i, j\}}}^\varepsilonll k_l}{ \rhorod\lambdaimits_{\substack{j=1\\ j\nablae i}}^\varepsilonll k_j}\right] \mathbf{R}_i = \\
= \varphirac{\rhorod\lambdaimits_{\substack{j=1\\ j\nablae i}}^\varepsilonll k_j + \sum\lambdaimits_{\substack{j=1\\ j\nablae i}}^\varepsilonll \rhorod\lambdaimits_{\substack{l=1\\ l\nablae j}}^\varepsilonll k_l}{\rhorod\lambdaimits_{\substack{j=1\\ j\nablae i}}^\varepsilonll k_j} \mathbf{R}_i = \varphirac{M_i + \sum\lambdaimits_{\substack{j=1\\ j\nablae i}}^\varepsilonll M_j}{M_i} \mathbf{R}_i = \varphirac{\lambda}{M_i} \mathbf{R}_i.
\varepsilonnd{multline*}
Therefore $\lambda \nablae 0$ and $\mathbf{R}_i = \varphirac{M_i}{\lambda} \id$. {\scriptsize\mathrm{t}extbullet}
\nablaoindent \mathrm{t}extbf{\mathrm{t}extsl{(III)}:} Let $X \in T M$ with $X \nablae 0$. Hence
\[\interno{\mathbf{S}_i X}{\mathbf{S}_i X} = \interno{\mathbf{S}_i^\mathrm{t}\mathbf{S}_i X}{X} \stackrel{\varepsilonqref{eq: RST}}{=} \interno{\mathbf{R}_i(\id - \mathbf{R}_i)X}{X} \stackrel{\mathrm{t}ext{\mathrm{t}extsl{(II)}}}{=} \lambda_i(1-\lambda_i) \|X\|^2 \gammaeq 0.\]
Since $\mathbf{R}_i \nablae 0$, then $\lambda_i \in (0,1]$, and it follows that $\lambda$ and $M_i$ have the same sign. Therefore, $M_1$, $\cdots$, $M_\varepsilonll$ have all the same sign and, by Lemma \ref{lem: sinal}, $k_ik_j >0$, for all $i,j \in \{1, \cdots, \varepsilonll\}$. {\scriptsize\mathrm{t}extbullet}
\nablaoindent \mathrm{t}extbf{\mathrm{t}extsl{(IV)}:} Since $\mathbf{L}_i^\mathrm{t}\mathbf{L}_i = \mathbf{R}_i = \lambda_i \id$, then $\interno{\mathbf{L}_i X}{\mathbf{L}_i Y} = \lambda_i\interno{X}{Y}$. Therefore $\mathbf{L}_i$ is a similarity of ratio $a_i = \hksqrt{\lambda_i}$. {\scriptsize\mathrm{t}extbullet}
\nablaoindent \mathrm{t}extbf{\mathrm{t}extsl{(V)}:} $\|\rhoi_i F\|^2 = \varphirac{1}{k_i} = \varphirac{\rhorod\lambdaimits_{\substack{j=1\\ j\nablae i}}^\varepsilonll k_j}{\rhorod\lambdaimits_{j=1}^\varepsilonll k_j} = \varphirac{\rhorod\lambdaimits_{\substack{j=1\\ j\nablae i}}^\varepsilonll k_j}{\lambda}\cdot \varphirac{\lambda}{\rhorod\lambdaimits_{j=1}^\varepsilonll k_j} = \lambda_i \sum\lambdaimits_{i=1}^\varepsilonll \varphirac{1}{k_i} = \lambda_i \|F\|^2$. {\scriptsize\mathrm{t}extbullet}
\nablaoindent \mathrm{t}extbf{\mathrm{t}extsl{(VI)}:} Lets consider the trilinear application $\betaeta \colon T_xM\mathrm{t}imes T_xM \mathrm{t}imes T_xM \mathrm{t}o \mathbb{R}$ given by $\betaeta(X,Y,Z) = \interno{\al{X}{Y}}{\mathbf{S}_i Z}$.
Using equation \varepsilonqref{eq: derivada R}, we conclude that $A_{\mathbf{S}_i Y} X = - \mathbf{S}_i^\mathrm{t} \al{X}{Y}$. Thus,
\[\interno{\al{X}{Z}}{\mathbf{S}_i Y} = \interno{A_{\mathbf{S}_i Y} X}{Z} = - \interno{\mathbf{S}_i^\mathrm{t} \al{X}{Y}}{Z} = -\interno{\al{X}{Y}}{\mathbf{S}_i Z}.\]
Hence $\betaeta$ is symmetric in the first 2 variables and antisymmetric in the last two, thus $\betaeta = 0$ and $\al{X}{Y} \rhoerp \mathbf{S}_i(T_x M)$. {\scriptsize\mathrm{t}extbullet}
\nablaoindent \mathrm{t}extbf{\mathrm{t}extsl{(VII)}:} Lets suppose that $\mathcal{N}_1$ is parallel in the normal connexion. Hence, by equation \varepsilonqref{eq: derivada S},
\betaegin{align*}
& \interno{(\nabla_X \mathbf{S}_i)Y}{\al{W}{Z}} = \interno{\mathbf{T}_i \al{X}{Y} - \al{X}{\mathbf{R}_i Y}}{\al{W}{Z}} \ \mathbb{R}ightarrow \\
& \mathbb{R}ightarrow \ \interno{\nablaperp_X \mathbf{S}_i Y}{\al{W}{Z}} - \interno{\mathbf{S}_i \nabla_X Y}{\al{W}{Z}} = \interno{\mathbf{T}_i \al{X}{Y}}{\al{W}{Z}} - \\
& \quad - \interno{\lambda_i \al{X}{Y}}{\al{W}{Z}} \ \mathbb{R}ightarrow \\
& \stackrel{\mathrm{t}extsl{(VI)}}{\mathbb{R}ightarrow} \ \interno{\nablaperp_X \mathbf{S}_i Y}{\al{W}{Z}} = \interno{\mathbf{T}_i \al{X}{Y}}{\al{W}{Z}} - \\
& \quad \lambda_i \interno{\al{X}{Y}}{\al{W}{Z}} \ \mathbb{R}ightarrow \\
& \mathbb{R}ightarrow \ - \cancel{\interno{\mathbf{S}_i Y}{\nablaperp_X \al{W}{Z}}} = \interno{\mathbf{T}_i \al{X}{Y}}{\al{W}{Z}} - \\
& \quad - \lambda_i \interno{\al{X}{Y}}{\al{W}{Z}} \ \mathbb{R}ightarrow \\
& \mathbb{R}ightarrow \ \interno{\mathbf{T}_i \al{X}{Y}}{\al{W}{Z}} = \lambda_i \interno{\al{X}{Y}}{\al{W}{Z}} \ \mathbb{R}ightarrow \\
& \mathbb{R}ightarrow \ \interno{\rhoi_i\al{X}{Y}}{\rhoi_i\al{W}{Z}} = \lambda_i \interno{\al{X}{Y}}{\al{W}{Z}}.
\varepsilonnd{align*}
Therefore $\lambdaeft.\rhoi_1\right|_{\mathcal{N}_1}$ is a similarity with ratio $a_i = \hksqrt{\lambda_i}$.
\varepsilonnd{prova}
\betaegin{teo}\lambdaabel{teo: Phi=0}
Let $f \colon M^m \mathrm{t}o \hat{\mathbb{O}}$ be an isometric immersion with $k_1$, $\cdots$, $k_\varepsilonll \in \mathbb{R}^*$ and lets assume that $\mathcal{N}_1$ is a parallel vector sub-bundle of $T^\rhoerp M$ with dimension $\betaar n$. If $k_i \mathbf{R}_i = k_j \mathbf{R}_j$, for every $i, j \in \{1, \cdots, \varepsilonll\}$, then $k_ik_j > 0$ for all pairs $i, j$, $m+ \betaar n \lambdaeq \min\{n_1, \cdots, n_\varepsilonll\}$ and $f = (\jmath_1\mathrm{t}imes\cdots \mathrm{t}imes \jmath_\varepsilonll) \circ g \circ \betaar f$, where each $\jmath_i \colon \mathbb{O}_{k_i}^{m+\betaar n} \hookrightarrow \o{i}$ is a totally geodesic inclusion, $g \colon \mathbb{O}_k^{m+\betaar n} \mathrm{t}o \rhorod\lambdaimits_{i=1}^\varepsilonll \mathbb{O}_{k_1}^{m+\betaar n}$ is a totally geodesic immersion like the one given at the beginning of this section and $\betaar f \colon M^m \mathrm{t}o \mathbb{O}_k^{m+\betaar n}$ is a isometric immersion.
\varepsilonnd{teo}
\betaegin{proof}
With our assumptions, all conclusions of Proposition \ref{prop: Phi=0} are true, hence $k_ik_j > 0$, for every $i$ and $j$.
Let $\imath \colon \hat{\mathbb{O}} \hookrightarrow \mathbb{R}N$ be the canonical inclusion and $F = \imath \circ f$. Thus $V := F_*TM \obot \imath_* \mathcal{N}_1^f \obot \spa{F}$ is a vector bundle with dimension $m+\betaar n + 1$.
\betaegin{afi}{$V$ is a constant vector subspace of $\mathbb{R}N$.}
Let $X,Y,Z \in \Gamma(TM)$. Thus
\betaegin{align*}
& \nablatil_X F_* Y \stackrel{\mathrm{t}ext{Lemma \ref{lem: aF}}}{=} F_* \nabla_X Y + \imath_* \af{X}{Y} + \sum_{i=1}^\varepsilonll \interno{\mathbf{R}_i X}{Y} \nablau_i = \\
& = F_* \nabla_X Y + \imath_* \af{X}{Y} + \sum_{i=1}^\varepsilonll \lambda_i \interno{X}{Y} k_i (\rhoi_i \circ F) = \\
& = F_* \nabla_X Y + \imath_* \af{X}{Y} + \sum_{i=1}^\varepsilonll \varphirac{\rhorod\lambdaimits_{\substack{j=1\\ j\nablae i}}^\varepsilonll k_j}{\lambda} \interno{X}{Y} k_i (\rhoi_i \circ F) = \\
& = F_* \nabla_X Y + \imath_* \af{X}{Y} + \varphirac{\rhorod\lambdaimits_{j=1}^\varepsilonll k_j}{\lambda} \interno{X}{Y} F \in \Gamma(V); \\
& \nablatil_X \imath_* \af{Y}{Z} = \imath_* \nablabar_X \af{Y}{Z} + \ai{f_* X}{\af{Y}{Z}} = \\
&= -F_* A^f_{\af{Y}{Z}} X + \imath_* \nablaperp_X \af{Y}{Z} + \sum_{i=1}^\varepsilonll \interno{\rhoi_i f_*X}{\af{Y}{Z}}\nablau_i = \\
&= -F_* A^f_{\af{Y}{Z}} X + \imath_* \nablaperp_X \af{Y}{Z} + \sum_{i=1}^\varepsilonll \cancel{\interno{\mathbf{S}_i f_*X}{\af{Y}{Z}}}\nablau_i = \\
&= -F_* A^f_{\af{Y}{Z}} X + \imath_* \nablaperp_X \af{Y}{Z}.
\varepsilonnd{align*}
Since $\mathcal{N}_1$ is parallel in the normal connexion of $f$, $\nablatil_X \imath_* \af{Y}{Z} \in \Gammaamma(V)$.
Last, $\nablatil_X F = F_* X \in V$. Thus $V$ is a parallel sub-bundle of $\mathbb{R}N$, therefore $V$ is a constant subspace on $\mathbb{R}N$.
\varepsilonnd{afi}
Because $V$ is a constant subspace of $\mathbb{R}_y^N$ and $F_*(TM) \subset V$, then $F(M) \subset F(x_0) + V$, for any fixed $x_0 \in M$, because $X \interno{F - F(x_0)}{\mathrm{t}imesi} = \interno{F_* X}{\mathrm{t}imesi} = 0$, for all constant $\mathrm{t}imesi \in V^\rhoerp$. But $\spa\{F(x_0)\} \subset V$, hence $F(M) \subset V$.
Let $x_0 \in M$ be a fixed point, then $V = \spa\lambdaeft\{\varphirac{F(x_0)}{\|F(x_0)\|}\right\}\obot F_*T_{x_0}M \obot \imath_* \mathcal{N}_1(x_0)$.
Since $k_1, \cdots, k_\varepsilonll$ have the same sign, $\|f(x)\|^2 = \sum\lambdaimits_{i=1}^\varepsilonll \varphirac{1}{k_i} \nablae 0$, for all $x \in M$. Thus $V$ is isomorphic to $\mathbb{R}_{\mathrm{t}au(k_1)}^{m+\betaar n +1}$.
\betaegin{afi}{Each $\lambdaeft.\rhoi_i\right|_V$ is a similarity of ratio $a_i = \hksqrt{\lambda_i}$.}
Let $X,Y,Z \in T_{x_0}M$. Hence
\betaegin{align*}
& \interno{\rhoi_iF(x_0)}{\rhoi_iF_*X} = \interno{\nablau_i(x_0)}{F_*X} = 0; \\
& \interno{\rhoi_iF(x_0)}{\rhoi_i\imath_* \af{X}{Y}} = \interno{\nablau_i(x_0)}{\imath_* \af{X}{Y}} = 0; \\
& \interno{\rhoi_iF_*X}{\rhoi_i\imath_*\af{Y}{Z}} = \interno{-\mathbf{S}_i X}{\af{Y}{Z}} = 0.
\varepsilonnd{align*}
Thus $\lambdaeft.\rhoi_i\right|_V = \lambdaeft.\rhoi_i\right|_{\spa\{F(x_0)\}}\mathrm{t}imes\lambdaeft.\rhoi_i\right|_{F_*T_{x_0}M}\mathrm{t}imes\lambdaeft.\rhoi_i\right|_{\mathcal{N}_1^f(x_0)}$, and, by items \mathrm{t}extsl{(IV)}, \mathrm{t}extsl{(V)} and \mathrm{t}extsl{(VII)} of Proposition \ref{prop: Phi=0}, we know that $\lambdaeft.\rhoi_i\right|_{\spa\{F(x_0)\}}$, $\lambdaeft.\rhoi_i\right|_{F_* T_{x_0} M}$ and $\lambdaeft.\rhoi_i\right|_{\mathcal{N}_1^f(x_0)}$ are similarities of ratio $a_i$. Therefore $\lambdaeft.\rhoi_i\right|_V$ is a similarity of ratio $a_i$.
\varepsilonnd{afi}
Given that each $\lambdaeft.\rhoi_i\right|_V$ is a similarity of ratio $a_i$, then $\rhoi_i(V) = \{(0, \cdots, 0)\} \mathrm{t}imes \mathbb{R}_{\mathrm{t}au(k_1)}^{m+\betaar n+1}\mathrm{t}imes\{(0, \cdots, 0)\}$. Hence $V \subset \rhorod\lambdaimits_{i=1}^\varepsilonll\mathbb{R}_{\mathrm{t}au(k_1)}^{m+\betaar n+1}$. But $f(M) \subset V \cap \lambdaeft( \o{1} \x \cdots \x \o{\ell} \right)$, hence, $f(M) \subset \rhorod\lambdaimits_{i=1}^\varepsilonll \mathbb{O}_{k_i}^{m+\betaar n}$. Therefore, $f = (\jmath_1\mathrm{t}imes \cdots, \mathrm{t}imes \jmath_\varepsilonll) \circ \mathrm{t}ilde{f}$, where each $\jmath_i \colon \mathbb{O}_{k_1}^{m+\betaar n} \mathrm{t}o \o{i}$ is a inclusion and $\mathrm{t}ilde{f} \colon M \mathrm{t}o \mathbb{O}_{k_1}^{m+\betaar n}\mathrm{t}imes \cdots \mathrm{t}imes \mathbb{O}_{k_\varepsilonll}^{m+\betaar n}$ is a isometric immersion with $\mathrm{t}ilde{f}(M) \subset V$.
Since $f(M) \subset V \cap \rhorod\lambdaimits_{i=1}^\varepsilonll \mathbb{O}_{k_i}^{m+\betaar n}$, then there is a connected component $C$ of $V \cap \mathbb{S}_k^{(m+\betaar n)\varepsilonll + \varepsilonll -1}$ such that $\rhoi(C) \subset \mathbb{O}_{k_i}^{m+\betaar n}$. Thus by Lemma \ref{lem: V} (and Corollary \ref{cor: V},if $k < 0$) there are linear isometries $T_1, \cdots, T_\varepsilonll$ such that $T_i\o{i} = \o{i}$ and
\[V = \set{G(x)}{x \in \mathbb{R}_{\mathrm{t}au(k)}^{n+1}} = \set{\betaig(a_1 T_1(x), \cdots, a_\varepsilonll T_\varepsilonll(x) \betaig)}{x \in \mathbb{R}_{\mathrm{t}au(k)}^{n+1}}.\]
Let $g \colon \mathbb{O}_k^{m+\betaar n} \mathrm{t}o \mathbb{O}_{k_1}^{m+\betaar n}\mathrm{t}imes \cdots \mathrm{t}imes \mathbb{O}_{k_\varepsilonll}^{m+\betaar n}$ be the totally geodesic isometric immersion given at the beginning of this section.
\betaegin{afi}{$\mathrm{t}ilde f(M) \subset g\lambdaeft( \mathbb{O}_k^{m+\betaar n} \right)$.}
\betaegin{align*}
& y \in \mathrm{t}ilde f(M) \mathbb{R}ightarrow y \in V \cap \rhorod_{i=1}^\varepsilonll \mathbb{O}_{k_i}^{m+\betaar n} \mathbb{R}ightarrow \\
& \mathbb{R}ightarrow y = (a_1T_1(x), \cdots, a_\varepsilonll T_\varepsilonll(x)), \ \mathrm{t}ext{for some} \ x \in \mathbb{R}^{m+\betaar n+1}_{\mathrm{t}au(k)} \ \mathrm{t}ext{and} \ a_iT_i(x) \in \mathbb{O}_{k_i}^{m+\betaar n} \mathbb{R}ightarrow \\
& \mathbb{R}ightarrow \rhoi_i(y) = a_i T_i(x) \ \mathrm{t}ext{and} \ x \in \mathbb{O}_{\mathrm{t}ilde k_i}^{m + \betaar n}.
\varepsilonnd{align*}
Therefore $y \in g\lambdaeft( \mathbb{O}_k^{m+\betaar n} \right)$.
\varepsilonnd{afi}
Let $\betaar{f} := g^{-1} \circ \mathrm{t}ilde{f} \colon M \mathrm{t}o \mathbb{O}_k^{m+\betaar n}$. Thus $\betaar{f}$ is a isometric immersion and $f = (\jmath_1\mathrm{t}imes \cdots \mathrm{t}imes \jmath_\varepsilonll) \circ g \circ \betaar{f}$.
\varepsilonnd{proof}
\section{Reduction of codimension}
We say that the codimension of $f \colon M^m \mathrm{t}o \hat{\mathbb{O}}$ \mathrm{t}extbf{is reduced by $\betaar n$ at the $i$th coordinate}, if there is a totally geodesic submanifold $\mathbb{O}_{k_i}^{m_i} \subset \o{i}$ such that $n_i - m_i = \betaar n$ and $f(M) \subset \o{1}\mathrm{t}imes \cdots \mathrm{t}imes \mathbb{O}_{k_i}^{m_i} \mathrm{t}imes \cdots \mathrm{t}imes \o{\varepsilonll}$.
For each $i \in \{1, \cdots, \varepsilonll\}$, let $\mathbb{O}_{k_i}^{m_i} \subset \o{i}$ be a totally geodesic submanifold. Let also $\mathbb{R}_{\mathrm{t}au(k_i)}^{m_i+\upsilon(k_i)} \subset \mathbb{R}^{N_i}$ be such that $\mathbb{O}_{k_i}^{m_i} = \mathbb{R}_{\mathrm{t}au(k_i)}^{m_i+\upsilon(k_i)} \cap \o{i}$. Lets consider the orthogonal projections
\[\betaegin{matrix}
\Pi_i : & \mathbb{R}_{\mathrm{t}au(k_1)}^{m_1+\upsilon(k_1)}\mathrm{t}imes \cdots \mathrm{t}imes \mathbb{R}_{\mathrm{t}au(k_\varepsilonll)}^{m_\varepsilonll+\upsilon(k_\varepsilonll)} & \lambdaongrightarrow & \mathbb{R}_{\mathrm{t}au(k_1)}^{m_1+\upsilon(k_1)}\mathrm{t}imes \cdots \mathrm{t}imes \mathbb{R}_{\mathrm{t}au(k_\varepsilonll)}^{m_\varepsilonll+\upsilon(k_\varepsilonll)} \\
& (x_1, \cdots, x_n) & \lambdaongmapsto & (0, \cdots, x_i, \cdots, 0).
\varepsilonnd{matrix}\]
Thus each $\Pi_i$ is the restriction of $\rhoi_i$ to $\mathbb{R}_{\mathrm{t}au(k_1)}^{m_1+\upsilon(k_1)}\mathrm{t}imes \cdots \mathrm{t}imes \mathbb{R}_{\mathrm{t}au(k_\varepsilonll)}^{m_\varepsilonll+\upsilon(k_\varepsilonll)}$.
Now, let $\mathrm{t}ilde{f} \colon M^m \mathrm{t}o \mathbb{O}_{k_1}^{m_1}\mathrm{t}imes\cdots\mathrm{t}imes\mathbb{O}_{k_\varepsilonll}^{m_\varepsilonll}$ be a isometric immersion and $f \colon M^m \mathrm{t}o \hat{\mathbb{O}}$ be given by $f := (\jmath_1 \mathrm{t}imes \cdots \mathrm{t}imes \jmath_\varepsilonll) \circ \mathrm{t}ilde{f}$, where each $\jmath_i \colon \mathbb{O}_{k_i}^{m_i} \hookrightarrow \o{i}$ is a totally geodesic inclusion. We will also use $\jmath_i$ denote the inclusion $\jmath_i \colon \mathbb{R}_{\mathrm{t}au(k_i)}^{m_i} \mathrm{t}o \mathbb{R}_{\mathrm{t}au(k_i)}^{n_i}$, hence, $(\jmath_1 \mathrm{t}imes \cdots \mathrm{t}imes \jmath_\varepsilonll) \circ \Pi_i = \rhoi_i \circ (\jmath_1 \mathrm{t}imes \cdots \mathrm{t}imes \jmath_\varepsilonll)$ and
\betaegin{align*}
& \rhoi_i f_* X = \lambdaeft(\jmath_1\mathrm{t}imes \cdots \mathrm{t}imes\jmath_\varepsilonll\right)_*\Pi_i \mathrm{t}ilde{f}_* X \mathbb{R}ightarrow \\
& \quad \mathbb{R}ightarrow f_* \mathbf{R}_i^f X + \mathbf{S}_i^f X = f_* \mathbf{R}_i^{\mathrm{t}ilde f} X + (\jmath_1 \mathrm{t}imes \cdots \mathrm{t}imes \jmath_\varepsilonll)_* \mathbf{S}_i^{\mathrm{t}ilde f} X. \\
& \rhoi_i (\jmath_1 \mathrm{t}imes \cdots \mathrm{t}imes \jmath_\varepsilonll)_* \mathrm{t}imesi = (\jmath_1 \mathrm{t}imes \cdots \mathrm{t}imes \jmath_\varepsilonll)_*\Pi_i \mathrm{t}imesi \ \mathbb{R}ightarrow \\
& \mathbb{R}ightarrow \ f_*\lambdaeft(\mathbf{S}_i^f\right)^\mathrm{t} (\jmath_1 \mathrm{t}imes \cdots \mathrm{t}imes \jmath_\varepsilonll)_* \mathrm{t}imesi + \mathbf{T}_i^f (\jmath_1 \mathrm{t}imes \cdots \mathrm{t}imes \jmath_\varepsilonll)_* \mathrm{t}imesi = \\
& \quad = f_* \betaig(\mathbf{S}_i^{\mathrm{t}ilde f}\betaig)^\mathrm{t}\mathrm{t}imesi + (\jmath_1 \mathrm{t}imes \cdots \mathrm{t}imes \jmath_\varepsilonll)_* \mathbf{T}_i^{\mathrm{t}ilde f} \mathrm{t}imesi.
\varepsilonnd{align*}
Therefore
\betaegin{equation}\lambdaabel{eq: tensores composta}
\betaegin{aligned}
& \mathbf{R}_i^f = \mathbf{R}_i^{\mathrm{t}ilde f}, \quad \mathbf{S}_i^f = (\jmath_1 \mathrm{t}imes \cdots \mathrm{t}imes \jmath_\varepsilonll)_*\mathbf{S}_i^{\mathrm{t}ilde f}, \\
& \lambdaeft(\mathbf{S}_i^f\right)^\mathrm{t} \lambdaeft.(\jmath_1 \mathrm{t}imes \cdots \mathrm{t}imes \jmath_\varepsilonll)_*\right|_{T_{\mathrm{t}ilde f}^\rhoerp M} = \betaig(\mathbf{S}_i^{\mathrm{t}ilde f}\betaig)^\mathrm{t},\\
& \mathbf{T}_i^f \lambdaeft.(\jmath_1 \mathrm{t}imes \cdots \mathrm{t}imes \jmath_\varepsilonll)_*\right|_{T_{\mathrm{t}ilde f}^\rhoerp M} = (\jmath_1 \mathrm{t}imes \cdots \mathrm{t}imes \jmath_\varepsilonll)_* \mathbf{T}_i^{\mathrm{t}ilde f}.
\varepsilonnd{aligned}
\varepsilonnd{equation}
In the rest of this section we are going to prove some results about reduction of codimension of $f$ at the $i$th coordinate.
\betaegin{lem}\lambdaabel{lem: reducao}
If $f\colon M^m \mathrm{t}o \hat{\mathbb{O}}$ is a isometric immersion, then:
\betaegin{enum}
\item $\mathbf{S}_i(TM)^\rhoerp$ is invariant under $\mathbf{T}_i$ and $\mathbf{S}_i(TM)^\rhoerp = \mathcal{U}_i \obot \mathcal{V}_i$, where $\mathcal{U}_i := \ker \mathbf{T}_i$ and $\mathcal{V}_i := \ker(\id - \mathbf{T}_i)$.
\item $\nablaperp \lambdaeft( \mathcal{U}_i \cap \mathcal{N}_1^\rhoerp \right)\subset \mathcal{U}_i$ and $\nablaperp \lambdaeft(\mathcal{V}_i \cap \mathcal{N}_1^\rhoerp \right)\subset \mathcal{V}_i$.
\item $\rhoi_i$ fixes the points of $\mathcal{V}_i$ and $\id - \rhoi_i$ fixes the points of $\mathcal{U}_i$.
\varepsilonnd{enum}
\varepsilonnd{lem}
\betaegin{prova}
\rhoar \nablaoindent \mathrm{t}extbf{\mathrm{t}extsl{(I)}:} It follows from the second equation in \varepsilonqref{eq: RST} that $\mathbf{T}_i \lambdaeft[ \mathbf{S}_i(TM) \right] \subset \mathbf{S}_i(TM)$. Hence, if $\mathrm{t}imesi \in \mathbf{S}_i(TM)$ and $\zeta \in \mathbf{S}_i(TM)^\rhoerp$, then $\interno{\mathrm{t}imesi}{\mathbf{T}_i \zeta} = \interno{\mathbf{T}_i \mathrm{t}imesi}{\zeta} = 0$. Therefore $\mathbf{T}_i$ leaves $\mathbf{S}_i(TM)^\rhoerp$ invariant.
From the third equation in \varepsilonqref{eq: RST}, it follows that $\lambdaeft.\mathbf{T}_i\right|_{\mathbf{S}_i(TM)^\rhoerp}^2 = \lambdaeft.\mathbf{T}_i\right|_{\mathbf{S}_i(TM)^\rhoerp}$, thus $\mathbf{T}_i|_{\mathbf{S}_i(TM)^\rhoerp}$ is a orthogonal projection and $\mathbf{S}_i(TM)^\rhoerp = {\mathcal{U}_i \obot \mathcal{V}_i}$, where $\mathcal{U}_i := \ker \mathbf{T}_i|_{\mathbf{S}_i(TM)^\rhoerp}$ and $\mathcal{V}_i := \lambdaeft.\ker(\id -\mathbf{T}_i)\right|_{\mathbf{S}_i(TM)^\rhoerp}$.
By another side, using the third equation \varepsilonqref{eq: RST},
\[\ker[\mathbf{T}_i(\id-\mathbf{T}_i)] = \ker \mathbf{S}_i\mathbf{S}_i^\mathrm{t} = \ker\mathbf{S}_i^\mathrm{t} = [\mathbf{S}_i(TM)]^\rhoerp.\]
Besides, $\ker \mathbf{T}_i$ and $\ker(\id-\mathbf{T}_i)$ are subsets of $\ker[\mathbf{T}_i(\id-\mathbf{T}_i)] = \mathbf{S}_i(TM)^\rhoerp$, therefore $\ker \mathbf{T}_i = \ker\mathbf{T}_i|_{\mathbf{S}_i(TM)^\rhoerp}$ and $\ker(\id-\mathbf{T}_i) = \ker(\id-\mathbf{T}_i)|_{\mathbf{S}_i(TM)^\rhoerp}$. {\scriptsize\mathrm{t}extbullet}
\nablaoindent \mathrm{t}extbf{\mathrm{t}extsl{(II)}:} If $\mathrm{t}imesi \in \Gamma \lambdaeft(\mathcal{V}_i \cap \mathcal{N}_1^\rhoerp \right)$, then
\[\nablaperp_X \mathrm{t}imesi - \mathbf{T}_i \nablaperp_X \mathrm{t}imesi = \nablaperp_X \mathbf{T}_i \mathrm{t}imesi - \mathbf{T}_i \nablaperp_X \mathrm{t}imesi \stackrel{\varepsilonqref{eq: derivada T}}{=} 0, \ \varphiorall X \in \Gamma(TM).\]
Hence $\nablaperp_X \mathrm{t}imesi \in \ker(\id - \mathbf{T}_i) = \mathcal{V}_i$.
If $\mathrm{t}imesi \in \mathcal{U}_i \cap \mathcal{N}_1^\rhoerp$, then
\[-\mathbf{T}_i \nablaperp_X \mathrm{t}imesi = \nablaperp_X \mathbf{T}_i \mathrm{t}imesi - \mathbf{T}_i \nablaperp_X \mathrm{t}imesi \stackrel{\varepsilonqref{eq: derivada T}}{=} 0, \ \varphiorall X \in \Gamma(TM).\]
Therefore $\nablaperp_X \mathrm{t}imesi \in \ker \mathbf{T}_i = \mathcal{U}_i$. {\scriptsize\mathrm{t}extbullet}
\nablaoindent \mathrm{t}extbf{\mathrm{t}extsl{(III)}:} Let $\mathrm{t}imesi \in \mathcal{U}_i = \ker \mathbf{T}_i$ and $\zeta \in \mathcal{V}_i = \ker(\id - \mathbf{T}_i)$ . From $\mathbf{S}_i(TM)^\rhoerp = \mathcal{U}_i \obot \mathcal{V}_i$, it follows that
\betaegin{align*}
& (\id - \rhoi_i) \mathrm{t}imesi = \mathrm{t}imesi - \mathbf{S}_i^\mathrm{t} \mathrm{t}imesi - \mathbf{T}_i \mathrm{t}imesi = \mathrm{t}imesi; \\
& \rhoi_i \zeta = \mathbf{S}_i^\mathrm{t} \zeta + \mathbf{T}_i \zeta = \zeta.
\varepsilonnd{align*}
Therefore $\rhoi_i$ fixes the points of $\mathcal{V}_i$, and $\id-\rhoi_i$ fixes the points of $\mathcal{U}_i$.
\varepsilonnd{prova}
\betaegin{teo}\lambdaabel{teo: reducao}
If $f\colon M^m\mathrm{t}o \hat{\mathbb{O}}$ is a isometric immersion, then the following claims are equivalent:
\betaegin{enum}
\item The codimension of $f$ is reduced by $\betaar n$ at the $i$th coordinate.
\item There is a vector sub-bundle $L^{\betaar n} \subset T^\rhoerp M$ such that $L^{\betaar n} \subset \mathcal{V}_i \cap \mathcal{N}_1^\rhoerp$ and $L^{\betaar n}$ is parallel in the normal connection of $f$.
\varepsilonnd{enum}
\varepsilonnd{teo}
\betaegin{prova}
\rhoar \nablaoindent \mathrm{t}extbf{\mathrm{t}extsl{(I)} $\mathbb{R}ightarrow$ \mathrm{t}extsl{(II)}:} Because the codimension of $f$ is reduced by $\betaar n$ at the $i$th coordinate, there is a totally geodesic submanifold $\mathbb{O}_{k_i}^{m_i} \subset \o{i}$, with $n_i - m_i = \betaar n$, and such that $f(M) \subset \o{1}\mathrm{t}imes\cdots\mathrm{t}imes\mathbb{O}_{k_i}^{m_i}\mathrm{t}imes\cdots \mathrm{t}imes\o{\varepsilonll}$.
Hence, there is a isometric immersion $\mathrm{t}ilde{f} \colon M^m \mathrm{t}o \o{1}\mathrm{t}imes\cdots\mathrm{t}imes\mathbb{O}_{k_i}^{m_i}\mathrm{t}imes\cdots \mathrm{t}imes\o{\varepsilonll}$ such that $f = (\id \mathrm{t}imes \cdots \mathrm{t}imes\jmath_i\mathrm{t}imes \cdots \mathrm{t}imes \id) \circ \mathrm{t}ilde{f}$, where $\jmath_i \colon \mathbb{O}_{k_i}^{m_i} \hookrightarrow \o{i}$ is the totally geodesic inclusion and each $\id \colon \o{j} \mathrm{t}o \o{j}$ is the identity map.
Let $L \subset T_f^\rhoerp M$ be the vector sub-bundle whose fiber in $x$ is given by
\betaegin{align*}
& L(x) := \lambdaeft(( \id \mathrm{t}imes \cdots \mathrm{t}imes\jmath_i \mathrm{t}imes \cdots \mathrm{t}imes\id )_* T_{\mathrm{t}ilde{f}(x)}\o{1} \mathrm{t}imes\cdots\mathrm{t}imes \mathbb{O}_{k_i}^{m_i}\mathrm{t}imes \cdots\mathrm{t}imes\o{\varepsilonll}\right)^\rhoerp = \\
&= \{(0, \cdots, 0)\} \mathrm{t}imes \lambdaeft( T_{\mathrm{t}ilde{f}_i(x)}\mathbb{O}_{k_i}^{m_i} \right)^\rhoerp \mathrm{t}imes \{(0,\cdots, 0)\}.
\varepsilonnd{align*}
Obviously, $\mathrm{d}im L = n_i - m_i = \betaar n$ and $\rhoi_i(L) = L$.
From \varepsilonqref{eq: tensores composta}, we know that $\mathbf{S}_i^f = (\id \mathrm{t}imes \cdots \mathrm{t}imes \jmath_i \mathrm{t}imes \cdots \mathrm{t}imes \id)_*\mathbf{S}_i^{\mathrm{t}ilde f}$, thus
\betaegin{align*}
& \mathbf{S}_i(TM) \subset (\id\mathrm{t}imes\cdots\mathrm{t}imes \jmath_i \mathrm{t}imes\cdots\mathrm{t}imes\id)_*T_{\mathrm{t}ilde{f}} \o{1}\mathrm{t}imes\cdots\mathrm{t}imes \mathbb{O}_{k_i}^{m_i}\mathrm{t}imes\cdots\mathrm{t}imes\o{\varepsilonll} \mathbb{R}ightarrow \\
& \mathbb{R}ightarrow L \subset \mathbf{S}_i(TM)^\rhoerp \ \mathbb{R}ightarrow L \subset \rhoi_i \lambdaeft[ \mathbf{S}_i(TM)^\rhoerp \right].
\varepsilonnd{align*}
On the oder side, by Lemma \ref{lem: reducao}, $\mathbf{S}_i(TM)^\rhoerp = \mathcal{U}_i\obot\mathcal{V}_i$ and $\mathcal{V}_i = \rhoi_i(\mathcal{V}_i) = \rhoi_i\lambdaeft[ \mathbf{S}_i(TM)^\rhoerp \right]$. Hence $L \subset \mathcal{V}_i$.
Because $(\id\mathrm{t}imes\cdots\mathrm{t}imes\jmath_i\mathrm{t}imes\cdots\mathrm{t}imes\id) \colon \o{1}\mathrm{t}imes\cdots\mathrm{t}imes\mathbb{O}_{k_i}^{m_i}\mathrm{t}imes\cdots\mathrm{t}imes\o{\varepsilonll} \mathrm{t}o \hat{\mathbb{O}}$ is totally geodesic, $\mathcal{N}_1^f = (\id\mathrm{t}imes\cdots\mathrm{t}imes\jmath_i\mathrm{t}imes\cdots\mathrm{t}imes \id)_* \mathcal{N}_1^{\mathrm{t}ilde{f}}$. Hence
\betaegin{multline*}
L(x) = \lambdaeft[ (\id\mathrm{t}imes\cdots\mathrm{t}imes\jmath_i\mathrm{t}imes\cdots\mathrm{t}imes\id)_* T_{\mathrm{t}ilde f(x)}\o{1}\mathrm{t}imes\cdots\mathrm{t}imes\mathbb{O}_{k_i}^{m_i}\mathrm{t}imes\cdots\mathrm{t}imes\o{\varepsilonll} \right]^\rhoerp \subset \\
\subset \lambdaeft[\lambdaeft(\id \mathrm{t}imes \cdots \mathrm{t}imes \jmath_i \mathrm{t}imes \cdots\mathrm{t}imes \id\right)_* \mathcal{N}_1^{\mathrm{t}ilde f}(x) \right]^\rhoerp = {\mathcal{N}_1^f(x)}^\rhoerp.
\varepsilonnd{multline*}
Therefore $L \subset \mathcal{V}_i \cap {\mathcal{N}^f_1}^\rhoerp$.
Now, let $\mathrm{t}imesi \in \Gamma(L)$ and
\betaegin{multline*}
\zeta = (\id \mathrm{t}imes\cdots\mathrm{t}imes\jmath_i\mathrm{t}imes\cdots\mathrm{t}imes\id)_* \mathrm{t}ilde \zeta \in \\
\in \Gamma \lambdaeft( (\id\mathrm{t}imes\cdots\mathrm{t}imes\jmath_i \mathrm{t}imes\cdots\mathrm{t}imes \id)_*T_{\mathrm{t}ilde{f}} \o{1}\mathrm{t}imes\cdots\mathrm{t}imes\mathbb{O}_{k_i}^{m_i}\mathrm{t}imes\cdots\mathrm{t}imes\o{\varepsilonll} \right).
\varepsilonnd{multline*}
Thus
\betaegin{multline*}
\interno{\nablaperp_X \mathrm{t}imesi}{\zeta} = - \interno{\mathrm{t}imesi}{\nablabar_X (\id \mathrm{t}imes \cdots \mathrm{t}imes\jmath_i\mathrm{t}imes\cdots\mathrm{t}imes \id)_* \mathrm{t}ilde{\zeta}} = \\
= -\interno{\mathrm{t}imesi}{(\id\mathrm{t}imes\cdots\mathrm{t}imes\jmath_i\mathrm{t}imes\cdots\mathrm{t}imes \id)_*\nablabar_X \mathrm{t}ilde{\zeta}} = 0.
\varepsilonnd{multline*}
Therefore $\nablaperp L \subset L$. {\scriptsize\mathrm{t}extbullet}
\nablaoindent\mathrm{t}extbf{\mathrm{t}extsl{(II)} $\mathbb{R}ightarrow$ \mathrm{t}extsl{(I)}:} Let $\imath \colon \hat{\mathbb{O}} \hookrightarrow \mathbb{R}N$ be the canonical inclusion, $F := \imath \circ f$ and $\mathrm{t}ilde{L} := \imath_*L$.
\betaegin{afi}{$\mathrm{t}ilde{L}$ is a constant subspace of $\mathbb{R}N$.}
Let $\mathrm{t}imesi \in\Gamma \lambdaeft( L \right)$. Because $L \subset \mathcal{V}_i \cap \mathcal{N}_1^\rhoerp$, then $A_\mathrm{t}imesi = 0$ and $\rhoi_i (\mathrm{t}imesi) = \mathrm{t}imesi$. Thus
\betaegin{align*}
& \nablatil_X \imath_* \mathrm{t}imesi = \imath_* \nablabar_X \mathrm{t}imesi + \ai{f_*X}{\mathrm{t}imesi} \stackrel{\mathrm{t}ext{Lemma \ref{lem: ai}}}{=} \\
& = - F_* \cancel{A_\mathrm{t}imesi X} + \imath_* \nablaperp_X \mathrm{t}imesi + \sum_{j=1}^\varepsilonll\cancel{\interno{f_*X}{\rhoi_j \mathrm{t}imesi}}\nablau_j = \imath_* \nablaperp_X \mathrm{t}imesi.
\varepsilonnd{align*}
But $L$ is parallel in the normal connexion of $f$, so $\nablatil_X \imath_* \mathrm{t}imesi \in \mathrm{t}ilde{L}$.
\varepsilonnd{afi}
\betaegin{afi}{$\rhoi_i \mathrm{t}ilde{L} = \mathrm{t}ilde{L}$.}
Because $\rhoi_i(L) = L$ and $\rhoi_i \imath_* = \imath_* \rhoi_i$, the claim is true.
\varepsilonnd{afi}
Lets consider the projection $P_i \colon \mathbb{R}^{N_1}\mathrm{t}imes\cdots\mathrm{t}imes\mathbb{R}^{N_\varepsilonll} \mathrm{t}o \mathbb{R}^{N_i}$, hence $\rhoi_i(x) = (0 \cdots, P_i(x), \cdots, 0)$. Lets also define $\betaar L_i := P_i\lambdaeft(\mathrm{t}ilde L^\rhoerp\right) = \lambdaeft[P_i\lambdaeft(\mathrm{t}ilde L \right)\right]^\rhoerp$ and $F_i := P_i \circ f$.
\betaegin{afi}{$f(M) \subset \mathbb{R}^{N_1} \mathrm{t}imes \cdots \mathrm{t}imes \lambdaeft( \betaar L_i + F_i(x_0)\right) \mathrm{t}imes \cdots \mathrm{t}imes \mathbb{R}^{N_\varepsilonll}$, where $x_0 \in M$ is any fixed point.}
It is enough to prove that $F_i(M) \subset \betaar L_i + F_i(x_0)$. But $f_* X \rhoerp \mathrm{t}ilde L$, then, if $\mathrm{t}imesi \in \mathrm{t}ilde L$ is a fixed vector, we have that
\[X\interno{F_i}{P_i \mathrm{t}imesi} = \interno{P_i f_*X}{P_i \mathrm{t}imesi} = \interno{\rhoi_if_* X}{\rhoi_i \mathrm{t}imesi} = \interno{f_* X}{\rhoi_i \mathrm{t}imesi} = \interno{f_*X}{\mathrm{t}imesi} = 0.\]
Thus $\interno{F_i(x)}{P_i \mathrm{t}imesi}$ is constant in $M$.
Now, if $\{\mathrm{t}imesi_1, \cdots, \mathrm{t}imesi_{\betaar n}\}$ is an orthogonal base of $\mathrm{t}ilde L$, then $\{P_i\mathrm{t}imesi_1, \cdots, P_i\mathrm{t}imesi_{\betaar n}\}$ is an orthogonal base of $P_i \lambdaeft(\mathrm{t}ilde L\right)$. Thus, $\interno{F_i(x)}{P_i \mathrm{t}imesi_j} = \interno{F_i(x_0)}{P_i \mathrm{t}imesi_j}$, where $x_0$ is a fixed point. Hence $F_i(x) - F_i(x_0) \rhoerp P_i\lambdaeft(\mathrm{t}ilde L\right)$. Therefore $F_i(x) - F_i(x_0) \in \lambdaeft[P_i\lambdaeft(\mathrm{t}ilde L\right)\right]^\rhoerp = \betaar L_i$.
\varepsilonnd{afi}
Now we have two cases to consider: the case $k_i \nablae 0$ and the case $k_i =0$. If $k_i = 0$, then $\mathbb{R}^{N_i} = \mathbb{E}^{n_i}$ and $F_i(M) \subset \betaar{L}_i + F_i(x_0)$. Hence, if we identify $\betaar{L}_i + F_i(x_0)= \mathbb{E}^{m_i}$, then $F_i(M) \subset \mathbb{E}^{m_i}$, with $m_i = n_i - \betaar n$.
If $k_i\nablae 0$, since $\nablau_i = k_i (\rhoi_i \circ F) \rhoerp \mathrm{t}ilde L$, then $F_i(x_0) \rhoerp P_i\lambdaeft( \mathrm{t}ilde L(x_0) \right) = P_i\lambdaeft( \mathrm{t}ilde L\right)$, hence $F_i(x_0) \in \betaar{L}_i$. Thus $F_i(M) \subset \betaar{L}_i \cap \o{i}$.
If $k_i > 0$, then $\o{i} = \ss{i}$ and $\betaar{L}_i \cap \o{i} = \mathbb{S}_{k_i}^{m_i} = \mathbb{O}_{k_i}^{m_i}$, where $m_i = n_i - \betaar n$. If $k_i < 0$, then $F_i(x_0)$ is a timelike and $\betaar L_i$ is also timelike, cause $F_i(x_0) \in \betaar L_i$. Thus $\betaar L_i \cap \o{i} = \betaar L_i \cap \h{i} = \mathbb{H}_{k_i}^{m_i} = \mathbb{O}_{k_i}^{m_i}$, where $m_i = n_i - \betaar n$.
Our conclusion is: $f(M) \subset \o{1} \mathrm{t}imes \cdots \mathrm{t}imes \mathbb{O}_{k_i}^{m_i}\mathrm{t}imes \cdots \mathrm{t}imes \o{n} \subset \hat{\mathbb{O}}$.
\varepsilonnd{prova}
Here we want to remark that, if the codimension at the $i$th coordinate of $f \colon M \mathrm{t}o \hat{\mathbb{O}}$ can be reduced by $\betaar n_i$ and the codimension at the $j$th coordinate can be reduced by $\betaar n_j$, then
\[f(M) \subset \o{1} \mathrm{t}imes \cdots \mathrm{t}imes \mathbb{O}_{k_i}^{n_i-\betaar n_i}\mathrm{t}imes \cdots \mathrm{t}imes \mathbb{O}_{k_j}^{n_j-\betaar n_j} \mathrm{t}imes \cdots \mathrm{t}imes \o{n},\]
for some totally geodesic submanifolds $\mathbb{O}_{k_i}^{n_i-\betaar n_i} \subset \o{i}$ and $\mathbb{O}_{k_j}^{n_j-\betaar n_j} \subset \o{j}$.
\betaegin{cor}\lambdaabel{cor: reducao}
Let $f\colon M^m\mathrm{t}o \hat{\mathbb{O}}$ be an isometric immersion and lets suppose that $\mathcal{V}_i \cap \mathcal{N}_1^\rhoerp$ be a vector sub-bundle of $T^\rhoerp M$ with dimension $\betaar n$. If $\nablaperp \lambdaeft(\mathcal{V}_i \cap \mathcal{N}_1^\rhoerp \right) \subset \mathcal{N}_1^\rhoerp$, then the codimension of $f$ is reduced by $\betaar n$ at the $i$th coordinate.
\varepsilonnd{cor}
\betaegin{proof}
By Theorem \ref{teo: reducao}, it is enough to show that $L := \mathcal{V}_i \cap \mathcal{N}_1^\rhoerp$ is parallel in the normal connexion of $f$. On the other side, we know by \mathrm{t}extsl{(II)} of Lemma \ref{lem: reducao} that $\nablaperp L \subset \mathcal{V}_i$. But $\nablaperp L \subset \mathcal{N}_1^\rhoerp$, therefore $L$ is parallel.
\varepsilonnd{proof}
\betaegin{teo} \lambdaabel{teorema: reducao de codimension 2}
Let $f\colon M^m \mathrm{t}o \hat{\mathbb{O}}$ be an isometric immersion and lets suppose that $\mathcal{V}_i \cap \mathcal{N}_1^\rhoerp$ be a vector sub-bundle of $T_f^\rhoerp M$. Thus $\nablaperp \lambdaeft(\mathcal{V}_i\cap \mathcal{N}_1^\rhoerp \right) \subset \mathcal{N}_1^\rhoerp$ if, and only if,
\betaegin{enum}
\item $\lambdaeft(\nablaperp_Z \mathcal{R}^\rhoerp \right)(X,Y,\mathrm{t}imesi) = 0$, for all $\mathrm{t}imesi \in\mathcal{V}_i \cap \mathcal{N}_1^\rhoerp$, and
\item $\nablaperp \lambdaeft(\mathcal{V}_i \cap \mathcal{N}_1^\rhoerp\right) \subset \{\varepsilonta\}^\rhoerp$, where $\varepsilonta$ is the mean curvature vector of $f$.
\varepsilonnd{enum}
\varepsilonnd{teo}
\betaegin{prova}
\rhoar \nablaoindent ($\mathbb{R}ightarrow$): Since $\mathcal{N}_1^\rhoerp \subset \{\varepsilonta\}^\rhoerp$ and $\nablaperp \lambdaeft(\mathcal{V}_i \cap\mathcal{N}_1^\rhoerp \right) \subset \mathcal{N}_1^\rhoerp$, then $\nablaperp \lambdaeft(\mathcal{V}_i \cap \mathcal{N}_1^\rhoerp\right) \subset \{\varepsilonta\}^\rhoerp$. We still have to show item \mathrm{t}extsl{(I)}.
By item \mathrm{t}extsl{(I)} of Lemma \ref{lem: reducao}, $\mathbf{S}_i(TM)^\rhoerp = \mathcal{U}_i \obot \mathcal{V}_i$. So, if $\mathrm{t}imesi \in \mathcal{V}_i \cap \mathcal{N}_1^\rhoerp$ and $X, Y \in TM$, then, by Ricci equation \varepsilonqref{eq: Ricci},
\betaegin{equation}\lambdaabel{eq: R perp}
\mathcal{R}^\rhoerp(X,Y)\mathrm{t}imesi = \al{X}{A_\mathrm{t}imesi Y} - \al{A_\mathrm{t}imesi X}{Y} + \sum_{i=1}^\varepsilonll k_i \lambdaeft( \mathbf{S}_i X \wedge \mathbf{S}_i Y \right) \mathrm{t}imesi = 0.
\varepsilonnd{equation}
Now, given $\mathrm{t}imesi \in \Gamma\lambdaeft( \mathcal{V}_i \cap \mathcal{N}_1^\rhoerp \right)$, and because $\nablaperp_Z \mathrm{t}imesi \in \mathcal{V}_i \cap \mathcal{N}_1^\rhoerp$, it holds that:
\betaegin{multline*}
\lambdaeft(\nablaperp_Z \mathcal{R}^\rhoerp \right)(X,Y,\mathrm{t}imesi) = \nablaperp_Z \mathcal{R}^\rhoerp(X,Y) \mathrm{t}imesi - \mathcal{R}^\rhoerp(\nabla_Z X, Y) \mathrm{t}imesi - \\
- \mathcal{R}^\rhoerp(X, \nabla_Z Y) \mathrm{t}imesi - \mathcal{R}^\rhoerp(X, Y) \nablaperp_Z \mathrm{t}imesi = 0. \ \mathrm{t}ext{{\scriptsize\mathrm{t}extbullet}
}
\varepsilonnd{multline*}
\nablaoindent ($\mathbb{L}eftarrow$): Lets suppose now that we have items \mathrm{t}extsl{(I)} and \mathrm{t}extsl{(II)}. We have to show that $\nablaperp \lambdaeft( \mathcal{V}_i \cap \mathcal{N}_1^\rhoerp \right) \subset \mathcal{N}_1^\rhoerp$.
By the same arguments used above, equation \varepsilonqref{eq: R perp} holds. So, let $\mathrm{t}imesi \in \Gamma \lambdaeft(\mathcal{V}_i \cap \mathcal{N}_1^\rhoerp\right) \subset \Gamma \lambdaeft(\{\varepsilonta\}^\rhoerp\right)$, then
\betaegin{align*}
& 0 = \lambdaeft( \nablaperp_Z \mathcal{R}^\rhoerp \right)(X,Y,\mathrm{t}imesi) = \cancel{\nablaperp_Z \mathcal{R}^\rhoerp (X,Y)\mathrm{t}imesi} - \cancel{\mathcal{R}^\rhoerp\lambdaeft(\nabla_Z X,Y\right) \mathrm{t}imesi} - \\
& \quad - \cancel{\mathcal{R}^\rhoerp\lambdaeft(X, \nabla_Z Y\right)\mathrm{t}imesi} - \mathcal{R}^\rhoerp(X,Y)\nablaperp_Z\mathrm{t}imesi = - \mathcal{R}^\rhoerp(X,Y)\nablaperp_Z\mathrm{t}imesi.
\varepsilonnd{align*}
On the other side, as a result of Lemma \ref{lem: reducao}, $\nablaperp_X \mathrm{t}imesi \in \mathcal{V}_i \subset \mathbf{S}_i(TM)^\rhoerp$. As a consequence, and by Ricci's equation \varepsilonqref{eq: Ricci},
\betaegin{multline*}
0 = \mathcal{R}^\rhoerp(X,Y)\nablaperp_Z\mathrm{t}imesi = \al{X}{A_{\nablaperp_Z \mathrm{t}imesi} Y} - \al{A_{\nablaperp_Z \mathrm{t}imesi}X}{Y} + \\
+ \sum_{i=1}^\varepsilonll k_i \cancel{\lambdaeft( \mathbf{S}_i X \wedge \mathbf{S}_i Y \right) \nablaperp_Z} \mathrm{t}imesi.
\varepsilonnd{multline*}
Hence, if $\zeta \in T^\rhoerp M$, then
\[0 = \interno{\al{X}{A_{\nablaperp_Z \mathrm{t}imesi} Y}}{\zeta} - \interno{\al{A_{\nablaperp_Z \mathrm{t}imesi}X}{Y}}{\zeta} = \interno{\lambdaeft[A_{\nablaperp_Z \mathrm{t}imesi}, A_\zeta \right]X}{Y}.\]
Therefore $\lambdaeft[A_{\nablaperp_Z \mathrm{t}imesi}, A_{\nablaperp_W \mathrm{t}imesi} \right] = 0$, for all $W, Z \in \Gamma(TM)$. Hence for each $x \in M$ there is an orthonormal base $\lambdaeft\{E_1(x), \cdots, E_m(x)\right\}$ of $T_x M$ that diagonalizes all elements of the set $\set{A_{\nablaperp_X \mathrm{t}imesi}}{X \in T_x M}$.
Since $\lambdaeft[A_{\nablaperp_Z \mathrm{t}imesi}, A_{\nablaperp_W \mathrm{t}imesi} \right] = 0$, for all $W, Z \in \Gamma(TM)$, then, for each $x \in M$ there is an orthonormal base $\lambdaeft\{E_1(x), \cdots, E_m(x)\right\}$ of $T_x M$ that diagonalizes all elements of the set $\set{A_{\nablaperp_X \mathrm{t}imesi}}{X \in T_x M}$.
Lets show that $\nablaperp_X \mathrm{t}imesi \in \mathcal{V}_i \cap \mathcal{N}_1^\rhoerp$. For this, let $\varepsilonll_{k,i}$ be the eigenvalue of $A_{\nablaperp_{E_k} \mathrm{t}imesi}$ associated with the eigenvector $E_i(x)$, for each pair $i,k \in \{1, \cdots, m\}$. Thus
\[\interno{\al{E_i}{E_j}}{\nablaperp_{E_k} \mathrm{t}imesi} = \interno{A_{\nablaperp_{E_k}\mathrm{t}imesi}E_i}{E_j} = \varepsilonll_{k,i}\interno{E_i}{E_j} = \varepsilonll_{k,i}\rhoartialta_{ij}.\]
Since $\mathrm{t}imesi \in \Gammaamma \lambdaeft(\mathcal{V}_i \cap \mathcal{N}_1^\rhoerp \right)$ and $\mathcal{V}_i \subset \mathbf{S}_i(TM)^\rhoerp$, it follows from the second equation in \varepsilonqref{eq: Codazzi} and that
\betaegin{align*}
& 0 = (\nabla_Y A)(X,\mathrm{t}imesi) - (\nabla_X A)(Y, \mathrm{t}imesi) = \\
& = \nabla_Y \cancel{A_\mathrm{t}imesi X} - \cancel{A_\mathrm{t}imesi \nabla_Y X} - A_{\nablaperp_Y \mathrm{t}imesi} X - \nabla_X \cancel{A_\mathrm{t}imesi Y} + \cancel{A_\mathrm{t}imesi \nabla_X Y} + A_{\nablaperp_X \mathrm{t}imesi} Y \mathbb{R}ightarrow \\
& \mathbb{R}ightarrow A_{\nablaperp_X \mathrm{t}imesi} Y = A_{\nablaperp_Y \mathrm{t}imesi} X, \ \mathrm{t}ext{for any $X, Y \in \Gamma(TM)$.}
\varepsilonnd{align*}
So, $A_{\nablaperp_{E_i} \mathrm{t}imesi} E_j = A_{\nablaperp_{E_j} \mathrm{t}imesi} E_i \mathbb{R}ightarrow \varepsilonll_{i,j} E_j = \varepsilonll_{j,i} E_i \mathbb{R}ightarrow \varepsilonll_{i,j} = 0$, if $i \nablae j$. Consequently
\[\interno{\al{E_i}{E_j}}{\nablaperp_{E_k} \mathrm{t}imesi} =
\betaegin{cases}
0, &\mathrm{t}ext{if $i \nablae j$ or $i \nablae k$,} \\
\varepsilonll_{i,i}, &\mathrm{t}ext{if $i=j=k$}.
\varepsilonnd{cases}\]
On the other side, given that $\nablaperp \lambdaeft(\mathcal{V}_i \cap \mathcal{N}_1^\rhoerp\right) \subset \{\varepsilonta\}^\rhoerp$, $\interno{\al{E_i}{E_i}}{\nablaperp_{E_i}\mathrm{t}imesi} = \sum\lambdaimits_{j=1}^{m} \interno{\al{E_j}{E_j}}{\nablaperp_{E_i}\mathrm{t}imesi} = m \interno{\varepsilonta}{\nablaperp_{E_i}\mathrm{t}imesi} = 0$. Therefore $\nablaperp_{E_i} \mathrm{t}imesi \in \mathcal{N}_1^\rhoerp$.
\varepsilonnd{prova}
\section{A Bonnet theorem for isometric immersions in $\o{1} \x \cdots \x \o{\ell}$}
\betaegin{teo}\lambdaabel{Bonnet}
Let $\mathrm{t}au_i : = \mathrm{t}au(k_i)$, $\rho := \mathrm{t}au_1 + \cdots + \mathrm{t}au_\varepsilonll$ and $m' := \sum\lambdaimits_{i=1}^\varepsilonll n_i -m$.
\betaegin{enum}
\item \mathrm{t}extbf{Existence:} Let $M^m$ be a connected and simply connected riemannian manifold and let $\mathcal{E}$ be a vector bundle with dimension $m'$ and index $\rho$ on $M$ with compatible connection $\nabla^\mathcal{E}$, curvature tensor $\mathcal{R}^\mathcal{E}$ and symmetric tensor $\alpha^\mathcal{E} \colon TM \obotlus TM \mathrm{t}o \mathcal{E}$. Lets also consider, for each $i \in \{1, \cdots, \varepsilonll\}$, the tensors $\mathbf{R}_i \colon TM \mathrm{t}o TM$, $\mathbf{S}_i \colon TM \mathrm{t}o \mathcal{E}$ and $T_i \colon \mathcal{E} \mathrm{t}o \mathcal{E}$, with $\mathbf{R}_i$ and $\mathbf{T}_i$ symmetric. For each $\varepsilonta\in \mathcal{E}$, let $A_\varepsilonta^\mathcal{E} \colon TM \mathrm{t}o TM$ be given by $\interno{A_\varepsilonta^\mathcal{E} X}{Y} = \interno{\alpha^\mathcal{E}(X,Y)}{\varepsilonta}$. With these assumptions, if equations \varepsilonqref{somas} until \varepsilonqref{eq: Ricci} hold, then there is an isometric immersion $f \colon M \mathrm{t}o \mathbb{O}$ and an vector bundle isometry $\Phi \colon \mathcal{E} \mathrm{t}o T^\rhoerp M$ such that
\betaegin{align*}
& \alpha^f = \Phi \alpha^\mathcal{E}, && \nablaperp \Phi = \Phi \nabla^\mathcal{E}, \\
& \rhoi_i \circ f_* = f_* \mathbf{R}_i + \Phi \mathbf{S}_i, && \lambdaeft.\rhoi_i\right|_{T^\rhoerp M} = f_* \mathbf{S}_i^\mathrm{t} \Phi^{-1} + \Phi \mathbf{T}_i \Phi^{-1}.
\varepsilonnd{align*}
\item \mathrm{t}extbf{Uniqueness:} Let $f, g \colon M \mathrm{t}o \mathbb{O}$ be isometric immersions such that $\mathbf{R}_i^f= \mathbf{R}_i^g$, for every $i \in \{1, \cdots, \varepsilonll\}$. Lets suppose that there is a vector bundle isometry $\Phi \colon T_f^\rhoerp M \mathrm{t}o T_g^\rhoerp M$ such that
\betaegin{align*}
& \Phi \alpha_f = \alpha_g, && \Phi {^f\nablaperp} = {^g\nablaperp} \Phi, \\
& \Phi \mathbf{S}_i^f = \mathbf{S}_i^g, && \Phi\mathbf{T}_i^f = \mathbf{T}_i^g\Phi, \ \varphiorall i \in\{1, \cdots, \varepsilonll\}.
\varepsilonnd{align*}
With the above conditions, there is an isometry $\varphi \colon \hat{\mathbb{O}} \mathrm{t}o \hat{\mathbb{O}}$ such that $\varphi \circ f = g$ and $\lambdaeft.\varphi_*\right|_{T_f^\rhoerp M} = \Phi$.
\varepsilonnd{enum}
\varepsilonnd{teo}
\subsection{Existence}
Let $\betaar \varepsilonll$ be the number of elements of the set $J = \set{i \in \{1, \cdots \varepsilonll\}}{k_i \nablae 0}$ and let $\mathcal{F} = \mathcal{E} \obotlus \mathcal{U}psilon$ be the Whitney sum, where $\mathcal{U}psilon$ is a semi-riemannian vector bundle on $M$, with dimension $\betaar\varepsilonll$ and index $\rho$. We can choose an (local) orthogonal frame $\set{\nablau_i}{ i \in J}$, of $\mathcal{U}psilon$ with $\|\nablau_i\|^2 = k_i$. If $k_i = 0$, that is, $i \nablaotin J$, let $\nablau_i = 0 \in \mathcal{U}psilon$.
Now, we can define a compatible connection in $\mathcal{F}$ and a symmetric section $\alpha^{\mathcal F} \in \Gamma \lambdaeft( T^*M \otimes T^*M \otimes \mathcal{F} \right)$ by
\betaegin{equation}\lambdaabel{eq: F}
\betaegin{aligned}
& \nabla^{\mathcal F}_X \nablau_i = -k_i \mathbf{S}_i X, \quad \nabla^{\mathcal F}_X \mathrm{t}imesi = \nabla_X^{\mathcal E} \mathrm{t}imesi + \sum_{i=1}^\varepsilonll \interno{\mathbf{S}_i X}{\mathrm{t}imesi} \nablau_i \quad \mathrm{t}ext{and} \\
& \alpha^{\mathcal F}(X,Y) = \alpha^{\mathcal E}(X,Y) + \sum_{i=1}^\varepsilonll \interno{\mathbf{R}_i X}{Y} \nablau_i,
\varepsilonnd{aligned}
\varepsilonnd{equation}
for all $X,Y \in \Gamma(TM)$ and all $\mathrm{t}imesi \in \Gamma(\mathcal{E})$.
Now lets define, for each $\varepsilonta \in \Gamma(\mathcal{U}psilon)$, the shape operator $A^{\mathcal F}_\varepsilonta \colon TM \mathrm{t}o TM$ by $\interno{A^{\mathcal F}_\varepsilonta X}{Y} = \interno{\alpha^{\mathcal F}(X,Y)}{\varepsilonta}$. So, if $\mathrm{t}imesi \in \Gamma(\mathcal E)$, then
\betaegin{multline*}
\interno{A^{\mathcal F}_\mathrm{t}imesi X}{Y} = \interno{\alpha^{\mathcal F}(X,Y)}{\mathrm{t}imesi} = \interno{\alpha^{\mathcal E}(X,Y) + \sum_{i=1}^\varepsilonll \interno{\mathbf{R}_i X}{Y} \nablau_i}{\mathrm{t}imesi} = \\
= \interno{\alpha^{\mathcal E}(X,Y)}{\mathrm{t}imesi} = \interno{A^{\mathcal E}_\mathrm{t}imesi X}{Y}.
\varepsilonnd{multline*}
On the other side,
\betaegin{multline*}
\interno{A^{\mathcal F}_{\nablau_i} X}{Y} = \interno{\alpha^{\mathcal F}(X,Y)}{\nablau_i} = \interno{\alpha^{\mathcal E}(X,Y) + \sum_{j=1}^\varepsilonll \interno{\mathbf{R}_j X}{Y} \nablau_j }{\nablau_i} = \\
= \interno{\mathbf{R}_i X}{Y} \|\nablau_i\|^2 = k_i \interno{\mathbf{R}_i X}{Y}.
\varepsilonnd{multline*}
Therefore
\betaegin{equation}\lambdaabel{AF}
A^{\mathcal F}_\mathrm{t}imesi = A^{\mathcal E}_\mathrm{t}imesi \quad \mathrm{t}ext{and} \quad A^{\mathcal F}_{\nablau_i} = k_i \mathbf{R}_i.
\varepsilonnd{equation}
\betaegin{afi}{$\mathcal{R}(X,Y)Z = A^{\mathcal F}_{\alpha^{\mathcal F}(Y,Z)}X - A^{\mathcal F}_{\alpha^{\mathcal F}(X,Z)}Y$, for any $X, Y, Z \in \Gamma(TM)$.}
\betaegin{align*}
& \mathcal{R}(X,Y)Z \stackrel{\varepsilonqref{eq: Gauss}}{=} \sum_{i=1}^\varepsilonll k_i \lambdaeft( \mathbf{R}_i X \wedge \mathbf{R}_i Y \right)Z + A^{\mathcal E}_{\alpha^{\mathcal E}(Y,Z)} X - A^{\mathcal E}_{\alpha^{\mathcal E}(X,Z)} Y =\\
& \stackrel{\varepsilonqref{eq: F}, \varepsilonqref{AF}}{=} \cancel{\sum_{i=1}^\varepsilonll k_i \interno{\mathbf{R}_i Y}{Z}\mathbf{R}_i X } - \betacancel{ \sum_{i=1}^\varepsilonll k_i\interno{\mathbf{R}_i X}{Z}\mathbf{R}_i Y } + A^{\mathcal F}_{\alpha^{\mathcal F}(Y,Z)} X - \\
& \rhohantom{\stackrel{\varepsilonqref{eq: F}, \varepsilonqref{AF}}{=}} - \cancel{ \sum_{i=1}^\varepsilonll \interno{\mathbf{R}_i Y}{Z} k_i \mathbf{R}_i X } - A^{\mathcal F}_{\alpha^{\mathcal F}(X,Z)} Y + \betacancel{\sum_{i=1}^\varepsilonll \interno{\mathbf{R}_i X}{Z} k_i \mathbf{R}_i Y} = \\
& = A^{\mathcal F}_{\alpha^{\mathcal F}(Y,Z)} X - A^{\mathcal F}_{\alpha^{\mathcal F}(X,Z)} Y.
\varepsilonnd{align*}
Therefore the claim holds.
\varepsilonnd{afi}
\betaegin{afi}{$\lambdaeft(\nabla_X \alpha^{\mathcal F}\right)(Y,Z) - \lambdaeft(\nabla_Y \alpha^{\mathcal F}\right)(X,Z) = 0$, for any $X,Y,Z \in \Gamma(TM)$.}
\betaegin{align*}
& \lambdaeft(\nabla_X \alpha^{\mathcal F}\right)(Y,Z) = \nabla^{\mathcal F}_X \alpha^{\mathcal F}(Y,Z) - \alpha^{\mathcal F}\lambdaeft(\nabla_X Y,Z\right) - \alpha^{\mathcal F}\lambdaeft(Y,\nabla_X Z\right) = \\
& \stackrel{\varepsilonqref{eq: F}}{=} \nabla^{\mathcal F}_X \lambdaeft[ \alpha^{\mathcal E}(Y,Z) + \sum_{i=1}^\varepsilonll \interno{\mathbf{R}_i Y}{Z} \nablau_i \right] - \alpha^{\mathcal E}\lambdaeft(\nabla_X Y,Z\right) - \sum_{i=1}^\varepsilonll \interno{\mathbf{R}_i \nabla_X Y}{Z} \nablau_i - \\
& \rhohantom{\stackrel{\varepsilonqref{eq: F}}{=}} - \alpha^{\mathcal E}\lambdaeft(Y,\nabla_X Z\right) - \sum_{i=1}^\varepsilonll \interno{\mathbf{R}_i Y}{\nabla_X Z}\nablau_i =\\
& = \nabla^{\mathcal F}_X \alpha^{\mathcal E}(Y,Z) + \sum_{i=1}^\varepsilonll \lambdaeft[ (\interno{\nabla_X \mathbf{R}_i Y}{Z} + \cancel{\interno{\mathbf{R}_i Y}{\nabla_X Z}}) \nablau_i + \interno{\mathbf{R}_i Y}{Z} \nabla^{\mathcal F}_X \nablau_i \right] - \\
& \quad - \alpha^{\mathcal E}\lambdaeft(\nabla_X Y,Z\right) - \sum_{i=1}^\varepsilonll \interno{\mathbf{R}_i \nabla_X Y}{Z} \nablau_i - \alpha^{\mathcal E}\lambdaeft(Y,\nabla_X Z\right) - \sum_{i=1}^\varepsilonll \cancel{\interno{\mathbf{R}_i Y}{\nabla_X Z}}\nablau_i = \\
& \stackrel{\varepsilonqref{eq: F}}{=} \nabla^{\mathcal E}_X \alpha^{\mathcal E}(Y,Z) + \sum_{i=1}^\varepsilonll \interno{\mathbf{S}_i X}{\alpha^{\mathcal E}(Y,Z)} \nablau_i + \sum_{i=1}^\varepsilonll \interno{\lambdaeft(\nabla_X \mathbf{R}_i\right) Y}{Z} \nablau_i - \\
& \quad -\sum_{i=1}^\varepsilonll \interno{\mathbf{R}_i Y}{Z} k_i \mathbf{S}_i X - \alpha^{\mathcal E}\lambdaeft(\nabla_X Y,Z\right) - \alpha^{\mathcal E}\lambdaeft(Y,\nabla_X Z\right).
\varepsilonnd{align*}
Hence
\betaegin{multline}\lambdaabel{eq: *}
\lambdaeft(\nabla^{\mathcal F}_X \alpha^{\mathcal F}\right)(Y,Z) = \lambdaeft( \nabla_X \alpha^{\mathcal E} \right)(Y,Z) + \\
+ \sum_{i=1}^\varepsilonll \lambdaeft(\interno{\mathbf{S}_i X}{\alpha^{\mathcal E}(Y,Z)} + \interno{\lambdaeft(\nabla_X \mathbf{R}_i\right) Y}{Z} \right)\nablau_i - \sum_{i=1}^\varepsilonll \interno{\mathbf{R}_i Y}{Z} k_i \mathbf{S}_i X.
\varepsilonnd{multline}
Thus
\betaegin{align*}
& \lambdaeft(\nabla^{\mathcal F}_Y \alpha^{\mathcal F}\right)(X,Z) \stackrel{\varepsilonqref{eq: *}}{=} \lambdaeft( \nabla_Y \alpha^{\mathcal E} \right)(X,Z) + \\
& \quad + \sum_{i=1}^\varepsilonll \lambdaeft(\interno{\mathbf{S}_i Y}{\alpha^{\mathcal E}(X,Z)} + \interno{\lambdaeft(\nabla_Y \mathbf{R}_i\right) X}{Z} \right)\nablau_i -\sum_{i=1}^\varepsilonll \interno{\mathbf{R}_i X}{Z} k_i \mathbf{S}_i Y = \\
& \stackrel{\varepsilonqref{eq: Codazzi}}{=} \lambdaeft( \nabla_X \alpha^{\mathcal E} \right)(Y,Z) + \sum_{i=1}^\varepsilonll k_i\lambdaeft[ \cancel{\interno{\mathbf{R}_i X}{Z}\mathbf{S}_i Y} - \interno {\mathbf{R}_i Y}{Z} \mathbf{S}_i X\right] + \\
& \quad + \sum_{i=1}^\varepsilonll \lambdaeft(\interno{\mathbf{S}_i Y}{\alpha^{\mathcal E}(X,Z)} + \interno{\lambdaeft(\nabla_Y \mathbf{R}_i\right) X}{Z} \right)\nablau_i - \sum_{i=1}^\varepsilonll \cancel{\interno{\mathbf{R}_i X}{Z} k_i \mathbf{S}_i Y} = \\
& \stackrel{\varepsilonqref{eq: derivada R}}{=} \lambdaeft( \nabla_X \alpha^{\mathcal E} \right)(Y,Z) + \sum_{i=1}^\varepsilonll \lambdaeft(\interno{\mathbf{S}_i Y}{\alpha^{\mathcal E}(X,Z)} + \interno{A^{\mathcal E}_{\mathbf{S}_i X} Y + \mathbf{S}_i^\mathrm{t} \alpha^{\mathcal E}(Y,X)}{Z} \right)\nablau_i - \\
& \quad - \sum_{i=1}^\varepsilonll k_i \interno {\mathbf{R}_i Y}{Z} \mathbf{S}_i X = \\
& = \lambdaeft( \nabla_X \alpha^{\mathcal E} \right)(Y,Z) + \sum_{i=1}^\varepsilonll \interno{A^{\mathcal E}_{\mathbf{S}_i Y}X + A^{\mathcal E}_{\mathbf{S}_i X} Y + \mathbf{S}_i^\mathrm{t} \alpha^{\mathcal E}(X,Y)}{Z} \nablau_i - \\
& \quad - \sum_{i=1}^\varepsilonll k_i \interno {\mathbf{R}_i Y}{Z} \mathbf{S}_i X = \\
& \stackrel{\varepsilonqref{eq: derivada R}}{=} \lambdaeft( \nabla_X \alpha^{\mathcal E} \right)(Y,Z) + \sum_{i=1}^\varepsilonll \lambdaeft( \interno{ (\nabla_X \mathbf{R}_i) Y}{Z} + \interno{\mathbf{S}_i X}{\alpha^{\mathcal E}(Y,Z)} \right)\nablau_i - \\
& \quad - \sum_{i=1}^\varepsilonll k_i \interno {\mathbf{R}_i Y}{Z} \mathbf{S}_i X \stackrel{\varepsilonqref{eq: *}}{=} \lambdaeft(\nabla^{\mathcal F}_X \alpha^{\mathcal F}\right)(Y,Z).
\varepsilonnd{align*}
Therefore the claim holds.
\varepsilonnd{afi}
\betaegin{afi}{${\mathcal{R}^\mathcal{F}}(X,Y) \varepsilonta = \alpha^{\mathcal F}\lambdaeft(X, A^{\mathcal F}_\varepsilonta Y\right) - \alpha^{\mathcal F}\lambdaeft(A^{\mathcal F}_\varepsilonta X, Y \right)$, for all $X,Y \in \Gamma(TM)$ and all $\varepsilonta\in \Gamma(\mathcal F)$.}
It is enough to prove the claim in the cases $\varepsilonta \in \Gamma(\mathcal E)$ and $\varepsilonta = \nablau_i$, for some $i \in \{1, \cdots, \varepsilonll\}$. So, let $\mathrm{t}imesi \in \Gamma( \mathcal E)$, hence
\betaegin{align*}
& {\cal R^F}(X,Y) \mathrm{t}imesi = \nabla^{\mathcal F}_X \nabla^{\mathcal F}_Y \mathrm{t}imesi - \nabla^{\mathcal F}_Y \nabla^{\mathcal F}_X \mathrm{t}imesi - \nabla^{\mathcal F}_{[X,Y]} \mathrm{t}imesi = \\
& \stackrel{\varepsilonqref{eq: F}}{=} \nabla^{\mathcal F}_X \lambdaeft( \nabla^{\mathcal E}_Y \mathrm{t}imesi + \sum_{i=1}^\varepsilonll \interno{\mathbf{S}_i Y}{\mathrm{t}imesi} \nablau_i\right) - \nabla^{\mathcal F}_Y \lambdaeft( \nabla^{\mathcal E}_X \mathrm{t}imesi + \sum_{i=1}^\varepsilonll \interno{\mathbf{S}_i X}{\mathrm{t}imesi} \nablau_i\right) - \\
& \quad - \nabla^{\mathcal E}_{[X,Y]} \mathrm{t}imesi - \sum_{i=1}^\varepsilonll \interno{\mathbf{S}_i [X,Y]}{\mathrm{t}imesi} \nablau_i = \\
& \stackrel{\varepsilonqref{eq: F}}{=} {\cal R^E}(X,Y) \mathrm{t}imesi + \sum_{i=1}^\varepsilonll \interno{\mathbf{S}_i X}{\nabla^{\mathcal E}_Y \mathrm{t}imesi} \nablau_i - \sum_{i=1}^\varepsilonll \interno{\mathbf{S}_i Y}{\nabla^{\mathcal E}_X \mathrm{t}imesi} \nablau_i - \sum_{i=1}^\varepsilonll \interno{\mathbf{S}_i [X,Y]}{\mathrm{t}imesi} \nablau_i + \\
& \quad + \sum_{i=1}^\varepsilonll (X\interno{\mathbf{S}_i Y}{\mathrm{t}imesi} - Y\interno{\mathbf{S}_i X}{\mathrm{t}imesi}) \nablau_i - \sum_{i=1}^\varepsilonll \interno{\mathbf{S}_i Y}{\mathrm{t}imesi} k_i \mathbf{S}_i X + \sum_{i=1}^\varepsilonll \interno{\mathbf{S}_i X}{\mathrm{t}imesi} k_i \mathbf{S}_i Y = \\
& = {\cal R^E}(X,Y) \mathrm{t}imesi + \sum_{i=1}^\varepsilonll \lambdaeft( \cancel{\interno{\mathbf{S}_i X}{\nabla^{\mathcal E}_Y \mathrm{t}imesi}} - \betacancel{\interno{\mathbf{S}_i Y}{\nabla^{\mathcal E}_X \mathrm{t}imesi}} - \interno{\mathbf{S}_i \nabla_X Y - \mathbf{S}_i \nabla_Y X}{\mathrm{t}imesi}\right) \nablau_i + \\
& \quad + \sum_{i=1}^\varepsilonll \lambdaeft(\interno{\nabla^{\mathcal E}_X \mathbf{S}_i Y}{\mathrm{t}imesi} + \betacancel{\interno{\mathbf{S}_i Y}{\nabla^{\mathcal E}_X \mathrm{t}imesi}} - \interno{\nabla^{\mathcal E}_Y \mathbf{S}_i X}{\mathrm{t}imesi} - \cancel{\interno{\mathbf{S}_i X}{\nabla^{\mathcal E}_Y \mathrm{t}imesi}} \right) \nablau_i -\\
& \quad - \sum_{i=1}^\varepsilonll k_i ( \mathbf{S}_i X \wedge \mathbf{S}_i Y) \mathrm{t}imesi = \\
& = {\cal R^E}(X,Y) \mathrm{t}imesi - \sum_{i=1}^\varepsilonll k_i ( \mathbf{S}_i X \wedge \mathbf{S}_i Y) \mathrm{t}imesi + \sum_{i=1}^\varepsilonll [ \interno{(\nabla_X \mathbf{S}_i) Y}{\mathrm{t}imesi} - \interno{(\nabla_Y \mathbf{S}_i) X}{\mathrm{t}imesi}]\nablau_i = \\
& \stackrel{\varepsilonqref{eq: Ricci}, \varepsilonqref{eq: derivada S}}{=} \alpha^{\mathcal E}\lambdaeft(X, A^{\mathcal E}_\mathrm{t}imesi Y\right) - \alpha^{\mathcal E}\lambdaeft(A^{\mathcal E}_\mathrm{t}imesi X, Y\right) + \\
& + \sum_{i=1}^\varepsilonll \interno{\cancel{\mathbf{T}_i \alpha^{\mathcal E}(X,Y)} - \alpha^{\mathcal E}(X,\mathbf{R}_i Y) - \cancel{\mathbf{T}_i \alpha^{\mathcal E}(Y,X)} + \alpha^{\mathcal E}(Y,\mathbf{R}_i X)}{\mathrm{t}imesi} \nablau_i = \\
& = \alpha^{\mathcal E}\lambdaeft(X, A^{\mathcal E}_\mathrm{t}imesi Y\right) - \alpha^{\mathcal E}\lambdaeft(A^{\mathcal E}_\mathrm{t}imesi X, Y\right) - \sum_{i=1}^\varepsilonll \interno{A^{\mathcal E}_\mathrm{t}imesi X}{\mathbf{R}_i Y} \nablau_i + \sum_{i=1}^\varepsilonll \interno{A^{\mathcal E}_\mathrm{t}imesi Y}{\mathbf{R}_i X}\nablau_i = \\
& \stackrel{\varepsilonqref{eq: F},\varepsilonqref{AF}}{=} \alpha^{\mathcal F}\lambdaeft(X, A^{\mathcal E}_\mathrm{t}imesi Y\right) - \alpha^\mathcal{F}\lambdaeft(A^\mathcal{F}_\mathrm{t}imesi X, Y\right).
\varepsilonnd{align*}
Now, for $i \in \{1, \cdots, \varepsilonll\}$,
\betaegin{align*}
& {\cal R^F}(X,Y) \nablau_i = \nabla^{\mathcal F}_X \nabla^{\mathcal F}_Y \nablau_i - \nabla^{\mathcal F}_Y \nabla^{\mathcal F}_X \nablau_i - \nabla^{\mathcal F}_{[X,Y]} \nablau_i = \\
& \stackrel{\varepsilonqref{eq: F}}{=} - \nabla^{\mathcal F}_X k_i \mathbf{S}_i Y + \nabla^{\mathcal F}_Y k_i \mathbf{S}_i X + k_i \mathbf{S}_i [X,Y] = \\
& \stackrel{\varepsilonqref{eq: F}}{=} - k_i \nabla^{\mathcal E}_X \mathbf{S}_i Y - k_i \sum_{j=1}^\varepsilonll \interno{\mathbf{S}_j X}{\mathbf{S}_i Y} \nablau_j + k_i \nabla^{\mathcal E}_Y \mathbf{S}_i X + k_i \sum_{j=1}^\varepsilonll \interno{\mathbf{S}_j Y}{\mathbf{S}_i X} \nablau_j + \\
& \quad + k_i \mathbf{S}_i \nabla_X Y - k_i \mathbf{S}_j \nabla_Y X = \\
& = -k_i \lambdaeft( \nabla_X \mathbf{S}_i \right)Y + k_i (\nabla_Y \mathbf{S}_i) X - k_i \sum_{j=1}^\varepsilonll \lambdaeft( \interno{\mathbf{S}_i^\mathrm{t}\mathbf{S}_j X}{Y} - \interno{\mathbf{S}_i^\mathrm{t}\mathbf{S}_j Y}{X} \right) \nablau_j = \\
& \stackrel{\varepsilonqref{eq: derivada S}, \varepsilonqref{eq: RST}, \varepsilonqref{eq: RST2}}{=} -k_i \lambdaeft[ \cancel{\mathbf{T}_i \alpha^{\mathcal E}(X,Y)} - \alpha^{\mathcal E}(X, \mathbf{R}_i Y) \right] + k_i \lambdaeft[ \cancel{\mathbf{T}_i \alpha^{\mathcal E}(Y,X)} - \alpha^{\mathcal E}(Y, \mathbf{R}_i X)\right] + \\
& \quad + k_i\sum_{\substack{j=1\\ j \nablae i}}^\varepsilonll \lambdaeft( \interno{\mathbf{R}_i\mathbf{R}_j X}{Y} - \interno{\mathbf{R}_i\mathbf{R}_j Y}{X} \right)\nablau_j - \interno{k_i\mathbf{R}_i(\id - \mathbf{R}_i)X}{Y} \nablau_i + \\
& \quad + k_i \interno{\mathbf{R}_i(\id - \mathbf{R}_i)Y}{X} \nablau_i = \\
& = \alpha^{\mathcal E}(X, k_i\mathbf{R}_i Y) - \alpha^{\mathcal E}(Y, k_i\mathbf{R}_i X) + \sum_{j=1}^\varepsilonll \lambdaeft( \interno{k_i\mathbf{R}_i\mathbf{R}_j X}{Y} - \interno{k_i \mathbf{R}_i\mathbf{R}_j Y}{X} \right) \nablau_j - \\
& \quad - \cancel{k_i\interno{\mathbf{R}_i X}{Y} \nablau_i} - \cancel{k_i \interno{\mathbf{R}_i Y}{X} \nablau_i} = \\
& = \alpha^{\mathcal E}(X, k_i\mathbf{R}_i Y) + \sum_{j=1}^\varepsilonll \interno{\mathbf{R}_j X}{k_i\mathbf{R}_i Y} \nablau_j - \\
& \quad - \alpha^{\mathcal E}(Y, k_i\mathbf{R}_i X) - \sum_{j=1}^\varepsilonll \interno{\mathbf{R}_j Y}{k_i\mathbf{R}_i X} \nablau_j = \\
& \stackrel{\varepsilonqref{AF}, \varepsilonqref{eq: F}}{=} \alpha^{\mathcal F}\lambdaeft(X, A_{\nablau_i}^{\mathcal F} Y \right) - \alpha^{\mathcal F}\lambdaeft(Y, A_{\nablau_i}^{\mathcal F} X\right).
\varepsilonnd{align*}
Therefore the claim holds.
\varepsilonnd{afi}
Now, let $n = \sum\lambdaimits_{i=1}^\varepsilonll \lambdaeft(n_i + \upsilon(k_i)\right)$ and $t = \sum\lambdaimits_{i=1}^\varepsilonll \mathrm{t}au(k_i)$. Since Claims 1, 2 and 3 hold, and because of Bonnet Theorem for isometric immersions in $\mathbb{R}_t^n$ (see \cite{LTV}), there is an isometric immersion $F \colon M \mathrm{t}o \mathbb{R}_t^n$ and a vector bundle isometry $\mathrm{t}ilde\Phi \colon \mathcal{F} \mathrm{t}o T_F^\rhoerp M$ such that $\alpha_F = \mathrm{t}ilde\Phi \alpha^{\mathcal F}$ and $\nablabarperp \mathrm{t}ilde \Phi = \mathrm{t}ilde \Phi \nabla^{\mathcal F}$, where $\nablabarperp$ is the normal connection in $T_F^\rhoerp M$.
Let $\mathcal G := TM \obotlus \mathcal F$ be the Whitney sum and lets define the following connection in $\cal G$:
\betaegin{align*}
&\nabla^{\cal G}_X Y = \nabla_X Y + \alpha^{\mathcal F}(X,Y), && \varphiorall X,Y \in \Gamma(TM); \\
&\nabla^{\cal G}_X \varepsilonta = - A^{\mathcal F}_\varepsilonta X + \nabla^{\mathcal F}_X \varepsilonta, && \varphiorall X \in \Gamma(TM), \ \mathrm{t}ext{and all} \ \varepsilonta \in \Gamma \lambdaeft( \cal F \right).
\varepsilonnd{align*}
Besides, for each $i \in \{1, \cdots \varepsilonll\}$, let $P_i \colon \mathcal{G} \mathrm{t}o \mathcal{G}$ be given by
\betaegin{align}\lambdaabel{Pi}
\lambdaeft.P_i\right|_{TM} &= \mathbf{R}_i + \mathbf{S}_i, & \lambdaeft.P_i\right|_{\mathcal E} &= \mathbf{S}_i^\mathrm{t} + \mathbf{T}_i, & P_i (\nablau_j) &= \rhoartialta_{ij} \nablau_i.
\varepsilonnd{align}
\betaegin{afi}{$P_iP_j = \rhoartialta_{ij} P_i$.}
Let $X \in TM$ and $\mathrm{t}imesi \in \mathcal{E}$. If $i \nablae j$, then
\betaegin{align*}
P_iP_j X &= P_i (\mathbf{R}_j X + \mathbf{S}_j X) = P_i \mathbf{R}_j X + P_i \mathbf{S}_j X = \\
&= \cancel{\mathbf{R}_i\mathbf{R}_j X} + \betacancel{\mathbf{S}_i\mathbf{R}_j X} + \cancel{\mathbf{S}_i^\mathrm{t}\mathbf{S}_j X} + \betacancel{\mathbf{T}_i\mathbf{S}_j X} \stackrel{\varepsilonqref{eq: RST2}}{=} 0; \\
P_i P_j \mathrm{t}imesi &= P_i(\mathbf{S}_j^\mathrm{t} \mathrm{t}imesi + \mathbf{T}_j \mathrm{t}imesi) = \cancel{\mathbf{R}_i\mathbf{S}_j^\mathrm{t} \mathrm{t}imesi} + \betacancel{\mathbf{S}_i\mathbf{S}_j^\mathrm{t} \mathrm{t}imesi} + \cancel{\mathbf{S}_i^\mathrm{t}\mathbf{T}_j \mathrm{t}imesi} + \betacancel{\mathbf{T}_i\mathbf{T}_j \mathrm{t}imesi} \stackrel{\varepsilonqref{eq: RST2}}{=} 0; \\
P_i P_j \nablau_k &= P_i \rhoartialta_{jk} \nablau_j = \rhoartialta_{jk} \rhoartialta_{ij} \nablau_i = 0.
\varepsilonnd{align*}
If $i = j$, then
\betaegin{align*}
{P_i}^2 X &= P_i (\mathbf{R}_i X + \mathbf{S}_i X) = \mathbf{R}_i^2 X + \mathbf{S}_i\mathbf{R}_i X + \mathbf{S}_i^\mathrm{t}\mathbf{S}_i X + \mathbf{T}_i\mathbf{S}_i X = \\
& \stackrel{\varepsilonqref{eq: RST}}{=} \mathbf{R}_i^2 X + \mathbf{S}_i \mathbf{R}_i X + \mathbf{R}_i(\id - \mathbf{R}_i) X + \mathbf{S}_i(\id - \mathbf{R}_i) X = \\
&=\mathbf{R}_i X + \mathbf{S}_i X = P_i X; \\
{P_i}^2 \mathrm{t}imesi &= P_i(\mathbf{S}_i^\mathrm{t} \mathrm{t}imesi + \mathbf{T}_i \mathrm{t}imesi) = \mathbf{R}_i\mathbf{S}_i^\mathrm{t} \mathrm{t}imesi + \mathbf{S}_i\mathbf{S}_i^\mathrm{t} \mathrm{t}imesi + \mathbf{S}_i^\mathrm{t}\mathbf{T}_i \mathrm{t}imesi + \mathbf{T}_i^2 \mathrm{t}imesi = \\
& \stackrel{\varepsilonqref{eq: RST}}{=} \mathbf{R}_i\mathbf{S}_i^\mathrm{t} \mathrm{t}imesi + \mathbf{T}_i(\id - \mathbf{T}_i) \mathrm{t}imesi + (\id - \mathbf{R}_i)\mathbf{S}_i^\mathrm{t} \mathrm{t}imesi + \mathbf{T}_i^2 \mathrm{t}imesi = \\
&= \mathbf{S}_i^\mathrm{t} \mathrm{t}imesi + \mathbf{T}_i \mathrm{t}imesi = P_i \mathrm{t}imesi; \\
{P_i}^2 \nablau_j &= P_i \rhoartialta_{ij} \nablau_i = \rhoartialta_{ij} \nablau_i = P_i \nablau_j.
\varepsilonnd{align*}
Therefore the claim holds.
\varepsilonnd{afi}
\betaegin{afi}{Each $P_i$ is a parallel tensor, that is, $\nabla^{\cal G}P_i = P_i \nabla^{\cal G}$.}
Let $X, Y \in \Gamma(TM)$ and $\mathrm{t}imesi \in \Gamma(\mathcal{F})$, thus
\betaegin{align*}
&\nabla^{\cal G}_X P_i Y = \nabla^{\cal G}_X (\mathbf{R}_i Y + \mathbf{S}_i Y) = \\
& = \nabla_X \mathbf{R}_i Y + \alpha^{\mathcal F}(X, \mathbf{R}_i Y) - A_{\mathbf{S}_i Y}^{\mathcal F} X + \nabla^{\mathcal F}_X \mathbf{S}_i Y = \\
& \stackrel{\varepsilonqref{eq: F}, \varepsilonqref{AF}}{=} \nabla_X \mathbf{R}_i Y + \alpha^{\mathcal E}(X, \mathbf{R}_i Y) + \sum_{j=1}^\varepsilonll \interno{\mathbf{R}_j X}{\mathbf{R}_i Y} \nablau_j - A_{\mathbf{S}_i Y}^{\mathcal E} X + \\
& \quad + \nabla^{\mathcal E}_X \mathbf{S}_i Y + \sum_{j=1}^\varepsilonll \interno{\mathbf{S}_j X}{\mathbf{S}_i Y}\nablau_j = \\
&\stackrel{\varepsilonqref{eq: RST}, \varepsilonqref{eq: RST2}}{=} \nabla_X \mathbf{R}_i Y + \alpha^{\mathcal E}(X, \mathbf{R}_i Y) + \sum_{\substack{j=1\\j \nablae i}}^\varepsilonll \cancel{\interno{\mathbf{R}_i\mathbf{R}_j X}{Y} \nablau_j} + \interno{\mathbf{R}_i^2 X}{Y} \nablau_i - \\
& \quad - A_{\mathbf{S}_i Y}^{\mathcal E} X + \nabla^{\mathcal E}_X \mathbf{S}_i Y - \sum_{\substack{j=1\\j \nablae i}}^\varepsilonll \cancel{\interno{\mathbf{R}_i\mathbf{R}_j X}{Y}\nablau_j} + \interno{\mathbf{R}_i(\id - \mathbf{R}_i) X}{Y} \nablau_i = \\
&= \nabla_X \mathbf{R}_i Y + \alpha^{\mathcal E}(X, \mathbf{R}_i Y) - A_{\mathbf{S}_i Y}^{\mathcal E} X + \nabla^{\mathcal E}_X \mathbf{S}_i Y + \interno{\mathbf{R}_i X}{Y} \nablau_i = \\
& \stackrel{\varepsilonqref{eq: derivada R}, \varepsilonqref{eq: derivada S}}{=} \mathbf{R}_i \nabla_X Y + \cancel{A^{\mathcal E}_{\mathbf{S}_i Y} X} + \mathbf{S}_i^\mathrm{t}\alpha^{\mathcal E}(X, Y) + \betacancel{\alpha^{\mathcal E}(X, \mathbf{R}_i Y)} - \cancel{A_{\mathbf{S}_i Y}^{\mathcal E} X} + \\
& \quad + \mathbf{S}_i \nabla_X Y + \mathbf{T}_i \alpha^{\mathcal E}(X,Y) - \betacancel{\alpha^{\mathcal E}(X, \mathbf{R}_i Y)} + \interno{\mathbf{R}_i X}{Y} \nablau_i = \\
& = \mathbf{R}_i \nabla_X Y + \mathbf{S}_i^\mathrm{t}\alpha^{\mathcal E}(X, Y) + \mathbf{S}_i \nabla_X Y + \mathbf{T}_i \alpha^{\mathcal E}(X,Y) + \interno{\mathbf{R}_i X}{Y} \nablau_i = \\
&= P_i \lambdaeft( \nabla_X Y + \alpha^{\mathcal E}(X,Y) + \sum_{j=1}^\varepsilonll \interno{\mathbf{R}_j X}{Y} \nablau_j \right) = P_i \nabla^{\cal G}_X Y.
\varepsilonnd{align*}
\betaegin{align*}
&\nabla^{\cal G}_X P_i \mathrm{t}imesi = \nabla^{\cal G}_ X \lambdaeft( \mathbf{S}_i^\mathrm{t} \mathrm{t}imesi + \mathbf{T}_i \mathrm{t}imesi \right) = \nabla_X \mathbf{S}_i^\mathrm{t} \mathrm{t}imesi + \alpha^{\mathcal F}\lambdaeft(X, \mathbf{S}_i^\mathrm{t} \mathrm{t}imesi\right) - A^{\mathcal F}_{\mathbf{T}_i \mathrm{t}imesi} X + \nabla^{\mathcal F}_X \mathbf{T}_i\mathrm{t}imesi = \\
&\stackrel{\varepsilonqref{eq: F},\varepsilonqref{AF}}{=} \nabla_X \mathbf{S}_i^\mathrm{t} \mathrm{t}imesi + \alpha^{\mathcal E}\lambdaeft(X, \mathbf{S}_i^\mathrm{t} \mathrm{t}imesi \right) + \sum_{j=1}^\varepsilonll \interno{\mathbf{R}_j X}{\mathbf{S}_i^\mathrm{t} \mathrm{t}imesi}\nablau_j - A^{\mathcal E}_{\mathbf{T}_i \mathrm{t}imesi} X + \nabla^{\mathcal E}_X \mathbf{T}_i\mathrm{t}imesi + \\
&\quad + \sum_{j=1}^\varepsilonll \interno{\mathbf{S}_j X}{\mathbf{T}_i\mathrm{t}imesi}\nablau_j = \\
&= \nabla_X \mathbf{S}_i^\mathrm{t} \mathrm{t}imesi + \alpha^{\mathcal E}\lambdaeft(X, \mathbf{S}_i^\mathrm{t} \mathrm{t}imesi \right) + \sum_{\substack{j=1\\j \nablae i}}^\varepsilonll \cancel{\interno{\mathbf{S}_i\mathbf{R}_j X}{\mathrm{t}imesi}\nablau_j} + \interno{\mathbf{S}_i\mathbf{R}_i X}{\mathrm{t}imesi} \nablau_i - A^{\mathcal E}_{\mathbf{T}_i \mathrm{t}imesi} X + \\
& \quad + \nabla^{\mathcal E}_X \mathbf{T}_i\mathrm{t}imesi + \sum_{\substack{j=1\\j \nablae i}}^\varepsilonll \cancel{\interno{\mathbf{T}_i\mathbf{S}_j X}{\mathrm{t}imesi}\nablau_j} + \interno{\mathbf{T}_i\mathbf{S}_i X}{\mathrm{t}imesi} \nablau_i = \\
& \stackrel{\varepsilonqref{eq: RST}, \varepsilonqref{eq: RST2}}{=} \nabla_X \mathbf{S}_i^\mathrm{t} \mathrm{t}imesi + \alpha^{\mathcal E}\lambdaeft(X, \mathbf{S}_i^\mathrm{t} \mathrm{t}imesi \right) - A^{\mathcal E}_{\mathbf{T}_i \mathrm{t}imesi} X + \nabla^{\mathcal E}_X \mathbf{T}_i\mathrm{t}imesi + \interno{\mathbf{S}_i X}{\mathrm{t}imesi} \nablau_i = \\
& \stackrel{\varepsilonqref{eq: derivada S},\varepsilonqref{eq: derivada T}}{=} \mathbf{S}_i^\mathrm{t} \nabla^{\mathcal E}_X \mathrm{t}imesi + \cancel{A^{\mathcal E}_{\mathbf{T}_i \mathrm{t}imesi} X} - \mathbf{R}_i A^{\mathcal E}_\mathrm{t}imesi X + \betacancel{\alpha^{\mathcal E}\lambdaeft(X, \mathbf{S}_i^\mathrm{t} \mathrm{t}imesi \right)} - \cancel{A^{\mathcal E}_{\mathbf{T}_i \mathrm{t}imesi} X} + \mathbf{T}_i \nabla^{\mathcal E}_X \mathrm{t}imesi - \\
& \quad - \mathbf{S}_i A_\mathrm{t}imesi X - \betacancel{\alpha^{\mathcal E}\lambdaeft(X,\mathbf{S}_i^\mathrm{t}\mathrm{t}imesi\right)} + \interno{\mathbf{S}_i X}{\mathrm{t}imesi} \nablau_i = \\
& = P_i \lambdaeft(-A_\mathrm{t}imesi^{\mathcal E} X + \nabla_X^{\mathcal E} \mathrm{t}imesi + \sum_{j=1}^\varepsilonll \interno{\mathbf{S}_j X}{\mathrm{t}imesi} \nablau_j \right) = P_i \lambdaeft( \nabla^{\cal G}_X \mathrm{t}imesi\right).
\varepsilonnd{align*}
\betaegin{align*}
\nabla_X^{{\cal G}} P_i \nablau_j & = \nabla_X^{{\cal G}} \rhoartialta_{ij} \nablau_i = - \rhoartialta_{ij} A_{\nablau_i}^{\mathcal F} X + \rhoartialta_{ij} \nabla^{\mathcal F}_X \nablau_i =\\
&\stackrel{\varepsilonqref{eq: F},\varepsilonqref{AF}}{=} -\rhoartialta_{ij}k_i \mathbf{R}_i X - \rhoartialta_{ij} k_i \mathbf{S}_i X = -\rhoartialta_{ij}k_i P_i(X). \\
P_i \nabla_X^{{\cal G}} \nablau_j &= P_i(-k_j A_{\nablau_j}^{\mathcal F} X + \nabla^{\mathcal F}_X \nablau_j) = P_i (-k_j \mathbf{R}_j X - k_j \mathbf{S}_j X) = \\
& = P_i (-k_j P_j (X)) = -\rhoartialta_{ij}k_j P_i(X).
\varepsilonnd{align*}
Since $\nabla^{\cal G}_X P_i Y = P_i \nabla^{\cal G}_X Y$, $\nabla^{\cal G}_X P_i \mathrm{t}imesi = P_i \nabla^{\cal G}_X \mathrm{t}imesi$ and $\nabla_X^{{\cal G}} P_i \nablau_j = P_i \nabla_X^{{\cal G}} \nablau_j$, for all $X, Y \in \Gamma(TM)$, all $\mathrm{t}imesi \in \Gamma(\mathcal{E})$ and all $i,j \in \{1, \cdots, \varepsilonll\}$, then $P_i$ is a parallel tensor.
\varepsilonnd{afi}
\betaegin{afi}{$P_i = P_i^\mathrm{t}$.}
It follows from straightforward calculations.
\varepsilonnd{afi}
Lets consider $\hat\Phi \colon \mathcal{G} \mathrm{t}o F^*T \mathbb{R}_t^N$ given by $\hat\Phi|_{TM} = F_* \colon TM \mathrm{t}o F_*TM$ and $\hat\Phi|_{\mathcal F} = \mathrm{t}ilde \Phi \colon \mathcal{F} \mathrm{t}o T_F^\rhoerp M$.
\betaegin{afi}{$\hat\Phi$ is parallel vector bundle isometry, that is, $\hat\Phi \nabla^{\cal G} = \nablatil \hat\Phi$, where $\nablatil$ is the connection in $\mathbb{R}_t^N$.} \lambdaabel{Pi parallel}
Let $X,Y \in \Gamma(TM)$ and $\varepsilonta \in \Gammaamma(\mathcal{F})$. Thus,
\betaegin{multline*}
\interno{A^{\mathcal F}_\varepsilonta X}{Y} = \interno{\alpha^{\mathcal F}(X, Y)}{\varepsilonta} = \interno{\mathrm{t}ilde\Phi \alpha^{\mathcal F}(X, Y)}{\mathrm{t}ilde\Phi \varepsilonta} = \\
= \interno{\alpha_F(X,Y)}{\mathrm{t}ilde\Phi \varepsilonta} = \interno{A^F_{\mathrm{t}ilde\Phi \varepsilonta} X}{Y}.
\varepsilonnd{multline*}
Hence $A^{\mathcal F}_\varepsilonta = A^F_{\mathrm{t}ilde\Phi\varepsilonta}$ and
\betaegin{align*}
\hat\Phi \nabla^{\cal G}_X Y &= \hat\Phi\lambdaeft( \nabla_X Y + \alpha^{\mathcal F}(X,Y) \right) = F_* \nabla_X Y + \mathrm{t}ilde\Phi \lambdaeft(\alpha^{\mathcal F}(X,Y) \right) = \\
&= F_* \nabla_X Y + \alpha_F(X,Y) = \nablatil_X F_* Y = \nablatil_X \hat\Phi Y. \\
\hat\Phi \nabla^{\cal G}_X \varepsilonta &= \hat\Phi \lambdaeft( -A^{\mathcal F}_\varepsilonta X + \nabla^{\mathcal F}_X \varepsilonta \right) = -F_*A^{\mathcal F}_\varepsilonta X + \mathrm{t}ilde\Phi \nabla^{\mathcal F}_X \varepsilonta = \\
&= -F_*A^F_{\mathrm{t}ilde\Phi\varepsilonta} X + \nablaperp_X \mathrm{t}ilde\Phi \varepsilonta = \nablatil_X \mathrm{t}ilde\Phi\varepsilonta = \nablatil_X \hat\Phi\varepsilonta.
\varepsilonnd{align*}
Thus $\hat \Phi$ is parallel.
\varepsilonnd{afi}
Since each $P_i$ is parallel, $P_i(\mathcal{G})$ is a parallel vector sub-bundle of $\cal G$. Besides, $\interno{P_i X}{P_j Y} = \interno{P_jP_i X}{Y} = \rhoartialta_{ij} \interno{P_i X}{Y}$. Then $P_i(\mathcal{G})$ and $P_j(\mathcal{G})$ are orthogonal, if $i \nablae j$. Hence each $\hat\Phi[P_i(\mathcal{G})]$ is a constant vector subspace of $\mathbb{R}_t^N$ and they are orthogonal. Besides, because $P_i(\nablau_j) = \rhoartialta_{ij} \nablau_i$ and $\|\nablau_i\|^2 = k_i$, then $\hat\Phi[P_i(\mathcal{G})] = \mathbb{R}^{N_i} = \mathbb{R}_{\mathrm{t}au(k_i)}^{N_i}$, where $N_i$ is the dimension of $P_i(\mathcal{G})$.
\betaegin{afi}{$\mathbb{R}_t^N = \mathbb{R}^{N_1} \obot \cdots \obot \mathbb{R}^{N_\varepsilonll}$}
Since $\mathbb{R}N = F_* TM \obot T_F^\rhoerp M$ and $\hat\Phi \colon \mathcal{G} \mathrm{t}o F^*T\mathbb{R}_t^N$ is a vector bundle isometry, we just have to show that $\mathcal{G} = P_1(\mathcal{G}) \obot \cdots \obot P_\varepsilonll(\mathcal{G})$. On the other side, we know that each pair $P_i(\mathcal{G})$ and $P_j(\mathcal{G})$ are orthogonal sub-fiber bundles. So we just have to show that $\zeta = P_1(\zeta) + \cdots + P_\varepsilonll(\zeta)$, for any $\zeta \in \mathcal{G}$.
Let $X \in TM$ and $\mathrm{t}imesi \in \cal E$, thus
\betaegin{align*}
& X = \id(X) + 0(x) \stackrel{\varepsilonqref{somas}}{=} \sum_{i=1}^\varepsilonll \mathbf{R}_i X + \sum_{i=1}^\varepsilonll \mathbf{S}_i X = \sum_{i=1}^\varepsilonll P_i(X), \\
& \mathrm{t}imesi = 0(\mathrm{t}imesi) + \id(\mathrm{t}imesi) \stackrel{\varepsilonqref{somas}}{=} \sum_{i=1}^\varepsilonll \mathbf{S}_i^\mathrm{t} \mathrm{t}imesi + \sum_{i=1}^\varepsilonll \mathbf{T}_i \mathrm{t}imesi = \sum_{i=1}^\varepsilonll P_i(\mathrm{t}imesi), \\
& \nablau_j \stackrel{\varepsilonqref{Pi}}{=} \sum_{i=1}^\varepsilonll P_i(\nablau_j).
\varepsilonnd{align*}
Therefore $\zeta = P_1(\zeta) + \cdots + P_\varepsilonll(\zeta)$, for any $\zeta \in \mathcal{G}$.
\varepsilonnd{afi}
For each $i \in \{ 1, \cdots, \varepsilonll\}$, let $\rhoi_i \colon \mathbb{R}_t^N \mathrm{t}o \mathbb{R}^{N_i}$ be the orthogonal projection.
\betaegin{afi}{$\rhoi_i \circ \hat\Phi = \hat\Phi \circ P_i$.}
If $\zeta \in \mathcal{G}$, then $\zeta = \zeta_i + \zeta_i^\rhoerp$, with $\zeta_i \in P_i(\mathcal{G})$ and $\zeta_i^\rhoerp \in P_i(\mathcal{G})^\rhoerp$. Thus $\zeta_i = P_i \zeta$ and $P_i\lambdaeft(\zeta_i^\rhoerp\right) = 0$, therefore $\lambdaeft(\hat\Phi \circ P_i\right)(\zeta) = \hat\Phi\lambdaeft(P_i(\zeta_i)\right) + \hat\Phi \lambdaeft(P_i\lambdaeft(\zeta_i^\rhoerp\right)\right) = \hat\Phi\lambdaeft(P_i(\zeta_i)\right)$.
On the other side, since $\hat\Phi[P_i(\mathcal{G})] = \mathbb{R}^{N_i}$, $\lambdaeft(\rhoi_i \circ \hat\Phi\right)(\zeta) = \rhoi_i \lambdaeft(\hat\Phi(\zeta_i)\right) + \rhoi_i\lambdaeft(\hat\Phi \lambdaeft(\zeta_i^\rhoerp \right) \right) = \rhoi_i \lambdaeft(\hat\Phi\lambdaeft(P_i(\zeta_i)\right)\right) = \hat\Phi(P_i(\mathrm{t}imesi))$.
\varepsilonnd{afi}
So
\betaegin{align}
& \rhoi_i\circ F_* = \rhoi_i \circ \hat\Phi|_{TM} = \hat\Phi\circ \lambdaeft.P_i\right|_{TM} = \hat\Phi (\mathbf{R}_i + \mathbf{S}_i) = F_* \mathbf{R}_i + \hat\Phi \mathbf{S}_i, \lambdaabel{piPhi1}\\
& \rhoi_i \circ \hat\Phi|_{\mathcal E} = \hat\Phi \circ \lambdaeft.P_i\right|_{\mathcal E} = \hat\Phi\lambdaeft(\mathbf{S}_i^\mathrm{t} + \mathbf{T}_i\right) = F_*\mathbf{S}_i^\mathrm{t} + \hat\Phi \mathbf{T}_i, \lambdaabel{piPhi2}\\
& \rhoi_i \hat\Phi (\nablau_j) = \hat\Phi P_i (\nablau_j) = \rhoartialta_{ij} \hat\Phi(\nablau_j). \lambdaabel{piPhi3}
\varepsilonnd{align}
By Claim \ref{Pi parallel},
\betaegin{align*}
& \nablatil_X \mathrm{t}ilde\Phi(\nablau_i) = \nablatil_X \hat\Phi(\nablau_i) = \hat\Phi\nabla_X^{\cal G} \nablau_i = \hat\Phi\lambdaeft(-A^{\mathcal F}_{\nablau_i} X + \nabla^{\mathcal F}_X \nablau_i \right) =\\
& \stackrel{\varepsilonqref{AF},\varepsilonqref{eq: F}}{=} \hat\Phi\lambdaeft( -k_i\mathbf{R}_i X - k_i\mathbf{S}_i X \right) = -k_i\lambdaeft(\hat\Phi\circ P_i\right)(X) = \\
& = - k_i\lambdaeft( \rhoi_i \circ \hat\Phi \right)(X) = -k_i \rhoi_i (F_* X).
\varepsilonnd{align*}
Therefore $\nablatil_X \mathrm{t}ilde\Phi(\nablau_i) = - k_i\rhoi_i(F_* X)$.
Let $\zeta_i := \rhoi_i \circ F + \varphirac{1}{k_i} \mathrm{t}ilde\Phi(\nablau_i)$. Because of \varepsilonqref{piPhi3}, $\zeta_i \in \mathbb{R}^{N_i}$, and since $\nablatil_X \mathrm{t}ilde\Phi(\nablau_i) = - k_i\rhoi_i(F_* X)$, $\nablatil_X \zeta_i = 0$. Hence each $\zeta_i$ is constant in $\mathbb{R}N$.
Let $P := \sum\lambdaimits_{i=1}^\varepsilonll \zeta_i = F + \sum\lambdaimits_{i=1}^\varepsilonll \varphirac{1}{k_i}\mathrm{t}ilde \Phi(\nablau_i)$, $\mathrm{t}ilde F := F - P$ and $\mathrm{t}ilde{\zeta}_i := \rhoi_i \circ \mathrm{t}ilde F + \varphirac{1}{k_i} \mathrm{t}ilde \Phi(\nablau_i)$. With the same calculations made for $\zeta_i$, we can show that $\mathrm{t}ilde{\zeta}_i$ is constant.
\betaegin{afi}{$\mathrm{t}ilde{\zeta}_i = 0$.}
In deed,
\betaegin{multline*}
\mathrm{t}ilde{\zeta}_i = \rhoi_i\lambdaeft(\mathrm{t}ilde F(x_0)\right) + \varphirac{1}{k_i}\mathrm{t}ilde \Phi(\nablau_i(x_0)) = \\
= \rhoi_i(F(x_0)) - \rhoi_i(P) + \varphirac{1}{k_i}\mathrm{t}ilde \Phi(\nablau_i(x_0)) = \zeta_i - \rhoi_i(P).
\varepsilonnd{multline*}
But $\rhoi_i(P) = \rhoi_i \lambdaeft(\sum\lambdaimits_{j=1}^\varepsilonll \zeta_j\right) = \zeta_i$. Therefore $\mathrm{t}ilde{\zeta}_i = 0$.
\varepsilonnd{afi}
\betaegin{afi}{Replacing $F$ by $\mathrm{t}ilde F$, if necessary, we can assume that $F(M) \subset \o{1} \x \cdots \x \o{\ell}$, with $\o{i} \subset \mathbb{R}^{N_i}$ and $n_i + \upsilon(k_i) = N_i$.}
Replacing $F$ by $\mathrm{t}ilde F$, if necessary, we can assume that each $\zeta_i = 0$, that is,
\[\rhoi_i \circ F + \varphirac{1}{k_i}\mathrm{t}ilde\Phi(\nablau_i) = 0 \mathbb{R}ightarrow \lambdaeft\|\rhoi_i \circ F \right\|^2 = \varphirac{\lambdaeft\| \mathrm{t}ilde\Phi(\nablau_i)\right\|^2}{k_i^2} = \varphirac{k_i}{k_i^2} = \varphirac{1}{k_i}.\]
Therefore, if $\o{i}$ is the connected component of $\ss{i} \subset \mathbb{R}^{N_i}$ such that $\rhoi_i(F(M))\subset \o{i}$, then $(\rhoi_i\circ F)(M) \subset \o{i}$ and $F(M) \subset \o{1} \x \cdots \x \o{\ell}$.
\varepsilonnd{afi}
Since $F(M) \subset \o{1} \x \cdots \x \o{\ell}$, there is an isometric immersion $f \colon M \mathrm{t}o \o{1} \x \cdots \x \o{\ell}$ such that $F = \imath \circ f$, where $\imath \colon \o{1} \x \cdots \x \o{\ell} \mathrm{t}o \mathbb{R}N$ is the canonical inclusion. Besides, because $\rhoi_i \circ F + \varphirac{1}{k_i}\mathrm{t}ilde\Phi(\nablau_i) = 0$, $\mathrm{t}ilde\Phi(\nablau_i) = -k_i(\rhoi_i\circ F) = \nablau_i^F$.
Let $\nablaperp$ be the normal connection of $f$ in $T_f^\rhoerp M$. We must to show that $\Phi = \mathrm{t}ilde\Phi|_{\mathcal E}$ is a vector bundle isometry such $\Phi(\mathcal{E}) = T_f^\rhoerp M$ and that
\betaegin{align*}
& \alpha^f = \Phi \alpha^\mathcal{E}, && \nablaperp \Phi = \Phi \nabla^\mathcal{E}, \\
& \rhoi_i \circ f_* = f_* \mathbf{R}_i + \Phi \mathbf{S}_i, && \lambdaeft.\rhoi_i\right|_{T^\rhoerp M} = f_* \mathbf{S}_i^\mathrm{t} \Phi^{-1} + \Phi \mathbf{T}_i \Phi^{-1}.
\varepsilonnd{align*}
Since $T_F^\rhoerp M = T_f^\rhoerp M \obot \spa\set{\rhoi_i\circ F}{k_i \nablae 0}$ and $\mathrm{t}ilde\Phi(\nablau_i) = - k_i(\rhoi_i\circ F)$, then $\spa\set{\rhoi_i\circ F}{k_i \nablae 0} = \mathrm{t}ilde\Phi(\spa[\nablau_1, \cdots, \nablau_\varepsilonll])$, thus $\mathrm{t}ilde\Phi(\mathcal{E}) = T_f^\rhoerp M$.
Because $J = \set{i \in \{1, \cdots, \varepsilonll\}}{k_i \nablae 0}$,
\betaegin{align*}
& \af{X}{Y} = \aF{X}{Y} - \sum_{i \in J} \interno{\aF{X}{Y}}{\nablau_i^F}\varphirac{\nablau_i^F}{\lambdaeft\|\nablau_i^F\right\|^2} = \\
& = \mathrm{t}ilde\Phi\lambdaeft( \alpha^{\mathcal F}(X,Y) \right) - \sum_{i\in J} \interno{\mathrm{t}ilde\Phi\lambdaeft(\alpha^{\mathcal F}(X,Y)\right)}{\mathrm{t}ilde\Phi(\nablau_i)}\varphirac{\mathrm{t}ilde\Phi(\nablau_i)}{k_i} = \\
& = \mathrm{t}ilde\Phi\lambdaeft( \alpha^{\mathcal F}(X,Y) - \sum_{i=1}^\varepsilonll\interno{\alpha^{\mathcal F}(X,Y)}{\nablau_i}\varphirac{\nablau_i}{k_i} \right) \stackrel{\varepsilonqref{eq: F},\varepsilonqref{AF}}{=} \mathrm{t}ilde\Phi\lambdaeft( \alpha^{\mathcal E}(X,Y) \right). \\
& \nablaperp_X \Phi \mathrm{t}imesi = \nablabarperp_X \mathrm{t}ilde\Phi\mathrm{t}imesi - \sum_{i \in J} \interno{\nablabarperp_X \mathrm{t}ilde\Phi\mathrm{t}imesi}{\nablau_i^F}\varphirac{\nablau_i^F}{\lambdaeft\|\nablau_i^F\right\|^2} = \\
& = \mathrm{t}ilde\Phi \nabla_X^{\mathcal F} \mathrm{t}imesi - \sum_{i \in J} \interno{\mathrm{t}ilde\Phi \nabla_X^{\mathcal F} \mathrm{t}imesi}{\mathrm{t}ilde\Phi \nablau_i}\varphirac{\mathrm{t}ilde\Phi \nablau_i}{k_i} = \\
& = \mathrm{t}ilde\Phi \lambdaeft( \nabla_X^{\mathcal F} \mathrm{t}imesi - \sum_{i \in J} \interno{\nabla_X^{\mathcal F} \mathrm{t}imesi}{\nablau_i}\varphirac{\nablau_i}{k_i} \right) \stackrel{\varepsilonqref{eq: F}}{=} \mathrm{t}ilde\Phi \lambdaeft( \nabla_X^{\mathcal E} \mathrm{t}imesi \right). \\
& \rhoi_i (f_* X) = \rhoi_i (F_* X) = \lambdaeft(\rhoi_i\circ\hat\Phi\right)(X) = \lambdaeft(\hat\Phi\circ P_i\right)(X) = \hat\Phi (\mathbf{R}_i X + \mathbf{S}_i X) = \\
& = F_*\mathbf{R}_i X + \mathrm{t}ilde\Phi(\mathbf{S}_i X) = f_* \mathbf{R}_i X + \Phi (\mathbf{S}_i X).
\varepsilonnd{align*}
Last, if $\zeta \in T^\rhoerp M = \Phi(\mathcal{E})$, then $\zeta = \Phi \mathrm{t}imesi$, for some $\mathrm{t}imesi \in \mathcal{E}$. Thus
\betaegin{align*}
& \rhoi_i(\zeta) = \rhoi_i(\Phi\mathrm{t}imesi) = \lambdaeft(\rhoi_i\circ\hat\Phi\right)(\mathrm{t}imesi) = \lambdaeft(\hat\Phi\circ P_i\right)(\mathrm{t}imesi) = \hat\Phi\lambdaeft(\mathbf{S}_i^\mathrm{t}\mathrm{t}imesi + \mathbf{T}_i\mathrm{t}imesi \right) =\\
& = F_*\mathbf{S}_i^\mathrm{t}\mathrm{t}imesi + \mathrm{t}ilde\Phi\mathbf{T}_i\mathrm{t}imesi = f_*\mathbf{S}_i^\mathrm{t} \Phi^{-1}(\zeta) + \Phi\mathbf{T}_i\Phi^{-1}(\zeta).
\varepsilonnd{align*}
Therefore $\lambdaeft.\rhoi_i\right|_{T^\rhoerp M} = f_*\mathbf{S}_i^\mathrm{t}\Phi^{-1} + \Phi\mathbf{T}_i\Phi^{-1}$.
$\Box$
\subsection{Uniqueness}
Let $f$, $g$, and $\Phi$ be like in \mathrm{t}extsl{(II)} of Theorem \ref{Bonnet}. So, lets consider $F := \imath\circ f$, $G := \imath\circ g$ and $\mathrm{t}ilde\Phi \colon T_F^\rhoerp M \mathrm{t}o T_G^\rhoerp M$ given by $\mathrm{t}ilde\Phi \imath_* \mathrm{t}imesi = \imath_*\Phi \mathrm{t}imesi$, for all $\mathrm{t}imesi \in T_f^\rhoerp M$, and $\mathrm{t}ilde\Phi \lambdaeft(\nablau_i^F\right) = \nablau_i^G$, with $\nablau_i^F := -k_i (\rhoi\circ F)$ and $\nablau_i^G := -k_i (\rhoi_i\circ G)$. Hence
\betaegin{align*}
& \mathrm{t}ilde\Phi \lambdaeft(\aF{X}{Y}\right) \stackrel{\mathrm{t}ext{Lemma \ref{lem: aF}}}{=} \mathrm{t}ilde\Phi\lambdaeft(\imath_*\af{X}{Y} + \sum_{i=1}^\varepsilonll \interno{\mathbf{R}_i^f X}{Y} \nablau_i\right) = \\
& = \imath_*\Phi \lambdaeft(\af{X}{Y}\right) + \sum_{i=1}^\varepsilonll \interno{\mathbf{R}_i^g X}{Y}\mathrm{t}ilde\Phi \lambdaeft(\nablau_i^F\right) = \\
& = \imath_*\alpha_g(X,Y) + \sum_{i=1}^\varepsilonll \interno{\mathbf{R}_i^g X}{Y} \nablau_i^G \stackrel{\mathrm{t}ext{Lemma \ref{lem: aF}}}{=} \alpha_G(X,Y).
\varepsilonnd{align*}
On the other side, if $\mathrm{t}imesi \in \Gamma\lambdaeft(T_f^\rhoerp M\right)$, then
\betaegin{align*}
& \mathrm{t}ilde\Phi \lambdaeft(^F\nablabarperp_X \imath_*\mathrm{t}imesi\right) \stackrel{\varepsilonqref{nbarperpxi}}{=} \mathrm{t}ilde\Phi \lambdaeft( \imath_* \lambdaeft(^f\nablaperp_X \mathrm{t}imesi\right) + \sum_{i=1}^\varepsilonll \interno{\mathbf{S}^f_i X}{\mathrm{t}imesi}\nablau^F_i\right) = \\
& = \imath_* \Phi \lambdaeft(^f\nablaperp_X \mathrm{t}imesi\right) + \sum_{i=1}^\varepsilonll \interno{\mathbf{S}^f_i X}{\mathrm{t}imesi}\nablau^G_i = \imath_* \lambdaeft(^g\nablaperp_X \Phi\mathrm{t}imesi\right) + \sum_{i=1}^\varepsilonll \interno{\Phi\mathbf{S}^f_i X}{\Phi\mathrm{t}imesi}\nablau^G_i = \\
& = \imath_* \lambdaeft(^g\nablaperp_X \Phi\mathrm{t}imesi\right) + \sum_{i=1}^\varepsilonll \interno{\mathbf{S}_i^g X}{\Phi\mathrm{t}imesi}\nablau^G_i = {^G\nablabarperp_X \imath_*\Phi\mathrm{t}imesi}.
\varepsilonnd{align*}
Therefore $\mathrm{t}ilde\Phi\alpha_F = \alpha_G$ and $\mathrm{t}ilde\Phi{^F\nablabarperp} = {^G\nablabarperp}\mathrm{t}ilde\Phi$.
From the uniqueness part of Bonnet Theorem for isometric immersions in $\mathbb{R}N = \mathbb{R}_t^n$, there is an isometry $\mathrm{t}au \colon \mathbb{R}N \mathrm{t}o\mathbb{R}N$ such that $G = \mathrm{t}au \circ F$ and $\lambdaeft.\mathrm{t}au_*\right|_{T_F^\rhoerp M} = \mathrm{t}ilde\Phi$. Lets denote $\mathrm{t}au(Z) = B(Z) + C$, where $C \in \mathbb{R}N$ is constant and $B = \mathrm{t}au_* \in \mathbb{O}_t(n)$ is an orthogonal transformation. Thus $B\lambdaeft(\nablau_i^F\right) = \mathrm{t}au_*\lambdaeft(\nablau_i^F\right) = \mathrm{t}ilde\Phi\lambdaeft(\nablau_i^F\right) = \nablau_i^G$, that is, $B(\rhoi_i \circ F) = \rhoi_i \circ G$, if $k_i \nablae 0$.
On the other side, since $G = \mathrm{t}au \circ F$, then $G_* = \mathrm{t}au_*F_* = BF_*$.
Now, if $X \in TM$ and $\mathrm{t}imesi \in T_f^\rhoerp M$, then
\betaegin{align*}
& \interno{\Phi\mathbf{S}_i^f X}{\mathrm{t}imesi} = \interno{\mathbf{S}_i^f X}{\Phi^{-1} \mathrm{t}imesi} = \interno{X}{{^f\mathbf{S}_i^\mathrm{t}} \Phi^{-1} \mathrm{t}imesi}; \\
& \interno{\Phi\mathbf{S}_i^f X}{\mathrm{t}imesi} = \interno{\mathbf{S}_i^g X}{\mathrm{t}imesi} = \interno{X}{{^g\mathbf{S}_i^\mathrm{t}} \mathrm{t}imesi}.
\varepsilonnd{align*}
Therefore $^f\mathbf{S}_i^\mathrm{t} = {^g\mathbf{S}_i^t}\Phi$. So, if $\mathrm{t}imesi \in T_f^\rhoerp M$, then
\betaegin{align*}
& B(\imath_*\mathrm{t}imesi) = B\lambdaeft( \imath_*f_*{^f\mathbf{S}_i^\mathrm{t}}\mathrm{t}imesi + \imath_*\mathbf{T}_i^f\mathrm{t}imesi \right) = \mathrm{t}au_*F_*{^g\mathbf{S}_i^\mathrm{t}}(\Phi\mathrm{t}imesi) + \mathrm{t}au_*\imath_*\mathbf{T}_i^f\mathrm{t}imesi = \\
& = G_*{^g\mathbf{S}_i^\mathrm{t}}(\Phi\mathrm{t}imesi) + \mathrm{t}ilde \Phi \imath_*\mathbf{T}_i^f\mathrm{t}imesi = G_*{^g\mathbf{S}_i^\mathrm{t}}(\Phi\mathrm{t}imesi) + \imath_* \Phi \mathbf{T}_i^f\mathrm{t}imesi = \\
& = G_*{^g\mathbf{S}_i^\mathrm{t}}(\Phi\mathrm{t}imesi) + \imath_*\mathbf{T}_i^g(\Phi\mathrm{t}imesi) = \imath_*\Phi\mathrm{t}imesi.
\varepsilonnd{align*}
As a result, $B\circ\imath_* = \imath_*\Phi$.
Given $\zeta \in \mathbb{R}N$ and $p \in M$, we know that there are $X \in T_p^f M$ and $\mathrm{t}imesi \in {^fT_p^\rhoerp M}$ such that $\zeta = F_* X + \imath_* \mathrm{t}imesi + \sum\lambdaimits_{i \in J} \interno{\zeta}{\nablau_i^F}\varphirac{\nablau_i^F(p)}{k_i}$. In this way,
\betaegin{align*}
& B(\rhoi_i \zeta) = B\lambdaeft(\rhoi_i\lambdaeft( F_*X + \imath_*\mathrm{t}imesi + \sum_{j \in J} \interno{\zeta}{\nablau_j^F}\varphirac{\nablau_j^F(p)}{k_j} \right)\right) = \\
& = B\lambdaeft( F_*\mathbf{R}_i^f X + F_*{^f\mathbf{S}_i^\mathrm{t}}\mathrm{t}imesi + \imath_*\mathbf{S}_i^f X + \imath_*\mathbf{T}_i^f \mathrm{t}imesi \right) + \interno{\zeta}{\nablau_i^F} \varphirac{B\lambdaeft(\nablau_i^F(p)\right)}{k_i} = \\
& = G_*\lambdaeft( \mathbf{R}_i^f X + {^f\mathbf{S}_i^\mathrm{t}} \mathrm{t}imesi\right) + \imath_*\Phi\lambdaeft( \mathbf{S}_i^f X + \mathbf{T}_i^f \mathrm{t}imesi\right) + \interno{\zeta}{\nablau_i^F}\varphirac{\nablau_i^G(p)}{k_i} = \\
& = G_*\lambdaeft( \mathbf{R}_i^g X + {^g\mathbf{S}_i^\mathrm{t}} \Phi(\mathrm{t}imesi)\right) + \imath_*\lambdaeft( \mathbf{S}_i^g X + \mathbf{T}_i^g \Phi(\mathrm{t}imesi)\right) + \interno{\zeta}{\nablau_i^F} \varphirac{\nablau_i^G(p)}{k_i} = \\
& = \imath_*\lambdaeft( g_* \mathbf{R}_i^g X + \imath_* \mathbf{S}_i^g X \right) + \imath_* \lambdaeft( g_*{^g\mathbf{S}_i^\mathrm{t}} \Phi(\mathrm{t}imesi) + \mathbf{T}_i^g \Phi(\mathrm{t}imesi)\right) + \interno{\zeta}{\nablau_i^F} \varphirac{\nablau_i^G(p)}{k_i} = \\
& = \rhoi_i G_* X + \imath_* \rhoi_i (\Phi\mathrm{t}imesi) + \interno{\zeta}{\nablau_i^F}\varphirac{\nablau_i^G{p}}{k_i} = \\
& = \rhoi_i (B(F_*X)) + \rhoi_i(\imath_*\Phi(\mathrm{t}imesi)) + \interno{\zeta}{\nablau_i^F}\varphirac{\nablau_i^G{p}}{k_i} = \\
& = \rhoi_i \lambdaeft( B(F_*X) + B(\imath_*\mathrm{t}imesi) + \sum_{j \in J}^\varepsilonll \interno{\zeta}{\nablau_j^F}\varphirac{B(\nablau_j^F(p))}{k_j}\right) = \rhoi_i (B(\zeta)).
\varepsilonnd{align*}
We have just showed that $B \circ \rhoi_i = \rhoi_i \circ B$, if $k_i \nablae 0$. If $k_i = 0$, the calculations above are simpler.
Let $C_i := \rhoi_i(C)$. Since $B(\rhoi_i \circ F) = \rhoi_i \circ G$, then
\betaegin{multline*}
B(\rhoi_i(F(p))) = \rhoi_i(G(p)) = \rhoi_i (B(F(p)) + C) = \\
= \rhoi_i(B(F(p))) + C_i = B(\rhoi_i(F(p))) + C_i \mathbb{R}ightarrow C_i = 0.
\varepsilonnd{multline*}
Thus $\mathrm{t}ilde\Phi\lambdaeft(\o{i}\right) = B(\o{i}) = \o{i}$, for each $i \in \{1, \cdots, \varepsilonll\}$. Therefore $g = \varphi \circ f$, where $\varphi = \mathrm{t}ilde\Phi|_{\hat{\mathbb{O}}}$ is a isometry of $\hat{\mathbb{O}}$ which fix each $\o{i}$.
$\Box$
\betaibliographystyle{acm}
\addcontentsline{toc}{section}{References}
\betaibliography{/home/bruno/Dropbox/Matematica/bibliografia}
\varepsilonnd{document} | math | 182,445 |
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